Machinery's Handbook [31 ed.] 083114131X, 9780831141318

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Machinery's Handbook, 31st Edition

A REFERENCE BOOK M anufacturing and Mechanical Engineer, Designer, Drafter, Metalworker, Toolmaker, M achinist, Hobbyist, Educator, and Student

for the

Machinery’s Handbook 31st Edition

By Erik Oberg, Franklin D. Jones, Holbrook L. Horton, Henry H. Ryffel, and Christopher J. McCauley

Laura Brengelman, Editor

2020

Industrial Press, Inc.

Copyright 2020, Industrial Press, Inc.

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Machinery's Handbook, 31st Edition

INDUSTRIAL PRESS, INC. 32 Haviland Street, Suite 3 South Norwalk, Connecticut 06854 U.S.A. Phone: 203-956-5593 Toll-Free: 888-528-7852 Fax: 203-354-9391 Email: [email protected] Title:  Machinery’s Handbook, 31st Edition Authors: Erik Oberg, Franklin D. Jones, Holbrook L. Horton, Henry H. Ryffel, and Christopher J. McCauley Library of Congress Control Number: 2019954863

COPYRIGHT 1914, 1924, 1928, 1930, 1931, 1934, 1936, 1937, 1939, 1940, 1941, 1942, 1943, 1944, 1945, 1946, 1948, 1950, 1951, 1952, 1953, 1954, 1955, 1956, 1957, 1959, 1962, 1964, 1966, 1968, 1971, 1974, 1975, 1977, 1979, 1984, 1988, 1992, 1996, 1997, 1998, 2000, 2004, 2008, 2012, 2016, 2020 © by Industrial Press, Inc. ISBN 978-0-8311-3731-1 (Toolbox, Thumb-Indexed, 4.6 × 7 in., 11.7 × 17.8 cm) ISBN 978-0-8311-3631-4 (Large Print, Thumb-Indexed, 7 × 10 in., 17.8 × 25.4 cm) ISBN 978-0-8311-3831-8 (Digital Edition) ISBN 978-0-8311-3931-5 (Digital Edition Upgrade) ISBN 978-0-8311-4131-8 (Toolbox and Digital Edition Combo, 4.6 × 7 in., 11.7 × 17.8 cm) ISBN 978-0-8311-4031-1 (Large Print and Digital Edition Combo, 7 × 10 in., 17.8 × 25.4 cm) No part of this book may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or by any information storage and retrieval system, without written permission from the publisher. Limits of Liability and Disclaimer of Warranty While every possible effort has been made to ensure the accuracy of all information presented herein, the publisher expresses no guarantee of the same, does not offer any warrant or guarantee that omissions or errors have not occurred, and may not be held liable for any damages resulting from use of this text. Readers accept full responsibility for their own safety and that of the equipment used in conjunction with this text.

Printed and bound by Thomson Press. MACHINERY’S HANDBOOK 31ST EDITION First Printing

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Machinery's Handbook, 31st Edition

MACHINERY’S HANDBOOK, 31ST EDITION, TEAM Editorial Advisory Board Steve Heather, an acclaimed mechanical engineer, worked in the defense, aircraft, automobile, and lighting industries for more than 30 years. More recently, he taught mechanical engineering and computer-aided design (CAD) at the college level to engineering and architectural students. He is an expert in AutoCAD®, CNC programming, multi-stage press tool design, and precision machining, as well as author of AutoCAD ® 3D Modeling and Engineers Precision Data Pocket Reference and coauthor of Beginning AutoCAD ® Exercise Workbook, Advanced AutoCAD ® Exercise Workbook, and AutoCAD ® Pocket Reference, all published by Industrial Press. He is an invaluable engineering advisor and contributor to material throughout the Machinery’s Handbook, as well as a skilled technical illustrator, whose work enhances the current edition. David O. Kazmer is professor of plastics engineering at the University of Massachusetts Lowell, as well as associate research professor at the University of Massachusetts Amherst. He is the recipient of 19 recognition awards, including the Ishii-Toshiba Design for Manufacturing Award, an inventor with over 20 patents, and author of more than 200 publications and vital text in the Machinery’s Handbook. His academic work is motivated by industry experiences, most recently as director of research and development at Dynisco HotRunners. His teaching and research encompass process development, product and machine design, and design methodologies, including polymer processing, design for manufacturing, optimization, simulation, process control, and technology strategy. His ongoing research has contributed to development of new manufacturing processes with improved real-time control and robust design tools. Howard Kuhn most recently served as adjunct professor at the University of Pittsburgh, Swanson School of Engineering, where he taught courses in manufacturing, additive manufacturing, product realization, and engineering entrepreneurship and performed research on additive manufacturing. He currently is a technical advisor to America Makes (National Additive Manufacturing Innovation Institute), where he previously was acting deputy director. Specializing in advanced technology implementation, he has engaged in the design and application of multiple additive manufacturing technologies for major government clients and private industry. He has developed undergraduate and graduate courses in engineering design, failure analysis, deformation processing, and powder and mechanical metallurgy; conducts tailored training courses on additive manufacturing for government agencies, trade organizations, and companies; and contributed authoritative revisions to this edition. Jennifer Marrs has worked as a mechanical engineer for more than 20 years. An accredited Professional Engineer, she earned a master of science degree in mechanical engineering at Northeastern University, holds a patent, and is a registered US patent agent. Her areas of expertise include manufacturing engineering, design and analysis of machinery, tool and fixture design, machinery safety, fluid systems, industry standards, and intellectual property issues. She has worked with companies of all sizes and ran a successful consulting practice for nearly a decade. She is the author of the Industrial Press text Machine Designers Reference, as well as key technical material in the Machinery’s Handbook.

Contributors Viktor P. Astakhov earned his Ph.D. in mechanical engineering from Tula State Polytechnic University, Tula–­Moscow, USSR–­Russia, in 1983. He was awarded a DSci designation (Dr. Habilitation, Docteur d’État) in 1991 and the title “State Professor of Ukraine” in 1991 for outstanding service rendered during his teaching career and the profound impact his work had on science and technology. An internationally recognized educator, researcher, and mechanical engineer, he has won a number of prestigious awards. In 2011, he was elected to the SME College of Fellows, and, in 2018, he became a member of the European Union

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Machinery's Handbook, 31st Edition MACHINERY’S HANDBOOK, 31ST EDITION, TEAM Academy of Science. As a professor, he has supervised graduate students at Michigan State University. He currently serves as a research professor at St. Petersburg State Polytechnic University in Russia, as well as the tool research and application manager of the General Motors Business Unit of PSMI.

Vukota Boljanovic received his B.S., M.S., and Ph.D. in mechanical engineering and has more than 45 years of experience in applied engineering in the aircraft and automotive industries, including serving as vice president for research and development with a major aircraft company. He has taught aerospace engineering, among other subjects, and has performed extensive research in development and manufacturing engineering, including the impact of design modification on tools, dies, and processes selection; aircraft assembly; and inspection. He is the author of numerous technical papers and books, including the Industrial Press titles Applied Mathematical and Physical Formulas, Metal Shaping Processes, Sheet Metal Forming Processes and Die Design, and Sheet Metal Stamping Die Designs, and has been widely recognized by both academia and industry for his contributions to manufacturing.

Charles “Wes” Cross has been a weld engineer for more than 25 years. He holds a degree in weld engineering from Le Tourneau University in Longview, Texas, and has held many positions in local American Welding Society (AWS) chapters. His industry experience encompasses weld applications, quality control programs, shop supervision, and weld program implementation and maintenance. He is an expert in weld processes and adhering to ASME, AWS, and NAVSEA standards and codes, as well as an AWS Certified Welding Inspector and Certified Welding Educator. He recently established All Welding Services, a consulting company for weld shops and weld code programs, working with companies around the world. Brad Dulin is a senior metrologist and has worked in the research and development in aerospace, biomedical, and other technical fields for more than 30 years. He has collaborated with such technological giants as Hughes Helicopters and Space and Communications divisions, the Jet Propulsion Laboratory, and Raytheon, as well as technology companies worldwide. He has served as a metrology consultant with BAE for the Kuwait Air Force and, most recently, as operations manager for the Kuwait Green Energy Company. He currently lives in Kuwait and is collaborating on the development of advanced hydrogen/solar-based alternative energy solutions. Arief Era is a graduate of Columbia University, where he received his master’s degree in mechanical engineering. As a structural analysis engineer, he worked on various commercial aircrafts for the Boeing Company. At Consolidated Edison of New York in the Gas Engineering department, he has developed fittings, valves, and piping systems for gas delivery while performing root cause analysis for field failures. More recently, he has led the Maps and Records Team for the gas delivery infrastructure of Manhattan. His experience and extensive work on developing, refining, and working with industry standard and related technical information has been crucial to the current edition of the Handbook.

Charles Gillis has over 24 years of machine design experience. He received his bachelor of science degree from Worcester Polytechnic Institute and a master’s degree in mechanical engineering from Northeastern University. Currently serving as a mechanical design engineer for the Gillette Company, designing automated machinery for manufacturing blade and razor products, he has been training practicing engineers in geometric dimensioning and tolerancing, print reading, and related mechanical analysis, design, and documentation topics for the last decade. A licensed Professional Engineer in Massachusetts, he holds a Geometric Dimensioning and Tolerancing Professional Certificate–Senior Level (GDTP-S) from ASME. In addition to authoring the bestselling Hammer’s Blueprint Reading Basics, 4th Edition, he also contributed to the Industrial Press titles Machine Designers Reference and The CAM Design and Manufacturing Handbook. Edmund Isakov earned his Ph.D. in technical sciences at the Novocherkassk Polytechnic Institute (Soviet Union). He is known for his work in research, development and applications of cutting tools for milling, turning, and boring. During nearly two decades at Kennametal, he became a noted authority on the technical analysis of cutting tools and processes. He holds 7 U.S. patents and 10 U.S.S.R Inventor’s Certificates pertaining to carbide and diamond

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Machinery's Handbook, 31st Edition MACHINERY’S HANDBOOK, 31ST EDITION, TEAM tools. A senior member of SME, he has authored numerous articles and papers, as well as the Industrial Press titles Cutting Data for Turning of Steel, Engineering Formulas for Metalcutting, and International System of Units (SI).

Melissa Klingenberg holds a master’s degree in manufacturing systems engineering and a Ph.D. in materials engineering. She has more than 25 years of experience in inorganic finishing operations, specializing in development, technology evaluation, and implementation of innovative coatings and surface finishing processes to improve engineering properties and address environmental issues. Known for research in wear-resistant coatings and replacement technologies for defense applications, she has co-authored many publications. A research and development engineer at Pennsylvania State University’s Applied Research Laboratory, she also has served as AESF Foundation president and on the AESF Council, Emerging Technologies Committee, NASF Research Board, and Sur/Fin Technical Committee, and as organizer and chair for the Surface Engineering for Defense and Aerospace Applications Conference. While a principal advising engineer at Concurrent Technologies Corporation, she received the 2015 National Association for Surface Finishing’s Scientific Achievement Award. 

Kathleen McKenzie is an educator, writer, editor, and copy editor specializing in science and mathematics. With a bachelor’s degree in chemistry and a master’s in mathematics, she has been a career mathematics educator—primarily at Binghamton University, SUNY, where she teaches undergraduate calculus. She has worked for major publishers and composition houses nationwide, and has helped develop and add materially to a number of best­ selling Industrial Press texts. Among these, she contributed to Technical Shop Mathematics, and served as development editor on The Handbook of PVC Pipe Design and Construction. In the last and particularly the current edition of the Machinery’s Handbook, she helped make significant revisions in the mathematics and measuring units sections, as well as refining other key topics. Merwan Mehta is a professor in the College of Engineering and Technology at East Carolina University in Greenville, North Carolina. He has taught engineering economics at the undergraduate and graduate levels since 2004. Prior to joining academia, he spent more than 20 years in the manufacturing industry as a partner in business, vice president, project director, manager, industrial and manufacturing engineer, and machine tool design engineer. A Certified Manufacturing Engineer and Certified Six Sigma Black Belt, he conducts workshops internationally on various engineering and operational excellence topics and has served as an examiner for the Missouri Quality Award, based on the Baldrige Criteria. His Applied Engineering Economics Using Excel is a top Industrial Press text.

David R. Quinonez has over 25 years of experience in welding, welding inspection, and nondestructive testing. His impressive experience in nondestructive testing began with nuclear submarines and aircraft carriers. Subsequent positions included nondestructive testing (NDT) and welding inspection on Rolls-Royce gas turbine engines, F-22 stealth fighter airframes, missile defense, commercial/military rockets, pipeline, and structural steel for private sector and public works projects. He is a Certified Welding Inspector performing visual welding inspection, dimensional verification, and NDT. His Level II certifications include UT, MT, and PT; past certifications have included Level II RT, ET, and ASNT Level III MT and PT. He is the author of the highly instructive Industrial Press title 1,001 Questions & Answers for the CWI Exam: Welding Metallurgy and Visual Inspection Study Guide. Peter Smid is a professional consultant, educator and speaker, with many years of practical, hands-­on experience. He consults to manufacturing industry and educational institutions on use of CAD/CAM software, CNC technology, part programming, advanced machining, tooling, and setup. His comprehensive industrial background in CNC programming, machining, and company-oriented training has assisted hundreds of companies, and he has developed and delivered thousands of customized educational and industrial programs to instructors, students, manufacturing companies, private sector organizations, and others. He is author of numerous, definitive articles and publications on the subject of CNC and CAD/CAM, with six titles for Industrial Press, including the CNC Programming Handbook and the CNC section of the Handbook.

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Machinery's Handbook, 31st Edition

PREFACE In continuous publication since 1914, Machinery’s Handbook has served as the principal reference work in metalworking, engineering, design, and manufacturing facilities, and in technical schools, colleges, and universities throughout the world for more than a century. Throughout this period, the Handbook editors have strived to create a comprehensive and practical resource, combining the most basic and crucial aspects of sophisticated manufacturing practice. The Handbook is an invaluable tool, to be used in much the same way as other tools, to design, make, repair, and maintain products of highest quality, at the lowest cost, and in the shortest time possible. The essential basics, material of proven and everlasting worth, always must be included, if the Handbook is to continue to serve the needs of the manufacturing and mechanical engineering community. Yet it is difficult to select suitable material from the vast supply of data related to these traditional yet rapidly evolving fields. An ongoing challenge is to provide valuable information for design and production departments in manufacturing plants and workshops of all sizes, as well as for product and system designers, job shops, hobbyists, and instructors and students in general, trade, technical, and engineering schools. The editorial team relies on conversations and written communications with users of the Handbook and experts in technical fields for guidance on topics to be introduced, revised, lengthened, shortened, or omitted. The original Handbook was designed to fit inside a standard toolbox. At the request of users, in 1997, the large print or “desktop” edition of the Handbook was introduced. The large print edition is identical to the traditional toolbox edition, only its size is increased by 140 percent, making it an easier-to-read reference. (Note that the type is standard reference size, not a larger font designed for visually impaired readers.) Other than size, there is no difference between the toolbox and large print editions. In 1998, Christopher McCauley developed and launched the first Machinery’s Handbook CD-ROM, containing the complete content of the printed book, with added indexes and hundreds of pages of archival material restored from earlier editions. Continued as the Machinery’s Handbook 31st Digital Edition, this versatile format offers rapid searching and navigation aids in the form of clickable links and cross references that take you quickly to pages referenced. The growing family of Machinery’s Handbook products also includes the Guide, Pocket Companion, and attractive combination packages. Longtime users of the Handbook will note many changes in recent editions, but an enduring goal of the editors is to make this encyclopedic reference easier to use. The Handbook continues to incorporate time-saving thumb tabs, much requested by users. In addition to the front table of contents, sectional contents beginning each major section, introduced in the 25th edition, also have proven useful to readers. In the 31st edition, these sectional contents have been expanded to provide even more detailed navigation aids. Overall, this edition has been edited, updated, and reset. Incorporating thousands of indi­ vidual changes and more than 250 new and revised tables and figures, it has expanded by nearly 100 pages, to 2,992 pages. Among major revisions of existing content and new material are the following: First and foremost, hundreds of specific references and pieces of key information based on the most current ANSI, ASME, and ASTM standards have been updated throughout the Handbook. Of all the reasons to purchase the 31st edition, these timely updates make this a must-have resource. To examine other improvements starting at page 1, while the core concepts of MATHEMATICS remain unchanged, this baseline material has benefited from reorganization, expansion, and elucidation by subject experts and educators to reflect current terminology and teaching. Among other formula additions in the Handbook, new calculations for Tolerance Analysis and Assignment can be found on page 684. Also see MEASURING UNITS, on page 2827, for added information on International System of Units (SI), related, base, and derived units, names, and prefixes. Expert revisions have been made throughout MECHANICS AND STRENGTH OF MATERIALS, beginning on page 156, regarding forces, strength, testing, and related analysis. The subsection PROPERTIES OF BODIES is now more specifically defined as RIGID BODY PARAMETERS.

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Machinery's Handbook, 31st Edition PREFACE The PLASTICS section, beginning on page 555, has received important updates, with new information on material characteristics, properties, and costs; calculations for elasticity, stress, strain, loads, and temperature effects; manufacturing and machining processes; design considerations; use of plastics for prototyping; and additive manufacturing (AM). Metal Additive Manufacturing, beginning on page 1555, has been redone, with illustrations and tables and updated text addressing the ongoing evolution of this dynamic field; key trends and recent developments in materials, processes and workflow, production considerations, and finishing; comparisons of AM costs, efficiencies, and capabilities with conventional processes for producing parts; and applicable ASTM and ISO standards. The related topic of POWDER METALLURGY, beginning on page 1522, also has received updates and added dimensional information. A new CORROSION section on page 548 discusses forms, causes, and methods of prevention; galvanic series, compatibility, and coupling; and common effects and mitigation methods. Other surface-related information includes updates in FINISHING OPERATIONS, starting on page 1632, on electropolishing, passivation, and plating, and added Sheet Metal Mill Finishes on page 1448. Related sheet metal additions include Blanking Pressure on page 1413, Three-Roll Bending on page 1423, and Sheet Metal Gauge Sizes on page 1453. In MACHINING OPERATIONS, see page 1154 for new material on boring and indexable boring bars. The new NONTRADITIONAL MACHINING AND CUTTING section on page 1344 describes mechanical erosion and electro-thermal processes. And in CNC NUMERICAL CONTROL PROGRAMMING, the topic CAD/CAM, on page 1390, has been reintroduced and revised. IRON AND STEEL CASTING has been renamed METAL CASTING, MOLDING, AND EXTRUSION, and this section, starting on page 1480, has been rewritten and expanded. Look for new information on working with iron, steel, and nonferrous casting metals; industry casting and molding processes; heating, pouring, flow, and fluidity; heat transfer, solidification, and cooling; and materials, applications, casting defects, design considerations, and computer modeling. The SOLDERING AND BRAZING and WELDING sections also have received important updates, including new and revised figures and tables, and an expanded section on Non­ destructive Testing, on page 1627. O-­R INGS, on page 2666, has been rewritten and expanded, with 13 tables and details of standards, designations, selection, clearance gap, cramping, face seals, glands, grooves, installation, lubrication, squeeze, stretch, and equations. In UNIFIED SCREW THREADS, the latest Standard Series and Selected Combination tables, beginning on page 1951, incorporate numerous updates. TRANSMISSION CHAINS, on page 2616, has been revised, with meticulous updates to the Horsepower Ratings for Roller Chains tables, based on the latest standards. There are the countless other changes in tables, figures, calculations, and text resulting from the Handbook team’s extensive review of current industry standards. As in previous editions, we continue to include expanded, parallel US Customary and metric figures. Where possible, formulas are presented with equivalent metric expressions (some in bold). Addition of new and revised Handbook topics often requires removal of older topics to gain space. Materials removed from the print book generally appear in the Digital Edition, which contains added material not in the current print edition. Included in the Digital Edition are mathematical tables on topics such as logarithms and trigonometry; material on cement and concrete, adhesives and sealants, coloring and etching metals, forge shop equipment, silent chain, worm and other gears, keys and keyways; and many other extracts from past editions that may be of use and interest. Absent in the 31st print edition is a conversion table that previously appeared on page 3 on fractional and decimal inch to millimeter conversion; the same information can be found in tables in the MEASURING UNITS section, which starts on page 2831. Other material moved to the Digital Edition includes tabular data, such as constants involving uses of π, moments of inertia and section moduli for shafts, and an older screw thread table on European systems, as well as a short section on Bakelite, and Change Gears for Helical Milling. Users requiring this information, or wishing to comment on these or other topics moved to the Digital Edition, are urged to contact the editors.

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Machinery's Handbook, 31st Edition PREFACE The editors are greatly indebted to readers who call attention to possible errors and defects in the Handbook, offer suggestions for including new or revised material, or have technical questions about the content and applicability to manufacturing problems encountered in the shop. Such dialog helps identify topics that require clarification or expansion. Queries involving Handbook material usually entail an in-depth review of the topic in question, frequently resulting in improved or new material. We also welcome new contributors to each edition, joining the long line of erudite industry experts who have made the Handbook what it is today, and we invite topical experts to contact the editors. Our perpetual goal is to increase the usefulness of the Handbook as much as possible. We welcome your input and look forward to hearing from you.

Acknowledgments and Special Thanks Machinery’s Handbook is indebted to the whole mechanical field for the data contained in this master reference work. On behalf of the Handbook editors past and present, we wish to express our appreciation to all who have assisted in furnishing data and contributed ideas, corrections, and other commentary on the Handbook. Most importantly, we thank the thousands of readers who have contacted us over the years with constructive criticism and suggestions regarding Handbook topics and presentation. Your comments on this edition, as well as past and future ones, are invaluable. Many of the American National Standards Institute (ANSI) standards that deal with mechanical engineering, extracts from which are included in the Handbook, are published by the American Society of Mechanical Engineers (ASME). The editors thank ASME for its exceptional collaboration in helping identify and bring essential data up to date, according to the latest, definitive, industry standards. Information concerning other standards and nomenclature also is included in the Handbook. Official standards and related publications are copyrighted by the issuing organizations; contact them directly for further information regarding current editions of standards and to purchase copies. On the following pages are brief biographies for the Machinery’s Handbook, 31st Edition team—an impressive roster of Editorial Advisory Board members and contributors. These esteemed colleagues have played a crucial role in guiding content decisions and advising on specific engineering questions and content challenges. Their lifelong educational and industry experience, impressive technical knowledge and expertise, and meticulous research have immeasurably enhanced the content of this edition. We also wish to thank those behind the scenes, our tremendous editorial and production team, without whom this edition would not be possible: the incomparable Jason Hughes, Abigail Parker, Billie Rothstein, and the rest of the remarkable team at Scribe; math maven Dan McKinney; eagle-eyed editorial team members Teresa Barensfeld, Cara Chamberlain, Gerald Murray, Julia Phelps, and Deborah Ring; and our masterful printing and binding partners at Thomson Press. In addition, longtime editor, mastermind, and retired leader of the Handbook team Christopher McCauley is recognized for his extensive additions to essential content over multiple editions. Accordingly, he has been added to the masthead of illustrious authors of the Handbook. The Handbook is indebted to many others in industry organizations and associations, educational institutions, commercial enterprises, and private practice. In this context, we cannot thank everyone who has written in, helped resolve a question, or has otherwise spent valuable time and effort assisting us improve past and present editions. Therefore, we have added a “Handbook Hall of Fame” to our website, where we can acknowledge and acclaim the most important participants in the community surrounding this legendary product. Again, we encourage you to send us your thoughts and feedback, and to share with us how the Machinery’s Handbook product family supports and enhances your involvement in this endlessly fascinating field. Laura Brengelman Editor

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Machinery's Handbook, 31st Edition

TABLE OF CONTENTS PREFACE ACKNOWLEDGMENTS AND SPECIAL THANKS MACHINERY’S HANDBOOK TEAM

vii ix x

MATHEMATICS • • • •

1

REAL NUMBERS AND THEIR OPERATIONS • ALGEBRA • GEOMETRY TRIGONOMETRY: SOLUTION OF TRIANGLES  •  MATRICES CALCULUS  •  STATISTICAL ANALYSIS OF MANUFACTURING DATA  ENGINEERING ECONOMICS

MECHANICS AND STRENGTH OF MATERIALS

154

PROPERTIES, TREATMENT, AND TESTING OF MATERIALS

365

DIMENSIONING, GAGING, AND MEASURING

618

TOOLING AND TOOLMAKING

829

• • • •

MECHANICS  •  VELOCITY, ACCELERATION, WORK, AND ENERGY STRENGTH OF MATERIALS • RIGID BODY PARAMETERS • BEAMS COLUMNS • PLATES, SHELLS, AND CYLINDERS • SHAFTS SPRINGS • DISC SPRINGS

• THE ELEMENTS, HEAT, MASS, AND WEIGHT  •  PROPERTIES OF WOOD, CERAMICS, PLASTICS, METALS  •  STANDARD STEELS  • TOOL STEELS  •  HARDENING, TEMPERING, AND ANNEALING • NONFERROUS ALLOYS • CORROSION • PLASTICS

• DRAFTING PRACTICES  •  ALLOWANCES AND TOLERANCES FOR FITS  •  MEASURING, INSTRUMENTS, AND INSPECTION METHODS • MICROMETER, VERNIER AND DIAL CALIPERS  •  SURFACE TEXTURE • CUTTING TOOLS • CEMENTED CARBIDES • MILLING CUTTERS • REAMERS • TWIST DRILLS AND COUNTERBORES • TAPS • STANDARD TAPERS  •  ARBORS, CHUCKS, AND SPINDLES • BROACHES AND BROACHING  •  FILES AND BURS  •  KNURLS AND KNURLING  •  TOOL WEAR AND SHARPENING

MACHINING OPERATIONS

1073

MANUFACTURING PROCESSES

1397

• CUTTING SPEEDS AND FEEDS  •  SPEEDS AND FEEDS TABLES • ESTIMATING SPEEDS AND MACHINING POWER  •  MICROMACHINING • MACHINING ECONOMETRICS  •  SCREW MACHINES, BAND SAWS, CUTTING FLUIDS  •  MACHINING NONFERROUS METALS AND NONMETALLIC MATERIALS  •  GRINDING FEEDS AND SPEEDS  •  GRINDING AND OTHER ABRASIVE PROCESSES  •  NONTRADITIONAL MACHINING AND CUTTING  • CNC NUMERICAL CONTROL PROGRAMMING • SHEET METAL WORKING AND PRESSES  •  ELECTRICAL DISCHARGE MACHINING  •  METAL CASTING, MOLDING, AND EXTRUSION  • POWDER METALLURGY •  SOLDERING AND BRAZING  •  WELDING • FINISHING OPERATIONS

Each section has a detailed Table of Contents located on the page indicated.

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Machinery's Handbook, 31st Edition TABLE OF CONTENTS

FASTENERS

1649

THREADS AND THREADING

1936

GEARS, SPLINES, AND CAMS

2201

MACHINE ELEMENTS

2389

MEASURING UNITS

2826



2876

• TORQUE AND TENSION IN FASTENERS  •  INCH THREADED FASTENERS • METRIC THREADED FASTENERS  •  HELICAL COIL SCREW THREAD INSERTS  •  BRITISH FASTENERS  •  MACHINE SCREWS AND NUTS  • CAP AND SET SCREWS  •  SELF-THREADING SCREWS  •  T-SLOTS, BOLTS, AND NUTS  •  RIVETS AND RIVETED JOINTS  •  PINS AND STUDS  • RETAINING RINGS  •  WING NUTS, WING SCREWS, AND THUMB SCREWS • NAILS, SPIKES, AND WOOD SCREWS

• SCREW THREAD SYSTEMS  •  UNIFIED SCREW THREADS • CALCULATING THREAD DIMENSIONS  •  METRIC SCREW THREADS • ACME SCREW THREADS • BUTTRESS THREADS • WHITWORTH THREADS • PIPE AND HOSE THREADS • OTHER THREADS • MEASURING SCREW THREADS  •  TAPPING AND THREAD CUTTING  •  THREAD ROLLING • THREAD GRINDING • THREAD MILLING • GEARS AND GEARING  •  HYPOID AND BEVEL GEARING  •  WORM GEARING • HELICAL GEARING • OTHER GEAR TYPES • CHECKING GEAR SIZES • GEAR MATERIALS • SPLINES AND SERRATIONS • CAMS AND CAM DESIGN • PLAIN BEARINGS  •  BALL, ROLLER, AND NEEDLE BEARINGS • LUBRICATION • COUPLINGS, CLUTCHES, BRAKES • KEYS AND KEYSEATS • FLEXIBLE BELTS AND SHEAVES • TRANSMISSION CHAINS  •  BALL AND ACME LEADSCREWS  •  ELECTRIC MOTORS • ADHESIVES AND SEALANTS • O-RINGS • ROLLED STEEL, WIRE, SHEET METAL, WIRE ROPE  •  SHAFT ALIGNMENT  •  FLUID POWER • SYMBOLS AND ABBREVIATIONS  •  MEASURING UNITS • US SYSTEM AND METRIC SYSTEM CONVERSIONS

Each section has a detailed Table of Contents located on the page indicated.

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Machinery's Handbook, 31st Edition TABLE OF CONTENTS MATHEMATICS

REAL NUMBERS AND THEIR OPERATIONS

(Continued)

3 Real Numbers 3 Properties of Real Numbers 3 Integers (Signed Numbers) 4 Order of Operations 5 Fractions and Mixed Numbers 6 Adding and Subtracting 7 Multiplying 8 Dividing 8 Decimal Numbers 9 Ratio and Proportion 10 Percentage 11 Powers and Roots 11 Properties of Exponents 12 Scientific Notation 13 Factorial Notation 13 Permutation 13 Combination 13 Prime Factorization of Numbers

GEOMETRY 42 Analytic Geometry 42 Rectangular Coordinate System 42 Slope of a Line 43 Lines and Line Segments 44 Equation Forms of a Line 49 Circle 51 Ellipse 54 Four-Arc Oval Approximating an Ellipse 55 Sphere 57 Parabola 58 Hyperbola 59 Complex Numbers 59 Imaginary Number 59 Forms of a Complex Number 61 Pure Geometry 61 Propositions of Geometry 66 Geometric Constructions 71 Area and Volume 71 Prismoidal Formula 71 Pappus-Guldinus Rules 72 Finding Area of a Surface of Revolution 72 Area of Irregular Plane Figure 73 Areas Enclosed by Cycloidal Curves 73 Contents of Cylindrical Tanks at Different Levels 75 Dimensions of Plane Figures 81 Polygons 83 Segments of a Circle 84 Segments of a Circle for Radius = 1 86 Diameters of Circles and Sides of Squares of Equal Area 87 Diagonals of Squares and Hexagons 88 Volumes of Solids

ALGEBRA 24 Definitions 24 Evaluating Algebraic Expressions 24 Combining Like Terms 25 Solving an Equation for an Unknown 26 Rearrangement and Transposition of Terms in Formulas 27 Algebraic Operations 27 Properties of Monomials and Exponents 27 Properties of Radicals 28 Polynomials 28 Operations on Polynomials 29 Factoring Polynomials 31 Equation Solving 31 System of Linear Equations 32 Second-Degree (Quadratic) Equation 34 Completing the Square 34 Using the Quadratic Formula 34 Cubic Equation 35 Functions 35 Graphs of Functions 36 Logarithms 36 Meaning 36 Properties 37 Common

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ALGEBRA

37 Natural 38 Using Calculators to Solve Logarithms 38 Solving an Equation Using Logarithms 39 Arithmetic Sequence 39 Geometric Sequence

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Machinery's Handbook, 31st Edition TABLE OF CONTENTS MATHEMATICS

TRIGONOMETRY: SOLUTION OF TRIANGLES

CALCULUS 130 Derivatives 130 Formulas 131 Rules 131 Integrals (Antiderivatives) 132 Integral Rules 132 Newton’s Method for Solving Equations 133 Formulas for Differential and Integral Calculus 135 Series Representation of a Function

94 Terminology 94 Degree and Radian Angle Measure 94 Trigonometric Ratios of Essential Angles 95 Functions of Angles 95 Right Triangle Ratios 96 Law of Sines 96 Law of Cosines 96 Trigonometric Identities 98 Solution of Right Triangles 99 Solution and Examples of Right Triangles 100 Solution and Examples of Oblique Triangles 102 Rapid Solution of Triangles 103 Conversion Tables of Angular Measure 105 Trigonometric Functions 106 Trigonometry Tables 111 Using a Calculator to Find Trigonometric Function Values 111 Versed Sine and Cosine 111 Sevolute Functions 111 Involute Functions 116 Spherical Trigonometry 116 Right-Angle Spherical Trigonometry 118 Oblique Spherical Trigonometry 120 Compound Angles 122 Interpolation

STATISTICAL ANALYSIS OF MANUFACTURING DATA 136 Statistics Theory in Brief 136 Probability 137 Normal Distribution Analysis 139 Applying Statistics 139 Minimum Number of Test or Data Points 139 Comparing Products with Respect to Average Performance ENGINEERING ECONOMICS 143 Interest 143 Variables 144 Simple Interest 144 Compound Interest 144 Determining Principal, Rate, or Time 145 Nominal versus Effective Interest Rates 146 Cash Flow and Equivalence 146 Present Value and Discount 146 Annuities 147 Sinking Funds 147 Cash Flow Diagrams 149 Depreciation 149 Straight Line 149 Sum of the Years Digits 149 Double Declining Balance 149 Statutory Depreciation 150 Evaluating Alternative Investments 150 Net Present Value 151 Capitalized Cost 152 Equivalent Uniform Annual Cost 153 Rate of Return 153 Benefit-Cost Ratio 153 Payback Period

MATRICES 124 Matrix Operations 124 Addition and Subtraction 124 Multiplication 125 Transpose 125 Determinant of a Square Matrix 125 Minors and Cofactors 126 Adjoint of a Matrix 126 Singularity and Rank 126 Inverse 127 Solving a System of Equations

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Machinery's Handbook, 31st Edition REAL NUMBERS

3

REAL NUMBERS AND THEIR OPERATIONS Real Numbers

Most mathematical computation is performed in the real number system. The universal set of the “reals” includes the subsets: naturals, whole numbers, integers, rationals, and irrationals. The naturals (also called counting numbers): {1, 2, 3, . . .} are included in the whole numbers: {0, 1, 2, 3, . . .}, which are included in the integers (or signed whole numbers): {. . . ,–2, –1, 0, 1, 2, . . .}. And all of these subsets are included in the rationals. Rational numbers, including integers, can be written in fraction form. Since all fractions can be divided numerator by denominator, their decimal form either terminates or repeats. Examples of rational numbers: –4/1, 3/5 = 0.6, 1/3 = 0.333. . . . The only set in the real numbers larger than the naturals that does not contain any of the other sets is the irrationals. These are not expressible as ratios. An irrational number’s decimal representation does not terminate and it has no pattern of repetition. Examples of irrational numbers are roots that cannot be simplified, such as 6 and 3 70 , as well as quantities like π and the natural log base e. The entire real number set is the union of the rationals and the irrationals. Properties of Real Numbers.—Though often obvious and followed almost automatically, the properties of real numbers are critical to mathematical reasoning. These properties justify various steps in solving algebraic problems, such as those in this Handbook. Equivalence properties (symmetry, reflexivity, transitivity) and operational properties of numbers are summarized here.

Equivalence Properties: The properties of equivalence relations are the basis of equation solving. Reflexive: a = a. Symmetric: If a = b, then b = a. Transitive: If a = b and b = c, then a = c. Substitution: If a = b, then a may be replaced by b in any equation or expression.

Operational Properties: These concern addition, subtraction, multiplication, and division, as summarized in the table below. Addition

Multiplication

Commutative:

Property

a+b=b+a

a×b=b×a

Associative:

(a + b) + c = a + (b + c)

(a × b) × c = a × (b × c)

Identity:

a+0=0+a=a

1×a=a×1=a

Inverse:

a + (–­a) = 0

a × 1/a = 1

Other Properties: Distributive of multiplication over addition: a × (b + c) = (a × b) + (a × c) (a + b) × c = (a × c) + (b × c) Zero property of multiplication:

If a × b = 0, then either a = 0 or b = 0

Zero property of division:

If a/b = 0, then a = 0 (b ≠ 0)

Integers (Signed Numbers).—Positive whole numbers extend to the right of zero on the number line. Negative whole numbers extend to the left of zero. Together with zero, these make up the integers (sometimes called signed numbers): {. . . ,–2, –1, 0, 1, 2, . . .}. The sciences (as well as economics and other fields) deal with negative as well as non-­ negative quantities. Temperature is an obvious example; so is land altitude, which can be

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Machinery's Handbook, 31st Edition INTEGERS

4

above, at, or below sea level. Angles can be negative, too, as explained in TRIGONOMETRY. Calculators facilitate computation that involves integers (signed numbers). Knowing the rules of integer operations prevents errors that might occur when a calculator is used. Absolute Value: A number’s absolute value, sometimes called its magnitude, is the number’s distance from zero on the number line. Whether a number is positive or negative, its absolute value is positive. For example, the absolute value of both 5 and –5 is 5. The absolute value of n is notated |n|; thus, |5| = 5 and |–5| = 5. Absolute value helps explain the rules of signed number addition and subtraction. Real Number Line: The real number line is generally shown with only the integers marked off (though all numbers are included). A number line is useful for conveying how signed numbers are added or subtracted. Operations on Signed Numbers: The following rules of operations apply to rational and irrational numbers as well. For simplicity, only integers are given as examples. Addition and Subtraction: Adding a negative number is equivalent to subtracting its absolute value. When a larger number is subtracted from a smaller number, the result is negative. The rules for adding and subtracting integers are illustrated with an example using four values: 7, 11, –7, and –11. The following examples illustrate the rules: Examples, Addition

7 + 11 = 18 7 + (–11) = 7 – 11 = –4 (–7) + 11 = 11 + (–7) = 11 – 7 = 4 (–7) + (–11) = –18

Examples, Subtraction

7 – 11 = –4 7 – (–11) = 7 + 11 = 18 (–7) – (–11) = (–7) + 11 = 11 + (–7) = 11 – 7 = 4 –7 – 11 = –18

Multiplication and Division: Multiplication or division of numbers with the same sign results in a positive answer. Opposite signed numbers result in negative answers when multiplied or divided. The following examples illustrate the rules: Examples, Multiplication Examples, Division 5 × 2 = 10 5 × (–2) = –10 (–5) × 2 = –10 (–5) × (–2) = 10

12 ÷ 3 = 4 (–12) ÷ 3 = –4 (12) ÷ (–3) = –4 (–12) ÷ (–3) = 4

Order of Operations.—­Mathematical operations are performed on numbers in a particular order, commonly referred to as PEMDAS, which stands for “Parentheses, Exponents, Multiplication, Division, Addition, Subtraction.” First, when there are no parentheses or other grouping symbols, multiplication and division are done before addition and subtraction. Then, proceeding from left to right, the addition and subtraction are done in the order they appear. For example: 100 –­26 + 7 × 2 –­100 ÷ 4 = 100 –­26 + 14 –­25 = 74 + 14 –­25 = 88 –­25 = 63

Parentheses ( ) and brackets [ ]—­called grouping symbols—­indicate if addition and subtraction are to occur before multiplication and division. The operations are performed from the innermost to the outermost grouping symbols. For example: [6 × (15 –­7)] ÷ 2 = [6 × 8] ÷ 2 = 48 ÷ 2 = 24

Exponents are a multiplication operation, but unless parentheses or brackets are present, exponents are applied before multiplication. For example: 4 × 92 = 4 × 81 = 324

Also, when parentheses are present next to a multiplication, the × can be omitted: 5(8 –­3) = 5(5) = 25

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Machinery's Handbook, 31st Edition Order of Operations

5

As explained in Fractions, the horizontal line in a fraction implies division. The top number (called the numerator) is divided by the bottom number (called the denominator). For example, 50 ------ = 50 ÷ 10 = 5 10 In formulas, the multiplication sign (×) may be omitted (when letters—called variables—are multiplied) or replaced by parentheses, which serve the same purpose. A × B = AB,  6 × 4 = (6)(4),   8 × a = 8a

A multiplication dot (⋅) is also sometimes used.

Fractions

Rational numbers can be written as common fractions or as decimal fractions. Com-

mon fractions are written as ba or a/b, where a (the numerator) and b (the denominator)

are integers (but b cannot be 0, since division by zero is not defined). The denominator represents the number of equal parts that a whole quantity is broken into. The numerator 2 is the number of these parts under consideration. For example, 5 indicates the whole of something is broken into 5 equal parts, and 2 of these parts are being considered. Any integer is a fraction with a denominator of 1. For example, 16 = 6. The implied operation in a fraction is division. Thus, ba means a ÷ b. Multiple: A multiple of a number n is the result of multiplying n by positive integer 1, 2, 3, . . . Thus, the multiples of 3 are 3, 6, 9, 12, . . . The least common multiple (LCM) of two or more numbers is the smallest multiple the numbers have in common. In the example below, the first few multiples of 6 and 20 are shown, with the LCM indicated in bold: 6: 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66, . . . 20: 20, 40, 60, 80, . . . Thus, 60 is the LCM of 6 and 20. Factor: An integer a is a factor of n if there is no remainder when n is divided by a. That is, if the result of n ÷ a is an integer. For example, 3 is a factor of 12 because 12 ⁄ 3 = 4. The greatest common factor (GCF) of two or more numbers is the largest of their common factors. Thus, the common factors of 12 and 18 are 2, 3, and 6; 6 is the GCF. Unit Fraction: A fraction having the same numerator and denominator is the unit fraction, 1 (or “one whole”). For example, 2 ⁄ 2 , 4 ⁄4 , 8 ⁄ 8, 16 ⁄ 16 , 32 ⁄ 32 , and 64 ⁄ 64 all equal 1. Proper Fraction: A fraction whose numerator is less than its denominator. 1 ⁄4 , 1 ⁄ 2 , and 47⁄ are examples of proper fractions. The value of any proper fraction is less than 1. 64 Improper Fraction: A fraction whose numerator is greater than its denominator. 3⁄ 2, 5⁄4, and –17⁄ 8 are examples of improper fractions. The absolute value of any improper fraction is greater than 1. Reducible Fraction: A reducible fraction is a common fraction in which numerator and denominator have a common factor and so can be reduced to lowest terms by dividing both numerator and denominator by this common factor. For example, in the fraction 12 ⁄ 18, the numerator and denominator have a GCF of 6. Thus, 12 ⁄ 18 reduces to 2 ⁄ 3 by dividing each part of the fraction by 6. A fraction such as 16 ⁄ 21 cannot be reduced, since 16 and 21 do not have a common factor. Mixed Number: A mixed number is a combination of a whole number and a proper fraction. The implied operation between them is addition. For example, 4 92 means 4 + 92 . A mixed number is converted to an improper fraction by multiplying the whole number part with the denominator and adding the numerator to obtain the numerator of the final fraction; the denominator remains the same.

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Machinery's Handbook, 31st Edition Fractions

6 Examples: 5

2 15 2 17 2 + = = 5+ = 3 3 3 3 3

9

1 18 1 19 1 = + = = 9+ 2 2 2 2 2

To convert mixed numbers to improper fractions, multiply the whole number by the denominator and add the numerator to obtain the new numerator. The denominator remains the same. For example, 1 2 × 2 + 1- = 5 --2 --- = -------------------2 2 2 7 3 × 16 + 7 55 3 ------ = ------------------------ = -----16 16 16

An improper fraction is converted to its mixed number form by dividing the numerator by denominator and placing the remainder over the denominator. Sometimes the fraction part can be reduced, as the second example shows. 17 1 = 17 ÷ 8 = 2 8 8

26 5 10 = 26 ÷ 16 = 1 = 1 16 16 8

Equivalent Fractions: A fraction raised to its equivalent form (“higher terms”) by multiplying numerator and denominator by the same number (that is, by multiplying by a form of 1). For example, 1 ⁄4 × 4 ⁄4 = 4 ⁄ 16 and 3 ⁄ 8 × 4 ⁄4 = 12 ⁄ 32 . Any integer n can be expressed as a fraction with a chosen denominator value of m by simply writing n as n/1 and multiplying by m/m. Example: To express 4 as an equivalent fraction with a denominator of 16, write 4 ⁄ 1 × 16 ⁄ = 64 ⁄ 16 16 Reciprocal: The reciprocal of any number a other than 0 is 1/a. (0 has no reciprocal, since 1/0 is undefined.) The reciprocal also is called the multiplicative inverse, since a × 1/a = 1. For example, the recip­rocal of 8 is 1 ⁄ 8; the reciprocal of 4 ⁄ 7 is 7⁄4. Least Common Denominator: Fractions cannot be added or subtracted without a common denominator. For example, 52 + 15 = 25+1 = 53 , a simple computation, since the denominator in the answer is the same denominator seen in the fractions. In general, ac + bc = a +c b . But fractions with different denominators cannot be added or subtracted until they are converted to equivalent forms that have common denominators. This is done by raising the fractions to higher terms (as explained previously). While any common multiple serves as a common denominator, it is preferable to use the least common multiple (LCM) of the denominator, referred to as the least common denominator (LCD). For example, 36 is the LCD of 92 and 56 , since the LCM of 9 and 6 is 36. Raising each fraction to its equivalent form having a denominator of 36 is shown: 2

9

×

4 4

=

8

36

and

5 6

×

6 6

=

30 36

9 7 and 10 Example: In the case of 11 the LCD is the product of the denominators, 11 × 10 = 110. Raising each fraction to its equivalent form is shown:

9

11

×

10 10

=

90

110

and

7

10

×

11 11

=

77

110

Adding and Subtracting Fractions and Mixed Numbers To Add or Subtract Common Fractions: 1) Convert each fraction to terms of the least common denominator; 2) add or subtract numerators; 3) if answer is an improper fraction, change it to a mixed number; and 4) reduce fraction part if necessary.

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Machinery's Handbook, 31st Edition FRACTIONS AND MIXED NUMBERS Example, Addition of Common Fractions 1 4 3 16 + 78

1 4 4×4 3 1 16 × 1 + 78 × 22

=

LCD = 16

= =

7

Example, Subtraction of Common Fractions 4 16 3 16 + 14 16

15 16 7 − 12

15 3 LCD = 48 16 × 3 7 × 44 − 12

45 48 = − 28 48

=

17 48

21 16

To Add Mixed Numbers: Two methods for adding mixed numbers are shown below the explanations. First method: 1) Raise fraction parts to the higher terms of the LCD; 2) add whole number parts and fraction parts separately; 3) if result has an improper fraction, convert it to a mixed number and add the whole number parts. Second method: 1) Convert mixed numbers to improper fractions; 2) raise resulting fractions to the higher terms of the LCD; 3) add fractions as usual and convert back to a mixed number; reduce, if needed. Method 1

2

1 2

2

1 × 16 2 16

4

1 4

→ 4

1 ×8 4 8

+1 15 32

=

2 16 32

=

8 4 32

Examples,  Addition of Mixed Numbers

2 4

1 2

→

1 4

+ 1 15 32

15 ×1 + 1 15 32 1 = + 1 32

Method 2

5 2

=

5 2

× 16

=

80 32

17 4

=

17 4

×8 8

=

136 32

16

47 47 47 1 × = 32 = + 32 + 32 1

7 7 7 3392 = 7 + 1 32 = 8 32

263 32

= 8 327

To Subtract Mixed Numbers: The meth­ods are similar to those for adding, except the fraction part may need to “borrow” from the whole number. The examples show the details. 1) Convert fraction parts to equivalent fractions with LCD; 2) subtract whole number and fraction parts separately, unless the first fraction’s numerator is smaller than the second. In that case, proceed as shown in the second and third examples below, borrowing 1 in the form of a fraction and then subtracting. Example 1 12 54

Examples,  Subtraction of Mixed Numbers

Example 2 43 14 = 15

Example 3

43 14 15

20 = 19 + 1 = 19 22

− 4 15

−19 53 = −19 195

−7 12 = −7 12 = −7 12

8 53

24 155

= 24 13

Example 4

8 92 = 7 + 1 92 = 7 119 −1 94 = −1 94 = −1 94

6 79

12 12

Multiplying Fractions and Mixed Numbers To Multiply Common Fractions: 1) Multiply numerators; 2) multiply denomina­tors; and 3) convert improper fractions to mixed numbers, if necessary. To Multiply Mixed Numbers: 1) Convert mixed numbers to improper fractions; 2) multiply numerators; 3) multiply denominators; and 4) convert improper fractions to mixed numbers, if necessary. Examples, Multiplication of Fractions

2×28×8 1616 22 88 ×15==4545 1515==3×315 3 3××

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4 45 × 1 13 3==5 5××3 3== 5×53×3==1515 5×     22

1 1

2222 4 4

2222 ×4×4

8888

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Machinery's Handbook, 31st Edition FRACTIONS AND MIXED NUMBERS

8

Dividing Fractions and Mixed Numbers To Divide Common Fractions: 1) Take the reciprocal of the dividing fraction; 2) multiply the numerators and denominators; and 3) convert improper fractions to mixed numbers, if necessary. To Divide Mixed Numbers: 1) Convert the mixed numbers to improper fractions; 2) take the reciprocal of the dividing fraction; 3) multiply numerators and denominators; and 4) convert improper fractions to mixed numbers, if necessary. Examples, Division of Fractions

21 42 5 2 ÷ 25 =212 × 221 2 ×21= 2×42 = 6535 5 = 7 ÷ 217 = 721× 5 7 = 75×5 =7×35

25 5 = 65    3 13 ÷ 23 1354 ÷= 210354 ÷=145103=÷ 1014 = 510 ×50145 == 22150 42 = 21 35 × 143 = 42

Decimal Numbers.—Decimal fractions are fractional parts of a whole whose implied denominators are multiples of 10. A decimal fraction of 0.1 has a value of 1/10, 0.01 has a value of 1/100, 0.001 has a value of 1/1000, and so on. Thus, the value of the digit in the first place right of the decimal point is expressed in tenths, a digit two places to the right is expressed in hundredths, a digit three places to the right is expressed in thousandths, and so on. Because the denominator is implied, the number to the right of the decimal point indicates the numerator of the decimal fraction. For example, 0.125 is equivalent to 125/1000. In industry, most decimal fractions are expressed in terms of thousandths rather than tenths or hundredths. For example, a decimal fraction of 0.2 is written as 0.200 and read as “200 thousandths” rather than “2 tenths”; a value of 0.75 is written as 0.750, and read as “750 thousandths” rather than “75 hundredths.” In the case of four place decimals, the values are expressed in terms of ten-thousandths. So a value of 0.1875 is read as “1875 ten-thousandths.” Just as a mixed number is the sum of a whole number and a fraction, a decimal number greater than 1 has a whole and a decimal part. For example, 10.125 = 10125 ⁄ 1000, which is read as “10 and 125 thousandths.” Adding or Subtracting Decimal Numbers: To add or subtract decimal numbers, align the decimal points and add or subtract the digits as usual. The decimal point in the answer is aligned with the decimal points in the numbers added or subtracted. Examples, Adding Decimal Fractions

0.125 1.0625 2.50 + 0.1875

1.750 0.875 0.125 + 2.0005

3.8750

4.7505

Examples, Subtracting Decimal Fractions

1.750 – 0.250

2.625 – 1.125

1.500

1.500

Multiplying Decimal Numbers: In setting up decimal multiplication, the decimal points do not have to be aligned. Long multiplication is done as usual, but the decimal point in the answer is placed so that the number of digits on its right is the same as the total number of digits on the right of the numbers multiplied. Examples, Decimal Number Multiplication

6.002

24.035

three decimal places

× 0.08

two decimal places

× 41.3

five decimal places

18006

1.92280

three decimal places one decimal place

60020

+ 2400800 247.8826

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four decimal places

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Machinery's Handbook, 31st Edition Decimal Numbers

9

Dividing Decimal Numbers: There are several types of decimal division problems: (1) a whole number divided by a decimal number; (2) a decimal number divided by a whole number; and (3) a decimal number divided by a decimal number. For all situations, if the divisor is a decimal, its decimal point must first be moved right to make it a whole number, and the dividend’s decimal likewise moved, before the operation is performed. Examples of each type are: 18 ÷ 0.3 = 180 ÷ 3 = 60; 1.8 ÷ 3 = 0.6; 1.8 ÷ 0.003 = 1800 ÷ 3 = 600.

Ratio and Proportion.—A ratio of quantities a to b is written a:b or as a fraction a/b. For example, the ratio of 12 to 3 is written 12:3 or 12/3. Ratios, like fractions, can be reduced: 12:3 is 4:1. The inverse (or reciprocal) ratio of a:b is b:a. Thus, the inverse ratio of 12:3 is 3:12. When two or more ratios are multiplied, the ratio obtained is a called a compound ratio. The compound ratio of a:b, c:d, and e:f is the ratio ace:bdf. For example, the compound ratio of 8:2, 9:3, and 10:5 is 8 × 9 × 10: 2 × 3 × 5, or 720:30. An equality of ratios, a/b = c/d, is called a proportion, which can be written as a:b::c:d, read as “a is to b as c is to d.” Thus, 6:3::10:5 because 6/3 and 10/5 both reduce to 2/1 or 2. In a proportion a:b::c:d, the first and last terms (which can be variables or numbers) are called the extremes, and the second and third are the means. Note that if a/b = c/d, then the rules of algebra show that ad = bc. Thus, the proportion a:b::c:d is equivalent to a × d = b × c. So the proportion 6:3::10:5 is equivalent to 6 × 5 = 3 × 10. Often, some part of a proportion is an unknown. For example, in the proportion 2:3::n:4 (2 is to 3 as n is to 4), n is found by setting up a proportion. According to the basic rules of algebra, 2:3::n:4 means (2)(4) = 3n, and hence, 8 = 3n, so n = 8/3. A full discussion of the rules for solving equations can be found in ALGEBRA. If the second and third terms in a proportion are the same, that term is the mean proportional of the other two. Thus, in the proportion 8:4::4:2, 4 is the mean proportional of 8 and 2. The mean proportional of any two numbers may be found by multiplying them and extracting the square root of the product. Thus, the mean proportional of 3 and 12 is 6, because 3 × 12 = 36, which is 62. Example 1, Involving Proportion: If it takes 18 days to assemble 4 lathes, how many days would it take to assemble 14 lathes? Solution: Let x be the number of days to be found. The proportion is written 4:18 :: 14:x, where x is the number of days to be found. Setting this up as an equation and solving: 4 14 = 18 x × 14- = 63 days ----------------x = 18 4

Example 2, Involving Direct (Simple) Proportion: 10 linear meters (32.81 feet) of bar stock are required as blanks for 100 clamping bolts. What total length x of stock, in meters and feet, is required for 912 bolts? Solution: The setup to solve the proportional meters-to-bolts problem comes from the way this proportion is read: “10 meters is to 100 bolts as how many meters is to 912 bolts.” It is solved accordingly: 10 x 10 × 912 9120 = = 91.2 meters = Solving for x : x = 100 912 100 100 10 x 10 × 912 9120 10 : 100 :: xthe : 912 , that to is,32 Solving for x : x = comes =from = 91 .2 meters Likewise, setup solve problem it as: “32.81 .81 = the x feet-to-bolts 32.81 × 912 reading 29,922.72 100 = 32 .81is : 100 :: x : bolts 912, that is, 100 = 912 for x: Thus: x 1=00 = 299.2 feet feet to 100 as how many feet is toSolving 912 bolts.” 100 912 100 100 32.81 × 912 29,922.72 32.81 x 32.81 : 100 :: x : 912, that is, = Solving for x: x= = = 299.2 feet 100 912 100 100 10 : 100 :: x : 912, that is,

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Machinery's Handbook, 31st Edition RATIO AND PROPORTION

Inverse Proportion: Quantities with an inversely proportional relationship behave in such a way that as one increases the other decreases. For example, a factory employing 270 workers completes a given number of automotive components weekly, the number of working hours being 44 per week. If the hours are reduced, then more workers will be required to do the same amount of work. How many employees would be required for the same production if the working hours were reduced to 40 per week? The hours per week is inversely proportional to the number of workers; fewer hours per worker means more workers are required. Letting x be the number of workers needed when time is reduced, the inverse proportion is written: Thus

270 : x :: 40 : 44

270 270 × 44- = 297 workers --------- = 40 -----and x = -------------------40 x 44 Problems Involving Both Direct and Inverse Proportions: If two groups of data are related by both direct (simple) and inverse proportions among the various quantities, a simple mathematical relation may be used to solve the problem as follows: Product of all directly proportional items in first group------------------------------------------------------------------------------------------------------------------------------------Product of all inversely proportional items in first group Product of all directly proportional items in second group = --------------------------------------------------------------------------------------------------------------------------------------------Product of all inversely proportional items in second group

Example: If a worker capable of turning 65 studs in a 10-hour day is paid $13.50 per hour, how much per hour should a worker be paid who turns 72 studs in a 9-hour day if compensated in the same proportion as the first worker? Solution: The first group of data in this problem consists of the number of hours worked, the hourly wage of the first worker, and the number of studs produced per day; the second group contains similar data for the second worker, except the hourly wage is unknown, so it is indicated by x. The labor cost per stud, as may be seen, is directly proportional to the number of hours worked and the hourly wage. These quantities, therefore, are used in the numerators of the fractions in the formula. The labor cost per stud is inversely proportional to the number of studs produced per day. (The greater the number of studs produced in a given time the less the cost per stud.) The numbers of studs per day, therefore, are placed in the denominators of the fractions in the formula. Thus, (---------------------------10 ) ( 13.50 )- = 9x -----65 72 ( 10 ) ( 13.50 ) ( 72 -) = $16.62 per hour x = ---------------------------------------( 65 ) ( 9 )

Percentage.—A percentage is a ratio expressed as a part of 100. For example, if out of 100 manufactured parts, 12 do not pass inspection, then 12 percent (12 of the 100) are rejected. The symbol % indicates percentage. The percent of gain (or loss) with respect to a base (original) quantity is found by divid­ ing the amount of gain (or loss) by the base quantity and multiplying the quotient by 100. For example, if a quantity of steel is bought for $2000 and sold for $2500, the profit is $500/2000 × 100, or 25 percent of the invested amount. Example: Out of a total output of 280 castings a day, 30 castings are, on average, rejected. What is the percentage of bad castings? 30-------× 100 = 10.71 percent 280

Percent Change: Any increase or decrease in some measured quantity can be expressed as a percent change using the formula:

final − original amount

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original

× 100 = percent change. If in the

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Machinery's Handbook, 31st Edition PERCENTAGE

11

previous example, production increased from 280 to 300, then the percent change would be: final − original amount original

× 100 =

300 − 280 280

× 100 =

20

0 280

× 100 = 7.14%.

The denominator is always the original amount. Percent change also can be negative. If production decreased from 280 to 245, the percent change would be: 245 − 280 280

× 100 =

−35 280

× 100 = −12.5% .

Powers and Roots

Powers: The square or second power of a number (or quantity) is the product of that number multiplied by itself. Thus, the square of 9 is 9 × 9. The square of a number is indicated by the exponent 2, thus: 92 = 9 × 9 = 81.

The cube or third power of a number n is the product n × n × n, or n3. Thus, the cube of 4 is 4 × 4 × 4 = 64, and is written 43. In general, the nth power of a is written an, where a is the base and n is the exponent. Roots: The square root of a given number is the positive number which, when multiplied by itself, will produce the given number. The square root of 16 (written 16 ) is 4 because 4 × 4 = 16. The other root of 16 is –4, but the use of the square root symbol indicates the positive (principal) square root only. Similarly, the cube root of a given number is the number which, when used as a factor three times, will produce the given number. Thus, the cube root of 64 (written 3 64 ) is 4 because 4 × 4 × 4 = 64. In general, the nth root of a is written as n a or a1/n.

Properties of Exponents a n- = a ( n – m ) ----( a m ) n = a mn am

an am = an + m

(b) a

n

=

n

a bn

a m / n = ( a 1 / n ) m or (am)1/n

a0 = 1 ( a ≠ 0 )

a1 / n =

n

a

( ab ) m = a m b m

1a – n = ---an

m

am / n = ( n a ) =

(a) n

m

1 - = an ------a –n unless a < 0, and m and n are both even

Examples  :

3 1 3 2 = 3 1 + 2 = 3 3 = 27

( x ) ( x3 ) = x(1 + 3) = x4

5 4- = 5 4 – 2 = 5 2 = 25 ---52

x 9- = x ( 9 – 6 ) = x 3 ---x6

( 2 4 ) 2 = 2 ( 4 ) ( 2 ) = 2 8 = 256

( x3 )3 = x(3)(3) = x9

( 9x ) 2 = 9 2 x 2 = 81x 2

( ab 4 ) 2 = a 2 b 8 3

32 3 / 5 = ( 32 1 / 5 ) 3 = ( 5 32 ) = 23 = 8

1- = ----14 – 3 = ---64 43

1 - = 2 5 = 32 ------2 –5

9x 0 = 9 ( 1 ) = 9

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12

Machinery's Handbook, 31st Edition POWERS AND ROOTS

Using logarithms can greatly facilitate the process of raising a number to a power or extracting its root. As shown in Logarithms on page 36, this is especially true if the power is not an integer. For example, the square root of 137.1 can only be found with a degree of accuracy through logarithms, a scientific calculator, or Taylor series polynomials. Scientific Notation.—Calculations involving both large and small magnitude numbers are facilitated by scientific notation. In this system, a number is expressed by two factors: (1) an integer from 1 to 9, possibly followed by a decimal, and (2) a power of 10. Large numbers in standard form are converted to scientific notation as shown in the following examples: 273.15 = 2.7315 × 102 50,000 = 5 × 104    4 In the example, 50,000 becomes 5 × 10 because the positive exponent on 10 is the number of places to the right that the decimal point moves so that the first factor falls between 1 and 10. Numbers less than 1 are converted to scientific notation as shown in the following examples: 0.0000001 = 1 × 10–7 0.840 = 8.40 × 10–1    The negative exponent shows the number of places to the left that the decimal point moves, so that the first factor falls between 1 and 10. Science and engineering quantities—which are often quite large or small—lend themselves to representation in scientific notation. For instance, Avogadro’s number, which is the number of particles in one mole of a substance, is 6.024 × 1023. The metric (SI) pressure unit of 1 pascal (Pa) is equivalent to 0.00000986923 atmosphere (atm) or 0.0001450377 pound/square inch (psi). In scientific notation, these figures are 9.86923 × 10 –6 atm and 1.450377 × 10 –4 psi, respectively. Engineering notation is a version of scientific notation in which the exponent of 10 is always a multiple of 3. (See MEASURING UNITS on page 2827 for a table of this system.) Multiplication in Scientific Notation: The procedure is as follows: 1) Multiply the first factors of the numbers to obtain the first factor of the product. 2) Add the exponents of the factors of 10 to obtain the product’s factor of 10. Thus: ( 4.31 × 10 – 2 ) × ( 9.01 × 10 ) = ( 4.31 × 9.01) × 10 – 2 + 1 = 38.8331 × 10 – 1 ( 5.98 × 10 4 ) × ( 4.37 × 10 3 ) = ( 5.98 × 4.37 ) × 10 4 + 3 = 26.1326 × 10 7 3) Write the final in conventional scientific notation, as explained in the previous section. So, for the two examples: 38.8331 × 10 –1 = 3.88331 × 10 0 = 3.88331, because 10 0 = 1, and 26.1326 × 107 = 2.61326 × 108. When multiplying several numbers written in this notation, the procedure is the same. Thus, (4.02 × 10 –3) × (3.987 × 10) × (4.863 × 105) = (4.02 × 3.987 × 4.863) × 10(–3+1+5) = 77.94 × 103 = 7.79 × 104, rounding off the first factor to two decimal places. Division in Scientific Notation: The procedure is as follows: 1) Divide the first factor of the dividend (the first number) by the first factor of the divisor (the second number) to get the first factor of the quotient. 2) Subtract the exponents of the factors of 10 to obtain the product’s factor of 10: ( 4.31 × 10 – 2 ) ÷ ( 9.0125 × 10 ) =

( 4.31 ÷ 9.0125 ) × ( 10 – 2 – 1 ) = 0.4782 × 10 – 3 = 4.782 × 10 – 4

It can be seen that this system of notation is helpful where several numbers of different magnitudes are to be multiplied and divided. 250 × 4698 × 0.00039 Example: Find the solution of --------------------------------------------------------43678 × 0.002 × 0.0147 Solution: Changing all these numbers to powers of 10 notation and performing the oper­ ations indicated:

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13

( 2.5 × 10 2 ) × ( 4.698 × 10 3 ) × ( 3.9 × 10 – 4 )---------------------------------------------------------------------------------------------------------( 4.3678 × 10 4 ) × ( 2 × 10 – 3 ) × ( 1.47 × 10 – 2 ) ( 2.5 × 4.698 × 3.9 ) ( 10 2 + 3 – 4 )- = -----------------------------------45.8055 × 10 = -------------------------------------------------------------------------( 4.3678 × 2 × 1.47 ) ( 10 4 – 3 – 2 ) 12.8413 × 10 – 1 = 3.5670 × 10 1 – ( – 1 ) = 3.5670 × 10 2 = 356.70 (rounded)

Factorial Notation.—A factorial is a mathematical shortcut denoted by the symbol ! fol­ lowing a number (for example, 3! is “three factorial”). n! is found by multiplying together all the positive integers less than or equal to the factorial number n. Zero factorial (0!) is defined as 1. For example: 3! = 1 × 2 × 3 = 6; 4! = 1 × 2 × 3 × 4 = 24; 7! = 1 × 2 × 3 × 4 × 5 × 6 × 7 = 5040; etc. Factorial notation is used in certain areas, including probability and analysis. The following two topics (permutations and combinations) relate to probability and statistics. Permutation.—A permutation is an arrangement of objects of a set into a sequence or order. In mathematics, the number of arrangements of n objects is given by n!. For example, 4 objects can be arranged 4! ways, that is, 4 × 3 × 2 × 1 = 24 ways. The number of ways r objects can be arranged (that is, ordered) from a set of n is given by the permutation n! formula n Pr = ----------------( n – r )! Example: How many ways can the letters X, Y, and Z be arranged? Solution: Three objects (r = 3) out of a set of 3 (n = 3) are being arranged. The numbers of possible arrangements for the three letters are 3!/(3 – 3)!= (3 × 2 × 1)/1 = 6. Listing them is not difficult, since there are so few: XYZ, XZY, YXZ, YZX, ZXY, ZYX. Example: There are 10 people participating in a foot race. How many arrangements of first, second, and third place winners are there? Solution: Here r is 3 and n is 10. The number of possible arrangements of winners are: 10! 10! P 10 3 = --------------------- = -------- = 10 × 9 × 8 = 720 ( 10 – 3 )! 7! Combination.—This is the number of ways r objects can be chosen from n in a way that order does not matter. It is expressed as “n choose r.” There are fewer combinations than permutations of r objects out of n, since it does not matter in what order the three objects are chosen. So in a combination, choosing ABC is the same as choosing ACB or BAC and n! so on. The formula is nCr = ---------------------( n – r )!r! Example: How many possible sets of 6 numbers can be picked with no regard for order from the numbers 1 to 52? Solution: Here r is 6 and n is 52. So the possible number of combinations is:

52! - = -----------52! - = 52 × 51 × 50 × 49 × 48 × 47- = -----------------------------------------------------------------= -------------------------20,358,520 ( 52 – 6 )!6! 46!6! 1×2×3×4×5×6 Prime Factorization of Numbers.—Tables of prime numbers and factors of numbers are particularly useful for calculations involving change-gear ratios for compound gearing, dividing heads, gear-generating machines, and mechanical designs having gear trains. Definition: p is a factor of a number n if the division n/p leaves no remainder. Thus, any number n has factors of itself and 1, because n/n = 1 and n/1 = n. Other factors of a number are found as follows: 2 is a factor of any even number. Thus, 28 = 2 × 14, and 210 = 2 × 105. 3 is a factor of any number where the sum of its digits is divisible by 3. Thus, 3 is a factor of 1869, because 1 + 8 + 6 + 9 = 24, and 24 ÷ 3 = 8. 52C6

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Machinery's Handbook, 31st Edition PRIME FACTORIZATION OF NUMBERS

14

4 is a factor of any number in which the last two digits are a number divisible by 4. Thus, 1844 has a factor 4, because 44 ÷ 4 = 11. 761 does not have a factor of 4, since 61 is not divisible by 4. 5 is a factor of any number that has a ones digit that is either 0 or 5. A prime number is one that has no factors except itself and 1. Thus, 2, 3, 5, 7, 11, etc. are prime numbers. 2 is the only even prime number. A factor which itself is a prime number is called a prime factor. All numbers can be expressed as a product of their prime factors. It can be determined if 7 is a factor of a number according to this process: Remove the last digit from the number, double it, and subtract it from the remaining number. If the result is divisible by 7 (e.g., 14, 7, 0, –7, etc.), then the number is divisible by 7. The prime factorization of a number is done by expressing the number as a product of its primes. For example, the prime factors of 20 are 2 and 5; the prime factorization is 2 × 2 × 5 = 20. The Prime Number and Factor Table, starting on page 15, give the smallest prime fac­tor of all odd numbers from 1 to 9600, and can be used for finding all the factors for num­bers up to this odd number. Where no factor is given for a number in the table, the letter P indicates that the number is a prime number. The last page of the tables lists prime numbers from 9551 through 18691; it can be used to identify unfactorable numbers in that range. Example 1: Find the factors of 833. Use the table on page 15 as illustrated below. Solution: The table on page 15 indicates that 7 is the smallest prime factor of 833, shown at the row-column intersection for 833. This leaves another factor, because 833 ÷ 7 = 119. From To …

33

…

0 100

100 200

200 300

300 400

400 500

500 600

600 700

700 800

800 900

900 1000

1000 1100

…

…

…

…

…

…

…

…

…

…

…

3 …

7 …

P …

3 …

P …

13 …

3 …

P …

7 …

3 …

It also shows that 7 is a prime factor of 119, leaving a factor 119 ÷ 7 = 17. From To 19

…

…

11

1100 1200

…

100 200

200 300

300 400

400 500

500 600

600 700

700 800

800 900

900 1000

1000 1100

…

…

…

…

…

…

…

…

…

…

…

…

7 …

3 …

11 …

P …

3 …

P …

P …

3 …

P …

P …

P indicates that 17 is a prime number and no other prime factors of 833 exist. From To 17

…

…

0 100

100 200

200 300

300 400

400 500

500 600

600 700

700 800

800 900

900 1000

1000 1100

…

…

…

…

…

…

…

…

…

…

…

P …

3 …

7 …

P …

3 …

11 …

P …

3 …

19 …

7 …

…

P

0 100 P

1100 1200

3 …

…

…

3 …

1100 1200 …

P …

Hence, the prime factorization of 833 is 7 × 7 × 17. Example 2: A set of four gears is required in a mechanical design to provide an overall gear ratio of 4104 ÷ 1200. Furthermore, no gear in the set is to have more than 120 teeth or less than 24 teeth. Determine the tooth numbers. Solution: The prime factorization of 4104 is determined to be 2 × 2 × 2 × 3 × 3 × 57 = 4104. The prime factorization of 1200 is determined to be 2 × 2 × 2 × 2 × 5 × 5 × 3 = 4104 2 × 2 × 2 × 3 × 3 × 57 72 × 57 1200. Therefore, ------------ = ---------------------------------------------------------- = ------------------ . Each resulting factor 1200 2×2×2×2×5×5×3 24 × 50 represents the number of teeth that fulfill the requirement. If the factors had been com­ 72 × 57 bined differently, say, to give ------------------ , then the 16-tooth gear in the denominator would 16 × 75 not satisfy the requirement of having no less than 24 teeth. Example 3: Factor 25,078 into two numbers, neither of which is larger than 200. Solution: The smallest factor of 25,078 is obviously 2, leaving 25,078 ÷ 2 = 12,539 to be factored further. However, from the last table, Prime Numbers from 9551 to 18691, on page 23, it is seen that 12,539 is a prime number; therefore, no other factors exist. So the factorization named is not possible.

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Machinery's Handbook, 31st Edition PRIME NUMBER AND FACTOR TABLES

15

Prime Number and Factor Table for 1 to 1199 From To 1 2 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 67 69 71 73 75 77 79 81 83 85 87 89 91 93 95 97 99

0 100 P

P P P P 3 P P 3 P P 3 P 5 3 P P 3 5 P 3 P P 3 P 7 3 P 5 3 P P 3 5 P 3 P P 3 7 P 3 P 5 3 P 7 3 5 P 3

100 200 P 2 P 3 P P 3 P 5 3 7 11 3 5 P 3 P 7 3 P P 3 11 5 3 P P 3 5 P 3 7 P 3 P 13 3 P 5 3 P P 3 5 11 3 P P 3 P P

200 300 3 2 7 5 3 11 P 3 5 7 3 13 P 3 P P 3 P 5 3 P P 3 5 13 3 P 11 3 P 7 3 P 5 3 P P 3 5 P 3 P P 3 7 17 3 P 5 3 13

300 400 7 2 3 5 P 3 P P 3 P 11 3 17 5 3 7 P 3 5 P 3 11 7 3 P P 3 P 5 3 P 19 3 5 P 3 7 P 3 13 P 3 P 5 3 P 17 3 5 P 3

400 500 P 2 13 3 11 P 3 7 5 3 P P 3 5 7 3 P P 3 19 P 3 P 5 3 P 11 3 5 P 3 P P 3 P 7 3 11 5 3 P 13 3 5 P 3 P 17 3 7 P

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500 600 3 2 P 5 3 P 7 3 5 11 3 P P 3 17 23 3 13 5 3 7 P 3 5 P 3 19 7 3 P 13 3 P 5 3 P P 3 5 P 3 7 11 3 P 19 3 P 5 3 P

600 700 P 2 3 5 P 3 13 P 3 P P 3 7 5 3 17 P 3 5 7 3 P P 3 P 11 3 P 5 3 P P 3 5 23 3 11 P 3 P 7 3 P 5 3 13 P 3 5 17 3

700 800 P 2 19 3 7 P 3 23 5 3 P 7 3 5 P 3 17 P 3 11 P 3 P 5 3 7 P 3 5 P 3 P 7 3 13 P 3 P 5 3 19 11 3 5 P 3 7 13 3 P 17

800 900 3 2 11 5 3 P P 3 5 19 3 P P 3 P P 3 7 5 3 P 29 3 5 7 3 23 P 3 P P 3 P 5 3 11 13 3 5 P 3 P P 3 P 7 3 19 5 3 29

900 1000 17 2 3 5 P 3 P 11 3 7 P 3 13 5 3 P 7 3 5 P 3 P 23 3 P 13 3 P 5 3 7 31 3 5 P 3 P 7 3 P 11 3 P 5 3 23 P 3 5 P 3

1000 1100 7 2 17 3 19 P 3 P 5 3 P P 3 5 13 3 P P 3 17 P 3 7 5 3 P P 3 5 7 3 P P 3 11 P 3 29 5 3 13 23 3 5 P 3 P P 3 P 7

1100 1200 3 2 P 5 3 P 11 3 5 P 3 19 P 3 7 P 3 11 5 3 17 7 3 5 31 3 P P 3 13 19 3 P 5 3 7 P 3 5 11 3 P 7 3 P 29 3 P 5 3 11

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Prime Number and Factor Table for 1201 to 2399 From To 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 67 69 71 73 75 77 79 81 83 85 87 89 91 93 95 97 99

1200 1300 P 3 5 17 3 7 P 3 P 23 3 P 5 3 P P 3 5 P 3 17 11 3 29 P 3 7 5 3 P 13 3 5 7 3 31 19 3 P P 3 P 5 3 P P 3 5 P 3

1300 1400 P P 3 P 7 3 13 5 3 P P 3 5 P 3 11 31 3 7 13 3 17 5 3 19 7 3 5 23 3 P 29 3 P 37 3 P 5 3 7 P 3 5 19 3 13 7 3 11 P

1400 1500 3 23 5 3 P 17 3 5 13 3 7 P 3 P P 3 P 5 3 P 11 3 5 P 3 P P 3 31 P 3 7 5 3 13 P 3 5 7 3 P P 3 P P 3 P 5 3 P

1500 1600 19 3 5 11 3 P 17 3 37 7 3 P 5 3 11 P 3 5 29 3 23 P 3 7 P 3 P 5 3 P 7 3 5 P 3 P 11 3 19 P 3 P 5 3 7 37 3 5 P 3

1600 1700 P 7 3 P P 3 P 5 3 P P 3 5 P 3 7 23 3 P 11 3 31 5 3 17 13 3 5 P 3 11 P 3 P P 3 7 5 3 23 41 3 5 7 3 19 P 3 P P

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1700 1800 3 13 5 3 P 29 3 5 17 3 P P 3 11 7 3 P 5 3 37 P 3 5 P 3 17 P 3 7 P 3 41 5 3 29 7 3 5 P 3 13 P 3 P P 3 11 5 3 7

1800 1900 P 3 5 13 3 P 7 3 23 17 3 P 5 3 31 P 3 5 11 3 7 19 3 P 43 3 17 5 3 11 P 3 5 P 3 P P 3 P P 3 7 5 3 P 31 3 5 7 3

1900 2000 P 11 3 P 23 3 P 5 3 19 17 3 5 41 3 P P 3 13 7 3 29 5 3 P P 3 5 19 3 37 13 3 7 11 3 P 5 3 P 7 3 5 P 3 11 P 3 P P

2000 2100 3 P 5 3 7 P 3 5 P 3 43 7 3 P P 3 19 5 3 P 13 3 5 23 3 7 P 3 11 29 3 P 5 3 P 19 3 5 31 3 P P 3 P P 3 7 5 3 P

2100 2200 11 3 5 7 3 P P 3 29 13 3 11 5 3 P P 3 5 P 3 P P 3 19 7 3 P 5 3 17 P 3 5 11 3 13 41 3 7 P 3 37 5 3 11 7 3 5 13 3

2200 2300 31 P 3 P 47 3 P 5 3 7 P 3 5 17 3 23 7 3 P P 3 P 5 3 13 P 3 5 37 3 7 31 3 P P 3 P 5 3 43 P 3 5 P 3 29 P 3 P 11

2300 2400 3 7 5 3 P P 3 5 7 3 11 23 3 13 17 3 P 5 3 P P 3 5 P 3 P 13 3 P 7 3 17 5 3 23 P 3 5 P 3 P P 3 7 P 3 P 5 3 P

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Machinery's Handbook, 31st Edition PRIME NUMBER AND FACTOR TABLES

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Prime Number and Factor Table for 2401 to 3599 From To 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 67 69 71 73 75 77 79 81 83 85 87 89 91 93 95 97 99

2400 2500 7 3 5 29 3 P 19 3 P 41 3 P 5 3 7 11 3 5 P 3 P 7 3 P 31 3 11 5 3 P 23 3 5 P 3 7 P 3 P 37 3 13 5 3 19 47 3 5 11 3

2500 2600 41 P 3 23 13 3 7 5 3 11 P 3 5 7 3 P 17 3 43 P 3 P 5 3 P P 3 5 P 3 13 11 3 17 7 3 31 5 3 P 29 3 5 13 3 P P 3 7 23

2600 2700 3 19 5 3 P 7 3 5 P 3 P 43 3 37 11 3 P 5 3 7 19 3 5 P 3 11 7 3 P P 3 P 5 3 17 P 3 5 P 3 7 P 3 P P 3 P 5 3 P

2700 2800 37 3 5 P 3 P P 3 11 P 3 7 5 3 P P 3 5 7 3 P 13 3 41 P 3 P 5 3 31 11 3 5 P 3 17 47 3 P 7 3 11 5 3 P P 3 5 P 3

2800 2900 P P 3 7 53 3 29 5 3 P 7 3 5 11 3 19 P 3 P 17 3 P 5 3 7 P 3 5 P 3 P 7 3 47 19 3 13 5 3 P 43 3 5 P 3 7 11 3 P 13

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2900 3000 3 P 5 3 P 41 3 5 P 3 23 37 3 P 29 3 7 5 3 P 17 3 5 7 3 13 P 3 P 11 3 P 5 3 P P 3 5 13 3 11 19 3 29 7 3 41 5 3 P

3000 3100 P 3 5 31 3 P 23 3 7 P 3 P 5 3 13 7 3 5 P 3 P 17 3 11 P 3 43 5 3 7 P 3 5 P 3 37 7 3 17 P 3 P 5 3 P 11 3 5 19 3

3100 3200 7 29 3 13 P 3 11 5 3 P P 3 5 53 3 31 13 3 P 43 3 7 5 3 47 23 3 5 7 3 29 P 3 P P 3 19 5 3 11 P 3 5 P 3 P 31 3 23 7

3200 3300 3 P 5 3 P 13 3 5 P 3 P 11 3 7 P 3 53 5 3 41 7 3 5 17 3 P P 3 P P 3 13 5 3 7 P 3 5 29 3 17 7 3 19 11 3 37 5 3 P

3300 3400 P 3 5 P 3 7 P 3 31 P 3 P 5 3 P P 3 5 47 3 13 P 3 P 17 3 7 5 3 P P 3 5 7 3 P P 3 11 31 3 17 5 3 P P 3 5 43 3

3400 3500 19 41 3 P 7 3 P 5 3 13 11 3 5 23 3 47 P 3 7 19 3 11 5 3 P 7 3 5 P 3 P P 3 P P 3 23 5 3 7 59 3 5 11 3 P 7 3 13 P

3500 3600 3 31 5 3 11 P 3 5 P 3 7 13 3 P P 3 P 5 3 P P 3 5 P 3 53 11 3 P P 3 7 5 3 43 P 3 5 7 3 P P 3 17 37 3 P 5 3 59

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Prime Number and Factor Table for 3601 to 4799 From To 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 67 69 71 73 75 77 79 81 83 85 87 89 91 93 95 97 99

3600 3700 13 3 5 P 3 23 P 3 P 7 3 P 5 3 19 P 3 5 P 3 11 P 3 7 41 3 13 5 3 P 7 3 5 19 3 P P 3 P 13 3 29 5 3 7 P 3 5 P 3

3700 3800 P 7 3 11 P 3 47 5 3 P 61 3 5 P 3 7 P 3 37 P 3 19 5 3 23 11 3 5 13 3 P 53 3 P P 3 7 5 3 P 19 3 5 7 3 17 P 3 P 29

3800 3900 3 P 5 3 13 37 3 5 11 3 P P 3 43 7 3 P 5 3 11 23 3 5 P 3 P P 3 7 17 3 P 5 3 53 7 3 5 P 3 P 11 3 13 P 3 17 5 3 7

3900 4000 47 3 5 P 3 P 7 3 P P 3 P 5 3 P P 3 5 31 3 7 P 3 P 11 3 59 5 3 37 17 3 5 P 3 11 29 3 41 23 3 7 5 3 P 13 3 5 7 3

4000 4100 P P 3 P 19 3 P 5 3 P P 3 5 P 3 29 37 3 11 7 3 13 5 3 P P 3 5 P 3 31 17 3 7 13 3 P 5 3 P 7 3 5 61 3 P P 3 17 P

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4100 4200 3 11 5 3 7 P 3 5 23 3 13 7 3 P P 3 P 5 3 P 41 3 5 11 3 7 P 3 P P 3 23 5 3 11 43 3 5 P 3 37 47 3 53 59 3 7 5 3 13

4200 4300 P 3 5 7 3 P 11 3 P P 3 41 5 3 P P 3 5 19 3 P P 3 31 7 3 P 5 3 P P 3 5 17 3 P P 3 7 11 3 P 5 3 P 7 3 5 P 3

4300 4400 11 13 3 59 31 3 19 5 3 7 29 3 5 P 3 61 7 3 P P 3 43 5 3 P 19 3 5 P 3 7 P 3 11 17 3 P 5 3 29 13 3 5 41 3 P 23 3 P 53

4400 4500 3 7 5 3 P 11 3 5 7 3 P P 3 19 43 3 11 5 3 23 P 3 5 P 3 P 61 3 P 7 3 P 5 3 41 17 3 5 11 3 P P 3 7 67 3 P 5 3 11

4500 4600 7 3 5 P 3 13 P 3 P P 3 P 5 3 7 23 3 5 13 3 19 7 3 P P 3 29 5 3 47 P 3 5 P 3 7 17 3 23 19 3 P 5 3 13 P 3 5 P 3

4600 4700 43 P 3 17 11 3 7 5 3 31 P 3 5 7 3 11 41 3 P P 3 P 5 3 P P 3 5 P 3 59 P 3 13 7 3 P 5 3 P 31 3 5 43 3 P 13 3 7 37

4700 4800 3 P 5 3 17 7 3 5 53 3 P P 3 29 P 3 P 5 3 7 11 3 5 47 3 P 7 3 67 P 3 11 5 3 19 13 3 5 17 3 7 P 3 P P 3 P 5 3 P

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Prime Number and Factor Table for 4801 to 5999 From To 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 67 69 71 73 75 77 79 81 83 85 87 89 91 93 95 97 99

4800 4900 P 3 5 11 3 17 P 3 P 61 3 7 5 3 11 P 3 5 7 3 47 29 3 37 13 3 23 5 3 43 P 3 5 31 3 P 11 3 P 7 3 19 5 3 P 67 3 5 59 3

4900 5000 13 P 3 7 P 3 17 5 3 P 7 3 5 13 3 P P 3 P 11 3 P 5 3 7 P 3 5 P 3 11 7 3 P P 3 P 5 3 13 17 3 5 P 3 7 P 3 19 P

5000 5100 3 P 5 3 P P 3 5 29 3 P P 3 11 47 3 7 5 3 P 71 3 5 7 3 P 31 3 13 P 3 61 5 3 37 11 3 5 P 3 P 13 3 P 7 3 11 5 3 P

5100 5200 P 3 5 P 3 19 P 3 7 P 3 47 5 3 23 7 3 5 11 3 53 37 3 P 19 3 P 5 3 7 13 3 5 P 3 P 7 3 31 P 3 71 5 3 P 29 3 5 P 3

5200 5300 7 11 3 41 P 3 13 5 3 17 23 3 5 P 3 P P 3 P 13 3 7 5 3 29 59 3 5 7 3 P 19 3 23 11 3 P 5 3 P P 3 5 17 3 11 67 3 P 7

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5300 5400 3 P 5 3 P 47 3 5 13 3 17 P 3 7 73 3 P 5 3 19 7 3 5 P 3 P 53 3 11 23 3 31 5 3 7 41 3 5 19 3 P 7 3 P 17 3 P 5 3 P

5400 5500 11 3 5 P 3 7 P 3 P P 3 11 5 3 61 P 3 5 P 3 P P 3 13 P 3 7 5 3 53 43 3 5 7 3 P 13 3 P P 3 P 5 3 11 17 3 5 23 3

5500 5600 P P 3 P 7 3 37 5 3 P P 3 5 P 3 P 11 3 7 29 3 23 5 3 31 7 3 5 P 3 67 P 3 19 P 3 P 5 3 7 P 3 5 37 3 P 7 3 29 11

5600 5700 3 13 5 3 71 31 3 5 41 3 7 P 3 17 13 3 43 5 3 P P 3 5 P 3 P P 3 P P 3 7 5 3 P 53 3 5 7 3 13 P 3 11 P 3 P 5 3 41

5700 5800 P 3 5 13 3 P 29 3 P 7 3 59 5 3 17 11 3 5 P 3 P P 3 7 P 3 11 5 3 13 7 3 5 73 3 29 23 3 53 P 3 P 5 3 7 P 3 5 11 3

5800 5900 P 7 3 P 37 3 P 5 3 11 P 3 5 P 3 7 19 3 13 P 3 P 5 3 P P 3 5 P 3 P 11 3 P P 3 7 5 3 P P 3 5 7 3 43 71 3 P 17

5900 6000 3 P 5 3 19 23 3 5 61 3 31 P 3 P 7 3 17 5 3 P 13 3 5 19 3 11 P 3 7 59 3 67 5 3 47 7 3 5 43 3 P 31 3 P 53 3 13 5 3 7

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Prime Number and Factor Table for 6001 to 7199 From To 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 67 69 71 73 75 77 79 81 83 85 87 89 91 93 95 97 99

6000 6100 17 3 5 P 3 P 7 3 11 13 3 19 5 3 P 37 3 5 P 3 7 P 3 P 23 3 P 5 3 73 11 3 5 P 3 13 P 3 59 P 3 7 5 3 P P 3 5 7 3

6100 6200 P 17 3 31 41 3 P 5 3 29 P 3 5 11 3 P P 3 17 7 3 P 5 3 11 P 3 5 47 3 61 P 3 7 31 3 P 5 3 37 7 3 5 23 3 41 11 3 P P

6200 6300 3 P 5 3 7 P 3 5 P 3 P 7 3 13 P 3 23 5 3 17 79 3 5 P 3 7 13 3 P 11 3 P 5 3 P P 3 5 P 3 11 61 3 P 19 3 7 5 3 P

6300 6400 P 3 5 7 3 P 59 3 P 71 3 P 5 3 P 13 3 5 P 3 17 P 3 11 7 3 P 5 3 P P 3 5 P 3 23 P 3 7 P 3 13 5 3 P 7 3 5 P 3

6400 6500 37 19 3 43 13 3 11 5 3 7 P 3 5 P 3 59 7 3 41 47 3 17 5 3 P P 3 5 11 3 7 23 3 29 P 3 P 5 3 11 P 3 5 13 3 P 43 3 73 67

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6500 6600 3 7 5 3 23 17 3 5 7 3 P 11 3 61 P 3 47 5 3 13 31 3 5 P 3 P P 3 79 7 3 P 5 3 P P 3 5 P 3 P 29 3 7 11 3 19 5 3 P

6600 6700 7 3 5 P 3 11 17 3 13 P 3 37 5 3 7 19 3 5 P 3 29 7 3 17 61 3 P 5 3 P P 3 5 59 3 7 P 3 11 P 3 41 5 3 P P 3 5 37 3

6700 6800 P P 3 19 P 3 7 5 3 P 11 3 5 7 3 53 P 3 P 23 3 11 5 3 17 43 3 5 29 3 P P 3 67 7 3 13 5 3 P P 3 5 11 3 P P 3 7 13

6800 6900 3 P 5 3 11 7 3 5 17 3 19 P 3 P P 3 P 5 3 7 P 3 5 41 3 13 7 3 P 19 3 P 5 3 P P 3 5 13 3 7 P 3 71 83 3 61 5 3 P

6900 7000 67 3 5 P 3 P 31 3 P 11 3 7 5 3 13 29 3 5 7 3 11 53 3 P P 3 17 5 3 P P 3 5 P 3 P 19 3 P 7 3 P 5 3 29 P 3 5 P 3

7000 7100 P 47 3 7 43 3 P 5 3 P 7 3 5 P 3 79 13 3 31 P 3 P 5 3 7 11 3 5 P 3 23 7 3 37 P 3 11 5 3 P 73 3 5 19 3 7 41 3 47 31

7100 7200 3 P 5 3 P 13 3 5 11 3 P 17 3 P P 3 7 5 3 11 37 3 5 7 3 P 23 3 17 P 3 13 5 3 67 71 3 5 P 3 43 11 3 P 7 3 P 5 3 23

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Machinery's Handbook, 31st Edition PRIME NUMBER AND FACTOR TABLES

21

Prime Number and Factor Table for 7201 to 8399 From To 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 67 69 71 73 75 77 79 81 83 85 87 89 91 93 95 97 99

7200 7300 19 3 5 P 3 P P 3 7 P 3 31 5 3 P 7 3 5 P 3 13 P 3 P 11 3 P 5 3 7 53 3 5 13 3 11 7 3 19 29 3 P 5 3 37 23 3 5 P 3

7300 7400 7 67 3 P P 3 71 5 3 13 P 3 5 17 3 P P 3 11 41 3 7 5 3 P P 3 5 7 3 17 37 3 53 P 3 73 5 3 47 11 3 5 83 3 19 P 3 13 7

7400 7500 3 11 5 3 31 P 3 5 P 3 41 13 3 7 17 3 P 5 3 43 7 3 5 11 3 P 29 3 P P 3 17 5 3 7 31 3 5 P 3 P 7 3 P P 3 59 5 3 P

7500 7600 13 3 5 P 3 7 11 3 P 73 3 P 5 3 P 17 3 5 P 3 P 19 3 P P 3 7 5 3 P P 3 5 7 3 67 P 3 P 11 3 P 5 3 P P 3 5 71 3

7600 7700 11 P 3 P 7 3 23 5 3 19 P 3 5 29 3 13 17 3 7 P 3 P 5 3 P 7 3 5 13 3 47 79 3 11 P 3 P 5 3 7 P 3 5 P 3 P 7 3 43 P

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7700 7800 3 P 5 3 13 11 3 5 P 3 7 P 3 P 59 3 11 5 3 71 P 3 5 61 3 23 P 3 P P 3 7 5 3 17 19 3 5 7 3 31 43 3 13 P 3 P 5 3 11

7800 7900 29 3 5 37 3 73 13 3 P 7 3 P 5 3 P 41 3 5 17 3 P 11 3 7 47 3 P 5 3 29 7 3 5 P 3 17 P 3 P P 3 P 5 3 7 13 3 5 53 3

7900 8000 P 7 3 P 11 3 41 5 3 P 89 3 5 P 3 7 P 3 P 17 3 13 5 3 P P 3 5 73 3 19 P 3 31 13 3 7 5 3 79 23 3 5 7 3 61 P 3 11 19

8000 8100 3 53 5 3 P P 3 5 P 3 13 71 3 23 7 3 29 5 3 P 11 3 5 13 3 83 P 3 7 P 3 11 5 3 P 7 3 5 41 3 P 59 3 P P 3 P 5 3 7

8100 8200 P 3 5 11 3 P 7 3 P 23 3 P 5 3 11 47 3 5 79 3 7 17 3 P 29 3 31 5 3 41 P 3 5 P 3 P 11 3 13 P 3 7 5 3 19 P 3 5 7 3

8200 8300 59 13 3 29 P 3 43 5 3 P P 3 5 19 3 P P 3 P 7 3 P 5 3 73 37 3 5 23 3 11 P 3 7 P 3 P 5 3 17 7 3 5 P 3 P P 3 P 43

8300 8400 3 19 5 3 7 P 3 5 P 3 53 7 3 11 P 3 13 5 3 31 19 3 5 17 3 7 P 3 61 13 3 P 5 3 P 11 3 5 P 3 17 83 3 P P 3 7 5 3 37

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Machinery's Handbook, 31st Edition PRIME NUMBER AND FACTOR TABLES

22

Prime Number and Factor Table for 8401 to 9599 From To 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 67 69 71 73 75 77 79 81 83 85 87 89 91 93 95 97 99

8400 8500 31 3 5 7 3 13 47 3 19 P 3 P 5 3 P P 3 5 11 3 23 P 3 P 7 3 79 5 3 11 P 3 5 P 3 43 37 3 7 61 3 17 5 3 13 7 3 5 29 3

8500 8600 P 11 3 47 67 3 P 5 3 7 P 3 5 P 3 19 7 3 P P 3 P 5 3 83 17 3 5 43 3 7 P 3 13 11 3 P 5 3 23 P 3 5 31 3 11 13 3 P P

8600 8700 3 7 5 3 P 79 3 5 7 3 37 P 3 P P 3 89 5 3 53 P 3 5 P 3 41 17 3 11 7 3 P 5 3 P 13 3 5 P 3 P 19 3 7 P 3 P 5 3 P

8700 8800 7 3 5 P 3 31 P 3 23 P 3 11 5 3 7 P 3 5 P 3 P 7 3 P 13 3 P 5 3 19 P 3 5 11 3 7 31 3 67 P 3 P 5 3 11 59 3 5 19 3

8800 8900 13 P 3 P 23 3 7 5 3 P P 3 5 7 3 P 11 3 P P 3 37 5 3 P 53 3 5 17 3 P P 3 P 7 3 19 5 3 13 83 3 5 P 3 17 P 3 7 11

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8900 9000 3 29 5 3 59 7 3 5 37 3 11 P 3 79 P 3 P 5 3 7 P 3 5 23 3 P 7 3 13 17 3 P 5 3 P P 3 5 47 3 7 13 3 11 89 3 17 5 3 P

9000 9100 P 3 5 P 3 P P 3 71 29 3 7 5 3 P 11 3 5 7 3 P P 3 83 P 3 11 5 3 P 13 3 5 P 3 47 43 3 29 7 3 31 5 3 61 P 3 5 11 3

9100 9200 19 P 3 7 P 3 13 5 3 11 7 3 5 P 3 23 P 3 P 13 3 41 5 3 7 P 3 5 P 3 P 7 3 89 53 3 P 5 3 67 P 3 5 P 3 7 29 3 17 P

9200 9300 3 P 5 3 P 61 3 5 13 3 P 23 3 P 11 3 7 5 3 P P 3 5 7 3 11 19 3 P 47 3 59 5 3 13 73 3 5 P 3 P P 3 37 7 3 P 5 3 17

9300 9400 71 3 5 41 3 P 67 3 7 P 3 P 5 3 19 7 3 5 P 3 P P 3 13 P 3 47 5 3 7 11 3 5 17 3 P 7 3 P 83 3 11 5 3 41 P 3 5 P 3

9400 9500 7 P 3 23 97 3 P 5 3 P P 3 5 11 3 P P 3 P P 3 7 5 3 11 13 3 5 7 3 P P 3 P 17 3 P 5 3 P 19 3 5 53 3 P 11 3 P 7

9500 9600 3 13 5 3 37 P 3 5 31 3 P 89 3 7 13 3 P 5 3 P 7 3 5 P 3 P 41 3 19 11 3 73 5 3 7 17 3 5 61 3 11 7 3 P 43 3 53 5 3 29

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Machinery's Handbook, 31st Edition PRIME NUMBER AND FACTOR TABLES

23

Prime Numbers from 9551 to 18691 9551 9587 9601 9613 9619 9623 9629 9631 9643 9649 9661 9677 9679 9689 9697 9719 9721 9733 9739 9743 9749 9767 9769 9781 9787 9791 9803 9811 9817 9829 9833 9839 9851 9857 9859 9871 9883 9887 9901 9907 9923 9929 9931 9941 9949 9967 9973 10007 10009 10037 10039 10061 10067 10069 10079 10091 10093 10099 10103 10111 10133 10139 10141 10151 10159 10163 10169 10177

10181 10193 10211 10223 10243 10247 10253 10259 10267 10271 10273 10289 10301 10303 10313 10321 10331 10333 10337 10343 10357 10369 10391 10399 10427 10429 10433 10453 10457 10459 10463 10477 10487 10499 10501 10513 10529 10531 10559 10567 10589 10597 10601 10607 10613 10627 10631 10639 10651 10657 10663 10667 10687 10691 10709 10711 10723 10729 10733 10739 10753 10771 10781 10789 10799 10831 10837 10847

10853 10859 10861 10867 10883 10889 10891 10903 10909 10937 10939 10949 10957 10973 10979 10987 10993 11003 11027 11047 11057 11059 11069 11071 11083 11087 11093 11113 11117 11119 11131 11149 11159 11161 11171 11173 11177 11197 11213 11239 11243 11251 11257 11261 11273 11279 11287 11299 11311 11317 11321 11329 11351 11353 11369 11383 11393 11399 11411 11423 11437 11443 11447 11467 11471 11483 11489 11491

11497 11503 11519 11527 11549 11551 11579 11587 11593 11597 11617 11621 11633 11657 11677 11681 11689 11699 11701 11717 11719 11731 11743 11777 11779 11783 11789 11801 11807 11813 11821 11827 11831 11833 11839 11863 11867 11887 11897 11903 11909 11923 11927 11933 11939 11941 11953 11959 11969 11971 11981 11987 12007 12011 12037 12041 12043 12049 12071 12073 12097 12101 12107 12109 12113 12119 12143 12149

12157 12161 12163 12197 12203 12211 12227 12239 12241 12251 12253 12263 12269 12277 12281 12289 12301 12323 12329 12343 12347 12373 12377 12379 12391 12401 12409 12413 12421 12433 12437 12451 12457 12473 12479 12487 12491 12497 12503 12511 12517 12527 12539 12541 12547 12553 12569 12577 12583 12589 12601 12611 12613 12619 12637 12641 12647 12653 12659 12671 12689 12697 12703 12713 12721 12739 12743 12757

12763 12781 12791 12799 12809 12821 12823 12829 12841 12853 12889 12893 12899 12907 12911 12917 12919 12923 12941 12953 12959 12967 12973 12979 12983 13001 13003 13007 13009 13033 13037 13043 13049 13063 13093 13099 13103 13109 13121 13127 13147 13151 13159 13163 13171 13177 13183 13187 13217 13219 13229 13241 13249 13259 13267 13291 13297 13309 13313 13327 13331 13337 13339 13367 13381 13397 13399 13411

13417 13421 13441 13451 13457 13463 13469 13477 13487 13499 13513 13523 13537 13553 13567 13577 13591 13597 13613 13619 13627 13633 13649 13669 13679 13681 13687 13691 13693 13697 13709 13711 13721 13723 13729 13751 13757 13759 13763 13781 13789 13799 13807 13829 13831 13841 13859 13873 13877 13879 13883 13901 13903 13907 13913 13921 13931 13933 13963 13967 13997 13999 14009 14011 14029 14033 14051 14057

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14071 14081 14083 14087 14107 14143 14149 14153 14159 14173 14177 14197 14207 14221 14243 14249 14251 14281 14293 14303 14321 14323 14327 14341 14347 14369 14387 14389 14401 14407 14411 14419 14423 14431 14437 14447 14449 14461 14479 14489 14503 14519 14533 14537 14543 14549 14551 14557 14561 14563 14591 14593 14621 14627 14629 14633 14639 14653 14657 14669 14683 14699 14713 14717 14723 14731 14737 14741

14747 14753 14759 14767 14771 14779 14783 14797 14813 14821 14827 14831 14843 14851 14867 14869 14879 14887 14891 14897 14923 14929 14939 14947 14951 14957 14969 14983 15013 15017 15031 15053 15061 15073 15077 15083 15091 15101 15107 15121 15131 15137 15139 15149 15161 15173 15187 15193 15199 15217 15227 15233 15241 15259 15263 15269 15271 15277 15287 15289 15299 15307 15313 15319 15329 15331 15349 15359

15361 15373 15377 15383 15391 15401 15413 15427 15439 15443 15451 15461 15467 15473 15493 15497 15511 15527 15541 15551 15559 15569 15581 15583 15601 15607 15619 15629 15641 15643 15647 15649 15661 15667 15671 15679 15683 15727 15731 15733 15737 15739 15749 15761 15767 15773 15787 15791 15797 15803 15809 15817 15823 15859 15877 15881 15887 15889 15901 15907 15913 15919 15923 15937 15959 15971 15973 15991

16001 16007 16033 16057 16061 16063 16067 16069 16073 16087 16091 16097 16103 16111 16127 16139 16141 16183 16187 16189 16193 16217 16223 16229 16231 16249 16253 16267 16273 16301 16319 16333 16339 16349 16361 16363 16369 16381 16411 16417 16421 16427 16433 16447 16451 16453 16477 16481 16487 16493 16519 16529 16547 16553 16561 16567 16573 16603 16607 16619 16631 16633 16649 16651 16657 16661 16673 16691

16693 16699 16703 16729 16741 16747 16759 16763 16787 16811 16823 16829 16831 16843 16871 16879 16883 16889 16901 16903 16921 16927 16931 16937 16943 16963 16979 16981 16987 16993 17011 17021 17027 17029 17033 17041 17047 17053 17077 17093 17099 17107 17117 17123 17137 17159 17167 17183 17189 17191 17203 17207 17209 17231 17239 17257 17291 17293 17299 17317 17321 17327 17333 17341 17351 17359 17377 17383

17387 17389 17393 17401 17417 17419 17431 17443 17449 17467 17471 17477 17483 17489 17491 17497 17509 17519 17539 17551 17569 17573 17579 17581 17597 17599 17609 17623 17627 17657 17659 17669 17681 17683 17707 17713 17729 17737 17747 17749 17761 17783 17789 17791 17807 17827 17837 17839 17851 17863 17881 17891 17903 17909 17911 17921 17923 17929 17939 17957 17959 17971 17977 17981 17987 17989 18013 18041

18043 18047 18049 18059 18061 18077 18089 18097 18119 18121 18127 18131 18133 18143 18149 18169 18181 18191 18199 18211 18217 18223 18229 18233 18251 18253 18257 18269 18287 18289 18301 18307 18311 18313 18329 18341 18353 18367 18371 18379 18397 18401 18413 18427 18433 18439 18443 18451 18457 18461 18481 18493 18503 18517 18521 18523 18539 18541 18553 18583 18587 18593 18617 18637 18661 18671 18679 18691

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24

Machinery's Handbook, 31st Edition ALGEBRA ALGEBRA

In engineering, manufacturing, and industrial applications, physical laws govern the behavior of all quantities. Algebraic formulas (equations) are the models for these laws. They usually consist of algebraic expressions, the most common being polynomials, rational expressions, and radicals. Most of the formulas used in this Handbook contain one or more of these. This section gives a foundation for understanding the algebra indispensable to solving equations. Definitions.—The vocabulary of algebra extends to all mathematics. The essential definitions are given here. Operation: Addition, subtraction, multiplication, division, root-­taking, raising to a power, taking a logarithm. Inverse Operation: An operation that reverses another operation. Addition and subtraction are inverse operations, as are multiplication and division. Taking the nth root is the inverse of raising a number to a power. Finding an antilogarithm is the inverse of finding a logarithm. Constant: A known quantity, either a number standing alone or a letter that is assumed to be given or known in an application. In 5x + 14, 14 is the constant. Usually, the letters a, b, and c are used to represent constants, as in the linear equation ax + by = c. Note: e and π are commonly seen constants. Variable: An unknown quantity, represented by a letter such as n, x, y, x, t. Note: e and π are not variables. Exponent: The power to which a variable or number is raised. Monomial: A single variable or number or a product of numbers and variables. Examples: 5, x, –4y2, 12xy2z3. Exponents in monomials are whole numbers, 0, 1, 2, 3, . . . , so x –­1 = 1/x and x1/2 = x are not monomials. Coefficient: The numerical factor in a term. Examples (in bold): 5x, 16n, –­2r. The coefficient of a variable standing alone is understood to be 1, for example, x = 1x; the coefficient of –x is –1. Term: Monomials are terms, but so are expressions that are not monomials: 1/x, x , 8x1/3, log x, and so on. Like Terms: This usually refers to monomials with the same variable and exponent, and having any real number coefficients, such as x and 7x; 2n2 and n2/4; 2rst/5 and 14rst, and so on. Any constant a can be written as ax 0, so all constants are like terms. But x1/2 and 4x1/2 also are like terms. Expression: Numbers and variables with operators (addition: +, multiplication: × or ⋅, etc.). Equation: Two expressions set equal to one another with the equal sign = .

Examples: 5x = x 2 –­6; 3 x = 14 . Solving equations for the unknown is the foundation of algebra. Inequality: Two expressions set against one another by >, 0, then log x = log y.

If x = y , then a x = a y for a > 0, a ≠ 1.

Both statements are true in the other direction, too: If log x = log y , then x = y.

Example 1: Find the square root of 754. Solution : Let x =

1/ 2

1.4387

Example 2: Solve 4x = 7x –­3 for x. 4 x = 7 x −3

→

distribute on the right

→

→

apply property

≈

27.460. That is,

=

1 2.8774 log 754 ≈ ≈ 1.4387. 2 2

754 ≈ 27.460.

l og 4 x = log 7 x −3 → x log 4 = ( x − 3)(log 7)

x log 4 = x log 7 − 3 log 7

proceed with algebra

y

754 . Then log x = log 754 = log 754

So, log x ≈ 1.4387, hence, x = 10

Solution :

x

If a = a then x = y.

→

by calculator

0.6021x = 0.8451x − 3(0.8451)

3(0.8451) = 0.8451x − 0.6021x → 2.5353 = 0.243 x, so x ≈ 10.433

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Machinery's Handbook, 31st Edition LOGARITHMS

39

The technique of taking the log (either common, natural, or other) of both sides of an equation is used often to solve for unknown exponents, as happens with compounding of interest (see ENGINEERING ECONOMICS on page 143). Arithmetic Sequence An arithmetic sequence (also called an arithmetic progression) is a sequence of numbers in which each term differs from the preceding one by a fixed amount, called the common dif­ference, d. Thus, 1, 3, 5, 7, etc. is an arithmetic sequence where the difference d is 2. Here, the consecutive terms of the sequence are increasing by 2. In the sequence 13, 10, 7, 4, etc., the difference is -3, and the sequence is decreasing. In any arithmetic progression (or portion of one): a = first term of the sequence, also called the a1 term l =  last term considered, also called an for the nth term n =  number of terms d =  common difference Sn =  sum of n terms The formula for the last term is l = a + (n - 1)d, or an = a1 + (n - 1)d. The sum of an arithn(a1 + an) n(a + l) or metic sequence with n terms is  Sn = . 2 2 In these formulas, d is positive when the progression is increasing and negative when it is decreasing. When any three of the preceding five quantities are given, the other two can be found by the formulas in the table Arithmetic Sequence Formulas on page 40. Often, however, the desired quantity can be determined by working with the information given. Example 1: In a given arithmetic progression, the first term is 5 and the last term 40, and the difference between terms is 7. To find the sum of the progression, first the number of terms has to be found. This is done by considering the difference between the first and last: 40 - 5 = 35. Dividing this by the difference between terms gives the number of intervals between the terms: 35 ÷ 7 = 5. Finally, adding 1 gives the number of terms in the sequence: n = 5 + 1 = 6. The sum of the sequence is: 6 n S = (a + l) = (5 + 40) = 3(45) = 135 2 2 Geometric Sequence A geometric sequence or progression is a sequence of numbers in which each term is derived by multiplying the preceding term by a constant multiplier called the ratio. When this ratio is greater than 1, the progression is increasing; when less than 1, it is decreasing. Thus, the sequence 2, 6, 18, 54, etc. is an increasing geometric sequence with a ratio of 3, and the sequence 24, 12, 6, etc. is a decreasing sequence with a ratio of 1 ⁄ 2. In any geometric progression (or part of progression): a =  first term of the sequence l =  last (or nth) term of the sequence n =  number of terms r =  ratio of the progression Sn =  sum of n terms rl – aand S = -----------The general formulas for the nth term: l = ar n – 1 r–1 When any three of the preceding five quantities are given, the other two can be found by the formulas in the table Geometric Sequence Formulas on page 41. Geometric pro­gressions are used for finding the successive speeds in machine tool drives, and in interest calculations. Example 2: The lowest speed of a lathe is 20 rpm. The highest speed is 225 rpm. There are 18 speeds. Find the ratio between successive speeds. 225- = 17 11.25 = 1.153 Ratio r = n – 1 --l- = 17 -------a 20

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Machinery's Handbook, 31st Edition ARITHMETIC SEQUENCE FORMULAS

40 To Find

a

d

l

n

S

Given

Arithmetic Sequence Formulas

Use Formula

d

l

n

a = l – ( n – 1 )d

d

n

S

d

l

S

n–1 --- – ------------ × d a = S 2 n d 1 a = --- ± --- ( 2l + d ) 2 – 8dS 2 2

l

n

S

a

l

n

a

n

S

a

l

S

l

n

S

a

d

n

l = a + ( n – 1 )d

a

d

S

a

n

S

d

n

S

a

d

l

a

d

S

a

l

S

d

l

S

a

d

n

1 --- ± --- 8dS + ( 2a – d ) 2 l = –d 2 2 ------ – a l = 2S n n–1 S l = --- + ------------ × d 2 n l – n = 1 + ---------ad d – 2a 1 n = --------------- ± ------ 8dS + ( 2a – d ) 2 2d 2d 2S n = ---------a+l 2l + d 1 n = -------------- ± ------ ( 2l + d ) 2 – 8dS 2d 2d n S = --- [ 2a + ( n – 1 )d ] 2

a

d

l

a

l

n

d

l

n

------ – l a = 2S n l ----------d = – an–1 – 2an--------------------d = 2S n(n – 1) l2 – a2 d = ---------------------2S – l – a – 2S-------------------d = 2nl n(n – 1)

2 – a2 a+l + -l + l--------------S = a---------= ----------- ( l + d – a ) 2d 2 2d n(a + l) Sn = 2 n S = --- [ 2l – ( n – 1 )d ] 2

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Machinery's Handbook, 31st Edition GEOMETRIC SEQUENCE FORMULAS To Find

a

l

n

r

S

Given

41

Geometric Sequence Formulas

Use Formula

l a = ----------rn – 1 r – 1 )Sa = (-----------------rn – 1

l

n

r

n

r

S

l

r

S

a = lr – ( r – 1 )S

l

n

S

a ( S – a )n – 1 = l ( S – l )n – 1

a

n

r

l = ar n – 1

a

r

S

1 l = --- [ a + ( r – 1 )S ] r

a

n

S

l ( S – l )n – 1 = a ( S – a )n – 1

n

r

S

a

l

r

a

r

S

a

l

S

l

r

S

S ( r – 1 )r n – 1l = ------------------------------rn – 1 l – log a- + 1 -------------------------n = log log r log [ a + ( r – 1 )S ] – log an = ---------------------------------------------------------log r log l – log a n = -----------------------------------------------------+1 log ( S – a ) – log ( S – l ) l – log [ lr – ( r – 1 )S ]- + 1 ---------------------------------------------------------n = log log r

a

l

n

r =

a

n

S

a

l

S

Sr + a----------– Sr n = ----a a S – a r = -----------S–l

l

n

S

a

n

r

a

l

r

a

l

n

l

n

r

n–1

--ln

n–1 l --------------- – --------r n = Sr S–l S–l ( r n – 1 )S = a--------------------r–1 lr – aS = -----------r–1 n–1 n n–1 n al – S = -------------------------------------n–1 l–n–1 a l ( rn – 1 ) S = --------------------------( r – 1 )r n – 1

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42

Machinery's Handbook, 31st Edition GEOMETRY GEOMETRY

Geometry is the branch of mathematics that studies the features of two- and threedimensional figures. (The word “geometry” means “measure of the earth.”) This branch can be separated into pure geometry and analytic geometry. Pure geometry is concerned with the propositions of shape, size, and relative position of figures, as well as their constructions. Analytic geometry studies geometry using the coordinate system, relying heavily on algebraic principles. The results of geometry apply to many areas of industry. The first part of this section, addressing analytic geometry, may be considered a continuation of the material in ALGEBRA, which begins on page 24. The rest of the section focuses on pure geometry, particularly the formulas for figure dimensions and figure construction; these are based on the concepts introduced by the Greek mathematician Euclid, in the fourth century BCE. Included are examples showing how measures of diameter, perimeter, area, surface area, volume, angle, and more are determined. In the case of angle measure, use of trigonometry is often necessary. Explanations of trigonometric relations are found in the next section, TRIGONOMETRY, beginning on page 94. Analytic Geometry Analytic geometry uses algebra to model geometric objects, such as points, lines, and circles, on the rectangular coordinate system.

Rectangular Coordinate System.—The rectangular coordinate system (also called the xy-plane or the Cartesian plane) is a grid formed by intersecting two real number lines at right angles (Fig. 1a). The horizontal x-axis (labeled X) intersects the vertical y-axis (labeled Y) at the point (0, 0), the origin. Any point P on the plane can be so identified by its x-coordinate and its y-coordinate in the ordered pair (x, y). The four quadrants formed by the x- and y-axes are numbered counterclockwise (see Fig. 1a). In Quadrant 1, both x and y coordinates are positive; in Quadrant 2, the x is negative and y is positive; in Quadrant 3, both coordinates are negative; and in Quadrant 4, the x-coordinate is positive and the y negative. Several representative points are pictured in Fig. 1b.

Fig. 1. (a) Rectangular Coordinate System;      (b) Examples of Points in Each Quadrant.

The rectangular coordinate system is used to illustrate ideas in algebra, analytic geometry, and trigonometry.

Slope of a Line.—The slope of the line passing through any two points (x1, y1) and (x 2 , y 2) in the plane is given by m =

Δy

Δx

=

y2 − y1 x2 − x1

, where Δ represents difference. Fig. 2a shows a

line with a positive slope, Fig. 2b a line with a negative slope. A horizontal line (Fig. 2c)

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Machinery's Handbook, 31st Edition ANALYTIC GEOMETRY

43

has a no slope, since ∆y = 0. Hence, m = 0/∆x = 0. Between any two points on a vertical line, ∆x = 0 (Fig. 2d). So, m = ∆y/0, which is undefined.

Fig. 2. Lines with (a) Positive Slope, m > 0; (b) Negative Slope, m < 0; (c) No Slope, m = 0; (d) Undefined Slope.

Lines and Line Segments.—A line in the plane is the shortest path between two known points, extending indefinitely in both directions. “Line AB” is notated AB. A line segment is the portion of the line between A and B. The line segment AB is notated AB; its length is indicated without the bar, as AB. The distinction between the actual line and its length is helpful to keep in mind when referring to the formulas for distance, midpoint, and the other concepts in this section. Distance Between Two Points: The distance d between two points A and B is the length of the line segment connecting them. The formula comes from the Pythagorean theorem, which says that the sum of the squares of the leg measures is the square of the hypotenuse. As labeled in Fig. 3, the legs are lengths x2 – x1 and y2 – y1, and the hypotenuse is the distance d between these points. Its formula is:

d(A, B) = AB =

2

( x2 – x1 ) + ( y2 – y1 )

2

Fig. 3. Distance between points A and B.

The order in which x and y are subtracted actually does not matter, since the squared difference is the same. Example 1: Find AB, the distance between points A(4, 5) and B(7, 8). Solution: The length of line segment AB is:

d =

2

2

(7 – 4) + (8 – 5) =

2

2

3 +3 =

18 = 3 2

Midpoint of a Line Segment: The midpoint M(x, y) of line segment AB is found by the coordinate formulas (Fig. 4a): x + x2 y + y2 x= 1 and y = 1 2 2 Internal Division of a Line Segment: Point P divides line segment AB (Fig. 4b) in the ratio m:n. That is, P is such a point that AP:PB as m:n. Then, the coordinates of P are given by: mx1 + nx2 my1 + ny2 x = and y = m+n m+n

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Machinery's Handbook, 31st Edition LINES AND LINE SEGMENTS

44

If the desired point is the midpoint of AB, then m = n= 1, and the coordinates of P are: x1 + x2 y1 + y2 x = ---------------and y = ---------------2 2 Example 2: Find the coordinates of a point P that divides the line segment defined by A(0, 0) and B(8, 6) at a ratio of 5:3.

5 × 0 + 3 × 8 = ----24- = 3 Solution:   x = -----------------------------5+3 8

(5)(0) + (3)(6) 18- = 2.25 y = ------------------------------ = ----8 5+3

External Division of a Line Segment: If the line segment AB is extended to point Q (Fig. 4c), and Q is such a point that AQ:QB as m:n, then the coordinates of Q are given by:

x =

mx1 – nx2 m–n

and

y =

my1 – ny2 m–n

Fig. 4. Divisions of Line Segment AB: (a) Midpoint M; (b) Internal Division Point P; (c) External Division Point Q.

Equation Forms of a Line.—Given any two known points, a linear equation can be expressed in either point-slope or slope-intercept form. y2 – y1 - for any two points (x, y) on a line. If the Point-Slope Form: Consider that m = --------------x2 – x1 known point (x 2 , y2) is replaced with an arbitrary (not specified) point (x, y), the slope y –y becomes m = --------------1-, which rearranges to y – y1 = m(x – x1) . This is the point-slope form x – x1 of the line. If x1 = y1 = 0 (that is, the line passes through the origin (0, 0)), then the equation becomes y – 0 = m(x – 0), or y = mx. For any line, the y-intercept is the point of intersection of the line with the y-axis, and the x-intercept is the point of intersection of the line with the x-axis. Thus, the points (0, y1) and (x1, 0) are the y- and x-intercepts, respectively. Suppose AB intersects the x-axis at point A(a, 0) and the y-axis at point B(0, b); then

m=

0−b

a−0

=

−b a

. Substitution of either (a, 0) or (0, b) into the formula and rearranging terms y

x gives the equation for AB: y − 0 = − ba ( x − a) ay −bx + ab ay bx ab . b + a =1 Generally, two points are known, and from these, m is determined. Then, either of the points are substituted into the point-slope form along with the slope to get the equation of the line.

Example 3: Find the equation of the line that passes through (4, 1) and (–2, 5). So,

m = 4−1(−−52) = −64 = −32 . Substituting this value for m and the point (4, 1) for (x1, y1) into the point-slope form gives y − 1 = − 23 ( x − 4).

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Machinery's Handbook, 31st Edition EQUATION FORMS OF A LINE

45

y2 – y1 - into the point-slope form and rearranging Point-Point Form: Substituting m = --------------x2 – x1 y −y terms gives y − y1 = x1 − x 2 ( x − x1 ) , the point-point form. 1 2 Slope-Intercept Form: Another often-used form is y = mx + b, where m is the slope and b is the y-intercept. (That is, when x = 0, then y = m(0) + b = b.) If the previous example is rearranged and solved for y, then the slope-intercept form is arrived

at: y = − 23 ( x − 4) + 1 = − 23 x + 83 + 1 = − 23 x + 11 3. Standard Form of a Line: In vector representations (see Complex Numbers on page 59), lines usually take the standard form, Ax + By = C, where A, B, C. For example, 7x – 3y = 4 is a line in standard form, with A = 7, B = –3, and C = 4. Example 4: What is the standard form equation of a line AB between points A(4, 5) and B(7, 8)? Solution: Using the point-point form of the line, where (4, 5) is (x1, y1) and (7, 8) is (x 2 , y 2), y –y y – y1 = x1 – x2 (x – x1) 1

y–5=

2

5–8 (x – 4) 4–7

y–5=x–4 x – y = –1 Example 5: Find the slope-intercept equation of a line passing through the points (3, 2) and (5, 6). The y-intercept is the intersection point of the line with the y-axis. Solution: First, find the slope: – 2- = 4--- = ------ = 6----------m = ∆y 2 ∆x 5–3 2 The slope-intercept form of the line is y = 2x + b, and the value of the constant b can be determined by substituting the coordinates of a point on the line into the general form. Using the coordinates of the point (3, 2) gives 2 = (2)(3) + b and rearranging, b = 2 - 6 = -4. As a check, using another point on the line, (5, 6), yields the same result, y = 6 = (2)(5) + b and b = 6 - 10 = -4. The equation of the line, therefore, is y = 2x - 4, indicating that line y = 2x - 4 intersects the y-axis at point (0,-4), the y-intercept. Example 6: Use the point-slope form to find the equation of the line passing through the point (3, 2) and having a slope of 2. y – 2 = 2(x – 3)

y = 2x – 6 + 2 y = 2x – 4 The slope of this line is positive and crosses the y-axis at the y-intercept, point (0,-4). Parallel Lines: The two lines, l1 and l2 , are parallel if they have the same slope, that is, if m1= m2 .

Fig. 5. (a) Parallel lines, l1 and l2 .       (b) Perpendicular lines, l1 and l2 .

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Machinery's Handbook, 31st Edition EQUATION FORMS OF A LINE

Perpendicular Lines: The two lines, l1 and l2 , are perpendicular if the product of their slopes is -1, that is, if m1 m2 = -1, since m1 = –1/m2 . (The slopes are negative reciprocals of one another.) Example 7: (a) Find the equation of the line that passes through the point (3, 4) and is par­allel to line 2x - 3y = 16. (b) Find the equation of the line perpendicular to the given line and through the same point. Solution (a): Line 2x - 3y = 16 in slope-intercept form is y = 2 ⁄ 3  x - 16 ⁄ 3  , so the equation of a line passing through (3, 4) is y – 4 = m ( x – 3 ) . Parallel lines have equal slope. Thus, from the point-slope form, y – 4 = 2 ⁄ 3 (x – 3) is parallel to line 2x - 3y = -6 and passes through point (3, 4). Solution (b): As illustrated in part (a), line 2x - 3y = -6 has a slope of 2 ⁄ 3. The product of the slopes of perpendicular lines is -1, thus the slope m of a line passing through point (3, 4) and perpendicular to 2x - 3y = -6 must satisfy the following:

– 1- = – 3 – 1 = ------m = -----2 ⁄3 m1 2

The equation of a line passing through point (3, 4) and perpendicular to the line 2x - 3y = 16 is y - 4 = -3 ⁄ 2 (x - 3), which rewritten is 3x + 2y = 17. Angle Between Two Lines: For two non-perpendicular lines with slopes m1 and m2 , the angle θ between the two lines is found by first applying trigonometric equation:

m1 – m2 tan θ = ---------------------1 + m1 m2

The discussion of how the angle is determined by this relation is found in TRIGONOMETRY, which begins on page 94. Example 8: Find the angle between the lines: 2x - y = 4 and 3x + 4y = 12. Solution: Rearranging each to be in the slope-intercept form shows the slopes are 2 and -3 ⁄4, respectively. The angle between two lines is given by 3 8----------+ 32 –  – 3--- 2 + -- 4 m1 – m2 4 - = ----11- = 11 -----tan θ = ---------------------- = ------------------------ = -----------4- = ----------2 4----------– 61 + m1 m2 –2 1 – 6--1 + 2  – --3-  4 4 4 11 θ = tan – 1  ------ = 79.70°, by trigonometry.  2

Distance Between a Point and a Line: The distance between a point (x1, y1) and a line given in the standard form Ax + By + C = 0 is Ax 1 + By 1 + C d = ------------------------------------2 2 A +B Example 9: Find the distance between the point (4,6) and the line 2x + 3y - 9 = 0. Solution: Using the formula:

Ax 1 + By 1 + C 2 × 4 + 3 × 6 – 9 = -------------------------8 + 18 – 9- = --------17- = -----------------------------------------d = ------------------------------------2 2 2 2 4 + 9 13 A +B 2 +3 Changing Coordinate Systems.—For simplicity it may be assumed that the origin in the Cartesian coordinate system coincides with the pole on a polar coordinate system and its x-axis with the polar horizontal axis. Then, if point P has polar coordinates of (r, q) and Cartesian coordi­nates of (x, y), by trigonometry x = r cosq and y = r sinq.

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47

()

y 2 2 Furthermore, by the Pythagorean theorem and trigonometry, r = x + y and θ = tan–1 -x –1 y tan . See TRIGONOMETRY on page 94 for a discussion of the related princix + y and θ = x ples of trigonometry. Example 1: Convert the Cartesian coordinate (3, 2) into polar coordinates. 2

()

2

r =

2

2

3 +2 =

9+4 =

()

2 θ = tan–1 --- = 33.69° 3

13 ≈ 3.6

Therefore the point is located approximately 3.6 units from the origin at an angle of about 33.69°. Thus, (3.6, 33.69º) is the polar form of the Cartesian point (3, 2). Graphically, the polar and Cartesian coordinates are related in the following figure: Y

(3, 2) P

x

2

r = 3.6

1

y

θ = 33.69° 0

0

1

2

X

3

Example 2: Convert the polar form (5, 608°) to Cartesian coordinates. First note that this point lies 5 units from the origin at an angle of 608°. As explained on page 105, in Trigonometric Functions, this locates the point in Quadrant IV. By trigonometry, x = r cosq and y = r sinq. Then, x = 5cos(608º) = -1.873 and y = 5sin(608°) = -4.636. There­fore, the Cartesian point equivalent is (-1.873, -4.636). This point lies in the fourth quadrant, where both coordinates are negative.

Spherical Coordinates.—It is convenient in certain problems, for example, those con­ cerned with spherical surfaces and therefore three-dimensional, to introduce spherical coordinates. In three-dimensional space, as the figure on the right shows, the x,y-plane is like a floor in a room, and the third dimension is given by the z-axis, which is where the walls of the room meet. An arbitrary point P in this space is described by three rectangular coordinates (x, y, z), converted to the following spherical coordinates: the distance r between point P and the origin O, the angle f that OP ′ makes with the x, y-plane, and the angle l that the projection OP ′ (the “shadow” of the segment OP on the x, y-plane) makes with the positive x-axis. z

m

=

Machinery's Handbook, 31st Edition CHANGING COORDINATE SYSTEMS

er

ia id

z

pole

n

P

r

P O

e q u ato r

x

y

x

r

O

y

The rectangular coordinates of a point in space can therefore be calculated from the spherical coordinates, and vice versa, by use of the formulas in the following table.

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Machinery's Handbook, 31st Edition RELATIONSHIP BETWEEN COORDINATE SYSTEMS Relationship Between Spherical and Rectangular Coordinates

Spherical to Rectangular

Rectangular to Spherical

r =

2

2

x +y +z

2

z φ = tan–1 --------------------2 2 x +y

x = r cos φ cos λ y = r cos φ sin λ z = r sin φ

for x2 + y2 ≠ 0

y λ = tan–1  --  x

for x > 0, y > 0

y λ = π + tan–1  --  x

for x < 0

y λ = 2π + tan–1  --  x

for x > 0, y < 0

Example 3: Find the spherical coordinates of the point P(3, -4, -12).

r =

2

2

2

3 + ( – 4 ) + ( – 12 ) = 13

– 12 12- φ = tan–1 ----------------------------- = tan–1  – ----= – 67.38°  5 2 2 3 + ( –4 ) --- = 360° – 53.13° = 306.87° λ = 360° + tan–1  – 4  3

The spherical coordinates of P are therefore r = 13, f = - 67.38°, and l = 306.87°. Cylindrical Coordinates: For problems in which points lie on the surface of a cylinder it is convenient to use cylindrical coordinates. The cylindrical coordinates r, q, z of P coincide with the polar coordinates of the point P ′ in the x, y-plane and the rectangular z-coordinate of P. Formulas for q hold only if x2 + y2 ≠ 0; q is undetermined if x = y = 0. Cylindrical to Rectangular Rectangular to Cylindrical

z 2

x = r cos θ y = r sin θ z = z

2

x +y r = --------------------1 x cos θ = -------------------2 2 x +y

P

y sin θ = -------------------2 2 x +y z = z

O θ

x

r

P

y

Example 4: Given the cylindrical coordinates of a point P, r = 3, q = -30°, z = 51, find the rectangular coordinates. Using the above formulas x = 3cos(-30°) = 3cos(30°) = 2.598; y = 3sin(-30°) = -3sin(30°) = -1.5; and z = 51. Therefore, the rectangular coordinates of point P are x = 2.598, y = -1.5, and z = 51.

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Machinery's Handbook, 31st Edition CIRCLE

49

Circle.—A circle is the set of points equidistant from a given point in the plane. Another name for this set of points is locus, which is a curve formed by all the points satisfying a particular equation. The general form for the equation of a circle is x 2 + y 2 + 2gx + 2fy + c

g2 + f 2 – c .

= 0, where -g and -f are the coordinates of the center and the radius is r = The standard form of a circle (center-radius form) is Y 2

2

2

(x – h) + (y – k) = r where r = length of the radius and point (h, k) is the center. When the center of circle is at point (0, 0), the equation x2

y2

r2

x2

Center (h, k) r

y2

r = = or + reduces to + Example 1: Point (4, 6) lies on a circle whose center (h, k) is at point (-2, 3). Find the circle’s equation. Solution: The radius is the distance from the center point (-2, 3) to point (4, 6), found using the method of Example 1 on page 43. 2

2

[ 4 – ( –2 ) ] + ( 6 – 3 ) =

r = 2

2

2

X

6 +3 =

45

2

Using the form ( x – h ) + ( y – k ) = r 2 ( x – h )2 + ( y – k )2 = r 2 and substituting h = –2, k = 3, and r 2 = 45: ( x + 2 ) + ( y – 3 )2 = x 2 + 4x + 4 + y 2 – 6y + 9 = 45 2 2 ( x + 2 ) + ( y – 3 ) = x 2 + 4x + 4 + y 2 – 6y + 9 = 45 x 2 + y 2 + 4x – 6y – 32 = 0 x 2 + y 2 + 4x – 6y – 32 = 0 2

Additional Formulas: Listed below are additional formulas for determining the geome­ try of plane circles and arcs. Although trigonometry and circular measure are related, they deal with angles in entirely different ways. In each of these formulas, the entered measure of the angles are in degrees, and the formulas convert them to radian measure (see page 94, in TRIGONOMETRY). C = πD = 2πR

----R == X + Y RadiusN



Diameter D = diameter of circle = 2R = ---



Area A = πR



X=



Y=



Area of complement sector M



2 πR 2 = R – --------- = 0.2146R 4



2

2

π

C π

2

2

2

2

2

R –Y

R –X

2

I = distance to section T H = height of section T Q = chord length for segment S P = chord length for segment section T

2 –1 P IP T + S = area of segment = R sin  ------- – -----2R 2

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Fig. 6a.

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Machinery's Handbook, 31st Edition CIRCLE FORMULAS

50

Central angle φ, in degrees = π

Arc length L = Rφ · Radius R =

L

·

180 π 2K

φ 180

=

L

Area of segment S =

·

π

2

= π

Area of sector K = φR 2 ·

L

R 180

E + 4F

2

8F RL

=

360 2 RL E ( R − F ) 2

–

2

() ( ( ))

Chord length E = 2 F ( 2 R − F ) = D sin

Depth F = R −

() ()

2

4R − E 2

2

= R 1 − cos

φ 2

φ 2

φ E/2 tan = 2 R−F sin

Fig. 6b.

E φ = 2 2R

Annulus R1 = radius of outer circle R2 = radius of inner circle

(

2

2

Area of annulus W = π R1 − R2 Area of annulus segment U =

)

φ 360

(

2

2

π R1 − R2

) Fig. 6c.

Example 2: Find the area of a sector of a circle having a central angle of 30° and a radius of 7 cm. 30 φ° 2 2 Solution: Referring to Fig. 6b, K =  --------- π × R =  --------- π × 7 = 12.83 cm 2 360 360 Example 3: Find the chord length E of a circular segment (Fig. 6b) with a depth of 2 cm at the center that is formed in a circle whose radius 12 cm. Solution: The chord length is

E = 2 F ( 2R – F ) = 2 2[(2)(12) – 2] = 2 44 = 4 11 = 13.27 cm Example 4: Find the area S of the segment in Example 3. Solution: First determine angle f, then find arc length L of the segment, and then solve for area S, as follows:

φ φ E ⁄ 2- = 13.27 ⁄ 2- = ------------------tan  --- = -----------0.6635, --- = tan–10.6635 =  2 12 – 2 2 R–F π π L = Rφ× 12 × 67.13° × = 14.06 cm = 180 180

33.56°,

φ = 67.13°

Copyright 2020,E (Industrial Inc.) 13.27 ( 10 ) ebooks.industrialpress.com R – F )- = Press, 12 ( 14.06 ------- – -------------------------------------------Area S = RL – ------------------------ = 84.36 – 66.35 = 18.01 cm 2 2 2 2 2

Machinery's Handbook, 31st Edition

φ φ E ⁄ 2⁄ 2- =CIRCLE FORMULAS ------------------tan  --- = -----------= 13.27 0.6635, --- = tan–10.6635 =  2 R–F 12 – 2 2 π π L = Rφ× 12 × 67.13° × = 14.06 cm = 180 180

51

33.56°,

φ = 67.13°

RL E ( R – F ) 12 ( 14.06 ) 13.27 ( 10 ) Area S = ------- – ---------------------- = ------------------------ – ------------------------ = 84.36 – 66.35 = 18.01 cm 2 2 2 2 2 Another way to find angle f is divide the chord length by twice the radius to obtain φ E- = length- = ---------------------------------sin  --- = chord 0.5529,  2 2R 2R

φ --- = sin–1 0.5529 = 33.5662°, 2

φ = 67.13°

Ellipse.—As a circle is the locus of points equidistant from a single point in the plane, an ellipse is the set of points whose location is established by two points in the plane. Referring to the figure, these two points are the foci, F1 and F2 , which lie on the longer of the two diameters of the ellipse. The longer diameter a is the major axis; the shorter b is the minor axis. The sum of the distances from the foci to any point P on the ellipse is constant. That is PF1 + PF2 = 2a. The latus rectum is the chord through the focus and perpendicular to the major axis. V1 and V2 , are the vertices. Like the circle, there is a general form and a standard form of the equation of an ellipse. The general form is: 2

2

Ax + Cy + Dx + Ey + F = 0

AC > 0

and A ≠ C

The constant F in this equation is not related to the foci, which are not numbers, but labels for points.

Ellipse

2

2

(x – h) (y – k) If (h, k) is the center, the standard equation of an ellipse is ------------------- + ------------------ = 1. 2 2 a b 2 2 2 The eccentricity e of the ellipse is given by e = c/a, where c = a – b . This is not the same e as the exponential base, which is a constant. Rather, eccentricity varies with 2

2

a –b c the figure. e = --------------------- = --- is a measure of the elongation of the ellipse and is ala a 2

2

ways less than 1. The distance between the two foci is 2c = 2 a – b . The aspect ratio of the ellipse is a/b. The equation of an ellipse centered at (h, k) = (0, 0) with foci 2

2

x y at (±c, 0) is ----- + ----- = 1 , and the ellipse is symmetric about both coordinate axes. Its 2 2 a b x-intercepts are (±a, 0) and y-intercepts are (0, ±b). The line segment joining (0, b) and (0, -b) is called the minor axis. The major vertices of the ellipse are (±a, 0), and the line segment joining vertices V1 and V2 is the major axis of the ellipse.

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52

Example 1: Determine the values of h, k, a, b, c, and e of the following ellipse: 2 2 3x + 5y – 12x + 30y + 42 = 0 Solution: Rearrange the ellipse equation into the standard form as follows, using the method of completing the square (see page 34): 2

2

2

2

3x + 5y – 12x + 30y + 42 = 3x – 12x + 5y + 30y + 42 = 0 2

2

2

2

3 ( x – 4x + 2 ) + 5 ( y + 6y + 3 ) = 15 2

2

2

2

3 ( x – 2 ) - + 5---------------------y + 3) - = 1 ( y + 3 ) = (-----------------x – 2 ) - + (-------------------------------------2 2 15 15 ( 5) ( 3) 2

2

(x – h) (y – k) Comparing to the form ------------------- + ------------------ = 1 and solving for c and e gives: 2 2 a b h = 2,

k = –3 ,

a =

5,

b =

3,

c =

2,

e =

2--5

Additional Formulas: An ellipse can be represented “parametrically” by the equations x = acosq and y = bsinq, where x and y are the rectangular coordinates of any point on the ellipse, and the parameter q is the angle at the center measured from the x-axis counterclockwise. The following figures correspond to the formulas below them for other measurements of the ellipse.

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Machinery's Handbook, 31st Edition ELLIPSE CALCULATIONS

53

Length, Point, and Angle Calculations R1 = radius of director circle =

2

R2 = radius of equivalent circle = P = center to focus distance =

2

A = major radius =

B +P

AB ,

B = minor radius =

A –P

A +B ,

2

2

A –B ,

2

2

2

2

2

2B -------A = distance, origin to latus rectum

J = any point (X, Y) on curve where X = A sin θ = A cos φ , and Y = B cos θ = B sin φ Y X f = angle with major axis = sin –1  --- = cos –1  --- ,  q = angle with minor axis = 90° – φ  A  B B 2 B L = total perimeter (approximate) = A 1.2  --- + 1.1  --- + 4 A A π L = perimeter (sections) =  --------- 2φ AB 180

Area Calculations

N = total surface area of ellipse = πAB

W = area between outer and inner ellipse = π ( A 1 B 1 – A 2 B 2 ) πAB M = area of complement section M = AB – ----------4 – 1  X 1 ----AB cos – Y X S = area of segment S = 1 1, where the angle that results from  A the inverse cosine is in radian measure –1 X 2 T+S = combined area of segment S + area T = ABcos  ------ – X 2 Y 2, where the angle A that results from the inverse cosine is in radian measure 2 –1 X V = area of segment V = (R2 ) sin  --- – XY, where the angle that results from the  A inverse sine is in radian measure –1 X K = area of sector K = ABcos  --- , where the angle that results from the inverse  A cosine is in radian measure

Example 2: Find area of sector K and complement area M, given the major radius of ellipse is 10 cm, minor radius of ellipse is 7 cm, dimension X = 8.2266 cm. Solution: Sectional area K: –1 X 1 – 1 8.2266 2 Area K = ABcos  ------ = 10 × 7 × cos  ---------------- = 70 × 0.6047 rad = 42.33 cm  A  10 

Solution: Complement area M: 2 πAB π × 10 × 7 Area M = AB – ----------- = 10 × 7 – ------------------------ = 15.0221 cm 4 4 Example 3: Find the area of elliptical segments S, T + S, provided that major radius A of ellipse is 10 cm, minor radius B of ellipse is 7 cm, dimension X1 = 8.2266 cm, dimension Y1 = 4.4717 cm, and dimension X2 = 6.0041 cm. Solution: Segment area S is found: –1 X 1 – 1 8.2266 2 S = ABcos  ------ – X 1 Y 1 = 10 × 7 × cos  ---------------- – 8.2266 × 4.4717 = 5.5437 cm  A  10 

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54

Solution: Segment area T + S: –1 X 2 φ = cos  ------ = 1.51072 rad,  A

Y 2 = B sin φ = 7 sin ( 1.51072 rad ) = 10.57504

–1 X 2 T + S = AB cos  ------ – X 2 Y 2 = 10 × 7 × 0.9268 – ( 6.0041 × 5.5978 )  A

= 64.876 – 33.6097 = 31.266 cm

2

Example 4: Find the area of elliptical segment V if major radius of ellipse is 4 inches, minor radius is 3 inches, dimension X = 2.3688 inches, dimension Y = 2.4231 inches. Solution: Segment area V: R2 =

AB ,

( R 2 ) 2 = AB = 3 × 4 = 12

–1 X – 1 2.3688 2 V = ( R 2) sin  --- – XY = 12sin  ---------------- – ( 2.3688 × 2.4231 )  4   A

= 7.6054 – 5.7398 = 1.8656 in2

Four-Arc Oval Approximating an Ellipse*.—The method of constructing an approximate ellipse by circular arcs, described on page 69, fails when the ratio of the major to minor diameter is 4 or greater. Additionally, it is reported that the method always pro­ duces a somewhat larger minor axis than intended. The method described below presents an alternative. An oval that approximates an ellipse, illustrated in Fig. 7, can be constructed from the fol­lowing equations: B 2 A 0.38 (1) r = -------  --- 2A  B where A and B are dimensions of the major and minor axis, respectively, and r is the radius of the curve at the long ends. The radius R and its location are found from Equations (2) and (3): A 2- – Ar + Br – ----B 2----4 4X = ------------------------------------------B – 2r

--- + X R = B 2

(2)

(3)

A

r

B R X

Fig. 7. Four-Arc Oval Ellipse

To make an oval thinner or fatter than what is given, select a smaller or larger radius r is chosen than that calculated by Equation (1), and then X and R are found using Equations (2) and (3). * Four-Arc Oval material contributed by Manfred K. Brueckner

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Machinery's Handbook, 31st Edition SPHERE

55

Sphere.—A sphere is the locus of points equidistant from a given point (the sphere’s cen­ ter) in three-dimensional space. Similar to the circle, the standard form for the equation of a sphere of radius R and centered at point (h, k, l) is: 2

2

2

(x – h) + (y – k) + (z – l) = R

2

The general form for the equation of a sphere can be written as follows, where A ≠ 0: 2

2

2

Ax + Ay + Az + Bx + Cy + Dz + E = 0 The general and standard forms of the sphere equations are related as follows:

– Bh = -----2A

– Ck = -----2A

R = radius of sphere D = diameter of sphere Ns = total surface area of sphere Nv = total volume of sphere

D-----l = – 2A

2

R =

2

2

B + C + D- – E --------------------------------2 A 4A

R1 = radius of outer sphere R2 = radius of inner sphere Ga , Ka , Sa , Ta , Ua , Wa , Z a = sectional surface areas Gv, Kv, Sv, Tv, Uv, Wv, Zv = sectional volumes

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Machinery's Handbook, 31st Edition SPHERE FORMULAS

56

Formulas for Spherical Radius

To Find

Formula

Radius of sphere from volume Nv

RN =

3

To Find

Radius of section T

3N --------v4π

Formula 2

RT =

2

2 2

2 P – Q – 4H - P ---------------------------------+ ----  8H 4

Formulas for Spherical Areas and Volumes Area Formula Volume Formula

Section

()

π 3 4π 3 N v = --- D = ------ R 6 3

2

Entire sphere

N a = 4πR

Section G

G a = 4πR 1 + 4πR 2

4π 3 3 G v = ------ ( R 1 – R 2 ) 3

Section K

2 K a = 2πR  1 – cos --φ-   2

F---------------K v = 2πR 3

Section S

2

2

2

2

2

2 E  S a = π  F + ---- 4

2

E- + F ------ S v = πF  ----8 6 2

2

Section T

T a = 2πRH

π 2 ---------- + 3P --------- T v = H --- H + 3Q  6  4 4 

Section U

2 2 φ U a = 2π ( R 1 + R 2 )  1 – cos  ---   2

3 3 U v = 2π ( R 1 – R 2 )  1 – cos  φ ---   2 

Section W

W a = 4π R 1 R 2

W v = 2π R 1 R 2

Section Z

φ 2 Z a = ( 4π R 1 R 2 ) --------360

φ 2 2 Z v = ( 2π R 1 R 2 ) --------- 360

2

2

2

Example 1: Find the inside and outside surface area Ga and volume Gv of wall G, pro­vided that R1 is 12.5 cm and R2 is 10.0 cm. Solution: Sectional area Ga and sectional volume Gv: 2

2

2

2

G a = 4πR 1 + 4πR 2 = 4π ( 12.5 ) + 4π (10) = 3220.13 cm

2

4π 3 4π 3 3 3 3 G v = ------ ( R 1 – R 2 ) = ------ ( 12.5 – 10 ) = 3992.44 cm 3 3

Example 2: Find the surface area Ka and volume Kv of section K of a sphere of radius 15.0 cm, if included angle φ = 90° and depth F = 5.0 cm. Solution: Sectional area Ka and sectional volume Kv: 2 2 φ 90° K a = 2πR  1 – cos  --- = 2π (15 )  1 – cos  --------  = 414.07 cm 2    2  2  2

2

2πR F 2π (15) (5) K v = ----------------- = --------------------------3 3

= 2356.19 cm 3

Example 3: Find the outside surface area Sa and sectional volume Sv of section S of a sphere if E = 20.0 cm and F = 5.0 cm. Solution: Sectional area Sa and sectional volume Sv:

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Machinery's Handbook, 31st Edition SPHERE FORMULAS 2

57

2

2 E 2 20 2 S a = π  F + ------ = π  5 + -------- = 392.70 cm   4 4  2

2

2

2

3 E F 20 5 S v = πF  ------ + ------ = π × 5  -------- + ----- = 850.85 cm 8  8 6 6

Example 4: Find the outside and inside surface area Ua and volume Uv of section U of a sphere if R1 = 5.0 inches, R2 = 4.0 inches, and included angle f = 30°. Solution: Sectional area Ua and sectional volume Uv: 2 2 2 2 φ 30° U a = 2π ( R 1 + R 2 )  1 – cos  ---  = 2π ( 5 + 4 )  1 – cos  --------  = 8.78 in 2  2   2    3 3 3 3 φ 30° U v = 2π ( R 1 – R 2 )  1 – cos  ---  = 2π ( 5 – 4 )  1 – cos  --------  = 13.06 in 3  2   2    Example 5: Find the total surface area Wa and volume Wv of ring W, if R1 = 5.0 inches and R2 = 4.0 inches. Solution: Sectional area Wa and sectional volume Wv: 2

2

W a = 4π R 1R 2 = 4π × 5 × 4 = 789.57 in 2 2

2

2

2

W v = 2π R 1R 2 = 2π × 5 × 4 = 1579.14 in 3

Parabola.—A parabola is the set of all points P in the plane that are equidistant from a focus F and a line called the directrix. The parts of the parabola are labeled in the figure below. 2 The general equation of a parabola horizontal axis is given by ( y – k ) = 4p ( x – h ), where the vertex is located at point (h, k), the focus F at point (h + p, k), and the directrix is the line x-axis at x = h - p. The latus rectum is the vertical line segment through the focus; its endpoints are on the parabola. Hence, it lies on the line x = h + p. The length of the latus rectum is four times the absolute value of the x-coordinate of the focus, hence |4(h + p)|. Example: Determine the focus, directrix, parabolic axis, vertex, and length of the latus rectum of the parabola 2 4y – 8x – 12y + 1 = 0 Solution: Rewrite the equation in the general form of a parabolic equation (see Solving by Completing the Square on page 34). 2

4y – 8x – 12y + 1 = 0 2

4y – 12y = 8x – 1 2 --y – 3y = 2x – 1 4

3 2 3 2 y – 3y --- +  --- = 2x – --1- + --92  2 4 4 2 y – 3 --- = 2 ( x + 1 )  2

Parabola

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Machinery's Handbook, 31st Edition HYPERBOLA

58

Thus, k = 3 ⁄ 2, h = -1 and p = 1 ⁄ 2. Focus F is located at point (h + p, k) = (1 ⁄ 2, 3 ⁄ 2); the directrix is located at x = h - p = -1 - 1 ⁄ 2 = - 3 ⁄ 2; the parabolic axis is the horizontal line y = 3 ⁄ 2; the vertex V(h, k) is located at point (-1, 3 ⁄ 2); and the latus rectum lies on the line x = h + p = -1 ⁄ 2. Its length is 4|-1 ⁄ 2| = 4(1 ⁄ 2) = 2.

Hyperbola.—Referring to the figure on the left, below, a hyperbola is the set of all points such that |d1 – d2 | is constant. That is, the difference between the distances from any point (x, y) to the foci, marked F1(–c, 0) and F2 (c, 0) does not change. The distance between the vertices, V1(–a, 0) and V2 (a, 0) (the turning points of the hyperbola) is 2a. Therefore, |d1 – d2 | = 2a for any two points on the hyperbola.

The figure on the right shows more detail. The slopes of the asymptotes (lines of approach) relate to the transverse and conjugate axis lengths, 2a and 2b. The center of a hyperbola is the point of intersection of the asymptotes. In the figure, the center is shown as the origin, (0, 0). The standard form of the hyperbola, as derived from the foci with center at the origin, is

is

( x − h) 2 a2

−

( y − k )2 b2

= 1.

x2

a

2

−

y2

b2

= 1. For any center (h, k), the equation

The general form of the hyperbola is given by Ax 2 + By 2 + Cx + Dy + E = 0, where AB < 0 and AB ≠ 0. 2

2

a +b Also, the eccentricity of a hyperbola, e = --------------------- , is always less than 1. a 2

2

The distance 2c between the two foci is given by the relation: 2c = 2 a + b . Example: Determine the values of h, k, a, b, c, and e of the hyperbola general form: 2

2

9x – 4y – 36x + 8y – 4 = 0 Solution: Convert the hyperbola equation into the standard form (see Solving by Completing the Square on page 34):

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Machinery's Handbook, 31st Edition HYPERBOLA 2

2

2

59

2

9x – 4y – 36x + 8y – 4 = ( 9x – 36x ) – ( 4y – 8y ) = 4 2

2

2

9 ( x – 4x + 4 ) – 4 ( y – 2y + 1 ) = 36

2 2 9----(-----------------x – 2 ) - 4 ( y – 1 )2 y – 1) - = 1 x – 2 ) - – (-----------------– ---------------------- = (-----------------2 2 36 36 2 3 2

2

(x – h) (y – k) Comparing the results above with the form ------------------- – ------------------ = 1 and calculating 2 2 a b 2

2

a +b eccentricity from e = --------------------- and distance c from c = a h = 2,

k = 1,

a = 2,

b = 3,

2

2

a + b gives

c =

13 ,

13e = --------2

Complex Numbers

Imaginary Number.—The square root of a negative number cannot be expressed with real numbers, since any negative number multiplied by itself is positive. But technical mathematics often relies on computation involving the square root of –1. For this, imaginary number i is defined as follows: i=

− 1 , so

(

−1

)

2

= −1

Imaginary numbers are not real numbers; they belong to the set of complex numbers. An example of an equation that cannot be solved with real numbers is x 2 + 1 = 0. Re­ arranging gives x 2 = –1, and taking the square root of both sides gives x = ±

−1 = ± i.

(Note: The letter j is also used to represent the imaginary number −1.) Forms of a Complex Number.—Complex numbers can be expressed in several forms, all of which are based on the complex coordinate system, as seen in Fig. 1 and Fig. 2. Operations on complex numbers: Complex numbers are added and subtracted much like real numbers, but with real parts added to real parts and imaginary to imaginary: (a + bi) + (c + di) = (a + c) + (bi + di) = (a + c) + (b + d)i (a + bi) – (c + di) = (a – c) + (bi – di) = (a – c) + (b – d)i where coefficients a, b, c, and d are real numbers. Example 1: (3 + 4i) + (2 – i) = (3 + 2) + (4i – i) = 5 + 3i Complex numbers are multiplied as binomials are, by FOIL: (a + bi)(c + di) = ac +adi + bci + bdi2 = ac + (ad + bc)i + bd(–1) = ac + (ad + bc)i – bd Example 2: (1 + 2i)(5 – 7i) = 5 – 7i + 10i –14i2 = 5 + 3i – (14)(–1) = 5 +3i + 14 = 19 + 3i Standard (rectangular) form of a complex number: A complex number z has a real part and an imaginary part. Its standard form is z = a + bi, where a is the real part and bi the imaginary part. Fig. 1 shows how the complex plane is similar to the real plane (see page 42 in Analytic Geometry), except here only the horizontal axis, x, is real, whereas the vertical axis, yi, is imaginary. Fig. 2 shows examples of complex numbers in rectangular form.

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Machinery's Handbook, 31st Edition FORMS OF COMPLEX NUMBERS

60

Fig. 1. Complex coordinate system

Fig. 2. Examples of standard (rectangular) form, z = a + bi

z = 5 + 3i lies in the first quadrant, with a and b both positive; z = –1 –i is in the third quadrant. Complex numbers can be converted from the standard form to any of three vector forms: polar, trigonometric, and exponential. Polar form of a complex number: Vectors are objects that have both magnitude (r) and direction (θ). Vectors are essential in electrical engineering and other fields for representing many processes such as alternating current and voltage. They are represented graphically by an arrow, as seen in Fig. 3.

z = rθ

Fig. 3. (a) Polar form of a complex number; (b) Magnitude r = |z| relationship to a and b; (c) Angle θ to a, b.

The polar form of a complex number is z = r θ. z is a vector in the sense of its magnitude (length) r and direction angle θ from the horizontal (Fig. 3a). By Pythagorean theorem, r 2 = a2 + b2; thus, magnitude r = a + b , which is called the modulus, is denoted |z|. From trigonometry, tan θ = b/a; hence, θ = tan–1(b/a). Trigonometric form of a complex number: Another form that shows the directional nature of complex numbers is the trigonometric form of z. From trigonometry, cosθ = a/r and sinθ = b/r; hence, 2

2

a = r cosθ,  b = r sinθ,   so, a + bi = r cosθ + ir sinθ = r (a cosθ + i sinθ ) Exponential form of a complex number: Recall that the number e is the base of the natural logarithm (see page 37). A complex number is represented in exponential form through Euler’s formulas, which are used widely in electrical engineering applications:

e

iθ

= cos θ + i sin θ ,

iθ

cos θ =

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e +e 2

− iθ

,

iθ

sin θ =

e −e

−iθ

2i

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Machinery's Handbook, 31st Edition PURE GEOMETRY

61

Pure Geometry The labels that identify the parts of a figure (angle A, radius r, diameter d, and so on) are used in the formulas to indicate the measure of that feature. By definition, if any two geometric features A and B have equal measure, they are said to be congruent. So, if the measure of A equals the measure of B, then A ≅ cB. Polygons are congruent if they have the same shape and size, that is, if one can be superimposed on the other point for point. Triangles are congruent if any of the propositions for triangle congruence hold, as summarized below. Table 1a. Propositions of Geometry

A triangle is a three-sided polygon. It is, in fact, the polygon with the least number of sides. The sides of a triangle meet at its vertices (singular vertex). The sum of the measures of all three angles of a triangle is 180 degrees. Hence, if the measures of any two angles are known, the third angle measure can always be found.

A C

B

A + B + C = 180° B = 180° – ( A + C )

A

B1

a

a1

b

b1

A

A a

B

e

c

b

c B

D

F E d

C a

e

SSS Proposition: If all three sides of one triangle are congruent (equal in measure) to all three sides of another triangle, then the triangles are congruent. If the three sides in one triangle are equal in measure to the three sides of another triangle, then the angles in the two triangles are equal in measure.

a1

c1

f

A

Hence, in the figure, if a = a1, b = b1, and A = A1, then the remaining side and angles also are equal in measure, and thus the triangles are congruent.

b1

a

C

SAS Proposition: If two sides and the included angle (the angle between the sides) of one triangle are congruent (equal in measure) to the corresponding (similarly located) sides and angle of another triangle, then the triangles are congruent.

a1

b

b

Hence, if a = a1, A = A1, and B = B1, the other corresponding side and angle are equal in measure, and thus the triangles are congruent.

A1

a

c

AAS Proposition: If two angles and the non-included side of one triangle are congruent to the corresponding (similarly located) angles and sides of another triangle, the triangles are congruent.

A1

B

D F

E

d

A = 180° – ( B + C ) C = 180° – ( A + B )

If a = a1, b = b1, and c = c1, then the corresponding angles are also equal in measure, and thus the triangles are congruent.

f

If the three sides of a triangle are proportional to corresponding sides of another triangle, then the triangles are similar, and the angles in the one are congruent (equal in measure) to the angles in the other. Hence, if a ⁄ d = b ⁄ e = c ⁄ f then A = D, B = E, C = F

Similar triangles are ones whose corresponding angles are congruent. If this is true then the corresponding sides are proportional. If the angles of one triangle are congruent (equal in measure) to the angles of another triangle, then the triangles are similar and their corre­sponding sides are proportional.

Hence, if A = D, B = E, and C = F then a ⁄ d = b ⁄ e = c ⁄ f

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Machinery's Handbook, 31st Edition PROPOSITIONS OF GEOMETRY

62

Table 1b. Propositions of Geometry

a

In an isosceles triangle, two sides a, b are congruent, hence the two angles opposite them (the base angles) are congruent. An equilateral (equiangular) triangle is also an isosceles triangle.

b B

A

60

a

In an equilateral (equiangular) triangle, all three sides (angles) are congruent. Since the sum of angle measures in any triangle is 180 degrees, then each angle in an equilateral (equiangular) triangle measures 60 degrees.

a

60

60

a A

30

A line in an equilateral triangle that bisects any of the angles (that is, divides it into two 30-degree angles), also bisects the side opposite the bisected angle and is perpendicular (at right angles) to it.

30

90

C

1/ 2 a

a B

1/ 2 b

B

1/ 2 a

b

1/ 2 B

90

b

1/ 2 b

a

b

A

B

D

Thus, if line AB bisects angle CAD, it also bisects line CD into two equal parts and is perpendicular to it.

If a line in an isosceles triangle drawn from the vertex where the two congruent sides meet in such a way that is bisects the third side (or base), then it also bisects the angle at the vertex from which it is drawn.

In every triangle, the greatest angle is opposite the longest side. And, the longest side is opposite the greatest angle.

Thus, if a is longer than b, then the measure of angle A is greater than that of angle B. And, if angle A measure is greater than B, then side a is longer than side b. According to the triangle inequality, for any triangle, the sum of the lengths of any two sides must be greater than or equal to the length of the remaining side. Thus, a + b ≥ c.

The Pythagorean theorem states that in a right-angle triangle, the square of the hypotenuse, that is, the side opposite the right angle, is equal to the sum of the squares on the two sides that form the right angle. These sides are the legs of the right triangle. Thus, a2 + b2 = c2

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Machinery's Handbook, 31st Edition PROPOSITIONS OF GEOMETRY

63

Table 1c. Propositions of Geometry In the figure, angle D is an exterior angle. The measure of an exterior angle is equal to the sum of the measures of the non-adjacent interior angles. Thus, A+B=D

A D

B

Intersecting lines form congruent vertical angles, that is, the angles opposite one another. In the figure, A and B are vertical angles and therefore congruent, and C and D are vertical angles and therefore congruent. That is,

D

B

A

A≅B C≅D

C B

l1

A

B

l2

A

A

Lines l1 and l2 are intersected by s transversal, so all angles A are congruent, and all angles B are congruent.

B

A quadrilateral is a four-sided polygon. In any quadrilateral, the sum of the measures of the interior angles is 360 degrees.

C D

1 /2

A

Thus, A + B + C + D = 360 degrees A parallelogram is a quadrilateral in which each pair of opposite sides is parallel and thus congruent. Opposite angles are also congruent. Each diagonal divides the parallelogram into congruent triangles, and the diagonals bisect each other. In the figure, diagonal D bisects diagonal d at their midpoints.

D

1 /2

B

B

A

A B

b

Corresponding angles formed when parallel lines are intersected by another line (a transversal) have equal measure (are congruent).

d

a

A

A1

h

b

h1

b1

h

Thus, if b = b1 and h and h1, then area A = area A1.

A1

1/ 2

c

90

c

b1

b

1/ 2

A

A rectangle is a parallelogram in which all four angles are right angles. In the figure, the rectangle’s base is marked b; its height is marked h; the parallelogram’s base is marked b1, and its height is marked h1. The area of a parallelogram is A = bh. If two parallelograms have a corresponding base and height of equal measure, their areas are equal.

h1

Triangles having equal base and equal height have equal area. Thus, if b = b1 and h and h1, then area A = area A1.

If a diameter of a circle is perpendicular (at right angles) to a chord, then it bisects the chord, dividing it into two equal parts. In the figure, the chord has length c, so its parts have length c ⁄2 .

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Machinery's Handbook, 31st Edition PROPOSITIONS OF GEOMETRY

64

Table 1d. Propositions of Geometry

A line tangent to a circle lies perpendicular (at right angles) to a radius drawn to meet it at the point of tangency.

90

Point of Tangency The figure shows two ways that circles can be tangent. A line drawn through the center of each circle (the diameter) will pass through the point of tangency.

a A A

Two tangents drawn from a single point outside a circle will be equal in length (a), and the angles they make with the chord that connects the points of tangency (A) will be equal in measure. The figure shows this congruency.

a

d A

The angle formed by a tangent and a chord drawn from the point of tangency measures one-half the central angle subtended by that chord.

B

That is, B = A/2

d A

B

b

c

a

A

B

A B

That is, B = A.

All inscribed angles sub­tended by the same chord in a circle are congruent (equal in measure).

C

d

c

The angle formed by a tangent and a chord drawn from the point of tangency has measure equal to the inscribed angle subtended by the chord.

In the figure, A, B, and C are subtended by chord cd, and so are equal in measure.

Referring to the figure, inscribed angle A and central angle B are subtended by the same arc. The measure of inscribed A is half the measure of central angle B. In the figure,

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A = B/2

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Machinery's Handbook, 31st Edition PROPOSITIONS OF GEOMETRY

65

Table 1e. Propositions of Geometry A

B

A major arc of a circle is one that measures more than half the circumference of the circle. A minor arc measures less than half the circumference of the circle.

In the figure, angle A is subtended by a minor arc, so it is an acute angle (it measures less than 90 degrees). Angle B is subtended by a major arc, so it is an obtuse angle (it measures more than 90 degrees). An angle subtended by a circle’s diameter is a right angle; the arc described by the diameter is a semicircle. Referring to the figure, right angle A is subtended by diameter d. Angle C = 90°

In the figure, the product of line segment lengths formed by intersecting chords in a circle are equal. Thus:

c

d

a

ab = cd

b

a c b

When two lines are drawn from a point outside a circle, one tangent and one through the circle intersecting it at two points, the line segment lengths are such that the square of the tangent segment is equal to the product of the segments formed by the other line. Thus: a 2 = bc

b

a B A

Arc lengths of a circle are proportional to the corresponding central angle measures. Thus: A:B = a:b

a

b A r

Circum. = c Area = a

r

B

R

Circum. = C Area = A

R

The lengths of circular arcs having the same central angle are pro­portional to the lengths of the radii. Thus, if A = B, then a/b = r/R

The ratio of the circumferences of two circles is proportional to the ratio of their radii.

c:C :: r:R so c/C = r/R The ratio of the areas of two circles is proportional to the ratio of the squares of their radii. a:A :: r2:R2 so a/A = r2/R2

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Machinery's Handbook, 31st Edition PROPOSITIONS OF GEOMETRY

66

The geometric constructions below are produced with a compass, protractor, and straight edge. The compass point marks the “centers” in the constructions that follow. The techniques described also apply for electronic drafting programs. As the section progresses, some constructions rely on previous ones. Table 2a. Geometric Constructions C

To bisect a line segment (divide it into two equal parts):

A

With the ends A and B as centers and a radius greater than one-half the line segment, use the compass to draw circular arcs. Through intersections C and D, use the straight edge draw line CD. This line bisects AB and is also perpendicular to AB.

B D

To draw a perpendicular to a line from a point A on that line:

D

B

With A as the center and with any radius, use the compass to draw circular arcs inter­secting the given line at B and C. Then, with B and C as centers and a radius longer than AB, draw circular arcs intersecting at D. With the straight edge, draw line DA. This line is perpendicular to BC.

C

A

To draw a perpendicular line from a point A to C at the end of line AB:

C D A

Draw line AB, and then with any point D not on AB as center, use the compass to draw a circle intersecting AB at E. Draw a line through E and D intersecting the circle at C; then join AC. This line is the required perpendicular.

E B

To draw a perpendicular to a line AB from a point C at a distance from it:

C A

E

F

B

D

1 A

2

3

4

5

With C as center, use the compass to draw an arc intersecting the given line at E and F. With E and F as centers, draw circular arcs with a radius longer than one-half the distance between E and F. These arcs intersect at D. Line CD, drawn with the straight edge, is the required perpendicular.

To divide a line segment AB into a number of equal parts:

C

B

Assume line segment AB is to be divided into five smaller segments of equal measure. Draw line AC at an acute angle with AB. Mark off on AC five parts of equal measure, any convenient length. With the straight edge, draw the segment B-5 and then draw lines parallel to B-5 through the other division points on AC. The points where these lines intersect AB are the required division points.

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Machinery's Handbook, 31st Edition PROPOSITIONS OF GEOMETRY

67

Table 2b. Geometric Constructions E

To draw a line parallel to a given line AB, at a given dis­tance from it:

F

A C

Using the compass, with any points C and D on AB as centers, draw with the compass circular arcs with the given distance as radius. Line EF, drawn with the straight edge to be tangent to (that is, touch without intersecting) the cir­cular arcs, is the required parallel line.

D B

D

B

To bisect an angle BAC:

With A as a center and any radius, use the compass to draw arc DE. With D and E as centers and a radius greater than one-half DE, use the compass to draw circular arcs intersecting at F. Line AF, drawn with the straight edge, divides the angle into two equal parts.

A F C

E H

C

A

To draw an angle upon a line AB equal to a given angle FGH:

L

E B

D

G

F

K

E

C

With point G as a center and with any radius, draw arc KL. With A as a center and with the same radius, draw arc DE. Make arc DE equal in length to KL and draw AC through E. Angle BAC then is equal in length to angle FGH.

To lay out a 60-degree angle:

With A as a center and any radius AB, draw an arc BC. With point B as a center and AB as a radius, draw an arc intersecting at E the arc just drawn. EAB is a 60-degree angle.

A

G

A 30-degree angle may be obtained either by bisecting a 60-degree angle or by drawing a line EG perpen­dicular to AB. Angle AEG is then 30 degrees.

B

D

E

To draw a 45-degree angle:

From point A on line AB, set off a distance AC. With a straight edge, draw perpen­dicular DC and set off a distance CE equal to AC. Draw AE. Angle EAC is a 45-degree angle.

A

C

B

C

A

To draw an equilateral triangle, with the length of the sides equal to AB:

B

Draw AB with straight edge. With A and B as centers and AB as radius, use the compass to draw circular arcs intersecting at C. With the straight edge, draw AC and BC. Then ABC is an equilateral triangle.

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Machinery's Handbook, 31st Edition PROPOSITIONS OF GEOMETRY

68

Table 2c. Geometric Constructions C

To draw a circular arc with a given radius through two given points A and B:

A

With A and B as centers, with the compass draw two circular arcs with the given radius intersecting at C. With C as center and the same radius, draw a circular arc through A and B.

B

To locate the center of an arc of a circle:

H C

D G A

B

E

E

F

C A F B

C A

To draw a tangent to a circle through a given point on the circum­ference:

Through a chosen point of tangency A on a circle with center B, draw radius BC. At point A, draw a line EF at right angles to BC. This line is the required tangent.

To divide a circular arc AB into two equal parts:

B

E D C F A

Select three points on the periphery of the circle, as A, B, and C. With each of these points as a center and setting the same radius with the compass, draw arcs intersecting each other. Through the points of inter­section, draw lines DE and GF. Point H, where these lines inter­sect, is the center of the circle.

To circumscribe a circle about a triangle ABC:

G B

E

With A and B as centers, and a radius larger than half the distance between A and B, with the compass draw circular arcs intersecting at C and D. Line CD divides arc AB into two equal parts at E.

Bisect the sides AB and AC, and from the midpoints E and F draw lines at right angles to the sides. These lines intersect at G. With G as a center and GA as a radius, draw circle ABC.

B E A

F D

To inscribe a circle in a triangle ABC:

Bisect two of the angles, A and B, by lines intersecting at D. From D, draw a line DE perpendicular to one of the sides, and with DE as a radius, use the compass to draw circle EFG.

G

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Machinery's Handbook, 31st Edition PROPOSITIONS OF GEOMETRY

69

Table 2d. Geometric Constructions A

B F

To circumscribe a circle about a square and to inscribe a circle in a square ABCD: Draw the square’s diagonals AC and BD with the straight edge. The centers of both the circumscribed and inscribed circles are located at the point E, where the two diagonals of the square intersect. The radius of the circumscribed circle is AE, and of the inscribed circle, EF.

E D

C

D

E To inscribe a hexagon in a circle with center C:

A

B

C F

Draw diameter AB. With A and B as centers and with the circle’s radius as radius, describe circular arcs intersecting the given circle at D, E, F, and G. Draw chords AD, DE, etc., forming the required hexagon ABCDEFG.

G

To circumscribe a hexagon about a circle with center C:

A

F

C

E G

Draw the circle’s diameter AB, and with A as center and the radius of the circle as radius, cut the circumference of the given circle at D. Draw chord AD and bisect it with radius CE. Through E, draw FG paral­lel to AD and intersecting diameter AB at F. With C as center and CF as radius, draw a circle. Within this circle, inscribe the hexagon as in the preceding problem. This is the circumscribed hexagon about the first circle.

B

D

E

D

e

A

f

F

g

G B

O

C

D K

L

A M

O N P

Describe circles with O as center and AB and CD as diameters. From a number of points, E, F, G, etc., on the outer circle, draw radii intersecting the inner circle at e, f, and g. From E, F, and G, draw lines perpendicular to AB, and from e, f, and g, draw line segments parallel to AB. The intersections of these perpendicular and paral­lel lines are points on the curve of the ellipse.

To construct an approximate ellipse by circular arcs:

B

E

To describe an ellipse with the given axes AB and CD:

F

G

C H

Let AC be the major axis and BN the minor. With the compass, draw semicircle ADC with O as center. Divide BD into three equal parts and set off BE equal to one of these parts. With A and C as centers and OE as radius, describe circular arcs KLM and FGH; with G and L as centers, and with the same radius, describe arcs FCH and KAM. Through F and G, draw line FP, and with P as center draw arc FBK. Arc HNM is drawn in the same manner.

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Machinery's Handbook, 31st Edition PROPOSITIONS OF GEOMETRY

70

Table 2e. Geometric Constructions

To construct a parabola:

Divide line segment AB into a number of equal parts and divide BC into the same number of parts. From the division points on AB, draw horizontal lines. From the division points on BC, draw lines to point A. The points of intersection of the lines drawn from points numbered alike are points on the parabola.

To construct a hyperbola:

From focus F, lay off a distance FD to be the transverse axis, or the distance AB between the two branches of the curve. With F as center and any distance FE greater than FB as a radius, describe a circular arc. Then with F1 as center and DE as radius, describe arcs intersecting at C and G the arc just described. C and G are points on the hyperbola. Any number of points can be found in a similar manner; when a sufficient number of points are found, draw a smooth curve through them.

C A

B

F

F1 E

D

G

To construct an involute:

F 2

E

3

1 D A

C

Divide the circumference of the base circle ABC into a number of equal parts. Through the division points 1, 2, 3, etc., draw tan­gents to the circle and make the lengths D-1, E-2, F-3, etc., of these tangents equal to the actual length of the arcs A-1, A-2, A-3, etc. Connect the ends of these tangents with a smooth curve.

B

1/ 2

Lead

6 5 4 3 2 1 0

1 0

2

3

4

5 6

To construct a helix:

Divide half the circumference of the cylinder on the surface of which the helix is to be described into a number of equal parts. Divide half the lead of the helix into the same number of equal parts. From the division points on the circle representing the cyl­inder, draw vertical lines using the straight edge, and from the division points on the lead, draw horizontal lines as shown. The intersections between lines numbered alike are points on the helix. Connect these points with a smooth curve.

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Machinery's Handbook, 31st Edition AREA AND VOLUME

71

Area and Volume The Prismoidal Formula.—A right prism is a three-dimensional figure composed of (two-dimensional) polygons that form the faces of the prism. The end faces are the bases, the other faces are the sides. The prismoidal formula is a general formula by which the volume of any prism, pyramid, or frustum of a pyramid may be found. A1 , A 2 =  end areas of the body A m =  area of faces between the two end surfaces h =  height of body h Then, volume of the body is calculated as V = --- ( A 1 + 4A m + A 2 ) . 6 Pappus-Guldinus Rules.—A surface of revolution is generated when a curve is revolved about an external axis. (The curve must lie wholly on one side of the axis of revolution and in the same plane.) The mathematics for finding the curve’s length and the length of the path of the centroid involves calculus. Some surface areas and volumes of solids of revolution can then be determined by the rules of the Pappus-Guldinus theorems. The area of the resulting surface is equal to the product of the length of the generating curve and the distance traveled by the curve’s center of gravity, or centroid (see figure below).

The volume of a solid body formed by the revolution of a surface FGHJ about axis KL equals the product of the surface area and the length of the path of its center of gravity about axis KL. C

A

5” 3”

B D

Example: By means of the Pappus-Guldinus rules, the area and volume of a cylindrical ring, or torus, may be found. A torus is formed when a circle is rotated about an axis. The center of grav­ity of the circle is its center. Hence, with the dimensions given in the illustration, the length of the path of the center of gravity of the circle it travels is the circumference 2πr = 2 × 3.1416 × 5 = 31.416 inches. Multiplying this path length by the circumference of the circle, which is 3.1416 × 3 = 9.4248 inches, gives 31.416 × 9.4248 = 296.089 square inches. This is the surface area of the torus.  d 2- = 0.7854 × 9 = 7.0686 square The volume of the torus equals the area of the circle, ----------4 inches, multiplied by the path length of the center of gravity, which is 31.416, as before; hence,

Volume = 7.0686 × 31.416 = 222.067 cubic inches

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Machinery's Handbook, 31st Edition AREA OF SURFACE OF REVOLUTION

72

0.41”

Approximate Method for Finding the Area of a Surface of Revolution.—The illustration below is an example of the approximate method based on Guldinus rule for finding the surface area of a symmetrical body. In the illustration, the dimensions in common fractions are the known dimensions; those in deci­mals are found by actual measurements on a figure drawn to scale. The surface area is found as fol- G lows: First, the entire form is sepa1 2” rated into such areas as are cylin­ 0.03” E F drical, conical, or spherical, since 0.51” their surface areas can be found 0.55” by exact formulas. In the illustra0.61” 1” tion, the three-dimensional portion 0.70” marked in the plane by ABCD is a 0.84” cylinder, the area of the surface of 0.99” 11 2 ” which can be easily found. The top 1.15” 1.36” area EF is simply a circular area and 1.52” 5 can thus be com­puted separately. 16 ” 1.62” The remainder of the surface generB A ated by rotating line AF about the 58 1 ” axis GH is found by the approxiT mate method. From point A, equal C D dis­tances are set off on line AF. In H the illus­tra­tion, each division indicated is 1 ⁄ 8 inch long. From the central or mid­dle point of each of these parts a line is drawn at right angles to the axis of rotation GH, the length of these lines or diameters (the length of each is given in decimals) is measured, all these lengths are added together and the sum is multiplied by the length of one division set off on line AF (in this case, 1 ⁄ 8 inch), and this product is multiplied by p to give the approximate area of the sur­face of revolution. In setting off divisions 1 ⁄8 inch long along line AF, the last division does not reach all the way to point F, but only to a point 0.03 inch below it. The part 0.03 inch high at the top of the cup can be considered as a cylinder of 1 ⁄2 -inch diameter and 0.03-inch height, the area of the cylindrical surface of which is easily computed. By adding the various surfaces together, the total surface of the cup is found as follows: Cylinder, 15 ⁄8 in. diameter, 0.41 in. height

2.093 in2

Circle, 1 ⁄2 in. diameter

0.196 in2

Cylinder, 1 ⁄2 in. diameter, 0.03 in. height

0.047 in2

Irregular surface

3.868 in2

Total

6.204 in2

Area of Irregular Plane Figure.—One of the most useful and accurate methods for determining the approximate area of a plane figure or irregular outline is known as Simpson’s rule. In applying Simpson’s rule to find an area, the work is done in four steps: 1) The area is divided into an even number N of parallel strips of equal width W; for exam­ple, in the accompanying diagram, the area has been divided into 8 strips of equal width. 2) The sides of the strips are labeled V0, V1, V2 , etc., up to V N. 3) These heights V0, V1, V2 ,…, V N are measured. 4) The values V0, V1, etc. are substituted in the following formula to find the area A of the figure:

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Machinery's Handbook, 31st Edition AREA OF IRREGULAR PLANE FIGURE

73

W A = ---- [ ( V 0 + V N ) + 4 ( V 1 + V 3 + … + V N – 1 ) + 2 ( V 2 + V 4 + … + V N – 2 ) ] 3 Example: The area of the accompanying figure was divided into 8 strips on a full-size drawing and the following data obtained. Calculate the area using Simpson’s rule. W = 1 cm V0 = 0  cm V1 = 1.91  cm V2 = 3.18  cm V3 = 3.81  cm V4 = 4.13  cm V5 = 5.27  cm V6 = 6.35  cm V7 = 4.45 cm V8 = 1.27  cm

W

v0

v1

v2

v3

v4

v5

v6

v7

v8

Substituting the given data in the Simpson’s formula, 1 A = --- [ ( 0 + 1.27 ) + 4 ( 1.91 + 3.81 + 5.27 + 4.45 ) + 2 ( 3.18 + 4.13 + 6.35 ) ] 3

1 = --- [ 1.27 + 4 ( 15.44 ) + 2 ( 13.66 ) ] = 30.12 cm 2 3 In applying Simpson’s rule, it should be noted that the larger the number of strips into which the area is divided the more accurate the results obtained. Areas Enclosed by Cycloidal Curves.—The area between a cycloid and the line upon which the generating circle rolls equals three times the area of the generating circle (see diagram, page 79). The areas between epicycloidal and hypocycloidal curves and the “fixed circle” upon which the generating circle is rolled may be determined by the follow­ ing formulas, in which a = radius of the fixed circle upon which the generating circle rolls, b = radius of the generating circle, A = the area for the epicycloidal curve, and A1 = the area for the hypocycloidal curve.  b 2 ( 3a – 2b )  b 2 ( 3a + 2b )A = --------------------------------A 1 = --------------------------------a a Contents of Cylindrical Tanks at Different Levels.—In conjunction with the table Segments of Circles for Radius = 1 starting on page 84, the following relations can give a close approximation of the liquid contents, at any level, in a cylindrical tank. Measuring Stick x

r x Less Than Half-Filled

y

y

r

d

L More Than Half-Filled Less Than Half-Filled

More Than Half-Filled

A long measuring rule calibrated in length units or a plain stick can be used for measuring contents at a particular level. In turn, the rule or stick can be graduated to serve as a volume gauge for the tank in question. The requirements are that the tank must have a circular cross section; the dimensions of the tank must be known; the gauge rod has to be inserted vertically through the top center of the tank so that it rests precisely in the center at the bottom of the tank; and the calculations must be done using consistent metric or US customary (also called English) units. The formulas and parameters are:

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Machinery's Handbook, 31st Edition CONTENTS OF CYLINDRICAL TANKS

74

K =  Cr2 L = tank constant (the same for any given tank) V T =  pK, for a tank that is completely full

(1) (2)

Vs =  KA

(3)

V =  VT - Vs = VT - KA, for a tank that is more than half full

(5)

(4)

V =  Vs, for a tank that is less than half full

where: C = liquid volume conversion factor; the exact value depends on the length and liquid volume units used during measurement: 0.00433 US gal/in3; 7.48 US gal/ft3; 0.00360 UK gal/in3; 6.23 UK gal/ft3; 0.001 liter/cm3; or 1000 liters/m3 V T = total volume of liquid tank can hold

Vs = volume formed by segment of circle having depth x in the given tank (see dia­gram)

V = volume of liquid at particular level in tank



A = segment area of a corresponding unit circle taken from the table starting on page 84.

L = length of tank; r = radius of tank ( = 1 ⁄2 diameter)

y = actual depth of contents in tank as shown on a gauge rod or stick

x = depth of the segment of a circle to be considered in given tank. As can be seen in the diagram, both x and y are the actual depth of contents when the tank is less than half full, but x is the depth of the void (d - y) above the contents when the tank is more than half full. In the discussion of the unit circle, page 84, r = 1, and so the height of a segment of a corresponding unit circle is x/r.

Example: A tank is 20 feet long and 6 feet in diameter. Convert a long stick graduated in inches into a gauge graduated at two points, 1000 and 3000 US gallons. Solution: L = 20 × 12 = 240 in.

r = 6⁄2 × 12 = 36 in.

From Formula (1): K = 0.00433 × 362 × 240 = 1346.80 From Formula (2): V T = 3.1416 × 1347 = 4231.7 US gal. The 72-inch mark from the bottom on the inch-stick can be graduated for the rounded full volume of 4230 and the halfway point 36 in. for 4230 ⁄ 2 or 2115. It can be seen that the 1000-gal mark would be below the halfway mark. From Formulas (3) and (4):

------------ = 0.7424 ; from the table starting on page 84, h can be interpolated as A 1000 = 1000 1347 0.5724; and x = y = 36 ×  0.5724  =  20.61. If the desired level of accuracy permits, interpolation can be omitted by choosing h directly from the table on page 84 for the value of A nearest that calculated above. Therefore, the 1000-gal mark is graduated 205 ⁄8 in. from bottom of rod. It can be seen that the 3000 mark would be above the halfway mark. Therefore, the circu­ lar segment considered is the cross section of the void space at the top of the tank. From Formulas (3) and (5):

– 3000- = 0.9131, h = 0.6648 , x = 36 × 0.6648 = ----------------------------A 3000 = 4230 23.93 in. 1347 Therefore, the 3000-gal mark is 72.00 - 23.93 = 48.07, or at the 481 ⁄16 -in. mark from the bottom.

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Machinery's Handbook, 31st Edition DIMENSIONS OF PLANE FIGURES

75

Dimensions of Plane Figures

The following pages contain diagrams of plane figures, along with formulas for finding their dimensions. Illustrations are labeled with the variables given in the formulas. Formulas for one dimension are derived from another by rearranging terms or substituting the formula for one dimension with another, where appropriate. Formulas that include decimals are derived either from the square root of 2 or 1 ⁄ 2 . As such, they are approximations of the dimension being solved for. Formulas that include π are approximated as well. The dimensions generally are as follows, with distinctions shown as needed for each figure: A = area; s = side; d = diagonal or diameter; h, H = height; a, b, c = sides or other segments as shown; r, R = radius; l = arc length or other. Square: d

A = s 2 = 1⁄2 d 2

s

s = 0.7071d =

A

d = 1.414s = 1.414 A

s

Example: Side s of a square is 15 in. Find the area of the square and the length of its diagonal. A = s 2 = 15 2 = 225 in2 d = 1.414s = 1.414 × 15 = 21.21 in

Example: The area of a square is 625 cm 2. Find the length of side s and diagonal d. s =

A =

625 = 25 cm

d = 1.414 A = 1.414 × 25 = 35.35 cm

Rectangle: 2

2

2

A = ab = a d – a = b d – b

a

d b

d =

a2

a =

d 2 – b 2 = A /b

+

b =

d 2 – a 2 = A /a

2

b2

Example: Side a of a rectangle is 12 cm, and the area is 70.5 cm 2. Find the length of side b and diagonal d. b = A /a = 70.5 /12 = 5.875 cm d =

a2 + b2 =

12 2 + 5.875 2 =

178.516 = 13.361 cm

Example: The sides of a rectangle are 30.5 and 11 cm. Find the area. A = a b = 30.5 × 11 = 335.5 cm2

Parallelogram: a b

A = ab a = A /b b = A /a

Note: The dimension a is the length of the vertical drawn at a right angle to side b. Dimension a is also considered the height of the parallelogram.

Example: Base b of a parallelogram is 16 ft. Height a is 5.5 ft. Find the area. A = ab = 5.5 × 16 = 88 ft2

Example: The area of a parallelogram is 12 in 2. The height is 1.5 in. Find the length of the base b. b = A /a = 12 /1.5 = 8 in.

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Machinery's Handbook, 31st Edition DIMENSIONS OF PLANE FIGURES

76

Right Triangle (one angle is a 90-degree angle): ab 2

From the Pythagorean theorem, a2 + b2 = c2, thus A = ------

c

a

c =

b

a2 + b2

a =

c2 – b2

b =

c2 – a2

Example: Side a is 6 in. and side b is 8 in. Find side c and area A:

c =

a 2 + b 2 = 6 2 + 8 2 = 36 + 64 = 100 = 10 in. ab 6×8 48 A = ------------ = ------------ = ------ = 24 in2 2 2 2 Example: Side c = 10 and side a = 6. Find side b: b in. = c 2 – a 2 = 10 2 – 6 2 = 100 – 36 = 64 = 8 in.

Acute Triangle (all three angles measure less than 90 degrees): c

A=

a

h

bh 2

, h=

b

A=

2

a −

(

2

2

a +b −c

2

2b

)

2

, so A =

b 2

S ( S − a )( S − b )( S − c ) , where S =

2

a −

(

2

2

a +b −c

2

2b

)

2

a +b+ c 2

Example: Side b = 7 inches, h = 4 inches, so A = bh/2 = (7 in. × 4 in.)/2 = 28 in 2/2 = 14 in 2 Example: Side a = 10 cm, b = 9 cm, and c = 8 cm 2. Find the area.

b a2 + b2 – c2 2 9 10 2 + 9 2 – 8 2 2 117 2 A = --- a 2 –  ---------------------------- = --- 10 2 –  -------------------------------- = 4.5 100 –  ---------      18  2 2b 2 2×9 = 4.5 100 – 42.25 = 4.5 57.75 = 4.5 × 7.60 = 34.20 cm2

Obtuse Triangle (one angle measures greater than 90 degrees): A=

c

a

h

bh 2

, h=

A=

b

2

a −

(

2

2

a +b −c 2b

2

)

2

, so A =

b 2

S ( S − a )( S − b )( S − c ) , where S =

2

a −

(

2

2

a +b −c 2b

2

)

2

a +b+ c 2

Example: If b = 5 cm and h = 3 cm, then A = bh/2 = (5 cm × 3 cm)/2 = 15 cm2/2 = 7.5 cm2 Example: Side a = 5 in., side b = 4 in., and side c = 8 in. Find the area.

S = ( a + b + c )/2 = ( 5 + 4 + 8 )/2 = 17/2 = 8.5

A =

=

S(S – a)(S – b)(S – c) =

8.5 × 3.5 × 4.5 × 0.5 =

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8.5 ( 8.5 – 5 ) ( 8.5 – 4 ) ( 8.5 – 8 )

66.937 = 8.18 in2

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Machinery's Handbook, 31st Edition DIMENSIONS OF PLANE FIGURES

77

Trapezoid: ( a + b )h Area = A = -------------------2

a

Note: In Britain, this figure is called a trapezium and the figure below it is known as a trapezoid, which is the reverse of the US terms. Example: Side a = 23 meters, side b = 32 meters, and height h = 12 meters. Find the area.

h b

( a + b )h ( 23 + 32 ) × 12 55 × 12 A = -------------------- = --------------------------------- = ------------------ = 330 m2 2 2 2

Trapezium: ( H + h )a + bh + cH Area = A = -----------------------------------------------2

H

h b

a

c

The area of a trapezium also can be found by dividing it into two triangles, as indicated by the dashed line. Each area is added to give the total area of the trapezium.

Example: Let a = 10 in., b = 2, c = 3 in., h = 8 in., and H = 12 in. Find the area. ( H + h )a + bh + cH A = -----------------------------------------------2 (20 × 10) + 16 + 36 = --------------------------------------------- = 2

Regular Hexagon: R r

60° 120°

s

( 12 + 8 ) × 10 + ( 2 × 8 ) + ( 3 × 12 ) = ------------------------- -------------------------------------------------------2 252 --------- = 126 in2 2

A = 2.598s2 = 2.598R2 = 3.464r 2 R = s = radius of circumscribed circle = 1.155r r = radius of inscribed circle = 0.866s = 0.866R s = R = 1.155r Example: The side s of a regular hexagon is 40 millimeters. Find the area and the radius r of the inscribed (drawn inside) circle.

A = 2.598s 2 = 2.598 × 40 2 = 2.598 × 1600 = 4156.8 mm2 r = 0.866s = 0.866 × 40 = 34.64 mm

Example: What is the length of the side of a hexagon circumscribed on (drawn around) a circle of 50 millimeters radius? In this case, because the hexagon is circumscribed on the circle, the circle is inscribed (drawn within) the hexagon. Hence, r = 50 mm and s = 1.155r = 1.155 × 50 = 57.75 mm

Regular Octagon:

R r

45° 135°

s

A = area = 4.828s2 = 2.828R 2 = 3.314r 2 R = radius of circumscribed circle = 1.307s = 1.082r r = radius of inscribed circle = 1.207s = 0.924R s = 0.765R = 0.828r Example: Find the area and the length of the side of an octagon inscribed (drawn inside) in a circle of 12 inches diameter. Diameter of circumscribed (drawn around) circle = 12 inches; hence, R = 6 in. A = 2.828R 2 = 2.828 × 6 2 = 2.828 × 36 = 101.81 in2 s = 0.765R = 0.765 × 6 = 4.590 in.

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Machinery's Handbook, 31st Edition DIMENSIONS OF PLANE FIGURES

78 Regular Polygon:

A = area α = 360° ÷ n

α

R

s

r

β

n = number of sides β = 180° – α

ns s2 nsr A = -------- = ----- R 2 – ---2 4 2 R =

s2 r 2 + ---4

r =

s2 R 2 – ---4

s = 2 R2 – r2

Example: Find the area of a polygon having 12 sides, inscribed in a circle with radius of 8 centimeters. The length of the side s is 4.141 centimeters.

ns 12 × 4.141 s2 4.141 2 A = ----- R 2 – ---- = ------------------------- 8 2 – ---------------- = 24.846 59.713 2 2 4 4 = 24.846 × 7.727 = 191.98 cm2

Circle: Area = A = πr 2 = 3.1416r 2 = 0.7854d 2 Circumference = C = 2πr = 6.2832r = 3.1416d

d

r

r = C ÷ 6.2832 =

A ÷ 3.1416 = 0.564 A

d = C ÷ 3.1416 =

A ÷ 0.7854 = 1.128 A

Length of arc for center angle of 1° = 0.008727d Length of arc for center angle of n° = 0.008727nd

Example: Find area A and circumference C of a circle with a diameter of 23 ⁄4 inches.

A = 0.7854d 2 = 0.7854 × 2.75 2 = 0.7854 × 2.75 × 2.75 = 5.9396 in2 C = 3.1416d = 3.1416 × 2.75 = 8.6394 in Example: The area of a circle is 16.8 in 2. Find its diameter. d = 1.128 A = 1.128 16.8 = 1.128 × 4.099 = 4.624 in.

Sector of a Circle: l α

r

3.1416 r α 2A Length of arc = l = ----------------------------------- = 0.01745rα = ------180 r

Area = A = 1⁄2 rl = 0.008727αr 2 57.296 l 2A 57.296 l Central angle, in degrees = α = -------------------, r = ------- = -------------------r l α

Example: The radius of a circle is 35 millimeters, and angle a of a sector of the circle is 60 degrees. Find the area of the sector and the length of arc l. A = 0.008727αr 2 = 0.008727 × 60 × 35 2 = 641.41mm 2 = 6.41cm 2

l = 0.01745rα = 0.01745 × 35 × 60 = 36.645 mm

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Machinery's Handbook, 31st Edition DIMENSIONS OF PLANE FIGURES

79

Segment of a Circle: l

A = area

l = length of arc

c = 2 h ( 2r – h )

h

c 2 + 4h 2 r = -------------------8h

c α

r

α = angle, in degrees

A = 1⁄2 [ rl – c ( r – h ) ] l = 0.01745rα

h = r – 1⁄2 4r 2 – c 2 = r [ 1 – cos ( α ⁄ 2 ) ]

57.296 l α = -------------------r

See also Segments of a Circle starting on page 83. Example: The radius r is 60 inches and the height h is 8 inches. Find the length of the chord c. c = 2 h ( 2r – h ) = 2 8 × ( 2 × 60 – 8 ) = 2 896 = 2 × 29.93 = 59.86 in.

Example: If c = 16, and h = 6 inches, what is the radius of the circle of which the segment is a part? c 2 + 4h 2 16 2 + 4 × 6 2 256 + 144 400 r = -------------------- = ----------------------------- = ------------------------ = --------- = 8 1⁄3 in. 8h 8×6 48 48

Cycloid: l

d r

Area = A = 3πr 2 = 9.4248r 2 = 2.3562d 2 = 3 × area of generating circle Length of cycloid = l = 8r = 4d

See also Areas Enclosed by Cycloidal Curves on page 73. Example: The diameter of the generating circle of a cycloid is 6 inches. Find the length l of the cycloi­dal curve and the area enclosed between the curve and the base line. l = 4d = 4 × 6 = 24 in.

A = 2.3562d 2 = 2.3562 × 6 2 = 84.82 in.2

Circular Ring (Annulus): Area = A = π ( R 2 – r 2 ) = 3.1416 ( R 2 – r 2 ) = 3.1416 ( R + r ) ( R – r )

d r D

R

= 0.7854 ( D 2 – d 2 ) = 0.7854 ( D + d ) ( D – d )

Example: Let the outside diameter D = 12 centimeters and the inside diameter d = 8 centimeters. Find the area of the ring. A = 0.7854 ( D 2 – d 2 ) = 0.7854 ( 12 2 – 8 2 ) = 0.7854 ( 144 – 64 ) = 0.7854 × 80 = 62.83 cm2

By the alternative formula:

A = 0.7854 ( D + d ) ( D – d ) = 0.7854 ( 12 + 8 ) ( 12 – 8 ) = 0.7854 × 20 × 4 = 62.83 cm2

Sector of Circular Ring:

 d

R

r D

α = central angle, in degrees A = area, απ A = --------- ( R 2 – r 2 ) = 0.00873α ( R 2 – r 2 ) 360 απ = ------------------ ( D 2 – d 2 ) = 0.00218α ( D 2 – d 2 ) 4 × 360

Example: Find the area, if the outside radius R = 5 inches, the inside radius r = 2 inches, and a = 72 degrees. A = 0.00873α ( R 2 – r 2 ) = 0.00873 × 72 ( 5 2 – 2 2 ) = 0.6286 ( 25 – 4 ) = 0.6286 × 21 = 13.2 in.2

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Machinery's Handbook, 31st Edition DIMENSIONS OF PLANE FIGURES

80 Spandrel or Fillet:

r

The shaded region is the spandrel (fillet). πr 2 Area = A = r 2 – -------- = 0.215r 2 = 0.1075c 2 4

c

Example: Find the area of a spandrel, the radius of which is 0.7 inch.

A = 0.215r 2 = 0.215 × 0.7 2 = 0.105 in2 Example: If chord c were given as 2.2 inches, what would be the area?

A = 0.1075c 2 = 0.1075 × 2.2 2 = 0.520 in2

Parabola: Area = A = 2⁄3 xy

The area of the shaded portion is equal to two-thirds of a rectangle which has x for its base and y for its height. Example: Let x in the illustration be 15 centimeters, and y be 9 cen­timeters. Find the area of the shaded portion of the parabola.

A = 2⁄3 xy = 2⁄3 × 15 × 9 = 10 × 9 = 90 cm2

Parabola: l

p l = length of arc = --2

y

When x is small in proportion to y, the following is a close approximation:

x

p 2

2x  2x 2x ------ 1 + 2x ------ + ln  ------ + 1 + ------  p p p p

2 x 2 2 x 4 l = y 1 + ---  -- – ---  -- 5  y 3  y

or

l =

4 y 2 + --- x 2 3

Example: If x = 2 feet and y = 24 feet, what is the approximate length l of the parabolic curve?

2 x 2 2 x 4 2 2 2 2 2 4 l = y 1 + ---  -- – ---  -- = 24 1 + ---  ------ – ---  ------ 3  y 5  y 3  24 5  24 = 24 × 1.0046 = 24.04 ft

Segment of Parabola: F E

B

D G

C

Area BFC = A = 2⁄3

× Area of parallelogram BCDE

If FG is the height of the segment, measured at right angles to BC, then: Area of segment BFC = 2⁄3 (BC)(FG)

Example: Suppose the length of the chord BC = 19.5 inches, and the distance between lines BC and DE, mea­sured at right angles to BC, is 2.25 inches. Find the area of the segment.

Area = A = 2⁄3 (BC)(FG) = 2⁄3 × 19.5 × 2.25 = 29.25 in2

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Machinery's Handbook, 31st Edition POLYGONS

81

Hyperbola:

b B a

te pto

ym

As

C

x

D y

xy ab x y Area BCD = A = ----- – ------ ln  --- + --- 2  a b 2 Example: The half-axes a and b are 3 and 2 inches, respectively. Find the area shown shaded in the illustration for x = 8 inches and y = 5 inches. Inserting the known values in the formula: 8×5 3×2 8 5 Area = A = ------------ – ------------ × ln  --- + --- = 20 – 3 × ln (5.167)  3 2 2 2

= 20 – 3 × 1.6423 = 20 – 4.927 = 15.073 in2

Ellipse: Area = A = πab = 3.1416ab An approximate formula for the perimeter is

Perimeter = P = 3.1416 2 ( a 2 + b 2 )

a

b

( a – b )2 A closer approximation is P = 3.1416 2 ( a 2 + b 2 ) – ------------------2.2

Example: The larger, or major, axis is 200 millimeters. The smaller, or minor, axis is 150 millimeters. Find the area and the approximate circumference. Here, then, a = 100, and b = 75. A = 3.1416ab = 3.1416 × 100 × 75 = 23,562 mm 2 = 235.62 cm 2

P = 3.1416 2 ( a 2 + b 2 ) = 3.1416 2 ( 100 2 + 75 2 ) = 3.1416 2 × 15,625 = 3.1416 31,250 = 3.1416 × 176.78 = 555.37 mm = 55.537 cm

Polygons.—A polygon is a many-sided figure in a two-dimensional plane. A polygon is sometimes referred to as an n-gon, where n is the number of sides. Triangles are polygons with the least number of sides (n = 3), followed by quadrilaterals (n = 4), pentagons (n = 5), hexagons (n = 6), heptagons (n = 7), octagons (n = 8), and so on. A regular polygon has congruent sides (all sides are of equal measure) and, hence, its interior angles are congruent. In Fig. 4a, β is the measure of each interior angle, 180 – β is the exterior angle measure at each vertex. α is a measure of the central angle. Irregular polygons (Fig. 4b) are polygons whose sides are not all congruent. Both Fig. 4a and Fig. 4b show convex polygons, in which all exterior angles measure less than 180 degrees. A concave polygon has some interior angles that measure greater than 180 degrees (see Fig. 4c). All the formulas in this section concern convex regular polygons.

Fig. 4. Polygons: (a) Convex Regular; (b) Convex Irregular; (c) Concave Irregular.

Fig. 5 shows how a regular polygon is either inscribed (drawn inside) or circumscribed (drawn around) a circle. A polygon inscribed within a circle, as shown in Fig. 5a, is drawn so that all its vertices touch the circle. Its radius is marked r. A polygon that circumscribes a circle, as shown in Fig. 5b, is drawn so that the circle touches each of the sides of the polygon. Its radius is marked R.

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Machinery's Handbook, 31st Edition POLYGONS

82

Fig. 5. Hexagons: (a) Inscribed (Drawn Inside) a Circle; (b) Circumscribed (Drawn Around) a Circle.

Formulas and Table for Regular Polygons: The following formulas and table can be used to calculate area, side length, and radii of inscribed and circumscribed polygons on a circle.

2

A= R=

nS cot α 4 S 2 sin α

2

nS cot α 2 2 sin2αtan cosα ,α = nR tan α , A==nR 2 sin α cos nR= nR =α 4

rA A r S , = , =R = = = cosααcos α N sin α cos α cos 2αsin α N sin

cotαα S cot α AScot = r = R cos α = r = R cos= α = 2 n2

A cot α n

A tan α R sin S = 2 R sin α =S 2=r 2tan α =α2= 2 r tan α = 2 N

Area, Length of Side, and Inscribed and Circumscribed Radii of Regular Polygons No. of Sides, n

A---S2

A----R2

A ---r2

R --S

--Rr

--SR

S--r

--rR

--rS

3 4 5 6 7 8 9 10 12 16 20 24 32 48 64

0.4330 1.0000 1.7205 2.5981 3.6339 4.8284 6.1818 7.6942 11.196 20.109 31.569 45.575 81.225 183.08 325.69

1.2990 2.0000 2.3776 2.5981 2.7364 2.8284 2.8925 2.9389 3.0000 3.0615 3.0902 3.1058 3.1214 3.1326 3.1365

5.1962 4.0000 3.6327 3.4641 3.3710 3.3137 3.2757 3.2492 3.2154 3.1826 3.1677 3.1597 3.1517 3.1461 3.1441

0.5774 0.7071 0.8507 1.0000 1.1524 1.3066 1.4619 1.6180 1.9319 2.5629 3.1962 3.8306 5.1011 7.6449 10.190

2.0000 1.4142 1.2361 1.1547 1.1099 1.0824 1.0642 1.0515 1.0353 1.0196 1.0125 1.0086 1.0048 1.0021 1.0012

1.7321 1.4142 1.1756 1.0000 0.8678 0.7654 0.6840 0.6180 0.5176 0.3902 0.3129 0.2611 0.1960 0.1308 0.0981

3.4641 2.0000 1.4531 1.1547 0.9631 0.8284 0.7279 0.6498 0.5359 0.3978 0.3168 0.2633 0.1970 0.1311 0.0983

0.5000 0.7071 0.8090 0.8660 0.9010 0.9239 0.9397 0.9511 0.9659 0.9808 0.9877 0.9914 0.9952 0.9979 0.9988

0.2887 0.5000 0.6882 0.8660 1.0383 1.2071 1.3737 1.5388 1.8660 2.5137 3.1569 3.7979 5.0766 7.6285 10.178

Example 1: A regular hexagon is inscribed in a circle of 6 in. diameter. Find the area and the radius of an inscribed circle. Here, R = 3 in. and n = 6. From the table, area A = 2.5981R2 = 2.5981 × 9 = 23.3829 in2. Radius of inscribed circle, r = 0.866R = 0.866 × 3 = 2.598 in. Example 2: An octagon is inscribed in a circle of 100 mm diameter. Thus, R = 50 mm and n = 8. Find the area and radius of an inscribed circle. A = 2.8284R2 = 2.8284 × 2500 = 7071 mm2 = 70.7 cm2. Radius of inscribed circle, r = 0.9239R = 09239 × 50 = 46.195 mm. Example 3: Thirty-two bolts are to be equally spaced on the periphery of a 16-in. diameter bolt-circle. Find the chordal distance between the bolts. Chordal distance equals the side length S of a polygon with n = 32 sides and R = 8. Hence, S = 0.196R = 0.196 × 8 = 1.568 in. Example 4: Sixteen bolts are to be equally spaced on the periphery of a bolt-circle, 250 mm diameter. Find the chordal distance between the bolts. Chordal distance equals the side length S of a polygon with 16 sides. R = 125. Thus, S = 0.3902R = 0.3902 × 125 = 48.775 mm.

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A tan α N

Machinery's Handbook, 31st Edition SEGMENTS OF CIRCLES

83

Segments of a Circle.—The table that follows gives the principal formulas for dimensions of circle segments. The dimensions are illustrated in the figures on pages 79 and 84. When two of the dimensions found together in the first column are known, the other dimensions are found by using the formulas in the corresponding row. For example, if radius r and chord c are known, solve for angle a using Equation (13), then use Equations (14) and (15) to solve for h and l, respectively. In these formulas, the value of a is in degrees between 0 and 180°. Formulas for Segments of a Circle

Given

Formulas

a, r

--- c = 2r sin  α  2

(1)

α h = r 1 – cos  ---  (2)   2 

---------l = πrα 180

(3)

a, c

c -----r = -------------α 2 sin  ---  2

(4)

α c h = – --- tan --- 2  4

(5)

πcα -----l = -------------------α 360 sin  ---  2

(6)

a, h

h ----r = --------------------α 1 – cos ---  2

(7)

2h-----c = ----------tan α ---  4

(8)

πhα -----l = ----------------------------------- 180 1 – cos  --α-   2 

(9)

a, l

180 l r = --------- --π α

(10)

α 360l sin --- 2 c = ---------------------------πα

2  c  α = cos –1  1 – -------2  2r 

(13)

4r – c h = r – ----------------------2

r, h

h α = 2cos –1  1 – ---  r

(16)

c = 2 h ( 2r – h ) (17)

π l = ------ rcos – 1  1 – h--- (18)  90 r

r, l

180 l α = --------- π r

(19)

c = 2r sin  90l --------   πR 

(20)

h = r 1 – cos  90l -------- (21)  πr  

c, h

2h α = 4tan – 1 ------ (22) c

(23)

2h c 2 + 4h 2 l = π  -------------------- tan –1  ------  c  360h 

r, c

2

2

2

2

c + 4h r = ------------------8h

Given

Formula To Find

c, l

360 α------------- -l- = ----------π c sin α --- 2

Given

(25)

Solve Equation (25) for a by iterationa, then r = Equation (10) h = Equation (5)

h, l

(11)

180l 1 – cos --α-   2  (12) h = -------------------------------------------πα

(14)

π c l = ------ rsin – 1  -----  2r 90

(15)

(24)

Formula To Find

180 α ------------- --l- = --------------------π h 1 – cos --α- 2

(26)

Solve Equation (26) for a by iterationa, then r = Equation (10) c = Equation (11)

a Equations (25) and (26) cannot be easily solved by ordinary means. To solve these equations, test various values of α until the left side of the equation equals the right side. For example, if given c = 4 and l = 5, the left side of Equation (25) equals 143.24, and testing various values of α reveals the right side equals 143.24 when α = 129.62°. Angle a is in degrees, 0 < a < 180 Formulas for Circular Segments contributed by Manfred Brueckner.

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Machinery's Handbook, 31st Edition SEGMENTS OF CIRCLES

84

Segments of Circles for Radius = 1.—Formulas for segments of circles are given on pages 79 and 83. When central angle a and radius r are known, the following table can be used to find the length of arc l, height of segment h (x in the discussion on page 79), chord length c, and segment area A. Column A/π is the ratio of segment area A to the area of a circle with radius r = 1, in percent. When angle a and radius r are not known, but segment l height h and chord length c are known, ratio h/c can be used to find a, l, and A by linear interpolation. Radius r is h found by the formula on page 79 or 83. The value of c l is then multiplied by the radius r and the area A by r 2. Angle a can be found thus with an accuracy of about  0.001 degree; arc length l with an error of about 0.02 perr cent; and area A with an error ranging from about 0.02 percent for the highest entry value of h/c to about 1 percent for values of h/c of about 0.050. For lower values of h/c, and where greater accuracy is required, area A should be found by the formula on page 79. Example: A 3-foot diameter cylindrical tank, mounted horizontally, contains fuel. What is the fuel depth, in inches, when the tank is 20% full? Solution: Locate 20% in table column A/π%. The depth equals h multiplied by the radius: hr = 0.50758 × 1.5 × 12 = 9.14 inches Segments of Circles for Radius = 1 (US Customary or Metric Units)

a, Deg. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40

l

0.01745 0.03491 0.05236 0.06981 0.08727 0.10472 0.12217 0.13963 0.15708 0.17453 0.19199 0.20944 0.22689 0.24435 0.26180 0.27925 0.29671 0.31416 0.33161 0.34907 0.36652 0.38397 0.40143 0.41888 0.43633 0.45379 0.47124 0.48869 0.50615 0.52360 0.54105 0.55851 0.57596 0.59341 0.61087 0.62832 0.64577 0.66323 0.68068 0.69813

h

0.00004 0.00015 0.00034 0.00061 0.00095 0.00137 0.00187 0.00244 0.00308 0.00381 0.00460 0.00548 0.00643 0.00745 0.00856 0.00973 0.01098 0.01231 0.01371 0.01519 0.01675 0.01837 0.02008 0.02185 0.02370 0.02563 0.02763 0.02970 0.03185 0.03407 0.03637 0.03874 0.04118 0.04370 0.04628 0.04894 0.05168 0.05448 0.05736 0.06031

c

0.01745 0.03490 0.05235 0.06980 0.08724 0.10467 0.12210 0.13951 0.15692 0.17431 0.19169 0.20906 0.22641 0.24374 0.26105 0.27835 0.29562 0.31287 0.33010 0.34730 0.36447 0.38162 0.39874 0.41582 0.43288 0.44990 0.46689 0.48384 0.50076 0.51764 0.53448 0.55127 0.56803 0.58474 0.60141 0.61803 0.63461 0.65114 0.66761 0.68404

Area A

0.0000 0.0000 0.0000 0.0000 0.0001 0.0001 0.0002 0.0002 0.0003 0.0004 0.0006 0.0008 0.0010 0.0012 0.0015 0.0018 0.0022 0.0026 0.0030 0.0035 0.0041 0.0047 0.0053 0.0061 0.0069 0.0077 0.0086 0.0096 0.0107 0.0118 0.0130 0.0143 0.0157 0.0171 0.0186 0.0203 0.0220 0.0238 0.0257 0.0277

A⁄ π

%

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.2 0.2 0.2 0.2 0.3 0.3 0.3 0.4 0.4 0.5 0.5 0.5 0.6 0.6 0.7 0.8 0.8 0.9

h/c

0.00218 0.00436 0.00655 0.00873 0.01091 0.01309 0.01528 0.01746 0.01965 0.02183 0.02402 0.02620 0.02839 0.03058 0.03277 0.03496 0.03716 0.03935 0.04155 0.04374 0.04594 0.04814 0.05035 0.05255 0.05476 0.05697 0.05918 0.06139 0.06361 0.06583 0.06805 0.07027 0.07250 0.07473 0.07696 0.07919 0.08143 0.08367 0.08592 0.08816

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a, Deg. 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80

l

0.71558 0.73304 0.75049 0.76794 0.78540 0.80285 0.82030 0.83776 0.85521 0.87266 0.89012 0.90757 0.92502 0.94248 0.95993 0.97738 0.99484 1.01229 1.02974 1.04720 1.06465 1.08210 1.09956 1.11701 1.13446 1.15192 1.16937 1.18682 1.20428 1.22173 1.23918 1.25664 1.27409 1.29154 1.30900 1.32645 1.34390 1.36136 1.37881 1.39626

h

0.06333 0.06642 0.06958 0.07282 0.07612 0.07950 0.08294 0.08645 0.09004 0.09369 0.09741 0.10121 0.10507 0.10899 0.11299 0.11705 0.12118 0.12538 0.12964 0.13397 0.13837 0.14283 0.14736 0.15195 0.15661 0.16133 0.16611 0.17096 0.17587 0.18085 0.18588 0.19098 0.19614 0.20136 0.20665 0.21199 0.21739 0.22285 0.22838 0.23396

c

0.70041 0.71674 0.73300 0.74921 0.76537 0.78146 0.79750 0.81347 0.82939 0.84524 0.86102 0.87674 0.89240 0.90798 0.92350 0.93894 0.95432 0.96962 0.98485 1.00000 1.01508 1.03008 1.04500 1.05984 1.07460 1.08928 1.10387 1.11839 1.13281 1.14715 1.16141 1.17557 1.18965 1.20363 1.21752 1.23132 1.24503 1.25864 1.27216 1.28558

Area A

0.0298 0.0320 0.0342 0.0366 0.0391 0.0418 0.0445 0.0473 0.0503 0.0533 0.0565 0.0598 0.0632 0.0667 0.0704 0.0742 0.0781 0.0821 0.0863 0.0906 0.0950 0.0996 0.1043 0.1091 0.1141 0.1192 0.1244 0.1298 0.1353 0.1410 0.1468 0.1528 0.1589 0.1651 0.1715 0.1781 0.1848 0.1916 0.1986 0.2057

A⁄ π

%

0.9 1.0 1.1 1.2 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.4 2.5 2.6 2.7 2.9 3.0 3.2 3.3 3.5 3.6 3.8 4.0 4.1 4.3 4.5 4.7 4.9 5.1 5.3 5.5 5.7 5.9 6.1 6.3 6.5

h/c

0.09041 0.09267 0.09493 0.09719 0.09946 0.10173 0.10400 0.10628 0.10856 0.11085 0.11314 0.11543 0.11773 0.12004 0.12235 0.12466 0.12698 0.12931 0.13164 0.13397 0.13632 0.13866 0.14101 0.14337 0.14574 0.14811 0.15048 0.15287 0.15525 0.15765 0.16005 0.16246 0.16488 0.16730 0.16973 0.17216 0.17461 0.17706 0.17952 0.18199

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Machinery's Handbook, 31st Edition SEGMENTS OF CIRCLES

85

Segments of Circles for Radius = Customary 1 (US Customary or Metric Segments of Circles for Radius = 1 (US or Metric Units)Units) (Continued) a, Deg.

l

h

c

Area A

A⁄ π

%

h/c

a, Deg.

0.18694

132

81

1.41372

0.23959

1.29890 0.2130

6.8

0.18446

83

1.44862

0.25104

1.32524 0.2280

7.3

0.18943

85

1.48353

82

84

86

87 88

89

90

91

92

93 94

95

96

97

98

99

100 101

102

103

104

105

106

1.43117

1.46608

1.50098

1.51844 1.53589

1.55334

1.57080

1.58825

1.60570

1.62316

0.24529

0.25686

0.26272

0.28066

1.38932 0.2682

0.27463 0.28675

0.29289

0.29909

0.30534

1.74533 1.76278

1.78024

1.79769

1.81514

1.83260

1.85005

1.40182 0.2767

1.41421 0.2854

1.42650 0.2942

1.43868 0.3032

7.5

7.8

8.0

8.3 8.5

8.8

9.1

0.19193

0.19444

0.19696

0.19948 0.20201

0.20456 0.20711

9.4

0.20967

9.7

0.21224

c

Area A

A⁄ π

%

1.81992 0.7658 24.4

0.32161

133

2.32129

0.60125

1.83412 0.7950 25.3

0.32781

134

135

136

137

2.30383

2.33874

2.35619

2.37365 2.39110

138

2.40855

140

2.44346

139

141

142

2.42601

2.46091

2.47837

0.59326

0.60927

0.61732

0.62539 0.63350

0.64163

0.64979

0.65798

0.66619

0.67443

1.82709 0.7803 24.8

1.84101 0.8097 25.8

1.84776 0.8245 26.2

1.85437 0.8395 26.7 1.86084 0.8546 27.2

1.86716 0.8697 27.7

1.87334 0.8850 28.2

1.87939 0.9003 28.7

1.88528 0.9158 29.2

1.89104 0.9314 29.6

145

2.53073

0.69929

1.90743 0.9786 31.1

0.36662

0.71598

1.91764 1.0105 32.2

0.33738

0.34394

0.35055

0.35721 0.36392

0.37068

0.37749

0.38434

0.39124

0.39818

1.48629 0.3405 10.8 1.49791 0.3502

11.1

1.52081 0.3701

11.8

1.50942 0.3601

11.5

1.53209 0.3803 12.1 1.54325 0.3906 12.4

1.55429 0.4010 12.8 1.56522 0.4117

13.1

0.22261

0.22523

0.22786

0.23050

0.23315 0.23582

0.23849 0.24117

1.57602 0.4224 13.4

0.24387

1.59727 0.4444 14.1

0.24929

1.58671 0.4333 13.8

0.24657

144

146

147

148

149

150 151

152

153

154

155

156

2.51327

2.54818

2.56563

2.58309

2.60054

2.61799

0.69098

0.70763

0.72436

0.73276 0.74118

2.63545

0.74962

2.67035

0.76655

2.65290

2.68781

2.70526

2.72271

0.75808

0.77505

0.78356

0.79209

1.90211 0.9627 30.6

1.91261 0.9945 31.7

1.92252 1.0266 32.7

1.92726 1.0428 33.2 1.93185 1.0590 33.7

1.93630 1.0753 34.2

1.94059 1.0917 34.7

1.94474 1.1082 35.3

1.94874 1.1247 35.8

1.95259 1.1413 36.3

159

2.77507

0.81776

1.96651 1.2084 38.5

115

116

117

118

119

120

121

1.98968

2.00713

2.02458

2.04204

2.05949

2.07694

2.09440

0.46270

0.47008

0.47750

0.48496

0.49246

0.50000

1.67734 0.5381 17.1

1.68678 0.5504 17.5

1.69610 0.5629 17.9

1.70528 0.5755 18.3

1.71433 0.5883 18.7

1.72326 0.6012 19.1

1.73205 0.6142 19.6

0.26866

0.27148

0.27431

0.27715

0.28001

0.28289

0.28577

0.28868

162

163

164

165

166

167

168

169

170

2.80998

2.82743

2.84489

2.86234

2.87979

2.89725

2.91470

2.93215

2.94961

2.96706

0.82635

0.83495

0.84357

0.85219

0.86083

0.86947

0.87813

0.88680

0.89547

0.90415

0.91284

0.39417

0.39772

0.40129

1.96962 1.2253 39.0

1.97257 1.2422 39.5

1.97538 1.2592 40.1

1.97803 1.2763 40.6

1.98054 1.2934 41.2

1.98289 1.3105 41.7

1.98509 1.3277 42.3

1.98714 1.3449 42.8

1.98904 1.3621 43.4

1.99079 1.3794 43.9

1.99239 1.3967 44.5

0.40852

0.41585

0.41955

0.42328

0.42704

0.43083

0.43464

0.43849

0.44236

0.44627

0.45020

0.45417

0.45817

171

2.98451

0.92154

1.99383 1.4140 45.0

0.46220

2.14675

0.52284

1.75763 0.6540 20.8

0.29747

173

3.01942

0.93895

1.99627 1.4488 46.1

0.47035

2.18166 2.19911

127

2.21657

129

2.25147

130

0.45536

1.66777 0.5259 16.7

0.26585

161

2.79253

0.39064

0.29159

2.16421

128

0.44806

1.65808 0.5138 16.4

0.26306

160

0.38714

1.74071 0.6273 20.0

124

126

0.44081

1.64825 0.5019 16.0

0.26028

0.80919

0.38366

0.50758

2.12930

125

0.43359

1.63830 0.4901 15.6

2.75762

0.38021

2.11185

122

123

0.42642

158

0.37678

0.41217

0.25752

0.25476

0.37337

1.96325 1.1915 37.9

1.62823 0.4784 15.2

1.61803 0.4669 14.9

0.36998

0.40489

0.41930

0.41221

0.36327

1.95630 1.1580 36.9

1.90241

114

0.35665

0.22001

0.21741

109

1.97222

0.35337

1.47455 0.3309 10.5

1.46271 0.3215 10.2

1.95985 1.1747 37.4

113

0.35010

0.32441

0.80063

1.95477

0.34686

0.35995

2.74017

112

0.34364

1.89665 0.9470 30.1

157

1.93732

0.34044

0.68270

0.25202

111

0.33725

2.49582

1.60771 0.4556 14.5

1.91986

0.33409

143

0.40518

110

0.33094

0.21482

1.86750

1.88496

0.32470

9.9

107

108

h/c

0.58531

1.45075 0.3123

0.33087

1.72788

1.37671 0.2599

7.0

h

2.28638

0.31165

1.67552

1.71042

1.35118 0.2437

1.36400 0.2517

0.31800

1.69297

1.33826 0.2358

0.26865

1.64061

1.65806

1.31212 0.2205

l

131

2.23402

2.26893

0.51519

0.53053

0.53825

0.54601

0.55380

0.56163

0.56949

0.57738

1.74924 0.6406 20.4

1.76590 0.6676 21.3

1.77402 0.6813 21.7

1.78201 0.6950 22.1

1.78987 0.7090 22.6

1.79759 0.7230 23.0

1.80517 0.7372 23.5

1.81262 0.7514 23.9

0.29452

0.30043

0.30341

0.30640

0.30941

0.31243

0.31548

0.31854

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172

174

175

176

177

178

179

180

3.00197

3.03687

3.05433

3.07178

3.08923

3.10669

3.12414

3.14159

0.93024

0.94766

0.95638

0.96510

0.97382

0.98255

0.99127

1.00000

1.99513 1.4314 45.6

1.99726 1.4662 46.7

1.99810 1.4836 47.2

1.99878 1.5010 47.8

1.99931 1.5184 48.3

1.99970 1.5359 48.9

1.99992 1.5533 49.4

2.00000 1.5708 50.0

0.46626

0.47448

0.47865

0.48284

0.48708

0.49135

0.49566

0.50000

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Machinery's Handbook, 31st Edition EQUAL AREA CIRCLES AND SQUARES

86

Diameters of Circles and Sides of Squares of Equal Area (US Customary or Metric Units)

D

Dia. of Circle, D 1

1⁄ 2

11 ⁄2 2

21 ⁄2 3

31 ⁄2 4

41 ⁄2 5

51 ⁄2 6

61 ⁄2 7

71 ⁄2 8

81 ⁄2 9

91 ⁄2

10

101 ⁄2 11

111 ⁄2 12

121 ⁄2 13

131 ⁄2 14

141 ⁄2 15

151 ⁄2 16

161 ⁄2 17

171 ⁄2 18

181 ⁄2 19

191 ⁄2 20

The table below will be found useful for determining the diameter of a circle of an area equal to that of a square, the side of which is known, or for determining the side of a square which has an area equal to that of a circle, the area or diameter of which is known. For example, if the diam­eter of a circle is 171 ⁄2 inches, it is found from the table by reading across from the first column that the side of a square of the same area is 15.51 inches. And both have area 240.53 in2.

S

Side of Square, S 0.44

Area of Circle or Square 0.196

Dia. of Circle, D 201 ⁄2

0.89

0.785

21

1.77

3.142

22

2.66

7.069

23

1.33 2.22 3.10

1.767 4.909 9.621

211 ⁄2 221 ⁄2 231 ⁄2

3.54

12.566

24

4.43

19.635

25

5.32

28.274

26

6.20

38.485

27

7.09

50.265

28

7.98

63.617

29

3.99 4.87 5.76 6.65 7.53 8.42

15.904 23.758 33.183 44.179 56.745 70.882

241 ⁄2 251 ⁄2 261 ⁄2 271 ⁄2 281 ⁄2 291 ⁄2

8.86

78.540

30

9.75

95.033

31

10.63

113.10

9.31

10.19 11.08

11.52

11.96

86.590 103.87 122.72

132.73

143.14

301 ⁄2 311 ⁄2 32

321 ⁄2 33

331 ⁄2

12.41

153.94

34

13.29

176.71

35

12.85 13.74

165.13 188.69

341 ⁄2 351 ⁄2

14.18

201.06

36

15.07

226.98

37

15.95

254.47

38

16.84

283.53

39

17.72

314.16

40

14.62 15.51 16.40 17.28

213.82 240.53 268.80 298.65

361 ⁄2 371 ⁄2 381 ⁄2 391 ⁄2

Side of Square, S 18.17

Area of Circle or Square 330.06

Dia. of Circle, D 401 ⁄2

18.61

346.36

41

19.50

380.13

42

20.38

415.48

43

19.05 19.94 20.83

363.05 397.61 433.74

411 ⁄2 421 ⁄2 431 ⁄2

21.27

452.39

44

22.16

490.87

45

23.04

530.93

46

23.93

572.56

47

24.81

615.75

48

25.70

660.52

49

21.71 22.60 23.49 24.37 25.26 26.14

471.44 510.71 551.55 593.96 637.94 683.49

441 ⁄2 451 ⁄2 461 ⁄2 471 ⁄2 481 ⁄2 491 ⁄2

26.59

706.86

50

27.47

754.77

51

28.36

804.25

52

29.25

855.30

53

30.13

907.92

54

31.02

962.11

55

27.03 27.92 28.80 29.69 30.57 31.46

730.62 779.31 829.58 881.41 934.82 989.80

501 ⁄2 511 ⁄2 521 ⁄2 531 ⁄2 541 ⁄2 551 ⁄2

31.90

1017.88

56

32.79

1075.21

57

33.68

1134.11

58

1194.59

59

1256.64

60

32.35 33.23 34.12

34.56

35.01

35.45

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1046.35 1104.47 1164.16

1225.42

561 ⁄2 571 ⁄2 581 ⁄2

591 ⁄2

Side of Square, S 35.89

Area of Circle or Square 1288.25

36.34

1320.25

37.22

1385.44

38.11

1452.20

38.99

1520.53

39.88

1590.43

40.77

1661.90

41.65

1734.94

42.54

1809.56

43.43

1885.74

44.31

1963.50

45.20

2042.82

46.08

2123.72

46.97

2206.18

47.86

2290.22

48.74

2375.83

49.63

2463.01

50.51

2551.76

51.40

2642.08

52.29

2733.97

53.17

2827.43

36.78 37.66 38.55 39.44 40.32 41.21 42.10 42.98 43.87 44.75 45.64 46.53 47.41 48.30 49.19 50.07 50.96 51.84 52.73

1352.65 1418.63 1486.17 1555.28 1625.97 1698.23 1772.05 1847.45 1924.42 2002.96 2083.07 2164.75 2248.01 2332.83 2419.22 2507.19 2596.72 2687.83 2780.51

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Machinery's Handbook, 31st Edition DIAGONALS OF SQUARES AND HEXAGONS

87

Diagonals of Squares and Hexagons.—The table below gives values of dimensions D and E described in the figures and equations that follow.

D

d

2 3 D = ---------- d = 1.154701 d 3

E

E = d 2 = 1.414214 d A value not given in the table can be obtained from the equations above, or by the simple addition of two or more values taken directly from the table. Further values can be obtained by shifting the decimal point. Example 1: Find D when d = 2 5 ⁄ 16 inches. From the table, for d = 2, D = 2.3094, and for d = 5 ⁄ 16 , D = 0.3608. Therefore, D = 2.3094 + 0.3608 = 2.6702 inches. Example 2: Find E when d = 20.25 millimeters. From the table, for d = 20, E = 28.2843; for d = 0.25, that is, 1 ⁄4, E = 0.3536; Thus, E = 28.2843 + 0.3536 = 28.6379 millimeters. d

1⁄ 32 1⁄ 16

3⁄ 32 0.1 1⁄ 8

Diagonals of Squares and Hexagons (US Customary or Metric Units)

D 0.0361 0.0722 0.1083 0.1155

1⁄ 4

0.2887

0.3 5⁄ 16

0.3464 0.3608

0.3248

11 ⁄ 32

0.3969

0.4 13 ⁄ 32

0.4619 0.4691

3⁄ 8

0.4330

7⁄ 16 15 ⁄ 32

0.5052 0.5413

0.5774 0.6134

9⁄ 16 19 ⁄ 32

0.6495

21 ⁄ 32

0.7578

0.7 23 ⁄ 32

0.8083 0.8299

0.6 5⁄ 8

0.6856

0.6928 0.7217

11 ⁄ 16

0.7939

3⁄ 4 25 ⁄ 32

0.8660

0.8

13 ⁄ 16 27 ⁄ 32 7⁄ 8

0.9021

0.9238 0.9382 0.9743 1.0104

D 1.0392

38.1051

46.6691

68

78.5197

40.4145

49.4975

70

80.8291

2.3094

0.4243 0.4419 0.4861 0.5303

36

41.5692

3.0

3.4641

4.2426

38

43.8786

0.7955 0.8397

0.8485 0.8839 0.9281 0.9723

0.9899 1.0165 1.0607 1.1049

1.1314 1.1490 1.1932 1.2374

24 25 26 27

28 29 30 31

84.2932

103.238

72

83.1385

101.823

59.3970

77

88.9120

108.894

18.4752 19.6299

23

73

52.3259

48.4974

16 17

22

53.7401

42.7239

42

16.1658

20 21

100.409

9.8995

14

19

81.9838

104.652 106.066

15.5564

18

98.9950

71

85.4479 86.6026

12.7017

15

97.5808

50.9117

74 75

11

0.6629

96.1666

55.1543 56.5686

11.3137 12.7279

11.5470

79.6744

45.0333 46.1880

9.2376 10.3923

10

69

39 40

8.0 9.0

8.0829

37

48.0833

E 94.7523

5.6569 7.0711

8.4853

12 13

0.7071 0.7513

4.6188 5.7735

2.8284

6.9282

0.5657 0.5745 0.6187

39.2598

1.4142

7.0

D 77.3650

35

34

1.1547

6.0

d 67

33

1.3258

1.0

4.0 5.0

E 45.2548

1.3700

2.0

0.3977

D 36.9504

1.2816

0.2210

0.3536

d 32

1.1186

1.0825

0.2828 0.3094

E 1.2728

1.0464

15 ⁄ 16 31 ⁄ 32

0.2652

0.2309 0.2526

d 0.9

29 ⁄ 32

0.1414

0.2165

9⁄ 32

0.5

0.1326 0.1768

0.1804

17 ⁄ 32

0.0884

0.1443

5⁄ 32 3⁄ 16

0.2 7⁄ 32

E 0.0442

14.1421

41 43 44 45

46

47.3427 49.6521 50.8068 51.9615 53.1162

57.9828

76

87.7573

60.8112 62.2254

78 79

90.0667 91.2214

65.0538

81

93.5308

63.6396

80

92.3761

107.480 110.309 111.723

113.137

114.551

13.8564 15.0111

16.9706 18.3848

47 48

54.2709 55.4256

66.4681 67.8823

82 83

94.6855 95.8402

115.966 117.380

17.3205

21.2132

50

57.7351

70.7107

85

98.1496

120.208

101.614

124.451

20.7846 21.9393

23.0940 24.2487 25.4034

19.7990

22.6274 24.0416 25.4559 26.8701

28.2843 29.6985

35.7957

55 56

61.1992 62.3539

63.5086 64.6633

72.1249 73.5391 74.9533 76.3676

77.7818 79.1960

86 87 88 89

90 91

96.9949

99.3043 100.459 102.768

103.923 105.078

118.794

121.622 123.037 125.865

127.279 128.693

65.8180

80.6102

92

106.232

130.108

59 60

68.1274 69.2821

83.4386 84.8528

94 95

108.542 109.697

132.936 134.350

36.7696

34.6410

54

58.8898 60.0445

84

57

30.0222

32.3316 33.4863

53

69.2965

33.9411 35.3554

32.5269

31.1769

51 52

56.5803

31.1127

26.5581

27.7128 28.8675

49

38.1838

39.5980 41.0122 42.4264 43.8406

Copyright 2020, Industrial Press, Inc.

58

61 62

63 64 65 66

66.9727

70.4368 71.5915

72.7462 73.9009 75.0556 76.2103

82.0244

86.2671 87.6813

89.0955 90.5097 91.9239 93.3381

93

96 97

98 99

100 …

107.387

110.851 112.006

113.161 114.315 115.470 …

131.522

135.765 137.179

138.593 140.007 141.421 …

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Machinery's Handbook, 31st Edition VOLUMES OF SOLIDS

88

Volumes of Solids

Cube:

Diagonal of cube face = d = s 2

d

Diagonal of cube = D =

D s s s

Volume = V = s 3 Side s =

3

3d 2 --------- = s 3 = 1.732s 2

V

Example: The side of a cube s measures 9.5 centimeters. Find its volume. Volume = V = s 3 = 9.5 3 = 9.5 × 9.5 × 9.5 = 857.375 cm3 Example: The volume of cube is 231 cubic centimeters. What is the length of the side? s = 3 V = 3 231 = 6.136 cm

Rectangular Prism: Volume = V = abc

b

V a = -----bc

c

a

V b = -----ac

V c = -----ab

Example: In a rectangular prism, a = 6 in., b = 5 in., c = 4 in. Find the volume. V = abc = 6 × 5 × 4 = 120 in3 Example: What should the height of a box be if it is to contain 25 cubic feet and if it is 4 feet long and 21 ⁄2 feet wide? Here, a = 4, c = 2.5, and V = 25. Then, V 25 25 b = height = ------ = ---------------- = ------ = 2.5 ft ac 4 × 2.5 10

General Right Prism: h = edge length A = area of end surface V = Ah The area A of the end surface is found by the formulas for areas of plane figures on the preceding pages.

A h

Example: A right prism having for its base a regular hexagon with a side s of 7.5 centimeters is 25 centime­ters high. Find the volume. Area of hexagon = A = 2.598s 2 = 2.598 × 56.25 = 146.14 cm 2

Volume of prism = Ah = 25 × 146.14 = 3653.5 cm 3

Right Pyramid:

Volume = V =

h A

( )

1 (Base area × h) 3

A square pyramid is pictured. In general, any right pyramid of height h, whose base is a regular, n-sided polygon of side length s has volume: nsh s2 nsrh V = ------------ = --------- R 2 – ---6 6 4

where r is the radius of the circle inscribed in the base, and R is the radius of the circle circumscribed on the base.

Example: A pyramid having a height of 5 feet has a base formed by a square, the sides of which are 3 feet. Find the volume. Area of base = 3 × 3 = 9 square feet, h = 5 feet Volume = V = (1⁄3)(Base area × h) = 1⁄3 × 9 × 5

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Machinery's Handbook, 31st Edition VOLUMES OF SOLIDS

89

Frustum of Pyramid: A1 h Volume = V = --- A 1 + A 2 + A 1A 2 3

(

h A2

)

Example: The pyramid in the previous example is cut off 4.5 feet from the base, and the upper part removed. The sides of the rectangle forming the top surface of the frustum are, then, 1 and 1.5 feet long, respectively. Find the volume of the frustum. Area of top = A 1 = 1 × 1.5 = 1.5 ft 2

Area of base = A 2 = 2 × 3 = 6 ft 2

4×5 V = ---------- ( 1.5 + 6 + 1.5 × 6 ) = 1.5 ( 7.5 + 9 ) = 1.5 × 10.5 = 15.75 ft 3 3

Wedge:

( 2a + c )bh Volume = V = --------------------------6

c

A wedge has five faces, two of them triangles and three trapezoids. Example: Let a = 4 inches, b = 3 inches, and c = 5 inches. The height h = 4.5 inches. Find the volume in in3 and cm3.

h

( 2a + c )bh ( 2 × 4 + 5 ) × 3 × 4.5 175.5 V = --------------------------- = ------------------------------------------------- = ------------- = 29.25 in 3 6 6 6 cm 3 = 479.32162 cm 3 V = 29.25 in 3 × 16.387064 --------in 3

b

a

Cylinder: d h r

Volume = V = 3.1416r 2 h = 0.7854d 2 h Surface area of open-ended cylinder = S = 6.2832rh = 3.1416dh

Total surface area A of closed cylinder:

A = 6.2832r ( r + h ) = 3.1416d ( 1⁄2 d + h )

Example: Diameter d of a cylinder is 2.5 inches. Height h is 20 inches. Find the volume and the area of the open-ended cylindrical surface S. V = 0.7854d 2 h = 0.7854 × 2.5 2 × 20 = 0.7854 × 6.25 × 20 = 98.17 in3 S = 3.1416dh = 3.1416 × 2.5 × 20 = 157.08 in2

Cylinder Portion: Volume = V = 1.5708r 2 ( h 1 + h 2 ) h2

h1

= 0.3927d 2 ( h 1 + h 2 )

Surface area = S = 3.1416r ( h 1 + h 2 )

r

= 1.5708d ( h 1 + h 2 )

d

Example: A cylinder 125 millimeters in diameter is cut off at an angle, as shown in the illustration. Dimension h1 = 100 mm, and h2 = 150 mm. Find the volume and the surface area S of the cylinder. V = 0.3927d 2 ( h 1 + h 2 ) = 0.3927 × 125 2 × ( 100 + 150) = 0.3927 × 15 ,625 × 250 = 1 ,533 ,984 mm 3 = 1534 cm 3 S = 1.5708d ( h 1 + h 2 ) = 1.5708 × 125 × 250 = 49 ,087.5 mm 2 = 490.9 cm 2

Copyright 2020, Industrial Press, Inc.

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Machinery's Handbook, 31st Edition VOLUMES OF SOLIDS

90

Segment of a Cylinder: a b r d

In the formulas below, segment area A and arc length l refer to the shaded segment of the circle. The formulas of these dimensions, found on page 75, assume the central angle measure is known. L Volume = V = ( 2⁄3 a 3 ± b × segment area) ----------r±b L Surface area = S = ( ad ± b × arc length area ) ----------r±b

Use + when shaded segment is greater than or equal to one-half the base circle area, use – when it is less than the base circle area.

d

Example: Find the volume of a cylinder segment of a 2-inch high cylinder with a diameter of 5 inches when it is cut so that line AC passes through the center of the base circle—that is, the base area is a half-circle. In this case, b = 0, and a = r; that is, a = 2.5 inches. Thus, ABC is a semicircle, and area ABC = 3.1416 × r 2 × 1 ⁄ 2 = 3.1416 × (2.5)2 × 1 ⁄ 2 = 9.82 in 2. 2 2 2 V =  --- × 2.5 3 + 0 × 9.82 ---------------- = --- × 15.625 × 0.8 = 8.33 in3 3  2.5 + 0 3

Hollow Cylinder: D r

R

t

Volume = V = = = =

3.1416h ( R 2 – r 2 ) = 0.7854h ( D 2 – d 2 ) 3.1416ht ( 2R – t ) = 3.1416ht ( D – t ) 3.1416ht ( 2r + t ) = 3.1416ht ( d + t ) 3.1416ht ( R + r ) = 1.5708ht ( D + d )

Example: A cylindrical shell, 28 cm high, is 36 cm in outside diameter, and 4 cm thick. Find its vol­ume.

h d

V = 3.1416ht ( D – t ) = 3.1416 × 28 × 4 ( 36 – 4 ) = 3.1416 × 28 × 4 × 32 = 11 ,259.5 cm3

Cone:

2h ------------------------- = 1.0472r 2 h = 0.2618d 2 h Volume = V = 3.1416r 3

Conical surface area = A = 3.1416r r 2 + h 2 = 3.1416rs = 1.5708ds s =

r2 + h2 =

d2 ----- + h 2 4

Example: Find the volume and surface area of a cone, the base of which is a circle of 6 inches diameter and the height of which is 4 inches. V = 0.2618d 2 h = 0.2618 × 6 2 × 4 = 0.2618 × 36 × 4 = 37.7 in3 A = 3.1416r r 2 + h 2 = 3.1416 × 3 × 3 2 + 4 2 = 9.4248 × 25 = 47.124 in2

Frustum of Cone:

V = volume

A = area of conical surface

V = 1.0472h ( R 2 + Rr + r 2 ) = 0.2618h ( D 2 + Dd + d 2 ) A = 3.1416s ( R + r ) = 1.5708s ( D + d ) a = R–r

s =

a2 + h2 =

( R – r )2 + h2

Example: Find the volume of a frustum of a cone of the follow­ing dimensions: D = 8 centimeters; d = 4 centimeters; h = 5 centi­meters.

V = 0.2618 × 5 ( 8 2 + 8 × 4 + 4 2 ) = 0.2618 × 5 ( 64 + 32 + 16 ) = 0.2618 × 5 × 112 = 146.61 cm3

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Machinery's Handbook, 31st Edition VOLUMES OF SOLIDS

91

Sphere: πd 3 4πr 3 Volume = V = ------------ = --------- = 4.1888r 3 = 0.5236d 3 3 6

r

Surface area = A = 4πr 2 = πd 2 = 12.5664r 2 = 3.1416d 2 r =

d

3

3V ------- = 0.6204 3 V 4π

Example: Find the volume and the surface area of a sphere 6.5 centimeters diameter. V = 0.5236d 3 = 0.5236 × 6.5 3 = 0.5236 × 6.5 × 6.5 × 6.5 = 143.79 cm 3 A = 3.1416d 2 = 3.1416 × 6.5 2 = 3.1416 × 6.5 × 6.5 = 132.73 cm 2 Example: The volume of a sphere is 64 cubic centimeters. Find its radius. r = 0.6204 3 64 = 0.6204 × 4 = 2.4816 cm

Spherical Sector: h c

r

2πr 2 h V = --------------- = 2.0944r 2 h 3 = total area of conical and spherical surface

A = 3.1416r ( 2h + c⁄2 ) c = 2 h ( 2r – h )

Example: Find the volume of a sector of a sphere with a 6-inch diameter (r = 3 inches) and 1.5-inch height h. Also find the length of chord c. V = 2.0944r 2 h = 2.0944 × 3 2 × 1.5 = 2.0944 × 9 × 1.5 = 28.27 in3

c = 2 h ( 2r – h ) = 2 1.5 ( 2 × 3 – 1.5 ) = 2 6.75 = 2 × 2.598 = 5.196 in

Spherical Segment: V = volume

h c

A = area of spherical surface

h c2 h2 V = 3.1416h 2  r – --- = 3.1416h  ----- + -----  8 6  3

r

c2 A = 2πrh = 6.2832rh = 3.1416  ----- + h 2 4  c 2 + 4h 2 r = -------------------8h

c = 2 h ( 2r – h )

Example: A segment of a sphere has the following dimensions: h = 50 millimeters; c = 125 millime­ters. Find the volume V and the radius of the sphere of which the segment is a part. 125 2 50 2 15 ,625 2500 V = 3.1416 × 50 ×  ----------- + -------- = 157.08 ×  ---------------- + ------------ = 372 ,247 mm 3 = 372 cm 3  8  8 6  6  125 2 + 4 × 50 2 15 ,625 + 10 ,000 25 ,625 r = ----------------------------------- = ---------------------------------------- = ---------------- = 64 mm 8 × 50 400 400

Ellipsoid:

4π V = ------abc = 4.1888abc 3

In an ellipsoid of revolution, or spheroid, where c = b:

b a

c

V = 4.1888ab 2

Example: Find the volume of a spheroid in which a = 5 in., and b = c = 1.5 in. V = 4.1888 × 5 × 1.5 2 = 47.124 in3

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Machinery's Handbook, 31st Edition VOLUMES OF SOLIDS

92 Spherical Zone: c1

3c 2 3c 2 V = 0.5236h  --------1 + --------2 + h 2  4  4

h

c2

A = 2πrh = 6.2832rh = area of spherical surface

r

r =

c 22  c 22 – c 12 – 4h 2 2 ----- + ------------------------------ 8h 4 

Example: In a spherical zone, let c1 = 3; c 2 = 4; and h = 1.5 in. Find the volume. 3 × 32 3 × 42 27 48 V = 0.5236 × 1.5 ×  -------------- + -------------- + 1.5 2 = 0.5236 × 1.5 ×  ------ + ------ + 2.25 = 16.493 in 3  4  4  4 4

Spherical Wedge: V = volume A = area of spherical surface α = center angle in degrees

 r

α 4πr 3 V = --------- × ------------ = 0.0116αr 3 360 3 α A = --------- × 4πr 2 = 0.0349αr 2 360

Example: Find the area of the spherical surface and the volume of a wedge of a sphere. The diameter of the sphere is 100 mm, and the center angle a is 45 degrees. V = 0.0116 × 45 × 50 3 = 0.0116 × 45 × 125 ,000 = 65 ,250 mm 3 = 65.25 cm 3 A = 0.0349 × 45 × 50 2 = 3926.25 mm 2 = 39.26 cm 2

Hollow Sphere:

r

R

d D

V = volume of material used to make hollow sphere 4π V = ------ ( R 3 – r 3 ) = 4.1888 ( R 3 – r 3 ) 3 π 3 = --- ( D – d 3 ) = 0.5236 ( D 3 – d 3 ) 6

Example: Find the volume of a hollow sphere, 8 in. in outside diameter, with a thickness of mate­ rial of 1.5 in. Here R = 4; r = 4 - 1.5 = 2.5. V = 4.1888 ( 4 3 – 2.5 3 ) = 4.1888 ( 64 – 15.625 ) = 4.1888 × 48.375 = 202.63 in3

Paraboloid: Volume = V = 1⁄2 π r 2 h = 0.3927d 2 h

r h

d

2π Area = A = -----3p

3

d2  ---- + p 2 – p 3 4 

d2 where p = -----8h

Example: Find the volume of a paraboloid in which h = 300 millimeters and d = 125 millimeters. V = 0.3927d 2 h = 0.3927 × 125 2 × 300 = 1 ,840 ,781 mm 3 = 1 ,840.8 cm 3

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Machinery's Handbook, 31st Edition VOLUMES OF SOLIDS

93

Paraboloid Segment: R

r

d

D

h

π V = --- h ( R 2 + r 2 ) = 1.5708h ( R 2 + r 2 ) 2 π = --- h ( D 2 + d 2 ) = 0.3927h ( D 2 + d 2 ) 8 Example: Find the volume of a segment of a paraboloid in which D = 5 in., d = 3 in., and h = 6 in.

V = 0.3927h ( D 2 + d 2 ) = 0.3927 × 6 × ( 5 2 + 3 2 ) = 0.3927 × 6 × 34 = 80.11 in3

Torus: V = 2π 2 Rr 2 = 19.739Rr 2

R

D

π2 = -----Dd 2 = 2.4674Dd 2 4

r

Area of surface = A = 4π 2 Rr = 39.478Rr = π 2 Dd = 9.8696Dd

Example: Find the volume and area of surface of a torus in which d = 1.5 in. and D = 5 in. V = 2.4674 × 5 × 1.5 2 = 2.4674 × 5 × 2.25 = 27.76 in3 A = 9.8696 × 5 × 1.5 = 74.022 in2

Barrel:

h

D

d

V = approximate volume If the sides are bent to the arc of a circle: 1 V = ------ πh ( 2D 2 + d 2 ) = 0.262h ( 2D 2 + d 2 ) 12 If the sides are bent to the arc of a parabola: V = 0.209h ( 2D 2 + Dd + 3⁄4 d 2 )

Example: Find the approximate contents of a barrel with sides bent to an arc of a circle, the inside dimensions of which are D = 60 cm, d = 50 cm, h = 120 cm. V = 0.262h ( 2D 2 + d 2 ) = 0.262 × 120 × ( 2 × 60 2 + 50 2 ) = 0.262 × 120 × ( 7200 + 2500 ) = 0.262 × 120 × 9700 = 304 ,968 cm3 = 0.305 m3

Ratio of Volumes:

d

If d = base diameter and height of a cone, a paraboloid and a cyl­ inder, and the diameter of a sphere, then the proportions of the volumes of these solids are to each other as follows: Cone : Paraboloid : Sphere: Cylinder =

d

⅓:½:⅔:1

Example: Assume, as an example, that a cone, paraboloid, and cylinder each has a diameter of 2 inches, and that the height is 2 inches, and that the diameter of a sphere is 2 inches. Then the volumes, written in formula form, are as follows: Cone

Paraboloid

Sphere

Cylinder

3.1416 × 2 2 × 2 3.1416 × ( 2p ) 2 × 2 3.1416 × 2 3 3.1416 × 2 2 × 2 ------------------------------------- : ------------------------------------------- : -------------------------- : ------------------------------------- = 12 8 6 4

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94

Machinery's Handbook, 31st Edition TRIGONOMETRY TRIGONOMETRY: SOLUTION OF TRIANGLES

Terminology.—A triangle is a polygon with three sides. The sum of the angle measures in any triangle in a plane is 180 degrees. The triangle inequality states that the sum of any two side lengths of a triangle is always greater than the length of the third side. It is not possible to construct a triangle that violates the triangle inequality Triangles are either right or oblique. A right triangle has a right angle, which measures 90 degrees. Oblique triangles do not contain a right angle. As with any polygon (see page 60), parts with equal measure are called congruent. Thus, a triangle with congruent sides is one whose sides have the same measure. A triangle with two con­gruent sides is called an isosceles triangle; a triangle with all three sides congruent is equi­lateral and hence equiangular. Each angle measures 180 degrees/3 = 60 degrees. Two positive angles whose measures total 90 degrees are called complementary angles. The two acute angles in any right triangle are complements of each other. Two positive angles whose sum is 180 degrees are called supplementary angles. An isosceles triangle has at least two congruent sides and angles (an equilateral triangle is also isosceles). Angles opposite the congruent sides are congruent angles. An obtuse tri­ angle has one angle measuring greater than 90 degrees. An acute triangle has all three angles measuring less than 90 degrees; hence, an equilateral (equiangular) triangle is also acute.

Degree and Radian Angle Measure.—Two modes of measuring angles are degree measure and radian measure. 1 radian is the measure of a circle’s central angle whose arc is the same length as the radius of the circle. For any size circle, 1 radian is approximately 57.3 degrees. Conversion between degree and radian measure is based on the relation 360° = 2π radians, or 180° = π radians. (π is the ratio of circumference to diameter, C/d, and is approximately 22/7 or 3.1415926. The actual value of π is an irrational number.) Degree measure is converted to its equivalent radian measure by the formula: Degree measure × π/180 = Radian measure

For example, 45° × π/180 = π/4. Radian measure is converted to degree by the formula:

Radian measure × 180/π = Degree measure

For example, π/3 × 180° = 60°. Radian measure is actually unitless, but it is customary to write “rad” to indicate when radian measure is used. Conversions for the essential degree measures of the circle are shown in the chart on page 95.

Trigonometric Ratios of Essential Angles.—An acute angle can be any degree measure between 0° and 90°, but the trigonometric values for base angles (designated as θ in the table below the diagrams) 30°, 45°, and 60° are usually memorized. They are derived from right triangles constructed so that the shortest side has length = 1. The other dimensions follow from the geometric construction of the angles (see Geometric Constructions starting on page 66) and the Pythagorean theorem. The triangles and trigonometric values of these angle measures, as well as those with base angles 0° and 90° (envisioned by a horizontal and a vertical line segment, respectively) are given in the diagrams and table below. These five are the essential angles. In the development of the essential angles, the length of the shortest side in each of the triangles is designated as 1, so by construction the other lengths follow. Derivations of the main three functions for the five angle measures is shown. Decimal values of square roots are rounded to three decimal places. In the complete trigonometry tables (Table 2a, Table 2b, and Table 2c), irrational values are carried out to six decimal places, for greater accuracy. For quick estimates, it is useful to memorize the truncated values in the table below.

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Machinery's Handbook, 31st Edition TRIGONOMETRY

–

–

–

–

–

–

–

95

–

–

–

Note: Base angles are designated as θ in the table below. The values are approximated to three decimal places. If more accuracy is needed, then calculator values carried out further can be used.

Functions of Angles Right Triangle Ratios.—Trigonometry is the branch of mathematics that addresses the relations of the sides and angles of triangles and the resulting functions of angles. The properties of trigonometry are derived, however, from the relationships of angles and sides in the right triangle, as pictured here. In a right triangle, the 90° angle, C, is oppo- s­ ite the longest side, c, the hypotenuse. Since the sum of angle measures in any triangle is 180°, the sum of the two acute angles is 90°. The shorter sides, a and b, are called the legs. They are opposite to angles A and B, respectively. By the Pythagorean theorem, the sum of the squares of the legs is equal to the square of the hypot­enuse; that is,

a

c

B C = 90˚

b

A

a2 + b2 = c2

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96

Machinery's Handbook, 31st Edition FUNCTIONS OF ANGLES

Referring to the figure at the bottom of page 95, the ratios of the sides of a right triangle with respect to angle A are named sine A, cosine A, and tangent A (abbreviated sin A, sin A, and tan A). Ratios and reciprocal ratios, named cosecant A, secant A, and cotangent A (abbreviated cscA, secA, and cotA) are defined as:

opposite - = --asin A = -------------------------hypotenuse c adjacent --cos A = --------------------------- = b hypotenuse c opposite a tan A = -------------------- = --adjacent b

1--------------------------- = --c- = ---------csc A = hypotenuse opposite a sin A 1 --------------------------- = --c- = ----------sec A = hypotenuse adjacent b cos A adjacentb 1 cot A = ------------------= --- = ----------opposite a tan A

Similar ratios are defined for angle B:

opposite - = --bsin B = -------------------------hypotenuse c adjacent --cos B = --------------------------- = a hypotenuse c opposite b tan B = -------------------- = --adjacent a

1--------------------------- = --c- = ---------csc B = hypotenuse opposite b sin B 1 --------------------------- = --c- = ----------sec B = hypotenuse adjacent a cos B adjacenta 1 cot B = ------------------= --- = ----------opposite b tan B

Thus, in a given right triangle, sin A = cos B, cos A = sin B, tan A = cot B. Similarly, csc A = sec B, sec A = csc B, cot a = tan B.

Law of Sines.—In any triangle, if a, b, and c are the sides, and A, B, and C their opposite angles, respectively, then: c a - = ---------b - = -------------------sin C sin A sin B b sin Aa = -------------sin B a sin Bb = -------------sin A sin Cc = a-------------sin A

or or or

so that: c sin A a = -------------sin C c sin B b = -------------sin C b sin Cc = -------------sin B

Law of Cosines.—In any triangle, if a, b and c are the sides and A, B, and C are the opposite angles, respectively, then:

a 2 = b 2 + c 2 – 2bc cos A b 2 = a 2 + c 2 – 2ac cos B c 2 = a 2 + b 2 – 2ab cos C The sine and cosine laws together with the proposition that the sum of the measures of the three angles is 180 degrees are the basis of all formulas relating to the solution of triangles. Formulas and examples for the solution of right-angle and oblique-angle triangles, arranged in tabu­lar form, are given on the following pages.

Trigonometric Identities.—It is possible to express trigonometric ratios in terms of other ratios by way of trigonometric identities. For example, sin(A + B) = sin A cos B + cos A sin B. It may be helpful to use an identity to quickly evaluate a trigonometric function, as shown by the examples given below the trigonometric identities and formulas derived from them.

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Machinery's Handbook, 31st Edition FUNCTIONS OF ANGLES

97

Reciprocal Identities: sin A- = ----------1 1 1 tan A = ----------    sec A = ------------     csc A = ----------cos A cot A cos A sin A Negative-Angle Identities: sin ( – A ) = – sin A     cos ( – A ) = cos A     tan ( – A ) = – tan A Cofunction Identities:

sin ( 90 – A ) = cos A     tan ( 90 – A ) = cot A     sec ( 90 – A ) = csc A cos ( 90 – A ) = sin A     cot ( 90 – A ) = tan A     csc ( 90 – A ) = sec A

Pythagorean Identities:

sin2 A + cos2 A = 1     1 + tan2 A = sec2 A     1 + cot2 A = csc2 A

Sum- and Difference-of-Angles Formulas:

sin ( A + B ) = sin A cos B + cos A sin B   sin ( A – B ) = sin A cos B – cos A sin B cos ( A + B ) = cos A cos B – sin A sin B   cos ( A – B ) = cos A cos B + sin A sin B tan A + tan Btan A – tan B tan ( A + B ) = -------------------------------  tan ( A – B ) = --------------------------------1 – tan A tan B 1 + tan A tan B cot A cot B – 1cot A cot B + 1 cot ( A + B ) = -------------------------------  cot ( A – B ) = --------------------------------cot B + cot A cot B – cot A Double-Angle Formulas: cos(2A) = cos2 A – sin2 A = 2 cos2 A – 1 = 1 – 2 sin2 A 2 2 tan A - = ----------------------------sin(2A) = 2 sin A cos A  tan (2A) = ---------------------cot A – tan A 1 – tan2 A Half-Angle Formulas: sin(A/2) = ± (1 – cos A)/2   cos(A/2) = ± (1 + cos A)/2 – cos A- = --------------------sin A 1 – cos A- = 1-------------------tan(A/2) = ± --------------------sin A 1 + cos A 1 + cos A Product-to-Sum Formulas: sin A cos B = 1⁄2 [ sin ( A + B ) + sin ( A – B ) ] cos A cos B = 1⁄2 [ cos ( A + B ) + cos ( A – B ) ] sin A sin B = 1⁄2 [ cos ( A – B ) – cos ( A + B ) ] tan A + tan Btan A tan B = ----------------------------cot A + cot B Sum and Difference of Functions Formulas: A–B A+B sin A + sin B = 2 sin  ------------- cos  -------------  2   2 

sin ( A + B )tan A + tan B = ------------------------cos A cos B

A–B A+B sin A – sin B = 2 sin  ------------- cos  -------------  2   2 

sin ( A – B )tan A – tan B = ------------------------cos A cos B

A+B ( A – B -) cos A + cos B = 2 cos  ------------- cos ---------------- 2  2

sin ( B + A )cot A + cot B = ------------------------sin A sin B

A–B A+B cos A – cos B = 2 sin  ------------- sin  -------------  2   2 

sin ( B – A )cot A – cot B = ------------------------sin A sin B

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Machinery's Handbook, 31st Edition Solution of Right Triangles

98

Solution of Right Triangles Right triangles (one angle measures 90°) are the easiest to “solve” (that is, to find the missing sides or angles), thanks to the Pythagorean theorem and right triangle ratios.

a

B

c

C  90° A b

The figure to the left shows right triangle ABC. Sides opposite corresponding angles are labeled a, b, c. The formulas in the table are for finding an unknown side or angle from given information as shown. There are several ways to solve for the missing dimension through one of the three basic trigonometric functions. Right angle C is always known (90°), thus, A = 90 – B and B = 90 – A.

Sides and Angle Known

Formulas for Sides, Angles to be Found

Sides a and b

c =

a2 + b2

A = tan–1(a/b)

B = 90° - A

Side a, hypotenuse c

b =

c2 – a2

A = sin–1(a/c) or A = cos–1(b/c)

B = 90° - A

Side b, hypotenuse c

a =

c2 – b2

B = sin–1(b/c) or B = cos–1(a/c)

A = 90° - B

Hypotenuse c, angle B

b = c sin B or a = c cos B

a = c cos B or b = c sin B

A = 90° - B

Hypotenuse c, angle A

b = c cos A or a = c sin A

a = c sin A or b = c cos A

B = 90° - A

Side b, angle B

bc = ---------sin B or ac = ---------cosB

ba = ---------tan B

A = 90° - B

Side b, angle A

b c = ----------cos A or a c = ----------sin A

a = b tan A

B = 90° - A

Side a, angle B

a c = ----------cos B

b = a tan B

A = 90° - B

Side a, angle A

ac = ---------sin A

ab = ---------tan A

B = 90° - A

The solutions of triangles—both right and oblique—rely not only on the Pythagorean theorem and the laws of cosines and of sines, but also on correct use of the trigonometry

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Machinery's Handbook, 31st Edition Solution of Right Triangles

99

tables, either when given an angle and seeking a trigonometric ratio, or “in reverse,” when given a ratio (side lengths) and seeking an angle. A full explanation of trigonometric functions and the use of their tables begins on page 105.

Solution and Examples of Right Triangles (US Customary or Metric Units) Examples below show angle measure further divided into units of minutes ( ′ ) and seconds ( ″ ). 1º = 60′; 1′ = 60″, rather than decimal parts. c = 22 inches; B = 41° 36 ′

a = c cos B = 22 × cos 41 ° 36′ = 22 × 0.74780 = 16.4516 in.

B 22

6´

°3

41

c



a

C  90°

A

b Hypotenuse and One Acute Angle Known

b = c sin B = 22 × sin 41 ° 36′ = 22 × 0.66393 = 14.6065 in.

A = 90 ° – B = 90 ° – 41 ° 36′ = 48 ° 24′ c = 25 centimeters; a = 20 centimeters

a

c2 – a2 =

b =

20

90°

B

A

b

c  25 Hypotenuse and One Side Known

C 

= 225 = 15 cm ------ = 0.8 sin A = a--- = 20 c 25 Hence, A = sin–1(0.8) = 53°8′ B = 90° – A = 90° – 53°8′ = 36°52′

A

c =

a2 + b2 =

90

=

°

B

625 – 400

a = 36 mm; b = 15 mm

b  15

a  36

25 2 – 20 2 =

36 2 + 15 2 =

1296 + 225

1521 = 39 mm

--- = 36 ------ = 2.4 tan A = a b 15

c

Hence, A = tan–1(2.4) = 67 ° 23′

B = 90 ° – A = 90 ° – 67 ° 23′ = 22 ° 37′

Two Sides Known

c

°

C

65 b

a - = --------------12 - = -----------------12 - = 13.2405 c = ---------m sin A 0.90631 sin 65 ° a - = 12 × cot 65 ° = 12 × 0.46631 b = ---------tan A = 5.5957 m

90°

A

B

a  12

a = 12 meters; A = 65°

One Side and One Angle Known

B =

90 °

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– A = 90 ° – 65 ° = 25 °

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Machinery's Handbook, 31st Edition SOLUTION OF OBLIQUE TRIANGLES

100

Solution and Examples of Oblique Triangles (US Customary or Metric Units) One Side and Two Excluded (Not Between) Angles Known (Law of Sines): If side a, angle A opposite it, and angle B, are known: A (Known) c B C (Known)

b

a (Known) One Side and Two Excluded Angles Known

A

a sin C a sin B b = -------------c = --------------sin A sin A ab sin C Area = -----------------2 If angles B and C are known, but not A, then A = 180 - (B + C). Example: a = 5 cm; A = 80°; B = 62° C = 180° – ( 80° + 62° ) = 180° – 142° = 38°

b



c

C = 180° – ( A + B )

80

sin B 5 × sin 62 ° × 0.88295 ----------------------------------------b = a = -------------------------- = 5 sin A sin 80 ° 0.98481

°

B  62° C

a5 Side and Angles Known

= 4.483 cm a sin C 5 × sin 38 ° × 0.61566 c = --------------= -------------------------- = 5---------------------------sin A sin 80 ° 0.98481 = 3.126 cm

Two Sides and Included Angle Known: n)

ow Kn b

(

If sides a and b, and angle C between them are known:

A

C (Known)

c

B

a (Known) Two Sides and One Included Angle Known

a

9

C  35°

B

A

c

b8 Sides and Angle Known

a sin C a sin C , so A = tan–1 ----------------------------tan A = ----------------------------b – ( a cos C ) b – ( a cos C ) sin C --------------B = 180 ° – ( A + C ) c = a sin A Side c may also be found directly as below: c = a 2 + b 2 – ( 2ab cos C ) sin C -----------------Area = ab 2 Example: a = 9 inches; b = 8 inches; C = 35° 9 × sin 35 ° a sin C = ---------------------------------------tan A = -----------------------------b – ( a cos C ) 8 – ( 9 × cos 35 ° )

9 × 0.57358 ------------------- = 8.22468 = ----------------------------------------= 5.16222 8 – ( 9 × 0.81915 ) 0.62765 Hence, A = tan–1(8.22468) = 83°4′ B = 180° – ( A + C ) = 180° – 118°4′ = 61°56′ 9 × 0.57358 = 5.2 inches a sin C = ---------------------------c = --------------sin A 0.99269

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Machinery's Handbook, 31st Edition SOLUTION OF OBLIQUE TRIANGLES

101

Two Sides and the Angle Opposite One of the Sides Known: If angle A, opposite side a, and other side b are known: n)

ow Kn b

(

A (Known) c B

C a (Known)

C

17

B

A

c

b

61°

Two Sides and Angle Opposite One Side Known

a  20

Sides and Angle Known

All Three Sides Known:

A B

C

c

wn)

w no (K b

(Kno

n)

sin A sin B = b-------------a a sin C c = -------------sin A

9 b

10

C

c

A

B

a8

Sides Known

sin C -----------------Area = ab 2

If B > A but < 90°, a second solution, B2 , C2 , c 2 , exists: B2 = 180° - B, C2 = 180° - (A + B2) c 2 = (a sin C2)/sin A,  area = (ab sin C2)/2 If a ≥ b sin A, then only the first solution exists. If a < b sin A, then no solution exists. Example: a = 20 cm; b = 17 cm; A = 61° sin A = ---------------------------17 × sin 61 °sin B = b--------------a 20

× 0.87462 = 0.74343 ------------------------------= 17 20 Hence, B = sin–1(0.74343) = 48 °1′ C = 180° – ( A + B ) = 180° – 109°1′ = 70°59′ a sin C 20 × sin 70°59′ × 0.94542 ------------------------------c = -------------= ------------------------------------- = 20 sin A sin 61° 0.87462 = 21.62 cm

If all three sides a, b, and c are known, then any angle can be found: 2 + c2 – a2 cos A = b---------------------------2bc

C = 180° – ( A + B )

a (Known)

All Three Sides Known

C = 180° – ( A + B )

sin A sin B = b--------------a sin C ab Area = -----------------2

Example: a = 8 in.; b = 9 in.; c = 10 in. 2 + 10 2 – 8 2 b 2 + c 2 – a 2 = 9------------------------------cos A = ---------------------------2bc 2 × 9 × 10

+ 100 – 64- = 117 ---------------------------------------- = 0.65000 = 81 180 180 A = cos–1(0.65000) = 49°27′ Hence, sin A 9 × 0.75984 --------------sin B = b = ---------------------------- = 0.85482 a 8 Hence, B = sin–1(0.85482) = 58°44′ C = 180° – ( A + B ) = 180° – 108°11′ = 71°49′

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Machinery's Handbook, 31st Edition RAPID SOLUTION OF RIGHT AND OBLIQUE TRIANGLES

102

Rapid Solution of Right and Oblique Triangles 2

c = a  b2

b

E

a

b

90

90

c sin E = ---a

90

c

b

90

b a = ---------sin D b

a 90

90

D

c

c a = -----------cos D 90

b = cotc D b

90

D

D

a sin - F b = --------------sin D b

D

c

a

Area = ab sin E 2

b

E

D

b = c cot E

E

b

90

90

c

a sin E c = ---------------sin D

D

a

E a sin c = ---------------sin D

D

E

a

c a = ---------sin E a

E

F =180° (D + E ) D

c

E

c

F

90

c

D

a

b

a

c = b tan E

E

b

90

D=180°(E + F )

F

F

c

E

c

c

a

90

b a = ---------cos E b

E

b2 + c2a-2 cos D = --------------------2bc

c

D

---------D-sin F = b------sin a b

sin D sin F = b----------------a b

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a

tan D= a sin E b a cos E b E a

E = 180° ( D +F ) D F

a

F a

D

E

F

D

D

c = a sin E E

a

c = b cot D b

90 c

90

c

D

a

E

b

D

D

b

b = a cos E

a

90

b tan D = ---c

c

c = a cos D

a

c

90

D

b = a × sin D

90

D

a

b

E

90

a

a = b2  c2

D = 90 °  E

a

a2  c 2

b = b

90

D

c

E

E = 90 °  D

-

sin D = b a

a

E

F =180° (D + E ) D F

E

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Machinery's Handbook, 31st Edition ANGLE CONVERSIONS

103

Conversion Tables of Angular Measure.—The accompanying tables of degrees, min­ utes, and seconds into radians; radians into degrees, minutes, and seconds; radians into degrees and decimals of a degree; and minutes and seconds into decimals of a degree and vice versa facilitate the conversion of measurements. Example 1: The “Degrees, Minutes, and Seconds into Radians” table is used to find the number of radians in 324 degrees, 25 minutes, 13 seconds as follows: 300 degrees   20 degrees   4 degrees     25 minutes        13 seconds 324°25 ′13″

=  5.235988 radians =  0.349066 radian =  0.069813 radian =  0.007272 radian =  0.000063 radian  =  5.662202 radians

Example 2: The “Radians into Degrees and Decimals of a Degree,” and “Radians into Degrees, Minutes and Seconds” tables are used to find the number of decimal degrees or degrees, minutes and seconds in 0.734 radian as follows: 0.7   radian = 40.1070 degrees 0.03  radian =  1.7189 degrees 0.004 radian  =   0.2292 degree 0.734 radian  =  42.0551 degrees

0.7   radian = 40°6 ′25″ 0.03  radian =  1°43 ′8″ 0.004 radian  =   0°13 ′45″ 0.734 radian = 41°62 ′78″ or 42°3 ′18″

Degrees, Minutes, and Seconds into Radians (Based on 180 degrees = π radians) Deg.

1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 Min. 1 2 3 4 5 6 7 8 9 10

Sec.

1 2 3 4 5 6 7 8 9 10

Rad.

Deg.

Rad.

Min.

Rad.

Sec.

17.453293 100 34.906585 200 52.359878 300 69.813170 400 87.266463 500 104.719755 600 122.173048 700 139.626340 800 157.079633 900 174.532925 1000

0.000291 0.000582 0.000873 0.001164 0.001454 0.001745 0.002036 0.002327 0.002618 0.002909

0.000005 0.000010 0.000015 0.000019 0.000024 0.000029 0.000034 0.000039 0.000044 0.000048

11 12 13 14 15 16 17 18 19 20

11 12 13 14 15 16 17 18 19 20

Rad.

1.745329 3.490659 5.235988 6.981317 8.726646 10.471976 12.217305 13.962634 15.707963 17.453293 Rad.

0.003200 0.003491 0.003782 0.004072 0.004363 0.004654 0.004945 0.005236 0.005527 0.005818 Rad.

0.000053 0.000058 0.000063 0.000068 0.000073 0.000078 0.000082 0.000087 0.000092 0.000097

Deg.

10 20 30 40 50 60 70 80 90 100

Degrees into Radians Rad. Deg. Rad.

Deg.

0.174533 1 0.017453 0.349066 2 0.034907 0.523599 3 0.052360 0.698132 4 0.069813 0.872665 5 0.087266 1.047198 6 0.104720 1.221730 7 0.122173 1.396263 8 0.139626 1.570796 9 0.157080 1.745329 10 0.174533 Minutes into Radians Min. Rad. Min. Rad.

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

21 22 23 24 25 26 27 28 29 30

0.006109 31 0.009018 0.006400 32 0.009308 0.006690 33 0.009599 0.006981 34 0.009890 0.007272 35 0.010181 0.007563 36 0.010472 0.007854 37 0.010763 0.008145 38 0.011054 0.008436 39 0.011345 0.008727 40 0.011636 Seconds into Radians Sec. Rad. Sec. Rad. 21 22 23 24 25 26 27 28 29 30

0.000102 0.000107 0.000112 0.000116 0.000121 0.000126 0.000131 0.000136 0.000141 0.000145

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31 32 33 34 35 36 37 38 39 40

0.000150 0.000155 0.000160 0.000165 0.000170 0.000175 0.000179 0.000184 0.000189 0.000194

Min. 41 42 43 44 45 46 47 48 49 50

Sec. 41 42 43 44 45 46 47 48 49 50

Rad.

Deg.

0.001745 0.003491 0.005236 0.006981 0.008727 0.010472 0.012217 0.013963 0.015708 0.017453

0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10

Rad.

Min.

Rad.

Sec.

0.011926 0.012217 0.012508 0.012799 0.013090 0.013381 0.013672 0.013963 0.014254 0.014544

0.000199 0.000204 0.000208 0.000213 0.000218 0.000223 0.000228 0.000233 0.000238 0.000242

51 52 53 54 55 56 57 58 59 60

51 52 53 54 55 56 57 58 59 60

Rad.

0.000175 0.000349 0.000524 0.000698 0.000873 0.001047 0.001222 0.001396 0.001571 0.001745 Rad.

0.014835 0.015126 0.015417 0.015708 0.015999 0.016290 0.016581 0.016872 0.017162 0.017453 Rad.

0.000247 0.000252 0.000257 0.000262 0.000267 0.000271 0.000276 0.000281 0.000286 0.000291

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Machinery's Handbook, 31st Edition ANGLE CONVERSIONS

104

Radians into Degrees and Decimals of a Degree (Based on π radians = 180 degrees) Rad. 10 20 30 40 50 60 70 80 90 100

Deg. 572.9578 1145.9156 1718.8734 2291.8312 2864.7890 3437.7468 4010.7046 4583.6624 5156.6202 5729.5780

Rad. 1 2 3 4 5 6 7 8 9 10

Deg. 57.2958 114.5916 171.8873 229.1831 286.4789 343.7747 401.0705 458.3662 515.6620 572.9578

Rad. 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Deg. 5.7296 11.4592 17.1887 22.9183 28.6479 34.3775 40.1070 45.8366 51.5662 57.2958

Rad. 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10

Deg. 0.5730 1.1459 1.7189 2.2918 2.8648 3.4377 4.0107 4.5837 5.1566 5.7296

Rad. 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.010

Deg. 0.0573 0.1146 0.1719 0.2292 0.2865 0.3438 0.4011 0.4584 0.5157 0.5730

Rad. 0.0001 0.0002 0.0003 0.0004 0.0005 0.0006 0.0007 0.0008 0.0009 0.0010

Deg. 0.0057 0.0115 0.0172 0.0229 0.0286 0.0344 0.0401 0.0458 0.0516 0.0573

Radians into Degrees, Minutes, and Seconds (Based on π radians = 180 degrees) Rad.

10 20 30 40 50 60 70 80 90 100

Angle

572°57 ′28″ 1145°54 ′56″ 1718°52 ′24″ 2291°49 ′52″ 2864°47 ′20″ 3437°44 ′48″ 4010°42 ′16″ 4583°39 ′44″ 5156°37 ′13″ 5729°34 ′41″

Rad. 1 2 3 4 5 6 7 8 9 10

Angle

57°17 ′45″ 114°35 ′30″ 171°53 ′14″ 229°10 ′59″ 286°28 ′44″ 343°46 ′29″ 401°4 ′14″ 458°21 ′58″ 515°39 ′43″ 572°57 ′28″

Rad. 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Angle

5°43 ′46″ 11°27 ′33″ 17°11 ′19″ 22°55 ′6″ 28°38 ′52″ 34°22 ′39″ 40°6 ′25″ 45°50 ′12″ 51°33 ′58″ 57°17 ′45″

Rad.

0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10

Angle

0°34 ′23″ 1°8 ′45″ 1°43 ′8″ 2°17 ′31″ 2°51 ′53″ 3°26 ′16″ 4°0 ′39″ 4°35 ′1″ 5°9 ′24″ 5°43 ′46″

Rad.

0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.010

Angle

0°3 ′26″ 0°6 ′53″ 0°10 ′19″ 0°13 ′45″ 0°17 ′11″ 0°20 ′38″ 0°24 ′4″ 0°27 ′30″ 0°30 ′56″ 0°34 ′23″

Rad.

0.0001 0.0002 0.0003 0.0004 0.0005 0.0006 0.0007 0.0008 0.0009 0.0010

Angle

0°0 ′21″ 0°0 ′41″ 0°1 ′ 2″ 0°1 ′23″ 0°1 ′43″ 0°2 ′ 4″ 0°2 ′24″ 0°2 ′45″ 0°3 ′6″ 0°3 ′26″

Minutes and Seconds into Decimal of a Degree and Vice Versa (Based on 1 second = 0.00027778 degree) Min. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Minutes into Decimals of a Degree

Deg. 0.0167 0.0333 0.0500 0.0667 0.0833 0.1000 0.1167 0.1333 0.1500 0.1667 0.1833 0.2000 0.2167 0.2333 0.2500 0.2667 0.2833 0.3000 0.3167 0.3333

Min. 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40

Deg. 0.3500 0.3667 0.3833 0.4000 0.4167 0.4333 0.4500 0.4667 0.4833 0.5000 0.5167 0.5333 0.5500 0.5667 0.5833 0.6000 0.6167 0.6333 0.6500 0.6667

Min. 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Deg. 0.6833 0.7000 0.7167 0.7333 0.7500 0.7667 0.7833 0.8000 0.8167 0.8333 0.8500 0.8667 0.8833 0.9000 0.9167 0.9333 0.9500 0.9667 0.9833 1.0000

Sec. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Seconds into Decimals of a Degree

Deg. 0.0003 0.0006 0.0008 0.0011 0.0014 0.0017 0.0019 0.0022 0.0025 0.0028 0.0031 0.0033 0.0036 0.0039 0.0042 0.0044 0.0047 0.0050 0.0053 0.0056

Sec. 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40

Deg. 0.0058 0.0061 0.0064 0.0067 0.0069 0.0072 0.0075 0.0078 0.0081 0.0083 0.0086 0.0089 0.0092 0.0094 0.0097 0.0100 0.0103 0.0106 0.0108 0.0111

Sec. 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Deg. 0.0114 0.0117 0.0119 0.0122 0.0125 0.0128 0.0131 0.0133 0.0136 0.0139 0.0142 0.0144 0.0147 0.0150 0.0153 0.0156 0.0158 0.0161 0.0164 0.0167

Example 3: Convert 11 ′37″ to decimals of a degree. From the “Min. into Dec. Deg.” table, 11 ′ = 0.1833 degree. From the “Sec. into Dec. Deg.” table, 37″ = 0.0103 degree. Adding, 11 ′37″ = 0.1833 + 0.0103 = 0.1936 degree. Example 4: Convert 0.1234 degree to minutes and seconds. From the “Min. into Dec. Deg.” table, 0.1167 degree = 7 ′. Subtracting 0.1167 from 0.1234 gives 0.0067. From the “Sec. into Dec. Deg.” table, 0.0067 = 24″ so that 0.1234 = 7 ′24″.

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Machinery's Handbook, 31st Edition TRIGONOMETRIC FUNCTIONS

105

Trigonometric Functions.—­Like algebraic functions, trigonometric functions define a relationship between input values (angle measure, expressed in radians) and output values (the trigonometric ratios associated with the angles). Recalling function notation, y = f(x) (see ALGEBRA on page 24), the main trigonometric functions are: f(x) = sin x, f(x) = cos x, and f(x) = tan x. The trigonometric graphs are derived from the points on the unit circle (circle of radius = 1) “translated” onto the (x, y)-­coordinate system. Because the radius is 1, cos θ = x/1 = x, and sin θ = y/1 = y. Thus, the rectangular coordinate ordered pair (x, y) on the unit circle corresponds to the trigonometric coordinates (cos θ, sin θ). Possible confusion arises from the different roles of x and y in the two graphs. On the unit circle (Fig. 1), x and y are the coordinates of the points that correspond to the adjacent (cosine) and opposite (sine) sides of the right triangle with radius 1. But on the function graphs (Fig. 2a), x is the angle measure, the same as θ, and y is the value of the function being plotted. So, “sin x” is the same as “sin θ.” Angle measure is marked in radians (not degrees). The graphs of y = sin x and y = cos x are seen in the figure on the same set of axes (Fig. 2a). y = tan x is graphed on its own set of axes (Fig. 2b). The domain of a trigonometric function is the set of angle measures for which the function is defined. The domain of both y = sin x and y = cos x is the set of all real numbers, since it makes sense to substitute any angle—­positive or negative—­into these functions. The answer is always a real number. Negative angles indicate an angle measured clockwise from the horizontal, whereas a positive angle (the usual situation) is measured counterclockwise. The domain of y = tan x, however, does not include the odd multiples of π/2, since tan x = sin x/cos x, and so, when cos x = 0, tan x is undefined. This happens at ±π/2, ±3π/2, ±5π/2, . . . As Fig. 2b shows, the tangent function approaches the dotted lines increasingly closer, never meeting them.

Fig. 1. The Unit Circle, which Gives the Sine and Cosine Relationship for All Angle Measures.

Fig. 2. (a) Graph of y = sin x and y = cos x; (b) Graph of y = tan x. Note: For any real number x, sin x (for example) is defined to be sin(x radians). Equivalent degree measures are shown here for reference.

The signs of the three main trigonometric functions (positive or negative) are shown in the diagram at the top of page 106. The names of the positive-signed functions of angles that lie in a particular quadrant are shown. All the functions are positive in the first, only sin x in the second, only tan x in the third, and only cos x is positive in the fourth. The mnemonic device “ASTC” for “All Students Take Calculus” is a simple way to remember the signs. csc x, sec x, and cot x have the same signs as their respective reciprocal functions.

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106

Machinery's Handbook, 31st Edition TRIGONOMETRIC FUNCTIONS

Fig. 3. Signs of the Three Main Trigonometric Functions

Fig. 3 shows the values of angles and ratios on the unit circle in great detail, as well as the signs of the functions in each of the four quadrants. Radian and degree measure are also marked off. The figure also shows the sign (+ or –) and the values of trigonometric functions for angles in each of the four quadrants, 0 to 90, 90 to 180, 180 to 270, and 270 to 360 degrees. A π radian is approximately 3.14, and 2π is approximately 6.3 radian measure, as the completed circle indicates. The corresponding degree measures are marked as well, as are the pure radian equivalents. The chart indi­cates, for example, that all the functions are positive for angles between 0 and 90 degrees. This is because x and y are both positive in the first quadrant, and all the trigonometric ratios are therefore also positive. In the same way, the cotangent of an angle between 180 and 270 degrees is positive and has a value between infinity and 0; in other words, the cotangent for 180 degrees is infinitely large and then the cotangent gradually decreases for increasing angles, so that the cotan­gent for 270 degrees equals 0. The cosine, tangent, and cotangent for angles between 90 and 180 degrees are negative, although they have the same absolute values as their respective angles from 0 to 90 degrees. Negative trigonometric values are preceded by a minus sign; thus, tan 123° 20′ = -1.5204. Inverse Trigonometric Functions: If the value of the sides of a triangle are known but an angle is unknown, the trigonometric function inverse is used to work backwards to find the angle measure. Inverse trigonometric functions are known by either the prefix “arc” before the name, or by the superscript –1 after the name. The notation is not to be confused with the meaning of a negative exponent. Trigonometric functions and their inverses are shown below. The roles of x and y are reversed in the inverse functions: x is the ratio given and y is the angle measure that is sought: Function Inverse Function y = sin x y = arcsin x, or y = sin–1 x y = cos x y = arccos x, or y = cos–1 x y = tan x y = arctan x, or y = tan–1 x The value of the angle that corresponds to the ratio is intended. Any angle can be found using the inverse trig function and relying on trig tables or a calculator. The examples give a few of the essential, familiar ratios for which the angles are readily known. Examples: Given sin x = 1, arcsin(1), or sin–1(1) = 90°, or π/2. Given cos x = 0.866, arccos(0.866), or cos –1(0.866) = 60°, or π/3. Given tan x = 1, arctan(1), or tan–1(1) = 45°, or π/4. Trigonometry Tables.—Table 2a, Table 2b, and Table 2c, starting on page 108, contain the values of the sine, cosine, tangent, and cotangent functions of angles from 0 to 90 degrees. Commonly referred to as “trig tables,” these values also are accessible on standard scientific calculators. Function values for all other angles can be obtained from the trig tables by applying the rules for signs of trigonometric functions and the useful relation­ ships among angles given in the following. Secant and cosecant functions can be found from sec A = 1/cos A and csc A = 1/sin A. The trig tables are divided by a double line. The body of each half table consists of four labeled columns of data between the columns that contain the angles. The angles left of the data increase moving down the table, and angles right of the data increase moving up the table. Column labels above the data identify the trig functions for angles listed in the left column of each half table. Columns labels below the data are for angles listed in the right column of each half table. To find the value of a function for a particular angle, first the angle is located in the table, then the appropriate function label across the top or bottom row of the table is located. At the intersection of the angle row and label column is the function value. Angles on opposite sides of each table are complementary angles (i.e.,

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Machinery's Handbook, 31st Edition Trigonometry Tables

107

their sum is 90°) and are related (see Cofunction Identities:, page 97). For example, sin 10° = cos 80° and cos 10° = sin 80°. Expanded trig tables are also available in the ADDITIONAL material in the Machinery’s Handbook 31 Digital Edition. Angle measures greater than 90 degrees are converted to their reference angle (that is, its angle equivalent) before a trigonometric value can be found. If the angle θ is between 90 and 180, its reference angle is 180 – θ; if θ is between 180 and 270, then θ – 180 is its reference angle; and if it is between 270 and 360 degrees, 360 – θ is its acute angle equivalent. To determine trigonometric values of functions of angles greater than 90°, subtract 90, 180, 270, or 360 from the angle to get the reference angle less than 90° and use Table 1, Useful Relationships Among Angles, to find the equivalent first-quadrant function and angle to look up in the trig tables. Radians

3π ------ 2.3 4 2.4 2.5 5π -----2.6 6

2.7 2.8 2.9 3.0

1.7 1.6 1.5 1.4 1.3 π 2π 1.9 1.8 -------π 1.2 --3 2.0 1.1 3 2 2.1 1.0 2.2

90 80 110 100 70 120 60 Degrees 50 130 (1 to 0) sin + sin + (0 to 1) 40 140 (0 to −1) cos − cos + (1 to 0) 30 150 ( to 0) tan − tan + (0 to ) 160

170

3.1 π 3.2

180

3.3

190

3.4

0.9

200

(0 to ) cot − ( to −1) sec − (1 to ) csc +

cot + ( to 0) sec + (1 to ) csc + ( to 1)

II I III IV

(0 to −1) sin − (−1 to 0) cos − (0 to ) tan + ( to 0) cot + (−1 to ) sec − ( to −1) csc −

π --4 0.8 0.7 0.6 --π6 0.5

0.4 0.3

20

0.2

10

0.1

0 and 360

sin − (−1 to 0) cos + (0 to 1) tan − ( to 0) cot − (0 to ) sec + ( to 1) csc − (−1 to )

6.3 2π 6.2 6.1

350 340

6.0 210 330 5.9 220 320 3.6 5.8 310 230 11π 7π 3.7 240 300 5.7 -------------250 6 290 3.8 6 260 270 280 5.6 3.9 5.5 5π 4.0 7π ----------5.4 4 4.1 4 5.3 4.2 5.2 4π 4.3 5π 5.1 ----------4.4 4.5 5.0 3 3 4.6 4.7 4.8 4.9 3π -----2 3.5

Fig. 4. Signs of Trigonometric Functions, Fractions of p, and Degree-Radian Conversion

Table 1. Useful Relationships Among Angles 180° ± q sin q -cos q ±tan q ±cot q -sec q csc q ±

±

90° ± q q -q sine -sin q sin q +cos q cosine sin q +cos q cos q tangent -tan q cot q tan q cotangent -cot q tan q cot q secant csc q sec q +sec q cosecant -csc q csc q +sec q Examples: cos (270° - q) = -sin q; tan (90° + q) = -cot q.

270° ± q

360° ± q

-cos q ±sin q cot q tan q ±csc q -sec q

+cos q ±tan q ±cot q +sec q ±csc q

± ±

Angle Function

±sin q

± ± ± ±

Example: Find the cosine of 336°40 ′. Fig. 4 shows that the cosine of every angle in Quadrant IV (270° to 360°) is positive. To find the angle and trig function to use when entering the trig table, subtract 270 from 336 to get cos 336°40 ′ = cos (270° + 66°40 ′) and then find the intersection of the “cos row” and the 270 ± q column in Table 1. Because cos (270 ± q) in the fourth quadrant is equal to ± sin q in the first quadrant, find sin 66°40 ′ in the trig table. Therefore, cos 336°40 ′ = sin 66°40 ′ = 0.918216.

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Machinery's Handbook, 31st Edition Trigonometry Tables

108

Table 2a. Trigonometric Values of Angles from 0° to 15° and 75° to 90° Angle

sin

0° 0 ′ 0.000000 10 0.002909

20 0.005818

30 0.008727

40 0.011635

50 0.014544

1° 0 ′ 0.017452 10 0.020361 20 0.023269

30 0.026177

40 0.029085

50 0.031992

2° 0 ′ 0.034899 10 0.037806 20 0.040713

30 0.043619

40 0.046525

50 0.049431

3° 0 ′ 0.052336 10 0.055241 20 0.058145

30 0.061049

40 0.063952

50 0.066854

4° 0 ′ 0.069756 10 0.072658 20 0.075559

30 0.078459

40 0.081359

50 0.084258

5° 0 ′ 0.087156 10 0.090053 20 0.092950

30 0.095846

40 0.098741

50 0.101635

6° 0 ′ 0.104528 10 0.107421 20 0.110313

30 0.113203

40 0.116093

50 0.118982

7° 0 ′ 0.121869 10 0.124756 20 0.127642

7° 30 ′ 0.130526 cos

cos

tan

cot

1.000000

0.000000

—

0.999996

0.999983

0.999962

0.999932

0.999894

0.999848

0.999793

0.999729

0.999657

0.999577

0.999488

0.999391

0.999285

0.999171

0.999048

0.998917

0.998778

0.998630

0.998473

0.002909

0.005818

171.8854

0.011636

85.93979

0.008727 0.014545

0.017455

0.020365

0.023275

0.026186

0.029097

0.032009

0.034921

0.037834

0.040747

0.043661

0.046576

0.049491

0.052408

0.055325

0.998308

0.058243

0.997953

0.064083

0.998135

0.997763

0.997564

0.997357

0.997141

0.996917

0.996685

0.996444

0.996195

0.995937

0.995671

0.995396 0.995113

0.994822

0.994522

0.994214

0.993897

0.993572

0.993238

0.992896

0.992546

0.992187

0.991820

0.991445 sin

343.7737

0.061163

0.067004

0.069927

0.072851

0.075775

0.078702

0.081629

0.084558

0.087489

0.090421

0.093354

0.096289

0.099226

0.102164

0.105104

0.108046 0.110990

0.113936

0.116883

0.119833

0.122785

0.125738

0.128694

0.131652 cot

Angle

7° 30 ′ 0.130526

40

50 0.136292

50

30

114.5887

20

68.75009

10

49.10388

89° 0 ′ 50

57.28996

42.96408

40

30

38.18846

20

34.36777

31.24158

10

26.43160

88° 0 ′ 50

28.63625

24.54176

40

30

22.90377

20

21.47040

20.20555

10

18.07498

87° 0 ′ 50

19.08114

17.16934

40

30

16.34986

20

15.60478

14.92442

10

13.72674

86° 0 ′ 50

12.70621

30

14.30067

13.19688

40

20

12.25051 11.82617

10

11.05943

85° 0 ′ 50

11.43005

10.71191

40

30

10.38540

20

10.07803

9.788173

10

9.255304

84° 0 ′ 50

9.514364

9.009826

40

30

8.776887

20

8.555547

8.344956

10

7.953022

83° 0 ′ 50

8.144346

7.770351

7.595754 tan

sin

90° 0 ′

40

82° 30 Angle

40 0.133410

8° 0 ′ 0.139173 10 0.142053 20 0.144932

30 0.147809

40 0.150686

50 0.153561

9° 0 ′ 0.156434 10 0.159307 20 0.162178

30 0.165048

40 0.167916

50 0.170783

10° 0 ′ 0.173648 10 0.176512 20 0.179375

30 0.182236

40 0.185095

50 0.187953

11° 0 ′ 0.190809 10 0.193664 20 0.196517

30 0.199368

40 0.202218

50 0.205065

12° 0 ′ 0.207912 10 0.210756 20 0.213599

30 0.216440

40 0.219279

50 0.222116

13° 0 ′ 0.224951 10 0.227784 20 0.230616

30 0.233445

40 0.236273

50 0.239098

14° 0 ′ 0.241922 10 0.244743 20 0.247563

30 0.250380

40 0.253195

50 0.256008

15° 0 ′ 0.258819 cos

cos

tan

cot

0.991445

0.131652

7.595754

82° 30 ′

0.137576

7.268725

10

0.143508

6.968234

82° 0 ′ 50

6.691156

30

0.991061

0.990669

0.990268

0.989859

0.989442

0.989016

0.988582

0.988139

0.987688

0.987229

0.986762

0.986286

0.985801

0.985309

0.984808

0.984298

0.983781

0.983255

0.982721

0.982178

0.981627

0.981068

0.980500

0.979925

0.979341

0.134613

0.140541

0.146478

0.149451

0.152426

0.155404

0.158384

0.161368

0.164354

0.167343

0.170334

0.173329

0.176327

0.179328

0.182332

0.185339

0.188349

0.191363

0.194380

0.197401

0.200425

0.203452

0.206483

7.428706 7.115370

6.826944 6.560554

20

40

20

6.434843

10

6.197028

81° 0 ′ 50

5.975764

30

6.313752

6.084438

5.870804

40

20

5.769369

10

5.576379

80° 0 ′ 50

5.395517

30

5.671282

5.484505

5.309279

40

20

5.225665

10

5.065835

79° 0 ′ 50

4.915157

30

5.144554

4.989403

4.843005

40

20

0.978748

0.209518

4.772857

10

0.977539

0.215599

4.638246

78° 0 ′ 50

4.510709

30

0.978148

0.976921

0.976296

0.975662

0.975020

0.974370

0.973712

0.973045

0.972370

0.971687

0.970995

0.970296

0.969588

0.968872

0.968148

0.967415

0.966675

0.965926 sin

0.212557

0.218645

0.221695

0.224748

4.704630

4.573629

4.449418

40

20

4.389694

10

4.274707

77° 0 ′ 50

0.240079

4.165300

30

0.246241

4.061070

10

3.961652

76° 0 ′ 50

3.866713

30

0.227806

0.230868

0.233934

0.237004

0.243157

0.249328

0.252420

0.255516

0.258618

0.261723

0.264834

0.267949 cot

4.331476

4.219332 4.112561

4.010781

3.913642

3.820828

3.775952

3.732051 tan

40

20

40

20

10 75° 0 ′ Angle

For angles 0° to 15° 0 ′ (angles found in a column to the left of the data), use the column labels at the top of the table; for angles 75° to 90° 0 ′ (angles found in a column to the right of the data), use the column labels at the bottom of the table.

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Machinery's Handbook, 31st Edition Trigonometry Tables

109

Table 2b. Trigonometric Values of Angles from 15° to 30° and 60° to 75° Angle

sin

cos

tan

cot

15° 0 ′ 0.258819 0.965926 0.267949 3.732051

10 0.261628 0.965169 0.271069 3.689093

Angle

sin

cos

tan

cot

75° 0 ′ 22° 30 ′ 0.382683 0.923880 0.414214 2.414214

20 0.264434 0.964404 0.274194 3.647047

40 0.385369 0.922762 0.417626 2.394489

30

23° 0 ′ 0.390731 0.920505 0.424475 2.355852 10 0.393407 0.919364 0.427912 2.336929

67° 0 ′ 50 30

40

30 0.267238 0.963630 0.277325 3.605884

40 0.270040 0.962849 0.280460 3.565575

20

50 0.272840 0.962059 0.283600 3.526094

67° 30

50

10

50 0.388052 0.921638 0.421046 2.375037

20 0.396080 0.918216 0.431358 2.318261

20 10

40

16° 0 ′ 0.275637 0.961262 0.286745 3.487414 10 0.278432 0.960456 0.289896 3.449512

74° 0 ′ 50

30 0.398749 0.917060 0.434812 2.299843

30 0.284015 0.958820 0.296213 3.375943

30

24° 0 ′ 0.406737 0.913545 0.445229 2.246037 10 0.409392 0.912358 0.448719 2.228568

66° 0 ′ 50 30

20 0.281225 0.959642 0.293052 3.412363

40

40 0.286803 0.957990 0.299380 3.340233

20

50 0.289589 0.957151 0.302553 3.305209

10

40 0.401415 0.915896 0.438276 2.281669 50 0.404078 0.914725 0.441748 2.263736

20 0.412045

0.911164

0.452218 2.211323

20 10

40

17° 0 ′ 0.292372 0.956305 0.305731 3.270853 10 0.295152 0.955450 0.308914 3.237144

73° 0 ′ 50

30 0.414693 0.909961 0.455726 2.194300

30 0.300706 0.953717 0.315299 3.171595

30

25° 0 ′ 0.422618 0.906308 0.466308 2.144507 10 0.425253 0.905075 0.469854 2.128321

65° 0 ′ 50 30

20 0.297930 0.954588 0.312104 3.204064

40

40 0.303479 0.952838 0.318500 3.139719

20

50 0.306249 0.951951 0.321707 3.108421

10

40 0.417338 0.908751 0.459244 2.177492 50 0.419980 0.907533 0.462771 2.160896

20 0.427884 0.903834 0.473410 2.112335

20 10

40

18° 0 ′ 0.309017 0.951057 0.324920 3.077684 10 0.311782 0.950154 0.328139 3.047492

72° 0 ′ 50

30 0.430511 0.902585 0.476976 2.096544

30 0.317305 0.948324 0.334595 2.988685

30

26° 0 ′ 0.438371 0.898794 0.487733 2.050304 10 0.440984 0.897515 0.491339 2.035256

64° 0 ′ 50 30

20 0.314545 0.949243 0.331364 3.017830

40

40 0.320062 0.947397 0.337833 2.960042

20

50 0.322816 0.946462 0.341077 2.931888

10

40 0.433135 0.901329 0.480551 2.080944 50 0.435755 0.900065 0.484137 2.065532

20 0.443593 0.896229 0.494955 2.020386

20 10

40

19° 0 ′ 0.325568 0.945519 0.344328 2.904211 10 0.328317 0.944568 0.347585 2.876997

71° 0 ′ 50

30 0.446198 0.894934 0.498582 2.005690

30 0.333807 0.942641 0.354119 2.823913

30

27° 0 ′ 0.453990 0.891007 0.509525 1.962611 10 0.456580 0.889682 0.513195 1.948577

63° 0 ′ 50 30

20 0.331063 0.943609 0.350848 2.850235

40

40 0.336547 0.941666 0.357396 2.798020

20

50 0.339285 0.940684 0.360679 2.772545

10

40 0.448799 0.893633 0.502219 1.991164

50 0.451397 0.892323 0.505867 1.976805

20 0.459166 0.888350 0.516875 1.934702

20 10

40

20° 0 ′ 0.342020 0.939693 0.363970 2.747477 10 0.344752 0.938694 0.367268 2.722808

70° 0 ′ 50

30 0.461749 0.887011 0.520567 1.920982

30 0.350207 0.936672 0.373885 2.674621

30

28° 0 ′ 0.469472 0.882948 0.531709 1.880726 10 0.472038 0.881578 0.535446 1.867600

62° 0 ′ 50 30

20 0.347481 0.937687 0.370573 2.698525

40

40 0.352931 0.935650 0.377204 2.651087

20

50 0.355651 0.934619 0.380530 2.627912

10

40 0.464327 0.885664 0.524270 1.907415 50 0.466901 0.884309 0.527984 1.893997

20 0.474600 0.880201 0.539195 1.854616

20 10

40

21° 0 ′ 0.358368 0.933580 0.383864 2.605089 10 0.361082 0.932534 0.387205 2.582609

69° 0 ′ 50

30 0.477159 0.878817 0.542956 1.841771

30 0.366501 0.930418 0.393910 2.538648

30

29° 0 ′ 0.484810 0.874620 0.554309 1.804048 10 0.487352 0.873206 0.558118 1.791736

61° 0 ′ 50 30

20 0.363793 0.931480 0.390554 2.560465

40

40 0.369206 0.929348 0.397275 2.517151

20

50 0.371908 0.928270 0.400646 2.495966

10

40 0.479713 0.877425 0.546728 1.829063 50 0.482263 0.876026 0.550513 1.816489

20 0.489890 0.871784 0.561939 1.779552

22° 0 ′ 0.374607 0.927184 0.404026 2.475087 10 0.377302 0.926090 0.407414 2.454506

68° 0 ′ 50

30 0.492424 0.870356 0.565773 1.767494

22° 30 0.382683 0.923880 0.414214 2.414214 cos sin cot tan

67° 30 Angle

30° 0 ′ 0.500000 0.866025 0.577350 1.732051 cos sin cot tan

20 0.379994 0.924989 0.410810 2.434217

40

40 0.494953 0.868920 0.569619 1.755559 50 0.497479 0.867476 0.573478 1.743745

20 10

40 20 10

60° 0 ′ Angle

For angles 15° to 30° 0 ′ (angles found in a column to the left of the data), use the column labels at the top of the table; for angles 60° to 75° 0 ′ (angles found in a column to the right of the data), use the column labels at the bottom of the table.

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Machinery's Handbook, 31st Edition Trigonometry Tables

110

Table 2c. Trigonometric Values of Angles from 30° to 60° Angle

sin

cos

tan

cot

30° 0 ′ 0.500000 0.866025 0.577350 1.732051 10 0.502517 0.864567 0.581235 1.720474

Angle

20 0.505030 0.863102 0.585134 1.709012

30 0.507538 0.861629 0.589045 1.697663

30 0.622515 0.782608 0.795436 1.257172

30

30

20

10 58° 0 ′ 50 40

30 0.537300 0.843391 0.637070 1.569686

30

40 0.539751 0.841825 0.641167 1.559655

20

50 0.542197 0.840251 0.645280 1.549715

20 0.549509 0.835488 0.657710 1.520426

52° 0 ′ 50

40

50 0.527450 0.849586 0.620832 1.610742

33° 0 ′ 0.544639 0.838671 0.649408 1.539865 10 0.547076 0.837083 0.653551 1.530102

38° 0 ′ 0.615661 0.788011 0.781286 1.279942 10 0.617951 0.786217 0.785981 1.272296

10

30 0.522499 0.852640 0.612801 1.631852

20 0.534844 0.844951 0.632988 1.579808

10 57° 0 ′ 50 40

30 0.551937 0.833886 0.661886 1.510835

30

40 0.554360 0.832277 0.666077 1.501328

20

50 0.556779 0.830661 0.670284 1.491904

10

34° 0 ′ 0.559193 0.829038 0.674509 1.482561 10 0.561602 0.827407 0.678749 1.473298

56° 0 ′ 50

30 0.566406 0.824126 0.687281 1.455009

30

20 0.564007 0.825770 0.683007 1.464115

40

40 0.568801 0.822475 0.691572 1.445980

20

50 0.571191 0.820817 0.695881 1.437027

35° 0 ′ 0.573576 0.819152 0.700208 1.428148 10 0.575957 0.817480 0.704551 1.419343 20 0.578332 0.815801 0.708913 1.410610

10 55° 0 ′ 50 40

30 0.580703 0.814116 0.713293 1.401948

30

40 0.583069 0.812423 0.717691 1.393357

20

50 0.585429 0.810723 0.722108 1.384835

36° 0 ′ 0.587785 0.809017 0.726543 1.376382 10 0.590136 0.807304 0.730996 1.367996 20 0.592482 0.805584 0.735469 1.359676

cot

30

59° 0 ′ 50

40 0.524977 0.851117 0.616809 1.621247

32° 0 ′ 0.529919 0.848048 0.624869 1.600335 10 0.532384 0.846503 0.628921 1.590024

tan

40 0.611067 0.791579 0.771959 1.295406

20

50 0.512543 0.858662 0.596908 1.675299

20 0.520016 0.854156 0.608807 1.642558

cos

50 40

40 0.510043 0.860149 0.592970 1.686426

31° 0 ′ 0.515038 0.857167 0.600861 1.664279 10 0.517529 0.855665 0.604827 1.653366

sin

60° 0 ′ 37° 30 ′ 0.608761 0.793353 0.767327 1.303225 52° 30 ′

10 54° 0 ′ 50 40

30 0.594823 0.803857 0.739961 1.351422

30

40 0.597159 0.802123 0.744472 1.343233

20

50 0.599489 0.800383 0.749003 1.335108

10

50 0.613367 0.789798 0.776612 1.287645

20 0.620235 0.784416 0.790697 1.264706

40 0.624789 0.780794 0.800196 1.249693

50 0.627057 0.778973 0.804979 1.242268

20

10

40

20

10

39° 0 ′ 0.629320 0.777146 0.809784 1.234897 10 0.631578 0.775312 0.814612 1.227579

51° 0 ′ 50

30 0.636078 0.771625 0.824336 1.213097

30

20 0.633831 0.773472 0.819463 1.220312

40 0.638320 0.769771 0.829234 1.205933

50 0.640557 0.767911 0.834155 1.198818

40

20

10

40° 0 ′ 0.642788 0.766044 0.839100 1.191754 10 0.645013 0.764171 0.844069 1.184738

50° 0 ′ 50

30 0.649448 0.760406 0.854081 1.170850

30

20 0.647233 0.762292 0.849062 1.177770

40 0.651657 0.758514 0.859124 1.163976

50 0.653861 0.756615 0.864193 1.157149

40

20

10

41° 0 ′ 0.656059 0.754710 0.869287 1.150368 10 0.658252 0.752798 0.874407 1.143633

49° 0 ′ 50

30 0.662620 0.748956 0.884725 1.130294

30

20 0.660439 0.750880 0.879553 1.136941

40 0.664796 0.747025 0.889924 1.123691

50 0.666966 0.745088 0.895151 1.117130

40

20

10

42° 0 ′ 0.669131 0.743145 0.900404 1.110613 10 0.671289 0.741195 0.905685 1.104137

48° 0 ′ 50

30 0.675590 0.737277 0.916331 1.091309

30

20 0.673443 0.739239 0.910994 1.097702

40 0.677732 0.735309 0.921697 1.084955

50 0.679868 0.733334 0.927091 1.078642

40

20

10

43° 0 ′ 0.681998 0.731354 0.932515 1.072369 10 0.684123 0.729367 0.937968 1.066134

47° 0 ′ 50

30 0.688355 0.725374 0.948965 1.053780

30

20 0.686242 0.727374 0.943451 1.059938

40 0.690462 0.723369 0.954508 1.047660

50 0.692563 0.721357 0.960083 1.041577

40

20

10

44° 0 ′ 0.694658 0.719340 0.965689 1.035530 10 0.696748 0.717316 0.971326 1.029520

46° 0 ′ 50

30 0.700909 0.713250 0.982697 1.017607

30

20 0.698832 0.715286 0.976996 1.023546

40

37° 0 ′ 0.601815 0.798636 0.753554 1.327045 10 0.604136 0.796882 0.758125 1.319044

53° 0 ′ 50

40 0.702981 0.711209 0.988432 1.011704

20

37° 30 0.608761 0.793353 0.767327 1.303225 cos sin cot tan

52° 30 Angle

45° 0 ′ 0.707107 0.707107 1.000000 1.000000 cos sin cot tan

45° 0 ′ Angle

20 0.606451 0.795121 0.762716 1.311105

40

50 0.705047 0.709161 0.994199 1.005835

10

For angles 30° to 45° 0 ′ (angles found in a column to the left of the data), use the column labels at the top of the table; for angles 45° to 60° 0 ′ (angles found in a column to the right of the data), use the column labels at the bottom of the table.

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Machinery's Handbook, 31st Edition TRIGONOMETRIC FUNCTIONS

111

Using a Calculator to Find Trigonometric Function Values.—Scientific calculators are quicker and more accurate than tables for finding trigonometric ratios or angles than relying on trigonometric tables. Inputting an angle, in either degree (DEG) or radian (RAD) measure, and pressing SIN, COS, or TAN key will produce the ratio value to a many decimal places, which can be rounded to the desired accuracy. Though reciprocal function keys are not usually included on the calculator, using the main three functions and the 1/x key will produce these ratios as well, since csc x = 1/sin x, sec x = 1/cos x, and cot x = 1/tan x. If the triangle’s side dimensions are known and the angle measure is sought, then ratio is entered using the keys labeled sin –1, cos –1, and tan –1. If these keys are not present, then INV is used with SIN, COS, or TAN. Again, the correct units, whether degree or radian measure, can be chosen. An advantage of using a calculator instead of a trigonometry table to find function values is that both positive and negative degree measures can be entered into a calculator; also, angles greater than 90 degrees do not need to be converted to their acute angle equivalent. Interpolation for angles whose measures fall between the values available in the tables is also not necessary. Example: Use a calculator to find all six of the trigonometry function values, to four decimal places, of 172°. Solution: The degree measure can be entered as is, without having to first inspect its quadrant and reference (acute) angle. Enter 172 in DEG mode, then, for the three main values, choose SIN, COS, TAN, each time rounding answer to four decimal places. sin(172°) = 0.1392 csc(172°) = 1/sin(172°) = 7.1853 cos(172°) = –0.9903 sec(172°) = 1/cos(172°) = –1.0098 tan(172°) = –0.1405 cot(172°) = 1/tan(172°) = –7.1153 If a scientific calculator or computer is not available, tables are the easiest way to find trig values. However, trigonometric function values can be calculated very accurately without a scien­tific calculator by using the infinite series formulas (see CALCULUS): A 7- … A 3- + ----A 5- – ----sin A = A – ----± 3! 5! 7!

6 A 2- + ----A 4- – A ------ ± … cos A = 1 – ----2! 4! 6!

3 1 3 A5 3 5 A7 1 ------ + --- × --- × ----- + … tan– 1 A = A – A sin– 1 A = A + --- × A ------ + A ------ – ------ ± … 2 3 2 4 5 7 3 5

where angle A is expressed in radians (multiplying degrees by p/180 = 0.0174533 gives radian measure). Generally, calculating just three or four terms of the expression is sufficient for accuracy. In these formulas, a number followed by the symbol ! is called a factorial (see Factorial Notation on page 13). Except for 0!, which equals 1, n! = n(n – 1)(n – 2) . . . down to 1. For example, 4! = 4 × 3 × 2 × 1 = 24. As an example, sin 42° = sin (42 × 0.0174533) = sin (0.733) = 0.733 – (0.733)3/3! + (0.733)5/5! – . . . ≈ 0.66912, which is close to the calculator answer of higher accuracy, 0.669130606. Versed Sine and Versed Cosine.—These functions are sometimes used in formulas for segments of a circle and may be obtained using the relationships: versed sin θ = 1 – cos θ; versed cos θ = 1 – sin θ . Sevolute Functions.—Sevolute functions are used in calculating the form diameter of involute splines. They are computed by subtracting the involute function of an angle from the secant of the angle (1/cos θ = sec θ). For example, sevolute of 20° = secant of 20° - involute function of 20° = 1.064178 - 0.014904 = 1.049274. Involute Functions.—Involute functions are used in certain formulas relating to the design and measurement of gear teeth as well as measurement of threads over wires. (See, for example, pages 2130 through 2133, 2286, and 2350). The value of an involute function is calculated from the following formulas: Involute of θ = (tan θ) – θ for θ in radians, and Involute of θ = (tan θ) – π × θ/180 for θ in degrees. Example: For an angle of 14 degrees and 10 minutes (14° 10´), the involute function is found as follows: 10 minutes = 10 ⁄ 60 = 0.166666°, 14 + 0.166666 = 14.166666°, so that the involute of 14.166666° = (tan 14.166666) – π × 14.166666 ⁄ 180 = 0.252420 – 0.247255 = 0.005165. The same result would be obtained by using the conversion tables beginning on page 112 to convert 14° 10´ to radians and then applying the first involute formula for radian measure given above.

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Machinery's Handbook, 31st Edition Involute Function Tables

112

Involute Function Values for Angles from 14 to 23 Degrees Minutes 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

14

15

16

0.004982 0.005000 0.005018 0.005036 0.005055 0.005073 0.005091 0.005110 0.005128 0.005146 0.005165 0.005184 0.005202 0.005221 0.005239 0.005258 0.005277 0.005296 0.005315 0.005334 0.005353 0.005372 0.005391 0.005410 0.005429 0.005448 0.005467 0.005487 0.005506 0.005525 0.005545 0.005564 0.005584 0.005603 0.005623 0.005643 0.005662 0.005682 0.005702 0.005722 0.005742 0.005762 0.005782 0.005802 0.005822 0.005842 0.005862 0.005882 0.005903 0.005923 0.005943 0.005964 0.005984 0.006005 0.006025 0.006046 0.006067 0.006087 0.006108 0.006129 0.006150

0.006150 0.006171 0.006192 0.006213 0.006234 0.006255 0.006276 0.006297 0.006318 0.006340 0.006361 0.006382 0.006404 0.006425 0.006447 0.006469 0.006490 0.006512 0.006534 0.006555 0.006577 0.006599 0.006621 0.006643 0.006665 0.006687 0.006709 0.006732 0.006754 0.006776 0.006799 0.006821 0.006843 0.006866 0.006888 0.006911 0.006934 0.006956 0.006979 0.007002 0.007025 0.007048 0.007071 0.007094 0.007117 0.007140 0.007163 0.007186 0.007209 0.007233 0.007256 0.007280 0.007303 0.007327 0.007350 0.007374 0.007397 0.007421 0.007445 0.007469 0.007493

0.007493 0.007517 0.007541 0.007565 0.007589 0.007613 0.007637 0.007661 0.007686 0.007710 0.007735 0.007759 0.007784 0.007808 0.007833 0.007857 0.007882 0.007907 0.007932 0.007957 0.007982 0.008007 0.008032 0.008057 0.008082 0.008107 0.008133 0.008158 0.008183 0.008209 0.008234 0.008260 0.008285 0.008311 0.008337 0.008362 0.008388 0.008414 0.008440 0.008466 0.008492 0.008518 0.008544 0.008571 0.008597 0.008623 0.008650 0.008676 0.008702 0.008729 0.008756 0.008782 0.008809 0.008836 0.008863 0.008889 0.008916 0.008943 0.008970 0.008998 0.009025

Degrees 17 18 19 Involute Function Values 0.009025 0.010760 0.012715 0.009052 0.010791 0.012750 0.009079 0.010822 0.012784 0.009107 0.010853 0.012819 0.009134 0.010884 0.012854 0.009161 0.010915 0.012888 0.009189 0.010946 0.012923 0.009216 0.010977 0.012958 0.009244 0.011008 0.012993 0.009272 0.011039 0.013028 0.009299 0.011071 0.013063 0.009327 0.011102 0.013098 0.009355 0.011133 0.013134 0.009383 0.011165 0.013169 0.009411 0.011196 0.013204 0.009439 0.011228 0.013240 0.009467 0.011260 0.013275 0.009495 0.011291 0.013311 0.009523 0.011323 0.013346 0.009552 0.011355 0.013382 0.009580 0.011387 0.013418 0.009608 0.011419 0.013454 0.009637 0.011451 0.013490 0.009665 0.011483 0.013526 0.009694 0.011515 0.013562 0.009722 0.011547 0.013598 0.009751 0.011580 0.013634 0.009780 0.011612 0.013670 0.009808 0.011644 0.013707 0.009837 0.011677 0.013743 0.009866 0.011709 0.013779 0.009895 0.011742 0.013816 0.009924 0.011775 0.013852 0.009953 0.011807 0.013889 0.009982 0.011840 0.013926 0.010011 0.011873 0.013963 0.010041 0.011906 0.013999 0.010070 0.011939 0.014036 0.010099 0.011972 0.014073 0.010129 0.012005 0.014110 0.010158 0.012038 0.014148 0.010188 0.012071 0.014185 0.010217 0.012105 0.014222 0.010247 0.012138 0.014259 0.010277 0.012172 0.014297 0.010307 0.012205 0.014334 0.010336 0.012239 0.014372 0.010366 0.012272 0.014409 0.010396 0.012306 0.014447 0.010426 0.012340 0.014485 0.010456 0.012373 0.014523 0.010486 0.012407 0.014560 0.010517 0.012441 0.014598 0.010547 0.012475 0.014636 0.010577 0.012509 0.014674 0.010608 0.012543 0.014713 0.010638 0.012578 0.014751 0.010669 0.012612 0.014789 0.010699 0.012646 0.014827 0.010730 0.012681 0.014866 0.010760 0.012715 0.014904

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21

22

0.014904 0.014943 0.014982 0.015020 0.015059 0.015098 0.015137 0.015176 0.015215 0.015254 0.015293 0.015333 0.015372 0.015411 0.015451 0.015490 0.015530 0.015570 0.015609 0.015649 0.015689 0.015729 0.015769 0.015809 0.015850 0.015890 0.015930 0.015971 0.016011 0.016052 0.016092 0.016133 0.016174 0.016215 0.016255 0.016296 0.016337 0.016379 0.016420 0.016461 0.016502 0.016544 0.016585 0.016627 0.016669 0.016710 0.016752 0.016794 0.016836 0.016878 0.016920 0.016962 0.017004 0.017047 0.017089 0.017132 0.017174 0.017217 0.017259 0.017302 0.017345

0.017345 0.017388 0.017431 0.017474 0.017517 0.017560 0.017603 0.017647 0.017690 0.017734 0.017777 0.017821 0.017865 0.017908 0.017952 0.017996 0.018040 0.018084 0.018129 0.018173 0.018217 0.018262 0.018306 0.018351 0.018395 0.018440 0.018485 0.018530 0.018575 0.018620 0.018665 0.018710 0.018755 0.018800 0.018846 0.018891 0.018937 0.018983 0.019028 0.019074 0.019120 0.019166 0.019212 0.019258 0.019304 0.019350 0.019397 0.019443 0.019490 0.019536 0.019583 0.019630 0.019676 0.019723 0.019770 0.019817 0.019864 0.019912 0.019959 0.020006 0.020054

0.020054 0.020101 0.020149 0.020197 0.020244 0.020292 0.020340 0.020388 0.020436 0.020484 0.020533 0.020581 0.020629 0.020678 0.020726 0.020775 0.020824 0.020873 0.020921 0.020970 0.021019 0.021069 0.021118 0.021167 0.021217 0.021266 0.021316 0.021365 0.021415 0.021465 0.021514 0.021564 0.021614 0.021665 0.021715 0.021765 0.021815 0.021866 0.021916 0.021967 0.022018 0.022068 0.022119 0.022170 0.022221 0.022272 0.022324 0.022375 0.022426 0.022478 0.022529 0.022581 0.022633 0.022684 0.022736 0.022788 0.022840 0.022892 0.022944 0.022997 0.023049

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Machinery's Handbook, 31st Edition Involute Function Tables

113

Involute Function Values for Angles from 23 to 32 Degrees Minutes 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

23

24

25

0.023049 0.023102 0.023154 0.023207 0.023259 0.023312 0.023365 0.023418 0.023471 0.023524 0.023577 0.023631 0.023684 0.023738 0.023791 0.023845 0.023899 0.023952 0.024006 0.024060 0.024114 0.024169 0.024223 0.024277 0.024332 0.024386 0.024441 0.024495 0.024550 0.024605 0.024660 0.024715 0.024770 0.024825 0.024881 0.024936 0.024992 0.025047 0.025103 0.025159 0.025214 0.025270 0.025326 0.025382 0.025439 0.025495 0.025551 0.025608 0.025664 0.025721 0.025778 0.025834 0.025891 0.025948 0.026005 0.026062 0.026120 0.026177 0.026235 0.026292 0.026350

0.026350 0.026407 0.026465 0.026523 0.026581 0.026639 0.026697 0.026756 0.026814 0.026872 0.026931 0.026989 0.027048 0.027107 0.027166 0.027225 0.027284 0.027343 0.027402 0.027462 0.027521 0.027581 0.027640 0.027700 0.027760 0.027820 0.027880 0.027940 0.028000 0.028060 0.028121 0.028181 0.028242 0.028302 0.028363 0.028424 0.028485 0.028546 0.028607 0.028668 0.028729 0.028791 0.028852 0.028914 0.028976 0.029037 0.029099 0.029161 0.029223 0.029285 0.029348 0.029410 0.029472 0.029535 0.029598 0.029660 0.029723 0.029786 0.029849 0.029912 0.029975

0.029975 0.030039 0.030102 0.030166 0.030229 0.030293 0.030357 0.030420 0.030484 0.030549 0.030613 0.030677 0.030741 0.030806 0.030870 0.030935 0.031000 0.031065 0.031130 0.031195 0.031260 0.031325 0.031390 0.031456 0.031521 0.031587 0.031653 0.031718 0.031784 0.031850 0.031917 0.031983 0.032049 0.032116 0.032182 0.032249 0.032315 0.032382 0.032449 0.032516 0.032583 0.032651 0.032718 0.032785 0.032853 0.032920 0.032988 0.033056 0.033124 0.033192 0.033260 0.033328 0.033397 0.033465 0.033534 0.033602 0.033671 0.033740 0.033809 0.033878 0.033947

Degrees 26 27 28 Involute Function Values 0.033947 0.038287 0.043017 0.034016 0.038362 0.043100 0.034086 0.038438 0.043182 0.034155 0.038514 0.043264 0.034225 0.038590 0.043347 0.034294 0.038666 0.043430 0.034364 0.038742 0.043513 0.034434 0.038818 0.043596 0.034504 0.038894 0.043679 0.034574 0.038971 0.043762 0.034644 0.039047 0.043845 0.034714 0.039124 0.043929 0.034785 0.039201 0.044012 0.034855 0.039278 0.044096 0.034926 0.039355 0.044180 0.034997 0.039432 0.044264 0.035067 0.039509 0.044348 0.035138 0.039586 0.044432 0.035209 0.039664 0.044516 0.035280 0.039741 0.044601 0.035352 0.039819 0.044685 0.035423 0.039897 0.044770 0.035494 0.039974 0.044855 0.035566 0.040052 0.044940 0.035637 0.040131 0.045024 0.035709 0.040209 0.045110 0.035781 0.040287 0.045195 0.035853 0.040366 0.045280 0.035925 0.040444 0.045366 0.035997 0.040523 0.045451 0.036069 0.040602 0.045537 0.036142 0.040680 0.045623 0.036214 0.040759 0.045709 0.036287 0.040839 0.045795 0.036359 0.040918 0.045881 0.036432 0.040997 0.045967 0.036505 0.041077 0.046054 0.036578 0.041156 0.046140 0.036651 0.041236 0.046227 0.036724 0.041316 0.046313 0.036798 0.041395 0.046400 0.036871 0.041475 0.046487 0.036945 0.041556 0.046575 0.037018 0.041636 0.046662 0.037092 0.041716 0.046749 0.037166 0.041797 0.046837 0.037240 0.041877 0.046924 0.037314 0.041958 0.047012 0.037388 0.042039 0.047100 0.037462 0.042120 0.047188 0.037537 0.042201 0.047276 0.037611 0.042282 0.047364 0.037686 0.042363 0.047452 0.037761 0.042444 0.047541 0.037835 0.042526 0.047630 0.037910 0.042608 0.047718 0.037985 0.042689 0.047807 0.038060 0.042771 0.047896 0.038136 0.042853 0.047985 0.038211 0.042935 0.048074 0.038287 0.043017 0.048164

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30

31

0.048164 0.048253 0.048343 0.048432 0.048522 0.048612 0.048702 0.048792 0.048883 0.048973 0.049064 0.049154 0.049245 0.049336 0.049427 0.049518 0.049609 0.049701 0.049792 0.049884 0.049976 0.050068 0.050160 0.050252 0.050344 0.050437 0.050529 0.050622 0.050715 0.050808 0.050901 0.050994 0.051087 0.051181 0.051274 0.051368 0.051462 0.051556 0.051650 0.051744 0.051838 0.051933 0.052027 0.052122 0.052217 0.052312 0.052407 0.052502 0.052597 0.052693 0.052788 0.052884 0.052980 0.053076 0.053172 0.053268 0.053365 0.053461 0.053558 0.053655 0.053752

0.053752 0.053849 0.053946 0.054043 0.054140 0.054238 0.054336 0.054433 0.054531 0.054629 0.054728 0.054826 0.054924 0.055023 0.055122 0.055221 0.055320 0.055419 0.055518 0.055617 0.055717 0.055817 0.055916 0.056016 0.056116 0.056217 0.056317 0.056417 0.056518 0.056619 0.056720 0.056821 0.056922 0.057023 0.057124 0.057226 0.057328 0.057429 0.057531 0.057633 0.057736 0.057838 0.057940 0.058043 0.058146 0.058249 0.058352 0.058455 0.058558 0.058662 0.058765 0.058869 .058973 0.059077 0.059181 0.059285 0.059390 0.059494 0.059599 0.059704 0.059809

0.059809 0.059914 0.060019 0.060124 0.060230 0.060335 0.060441 0.060547 0.060653 0.060759 0.060866 0.060972 0.061079 0.061186 0.061292 0.061400 0.061507 0.061614 0.061721 0.061829 0.061937 0.062045 0.062153 0.062261 0.062369 0.062478 0.062586 0.062695 0.062804 0.062913 0.063022 0.063131 0.063241 0.063350 0.063460 0.063570 0.063680 0.063790 0.063901 0.064011 0.064122 0.064232 0.064343 0.064454 0.064565 0.064677 0.064788 0.064900 0.065012 0.065123 0.065236 0.065348 0.065460 0.065573 0.065685 0.065798 0.065911 0.066024 0.066137 0.066251 0.066364

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Machinery's Handbook, 31st Edition Involute Function Tables

114

Involute Function Values for Angles from 32 to 41 Degrees Minutes 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

32

33

34

0.066364 0.066478 0.066591 0.066705 0.066820 0.066934 0.067048 0.067163 0.067277 0.067392 0.067507 0.067622 0.067738 0.067853 0.067969 0.068084 0.068200 0.068316 0.068432 0.068549 0.068665 0.068782 0.068899 0.069016 0.069133 0.069250 0.069367 0.069485 0.069602 0.069720 0.069838 0.069956 0.070075 0.070193 0.070312 0.070430 0.070549 0.070668 0.070788 0.070907 0.071026 0.071146 0.071266 0.071386 0.071506 0.071626 0.071747 0.071867 0.071988 0.072109 0.072230 0.072351 0.072473 0.072594 0.072716 0.072838 0.072960 0.073082 0.073204 0.073326 0.073449

0.073449 0.073572 0.073695 0.073818 0.073941 0.074064 0.074188 0.074312 0.074435 0.074559 0.074684 0.074808 0.074932 0.075057 0.075182 0.075307 0.075432 0.075557 0.075683 0.075808 0.075934 0.076060 0.076186 0.076312 0.076439 0.076565 0.076692 0.076819 0.076946 0.077073 0.077200 0.077328 0.077455 0.077583 0.077711 0.077839 0.077968 0.078096 0.078225 0.078354 0.078483 0.078612 0.078741 0.078871 0.079000 0.079130 0.079260 0.079390 0.079520 0.079651 0.079781 0.079912 0.080043 0.080174 0.080306 0.080437 0.080569 0.080700 0.080832 0.080964 0.081097

0.081097 0.081229 0.081362 0.081494 0.081627 0.081760 0.081894 0.082027 0.082161 0.082294 0.082428 0.082562 0.082697 0.082831 0.082966 0.083101 0.083235 0.083371 0.083506 0.083641 0.083777 0.083913 0.084049 0.084185 0.084321 0.084458 0.084594 0.084731 0.084868 0.085005 0.085142 0.085280 0.085418 0.085555 0.085693 0.085832 0.085970 0.086108 0.086247 0.086386 0.086525 0.086664 0.086804 0.086943 0.087083 0.087223 0.087363 0.087503 0.087644 0.087784 0.087925 0.088066 0.088207 0.088348 0.088490 0.088631 0.088773 0.088915 0.089057 0.089200 0.089342

Degrees 35 36 37 Involute Function Values 0.089342 0.098224 0.107782 0.089485 0.098378 0.107948 0.089628 0.098532 0.108113 0.089771 0.098686 0.108279 0.089914 0.098840 0.108445 0.090058 0.098994 0.108611 0.090201 0.099149 0.108777 0.090345 0.099303 0.108943 0.090489 0.099458 0.109110 0.090633 0.099614 0.109277 0.090777 0.099769 0.109444 0.090922 0.099924 0.109611 0.091067 0.100080 0.109779 0.091211 0.100236 0.109947 0.091356 0.100392 0.110114 0.091502 0.100549 0.110283 0.091647 0.100705 0.110451 0.091793 0.100862 0.110619 0.091938 0.101019 0.110788 0.092084 0.101176 0.110957 0.092230 0.101333 0.111126 0.092377 0.101490 0.111295 0.092523 0.101648 0.111465 0.092670 0.101806 0.111635 0.092816 0.101964 0.111805 0.092963 0.102122 0.111975 0.093111 0.102280 0.112145 0.093258 0.102439 0.112316 0.093406 0.102598 0.112486 0.093553 0.102757 0.112657 0.093701 0.102916 0.112829 0.093849 0.103075 0.113000 0.093998 0.103235 0.113172 0.094146 0.103395 0.113343 0.094295 0.103555 0.113515 0.094443 0.103715 0.113688 0.094593 0.103875 0.113860 0.094742 0.104036 0.114033 0.094891 0.104196 0.114205 0.095041 0.104357 0.114378 0.095190 0.104518 0.114552 0.095340 0.104680 0.114725 0.095490 0.104841 0.114899 0.095641 0.105003 0.115073 0.095791 0.105165 0.115247 0.095942 0.105327 0.115421 0.096093 0.105489 0.115595 0.096244 0.105652 0.115770 0.096395 0.105814 0.115945 0.096546 0.105977 0.116120 0.096698 0.106140 0.116296 0.096850 0.106304 0.116471 0.097002 0.106467 0.116647 0.097154 0.106631 0.116823 0.097306 0.106795 0.116999 0.097459 0.106959 0.117175 0.097611 0.107123 0.117352 0.097764 0.107288 0.117529 0.097917 0.107452 0.117706 0.098071 0.107617 0.117883 0.098224 0.107782 0.118061

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40

0.118061 0.118238 0.118416 0.118594 0.118773 0.118951 0.119130 0.119309 0.119488 0.119667 0.119847 0.120027 0.120207 0.120387 0.120567 0.120748 0.120929 0.121110 0.121291 0.121473 0.121655 0.121837 0.122019 0.122201 0.122384 0.122567 0.122750 0.122933 0.123117 0.123300 0.123484 0.123668 0.123853 0.124037 0.124222 0.124407 0.124592 0.124778 0.124964 0.125150 0.125336 0.125522 0.125709 0.125896 0.126083 0.126270 0.126457 0.126645 0.126833 0.127021 0.127209 0.127398 0.127587 0.127776 0.127965 0.128155 0.128344 0.128534 0.128725 0.128915 0.129106

0.129106 0.129297 0.129488 0.129679 0.129870 0.130062 0.130254 0.130446 0.130639 0.130832 0.131025 0.131218 0.131411 0.131605 0.131799 0.131993 0.132187 0.132381 0.132576 0.132771 0.132966 0.133162 0.133358 0.133553 0.133750 0.133946 0.134143 0.134339 0.134537 0.134734 0.134931 0.135129 0.135327 0.135525 0.135724 0.135923 0.136122 0.136321 0.136520 0.136720 0.136920 0.137120 0.137320 0.137521 0.137722 0.137923 0.138124 0.138326 0.138528 0.138730 0.138932 0.139134 0.139337 0.139540 0.139743 0.139947 0.140151 0.140355 0.140559 0.140763 0.140968

0.140968 0.141173 0.141378 0.141584 0.141789 0.141995 0.142201 0.142408 0.142614 0.142821 0.143028 0.143236 0.143443 0.143651 0.143859 0.144068 0.144276 0.144485 0.144694 0.144903 0.145113 0.145323 0.145533 0.145743 0.145954 0.146165 0.146376 0.146587 0.146799 0.147010 0.147222 0.147435 0.147647 0.147860 0.148073 0.148286 0.148500 0.148714 0.148928 0.149142 0.149357 0.149572 0.149787 0.150002 0.150218 0.150434 0.150650 0.150866 0.151083 0.151299 0.151517 0.151734 0.151952 0.152169 0.152388 0.152606 0.152825 0.153044 0.153263 0.153482 0.153702

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Machinery's Handbook, 31st Edition Involute Function Tables

115

Involute Function Values for Angles from 41 to 50 Degrees Minutes 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

41

42

43

0.153702 0.153922 0.154142 0.154362 0.154583 0.154804 0.155025 0.155247 0.155469 0.155691 0.155913 0.156135 0.156358 0.156581 0.156805 0.157028 0.157252 0.157476 0.157701 0.157925 0.158150 0.158375 0.158601 0.158826 0.159052 0.159279 0.159505 0.159732 0.159959 0.160186 0.160414 0.160642 0.160870 0.161098 0.161327 0.161555 0.161785 0.162014 0.162244 0.162474 0.162704 0.162934 0.163165 0.163396 0.163628 0.163859 0.164091 0.164323 0.164556 0.164788 0.165021 0.165254 0.165488 0.165722 0.165956 0.166190 0.166425 0.166660 0.166895 0.167130 0.167366

0.167366 0.167602 0.167838 0.168075 0.168311 0.168548 0.168786 0.169023 0.169261 0.169500 0.169738 0.169977 0.170216 0.170455 0.170695 0.170935 0.171175 0.171415 0.171656 0.171897 0.172138 0.172380 0.172621 0.172864 0.173106 0.173349 0.173592 0.173835 0.174078 0.174322 0.174566 0.174811 0.175055 0.175300 0.175546 0.175791 0.176037 0.176283 0.176529 0.176776 0.177023 0.177270 0.177518 0.177766 0.178014 0.178262 0.178511 0.178760 0.179009 0.179259 0.179509 0.179759 0.180009 0.180260 0.180511 0.180763 0.181014 0.181266 0.181518 0.181771 0.182024

0.182024 0.182277 0.182530 0.182784 0.183038 0.183292 0.183547 0.183801 0.184057 0.184312 0.184568 0.184824 0.185080 0.185337 0.185594 0.185851 0.186109 0.186367 0.186625 0.186883 0.187142 0.187401 0.187661 0.187920 0.188180 0.188440 0.188701 0.188962 0.189223 0.189485 0.189746 0.190009 0.190271 0.190534 0.190797 0.191060 0.191324 0.191588 0.191852 0.192116 0.192381 0.192646 0.192912 0.193178 0.193444 0.193710 0.193977 0.194244 0.194511 0.194779 0.195047 0.195315 0.195584 0.195853 0.196122 0.196392 0.196661 0.196932 0.197202 0.197473 0.197744

Degrees 44 45 46 Involute Function Values 0.197744 0.214602 0.232679 0.198015 0.214893 0.232991 0.198287 0.215184 0.233304 0.198559 0.215476 0.233616 0.198832 0.215768 0.233930 0.199104 0.216061 0.234243 0.199377 0.216353 0.234557 0.199651 0.216646 0.234871 0.199924 0.216940 0.235186 0.200198 0.217234 0.235501 0.200473 0.217528 0.235816 0.200747 0.217822 0.236132 0.201022 0.218117 0.236448 0.201297 0.218412 0.236765 0.201573 0.218708 0.237082 0.201849 0.219004 0.237399 0.202125 0.219300 0.237717 0.202401 0.219596 0.238035 0.202678 0.219893 0.238353 0.202956 0.220190 0.238672 0.203233 0.220488 0.238991 0.203511 0.220786 0.239310 0.203789 0.221084 0.239630 0.204067 0.221383 0.239950 0.204346 0.221682 0.240271 0.204625 0.221981 0.240592 0.204905 0.222281 0.240913 0.205185 0.222581 0.241235 0.205465 0.222881 0.241557 0.205745 0.223182 0.241879 0.206026 0.223483 0.242202 0.206307 0.223784 0.242525 0.206588 0.224086 0.242849 0.206870 0.224388 0.243173 0.207152 0.224690 0.243497 0.207434 0.224993 0.243822 0.207717 0.225296 0.244147 0.208000 0.225600 0.244472 0.208284 0.225904 0.244798 0.208567 0.226208 0.245125 0.208851 0.226512 0.245451 0.209136 0.226817 0.245778 0.209420 0.227123 0.246106 0.209705 0.227428 0.246433 0.209991 0.227734 0.246761 0.210276 0.228041 0.247090 0.210562 0.228347 0.247419 0.210849 0.228654 0.247748 0.211136 0.228962 0.248078 0.211423 0.229270 0.248408 0.211710 0.229578 0.248738 0.211998 0.229886 0.249069 0.212286 0.230195 0.249400 0.212574 0.230504 0.249732 0.212863 0.230814 0.250064 0.213152 0.231124 0.250396 0.213441 0.231434 0.250729 0.213731 0.231745 0.251062 0.214021 0.232056 0.251396 0.214311 0.232367 0.251730 0.214602 0.232679 0.252064

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47

48

49

0.252064 0.252399 0.252734 0.253069 0.253405 0.253742 0.254078 0.254415 0.254753 0.255091 0.255429 0.255767 0.256106 0.256446 0.256786 0.257126 0.257467 0.257808 0.258149 0.258491 0.258833 0.259176 0.259519 0.259862 0.260206 0.260550 0.260895 0.261240 0.261585 0.261931 0.262277 0.262624 0.262971 0.263318 0.263666 0.264014 0.264363 0.264712 0.265062 0.265412 0.265762 0.266113 0.266464 0.266815 0.267167 0.267520 0.267872 0.268225 0.268579 0.268933 0.269287 0.269642 0.269998 0.270353 0.270709 0.271066 0.271423 0.271780 0.272138 0.272496 0.272855

0.272855 0.273214 0.273573 0.273933 0.274293 0.274654 0.275015 0.275376 0.275738 0.276101 0.276464 0.276827 0.277191 0.277555 0.277919 0.278284 0.278649 0.279015 0.279381 0.279748 0.280115 0.280483 0.280851 0.281219 0.281588 0.281957 0.282327 0.282697 0.283067 0.283438 0.283810 0.284182 0.284554 0.284927 0.285300 0.285673 0.286047 0.286422 0.286797 0.287172 0.287548 0.287924 0.288301 0.288678 0.289056 0.289434 0.289812 0.290191 0.290570 0.290950 0.291330 0.291711 0.292092 0.292474 0.292856 0.293238 0.293621 0.294004 0.294388 0.294772 0.295157

0.295157 0.295542 0.295928 0.296314 0.296701 0.297088 0.297475 0.297863 0.298251 0.298640 0.299029 0.299419 0.299809 0.300200 0.300591 0.300983 0.301375 0.301767 0.302160 0.302553 0.302947 0.303342 0.303736 0.304132 0.304527 0.304924 0.305320 0.305718 0.306115 0.306513 0.306912 0.307311 0.307710 0.308110 0.308511 0.308911 0.309313 0.309715 0.310117 0.310520 0.310923 0.311327 0.311731 0.312136 0.312541 0.312947 0.313353 0.313759 0.314166 0.314574 0.314982 0.315391 0.315800 0.316209 0.316619 0.317029 0.317440 0.317852 0.318264 0.318676 0.319089

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Machinery's Handbook, 31st Edition SPHERICAL TRIGONOMETRY

116

Spherical Trigonometry

Spherical trigonometry deals with the measurement of angles of polygons—triangles in particular—that lie on the surface of spheres. The sides of a spherical triangle conform to the surface of the sphere, and, unlike a plane triangle, the sum of angle measures of a spherical triangle ranges from 180° to 540°. Right-Angle Spherical Trigonometry.—The solid black lines A, B, and C of Fig. 1 rep­ resent the sides of a right spherical triangle. The dashed lines J and K are radii of the sphere extending from the center of the sphere to the triangle’s vertices. The several plane trian­ gles, indicated by the various broken lines, are formed from the radii and vertices of the spherical triangle. J and K are radii and thus have the same value.

Fig. 1. Right Spherical Triangle

Formulas for Right Spherical Triangles Formulas for Lengths π A = K × --------- × F° 180

π B = J × --------- × G° 180

π C = J × --------- × H° 180

180 B J = --------- × ------G° π

180 A K = --------- × -----F° π

Formulas for Angles 180 A F° = --------- × ---π K

Angle D

sin D = sin F × csc H

E F

sin F = tan G × cot E

G H

cos H = cos G × cos F

180 B G° = --------- × --π J

Angular Relationships

cos D = tan G × cot H

180 C H° = --------- × ---π J

tan D = tan F × csc G

cos E = cos G × sin D

tan E = tan G × csc F

cos F = sec G × cos H

tan F = tan D × sin G

cos G = cos H × sec F

tan G = sin F × tan E

cos H = cot D × cot E

Area Formula π π 2 2 Area = K × --------- ( D° + E° + 90° – 180° ) = K × --------- ( D° + E° – 90° ) 180 180

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Machinery's Handbook, 31st Edition SPHERICAL TRIGONOMETRY

117

The side and angle labels in examples that follow refer to those of the right spherical tri­angle in Fig. 1. Example 1: Find the length of arc A of a right spherical triangle on the surface of a sphere where radius K = 30.00 inches and angle F = 10°. π 180

π 180

Solution: A = K × --------- × F = 30 × --------- × 10 = 5.2359 in. Example 2: Find length of arc B on a sphere of radius J = 11.20 inches if angle G = 10°. π 180

π 180

Solution: B = J × --------- × G = 11.20 × --------- × 10 = 1.9547 in. Example 3: A right spherical triangle is to be constructed on the surface of a sphere 22.400 inches in diameter. Side A is 7.125 inches and angle E is 57° 59′ 19″. Determine the lengths of sides B and C, and the measure of angle D, and the area of the triangle. Solution: The radius of the sphere, J = K = 11.200, and the length of side A is used to find the value of angle F. Angle E is converted to decimal degree format for simplicity; then angles E and F are used to solve the equation for angle tan G. Side B and angle D can then be found. Angle H can be calculated using either of the two equations given for cos H, and finally the length of side C can be found. Notice that the sum of angles D + E + 90° is not equal to 180°, but 194.98°. Calculation details are as follows: 180 A 180 7.125 F° = --------- × ---- = --------- × ---------------- = 36.449324° π K π 11.200 59 19 E = 57°59′19″ = 57 + ------ + ------------ = 57.988611° 60 3600 tan G = sin F × tan E = sin ( 36.449324° ) × tan ( 57.988611° ) = 0.950357 G = tan–1 (0.950357) = 43.541944° π π B = J × --------- × G° = 11.200 × --------- × 43.541944 = 8.511443 180 180 tan D = tan F × csc G = tan ( 36.449324° ) × csc ( 43.541944° ) = 1.0721569 180 D = --------- × tan–1( 1.0721569 ) = 46.994354° π cos H = cos G × cos F = cos ( 43.541944° ) × cos ( 36.449324° ) = 0.58307306 180 H = --------- × cos–1( 0.58307306 ) = 54.333023° π π π C = J × --------- × H° = 11.200 × --------- × 54.333023 ° = 10.62085 180 180 Angles ( D + E + 90° ) = 46.994354° + 57.988611° + 90° = 194.98297° 2

Area = 11.200 × ( 194.98297 – 180 ) = 50.142591 in

2

Example 4: A right spherical triangle on a 20-mm diameter sphere has two 90° angles, and the distance B between the 90° angles is 1 ⁄3 the circumference of the sphere. Find angle E, the area of the triangle, and check using the conventional formula for area of a sphere. Solution: By inspection, angle G is 360°/3 = 120°. Because angles D and G are known, angle E can be calculated using cos E = cos G × sin D . Therefore, cos E = cos G × sin D = cos ( 120° ) × sin ( 90° ) = – 0.5 E = cos–1 ( – 0.5 ) = 120° π 2 2 Area = 10 × --------- ( 120° + 90° + 90° – 180° ) = 100 × 2.0943951 = 209.4 mm 180 2

( 100 )- = 209.4 mm 2 ------------- = 4π -------------------Check: Total area of 20-mm diameter sphere/6 = 4πR 6 6

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Machinery's Handbook, 31st Edition Spherical Trigonometry

118

Oblique Spherical Trigonometry.—The heavy solid lines B, C, and S of Fig. 2 represent the sides of an oblique spherical triangle. The dashed lines J and L are radii of the sphere extending from the center of the sphere to the vertices of the triangle. The several plane tri­angles, indicated by the various broken lines, are formed from the radii and vertices of the spherical triangle. J and L are radii and thus have the same value.

Fig. 2. Oblique Spherical Triangle

Formulas for Oblique Spherical Triangles Formulas for Lengths π B = J × --------- × G° 180

π C = J × --------- × H° 180

π S = L × --------- × R° 180

180 B J = --------- × ------π G°

180 S L = --------- × -----π R°

Formulas for Angles 180 B G° = --------- × --π J

Angle D

180 C H° = --------- × ---π J

Relationships

180 S R° = --------- × --π L

Angular Relationships Angle

Relationships

sin D = sin R × sin E × cscG

E

sin E = sin D × sin G × cscR

G

sin G = sin R × sin E × cscD

E1

cot E 1 = tan D × cos H

N

cos N = cos D × cscE 1 × sin E 2

E2

cot E 2 = tan N × cos R

N

R+G sin  --------------  2  N D–E cot  ---- = ---------------------------- × tan  --------------  2   2 R–G sin  --------------  2 

R

sin R = sin D × sin G × csc E

H

D+E sin  --------------  2  H R–G tan  ---- = ---------------------------- × tan  --------------  2   2 D–E sin  --------------  2 

Area Formula π 2 Area = L × --------- ( D + E + N – 180° ) 180

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Machinery's Handbook, 31st Edition Spherical Trigonometry

119

The side and angle labels in the examples that follow refer to those of the oblique spherical triangle in Fig. 2.

Example: An oblique spherical triangle is to be constructed on the surface of a sphere of unknown size. The length of side S will be 5.470 inches; the spherical angle of arc S must be 51° 17′ 31″ (angle R in Fig. 2). Angle D must be 59° 55′ 10″, and angle E must be 85° 36′ 32″. Find the size of the sphere, lengths of sides B and C, and the value of angle N. Solution: Convert known angles to decimal degree format to simplify calculations: 31 - = 51.291944° ------ + ----------R = 51° + 17 60 3600 10 - = ------ + ----------D = 59° + 55 59.919444° 60 3600 32 - = ------ + ----------E = 85° + 36 85.608889° 60 3600

Find the radius of the sphere:

180 S 180 5.470 L = --------- × ------ = --------- × ---------------------------- = 6.11 inches π R° π 51.291944°

Find the values of angles of G and H in order to get lengths of sides B and C. Then solve for the value of angle N, and finally the area. Remember that both J and L are radii, thus J = L. sin G = sin R × sin E × csc D = 0.780342 × 0.997065 × 1.15564 = 0.899148 G = sin–1 ( 0.899148 ) = 64.046301° π π B = J × --------- × G° = 6.11 × --------- × 64.046301° = 6.829873 inches 180 180

D+E sin  --------------  2  R–G sin ( 72.76417 ) H tan  ---- = ---------------------------- × tan  -------------- = ----------------------------------------- × tan ( – 6.377185 )  2   2 sin ( – 12.844723 ) D–E sin  --------------  2  0.955093 = ------------------------- ( – 0.111765 ) = 0.480167 – 0.222310

H H = 51.297543° ---- = tan–1 ( 0.480167 ) = 25.648772°, 2 π π C = J × --------- × H° = 6.11 × --------- × 51.297543° = 5.470350 inches 180 180

R+G sin  --------------  2  N D–E sin ( 57.669123 ) cot  ---- = ---------------------------- × tan  -------------- = -------------------------------------- × tan ( – 12.844723 )  2  2  sin ( – 6.377185 ) R–G sin  --------------  2  0.844974 = ------------------------- ( – 0.228015 ) = 1.7345957 – 0.111073

N ---- = cot–1 ( 1.7345957 ) = 29.963587°, 2

N = 59.927175°

π 2 Area = L × --------- ( D + E + N – 180° ) = 16.585 in 180 2

The triangle is an isosceles spherical triangle with legs B and C each being 5.470 inches. If angle E1 or E2 is known, then any problem involving oblique spherical triangles can be solved as two right spherical triangles; in that case, the equations for right spherical trian­gles are used.

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120

Machinery's Handbook, 31st Edition COMPOUND ANGLES Compound Angles

Referring to the Formulas for Compound Angles on page 121, in Fig. 1 is shown what might be considered as a thread-cutting tool without front clear­ance. A is a known angle in plane y-y of the top surface. C is the corresponding angle in plane x-x that is at some given angle B with plane y-y. Thus, angles A and B are components of the compound angle C. Example Referring to Fig. 1: Angle 2A in plane y-y is known, as is angle B between planes x-x and y-y. It is required to find compound angle 2C in plane x-x. Solution: Let 2A = 60 and B = 15; then tan C = tan A × cos B = tan 30° × cos 15° tan C = 0.57735 × 0.96592 = 0.55767   C = 29° 8.8 ′ 2C = 58° 17.6 ′ Fig. 2 shows a thread-cutting tool with front clearance angle B. Angle A is one-half the angle between the cutting edges in plane y-y of the top surface, and compound angle C is one-half the angle between the cutting edges in a plane x-x at right angles to the inclined front edge of the tool. The angle between planes y-y and x-x is, therefore, equal to clearance angle B. Example Referring to Fig. 2: Find the angle 2C between the front faces of a threadcutting tool having a known clearance angle B that will permit the grinding of these faces so their top edges form the desired angle 2A for cutting the thread. Solution: Let 2A = 60 and B = 15; then

tan 30° tan A------------------tan C = ----------= ----------------= 0.57735 cos 15° cos B 0.96592 tan C = 0.59772   C = 30° 52 ′

2C = 61° 44 ′

In Fig. 3 is shown a form-cutting tool in which the angle A is one-half the angle between the cutting edges in plane y-y of the top surface; B is the front clearance angle; and C is one-half the angle between the cutting edges in plane x-x at right angles to the front edges of the tool. The formula for finding angle C when angles A and B are known is the same as that for Fig. 2. Example Referring to Fig. 3: Find the angle 2C between the front faces of a form-cutting tool having a known clearance angle B that will permit the grinding of these faces so their top edges form the desired angle 2A for form cutting. Solution: Let 2A = 46 and B = 12; then

tan 23° tan A0.42447 tan C = ----------= ----------------= ------------------cos 12° cos B 0.97815 tan C = 0.43395   C = 23° 27.5 ′ 2C = 46° 55 ′ In Fig. 4 is shown a wedge-shaped block, the top surface of which is inclined at com­ pound angle C with the base in a plane at right angles with the base and at angle R with the front edge. Angle A, in the vertical plane of the front of the plate, and angle B, in the vertical plane of one side that is at right angles to the front, are components of angle C.

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Machinery's Handbook, 31st Edition COMPOUND ANGLES

121

Formulas for Compound Angles y C

x

y

y

A

A

y

y

C

C

x B

B

x

x

B

B

x

Fig. 2.

Fig. 1.

B Fig. 3.

x

C

B

For given angles A and B, find the y resultant angle C in plane x-x. Angle B is measured in vertical plane y-y of midsection.

A

x A Fig. 4.

C x

R x A Fig. 5.

x R C1

A1

Fig. 1  tan C = tan A × cos B

tan A Fig. 2  tan C = -----------cos B Fig. 3  (Same formula as for Fig. 2) Fig. 4. In machining a plate to angles A and B, the plate is held at angle C in plane x-x. Angle of rotation R in plane parallel to base (or complement of R) is for locating plate so that plane x-x is per­pendicular to axis of pivot on angle-plate or work-holding vise. tan B tan Atan R = ------------ , tan C = ----------tan A cos R

R

B

x

B1 x

Fig. 6.

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Fig. 5. Angle R in horizontal plane parallel to base is angle from plane x-x to side having angle A. tan Atan R = ----------tan B tan C = tan A × cos R = tan B × sin R Compound angle C is angle in plane x-x from base to corner formed by inter­section of planes inclined to angles A and B. This formula for C may be used to find cotangent of complement of C1, Fig. 6. Fig. 6. Angles A1 and B1 are mea­sured in vertical planes of front and side elevations. Plane x-x is located by angle R from center-line or from plane of angle B1. tan A tan R = -------------1tan B 1

tan A tan B tan C 1 = -------------1- = -------------1sin R cos R The resultant angle C1 would be required in drilling hole for pin.

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Machinery's Handbook, 31st Edition COMPOUND ANGLES

122

Example Referring to Fig. 4: Find the compound angle C of a wedge-shaped block having known component angles A and B in sides at right angles to each other. C = compound angle in plane x-x, which is the resultant of angles A and B Solution: Let A = 47°14 ′ and B = 38°10 ′; then tan B- = -----------------------tan 38°10′- = -----------------0.78598- = 0.72695 tan R = ----------R = 36°0.9′ tan 47°14′ 1.0812 tan A tan 47°14′ = -----------------tan A- = --------------------------1.0812- = 1.3367 C = 53°12′ tan C = ----------cos R cos 36 °0.9′ 0.80887 In Fig. 5 is shown a four-sided block, two sides of which are at right angles to each other and to the base of the block. The other two sides are inclined at an oblique angle with the base. Angle C is a compound angle formed by the intersection of these two inclined sides and the intersection of a vertical plane passing through x-x, and the base of the block. The components of angle C are angles A and B, and angle R is the angle in the base plane of the block between the plane of angle C and the plane of angle A.

Example Referring to Fig. 5: Find the angles C and R in the block shown in Fig. 5 when angles A and B are known. Solution: Let angle A = 27° and B = 36°; then cot C = cot2 A + cot2 B = 1.9626 2 + 1.3764 2 = 5.74627572 = 2.3971

C = 22°38.6′ cot B- = ---------------cot 36°- = 1.3764 ---------------- = 0.70131 tan R = ----------R = 35°2.5′ cot A cot 27° 1.9626 Example Referring to Fig. 6: A rod or pipe is inserted into a rectangular block at an angle. Angle C1 is the compound angle of inclination (measured from the vertical) in a plane passing through the center line of the rod or pipe and at right angles to the top sur­ face of the block. Angles A1 and B1 are the angles of inclination of the rod or pipe when viewed respectively in the front and side planes of the block. Angle R is the angle between the plane of angle C1 and the plane of angle B1. Find angles C1 and R when a rod or pipe is inclined at known angles A1 and B1. Solution: Let A1 = 39° and B1 = 34°; then tan C 1 =

tan2 A 1 + tan2 B 1 =

0.80978 2 + 0.67451 2 = 1.0539

C 1 = 46°30.2′ tan A ------------------- = 1.2005 tan R = -------------1- = 0.80978 tan B 1 0.67451

R = 50°12.4′

Interpolation.—In mathematics, interpolation is the process of finding a value in a table or in a mathematical expression which falls between two given tabulated or known values. In engineering handbooks, the values of trigonometric functions are usually given only in degrees and minutes; hence, if the angle is given in degrees, minutes and seconds, the value of the function is determined from the nearest given values by interpolation. Interpolation to Find Functions of an Angle: Assume that the sine of 14°22 ′26″ is to be determined. It is evident that this value lies between the sin 14° 22 ′ and the sin 14° 23 ′. sin 14° 23 ′ = 0.24841 and sin 14° 22 ′ = 0.24813. The difference is 0.24841 - 0.24813 = 0.00028. Consider this difference as a whole number (28) and multiply it by a fraction hav­ ing as its numerator the number of seconds (26) in the given angle, and as its denominator 60 (number of seconds in one minute). Thus 26 ⁄ 60 × 28 = 12 (nearly); hence, by adding 0.00012 to sin 14° 22 ′ we find that sin 14°22 ′26″ ≈ 0.24813 + 0.00012 = 0.24825. The correction value (represented in this example by 0.00012) is added to the function of the smaller angle nearest the given angle in dealing with sines or tangents, but this correction value is sub­t racted in dealing with cosines or cotangents.

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Machinery's Handbook, 31st Edition MATRICES

123

MATRICES Multiple variables are often present in technology and engineering scenarios; some examples include electrical circuitry, in which there is a long series of resistors, and cost analysis (labor, materials, capital) in industrial economics. A system of linear equations can be solved by several methods, such as substitution and elimination (see pages 32 to 33, Solving a System of Linear Equations in ALGEBRA). Another way to solve a system, useful especially when more than two variables are involved, is to set up a matrix of the equations’ coefficients. A matrix consists of real numbers arranged in horizontal rows and vertical columns to form a rectangular array. An array of m rows and n columns is an m × n matrix (read as “m by n”) and is written as

. . 0 a 11 . … . . . .. .. . .. ... .. . . . . . a mn 0 . … The aij terms are called the entries or elements of the matrix. The subscript i identi­ fies the row position of an entry, and the subscript j identifies its column position in the matrix. For example, in the matrix below, a11 = 3, a12 = 4, a21 = –1, and a22 = 2:

3 4 –1 2 Special matrices used in matrix operations include: a1 . … . . a 1n

a 11 .

(a 11 . . . a 1n )

. .

.

..

.. .

..

...

.

.

Column Matrix (m × 1)

Row Matrix (1 × n)

0

.

. . 0 …

... .

.

.. .

.

. . a nn a n1 . …

a m1

... ..

0

.

.

.

.

...

.

.

. . 0 …

Square Matrix Zero, or Null, (n × n) Matrix (n × n)

The two types of special square matrices include: . a1 . …

... ..

.

.

.

0

..

.

.

.

0

... .. .

.

. . ann …

Diagonal Matrix (n × n) All entries are 0, except possibly those on the diagonal.

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1

.

... ..

.

.

… . . a1n

.. .

..

.

.

an1 . … .

..

.

.

.

1

Identity Matrix (n × n) All diagonal entries are 1, all others are 0.

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Machinery's Handbook, 31st Edition Matrix Operations

124

Matrix Operations Matrix Addition and Subtraction.—The sum or difference of matrices is determined simply by adding or subtracting the corre­sponding elements of each matrix. So, matrix C is the result of adding or subtracting matrices A and B; the entries are combined as follows: aij + bij = cij  or aij - bij = cij

Matrices must be the same size, m × n, to be combined this way. That is, A mn + Bmn = Cmn ; A mn - Bmn = Cmn . An efficient way to indicate both + and - is with the symbol ±. Thus, in the matrix display below, aij ± bij covers both matrix operations. Example 1: 4 6 –5 8 –2 6 ( 4 + 8 ) ( 6 – 2 ) ( –5 + 6 ) 12 4 1 5 – 7 8 + – 6 9 5 = ( 5 – 6 ) ( – 7 + 9 ) ( 8 + 5 ) = – 1 2 13 –8 6 –7 9 –2 2 ( –8 + 9 ) ( 6 – 2 ) ( –7 + 2 ) 1 4 –5

Matrix Multiplication.—Two matrices can be multiplied only if the number of col­umns in the first matrix equals the number of rows of the second matrix. For example, a 1 × 3 matrix can multiplied by a 3 × 2 matrix, but not the other way around. Or a 2 × 4 by a 4 × 3, in that order. In general, an m × n and an n × p can be multiplied; the product matrix is an m × p matrix. Matrix multi­plication is not commutative, that is, A × B is not necessarily equal to B × A. The steps in matrix multiplication are shown in the instructive example below for a general 3 × 2 matrix multiplied by a 2 × 1 matrix. The result is a 3 × 1 matrix. 1

4

–3

1

2

×

–1

1 · 0 + 4 ∙ –2

0

=

–2

–8

–3 ∙ 0 + 1 ∙ –2

–2

=

2 ∙ 0 + –1 ∙ –2

2

In general, each entry (AB)ij in the product matrix (where i is the row number and j is the column number) is equal to ai1 b1j + ai2 b2j + … + ain bnj. For example: a11

a12

a21

a22

a31

a32

3×2 matrix

×

b11

b12

b21

b22

=

a11b11 + a12b21

a11b12 + a12b22

a21b11 + a22b21

a21b12 + a22b22

a31b11 + a32b21

a31b12 + a32b22

2×2 matrix

3×2 matrix

Example 2: 1 23 789 (1 ⋅ 7 + 2 ⋅ 1 + 3 ⋅ 4) 4 5 6 × 1 2 3 = (4 ⋅ 7 + 5 ⋅ 1 + 6 ⋅ 4) (3 ⋅ 7 + 2 ⋅ 1 + 1 ⋅ 4) 3 21 457 ( 7 + 2 + 12 ) = ( 28 + 5 + 24 ) ( 21 + 2 + 4 )

(1 ⋅ 8 + 2 ⋅ 2 + 3 ⋅ 5) (4 ⋅ 8 + 5 ⋅ 2 + 6 ⋅ 5) (3 ⋅ 8 + 2 ⋅ 2 + 1 ⋅ 5)

( 8 + 4 + 15 ) ( 32 + 10 + 30 ) ( 24 + 4 + 5 )

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(1 ⋅ 9 + 2 ⋅ 3 + 3 ⋅ 7) (4 ⋅ 9 + 5 ⋅ 3 + 6 ⋅ 7) (3 ⋅ 9 + 2 ⋅ 3 + 1 ⋅ 7)

21 27 36 ( 9 + 6 + 21 ) ( 36 + 15 + 42 ) = 57 72 93 ( 27 + 6 + 7 ) 27 33 40

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Machinery's Handbook, 31st Edition Matrix Operations

125

Transpose of a Matrix.—If the rows of a matrix A mn are interchanged with its columns, the new matrix is called the transpose of matrix A, or AT. The first row of the matrix becomes the first column in the transposed matrix, the second row of the matrix becomes the second column, and the third row of the matrix becomes the third column. Example 3: 1 2 3 1 4 7 T A = 4 5 6 A = 2 5 8 7 8 9 3 6 9 Determinant of a Square Matrix.—Every square matrix A is associated with a real num­ber, its determinant, which may be written det A or  A . a b , c d det A = A = ad – bc.

For A = Example 4: 

A = 2 –1 1 –3

2 –1 1 –3

det A =

= ( 2 ) ( –3 ) – ( 1 ) ( –1 ) = –5

The process for taking the determinant of a 3 × 3 matrix is shown next. It entails multiplying the first entry of each column by the determinant of the remaining 2 × 2 matrix and alternately adding or subtracting the product. B=

a

b

c

d

e

f

g

h

i

,

a

b

c

det B = d

e

f =a

g

h

i

e h

f

d –b

i

g

f i

+c

d

e

g

h

Example 5: Find the determinant of the following matrix. 567 A = 123 456

Solution:



det A = 5 ( 12 – 15 ) – 6 ( 6 – 12 ) + 7 ( 5 – 8 )

= 5 ( – 3 ) – 6 ( – 6 ) + 7 ( – 3 ) = –15 + 36 – 21 = 0

Minors and Cofactors.—The minor Mij of a matrix A is the determinant of a submatrix resulting from the elimination of row i and column j. If A is a square matrix, the minor Mij of the entry aij is the determinant of the matrix obtained by deleting the ith row and jth col­umn of A. The cofactor Cij of the entry aij is given by Cij = (-1)(i+j)Mij. Thus, the sign of cofactor aij alternates across the row it lies in. The matrix formed by its cofactors is called the cofactor matrix. 123

Example 6: Find the minors and cofactors of A = 4 5 6 321

Solution: To determine the minor M11, delete the first row and first column of A and find the determinant of the resulting matrix. M 11 =

56 21

= ( 5 × 1 ) – ( 6 × 2 ) = 5 – 12 = – 7

Similarly to find M12 , delete the first row and second column of A and find the determi­ nant of the resulting matrix.

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126

Machinery's Handbook, 31st Edition Matrix Operations 46 31

M 12 =

= ( 4 × 1 ) – ( 6 × 3 ) = 4 – 18 = – 14

Continuing this way, we obtain the following minors: M 11 = – 7

M 12 = – 14

M 13 = – 7

M 31 = – 3

M 32 = – 6

M 33 = – 3

M 21 = – 4

M 22 = – 8

M 23 = – 4

To find the cofactor, calculate Cij = (-1)(i+j) × Mij, thus C11 = (-1)(1+1) × M11 = 1 × (-7) = -7. Similarly C12 = (-1)(1+2) × M12 = (-1)(-14) = 14, and continuing this way we obtain the following cofactors C 11 = – 7

C 12 = 14

C 13 = – 7

C 21 = 4

C 22 = – 8

C 23 = 4

C 31 = – 3

C 32 = 6

C 33 = – 3

–7 14 –7

Thus, the cofactor matrix is

4 –8 4 –3 6 –3

Adjoint of a Matrix.—The transpose of cofactor matrix is called the adjoint matrix. To obtain the adjoint matrix, the cofactor matrix is determined and then transposed. Example 7: Find the adjoint matrix of A: 123 A = 456 321

Solution: The cofactor matrix from the above example is shown below at the left, and the adjoint matrix is shown on the right. cofactor ( A ) =

– 7 14 – 7 4 –8 4 –3 6 –3

adj ( A ) =

– 7 14 – 7 4 –8 4 –3 6 –3

T

–7 4 –3 = 14 – 8 6 –7 4 –3

Singularity and Rank of a Matrix.—A singular matrix is one whose determinant is zero. The rank of a matrix is the maximum number of linearly independent row or column vectors it contains. In ALGEBRA, Solving a System of Linear Equations, it is explained that a system with a unique solution is a linearly independent system. Such systems are vital to depicting many real-life processes. Dependent systems are those with infinite solutions (the lines are collinear), and their determinant is zero. Linearly independent vectors have a non-zero determinant. Inverse of a Matrix.—A square non-singular matrix A has an inverse A -1 such that the product of matrix A and inverse matrix A -1 is the identity matrix I. The operation is commutative as well. Thus, AA -1 = A -1 A = I. The inverse is the ratio of adjoint of the matrix and the determinant of that matrix. A

–1

adj ( A ) = ---------------A

Example 8: What is the inverse of the following matrix? 235 A = 416 140

Solution: The basic formula of an inverse of a matrix is A

–1

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adj ( A ) = ---------------A

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Machinery's Handbook, 31st Edition Matrix Operations

127

The determinant of A is

The cofactors are

A = 2(1 × 0 – 4 × 6) – 3(4 × 0 – 1 × 6) + 5(4 × 4 – 1 × 1) = 2 ( 0 – 24 ) – 3 ( 0 – 6 ) + 5 ( 16 – 1 ) = – 48 + 18 + 75 = 45

a 11 = ( – 1 )

1+1

1 6 4 0

= – 24

a 12 = ( – 1 )

1+2

46 10

= 6

a 13 = ( – 1 )

1+3

41 14

= 15

a 21 = ( – 1 )

2+1

3 5 4 0

= 20

a 22 = ( – 1 )

2+2

25 10

= –5

a 23 = ( – 1 )

2+3

23 14

= –5

a 31 = ( – 1 )

3+1

3 5 1 6

= 13

a 32 = ( – 1 )

3+2

25 46

= 8

a 33 = ( – 1 )

3+3

23 41

= – 10

– 24 6 15

The matrix of cofactors is 20 – 5 – 5 and the adjoint matrix is 13 8 – 10

– 24 20 13 6 –5 8 15 – 5 – 10

Then the inverse of matrix A is A

–1

1 adj ( A ) = ---------------- = -----45 A

– 24 20 13 6 –5 8 15 – 5 – 10

Multiplying A -1 by A results in the identity matrix: 1 45

–24 20

13

2

3

5

6 –5

8

4

1

6

15 –5 –10

1

4

0

1

= 45

45

0

0

0

45

0

0

0

45

=

1

0

0

0

1

0

0

0

1

Solving a System of Equations.—Matrices can be used to solve systems of simultaneous equa­tions with a large number of unknowns. Geometrically, the solution of a system is the point in the plane or in space where the lines intersect. Variables may represent the unknowns in industrial and other engineering applications: a series of resistances in a circuit, for example, or, perhaps, factors in a manufacturing process, such as labor costs, materials, and equipment. Generally, this method is less cumbersome than using substitution methods (see Solving a System of Linear Equations in ALGEBRA). The coefficients of the equations are placed in matrix form. The matrix is then transformed by row and column operations into the identity matrix to yield a solution. The process, as described in the example, is called matrix reduction to row echelon form. It is done using any or all of three valid vector operations: 1) multiplying each entry in a row by a constant; 2) adding or subtracting two rows; 3) changing the order of rows. Example 9: Solve the system of linear equations in three dimensions using matrix operations. – 4x 1 + 8x 2 + 12x 3 = 16 3x 1 – x 2 + 2x 3 = 5

x 1 + 7x 2 + 6x 3 = 10

Solution: First, the equation coefficients and constants are placed into what is called an augmented matrix. The object is to transform the matrix of the original coefficients into the following form, thereby obtaining a solu­tion (x1, x 2 , x 3) to the system of equations.

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128

– 4 8 12 3 –1 2 1 7 6

1 0 0

16 5 ⇔ 10

0 1 0 0 0 1

x1 x2 x3

The coefficient matrix is transformed so that element c11 is 1 and all other elements in the first column are 0, as follows: a) divide row 1 (R1) by -4; b) multiply new R1 by -3, then add to R2 ; and c) multiply R1 by -1, then add to R3. 4 – -----–4 3 1

8 -----–4 –1 7

12 -----–4 2 6

16 -----–4 ⇒ 5 10

1 –2 –3 (3 – 3) (– 1 + 6) (2 + 9) (1 – 1) (7 + 2) (6 + 3)

–4 1 –2 –3 ( 5 + 12 ) ⇒ 0 5 11 ( 10 + 4 ) 0 9 9

–4 17 14

The resulting matrix is transformed so that element c 22 is 1 and all other elements in the sec­ond column are 0, as follows: a) divide R3 by 9; b) multiply new R3 by -5, then add to R2 ; c) multiply R3 by 2, then add to R1; and d) swap R2 and R3. 1 0 0 --9

–2 5 9 --9

–3 –4 11 17 ⇒ 9 14 --- - ----9 9

1 ( –2 + 2 ) ( –3 + 2 )

 – 4 + 28 ------  9

0 ( 5 – 5 ) ( 11 – 5 )

 17 – 70 ------  9

0

1

1 0 –1 ⇒

14 -----9

1

00 6 01 1

8 – --9 83 ⇒ -----9 14 -----9

1 0 –1 01 1 00 6

8 – --9 14 -----9 83 -----9

The resulting matrix is finally reduced so that element c33 is 1 and all other elements in the third column are 0, as follows: a) divide R3 by 6; b) multiply new R3 by –1, then add to R2 ; and c) add R3 to R1. 1 0 –1 0 1 1 6 0 0 --6

8 1 0 ( –1 + 1 ) – --9 14 ⇒ -----0 1 (1 – 1) 9 83 ----------00 1 9(6)

83 – 8 --- + ------  9 54

100

14 83  ----- – ----- 9 54

⇒ 010

83 -----54

001

35 -----54 1 -----54 83 -----54

When the identity matrix has been formed, the last column contains the values of x1, x 2 , and x 3 that satisfy the original equations. 35 x 1 = -----54

1 x 2 = -----54

83 x 3 = -----54

Checking that the solutions satisfy the original system:

() () ()

35 1 83 864 +8 + 12 = = 16 54 54 54 54

–4

() () ()

3

1 83 270 35 –1 +2 = =5 54 54 54 54

() () ()

1

35 1 83 540 +7 +6 = = 10 54 54 54 54

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Machinery's Handbook, 31st Edition USING MATRICES TO SOLVE A SYSTEM OF EQUATIONS

129

Example 10: Use matrix operations to find the amperages of the currents (I1, I2 , I3) in the following electri­cal network.

By Kirchoff’s first law, concerning the sum of currents: I 1 + I 2 = I 3    that is, I 1 + I 2 – I 3 = 0

By Kirchoff’s second law, concerning the sum of voltages, and Ohm’s law, that voltage is the product of current and resistance:

2Ω

A I2

I1 40V

5Ω

I3

+

10Ω

− B

2I 1 + 5I 3 – 40 = 0

− + 30V

10I 2 + 5I 3 – 30 = 0

By combining the three equations, a linear system of independent equations is formed. A system is linearly independent if no one equation is a constant multiple of any other. That is, the equations represent distinct lines, but they are not parallel (hence, they have a solution, that is, a point of intersection). Solve the system for the currents I1, I2 , and I3: I1 + I2 – I3 = 0 2I 1 + 5I 3 = 40 10I 2 + 5I 3 = 30

Solution: If A is the matrix of coefficients of the currents, B is the matrix of the currents themselves (the vari­ables), and C is the matrix of constants from the right side of the equations (the voltages), then the prob­lem can be written in the following form: AB = C, or equivalently, B = A -1C, where A -1 is the inverse of matrix A. Thus, I1

1 1 –1 A = 2 0 5 0 10 5

0 C = 40 30

B = I2 I3

I1

1 1 –1 I2 = 2 0 5 0 10 5 I3

and

–1

0 40 30

Using the method of Example 8, the inverse of matrix A is

A

–1

1 1 –1 = 2 0 5 0 10 5

–1

5 3 1 --- ------ – ----8 16 16 50 15 – 5 1 7 1 ----= – ------ 10 – 5 7 = 1--- – ----- 80 8 16 80 – 20 10 2 1 1 1 – --- --- -----4 8 40

and finally, matrix B can be found by matrix multiplication: 5 3 1 --- ------ – ----8 16 16 0 5.625 –1 1 7 1 B = A C = --- – ------ ------ 40 = 0.125 8 16 80 30 5.75 1 1 1 – --- --- -----4 8 40

Thus, I1 = 5.625 amps, I2 = 0.125 amp, and I3 = 5.75 amps.

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130

Machinery's Handbook, 31st Edition Calculus CALCULUS

Problems in engineering and other sciences are often modeled by functions other than simple first-­ and second-­degree polynomials. The features of higher order polynomials, as well as trigonometric and other non-­algebraic functions, are found using techniques of calculus. The essential operations of calculus are differentiation and antidifferentiation (or integration). Brief explanations of these processes, as well as detailed formulas, are given here. Derivatives

Between any two points of a linear function y = mx + b, the change in y with respect to x is the constant m. Most functions of interest, however, are nonlinear. For example, the velocity of a projectile is not modeled by a line, since objects are subject to gravity, among other outside forces, and therefore accelerate as they fall. The instantaneous rate of change of a function y = f(x) at a single point of a curve is a critical feature for any model. This quantity is called the derivative of f with respect to x (or other independent variable). A derivative is notated in one of three ways: y′, f′(x), or dy/dx. As an example, the instantaneous rate of change of displacement s of an object at any instant in time t is its derivative function, velocity; that is, s′(t) or ds/dt = v(t). Graphically, the derivative function gives the slope of the line tangent to a point of the graph of f. That is, y ´ = mtan . Any constant function y = c has a slope of zero, since its graph is a horizontal line. Hence, y′ = 0 at every point of a line. Fig. 1a shows a portion of a nonlinear function. The tangent line drawn to the point of the curve at x has a slope equal to the derivative f ´( x ). Any group of curves that represent a family of functions, such as the parabolas f 1, f 2 , and f 3 in Fig. 1b, have the same derivative function, since the slope of the tangent lines at any given value of x is the same for each curve. The slope of the tangent line drawn to each of the curves at x = 0 is f ´(0) = 2(0) + 2 = 2.

 

(a)

(b)

Fig. 1. (a) f ´( x ) gives the slope of the tangent line to a curve f (x) for any x in the domain; (b) A family of parabola functions, all with the same derivative function, f ´( x) = 2 x + 2.

Any continuous, smooth function is differentiable (that is, it has a derivative) at each point on its domain. (Roughly speaking, a “continuous” function has no breaks, and a “smooth” function has no sharp corners.) Polynomial, trigonometric, exponential, and logarithmic functions are differentiable everywhere on their domains. To “differentiate” a function means to find (or “take”) its derivative. Derivative Formulas.—­The formulas used most often in derivative applications are: Constant: If y = c, then y´ = 0.

Coefficient: If y = cx, then y´ = c.

Power: If y = x n , then y´ = nx n −1 , for any n in the set of real numbers.

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131

Note: The power rules when n = –­1 and when n = ½ are often stated as their own rules: 1 1 −1 1 1 y ´ = x −1 , y ´ = − x −2 = 2 y = x = x 2 , y ´ = 12 x − 2 = 1 = and 2 x 2 x 2x Logarithmic: 1 ; for natural base e, y = ln x, then y = 1 . For any base a, if y = log a x, then y ´ = x ln ´ x a

Exponential: If y = a x , then y ´ = a x ln a. If y = e x , then y ´ = e x . Trigonometric: If y = sin x, then y ´ = cos x. If y = cos x, then y ´ = − sin x. If y = tan x, then y ´ = sec 2 x. For a complete list of the differentiation formulas, see Table of Derivatives and Integrals. Derivative Rules.—­Just like other functions, derivatives have certain properties: Rule

Rule

Example

Sum or difference Sum ( f ±orgdifference )´( x ) = f ´( x() ± f ±g´g( x)´)( x ) = f ´( x ) ± g´( x )

yy´ == 5xx +−1x 1−2 9,,

y = 2 x ln x,

yy´ == 22ln x lnx + x, ( 2 x ) 1xy ´ == 22ln ln xx ++ (22 x ) 1x = 2 ln x + 2 x)x 4 x 2 − 8 x 2 4 x 9− ( 4 x −9)( 2 x) 4 x 4−x ( 4−x8 x−9)(+218 , 2 y´= = yy´ == 4 x − = 4 x2 2 2 x24 x x x

Product

Product ( fg )´( x ) = f ´( x ) g ((xfg ) +)´(gx´() x=) ff(´x( )x ) g ( x ) + g´( x ) f ( x )

Quotient

f ( x ) ´ f ´( x ) g ( xf)(−x )g´´( x ) ff(´x( )x ) g ( x ) − g´( x ) f ( x ) Quotient = = g ( x) [ g ( x) ]2 [ g ( x) ]2

9 y = 4 x− x2

Chain rule

Chain x )) ]´ = f ´(u ([xf))(u´((xx)) ) ]´ = f ´(u ( x ))u´( x ) [ f (u (rule

y=

( )

( )

 

Example

5 y = x + 1x − 9,,

y´=

,

54

y´= 5x

x

4

− x12

( )

( )

2

2

( )

(

( )

)

(

1/ 2

x 2 + 5 x − 1 =y =x 2 +x52 x+−51x − 1 = x 2 + 5 x − 1 1 2

(x

2

)

−1/ 2

(

2

2

2 + 5 x − 1 y ´ = (122 xx+ +5 )5=x − 1

)

2x + 5

−1/ 2

)

1/ 2

( 2x + 5 ) =

2x + 5

2 (x2 + 5x – 1)1/22 (x2 + 5x – 1)1/2

Integrals (Antiderivatives)

The other fundamental calculus operation, anti­differentiation, also called integration, is the reverse process of differentiation. Whereas derivatives are rate-­of-­change functions, antiderivatives are accumulation functions. We define F to be an antiderivative of f if F´(x) = f(x). Each function in a family of antiderivatives F(x) + C, where c is a constant, has the same derivative f, since (F + C)´ = F´ + C´ = F´ + 0 = F´. For this reason, an antiderivative function F(x) also is called an indefinite integral. The operation of antidifferentiation is indicated by the notation F ( x) = ∫ f ( x)dx.

A definite integral indicates the area under f on a closed interval [a, b] of its domain b (Fig. 2). It is denoted by F (b) − F (a) = ∫a f ( x) dx, where a and b are the bounds of integration, and F(a) and F(b) are the antiderivative values of f at these bounds. Just as velocity is the derivative of displacement, so displacement is the antiderivative t2 of velocity. On a time interval [t1, t 2], displacement is found by s (t2 ) − s (t1 ) = ∫t v(t ) dt. 1

Fig. 2. The definite integral on [a, b] is the area under the curve from x = a to x = b.

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+ 18 x

132

Machinery's Handbook, 31st Edition Calculus

Integral Rules.—­Integration rules are somewhat similar to differentiation rules, since they undo what the derivative does. The analogous processes for the chain, product, and quotient rules of derivatives are covered by either u-­substitution or integration by parts for integrals. Table of Derivatives and Integrals contains integration formulas. Newton’s Method for Solving Equations.—­A lgebraic (polynomial, rational, and root) and transcendental (trigonometric, exponential and logarithmic) equations often can be quickly solved directly, using the processes described in ALGEBRA and TRIGONOMETRY (for example, cos x = 1 or log x = 4). But, long before there were calculators that could solve less straightforward equations to a high degree of accuracy, approximation methods were developed. One such method is Newton’s (or the Newton-­Raphson) method, which produces excellent approximations of the solution of more difficult equations, with the help of differentiation. Some equations that can be solved by Newton’s method are: x2 = 101   x3 – 2x2 = 5   cos x = x Rewriting any of these equations as a function f(x) = 0 converts the problem into one of finding the roots of f(x)—that is, those values where the function crosses the x-axis. For example, x 2 = 101 is rewritten as f(x) = x 2 – 101 = 0. This function has two real roots (see ALGEBRA). A good first estimate of the positive root is 10. Rewriting the other two equations, x 3 – 2x 2 = 5 and cos x = x in the f(x) = 0 form does not give as obvious an estimate of the root(s), but by inspection (trial and error) or a rough graph, an estimate can be made. In each case, this first estimate is called r1. From these, successive estimates r 2, r 3, . . . are made, each progressively closer to the exact value of the root. After estimating r1, the first derivative of the function, f ′(x), is found. This is the equa­tion for the function’s instantaneous rate of change at any value of x. f ′(x) is the equation that gives the slope of the line tan­gent to the function’s curve at a given x. In the above examples, f  ′(x) is, respectively, 2x, 3x 2 - 4x, and –sin x + 1. These were found by the methods described in Table of Derivatives and Integrals on page 133. Starting with the first estimate, the steps of Newton’s method are as follows: r1 is the first estimate of the value of the root of f(x) = 0. Find f(r1), the value of f(x) at x = r1.

Find f  ′(x), the first derivative of f(x). Find f  ′(r1), the value of f  ′(x) at x = r1.

Get the second approximation of the root of f(x) = 0, r 2 , by calculating

r 2 = r 1 – f ( r 1 ) ⁄ f ′( r 1 ) and, further approximations,

r n = r n – 1 – f ( r n – 1 ) ⁄ f ′( r n – 1 ) Example: Find the square root of 101 using the Newton-Raphson method. Solution: The problem is restated as the algebraic equation x 2 = 101, rewritten as x 2 – 101 = 0, and solved for the positive root. r1 = 10 is a good first estimate. Then, apply the steps of the algorithm:

f ( r 1 ) = f ( 10 ) = 10 2 – 101 = – 1

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Machinery's Handbook, 31st Edition Calculus

133

Step 2. The first derivative, f  ′(x), of x 2 - 101 is 2x, as stated previously, so that f  ′(10) = 2(10) = 20. Then, r2 = r1 - f(r1)/f  ′(r1) = 10 - (-1)/20 = 10 + 0.05 = 10.05

Check: 10.052 = 101.0025; a calculator determination of 101 gives 10.0498756; the error using Newton’s method is 0.0001244. Step 3. The next, better approximation is r 3 = r 2 – f ( r 2 ) ⁄ f ′( r 2 ) = 10.05 – f ( 10.05 ) ⁄ f ′( 10.05 )

= 10.05 – ( 10.05 2 – 101 ) ⁄ 2 ( 10.05 ) = 10.049875

Check:  10.049875 2 = 100.9999875 ; error = 0.0000125 Closer approximations result from subsequent applications of the algorithm. Formulas for Differential and Integral Calculus.—The following are formulas for obtaining the derivatives and integrals of basic mathematical functions. In these formulas, the letters a, b, and c denote constants; the letter x denotes a variable; and the letters u and v denote func­tions of the variable x. The expression d/dx means the derivative with respect to x, and as such applies to whatever expression in parentheses follows it. Thus, d(cx)/dx means the derivative with respect to the variable x of the product cx where c is a constant. Derivatives

Table of Derivatives and Integrals

Integrals

d (c) dx

=

0

∫ cdx

=

cx + constant

d (x) dx

=

1

∫ 1 dx

=

x +C

∫ x n dx

=

x ------------ + C, for all real numbers n+1

dx

=

1 --- ln ax + b a

1

= ln|x| + C

x

=

e +C

eaxdx =

eax a +C

d n (x ) dx

( )

d 1 dx x

d ( loga x ) dx

=

nx

n–1

, for all real numbers n

1 = – ----2 +C x

∫ -------------ax + b

1 = -----------

∫ --x- dx

d x (e ) dx

=

∫ e dx

d ax (e ) dx

= aeax

d x (a ) dx

= a x ln a

d ( x) dx

=

d (ln x) dx

= --1

d ( sin x ) dx

x ln a e

x

1 ---------2 x x

=

cos x

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x

∫ a dx dx

∫ ------x

∫ ln x dx ∫ cos x dx

n+1

n ≠ –1 +C

x

x

a +C = ---------ln a

=

2 x +C

= x ln x – x + C =

sin x + C

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Machinery's Handbook, 31st Edition DERIVATIVE AND INTEGRAL TABLES

134

TableTable of Derivatives and Integrals (Continued) of Derivatives and Integrals

Derivatives

Integrals

d ( cos x ) dx

=

– sin x

d ( tan x ) dx

=

d ( cot x ) dx

=

d ( sec x ) dx

∫ sin x dx

=

sec x

∫ tan x dx

= – ln | cos x | + C

– csc x

2

∫ cot x dx

= ln | sin x | + C

=

sec x tan x

∫ sin

2

x dx

=

1 1 – --- sin ( 2x ) + --- x + C 2 4

d ( csc x ) dx

=

– csc x cot x

∫ cos

2

x dx

=

1 1 --- sin ( 2x ) + --- x + C 4 2

d ( sin–1 x ) dx

=

1 ------------------ x ≠ 1, –1 2 1–x

∫ -------------------2 2

=

x sin–1  --- + C  b

d ( cos–1 x ) dx

=

–1 ------------------ x ≠ 1, –1 2 1–x

∫ -------------------2 2

d ( tan–1 x ) dx

=

1 -------------2 1+x

∫ b---------------2 2 +x

d ( cot–1 x ) dx

=

–1 -------------2 1+x

∫ b--------------2 2 –x

d ( sec–1 x ) dx

=

1 --------------------- x ≠ 1, –1 x x2 – 1

∫ x--------------2 2 –b

–1 d ( csc x ) dx

=

–1 --------------------- x ≠ 1, –1 x x2 – 1

dx ∫ ----------------------------ax 2 + bx + c

d ( ln ( sin x )) dx

=

cot x

∫e

d (ln ( cos x )) dx

=

– tan x

∫e

d ( ln ( tan x )) dx

2 = ----------------sin (2x)

- dx ∫ --------sin x

d ( ln ( cot x )) dx

–2 = ---------------sin(2x )

- dx ∫ ---------cos x

d ( ln ( ax + b)) dx

2

dx

b –x dx

x –b

a ax + b

= ---------------

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(

)

2 2 = ln x + x – b + C b b

dx

=

dx

–1 x–b = 1--- tan –1  --x- = ----- log --------------+C  b 2b b x+b

dx

1 x–b = – --1- cos –1  --x- = ----- log --------------+C  

1 – 1  x --- tan --- + C  b b

b

2b

b

[

[ ] [ ] x+b

]

2 2ax + b = ------------------------ tan–1 ------------------------ +C 2 2 4ac – b

4ac – b

sin ( bx ) dx

=

a sin ( bx ) – b cos ( bx ) ax ---------------------------------------------------- e + C 2 2 a +b

cos ( bx ) dx

=

a cos ( bx ) + b sin ( bx ) ax ---------------------------------------------------- e + C 2 2 a +b

ax

ax

– cos x + C

1

|

|

|

|

1

= ln csc x – cot x + C

1

= ln sec x + tan x + C

- dx ∫ -------------------1 + cos x

= csc x – cot x + C

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Machinery's Handbook, 31st Edition Series

135

Series Representation of a Function Some hand calculations, as well as computer programs of certain types of math­ematical problems, may be facilitated by the use of an appropriate series. For example, in some gear problems, the angle corresponding to a given or calculated involute function (see TRIGONOMETRY) is found by using a series together with an iterative procedure such as the Newton-Raphson method described in the previous section. The following are those series most commonly used for such purposes. In the series for trigonometric functions, the angles x are in radians (1 radian = 180/p degrees, or about 57.3 degrees). The expression exp(-x 2) means that the base e of the natural logarithm system is raised to the -x 2 power, where e = 2.7182818. A sum of terms in a sequence of terms is called a series. In its simplest notation, the in∞

finite sum of the sequence a 0, a1, a2, . . . is the series ∑ an = a0 + a1 + a2 + .... For example, n= 0

∞

n 1 2 1 2 for the sequence of x, x2, x3, . . . , the series is given as ∑ x = x + x + ... = x + x + ... . n=1

In calculus, it is helpful to represent certain functions by a special series called the Taylor series. In a manner similar to the Newton-Raphson method, the terms of a Taylor series include the derivatives of the function being approximated. In the table below, common functions and their corresponding infinite series are shown. For any x in the domain (shown to the right of each series), the value of the function can be found. Common Series

x3 + x5 − x 7 +  3! 5! 7! 2 4 6 cos x = 1− x2! + x4! − x6! +  3 5 x7 +  tan x = x + x3 + 215x − 17 315 3 5 −1 x x x5 +  3 15 sin x = x + 6 + 40 + 336 3 5 −1 π x x x5 + ) 3 cos x = 2 − (x + 6 + 40 + 15 336 3 5 7 −1 x x x tan x = x − 3 + 5 − 7 + 

sin x = x −

π 4

for all real x for all real x for x
0.5858l, the second is the maximum stress. Stress is zero at

n x= m

Wx 2 b ^ 3n − m x h 12EIl 3

Between support and load,

y =

Wa 2 v 6 2 3l b − v 2 ^3l − a h@ 12EIl3

Deflections at Critical Pointsa Maximum deflection is at v = 0.4472l, and is

Wl 3 107.33EI

Deflection at load,

7 Wl 3 768 EI

Deflection at load,

Wa3 b 2^ 3l + bh 12EIl 3

If a < 0.5858l, maximum deflec­tion is

Wa 2 b 6EI

b 2l + b

and located between load and support, at

v=l

b 2l + b

If a = 0.5858l, maximum deflec­tion is at load and is

Wl 3 101.9EI

If a > 0.5858l, maximum

deflec­tion is

Wbn3 and 3E Im 2 l 3

located between load and point of fixture, at

2n x= m

Machinery's Handbook, 31st Edition Beam Stress and Deflection Tables

3 Wl 16

262

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Table Table 1. (Continued) Stresses and Deflections in Beams 1. Stresses and Deflections in Beams

Deflections Stresses General Formula for Stress at any Point Stresses at Critical Points General Formula for Deflection at any Pointa Case 15. — Fixed at One End, Supported at the Other, Uniform Load

Type of Beam

s=

3 Wl 16

Total Load W x

Maximum stress at point of

fixture,

Wl 8Z

y=

Wx 2 ^l − x h ^3l − 2x h 48EIl

Stress is zero at x = 1∕4 l. Greatest negative stress is

at x = 5∕8 l and is

5 W 16

l

5 W 8

W ^l − x h 1 a l − xk 4 2Zl

x

l

Wl 6

Maximum stress, at

sup­port,

Wl 3Z

Wx 2 y = 24EIl ^2l − x h2

free end,

W

l

Wl 2

Wl 3 24EI

Wl 2Z Stress at free end, − Wl 2Z Stress at support,

Wx 2 y = 12EI ^3l − 2x h

Maximum deflection, at free end,

Wl 3 12EI

These are the maximum stresses and are equal and opposite. Stress is zero at center.

263

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x

W 1 s = Z a 2 l − xk

Maximum deflection, at free end,

Wl − 6Z

Case 17. — Fixed at One End, Free but Guided at the Other, with Load

W

Wl 3 192EI

Deflection at point of greatest negative stress, at x = 5∕8  l is

Stress is zero at x = 0.4227l Greatest negative stress, at

W

Wl 2

Wl 3 185EI

Deflection at center,

9 Wl − 128 Z

Case 16. — Fixed at One End, Free but Guided at the Other, Uniform Load

Wl 3

Maximum deflection is at x = 0.5785l, and is

Wl 3 187EI

Wl 1 x 1 x 2 s = Z &3 − l + 2` l j 0

Total Load W

Deflections at Critical Pointsa

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TableTable 1. (Continued) Stresses and Deflections in Beams 1. Stresses and Deflections in Beams

Stresses Deflections General Formula for Stress at any Point Stresses at Critical Points General Formula for Deflection at any Pointa Case 18. — Fixed at Both Ends, Load at Center

Type of Beam

Between each end and load,

x

W 2

W 12

l

12

W 1 s = 2Z a 4 l − x k

Wl 8

x

Stress at ends, Wl

Case 19. — Fixed at Both Ends, Load at any Point

s=

Wb 2 6 al − x ^l + 2a h@ Zl 3

For segment of length b,

s=

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x a

Wb 2 (l + 2a) l3

W l

b

v

Wa 2b l2

Wa2 (l + 2b) l3

Maximum deflection, at load,

Wl 3 192EI

These are the maximum stresses and are equal and opposite. Stress is zero at x = 1∕4  l

W 2

For segment of length a,

Wab 2 l2

Wx 2 y = 48EI ^3l − 4x h

8Z Stress at load, − Wl 8Z

Deflections at Critical Pointsa

Wa 2 6bl − v ^ l + 2bh@ Zl 3

Stress at end next to

segment of length a,

Wa b2 Zl 2

Stress at end next to segment of length b,

Wa 2 b Zl 2

Maximum stress is at end next to shorter segment. Stress is zero at and

al x = l + 2a bl v = l + 2b

Greatest negative stress, at load,

−

2Wa2 b 2 Zl 3

For segment of length a,

y=

Wx 2 b2 6 ^ 2a l − x h + l ^a − x h@ 6EIl 3

For segment of length b,

y=

Wv 2 a2 6 ^ 2b l − vh + l^b − vh@ 6EIl3

Deflection at load,

Wa3 b 3 3EIl 3

Let b be the length of the longer segment and a of the shorter one. The maximum deflection is in the longer segment, at

and is

2bl v = l + 2b

2Wa 2 b3 3EI ^l + 2bh2

Machinery's Handbook, 31st Edition Beam Stress and Deflection Tables

Wl 8

264

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Table Table 1. (Continued) Stresses and Deflections in Beams 1. Stresses and Deflections in Beams

Deflections Stresses General Formula for Stress at any Point Stresses at Critical Points General Formula for Deflection at any Pointa Case 20. — Fixed at Both Ends, Uniform Load

Type of Beam

Wl 1 x x 2 s = 2Z & 6 − l + ` l j 0

Total Load W

Wl 12

x

Wl 12

l

W 2

Maximum stress, at ends,

Wx 2 y = 24EIl ^l − x h2

Wl 12Z

center, −

Wl 24Z

Case 21. — Continuous Beam, with Two Unequal Spans, Unequal, Uniform Loads

Between R1 and R,

l − x ^l1 − x h W1 s = 1Z ( − R1 2 2l1



W1 + W2 1 + 2 8





W1l1 W2l2 + l2 l1



s=

l2 − u ^l2 − u h W2 ( − R 22 Z 2l2

Stress at support R,

W1 l 12 + W2 l 22 8Z ^l1 + l2h Greatest stress in the first span is at

x=

l1 ^W1 − R1 h W1

and is −

R21 l1 2ZW1

Greatest stress in the sec­ond span is at

u =

l2 ^ W − R2 h W2 2

and is, −

Between R1 and R,

"

x ^l1 − x h ^2l1 − x h^4R 1 − W1 h y = 24 EI W ^l − x h2 − 1 1 l1 Between R2 and R,

y=

"

"

u ^l2 − u h ^2l2 − u h^4R 2 − W2 h 24EI W ^l − u h2 − 2 2 l2

"

This case is so complicated that convenient general expressions for the critical deflections cannot be obtained.

R22 l 2 2ZW2

265

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Total Load W1 Total Load W2 R R1 R2 x u l1 l2 l1 W1 (3l1 + 4l2) – W2l 22 l2 W2 (3l2 + 4l1) – W1l 21 8l2(l1 + l2) 8l1(l1 + l2)

Between R2 and R,

Maximum deflection, at center,

Wl 3 384EI

Stress is zero at x = 0.7887l and at x = 0.2113l Greatest negative stress, at

W 2

Deflections at Critical Pointsa

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TableTable 1. (Continued) Stresses and Deflections in Beams 1. Stresses and Deflections in Beams

Stresses Deflections General Formula for Stress at any Point Stresses at Critical Points General Formula for Deflection at any Pointa Case 22. — Continuous Beam, with Two Equal Spans, Uniform Load

Type of Beam

s=

W ^l − x h 1 a l − xk 4 2Zl

Maximum stress at

point A,

y=

Wx 2 ^l − x h ^3l − 2x h 48EIl

Stress is zero at x = 1∕4 l

3 W 8

l

x

5 W 4

x

9 Wl − 128 Z

Deflection at point of greatest negative stress, at x = 5∕8 l is

Wl 3 187EI

Case 23. — Continuous Beam, with Two Equal Spans, Equal Loads at Center of Each

Between point A and load,

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W s = 16Z ^3l − 11x h

W

B 12

5 W 16

 l

x

A

12

11 W 16

W

x

12

 l

B

12

5 W 16

Wl 3 185EI

Wl 3 192EI

3 W 8

l

Maximum deflection is at x = 0.5785l, and is

Deflection at center of span,

Greatest negative stress is at x = 5∕8 l and is,

Total Load on Each Span, W A

Deflections at Critical Pointsa

Between point B and load,

5 Wv s = − 16 Z

Maximum stress at

point A,

3 Wl 16 Z

Stress is zero at

3 x = 11 l

Greatest negative stress at center of span,

5 Wl − 32 Z

Between point A and load,

Wx 2 y = 96EI ^9l − 11x h

Between point B and load,

2 2 Wv y = 96EI ^3l − 5v h

Maximum deflection is at v = 0.4472l, and is

Wl 3 107.33EI

Deflection at load,

7 Wl 3 768 EI

Machinery's Handbook, 31st Edition Beam Stress and Deflection Tables

Wl 8Z

266

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Table Table 1. (Continued) Stresses and Deflections in Beams 1. Stresses and Deflections in Beams

Deflections Stresses General Formula for Stress at any Point Stresses at Critical Points General Formula for Deflection at any Pointa Case 24. — Continuous Beam, with Two Unequal Spans, Unequal Loads at any Point of Each

Type of Beam

Between R1 and W1,

wr s = − Z1

m=

1 2(l1 + l 2) R1

w a1

W1a1b1 Wab (l1 + a1) + 2 2 2 (l2 + a2) l1 l2

W1

l1

W2

R

u b1

x b2

l2

a2

v

R2

=r

1 6 ^ m l1 − u h − W1 a1 u @ l1 Z

Between R and W2 , s =

1 6 ^ m l 2 − x h − W2 a2 x @ l2 Z

Between R2 and W2 ,

W1b1 – m W1a1 + m W2a2 + m W2b2 – m + l1 l1 l2 l2 = r1

Between R and W1, s =

= r2

vr s = − Z2

Stress at load W1,

−

a1 r1 Z

Stress at support R,

m Z

Stress at load W2 ,

ar − 2Z 2

The greatest of these is the maximum stress.

Between R1 and W1,

W b3 w y = 6EI '^l1 − wh^l1 + wh r1 − 1 1 1 l1 Between R and W1,

u 6 W a b ^l + a h 6EIl1 1 1 1 1 1 − W1 a 1 u2 − m ^2l1 − u h^l1 − u h@

y=

Between R and W2

x 6 W a b ^l + a h 6EIl2 2 2 2 2 2 − W2 a 2 x 2 − m ^2l2 − x h^l2 − x h@

y=

Between R2 and W2 ,

Deflections at Critical Pointsa Deflection at load W1,

a1 b1 62a1 b1W1 6EIl1 − m ^l1 + a1 h @

Deflection at load W2 ,

a2 b2 62a b W 6EIl2 2 2 2 − m ^l2 + a 2h@

This case is so complicated that convenient general expressions for the maximum deflections cannot be obtained.

W b3 v y = 6EI '^l2 − vh^l2 + vh r2 − 2 2 1 l2

267

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a The deflections apply only to cases where the cross section of the beam is constant for its entire length. In the diagrammatical illustrations of the beams and their loading, the values indicated near, but below, the supports are the “reactions” or upward forces at the sup­ports. For Cases 1 to 12, inclusive, the reactions, as well as the formulas for the stresses, are the same whether the beam is of constant or variable cross section. For the other cases, the reactions and the stresses given are for constant cross section beams only. The bending moment at any point in inch-pounds (newton-meters if metric units are used) is s 3 Z and can be found by omitting the divisor Z in the formula for the stress given in the tables. A positive value of the bending moment denotes tension in the upper fibers and compression in the lower ones. A negative value denotes the reverse, The value of W corresponding to a given stress is found by transposition of the formula. For example, in Case 1, the stress at the critical point is s = − Wl ÷ 8Z. From this formula we find W = − 8Zs ÷ l. Of course, the negative sign of W may be ignored.

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TableTable 1. (Continued) Stresses and Deflections in Beams 1. Stresses and Deflections in Beams

Machinery's Handbook, 31st Edition Rectangular and Round Solid Beams

268

In Table 1, if there are several kinds of loads, as, for instance, a uniform load and a load at any point, or separate loads at different points, the total stress and the total deflection at any point is found by adding together the various stresses or deflections at the point considered due to each load acting by itself. If the stress or deflection due to any one of the loads is negative, it must be subtracted instead of added. Table 2a and Table 2b give expressions for determining dimensions of rectangular and round beams in terms of beam stresses and load. Table 2a. Rectangular Solid Beams

Style of Loading and Support

h

Stress in Extreme Beam Height, h Fibers, f Beam Length, l inch (mm) inch (mm) lb/in2  (N/mm2)

Breadth of Beam, b inch (mm)

Beam fixed at one end, loaded at the other

6lW =b fh 2

l

6lW bf = h

6lW =f bh 2

Total Load, W lb (N)

bfh 2 6W = l

bfh 2 6l = W

Beam fixed at one end, uniformly loaded

h

3lW =b fh2

l

3lW bf = h

3lW =f bh 2

bfh 2 3W = l

bfh 2 3l = W

Beam supported at both ends, single load in middle

h

3lW =b 2fh2

l

3lW 2bf = h

3lW =f 2bh 2

2bfh 2 3W = l

2bfh 2 3l = W

Beam supported at both ends, uniformly loaded

h

3lW =b 4fh2

l a

h l

6Wa c =b fh 2l

l

4bfh 2 3W = l

4bfh 2 3l = W

6Wa c bfl = h

6Wa c =f bh 2l

a+c=l

bh2fl 6ac = W

Beam supported at both ends, two symmetrical loads

a h

3lW =f 4bh2

Beam supported at both ends, single unsymmetrical load

c

a

3lW 4bf = h

3Wa =b fh 2

3Wa bf = h

3Wa =f bh 2

l, any length

bh2f 3W = a

bh2f 3a = W

Deflection of Beam Uniformly Loaded for Part of Its Length.—In the following for­ mulas, lengths are in inches, weights in pounds. W = total load; L = total length between supports; E = modulus of elasticity; I = moment of inertia of beam section; a = fraction of length of beam at each end, that is not loaded = b ÷ L; and f = deflection.

f =

WL3 ^5 − 24a2 + 16a4 h 384EI ^1 − 2a h

The expression for maximum bending moment is: Mmax = 1∕8WL (1 + 2a).

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Machinery's Handbook, 31st Edition RECTANGULAR AND ROUND SOLID BEAMS

269

Table 2b. Round Solid Beams Style of Loading and Support

d

Diameter of Beam, d inch (mm)

Stress in Extreme Fibers, f lb/in2  (N/mm2)

Beam Length, l inch (mm)

Total Load, W lb (N)

10.18lW =f d3

d 3f 10.18W = l

d 3f 10.18l = W

Beam fixed at one end, loaded at the other

3

l

10.18lW =d f

Beam fixed at one end, uniformly loaded

d

3

l

5.092Wl =d f

5.092Wl =f d3

d 3f 5.092W = l

d 3f 5.092l = W

Beam supported at both ends, single load in middle

d

3

l

2.546Wl =d f

2.546Wl =f d3

d 3f 2.546W = l

d 3f 2.546l = W

Beam supported at both ends, uniformly loaded

d

3

l a

1.273Wl =d f

d

3

l

10.18Wa c =d fl

l

d 3f 1.273l = W

10.18Wa c =f d 3l

a+c=l

d 3fl 10.18ac = W

Beam supported at both ends, two symmetrical loads

a d

d 3f 1.273W = l

Beam supported at both ends, single unsymmetrical load

c

a

1.273Wl =f d3

3

5.092Wa =d f

5.092Wa =f d3

l, any length

d 3f 5.092W = a

d 3f 5.092a = W

These formulas apply to simple beams resting on supports at the ends. b

W

b

L

If the formulas are used with metric SI units, W = total load in newtons (N); L = total length between supports in millimeters; E = modulus of elasticity in newtons per mil­limeter2 (N/mm 2); I = moment of inertia of beam section in mm4; a = fraction of length of beam at each end, that is not loaded = b ÷ L; and f = deflection in mm. The bending moment Mmax is in newton-millimeters (N·mm).

Note: A load due to the weight of a mass of M kilograms is Mg newtons, where g = approximately 9.81 m/s2.

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270

Machinery's Handbook, 31st Edition BENDING STRESS AND OBLIQUE FORCE

Bending Stress Due to an Oblique Transverse Force.—The following illustration shows a beam and a channel being subjected to a transverse force acting at an angle φ to the center of gravity. To find the bending stress, the moments of inertia I around axes 3-3 and 4-4 are computed from the following equations: I3 = Ix sin2φ + Iy cos2φ, and I4 = Ix cos2φ + Iy sin2φ. y The computed bending stress fb is then found from fb = M c sin z + x cosz m where Iy I x M is the bending moment due to force F. F

Y x



X



3

3 X

F

Y x 3

y

X

y 4

Y

X

4

3

Y

Beams of Uniform Strength Throughout Their Length.—The bending moment in a beam is generally not uniform throughout its length, but varies. Therefore, a beam of uni­form cross section which is made strong enough at its most strained section will have an excess of material at every other section. Sometimes it may be desirable to have the cross section uniform, but at other times the metal can be more advantageously distributed if the beam is so designed that its cross section varies from point to point, so that it is at every point just great enough to take care of the bending stresses at that point. Table 3a and Table 3b are given showing beams in which the load is applied in different ways and which are sup­ported by different methods, and the shape of the beam required for uniform strength is indicated. It should be noted that the shape given is the theoretical shape required to resist bending only. It is apparent that sufficient cross section of beam must also be added either at the points of support (in beams supported at both ends), or at the point of application of the load (in beams loaded at one end), to take care of the vertical shear. It should be noted that the theoretical shapes of the beams given in the two tables that fol­low are based on the stated assumptions of uniformity of width or depth of cross section, and unless these are observed in the design the theoretical outlines do not apply without modifications. For example, in a cantilever with the load at one end, the outline is a parab­ola only when the width of the beam is uniform. It is not correct to use a strictly parabolic shape when the thickness is not uniform, as, for instance, when the beam is made of an I- or T-section. In such cases, some modification may be necessary; but it is evident that what­ever the shape adopted, the correct depth of the section can be obtained by an investigation of the bending moment and the shearing load at a number of points, and then a line can be drawn through the points thus ascertained, which will provide for a beam of practically uniform strength whether the cross section be of uniform width or not.

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Machinery's Handbook, 31st Edition Beams of Uniform Strength

271

Table 3a. Beams of Uniform Strength Throughout Their Length Type of Beam

Description

d l h

P

d l h

P

b

Formulaa

Load at one end. Width of beam uniform. Depth of beam decreasing towards loaded end. Outline of beam-shape, parabola with vertex at loaded end.

Load at one end. Width of beam uniform. Depth of beam decreasing towards loaded end. Outline of beam, one-half of a parabola with vertex at loaded end. Beam may be reversed so that upper edge is parabolic.

Load at one end. Depth of beam uniform. Width of beam decreasing towards loaded end. Outline of beam triangular, with apex at loaded end.

l h

P=

Sbh 2 6l

P=

Sbh 2 6l

P=

Sbh 2 6l

P=

Sbh 2 6l

P=

Sbh 2 3l

P=

Sbh 2 3l

P b

1 h Beam of approximately uniform strength. 2 Load at one end. Width of beam uniform. Depth

l h

b h

P

l Total Load = P

b l Total Load = P h

of beam decreasing towards loaded end, but not tapering to a sharp point.

Uniformly distributed load. Width of beam uniform. Depth of beam decreasing towards outer end. Outline of beam, right-angle trian­gle.

Uniformly distributed load. Depth of beam uniform. Width of beam gradually decreasing towards outer end. Outline of beam is formed by two parabolas which are tangent to each other at their vertexes at the outer end of the beam.

a In the formulas, P = load in pounds; S = safe stress in lb/in 2; and a, b, c, h, and l are in inches. If metric SI units are used, P is in newtons (N); S = safe stress in N/mm 2; and a, b, c, h, and l are in millimeters.

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Machinery's Handbook, 31st Edition Beams of Uniform Strength

272

Table 3b. Beams of Uniform Strength Throughout Their Length Type of Beam

b l h

a

c P

l

b

a

c b P

b l

h P

b l

h P

b l

h

Description

Formulaa

Beam supported at both ends. Load concentrated at any point. Depth of beam uniform. Width of beam maximum at point of loading. Outline of beam, two triangles with apexes at points of sup­port.

Beam supported at both ends. Load concentrated at any point. Width of beam uniform. Depth of beam maximum at point of loading. Outline of beam is formed by two parabolas with their ver­texes at points of support.

Beam supported at both ends. Load concentrated in the middle. Depth of beam uniform. Width of beam maximum at point of loading. Outline of beam, two triangles with apexes at points of sup­port.

Beam supported at both ends. Load concentrated at center. Width of beam uniform. Depth of beam maximum at point of loading. Outline of beam, two parabolas with vertices at points of support.

Beam supported at both ends. Load uniformly distributed. Depth of beam uniform. Width of beam maximum at center. Outline of beam, two parabolas with vertexes at middle of beam.

Sbh 2l P = 6ac

Sbh 2l P = 6ac

P=

2Sbh 2 3l

P=

2Sbh 2 3l

P=

4Sbh 2 3l

P=

4Sbh 2 3l

Total Load = P

l

b

h

Beam supported at both ends. Load uniformly distributed. Width of beam uniform. Depth of beam maximum at center. Outline of beam one-half of an ellipse.

Total Load = P a For details of English and metric SI units used in the formulas, see footnote on page 271.

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Machinery's Handbook, 31st Edition Deflection in Beam Design

273

Deflection as a Limiting Factor in Beam Design.—For some applications, a beam must be stronger than required by the maximum load it is to support in order to prevent exces­ sive deflection. Maximum allowable deflections vary widely for different classes of ser­ vice, so a general formula for determining them cannot be given. When exceptionally stiff girders are required, one rule is to limit the deflection to 1 inch per 100 feet of span; hence, if l = length of span in inches, deflection = l ÷ 1200. According to another formula, deflec­tion limit = l ÷ 360 where beams are adjacent to materials like plaster, which would be bro­ken by excessive beam deflection. Some machine parts of the beam type must be very rigid to maintain alignment under load. For example, the deflection of a punch press column may be limited to 0.010 inch or less. These examples merely illustrate variations in prac­tice. It is impracticable to give general formulas for determining the allowable deflection in any specific application because the allowable amount depends on the conditions gov­erning each class of work.

Procedure in Designing for Deflection: Assume that a deflection equal to l ÷ 1200 is to be the limiting factor in selecting a wide-flange (W-shape) beam having a span length of 144 inches. Supports are at both ends and load at center is 15,000 pounds. Deflection y is to be limited to 144 ÷ 1200 = 0.12 inch. According to the formula on page 257 (Case 2), in which W = load on beam in pounds, l = length of span in inches, E = modulus of elasticity of material in psi, and I = moment of inertia of cross section in inches4:

15, 000 # 1443 Wl 3 Wl 3 Deflection y = 48EI hence, I = 48yE = 48 # 0.12 # 29, 000, 000 = 268.1

A structural wide-flange beam, see Steel Wide-Flange Sections on page 2694, having a depth of 12 inches and weighing 35 pounds per foot has a moment of inertia I of 285 and a section modulus (Z or S) of 45.6. Checking now for maximum stress s (Case 2, page 257): 15, 000 # 144 Wl s = 4Z = = 11, 842 lbs/in2 4 # 46.0 Although deflection is the limiting factor in this case, the maximum stress is checked to make sure that it is within the allowable limit. As the limiting deflection is decreased, for a given load and length of span, the beam strength and rigidity must be increased, and, con­sequently, the maximum stress is decreased. Thus, in the preceding example, if the maxi­mum deflection is 0.08 inch instead of 0.12 inch, then the calculated value for the moment of inertia I will be 402; hence a W 12 3 53 beam having an I value of 426 could be used (nearest value above 402). The maximum stress then would be reduced to 7640 lb/in2 and the calculated deflection is 0.076 inch. A similar example using metric SI units is as follows. Assume that a deflection equal to l ÷ 1000 mm is to be the limiting factor in selecting a W-beam having a span length of 5 meters. Supports are at both ends and the load at the center is 30 kN. Deflection y is to be limited to 5000 ÷ 1000 = 5 mm. The formula on page 257 (Case 2) is applied, and W = load on beam in N; l = length of span in mm; E = modulus of elasticity (assume 200,000 N/mm 2 in this example); and I = moment of inertia of cross section in mm4. Thus,

hence

Wl 3 Deflection y = 48EI 30, 000 # 5000 3 Wl 3 I = 48yE = 48 # 5 # 200, 000 = 78, 125, 000 mm4

Although deflection is the limiting factor in this case, the maximum stress is checked to make sure that it is within the allowable limit using the formula from page 257 (Case 2):

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274

Machinery's Handbook, 31st Edition Curved Beams

Wl s = 4Z The units of s are newtons per square millimeter; W is the load in newtons; l is the length in mm; and Z = section modulus of the cross section of the beam = I ÷ dis­ tance in mm from neutral axis to extreme fiber. Curved Beams.—The formula S = Mc/I used to compute stresses due to bending of beams is based on the assumption that the beams are straight before any loads are applied. In beams having initial curvature, however, the stresses may be considerably higher than pre­dicted by the ordinary straight-beam formula because the effect of initial curvature is to shift the neutral axis of a curved member in from the gravity axis toward the center of cur­vature (the concave side of the beam). This shift in the position of the neutral axis causes an increase in the stress on the concave side of the beam and decreases the stress at the outside fibers. Hooks, press frames, and other machine members which as a rule have a rather pro­ nounced initial curvature may have a maximum stress at the inside fibers of up to about 1 3 ∕2 times that predicted by the ordinary straight-beam formula.

Stress Correction Factors for Curved Beams: A simple method for determining the maximum fiber stress due to bending of curved members consists of 1) calculating the maximum stress using the straight-beam formula S = Mc/I; and; and 2) multiplying the calculated stress by a stress correction factor. Table 4 on page 275 gives stress correction factors for some of the common cross sections and proportions used in the design of curved members.

An example in the application of the method using English units of measurement is given at the bottom of the table. A similar example using metric SI units is as follows: The fiber stresses of a curved rectangular beam are calculated as 40 newtons per millime­ter2, using the straight beam formula, S = Mc/I. If the beam is 150 mm deep and its radius of curvature is 300 mm, what are the true stresses? R/c = 300 ∕ 75 = 4. From Table 4 on page 275, the K factors corresponding to R/c = 4 are 1.20 and 0.85. Thus, the inside fiber stress is 40 3 1.20 = 48 N/mm 2 = 48 megapascals (MPa); and the outside fiber stress is 40 3 0.85 = 34 N/mm 2 = 34 MPa. Approximate Formula for Stress Correction Factor: The stress correction factors given in Table 4 on page 275 were determined by Wilson and Quereau and published in the Uni­versity of Illinois Engineering Experiment Station Circular No. 16, “A Simple Method of Determining Stress in Curved Flexural Members.” In this same publication the authors indicate that the following empirical formula may be used to calculate the value of the stress correction factor for the inside fibers of sections not covered by the tabular data to within 5 percent accuracy, except in triangular sections where up to 10 percent deviation may be expected. However, for most engineering calculations, this formula should prove satisfactory for general use in determining the factor for the inside fibers.

K = 1.00 + 0.5

I : 1 1 + D bc 2 R − c R

(Use 1.05 instead of 0.5 in this formula for circular and elliptical sections.) I =  moment of inertia of section about centroidal axis b =  maximum width of section c =  distance from centroidal axis to inside fiber, i.e., to the extreme fiber nearest the center of curvature R =  radius of curvature of centroidal axis of beam

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Machinery's Handbook, 31st Edition Curved Beams

275

Table 4. Values of Stress Correction Factor K for Various Curved Beam Sections Section

R∕c

R

h

c

c R

b

c 2b

b

R 3b

c 2b

b R

5b

c

b

4b R 3 5b c b R

1.2 1.4 1.6 1.8 2.0 3.0 4.0 6.0 8.0 10.0 1.2 1.4 1.6 1.8 2.0 3.0 4.0 6.0 8.0 10.0 1.2 1.4 1.6 1.8 2.0 3.0 4.0 6.0 8.0 10.0 1.2 1.4 1.6 1.8 2.0 3.0 4.0 6.0 8.0 10.0 1.2 1.4 1.6 1.8 2.0 3.0 4.0 6.0 8.0 10.0 1.2 1.4 1.6 1.8 2.0 3.0 4.0 6.0 8.0 10.0

Factor K Inside Outside Fiber Fiber 3.41 .54 2.40 .60 1.96 .65 1.75 .68 1.62 .71 1.33 .79 1.23 .84 1.14 .89 1.10 .91 1.08 .93 2.89 .57 2.13 .63 1.79 .67 1.63 .70 1.52 .73 1.30 .81 1.20 .85 1.12 .90 1.09 .92 1.07 .94 3.01 .54 2.18 .60 1.87 .65 1.69 .68 1.58 .71 1.33 .80 1.23 .84 1.13 .88 1.10 .91 1.08 .93 3.09 .56 2.25 .62 1.91 .66 1.73 .70 1.61 .73 1.37 .81 1.26 .86 1.17 .91 1.13 .94 1.11 .95 3.14 .52 2.29 .54 1.93 .62 1.74 .65 1.61 .68 1.34 .76 1.24 .82 1.15 .87 1.12 .91 1.10 .93 3.26 .44 2.39 .50 1.99 .54 1.78 .57 1.66 .60 1.37 .70 1.27 .75 1.16 .82 1.12 .86 1.09 .88

a

y0 .224R .151R .108R .084R .069R .030R .016R .0070R .0039R .0025R .305R .204R .149R .112R .090R .041R .021R .0093R .0052R .0033R .336R .229R .168R .128R .102R .046R .024R .011R .0060R .0039R .336R .229R .168R .128R .102R .046R .024R .011R .0060R .0039R .352R .243R .179R .138R .110R .050R .028R .012R .0060R .0039R .361R .251R .186R .144R .116R .052R .029R .013R .0060R .0039R

Section

R∕c

32

41 2 t

t

4t

t c

t

R

3t 2t t 6t

4t c

R

4t

t 3t

t c

t

3t R

2d d c R t 2

4t

4t

t 2t t 2

t c R

1.2 1.4 1.6 1.8 2.0 3.0 4.0 6.0 8.0 10.0 1.2 1.4 1.6 1.8 2.0 3.0 4.0 6.0 8.0 10.0 1.2 1.4 1.6 1.8 2.0 3.0 4.0 6.0 8.0 10.0 1.2 1.4 1.6 1.8 2.0 3.0 4.0 6.0 8.0 10.0 1.2 1.4 1.6 1.8 2.0 3.0 4.0 6.0 8.0 10.0

Factor K Inside Outside Fiber Fiber 3.63 .58 2.54 .63 2.14 .67 1.89 .70 1.73 .72 1.41 .79 1.29 .83 1.18 .88 1.13 .91 1.10 .92 3.55 .67 2.48 .72 2.07 .76 1.83 .78 1.69 .80 1.38 .86 1.26 .89 1.15 .92 1.10 .94 1.08 .95 2.52 .67 1.90 .71 1.63 .75 1.50 .77 1.41 .79 1.23 .86 1.16 .89 1.10 .92 1.07 .94 1.05 .95 3.28 .58 2.31 .64 1.89 .68 1.70 .71 1.57 .73 1.31 .81 1.21 .85 1.13 .90 1.10 .92 1.07 .93 2.63 .68 1.97 .73 1.66 .76 1.51 .78 1.43 .80 1.23 .86 1.15 .89 1.09 .92 1.07 .94 1.06 .95

y0 a .418R .299R .229R .183R .149R .069R .040R .018R .010R .0065R .409R .292R .224R .178R .144R .067R .038R .018R .010R .0065R .408R .285R .208R .160R .127R .058R .030R .013R .0076R .0048R .269R .182R .134R .104R .083R .038R .020R .0087R .0049R .0031R .399R .280R .205R .159R .127R .058R .031R .014R .0076R .0048R

Example: The fiber stresses of a curved rectangular beam are calculated as 5000 psi using the straight beam for­mula, S = Mc/I. If the beam is 8 inches deep and its radius of curvature is 12 inches, what are the true stresses? Solution: R/c = 12 ∕ 4 = 3. The factors in the table corresponding to R/c = 3 are 0.81 and 1.30. Outside fiber stress = 5000 3 0.81 = 4050 psi; inside fiber stress = 5000 3 1.30 = 6500 psi.

a y is the distance from the centroidal axis to the neutral axis of curved beams subjected to pure 0 bending and is measured from the centroidal axis toward the center of curvature.

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Machinery's Handbook, 31st Edition Curved Beams

276

The accompanying diagram shows the dimensions of a clamp frame of rectan­gular cross section. Determine the maximum stress at points A and B due to a clamping force of 1000 pounds. 1,000 lbs 4 A

R

2

B6

24 c 1,000 lbs

The cross-sectional area = 2 3 4 = 8 square inches; the bending moment at section AB is 1000 (24 + 6 + 2) = 32,000 inch pounds; the distance from the center of gravity of the sec­tion at AB to point B is c = 2 inches; and using the formula on page 242, the moment of iner­tia of the section is 2 3 (4)3 ÷ 12 = 10.667 inches4. Using the straight-beam formula, page 274, the stress at points A and B due to the bend­ ing moment is: Mc 32, 000 # 2 S = I = 10.667 = 6000 psi The stress at A is a compressive stress of 6000 psi and that at B is a tensile stress of 6000 psi. These values must be corrected to account for the curvature effect. In Table 4 on page 275 for R/c = (6 + 2)/2 = 4, the value of K is found to be 1.20 and 0.85 for points B and A respectively. Thus, the actual stress due to bending at point B is 1.20 3 6000 = 7200 psi in tension, and the stress at point A is 0.85 3 6000 = 5100 psi in compression. To these stresses at A and B must be added, algebraically, the direct stress at section AB due to the 1000-pound clamping force. The direct stress on section AB will be a tensile stress equal to the clamping force divided by the section area. Thus 1000 ÷ 8 = 125 psi in tension. The maximum unit stress at A is, therefore, 5100 − 125 = 4975 psi in compression, and the maximum unit stress at B is 7200 + 125 = 7325 psi in tension. The following is a similar calculation using metric SI units, assuming that it is required to determine the maximum stress at points A and B due to clamping force of 4 kN acting on the frame. The frame cross section is 50 by 100 mm, the radius R = 200 mm, and the length of the straight portions is 600 mm. Thus, the cross-sectional area = 50 3 100 = 5000 mm2; the bending moment at AB is 4000 (600 + 200) = 3,200,000 newton-millimeters; the distance from the center of gravity of the section at AB to point B is c = 50 mm; and the moment of inertia of the section is (from the formula on page 242), 50(100 3)⁄12 = 4,170,000 mm4. Using the straight-beam formula, page 274, the stress at points A and B due to the bending moment is: Mc 3, 200, 000 # 50 s= I = 4, 170, 000 = 38.4 newtons per millimeter2 = 38.4 megapascals The stress at A is a compressive stress of 38.4 N/mm2, while that at B is a tensile stress of 38.4 N/mm 2. These values must be corrected to account for the curvature

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Machinery's Handbook, 31st Edition Size of Rail to Carry Load

277

effect. From the table on page 275, the K factors are 1.20 and 0.85 for points A and B, respectively, derived from R/c = 200 ∕ 50 = 4. Thus, the actual stress due to bending at point B is 1.20 3 38.4 = 46.1 N/mm 2 (46.1 MPa) in tension; and the stress at point A is 0.85 3 38.4 = 32.6 N/mm 2 (32.6 MPa) in compression.

To these stresses at A and B must be added, algebraically, the direct stress at section AB due to the 4 kN clamping force. The direct stress on section AB will be a tensile stress equal to the clamping force divided by the section area. Thus, 4000 ∕ 5000 = 0.8 N/mm 2. The maximum unit stress at A is, therefore, 32.61 − 0.8 = 31.8 N/mm 2 (31.8 MPa) in compression, and the maximum unit stress at B is 46.1 + 0.8 = 46.9 N/ mm2 (46.9 MPa) in tension.

Size of Rail Necessary to Carry a Given Load.—The following formulas may be employed for determining the size of rail and wheel suitable for carrying a given load. Let A = the width of the head of the rail in inches; B = width of the tread of the rail in inches; C = the wheel-load in pounds; D = the diameter of the wheel in inches. A B 5 16

”

R

Then the width of the tread of the rail in inches is found from the formula: C B = 1250D

(1)

The width A of the head equals B + ∕8 inch. The diameter D of the smallest track wheel that will safely carry the load is found from the formula: C D = A# K (2) 5

in which K = 600 to 800 for steel castings; K = 300 to 400 for cast iron.

As an example, assume that the wheel-load is 10,000 pounds; the diameter of the wheel is 20 inches; and the material is cast steel. Determine the size of rail necessary to carry this load. From Formula (1):

10,000 B = 1250 # 20 = 0.4 inch

The width of the rail required equals 0.4 + 5∕8 inch = 1.025 inch. Determine also whether a wheel 20 inches in diameter is large enough to safely carry the load. From Formula (2):

10,000 1 D = 1.025 # 600 = 16 4 inches

This is the smallest diameter of track wheel that will safely carry the load; hence a 20-inch wheel is ample.

American Railway Engineering Association Formulas.—The American Railway Engineering Association recommends for safe operation of steel cylinders rolling on steel plates that the allowable load p in pounds per inch of length of the cylinder should not exceed the value calculated from the formula

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Machinery's Handbook, 31st Edition Stresses Produced by Shocks

278 p=

y.s. − 13,000 20,000 600d for diameter d less than 25 inches

This formula is based on steel having a yield strength, y.s., of 32,000 pounds per square inch. For roller or wheel diameters of up to 25 inches, the Hertz stress (contact stress) resulting from the calculated load p will be approximately 76,000 pounds per square inch. For a 10-inch diameter roller the safe load per inch of roller length is

p=

32,000 − 13,000 600 # 10 = 5700 lbs per inch of length 20,000

Therefore, to support a 10,000 pound load the roller or wheel would need to be 10,000 ∕ 5700 = 1.75 inches wide. Stresses Produced by Shocks

Stresses in Beams Produced by Shocks.—Any elastic structure subjected to a shock will deflect until the product of the average resistance developed by the deflection and the dis­ tance through which it has been overcome has reached a value equal to the energy of the shock. It follows that for a given shock, the average resisting stresses are inversely propor­ tional to the deflection. If the structure were perfectly rigid, the deflection would be zero and the stress infinite. The effect of a shock is, therefore, to a great extent dependent upon the elastic property (the springiness) of the structure subjected to the impact.

The energy of a body in motion, such as a falling body, may be spent in each of four ways:

1) In deforming the body struck as a whole.

2) In deforming the falling body as a whole.

3) In partial deformation of both bodies on the surface of contact (most of this energy will be transformed into heat).

4) Part of the energy will be taken up by the supports, if these are not perfectly rigid and inelastic.

How much energy is spent in the last three ways is usually difficult to determine, and for this reason it is safest to figure as if the whole amount were spent as in Case 1. If a reli­able judgment is possible as to what percentage of the energy is spent in other ways than the first, a corresponding fraction of the total energy can be assumed as developing stresses in the body subjected to shocks. One investigation into the stresses produced by shocks led to the following conclusions:

1) A suddenly applied load will produce the same deflection, and, therefore, the same stress as a static load twice as great; and 2) the unit stress p (see formulas in Table 1, Stresses Produced in Beams by Shocks) for a given load producing a shock varies directly as the square root of the modulus of elasticity E and inversely as the square root of the length L of the beam and the area of the section. Thus, for instance, if the sectional area of a beam is increased by four times, the unit stress will diminish only by half. This result is entirely different from those produced by static loads where the stress would vary inversely with the area, and within certain limits be practically independent of the modulus of elasticity.

In Table 1, the expression for the approximate value of p, which is applicable whenever the deflection of the beam is small as compared with the total height h through which the body producing the shock is dropped, is always the same for beams supported at both ends and subjected to shock at any point between the supports. In the formulas all dimensions are in inches and weights in pounds.

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Machinery's Handbook, 31st Edition Stresses Produced by Shocks

279

Table 1. Stresses Produced in Beams by Shocks Method of Support and Point Struck by Falling Body

Fiber (Unit) Stress p Produced by Weight Q Dropped Through a Distance h

Approximate Value of p

Supported at both ends; struck in center.

QaL p = 4I c 1 +

1+

96hEI m QL3

p= a

6QhE LI

Fixed at one end; struck at the other.

p=

QaL c I 1+

1+

6hEI m QL3

p= a

6QhE LI

384hEI m QL3

p= a

6QhE LI

Fixed at both ends; struck in center.

QaL p = 8I c 1 +

1+

I = moment of inertia of section; a = distance of extreme fiber from neutral axis; L = length of beam; E = modulus of elasticity.

If metric SI units are used, p is in newtons per square millimeter; Q is in newtons; E = modulus of elasticity in N/mm2; I = moment of inertia of section in mm4; and h, a, and L in mm. Note: If Q is given in kilograms, the value referred to is mass. The weight Q of a mass M kilograms is Mg newtons, where g = approximately 9.81 m/s2. Examples of How Formulas for Stresses Produced by Shocks are Derived: The general formula from which specific formulas for shock stresses in beams, springs, and other machine and structural members are derived is:

2h (1) 1+ y m In this formula, p = stress in psi due to shock caused by impact of a moving load; ps = stress in psi resulting when moving load is applied statically; h = distance in inches that load falls before striking beam, spring, or other mem­ber; y = deflection in inches resulting from static load. As an example of how Formula (1) may be used to obtain a formula for a specific ap­ pli­cation, suppose that the load W shown applied to the beam in Case 2 on page 257 were dropped on the beam from a height of h inches instead of being gradually applied (static loading). The maximum stress ps due to load W for Case 2 is given as Wl ÷ 4Z and the max­imum deflection y is given as Wl3 ÷ 48 EI. Substituting these values in Formula (1), p = ps c 1 +

2h m = Wl c 1 + 1 + 96hEI m (2) 4Z Wl 3 ' 48EI Wl 3 If in Formula (2) the letter Q is used in place of W and if Z, the section modulus, is replaced by its equivalent, I ÷ distance a from neutral axis to extreme fiber of beam, then Formula (2) becomes the first formula given in the accompanying Table 1, Stresses Produced in Beams by Shocks. Stresses in Helical Springs Produced by Shocks.—A load suddenly applied on a spring will produce the same deflection, and, therefore, also the same unit stress, as a static load twice as great. When the load drops from a height h, the stresses are as given in the accom­ panying Table 2. The approximate values are applicable when the deflection is small as compared with the height h. The formulas show that the fiber stress for a given shock will be greater in a spring made from a square bar than in one made from a round bar, if the diameter of coil is the same and the side of the square bar equals the diameter of the round Wl p = 4Z c 1 +

1+

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Machinery's Handbook, 31st Edition Stresses Produced by Shocks

280

bar. It is, therefore, more economical to use round stock for springs which must withstand shocks, due to the fact that the deflection for the same fiber stress for a square bar spring is smaller than that for a round bar spring, the ratio being as 4 to 5. The round bar spring is therefore capable of storing more energy than a square bar spring for the same stress. Table 2. Stresses Produced in Springs by Shocks

Form of Bar from Which Spring is Made Round

Square

Fiber (Unit) Stress f Produced by Weight Q Dropped a Height h on a Helical Spring

f = f =

8QD c1 + π d3

9QD c1 + 4d 3

Ghd 4 m 4QD3n

f = 1.27

QhG Dd 2n

Ghd 4 m 0.9 π QD3n

f = 1.34

QhG Dd 2n

1+ 1+

Approximate Value of f

G = modulus of elasticity for torsion; d = diameter or side of bar; D = mean diameter of spring; n = number of coils in spring.

Shocks from Bodies in Motion.—The formulas given can be applied, in general, to shocks from bodies in motion. A body of weight W moving horizontally with the velocity of v feet per second has a stored-up energy: 1 Wv 2 6Wv 2 EK = 2 # g foot-pounds or g inch-pounds This expression may be substituted for Qh in the tables in the equations for unit stresses containing this quantity, and the stresses produced by the energy of the moving body thereby determined. The formulas in the tables give the maximum value of the stresses, providing the designer with some definitive guidance even where there may be justification for assuming that only a part of the energy of the shock is taken up by the member under stress. The formulas can also be applied using metric SI units. The stored-up energy of a body of mass M kilograms moving horizontally with the velocity of v meters per sec­ond is: 1 EK = 2 Mv 2 newton-meters This expression may be substituted for Qh in the appropriate equations in the ta­ bles. For calculation in millimeters, Qh = 1000 EK newton-millimeters. Fatigue Stresses.—So-called “fatigue ruptures” occur in parts that are subjected to con­ tinually repeated shocks or stresses of small magnitude. Machine parts that are subjected to continual stresses in varying directions, or to repeated shocks, even if of comparatively small magnitude, may fail ultimately if designed from a mere knowledge of the behavior of the material under a steady stress, such as is imposed upon it by ordinary tensile stress testing machines. Examinations of numerous cases of machine parts, broken under actual working conditions, indicate that at least 80 percent of these ruptures are caused by fatigue stresses. Most fatigue ruptures are caused by bending stresses, and frequently by a revolv­ing bending stress. Hence, to test materials for this class of stress the tests should be made to stress the material in a manner similar to that in which it will be stressed under actual working conditions. See Fatigue Properties on page 209 for more on this topic.

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Machinery's Handbook, 31st Edition Strength of Columns OR STRUTS

281

COLUMNS Strength of Columns or Struts Structural members which are subject to compression may be so long in proportion to their diameter or lateral dimensions that failure may be the result 1) of both compression and bending; and 2) of bending or buckling to such a degree that compression stress may be ignored. In such cases, the slenderness ratio is important. This ratio equals the length l of the col­umn in inches or millimeters, according to the unit system in use, divided by the least radius of gyration r of the cross section. Various formulas have been used for designing columns which are too slender to be designed for compression only. Rankine or Gordon Formula.—This formula is generally applied when slenderness ratios range between 20 and 100, and sometimes for ratios up to 120. The notation, in English and metric SI units of measurement, is given on page 283.

p=

S 2 2 l 2 = ultimate load, lb/in or N/mm 1 + K` r j

Factor K may be established by tests with a given material and end condition, and for the probable range of l/r. If determined by calculation, K = S/Cπ2 E. Factor C equals 1 for either rounded or pivoted column ends, 4 for fixed ends, and 1 to 4 for square flat ends. The fac­ tors 25,000, 12,500, etc., in the Rankine formulas, arranged as on page 283, equal 1/K, and have been used extensively. Straight-Line Formula.—This general type of formula is often used in designing com­ pression members for buildings, bridges, or similar structural work. It is convenient espe­ cially in designing a number of columns that are made of the same material but vary in size, assuming that factor B is known. This factor is determined by tests.

l p = Sy − B ` r j = ultimate load, lb/in2 Sy equals yield point, lb/in 2, and factor B ranges from 50 to 100. Safe unit stress = p ÷ factor of safety. Formulas of American Railway Engineering Association.—The formulas that follow apply to structural steel having an ultimate strength of 60,000 to 72,000 lb/in2. For building columns having l/r ratios not greater than 120: allowable unit stress = 17,000 − 0.485 l2/r2 For columns having l/r ratios greater than 120:

allowable unit stress =

18, 000 1 + l 2 ⁄ 18, 000r2

For bridge compression members centrally loaded and values of l/r not greater than 140:

1 Allowable unit stress, riveted ends = 15, 000 − 4 1 Allowable unit stress, pin ends = 15, 000 − 3

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Machinery's Handbook, 31st Edition FORMULAS FOR COLUMNS

282

Euler Formula.—This formula is for columns that are so slender that bending or buckling action predominates and compressive stresses are not taken into account.

P=

Cπ 2 IE = total ultimate load, in pounds or newtons l2

The notation, in English and metric SI units of measurement, is given in the table Rankine and Euler Formulas for Columns on page 283. Factors C for different end condi­ tions are included in the Euler formulas at the bottom of the table. According to a series of experiments, Euler formulas should be used if the values of l/r exceed the following ratios: Structural steel and flat ends, 195; hinged ends, 155; round ends, 120; cast iron with flat ends, 120; hinged ends, 100; round ends, 75; oak with flat ends, 130. The critical slender­ness ratio, which marks the dividing line between the shorter columns and those slender enough to warrant using the Euler formula, depends upon the column material and its end conditions. If the Euler formula is applied when the slenderness ratio is too small, the cal­culated ultimate strength will exceed the yield point of the material and, obviously, will be incorrect. Eccentrically Loaded Columns.—In the application of the column formulas previously referred to, it is assumed that the action of the load coincides with the axis of the column. If the load is offset relative to the column axis, the column is said to be eccentrically loaded, and its strength is then calculated by using a modification of the Rankine formula, the quantity cz/r 2 being added to the denominator, as shown in the table on the next page. This modified formula is applicable to columns having a slenderness ratio varying from 20 or 30 to about 100. Machine Elements Subjected to Compressive Loads.—As in structural compression members, an unbraced machine member that is relatively slender (i.e., its length is more than, say, six times the least dimension perpendicular to its longitudinal axis) is usually designed as a column because failure due to overloading (assuming a compressive load centrally applied in an axial direction) may occur by buckling or a combination of buckling and compression rather than by direct compression alone. In the design of unbraced steel machine “columns” which are to carry compressive loads applied along their longitudinal axes, two formulas are in general use:

Sy Ar 2 Q

(Euler)

Pcr =

(J. B. John­son)

Pcr = ASy a 1 −

(1)

Q k 4r 2

(2)

where

Q=

Sy l 2 n π 2E

(3)

In these formulas, Pcr = critical load in pounds that would result in failure of the column; A = cross sectional area, inches2; Sy = yield point of material, psi; r = least radius of gyration of cross section, inches; E = modulus of elasticity, psi; l = column length, inches; and n = coefficient for end conditions. For both ends fixed, n = 4; for one end fixed, one end free, n = 0.25; for one end fixed and the other end free but guided, n = 2; for round or pinned ends, free but guided, n = 1; and for flat ends, n = 1 to 4. It should be noted that these values of n represent ideal conditions that are seldom attained in practice; for example, for both ends fixed, a value of n = 3 to 3.5 may be more realistic than n = 4. If metric SI units are used in these formulas, Pcr = critical load in newtons that would result in failure of the column; A = cross-sectional area, mm 2; Sy = yield point of the material, N/mm 2; r = least radius of gyration of cross section, mm; E = modulus of elasticity, N/mm2; l = column length, mm; and n = a coefficient for end conditions. The coefficients given are valid for calculations in metric units.

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Machinery's Handbook, 31st Edition FORMULAS FOR COLUMNS

283

Rankine and Euler Formulas for Columns Symbol p P S l r I r2 E c z

Quantity Ultimate unit load Total ultimate load Ultimate compressive strength of material Length of column or strut Least radius of gyration Least moment of inertia Moment of inertia/area of section Modulus of elasticity of material Distance from neutral axis of cross section to side under compression Distance from axis of load to axis coinciding with center of gravity of cross section

English Unit Lbs/in2 Pounds Lbs/in2 Inches Inches Inches4 Inches2 Lbs/in2

Metric SI Units Newtons/mm2 Newtons Newtons/mm2 Millimeters Millimeters Millimeters4 Millimeters2 Newtons/mm2

Inches

Millimeters

Inches

Millimeters

Rankine Formulas Both Ends of One End Fixed and Column Fixed One End Rounded

Material

p=

Steel

p=

Cast Iron

p=

Wrought Iron

p=

Timber

1+ 1+ 1+ 1+

S

l2 25, 000r 2 S

l2 5000r 2 S

l2 35, 000r 2 S

l2 3000r 2

p= p= p= p=

1+ 1+ 1+ 1+

S

l2 12, 500r 2 S

l2 2500r 2 S

l2 17, 500r 2 S

l2 1500r 2

Both Ends Rounded

p= p= p= p=

1+ 1+ 1+ 1+

S l2 6250r 2 S

l2 1250r 2 S

l2 8750r 2 S

l2 750r 2

Formulas Modified for Eccentrically Loaded Columns

Both Ends of Column Fixed

Material Steel

p=

1+

One End Fixed and One End Rounded

S l2 cz + 25, 000r 2 r 2

p=

1+

Both Ends Rounded

S l2 cz + 12, 500r2 r2

p=

1+

S l2 cz + 6250r 2 r 2

For materials other than steel, such as cast iron, use the Rankine formulas given in the upper table and add to the denominator the quantity cz/r2 Both Ends of Column Fixed

P=

4π 2IE l2

Euler Formulas for Slender Columns One End Fixed and Both Ends One End Rounded Rounded

P=

2π 2IE l2

P=

π 2 IE l2

One End Fixed and One End Free

P=

π 2IE 4l 2

Allowable Working Loads for Columns: To find the total allowable working load for a given sec­ tion, divide the total ultimate load P (or p 3 area), as found by the appropriate formula above, by a suitable factor of safety.

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Machinery's Handbook, 31st Edition FORMULAS FOR COLUMNS

284

Factor of Safety for Machine Columns: When the conditions of loading and the physical qualities of the material used are accurately known, a factor of safety as low as 1.25 is sometimes used when minimum weight is important. Usually, however, a factor of safety of 2 to 2.5 is applied for steady loads. The factor of safety represents the ratio of the critical load Pcr to the working load. Application of Euler and Johnson Formulas: To determine whether the Euler or John­son formula is applicable in any particular case, it is necessary to determine the value of the quantity Q ÷ r 2. If Q ÷ r 2 is greater than 2, then the Euler Formula (1) should be used; if Q ÷ r 2 is less than 2, then the J. B. Johnson formula is applicable. Most compression members in machine design are in the range of proportions covered by the Johnson formula. For this reason a good procedure is to design machine elements on the basis of the Johnson formula and then as a check calculate Q ÷ r 2 to determine whether the Johnson formula applies or the Euler formula should have been used.

Example 1, Compression Member Design: A rectangular machine member 24 inches long and 1∕2 3 1 inch in cross section is to carry a compressive load of 4000 pounds along its axis. What is the factor of safety for this load if the material is machinery steel having a yield point of 40,000 psi, the load is steady, and each end of the rod has a ball connection so that n = 1? Solution: From Formula (3)

40, 000 # 24 # 24 Q = 1 # 3.1416 # 3.1416 # 30, 000, 000 = 0.0778 (The values 40,000 and 30,000,000 were obtained from the table Strength Data for Iron and Steel on page 429.) The radius of gyration r for a rectangular section (page 242) is 0.289 3 the dimension in the direction of bending. In columns, bending is most apt to occur in the direction in which the section is the weakest, the 1∕2 -inch dimension in this example. Hence, least radius of gyration r = 0.289 3 1∕2 = 0.145 inch.

Q 0.0778 = = 3.70 r 2 ^0.145h2 which is more than 2, so the Euler formula will be used.

Pcr =

1 sy Ar 2 40, 000 # 2 # 1 Q = 3.70

= 5400 pounds so that the factor of safety is 5400 ' 4000 = 1.35 Example 2, Compression Member Design: In the preceding example, the column formu­ las were used to check the adequacy of a column of known dimensions. The more usual problem involves determining what the dimensions should be to resist a specified load. For example: A 24-inch long bar of rectangular cross section with width w twice its depth d is to carry a load of 4000 pounds. What must the width and depth be if a factor of safety of 1.35 is to be used? Solution: First determine the critical load Pcr:

Pcr = working load # factor of safety = 4000 # 1.35 = 5400 pounds

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Machinery's Handbook, 31st Edition FORMULAS FOR COLUMNS

285

Next determine Q, which is found as in Example 1, and will again be 0.0778. Assume Formula (2) applies:

Pcr = Asy a 1 −

Q k 4r 2

5400 = w # d # 40, 000 a1 − = 2d 2 # 40, 000 a1 − 5400 0.01945 k 2 a 40, 000 # 2 = d 1 − r2

0.0778 k 4r 2

0.01945 k r2

As mentioned in Example 1 the least radius of gyration r of a rectangle is equal to 0.289 times the least dimension, d, in this case. Therefore, substituting for d the value r ÷ 0.289, 2 5400 0.01945 k ` r ja 40, 000 # 2 = 0.289 1 − r2

5400 # 0.289 # 0.289 = r 2 − 0.01945 40, 000 # 2

0.005638 = r 2 − 0.01945 r 2 = 0.0251

Checking to determine if Q ÷ r 2 is greater or less than 2,

Q 0.0778 = = 3.1 r 2 0.0251 Therefore, Formula (1) should have been used to determine r and dimensions w and d. Using Formula (1),

5400 =

2 r 2 40, 000 # 2d 2 # r 2 40, 000 # 2 # ` 0.289 j r = Q 0.0778

5400 # 0.0778 # 0.289 # 0.289 = 0.0004386 40, 000 # 2 0.145 d = 0.289 = 0.50 inch

r4 =

and w = 2d = 1 inch as in the previous example. American Institute of Steel Construction.—For main or secondary compression mem­ bers with l/r ratios up to 120, safe unit stress = 17,000 − 0.485l2/r 2. For columns and brac­ ing or other secondary members with l/r ratios above 120, For bracing and secondary members, safe unit stress, psi =

18, 000 1 + l 2 ⁄ 18, 000r 2

l⁄r 18, 000 o # e1.6 – 200 1 + l 2 ⁄ 18, 000r 2 Pipe Columns: Allowable concentric loads for steel pipe columns based on the above formulas are given in the table on page 286. For main members, safe unit stress, psi =

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Machinery's Handbook, 31st Edition Loads for Steel Pipe Columns

286

Allowable Concentric Loads for Steel Pipe Columns STANDARD STEEL PIPE Nominal Diameter, Inches Wall Thickness, Inch

Weight per Foot, Pounds

Effective Length (KL), Feeta 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 22 24 25 26

Nominal Diameter, Inches

Wall Thickness, Inch Weight per Foot, Pounds Effective Length (KL), Feeta 6 7 8 9 10 11 12 13 14 15 16 18 19 20 21 22 24 26 28

12

0.375

49.56

10

0.365

40.48

8

0.322

28.55

6

0.280

18.97

14.62

4

0.237

10.79

31∕2

0.226 9.11

Allowable Concentric Loads in Thousands of Pounds

303 301 299 296 293 291 288 285 282 278 275 272 268 265 261 254 246 242 238

246 243 241 238 235 232 229 226 223 220 216 213 209 205 201 193 185 180 176

171 168 166 163 161 158 155 152 149 145 142 138 135 131 127 119 111 106 102

12 0.500 65.42

10 0.500 54.74

8 0.500 43.39

110 108 106 103 101 98 95 92 89 86 82 79 75 71 67 59 51 47 43

EXTRA STRONG STEEL PIPE

400 397 394 390 387 383 379 375 371 367 363 353 349 344 337 334 323 312 301

5

0.258

6 0.432 28.57

3

0.216 7.58

83 81 78 76 73 71 68 65 61 58 55 51 47 43 39 32 27 25 23

59 57 54 52 49 46 43 40 36 33 29 26 23 21 19 15 13 12

48 46 44 41 38 35 32 29 25 22 19 17 15 14 12 10

38 36 34 31 28 25 22 19 16 14 12 11 10 9

5 0.375 20.78

4 0.337 14.98

3 1∕2 0.318 12.50

3 0.300 10.25

Allowable Concentric Loads in Thousands of Pounds 332 259 166 118 81 66 328 255 162 114 78 63 325 251 159 111 75 59 321 247 155 107 71 55 318 243 151 103 67 51 314 239 146 99 63 47 309 234 142 95 59 43 305 229 137 91 54 38 301 224 132 86 49 33 296 219 127 81 44 29 291 214 122 76 39 25 281 203 111 65 31 20 276 197 105 59 28 18 271 191 99 54 25 16 265 185 92 48 22 14 260 179 86 44 21 248 166 73 37 17 236 152 62 32 224 137 54 27

52 48 45 41 37 33 28 24 21 18 16 12 11

a With respect to radius of gyration. The effective length (KL) is the actual unbraced length, L, in feet, multiplied by the effective length factor (K), which is dependent upon the restraint at the ends of the unbraced length and the means available to resist lateral movements. K may be determined by referring to the last portion of this table.

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Machinery's Handbook, 31st Edition Loads for Steel Pipe Columns

287

Allowable Concentric Loads for Steel Pipe Columns (Continued) DOUBLE-EXTRA STRONG STEEL PIPE 8

Nominal Diameter, Inches

6

0.875

Wall Thickness, Inch

72.42

Weight per Foot, Pounds

Effective Length (KL), Feeta 6

431

8

7

53.16

11

91

417

292

202

133

77

403

275

16 17

237

178

109

160

91

216

55

97

44

193

108

20

310

168

87

181

288

24

142

264

26

84 69 60

51 43 37 32 28 24

22

40

33

61

102

213

49

72

119

240

28

62

119

331

22

70

130

205

321

81

141

18

19

100

151

227

341

118

170

247

351

126

187

257

360

140

195

266

369

15

209

284

378

14

18.58

147

387

13

27.54

3

0.600

216

299

395

12

38.55

4

0.674

306

410

10

0.750

Allowable Concentric Loads in Thousands of Pounds

424

9

5

0.864

52

88

44

EFFECTIVE LENGTH FACTORS (K) FOR VARIOUS COLUMN CONFIGURATIONS (a)

(b)

(c)

(d)

(e)

(f)

Buckled shape of column is shown by dashed line

Theoretical K value

Recommended design value when ideal conditions are approximated

0.5

0.7

1.0

1.0

2.0

2.0

0.65

0.80

1.2

1.0

2.10

2.0

Rotation fixed and translation fixed End condition code

Rotation free and translation fixed Rotation fixed and translation free Rotation free and translation free

Load tables are given for 36 ksi yield stress steel. No load values are given below the heavy hori­ zontal lines, because the Kl/r ratios (where l is the actual unbraced length in inches and r is the gov­erning radius of gyration in inches) would exceed 200. Data from “Manual of Steel Construction,” 8th ed., 1980, with permission of the American Insti­ tute of Steel Construction (AISC).

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288

Machinery's Handbook, 31st Edition Plates, Shells, and Cylinders PLATES, SHELLS, AND CYLINDERS

Flat Stayed Surfaces.—Large flat areas are often held against pressure by stays distrib­ uted at regular intervals over the surface. In boiler work, these stays are usually screwed into the plate and the projecting end riveted over to insure steam tightness. The US Board of Supervising Inspectors and the American Boiler Manufacturers Association (ABMA) rules give the fol­lowing formula for flat stayed surfaces: C # t2 P= S2

in which P =  pressure in pounds per square inch C =  a constant, which equals 112 for plates 7∕16 inch and under 120, for plates over 7∕16 inch thick 140, for plates with stays having a nut and bolt on the inside and outside 160, for plates with stays having washers of at least one-half the plate thickness, and with a diameter at least one-half of the greatest pitch t =  thickness of plate in 16ths of an inch (thickness = 7∕16 , t = 7) S =  greatest pitch of stays in inches

Strength and Deflection of Flat Plates.—Generally, the formulas used to determine stresses and deflections in flat plates are based on certain assumptions that can be closely approximated in practice. These assumptions are: 1) the thickness of the plate is not greater than one-quarter the least width of the plate; 2) the greatest deflection when the plate is loaded is less than one-half the plate thickness; 3) the maximum tensile stress resulting from the load does not exceed the elastic limit of the material; and 4) all loads are perpendicular to the plane of the plate.

Plates of ductile materials fail when the maximum stress resulting from deflection under load exceeds the yield strength; for brittle materials, failure occurs when the maximum stress reaches the ultimate tensile strength of the material involved. Square and Rectangular Flat Plates.—The formulas that follow give the maximum stress and deflection of flat steel plates supported in various ways and subjected to the loading indicated. These formulas are based upon a modulus of elasticity for steel of 30,000,000 psi and a value of Poisson’s ratio of 0.3. If the formulas for maximum stress, S, are applied without modification to other materials, such as cast iron, aluminum, and brass, for which the range of Poisson’s ratio is about 0.26 to 0.34, the maxi­mum stress calculations will be in error by not more than about 3 percent. The deflection formulas may also be applied to materials other than steel by substituting in these formulas the appropriate value for E, the modulus of elasticity of the material (see pages 429 and 510). The deflections thus obtained will not be in error by more than about 3 percent. In the stress and deflection formulas that follow, p =  uniformly distributed load acting on plate, psi W =  total load on plate, pounds; W = p 3 area of plate L =  distance between supports (length of plate), inches. For rectangular plates, L = long side, l = short side t =  thickness of plate, inches S =  maximum tensile stress in plate, psi d =  maximum deflection of plate, in. E =  modulus of elasticity in tension. E = 30,000,000 psi for steel

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Machinery's Handbook, 31st Edition Plates, Shells, and Cylinders

289

If metric SI units are used in the formulas, then, W =  total load on plate, newtons L =  distance between supports (length of plate), mm. For rectangular plates, L = long side, l = short side t =  thickness of plate, mm S =  maximum tensile stress in plate, N/mm2 d =  maximum deflection of plate, mm E =  modulus of elasticity, N/mm2

a) Square flat plate supported at top and bottom of all four edges and a uniformly distrib­ uted load over the surface of the plate.

S=

0.29W t2

d=

(1)

0.0443WL2 Et 3

(2)

b) Square flat plate supported at the bottom only of all four edges and a uniformly distrib­uted load over the surface of the plate.

S=

0.28W t2

d=

(3)

0.0443WL2 Et 3

(4)

c) Square flat plate with all edges firmly fixed and a uniformly distributed load over the surface of the plate.

S=

0.31W t2

d=

(5)

0.0138WL2 Et 3

(6)

d) Square flat plate with all edges firmly fixed and a uniform load over small circular area at the center. In Equations (7) and (9), r 0 = radius of area to which load is applied. If r 0   <   1.7t, use rs where rs = 1.6r02 + t 2 − 0.675t.

S=

0.62W L log e a 2r k t2 0

d=

(7)

0.0568WL2 Et 3

(8)

e) Square flat plate with all edges supported above and below, or below only, and a con­ centrated load at the center. See Item d), above, for definition of r 0.

S=

0.62W : L loge a 2r k + 0.577D (9) t2 0

d=

0.1266WL2 Et 3

(10)

f) Rectangular plate with all edges supported at top and bottom and a uniformly distrib­ uted load over the surface of the plate.

S=

0.75W L l2 t 2 a l + 1.61 2 k L

(11)

d=

0.1422W L 2.21 Et 3 a 3 + 2 k l L

(12)

g) Rectangular plate with all edges fixed and a uniformly distributed load over the sur­ face of the plate.

S=

0.5W L 0.623l5 k t2a l + L5

(13)

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d=

0.0284W L 1.056l 2 k + L4 l3

Et 3 a

(14)

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Machinery's Handbook, 31st Edition Plates, Shells, and Cylinders

290

Circular Flat Plates.—In the following formulas, R = radius of plate to supporting edge in inches; W = total load in pounds; and other symbols are the same as used for square and rectangular plates. If metric SI units are used, R = radius of plate to supporting edge in mm, and the values of other symbols are the same as those used for square and rectangular plates. a) Edge supported around the circumference and a uniformly distributed load over the surface of the plate.

S=

0.39W t2

(15)

d=

0.221WR2 Et 3

(16)

b) Edge fixed around circumference and a uniformly distributed load over the surface of the plate.

S=

0.24W t2

(17)

d=

0.0543WR2 Et 3

(18)

c) Edge supported around the circumference and a concentrated load at the center.

S =

0.48W : R t2 1 + 1.3 log e 0.325t − 0.0185 2 D (19) t2 R

d=

0.55WR 2 Et 3

(20)

d) Edge fixed around circumference and a concentrated load at the center.

S=

0.62W : R t2 log e 0.325t + 0.0264 2 D (21) R t2

d=

0.22WR2 Et 3

(22)

Strength of Cylinders Subjected to Internal Pressure.—In designing a cylinder to withstand internal pressure, the choice of formula to be used depends on 1) the kind of material of which the cylinder is made (whether brittle or ductile); 2) the construction of the cylinder ends (whether open or closed); and 3) whether the cylinder is classed as a thin- or a thick-walled cylinder. A cylinder is considered to be thin-walled when the ratio of wall thickness to inside diam­eter is 0.1 or less and thick-walled when this ratio is greater than 0.1. Materials such as cast iron, hard steel, and cast aluminum are considered to be brittle materials; lowcarbon steel, brass, bronze, etc. are considered to be ductile. In the formulas that follow, p = internal pressure, psi; D = inside diameter of cylinder, inches; t = wall thickness of cylinder, inches; μ = Poisson’s ratio, = 0.3 for steel, 0.26 for cast iron, 0.34 for aluminum and brass; and S = allowable tensile stress, psi. Metric SI units can be used in Formulas (23), (25), (26), and (27), where p = internal pressure in N/mm2; D = inside diameter of cylinder, mm; t = wall thickness, mm; μ = Poisson’s ratio, = 0.3 for steel, 0.26 for cast iron, and 0.34 for aluminum and brass; and S = allowable tensile stress, N/mm 2. For the use of metric SI units in Formula (24), see below. Thin-walled Cylinders: Dp t = 2S (23) For low-pressure cylinders of cast iron such as are used for certain engine and press applications, a formula in common use is

Dp t = 2500 + 0.3

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Machinery's Handbook, 31st Edition Plates, Shells, and Cylinders

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This formula is based on allowable stress of 1250 pounds per square inch (psi) and will give a wall thickness 0.3 inch greater than Formula (23) to allow for variations in metal thickness that may result from the casting process. If metric SI units are used in Formula (24), t = cylinder wall thickness in mm; D = inside diameter of cylinder, mm; and the allowable stress is in N/mm2. The value of 0.3 inches additional wall thickness is 7.62 mm, and the next highest number in preferred metric basic sizes is 8 mm. Thick-walled Cylinders of Brittle Material, Ends Open or Closed: Lamé’s equation is used when cylinders of this type are subjected to internal pressure.

S+p D (25) t = 2 d S − p − 1n The table Ratio of Outside Radius to Inside Radius, Thick Cylinders on page 292 is for convenience in calculating the dimensions of cylinders under high internal pressure with­out the use of Formula (25). Example, Use of the Table: Assume that a cylinder of 10 inches inside diameter is to withstand a pressure of 2500 psi; the material is cast iron and the allow­able stress is 6000 psi. To solve the problem, locate the allowable stress per square inch in the left-hand column of the table and the working pressure at the top of the columns. Then find the ratio between the outside and inside radii in the body of the table. In this example, the ratio is 1.558, and hence the outside diameter of the cylinder should be 10 3 1.558, or about 155∕8 inches. The thickness of the cylinder wall will therefore be (15.58 − 10)⁄2 = 2.79 inches. Unless very high-grade material is used and sound castings assured, cast iron should not be used for pressures exceeding 2000 pounds psi (13.75 N/mm 2). It is well to leave more metal in the bottom of a hydraulic cylinder than is indicated by the results of calculations because a hole of some size must be cored in the bottom to permit the entrance of a boring bar when finishing the cylinder, and when this hole is subsequently tapped and plugged it often gives trouble if there is too little thickness. For steady or gradually applied stresses, the maximum allowable fiber stress S may be assumed to be from 3500 to 4000 psi (24–27 N/mm 2) for cast iron; from 6000 to 7000 psi (41–48 N/mm 2) for brass; and 12,000 psi (82 N/mm 2) for steel castings. For intermittent stresses, such as in cylinders for steam and hydraulic work, 3000 psi (20 N/mm 2) for cast iron; 5000 psi (34 N/mm 2) for brass; and 10,000 psi (69 N/mm 2) for steel castings is ordi­ narily used. These values give ample factors of safety. Note: In metric SI units, 1000 pounds per square inch equals 6.895 newtons per square millimeter (1000 lb/in2 = 6.895 N/mm2). Also, one newtons per square mil­ limeter equals one megapascal (1 N/mm 2 = 1 MPa). Thick-walled Cylinders of Ductile Material, Closed Ends: Clavarino’s equation is used: S + ^1 − 2µ hp D t= 2= − 1G (26) S − ^1 + µ hp Thick-walled Cylinders of Ductile Material, Open Ends: Birnie’s equation is used: S + ^1 − µ hp D (27) t= 2= − 1G S − ^1 + µ hp

Spherical Shells Subjected to Internal Pressure.—Let: D =  internal diameter of shell in inches p =  internal pressure in psi S =  safe tensile stress per square inch t = thickness of metal in the shell, in inches. Then, t = pD ÷ 4S

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Ratio of Outside Radius to Inside Radius, Thick Cylinders Allowable Stress per Sq. In. of Section

1000

1500

2000

2500

Working Pressure in Cylinder, Pounds per Square Inch 3000

3500

4000

4500

5000

5500

6000

6500

2000

1.732

…

…

…

…

…

…

…

…

…

…

…

…

2500

1.528

2.000

…

…

…

…

…

…

…

…

…

…

…

7000

3000

1.414

1.732

2.236

…

…

…

…

…

…

…

…

…

…

3500

1.342

1.581

1.915

2.449

…

…

…

…

…

…

…

…

…

4000

1.291

1.483

1.732

2.082

2.646

…

…

…

…

…

…

…

…

4500

1.254

1.414

1.612

1.871

2.236

2.828

…

…

…

…

…

…

…

5000

1.225

1.363

1.528

1.732

2.000

2.380

3.000

…

…

…

…

…

…

5500

1.202

1.323

1.464

1.633

1.844

2.121

2.517

3.162

…

…

…

…

…

6000

1.183

1.291

1.414

1.558

1.732

1.949

2.236

2.646

3.317

…

…

…

…

6500

…

1.265

1.374

1.500

1.648

1.826

2.049

2.345

2.769

3.464

…

…

…

7000

…

1.243

1.342

1.453

1.581

1.732

1.915

2.145

2.449

2.887

3.606

…

…

7500

…

1.225

1.314

1.414

1.528

1.658

1.813

2.000

2.236

2.550

3.000

3.742

…

8000

…

1.209

1.291

1.382

1.483

1.599

1.732

1.890

2.082

2.324

2.646

3.109

3.873

8500

…

1.195

1.271

1.354

1.446

1.549

1.667

1.803

1.964

2.160

2.408

2.739

3.215

9000

…

1.183

1.254

1.330

1.414

1.508

1.612

1.732

1.871

2.035

2.236

2.490

2.828

9500

…

…

1.238

1.309

1.387

1.472

1.567

1.673

1.795

1.936

2.104

2.309

2.569

10,000

…

…

1.225

1.291

1.363

1.441

1.528

1.624

1.732

1.856

2.000

2.171

2.380

10,500

…

…

1.213

1.275

1.342

1.414

1.494

1.581

1.679

1.789

1.915

2.062

2.236

11,000

…

…

1.202

1.260

1.323

1.390

1.464

1.544

1.633

1.732

1.844

1.972

2.121

11,500

…

…

1.192

1.247

1.306

1.369

1.438

1.512

1.593

1.683

1.784

1.897

2.028

12,000

…

…

1.183

1.235

1.291

1.350

1.414

1.483

1.558

1.641

1.732

1.834

1.949

12,500

…

…

…

1.225

1.277

1.333

1.393

1.458

1.528

1.604

1.687

1.780

1.883

13,000

…

…

…

1.215

1.265

1.318

1.374

1.435

1.500

1.571

1.648

1.732

1.826

13,500

…

…

…

1.206

1.254

1.304

1.357

1.414

1.475

1.541

1.612

1.690

1.776

14,000

…

…

…

1.198

1.243

1.291

1.342

1.395

1.453

1.515

1.581

1.653

1.732

14,500

…

…

…

1.190

1.234

1.279

1.327

1.378

1.433

1.491

1.553

1.620

1.693

15,000

…

…

…

1.183

1.225

1.268

1.314

1.363

1.414

1.469

1.528

1.590

1.658

16,000

…

…

…

1.171

1.209

1.249

1.291

1.335

1.382

1.431

1.483

1.539

1.599

Formula (28) also applies to hemi-spherical shells, such as the hemi-spherical head of a cylindrical container subjected to internal pressure, etc. If metric SI units are used, then: D =  internal diameter of shell in millimeters p =  internal pressure in newtons per square millimeter S =  safe tensile stress in newtons per square millimeter t =  thickness of metal in the shell, in millimeters. Use Formula (28).

Meters can be used in the formula in place of millimeters, provided the treatment is consistent throughout.

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293

Example: Find the thickness of metal required in the hemi-spherical end of a cylindrical vessel, 2 feet in diameter, subjected to an internal pressure of 500 pounds per square inch. Solution: The material is mild steel and a tensile stress of 10,000 psi is allowable.

500 # 2 # 12 t = 4 # 10, 000 = 0.3 inch

Example: A similar example using metric SI units is as follows: find the thickness of metal required in the hemi-spherical end of a cylindrical vessel, 750 mm in diameter, subjected to an internal pressure of 3 newtons/mm 2. The material is mild steel and a tensile stress of 70 newtons/mm 2 is allowable.

3 # 750 t = 4 # 70 = 8.04 mm

If the radius of curvature of the domed head of a boiler or container subjected to internal pressure is made equal to the diameter of the boiler, the thickness of the cylindrical shell and of the spherical head should be made the same. For example, if a boiler is 3 feet in diameter, the radius of curvature of its head should also be 3 feet, if material of the same thickness is to be used and the stresses are to be equal in both the head and cylindrical por­tion. Collapsing Pressure of Cylinders and Tubes Subjected to External Pressures.—The following formulas may be used for finding the collapsing pressures of lap-welded Besse­ mer steel tubes:

t P = 86, 670 D − 1386

t 3 P = 50, 210, 000 ` D j

(29) (30)

in which P = collapsing pressure in psi; D = outside diameter of tube or cylinder in inches; t = thickness of wall in inches.

Formula (29) is for values of P greater than 580 pounds per square inch, and Formula (30) is for values of P less than 580 pounds per square inch. These formulas are substantially correct for all lengths of pipe greater than six diameters between transverse joints that tend to hold the pipe to a circular form. The pressure P found is the actual collapsing pressure, and a suitable factor of safety must be used. Ordinarily, a factor of safety of 5 is sufficient. In cases where there are repeated fluctuations of the pressure, vibration, shocks and other stresses, a factor of safety of from 6 to 12 should be used. If metric SI units are used the formulas are:

t P = 597. 6 D − 9. 556 t 3 P = 346, 200 ` D j

(31) (32)

where P = collapsing pressure in newtons per square millimeter; D = outside diame­ ter of tube or cylinder in millimeters; and t = thickness of wall in millimeters. For­ mula (31) is for values of P greater than 4 N/mm 2, and Formula (32) is for values of P less than 4 N/mm 2. The table Tubes Subjected to External Pressure is based upon the requirements of the Steam Boat Inspection Service of the Department of Commerce and Labor and gives the permissible working pressures and corresponding minimum wall thickness for long, plain, lap-welded and seamless steel flues subjected to external pressure only. The table thick­nesses have been calculated from the formula:

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294

t =

6^ F # p h + 1386@ D 86, 670

in which D = outside diameter of flue or tube in inches; t = thickness of wall in inches; p = working pressure in pounds per square inch; F = factor of safety. The formula is applicable to working pressures greater than 100 pounds per square inch, to outside diameters from 7 to 18 inches, and to temperatures less than 650°F. The preceding Formulas (29) and (30) were determined by Prof. R. T. Stewart, Dean of the Mechanical Engineering Department of the University of Pittsburgh, in a series of experiments carried out at the plant of the National Tube Co., McKeesport, Pennsylvania. The apparent fiber stress under which the different tubes failed varied from about 7000 pounds per square inch for the relatively thinnest to 35,000 pounds per square inch for the relatively thickest walls. The average yield point of the material tested was 37,000 pounds and the tensile strength 58,000 pounds per square inch, so it is evident that the strength of a tube subjected to external fluid collapsing pressure is not dependent alone upon the elastic limit or ultimate strength of the material from which it is made. Tubes Subjected to External Pressure Outside Diameter of Tube, Inches

Working Pressure in Pounds per Square Inch 100

120

140

160

180

200

7

0.152

Thickness of Tube in Inches. Safety Factor, 5 0.160 0.168 0.177 0.185 0.193

9

0.196

0.206

8

0.174

10

0.218

11

0.239

12

0.265

0.289

0.313

0.337

0.361

0.366

0.366

18

0.252

0.343

0.344

16

0.241

0.320

0.323

16

0.229

0.217

0.298

0.301

15

0.202

0.275

0.283

14

0.193

0.252

0.261

13

0.183

0.385

0.389

0.387

0.409

0.412

0.433

0.211

0.220

0.264

0.275

0.227

0.237

0.277

0.290

0.303

0.328

0.353

0.378

0.404

0.429

0.454

0.317

0.343

0.369

0.396

0.422

0.448

0.475

0.248

0.303

0.330

0.358

0.385

0.413

0.440

0.468

0.496

220 0.201

0.258

0.229

0.287

0.316

0.344

0.373

0.402

0.430

0.459

0.488

0.516

Dimensions and Maximum Allowable Pressure of Tubes Subjected to External Pressure

Outside Dia., Inches 2

Thick­ Max. ness Pressure of Allowed, Material, psi Inches 0.095

21∕4

0.095

21∕2

0.109

23∕4

0.109

427

380

Outside Dia., Inches 3

Thick­ Max. ness Pressure of Allowed, Material, psi Inches 0.109

31∕4

0.120

392

31∕2

0.120

356

33∕4

0.120

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Outside Dia., Inches

0.134

0.134

303

41∕2

308

5

0.148

235

282

6

0.165

199

332

4

Thick­ Max. ness Pressure of Allowed, Material, psi Inches 238

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Machinery's Handbook, 31st Edition Shafts

295

SHAFTS Shaft Calculations Torsional Strength of Shafting.—In the formulas that follow, α =  angular deflection of shaft in degrees c =  distance from center of gravity to extreme fiber D =  diameter of shaft in inches G =  torsional modulus of elasticity = 11,500,000 pounds per square inch for steel J =  polar moment of inertia of shaft cross section (see table) l =  length of shaft in inches N =  angular velocity of shaft in revolutions per minute P =  power transmitted in horsepower Ss =  allowable torsional shearing stress in pounds per square inch T =  torsional or twisting moment in inch-pounds Zp =  polar section modulus (see table page 252)

The allowable twisting moment for a shaft of any cross section such as circular, square, etc., is:

T = Ss # Z p

(1)

For a shaft delivering P horsepower at N revolutions per minute the twisting moment T being transmitted is: 63, 000P T= (2) N

The twisting moment T as determined by Formula (2) should be less than the value deter­m ined by using Formula (1) if the maximum allowable stress Ss is not to be exceeded. The diameter of a solid circular shaft required to transmit a given torque T is:

D=

3

5.1T Ss

(3a)

or

D=

3

321, 000P NSs

(3b)

The allowable stresses that are generally used in practice are: 4000 pounds per square inch for main power-transmitting shafts; 6000 pounds per square inch for lineshafts carry­ing pulleys; and 8500 pounds per square inch for small, short shafts, countershafts, etc. Using these allowable stresses, the horsepower P transmitted by a shaft of diameter D, or the diameter D of a shaft to transmit a given horsepower P may be determined from the fol­lowing formulas: For main power-transmitting shafts:

D 3N P = 80

(4a)

or

D=

or

D=

3

80P N

(4b)

53.5P N

(5b)

For lineshafts carrying pulleys:

D 3N P = 53.5

(5a)

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For small, short shafts:

D 3N P = 38

(6a)

3

D=

or

38P N

(6b)

Shafts that are subjected to shocks, such as sudden starting and stopping, should be given a greater factor of safety resulting in the use of lower allowable stresses than those just mentioned. Example: What should be the diameter of a lineshaft to transmit 10 horsepower if the shaft is to make 150 revolutions per minute? Using Formula (5b),

D=

3

53.5 # 10 9 = 1.53 or, say, 1 16 inches 150

Example: What horsepower would be transmitted by a short shaft, 2 inches in diameter, carrying two pulleys close to the bearings, if the shaft makes 300 revolutions per minute? Using Formula (6a), 2 3 # 300 P= = 63 horsepower 38 Torsional Strength of Shafting, Calculations in Metric SI Units.—The allowable twist­ ing moment for a shaft of any cross section such as circular, square, etc. can be cal­ culated from: T = Ss # Zp (7)

where T = torsional or twisting moment in N-mm; Ss = allowable tor­sional shearing stress in N/mm 2; and Zp = polar section mod­ulus in mm3. For a shaft delivering power of P kilowatts at N revolutions per minute, the twisting moment T being transmitted is:

T=

9.55 # 10 6P N

(8a)

or

T=

10 6P ω

(8b)

where T is in newton-millimeters, and ω = angular velocity in radians per second. The diameter D of a solid circular shaft required to transmit a given torque T is:

D=

3

5.1T Ss

(9a)

or

D=

or

D=

3

3

48.7 # 10 6P NSs

(9b)

5.1 # 10 6P ω Ss

(9c)

where D is in millimeters; T is in newton-millimeters; P is power in kilowatts; N = rev­olutions per minute; Ss = allowable torsional shearing stress in newtons per square millimeter, and ω = angular velocity in radians per second. If 28 newtons/mm 2 and 59 newtons/mm 2 are taken as the generally allowed stresses for main power-transmitting shafts and small short shafts, respectively, then using these allowable stresses, the power P transmitted by a shaft of diameter D, or the diameter D of a shaft to transmit a given power P may be determined from the follow­ ing formulas:

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297

For main power-transmitting shafts:

P=

D3 N 1.77 # 10 6

(10a)

or

D=

3

(11a)

or

D=

3

1.77 # 10 6P (10b) N

For small, short shafts:

P=

D3 N 0.83 # 10 6

0.83 # 10 6P (11b) N

where P is in kilowatts, D is in millimeters, and N = revolutions per minute. Example: What should be the diameter of a power-transmitting shaft to transmit 150 kW at 500 rpm? 3

D=

1.77 # 10 6 # 150 = 81 millimeters 500

Example: What power would a short shaft, 50 millimeters in diameter, transmit at 400 rpm?

P=

50 3 # 400 = 60 kilowatts 0.83 # 10 6

Torsional Deflection of Circular Shafts.—Shafting must often be proportioned not only to provide the strength required to transmit a given torque, but also to prevent torsional deflection (twisting) through a greater angle than has been found satisfactory for a given type of service. For a solid circular shaft the torsional deflection in degrees is given by: 584Tl (12) α= D4 G

Example: Find the torsional deflection for a solid steel shaft 4 inches in diameter and 48 inches long, subjected to a twisting moment of 24,000 inch-pounds. By Formula (12),

α=

584 # 24, 000 # 48 = 0.23 degree 44 # 11, 500, 000

Formula (12) can be used with metric SI units, where α = angular deflection of shaft in degrees; T = torsional moment in newton-millimeters; l = length of shaft in millime­ters; D = diameter of shaft in millimeters; and G = torsional modulus of elasticity in newtons per square millimeter. Example: Find the torsional deflection of a solid steel shaft, 100 mm in diameter and 1300 mm long, subjected to a twisting moment of 3 3 10 6 newton-millimeters. The torsional modulus of elasticity is 80,000 newtons/mm 2. By Formula (12)

α=

584 # 3 # 10 6 # 1300 = 0.285 degree 100 4 # 80, 000

The diameter of a shaft that is to have a maximum torsional deflection α is given by:

Tl (13) Gα Formula (13) can be used with metric SI units, where D = diameter of shaft in milli­meters; T = torsional moment in newton-millimeters; l = length of shaft in D = 4.9 #

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298

millime­ters; G = torsional modulus of elasticity in newtons per square millimeter; and α = angular deflection of shaft in degrees. According to some authorities, the allowable twist in steel transmission shafting should not exceed 0.08 degree per foot length of the shaft. The diameter D of a shaft that will per­ mit a maximum angular deflection of 0.08 degree per foot of length for a given torque T or for a given horsepower P can be determined from the formulas:

D = 0.29 4 T

(14a)

or

D = 4.6 #

4

P N

(14b)

Using metric SI units and assuming an allowable twist in steel transmission shaft­ ing of 0.26 degree per meter length, Formulas (14a) and (14b) become: 4 P D = 125.7 # N D = 2.26 4 T or where D = diameter of shaft in millimeters; T = torsional moment in newtonmillime­ters; P = power in kilowatts; and N = revolutions per minute. Another rule that has been generally used in mill practice limits the deflection to 1 degree in a length equal to 20 times the shaft diameter. For a given torque or horsepower, the diam­eter of a shaft having this maximum deflection is given by:

D = 0.1 3 T

(15a)

or

D = 4.0 #

3

P N

(15b)

Example: Find the diameter of a steel lineshaft to transmit 10 horsepower at 150 revolu­ tions per minute with a torsional deflection not exceeding 0.08 degree per foot of length. Solution: By Formula (14b),

D = 4.6 #

4

10 150 = 2.35 inches

This diameter is larger than that obtained for the same horsepower and rpm in the exam­ ple given for Formula (5b) in which the diameter was calculated for strength consider­ ations only. The usual procedure in the design of shafting which is to have a specified maximum angular deflection is to compute the diameter first by means of Formulas (13), (14a), (14b), (15a), or (15b) and then by means of Formulas (3a), (3b), (4b), (5b), or (6b), using the larger of the two diameters thus found. Linear Deflection of Shafting.—For steel line shafting, it is considered good practice to limit the linear deflection to a maximum of 0.010 inch per foot of length. The maximum distance in feet between bearings for average conditions, in order to avoid excessive linear deflection, is determined by the formulas: 3

L =   8.95 D 2 for shafting subject to no bending action except its own weight 3 L =   5.2 D 2 for shafting subject to bending action of pulleys, etc.

in which D = diameter of shaft in inches and L = maximum distance between bearings in feet. Pulleys should be placed as close to the bearings as possible. In general, shafting up to 3 inches in diameter is almost always made from cold-rolled steel. This shafting is true and straight and needs no turning, but if keyways are cut in the shaft, it must usually be straightened afterwards, as the cutting of the keyways relieves the tension on the surface of the shaft produced by the cold-rolling process. Sizes of shafting from 3 to 5 inches in diameter may be either cold-rolled or turned, more frequently the latter, and all larger sizes of shafting must be turned because cold-rolled shafting is not available in diameters larger than 5 inches.

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Machinery's Handbook, 31st Edition Shafts

299

Diameters of Finished Shafting (former American Standard ASA B17.1) Diameters, Inches Transmis­ Machinery sion Shafting Shafting 1∕2

9∕16 5∕8 11∕16 3∕4 13∕16 7∕8 15∕16

13∕16

17∕16

111∕16

15∕16

1

11∕16 11∕8 13∕16 11∕4 15∕16 13∕8 17∕16 11∕2 19∕16 15∕8 111∕16 13∕4

Minus Toler­ ances, Inchesa 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.003 0.003 0.003 0.003 0.003 0.003 0.003 0.003 0.003 0.003 0.003 0.003

Diameters, Inches Transmis­ Machinery sion Shafting Shafting

1 15∕16

23∕16

27∕16

215∕16

37∕16

113∕16 17∕8 115∕16 2 21∕16 21∕8 23∕16 21∕4 25∕16 23∕8 27∕16 21∕2 25∕8 23∕4 27∕8 3 31∕8 31∕4 33∕8 31∕2 35∕8

Minus Toler­ ances Inchesa 0.003 0.003 0.003 0.003 0.004 0.004 0.004 0.004 0.004 0.004 0.004 0.004 0.004 0.004 0.004 0.004 0.004 0.004 0.004 0.004 0.004

Diameters, Inches Transmis­ Machinery sion Shafting Shafting

3 15∕16 47∕16 415∕16 57∕16 515∕16 61∕2 7 71∕2 8 … …

33∕4 37∕8 4

41∕4 41∕2 43∕4 5 51∕4 51∕2 53∕4 6 61∕4 61∕2 63∕4 7

71∕4 71∕2 73∕4 8 … …

Minus Tolerances, Inchesa 0.004 0.004 0.004 0.005 0.005 0.005 0.005 0.005 0.005 0.005 0.005 0.006 0.006 0.006 0.006 0.006 0.006 0.006 0.006 … …

a Note:—These tolerances are negative or minus and represent the maximum allowable variation below the exact nominal size. For instance the maximum diameter of the 115 ⁄16 inch shaft is 1.938 inch and its minimum allowable diameter is 1.935 inch. Stock lengths of finished transmis­ sion shafting shall be: 16, 20 and 24 feet.

Design of Transmission Shafting.—The following guidelines for the design of shafting for transmitting a given amount of power under various conditions of loading are based upon formulas given in the former American Standard ASA B17c Code for the Design of Transmission Shafting. These formulas are based on the maximum-shear theory of failure which assumes that the elastic limit of a ductile ferrous material in shear is practically one-half its elastic limit in tension. This theory agrees, very nearly, with the results of tests on ductile materials and has gained wide acceptance in practice. The formulas given apply in all shaft designs including shafts for special machinery. The limitation of these formulas is that they provide only for the strength of shafting and are not concerned with the torsional or lineal deformations which may, in shafts used in machine design, be the controlling factor (see Torsional Deflection of Circular Shafts on page 297 and Linear Deflection of Shafting on page 298 for deflection considerations). In the formu­las that follow, 3 B =   1 ' ^1 − K 4h (see Table 3) D =  outside diameter of shaft in inches D1 =  inside diameter of a hollow shaft in inches Km =  shock and fatigue factor to be applied in every case to the computed bending moment (see Table 1) Kt =  combined shock and fatigue factor to be applied in every case to the computed torsional moment (see Table 1) M =  maximum bending moment in inch-pounds N =  revolutions per minute P =  maximum power to be transmitted by the shaft in horsepower

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pt = maximum allowable shearing stress under combined loading conditions in pounds per square inch (see Table 2) S =  maximum allowable flexural (bending) stress, in either tension or compres­sion in pounds per square inch (see Table 2) Ss =  maximum allowable torsional shearing stress in pounds per square inch (see Table 2) T =  maximum torsional moment in inch-pounds V =  maximum transverse shearing load in pounds For shafts subjected to pure torsional loads only,

D=B

3

5.1K t T Ss

(16a)

or

D=B

321, 000Kt P (16b) Ss N

3

For stationary shafts subjected to bending only, 3 10.2Km M D=B S For shafts subjected to combined torsion and bending,

D= B or

3

D=B #

(17)

5.1 ^ Km M h2 + ^ Kt T h2 pt 3

(18a)

63, 000Kt P m ^ Km M h2+ c

2

5.1 pt

N

(18b)

Formulas (16a) to (18b) may be used for solid shafts or for hollow shafts. For solid shafts the factor B is equal to 1, whereas for hollow shafts the value of B depends on the value of K which, in turn, depends on the ratio of the inside diameter of the shaft to the outside diam­ eter (D1 ÷ D = K). Table 3 gives values of B corresponding to various values of K. For short solid shafts subjected only to heavy transverse shear, the diameter of shaft required is: 1.7V D= (19) S s

Formulas (16a), (17), (18a) and (19), can be used unchanged with metric SI units. Formula (16b) becomes:

D=B

48.7K t P  and Formula (18b) becomes: Ss N

3

D=B

3

5.1 pt

^ Km M h2 + a 9.55Kt P k

N

2

Throughout the formulas, D = outside diameter of shaft in millimeters; T = maxi­ mum torsional moment in newton-millimeters; Ss = maximum allowable torsional shearing stress in newtons per millimeter squared (see Table 2); P = maximum power to be transmitted in milliwatts; N = revolutions per minute; M = maximum bending moment in newton-millimeters; S = maximum allowable flexural (bending) stress, either in tension or compression in newtons per millimeter squared (see Table 2); p t = maximum allowable shearing stress under combined loading conditions in newtons per millimeter squared; and V = maximum transverse shearing load in

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kilograms. The factors Km , Kt , and B are unchanged, and D1 = the inside diameter of a hollow shaft in millimeters. Table 1. Recommended Values of the Combined Shock and Fatigue Factors for Various Types of Load Stationary Shafts Kt Km 1.0 1.0 1.5–2.0 1.5–2.0 … …

Type of Load Gradually applied and steady Suddenly applied, minor shocks only Suddenly applied, heavy shocks

Rotating Shafts Km Kt 1.5 1.0 1.5–2.0 1.0–1.5 2.0–3.0 1.5–3.0

Table 2. Recommended Maximum Allowable Working Stresses for Shafts Under Various Types of Load Type of Load Simple Bending S = 16,000 S = 12,000 (See note a)

Material “Commercial Steel” shafting without keyways “Commercial Steel” shafting with keyways Steel purchased under definite physical specs.

Pure Torsion Ss = 8000 Ss = 6000 (See note b)

Combined Stress pt = 8000 pt = 6000 (See note b)

a S = 60 percent of the elastic limit in tension but not more than 36 percent of the ultimate tensile strength. b S and p = 30 percent of the elastic limit in tension but not more than 18 percent of the ultimate s t tensile strength.

If the values in the Table are converted to metric SI units, note that 1000 pounds per square inch = 6.895 newtons per square millimeter.

Table 3. Values of the Factor B Corresponding to Various Values of K for Hollow Shafts K= B=

D1 D 3

4

1 ' ^1 − K h

0.95

0.90

0.85

0.80

1.75

1.43

1.28

1.19

For solid shafts, B = 1 because K = 0, as follows: B =

0.75

0.70

0.65

0.60

0.55

0.50

1.14

1.10

1.07

1.05

1.03

1.02

3

4

3

1 ' ^1 − K h = 1 ' ^1 − 0h = 1

Effect of Keyways on Shaft Strength.—Keyways cut into a shaft reduce its load carry­ing ability, particularly when impact loads or stress reversals are involved. To ensure an adequate factor of safety in the design of a shaft with standard keyway (width, one-quarter, and depth, one-eighth of shaft diameter), the former Code for Transmission Shafting tenta­tively recommended that shafts with keyways be designed on the basis of a solid circular shaft using not more than 75 percent of the working stress recommended for the solid shaft. See also page 2539. Formula for Shafts of Brittle Materials.—The preceding formulas are applicable to ductile materials and are based on the maximum-shear theory of failure which assumes that the elastic limit of a ductile material in shear is one-half its elastic limit in tension. Brittle materials are generally stronger in shear than in tension; therefore, the maximumshear theory is not applicable. The maximum-normal-stress theory of failure is now gener­ally accepted for the design of shafts made from brittle materials. A material may be con­sidered brittle if its elongation in a 2-inch gage length is less than 5 percent. Materials such as cast iron, hardened tool steel, hard bronze, etc. conform to this rule. The diameter of a shaft made of a brittle material may be determined from the following formula which is based on the maximum-normal-stress theory of failure: D = B 3 5.1 7^ Km M h + ^ Km M h2 + ^ K t T h2A St

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where St is the maximum allowable tensile stress in pounds per square inch and the other quantities are as previously defined. The formula can be used unchanged with metric SI units, where D = outside diame­ ter of shaft in millimeters; St = the maximum allowable tensile stress in newtons per millimeter squared; M = maximum bending moment in newton-millimeters; and T = maximum torsional moment in newton-millimeters. The factors Km , Kt , and B are unchanged. Critical Speed of Rotating Shafts.—At certain speeds, a rotating shaft will become dynamically unstable and the resulting vibrations and deflections can result in damage not only to the shaft but to the machine of which it is a part. The speeds at which such dynamic instability occurs are called the critical speeds of the shaft. On page 205 are given formulas for the critical speeds of shafts subject to various conditions of loading and support. A shaft may be safely operated either above or below its critical speed, good practice indicating that the operating speed be at least 20 percent above or below the critical. The formulas commonly used to determine critical speeds are sufficiently accurate for general purposes. However, the torque applied to a shaft has an important effect on its crit­ical speed. Investigations have shown that the critical speeds of a uniform shaft are decreased as the applied torque is increased, and that there exist critical torques which will reduce the corresponding critical speed of the shaft to zero. A detailed analysis of the effects of applied torques on critical speeds may be found in a paper, “Critical Speeds of Uniform Shafts under Axial Torque,” by Golomb and Rosenberg, presented at the First US National Congress of Applied Mechanics in 1951. Shaft Couplings.—A shaft coupling is a device for fastening together the ends of two shafts, so that the rotary motion of one causes rotary motion of the other. One of the most simple and common forms of coupling is the flange coupling Fig. 1a and Fig. 1b. It consists of two flanged sleeves or hubs, each of which is keyed to the end of one of the two shafts to be connected. The sleeves are held together and prevented from rotating relative to each other by bolts through the flanges as indicated. Flange Coupling

Fig. 1a.

Fig. 1b.

Flexible Couplings: Flexible couplings are the most common mechanical means of com­pensating for unavoidable errors in alignment of shafts and shafting. When correctly applied, they are highly efficient for joining lengths of shafting without causing loss of power from bearing friction due to misalignment, and for use in direct motor drives for all kinds of machinery. Flexible couplings are not intended to be used for connecting a driven shaft and a driving shaft that are purposely placed in different planes or at an angle but are intended simply to overcome slight unavoidable errors in alignment that develop in ser­ vice. There is a wide variety of flexible coupling designs; most of them consist essentially of two flanged members or hubs, fastened to the shafts and connected by some yielding arrangement. Balance is an important factor in coupling selection or design; it is not suffi­ cient that the coupling be perfectly balanced when installed, but it must remain in balance after wear has taken place.

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303

Comparison of Hollow and Solid Shafting with Same Outside Diameter.—Table 4 that follows gives the percent decrease in strength and weight of a hollow shaft relative to the strength and weight of a solid shaft of the same diameter. The upper figures in each line give the percent decrease in strength and the lower figures give the percent decrease in weight. Example: A 4-inch shaft, with a 2-inch hole through it, has a weight 25 percent less than a solid 4-inch shaft, but its strength is decreased only 6.25 percent. Table 4. Comparative Torsional Strengths and Weights of Hollow and Solid Shafting with Same Outside Diameter

Dia. of Solid and Hollow Shaft, Inches 11∕2 13∕4 2 21∕4 21∕2 23∕4 3 31∕4 31∕2 33∕4 4 41∕4 41∕2 43∕4 5 51∕2 6 61∕2 7 71∕2 8

Diameter of Axial Hole in Hollow Shaft, Inches

1

11∕4

11∕2

13∕4

2

21∕2

3

31∕2

4

41∕2

19.76 44.44

48.23 69.44

… …

… …

… …

… …

… …

… …

… …

… …

32.66

51.02

73.49

…

…

…

…

…

…

…

10.67

6.25

25.00 3.91

26.04

53.98

15.26

31.65

58.62

9.53

19.76

36.60

62.43

…

…

…

…

…

6.25

12.96

24.01

40.96

…

…

…

…

…

4.28

8.86

16.40

27.98

68.30

…

…

…

…

6.25

11.58

19.76

48.23

…

…

…

…

39.07

56.25

76.54

…

…

…

…

…

…

…

…

…

…

20.66

29.74

11.11

17.36

25.00

34.01

9.46

14.80

21.30

29.00

8.16

12.76

18.36

25.00

7.11

11.11

16.00

21.77

28.45

6.25

9.77

14.06

19.14

25.00

8.65

12.45

16.95

22.15

34.61

7.72

11.11

15.12

19.75

30.87

6.93

9.97

13.57

17.73

27.70

8.10

12.25

16.00

25.00

7.43

10.12

13.22

20.66

29.76

6.25

8.50

11.11

17.36

25.00

34.02

7.24

9.47

14.79

21.30

28.99

8.16

12.76

18.36

25.00

7.11

11.11

16.00

21.77

28.45

6.25

9.77

14.06

19.14

25.00

1.24

0.87 0.67 0.51

0.40

3.01

2.19 1.63 1.24

0.96

0.31

0.74

0.25

0.70

0.20

0.50

0.16

0.40

5.54 4.94 4.43

4.00

6.25

4.54 3.38 2.56

1.98

1.56 1.24 1.00

0.81

0.11

0.27

0.55

0.09

0.19

0.40

3.30

2.77

5.17

4.34

40.48

6.25

10.67

26.04

53.98

…

…

…

4.75

8.09

19.76

40.96

75.89

…

…

6.25

15.26

31.65

58.62

…

…

3.68

2.89 2.29 1.85

1.51

1.03

0.73

0.40

6.25

37.87 32.66

4.91 3.91 3.15

2.56

1.75

1.24

0.90

0.67

0.04

0.08

0.16

0.30

0.51

0.03

0.06

0.13

0.23

0.40

1.56

2.77

2.44

4.00

3.51

…

…

0.22

1.77

…

…

0.11

4.59

…

…

…

0.05

3.19

…

…

…

72.61

0.59

2.04

…

…

…

…

35.02

0.29

5.32

69.44

…

…

…

14.35

0.14

3.70

44.44

82.63

…

…

8.41

0.06

2.36

52.89

…

…

…

13.22

1.75

…

…

…

25.00

64.00

…

…

30.87

49.00

79.00

…

16.00

36.00

60.49

…

19.75 2.56

44.44

…

5.44

4.78

59.17 51.02

44.44

39.07

85.22 73.49

64.00

56.25

… …

87.10

76.56

… …

…

…

… …

…

…

11.99

24.83

46.00

78.47

…

9.53

19.76

36.60

62.43

…

7.68

15.92

29.48

50.29

80.56

6.25

12.96

24.01

40.96

65.61

4.27

3.02

2.19

1.63

1.24

0.96

49.85 44.44 39.90

36.00 8.86

6.25

4.54

3.38

2.56

1.98

67.83 60.49 54.29

49.00

88.59 79.00 70.91

64.00

… …

89.75

81.00

16.40

27.98

44.82

11.58

19.76

31.65

40.48

52.89

44.44

66.94

56.25

8.41

14.35

23.98

6.25

10.67

17.08

4.75

3.68

37.87

32.66

47.93

41.33

8.09

12.96

6.25

10.02

36.00

31.64

The upper figures in each line give number of percent decrease in strength; the lower figures give percent decrease in weight.

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Machinery's Handbook, 31st Edition Springs SPRINGS Introduction to Spring Design

Many advances have been made in the spring industry in recent years. For example: developments in materials permit longer fatigue life at higher stresses; simplified design procedures reduce the complexities of design, and improved methods of manufacture help to speed up some of the complicated fabricating procedures and increase production. New types of testing instruments and revised tolerances also permit higher standards of accu­racy. Designers should also consider the possibility of using standard springs now avail­able from stock. They can be obtained from spring manufacturing companies located in different areas, and small shipments usually can be made quickly. Designers of springs require information in the following order of precedence to simplify design procedures. 1) Spring materials and their applications 2) Allowable spring stresses 3) Spring design data with tables of spring characteristics, tables of formulas, and toler­ances. Only the more commonly used types of springs are covered in detail here. Special types and designs rarely used such as torsion bars, volute springs, Belleville washers, constant force, ring and spiral springs and those made from rectangular wire are only described briefly. Belleville and disc springs are discussed in the section DISC SPRINGS starting on page 350. Notation.—The following symbols are used in spring equations: AC =  Active coils b =  Widest width of rectangular wire, inches CL =  Compressed length, inches D =  Mean coil diameter, inches = OD − d d =  Diameter of wire or side of square, inches E =  Modulus of elasticity in tension, pounds per square inch F =  Deflection, for N coils, inches F° =  Deflection, for N coils, rotary, degrees f =  Deflection, for one active coil FL =  Free length, unloaded spring, inches G =  Modulus of elasticity in torsion, pounds per square inch IT =  Initial tension, pounds K =  Curvature stress correction factor L =  Active length subject to deflection, inches N =  Number of active coils, total P =  Load, pounds p =  Pitch, inches R =  Distance from load to central axis, inches S or St =  Stress, torsional, pounds per square inch Sb =  Stress, bending, pounds per square inch SH =  Solid height Sit =  Stress, torsional, due to initial tension, pounds per square inch T =  Torque = P 3 R, pound-inches TC =  Total coils t =  Thickness, inches U =  Number of revolutions = F °⁄360°

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305

Spring Materials The spring materials most commonly used include high-carbon spring steels, alloy spring steels, stainless spring steels, copper-base spring alloys, and nickel-base spring alloys. High-Carbon Spring Steels in Wire Form.—These spring steels are the most com­monly used of all spring materials because they are the least expensive, are easily worked, and are readily available. However, they are not satisfactory for springs operating at high or low temperatures or for shock or impact loading. The following wire forms are avail­able: Music Wire, ASTM A228: (0.80–0.95 percent carbon) This is the most widely used of all spring materials for small springs operating at temperatures up to about 250°F. It is tough, has a high tensile strength, and can withstand high stresses under repeated load­ing. The material is readily available in round form in diameters ranging from 0.005 to 0.125 inch and in some larger sizes up to 3∕16 inch. It is not available with high tensile strengths in square or rectangular sections. Music wire can be plated easily and is obtain­ able pretinned or preplated with cadmium, but plating after spring manufacture is usually preferred for maximum corrosion resistance. Oil-Tempered MB Grade, ASTM A229: (0.60–0.70 percent carbon) This generalpur­pose spring steel is commonly used for many types of coil springs where the cost of music wire is prohibitive and in sizes larger than are available in music wire. It is readily available in diameters ranging from 0.125 to 0.500 inch, but both smaller and larger sizes may be obtained. The material should not be used under shock and impact loading conditions, at temperatures above 350°F, or at temperatures in the sub-zero range. Square and rectangular sections of wire are obtainable in fractional sizes. Annealed stock also can be obtained for hardening and tempering after coiling. This material has a heat-treating scale that must be removed before plating. Oil-Tempered HB Grade, SAE 1080: (0.75–0.85 percent carbon) This material is simi­lar to the MB Grade except that it has a higher carbon content and a higher tensile strength. It is obtainable in the same sizes and is used for more accurate requirements than the MB Grade, but is not so readily available. In lieu of using this material it may be better to use an alloy spring steel, particularly if a long fatigue life or high endurance properties are needed. Round and square sections are obtainable in the oil-tempered or annealed condi­tions. Hard-Drawn MB Grade, ASTM A227: (0.60–0.70 percent carbon) This grade is used for general-purpose springs where cost is the most important factor. Although increased use in recent years has resulted in improved quality, it is best not to use this grade where long life and accuracy of loads and deflections are important. It is available in diameters ranging from 0.031 to 0.500 inch and in some smaller and larger sizes also. The material is avail­able in square sections but at reduced tensile strengths. It is readily plated. Applications should be limited to those in the temperature range of 0 to 250°F. High-Carbon Spring Steels in Flat Strip Form.—Two types of thin, flat, high-carbon spring steel strip are most widely used although several other types are obtainable for spe­cific applications in watches, clocks, and certain instruments. These two compositions are used for over 95 percent of all such applications. Thin sections of these materials under 0.015 inch having a carbon content of over 0.85 percent and a hardness of over 47 RC (Rockwell C scale) are susceptible to hydrogen-embrittlement, even though special plating and heating operations are employed. The two types are described as follows: Cold-Rolled Spring Steel, Blue-Tempered or Annealed, SAE 1074, also 1064, and 1070: (0.60 to 0.80 percent carbon) This very popular spring steel is available in thicknesses ranging from 0.005 to 0.062 inch and in some thinner and thicker sections. The material is available in the annealed condition for forming in 4-slide machines and in presses, and can readily be hardened and tempered after forming. It is also available in the heat-treated

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Machinery's Handbook, 31st Edition Spring Materials

or blue-tempered condition. The steel is obtainable in several finishes, such as straw color, blue color, black, or plain. Hardnesses ranging from 42 to 46 RC (Rockwell C scale) are recom­mended for spring applications. Uses include spring clips, flat springs, clock springs, and motor, power, and spiral springs. Cold-Rolled Spring Steel, Blue-Tempered Clock Steel, SAE 1095: (0.90 to 1.05 percent carbon) This popular type should be used principally in the blue-tempered condition. Although obtainable in the annealed condition, it does not always harden properly during heat treatment as it is a “shallow” hardening type. It is used principally in clocks and motor springs. End sections of springs made from this steel are annealed for bending or piercing operations. Hardnesses usually range from 47 to 51 RC. Other materials available in strip form and used for flat springs are brass, phosphorbronze, beryllium-copper, stainless steels, and nickel alloys. Alloy Spring Steels.—These spring steels are used for conditions of high stress, and shock or impact loadings. They can withstand both higher and lower temperatures than the highcarbon steels and are obtainable in either the annealed or pretempered conditions. Chromium Vanadium, ASTM A231: This very popular spring steel is used under condi­ tions involving higher stresses than those for which the high-carbon spring steels are rec­ommended and is also used where good fatigue strength and endurance are needed. It behaves well under shock and impact loading. The material is available in diameters rang­ ing from 0.031 to 0.500 inch and in some larger sizes also. In square sections it is available in fractional sizes. Both the annealed and pretempered types are available in round, square, and rectangular sections. It is used extensively in aircraft-engine valve springs and for springs operating at temperatures up to 425°F. Silicon Manganese: This alloy steel is quite popular in Great Britain. It is less expensive than chromium-vanadium steel and is available in round, square, and rectangular sections in both annealed and pretempered conditions in sizes ranging from 0.031 to 0.500 inch. It was formerly used for knee-action springs in automobiles. It is used in flat leaf springs for trucks and as a substitute for more expensive spring steels. Chromium Silicon, ASTM A401: This alloy is used for highly stressed springs that require long life and are subjected to shock loading. It can be heat treated to higher hard­ nesses than other spring steels so that high tensile strengths are obtainable. The most pop­ular sizes range from 0.031 to 0.500 inch in diameter. Very rarely are square, flat, or rectangular sections used. Hardnesses ranging from 50 to 53 RC are quite com­mon and the alloy may be used at temperatures up to 475°F. This material is usually ordered specially for each job. Stainless Spring Steels.—The use of stainless spring steels has increased and several compositions are available all of which may be used for temperatures up to 550°F. They are all corrosion resistant. Only the stainless 18-8 compositions should be used at subzero temperatures. Stainless Type 302, ASTM A313: (18 percent chromium, 8 percent nickel) This stain­less spring steel is very popular because it has the highest tensile strength and quite uni­form properties. It is cold-drawn to obtain its mechanical properties and cannot be hardened by heat treatment. This material is nonmagnetic only when fully annealed and becomes slightly magnetic due to the cold-working performed to produce spring proper­ties. It is suitable for use at temperatures up to 550°F. and for sub-zero tempera­tures. It is very corrosion resistant. The material best exhibits its desirable mechanical properties in diameters ranging from 0.005 to 0.1875 inch although some larger diameters are available. It is also available as hard-rolled flat strip. Square and rectangular sections are available but are infrequently used.

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Machinery's Handbook, 31st Edition Spring Materials

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Stainless Type 304, ASTM A313: (18 percent chromium, 8 percent nickel) This mate­r ial is quite similar to Type 302, but has better bending properties and about 5 percent lower tensile strength. It is a little easier to draw, due to the slightly lower carbon content. Stainless Type 316, ASTM A313: (18 percent chromium, 12 percent nickel, 2 percent molybdenum) This material is quite similar to Type 302 but is slightly more corrosion resistant because of its higher nickel content. Its tensile strength is 10 to 15 percent lower than Type 302. It is used for aircraft springs. Stainless Type 17-7 PH ASTM A313: (17 percent chromium, 7 percent nickel) This alloy, which also contains small amounts of aluminum and titanium, is formed in a moder­ ately hard state and then precipitation hardened at relatively low temperatures for several hours to produce tensile strengths nearly comparable to music wire. This material is not readily available in all sizes, and has limited applications due to its high manufacturing cost. Stainless Type 414, SAE 51414: (12 percent chromium, 2 percent nickel) This alloy has tensile strengths about 15 percent lower than Type 302 and can be hardened by heattreatment. For best corrosion resistance it should be highly polished or kept clean. It can be obtained hard drawn in diameters up to 0.1875 inch and is commonly used in flat cold-rolled strip for stampings. The material is not satisfactory for use at low temperatures. Stainless Type 420, SAE 51420: (13 percent chromium) This is the best stainless steel for use in large diameters above 0.1875 inch and is frequently used in smaller sizes. It is formed in the annealed condition and then hardened and tempered. It does not exhibit its stainless properties until after it is hardened. Clean bright surfaces provide the best corro­sion resistance, therefore the heat-treating scale must be removed. Bright hardening meth­ods are preferred. Stainless Type 431, SAE 51431: (16 percent chromium, 2 percent nickel) This spring alloy acquires high tensile properties (nearly the same as music wire) by a combination of heattreatment to harden the wire plus cold-drawing after heat treatment. Its corrosion resistance is not equal to Type 302. Copper-Base Spring Alloys.—Copper-base alloys are important spring materials because of their good electrical properties combined with their good resistance to corro­ sion. Although these materials are more expensive than the high-carbon and the alloy steels, they nevertheless are frequently used in electrical components and in sub-zero tem­peratures. Spring Brass, ASTM B134: (70 percent copper, 30 percent zinc) This material is the least expensive and has the highest electrical conductivity of the copper-base alloys. It has a low tensile strength and poor spring qualities, but is extensively used in flat stampings and where sharp bends are needed. It cannot be hardened by heat treatment and should not be used at temperatures above 150°F, but is especially good at sub-zero tempera­tures. Available in round sections and flat strips, this hard-drawn material is usually used in the “spring hard” temper. Phosphor Bronze, ASTM B159: (95 percent copper, 5 percent tin) This alloy is the most popular of this group because it combines the best qualities of tensile strength, hard­ness, electrical conductivity, and corrosion resistance with the least cost. It is more expen­sive than brass, but can withstand stresses 50 percent higher. The material cannot be hardened by heat treatment. It can be used at temperatures up to 212°F and at sub-zero temperatures. It is available in round sections and flat strip, usually in the “extra-hard” or “spring hard” tempers. It is frequently used for contact fingers in switches because of its low arcing properties. An 8 percent tin composition is used for flat springs and a superfine grain composition called “Duraflex” has good endurance properties. Beryllium Copper, ASTM B197: (98 percent copper, 2 percent beryllium) This alloy can be formed in the annealed condition and then precipitation hardened after forming at

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Machinery's Handbook, 31st Edition Spring Materials

temperatures around 600°F, for 2 to 3 hours. This treatment produces a high hard­ness combined with a high tensile strength. After hardening, the material becomes quite brittle and can withstand very little or no forming. It is the most expensive alloy in the group and heat treating is expensive due to the need for holding the parts in fixtures to pre­vent distortion. The principal use of this alloy is for carrying electric current in switches and in electrical components. Flat strip is frequently used for contact fingers.

Nickel-Base Spring Alloys.—Nickel-base alloys are corrosion resistant, withstand both elevated and sub-zero temperatures, and their non-magnetic characteristic makes them useful for such applications as gyroscopes, chronoscopes, and indicating instruments. These materials have a high electrical resistance and therefore should not be used for con­ ductors of electrical current.

Monel*: (67 percent nickel, 30 percent copper) This material is the least expensive of the nickel-base alloys. It also has the lowest tensile strength but is useful due to its resis­tance to the corrosive effects of sea water and because it is nearly non-magnetic. The alloy can be subjected to stresses slightly higher than phosphor bronze and nearly as high as beryllium copper. Its high tensile strength and hardness are obtained as a result of cold-drawing and cold-rolling only, since it can not be hardened by heat treatment. It can be used at temperatures ranging from −100 to +425°F at normal operating stresses and is available in round wires up to 3∕16 inch in diameter with quite high tensile strengths. Larger diameters and flat strip are available with lower tensile strengths.

“K” Monel*: (66 percent nickel, 29 percent copper, 3 percent aluminum) This mate­r ial is quite similar to Monel except that the addition of the aluminum makes it a precipita­tionhardening alloy. It may be formed in the soft or fairly hard condition and then hardened by a long-time age-hardening heat treatment to obtain a tensile strength and hardness above Monel and nearly as high as stainless steel. It is used in sizes larger than those usually used with Monel, is non-magnetic and can be used in temperatures ranging from −100 to +450°F at normal working stresses under 45,000 pounds per square inch.

Inconel*: (78 percent nickel, 14 percent chromium, 7 percent iron) This is one of the most popular of the non-magnetic nickel-base alloys because of its corrosion resistance and because it can be used at temperatures up to 700°F. It is more expensive than stainless steel but less expensive than beryllium copper. Its hardness and tensile strength is higher than that of “K” Monel and is obtained as a result of cold-drawing and cold-rolling only. It cannot be hardened by heat treatment. Wire diameters up to 1∕4 inch have the best tensile properties. It is often used in steam valves, regulating valves, and for springs in boil­ers, compressors, turbines, and jet engines.

Inconel “X”*: (70 percent nickel, 16 percent chromium, 7 percent iron) This material is quite similar to Inconel but the small amounts of titanium, columbium and aluminum in its composition make it a precipitation-hardening alloy. It can be formed in the soft or par­tially hard condition and then hardened by holding it at 1200°F for 4 hours. It is nonmagnetic and is used in larger sections than Inconel. This alloy is used at temperatures up to 850°F and at stresses up to 55,000 pounds per square inch.

Duranickel*: (“Z” Nickel) (98 percent nickel) This alloy is non-magnetic, corrosion resistant, has a high tensile strength and is hardenable by precipitation hardening at 900°F for 6 hours. It may be used at the same stresses as Inconel but should not be used at temperatures above 500°F.

Nickel-Base Spring Alloys with Constant Moduli of Elasticity.—Some special nickel alloys have a constant modulus of elasticity over a wide temperature range. These materi­als are especially useful where springs undergo temperature changes and must exhibit uni­ form spring characteristics. These materials have a low or zero thermo-elastic coefficient *Trade name of the International Nickel Company.

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and therefore do not undergo variations in spring stiffness because of modulus changes due to temperature differentials. They also have low hysteresis and creep values which makes them preferred for use in food-weighing scales, precision instruments, gyroscopes, mea­suring devices, recording instruments and computing scales where the temperature ranges from −50 to +150°F. These materials are expensive, none being regularly stocked in a wide variety of sizes. They should not be specified without prior discussion with spring manufacturers because some suppliers may not fabricate springs from these alloys due to the special manufacturing processes required. All of these alloys are used in small wire diameters and in thin strip only and are covered by US patents. They are more specifically described as follows: Elinvar*: (nickel, iron, chromium) This alloy, the first constant-modulus alloy used for hairsprings in watches, is an austenitic alloy hardened only by cold-drawing and coldroll­ing. Additions of titanium, tungsten, molybdenum and other alloying elements have brought about improved characteristics and precipitation-hardening abilities. These improved alloys are known by the following trade names: Elinvar Extra, Durinval, Modul­ var and Nivarox. Ni-Span C*: (nickel, iron, chromium, titanium) This very popular constant-modulus alloy is usually formed in the 50 percent cold-worked condition and precipitation-hard­ ened at 900°F for 8 hours, although heating up to 1250°F for 3 hours pro­duces hardnesses on the Rockwell C scale of 40 to 44 RC, permitting safe torsional stresses of 60,000 to 80,000 pounds per square inch. This material is ferromagnetic up to 400°F; above that temperature it becomes non-magnetic. Iso-Elastic†: (nickel, iron, chromium, molybdenum) This popular alloy is relatively easy to fabricate and is used at safe torsional stresses of 40,000 to 60,000 pounds per square inch and hardnesses of 30 to 36 RC. It is used principally in dynamometers, instru­ ments, and food-weighing scales. Elgiloy‡: (nickel, iron, chromium, cobalt) This alloy, also known by the trade names 8J Alloy, Durapower, and Cobenium, is a non-magnetic alloy suitable for subzero tempera­tures and temperatures up to about 1000°F, provided that torsional stresses are kept under 75,000 pounds per square inch. It is precipitation-hardened at 900°F for 8 hours to produce hardnesses of 48 to 50 RC. The alloy is used in watch and instrument springs. Dynavar§: (nickel, iron, chromium, cobalt) This alloy is a non-magnetic, corrosionresistant material suitable for sub-zero temperatures and temperatures up to about 750°F, provided that torsional stresses are kept below 75,000 pounds per square inch. It is precipitation-hardened to produce hardnesses of 48 to 50 RC and is used in watch and instrument springs. Spring Stresses Allowable Working Stresses for Springs.—The safe working stress for any particular spring depends to a large extent on the following items: 1) Type of spring — whether compression, extension, torsion, etc. 2) Size of spring — small or large, long or short 3) Spring material 4) Size of spring material 5) Type of service — light, average, or severe 6) Stress range — low, average, or high *Trade name of Soc. Anon. de Commentry Fourchambault et Decazeville, Paris, France. †Trade name of John Chatillon & Sons. ‡Trade name of Elgin National Watch Company. §Trade name of Hamilton Watch Company.

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7) Loading — static, dynamic, or shock 8) Operating temperature 9) Design of spring — spring index, sharp bends, hooks.

Consideration should also be given to other factors that affect spring life: corrosion, buckling, friction, and hydrogen embrittlement decrease spring life; manufacturing opera­tions such as high-heat stress-equalizing, presetting, and shot-peening increase spring life. Item 5, the type of service to which a spring is subjected, is a major factor in determining a safe working stress once consideration has been given to type of spring, kind and size of material, temperature, type of loading, and so on. The types of service are: Light Service: This includes springs subjected to static loads or small deflections and sel­dom-used springs, such as those in bomb fuses, projectiles, and safety devices. This service is for 1000 to 10,000 deflections. Average Service: This includes springs in general use in machine tools, mechanical products, and electrical components. Normal frequency of deflections not exceeding 18,000 per hour permit such springs to withstand 100,000 to 1,000,000 deflections. Severe Service: This includes springs subjected to rapid deflections over long periods of time and to shock loading, such as in pneumatic hammers, hydraulic controls and valves. This service is for 1,000,000 deflections, and above. Lowering the values 10 percent per­ mits 10,000,000 deflections. Fig. 1 through Fig. 6 show curves that relate the three types of service conditions to allow­able working stresses and wire sizes for compression and extension springs, and safe val­ues are provided. Fig. 7 through Fig. 10 provide similar information for helical torsion springs. In each chart, the values obtained from the curves may be increased by 20 percent (but not beyond the top curves on the charts if permanent set is to be avoided) for springs that are baked, and shot-peened, and compression springs that are pressed. Springs stressed slightly above the Light Service curves will take a permanent set. A curvature correction factor is included in all curves, and is used in spring design calcu­lations (see examples beginning page 317). The curves may be used for materials other than those designated in Fig. 1 through Fig. 10, by applying multiplication factors as given in Table 1. 160

Torsional Stress (corrected) Pounds per Square Inch (thousands)

150

Hard Drawn Steel Wire QQ-W-428, Type II; ASTM A227, Class II

140

Light Service

130 120

Average Service

110

Severe Service

100 90 80

0 .020 .040 .060 .080 .100 .120 .140 .160 .180 .200 .220 .240 .260 .280 .300 .320 .340 .360 .380 .400 .420 .440 .460 .480 .500

70

Wire Diameter (inch)

Fig. 1. Allowable Working Stresses for Compression Springs—Hard Drawn Steel Wirea

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220 210 200 190 180 170 160 150 140 130 120 110 100 90 80

311

MUSIC WIRE QQ-Q-470, ASTM A228 Light Service Average Service Severe Service

0 .010 .020 .030 .040 .050 .060 .070 .080 .090 .100 .110 .120 .130 .140 .150 .160 .170 .180 .190 .200 .210 .220 .230 .240 .250

Torsional Stress (Corrected) Pounds per Square Inch (thousands)

Machinery's Handbook, 31st Edition Stresses in Springs

Wire Diameter (inch)

Fig. 2. Allowable Working Stresses for Compression Springs—Music Wirea 160

Torsional Stress (corrected) Pounds per Square Inch (thousands)

150 140 130

Oil-tempered Steel Wire QQ-W-428, Type I; ASTM A229, Class II Light Service Average Service

120

Severe Service

110 100 90 80

0 .020 .040 .060 .080 .100 .120 .140 .160 .180 .200 .220 .240 .260 .280 .300 .320 .340 .360 .380 .400 .420 .440 .460 .480 .500

70

Wire Diameter (inch)

Fig. 3. Allowable Working Stresses for Compression Springs—Oil-Tempereda

Torsional Stress (corrected) Pounds per Square Inch (thousands)

190 180

Chrome-silicon Alloy Steel Wire QQ-W-412, comp 2, Type II; ASTM A401 Light Service Average Service

170

Severe Service

160 150 140 130 120

0 .020 .040 .060 .080 .100 .120 .140 .160 .180 .200 .220 .240 .260 .280 .300 .320 .340 .360 .380 .400 .420 .440 .460 .480 .500

110

Wire Diameter (inch)

Fig. 4. Allowable Working Stresses for Compression Springs—Chrome-Silicon Alloy Steel Wirea

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312 160

Corrosion-resisting Steel Wire QQ-W-423, ASTM A313

Torsional Stress (corrected) Pounds per Square Inch (thousands)

150 140

Light service Average service

130 120

Severe service

110 100 90 80

0 .020 .040 .060 .080 .100 .120 .140 .160 .180 .200 .220 .240 .260 .280 .300 .320 .340 .360 .380 .400 .420 .440 .460 .480 .500

70

Wire Diameter (inch)

190 180 170 160 150 140 130 120 110 100 90 80

Chrome-vanadium Alloy Steel Wire, ASTM A231 Light service Average service

Severe service

0 .020 .040 .060 .080 .100 .120 .140 .160 .180 .200 .220 .240 .260 .280 .300 .320 .340 .360 .380 .400 .420 .440 .460 .480 .500

Torsional Stress (corrected) Pounds per Square Inch (thousands)

Fig. 5. Allowable Working Stresses for Compression Springs—Corrosion-Resisting Steel Wirea

Wire Diameter (inch)

270 260 250 240 230 220 210 200 190 180 170 160 150 140 130 120

Music Wire, ASTM A228 Light service Average service

Severe service

0 .010 .020 .030 .040 .050 .060 .070 .080 .090 .100 .110 .120 .130 .140 .150 .160 .170 .180 .190 .200 .210 .220 .230 .240 .250

Stress, Pounds per Square Inch (thousands)

Fig. 6. Allowable Working Stresses for Compression Springs—Chrome-Vanadium Alloy Steel Wirea

Wire Diameter (inch)

Fig. 7. Recommended Design Stresses in Bending for Helical Torsion Springs—Round Music Wire

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260 250 240 230 220 210 200 190 180 170 160 150 140 130 120 110

313

Oil-tempered MB Grade, ASTM A229 Type I Light service Average service Severe service

0 .020 .040 .060 .080 .100 .120 .140 .160 .180 .200 .220 .240 .260 .280 .300 .320 .340 .360 .380 .400 .420 .440 .460 .480 .500

Stress, Pounds per Square Inch (thousands)

Machinery's Handbook, 31st Edition Stresses in Springs

Wire Diameter (inch)

220 210 200 190 180 170 160 150 140 130 120 110 100 90 80 70

Stainless Steel, “18-8,” Types 302 & 304, ASTM A313 Light Service

Average Service Severe Service

0 .020 .040 .060 .080 .100 .120 .140 .160 .180 .200 .220 .240 .260 .280 .300 .320 .340 .360 .380 .400 .420 .440 .460 .480 .500

Stress, Pounds per Square Inch (thousands)

Fig. 8. Recommended Design Stresses in Bending for Helical Torsion Springs— Oil-Tempered MB Round Wire

Wire Diameter (inch)

290 280 270 260 250 240 230 220 210 200 190 180 170 160 150 140

Chrome-silicon, ASTM A401 Light service Average service

Severe service

0 .020 .040 .060 .080 .100 .120 .140 .160 .180 .200 .220 .240 .260 .280 .300 .320 .340 .360 .380 .400 .420 .440 .460 .480 .500

Stress, Pounds per Square Inch (thousands)

Fig. 9. Recommended Design Stresses in Bending for Helical Torsion Springs— Stainless Steel Round Wire

Wire Diameter (inch)

Fig. 10. Recommended Design Stresses in Bending for Helical Torsion Springs— Chrome-Silicon Round Wire a Although Fig. 1 through Fig. 6 are for compression springs, they may also be used for extension springs; for extension springs, reduce the values obtained from the curves by 10 to 15 percent.

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Machinery's Handbook, 31st Edition Stresses in Springs

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Table 1. Correction Factors for Other Materials Compression and Tension Springs Material

Factor

Silicon-manganese Valve-spring quality wire Stainless Steel, 304 and 420

Multiply the values in the chro­mium-vanadium curves (Fig. 6) by 0.90

Use the values in the chromium-vanadium curves (Fig. 6)

Multiply the values in the corro­sion-resisting steel curves (Fig. 5) by 0.95

Material

Factor

Stainless Steel, 316

Multiply the values in the corro­sion-resisting steel curves (Fig. 5) by 0.90

Stainless Steel, 431 and 17-7PH

Multiply the values in the music wire curves (Fig. 2) by 0.90

Helical Torsion Springs Material

Hard Drawn MB

Factora

Over 1∕32 to 3∕16 inch Over 3∕16 to 1∕4 inch

0.70 0.65

Up to 1∕8 inch diameter

1.00

Stainless Steel, 17-7 PH

Over 1∕32 to 1∕16 inch Over 1∕16 to 1∕8 inch Over 1∕8 to 3∕16 inch Over 3∕16 inch

Over 1∕16 to 1∕8 inch Over 1∕8 inch

1.00

Up to 1∕16 inch diameter

1.05

Over 1∕16 inch

Phosphor Bronze

1.12

Up to 1∕8 inch diameter

0.70 0.75 0.80 0.90 1.00

0.85 0.95

Chromium-Vanadium

1.07

Stainless Steel, 420

Up to 1∕32 inch diameter

0.80

Over 1∕32 to 1∕16 inch

0.50

Over 3∕16 inch

Up to 1∕32 inch diameter

0.75

Over 1∕4 inch

Over 1∕8 to 3∕16 inch

Factora

Stainless Steel, 431

Stainless Steel, 316 Up to 1∕32 inch diameter

Material

0.70

1.10 0.45

Over 1∕8 inch

0.55

Up to 1∕32 inch diameter

0.55

Beryllium Copperb Over 1∕32 to 1∕16 inch Over 1∕16 to 1∕8 inch Over 1∕8 inch

0.60 0.70 0.80

a Multiply the values in the curves for oil-tempered MB grade ASTM A229 Type 1 steel (Fig. 8)

by these factors to obtain required values. b Hard drawn and heat treated after coiling. For use with design stress curves shown in Fig. 2, Fig. 5, Fig. 6, and Fig. 8.

Endurance Limit for Spring Materials.—When a spring is deflected continually it will become “tired” and fail at a stress far below its elastic limit. This type of failure is called fatigue failure and usually occurs without warning. Endurance limit is the highest stress, or range of stress, in pounds per square inch that can be repeated indefinitely without failure of the spring. Usually ten million cycles of deflection is called “infinite life” and is satisfac­tory for determining this limit. For severely worked springs of long life, such as those used in automobile or aircraft engines and in similar applications, it is best to determine the allowable working stresses by referring to the endurance limit curves seen in Fig. 11. These curves are based princi­ pally upon the range or difference between the stress caused by the first or initial load and the stress caused by the final load. Experience with springs designed to stresses within the limits of these curves indicates that they should have infinite or unlimited fatigue life. All values include Wahl curvature correction factor. The stress ranges shown may be increased 20 to 30 percent for springs that have been properly heated, pressed to remove set, and then shot peened, provided that the increased values are lower than the torsional elastic limit by at least 10 percent.

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Machinery's Handbook, 31st Edition Stresses in Springs

315

120

Final Stress, Including Curvature Correction, 1000 psi

110 0" 0.03 nder 5" ire u 0.12 o W t ic " 31 Mus e 0.0 ir ic W adium Mus Van 0%C ome el 0.8 ade Chr g Ste gr in B r p el M OT S g Ste 0.08%c prin l e S e e t T O gS grad Sprin teel mb S *HD g Sprin e 302 *HD 8 typ l 18Stee s s H.T. inle hard *Sta r full e p p o ard mC ng h ylliu spri % *Ber 5 e ronz ur B osph s *Ph s Bra ring *Sp d Lan irst F o t ue ss D

100 90 80 70 60 50 40 30 20 10 0 0

tial

Ini

e

Str

5 10 15 20 25 30 35 40 45 50 55 Initial Stress, Due to First Load, Corrected for Curvature, 1000 psi

60

Fig. 11. Endurance Limit Curves for Compression Springs Notes: For commercial spring materials with wire diameters up to 1∕4 inch except as noted. Stress ranges may be increased by approximately 30 percent for properly heated, preset, shot-peened springs.

Materials preceeded by * are not ordinarily recommended for long continued service under severe operating conditions.

Working Stresses at Elevated Temperatures.—Since modulus of elasticity decreases with increase in temperature, springs used at high temperatures exert less load and have larger deflections under load than at room temperature. The torsional modulus of elasticity for steel may be 11,200,000 pounds per square inch at room temperature, but it will drop to 10,600,000 pounds per square inch at 400°F. and will be only 10,000,000 pounds per square inch at 600°F. Also, the elastic limit is reduced, thereby lowering the permissible working stress. Design stresses should be as low as possible for all springs used at elevated temperatures. In addition, corrosive conditions that usually exist at high temperatures, especially with steam, may require the use of corrosion-resistant material. Table 2 shows the permissible elevated temperatures at which various spring materials may be operated, together with the maximum recommended working stresses at these temperatures. The loss in load at the temperatures shown is less than 5 percent in 48 hours; however, if the temperatures listed are increased by 20 to 40 degrees, the loss of load may be nearer 10 percent. Maximum stresses shown in the table are for compression and extension springs and may be

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Machinery's Handbook, 31st Edition STRESSES IN SPRINGS

316

increased by 75 percent for torsion and flat springs. In using the data in Table 2 it should be noted that the values given are for materials in the heat-treated or spring temper condition. Table 2. Recommended Maximum Working Temperatures and Corresponding Maximum Working Stresses for Springs Max. Working Temp., °F

Max. Working Stress, psi

Brass Spring Wire

150

30,000

Alloy Spring Steels

400

65,000

Monel K-Monel

425 450

40,000 45,000

Spring Material Phosphor Bronze Music Wire Beryllium-Copper Hard Drawn Steel Wire Carbon Spring Steels

225 250 300 325 375

35,000 75,000 40,000 50,000 55,000

Spring Material Permanickela

Stainless Steel 18-8 Stainless Chromium 431 Inconel High-Speed Steel Inconel X Chromium-MolybdenumVanadium Cobenium, Elgiloy

Max. Working Temp, °F

Max. Working Stress, psi

500

50,000

900

55,000

1000

75,000

550 600 700 775 850

55,000 50,000 50,000 70,000 55,000

a Formerly called Z-Nickel, Type B.

Loss of load at temperatures shown is less than 5 percent in 48 hours.

Spring Design Data Spring Characteristics.—This section provides tables of spring characteristics, tables of principal formulas, and other information of a practical nature for designing the more com­monly used types of springs. Standard wire gages for springs: Information on wire gages is given in the section beginning on page 2702, and gages in decimals of an inch are given in the table on page 2703. It should be noted that the range in this table extends from Number 7 ∕ 0 through Number 80. However, in spring design, the range most commonly used extends only from Gage Number 4 ∕ 0 through Number 40. When selecting wire use Steel Wire Gage or Wash­burn and Moen gage for all carbon steels and alloy steels except music wire; use Brown & Sharpe gage for brass and phosphor bronze wire; use Birmingham gage for flat spring steels, and cold rolled strip; and use piano or music wire gage for music wire. Spring index: The spring index is the ratio of the mean coil diameter of a spring to the wire diameter (D/d). This ratio is one of the most important considerations in spring design because the deflection, stress, number of coils, and selection of either annealed or tem­pered material depend to a considerable extent on this ratio. The best proportioned springs have an index of 7 through 9. Indexes of 4 through 7, and 9 through 16 are often used. Springs with values larger than 16 require tolerances wider than standard for manufactur­ing; those with values less than 5 are difficult to coil on automatic coiling machines. Direction of helix: Unless functional requirements call for a definite hand, the helix of compression and extension springs should be specified as optional. When springs are designed to operate, one inside the other, the helices should be opposite hand to prevent intermeshing. For the same reason, a spring that is to operate freely over a threaded mem­ber should have a helix of opposite hand to that of the thread. When a spring is to engage with a screw or bolt, it should, of course, have the same helix as that of the thread. Helical Compression Spring Design.—After selecting a suitable material and a safe stress value for a given spring, designers should next determine the type of end coil forma­ tion best suited for the particular application. Springs with unground ends are less expen­ sive but they do not stand perfectly upright; if this requirement has to be met, closed ground ends are used. Helical compression springs with different types of ends are shown in Fig. 12.

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Machinery's Handbook, 31st Edition Spring Design

Open Ends Not Ground, Right Hand Helix

Closed Ends Not Ground, Right Hand Helix

Closed Ends Not Ground, Left Hand Helix

Open Ends Not Ground, Left Hand Helix

317

Fig. 12. Types of Helical Compression Spring Ends

Spring design formulas: Table 3 gives formulas for compression spring dimensional characteristics, and Table 4 gives design formulas for compression and extension springs.

Curvature correction: In addition to the stress obtained from the formulas for load or deflection, there is a direct shearing stress and an increased stress on the inside of the sec­tion due to curvature. Therefore, the stress obtained by the usual formulas should be multi­plied by a factor K taken from the curve in Fig. 13. The corrected stress thus obtained is used only for comparison with the allowable working stress (fatigue strength) curves to determine if it is a safe stress and should not be used in formulas for deflection. The curva­ ture correction factor K is for compression and extension springs made from round wire. For square wire reduce the K value by approximately 4 percent. Design procedure: The limiting dimensions of a spring are often determined by the available space in the product or assembly in which it is to be used. The loads and deflec­ tions on a spring may also be known or can be estimated, but the wire size and number of coils are usually unknown. Design can be carried out with the aid of the tabular data that appears later in this section (see Table 5, which is a simple method, or by calculation alone using the formulas in Table 3 and Table 4.

Example: A compression spring with closed and ground ends is to be made from ASTM A229 high carbon steel wire, as shown in Fig. 14. Determine the wire size and number of coils.

Method 1, using table: Referring to Table 5, starting on page 321, locate the spring out­ side diameter (13∕16 inches, from Fig. 14 on page 319) in the left-hand column. Note from the drawing that the spring load is 36 pounds. Move to the right in the table to the figure nearest this value, which is 41.7 pounds. This is somewhat above the required value but safe. Immediately above the load value, the deflection f is given, which in this instance is 0.1594 inch. This is the deflection of one coil under a load of 41.7 pounds with an uncorrected tor­sional stress S of 100,000 pounds per square inch for ASTM A229 oil-tempered MB steel. For other spring materials, see the footnotes to Table 5. Moving vertically in Table 5 from the load entry, the wire diameter is found to be 0.0915 inch. The remaining spring design calculations are completed as follows:

Step 1: The stress with a load of 36 pounds is obtained by proportion, as follows: The 36 pound load is 86.3 percent of the 41.7 pound load; therefore, the stress S at 36 pounds = 0.863 3 100,000 = 86,300 pounds per square inch.

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Machinery's Handbook, 31st Edition Spring Design

318

Table 3. Formulas for Compression Springs Type of End

Open or Plain (not ground)

Open or Plain (with ends ground)

Feature

Squared or Closed (not ground)

Closed and Ground

Formulaa

Pitch (p)

FL − d N

FL TC

FL − 3d N

FL − 2d N

Solid Height (SH)

(TC + 1)d

TC 3 d

(TC + I)d

TC 3 d

Number of Active Coils (N)

N = TC FL − d = p

N = TC − 2 =

FL − 3d p

N = TC − 2 FL − 2d = p

FL − d p

FL p

FL − 3d +2 p

FL − 2d +2 p

(p 3 TC) + d

p 3 TC

(p 3 N) + 3d

(p 3 N) + 2d

Total Coils (TC) Free Length (FL)

N = TC − 1 FL = p − 1

a The symbol notation is given on page 304.

Table 4. Formulas for Compression and Extension Springs Formulaa, b

Feature

Springs made from round wire 3

4

Springs made from square wire

Load, P Pounds

P=

0.393Sd Gd F = D 8ND 3

P=

0.416Sd 3 Gd 4F = D 5.58ND3

Stress, Torsional, S Pounds per square inch

S=

GdF PD = π ND2 0.393d 3

S =

GdF D =P 2.32ND2 0.416d 3

Deflection, F Inch

F=

8PND3 π SN D 2 = Gd Gd 4

F=

5.58PND3 2.32SN D 2 = Gd Gd 4

Number of Active Coils, N

N =

Gd 4F GdF = π SD 2 8PD3

N=

Gd 4F GdF = 5.58PD3 2.32SD 2

Wire Diameter, d Inch

d=

π SN D2 GF =

d=

2.32SN D2 = GF

Stress due to Initial Tension, Sit

S Sit = P # IT

3

2.55PD S

3

PD 0.416S

S Sit = P # IT

a The symbol notation is given on page 304.

b Two formulas are given for each feature, and designers can use the one found to be appropriate for a given design. The end result from either of any two formulas is the same.

Step 2: The 86.3 percent figure is also used to determine the deflection per coil f at 36 pounds load: 0.863 3 0.1594 = 0.1375 inch. F 1.25 Step 3: The number of active coils AC = f = 0.1375 = 9.1

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319

2.1 2.0 1.9 Correction Factor, K

1.8 1.7 1.6 1.5 1.4 1.3 1.2 1.1 1.0

1

2

3

4

5 6 7 Spring Index

8

9

10

11

12

Fig. 13. Compression and Extension Spring-Stress Correction for Curvaturea a For

springs made from round wire. For springs made from square wire, reduce the K factor values by approximately 4 percent.

13 16

”

- 36 Pounds (P)

11 16 ” Max (SH) 11 4 ” (CL)

11 4 ” (CL)

2 1 16 ” (to inside Diameter of Hooks) Fig. 14. Compression Spring Design Example

Step 4: Total Coils TC = AC + 2  (Table 3) = 9 + 2 = 11

Therefore, a quick answer is: 11 coils of 0.0915 inch diameter wire. However, the design procedure should be completed by carrying out these remaining steps: Step 5: From Table 3, Solid Height = SH = TC 3 d = 11 3 0.0915 @ 1 inch

Therefore, Total Deflection = FL − SH = 1.5 inches

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Machinery's Handbook, 31st Edition Spring Design

320

86, 300 Step 6: Stress Solid = 1.25 # 1.5 = 103, 500 pounds per square inch O.D. 0.8125 Step 7: Spring Index = d − 1 = 0.0915 − 1 = 7.9

Step 8: From Fig. 13 on page 319, the curvature correction factor K = 1.185 Step 9: Total Stress at 36 pounds load = S 3 K = 86,300 3 1.185 = 102,300 pounds per square inch. This stress is below the 117,000 pounds per square inch permitted for 0.0915 inch wire shown on the middle curve in Fig. 3 on page 311, so it is a safe working stress. Step 10: Total Stress at Solid = 103,500 3 1.185 = 122,800 pounds per square inch. This stress is also safe, as it is below the 131,000 pounds per square inch shown on the top curve of Fig. 3, and therefore the spring will not set. Method 2, using formulas: The procedure for design using formulas is as follows (the design example is the same as in Method 1, and the spring is shown in Fig. 14): Step 1: Select a safe stress S below the middle fatigue strength curve Fig. 3 on page 311 for ASTM A229 steel wire, say 90,000 pounds per square inch. Assume a mean diameter D slightly below the 13∕16 -inch O.D., say 0.7 inch. Note that the value of G is 11,200,000 pounds per square inch (Table 20 on page 346). Step 2: A trial wire diameter d and other values are found by formulas from Table 4 as follows:

d =

3

2.55PD 3 2.55 # 36 # 0.7 = S 90, 000

= 3 0.000714 = 0.0894 inch

Note: Table 21 on page 347 can be used to avoid solving the cube root. Step 3: From Table 21 (also see the table on page 2703), select the nearest wire gauge size, which is 0.0915 inch diameter. Using this value, the mean diameter D = 13∕16 inch − 0.0915 = 0.721 inch. Step 4: The stress S =

PD 36 # 0.721 = = 86, 300 lb / in2 0.393d 3 0.393 # 0.09153

Step 5: The number of active coils is

N =

11, 200, 000 # 0.0915 # 1.25 GdF = = 9.1 ( say 9) π SD 2 3.1416 # 86, 300 # 0.7212

The answer is the same as before, which is to use 11 total coils of 0.0915-inch diameter wire. The total coils, solid height, etc., are determined in the same manner as in Method 1. Table of Spring Characteristics.—Table 5 gives characteristics for compression and extension springs made from ASTM A229 oil-tempered MB spring steel having a tor­ sional modulus of elasticity G of 11,200,000 pounds per square inch, and an uncorrected torsional stress S of 100,000 pounds per square inch. The deflection f for one coil under a load P is shown in the body of the table. The method of using these data is explained in the problems for compression and extension spring design. The table may be used for other materials by applying factors to f. The factors are given in a footnote to the table.

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Spring Outside Dia.

Nom.

Dec.

7∕6 4

.1094

1∕8

.125

9∕6 4

.1406 .1563 .1719

3∕16

.1875

13∕6 4

.2031

7∕3 2

.2188

15∕6 4

.2344

1∕4

.250

9∕3 2

.2813

5∕16

.3125

11∕3 2

.3438

3∕8

.375

.012

.014

.016

.018

.020

.022

.024

.026

.028

.030

.032

.034

.036

.038

19 .041

18 .0475

17 .054

16 .0625

.0277 .395 .0371 .342 .0478 .301 .0600 .268 .0735 .243 .0884 .221 .1046 .203 … … … … … … … … … … … … … …

.0222 .697 .0299 .600 .0387 .528 .0487 .470 .0598 .424 .0720 .387 .0854 .355 .1000 .328 .1156 .305 … … … … … … … … … …

.01824 1.130 .0247 .971 .0321 .852 .0406 .758 .0500 .683 .0603 .621 .0717 .570 .0841 .526 .0974 .489 .1116 .457 .1432 .403 … … … … … …

.01529 1.722 .0208 1.475 .0272 1.291 .0345 1.146 .0426 1.031 .0516 .938 .0614 .859 .0721 .793 .0836 .736 .0960 .687 .1234 .606 .1541 .542 … … … …

.01302 2.51 .01784 2.14 .0234 1.868 .0298 1.656 .0369 1.488 .0448 1.351 .0534 1.237 .0628 1.140 .0730 1.058 .0839 .987 .1080 .870 .1351 .778 .1633 .703 … …

.01121 3.52 .01548 2.99 .0204 2.61 .0261 2.31 .0324 2.07 .0394 1.876 .0470 1.716 .0555 1.580 .0645 1.465 .0742 1.366 .0958 1.202 .1200 1.074 .1470 .970 .1768 .885

.00974 4.79 .01353 4.06 .01794 3.53 .0230 3.11 .0287 2.79 .0349 2.53 .0418 2.31 .0494 2.13 .0575 1.969 .0663 1.834 .0857 1.613 .1076 1.440 .1321 1.300 .1589 1.185

.00853 6.36 .01192 5.37 .01590 4.65 .0205 4.10 .0256 3.67 .0313 3.32 .0375 3.03 .0444 2.79 .0518 2.58 .0597 2.40 .0774 2.11 .0973 1.881 .1196 1.697 .1440 1.546

.00751 8.28 .01058 6.97 .01417 6.02 .01832 5.30 .0230 4.73 .0281 4.27 .0338 3.90 .0401 3.58 .0469 3.21 .0541 3.08 .0703 2.70 .0886 2.41 .1090 2.17 .1314 1.978

.00664 10.59 .00943 8.89 .01271 7.66 0.1649 6.72 .0208 5.99 .0255 5.40 .0307 4.92 .0365 4.52 .0427 4.18 .0494 3.88 .0643 3.40 .0811 3.03 .0999 2.73 .1206 2.48

.00589 13.35 .00844 11.16 .01144 9.58 .01491 8.39 .01883 7.47 .0232 6.73 .0280 6.12 .0333 5.61 .0391 5.19 .0453 4.82 .0591 4.22 .0746 3.75 .0921 3.38 .1113 3.07

… … .00758 13.83 .01034 11.84 .01354 10.35 .01716 9.19 .0212 8.27 .0257 7.52 .0306 6.88 .0359 6.35 .0417 5.90 .0545 5.16 .0690 4.58 .0852 4.12 .1031 3.75

… … .00683 16.95 .00937 14.47 .01234 12.62 .01569 11.19 .01944 10.05 .0236 9.13 .0282 8.35 .0331 7.70 .0385 7.14 .0505 6.24 .0640 5.54 .0792 4.98 .0960 4.53

… … .00617 20.6 .00852 17.51 .01128 15.23 .01439 13.48 .01788 12.09 .0218 10.96 .0260 10.02 .0307 9.23 .0357 8.56 .0469 7.47 .0596 6.63 .0733 5.95 .0895 5.40

… … … … .00777 21.0 .01033 18.22 .01324 16.09 .01650 14.41 .0201 13.05 .0241 11.92 .0285 10.97 .0332 10.17 .0437 8.86 .0556 7.85 .0690 7.05 .0839 6.40

… … … … … … .00909 23.5 .01172 21.8 .01468 18.47 .01798 16.69 .0216 15.22 .0256 13.99 .0299 12.95 .0395 11.26 .0504 9.97 .0627 8.94 .0764 8.10

… … … … … … … … .00914 33.8 .01157 30.07 .01430 27.1 .01733 24.6 .0206 22.5 .0242 20.8 .0323 18.01 .0415 15.89 .0518 14.21 .0634 12.85

… … … … … … … … … … .00926 46.3 .01155 41.5 .01411 37.5 .01690 34.3 .01996 31.6 .0268 27.2 .0347 23.9 .0436 21.3 .0535 19.27

… … … … … … … … … … … … … … .01096 61.3 .01326 55.8 .01578 51.1 .0215 43.8 .0281 38.3 .0355 34.1 .0438 30.7

Deflection f (inch) per coil, at Load P (pounds) c

a This table is for ASTM A229 oil tempered spring steel with a torsional modulus G of 11,200,000 psi, and an uncorrected torsional stress of 100,000 psi. For other

materials use the following factors: stainless steel, multiply f by 1.067; spring brass, multiply f by 2.24; phosphor bronze, multiply f by 1.867; Monel metal, multiply f by 1.244; beryllium copper, multiply f by 1.725; Inconel (non-magnetic), multiply f by 1.045. b Round wire. For square wire, multiply f by 0.707, and p, by 1.2 c The upper figure is the deflection and the lower figure the load as read against each spring size. Note: Intermediate values can be obtained within reasonable accuracy by interpolation.

321

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5∕3 2 11∕6 4

Wire Size or Washburn and Moen Gauge, and Decimal Equivalent b

.010

Machinery's Handbook, 31st Edition Spring Design

Copyright 2020, Industrial Press, Inc.

Table 5. Compression and Extension Spring Deflectionsa

Spring Outside Dia.

Dec.

13∕3 2

.4063

7∕16

.4375

15∕3 2

.4688

1∕2

.500

17∕3 2

.5313

9∕16

.5625

19∕3 2

.5938

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5∕8

.625

21∕3 2

.6563

11∕16

.6875

23∕3 2

.7188

3∕4

.750

25∕3 2

.7813

13∕16

.8125

Wire Size or Washburn and Moen Gauge, and Decimal Equivalent

.026

.028

.030

.032

.034

.036

.038

.1560 1.815 .1827 1.678 .212 1.559 .243 1.456 .276 1.366 … … … … … … … … … … … … … … … … … …

.1434 2.28 .1680 2.11 .1947 1.956 .223 1.826 .254 1.713 .286 1.613 … … … … … … … … … … … … … … … …

.1324 2.82 .1553 2.60 .1800 2.42 .207 2.26 .235 2.12 .265 1.991 .297 1.880 .331 1.782 … … … … … … … … … … … …

.1228 3.44 .1441 3.17 .1673 2.94 .1920 2.75 .219 2.58 .247 2.42 .277 2.29 .308 2.17 .342 2.06 … … … … … … … … … …

.1143 4.15 .1343 3.82 .1560 3.55 .1792 3.31 .204 3.10 .230 2.92 .259 2.76 .288 2.61 .320 2.48 .352 2.36 … … … … … … … …

.1068 4.95 .1256 4.56 .1459 4.23 .1678 3.95 .1911 3.70 .216 3.48 .242 3.28 .270 3.11 .300 2.95 .331 2.81 .363 2.68 … … … … … …

.1001 5.85 .1178 5.39 .1370 5.00 .1575 4.67 .1796 4.37 .203 4.11 .228 3.88 .254 3.67 .282 3.49 .311 3.32 .342 3.17 .374 3.03 … … … …

19

18

17

16

15

14

13

3∕3 2

12

11

1∕8

.041

.0475

.054

.0625

.072

.080

.0915

.0938

.1055

.1205

.125

.0913 7.41 .1075 6.82 .1252 6.33 .1441 5.90 .1645 5.52 .1861 5.19 .209 4.90 .233 4.63 .259 4.40 .286 4.19 .314 3.99 .344 3.82 .375 3.66 .407 3.51

.0760 11.73 .0898 10.79 .1048 9.99 .1209 9.30 .1382 8.70 .1566 8.18 .1762 7.71 .1969 7.29 .219 6.92 .242 6.58 .266 6.27 .291 5.99 .318 5.74 .346 5.50

.0645 17.56 .0764 16.13 .0894 14.91 .1033 13.87 .1183 12.96 .1343 12.16 .1514 11.46 .1693 10.83 .1884 10.27 .208 9.76 .230 9.31 .252 8.89 .275 8.50 .299 8.15

.0531 27.9 .0631 25.6 .0741 23.6 .0859 21.9 .0987 20.5 .1122 19.17 .1267 18.04 .1420 17.04 .1582 16.14 .1753 15.34 .1933 14.61 .212 13.94 .232 13.34 .253 12.78

.0436 43.9 .0521 40.1 .0614 37.0 .0714 34.3 .0822 31.9 .0937 29.9 .1061 28.1 .1191 26.5 .1330 25.1 .1476 23.8 .1630 22.7 .1791 21.6 .1960 20.7 .214 19.80

.0373 61.6 .0448 56.3 .0530 51.7 .0619 47.9 .0714 44.6 .0816 41.7 .0926 39.1 .1041 36.9 .1164 34.9 .1294 33.1 .1431 31.5 .1574 30.0 .1724 28.7 .1881 27.5

.0304 95.6 .0367 86.9 .0437 79.7 .0512 73.6 .0593 68.4 .0680 63.9 .0774 60.0 .0873 56.4 .0978 53.3 .1089 50.5 .1206 48.0 .1329 45.7 .1459 43.6 .1594 41.7

.0292 103.7 .0353 94.3 .0420 86.4 .0494 80.0 .0572 74.1 .0657 69.1 .0748 64.8 .0844 61.0 .0946 57.6 .1054 54.6 .1168 51.9 .1288 49.4 .1413 47.1 .1545 45.1

.0241 153.3 .0293 138.9 .0351 126.9 .0414 116.9 .0482 108.3 .0555 100.9 .0634 94.4 .0718 88.7 .0807 83.7 .0901 79.2 .1000 75.2 .1105 71.5 .1214 68.2 .1329 65.2

… … .0234 217. .0282 197.3 .0335 181.1 .0393 167.3 .0455 155.5 .0522 145.2 .0593 136.2 .0668 128.3 .0748 121.2 .0833 114.9 .0923 109.2 .1017 104.0 .1115 99.3

… … .0219 245. .0265 223. .0316 205. .0371 188.8 .0430 175.3 .0493 163.6 .0561 153.4 .0634 144.3 .0710 136.3 .0791 129.2 .0877 122.7 .0967 116.9 .1061 111.5

Deflection f (inch) per coil, at Load P (pounds)

a This table is for ASTM A229 oil tempered spring steel with a torsional modulus G of 11,200,000 psi, and an uncorrected torsional stress of 100,000 psi. For other

materials, and other important footnotes, see page 321.

Machinery's Handbook, 31st Edition Spring Design

Nom.

322

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Table 5. (Continued) Compression and Extension Spring Deflectionsa

Spring Outside Dia.

Nom.

Dec. .875

7∕8 29∕3 2

.9063

15∕16

.9375

31∕3 2

.9688 1.000

11∕32

1.031

11∕16

1.063

11∕32

1.094

11∕8

1.125

13∕16

1.188

11∕4

1.250

15∕16

1.313

13∕8

1.375

17∕16

1.438

14

13

3∕3 2

12

11

1∕8

10

9

5∕3 2

8

7

3∕16

6

5

7∕3 2

4

.072

.080

.0915

.0938

.1055

.1205

.125

.135

.1483

.1563

.162

.177

.1875

.192

.207

.2188

.2253

.251 18.26 .271 17.57 .292 16.94 .313 16.35 .336 15.80 .359 15.28 .382 14.80 .407 14.34 .432 13.92 .485 13.14 .541 12.44 .600 11.81 .662 11.25 .727 10.73

.222 25.3 .239 24.3 .258 23.5 .277 22.6 .297 21.9 .317 21.1 .338 20.5 .360 19.83 .383 19.24 .431 18.15 .480 17.19 .533 16.31 .588 15.53 .647 14.81

.1882 39.4 .204 36.9 .219 35.6 .236 34.3 .253 33.1 .271 32.0 .289 31.0 .308 30.0 .328 29.1 .368 27.5 .412 26.0 .457 24.6 .506 23.4 .556 22.3

.1825 41.5 .1974 39.9 .213 38.4 .229 37.0 .246 35.8 .263 34.6 .281 33.5 .299 32.4 .318 31.4 .358 29.6 .400 28.0 .444 26.6 .491 25.3 .540 24.1

.1574 59.9 .1705 57.6 .1841 55.4 .1982 53.4 .213 51.5 .228 49.8 .244 48.2 .260 46.7 .277 45.2 .311 42.6 .349 40.3 .387 38.2 .429 36.3 .472 34.6

.1325 91.1 .1438 87.5 .1554 84.1 .1675 81.0 .1801 78.1 .1931 75.5 .207 73.0 .221 70.6 .235 68.4 .265 64.4 .297 60.8 .331 57.7 .367 54.8 .404 52.2

.1262 102.3 .1370 98.2 .1479 94.4 .1598 90.9 .1718 87.6 .1843 84.6 .1972 81.8 .211 79.2 .224 76.7 .254 72.1 .284 68.2 .317 64.6 .351 61.4 .387 58.4

.1138 130.5 .1236 125.2 .1338 120.4 .1445 115.9 .1555 111.7 .1669 107.8 .1788 104.2 .1910 100.8 .204 97.6 .231 91.7 .258 86.6 .288 82.0 .320 77.9 .353 74.1

.0999 176.3 .1087 169.0 .1178 162.3 .1273 156.1 .1372 150.4 .1474 145.1 .1580 140.1 .1691 135.5 .1804 131.2 .204 123.3 .230 116.2 .256 110.1 .285 104.4 .314 99.4

.0928 209. .1010 199.9 .1096 191.9 .1183 184.5 .1278 177.6 .1374 171.3 .1474 165.4 .1578 159.9 .1685 154.7 .1908 145.4 .215 137.0 .240 129.7 .267 123.0 .295 117.0

.0880 234. .0959 224. .1041 215. .1127 207. .1216 198.8 .1308 191.6 .1404 185.0 .1503 178.8 .1604 173.0 .1812 162.4 .205 153.1 .229 144.7 .255 137.3 .282 130.6

.0772 312. .0843 299. .0917 286. .0994 275. .1074 264. .1157 255. .1243 246. .1332 238. .1424 230. .1620 215. .1824 203. .205 191.6 .227 181.7 .252 172.6

.0707 377. .0772 360. .0842 345. .0913 332. .0986 319. .1065 307. .1145 296. .1229 286. .1315 276. .1496 259. .1690 244. .1894 230. .211 218. .234 207.

.0682 407. .0746 389. .0812 373. .0882 358. .0954 344. .1029 331. .1107 319. .1188 308. .1272 298. .1448 279. .1635 263. .1836 248. .204 235. .227 223.

.0605 521. .0663 498. .0723 477. .0786 457. .0852 439. .0921 423. .0993 407. .1066 393. .1142 379. .1303 355. .1474 334. .1657 315. .1848 298. .205 283.

.0552 626. .0606 598. .0662 572. .0721 548. .0783 526. .0845 506. .0913 487. .0982 470. .1053 454. .1203 424. .1363 399. .1535 376. .1713 356. .1905 337.

.0526 691. .0577 660. .0632 631. .0688 604. .0747 580. .0809 557. .0873 537. .0939 517. .1008 499. .1153 467. .1308 438. .1472 413. .1650 391 .1829 371.

Deflection f (inch) per coil, at Load P (pounds)

a This table is for ASTM A229 oil tempered spring steel with a torsional modulus G of 11,200,000 psi, and an uncorrected torsional stress of 100,000 psi. For other

materials, and other important footnotes, see page 321.

323

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1

Wire Size or Washburn and Moen Gauge, and Decimal Equivalent

15

Machinery's Handbook, 31st Edition Spring Design

Copyright 2020, Industrial Press, Inc.

Table 5. (Continued) Compression and Extension Spring Deflectionsa

Spring Outside Dia.

Nom.

Dec.

11∕2

1.500 1.625 1.750

17∕8

1.875

115∕16

1.938

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2

2.000

21∕16

2.063

21∕8

2.125

23∕16

2.188

21∕4

2.250

25∕16

2.313

23∕8

2.375

27∕16

2.438

21∕2

2.500

Wire Size or Washburn and Moen Gauge, and Decimal Equivalent

11

1∕8

10

9

5∕3 2

8

7

.1205

.125

.135

.1483

.1563

.162

.177

.443 49.8 .527 45.7 .619 42.2 .717 39.2 .769 37.8 .823 36.6 .878 35.4 .936 34.3 .995 33.3 1.056 32.3 1.119 31.4 1.184 30.5 … … … …

.424 55.8 .505 51.1 .593 47.2 .687 43.8 .738 42.3 .789 40.9 .843 39.6 .898 38.3 .955 37.2 1.013 36.1 1.074 35.1 1.136 34.1 1.201 33.2 1.266 32.3

.387 70.8 .461 64.8 .542 59.8 .629 55.5 .676 53.6 .723 51.8 .768 50.1 .823 48.5 .876 47.1 .930 45.7 .986 44.4 1.043 43.1 1.102 42.0 1.162 40.9

.350 94.8 .413 86.7 .485 80.0 .564 74.2 .605 71.6 .649 69.2 .693 66.9 .739 64.8 .786 62.8 .835 60.9 .886 59.2 .938 57.5 .991 56.0 1.046 54.5

.324 111.5 .387 102.0 .456 94.0 .530 87.2 .569 84.2 .610 81.3 .652 78.7 .696 76.1 .740 73.8 .787 71.6 .834 69.5 .884 67.6 .934 65.7 .986 64.0

.310 124.5 .370 113.9 .437 104.9 .508 97.3 .546 93.8 .585 90.6 .626 87.6 .667 84.9 .711 82.2 .755 79.8 .801 77.5 .848 75.3 .897 73.2 .946 71.3

.277 164.6 .332 150.3 .392 138.5 .457 128.2 .492 123.6 .527 119.4 .564 115.4 .602 111.8 .641 108.3 .681 105.7 .723 101.9 .763 99.1 .810 96.3 .855 93.7

3∕16

6

5

7∕3 2

4

3

1∕4

2

9∕3 2

0

5∕16

.1875

.192

.207

.2188

.2253

.2437

.250

.2625

.2813

.3065

.3125

.258 197.1 .309 180.0 .366 165.6 .426 153.4 .458 147.9 .492 142.8 .526 138.1 .562 133.6 .598 129.5 .637 125.5 .676 121.8 .716 118.3 .757 115.1 .800 111.6

.250 213. .300 193.9 .355 178.4 .414 165.1 .446 159.2 .478 153.7 .512 148.5 .546 143.8 .582 139.2 .619 135.0 .657 131.0 .696 127.3 .737 123.7 .778 120.4

.227 269. .273 246. .323 226. .377 209. .405 201. .436 194.3 .467 187.7 .499 181.6 .532 175.8 .566 170.5 .601 165.4 .637 160.7 .674 156.1 .713 151.9

.210 321. .254 292. .301 269. .351 248. .379 239. .407 231. .436 223. .466 216. .497 209. .529 202. .562 196.3 .596 190.7 .631 185.3 .667 180.2

.202 352. .244 321. .290 295. .339 272. .365 262. .392 253. .421 245. .449 236. .479 229. .511 222. .542 215. .576 209. .609 203. .644 197.5

.1815 452. .220 411. .261 377. .306 348. .331 335. .355 324. .381 312. .407 302. .435 292. .463 283. .493 275. .523 267. .554 259. .586 252.

.1754 499. .212 446. .253 409. .296 378. .320 364. .344 351. .369 339. .395 327. .421 317. .449 307. .478 298. .507 289. .537 281. .568 273.

.1612 574. .1986 521. .237 477. .278 440. .300 425. .323 409. .346 395. .371 381. .396 369. .423 357. .449 347. .477 336. .506 327. .536 317.

.1482 717. .1801 650. .215 595. .253 548. .273 528. .295 509. .316 491. .339 474. .362 459. .387 444. .411 430. .437 417. .464 405. .491 394.

.1305 947. .1592 858. .1908 783. .225 721. .243 693. .263 668. .282 644. .303 622. .324 601. .346 582. .368 564. .392 547. .416 531. .441 516.

.1267 1008. .1547 912. .1856 833. .219 767. .237 737. .256 710. .275 685. .295 661. .316 639. .337 618. .359 599. .382 581. .405 564. .430 548.

Deflection f (inch) per coil, at Load P (pounds)

a This table is for ASTM A229 oil tempered spring steel with a torsional modulus G of 11,200,000 psi, and an uncorrected torsional stress of 100,000 psi. For other

materials, and other important footnotes, see page 321.

Machinery's Handbook, 31st Edition Spring Design

15∕8 13∕4

324

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Table 5. (Continued) Compression and Extension Spring Deflectionsa

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325

Extension Springs.—About 10 percent of all springs made by many companies are of this type, and they frequently cause trouble because insufficient consideration is given to stress due to initial tension, stress and deflection of hooks, special manufacturing methods, secondary operations and overstretching at assembly. Fig. 15 shows types of ends used on these springs.

Machine loop and machine hook shown in line

Machine loop and machine hook shown at right angles

Hand loop and hook at right angles

Full loop on side and small eye from center

Double twisted full loop over center

Single full loop centered

Full loop at side

Small off-set hook at side

Machine half-hook over center

Small eye at side

Small eye over center

Reduced loop to center

Hand half-loop over center

Plain squarecut ends

All the Above Ends are Standard Types for Which No Special Tools are Required

Long round-end hook over center

Long square-end hook over center

Extended eye from either center or side

V-hook over center

Straight end annealed to allow forming

Coned end with short swivel eye

Coned end to hold long swivel eye

Coned end with swivel bolt

Coned end with swivel hook

This Group of Special Ends Requires Special Tools Fig. 15. Types of Helical Extension Spring Ends

Initial tension: In the spring industry, the term “Initial tension” is used to define a force or load, measurable in pounds or ounces, which presses the coils of a close wound extension spring against one another. This force must be overcome before the coils of a spring begin to open up. Initial tension is wound into extension springs by bending each coil as it is wound away from its normal plane, thereby producing a slight twist in the wire which causes the coil to spring back tightly against the adjacent coil. Initial tension can be wound into cold-coiled

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extension springs only. Hot-wound springs and springs made from annealed steel are hard­ened and tempered after coiling, and therefore initial tension cannot be produced. It is pos­sible to make a spring having initial tension only when a high tensile strength, obtained by cold drawing or by heat treatment, is possessed by the material as it is being wound into springs. Materials that possess the required characteristics for the manufacture of such springs include hard-drawn wire, music wire, pre-tempered wire, 18-8 stainless steel, phosphor-bronze, and many of the hard-drawn copper-nickel, and nonferrous alloys. Per­m issible torsional stresses resulting from initial tension for different spring indexes are shown in Fig. 16. 44 42

The values in the curves in the chart are for springs made from spring steel. They should be reduced 15 percent for stainless steel. 20 percent for copper-nickel alloys and 50 percent for phosphor bronze.

40 Torsional Stress, Pounds per Square Inch (thousands)

38 36 34 32 30 28

Initial tension in this area is readily obtainable. Use whenever possible.

26 24 22

Maximum initial tension

20 18

Pe

rm

16 14

iss

ibl

12 10 8

ors

ion

al

str ess

Inital tension in this area is difficult to maintain with accurate and uniform results.

6 4

et

3

4

5

6

7

8 9 10 11 12 13 14 15 16 Spring Index

Fig. 16. Permissible Torsional Stress Caused by Initial Tension in Coiled Extension Springs for Different Spring Indexes

Hook failure: The great majority of breakages in extension springs occurs in the hooks. Hooks are subjected to both bending and torsional stresses and have higher stresses than the coils in the spring. Stresses in regular hooks: The calculations for the stresses in hooks are quite compli­cated and lengthy. Also, the radii of the bends are difficult to determine and frequently vary between specifications and actual production samples. However, regular hooks are more highly stressed than the coils in the body and are subjected to a bending stress at section B

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(see Table 6.) The bending stress Sb at section B should be compared with allowable stresses for torsion springs and with the elastic limit of the material in tension (See Fig. 7 through Fig. 10.) Stresses in cross-over hooks: Results of tests on springs having a normal average index show that the cross-over hooks last longer than regular hooks. These results may not occur on springs of small index or if the cross-over bend is made too sharply. In as much as both types of hooks have the same bending stress, it would appear that the fatigue life would be the same. However, the large bend radius of the regular hooks causes some torsional stresses to coincide with the bending stresses, thus explaining the earlier breakages. If sharper bends were made on the regular hooks, the life should then be the same as for cross-over hooks. Table 6. Formula for Bending Stress at Section B Type of Hook

P

Stress in Bending

d B

P

d

Sb =

ID D Regular Hook

B

D

5PD2 IDd 3

Cross-over Hook

Stresses in half hooks: The formulas for regular hooks can also be used for half hooks, because the smaller bend radius allows for the increase in stress. It will therefore be observed that half hooks have the same stress in bending as regular hooks. Frequently overlooked facts by many designers are that one full hook deflects an amount equal to one half a coil and each half hook deflects an amount equal to one tenth of a coil. Allowances for these deflections should be made when designing springs. Thus, an exten­ sion spring, with regular full hooks and having 10 coils, will have a deflection equal to 11 coils, or 10 percent more than the calculated deflection. Extension Spring Design.—The available space in a product or assembly usually deter­ mines the limiting dimensions of a spring, but the wire size, number of coils, and initial ten­sion are often unknown. Example: An extension spring is to be made from spring steel ASTM A229, with regular hooks as shown in Fig. 17. Calculate the wire size, number of coils and initial tension. Note: Allow about 20 to 25 percent of the 9 pound load for initial tension, say 2 pounds, and then design for a 7 pound load (not 9 pounds) at 5∕8 inch deflection. Also use lower stresses than for a compression spring to allow for overstretching during assembly and to obtain a safe stress on the hooks. Proceed as for compression springs, but locate a load in the tables somewhat higher than the 9 pound load. Method 1, using table: From Table 5 locate 3∕4 inch outside diameter in the left column and move to the right to locate a load P of 13.94 pounds. A deflection f of 0.212 inch appears above this figure. Moving vertically from this position to the top of the column a suitable wire diameter of 0.0625 inch is found. The remaining design calculations are completed as follows: Step 1: The stress with a load of 7 pounds is obtained as follows: The 7 pound load is 50.2 percent of the 13.94 pound load. Therefore, the stress S at 7 pounds = 0.502 percent 3 100,000 = 50,200 pounds per square inch. Step 2: The 50.2 percent figure is also used to determine the deflection per coil f: 0.502 percent 3 0.212 = 0.1062 inch.

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75 to 85 Percent of Inside Diameter

2 Pounds (Initial Tension) 9 Pounds (P)

34

”

17 16” (CL)

58

” (F)

2 1 16 ” (to Inside Diameter of Hooks) Fig. 17. Extension Spring Design Example

Step 3: The number of active coils (say 6)

F 0.625 AC = f = 0.1062 = 5.86 This result should be reduced by 1 to allow for deflection of 2 hooks (see notes 1 and 2 that follow these calculations.) Therefore, a quick answer is: 5 coils of 0.0625 inch diameter wire. However, the design procedure should be completed by carrying out the following steps: Step 4: The body length = (TC + 1) 3 d = (5 + 1) 3 0.0625 = 3∕8 inch. Step 5: The length from the body to inside hook

FL − Body 1.4375 − 0.375 = = 0.531 inch 2 2 0.531 0.531 Percentage of ID = ID = 0.625 = 85 percent =

This length is satisfactory, see Note 3 following this procedure. Step 6:

The spring index =

OD 0.75 d − 1 = 0.0625 − 1 = 11

Step 7: The initial tension stress is

Sit =

50, 200 # 2 S # IT = 14, 340 pounds per square inch 7 P =

This stress is satisfactory, as checked against curve in Fig. 16. Step 8: The curvature correction factor K = 1.12 (Fig. 13). Step 9: The total stress = (50,200 + 14,340) 3 1.12 = 72.285 pounds per square inch This result is less than 106,250 pounds per square inch permitted by the middle curve for 0.0625 inch wire in Fig. 3 and therefore is a safe working stress that permits some addi­ tional deflection that is usually necessary for assembly purposes.

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Step 10: The large majority of hook breakage is due to high stress in bending and should be checked as follows: From Table 6, stress on hook in bending is:

Sb =

5PD2 5 × 9 × 0.6875 2 = = 139, 200 pounds per square inch 0.625 × 0.0625 3 IDd 3

This result is less than the top curve value, Fig. 8, for 0.0625 inch diameter wire, and is therefore safe. Also see Note 5 that follows. Notes: The following points should be noted when designing extension springs: 1) All coils are active and thus AC = TC. 2) Each full hook deflection is approximately equal to 1∕2 coil. Therefore for 2 hooks, reduce the total coils by 1. (Each half hook deflection is nearly equal to 1∕10 of a coil.) 3) The distance from the body to the inside of a regular full hook equals 75 to 85 percent (90 percent maximum) of the ID. For a cross-over center hook, this distance equals the ID. 4) Some initial tension should usually be used to hold the spring together. Try not to exceed the maximum curve shown on Fig. 16. Without initial tension, a long spring with many coils will have a different length in the horizontal position than it will when hung ver­tically. 5) The hooks are stressed in bending, therefore their stress should be less than the maxi­ mum bending stress as used for torsion springs—use top fatigue strength curves Fig. 7 through Fig. 10.

Method 2, using formulas: The sequence of steps for designing extension springs by for­ mulas is similar to that for compression springs. The formulas for this method are given in Table 3. Tolerances for Compression and Extension Springs.—Tolerances for coil diameter, free length, squareness, load, and the angle between loop planes for compression and extension springs are given in Table 7 through Table 12. To meet the requirements of load, rate, free length, and solid height, it is necessary to vary the number of coils for compression springs by ± 5 percent. For extension springs, the tolerances on the numbers of coils are: for 3 to 5 coils, ± 20 percent; for 6 to 8 coils, ± 30 percent; for 9 to 12 coils, ± 40 percent. For each additional coil, a further 11∕2 percent tolerance is added to the extension spring val­ues. Closer tolerances on the number of coils for either type of spring lead to the need for trimming after coiling, and manufacturing time and cost are increased. Fig. 18 shows devi­ations allowed on the ends of extension springs, and variations in end alignments. Table 7. Compression and Extension Spring Coil Diameter Tolerances

Wire Diameter, Inch

4

0.015 0.023 0.035 0.051 0.076 0.114 0.171 0.250 0.375 0.500

0.002 0.002 0.002 0.003 0.004 0.006 0.008 0.011 0.016 0.021

6 0.002 0.003 0.004 0.005 0.007 0.009 0.012 0.015 0.020 0.030

8 0.003 0.004 0.006 0.007 0.010 0.013 0.017 0.021 0.026 0.040

Spring Index 10

Tolerance, ± inch 0.004 0.006 0.007 0.010 0.013 0.018 0.023 0.028 0.037 0.062

12

14

16

0.005 0.007 0.009 0.012 0.016 0.021 0.028 0.035 0.046 0.080

0.006 0.008 0.011 0.015 0.019 0.025 0.033 0.042 0.054 0.100

0.007 0.010 0.013 0.017 0.022 0.029 0.038 0.049 0.064 0.125

Courtesy of the Spring Manufacturers Institute

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.05 inch × Outside diameter

± .05 inch × Outside diameter

5 degrees

.05 inch × Outside diameter

d 2

or

1 64

inch.

Whichever is greater

45 degrees

Maximum Opening for Closed Loop

Maximum Overlap for Closed Loop

Fig. 18. Maximum Deviations Allowed on Ends and Variation in Alignment of Ends (Loops) for Extension Springs

Table 8. Compression Spring Normal Free-Length Tolerances, Squared and Ground Ends Number of Active Coils per Inch 0.5 1 2 4 8 12 16 20

4 0.010 0.011 0.013 0.016 0.019 0.021 0.022 0.023

6

8

0.011 0.013 0.015 0.018 0.022 0.024 0.026 0.027

Spring Index 10

12

Tolerance, ± Inch per Inch of Free Lengtha 0.012 0.015 0.017 0.021 0.024 0.027 0.029 0.031

0.013 0.016 0.019 0.023 0.026 0.030 0.032 0.034

0.015 0.017 0.020 0.024 0.028 0.032 0.034 0.036

14

16

0.016 0.018 0.022 0.026 0.030 0.034 0.036 0.038

0.016 0.019 0.023 0.027 0.032 0.036 0.038 0.040

a For springs less than 0.5 inch long, use the tolerances for 0.5 inch long springs. For springs with unground closed ends, multiply the tolerances by 1.7. Courtesy of the Spring Manufacturers Institute

Table 9. Extension Spring Normal Free-Length and End Tolerances

Free-Length Tolerances

Spring Free Length (inch)

Up to 0.5 Over 0.5 to 1.0 Over 1.0 to 2.0 Over 2.0 to 4.0

End Tolerances

Tolerance (inch)

Total Number of Coils

Angle Between Loop Planes

±0.020 ±0.030 ±0.040 ±0.060

3 to 6 7 to 9 10 to 12

±25° ±35° ±45°

Free-Length Tolerances

Spring Free Length (inch)

Over 4.0 to 8.0 Over 8.0 to 16.0 Over 16.0 to 24.0

Tolerance (inch) ±0.093 ±0.156 ±0.218

End Tolerances

Total Number of Coils 13 to 16 Over 16

Angle Between Loop Planes ±60° Random

Courtesy of the Spring Manufacturers Institute

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Table 10. Compression Spring Squareness Tolerances Slenderness Ratio FL/Da 0.5 1.0 1.5 2.0 3.0 4.0 6.0 8.0 10.0 12.0

4

6

3.0 2.5 2.5 2.5 2.0 2.0 2.0 2.0 2.0 2.0

3.0 3.0 2.5 2.5 2.5 2.0 2.0 2.0 2.0 2.0

Spring Index 8 10 12 Squareness Tolerances (± degrees) 3.5 3.5 3.5 3.0 3.0 3.0 2.5 3.0 3.0 2.5 2.5 3.0 2.5 2.5 2.5 2.5 2.5 2.5 2.0 2.5 2.5 2.0 2.0 2.5 2.0 2.0 2.0 2.0 2.0 2.0

14

16

3.5 3.5 3.0 3.0 2.5 2.5 2.5 2.5 2.5 2.0

4.0 3.5 3.0 3.0 3.0 2.5 2.5 2.5 2.5 2.5

a Slenderness Ratio =

FLD Springs with closed and ground ends, in the free position. Squareness tolerances closer than those shown require special process techniques which increase cost. Springs made from fine wire sizes, and with high spring indices, irregular shapes or long free lengths, require special attention in deter­m ining appropriate tolerance and feasibility of grinding ends.

Table 11. Compression Spring Normal Load Tolerances

Length Tolerance, ± inch

0.05

0.10

0.15

0.20

0.25

0.005 0.010 0.020 0.030 0.040 0.050 0.060 0.070 0.080 0.090 0.100 0.200 0.300 0.400 0.500

12 … … … … … … … … … … … … … …

7 12 22 … … … … … … … … … … … …

6 8.5 15.5 22 … … … … … … … … … … …

5 7 12 17 22 … … … … … … … … … …

… 6.5 10 14 18 22 25 … … … … … … … …

a From free length to loaded position.

0.30

Deflection (inch)a

0.40

0.50

0.75

1.00

Tolerance, ± Percent of Load … … … … … 5.5 5 … … … 8.5 7 6 5 … 12 9.5 8 6 5 15.5 12 10 7.5 6 19 14.5 12 9 7 22 17 14 10 8 25 19.5 16 11 9 … 22 18 12.5 10 … 25 20 14 11 … … 22 15.5 12 … … … … 22 … … … … … … … … … … … … … … …

1.50

2.00

3.00

4.00

6.00

… … … … 5 5.5 6 6.5 7.5 8 8.5 15.5 22 … …

… … … … … … 5 5.5 6 6 7 12 17 21 25

… … … … … … … … 5 5 5.5 8.5 12 15 18.5

… … … … … … … … … … … 7 9.5 12 14.5

… … … … … … … … … … … 5.5 7 8.5 10.5

Torsion Spring Design.—Fig. 19 shows the types of ends most commonly used on tor­ sion springs. To produce them requires only limited tooling. The straight torsion end is the least expensive and should be used whenever possible. After determining the spring load or torque required and selecting the end formations, the designer usually estimates suitable space or size limitations. However, the space should be considered approximate until the wire size and number of coils have been determined. The wire size is dependent principally upon the torque. Design data can be developed with the aid of the tabular data, which is a simple method, or by calculation alone, as shown in the following sections. Many other factors affecting the design and operation of torsion springs are also covered in the section, Torsion Spring Design Recommendations on page 337. Design formulas are shown in Table 13. Curvature correction: In addition to the stress obtained from the formulas for load or deflection, there is a direct shearing stress on the inside of the section due to curva­ture. Therefore, the stress obtained by the usual formulas should be multiplied by the factor K obtained from the curve in Fig. 20. The corrected stress thus obtained is used only for com­parison with the allowable working stress (fatigue strength) curves to determine if it is a safe value, and should not be used in the formulas for deflection.

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Table 12. Extension Spring Normal Load Tolerances Spring Index

4

6

8

10

12

14

16

FL F

0.015

8 6 4.5 2.5 1.5 0.5 12 8 6 4.5 2.5 1.5 0.5 12 8 6 4.5 2.5 1.5 0.5 12 8 6 4.5 2.5 1.5 0.5 12 8 6 4.5 2.5 1.5 0.5 12 8 6 4.5 2.5 1.5 0.5 12 8 6 4.5 2.5 1.5 0.5

18.5 16.8 15.0 13.1 10.2 6.2 17.0 16.2 15.2 13.7 11.9 9.9 6.3 15.8 15.0 14.2 12.8 11.2 9.5 6.3 14.8 14.2 13.4 12.3 10.8 9.2 6.4 14.0 13.2 12.6 11.7 10.5 8.9 6.5 13.1 12.4 11.8 11.1 10.1 8.6 6.6 12.3 11.7 11.0 10.5 9.7 8.3 6.7

12

20.0

0.022 18.5

17.5 16.1 14.7 12.4 9.9 5.4 15.5 14.7 14.0 12.4 10.8 9.0 5.5 14.3 13.7 13.0 11.7 10.2 8.6 5.6 13.3 12.8 12.1 10.8 9.6 8.3 5.7 12.3 11.8 11.2 10.2 9.2 8.0 5.8 11.3 10.9 10.4 9.7 8.8 7.7 5.9 10.3 10.0 9.6 9.1 8.4 7.4 5.9

0.032 17.6

16.7 15.5 14.1 12.1 9.3 4.8 14.6 13.9 12.9 11.5 10.2 8.3 4.9 13.1 12.5 11.7 10.7 9.5 7.8 5.0 12.0 11.6 10.8 10.0 9.0 7.5 5.1 11.1 10.7 10.2 9.4 8.5 7.2 5.3 10.2 9.8 9.3 8.7 8.1 7.0 5.4 9.2 8.9 8.5 8.1 7.6 6.6 5.5

Wire Diameter (inch)

0.044

0.062

0.092

0.125

0.187

0.250

0.375

0.437

15.8 14.7 13.5 11.8 8.9 4.6 14.1 13.4 12.3 11.0 9.8 7.7 4.7 13.0 12.1 11.2 10.1 8.8 7.1 4.8 11.9 11.2 10.5 9.5 8.4 6.9 4.9 10.8 10.2 9.7 9.0 8.0 6.8 5.1 9.7 9.2 8.8 8.2 7.6 6.7 5.2 8.6 8.3 8.0 7.5 7.0 6.2 5.3

15.0 13.8 12.6 10.6 8.0 4.3 13.5 12.6 11.6 10.5 9.4 7.3 4.5 12.1 11.4 10.6 9.7 8.3 6.9 4.5 11.1 10.5 9.8 9.0 8.0 6.7 4.7 10.1 9.6 9.0 8.4 7.8 6.5 4.9 9.1 8.7 8.3 7.8 7.1 6.3 5.0 8.1 7.8 7.5 7.2 6.7 6.0 5.1

14.5 13.2 12.0 10.0 7.5 4.1 13.1 12.2 10.9 10.0 9.0 7.0 4.3 12.0 11.0 10.0 9.0 7.9 6.7 4.4 10.9 10.2 9.3 8.5 7.7 6.5 4.5 9.8 9.3 8.5 8.0 7.4 6.3 4.7 8.8 8.3 7.7 7.2 6.7 6.0 4.8 7.7 7.4 7.1 6.8 6.3 5.8 5.0

14.0 12.7 11.5 9.1 7.0 4.0 12.7 11.7 10.7 9.6 8.5 6.7 4.1 11.5 10.6 9.7 8.7 7.7 6.5 4.2 10.5 9.7 8.9 8.1 7.3 6.3 4.3 9.5 8.9 8.2 7.6 7.0 6.1 4.5 8.4 8.0 7.5 7.0 6.5 5.8 4.6 7.4 7.2 6.9 6.5 6.1 5.6 4.8

13.2 11.8 10.3 8.5 6.5 3.8 12.0 11.0 10.0 9.0 7.9 6.4 4.0 10.8 10.1 9.3 8.3 7.4 6.2 4.1 9.9 9.2 8.6 7.8 7.0 6.0 4.2 9.0 8.4 7.9 7.2 6.6 5.7 4.3 8.1 7.6 7.2 6.7 6.2 5.5 4.4 7.2 6.8 6.5 6.2 5.7 5.3 4.6

12.5 11.2 9.7 8.0 6.1 3.6 11.5 10.5 9.4 8.3 7.2 6.0 3.7 10.2 9.4 8.6 7.8 6.9 5.8 3.9 9.3 8.6 8.0 7.3 6.5 5.6 4.0 8.5 7.9 7.4 6.8 6.1 5.4 4.2 7.6 7.2 6.8 6.3 5.7 5.2 4.3 6.8 6.5 6.2 5.8 5.4 5.1 4.5

13.8

13.0

12.6

Tolerance, ± Percent of Load 16.9 16.2 15.5 15.0 14.3

11.5 9.9 8.4 6.8 5.3 3.3 11.2 10.0 8.8 7.6 6.2 4.9 3.5 10.0 9.0 8.1 7.2 6.1 4.9 3.6 9.2 8.3 7.6 6.8 5.9 5.0 3.8 8.2 7.5 6.9 6.3 5.6 4.8 4.0 7.2 6.8 6.3 5.8 5.2 4.7 4.2 6.3 6.0 5.7 5.3 4.9 4.6 4.3

11.0 9.4 7.9 6.2 4.8 3.2 10.7 9.5 8.3 7.1 6.0 4.7 3.4 9.5 8.6 7.6 6.6 5.6 4.5 3.5 8.8 8.0 7.2 6.4 5.5 4.6 3.7 7.9 7.2 6.4 5.8 5.2 4.5 3.3 7.0 6.4 5.9 5.4 5.0 4.5 4.0 6.1 5.7 5.4 5.1 4.7 4.4 4.1

FL ⁄ F = the ratio of the spring free length FL to the deflection F.

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Hook

Hinged

Straight Offset

Straight Torsion

Fig. 19. The Most Commonly Used Types of Ends for Torsion Springs

Correction Factor, K

1.3

1.2

Round Wire Square Wire and Rectangular Wire K × S = Total Stress

1.1

1.0

3

4

5

6

7

8 9 10 Spring Index

11

12

13

14

15

16

Fig. 20. Torsion Spring Stress Correction for Curvature

Torque: Torque is a force applied to a moment arm and tends to produce rotation. Tor­ sion springs exert torque in a circular arc and the arms are rotated about the central axis. It should be noted that the stress produced is in bending, not in torsion. In the spring industry it is customary to specify torque in conjunction with the deflection or with the arms of a spring at a definite position. Formulas for torque are expressed in pound-inches. If ounceinches are specified, it is necessary to divide this value by 16 in order to use the formulas. When a load is specified at a distance from a centerline, the torque is, of course, equal to the load multiplied by the distance. The load can be in pounds or ounces with the distances in inches or the load can be in grams or kilograms with the distance in centimeters or milli­meters, but to use the design formulas, all values must be converted to pounds and inches. Design formulas for torque are based on the tangent to the arc of rotation and presume that a rod is used to support the spring. The stress in bending caused by the moment P 3 R is identical in magnitude to the torque T, provided a rod is used. Theoretically, it makes no difference how or where the load is applied to the arms of tor­sion springs. Thus, in Fig. 21, the loads shown multiplied by their respective distances pro­duce the same torque; i.e., 20 3 0.5 = 10 pound-inches; 10 3 1 = 10 pound-inches; and 5 3 2 = 10 pound-inches. To further simplify the understanding of torsion spring torque, observe in both Fig. 22 and Fig. 23 that although the turning force is in a circular arc the torque is not equal to P times the radius. The torque in both designs equals P 3 R because the spring rests against the support rod at point a.

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Machinery's Handbook, 31st Edition Spring Design

334

Table 13. Formulas for Torsion Springs Springs made from round wire

Feature d= Wire diameter, Inches

Sb = Stress, bending pounds per square inch

N= Active Coils

F° = Deflection

T= Torque Inch lbs. (Also = P 3 R)

I D1 = Inside Diameter After Deflection, Inches

Formula a, b

Springs made from square wire

3

10.18T Sb

3

6T Sb

4

4000TN D EF °

4

2375TN D EF °

10.18T d3

6T d3

EdF ° 392ND

EdF ° 392ND

EdF ° 392Sb D

EdF ° 392Sb D

Ed 4F ° 4000TD

Ed 4F ° 2375TD

392Sb N D Ed

392Sb N D Ed

4000TN D Ed 4

2375TN D Ed 4

0.0982Sb d 3

0.1666Sb d 3

Ed 4F ° 4000ND

Ed 4F ° 2375ND

N ^ID free h F° N + 360

N ^ID free h F° N + 360

a Where two formulas are given for one feature, the designer should use the one found to be appropriate for the given design. The end result from either of any two formulas is the same. b The symbol notation is given on page 304.

Design Procedure: Torsion spring designs require more effort than other kinds because consideration has to be given to more details, such as the proper size of a supporting rod, reduction of the inside diameter, increase in length, deflection of arms, allowance for fric­ tion, and method of testing.

Example: What music wire diameter and how many coils are required for the torsion spring shown in Fig. 24, which is to withstand at least 1000 cycles? Determine the cor­ rected stress and the reduced inside diameter after deflection.

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Machinery's Handbook, 31st Edition Spring Design 5 lbs.

2” 20 lbs. 12

20 lbs.

1”

”

335

Fig. 21. Right-Hand Torsion Spring

P

R

s

diu

Ra a

P

Left Hand Torsion Springs Torque T = P  R, not P  radius.

Rod

Fig. 22. Left-Hand Torsion Spring The Torque is T = P 3 R, Not P 3 Radius, because the Spring is Resting Against the Support Rod at Point a.

P

R

Radiu s

P

a Rod

Fig. 23. Left-Hand Torsion Spring As with the Spring in Fig. 22, the Torque is T = P 3 R, Not P 3 Radius, Because the Support Point Is at a.

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Machinery's Handbook, 31st Edition Spring Design

336

Free Position

90°

Deflection F° 16 lbs. = P 58

d

”=R Loaded Position

12

21 32

”

To Fit Over

7 16

” rod

”

Left Hand

Fig. 24. Torsion Spring Design Example. The Spring is to be Assembled on a 7∕16 -Inch Support Rod.

Method 1, using table: From Table 14, page 339, locate the 1∕2 inch inside diameter for the spring in the left-hand column. Move to the right and then vertically to locate a torque value nearest to the required 10 pound-inches, which is 10.07 pound-inches. At the top of the same column, the music wire diameter is found, which is Number 31 gauge (0.085 inch). At the bottom of the same column the deflection for one coil is found, which is 15.81 degrees. As a 90-degree deflection is required, the number of coils needed is 90 ∕15.81 = 5.69 (say 53∕4 coils).

D 0.500 + 0.085 = 6.88 and thus the curvature correction The spring index d = 0.085 factor K from Fig. 20 = 1.13. Therefore the corrected stress equals 167,000 3 1.13 = 188,700 pounds per square inch which is below the Light Service curve (Fig. 7) and therefore should provide a fatigue life of over 1,000 cycles. The reduced inside diameter due to deflection is found from the formula in Table 13: ID1 =

N ^ID free h 5.75 # 0.500 = F 90 = 0. 479 in. N + 360 5.75 + 360

This reduced diameter easily clears a suggested 7∕16 inch diameter supporting rod: 0.479 − 0.4375 = 0.041 inch clearance, and it also allows for the standard tolerance. The overall length of the spring equals the total number of coils plus one, times the wire diameter. Thus, 63∕4 3 0.085 = 0.574 inch. If a small space of about 1∕64 in. is allowed between the coils to eliminate coil friction, an overall length of 21∕32 inch results.

Although this completes the design calculations, other tolerances should be applied in accordance with the Torsion Spring Tolerance Table 16 through Table 17 shown at the end of this section.

Longer fatigue life: If a longer fatigue life is desired, use a slightly larger wire diameter. Usually the next larger gage size is satisfactory. The larger wire will reduce the stress and still exert the same torque, but will require more coils and a longer overall length.

Percentage method for calculating longer life: The spring design can be easily adjusted for longer life as follows:

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Machinery's Handbook, 31st Edition Spring Design

337

1) Select the next larger gage size, which is Number 32 (0.090 inch) from Table 14. The torque is 11.88 pound-inches, the design stress is 166,000 pounds per square inch, and the deflection is 14.9 degrees per coil. As a percentage the torque is 10 ∕ 11.88 3 100 = 84 percent. 2) The new stress is 0.84 3 166,000 = 139,440 pounds per square inch. This value is under the bottom or Severe Service curve, Fig. 7, and thus assures longer life. 3) The new deflection per coil is 0.84 3 14.97 = 12.57 degrees. Therefore, the total num­ ber of coils required = 90 ∕ 12.57 = 7.16 (say 71∕8). The new overall length = 81∕8 3 0.090 = 0.73 inch (say 3∕4 inch). A slight increase in the overall length and new arm location are thus necessary.

Method 2, using formulas: When using this method, it is often necessary to solve the for­ mulas several times because assumptions must be made initially either for the stress or for a wire size. The procedure for design using formulas is as follows (the design example is the same as in Method 1, and the spring is shown in Fig. 24): Step 1: Note from Table 13, page 334 that the wire diameter formula is: 10.18T d= 3 Sb

Step 2: Referring to Fig. 7, select a trial stress, say 150,000 pounds per square inch. Step 3: Apply the trial stress, and the 10 pound-inches torque value in the wire diameter formula: 10.18T 10.18 # 10 = 3 150, 000 = 3 0.000679 = 0.0879 inch d=3 Sb The nearest gauge sizes are 0.085 and 0.090 inch diameter. Note: Table 21, page 347, can be used to avoid solving the cube root. Step 4: Select 0.085 inch wire diameter and solve the equation for the actual stress: 10.18T 10.18 # 10 = = 165, 764 pounds per square inch Sb = d3 0.085 3 Step 5: Calculate the number of coils from the equation, Table 13: 28, 500, 000 # 0.085 # 90 3 EdF ° N= = 392 # 165, 764 # 0.585 = 5.73 ( say 5 4 ) 392Sb D Step 6: Calculate the total stress. The spring index is 6.88, and the correction factor K is 1.13 therefore total stress = 165,764 3 1.13 = 187,313 pounds per square inch. Note: The corrected stress should not be used in any of the formulas as it does not determine the torque or the deflection. Torsion Spring Design Recommendations.—The following recommendations should be taken into account when designing torsion springs: Hand: The hand or direction of coiling should be specified and the spring designed so deflection causes the spring to wind up and to have more coils. This increase in coils and overall length should be allowed for during design. Deflecting the spring in an unwinding direction produces higher stresses and may cause early failure. When a spring is sighted down the longitudinal axis, it is “right hand” when the direction of the wire into the spring takes a clockwise direction or if the angle of the coils follows an angle similar to the threads of a standard bolt or screw, otherwise it is “left hand.” A spring must be coiled right-handed to engage the threads of a standard machine screw. Rods: Torsion springs should be supported by a rod running through the center whenever possible. If unsupported, or if held by clamps or lugs, the spring will buckle and the torque will be reduced or unusual stresses may occur.

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338

Machinery's Handbook, 31st Edition Spring Design

Diameter Reduction: The inside diameter reduces during deflection. This reduction should be computed and proper clearance provided over the supporting rod. Also, allow­ ances should be considered for normal spring diameter tolerances. Winding: The coils of a spring may be closely or loosely wound, but they seldom should be wound with the coils pressed tightly together. Tightly wound springs with initial tension on the coils do not deflect uniformly and are difficult to test accurately. A small space between the coils of about 20 to 25 percent of the wire thickness is desirable. Square and rectangular wire sections should be avoided whenever possible as they are difficult to wind, expensive, and are not always readily available. Arm Length: All the wire in a torsion spring is active between the points where the loads are applied. Deflection of long extended arms can be calculated by allowing one third of the arm length, from the point of load contact to the body of the spring, to be converted into coils. However, if the length of arm is equal to or less than one-half the length of one coil, it can be safely neglected in most applications. Total Coils: Torsion springs having less than three coils frequently buckle and are diffi­ cult to test accurately. When thirty or more coils are used, light loads will not deflect all the coils simultaneously due to friction with the supporting rod. To facilitate manufacturing it is usually preferable to specify the total number of coils to the nearest fraction in eighths or quarters, such as 5 1∕8 , 5 1∕4 , 5 1∕2 , etc. Double Torsion: This design consists of one left-hand-wound series of coils and one series of right-hand-wound coils connected at the center. These springs are difficult to manufacture and are expensive, so it often is better to use two separate springs. For torque and stress calculations, each series is calculated separately as individual springs; then the torque values are added together, but the deflections are not added. Bends: Arms should be kept as straight as possible. Bends are difficult to produce and often are made by secondary operations, so they are therefore expensive. Sharp bends raise stresses that cause early failure. Bend radii should be as large as practicable. Hooks tend to open during deflection; their stresses can be calculated by the same procedure as that for tension springs. Spring Index: The spring index must be used with caution. In design formulas it is D/d. For shop measurement it is OD/d. For arbor design it is ID/d. Conversions are easily per­ formed by either adding or subtracting 1 from D/d. Proportions: A spring index between 4 and 14 provides the best proportions. Larger ratios may require more than average tolerances. Ratios of 3 or less often cannot be coiled on automatic spring coiling machines because of arbor breakage. Also, springs with smaller or larger spring indexes often do not give the same results as are obtained using the design formulas. Table of Torsion Spring Characteristics.—Table 14 shows design characteristics for the most commonly used torsion springs made from wire of standard gauge sizes. The deflection for one coil at a specified torque and stress is shown in the body of the table. The figures are based on music wire (ASTM A228) and oil-tempered MB grade (ASTM A229), and can be used for several other materials which have similar values for the mod­ulus of elasticity E. However, the design stress may be too high or too low, and the design stress, torque, and deflection per coil should each be multiplied by the appropriate correc­tion factor in Table 15 when using any of the materials given in that table.

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1 .010

2 .011

3 .012

4 .013

5 .014

6 .016

7 .018

8 .020

9 .022

10 .024

11 .026

12 .029

13 .031

14 .033

15 .035

Design Stress, kpsi

232

229

226

224

221

217

214

210

207

205

202

199

197

196

194

192

Torque, pound-inch

.0228

.0299

.0383

.0483

.0596

.0873

.1226

.1650

.2164

.2783

.3486

.4766

.5763

.6917

.8168

.9550 …

Inside Diameter, inch

Deflection, degrees per coil

16 .037

0.0625

22.35

20.33

18.64

17.29

16.05

14.15

18.72

11.51

10.56

9.818

9.137

8.343

7.896

…

…

5 ∕ 64

0.078125

27.17

24.66

22.55

20.86

19.32

16.96

15.19

13.69

12.52

11.59

10.75

9.768

9.215

…

…

…

3 ∕ 32

0.09375

31.98

28.98

26.47

24.44

22.60

19.78

17.65

15.87

14.47

13.36

12.36

11.19

10.53

10.18

9.646

9.171

7 ∕ 64

0.109375

36.80

33.30

30.38

28.02

25.88

22.60

20.12

18.05

16.43

15.14

13.98

12.62

11.85

11.43

10.82

10.27

1 ∕ 8

0.125

41.62

37.62

34.29

31.60

29.16

25.41

22.59

20.23

18.38

16.91

15.59

14.04

13.17

12.68

11.99

11.36

9 ∕ 64

0.140625

46.44

41.94

38.20

35.17

32.43

28.23

25.06

22.41

20.33

18.69

17.20

15.47

14.49

13.94

13.16

12.46

5 ∕ 32

0.15625

51.25

46.27

42.11

38.75

35.71

31.04

27.53

24.59

22.29

20.46

18.82

16.89

15.81

15.19

14.33

13.56

3 ∕ 16

0.1875

60.89

54.91

49.93

45.91

42.27

36.67

32.47

28.95

26.19

24.01

22.04

19.74

18.45

17.70

16.67

15.75

7 ∕ 32

0.21875

70.52

63.56

57.75

53.06

48.82

42.31

37.40

33.31

30.10

27.55

25.27

22.59

21.09

20.21

19.01

17.94

1 ∕ 4

0.250

80.15

72.20

65.57

60.22

55.38

47.94

42.34

37.67

34.01

31.10

28.49

25.44

23.73

22.72

21.35

20.13

AMW Wire Gauge Decimal Equivalenta

17 .039

18 .041

19 .043

20 .045

21 .047

22 .049

23 .051

24 .055

25 .059

26 .063

27 .067

28 .071

29 .075

30 .080

31 .085

Design Stress, kpsi

190

188

187

185

184

183

182

180

178

176

174

173

171

169

167

Torque, pound-inch

1.107

1.272

1.460

1.655

1.876

2.114

2.371

2.941

3.590

4.322

5.139

6.080

7.084

8.497

10.07

10.80

10.29

9.876

9.447

9.102

8.784

Inside Diameter, inch 1 ∕ 8 0.125

Deflection, degrees per coil …

…

…

…

…

…

…

…

…

9 ∕ 64

0.140625

11.83

11.26

10.79

10.32

9.929

9.572

9.244

8.654

8.141

…

…

…

…

…

…

5 ∕ 32

0.15625

12.86

12.23

11.71

11.18

10.76

10.36

9.997

9.345

8.778

8.279

7.975

…

…

…

…

3 ∕ 16

0.1875

14.92

14.16

13.55

12.92

12.41

11.94

11.50

10.73

10.05

9.459

9.091

8.663

8.232

7.772

7.364

7 ∕ 32

0.21875

16.97

16.10

15.39

14.66

14.06

13.52

13.01

12.11

11.33

10.64

10.21

9.711

9.212

8.680

8.208

1 ∕ 4

0.250

19.03

18.04

17.22

16.39

15.72

15.09

14.52

13.49

12.60

11.82

11.32

10.76

10.19

9.588

9.053

a For sizes up to 13 gauge, the table values are for music wire with a modulus E of 29,000,000 psi; and for sizes from 27 to 31 gauge, the values are for oil-tempered

MB with a modulus of 28,500,000 psi.

339

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1 ∕ 16

Machinery's Handbook, 31st Edition Spring Design

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Table 14. Torsion Spring Deflections

AMW Wire Gauge Decimal Equivalenta

8 .020

9 .022

10 .024

Table 14. (Continued) Torsion Spring Deflections 11 .026

12 .029

13 .031

14 .033

15 .035

16 .037

17 .039

18 .041

19 .043

20 .045

21 .047

22 .049

23 .051

Design Stress, kpsi

210

207

205

202

199

197

196

194

192

190

188

187

185

184

183

182

Torque, pound-inch

.1650

.2164

.2783

.3486

.4766

.5763

.6917

.8168

.9550

1.107

1.272

1.460

1.655

1.876

2.114

2.371

Inside Diameter, inch

Deflection, degrees per coil

0.28125

42.03

37.92

34.65

31.72

28.29

26.37

25.23

23.69

22.32

21.09

19.97

19.06

18.13

17.37

16.67

16.03

5 ∕ 16

0.3125

46.39

41.82

38.19

34.95

31.14

29.01

27.74

26.04

24.51

23.15

21.91

20.90

19.87

19.02

18.25

17.53

11 ∕ 32

0.34375

50.75

45.73

41.74

38.17

33.99

31.65

30.25

28.38

26.71

25.21

23.85

22.73

21.60

20.68

19.83

19.04

0.375

55.11

49.64

45.29

41.40

36.84

34.28

32.76

30.72

28.90

27.26

25.78

24.57

23.34

22.33

21.40

20.55

13 ∕ 32

0.40625

59.47

53.54

48.85

44.63

39.69

36.92

35.26

33.06

31.09

29.32

27.72

26.41

25.08

23.99

22.98

22.06

7 ∕ 16

0.4375

63.83

57.45

52.38

47.85

42.54

39.56

37.77

35.40

33.28

31.38

29.66

28.25

26.81

25.64

24.56

23.56

15 ∕ 32

0.46875

68.19

61.36

55.93

51.00

45.39

42.20

40.28

37.74

35.47

33.44

31.59

30.08

28.55

27.29

26.14

25.07

0.500

72.55

65.27

59.48

54.30

48.24

44.84

42.79

40.08

37.67

35.49

33.53

31.92

30.29

28.95

27.71

26.58

3 ∕ 8

1 ∕ 2

AMW Wire Gauge Decimal Equivalenta

24 .055

25 .059

26 .063

27 .067

28 .071

29 .075

30 .080

31 .085

32 .090

33 .095

34 .100

35 .106

36 .112

37 .118

1∕8 125

Design Stress, kpsi

180

178

176

174

173

171

169

167

166

164

163

161

160

158

156

Torque, pound-inch

2.941

3.590

4.322

5.139

6.080

7.084

8.497

10.07

11.88

13.81

16.00

18.83

22.07

25.49

29.92

Inside Diameter, inch

Deflection, degrees per coil

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9 ∕ 32

0.28125

14.88

13.88

13.00

12.44

11.81

11.17

10.50

9.897

9.418

8.934

8.547

8.090

7.727

7.353

6.973

5 ∕ 16

0.3125

16.26

15.15

14.18

13.56

12.85

12.15

11.40

10.74

10.21

9.676

9.248

8.743

8.341

7.929

7.510

11 ∕ 32

0.34375

17.64

16.42

15.36

14.67

13.90

13.13

12.31

11.59

11.00

10.42

9.948

9.396

8.955

8.504

8.046

0.375

19.02

17.70

16.54

15.79

14.95

14.11

13.22

12.43

11.80

11.16

10.65

10.05

9.569

9.080

8.583

13 ∕ 32

0.40625

20.40

18.97

17.72

16.90

15.99

15.09

14.13

13.28

12.59

11.90

11.35

10.70

10.18

9.655

9.119

7 ∕ 16

0.4375

21.79

20.25

18.90

18.02

17.04

16.07

15.04

14.12

13.38

12.64

12.05

11.35

10.80

10.23

9.655

15 ∕ 32

0.46875

23.17

21.52

20.08

19.14

18.09

17.05

15.94

14.96

14.17

13.39

12.75

12.01

11.41

10.81

10.19

0.500

24.55

22.80

21.26

20.25

19.14

18.03

16.85

15.81

14.97

14.13

13.45

12.66

12.03

11.38

10.73

3 ∕ 8

1 ∕ 2

a For sizes up to 13 gauge, the table values are for music wire with a modulus E of 29,000,000 psi; and for sizes from 27 to 31 gauge, the values are for oil-tempered

MB with a modulus of 28,500,000 psi.

Machinery's Handbook, 31st Edition Spring Design

9 ∕ 32

340

Copyright 2020, Industrial Press, Inc.

AMW Wire Gauge Decimal Equivalenta

16 .037

17 .039

18 .041

Table 14. (Continued) Torsion Spring Deflections 19 .043

20 .045

21 .047

22 .049

23 .051

24 .055

25 .059

26 .063

27 .067

28 .071

29 .075

30 .080

Design Stress, kpsi

192

190

188

187

185

184

183

182

180

178

176

174

173

171

169

Torque, pound-inch

.9550

1.107

1.272

1.460

1.655

1.876

2.114

2.371

2.941

3.590

4.322

5.139

6.080

7.084

8.497

Inside Diameter, inch

Deflection, degrees per coil

17 ∕ 32

0.53125

39.86

37.55

35.47

33.76

32.02

30.60

29.29

28.09

25.93

24.07

22.44

21.37

20.18

19.01

17.76

9 ∕ 16

0.5625

42.05

39.61

37.40

35.59

33.76

32.25

30.87

29.59

27.32

25.35

23.62

22.49

21.23

19.99

18.67

19 ∕ 32

0.59375

44.24

41.67

39.34

37.43

35.50

33.91

32.45

31.10

28.70

26.62

24.80

23.60

22.28

20.97

19.58

0.625

46.43

43.73

41.28

39.27

37.23

35.56

34.02

32.61

30.08

27.89

25.98

24.72

23.33

21.95

20.48

21 ∕ 32

0.65625

48.63

45.78

43.22

41.10

38.97

37.22

35.60

34.12

31.46

29.17

27.16

25.83

24.37

22.93

21.39

11 ∕ 16

0.6875

50.82

47.84

45.15

42.94

40.71

38.87

37.18

35.62

32.85

30.44

28.34

26.95

25.42

23.91

22.30

23 ∕ 32

0.71875

53.01

49.90

47.09

44.78

42.44

40.52

38.76

37.13

34.23

31.72

29.52

28.07

26.47

24.89

23.21

0.750

55.20

51.96

49.03

46.62

44.18

42.18

40.33

38.64

35.61

32.99

30.70

29.18

27.52

25.87

24.12

6 .192

5 .207

5 ∕ 8

3 ∕ 4

Wire Gaugea, b or Size and Decimal Equivalent

31 .085

32 .090

33 .095

34 .100

35 .106

36 .112

37 .118

1∕8 .125

10 .135

9 .1483

5∕3 2 .1563

8 .162

7 .177

3∕16 .1875

Design Stress, kpsi

167

166

164

163

161

160

158

156

161

158

156

154

150

149

146

143

Torque, pound-inch

10.07

11.88

13.81

16.00

18.83

22.07

25.49

29.92

38.90

50.60

58.44

64.30

81.68

96.45

101.5

124.6 7.015

Inside Diameter, inch

Deflection, degrees per coil

0.53125

16.65

15.76

14.87

14.15

13.31

12.64

11.96

11.26

10.93

9.958

9.441

9.064

8.256

7.856

7.565

9 ∕ 16

0.5625

17.50

16.55

15.61

14.85

13.97

13.25

12.53

11.80

11.44

10.42

9.870

9.473

8.620

8.198

7.891

7.312

19 ∕ 32

0.59375

18.34

17.35

16.35

15.55

14.62

13.87

13.11

12.34

11.95

10.87

10.30

9.882

8.984

8.539

8.218

7.609

0.625

19.19

18.14

17.10

16.25

15.27

14.48

13.68

12.87

12.47

11.33

10.73

10.29

9.348

8.881

8.545

7.906

21 ∕ 32

0.65625

20.03

18.93

17.84

16.95

15.92

15.10

14.26

13.41

12.98

11.79

11.16

10.70

9.713

9.222

8.872

8.202

11 ∕ 16

0.6875

20.88

19.72

18.58

17.65

16.58

15.71

14.83

13.95

13.49

12.25

11.59

11.11

10.08

9.564

9.199

8.499

23 ∕ 32

0.71875

21.72

20.52

19.32

18.36

17.23

16.32

15.41

14.48

14.00

12.71

12.02

11.52

10.44

9.905

9.526

8.796

0.750

22.56

21.31

20.06

19.06

17.88

16.94

15.99

15.02

14.52

13.16

12.44

11.92

10.81

10.25

9.852

9.093

5 ∕ 8

3 ∕ 4

a For sizes up to 26 gauge, the table values are for music wire with a modulus E of 29,500,000 psi; for sizes from 27 gauge to 1 ⁄

inch diameter the table values are for music wire with a modulus of 28,500,000 psi; for sizes from 10 gauge to 1 ⁄ 8 inch diameter, the values are for oil-tempered MB with a modulus of 28,500,000 psi. b Gauges 31 through 37 are AMW gauges. Gauges 10 through 5 are Washburn and Moen. 8

341

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17 ∕ 32

Machinery's Handbook, 31st Edition Spring Design

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AMW Wire Gauge Decimal Equivalenta

24 .055

25 .059

26 .063

27 .067

28 .071

29 .075

30 .080

31 .085

32 .090

33 .095

34 .100

35 .106

36 .112

37 .118

1∕8 .125

180

178

176

174

173

171

169

167

166

164

163

161

160

158

156

2.941

3.590

4.322

5.139

6.080

7.084

8.497

10.07

11.88

13.81

16.00

18.83

22.07

25.49

29.92

0.8125

38.38

35.54

33.06

31.42

29.61

27.83

25.93

24.25

22.90

21.55

20.46

19.19

18.17

17.14

16.09

0.875

41.14

38.09

35.42

33.65

31.70

29.79

27.75

25.94

24.58

23.03

21.86

20.49

19.39

18.29

17.17

15 ∕ 16

0.9375

43.91

40.64

37.78

35.88

33.80

31.75

29.56

27.63

26.07

24.52

23.26

21.80

20.62

19.44

18.24

1

1.000 1.0625

46.67 49.44

43.19 45.74

40.14 42.50

38.11 40.35

35.89 37.99

33.71 35.67

31.38 33.20

29.32 31.01

27.65 29.24

26.00 27.48

24.66 26.06

23.11 24.41

21.85 23.08

20.59 21.74

19.31 20.38

11 ∕ 8

1.125

52.20

48.28

44.86

42.58

40.08

37.63

35.01

32.70

30.82

28.97

27.46

25.72

24.31

22.89

21.46

1.1875

54.97

50.83

47.22

44.81

42.18

39.59

36.83

34.39

32.41

30.45

28.86

27.02

25.53

24.04

22.53

1.250

57.73

53.38

49.58

47.04

44.27

41.55

38.64

36.08

33.99

31.94

30.27

28.33

26.76

25.19

23.60

Inside Diameter, inch 13 ∕ 16 7 ∕ 8

11 ∕ 16 13 ∕ 16 11 ∕ 4

Deflection, degrees per coil

Washburn and Moen Gauge or Size and Decimal Equivalenta

10 .135

9 .1483

5∕3 2 .1563

8 .162

7 .177

3∕16 .1875

6 .192

5 .207

7∕3 2 .2188

4 .2253

3 .2437

1∕4 .250

9∕3 2 .2813

5∕16 .3125

11∕3 2 .3438

3∕8 .375

Torque, pound-inch

38.90

50.60

58.44

64.30

81.68

96.45

101.5

124.6

146.0

158.3

199.0

213.3

301.5

410.6

542.5

700.0

0.8125

15.54

14.08

13.30

12.74

11.53

10.93

10.51

9.687

9.208

8.933

8.346

8.125

7.382

6.784

6.292

5.880

0.9375

17.59

15.91

15.02

14.38

12.99

12.30

11.81

10.87

10.32

10.01

9.333

9.081

8.225

7.537

6.972

6.499

15.18

14.35

Design Stress, kpsi

Inside Diameter, inch 13 ∕ 16

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7 ∕ 8 15 ∕ 16

1

0.875

11 ∕ 16

1.000 1.0625

13 ∕ 16

1.1875

11 ∕ 8

1.125

11 ∕ 4

1.250

161

16.57 18.62 19.64 20.67 21.69 22.72

158

15.00 16.83 17.74 18.66 19.57 20.49

156

14.16 15.88 16.74 17.59 18.45 19.31

154

150

13.56

12.26

15.19 16.01

13.72 14.45

16.83 17.64 18.46

15.90 16.63

149

11.61

146

142

141

Deflection, degrees per coil

11.16

12.98 13.66

12.47 13.12

15.03

14.43

15.71

143

13.77 15.08

10.28 11.47 12.06

9.766 10.88 11.44

12.66

12.00

13.84

13.11

13.25

12.56

9.471 10.55 11.09 11.62

12.16

12.70

140

8.840 9.827 10.32 10.81 11.31

11.80

139

8.603 9.559 10.04 10.52 10.99 11.47

138

7.803 8.647 9.069 9.491 9.912

10.33

137

7.161 7.914 8.291 8.668 9.045

9.422

136

6.632 7.312 7.652

135

6.189 6.808 7.118

7.993

7.427

8.673

8.046

8.333

7.737

a For sizes up to 26 gauge, the table values are for music wire with a modulus E of 29,500,000 psi; for sizes from 27 gauge to 1 ⁄ inch diameter the table values are 8 for music wire with a modulus of 28,500,000 psi; for sizes from 10 gauge to 1 ⁄ 8 inch diameter, the values are for oil-tempered MB with a modulus of 28,500,000 psi. For an example in the use of the table, see the example starting on page 334. Note: Intermediate values may be interpolated within reasonable accuracy.

Machinery's Handbook, 31st Edition Spring Design

Design Stress, kpsi Torque, pound-inch

342

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Table 14. (Continued) Torsion Spring Deflections AMW Wire Gauge Decimal Equivalenta

Machinery's Handbook, 31st Edition Spring Design

343

Table 15. Correction Factors for Other Materials

Materiala Hard Drawn MB Chrome-vanadium

Factor 0.75 1.10

Chrome-silicon

1.20

Stainless 302 and 304

0.85

  Up to 1∕8 inch diameter

0.75

  Over 1∕8 to 1∕4 inch diameter

 

Over 1∕4

0.65

inch diameter

Stainless 431

0.80

Stainless 420

0.85

Materiala Stainless 316

Factor

  Up to 1∕8 inch diameter

0.75

  Over 1∕4 inch diameter

0.65

  Up to 1∕8 inch diameter

1.00

  Over 3∕16 inch diameter

1.12

0.65

  Over 1∕8 to 1∕4 inch diameter Stainless 17-7 PH

1.07

  Over 1∕8 to 3∕16 inch diameter …

…

a For use with values in Table 14. Note: The figures in Table 14 are for music wire (ASTM A228)

and oil-tempered MB grade (ASTM A229) and can be used for several other materials that have a similar modulus of elasticity E. However, the design stress may be too high or too low, and therefore the design stress, torque, and deflection per coil should each be multiplied by the appropriate correction factor when using any of the materials given in this table (Table 15).

Torsion Spring Tolerances.—Torsion springs are coiled in a different manner from other types of coiled springs and therefore different tolerances apply. The commercial tol­erance on loads is ±10 percent and is specified with reference to the angular deflection. For example: 100 pound-inches ±10 percent at 45 degrees deflection. One load specified usually suffices. If two loads and two deflections are specified, the manufacturing and test­ing times are increased. Tolerances smaller than ±10 percent require each spring to be individually tested and adjusted, which adds considerably to manufacturing time and cost. Table 16, Table 17, and Table 18 give, respectively, free angle tolerances, tolerances on the number of coils, and coil diameter tolerances. Table 16. Torsion Spring Tolerances for Angular Relationship of Ends Number of Coils (N) 1 2 3 4 5 6 8 10 15 20 25 30 50

4

6

2 4 5.5 7 8 9.5 12 14 20 25 29 32 45

3 5 7 9 10 12 15 19 25 30 35 38 55

8 3.5 6 8 10 12 14.5 18 21 28 34 40 44 63

10

Spring Index 12

14

Free Angle Tolerance, ± degrees 4 4.5 5 7 8 8.5 9.5 10.5 11 12 14 15 14 16 18 16 19 20.5 20.5 23 25 24 27 29 31 34 36 37 41 44 44 48 52 50 55 60 70 77 84

16

18

5.5 9 12 16 20 21 27 31.5 38 47 56 65 90

5.5 9.5 13 16.5 20.5 22.5 28 32.5 40 49 60 68 95

20 6 10 14 17 21 24 29 34 42 51 63 70 100

Table 17. Torsion Spring Tolerance on Number of Coils Number of Coils

Tolerance

Number of Coils

Tolerance

up to 5

±5°

over 10 to 20

±15°

over 5 to 10

±10°

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over 20 to 40

±30°

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Machinery's Handbook, 31st Edition Spring Design

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Table 18. Torsion Spring Coil Diameter Tolerances Wire Diameter, Inch 0.015 0.023 0.035 0.051 0.076 0.114 0.172 0.250

4

6

10

12

Coil Diameter Tolerance, ± inch 0.002 0.002 0.003 0.002 0.003 0.004 0.003 0.004 0.006 0.005 0.007 0.008 0.007 0.009 0.012 0.010 0.013 0.018 0.013 0.020 0.027 0.022 0.030 0.040

0.002 0.002 0.002 0.003 0.005 0.007 0.010 0.014

0.002 0.002 0.002 0.002 0.003 0.004 0.006 0.008

Spring Index

8

14

16

0.003 0.005 0.007 0.010 0.015 0.022 0.034 0.050

0.004 0.006 0.009 0.012 0.018 0.028 0.042 0.060

Miscellaneous Springs.—This section provides information on various springs, some in common use, some less commonly used. Conical compression: These springs taper from top to bottom and are useful where an increasing (instead of a constant) load rate is needed, where solid height must be small, and where vibration must be damped. Conical springs with a uniform pitch are easiest to coil. Load and deflection formulas for compression springs can be used—using the average mean coil diameter, and provided the deflection does not cause the largest active coil to lie against the bottom coil. When this happens, each coil must be calculated separately, using the standard formulas for compression springs. Constant force springs: Those springs are made from flat spring steel and are finding more applications each year. Complicated design procedures can be eliminated by select­ ing a standard design from thousands now available from several spring manufacturers. Spiral, clock, and motor springs: Although often used in wind-up type motors for toys and other products, these springs are difficult to design and results cannot be calculated with precise accuracy. However, many useful designs have been developed and are avail­ able from spring manufacturing companies. Flat springs: These springs are often used to overcome operating space limitations in various products such as electric switches and relays. Table 19 lists formulas for designing flat springs. The formulas are based on standard beam formulas where the deflection is small. Table 19. Formulas for Flat Springs y

P

L

L

Feature

Plan Deflection, y Inches

y =

Load, P, Pounds

P=

PL3 4Ebt 3

S L2 = 6bEt

=

2Sb bt 2 3L

4Ebt 3y L3

b

P

Plan y =

4PL3 Ebt 3

y b

y= =

P =

Sb bt 2 6L

P=

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=

6PL3 Ebt 3

SbL2 Et

Sb bt 2 6L

Ebt 3y 6L3

y= = P= =

P y b 4

b

2Sb L2 3Et

Ebt 3y 4L3

L

y

b

=

=

P

L

5.22PL3 Ebt 3

0.87Sb L2 Et Sb bt 2 6L

Ebt 3y 5.22L3

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Machinery's Handbook, 31st Edition Spring Design

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Table Table 19. (Continued) Formulas forSprings Flat Springs 19. Formulas for Flat y

P L

Feature

Plan Stress, Sb Bending psi Thickness, t Inches

3PL 2bt 2 6Ety = L2 Sb L2 t = 6Ey Sb =

=

3

PL3 4Eby

P

L Plan

b

6PL bt 2

Sb =

3Ety 2L2 2SbL2 t = 3Ey =

=

3

4PL3 Eby

b

P

L

y

L

y

b

P

b 4

b 6PL bt 2 Ety = L2

Sb =

Sb L Ey 6PL3 = 3 Eby

t =

2

y

Sb =

6PL bt 2

Ety 0.87L 2 0.87Sb L2 t = Ey =

=

3

5.22PL3 Eby

Based on standard beam formulas where the deflection is small. y is deflection, see page 304 for other notation. Note: Where two formulas are given for one feature, the designer should use the one found to be appropriate for the given design. The result from either of any two formulas is the same.

Belleville washers or disc springs: These washer type springs can sustain relatively large loads with small deflections, and the loads and deflections can be increased by stack­ing the springs. Information on springs of this type is given in the section DISC SPRINGS starting on page 350. Volute springs: These springs are often used on army tanks and heavy field artillery, and seldom find additional uses because of their high cost, long production time, difficulties in manufacture, and unavailability of a wide range of materials and sizes. Small volute springs are often replaced with standard compression springs. Torsion bars: Although the more simple types are often used on motor cars, the more complicated types with specially forged ends are finding fewer applications. Moduli of Elasticity of Spring Materials.—The modulus of elasticity in tension, denoted by the letter E, and the modulus of elasticity in torsion, denoted by the letter G, are used in formulas relating to spring design. Values of these moduli for various ferrous and nonferrous spring materials are given in Table 20. General Heat Treating Information for Springs.—The following is general informa­tion on the heat treatment of springs, and is applicable to pre-tempered or hard-drawn spring materials only. Compression springs are baked after coiling (before setting) to relieve residual stresses and thus permit larger deflections before taking a permanent set. Extension springs also are baked, but heat removes some of the initial tension. Allow­ ance should be made for this loss. Baking at 500 degrees F for 30 minutes removes approx­ imately 50 percent of the initial tension. The shrinkage in diameter however, will slightly increase the load and rate. Outside diameters shrink when springs of music wire, pretempered MB, and other car­ bon or alloy steels are baked. Baking also slightly increases the free length and these changes produce a little stronger load and increase the rate. Outside diameters expand when springs of stainless steel (18-8) are baked. The free length is also reduced slightly and these changes result in a little lighter load and a decrease in the spring rate. Inconel, Monel, and nickel alloys do not change much when baked.

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Machinery's Handbook, 31st Edition Spring Design

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Beryllium-copper shrinks and deforms when heated. Such springs usually are baked in fixtures or supported on arbors or rods during heating.

Brass and phosphor bronze springs should be given a light heat only. Baking above 450°F will soften the material. Do not heat in salt pots.

Torsion springs do not require baking because coiling causes residual stresses in a direc­tion that is helpful, but such springs frequently are baked so that jarring or handling will not cause them to lose the position of their ends. Table 20. Moduli of Elasticity in Torsion and Tension of Spring Materials Ferrous Materials

Material (Commercial Name)

Hard Drawn MB Up to 0.032 inch 0.033 to 0.063 inch 0.064 to 0.125 inch 0.126 to 0.625 inch Music Wire Up to 0.032 inch 0.033 to 0.063 inch 0.064 to 0.125 inch 0.126 to 0.250 inch Oil-Tempered MB Chrome-Vanadium Chrome-Silicon Silicon-Manganese Stainless Steel Types 302, 304, 316 Type 17-7 PH Type 420 Type 431

Modulus of Elasticitya, psi

In Torsion, G

In Tension, E

11,700,000 11,600,000 11,500,000 11,400,000

28,800,000 28,700,000 28,600,000 28,500,000

12,000,000 11,850,000 11,750,000 11,600,000 11,200,000 11,200,000 11,200,000 10,750,000

29,500,000 29,000,000 28,500,000 28,000,000 28,500,000 28,500,000 29,500,000 29,000,000

10,000,000 10,500,000 11,000,000 11,400,000

28,000,000c 29,500,000 29,000,000 29,500,000

Nonferrous Materials

Material (Commercial Name)

Spring Brass Type 70-30 Phosphor Bronze 5 percent tin Beryllium-Copper Cold Drawn 4 Nos. Pretempered, fully hard Inconelb 600 Inconelb X 750 Monelb 400 Monelb K 500 Duranickelb 300 Permanickelb Ni Spanb C 902 Elgiloyd Iso-Elastice

Modulus of Elasticitya, psi

In Torsion, G

In Tension, E

5,000,000

15,000,000

6,000,000

15,000,000

7,000,000 7,250,000 10,500,000 10,500,000 9,500,000 9,500,000 11,000,000 11,000,000 10,000,000 12,000,000 9,200,000

17,000,000 19,000,000 31,000,000c 31,000,000c 26,000,000 26,000,000 30,000,000 30,000,000 27,500,000 29,500,000 26,000,000

a Note: Modulus G (shear modulus) is used for compression and extension springs; modulus E (Young’s modulus) is used for torsion, flat, and spiral springs. b Trade name of International Nickel Company. c May be 2,000,000 pounds per square inch less if material is not fully hard. d Trade name of Hamilton Watch Company. e Trade name of John Chatillon & Sons.

Spring brass and phosphor bronze springs that are not very highly stressed and are not subject to severe operating use may be stress relieved after coiling by immersing them in boiling water for a period of 1 hour.

Positions of loops will change with heat. Parallel hooks may change as much as 45 degrees during baking. Torsion spring arms will alter position considerably. These changes should be allowed for during looping or forming.

Quick heating after coiling either in a high-temperature salt pot or by passing a spring through a gas flame is not good practice. Samples heated in this way will not conform with production runs that are properly baked. A small, controlled-temperature oven should be used for samples and for small lot orders. Plated springs should always be baked before plating to relieve coiling stresses and again after plating to relieve hydrogen embrittlement.

Hardness values fall with high heat—but music wire, hard drawn, and stainless steel will increase 2 to 4 points on the Rockwell C scale.

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Machinery's Handbook, 31st Edition Spring Design

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Table 21. Squares, Cubes, and Fourth Powers of Wire Diameters Steel Wire Gage (US) 7-0 6-0 5-0 4-0 3-0 2-0 1-0 1 2 3 4 5 6 … 7 … 8 … 9 … … 10 … … 11 … … … 12 … … 13 … … 14 … 15 … … … 16 … … 17 … … 18 … … … 19 … … … 20 … 21 … … 22 … 23 … 24 …

Music or Piano Wire Gage … … … … … … … … … … … … … 45 … 44 43 42 … 41 40 … 39 38 … 37 36 35 … 34 33 … 32 31 30 29 … 28 27 26 … 25 24 … 23 22 … 21 20 19 18 17 16 15 … 14 … 13 12 … 11 … 10 … 9

Diameter, Inch 0.4900 0.4615 0.4305 0.3938 0.3625 0.331 0.3065 0.283 0.2625 0.2437 0.2253 0.207 0.192 0.180 0.177 0.170 0.162 0.154 0.1483 0.146 0.138 0.135 0.130 0.124 0.1205 0.118 0.112 0.106 0.1055 0.100 0.095 0.0915 0.090 0.085 0.080 0.075 0.072 0.071 0.067 0.063 0.0625 0.059 0.055 0.054 0.051 0.049 0.0475 0.047 0.045 0.043 0.041 0.039 0.037 0.035 0.0348 0.033 0.0317 0.031 0.029 0.0286 0.026 0.0258 0.024 0.023 0.022

Section Area 0.1886 0.1673 0.1456 0.1218 0.1032 0.0860 0.0738 0.0629 0.0541 0.0466 0.0399 0.0337 0.0290 0.0254 0.0246 0.0227 0.0206 0.0186 0.0173 0.0167 0.0150 0.0143 0.0133 0.0121 0.0114 0.0109 0.0099 0.0088 0.0087 0.0078 0.0071 0.0066 0.0064 0.0057 0.0050 0.0044 0.0041 0.0040 0.0035 0.0031 0.0031 0.0027 0.0024 0.0023 0.0020 0.00189 0.00177 0.00173 0.00159 0.00145 0.00132 0.00119 0.00108 0.00096 0.00095 0.00086 0.00079 0.00075 0.00066 0.00064 0.00053 0.00052 0.00045 0.00042 0.00038

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Square 0.24010 0.21298 0.18533 0.15508 0.13141 0.10956 0.09394 0.08009 0.06891 0.05939 0.05076 0.04285 0.03686 0.03240 0.03133 0.02890 0.02624 0.02372 0.02199 0.02132 0.01904 0.01822 0.01690 0.01538 0.01452 0.01392 0.01254 0.01124 0.01113 0.0100 0.00902 0.00837 0.00810 0.00722 0.0064 0.00562 0.00518 0.00504 0.00449 0.00397 0.00391 0.00348 0.00302 0.00292 0.00260 0.00240 0.00226 0.00221 0.00202 0.00185 0.00168 0.00152 0.00137 0.00122 0.00121 0.00109 0.00100 0.00096 0.00084 0.00082 0.00068 0.00067 0.00058 0.00053 0.00048

Cube 0.11765 0.09829 0.07978 0.06107 0.04763 0.03626 0.02879 0.02267 0.01809 0.01447 0.01144 0.00887 0.00708 0.00583 0.00555 0.00491 0.00425 0.00365 0.00326 0.00311 0.00263 0.00246 0.00220 0.00191 0.00175 0.00164 0.00140 0.00119 0.001174 0.001000 0.000857 0.000766 0.000729 0.000614 0.000512 0.000422 0.000373 0.000358 0.000301 0.000250 0.000244 0.000205 0.000166 0.000157 0.000133 0.000118 0.000107 0.000104 0.000091 0.0000795 0.0000689 0.0000593 0.0000507 0.0000429 0.0000421 0.0000359 0.0000319 0.0000298 0.0000244 0.0000234 0.0000176 0.0000172 0.0000138 0.0000122 0.0000106

Fourth Power 0.05765 0.04536 0.03435 0.02405 0.01727 0.01200 0.008825 0.006414 0.004748 0.003527 0.002577 0.001836 0.001359 0.001050 0.000982 0.000835 0.000689 0.000563 0.000484 0.000455 0.000363 0.000332 0.000286 0.000237 0.000211 0.000194 0.000157 0.000126 0.0001239 0.0001000 0.0000815 0.0000701 0.0000656 0.0000522 0.0000410 0.0000316 0.0000269 0.0000254 0.0000202 0.0000158 0.0000153 0.0000121 0.00000915 0.00000850 0.00000677 0.00000576 0.00000509 0.00000488 0.00000410 0.00000342 0.00000283 0.00000231 0.00000187 0.00000150 0.00000147 0.00000119 0.00000101 0.000000924 0.000000707 0.000000669 0.000000457 0.000000443 0.000000332 0.000000280 0.000000234

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Machinery's Handbook, 31st Edition Spring Design

348

Spring Failure.—Spring failure may be breakage, high permanent set, or loss of load. The causes are listed in groups in Table 22. Group 1 covers causes that occur most frequently; Group 2 covers causes that are less frequent; and Group 3 lists causes that occur occasion­ally. Table 22. Causes of Spring Failure Cause High stress

Group 1

Group 2

Group 3

Comments and Recommendations The majority of spring failures are due to high stresses caused by large deflections and high loads. High stresses should be used only for statically loaded springs. Low stresses lengthen fatigue life.

Improper electroplating methods and acid cleaning of springs, without Hydrogen proper baking treatment, cause spring steels to become brittle, and are a embrittlement frequent cause of failure. Nonferrous springs are immune. Sharp bends and holes

Sharp bends on extension, torsion, and flat springs, and holes or notches in flat springs, cause high concentrations of stress, resulting in failure. Bend radii should be as large as possible, and tool marks avoided.

Fatigue

Repeated deflections of springs, especially above 1,000,000 cycles, even with medium stresses, may cause failure. Low stresses should be used if a spring is to be subjected to a very high number of operating cycles.

Shock loading

Impact, shock, and rapid loading cause far higher stresses than those com­puted by the regular spring formulas. High-carbon spring steels do not withstand shock loading as well as do alloy steels.

Corrosion

Slight rusting or pitting caused by acids, alkalis, galvanic corrosion, stress corrosion cracking, or corrosive atmosphere weakens the material and causes higher stresses in the corroded area.

Faulty heat treatment

Keeping spring materials at the hardening temperature for longer periods than necessary causes an undesirable growth in grain structure, resulting in brittleness, even though the hardness may be correct.

Faulty material

Poor material containing inclusions, seams, slivers, and flat material with rough, slit, or torn edges is a cause of early failure. Overdrawn wire, improper hardness, and poor grain structure also cause early failure.

High temperature

High operating temperatures reduce spring temper (or hardness) and lower the modulus of elasticity, thereby causing lower loads, reducing the elastic limit, and increasing corrosion. Corrosion-resisting or nickel alloys should be used.

Low temperature

Temperatures below −40 degrees F reduce the ability of carbon steels to withstand shock loads. Carbon steels become brittle at −70 degrees F. Cor­rosion-resisting, nickel, or nonferrous alloys should be used.

Friction

Close fits on rods or in holes result in a wearing away of material and occasional failure. The outside diameters of compression springs expand during deflection but they become smaller on torsion springs.

Other causes

Enlarged hooks on extension springs increase the stress at the bends. Car­rying too much electrical current will cause failure. Welding and soldering frequently destroy the spring temper. Tool marks, nicks, and cuts often raise stresses. Deflecting torsion springs outwardly causes high stresses and winding them tightly causes binding on supporting rods. High speed of deflection, vibration, and surging due to operation near natural periods of vibration or their harmonics cause increased stresses.

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349

Table 23. Arbor Diameters for Springs Made from Music Wire Spring Outside Diameter (inch)

Wire Dia. (inch)

1∕16

0.008

0.039

0.060

0.078

0.093

0.107

0.012

0.034

0.059

0.081

0.101

0.119

0.010

0.014

0.016

0.018

0.020

0.022

0.024

0.026

0.028

0.030

0.032

0.034

0.036

0.038

0.041

0.0475 0.054

0.0625 0.072

0.080

0.0915

0.1055

0.1205 0.125

Wire Dia. (inch) 0.022

0.024

0.026

0.028

0.030

0.032

0.034

0.036

0.038

0.041

0.0475

0.054

0.0625

0.072

0.080

0.0915

0.1055

0.1205

0.125

0.037

0.031

0.028 …

…

…

…

…

…

…

…

…

…

…

…

…

…

…

…

…

…

…

…

…

3∕3 2

0.060

0.057

0.055

0.053

0.049

0.046

0.043 …

…

…

…

…

…

…

…

…

…

…

…

…

…

…

…

…

1∕8

0.080

0.081

0.079

0.077

0.075

0.072

0.070

0.067

0.064

0.061

0.057 …

…

…

…

5∕3 2

0.099

3∕16

7∕3 2

1∕4

0.119

0.129

0.135

0.150

0.115

0.129

0.102

0.121

0.140

0.101

0.124

0.102

0.096

0.097

0.095

0.093

0.091

0.088

0.085

0.123

0.123

0.122

0.120 0.118

0.115

0.075

0.103

…

…

…

…

…

…

…

…

…

…

…

…

…

…

…

0.177

0.189

0.200

…

…

0.194

0.209

0.224

…

…

…

0.206

0.179

0.205

0.176

0.201

0.168

0.194

0.160

0.187

0.146

0.169

0.129

0.158

…

…

0.144

…

…

…

…

…

…

0.208

0.182

…

…

0.209

0.184

…

…

0.209

0.185

0.108

…

0.208

0.187

0.132

…

0.208

0.187

0.142

…

0.207

0.187

0.151

0.115

0.206

0.187

0.154

0.125

0.203

0.186

0.156

0.129

…

…

…

0.200

0.184

0.159

0.131

0.103

0.182

0.161

0.134

…

0.178

0.163

0.136

…

…

…

0.187

0.165

0.138

…

…

1∕2

0.172

0.166

0.141

…

…

7∕16

0.156

0.163

0.166

0.143

…

…

3∕8

0.164

0.165

0.144

0.087

…

0.165

0.145

0.098

…

11∕3 2

0.154

0.161

0.144

0.106

…

5∕16

0.142

0.161

0.144

0.111

0.109

0.078

0.142

0.113

0.082

9∕3 2

Arbor Diameter (inch)

…

…

…

0.200

0.215

0.220

0.224

0.226

0.228

0.229

0.229

0.229

0.229

0.229

0.227

0.226

0.220

0.212

0.201

0.186

0.173 …

…

…

…

…

…

…

…

…

…

0.213

0.234

0.231

0.259

0.237

0.242

0.245

0.248

0.250

0.251

0.251

0.251

0.250

0.251

0.250

0.244

0.245

0.228

0.214

0.201

0.181 …

…

…

…

…

0.250

0.271

0.268

0.296

0.275

0.280

0.285

0.288

0.291

0.292

0.292

0.294

0.293

0.294

0.293

0.287

0.280

0.268

0.256

0.238

0.215 …

…

0.284

0.305

0.312

0.318

0.323

0.328

0.331

0.333

0.333

0.335

0.336

0.337

0.336

0.330

0.319

0.308

0.293

0.271

0.215

0.239

Spring Outside Diameter (inch) 9∕16

5∕8

11∕16

3∕4

13∕16

7∕8

15∕16

1

11∕8

11∕4

13∕8

11∕2

13∕4

2

Arbor Diameter (inches) 0.332

0.341

0.350

0.356

0.362

0.367

0.370

0.372

0.375

0.378

0.380

0.381

0.379

0.357

0.367

0.380

0.387

0.395

0.400

0.404

0.407

0.412

0.416

0.422

0.425

0.426

0.370

0.418

0.347

0.398

0.360

0.327

0.303

0.295

0.411

0.381

0.358

0.351

0.380

0.393

0.406

0.416

0.426

0.432

0.437

0.442

0.448

0.456

0.464

0.467

0.468

0.466

0.461

0.448

0.433

0.414

0.406

…

0.415

0.430

0.442

0.453

0.462

0.469

0.474

0.481

0.489

0.504

0.509

0.512

0.512

0.509

0.500

0.485

0.468

0.461

…

…

…

0.467

0.481

0.490

0.498

0.506

0.512

0.522

0.541

0.550

0.556

0.555

0.554

0.547

0.535

0.520

0.515

…

…

…

…

0.506

0.516

0.526

0.536

0.543

0.554

0.576

0.589

0.597

0.599

0.599

0.597

0.586

0.571

0.567

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…

…

…

…

…

0.540

0.552

0.562

0.572

0.586

0.610

0.625

0.639

0.641

0.641

0.640

0.630

0.622

0.617

…

…

…

…

…

…

0.557

0.589

0.600

0.615

0.643

0.661

0.678

0.682

0.685

0.685

0.683

0.673

0.671

…

…

…

…

…

…

…

…

…

…

…

…

…

…

…

0.650

0.670

0.706

0.727

0.753

0.765

0.772

0.776

0.775

0.772

0.770

…

…

0.718

0.763

0.792

0.822

0.840

0.851

0.860

0.865

0.864

0.864

…

…

…

…

…

…

…

…

…

…

0.812

0.850

0.889

…

…

…

…

…

…

…

…

…

…

…

0.906

0.951

0.911

0.980

0.942

1.02

0.930

0.952

0.955

0.955

1.00

1.04

1.04

1.05

…

…

…

…

…

…

…

…

…

…

…

…

1.06 1.11

1.13

1.16

1.20

1.22

1.23

…

…

…

…

…

…

…

…

…

…

…

…

1.17

1.22

1.26

1.30

1.35

1.38

1.39

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Machinery's Handbook, 31st Edition Disc Springs

350

DISC SPRINGS Performance of Disc Springs Introduction.—Disc springs, also known as Belleville springs, are conically formed from washers and have rectangular cross section. The disc spring concept was invented by a Frenchman Louis Belleville in 1865. His springs were relatively thick and had a small amount of cone height or “dish”, which determined axial deflection. At that time, these springs were used in the buffer parts of railway rolling stock, for recoil mechanisms of guns, and some other applications. The use of disc springs will be advantageous when space is limited and high force is required, as these conditions cannot be satisfied by using coil springs. Load-deflection characteristics of disc springs are linear and regressive depending on their dimensions and the type of stacking. A large number of standard sizes are available from disc spring manufacturers and distributors, so that custom sizes may not be required. Therefore, disc springs are widely used today in virtually all branches of engi­neering with possibilities of new applications. Disc Spring Nomenclature.—Disc spring manufacturers assign their own part number for each disc spring, but the catalog numbers for disc springs are similar, so each item can often be identified regardless of manufacturer. The disc spring identification number is a numerical code that provides basic dimensions in millimeters. Identification numbers represent the primary dimensions of the disc spring and consist of one, two, or three numbers separated from each other by dash marks or spaces. Disc spring manufacturers in the United States also provide dimensions in inches. Dimensions of several typical disc springs are shown in the following table. Basic nomenclature is illustrated in Fig. 1. Catalog Number (mm) 8-4.2-0.4 50-25.4-2 200-102-12

Outside Diameter D (mm) 8 50 200

Inside Diameter d (mm) 4.2 25.4 102

Thickness t (mm) 0.4 2 12

Equivalent Catalog Number (inch) 0.315-0.165- 0.0157 1.97-1.00-0.0787 7.87-4.02-0.472

Additional dimensions shown in catalogs are cone (dish) height h at unloaded condition, and overall height H = h + t, that combines the cone height and the thickness of a disc spring. d

H t

h

D

Fig. 1. Disc Spring Nomenclature

Disc Spring Group Classification.—Forces and stresses generated by compression depend on disc spring thickness much more than on any other dimensions. Standard DIN 2093 divides all disc springs into three groups in accordance with their thickness: Group 1 includes all disc springs with thickness less than 1.25 mm (0.0492 inch). Group 2 includes all disc springs with thickness between 1.25 mm and 6.0 mm (0.0492 inch and 0.2362 inch). Group 3 includes disc springs with thickness greater than 6.0 mm (0.2362 inch). There are 87 standard disc spring items, which are manufactured in accordance with DIN 2093 specifications for dimensions and quality requirements. There are 30 standard disc spring items in Group 1. The smallest and the largest disc springs in this group are

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Machinery's Handbook, 31st Edition Disc Springs

351

8-4.2-0.2 and 40-20.4-1, respectively. Group 2 has 45 standard disc spring items. The smallest and the largest disc springs are 22.5-11.2-1.25 and 200-102-5.5, respectively. Group 3 includes 12 standard disc spring items. The smallest and the largest disc springs of this group are 125-64-8 and 250-127-14, respectively. Summary of Disc Spring Sizes Specified in DIN 2093 Classification

OD

Min.

40 mm (1.575 in) 225 mm (8.858 in) 250 mm (9.843 in)

6 mm (0.236 in) 20 mm (0.787 in) 125 mm (4.921 in)

Group 1 Group 2 Group 3

Max

ID

Min.

3.2 mm (0.126 in) 10.2 mm (0.402 in) 61 mm (2.402 in)

Max

Min.

20.4 mm (0.803 in) 112 mm (4.409 in) 127 mm (5.000 in)

Thickness

0.2 mm (0.008 in) 1.25 mm (0.049 in) 6.5 mm (0.256 in)

Max

1.2 mm (0.047 in) 6 mm (0.236 in) 16 mm (0.630 in)

The number of catalog items by disc spring dimensions depends on the manufacturer. Currently, the smallest disc spring is 6-3.2-0.3 and the largest is 250-127-16. One of the US disc spring manufacturers, Key Bellevilles, Inc. offers 190 catalog items. The greatest number of disc spring items can be found in Christian Bauer GmbH + Co. catalog. There are 291 disc spring catalog items in all three groups. Disc Spring Contact Surfaces.—Disc springs are manufactured with and without con­ tact (also called load-bearing) surfaces. Contact surfaces are small flats at points 1 and 3 in Fig. 2, adjacent to the corner radii of the spring. The width of the contact surfaces w depends on the outside diameter D of the spring, and its value is approximately w = D / 150. F

d

w

1

H t' 3

w F

D Fig. 2. Disc Spring with Contact Surfaces

Disc springs of Group 1 and Group 2, that are contained in the DIN 2093 standard, do not have contact surfaces, although some Group 2 disc springs not included in DIN 2093 are manufactured with contact surfaces. All disc springs of Group 3 (standard and nonstan­dard) are manufactured with contact surfaces. Almost all disc springs with contact surfaces are manufactured with reduced thickness. Disc springs without contact surfaces have a corner radii r whose value depends on the spring thickness t. One disc spring manufacturer recommends the following relationship: r=t/6

Disc Spring Materials .—A wide variety of materials are available for disc springs, but selection of the material depends mainly on application. High-carbon steels are used only for Group 1 disc springs. AISI 1070 and AISI 1095 carbon steels are used in the United States. Sim­ilar high-carbon steels such as DIN 1.1231 and DIN 1.1238 (Germany), and BS 060 A67 and BS 060 A78 (Great Britain) are used in other countries. The most common materials for Groups 2 and 3 springs operating under normal conditions are

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352

Machinery's Handbook, 31st Edition Disc Springs

chromium-vanadium alloy steels such as AISI 6150 used in the United States. Similar alloys such as DIN 1.8159 and DIN 1.7701 (Germany) and BS 735 A50 (Great Britain) are used in foreign countries. Some disc spring manufacturers in the United States also use chromium alloy steel AISI 5160. The hard­ness of disc springs in Groups 2 and 3 should be 42 to 52 RC. The hardness of disc springs in Group 1 tested by the Vickers method should be 412 to 544 HV. If disc springs must withstand corrosion and high temperatures, stainless steels and heat-resistant alloys are used. Most commonly used stainless steels in the United States are AISI types 301, 316, and 631, which are similar to foreign material numbers DIN 1.4310, DIN 1.4401, and DIN 1.4568, respectively. The operating temperature range for 631 stainless steel is −330 to 660ºF (−200 to 350ºC). Among heat-resistant alloys, Inconel 718 and Inc­onel X750 (similar to DIN 2.4668 and DIN 2.4669, respectively) are the most popular. Operating temperature range for Inconel 718 is −440 to 1290ºF (−260 to 700ºC). When disc springs are stacked in large numbers and their total weight becomes a major concern, titanium α-β alloys can be used to reduce weight. In such cases, Ti-6Al-4V alloy is used. If nonmagnetic and corrosion resistant properties are required and material strength is not an issue, phosphor bronzes and beryllium-coppers are the most popular copper alloys for disc springs. Phosphor bronze C52100, which is similar to DIN material number 2.1030, is used at the ordinary temperature range. Beryllium-coppers C17000 and C17200, similar to material numbers DIN 2.1245 and DIN 2.1247 respectively, works well at very low temperatures. Strength properties of disc spring materials are characterized by moduli of elasticity and Poisson’s ratios. These are summarized in Table 1. Table 1. Strength Characteristics of Disc Spring Materials Modulus of Elasticity

Material

All Steels Heat-resistant Alloys α-β Titanium Alloys (Ti-6Al-4V) Phosphor Bronze (C52100) Beryllium-copper (C17000) Beryllium-copper (C17200)

106 psi

N/mm2

28-31

193,000–213,700

17 16 17 18

117,200 110,300 117,200 124,100

Poisson’s Ratio 0.30 0.28–0.29 0.32 0.35 0.30 0.30

Stacking of Disc Springs.—Individual disc springs can be arranged in series and parallel stacks. Disc springs in series stacking, Fig. 3, provide larger deflection Stotal under the same load F as a single disc spring would generate. Disc springs in parallel stacking, Fig. 4, gen­erate higher loads Ftotal with the same deflection s than a single disc spring would have. n =  number of disc springs in stack s =  deflection of single spring Stotal =  total deflection of stack of n springs F =  load generated by a single spring Ftotal =  total load generated by springs in stack L0 =  length of unloaded spring stack Series: For n disc springs arranged in series, Fig. 3, the following equations are applied: Ftotal = F

Stotal = s # n L 0 = H # n = ^t + hh # n

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(1)

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Machinery's Handbook, 31st Edition Disc Spring Stacking

353

F

L0

L1,2

t

H

h F

d D

Fig. 3. Disc Springs in Series Stacking L1,2 Indices Indicate Length of Spring Stack under Minimum and Maximum Load.

Parallel: Parallel stacking generates a force that is directly proportional to the number of springs arranged in parallel. Two springs in parallel will double the force, three springs in parallel will triple the force, and so on. However, it is a common practice to use two springs in parallel in order to keep the frictional forces between the springs as low as possible. Oth­erwise, the actual spring force cannot be accurately determined due to deviation from its theoretical value. For n disc springs arranged in parallel as in Fig. 4, the following equations are applied: Ftotal = F # n

Stotal = s L0 = H + t ^n − 1h = ^h + th + tn − t = h + tn

(2)

d

L0

t h

D

H

Fig. 4. Disc Springs in Parallel Stacking

Parallel-Series: When both higher force and greater deflection are required, disc springs must be arranged in a combined parallel-series stacking as illustrated in Fig. 5. F

L0

L 1,2 H t

h d D

F

Fig. 5. Disc Springs in Parallel-Series Stacking

Normally, two springs in parallel are nested in series stacking. Two springs in parallel, called a pair, double the force, and the number of pairs, n p, determines the total deflection, Stotal .

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Machinery's Handbook, 31st Edition Disc Spring Forces and Stresses

354

For n p disc spring pairs arranged in series, the following equations are applied:

Ftotal = 2 # F Stotal = s # np L 0 = H # np = ^2t + hh # np

(3)

Disc Spring Forces and Stresses Several methods of calculating forces and stresses for given disc spring configurations exist, some very complicated, others of limited accuracy. The theory which is widely used today for force and stress calculations was developed more than 65 years ago by Almen and Laszlo.

The theory is based on the following assumptions: cross sections are rectangular without radii over the entire range of spring deflection; no stresses occur in the radial direction; disc springs are always under elastic deformation during deflection; and due to small cone angles of unloaded disc springs (between 3.5° and 8.6°), mathematical simplifica­tions are applied.

The theory provides accurate results for disc springs with the following ratios: outsideto-inside diameter, D / d = 1.3 to 2.5; and cone height-to-thickness, h/t is up to 1.5.

Force Generated by Disc Springs Without Contact Surfaces.—Disc springs in Group 1 and most of disc springs in Group 2 are manufactured without contact (load-bearing) sur­faces, but have corner radii. A single disc spring force applied to points 1 and 3 in Fig. 6 can be found from Equation (4) in which corner radii are not considered: 4 · E· s 8` h − s j · ^h − sh · t + t 3B F= 2 (4) ^1 − µ 2h · K1 · D 2

where F = disc spring force; E = modulus of elasticity of spring material; μ = Pois­son’s ratio of spring material; K1 = constant depending on outside-to-inside diameter ratio; D = disc spring nominal outside diameter; h = cone (dish) height; s = disc spring deflec­ tion; and, t = disc spring thickness. D F 1

H

2

t

3

F

h

d

Fig. 6. Schematic of Applied Forces

It has been found that the theoretical forces calculated using Equation (4) are lower than the actual (measured) spring forces, as illustrated in Fig. 7. The difference between theo­ retical (trace 1) and measured force values (trace 3) was significantly reduced (trace 2) when the actual outside diameter of the spring in loaded condition was used in the calcula­tions.

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Machinery's Handbook, 31st Edition Disc Spring Forces and Stresses 6000

355 3

2

5500

1

5000 4500

Force (pounds)

4000 3500 3000 2500 2000 1500 1000 500 0 0.01

0

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.10

Deflection (inch)

Fig. 7. Force-Deflection Relationships (80-36-3.6 Disc Springs) 1 - Theoretical Force Calculated by Equation (4) 2 - Theoretical Force Calculated by Equation (10) 3 - Measured Force

The actual outside diameter Da of a disc spring contact circle is smaller than the nominal outside diameter D due to cone angle α and corner radius r, as shown in Fig. 8. Diameter Da cannot be measured, but can be calculated by Equation (9) developed by Dr. Isacov. D/2 d/2

t r r h Da / 2 D/2

t r

r

a b

Da / 2

Fig. 8. Conventional Shape of Disc Spring

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Machinery's Handbook, 31st Edition Disc Spring Forces and Stresses

356 From Fig. 8,

Da D = − ^a + bh 2 2

(5)

where a = tsinα and b = rcosα. Substitution of a and b values into Equation (5) gives:

Da D ^ = − t sin α + r cos αh 2 2

The cone angle α is found from: h 2h tan α = D d = D − d 2 −2

(6)

2h α = atan a D − d k

(7)

t Substituting α from Equation (7) and r = 6 into Equation (6) gives: Da D 2h 1 2h = − t & sin :ata n a D − d kD + 6 cos :atan a D − d kD0 2 2

Finally,

(8)

2h 1 2h Da = D − 2t &sin :atan a D − d kD + 6 cos :atan a D − d kD0

(9)

Substituting Da from Equation (9) for D in Equation (4) yields Equation (10), that pro­ vides better accuracy for calculating disc spring forces. 4 ·E ·s 8` h − s j · ^h − sh · t + t 3B F= 2 (10) ^1 − µ 2h K1 · D a2 The constant K1 depends on disc spring outside diameter D, inside diameter d, and their ratio δ = D∕d : 2 aδ − 1k δ K1 = (11) δ+1 2 k πa − δ − 1 ln δ

Table 2 compares the spring force of a series of disc springs deflected by 75 percent of their cone height, i.e., s = 0.75h, as determined from manufacturers catalogs calculated in accor­dance with Equation (4), calculated forces by use of Equation (10), and measured forces. Table 2. Comparison Between Calculated and Measured Disc Spring Forces Disc Spring Catalog Item 50 – 22.4 – 2.5 S = 1.05 mm

Schnorr Handbook for Disc Springs 8510 N 1913 lbf

Christian Bauer Disc Spring Handbook 8510 N 1913 lbf

Key Bellevilles Disc Spring Catalog 8616 N 1937 lbf

Spring Force Calculated by Equation (10)

Measured Disc Spring Force

9020 N 2028 lbf

9563 N 2150 lbf

60 – 30.5 – 2.5 S = 1.35 mm

8340 N 1875 lbf

8342 N 1875 lbf

8465 N 1903 lbf

8794 N 1977 lbf

8896 N 2000 lbf

60 – 30.5 – 3 S = 1.275 mm

13200 N 2967 lbf

13270 N 2983 lbf

13416 N 3016 lbf

14052 N 3159 lbf

13985 N 3144 lbf

70 – 35.5 – 3 S = 1.575 mm

12300 N 2765 lbf

12320 N 2770 lbf

12397 N 2787 lbf

12971 N 2916 lbf

13287 N 2987 lbf

17170 N 3860 lbf

17304 N 3890 lbf

70 – 35.5 – 3.5 S = 1.35 mm

16180 N 3637 lbf

Comparison made at 75 percent deflection, in newtons (N) and pounds (lbf)

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Machinery's Handbook, 31st Edition Disc Spring Forces and Stresses

357

The difference between disc spring forces calculated by Equation (10) and the measured forces varies from −5.7 percent (maximum) to +0.5 percent (minimum). Disc spring forces calculated by Equation (4) and shown in manufacturers catalogs are less than measured forces by −11 percent (maximum) to −6 percent (minimum). Force Generated by Disc Spring with Contact Surfaces.—Some of disc springs in Group 2 and all disc springs in Group 3 are manufactured with small contact (load-bear­ing) surfaces or flats in addition to the corner radii. These flats provide better contact between disc springs, but, at the same time, they reduce the springs outside diameter and generate higher spring force because in Equation (4) force F is inversely proportional to the square of outside diameter D 2. To compensate for the undesired force increase, the disc spring thickness is reduced from t to t ′. Thickness reduction factors t ′ / t are approximately 0.94 for disc spring series A and B, and approximately 0.96 for series C springs. With such reduction factors, the disc spring force at 75 percent deflection is the same as for equivalent disc spring without contact surfaces. Equation (12), which is similar to Equation (10), has an additional constant K4 that correlates the increase in spring force due to contact surfaces. If disc springs do not have contact surfaces, then K4 2 = K4 = 1. 4 · E · K 24 · s 8 2 · ` s 3 F= K 4 h l − 2 j · ^hl − sh · ^ tl + tlh B (12) ^1 − µ2h · K1 · D 2a where t ′ =  reduced thickness of a disc spring h  ′ =   cone height adjusted to reduced thickness: h ′= H − t ′ (h ′ > h) K4 =  constant applied to disc springs with contact surfaces. K4 2 can be calculated as follows:

K 24 =

−b+

b2 − 4ac 2a

(13)

where a = t ′(H − 4t ′ + 3t) (5H − 8 t ′ + 3t); b = 32(t ′) ; and, c = −t [5(H − t) + 32t ]. Disc Spring Functional Stresses.—Disc springs are designed for both static and dynamic load applications. In static load applications, disc springs may be under constant or fluctuating load conditions that change up to 5,000 or 10,000 cycles over long time intervals. Dynamic loads occur when disc springs are under continuously changing deflec­tion between pre-load (approximately 15 to 20 percent of the cone height) and the maximum deflection values over short time intervals. Both static and dynamic loads cause compres­ sive and tensile stresses. The position of critical stress points on a disc spring cross section are shown in Fig. 9. 3

Do

F 0

t

F

0

2

2

3

h s

2

F 1

1

2

3

H

F d D

Fig. 9. Critical Stress Points s is Deflection of Spring by Force F; h − s is a Cone Height of Loaded Disc Spring.

Compressive stresses are acting at points 0 and 1, that are located on the top surface of the disc spring. Point 0 is located on the cross-sectional midpoint diameter, and point 1 is located on the top inside diameter. Tensile stresses are acting at points 2 and 3, which are located on the bottom surface of the disc spring. Point 2 is on the bottom inside diameter, and point 3 is on the bottom outside diameter. The following equations are used to

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Machinery's Handbook, 31st Edition DISC SPRING FORCES AND STRESSES

358

calcu­late stresses. The minus sign “−” indicates that compressive stresses are acting in a direc­tion opposite to the tensile stresses.

Point 0 :

3 v0 = - r ·

Point 1 :

v1 =

Point 2 :

v2 =

Point 3 :

v3 =

4E · t · s ·K4

(14)

(1 – µ 2 ) · K1 · D2a

s 4E · K4 · s · 8 K 4 · K2 · `h – 2 j + K3 · t B

(15)

(1 – µ 2 ) · K1 · D2a

s 4E · K4 · s · 8 K 3 · t – K2 · K4 · `h – 2 j B

(16)

(1 – µ 2 ) · K1 · D2a

s 4E · K4 · s · 8 K 4 · (2K3 – K2) · ` h – 2 j B + K3 · t

(17)

(1 – µ 2 ) · K1 · D2a · δ

K2 and K3 are disc spring dimensional constants, defined as follows:

K2 =

6a

δ−1 − 1k lnδ π · lnδ

(18)

K3 =

3 · ^δ − 1h π · lnδ

(19)

where δ = D/d is the outside-to-inside diameter ratio.

In static application, if disc springs are fully flattened (100 percent deflection), compressive stress at point 0 should not exceed the tensile strength of disc spring materials. For most spring steels, the permissible value is σ0  1600 N/mm 2 or 232,000 psi.

In dynamic applications, certain limitations on tensile stress values are recommended to obtain controlled fatigue life of disc springs utilized in various stacking. Maximum tensile stresses at points 2 and 3 depend on the Group number of the disc springs. Stresses σ2 and σ3 should not exceed the following values: Maximum allowable tensile stresses at points 2 and 3

Group 1

1300 N/mm 2 (188,000 psi)

Group 2

1250 N/mm 2 (181,000 psi)

Group 3

1200 N/mm 2 (174,000 psi)

Fatigue Life of Disc Springs.—Fatigue life is measured in terms of the maximum num­ber of cycles that dynamically loaded disc springs can sustain prior to failure. Dynamically loaded disc springs are divided into two groups: disc springs with unlimited fatigue life, which exceeds 2 3 106 cycles without failure, and disc springs with limited fatigue life between 104 cycles and less than 2 3 106 cycles.

Typically, fatigue life is estimated from three diagrams, each representing one of the three Groups of disc springs (Fig. 10, Fig. 11, and Fig. 12). Fatigue life is found at the intersection of the vertical line representing minimum tensile stress σmin with the horizontal line, which represents maximum tensile stress σmax. The point of intersection of these two lines defines fatigue life expressed in number of cycles N that can be sustained prior to failure. Example: For Group 2 springs in Fig. 11, the intersection point of the σmin = 500 N/mm 2 line with the σmax = 1200 N/mm 2 line, is located on the N = 105 cycles line. The estimated fatigue life is 105 cycles.

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Machinery's Handbook, 31st Edition Disc Spring Fatigue Life

359

1400

A

B

C

Maximun Tensile Stress (N /mm2)

1200

1000

800

600

Number of Loading Cycles 400

A B C

200

0

0

200

400

600

800

1000

1200

100,000 500,000 2,000,000

1400

Minimum Tensile Stress (N / mm2)

Fig. 10. Group 1 Diagram for Estimating Fatigue Life of Disc Springs (0.2  t < 1.25 mm) 1400

A

B

C

Maximun Tensile Stress (N /mm2)

1200

1000

800

600

Number of Loading Cycles 400

A B C

200

0

0

200

400

600

800

1000

1200

100,000 500,000 2,000,000

1400

Minimum Tensile Stress (N / mm2)

Fig. 11. Group 2 Diagram for Estimating Fatigue Life of Disc Springs (1.25  t  6 mm) 1400

A

Maximun Tensile Stress (N /mm2)

1200

B

C

1000

800

600

Number of Loading Cycles 400

A B C

200

0

0

200

400

600

800

1000

1200

100,000 500,000 2,000,000

1400

Minimum Tensile Stress (N / mm2)

Fig. 12. Group 3 Diagram for Estimating Fatigue Life of Disc Springs (6 < t  16 mm)

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360

Machinery's Handbook, 31st Edition Disc Spring Fatigue Life

When the intersection points of the minimum and maximum stress lines fall inside the areas of each cycle line, only the approximate fatigue life can be estimated by extrapolating the distance from the point of intersection to the nearest cycle line. The extrapolation can­not provide accurate values of fatigue life, because the distance between the cycle lines is expressed in logarithmic scale, and the distance between tensile strength values is expressed in linear scale (Figs. 10, 11, and 12), therefore linear-to-logarithmic scales ratio is not applicable.

When intersection points of minimum and maximum stress lines fall outside the cycle lines area, especially outside the N = 105 cycles line, the fatigue life cannot be estimated.

Thus, the use of the fatigue life diagrams should be limited to such cases when the mini­ mum and maximum tensile stress lines intersect exactly with each of the cycle lines. To calculate fatigue life of disc springs without the diagrams, the following equations developed by the author can be used.

Disc Springs in Group 1

N = 10 10.29085532 – 0.00542096 (σmax – 0.5σmin)

(20)

Disc Springs in Group 2

N = 10

10.10734911 – 0.00537616 (σmax – 0.5σmin)

(21)

Disc Springs in Group 3

N = 10 13.23985664 – 0.01084192 (σmax – 0.5σmin)

(22)

As can be seen from Equations (20), (21), and (22), the maximum and minimum tensile stress range affects the fatigue life of disc springs. Since tensile stresses at Points 2 and 3 have different values, see Equations (16) and (17), it is necessary to determine at which critical point the minimum and maximum stresses should be used for calculating fatigue life. The general method is based on the diagram, Fig. 9 from which Point 2 or Point 3 can be found in relationship with disc spring outside-to-inside diameters ratio D/d and disc spring cone height-to-thickness ratio h/r. This method requires intermediate calculations of D/d and h/t ratios and is applicable only to disc springs without contact surfaces. The method is not valid for Group 3 disc springs or for disc springs in Group 2 that have contact surfaces and reduced thickness. A simple and accurate method, that is valid for all disc springs, is based on the following statements: if (σ2 max − 0.5 σ2 min) > (σ3 max − 0.5 σ3 min), then Point 2 is used, otherwise if (σ3 max − 0.5 σ3 min) > (σ2 max − 0.5 σ2 min), then Point 3 is used

The maximum and minimum tensile stress range for disc springs in Groups 1, 2, and 3 is found from the following equations. For disc springs in Group 1:

σmax − 0.5σmin = For disc springs in Group 2:

σmax − 0.5σmin = For disc springs in Group 3:

σmax − 0.5σmin =

10.29085532 − log N 0.00542096

(23)

10.10734911 − log N 0.00537616

(24)

13.23985664 − log N 0.01084192

(25)

Thus, Equations (23), (24), and (25) can be used to design any spring stack that provides required fatigue life. The following example illustrates how a maximum-minimum stress range is calculated in relationship with fatigue life of a given disc spring stack.

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Machinery's Handbook, 31st Edition DISC SPRING EXAMPLES

361

Example: A dynamically loaded stack, which utilizes disc springs in Group 2, must have the fatigue life of 5 3 105 cycles. The maximum allowable tensile stress at Points 2 or 3 is 1250 N/mm 2. Find the minimum tensile stress value to sustain N = 5 3 105 cycles. Solution: Substitution of σmax = 1250 and N = 5 3 105 in Equation (24) gives:

1250 − 0.5σmin = from which σmin =

10.10734911 − log ^5 # 105h 10.10734911 − 5.69897 = = 820 0.00537616 0.00537616

1250 − 820 = 860 N/mm 2 ^124, 700 psih 0.5

Recommended Dimensional Characteristics of Disc Springs.—Dimensions of disc springs play a very important role in their performance. It is imperative to check selected disc springs for dimensional ratios, that should fall within the following ranges: 1) Diameters ratio, δ = D∕d = 1.7 to 2.5. 2) Cone height-to-thickness ratio, h∕t = 0.4 to 1.3. 3) Outside diameter-to-thickness ratio, D∕t = 18 to 40.

Small values of δ correspond with small values of the other two ratios. The h∕t ratio deter­ mines the shape of force-deflection characteristic graphs, that may be nearly linear or strongly curved. If h∕t = 0.4 the graph is almost linear during deflection of a disc spring up to its flat position. If h∕t = 1.6 the graph is strongly curved and its maximum point is at 75 percent deflection. Disc spring deflection from 75 to 100 percent slightly reduces spring force. Within the h∕t = 0.4 − 1.3 range, disc spring forces increase with the increase in deflection and reach maximum values at 100 percent deflection. In a stack of disc springs with a ratio h∕t > 1.3 deflec­tion of individual springs may be unequal, and only one disc spring should be used if pos­sible. Example Applications of Disc Springs Example 1, Disc Springs in Group 2 (no contact surfaces): A mechanical device that works under dynamic loads must sustain a minimum of 1,000,000 cycles. The applied load varies from its minimum to maximum value every 30 seconds. The maximum load is approximately 20,000N (4,500 lbf). A 40-mm diameter guide rod is a receptacle for the disc springs. The rod is located inside a hollow cylinder. Deflection of the disc springs under minimum load should not exceed 5.5 mm (0.217 inch) including a 20 percent pre­load deflection. Under maximum load, the deflection is limited to 8 mm (0.315 inch) max­imum. Available space for the disc spring stack inside the cylinder is 35 to 40 mm (1.38 to 1.57 inch) in length and 80 to 85 mm (3.15 to 3.54 inch) in diameter. Select the disc spring catalog item, determine the number of springs in the stack, the spring forces, the stresses at minimum and maximum deflection, and actual disc spring fatigue life. Solution: 1) Disc spring standard inside diameter is 41 mm (1.61 inch) to fit the guide rod. The outside standard diameter is 80 mm (3.15 in) to fit the cylinder inside diameter. Disc springs with such diameters are available in various thickness: 2.25, 3.0, 4.0, and 5.0 mm (0.089, 0.118, 0.157, and 0.197 inch). The 2.25- and 3.0-mm thick springs do not fit the applied loads since the maximum force values for disc springs with such thickness are 7,200N and 13,400N (1,600 lbf and 3,000 lbf) respectively. A 5.0-mm thick disc spring should not be used because its D∕t ratio, 80∕5 = 16, is less than 18 and is considered as unfavor­ able. Disc spring selection is narrowed to an 80-41-4 catalog item. 2) Checking 80 - 41 - 4 disc spring for dimensional ratios: . δ = D∕d = 80∕41 = 1.95   h∕t = 2 2∕ 4 = 0.55   D∕t = 80∕4 = 20 Because the dimensional ratios are favorable, the 80-41-4 disc springs are selected.

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362

3) The number of springs in the stack is found from Equation (1): n = L 0 / (t + h) = 40/(4 + 2.2) = 40∕6.2 = 6.45.

Rounding n to the nearest integer gives n = 6. The actual length of unloaded spring stack is L 0 = 6.2 3 6 = 37.2 mm (1.465 inch) and it satisfies the L 0 < 40 mm condition.

4) Calculating the cone angle α from Equation (7) and actual outside diameter Da from Equation (9) gives:

2 # 2.2 α = atan a 80 − 41 k = atan ^0.11282h = 6.4 °

1 Da = 80 − 2 # 4 asin 6 atan ^0.11282h@ + 6 cos 6atan ^0.11282h@k Da = 77.78 mm ( 3. 062 in )

5) Calculating constant K1 from Equation (11): D δ = d = 1.95122

a 1.95122 − 1 k

2

1.95122 = 0.6841 K1 = 1.95122 + 1 2 E π · ; 1.95122 − 1 − ln ^1.95122h 6) Calculating minimum and maximum forces, Fmin and Fmax from Equation (10):

Based on the design requirements, the disc spring stack is deflecting by 5.5 mm (0.217 in) under minimum load, and each individual disc spring is deflecting by 5.5  /  6 @ 0.92 mm (0.036 in). A single disc spring deflection smin = 0.9 mm (0.035 in) is used to calculate Fmin . Under maximum load, the disc spring stack is permitted maximum deflection of 8 mm (0.315 in), and each individual disc spring deflects by 8  /  6 @ 1.33 mm (0.0524 in). A disc spring deflection smax = 1.32 mm (0.052 in) will be used to calculate Fmax . If disc springs are made of AISI 6150 alloy steel, then modulus of elasticity E = 206,000 N/mm 2 (30 3 106 psi) and Poisson’s ratio μ = 0.3.

Fmin =

4 · 206000

^1 − 0.3 2h^0.6841h^77.78h2

Fmin = 14390N ( 3235 lbf) Fmax =

4 · 206000

^1 − 0.3 2h^0.6841h^77.78h2

:a 2.2 − 0.9 k · ^2.2 − 0.9 h · 4 + 4 3D 0.9 2

:a 2.2 − 1.32 k · ^2.2 − 1.32h · 4 + 4 3D 1.32 2

Fmax = 20050N ( 4510 lbf)

7) Calculating constant K2 , Equation (18):

D 80 δ = d = 41 = 1.95122

K2 =

6a

1.95122 − 1 δ −1 − 1 k 6 c ln ^1.95122h − 1 m lnδ = = 1.2086 π · lnδ π · ln ^1.95122h

8) Calculating constant K3 (Equation (19)):

K3 =

3 · ^δ − 1h 3 · ^1.95122 − 1h = = 1.3589 π · lnδ π · ln ^1.95122h

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363

9) Compressive stress σ0 at point 0 due to maximum deflection, Equation (14): 4E · t · s · K4 4 · 206000 · 4 · 1.32 · 1 3 = – π3 · σ0 = – π · (1 – 0.32) · 0.6841 · 77.782 (1 – µ 2 ) · K · D2 1

a

σ 0 = 1103 N/mm2 = 160000 psi

Because the compressive stress at point 0 does not exceed 1600 N/mm 2, its current value satisfies the design requirement. 10) Tensile stress σ2 at point 2 due to minimum deflection s = 0.9 mm, Equation (16): s 4E · K4 · s · 8 K 3 · t – K2 · K4 · ` h – 2 j B = σ2 min = (1 – µ 2 ) · K1 · D2a

4 · 206000 · 1 · 0.9 : 1.3589 · 4 – 1.2086 · 1 · a 2.2 – 02.9 k D (1 – 0.32) · 0.6841 · 77.782

= 654 N / mm2

11) Tensile stress σ2 at point 2 due to maximum deflection s = 1.32 mm, Equation (16): s 4E · K4 · s · 8 K 3 · t – K2 · K4 · ` h – 2 j B = σ2 max = (1 – µ 2 ) · K1 · D2a

4 · 206000 · 1 · 1.32 · : 1.3589 · 4 – 1.2086) · 1 · a 2.2 – 1.232 k D (1 – 0.32) · 0.6841 · 77.782

= 1032N / mm2

Thus, σ2 min = 654 N/mm 2 (94,850 psi) and σ2 max = 1032 N/mm 2 (149,700 psi). 12) Tensile stress σ3 at point 3 due to minimum deflection s = 0.9 mm, Equation (17): σ 3min =

s 4E · K4 · s · 8 K 4 · (2K3 – K2) · `h – 2 j + K 3 · t B (1 – µ 2 ) · K1 · D2a · δ

=

4 · 206000 · 1 · 0.9 · :1 · (2 · 1.3589 – 1.2086) · a 2.2 – 02.9 k + 1.3589 · 4D (1 – 0.32) · 0.6841 · 77.782 · 1.95122

= 815 N / mm2

13) Tensile stress σ3 at point 3 due to maximum deflection s = 1.32 mm, Equation (17): σ 3max =

s 4E · K4 · s · 8 K 4 · (2K3 – K2) · `h – 2 j + K 3 · t B (1 – µ 2 ) · K1 · D2a · δ

=

4 · 206000 · 1 · 1.32 · :1 · (2 · 1.3589 – 1.2086) · a 2.2 – 1.232 k + 1.3589 · 4D (1 – 0.32) · 0.6841 · 77.782 · 1.95122

= 1149N / mm2

Thus, σ3 min = 815 N/mm 2 (118,200 psi) and σ3 max = 1149 N/mm 2 (166,600 psi). 14) Functional tensile stress range at critical points 2 and 3. Point 2: σ2 max − 0.5σ2 min = 1032 − 0.5 3 654 = 705 N/mm 2 Point 3: σ3 max − 0.5σ3 min = 1149 − 0.5 3 815 = 741.5 N/mm 2 Because σ3 max − 0.5σ3 min > σ2 max − 0.5 σ2 min , the tensile stresses at point 3 are used for fatigue life calculations.

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Machinery's Handbook, 31st Edition DISC SPRING EXAMPLES

15) Fatigue life of selected disc springs, Equation (21): N = 10[10.10734911 – 0.00537616 (1149 – 0.5 3 815)] = 1010.10734911 – 3.98642264 = 10 6.12092647 N = 1,321,000 cycles. Thus, the calculated actual fatigue life exceeds required minimum number of cycles by 31 percent. In conclusion, the six 80-41-4 disc springs arranged in series stacking, satisfy the require­ments and will provide a 32 percent longer fatigue life than required by the design criteria. Example 2: A company wishes to use Group 3 disc springs with contact surfaces on cou­ plings to absorb bumping impacts between railway cars. Given: D =  200 mm, disc spring outside diameter d =  102 mm, disc spring inside diameter t =  14 mm, spring standard thickness t  ′ =  13.1 mm, spring reduced thickness h =  4.2 mm, cone height of unloaded spring n =  22, number of springs in series stacking Si =  33.9 mm, initial deflection of the pack Sa =  36.0 mm, additional deflection of the pack

Find the fatigue life in cycles and determine if the selected springs are suitable for the application. The calculations are performed in the following sequence: 1) Determine the minimum smin and maximum smax deflections of a single disc spring: ^ S + S h ^33.9 + 36h smax = i n a = = 3.18 mm 22 Si 33.9 smin = n = = 1.54 mm 22

2) Use Equations (16) and (17) to calculate tensile stresses σ2 and σ3 at smin and smax deflections: σ2min = 674 N/mm 2, σ2max = 1513 N/mm 2, σ3min = 707 N/mm 2, σ3max = 1379 N/mm 2 3) Determine critical stress points: σ2max − 0.5σ2min = 1513 − 0.5 3 674 = 1176 N/mm 2 σ3max − 0.5σ3min = 1379 − 0.5 3 707 = 1025.5 N/mm 2 Because (σ2max − 0.5σ2min) > (σ3max − 0.5σ3min), then tensile stresses at point 2 are used to calculate fatigue life. 4) Fatigue life N is calculated using Equation (22): N = 10 [13.23985664 − (0.01084192  3 1176)] = 10 0.49 = 3 cycles The selected disc springs at the above-mentioned minimum and maximum deflection values will not sustain any number of cycles. It is imperative to check the selected disc springs for dimensional ratios: Outside-to-inside diameters ratio, 200/102 = 1.96; within recommended range. Cone height-to-thickness ratio is 4.2/13.1 = 0.3; out of range, the minimum ratio is 0.4. Outside diameter-to-thickness ratio is 200/13.1 = 15; out of range, the minimum ratio is 18. Thus, only one of the dimensional ratios satisfies the requirements for the best disc spring performance.

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Machinery's Handbook, 31st Edition TABLE OF CONTENTS

PROPERTIES, TREATMENT, AND TESTING OF MATERIALS THE ELEMENTS, HEAT, MASS, AND WEIGHT

(Continued)

368 The Elements 369 Latent Heat 369 Specific Heat 370 Heat Loss from Steam Pipes 371 Thermal Properties 375 Adjusting Length for Temperature 375 Changes Due to Temperature 377 Properties of Mass and Weight 377 Density 377 Specific Gravity 379 Weight of Natural Piles 381 Molecular Weight 381 Mole or Mol 381 Air 381 Alligation 381 Earth or Soil Weight

382 Properties of Wood 382 Mechanical Properties 383 Weight 383 Effect of Pressure Treatment 384 Density 384 Machinability 386 Tabulated Properties of 386 Ceramics 387 Plastics 388 Investment Casting Alloys 390 Powdered Metal Alloys 391 Elastic Properties of Materials 392 Tensile Strength of Spring Wire 392 Effect of Temperature on Elasticity and Strength of Metals STANDARD STEELS

396 397 398 399

Property, Composition, Application Standard Steel Classification Numbering Systems Identifying Metals Standard Steel Numbering System Binary, Ternary, and Quarternary Alloys Damascus Steel AISI and SAE Numbers for Steels AISI and SAE Designations Composition of Carbon Steels

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Composition of Alloy Steels Composition of Stainless Steels Thermal Treatments of Steel Applications of Steels Carbon Steels Carburizing GradeAlloy Steels Hardenable Grade Alloy Steels Characteristics of Stainless Steels Chromium-Nickel Austenitic Steels Stainless Chromium Irons and Steels High-Strength, Low-Alloy Steels Mechanical Properties of Steels

430 430 433 434 436 440 443 443 445 446 446 448 448 449 449

Overview Properties Common Tool Faults and Failures Effect of Alloying Elements Classification Selection High-Speed Tool Steels Molybdenum-Type Tungsten-Type Hot-Work Tool Steels Chromium Types Tungsten-Types Molybdenum-Types Cold-Work Tool Steels High-Carbon, High-Chromium Types Oil-Hardening Types Medium-Alloy, Air-Hardening Types Shock-Resisting Tool Steels Mold Steels Special-Purpose Tool Steels Water-Hardening Tool Steels Mill Production Forms Tolerances of Dimensions Allowances for Machining Decarburization Limits Advances in Tool Steel-Making

TOOL STEELS

PROPERTIES OF WOOD, CERAMICS, PLASTICS, METALS

393 393 395 396 396 396

STANDARD STEELS

401 403 404 405 407 410 411 412 415 416 417 418

449 450 452 452 454 454 456 457 457 457 457 

HARDENING, TEMPERING, AND ANNEALING

458 Heat Treatment of Standard Steels 458 Heat Treating Definitions 462 Hardness and Hardenability

365

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Machinery's Handbook, 31st Edition TABLE OF CONTENTS

PROPERTIES, TREATMENT, AND TESTING OF MATERIALS HARDENING, TEMPERING, AND ANNEALING

(Continued)

464

465 466 466 467 468 469 469 470 470 471 471 472 472 472 472 473 474 474 475 476 476 477 478 478 478 478 479 479 479 480 481 481 481 481 482 482 482 483 483 483 483 483

Surface Hardening Treatment (Case Hardening) Fully Annealed Carbon Steel Slow Cooling Rapid Cooling or Quenching Steel Heat Treating Furnaces Physical Properties of Heat Treated Steels Hardening Basic Steps Critical Points of Decalescence and Recalescence Hardening Temperatures Heating Steel in Liquid Baths Salt Baths Lead Bath Defects in Hardening Scale on Hardened Steel Hardening or Quenching Baths In Water In a Molten Salt Bath Tanks for Quenching Baths Interrupted Quenching Laser and Electron-Beam Surface Hardening Tempering Temperatures In Oil In Salt Baths In a Lead Bath To Prevent Lead from Sticking In Sand Double Tempering Annealing, Spheroidizing, and Normalizing Annealing Spheroidizing Normalizing Case Hardening Carburization Pack-Hardening Cyanide Hardening Nitriding Liquid Carburizing Gas Carburizing Vacuum Carburizing Carburizing Steels Clean after Case Hardening

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HARDENING, TEMPERING, AND ANNEALING

(Continued)

484 484 485 485 485 486

486 487 489 493 493 496 498 498 499 499 499 501 501 502 503 503 503 503 504 504 504 504 506 506 506

Flame Hardening Induction Hardening Induction Heating Equipment Depth of Heat Penetration Steel for Induction Hardening Through Hardening, Annealing, and Normalizing by Induction Induction Surface Hardening Typical Heat Treatments Metallography Heat Treating High-Speed Steels Tungsten High-Speed Steel Molybdenum High-Speed Steels Nitriding High-Speed Steel Tools Equipment for Hardening Heating for Forging Subzero Treatment of Steel Resulting Changes Carburized Parts Application to High-Speed Steel Testing the Hardness of Metals Brinell Hardness Test Rockwell Hardness Test Shore’s Scleroscope Vickers Hardness Test Knoop Hardness Numbers Monotron Hardness Indicator Keep’s Test Comparison of Hardness Scales Turner’s Sclerometer Mohs’s Hardness Scale Durometer Tests

NONFERROUS ALLOYS 510 Strength Data for Nonferrous Metals 511 Coppers and Copper Alloys 511 Cast 516 Wrought 525 Copper-Silicon and CopperBeryllium Alloys 527 Aluminum and Aluminum Alloys 528 Temper Designations 531 Alloy Designation Systems 539 Heat-Treatability 540 Clad Aluminum Alloys 540 Aluminum Alloys, Wrought, Sheet 540 Principal Alloy Groups

366

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Machinery's Handbook, 31st Edition TABLE OF CONTENTS

PROPERTIES, TREATMENT, AND TESTING OF MATERIALS NONFERROUS ALLOYS

(Continued)

(Continued)



541 542 545 545

569 572 576 578 578 578 580 582 582 583 584 584 586 586 587 587 592 594 595 600 601 606 609 610 611 611 612 613 614 614 615 615 615

Type Metal Magnesium Alloys Nickel and Nickel Alloys Titanium and Titanium Alloys CORROSION



548 Types and Methods of Prevention 548 Uniform (General) Corrosion 548 Chemical Corrosion 548 Dry and High-Temperature Corrosion 548 Electrochemical Corrosion Galvanic Corrosion 549 553 Effects of Corrosion Crevice Corrosion 553 553 Pitting Corrosion Microbiologically Influenced 553 Corrosion 553 Intergranular Corrosion Selective Leaching (Dealloying) 553 553 Stress Corrosion Cracking 554 Tribocorrosion 554 Hydrogen Embrittlement 554 Corrosion Fatigue PLASTICS



555 Properties of Plastics 555 Plastics Families 556 Plastics Materials 558 Application and Physical Properties 563 Stress and Strain

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PLASTICS

Strength and Modulus Thermal Properties Electrical Properties Mechanical Properties Design Analysis Structural Analysis Design Stresses Thermal Stresses Designing for Stiffness Manufacture of Plastics Products Sheet Thermoforming Blow Molding Processing of Thermosets Polyurethanes Reinforced Plastics Injection Molding Load-Bearing Parts Melt Flow in the Mold Design for Assembly Assembly with Fasteners Machining Plastics Plastics Gears Polishing Plastics Finishing and Decorating Plastics Development of Plastics Prototypes Additive Manufacturing Plastics Stereolithography Selective Laser Sintering Fused Deposition Modeling Binder Jetting Material Jetting Multijet Fusion Additive Manufacturing Considerations for Tooling and Parts

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368

Machinery's Handbook, 31st Edition Properties, Treatment, and Testing of Materials THE ELEMENTS, HEAT, MASS, AND WEIGHT

Table 1. The Elements — Symbols, Atomic Numbers and Weights, Melting Points Name of Element

Symbol

Actinium Aluminum Americium Antimony Argon Arsenic Astatine Barium Berkelium Beryllium Bismuth Boron Bromine Cadmium Calcium Californium Carbon Cerium Cesium Chlorine Chromium Cobalt Copper Curium Dysprosium Einsteinium Erbium Europium Fermium Fluorine Francium Gadolinium Gallium Germanium Gold Hafnium Helium Holmium Hydrogen Indium Iodine Iridium Iron Krypton Lanthanum Lawrencium Lead Lithium Lutetium Magnesium Manganese Mendelevium Mercury Molybdenum Neodymium

Ac Al Am Sb A As At Ba Bk Be Bi B Br Cd Ca Cf C Ce Cs Cl Cr Co Cu Cm Dy Es Er Eu Fm F Fr Gd Ga Ge Au Hf He Ho H In I Ir Fe Kr La Lw Pb Li Lu Mg Mn Md Hg Mo Nd

Atomic Num. Weight 89 13 95 51 18 33 85 56 97 4 83 5 35 48 20 98 6 58 55 17 24 27 29 96 66 99 68 63 100 9 87 64 31 32 79 72 2 67 1 49 53 77 26 36 57 103 82 3 71 12 25 101 80 42 60

227.028 26.9815 (243) 121.75 39.948 74.9216 (210) 137.33 (247) 9.01218 208.980 10.81 79.904 112.41 40.08 (251) 12.011 140.12 132.9054 35.453 51.996 58.9332 63.546 (247) 162.5 (252) 167.26 151.96 (257) 18.9984 (223) 157.25 69.72 72.59 196.967 178.49 4.00260 164.930 1.00794 114.82 126.905 192.22 55.847 83.80 138.906 (260) 207.2 6.941 174.967 24.305 54.9380 (258) 200.59 95.94 144.24

a At 28 atm (2.837 MPa).

Melting Point, °C

Name of Element

Symbol

1050 660.37 994 ± 4 630.74 −189.2 817a 302 725 … 1278 ± 5 271.3 2079 −7.2 320.9 839 ± 2 … 3652c 798 ± 2 28.4 ± 0.01 −100.98 1857 ± 20 1495 1083.4 ± 0.2 1340 ± 40 1409 … 1522 822 ± 5 … −219.62 27b 1311 ± 1 29.78 937.4 1064.434 2227 ± 20 −272.2d 1470 −259.14 156.61 113.5 2410 1535 −156.6 920 ± 5 … 327.502 180.54 1656 ± 5 648.8 ± 0.5 1244 ± 2 … −38.87 2617 1010

Neon Neptunium Nickel Niobium Nitrogen Nobelium Osmium Oxygen Palladium Phosphorus Platinum Plutonium Polonium Potassium Praseodymium Promethium Protactinium Radium Radon Rhenium Rhodium Rubidium Ruthenium Samarium Scandium Selenium Silicon Silver Sodium Strontium Sulfur Tantalum Technetium Tellurium Terbium Thallium Thorium Thulium Tin Titanium Tungsten Unnilhexium Unnilnonium Unniloctium Unnilpentium Unnilquadium Unnilseptium Uranium Vanadium Xenon Ytterbium Yttrium Zinc Zirconium

Ne Np Ni Nb N No Os O Pd P Pt Pu Po K Pr Pm Pa Ra Rn Re Rh Rb Ru Sm Sc Se Si Ag Na Sr S Ta Tc Te Tb Tl Th Tm Sn Ti W Unh Unn Uno Unp Unq Uns U V Xe Yb Y Zn Zr

Num. 10 93 28 41 7 102 76 8 46 15 78 94 84 19 59 61 91 88 86 75 45 37 44 62 21 34 14 47 11 38 16 73 43 52 65 81 90 69 50 22 74 106 109 108 105 104 107 92 23 54 70 39 30 40

Atomic Weight

20.1179 237.048 58.69 92.9064 14.0067 (259) 190.2 15.9994 106.42 30.9738 195.08 (244) (209) 39.0938 140.908 (145) 231.0359 226.025 (222) 186.207 102.906 85.4678 101.07 150.36 44.9559 78.96 28.0855 107.868 22.9898 87.62 32.06 180.9479 (98) 127.60 158.925 204.383 232.038 168.934 118.71 47.88 183.85 (266) (266) (265) (262) (261) (261) 238.029 50.9415 131.29 173.04 88.9059 65.39 91.224

Melting Point, °C −248.67 640 ± 1 1453 2468 ± 10 −209.86 … 3045 ± 30 −218.4 1554 44.1 1772 641 254 63.25 931 ± 4 1080b 1600 700 −71 3180 1965 ± 3 38.89 2310 1072 ± 5 1539 217 1410 961.93 97.81 ± 0.03 769 112.8 2996 2172 449.5 ± 0.3 1360 ± 4 303.5 1750 1545 ± 15 231.9681 1660 ± 10 3410 ± 20 … … … … … … 1132 ± 0.8 1890 ± 10 −111.9 824 ± 5 1523 ± 8 419.58 1852 ± 2

b Approximate. c Sublimates.

d At 26 atm (2.635 MPa).

Notes: Values in parentheses are atomic weights of the most stable known isotopes. Melting points are at standard pressure except as noted.

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369

Heat and Combustion Related Properties

Latent Heat.—When a body changes from the solid to the liquid state or from the liquid to the gaseous state, a certain amount of heat is used to accomplish this change. This heat does not raise the temperature of the body and is called latent heat. When the body changes again from the gaseous to the liquid, or from the liquid to the solid state, it gives out this quantity of heat. The latent heat of fusion is the heat supplied to a solid body at the melting point; this heat is absorbed by the body although its temperature remains nearly stationary during the whole operation of melting. The latent heat of evaporation is the heat that must be supplied to a liquid at the boiling point to transform the liquid into a vapor. The latent heat is generally given in British thermal units per pound, or kilojoules per kilogram. The latent heat of evaporation of water is 966.6 Btu/pound, or 2248 kJ/kg. This means that it takes 966.6 Btu to evaporate 1 pound, or 2248 kJ to evaporate 1 kilogram, of water after it has been raised to the boiling point, 212°F or 100°C. When a body changes from the solid to the gaseous state without passing through the liq­uid stage, as solid carbon dioxide does, the process is called sublimation. Table 2. Latent Heat of Fusion

Substance

Btu per Pound

Liquid

Btu per Pound

Bismuth Beeswax Cast iron, gray Cast iron, white

22.75 76.14 41.40 59.40

kJ/kg

52.92 177.10 96.30 138.16

Substance

Paraffine Phosphorus Lead Silver

Btu per Pound

63.27 9.06 10.00 37.92

kJ/kg

147.17   21.07   23.26 88.20

Substance

Sulfur Tin Zinc Ice

Btu per Pound

16.86 25.65 50.63 144.00

kJ/kg

39.22 59.66 117.77 334.94

Table 3. Latent Heat of Evaporation Alcohol, ethyl Alcohol, methyl Ammonia

371.0 481.0 529.0

kJ/kg 863 1119 1230

Liquid

Carbon bisulfide Ether Sulfur dioxide

Btu per Pound 160.0 162.8 164.0

kJ/kg 372 379 381

Liquid

Turpentine Water

Btu per Pound 133.0 966.6

kJ/kg 309 2248

Table 4. Boiling Points of Various Substances at Atmospheric Pressure Substance

Aniline Alcohol Ammonia Benzine Bromine Carbon bisulfide

Boiling Point °F

363 173 −28 176 145 118

°C

183.9 78.3 −33.3 80.0 62.8 47.8

Substance

Chloroform Ether Linseed oil Mercury Napthaline Nitric acid Oil of turpentine

Boiling Point °F

140 100 597 676 428 248 315

°C

60.0 37.8 313.9 357.8 220.0 120.0 157.2

Substance

Saturated brine Sulfur Sulfuric acid Water, pure Water, sea Wood alcohol

Boiling Point °F

226 833 590 212 213.2 150

°C

107.8 445.0 310.0 100.0 100.7 65.6

Specific Heat.—The specific heat of a substance is the ratio of the heat required to raise the temperature of a certain weight of the given substance 1°F to the heat required to raise the temperature of the same weight of water 1°F. As the specific heat is not constant at all temperatures, it is generally assumed that it is determined by raising the temperature from 62 to 63°F. For most substances, however, specific heat is practically constant for tempera­tures up to 212°F. In metric units, specific heat is defined as the ratio of the heat needed to raise the tempera­ture of a mass by 1°C to the heat needed to raise the temperature of the same mass of water by 1°C. In the metric system, heat is measured in joules (J), mass is in grams (g), and mea­surements are usually taken at 15°C. Because specific heat is a dimensionless ratio, the values given in Table 5 and Table 6 that fol­low are valid in both the US system and the metric system.

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Machinery's Handbook, 31st Edition Heat Table 5. Average Specific Heats of Various Substances

Substance Alcohol (absolute) Alcohol (density 0.8) Aluminum Antimony Benzine Brass Brickwork Cadmium Carbon Charcoal Chalk Coal Coke Copper, 32° to 212°F (0–100°C) Copper, 32° to 572°F (0–100°C) Corundum Ether Fusel oil Glass Gold Graphite Ice Iron, cast Iron, wrought, 32° to 212°F (0–100°C)    32° to 392°F (0–200°C)    32° to 572°F (0–300°C)    32° to 662°F (0–350°C) Iron, at high temperatures:    1382° to 1832°F (750–1000°C)    1750° to 1840°F (954–1004°C)    1920° to 2190°F (1049–1199°C) Kerosene

Specific Heat 0.700 0.622 0.214 0.051 0.450 0.094 0.200 0.057 0.204 0.200 0.215 0.240 0.203 0.094 0.101 0.198 0.503 0.564 0.194 0.031 0.201 0.504 0.130 0.110 0.115 0.122 0.126 0.213 0.218 0.199 0.500

Substance

Lead Lead (fluid) Limestone Magnesia Marble Masonry, brick Mercury Naphtha Nickel Oil, machine Oil, olive Paper Phosphorus Platinum Quartz Sand Silica Silver Soda Steel, high carbon Steel, mild Stone (generally) Sulfur Sulfuric acid Tin (solid) Tin (fluid) Turpentine Water Wood, fir Wood, oak Wood, pine Zinc

Specific Heat 0.031 0.037 0.217 0.222 0.210 0.200 0.033 0.310 0.109 0.400 0.350 0.32 0.189 0.032 0.188 0.195 0.191 0.056 0.231 0.117 0.116 0.200 0.178 0.330 0.056 0.064 0.472 1.000 0.650 0.570 0.467 0.095

Table 6. Specific Heat of Gases Gas Acetic acid Air Alcohol Ammonia Carbonic acid Carbonic oxide Chlorine

Constant Pressure 0.412 0.238 0.453 0.508 0.217 0.245 0.121

Constant Volume … 0.168 0.399 0.399 0.171 0.176 …

Gas Chloroform Ethylene Hydrogen Nitrogen Oxygen Steam

Constant Pressure 0.157 0.404 3.409 0.244 0.217 0.480

Constant Volume … 0.332 2.412 0.173 0.155 0.346

Heat Loss from Uncovered Steam Pipes.—The loss of heat from a bare steam or hotwater pipe varies with the temperature difference between the inside of the pipe and the sur­rounding air. The loss is 2.15 Btu per hour, per square foot of pipe surface, per degree F of temperature difference when the latter is 100 degrees; for a difference of 200 degrees, the loss is 2.66 Btu; for 300 degrees, 3.26 Btu; for 400 degrees, 4.03 Btu; for 500 degrees, 5.18 Btu. Thus, if the pipe area is 1.18 square feet per foot of length, and the temperature differ­ence 300°F, the loss per hour per foot of length = 1.18 3 300 3 3.26 = 1154 Btu.

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371

Table 7. Values of Thermal Conductivity (k) and of Conductance (C) of Common Building and Insulating Materials Type of Material

BUILDING Batt:   Mineral Fiber   Mineral Fiber

  Mineral Fiber   Mineral Fiber   Mineral Fiber Block:   Cinder  Cinder  Cinder Block:   Concrete  Concrete  Concrete Board:   Asbestos Cement  Plaster

  Plywood Brick:   Common   Face Concrete (poured) Floor:   Wood Subfloor

  Hardwood Finish   Tile Glass:  Architectural Mortar:   Cement Plaster:   Sand

  Sand and Gypsum   Stucco Roofing:   Asphalt Roll   Shingle, asb. cem.   Shingle, asphalt   Shingle, wood

Thick­ ness, in.

k or C a

…

… 0.14

BUILDING (Continued) Siding:   Metalb

0.05 0.04 0.03 … 0.90 0.58 0.53 … 1.40 0.90 0.78 … 16.5

Stone: Lime or Sand Wall Tile:   Hollow Clay, 1-Cell   Hollow Clay, 2-Cell   Hollow Clay, 3-Cell   Hollow Gypsum INSULATING Blanket, Mineral Fiber:   Felt Rock or Slag   Glass   Textile

1.07 … 5.0 9.0 12.0 …

Board, Block and Pipe Insulation:   Amosite   Asbestos Paper   Glass or Slag (for Pipe)   Glass or Slag (for Pipe)

1.47 20.0 … 10.00 … 5.0 … 13.30

  Magnesia (85%)   Mineral Fiber   Polystyrene, Beaded   Polystyrene, Rigid   Rubber, Rigid Foam   Wood Felt Loose Fill:   Cellulose

2–23⁄4

3–31⁄2

31⁄2 –61⁄2 6–7 81⁄2 … 4 8 12 … 4 8 12 … 1⁄4 1⁄2 3⁄4

… 1 1 1 … 3⁄4 3⁄4

Avg. … … … 1 … 3⁄8

1⁄2

1 … Avg. Avg. Avg. Avg.

0.09

2.22

1.06

11.10 5.0 … 6.50 4.76 2.27 1.06

Type of Material

  Wood, Med. Density

Blanket, Hairfelt

  Glass, Cellular

  Mineral Fiber   Perlite   Silica Aerogel   Vermiculite Mineral Fiber Cement:   Clay Binder   Hydraulic Binder

Thick­ ness, in. … Avg. 7⁄16

k or C a

… 1.61

1.49

… 1 … 4 8 12 Avg.

… 12.50 … 0.9 0.54 0.40 0.7

… … … … …

… … … … …

… … … … … …

…

…

… … … … … … … …

… … … … … … …

Max. Temp.,°F

Density, lb per cu. ft.

k a

… …

… …

… …

…

…

…

… … … … … … …

… … … … … … …

… 400 1200 350 350

… 3 to 8 6 to 12 0.65 0.65

… 0.26 0.26c 0.33 0.31

… … … … … …

… … 1500 700 350 1000

… … 15 to 18 30 3 to 4 10 to 15

… … 0.32c 0.40c 0.23 0.33c

… … … … … … … …

600 100 170 170 150 180 … …

11 to 12 15 1 1.8 4.5 20 … 2.5 to 3

0.35c 0.29 0.28 0.25 0.22 0.31 … 0.27

…

…

… … … … … … …

… … … … … … …

180

800

… … … … … 1800 1200

10

9

2 to 5 5 to 8 7.6 7 to 8.2 … 24 to 30 30 to 40

0.29

0.40

0.28 0.37 0.17 0.47 … 0.49c 0.75c

a Units are in Btu/hr-ft 2 -°F. Where thickness is given as 1 inch, the value given is thermal conductivity (k); for other thicknesses the value given is thermal conductance (C). All values are for a test mean temperature of 75°F, except those designated with c, which are for 100°F. b Over hollowback sheathing. c Test mean temperature 100°F, see footnote a.

Source: American Society of Heating, Refrigerating and Air-Conditioning Engineers, Inc.: Hand­book of Fundamentals.

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Table 8. Typical Values of Coefficient of Linear Thermal Expansion for Thermoplastics and Other Commonly Used Materials Materiala

Liquid Crystal—GR Glass Steel

in/in/deg F 3 10−5 0.3 0.4 0.6

cm/cm/deg C 3 10−5 0.6 0.7 1.1

Concrete

0.8

1.4

Copper Bronze Brass Aluminum Polycarbonate—GR Nylon—GR TP polyester—GR Magnesium Zinc ABS—GR

0.9 1.0 1.0 1.2 1.2 1.3 1.4 1.4 1.7 1.7

1.6 1.8 1.8 2.2 2.2 2.3 2.5 2.5 3.1 3.1

in/in/deg F 3 10−5

Materiala

ABS—GR Polypropylene—GR Epoxy—GR Polyphenylene sulfide—GR Acetal—GR Epoxy Polycarbonate Acrylic ABS Nylon Acetal Polypropylene TP Polyester Polyethylene

1.7 1.8 2.0

cm/cm/deg C 3 10−5 3.1 3.2 3.6

2.0

3.6

2.2 3.0 3.6 3.8 4.0 4.5 4.8 4.8 6.9 7.2

4.0 5.4 6.5 6.8 7.2 8.1 8.5 8.6 12.4 13.0

a GR = Typical glass fiber-reinforced material. Other plastics materials shown are unfilled.

Table 9. Linear Expansion of Various Substances between 32 and 212°F Expansion of Volume = 3 3 Linear Expansion Linear Expansion

Substance

for 1°F

Brick Cement, Portland Concrete Ebonite Glass, thermometer Glass, hard Granite Marble, from to

0.0000030 0.0000060 0.0000080 0.0000428 0.0000050 0.0000040 0.0000044 0.0000031 0.0000079

for 1°C

0.0000054 0.0000108 0.0000144 0.0000770 0.0000090 0.0000072 0.0000079 0.0000056 0.0000142

Linear Expansion

Substance

for 1°F

Masonry, brick from    to Plaster Porcelain Quartz, from to Slate Sandstone Wood, pine

0.0000026 0.0000050 0.0000092 0.0000020 0.0000043 0.0000079 0.0000058 0.0000065 0.0000028

for 1°C

0.0000047 0.0000090 0.0000166 0.0000036 0.0000077 0.0000142 0.0000104 0.0000117 0.0000050

Table 10. Coefficients of Heat Transmission Metal

Aluminum Antimony Brass, yellow Brass, red Copper

Btu per Second 0.00203 0.00022 0.00142 0.00157 0.00404

Metal

German silver Iron Lead Mercury Steel, hard

Btu per Second 0.00050 0.00089 0.00045 0.00011 0.00034

Metal

Steel, soft Silver Tin Zinc …

Btu per Second 0.00062 0.00610 0.00084 0.00170 …

Heat transmitted, in British thermal units, per second, through metal 1 inch thick, per square inch of surface, for a temperature difference of 1°F.

Table 11. Coefficients of Heat Radiation

Surface

Cast-iron, new Cast-iron, rusted Copper, polished Glass Iron, ordinary Iron, sheet-, polished Oil

Btu per Hour 0.6480 0.6868 0.0327 0.5948 0.5662 0.0920 1.4800

Surface

Sawdust Sand, fine Silver, polished Tin, polished Tinned iron, polished Water …

Btu per Hour 0.7215 0.7400 0.0266 0.0439 0.0858 1.0853 …

Heat radiated, in British thermal units, per square foot of surface per hour, for a temperature differ­ence of 1° F.

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Table 12. Freezing Mixtures Temperature Change,°F From To 32 ±0 32 −0.4 32 −27 32 −44

Mixture

Common salt (NaCl), 1 part; snow, 3 parts Common salt (NaCl), 1 part; snow, 1 part Calcium chloride (CaCl2), 3 parts; snow, 2 parts

Calcium chloride (CaCl2), 2 parts; snow, 1 part

Sal ammoniac (NH4 Cl), 5 parts; saltpeter (KNO3), 5 parts; water, 16 parts

Sal ammoniac (NH4 Cl), 1 part; saltpeter (KNO3), 1 part; water, 1 part

50

+10

50 32

−11 +3 −35

46

Ammonium nitrate (NH4 NO3), 1 part; water, 1 part Potassium hydrate (KOH), 4 parts; snow, 3 parts

Temperature Change,°C From To 0 −17.8 0 −18 0 −32.8 0 −42.2 10

7.8

10 0

−12.2 −23.9 −16.1 −37.2

Ignition Temperatures.—The following temperatures are required to ignite the different substances specified: Phosphorus, transparent, 120°F (49°C); bisulfide of carbon, 300°F (149°C); gun cotton, 430°F (221°C); nitroglycerine, 490°F (254°C); phosphorus, amor ­phous, 500°F (260°C); rifle powder, 550°F (288°C); charcoal, 660°F (349°C); dry pine wood, 800°F (427°C); dry oak wood, 900°F (482°C). Table 13. Typical Thermal Properties of Various Metals

Material and Alloy Designation a

Melting Point, °F

Density, r lb/in3

g/cc

solidus

liquidus

Conduc­ tivity, k

Specific Heat, C

Coeff. of Expansion, a

Btu/ hr-ft-°F

Btu/lb/°F

min/in-°F

mm/m-°C

Aluminum Alloys 2011 2017 2024 3003 5052 5086 6061 7075

0.102 0.101 0.100 0.099 0.097 0.096 0.098 0.101

Manganese Bronze C11000 (Electrolytic tough pitch) C14500 (Free machining Cu) C17200, C17300 (Beryllium Cu) C18200 (Chromium Cu) C18700 (Leaded Cu) C22000 (Commercial bronze, 90%) C23000 (Red brass, 85%) C26000 (Cartridge brass, 70%) C27000 (Yellow brass) C28000 (Muntz metal, 60%) C33000 (Low-leaded brass tube) C35300 (High-leaded brass) C35600 (Extra-high-leaded brass) C36000 (Free machining brass) C36500 (Leaded Muntz metal) C46400 (Naval brass) C51000 (Phosphor bronze, 5% A) C54400 (Free cutting phos. bronze) C62300 (Aluminum bronze, 9%) C62400 (Aluminum bronze, 11%) C63000 (Ni-Al bronze) Nickel-Silver

0.302 0.321 0.323 0.298 0.321 0.323 0.318 0.316 0.313 0.306 0.303 0.310 0.306 0.307 0.307 0.304 0.304 0.320 0.321 0.276 0.269 0.274 0.314

2.823

2.796 2.768 2.740 2.685 2.657 2.713 2.796

995 995 995 1190 1100 1085 1080 890

1190 1185 1180 1210 1200 1185 1200 1180

82.5 99.4 109.2 111 80 73 104 70

0.23 0.22 0.22 0.22 0.22 0.23 0.23 0.23

12.8 13.1 12.9 12.9 13.2 13.2 13.0 13.1

23.0 23.6 23.2 23.2 23.8 23.8 23.4 23.6

1590 1941 1924 1590 1958 1750 1870 1810 1680 1660 1650 1660 1630 1630 1630 1630 1630 1750 1700 1905 1880 1895 1870

1630 1981 1967 1800 1967 1975 1910 1880 1750 1710 1660 1720 1670 1660 1650 1650 1650 1920 1830 1915 1900 1930 2030

61 226 205 62 187 218 109 92 70 67 71 67 67 67 67 71 67 40 50 31.4 33.9 21.8 17

0.09 0.09 0.09 0.10 0.09 0.09 0.09 0.09 0.09 0.09 0.09 0.09 0.09 0.09 0.09 0.09 0.09 0.09 0.09 0.09 0.09 0.09 0.09

11.8 9.8 9.9 9.9 9.8 9.8 10.2 10.4 11.1 11.3 11.6 11.2 11.3 11.4 11.4 11.6 11.8 9.9 9.6 9.0 9.2 9.0 9.0

17.6 17.8 17.8 17.6 17.6 18.4 18.7 20.0 20.3 20.9 20.2 20.3 20.5 20.5 20.9 21.2 17.8 17.3 16.2 16.6 16.2 16.2

Copper-Base Alloys

8.359

8.885 8.941 8.249 8.885 8.941 8.802 8.747 8.664 8.470 8.387 8.581 8.470 8.498 8.498 8.415 8.415 8.858 8.885 7.640 7.446 7.584 8.691

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Machinery's Handbook, 31st Edition Thermal Properties of Materials

374

Table Table 13. (Continued) Typical Thermal Properties of Various 13. Typical Thermal Properties of Various MetalsMetals Material and Alloy Designation a

Density, r lb/in3

g/cc

Melting Point, °F

solidus

liquidus

Nickel 200, 201, 205 Hastelloy C-22 Hastelloy C-276 Inconel 718 Monel Monel 400 Monel K500 Monel R405

0.321 0.314 0.321 0.296 0.305 0.319 0.306 0.319

Nickel-Base Alloys 8.885 2615 2635 8.691 2475 2550 8.885 2415 2500 8.193 2300 2437 8.442 2370 2460 8.830 2370 2460 8.470 2400 2460 8.830 2370 2460

S30100 S30200, S30300, S30323 S30215 S30400, S30500 S30430 S30800 S30900, S30908 S31000, S31008 S31600, S31700 S31703 S32100 S34700 S34800 S38400 S40300, S41000, S41600, S41623 S40500 S41400 S42000, S42020 S42200 S42900 S43000, S43020, S43023 S43600 S44002, S44004 S44003 S44600 S50100, S50200

0.290 0.290 0.290 0.290 0.290 0.290 0.290 0.290 0.290 0.290 0.290 0.290 0.290 0.290 0.280 0.280 0.280 0.280 0.280 0.280 0.280 0.280 0.280 0.280 0.270 0.280

8.027 8.027 8.027 8.027 8.027 8.027 8.027 8.027 8.027 8.027 8.027 8.027 8.027 8.027 7.750 7.750 7.750 7.750 7.750 7.750 7.750 7.750 7.750 7.750 7.474 7.750

Malleable Iron, A220 (50005, 60004, 80002) Gray Cast Iron Ductile Iron, A536 (120-90-02) Ductile Iron, A536 (100-70-03) Ductile Iron, A536 (80-55-06) Ductile Iron, A536 (65-45-120) Ductile Iron, A536 (60-40-18) Cast Steel, 3%C

0.265

7.335

0.25 0.25 0.25 0.25 0.25 0.25 0.25

6.920 6.920 6.920 6.920 6.920 6.920 6.920

Commercially Pure Ti-5Al-2.5Sn Ti-8Mn

0.163 0.162 0.171

4.512 4.484 4.733

Stainless Steels 2550 2550 2500 2550 2550 2550 2550 2550 2500 2500 2550 2550 2550 2550 2700 2700 2600 2650 2675 2650 2600 2600 2500 2500 2600 2700

2590 2590 2550 2650 2650 2650 2650 2650 2550 2550 2600 2650 2650 2650 2790 2790 2700 2750 2700 2750 2750 2750 2700 2750 2750 2800

Cast Iron and Steel

Conduc­ tivity, k

Specific Heat, C

Coeff. of Expansion, a

Btu/ hr-ft-°F

Btu/lb/°F

min/in-°F

mm/m-°C

43.3 7.5 7.5 6.5 10 12.6 10.1 10.1

0.11 0.10 0.10 0.10 0.10 0.10 0.10 0.10

8.5 6.9 6.2 7.2 8.7 7.7 7.6 7.6

15.3 12.4 11.2 13.0 15.7 13.9 13.7 13.7

9.4 9.4 9.2 9.4 6.5 8.8 9.0 8.2 9.4 8.3 9.3 9.3 9.3 9.4 14.4 15.6 14.4 14.4 13.8 14.8 15.1 13.8 14.0 14.0 12.1 21.2

0.12 0.12 0.12 0.12 0.12 0.12 0.12 0.12 0.12 0.12 0.12 0.12 0.12 0.12 0.11 0.12 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.12 0.11

9.4 9.6 9.0 9.6 9.6 9.6 8.3 8.8 8.8 9.2 9.2 9.2 9.3 9.6 5.5 6.0 5.8 5.7 6.2 5.7 5.8 5.2 5.7 5.6 5.8 6.2

16.9 17.3 16.2 17.3 17.3 17.3 14.9 15.8 15.8 16.6 16.6 16.6 16.7 17.3 9.9 10.8 10.4 10.3 11.2 10.3 10.4 9.4 10.3 10.1 10.4 11.2

29.5

0.12

7.5

13.5

liquidus approxi­mately, 2100 to 2200, depending on composition

28.0

liquidus, 2640

28.0

0.25 0.16 0.16 0.15 0.15 0.12 0.12

5.8 5.9–6.2 5.9–6.2 5.9–6.2 5.9–6.2 5.9–6.2 7.0

10.4 10.6–11.16 10.6–11.16 10.6–11.16 10.6–11.16 10.6–11.16 12.6

9.0 4.5 6.3

0.12 0.13 0.19

5.1 5.3 6.0

9.2 9.5 10.8

Titanium Alloys 3000 2820 2730

3040 3000 2970

20.0 18.0 20.8

a Alloy designations correspond to the AluminumAssociation numbers for aluminum alloys and to the unified numbering system (UNS) for copper and stainless steel alloys. A220 and A536 are ASTM specified irons.

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Machinery's Handbook, 31st Edition DIMENSIONAL–TEMPERATURE CHANGE

375

Adjusting Lengths for Reference Temperature.—The standard reference temperature for industrial length measurements is 20 degrees Celsius (68 degrees Fahrenheit). For other temperatures, corrections should be made in accordance with the difference in ther­ mal expansion for the two parts, especially when the gage is made of a different material than the part to be inspected. Example: An aluminum part is to be measured with a steel gage when the room tempera­ ture is 30 °C. The aluminum part has a coefficient of linear thermal expansion, aPart = 24.7 3 10−6 mm/mm-°C, and for the steel gage, aGage = 10.8 3 10−6 mm/mm-°C.

At the reference temperature, the specified length of the aluminum part is 20.021 mm. What is the length of the part at the measuring (room) temperature? DL, the change in the measured length due to temperature, is given by:

DL = L ^TR − T0 h^αPart − αGageh = 20.021 ^30 − 20h^24.7 − 10.8h # 10 − 6mm = 2782.919 # 10 − 6 . 0.003 mm where L = length of part at reference temperature; TR = room temperature (temperature of part and gage) and T0 = reference temperature. Thus, the temperature-corrected length at 30°C is L + DL = 20.021 + 0.003 = 20.024 mm.

Length Change Due to Temperature.—Table 14 gives changes in length for variations from the standard reference temperature of 68°F (20°C) for materials of known coeffi­ cients of expansion, a. Coefficients of expansion are given in tables on pages 372, 373, 374, 386, 387, and elsewhere. Example: In Table 14, for coefficients between those listed, add appropriate listed val­ ues. For example, a length change for a coefficient of 7 is the sum of values in the 5 and 2 columns. Fractional interpolation also is possible. Thus, in a steel bar with a coefficient of thermal expansion of 6.3 3 10 −6 = 0.0000063 in/in = 6.3 min/in of length/°F, the increase in length at 73°F is 25 + 5 + 1.5 = 31.5 min/in of length. For a steel with the same coefficient of expansion, the change in length, measured in degrees C, is expressed in microns (micrometers)/meter (mm/m) of length. Alternatively, and for temperatures beyond the scope of the table, the length difference due to a temperature change is equal to the coefficient of expansion multiplied by the change in temperature, i.e., DL = aDT. Thus, for the previous example, DL = 6.3 3 (73 − 68) = 6.3 3 5 = 31.5 min/in. Change in Radius of Thin Circular Ring with Temperature.—Consider a circular ring of initial radius r, that undergoes a temperature change DT. Initially, the circumference of the ring is c = 2pr. If the coefficient of expansion of the ring material is a, the change in circumference due to the temperature change is Dc = 2pr aDT. The new circumference of the ring will be: cn = c + Dc = 2pr + 2praDT = 2pr (1 + aDT ).

Note: An increase in temperature causes Dc to be positive, and a decrease in temperature causes Dc to be negative. As the circumference increases, the radius of the circle also increases. If the new radius is R, the new circumference is 2pR. For a given change in temperature, DT, the change in radius of the ring is found as follows:

cn = 2πR = 2πr(1 + α∆T)

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R = r + rα∆T

∆r = R  r = rα∆T

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Machinery's Handbook, 31st Edition DIMENSIONAL–TEMPERATURE CHANGE

376

Table 14. Differences in Length in Microinches/Inch (Microns/Meter) for Changes from the Standard Temperature of 68°F (20°C) Temperature Deg. F C 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98

−10 −9 −8 −7 −6 −5 −4 −3 −2 −1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50

Coefficient of Thermal Expansion of Material per Degree F (C) 3 106 3 4 5 10 15 20 25 for °F in microinches/inch of length (min/in) Total Change in Length from Standard Temperature { for °C or °K in microns/meter of length (mm/m) 1

2

−30 −29 −28 −27 −26 −25 −24 −23 −22 −21 −20 −19 −18 −17 −16 −15 −14 −13 −12 −11 −10 −9 −8 −7 −6 −5 −4 −3 −2 −1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

−60 −58 −56 −54 −52 −50 −48 −46 −44 −42 −40 −38 −36 −34 −32 −30 −28 −26 −24 −22 −20 −18 −16 −14 −12 −10 −8 −6 −4 −2 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54 56 58 60

−90 −87 −84 −81 −78 −75 −72 −69 −66 −63 −60 −57 −54 −51 −48 −45 −42 −39 −36 −33 −30 −27 −24 −21 −18 −15 −12 −9 −6 −3 0 3 6 9 12 15 18 21 24 27 30 33 36 39 42 45 48 51 54 57 60 63 66 69 72 75 78 81 84 87 90

−120 −116 −112 −108 −104 −100 −96 −92 −88 −84 −80 −76 −72 −68 −64 −60 −56 −52 −48 −44 −40 −36 −32 −28 −24 −20 −16 −12 −8 −4 0 4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68 72 76 80 84 88 92 96 100 104 108 112 116 120

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−150 −145 −140 −135 −130 −125 −120 −115 −110 −105 −100 −95 −90 −85 −80 −75 −70 −65 −60 −55 −50 −45 −40 −35 −30 −25 −20 −15 −10 −5 0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 135 140 145 150

−300 −290 −280 −270 −260 −250 −240 −230 −220 −210 −200 −190 −180 −170 −160 −150 −140 −130 −120 −110 −100 −90 −80 −70 −60 −50 −40 −30 −20 −10 0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200 210 220 230 240 250 260 270 280 290 300

−450 −435 −420 −405 −390 −375 −360 −345 −330 −315 −300 −285 −270 −255 −240 −225 −210 −195 −180 −165 −150 −135 −120 −105 −90 −75 −60 −45 −30 −15 0 15 30 45 60 75 90 105 120 135 150 165 180 195 210 225 240 255 270 285 300 315 330 345 360 375 390 405 420 435 450

−600 −580 −560 −540 −520 −500 −480 −460 −440 −420 −400 −380 −360 −340 −320 −300 −280 −260 −240 −220 −200 −180 −160 −140 −120 −100 −80 −60 −40 −20 0 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300 320 340 360 380 400 420 440 460 480 500 520 540 560 580 600

−750 −725 −700 −675 −650 −625 −600 −575 −550 −525 −500 −475 −450 −425 −400 −375 −350 −325 −300 −275 −250 −225 −200 −175 −150 −125 −100 −75 −50 −25 0 25 50 75 100 125 150 175 200 225 250 275 300 325 350 375 400 425 450 475 500 525 550 575 600 625 650 675 700 725 750

30

−900 −870 −840 −810 −780 −750 −720 −690 −660 −630 −600 −570 −540 −510 −480 −450 −420 −390 −360 −330 −300 −270 −240 −210 −180 −150 −120 −90 −60 −30 0 30 60 90 120 150 180 210 240 270 300 330 360 390 420 450 480 510 540 570 600 630 660 690 720 750 780 810 840 870 900

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Machinery's Handbook, 31st Edition Specific Gravity

377

Properties of Mass and Weight Density.—The density of any solid, fluid or gaseous substance is the mass of that sub­ stance per unit volume. If weight is used in the ordinary sense as being equivalent to mass, then density may be defined as the weight per unit volume. The density depends upon the unit in which the mass or weight is expressed, and upon the unit of volume used. In engi­ neering and scientific work, density is generally expressed in grams per cubic centimeter, without naming the units, because the density will be equal to the specific gravity.

Specific Gravity.—Specific gravity is a number indicating how many times a volume of material is heavier than an equal volume of water. Density of water varies slightly at differ­ent temperatures. In exacting scientific studies, a reference temperature of 4°C (39.2°F) is often used, and the weight of 1 cubic meter of pure water at 4°C is 1000 kg. In engineering practice, the usual custom is to measure specific gravity at water temperature of 60 or 62°F (15.5 or 16.6°C); 1 cubic foot of pure water at 62°F weighs 62.355 pounds. Given

Specific Gravity (S.G.)

Rule to find density

weight/cm3 = S.G. weight/m3 = S.G. 3 1000 weight/in3 = S.G. 3 0.0361 weight/ft3 = S.G. 4 0.01604

Given

Rule to find specific gravity

weight/cm3 weight/m3 weight/in3 weight/ft3

S.G. = weight/cm3 S.G. = weight/m3 4 1000 S.G. = weight/in3 4 0.0361 S.G. = weight/ft3 3 0.01604

When specific gravity is known, the weight per cubic centimeter is equal to its specific gravity. The weight per cubic meter equals the specific gravity 3 1000. The weight per cubic inch equals the specific gravity 3 0.0361. The weight of a cubic foot equals the specific gravity divided by 0.01604. When weight per cubic centimeter is known, the specific gravity is equal to the weight per cubic centimeter. If weight per cubic meter is known, the specific gravity equals this weight divided by 1000. If density is given in lb/in3, specific gravity may be determined by dividing the density by 0.0361. If weight per cubic foot is known, specific gravity equals this weight multiplied by 0.01604.

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2.6 162 … 74 17.5 1091 19.3 1204 2.7 168 … 109 2.4 150 0.9 56 7.2 447 7.7 479 2.7 168 11.4 711 2.6 162 2.7 168 2.4 150 13.56 845.3 2.8 175 1.5 94 8.3 517 8.7 542 1.16 73 2.19 137 1.8 112 1.8 112

2595 1185 17476 19286 2691 1746 2403 897 7160 7673 2691 11389 2595 2691 2403 13540 2803 1506 8282 8682 1169 2195 1794 1794

Substance

Platinum 21.5 1342 Polycarbonate 1.19 74 Polyethylene 0.97 60 Polypropylene 0.91 57 Polyurethane 1.05 66 Quartz 2.6 162 Salt, common … 48 Sand, dry … 100 Sand, wet … 125 Sandstone 2.3 143 Silver 10.5 656 Slate 2.8 175 Soapstone 2.7 168 Steel 7.9 491 Sulfur 2.0 125 Tar, bituminous 1.2 75 Tile 1.8 112 Trap rock 3.0 187 1.0 62.43 Water at 4°C 1.0 62.355 Water at 62°F White metal 7.3 457 Zinc, cast 6.9 429 Zinc, sheet 7.2 450 … … …

kg/m3

Substance

Glass Glass, crushed Gold, 22 carat fine Gold, pure Granite Gravel Gypsum Ice Iron, cast Iron, wrought Iron slag Lead Limestone Marble Masonry Mercury Mica Mortar Nickel, cast Nickel, rolled Nylon 6, Cast PTFE Phosphorus Plaster of Paris

aWeight

lb/ft3

1057 1185 7785 2563 2675 2403 1394 1794 1794 2291 2002 2195 1794 1602 1554 3092 2291 400 1506 1297 2195 1201 1602 3989

Specific Gravity

66 74 486 160 167 150 87 112 112 143 125 137 112 100 97 193 143 25 94 81 137 75 100 249

kg/m3

lb/ft3

kg/m3

1.05 1.19 7.8 2.6 2.7 2.4 1.4 1.8 1.8 2.3 2.0 2.2 1.8 1.6 1.55 3.1 2.3 0.4 1.5 1.3 2.2 … … 4.0

aWeight

lb/ft3

Specific Gravity

Substance

ABS Acrylic Aluminum bronze Aluminum, cast Aluminum, wrought Asbestos Asphaltum Borax Brick, common Brick, fire Brick, hard Brick, pressed Brickwork, in cement Brickwork, in mortar CPVC Cement, Portland (set) Chalk Charcoal Coal, anthracite Coal, bituminous Concrete Earth, loose Earth, rammed Emery

aWeight

Specific Gravity

Examples: The specific gravity of cast iron is 7.2. The weight of 80 cm 3 of cast iron = 7.2 3 80 = 5.6 kg. The weight of 5 in3 of cast iron = 7.2 3 0.0361 3 5 = 1.2996 pounds. Examples: The weight of a cubic centimeter of gold is 19.31 grams. The specific gravity of gold = weight of a cubic centimeter of gold = 19.31. A cubic inch of gold weighs 0.697 pound. The specific gravity of gold = 0.697  0.0361 = 19.31 Table 15. Average Specific Gravity of Various Substances

21497 1185 961 913 1057 2595 769 1602 2002 2291 10508 2803 2691 7865 2002 1201 1794 2995 1000 7320 6872 7208   …

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Machinery's Handbook, 31st Edition Specific Gravity

378

a The weight per cubic foot or cubic meter is calculated on the basis of the specific gravity except for those substances that occur in bulk, heaped, or loose form. In these instances, only the weights per cubic foot and cubic meter are given because the voids present in representative samples make the values of the specific gravities inaccurate.

Material

Anthracite coal Bituminous coal Charcoal Coke

Average Weights and Volumes of Solid Fuels

lb/ft3

ft3/ton (2240 lb)

55–65 50–55 18–18.5 28

34–41 41–45 121–124 80

kg/m3

lb/bushel

881–1041 801–881 288–296 449

67 (80a) 78–86 (80a) 22–23 (20a) 35 (40a)

m3/t

0.96–1.14 1.14–1.25 3.37–3.47 2.23

a Legal commodities weight/bushel defined by statute in some states.

Note: t = metric ton = 1000 kg; ton = US ton of 2000 lbs; a gross or long ton = 2240 lbs.

Specific Gravity of Gases.—The specific gravity of gases is the number that indicates their weight in comparison with that of an equal volume of air. The specific gravity of air is 1, and the comparison is made at 32°F (0°C). Values are given in Table 16. Gas

Aira Acetylene Alcohol vapor Ammonia Carbon dioxide Carbon monoxide Chlorine

Table 16. Specific Gravity of Gases At 32°F (0°C) Sp. Gr. 1.000 0.920 1.601 0.592 1.520 0.967 2.423

Gas

Ether vapor Ethylene Hydrofluoric acid Hydrochloric acid Hydrogen Illuminating gas Mercury vapor

Gas

Sp. Gr. 2.586 0.967 2.370 1.261 0.069 0.400 6.940

Sp. Gr.

Marsh gas Nitrogen Nitric oxide Nitrous oxide Oxygen Sulfur dioxide Water vapor

0.555 0.971 1.039 1.527 1.106 2.250 0.623

a 1 cubic foot of air at 32°F and atmospheric pressure weighs 0.0807 pound. 1 cubic meter of air at 0°C and atmospheric pressure weighs 1.29 kg.

Specific Gravity of Liquids.—The specific gravity of liquids is the number that indicates how much a certain volume of the liquid weighs compared with an equal volume of water, the same as with solid bodies. Specific gravity of various liquids is given in Table 17. The density of liquid is often expressed in degrees on the hydrometer, an instrument for determining the density of liquids that is provided with graduations made to an arbitrary scale. The hydrometer consists of a glass tube with a bulb at one end containing air and arranged with a weight at the bottom so as to float in an upright position in the liquid, the density of which is to be measured. The depth to which the hydrometer sinks in the liquid is read off on the graduated scale. The most commonly used hydrometer is the Baumé, see Table 18. The value of the degrees of the Baumé scale differs according to whether the liquid is heavier or lighter than water. The specific gravity for liquids heavier than water equals 145 4 (145 − degrees Baumé). For liquids lighter than water, the specific gravity equals 140 4 (130 + degrees Baumé). Liquid

Acetic acid Alcohol, commercial Alcohol, pure Ammonia Benzine Bromine Carbolic acid Carbon disulfide

Sp. Gr. 1.06 0.83 0.79 0.89 0.69 2.97 0.96 1.26

Table 17. Specific Gravity of Liquids Liquid

Cotton-seed oil Ether, sulfuric Fluoric acid Gasoline Kerosene Linseed oil Mineral oil Muriatic acid

Sp. Gr. 0.93 0.72 1.50 0.70 0.80 0.94 0.92 1.20

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Liquid

Naphtha Nitric acid Olive oil Palm oil Petroleum oil Phosphoric acid Rape oil Sulfuric acid

Sp. Gr. 0.76 1.50 0.92 0.97 0.82 1.78 0.92 1.84

Liquid

Tar Turpentine oil Vinegar Water Water, sea Whale oil

Sp. Gr. 1.00 0.87 1.08 1.00 1.03 0.92

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Machinery's Handbook, 31st Edition Specific Gravity

379

Table 18. Degrees on Baumé’s Hydrometer Converted to Specific Gravity Deg. Baumé 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26

Specific Gravity for Liquids Heavier than Lighter than Water Water 1.000 1.007 1.014 1.021 1.028 1.036 1.043 1.051 1.058 1.066 1.074 1.082 1.090 1.099 1.107 1.115 1.124 1.133 1.142 1.151 1.160 1.169 1.179 1.189 1.198 1.208 1.219

… … … … … … … … … … 1.000 0.993 0.986 0.979 0.972 0.966 0.959 0.952 0.946 0.940 0.933 0.927 0.921 0.915 0.909 0.903 0.897

Deg. Baumé 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53

Specific Gravity for Liquids Heavier than Lighter Water than Water 1.229 1.239 1.250 1.261 1.272 1.283 1.295 1.306 1.318 1.330 1.343 1.355 1.368 1.381 1.394 1.408 1.422 1.436 1.450 1.465 1.480 1.495 1.510 1.526 1.542 1.559 1.576

0.892 0.886 0.881 0.875 0.870 0.864 0.859 0.854 0.849 0.843 0.838 0.833 0.828 0.824 0.819 0.814 0.809 0.805 0.800 0.796 0.791 0.787 0.782 0.778 0.773 0.769 0.765

Deg. Baumé 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80

Specific Gravity for Liquids Heavier Lighter than Water than Water 1.593 1.611 1.629 1.648 1.667 1.686 1.706 1.726 1.747 1.768 1.790 1.813 1.836 1.859 1.883 1.908 1.933 1.959 1.986 2.014 2.042 2.071 2.101 2.132 2.164 2.197 2.230

0.761 0.757 0.753 0.749 0.745 0.741 0.737 0.733 0.729 0.725 0.721 0.718 0.714 0.710 0.707 0.704 0.700 0.696 0.693 0.689 0.686 0.683 0.679 0.676 0.673 0.669 0.666

How to Estimate the Weight of Natural Piles.—To calculate the upper and lower limits of weight of a substance piled naturally on a plate, use the following: For a substance piled naturally on a circular plate, form­ing a cone of material,

W = M D3

(1)

where W = weight, lb (kg); D = diameter of plate in Fig. 1a, in feet (meters); and, M = a materials factor, whose upper and lower limits are given in Table 19b.

D

Fig. 1a. Conical Pile For a substance piled naturally on a rectangular plate,

W = MRA3

A

B

Fig. 1b. Rectangular Pile

(2)

where W = weight, lb (kg); A and B = the length and width in feet (meters), respectively, of the rectangular plate in Fig. 1b, with B ≤ A; M = a materials factor, whose upper and lower limits are given in Table 19b; and, R = is a fac­tor given in Table 19a as a function of the ratio B/A.

Example: Find the upper and lower limits of the weight of dry ashes piled naturally on a plate 10 ft. in diameter. Using Equation (1), M = 4.58 from Table 19b, the lower limit W = 4.58 3 103 = 4,580 lb. For M = 5.89, the upper limit W = 5.89 3 103 = 5,890 lb. Example: What weight of dry ashes rests on a rectangular plate 10 ft. by 5 ft.?

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Machinery's Handbook, 31st Edition Weight of Piles

380

For B/A = 5⁄10 = 0.5, R = 0.39789 from Table 19a. Using Equation (2), for M = 4.58, the lower limit W = 4.58 3 0.39789 3 103 = 1,822 lb. For M = 5.89, the upper limit W = 5.89 3 0.39789 3 103 = 2,344lb. Example: What is the weight of a pile of cast iron chips resting on a rectangular plate 4 m by 2 m? For B/A = 2⁄4, R = 0.39789 from Table 19a. Using Equation (2), for M = 17.02, the lower limit W = 17.02 3 0.39789 3 43 = 433 kg. For M = 26.18, the upper limit W = 26.18 3 0.39789 3 43 = 667 kg. Table 19a. Factor R as a Function of B/A (B ≤ A)

B/A 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.11 0.12 0.13 0.14 0.15 0.16 0.17

R 0.00019 0.00076 0.00170 0.00302 0.00470 0.00674 0.00914 0.01190 0.01501 0.01846 0.02226 0.02640 0.03088 0.03569 0.04082 0.04628 0.05207

Material

B/A 0.18 0.19 0.20 0.21 0.22 0.23 0.24 0.25 0.26 0.27 0.28 0.29 0.30 0.31 0.32 0.33 0.34

R 0.05817 0.06458 0.07130 0.07833 0.08566 0.09329 0.10121 0.10942 0.11792 0.12670 0.13576 0.14509 0.15470 0.16457 0.17471 0.18511 0.19576

B/A 0.35 0.36 0.37 0.38 0.39 0.40 0.41 0.42 0.43 0.44 0.45 0.46 0.47 0.48 0.49 0.50 0.51

R 0.20666 0.21782 0.22921 0.24085 0.25273 0.26483 0.27717 0.28973 0.30252 0.31552 0.32873 0.34216 0.35579 0.36963 0.38366 0.39789 0.41231

B/A 0.52 0.53 0.54 0.55 0.56 0.57 0.58 0.59 0.60 0.61 0.62 0.63 0.64 0.65 0.66 0.67 0.68

R 0.42691 0.44170 0.45667 0.47182 0.48713 0.50262 0.51826 0.53407 0.55004 0.56616 0.58243 0.59884 0.61539 0.63208 0.64891 0.66586 0.68295

B/A 0.69 0.70 0.71 0.72 0.73 0.74 0.75 0.76 0.77 0.78 0.79 0.80 0.81 0.82 0.83 0.84 0.85

R 0.70015 0.71747 0.73491 0.75245 0.77011 0.78787 0.80572 0.82367 0.84172 0.85985 0.87807 0.89636 0.91473 0.93318 0.95169 0.97027 0.98891

B/A 0.86 0.87 0.88 0.89 0.90 0.91 0.92 0.93 0.94 0.95 0.96 0.97 0.98 0.99 1.00 … …

Table 19b. Limits of Factor M for Various Materials

Almonds, whole Aluminum chips Aluminum silicate Ammonium chloride Asbestos, shredded Ashes, dry Ashes, damp Asphalt, crushed Bakelite, powdered Baking powder Barium carbonate Bauxite, mine run Beans, navy, dry Beets, sugar, shredded Bicarbonate of soda Borax Boric acid Bronze chips Buckwheat Calcium lactate Calcium oxide (lime) Carbon, ground Casein Cashew nuts Cast iron chips Cement, Portland Cinders, coal Clay, blended for tile Coal, anthracite, chestnut Coal, bituminous, sized Coal, ground Cocoa, powdered Coconut, shredded Coffee beans

Factor M

2.12–3.93 0.92–1.96 3.7–6.41 3.93–6.81 2.62–3.27 4.58–5.89 6.24–7.80 3.4–5.89 3.93–5.24 3.1–5.37 9.42 5.9–6.69 3.63 0.47–0.55 3.10 3.78–9.16 4.16–7.20 3.93–6.54 2.8–3.17 3.4–3.8 3.30 2.51 2.72–4.71 4.19–4.84 17.02–26.18 6.8–13.09 3.02–5.24 5.89 2.43 2.64–4.48 2.90 3.93–4.58 2.62–2.88 2.42–5.89

Material

Coffee, ground Coke, pulverized Copper oxide, powdered Cork, granulated Corn on cob Corn sugar Cottonseed, dry, de-linted Diatomaceous earth Dicalcium phosphate Ebonite, crushed Epsom salts Feldspar, ground Fish scrap Flour Flue dust Fluorspar (Fluorite) Graphite, flake Gravel Gypsum, calcined Hominy Hops, dry Kaolin clay Lead silicate, granulated Lead sulphate, pulverized Lime, ground Limestone, crushed Magnesium chloride Malt, dry, ground Manganese sulphate Marble, crushed Mica, ground Milk, whole, powdered Oats Orange peel, dry

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Factor M

1.89–3.27 2.21 20.87 1.57–1.96 1.29–1.33 2.34–4.06 1.66–5.24 0.83–1.83 5.63 4.91–9.16 3.02–6.54 8.51–9.16 5.24–6.54 5.61–10.43 2.65–3.40 10.73–14.40 3.02–5.24 6.8–13.18 6.04–6.59 2.8–6.54 4.58 12.32–21.34 25.26 24.09 7.85 6.42–11.78 4.32 1.66–2.88 5.29–9.16 6.8–12.44 1.24–1.43 2.62 1.74–2.86 1.96

Material

Peanuts, unshelled Peanuts, shelled Peas, dry Potassium carbonate Potassium sulphate Pumice Rice, bran Rubber, scrap, ground Salt, dry, coarse Salt, dry, fine Saltpeter Salt rock, crushed Sand, very fine Sawdust, dry Sesame seed Shellac, powdered Slag, furnace, granular Soap powder Sodium nitrate Sodium sulphite Sodium sulphate Soybeans Steel chips, crushed Sugar, refined Sulphur Talcum powder Tin oxide, ground Tobacco stems Trisodium phosphate Walnut shells, crushed Wood chips, fir Zinc sulphate … …

R 1.00761 1.02636 1.04516 1.06400 1.08289 1.10182 1.12078 1.13977 1.15879 1.17783 1.19689 1.21596 1.23505 1.25414 1.27324 … …

Factor M

1.13–3.14 2.65–5.89 2.75–3.05 3.85–6.68 5.5–6.28 5.24–5.89 1.51–2.75 2.11–4.58 3.02–8.38 5.29–10.47 6.05–10.47 4.58 7.36–9 0.95–2.85 2.04–4.84 2.34–4.06 4.53–8.51 1.51–3.27 3.96–4.66 10.54 6.92 3.48–6.28 7.56–19.63 3.78–7.2 4.5–6.95 4.37–5.9 9.17 1.96–3.27 4.53–7.85 2.65–5.24 2.49–2.88 8.85–11.12 … …

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Machinery's Handbook, 31st Edition Weight of Piles

381

Molecular Weight.—The smallest mass of a chemical combination which can be con­ ceived of as existing and yet preserving its chemical properties is known as a molecule. The molecular weight of a chemical compound is equal to the sum of the atomic weights of the atoms contained in the molecule, and is calculated from the atomic weights, when the symbol of the compound is known. See Table 1 on page 368 for atomic weights. The atomic weight of silver is 107.88; of nitrogen, 14.01; and of oxygen, 16; hence, the molec­ ular weight of silver-nitrate, the chemical formula of which is AgNO3 equals 107.88 + 14.01 + (3 3 16) = 169.89. Mole or Mol.—This term is used as a designation of quantity in electro-chemistry to indicate the number of grams of a substance equal to its molecular weight. For example, one mol of silver-nitrate equals 169.89 grams, the molecular weight of silver-nitrate being 169.89. Air.—Air is a mechanical mixture composed of 78 percent, by volume, of nitrogen, 21 percent of oxygen, and 1 percent of argon. The weight of pure air at 32 °F (0 °C), at an atmo­spheric pressure of 29.92 inches of mercury (760 mm mercury or 760 torr) or 14.70 pounds per square inch, is 0.08073 pound per cubic foot. The volume of a pound of air at the same temperature and pressure is 12.387 cubic feet. The weight of air, in pounds per cubic foot, at any other temperature or pressure may be determined by first multiplying the barometer reading (atmospheric pressure in inches of mercury) by 1.325 and then dividing the prod­uct by the absolute temperature in degrees F. The absolute zero from which all tempera­tures must be derived in dealing with the weight and volume of gases, is assumed to be minus 459.67 °F (273.15 °C). Hence, to obtain the absolute temperature, add to the tem­perature observed on a regular Fahrenheit thermometer the value 459.67. Alligation.—Alligation or “the rule of mixtures” are names applied to several rules of arithmetical processes for determining the relation between proportions and prices of the ingredients of a mixture and the cost of the mixture per unit of weight or volume. For exam­ple, if an alloy is composed of several metals varying in price, the price per pound of the alloy can be found as in the following example: An alloy is composed of 50 pounds of cop­per at $1.70 a pound, 10 pounds of tin at $4.05 a pound, 20 pounds of zinc at $0.99 a pound, and 5 pounds of lead at $1.10 cents a pound. What is the cost of the alloy per pound, no account being taken of the cost of mixing it? Multiply the number of pounds of each of the ingredients by its price per pound, add these products together, and divide the sum by the total weight of all the ingredients. The quotient is the price per pound of the alloy. Example: The foregoing example would be worked out numerically as follows: Total cost of materials: 50 × 1.70 +10 × 4.05 + 20 × 0.99 + 5 × 1.10 = $150.80 Total weight of metal in alloy: 50 + 10 + 20 + 5 = 85 lbs. Price per pound of alloy = 150.80 ÷ 85= $1.77, approximately. Earth or Soil Weight.—Loose earth has a weight of approximately 75 lb/ft 3 (1200 kg/m3) and rammed earth, 100 lb/ft 3 (1600 kg/m3). Composition of Earth Crust: The solid crust of the earth, according to an estimate, is composed approximately of the following elements: oxygen 44 to 48.7 percent; silicon 22.8 to 36.2 percent; aluminum 6.1 to 9.9 percent; iron 2.4 to 9.9 percent; calcium 0.9 to 6.6 percent; magnesium 0.1 to 2.7 percent; sodium 2.4 to 2.5 percent; potassium 1.7 to 3.1 percent. Loads on Soils and Rocks: The bearing capacities of soils and rocks is useful in structural engineering and also of value under certain conditions in connection with the installation of very heavy machinery requiring foundations. The ultimate resistance of various soils and rocks will be given in tons per square foot: natural earth that is solid and dry, 4 to 6 tons; thick beds of absolutely dry clay, 4 tons; thick beds of moderately dry clay, 2 tons; soft clay, 1 ton; gravel that is dry, coarse, and well packed, 6 to 8 tons; soft, friable rock and shales, 5 to 10 tons; sand that is compact, dry, and well cemented, 4 tons; natural sand in a clean dry condition, 2 to 4 tons; compact bedrock, northern red sandstone, 20 tons; com­pact bedrock, northern sound limestone, 25 tons; compact bedrock granite, 30 tons.

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Machinery's Handbook, 31st Edition Wood PROPERTIES OF WOOD, CERAMICS, PLASTICS, METALS Properties of Wood

Mechanical Properties of Wood.—Wood is composed of cellulose, lignin, ash-forming minerals, and extractives formed into a cellular structure. (Extractives are substances that can be removed from wood by extraction with such solvents as water, alcohol, acetone, benzene, and ether.) Variations in the characteristics and volumes of the four components and differences in the cellular structure result in some woods being heavy and some light, some stiff and some flexible, and some hard and some soft. For a single species, the prop­ erties are relatively constant within limits; therefore, selection of wood by species alone may sometimes be adequate. However, to use wood most effectively in engineering appli­ cations, the effects of physical properties or specific characteristics must be considered. The mechanical properties listed in the accompanying Table 1 were obtained from tests on small pieces of wood termed “clear” and “straight grained” because they did not contain such characteristics as knots, cross grain, checks, and splits. However, these test pieces did contain such characteristics as growth rings that occur in consistent patterns within the piece. Since wood products may contain knots, cross grain, etc., these characteristics must be taken into account when assessing actual properties or when estimating actual perfor­mance. In addition, the methods of data collection and analysis have changed over the years during which the data in Table 1 have been collected; therefore, the appropriateness of the data should be reviewed when used for critical applications such as stress grades of lumber. Wood is an orthotropic material; that is, its mechanical properties are unique and inde­ pendent in three mutually perpendicular directions—longitudinal, radial, and tangential. These directions are illustrated in the following figure.

Radial

Longitudinal

in Gra ction e r i D

Tangential

Modulus of Rupture: The modulus of rupture in bending reflects the maximum loadcar­r ying capacity of a member and is proportional to the maximum moment borne by the member. The modulus is an accepted criterion of strength, although it is not a true stress because the formula used to calculate it is valid only to the proportional limit.

Work to Maximum Load in Bending: The work to maximum load in bending represents the ability to absorb shock with some permanent deformation and more or less injury to a specimen; it is a measure of the combined strength and toughness of the wood under bend­ing stress.

Maximum Crushing Strength: The maximum crushing strength is the maximum stress sustained by a compression parallel-to-grain specimen having a ratio of length to least diameter of less than 11.

Compression Perpendicular to Grain: Strength in compression perpendicular to grain is reported as the stress at the proportional limit because there is no clearly defined ultimate stress for this property.

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Machinery's Handbook, 31st Edition Wood

383

Shear Strength Parallel to Grain: Shear strength is a measure of the ability to resist inter­nal slipping of one part upon another along the grain. The values listed in the table are aver­ages of the radial and tangential shears. Tensile Strength Perpendicular to Grain: The tensile strength perpendicular to the grain is a measure of the resistance of wood to forces acting across the grain that tend to split the material. Averages of radial and tangential measurements are listed. Table 1. Mechanical Properties of Commercially Important US Grown Woods

Use the first number in each column for GREEN wood; use the second num­ber for DRY wood.

Basswood, American Cedar, N. white Cedar, W. red Douglas Fir, coasta Douglas Fir, interior W. Douglas Fir, interior N. Douglas Fir, interior S. Fir, balsam Hemlock, Eastern Hemlock, Mountain Hemlock, Western Pine, E. white Pine, Virginia Pine, W. white Redwood, old-growth Redwood, young-growth Spruce, Engelmann Spruce, red Spruce, white

Static Bending

Modulus of Rupture (103 psi) 5.0 4.2 5.2 7.7 7.7 7.4 6.8 5.5 6.4 6.3 6.6 4.9 7.3 4.7 7.5 5.9 4.7 6.0 5.0

Work to Max Load (in-lb/in3)

8.7 5.3 6.5 5.7 7.5 5.0 12.4 7.6 12.6 7.2 13.1 8.1 11.9 8.0 9.2 4.7 8.9 6.7 11.5 11.0 11.3 6.9 9.9 5.2 13.0 10.9 9.7 5.0 10.0 7.4 7.9 5.7 9.3 5.1 10.8 6.9 9.4 6.0

7.2 4.8 5.8 9.9 10.6 10.5 9.0 5.1 6.8 10.4 8.3 8.3 13.7 8.8 6.9 5.2 6.4 8.4 7.7

Maximum Crushing Strength (103 psi)

2.22 1.90 2.77 3.78 3.87 3.47 3.11 2.63 3.08 2.88 3.36 2.44 3.42 2.43 4.20 3.11 2.18 2.72 2.35

4.73 3.96 4.56 7.23 7.43 6.90 6.23 5.28 5.41 6.44 7.20 5.66 6.71 5.04 6.15 5.22 4.48 5.54 5.18

Compression Strength Perpendicular to Grain (psi) 170 230 240 380 420 360 340 190 360 370 280 220 390 190 420 270 200 260 210

370 310 460 800 760 770 740 404 650 860 550 580 910 470 700 520 410 550 430

Shear Strength Parallel to Grain (psi)

600 620 770 900 940 950 950 662 850 930 860 680 890 680 800 890 640 750 640

990 850 990 1,130 1,290 1,400 1,510 944 1,060 1,540 1,290 1,170 1,350 1,040 940 1,110 1,200 1,290 970

Tensile Strength Perp. to Grain (psi)

280 240 230 300 290 340 250 180 230 330 290 250 400 260 260 300 240 220 220

350 240 220 340 350 390 330 180 … … 340 420 380 … 240 250 350 350 360

a Coast: grows west of the summit of the Cascade Mountains in OR and WA. Interior west: grows in CA and all counties in OR and WA east of but adjacent to the Cascade summit. Interior north: grows in remainder of OR and WA and ID, MT, and WY. Interior south: grows in UT, CO, AZ, and NM. Results of tests on small, clear, straight-grained specimens. Data for dry specimens are from tests of seasoned material adjusted to a moisture content of 12%. Source: US Department of Agriculture: Wood Handbook.

Weight of Wood.—The weight of seasoned wood per cord is approximately as follows, assuming about 70 cubic feet of solid wood per cord: beech, 3300 pounds; chestnut, 2600 pounds; elm, 2900 pounds; maple, 3100 pounds; poplar, 2200 pounds; white pine, 2200 pounds; red oak, 3300 pounds; white oak, 3500 pounds. For additional weights of green and dry woods, see Table 2. Weight per Foot of Wood, Board Measure.—The following is the weight in pounds of various kinds of woods, commercially known as dry timber, per foot board measure: white oak, 4.16; white pine, 1.98; Douglas fir, 2.65; short-leaf yellow pine, 2.65; red pine, 2.60; hemlock, 2.08; spruce, 2.08; cypress, 2.39; cedar, 1.93; chestnut, 3.43; Georgia yellow pine, 3.17; California spruce, 2.08. For other woods, divide the weight/ft3 from Table 2 by 12 to obtain the approximate weight per board foot. Effect of Pressure Treatment on Mechanical Properties of Wood.—The strength of wood preserved with creosote, coal-tar, creosote-coal-tar mixtures, creosotepetroleum mixtures, or pentachlorophenol dissolved in petroleum oil is not reduced. However, water­borne salt preservatives contain chemicals, such as copper, arsenic, chromium, and ammo­nia, which have the potential of affecting mechanical properties

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Machinery's Handbook, 31st Edition Wood

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of treated wood and causing mechanical fasteners to corrode. Preservative salt-retention levels required for marine protection may reduce bending strength by 10 percent or more. Density of Wood.—The following formula can be used to find the density of wood in lb/ft 3 as a function of its moisture content. G M ρ = 62.4 a 1 + G # 0.009 # M ka 1 + 100 k where r is the density, G is the specific gravity of wood, and M is the moisture content expressed in percent.

Basswood

Beech

Birch

Birch, paper

Cedar, Alaska

Cedar, eastern red

Cedar, northern white

Cedar, southern white

Cedar, western red

Cherry, black

Chestnut

Cottonwood, eastern

Cottonwood, northern black

Cypress, southern

Douglas fir, coast region

42

54

57

50

36

37

28

26

27

45

55

49

46

51

38

41

38

26

26

45

44

38

31

33

22

23

23

35

30

28

24

32

34

Elm, rock

Elm, slippery

Fir, balsam

Fir, commercial white

Gum, black

Gum, red

Hemlock, eastern

Hemlock, western

Hickory, pecan

Hickory, true

Honeylocust

Larch, western

Locust, black

Maple, bigleaf

Maple, black

Maple, red

Maple, silver

Maple, sugar

Source: United States Department of Agriculture

35 54

30 35

53

56

45

46

45

50

50

41

62

63

61

48

58

47

54

50

45

56

44

37

25

27

35

34

28

29

45

51

…

36

48

34

40

38

33

44

Species Oak, red Oak, white

Pine, lodgepole

Pine, northern white

Pine, Norway

Pine, ponderosa

Pines, southern yellow:

  Pine, loblolly

  Pine, longleaf

  Pine, shortleaf

Pine, sugar

Pine, western white

Poplar, yellow

Redwood

Spruce, eastern

Spruce, Engelmann

Spruce, Sitka

Sycamore

Tamarack

Walnut, black

Green

48

46

43

Species Douglas fir, Rocky Mt. region Elm, American

Airdry

28 34

Green

46 52

Airdry

Ash, commercial white

Ash, Oregon

Aspen

Green

Species Alder, red Ash, black

Airdry

Table 2. Weights of American Woods, in Pounds per Cubic Foot

64 63

44 47

39

36

42

45 53

55

52

52

35

38

50

34

39

33

52

47

58

29

25

34

28 36

41

36

25

27

28

28

28

23

28

34

37

38

Machinability of Wood.—The ease of working wood with hand tools generally varies directly with the specific gravity of the wood; the lower the specific gravity, the easier the wood is to cut with a sharp tool. A rough idea of the specific gravity of various woods can be obtained from the preceding table by dividing the weight of wood in lb/ft 3 by 62.355. A wood species that is easy to cut does not necessarily develop a smooth surface when it is machined. Three major factors, other than specific gravity, influence the smoothness of the surface obtained by machining: interlocked and variable grain, hard deposits in the grain, and reaction wood. Interlocked and variable grain is a characteristic of many tropical and some domestic species; this type of grain structure causes difficulty in planing quarter sawn boards unless careful attention is paid to feed rates, cutting angles, and sharpness of the knives. Hard deposits of calcium carbonate, silica, and other minerals in the grain tend to dull cutting edges quickly, especially in wood that has been dried to the usual in service moisture content. Reaction wood results from growth under some physical stress, such as occurs in leaning trunks and crooked branches. Generally, reaction wood occurs as tension wood in hardwoods and as compression wood in softwoods. Tension wood is particularly troublesome, often resulting in fibrous and fuzzy surfaces, especially in woods of lower density. Reaction wood may also be responsible for pinching saw blades, resulting in burn­ing and dulling of teeth. The Table 3 rates the suitability of various domestic hardwoods for machining. The data for each species represent the percentage of pieces machined that successfully met

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Machinery's Handbook, 31st Edition Wood

385

the listed quality requirement for the processes. For example, 62 percent of the black walnut pieces planed came out perfect, but only 34 percent of the pieces run on the shaper achieved good to excellent results. Table 3. Machinability and Related Properties of Various Domestic Hardwoods

Type of Wood

Alder, red Ash Aspen Basswood Beech Birch Birch, paper Cherry, black Chestnut Cottonwood Elm, soft Hackberry Hickory Magnolia Maple, bigleaf Maple, hard Maple, soft Oak, red Oak, white Pecan Sweetgum Sycamore Tanoak Tupelo, black Tupelo, water Walnut, black Willow Yellow-poplar

Planing

Shaping

Perfect

Good to Excellent

61 75 26 64 83 63 47 80 74 21 33 74 76 65 52 54 41 91 87 88 51 22 80 48 55 62 52 70

Turning Boring Quality Required Fair to Good to Excellent Excellent

20 55 7 10 24 57 22 80 28 3 13 10 20 27 56 72 25 28 35 40 28 12 39 32 52 34 5 13

88 79 65 68 90 80 … 88 87 70 65 77 84 79 8 82 76 84 85 89 86 85 81 75 79 91 58 81

64 94 78 76 99 97 … 100 91 70 94 99 100 71 100 99 80 99 95 100 92 98 100 82 62 100 71 87

Mortising

Sanding

Fair to Excellent

Good to Excellent

52 58 60 51 92 97 … 100 70 52 75 72 98 32 80 95 34 95 99 98 53 96 100 24 33 98 24 63

… 75 … 17 49 34 … … 64 19 66 … 80 37 … 38 37 81 83 … 23 21 … 21 34 … 24 19

The data above represent the percentage of pieces attempted that meet the quality requirement listed.

Nominal and Minimum Sizes of Sawn Lumber

Type of Lumber Boards

Thickness (inches)

Nominal, Tn 1

11⁄4

11⁄2 2

Dimension Lumber

21⁄2 3

31⁄2

Green

3⁄4

25⁄3 2

1

11⁄4 11⁄2

2

21⁄2

11⁄32

5 to 7

19⁄16

2 to 4

29⁄16

8 to 16

39⁄16

…

21⁄16

4

41⁄16

…

Tn − 1⁄2

5 and up

2 to 4 8 to 16

31⁄16

31⁄2

Nominal, Wn

19⁄32

3

4

41⁄2 Timbers

Dry

5 to 6 … …

5 and up

Face Widths (inches) Dry

Green

Wn − 1⁄2

Wn − 7⁄16

Wn − 3⁄4

Wn − 1⁄2

Wn − 1⁄2

Wn − 7⁄16

Wn − 3⁄4

Wn − 1⁄2

…

…

Wn − 1⁄2

Wn − 1⁄2 … … …

Wn − 3⁄8

Wn − 3⁄8 … …

Wn − 1⁄2

Source: National Forest Products Association: Design Values for Wood Construction. Moisture content: dry lumber ≤ 19%; green lumber > 19%. Dimension lumber refers to lumber 2 to 4 inches thick (nominal) and 2 inches or greater in width. Timbers refers to lumber of approximately square cross section, 5 3 5 inches or larger, and a width no more than 2 inches greater than the thickness.

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Typical Properties of Ceramics Materials Densitya Material

lb/in3

Machinable Glass Ceramic

Glass-Mica

Machining Grades Molding Grades

Aluminum Silicate Alumina Silicate Silica Foam TiO2 (Titania) Lava (Grade A) Zirconium Phosphate ZrO2 ZrO2 ·SiO2 (Zircon)

0.11

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MgO·SiO2 (Steatite)

2MgO·2Al2 O3 ·5SiO2  (Cordierite) Al2 O3 (Alumina)

2.49 0.09 0.11 3.04 0.10 2.77 0.09–0.10 2.49–2.77 0.10 2.77 0.13–0.17 3.60–4.70 0.14 3.88 0.10 2.77 0.08 2.21 0.08 2.21 0.03 0.83 0.14 3.88 0.08 2.21 0.11 3.04 0.21 5.81 0.11

2MgO·SiO2 (Forsterite)

94% 96% 99.5% 99.9%

g/cm3

3.04

3.04

0.09–0.10 2.49–2.77 0.06 1.66 0.08 2.21 0.09 2.49 0.13 3.60 0.13–0.14 3.60–3.88 0.14 3.88 0.14 3.88

Dielectric Strength (V/mil) 1000 400 380 400 380 300–325 350 80 100 70 80 100 80 NA … 220

240

210–240 60 100–172 200 210 210 200 …

Coeff. of Expansionb 10−6 in/in-°F 4.1–7.0 6 5.2 10.5–11.2 9.4 11–11.5 10.3 2.5 2.9 … 0.3 4.61 1.83 0.5 6.1 1.94

5.56

3.83–5.44 0.33 1.22–1.28 1.33 3.33 3.5–3.7 3.72 3.75

10−6 m/m-°C 7.38–12.6 10.8 9.4 18.9–20.2 16.9 19.8–20.7 18.5 4.5 5.2 … 0.5 8.3 3.3 0.9 11.0 3.5

10.0

6.89–9.79 0.6 2.20–2.30 2.4 6.0 6.3–6.6 6.7 6.8

Flexural Strength (103 psi)

Mohs’s Hardnessc

15 14 12.5–13 11 9–10 9 4.5 10 … 0.4 20 9 7.5 102

48 RA 5.5 5.0 90 RH 90 RH 90 RH 90 RH 1–2 6.0 … NA 8 6 NA 1300 HV

20

7.5

16

18–21 3.4 8–12 15 44 48–60 70 72

7.5

7.5 6.5 7–7.5 8 9 9 9 9

Operating Temp. (°F) 1472 700 1100 750 1100 700–750 1300 1000 2100 2370 2000 1800 2000 2800 … 1825

1825

Tensile Strength 103 psi

MPa

… … … 6 5 6–6.5 6 … … … … 7.5 2.5 … …

… … … 41 34 41–45 41 … … … … 52 17 … …

10

69

10

69

1825 8.5–10 59–69 2000 2.5 17 2000 3.5–3.7 24–25 2000 4 28 2700 20 138 2600–2800 25 172 2700 28 193 2900 … …

Compressive Strength (103 psi) 50 40 32 40–45 32 33–35 30 12 25 … 1.4 100 40 30 261 90

85

80–90 18.5 30–40 50 315 375 380 400

Thermal Conductivityd (Btu-ft/hr-ft2-°F)

W/(m-k)

0.85 0.24 0.34 0.24–0.29 0.34 0.29–0.31 0.3 0.92 0.75 0.38 0.10 … 0.92 0.4 (approx.) 1.69

1.47 0.42 0.59 0.41–0.50 0.59 0.50–0.54 0.52 1.59 1.30 0.66 0.17 … 1.59 0.69 2.92

4.58

7.93

…

3.17–3.42 1.00 1.00 1.83 16.00 20.3–20.7 21.25 …

…

5.49–5.92 1.73 1.73 3.17 27.69 35.13–35.8 36.78 …

a Obtain specific gravity by dividing density in lb/in 3 by 0.0361; for density in lb/ft 3, multiply lb/in 3 by 1728; for kg/m 3, multiply density in lb/in 3 by 27,679.9. b To convert coefficient of expansion to 10 –6 in/in-°C, multiply table value by 1.8.

c Mohs’s Hardness scale is used unless otherwise indicated as follows: RA and RH for Rockwell A and H scales, respectively; HV for Vickers hardness. d To convert conductivity from Btu-ft/hr-ft 2 -°F to cal-cm/sec-cm 2 -°C, divide by 241.9.

Machinery's Handbook, 31st Edition Properties of Ceramics

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Tabulated Properties of Ceramics, Plastics, and Metals

Densitya Material

lb/in3

0.038 0.037 0.056 0.051 0.051 0.043 0.043 0.056 0.067 0.050 0.042 0.047 0.041 0.042 0.049 0.079 0.050 0.064 0.050 0.043 0.046 0.035 0.034 0.030 0.051

g/cm3 1.052 1.024 1.550 1.412 1.412 1.190 1.190 1.550 1.855 1.384 1.163 1.301 1.135 1.163 1.356 2.187 1.384 1.772 1.384 1.190 1.273 0.969 0.941 0.830 1.412

1.05 1.03 1.55 1.41 1.41 1.19 1.19 1.55 1.87 1.39 1.16 1.30 1.14 1.16 1.36 2.19 1.39 1.77 1.38 1.19 1.27 0.97 0.94 0.83 1.41

Dielectric Strength V/mil

… … … 380 … 500 500 … … … 295 … 600 … 1300 480 500 260 … 380 480 475 710 … 560

MV/m … … … 14.96 … 19.69 19.69 … … … 11.61 … 23.62 … 51.18 18.90 19.69 10.24 … 14.96 18.90 18.70 27.95 … 22.05

Coeff. of Expansionb 10−6 in/in-°F 53.0 … … 47.0 58.0 35.0 15.0 34.0 11.1 … 45.0 … 45.0 … 39.0 50.0 29.5 60.0 11.1 37.5 … 20.0 19.0 … …

Tensile Modulus

10−6 m/m-°C 95.4 … … 84.6 104.4 63.0 27.0 61.2 20.0 … 81.0 … 81.0 … 70.2 90.0 53.1 108.0 20.0 67.5 … 36.0 34.2 … …

103 psi 275 200 1000 437 310 400 750 400 … 1350 380 … 390 … 500 225 550 320 … 345 430 156 110 220 300

MPa

1896 1379 6895 3013 2137 2758 5171 2758 … 9308 2620 … 2689 … 3447 1551 3792 2206 … 2379 2965 1076 758 1517 2068

Izod Impact ft-lb/in of J/m of notch notch 7 … 0.9 2 … 0.5 14 3 8 2.8 1.4 … 1 2.2 0.5 3 0.8 3 2.4 14 1.1 6 No Break 2.5 1.5

373.65 … 48.04 106.76 … 26.69 747.30 160.14 427.03 149.46 74.73 … 53.38 117.43 26.69 160.14 42.70 160.14 128.11 747.30 58.72 320.27 … 133.45 80.07

Flexural Modulus ksi at MPa at % 73°F 23°C Elongation Hardnessc

300 330 715 400 320 400 800 400 1 1400 450 … … … 400 80 400 200 1000 340 480 160 130 … …

2068 2275 4930 2758 2206 2758 5516 2758 7 9653 3103 … … … 2758 552 2758 1379 6895 2344 3309 1103 896 … …

… … … 13 … 2.7 2.1 4 … … 20 … 240 … 70 350 31–40 80 … 110 … 900 450 … …

105 Rr 105 Rr 94 Rm 94 Rm 94 Rm 94 Rm 94 Rm … 101 Rm 119 Rr 100 Rr … 118 Rr … … … 110 Rr 100 Rr 100 Rm 74 Rm … … 64 Rr … …

Max. Operating Temp. °F

°C

200 … … … 200 180 311 212 260 … 210 … 230 … 230 … 170 180 248 290 … 180 176 … …

93 … … … 93 82 155 100 127 … 99 … 110 … 110 … 77 82 120 143 … 82 80 … …

Polyphenylene Sul­fide

0.047

1.301

1.30

380

14.96

…

…

…

…

0.5

26.69

550

3792

…

…

…

…

Polypropylene Polysulfone Polyurethane

0.033 0.045 0.038

0.913 1.246 1.052

0.91 1.25 1.05

600 425 …

23.62 16.73 …

96.0 31.0 …

172.8 55.8 …

155 360 …

1069 2482 …

0.75 1.2 …

40.03 64.05 …

200 390 …

1379 2689 …

120 50 465–520

92 Rr 120 Rr …

150 325 …

66 163 …

a To obtain specific gravity, divide density in lb/in 3 by 0.0361; for density in lb/ft 3, multiply lb/in 3 by 1728; for kg/m 3, multiply density in lb/in 3 by 27,679.9. b To convert coefficient of expansion to 10 –6 in/in-°C, multiply table value by 1.8.

c Hardness value scales are: RM and RR for Rockwell M and R scales, respectively.

387

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ABS, Extrusion Grade ABS, High Impact Acetal, 20% Glass Acetal, Copolymer Acetyl, Homopolymer Acrylic Azdel CPVC Fiberglass Sheet Nylon 6, 30% Glass Nylon 6, Cast Nylon 6 / 6, Cast Nylon 6 / 6, Extruded Nylon 60L, Cast PET, unfilled PTFE (Teflon) PVC PVDF Phenolics Polycarbonate Polyetherimide Polyethylene, HD Polyethylene, UHMW Polymethylpentene Polymid, unfilled

Specific Gravity

Machinery's Handbook, 31st Edition Properties of Plastics

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Typical Properties of Plastics Materials

Machinery's Handbook, 31st Edition Properties of Investment Casting Alloys

388

Mechanical Properties of Various Investment Casting Alloys Alloy Designation 356 A356 A357 355, C355 D712 (40E) A354 RR-350 Precedent 71 KO-1 Al Bronze C (954) Al Bronze D (955) Manganese Bronze, A Manganese Bronze, C Silicon Bronze Tin Bronze Leaded Yellow Brass (854) Red Brass Silicon Brass Pure Copper Beryllium Cu 10C (820) Beryllium Cu 165C (824) Beryllium Cu 20C (825) Beryllium Cu 275C (828) Chrome Copper IC 1010 IC 1020 IC 1030 IC 1035 IC 1045 IC 1050 IC 1060 IC 1090 IC 2345 IC 4130 IC 4140 IC 4150 IC 4330 IC 4340 IC 4620 IC 6150, IC 8740 IC 8620 IC 8630 IC 8640

Tensile Strength (103 psi)

0.2% Yield Strengtha (103 psi)

% Elonga­tion

Hardnessc

As Cast As Cast As Cast As Cast As Cast As Cast As Cast As Cast As Cast

32–40 38–40 33–50 35–50 34–40 47–55 32–45 35–55 56–60

22–30 28–36 27–40 28–39 25–32 36–45 24–38 25–45 48–55

3–7 3–10 3–9 1–8 4–8 2–5 1.5–5 2–5 3–5

… … … … … … … … …

As Cast Heat Treated As Cast Heat Treated … … … … … … … … As Cast Hardened … As Cast Hardened As Cast …

75–85 90–105 90–100 110–120 65–75 110–120 45 40–50 30–50 30–40 70 20–30 45–50 90–100 70–155 70–80 110–160 80–90 33–50

30–40 45–55 40–50 60–70 25–40 60–70 18 18–30 11–20 14–25 32 … 40–45 90–130 40–140 50–55 … … 20–40

10–20 6–10 6–10 5–8 16–24 8–16 20 20–35 15–25 20–30 24 4–50 15–20 3–8 1–15 18–23 1–4 15–20 20–30

80–85 RB 91–96 RB 91–96 RB 93–98 RB 60–65 RB 95–100 RB … 40–50 RB … 30–35 RB … 35–42 RB 50–55 RB 90–95 RB 60 RB–38 RC 75–80 RB 25–44 RC 80–85 RB 70–78 RB

Annealed Annealed Annealed Hardened Annealed Hardened Annealed Hardened Annealed Hardened Annealed Hardened Annealed Hardened Hardened Hardened Hardened Hardened Hardened Hardened Hardened Hardened Hardened Hardened Hardened

50–60 60–70 65–75 85–150 70–80 90–150 80–90 100–180 90–110 125–180 100–120 120–200 110–150 130–180 130–200 130–170 130–200 140–200 130–190 130–200 110–150 140–200 100–130 120–170 130–200

30–35 40–45 45–50 60–150 45–55 85–150 50–60 90–180 50–65 100–180 55–70 100–180 70–80 130–180 110–180 100–130 100–155 120–180 100–175 100–180 90–130 120–180 80–110 100–130 100–180

30–35 25–40 20–30 0–15 20–30 0–15 20–25 0–10 20–25 0–10 5–10 0–3 12–20 0–3 5–10 5–20 5–20 5–10 5–20 5–20 10–20 5–10 10–20 7–20 5–20

50–55 RB 80 RB 75 RB 20–50 RC 80 RB 25–52 RC 100 RB 25–57 RC 100 RB 30–60 RC 25 RC 30–60 RC 30 RC 37–50 RC 30–58 RC 23–49 RC 29–57 RC 25–58 RC 25–48 RC 20–55 RC 20–32 RC 30–60 RC 20–45 RC 25–50 RC 30–60 RC

Material Condition

Aluminum

Copper-Base Alloysa

Carbon and Low-Alloy Steels and Iron

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Machinery's Handbook, 31st Edition Properties of Investment Casting Alloys

389

Mechanical Properties of Various Investment Casting AlloysAlloys (Continued) Mechanical Properties of Various Investment Casting Material Condition

Tensile Strength (103 psi)

0.2% Yield Strengtha (103 psi)

% Elonga­tion

Hardnessc

Hardened Hardened Hardened Hardened … Annealed Normalized

170–220 120–170 180–230 130–170 50–60 60–80 100–120

140–200 110–150 140–180 100–140 37–43 40–50 70–80

0–10 7–20 1–7 6–12 30–35 18–24 3–10

… … 30–65 RC 25–48 RC 55 RB 143–200 BHN 243–303 BHN

CA-15 IC 416 CA-40 IC 431 IC 17-4 Am-355 IC 15-5 CD-4M Cu

Hardened Hardened Hardened Hardened Hardened Hardened Hardened Annealed Hardened

95–200 95–200 200–225 110–160 150–190 200–220 135–170 100–115 135–145

75–160 75–160 130–210 75–105 140–160 150–165 110–145 75–85 100–120

5–12 3–8 0–5 5–20 6–20 6–12 5–15 20–30 10–25

94 RB–45 RC 94 RB–45 RC 30–52 RC 20–40 RC 34–44 RC … 26–38 RC 94–100 RB 28–32 RC

CF-3, CF-3M, CF-8, CF-8M, IC 316F CF-8C CF-16F CF-20 CH-20 CN-7M IC 321, CK-20

Annealed Annealed Annealed Annealed Annealed Annealed Annealed

70–85 70–85 65–75 65–75 70–80 65–75 65–75

40–50 32–36 30–35 30–45 30–40 25–35 30–40

35–50 30–40 35–45 35–60 30–45 35–45 35–45

90 RB (max) 90 RB (max) 90 RB (max) 90 RB (max) 90 RB (max) 90 RB (max) 90 RB (max)

Annealed As Cast Annealed AC to 24°C

RH Monel Monel E M-35 Monel

AC to 816°C As Cast As Cast Annealed As Cast Annealed Hardened As Cast As Cast As Cast

75–85 80–95 75–95 63–70 35–45 50–60 65–75 80–100 65–75 100–110 120–140 100–110 65–80 65–80

50–60 45–55 45–55 41–45 … 25–30 35–40 40–55 32–38 55–65 85–100 60–80 33–40 25–35

8–12 8–12 8–12 10–15 12–20 30–40 10–20 15–30 25–35 5–10 0 10–20 25–35 25–40

90–100 RB 90–100 RB 90 RB–25 RC 85–96 RB … 50–60 RB 80–90 RB 10–20 RC 65–75 RB 20–28 RC 32–38 RC 20–30 RC 67–78 RB 65–85 RB

Cobalt 21 Cobalt 25 Cobalt 31 Cobalt 36 F75 N-155

As Cast As Cast As Cast As Cast As Cast Sol. Anneal

95–130 90–120 105–130 90–105 95–110 90–100

65–95 60–75 75–90 60–70 70–80 50–60

8–20 15–25 6–10 15–20 8–15 15–30

24–32 RC 20–25 RC 20–30 RC 30–36 RC 25–34 RC 90–100 RB

Alloy Designation IC 8665 IC 8730 IC 52100 IC 1722AS 1.2% Si Iron Ductile Iron, Ferritic Ductile Iron, Pearlitic

Carbon and Low-Alloy Steels and Iron (Continued)

Hardenable Stainless Steel

Austenitic Stainless Steels

Alloy B Alloy C Alloy Xb Invar (Fe-Ni alloy) In 600 (Inconel) In 625 (Inconel) Monel 410 S Monel

Nickel-Base Alloys

Cobalt-Base Alloys

a For copper alloys, yield strength is determined by 0.5% extension under load or 0.2% offset method. A number in parentheses following a copper alloy indicates the UNS designation of that alloy (for example, Al Bronze C (954) identifies the alloy as UNS C95400). b AC = air cooled to temperature indicated. c Hardness value scales are: RB and RC for Rockwell B and C scales, respectively, and BHN for Brinell Hardness Numbers. Source: Investment Casting Institute. Mechanical properties are average values of separately cast test bars and are for reference only. For items marked … data are not available. Alloys identi­ fied by IC followed by an SAE designation number (IC 1010 steel, for example) are generally similar to the SAE material although properties and chemical composition may be different.

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Machinery's Handbook, 31st Edition Properties of Powdered Metal Alloys

390

Typical Properties of Compressed and Sintered Powdered Metal Alloys Strength (103 psi)

Alloy Numbera  and Nominal Composition (%)

Density (g/cc)

Hardnessb

Transverse Rupture

Ultimate Tensile

Yield

% Elongation

Copper Base …

7.7–7.9

81–82 RH

54–68

24–34

…

10–26

CNZ-1818 63Cu, 17.5Ni, Bal. Zn

7.9

90 RH

73

34

20

11

CTG-1001 10Sn, 1C, Bal. Cu

6.5

CZP-3002

100Cu

70Cu, 1.5Pb, Bal. Zn

8

CTG-1004 10Sn, 4.4C, Bal. Cu

7

75 RH 67 RH 45 RH

… 20

25.8

33.9 9.4

15.1

… 6.5 9.6

24 6

9.7

Iron Base (Balance of composition, Fe) FC-2015

23.5Cu, 1.5C

FX-2008

20Cu, 1C

FC-0800

6.5

65 RB

80

52.4

48.5

0

6.3–6.8

39–55 RB

75–100

38–54

32–47

1 or less

4Ni, 1-2Cu, 0.75C

6.3–7

64–84 RB

70–107

37–63

30–47

1–1.6

FN-0005

0.45C, 0.50 MnS

6.4–6.8

66–78 RF

44–61

…

…

F-0008

0.6–0.9C

FN-0408 F-0000

F-0000

FC-0508

FN-0405

FN-0208

FN-0205

FN-0200 FC-0208

FC-2008 …

FL-4605

FL-4605

SS-316L … SS-410

FL-4608

SS-303N1

SS-304N1

8Cu, 0.4C

100Fe

0.02C, 0.45P

7.3

6.5

93 RB 26 RF

164.2 37.7

6.6–7.2

35–50 RB

90–125

0.6–0.9C, 4–6Cu

5.9–6.8

60–80 RB

100–145

2Ni, 0.8C

6.6–7.0

50–70 RB

70–108

4Ni, 0.5C

2Ni, 0.5C

2Ni, 0.25C

2Cu, 0.75C 20Cu, 1C

4Ni, 0.6C, 1.6Cu, 0.55Mo

1.8Ni, 0.6C, 1.6Cu, 0.55Mo 1.8Ni, 0.6C, 0.55Mo

17Cr, 13Ni, 2.2Mo, 0.9Si

17Cr, 13Ni, 2.2Mo, 0.9Si, 15-20Cu

6.2–7

6.6–7.0

6.6–7.0 6.6

50–70 RB

73–82 RB

51–61 RB 29 RB

72.3

15.7

29–38

58–82

50–70

35–57

90–100

47–50

57.5

11

…

61–100

72–93

57.7

47–58

35–45 25.8

38–40

7

≤7

Machinery's Handbook, 31st Edition British Standard Metric ISO Limits and Fits

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Table 6b. British Standard Fundamental Deviations for Holes BS 4500:1969

Machinery's Handbook, 31st Edition TOLERANCE ANALYSIS AND ASSIGNMENT

684

Tolerance Analysis and Assignment

All parts are produced with some amount of variation, or difference, between the intended geometry and features of the part produced. While variation is unavoidable and expected, the challenge is determining an acceptable amount. The amount and type of variation is called tolerance. Part feature tolerances are not assigned arbitrarily, but rather are specified by designers with good understanding of manufacturing process capabilities, cost limitations, and assembly and functional requirements. Good design allows for the broadest tolerances possible that meet all requirements, resulting in parts that are easier and less costly to produce. Overly restrictive tolerances limit manufacturers to unnecessarily precise methods, driving up costs, which can be impacted dramatically by small differences in tolerances. Tolerance analysis and tolerance assignment are bottom-up and top-down approaches, respectively, to calculating tolerances, and typically they are used in iterative fashion. Tolerance assignment is done at the design stage, to meet assembly and functional requirements. Tolerance analysis is performed later, on preliminary or final part tolerances, to detect any needed design improvements and changes in tolerance assignments. Predicting the results of variations may be an end in itself, but more often, the goal is to select the best tolerances throughout an assembly to meet design requirements. The Tolerance Stack-Up Chain.—Tolerance stack-up calculations analyze the effects that tolerances have on function and guide tolerance assignment. A tolerance stack-up chain diagram is created first to clearly identify the assembly dimension to be solved and each part dimension contributing to the stack-up, as is shown in Fig. 5. A

G

F

E

B

C

D

Mean (μ) (from mfg. data)

Standard Deviation (σ) (from mfg. data)

Dim

Max

Min

B

0.376

0.375

0.375464

0.000106

C

0.5906

0.5858

0.588126

0.001004

D

1.751

1.749

1.750128

0.000203

E

0.1875

0.1825

0.185147

0.000874

F

1.755

1.745

1.750748

0.001200

G

0.9395

0.9355

0.937331

0.000517

Fig. 5. Tolerance Stack-Up Chain Diagram, Related Specifications, and Dimension Data

From the diagram, a tolerance stack-up equation is derived. In dimension labeling, a sign convention denotes direction: positive for up or right; negative for down or left. The assembly dimension can then be solved, as in the following example, written in general form: A=–B–C–D+E+F+G (1) Clear, unambiguous part definition, as provided by geometric dimensioning and tolerancing, is required for accurate tolerance analysis. Care must be taken to ensure the effects of geometric variation are included in dimensions expressed numerically. Any sources of variation allowed by referenced drawings must be included in the stack-up equation. Tolerance Analysis Calculation Methods.—For solving analysis or assignment problems, four methods of tolerance stack-up calculation are common. Worst-Case: During analysis, the worst-case method predicts the largest variation in the assembly dimension and requires the smallest tolerances be assigned. Worst-case calculations are appropriate when every feature is inspected to screen from use any out-of-specification parts, and/or when there are few (approximately seven or fewer) dimensions in the chain. Benefits of this analytical method are its simplicity and speed. A tolerance stack-up calculation is performed twice, resulting in the maximum value on the first pass and the minimum value on the second pass. Thus, the general stack-up equation is expressed as two separate equations, with each dimension represented by its maximum and minimum values. The default is to put maximum values on the first line and minimum on the second; this is reversed for contributing dimensions with negative signs.

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Machinery's Handbook, 31st Edition TOLERANCE ANALYSIS AND ASSIGNMENT A max = – Bmin – Cmin – Dmin + Emax + Fmax + Gmax

685 (2a)

(2b) Example: Predict the minimum and maximum assembly dimensions for the stack-up diagram and dimensions given in Fig. 5 using the worst-case method. Solution: Using Equations (2a) and (2b): A max = – 0.375 – 0.5858 – 1.749 + 0.1875 + 1.755 + 0.9395 = 0.1722 A min = – 0 .376 – 0.5906 – 1.751 + 0.1825 + 1.745 + 0.9355 = 0.1454

A min = – Bmax – Cmax – Dmax + Emin + Fmin + G min

Statistical: This approach produces likely values and ranges, predicts smaller assembly variations, and permits larger tolerances to be assigned. While it does not guarantee conformity to requirements, conformance and defect rates can be predicted. Statistical calculations are appropriate when many dimensions are contributing to a stack-up chain or when it can be assumed that the contributing dimensions follow normal distributions, such as when statistical process control (SPC) is used (see STATISTICAL ANALYSIS OF MANUFACTURING DATA on page 136 for statistics terminology used here). The assembly dimension A is modeled as a normal distribution, with assembly mean (µA) and assembly standard deviation (σA) calculated using the general stack-up equations written in statistical form: Assembly mean (µA): µA = – µB – µC – µD + µE + µF + µG (3) Assembly standard deviation (σA):

σA =

σB2 + σC2 + σD2 + σE2 + σF2 + σG2

(4)

Example: Use the statistical method to determine the range of assembly dimension values, between which 99 percent of assembly dimensions are expected to fall, for the stack-up diagram and dimensions given in Fig. 5. Manufacturing data show the contributing dimensions are well modeled as normal distributions using the parameters included in the figure. Solution: Model assembly dimension A as a normal distribution with parameters calculated using Equations (3) and (4): µA = – 0. 375464 – 0. 588126 – 1.750128 + 0.185147 + 1.750748 + 0.937331 = 0.159508

σ A = 0.000106 2 + 0.001004 2 + 0.0002032 + 0.000874 2 + 0.001200 2 + 0.0005172 = 0.001879

99 percent of the area under the normal distribution curve falls within the range of µ±2.5758σ. Upper and lower limits of this range for assembly dimension A are calculated as: A upper = 0.159508 + ( 2.5758 )( 0.001879 ) = 0.164348

A lower = 0.159508 −

( 2.5758 )( 0.001879 )

= 0.154668

Therefore, 99 percent of assembly dimensions are expected to measure between 0.154668 and 0.164348. Root Sum of Squares (RSS): This method belongs somewhere between the worst-case and statistical methods in terms of when applicable and the desired characteristics of the result. RSS is technically a statistical method with several simplifying assumptions applied. Each contributing dimension is modeled as a normal distribution, with mean (μ) and standard deviation (σ) calculated from the maximum and minimum values. Each contributing dimension mean (μ) is calculated as: Max + Min

µ=

2

(5)

Each contributing dimension standard deviation (σ) is calculated as: σ=

Max − Min 6

(6)

The calculations for assembly mean (µA) and assembly standard deviation (σA) are the same as in the statistical method; see Equations (3) and (4). The assembly dimension is then predicted to fall between the limits:

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Machinery's Handbook, 31st Edition TOLERANCE ANALYSIS AND ASSIGNMENT

Amax = µA + 3σA (7a) Amin = µA − 3σA (7b) Example: Predict the minimum and maximum assembly dimension for the stack-up diagram and dimensions given in Fig. 5. Assume manufacturing data are not available for this example and apply the RSS method. Solution: Using Equations (5) and (6), calculate a mean and standard deviation to use as model parameters for each of the contributing dimensions in Fig. 5. Starting with B: µ = 0.376 + 0.375 = 0.3755 2

0.376 − 0.375 = 0.000167 σ= 6

Dim B C D E F G

Mean (μ) 0.3755 0.5882 1.75 0.185 1.75 0.9375

Standard Deviation (σ) 0.000167 0.000800 0.000333 0.000833 0.001667 0.000667

Repeating this calculation for dimensions C through G produces the table values shown. Next, the assembly dimension parameters are calculated using Equations (3) and (4): μA =

–0.3755 – 0.5882 – 1.75 + 0.185 + 1.75 + 0.9375 = 0.1588

σA = 0.0001672 + 0.000800 2 + 0.0003332 + 0.0008332 + 0.0016672 + 0.0006672 = 0.002167 Finally, the assembly limits are calculated using Equations (7a) and (7b): Amax = 0.1588 + ( 3 )( 0.002167 ) = 0.165301

Amin = 0.1588 − (3) ( 0.002167 ) = 0.152299

Therefore, assemblies are expected to measure between 0.152299 and 0.165301. Monte Carlo Simulation: This random-number-based calculation method attempts to model variation realistically. The simulation runs numerous cases and builds a probability distribution for the stack-up, which is used to predict results, including defect rates. For each case, contributing dimensions in the stack-up chain are randomly chosen according to their statistical distribution (normal, uniform, or otherwise). The dimensions are combined according to the stack-up equation, which may be nonlinear. Tolerance Assignment.—This is the process of distributing the “tolerance budget” among contributing dimensions of the stack-up chain to manage costs and meet target quality levels. As explained above, analysis produces one solution (assembly dimension) from several inputs (dimensions of contributing parts), whereas assignment attempts to produce several contributing part dimensions from one assembly dimension. There are infinite ways to assign part tolerances to achieve a desired assembly tolerance (thus, analytical solutions usually are not possible), with several methods for assigning tolerances being most common. Even Assignment: Part feature tolerances are assigned equally among assignable tolerances to stay within the required assembly tolerance. Proportional Scaling: Preliminary tolerances are assigned and analyzed. If the assembly tolerance exceeds the allowed amount, assignable part feature tolerances are scaled down proportionally as needed. Weight Factors: Weight factors, determined on the basis of manufacturing difficulty, are allocated to each assignable tolerance. The assembly tolerance budget is divided across each tolerance proportional to its weight factor. Cost Minimization: A cost-versus-tolerance relationship is determined for each contributing dimension. The optimization algorithm varies and compares the tolerances of each dimension to arrive at the combination that minimizes total cost. Total Cost Minimization: This method loosens tolerances to reduce cost while considering the cost of lower yields due to production of some number of defective parts.

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Machinery's Handbook, 31st Edition Preferred Numbers

687

Preferred Numbers

Preferred numbers are series of numbers selected to be used for standardization purposes in preference to any other numbers. Their use will lead to simplified practice, and they should be employed whenever possible for individual standard sizes and ratings, or for a series, in applications similar to the following: 1) Important or characteristic linear dimensions such as diameters and lengths, areas, volume, weights, capacities. 2) Ratings of machinery and apparatus in horsepower, kilowatts, kilovolt-amperes, volt­ ages, currents, speeds, power-factors, pressures, heat units, temperatures, gas or liquidflow units, weight-handling capacities, etc. 3) Characteristic ratios of figures for all kinds of units. American National Standard for Preferred Numbers.—ANSI Standard Z17.1-1973 covers basic series of preferred numbers which are independent of any measurement sys­ tem and therefore can be used with metric or customary units. This standard has been with­drawn with no superseding standard specified. The numbers are rounded values of the following five geometric series of numbers: 10N/5, 10N/10, 10N/20, 10N/40, and 10N/80, where N is an integer in the series 0, 1, 2, 3, etc. The designations used for the five series are respectively R5, R10, R20, R40, and R80, where R stands for Renard (Charles Renard, originator of the first preferred number system) and the number indicates the root of 10 on which the particular series is based. The R5 series gives 5 numbers approximately 60 percent apart, the R10 series gives 10 num­bers approximately 25 percent apart, the R20 series gives 20 numbers approximately 12 percent apart, the R40 series gives 40 numbers approximately 6 percent apart, and the R80 series gives 80 numbers approximately 3 percent apart. The number of sizes for a given purpose can be minimized by using the R5 series first and then adding sizes from the R10 and R20 series as needed. The R40 and R80 series are used principally for expressing tol­erances in sizes based on preferred numbers. Preferred numbers below 1 are formed by dividing the given numbers by 10, 100, etc., and numbers above 10 are obtained by multi­plying the given numbers by 10, 100, etc. Sizes graded according to the system may not be exactly proportional to one another because preferred numbers may differ from calculated values by +1.26 percent to –1.01 percent. Deviations from preferred numbers are used in some instances — for example, where whole numbers are needed, such as 32 instead of 31.5 for the number of teeth in a gear. Basic Series of Preferred Numbers ANSI Z17.1-1973

R5

R10

R20

R40

1.00 1.60 2.50 4.00 6.30 … … … … … … … … … … … … … … …

1.00 1.25 1.60 2.00 2.50 3.15 4.00 5.00 6.30 8.00 … … … … … … … … … …

1.00 1.12 1.25 1.40 1.60 1.80 2.00 2.24 2.50 2.80 3.15 3.55 4.00 4.50 5.00 5.60 6.30 7.10 8.00 9.00

1.00 1.06 1.12 1.18 1.25 1.32 1.40 1.50 1.60 1.70 1.80 1.90 2.00 2.12 2.24 2.36 2.50 2.65 2.80 3.00

Series Designation R40 Preferred Numbers

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3.15 3.35 3.55 3.75 4.00 4.25 4.50 4.75 5.00 5.30 5.60 6.00 6.30 6.70 7.10 7.50 8.00 8.50 9.00 9.50

R80

R80

R80

R80

1.00 1.03 1.06 1.09 1.12 1.15 1.18 1.22 1.25 1.28 1.32 1.36 1.40 1.45 1.50 1.55 1.60 1.65 1.70 1.75

1.80 1.85 1.90 1.95 2.00 2.06 2.12 2.18 2.24 2.30 2.36 2.43 2.50 2.58 2.65 2.72 2.80 2.90 3.00 3.07

3.15 3.25 3.35 3.45 3.55 3.65 3.75 3.87 4.00 4.12 4.25 4.37 4.50 4.62 4.75 4.87 5.00 5.15 5.20 5.45

5.60 5.80 6.00 6.15 6.30 6.50 6.70 6.90 7.10 7.30 7.50 7.75 8.00 8.25 8.50 8.75 9.00 9.25 9.50 9.75

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Machinery's Handbook, 31st Edition Preferred Metric Sizes

688

Preferred Metric Sizes.—American National Standard ANSI B32.4M-1980 (R1994), presents series of preferred metric sizes for round, square, rectangular, and hexagonal metal products. Table 1 gives preferred metric diameters from 1 to 320 millimeters for round metal products. Wherever possible, sizes should be selected from the Preferred Series shown in the table. A Second Preference Series is also shown. A Third Preference Series not shown in the table is: 1.3, 2.1, 2.4, 2.6, 3.2, 3.8, 4.2, 4.8, 7.5, 8.5, 9.5, 36, 85, and 95. This standard has now been consolidated into ANSI/ASME B32.100-2016, see Metric Sizes for Flat Metal Products on page 2708. ANSI/ASME B4.2-1978 (R2009) states that the basic size of mating parts should be chosen from the first choice sizes listed in Table 1. Most of the Preferred Series sizes are derived from the American National Standard “10 series” of preferred numbers (see American National Standard for Preferred Numbers on page 687). Most of the Second Preference Series are derived from the “20 series” of preferred numbers. Third Preference sizes are generally from the “40 series” of preferred numbers. For preferred metric diameters less than 1 millimeter, preferred across flat metric sizes of square and hexagon metal products, preferred across flat metric sizes of rectangular metal products, and preferred metric lengths of metal products, reference should be made to the Standard. Table 1. American National Standard Preferred Metric Sizes ANSI/ASME B4.2-1978 (R2009)

Basic Size, mm

1st Choice 1 … 1.2 … 1.6 … 2 … 2.5 … 3 … 4 … 5 …

2nd Choice … 1.1 … 1.4 … 1.8 … 2.2 … 2.8 … 3.5 … 4.5 … 5.5

Basic Size, mm

1st Choice 6 … 8 … 10 … 12 … 16 … 20 … 25 … 30 …

2nd Choice … 7 … 9 … 11 … 14 … 18 … 22 … 28 … 35

Basic Size, mm

1st Choice 40 … 50 … 60 … 80 … 100 … 120 … 160 … 200 …

2nd Choice … 45 … 55 … 70 … 90 … 110 … 140 … 180 … 220

Basic Size, mm

1st Choice 250 … 300 … 400 … 500 … 600 … 800 … 1000 … … …

2nd Choice … 280 … 350 … 450 … 550 … 700 … 900 … … … …

Preferred Metric Sizes for Metal Products.—See Metric Sizes for Flat Metal Products on page 2708.

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Machinery's Handbook, 31st Edition British Standard Preferred Numbers and Sizes

689

British Standard Preferred Numbers and Preferred Sizes.—This British Standard, PD 6481:1977 1983, gives recommendations for the use of preferred numbers and pre­ ferred sizes for functional characteristics and dimensions of various products. The preferred number system is internationally standardized in ISO 3. It is also referred to as the Renard, or R, series (see American National Standard for Preferred Numbers on page 687). The series in the preferred number system are geometric series, that is, there is a constant ratio between each figure and the succeeding one, within a decimal framework. Thus, the R5 series has five steps between 1 and 10, the R10 series has 10 steps between 1 and 10, the R20 series, 20 steps, and the R40 series, 40 steps, giving increases between steps of approximately 60, 25, 12, and 6 percent, respectively. The preferred size series have been developed from the preferred number series by rounding off the inconvenient numbers in the basic series and adjusting for linear measure­ment in millimeters. These series are shown in Table 2. After taking all normal considerations into account, it is recommended that (a) for ranges of values of the primary functional characteristics (outputs and capacities) of a series of products, the preferred number series R5 to R40 (see page 687) should be used, and (b) whenever linear sizes are concerned, the preferred sizes as given in the following table should be used. The presentation of preferred sizes gives designers and users a logical selection and the benefits of rational variety reduction. The second-choice size given should only be used when it is not possible to use the first choice, and the third choice should be applied only if a size from the second choice cannot be selected. With this procedure, common usage will tend to be concentrated on a limited range of sizes, and a contribution is thus made to variety reduction. However, the decision to use a particular size cannot be taken on the basis that one is first choice and the other not. Account must be taken of the effect on the design, the availability of tools, and other rele­vant factors. Table 2. British Standard Preferred Sizes, PD 6481: 1977 (1983)

Choice 1st 2nd 3rd 1

1.2

1.1 1.4

1.6 1.8 2 2.2 2.5 2.8 3 3.5 4 4.5 5

1st

Choice 2nd 3rd 5.5

1.3

6

1.5

6.5

1.7

7

1.9

8

2.1

9

2.4

10

2.6

12

3.2 3.8 4.2 4.8

5.2

5.8 6.2 6.8 7.5 8.5 9.5

11 14

16 18 20 22

1st

Choice 2nd 3rd 23 24

25 30

28 32

35 40

38 42

45 13 15 17 19 21

50

48 52

55 60

58 62

26

34 36

1st 65

70

Choice 2nd 3rd 68 72

75 80

78 85

44

90

46

95 100

54

105

56

110

64

120

66

Choice 2nd 3rd 125

130 74 76 82 88 92 98 102 108

115

1st

112 118

135 140 145 150 155 160 165 170 175 180 185

122

128 132 138 142 148 152 158 162 168 172 178 182

1st

Choice 2nd 3rd

188

190 195 200 210 220 230 240 250 260 270 280 290 300

192 198 205 215 225 235 245 255 265 275 285 295

For dimensions above 300, each series continues in a similar manner, i.e., the intervals between each series number are the same as between 200 and 300.

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Machinery's Handbook, 31st Edition Measuring Instruments

690

MEASURING, INSTRUMENTS, AND INSPECTION METHODS Reading Verniers and Micrometers

Reading a Vernier.—A general rule for taking readings with a vernier scale is as follows: Note the number of inches and subdivisions of an inch that the zero mark of the vernier scale has moved along the true scale, and then add to this reading as many thousandths, hundredths, or whatever fractional part of an inch the vernier reads to, as there are spaces between the vernier zero and that line on the vernier coinciding with one on the true scale. For example, if the zero line of a vernier that reads to thousandths is slightly beyond the 0.5 inch division on the main or true scale, as shown in Fig. 1, and graduation line 10 on the vernier exactly coincides with one on the true scale, the reading is 0.5 + 0.010 or 0.510 inch. In order to determine the reading or fractional part of an inch that can be obtained by a vernier, multiply the denominator of the finest subdivision given on the true scale by the total number of divisions on the vernier. For example, if one inch on the true scale is divided into 40 parts or fortieths (as in Fig. 1), and the vernier into twenty-five parts, the vernier will read to thousandths of an inch, as 25 3 40 = 1000. Similarly, if there are sixteen divisions to the inch on the true scale and a total of eight on the vernier, the latter will enable readings to be taken within 1⁄128 of an inch, as 8 3 16 = 128.

0

1

2

5

0

6

7

5

8

10

9

15

1

20

1

2

5

25

Fig. 1. Inch Vernier

3

0

20

10

0

10

15 0 45 30 60

20

30

15 30 45

40

60

50

6

0

Fig. 2. Protractor with Vernier Scale

If the vernier is on a protractor, note the whole number of degrees passed by the vernier zero mark and then count the spaces between the vernier zero and the line coinciding with a graduation on the protractor scale. If the vernier indicates angles within five minutes or one-twelfth degree (as in Fig. 2), the number of spaces multiplied by 5 will, of course, give the number of minutes to be added to the whole number of degrees. The reading of the protractor set as illustrated would be 14 whole degrees (the number passed by the zero mark on the vernier) plus 30 minutes, as the graduation 30 on the vernier is the only one to the right of the vernier zero that exactly coincides with a line on the protractor scale. It will be

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Machinery's Handbook, 31st Edition Measuring Instruments

691

noted that there are duplicate scales on the vernier, one being to the right and the other to the left of zero. The left-hand scale is used when the vernier zero is moved to the left of the zero of the protractor scale, whereas the right-hand graduations are used when the movement is to the right. Reading a Metric Vernier.—The smallest graduation on the bar (true or main scale) of the metric vernier gage shown in Fig. 1, is 0.5 millimeter. The scale is numbered at each twentieth division, and thus increments of 10, 20, 30, 40 millimeters, etc., are indicated. There are 25 divisions on the vernier scale, occupying the same length as 24 divisions on the bar, which is 12 millimeters. Therefore, one division on the vernier scale equals one twenty-fifth of 12 millimeters = 0.04 3 12 = 0.48 millimeter. Thus, the difference between one bar division (0.50 mm) and one vernier division (2.48 mm) is 0.50 – 0.48 = 0.02 milli­meter, which is the minimum measuring increment that the gage provides. To permit direct readings, the vernier scale has graduations to represent tenths of a millimeter (0.1 mm) and fiftieths of a millimeter (0.02 mm).

30

40

0

50

70

10 20 30 40 50

Fig. 1. Metric Vernier

To read a vernier gage, first note how many millimeters the zero line on the vernier is from the zero line on the bar. Next, find the graduation on the vernier scale that exactly coincides with a graduation line on the bar, and note the value of the vernier scale gradua­tion. This value is added to the value obtained from the bar, and the result is the total read­ing. In the example shown in Fig. 1, the vernier zero is just past the 40.5 millimeters gradua­ tion on the bar. The 0.18 millimeter line on the vernier coincides with a line on the bar, and the total reading is therefore 40.5 + 0.18 = 40.68 mm. Dual Metric-Inch Vernier.—The vernier gage shown in Fig. 2 has separate metric 50-division, and inch 25-division vernier scales to permit measurements in either system. On the bar, the smallest metric graduation is 1 millimeter, and the 50 divisions of the vernier occupy the same length as 49 divisions on the bar, which is 49 mm. Therefore, one division on the vernier scale equals one-fiftieth of 49 millimeters = 0.02 × 49 = 0.98 mm. Thus, the difference between one bar division (1.0 mm) and one vernier division (0.98 mm) is 0.02 mm, which is the minimum measuring increment the gage provides. The vernier scale is graduated for direct reading to 0.02 mm. In Fig. 2, the vernier zero is just past the 27 mm graduation on the bar, and the 0.42 mm graduation on the vernier coincides with a line on the bar. The total reading is therefore 27.42 mm.

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Machinery's Handbook, 31st Edition Measuring Instruments

692

The smallest inch graduation on the bar is 0.025 inch, and the 25 vernier divisions occupy the same length as 49 bar divisions, which is 1.225 inches. Therefore, one vernier division equals one twenty-fifth of 1.225 inches = 0.04 × 1.225 = 0.049 inch. Thus, the difference between the length of two bar divisions and a vernier division is 0.050 − 0.049 = 0.001 inch. The vernier scale is graduated for direct reading to 0.001 inch. In the example, the vernier zero is past the 1.075 graduation on the bar, and the 0.004 graduation on the vernier coincides with a line on the bar. Thus, the total reading is 1.079 inches.

Fig. 2. Dual Metric-Inch Vernier

15 0

1

2

10

0 1 23

Reading a Micrometer.—The spindle of an inch-system micrometer has 40 threads per inch, so that one turn moves the spindle axially 0.025 inch (1 4 40 = 0.025), equal to the distance between two graduations on the frame. The 25 graduations on the thimble allow the 0.025 inch to be further divided, so that turning the thimble through one division moves the spindle axially 0.001 inch (0.025 4 25 = 0.001). To read a micrometer, count the num­b er of whole divisions visible on the scale of the frame, multiply this number by 25 (the number of thousandths of an inch that each division represents) and add to the product the number of that division on the thimble coinciding with the axial zero line on the frame. The result will be the diameter expressed in thousandths of an inch. As the numbers 1, 2, 3, etc., opposite every fourth subdivision on the frame indicate hundreds of thou­sandths, the reading can easily be taken mentally. Suppose the thimble were screwed out so that graduation 2 and three additional subdivisions were visible (as shown in Fig. 3), and that graduation 10 on the thimble coincided with the axial line on the frame. The reading then would be 0.200 + 0.075 + 0.010, or 0.285 inch.

0

0 1

2

20

5 Fig. 3. Inch Micrometer

Fig. 4. Inch Micrometer with Vernier

Some micrometers have a vernier scale on the frame in addition to the regular gradua­ tions so that measurements within 0.0001 part of an inch can be taken. Micrometers of this type are read as follows: First determine the number of thousandths, as with an ordinary micrometer, and then find a line on the vernier scale that exactly coincides with one on the thimble; the number of this line represents the number of ten-thousandths to be added to the number of thousandths obtained by the regular graduations. The reading shown in the illustration, Fig. 4, is 0.275 + 0.020 + 0.0003 = 0.2953 inch.

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Machinery's Handbook, 31st Edition Sine-Bars

693

Micrometers graduated according to the English system of measurement ordinarily have a table of decimal equivalents stamped on the sides of the frame so that fractions such as sixty-fourths, thirty-seconds, etc., can readily be converted into decimals. Reading a Metric Micrometer.—The spindle of an ordinary metric micrometer has 2 threads per millimeter, and thus one complete revolution moves the spindle through a dis­tance of 0.5 millimeter. The longitudinal line on the frame is graduated with 1 millimeter divisions and 0.5 millimeter subdivisions. The thimble has 50 graduations, each being 0.01 millimeter (one-hundredth of a millimeter). To read a metric micrometer, note the number of millimeter divisions visible on the scale of the sleeve, and add the total to the particular division on the thimble that coincides with the axial line on the sleeve. Suppose that the thimble were screwed out so that gradu­ ation 5 and one additional 0.5 subdivision were visible (as shown in Fig. 5), and that grad­ uation 28 on the thimble coincided with the axial line on the sleeve. The reading then would be 5.00 + 0.5 + 0.28 = 5.78 mm. 0

5

30 25

Fig. 5. Metric Micrometer

To permit measurements within 0.002 millimeter to be made, some micrometers are provided with a vernier scale on the sleeve in addition to regular graduations. Micrometers of this type are read as follows: First, as with an ordinary micrometer, determine the number of whole millimeters (if any) and the number of hundredths of a millimeter, and then find a line on the sleeve vernier scale that exactly coincides with one on the thimble. The number of this coinciding vernier line represents the number of two-thousandths of a mil­limeter to be added to the reading already obtained. Thus, for example, a measurement of 2.958 millimeters would be obtained by reading 2.5 millimeters on the sleeve, adding 0.45 millimeter read from the thimble, and then adding 0.008 millimeter as determined by the vernier. Note: 0.01 millimeter = 0.000393 inch, and 0.002 millimeter = 0.000078 inch (78 mil­ lionths). Therefore, metric micrometers provide smaller measuring increments than com­parable inch unit micrometers—the smallest graduation of an ordinary inch-reading micrometer is 0.001 inch; the vernier type has graduations down to 0.0001 inch. When using either a metric or inch micrometer without a vernier, smaller readings than those graduated may, of course, be obtained by visual interpolation between graduations. Sine-Bar Use The sine-bar is used either for very accurate angular measurements or for locating work at a given angle as, for example, in surface grinding templets, gages, etc. The sine-bar is especially useful in measuring or checking angles when the limit of accuracy is 5 minutes or less. Some bevel protractors are equipped with verniers which read to 5 minutes, but the setting depends upon the alignment of graduations, whereas a sine-bar usually is located by positive contact with precision gage-blocks selected for whatever dimension is required for obtaining a given angle. Types of Sine-Bars.—A sine-bar consists of a hardened, ground and lapped steel bar with very accurate cylindrical plugs of equal diameter attached to or near each end. The form illustrated by Fig. 1 has notched ends for receiving the cylindrical plugs so that they are held firmly against both faces of the notch. The standard center-to-center distance C between the plugs is either 5 or 10 inches. The upper and lower sides of sine-bars are paral­lel to the center line of the plugs within very close limits. The body of the sine-bar ordi­narily

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Machinery's Handbook, 31st Edition Sine-Bars

694

has several through holes to reduce the weight. In the making of the sine-bar shown in Fig. 2, if too much material is removed from one locating notch, regrinding the shoulder at the opposite end would make it possible to obtain the correct center distance. That is the reason for this change in form. The type of sine-bar illustrated by Fig. 3 has the cylindrical disks or plugs attached to one side. These differences in form or arrangement do not, of course, affect the principle governing the use of the sine-bar. An accurate surface plate or master flat is always used in conjunction with a sine-bar in order to form the base from which the vertical measurements are made. F

D A

C H

A

E

H E

Fig. 1.

Fig. 2.

H

G y

x Fig. 3.

x

a b

y

H Fig. 4.

Setting a Sine-Bar to a Given Angle.—To find the vertical distance H for setting a sine-bar to the required angle, convert the angle to decimal form on a pocket calculator, take the sine of that angle, and multiply by the distance between the cylinders. For example, if an angle of 31 degrees, 30 minutes is required, the equivalent angle is 31 degrees plus 30 ⁄60 = 31 + 0.5, or 31.5 degrees. (For conversions from minutes and seconds to decimals of degrees and vice versa, see page 103). The sine of 31.5 degrees is 0.5225, and multiplying this value by the sine-bar length gives 2.613 inches for the height H, Fig. 1 and Fig. 3, of the gage blocks. Finding Angle when Height H of Sine-Bar is Known.—To find the angle equivalent to a given height H, reverse the above procedure. Thus, if the height H is 1.4061 in., dividing by 5 gives a sine of 0.28122, which corresponds to an angle of 16.333 degrees, or 16 degrees 20 minutes. Checking Angle of Templet or Gage by Using Sine-Bar.—Place templet or gage on sine-bar as indicated by dotted lines, Fig. 1. Clamps may be used to hold work in place. Place upper end of sine-bar on gage blocks having total height H corresponding to the required angle. If upper edge D of work is parallel with surface plate E, then angle A of work equals angle A to which sine-bar is set. Parallelism between edge D and surface plate may be tested by checking the height at each end with a dial gage or some type of indicating comparator. Measuring Angle of Templet or Gage with Sine-Bar.—To measure such an angle, adjust height of gage blocks and sine-bar until edge D, Fig. 1, is parallel with surface plate E; then find angle corresponding to height H of gage blocks. For example, if height H is

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Machinery's Handbook, 31st Edition Sine-Bars

695

2.5939 inches when D and E are parallel, the calculator will show that the angle A of the work is 31 degrees, 15 minutes. Checking Taper per Foot with Sine-Bar.—As an example, assume that the plug gage in Fig. 2 is supposed to have a taper of 61 ⁄8 inches per foot and taper is to be checked by using a 5-inch sine-bar. The table of Tapers per Foot and Corresponding Angles on page 698 shows that the included angle for a taper of 6 1 ⁄8 inches per foot is 28 degrees 38 minutes 1 second, or 28.6336 degrees from the calculator. For a 5-inch sine-bar, the calculator gives a value of 2.396 inch for the height H of the gage blocks. Using this height, if the upper surface F of the plug gage is parallel to the surface plate, the angle corresponds to a taper of 61 ⁄8 inches per foot. Setting Sine-Bar having Plugs Attached to Side.—If the lower plug does not rest directly on the surface plate, as in Fig. 3, the height H for the sine-bar is the difference between heights x and y, or the difference between the heights of the plugs; otherwise, the procedure in setting the sine-bar and checking angles is the same as previously described. Checking Templets Having Two Angles.—Assume that angle a of templet, Fig. 4, is 9 degrees, angle b is 12 degrees, and that edge G is parallel to the surface plate. For an angle b of 12 degrees, the calculator shows that the height H is 1.03956 inches. For an angle a of 9 degrees, the difference between measurements x and y when the sine-bar is in contact with the upper edge of the templet is 0.78217 inch. Sine-Bar Tables to Set Sine-Bars to Given Angle.—Machinery’s Handbook 31 Digital Edition con­tains tables that give constants for sine-bars of 2.5 to 10 inches and 75 to 150 mm length. These constants represent the vertical height H for setting a sine-bar of the corresponding length to the required angle. Using Sine-Bar Tables with Sine-Bars of Other Lengths.—A sine-bar may sometimes be preferred that is longer (or shorter) than that given in available tables because of its lon­ger working surface or because the longer center distance is conducive to greater precision. To use the sine-bar tables with a sine-bar of another length to obtain the vertical distances H, multiply the value obtained from the table by the fraction (length of sine-bar used 4 length of sine-bar specified in table). Example: Use the 5-inch sine-bar table to obtain the vertical height H for setting a 10-inch sine-bar to an angle of 39 degrees. The sine of 39 degrees is 0.62932, hence the vertical height H for setting a 10-inch sine-bar is 6.2932 inches. Solution: The height H given for 39 degrees in the 5-inch sine-bar table (Constants for 5-Inch Sine-Bar in the ADDITIONAL material in the Machinery’s Handbook 31 Digital Edition) is 3.14660. The corresponding height for a 10-inch sine-bar is 10 ⁄5 3 3.14660 = 6.2932 inches. Using a Calculator to Determine Sine-Bar Constants for a Given Angle.—The con­ stant required to set a given angle for a sine-bar of any length can be quickly determined by using a scientific calculator. The required formulas are as follows: a) angle A given in degrees and calculator is set to measure angles in radian

π H = L # sin ` A # 180 j

or

a) angle A is given in radian, or b) angle A is given in degrees and calculator is set to measure angles in degrees

H = L # sin ^ Ah

where L = length of the sine-bar   A = angle to which the sine-bar is to be set H = vertical height to which one end of sine-bar must be set to obtain angle A π = 3.141592654

In the previous formulas, the height H and length L must be given in the same units but may be in either metric or US units. Thus, if L is given in mm, then H is in mm; and, if L is given in inches, then H is in inches.

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Machinery's Handbook, 31st Edition Tapers

696

Measuring Tapers with V-Block and Sine-Bar.—The taper on a conical part may be checked or found by placing the part in a V-block that rests on the surface of a sine-plate or sine-bar as shown in the accompanying diagram. The advantage of this method is that the axis of the V-block may be aligned with the sides of the sine-bar. Thus, when the tapered part is placed in the V-block, it will be aligned perpendicular to the transverse axis of the sine-bar.

A

D D 2

C

B

The sine-bar is set to angle B = (C + A/2) where A/2 is one-half the included angle of the tapered part. If D is the included angle of the precision V-block, the angle C is calculated from the formula:

sin C =

sin (A ⁄ 2) sin (D ⁄ 2)

If dial indicator readings show no change across all points along the top of the taper sur­ face, then this checks that the angle A of the taper is correct. If the indicator readings vary, proceed as follows to find the actual angle of taper: 1) Adjust the angle of the sine-bar until the indicator reading is constant. Then find the new angle B ′ as explained in the paragraph Measuring Angle of Templet or Gage with Sine-Bar on page 694; and Using the angle B ′ calculate the actual half-angle A ′/2 of the taper from the formula:

Al sin B l ta n 2 = D csc 2 + cos B l

The taper per foot corresponding to certain half-angles of taper may be found in the table on page 698. Dimensioning Tapers.—At least three methods of dimensioning tapers are in use. Standard Tapers: Give one diameter or width, the length, and insert note on drawing des­ignating the taper by number. Special Tapers: In dimensioning a taper when the slope is specified, the length and only one diameter should be given or the diameters at both ends of the taper should be given and length omitted. Precision Work: In certain cases where very precise measurements are necessary, the taper surface, either external or internal, is specified by giving a diameter at a certain dis­ tance from a surface and the slope of the taper.

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Machinery's Handbook, 31st Edition Angles and Tapers

697

Accurate Measurement of Angles and Tapers

When great accuracy is required in the measurement of angles, or when originating tapers, disks are commonly used. The principle of the disk method of taper measurement is that if two disks of unequal diameters are placed either in contact or a certain distance apart, lines tangent to their peripheries will represent an angle or taper, the degree of which depends upon the diameters of the two disks and the distance between them. A C

B

B1

α

A1

The gage shown in the accompanying illustration, which is a form commonly used for originating tapers or measuring angles accurately, is set by means of disks. This gage con­ sists of two adjustable straight edges A and A1, which are in contact with disks B and B1. The angle α or the taper between the straight edges depends, of course, upon the diameters of the disks and the center distance C, and, as these three dimensions can be measured accu­rately, it is possible to set the gage to a given angle within very close limits. Moreover, if a record of the three dimensions is kept, the exact setting of the gage can be reproduced quickly at any time. The following rules may be used for adjusting a gage of this type and cover all problems likely to arise in practice. Disks are also occasionally used for the set­ ting of parts in angular positions when they are to be machined accurately to a given angle: the rules are applicable to these conditions also. Measuring Dovetail Slides.—Dovetail slides that must be machined accurately to a given width are commonly gaged by using pieces of cylindrical rod or wire and measuring as indicated by the dimensions x and y of the accompanying illustrations. y

x

e 

h

D a

D



c

b

The rod or wire used should be small enough so that the point of contact e is somewhat below the corner or edge of the dovetail. To obtain dimension x for measuring male dovetails, add 1 to the cotangent of one-half the dovetail angle α, multiply by diameter D of the rods used, and add the product to dimension α. 1 x = D a 1 + cot 2 αk + a c = h # cot α To obtain dimension y for measuring a female dovetail, add 1 to the cotangent of onehalf the dovetail angle α, multiply by diameter D of the rod used, and subtract the result from dimension b. Expressing these rules as formulas: 1 y = b − Da 1 + cot 2 αk

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Machinery's Handbook, 31st Edition Angles and Tapers

698 Taper per Foot 1 ⁄6 4 1 ⁄3 2 1 ⁄16 3 ⁄3 2 1 ⁄8 5 ⁄3 2 3 ⁄16 7⁄3 2 1 ⁄4 9 ⁄3 2 5 ⁄16 11 ⁄3 2 3 ⁄8 13 ⁄3 2 7⁄16 15 ⁄3 2 1 ⁄2 17⁄3 2 9 ⁄16 19 ⁄3 2 5 ⁄8 21 ⁄3 2 11 ⁄16 23 ⁄3 2 3 ⁄4 25 ⁄3 2 13 ⁄16 27⁄3 2 7⁄8 29 ⁄3 2 15 ⁄16 31 ⁄3 2

1

11 ⁄16 11 ⁄8

13⁄16

11 ⁄4

15⁄16

13⁄8

17⁄16

11 ⁄2

19⁄16

15⁄8

111 ⁄16 13⁄4

113⁄16

Tapers per Foot and Corresponding Angles Included Angle 0.074604° 0.149208° 0.298415 0.447621 0.596826 0.746028 0.895228 1.044425 1.193619

0° 0

Angle with Center Line 4 ′ 8

0

17

0

35

0 0 0 1 1

26 44 53 2

11

29″ 0° 57 0 54

0

49

0

51 46 43 40

37

0 0 0 0

0

Taper per Foot 17⁄8

2 ′ 4

14″ 29

13

26

21 ⁄8

23

23⁄8

8

17 22 26 31

35

57

54

115⁄16 2 21 ⁄4

51

21 ⁄2

49

23⁄4

20

25⁄8

1

20

34

0

40

17

27⁄8

1.641173

1

38

28

0

49

14

31 ⁄8

1.939516

1

56

22

0

58

1.342808 1.491993 1.790347 2.088677 2.237832 2.386979 2.536118

2.685248 2.834369 2.983481 3.132582 3.281673 3.430753 3.579821 3.728877 3.877921 4.026951 4.175968 4.324970 4.473958 4.622931

1 1 2 2 2

2

2 2 2 3

29 47 5

14 23

32

41 50 59

4 4 4 4

13

10

7 4 1

51

34

4

16

25

3 3

19

57

16

3

25

7

3 3

31

43 52 1

10 19 28 37

54 47 44 41 37 33 30 26 23

0 0 1 1 1

1

1 1 1 1 1 1 1 1 1 2 2 2

44 53

11

37

20

33

29

30

38

27

16

25 33 42 47 51 56 0 5 9

5.367550

5

22

3

2

41

5

57

6.260490 6.557973 6.855367 7.152669 7.449874 7.746979 8.043980 8.340873 8.637654

5 6 6 6 7 7 7 8 8 8

39 15 33 51 9

27

55 47 38 29 19 10 0

44

49

20

27

2

38

38 16

2

41 ⁄8

59

2 2 2 3

32 49 58 7

4

41 ⁄4

43⁄8

41 ⁄2

55

45⁄8

52

47⁄8 5

24 20

43⁄4

49

51 ⁄8

45

53⁄8

17

9

18

23

5.962922

5

37⁄8

13

2

5.665275

35⁄8

14

19

11

8

2 2

33⁄8

33⁄4

7

46

4

11

31 ⁄4

31 ⁄2

4

5

43

3

40

2

4.771888

5.069753

46

51 ⁄4

51 ⁄2

41

55⁄8

6

57⁄8 6

2

53⁄4

57

61 ⁄8

49

63⁄8

53

61 ⁄4

3

16

44

61 ⁄2

3

34

35

63⁄4

3 3 3 4 4 4

25 43 52 1

10 19

40

65⁄8

30

67⁄8

19

71 ⁄8

8

73⁄8

25 14

7

71 ⁄4

Included Angle 8.934318° 9.230863° 9.527283

8° 9 9

10.119738

10

11.302990

11

10.711650 11.893726

Angle with Center Line 56 ′ 4″ 13 51 31 38 7

11

18

11

10

42 42

11

53 37

4° 4

28 ′ 36

5

3

4

45

49

5

21

21

5

56

5

39

12.483829

12

29

2

6

14

13.662012

13

39 43

6

49

13.073267

13

4 24

6

14.250033

14

15

0

7

15.423785

15

25 26

7

14.837300 16.009458 16.594290 17.178253 17.761318 18.343458

14 16

35 39

17

45 41

17 18

18

20.084047

20

20.662210

0 34

16

18.924644

19.504850

50 14

19

10 42 20 36

55 29

30 17 5

3

2″ 56

32

36 5

49

31 12 52

7

30

42

43

8

17

50

8

52

7 8 8 9

25 0

35 10

7

17 21 50 18

9

27

44

10

2

31

9

45

9

20

39 44

10

19

52

21.815324

21

48 55

10

54

28

22.963983

22

57 50

11

28

55

21.239311

22.390223

21

22

14 22

23 25

23.536578

23

32 12

24.678175

24 25

24.107983 25.247127 25.814817 26.381221 26.946316 27.510079

24 25 26 26 27

28.072487

28

29.193151

29

28.633518 29.751364 30.308136 30.863447 31.417276 31.969603

40 41

12

20

48 53

12

54

6 29

14 50 22 52 56 47

12

12 13

3

37 11

11

42 6

14

21 25 27

26

13

28

4 21

14

2

10

11 35

14

35

48

18 29

15

9

15

25

15

30 36 1

29

45

5

30

51 48

31

11

46

38

31

11

37

11

28

30

10

2

58

11

4

11

13 14 14

45 19 52

15

25

15

59

42

23 18 0

32

54 31

5

32.520409

32

31 13

16

15

37

33.617383

33

37

16

48

31

33.069676 34.163514

33 34

3

9 49

16 17

32

4

5

54

Taper per foot represents inches of taper per foot of length. For conversions into decimal degrees and radians see Conversion Tables of Angular Measure on page 103.

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1⁄

16

3⁄

32

1⁄ 8

1⁄ 4

3⁄ 8

Taper per Foot 1⁄ 2

0.600

5⁄ 8

3⁄ 4

1

1 1 ⁄4

0.0033 0.0065 0.0130 0.0195 0.0260 0.0326 0.0391 0.0456 0.0521 0.0586 0.0651 0.0716 0.0781 0.0846 0.0911 0.0977 0.1042 0.2083 0.3125 0.4167 0.5208 0.6250 0.7292 0.8333 0.9375 1.0417 1.1458 1.2500

699

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0.0002 0.0002 0.0003 0.0007 0.0010 0.0013 0.0016 0.0016 0.0020 0.0026 0.0003 0.0005 0.0007 0.0013 0.0020 0.0026 0.0031 0.0033 0.0039 0.0052 0.0007 0.0010 0.0013 0.0026 0.0039 0.0052 0.0063 0.0065 0.0078 0.0104 0.0010 0.0015 0.0020 0.0039 0.0059 0.0078 0.0094 0.0098 0.0117 0.0156 0.0013 0.0020 0.0026 0.0052 0.0078 0.0104 0.0125 0.0130 0.0156 0.0208 0.0016 0.0024 0.0033 0.0065 0.0098 0.0130 0.0156 0.0163 0.0195 0.0260 0.0020 0.0029 0.0039 0.0078 0.0117 0.0156 0.0188 0.0195 0.0234 0.0313 0.0023 0.0034 0.0046 0.0091 0.0137 0.0182 0.0219 0.0228 0.0273 0.0365 0.0026 0.0039 0.0052 0.0104 0.0156 0.0208 0.0250 0.0260 0.0313 0.0417 0.0029 0.0044 0.0059 0.0117 0.0176 0.0234 0.0281 0.0293 0.0352 0.0469 0.0033 0.0049 0.0065 0.0130 0.0195 0.0260 0.0313 0.0326 0.0391 0.0521 0.0036 0.0054 0.0072 0.0143 0.0215 0.0286 0.0344 0.0358 0.0430 0.0573 0.0039 0.0059 0.0078 0.0156 0.0234 0.0313 0.0375 0.0391 0.0469 0.0625 0.0042 0.0063 0.0085 0.0169 0.0254 0.0339 0.0406 0.0423 0.0508 0.0677 0.0046 0.0068 0.0091 0.0182 0.0273 0.0365 0.0438 0.0456 0.0547 0.0729 0.0049 0.0073 0.0098 0.0195 0.0293 0.0391 0.0469 0.0488 0.0586 0.0781 1 0.0052 0.0078 0.0104 0.0208 0.0313 0.0417 0.0500 0.0521 0.0625 0.0833 2 0.0104 0.0156 0.0208 0.0417 0.0625 0.0833 0.1000 0.1042 0.1250 0.1667 3 0.0156 0.0234 0.0313 0.0625 0.0938 0.1250 0.1500 0.1563 0.1875 0.2500 4 0.0208 0.0313 0.0417 0.0833 0.1250 0.1667 0.2000 0.2083 0.2500 0.3333 5 0.0260 0.0391 0.0521 0.1042 0.1563 0.2083 0.2500 0.2604 0.3125 0.4167 6 0.0313 0.0469 0.0625 0.1250 0.1875 0.2500 0.3000 0.3125 0.3750 0.5000 7 0.0365 0.0547 0.0729 0.1458 0.2188 0.2917 0.3500 0.3646 0.4375 0.5833 8 0.0417 0.0625 0.0833 0.1667 0.2500 0.3333 0.4000 0.4167 0.5000 0.6667 9 0.0469 0.0703 0.0938 0.1875 0.2813 0.3750 0.4500 0.4688 0.5625 0.7500 10 0.0521 0.0781 0.1042 0.2083 0.3125 0.4167 0.5000 0.5208 0.6250 0.8333 11 0.0573 0.0859 0.1146 0.2292 0.3438 0.4583 0.5500 0.5729 0.6875 0.9167 12 0.0625 0.0938 0.1250 0.2500 0.3750 0.5000 0.6000 0.6250 0.7500 1.0000 The amount of taper in a certain length of inches is equal to the taper per foot divided by 12 then multiplied by the given length of the tapered part. 1⁄ 32 1⁄ 16 1⁄ 8 3⁄ 16 1⁄ 4 5⁄ 16 3⁄ 8 7⁄ 16 1⁄ 2 9⁄ 16 5⁄ 8 11 ⁄ 16 3⁄ 4 13 ⁄ 16 7⁄ 8 15 ⁄ 16

Machinery's Handbook, 31st Edition Angles and Tapers

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Amount of Taper in a Given Length, When the Taper per Foot is Known Length of Taper

Machinery's Handbook, 31st Edition Angles and Tapers

700

Rules for Figuring Tapers DL = diameter of taper, large end DS = diameter of taper, small end L = length of taper (inches) TPF = taper per foot TPI = taper per inch T = taper in a certain length, in inches Given

DS

DL L

To Find

Rule

TPF

TPI

TPI = TPF ⁄ 12

TPI

TPF

TPF = TPI × 12

DL , D S , L

TPF

TPF = 12

DL , L, TPF

DS

D S = DL –

TPF L 12

DS , L, TPF

DL

D L = DS +

TPF L 12

DL , DS , TPF

L

L, TPF

T

DL – DS L

L = (DL – DS) 12 TPF

T =

TPF ×L 12

To find angle α for given taper T in inches per foot.—

d

C

D

a = 2 arctan (T / 24)

Example: What angle α is equivalent to a taper of 1.5 inches per foot?

α = 2 # arctan (1.5/24) = 7.153°

To find taper per foot T given angle α in degrees.—

T = 24 tan `α ⁄ 2j inches per foot Example: What taper T is equivalent to an angle of 7.153°?

T = 24 ta n a 7.153 ⁄ 2 k = 1. 5 inches per foot To find angle α given dimensions D, d, and C.—Let K be the difference in the disk diameters divided by twice the center distance. K = (D – d)/(2C), then a = 2 arcsin K Example: If the disk diameters d and D are 1 and 1.5 inches, respectively, and the center distance C is 5 inches, find the included angle α.

K = ^1.5 − 1h ⁄ ^2 # 5h = 0.05

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a = 2 # arcsin0.05 = 5.732°

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Machinery's Handbook, 31st Edition Angles and Tapers

701

To find taper T measured at right angles to a line through the disk centers given di­men­sions D, d, and distance C.—Find K using the formula in the previous example, then T = 24K ⁄ 1 − K 2 inches per foot. Example: If disk diameters d and D are 1 and 1.5 inches, respectively, and the center dis­ tance C is 5 inches, find the taper per foot.

K = (1.5 – 1) ⁄ (2 # 5) = 0.05

T =

24 # 0.05 = 1.2015 inches per foot 1 − ^0.05h2

To find center distance C for a given taper T in inches per foot.—

C=

1 + ( T ⁄ 24 )

D−d 2 #

2

T ⁄ 24

inches

Example: Gage is to be set to 3⁄4 inch per foot, and disk diameters are 1.25 and 1.5 inches, respectively. Find the required center distance for the disks.

C=

1.5 − 1.25 # 2

1 + (0.75 ⁄ 24)2 0.75 ⁄ 24

= 4.002 inches

To find center distance C for a given angle α and dimensions D and d. C = ^ D − d h ⁄ 2 sin (a ⁄ 2) inches Example: If an angle α of 20° is required, and the disks are 1 and 3 inches in diameter, respectively, find the required center distance C. C = ^3 − 1h ⁄ ^2 # sin 10°h = 5.759 inches To find taper T measured at right angles to one side.—When one side is taken as a base line and the taper is measured at right angles to that side, calculate K as explained above and use the following formula for determining the taper T: D d

C

T = 24K

1 − K2 inches per foot 1 − 2K 2

Example: If the disk diameters are 2 and 3 inches, respectively, and the center distance is 5 inches, what is the taper per foot measured at right angles to one side?

K=

3– 2 = 0.1 2#5

T = 24 # 0.1 #

1 – ^0.1h2 = 2.4367 in . per ft. 1 – 62 # ^0.1h2@

To find center distance C when taper T is measured from one side.— D−d C= inches 2 − 2 ⁄ 1 + (T ⁄ 12)2 Example: If the taper measured at right angles to one side is 6.9 inches per foot, and the disks are 2 and 5 inches in diameter, respectively, what is center distance C? 5−2 C= = 5.815 inches. 2 − 2 1 = (6.9 ⁄ 12)2

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Machinery's Handbook, 31st Edition ANGLES AND TAPERS

702

To find diameter D of a large disk in contact with a small disk of diameter d given angle α.—

D = d#

D

d

1 + sin ( a ⁄ 2) inches 1 − sin ( a ⁄ 2)

Example: The required angle α is 15°. Find diameter D of a large disk that is in contact with a standard 1-inch reference disk. 1 + sin 7.5° D = 1 # 1 − sin 7.5° = 1.3002 inches Measurement over Pins and Rolls Measurement over Pins.—When the distance across a bolt circle is too large to measure using ordinary measuring tools, then the required distance may be found from the distance across adjacent or alternate holes using one of the methods that follow: c

c θ

y

θ

= 3 ---- 60 n -----

x

d

= 3 ---- 60 n -----

x

θ = 3 ------6---0 n

d

x

Fig. 1a.

d

Fig. 1c.

Fig. 1b.

Even Number of Holes in Circle: To measure the unknown distance x over opposite plugs in a bolt circle of n holes (n is even and greater than 4), as shown in Fig. 1a, where y is the distance over alternate plugs, d is the diameter of the holes, and q = 360/n is the angle between adjacent holes, use the following general equation for obtaining x: y−d x= +d sin i Example: In a die that has six 3/4-inch diameter holes equally spaced on a circle, where the distance y over alternate holes is 41 ⁄2 inches, and the angle q between adjacent holes is 60, then 4.500 − 0.7500 x= + 0.7500 = 5.0801 sin 60° In a similar problem, the distance c over adjacent plugs is given, as shown in Fig. 1b. If the number of holes is even and greater than 4, the distance x over opposite plugs is given in the following formula:

f sin x = 2 ^c − dh

a 180 − i k p 2 sin i

+d

where d and q are as defined above.

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Machinery's Handbook, 31st Edition Measurement OVER PINS AND Rolls

703

Odd Number of Holes in Circle: In a circle as shown in Fig. 1c, where the number of holes n is odd and greater than 3, and the distance c over adjacent holes is given, then q equals 360/n and the distance x across the most widely spaced holes is given by: c−d x= 2 +d i sin 4 Checking a V-Shaped Groove by Measurement Over Pins.—In checking a groove of the shape shown in Fig. 2, it is necessary to measure the dimension X over the pins of radius R. If values for the radius R, dimension Z, and the angles α and b are known, the problem is to determine the distance Y, to arrive at the required overall dimension for X. If a line AC is drawn from the bottom of the V to the center of the pin at the left in Fig. 2, and a line CB from the center of this pin to its point of tangency with the side of the V, a right-angle tri­angle is formed in which one side, CB, is known and one angle, CAB, can be determined. A line drawn from the center of a circle to the point of intersection of two tangents to the cir­cle bisects the angle made by the tangent lines, and angle CAB therefore equals 1 ⁄2 (α + b). The length AC and the angle DAC can now be found, and with AC known in the right-angle triangle ADC, AD, which is equal to Y, can be found. X

Y

R

B C D

Z

Y

R

2

A

2

Fig. 2.

The value for X can be obtained from the formula α+β α−β X = Z + 2R a csc 2 cos 2 + 1 k For example, if R = 0.500, Z = 1.824, α = 45 degrees, and b = 35 degrees, 45° + 35 ° 45 − 35 X = 1.824 + ^2 # 0.5ha csc cos ° 2 ° + 1 k 2 X = 1.824 + csc 40° cos 5 ° + 1

X = 1.824 + 1.5557 # 0.99619 + 1 X = 1.824 + 1.550 + 1 = 4.374

Checking Radius of Arc by Measurement Over Rolls.—The radius R of large-radius concave and convex gages of the type shown in Fig. 3a, Fig. 3b and Fig. 3c can be checked by mea­surement L over two rolls with the gage resting on the rolls as shown. If the diameter of the rolls D, the length L, and the height H of the top of the arc above the surface plate (for the concave gage, Fig. 3a) are known or can be measured, the radius R of the workpiece to be checked can be calculated trigonometrically, as follows.

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Machinery's Handbook, 31st Edition Measurement OVER PINS AND Rolls

704

Referring to Fig. 3a for the concave gage, if L and D are known, cb can be found, and if H and D are known, ce can be found. With cb and ce known, ab can be found by means of a diagram as shown in Fig. 3c. In diagram Fig. 3c, cb and ce are shown at right angles as in Fig. 3a. A line is drawn con­ necting points b and e, and line ce is extended to the right. A line is now drawn from point b perpendicular to be and intersecting the extension of ce at point f. A semicircle can now be drawn through points b, e, and f with point a as the center. Triangles bce and bcf are similar and have a common side. Thus ce:bc::bc:cf. With ce and bc known, cf can be found from this proportion and, hence, ef, which is the diameter of the semicircle and radius ab. Then R = ab + D/2. a

L e D

R

H

b

c

D

R

c

b e L

a Fig. 3a.

Fig. 3b.

b

e

c

a

f

Fig. 3c.

The procedure for the convex gage is similar. The distances cb and ce are readily found, and from these two distances ab is computed on the basis of similar triangles as before. Radius R is then readily found. The derived formulas for concave and convex gages are as follows: Formulas:

R=

^ L − D h2 H + 8 ^ H − Dh 2

R=

^ L − D h2

8D

(Concave gage Fig. 3a) (Convex gage Fig. 3b)

For example: For Fig. 3a, let L = 17.8, D = 3.20, and H = 5.72, then ^17.8 − 3.20 h2 5.72 ^14.60 h 2 R= + 2 = 8 # 2.52 + 2.86 8 ^5.72 − 3.20 h 213.16 R = 20.16 + 2.86 = 13.43

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Machinery's Handbook, 31st Edition Checking Shaft Conditions

705

For Fig. 3b, let L = 22.28 and D = 3.40, then

R=

^22.28 − 3.40 h2

8 # 3.40

356.45 = 27.20 = 13.1

Checking Shaft Conditions Checking for Various Shaft Conditions.—An indicating height gage, together with V-blocks can be used to check shafts for ovality, taper, straightness (bending or curving), and concentricity of features (as shown exaggerated in Fig. 4). If a shaft on which work has been completed shows lack of concentricity, it may be due to the shaft having become bent or bowed because of mishandling or oval or tapered due to poor machine conditions. In checking for concentricity, the first step is to check for ovality, or out-of-roundness, as in Fig. 4a. The shaft is supported in a suitable V-block on a surface table, and the dial indicator plunger is placed over the workpiece, which is then rotated beneath the plunger to obtain readings of the amount of eccentricity. This procedure (sometimes called clocking, owing to the resemblance of the dial indica­ tor to a clock face) is repeated for other shaft diameters as necessary, and, in addition to making a written record of the measurements, the positions of extreme conditions should be marked on the workpiece for later reference.

Geometrical Form Desired

Bent or Crooked

Check for Ovality a

Check for Taper b

Check for Bent Shaft c

Curved or Bowed Check for Curved Shaft d Tapered

Check for Concentricity e

Eccentric Fig. 4.

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706

Machinery's Handbook, 31st Edition CHECKING SHAFT CONDITIONS

To check for taper, the shaft is supported in the V-block and the dial indicator is used to measure the maximum height over the shaft at various positions along its length, as shown in Fig. 4b, without turning the workpiece. Again, the shaft should be marked with the read­ing positions and values, and with the direction of the taper, and a written record should be made of the amount and direction of any taper discovered. Checking for a bent shaft requires that the shaft be clocked at the shoulder and at the far­ ther end, as shown in Fig. 4c. For a second check, the shaft is rotated only 90° or a quarter turn. When the recorded readings are compared with those from the ovality and taper checks, the three conditions can be distinguished. To detect a curved or bowed condition, the shaft should be suspended in two V-blocks with only about 1 ⁄8 inch of each end in each V. Alternatively, the shaft can be placed between centers. The shaft is then clocked at several points, as shown in Fig. 4d, but pref­erably not at those locations used for the ovality, taper, or crookedness checks. If the single element due to curvature is to be distinguished from the effects of ovality, taper, and crook­edness, and its value assessed, great care must be taken to differentiate between the condi­tions detected by the measurements. Finally, the amount of eccentricity between one shaft diameter and another may be tested by the setup shown in Fig. 4e. With the indicator plunger in contact with the smaller diam­eter, close to the shoulder, the shaft is rotated in the V-block and the indicator needle posi­tion is monitored to find the maximum and minimum readings. Curvature, ovality, or crookedness conditions may tend to cancel each other, as shown in Fig. 5, and one or more of these degrees of defectiveness may add themselves to the true eccentricity readings, depending on their angular positions. Fig. 5a shows, for instance, how crookedness and ovality tend to cancel each other, and also shows their effect in falsi­fying the reading for eccentricity. As the same shaft is turned in the V-block to the position shown in Fig. 5b, the maximum curvature reading could tend to cancel or reduce the max­imum eccentricity reading. Where maximum readings for ovality, curvature, or crooked­ness occur at the same angular position, their values should be subtracted from the eccentricity reading to arrive at a true picture of the shaft condition. Confirmation of eccen­tricity readings may be obtained by reversing the shaft in the V-block, as shown in Fig. 5c, and clocking the larger diameter of the shaft. Maximum Eccentricity

Maximum Maximum Ovality Eccentricity Maximum Crookedness

Maximum Curvature Maximum Crookedness Position a

Maximum Ovality

Position b

c Fig. 5.

Out-of-Roundness—Lobing.—With the imposition of finer tolerances and the develop­ ment of improved measurement methods, it has become apparent that no hole, cylinder, or

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Machinery's Handbook, 31st Edition Out Of Roundness—Lobing

707

sphere can be produced with a perfectly symmetrical round shape. Some of the conditions are diagrammed in Fig. 6, where Fig. 6a shows simple ovality and Fig. 6b shows ovality occurring in two directions. From the observation of such conditions have come the terms lobe and lobing. Fig. 6c shows the three-lobed shape common with centerless-ground components, and Fig. 6d is typical of multi-lobed shapes. In Fig. 6e are shown surface waviness, surface roughness, and out-of-roundness, which often are combined with lob­ing. Ring Cage Part a

b

c

d

e

Fig. 6.

In Fig. 6a through Fig. 6d, the cylinder (or hole) diameters are shown at full size, but the lobes are magnified some 10,000 times to make them visible. In precision parts, the deviation from the round condition is usually only in the range of millionths of an inch, although it occasionally can be 0.0001 inch, 0.0002 inch, or more. For instance, a 3-inch-diameter part may have a lobing condition amounting to an inaccuracy of only 30 millionths (0.000030 inch). Even if the distortion (ovality, waviness, roughness) is small, it may cause hum, vibration, heat buildup, and wear, possibly leading to eventual failure of the component or assembly. Plain elliptical out-of-roundness (two lobes), or any even number of lobes, can be detected by rotating the part on a surface plate under a dial indicator of adequate resolution, or by using an indicating caliper or snap gage. However, supporting such a part in a V-block during measurement will tend to conceal roundness errors. Ovality in a hole can be detected by a dial-type bore gage or internal measuring machine. Parts with odd numbers of lobes require an instrument that can measure the envelope or complete circumference. Plug and ring gages will tell whether a shaft can be assembled into a bearing but not whether there will be a good fit, as illustrated in Fig. 6e. A standard, 90-degree included-angle V-block can be used to detect and count the num­ ber of lobes, but to measure the exact amount of lobing indicated by R-r in Fig. 7 requires a V-block with an angle α, which is related to the number of lobes. This angle α can be cal­ culated from the formula 2α = 180° – 360°/N, where N is the number of lobes. Thus, for a threelobe form, α becomes 30 degrees, and the V-block used should have a 60-degree included angle. The distance M, which is obtained by rotating the part under the compara­tor plunger, is converted to a value for the radial variation in cylinder contour by the for­mula M = (R – r) (1 + csc α). M R r

R r 90°



Fig. 7.

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Machinery's Handbook, 31st Edition Out Of Roundness—Lobing

708

Using a V-block (even of appropriate angle) for parts with odd numbers of lobes will give exaggerated readings when the distance R – r (Fig. 7) is used as the measure of the amount of out-of-roundness. The accompanying table shows the appropriate V-block angles for various odd numbers of lobes and the factors (1 + csc α) by which the readings are increased over the actual out-of-roundness values. Table of Lobes, V-Block Angles and Exaggeration Factors in Measuring Out-of-Round Conditions in Shafts Included Angle of V-Block (deg) 60 108 128.57 140

Number of Lobes 3 5 7 9

Exaggeration Factor (1 + csc α) 3.00 2.24 2.11 2.06

Measurement of a complete circumference requires special equipment, often incorporat­ ing a precision spindle running true within two millionths (0.000002) inch. A stylus attached to the spindle is caused to traverse the internal or external cylinder being inspected, and its divergences are processed electronically to produce a polar chart similar to the wavy outline in Fig. 6e. Electronic circuits provide for the variations due to sur­face effects to be separated from those of lobing and other departures from the “true” cyl­inder traced out by the spindle. Coordinates for Hole Circles Type “A” Hole Circles.—Type “A” hole circles can be identified by hole number 1 at the top of the hole circle, as shown in Fig. 1a and Fig. 1b. The x, y coordinates for hole circles of from 3 to 33 holes corresponding to the geometry of Fig. 1a are given in Table 1a, and cor­ responding to the geometry of Fig. 1b in Table 1b. Holes are numbered in a counter­clock­ wise direction as shown. Coordinates given are based upon a hole circle of (1) unit diameter. For other diameters, multiply the x and y coordinates from the table by the hole circle diameter. For example, for a 3-inch or 3-centimeter hole circle diameter, multiply table values by 3. Coordinates are valid in any unit system. X 1

–Y –X 3

Ref 5

Ref

Y 2

5

+X

+Y

Ref

2

1

4

Fig. 1a. Type “A” Circle

3

4

Fig. 1b. Type “A” Circle

The origin of the coordinate system in Fig. 1a, marked “Ref”, is at the center of the hole circle at position x = 0, y = 0. The equations for calculating hole coordinates for type “A” circles with the coordinate system origin at the center of the hole circle are as follows: 360 2r i= n = n

D xH = − 2 sin ^^ H − 1h ih

D yH = − 2 cos ^^ H − 1h ih

(1a)

where n = number of holes in circle; D = diameter of hole circle; q = angle between adjacent holes in circle; H = number (from 1 to n) of the current hole; xH = x coordinate at position of hole number H; and, yH = y coordinate at position of hole number H.

Example 1(a): Calculate the hole coordinates for the 5-hole circle shown in Fig. 1a when circle diameter = 1. Compare the results to the data tabulated in Table 1a.

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Machinery's Handbook, 31st Edition Determining Hole Circle Coordinates Hole



1 2 3 4 5



q = 360/5 = 72°  D = 1

xH   =  x1   =  –1 ⁄2 3 sin(0) = 0.00000 x 2   =  –1 ⁄2 3 sin(72) = –0.47553 x 3   =  –1 ⁄2 3 sin(144) = –0.29389 x 4   =  –1 ⁄2 3 sin(216) = 0.29389 x5   =  –1 ⁄2 3 sin(288) = 0.47553



709

y1   =  –1 ⁄2 3 cos(0) = –0.50000 y 2   =  –1 ⁄2 3 cos(72) = –0.15451 y3   =  –1 ⁄2 3 cos(144) = 0.40451 y4   =  –1 ⁄2 3 cos(216) = 0.40451 y5   =  –1 ⁄2 3 cos(288) = –0.15451

In Fig. 1b, the origin of the coordinate system (point 0,0) is located at the top left of the figure at the intersection of the two lines labeled “Ref.” The center of the hole circle is off­ set from the coordinate system origin by distance XO in the +x direction, and by distance YO in the +y direction. In practice the origin of the coordinate system can be located at any con­venient distance from the center of the hole circle. In Fig. 1b, it can be determined by inspection that the distances XO = YO = D ⁄2 . The equations for calculating hole positions of type “A” circles of the Fig. 1b type are almost the same as in Equation (1a), but with the addition of XO and YO terms, as follows: 360 2π i= n = n

D xH = − 2 sin ^^ H − 1h ih + XO

D yH = − 2 cos ^^ H − 1h i h + YO (1b)

Example 1(b): Use results of Example 1 to determine hole coordinates of Fig. 1b for circle diameter = 1, and compare results with Table 1b. Hole 1 2 3 4 5



q  =  360/5 = 72°  D  =  1  XO  =  D⁄2   =  0.50000  YO  =  D⁄2 = 0.50000

x1   =  0.00000 + XO = 0.50000 x2   =  –0.47553 + XO = 0.02447 x3   =  –0.29389 + XO = 0.20611 x4   =  0.29389 + XO = 0.79389 x5   =  0.47553 + XO = 0.97553





y1   =  –0.50000 + YO = 0.00000 y2   =  –0.15451 + YO = 0.34549 y3   =  0.40451 + YO = 0.90451 y4   =  0.40451 + YO = 0.90451 y5   =  –0.15451 + YO = 0.34549

Type “B” Hole Circles.—Compared to type “A” hole circles, type “B” hole circles, Fig. 2a and Fig. 2b, are arranged such that the circle of holes is rotated about the center of the circle by q ⁄2 degrees, that is, 1⁄2 of the angle between adjacent holes. The x,y coordinates for type “B” hole circles of from 3 to 33 holes are given in Table 2a for geometry corresponding to Fig. 2a, and in Table 2b for geometry corresponding to Fig. 2b. Holes are numbered in a counterclockwise direction as shown. Coordinates given are based upon a hole circle of (1) unit diameter. For other diameters, multiply the x and y coordinates from the table by the hole circle diameter. For example, for a 3-inch or 3-centimeter hole circle diameter, multi­ply table values by 3. Coordinates are valid in any unit system. X

2

–Y

–X

5

+X

+Y

Y

Ref

4

3

Fig. 2a. Type “B” Circle

Ref

1

5

4

2

Ref

1

3

Fig. 2b. Type “B” Circle

In Fig. 2a, the coordinate system origin, marked “Ref”, is at the center of the hole circle at position x = 0, y = 0. Equations for calculating hole coordinates for type “B” circles with the coordinate system origin at the center of the hole circle as in Fig. 2a are as follows: 360 2π i= n = n

D i xH = − 2 sin a^ H − 1h i + 2 k

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D i yH = − 2 cos a^ H − 1h i + 2 k (2a)

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Machinery's Handbook, 31st Edition Determining Hole Circle Coordinates

710

where n = number of holes in circle; D = diameter of hole circle; q = angle between adjacent holes; xH = x coordinate at position of hole number H; and, yH = y coordinate at position of hole number H. Example 2(a): Calculate the hole coordinates for the 5-hole circle shown in Fig. 2a when circle diameter = 1. Compare the results to the data in Table 2a. Hole 1 2 3 4 5





q  =  360/5 = 72° 

q⁄   2

=  36°  D  =  1

x1   =  –1 ⁄2 3 sin(36) = –0.29389 x 2   =  –1 ⁄2 3 sin(108) = –0.47553 x 3   =  –1 ⁄2 3 sin(180) = 0.00000 x 4   =  –1 ⁄2 3 sin(252) = 0.47553 x5   =  –1 ⁄2 3 sin(324) = 0.29389



y1   =  –1 ⁄2 3 cos(36) = –0.40451 y 2   =  –1 ⁄2 3 cos(108) = 0.15451 y3   =  –1 ⁄2 3 cos(180) = 0.50000 y4   =  –1 ⁄2 3 cos(252) = 0.15451 y5   =  –1 ⁄2 3 cos(324) = –0.40451

In Fig. 2b, the origin of the coordinate system (point 0,0) is located at the top left of the figure at the intersection of the two lines labeled “Ref.” The center of the hole circle is off­set from the coordinate system origin by distance XO in the +x direction and by distance YO in the +y direction. In practice, the origin of the coordinate system can be chosen at any con­venient distance from the hole circle origin. In Fig. 2b, it can be determined by inspection that distance XO = YO = D⁄2 . The equations for calculating hole positions of type “B” circles of the Fig. 2b type are the same as in Equation (2a) but with the addition of XO and YO terms, as follows: 360 2π i= n = n

D i xH = − 2 sin a^ H − 1h i + 2 k + X O

D i yH = − 2 cos a^ H − 1h i + 2 k + YO

(2b)

Example 2(b):Use the coordinates obtained in Example 2(a) to determine the hole coor­ dinates of a 5-hole circle shown in Fig. 1b with circle diameter = 1. Compare the results to the data in Table 2b.

Hole 1 2 3 4 5





q ⁄2   =  36°  D  =  1 XO   1 – ⁄2 3 sin(36) + 0.50000 = 0.20611 –1 ⁄2 3 sin(108) + 0.50000 = 0.02447 –1 ⁄2 3 sin(180) + 0.50000 = 0.50000 –1 ⁄2 3 sin(252) + 0.50000 = 0.97553 –1 ⁄2 3 sin(324) + 0.50000 = 0.79389

q  =  360/5 = 72° 

x1   =  x 2   =  x 3   =  x 4   =  x5   = 

=  D ⁄2 = 0.50000  YO   =  D ⁄2 = 0.50000

y1   =  –1 ⁄2 3 cos(36) + 0.50000 = 0.09549 y 2   =  –1 ⁄2 3 cos(108) + 0.50000 = 0.65451 y3   =  –1 ⁄2 3 cos(180) + 0.50000 = 1.00000 y4   =  –1 ⁄2 3 cos(252) + 0.50000 = 0.65451 y5   =  –1 ⁄2 3 cos(324) + 0.50000 = 0.09549

Adapting Hole Coordinate Equations for Different Geometry.—Hole coordinate val­ ues in Table 1a through Table 2b are obtained using the equations given previously, along with the geometry of the corresponding figures. If the geometry does not match that given in one of the previous figures, hole coordinate values from the tables or equations will be incor­rect. Fig. 3 illustrates such a case. Fig. 3 resembles a type “A” hole circle (Fig. 1b) with hole number 2 at the top, and it also resembles a type “B” hole circle (Fig. 2b) in which all holes have been rotated 90° clockwise. A closer look also reveals that the positive y direction in Fig. 3 is opposite that used in Fig. 1b and Fig. 2b. Therefore, to determine the hole coordinates of Fig. 3 it is necessary to create new equations that match the given geometry or to modify the previous equations to match the Fig. 3 geometry. +y 6.0

2

3 4

5

7.5

1

30°

Ø10.0

6 +x

Fig. 3.

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Machinery's Handbook, 31st Edition Lengths of Chords

711

Example 3(a), Determining Hole Coordinates for Fig. 3: Write equations for the coordi­ nates of holes 1, 2, and 3 of Fig. 3. x1 = 7.5 + 5cos(30°) = 11.8301 x 2 = 7.5 + 5cos(30°+ 60°) = 7.5 x 3 = 7.5 + 5cos(30°+ 120°) = 3.1699

y1 = 6.0 + 5sin(30°) = 8.5000 y 2 = 6.0 + 5sin(30°+ 60°) = 11.0000 y 3 = 6.0 + 5sin(30°+ 120°) = 8.5000

Example 3(b), Modify Equation (2b) for Fig. 3: In Fig. 3, hole numbering is rotated 90° (p/2 radian) in the clockwise (negative) direction relative to Fig. 2b, and the direction of the +y coordinate axis is the reverse, or negative, of that given in Fig. 2b. Equations for Fig. 3 can be obtained from Equation (2b) by 1) subtracting 90° from the angle of each hole in the x and y equations of Equation (2b) 2) multiplying the yH equation by –1 to reverse the orientation of the y axis D i xH = − 2 sin a^ H − 1h i + 2 − 90 k + X 0

D i yH = − a − 2 cos a^ H − 1h i + 2 − 90 k + Y 0 k

In Fig. 3, q = 360/n = 60° for 6 holes, XO = 7.5, and YO = –6.0. x1 = –5sin(30 – 90) + 7.5 = 11.8301 x 2 = –5sin(60° + 30°– 90°) + 7.5 = 7.5 x 3 = –5sin(120° + 30°– 90°) + 7.5 = 3.1699

y1 = 5cos(30 – 90) + 6 = 8.5000 y 2 = 5cos(60° + 30°– 90°) + 6 = 11.0000 y 3 = 5cos(120° + 30°– 90°) + 6 = 8.5000

Lengths of Chords on Hole Circle Circumference.—Table 3 on page 720 gives the lengths of chords for spacing off the circumferences of circles. The object of this table is to make possible the division of the periphery into a number of equal parts without trials with the dividers. Table 3 is calculated for circles having a diameter equal to 1. For circles of other diameters, the length of chord given in the table should be multiplied by the diameter of the circle. Table 3 may be used by toolmakers when setting “buttons” in circular forma­ tion and with inch or metric dimensions. See Coordinates for Hole Circles on page 708 for more information on this topic. Example: Assume that it is required to divide the periphery of a circle of 20 inches diam­ eter into thirty-two equal parts. Solution: From the table the length of the chord is found to be 0.098017 inch, if the diam­ eter of the circle were 1 inch. With a diameter of 20 inches the length of the chord for one division would be 20 3 0.098017 = 1.9603 inches. Another example, in metric units: For a 100 millimeter diameter requiring 5 equal divisions, the length of the chord for one divi­sion would be 100 × 0.587785 = 58.7785 millimeters. Example: Assume that it is required to divide a circle of 61 ⁄2 millimeter diameter into seven equal parts. Find the length of the chord required for spacing off the circumference. Solution: In Table 3, the length of the chord for dividing a circle of 1 millimeter diameter into 7 equal parts is 0.433884 mm. The length of chord for a circle of 1 6 ⁄2 mm diameter is 61 ⁄2 × 0.433884 = 2.820246 mm.

Example: Assume that it is required to divide a circle having a diameter of 923⁄32 inches into 15 equal divisions. Solution: In Table 3, the length of the chord for dividing a circle of 1 inch diameter into 15 equal parts is 0.207912 inch. The length of chord for a circle of 9 inches diameter is 923⁄32 3 0.207912 = 2.020645 inches.

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1

2

–Y –X

# 1 2 3 # 1 2 3 4

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# 1 2 3 4 5 # 1 2 3 4 5 6

+X

+Y

3 holes x y # 0.00000 –0.50000 1 –0.43301 0.25000 2 0.43301 0.25000 3 4 4 holes x y 5 0.00000 –0.50000 6 –0.50000 0.00000 7 0.00000 0.50000 0.50000 0.00000 # 1 5 holes x y 2 0.00000 –0.50000 3 –0.47553 –0.15451 4 –0.29389 0.40451 5 0.29389 0.40451 6 0.47553 –0.15451 7 8 6 holes x y 0.00000 –0.50000 # –0.43301 –0.25000 1 –0.43301 0.25000 2 0.00000 0.50000 3 0.43301 0.25000 4 0.43301 –0.25000 5 6 7 8 9

4

# 1 2 3 4 5 6 7 8 9 10 11

7 holes x y 0.00000 –0.50000 –0.39092 –0.31174 –0.48746 0.11126 –0.21694 0.45048 0.21694 0.45048 0.48746 0.11126 0.39092 –0.31174 8 holes x y # 0.00000 –0.50000 1 –0.35355 –0.35355 2 –0.50000 0.00000 3 –0.35355 0.35355 4 0.00000 0.50000 5 0.35355 0.35355 6 0.50000 0.00000 7 0.35355 –0.35355 8 9 9 holes x y 10 0.00000 –0.50000 11 –0.32139 –0.38302 12 –0.49240 –0.08682 –0.43301 0.25000 # –0.17101 0.46985 1 0.17101 0.46985 2 0.43301 0.25000 3 0.49240 –0.08682 4 0.32139 –0.38302 5

10 holes x y 0.00000 –0.50000 –0.29389 –0.40451 –0.47553 –0.15451 –0.47553 0.15451 –0.29389 0.40451 0.00000 0.50000 0.29389 0.40451 0.47553 0.15451 0.47553 –0.15451 0.29389 –0.40451 11 holes x y 0.00000 –0.50000 –0.27032 –0.42063 –0.45482 –0.20771 –0.49491 0.07116 –0.37787 0.32743 –0.14087 0.47975 0.14087 0.47975 0.37787 0.32743 0.49491 0.07116 0.45482 –0.20771 0.27032 –0.42063 12 holes x y 0.00000 –0.50000 –0.25000 –0.43301 –0.43301 –0.25000 –0.50000 0.00000 –0.43301 0.25000 –0.25000 0.43301 0.00000 0.50000 0.25000 0.43301 0.43301 0.25000 0.50000 0.00000 0.43301 –0.25000 0.25000 –0.43301 13 holes x y 0.00000 –0.50000 –0.23236 –0.44273 –0.41149 –0.28403 –0.49635 –0.06027 –0.46751 0.17730

13 holes (Continued) # x y 6 –0.33156 0.37426 7 –0.11966 0.48547 8 0.11966 0.48547 9 0.33156 0.37426 10 0.46751 0.17730 11 0.49635 –0.06027 12 0.41149 –0.28403 13 0.23236 –0.44273 14 holes # x y 1 0.00000 –0.50000 2 –0.21694 –0.45048 3 –0.39092 –0.31174 4 –0.48746 –0.11126 5 –0.48746 0.11126 6 –0.39092 0.31174 7 –0.21694 0.45048 8 0.00000 0.50000 9 0.21694 0.45048 10 0.39092 0.31174 11 0.48746 0.11126 12 0.48746 –0.11126 13 0.39092 –0.31174 14 0.21694 –0.45048 15 holes # x y 1 0.00000 –0.50000 2 –0.20337 –0.45677 3 –0.37157 –0.33457 4 –0.47553 –0.15451 5 –0.49726 0.05226 6 –0.43301 0.25000 7 –0.29389 0.40451 8 –0.10396 0.48907 9 0.10396 0.48907 10 0.29389 0.40451 11 0.43301 0.25000 12 0.49726 0.05226 13 0.47553 –0.15451 14 0.37157 –0.33457 15 0.20337 –0.45677

# 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 # 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 # 1 2 3 4 5 6 7

16 holes x y 0.00000 –0.50000 –0.19134 –0.46194 –0.35355 –0.35355 –0.46194 –0.19134 –0.50000 0.00000 –0.46194 0.19134 –0.35355 0.35355 –0.19134 0.46194 0.00000 0.50000 0.19134 0.46194 0.35355 0.35355 0.46194 0.19134 0.50000 0.00000 0.46194 –0.19134 0.35355 –0.35355 0.19134 –0.46194 17 holes x y 0.00000 –0.50000 –0.18062 –0.46624 –0.33685 –0.36950 –0.44758 –0.22287 –0.49787 –0.04613 –0.48091 0.13683 –0.39901 0.30132 –0.26322 0.42511 –0.09187 0.49149 0.09187 0.49149 0.26322 0.42511 0.39901 0.30132 0.48091 0.13683 0.49787 –0.04613 0.44758 –0.22287 0.33685 –0.36950 0.18062 –0.46624 18 holes x y 0.00000 –0.50000 –0.17101 –0.46985 –0.32139 –0.38302 –0.43301 –0.25000 –0.49240 –0.08682 –0.49240 0.08682 –0.43301 0.25000

18 holes (Continued) # x y 8 –0.32139 0.38302 9 –0.17101 0.46985 10 0.00000 0.50000 11 0.17101 0.46985 12 0.32139 0.38302 13 0.43301 0.25000 14 0.49240 0.08682 15 0.49240 –0.08682 16 0.43301 –0.25000 17 0.32139 –0.38302 18 0.17101 –0.46985 19 holes # x y 1 0.00000 –0.50000 2 –0.16235 –0.47291 3 –0.30711 –0.39457 4 –0.41858 –0.27347 5 –0.48470 –0.12274 6 –0.49829 0.04129 7 –0.45789 0.20085 8 –0.36786 0.33864 9 –0.23797 0.43974 10 –0.08230 0.49318 11 0.08230 0.49318 12 0.23797 0.43974 13 0.36786 0.33864 14 0.45789 0.20085 15 0.49829 0.04129 16 0.48470 –0.12274 17 0.41858 –0.27347 18 0.30711 –0.39457 19 0.16235 –0.47291 20 holes # x y 1 0.00000 –0.50000 2 –0.15451 –0.47553 3 –0.29389 –0.40451 4 –0.40451 –0.29389 5 –0.47553 –0.15451 6 –0.50000 0.00000 7 –0.47553 0.15451 8 –0.40451 0.29389 9 –0.29389 0.40451 10 –0.15451 0.47553

20 holes (Continued) # x y 11 0.00000 0.50000 12 0.15451 0.47553 13 0.29389 0.40451 14 0.40451 0.29389 15 0.47553 0.15451 16 0.50000 0.00000 17 0.47553 –0.15451 18 0.40451 –0.29389 19 0.29389 –0.40451 20 0.15451 –0.47553 21 holes # x y 1 0.00000 –0.50000 2 –0.14738 –0.47779 3 –0.28166 –0.41312 4 –0.39092 –0.31174 5 –0.46544 –0.18267 6 –0.49860 –0.03737 7 –0.48746 0.11126 8 –0.43301 0.25000 9 –0.34009 0.36653 10 –0.21694 0.45048 11 –0.07452 0.49442 12 0.07452 0.49442 13 0.21694 0.45048 14 0.34009 0.36653 15 0.43301 0.25000 16 0.48746 0.11126 17 0.49860 –0.03737 18 0.46544 –0.18267 19 0.39092 –0.31174 20 0.28166 –0.41312 21 0.14738 –0.47779 22 holes # x y 1 0.00000 –0.50000 2 –0.14087 –0.47975 3 –0.27032 –0.42063 4 –0.37787 –0.32743 5 –0.45482 –0.20771 6 –0.49491 –0.07116 7 –0.49491 0.07116 8 –0.45482 0.20771 9 –0.37787 0.32743

22 holes (Continued) # x y 10 –0.27032 0.42063 11 –0.14087 0.47975 12 0.00000 0.50000 13 0.14087 0.47975 14 0.27032 0.42063 15 0.37787 0.32743 16 0.45482 0.20771 17 0.49491 0.07116 18 0.49491 –0.07116 19 0.45482 –0.20771 20 0.37787 –0.32743 21 0.27032 –0.42063 22 0.14087 –0.47975 23 holes # x y 1 0.00000 –0.50000 2 –0.13490 –0.48146 3 –0.25979 –0.42721 4 –0.36542 –0.34128 5 –0.44394 –0.23003 6 –0.48954 –0.10173 7 –0.49883 0.03412 8 –0.47113 0.16744 9 –0.40848 0.28834 10 –0.31554 0.38786 11 –0.19920 0.45861 12 –0.06808 0.49534 13 0.06808 0.49534 14 0.19920 0.45861 15 0.31554 0.38786 16 0.40848 0.28834 17 0.47113 0.16744 18 0.49883 0.03412 19 0.48954 –0.10173 20 0.44394 –0.23003 21 0.36542 –0.34128 22 0.25979 –0.42721 23 0.13490 –0.48146 24 holes # x y 1 0.00000 –0.50000 2 –0.12941 –0.48296 3 –0.25000 –0.43301 4 –0.35355 –0.35355

Machinery's Handbook, 31st Edition Hole Circle Coordinates

3

Ref 5

# 1 2 3 4 5 6 7 8 9 10

712

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Table 1a. Hole Coordinate Dimension Factors for Type “A” Hole Circles

Ref

27 holes (Continued) # x y 24 0.40106 –0.29858 25 0.32139 –0.38302 26 0.22440 –0.44682 27 0.11531 –0.48652 28 holes # x y 1 0.00000 –0.50000 2 –0.11126 –0.48746 3 –0.21694 –0.45048 4 –0.31174 –0.39092 5 –0.39092 –0.31174 6 –0.45048 –0.21694 7 –0.48746 –0.11126 8 –0.50000 0.00000 9 –0.48746 0.11126 10 –0.45048 0.21694 11 –0.39092 0.31174 12 –0.31174 0.39092 13 –0.21694 0.45048 14 –0.11126 0.48746 15 0.00000 0.50000 16 0.11126 0.48746 17 0.21694 0.45048 18 0.31174 0.39092 19 0.39092 0.31174 20 0.45048 0.21694 21 0.48746 0.11126 22 0.50000 0.00000 23 0.48746 –0.11126 24 0.45048 –0.21694 25 0.39092 –0.31174 26 0.31174 –0.39092 27 0.21694 –0.45048 28 0.11126 –0.48746 29 holes # x y 1 0.00000 –0.50000 2 –0.10749 –0.48831 3 –0.20994 –0.45379 4 –0.30259 –0.39805 5 –0.38108 –0.32369 6 –0.44176 –0.23420 7 –0.48177 –0.13376 8 –0.49927 –0.02707

29 holes (Continued) # x y 9 –0.49341 0.08089 10 –0.46449 0.18507 11 –0.41384 0.28059 12 –0.34385 0.36300 13 –0.25778 0.42843 14 –0.15965 0.47383 15 –0.05406 0.49707 16 0.05406 0.49707 17 0.15965 0.47383 18 0.25778 0.42843 19 0.34385 0.36300 20 0.41384 0.28059 21 0.46449 0.18507 22 0.49341 0.08089 23 0.49927 –0.02707 24 0.48177 –0.13376 25 0.44176 –0.23420 26 0.38108 –0.32369 27 0.30259 –0.39805 28 0.20994 –0.45379 29 0.10749 –0.48831 30 holes # x y 1 0.00000 –0.50000 2 –0.10396 –0.48907 3 –0.20337 –0.45677 4 –0.29389 –0.40451 5 –0.37157 –0.33457 6 –0.43301 –0.25000 7 –0.47553 –0.15451 8 –0.49726 –0.05226 9 –0.49726 0.05226 10 –0.47553 0.15451 11 –0.43301 0.25000 12 –0.37157 0.33457 13 –0.29389 0.40451 14 –0.20337 0.45677 15 –0.10396 0.48907 16 0.00000 0.50000 17 0.10396 0.48907 18 0.20337 0.45677 19 0.29389 0.40451 20 0.37157 0.33457 21 0.43301 0.25000

30 holes (Continued) # x y 22 0.47553 0.15451 23 0.49726 0.05226 24 0.49726 –0.05226 25 0.47553 –0.15451 26 0.43301 –0.25000 27 0.37157 –0.33457 28 0.29389 –0.40451 29 0.20337 –0.45677 30 0.10396 –0.48907 31 holes # x y 1 0.00000 –0.50000 2 –0.10065 –0.48976 3 –0.19718 –0.45948 4 –0.28563 –0.41038 5 –0.36240 –0.34448 6 –0.42432 –0.26448 7 –0.46888 –0.17365 8 –0.49423 –0.07571 9 –0.49936 0.02532 10 –0.48404 0.12533 11 –0.44890 0.22020 12 –0.39539 0.30605 13 –0.32569 0.37938 14 –0.24265 0.43717 15 –0.14968 0.47707 16 –0.05058 0.49743 17 0.05058 0.49743 18 0.14968 0.47707 19 0.24265 0.43717 20 0.32569 0.37938 21 0.39539 0.30605 22 0.44890 0.22020 23 0.48404 0.12533 24 0.49936 0.02532 25 0.49423 –0.07571 26 0.46888 –0.17365 27 0.42432 –0.26448 28 0.36240 –0.34448 29 0.28563 –0.41038 30 0.19718 –0.45948 31 0.10065 –0.48976

32 holes x y 0.00000 –0.50000 –0.09755 –0.49039 –0.19134 –0.46194 –0.27779 –0.41573 –0.35355 –0.35355 –0.41573 –0.27779 –0.46194 –0.19134 –0.49039 –0.09755 –0.50000 0.00000 –0.49039 0.09755 –0.46194 0.19134 –0.41573 0.27779 –0.35355 0.35355 –0.27779 0.41573 –0.19134 0.46194 –0.09755 0.49039 0.00000 0.50000 0.09755 0.49039 0.19134 0.46194 0.27779 0.41573 0.35355 0.35355 0.41573 0.27779 0.46194 0.19134 0.49039 0.09755 0.50000 0.00000 0.49039 –0.09755 0.46194 –0.19134 0.41573 –0.27779 0.35355 –0.35355 0.27779 –0.41573 0.19134 –0.46194 0.09755 –0.49039 33 holes # x y 1 0.00000 –0.50000 2 –0.09463 –0.49096 3 –0.18583 –0.46418 4 –0.27032 –0.42063 5 –0.34504 –0.36187 6 –0.40729 –0.29003 7 –0.45482 –0.20771 8 –0.48591 –0.11788 9 –0.49943 –0.02379 10 –0.49491 0.07116

# 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32

33 holes (Continued) # x y 11 –0.47250 0.16353 12 –0.43301 0.25000 13 –0.37787 0.32743 14 –0.30908 0.39303 15 –0.22911 0.44442 16 –0.14087 0.47975 17 –0.04753 0.49774 18 0.04753 0.49774 19 0.14087 0.47975 20 0.22911 0.44442 21 0.30908 0.39303 22 0.37787 0.32743 23 0.43301 0.25000 24 0.47250 0.16353 25 0.49491 0.07116 26 0.49943 –0.02379 27 0.48591 –0.11788 28 0.45482 –0.20771 29 0.40729 –0.29003 30 0.34504 –0.36187 31 0.27032 –0.42063 32 0.18583 –0.46418 33 0.09463 –0.49096

713

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26 holes (Continued) # x y 8 –0.49635 0.06027 9 –0.46751 0.17730 10 –0.41149 0.28403 –Y 11 –0.33156 0.37426 12 –0.23236 0.44273 5 2 13 –0.11966 0.48547 14 0.00000 0.50000 –X +X 15 0.11966 0.48547 16 0.23236 0.44273 17 0.33156 0.37426 +Y 18 0.41149 0.28403 3 4 19 0.46751 0.17730 20 0.49635 0.06027 21 0.49635 –0.06027 22 0.46751 –0.17730 24 holes (Continued) 25 holes (Continued) # x y # x y 23 0.41149 –0.28403 5 –0.43301 –0.25000 7 –0.49901 –0.03140 24 0.33156 –0.37426 6 –0.48296 –0.12941 8 –0.49114 0.09369 25 0.23236 –0.44273 7 –0.50000 0.00000 9 –0.45241 0.21289 26 0.11966 –0.48547 8 –0.48296 0.12941 10 –0.38526 0.31871 27 holes x y 9 –0.43301 0.25000 11 –0.29389 0.40451 # 10 –0.35355 0.35355 12 –0.18406 0.46489 1 0.00000 –0.50000 11 –0.25000 0.43301 13 –0.06267 0.49606 2 –0.11531 –0.48652 12 –0.12941 0.48296 14 0.06267 0.49606 3 –0.22440 –0.44682 13 0.00000 0.50000 15 0.18406 0.46489 4 –0.32139 –0.38302 14 0.12941 0.48296 16 0.29389 0.40451 5 –0.40106 –0.29858 15 0.25000 0.43301 17 0.38526 0.31871 6 –0.45911 –0.19804 16 0.35355 0.35355 18 0.45241 0.21289 7 –0.49240 –0.08682 17 0.43301 0.25000 19 0.49114 0.09369 8 –0.49915 0.02907 18 0.48296 0.12941 20 0.49901 –0.03140 9 –0.47899 0.14340 19 0.50000 0.00000 21 0.47553 –0.15451 10 –0.43301 0.25000 20 0.48296 –0.12941 22 0.42216 –0.26791 11 –0.36369 0.34312 21 0.43301 –0.25000 23 0.34227 –0.36448 12 –0.27475 0.41774 22 0.35355 –0.35355 24 0.24088 –0.43815 13 –0.17101 0.46985 23 0.25000 –0.43301 25 0.12434 –0.48429 14 –0.05805 0.49662 24 0.12941 –0.48296 15 0.05805 0.49662 26 holes # x y 16 0.17101 0.46985 25 holes # x y 1 0.00000 –0.50000 17 0.27475 0.41774 1 0.00000 –0.50000 2 –0.11966 –0.48547 18 0.36369 0.34312 2 –0.12434 –0.48429 3 –0.23236 –0.44273 19 0.43301 0.25000 3 –0.24088 –0.43815 4 –0.33156 –0.37426 20 0.47899 0.14340 4 –0.34227 –0.36448 5 –0.41149 –0.28403 21 0.49915 0.02907 5 –0.42216 –0.26791 6 –0.46751 –0.17730 22 0.49240 –0.08682 6 –0.47553 –0.15451 7 –0.49635 –0.06027 23 0.45911 –0.19804

1

Machinery's Handbook, 31st Edition Hole Circle Coordinates

Copyright 2020, Industrial Press, Inc.

TableTable 1a. (Continued) Hole Coordinate Dimension Factors for“A” Type “A”Circles Hole Circles 1a. Hole Coordinate Dimension Factors for Type Hole

X

Ref

1

# 1 2 3 # 1 2 3 4

ebooks.industrialpress.com

# 1 2 3 4 5 # 1 2 3 4 5 6

5

3

3 holes x y 0.50000 0.00000 0.06699 0.75000 0.93301 0.75000 4 holes x y 0.50000 0.00000 0.00000 0.50000 0.50000 1.00000 1.00000 0.50000 5 holes x y 0.50000 0.00000 0.02447 0.34549 0.20611 0.90451 0.79389 0.90451 0.97553 0.34549 6 holes x y 0.50000 0.00000 0.06699 0.25000 0.06699 0.75000 0.50000 1.00000 0.93301 0.75000 0.93301 0.25000

4 # 1 2 3 4 5 6 7 # 1 2 3 4 5 6 7 8 # 1 2 3 4 5 6 7 8 9

7 holes x y 0.50000 0.00000 0.10908 0.18826 0.01254 0.61126 0.28306 0.95048 0.71694 0.95048 0.98746 0.61126 0.89092 0.18826 8 holes x y 0.50000 0.00000 0.14645 0.14645 0.00000 0.50000 0.14645 0.85355 0.50000 1.00000 0.85355 0.85355 1.00000 0.50000 0.85355 0.14645 9 holes x y 0.50000 0.00000 0.17861 0.11698 0.00760 0.41318 0.06699 0.75000 0.32899 0.96985 0.67101 0.96985 0.93301 0.75000 0.99240 0.41318 0.82139 0.11698

# 1 2 3 4 5 6 7 8 9 10 11 # 1 2 3 4 5 6 7 8 9 10 11 12 # 1 2 3 4 5

10 holes 13 holes (Continued) x y # x y 0.50000 0.00000 6 0.16844 0.87426 0.20611 0.09549 7 0.38034 0.98547 0.02447 0.34549 8 0.61966 0.98547 0.02447 0.65451 9 0.83156 0.87426 0.20611 0.90451 10 0.96751 0.67730 0.50000 1.00000 11 0.99635 0.43973 0.79389 0.90451 12 0.91149 0.21597 0.97553 0.65451 13 0.73236 0.05727 0.97553 0.34549 14 holes x y 0.79389 0.09549 # 1 0.50000 0.00000 11 holes x y 2 0.28306 0.04952 0.50000 0.00000 3 0.10908 0.18826 0.22968 0.07937 4 0.01254 0.38874 0.04518 0.29229 5 0.01254 0.61126 0.00509 0.57116 6 0.10908 0.81174 0.12213 0.82743 7 0.28306 0.95048 0.35913 0.97975 8 0.50000 1.00000 0.64087 0.97975 9 0.71694 0.95048 0.87787 0.82743 10 0.89092 0.81174 0.99491 0.57116 11 0.98746 0.61126 0.95482 0.29229 12 0.98746 0.38874 0.77032 0.07937 13 0.89092 0.18826 14 0.71694 0.04952 12 holes x y 15 holes x y 0.50000 0.00000 # 0.25000 0.06699 1 0.50000 0.00000 0.06699 0.25000 2 0.29663 0.04323 0.00000 0.50000 3 0.12843 0.16543 0.06699 0.75000 4 0.02447 0.34549 0.25000 0.93301 5 0.00274 0.55226 0.50000 1.00000 6 0.06699 0.75000 0.75000 0.93301 7 0.20611 0.90451 0.93301 0.75000 8 0.39604 0.98907 1.00000 0.50000 9 0.60396 0.98907 0.93301 0.25000 10 0.79389 0.90451 0.75000 0.06699 11 0.93301 0.75000 12 0.99726 0.55226 13 holes x y 13 0.97553 0.34549 0.50000 0.00000 14 0.87157 0.16543 0.26764 0.05727 15 0.70337 0.04323 0.08851 0.21597 0.00365 0.43973 0.03249 0.67730

# 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 # 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 # 1 2 3 4 5 6 7

16 holes x y 0.50000 0.00000 0.30866 0.03806 0.14645 0.14645 0.03806 0.30866 0.00000 0.50000 0.03806 0.69134 0.14645 0.85355 0.30866 0.96194 0.50000 1.00000 0.69134 0.96194 0.85355 0.85355 0.96194 0.69134 1.00000 0.50000 0.96194 0.30866 0.85355 0.14645 0.69134 0.03806 17 holes x y 0.50000 0.00000 0.31938 0.03376 0.16315 0.13050 0.05242 0.27713 0.00213 0.45387 0.01909 0.63683 0.10099 0.80132 0.23678 0.92511 0.40813 0.99149 0.59187 0.99149 0.76322 0.92511 0.89901 0.80132 0.98091 0.63683 0.99787 0.45387 0.94758 0.27713 0.83685 0.13050 0.68062 0.03376 18 holes x y 0.50000 0.00000 0.32899 0.03015 0.17861 0.11698 0.06699 0.25000 0.00760 0.41318 0.00760 0.58682 0.06699 0.75000

18 holes (Continued) # x y 8 0.17861 0.88302 9 0.32899 0.96985 10 0.50000 1.00000 11 0.67101 0.96985 12 0.82139 0.88302 13 0.93301 0.75000 14 0.99240 0.58682 15 0.99240 0.41318 16 0.93301 0.25000 17 0.82139 0.11698 18 0.67101 0.03015 19 holes # x y 1 0.50000 0.00000 2 0.33765 0.02709 3 0.19289 0.10543 4 0.08142 0.22653 5 0.01530 0.37726 6 0.00171 0.54129 7 0.04211 0.70085 8 0.13214 0.83864 9 0.26203 0.93974 10 0.41770 0.99318 11 0.58230 0.99318 12 0.73797 0.93974 13 0.86786 0.83864 14 0.95789 0.70085 15 0.99829 0.54129 16 0.98470 0.37726 17 0.91858 0.22653 18 0.80711 0.10543 19 0.66235 0.02709 20 holes # x y 1 0.50000 0.00000 2 0.34549 0.02447 3 0.20611 0.09549 4 0.09549 0.20611 5 0.02447 0.34549 6 0.00000 0.50000 7 0.02447 0.65451 8 0.09549 0.79389 9 0.20611 0.90451 10 0.34549 0.97553

20 holes (Continued) 22 holes (Continued) # x y # x y 11 0.50000 1.00000 10 0.22968 0.92063 12 0.65451 0.97553 11 0.35913 0.97975 13 0.79389 0.90451 12 0.50000 1.00000 14 0.90451 0.79389 13 0.64087 0.97975 15 0.97553 0.65451 14 0.77032 0.92063 16 1.00000 0.50000 15 0.87787 0.82743 17 0.97553 0.34549 16 0.95482 0.70771 18 0.90451 0.20611 17 0.99491 0.57116 19 0.79389 0.09549 18 0.99491 0.42884 20 0.65451 0.02447 19 0.95482 0.29229 20 0.87787 0.17257 21 holes # x y 21 0.77032 0.07937 1 0.50000 0.00000 22 0.64087 0.02025 2 0.35262 0.02221 23 holes x y 3 0.21834 0.08688 # 4 0.10908 0.18826 1 0.50000 0.00000 5 0.03456 0.31733 2 0.36510 0.01854 6 0.00140 0.46263 3 0.24021 0.07279 7 0.01254 0.61126 4 0.13458 0.15872 8 0.06699 0.75000 5 0.05606 0.26997 9 0.15991 0.86653 6 0.01046 0.39827 10 0.28306 0.95048 7 0.00117 0.53412 11 0.42548 0.99442 8 0.02887 0.66744 12 0.57452 0.99442 9 0.09152 0.78834 13 0.71694 0.95048 10 0.18446 0.88786 14 0.84009 0.86653 11 0.30080 0.95861 15 0.93301 0.75000 12 0.43192 0.99534 16 0.98746 0.61126 13 0.56808 0.99534 17 0.99860 0.46263 14 0.69920 0.95861 18 0.96544 0.31733 15 0.81554 0.88786 19 0.89092 0.18826 16 0.90848 0.78834 20 0.78166 0.08688 17 0.97113 0.66744 21 0.64738 0.02221 18 0.99883 0.53412 19 0.98954 0.39827 22 holes # x y 20 0.94394 0.26997 1 0.50000 0.00000 21 0.86542 0.15872 2 0.35913 0.02025 22 0.75979 0.07279 3 0.22968 0.07937 23 0.63490 0.01854 4 0.12213 0.17257 24 holes x y 5 0.04518 0.29229 # 6 0.00509 0.42884 1 0.50000 0.00000 7 0.00509 0.57116 2 0.37059 0.01704 8 0.04518 0.70771 3 0.25000 0.06699 9 0.12213 0.82743 4 0.14645 0.14645

Machinery's Handbook, 31st Edition Hole Circle Coordinates

Ref

Y 2

# 1 2 3 4 5 6 7 8 9 10

714

Copyright 2020, Industrial Press, Inc.

Table 1b. Hole Coordinate Dimension Factors for Type “A” Hole Circles

X

1

Ref

Ref

Y 2

5

3

4

29 holes (Continued) # x y 9 0.00659 0.58089 10 0.03551 0.68507 11 0.08616 0.78059 12 0.15615 0.86300 13 0.24222 0.92843 14 0.34035 0.97383 15 0.44594 0.99707 16 0.55406 0.99707 17 0.65965 0.97383 18 0.75778 0.92843 19 0.84385 0.86300 20 0.91384 0.78059 21 0.96449 0.68507 22 0.99341 0.58089 23 0.99927 0.47293 24 0.98177 0.36624 25 0.94176 0.26580 26 0.88108 0.17631 27 0.80259 0.10195 28 0.70994 0.04621 29 0.60749 0.01169 30 holes # x y 1 0.50000 0.00000 2 0.39604 0.01093 3 0.29663 0.04323 4 0.20611 0.09549 5 0.12843 0.16543 6 0.06699 0.25000 7 0.02447 0.34549 8 0.00274 0.44774 9 0.00274 0.55226 10 0.02447 0.65451 11 0.06699 0.75000 12 0.12843 0.83457 13 0.20611 0.90451 14 0.29663 0.95677 15 0.39604 0.98907 16 0.50000 1.00000 17 0.60396 0.98907 18 0.70337 0.95677 19 0.79389 0.90451 20 0.87157 0.83457 21 0.93301 0.75000

30 holes (Continued) # x y # 22 0.97553 0.65451 1 23 0.99726 0.55226 2 24 0.99726 0.44774 3 25 0.97553 0.34549 4 26 0.93301 0.25000 5 27 0.87157 0.16543 6 28 0.79389 0.09549 7 29 0.70337 0.04323 8 30 0.60396 0.01093 9 10 31 holes # x y 11 1 0.50000 0.00000 12 2 0.39935 0.01024 13 3 0.30282 0.04052 14 4 0.21437 0.08962 15 5 0.13760 0.15552 16 6 0.07568 0.23552 17 7 0.03112 0.32635 18 8 0.00577 0.42429 19 9 0.00064 0.52532 20 10 0.01596 0.62533 21 11 0.05110 0.72020 22 12 0.10461 0.80605 23 13 0.17431 0.87938 24 14 0.25735 0.93717 25 15 0.35032 0.97707 26 16 0.44942 0.99743 27 17 0.55058 0.99743 28 18 0.64968 0.97707 29 19 0.74265 0.93717 30 20 0.82569 0.87938 31 21 0.89539 0.80605 32 22 0.94890 0.72020 23 0.98404 0.62533 # 24 0.99936 0.52532 1 25 0.99423 0.42429 2 26 0.96888 0.32635 3 27 0.92432 0.23552 4 28 0.86240 0.15552 5 29 0.78563 0.08962 6 30 0.69718 0.04052 7 31 0.60065 0.01024 8 9 10

32 holes 33 holes (Continued) x y # x y 0.50000 0.00000 11 0.02750 0.66353 0.40245 0.00961 12 0.06699 0.75000 0.30866 0.03806 13 0.12213 0.82743 0.22221 0.08427 14 0.19092 0.89303 0.14645 0.14645 15 0.27089 0.94442 0.08427 0.22221 16 0.35913 0.97975 0.03806 0.30866 17 0.45247 0.99774 0.00961 0.40245 18 0.54753 0.99774 0.00000 0.50000 19 0.64087 0.97975 0.00961 0.59755 20 0.72911 0.94442 0.03806 0.69134 21 0.80908 0.89303 0.08427 0.77779 22 0.87787 0.82743 0.14645 0.85355 23 0.93301 0.75000 0.22221 0.91573 24 0.97250 0.66353 0.30866 0.96194 25 0.99491 0.57116 0.40245 0.99039 26 0.99943 0.47621 0.50000 1.00000 27 0.98591 0.38212 0.59755 0.99039 28 0.95482 0.29229 0.69134 0.96194 29 0.90729 0.20997 0.77779 0.91573 30 0.84504 0.13813 0.85355 0.85355 31 0.77032 0.07937 0.91573 0.77779 32 0.68583 0.03582 0.96194 0.69134 33 0.59463 0.00904 0.99039 0.59755 1.00000 0.50000 0.99039 0.40245 0.96194 0.30866 0.91573 0.22221 0.85355 0.14645 0.77779 0.08427 0.69134 0.03806 0.59755 0.00961 33 holes x y 0.50000 0.00000 0.40537 0.00904 0.31417 0.03582 0.22968 0.07937 0.15496 0.13813 0.09271 0.20997 0.04518 0.29229 0.01409 0.38212 0.00057 0.47621 0.00509 0.57116

715

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24 holes (Continued) 25 holes (Continued) # x y # x y 5 0.06699 0.25000 7 0.00099 0.46860 6 0.01704 0.37059 8 0.00886 0.59369 7 0.00000 0.50000 9 0.04759 0.71289 8 0.01704 0.62941 10 0.11474 0.81871 9 0.06699 0.75000 11 0.20611 0.90451 10 0.14645 0.85355 12 0.31594 0.96489 11 0.25000 0.93301 13 0.43733 0.99606 12 0.37059 0.98296 14 0.56267 0.99606 13 0.50000 1.00000 15 0.68406 0.96489 14 0.62941 0.98296 16 0.79389 0.90451 15 0.75000 0.93301 17 0.88526 0.81871 16 0.85355 0.85355 18 0.95241 0.71289 17 0.93301 0.75000 19 0.99114 0.59369 18 0.98296 0.62941 20 0.99901 0.46860 19 1.00000 0.50000 21 0.97553 0.34549 20 0.98296 0.37059 22 0.92216 0.23209 21 0.93301 0.25000 23 0.84227 0.13552 22 0.85355 0.14645 24 0.74088 0.06185 23 0.75000 0.06699 25 0.62434 0.01571 24 0.62941 0.01704 26 holes # x y 25 holes # x y 1 0.50000 0.00000 1 0.50000 0.00000 2 0.38034 0.01453 2 0.37566 0.01571 3 0.26764 0.05727 3 0.25912 0.06185 4 0.16844 0.12574 4 0.15773 0.13552 5 0.08851 0.21597 5 0.07784 0.23209 6 0.03249 0.32270 6 0.02447 0.34549 7 0.00365 0.43973

26 holes (Continued) 27 holes (Continued) # x y # x y 8 0.00365 0.56027 24 0.90106 0.20142 9 0.03249 0.67730 25 0.82139 0.11698 10 0.08851 0.78403 26 0.72440 0.05318 11 0.16844 0.87426 27 0.61531 0.01348 12 0.26764 0.94273 28 holes x y 13 0.38034 0.98547 # 14 0.50000 1.00000 1 0.50000 0.00000 15 0.61966 0.98547 2 0.38874 0.01254 16 0.73236 0.94273 3 0.28306 0.04952 17 0.83156 0.87426 4 0.18826 0.10908 18 0.91149 0.78403 5 0.10908 0.18826 19 0.96751 0.67730 6 0.04952 0.28306 20 0.99635 0.56027 7 0.01254 0.38874 21 0.99635 0.43973 8 0.00000 0.50000 22 0.96751 0.32270 9 0.01254 0.61126 23 0.91149 0.21597 10 0.04952 0.71694 24 0.83156 0.12574 11 0.10908 0.81174 25 0.73236 0.05727 12 0.18826 0.89092 26 0.61966 0.01453 13 0.28306 0.95048 14 0.38874 0.98746 27 holes # x y 15 0.50000 1.00000 1 0.50000 0.00000 16 0.61126 0.98746 2 0.38469 0.01348 17 0.71694 0.95048 3 0.27560 0.05318 18 0.81174 0.89092 4 0.17861 0.11698 19 0.89092 0.81174 5 0.09894 0.20142 20 0.95048 0.71694 6 0.04089 0.30196 21 0.98746 0.61126 7 0.00760 0.41318 22 1.00000 0.50000 8 0.00085 0.52907 23 0.98746 0.38874 9 0.02101 0.64340 24 0.95048 0.28306 10 0.06699 0.75000 25 0.89092 0.18826 11 0.13631 0.84312 26 0.81174 0.10908 12 0.22525 0.91774 27 0.71694 0.04952 13 0.32899 0.96985 28 0.61126 0.01254 14 0.44195 0.99662 29 holes x y 15 0.55805 0.99662 # 16 0.67101 0.96985 1 0.50000 0.00000 17 0.77475 0.91774 2 0.39251 0.01169 18 0.86369 0.84312 3 0.29006 0.04621 19 0.93301 0.75000 4 0.19741 0.10195 20 0.97899 0.64340 5 0.11892 0.17631 21 0.99915 0.52907 6 0.05824 0.26580 22 0.99240 0.41318 7 0.01823 0.36624 23 0.95911 0.30196 8 0.00073 0.47293

Machinery's Handbook, 31st Edition Hole Circle Coordinates

Copyright 2020, Industrial Press, Inc.

TableTable 1b. (Continued) Hole Coordinate Dimension Factors for“A” Type “A”Circles Hole Circles 1b. Hole Coordinate Dimension Factors for Type Hole

1

2

–Y

–X

5

Ref

+X

4

3 # 1 2 3 # 1 2 3 4

ebooks.industrialpress.com

# 1 2 3 4 5 # 1 2 3 4 5 6

3 holes x y –0.43301 –0.25000 0.00000 0.50000 0.43301 –0.25000 4 holes x y –0.35355 –0.35355 –0.35355 0.35355 0.35355 0.35355 0.35355 –0.35355 5 holes x y –0.29389 –0.40451 –0.47553 0.15451 0.00000 0.50000 0.47553 0.15451 0.29389 –0.40451 6 holes x y –0.25000 –0.43301 –0.50000 0.00000 –0.25000 0.43301 0.25000 0.43301 0.50000 0.00000 0.25000 –0.43301

# 1 2 3 4 5 6 7 # 1 2 3 4 5 6 7 8 # 1 2 3 4 5 6 7 8 9

7 holes x y –0.21694 –0.45048 –0.48746 –0.11126 –0.39092 0.31174 0.00000 0.50000 0.39092 0.31174 0.48746 –0.11126 0.21694 –0.45048 8 holes x y –0.19134 –0.46194 –0.46194 –0.19134 –0.46194 0.19134 –0.19134 0.46194 0.19134 0.46194 0.46194 0.19134 0.46194 –0.19134 0.19134 –0.46194 9 holes x y –0.17101 –0.46985 –0.43301 –0.25000 –0.49240 0.08682 –0.32139 0.38302 0.00000 0.50000 0.32139 0.38302 0.49240 0.08682 0.43301 –0.25000 0.17101 –0.46985

# 1 2 3 4 5 6 7 8 9 10 11 # 1 2 3 4 5 6 7 8 9 10 11 12 # 1 2 3 4 5

10 holes x y –0.15451 –0.47553 –0.40451 –0.29389 –0.50000 0.00000 –0.40451 0.29389 –0.15451 0.47553 0.15451 0.47553 0.40451 0.29389 0.50000 0.00000 0.40451 –0.29389 0.15451 –0.47553 11 holes x y –0.14087 –0.47975 –0.37787 –0.32743 –0.49491 –0.07116 –0.45482 0.20771 –0.27032 0.42063 0.00000 0.50000 0.27032 0.42063 0.45482 0.20771 0.49491 –0.07116 0.37787 –0.32743 0.14087 –0.47975 12 holes x y –0.12941 –0.48296 –0.35355 –0.35355 –0.48296 –0.12941 –0.48296 0.12941 –0.35355 0.35355 –0.12941 0.48296 0.12941 0.48296 0.35355 0.35355 0.48296 0.12941 0.48296 –0.12941 0.35355 –0.35355 0.12941 –0.48296 13 holes x y –0.11966 –0.48547 –0.33156 –0.37426 –0.46751 –0.17730 –0.49635 0.06027 –0.41149 0.28403

13 holes (Continued) # x y 6 –0.23236 0.44273 7 0.00000 0.50000 8 0.23236 0.44273 9 0.41149 0.28403 10 0.49635 0.06027 11 0.46751 –0.17730 12 0.33156 –0.37426 13 0.11966 –0.48547 14 holes # x y 1 –0.11126 –0.48746 2 –0.31174 –0.39092 3 –0.45048 –0.21694 4 –0.50000 0.00000 5 –0.45048 0.21694 6 –0.31174 0.39092 7 –0.11126 0.48746 8 0.11126 0.48746 9 0.31174 0.39092 10 0.45048 0.21694 11 0.50000 0.00000 12 0.45048 –0.21694 13 0.31174 –0.39092 14 0.11126 –0.48746 15 holes # x y 1 –0.10396 –0.48907 2 –0.29389 –0.40451 3 –0.43301 –0.25000 4 –0.49726 –0.05226 5 –0.47553 0.15451 6 –0.37157 0.33457 7 –0.20337 0.45677 8 0.00000 0.50000 9 0.20337 0.45677 10 0.37157 0.33457 11 0.47553 0.15451 12 0.49726 –0.05226 13 0.43301 –0.25000 14 0.29389 –0.40451 15 0.10396 –0.48907

# 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 # 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 # 1 2 3 4 5 6 7

16 holes x y –0.09755 –0.49039 –0.27779 –0.41573 –0.41573 –0.27779 –0.49039 –0.09755 –0.49039 0.09755 –0.41573 0.27779 –0.27779 0.41573 –0.09755 0.49039 0.09755 0.49039 0.27779 0.41573 0.41573 0.27779 0.49039 0.09755 0.49039 –0.09755 0.41573 –0.27779 0.27779 –0.41573 0.09755 –0.49039 17 holes x y –0.09187 –0.49149 –0.26322 –0.42511 –0.39901 –0.30132 –0.48091 –0.13683 –0.49787 0.04613 –0.44758 0.22287 –0.33685 0.36950 –0.18062 0.46624 0.00000 0.50000 0.18062 0.46624 0.33685 0.36950 0.44758 0.22287 0.49787 0.04613 0.48091 –0.13683 0.39901 –0.30132 0.26322 –0.42511 0.09187 –0.49149 18 holes x y –0.08682 –0.49240 –0.25000 –0.43301 –0.38302 –0.32139 –0.46985 –0.17101 –0.50000 0.00000 –0.46985 0.17101 –0.38302 0.32139

18 holes (Continued) 20 holes (Continued) # x y # x y 8 –0.25000 0.43301 11 0.07822 0.49384 9 –0.08682 0.49240 12 0.22700 0.44550 10 0.08682 0.49240 13 0.35355 0.35355 11 0.25000 0.43301 14 0.44550 0.22700 12 0.38302 0.32139 15 0.49384 0.07822 13 0.46985 0.17101 16 0.49384 –0.07822 14 0.50000 0.00000 17 0.44550 –0.22700 15 0.46985 –0.17101 18 0.35355 –0.35355 16 0.38302 –0.32139 19 0.22700 –0.44550 17 0.25000 –0.43301 20 0.07822 –0.49384 18 0.08682 –0.49240 21 holes # x y 19 holes # x y 1 –0.07452 –0.49442 1 –0.08230 –0.49318 2 –0.21694 –0.45048 2 –0.23797 –0.43974 3 –0.34009 –0.36653 3 –0.36786 –0.33864 4 –0.43301 –0.25000 4 –0.45789 –0.20085 5 –0.48746 –0.11126 5 –0.49829 –0.04129 6 –0.49860 0.03737 6 –0.48470 0.12274 7 –0.46544 0.18267 7 –0.41858 0.27347 8 –0.39092 0.31174 8 –0.30711 0.39457 9 –0.28166 0.41312 9 –0.16235 0.47291 10 –0.14738 0.47779 10 0.00000 0.50000 11 0.00000 0.50000 11 0.16235 0.47291 12 0.14738 0.47779 12 0.30711 0.39457 13 0.28166 0.41312 13 0.41858 0.27347 14 0.39092 0.31174 14 0.48470 0.12274 15 0.46544 0.18267 15 0.49829 –0.04129 16 0.49860 0.03737 16 0.45789 –0.20085 17 0.48746 –0.11126 17 0.36786 –0.33864 18 0.43301 –0.25000 18 0.23797 –0.43974 19 0.34009 –0.36653 19 0.08230 –0.49318 20 0.21694 –0.45048 21 0.07452 –0.49442 20 holes # x y 22 holes x y 1 –0.07822 –0.49384 # 2 –0.22700 –0.44550 1 –0.07116 –0.49491 3 –0.35355 –0.35355 2 –0.20771 –0.45482 4 –0.44550 –0.22700 3 –0.32743 –0.37787 5 –0.49384 –0.07822 4 –0.42063 –0.27032 6 –0.49384 0.07822 5 –0.47975 –0.14087 7 –0.44550 0.22700 6 –0.50000 0.00000 8 –0.35355 0.35355 7 –0.47975 0.14087 9 –0.22700 0.44550 8 –0.42063 0.27032 10 –0.07822 0.49384 9 –0.32743 0.37787

22 holes (Continued) # x y 10 –0.20771 0.45482 11 –0.07116 0.49491 12 0.07116 0.49491 13 0.20771 0.45482 14 0.32743 0.37787 15 0.42063 0.27032 16 0.47975 0.14087 17 0.50000 0.00000 18 0.47975 –0.14087 19 0.42063 –0.27032 20 0.32743 –0.37787 21 0.20771 –0.45482 22 0.07116 –0.49491 23 holes # x y 1 –0.06808 –0.49534 2 –0.19920 –0.45861 3 –0.31554 –0.38786 4 –0.40848 –0.28834 5 –0.47113 –0.16744 6 –0.49883 –0.03412 7 –0.48954 0.10173 8 –0.44394 0.23003 9 –0.36542 0.34128 10 –0.25979 0.42721 11 –0.13490 0.48146 12 0.00000 0.50000 13 0.13490 0.48146 14 0.25979 0.42721 15 0.36542 0.34128 16 0.44394 0.23003 17 0.48954 0.10173 18 0.49883 –0.03412 19 0.47113 –0.16744 20 0.40848 –0.28834 21 0.31554 –0.38786 22 0.19920 –0.45861 23 0.06808 –0.49534 24 holes # x y 1 –0.06526 –0.49572 2 –0.19134 –0.46194 3 –0.30438 –0.39668 4 –0.39668 –0.30438

Machinery's Handbook, 31st Edition Hole Circle Coordinates

+Y

# 1 2 3 4 5 6 7 8 9 10

716

Copyright 2020, Industrial Press, Inc.

Table 2a. Hole Coordinate Dimension Factors for Type “B” Hole Circles

5

Ref

27 holes (Continued) # x y 24 0.36369 –0.34312 25 0.27475 –0.41774 26 0.17101 –0.46985 27 0.05805 –0.49662 28 holes # x y 1 –0.05598 –0.49686 2 –0.16514 –0.47194 3 –0.26602 –0.42336 4 –0.35355 –0.35355 5 –0.42336 –0.26602 6 –0.47194 –0.16514 7 –0.49686 –0.05598 8 –0.49686 0.05598 9 –0.47194 0.16514 10 –0.42336 0.26602 11 –0.35355 0.35355 12 –0.26602 0.42336 13 –0.16514 0.47194 14 –0.05598 0.49686 15 0.05598 0.49686 16 0.16514 0.47194 17 0.26602 0.42336 18 0.35355 0.35355 19 0.42336 0.26602 20 0.47194 0.16514 21 0.49686 0.05598 22 0.49686 –0.05598 23 0.47194 –0.16514 24 0.42336 –0.26602 25 0.35355 –0.35355 26 0.26602 –0.42336 27 0.16514 –0.47194 28 0.05598 –0.49686 29 holes # x y 1 –0.05406 –0.49707 2 –0.15965 –0.47383 3 –0.25778 –0.42843 4 –0.34385 –0.36300 5 –0.41384 –0.28059 6 –0.46449 –0.18507 7 –0.49341 –0.08089 8 –0.49927 0.02707

29 holes (Continued) # x y 9 –0.48177 0.13376 10 –0.44176 0.23420 11 –0.38108 0.32369 12 –0.30259 0.39805 13 –0.20994 0.45379 14 –0.10749 0.48831 15 0.00000 0.50000 16 0.10749 0.48831 17 0.20994 0.45379 18 0.30259 0.39805 19 0.38108 0.32369 20 0.44176 0.23420 21 0.48177 0.13376 22 0.49927 0.02707 23 0.49341 –0.08089 24 0.46449 –0.18507 25 0.41384 –0.28059 26 0.34385 –0.36300 27 0.25778 –0.42843 28 0.15965 –0.47383 29 0.05406 –0.49707 30 holes # x y 1 –0.05226 –0.49726 2 –0.15451 –0.47553 3 –0.25000 –0.43301 4 –0.33457 –0.37157 5 –0.40451 –0.29389 6 –0.45677 –0.20337 7 –0.48907 –0.10396 8 –0.50000 0.00000 9 –0.48907 0.10396 10 –0.45677 0.20337 11 –0.40451 0.29389 12 –0.33457 0.37157 13 –0.25000 0.43301 14 –0.15451 0.47553 15 –0.05226 0.49726 16 0.05226 0.49726 17 0.15451 0.47553 18 0.25000 0.43301 19 0.33457 0.37157 20 0.40451 0.29389 21 0.45677 0.20337

30 holes (Continued) # x y 22 0.48907 0.10396 23 0.50000 0.00000 24 0.48907 –0.10396 25 0.45677 –0.20337 26 0.40451 –0.29389 27 0.33457 –0.37157 28 0.25000 –0.43301 29 0.15451 –0.47553 30 0.05226 –0.49726 31 holes # x y 1 –0.05058 –0.49743 2 –0.14968 –0.47707 3 –0.24265 –0.43717 4 –0.32569 –0.37938 5 –0.39539 –0.30605 6 –0.44890 –0.22020 7 –0.48404 –0.12533 8 –0.49936 –0.02532 9 –0.49423 0.07571 10 –0.46888 0.17365 11 –0.42432 0.26448 12 –0.36240 0.34448 13 –0.28563 0.41038 14 –0.19718 0.45948 15 –0.10065 0.48976 16 0.00000 0.50000 17 0.10065 0.48976 18 0.19718 0.45948 19 0.28563 0.41038 20 0.36240 0.34448 21 0.42432 0.26448 22 0.46888 0.17365 23 0.49423 0.07571 24 0.49936 –0.02532 25 0.48404 –0.12533 26 0.44890 –0.22020 27 0.39539 –0.30605 28 0.32569 –0.37938 29 0.24265 –0.43717 30 0.14968 –0.47707 31 0.05058 –0.49743

32 holes x y –0.04901 –0.49759 –0.14514 –0.47847 –0.23570 –0.44096 –0.31720 –0.38651 –0.38651 –0.31720 –0.44096 –0.23570 –0.47847 –0.14514 –0.49759 –0.04901 –0.49759 0.04901 –0.47847 0.14514 –0.44096 0.23570 –0.38651 0.31720 –0.31720 0.38651 –0.23570 0.44096 –0.14514 0.47847 –0.04901 0.49759 0.04901 0.49759 0.14514 0.47847 0.23570 0.44096 0.31720 0.38651 0.38651 0.31720 0.44096 0.23570 0.47847 0.14514 0.49759 0.04901 0.49759 –0.04901 0.47847 –0.14514 0.44096 –0.23570 0.38651 –0.31720 0.31720 –0.38651 0.23570 –0.44096 0.14514 –0.47847 0.04901 –0.49759 33 holes # x y 1 –0.04753 –0.49774 2 –0.14087 –0.47975 3 –0.22911 –0.44442 4 –0.30908 –0.39303 5 –0.37787 –0.32743 6 –0.43301 –0.25000 7 –0.47250 –0.16353 8 –0.49491 –0.07116 9 –0.49943 0.02379 10 –0.48591 0.11788

# 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32

33 holes (Continued) # x y 11 –0.45482 0.20771 12 –0.40729 0.29003 13 –0.34504 0.36187 14 –0.27032 0.42063 15 –0.18583 0.46418 16 –0.09463 0.49096 17 0.00000 0.50000 18 0.09463 0.49096 19 0.18583 0.46418 20 0.27032 0.42063 21 0.34504 0.36187 22 0.40729 0.29003 23 0.45482 0.20771 24 0.48591 0.11788 25 0.49943 0.02379 26 0.49491 –0.07116 27 0.47250 –0.16353 28 0.43301 –0.25000 29 0.37787 –0.32743 30 0.30908 –0.39303 31 0.22911 –0.44442 32 0.14087 –0.47975 33 0.04753 –0.49774

717

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26 holes (Continued) # x y 8 –0.48547 0.11966 9 –0.44273 0.23236 –Y 10 –0.37426 0.33156 11 –0.28403 0.41149 12 –0.17730 0.46751 –X +X 13 –0.06027 0.49635 14 0.06027 0.49635 4 2 15 0.17730 0.46751 16 0.28403 0.41149 +Y 17 0.37426 0.33156 18 0.44273 0.23236 3 19 0.48547 0.11966 20 0.50000 0.00000 21 0.48547 –0.11966 22 0.44273 –0.23236 24 holes (Continued) 25 holes (Continued) # x y # x y 23 0.37426 –0.33156 5 –0.46194 –0.19134 7 –0.49901 0.03140 24 0.28403 –0.41149 6 –0.49572 –0.06526 8 –0.47553 0.15451 25 0.17730 –0.46751 7 –0.49572 0.06526 9 –0.42216 0.26791 26 0.06027 –0.49635 8 –0.46194 0.19134 10 –0.34227 0.36448 27 holes x y 9 –0.39668 0.30438 11 –0.24088 0.43815 # 10 –0.30438 0.39668 12 –0.12434 0.48429 1 –0.05805 –0.49662 11 –0.19134 0.46194 13 0.00000 0.50000 2 –0.17101 –0.46985 12 –0.06526 0.49572 14 0.12434 0.48429 3 –0.27475 –0.41774 13 0.06526 0.49572 15 0.24088 0.43815 4 –0.36369 –0.34312 14 0.19134 0.46194 16 0.34227 0.36448 5 –0.43301 –0.25000 15 0.30438 0.39668 17 0.42216 0.26791 6 –0.47899 –0.14340 16 0.39668 0.30438 18 0.47553 0.15451 7 –0.49915 –0.02907 17 0.46194 0.19134 19 0.49901 0.03140 8 –0.49240 0.08682 18 0.49572 0.06526 20 0.49114 –0.09369 9 –0.45911 0.19804 19 0.49572 –0.06526 21 0.45241 –0.21289 10 –0.40106 0.29858 20 0.46194 –0.19134 22 0.38526 –0.31871 11 –0.32139 0.38302 21 0.39668 –0.30438 23 0.29389 –0.40451 12 –0.22440 0.44682 22 0.30438 –0.39668 24 0.18406 –0.46489 13 –0.11531 0.48652 23 0.19134 –0.46194 25 0.06267 –0.49606 14 0.00000 0.50000 24 0.06526 –0.49572 15 0.11531 0.48652 26 holes # x y 16 0.22440 0.44682 25 holes # x y 1 –0.06027 –0.49635 17 0.32139 0.38302 1 –0.06267 –0.49606 2 –0.17730 –0.46751 18 0.40106 0.29858 2 –0.18406 –0.46489 3 –0.28403 –0.41149 19 0.45911 0.19804 3 –0.29389 –0.40451 4 –0.37426 –0.33156 20 0.49240 0.08682 4 –0.38526 –0.31871 5 –0.44273 –0.23236 21 0.49915 –0.02907 5 –0.45241 –0.21289 6 –0.48547 –0.11966 22 0.47899 –0.14340 6 –0.49114 –0.09369 7 –0.50000 0.00000 23 0.43301 –0.25000

1

Machinery's Handbook, 31st Edition Hole Circle Coordinates

Copyright 2020, Industrial Press, Inc.

TableTable 2a. (Continued) Hole Coordinate Dimension Factors for“B” Type “B”Circles Hole Circles 2a. Hole Coordinate Dimension Factors for Type Hole

1

Y

# 1 2 3 # 1 2 3 4

ebooks.industrialpress.com

# 1 2 3 4 5 # 1 2 3 4 5 6

3 # 1 2 3 4 5 6 7 # 1 2 3 4 5 6 7 8 # 1 2 3 4 5 6 7 8 9

# 1 2 3 4 5 6 7 8 9 10

# 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 # 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 # 1 2 3 4 5 6 7

16 holes x y 0.40245 0.00961 0.22221 0.08427 0.08427 0.22221 0.00961 0.40245 0.00961 0.59755 0.08427 0.77779 0.22221 0.91573 0.40245 0.99039 0.59755 0.99039 0.77779 0.91573 0.91573 0.77779 0.99039 0.59755 0.99039 0.40245 0.91573 0.22221 0.77779 0.08427 0.59755 0.00961 17 holes x y 0.40813 0.00851 0.23678 0.07489 0.10099 0.19868 0.01909 0.36317 0.00213 0.54613 0.05242 0.72287 0.16315 0.86950 0.31938 0.96624 0.50000 1.00000 0.68062 0.96624 0.83685 0.86950 0.94758 0.72287 0.99787 0.54613 0.98091 0.36317 0.89901 0.19868 0.76322 0.07489 0.59187 0.00851 18 holes x y 0.41318 0.00760 0.25000 0.06699 0.11698 0.17861 0.03015 0.32899 0.00000 0.50000 0.03015 0.67101 0.11698 0.82139

18 holes (Continued) # x y 8 0.25000 0.93301 9 0.41318 0.99240 10 0.58682 0.99240 11 0.75000 0.93301 12 0.88302 0.82139 13 0.96985 0.67101 14 1.00000 0.50000 15 0.96985 0.32899 16 0.88302 0.17861 17 0.75000 0.06699 18 0.58682 0.00760 19 holes # x y 1 0.41770 0.00682 2 0.26203 0.06026 3 0.13214 0.16136 4 0.04211 0.29915 5 0.00171 0.45871 6 0.01530 0.62274 7 0.08142 0.77347 8 0.19289 0.89457 9 0.33765 0.97291 10 0.50000 1.00000 11 0.66235 0.97291 12 0.80711 0.89457 13 0.91858 0.77347 14 0.98470 0.62274 15 0.99829 0.45871 16 0.95789 0.29915 17 0.86786 0.16136 18 0.73797 0.06026 19 0.58230 0.00682 20 holes # x y 1 0.42178 0.00616 2 0.27300 0.05450 3 0.14645 0.14645 4 0.05450 0.27300 5 0.00616 0.42178 6 0.00616 0.57822 7 0.05450 0.72700 8 0.14645 0.85355 9 0.27300 0.94550 10 0.42178 0.99384

20 holes (Continued) 22 holes (Continued) # x y # x y 11 0.57822 0.99384 10 0.29229 0.95482 12 0.72700 0.94550 11 0.42884 0.99491 13 0.85355 0.85355 12 0.57116 0.99491 14 0.94550 0.72700 13 0.70771 0.95482 15 0.99384 0.57822 14 0.82743 0.87787 16 0.99384 0.42178 15 0.92063 0.77032 17 0.94550 0.27300 16 0.97975 0.64087 18 0.85355 0.14645 17 1.00000 0.50000 19 0.72700 0.05450 18 0.97975 0.35913 20 0.57822 0.00616 19 0.92063 0.22968 20 0.82743 0.12213 21 holes # x y 21 0.70771 0.04518 1 0.42548 0.00558 22 0.57116 0.00509 2 0.28306 0.04952 23 holes x y 3 0.15991 0.13347 # 4 0.06699 0.25000 1 0.43192 0.00466 5 0.01254 0.38874 2 0.30080 0.04139 6 0.00140 0.53737 3 0.18446 0.11214 7 0.03456 0.68267 4 0.09152 0.21166 8 0.10908 0.81174 5 0.02887 0.33256 9 0.21834 0.91312 6 0.00117 0.46588 10 0.35262 0.97779 7 0.01046 0.60173 11 0.50000 1.00000 8 0.05606 0.73003 12 0.64738 0.97779 9 0.13458 0.84128 13 0.78166 0.91312 10 0.24021 0.92721 14 0.89092 0.81174 11 0.36510 0.98146 15 0.96544 0.68267 12 0.50000 1.00000 16 0.99860 0.53737 13 0.63490 0.98146 17 0.98746 0.38874 14 0.75979 0.92721 18 0.93301 0.25000 15 0.86542 0.84128 19 0.84009 0.13347 16 0.94394 0.73003 20 0.71694 0.04952 17 0.98954 0.60173 21 0.57452 0.00558 18 0.99883 0.46588 19 0.97113 0.33256 22 holes # x y 20 0.90848 0.21166 1 0.42884 0.00509 21 0.81554 0.11214 2 0.29229 0.04518 22 0.69920 0.04139 3 0.17257 0.12213 23 0.56808 0.00466 4 0.07937 0.22968 24 holes x y 5 0.02025 0.35913 # 6 0.00000 0.50000 1 0.43474 0.00428 7 0.02025 0.64087 2 0.30866 0.03806 8 0.07937 0.77032 3 0.19562 0.10332 9 0.17257 0.87787 4 0.10332 0.19562

Machinery's Handbook, 31st Edition Hole Circle Coordinates

Ref

2

3 holes x y 0.06699 0.25000 0.50000 1.00000 0.93301 0.25000 4 holes x y 0.14645 0.14645 0.14645 0.85355 0.85355 0.85355 0.85355 0.14645 5 holes x y 0.20611 0.09549 0.02447 0.65451 0.50000 1.00000 0.97553 0.65451 0.79389 0.09549 6 holes x y 0.25000 0.06699 0.00000 0.50000 0.25000 0.93301 0.75000 0.93301 1.00000 0.50000 0.75000 0.06699

10 holes 13 holes (Continued) x y # x y 0.34549 0.02447 6 0.26764 0.94273 0.09549 0.20611 7 0.50000 1.00000 5 0.00000 0.50000 8 0.73236 0.94273 0.09549 0.79389 9 0.91149 0.78403 0.34549 0.97553 10 0.99635 0.56027 0.65451 0.97553 11 0.96751 0.32270 0.90451 0.79389 12 0.83156 0.12574 1.00000 0.50000 13 0.61966 0.01453 4 0.90451 0.20611 14 holes x y 0.65451 0.02447 # 1 0.38874 0.01254 11 holes # x y 2 0.18826 0.10908 1 0.35913 0.02025 3 0.04952 0.28306 2 0.12213 0.17257 4 0.00000 0.50000 3 0.00509 0.42884 5 0.04952 0.71694 7 holes x y 4 0.04518 0.70771 6 0.18826 0.89092 0.28306 0.04952 5 0.22968 0.92063 7 0.38874 0.98746 0.01254 0.38874 6 0.50000 1.00000 8 0.61126 0.98746 0.10908 0.81174 7 0.77032 0.92063 9 0.81174 0.89092 0.50000 1.00000 8 0.95482 0.70771 10 0.95048 0.71694 0.89092 0.81174 9 0.99491 0.42884 11 1.00000 0.50000 0.98746 0.38874 10 0.87787 0.17257 12 0.95048 0.28306 0.71694 0.04952 11 0.64087 0.02025 13 0.81174 0.10908 14 0.61126 0.01254 8 holes 12 holes 15 holes x y # x y x y 0.30866 0.03806 1 0.37059 0.01704 # 0.03806 0.30866 2 0.14645 0.14645 1 0.39604 0.01093 0.03806 0.69134 3 0.01704 0.37059 2 0.20611 0.09549 0.30866 0.96194 4 0.01704 0.62941 3 0.06699 0.25000 0.69134 0.96194 5 0.14645 0.85355 4 0.00274 0.44774 0.96194 0.69134 6 0.37059 0.98296 5 0.02447 0.65451 0.96194 0.30866 7 0.62941 0.98296 6 0.12843 0.83457 0.69134 0.03806 8 0.85355 0.85355 7 0.29663 0.95677 9 0.98296 0.62941 8 0.50000 1.00000 9 holes x y 10 0.98296 0.37059 9 0.70337 0.95677 0.32899 0.03015 11 0.85355 0.14645 10 0.87157 0.83457 0.06699 0.25000 12 0.62941 0.01704 11 0.97553 0.65451 0.00760 0.58682 12 0.99726 0.44774 13 holes x y 0.17861 0.88302 # 13 0.93301 0.25000 0.50000 1.00000 1 0.38034 0.01453 14 0.79389 0.09549 0.82139 0.88302 2 0.16844 0.12574 15 0.60396 0.01093 0.99240 0.58682 3 0.03249 0.32270 0.93301 0.25000 4 0.00365 0.56027 0.67101 0.03015 5 0.08851 0.78403

Ref

718

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Table 2b. Hole Coordinate Dimension Factors for Type “B” Hole Circles

X

Ref

5

1

Y

Ref

2

4 3

29 holes (Continued) # x y 9 0.01823 0.63376 10 0.05824 0.73420 11 0.11892 0.82369 12 0.19741 0.89805 13 0.29006 0.95379 14 0.39251 0.98831 15 0.50000 1.00000 16 0.60749 0.98831 17 0.70994 0.95379 18 0.80259 0.89805 19 0.88108 0.82369 20 0.94176 0.73420 21 0.98177 0.63376 22 0.99927 0.52707 23 0.99341 0.41911 24 0.96449 0.31493 25 0.91384 0.21941 26 0.84385 0.13700 27 0.75778 0.07157 28 0.65965 0.02617 29 0.55406 0.00293 30 holes # x y 1 0.44774 0.00274 2 0.34549 0.02447 3 0.25000 0.06699 4 0.16543 0.12843 5 0.09549 0.20611 6 0.04323 0.29663 7 0.01093 0.39604 8 0.00000 0.50000 9 0.01093 0.60396 10 0.04323 0.70337 11 0.09549 0.79389 12 0.16543 0.87157 13 0.25000 0.93301 14 0.34549 0.97553 15 0.44774 0.99726 16 0.55226 0.99726 17 0.65451 0.97553 18 0.75000 0.93301 19 0.83457 0.87157 20 0.90451 0.79389 21 0.95677 0.70337

30 holes (Continued) # x y # 22 0.98907 0.60396 1 23 1.00000 0.50000 2 24 0.98907 0.39604 3 25 0.95677 0.29663 4 26 0.90451 0.20611 5 27 0.83457 0.12843 6 28 0.75000 0.06699 7 29 0.65451 0.02447 8 30 0.55226 0.00274 9 10 31 holes # x y 11 1 0.44942 0.00257 12 2 0.35032 0.02293 13 3 0.25735 0.06283 14 4 0.17431 0.12062 15 5 0.10461 0.19395 16 6 0.05110 0.27980 17 7 0.01596 0.37467 18 8 0.00064 0.47468 19 9 0.00577 0.57571 20 10 0.03112 0.67365 21 11 0.07568 0.76448 22 12 0.13760 0.84448 23 13 0.21437 0.91038 24 14 0.30282 0.95948 25 15 0.39935 0.98976 26 16 0.50000 1.00000 27 17 0.60065 0.98976 28 18 0.69718 0.95948 29 19 0.78563 0.91038 30 20 0.86240 0.84448 31 21 0.92432 0.76448 32 22 0.96888 0.67365 23 0.99423 0.57571 # 24 0.99936 0.47468 1 25 0.98404 0.37467 2 26 0.94890 0.27980 3 27 0.89539 0.19395 4 28 0.82569 0.12062 5 29 0.74265 0.06283 6 30 0.64968 0.02293 7 31 0.55058 0.00257 8 9 10

32 holes 33 holes (Continued) x y # x y 0.45099 0.00241 11 0.04518 0.70771 0.35486 0.02153 12 0.09271 0.79003 0.26430 0.05904 13 0.15496 0.86187 0.18280 0.11349 14 0.22968 0.92063 0.11349 0.18280 15 0.31417 0.96418 0.05904 0.26430 16 0.40537 0.99096 0.02153 0.35486 17 0.50000 1.00000 0.00241 0.45099 18 0.59463 0.99096 0.00241 0.54901 19 0.68583 0.96418 0.02153 0.64514 20 0.77032 0.92063 0.05904 0.73570 21 0.84504 0.86187 0.11349 0.81720 22 0.90729 0.79003 0.18280 0.88651 23 0.95482 0.70771 0.26430 0.94096 24 0.98591 0.61788 0.35486 0.97847 25 0.99943 0.52379 0.45099 0.99759 26 0.99491 0.42884 0.54901 0.99759 27 0.97250 0.33647 0.64514 0.97847 28 0.93301 0.25000 0.73570 0.94096 29 0.87787 0.17257 0.81720 0.88651 30 0.80908 0.10697 0.88651 0.81720 31 0.72911 0.05558 0.94096 0.73570 32 0.64087 0.02025 0.97847 0.64514 33 0.54753 0.00226 0.99759 0.54901 0.99759 0.45099 0.97847 0.35486 0.94096 0.26430 0.88651 0.18280 0.81720 0.11349 0.73570 0.05904 0.64514 0.02153 0.54901 0.00241 33 holes x y 0.45247 0.00226 0.35913 0.02025 0.27089 0.05558 0.19092 0.10697 0.12213 0.17257 0.06699 0.25000 0.02750 0.33647 0.00509 0.42884 0.00057 0.52379 0.01409 0.61788

719

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24 holes (Continued) 25 holes (Continued) # x y # x y 5 0.03806 0.30866 7 0.00099 0.53140 6 0.00428 0.43474 8 0.02447 0.65451 7 0.00428 0.56526 9 0.07784 0.76791 8 0.03806 0.69134 10 0.15773 0.86448 9 0.10332 0.80438 11 0.25912 0.93815 10 0.19562 0.89668 12 0.37566 0.98429 11 0.30866 0.96194 13 0.50000 1.00000 12 0.43474 0.99572 14 0.62434 0.98429 13 0.56526 0.99572 15 0.74088 0.93815 14 0.69134 0.96194 16 0.84227 0.86448 15 0.80438 0.89668 17 0.92216 0.76791 16 0.89668 0.80438 18 0.97553 0.65451 17 0.96194 0.69134 19 0.99901 0.53140 18 0.99572 0.56526 20 0.99114 0.40631 19 0.99572 0.43474 21 0.95241 0.28711 20 0.96194 0.30866 22 0.88526 0.18129 21 0.89668 0.19562 23 0.79389 0.09549 22 0.80438 0.10332 24 0.68406 0.03511 23 0.69134 0.03806 25 0.56267 0.00394 24 0.56526 0.00428 26 holes # x y 25 holes # x y 1 0.43973 0.00365 1 0.43733 0.00394 2 0.32270 0.03249 2 0.31594 0.03511 3 0.21597 0.08851 3 0.20611 0.09549 4 0.12574 0.16844 4 0.11474 0.18129 5 0.05727 0.26764 5 0.04759 0.28711 6 0.01453 0.38034 6 0.00886 0.40631 7 0.00000 0.50000

26 holes (Continued) 27 holes (Continued) # x y # x y 8 0.01453 0.61966 24 0.86369 0.15688 9 0.05727 0.73236 25 0.77475 0.08226 10 0.12574 0.83156 26 0.67101 0.03015 11 0.21597 0.91149 27 0.55805 0.00338 12 0.32270 0.96751 28 holes x y 13 0.43973 0.99635 # 14 0.56027 0.99635 1 0.44402 0.00314 15 0.67730 0.96751 2 0.33486 0.02806 16 0.78403 0.91149 3 0.23398 0.07664 17 0.87426 0.83156 4 0.14645 0.14645 18 0.94273 0.73236 5 0.07664 0.23398 19 0.98547 0.61966 6 0.02806 0.33486 20 1.00000 0.50000 7 0.00314 0.44402 21 0.98547 0.38034 8 0.00314 0.55598 22 0.94273 0.26764 9 0.02806 0.66514 23 0.87426 0.16844 10 0.07664 0.76602 24 0.78403 0.08851 11 0.14645 0.85355 25 0.67730 0.03249 12 0.23398 0.92336 26 0.56027 0.00365 13 0.33486 0.97194 14 0.44402 0.99686 27 holes # x y 15 0.55598 0.99686 1 0.44195 0.00338 16 0.66514 0.97194 2 0.32899 0.03015 17 0.76602 0.92336 3 0.22525 0.08226 18 0.85355 0.85355 4 0.13631 0.15688 19 0.92336 0.76602 5 0.06699 0.25000 20 0.97194 0.66514 6 0.02101 0.35660 21 0.99686 0.55598 7 0.00085 0.47093 22 0.99686 0.44402 8 0.00760 0.58682 23 0.97194 0.33486 9 0.04089 0.69804 24 0.92336 0.23398 10 0.09894 0.79858 25 0.85355 0.14645 11 0.17861 0.88302 26 0.76602 0.07664 12 0.27560 0.94682 27 0.66514 0.02806 13 0.38469 0.98652 28 0.55598 0.00314 14 0.50000 1.00000 29 holes x y 15 0.61531 0.98652 # 16 0.72440 0.94682 1 0.44594 0.00293 17 0.82139 0.88302 2 0.34035 0.02617 18 0.90106 0.79858 3 0.24222 0.07157 19 0.95911 0.69804 4 0.15615 0.13700 20 0.99240 0.58682 5 0.08616 0.21941 21 0.99915 0.47093 6 0.03551 0.31493 22 0.97899 0.35660 7 0.00659 0.41911 23 0.93301 0.25000 8 0.00073 0.52707

Machinery's Handbook, 31st Edition Hole Circle Coordinates

Copyright 2020, Industrial Press, Inc.

TableTable 2b. (Continued) Hole Coordinate Dimension Factors for“B” Type “B”Circles Hole Circles 2b. Hole Coordinate Dimension Factors for Type Hole

X

Machinery's Handbook, 31st Edition Lengths of Chords

720

Table 3. Lengths of Chords for Spacing Off the Circumferences of Circles with a Diameter Equal to 1 (English or Metric units) No. of Spaces 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40

Length of Chord 0.866025 0.707107 0.587785 0.500000 0.433884 0.382683 0.342020 0.309017 0.281733 0.258819 0.239316 0.222521 0.207912 0.195090 0.183750 0.173648 0.164595 0.156434 0.149042 0.142315 0.136167 0.130526 0.125333 0.120537 0.116093 0.111964 0.108119 0.104528 0.101168 0.098017 0.095056 0.092268 0.089639 0.087156 0.084806 0.082579 0.080467 0.078459

No. of Spaces 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78

Length of Chord 0.076549 0.074730 0.072995 0.071339 0.069756 0.068242 0.066793 0.065403 0.064070 0.062791 0.061561 0.060378 0.059241 0.058145 0.057089 0.056070 0.055088 0.054139 0.053222 0.052336 0.051479 0.050649 0.049846 0.049068 0.048313 0.047582 0.046872 0.046183 0.045515 0.044865 0.044233 0.043619 0.043022 0.042441 0.041876 0.041325 0.040789 0.040266

No. of Spaces 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116

Length of Chord 0.039757 0.039260 0.038775 0.038303 0.037841 0.037391 0.036951 0.036522 0.036102 0.035692 0.035291 0.034899 0.034516 0.034141 0.033774 0.033415 0.033063 0.032719 0.032382 0.032052 0.031728 0.031411 0.031100 0.030795 0.030496 0.030203 0.029915 0.029633 0.029356 0.029085 0.028818 0.028556 0.028299 0.028046 0.027798 0.027554 0.027315 0.027079

No. of Spaces 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154

Length of Chord 0.026848 0.026621 0.026397 0.026177 0.025961 0.025748 0.025539 0.025333 0.025130 0.024931 0.024734 0.024541 0.024351 0.024164 0.023979 0.023798 0.023619 0.023443 0.023269 0.023098 0.022929 0.022763 0.022599 0.022438 0.022279 0.022122 0.021967 0.021815 0.021664 0.021516 0.021370 0.021225 0.021083 0.020942 0.020804 0.020667 0.020532 0.020399

For circles of other diameters, multiply length given in table by diameter of circle.

Example: In a drill jig, 8 holes, each 1 ⁄2 inch diameter, were spaced evenly on a 6-inch diameter circle. To test the accuracy of the jig, plugs were placed in adjacent holes, and the distance over the plugs was measured with a micrometer. What should be the micrometer reading? Solution: The micrometer reading equals the diameter of one plug plus 6 times the chordal distance between adjacent hole centers given in the table above. Thus, the reading should be 1 ⁄2 + (6 3 0.382683) = 2.796098 inches.

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Machinery's Handbook, 31st Edition Gage Blocks

721

Gage Blocks The primary standard for linear measurement is the gage block. Gage blocks were origi­ nally called Jo-Blocks after their Swedish inventor, Carl Edvard Johansson. These preci­ sion lapped blocks are the primary means of establishing measurement traceability to the prime standards located in the national laboratories of every country. The primary length standards are themselves defined by the distance traveled by light in a vacuum over a fixed time period.

The gage block is critical in establishing true traceability and measurement assur­ance in the dimensional discipline. There are several materials and grades of gage blocks to select from. The most common material in use today is steel. However, there is also Croblox, made by Mitutoyo, a thermally stable material with a very low coefficient of thermal expansion, and ceramic blocks, with extremely good wear capabilities. The average life span of a gage block is approximately 3 years. With proper care and cleaning the gage block may last many years longer but will eventually wear beyond the limits of the allow­ able tolerances and will need to be replaced.

The size tolerances applied to gage blocks, defined in the ANSI/ASME B89.1.9-2002, are shown in Table 1a and Table 1b, for inch and metric units, respectively. Nearly all gage blocks are manufactured and calibrated to this standard. B89.1.9 establishes the allowable deviations for size variance as well as flatness and parallelism. It is these controlled dimen­sions that give the gage block the properties necessary for use as a dimensional standard. Table 1a. Maximum Permitted Deviations of Length at Any Pointa and Tolerance on Variation in Length, Inchb ANSI/ASME B89.1.9-2002 (R2012) Limit on Deviations of Length a

Tolerance for Variation in Length b

Limit on Deviations of Length a

Tolerance for Variation in Length b

Limit on Deviations of Length a

Tolerance for Variation in Length b

Grade AS-2

Tolerance for Variation in Length b

Grade AS-1

Limit on Deviations of Length a

Grade 0

Tolerance for Variation in Length b

Nominal Length Range, ln inches ≤ 0.05

Grade 00

Limit on Deviations of Length a

Calibration Grade K

± te μin.

tv μin.

± te μin.

tv μin.

± te μin.

tv μin.

± te μin.

tv μin.

± te μin.

tv μin.

0.05 < ln ≤ 0.4 0.55 < ln ≤ 1 1 < ln ≤ 2 2 < ln ≤ 3 3 < ln ≤ 4 4 < ln ≤ 5 5 < ln ≤ 6 6 < ln ≤ 7 7 < ln ≤ 8 8 < ln ≤ 10 10 < ln ≤ 12 12 < ln ≤ 16 16 < ln ≤ 20 20 < ln ≤ 24 24 < ln ≤ 28 28 < ln ≤ 32 32 < ln ≤ 36 36 < ln ≤ 40

12 10 12 16 20 24 32 32 40 40 48 56 72 88 104 120 136 152 160

2

3

4

5 6 7 8 10

4 3 3 4 5 6 8 8 10 10 12 14 18 20 25 30 34 38 40

2

3

4

5 6 7 8 10

6 5 6 8 10 12 16 16 20 20 24 28 36 44 52 60 68 76 80

a Maximum permitted deviations of length at any point, ±t b Tolerance, t

v μinch, for the variation in length.

4

5

6 7 8 10 12 14 16

12 8 12 16 20 24 32 40

48 56 72 88 104 120 136 152 168

6

8

10 12 14 16 18 20 24

24 18 24 32 40 48 64 80

104 112 144 176 200 240 260 300 320

12

14

16 18 20 24 28 32 36 40

e μinch, from nominal length, l e inches.

Care of Gage Blocks.—Through proper care and handling of gage blocks, their functional life span can be maximized and many years of use can be realized from your investment. The basic care and cleaning of gage blocks should follow these simple guidelines.

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Machinery's Handbook, 31st Edition Gage Blocks

722

Table 1b. Maximum Permitted Deviations of Length at Any Pointa and Tolerance on Variation in Length, Metricb ANSI/ASME B89.1.9-2002 (R2012)

0.5 < ln ≤ 10 10 < ln ≤ 25 25 < ln ≤ 50 50 < ln ≤ 75 75 < ln ≤ 100 100 < ln ≤ 150 150 < ln ≤ 200 200 < ln ≤ 250 250 < ln ≤ 300 300 < ln ≤ 400 400 < ln ≤ 500 500 < ln ≤ 600 600 < ln ≤ 700 700 < ln ≤ 800 800 < ln ≤ 900 900 < ln ≤ 1000

Tolerance for Variation in Length  b

Limit on Deviations of Length a

Tolerance for Variation in Length  b

Limit on Deviations of Length a

Tolerance for Variation in Length  b

Limit on Deviations of Length a

Tolerance for Variation in Length  b

Grade AS-2

± te μm

tv μm

± te μm

tv μm

± te μm

tv μm

± te μm

tv μm

± te μm

tv μm

0.30 0.20 0.30 0.40 0.50 0.60 0.80 1.00 1.20 1.40 1.80 2.20 2.60 3.00 3.40 3.80 4.20

0.05 0.06 0.07 0.08 0.09 0.10 0.12 0.14 0.16 0.18 0.20 0.25

0.10 0.07 0.10 0.12 0.15 0.20 0.25 0.30 0.35 0.45 0.50 0.65 0.75 0.85 0.95 1.00

0.05 0.06 0.07 0.08 0.09 0.10 0.12 0.14 0.16 0.18 0.20 0.25

0.14 0.12 0.14 0.20 0.25 0.30 0.40 0.50 0.60 0.70 0.90 1.10 1.30 1.50 1.70 1.90 2.00

a Maximum permitted deviations of length at any point, ± t b Tolerance, t

Grade AS-1

Limit on Deviations of Length a

Grade 0

Tolerance for Variation in Length  b

Nominal Length Range, ln mm ≤ 0.5

Grade 00

Limit on Deviations of Length a

Calibration Grade K

v μm, for the variation in length.

0.10

0.12 0.14 0.16 0.18 0.20 0.25 0.30 0.35 0.40

0.30 0.20 0.30 0.40 0.50 0.60 0.80 1.00 1.20 1.40 1.80 2.20 2.60 3.00 3.40 3.80 4.20

0.16 0.18 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.60

0.60 0.45 0.60 0.80 1.00 1.20 1.60 2.00 2.40 2.80 3.60 4.40 5.00 6.00 6.50 7.50 8.00

0.30

0.35 0.40 0.45 0.50 0.60 0.70 0.80 0.90 1.00

e μm, from nominal length, l e mm.

1) Always keep gage blocks clean and well oiled when not in direct use. Alco­hol is acceptable as a cleaner, but it is always advisable to coat the gage block with a rust inhibitor when placing it back in the case. A very light machine oil is recommended. 2) Take great care when removing gage blocks from the case so as not to nick or damage the working surface. Clean with a soft cloth or chamois and isopropanol. Never touch gage blocks with bare hands. Oil from fingers will cause corrosion on the bare metal surface. 3) Always keep the gage blocks over a soft cloth or chamois when handling or wringing them together. Dropping the gage blocks onto a hard surface or other gage blocks will dam­age the working surface and cause an error beyond the limits of the tolerance. Always treat gage blocks as highly accurate precision instruments. 4) Should the gage block surface show signs of degradation and the wringing together of blocks become difficult, the surface may need to be deburred. The use of a serrated sintered aluminum oxide deburring stone is recommended to recondition the surface and renew the ability to join the gage blocks through wringing. Caution must be exercised when deburr­ ing the surface of gage blocks so it is not damaged instead of repaired. 5) Gently place the gage block flat on the serrated block. With two fingers (using gloves), press down firmly, but not hard, on the gage block and slide it lengthwise over the serra­ tions on the block for three or four strokes until the surface feels very smooth. Turn the gage block over and repeat the movement. Remove and clean the gage block thoroughly. 6) It is important that the serrated sintered aluminum oxide deburring stone is cleaned as well, and metal deposits, oils, and dirt are not allowed to build up on the surface. A cotton packing impregnated with a metal solvent will clean the serrated sintered aluminum oxide deburring stone. Calibration and Verification of Gage Blocks.—The calibration and verification of gage blocks should be completed on a regular basis. This is done to maintain measurement assurance in every good quality program. The quality assurance program will determine

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the optimal interval for recalibration of the gage block sets to maintain the appropriate level of measurement assurance. Calibration of the gage blocks should be done by an approved and, if necessary, certified calibration laboratory that provides impartial third party confirmation of the calibrated features of the gage block set. Precision Gage Blocks.—Precision gage blocks are usually purchased in sets containing a specific number of blocks of different sizes. The nominal gage lengths of individual blocks in a set are determined mathematically so that particular desired lengths can be obtained by combining selected blocks. They are made to several different tolerance grades that categorize them as master blocks, calibration blocks, inspection blocks, and workshop blocks. Master blocks are employed as basic reference standards; calibration blocks are used for high precision gaging work and calibrating inspection blocks; inspec­tion blocks are used as toolroom standards and for checking and setting limit and compar­ator gages, for example. The workshop blocks are working gages used as shop standards for direct precision measurements and gaging applications, including sine-bars. Federal Specification GGG-G-15C, Gage Blocks (see below), lists typical sets and gives details of materials, design, and manufacturing requirements, and tolerance grades. When there is in a set no single block of the exact size wanted, two or more blocks are com­ bined by “wringing” them together. Wringing is achieved by first placing one block cross­ wise on the other and applying some pressure. Then a swiveling motion is used to twist the blocks to a parallel position, causing them to adhere firmly to one another. When combining blocks for a given dimension, the object is to use as few blocks as pos­ sible to obtain the dimension. The procedure for selecting blocks is based on successively eliminating the right-hand figure of the desired dimension. Example: Referring to inch-size gage block Set Number 1 below, determine the blocks required to obtain 3.6742 inches. Step 1: Eliminate 0.0002 by selecting a 0.1002 block. Subtract 0.1002 from 3.6743 = 3.5740. Step 2: Eliminate 0.004 by selecting a 0.124 block. Subtract 0.124 from 3.5740 = 3.450. Step 3: Eliminate 0.450 with a block this size. Sub­ tract 0.450 from 3.450 = 3.000. Step 4: Select a 3.000 inch block. The combined blocks are 0.1002 + 0.124 + 0.450 + 3.000 = 3.6742 inches. Gage Block Sets, Inch Sizes (Federal Specification GGG-G-15C).—Set Number 1 (81 Blocks): First Series: 0.0001-Inch Increments (9 Blocks), 0.1001 to 0.1009; Second Series: 0.001-Inch Increments (49 Blocks), 0.101 to 0.149; Third Series: 0.050-Inch Increments (19 Blocks), 0.050 to 0.950; Fourth Series: 1.000-Inch Increments (4 Blocks), 1.000- to 4.000-inch. Set Numbers 2, 3, and 4: The specification does not list a set 2 or 3. Gage block Set Num­­ ber 4 (88 Blocks), listed in the Specification, is not given here; it is the same as Set Number 1 (81 Blocks) but contains seven additional blocks measuring 0.0625, 0.078125, 0.093750, 0.100025, 0.100050, 0.100075, and 0.109375 inch. Set Number 5 (21 Blocks): First Series: 0.0001-Inch Increments (9 Blocks), 0.0101 to 0.0109; Second Series: 0.001-Inch Increments (11 Blocks), 0.010 to 0.020; One Block 0.01005 inch. Set Number 6 (28 Blocks): First Series: 0.0001-Inch Increments (9 Blocks), 0.0201 to 0.0209; Second Series: 0.001-Inch Increments (9 Blocks). 0.021 to 0.029; Third Series: 0.010-Inch Increments (9 Blocks), 0.010 to 0.090; One Block 0.02005 Inch. Long Gage Block Set Number 7 (8 Blocks): Whole-Inch Series (8 Blocks), 5, 6, 7, 8, 10, 12, 16, 20 inches. Set Number 8 (36 Blocks): First Series: 0.0001-Inch Increments (9 Blocks), 0.1001 to 0.1009; Second Series: 0.001-Inch Increments (11 Blocks), 0.100 to 0.110; Third Series: 0.010-Inch Increments (8 Blocks), 0.120 to 0.190; Fourth Series: 0.100-Inch Increments (4 Blocks), 0.200 to 0.500; Whole-Inch Series (3 Blocks), 1, 2, 4 Inches; One Block 0.050 inch.

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Machinery's Handbook, 31st Edition Gage Blocks

Set Number 9 (20 Blocks): First Series: 0.0001-Inch Increments (9 Blocks), 0.0501 to 0.0509; Second Series: 0.001-Inch Increments (10 Blocks), 0.050 to 0.059; One Block 0.05005 inch. Gage Block Sets, Metric Sizes (Federal Specification GGG-G-15C).—Set Number 1M (45 Blocks): First Series: 0.001-Millimeter Increments (9 Blocks), 1.001 to 1.009; Sec­ond Series: 0.01-Millimeter Increments (9 Blocks), 1.01 to 1.09; Third Series: 0.10-Milli­meter Increments (9 Blocks), 1.10 to 1.90; Fourth Series: 1.0-Millimeter Increments (9 Blocks), 1.0 to 9.0; Fifth Series: 10-Millimeter Increments (9 Blocks), 10 to 90 mm. Set Number 2M (88 Blocks): First Series: 0.001-Millimeter Increments (9 Blocks), 1.001 to 1.009; Second Series: 0.01-Millimeter Increments (49 Blocks), 1.01 to 1.49; Third Series: 0.50-Millimeter Increments (19 Blocks), 0.5 to 9.5; Fourth Series: 10-Millimeter Increments (10 Blocks), 10 to 100; One Block 1.0005 mm. Set Number 3M: Gage block Set Number 3M (112 Blocks) is not given here. It is similar to Set Number 2M (88 Blocks), and the chief difference is the inclusion of a larger number of blocks in the 0.5-millimeter increment series up to 24.5 mm. Set Number 4M (45 Blocks): First Series: 0.001-Millimeter Increments (9 Blocks), 2.001 to 2.009; Second Series: 0.01-Millimeter Increments (9 Blocks), 2.01 to 2.09; Third Series: 0.10-Millimeter Increments (9 Blocks), 2.1 to 2.9; Fourth Series: 1-Millimeter Increments (9 Blocks), 1.0 to 9.0; Fifth Series: 10-Millimeter Increments (9 Blocks), 10 to 90 mm. Set Numbers 5M, 6M, 7M: Set Numbers 5M (88 Blocks), 6M (112 Blocks), and 7M (17 Blocks) are not listed here. Long Gage Block Set Number 8M (8 Blocks): Whole-Millimeter Series (8 Blocks), 125, 150, 175, 200, 250, 300, 400, 500 mm. Surface Plates The surface plate is the primary plane from which all vertical measurements are made. The quality and dependability of this surface is one of the most critical elements in dimen­ sional inspection measurement. Originally made from cast iron, the present-day granite plate was first developed during World War II because most metal was being used in the war effort. Faced with a need to check precision parts, Mr. Wallace Herman, a metal working and monument shop owner, decided to investigate the use of granite as a suitable replacement for the then-common cast-iron surface plate and manufactured the first granite surface plate in his shop in Dayton, Ohio. Although surface plates have changed in their design and materials, the basic concept has remained the same. The stability and precision that can be achieved with granite is actually far superior to cast iron and is much easier to maintain. With the proper care and mainte­nance, a well-made surface plate can last for generations and always remain within the parameters of the grade to which it was originally made, or even better. Materials and Grades of Surface Plates.—The selection of a surface plate is driven directly by the specific application the plate will be used for. A plate, for instance, that will be used in a very large machining facility would be primarily concerned with the load-bear­ing properties and secondarily in surface flatness accuracy, although both are important concerns. A surface plate that will be used in a metrology laboratory or high precision inspection department with a high volume of work would be concerned with high accuracy and surface wear properties. In each case the material and design would be considered for the application in mind before a selection is made and a purchase initiated. The material properties of the granite is what makes the difference in the performance of surface plates. The differences in the various types of granite are considered in Table 1, based on Federal Specification GGG-P-463c, Plate, Surface (Granite) (Inch and Metric).

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Table 1. Granite Rock Types, Physical Properties, and Mineral Components GGG-P-463c Rock Type

Biotite gran­ite

Natural Color

Texture

Bluish gray

Finegrained

Light gray Pink

Biotite hornblende granite Biotitemus­covite Diabase Hypersthene Gabbro Muscovitebiotite gran­ite-gneiss

Mediumgrained

Reddish brown

Finegrained

Light gray

Medium- to fine-grained

Dark gray

Finegrained

Light gray

Mediumgrained

Mineral Constituents, Descending Order Of Abundance

Orthoclase, smokey quartz,a oligoclase, albite, biotite, muscovite, magnetite and zircon Oligoclase, orthoclase and microcline, quartz, biotite, apatite and zircon Orthoclase with a small amount of microcline, plagioclase, quartza, biotite, magnetite, and garnet Orthoclase and microcline, quartza, hornblende, biotite, plagioclase and magnetite Microcline, quartz, plagioclase, biotite, muscovite and magnetite Plagioclase, pyroxene and magnetite Plagioclase, pryoxene, hornblende, mag­netite and biotite Microcline and orthoclase, oligoclase, quartz, rutile, muscovite

Modulus of Elasticity 106 psi

GPa

3.5–7.0

24.1–48.2

3.5–7.0

24.1–48.2

5.0–9.0

34.4–62.0

6.0–9.0

41.3–62.0

5.0–7.0

34.4–48.2

9.0–12.0

62.0–82.7

10.0–12.0

68.9–82.7

3.5–8.0

24.1–55.1

a 28 to 32% quartz by volume. In certain conditions, high quartz content tends to increase wear life.

As indicated in Table 1, fine-grained pink granite containing a small amount of quartz has a lower modulus of elasticity and therefore a lower load bearing per square foot capac­ ity. The presence of large quartz crystals, however, results in a high degree of wearability by providing an ultra-smooth surface finish with increased surface hardness that resists wear on the granite and the precision ground and lapped instruments used on it. Although the load-bearing properties are lower than those of black or dark gray granite, this can be compensated for by increasing the thickness of the plate.

As a result of the increased wearability of pink granite, the interval between lapping com­pared to fine-grained black granite can be as much as five times as long. This is an import­ant consideration when planning the maintenance costs and downtime involved in maintaining a production schedule.

The precision lapped and calibrated granite surface plate is a high-precision piece of equipment and must be maintained as such. Great care should be taken at all times to protect the surface, and attention to cleanliness is critical in the life span of the surface plate. It is essential that the surface be protected from the buildup of dirt, grease, airborne grime and oils. The plate should be covered when not in use to avoid accidentally dropping objects on the surface and chipping or cracking the precision finished surface.

All surface plates should be installed and supported according to the manufacturer’s design and recommendations. Plates up to and including 6–12 feet are supported in a three-point non-distortable support system of hard rubber pads that are installed during manufacturing and remain in place during lapping and finishing. These pads are critical in the correct sup­port of the surface plate and must never be removed or repositioned. Always make sure the surface plate is resting on these pads, and never support the plate by its ledges or under the four corners as this will cause deformation of the surface and introduce errors beyond the tolerance limits. Surface plates made to meet standard guidelines and accuracy parameters established over the last fifty years and published in documents such as the Federal Specification GGG-P-463c will be manufactured in certain pre-designed sizes. These designs have been analyzed for dimensional stability and dependability and will, with proper care, provide dependable measurement assurance for many years. There are manufacturers that will spe­cial order surface plates in a wide variety of sizes and configurations to meet the needs

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of specialized applications, and have engineering staffs that will design a surface plate to meet those needs. However, most applications and manufacturers’ products fall under the design guidelines of the tables in the GGG-P-463c Federal Specification. Table 2a. Standard Sizes for Rectangular Granite Surface Plates GGG-P-463c Width (inch) 12 18 24

36

48 60 72

Length (inch) 12 18 18 24 24 36 48 36 48 60 72 48 60 72 96 120 120 96 144

Inch

Calculated Diagonal (inch) 17.0 21.6 25.5 30.0 33.9 43.3 53.7 50.9 60.0 70.0 80.5 67.9 76.9 86.5 107.3 129.2 134.2 120.0 161.0

Area (ft2) 1 1.5 2.25 3 4 6 8 9 12 15 18 16 20 24 32 40 50 48 72

Width (mm) 300 450 600

900

1200 1500 1800

Length (mm) 300 350 450 600 600 900 1200 900 1200 1500 1800 1200 1500 1800 2400 3000 300 2400 3600

Metric

Calculated Diagonal (mm) 424 541 636 750 849 1082 1342 1273 1500 1749 2012 1697 1921 2163 2683 3231 3304 3000 4025

Area (m2)

0.090 0.135 0.202 0.270 0.360 0.540 0.720 0.810 1.080 1.350 1.620 1.440 1.800 2.160 2.880 3.600 4.500 4.320 6.480

Table 2b. Standard Sizes for Round Granite Surface Plates GGG-P-463c Diameter (inch) 12 18 24 36 48

Inch

Area (ft2) 0.8 1.8 3.1 7.1 48

Diameter (mm) 300 450 600 900 1200

Millimeter

Area (m2) 0.071 0.159 0.283 0.636 1.131

Thickness: For rectangular and round surface plates, specify thickness only if essential; see Appendix 30 and inch and metric versions of Tables XI and XII of GGG-P-463c.

Surface plate grades are established in the Federal Specifications and are the guidelines by which the plates are calibrated. The flatness tolerances in microinches for standard inch-dimension plates, listed in Table 3a., are obtained through the standard formula: D2 Total flatness tolerance for inch − dimension grade AA plates = 40 + 25 µ in. where D = diagonal or diameter of the plate in inches. The calculated flatness tolerance for grade AA is rounded to the nearest 25µin. For metric plate sizes, the total flatness tolerance of grade AA plates in micrometers is: Total flatness tolerance for metric grade AA plates = 1 + 1.62D210 – 6µ m

where D = diagonal or diameter of the plate in millimeters. For both the inch and metric plates, the tolerances of the A and B grades are 2 and 4 times, respectively, those for grade AA. Table 3a and Table 3b, adapted from Federal Specification GGG-P-463c, contain the calcu­lated tolerances for the standard size and grades of rectangular and round surface plates.

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Table 3a. Total Flatness Tolerance, Rectangular Surface Plates GGG-P-463c Rectangular Plates

Inch Sizes, Tolerances in Microinches Width (inch)

Length (inch) 12 18 18 24 24 36 48 36 48 60 72 48 60 72 96 120 120 96 144

12 18 24

36

48 60 72

Grade AA 50 50 50 75 75 100 150 150 200 250 300 200 300 350 500 700 750 600 1100

Grade A 100 100 100 150 150 200 300 300 400 500 600 400 600 700 1000 1400 1500 1200 2200

Millimeter Sizes, Tolerances in Micrometers Grade B 200 200 200 300 300 400 600 600 800 1000 1200 800 1200 1400 2000 2800 3000 2400 4400

Width (mm) 300 450 600

900

1200 1500 1800

Length (mm) 300 450 450 600 600 900 1200 900 1200 1500 1800 1200 1500 1800 2400 3000 3000 2400 3600

Grade AA 1.3 1.5 1.6 1.9 2.2 2.9 3.9 3.6 4.6 5.9 7.5 5.6 6.9 8.5 12.5 17.7 18.5 15.4 26.9

Grade A 2.6 2.9 3.3 3.8 4.3 5.7 7.8 7.2 9.2 11.8 15.0 11.2 13.8 17.0 25.0 35.4 36.9 30.8 53.8

Grade B 5.2 5.9 6.6 7.6 8.6 11.5 15.5 14.4 18.4 23.6 29.9 22.4 17.6 33.9 50.0 70.8 73.9 61.6 107.7

Table 3b. Total Flatness Tolerance, Round Surface Plates GGG-P-463c Round Plates

Inch Sizes, Tolerances in Microinches Diameter (inch) 12 18 24 36 48

Grade AA 50 50 75 100 125

Grade A 100 100 150 200 250

Grade B 200 200 300 400 500

Millimeter Sizes, Tolerances in Micrometers Diameter (mm) 300 450 600 900 1200

Grade AA 1.1 1.3 1.6 2.3 3.3

Grade A 2.3 2.5 3.2 4.6 6.6

Grade B 4.6 5.3 6.3 9.2 13.2

Calibration of Surface Plates.—Surface plates, like other precision instruments, will drift out of tolerance in time and need to be periodically checked and even adjusted to maintain accuracy. For a surface plate, this adjustment involves lapping the surface and physically removing material until the entire surface is once again flat to within the limits of the grade to which it was made. This is a labor-intensive adjustment that should only be attempted by a trained technician with the appropriate tools. An untrained technician, even with the proper lapping tools and compounds, may cause more harm than good when attempting such an adjustment. However, with proper care and cleaning, a surface plate may not need lapping at every calibration. Completing the calibration process will give the quality assurance program the data needed to identify wear patterns on a surface plate and have the proper maintenance completed before an out of tolerance condition occurs, thereby avoiding a costly failure impact analysis and reinspection of parts, or even a recall of finished parts from a cus­tomer. The calibration process is a cost-effective alternative to these undesirable effects. Fig. 1 illustrates the “Union Jack” or eight line pattern that is used to analyze the overall flatness of the surface plate working area. Each line is measured independently, and the positive and negative elevations are recorded at predetermined intervals. The peak to val­ley

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flatness is calculated from the recorded data. This method was developed by JC Moody while working for Sandia Corporation, and the resulting plot of the data points is referred to as a Moody Plot. There are many software packages available that simplify the process of cal­culating this plot and provide a detailed printout of the data in a graphical format. TI MFG Label or Certification Label (North Reference) 4

5

8

TI

1 7 2

3

6 Fig. 1. Union Jack Pattern

Repeat-o-Meter Method (Fed. Spec. GGG-P-463c) The easiest and quickest method of monitoring for surface plate wear and tolerance adherence is the Repeat-o-Meter method. A repeat reading gage similar in design to the one shown in Fig. 2 is used to estab­lish the variation in flatness of the surface plate. Indicating System

Handle

Two Fixed Feet 3/8´´

3/8´´ 5´´

Fixed Foot 5´´

Floating Contact Flexure Spring 3/8´´

Fig. 2. Repeat Reading Gage from Federal Specification GGG-P-463c

The gage is placed at the center of the surface plate, and the indicator is set to zero. The eight-line Union Jack pattern, Fig. 1, is then scanned in the sequence defined in the Federal Specification GGG-P-463c. The result of the maximum reading minus the minimum read­ing shall not exceed the flatness tolerance expressed in Table 4a and Table 4b for inchand milli­meter-sized plates, respectively. Table 4a. Tolerance for Repeat Reading of Measurement (microinches)

Diagonal / Diameter Range Over Thru … 30 60 90 120 150

All Sizes

30 60 90 120 150 …

Grade AA Grade A Grade B Full Indicator Movement (FIM), microinches 35 45 60 75 90 100 25

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60 70 80 100 120 140 50

110 120 160 200 240 280 100

Obtained

When not specified When specified

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Table 4b. Tolerance for Repeat Reading of Measurement (micrometers)

Diagonal / Diameter Range (mm) Over Thru

Grade AA Grade A Grade B Full Indicator Movement (FIM), micrometers

… 800 800 1500 1500 2200 2200 3000 3000 3800 3800 … All Sizes

35 45 60 75 90 100 0.6

60 70 80 100 120 140 1.3

Obtained

110 120 160 200 240 280 2.5

When not specified When specified

Note for Table 4a and Table 4b: If it is intended that small objects be measured on large surface plates, it should be noted that a larger tolerance in flatness over small areas is permitted on larger plates.

All points on the work surface shall be contained between two parallel planes, the roof plane and the base plane. The distance between these planes shall be no greater than that specified in the tolerance table for the respective grades. Autocollimator Calibration (Ref NAVAIR 17-20MD-14, 1 MAY 1995).—Calibra­tion of a surface plate by autocollimator is one of the most accurate and relatively quickest methods in use today. The method has been in use since the advent of the autocollimator in the 1940s but was refined by JC Moody with the development of the Moody Plot analysis method that utilized the data collected on perimeters, diagonals and bisectors. The result­ ing pattern resembles the “Union Jack” and is referred to as such. Autocollimators today have been enhanced with the inclusion of CCD devices, digital readouts, computer interfacing, and automatic data collection that calculates the devia­ tions and analyzes the results in a fraction of the time previously required, producing a full color graphical diagram of the surface plate variances. Fig. 3 and Fig. 4 show the positioning of the turning mirrors used when performing the sur­face plate calibration with an autocollimator. North Side

Standard mirror positions lines 1, 3, 6

0

5

10

Line 3 Movable mirror positions for lines 1, 3, 6 15 Line 8

20 e2 Lin

25

Arrow heads on lines indicate direction of movement of mirrors

e2 Lin

Line 8 30

Autocollimator I-3-6 position 1 SW

NE

Line 4

35

40

Lin e

45

Line 7

Numbers indicate lines that are read with autocollimator in this position

NW

Line 6

Note: Example uses mirror base feed with 5-inch length

MFG nameplate

1

50

First reading must be with mirror nearest autocollimator

55

SE

Fig. 3. Placement of the First Turning Mirror and Retroreflector: TO 33K6-4-137-1

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Data Table VII

Data Table IV 2–4 Autocollimator position 2 Da ta T abl e Data Table IX

I

e II

l Tab ata

D 2

4

5

Line 3

II

Data Table V

7–8

e2

Lin

Data Table VI

Line 5

Line 6

North Side

Data Table VIII

Line 8

Lin

Typical stationary mirror position

e1

5

Line 4

8

7

Surface Plate

Fig. 4. Turning Mirror Placement for Remaining Lines: TO 33K6-4-137-1

Interferometer Calibration (Ref TO 33K6-4-10-1 30 April 2006).—Calibration of a surface plate can also be accomplished using a laser interferometer. The laser offers the absolute lowest uncertainty in the calibration process, and, although it takes more time than either of the other methods, it is sometimes required by some customers doing military or government contract work. There are several models and methods available on the market today. The general process will be covered, but familiarity with the particular model being used is necessary to complete the full calibration procedure. The process is much the same as the autocollimator calibration method using turning mirrors. The laser tripod and laser head are located as close to the line being shot as possi­ ble. Move the tripod and realign the laser interferometer for each line during calibration. If available, use the turning mirrors to facilitate the alignment of the beam. As with any highprecision physical measurements, air currents and vibrations will affect the indications. Take precautions to minimize these affects. Clean the surface plate twice with surface plate cleaner and once again with ethyl alcohol prior to beginning the calibration process.

Mark the surface with lines as shown in Fig. 5 below. The lines will be either 3 or 4 inches away from the edge of the surface plate, depending on the size of the plate. The North side of the plate will always be the side bearing the manufacturer’s label or the previous calibra­tion label. TI MFG or Certification Label (North Reference)

Fig. 5. Reference Lines for Union Jack 1

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Clamp the straight edge to the surface plate in the position shown in Fig. 6 below. The straightedge will always be offset from the center of the line so that the centerline of the mirror will travel directly above the center of the reference line. Place the interferometer and retroreflector on the reference line closest to the laser head. Direct the beam through the interferometer optics and, using the vertical and horizontal adjustments on the tripod head, adjust until the beam returns through the optics and regis­ ters a strong beam strength on the beam strength indicator. Slide the retroreflector to the far end of the reference line and use the rotation adjustment on the tripod head to swing the beam back into alignment with the laser head until once again the beam strength indicator shows a strong beam strength. Using this method of Translate Near / Rotate Far you will be able to align the beam with the path of the reference line. Patience will be required, and the manufacturer will also have information regarding the best method for the specific model being used. Clamp

North

Straightedge

Movable Mirror West

East

TI Straightedge Face

MFG’s Namplate W/SER. No.

Clamp

South

Fig. 6. Placement of the Straightedge for Measurement of the First Reference Line

Once the alignment of the first reference line has been completed, take the measurements at the stations recommended by the software calculations. The number of stations will change according to the size of the surface plate and the length of the footpad used for the retroreflector. Record the values at each station along the first reference line then return to the first position and verify that the reading repeats before proceeding to the next line. Great care must be taken to ensure a clean plate and footpad, free of residue or buildup. A very small amount of dirt or dust can cause an out of tolerance condition and result in many wasted hours of failure impact analysis and possibly recall of parts to be reinspected unnecessarily. When the data from all the reference lines has been collected, the interferometer software will calculate the overall flatness of the plate and make the determination as to whether or not the plate meets the specific grade to which it has been assigned or needs to be lapped to meet tolerance limits. If there is a failure determination, the plate will need to be lapped and reshot to collect the “As Left” data and close the calibration event with an acceptance of the calibration data. Lapping a surface plate is a very specific skill and should not be attempted by an amateur or enthusiastic technician however good the intentions may be. Lapping requires special­ ized tools and compounds as well as skill. An inexperi­enced technician attempting to lap a surface plate may well cause more damage than is already present. Always consult an experienced and proven professional in this field.

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732

Machinery's Handbook, 31st Edition V-Blocks

V-Blocks The V-block is a precision holding jig or fixture devised as a tool or to enable the preci­ sion and repeatable positioning of a cylindrical work piece. The V-block enables drilling, grinding, or milling operations on a robust and precision platform as compared to the vise or other simple holding devices. In 1902, Elmer Cobb and Eugene Spaulding of Portland, Maine patented a Machinists V-block to expand their capability to hold and drill extended workpieces. The V-blocks were machined with a large channel on one side and a pair of smaller channels on the other to accommodate the various diameters that were encountered in the basic work done in the machine shop of the era. Each block had a hole in the center to allow the passage of a drill through the work piece. The blocks were a matched set that were secured together by two rods that aligned the blocks together and allowed them to slide open or to be closed together to support work of varying lengths.

Fig. 1. Machinists’ V-Block, Patented 1902 by Cobb and Spaulding

In 1923, Gustave Hines invented a device that incorporated a scale along the linear side of the work piece, and locked the work into a scaled collar that could be rotated accurately throughout 180°. The device provided a drill guide that would accurately measure dis­ tances between centerlines of holes, and provide rotational accuracy.

Gustave Hines 1923 Innovation

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Machinery's Handbook, 31st Edition V-BLOCK CLASSIFICATION

733

Classification.—The Federal Specification GGG-V-191D 1971 “V Blocks and Clamps” was developed to provide the acceptable criteria for items to be purchased under contract by the US Gov­ernment or Department of Defense. This specification has been superseded by Federal Specification A-A-51150B and since canceled. However, these and related standards are still referenced by many manufacturers of precision V-blocks and were the initial guide­lines for the manufacturing and specification of the tool. Federal Specification Coverage: This specification covers only the types, styles and grades generally purchased by the Federal Government, and does not include all types, styles, and grades which are commercially available. Type, Style, and Grade Type I – Single V-Groove (page 735) Style 2: Double level clamping Style 3: Reversible clamping yoke Style 4: Magnetic clamping (mild steel and brass laminated) Style 5: Ribbed, without clamping Style 6: Plain without clamping Grades: Grade A: Steel (alloy); Grade B: Mild Steel (and brass laminated for style 4); and Grade C: Cast Steel. Type II – Double V-Groove (page 738) Style 1: Single level clamping Grades: Grade A: Steel (alloy); and Grade B: Cast Iron (class 40 iron). Type III – Quadruple V-Groove (page 739) Style 1: Single level clamping Grades: Grade A: Steel (alloy); and Grade B: Cast Iron (class 40 iron). Type IV – Combination, Top and End V-Grooves (page 741) Style 1: Single level clamping Grades: Grade A: Steel (alloy) Material.—Note: Grade and surface finish specifications are a representation of all gov­ erning standards in use today. Grade A Blocks: Grade A blocks shall be steel or steel alloy and shall have a surface hardness of 58 to 65 RC on the Rockwell C scale. Grade B Blocks: Grade B blocks shall be a good quality mild steel, free from porosity and other injurious defects and have a surface hardness of 40 to 45 on the Rockwell C scale. Grade C Blocks: Grade C blocks shall be a good quality cast steel free of porosity or other injurious defects and shall have a surface hardness of 150 to 180 BHN (Brinell Hardness Number). Precision Granite Blocks: Precision granite V-blocks shall be made from a close grained uniform texture granite free from flaws or fissures and inclusions of softer material. Gran­ ite V-blocks shall adhere to either Grade A or B accuracy. Grade A and matched sets are to be provided in a protective case. Finished Surfaces: Surfaces of the V-grooves, tops sides, bottom, V-groove relief slots, and clamp engaging grooves shall have finished surfaces measured in micrometers arith­metical average (A.A.) or roughness average (Ra) in accordance with ANSI/ASME B46.1-2009. All other surfaces shall be finished in accordance with normal manufac­ turer’s standard commercial practice. Surface Finish, Grades A and B, Style 4, Blocks: Surface of V-groove relief slots and clamp engaging grooves (when applicable) on blocks shall have a surface roughness not greater than 125 µin (3.175 µm). All other finished surfaces shall have a surface roughness not greater than 32 µin (0.81 µm) Ra.

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734

Machinery's Handbook, 31st Edition V-BLOCK Material

Surface Finish, Grade B (except Style 4), and Grade C Blocks: Surface of V-groove(s). The V-grooves relief slots and clamp engaging grooves (when applicable) on blocks shall have a surface roughness not greater than 125 µin (3.175 µm) Ra. All other finished surfaces shall have a sur­face roughness not greater than 63 µin (1.600 µm) Ra. Surface Edges: Edges of all exterior surfaces, including corners, shall be rounded or bev­eled not less than 0.010 inch and shall be free from fins and burrs. Tolerance.—All Grade A; and Style 4, Grade B, mild steel laminated with brass, shall have a ground finish on the ends, sides, top, bottom, and the V-groove(s). The V-grooves shall be centered with the sides, square with the sides and ends, and parallel to the bottom within 0.0003 in. All surfaces shall be square with each other within 0.0003 in. Grade B (except Style 4) and Grade C shall have a machined or ground finish. The V-groove(s) shall be parallel and centered with the sides to within 0.002 in. The ends shall be square with the sides within 0.002 in. and the sides shall be square with the top and the bot­tom within 0.002 in. The V-grooves shall be 45° each side of the center perpendicular plane for all grades. When specified, Grade B or C blocks shall be furnished with tolerances as specified for Grade A blocks. When the V-blocks are provided in matched sets, they shall be clearly marked to identify them as such and shall be identical to each other within 0.0002 in. on all critical features. V-Block Design.—Each V-block shall have a 90° V-groove 45° from center perpendicular plane cut lengthwise on the top, or top and end, or top and bottom, for the type specified. A slot shall be provided at the apex of the V-groove for chip relief. The exception being with the Type III V-blocks which shall be furnished without the chip relief unless otherwise specified. When V-blocks are furnished in “matched pairs” they shall be marked to readily identify them as a pair. Clamping grooves shall be provided along the full length of the sides of Styles 1 and 2 blocks for engaging the clamps. Clamps.—The clamps shall consist essentially of a yoke and one or two screws for the style specified, and shall be of a size to accommodate the maximum size round stock spec­ified in the respective tables. The Clamp shall be made of good quality steel or iron and suf­ficiently robust to provide adequate support and security of the work piece in the V-block during drilling and grinding operations. When engaged in the clamping grooves, the clamp should not extend past the side faces of the block. When provided with matched sets the clamp shall be applicable to either member of the set. The screws for the clamping device shall be right handed thread conforming to ANSI/ASME B1.2 or FED-STD-H28. Yokes.—Yokes shall be made of good quality steel or iron and sufficiently robust to pro­vide adequate support and security of the work piece in the V-block during drilling and grinding operations and should not extend past the side faces of the block. V-block and yoke screws for Type 1 Style 3 shall be threaded in accordance with ANSI/ASME B1.2 or FED-STD-H28 or the equivalent and shall be a Class 2 fit. When the yoke is in place it shall bisect the 90° V-groove. The yokes when engaged in the clamping grooves shall stand in an upright position. Yokes for the style 3 blocks shall have a slot on each end to engage the two adjusting screws. The yoke shall be designed so that the pads bisect the 90° V-groove to allow the yoke to secure stock that does not extend above the V-groove. When inverted the yoke shall be able to secure stock of the maximum size specified. Screws.—All screws shall be made of steel of such quality as to withstand hard usage. Screw heads shall be knurled and provided with a hole through the center for tightening with a pin or rod. All screws shall conform to the ANSI/ASME B1.2 or FED-STD-H28 or equiva­lent and shall be a class 2 fit.

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Machinery's Handbook, 31st Edition V-Block Types

735

Type 1, Style 2.—Type 1 V-blocks are of the single V type. The Style 2 V-block shall have two clamping grooves on each side to accommodate clamping from minimum to the full range of the capacity of the V-groove. The block shall be similar to Fig. 2 and conform to the dimensions shown in Table 1. Style 2 V-blocks can be furnished in matched pairs and the clamps shall be suitable for use with either V-block.

Fig. 2. Type 1, Style 2

Table 1. Accuracy and Dimensions, Type 1, Style 2 Grade and Designating Size Grade A

Grade B

1 2 1 2

Flatness of Working Faces 0.00015 0.0001 0.0004 0.0004

Grades A & B Designating Size 1 2

Permitted Deviations (inch)

End/Side Square to Base 0.0002 0.0002 0.002 0.002

Parallelism of Opposite Faces 0.0002 0.0002 0.001 0.001

Parallelism of “V” to Base and Sides 0.0003 0.0003 0.002 0.002

Nominal Dimensions (inch) Length 3.000 2.500

All dimensions are in inches.

Width 2.500 3.000

Height 2.500 2.000

“V” Centrality 0.0003 0.0003 0.002 0.002

Matching Tolerance over “V” Length 0.0002 0.0002 0.001 0.001

Maximum Capacity 2.500 2.000

Type 1, Style 3.—Style 3, reversible clamping yoke. Style 3 blocks shall be similar to Fig. 3, conform to dimensions and accuracies shown in Table 2 for the sizes specified and be designed to facilitate clamping to a machine table or plate. The top surface shall have 4 drilled and tapped holes, one near each corner, to accommodate the yoke screws. If the blocks are furnished in matched pairs the yokes and screws shall be suitable for use with either block of the matched pair.

Fig. 3. Type 1, Style 3 Machinists’ Table Block with Yoke

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Machinery's Handbook, 31st Edition V-Block Types

736

Table 2. Accuracy and Dimensions, Type 1, Style 3 Machinists’ Table Block Designating Size Grade A Grade B Grade A Grade B

Flatness of Working Faces 0.0001 0.0004

Permitted Deviations (inch) Parallelism Parallelism of “V” to of Opposite Base and faces Side 0.0002 0.0002 0.0003 0.002 0.001 0.002 Grade Characteristics

End/Side Square to Base

“V” Centrality

Matching Tolerance

0.0003 0.002

0.0002 0.001

•  Up to 6.375”        •  Steel alloy or mild steel •  Surface finish NTE 32 µin   •  Material hardness 40 to 62 RC

•  Up to 6.375”        •  Mild steel or fine grained cast steel •  Surface finish NTE 63 µin   •  Material hardness 40–45 RC steel / 200–280 BHN cast steel

Grades A & B Designating Size Length 1 1.375 2 2.125 3 2.250 4 2.875 5 6.000 6 8.000 All dimensions are in inches.

Nominal Dimensions (inch) Width 1.375 2.375 2.9375 4.000 6.000 8.000

Height 1.125 1.875 1.9375 2.875 6.000 8.000

Maximum Capacity 0.75 1.500 2.000 3.000 4.500 6.375

Type 1, Style 4, Magnetic Clamping Blocks.—Style 4, magnetic clamping blocks shall be similar to Fig. 4 and conform to the dimensions and accuracies shown in Table 3. Mag­ netic clamping V-blocks shall be either made of mild steel laminated with brass or an assembly of mild steel with an embedded magnetic core that can be disengaged to allow positioning of work piece. Style 4 magnetic clamping blocks shall have a ground finish on the sides, top, bottom, and V-grooves.

Fig. 4. Type 1, Style 4 Magnetic Clamping V Blocks

Fig. 5. Type 1, Style 5 Ribbed without Clamps

Type 1, Style 5, Ribbed without Clamps.—The Style 5 block shall be designed without clamping capability for the work piece. The block shall be similar in design to Fig. 5 and conform to the dimensions and accuracies shown in Table 4. The Style 5 block shall be of ribbed design to provide strength with light weight. There may be lightening holes in the webbing section to reduce weight without affecting the performance capabilities. The Style 5 block shall be of a design as to facilitate clamping to a machine table or plate.

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Machinery's Handbook, 31st Edition V-Block Types

737

Table 3. Accuracy and Dimensions for Type 1, Style 4 Magnetic Clamping V-Block

Grade Grades A, B Grade C Grade A, B

Permitted Deviations (inch) Parallelism Parallelism “V” of “V” to Matching of opposite Base and tolerance Centrality faces Side 0.0001 0.0002 0.0002 0.0003 0.0003 0.0002 0.0004 0.002 0.001 0.002 0.002 0.001 Grade Characteristics •  Up to 2.125” •  Mild steel with embedded magnetic core, or laminated mild steel with brass •  Surface finish NTE 32µin   •   Material hardness for mild steel 50–62 RC

Flatness of Working Faces

End/Side Square to Base

•  Up to 2.125” •  Mild steel with embedded magnetic core, or laminated mild steel with brass •  Surface finish NTE 63µin   •   Material hardness 40–45 RC Steel

Grade C

All Grades, Designating Size Length 1 1.750 2 2.375 3 2.500 4 3.750 All dimensions in inches.

Nominal Dimensions Width 2.375 1.875 2.500 2.375

Height 1.875 1.750 1.750 1.750

Maximum Capac­ity 2.125 1.625 2.250 2.125

Due to the nature of the material, special consideration has been given to the granite V-blocks and the general recommended dimensions are given in Table 4. Higher accuracy granite V-blocks are commercially available and manufacturers’ tolerances apply. Table 4. Accuracy and Dimensions for Type 1, Style 5 Ribbed without Clamps

Grade Grades A, B Grade C Grades A, B Grade C

Permitted Deviations (inch) Parallelism Parallelism “V” of “V” to Matching of Opposite Base and Tolerance Centrality Faces Side 0.0001 0.0002 0.0002 0.0003 0.0003 0.0002 0.0004 0.002 0.001 0.002 0.002 0.001 Grade Characteristics •  Up to 10.500” •  Mild Steel or close grained cast steel free from porosity and defects •  Surface finish NTE 63 µin    •   Material hardness 40–45 RC steel

Flatness of Working Faces

End/Side Square to Base

•  Up to 10.500” •  Mild Steel or close grained cast steel free from porosity and defects •  Surface finish NTE 63 µin    •   Material hardness 40–45 RC steel

All Grades, Designating Size Length 1 2.250 2 3.500 3 5.000 4 8.000 5 10.000 6 12.000 All dimensions in inches.

Nominal Dimensions (inch) Width 4.000 4.000 6.000 8.000 10.000 12.000

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Height 4.000 4.000 6.000 8.000 10.000 12.000

Maximum Capacity 3.000 3.000 4.875 6.500 8.500 10.500

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Machinery's Handbook, 31st Edition V-Block Types

738

Type 1, Style 6, Plain, without Clamps.—Type 1, Style 6 V-block shall be designed sim­ ilar to the style shown in Fig. 6 and conform to the dimensions and accuracies described in Table 5. The Style 6 block shall be designed plain with a chip relief at the apex of the V-groove. The block shall be furnished in a set of three suitable for general machine shop use.

Fig. 6. Type 1, Style 6 General Machine Shop Support V-Block

Table 5. Type 1, Style 6 General Machine Shop Support V-Block Grade and Designating Size Grade A 1 2 Grade B 1 2

Flatness of Work­ing Faces

Permitted Deviations (inch)

End/Side Parallelism of Parallelism of Square to Opposite “V” to Base and Base Faces Sides

“V” Centrality

Matching Tolerance over “V” Length

0.0001 0.0001

0.0002 0.0002

0.0002 0.0002

0.0003 0.0003

0.0003 0.0003

0.0002 0.0002

0.0004 0.0004

0.002 0.002

0.001 0.001

0.002 0.002

0.002 0.002

0.001 0.001

All Grades, Designating Size 1 2 3 4 5 6

Nominal Dimensions (inch) Length 2.92 3.93 5.90 7.87 9.84 11.81

Width 1.57 1.77 2.16 2.75 3.34 4.13

Height 1.57 1.77 2.16 2.75 3.34 4.13

Maximum Capacity

0.160 to 1.50 0.236 to 1.70 0.315 to 2.10 0.315 to 2.70 0.475 to 3.30 0.600 to 4.10

All dimensions in inches.

Type II, Double V-Blocks.—The Type II, Double V-Blocks shall be designed to incorpo­ rate two V-grooves, as in Fig. 7a, Fig. 7b, and Fig. 7c, on one block machined on opposite sides, and both conforming to the dimensions and accuracies shown in Table 6. Nominal dimen­sions shown in Table 6 do not encompass all configurations available but define the toler­ances that need to be adhered to and define the design parameters. In the case that the design furnishes two V-grooves of different sizes, the accuracy tolerances of the larger size shall apply to both.

The Type II, Double V-Block shall be furnished individually or in matched pairs and shall conform to Grade A or Grade B accuracy tolerances and shall come furnished with a clamp or clamps sufficiently robust to enable adequate security of work piece during average drilling or grinding operations. If furnished in matched pairs the clamps shall be usable on either member of the matched set. The clamps shall be designed to engage a single clamp­ing groove along the outer face of the block.

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Machinery's Handbook, 31st Edition V-Block Types

739

Type II, Style 1 Double Groove Examples

Fig. 7a. Double Groove Single Level Clamping

Fig. 7b. Double Groove Double Level Clamping

Fig. 7c. Double Groove Single Level Clamping

Table 6. Type II, Style 1 Double Groove Single Level Clamping Grade Grade A Grade B Grade A Grade B

Flatness of Working Faces 0.0001 0.0004 •  •  •  •  •  • 

End/Side Square to Base 0.0002 0.002

Permitted Deviations (inch) Parallelism of Parallelism of Opposite “V” to Base Faces and Side 0.0002 0.0003 0.001 0.002 Grade Characteristics

“V” Centrality

Matching Tolerance

0.0003 0.002

0.0002 0.001

Steel alloy Surface finish NTE 32µin Material hardness 50–62 RC Mild steel or high grade cast steel Surface finish NTE 63µin Material hardness 40–45 RC steel / 200–280 BHN cast steel

Designating Size

Nominal Dimensions (inch)

Length

1 1.625 2 2.000 All dimensions in inches.

Width 1.250 1.500

Height 1.250 1.500

Maximum Capacity 1.000 1.500

Type III, Quadruple V-Groove Block.—The Type III, Quadruple Groove Block shall have four V-grooves, one on each of the four sides. This type shall be furnished in Grade A or B. Grade A shall be of a steel alloy and conform to the dimensions and accuracy toler­ ances defined in Table 7.

Fig. 8. Type III, Style 1 Quadruple V-Block

Fig. 9. Type III, Style 2 Quadruple V-Block

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Fig. 10. Type IV, Style 1 Combination Top and End V-Block

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Machinery's Handbook, 31st Edition V-Block Types

740

The Grade B Type III shall be of fine grained cast steel and conform to the dimensions and tolerances defined in Table 8. The nominal dimensions referenced in Table 7 and Table 8 do not represent the full extent of options available. Tolerances are defined as the minimum acceptable. Higher accuracy V-blocks of the Type III Style 1 and 2 may be available. Table 7. Type III, Style 1 Quadruple V-Groove Block

Permitted Deviations (inch) Parallelism Parallelism “V” of “V” to Matching of Opposite Base and Tolerance Centrality Faces Side 0.0001 0.0002 0.0002 0.0003 0.0003 0.0002 0.0004 0.002 0.001 0.002 0.002 0.001 Grade Characteristics •  Steel alloy  •   Surface finish NTE 32µin  •   Material hardness 50–62 RC •  Mild steel or high grade cast steel  •  Surface finish NTE 63µin •  Material hardness 40–45 RC steel / 200–280 BHN cast steel

Flatness of Working Faces

Grade Grade A Grade B Grade A Grade B

Designating Size 1 All dimensions in inches

End/Side Square to Base

Nominal Dimensions (inch) Length

Width

Height

2.000

1.500

1.500

Maximum Capacity 1.125

Type III, Style 2, Quadruple V-Block.—Type III, Style 2, Quadruple V-Block is designed for a robust precision support of the work piece. This style V-block is not fur­nished with clamp or yoke and may be made of either mild steel or fine grain cast steel. The Type III, Style 2 block may be designed with uniform V-groove capability or a range of capacities. The tolerance of the Style 2 V-block will conform to those shown in Table 8, however the nominal dimensions and accuracies referenced in Table 8 do not represent the full extent of options available. Manufacturers’ tolerances are always applied. Table 8. Type III, Style 2 Quadruple V-Block

Grade Grade A Grade B Grade A Grade B All Grades, Designating Size 1 2 3 4 5 6

Permitted Deviations (inch) Parallelism Parallelism “V” of “V” to of Opposite Base and Centrality Faces Side 0.0003 0.0005 0.0005 0.0003 0.0003 0.0005 0.001 0.001 0.0012 0.0012 Grade Characteristics •  Steel alloy  •  Surface finish NTE 32µin •  Material hardness 50–62 RC •  Mild steel or high grade cast steel  •  Surface finish NTE 63 µin •  Material hardness 40–45 RC steel / 200–280 BHN cast steel

Flatness of Working Faces

End/Side Square to Base

Length (mm)

Height (mm)

95 100 120 120 150 200

Matching Tolerance 0.0002 0.001

Nominal Dimensions

Width (mm)

Maximum Capacity (mm)

35 30 35 60 75 100

20 × 1 – 30 × 1 – 40 × 1 – 50 × 1 20 × 1 – 30 × 1 – 40 × 1 – 50 × 1 30 × 1 – 40 × 1 – 50 × 1 – 60 × 1 30 × 1 – 40 × 1 – 50 × 1 – 60 × 1 40 × 1 – 50 × 1 – 60 × 1 – 70 × 1 60 × 1 – 70 × 1 – 80 × 1 – 100 × 1

70 100 120 120 150 200

Dimensions of deviations are in inches; Nominal dimensions are in mm.

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Machinery's Handbook, 31st Edition V-Block Types

741

Type IV, Combination Top and End V-Groove Block.—The Type IV, Combination Top and End V-Groove Block shall be furnished in individual blocks consisting of one block, one clamp and one securing screw which is similar to the design illustrated in Fig. 10. The V-block shall be so designed that when turned on side or end that there shall be no interference from the sliding clamp. One end of the block shall be stepped and this end shall contain a V-groove and clamping grooves at right angle to the base. The blocks shall be designed to facilitate clamping to a machine bed or plate and have two tapped holes in the base and one on each side for mounting purposes. Style 1, Single Level Clamping.—The Type IV, Style 1 Block shall be similar in design to the one shown in Fig. 10 and shall conform to the dimensions and tolerances as defined in Table 9. The Type IV, Style 1 shall be offered in Grade A configuration. Grade A tolerance shall be applied to the perpendicular secondary V-groove as it is to the primary V-groove. Table 9. Combination Top and End V-Groove Block

Permitted Deviations (inch) Parallelism Parallelism “V” of “V” to of Opposite Base and Centrality Faces Side 0.0001 0.0002 0.0002 0.0003 0.0003 •  Steel alloy  •  Surface finish NTE 32µin •  Material hardness 50–62 RC

Flatness of Working Faces

Grade Grade A Characteristics

End/Side Square to Base

Nominal Dimensions (inch)

Designating Size

Length

1

3.625

Width 1.875

Height 1.875

Matching Tolerance N/A

Maximum Capacity 1.3125

All dimensions in inches

Metric Products.—Products manufactured to metric dimensions (SI units) shall be con­ sidered on an equal basis with those manufactured to the inch units, providing they fall within the applicable tolerance conversions. The British Standard BS 3731:1987 provides specific requirements for hollow or solid, single or double V-groove blocks made from cast iron or steel, and solid V-blocks made from granite. Two grades of accuracy are defined, Grade 1 and Grade 2. Provisions are made for matched sets as well. British Standard V-Block Designs, BS 3731 Maximum Diameter

Maximum Diameter

Maximum Diameter

B B A

Lightning Holes

B

Cast Iron Hollow V-Blocka

A Solid V-Blocka

A

Max. Dia.

Double V-Blocka

a Length C (side view) not shown.

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Machinery's Handbook, 31st Edition V-Block Types

742

Table 10. Accuracy of Grade 1 and Grade 2 V-Blocks, BS 3731:1987 Permitted Deviations

Designating Size, mm

Flatness of Working Faces, µm

End and Side Faces, Squareness to the Base, µm

Parallelism of Opposite Faces, µm

20 25 40 50 63 75 80 85 100 125 160 200

2 2 3 3 4 5 5 5 6 8 10 12

4 4 6 6 8 10 10 10 12 16 20 24

4 4 6 6 8 10 10 10 12 16 20 24

20 25 40 50 63 75 80 85 100 125 160 200

6 6 8 8 12 16 16 16 18 24 26 28

8 10 14 14 26 34 34 34 38 40 42 46

8 10 14 14 26 34 34 34 38 40 42 46

Parallelism of “V” Axis to Base and Side Faces, µm Grade 1 2 2 4 4 6 8 8 8 10 14 18 22 Grade 2 8 10 14 14 26 34 34 34 38 40 42 46

Squareness Equality of “V” Axis of “V” to End Face, Flank Angle, µm minutes

“V” Centrality, µm

Matching Tolerance over “V” Length, µm

2 2 4 4 6 8 8 8 10 14 18 22

1 1 1 1 1 1 1 1 1 1 1 1

4 4 6 6 8 10 10 10 12 16 20 24

2 2 3 3 4 5 5 5 6 8 10 12

8 10 14 14 26 34 34 34 38 40 42 46

2 2 2 2 2 2 2 2 2 2 2 2

6 8 10 10 18 22 22 22 24 24 26 28

4 4 6 6 8 10 10 10 12 16 20 24

Note: Intermediate sizes should be made to the accuracy specified for the next smaller designated size. Double V-blocks should be made to the accuracy specified for the larger designating size.

Max Dia., mm

Recommended General Dimensions for Steel, Cast Iron, and Granite V-Blocks BS 3731

Width A, mm

Height B, Length C, Max Dia., mm mm mm

Cast Iron Hollow V-Block 80 60 100 75 130 90 150 100 180 130 220 160 Double V-Block 20 and 25 32 32 20 and 40 40 40 63 80 100 125 160 200

35 40 45 50 60 70 42 50

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40 85 50 75 85 125

Width A, mm

Height B, Length C, mm mm

Solid Steel or Cast Iron 50 40 100 70 Solid Granite V-Block 75 75 100 100 150 150 200 200

25 50 75 100 150 200

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Machinery's Handbook, 31st Edition V-BLOCK CALIBRATION

743

Calibration of V-Blocks Calibration Description.—Calibration of the V-block shall be accomplished at intervals determined by either National Conference of Standards Laboratories (NCSL) Recom­mended Practices found in RP-1 “Establishment and Adjustment of Calibration Intervals” or the responsible facilities Quality Assurance Manual. Calibration intervals are designed to provide an acceptable level of assurance to adherence to original manufacturer’s speci­fications. Calibration of all precision measurement equipment is critical to providing mea­surement assurance to the quality system and maintaining a sound, robust manufacturing process and quality system. Table 11. Tolerance Limits per Grade

Test Instrument (TI) Characteristics

Range

Ends/sides Squareness To Each Other

Up to 5 in.

Performance Specification for Accuracy Grade A – 200 µin Grade B – 0.002 in.

Over 5 in. Up to 5 in.

Parallelism of V to Base

Over 5 in. Up to 5 in.

V Centrality

Over 5 in.

Flatness of Working Faces Matching Tolerance

Less than 6.00 in. Over 6.00 in.

Test Method

Grade A – 500 µin Grade B – 0.002 in.

Grade A – 300 µin Grade B – 0.002 in. Grade A – 500 µin Grade B – 0.002 in.

Grade A – 300 µin Grade B – 0.002 in. Grade A – 500 µin Grade B – 0.002 in.

Verified with Electronic Indicator

Grade A – 100 µin Grade B – 500 µin Grade A – 200 µin Grade A – 300 µin Grade B – 500 µin

Note: Tolerances listed in Table 11 are a representation of the standard tolerances recommended by the GGG-V-191D -1971, AFMETCAL (Air Force Metrology & Calibration Program) and gen­ eral commercial manufacturing. Be advised that not all features are calibrated on all types and styles of V-blocks. Manufacturer’s tolerances shall be applied when provided. The actual use of the V-block should be taken into consideration when selecting the features to be calibrated. Item

Table 12. Equipment Requirements

Minimum Use Specifications

Calibration Equipment Examples

Surface Plate

Range: 24 × 36 inch, Grade AA, 100 µin overall Flatness

Per Fed Spec GGG-P-463

Electronic Indicator

Range: ±0.010, Accuracy: ±0.00001 in.

Mahr-Federal, Model 832 Digital Electronic Amplifier w/ EHE-2056 LVDT

Range: 0.5 in. dia. × 3 in L; Accuracy: Class XX

Van Keuren, Extra Length Reversible Plug Gage

Range: 1.00 in dia. × 2 in. L; Accuracy: Class XX

Van Keuren, Reversible Plug Gage

Range: 9 in. L × 12 in. H × 3 in. W Accuracy: Grade AA 25µin/6 in.

L.S.Starrett Granite Tri-Square

Range: 6 × 6 inch; Accuracy: ±200 µin

Taft Pierce Model 9123-1

Precision Cylindrical Gage

Granite Master Square Squarol Sine Plate

Range: 0.375 in. to 5.00 in

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Squarol Squareness Checker Model #700

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Machinery's Handbook, 31st Edition V-BLOCK CALIBRATION

744 Item SquareMaster Master Square Cylindrical Square

Table Table 12. (Continued) Equipment Requirements 12. Equipment Requirements

Minimum Use Specifications Calibration Equipment Examples Alternative Equipment Requirements

Range: 0 in. to 12 in.; Accuracy ±50 µin over 12.00 in.

Range 3 in. W × 5 in. L × 12 in. H Range to fit max diameter of UUT. (Example: 5 in. × 12 in.); Accuracy: 100 µin

PMC/Mercury SquareMaster Model VSQ-12 SquareMaster Master Square Taft Pierce Model 9143

Preliminary Operations.—Review and become familiar with the entire procedure before proceeding with the cali­bration process. Alternative standards may be substituted for the ones listed in Table 12. Equipment Requirements. Thorough consideration must be paid to the Accuracy Ratio, Total Accuracy Ratio (TAR), and Measurement Uncertainties introduced when substitut­ing standards. Some standards represented do not offer a 4:1 TAR. If this is an unaccept­able condition according to your particular quality assurance system then measures must be taken to calculate the uncertainty and introduce this into your uncertainty budget, or determine the acceptable tolerance limits through Guard Banding or similar method to sat­isfy the requirements of your quality system. •  Allow all the components to stabilize a minimum of 4 hours to the temperature of the calibration area before beginning the procedure. •  Clean and closely inspect all critical surfaces for nicks and burrs. Carefully stone with fine hard Arkansas stone to remove any incursions. •  Turn on electronic indicator and allow to warm up for a minimum of 30 minutes before proceeding. Calibration Procedure

Note: Document the results of all measurements and tests for future reference, Trend Analysis, Root Cause Analysis or Calibration Interval Adjustment. Sides Square To Each Other.—1) Set up equipment as shown in Fig. 11. a) Special attention must be paid to establishing ZERO indication at the height of the V-block (UUT or Unit Under Test) sides to be measured. The SQUAROL (squareness checker) is rotated so the crest of the radius is indicated by the lowest reading on the Electronic Indicator. Plus and Minus travel must be available on the indicator head. Squareness Checker: Set to Height of V-Block

Granite Square Master

Surface Plate Fig. 11.

2) Set the electronic indicator to the appropriate range to measure the Squareness of the Side A of the UUT. a) ±0.0002 in. for Grade A. (±0.0005 over 5 in.) b) ±0.002 in. for Grade B (or C) 3) Set the gage head of the electronic indicator against the granite square master at the appropriate height to measure side A of the UUT at approximately 90% of it vertical height. Rotate the Squarol to seek the lowest indication. Using the fine adjust of the Squarol to bring the reading close to the zero indication. Assure that plus and minus travel can be realized.

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Machinery's Handbook, 31st Edition V-BLOCK CALIBRATION

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4) Zero the indicator with the electronic fine adjust to set the final zero indication.

5) Remove the granite square master and replace with the UUT (Side A) as shown in Fig. 12. Side A is determined arbitrarily, but once established, must be maintained throughout the procedure to determine the relationship of all sides to each other. Squareness Checker: Set to Height of V-Block

V-Block (UUT)

Electronic Indicator

Side B Surface Plate Side A Fig. 12.

6) Rotate the Squarol to seek the lowest indication. Verify that the indication is within the tolerance limits for the UUT as defined in Table 11. Record the indication as Side A to Base Squareness.

7) It may be necessary to place a parallel bar or right-angle iron behind the UUT to secure the UUT in place while obtaining indication. This will not affect the accuracy of the mea­surement as long as the measurement is repeatable and consistent.

8) Reverse the UUT and repeat Items 1) through 6) for Side B. Record indication as Side B to Base Squareness.

9) Rotate UUT 90° clockwise and repeat Item 6) and record indication as End A to Base Squareness. 10) Rotate UUT 180° clockwise and repeat Item 6) and record indication as End B to Base Squareness.

11) The End measurements will not be applicable to all styles and types due to design fea­tures. In that case disregard Items 8) and 9). 12) Rotate the UUT so that Side A is down and the indication is of Base to Side A. 13) Repeat Items 6) through 9) for remaining three sides (End A, End B, Top)

14) Rotate the UUT so that Side B is down and the indication is of Base to Side B.

15) Rotate the UUT so End A is down and the indication is Base to End A.

16) Repeat Items 6) through 9)for remaining three sides. (Side A, Side B Top) 17) Repeat process for End B if applicable.

18) All readings must be within limits defined in Table 11.

Parallelism of V to Base.—1) Select the appropriate size precision cylindrical gage or cylindrical square for the UUT being calibrated. The appropriate size gage should contact the V-groove near the center of the working surface. Leave approximately half the length of the cylindrical gage extended over one end of the UUT as shown in Fig. 13. Note: Larger V-blocks will require the use of cylindrical squares of various diameters to accomplish the engagement at the center of the working surface. Special care should be taken in handling these precision standards to avoid nicking or damaging the fine lapped surfaces. Adaptation of the methods used herein can be made to accomplish the calibration of larger V-blocks without reducing confidence levels associated with the readings taken.

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Machinery's Handbook, 31st Edition V-BLOCK CALIBRATION

Indication over Crest of Precision Cylindrical Gage Surface Plate

Fig. 13. V-Block with Cylindrical Gage Installed

2) Place the contact point on top of the cylindrical gage as close as possible and directly above the end of the V-block. Seek the top of the cylinder by sweeping across the radius and observing the maximum indication. 3) Zero the electronic indicator at the high point of the sweep. 4) Move the gage head to the end of the cylindrical gage. Indicate the peak of the radius and note the reading. 5) Rotate the cylindrical gage or square 180° and repeat Items 2) through 4). 6) Do not adjust the zero setting of the electronic indicator from Item 5). Note: If the cylindrical gage or cylindrical square being used covers 3⁄4 of the V-groove then Gage or Square does not have to be moved to the other end. 7) Remove the cylindrical gage or cylindrical square being used and place it on the oppo­ site end of the V-groove with the same end facing out. 8) Place the contact point on top of the cylindrical gage as close as possible and directly above the end of the V-block. Seek the top of the Cylinder by sweeping across the radius and observing the maximum indication. Note this reading. 9) Move the gage head to the end of the cylindrical gage. Indicate the peak of the radius and note the reading. 10) Rotate the cylindrical gage or square 180° and repeat Items 8) through 9) retaining the original zero setting position. 11) Record the largest reading found as the maximum error of Parallelism of V to Base. Reading shall fall within the Tolerance Limits defined in Table 11. 12) Lay UUT on Side A and perform Items 1) through 11) to calibrate Parallelism of V-Groove to Side A. 13) Lay UUT on Side B and perform Items 1) through 11) to calibrate Parallelism of V-Groove to Side B. 14) Repeat Items 1) through 13) for V-groove opposite and all remaining V-grooves if unit is Type 2, Style 1; Type 3, Style 1 or 2; or Type 4, Style 1. a) Double V-Groove Block Tolerance is to be derived from the tolerance of the larger of the two V-grooves. b) The Type III Style 2 Quadruple V-Block is generally produced in the B Grade and the applicable tolerance shall be applied to all V-grooves inclusive. Grade A tolerances shall apply to all V-grooves in a Grade A Quadruple Type III Style 2 regardless of the size of the V-groove unless otherwise specified by the manufacturer. 15) If V-blocks are a matched pair the maximum deviation between the members may not exceed the tolerance limits as defined in Table 11. 16) All dimensions must be indicated in matched sets. 17) Place both members of the set side by side on the surface plate with the V-groove up as shown in Fig. 14. The clamp or yoke is not installed for this measurement. 18) Place the contact point at the upper left corner of V-Block 1 and zero the indicator at that point.

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Machinery's Handbook, 31st Edition V-BLOCK CALIBRATION

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19) Indicate the top surface of the V-block and confirm the flatness of the block. If there are any erroneous readings, immediately check the V-block base for nicks or burrs that could be affecting the reading. Gently remove nicks or burrs with a hard Arkansas stone. 20) Move the contact point to V-Block 2. Do not change the zero setting and indicate the top surface of V-Block 2 and confirm that the readings are within the tolerance limits as defined in Table 11 for matched sets. 21) Rotate the UUT set so that Side A is now on the surface plate and place the contact point in the same relative position as with the TOP measurement. Starting Point

V-Block 1

V-Block 2

Fig. 14. Matched V-Block Setup TOP VIEW

22) Indicate the entire surface of V-Block 1, once again observing for any abnormalities that may indicate a nick or burr on the side opposite. 23) Move the contact point to the starting point on V-Block 2 and observe the deviation from the zero setting. Indicate the entire surface of V-Block 2 and note the maximum devi­ation from zero. 24) Verify that all readings are within the tolerance limits as defined in Table 11. 25) Rotate the UUT set so that the TOP is now on the surface plate. 26) Place the contact point in the same relative starting position as previously used. If the V-Block Set has a lapped or flat ground surface, indicate the entire surface observing any abnormalities and deviations in accordance with the tolerance table. If the UUT has a dou­ble rail supporting system, only indicate the sections that come into contact with the sur­face plate under normal use. 27) Move the contact point to V-Block 2 starting point and observe the initial deviation from zero. Indicate the applicable surface area and verify that all deviations are within the tolerance limits as defined in Table 11. 28) Rotate the UUT set to the final side and repeat Items 21) through 24). 29) Place UUT set on End A and repeat Items 21) through 24). 30) Place UUT set on End B and repeat Items 21) through 24). V-Groove Centrality.—1) The V-Groove Centrality is not calibrated in all types and styles. Centrality of the V-Groove is only applicable if the V-block can be set on its side with no interference when the clamp or yoke is installed. a) Precision granite V-blocks are generally produced in laboratory grade accuracies and require special attention in handling to avoid damage to the standards and V-blocks. b) Type 1 Style 5 are sometimes large and require special care in handling due to the weight of the UUT and the size of the standard use in this calibration. Care must be taken when securing a large cylindrical square into the V-groove so no nicks or burrs are raised in the lapped surface. 2) Using the same cylindrical gage or cylindrical square as in Parallelism of V to Base: Items through 13), lay the UUT on Side A as shown in Fig. 15. 3) Place the contact point on top of the cylindrical gage close to the end of the V-block. Seek the top of the Cylinder by sweeping across the radius and observing the maximum indication. Zero the electronic indicator.

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Machinery's Handbook, 31st Edition V-BLOCK CALIBRATION

4) Without disturbing the position of the cylindrical gage mounted in the V-groove, rotate the setup over and lay the UUT on Side B. 5) Place the contact point on top of the cylindrical gage as close as possible to the same distance from the end of the V-block as in Item 3). Seek the top of the cylinder by sweeping across the radius and observing the maximum indication. This reading must be within the tolerance limits defined in Table 11.

Indication over Crest of Precision Cylindrical Gage Surface Plate

Fig. 15. V-Block Indicate Side A

6) Rotate the cylindrical gage 180° and repeat Items 2) through 5). This reading must be within the tolerance limits defined in Table 11. 7) Repeat the process for the opposite end, and for each remaining V-groove on the UUT. 8) If the UUT is a matched set, the maximum deviation must not exceed the limits for matched sets as defined in Table 11. Flatness of Working Surfaces.—1) Place UUT onto sine plate as shown in Fig. 16 and seat well against the end bar. Assure that no nicks or burrs are present on either the plate or end bar that could affect the reading. 2) Set up the Sine Plate to an angle of 45° 3) Place the contact point of the electronic indicator into the V-groove at the upper left position of the first line as shown in the Top View of Surface of Fig. 16 and zero the electronic indicator at this position. Top View of Surface Surface Flat and Parallel

Meter

Gage Block Stack to Sine of °45 Electronic Height Gage

Sine Plate Surface Plate

Fig. 16. Flatness of Working Surface Set Up

4) Sweep the surface along the lines shown from the throat of the V-groove to the outer edge Left to Right (or End A to End B) 5) Verify that all readings are within the tolerance limits as defined in Table 11. 6) Rotate the V-block so the side opposite is now parallel to the surface plate and repeat Items 3) through 5), working from End B to End A. 7) Record all readings on a permanent record and retain for use by the QA system. 8) Remove all standards. Clean and return to protected storage condition. Remove UUT from set up and store in appropriate protective containers or stations.

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Machinery's Handbook, 31st Edition Parallel Bars

749

Parallel Bars

Parallel bars are used for workpiece support during layout, machining, and inspection operations. Parallel bars are made from either steel, cast iron, or granite, come in a wide range of sizes and are either used alone or in matched pairs. In general, there are four types of parallel bars: Type I, Solid; Type II, Ribbed; Type III, Box; Type IV, Adjustable. Although the parallel bar is a relatively simple tool, it is still considered a precision instru­ment and must be handled with the same attention to care and handling as other precision ground and finished supporting gages. Most precision parallels are made from heat-treated steel and hardened to 55 to 60 on the C Rockwell scale. Generally, they have a finish of 8 µin Ra with a fine finish free from all grind marks, chatter, or cracks. Granite parallel bars are also very desirable in an environment where thermal expansion is a consideration or where steel or cast iron would not be acceptable. Granite parallel bars are made to a very high degree of accuracy and can be used in a precision inspection setup. Type I, Solid.—Type I parallel bars are designed to be used independently as an individ­ ual bar to aid in inspection setups. They can be used as a reference surface or to establish a vertical plane perpendicular to a base plane. They can also be used as an extended support plane, when used in matched pairs, to establish an elevated plane that is parallel to the base plane. Type I parallel bars have four finished sides that shall not vary from a true plane by more than 0.0002 inch per foot (0.005mm/300mm). The adjacent sides will be square to each other within 0.0005 inch (0.012mm). Type I parallel bars will appear similar to Fig. 1 and will meet the tolerance specifications stated in Table 1.

Fig. 1. Parallel Bars, Type 1, Solid

Table 1. Steel and Cast-Iron Parallel Bar Tolerances, Inch Sizes Width/Height (in.)

(in.) 0.002

Parallelism (in.) Inch Sizes (in.) 0.0001

Width/Height Variance Matched Pairs (in.)

(in.) 0.0002

(in.) 0.0002

0.0002

0.0001

0.0002

0.0002

Tolerance Limits

Sizea (in.) to 3⁄16

1⁄ 8

1⁄ 4

3⁄ 4

to 3⁄4

to 11 ⁄8

11 ⁄8 to 11 ⁄2

Straightness (in.)

0.0005 0.0002

0.00015

0.0002 0.0003

0.0002 0.0003

0.0002 0.0004 0.0004 Millimeter Sizes (mm) (mm) (mm) (mm) (mm) 3 to 5 0.0500 0.0025 0.0050 0.0050 6 to 20 0.0130 0.0025 0.0050 0.0050 20 to 25 0.0050 0.0025 0.0050 0.0050 25 to 35 0.0050 0.0040 0.0080 0.0080 35 to 75 0.0050 0.0050 0.0100 0.0100 a Size as applicable to either width or height dimension specified by the manufacturer. 11 ⁄2 to 3

0.0002

0.0001

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Machinery's Handbook, 31st Edition Parallel Bars

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Some common size configurations available from most manufacturers of Type I preci­ sion parallel bars appear in Table 2. This is a representation of sizes that are available; how­ever, it is possible to have parallel bars custom made to a special order for a particular application. In that case, either the tolerance specifications of the engineering drawing or the specifications defined in the Federal Specification GGG-P-61a shall apply. It is at the discretion of the designing engineer which tolerances will apply in this case. Table 2. Common Sizes of Parallel Bars

Width and Height (in.) –1 – 13⁄16

1⁄ 8

1⁄ 8

3⁄ 16

3⁄ 16

1⁄ 4

– 7⁄8

– 11 ⁄8 – 3⁄8

1⁄ 4

– 1 ⁄2

1⁄ 4

– 3⁄4

1⁄ 4

1⁄ 4

– 5⁄8 –1

Length ± 0.002 (in.) 6 6 6 6 6 6

6 6

6

Width and Height (in.) 7⁄ – 7⁄ 16 8 1 ⁄ – 5⁄ 2 8 1⁄ 2

1⁄ 2

– 3⁄4

– 13⁄16

1⁄ 2

– 5⁄8

9

9

11 ⁄4 – 21 ⁄2

9

11 ⁄2 – 3

3⁄ 8

– 7⁄8

6

3⁄ 4

– 11 ⁄2

6

12

–2

9

– 11 ⁄4

3⁄ 4

–1

3⁄ 8

–1

1–3

9 9

12

12

1 – 11 ⁄2 1–2

–1

1⁄ 2

12 12

12

3⁄ 4

9

– 11 ⁄4

Length ± 0.002 (in.)

–1

3⁄ 4

1⁄ 2

1⁄ 2

– 11 ⁄2

–1

11 ⁄ 16

6

6

1⁄ 2

3⁄ 8

1 ⁄ – 11 ⁄ 2 2 1⁄ – 2 2

6

–1

6 6

6 6

– 3⁄8

– 1 ⁄2 – 3⁄4

Width and Height (in.)

1⁄ 4

3⁄ 8

3⁄ 8

Length ± 0.002 (in.)

12

12

12

11 ⁄4 – 13⁄4

12

11 ⁄2 – 2

12

12

12

Type II, Ribbed.—The Type II parallel bars are rectangular in cross section and made from a cast gray iron. The casting is ribbed to provide lightness while maintaining strength and rigidity. The ribbing on the Type II parallel bar extends to the outer surfaces of the bar and becomes an integral part of the working surface. The working surfaces of the Type II have a fine ground surface that will not exceed a value of 16 µin Ra. All the sharp edges are removed, and the surface is free from all machining marks such as grind chatter or burn. The Type II parallel bar is finished on the four sides of its length. The ends not being con­sidered working surfaces are not finished for work applications. The Type II parallel bar is a much more rugged design that the Type I and is designed to support workpieces of larger dimensions. Type II parallel bars are commonly available in sizes up to 4 by 8 inches with a length of 36 inches. Although larger in overall size, the Type II still maintains a very close tolerance in straightness, parallelism, and squareness.

Fig. 2. Parallel Bars, Type II, Ribbed

After each bar is cast and rough ground, it is subject to a seasoning or aging process. This process can either be natural or artificial. The aging process is necessary to provide long term stability after finish grinding. Once the bar is aged and seasoned, the final dimen­ sions will remain stable and not warp or twist for many years. Well-made and seasoned castings have been known to be in service for nearly 100 years and still hold their original geometry. Type II parallel bars will appear similar to Fig. 2 and will conform to the

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Machinery's Handbook, 31st Edition Parallel Bars

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toler­ances defined in Table 3. Squareness and parallelism tolerances are Total Indicated Read­ings (TIR) over the full length of the bar. The following table shows some common sizes that are readily available from most man­ufacturers of precision parallel bars. These are representative of the sizes that may be avail­able in the general market, and almost any size or combination may be specially made for a specific application. Table 3. Tolerance Limits for Type II - Ribbed Parallel Bars

Size

Tolerance Limits

Straightness

(inch) × 3 × 24 2 × 4 × 24 21 ⁄2 × 5 × 24 3 × 6 × 36 4 × 8 × 36

(inch) 0.0002 0.0002 0.0002 0.0002 0.0002

(mm) 35 × 75 × 600 50 × 100 × 600 50 × 152 × 600 75 × 150 × 1000 100 × 200 × 1000

(mm) 0.005 0.005 0.005 0.005 0.005

11 ⁄2

Parallelism Inch Sizes (inch) 0.0005 0.0005 0.0005 0.0005 0.0005 Millimeter Sizes (mm) 0.013 0.013 0.013 0.013 0.013

Squareness

Width/Height Variance Matched Pairs

(inch) 0.0005 0.0005 0.0005 0.0005 0.0005

(inch) 0.0005 0.0005 0.0005 0.0005 0.0005

(mm) 0.013 0.013 0.013 0.013 0.013

(mm) 0.013 0.013 0.013 0.013 0.013

Type III, Box Parallel.—The Type III “Box” Parallel is designed to provide a wide working surface. The Type III can either be square or rectangular in cross section. The rect­ angular design will have a rib running through the middle of the block the entire length. This provides support to the two larger surfaces. The surfaces of all six sides will be finish ground to an average of 16 µinch Ra and all sharp edges will be removed. The ground sur­ faces will all be free from grind chatter marks and grind burn marks.

Fig. 3. Type III, Box Parallel

The Type III parallel bar, after casting and rough grinding, will be subject to a seasoning or aging process. This process can either be artificial or natural, and insures the stability of the material after the grinding process. The finished Type III parallel bars will have a material hardness of 180 BHN (Brinell Hardness Number) checked with a 10mm ball and a 3000 Kg load. Each of the six working surfaces will not vary from a true plane (straightness) by more than 0.0002 inch per foot. The opposite sides and ends will be parallel to each other within 0.0005 inch. All adjacent sides will be square to each other within 0.0005 inch as well. The

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size as specified will also be within 0.0005 inch. The Type III box parallel will appear sim­ilar to Fig. 3. Tolerances are given in Table 4. Table 4. Tolerance Limits for Type III Box Parallel

Size (in.)

4×4×6 4×6×6 5 × 8 × 12 10 × 10 × 10

Straightness (in.)

Squareness (in.)

0.0001 0.0001 0.0002 0.0001

0.0005 0.0005 0.0005 0.0005

Parallelism (in.) 0.0005 0.0005 0.0005 0.0005

Type IV, Adjustable Parallel.—The adjustable parallel is a precision parallel that is adjustable to any height within a specified range. This design allows a flexibility in the use of the parallel that the other designs do not possess. The adjustable parallel is made of closegrained, seasoned cast iron. After initial casting and rough machining, the members of the parallel are subject to an aging or seasoning process that can either be natural or artificial. When complete, the member will have a hardness value of no less than 87 RB on the Rock­ well B scale. The final machining on the adjustable parallel shall have a finish of 16 µin Ra on the working surfaces and a 32 µin Ra on the sides. The sides of the adjustable par­allel are not designed to be working surfaces.

Fig. 4. Type IV, Adjustable Parallel

The adjustable parallel is made of two pieces. One piece is a fixed member machined with a dovetail slot, and the other a sliding member machined with a dovetail that fits into the slot and slides smoothly. It is this feature that gives the adjustable parallel the ability to be adjusted to any height within the range specified. The width of the bearing surface between the two members will be 1 ⁄2 the total thickness of the parallel. The two members are held in place and fixed by a locking screw arrangement that secures the two members and prevents them from moving or slipping after they have been set. Adjustable parallels are supplied in either individual pieces or in sets that cover a wide range. The adjustable parallel will conform to the tolerances defined in Table 5 and will appear similar to Fig. 4. A bilateral tolerance will be applied to the parallelism and flatness Table 5. Tolerance Limits for Type IV - Adjustable Parallel

Size (range) (in.)

0.375–0.500 0.500–0.6875 0.6875–0.9375 0.9375–1.3125 1.3125–1.750 1.750–2.250

Length (in.) 1.750 2.125 2.6875 3.5625 4.1875 5.0625

Thickness (in.) 9 ⁄ 32 9 ⁄ 32 9 ⁄ 32 9 ⁄ 32 9 ⁄ 32 9 ⁄ 32

Parallelism (in.) 0.0005 0.0005 0.0007 0.0007 0.0009 0.0009

Straightness (flatness) (in.) 0.00005 0.00005 0.00007 0.00009 0.00010 0.00012

Lock Screws 1 1 1 2 2 2

Granite Parallel, High-Precision.—The granite parallel is designed for the very high precision applications where thermal or magnetic properties must be taken into consider­ ation. The granite parallel is made with a much tighter tolerance than the cast-iron or steel

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parallels. The configuration is basically the same as the Type I solid parallel; the only dif­ ference is in the material and the tolerance limits. The granite parallel is supplied in either individual members or matched sets. Granite parallels must meet the tolerances defined in the table below and will appear similar to Fig. 1. Table 6. Tolerance Limits - Granite Parallel

Size (in.)

0.5 × 1 × 6 0.750 × 1 × 6 0.750 × 1.5 × 9 1 × 2 × 12 1.5 × 3 × 18 2 × 4 × 24 1.5 × 4 × 30

Grade AA Flatness and Parallelism 0.000025 0.00003 0.00004 0.00006 0.00015 0.00020 0.00025

Grade A Flatness and Parallelism 0.00005 0.00006 0.00008 0.00010 0.00030 0.00040 0.00040

Grade B Flatness and Parallelism 0.00010 N/A 0.00010 N/A N/A N/A N/A

Calibration, Precision Parallel Bars, Naval Air Systems and Air Force Metrology.— It is com­monly understood that wear and naturally occurring damage will affect the per­ formance of all precision measuring equipment. Identifying an out of tolerance condition before it can impact a critical measurement is the primary goal of calibration. Parallel bars are no exception to this rule. A precision-ground steel or cast-iron parallel bar may become worn or even deformed from daily wear or excessive forces applied. A granite parallel can also show evidence of wear, and, due to the extremely close tolerances applied to granite parallel bars, the monitoring of these instruments is even more critical. It is not extremely difficult to perform the calibration of the parallel bar, and it can be achieved in a relatively short period of time in a temperature-controlled environment on a clean surface plate with an electronic indicator, a height transfer standard, and a few gage blocks. The features that will be observed are flatness, parallelism, and height and width of matched pairs. The exact height and width of individual parallel bars is not a critical feature, but in this status they can only be used independently. As an additional note, a Pratt & Whitney Supermicrometer can be used to measure the exact height and width of a set of matched parallel bars, but, in the absence of this instru­ ment, the same results can be achieved with the instruments listed above. The uncertainty of the measurement is effectively the same, and the confidence in the results is just as high with either method. Flatness (Straightness) Calibration Method.—In any calibration, the first steps are to ensure a clean working surface and a proper setup of the standards and measuring instru­ ments. To that end, the first step is to clean the surface plate thoroughly with an approved surface plate cleaner and place the electronic indicator next to the working surface. The indicator should be powered up and allowed to warm up. The surface plate should be of an accuracy grade to provide a surface flatness deviation no greater than 0.00005" within the working surface that will be used. The UUT or Unit Under Test should be cleaned and, as a prelim­inary step to calibration, deburred with a clean ultra-fine hard Arkansas stone or gage block deburring stone. Only the edges should be stoned as this is where most burrs will occur. However, all edges, corners and surfaces should be observed to ensure that no dam­age or burring is present before proceeding. For precision cast-iron or steel parallel bars, the UUT will be set up on two gage blocks of the same size and accuracy grade. A 2.000" gage block is recommended as a minimum. This will allow access to both sides of the parallel bar when taking data to determine the flat­ness of each side. It is important that the gage blocks are placed at the correct points to sup­port the UUT without any sagging effect on the bar. These are called Airy points and are calculated by a simple formula: Airy point separation distance = 0.554L,where L is the length of the UUT.

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Machinery's Handbook, 31st Edition Parallel Bars

Example: For a 12.00" UUT, the gage blocks should be placed 0.554 × 12.00 = 6.650 inches apart and located (12.00 – 6.650) ⁄ 2 = 2.675 inches from each end, as shown in Fig. 5. This distance should be very close to the calculated distance, but it does not have to be exact. Establishing the distance using a machinists’ rule within 0.100" is sufficient. Top View of Surface A

Meter Surface A Gage Block

Surface B

Surface Plate

Electronic Height Gage

Fig. 5. Parallel Bar Flatness Setup

The UUT will be marked at five points equally spaced along the distance of each surface. The marks should be done with a felt-tipped marker or other suitable method. By general convention, the largest parallel surfaces are designated A and B. The two smaller surfaces are C and D. The markings A1 through D1 and A2 through D2 should be aligned with one another. A datasheet similar to the one shown in Table 7 should be created to collect the data and aid in the calculations of the deviations. With the UUT placed on the gage blocks as shown, configure the electronic indicator to measure “over” and set the indicator amplifier range to ±0.0002 with a resolution of 0.00001". It is recommended, however, to begin the readings at a slightly higher range and dial the scale in to a more sensitive resolution to determine the magnitude of the error. This will keep the readings on scale. Take the final readings in the highest sensitivity scale. Place the indicator contact point on the UUT at the A1 position and zero the amplifier. All readings will have the x1 position as the zero point (A1, B1, C1, D1). All deviations will be from this point. Sweep the surface of the UUT, recording on the datasheet the deviation as indicated at each of the calibration points. Ensure that each point does not exceed the documented tol­ erance for the type, style, grade, and size of the UUT. When Side A has been scanned, adjust the electronic indicator contact point for “Under” measurements, place the contact point at the B1 calibration point and zero the indicator amplifier. This measurement is taken from the underside of the UUT, and the data collected will be used to calculate the parallelism of the two sides A to B. Once again, sweep the sur­face and record the deviations of the surface at each of the designated calibration points. Verify that all readings for sides A and B are within the tolerance limits assigned to the UUT. Parallelism will be calculated from the sum of the deviations of each side. Calculate the sum of the deviations at each calibration point for surfaces A and B. The sum of the devia­ tions will reveal the parallelism of the two surfaces. Verify that the values calculated are within the assigned tolerances for the UUT. Carefully move the UUT to the adjacent sides (C and D) with side C up and repeat the sequence to determine the flatness and parallelism for sides C and D. Once complete and the values have been determined to be within the assigned tolerance limits, remove the UUT from the gage blocks and place on the surface pate. If this is a matched set of parallel bars, place the second member on the gage blocks as for the first member and complete the sequence for the second member.

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Table 7. Calibration of Parallel Bar and Matched Parallel Bar Sets Surface A Zero Setting (Point 1) Cal Point 2 Cal Point 3 Cal Point 4 Cal Point 5 Surface B (under) Zero Setting (Point 1) Cal Point 2 Cal Point 3 Cal Point 4 Cal Point 5 Surface C Zero Setting (Point 1) Cal Point 2 Cal Point 3 Cal Point 4 Cal Point 5 Surface D (under) Zero Setting (Point 1) Cal Point 2 Cal Point 3 Cal Point 4 Cal Point 5 Sum Surfaces A + B A2 + B2 A3 + B3 A4 + B4 A5 + B5 Sum Surfaces C + D C2 + D2 C3 + D3 C4 + D4 C5 + D5

Nominal

Straightness

As Found

As Left

Tolerance

Nominal

As Found

As Left

Tolerance

Nominal

As Found

As Left

Tolerance

Nominal

As Found

As Left

Tolerance

Parallelism Calculations Nominal As Found

As Left

Tolerance

Nominal

As Left

Tolerance

As Found

Height and Width, Matched Sets.—The calibration of height and width of matched sets of steel or cast-iron parallel bars is a relatively simple process. Place the two parallel bars side by side on the surface plate in the same orientation, that is, with the same side up and the predefined calibration points adjacent to each other. One parallel bar will be designated as “A” and the other as “B”. Position the height transfer gage next to the parallel bars and dial the transfer gage to the dimension of the side under test. Zero the electronic indicator on the transfer gage. Pull back the indicator probe off the land of the transfer gage and then replace it onto the land and ensure the zero setting repeats. If it does not repeat, re-zero the indicator amplifier and repeat the reading. Verify the zero setting in this manner until the reading repeats three times in a row with no variation. Then move the indicator probe to the surface of the “A” parallel bar and observe the deviation from zero reading. Sweep the entire surface and verify that all readings fall within the tolerance limits defined for the type and grade under test. Return the indicator probe to the transfer gage land and confirm that the zero point has not shifted before accepting the readings observed. If the zero point has shifted, it will be necessary to repeat the zero point setup and take the readings from the parallel bar again. If the readings are acceptable, proceed to the “B” parallel bar and verify that the deviation from zero reading is within the tolerance limits defined and the deviation from the “A” block is not beyond the limits for the type and grade under test.

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Machinery's Handbook, 31st Edition Parallel Bars Top View of Surface C Meter Surface C A

B

Surface D

Electronic Height Gage

Surface Plate

Fig. 6. Parallel Bar Matched Set, Height and Width Calibration

Size limitations for granite parallel bars are not as rigid as those for individual bars, acceptable limits for size being ±1 ⁄32 inch. However, the tolerance for size of matched sets is the same as that for steel and cast iron. The calibration procedure for granite parallel bars is the same as that for steel and cast iron. As a final note, it should be mentioned that a Pratt & Whitney Supermicrometer can also be used to calibrate the height and width of a parallel bar or a matched set of parallel bars. The Supermicrometer is set up using a gage block or gage block stack to the precise dimen­ sion of either the height or width of the parallel bar and zeroed at that point. The gage block is then removed, and the parallel bar is set on the elevating table between the anvils of the Supermicrometer and supported by two flatted rounds. The parallel bar is then measured at the first calibration point and the reading compared to the tolerance limits to assure compli­ ance. The bar is then measured at each remaining calibration point, verifying at each point the compliance to the tolerance limits. This same process is then completed for the “B” par­allel bar in the matched set, and the values obtained in the measurement of the “A” bar are compared to assure that deviations in the matched sizes are within the tolerance limits. This method is very good for smaller parallel bar matched sets if a Supermicrometer is avail­able. Right-Angle Plates Right-angle plates or knees, as they are more generally referred to, are divided into six different types as defined in Federal Specification GGG-P-441A. Right-angle plates are used in both machining applications and the setup and inspection of machined parts and assemblies. The plate material is usually high grade, fine-grained, controlled-process iron or iron alloy castings. These castings are always of uniform quality, free from blow holes, porosity or other material inconsistencies, and defects. All castings after being rough cut are stress-relieved to assure dimensional stability and become more stable as they age. With proper care and surface maintenance, angle plates can perform for many years. Fine angle plates over 50 years old are still in service. Right-Angle Plate Grades.—Most right-angle plates are manufactured to meet the guide­lines and tolerances laid out in the GGG-P-441A or IS 2554:1971 (India Standards Bureau). While these two governing documents may differ slightly, they both achieve the goal of establishing dimensional guidelines and tolerances that enable the manufacturer to produce a dependably consistent product with values and accuracies that customers require to maintain a high degree of measurement confidence in their quality systems. Angle plates come in grades according to the surface finish that is applied and the dimen­sional accuracies that they are manufactured to. The assigned grade is in direct

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correlation to the quality and accuracy of the angle plate. Federal Specification GGG-P441A speci­fies three Grades of surface finish applied to right-angle plates. In addition, there are six distinct Types of angle plates that have unique properties according to their specifically designed target uses in the manufacturing process. Grade A angle plates are the highest grade and are used in precise applications where the smallest deviation from square (90°) is required. These plates are used in various industries for clamping and holding work in a vertical position. The scraped surface is ben­ eficial for supporting work in a very precise vertical plane and facilitating the dispersion of lubricants and coolants during the manufacturing process. Grade A plates are required to have working surfaces that are fine precision-scraped with relief spots to prevent sticking of gage blocks or precision-lapped instruments. These relief spots allow the surface to be controlled to a high degree of flatness. Each square inch of surface on a Grade A plate has 15 to 18 spotting cavities, from 0.0002" to 0.0005" deep, and a bearing surface of 20 to 40 percent. The bearing area does not deviate from the mean plane by more than 0.0001 inch in over 24 inches, and no more than 0.0002 inch in up to 60 inches, as per Table 1. No square inch may vary from the adjacent area by more than 0.0001" per square foot. The adjacent working surfaces on any Grade A angle plate shall not vary from square by more than 0.0001" in every 8". This equates to 2.5 arc seconds. These angle plates make excellent fix­turing for high precision shaping, milling, grinding, drilling, or boring operations. Table 1. Maximum Permissible Deviation of Bearing Area from Main Plane ( GGG-P-441A)

Maximum dimension of working surface, inches

Above 21 ⁄2 12 24 36 48

To and including 12 24 36 48 60

Maximum permissible deviation of bearing areas from mean plane, inch Grades A and B

Grade C

0.0001

0.0002

0.0002

0.0004 0.0006 0.0008

Grade B plates are made to meet requirements of precision inspection and calibration operations. Although the style of the plate can accommodate uses in any of the manufactur­ ing practices, the precision-ground surface finish lends itself to inspection applications on the surface plate and use in calibration laboratories, high precision inspection stations, or situations where a fine precision-ground finish is desired and the highest degree of accu­ racy necessary. The Grade B angle plate has a working surface that is precision ground to a roughness average value of not more that 16 microinches for plates less than 64 square inches and 32 microinches for larger plates. The precision-ground surface of the Grade B plate is well suited to the applications involved in the inspection process of high accuracy, tight tolerance, precision-made parts. The fine-ground finish offers a precise surface that does not deviate from the mean plane by more than 0.0001 inch in over 24 inches, and no more than 0.0002 inch in up to 60 inches, as per Table 1. No adjacent areas shall vary with each other by more than 0.0001 inch per square foot. The aspect that is most critical about the angle plate is the control of squareness and the ability to pass on the accuracy to the parts being machined or inspected. All adjacent working surfaces will not vary from square by more than 0.0001 inch in every 8 inches. This high degree of accuracy in the Grade A and Grade B angle plates provides the measurement assurance necessary to inspect precision parts and calibrate precision measuring instruments. Grade C angle plates are the workhorses of the plate grades. They are designed to have a smooth machined finish and controlled to not exceed a surface roughness of 32 r.m.s. The bearing area of the working surface shall not exceed a deviation from the mean plane any greater than between 0.0002 to 0.0008 inch depending on the size of the bearing area, as per Table 1. Furthermore, no adjacent square foot may deviate from another by more than

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0.0003 inch per square foot. This ensures that the bearing surface is tightly controlled in the graduation of allowable error and will not deviate from the mean plane beyond the allowable limits. Deviation beyond the allowable graduation limits will introduce an error in the surface flatness beyond the functional limits. This flatness deviation control enables the surface of the angle plate to support the workpiece in such a manner as not to introduce errors due to nonuniform support beneath the workpiece or squareness errors from the adjacent sides.

IS 2554:1971 Grades: Standard sizes and accuracy of angle plates specified by IS 2554:1971 (India Standards Bureau) are given in Table 2a and Table 2b. Table 2a. Sizes and Accuracy of Slotted Angle Plates Metric Sizes ( IS 2554:1971) Accuracy in Microns, μm

Flatness of Working Faces

Squareness of Working Faces over Dimension H

Parallelism of Opposite Faces & Edges over their Total Length

Squareness of End Faces with Respect to Exterior Faces as Measured over Dimension L

150 × 100 × 125 150 × 150 × 150 175 × 100 × 125 200 × 150 × 125 250 × 150 × 175 300 × 200 × 225 300 × 300 × 300 350 × 200 × 250 450 × 300 × 350 400 × 400 × 400 600 × 400 × 450

5 5 5 8 8 8 8 8 10 10 10

10 10 13 15 15 18 18 18 18 18 20

13 13 15 18 18 20 20 20 20 20 23

13 13 15 18 18 20 20 20 20 20 23

700 × 420 × 700 900 × 600 × 700 1000 × 700 × 500

50

140

140

140

Size (mm) L×B×H

Grade 1

Grade 2

Table 2b. Sizes and Accuracy of Slotted Angle Plates Inch Sizes ( IS 2554:1971) Accuracy in Inches

Size (in.) L×B×H 6×4×5 6×6×6 7×6×5 8×6×5 10 × 6 × 7 12 × 8 × 9 12 × 12 × 12 14 × 8 × 10 18 × 12 × 14 16 × 16 × 16 24 × 16 × 18 28 × 16 × 28 36 × 24 × 28 24 × 16 × 18

Flatness of Working Faces 0.0002 0.0002 0.0002 0.0003 0.0003 0.0003 0.0003 0.0003 0.0004 0.0004 0.0004

0.002

Squareness of Working Faces over Dimension H Grade 1

0.0004 0.0004 0.0005 0.0006 0.0006 0.0007 0.0007 0.0007 0.0007 0.0007 0.0008

Grade 2

0.0055

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Parallelism of Opposite Faces & Edges over their Total Length 0.0005 0.0005 0.0006 0.0007 0.0007 0.0008 0.0008 0.0008 0.0008 0.0008 0.0009

0.0055

Squareness of End Faces with Respect to Exterior Faces as Measured over Dimension L 0.0005 0.0005 0.0006 0.0007 0.0007 0.0008 0.0008 0.0008 0.0008 0.0008 0.0009

0.0055

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Machinery's Handbook, 31st Edition Right-Angle Plates

759

Angle Plate Types.—The right-angle plate is further divided into Types and Classes, each type having unique characteristics and uses. Types I, II, and III angle plates or knees are intended to be used by craftsmen for ordinary machine shop operations; Types IV, V, and VI are intended to be used principally by toolmakers and for precision type work. Type I: Class 1-Plain Right-Angle Plate and Class 2-Slotted Right-Angle Plate: The Type I Class 1 angle plate has two members with the outside surfaces at right angles to each other. Both the working surfaces and the ends are machined square to within the tolerance limits assigned to their grade, (Fig. 1a). Type I Class 2 plates are similar to Class 1 with the vertical and horizontal slots provided on one or both faces, (Fig. 1b). X Y Y

Z Z

Y X

Y

Z

Fig. 1a. Type 1 Class 1-Plain

Fig. 1b. Type 1 Class 2-Slotted

Fig. 1c. Type 1 Class 3, Measuring-Plain

All the right-angle plates of the various types have ribbing of sufficient thickness to assure maximum support for flatness, squareness, and stability. Class 1 plates of approxi­ mately 10 to 12 inches will have at least 2 ribs, those of 16 to 24 inches at least 3 ribs, and those of 36 inches and above a minimum of 4 ribs in their design. Plates weighing over 75 pounds are designed with lifting holes in the ribbing or threaded holes for the placement of lifting eyes.

Type I, Class 3, Measuring Plane Right-Angle Plate: The Type I Class 3 angle plate, Fig. 1c, is designed to establish a measuring plane for production or inspection functions. It has two outside working surfaces that are at right angles to each other. The working surfaces are finished according to the grade specified and are the only two faces designed for func­tional operation. Unless specifically ordered otherwise, the sides and ends will be finish ground only, and these surfaces are not to be used to determine or establish squareness. The rib provides support between the ends of the plate and is cored out to provide a good hand gripping surface without sacrificing the stabilizing effect of the rib. The short base member “Y” is provided with a suitable hole so the plate may be bolted to the table of a machine or a tapped hole in a surface plate.

Type II, Inside Right-Angle Plate: The Type II right-angle plate is an inside right angle that is constructed with the same adherence to squareness and finish as the outside angle plates, Fig. 2. The inside angle plate is an inverted T-shaped casting formed by a base and a perpendicular member of approximately the same thickness. One side of the perpendicu­lar, the adjacent top side of the base, and the bottom of the base are the three working sur­faces and are all finished in accordance to the grade specified. The opposite ends and opposite sides are machined parallel and square to their respective working surfaces. X Y X

Z

Fig. 2. Type II, Inside Right-Angle Plate

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Z

Y

Fig. 3. Type III, Universal Right-Angle Plates

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Machinery's Handbook, 31st Edition Right-Angle Plates

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Type III, Universal Right-Angle Plate: By far the most versatile and useful right-angle plate in the machine shop is the Type III universal right-angle plate, Fig. 3. The Type III has two outside working surfaces at right angles and all adjacent sides are ground and finished to the same tolerance assigned to the main working surfaces. This provides a full range of positioning capabilities and makes the universal right angle just as accurate on its side as it is resting on its main working surface. All opposite sides are parallel within 0.0002 inch per foot. The universal right angle has a rib located between the ends that is square in con­figuration and provides support and stability to the main working surfaces while providing additional working surfaces that can be utilized for work support due to the accuracy assigned to the entire plate. The rib is hollowed out to provide good hand grip without sac­r ificing the stabilizing characteristics. Type IV: Machinist’s Adjustable Angle Plate: The Type IV adjustable angle plate, Fig. 4, is designed to allow rotation of the workpiece through 360°. The workpiece support table is adjustable from zero horizontal to 90° vertical in 10° increments. Basically, the Type IV machinist’s adjustable angle plate consists of a tilting table mounted on a rotary table base. The base has heavy lugs or bolt slots to accommodate mounting the plate on machine tools. The bottom of the base is machined true flat, parallel to the top plate and square to the ver­tical axis. The base is the true foundation of the plate and must be manufactured to the tightest of tolerances. All accuracies extend from the craftsmanship that is put into the base work of the Type IV plate. The top of the rotary base is graduated in one degree increments for the full 360° and is marked at least every 10° for a minimum of plus and minus 90° from zero (0°). Most plates are marked around the full 360° in 10° major divisions with 1× minor divisions, permanently engraved or etched on a chrome or brightly finished face for ease of reading and maintenance. The standard rotary accuracy to which all Type IV plates are made is ±2.5 minutes, noncumulative. This tolerance maintains the accuracy ratio neces­sary to maintain a good measurement assurance level in your quality system. The tilting table consists of two plates joined together at one end by a precision hinge and provided with an adjustable and positive locking mechanism. The bottom plate is attached to the base at the vertical axis. This plate will swivel through 360° freely and will also have a positive locking mechanism. The bottom of the plate is marked with a permanent line for accurate positioning in relation to the graduations on the base. The top plate is made in one piece as a casting and has T-slots milled in from the solid plate, as shown in Fig. 4. The top plate is adjustable up to 90° and the positive locking mechanism is robust enough to secure the workpiece into position during milling, drilling, or boring operations without movement. The top plate is made with a protractor integrated into the design that is marked from 0° (horizontal) to 90° (vertical) with major scale indications at every 10°. Minor scale graduations may be indicated at 1° increments. The noncumulative error between gradua­tions of the protractor is no greater than ±2.5 minutes, which allows for a high level of con­fidence in the measurements and operations made with this plate. X

X

Y

Y A

B

Fig. 4. Type IV, Machinist’s Adjustable Angle Plate

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C

Fig. 5. Type V, Toolmaker’s Adjustable Knee

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Type V, Toolmaker’s Adjustable Angle Plate: The toolmaker’s adjustable knee or adjust­ able angle plate extends the capabilities of the right-angle plate and allows for precise adjustment of the support table or bearing surface from 0° (horizontal) through 90° (verti­ cal), Fig. 5. The same tolerances in regard to flatness and squareness apply to the bearing surface and the base. The toolmaker’s adjustable knee is constructed of a plate, finished to the tolerances deter­m ined by the class specified, and a right-angle iron with two ribs. The right-angle iron and the plate are joined together by a precision hinge that has the capability to be secured at any position within the arc from 0° to 90°. The two ribs are of sufficient thickness to provide stability and support enough to withstand the forces applied during boring, milling, drill­ing, or grinding operations. The locking mechanism must be capable of maintaining its locked position under the same circumstances. The table is provided with T-slots for the mounting of the workpiece. The quadrant scale of the adjustable angle plate is divided into 90 divisions and a ver­nier scale. The vernier scale is graduated into 5 minute divisions from 0 to 60 minutes on either side of the zero index. The 0 (zero index) and every 15 minute index thereafter on both sides of 0 is clearly marked with the corresponding number, and the 5 minute divisions are clearly readable. The quadrant scale has 1 degree divisions from 0° to 90°. The zero index and every 10° index following will be clearly marked with the corresponding num­ber. A tolerance not to exceed plus or minus 2 minutes, noncumulative, is given for the error between any two graduations. Due to its specialized nature, the toolmaker’s adjustable angle plate usually is only man­ ufactured in one size according to the GGG-P-441A Federal Specification.

Type VI: Toolmaker’s Non-Adjustable Angle Plate: The Type VI toolmaker’s nonadjust­ able right-angle plate is different from the other types in that it has two members. These members have two working surfaces at right angles to each other as well as a finished pad on the back of both working surfaces. The design is of sufficient thickness along with the rib­bing to insure rigidity and stability in all operations and applications. X

Y

Z

Fig. 6. Type VI, Toolmaker’s Non-Adjustable Knee

Both front sides and the back side pads are considered functional working surfaces and will be manufactured according to the grade specified. The working surfaces, the opposite sides, and the opposite ends are all machined parallel to each other and square to their respective working surfaces. All machined surfaces will meet the tolerance limits assigned to the specified grade for finish, flatness, squareness, and parallelism. The Type VI tool­maker’s nonadjustable right-angle plate will appear similar to Fig. 6.

Calibration and Maintenance of Right-Angle Plates.—In many machine shops and quality assurance programs, right-angle plates are not calibrated and are considered “Cali­bration Not Required” or “Reference Only” instruments that are not placed into the cali­bration recall system. Inevitably, this practice may lead to unknown error in a manufacturing operation or inspection procedure. However, calibration of a right-angle plate may be an investment in quality if it becomes evident that an angle plate that has worn

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Machinery's Handbook, 31st Edition Right-Angle Plates

out of squareness or flatness is introducing error in a machining operation or inspection process. There are several calibration procedures and methods for verifying the accuracy of the right-angle plate. Flatness Calibration: The first feature that should be calibrated is the flatness of the working surfaces. The tolerances for the calibration of any individual right-angle plate is defined by the manufacturer and in their absence the GGG-P-441A shall provide an acceptable calibration tolerance. For Type I, II and VI angle plates, the setup for the flat­ ness calibration will appear similar to Fig. 7.

TI Measuring Planes Detail A TI

Electronic Height Gage

Surface Plate

Fig. 7. Calibration of Surface Flatness

Place the angle plate (the Test Instrument, or TI) on the leveling plate, as shown, and use a jack stand or similar device to support the TI in position. A bubble level can assist in lev­ eling the plate. When the plate has been leveled, use the electronic indicator to measure the surface following the pattern shown in the Detail A of Fig. 7. This pattern is generally called the Union Jack in that it resembles the stripes on the British flag. Observe the readings and record the maximum plus and the maximum minus readings overall. Subtract the maximum minus reading from the maximum plus reading; the result is the overall flatness of the surface. This value must meet the manufacturer’s toler­ance limits for the model under test or the tolerances specified in the GGG-P-441A. The same process is performed for the side opposite and the sides of the TI. All readings must meet the tolerances stated by the manufacturer or the GGG-P-441A standard. Flatness is only checked on TIs that require external hardware for support and leveling on the surface plate. On units not requiring external support, such as Type III plates, flatness is checked as a function of the parallelism calibration. Parallelism Calibration (Type III Right-Angle) The Type III universal right-angle plate has a parallelism tolerance due to the unique design of this unit. As discussed previ­ously, the Type III is finished on all sides and is square and parallel to all working surfaces and sides opposite. General Parallelism Tolerances Size Range Total Indicator Reading > 2.5 to 24 inches 0.0004 inch TIR > 24 to 36 inches 0.0008 inch TIR > 36 to 48 inches 0.0012 inch TIR > 48 to 60 inches 0.0016 inch TIR

Before beginning the calibration process, attention must be paid to the surface condition to be certain that no burrs or damage to the edges or surface is present that might affect the outcome of the measurements. A super fine-grained “hard Arkansas stone” may be used

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Machinery's Handbook, 31st Edition Right-Angle Plates

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to remove any burrs on the edges or surfaces prior to beginning the calibration process. After the right-angle plate is clean and free from burrs, the TI is placed on the surface plate as shown in Fig. 8. There is no need for a leveling plate as the side opposite is finished and parallel to the working surface, and the parallelism feature is what is being indicated.

TI Measuring Planes Detail A TI

Electronic Height Gage

Leveling Plate

Surface Plate

Fig. 8. Measuring Flatness/Parallelism of Type III Right-Angle Plate

Using the electronic indicator as before, indicate the surface in the Union Jack pattern and record the maximum plus and the maximum minus readings overall. In general, it is a good idea to zero the electronic indicator in the center of the Union Jack and take the read­ings from that point. This provides an easily repeated starting point and an overall numerical picture of the parallelism of the surface. As before, subtract the minimum read­ ing from the maximum; the result must meet the tolerance limits specified by the man­ ufacturer or GGG-P-441A. Repeat this process for the remaining sides. All results must meet the stated tolerances. Squareness Calibration (Preferred Method) The simplest method that can be used to verify the squareness of the Type III universal right-angle plate, as well as the squareness of the other types, is quite sufficient when done properly to meet the accuracy ratio that is required for a high confidence level measurement. This method provides quantifiable data that can be assigned an uncertainty should the Quality System determine this necessary. This method provides a good measurement to verify the squareness value and identify any out of tolerance conditions before the error is passed on into the manufacturing or inspec­tion process. Place the angle plate (TI) on the surface plate with two 0.1005-inch gage blocks between the face of the angle plate and the face of the granite angle block, as shown in Fig. 9a.

Granite Angle Block

TI

Gage Blocks

Surface Plate

Fig. 9a. Gage Blocks Between TI and Granite Angle Block

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Granite Angle Block

TI

Gage Blocks

Surface Plate

Fig. 9b. Gage Blocks Between Rotated TI and Granite Angle Block

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Machinery's Handbook, 31st Edition Right-Angle Plates

Then, beginning with a 0.1000-inch gage block, gently insert the gage block between the TI and the granite angle block at the top edge of the TI. Continue placing increasingly larger gage blocks between the TI and the granite angle block until the next larger gage block cannot be inserted without moving the TI. A gentle force should be applied to make sure the gage block is in direct contact with the granite surface, and as the gage block slides between the TI and the granite there should be no resistance until reaching the gage block size that is too large to go between without moving the granite. Record the size of this block as Block A. Rotate the TI and place the side opposite against the granite angle block as shown in Fig. 9b. Repeat the gage block sequence as above and record the size of the gage block as Block B. To calculate the squareness of the working faces of the rotated TI follow the formula: A−B Squareness = 2

Example: Block A = 0.1001 inch and Block B = 0.1009 inch. Solution: The calculation would be 0.1001 minus 0.1009 divided by 2. The result being (–)0.0004 in overall squareness of the working surfaces to a perfect perpendicular plane. Repeat the sequence for all remaining sides opposite. All results must meet the tolerances specified by the manufacturer or the GGG-P-441A standard. If for any reason the results do not meet the specified tolerances, it is possible to have the right-angle plates refurbished at a fraction of the cost of purchasing a new one. It is also possible that the deviations may be accept­able to the quality system, and, if the results are discussed with the Quality Engineer, the decision may be made to accept the unit “As Is” and annotate the deviation for future refer­ence. The unit may also be downgraded to a less critical operation and the previous appli­cation be taken over by a new unit of acceptable condition. These decisions must be made by the quality engineers or quality managers and follow the guidelines of the quality sys­tem in place. Measurements Using Light Measuring by Light-wave Interference Bands.—Surface variations as small as two millionths (0.000002) inch can be detected by light-wave interference methods, using an optical flat. An optical flat is a transparent block, usually of plate glass, clear fused quartz, or borosilicate glass, the faces of which are finished to extremely fine limits (of the order of 1 to 8 millionths [0.000001 to 0.000008] inch, depending on the application) for flatness. When an optical flat is placed on a “flat” surface, as shown in Fig. 1, any small departure from flatness will result in formation of a wedge-shaped layer of air between the work sur­face and the underside of the flat. Light rays reflected from the work surface and underside of the flat either interfere with or reinforce each other. Interference of two reflections results when the air gap measures exactly half a wavelength of the light used and produces a dark band across the work sur­ face when viewed perpendicularly, under monochromatic helium light. A light band is produced halfway between the dark bands when the rays reinforce each other. With the 0.0000232-inch-wavelength helium light used, the dark bands occur where the optical flat and work surface are separated by 11.6 millionths (0.0000116) inch, or multiples thereof. 7 fringes × .0000116 = .0000812″

.0000812″

.0000116″ Fig. 1.

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Machinery's Handbook, 31st Edition Measurements Using Light

765

For instance, at a distance of seven dark bands from the point of contact, as shown in Fig. 1, the underface of the optical flat is separated from the work surface by a distance of 7 3 0.0000116 inch or 0.0000812 inch. The bands are separated more widely and the indica­ tions become increasingly distorted as the viewing angle departs from the perpendicular. If the bands appear straight, equally spaced and parallel with each other, the work surface is flat. Convex or concave surfaces cause the bands to curve correspondingly, and a cylindri­cal tendency in the work surface will produce unevenly spaced, straight bands. See also Interpreting Optical Flat Fringe Patterns on page 791. Interferometer.—The interferometer is an instrument of great precision for measuring exceedingly small movements, distances, or displacements, by means of the interference of two beams of light. Instruments of this type are used by physicists and by the makers of astronomical instruments requiring great accuracy. Prior to the introduction of the interfer­ometer, the compound microscope had to be used in connection with very delicate mea­surements of length. The microscope, however, could not be used for objects smaller than one-half a wavelength of light. Two physicists (Professors Michelson and Morley) devel­oped the interferometer for accomplishing in the labora­tory what was beyond the range of the compound microscope. This instrument consisted principally of a system of optical mirrors arranged in such a way as to let the waves of light from a suitable source pass between and through them, the waves in the course of their travel being divided and reflected a certain number of times, thus making it possible to measure objects ten times smaller than was possible with the best compound microscope obtainable. Professor C.W. Chamberlain of Denison University invented another instru­ment known as the compound interferometer, which is much more sensitive than the one previously referred to; in fact, it is claimed that it will measure a distance as small as one twenty-millionth of an inch. These compound interferometers have been constructed in several different forms. An important practical application of the interferometer is in measuring precision gages by a fundamental method of measurement. The use of this optical apparatus is a scientific undertaking, requiring considerable time and involving complex calculations. For this rea­son all commercial methods of checking accuracy must be comparative, and the taking of fundamental measurements is necessarily confined to the basic or primary standards, such as are used to a very limited extent for checking working masters, where the greatest possi­ble degree of accuracy is required. The interferometer is used to assist in determining the number of light waves of known wavelength (or color) which at a given instant are between two planes coinciding with the opposite faces of a gage block or whatever part is to be measured. When this number is known, the thickness can be computed because the lengths of the light waves used have been determined with almost absolute precision. The light, therefore, becomes a scale with divisions — approximately two hundred-thou­sandths inch apart.

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766

Machinery's Handbook, 31st Edition MICROMETER CALIPERS MICROMETER, VERNIER, AND DIAL CALIPERS Micrometer Calipers

Classification.—The definition of the various classifications of micrometers was estab­ lished in the Federal Specification GGG-C-105 and provided the basic requirements that were to be met by manufactures for sale to the Federal Supply Service, General Services Administration. As stated in the GGG-C-105C (1987); Federal Specifications do not include all varieties of the commodity as indicated by the title of this specification or which are commercially available, but intend to cover only those generally used by the Federal Government. This Federal Specification would become the guideline for manufacturers throughout the United States. Not only was the Federal Government a large consumer of the preci­sion measurement instruments produced by domestic manufacturers, but the establishment of the design and accuracies provided a direction for further development and responded to the needs of the growing quality systems that were, by necessity, playing a greater and greater role in the manufacturing of high quality parts and systems. As the complexity of the systems grew, the greater the need for precision measurement instruments that could provide the measurement assurance needed to support those sys­tems. Establishing the design and accuracies for these instruments was a necessary step and provided the foundation for all future developments. Types, Classes, and Styles Type I – Caliper, Micrometer, Outside (page 767) Class 1: Enameled (I-Beam or Hollow Tubular) Frame Style A: Solid Anvil Style B: Interchangeable Anvil Style C: Tubing, Ball Anvil Style D: Tubing, Upright Anvil Class 2: Finished Frame Style A: Solid Anvil Style B: Paper Gage Style C: Tubing, Ball Anvil Style D: Thread Type II – Caliper, Micrometer, Inside (page 776) Class 1: With Jaws Class 2: Rod and Sleeve Class 3: Tubular Style A: Interchangeable Head Style B: Fixed Head Type III – Gage, Depth, Micrometer (page 780) Class 1: Interchangeable Measuring Rods Style A: Full base Style B: Half base Type IV – Micrometer Heads Class 1: High Accuracy Mechanical Class 2: High Accuracy Digital Class 3: Extended Travel Construction Requirements.—Materials: The micrometer spindles and anvils and com­ponent parts of the Type II and Type III caliper micrometers and gages having utile mea­suring surfaces shall be made of high grade tool steel or alloy steel. The source stock shall be assured of a refined crystal structure with a uniform carbide distribution, and shall be sound and free of injurious porosity, soft spots, hard spots and other defects. All materials shall meet the minimum requirements for hardness, stability and strength.

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Machinery's Handbook, 31st Edition MICROMETER CALIPERS

767

Resolution: Micrometers using the inch system shall have a least significant digit of 0.001 in., 0.0001 in., or 0.00005 in. Micrometers using the SI (metric) system shall have a least significant digit of 0.01 mm, 0.002 mm, or 0.001 mm. Finish: Unless otherwise specified, the frames and graduated components of Type I Class 2 micrometers and the thimble and barrels of all inside, depth, and micrometer heads shall have either a dull (non-glare) or satin chrome finish. All other exposed surfaces, excepting Type I Class 1 frames, shall have a ground finish. The surface roughness of all ground surfaces excepting direct measuring surfaces shall not exceed 32 min (0.8 mm) Ra, and shall be coated to prevent corrosion. The Type I Class 1 frames shall have either a smooth or ripple enamel coating. Identification: The Type I, Class 1 frame shall have a manufacturer’s pad of smooth machined finish. All instruments shall be permanently and legibly marked with the manu­ facturer’s name or trademark, so the source of manufacture can be easily determined, part number, and range of the instrument. Adjustments: Each type, style and class shall provide a method for adjustment due to wear of the measuring faces and/or wear between the spindle screw and the nut. Regular calibration and adjustment of the nut will maintain the accuracy of the micrometer. This is especially true in higher accuracy instruments. Micrometer Screw Spindle: The screw spindle is a critical component of the micrometer. The true accuracy of the instrument derives from the accuracy of the manufacturing of the threaded (screw) section of the spindle. The screw spindle material will be a stabilized high grade tool steel or alloy steel. The spindle screw section will be precision ground and have a hardness of no less than 62 RC on the Rockwell C scale. The exposed spindle section will have a fine ground finish with a surface finish of no greater than 32 min (0.08 mm) Ra. The fit between the spindle and the bearing including the nut, shall be freeturning and exhibit­ing no side shake, end shake or backlash. Measuring Faces: The spindle and anvil of all Type I, Class 1 and Class 2, micrometers shall be at minimum made of a stabilized high grade tool steel or alloy steel and have a hardness of no less than 62 RC. Alternatively, the measuring faces can be made of tungsten carbide with a hardness value of 92 RA on the Rockwell A scale. The tungsten carbide surfaces are preferred and provide a greatly improved life span for the measuring surfaces. Graduations: The graduations shall have a depth reduced below the surface of the barrel and thimble and shall be of contrasting color. Variations in the width of the graduated lines on the barrel, thimble or vernier shall not exceed 0.001 inch. Outside Micrometers Type I, Caliper Micrometer.—Size and Range: The size of an outside micrometer cali­ per is defined as the maximum cylindrical diameter measurement that can be obtained by that particular instrument. The range is defined as the span of obtainable measurements from the minimum to maximum capacity of the particular micrometer. Design: The basic features of the micrometer design shall consist of a frame, micrometer screw spindle, barrel and thimble, adjusting nut, either a lock-ring or lever type spindle lock, and a ratchet or friction stop, as shown in Fig. 1. Higher accuracy micrometers will include a vernier scale on the barrel that enables the reading of measurements to 1/10,000 of an inch (0.002 mm).

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Machinery's Handbook, 31st Edition Outside Micrometer Calipers

768

Clamp Ring or Lock Nut Ratchet or Vernier Friction Stop

Anvil

0 20

Spindle Barrel

Thimble

Vernier

Frame

0

0

1

2

3

4

20

Graduations

Reference Line Fig. 1. Design Features of Type I, Class 1 Micrometer

Frame: Frames of all outside micrometers in sizes from 1 ⁄2 inch to 4 inch shall be of steel. Frames of micrometers from 5 inch and greater shall be of steel, tubular steel or malleable iron. The Type I, Class 1 enameled frame shall have a cutaway back (Fig. 2a), long tapered back (Fig. 2b), or C-type Back (Fig. 2c), of either I-beam or hollow tubular steel frame con­ struction, and shall have either a smooth or rippled enameled finish.

Fig. 2a. Cut-Away Back

Fig. 2b. Long Taper Back

Fig. 2c. C-Type Back

Flexure: The cross section design and material used in the frame shall be, in combina­ tion, of sufficient design, that when a force of 2 pounds (10 N) is applied in parallel to the spindle axis upon the anvil, that a flexure of the frame shall not exceed the tolerance as specified in Table 8a and Table 8b. Micrometer Screw Spindle: The entire micrometer screw spindle shall be hardened and stabilized to no less than 62 RC from a high grade or alloy steel. The thread form and all surfaces shall be precision ground and smoothly lapped. The thread form shall conform to ANSI/ASME B1.3-2007 (R2017), “Screw Thread Gaging Systems for Acceptability: Inch and Metric Screw Threads”, and be a UN 40 pitch for English measurements and a M 0.5 mm pitch for SI (metric) measurements. The cylindrical portion of micrometer screw spindle shall be a free turning fit to the frame bearing without bind or shake and move smoothly and freely through the entire length of travel. There shall be no backlash between the threaded portion and the nut and at the maximum reading, there shall be no less than 0.240 inch engagement of the nut on the threaded portion of the screw spindle. The thread lead shall be sufficiently accurate throughout the entire length so as to permit compliance to the tolerances defined in Table 8a and Table 8b. The diameter of the spindle shall be no less than 0.1875 (3⁄16) inch for 1 ⁄2 inch micrometers, and no less than 0.2343 (15⁄64) inch

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Machinery's Handbook, 31st Edition Outside Micrometer Calipers

769

(5.9 mm) for 1.00 inch (25.0 mm) and larger micrometers. The diameter of the anvil and spindle measur­ing face shall be equal in diameter to within 0.0015 inch (0.038 mm). Barrel and Thimble: The diameter of the graduated surface of the barrel shall be not less than 0.406 (13⁄32) inch (10.31 mm). The angle of the bevel at the graduated end of the thimble shall not be more than 20 degrees as measured from the barrel. The beveled end of the thim­ble at the reading end shall be sharp but dulled to prevent injury to the operator. The major diameter opposite the reading end shall be diamond knurled to facilitate ease of operation. The radial clearance from the barrel to the reading end of the thimble shall not exceed 0.020 and shall be concentric and constant throughout the range of travel. When set at 0.000, 0.025, 0.050 inch …, etc., the reading edge shall bisect the graduation line exactly. Throughout the range of travel, there shall be no more than 2 ⁄3 of the line visible at any point. The barrel shall be constructed so as to prevent dust and foreign matter from coming into contact with the spindle screw. Adjusting Nut: The micrometer design shall provide an adjusting nut to compensate for wear between the spindle screw and the fixed nut. There shall as well be a means provided to compensate as well for wear of the measuring surfaces. This is generally accomplished through adjustment of the barrel or by disengaging the spindle screw from the thimble and resetting the zero indication of the micrometer. Both adjustment of the barrel and the adjusting nut shall be such that they remain secure after resetting and will not impair the original accuracy of the instrument. Disengaging the spindle screw and resetting the zero indication are more advanced adjustments and would require a recalibration of the micrometer before returning to service. Micrometers with a friction-adjusting sleeve on the barrel shall have no less than 31 ⁄2 and no more than 15 in-lb torque on the friction sleeve. Spanner wrench, keys or other specialized devices shall be provided by the manufacturer of each micrometer. Spindle Lock: The micrometer shall have a spindle lock which will arrest the spindle at the selected point and not alter the position when engaged. The spindle lock will be either a diamond knurled ring friction lock or a lever operated lock. Ratchet or Friction Stop: The ratchet or friction stop is a device that shall exert a constant and even measuring force on the spindle and effectively removes subjective “feel” of mea­ surements. This device is critical to obtaining an objective measurement and removes a variable in uncertainty analysis by delivering the same force between the spindle and anvil regardless of the operator. The design shall be either the ratchet or friction stop design. It shall be sufficiently knurled to enable satisfactory operation and shall exert a compressive force between the measuring faces of 1 to 11 ⁄2 lbs. up to the 6 inch size and 1 to 3 lbs. for larger sizes. Measuring Faces of Micrometer Screw Spindle: The face of the micrometer screw spin­ dle shall be at right angles to the axis at all positions of rotation of the screw spindle. Shall be square and flat and parallel to the anvil and shall maintain this orientation throughout the entire range of travel. The measuring face shall meet the tolerance limits as defined in Table 8a and Table 8b. Measuring Faces: All measuring faces shall be hardened, ground and lapped. The hard­ ness shall be no less than 62 RC and the lapped finish shall be no greater than 4 min Ra. Utile measuring faces can be lapped to recondition them should wear from use cause the surface to exceed the tolerance limits. Carbide Measuring Faces: Some micrometer calipers are furnished with carbide mea­ suring faces. The carbide measuring face tip shall not be less than 0.030 inch thick and shall be an integral part of the spindle and anvil. When provided with carbide measuring faces, both spindle and anvil shall be faced with the carbide tip.

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770

Machinery's Handbook, 31st Edition Outside Micrometer Calipers

Graduations: Reference, Graduation and Vernier lines on the barrel and thimble shall be clearly cut to a width of not more than 0.010 inch and all vernier and graduation lines shall be within 0.001 inch in uniform size. The lines and numbers shall have distinct and clean edges and shall be clearly legible under ordinary service conditions. The barrel and thim­ble shall be graduated as follows: Inch Reading Micrometer: The graduations on the thimble of the English (inch) reading micrometer shall be divided into 25 divisions each representing 0.001 inch. They shall be numbered as 0, 1, 2, 3, …, 23, and 24. Zero, then, represents 25 graduations or one full rev­ olution. The barrel is graduated with a longitudinal reference line that runs parallel to the axis of travel of the screw spindle. This reference line is graduated into 40 divisions each representing 0.025 or one full revolution of the thimble. The first graduation shall be num­ bered 0 (zero) and shall be the zeroing point for all consecutive measurements. After that every fourth division shall be numbered 1, 2, 3, and so on to 9, and 0. These divisions rep­ resent 0.100, 0.200, 0.300 and so on throughout the 1.000 inch length of travel. The 0.100 in. lines shall meet the reference line and extend above the line in a manner to designate them as the major 0.100 divisions. The 0.050 in. division lines shall meet the reference line and extend above the line but shall be half the length of the major 0.100 in. lines. The 0.025 in. graduation lines shall also meet the reference line and extend above the line to half the length of the 0.050 division lines. This method establishes a readily visible reference to the exact cumulative value of the reading. The vernier lines for the measurement of 1/ 10,000th inch (0.0001) shall be engraved on the upper part of the barrel parallel to the axis of travel of the spindle and extend a sufficient length along the barrel as to provide a reading at any point throughout the entire length of travel. The 10 vernier spaces are equal to nine divi­sions on the thimble and are numbered 0, 1, 2, 3, …, 9, and 0. Metric Reading Micrometer: The graduations of metric (SI) reading micrometers are configured in the same manner as the English reading micrometers. However, it must be stressed that manufacturers have developed variations of the basic principal that are equally accurate, and in some cases may be considered a genuine improvement. Slanted minor graduation lines, and split major minor graduations, have both been employed with great success. However, the principle for division of the scale is the same, and provides the amplification of the spindle screw movement necessary to obtain precision and repeatable measurements with the micrometer. The metric micrometer thimble is graduated into 50 divisions, each representing 0.01 mm. These are numbered as 0, 5, 10, 15, …, 45. The bar­rel is graduated along the reference line at 0.5 mm intervals. The reference line runs paral­lel to the axis of travel of the spindle screw. The first of the graduations lines is designated as 0 (zero) and every 10th graduation thereafter is numbered as 5, 10, 15, 20, and 25. The graduation lines representing 1.0 mm shall be meet the reference line and be distinguished by either length or position. The graduation lines representing 0.05 mm shall be distin­guished from all other lines again by either length or position, as in the English scale sys­tem. The vernier scale shall be as described previously in the section Inch Reading Micrometer. Accuracy: All micrometers shall be calibrated at a controlled temperature in a manner and method developed to assure performance parameters are acceptable and within the limits of tolerances as defined in Table 21a and Table 21b. Proper maintenance and periodic cal­ibration are mandatory to maintaining a high confidence level in all precision measuring instruments. Reference Setting Standards: The Type I, Class 1, Styles A and B, and Type I, Class 2, Style A micrometers are furnished with a reference standard for checking and setting the zero reading between the regularly scheduled calibration. These reference standards do not constitute a complete and thorough calibration but may be used to verify the zero setting of the instrument. These setting standards are themselves calibrated to assure adherence to tolerance, and act as a transfer standard during the calibration interval. The

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Machinery's Handbook, 31st Edition ANVIL MICROMETERS

771

setting stan­dards for the 1.000 inch (25 mm) micrometer shall be either a disk or rod of 1 inch, or 25 mm, respectively. Standards for larger sizes shall be rods. The rods may have either flat and parallel, or spherical ends. The disk, and the ends of the rods, shall have a hardness of no less than 58 RC. The disk shall be concentric in diameter to within 0.00005 inch (0.0012 mm) and shall be within 0.00005 inch (0.0012 mm) of the nominal size. The tolerances for other lengths shall conform to those defined in Table 9. Tolerances and instructions for calibration of reference standards are given in the section Calibration of Reference Setting Standards starting on page 787. Type I, Class 1, Style A.—Type I, Class 1, Style A, Solid Anvil Micrometers shall follow a design similar to Fig. 3 and conform to tolerances and requirements of Table 8a and Table 8b and be finished in either a smooth or rippled enamel. Spindle Adjustable Sleeve

Measuring Faces

Graduated Thimble

Anvil 0 123456

Friction Thimble (optional)

Fiducial Graduation

Ratchet Assembly (optional)

Locking Device

Frame

Fig. 3. Type I, Class 1, Style A: Solid Anvil Micrometer

Type I, Class 1, Style B.—Interchangeable Anvil Micrometer shall be in design similar to Fig. 4 and conform to the tolerances and requirements in Table 8a and Table 8b for accuracy. The anvils of the Type I, Class 1, Style B micrometer shall be designed so as to permit a secure and solid seating that allows no side movement and provides accurate alignment with the spindle measuring face. The anvils will provide a means to compensate for wear of the anvil faces and will be provided in the size ranges defined in Table 1. 0

1

0 20

2–3 Standard

Fig. 4. Type I, Class 1, Style B, Interchangeable Anvil, Enameled (I-beam or Hollow Tubular) Frame. Provided with reference setting standards. Shown with four 1-inch incremental anvils.

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Machinery's Handbook, 31st Edition ANVIL MICROMETERS

772

Table 1. Standard Sizes, Type I Class 1 Style B, Interchangeable Anvil Micrometers 0–4 2–6 6–9 6–12 9–12 12–16

Size Range, Inch

12–18 16–20 18–24 20–24 24–96 (in 6 inch increments)

0–100 50–150 150–225 150–300 225–300 300–400

Size Range, mm

300–450 400–500 450–600 500–600 600–2400 (in 150 mm increments)

Type I, Class 1, Style C.—Tubing Ball Anvil Micrometer shall appear similar in design to Fig. 5. Anvil

Measuring Range

Ratchet or Friction Stop

Barrel 0 1

0 20

Measuring Faces

Spindle

Thimble Graduations

Frame

Fig. 5. Type I, Class 1, Style C: Tubing Ball Anvil, Enameled (I-Beam or Hollow Tubular) Frame, Fixed Ball Anvil

Size and Range: The Style C Tubing Ball Anvil Micrometer is generally provided in the ⁄2 inch or 1 inch (13 mm or 25 mm) range. The range and accuracy of the micrometer is defined in Table 8a and Table 8b. The 1 ⁄2 inch (13 mm) size micrometer shall conform to the tol­erance limits expressed for the 0–1 inch micrometer. Anvil: The measuring face of the anvil shall be a bisected sphere permanently affixed to the frame with the radial center aligned to the projected axis of the spindle. The anvil shall be of such a radius as to permit accurate measurements of the wall thickness of tubing hav­ing a minimum inside diameters of 0.3125 inch for the 0 to 1 ⁄2 inch range. The radius of the anvil for the 0 to 1 inch micrometer shall permit accurate measurement of minimum inside wall thickness diameters of 0.375 inch. For the metric calibrated micrometer, the anvil radius shall be adequate to accurately measure a minimum internal wall diameter of 8 mm for the 13 mm micrometer, and a minimum internal wall diameter of 10 mm for the 25 mm range micrometers. Type I, Class 1, Style D.—The Style D, Tubing Upright Anvil Micrometer, shall have the half-C type frame, of I-Beam construction, with the diametrical axis of the anvil perpen­ dicular to the axis of the spindle and in the same plane. The Style D micrometer shall fol­low in design the example in Fig. 6. Micrometer is provided in 1 inch or 25 mm range. Spindle and Anvil: The measuring face of the spindle will be no less than 0.2343-inch diameter and be flat and parallel to the axis of the anvil. The spindle face shall be perpen­ dicular to the axis of the anvil throughout the rotation and travel of the spindle. The diame­ ter of the anvil shall permit the anvil to freely enter a 0.375-inch diameter hole to a depth of 0.750 inch. Type I, Class 2, Finished Frame.—The Class 2 Fixed Anvil, Finished Frame Microme­ ter shall have a long taper back, cutaway, or C-type frame (see Fig. 2a, Fig. 2b, and Fig. 2c) of a solid rectangular construction. The sides of the frame shall be stamped, engraved or per­manently marked in a practicable location in the decimal equivalent of common fractions; 8ths, 16ths, 32nds, 64ths. Stamping, engraving or etching shall be sufficiently deep enough to make characters readily legible in normal operating conditions. 1

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Machinery's Handbook, 31st Edition ANVIL MICROMETERS Measuring Range

Measuring Faces

773

Barrel 0 1

0 20

Anvil

Spindle

Thimble Graduations

Frame Fig. 6. Type I, Class 1, Style D: Upright Anvil Tubing Micrometer. Enameled (I-Beam or Hollow Tubular) Frame.

Finish: The finish of the Type I, Class 2 frames shall be a dull, non-reflective chrome plating. The utile measuring face of the anvil and spindle shall have a finish not to exceed 4 min Ra. The finish of the outer surfaces of the spindle shall not exceed 32 min Ra. Type I, Class 2, Style A: The Style A Solid Anvil Micrometer is provided in 0–1 inch, 1–2 inch, and 2–3 inch only. The Style A micrometer shall conform to the tolerance limits as defined in Table 8a and Table 8b and in design appear similar to Fig. 7.

Anvil

Measuring Range Measuring Faces

Locking Nut or Clamp Ring Ratchet or Vernier Friction Stop 0 0 1 20

Spindle

Thimble Graduations Barrel Vernier 543210

Frame

1-8.125 1-4.250 3-8.375 1-2.500 5-8.625 3-4.750 7-8.875 16THS. 1.0625 3.1875 5.3125 7.4375 9.56.25 11.6875 13.8125 15.9375

32NDS. 1.0312 3.0937 5.1625 7.2187 9.2812 11.3437 13.4062 15.4687 17. 5312 19.5937 21.6562 23.7187 25. 7812 27.8437 29.9062 31.9678

0 0

Reference Line

1

2

3

4

20

Graduations

Fig. 7. Type I, Class 2, Style A: Solid Anvil, Finished Frame Micrometer

Type I, Class 2, Style B: The Style B Paper Gage (Disk) Micrometer shall be designed with a disk attached to the anvil and spindle of approximately 0.4375 inch diameter. The Style B micrometer is provided in ranges from 0–1 ⁄2 inch, 0–1 inch, and 1–2 inch. The attached disk may be larger or smaller than recommended diameter. All variations in design shall adhere to the tolerance limits defined in Table 8a and Table 8b. The Style B micrometer shall follow in design and appear similar to Fig. 8.

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Machinery's Handbook, 31st Edition ANVIL MICROMETERS

774

5 0

1 0

1-8.125 1-4.250 3-8.375 16 THS. 1.0625 3.1875 0.8125 7.9375

32NDS. 1.0312 3.0937 5.1625 7.2187 9.2812 11.3437 13.4062 15.4687

Fig. 8. Type I, Class 2, Style B: Paper (Disk) Micrometer

Type I, Class 2, Style C: The Solid Ball Anvil Tubing Micrometer is similar in all aspects to the Type I, Class 1, Style C design with the exception that the frame is a solid rectangular construction. Style C shall adhere to the tolerance limits defined in Table 8a and Table 8b. The Style C micrometer shall by design appear similar to Fig. 9.

Measuring Range Anvil

Barrel 0 1

0 20

Measuring Faces

Frame

1-8.125 1-4.250 3-8.375 1-2.500 5-8.625 3-4.750 7-8.875 16THS. 1.0625 3.1875 5.3125 7.4375 9.56.25 11.6875 13.8125 15.9375

Spindle

32NDS. 1.0312 3.0937 5.1625 7.2187 9.2812 11.3437 13.4062 15.4687 17.5312 19.5937 21.6562 23.7187 25.7812 27.8437 29.9062 31.9678

Thimble Graduations

Fig. 9. Type I, Class 2, Style C: Solid Ball Anvil Tubing Micrometer

Type I, Class 2, Style D, Thread Pitch Micrometer: The Type I, Class 2, Style D pitch micrometer shall appear similar in design to Fig. 10 for fixed pitch micrometers. The Style D interchangeable anvil pitch micrometers vary in that the design incorporates an adjust­able anvil with an anvil lock to allow for resetting the zero indication of the micrometer when changing the matched anvil/spindle pitch sets. The interchangeable anvil pitch micrometer shall appear similar to Fig. 11. Both designs adhere to the tolerance limits defined in Table 2 for the frame design and pitch indication accuracy. The micrometer sec­tion of the pitch micrometer shall adhere to the tolerance limits of Table 8a and Table 8b for accuracy of the micrometer indication. Design: The design of the Type I, Class 2, Style D pitch micrometer shall be similar to that shown in Fig. 10 or Fig. 11. The frame design, either C-type or rectangular, shall be shaped to permit the measurement of screw thread pitch diameters of Unified thread form threads within size and range of the specific pitch micrometer. The measuring faces of the pitch micrometer consist of a cone (spindle) and a V-groove (anvil) with an inclusive angle the same as the angle of the thread to be measured. The apex of the spindle cone is to be centered and concentric to the axis of the spindle.

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Machinery's Handbook, 31st Edition ANVIL MICROMETERS

775

Table 2. Type I, Class 2, Style D Pitch Micrometer Design Specifications

Size (in.) 1⁄ 2

1 1 1 1 2 2 2 2

Range (in.)

0–0.500 0–1 0–1 0–1 0–1 1–2 1–2 1–2 1–2

Number of Threads per inch (1/pitch) 48 to 64 8–13 14–20 22–30 32–40 4.5–7 8–13 14–20 22–30

Permissible Flexure (in.)

Measuring Range Anvil

0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001

Tolerance on Half Angle of Measuring Faces (±) Degree Minutes 0 0 0 0 0 0 0 0 0

18 5 6 8 12 4 5 6 8

Error in Indicated Measurement (max.) (in.) 0.00010 0.00020 0.00015 0.00015 0.00015 0.00025 0.00025 0.00020 0.00020

Locking Nut or Clamp Ring Thimble 0 1

Error in Spindle/Anvil Alignment (max.) (in.) 0.00010 0.00020 0.00015 0.00015 0.00015 0.00030 0.00030 0.00015 0.00015

Ratchet or Friction Stop

0 20

Spindle

Graduations Barrel

Frame Fig. 10. Type I, Class 2, Style D: Thread (Fixed Pitch) Micrometer

Anvil: The anvil shall be secured in the frame with it axis aligned and concentric to the axis of the micrometer screw spindle. The anvil shall be well seated and secured but allow for a free turning fit with no shake or lateral movement. The end of the anvil facing the spindle cone shall have a V-groove of the same angle as the thread form to be measured. Both the spindle cone and anvil V-groove shall comply with the parameters set forth in Table 2. Zero Adjust

Anvil Lock

Fig. 11. Type I, Class 2, Style D: Interchangeable Anvil Thread (Pitch) Micrometer

Measuring Surfaces: The conical surface of the screw spindle and the angular surfaces of the V-groove anvil shall conform to the angular requirements defined in Table 2. The inter­ section of the apex of the conical screw spindle and the V-groove anvil shall be centered, aligned and in the same plane as the axis of the micrometer screw spindle. The surfaces of the conical spindle and the V-groove anvil shall be ground and lapped and hardened to no less than 62 RC. The surface roughness of these surfaces shall not exceed 4 min Ra.

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Machinery's Handbook, 31st Edition INSIDE MICROMETERs WITH JAWS

776

Inside Micrometers

Type II, Class 1, Inside.—The Type II, Class 1, Inside Micrometer with Jaws shall appear similar in design to Fig. 12 and conform to parameters and accuracy as defined in Table 3. Size (in.) 1 11 ⁄2 2

Table 3. Type II, Class 1, Inside Micrometer with Jaws Range (in.)

Tolerance (in.)

0.200 to 1 0.500 to 11 ⁄2 1 to 2

Radial Alignment (in.)

0.0002 0.0002 0.0002

0.003 0.003 0.003

Vernier Type Jaws Spindle Barrel Jaw Adjusment Screw

Thimble 0 9 8 7

20 0

Graduations Jaw Lock Screw

Reference Line Spindle Lock Screw Positioning and Guide Slot

Fig. 12. Type II, Class 1: Inside Micrometer with Jaws

Design : The inside micrometer with jaws shall have a vernier-caliper style measuring jaw, micrometer screw, spindle, barrel and thimble, knurled lock screw, lock device and adjusting screw. Jaws : The inside micrometer vernier-caliper style jaws shall be designed so the measur­ ing surfaces remain parallel throughout the entire range of measurement. The jaws shall travel parallel to the axis of the spindle travel and in the same plane. Micrometer Screw: The Type II, Class 1 inside micrometer shall be driven by a micrometer screw with a lead accuracy throughout its entire length of travel that will per­m it conformance to the tolerance defined in Table 3. The micrometer screw shall in all aspects adhere to the requirements and performance parameters previously defined for the Micrometer Screw Spindle on page 767. Spindle: The Type II, Class 1 inside micrometer shall have a spindle that is not an integral part of the Micrometer Screw. The spindle shall be designed to have only longitudinal movement in the barrel while the micrometer screw is secured to the thimble and free to rotate on the connection to the spindle. The spindle shall have a good free moving fit in the barrel bearing with no shake or bind and shall move smoothly throughout its entire length of travel. The spindle is secured from rotation and guided by the positioning guide and slot. Barrel and Thimble: The barrel and thimble of the Type II, Class 1 inside micrometer shall conform to the design requirements as defined in the previous section Type I, Caliper Micrometer. The design requirements for the applicable graduations shall as well be adhered to with the exception of the vernier scale. The numerals for graduations on the bar­rel shall read in the opposite direction from those on the conventional Type I micrometers as the measurements are reading an internal measurement that will become smaller as the micrometer spindle is drawn outward and larger as it is turned in. The first and every fourth graduation on the barrel along the longitudinal reference line shall

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Machinery's Handbook, 31st Edition INSIDE MICROMETERs WITH JAWS

777

be numbered 0, 9, 8, … 3, and 2 for the 0.200 to 1 inch range inside micrometer; and 15, 14, 13, … 6, and 5 for the 0.500 inch through 1.5 inch range inside micrometer. The numerals for graduations on the thimble shall read in the opposite direction from those on a conventional Type I OD micrometer. Measuring Faces: The measuring faces on the Type II, Class 1 inside micrometer shall two jaws in the vernier-caliper style designed to be perpendicular to the axis of travel of the micrometer spindle and shall have a radius of curvature no greater than 1 ⁄2 the diameter if the smallest value of the measuring range. The measuring surfaces shall conform to all design and accuracy requirements defined in the previous section, Type I, Caliper Micrometer. Type II, Class 2, Rod and Sleeve.—The Type II, Class 2, Rod and Sleeve Inside Micrometer shall follow in design as shown in Fig. 13. The ranges and accuracy shall fol­low Table 4. The micrometer head is calibrated separately from the length rods and main­tains a separate calibration tolerance as shown in Table 4. 3

4

Rod 01

0 20

Spacing Collar Micrometer Head Means for Attaching to Head

Handle Fig. 13. Type II, Class 2: Inside Micrometer: Rod and Sleeve Design

Table 4. Type II, Class 2, Inside Micrometer Accuracy Tolerance Range (inch)

1 to 2 2 to 8 2 to 12 8 to 32 8 to 36

Inch Tolerance Micrometer Length Head (inch) Rods (inch) 0.0002 0.0002 0.0002 0.0002 0.0002

0.0003 0.0004 0.0005 0.001 0.001

Metric

Range (mm)

1 to 2 25 to 50 50 to 200 50 to 300 200 to 900

Tolerance Micrometer Length Head (mm) Rods (mm) 0.0002 0.005 0.005 0.005 0.005

0.0003 0.0075 0.010 0.0125 0.025

Design: The Type II, Class 2 inside micrometer shall essentially consist of a micrometer head, measuring (length) rods, spacing collars, a knurled locking screw to secure micrometer head in place after obtaining measurement and an extension handle in some designs to optimize handling of unit in deeper bores or smaller diameters. Micrometer Head : The micrometer head consists of a micrometer screw barrel and thimble graduated in design as defined for the Type I outside micrometer (see Graduations on page 767) with the exception of the vernier lines.

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Machinery's Handbook, 31st Edition ROD AND SLEEVE INSIDE MICROMETERS

For Type II, Class 2 inside micrometers designed to inch scale: •  in the 1 to 2 inch range, graduations shall be from 0 to 0.250 inch on the barrel •  in the 2 to 8 inch, and 2 to 12 inch range, graduations will be from 0 to 0.500 inch •  in the 8 to 32 or 36 inch range, graduations will be from 0 to 1.000 inch distance

For Type II, Class 2 inside micrometer designed to the metric scale: •  in the 25 to 50mm range, graduations shall be from 0 to 7mm on the barrel •  in the 50 to 200mm, and 50 to 300mm range, graduations will be from 0 to 13mm •  in the 200 to 800 or 900mm range, graduations will be from 0 to 25mm on the barrel

All Type II, Class 2 micrometer heads shall conform in feature design and finish as pre­ viously defined in the Type I, Caliper Micrometer general definitions.

Micrometer Screw, Barrel and Thimble: The lead of the micrometer screw shall be accu­rate to within the performance limits as defined in the Manufacturers’ Production Tolerance Tables on page 781 throughout the entire length of travel. The barrel and thimble shall conform to the design definitions as previously outlined in the Type I, Caliper Micrometer section.

Measuring Rods and Spacing Collars: A sufficient number of Measuring (length) Rods and Spacing Collars shall be provided to enable the measurement of all distances within the range of each Inside Micrometer Set. The Rods shall be of such design as to provide positive seating with no side motion when assembled with the micrometer head. Length Rods shall provide an adjustable contact point to enable compensation for wear of the mea­suring face. Each Length Rod shall be plainly, clearly and permanently marked for mea­surement range.

Measuring Faces: The measuring face shall have a radius of curvature less than onehalf of the smallest measuring range of the inside micrometer with a surface roughness not to exceed 4 minch Ra. The locating faces on all spacing collars and length rods and microme­ter head ends shall have a surface finish not to exceed 8 minch Ra.

Type II, Class 3, Style A.—The Type II, Class 3, Style A, Tubular Inside Micrometer, Interchangeable Head shall conform to all design feature definitions of the Class 2 inside micrometer with the exception that the length rods shall be tubular in design and conform to the accuracy and length requirements defined in Table 5. The Style A inside micrometer shall follow in design and appear similar to Fig. 14.

0 1 2

0 20

Fig. 14. Type II, Class 3, Style A: Tubular Inside Micrometer with Interchangeable Heads

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Machinery's Handbook, 31st Edition TUBULAR INSIDE MICROMETERs

779

Table 5. Type II, Class 3, Style A, Range and Tolerance Inch

Range of Micrometer Head (inch)

Range (inch)

1 to 8 1 to 12 4 to 24 4 to 32 4 to 40

0 to 0.500 0 to 0.500 0 to 1.000 0 to 1.000 0 to 1.000 (two heads) 0 to 0.500 / 0 to 1.000

1 to 32

Range (mm)

40 to 200 40 to 300 100 to 600 100 to 800 100 to 1000 40 to 800

Range of Micrometer Head (mm) 0 to 13 0 to 13 0 to 25 0 to 25 0 to 25 (two heads) 0 to 13 / 0 to 25

Tolerance Micrometer Head (inch) 0.0002 0.0002 0.0002 0.0002 0.0002

Length Rods (inch) 0.0005 0.0005 0.0007 0.009 0.001

0.0002 Metric

0.0015 Tolerance

Micrometer Head (mm) 0.005 0.005 0.005 0.005 0.005

Length Rods (mm) 0.012 0.012 0.018 0.022 0.022

0.005

0.025

Measuring (Length) Rods: A sufficient number of length rods of hollow steel tubular construction shall be provided to enable measurement of any dimension within the range of the inside micrometer set. The length rods shall be so designed that they may be mounted on either or both ends of the micrometer head. A cap head measuring face shall be provided to mount to either or both ends of the micrometer head in the case where no length rod is being used. Each rod is to be adjustable by means of a hardened and ground plug at one end that can be turned either into or out of the length rod. Type II, Class 3, Style B.—The Type II, Class 3, Style B, Tubular Inside Micrometer, Fixed Head shall be of hollow steel tubular construction with a micrometer head perma­ nently attached. Each unit will be equipped with a thermal isolation grip to negate the effects of thermal expansion from the heat of the hand. The Style B shall follow in design and appear similar to Fig. 15. Accuracy and length requirements are defined in Table 6. 02

4–41 2

5 0

Fig. 15. Type II, Class 3, Style B: Inside Tubular Fixed Micrometer Head

Type II, Micrometer Head, Adjustments, Measuring Faces: The Type II, Class 2, Style B inside micrometer uses a micrometer head that adheres in design and function to the fea­ tures defined in the Type I, Caliper Micrometer section. The measuring faces shall have a spherical radius not to exceed 1 ⁄2 the radius of the smallest measuring range but otherwise adhere to the parameters defined in that section. The adjustment shall be as defined in the Type I, Caliper Micrometer section.

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Machinery's Handbook, 31st Edition TUBULAR INSIDE MICROMETERs

780

Table 6. Type II, Class 3, Style B, Inside Tubular Fixed Micrometer Head Inch

Range (inch)

2 to 2 2 to 3 3 to 3 3 to 4 4 to 4 4 to 5 5 to 12 (1 inch increments)

Range (mm)

50 to 63 63 to 75 75 to 88 88 to 100 100 to 113 113 to 125 125 to 300 (25mm increments)

Tolerance Micrometer Head Length Rods (inch) (inch)

Range of Micrometer Head (inch) 0 to 0.500 0 to 0.500 0 to 0.500 0 to 0.500 0 to 0.500 0 to 0.500

0 to 1.000

0.0002 0.0002 0.0002 0.0002 0.0002 0.0002

0.0003 0.0003 0.0003 0.0003 0.0003 0.0003

0.0002

0.0004

Metric

Tolerance

Range of Micrometer Head (mm) 0 to 13 0 to 13 0 to 13 0 to 13 0 to 13 0 to 13

Micrometer Head (mm) 0.005 0.005 0.005 0.005 0.005 0.005

Length Rods (mm) 0.007 0.007 0.007 0.007 0.007 0.007

0 to 25

0.005

0.010

Type III, Micrometer Depth Gage.—The Type III micrometer depth gage shall be sim­ ilar in design and function to Fig. 16 below. The Type III depth micrometer consists of a micrometer head, base and interchangeable depth rods. The unit shall be further provided with a ratchet or friction thimble, a clamp ring, spindle lock nut or lever-type spindle lock.

0

2

5 0 20

Fig. 16. Type III: Micrometer Depth Gage

Micrometer Head: The micrometer head shall be provided in a measuring range of 0–1.000 inch or 0–25 mm. The resolution of the instrument shall be 0.001 inch / 0.01 mm, 0.0001 inch / 0.002 mm (with vernier scale) or 0.00005 inch / 0.001 mm (units equipped

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Machinery's Handbook, 31st Edition MICROMETER DEPTH GAGES

781

with digital encoders). The graduations on the barrel shall read from the thimble end toward the base to indicate extension of the measuring rod from the base position of zero. Base: The base shall be designed with a measuring surface of no less than 3⁄8 inch, no more than 7⁄8 inch in width, and a length of either 2, 21 ⁄2, 3, 4, 5, or 6 inches. The surface finish shall not exceed 4 min Ra with a surface hardness of not less than 62 RC on the Rockwell C scale. The measuring surface of the base shall be flat to within 0.0001 inch per inch of length. Measuring Rods: The basic micrometer depth gage shall be equipped with three depth measuring rods to provide for the measurement of ranges of 0 to 1, 1 to 2, and 2 to 3 inches or 0 to 25, 25 to 50, and 50 to 75 mm. The depth measuring rods shall be of a design so that one rod at a time can be inserted through a hole in the micrometer screw spindle. The rod seat on the barrel end of the base shall provide a mounting surface parallel to the base mea­ suring surface. When inserted into position the measuring rod shall be held securely into place and pass freely and smoothly throughout the entire range of travel. The measuring rod shall have a means of compensation for wear. The measuring rods shall be straight with a runout when fully extended of no more than 0.003 inch. Additional Depth Measuring Rods: While the basic design requires that the set be pro­ vided with three depth measuring rods to reach the range of 0 to 3 inches, additional rods are available in expanded sets. All additional measuring rods shall be of design and perfor­mance as defined in this section. Accuracy: All micrometer depth gage micrometer heads and depth measuring rods shall meet the accuracy limits as defined in Table 7. Adjustment to out of tolerance conditions shall be performed before instrument is returned to service. Table 7. Type III, Micrometer Depth Gage Inch

Range (inch)

Range of Micrometer Head (inch)

Range (mm)

Range of Micrometer Head (mm)

0 to 3 3 to 6 6 to 9

0 to 1.000 0 to 1.000 0 to 1.000

0.0001 0.00015 0.0002

Metric

0 to 75 75 to 175 175 to 275

0 to 25 0 to 25 0 to 25

Tolerance Micrometer Head Depth Rods (inch) (inch) 0.0003 0.0004 0.00045

Tolerance Micrometer Head Depth Rods (mm) (mm) 0.002 0.003 0.004

0.006 0.008 0.009

Manufacturers’ Production Tolerance Tables

The Manufacturers’ Production Tolerance Tables, Table 8a and Table 8b, represent the parameters that are utilized by the original manufacturers of the instruments. These values were defined by the Project Team B89.1.13 of the ASME Standards Committee on Dimen­sional Metrology. This committee is a representative group of the manufacturing industry that establishes the standards used in the production of micrometers. These standards address the requirements of the American industry and the ISO efforts in the support of international commerce. While these tolerance tables and values are utilized in the manu­facturing of the instruments, the calibration of the individual instruments is slightly differ­ent and does not always include all the geometrical parameters that are taken into consideration during the manufacturing process.

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Machinery's Handbook, 31st Edition MANUFACTURERS’ PRODUCTION TOLERANCE TABLES

782

Table 8a. Manufacturers’ Production Tolerance Tables, Inch ASME B89.1.13-2001 Maximum Parallelism

Range

Permissible Frame Flexure (2 lbf)

Fixed Anvil

Interchange Anvil

Indicated Measurement Error (max.)

Spindle/Anvil Alignment Error (max.)

0–1 1–2 2–3 3–4 4–5 5–6 6–7 7–9 9–12 12–18 18–24 24–30 30–36

0.00010 0.00010 0.00010 0.00015 0.00015 0.00015 0.00020 0.00020 0.00030 0.00040 0.00050 0.00060 0.00070

0.00005 0.00010 0.00020 0.00020 0.00020 0.00020 0.00020 0.00025 0.00030 0.00040 0.00050 0.00060 0.00070

0.00040 0.00040 0.00040 0.00040 0.00040 0.00040 0.00040 0.00060 0.00060 0.00080 0.00080 0.00100 0.00010

0.00010 0.00020 0.00020 0.00020 0.00020 0.00020 0.00020 0.00020 0.00030 0.00040 0.00050 0.00060 0.00070

0.0020 0.0030 0.0045 0.0060 0.0080 0.0100 0.0100 0.0100 0.0100 0.0150 0.0150 0.0150 0.0150

(in.)

(in.)

(in.)

(in.)

(in.)

(in.)

Table 8b. Manufacturers’ Production Tolerance Tables, Metric ASME B89.1.13 Range

Permissible Frame Flexure (10N)

0–25 25–50 50–75 75–100 100–125 125–150 150–175 175–200 200–225 225–250 250–275 275–300 300–350 350–375 375–400 400–425 425–450 450–475 475–500 500–600 600–700 700–800 800–900

0.002 0.002 0.003 0.003 0.004 0.005 0.006 0.006 0.007 0.008 0.008 0.009 0.010 0.011 0.012 0.012 0.013 0.014 0.015 0.017 0.020 0.022 0.025

(mm)

(mm)

Maximum Parallelism

Fixed Anvil

Interchange Anvil

Indicated Measurement Error (max.)

Spindle/Anvil Alignment Error (max.)

0.002 0.002 0.003 0.003 0.004 0.004 0.005 0.005 0.006 0.006 0.007 0.007 0.008 0.009 0.009 0.010 0.010 0.011 0.011 0.013 0.015 0.017 0.019

0.010 0.010 0.010 0.010 0.010 0.015 0.015 0.015 0.015 0.015 0.015 0.020 0.020 0.020 0.020 0.020 0.020 0.025 0.025 0.030 0.030 0.035 0.035

0.004 0.004 0.005 0.005 0.006 0.006 0.007 0.007 0.008 0.008 0.009 0.009 0.010 0.011 0.011 0.012 0.012 0.013 0.013 0.015 0.017 0.019 0.021

0.05 0.10 0.10 0.15 0.20 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.38 0.38 0.38 0.38 0.38 0.38

(mm)

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(mm)

(mm)

(mm)

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Machinery's Handbook, 31st Edition CALIPERS, VERNIER AND DIAL

783

Calipers, Vernier and Dial

Classification.—The definition of the classifications of vernier calipers was established in the GGG-C-111 standard and provides the basic requirements that were to be met by manufacturers supplying such calipers to the Federal Supply Service, General Services Administration. While the Federal Specifications do not include all varieties of the calipers commercially available, it does provide a guideline for manufacturers. Dial and digital cal­ipers follow the general design features as illustrated in Fig. 17, Fig. 18, and Fig. 20. Any variation in designs must incorporate all aspects and necessary measure­ ment capabilities as defined in the original government specification. Centers for Setting Dividers 2 3 4 5 6 7 8

Lock Screw

Carrier

2 3 4 5

2 3 4 5 6 7 8 9

Graduation Bar

Gib

3

1

0 5 10 15 20 25 30

Adjusting Screw

Vernier Plate

Adjusting Nut

Adjusting Jaw Fig. 17. Original Design: Type 1, Class 1 with Fine Adjust Carrier, Gib, Lock Screw and Fine Adjust Screw and Nut

Type and Classes.—Calipers and gages shall be of the following types, classes, and styles, as specified: Type 1 – Calipers, vernier Class 1: Inch measure Class 2: Metric measure Class 3: Inch-metric measure Internal Jaws

Locking Screw Imperial Scale 0

0

2

4 1

5

10

6 8 2

0

1

2

3

15

20

1 3

2

4

5

25

0.001 in

4

6 8 5

4 6

7

8

2 6

2

7

4

6 8 8

3 9

2

4 10

6 8 4 2 4 11 12 13

6 8 5 2 14 15

4

6

8

6

9 10

Metric Scale

Depth Measuring Blade

External Jaws Fig. 18. Type 1, Class 3 – Inch/Metric Dual Scale with Vernier Scale Flush to Edge of Main Scale

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Machinery's Handbook, 31st Edition CALIPERS, VERNIER AND DIAL

Construction Requirements.—Materials: The beam and all measuring surfaces having direct contact with the work shall be made of high grade tool steel. The measuring faces may be made of Tool Steel with Carbide inserts. All materials shall be sound and free of injurious porosity, soft spots, hard spots and other defects. All parts subject to wear, break­age or distortion shall be replaceable. Main-scale Member (Beam): The main-scale member, or beam, of the Type 1 caliper shall be graduated as defined herein for each design variant. The main-scale shall be of a cross sectional area as to provide stability without undue weight when the instrument is used to make measurements at the maximum range. The measuring face and slide locating edge shall be straight within 0.0003 inch for each linear foot of the beam length. The total tolerance for the instrument (if over 12 inch) shall not be accumulated in any 12 inch incre­ment and all deviation in straightness shall be evenly distributed along the entire length of the instrument. Slide: The slide or moveable jaw shall be machined to receive the main-scale beam and shall have a fit and surface finish so as to allow the slide to move through the entire length of the beam without lateral shake or bind when not locked into position. The locating side of the slot shall be square with the measuring plane within 0.0003 inch along the full length of the contact surface. The opposite side of the moveable jaw shall be fitted with a nib or knife edge inside measuring jaw and a locking screw. Vernier Plate: The vernier plate shall either overlap or be flush with the edge of the main scale. The overlap of the main scale shall be no less than 1/64 inch and straight with 0.002 inch in the full length. The vernier plate shall be beveled to prevent parallax error. If the scale is installed flush along the edge of the main scale, the edge of the Vernier Plate shall slide easily along the main scale. The edges of both the main scale and the vernier plate shall extend fully to the edges. The Plate shall be held in place with two screws through elongated mounting holes to allow no less than 0.010 inch longitudinal adjustment. Measuring Contacts: The measuring contacts shall have a surface finish of 8 min Ra or better and have a hardness value of no less than 60 to 65 RC (Rockwell C scale). Scale and Graduations.—All calipers regardless of Type or Class will have a main scale and a Vernier scale or dial. The vernier scale shall be a direct reading scale so that when aligned with the zero point of the data scale, the first division of the vernier scale will appear below the first mark on the main scale. All vernier scale graduation spacing is a pre­cise fraction smaller than the marks on the main scale. All graduations shall be machine cut or etched between 0.003 to 0.006 in width and from 0.001 to 0.003 inch in depth and shall not vary in width on the main scale or the vernier scale by more than 0.001 inch. The edges of the graduations shall be straight and not vary more than 0.0002 inch throughout the full length of the graduations along the full length of the instrument. The graduations shall be sharp and well defined and contrast with the surrounding area through chemical etching or other similar methods so that the graduations are distinct and easily readable to enable the use of the instrument under normal working conditions. The graduations will be parallel to each other and square to the locating edge of the main scale within 0.0004 inch. The vernier scale graduations shall be parallel and square to the reading edge of the vernier plate or scale within 0.0002 inch. The index of both the main and vernier scales will be located ref­erencing the measuring contacts planes so as to be direct reading. Inch Graduated Instruments.—Main Scale: The inch graduated instruments will be graduated to read in thousandths of an inch increments. Each inch of the main scale shall be represented with a graduation mark representing the inch incremental value. Each inch segment shall be subdivided into ten increments of 0.100 inch. Each 0.100 inch increment shall be further divided into 4 increments of 0.025 inch each for 100 divisions per inch instruments or two divisions of 0.050 inch for 50 divisions per inch instruments. Each graduation of the main scale within the first foot shall be accurately located to the reference zero index within 0.0003 inch. With each additional foot length of the main scale and

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Machinery's Handbook, 31st Edition CALIPERS, VERNIER AND DIAL

785

addi­tional 0.0003 inch shall be acceptable with no total tolerance accumulated within any sin­gle foot of main scale length. All graduations within any 1 inch increment shall be accurately located in that inch increment within 0.0002 inch.

Vernier Scale : The inch vernier scale shall be divided into either 25 minor divisions each equivalent to one twenty-fifth of the main scale minor division or twenty five thousandths of an inch or 50 divisions each equivalent to one fiftieth of the main scale minor division or fifty thousandths of an inch. On either scale each alignment of the vernier scale mark to the main scale is equivalent to one thousandths of an inch. The graduated length for the 25 division inch graduated vernier scale shall be either 0.6000 or 1.2250 inch ±0.0002 and are divided into 25 equal increments accurately spaced to within ±0.0002 inch. All gradua­tions shall be of a length so as to be easily discernible with the graduations for 0 (zero) and each 0.005 inch having extra length to enable them to be easily distinguished. The 0 (zero) index and each 0.005 inch increment shall be clearly marked with the incremental value. The vernier scales that are divided into 50 increments shall have a length of either 1.225 or 2.450 ±0.0002 with each increment accurately spaced to within 0.0002 inch. Each instru­ment shall be marked with the scaled value.

Metric Graduated Instruments.—Main Scale: The metric graduated instruments will be graduated to read in 1/50 mm (0.02mm) increments. Each centimeter of the main scale shall be represented with a graduation mark representing the incremental value. For instru­ments with 25 increment vernier scales, the main scale shall be graduated in centimeters, millimeters and half-millimeters. Each centimeter segment shall be subdivided into ten increments of 1 millimeter each. Each 1 millimeter increment shall be further divided into 2 increments of 1 ⁄2 millimeter each for 20 divisions per centimeter. Centimeters shall be numbered from zero with easily read numerals. The graduation lines for cm, 1 ⁄2 cm, mm, and 1 ⁄2 mm shall have length of descending order. All graduation lines shall be clearly marked as described in the previous Scale and Graduations and Inch Graduated Instruments sections, and be easily readable under normal operating conditions. Instruments with 50 increment vernier scales shall have the main scale graduated in cen­timeters and millimeters with half-millimeters at the manufacturers’ option. Each graduation of the main scale within the first 30 cm shall be accurately located to the reference zero index within 0.008 mm. With each additional 30 cm length of the main scale and additional 0.008 mm shall be acceptable with no total tolerance accumulated within any single 30 cm of main scale length. All graduations within any 1 cm increment shall be accurately located in that centimeter increment within 0.005 mm.

Vernier Scale: Vernier scales for metric instruments divided into 25 increments shall have a graduated length of 12 mm or 24.5 mm, ±0.005 mm and scales divided into 50 incre­ments shall have a graduated length of 24.5 mm or 49 mm, ±0.005 mm.

3 0

4 1

2

5 3

4

6 5

6

7 7

8 9 0.02 mm

8 0

Fig. 19. Common Metric Vernier Caliper Scale Markings

The metric vernier scale most generally found in the current market shall be divided into 10 divisions or 0.1 mm, each division further divided into 5 minor divisions equivalent to a resolution of 0.02 mm. The spacing and location from zero shall be accurate to within 0.005 mm. All graduations shall be of a length so as to be easily discernible with the grad­uations for 0 (zero) and each 0.1 mm having extra length to enable them to be easily

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Machinery's Handbook, 31st Edition CALIPERS, VERNIER AND DIAL

786

distin­guished. The 0 (zero) index and each major division shall be clearly marked with the numerical incremental value. Dial Caliper.—The dial caliper uses a precise rack and pinion movement that drives an indicating hand to provide a direct reading on a circular dial. This precision movement eliminates the necessity to visually read the vernier scale and negates the effects of subjec­tive observation. Inside Jaws

Gib Slide Screws

Depth Rod

Lock Screw 80 70

90

0

Beam

10 20 30

Fine English Range 6”. Bar Graduatons 0.100” Adjusting Dial Indicator 0.001” Graduations Roll 0.100” Range per Revolution Dial Indicator Metric Range 150 mm. Bar Graduations 2mm Bezel Clamp Dial Indicator 0.02 mm Graduations Outside Jaws 2 mm Range per Revolution 60

50

80 70

40

90

60

0

50

10 20 40

30

80 70

90

60

0

10 20

50

40

30

Inside Measurement

Outside Measurement

Depth Measurement

Fig. 20. Dial Caliper – Fine Adjust Wheel and Gib Slide Screws for Accuracy Adjustment

The movement, when adjusted properly and calibrated will provide a measurement with a higher level of confidence than the vernier scale. Typically the indicating hand rotates once every 0.100 in or 1 mm and provides a resolution of 0.001 inch or 0.02 millimeter. The main scale still indicates the major divisions that are added to the reading taken from the dial indication. An additional feature of the dial caliper is that the dial is designed to be rotatable beneath the pointer allowing the operator to “set” the caliper at a known value and zero the indicator and make “differential measurements” from the known value. This func­tion was not possible with the original vernier caliper. Digital Caliper.—The most current refinement in the instrument is in the replacement of the dial movement or vernier scale with a digital display. The dial rack and pinion move­ment or vernier scale has been replaced with the application of a linear encoder that allows switching between millimeters and inches on demand, zeroing the scale at any point along the scale for differential measurements and electronically “holding” a measurement possi­bly taken in an awkward position where the digital display may not be visible. Many digital calipers offer a serial data output function that allows the operator to capture collected data in a spreadsheet or dedicated controller for analysis and more accurate records collection. Inside Measurement Jaws

Step Measurement Land

In/mm Conversion Button Lock Screw Digital Display IEEE Output Connector inch/mm

18.80

mm

OFF

ON

ZERO

Thumb Roller/Fine Adjustment

Battery Compartment Power ON/ZERO Button Power OFF Button

Depth Measurement Rod

Slider

Outside Measurement Jaws

Fig. 21. Digital Caliper – Average Feature Configuration

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Machinery's Handbook, 31st Edition CALIBRATION OF REFERENCE SETTING STANDARDS

787

Calibration of Reference Setting Standards

Preliminary Operations 1)  Review and become familiar with the entire calibration procedure before beginning the calibration process. Ensure familiarity with setup and operation of all standards involved in the procedure. 2)  Always observe safe operating methods for all standards, whether electrical, physical, mechanical, or dimensional; exercise caution in handling of all precision standards. 3)  Ensure measuring surfaces are clean and free of nicks and burrs that could affect the accuracy of the measurements. Observe the rod ends for any possible damage and observe the unit for bending or any condition that may affect the parallelism of the measuring surfaces. If any condition of concern is observed, light stoning with a hard Arkansas stone can be used with care to remove any nicks or burrs. 4)  Bring the UUT (Unit Under Test) into the controlled environment 12 hours prior to beginning the calibration process. Allow all standards and UUT to stabilize to com­mon temperature. Typical Equipment Requirements

Minimum Use Name Specifications Universal Horizontal Range: 0 to 24 inch Measuring Machine Accuracy: ±30min

Recommended Measurement Standards

Mahr-Federal 828

Alternative Mahr Linear 1200

Gage Block Set

Range: 0.050 to 4.000 inch 2.0 to 450 mm

Gage Block Set (Long Block)

5 to 84 inch 125 to 500 mm

Starrett Webber SS 8.A1X Mitutoyo 516-762-16 Starrett Webber SS 8. Mitutoyo 516-753-10 MA1X

Height Transfer Gage

Range: 1 to 48 inch

Starrett DIGI-CHECK II DHG 49

Electronic Height Indicator Precision V-Block Surface Plate

Starrett Webber SC 81.A1 Mitutoyo 516-401-16 Starrett Webber S2C Mitutoyo 516-442-10 88.MA1

Mitutoyo Universal Height Master Model 515-513 Mitutoyo Analog MU-Checker 519-420A with 519-327 Lever Head

Range: ±0.032 in. (min.) Mahr Federal Accuracy: Model 832 w/ LVDT ±50 min ±0.001 mm Range: Adequate to support UUT in vertical position. Accuracy: Calibrated to specifications of GGG-V-191D (or equivalent) Range: 24 x 36 Starrett Crystal Pink Mitutoyo 517-808 Accuracy: Grade A EDP 80655

Reference Setting Standard Calibration Process.—Note: Special care should be taken to clean and apply a protective coating when handling gage blocks or other precision stan­ dards with bare hands. The use of clean cotton or vinyl gloves is recommended to prevent corrosive oils from damaging precision surfaces. The temperature of the calibration environment must be maintained at approximately 23° C (73.4° F) with a fluctuation of no more than ±0.5° C (±0.9 °F) per hour. 1) Calibration of UUT up to 4.000 inch shall be performed on the horizontal measuring machine. All other measurements shall be performed in the vertical position as described in this procedure. 2) Select gage block equal to the size of the UUT. Wringing blocks together to achieve the desired size should be avoided if possible as this method increases possibility of error. 3) Insert the gage block between the anvil and spindle of the universal horizontal measur­ing machine resting against the anvil. Bring the head (spindle) into contact with

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the gage block and gently bring the indicator to the zero indication. Lock the head into position. Seat the gage block, re-zero, and assure the measurement is repeatable. 4) Remove the gage block and replace with the UUT. Assure that the UUT is seated and the measurement is being read at the peak (highest reading) for radius rod ends, or the par­allel point (lowest reading) for flat rod ends. Assure radius is centered on the anvil and spindle. Verify the reading is within the tolerance limits as defined in Table 9. 5) Setup the universal horizontal measuring machine for the next size UUT as described in Items 2) and 3). 6) Repeat measurement as described in Item 4). 7) Continue the process through the range of the universal horizontal measuring machine or until all UUT in the set have been measured. If the UUT length in the set exceeds the length of the universal horizontal measuring machine, proceed to Item 8) for the process to measure the remainder of the set up to 40 inch (1000 mm). Heigth Transfer Gage Gage Block Stack Meter

Reference Standard

Electronic Height Gage V-Block

Surface Handle

Fig. 22. Vertical Measurement of Reference Setting Standard

8) Using a precision V-block, clamp the UUT into the V-block ensuring the rod end is in contact with the surface plate and the UUT is secured in a vertical orientation. 9) Using either a height transfer gage or a gage block, set the electronic indicator to a zero indication. Set the range and resolution of the indicator so as to adequately indicate the deviation from zero in relation to the tolerance limits assigned to the UUT. 10) Repeat the zero indication several times to assure no drift or error has occurred. 11) Move the indicator from the standard to UUT and seek the highest point (for radius ends) or sweep the flat rod end for deviation from nominal and indication of parallelism. 12) Confirm that the indicated deviation is within the tolerance limits as assigned to the UUT in Table 9. Table 9. Size and Tolerance for Inch and Metric Reference Standards ( ASME B89.1.13-2001)

Length (in.) 1 2 to 4 5 to 8 9 to 11 12 to 18 19 to 24 25 to 36 37 to 40

English Tolerance (in.) ±0.00005 ±0.0001 ±0.00015 ±0.0002 ±0.00025 ±0.0003 ±0.00035 ±0.0004

Length (mm) 25 50 to 125 150 to 200 225 to 275 300 to 425 450 to 575 600 to 750 775 to 900

Metric

Tolerance (mm) ±0.0012 ±0.002 ±0.003 ±0.004 ±0.005 ±0.006 ±0.007 ±0.008

13) Repeat Items 8) through 12) for the remaining reference setting standards in the set.

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Calibration of Outside Micrometers

Typical equipment requirements for calibration procedures are shown in Table 10; a description of the calibration is given in Table 11, and various types of monochromatic light sources are given in Table 12. Table 10. Calibration Equipment Standards Recommended Equipment Minimum Use Specifications

Item

 Helium Discharge:  11.6 min Length/Fringe Van Keuren L-1A or L-1 a

Monochromatic Light

Range: N/A Accuracy: N/A

Optical Parallel Set

Range: Two optical parallels dif­fering in thickness by 0.0125 in. Van Keuren: OP-2B Accuracy:  10 min (flat and parallel)

Gage Block Set Gage Block Set   (Long Block Set)

Range:   0.05 to 4 inch   2.0 to 450 mm Range:   5 to 84 inch   125 to 500 mm

Optional Equipment

 U6 Monochromatic Range: 6 in diameter Reflex Interference Accuracy: N/A Viewer

b

Starrett Webber SC 81.A1 Starrett Webber S2C 88. MA1X Starrett Webber SS 8.A1X Starrett Webber SS 8.MA1X  Helium Discharge:  11.6 min Length/Fringe Van Keuren V760300 a

a While the Helium Discharge Monochromatic Light is one of the most common frequencies in use, other monochromatic light sources are available (see Table 12) and all will produce the same results. The band count is multiplied by the fringe length value to obtain the flatness of the measured surface. See examples in the table Interpreting Optical Flat Fringe Patterns. b The U6 Monochromatic Reflex Interference Viewer is used to observe the base flatness of the depth micrometer directly through an optical flat. The same technique of fringe count can be utilized to obtain the flatness as is used for the OD micrometer anvil and spindle.

UUT Characteristics Flatness

Parallelism

Length and Linearity

Table 11. Calibration Description Performance Test Method Specifications Range: Measured with Optical Flat/Parallel   0 to 54 in. converting fringe curvature to an   0 to 300 mm equiva­lent linear value. Tolerance: See Fig. 12 Range:   0 to 1 inch For UUT with 0–1 inch (0–25 mm) range,   0 to 25 mm measured at two positions with optical Tolerance: parallel set.  50 min.   0.0013 mm Range: Comparative measurement of UUT   0 to 54 inch read­ing to gage block test setup to test   0 to 1400 mm the basic length and micrometer head Tolerance: See Table 21a linear­ity. and Table 21b

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Machinery's Handbook, 31st Edition CALIBRATION OF OUTSIDE MICROMETERS

Note: The Minimum Use Specifications column of Table 10 represents the principal parameters required for performance of the calibration. References to specific equipment are recommendations only, and in no way constitute an endorsement other than as examples of the appropriate accuracy required to maintain the correct Test Accuracy Ratio (TAR) and confidence level in the calibration procedure. It is understood that equivalent equipment produced by other manufacturers is capable of equally satisfactory performance of this procedure. Note: If the UUT uses the metric scale, use test points equivalent to those defined for the English scale so as to test the micrometer at five places in the circumference of the drum and five equally spaced intervals throughout the range of travel of the micrometer spindle screw.

Table 12. Types of Monochromatic Light Source Wave Length Length/Fringe

Light Source

Helium-Neon Laser (bright red) Sodium Vapor (yellow) Mercury Vapor (green) Helium Discharge (yellow-orange)

µin

24.9 23.2 21.5 23.1

nm

632.8 589.3 546.1 587.6

µin

12.5 11.6 10.8 11.6

nm

316.4 294.7 273.1 293.8

Preliminary Operations.—Ensure a clean and well illuminated work area, free from drafts, excessive humidity with a stable temperature that does not fluctuate more than ±2° per hour. It is recommended that a stable temperature of 68° F is maintained in the calibra­tion work area. Note: The calibration of OD Micrometers and similar precision measurement devices is accomplished through the use of gage blocks as the length standard. Due to the thermal reactivity of gage blocks, environmental stability is a major consideration in maintaining the confidence in the accuracy of the measurements taken during the calibration. Calibration of UUT with an accuracy of ±0.001 inch or greater can be accurately cali­ brated in an environment with a temperature fluctuation of ±4°F. Calibration of UUT with a higher accuracy, such as ±0.0001 inch, must be accomplished in a stable environment of no greater than 68°F with a temperature fluctuation of no more than ± 2°F per hour. Ensure that the UUT and all associated standards have been allowed to stabilize in the controlled environment for a minimum of 8 hours prior beginning the calibration proce­ dure. This is especially critical for UUT in excess of 10 inches. Closely examine the UUT for any burrs or nicks on the measuring surfaces that could in any way damage the highly polished precision surface finishes of the gage blocks and opti­cal flats. Damage to these surfaces could degrade or potentially ruin the standard. Thoroughly clean the UUT and rotate the micrometer through the full range of travel, ensuring smooth movement with no binding or indication of drag or misalignment of the spindle through the support bearing. Note: It is recommended that well-fitting cotton or nitrile gloves be used to reduce ther­ mal transfer and protect from corrosive elements of direct contact. Calibration Procedure, Flatness Test (Outside Micrometers).—UUT used in preci­ sion work or any re-lapped anvil or spindle require a flatness test. If the instrument is known to be for general purpose; i.e. tool box issue, diameter measurement where flatness is not a factor, measurement is a non-quantified value, etc., a flatness test would not be required. Calibration records should reflect “flatness not calibrated”. Ball anvil and tubing micrometers require a flatness test on the spindle only. •  Connect the Monochromatic Light to a power source and turn on. Allow a few moments to warm up. •  Position the UUT in the monochromatic light field and place the optical flat on the anvil. Apply slight pressure and gently seat the optical flat until 3 to 5 bands are dis­played on the surface. It is important to not apply greater than necessary force or to cause damage to the

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optical flat by pressing down if a burr is detected. Scratching the surface of the optical flat can ruin the flat. Caution is required. •  Using the optical flat and monochromatic light, observe interference bands on anvil face. •  Count the number of bands between the lowest ends and the crest of 1 band. Multiply this number by the frequency of the monochromatic light being used. Confirm that the calculated deviation is within the tolerance limits defined in Table 13. 1 band = 11.6 millionths (µin) using helium discharge light source (Table 12). Record the band count, frequency and result.

Interpreting Optical Flat Fringe Patterns

Fig. 23a.

Fig. 23b.

Fig. 23c.

Fig. 23d.

Example, Optical Flat Fringe Patterns: Fig. 23a through Fig. 23d illustrate fringe pat­terns produced by an optical flat in contact with several test surfaces. The following illustrates how to interpret the fringe patterns observed to determine deviations in flatness of the unit under test (UUT). a) Convex surface (Fig. 23a): side edges low 11.6 µin b) Convex surface (Fig. 23b): side edges low 5.7 µin. c) Nearly flat (Fig. 23c): side edges low 5.7 µin. d) Complex surface (Fig. 23d) both convex and concave: hollow in center, higher each side of center and lower at side edges: error 3 µin. •  Turn over the UUT and repeat preceding steps for the spindle. If this is inconvenient due to size of UUT, remove the spindle to take the reading. Calculate in the same man­ner as above. Record the reading. •  If UUT is an interchangeable anvil micrometer, repeat measurement for each anvil. •  Note: If anvil or spindle do not create an image, or are too worn to produce an image, they will need to be sent out to the appropriate vendor for lapping.

Table 13. Anvil/Spindle Flatness Tolerances per USAF Range Tolerance English UUT Metric UUT Inches Millimeters (inch) (millimeters) 0 to 3 0 to 75 0.00005 0.001 4 to 9 100 to 225 0.00008 0.002 10 and above 250 and above 0.0001 0.0025 All with a resolution of 50 in. or 0.001 mm 0.000025 0.0006

Parallelism Test (Outside Micrometers).—The anvil and spindle parallelism test is per­formed on 0–1 inch / 0–25mm micrometers where a high degree of accuracy is required to be maintained. Prior to the parallelism test, the flatness test must be performed to the satis­faction of the tolerance of the UUT. •  It is critical to the accurate measurement that no dirt or dust particles are present. It is highly recommended that several measurements be taken to verify the results of the measurement. Cleaning the anvil and spindle and optical parallel with lint free lens tissue will aid in assuring the accuracy of the measurement.

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Machinery's Handbook, 31st Edition CALIBRATION OF OUTSIDE MICROMETERS

•  Position the thinner optical parallel between the anvil and spindle of the UUT and adjust the UUT until both surfaces are in contact with the optical parallel. Do not exert greater than necessary force in attempting to minimize interference band count. Adjust the optical flat on the anvil end of the UUT to obtain the minimum band image. The optimum number of bands anticipated here is four. This provides an easily read image to calculate the parallelism of the anvil and spindle. Note the number of bands on the anvil face as count A. •  Without changing the position of the UUT, carefully turn and observe the number of bands on the spindle face. Note this as count B. •  Compute the parallelism of the anvil and spindle over the smaller optical flat as follows. Use the band width of the monochromatic light that is being used; 11.6 × (A + B) = Parallelism (microinches) where A = count A; B = count B; and 11.6 is the fringe/length of the light source in microinches (Table 12). •  Remove the thinner optical flat and replace with the 0.0125 inch thicker one and repeat the measurement process. To calculate the anvil to spindle parallelism inclusive of the rotation of the spindle, calculate the difference between the two optical flat measurements. •  Verify that the difference between the two sets of measurements is within ±50 μinch or ±12 μm. •  If only one optical flat is available the same measurement can be achieved by using the measurement technique described above but calculated as 11.6 × (A − B) = Parallelism (microinches). The exception being that this method will not take into consideration the rotation of the spindle. Linearity Test (Outside Micrometers).—1) Set the UUT to indicate zero reading using the ratchet or friction thimble, and assure that the reading meets the applicable tolerance limits. If an Out of Tolerance (OOT) reading is identified at this time, assure there is no for­eign particle or burr on the anvil or spindle. An accurate zero indication must be estab­ lished at this time as all other measurements will be affected by any error at zero. Note: The micrometer anvil and spindle can be closed over a sheet of clean white paper damp with isopropyl alcohol, which can then be drawn out slowly to remove any obstruct­ ing matter. Repeat with a dry paper and observe any smear or smudge that would indicate dirt or foreign matter. Clean until paper comes through with no residue. 2) For Interchangeable Anvil Micrometers the appropriate length gage block will be used to establish the zero indication. 3) The recommended size gage blocks used to test the micrometer spindle thread are very important. The recommended test points achieve the testing of the accuracy of the micrometer throughout the entire length and at points evenly distributed around the diam­ eter of the spindle thread. This verifies the accuracy of the lead error of the thread as well as identifying any sections that may have experienced excessive wear or damage. The recom­mended sizes are as follows: a) X.210, X.420, X.605, X.815 and 1(X).000 (inch) and 5.10, 10.20, 15.30, 20.40, and 25.00 (mm). The micrometer should be tested at zero and full scale. X represents the applicable length for the UUT range. b) When wringing gage blocks together to achieve the desired size, the stack must soak for 1 hour when calibrating UUT with 0.0001 inch (0.002 mm) resolution in a range over 8.000 inches (200 mm). The soak time does not apply to sizes over 8.000 in (200 mm) that do not require wringing. 4) Select the first gage block stack and measure it using the ratchet or friction thimble. Do not rely on feel as the correct and controlled loading of measuring force by the ratchet or friction thimble will give the most accurate and repeatable reading. All other methods are subjective and open to operator influence. Several clicks of the ratchet or applying force until the friction thimble slips and assuring the micrometer is gently seated on the

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gage block stack will give an accurate reading. Assure the reading is repeatable. The reading must meet the tolerance limits as defined in Table 21a and Table 21b for Indicated Measurement Error. 5) Repeat these steps for each remaining gage block stack and record readings for evalu­ation or trend analysis. Follow all guidelines of the quality assurance programs in place in regards to recording and retaining all data collected and reporting OOT (out of tolerance) conditions. 6) For Interchangeable Anvil Micrometers, after calibration of the micrometer spindle thread, repeat Items 1) and 2) for each anvil in the set. Once the micrometer spindle thread accuracy has been confirmed, only the zero setting of each interchangeable anvil needs to be calibrated. Verify that each interchangeable anvil meets the tolerance limits assigned and record any OOT conditions before adjusting to meet tolerance limits if neces­sary. Calibration of Depth Micrometers

Note: Do not lock the micrometer head without the depth rod installed. This could result in damage to the split ring locking mechanism causing the rods to bind. Flatness Calibration.—1) For Depth Micrometers with a resolution of 100 μin (or metric equivalent) or better, the Van Keuren model U6 Monochromatic Reflex Interference Viewer is used to observe the base flatness of the depth micrometer directly through an optical flat. The same technique of fringe count can be utilized to obtain the flatness as is used for the OD micrometer anvil and spindle (Interpreting Optical Flat Fringe Patterns). 2) For Depth Micrometers with 0.001 inch (0.01 mm) resolution per division, place the UUT on an individual 1.000 in (25 mm) gage block with the depth rod extended through the center relief hole as shown in Fig. 24. Bring the depth rod into contact using the ratchet stop or friction thimble and observe the reading. Repeat this measurement three times ensuring repeatability of the reading. Record the reading as reading A. 3) Remove the UUT from the setup and place it on two gage blocks (or stacks) of 1.000 in. (25 mm) as shown in Fig. 24. Bring the depth rod into contact using the ratchet stop or friction thimble as in Item 2) and observe the reading. Repeat this measurement three times ensuring repeatability of the reading. Record the reading as reading B. Subtract reading B from reading A and ensure that the result is within the tolerance limits as listed in Table 14. Table 14. Depth Micrometer Base Flatness Range Tolerance Inches Millimeters (inch) (millimeters) All with a resolution of All with a resolution of 0.0001 0.0025 0.0001 in. or 50min 0.0025mm or 0.001mm All with a resolution of All with a resolution of 0.001 0.01 0.001 in. 0.01 mm

Note: The Linearity of the UUT is checked using the 0–1 inch (0–25 mm) depth rod only. Once the accuracy of the Micrometer Head has been verified, the remaining depth rod extensions are calibrated at the zero position only. It is critical to the outcome of the cali­ bration of the entire set that the base zero setting for the micrometer sleeve and thimble be accurate and without error or deviation. Any error in the base zero setting will cause errors to be translated to the depth rod extensions. Linearity Calibration (Depth Micrometers).—1) Select the appropriate gage blocks to make stacks to measure 0.210, 0.420, 0.605, and 0.815 inch, or 5.10, 10.20, 15.30, 20.40, and 25.00 mm. The 1.000 inch (25.00 mm) block will also be used as the micrometer head will be calibrated at zero to full scale. Clean and wring gage blocks assuring no dirt or for­eign matter is present.

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Machinery's Handbook, 31st Edition CALIBRATION OF DEPTH MICROMETERS View A

TI Gage Block

Surface Plate View B

TI

Gage Block

Gage Block Surface Plate

Fig. 24. Figure 4: Depth Micrometer Base Flatness Measurement

2) Place the UUT on the surface plate and zero against the plate. A calibrated surface plate must be used as it is used to establish the base zero setting and is considered as one of the standards used in the calibration procedure. Record any deviation and assure the read­ing is within the tolerance limits as defined. 3) Place the first gage block stack (0.210 inch or metric equivalent) on the surface plate and place the UUT on top of the stack. Measure the stack through the center relief hole in the gage block. Observe the reading. Repeat the measurement three times and assure repeatability of the measurement. Verify that the reading is within the tolerance limits. 4) Repeat Item 3) for all remaining gage block stacks. Record all readings. If an OOT condition is discovered, adjust the UUT and repeat the linearity calibration to verify the adjusted accuracy. 5) Repeat Item 2) for all remaining depth rod extensions. For UUT with resolution of 0.0001 in or better, allow gage block stacks that require wringing to assemble to stabilize for one hour for lengths over 8.00inch. Stabilization is not required for single block lengths. Calibration of Micrometer Heads Linearity Calibration.—1) Select the appropriate gage blocks as defined in the Linearity Test (Outside Micrometers). As previously the micrometer head will be calibrated at zero to full scale. Clean and wring gage blocks, as appropriate, assuring no dirt or foreign matter is present.

012

4 3 2 1 0 24 23 22 21

Micrometer Head TI

V-Block

Surface Plate

Fig. 25. Micrometer Head Set Up — Vertical in V-Block

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2) Mount the micrometer head vertically by the shank into a precision V-block as shown in Fig. 25, and adjust so that the zero position can be achieved making contact with the cal­ ibrated surface plate. If a zero position cannot be achieved with the V-block available, use an intermediary gage block of the necessary size so as to achieve contact with the spindle face and set zero on that intermediary block. This is the zero indication and will not be included in the linear measurement. Repeat the zero measurement three time to assure repeatability of the measurement (zero setting) Record the reading as zero. Verify that the reading is within the tolerance limits for the UUT. Raise the micrometer head spindle and insert the first step gage block (0.210 in or metric equivalent) lower the micrometer head until the spindle measuring face makes contact with the gage block stack. The micrometer head has no ratchet stop or friction thimble to control the force applied. It is important to not overdrive the spindle and introduce an error caused by disturbing the setup. It does not require a great amount of force to lift the setup. Care must be taken to apply the measure­ ment gently but firmly. Repeat the measurement three times to assure the repeatability and accuracy of the measurement. 3) Repeat Item 2) for all remaining gage block stacks throughout the full range of the UUT. Record all readings and verify that all readings are within tolerance limits. Adhere to manufacturers’ tolerances if different than the tolerance limits listed in Calibration Toler­ance Tables, Table 21a and Table 21b. Calibration of Thread Micrometers

Note: The Thread (Pitch) Micrometer uses a dedicated set of spindle and anvil contacts that measure the pitch diameter of a thread of a given pitch range. There are fixed anvil micrometers as well as interchangeable anvil/spindle sets. The interchangeable sets come with matched spindle/anvil sets that can be installed into the micrometer frame and then set to zero. Linearity Calibration (Fixed Pitch).—The fixed anvil micrometers usually do not have the capability to reset the zero indication other than the standard barrel adjustment. The calibration of these instruments is achieved by the measurement of master setting thread plug members that are selected to cover, as close as possible, 25, 50 and 75 percent of the range of the UUT. 1) Three calibrated Master Setting Thread Plug Gages are selected that cover 25, 50 and 75 percent of the measuring range of the UUT in the range of pitch that the UUT is designed for. (i.e., 8–13 pitch, 16–18 pitch, 18–24 pitch). 2) Mount the pitch micrometer in a micrometer stand to enable easier operation of the UUT and to provide better control of the standard. 3) Gently bring the Anvil and Spindle together to achieve a zero indication. If zero read­ ing is not exact, adjust the barrel so no error from zero exists. Repeat the zero indication to assure repeatability. 4) Open the micrometer anvil and spindle beyond the point of the first standard pitch diameter and seat the standard against the anvil. 5) Slowly close the micrometer until the spindle cone seats into the thread opposite the supported anvil. 6) Gently roll the standard up and down against the spindle cone to seek the highest point on the diameter. When this point is located, take a measurement of the pitch diameter. 7) Record the measurement and assure that the measurement is within the tolerance lim­its for the UUT. Note: UUT with the resolution of ±0.001 require a Class X master thread setting plug gage. UUT with a resolution of ±0.0001 require a class W master thread setting plug gage. 8) Remove the first standard and repeat measurement throughout the remainder of the master thread setting plug gages selected for the UUT. Interchangeable Anvil Pitch Micrometer Sets.—1) Interchangeable Anvil Pitch Micrometer sets include several sets of Anvil and Spindle Cone, and V Anvils. These sets are

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Machinery's Handbook, 31st Edition CALIBRATION OF THREAD MICROMETERS

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dedicated to a range of thread pitch. Most sets also include a set of flat anvil and spindle inserts that can be used to calibrate the micrometer exactly the same as a standard micrometer. If this is the case proceed as follows. 2) Select the flat anvil and spindle set and install. 3) Zero the anvil and spindle and assure repeatability. 4) Perform calibration as described in Linearity Test (Outside Micrometers), Item 3) on page 792 using the defined test points to verify the accuracy of the micrometer head. 5) Remove the flat anvil and spindle set and install the smallest (highest pitch) dedicated Anvil V and Spindle Cone set. Close anvil and spindle and re zero the micrometer to the new set. 6) Select one master thread setting plug gage for the pitch range, preferably at midrange of travel, and measure as described in Linearity Calibration (Fixed Pitch) section. Verify that the measurement is within the tolerance limits of the UUT. This measurement verified the accuracy of the flank angle of the cone and V anvil. This measurement can also be per­formed on an optical comparator. However, the physical measurement should be per­formed as the final step in the calibration of the interchangeable cone and anvil sets. 7) Repeat for all remaining Anvil V and Spindle Cone sets. Calibration of Inside Micrometers

Caliper Type Inside Micrometer.—A description of the calibration and equipment requirements for the Type II, Class 1, Inside Micrometer with Jaws are given in Table 15. Table 15. Calibration Requirements of the Type II, Class 1, Inside Micrometer with Jaws

UUT Characteristics

Calibration Description

Performance Specifications Range: 0.200 to 1.000 inch UUT Length 0.500 to 1.500 inch and Linearity 1.000 to 2.000 inch (or metric equivalent)

Test Method Comparative measurement of UUT reading to gage block test setup to test the basic length and micrometer head linearity.

Equipment Requirements

Description

Minimum Use Specifications

Calibration Equipment

Sub Item

Gage Block Range: ±0.05 to 4 inch L.S. Starrett Co. L.S. Starrett Co. Accuracy: ±12 min from stated value Set SS 36.A1 SS 81.A1 Calibration Process.—1) Set the UUT to indicate the minimum reading for the UUT using the ratchet or friction thimble. Assure that the reading indicates the minimum for the UUT model and range. If the reading is observed as off the minimum at this time, assure there is no foreign particle or burr on the anvil or spindle. An accurate zero indication must be established at this time as all other measurements will be affected by any error at zero. 2) The recommended size gage blocks used to test the micrometer spindle thread are very important. The recommended test points achieve the testing of the accuracy of the micrometer throughout the entire length and at points evenly distributed around the diam­ eter of the spindle thread. This verifies the accuracy of the lead error of the thread as well as identifying any sections that may have experienced excessive wear or damage. The recom­mended sizes are given in Table 16:

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Machinery's Handbook, 31st Edition CALIBRATION OF INSIDE MICROMETERS

(in.)

Size (mm)

797

Table 16. Recommended Test Points for Type II, Class 1, Inside Caliper Micrometer (in.)

Range

(mm)

1

25

0.200 to 1

5 to 25

11 ⁄2

30

0.500 to 11 ⁄2

5 to 30

2

50

1 to 2

25 to 50

Recommended Test Points (in.) (mm) 0.210, 0.420, 0.605, 0.815 5.10, 10.20, 15.30, 20.40 and 1.000 and 25.00 0.510, 0.720, 0.905, 1.115, 7.10, 12.20, 17.30, 23.40, and 1.500 and 30.00 1.210, 1.420, 1.605, 1.815, 25.10, 30.20, 35.30, and 2.000 40.40, and 50.00

Note:

a) The micrometer should be tested at zero and full scale.

b) Care must be taken when wringing gage blocks together to achieve the desired size. It is critical to clean the gage blocks thoroughly before wringing and assembling the end blocks to avoid damaging the superior finish of the standard.

3) Assemble the first gage block stack and measure it using the ratchet or friction thimble as described in the Linearity Test (Outside Micrometers) on page 792. Do not rely on feel as the correct and controlled loading of measuring force by the ratchet or friction thimble will give the most accurate and repeatable reading. Several clicks of the ratchet or applying force until the friction thimble slips and assuring the micrometer is gently seated on the gage block stack will give an accurate reading. Gently rock the micrometer to assure that the Nibs are parallel to the gage block end block faces. Repeat the measurement several times to assure the reading is repeatable. The reading must meet the tolerance limits as defined in Table 17, Tolerance column. Table 17. Tolerance Table: Type II, Class 1 Inside Micrometer with Jaws Size Range Tolerance (in.) (in.) (in.) 1 0.200 to 1 ±0.001 ±0.001 11 ⁄2 0.500 to 11 ⁄2 2 1 to 2 ±0.001 4) Repeat these steps for each remaining gage block stack and record readings for evalu­ ation or trend analysis. Follow all guidelines of the quality assurance programs in place in regards to recording and retaining all data collected and reporting OOT conditions. Gage Block Stack with End Blocks

0 9 8 7

20 0

Fig. 26. Measure Gage Block Stack Assembled with End Blocks

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Machinery's Handbook, 31st Edition CALIBRATION OF INSIDE MICROMETERS

798

Tubular Type Inside Micrometer.—A description of the calibration and equipment requirements for the tubular type inside micrometer are given in Table 18. Table 18. Calibration Requirements of Tubular Type Inside Micrometer

Test Instrument (TI) Characteristics Length End caps Extension Rods

Calibration Description

Performance Specifications

Range: All Accuracy: As listed in Tables 4, 5, and 6 Range: All Accuracy: ±0.00025 in Range: All Accuracy: As listed in Tables 4, 5, and 6

Test Method

Measured using a standard measuring machine or supermicrometer referenced to gage blocks

Equipment Requirements

Description Gage Block Set Gage Block Set Supermicrometer Standard Measuring Machine

Minimum Use Specifications

Range: 0.05 to 4 in Accuracy: ±12 min from stated value Range: 5 to 20 in Accuracy: ±5 min/in Range: 0 to 10.00 inch 0 to 250 mm Accuracy: 20 min Range: 0 to 148 inch 0 to 1200 mm Accuracy: 50 min

Calibration Equipment

L.S. Starrett Co. SC 36.A1 SC 36.MA1X L.S. Starrett Co. SS8.A1X

Alternate Selection

L.S. Starrett Co. SC 88.A1X SC 88.MA1X

Pratt & Whitney Model PC-250 Pratt & Whitney Model DMM-1219

Preliminary Operations.—1) UUT shall be brought into the calibration area at least 4 hours prior to beginning the calibration process. 2) Ensure that all working surfaces are clean and free from dust and dirt. 3) The UUT must be free from all rust, nicks and burrs that would affect the results of the measurements and operation of the UUT. 4) Linear measurements are affected by temperature and all measurements must be con­ ducted in a temperature controlled environment. The preferred environment is 68°F with no greater than ±1° per hour deviation from nominal. The temperature must have been sta­ble and within the preferred range for a minimum of 8 hours prior to beginning the calibra­tion soak time. Length Calibration Process.—Preferred Method: 1) Install the shortest extension rod available for the UUT set being calibrated. If the UUT has end caps (Type II Class 3 Style A) install just the end caps. This section will calibrate the micrometer head accuracy. Note: If the TI being calibrated is equipped with two end caps that can be removed and interchanged (Type II Class 3 Style A), the end caps must be checked for wear and must be the same length (nominal 0.250 inch). If the end caps differ in length by more than ±0.00025 inch (as measured on the Supermicrometer or Standard Measuring Machine, both TI end caps must be replaced. 2) Select a gage block of a length close to the nominal length of the UUT when set to zero on the micrometer head and the extension rod or end cap for minimum length indication is installed.

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Machinery's Handbook, 31st Edition CALIBRATION OF INSIDE MICROMETERS

799

3) For UUT with greater than 0.0001 inch (0.001 mm) resolution, gage block stacks that are wrung together to achieve the nominal setting dimension over 8 inches, must soak for one hour before establishing the setting dimension. This does not apply to single gage blocks used as setting standards. 4) Set the Reference of the supermicrometer or standard measuring machine to zero. 5) With the UUT set to zero, place into the measuring plane of the supermicrometer or standard measuring machine and measure the UUT. Assure that the UUT is supported on either a flatted round or adequate support to allow it to maintain alignment with the axis of measurement of the standard. 6) Verify that the measurement is within the tolerance limits for the UUT. Refer to the tolerance tables for the Type II Class 2 (Table 4), Type II Class 3 Style A (Table 5), and Type II Class 3 Style B (Table 6) for applicable tolerance limits. 7) Remove the UUT from the standard and set to the next test point as defined in Table 12 for the 1 inch (25 mm) micrometer head. 8) Repeat Items 5) and 6) for all remaining test points. Alternate Method: 1) Install the shortest extension rod or end caps and set UUT to zero on the micrometer head. Wring together appropriate size gage blocks to measure the zero indication assembly. 2) Wring or attach the Caliper Jaw End Blocks from the gage block accessories and assure the setup is square and true. If using the connecting rods, minimal force is required to secure the end blocks to the gage block stack. Note: For UUT with greater than 0.0001 inch resolution, gage block assemblies longer than 8.00 in (200 mm) will require 1 hour stabilization time before taking measurement. 3) Use the UUT to measure the gage block stack. Care must be taken to assure that the measurement is taken at the shortest distance between the caliper end block faces. 0 9 8 7

20 0

Gage Block Stack Fig. 27. Inside Micrometer Gage Block Stack Measurement

4) Repeat Items to 3) for each remaining test point. Measure each gage block setup and assure that the indication is within the tolerance limits as defined in the Table 4, Table 5, and Table 6 for the Type II Class 2, Type II Class 3 Style A, and Type II Class 3 Style B inside micrometers. Extension Rod Calibration.—Preferred Method: 1) Remove the shortest extension rod or one end cap used in the linearity calibration and replace with the next UUT extension rod in the set. 2) Ensure the Standard Measuring Machine has retained its reference. Verify the zero setting if necessary. The Supermicrometer will require re-mastering every 1 inch and can only be used up to 10 inches (generally). The standard measuring machine uses a continu­ ous measurement path but must be re-mastered every 8 inches. 3) For UUT with 0.0001 inch (0.001 mm) resolution, allow for 1 hour stabilization time for gage block stacks that are wrung to achieve nominal length over 8 inches (200 mm). 4) Using a flatted round or appropriate supporting device, mount the UUT into the stan­ dard. Assure the alignment of the measuring path and measure the UUT. 5) Confirm that the indication is within the tolerance limits as listed in the Table 4, Table 5, and Table 6 for the Type II Class 2, Type II Class 3 Style A, and Type II Class 3 Style B inside micrometer.

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Machinery's Handbook, 31st Edition CALIBRATION OF INSIDE MICROMETERS

800

6) Repeat Items to 5) for all remaining extension rods, sleeves and end caps in the set.

Alternate Method: 1) Remove the shortest extension rod or one end cap used in the lin­ earity calibration, and replace with the next UUT extension rod in the set. 2) Wring together a gage block stack to achieve the nominal desired length of the micrometer head at zero indication and the included extension rod. 3) Wring, or attach using the accessory attachment rods, the outside caliper jaws to the gage block stack. Assure stack is square and true. Minimal force is required to secure the outside caliper jaws to the gage block stack. 4) For UUT with 0.0001 inch (0.001 mm) resolution, allow for 1 hour stabilization time for gage block stacks that are wrung to achieve nominal length over 8 inches (200 mm). 5) Use the UUT to measure the gage block stack. 6) Confirm that the measured value is within the tolerance limits as listed in the Table 4, Table 5, and Table 6 for the Type II Class 2, Type II Class 3 Style A, and Type II Class 3 Style B inside micrometer. 7) Repeat Items to 6) for all remaining extension rods, sleeves and end caps in the set. Calibration of Vernier, Dial and Digital Calipers

Preliminary Operations.—1) Review and become familiar with the entire calibration procedure before beginning calibration process. Ensure familiarity with setup and opera­ tion of all standards involved in the procedure. 2) Always observe safe handling methods for all standards and exercise caution in the handling of all precision standards. 3) Ensure the work area is clean, well illuminated, and free from excessive drafts and humidity. 4) Ensure measuring surfaces are clean and free of nicks and burrs that could affect the accuracy of the measurements. Observe the Outside Measuring Jaws and Inside Measur­ ing Jaws for any possible damage and observe the unit for wear or deformation or any con­ dition that may affect the parallelism of the measuring surfaces. Close the measuring jaws completely and tighten the lock screw. Hold the UUT in front of a strong light source and verify that there are no burrs on the OD or ID measuring jaws. If any condition of concern is observed in regards to nicks or burrs along the beam, light stoning with a hard Arkansas stone can be used with care to remove these incursions. 5) Observe the condition of the rack along the full length of a dial caliper to assure there is no damage to the rack gear tooth or embedded chips or dirt that will affect or damage the movement. Look for bent or damaged gearing. Open and close the slide slowly and listen and feel for evidence or broken or damaged drive pinion gear. 6) Assure that the movement along the length of the beam does not bind or indicate the presence of any nicks or burrs along the beam. Assure the movement is not too loose. Observe the position of the gib slide screws that they have not been removed or loosened. 7) Bring the UUT (Unit Under Test) into the controlled environment a minimum of 4 hours prior to beginning the calibration process. Allow all standards and UUT to stabilize to common temperature before proceeding. Table 19. Equipment Requirements and Minimum Accuracy Specifications of Required Standards

Noun Gage Block Set Gage Block Set (Long Block)

Recommended Measurement Standards

Alternative

Starrett Webber SC 81.A1 Starrett Webber S2C 88.MA1

Mitutoyo 516-401-16 Mitutoyo 516-442-10

Starrett Webber SS 8.A1X Starrett Webber SS 8.MA1X

Mitutoyo 516-762-16 Mitutoyo 516-753-10

Minimum Use Specifications

Range: 0.050 to 4.000 inch 2.0 to 450 mm 5 to 84 inch 125 to 500 mm

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Machinery's Handbook, 31st Edition CALIBRATION OF VERNIER, DIAL AND DIGITAL CALIPERS

801

Table Table 19. (Continued) Equipment Requirements 19. Equipment Requirements and and Minimum Accuracy Accuracy Specifications Specifications of of Required Required Standards Standards Minimum Noun Precision Micrometer

Minimum Use Specifications Range: 0–1.0000 inch 0–25 mm

Recommended Measurement Standards

Alternative

Mitutoyo Series 293

Starrett No.3732 Precision Ground Pin or Ball ±40 in / 0.001 mm

Precision Pin or Ball

0.500 inch / 12 mm

Van Keuren Class XX

Surface Plate

Range: 24 x 36 Accuracy: Grade A

Starrett Crystal Pink EDP 80655

Mitutoyo 517-808

Note: Standards listed are recommended and are determined to meet the necessary accuracy requirements of the calibration process. Substitute or alternative standards may be used as long as they have a current in tolerance calibration status and have been determined to meet or exceed the minimum accuracy specifications as listed in Table 19. Note: Special care should be taken to clean and apply protective coating when handling gage blocks or other precision standards with bare hands. The use of clean cotton or vinyl gloves is recom­mended to prevent corrosive oils from damaging precision surfaces.

Vernier, Dial and Digital Caliper Calibration Process.—The temperature of the cali­ bration environment must be maintained at approximately 23° C with a fluctuation in tem­perature of no more than ±0.1° per hour. Table 20. Range and Tolerance Limits Range (inch)

Accuracy (inch)

Scale Zero

Set at Zero

±0.0005

Jaw Parallelism

0.00–0.250

±0.001 inch

ID Jaws (Nibs) Set at Scale Zero

0 to 6.000–0.250 0 to 9.000–0.300 0 to 12.000–0.300 0 to 24.000–0.300 0 to 36.000–0.500 0 to 48.000–0.500

Measured with Precision cylin­drical plug gage

(inch) +0.000 -0.005

Measured with precision micrometer

Feature

Outside (OD) Jaws

0–6.00 inch 6–24 inch 24–48 inch

ID Jaws (Knife Edge)

0–6.00 inch 6–24 inch 24–48 inch

Depth Rod

0.000–6.000 inch

Method Determined by observation of scale

±0.0010 inch ±0.0020 inch ±0.0030 inch

Determined at 4 equal points along the main scale by direct comparison with gage blocks

±0.0010 inch ±0.0020 inch ±0.0030 inch

Measured with gage blocks

±0.001 inch

Measured with gage blocks

Note: Metric equivalents apply to all tolerance limits stated in Table 20. All tolerances based on AFMETCAL, NAVAIR and GGG-C-111guidelines and specifications. Default to manufacturers specifications if different from those represented here.

Nib Calibration, Inside Measurement, Type 1.—Class 1, 2, and 3: The Total Nib Width (measured to the fourth place) shall be measured over both nibs with the jaws fully closed as shown in Fig. 28. The offset value shall be marked on the Inside Measurement side of the sliding jaw or main beam and on both sides of the main beam of the Type 1, Class 3.

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Machinery's Handbook, 31st Edition CALIBRATION OF VERNIER, DIAL AND DIGITAL CALIPERS a) Alternately the vernier scale can be adjusted to directly compensate for and deviation from nominal dimension of the nib. This can be accomplished by adjusting the inside measurement vernier scale to zero on a known standard. Measurement of the nib nominal dimension must still be performed as excessive wear to the nib will result in adjustment in excess of the available range of the vernier scale. b) Assemble a 1.000 inch gage block with accessory end blocks as shown in Fig. 30 of the Calibration of Inside Dimension Jaws (Nibs) section. c) Measure the gage block stack and adjust the vernier scale to zero at this known point. d) This method is applied to the Type 1 style caliper only.

Inside 1

5

Inside – 0.003 inch

Nominal 0.500 inch Reading 0.467 inch Nib Dimension - Measured with Precision Micrometer

Fig. 28. Nib Dimension Measurement Location

Calibration of Jaw Parallelism.—1) Slide jaws together and tighten the sliding jaw lock screw. Hold the UUT in front of a strong light source. Observe that no light is visible between the jaw measuring surfaces. Look carefully for evidence of burrs, damage to the measuring surfaces, excessive wear or bowing. a) If any burrs or damage appears gently stone the jaws with a white hard Arkansas stone. Only use a super fine grain stone for this operation. 2) Loosen the sliding jaw lock screw and open the jaws. Place the precision cylindrical plug gage or ball in between the jaws as close as possible to the beam and measure the stan­ dard. Observe the scale, dial, or digital reading. 3) Move the plug gage or ball to the opposite end of the measuring jaws (tip) and remea­ sure. The difference between the two readings must be within tolerance stated in Table 20. Range and Tolerance Limits for “Outside Jaws”. 4) If the readings are within tolerance, proceed to Calibration of Outside Jaws. If adjust­ ment is needed, go to Items 5) through 7), in Adjustment of Gib Slide Set Screws. Adjustment of Gib Slide Set Screws.—5) Using a small precision screwdriver loosen the gib slide set screws then retighten evenly with a slight pressure. a) It is important that both set screws are tightened evenly and a slight resistance is felt in the sliding jaw full range movement. 6) Operate the UUT several times over its full range to ensure that the sliding jaw can be moved smoothly over the full length of the beam.

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Machinery's Handbook, 31st Edition CALIBRATION OF VERNIER, DIAL AND DIGITAL CALIPERS

803

7) Repeat steps 1 through 3 to verify the Jaw Parallelism in within tolerance as stated in Table 2 for Outside Jaws. 8) If the readings are within tolerance, proceed to Calibration of Outside Jaws. If read­ ings are not within tolerance, record all readings as “As Found” and consult the end user to determine the course of action that they desire. Excessive damage or wear to the Jaws, Nibs or ID measuring contacts can result in a high degree of uncertainty in all measurements taken with the instrument and is cause for concern. Consideration of these facts should be addressed before proceeding. Calibration of Outside Jaws.—1) Slide the UUT jaws together and tighten the sliding jaw set screw. 2) Set zero for this calibration measurement. a) For vernier calipers close the jaws and observe the alignment of the zero indica­ tion on the main scale and vernier scale. b) For dial calipers, adjust the bezel for proper zero indication. c) For digital calipers, press the ZERO set button on the face of the UUT. Verify the digital indication is between -0.0005 and +0.0005. 3) Loosen the sliding jaw lock screw. Open and close the sliding jaw to verify repeatabil­ ity of the zero setting. a) For digital calipers, loosen the jaw set screw and open the calipers to a mid-point location. Lock the jaw set screw then press the zero set switch on the front of the instrument. Verify the digital indication is between -0.0005 and +0.0005. Close the slide jaw and zero the instrument. 4) Select the gage blocks needed to check the caliper at the test points listed below: a) For dial calipers, select the proper gage blocks to check the caliper at the 3 o’clock, 6 o’clock, 9 o’clock and 12 o’clock positions of the dial face in the first inch. (0.125, 0.550, 0.675, 1.000) b) For digital and vernier calipers, select gage blocks equal to 25, 50, 75, 100 percent of the first inch (0.250, 0.500, 0.750 and 1.000 inches). 5) Select the gage blocks needed to check the UUT at 25, 50, 75, and 100 percent of full scale.Place the UUT and all gage blocks on the surface plate for a minimum of 15 minutes to stabilize. The gage blocks should have both working surfaces exposed, as in Fig. 29.

90

0

80

10 20

70

30

0.750

60

50

40

Outside Jaws

0.500

0.250

End Blocks 1.000 4.000

2.000

3.000

Surface Plate Fig. 29. Gage Blocks and End Blocks Staged for Calibration of 6-inch Dial Caliper

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Machinery's Handbook, 31st Edition CALIBRATION OF VERNIER, DIAL AND DIGITAL CALIPERS

6) Open the outside jaws and measure the gage block that equates to the first test point defined in step 4. Ensure that the jaws are squarely placed with a slight pressure against the gage block measuring surfaces. Use the thumb wheel or fine adjust to apply gentle but steady pressure against the gage block. Rock the caliper gently to ensure that the OD mea­suring faces are seated squarely. Seek the smallest reading obtainable with gentle pressure. Record the value. 7) Repeat Item 6) for remaining test points defined in Item 4), parts a) and b), and Item 5). 8) If the outside jaws are within tolerance, proceed to the next section: Calibration of Inside Dimension Jaws (Nibs). If the outside jaws are found Out of Tolerance, continue through each calibration section and record all data as “As Found” prior to any further adjustments. Upon completion of procedure, report any out of tolerance conditions that may have been discovered. Any out of tolerance conditions must be addressed before returning the UUT to service. Calibration of Inside Dimension Jaws (Nibs).—1) Select the gage block needed to check the UUT at the first test point as defined above in Item 4) of Calibration of Outside Jaws. Clean the surface of the gage block and end blocks with lint free wipe and alcohol and assemble the Accessory End Blocks and assure they are secured in position snugly. 2) Close the caliper jaws and assure the dial or digital indication or vernier scale indicate zero. 3) Measure the dimension of the gage block as illustrated in Fig. 30. Use a gentle rocking motion to assure that the jaws are seated squarely. Seek the lowest reading as this is the true distance between the parallel faces of the end blocks. Note: Excessive force must be avoided as this can cause deformation of the UUT jaws and result in erroneous readings. A gentle consistent feel must be used to provide an accu­ rate reading. If too much force is applied or the ID measuring jaws are not in line and per­pendicular to the parallel planes of the end blocks, a larger and unrepeatable reading will result. Repeat all measurements several times to assure that you have a true reading.

60

50

40 30

70

10 20

80 90

0

1.000 4.000

0.750

3.000

0.500

2.000

0.250

End Blocks

Surface Plate Fig. 30. Inside Dimension Measurement of Gage Blocks using Accessory End Blocks

4) Repeat the set up and readings at the 3 o’clock, 6 o’clock, 9 o’clock and 12 o’clock positions of the dial face in the first inch. For vernier and digital calipers measure at 0.250, 0.500, 0.750 and 1.000 inches. Record all readings.

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Machinery's Handbook, 31st Edition CALIBRATION OF VERNIER, DIAL AND DIGITAL CALIPERS

805

5) Measure the remainder of the length of the UUT at 25, 50, 75, and 100 percent of full scale. Select the appropriate gage blocks and assemble as illustrated in Fig. 30. Measure each test point and record all readings. Calibration of Depth Rod.—1) Select a 1.000 inch gage block and position it on the sur­ face plate with the measuring surface against the plate. 2) Close the caliper jaws completely and place the end of the mail beam flat on the surface plate in a vertical position. Apply slight downward pressure bringing the depth rod in con­tact with the surface plate. Zero the dial or display at this point. a) Type 1 Class 1 calipers do not have depth measurement capability. b) Type 1 Class 2 have a depth rod that is directly tied to the Vernier Scale. If the Zero Reading is OOT (out of tolerance) at this point the depth rod will require ser­vice or replacement.

80

70

90

60

0

50 30

10 20

40

Gage Blocks

Surface Plate Fig. 31. Caliper Depth Rod Extended through Center Relief Hole of Gage Block.

3) Place the caliper depth rod through the center relief hole of the gage block and extend the rod to make contact with the surface plate as shown in Fig. 31. Assure that the end of the main beam is seated squarely on the surface of the gage block. Apply gently downward pressure with the thumb wheel until contact on the surface plate is felt. Using the thumb wheel, lift the depth rod and re-apply contact to repeat the measurement. Confirm the mea­surement repeats before accepting. a) Alternately, the depth rod can be extended down the outside of the gage block while in direct contact with the outer surface. The end of the main beam is squarely in contact with the surface of the gage block. This method can be used with a rectangular gage block that has no center relief hole or to assure a square contact between the end of the main beam and the depth rod. 4) If the depth rod is determined to be within tolerance, proceed to the next section, Calibration of Step Height. If the depth rod is found out of tolerance, continue through

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Machinery's Handbook, 31st Edition CALIBRATION OF VERNIER, DIAL AND DIGITAL CALIPERS

each cal­ibration section and record all data as “As Found” prior to any further action. Upon completion of procedure, report any out of tolerance conditions that may have been dis­covered. Any out of tolerance conditions must be addressed before returning the UUT to service. Calibration of Step Height.—1) Close the caliper jaws completely and set zero on the dial or display. 2) Place the caliper head downward on the surface plate as shown in Fig. 32.

80

70

90

60

0

50 30

10 20

40

1.000

Surface Plate Fig. 32. Step Height Measurement with Gage Block in Position behind Main Beam Engaging Step Height Measurement Surface

3) Open the caliper jaws to accommodate the measurement of a 1.000 inch gage block by the Step Height measuring surface on the back side of the moveable jaw. 4) Bring the Step Height measuring surface into contact with the surface of the gage block. Assure the back of the main beam is in direct contact with the outside of the gage block to establish a good perpendicular condition. 5) Apply gently downward pressure with the thumb wheel. 6) Repeat the measurement several times to confirm repeatability before accepting the measurement. 7) Record the measurement. There is no adjustment for this feature. 8) If the Step Height is determined to be within tolerance, remove and store all standards in their appropriate containers and apply any corrosion protection methods as required. If the step height is found Out of Tolerance, record all data as “As Found” prior to any further action. Upon completion of procedure, report any out of tolerance conditions that may have been discovered. Any out of tolerance conditions must be addressed before returning the UUT to service.

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Machinery's Handbook, 31st Edition CALIBRATION TOLERANCE TABLES

807

Calibration Tolerance Tables Table 21a. Calibration Tolerance Tables, English Measuring Faces

Size (in.)

1 2 3 4 5 6 7 8 9 10 11 12 13 to 18 19 to 24 25 to 30 31 to 36

Range (in.)

0–1 1–2 2–3 3–4 4–5 5–6 6–7 7–8 8–9 9–10 10–11 11–12 Per 1 inch " " "

Permissible Flexure (in.) 0.0001 0.0001 0.0001 0.00015 0.00015 0.00015 0.0002 0.0002 0.0003 0.0003 0.0003 0.0003 0.0004 0.0005 0.0006 0.0007

Flatness, max. (in.) 0.00005 0.00005 0.00005 0.00008 0.00008 0.00008 0.00008 0.00008 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001

Parallelism, max. (in.) 0.00005 0.00010 0.00015 0.0002 0.0002 0.0002 0.00025 0.00025 0.00025 0.0003 0.0003 0.0003 0.0004 0.0005 0.0006 0.0007

Indicated Measurement Error, max. (in.) 0.0001 0.00015 0.00015 0.0002 0.0002 0.0002 0.00025 0.00025 0.00025 0.0003 0.0003 0.0003 0.0004 0.0005 0.0006 0.0007

Table 21b. Calibration Tolerance Tables, Metric Measuring Faces

Size (mm) 13 25 50 75 100 125 150 175 200 225 250 275 300

Range (mm)

0–13 0–25 25–50 50–75 75–100 100–125 125–150 150–175 175–200 200–225 225–250 250–275 275–300

Permissible Flexure (mm) 0.0025 0.0025 0.0025 0.0025 0.004 0.004 0.004 0.005 0.005 0.005 0.008 0.008 0.008

Flatness, max. (mm) 0.001 0.0012 0.0012 0.0012 0.002 0.002 0.002 0.002 0.002 0.002 0.0025 0.0025 0.0025

Parallelism, max. (mm) 0.0012 0.0012 0.0025 0.004 0.005 0.005 0.005 0.006 0.006 0.006 0.0075 0.0075 0.0075

Indicated Measurement Error, max. (mm) 0.003 0.004 0.004 0.004 0.005 0.005 0.005 0.006 0.006 0.006 0.0075 0.0075 0.0075

Spindle/Anvil Alignment Error max. (in.) 0.002 0.003 0.0045 0.006 0.007 0.009 0.010 0.010 0.010 0.010 0.010 0.010 0.010 0.010 0.010 0.010

Spindle/Anvil Alignment Error, max. (mm) 0.03 0.05 0.08 0.11 0.15 0.19 0.23 0.25 0.25 0.25 0.25 0.25 0.25

Note: Indicated Measurement Error tolerances for UUT with 0.001 inch graduations (no vernier scale) are set to ±0.001 up to 30 inch, and ±0.002 inch from 31 to 36 inch. Tolerances of UUT over 36 inch are subject to manufacturer’s specification. Tolerance for UUT with a resolution of 50 µin shall be ±50µin for 0–1 inch; ±0.0001 inch for 1 through 4 inches; ±0.0002 inch for 4 three 9 inches; ±0.0003 inch for 9 three 12 inches; ±0.0004 inch for 12 to 18 inches; ±0.0005 for 18 to 24 inches; ±0.0006 for 24 to 30 inches; and ±0.0007 for 30 to 36 inches. All metric equivalents shall apply to UUT of this resolution. Metric Scale UUT with 0.01 mm graduations are set at ±0.01 mm for up to 300 mm. Tolerances above 300 mm are subject to manufacturers specifications.

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808

Machinery's Handbook, 31st Edition Surface Texture SURFACE TEXTURE American National Standard Surface Texture (Surface Roughness, Waviness, and Lay)

American National Standard ANSI/ASME B46.1-2009 is concerned with the geometric irregularities of surfaces of solid materials, physical specimens for gaging roughness, and the characteristics of stylus instrumentation for measuring roughness. The standard defines surface texture and its constituents: roughness, waviness, lay, and flaws. A set of symbols for drawings, specifications, and reports is established. To ensure a uniform basis for measurements, the standard also provides specifications for Precision Reference Spec­imens, and Roughness Comparison Specimens, and establishes requirements for stylus-type instruments. The standard is not concerned with luster, appearance, color, corrosion resistance, wear resistance, hardness, subsurface microstructure, surface integrity, or many other characteristics that may govern considerations in specific applications. The standard is expressed in SI metric units, but US customary units may be used with­ out prejudice. The standard does not define the degrees of surface roughness and waviness or type of lay suitable for specific purposes, nor does it specify the means by which any degree of such irregularities may be obtained or produced. However, criteria for selection of surface qualities and information on instrument techniques and methods of producing, controlling and inspecting surfaces are included in Appendixes attached to the standard. The Appendix sections are not considered a part of the standard: they are included for clar­ification or information purposes only. Surfaces, in general, are very complex in character. The standard deals only with the height, width, and direction of surface irregularities because these characteristics are of practical importance in specific applications. Surface texture designations as delineated in this standard may not be a sufficient index to performance. Other part characteristics such as dimensional and geometrical relationships, material, metallurgy, and stress must also be controlled. Definitions of Terms Relating to the Surfaces of Solid Materials.—The terms and rat­ings in the standard relate to surfaces produced by such means as abrading, casting, coat­ing, cutting, etching, plastic deformation, sintering, wear, and erosion. Error of form is considered to be that deviation from the nominal surface caused by errors in machine tool ways, guides, insecure clamping or incorrect alignment of the work­piece or wear, none of which are included in surface texture. Out-of-roundness and out-of-flatness are examples of errors of form. See ANSI/ASME B89.3.1-1972 (R2003) for measurement of out-of-roundness. Flaws are unintentional, unexpected, and unwanted interruptions in the topography typ­ical of a part surface and are defined as such only when agreed upon by buyer and seller. If flaws are defined, the surface should be inspected specifically to determine whether flaws are present, and rejected or accepted prior to performing final surface roughness measure­ments. If defined flaws are not present, or if flaws are not defined, then interruptions in the part surface may be included in roughness measurements. Lay is the direction of the predominant surface pattern, ordinarily determined by the pro­duction method used. Roughness consists of the finer irregularities of the surface texture, usually including those irregularities that result from the inherent action of the production process. These irregularities are considered to include traverse feed marks and other irregularities within the limits of the roughness sampling length. Surface is the boundary of an object that separates that object from another object, sub­ stance or space. Surface, measured is the real surface obtained by instrumental or other means.

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809

Flaw

Lay

Waviness Spacing

Waviness Height

Roughness Average — Ra

Peaks

Valleys Mean Line

Roughness Spacing

Fig. 1. Pictorial Display of Surface Characteristics

Surface, nominal is the intended surface contour (exclusive of any intended surface roughness), the shape and extent of which is usually shown and dimensioned on a drawing or descriptive specification. Surface, real is the actual boundary of the object. Manufacturing processes determine its deviation from the nominal surface. Surface texture is repetitive or random deviations from the real surface that forms the three-dimensional topography of the surface. Surface texture includes roughness, wavi­ ness, lay and flaws. Fig. 1 is an example of a unidirectional lay surface. Roughness and waviness parallel to the lay are not represented in the expanded views. Waviness is the more widely spaced component of surface texture. Unless otherwise noted, waviness includes all irregularities whose spacing is greater than the roughness sampling length and less than the waviness sampling length. Waviness may result from

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810

such factors as machine or work deflections, vibration, chatter, heat treatment, or warping strains. Roughness may be considered as being superposed on a ‘wavy’ surface. Definitions of Terms Relating to the Measurement of Surface Texture.—Terms regarding surface texture pertain to the geometric irregularities of surfaces and include roughness, waviness and lay. Profile is the contour of the surface in a plane measured normal, or perpendicular, to the surface, unless another angle is specified. Graphical centerline. See Mean Line. Height (z) is considered to be those measurements of the profile in a direction normal, or perpendicular, to the nominal profile. For digital instruments, the profile Z(x) is approxi­ mated by a set of digitized values. Height parameters are expressed in micrometers (μm). Height range (z) is the maximum peak-to-valley surface height that can be detected accurately with the instrument. It is measurement normal, or perpendicular, to the nominal profile and is another key specification. Mean line (M) is the line about which deviations are measured and is parallel to the general direction of the profile within the limits of the sampling length. See Fig. 2. The mean line may be determined in one of two ways. The filtered mean line is the centerline established by the selected cutoff and its associated circuitry in an electronic roughness average measuring instrument. The least squares mean line is formed by the nominal pro­file, but by dividing into selected lengths the sum of the squares of the deviations minimizes the deviation from the nominal form. The form of the nominal profile could be a curve or a straight line. Peak is the point of maximum height on that portion of a profile that lies above the mean line and between two intersections of the profile with the mean line. Profile measured is a representation of the real profile obtained by instrumental or other means. When the measured profile is a graphical representation, it will usually be distorted through the use of different vertical and horizontal magnifications but shall otherwise be as faithful to the profile as technically possible. Profile, modified is the measured profile where filter mechanisms (including the instru­ ment datum) are used to minimize certain surface texture characteristics and emphasize others. Instrument users apply profile modifications typically to differentiate surface roughness from surface waviness. Profile, nominal is the profile of the nominal surface; it is the intended profile (exclusive of any intended roughness profile). Profile is usually drawn in an x-z coordinate system. See Fig. 2. Measure Profile

Z

X

Nominal Profile

Fig. 2. Nominal and Measured Profiles

Profile, real is the profile of the real surface. Profile, total is the measured profile where the heights and spacing may be amplified dif­ferently, but otherwise no filtering takes place. Roughness profile is obtained by filtering out the longer wavelengths characteristic of waviness. Roughness spacing is the average spacing between adjacent peaks of the measured pro­ file within the roughness sampling length.

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Roughness topography is the modified topography obtained by filtering out the longer wavelengths of waviness and form error. Sampling length is the nominal spacing within which a surface characteristic is deter­ mined. The range of sampling lengths is a key specification of a measuring instrument. Spacing is the distance between specified points on the profile measured parallel to the nominal profile. Spatial (x) resolution is the smallest wavelength that can be resolved to 50 percent of the actual amplitude. This also is a key specification of a measuring instrument. System height resolution is the minimum height that can be distinguished from back­ ground noise of the measurement instrument. Background noise values can be determined by measuring approximate rms roughness of a sample surface where actual roughness is significantly less than the background noise of the measuring instrument. It is a key instru­mentation specification. Topography is the three-dimensional representation of geometric surface irregularities. Topography, measured is the three-dimensional representation of geometric surface irregularities obtained by measurement. Topography, modified is the three-dimensional representation of geometric surface irregularities obtained by measurement but filtered to minimize certain surface character­istics and accentuate others. Valley is the point of maximum depth on that portion of a profile that lies below the mean line and between two intersections of the profile with the mean line. Waviness, evaluation length (L), is the length within which waviness parameters are determined. Waviness, long-wavelength cutoff (lcw) is the spatial wavelength above which the undula­tions of waviness profile are removed to identify form parameters. A digital Gaussian filter can be used to separate form error from waviness, but its use must be specified. Waviness profile is obtained by filtering out the shorter roughness wavelengths charac­ teristic of roughness and the longer wavelengths associated with the part form parameters. Waviness sampling length is a concept no longer used. See waviness long-wavelength cutoff and waviness evaluation length. Waviness short-wavelength cutoff (lsw) is the spatial wavelength below which rough­ ness parameters are removed by electrical or digital filters. Waviness topography is the modified topography obtained by filtering out the shorter wavelengths of roughness and the longer wavelengths associated with form error. Waviness spacing is the average spacing between adjacent peaks of the measured profile within the waviness sampling length. Sampling Lengths.—Sampling length is the normal interval for a single value of a sur­face parameter. Generally it is the longest spatial wavelength to be included in the profile measurement. Range of sampling lengths is an important specification for a measuring instrument. Sampling Length

l

l

l

l

l

Evaluation Length, L

Traverse Length Fig. 3. Traverse Length

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Roughness sampling length (l) is the sampling length within which the roughness aver­age is determined. This length is chosen to separate the profile irregularities des­ ignated as roughness from those irregularities designated as waviness. It is different from evaluation length (L) and the traversing length. See Fig. 3. Evaluation length (L) is the length the surface characteristics are evaluated. The evalua­ tion length is a key specification of a measuring instrument. Traversing length is the profile length traversed to establish a representative evaluation length. It is always longer than the evaluation length. See Section 4.4.4 of ANSI/ASME B46.1-2009 for values that should be used for different types of measurements. Cutoff is the electrical response characteristic of the measuring instrument which is selected to limit the spacing of the surface irregularities to be included in the assessment of surface texture. Cutoff is rated in millimeters. In most electrical averaging instruments, the cutoff can be user-selected and is a characteristic of the instrument rather than of the surface being measured. In specifying the cutoff, care must be taken to choose a value that will include all the surface irregularities to be assessed. Waviness sampling length (l) is a concept no longer used. See waviness long-wavelength cutoff and waviness evaluation length. Roughness Parameters.—Roughness refers to the fine irregularities of the surface texture resulting from the production process or material condition. Roughness average (Ra or Ra), also known as arithmetic average (AA), is the arithmetic aver­age of the absolute values of the measured profile height deviations divided by the evalua­tion length, L. This is shown as the shaded area of Fig. 4 and generally includes sampling lengths or cutoffs. For graphical determinations of roughness average, the height devia­tions are measured normal, or perpendicular, to the chart center line. Y'

Mean Line

X

a b

c

d

e

f

g

h

i

j

k

l

m n

o

p

q

r

s

t u

v

w

X'

Y

Fig. 4.

Roughness average is expressed in micrometers (μm). A micrometer is one millionth of a meter (0.000001 meter). A microinch (μin) is one millionth of an inch (0.000001 inch). One microinch equals 0.0254 micrometer (1 μin. = 0.0254 μm). Roughness average (Ra) value from continuously averaging meter reading may be made of readings from stylus-type instruments of the continuously averaging type. To ensure uniform interpretation, it should be understood that the reading that is considered significant is the mean reading around which the needle tends to dwell or fluctuate with a small amplitude. Roughness is also indicated by the root-mean-square (rms) average, which is the square root of the average value squared, within the evaluation length and measured from the mean line shown in Fig. 4, expressed in micrometers. A roughness-measuring instrument calibrated for rms average usually reads about 11 percent higher than an instrument cali­ brated for arithmetical average. Such instruments usually can be recalibrated to read arith­metical average. Some manufacturers consider the difference between rms and AA to be small enough that rms on a drawing may be read as AA for many purposes.

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Roughness evaluation length (L), for statistical purposes should, whenever possible, consist of five sampling lengths (l). Use of other than five sampling lengths must be clearly indicated. Waviness Parameters.—Waviness is the more widely spaced component of surface tex­ ture. Roughness may be thought of as superimposed on waviness. Waviness height (Wt) is the peak-to-valley height of the modified profile with roughness and part form errors removed by filtering, smoothing or other means. This value is typi­ cally three or more times the roughness average. The measurement is taken normal, or per­pendicular, to the nominal profile within the limits of the waviness sampling length. Waviness evaluation length (Lw) is the evaluation length required to determine waviness parameters. For waviness, the sampling length concept is no longer used. Rather, only waviness evaluation length (Lw) and waviness long-wavelength cutoff (lew) are defined. For better statistics, the waviness evaluation length should be several times the waviness long-wavelength cutoff. Relation of Surface Roughness to Tolerances.—Because the measurement of surface roughness involves the determination of the average linear deviation of the measured sur­face from the nominal surface, there is a direct relationship between the dimensional toler­ance on a part and the permissible surface roughness. It is evident that a requirement for the accurate measurement of a dimension is that the variations introduced by surface rough­ness should not exceed the dimensional tolerances. If this is not the case, the measurement of the dimension will be subject to an uncertainty greater than the required tolerance, as illustrated in Fig. 5. Roughness Height

Roughness Mean Line

Profile Height

Uncertainty in Measurement

Roughness Mean Line

Roughness Height

Profile Height

Fig. 5.

The standard method of measuring surface roughness involves the determination of the average deviation from the mean surface. On most surfaces the total profile height of the surface roughness (peak-to-valley height) will be approximately four times (43) the mea­sured average surface roughness. This factor will vary somewhat with the character of the surface under consideration, but the value of four may be used to establish approximate profile heights. From these considerations it follows that if the arithmetical average value of surface roughness specified on a part exceeds one eighth of the dimensional tolerance, the whole tolerance will be taken up by the roughness height. In most cases, a smaller roughness specification than this will be found; but, on parts where very small dimensional tolerances are given, it is necessary to specify a suitably small surface roughness so useful dimen­sional measurements can be made. The tables on pages 648 and 674 show the rela­tions between machining processes and working tolerances. Values for surface roughness produced by common processing methods are shown in Table 1. The ability of a processing operation to produce a specific surface roughness depends on many factors. For example, in surface grinding, the final surface depends on the peripheral speed of the wheel, the speed of the traverse, the rate of feed, the grit size, bonding material and state of dress of the wheel, the amount and type of lubrication at the

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Machinery's Handbook, 31st Edition Surface Texture

point of cutting, and the mechanical properties of the piece being ground. A small change in any of the above factors can have a marked effect on the surface produced. Table 1. Surface Roughness Produced by Common Production Methods Process

Roughness Average, Ra - Micrometers μm (Microinches μin) 50 25 12.5 6.3 3.2 1.6 0.80 0.40 0.20 (2000) (1000) (500) (250) (125) (63) (32) (16) (8)

Flame Cutting Snagging Sawing Planing, Shaping Drilling Chemical Milling Elect. Discharge Mach. Milling Broaching Reaming Electron Beam Laser Electro-Chemical Boring, Turning Barrel Finishing Electrolytic Grinding Roller Burnishing Grinding Honing Electro-Polish Polishing Lapping Superfinishing Sand Casting Hot Rolling Forging Perm. Mold Casting Investment Casting Extruding Cold Rolling, Drawing Die Casting The ranges shown above are typical of the processes listed. Higher or lower values may be obtained under special conditions.

KEY

0.10 (4)

0.05 (2)

0.025 (1)

0.012 (0.5)

Average Application Less Frequent Application

Instrumentation for Surface Texture Measurement.—Instrumentation used for mea­ sure­ment of surface texture, including roughness and waviness generally falls into six types. These include: Type I, Profiling Contact Skidless Instruments: Used for very smooth to very rough sur­ faces. Used for roughness and may measure waviness. Can generate filtered or unfiltered profiles and may have a selection of filters and parameters for data analysis. Examples include: 1) skidless stylus-type with LVDT (linear variable differential transformer) verti­cal trans­ducers; 2) skidless-type using an interferometric transducer; 3) skidless stylus-type using capacitance transducer. Type II, Profiling Non-contact Instruments: Capable of full profiling or topographical analysis. Non-contact operation may be advantageous for softness but may vary with sam­ple type and reflectivity. Can generate filtered or unfiltered profiles but may have diffi­ culty with steeply inclined surfaces. Examples include: 1) interferometric microscope; 2) optical focus sending; 3) Nomarski differential profiling; 4) laser triangulation; 5) scan­ning electron microscope (SEM) stereoscopy; 6) confocal optical microscope. Type III, Scanned Probe Microscope: Features high spatial resolution (at or near the atomic scale) but area of measurement may be limited. Examples include: 1) scanning tun­neling microscope (STM) and 2) atomic force microscope (AFM).

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815

Type IV, Profiling Contact Skidded Instruments: Uses a skid as a datum to eliminate lon­ ger wavelengths; thus cannot be used for waviness or errors of form. May have a selection of filters and parameters and generates an output recording of filtered and skid-modified profiles. Examples include: 1) skidded, stylus-type with LVDT vertical measuring trans­ ducer and 2) fringe-field capacitance (FFC) transducer.

Type V, Skidded Instruments with Parameters Only: Uses a skid as a datum to eliminate longer wavelengths; thus cannot be used for waviness or errors of form. Does not generate a profile. Filters are typically 2RC type and generate Ra, but other parameters may be avail­able. Examples include: 1) skidded, stylus-type with piezoelectric measuring transducer and 2) skidded, stylus-type with moving coil measuring transducer.

Type VI, Area Averaging Methods: Used to measure averaged parameters over defined areas, but do not generate profiles. Examples include: 1) parallel plate capacitance (PPC) method; 2) total integrated scatter (TIS); 3) angle resolved scatter (ARS)/bi-directional reflectance distribution function (BRDF).

Selecting Cutoff for Roughness Measurements.—In general, surfaces will contain irregularities with a large range of widths. Surface texture instruments are designed to respond only to irregularity spacings less than a given value, called cutoff. In some cases, such as surfaces in which actual contact area with a mating surface is important, the largest convenient cutoff will be used. In other cases, such as surfaces subject to fatigue failure, only the irregularities of small width will be important, and more significant values will be obtained when a short cutoff is used. In still other cases, such as identifying chatter marks on machined surfaces, information is needed on only the widely space irregularities. For such measurements, a large cutoff value and a larger radius stylus should be used.

The effect of variation in cutoff can be understood better by reference to Fig. 6. The pro­ file at the top is the true movement of a stylus on a surface having a roughness spacing of about 1 mm and the profiles below are interpretations of the same surface with cutoff value settings of 0.8 mm, 0.25 mm and 0.08 mm, respectively. It can be seen that the trace based on 0.8 mm cutoff includes most of the coarse irregularities and all of the fine irregularities of the surface. The trace based on 0.25 mm excludes the coarser irregularities but includes the fine and medium fine. The trace based on 0.08 mm cutoff includes only the very fine irregularities. In this example, the effect of reducing the cutoff has been to reduce the roughness average indication. However, had the surface been made up only of irregulari­ ties as fine as those of the bottom trace, the roughness average values would have been the same for all three cutoff settings.

In other words, all irregularities having a spacing less than the value of the cutoff used are included in a measurement. Obviously, if the cutoff value is too small to include coarser irregularities of a surface, the measurements will not agree with those taken with a larger cutoff. For this reason, care must be taken to choose a cutoff value which will include all of the surface irregularities it is desired to assess.

To become proficient in the use of continuously averaging stylus-type instruments, the inspector or machine operator must realize that for uniform interpretation the reading considered significant is the mean reading around which the needle tends to dwell or fluctuate under small amplitude.

Drawing Practices for Surface Texture Symbols.—American National Standard ANSI/ASME Y14.36-2018 establishes the method to designate symbolic con­trols for surface texture of solid materials. It includes methods for controlling roughness, waviness, and lay, and provides a set of symbols for use on drawings, specifications, or other documents. The standard is expressed in SI metric units, but US customary units may be used without prejudice. Units used (metric or non-metric) should be consistent with the other units used on the drawing or documents. Approximate non-metric equiva­lents are shown for reference.

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Machinery's Handbook, 31st Edition Surface Texture Measured Profile Without Electrical Filtering 1 mm

With 0.8 mm Cutoff = 3.5-4.2 m Ra

25 m

With 0.25 mm Cutoff = 1.8-2.2 m Ra

With 0.08mm Cutoff = 0.95-1.05 m Ra

Fig. 6. Effects of Various Cutoff Values

Surface Texture Symbol.—The symbol used to designate control of surface irregulari­ ties is shown in Fig. 7b and Fig. 7d. Where surface texture values other than roughness average are specified, the symbol must be drawn with the horizontal extension as shown in Fig. 7f. Use of Surface Texture Symbols: When required from a functional standpoint, the desired surface characteristics should be specified. Where no surface texture control is specified, the surface produced by normal manufacturing methods is satisfactory provided it is within the limits of size (and form) specified in accordance with ASME Y14.5-2018, Dimensioning and Tolerancing. It is considered good practice to always specify some maximum value, either specifically or by default (for example, in the manner of the note shown in Fig. 8 on page 818). Material Removal Required or Prohibited: The surface texture symbol is modified when necessary to require or prohibit removal of material. When it is necessary to indicate that a surface must be produced by removal of material by machining, specify the symbol shown in Fig. 7b. When required, the amount of material to be removed is specified, as shown in Fig. 7c, in millimeters for metric drawings and in inches for nonmetric drawings. Tolerance for material removal may be added to the basic value shown or specified in a general note. When it is necessary to indicate that a surface must be produced without material removal, specify the machining prohibited symbol as shown in Fig. 7d. Proportions of Surface Texture Symbols: The recommended proportions for drawing the surface texture symbol are shown in Fig. 7f. The letter height and line width should be the same as those for dimensions and dimension lines.

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817

Surface Texture Symbols and Construction

Symbol

Meaning

Basic Surface Texture Symbol. Surface may be produced by any method except when the bar or circle (Fig. 7b or Fig. 7d) is specified.

Fig. 7a.

Material Removal By Machining Is Required. The horizontal bar indicates that material removal by machining is required to produce the surface and that material must be provided for that purpose.

Fig. 7b.

Material Removal Allowance. The number indicates the amount of stock to be removed by machining in millimeters (or inches). Tolerances may be added to the basic value shown or in general note.

3.5

Fig. 7c.

Fig. 7d.

Material Removal Prohibited. The circle in the V-shape indicates that the surface must be produced by processes such as casting, forging, hot finishing, cold fin­ishing, die casting, powder metallurgy or injection molding without subsequent removal of material.

Fig. 7e.

Surface Texture Symbol. To be used when any surface characteristics are spec­ified above the horizontal line or the right of the symbol. Surface may be pro­duced by any method except when the bar or circle (Fig. 7b and Fig. 7d) is specified.

3X

3 Approx.

1.5 X

OO 60°

O.OO

OO OO

3X

1.5 X

60°

Letter Height = X Fig. 7f.

Applying Surface Texture Symbols.—The point of the symbol should be on a line repre­ senting the surface, an extension line of the surface, or a leader line directed to the surface, or to an extension line. The symbol may be specified following a diameter dimension. Although ASME Y14.5-2018, “Dimensioning and Tolerancing,” specifies that normally all textual dimensions and notes should be read from the bottom of the drawing, the surface texture symbol itself with its textual values may be rotated as required. Regardless, the long leg (and extension) must be to the right as the symbol is read. For parts requiring extensive and uniform surface roughness control, a general note may be added to the draw­ing that applies to each surface texture symbol specified without values as shown in Fig. 8. When the symbol is used with a dimension, it affects the entire surface defined by the dimension. Areas of transition, such as chamfers and fillets, shall conform with the rough­ est adjacent finished area unless otherwise indicated. Surface texture values, unless otherwise specified, apply to the complete surface. Draw­ ings or specifications for plated or coated parts shall indicate whether the surface texture values apply before plating, after plating, or both before and after plating. Only those values required to specify and verify the required texture characteristics should be included in the symbol. Values should be in metric units for metric drawing and non-metric units for non-metric drawings. Minority units on dual dimensioned drawings are enclosed in brackets.

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1.6

XX 1.6

1.6

Machinery's Handbook, 31st Edition Surface Texture

818

Unless Otherwise Specified: All Surfaces 3.2

Fig. 8. Application of Surface Texture Symbols

Roughness and waviness measurements, unless otherwise specified, apply in a direction which gives the maximum reading, generally across the lay. Cutoff or Roughness Sampling Length, (l): Standard values are listed in Table 2. When no value is specified, the value 0.8 mm (0.030 in.) applies. Table 2. Standard Roughness Sampling Length (Cutoff) Values mm

in.

0.08

0.003

0.80

0.030

0.25

0.010

mm

in.

2.5

0.1

25.0

1.0

8.0

0.3

Roughness Average (Ra or Ra): The preferred series of specified roughness average values is given in Table 3. Table 3. Preferred Series Roughness Average (Ra) Values μm 0.012 0.025a 0.050a 0.075a 0.10a 0.125 0.15 0.20a 0.25 0.32

μin 0.5 1a 2a 3 4a 5 6 8a 10 13

μm 0.40a 0.50 0.63 0.80a 1.00 1.25 1.60a 2.0 2.5 3.2a

μin 16a 20 25 32a 40 50 63a 80 100 125a

μm 4.0 5.0 6.3a 8.0 10.0 12.5a 15 20 25a …

μin 160 200 250a 320 400 500a 600 800 1000a …

a Recommended

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Waviness Height (Wt): The preferred series of maximum waviness height values is listed in Table 4. Waviness height is not currently shown in US or ISO Standards. It is included here to follow present industry practice in the United States. Table 4. Preferred Series Maximum Waviness Height Values mm 0.0005 0.0008 0.0012 0.0020 0.0025 0.005

inch 0.00002 0.00003 0.00005 0.00008 0.0001 0.0002

mm 0.008 0.012 0.020 0.025 0.05 0.08

inch 0.0003 0.0005 0.0008 0.001 0.002 0.003

mm 0.12 0.20 0.25 0.38 0.50 0.80

inch 0.005 0.008 0.010 0.015 0.020 0.030

Lay: Symbols for designating the direction of lay are shown and interpreted in Table 5.

Example Designations.—Table 6 illustrates examples of designations of roughness, waviness, and lay by insertion of values in appropriate positions relative to the symbol.

Where surface roughness control of several operations is required within a given area or on a given surface, surface qualities may be designated, as in Fig. 9a. If a surface must be produced by one particular process or a series of processes, they should be specified as shown in Fig. 9b. Where special requirements are needed on a designated surface, a note should be added at the symbol giving the requirements and the area involved. An example is illustrated in Fig. 9c.

Surface Texture of Castings.—Surface characteristics should not be controlled on a drawing or specification unless such control is essential to functional performance or appearance of the product. Imposition of such restrictions when unnecessary may increase production costs and in any event will serve to lessen the emphasis on the control specified for important surfaces. Surface characteristics of castings should never be considered on the same basis as machined surfaces. Castings are characterized by random distribution of nondirectional deviations from the nominal surface. Surfaces of castings rarely need control beyond that provided by the production method necessary to meet dimensional requirements. Comparison specimens are frequently used for evaluating surfaces having specific functional requirements. Surface texture control should not be specified unless required for appearance or function of the surface. Specifi­ cation of such requirements may increase cost to the user.

Engineers should recognize that different areas of the same castings may have different surface textures. It is recommended that specifications of the surface be limited to defined areas of the casting. Practicality of and methods of determining that a casting’s surface tex­ture meets the specification shall be coordinated with the producer. The Society of Auto­motive Engineers standard J435 “Automotive Steel Castings” describes methods of evaluating steel casting surface texture used in the automotive and related industries. Metric Dimensions on Drawings.—The length units of the metric system most generally used in connection with any work relating to mechanical engineering are the meter (39.37 inches) and the millimeter (0.03937 inch). One meter equals 1000 millime­ters. On mechanical drawings, all dimensions are generally given in millimeters, no matter how large the dimensions may be. In fact, dimensions of such machines as locomotives and large electrical apparatus are given exclusively in millimeters. This practice is adopted to avoid mistakes due to misplacing decimal points, or misreading dimensions as when other units are used as well. When dimensions are given in millimeters, many of them can be given without resorting to decimal points, as a millimeter is only a little more than 1 ⁄32 inch. Only dimensions of precision need be given in decimals of a millimeter; such dimen­sions are generally given in hundredths of a millimeter—for example, 0.02 millimeter, which is equal to 0.0008 inch. As 0.01 millimeter is equal to 0.0004 inch, dimensions are seldom given with greater accuracy than to hundredths of a millimeter.

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Machinery's Handbook, 31st Edition Surface Texture

820 Lay Symbol

Table 5. Lay Symbols Meaning

Example Showing Direction of Tool Marks

Lay approximately parallel to the line rep­resenting the surface to which the symbol is applied.

Lay approximately perpendicular to the line representing the surface to which the symbol is applied.

X

Lay angular in both directions to line repre­senting the surface to which the symbol is applied.

M

Lay multidirectional.

C

Lay approximately circular relative to the center of the surface to which the symbol is applied.

R

P

Lay approximately radial relative to the center of the surface to which the symbol is applied.

Lay particulate, nondirectional, or protu­berant.

X

M

C

R

P

Scales of Metric Drawings: Drawings made to the metric system are not made to scales of ⁄2  , 1 ⁄4  , 1 ⁄8  , etc., as with drawings made to the English system. If the object cannot be drawn full size, it may be drawn 1 ⁄2  , 1 ⁄5  , 1 ⁄10   , 1 ⁄20  , 1 ⁄50   , 1 ⁄100   , 1 ⁄200   , 1 ⁄500   , or 1 ⁄1000 size. If the object is too small and has to be drawn larger, it is drawn 2, 5, or 10 times its actual size. 1

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Machinery's Handbook, 31st Edition Surface Texture

821

Table 6. Application of Surface Texture Values to Symbol ANSI B46.1-1978

Roughness average rating is placed at the left of the long leg. The specification of only one rating shall indicate the maximum value, and any lesser value shall be acceptable. Specify in micrometers (microinch).

1.6

1.6

3.5

1.6 0.8

The specification of maximum and minimum roughness average values indicates permissible range of roughness. Specify in micrometers (microinch).

1.6

0.8

Removal of material is prohibited.

0.005-5 Maximum waviness height rating is the first rating place above the horizontal

extension. Any lesser rating shall be acceptable. Specify in millimeters (inch). Maximum waviness spacing rating is the second rating placed above the horizon­tal extension and to the right of the waviness height rating. Any lesser rating shall be acceptable. Specify in millimeters (inch).

0.8

0.8 0.8

Material removal by machining is required to produce the surface. The basic amount of stock provided for material removal is specified at the left of the short leg of the symbol. Specify in millimeters (inch).

Lay designation is indicated by the lay symbol placed at the right of the long leg.

2.5 0.5

Roughness sampling length or cutoff rating is placed below the horizontal exten­sion. When no value is shown, 0.80 mm (0.030 inch) applies. Specify in millime­ters (inch). Where required, maximum roughness spacing shall be placed at the right of the lay symbol. Any lesser rating shall be acceptable. Specify in millimeters (inch).

Examples of Special Designations XX.XX ± X.XX XX.XX ± X.XX

XX

XX.XX ± X.XX

XX

Fig. 9a.

Mill

J

Grind

0.8

Lap

Fig. 9b.

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Width J

Fig. 9c.

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Machinery's Handbook, 31st Edition ISO Surface Finish Standards

822

ISO Surface Finish Standards ISO surface finish standards are comprised of numerous individual standards that, taken as a whole, form a set of standards roughly comparable in scope to American National Standard ANSI/ASME Y14.36.

ISO Surface Finish (ISO 1302).—The primary standard dealing with surface finish, ISO 1302:2002 is concerned with the methods of specifying surface texture symbology and additional indications on engineering drawings. The parameters in ISO surface finish stan­dards relate to surfaces produced by abrading, casting, coating, cutting, etching, plastic deformation, sintering, wear, erosion, and some other methods. ISO 1302 defines how surface texture and its constituents, roughness, waviness, and lay, are specified on the symbology. Surface defects are specifically excluded from consider­ ation during inspection of surface texture, but definitions of flaws and imperfections are discussed in ISO 8785. Position of complementary requirements: all values in millimeters

Basic symbol for surface under consideration or to a specification explained elsewhere in a note. The textual indication is APA (any process allowed)

Manufacturing method, treatment, coating or other requirement

Machining allowance (as on casting and forgings)

c

e

d b

Lay and orientation

Second texture parameter with numerical limit and band and/or sampling length. For a third or subsequent texture requirement, positions “a” and “b” are moved upward to allow room

Basic symbol where material removal is not permitted. The textual indication is NMR (no material removed) Basic symbol with all round circle added to indicate the specification applies to all surfaces in the view shown in profile (outline)

Text height

Line width for symbols d and d '

d'

c a

x'

2.5

h (ISO 3098-2)

Single texture parameter with numerical limit and band and/or sampling length

a

Basic symbol for material removal is required, for example machining. The textual indication is MRR (material removal required)

x

e

3.5

5

h

d b

7

10

14

20

0.25

0.35

0.5

0.7

1

1.2

2

Height for segment

x

3.5

5

7

10

14

20

28

Height for symbol segment

x'

7.5

10.5

15

21

30

42

60

Fig. 1. ISO Surface Finish Symbols

Differences Between ISO and ANSI/ASME Surface Finish Symbology: ISO 1302, like ANSI/ASME Y14.36-2018, is not concerned with luster, appearance, color, corrosion resistance, wear resis­tance, hardness, sub-surface microstructure, surface integrity, or many other characteris­tics that may govern considerations in specific applications. Visually, ISO 1302 surface finish symbols are similar to the ANSI/ASME symbols; however, with the release of the 2002 edi­tion, the indication of some of the parameters has changed when compared to ANSI/ASME Y14.36. The proportions of the symbol in

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Machinery's Handbook, 31st Edition ISO FINISH STANDARDS

823

relationship to text height differs in each as well. There is now less harmonization between ANSI/ASME Y14.36 and ISO 1302 than has been the case previously. Table 1. Other ISO Standards Related to Surface Finish ISO 3274:1996

“Geometrical Product Specifications (GPS) — Surface texture: Profile method; Nominal characteristics of contact (stylus) instruments.”

ISO 4287:1997

“Geometrical Product Specifications (GPS) — Surface texture: Profile method; Terms, definitions and surface texture parameters.”

ISO 4288:1996

“Geometrical Product Specifications (GPS) — Surface texture: Profile method; Rules and procedures for the assessment of surface texture.”

ISO 8785:1998

“Geometrical Product Specifications (GPS) — Surface imperfections — Terms, definitions and parameters.”

ISO 12085:1996

“Geometrical Product Specifications (GPS) — Surface texture: Profile method — Motif parameters.”

ISO 13565-1:1996 ISO 13565-2:1996 ISO 13565-3:1998

“Geometrical Product Specifications (GPS) — Surface texture: Profile method; Surfaces having stratified functional properties Part 1: Filtering and general measurement conditions.” Part 2: Height characterization using the linear material ratio curve.” Part 3: Height characterization using the material probability curve.”

Table 2. ISO Surface Parameter Symbols ( ISO 4287:1997) Rp = max height profile Rv = max profile valley depth Rz* = max height of the profile Rc = mean height of the profile Rt = total height of the profile Ra = arithmetic mean deviation of the profile Rq = root mean square deviation of the profile Rsk = skewness of the profile Rku = kurtosis of the profile RSm = mean width of the profile RDq = root mean square slope of the profile Rmr = material ration of the profile

Rδc = profile section height difference Ip = sampling length - primary profile lw = sampling length - waviness profile lr = sampling length - roughness profile ln = evaluation length Z(x) = ordinate value dZ  / dX = local slope Zp = profile peak height Zv = profile valley depth Zt = profile element height Xs = profile element width Ml = material length of the profile

Graphic Symbology Textural Descriptions.—New to this version of ISO 1302:2002 is the ability to add textual descriptions of the graphic symbology used on drawing. This gives specifications writers a consistent means to describe surface texture specification from within a body of text without having to add illustrations. See Fig. 1 for textual appli­ cation definitions, then Fig. 2- Fig. 6 for applications of this concept. turned Rz 3.1

Rz 6

3 21±0.1

Ra 1.5 0.1

0.2 A B

Fig. 2. Indication of Texture Require­ ment on a “Final” Workpiece, Reflecting a 3 mm Machining Allowance

Fig. 3. Surface Texture Indications Combined with Geometric Dimensioning and Tolerancing

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824

Machinery's Handbook, 31st Edition ISO Textural Descriptions

ISO 1302:2002 does not define the degrees of surface roughness and waviness or type of lay for specific purposes, nor does it specify the means by which any degree of such irreg­ ularities may be obtained or produced. Also, errors of form such as out-of-roundness and out-of-flatness are not addressed in the ISO surface finish standards. This edition does bet­ ter illustrate how surface texture indications can be used on castings to reflect machining allowances (Fig. 2) and how symbology can be attached to geometric dimensioning and tolerancing symbology (See Fig. 3). U Rz 0.9 L Ra 0.3

MRR U Rz 0.9; L Ra 0.3

Fig. 4. Indication of Bilateral Surface Specification Shown Textually and as Indicated on a Drawing turned Rz 3.1

MRR turned Rz 3.1

Indication of a machining process and requirement for roughness shown textually and as indicated on a drawing. Fig. 5a. Indication of Manufacturing Processes or Related Information Fe/Ni15p Cr r Rz 0.6

NMR Fe/Ni15p Cr r; Rz 0.6

Indication of coating and roughness requirement shown textually and as indicated on a drawing.

Fig. 5b. Indication of Manufacturing Processes or Related Information

Upper (U) and lower (L) limits Filter type "X." Gaussian is the current standard (ISO 11562). Previously it was the 2RC-filter, and in the future it could change again. It is suggested that companies specify Gaussian or " 2RC" to avoid misinterpretation. Evaluation length (ln)

U "X" 0.08-0.8 / Rz8max 3.3 Transmission band as either short-wave and/or long-wave Surface texture parameter. First letter is Profile (R, W, P). Second character is Characteristic/parameter: p, v, z, c, t, a, q, sk, ku, m, ∆q, mr(c), δ c, mr. See ISO 4287

Limit value (in micrometers) Interpretation of spec limit: 16% or max Manufacturing process

ground U "X" 0.08-0.8 / Rz8max 3.3 Surface texture lay Material removal allowed or not allowed (APA, MRR, NMR) Fig. 6. Control Elements for Indication of Surface Texture Requirements on Drawings

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Machinery's Handbook, 31st Edition ISO Surface Finish Rules

825

ISO Profiles.—Profile parameters may be one of three types (ISO 4287). These include: R-profile: Defined as the evaluation length. The ISO default length ln consists of five sampling lengths lr   , thus ln = 5 3 lr   W-profile: This parameter indicates waviness. There is no default length. P-profile: Indicates the structure parameters. The default evaluation length is defined in ISO 4288: 1996. Rules for Comparing Measured Values to Specified Limits.— Max Rule: When a maximum requirement is specified for a surface finish parameter on a drawing (e.g. Rz1.5max), none of the inspected values may extend beyond the upper limit over the entire surface. The term “max” must be added to the parametric symbol in the sur­face finish symbology on the drawing. 16%-Rule: When upper and lower limits are specified, no more than 16% of all measured values of the selected parameter within the evaluation length may exceed the upper limit. No more than 16% of all measured values of the selected parameter within the evaluation length may be less than the lower limit. Exceptions to the 16%-Rule: Where the measured values of roughness profiles being inspected follow a normal distribution, the 16%-rule may be overridden. This is allowed when greater than 16% of the measured values exceeds the upper limit, but the total rough­ ness profile conforms with the sum of the arithmetic mean and standard deviation (μ + s). Effectively, this means that the greater the value of s, the further μ must be from the upper limit (see Fig. 7). 1 Upper limit of surface texture parameter

2

2

1

Fig. 7. Roughness Parameter Value Curves Showing Mean and Standard Deviation

Indications of Transmission Band and Sampling Length in Textual Format: With the “16%-rule” transmission band as default it is shown textually and in drawings as: Ra 0.7 Rz1 3.3

MRR Ra 0.7; Rz1 3.3

If the “max-rule” transmission band is applied, it is shown textually and in drawings as: 0.0025-0.8 / Rz 3.0

MRR 0.0025-0.8 / Rz 3.0

Transmission band and sampling length are specified when there is no default value. The transmission band is indicated with the cut-off value of the filters in millimeters separated by a hyphen (-) with the short-wave filter first and the long-wave filter second. Again, in textual format and on drawings: 0.0025-0.8 / Rz 3.0

MRR 0.0025-0.8 / Rz 3.0

A specification may indicate one or both of the two transmission band filters. If only one is indicated, the hyphen is maintained to indicate whether the indication is the shortwave or the long-wave filter. 0.008–

(short-wave filter indication)

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or

–0.25

(long-wave filter indication)

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Machinery's Handbook, 31st Edition ISO Surface Finish Rules

826

Determining Cut-off Wavelength: When the sampling length is specified on the drawing or in documentation, the cut-off wavelength lc is equal to the sample length. When no sampling length is specified, the cut-off wavelength is estimated using Table 3. Measurement of Roughness Parameters: For non-periodic roughness, the parameter Ra, Rz, Rz1max or RSm is first estimated using visual inspection, comparison to specimens, graphic analysis, etc. The sampling length is then selected from Table 3, based on the use of Ra, Rz, Rz1max or RSm. Then, with instrumentation, a representative sample is taken using the sampling length chosen above. The measured values are then compared to the ranges of values in Table 3 for the partic­ ular parameter. If the value is outside the range of values for the estimated sampling length, the measuring instrument is adjusted for the next higher or lower sampling length and the measurement repeated. If the final setting corresponds to Table 3, then both the sampling length setting and Ra, Rz, Rz1max or RSm values are correct and a representative measure­ment of the parameter can be taken. For periodic roughness, the parameter RSm is estimated graphically and the recom­ mended cut-off values selected using Table 3. If the value is outside the range of values for the estimated sampling length, the measuring instrument is adjusted for the next higher or lower sampling length and the measurement repeated. If the final setting corresponds to Table 3, then both the sampling length setting and RSm values are correct and a representa­ tive measurement of the parameter can be taken. Table 3. Sampling Lengths

For Ra, Rq, Rsk, Rku, RDq

For Rz, Rv, Rp, Rc, Rt

Ra, μm

Rz, Rz1max , μm

RSm, μm

Evaluation length, ln (mm)

Curves for Periodic and Non-periodic Profiles

Sampling length, lr (mm)

Curves for Non-periodic Profiles such as Ground Surfaces

(0.006) < Ra ≤ 0.02

(0.025) < Rz, Rz1max ≤ 0.1

0.013 < RSm ≤ 0.04

0.08

0.4

0.02 < Ra ≤ 0.1

0.1 < Rz, Rz1max ≤ 0.5

0.04 < RSm ≤ 0.13

0.25

1.25

0.1 < Ra ≤ 2

0.5 < Rz, Rz1max ≤ 10

0.13 < RSm ≤ 0.4

0.8

4

2 < Ra ≤ 10

10 < Rz, Rz1max ≤ 50

0.4 < RSm ≤ 1.3

2.5

12.5

10 < Ra ≤ 80

50 < Rz, Rz1max ≤ 200

1.3 < RSm ≤ 4

8

40

For R-parameters and RSm

Table 4. Preferred Roughness Values and Roughness Grades Roughness values, Ra

Roughness values, Ra

μin

Previous Grade Number from ISO 1302

50

2000

N12

0.8

32

N6

25

1000

N11

0.4

16

N5

12.5

500

N10

0.2

8

N4

6.3

250

N9

0.1

4

N3

3.2

125

N8

0.05

2

N2

1.6

63

N7

0.025

1

N1

μm

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μm

μin

Previous Grade Number from ISO 1302

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Machinery's Handbook, 31st Edition ISO Surface Texture Symbology Examples

827

Examples of ISO Applications of Surface Texture Symbology Example 1: Surface roughness in Fig. 8 is produced by milling with a bilateral tolerance between an upper limit of Ra = 55 mm and a lower limit of Ra = 6.2mm. Both apply the “16%-rule” default (ISO 4288). Both transmission bands are 0.008 - 4 mm, using default evaluation length (5 × 4 mm = 20 mm) (ISO 4288). The surface lay is circular about the center. U and L are omitted because it is obvious one is upper and one lower. Material removal is allowed.

C

Milled

Ground

0.008-4 / Ra 55

Ra 1.5

0.008-4 / Ra 6.2

-2.5 / Ramax 6.7

Fig. 8.

Fig. 9.

Example 2: Surface roughness in Fig. 9 is produced by grinding to two upper limit speci­ fications: Ra = 1.5 mm and limited to Rz = 6.7 mm max. The default “16%-rule,” default transmission band and default evaluation length apply to the Ra, while the “max-rule”, a -2.5 mm transmission band and default evaluation length, apply to the Rz. The surface lay is perpendicular relative to the plane of projection, and material removal is allowed.

Example 3: Fig. 10 indicates a simplified representation where surface roughness of Rz = 6.1 mm is the default for all surfaces as indicated by the Rz = 6.1 specification, plus basic symbol within parentheses. The default “16%-rule” applies to both as does the default transmission band (ISO 4288 and ISO 3274). Any deviating specification is called out with local notes such as the Ra =0.7 mm specification. There is no lay requirement, and material removal is allowed. Fe/Ni10b Cr r

Ra 0.7

Fe/Ni20p Cr r Rz 6.1

( )

Rz 1

Fig. 10.

Fig. 11.

-0.8 / Ra 3.1 U -2.5 / Rz 18 L -2.5 / Rz 6.5

Fig. 12.

Example 4: Surface treatment without any material removal allowed is indicated in Fig. 11, and to a single unilateral upper limit specification of Rz = 1 mm. The default “16%rule,” default transmission band and default evaluation length apply. The surface treat­ment is nickel-chrome plated to all surfaces shown in profile (outline) in the view where the symbol is applied. There is no lay requirement.

Example 5: In Fig. 12, surface roughness is produced by any material removal process to one unilateral upper limit and one bilateral specification: the unilateral Ra = 3.1 is to the default “16%-rule,” a transmission band of -0.8 mm and the default evaluation length (5 × 0.8 = 4 mm). The bilateral Rz has an upper limit of Rz = 18 mm and a lower limit of Rz = 6.5 mm. Both limits are to a transmission band of -2.5 mm with both to the default 5 × 2.5 = 12.5 mm. The symbols U and L may be indicated even if it is obvious. Surface treatment is nickel/chromium plating. There is no lay requirement. Example 6: Surface texture symbology may be combined with dimension leaders and witness (extension) lines, as in Fig. 13. Surface roughness for the side surfaces of the key­ way is produced by any material removal process to one unilateral upper limit specifica­ tion, Ra = 6.5 mm. It is to the default “16%-rule,” default transmission band and default evaluation length (5 × lc) (ISO 3274). There is no lay requirement.

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828

Machinery's Handbook, 31st Edition ISO Surface Texture Symbology Examples

Surface roughness for the chamfer is produced by any material removal process to one unilateral upper limit specification, Ra = 2.5 mm. It is to the default “16%-rule,” default transmission band and default evaluation length (5 × lc) (ISO 3274). There is no lay requirement.

Rz 50

Ø40

2 6.

Ra 6.5

Ra

2 x 45 A

R3

Ra

Ra

A

1.5

2.5

Fig. 13.

Fig. 14.

Example 7: Surface texture symbology may be applied to extended extension lines or on extended projection lines, Fig. 14. All feature surface roughness specifications shown are obtainable by any material removal process and are single unilateral upper limit specifica­ tions, respectively: Ra = 1.5 mm, Ra = 6.2 mm and Rz = 50 mm. All are to “16%-rule” default, default transmission band and default evaluation length (5 × lc). There is no lay requirement for any of the three. Example 8: Surface texture symbology and dimensions may be combined on leader lines, as in Fig. 15. The feature surface roughness specifications shown are obtainable by any material removal process and are single unilateral upper limit specifications, respectively: Rz = 1 mm, to the default “16%-rule,” default transmission band and default evaluation length (5 × lc). There is no lay requirement. Fe/Cr50 Rz 1.7

3x

Ø1 4

Rz

1

Ground Rz 6.5

14

Fig. 15.

Fig. 16.

Example 9: Symbology can be used for dimensional information and surface treatment. Fig. 16 illustrates three successive steps of a manufacturing process. The first step is a single unilateral upper limit Rz = 1.7 mm to the default “16%-rule,” default evaluation length (5 × lc) and default transmission band. It is obtainable by any material removal process, with no lay characteristics specified. Step two, indicated with a phantom line over the whole length of the cylinder, has no surface texture requirement other than chromium plating. The third step is a single unilateral upper limit of Rz = 6.5 mm applied only to the first 14 mm of the cylinder surface. The default “16%-rule” applies as does default evaluation length (5 × lc) and default transmission band. Material removal is to be by grinding, with no lay characteristics specified.

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Machinery's Handbook, 31st Edition TABLE OF CONTENTS

TOOLING AND TOOLMAKING CUTTING TOOLS

MILLING CUTTERS 871 Selection of Milling Cutters 871 Number of Teeth 872 American National Standard 873 Plain Milling Cutters 874 Side Milling Cutters 875 T-Slot Milling Cutters 876 Metal Slitting Saws 877 Single- and Double-Angle 878 Shell Mills 879 Helical End Mills 890 Arbor-Type Cutters 893 Roller Chain Sprocket 894 Keys and Keyways 895 Woodruff Keyseat Cutters 899 Spline-Shaft Milling Cutter 899 Cutter Grinding 900 Wheel Speeds and Feeds 900 Clearance Angles 901 Rake Angles for Milling Cutters 901 Eccentric-Type Radial Relief 904 Indicator Drop Method 906 Distance to Set Tooth 907 Counter Milling

832 Terms and Definitions 832 Tool Contour 835 Relief Angles 836 Rake Angles 837 Side Cutting Edge and Lead Angles 837 End Cutting Edge Angle 837 Nose Radius 838 Chipbreakers 839 Planing Tools 839 Indexable Inserts 840 Identification System 841 Indexable Insert Tool Holders 842 Standard Shank Sized for Holders 842 Identification System for Holders 843 Letter Symbols 844 Selecting Holders 847 Sintered Carbide Blanks and Cutting Tools 847 Sintered Carbide Blanks 847 Single Point Tools 847 Single-Point, Sintered Carbide-Tipped Tools 850 Carbide-Tipped Tools 850 Style A 851 Style B 852 Style C 852 Style D 853 Style E 853 Styles ER and EL 854 Style F 855 Style G 856 Chipbreaker 856 Chipless Machining 856 Indexable Insert Holders for NC 857 Insert Radius Compensation 859 Threading Tool Insert Radius

REAMERS 908 909 909 909 910 912 913 915

CEMENTED CARBIDES

915 916

860 Cemented Carbides and Other Hard Materials 860 Carbides and Carbonitrides 861 Properties of Tungsten CarbideBased Cutting Tool Hard Metals 865 Ceramics 866 ISO Classifications of Hardmetals 868 Superhard Materials 870 Hardmetal Tooling

917 918 919 920 921 922 923 926 928 929

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829

Hand Reamers Irregular Spacing of Teeth Threaded-End Hand Reamers Fluted Chucking Reamers Rose Chucking Reamers Vertical Adjustment of Tooth-Rest TermsApplying to Reamers Direction of Rotation and Helix Dimensions of Centers Calculating Countersink and Spot Drill Depths Reamer Difficulties Types and Sizes of Reamers Expansion Chucking Reamers Hand Reamers Expansion Hand Reamers Driving Slots and Lugs Chucking Reamers Shell Reamers Center Reamers Taper Pipe Reamers

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Machinery's Handbook, 31st Edition TABLE OF CONTENTS

TOOLING AND TOOLMAKING TWIST DRILLS AND COUNTERBORES 931 932 952 953 954 955 955 955 956 957 958 959 961 961 961 962 963 963 964 964 964 965 965 967 968 968

Definitions of Twist Drill Terms Types of Drills Split-Sleeve, Collet Drill Drivers Three- and Four-Flute Straight Shank Core Drills Twist Drills and Centering Tools British Standard Combined Drills and Countersinks Drill Drivers—Split-Sleeve, Collet-Type British Standard Metric Twist Drills Gauge and Letter Sizes Morse Taper-Shank Twist Drills Tolerance on Diameter Parallel Shank Jobber Twist Drills Stub Drills Steels for Twist Drills Accuracy of Drilled Holes Counterboring Interchangeable Cutters and Guides Three-Piece Counterbores Counterbore Sizes for Hex-Head Bolts and Nuts Sintered Carbide Boring Tools Style Designations Boring Tools Square Carbide-Tipped Square Solid Carbide Round Boring Machines, Origin TAPS

969 Thread Form, Styles, and Types 971 Standard System of Tap Marking 973 Thread Series Designations 974 Thread Limits, Ground Thread 975 Thread Limits, Cut Thread 976 M Profile Metric Taps 976 Thread Limits, Ground Thread 977 Definitions of Tap Terms 983 Tap Dimensions, Inch and Metric 986 Optional Neck and Thread Length 989 Extension Tap Dimensions

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(Continued)

990 991 992 993 994

995 995 996 997 998 1003 1005 1006 1009 1009 1009 1011 1012 1012

TAPS

Fine Pitch Tap Dimensions Standard Number of Flutes Pulley Taps Dimensions Straight and Taper Pipe Tap Runout and Locational Tolerance M Profile Tap D Limits (Inch) M Profile Tap D Limits (mm) Tap Sizes for Class 6H Threads Tap Sizes, Unified 2B and 3B Unified Threads Taps H Limits Straight Pipe Tap Thread Limits Taper Pipe Tap Thread Limits Screw Thread Insert Tap Limits Acme and Square-Threaded Taps Adjustable Taps Drill Hole Sizes for Acme Threads Proportions Tapping Square Threads Collapsible Taps STANDARD TAPERS

1013 1013 1013 1014 1015 1021 1022 1023 1024 1024 1025 1026 1027 1029 1030 1031 1032 1033 1033 1034 1036

830

Tapers for Machine Tool Spindles Morse Taper Brown & Sharpe Taper Jarno Taper American National Standard Machine Tapers British Standard Tapers Morse Taper Sleeves Brown & Sharpe Taper Shanks Jarno Taper Shanks Tapers for Machine Tool Spindles Plug and Ring Gages Jacobs Tapers and Threads Spindle Noses Tool Shanks Draw-In Bolt Ends Spindle Nose V-Flange Tool Shanks and Retention Knobs Collets R8 Collet Collets for Lathes, Mills, Grinders, and Fixtures ER Type Collets

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Machinery's Handbook, 31st Edition TABLE OF CONTENTS

TOOLING AND TOOLMAKING ARBORS, CHUCKS, AND SPINDLES

KNURLS AND KNURLING 1056 1056 1056 1057 1058 1058 1058 1059 1060 1060

1037 Portable Tool Spindles 1037 Circular SawArbors 1037 Spindles for Geared Chucks 1037 Spindle Sizes 1037 Straight Grinding Wheel Spindles 1038 Square Drives for Portable Air and Electric Tools 1039 Threaded and Tapered Spindles 1039 Abrasion Tool Spindles 1040 Hexagonal Chucks for Portable Air and Electric Tools 1041 Mounted Wheels and Points 1043 Shapes and Sizes

TOOL WEAR AND SHARPENING 1061 1062 1062 1062 1063 1063 1063

BROACHES AND BROACHING 1044 The Broaching Process 1044 Types of Broaches 1045 Pitch of Broach Teeth 1046 Data for Surface Broaches 1047 Depth of Cut per Tooth 1048 FaceAngle or Rake 1048 ClearanceAngle 1048 Land Width 1048 Depth of Broach Teeth 1048 Radius of Tooth Fillet 1048 Total Length of Broach 1048 Chipbreakers 1049 ShearAngle 1049 Types of Broaching Machines 1049 Ball-Broaching 1050 Broaching Difficulties

1064 1065 1065 1066 1066 1066 1066 1067 1067 1067 1068 1068 1068 1068 1070

FILES AND BURS 1051 1052 1052 1054 1055 1055

Definitions of File Terms File Characteristics Classes of Files Effectiveness of Rotary Files and Burs Speeds of Rotary Files and Burs Steel Wool

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Standard Knurls and Knurling Preferred Sizes Specifications Cylindrical Knurling Tools Flat Knurling Tools Specifications for Flat Dies Formulas for Knurled Work Recommended Tolerances Marking on Knurls and Dies Concave Knurls

1071 1072 1072

831

Flank Wear Cratering Cutting Edge Chipping Deformation Surface Finish Sharpening Twist Drills Relief Grinding of the Tool Flanks Drill Point Thinning Sharpening Carbide Tools Silicon Carbide Wheels Diamond Wheels Grit Sizes Wheel Grades Concentration Dry Versus Wet Grinding of Carbide Tools Coolants for Carbide Tool Grinding Peripheral versus Flat Side Grinding Lapping Carbide Tools Chipbreaker Grinding Miscellaneous Points Meshes, Sieves, and Screens Commercial Sieve Mesh Dimensions Standard Sieves and Mesh Sizes Typical Openings in Laboratory Sieve Series Abrasive Grit Number

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832

Machinery's Handbook, 31st Edition Tooling and Toolmaking CUTTING TOOLS Terms and Definitions

Tool Contour.—Tools for turning, planing, etc., are made in straight, bent, offset, and other forms to place the cutting edges in convenient positions for operating on differently located surfaces. The contour or shape of the cutting edge may also be varied to suit different classes of work. Tool shapes, however, are not only related to the kind of operation, but, in roughing tools particularly, the contour may have a decided effect upon the cutting effi­ ciency of the tool. To illustrate, an increase in the side cutting-edge angle of a roughing tool, or in the nose radius, tends to permit higher cutting speeds because the chip will be thinner for a given feed rate. Such changes, however, may result in chattering or vibrations unless the work and the machine are rigid; hence, the most desirable contour may be a compromise between the ideal form and one that is needed to meet practical requirements. Terms and Definitions.—The terms and definitions relating to single-point tools vary somewhat in different plants, but the following are in general use. Side Rake Angle

End Cutting Edge Angle

Side Cutting Edge Angle

Back Rake Angle Tool Point or Nose Radius

Side Relief Angle End Relief Angle

Fig. 1. Terms Applied to Single-Point Turning Tools

Single-Point Tool: This term is applied to tools for turning, planing, boring, etc., which have a cutting edge at one end. This cutting edge may be formed on one end of a solid piece of steel, or the cutting part of the tool may consist of an insert or tip which is held to the body of the tool by brazing, welding, or mechanical means. Shank: The shank is the main body of the tool. If the tool is an inserted cutter type, the shank supports the cutter or bit. (See diagram, Fig. 1.) Nose: A general term sometimes used to designate the cutting end but usually relating more particularly to the rounded tip of the cutting end. Face: The surface against which the chips bear, as they are severed in turning or planing operations, is called the face. Flank: The flank is that end surface adjacent to the cutting edge and below it when the tool is in a horizontal position as for turning. Base: The base is the surface of the tool shank that bears against the supporting toolholder or block. Side Cutting Edge: The side cutting edge is the cutting edge on the side of the tool. Tools such as that shown in Fig. 1 do the bulk of the cutting with this cutting edge and are, therefore, sometimes called side cutting edge tools. End Cutting Edge: The end cutting edge is the cutting edge at the end of the tool. On side cutting edge tools, the end cutting edge can be used for light plunging and facing cuts. Cutoff tools and similar tools have only one cutting edge located on the end. These

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tools and other tools that are intended to cut primarily with the end cutting edge are some­ times called end cutting edge tools. Rake: A metal-cutting tool is said to have rake when the tool face or surface against which the chips bear as they are being severed is inclined for the purpose of either increas­ ing or diminishing the keenness or bluntness of the edge. The magnitude of the rake is most conveniently measured by two angles called the back rake angle and the side rake angle. The tool shown in Fig. 1 has rake. If the face of the tool did not incline but was parallel to the base, there would be no rake; the rake angles would be zero. Positive Rake: If the inclination of the tool face is such as to make the cutting edge keener or more acute than when the rake angle is zero, the rake angle is defined as positive. Negative Rake: If the inclination of the tool face makes the cutting edge less keen or more blunt than when the rake angle is zero, the rake is defined as negative. Back Rake: The back rake is the inclination of the face toward or away from the end or the end cutting edge of the tool. When the inclination is away from the end cutting edge, as shown in Fig. 1, the back rake is positive. If the inclination is downward toward the end cutting edge, the back rake is negative. Side Rake: The side rake is the inclination of the face toward or away from the side cut­ ting edge. When the inclination is away from the side cutting edge, as shown in Fig. 1, the side rake is positive. If the inclination is toward the side cutting edge the side rake is nega­tive. Relief: The flanks below the side cutting edge and the end cutting edge must be relieved to allow these cutting edges to penetrate into the workpiece when taking a cut. If the flanks are not provided with relief, the cutting edges will rub against the workpiece and be unable to penetrate in order to form the chip. Relief is also provided below the nose of the tool to allow it to penetrate into the workpiece. The relief at the nose is usually a blend of the side relief and the end relief. End Relief Angle: The end relief angle is a measure of the relief below the end cutting edge. Side Relief Angle: The side relief angle is a measure of the relief below the side cutting edge. Back Rake Angle: The back rake angle is a measure of the back rake. It is measured in a plane that passes through the side cutting edge and is perpendicular to the base. Thus, the back rake angle can be defined by measuring the inclination of the side cutting edge with respect to a line or plane that is parallel to the base. The back rake angle may be positive, negative, or zero depending upon the magnitude and direction of the back rake. Side Rake Angle: The side rake angle is a measure of the side rake. This angle is always measured in a plane that is perpendicular to the side cutting edge and perpendicular to the base. Thus, the side rake angle is the angle of inclination of the face perpendicular to the side cutting edge with reference to a line or a plane that is parallel to the base. End Cutting Edge Angle: The end cutting edge angle is the angle made by the end cutting edge with respect to a plane perpendicular to the axis of the tool shank. It is provided to allow the end cutting edge to clear the finish machined surface on the workpiece. Side Cutting Edge Angle: The side cutting edge angle is the angle made by the side cut­ting edge and a plane that is parallel to the side of the shank. Nose Radius: The nose radius is the radius of the nose of the tool. The performance of the tool, in part, is influenced by nose radius so that it must be carefully controlled. Lead Angle: The lead angle, shown in Fig. 2, is not ground on the tool. It is a tool setting angle which has a great influence on the performance of the tool. The lead angle is bounded by the side cutting edge and a plane perpendicular to the workpiece surface when the tool is in position to cut; or, more exactly, the lead angle is the angle between the side cutting edge and a plane perpendicular to the direction of the feed travel.

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Machinery's Handbook, 31st Edition Cutting Tools Lead Angle

Lead Angle Equal to Side Cutting Angle

Side Cutting Edge Angle

Fig. 2. Lead Angle on Single-Point Turning Tool

Solid Tool: A solid tool is a cutting tool made from one piece of tool material. Brazed Tool: A brazed tool is a cutting tool having a blank of cutting-tool material per­ manently brazed to a steel shank. Blank: A blank is an unground piece of cutting-tool material from which a brazed tool is made. Tool Bit: A tool bit is a relatively small cutting tool that is clamped in a holder in such a way that it can readily be removed and replaced. It is intended primarily to be reground when dull and not indexed. Tool-Bit Blank: The tool-bit blank is an unground piece of cutting-tool material from which a tool bit can be made by grinding. It is available in standard sizes and shapes. Tool-Bit Holder: Usually made from forged steel, the tool-bit holder is used to hold the tool bit, to act as an extended shank for the tool bit, and to provide a means for clamping in the tool post. Straight-Shank Tool-Bit Holder: A straight-shank tool-bit holder has a straight shank when viewed from the top. The axis of the tool bit is held parallel to the axis of the shank. Offset-Shank Tool-Bit Holder: An offset-shank tool-bit holder has the shank bent to the right or left, as seen in Fig. 3. The axis of the tool bit is held at an angle with respect to the axis of the shank. Side Cutting Tool: A side cutting tool has its major cutting edge on the side of the cutting part of the tool. The major cutting edge may be parallel or at an angle with respect to the axis of the tool. Indexable Inserts: An indexable insert is a relatively small piece of cutting-tool material that is geometrically shaped to have two or several cutting edges that are used until dull. The insert is then indexed on the holder to apply a sharp cutting edge. When all the cutting edges have been dulled, the insert is discarded. The insert is held in a pocket or against other locating surfaces on an indexable insert holder by means of a mechanical clamping device that can be tightened or loosened easily. Indexable Insert Holder: Made of steel, an indexable insert holder is used to hold index­ able inserts. It is equipped with a mechanical clamping device that holds the inserts firmly in a pocket or against other seating surfaces. Straight-Shank Indexable Insert Holder: A straight-shank indexable insert tool-holder is essentially straight when viewed from the top, although the cutting edge of the insert may be oriented parallel, or at an angle to, the axis of the holder. Offset-Shank Indexable Insert Holder: An offset-shank indexable insert holder has the head end, or the end containing the insert pocket, offset to the right or left, as shown in Fig. 3.

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Fig. 3. Top: Right-Hand Offset-Shank, Indexable Insert Holder Bottom: Right-Hand Offset-Shank Tool-Bit Holder

End Cutting Tool: An end cutting tool has its major cutting edge on the end of the cutting part of the tool. The major cutting edge may be perpendicular or at an angle, with respect to the axis of the tool. Curved Cutting-Edge Tool: A curved cutting-edge tool has a continuously variable side cutting edge angle. The cutting edge is usually in the form of a smooth, continuous curve along its entire length, or along a large portion of its length. Right-Hand Tool: A right-hand tool has the major, or working, cutting edge on the righthand side when viewed from the cutting end with the face up. As used in a lathe, such a tool is usually fed into the work from right to left, when viewed from the shank end. Left-Hand Tool: A left-hand tool has the major or working cutting edge on the left-hand side when viewed from the cutting end with the face up. As used in a lathe, the tool is usu­ ally fed into the work from left to right, when viewed from the shank end. Neutral-Hand Tool: A neutral-hand tool is a tool to cut either left to right or right to left; or the cut may be parallel to the axis of the shank as when plunge cutting. Chipbreaker: A groove formed in or on a shoulder on the face of a turning tool back of the cutting edge to break up the chips and prevent the formation of long, continuous chips that would be dangerous to the operator and also bulky and cumbersome to handle. A chipbreaker of the shoulder type may be formed directly on the tool face, or it may consist of a separate piece that is held either by brazing or by clamping. Relief Angles.—The end relief angle and the side relief angle on single-point cutting tools are usually, though not invariably, made equal to each other. The relief angle under the nose of the tool is a blend of the side and end relief angles. The size of the relief angles has a pronounced effect on the performance of the cutting tool. If the relief angles are too large, the cutting edge will be weakened and in danger of breaking when a heavy cutting load is placed on it by a hard and tough material. On finish cuts, rapid wear of the cutting edge may cause problems with size control on the part. Relief angles that are too small will cause the rate of wear on the flank of the tool below the cutting edge to increase, thereby significantly reducing the tool life. In general, when cut­ting hard and tough materials, the relief angles should be 6 to 8 degrees for highspeed steel tools and 5 to 7 degrees for carbide tools. For medium steels, mild steels, cast iron, and other average work the recommended values of the relief angles are 8 to 12 degrees for high-speed steel tools and 5 to 10 degrees for carbides. Ductile materials having a rela­tively low modulus of elasticity should be cut using larger relief angles. For example, the relief angles recommended for turning copper, brass, bronze, aluminum,

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Machinery's Handbook, 31st Edition Cutting Tools

ferritic malleable iron, and similar metals are 12 to 16 degrees for high-speed steel tools and 8 to 14 degrees for carbides. Larger relief angles generally tend to produce a better finish on the finish machined sur­ face because less surface of the worn flank of the tool rubs against the workpiece. For this reason, single-point thread-cutting tools should be provided with relief angles that are as large as circumstances will permit. Problems encountered when machining stainless steel may be overcome by increasing the size of the relief angle. The relief angles used should never be smaller than necessary. Rake Angles.—Machinability tests have confirmed that when the rake angle along which the chip slides, called the true rake angle, is made larger in the positive direction, the cut­ting force and the cutting temperature will decrease. Also, the tool life for a given cutting speed will increase with increases in the true rake angle up to an optimum value, after which it will decrease again. For turning tools that cut primarily with the side cutting edge, the true rake angle corresponds rather closely with the side rake angle except when taking shallow cuts. Increasing the side rake angle in the positive direction lowers the cut­ting force and the cutting temperature, while at the same time it results in a longer tool life or a higher permissible cutting speed up to an optimum value of the side rake angle. After the optimum value is exceeded, the cutting force and the cutting temperature will continue to drop; however, the tool life and the permissible cutting speed will decrease. As an approximation, the magnitude of the cutting force will decrease about one percent per degree increase in the side rake angle. While not exact, this rule of thumb does corre­ spond approximately to test results and can be used to make rough estimates. Of course, the cutting force also increases about one percent per degree decrease in the side rake angle. The limiting value of the side rake angle for optimum tool life or cutting speed depends upon the work material and the cutting tool material. In general, lower values can be used for hard and tough work materials. Cemented carbides are harder and more brittle than high-speed steel; therefore, the rake angles usually used for cemented carbides are less positive than for high-speed steel. Negative rake angles cause the face of the tool to slope in the opposite direction from pos­itive rake angles and, as might be expected, they have an opposite effect. For side cutting edge tools, increasing the side rake angle in a negative direction will result in an increase in the cutting force and an increase in the cutting temperature of approximately one percent per degree change in rake angle. For example, if the side rake angle is changed from 5 degrees positive to 5 degrees negative, the cutting force will be about 10 percent larger. Usually the tool life will also decrease when negative side rake angles are used, although the tool life will sometimes increase when the negative rake angle is not too large and when a fast cutting speed is used. Negative side rake angles are usually used in combination with negative back rake angles on single-point cutting tools. The negative rake angles strengthen the cutting edges, enabling them to sustain heavier cutting loads and shock loads. They are recommended for turning very hard materials and for heavy interrupted cuts. There is also an economic advantage in favor of using negative rake indexable inserts and tool holders inasmuch as the cutting edges provided on both the top and bottom of the insert can be used. On turning tools that cut primarily with the side cutting edge, the effect of the back rake angle alone is much less than the effect of the side rake angle, although the direction of the change in cutting force, cutting temperature, and tool life is the same. The effect that the back rake angle has can be ignored unless, of course, extremely large changes in this angle are made. A positive back rake angle does improve the performance of the nose of the tool somewhat and is helpful in taking light finishing cuts. A negative back rake angle strength­ens the nose of the tool and is helpful when interrupted cuts are taken. The back rake angle has a very significant effect on the performance of end cutting edge tools, such as cut-off tools. For these tools, the effect of the back rake angle is very similar to the effect of the side rake angle on side cutting edge tools.

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Side Cutting Edge and Lead Angles.—These angles are considered together because the side cutting edge angle is usually designed to provide the desired lead angle when the tool is being used. The side cutting edge angle and the lead angle will be equal when the shank of the cutting tool is positioned perpendicular to the workpiece, or, more correctly, perpendicular to the direction of the feed. When the shank is not perpendicular, the lead angle is determined by the side cutting edge and an imaginary line perpendicular to the feed direction. The flow of the chips over the face of the tool is approximately perpendicular to the side cutting edge except when shallow cuts are taken. The thickness of the undeformed chip is measured perpendicular to the side cutting edge. As the lead angle is increased, the length of chip in contact with the side cutting edge is increased, and the chip will become longer and thinner. This effect is the same as increasing the depth of cut and decreasing the feed, although the actual depth of cut and feed remain the same and the same amount of metal is removed. The effect of lengthening and thinning the chip by increasing the lead angle is very beneficial as it increases the tool life for a given cutting speed or that speed can be increased. Increasing the cutting speed while the feed and the tool life remain the same leads to faster production. However, an adverse effect must be considered. Chatter can be caused by a cutting edge that is oriented at a high lead angle when turning, and sometimes, when turning long and slender shafts, even a small lead angle can cause chatter. In fact, an unsuitable lead angle of the side cutting edge is one of the principal causes of chatter. When chatter occurs, often simply reducing the lead angle will cure it. Sometimes, very long and slender shafts can be turned successfully with a tool having a zero degree lead angle (and having a small nose radius). Boring bars, being usually somewhat long and slender, are also susceptible to chatter if a large lead angle is used. The lead angle for boring bars should be kept small, and for very long and slender boring bars a zero-degree lead angle is recommended. It is impos­sible to provide a rule that will determine when chatter caused by a lead angle will occur and when it will not. The first consideration is the length to diameter ratio of the part to be turned, or of the boring bar. Then the method of holding the work­piece must be considered — a part that is firmly held is less apt to chatter. Finally, the over­all condition and rigidity of the machine must be considered because they may be the real cause of chatter. Although chatter can be a problem, the advantages gained from high lead angles are such that the lead angle should be as large as possible at all times. End Cutting Edge Angle.—The size of the end cutting edge angle is important when tool wear by cratering occurs. Frequently, the crater will enlarge until it breaks through the end cutting edge just behind the nose, and tool failure follows shortly. Reducing the size of the end cutting edge angle tends to delay the time of crater breakthrough. When cratering takes place, the recommended end cutting edge angle is 8 to 15 degrees. If there is no cratering, the angle can be made larger. Larger end cutting edge angles may be required to enable profile turning tools to plunge into the work without interference from the end cutting edge. Nose Radius.—The tool nose is a very critical part of the cutting edge since it cuts the fin­ ished surface on the workpiece. If the nose is made to a sharp point, the finish machined surface will usually be unacceptable and the life of the tool will be short. Thus, a nose radius is required to obtain an acceptable surface finish and tool life. The surface finish obtained is determined by the feed rate and by the nose radius if other factors such as the work material, the cutting speed, and cutting fluids are not considered. A large nose radius will give a better surface finish and will permit a faster feed rate to be used. Machinability tests have demonstrated that increasing the nose radius will also improve the tool life or allow a faster cutting speed to be used. For example, high-speed steel tools were used to turn an alloy steel in one series of tests where complete or catastrophic tool failure was used as a criterion for the end of tool life. The cutting speed for a 60-minute

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tool life was found to be 125 fpm (0.635 m/s) when the nose radius was 1 ⁄16 inch (1.59 mm) and 160 fpm (0.8.13 m/s) when the nose radius was 1 ⁄4 inch (6.35 mm). A very large nose radius can often be used but a limit is sometimes imposed because the tendency for chatter to occur is increased as the nose radius is made larger. A nose radius that is too large can cause chatter, and, when it does, a smaller nose radius must be used on the tool. It is always good practice to make the nose radius as large as is compatible with the operation being performed.

Chipbreakers.—Many steel turning tools are equipped with chipbreaking devices to pre­ vent the formation of long continuous chips in connection with the turning of steel at the high speeds made possible by high-speed steel and especially cemented carbide tools. Long steel chips are dangerous to the operator and cumbersome to handle, and they may twist around the tool and cause damage. Broken chips not only occupy less space, but per­ mit a better flow of coolant to the cutting edge. Several different forms of chipbreakers are illustrated in Fig. 4.

Angular Shoulder Type: The angular shoulder type shown at A is one of the commonly used forms. As the enlarged sectional view shows, the chipbreaking shoulder is located back of the cutting edge. The angle a between the shoulder and cutting edge may vary from 6 to 15 degrees or more, 8 degrees being a fair average. The ideal angle, width W and depth G, depend upon the speed and feed, the depth of cut, and the material. As a general rule, width W, at the end of the tool, varies from 3⁄32 to 7⁄32 inch (2.4–5.6 mm), and the depth G may range from 1 ⁄64 to 1 ⁄16 inch (0.4–1.6 mm). The shoulder radius equals depth G. If the tool has a large nose radius, the corner of the shoulder at the nose end may be beveled off, as illus­trated at B, to prevent it from coming into contact with the work. The width K for type B should equal approximately 1.5 times the nose radius. Parallel Shoulder Type: Diagram C shows a design with a chipbreaking shoulder that is parallel with the cutting edge. With this form, the chips are likely to come off in short curled sections. The parallel form may also be applied to straight tools that do not have a side cutting-edge angle. The tendency with this parallel shoulder form is to force the chips against the work and damage it. K

a x

45° x

x

W G

L

Section X–X Enlarged A

B

C

x

W

G

Section X–X Enlarged D

E

Fig. 4. Different Forms of Chipbreakers for Turning Tools

Groove Type: This type (diagram D) has a groove in the face of the tool produced by grinding. Between the groove and the cutting edge, there is a land L. Under ideal condi­ tions, this width L, the groove width W, and the groove depth G, would be varied to suit the feed, depth of cut and material. For average use, L and G are about 1 ⁄32 inch (0.79 mm), and W 1 ⁄16 inch (1.59 mm). There are differences of opinion concerning the relative merits of the groove type and the shoulder type. Both types have proved satisfactory when properly proportioned for a given class of work.

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Chipbreaker for Light Cuts: Diagram E illustrates a form of chipbreaker that is some­ times used on tools for finishing cuts having a maximum depth of about 1 ⁄32 inch (0.79 mm). This chipbreaker is a shoulder type, having an angle of 45 degrees and a maximum width of about 1 ⁄16 inch (1.59 mm). It is important in grinding all chipbreakers to give the chipbear­ing surfaces a fine finish, such as would be obtained by honing. This finish greatly increases the life of the tool. Planing Tools.—Many of the principles governing the shape of turning tools also apply in the grinding of tools for planing. The amount of rake depends upon the hardness of the material, and the direction of the rake should be away from the working part of the cut­ting edge. The angle of clearance should be about 4 or 5 degrees for planer tools, which is less than for lathe tools. This small clearance is allowable because a planer tool is held about square with the platen, whereas a lathe tool, the height and inclination of which can be varied, may not always be clamped in the same position. Carbide Tools: Carbide tools for planing usually have negative rake. Round-nose and square-nose end-cutting tools should have a “negative back rake” (or front rake) of 2 or 3 degrees. Side cutting tools may have a negative back rake of 10 degrees, a negative side rake of 5 degrees, and a side cutting-edge angle of 8 degrees. Indexable Inserts

Introduction.—A large proportion of cemented carbide, single-point cutting tools are indexable inserts and indexable insert tool holders. Dimensional specifications for solid sintered carbide indexable inserts are given in ANSI B212.12-1991 (R2002). Samples of the many insert shapes are shown in Table 3b. Most modern, cemented carbide, face mill­ ing cutters are of the indexable insert type. Larger size end milling cutters, side milling or slotting cutters, boring tools, and a wide variety of special tools are made to use indexable inserts. These inserts are primarily made from cemented carbide, although most of the cemented oxide cutting tools are also indexable inserts. The objective of this type of tooling is to provide an insert with several cutting edges. When an edge is worn, the insert is indexed in the tool holder until all the cutting edges are used up, after which it is discarded. The insert is not intended to be reground. The advan­ tages are that the cutting edges on the tool can be rapidly changed without removing the tool holder from the machine, tool-grinding costs are eliminated, and the cost of the insert is less than the cost of a similar, brazed carbide tool. Of course, the cost of the tool holder must be added to the cost of the insert; however, one tool holder will usually last for a long time before it, too, must be replaced. Indexable inserts and tool holders are made with a negative or positive rake. Negative rake inserts have the advantage of having twice as many cutting edges available as comparable positive rake inserts, because the cutting edges on both the top and bottom of negative rake inserts can be used, while only the top cutting edges can be used on posi­tive rake inserts. Positive rake inserts have a distinct advantage when machining long and slender parts, thin-walled parts, or other parts that are subject to bending or chatter when the cutting load is applied to them, because the cutting force is significantly lower as com­pared to that for negative rake inserts. Indexable inserts can be obtained in the following forms: utility ground, or ground on top and bottom only; precision ground, or ground on all surfaces; prehoned to produce a slight rounding of the cutting edge; and precision molded, which are unground. Positive-negative rake inserts also are available. These inserts are held on a negative-rake tool holder and have a chipbreaker groove that is formed to pro­duce an effective positive-rake angle while cutting. Cutting edges may be available on the top surface only, or on both top and bottom surfaces. The positive-rake chipbreaker surface may be ground or precision molded on the insert. Many materials, such as gray cast iron, form a discontinuous chip. For these materials an insert that has plain faces without chipbreaker grooves should always be used. Steels and

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Machinery's Handbook, 31st Edition INDEXABLE INSERTS

other ductile materials form a continuous chip that must be broken into small segments when machined on lathes and planers having single-point, cemented carbide and cemented-oxide cutting tools; otherwise, the chips can cause injury to the operator. In this case a chipbreaker must be used. Some inserts are made with chipbreaker grooves molded or ground directly on the insert. When inserts with plain faces are used, a cementedcarbide plate-type chipbreaker is clamped on top of the insert.

Identification System for Indexable Inserts.—The size of indexable inserts is deter­ mined by the diameter of an inscribed circle (I.C.), except for rectangular and parallelo­ gram inserts where the length and width are used. To describe an insert in its entirety, a standard ANSI B212.4-2002 identification system is used where each position number designates a feature of the insert. The ANSI Standard includes items now commonly used and facilitates identification of items not in common use. Identification consists of up to ten positions; each position defines a characteristic of the insert as shown below: 1 2 3 4 5 6 7 8a 9a 10a T N M G 5 4 3 A a Eighth, Ninth, and Tenth Positions are used only when required.

1) Shape: The shape of an insert is designated by a letter: R for round; S, square; T, trian­ gle; A, 85° parallelogram; B, 82° parallelogram; C, 80° diamond; D, 55° diamond; E, 75° diamond; H, hexagon; K, 55° parallelogram; L, rectangle; M, 86° diamond; O, octagon; P, pentagon; V, 35° diamond; and W, 80° trigon. 2) Relief Angle (Clearances): The second position is a letter denoting the relief angles: N for 0°; A, 3°; B, 5°; C, 7°; P, 11°; D, 15°; E, 20°; F, 25°; G, 30°; H, 0° & 11°∗; J, 0° & 14°*; K, 0° & 17°*; L, 0° & 20°*; M, 11° & 14°*; R, 11° & 17°*; S, 11° & 20°*. When mounted on a holder, the actual relief angle may be different from that on the insert. 3) Tolerances: The third position is a letter and indicates the tolerances controlling the indexability of the insert. Tolerances specified do not imply the method of manufacture.

Symbol A B C D E F G

Tolerance (± from nominal) Inscribed Thickness, Circle, Inch Inch 0.001 0.001 0.001 0.005 0.001 0.001 0.001 0.005 0.001 0.001 0.0005 0.001 0.001 0.005

Symbol H J K L M U N

Tolerance (± from nominal) Inscribed Thickness, Circle, Inch Inch 0.0005 0.001 0.002–0.005 0.001 0.002–0.005 0.001 0.002–0.005 0.001 0.005 0.002–0.004a 0.005 0.005–0.010a 0.001 0.002–0.004a

a Exact tolerance is determined by size of insert. See ANSI B212.12.

4) Type: The type of insert is designated by a letter: A, with hole; B, with hole and coun­ tersink; C, with hole and two countersinks; F, chip grooves both surfaces, no hole; G, same as F but with hole; H, with hole, one countersink, and chip groove on one rake surface; J, with hole, two countersinks and chip grooves on two rake surfaces; M, with hole and chip groove on one rake surface; N, without hole; Q, with hole and two countersinks; R, without hole but with chip groove on one rake surface; T, with hole, one countersink, and chip groove on one rake face; U, with hole, two countersinks, and chip grooves on two rake faces; and W, with hole and one countersink. Note: a dash may be used after * Second angle is secondary facet angle, which may vary by ± 1°.

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Machinery's Handbook, 31st Edition INDEXABLE INSERTS

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position 4 to separate the shape-describing portion from the following dimensional description of the insert and is not to be considered a position in the standard description. 5) Size: The size of the insert is designated by a one- or two-digit number. For regular polygons and diamonds, insert size is the number of eighths of an inch in the nominal size of the inscribed circle, and will be a one- or two-digit number when the number of eighths is a whole number. It will be a two-digit number, including one decimal place, when it is not a whole number. Rectangular and parallelogram inserts require two digits: the first digit indicates the number of eighths of an inch width and the second digit the number of quar­ters of an inch length. 6) Thickness: The thickness is designated by a one- or two-digit number, which indicates the number of sixteenths of an inch in the thickness of the insert. Thickness is a one-digit number when the number of sixteenths is a whole number; it is a two-digit number carried to one decimal place when the number of sixteenths of an inch is not a whole number. 7) Cutting Point Configuration: The cutting point, or nose radius, is designated by a number representing 1 ⁄64 ths of an inch; a flat at the cutting point or nose, is designated by a letter: 0 for sharp corner, 0.002 inch max. radius; 0.2 for 0.004 radius; 0.3 for 0.008 radius; 1, 1 ⁄64 inch radius; 2, 1 ⁄32 inch radius; 3, 3⁄64 inch radius; 4, 1 ⁄16 inch radius; 5, 5 ⁄64 inch radius; 6, 3⁄32 inch radius; 7, 7⁄64 inch radius; 8, 1 ⁄8 inch radius; X, any other radius; A, square insert with 45° chamfer; D, square insert with 30° chamfer; E, square insert with 15° cham­fer; F, square insert with 3° chamfer; K, square insert with 30° double chamfer; L, square insert with 15° double chamfer; M, square insert with 3° double chamfer; N, truncated tri­angle insert; and P, flatted corner triangle insert. 8) Special Cutting Point Definition: The eighth position, if it follows a letter in the 7th position, is a number indicating the number of 1 ⁄64 ths of an inch in the primary facet length measured parallel to the edge of the facet. 9) Hand: R, right; L, left; to be used when required in ninth position. 10) Other Conditions: Position ten defines special conditions (such as edge treatment, surface finish): A, honed, 0.0005 to less than 0.003 inch (0.0127 to 0.0762 mm); B, honed, 0.003 to less than 0.005 inch (0.0762 to 0.127 mm); C, honed, 0.005 to less than 0.007 inch (0.127 to 0.178 mm); J, polished, 4 µinch (0.1016 µm) arithmetic average (AA) on rake surfaces only; T, chamfered, manufacturer’s standard negative land, rake face only. Indexable Insert Tool Holders.—Indexable insert tool holders are made from a good grade of steel which is heat treated to a hardness of 44 to 48 RC (Rockwell C scale) for most normal applica­tions. Accurate pockets that serve to locate the insert in position and to provide surfaces against which the insert can be clamped are machined in the ends of tool holders. A cemented carbide seat usually is provided and is held in the bottom of the pocket by a screw or by the clamping pin, if one is used. The seat is necessary to provide a flat bearing surface upon which the insert can rest and, in so doing, it adds materially to the ability of the insert to withstand the cutting load. The seating surface of the holder may provide a positive-, negative-, or a neutral-rake orientation to the insert when it is in position on the holder. Holders, therefore, are classified as positive, negative, or neutral rake. Four basic methods are used to clamp the insert on the holder: 1) Clamping, usually top clamping; 2) Pin-lock clamping; 3) Multiple clamping using a clamp, usually a top clamp, and a pin lock; and 4) Clamping the insert with a machine screw. All top clamps are actuated by a screw that forces the clamp directly against the insert. When required, a cemented carbide, plate-type chipbreaker is placed between the clamp and the insert. Pin-lock clamps require an insert having a hole: the pin acts against the walls of the hole to clamp the insert firmly against the seating surfaces of the holder. Multiple or combination clamping, simultaneously using both a pin-lock and a top clamp, is recom­mended when taking heavier or interrupted cuts. Holders are available on which all the above-mentioned methods of clamping may be used. Other holders are made with only a top clamp or a pin lock. Screw-on type holders use a machine screw to hold the

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Machinery's Handbook, 31st Edition INDEXABLE INSERTS

842

insert in the pocket. Most standard indexable insert holders are either straight-shank or offset-shank, although special holders are made having a wide variety of configurations. The common shank sizes of indexable insert tool holders are shown in Table 1. Not all styles are available in every shank size. Positive- and negative-rake tools are also not avail­able in every style or shank size. Some manufacturers provide additional shank sizes for certain tool holder styles. For more complete details, the manufacturers’ catalogs must be consulted. Table 1. Standard Shank Sizes for Indexable Insert Holders

B

A C Basic Shank Size

1 ⁄2 5 ⁄8 5 ⁄8 3 ⁄4 3 ⁄4 3 ⁄4

Shank Dimensions for Indexable Insert Holders

A

× 11 ⁄4 × 6

0.625

15.87

1.250

31.75

6.000

152.40

×1×6

0.750

19.05

1.000

25.40

6.000

152.40

0.625

15.87

0.750

× 3⁄4 × 41 ⁄2

0.625

19.05

0.750

× 11 ⁄4 × 6

inch 0.500

inch 4.500

15.87

0.750

19.05

mm 12.70

Ca

× 1 ⁄2 × 41 ⁄2

× 5⁄8 × 41 ⁄2

mm 12.70

B

inch 0.500

19.05

1.250

31.75

4.500

mm 114.30 114.30

4.500

114.30

6.000

152.40

1×1×6 1 × 11 ⁄4 × 6

1.000 1.000

25.40 25.40

1.000 1.250

25.40 31.75

6.000 6.000

152.40 152.40

11 ⁄4 × 11 ⁄4 × 7

1.250

31.75

1.250

31.75

7.000

177.80

1.000

1 × 11 ⁄2 × 6

25.40

1.500

38.10

6.000

152.40

11 ⁄4 × 11 ⁄2 × 8

1.250

31.75

1.500

38.10

8.000

203.20

11 ⁄2 × 11 ⁄2 × 7

1.500

38.10

1.500

38.10

7.000

177.80

2×2×8

2.000

50.80

2.000

50.80

8.000

203.20

1.375

13⁄8 × 21 ⁄16 × 63⁄8

34.92

1.750

13⁄4 × 13⁄4 × 91 ⁄2

2.062

44.45

52.37

1.750

44.45

6.380

162.05

9.500

241.30

a Holder length; may vary by manufacturer. Actual shank length depends on holder style.

Identification System for Indexable Insert Holders.—The following identification system conforms to the American National Standard, ANSI B212.5-2002, Metric Holders for Indexable Inserts. Each position in the system designates a feature of the holder in the following sequence: 1

2

3

4

5

—

6

—

7

—

8a

—

9

—

10a

C

T

N

A

R

—

85

—

25

—

D

—

16

—

Q

1) Method of Holding Horizontally Mounted Insert: The method of holding or clamping is designated by a letter: C, top clamping, insert without hole; M, top and hole clamping, insert with hole; P, hole clamping, insert with hole; S, screw clamping through hole, insert with hole; W, wedge clamping. 2) Insert Shape: The insert shape is identified by a letter: H, hexagonal; O, octagonal; P, pentagonal; S, square; T, triangular; C, rhombic, 80° included angle; D, rhombic, 55° included angle; E, rhombic, 75° included angle; M, rhombic, 86° included angle; V, rhom­ bic, 35° included angle; W, hexagonal, 80° included angle; L, rectangular; A, parallelo­ gram, 85° included angle; B, parallelogram, 82° included angle; K, parallelogram, 55° included angle; R, round. The included angle is always the smaller angle. 3) Holder Style: The holder style designates the shank style and the side cutting edge angle, or end cutting edge angle, or the purpose for which the holder is used. It is desig­nated

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Machinery's Handbook, 31st Edition INDEXABLE INSERTS

843

by a letter: A for straight shank with 0° side cutting edge angle; B, straight shank with 15° side cutting edge angle; C, straight-shank end cutting tool with 0° end cutting edge angle; D, straight shank with 45° side cutting edge angle; E, straight shank with 30° side cutting edge angle; F, offset shank with 0° end cutting edge angle; G, offset shank with 0° side cutting edge angle; J, offset shank with negative 3° side cutting edge angle; K, offset shank with 15° end cutting edge angle; L, offset shank with negative 5° side cutting edge angle and 5° end cutting edge angle; M, straight shank with 40° side cutting edge angle; N, straight shank with 27° side cutting edge angle; R, offset shank with 15° side cut­ting edge angle; S, offset shank with 45° side cutting edge angle; T, offset shank with 30° side cutting edge angle; U, offset shank with negative 3° end cutting edge angle; V, straight shank with 171 ⁄2° side cutting edge angle; W, offset shank with 30° end cutting edge angle; Y, offset shank with 5° end cutting edge angle. 4) Normal Clearances: The normal clearances of inserts are identified by letters: A, 3°; B, 5°; C, 7°; D, 15°; E, 20°; F, 25°; G, 30°; N, 0°; P, 11°. 5) Hand of tool: The hand of the tool is designated by a letter: R for right-hand; L, lefthand; and N, neutral, or either hand. 6) Tool Height for Rectangular Shank Cross Sections: The tool height for tool holders with a rectangular shank cross section and the height of cutting edge equal to shank height are given as two-digit numbers representing these values in millimeters. For example, a height of 32 mm would be encoded as 32; 8 mm would be encoded as 08, where the onedigit value is preceded by a zero. 7) Tool Width for Rectangular Shank Cross Sections: The tool width for tool holders with a rectangular shank cross section is given as a two-digit number representing this value in millimeters. For example, a width of 25 mm would be encoded as 25; 8 mm would be encoded as 08, where the one-digit value is preceded by a zero. 8) Tool Length: The tool length is designated by a letter: A, 32 mm; B, 40 mm; C, 50 mm; D, 60 mm; E, 70 mm; F, 80 mm; G, 90 mm; H, 100 mm; J, 110 mm; K, 125 mm; L, 140 mm; M, 150 mm; N, 160 mm; P, 170 mm; Q, 180 mm; R, 200 mm; S, 250 mm; T, 300 mm; U, 350 mm; V, 400 mm; W, 450 mm; X, special length to be specified; Y, 500 mm. 9) Indexable Insert Size: The size of indexable inserts is encoded as follows: For insert shapes C, D, E, H, M, O, P, R, S, T, V, the side length (the diameter for R inserts) in milli­ meters is used as a two-digit number, with decimals being disregarded. For example, the symbol for a side length of 16.5 mm is 16. For insert shapes A, B, K, L, the length of the main cutting edge or of the longer cutting edge in millimeters is encoded as a two-digit number, disregarding decimals. If the symbol obtained has only one digit, then it should be preceded by a zero. For example, the symbol for a main cutting edge of 19.5 mm is 19; for an edge of 9.5 mm, the symbol is 09. 10) Special Tolerances: Special tolerances are indicated by a letter: Q, back and end qualified tool; F, front and end qualified tool; B, back, front, and end qualified tool. A qual­ ified tool is one that has tolerances of ± 0.08 mm for dimensions F, G, and C. (See Table 2.) Table 2. Letter Symbols for Qualification of Tool Holders Position 10 ANSI B212.5-2002

Qualification of Tool Holder

Q

Letter Symbol F G ±0.08

F ± 0.08 C ±0.08

C ±0.08

Back and end qualified tool

Front and end qualified tool

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B F ±0.08

G ±0.08

C ±0.08

Back, front, and end qualified tool

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844

Machinery's Handbook, 31st Edition INDEXABLE INSERTS

Selecting Indexable Insert Holders.—A guide for selecting indexable insert holders is provided by Table 3b. Some operations such as deep grooving, cut-off, and threading are not given in this table. However, tool holders designed specifically for these operations are available. The boring operations listed in Table 3b refer primarily to larger holes into which the holders will fit. Smaller holes are bored using boring bars. An examination of this table shows that several tool-holder styles can be and frequently are used for each operation. Selection of the best holder for a given job depends largely on the job, and there are certain basic facts that should be considered in making the selection. Rake Angle: A negative-rake insert has twice as many cutting edges available as a com­ parable positive-rake insert. Sometimes the tool life obtained when using the second face may be less than that obtained on the first face because the tool wear on the cutting edges of the first face may reduce the insert strength. Nevertheless, the advantage of negativerake inserts and holders is such that they should be considered first in making any choice. Posi­tive-rake holders should be used where lower cutting forces are required, as when machin­ing slender or small-diameter parts, when chatter may occur, and for machining some materials, such as aluminum, copper, and certain grades of stainless steel, when positive-negative rake inserts can sometimes be used to advantage. These inserts are held on nega­tive-rake holders that have their rake surfaces ground or molded to form a positive-rake angle. Insert Shape: The configuration of the workpiece, the operation to be performed, and the lead angle required often determine the insert shape. When these factors need not be con­sidered, the insert shape should be selected on the basis of insert strength and the maximum number of cutting edges available. Thus, a round insert is the strongest and has a maximum number of available cutting edges. It can be used with heavier feeds while producing a good surface finish. Round inserts are limited by their tendency to cause chatter, which may preclude their use. The square insert is the next most effective shape, providing good corner strength and more cutting edges than all other inserts except the round insert. The only limitation of this insert shape is that it must be used with a lead angle. Therefore, the square insert cannot be used for turning square shoulders or for back-facing. Triangle inserts are the most versatile and can be used to perform more operations than any other insert shape. The 80-degree diamond insert is designed primarily for heavy turning and facing operations using the 100-degree corners, and, for turning and back-facing square shoulders, using the 80-degree corners. The 55- and 35-degree diamond inserts are intended primarily for tracing. Lead Angle: Tool holders should be selected to provide the largest possible lead angle, although limitations are sometimes imposed by the nature of the job. For example, when tuning and back-facing a shoulder, a negative lead angle must be used. Slender or smalldiameter parts may deflect, causing difficulties in holding size, or chatter when the lead angle is too large. End Cutting Edge Angle: When tracing or contour turning, the plunge angle is deter­ mined by the end cutting edge angle. A 2-deg minimum clearance angle should be pro­ vided between the workpiece surface and the end cutting edge of the insert. Table 3a provides the maximum plunge angle for holders commonly used to plunge when tracing where insert shape identifiers are S = square, T = triangle, D = 55-deg diamond, V = 35-deg diamond. When severe cratering cannot be avoided, an insert having a small, end cutting edge angle is desirable to delay the crater breakthrough behind the nose. For very heavy cuts, a small, end cutting edge angle will strengthen the corner of the tool. Tool holders for numerical control machines are discussed beginning page 856.

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Machinery's Handbook, 31st Edition INDEXABLE INSERTS

845

Table 3a. Maximum Plunge Angle for Tracing or Contour Turning Tool Maximum Tool Maximum Holder Insert Plunge Holder Insert Plunge Style Shape Angle Style Shape Angle E T J D 58° 30° D and S S J V 43° 50° H D N T 71° 55° 58° to 60° J T N D 25°

15°

15°

15°

5° 15°

T

A

R

A

R

B

T

B

T

B

S

B

C

C

T

d

d

d

P

d

d

d

N

d

d

d

P

d

d

d

N

d

d

d

N

d

d

d

N

d

d

d

P

d

d

d

N

d

d

d

d

P

d

d

d

d

N

d

d

d

P

d

d

d

N

d

d

N

d

d

d

d

P

d

d

d

d

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Bore

N

Plane

Chamfer

Groove

Trace

Turn and Face

Turn and Backface

Face

A

N-Negative P-Positive

T

Turn

0°

A

Application

Rake

0°

Insert Shape

Tool

Tool Holder Style

Table 3b. Indexable Insert Holder Application Guide

d

d

d

d

d

d

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Machinery's Handbook, 31st Edition INDEXABLE INSERTS

846

0°

10° 0°

38° –3°

–3°

–3°

15°

G

T

G

R

G

C

H

D

J

T

J

D

J

V

K

S

Plane

d

d

d

d

d

d

P

d

d

d

d

d

d

d

N

d

d

d

d

d

P

d

d

d

d

d

N

d

d

Groove

d

Trace

Bore

T

Chamfer

F

T

Turn and Face

E

N

Turn and Backface

30°

Face

S

Turn

D

N-Negative P-Positive

Insert Shape

45°

Tool

Application

Rake

Tool Holder Style

Table Table 3b. (Continued) Indexable Holder Application 3b. Indexable InsertInsert Holder Application GuideGuide

d

P

d

d

d

N

d

d

d

P

d

d

d

N

d

d

d

N

d

d

d

P

d

d

d

N

d

d

d

N

d

d

P

d

d

N

d

d

N

d

d

N

d

d

d

P

d

d

d

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Machinery's Handbook, 31st Edition INDEXABLE INSERTS

847

5° –5°

33° 27°

27°

N

D

45°

S

S

10°

W

T

N

Plane

Bore

Chamfer

Groove

d

Trace

T

d

Turn and Face

N

N

Turn and Backface

C

Face

L

Turn

C

N-Negative P-Positive

K

Application

Rake

15°

Insert Shape

Tool

Tool Holder Style

Table Table 3b. (Continued) Indexable Holder Application 3b. Indexable InsertInsert Holder Application GuideGuide

d

d

d

N

d

d

d

P

d

d

d

N

d

d

d

N

d

d

d

d

d

d

d

P

d

d

d

d

d

d

d

N

d

d

Sintered Carbide Blanks and Cutting Tools

Sintered Carbide Blanks.—As shown in Table 4, American National Standard ANSI B212.1-2002 provides standard sizes and designations for eight styles of sintered carbide blanks. These blanks are the unground solid carbide from which either solid or tipped cut­ ting tools are made. Tipped cutting tools are made by brazing a blank onto a shank to pro­ duce the cutting tool; these tools differ from carbide insert cutting tools which consist of a carbide insert held mechanically in a tool holder. A typical single-point carbide-tipped cut­ting tool is shown in Fig. 1 on page 849. Single-Point, Sintered Carbide-Tipped Tools.—American National Standard ANSI B212.1-2002 covers eight different styles of single-point, carbide-tipped general purpose tools. These styles are designated by the letters A to G inclusive. Styles A, B, F, G, and E with offset point are either right- or left-hand cutting as indicated by the letters R or L. Dimensions of tips and shanks are given in Table 5 to Table 12. For dimensions and tolerances not shown, and for the identification system, dimensions, and tolerances of sintered car­bide boring tools, see the Standard. A number follows the letters of the tool style and hand designation, and, for square shank tools, represents the number of sixteenths of an inch of width, W, and height, H. With rect­ angular shanks, the first digit of the number indicates the number of eighths of an inch in the shank width, W, and the second digit the number of quarters of an inch in the shank height, H. One exception is the 11 ⁄2 × 2-inch size, which has been arbitrarily assigned the number 90.

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Machinery's Handbook, 31st Edition Carbide Tips and Tools

848

Table 4. American National Standard Sizes and Designations for Carbide Blanks ANSI B212.1-2002 (R2007) Blank Dimensionsa T W L 1 ⁄16

1 ⁄8

5 ⁄8

1 ⁄16

5 ⁄32

1 ⁄4

1 ⁄16

3 ⁄16

1 ⁄4

1 ⁄16

1 ⁄4

1 ⁄4

1 ⁄16

1 ⁄4

5 ⁄16

3 ⁄32

1 ⁄8

3 ⁄4

3 ⁄32

3 ⁄16

5 ⁄16

3 ⁄32

3 ⁄16

1 ⁄2

3 ⁄32

1 ⁄4

3 ⁄8

3 ⁄32

1 ⁄4

1 ⁄2

3 ⁄32

5 ⁄16

3 ⁄8

3 ⁄32

3 ⁄8

3 ⁄8

3 ⁄32

3 ⁄8

1 ⁄2

3 ⁄32

7⁄16

1 ⁄2

3 ⁄32

5 ⁄16

3 ⁄8

1 ⁄8

3 ⁄16

3 ⁄4

1 ⁄8

1 ⁄4

1 ⁄2

1 ⁄8

1 ⁄4

5 ⁄8

1 ⁄8

1 ⁄4

3 ⁄4

1 ⁄8

5 ⁄16

7⁄16

1 ⁄8

5 ⁄16

1 ⁄2

1 ⁄8

3 ⁄16

3 ⁄4

1 ⁄8

5 ⁄16

5 ⁄8

1 ⁄8

3 ⁄8

1 ⁄2

1 ⁄8

3 ⁄8

3 ⁄4

1 ⁄8

1 ⁄2

1 ⁄2

1 ⁄8

1 ⁄2

3 ⁄4

1 ⁄8

3 ⁄4

3 ⁄4

5 ⁄32

3 ⁄8

9 ⁄16

5 ⁄32

3 ⁄8

3 ⁄4

5 ⁄32

5 ⁄8

5 ⁄8

3 ⁄16

5 ⁄16

7⁄16

3 ⁄16

5 ⁄16

5 ⁄8

3 ⁄16

3 ⁄8

1 ⁄2

3 ⁄16

3 ⁄8

5 ⁄8

3 ⁄16

3 ⁄8

3 ⁄4

3 ⁄16

7⁄16

5 ⁄8

3 ⁄16

7⁄16

13 ⁄16

3 ⁄16

1 ⁄2

1 ⁄2

3 ⁄16

1 ⁄2

3 ⁄4

3 ⁄16

3 ⁄4

3 ⁄4

Styleb 1000 2000 Blank Designation

Blank Dimensionsa T W L

1010

1 ⁄4

3 ⁄8

9 ⁄16

1 ⁄4

3 ⁄8

3 ⁄4

1 ⁄4

7⁄16

5 ⁄8

1 ⁄4

1 ⁄2

3 ⁄4

1 ⁄4

9 ⁄16

1 ⁄4

5 ⁄8

5 ⁄8

1 ⁄4

3 ⁄4

3 ⁄4

1 ⁄4

3 ⁄4

1015 1020 1025 1030 1035 1040 1050 1060 1070 1080 1090 1100 1105

1080 1110

1120 1130 1140 1150 1160 1110

1170 1180 1190

1200 1210 1215 1220 1230 1240 1250 1260 1270 1280 1290 1300 1310 1320 1330 1340

2010 2015 2020 2025 2030 2035 2040 2050 2060 2070 2080 2090

2100 2105

2080 2110

2120 2130 2140 2150 2160 2110

2170 2180 2190

2200 2210 2215 2220 2230 2240 2250 2260 2270 2280 2290 2300 2310 2320 2330 2340

1 ⁄4

1

1

1

0000

0370

1370

3370

4370 4390

7⁄16

15 ⁄16

5 ⁄16

1 ⁄2

3 ⁄4

5 ⁄16

1 ⁄2

5 ⁄16

5 ⁄8

5 ⁄16

3 ⁄4

5 ⁄16

3 ⁄4

1

5 ⁄16

3 ⁄4

11 ⁄4

3 ⁄8

1 ⁄2

3 ⁄4

3 ⁄8

1 ⁄2

3 ⁄8

5 ⁄8

3 ⁄8

5 ⁄8

0405

1405

3405

3 ⁄8

3 ⁄4

3 ⁄8

3 ⁄4

1 ⁄2

3 ⁄4

1410

3400

4400

3410

4410

3415

4415

0430

1430

3430

4430

3450

4450

0440

1420 1440

0450

1450

0470

1470

1460

3420 3440

3460

4475

3470

1475

3475

0490

1490

3490

4470

3480

4480 4490

0500

1500

3500

4500

0515

1515

3515

4515

0525

1510

1520 1525

3510

4510

3520

4520

3525

4525

1530

3530

4530

0490

1490

3490

4490

3 ⁄4

11 ⁄4

1 ⁄2

3 ⁄4

1 ⁄2

3 ⁄4

0540 0550

11 ⁄2

1540 1550

F

1 ⁄4

5 ⁄16

…

5000

1 ⁄16 3 ⁄32

1 ⁄4

3 ⁄8

1 ⁄16

…

W

L

5030

3540

4540

3550

4550

6000

70000

…

7060

6100

…

Styleb …

3 ⁄32

5 ⁄16

3 ⁄8

…

5080

6080

3 ⁄32

3 ⁄8

1 ⁄2

3 ⁄32

7⁄16

1 ⁄2

…

5105

…

1 ⁄8

5 ⁄16

5 ⁄8

3 ⁄32

3 ⁄32

1 ⁄4

3 ⁄8

1 ⁄16

1 ⁄8

1 ⁄2

1 ⁄2

…

5 ⁄32

3 ⁄8

3 ⁄4

1 ⁄8

5 ⁄32

5 ⁄8

5 ⁄8

3 ⁄16

3 ⁄4

3 ⁄4

1

4440

4460

0475

1480

4420

0530

3 ⁄8

1 ⁄4

4405

11 ⁄2 1

1 ⁄2

T

4380

1415

0520

11 ⁄4

1400

4360

0415

0510

11 ⁄4

3380

3390

0480

1

1380

3360

1390

0460 3 ⁄4

1360

0390

0420

5 ⁄16

1

4350

0410

5 ⁄8

1

3350

0400

1

1

1350

0380

7⁄16

4000

0350 0360

5 ⁄16

Styleb 1000 3000 Blank Designation

3 ⁄4

…

5100 … …

… …

5200

6200

…

5240

6240

…

5410

…

…

…

5340

… … …

7170 7060 …

…

7230

6340

…

… …

a All dimensions are in inches. b See Fig. 1 on page 849 for a description of styles.

A typical single-point carbide-tipped cutting tool is shown in Fig. 2. The side rake, side relief, and the clearance angles are normal to the side-cutting edge, rather than the shank, to facilitate its being ground on a tilting-table grinder. The end-relief and clearance angles are normal to the end-cutting edge. The back-rake angle is parallel to the side-cutting edge.

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Machinery's Handbook, 31st Edition Carbide Tips and Tools

13° ± 1°

W

W T

T

L Style 0000

13° ± 1°

W

R

L

L

13° ± 1°

W T

Style 4000

Style 3000 1 64

L Style 2000

7° ± 1°

7° ± 1°

40° ± 1° Sharp to

R

W T

L Style 1000

R

T

849

30° ± 1° Sharp to

Flat

W

1 64

Flat

30° ± 1°

W

W

T L 40° ± 1° Style 5000 Point Central Within 0.015

L 30° ± 1°

+ 0.000 F – 0.020

T

Style 6000 Point Central Within 0.015

L 30° ± 1°

T Sharp to

Style 7000

1 64

Flat

Fig. 1. Eight styles of Sintered Carbide Blanks (see Table 4)

Side Rake

Side Relief Angle Side Clearance Angle

Tip Width

Tip Overhang Nose Radius

Tip Thickness Cutting Height

End Cutting Edge Angle (ECEA) Shank Width Side Cutting Edge Angle (SCEA) Overall length

Tip Length

Back Rake

Shank Height

Tip Overhang End Relief Angle End Clearance Angle Fig. 2. A Typical Single-Point Carbide-Tipped Cutting Tool

The tip of the brazed carbide blank overhangs the shank of the tool by either 1 ⁄32 or 1 ⁄16 inch, depending on the size of the tool. For tools in Table 5, Table 6, Table 7, Table 8, Table 11 and Table 12, the maximum overhang is 1 ⁄32 inch for shank sizes 4, 5, 6, 7, 8, 10, 12 and 44; for other shank sizes in these tables, the maximum overhang is 1 ⁄16 inch. In Table 9 and Table 10, all tools have maximum over­hang of 1 ⁄32 inch.

Single-point Tool Nose Radii: The tool nose radii recommended in the American National Standard are as follows: For square-shank tools up to and including 3⁄8 -inch

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850

square tools, 1 ⁄64 inch; for those over 3⁄8 -inch square through 11 ⁄4 -inches square, 1 ⁄32 inch; and for those above 11 ⁄4 -inches square, 1 ⁄16 inch. For rectangular-shank tools with shank sections of 1 ⁄2 × 1 inch through 1 × 11 ⁄2 inches, the nose radii are 1 ⁄32 inch, and for 1 × 2and 11 ⁄2 × 2-inch shanks, the nose radius is 1 ⁄16 inch. Single-point Tool Angle Tolerances: The tool angles shown on the diagrams in Table 5 through Table 12 are general recommendations. Tolerances applicable to these angles are ± 1 degree on all angles except end and side clearance angles; for these the tolerance is ± 2 degrees. Table 5. American National Standard Style A Carbide-Tipped Tools ANSI B212.1-2002 (R2007) 7° ± 1° 10° ± 2°

6° ± 1°

8° ± 1° W

Overhang

R

0° ± 1°

T

A

F

0° ± 1°

L

C Tool Designation and Carbide Grade

H

Overhang

B

7° ± 1° 10° ± 2° Designation Style ARa

Style ALa

Style: AR Right Hand (shown) Style: AL Left Hand (not shown)

Shank Dimensions

Tip Dimensions Tip Designationa

Width A

Height B

Length C

1 ⁄4

1 ⁄4

2

5 ⁄16

5 ⁄16

21 ⁄4

2070

7⁄16

3

2070

1 ⁄2

1 ⁄2

31 ⁄2

5 ⁄8

5 ⁄8

4

3 ⁄4

3 ⁄4

41 ⁄2

Thickness T

Width W

Length L

Square Shank AR 4

AR 5 AR 6 AR 7 AR 8

AR 10 AR 12 AR 16

AR 20 AR 24 AR 44

AR 54 AR 55 AR 64 AR 66 AR 85

AR 86 AR 88 AR 90

AL 4

AL 5 AL 6 AL 7 AL 8

AL 10 AL 12 AL 16

AL 20 AL 24 AL 44

AL 54 AL 55 AL 64 AL 66 AL 85

AL 86 AL 88 AL 90

3 ⁄8 7⁄16

3 ⁄8

1 11 ⁄4

11 ⁄2 1 ⁄2

1 11 ⁄4

11 ⁄2 1

5 ⁄8

1

5 ⁄8

11 ⁄4

2040

2070

21 ⁄2

6

7 8

2170

2230

{

{ {

2310

P3390, P4390

P3460, P4460 P3510, P4510

3 ⁄32

3 ⁄16

5 ⁄16

3 ⁄32

1 ⁄4

1 ⁄2

3 ⁄32

1 ⁄4

1 ⁄2

3 ⁄32

1 ⁄4

1 ⁄2

1 ⁄8

5 ⁄16

5 ⁄8

5 ⁄32

3 ⁄8

3 ⁄4

3 ⁄16

7⁄16

13 ⁄16

1 ⁄4

9 ⁄16

5 ⁄16

5 ⁄8

3 ⁄8

5 ⁄8

1 1 1

Rectangular Shank 6 6

7

{

P3360, P4360

P3360, P4360

5 ⁄16

5 ⁄8

3 ⁄8

3 ⁄4

1 ⁄4

3 ⁄8

3 ⁄4

1 ⁄4

1 ⁄2

3 ⁄4 15 ⁄16

3 ⁄4

11 ⁄2

8

{

P3430, P4430

5 ⁄16

7⁄16

5 ⁄16

5 ⁄8

11 ⁄2

8

{

P3510, P4510

3 ⁄8

5 ⁄8

3 ⁄8

5 ⁄8

1

1 ⁄2

3 ⁄4

11 ⁄4

1 11 ⁄2

11 ⁄4 2

2

7 10

10

{ {

{

P3380, P4380

3 ⁄16 1 ⁄4

1

1

{

P2260

3 ⁄4

1

6

{

P3460, P4460 P3510, P4510

P3540, P4540

1 1

a “A” is straight shank, 0 deg., SCEA (side-cutting-edge angle). “R” is right-cut. “L” is left-cut. Where a pair of tip numbers is shown, the upper number applies to AR tools, the lower to AL tools. All dimensions are in inches.

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Table 6. American National Standard Style B Carbide-Tipped Tools with 15-degree Side-cutting-edge Angle ANSI B212.1-2002 (R2007) 7° ±1° 10° ± 2°

Overhang

6° ± 1°

To sharp corner

W

T

15° ± 1°

F Ref

A

L

R

15° ± 1° C

0° ± 1° H

Tool designation and carbide grade

Overhang

7° ±1°

10° ± 2°

Designation Style Style BR BL BR 4 BR 5 BR 6 BR 7 BR 8 BR 10 BR 12 BR 16 BR 20 BR 24

BL 4 BL 5 BL 6 BL 7 BL 8 BL 10 BL 12 BL 16 BL 20 BL 24

BR 44 BR 54 BR 55 BR 64 BR 66 BR 85 BR 86 BR 88 BR 90

BL 44 BL 54 BL 55 BL 64 BL 66 BL 85 BL 86 BL 88 BL 90

B

Style GR right hand (shown) Style GE left hand (not shown)

Width A

Shank Dimensions Height Length B C

Square Shank

1 ⁄4

1 ⁄4

3 ⁄8

3 ⁄8

5 ⁄16

5 ⁄16

7⁄16 1 ⁄2

7⁄16 1 ⁄2

3 ⁄4

3 ⁄4

5 ⁄8

5 ⁄8

1 11 ⁄4 11 ⁄2

1 11 ⁄4 11 ⁄2

1 ⁄2

1 1 11 ⁄4 1 11 ⁄2 11 ⁄4 11 ⁄2 2 2

5 ⁄8 5 ⁄8 3 ⁄4 3 ⁄4

1 1 1 11 ⁄2

Tip Designationa

2 2015 2040 21 ⁄4 2070 21 ⁄2 3 2070 2170 31 ⁄2 4 2230 2310 41 ⁄2 6 { 3390, 4390 7 { 3460, 4460 8 { 3510, 4510 Rectangular Shank 6 2260 6 { 3360, 4360 7 { 3360, 4360 6 { 3380, 4380 8 { 3430, 4430 7 { 3460, 4460 8 { 3510, 4510 10 { 3510, 4510 10 { 3540, 4540

Tip Dimensions Thickness Width T W 1 ⁄16 3 ⁄32

5 ⁄32

3 ⁄32

3 ⁄16 1 ⁄4

5 ⁄32

5 ⁄16 3 ⁄8

3 ⁄32 1 ⁄8 3 ⁄16

1 ⁄4 5 ⁄16

1 ⁄4

7⁄16

9 ⁄16

Length L 1 ⁄4

5 ⁄16 1 ⁄2 1 ⁄2 5 ⁄8

3 ⁄4 13 ⁄16

1 1 1

3 ⁄8

5 ⁄8 5 ⁄8

3 ⁄16 1 ⁄4

5 ⁄16 3 ⁄8

5 ⁄8

1 ⁄4

1 ⁄2

5 ⁄16 3 ⁄8

5 ⁄8

3 ⁄4 15 ⁄16

1 ⁄2

3 ⁄4

1 ⁄4

5 ⁄16

3 ⁄8

3 ⁄8

7⁄16

5 ⁄8 5 ⁄8

3 ⁄4 3 ⁄4

1 1 1 11 ⁄4

Where a pair of tip numbers is shown, the upper number applies to BR tools, the lower to BL tools. All dimensions are in inches. a

Brazing Carbide Tips to Steel Shanks.—Sintered carbide tips or blanks are attached to steel shanks by brazing. Shanks usually are made of low-alloy steels having carbon contents ranging from 0.40 to 0.60 percent. Shank Preparation: The carbide tip usually is inserted into a milled recess or seat. When a recess is used, the bottom should be flat to provide a firm even support for the tip. The corner radius of the seat should be somewhat smaller than the radius on the tip to avoid contact and insure support along each side of the recess. Cleaning: All surfaces to be brazed must be absolutely clean. Surfaces of the tip may be cleaned by grinding lightly or by sand-blasting. Brazing Materials and Equipment: The brazing metal may be copper, naval brass such as Tobin bronze, or silver solder. A flux such as borax is used to protect the clean surfaces and prevent oxidation. Heating may be done in a furnace or by oxy-acetylene torch or an oxy-hydrogen torch. Copper brazing usually is done in a furnace, although an oxy-hydrogen torch with excess hydrogen is sometimes used. Brazing Procedure: One method using a torch is to place a thin sheet material, such as copper foil, around and beneath the carbide tip, the top of which is covered with flux. The flame is applied to the underside of the tool shank, and, when the materials melt, the tip is pressed firmly into its seat with tongs or with the end of a rod. Brazing material in the form of wire or rod may be used to coat or tin the surfaces of the recess after the flux

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Table 7. American National Standard Style C Carbide-Tipped Tools ANSI B212.1-2002 (R2007) 3°I2°

Overhang

5° ± 2° Both sides 0° ± 1°

0.015 × 45° Maximum permissible

W

A

F 90° ± 1°

0° ± 1°

T

L

H

Overhang

C4 C5 C6 C7 C8 C 10 C 12 C 16 C 20

Width, A

Shank Dimensions Height, B Length, C

C 44 C 54 C 55 C 64 C 66 C 86

1 11 ⁄4

1 1 11 ⁄4 1 11 ⁄2 11 ⁄2

6 6 7 6 8 8

1320 1400 1400 1405 1470 1475

5 ⁄16

5 ⁄16 3 ⁄8

7⁄16

7⁄16

1 ⁄2

1 ⁄2

5 ⁄8

5 ⁄8

3 ⁄4

3 ⁄4

5 ⁄8 5 ⁄8 3 ⁄4

1

3 ⁄4

Tip Designation 1030 1080 1090 1105 1200 1240 1340 1410 1480

3 ⁄8

1 ⁄2

Note – Tool must pass through slot of nominal width “A”

2 21 ⁄4 21 ⁄2 3 31 ⁄2 4 41 ⁄2 6 7

1 ⁄4

1 ⁄4

1 11 ⁄4

B

7° ± 1°

10° ± 2°

Designation

C Tool designation and carbide grade

Thickness, T

Tip Dimensions Width, W

Length, L

1 ⁄16

1 ⁄4

3 ⁄32

5 ⁄16

3 ⁄8

3 ⁄32

3 ⁄8

3 ⁄8

5 ⁄16

3 ⁄32

7⁄16

1 ⁄2

1 ⁄8

1 ⁄2

1 ⁄2

5 ⁄32

5 ⁄8

5 ⁄8

3 ⁄16

3 ⁄4

1 ⁄4 5 ⁄16

3 ⁄4

1 11 ⁄4

3 ⁄4 3 ⁄4

3 ⁄16

1 ⁄2

1 ⁄2

1 ⁄4

5 ⁄8

5 ⁄8

1 ⁄4

5 ⁄8

5 ⁄8

1 ⁄4

3 ⁄4

5 ⁄16

3 ⁄4

3 ⁄4 3 ⁄4

1

5 ⁄16

3 ⁄4

All dimensions are in inches. Square shanks above horizontal line, rectangular below.

Table 8. American National Standard Style D, 80-degree Nose-angle Carbide-Tipped Tools ANSI B212.1-2002 (R2007) 10° ± 2° Both sides 7° ± 1° Overhang W

0° ± 1°

Note – Tool must pass through slot of nominal width “A”

40° ± 1°

R

A

F

To sharp corner 0° ± 1°

T

40° ± 1° C± L

H Designation D4 D5 D6 D7 D8 D 10 D 12 D 16

Width, A

1

1 ⁄4

1 ⁄4

5 ⁄16

5 ⁄16

3 ⁄8

3 ⁄8

7⁄16

7⁄16

1 ⁄2

1 ⁄2

5 ⁄8

5 ⁄8

3 ⁄4

3 ⁄4

1

1 8

Tool designation and carbide grade

B

Shank Dimensions Height, B Length, C 2 21 ⁄4 21 ⁄2 3 31 ⁄2 4 41 ⁄2 6

+0.000 –0.010

Tip Designation 5030 5080 5100 5105 5200 5240 5340 5410

+0.000 –0.010

Thickness, T

Tip Dimensions Width, W

Length, L

1 ⁄16

1 ⁄4

3 ⁄32

5 ⁄16

3 ⁄8

3 ⁄32

3 ⁄8

1 ⁄2

3 ⁄32

7⁄16

1 ⁄2

1 ⁄8

1 ⁄2

1 ⁄2

5 ⁄32

5 ⁄8

5 ⁄8

3 ⁄16

3 ⁄4

1 ⁄4

1

5 ⁄16

3 ⁄4 3 ⁄4

All dimensions are in inches.

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Machinery's Handbook, 31st Edition Carbide Tips and Tools

853

Table 9. American National Standard Style E, 60-degree Nose-angle, Carbide-Tipped Tools ANSI B212.1-2002 (R2007) 6° ± 2° Both sides 3° ± 1°

30° ± 1° 60° ± 1°

W

0.000 A+ – 0.010

F

0° ± 1°

1 8

C±

To sharp corner

T

0° ± 1°

Tool designation and carbide grade

L

0.000 B + – 0.010

H

Overhang at point

E4

E5 E6 E8

E 10 E 12

Height B

1 ⁄4

1 ⁄4

5 ⁄16

5 ⁄16

3 ⁄8

3 ⁄8

1 ⁄2

1 ⁄2

5 ⁄8

5 ⁄8

3 ⁄4

3 ⁄4

Note – Tool must pass through slot of nominal width “A”

max

Shank Dimensions

Width A

Designation

1 32

Length C

Thickness T

6030

1 ⁄16

1 ⁄4

5 ⁄16

5 ⁄16

3 ⁄8

2

Width W

Length L

21 ⁄4

6080

3 ⁄32

31 ⁄2 4

6200

3 ⁄32

3 ⁄8

1 ⁄2

1 ⁄8

1 ⁄2

1 ⁄2

5 ⁄32

5 ⁄8

5 ⁄8

3 ⁄16

3 ⁄4

3 ⁄4

6100

21 ⁄2

6240

6340

41 ⁄2

All dimensions are in inches.

Tip Dimensions

Tip Designation

Table 10. American National Standard Styles ER and EL, 60-degree Nose-angle, Carbide-Tipped Tools with Offset Point ANSI B212.1-2002 (R2007) 6° ± 2° Both sides

F ± 0.010 to sharp corner

3° ± 1°

W 0° ± 1°

0.000 A+ – 0.010

60° ± 1°

Flush to 0.015

30° ± 1°

T

0° ± 1°

L

C±

1 8

Tool designation and carbide grade

0.000 B+ – 0.010

H

Overhang at point Designation Style Style ER EL ER 4

ER 5 ER 6 ER 8

ER 10 ER 12

EL 4

Width A

1 32

Shank Dimensions Height B

EL 5

1 ⁄4

1 ⁄4

5 ⁄16

5 ⁄16

EL 8

3 ⁄8

3 ⁄8

1 ⁄2

1 ⁄2

5 ⁄8

5 ⁄8

3 ⁄4

3 ⁄4

EL 6 EL 10 EL 12

max

All dimensions are in inches.

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Length C 2

21 ⁄4

21 ⁄2 31 ⁄2 4 41 ⁄2

Style ER right hand (shown) Style EL left hand (not shown) Tip Designa­ tion 1020

Thick. T

Tip Dimensions Width W

Length L

7060

1 ⁄16

3 ⁄16

1 ⁄4

3 ⁄32

1 ⁄4

3 ⁄8

7170

3 ⁄32

1 ⁄4

3 ⁄8

1 ⁄8

5 ⁄16

5 ⁄8

1 ⁄8

5 ⁄16

5 ⁄8

5 ⁄32

3 ⁄8

3 ⁄4

7060 7170

7230

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Machinery's Handbook, 31st Edition Carbide Tips and Tools

854

Table 11. American National Standard Style F, Offset, End-cutting Carbide-Tipped Tools ANSI B212.1-2002 (R2007) +0 F– 1

Overhang

10° ± 2°

1

L

16

0° ± 1° D min

7° ± 1°

G Ref

R +0 E– 1 8

6° ± 1°

Style FR right hand (shown) 10° ± 2° Style FL left hand (not shown)

7° ± 1°

Designation

Shank Dimensions

Style FL

Width A

Height B

FR 8 FR 10 FR 12 FR 16 FR 20 FR 24

FL 8 FL 10 FL 12 FL 16 FL 20 FL 24

1 ⁄2

1 ⁄2

3 ⁄4

3 ⁄4

FR 44 FR 55 FR 64 FR 66 FR 85 FR 86 FR 90

FL 44 FL 55 FL 64 FL 66 FL 85 FL 86 FL 90

Length C

Offset G

Length of Offset E

Square Shank

5 ⁄8

1

11 ⁄4 11 ⁄2 1 ⁄2 5 ⁄8 3 ⁄4 3 ⁄4

1 1

11 ⁄2

5 ⁄8

1

11 ⁄4 11 ⁄2 1

11 ⁄4 1

11 ⁄2 11 ⁄4 11 ⁄2 2

31 ⁄2 4

41 ⁄2 6 7 8 6 7 6 8 7 8 10

{ { 5 ⁄8 { 11 ⁄8 3 ⁄4 { 13⁄8 3 ⁄4 { 11 ⁄2 3 ⁄4 { 11 ⁄2 Rectangular Shank 7⁄8 1 ⁄2 { 5 ⁄8 { 11 ⁄8 5 ⁄8 { 13⁄16 3 ⁄4 { 11 ⁄4 3 ⁄4 { 11 ⁄2 3 ⁄4 { 11 ⁄2 3 ⁄4 { 15⁄8 1 ⁄4

3 ⁄8

C

B

T

Overhang

Style FR

8° ± 1°

Tool designation and carbide grade

W H

A ± 32

3 ⁄4

1

Tip Dimensions Tip Designation

P4170, P3170 P1230, P3230 P4310, P3310 P4390, P3390 P4460, P3460 P4510, P3510 P4260, P1260 P4360, P3360 P4380, P3380 P4430, P3430 P4460, P3460 P4510, P3510 P4540, P3540

Thickness T

Width W

1 ⁄8

5 ⁄16

5 ⁄8

3 ⁄16

7⁄16

13 ⁄16

5 ⁄32

3 ⁄8

Length L

3 ⁄4

5 ⁄8 5 ⁄8

1 1 1

5 ⁄16

5 ⁄8

1 ⁄4

1 ⁄2

3 ⁄4

5 ⁄16

5 ⁄8

1 ⁄2

3 ⁄4

1 ⁄4

5 ⁄16

3 ⁄8

3 ⁄16

1 ⁄4

5 ⁄16

3 ⁄8

9 ⁄16

3 ⁄8

7⁄16

5 ⁄8

3 ⁄4

15 ⁄16

1 1

11 ⁄4

All dimensions are in inches. Where a pair of tip numbers is shown, the upper number applies to FR tools, the lower to FL tools.

melts and runs freely. The tip is then inserted, flux is applied to the top, and heating is continued until the coatings melt and run freely. The tip, after coating with flux, is placed in the recess and the shank end is heated. Then a small piece of silver solder, having a melting point of 1325°F (718°C), is placed on top of the tip. When this solder melts, it runs over the nickel-coated surfaces while the tip is held firmly into its seat. The brazed tool should be cooled slowly to avoid cracking due to unequal contraction between the steel and carbide. Carbide Tools.—Cemented or sintered carbides are used in the machine-building and var­ious other industries, chiefly for cutting tools but also for certain other tools or parts subject to considerable abrasion or wear. Carbide cutting tools, when properly selected to obtain the right combination of strength and hardness, are very effective in machining all classes of iron and steel, nonferrous alloys, non-metallic materials, hard rubber, synthetic resins, slate, marble, and other materials which would quickly dull steel tools either because of hardness or abrasive action. Carbide cutting tools are not only durable, but capable of exceptionally high cutting speeds. See CEMENTED CARBIDES starting on page 860 for more on these materials. Tungsten carbide is used extensively in cutting cast iron, nonferrous metals that form short chips in cutting, plastics and various other non-metallic materials. A grade having

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Machinery's Handbook, 31st Edition Carbide Tips and Tools

855

Table 12. American National Standard Style G, Offset, Side-cutting, Carbide-Tipped Tools ANSI B212.1-2002 (R2007) 6° ± 1°

Overhang W 10° ± 2°

T

7° ± 1°

8° ± 1°

+0 F– 1

16

R

0° ± 1°

+0 E– 1 8

C

0° ± 1°

Tool designation and carbide grade

H Overhang

Style GR

Style GL

Width A

Height B

GR 8

GL 8

1 ⁄2

1 ⁄2

GR 10

GL 10

5 ⁄8

5 ⁄8

GR 12

GL 12

3 ⁄4

3 ⁄4

B 7° ± 1°

10° ± 2°

Shank Dimensions

A ± 32

L

D min

Designation

1

G Ref

Tip Dimensions

Length of Offset E

Length C

Offset G

31 ⁄2

1 ⁄4

4

11 ⁄16

3 ⁄8

13⁄8

41 ⁄2

3 ⁄8

11 ⁄2

Style GR right hand (shown) Style GL left hand (not shown)

Tip Designation

Thickness T

Width W

Length L

Square Shank

GR 16

GL 16

1

1

GR 20

GL 20

GR 24

GL 24

11 ⁄4

11 ⁄4

11 ⁄2

13 ⁄16

{

P3390, P4390

1 ⁄4

9 ⁄16

1

{

P3460, P4460

5 ⁄16

5 ⁄8

1

P3510, P4510

3 ⁄8

5 ⁄8

1

3 ⁄4

113⁄16

{ {

P3260, P4260

3 ⁄16

5 ⁄16

5 ⁄8

{

P3360, P4360

1 ⁄4

3 ⁄8

3 ⁄4

{

P3380, P4380

1 ⁄4

1 ⁄2

3 ⁄4

{

P3430, P4430

5 ⁄16

7⁄16

15 ⁄16

{

P3460, P4460

5 ⁄16

5 ⁄8

1

{

P3510, P4510

3 ⁄8

5 ⁄8

1

{

P3540, P4540

1 ⁄2

3 ⁄4

11 ⁄4

Rectangular Shank 6

1 ⁄4

7

11 ⁄16

3 ⁄8

GR 64

GL 64

3 ⁄4

1

6

13⁄8

1 ⁄2

GR 66

GL 66

3 ⁄4

11 ⁄2

8

17⁄16

1 ⁄2

7

15⁄8

1 ⁄2

8

111 ⁄16

1 ⁄2

10

111 ⁄16

3 ⁄4

21 ⁄16

11 ⁄2

7⁄16

8

11 ⁄4

GL 90

3 ⁄16

11 ⁄2 1

GR 90

P3310, P2310

113⁄16

5 ⁄8

1

3 ⁄4

{

5 ⁄8

111 ⁄16

1 ⁄2

1

3 ⁄8

3 ⁄4

GL 55

GL 85

5 ⁄16

5 ⁄32

1 ⁄2

GL 44

GL 86

1 ⁄8

P3230, P4230

6

GR 55

GR 85

P3170, P4170

{

7

GR 44

GR 86

{

11 ⁄4 11 ⁄2

2

All dimensions are in inches. Where a pair of tip numbers is shown, the upper number applies to GR tools, the lower to GL tools.

a hardness of 87.5 RA on the Rockwell A scale might be used where a strong grade is required, as for rough­ing cuts, whereas for light high-speed finishing or other cuts, a hardness of about 92 RA might be preferable. When tungsten carbide is applied to steel, craters or chip cavities are formed back of the cutting edge; hence other carbides have been developed that offer greater resistance to abrasion. Tungsten-titanium carbide (often called “titanium carbide”) is adapted to cutting either heat-treated or non-heat-treated steels, cast steel, or any tough material that might form chip cavities. It is also applicable to bronzes, monel metal, aluminum alloys, etc. Tungsten-tantalum carbide or “tantalum carbide” cutting tools are also applicable to steels, bronzes or other tough materials. A hardness of 86.8 RA is recommended by one manufacturer for roughing steel, whereas a grade for finishing might have a hard­ness ranging from 88.8 to 91.5 RA.

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856

Machinery's Handbook, 31st Edition Carbide Tips and Tools

Chipbreaker.—The term “chipbreaker” indicates a method of forming or grinding turn­ing tools which will cause the chips to break up into short pieces, thus preventing the forma­tion of long or continuous chips that would occupy considerable space and be difficult to handle. The chipbreaking form of cutting end is especially useful in turning with carbide-tipped steel turning tools because the cutting speeds are high and the chip formation rapid. The chipbreaker consists of a shoulder back of the cutting edge. As the chip encounters this shoulder it is bent and broken repeatedly into small pieces. Some tools have attached or “mechanical” chipbreakers, which serve the same purpose as the shoulder.

Chipless Machining.—Chipless machining is the term applied to methods of cold form­ ing metals to the required finished part shape (or nearly finished shape) without the pro­ duction of chips (or with a minimum of subsequent machining required). Cold forming of steel has long been performed in such operations as wire-, bar-, and tube-drawing; cold-heading; coining; and conventional stamping and drawing. However, newer methods of plastic deformation with greatly increased degrees of metal displacement have been devel­oped. Among these processes are the rolling of serrations, splines, and gears; power spin­ning; internal swaging; radial forging; the cold forming of multiple-diameter shafts; cold extrusion; and high-energy-rate forming, which includes explosive forming. The pro­cesses of cold heading, thread rolling and rotary swaging are also considered chipless machining processes.

Indexable Insert Holders for NC.—Indexable insert holders for numerical control lathes are usually made to more precise standards than ordinary holders. Where applicable, reference should be made to American National Standard B212.3-2002, Precision Holders for Indexable Inserts. This standard covers the dimensional specifications, styles, and des­ignations of precision holders for indexable inserts, which are defined as tool holders that locate the gage insert (a combination of shim and insert thicknesses) from the back or front and end surfaces to a specified dimension with a ± 0.003 inch (± 0.08 mm) tolerance. In NC programming, the programmed path is that followed by the center of the tool tip, which is the center of the point, or nose radius, of the insert. The surfaces produced are the result of the path of the nose and the major cutting edge, so it is necessary to compensate for the nose or point radius and the lead angle when writing the program. Table 1, from B212.3, gives the compensating dimensions for different holder styles. The reference point is determined by the intersection of extensions from the major and minor cutting edges, which would be the location of the point of a sharp pointed tool. The distances from this point to the nose radius are L1 and D1; L2 and D2 are the distances from the sharp point to the center of the nose radius. Threading tools have sharp corners and do not require a radius compensation. Other dimensions of importance in programming threading tools are also given in Table 2; the data were developed by Kennametal, Inc. The C and F characters are tool holder dimensions other than the shank size. In all instances, the C dimension is parallel to the length of the shank and the F dimension is par­ allel to the side dimension; actual dimensions must be obtained from the manufacturer. For all K style holders, the C dimension is the distance from the end of the shank to the tangent point of the nose radius and the end cutting edge of the insert. For all other holders, the C dimension is from the end of the shank to a tangent to the nose radius of the insert. The F dimension on all B, D, E, M, P, and V style holders is measured from the back side of the shank to the tangent point of the nose radius and the side cutting edge of the insert. For all A, F, G, J, K, and L style holders, the F dimension is the distance from the back side of the shank to the tangent of the nose radius of the insert. In all these designs, the nose radius is the standard radius corresponding to those given in the paragraph Cutting Point Configuration on page 841.

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Table 1. Insert Radius Compensation ANSI B212.3-2002 (R2011) Square Profile

C

B Stylea Also applies to R Style

D-1

F

C

.0191

.0009

3⁄ 64

.0105

.0574

.0028

Rad.

L-1 D-2

45°

D-2

.0038

.0442

Turning 45° Lead Angle D-1 L-1 L-2

3⁄ 64

.0194

.0663

.0194

.0129 .0259

.0442 .0884

.0129 .0259

Facing 15° Lead Angle L-1 L-2 D-1

1⁄ 64

.0009

3⁄ 64

.0028

1⁄ 16

C

.0221

.0065

1⁄ 32

L-1 L-2

.0765

D-2

.0110

.0019

.0221

Rad. D-1

.0140

.0383

.0065

1⁄ 16

15°

.0070

1⁄ 64 1⁄ 32

D-1 L-2

F

.0035

1⁄ 16

15°

Turning 15° Lead Angle L-1 L-2 D-1

1⁄ 64 1⁄ 32

D-2 L-2

D Stylea Also applies to S Style

K Stylea

Rad.

F L-1

.0331

D-2 0 0 0 0 D-2

.0110

.0035

.0191

.0331

.0105

.0574

.0019

.0221

.0038

.0442

.0070 .0140

.0383 .0765

Triangle Profile F

G Stylea

L-1

C

L-2 D-1 0°

1⁄ 64

D-2

B Stylea Also applies to R Style

F

L-1

F

.0114

.0271

0

D-2

.0156

.0229

.0541

0

.0312

1⁄ 16

.0458

.1082

0

.0625

.0343

.0812

0

.0469

Turning and Facing 15° Lead Angle Rad. L-1 L-2 D-1 D-2

C

1⁄ 64

L-2

15°

C

F Stylea

Turning 0° Lead Angle L-1 L-2 D-1

1⁄ 32 3⁄ 64

D-1

D-2

Rad.

.0078

1⁄ 16

.0582

.1207

.0156

3⁄ 64

Rad.

3⁄ 64

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.0039

.0604

1⁄ 32

D-2 D-1

.0302

.0291

1⁄ 64

90° L-2 L-1

.0146

1⁄ 32

1⁄ 16

.0437

.0906

.0117

Facing 90° Lead Angle L-1 L-2 D-1 0

.0156

0

.0469

0 0

.0081

.0162

.0243

.0324 D-2

.0114

.0271

.0343

.0812

.0312

.0229

.0625

.0458

.0541 .1082

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Table Table 1. (Continued) Radius Compensation ANSI B212.3-2002 (R2011) 1. InsertInsert Radius Compensation ANSI B212.3-2002 (R2011) Triangle Profile (continued)

C F

Rad. 1⁄ 64

D-2

J Stylea

3°

L-2 L-1

D-1

1⁄ 32 3⁄ 64 1⁄ 16

80° Diamond Profile C F

G Stylea

Rad. 1⁄ 64

D-2

1⁄ 32 3⁄ 64

D-1

0°

1⁄ 16

L-2

L-1

F

L-2 L-1

5°

C

F Stylea

L-2 L-1

15° C

1⁄ 16

.0170 .0340 .0511

.0681

Turning & Facing 0° Lead Angle L-1 L-2 D-1 D-2 .0030

.0186

0

.0156

.0090

.0559

0

.0469

.0060 .0120

.0312 .0745

0 0

.0312 .0625

D-1

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.0062

.0344 .0688

.0031 .0062

Facing 0° Lead Angle L-1 L-2 D-1

.0172 .0344 .0516

.0688 D-2

0

.0156

.0030

.0186

0

.0469

.0090

.0559

0 0

.0312 .0625

.0060 .0120

Turning 15° Lead Angle L-1 L-2 D-1

.0011

.0167

.0003

.0372 .0745 D-2

.0117

.0022

.0384

.0006

.0234

1⁄ 16

.0043

.0668

.0012

.0468

3⁄ 64

.0032

.0003

3⁄ 64

.0009

1⁄ 16

.0501

.0009

Facing 15° Lead Angle L-1 L-2 D-1

1⁄ 64 1⁄ 32

L-1 L-2 C

.0031

1⁄ 32

Rad.

15°

D-2

.0056

.0047

1⁄ 64

R Stylea

K Stylea

.1048

.0516

Rad.

D-1

F

.0423

.0047

1⁄ 16

F

L-2

.0042

3⁄ 64

1⁄ 32

L-1

.0786

.0016

3⁄ 64

D-2

.0318

.0028

.0172

1⁄ 64

D-2

D-1

.0014

.0524

.0016

Rad.

0°

F

.0262

.0212

1⁄ 64 1⁄ 32

D-2

D-1

.0106

Turning & Facing 5° Reverse Lead Angle Rad. L-1 L-2 D-1 D-2

C

L Stylea

Turning & Facing 3° Lead Angle L-1 L-2 D-1 D-2

.0117

D-2

.0011

.0167

.0032

.0501

.0006

.0234

.0022

.0012

.0468

.0043

.0351

.0351

.0334 .0668

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Table Table 1. (Continued) Radius Compensation ANSI B212.3-2002 (R2011) 1. InsertInsert Radius Compensation ANSI B212.3-2002 (R2011) 55° Profile

C

F

Rad. 1⁄ 64

D-2

J Stylea

1⁄ 32

L-2

D-1

3⁄ 64

3°

1⁄ 16

L-1

35° Profile C

J Stylea Negative rake holders have 6° back rake and 6° side rake

Rad.

F

1⁄ 64

D-1

1⁄ 32

3°

L-1

D-2

3⁄ 64 1⁄ 16

L-2

C

F

D-2

L Stylea

Rad.

D-1

+ 5°

.0135

.0292

.0015

.0406

.0875

.0046

.0271

.0583

.0541

.1166

.0172

.0031

.0343 .0519

.0062

.0687

Profiling 3° Reverse Lead Angle D-1 D-2 L-1 L-2 .0330

.0487

.0026

.0182

.0991

.1460

.0077

.0546

.0661

.0973

.1322

.1947

.0051

.0364

.0103

.0728

Profiling 5° Lead Angle D-1 L-1 L -2

D-2

1⁄ 64

.0324

.0480

.0042

.0198

3⁄ 64

.0971

.1440

.0128

.0597

1⁄ 32

L-1 L-2

Profiling 3° Reverse Lead Angle L-1 L-2 D-1 D-2

1⁄ 16

.0648

.0360

.1205

.1920

.0086

.0398

.0170

.0795

a L-1 and D-1 over sharp point to nose radius; and L-2 and D-2 over sharp point to center of nose

radius. The D-1 dimension for the B, E, D, M, P, S, T, and V style tools are over the sharp point of insert to a sharp point at the intersection of a line on the lead angle on the cutting edge of the insert and the C dimension. The L-1 dimensions on K style tools are over the sharp point of insert to sharp point intersection of lead angle and F dimensions. All dimensions are in inches.

Table 2. Threading Tool Insert Radius Compensation for NC Programming Threading

Insert Size 2 3 4 5

T

5 ⁄ Wide 32 3 ⁄ Wide 16  1 ⁄ Wide 4  3 ⁄ Wide 8

Buttress Threading

NTB-B

Z

X NTB-A

R .040 .046 .053 .099

U .075 .098 .128 .190

29° Acme

NA

R

Y .040 .054 .054 …

X .024 .031 .049 …

Z .140 .183 .239 …

60° V-Threading

NTF

Y

3°

U NT

T

All dimensions are given in inches. Courtesy of Kennametal, Inc.

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860

Machinery's Handbook, 31st Edition Cemented Carbides and Other Hard Materials CEMENTED CARBIDES Cemented Carbides and Other Hard Materials

Carbides and Carbonitrides.—Though high-speed steel retains its importance for such applications as drilling and broaching, most metal cutting is carried out with carbide tools. For materials that are very difficult to machine, carbide is now being replaced by carboni­trides, ceramics, and superhard materials. Cemented (or sintered) carbides and carboni­trides, known collectively in most parts of the world as hard metals, are a range of very hard, refractory, wear-resistant alloys made by powder metallurgy techniques. The minute carbide or nitride particles are “cemented” by a binder metal that is liquid at the sintering temperature. Compositions and properties of individual hardmetals can be as different as those of brass and high-speed steel. All hardmetals are cermets, combining ceramic particles with a metallic binder. It is unfortunate that (owing to a mistranslation) the term cermet has come to mean either all hardmetals with a titanium carbide (TiC) base or simply cemented titanium carbonitrides. Although no single element other than carbon is present in all hardmetals, it is no accident that the generic term is “tungsten carbide.” The earliest successful grades were based on carbon, as are the majority of those made today, as listed in Table 1. The outstanding machining capabilities of high-speed steel are due to the presence of very hard carbide particles, notably tungsten carbide, in the iron-rich matrix. Modern methods of making cutting tools from pure tungsten carbide were based on this knowl­ edge. Early pieces of cemented carbide were much too brittle for industrial use, but it was soon found that mixing tungsten carbide powder with up to 10 percent of metals such as iron, nickel, or cobalt, allowed pressed compacts to be sintered at about 1500°C to give a product with low porosity, very high hardness, and considerable strength. This combina­tion of properties made the materials ideally suitable for use as tools for cutting metal. Cemented carbides for cutting tools were introduced commercially in 1927, and, although the key discoveries were made in Germany, many of the later developments have taken place in the United States, Austria, Sweden, and other countries. Recent years have seen two “revolutions” in carbide cutting tools, one led by the United States and the other by Europe. These were the change from brazed to clamped carbide inserts and the rapid development of coating technology. When indexable tips were first introduced, it was found that so little carbide was worn away before they were discarded that a minor industry began to develop, regrinding the so-called “throwaway” tips and selling them for reuse in adapted toolholders. Hardmetal con­sumption, which had grown dramatically when indexable inserts were introduced, leveled off and began to decline. This situation was changed by the advent and rapid acceptance of carbide, nitride, and oxide coatings. Application of an even harder, more wear-resistant surface to a tougher, more shock-resistant substrate allowed production of new genera­tions of longer-lasting inserts. Regrinding destroyed the enhanced properties of the coat­ings and was, therefore, abandoned for coated tooling. Brazed tools have the advantage that they can be reground over and over again, until almost no carbide is left, but the tools must always be reset after grinding to maintain machining accuracy. However, all brazed tools suffer to some extent from the stresses left by the brazing process, which in unskilled hands or with poor design can shatter the carbide even before it has been used to cut metal. In present conditions, it is cheaper to use indexa­ble inserts, which are tool tips of precise size, clamped in similarly precise holders, needing no time-consuming and costly resetting but usable only until each cutting edge or corner has lost its initial sharpness (see Introduction and related topics starting on page 839 and Indexable Insert Holders for NC on page 856). The absence of brazing stresses and the “one-use” concept also means that harder, longer-lasting grades can be used.

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Table 1. Typical Properties of Tungsten Carbide-Based Cutting Tool Hard Metals ISO Application Code P01 P05 P10 P15 P20 P25 P30 P40 P50 M10 M20 M30 M40 K01 K05 K10 K20 K30 K40

Composition (%) WC 50 78 69 78 79 82 84 85 78 85 82 86 84 97 95 92 94 91 89

TiC 35 16 15 12 8 6 5 5 3 5 5 4 4

TaC 7 8 3 5 4 2 3 4 5 2 1 2

Co 6 6 8 7 8 8 9 10 16 6 8 10 10 3 4 6 6 9 11

Density (g/cm3) 8.5 11.4 11.5 11.7 12.1 12.9 13.3 13.4 13.1 13.4 13.3 13.6 14.0 15.2 15.0 14.9 14.8 14.4 14.1

Hardness (Vickers) 1900 1820 1740 1660 1580 1530 1490 1420 1250 1590 1540 1440 1380 1850 1790 1730 1650 1400 1320

Transverse Rupture Strength (N/mm2) 1100 1300 1400 1500 1600 1700 1850 1950 2300 1800 1900 2000 2100 1450 1550 1700 1950 2250 2500

A complementary development was the introduction of ever more complex chipbreak­ ers derived from computer-aided design and pressed and sintered to precise shapes and dimensions. Another advance was the application of hot isostatic pressing (HIP), which has moved hardmetals into applications that were formerly uneconomical. This method allows virtually all residual porosity to be squeezed out of the carbide by means of inert gas at high pressure applied at about the sintering temperature. Toughness, rupture strength, and shock resistance can be doubled or tripled by this method, and the reject rates of very large sintered components are reduced to a fraction of their previous levels. Further research has produced a substantial number of excellent cutting-tool materials based on titanium carbonitride. Generally called “cermets,” as noted previously, carboni­ tride-based cutting inserts offer excellent performance and considerable prospects for the future. Compositions and Structures: Properties of hardmetals are profoundly influenced by microstructure. The microstructure, in turn, depends on many factors, including basic chem­ical composition of the carbide and matrix phases; size, shape, and distribution of carbide particles; relative proportions of carbide and matrix phases; degree of intersolubility of carbides; excess or deficiency of carbon; variations in composition and structure caused by diffusion or segregation; production methods generally, but especially milling, carburiz­ing, and sintering methods, and the types of raw materials; post sintering treatments such as hot isostatic pressing; and coatings or diffusion layers applied after initial sintering. Tungsten Carbide/Cobalt (WC/Co): The first commercially available cemented car­ bides consisted of fine angular particles of tungsten carbide bonded with metallic cobalt. Intended initially for wire-drawing dies, this composition type is still considered to have the greatest resistance to simple abrasive wear and, therefore, to have many applications in machining. For maximum hardness to be obtained from closeness of packing, the tungsten carbide grains should be as small as possible, preferably below 1 mm (0.00004 inch) and consider­ ably less for special purposes. Hardness and abrasion resistance increase as the cobalt con­tent is lowered, provided that a minimum of cobalt is present (2 percent can be enough,

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Machinery's Handbook, 31st Edition Cemented Carbides and Other Hard Materials

although 3 percent is the realistic minimum) to ensure complete sintering. In general, as carbide grain size or cobalt content or both are increased—frequently in unison—tougher and less hard grades are obtained. No porosity should be visible, even under the highest optical magnification. WC/Co compositions used for cutting tools range from about 2 to 13 percent cobalt, and from less than 0.5 to more than 5 mm (0.00002–0.0002 in.) in grain size. For stamping tools, swaying dies, and other wear applications for parts subjected to moderate or severe shock, cobalt content can be as much as 30 percent, and grain size a maximum of about 10 mm (0.0004 in.). In recent years, “micrograin” carbides, combining submicron (less than 0.00004 in.) carbide grains with relatively high cobalt content have found increasing use for machining at low speeds and high feed rates. An early use was in high-speed wood­ working cutters such as are used for planing. For optimum properties, porosity should be at a minimum, carbide grain size as regular as possible, and carbon content of the tungsten carbide phase close to the theoretical (stoi­ chiometric) value. Many tungsten carbide/cobalt compositions are modified by small but important additions—from 0.5 to perhaps 3 percent of tantalum, niobium, chromium, vanadium, titanium, hafnium, or other carbides. The basic purpose of these additions is generally inhibition of grain growth so that a consistently fine structure is maintained. Tungsten-Titanium Carbide/Cobalt (WC/TiC/Co): These grades are used for tools to cut steels and other ferrous alloys, the purpose of the TiC content being to resist the hightemperature diffusive attack that causes chemical breakdown and cratering. Tungsten car­bide diffuses readily into the chip surface, but titanium carbide is extremely resistant to such diffusion. A solid solution or “mixed crystal” of WC in TiC retains the anticratering property to a great extent. Unfortunately, titanium carbide and TiC-based solid solutions are considerably more brittle and less abrasion resistant than tungsten carbide. TiC content, therefore, is kept as low as possible, only sufficient TiC being provided to avoid severe cratering wear. Even 2 or 3 percent of titanium carbide has a noticeable effect, and, as the relative content is sub­ stantially increased, the cratering tendency becomes more severe. In the limiting formulation, the carbide is tungsten-free and based entirely on TiC, but, generally, TiC content extends to no more than about 18 percent. Above this figure, the car­bide becomes excessively brittle and is very difficult to braze, although this drawback is not a problem with throwaway inserts. WC/TiC/Co grades generally have two distinct carbide phases, angular crystals of almost pure WC and rounded TiC/WC mixed crystals. Among progressive manufacturers, although WC/TiC/Co hardmetals are very widely used, in certain important respects they are obsolescent, having been superseded by the WC/TiC/Ta(Nb)C/Co series in the many applications where higher strength combined with crater resistance is an advantage. TiC, TiN, and other coatings on tough substrates have also diminished the attractions of highTiC grades for high-speed machining of steels and ferrous alloys. Tungsten-Titanium-Tantalum (-Niobium) Carbide/Cobalt: Except for coated carbides, tungsten-titanium-tantalum (-niobium) grades could be the most popular class of hardmet­als. Used mainly for cutting steel, they combine and improve upon most of the best features of the longer-established WC/TiC/Co compositions. These carbides compete directly with carbonitrides and silicon nitride ceramics, and the best cemented carbides of this class can undertake very heavy cuts at high speeds on all types of steels, including austenitic stain­less varieties. These tools also operate well on ductile cast irons and nickel-base superal­loys, where great heat and high pressures are generated at the cutting edge. However, they do not have the resistance to abrasive wear possessed by micrograin straight tungsten car­bide grades nor the good resistance to cratering of coated grades and titanium carbide-based cermets. Titanium Carbide/Molybdenum/Nickel (TiC/Mo/Ni): The extreme indentation hardness and crater resistance of titanium carbide, allied to the cheapness and availability of its

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Machinery's Handbook, 31st Edition Cemented Carbides and Other Hard Materials

863

main raw material (titanium dioxide, TiO2), provide a strong inducement to use grades based on this carbide alone. Although developed early in the history of hardmetals, these carbides were difficult to braze satisfactorily and consequently were little used until the advent of clamped, throwaway inserts. Moreover, the carbides were notoriously brittle and could take only fine cuts in minimal-shock conditions. Titanium carbide-based grades again came into prominence about 1960, when nickelmolybdenum began to be used as a binder instead of nickel. The new grades were able to perform a wider range of tasks, including interrupted cutting and cutting under shock con­ditions. The very high indentation hardness values recorded for titanium carbide grades are not accompanied by correspondingly greater resistance to abrasive wear, the apparently less hard tungsten carbide being considerably superior in this property. Moreover, carboni­ trides, advanced tantalum-containing multicarbides, and coated variants generally provide better all-around cutting performances. Titanium-Base Carbonitrides: Development of titanium-carbonitride-based cuttingtool materials predates the use of coatings of this type on more conventional hardmetals by many years. Appreciable, though uncontrolled, amounts of carbonitride were often pres­ent, if only by accident, when cracked ammonia was used as a less expensive substitute for hydrogen in some stages of the production process in the 1950s and perhaps for two decades earlier. Much of the recent, more scientific development of this class of materials has taken place in the United States, particularly by Teledyne Firth Sterling with its SD3 grade and in Japan by several companies. Many of the compositions currently in use are extremely complex, and their structures—even with apparently similar compositions—can vary enormously. For instance, Mitsubishi characterizes its Himet NX series of cermets as TiC/WC/Ta(Nb)C/Mo2 C/TiN/Ni/Co/Al, with a structure comprised of both large and medium-size carbide particles (mainly TiC according to the quoted density) in a superal­ loy-type matrix containing an aluminum-bearing intermetallic compound. Steel- and Alloy-Bonded Titanium Carbide: The class of material exemplified by FerroTic, as it is known, consists primarily of titanium carbide bonded with heat-treatable steel, but some grades also contain tungsten carbide or are bonded with nickel- or copper-base alloys. These cemented carbides are characterized by high binder contents (typically 50–60 percent by volume) and lower hardnesses, compared with the more usual hardmetals, and by the great variation in properties obtained by heat treatment. In the annealed condition, steel-bonded carbides have a relatively soft matrix and can be machined with little difficulty, especially by CBN (superhard cubic boron nitride) tools. After heat treatment, the degree of hardness and wear resistance achieved is considerably greater than that of normal tool steels, although understandably much less than that of tra­ditional sintered carbides. Microstructures are extremely varied, being composed of 40–50 percent TiC by volume and a matrix appropriate to the alloy composition and the stage of heat treatment. Applications include stamping, blanking and drawing dies, machine com­ponents, and similar items where the ability to machine before hardening reduces produc­tion costs substantially. Coating: As a final stage in carbide manufacture, coatings of various kinds are applied mainly to cutting tools, where for cutting steel in particular it is advantageous to give the rank and clearance surfaces characteristics that are quite different from those of the body of the insert. Coatings of titanium carbide, nitride, or carbonitride; of aluminum oxide; and of other refractory compounds are applied to a variety of hardmetal substrates by chemical or physical vapor deposition (CVD or PVD) or by newer plasma methods. The most recent types of coatings include hafnium, tantalum, and zirconium carbides and nitrides; alumina/titanium oxide; and multiple carbide/carbonitride/nitride/oxide, oxynitride or oxycarbonitride combinations. Greatly improved properties have been

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Machinery's Handbook, 31st Edition Cemented Carbides and Other Hard Materials

claimed for variants with as many as 13 distinct CVD coatings. A markedly sharper cutting edge compared with other CVD-coated hardmetals is claimed, permitting finer cuts and the successful machining of soft but abrasive alloys. The keenest edges on coated carbides are achieved by the techniques of physical vapor deposition. In this process, ions are deposited directionally from the electrodes, rather than evenly on all surfaces, so the sharpness of cutting edges is maintained and may even be enhanced. PVD coatings currently available include titanium nitride and carbonitride, their distinctive gold color having become familiar throughout the world on high-speed steel tooling. The high temperatures required for normal CVD tend to soften heat-treated high-speed steel. PVD-coated hardmetals have been produced commercially for several years, especially for precision milling inserts. Recent developments in extremely hard coatings, generally involving exotic techniques, include boron carbide, cubic boron nitride, and pure diamond. Almost the ultimate in wear resistance, the commercial applications of thin plasma-generated diamond surfaces at present are mainly in the manufacture of semiconductors, where other special properties are important. For cutting tools, the substrate is of equal importance to the coating in many respects, its critical properties including fracture toughness (resistance to crack propagation), elastic modulus, resistance to heat and abrasion, and expansion coefficient. Some manufacturers are now producing inserts with graded composition so that structures and properties are optimized at both surface and interior, and coatings are less likely to crack or break away. Specifications: Compared with other standardized materials, the world of sintered hard­metals is peculiar. For instance, an engineer who seeks a carbide grade for the finish-machining of a steel component may be told to use ISO Standard Grade P10 or Industry Code C7. If the composition and nominal properties of the designated tool material are then requested, the surprising answer is that, in basic composition alone, the tungsten car­bide content of P10 (or of the now superseded C7) can vary from zero to about 75, titanium carbide from 8 to 80, cobalt 0 to 10, and nickel 0 to 15 percent. There are other possible constituents, also, in this so-called standard alloy, and many basic properties can vary as much as the composition. All that these dissimilar materials have in common, and all that the so-called standards mean, is that their suppliers— and sometimes their suppliers alone—consider them suitable for one particular and illdefined machining application (which for P10 or C7 is the finish machining of steel). This peculiar situation arose because the production of cemented carbides in occupied Europe during World War II was controlled by the German Hartmetallzentrale, and no fac­tory other than Krupp was permitted to produce more than one grade. By the end of the war, all German-controlled producers were equipped to make the G, S, H, and F series to Ger­man standards. In the postwar years, this series of carbides formed the basis of unofficial European standardization. With the advent of the newer multicarbides, the previous iden­tities of grades were gradually lost. The applications relating to the old grades were retained, however, as a new German DIN standard, eventually being adopted, in somewhat modified form, by the International Standards Organization (ISO) and by ANSI in the United States. The American cemented carbides industry developed under diverse ownership and solid competition. The major companies actively and independently developed new varieties of hardmetals, and there was little or no standardization, although there were many attempts to compile equivalent charts as a substitute for true standardization. Around 1942, the Buick division of GMC produced a simple classification code that arranged nearly 100 grades derived from 10 manufacturers under only 14 symbols (TC-1 to TC-14). In spite of serious deficiencies, this system remained in use for many years as an American industry standard; that is, Buick TC-1 was equivalent to industry code C1. Buick itself went much further, using the tremendous influence, research facilities, and purchasing potential of its parent company to standardize the

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Machinery's Handbook, 31st Edition Cemented Carbides and Other Hard Materials

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products of each carbide manufacturer by properties that could be tested, rather than by the indeterminate recommended applications. Many large-scale carbide users have developed similar systems in attempts to exert some degree of in-house standardization and quality control. Small and medium-sized users, however, still suffer from so-called industry standards, which only provide a starting point for grade selection. ISO standard 513, summarized in Table 2, divides all machining grades into three color-coded groups: straight tungsten carbide grades (letter K, color red) for cutting gray cast iron, nonferrous metals, and nonmetallics; highly alloyed grades (letter P, color blue) for machining steel; and less alloyed grades (letter M, color yellow, generally with less TiC than the corresponding P series), which are multipurpose and may be used on steels, nickel-base superalloys, ductile cast irons, and so on. Each grade within a group is also given a number to represent its position in a range from maximum hardness to maximum toughness (shock resistance). Typical applications are described for grades at more or less regular numerical intervals. Although coated grades scarcely existed when the ISO stan­dard was prepared, it is easy to classify coated as uncoated carbides— or carbonitrides, ceramics, and superhard materials—according to this system. In this situation, it is easy to see how one plant will prefer one manufacturer’s carbide and a second plant will prefer that of another. Each has found the carbide most nearly ideal for the particular conditions involved. In these circumstances it pays each manufacturer to make grades that differ in hardness, toughness, and crater resistance, so that they can pro­vide a product that is near the optimum for a specific customer’s application. Although not classified as a hard metal, new particle or powder metallurgical methods of manufacture, coupled with new coating technology, have led in recent years to something of an upsurge in the use of high-speed steel. Lower cost is a big factor, and the development of such coatings as titanium nitride, cubic boron nitride, and pure diamond, has enabled some high-speed steel tools to rival tools made from tungsten and other carbides in their ability to maintain cutting accuracy and prolong tool life. Multiple layers may be used to produce optimum properties in the coating, with adhesive strength where there is contact with the substrate, combined with hardness at the cutting surface to resist abrasion. Total thickness of such coating, even with multiple layers, is seldom more than 15 microns (0.000060 in.). Importance of Correct Grades: A great diversity of hardmetal types is required to cope with all possible combinations of metals and alloys, machining operations, and working conditions. Tough, shock-resistant grades are needed for slow speeds and interrupted cut­ting, harder grades for high-speed finishing, heat-resisting alloyed grades for machining superalloys, and crater-resistant compositions, including most of the many coated variet­ies, for machining steels and ductile iron. Ceramics.—Moving up the hardness scale, ceramics provide increasing competition for cemented carbides, both in performance and in cost-effectiveness, though not yet in reli­ability. Hardmetals themselves consist of ceramics—nonmetallic refractory compounds, usually carbides or carbonitrides—with a metallic binder of much lower melting point. In such systems, densification generally takes place by liquid-phase sintering. Pure ceramics have no metallic binder, but may contain lower-melting-point compounds or ceramic mix­tures that permit liquid-phase sintering to take place. Where this condition is not possible, hot pressing or hot isostatic pressing can often be used to make a strong, relatively pore-free component or cutting insert. This section is restricted to those ceramics that compete directly with hardmetals, mainly in the cutting-tool category as shown in Table 3. Ceramics are hard, completely nonmetallic substances that resist heat and abrasive wear. Increasingly used as clamped indexable tool inserts, ceramics differ significantly from tool steels, which are completely metallic. Ceramics also differ from cermets such as cemented carbides and carbonitrides, which comprise minute ceramic particles held together by metallic binders.

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P Blue

Broad Categories of Materials to be Machined

Ferrous with long chips

Designation (Grade)

Steel, steel castings

P10

Steel, steel castings

P20

Steel, steel castings, ductile cast iron with long chips Steel, steel castings, ductile cast iron with long chips Steel, steel castings with sand inclusions and cavities

P40

Ferrous metals with long or short chips, and nonfer­rous metals

P50

Steel, steel castings of medium-  or low-tensile strength, with sand inclusions and cavities

M10

Steel, steel castings, manganese steel, gray cast iron, alloy cast iron Steel, steel castings, austenitic or manganese steel, gray cast iron Steel, steel castings, austenitic steel, gray cast iron, high-temperature-resistant alloys Mild, free-cutting steel, low-tensile steel, non­ferrous metals and light alloys Very hard gray cast iron, chilled castings over 85 Shore, high-silicon aluminum alloys, hard­ened steel, highly abrasive plastics, hard card­board, ceramics Gray cast iron over 220 BHN (Brinell), malleable cast iron with short chips, hardened steel, siliconaluminum and copper alloys, plastics, glass, hard rubber, hard cardboard, porcelain, stone Gray cast iron up to 220 BHN, nonferrous metals, copper, brass, aluminum Low-hardness gray cast iron, low-tensile steel, compressed wood Softwood or hardwood, nonferrous metals

M20 M30 M40

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K Red

Ferrous metals with short chips, nonferrous metals and non-metallic materials

Groups of Applications

P01

P30

M Yellow

Specific Material to be Machined

K01

K10

K20 K30 K40

Use and Working Conditions

Finish turning and boring; high cutting speeds, small chip sections, accu­rate dimensions, fine finish, vibration-free operations Turning, copying, threading, milling; high cutting speeds; small or medium chip sections Turning, copying, milling; medium cutting speeds and chip sections, plan­ing with small chip sections Turning, milling, planing; medium or large chip sections, unfavorable machining conditions Turning, planing, slotting; low cutting speeds, large chip sections, with possible large cutting angles, unfavorable cutting conditions, and work on automatic machines Operations demanding very tough carbides; turning, planing, slotting; low cutting speeds, large chip sections, with possible large cutting angles, unfavorable conditions and work on automatic machines Turning; medium or high cutting speeds, small or medium chip sections

Direction of Decrease in Characteristic of cut

of carbide ↑­ speed

↑ wear resistance

Turning, milling; medium cutting speeds and chip sections Turning, milling, planing; medium cutting speeds, medium or large chip sections Turning, parting off; particularly on automatic machines Turning, finish turning, boring, milling, scraping

Turning, milling, drilling, boring, broaching, scraping

Turning, milling, planing, boring, broaching, demanding very tough car­bide Turning, milling, planing, slotting, unfavorable conditions, and possibility of large cutting angles Turning, milling, planing, slotting, unfavorable conditions, and possibility of large cutting angles

↓ feed ↓ toughness

Machinery's Handbook, 31st Edition Cemented Carbides and Other Hard Materials

Symbol and Color

866

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Table 2. ISO Classifications of Hardmetals (Cemented Carbides and Carbonitrides) by Application Main Types of Chip Removal

Machinery's Handbook, 31st Edition Cemented Carbides and Other Hard Materials Table 3. Typical Properties of Cutting Tool Ceramics

Group

Typical composition types Density (g/cm3)

Transverse rupture strength (N/mm2)

Compressive strength (kN/mm2)

Hardness (HV)

Hardness HK (kN/mm2)

Young’s modulus (kN/mm2)

Modulus of rigidity (kN/mm2)

Alumina

Alumina/TiC

Silicon Nitride

Al2 O3 or Al2 O3 /ZrO2

70 ⁄ 30 Al2 O3 /TiC

Si3 N4 /Y2 O3 plus

4.0

Thermal expansion coefficient (10–6/K) Thermal conductivity (W/m K)

  

Fracture toughness (K1c MN/m3 ⁄ 2)

3.27

3.4

4.0

4.7

700

750

800

1750

1800

1600

380

370

300

4.0

150

Poisson’s ratio

4.25

0.24 8.5

23

2.3

4.5

160

0.22 7.8

17

3.3

PCD

150

0.20 3.2

22

5.0

50

925

430

0.09 3.8

120

7.9

867 PCBN

3.1

800

3.8

28

680

280

0.22 4.9

100

10

Alumina-based ceramics were introduced as cutting inserts during World War II and were for many years considered too brittle for regular machine-shop use. Improved machine tools and finer-grain, tougher compositions incorporating zirconia or silicon car­bide “whiskers” now permit their use in a wide range of applications. Silicon nitride, often combined with alumina (aluminum oxide), yttria (yttrium oxide), and other oxides and nitrides, is used for much of the high-speed machining of superalloys, and newer grades have been formulated specifically for cast iron—potentially a far larger market. In addition to improvements in toolholders, great advances have been made in machine tools, many of which now feature the higher powers and speeds required for the efficient use of ceramic tooling. Brittleness at the cutting edge is no longer a disadvantage, with the improvements made to the ceramics themselves, mainly in toughness, but also in other critical properties. Although very large numbers of useful ceramic materials are now available, only a few combinations have been found to combine such properties as minimum porosity, hardness, wear resistance, chemical stability, and resistance to shock to the extent necessary for cut­ting-tool inserts. Most ceramics used for machining are still based on high-purity, fine-grained alumina (aluminum oxide) but embody property-enhancing additions of other ceramics such as zirconia (zirconium oxide), titania (titanium oxide), titanium carbide, tungsten carbide, and titanium nitride. For commercial purposes, those more commonly used are often termed “white” (alumina with or without zirconia) or “black” (roughly 70 ¤ 30 alumina/titanium carbide). More recent developments are the distinctively green alumina ceramics strengthened with silicon carbide whiskers and the brown-tinged silicon nitride types. Ceramics benefit from hot isostatic pressing to remove the last vestiges of porosity and raise substantially the material’s shock resistance, even more than carbide-based hard­ metals. Significant improvements are derived by even small parts such as tool inserts, although, in principle, they should not need such treatment if raw materials and manufac­ turing methods are properly controlled. Oxide Ceramics: Alumina cutting tips have extreme hardness—more than HV 2000 or HRA 94—and give excellent service in their limited but important range of uses such as the machining of chilled iron rolls and brake drums. A substantial family of aluminabased materials has been developed, and fine-grained alumina-based composites now have suf­ficient strength for milling cast iron at speeds up to 2500 ft/min (800 m/min). Resistance to cratering when machining steel is exceptional. Oxide/Carbide Ceramics: A second important class of alumina-based cutting ceramics combines aluminum oxide or alumina-zirconia with a refractory carbide or carbides,

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Machinery's Handbook, 31st Edition Cemented Carbides and Other Hard Materials

nearly always 30 percent TiC. The compound is black and normally hot pressed or hot isostatically pressed (HIPed). As shown in Table 3, the physical and mechanical properties of this material are generally similar to those of the pure alumina ceramics, but strength and shock resistance are generally higher, being comparable with those of higher-toughness simple alumina-zirconia grades. Current commercial grades are even more complex, com­bining alumina, zirconia, and titanium carbide with the further addition of titanium nitride. Silicon Nitride Base: One of the most effective ceramic cutting-tool materials developed in the United Kingdom is Syalon (from SiAlON or silicon-aluminum-oxynitride) though it incorporates a substantial amount of yttria for efficient liquid-phase sintering). The material combines high strength with hot hardness, shock resistance, and other vital properties. Syalon cutting inserts are made by Kennametal and Sandvik and sold as Kyon 2000 and CC680, respec­tively. The brown Kyon 200 is suitable for machining highnickel alloys and cast iron, but a later development, Kyon 3000, has good potential for machining cast iron. Resistance to thermal stress and thermal shock of Kyon 2000 are comparable to those of sintered carbides. Toughness is substantially less than that of carbides, but roughly twice that of oxide-based cutting-tool materials at temperatures up to 850°C. Syon 200 can cut at high edge temperatures and is harder than carbide and some other ceramics at over 700°C, although softer than most at room temperature. Whisker-Reinforced Ceramics: To improve toughness, Greenleaf Corporation has reinforced alumina ceramics with silicon carbide single-crystal “whiskers” that impart a distinctive green color to the material, marketed as WG300. Typically as thin as human hairs, the immensely strong whiskers improve tool life under arduous conditions. Whisker-rein­forced ceramics and perhaps hardmetals are likely to become increasingly important as cutting and wear-resistant materials. Their only drawback seems to be the carcinogenic nature of the included fibers, which requires stringent precautions during manufacture.

Superhard Materials.—Polycrystalline synthetic diamond (PCD) and cubic boron nitride (PCBN), in the two columns at the right in Table 3, are almost the only cutting-insert materials in the “superhard” category. Both PCD and PCBN are usually made with the highest practicable concentration of the hard constituent, although ceramic or metallic binders can be almost equally important in providing overall strength and optimizing other properties. Variations in grain size are another critical factor in determining cutting char­acteristics and edge stability. Some manufacturers treat CBN in similar fashion to tungsten carbide, varying the composition and amount of binder within exceptionally wide limits to influence the physical and mechanical properties of the sintered compact. In comparing these materials, users should note that some inserts comprise solid poly­crystalline diamond or CBN and are double-sized to provide twice the number of cutting edges. Others consist of a layer, from 0.020- to 0.040-inch (0.5- to 1-mm) thick, on a tough carbide backing. A third type is produced with a solid superhard material almost sur­rounded by sintered carbide. A fourth type, used mainly for cutting inserts, is comprised of solid hard metal with a tiny superhard insert at one or more (usually only one) cutting corners or edges. Superhard cutting inserts are expensive—up to 30 times the cost of equivalent shapes or sizes in ceramic or cemented carbide—but their outstanding properties, excep­tional performance and extremely long life can make them by far the most cost-effective for certain applications. Diamond: Diamond is the hardest material found or made. As harder, more abrasive ceramics and other materials came into widespread use, diamond began to be used for grinding wheel grits. Cemented carbide tools virtually demanded diamond grinding wheels for fine edge finishing. Solid single-crystal diamond tools were and are used to a small extent for special purposes, such as microtomes, for machining of hard materials, and for exceptionally fine finishes. These diamonds are made from comparatively large,

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Machinery's Handbook, 31st Edition Cemented Carbides and Other Hard Materials

869

high-quality gem-type diamonds, have isotropic properties, and are very expensive. By comparison, diamond abrasive grits cost only a few dollars a carat. Synthetic diamonds are produced from graphite using high temperatures and extremely high pressures. The fine diamond particles produced are sintered together in the presence of a metal “catalyst” to produce high-efficiency anisotropic cuttingtool inserts. These tools comprise either a solid diamond compact or a layer of sintered diamond on a carbide backing, and are made under conditions similar to, though less severe than, those used in diamond synthesis. Both natural and synthetic diamond can be sintered in this way, although the latter method is the most frequently used. Polycrystalline diamond (PCD) compacts are immensely hard and can be used to machine many substances, from highly abrasive hardwoods and glass fiber to nonferrous metals, hardmetals, and tough ceramics. Important classes of tools that are also available with cubic boron nitride inserts include brazed-tip drills, single-point turning tools, and face-milling cutters. Boron Nitride: Polycrystalline diamond has one big limitation: it cannot be used to machine steel or any other ferrous material without rapid chemical breakdown. Boron nitride does not have this limitation. Normally soft and slippery like graphite, the soft hex­agonal crystals (HBN) become cubic boron nitride (CBN) when subjected to ultrahigh pressures and temperatures, with a structure similar to and hardness second only to dia­mond. As a solid insert of polycrystalline cubic boron nitride (PCBN), the compound machines even the hardest steel with relative immunity from chemical breakdown or cra­tering. Backed by sintered carbide, inserts of PCBN can readily be brazed, increasing the useful­ness of the material and the range of tooling in which it can be used. With great hardness and abrasion resistance, coupled with extreme chemical stability when in contact with fer­rous alloys at high temperatures, PCBN has the ability to machine both steels and cast irons at high speeds for long operating cycles. Only its currently high cost in relation to hardmet­als prevents its wider use in mass-production machining. Similar in general properties to PCBN, the recently developed “Wurbon” consists of a mixture of ultrafine (0.02 mm grain size) hexagonal and cubic boron nitride with a “wurtz­ite” structure, and is produced from soft hexagonal boron nitride in a microsecond by an explosive shockwave. Basic Machining Data: Most mass-production metal-cutting operations are carried out with carbide-tipped tools, but their correct application is not simple. Even apparently simi­lar batches of the same material vary greatly in their machining characteristics and may require different tool settings to attain optimum performance. Depth of cut, feed, surface speed, cutting rate, desired surface finish, and target tool life often need to be modified to suit the requirements of a particular component. For the same downtime, the life of an insert between indexings can be less than that of an equivalent brazed tool between regrinds, so a much higher rate of metal removal is possible with the indexable or throwaway insert. It is commonplace for the claims for a new coating to include increases in surface-speed rates of 200–300 percent, and for a new insert design to offer similar improvements. Many operations are run at metal removal rates that are far from optimum for tool life because the rates used maximize productivity and cost-effec­tiveness. Thus any recommendations for cutting speeds and feeds must be oversimplified or extremely complex, and must be hedged with many provisos, dependent on the technical and economic conditions in the manufacturing plant concerned. A preliminary grade selection should be made from the ISO-based tables and manufacturers’ literature con­ sulted for recommendations on the chosen grades and tool designs. If tool life is much greater than that desired under the suggested conditions, speeds, feeds, or depths of cut may be increased. If tools fail by edge breakage, a tougher (more shock-resistant) grade should be selected, with a numerically higher ISO code.

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Machinery's Handbook, 31st Edition Cemented Carbides and Other Hard Materials

Alternatively, increasing the surface speed and decreasing the feed may be tried. If tools fail prematurely from what appears to be abrasive wear, a harder grade with numerically lower ISO designation should be tried. If cratering is severe, use a grade with higher tita­ nium carbide content; that is, switch from an ISO K to M or M to P grade, use a P grade with lower numerical value, change to a coated grade, or use a coated grade with a (claimed) more-resistant surface layer. Built-Up Edge and Cratering: The big problem in cutting steel with carbide tools is asso­ ciated with the built-up edge and the familiar phenomenon called cratering. Research has shown that the built-up edge is continuous with the chip itself during normal cutting. Addi­ tions of titanium, tantalum, and niobium to the basic carbide mixture have a remarkable effect on the nature and degree of cratering, which is related to adhesion between the tool and the chip. Hardmetal Tooling for Wood and Nonmetallics.—Carbide-tipped circular saws are now conventional for cutting wood, wood products such as chipboard, and plastics, and tipped bandsaws of large size are also gaining in popularity. Tipped handsaws and mechanical equivalents are seldom needed for wood, but they are extremely useful for cut­ting abrasive building boards, glass-reinforced plastics, and similar material. Like the hardmetal tips used on most other woodworking tools, saw tips generally make use of straight (unalloyed) tungsten carbide/cobalt grades. However, where excessive heat is generated as with the cutting of high-silica hardwoods and particularly abrasive chip­boards, the very hard but tough tungsten-titanium-tantalum-niobium carbide solid-solu­tion grades, normally reserved for steel finishing, may be preferred. Saw tips are usually brazed and reground a number of times during service, so coated grades appear to have lit­tle immediate potential in this field. Cutting Blades and Plane Irons: These tools comprise long, thin, comparatively wide slabs of carbide on a minimal-thickness steel backing. Compositions are straight tungsten carbide, preferably micrograin (to maintain a keen cutting edge with an included angle of 30° or less), but with relatively high amounts of cobalt, 11–13 percent, for toughness. Con­siderable expertise is necessary to braze and grind these cutters without inducing or failing to relieve the excessive stresses that cause distortion or cracking. Other Woodworking Cutters: Routers and other cutters are generally similar to those used on metals and include many indexable-insert designs. The main difference with wood is that rotational and surface speeds can be the maximum available on the machine. High-speed routing of aluminum and magnesium alloys was developed largely from machines and techniques originally designed for work on wood. Cutting Other Materials: The machining of plastics, fiber-reinforced plastics, graphite, asbestos, and other hard and abrasive constructional materials mainly requires abrasion resistance. Cutting pressures and power requirements are generally low. With thermoplas­tics and some other materials, particular attention must be given to cooling because of soft­ening or degradation of the work material that might be caused by the heat generated in cutting. An important application of cemented carbides is the drilling and routing of printed circuit boards. Solid tungsten carbide drills of extremely small sizes are used for this work.

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Machinery's Handbook, 31st Edition Milling Cutters

871

MILLING CUTTERS Selection of Milling Cutters The most suitable type of milling cutter for a particular milling operation depends on such factors as the kind of cut to be made, the material to be cut, the number of parts to be machined, and the type of milling machine available. Solid cutters of small size will usu­ ally cost less, initially, than inserted blade types; for long-run production, inserted-blade cutters will probably have a lower overall cost. Depending on either the material to be cut or the amount of production involved, the use of carbide-tipped cutters in preference to high-speed steel or other cutting tool materials may be justified. Rake angles depend on both the cutter material and the work material. Carbide and cast alloy cutting tool materials generally have smaller rake angles than high-speed steel tool materials because of their lower edge strength and greater abrasion resistance. Soft work materials permit higher radial rake angles than hard materials; thin cutters permit zero or practically zero axial rake angles; and wide cutters operate more smoothly with high axial rake angles. See Rake Angles for Milling Cutters on page 901. Cutting edge relief or clearance angles are usually from 3 to 6 degrees for hard or tough materials, 4 to 7 degrees for average materials, and 6 to 12 degrees for easily machined materials. See Clearance Angles for Milling Cutter Teeth on page 900. The number of teeth in the milling cutter is also a factor that should be given consider­ ation, as explained in the next paragraph. Number of Teeth in Milling Cutters.—In determining the number of teeth a milling cut­ ter should have for optimum performance, there is no universal rule. There are, however, two factors that should be considered in making a choice: 1) The number of teeth should never be so great as to reduce the chip space between the teeth to a point where a free flow of chips is prevented; and 2) The chip space should be smooth and without sharp corners that would cause clogging of the chips in the space. For milling ductile materials that produce a continuous and curled chip, a cutter with large chip spaces is preferable. Such coarse tooth cutters permit an easier flow of the chips through the chip space than would be obtained with fine tooth cutters and help to eliminate cutter “chatter.” For cutting operations in thin materials, fine tooth cutters reduce cutter and workpiece vibration and the tendency for the cutter teeth to “straddle” the workpiece and dig in. For slitting copper and other soft nonferrous materials, teeth that are either chamfered or alternately flat and V-shaped are best. As a general rule, to give satisfactory performance the number of teeth in milling cutters should be such that no more than two teeth at a time are engaged in the cut. Based on this rule, the following formulas (valid in both SI and English system of units) are recom­mended: For face milling cutters, 6.3D T = W (1) For peripheral milling cutters,

T =

12.6DcosA D + 4d

(2)

where T = number of teeth in cutter; D = cutter diameter in inches (mm); W = width of cut in inches (mm); d = depth of cut in inches (mm); and A = helix angle of cutter. To find the number of teeth that a cutter should have when other than two teeth are in the cut at the same time, Formulas (1) and (2) should be divided by 2 and the result mul­tiplied by the number of teeth desired in the cut.

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Machinery's Handbook, 31st Edition Milling Cutters

Example: Determine the required number of teeth in a face mill where D = 6 inches and W = 4 inches. Using Formula (1),

6.3 # 6 = 10 teeth, approximately 4 Example: Determine the required number of teeth in a plain milling cutter where D = 4 inches and d = 1 ⁄4 inch. Using Formula (2), T =

12.6 # 4 # cos0° = 10 teeth, approximately 1 4 + a4 # 4 k In high-speed milling with sintered carbide, high-speed steel, and cast nonferrous cut­ ting tool materials, a formula that permits full use of the power available at the cutter but prevents overloading of the motor driving the milling machine is: K #H T = F # N #d #W (3) T =

where T = number of cutter teeth; H = horsepower (kilowatts) available at the cutter; F = feed per tooth in inches (mm); N = revolutions per minute of cutter; d = depth of cut in inches (mm); W = width of cut in inches (mm); and K = a constant that may be taken as 0.65 for average steel, 1.5 for cast iron, and 2.5 for aluminum. For metric units, K = 14278 for average steel, 32949 for cast iron, and 54915 for aluminum. These values are conserva­tive and take into account dulling of the cutter in service. Example: Determine the required number of teeth in a sintered carbide-tipped face mill for high-speed milling of 200 BHN (Brinell Hardness Number) alloy steel if H = 7.5 kilowatt; F = 0.2032 mm; N = 272 rpm; d  = 3.2 mm; W = 152.4 mm; and K for alloy steel is 14278. Using Formula (3),

14278 # 7.5 T = 0.2032 # 272 # 3.2 # 152.4 = 4 teeth, approximately

American National Standard Milling Cutters.—According to American National Standard ANSI/ASME B94.19-1997 (R2019), milling cutters may be classified in two general ways, given as follows: By Type of Relief on Cutting Edges: Milling cutters may be described on the basis of one of two methods of providing relief for the cutting edges. Profile sharpened cutters are those on which relief is obtained and which are resharpened by grinding a narrow land back of the cutting edges. Profile sharpened cutters may produce flat, curved, or irregular surfaces. Form relieved cutters are those which are so relieved that by grinding only the faces of the teeth the original form is maintained throughout the life of the cutters. Form relieved cutters may produce flat, curved or irregular surfaces. By Method of Mounting: Milling cutters may be described by one of two methods used to mount the cutter. Arbor type cutters are those which have a hole for mounting on an arbor and usually have a keyway to receive a driving key. These are sometimes called Shell type. Shank type cutters are those which have a straight or tapered shank to fit the machine tool spindle or adapter. Explanation of the “Hand” of Milling Cutters.—In the ANSI/ASME Standard, the terms “right hand” and “left hand” are used to describe hand of rotation, hand of cutter and hand of flute helix. Hand of Rotation or Hand of Cut is described as either “right hand” if the cutter revolves counterclockwise as it cuts when viewed from a position in front of a horizontal milling machine and facing the spindle, or “left hand” if the cutter revolves clockwise as it cuts when viewed from the same position.

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Machinery's Handbook, 31st Edition Milling Cutters

873

American National Standard Plain Milling Cutters ANSI/ASME B94.19-1997 (R2019) Cutter Diameter Max.

Min.

Range of Face Widths, Nom.a

21 ⁄2

2.515

2.485

1 ⁄ , 5 ⁄ , 3 ⁄ , 1, 11 ⁄ , 2 8 4 2

3

3.015

2.985

3

3.015

2.985

4

4.015

3.985

4

4.015

3.985

21 ⁄2

2.515

2.485

3.015

2.985

Nom.

21 ⁄2 3

4

3

4

2.515

4.015

3.015

4.015

2.485

3.985

2.985

3.985

Light-duty Cuttersb 3⁄ , 1 ⁄ , 5⁄ , 3⁄ , 16 4 16 8 2 and 3 3⁄ , 1 ⁄ , 5⁄ , 3⁄ , 16 4 16 8 5 ⁄ , 3 ⁄ , and 11 ⁄ 8 4 2 1 ⁄ , 5⁄ , 3⁄ , 2 8 4 1, 11 ⁄4 , 11 ⁄2 , 2

and 3

1 ⁄ , 5 ⁄ and 3 ⁄ 4 16 8 3⁄ , 1 ⁄ , 5⁄ , 3⁄ , 8 2 8 4 1, 11 ⁄2 , 2, 3

and 4 Heavy-duty Cuttersc 2 4

2, 21 ⁄2 , 3, 4 and 6 2, 3, 4 and 6 High-helix Cuttersd 4 and 6 8

Nom.

Hole Diameter Max.

Min.

1

1.00075

1.0000

1

1.00075

1.0000

11 ⁄4

1.2510

1.2500

1

1.00075

1.0000

11 ⁄4

1.2510

1.2500

1

1.00075

1.0000

11 ⁄4

1.2510

1.2500

11 ⁄4

1.2510

1.2500

1

11 ⁄2

11 ⁄2

1.0010

1.5010

1.5010

1.0000

1.5000

1.5000

a Tolerances on Face Widths: Up to 1 inch, inclusive, ± 0.001 inch; over 1 to 2 inches, inclusive,

+0.010, -0.000 inch; over 2 inches, +0.020, -0.000 inch. b Light-duty plain milling cutters with face widths under 3 ⁄ inch have straight teeth. Cutters with 4 3 ⁄ - inch face and wider have helix angles of not less than 15 degrees nor greater than 25 degrees. 4 c Heavy-duty plain milling cutters have a helix angle of not less than 25 degrees nor greater than 45 degrees. d High-helix plain milling cutters have a helix angle of not less than 45 degrees nor greater than 52 degrees. All dimensions are in inches. All cutters are high-speed steel. Plain milling cutters are of cylindri­cal shape, having teeth on the peripheral surface only.

Hand of Cutter: Some types of cutters require special consideration when referring to their hand. These are principally cutters with unsymmetrical forms, face-type cutters, or cutters with threaded holes. Symmetrical cutters may be reversed on the arbor in the same axial position and rotated in the cutting direction without altering the contour produced on the work-piece, and may be considered as either right or left hand. Unsymmetrical cutters reverse the contour produced on the work-piece when reversed on the arbor in the same axial position and rotated in the cutting direction. A single-angle cutter is considered to be a right-hand cutter if it revolves counterclockwise, or a left-hand cutter if it revolves clock­wise, when cutting as viewed from the side of the larger diameter. The hand of rotation of a single angle milling cutter need not necessarily be the same as its hand of cutter. A single corner rounding cutter is considered to be a right-hand cutter if it revolves counterclock­wise, or a left-hand cutter if it revolves clockwise, when cutting as viewed from the side of the smaller diameter.

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Machinery's Handbook, 31st Edition Milling Cutters

874

American National Standard Side Milling Cutters ANSI/ASME B94.19-1997 (R2019)

Cutter Diameter Nom.

Max.

Min.

Range of Face Widths Nom.a

Hole Diameter

Nom.

Max.

Min.

0.62575

0.6250

1.00075

1.0000

Side Cutters b 2

21 ⁄2 3

4

2.015

2.515

3.015

4.015

1.985

2.485

2.985

4.985

1 ⁄2 , 5 ⁄8 , 3 ⁄4

3.985

5

5.015

4.985

6 6 7 7 8 8

6.015

6.015

7.015 7.015 8.015 8.015

5.985

5.985

6.985

2.515

2.485

3

3.015

2.985

4

4.015

3.985

5

5.015

4.985

6

6.015

5.985

8 4

5 6

3.015

8.015 4.015

5.015 6.015

2.985

7.985 3.985

7⁄8

1

1.0000

1

1.00075

1.0000

1

1.00075

1.0000

11 ⁄4

1 ⁄2 , 5 ⁄8 , 3 ⁄4 , 1 3 ⁄4

11 ⁄4 11 ⁄4

3 ⁄4

11 ⁄2

3 ⁄4 , 1

11 ⁄2

3 ⁄4 , 1

Staggered-Tooth Side Cutters c 1 ⁄4 , 5 ⁄16 , 3 ⁄8 , 1 ⁄2

0.8750

1.00075

1 ⁄2 , 5 ⁄8 , 3 ⁄4 , 1 1 ⁄2

0.87575

1

11 ⁄4

7.985 7.985

5 ⁄8

1 ⁄2 , 5 ⁄8 , 3 ⁄4

6.985

21 ⁄2 3

1 ⁄4 , 5 ⁄16 , 3 ⁄8 , 7⁄16 , 1 ⁄2

1 ⁄4 , 3 ⁄8 , 1 ⁄2 , 5 ⁄8 , 3 ⁄4 , 7⁄8

4.015

5.015

1 ⁄4 , 3 ⁄8 , 1 ⁄2

3.985

4

5

3 ⁄16 , 1 ⁄4 , 3 ⁄8

11 ⁄4

7⁄8

1.2510

1.2510 1.2510 1.2510 1.5010 1.2510 1.5010

0.87575

1.2500

1.2500 1.2500 1.2500 1.5000 1.2500 1.5000 0.8750

1

1.00075

1.0000

11 ⁄4

1.2510

1.2500

1 ⁄2 , 5 ⁄8 , 3 ⁄4

11 ⁄4

1.2510

1.2500

3 ⁄8 , 1 ⁄2 , 5 ⁄8 , 3 ⁄4 , 7⁄8 , 1 3 ⁄8 , 1 ⁄2 , 5 ⁄8 , 3 ⁄4 , 1

11 ⁄4

11 ⁄2

1.2510

1.2500

3 ⁄4

11 ⁄4

1.2510

1.2500

11 ⁄4

1.2510

1.2500

3 ⁄16 , 1 ⁄4 , 5 ⁄16 , 3 ⁄8 1 ⁄2 , 5 ⁄8 , 3 ⁄4

1 ⁄4 , 5 ⁄16 , 3 ⁄8 , 7⁄16 , 1 ⁄2 , 5 ⁄8 , 3 ⁄4

and 7⁄8

Half Side Cutters d

4.985 5.985

3 ⁄4 3 ⁄4

11 ⁄4

11 ⁄4

1.2510

1.5010

1.2510

1.2500

1.5000

1.2500

Tolerances on Face Widths: For side cutters, +0.002, -0.001 inch; for staggered-tooth side cutters up to 3⁄4 -inch face width, inclusive, +0.000 -0.0005 inch, and over 3⁄4 to 1 inch, inclusive, +0.000 - 0.0010 inch; and for half side cutters, +0.015, -0.000 inch. b Side milling cutters have straight peripheral teeth and side teeth on both sides. c Staggered-tooth side milling cutters have peripheral teeth of alternate right- and left-hand helix and alternate side teeth. d Half side milling cutters have side teeth on one side only. The peripheral teeth are helical of the same hand as the cut. Made either with right-hand or left-hand cut. All dimensions are in inches. All cutters are high-speed steel. Side milling cutters are of cylindrical shape, having teeth on the periphery and on one or both sides. a

Hand of Flute Helix: Milling cutters may have straight flutes, which means that their cut­ting edges are in planes parallel to the cutter axis. Milling cutters with flute helix in one direction only are described as having a right-hand helix if the flutes twist away from the observer in a clockwise direction when viewed from either end of the cutter or as having a left-hand helix if the flutes twist away from the observer in a counterclockwise direction when viewed from either end of the cutter. Staggered tooth cutters are milling cutters with every other flute of opposite (right- and left-hand) helix. An illustration describing the various milling cutter elements of both a profile cutter and a form-relieved cutter is given on page 876.

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Machinery's Handbook, 31st Edition Milling Cutters

875

American National Standard Staggered Teeth, T-Slot Milling Cutters with Brown & Sharpe Taper and Weldon Shanks ANSI/ASME B94.19-1997 (R2019)

L

W N

D

L

W

S

Bolt Size

Cutter Dia., D

Face Width, W

Neck Dia., N

1 ⁄4

9 ⁄16

15 ⁄64

17⁄64

5 ⁄16

21 ⁄32

17⁄64

21 ⁄64

3 ⁄8

25 ⁄32

21 ⁄64

13 ⁄32

1 ⁄2

31 ⁄32

25 ⁄64

17⁄32

31 ⁄64

21 ⁄32

5 ⁄8

25 ⁄32

5 ⁄8 3 ⁄4

1

D

N

11 ⁄4

115⁄32

127⁄32

With Weldon Shank Length, Dia., L S

…

…

219⁄32

1 ⁄2

…

…

31 ⁄4

3 ⁄4

…

…

5

211 ⁄16

7

1

47⁄16

9

71 ⁄4

3 ⁄4 1

315⁄16

9

67⁄8

1 ⁄2

37⁄16

7

51 ⁄4

11 ⁄32

53 ⁄64

With B. & S. Tapera,b Taper Length, No. L

413⁄16

11 ⁄4

a For dimensions of Brown & Sharpe taper shanks, see information given on page 1023.

b Brown & Sharpe taper shanks have been removed from ANSI/ASME B94.19; they are included for reference only. All dimensions are in inches. All cutters are high-speed steel, and only right-hand cutters are stan­dard. Tolerances: On D, +0.000, -0.010 inch; on W, +0.000, -0.005 inch; on N, +0.000, -0.005 inch; on L, ± 1 ⁄16 inch; on S, -00001 to -0.0005 inch.

American National Standard Form Relieved Corner Rounding Cutters with Weldon Shanks ANSI/ASME B94.19-1997 (R2019)

L d

S

D R

Rad., R

Dia., D

Dia., d

S 3 ⁄8

1 ⁄16

7⁄16

1 ⁄4

3 ⁄32

1 ⁄2

1 ⁄4

3 ⁄8

1 ⁄8

5 ⁄8

1 ⁄4

1 ⁄2

5 ⁄32

3 ⁄4

5 ⁄16

1 ⁄2

3 ⁄16

7⁄8 1

5 ⁄16

1 ⁄2

3 ⁄8

1 ⁄2

3 ⁄8

1 ⁄2

1 ⁄4 5 ⁄16

11 ⁄8

L 21 ⁄2 21 ⁄2 3 3 3 3

31 ⁄4

Rad., R 3 ⁄8 3 ⁄16 1 ⁄4 5 ⁄16 3 ⁄8 7⁄16 1 ⁄2

Dia., D

Dia., d

S

11 ⁄4

3 ⁄8

1 ⁄2

1

5 ⁄16

3 ⁄4

3 ⁄8

3 ⁄4

3 ⁄8

7⁄8

3 ⁄8

7⁄8

7⁄8

11 ⁄8 11 ⁄4

13⁄8

11 ⁄2

3 ⁄8 3 ⁄8

1 1

L 31 ⁄2 31 ⁄8 31 ⁄4

31 ⁄2 33⁄4 4

41 ⁄8

All dimensions are in inches. All cutters are high-speed steel. Right-hand cutters are standard. Tolerances: On D, ±0.010 inch; on diameter of circle, 2R, ±0.001 inch for cutters up to and includ­ ing 1 ⁄8 -inch radius, +0.002, -0.001 inch for cutters over 1 ⁄8 -inch radius; on S, -0.0001 to -0.­0 005 inch; and on L, ± 1 ⁄16 inch.

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Machinery's Handbook, 31st Edition Milling Cutters

876

American National Standard Metal Slitting Saws ANSI/ASME B94.19-1997 (R2019) Cutter Diameter

Nom.

Max.

21 ⁄2

2.515

3

3.015

4

4.015

5 5 6 6 8 8

5.015 5.015 6.015 6.015 8.015 8.015

21 ⁄2 3 4 5 5 6 6 8 8

2.515 3.015 4.015 5.015 5.015 6.015 6.015 8.015 8.015

3 4 5 6 6 8 10 12

3.015 4.015 5.015 6.015 6.015 8.015 10.015 12.015

Hole Diameter Range of Face Widths Nom. Max. Nom.a Plain Metal Slitting Saws b 1 ⁄32 , 3 ⁄64 , 1 ⁄16 , 3 ⁄32 , 1 ⁄8 7⁄8 2.485 0.87575 1 ⁄32 , 3 ⁄64 , 1 ⁄16 , 3 ⁄32 , 2.985 1 1.00075 1 ⁄8 and 5 ⁄32 1 ⁄32 , 3 ⁄64 , 1 ⁄16 , 3 ⁄32 , 1 ⁄8 , 3.985 1 1.00075 5 ⁄32 and 3 ⁄16 1 ⁄16 , 3 ⁄32 , 1 ⁄8 1 1.00075 4.985 1 ⁄8 1.2510 4.985 11 ⁄4 1 ⁄16 , 3 ⁄32 , 1 ⁄8 1 1.00075 5.985 1 ⁄8 , 3 ⁄16 1.2510 5.985 11 ⁄4 1 ⁄8 1 1.00075 7.985 1 ⁄8 7.985 1.2510 11 ⁄4 c Metal Slitting Saws with Side Teeth 1 ⁄16 , 3 ⁄32 , 1 ⁄8 7⁄8 2.485 0.87575 1 ⁄16 , 3 ⁄32 , 1 ⁄8 , 5 ⁄32 2.985 1 1.00075 1 ⁄16 , 3 ⁄32 , 1 ⁄8 , 5 ⁄32 , 3 ⁄16 1 1.00075 3.985 1 3 1 5 3 1 1.00075 4.985 ⁄16 , ⁄32 , ⁄8 , ⁄32 , ⁄16 1 ⁄8 1.2510 4.985 11 ⁄4 1 ⁄16 , 3 ⁄32 , 1 ⁄8 , 3 ⁄16 1 1.00075 5.985 1 ⁄8 , 3 ⁄16 1.2510 5.985 11 ⁄4 1 ⁄8 1 1.00075 7.985 1 3 1 7.985 1.2510 ⁄8 , ⁄16 1 ⁄4 Metal Slitting Saws with Staggered Peripheral and Side Teeth d 3 ⁄16 1 1.00075 2.985 3 ⁄16 3.985 1 1.00075 3 1 1 1.00075 4.985 ⁄16 , ⁄4 3 ⁄16 , 1 ⁄4 1 1.00075 5.985 3 ⁄16 , 1 ⁄4 1.2510 5.985 11 ⁄4 3 ⁄16 , 1 ⁄4 1.2510 7.985 11 ⁄4 3 ⁄16 , 1 ⁄4 1.2510 9.985 11 ⁄4 1 5 1 11.985 1.5010 ⁄4 , ⁄16 1 ⁄2 Min.

a Tolerances on face widths are plus or minus 0.001 inch.

Min. 0.8750 1.0000 1.0000 1.0000 1.2500 1.0000 1.2500 1.0000 1.2500 0.8750 1.0000 1.0000 1.0000 1.2500 1.0000 1.2500 1.0000 1.2500 1.0000 1.0000 1.0000 1.0000 1.2500 1.2500 1.2500 1.5000

b Plain metal slitting saws are relatively thin plain milling cutters having peripheral teeth only. They are furnished with or without hub, and their sides are concaved to the arbor hole or hub. c Metal slitting saws with side teeth are relatively thin side milling cutters having both peripheral and side teeth. d Metal slitting saws with staggered peripheral and side teeth are relatively thin staggered-tooth milling cutters having peripheral teeth of alternate right- and left-hand helix and alternate side teeth. All dimensions are in inches. All saws are high-speed steel. Metal slitting saws are similar to plain or side milling cutters but are relatively thin.

Milling Cutter Terms

Face Width

Helical Teeth Helical Rake Angle L.H. Helix Shown

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Machinery's Handbook, 31st Edition Milling Cutters

877

Milling CutterCutter TermsTerms (Continued) Milling

Radial Rake Angle (Positive Shown)

Tooth Face

Radial Relief Tooth

Axial Relief

Flute Fillet

Offset

Peripheral Cutting Edge

Radial Relief Angle

Tooth Face

Tooth Face

Clearance Surface Land Heel Flute

Axial Relief Angle

Clearance Surface

Radial Rake Angle (Positive Shown) Offset

Fillet

Tooth Concavity

Lip

Lip Angle

American National Standard Single- and Double-Angle Milling Cutters ANSI/ASME B94.19-1997 (R2019) Cutter Diameter Nom.

Max.

Hole Diameter Min.

Nominal Face Widtha

Nom.

Max.

Min.

Single-Angle Cuttersb 3 ⁄8 -24 UNF-2B RH

c11 ⁄4

1.265

1.235

7⁄16

c15 ⁄8

1.640

1.610

9 ⁄16

23⁄4

2.765

2.735

1 ⁄2

1

1.00075

1.0000

3

3.015

2.985

1 ⁄2

11 ⁄4

1.2510

1.2500

23⁄4

2.765

2.735

1

1.00075

1.0000

3 ⁄8 -24 UNF-2B LH 1 ⁄2 -20 UNF-2B RH

Double-Angle Cuttersd 1 ⁄2

a Face width tolerances are plus or minus 0.015 inch.

b Single-angle milling cutters have peripheral teeth, one cutting edge of which lies in a conical surface and the other in the plane perpendicular to the cutter axis. There are two types: one has a plain keywayed hole and has an included tooth angle of either 45 or 60 degrees ±10 minutes; the other has a threaded hole and an included tooth angle of 60 degrees ±10 minutes. Cutters with a right-hand threaded hole have a right-hand hand of rotation and a right-hand hand of cutter. Cutters with a left-hand threaded hole have a left-hand hand of rotation and a left-hand hand of cutter. Cutters with plain keywayed holes are standard as either right-hand or left-hand cutters. c These cutters have threaded holes, the sizes of which are given under “Hole Diameter.” d Double-angle milling cutters have symmetrical peripheral teeth, both sides of which lie in conical surfaces. They are designated by the included angle, which may be 45, 60 or 90 degrees. Tolerances are ±10 minutes for the half angle on each side of the center. All dimensions are in inches. All cutters are high-speed steel.

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Machinery's Handbook, 31st Edition Milling Cutters

878

American National Standard Shell Mills ANSI/ASME B94.19-1997 (R2019) E L

D JK

C

H

F

W Dia., D inches 11 ⁄4

11 ⁄2 13⁄4 2

Width, W inches 1

11 ⁄8 11 ⁄4

13⁄8

21 ⁄4

11 ⁄2

23⁄4 3

15⁄8

21 ⁄2

31 ⁄2 4

41 ⁄2 5 6

15⁄8 13⁄4

17⁄8

21 ⁄4

21 ⁄4 21 ⁄4 21 ⁄4

Dia., H inches

B Length, B inches

Width, C inches

Depth, E inches

Radius, F inches

Dia., J inches

Dia., K degrees

1 ⁄2

5 ⁄8

1 ⁄4

5 ⁄32

1 ⁄64

11 ⁄16

5 ⁄8

1 ⁄2

5 ⁄8

1 ⁄4

5 ⁄32

1 ⁄64

11 ⁄16

5 ⁄8

3 ⁄4

3 ⁄4

5 ⁄16

3 ⁄16

1 ⁄32

15 ⁄16

7⁄8

3 ⁄4

3 ⁄4

5 ⁄16

3 ⁄16

1 ⁄32

15 ⁄16

7⁄8

3 ⁄4

3 ⁄8

7⁄32

1 ⁄32

3 ⁄4

3 ⁄8

7⁄32

1 ⁄32

3 ⁄4

3 ⁄8

7⁄32

1 ⁄32

3 ⁄4

1 ⁄2

9 ⁄32

1 ⁄32

3 ⁄4

1 ⁄2

9 ⁄32

1 ⁄32

5 ⁄8

3 ⁄8

1 ⁄16

5 ⁄8

3 ⁄8

1 ⁄16

5 ⁄8

3 ⁄8

1 ⁄16

3 ⁄4

7⁄16

1 ⁄16

1

1 1

11 ⁄4

11 ⁄4

11 ⁄2 11 ⁄2 11 ⁄2 2

1 1 1

1

11 ⁄4

21 ⁄32

17⁄8

213⁄16

0

5

11 ⁄2

29⁄16

0

13⁄16

121 ⁄32

21 ⁄16

0

0

13⁄16

111 ⁄16

0

13⁄16

13⁄8

11 ⁄2

Angle, L inches

11 ⁄2

0 5 5

5

17⁄8

10

21 ⁄2

15

17⁄8

10

All cutters are high-speed steel. Right-hand cutters with right-hand helix and square corners are standard. Tolerances: On D, +1 ⁄64 inch; on W, ±1 ⁄64 inch; on H, +0.0005 inch; on B, +1 ⁄64 inch; on C, at least +0.008 but not more than +0.012 inch; on E, +1 ⁄64 inch; on J, ±1 ⁄64 inch; on K, ±1 ⁄64 inch.

End Mill Terms

Radial Relief Angle

Radial Land

Radial Clearance Angle

Enlarged Section of End Mill Tooth

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Machinery's Handbook, 31st Edition Milling Cutters

879

End Mill (Continued) EndTerms Mill Terms End Cutting Edge Concavity Angle Tooth Face

Radial Rake Angle (Positive Shown) End Clearance

Tooth Face

Axial Relief Angle

Radial Cutting Edge

End Gash Flute

Helix Angle Enlarged Section of End Mill

American National Standard Multiple- and Two-Flute Single-End Helical End Mills with Plain Straight and Weldon Shanks ANSI/ASME B94.19-1997 (R2019) L

L

W S

D

S

Nom. 1 ⁄8 3 ⁄16 1 ⁄4 3 ⁄8 1 ⁄2

Cutter Diameter, D Max.

.130

.1925 .255 .380 .505

3 ⁄4

.755

1 ⁄8

.125

3 ⁄16 1 ⁄4 5 ⁄16 3 ⁄8 1 ⁄2 5 ⁄8 3 ⁄4 7⁄8

.1875 .250

.3125 .375 .500 .625 .750 .875

1

1.000

11 ⁄2

1.500

11 ⁄4

1.250

Min.

D

Shank Diameter, S

Max.

Min.

Multiple-Flute with Plain Straight Shanks .125

.1875 .250 .375 .500 .750

.125

.1875 .250 .375 .500 .750

.1245 .1870 .2495 .3745 .4995 .7495

Two-Flute for Keyway Cutting with Weldon Shanks .1235 .1860 .2485 .3110

.3735 .4985 .6235 .7485 .8735

.375 .375 .375 .375 .375 .500 .625 .750 .875

.9985

1.000

1.4985

1.250

1.2485

W

1.250

.3745 .3745 .3745 .3745 .3745

Length of Cut, W 5 ⁄16 1 ⁄2 5 ⁄8 3 ⁄4 15 ⁄16

11 ⁄4 3 ⁄8

7⁄16 1 ⁄2 9 ⁄16 9 ⁄16

.4995

1

.7495

15⁄16

.6245 .8745

11 ⁄4

13⁄8

111 ⁄16

113⁄16 21 ⁄4

25⁄8 25⁄16 25⁄16

25⁄16 25⁄16 25⁄16 3

15⁄16

37⁄16

11 ⁄2

33⁄4

15⁄8

41 ⁄8

.9995

15⁄8

1.2495

15⁄8

1.2495

Length Overall, L

39⁄16 41 ⁄8 41 ⁄8

All dimensions are in inches. All cutters are high-speed steel. Right-hand cutters with right-hand helix are standard.

The helix angle is not less than 10 degrees for multiple-flute cutters with plain straight shanks; the helix angle is optional with the manufacturer for two-flute cutters with Weldon shanks. Tolerances: On W, ±1 ⁄32 inch; on L, ±1 ⁄16 inch.

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Machinery's Handbook, 31st Edition Milling Cutters

880

ANSI Regular-, Long-, and Extra-Long-Length, Multiple-Flute Medium Helix Single-End End Mills with Weldon Shanks ANSI/ASME B94.19-1997 (R2019)

L W

D

S

Cutter Dia., D 1 ⁄8 b

As Indicated by the Dimensions Given Below, Shank Diameter S May Be Larger, Smaller, or the Same as the Cutter Diameter D Regular Mills

S

W

3 ⁄8

3 ⁄8

3 ⁄16 b

3 ⁄8

1 ⁄2

1 ⁄4 b

3 ⁄8

5 ⁄8

5 ⁄16 b

3 ⁄8

3 ⁄4

3 ⁄8 b

3 ⁄8

3 ⁄4

7⁄16

3 ⁄8

1 ⁄2

3 ⁄8

1 ⁄2 b

1 ⁄2

9 ⁄16

1 ⁄2

5 ⁄8

1 ⁄2

11 ⁄16

1 ⁄2

3 ⁄4

1 ⁄2

5 ⁄8 b

5 ⁄8

11 ⁄16

5 ⁄8

3 ⁄4 b

5 ⁄8

13 ⁄16

5 ⁄8

7⁄8

5 ⁄8

1

7⁄8

1

11 ⁄8

5 ⁄8 7⁄8 7⁄8 7⁄8

1 1

11 ⁄4

L

25⁄16

Long Mills

Na 4

23⁄8

4

21 ⁄2

4

27⁄16

21 ⁄2

4

4

211 ⁄16

4

31 ⁄4

4

211 ⁄16

4

13⁄8

33⁄8

4

15⁄8

35⁄8

4

15⁄8

33⁄4

4

17⁄8

33⁄4 4

4

17⁄8

4

13⁄8 15⁄8

15⁄8 15⁄8 17⁄8

33⁄8

35⁄8 33⁄4

4

S

…

… 3 ⁄8 3 ⁄8 3 ⁄8 1 ⁄2 1 ⁄2

…

…

W

…

…

11 ⁄4

L

…

…

31 ⁄16

13⁄8

31 ⁄8

13⁄4 2

33⁄4 4

11 ⁄2

…

…

31 ⁄4

…

…

4

5 ⁄8

4

3 ⁄4

3

51 ⁄4

4

…

…

…

6 6 6

…

… …

…

7⁄8

1

21 ⁄2 … … …

…

45⁄8 …

…

31 ⁄2 4

53⁄4

…

…

17⁄8 2

…

41 ⁄4

6

1

4

61 ⁄2

…

…

3 ⁄8 3 ⁄8

…

1 ⁄2

…

…

…

13⁄4 2

L

…

…

39⁄16 33⁄4

…

…

…

…

4

3 ⁄4

4

61 ⁄4

…

…

…

…

… 4

4

… 6

…

…

7⁄8

1

… …

…

…

… …

…

5

6

61 ⁄8 …

71 ⁄4

…

…

…

…

81 ⁄2 …

11 ⁄4

1

2

41 ⁄2

6

…

…

…

…

…

…

…

13⁄8

1

2

41 ⁄2

11 ⁄4

11 ⁄4

2

41 ⁄2

13⁄4 2

11 ⁄4

2

41 ⁄2

11 ⁄2

11 ⁄2

aN=

1

11 ⁄4

11 ⁄4

2

2 2

41 ⁄2

41 ⁄2 41 ⁄2

6 6

… 1

… 4

…

61 ⁄2

6

11 ⁄4

4

61 ⁄2

6

11 ⁄4

4

61 ⁄2

6 8

11 ⁄4 11 ⁄4

4 4

61 ⁄2 61 ⁄2

…

…

4

…

…

…

4

…

6

…

…

…

11 ⁄4 …

…

…

81 ⁄2 …

…

…

4

…

…

61 ⁄2 …

…

4

…

…

…

6

…

…

…

41 ⁄2

4

…

4

2

6

…

4

4

…

5

4

…

4

…

3

5 ⁄8

…

…

41 ⁄4 …

4

…

Na …

21 ⁄2 …

41 ⁄2

1

4

4

… 3 ⁄8

W

…

2

11 ⁄8

1

4

…

1

41 ⁄4

6

4

S

11 ⁄4 1

7⁄8

2

4

…

… …

4

…

4 4

…

…

41 ⁄8 41 ⁄8

…

…

61 ⁄2 …

17⁄8

Extra-Long Mills

Na

… …

… …

… 6

… … … …

6

…

…

…

…

6

11 ⁄4 …

8

101 ⁄2 …

…

6 6 8

…

…

… … …

…

…

… 6

…

Number of flutes. b In this size of regular mill, a left-hand cutter with left-hand helix is also standard. All dimensions are in inches. All cutters are high-speed steel. Helix angle is greater than 19 degrees but not more than 39 degrees. Right-hand cutters with right-hand helix are standard. Tolerances: On D, +0.003 inch; on S, -0.0001 to -0.0005 inch; on W, ±1 ⁄32 inch; on L, ±1 ⁄16 inch.

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Machinery's Handbook, 31st Edition Milling Cutters

881

ANSI Two-Flute, High Helix, Regular-, Long-, and Extra-Long-Length, Single-End End Mills with Weldon Shanks ANSI/ASME B94.19-1997 (R2019) L

W

D

S Regular Mill

Cutter Dia., D

1

S

W

1 ⁄4

3 ⁄8

5 ⁄8

5 ⁄16

3 ⁄8

3 ⁄4

3 ⁄8

3 ⁄8

3 ⁄4

7⁄16

3 ⁄8

1 ⁄2

1 ⁄2

5 ⁄8

5 ⁄8

3 ⁄4

3 ⁄4

7⁄8

7⁄8

1

11 ⁄4

1

11 ⁄4

11 ⁄4

Extra-Long Mill

S

W

L

S

W

L

27⁄16

3 ⁄8

11 ⁄4

31 ⁄16

3 ⁄8

39⁄16

21 ⁄2

3 ⁄8

31 ⁄4

13⁄4 2

3 ⁄8

33⁄4

3

5

21 ⁄2

3 ⁄8

211 ⁄16

11 ⁄4

31 ⁄4

15⁄8

37⁄8

15⁄8

11 ⁄4

11 ⁄2 2

Long Mill L

1 ⁄2 1 ⁄2

33⁄4

5 ⁄8 3 ⁄4

…

17⁄8 2

41 ⁄8

41 ⁄2

1

2

41 ⁄2

11 ⁄4

2

41 ⁄2

2

41 ⁄2

11 ⁄4 11 ⁄4

13⁄8

31 ⁄8

13⁄4 2

33⁄4 4

11 ⁄2

3 ⁄8

21 ⁄2 …

…

1 ⁄2

41 ⁄4 …

4

21 ⁄2 3

45⁄8

4

61 ⁄2

1

6

81 ⁄2

4

61 ⁄2

11 ⁄4 …

8

101 ⁄2 …

5 ⁄8

51 ⁄4 …

…

4

3 ⁄4

61 ⁄2

4

…

61 ⁄4 …

…

6

11 ⁄4

61 ⁄2

61 ⁄8

4

81 ⁄2

…

All dimensions are in inches. All cutters are high-speed steel. Right-hand cutters with right-hand helix are standard. Helix angle is greater than 39 degrees. Tolerances: On D, +0.003 inch; on S, -0.0001 to -0.0005 inch; on W, ±1 ⁄32 inch; and on L, ±1 ⁄16 inch.

Combination Shanks for End Mills ANSI/ASME B94.19-1997 (R2019) Right-Hand Cut

Left-Hand Cut .015

G K 1/2 K 90° H

E

B

F

C

12° Dia. A 11 ⁄2 2 21 ⁄2

.015

M

La

B

211 ⁄16

13⁄16

31 ⁄2

115⁄16

31 ⁄4

1/2 K

45°

123⁄32

a Length of shank.

90°

12°

D

A J

Central with “K”

J

45°

L

M

D H

B E

K

45°

C L F

45°

A

Central with “K”

G

C

D

E

F

G

H

J

K

.515

1.406

11 ⁄2

.515

1.371

9 ⁄16

1.302

.377

.700

.700

1.900

2.400

13⁄4 2

.700

.700

1.809

2.312

5 ⁄8 3 ⁄4

1.772

2.245

.440

.503

M 7⁄16 1 ⁄2 9 ⁄16

All dimensions are in inches. Modified for use as Weldon or Pin Drive shank.

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Machinery's Handbook, 31st Edition Milling Cutters

882

ANSI Roughing, Single-End End Mills with Weldon Shanks, High-Speed Steel ANSI/ASME B94.19-1997 (R2019) L W S Cutter D

Diameter

D Shank S

1 ⁄2

1 ⁄2

1 ⁄2

1 ⁄2

1 ⁄2

1 ⁄2

5 ⁄8

5 ⁄8

5 ⁄8

5 ⁄8

5 ⁄8

5 ⁄8

3 ⁄4

3 ⁄4

3 ⁄4

3 ⁄4

3 ⁄4

1 1

11 ⁄4 11 ⁄4 11 ⁄2 11 ⁄2 13⁄4 13⁄4

Cut W 1

11 ⁄4 2

11 ⁄4 15⁄8 21 ⁄2 11 ⁄2 15⁄8 3 2 4 2 4 2 4 2 4

3 ⁄4

1 1

11 ⁄4 11 ⁄4 11 ⁄4 11 ⁄4 11 ⁄4 11 ⁄4

Length

Overall L

Cutter D

3

Diameter

2 2 2 2 2 2 2 2 2

31 ⁄4 4

33⁄8 33⁄4 45⁄8 33⁄4 37⁄8 51 ⁄4 41 ⁄2 61 ⁄2 41 ⁄2 61 ⁄2 41 ⁄2 61 ⁄2 41 ⁄2 61 ⁄2

21 ⁄2 21 ⁄2 21 ⁄2 21 ⁄2 3 3 3 3

Shank S

Cut W

2 2 2 2 2 2 2 2 2 2 2 2 2

2 3 4 5 6 7 8 10 12 4 6 8 10 4 6 8 10

21 ⁄2 21 ⁄2 21 ⁄2 21 ⁄2

Length

All dimensions are in inches. Right-hand cutters with right-hand helix are standard. Tolerances: Outside diameter, +0.025, -0.005 inch; length of cut, +1 ⁄8 , -1 ⁄32 inch.

Overall L 53⁄4 63⁄4 73⁄4 83⁄4 93⁄4 103⁄4 113⁄4 133⁄4 153⁄4 73⁄4 93⁄4 113⁄4 133⁄4 73⁄4 93⁄4 113⁄4 133⁄4

American National Standard Heavy Duty, Medium Helix Single-End End Mills, 21 ⁄2 -inch Combination Shank, High-Speed Steel ANSI/ASME B94.19-1997 (R2019) L

W

2 1 2” Dia. of Cutter, D 21 ⁄2 21 ⁄2 21 ⁄2 21 ⁄2 21 ⁄2 21 ⁄2 21 ⁄2 3 3

No. of Flutes 3 3 6 6 6 6 6 2 2

D Length of Cut, W 8 10 4 6 8 10 12 4 6

Length Overall, L 12 14 8 10 12 14 16

73⁄4 93⁄4

Dia. of Cutter, D

No. of Flutes

Length of Cut, W

3 3 3 3 3 3 3 3 …

3 3 3 8 8 8 8 8 …

4 6 8 4 6 8 10 12 …

Length Overall, L 73⁄4 93⁄4 113⁄4 73⁄4 93⁄4 113⁄4 133⁄4 153⁄4

…

All dimensions are in inches. For shank dimensions, see page 881. Right-hand cutters with right-hand helix are standard. Helix angle is greater than 19 degrees but not more than 39 degrees. Tolerances: On D, +0.005 inch; on W, ±1 ⁄32 inch; on L, ±1 ⁄16 inch.

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Machinery's Handbook, 31st Edition Milling Cutters

883

ANSI Stub-, Regular-, and Long-Length, Four-Flute, Medium Helix, Plain-End, Double-End Miniature End Mills with 3 ⁄16 -Inch Diameter Straight Shanks ANSI/ASME B94.19-1997 (R2019)

W

L

B

D

B

W D

3 16

”

Stub Length

Dia. D

W

1⁄ 16 3⁄ 32 1⁄ 8 5⁄ 32 3⁄ 16

3⁄ 32 9⁄ 64 3⁄ 16 15 ⁄ 64 9⁄ 32

Dia. D

L 2 2 2 2 2

1⁄ 2

5⁄ 32

7⁄ 8

21 ⁄2

3⁄ 4

31 ⁄8

1

33⁄8

25⁄8 31 ⁄4

7⁄ 8

1

3⁄ 16

L

7⁄ 32

9⁄ 32

3⁄ 4

1⁄ 8

L 21 ⁄4 21 ⁄4 21 ⁄4 21 ⁄4 21 ⁄4

3⁄ 16 9⁄ 32 3⁄ 8 7⁄ 16 1⁄ 2

W

3⁄ 8

3⁄ 32

Regular Length

Long Length

B

1⁄ 16

W

All dimensions are in inches. All cutters are high-speed steel. Right-hand cutters with right-hand helix are standard. Helix angle is greater than 19 degrees but not more than 39 degrees. Tolerances: On D, + 0.003 inch (if the shank is the same diameter as the cutting portion, however, then the tolerance on the cutting diameter is - 0.0025 inch.); on W, + 1 ⁄32 , - 1 ⁄64 inch; and on L, 1 ± ⁄16 inch.

American National Standard 60-Degree Single-Angle Milling Cutters with Weldon Shanks ANSI/ASME B94.19-1997 (R2019) L W S

Dia., D 3⁄ 4

13⁄8

60°

D

S

W

L

Dia., D

S

W

L

3⁄ 8

5⁄ 16

21 ⁄8

17⁄8

7⁄ 8

13 ⁄ 16

31 ⁄4

5⁄ 8

9⁄ 16

27⁄8

21 ⁄4

1

11 ⁄16

33⁄4

All dimensions are in inches. All cutters are high-speed steel. Right-hand cutters are standard. Tolerances: On D, ± 0.015 inch; on S, - 0.0001 to - 0.0005 inch; on W, ± 0.015 inch; and on L, ±1 ⁄16 inch.

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Machinery's Handbook, 31st Edition Milling Cutters

884

American National Standard Stub-, Regular-, and Long-Length, Two-Flute, Medium Helix, Plain- and Ball-End, Double-End Miniature End Mills with 3 ⁄16 -Inch Diameter Straight Shanks ANSI/ASME B94.19-1997 (R2019) W

D

L

W

D

D

3 16

Dia., C and D

W

1 ⁄32

3 ⁄64

3 ⁄64

1 ⁄16

1 ⁄16

3 ⁄32

5 ⁄64

1 ⁄8

3 ⁄32

9 ⁄64

7⁄64

5 ⁄32

Plain-End

1 ⁄8

3 ⁄16

9 ⁄64

7⁄32

5 ⁄32

15 ⁄64

11 ⁄64

1 ⁄4

3 ⁄16

L

W

C

Stub Length L 2

W …

2

3 ⁄32

…

2

9 ⁄64

2

…

2

3 ⁄16

2

…

2

15 ⁄64

2

…

2

9 ⁄32

Long-Length, Plain-End

Dia., D

Ba

1 ⁄16

3 ⁄8

7⁄32

3 ⁄32

1 ⁄2

9 ⁄32

1 ⁄8

3 ⁄4

3 ⁄4

W

9 ⁄32

L

21 ⁄2 25⁄8 31 ⁄8

D C

3 16

Ball-End

…

2

2

W

L …

W

Plain-End

3 ⁄32

…

9 ⁄64

…

15 ⁄64

…

21 ⁄64

…

13 ⁄32

2

2

21 ⁄4

9 ⁄32

21 ⁄4

3 ⁄8 …

1 ⁄2

Dia., D

Ba

5 ⁄32

7⁄8

Ball-End

…

…

21 ⁄4 …

…

21 ⁄4

7⁄16

21 ⁄4

1 ⁄2

21 ⁄4 … 21 ⁄4 …

…

21 ⁄4

Long-Length, Plain-End W

7⁄8

1

1

L … 21 ⁄4 …

…

21 ⁄4

1 ⁄2

3 ⁄16

3 ⁄16

21 ⁄4

7⁄16

2

21 ⁄4

21 ⁄4

3 ⁄8

…

21 ⁄4

W …

21 ⁄4

9 ⁄32

2

L

21 ⁄4

3 ⁄16

2

Regular Length

L

31 ⁄4 33⁄8

a B is the length below the shank.

All dimensions are in inches. All cutters are high-speed steel. Right-hand cutters with right-hand helix are standard. Helix angle is greater than 19 degrees but not more than 39 degrees. Tolerances: On C and D, - 0.0015 inch for stub and regular length; + 0.003 inch for long length (if the shank is the same diameter as the cutting portion, however, then the tolerance on the cutting diam­eter is - 0.0025 inch.); on W, + 1 ⁄32 , - 1 ⁄64 inch; and on L, ± 1 ⁄16 inch.

American National Standard Multiple-Flute, Helical Series End Mills with Brown & Sharpe Taper Shanks L

W D

Dia., D

W

1 ⁄2

15 ⁄16

3 ⁄4

1

11 ⁄4 15⁄8

L

415⁄16 51 ⁄4 55⁄8

Taper No. 7 7 7

Dia., D 11 ⁄4 11 ⁄2 2

W 2

21 ⁄4 23⁄4

L

71 ⁄4 71 ⁄2 8

Taper No. 9 9 9

All dimensions are in inches. All cutters are high-speed steel. Right-hand cutters with right-hand helix are standard. Helix angle is not less than 10 degrees. No. 5 taper is standard without tang; Nos. 7 and 9 are standard with tang only. Tolerances: On D, +0.005 inch; on W, ±1 ⁄32 inch; and on L ±1 ⁄16 inch. For dimensions of B & S taper shanks, see information given on page 1023.

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Machinery's Handbook, 31st Edition Milling Cutters

885

American National Standard Stub- and Regular-Length, Two-Flute, Medium Helix, Plain- and Ball-End, Single-End End Mills with Weldon Shanks ANSI/ASME B94.19-1997 (R2019) L

W

S

D

L

W

D

S C

Regular Length — Plain End Dia., D 1 ⁄8

S

9 ⁄16

3 ⁄8 3⁄8 3 ⁄8 3 ⁄8 3 ⁄8 3 ⁄8 3 ⁄8 1 ⁄2 1 ⁄2

5 ⁄8

1 ⁄2

3 ⁄4

1 ⁄2

3 ⁄16 1 ⁄4

5 ⁄16 3 ⁄8

7⁄16 1 ⁄2

1 ⁄2

11 ⁄16 5 ⁄8

11 ⁄16

3 ⁄4

13 ⁄16 7⁄8

1

7⁄8

1 11 ⁄8 11 ⁄4 1 11 ⁄8 11 ⁄4 13⁄8 11 ⁄2 11 ⁄4 11 ⁄2 13⁄4 2

1 ⁄2 5 ⁄8 5 ⁄8 5 ⁄8 5 ⁄8 5 ⁄8 5 ⁄8 7⁄8 7⁄8 7⁄8 7⁄8

1 1 1 1 1 11 ⁄4 11 ⁄4 11 ⁄4 11 ⁄4

W 3 ⁄8

7⁄16

1 ⁄2

9 ⁄16 9 ⁄16

13 ⁄16 13 ⁄16 1 11 ⁄8

11 ⁄8 15⁄16 15⁄16 15⁄16 15⁄16 15⁄16 11 ⁄2 11 ⁄2 11 ⁄2 11 ⁄2 11 ⁄2 15⁄8 15⁄8 15⁄8 15⁄8 15⁄8 15⁄8 15⁄8 15⁄8 15⁄8 15⁄8 15⁄8

Stub Length — Plain End

L 25⁄16 25⁄16 25⁄16 25⁄16 25⁄16 21 ⁄2 21 ⁄2 3 31 ⁄8

31 ⁄8 35⁄16 35⁄16 37⁄16 37⁄16 37⁄16 35⁄8 35⁄8 35⁄8 33⁄4 33⁄4 37⁄8 37⁄8 41 ⁄8 41 ⁄8 41 ⁄8 41 ⁄8 41 ⁄8 41 ⁄8 41 ⁄8 41 ⁄8 41 ⁄8

Cutter Dia., D 1 ⁄8

3 ⁄16 1 ⁄4

Shank Dia., S 3 ⁄8 3 ⁄8 3 ⁄8

Length of Cut. W 3 ⁄16

9 ⁄32

3 ⁄8

Length Overall. L 21 ⁄8 23⁄16 21 ⁄4

Regular Length — Ball End

Dia., C and D 1 ⁄8 3 ⁄16 1 ⁄4

Shank Dia., S 3 ⁄8 3 ⁄8 3 ⁄8

1 ⁄2

1

21 ⁄2 21 ⁄2 3

1 ⁄2

1 11 ⁄8 11 ⁄8

3 31 ⁄8 31 ⁄8

13⁄8 15⁄16 15⁄8

31 ⁄2 35⁄16 37⁄8

3 ⁄8

7⁄16 1 ⁄2 5 ⁄8

1 ⁄2

9 ⁄16

5 ⁄8

Length Overall. L 25⁄16 3 2 ⁄8 27⁄16

3 ⁄4

5 ⁄16 3 ⁄8

Length of Cut. W 3 ⁄8 1 ⁄2 5 ⁄8

3 ⁄8

1 ⁄2

5 ⁄8

3 ⁄4

3 ⁄4

1 ⁄2

7⁄8

1 11 ⁄8

7⁄8

1 1

2 21 ⁄4 21 ⁄4

41 ⁄4 43⁄4 43⁄4

11 ⁄4 11 ⁄2

11 ⁄4 11 ⁄4

21 ⁄2 21 ⁄2

5 5

3 ⁄4

3 ⁄4

All dimensions are in inches. All cutters are high-speed steel. Right-hand cutters with right-hand helix are standard. Helix angle is greater than 19 degrees but not more than 39 degrees. Tolerances: On C and D, -0.0015 inch for stub-length mills, + 0.003 inch for regular-length mills; on S, -0.0001 to -0.0005 inch; on W, ± 1 ⁄32 inch; and on L, ± 1 ⁄16 inch. The following single-end end mills are available in premium high-speed steel: ball end, two flute, with D ranging from 1 ⁄8 to 11 ⁄2 inches; ball end, multiple flute, with D ranging from 1 ⁄8 to 1 inch; and plain end, two flute, with D ranging from 1 ⁄8 to 11 ⁄2 inches.

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Machinery's Handbook, 31st Edition Milling Cutters

886

American National Standard Long-Length Single-End and Stub-, and Regular-Length, Double-End, Plain- and Ball-End, Medium Helix, Two-Flute End Mills with Weldon Shanks ANSI/ASME B94.19-1997 (R2019) L

B

W D

S C

L

W

D

W D

S

C Dia., C and D

Single End Long Length — Plain End S

1 ⁄8

3 ⁄16 1 ⁄4

3 ⁄8

3 ⁄8

3 ⁄8

5 ⁄16 7⁄16

3 ⁄8

1 ⁄2

1 ⁄2

3 ⁄4

3 ⁄4

5 ⁄8

1 11 ⁄4

… …

…

5 ⁄8

1 11 ⁄4

Ba

W

… …

11 ⁄2 13⁄4 13⁄4 … 27⁄32 223⁄32 311 ⁄32 431 ⁄32 431 ⁄32

5 ⁄8

Dia., C and D

S

1 ⁄8

3 ⁄8

3 ⁄16

3 ⁄8

5 ⁄32

7⁄32

1 ⁄4

9 ⁄32

5 ⁄16

11 ⁄32 3 ⁄8

13 ⁄32

7⁄16

15 ⁄32 1 ⁄2

9 ⁄16

5 ⁄8

11 ⁄16 3 ⁄4

7⁄8

1

Stub Length — Plain End W 3 ⁄16

3 ⁄8

15 ⁄64

3 ⁄8

21 ⁄64

… … … … … … … … … … … … … …

… … … … … … … … … … … … … …

3 ⁄8

9 ⁄32

3 ⁄8

… …

3 ⁄4

… 1 13⁄8 15⁄8 21 ⁄2 3

L 23⁄4 23⁄4 23⁄4 27⁄8 27⁄8 … … … … … … … … … … … … … …

S

… … 1 3 ⁄16 35⁄16 35⁄16 … 4 45⁄8 53⁄8 71 ⁄4 71 ⁄4

3 ⁄4

a B is the length below the shank.

Long Length — Ball End

L

3 ⁄8 3 ⁄8 3 ⁄8

1 ⁄2 1 ⁄2 5 ⁄8 3 ⁄4

3 ⁄8

3 ⁄8

3 ⁄8

7⁄16

3 ⁄8 3 ⁄8 3 ⁄8 3 ⁄8 3 ⁄8 3 ⁄8

1 ⁄2 1 ⁄2 1 ⁄2 1 ⁄2 5 ⁄8 5 ⁄8 3 ⁄4 3 ⁄4

7⁄8

1

7⁄16 1 ⁄2 1 ⁄2

9 ⁄16 9 ⁄16 9 ⁄16 9 ⁄16

13 ⁄16 13 ⁄16 13 ⁄16 13 ⁄16

11 ⁄8 11 ⁄8 15⁄16 15⁄16 19⁄16 15⁄8

3 ⁄8

11 ⁄8 11 ⁄2 13⁄4 13⁄4 17⁄8 21 ⁄4 23⁄4 33⁄8 5 …

3 ⁄8

1

W

13 ⁄16

…

Double End Regular Length — Plain End S W 3 ⁄8

Ba

3 ⁄8

L 31 ⁄16 31 ⁄8 31 ⁄8 31 ⁄8 31 ⁄8 31 ⁄8 31 ⁄8 31 ⁄8 31 ⁄8 33⁄4 33⁄4 33⁄4 33⁄4 41 ⁄2 41 ⁄2 5 5 51 ⁄2 57⁄8

1 ⁄2 5 ⁄8 3 ⁄4

3 ⁄4 1 1 13⁄8 15⁄8 21 ⁄2 …

S 3 ⁄8 3 ⁄8 3 ⁄8 3 ⁄8 3 ⁄8 1 ⁄2 1 ⁄2 5 ⁄8 3 ⁄4

1

… … … … … … … … …

L 23⁄8 211 ⁄16 31 ⁄16 35⁄16 35⁄16 311 ⁄16 4 45⁄8 53⁄8 71 ⁄4 …

Regular Length — Ball End W L 3 ⁄8 … 7⁄16 … 1 ⁄2 … 9 ⁄16 … 9 ⁄16 … 13 ⁄16 … 13 ⁄16 … 11 ⁄8 … 15⁄16 … 15⁄8

31 ⁄16 … 31 ⁄8 … 31 ⁄8 … 31 ⁄8 … 31 ⁄8 … 33⁄4 … 33⁄4 … 41 ⁄2 … 5 … 57⁄8

All dimensions are in inches. All cutters are high-speed steel. Right-hand cutters with right-hand helix are standard. Helix angle is greater than 19 degrees but not more than 39 degrees. Tolerances: On C and D, + 0.003 inch for single-end mills, -0.0015 inch for double-end mills; on S, -0.0001 to -0.0005 inch; on W, ±1 ⁄32 inch; and on L, ±1 ⁄16 inch.

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Machinery's Handbook, 31st Edition Milling Cutters

887

American National Standard Regular-, Long-, and Extra-Long-Length, Three- and Four-Flute, Medium Helix, Center Cutting, Single-End End Mills with Weldon Shanks ANSI/ASME B94.19-1997 (R2019)

L

W

S

D L

S Dia., D 1 ⁄8

3 ⁄8

3 ⁄8

1 ⁄4

3 ⁄8

5 ⁄8

3 ⁄16

5 ⁄16 3 ⁄8

1 ⁄2 5 ⁄8

11 ⁄16 3 ⁄4

7⁄8

1 11 ⁄8 11 ⁄4 11 ⁄2 Dia., D

3 ⁄8

3 ⁄4

3 ⁄8

3 ⁄4

11 ⁄4 15⁄8 15⁄8 15⁄8 17⁄8 2 2 2 2

1 ⁄2 5 ⁄8 5 ⁄8 3 ⁄4

7⁄8

1 1 11 ⁄4 11 ⁄4

S W Regular Length 3 ⁄8

3 ⁄8

3 ⁄8

5 ⁄8

3 ⁄8

7⁄16 1 ⁄2 1 ⁄2

9 ⁄16

9 ⁄16

5 ⁄8

3 ⁄4

5 ⁄8 3 ⁄4

7⁄8

3 ⁄4

7⁄8

3 ⁄8 3 ⁄8 3 ⁄8 3 ⁄8 3 ⁄8

1 ⁄2 1 ⁄2 1 ⁄2 1 ⁄2 1 ⁄2 5 ⁄8 5 ⁄8 5 ⁄8 5 ⁄8 3 ⁄4 3 ⁄4 3 ⁄4

7⁄8

1

Four Flute Long Length S W

L

… …

25⁄16 23⁄8 27⁄16 21 ⁄2 21 ⁄2 31 ⁄4 33⁄4 33⁄4 37⁄8 41 ⁄8 41 ⁄2 41 ⁄2 41 ⁄2 41 ⁄2

1 ⁄2

1 ⁄8

5 ⁄16

1 1 1

3 ⁄8

1 ⁄4

3 ⁄16

1

D

Regular Length W

S

W

1 ⁄2 3 ⁄4 3 ⁄4 1 1 11 ⁄4 13⁄8 13⁄8 13⁄8 15⁄8 15⁄8 15⁄8 17⁄8 17⁄8 15⁄8 17⁄8 17⁄8 17⁄8 2

3 ⁄8 3 ⁄8 3 ⁄8

1 ⁄2 5 ⁄8

… 3 ⁄4 7⁄8 1 … 11 ⁄4 … L

Three Flute

25⁄16 23⁄8 27⁄16 21 ⁄2 21 ⁄2 211 ⁄16 211 ⁄16 31 ⁄4 33⁄8 33⁄8 33⁄8 35⁄8 33⁄4 33⁄4 4 4 37⁄8 41 ⁄8 41 ⁄8 41 ⁄8 41 ⁄2

… … 11 ⁄4 13⁄8 11 ⁄2 2 21 ⁄2 … 3 31 ⁄2 4 … 4 …

Dia., D 11 ⁄8 11 ⁄4 11 ⁄2 11 ⁄4 11 ⁄2 13⁄4 2

L

S

… … 33⁄16 31 ⁄8 31 ⁄4 4 45⁄8 … 51 ⁄4 53⁄4 61 ⁄2 … 61 ⁄2 …

… …

Extra-Long Length W L … … 13⁄4 2 21 ⁄2 3 4 … 4 5 6 … 6 …

3 ⁄8 3 ⁄8 3 ⁄8

1 ⁄2 5 ⁄8

…

3 ⁄4

7⁄8

1 … 11 ⁄4 …

S W Regular Length (cont.) 1 1 1 11 ⁄4 11 ⁄4 11 ⁄4 11 ⁄4

… … 39⁄16 33⁄4 41 ⁄4 5 61 ⁄8 … 61 ⁄4 71 ⁄4 81 ⁄2 … 81 ⁄2 … L

2 2 2 2 2 2 2

41 ⁄2 41 ⁄2 41 ⁄2 41 ⁄2 41 ⁄2 41 ⁄2 41 ⁄2

11 ⁄4 13⁄8 11 ⁄2 13⁄4 2 21 ⁄2 3 4 4 4 4 4

311 ⁄16 31 ⁄8 31 ⁄4 33⁄4 4 45⁄8 51 ⁄4 61 ⁄2 61 ⁄2 61 ⁄2 61 ⁄2 61 ⁄2

Long Length 1 ⁄4

3 ⁄8

3 ⁄8

3 ⁄8

5 ⁄16 7⁄16 1 ⁄2 5 ⁄8 3 ⁄4

1 11 ⁄4 11 ⁄2 13⁄4 2

3 ⁄8 1 ⁄2 1 ⁄2 5 ⁄8

3 ⁄4 1 11 ⁄4 11 ⁄4 11 ⁄4 11 ⁄4

All dimensions are in inches. All cutters are high-speed steel. Right-hand cutters with right-hand helix are standard. Helix angle is greater than 19 degrees but not more than 39 degrees. Tolerances: On D, +0.003 inch; on S, -0.0001 to -0.0005 inch; on W, ±1 ⁄32 inch; and on L, ±1 ⁄16 inch. The following center-cutting, single-end end mills are available in premium high-speed steel: reg­ular-length, multiple flute, with D ranging from 1 ⁄8 to 11 ⁄2 inches; long length, multiple flute, with D ranging from 3⁄8 to 11 ⁄4 inches; and extra-long-length, multiple flute, with D ranging from 3 ⁄ to 11 ⁄ inches. 8 4

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Machinery's Handbook, 31st Edition Milling Cutters

888

American National Standard Stub- and Regular-Length, Four-Flute, Medium Helix, Double-End End Mills with Weldon Shanks ANSI/ASME B94.19-1997 (R2019) L

W D

W

S

D

Dia., D

S

1 ⁄8

3 ⁄8

3 ⁄16

3 ⁄16

3 ⁄8

5 ⁄32

3 ⁄8

15 ⁄64

23⁄4

7⁄32

3 ⁄8

21 ⁄64

27⁄8

31 ⁄16

11 ⁄32

3 ⁄8

3 ⁄4

31 ⁄2

3 ⁄8 a

3 ⁄8

3 ⁄4

13 ⁄32

1 ⁄2

7⁄16

1 ⁄2

15 ⁄32

1 ⁄2

1 ⁄2 a

1 ⁄2

1

9 ⁄16

5 ⁄8

13⁄8

W

1 ⁄8 a

3 ⁄8

3 ⁄8

5 ⁄32 a

3 ⁄8

7⁄16

3 ⁄16 a

3 ⁄8

1 ⁄2

7⁄32

3 ⁄8

9 ⁄16

1 ⁄4 a

3 ⁄8

5 ⁄8

9 ⁄32

3 ⁄8

11 ⁄16

5 ⁄16 a

3 ⁄8

3 ⁄4

Dia., D

L 23⁄4

31 ⁄8 31 ⁄4

31 ⁄4

33⁄8 33⁄8

31 ⁄2

S

W

Dia., D

L

Stub Length

23⁄4

9 ⁄32

Regular Length

1

1 ⁄4

1

5 ⁄8 a

41 ⁄8

13 ⁄16

7⁄8

7⁄8

7⁄8

3 ⁄4 a

1

41 ⁄8 5

3 ⁄4

…

55⁄8

17⁄8

61 ⁄8

55⁄8

17⁄8 …

17⁄8

…

5

15⁄8 15⁄8

3 ⁄4

1

27⁄8

… 13⁄8

5 ⁄8

11 ⁄16

L

3 ⁄8

…

31 ⁄2

41 ⁄8

W

3 ⁄8

…

41 ⁄8

1

S

61 ⁄8 63⁄8

…

…

a In this size of regular mill, a left-hand cutter with a left-hand helix is also standard.

All dimensions are in inches. All cutters are high-speed steel. Right-hand cutters with right-hand helix are standard. Helix angle is greater than 19 degrees but not more than 39 degrees. Tolerances: On D, +0.003 inch (if the shank is the same diameter as the cutting portion, however, then the tolerance on the cutting diameter is -0.0025 inch); on S, -0.0001 to -0.0005 inch; on W, 1 ± ⁄32 inch; and on L, ±1 ⁄16 inch.

American National Standard Regular-Length, Three- and Four-Flute, Medium Helix, Double-End End Mills with Weldon Shanks ANSI/ASME B94.19-1997 (R2019) W

L

S

W D

D Dia., D

S

Three Flute

1 ⁄8

3 ⁄8

3 ⁄8

3 ⁄8

1 ⁄2

1 ⁄4

3 ⁄8

5 ⁄8

5 ⁄16

3 ⁄8

3 ⁄4

3 ⁄8

3 ⁄8

3 ⁄4

7⁄16

1 ⁄2

1

1 ⁄2

1 ⁄2

1

9 ⁄16

5 ⁄8

13⁄8

5 ⁄8

5 ⁄8

3 ⁄4

3 ⁄4

1

1

D

13⁄8

15⁄8

17⁄8

W D

S

L

Dia., D

S

31 ⁄16

1 ⁄8

3 ⁄8

3 ⁄8

3 ⁄16

3 ⁄8

1 ⁄2

1 ⁄4

3 ⁄8

5 ⁄8

5 ⁄16

3 ⁄8

3 ⁄4

3 ⁄8

3 ⁄8

1 ⁄2

1 ⁄2

5 ⁄8

5 ⁄8

3 ⁄4

3 ⁄4

W

3 ⁄16

L

W

31 ⁄4

33⁄8

31 ⁄2 31 ⁄2

41 ⁄8 41 ⁄8 5 5 55⁄8

63⁄8

7⁄8

1

…

7⁄8

1

…

Four Flute

W

3 ⁄4

1 13⁄8

15⁄8

L 31 ⁄16

31 ⁄4

33⁄8

31 ⁄2 31 ⁄2

41 ⁄8 5

55⁄8

17⁄8

61 ⁄8

…

…

17⁄8

63⁄8

All dimensions are in inches. All cutters are high-speed steel. Right-hand cutters with right-hand helix are standard. Helix angle is greater than 19 degrees but not more than 39 degrees. Tolerances: On D, +0.0015 inch; on S, -0.0001 to -0.0005 inch; on W, ±1 ⁄32 inch; and on L, ±1 ⁄16 inch.

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Machinery's Handbook, 31st Edition Milling Cutters

889

American National Standard Plain- and Ball-End, Heavy-Duty, Medium Helix, Sin­gleEnd End Mills with 2-Inch Diameter Shanks ANSI/ASME B94.19-1997 (R2019) L

W

Two

2”

Four

Six

Two

Three

Four

Six

D C L

W

2”

D

Dia., C and D

W

2

2

2

3

2

…

2

8

2

Plain End L

…

73⁄4 …

2, 3, 4, 6

…

4

5

73⁄4

83⁄4

6

113⁄4

6

8

93⁄4

113⁄4 …

6

21 ⁄2

…

73⁄4 …

21 ⁄2

8

113⁄4

6

21 ⁄2

No. of Flutes

…

2, 3

…

2, 3, 4, 6

93⁄4

4

21 ⁄2

Ball End L

2, 4, 6

63⁄4

6

W

53⁄4

4

2

No. of Flutes

93⁄4

6

2, 3, 4, 6

…

2, 4, 6

…

… 6

…

…

5

2, 4 6

… 4

83⁄4 …

…

…

…

…

…

All dimensions are in inches. All cutters are high-speed steel. Right-hand cutters with right-hand helix are standard. Helix angle is greater than 19 degrees but not more than 39 degrees. Tolerances: On C and D, + 0.005 inch for 2, 3, 4 and 6 flutes: on W, ± 1 ⁄16 inch; and on L, ± 1 ⁄16 inch.

Dimensions of American National Standard Weldon Shanks ANSI/ASME B94.19-1997 (R2019)

Dia.

Shank

3 ⁄8 1 ⁄2 5 ⁄8 3 ⁄4 7⁄8

Length 19⁄16

Flat Xa

Lengthb

0.325

0.280

Dia. 1

125⁄32

0.440

0.330

11 ⁄4

21 ⁄32

0.675

0.455

2

129⁄32

21 ⁄32

0.560

0.810

0.400

0.455

Shank

11 ⁄2

21 ⁄2

a X is distance from bottom of flat to opposite side of shank.

Length

Flat Xa

Lengthb

29⁄32

0.925

0.515

211 ⁄16

1.406

0.515

31 ⁄2

2.400

0.700

29⁄32 31 ⁄4

1.156

1.900

0.515

0.700

b Minimum.

All dimensions are in inches. Centerline of flat is at half-length of shank except for 11 ⁄2 -, 2- and 21 ⁄2 -inch shanks where it is 13⁄16 , 127⁄32 and 115⁄16 from shank end, respectively. Tolerance on shank diameter, - 0.0001 to - 0.0005 inch.

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Machinery's Handbook, 31st Edition Milling Cutters

890

Amerian National Standard Relieved, Concave, Convex, and Corner-Rounding Arbor-Type Cutters ANSI/ASME B94.19-1997 (R2019)

C

W H

H

D Concave

Nom.

Diameter C or Radius R Max.

C

W

H

D

Convex Min.

Cutter Dia. Da

R

W D

Corner-Rounding

Width W ± .010b

Nom.

Diameter of Hole H Max.

Min.

Concave Cuttersc 1 ⁄8 3 ⁄16 1 ⁄4 5 ⁄16 3 ⁄8 7⁄16 1 ⁄2 5 ⁄8 3 ⁄4 7⁄8

1

0.1270 0.1895 0.2520 0.3145 0.3770 0.4395 0.5040 0.6290 0.7540 0.8790

1.0040

1 ⁄8 3 ⁄16 1 ⁄4 5 ⁄16 3 ⁄8 7⁄16 1 ⁄2 5 ⁄8 3 ⁄4 7⁄8

1

0.1270

0.1240 0.1865 0.2490 0.3115

0.3740 0.4365 0.4980 0.6230 0.7480 0.8730

0.9980

0.1230

0.1895

0.1855

0.3145

0.3105

0.2520

0.2480

21 ⁄4

23⁄4 3 33⁄4

13⁄16

41 ⁄4

19⁄16

4

21 ⁄4

31 ⁄2

0.8770

1.0020

0.7480

0.8730

0.9980

4

41 ⁄4

1 ⁄4 3 ⁄8 1 ⁄2 5 ⁄8

0.1260

0.2520

0.3770

0.5020

0.6270

0.1240

0.2490

0.3740

0.4990

0.6240

21 ⁄2

1.250

11 ⁄4

1.251

11 ⁄4

1.251

1

1.00075

1

1.00075

1

1.00075

1

1.00075

1.250 1.250 1.250

1.00000

1.00000

1.00000 1.00000

1

1.00075

1.00000

1 ⁄2

1

1.00075

11 ⁄4

1.251

1.00000

5 ⁄8

11 ⁄4

1.251

1

7⁄16

3 ⁄4 7⁄8

1 1 ⁄4

1.251

11 ⁄4

1.251

1

1.250 1.250 1.250 1.250

1.00000

11 ⁄4

1.251

1.250

11 ⁄4

1.251

33⁄4

9 ⁄16

15 ⁄16

1.00000

1.00075

1

3 ⁄4

1.00075

11 ⁄4

13 ⁄32

41 ⁄4

1.00000

1.00000

1.251

3

41 ⁄4

1.00075

11 ⁄4

3 ⁄8

Corner-Rounding Cuttersd

1 ⁄8

1

1.00075

1.00000

1.00000

5 ⁄16

33⁄4

1.00000

1.00075

1.251

1 ⁄4

3

1.00075

1.00075

3 ⁄16

23⁄4

1

11 ⁄4

1 ⁄8

21 ⁄2

3

0.7520

Convex Cuttersc

21 ⁄4

0.4980

0.6230

13⁄8

1.00000

1

13 ⁄16

1

31 ⁄2

0.5020

0.6270

3 ⁄4

1.00000

1.00075

1

5 ⁄8

3

23⁄4

0.4355

9 ⁄16

1.00075

1

1

7⁄16

23⁄4

0.3730

0.4395

3 ⁄8

21 ⁄2

0.3770

1

1 ⁄4

21 ⁄4

11 ⁄4

1.00075 1.251

1.00000 1.250 1.250

a Tolerances on cutter diameter are +1 ⁄ , −1 ⁄ inch for all sizes. 16 16 b Tolerance does not apply to convex cutters.

c Size of cutter is designated by specifying diameter C of circular form.

d Size of cutter is designated by specifying radius R of circular form. All dimensions in inches. All cutters are high-speed steel and are form relieved. Right-hand corner rounding cutters are standard, but left-hand cutter for 1 ⁄4 -inch size is also stan­dard. For key and keyway dimensions for these cutters, see page 894.

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Machinery's Handbook, 31st Edition Milling Cutters

891

American National Standard Roughing and Finishing Gear Milling Cutters for Gears with 141 ⁄2 -Degree Pressure Angles ANSI/ASME B94.19-1997 (R2019)

D

D

H

H

ROUGHING

Diametral Pitch 1

11 ⁄4

11 ⁄2 13⁄4 2 2

21 ⁄2

21 ⁄2 3 1

11 ⁄4

Dia. of Cutter, D

Dia. of Hole, H

81 ⁄2

2

73⁄4 7

33⁄8

1

13⁄4

4

43⁄4

13⁄4

6

31 ⁄2

13⁄4

4

41 ⁄4

11 ⁄4 1

11 ⁄2

61 ⁄8

13⁄4

55⁄8

13⁄4

11 ⁄2

3

51 ⁄4

11 ⁄2

4

43⁄4

13⁄4

4

41 ⁄4

5 5 5

6

5

7

11 ⁄2

61 ⁄8

5

41 ⁄4

13⁄4

53⁄4

21 ⁄2

4

5

5

6

2

4

4

13⁄4

13⁄4

53⁄4

55⁄8 43⁄4

41 ⁄2 35⁄8

13⁄4

11 ⁄2 13⁄4

11 ⁄4

11 ⁄2

11 ⁄4 1

31 ⁄8 35⁄8

33⁄8

8

31 ⁄4

9

31 ⁄8

8 9

10

11 ⁄2

11 ⁄4 1

13⁄4

31 ⁄2 27⁄8 23⁄4 3

11 ⁄4 1

…

31 ⁄4

14

21 ⁄8

16

21 ⁄8

11 ⁄4 1

20

11 ⁄2

11 ⁄2

11 ⁄4 1

11 ⁄4 1

16

18

18

20

22

22

24

24

1

23⁄8 2

1

21 ⁄4 2

1

23⁄8 2

1

21 ⁄4

1

13⁄4

13⁄4

7⁄8

30

13⁄4

7⁄8

36

13⁄4

48

13⁄4 …

12

27⁄8

12

25⁄8

21 ⁄4

11 ⁄4 1

21 ⁄2

1

7⁄8

32

40

…

…

11 ⁄4 1

21 ⁄2

26

28

11 ⁄2

27⁄8

11 ⁄4 1

1

14

8

33⁄8

11 ⁄2

11

12

7

31 ⁄8

…

25⁄8

23⁄8

7

27⁄8 …

23⁄4

23⁄8

6

37⁄8

8

10 10

6

11 ⁄2

11 ⁄4 1

11 ⁄4 1 13⁄4

31 ⁄2

27⁄8

8

33⁄4

41 ⁄4

37⁄8

7

11

33⁄8

11 ⁄4

33⁄4 11 ⁄4 Finishing Gear Milling Cutters

7

13⁄4

11 ⁄2

43⁄8

6

43⁄8 41 ⁄4

41 ⁄2 35⁄8

6

2

43⁄4

4

2

61 ⁄2

3

3

81 ⁄2

11 ⁄2

21 ⁄2 3

Dia. of Hole, H

5

13⁄4

61 ⁄2

Dia. of Cutter, D

11 ⁄2

73⁄4 7

13⁄4 2

Diametral Pitch

51 ⁄4

53⁄4

53⁄4

FINISHING

3

2

61 ⁄2 61 ⁄2

Dia. of Dia. of Diametral Cutter, Hole, Pitch D H Roughing Gear Milling Cutters

13⁄4 13⁄4

13⁄4

…

7⁄8 7⁄8 7⁄8 7⁄8 7⁄8 7⁄8 7⁄8 7⁄8 7⁄8 7⁄8 7⁄8 7⁄8 7⁄8

…

…

All dimensions are in inches. All gear milling cutters are high-speed steel and are form relieved. For keyway dimensions see page 894. Tolerances: On outside diameter, + 1 ⁄16 , -1 ⁄16 inch; on hole diameter, through 1-inch hole diameter, +0.00075 inch, over 1-inch and through 2-inch hole diameter, +0.0010 inch. For cutter number relative to numbers of gear teeth, see page 2227. Roughing cutters are made with No. 1 cutter form only.

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Machinery's Handbook, 31st Edition Milling Cutters

892

American National Standard Gear Milling Cutters for Mitre and Bevel Gears with 141 ⁄2 -Degree Pressure Angles ANSI/ASME B94.19-1997 (R2019) Diameter of Cutter, D

Diametral Pitch 3 4 5 6 7 8

4

35⁄8 33⁄8 31 ⁄8 27⁄8 27⁄8

Diameter of Hole, H 11 ⁄4 11 ⁄4 11 ⁄4 1 1 1

Diameter of Cutter, D

Diametral Pitch 10 12 14 16 20 24

23⁄8 21 ⁄4 21 ⁄8 21 ⁄8 2

Diameter of Hole, H 7⁄8 7⁄8 7⁄8 7⁄8 7⁄8

13⁄4

7⁄8

All dimensions are in inches. All cutters are high-speed steel and are form relieved. For keyway dimensions see page 894. For cutter selection see page 2266. Tolerances: On outside diameter, +1 ⁄16 , -1 ⁄16 inch; on hole diameter, through 1-inch hole diameter, +0.00075 inch, for 11 ⁄4 -inch hole diameter, +0.0010 inch. To select the cutter number for bevel gears with the axis at any angle, double the back cone radius and multiply the result by the diametral pitch. This procedure gives the number of equivalent spur gear teeth and is the basis for selecting the cutter number from the table on page 2229. W

American National Standard Roller Chain Sprocket Milling Cutters

D H

Chain Pitch 1 ⁄4 1 ⁄4 1 ⁄4 1 ⁄4 1 ⁄4 1 ⁄4

3 ⁄8 3 ⁄8 3 ⁄8 3 ⁄8 3 ⁄8 3 ⁄8

1 ⁄2 1 ⁄2 1 ⁄2 1 ⁄2 1 ⁄2 1 ⁄2 5 ⁄8 5 ⁄8 5 ⁄8 5 ⁄8

5 ⁄8 5 ⁄8

American National Standard Roller Chain Sprocket Milling Cutters ANSI/ASME B94.19-1997 (R2019) Dia. of Roll 0.130 0.130 0.130 0.130 0.130 0.130 0.200 0.200 0.200 0.200 0.200 0.200 0.313 0.313 0.313 0.313 0.313 0.313 0.400 0.400 0.400 0.400 0.400 0.400

No. of Teeth in Sprocket 6 7-8 9-11 12-17 18-34 35 and over 6 7-8 9-11 12-17 18-34 35 and over 6 7-8 9-11 12-17 18-34 35 and over 6 7-8 9-11 12-17 18-34 35 and over

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Dia. of Cutter, D 23⁄4 23⁄4 23⁄4 23⁄4 23⁄4 23⁄4 23⁄4 23⁄4 23⁄4 23⁄4 23⁄4 23⁄4 3 3 31 ⁄8 31 ⁄8 31 ⁄8 31 ⁄8 31 ⁄8 31 ⁄8 31 ⁄4 31 ⁄4 31 ⁄4 31 ⁄4

Width of Cutter, W 5 ⁄16 5 ⁄16 5 ⁄16 5 ⁄16

9 ⁄32 9 ⁄32

15 ⁄32 15 ⁄32 15 ⁄32 7⁄16 7⁄16

13 ⁄32 3 ⁄4 3 ⁄4 3 ⁄4 3 ⁄4

23 ⁄32 11 ⁄16 3 ⁄4 3 ⁄4 3 ⁄4 3 ⁄4

23 ⁄32 11 ⁄16

Dia. of Hole, H 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

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Machinery's Handbook, 31st Edition Milling Cutters

Chain Pitch 3 ⁄4 3 ⁄4 3 ⁄4 3 ⁄4 3 ⁄4 3 ⁄4 1 1 1 1 1 11 ⁄4 11 ⁄4 11 ⁄4 11 ⁄4 11 ⁄4 11 ⁄2 11 ⁄2 11 ⁄2 11 ⁄2 11 ⁄2 11 ⁄2 13⁄4 13⁄4 13⁄4 13⁄4 13⁄4 13⁄4 2 2 2 2 2 2 21 ⁄4 21 ⁄4 21 ⁄4 21 ⁄4 21 ⁄4 21 ⁄4 21 ⁄2 21 ⁄2 21 ⁄2 21 ⁄2 21 ⁄2 21 ⁄2 3 3 3 3 3 3

American National Standard Roller Chain Sprocket Milling Cutters ANSI/ASME B94.19-1997 (R2019) (Continued) Milling Cutters ANSI/ASME B94.19-1997 (R2019) Dia. of Roll 0.469 0.469 0.469 0.469 0.469 0.469 0.625 0.625 0.625 0.625 0.625 0.750 0.750 0.750 0.750 0.750 0.875 0.875 0.875 0.875 0.875 0.875 1.000 1.000 1.000 1.000 1.000 1.000 1.125 1.125 1.125 1.125 1.125 1.125 1.406 1.406 1.406 1.406 1.406 1.406 1.563 1.563 1.563 1.563 1.563 1.563 1.875 1.875 1.875 1.875 1.875 1.875

No. of Teeth in Sprocket 6 7-8 9-11 12-17 18-34 35 and over 6 7-8 9-11 18-34 35 and over 6 7-8 9-11 18-34 35 and over 6 7-8 9-11 12-17 18-34 35 and over 6 7-8 9-11 12-17 18-34 35 and over 6 7-8 9-11 12-17 18-34 35 and over 6 7-8 9-11 12-17 18-34 35 and over 6 7-8 9-11 12-17 18-34 35 and over 6 7-8 9-11 12-17 18-34 35 and over

Dia. of Cutter, D 31 ⁄4 31 ⁄4 33⁄8 33⁄8 33⁄8 33⁄8 37⁄8 4 41 ⁄8 41 ⁄4 41 ⁄4 41 ⁄4 43⁄8 41 ⁄2 45⁄8 45⁄8 43⁄8 41 ⁄2 45⁄8 45⁄8 43⁄4 43⁄4 5 51 ⁄8 51 ⁄4 53⁄8 51 ⁄2 51 ⁄2 53⁄8 51 ⁄2 55⁄8 53⁄4 57⁄8 57⁄8 57⁄8 6 61 ⁄4 63⁄8 61 ⁄2 61 ⁄2 63⁄8 65⁄8 63⁄4 67⁄8 7 71 ⁄8 71 ⁄2 73⁄4 77⁄8 8 8 81 ⁄4

Width of Cutter, W 29 ⁄32 29 ⁄32 29 ⁄32 7⁄8 27⁄32 13 ⁄16 11 ⁄2 11 ⁄2 115⁄32 113⁄32 111 ⁄32 113⁄16 113⁄16 125⁄32 111 ⁄16 15⁄8 113⁄16 113⁄16 125⁄32 13⁄4 111 ⁄16 15⁄8 23⁄32 23⁄32 21 ⁄16 21 ⁄32 131 ⁄32 17⁄8 213⁄32 213⁄32 23⁄8 25⁄16 21 ⁄4 25⁄32 211 ⁄16 211 ⁄16 221 ⁄32 219⁄32 215⁄32 213⁄32 3 3 215⁄16 229⁄32 23⁄4 211 ⁄16 319⁄32 319⁄32 317⁄32 315⁄32 311 ⁄32 37⁄32

893

Dia. of Hole, H 1 1 1 1 1 1 11 ⁄4 11 ⁄4 11 ⁄4 11 ⁄4 11 ⁄4 11 ⁄4 11 ⁄4 11 ⁄4 11 ⁄4 11 ⁄4 11 ⁄4 11 ⁄4 11 ⁄4 11 ⁄4 11 ⁄4 11 ⁄4 11 ⁄2 11 ⁄2 11 ⁄2 11 ⁄2 11 ⁄2 11 ⁄2 11 ⁄2 11 ⁄2 11 ⁄2 11 ⁄2 11 ⁄2 11 ⁄2 11 ⁄2 11 ⁄2 11 ⁄2 11 ⁄2 11 ⁄2 11 ⁄2 13⁄4 13⁄4 13⁄4 13⁄4 13⁄4 13⁄4 2 2 2 2 2 2

All dimensions are in inches. All cutters are high-speed steel and are form relieved. For keyway dimensions see page 894. Tolerances: Outside diameter, +1 ⁄16 , -1 ⁄16 inch; hole diameter, through 1-inch diameter, + 0.00075 inch, above 1-inch diameter and through 2-inch diameter, + 0.0010 inch. For tooth form, see ANSI/ASME sprocket tooth form table on page 2633.

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A

C

Corner Radius

Corner Radius

894

E

H

B

ARBOR AND KEYSEAT Nom.Arbor and Cutter Hole Dia.

Nom. Size Key (Square)

1 ⁄2

3 ⁄32

5 ⁄8

1 ⁄8

3 ⁄4

1 ⁄8

7⁄8

1 ⁄8

1

11 ⁄4

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11 ⁄2 13⁄4 2

21 ⁄2 3

31 ⁄2 4

41 ⁄2 5

1 ⁄4 5 ⁄16 3 ⁄8 7⁄16 1 ⁄2 5 ⁄8 3 ⁄4 7⁄8

1

11 ⁄8 11 ⁄4

Arbor and Keyseat

CUTTER HOLE AND KEYWAY

A Max.

A Min.

B Max.

B Min.

C Max.

C Min.

0.0947

0.0937

0.4531

0.4481

0.106

0.099

0.1260 0.1260 0.1260 0.2510 0.3135 0.3760 0.4385 0.5010 0.6260 0.7510 0.8760

1.0010

1.1260 1.2510

0.1250 0.1250 0.1250 0.2500 0.3125 0.3750 0.4375 0.5000 0.6250 0.7500 0.8750

1.0000

1.1250 1.2500

a D max. is 0.010 inch larger than D min.

All dimensions given in inches.

0.5625 0.6875 0.8125 0.8438 1.0630 1.2810 1.5000 1.6870 2.0940 2.5000 3.0000

3.3750

3.8130 4.2500

F

D

0.5575 0.6825 0.8075 0.8388 1.0580 1.2760 1.4950 1.6820 2.0890 2.4950 2.9950

3.3700

3.8080 4.2450

0.137 0.137 0.137 0.262 0.343 0.410 0.473 0.535 0.660 0.785 0.910

1.035

1.160 1.285

0.130 0.130 0.130 0.255 0.318 0.385 0.448 0.510 0.635 0.760 0.885

1.010

1.135 1.260

Hole and Keyway

ARBOR AND KEY

Da Min.

H Nom.

Corner Radius

0.5578

3 ⁄64

0.020

1 ⁄16

1 ⁄32

1 ⁄16

1 ⁄32

0.6985 0.8225 0.9475 1.1040 1.3850 1.6660 1.9480 2.1980 2.7330 3.2650 3.8900

4.3900

4.9530 5.5150

1 ⁄16

1 ⁄32

3 ⁄32

3 ⁄64

1 ⁄8

1 ⁄16

5 ⁄32

1 ⁄16

3 ⁄16

1 ⁄16

3 ⁄16

1 ⁄16

7⁄32

1 ⁄16

1 ⁄4

3 ⁄32

3 ⁄8

3 ⁄32

3 ⁄8

3 ⁄32

7⁄16

1 ⁄8

1 ⁄2

1 ⁄8

E Max. 0.0932

Arbor and Key

E Min.

0.0927

F Max.

0.5468

F Min. 0.5408

0.1245

0.1240

0.6875

0.6815

0.1245

0.1240

0.9375

0.9315

0.1245 0.2495

0.3120

0.1240 0.2490

0.4365

1.9380

1.9320

0.4990

0.8745 0.9995 1.1245 1.2495

1.0880

1.3690

0.4995 0.7495

0.8065

1.3750

0.3740

0.6245

1.0940

0.3115

0.3745 0.4370

0.8125

0.6240 0.7490 0.8740 0.9990 1.1240 1.2490

1.6560 2.1880 2.7180 3.2500 3.8750 4.3750 4.9380 5.5000

1.6500 2.1820 2.7120 3.2440 3.8690 4.3690 4.9320 5.4940

Machinery's Handbook, 31st Edition Milling Cutters

Copyright 2020, Industrial Press, Inc.

American National Standard Keys and Keyways for Milling Cutters and Arbors ANSI/ASME B94.19-1997 (R2019)

Machinery's Handbook, 31st Edition Milling Cutters

895

American National Standard Woodruff Keyseat Cutters—Shank-Type StraightTeeth and Arbor-Type Staggered-Teeth ANSI/ASME B94.19-1997 (R2019) W

L

W

D 12

Cutter Number 202

202 1 ⁄2 302 1 ⁄2 203 303 403 204 304 404 305 405 505 605 406

Cutter Number 617 817 1017 1217 822

D

H

” Diam.

Nom. Dia.of Cutter, D

Width of Face, W

1 ⁄4

1 ⁄16

5 ⁄16

3 ⁄32

5 ⁄16 3 ⁄8 3 ⁄8 3 ⁄8

1 ⁄2 1 ⁄2 1 ⁄2 5 ⁄8 5 ⁄8 5 ⁄8 5 ⁄8 3 ⁄4

1 ⁄16

1 ⁄16

3 ⁄32 1 ⁄8

1 ⁄16

3 ⁄32 1 ⁄8

3 ⁄32 1 ⁄8

5 ⁄32 3 ⁄16 1 ⁄8

Length Over­ Cutter all, Number L 21 ⁄16 21 ⁄16 23⁄32 21 ⁄16 23⁄32 21 ⁄8 21 ⁄16 23⁄32 21 ⁄8 23⁄32 21 ⁄8 25⁄32 23⁄16 21 ⁄8

Nom. Dia.of Cutter, D

Width of Face, W

21 ⁄8 21 ⁄8 21 ⁄8 21 ⁄8 23⁄4

3 ⁄16

3 ⁄4

5 ⁄16

3 ⁄4

1 ⁄4

3 ⁄8 1 ⁄4

506 606 806 507 607 707 807 608 708 808 1008 1208 609 709

Dia. of Cutter Hole, Number H 3 ⁄4 3 ⁄4

1

1022

1222 1422 1622 1228

Shank-type Cutters Nom. Width Length Over­ Dia. of of Cutter all, Cutter, Face, Number L W D 3 ⁄4

5 ⁄32

3 ⁄4

1 ⁄4

3 ⁄4 7⁄8 7⁄8 7⁄8 7⁄8

1 1 1 1 1

3 ⁄16

5 ⁄32 3 ⁄16

7⁄32 1 ⁄4

3 ⁄16

7⁄32 1 ⁄4

5 ⁄16 3 ⁄8

3 ⁄16 11 ⁄8 7⁄32 11 ⁄8 Arbor-type Cutters

Nom. Dia.of Cutter, D

23⁄4 23⁄4 23⁄4 23⁄4 31 ⁄2

Width of Face, W 5 ⁄16 3 ⁄8

7⁄16 1 ⁄2 3 ⁄8

25⁄32 23⁄16 21 ⁄4 25⁄32 23⁄16 27⁄32 21 ⁄4 23⁄16 27⁄32 21 ⁄4 25⁄16 23⁄8 23⁄16 27⁄32

809 1009 610 710 810 1010 1210 811 1011 1211 812 1012 1212 …

Dia. of Cutter Hole, Number H 1 1 1 1 1

1628 1828 2028 2428 …

Nom. Dia.of Cutter, D 1 1 ⁄8 1 1 ⁄8 11 ⁄4 11 ⁄4 11 ⁄4 11 ⁄4 11 ⁄4 13⁄8 13⁄8 13⁄8 11 ⁄2 11 ⁄2 11 ⁄2 … Nom. Dia.of Cutter, D 31 ⁄2 31 ⁄2 31 ⁄2 31 ⁄2 …

Width of Face, W

Length Over­ all, L

…

2 1 ⁄4 2 5⁄16 23⁄16 27⁄32 21 ⁄4 25⁄16 23⁄8 21 ⁄4 25⁄16 23⁄8 21 ⁄4 25⁄16 23⁄8 …

Width of Face, W

Dia. of Hole, H

1 ⁄4

5 ⁄16

3 ⁄16

7⁄32 1 ⁄4

5 ⁄16 3 ⁄8 1 ⁄4

5 ⁄16 3 ⁄8 1 ⁄4

5 ⁄16 3 ⁄8

1 ⁄2

9 ⁄16 5 ⁄8

3 ⁄4 …

1 1 1 1 …

All dimensions are given in inches. All cutters are high-speed steel. Shank type cutters are standard with right-hand cut and straight teeth. All sizes have 1 ⁄2 -inch diam­eter straight shank. Arbor type cutters have staggered teeth. For Woodruff key and key-slot dimensions, see pages 2557 through 2559. Tolerances: Face with W for shank-type cutters: 1 ⁄16 - to 5⁄32 -inch face, + 0.0000, -0.0005; 3⁄16 to 7⁄32 , - 0.0002, - 0.0007; 1 ⁄4 , -0.0003, -0.0008; 5⁄16 , -0.0004, -0.0009; 3⁄8 , - 0.0005, -0.0010 inch. Face width W for arbor-type cutters; 3⁄16 inch face, -0.0002, -0.0007; 1 ⁄4 , -0.0003, -0.0008; 5 ⁄ , -0.0004, -0.0009; 3 ⁄ and over, -0.0005, -0.0010 inch. Hole size H: +0.00075, -0.0000 inch. 16 8 Diameter D for shank type cutters: 1 ⁄4 - through 3⁄4 -inch diameter, +0.010, +0.015, 7⁄8 through 11 ⁄8 , +0.012, +0.017; 11 ⁄4 through 11 ⁄2 , +0.015, +0.020 inch. These tolerances include an allowance for sharpening. For arbor-type cutters, diameter D is furnished 1 ⁄32 inch larger than listed, and a tolerance of ±0.002 inch applies to the oversize diameter.

Copyright 2020, Industrial Press, Inc.

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896

Machinery's Handbook, 31st Edition Milling Cutters

Setting Angles for Milling Straight Teeth of Uniform Land Width in End Mills, Angular Cutters, and Taper Reamers.—The accompanying tables give setting angles for the dividing head when straight teeth, having a land of uniform width throughout their length, are to be milled using single-angle fluting cutters. These setting angles depend upon three factors: the number of teeth to be cut; the angle of the blank in which the teeth are to be cut; and the angle of the fluting cutter. Setting angles for various combinations of these three factors are given in the tables. For example, assume that 12 teeth are to be cut on the end of an end mill using a 60-degree cutter. By following the horizontal line from 12 teeth, read in the column under 60 degrees that the dividing head should be set to an angle of 70 degrees and 32 minutes.

A

C

B

The following formulas, which were used to compile these tables, may be used to calcu­ late the setting-angles for combinations of number of teeth, blank angle, and cutter angle not covered by the tables. In these formulas, A = setting-angle for dividing head, B = angle of blank in which teeth are to be cut, C = angle of fluting cutter, N = number of teeth to be cut, and D and E are angles not shown on the accompanying diagram that are used only to simplify calculations. tan D = cos( 360° ⁄ N ) # cotB (1)

sin E = tan ( 360° ⁄ N ) # cotC # sinD Setting-angle A = D − E

(2) (3)

Example: Suppose 9 teeth are to be cut in a 35-degree blank using a 55-degree singleangle fluting cutter. Then, N = 9, B = 35°, and C = 55°.

tanD = cos ( 360° ⁄ 9) # cot35° = 0.76604 # 1.4281 = 1.0940; and D = 47°34′ sin E = tan ( 360° ⁄ 9) # cot55° # sin47°34′ = 0.83910 # 0.70021 # 0.73806 = 0.43365; and E = 25°42 ′

Setting angle A = 47°34 ′ − 25°42 ′ = 21° 52′ For end mills and side mills, the angle of the blank B is 0 degrees, and the following sim­ plified formula may be used to find the setting angle A:

cosA = tan a360° ⁄ N k # cotC

(4)

Example: If in the previous example the blank angle was 0 degrees,

cos A = tan (360°/9) × cot 55° = 0.83910 × 0.70021 = 0.58755

and setting-angle A = 54°1′.

Copyright 2020, Industrial Press, Inc.

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Machinery's Handbook, 31st Edition Milling Cutters

897

Angles of Elevation for Milling Straight Teeth in 0-, 5-, 10-, 15-, 20-, 25-, 30-, and 35-degree Blanks Using Single-Angle Fluting Cutters No. of Teeth

Angle of Fluting Cutter 90°

80°

70°

60°

50°

90°

80°

0° Blank (End Mill) 6

…

8

…

12

…

10

14

16

18

20

22

…

…

…

…

…

…

72°  13 ′ 79 51

82 38

84  9

85  8

85 49

86 19

86 43

87  2

24

…

87 18

6

70°  34 ′

53°  50 ′

68 39

…

…

54°  44 ′

32°  57 ′

77 52

70 32

61  2

50°  55 ′ 74 40

79 54

81 20

82 23

83 13

83 52

84 24

65 12

52 26

73 51

66 10

76 10

69 40

77 52

72 13

79 11

74 11

80 14

75 44

80°  4 ′

82 57

83 50

84 14

84 27

84 35

84 41

84 45

84 47

62°  34 ′ 72 52

76 31

78 25

79 36

80 25

81  1

81 29

81 50

81  6

77  0

84 49

82  7

…

…

61°  49 ′

46°  12 ′

10° Blank 8

10

12

14

16

18

20

22

76  0

77 42

78 30

78 56

79 12

79 22

79 30

79 35

34°  5 ′

55 19

41°  56 ′

20°  39 ′

72 46

66 37

59 26

49 50

74  9 75  

5

75 45

76 16

76 40

24

79 39

76 59

6

53°  57 ′

39°  39 ′

62 44

69  2

70 41

71 53

72 44

73 33

74  9

53 30

40 42

55 19

63  6

59  1

65 37

67 27

61 43

68 52

63 47

69 59

65 25

69 15

71 40

72 48

73 26

73 50

74  5

74 16

74 24

59 46

64 41

67 13

68 46

69 49

70 33

71  6

71 32

70 54

66 44

74 30

71 53

43 53

…

…

31°  53 ′

14°  31 ′

47°  0 ′

34°  6 ′

56  2

49 18

40 40

20° Blank 8

10

12

14

16

18

20

22

62 46

65 47

67 12

68  0

68 30

68 50

69  3

69 14

53 45

59  4

61 49

63 29

64 36

65 24

65 59

66 28

24

69 21

66 49

6

40°  54 ′

29°  22 ′

23°  18 ′ 51 50

58 39

60 26

61 44

62 43

63 30

64  7

43 18

32  1

53  4

46  0

55 39

49 38

57 32

52 17

58 58

54 18

60  7

55 55

56 36

60  2

61 42

62 38

63 13

63 37

63 53

64  5

48  8

53 40

56 33

58 19

59 29

60 19

60 56

61 25

57 12

64 14

61 47

34 24

…

…

24°  12 ′

10°  14 ′

35°  32 ′

25°  19 ′

46 14

40 12

32 32

30° Blank 8

10

12

14

16

18

20

22

24

50 46

54 29

56 18

57 21

58  0

58 26

58 44

58 57

59  8

42 55

48 30

51 26

53 15

54 27

55 18

55 55

56 24

56 48

42  3

48 52

50 39

51 57

52 56

53 42

54 20

50°

61 47

…

…

48°  0 ′

25°  40 ′

72 10

64 52

55  5

41°  41 ′ 68 35

74 24

75 57

77  6

77 59

78 40

79 15

59 11

68 23

70 49

72 36

73 59

75  4

75 57

46  4

60 28

64  7

66 47

68 50

70 26

71 44

49 21

…

…

36°  34 ′

17°  34 ′

61 13

54 14

45 13

28 °  4 ′ 57  8

63 46

65 30

66 46

67 44

68 29

69  6

48 12

57 59

60 33

62 26

63 52

65  0

65 56

36 18

50 38

54 20

57  0

59  3

60 40

61 59

25° Blank

61  2

16°  32 ′

60°

15° Blank

66  9

70 31

70° 5° Blank

38 55

…

…

27°  47 ′

11°  33 ′

51  2

44 38

36 10

19°  33 ′ 46 47

53 41

55 29

56 48

57 47

58 34

59 12

38 43

48 20

50 53

52 46 54 11

55 19

56 13

27 47

41 22

44 57

47 34

49 33

51  9

52 26

35° Blank

34 31

43 49

46 19

48  7

49 30

50 36

51 30

Copyright 2020, Industrial Press, Inc.

24 44

37 27

40 52

43 20

45 15

46 46 48 0

45 17

49  7

51  3

52  9

52 50

53 18

53 38

53 53

54  4

38  5

43 33

46 30

48 19

49 20

50 21

50 59

51 29

51 53

14°  3 ′

30 18

37 35

41 39

44 12

45 56

47 12

48 10

48 56

49 32

…

… 

21°  4 ′

8°  41 ′

30 38

36  2

39 28

21 40

28 55

33 33

41 51

36 45

44 57

40 57

43 36

46  1

46 52

39 

8 

42 24

43 35

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Machinery's Handbook, 31st Edition Milling Cutters

898

Angles of Elevation for Milling Straight Teeth in 40-, 45-, 50-, 55-, 60-, 65-, 70-, and 75-degree Blanks Using Single-Angle Fluting Cutters No. of Teeth

Angle of Fluting Cutter 90°

80°

70°

60°

50°

90°

80°

40° Blank 6

8

10

12

14

16

18

20

22

30°  48 ′ 40  7

43 57

45 54

47  3

47 45

48 14

48 35

48 50

21°  48 ′ 33 36

38 51

41 43

43 29

44 39

45 29

46  7

46 36

24

49  1

46 58

6

22°   45 ′

15°  58 ′

26 33

…

…

18°  16 ′

7°  23 ′

37 14

32  3

11°  58 ′ 33 32

39 41

41 21

42 34

43 30

44 13

44 48

27  3

35 19

37 33

39 13

40 30

41 30

18 55

25 33

29 51

32 50

35  5

36 47

38  8

26°  34 ′ 35 16

38 58

40 54

42  1

42 44

43 13

43 34

43 49

18°  43 ′ 29 25

34 21

37  5

38 46

39 54

40 42

41 18

41 46

42 19

39 15

44  0

42  7

…

… 5°  20 ′

19°  17 ′

13°  30 ′

50° Blank 8

10

12

14

16

18

20

22

30 41

34 10

36  0

37  5

37 47

38 15

38 35

38 50

8°  38 ′

19 59

13°  33 ′

32 34

28 53

24 42

34  9

35 13

35 58

36 32

36 58

24

39  1

37 19

6

16°   6 ′

11°  12 ′

25 39

31  1

32 29

33 33

34 21

34 59

35 30

20 32

27 26

29 22

30 46

31 52

32 44

14  9

19 27

22 58

25 30

27 21

28 47

29 57

26 21

29 32

31 14

32 15

32 54

33 21

33 40

33 54

21 52

25 55

28 12

29 39

30 38

31 20

31 51

32 15

33 25

30 52

34  5

32 34

…

…

9°  37 ′

3°  44 ′

13°  7 ′

9°   8 ′

60° Blank 8

10

12

14

16

18

20

22

24

22 13

25  2

26 34

27 29

28  5

18 24

14 19

23 57

21 10

21 56

25 14

26  7

28 29

26 44

29  0

27 34

28 46

29  9

6°  2 ′

27 11

27 50

18 37

22 51

24  1

24 52

25 30

26  2

26 26

8

10

12

14

16

18

20

22

24

10°  18 ′ 14 26

16 25

17 30

18  9

18 35

18 53

19  6

19 15

19 22

7°  9 ′

11 55

14 21

15 45

16 38

17 15

17 42

18  1

18 16

18 29

3°  48 ′

9  14

12  8

13 53

15  1

15 50

16 26

14 49

17 59

20  6

21 37

22 44

23 35

24 17

24 50

23  8

…

…

15°  48 ′

5°  58 ′

33  0

28 18

10°  11 ′ 29 24

35 17

36 52

38  1

38 53

39 34

40  7

23 40

31 18

33 24

34 56

36  8

37  5

37 50

16 10

22 13

26  9

28 57

30  1

32 37

34 53

35 55

17  3

…

…

11°  30 ′

4°  17 ′

24 59

21 17

16 32

7°  15 ′

22  3

26 53

28 12

29 10

29 54

30 29

30 57

17 36

23 43

25 26

26 43

27 42

28 28

29  7

11 52

19 40

21 54

23 35

24 53

25 55

26 46

10  5

14 13

16 44

18 40

20  6

21 14

22  8

22 52

18 15

20 40

21 59

22 48

23 18

23 40

23 55

24  6

24 15

15  6

18  4

19 48

20 55

21 39 22 11

22 35

22 53

23  8

4°  53 ′ 11 42

15 19

17 28

18 54

19 53

20 37

21 10

21 36

21 57

…

…

7°  50 ′

3°  1 ′

12  9

8  15

16 37

13 48

14 49

17 53

18 50

19 33

20  8

20 36

11 32

15 24

16 37

17 34

18 20

18 57

75° Blank …

…

6°  9 ′

2°  21 ′

9 37

11 45

13 11

14 13

14 59

16 53

15 35

17 33

16 25

17 15

50°

65° Blank

70° Blank 6

60°

55° Blank

25 31

30  2

70° 45° Blank

16 3

Copyright 2020, Industrial Press, Inc.

6  30

9   8

10 55

12 13

13 13

13 59

14 35

15  5

7°  38 ′

10 44

12 14

13  4

13 34

13 54

14  8

14 18

14 25

14 31

5°  19 ′

8  51

10 40

2°  50 ′

6  51

9   1

…

…

4°  34 ′

1°  45 ′

7   8

11 45

10 21

12 54

11 50

10 37

12 38

11 39

12 26

13 14

13 29

13 41

13 50

11 13

12 17

12 53

13  7

8  45

9  50 11 12

12  0

12 18

4  49

6  47

8   7

9   7

9  51

10 27

10 54 11 18

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Machinery's Handbook, 31st Edition Cutter Grinding

899

Angles of Elevation for Milling Straight Teeth in 80- and 85-degree Blanks Using Single-Angle Fluting Cutters No.of Teeth

90°

70°

80°

60°

Angle of Fluting Cutter 50°

90°

80°

70°

80° Blank 6 8 10 12 14 16 18 20 22 24

5°  2 ′ 7  6 8  7 8 41 9  2 9 15 9 24 9 31 9 36 9 40

3°  30 ′ 5 51 7  5 7 48 8 16 8 35 8 48 8 58 9  6 9 13

1°  52 ′ 4 31 5 59 6 52 7 28 7 51 8 10 8 24 8 35 8 43

60°

50°

… 1°  29 ′ 2 21 2 53 3 15 3 30 3 43 3 52 3 59 4  5

… 0°  34 ′ 1 35 2 15 2 42 3  1 3 16 3 28 3 37 3 45

85° Blank … 3°  2 ′ 4 44 5 48 6 32 7  3 7 26 7 44 7 59 8 11

… 1°  8 ′ 3 11 4 29 5 24 6  3 6 33 6 56 7 15 7 30

2°  30 ′ 3 32 4  3 4 20 4 30 4 37 4 42 4 46 4 48 4 50

1°  44 ′ 2 55 3 32 3 53 4  7 4 17 4 24 4 29 4 33 4 36

0°  55 ′ 2 15 2 59 3 25 3 43 3 56 4  5 4 12 4 18 4 22

Spline-Shaft Milling Cutter.—The most efficient method of forming splines on shafts is by hobbing, but special milling cutters may also be used. Since the cutter forms the space between adjacent splines, it must be made to suit the number of splines and the root diam­ eter of the shaft. The cutter angle B equals 360 degrees divided by the number of splines. The following formulas are for determining the chordal width C at the root of the splines or the chordal width across the concave edge of the cutter. In these formulas, A = angle between center line of spline and a radial line passing through the intersection of the root circle and one side of the spline; W = width of spline; d = root diameter of splined shaft; C = chordal width at root circle between adjacent splines; N = number of splines. B

C

W

W sinA = d

(

)

C = d  sin 180 – A N

A d

Splines of involute form are often used in preference to the straight-sided type. Dimen­ sions of the American Standard involute splines and hobs are given in the section on splines. Cutter Grinding

Wheels for Sharpening Milling Cutters.—Milling cutters may be sharpened by using either the periphery of a disk wheel or the face of a cup wheel. The latter grinds the lands of the teeth flat, whereas the periphery of a disk wheel leaves the teeth slightly concave back of the cutting edges. The concavity produced by disk wheels reduces the effective clear­ ance angle on the teeth, the effect being more pronounced for wheels of small diameter than for wheels of large diameter. For this reason, large diameter wheels are preferred when sharpening milling cutters with disk type wheels. Irrespective of what type of wheel is used to sharpen a milling cutter, any burrs resulting from grinding should be carefully removed by a hand stoning operation. Stoning also helps to reduce the roughness

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900

of grind­ing marks and improves the quality of the finish produced on the surface being machined. Unless done very carefully, hand stoning may dull the cutting edge. Stoning may be avoided and a sharper cutting edge produced if the wheel rotates toward the cutting edge, which requires that the operator maintain contact between the tool and the rest while the wheel rotation is trying to move the tool away from the rest. Though slightly more difficult, this method will eliminate the burr. Specifications of Grinding Wheels for Sharpening Milling Cutters

Cutter Material Carbon Tool Steel High-Speed Steel:   18-4-1   {   18-4-2   { Cast Nonferrous Tool Material Sintered Carbide Carbon Tool Steel and High-Speed Steelb

Operation Roughing Finishing

Roughing Finishing Roughing Finishing Roughing Finishing Roughing after Brazing Roughing Finishing Roughing Finishing

Abrasive Material

Grinding Wheel Grain Size

Bond

Vitrified Vitrified

60 100 80 100 46 100-120

K,H H F,G,H H H,K,L,N H

Vitrified Vitrified Vitrified Vitrified Vitrified Vitrified

60

G

Vitrified

Diamond Diamond

100 Up to 500

a

Resinoid Resinoid

Cubic Boron Nitride

80-100 100-120

R,P S,T

Aluminum Oxide

Aluminum Oxide Silicon Carbide

46-60 100

Grade K H

Aluminum Oxide

a

Resinoid Resinoid

a Not indicated in diamond wheel markings.

b For hardnesses above 56 RC on the Rockwell C scale.

Wheel Speeds and Feeds for Sharpening Milling Cutters.—Relatively low cutting speeds should be used when sharpening milling cutters to avoid tempering and heat check­ing. Dry grinding is recommended in all cases except when diamond wheels are employed. The surface speed of grinding wheels should be in the range of 4500–6500 ft/ min (22.8 to 33 m/s) for grinding milling cutters of high-speed steel or cast nonferrous tool material. For sintered carbide cutters, 5000–5500 ft/min (25.4 to 27.9 m/s) should be used. The maximum stock removed per pass of the grinding wheel should not exceed about 0.0004 inch (0.010 mm) for sintered carbide cutters; 0.003 inch (0.076 mm) for large highspeed steel and cast nonferrous tool material cutters; and 0.0015 inch (0.038 mm) for nar­row saws and slotting cutters of high-speed steel or cast nonferrous tool material. The stock removed per pass of the wheel may be increased for backing-off operations such as the grinding of secondary clearance behind the teeth since there is usually a sufficient body of metal to carry off the heat. Clearance Angles for Milling Cutter Teeth.—The clearance angle provided on the cut­ ting edges of milling cutters has an important bearing on cutter performance, cutting effi­ ciency, and cutter life between sharpenings. It is desirable in all cases to use a clearance angle as small as possible so as to leave more metal back of the cutting edges for better heat dissipation and to provide maximum support. Excessive clearance angles not only weaken the cutting edges, but also increase the likelihood of “chatter,” which will result in poor fin­ish on the machined surface and reduce the life of the cutter. According to The Cincinnati Milling Machine Co., milling cutters used for general purpose work and having diameters from 1 ⁄8 to 3 inches (3.18–76.2 mm) should have clearance angles from 13 to 5 degrees, respectively, decreasing proportionately as the diameter increases. General

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Machinery's Handbook, 31st Edition Cutter Grinding

901

purpose cut­ters over 3 inches (76.2 mm) in diameter should be provided with a clearance angle of 4 to 5 degrees. The land width is usually 1 ⁄64 , 1 ⁄32 , and 1 ⁄16 inch (0.4, 0.8, and 1.6 mm), respec­tively, for small, medium, and large cutters. The primary clearance or relief angle for best results varies according to the material being milled, as follows: low-carbon, high-carbon, and alloy steels, 3 to 5 degrees; cast iron and medium and hard bronze, 4 to 7 degrees; brass, soft bronze, aluminum, mag­nesium, plastics, etc., 10 to 12 degrees. When milling cutters are resharpened, it is custom­ary to grind a secondary clearance angle of 3 to 5 degrees behind the primary clearance angle to reduce the land width to its original value and thus avoid interference with the sur­face to be milled. The following is a general formula for plain milling cutters, face mills, and form relieved cutters, which gives the clearance angle C, in degrees, necessitated by the feed per revolu­ tion F, in inches; the width of land L, in inches; the depth of cut d, in inches (mm); the cutter diameter D, in inches; and the Brinell Hardness Number B of the work being cut: F 45860 C = DB a 1.5L + π D d ^D − d h k

Rake Angles for Milling Cutters.—In peripheral milling cutters, the rake angle is gener­ ally defined as the angle in degrees that the tooth face deviates from a radial line to the cut­ting edge. In face milling cutters, the teeth are inclined with respect to both the radial and axial lines. These angles are called radial and axial rake, respectively. The radial and axial rake angles may be positive, zero, or negative. Positive rake angles should be used whenever possible for all types of high-speed steel milling cutters. For sintered carbide-tipped cutters, zero and negative rake angles are fre­ quently employed to provide more material back of the cutting edge to resist shock loads. Rake Angles for High-Speed Steel Cutters: Positive rake angles of 10 to 15 degrees are satis­factory for milling steels of various compositions with plain milling cutters. For softer materials, such as magnesium and aluminum alloys, the rake angle may be 25 degrees or more. Metal slitting saws for cutting alloy steel usually have rake angles from 5 to 10 degrees, whereas zero and sometimes negative rake angles are used for saws to cut copper and other soft nonferrous metals to reduce the tendency to “hog in.” Form relieved cutters usually have rake angles of 0, 5, or 10 degrees. Commercial face milling cutters usually have 10 degrees positive radial and axial rake angles for general use in milling cast iron, forged and alloy steel, brass, and bronze. For milling castings and forgings of magnesium and free-cutting aluminum and their alloys, the rake angles may be increased to 25 degrees positive or more, depending on the operating conditions; a smaller rake angle is used for abrasive or difficult to machine aluminum alloys. Cast Nonferrous Tool Material Milling Cutters: Positive rake angles are generally pro­ vided on milling cutters using cast nonferrous tool materials, although negative rake angles may be used advantageously for some operations such as those where shock loads are encountered or where it is necessary to eliminate vibration when milling thin sections. Sintered Carbide Milling Cutters: Peripheral milling cutters, such as slab mills, slotting cutters, saws, etc., tipped with sintered carbide, generally have negative radial rake angles of 5 degrees for soft low-carbon steel and 10 degrees or more for alloy steels. Positive axial rake angles of 5 and 10 degrees, respectively, may be provided, and for slotting saws and cutters, 0 degree axial rake may be used. On soft materials, such as free-cutting aluminum alloys, positive rake angles of 10 to so degrees are used. For milling abrasive or difficult to machine aluminum alloys, small positive or even negative rake angles are used. Eccentric-Type Radial Relief.—When the radial relief angles on peripheral teeth of mill­ ing cutters are ground with a disc-type grinding wheel in the conventional manner, the ground surfaces on the lands are slightly concave, conforming approximately to the radius of the wheel. A flat land is produced when the radial relief angle is ground with a cup wheel. Another entirely different method of grinding the radial angle is by the eccentric method, which produces a slightly convex surface on the land. If the radial relief angle at

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902

the cutting edge is equal for all of the three types of land mentioned, it will be found that the land with the eccentric relief will drop away from the cutting edge a somewhat greater dis­tance for a given distance around the land than will the others. This is evident from a study of Table 1, Indicator Drops for Checking the Radial Relief Angle on Peripheral Teeth. This feature is an advantage of the eccentric-type relief, which also produces an excellent finish. Table 1. Indicator Drops for Checking the Radial Relief Angle on Peripheral Teeth Cutter Diameter, Inch 1 ⁄16 3 ⁄32 1 ⁄8 5 ⁄32 3 ⁄16 7⁄32 1 ⁄4 9 ⁄32 5 ⁄16 11 ⁄32 3 ⁄8 13 ⁄32 7⁄16 15 ⁄32 1 ⁄2 9 ⁄16 5 ⁄8 11 ⁄16 3 ⁄4 13 ⁄16 7⁄8

Indicator Drops, Inches For Flat and Concave Relief For Eccentric Relief Min. Max. Min. Max.

Rec. Range of Radial Relief Angles, Degrees

Checking Distance, Inch

20-25

.005

.0014

.0019

.0020

.0026

.007

15-19

.010

.0018

.0026

.0028

.0037

.015

12-16

.010

.0016

.0023

.0022

.0030

.015

10-14

.015

.0017

.0028

.0027

.0039

16-20 13-17 11-15

.005 .010

.010

.0012 .0017

.0015

.0015 .0024

.0022

.0015 .0024

.0020

.0019 .0032

.0028

Rec. Max. Primary Land Width, Inch .007 .015

.015

.020

10-14

.015

.0018

.0029

.0027

.0039

.020

10-13

.015

.0020

.0028

.0027

.0035

.020

10-13

10-13

.015

.015

.0019

.0020

.0027

.0029

.0027

.0027

.0035

.0035

.020

.020

9-12

.020

.0022

.0032

.0032

.0044

.025

9-12

.020

.0023

.0034

.0032

.0043

.025

9-12

9-12

.020

.020

.0022

.0024

.0033

.0034

.0032

.0032

.0043

.0043

9-12

.020

.0024

.0035

.0032

.0043

8-11

.030

.0029

.0045

.0043

.0059

8-11

.020

.0022

.0032

.0028

.0039

.025

.025

.025

.025

.035

8-11

.030

.0030

.0046

.0043

.0059

.035

8-11

.030

.0032

.0048

.0043

.0059

.035

8-11

.030

.0031

.0047

.0043

.0059

.035

1 11 ⁄8

7-10 7-10 7-10

.030 .030 .030

.0027 .0028 .0029

.0043 .0044 .0045

.0037 .0037 .0037

.0054 .0054 .0053

.035 .035 .035

13⁄8

6-9

.030

.0025

.0041

.0032

.0048

.035

15⁄8

6-9

.030

.0026

.0042

.0032

.0048

.035

17⁄8 2 21 ⁄4

6-9 6-9 5-8

.030 .030 .030

.0027 .0027 .0022

.0043 .0043 .0038

.0032 .0032 .0026

.0048 .0048 .0042

.035 .035 .040

23⁄4 3 31 ⁄2 4 5 6 7 8 10 12

5-8 5-8 5-8 5-8 4-7 4-7 4-7 4-7 4-7 4-7

.030 .030 .030 .030 .030 .030 .030 .030 .030 .030

.0023 .0023 .0024 .0024 .0019 .0019 .0020 .0020 .0020 .0020

.0039 .0039 .0040 .0040 .0035 .0035 .0036 .0036 .0036 .0036

.0026 .0026 .0026 .0026 .0021 .0021 .0021 .0021 .0021 .0021

.0042 .0042 .0042 .0042 .0037 .0037 .0037 .0037 .0037 .0037

.040 .040 .047 .047 .047 .047 .060 .060 .060 .060

15 ⁄16

11 ⁄4

11 ⁄2 13⁄4

21 ⁄2

6-9 6-9

6-9

5-8

.030 .030 .030

.030

.0024 .0026 .0026

.0023

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.0040 .0041 .0042

.0039

.0032 .0032 .0032

.0026

.0048 .0048 .0048

.0042

.035 .035 .035

.040

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Machinery's Handbook, 31st Edition Cutter Grinding

903

The setup for grinding an eccentric relief is shown in Fig. 1. In this setup, the point of con­tact between the cutter and the tooth rest must be in the same plane as the centers, or axes, of the grinding wheel and the cutter. A wide face is used on the grinding wheel, which is trued and dressed at an angle with respect to the axis of the cutter. An alternate method is to tilt the wheel at this angle. Then as the cutter is traversed and rotated past the grinding wheel while in contact with the tooth rest, an eccentric relief will be generated by the angu­lar face of the wheel. This type of relief can only be ground on the peripheral teeth on mill­ing cutters having helical flutes because the combination of the angular wheel face and the twisting motion of the cutter is required to generate the eccentric relief. Therefore, an eccentric relief cannot be ground on the peripheral teeth of straight-fluted cutters. Table 2 is a table of wheel angles for grinding an eccentric relief for different combina­ tions of relief angles and helix angles. When angles are required that cannot be found in this table, the wheel angle, W, can be calculated by using the following formula, in which R is the radial relief angle and H is the helix angle of the flutes on the cutter.

tanW = tanR # tanH Table 2. Grinding Wheel Angles for Grinding Eccentric-Type Radial Relief Angle Radial Relief Angle, R, Degrees 1

Helix Angle of Cutter Flutes, H, Degrees 12

18

0°13 ′

0°19 ′

0°22 ′

0°59 ′

1°06 ′

0°26 ′

0°39 ′

4

0°51 ′

1°18 ′

0°38 ′

2°53 ′

4°12 ′

5°00 ′

5°57 ′

3°28 ′

5°02 ′

6°00 ′

7°08 ′

1°43 ′

2°37 ′

10

2°09 ′

3°17 ′

11

2°22 ′

3°37 ′

4°03 ′

13

2°49 ′

4°17 ′

4°48 ′

9

1°56 ′

2°35 ′

2°17 ′

2°34 ′

4°03 ′

2°57 ′

3°18 ′

5°13 ′

3°57 ′

14

3°02 ′

4°38 ′

16

3°29 ′

5°19 ′

18

3°57 ′

6°02 ′

20

4°25 ′

6°45 ′

19

21

3°16 ′ 3°43 ′ 4°11 ′

4°59 ′

4°25 ′ 5°11 ′

5°34 ′ 5°57 ′

5°49 ′

6°24 ′

7°00 ′ 7°36 ′ 8°11 ′

8°48 ′ 9°24 ′

6°44 ′

7°34 ′

8°25 ′ 9°16 ′

4°00 ′

4°46 ′

5°07 ′

3°00 ′

7°00 ′

8°00 ′

9°00 ′

11°00 ′

13°03 ′

13°58 ′

13°00 ′

15°23 ′

16°28 ′

15°00 ′

17°43 ′

18°56 ′

18°52 ′

20°09 ′

21°10 ′

22°35 ′

14°00 ′ 16°00 ′

14°13 ′

16°33 ′

20°01 ′

6°23 ′

7°09 ′

11°15 ′

16°07 ′

19°00 ′

22°19 ′

12°30 ′

17°51 ′

7°51 ′

25

5°40 ′

7°33 ′ 7°57 ′

10°37 ′ 11°52 ′

7°29 ′

8°22 ′

13°08 ′

8°14 ′

9°12 ′

14°25 ′

8°37 ′

8°47 ′

9°38 ′

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13°46 ′

15°04 ′

15°15 ′

16°59 ′

18°00 ′ 20°00 ′

23°27 ′

21°00 ′

24°35 ′

19°36 ′

23°00 ′

26°50 ′

21°22 ′

25°00 ′

29°04 ′

18°44 ′

20°29 ′

10°12 ′

11°52 ′

17°00 ′

6°45 ′

7°40 ′

8°56 ′

10°00 ′

11°49 ′

13°32 ′

9°30 ′

6°23 ′

11°28 ′

12°00 ′

12°40 ′

8°19 ′

3°50 ′

10°41 ′

10°07 ′ 10°58 ′

3°34 ′

14°23 ′

5°09 ′ 5°24 ′

3°40 ′

4°38 ′

5°53 ′

2°34 ′

10°01 ′

23

4°55 ′

2°56 ′

2°31 ′

1°17 ′

2°23 ′

6°21 ′

7°07 ′

24

52

2°00 ′

5°40 ′

4°40 ′

22

1°12 ′

2°11 ′

1°44 ′

1°41 ′

1°00 ′

1°49 ′

1°57 ′

1°30 ′

0°50 ′

3°21 ′

1°38 ′

7

0°35 ′

2°19 ′

1°17 ′

17

50

1°27 ′

1°04 ′

15

45

1°09 ′

5

12

40

0°44 ′

6

8

30

Wheel Angle, W, Degrees

2

3

20

22°00 ′

24°00 ′

25°43 ′

27°57 ′

12°43 ′

15°13 ′ 17°42 ′

21°22 ′ 23°47 ′

24°59 ′ 26°10 ′

27°21 ′ 28°31 ′ 29°41 ′

30°50 ′

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Machinery's Handbook, 31st Edition Cutter Grinding

904

Indicator Drop Method of Checking Relief and Rake Angles.—The most convenient and inexpensive method of checking the relief and rake angles on milling cutters is by the indicator drop method. Three tables, Table 1, Table 3 and Table 4, of indicator drops are provided in this section for checking radial relief angles on the peripheral teeth, relief angles on side and end teeth, and rake angles on the tooth faces. A

R

W

Section A–A

H A

Fig. 1. Setup for Grinding Eccentric-Type Radial Relief Angle

Table 3. Indicator Drops for Checking Relief Angles on Side Teeth and End Teeth Checking Distance, Inch .005 .010 .015 .031 .047 .062

Given Relief Angle 1° .00009 .00017 .00026 .00054 .00082 .00108

2° .00017 .00035 .0005 .0011 .0016 .0022

3° .00026 .00052 .00079 .0016 .0025 .0032

4°

5°

6°

7°

8°

9°

.0005 .0011 .0016 .0033 .0049 .0065

.0006 .0012 .0018 .0038 .0058 .0076

.0007 .0014 .0021 .0044 .0066 .0087

.0008 .0016 .0024 .0049 .0074 .0098

Indicator Drop, inch

.00035 .0007 .0010 .0022 .0033 .0043

.0004 .0009 .0013 .0027 .0041 .0054

“Radial” Starting Position

Indicator Drop

Checking Distance Fig. 2. Setup for Checking the Radial Relief Angle by Indicator Drop Method

The setup for checking the radial relief angle is illustrated in Fig. 2. Two dial test indica­ tors are required, one of which should have a sharp-pointed contact point. This indicator is positioned so that the axis of its spindle is vertical, passing through the axis of the cutter. The cutter may be held by its shank in the spindle of a tool and cutter grinder workhead, or

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Machinery's Handbook, 31st Edition Cutter Grinding

905

between centers while mounted on a mandrel. The cutter is rotated to the position where the vertical indicator contacts a cutting edge. The second indicator is positioned with its spindle axis horizontal and with the contact point touching the tool face just below the cut­ ting edge. With both indicators adjusted to read zero, the cutter is rotated a distance equal to the checking distance, as determined by the reading on the second indicator. Then the indicator drop is read on the vertical indicator and checked against the values in the tables. The indicator drops for radial relief angles ground by a disc-type grinding wheel and those ground with a cup wheel are so nearly equal that the values are listed together; values for the eccentric-type relief are listed separately, since they are larger. A similar procedure is used to check the relief angles on the side and end teeth of milling cutters; however, only one indicator is used. Also, instead of rotating the cutter, the indicator or the cutter must be moved a distance equal to the checking distance in a straight line. Table 4. Indicator Drops for Checking Rake Angles on Milling Cutter Face

Indicator Drop

Set indicator to read zero on horizontal plane passing through cutter axis. Zero cutting edge against indicator. Rate Angle, Deg. 1 2 3 4 5 6 7 8 9 10

.031 .0005 .0011 .0016 .0022 .0027 .0033 .0038 .0044 .0049 .0055

Measuring Distance, inch .062

.094

Indicator Drop, inch .0011 .0016 .0022 .0033 .0032 .0049 .0043 .0066 .0054 .0082 .0065 .0099 .0076 .0115 .0087 .0132 .0098 .0149 .0109 .0166

.125 .0022 .0044 .0066 .0087 .0109 .0131 .0153 .0176 .0198 .0220

Measuring Distance

Move cutter or indicator measuring distance. Rate Angle, Deg. 11 12 13 14 15 16 17 18 19 20

.031 .0060 .0066 .0072 .0077 .0083 .0089 .0095 .0101 .0107 .0113

Measuring Distance, inch .062

.094

Indicator Drop, inch .0121 .0183 .0132 .0200 .0143 .0217 .0155 .0234 .0166 .0252 .0178 .0270 .0190 .0287 .0201 .0305 .0213 .0324 .0226 .0342

.125 .0243 .0266 .0289 .0312 .0335 .0358 .0382 .0406 .0430 .0455

Relieving Attachments.—A relieving attachment is a device applied to lathes (especially those used in tool-rooms) for imparting a reciprocating motion to the tool-slide and tool, in order to provide relief or clearance for the cutting edges of milling cutters, taps, hobs, etc. For example, in making a milling cutter of the formed type, such as is used for cutting gears, it is essential to provide clearance for the teeth and so form them that they may be ground repeatedly without changing the contour or shape of the cutting edge. This may be accomplished by using a relieving attachment. The tool for “backing off” or giving clear­ance to the teeth corresponds to the shape required, and it is given a certain amount of recip­rocating movement, so that it forms a surface back of each cutting edge, which is of uniform cross section on a radial plane but eccentric to the axis of the cutter sufficiently to provide the necessary clearance for the cutting edges.

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Machinery's Handbook, 31st Edition Cutter Grinding

906

Various Set-ups Used in Grinding the Clearance Angle on Milling Cutter Teeth Offset

Offset

Offset

Rest

Rest

Rest Wheel Above Center

Offset

Rest

In-Line Centers

Wheel Below Center

Cup Wheel

Distance to Set Center of Wheel Above the Cutter Center (Disk Wheel) Desired Clearance Angle, Degrees

Dia. of Wheel, Inches

1

3

.026

.052

.079

.105

.131

.157

.183

.209

5

.044

.087

.131

.174

.218

.261

.305

.348

4 6 7 8 9

10

2

3

4

5

6

7

8

9

10

11

12

.235

.260

.286

.312

.391

.434

.477

.520

a Distance to Offset Wheel Center Above Cutter Center, Inches

.035 .052 .061 .070 .079 .087

.070 .105 .122 .140 .157 .175

.105 .157 .183 .209 .236 .262

.140 .209 .244 .279 .314 .349

a Calculated from the formula: Offset =

.174 .261 .305 .349 .392 .436

.209 .314 .366 .418 .470 .523

.244 .366 .427 .488 .548 .609

.278 .417 .487 .557 .626 .696

.313 .469 .547 .626 .704 .782

.347 .521 .608 .695 .781 .868

.382 .572 .668 .763 .859 .954

Wheel Diameter × 1 ⁄2 × Sine of Clearance Angle.

.416 .624 .728 .832 .936

1.040

Distance to Set Center of Wheel Below the Cutter Center (Disk Wheel) Desired Clearance Angle, Degrees

Dia. of Cutter, Inches 2

1

2

3

4

5

6

7

8

9

10

11

12

.156

.174

.191

.208

.313

.347

.382

.416

a Distance to Offset Wheel Center Below Cutter Center, Inches

.017

.035

.052

.070

.087

.105

.122

.139

3

.026

.052

.079

.105

.131

.157

.183

.209

.235

5

.044

.087

.131

.174

.218

.261

.305

.348

.391

4 6 7 8 9

10

.035 .052 .061 .070 .079 .087

.070 .105 .122 .140 .157 .175

.105 .157 .183 .209 .236 .262

.140 .209 .244 .279 .314 .349

a Calculated from the formula: Offset =

.174 .261 .305 .349 .392 .436

.209 .314 .366 .418 .470 .523

.244 .366 .427 .488 .548 .609

.278 .417 .487 .557 .626 .696

.469 .547 .626 .704 .782

.260 .434 .521 .608 .695 .781 .868

.286 .477 .572 .668 .763 .859 .954

Cutter Diameter × 1 ⁄2 × Sine of Clearance Angle.

.312 .520 .624 .728 .832 .936

1.040

Distance to Set Tooth Rest Below Center Line of Wheel and Cutter.—When the clearance angle is ground with a disc-type wheel by keeping the center line of the wheel in line with the center line of the cutter, the tooth rest should be lowered by an amount given by the following formula:

Offset =

Wheel Diam. # Cutter Diam. # Sine of One -half the Clearance Angle Wheel Diam. + Cutter Diam.

Distance to Set Tooth Rest Below Cutter Center When Cup Wheel is Used.—When the clearance is ground with a cup wheel, the tooth rest is set below the center of the cutter the same amount as given in the table for Distance to Set Center of Wheel Below the Cutter Center (Disk Wheel).

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Machinery's Handbook, 31st Edition COuNTeR MILLING

907

Counter Milling.—Changing the direction of a linear milling operation by a specific angle requires a linear offset before changing the angle of cut. This compensates for the radius of the milling cutters, as illustrated in Fig. 1a and Fig. 1b. Radius Radius

Cutter Path

r

x

r

x

Cutter Path

Inside

θ Angle

θ

Fig. 1a. Inside Milling

Outside Angle

Fig. 1b. Outside Milling

For inside cuts the offset is subtracted from the point at which the cutting direction changes (Fig. 1a), and for outside cuts the offset is added to the point at which the cutting direction changes (Fig. 1b). The formula for the offset is

x = rM where x = offset distance; r = radius of the milling cutter; and, M = the multiplication factor (M = tan θ∕2  ). The value of M for certain angles can be found in Table 5. Table 5. Offset Multiplication Factors Deg°

M

Deg°

M

Deg°

M

Deg°

M

Deg°

M

1°

0.00873

19°

0.16734

37°

0.33460

55°

0.52057

73°

0.73996

2°

0.01746

20°

0.17633

38°

0.34433

56°

0.53171

74°

0.75355

3°

0.02619

21°

0.18534

39°

0.35412

57°

0.54296

75°

0.76733

4°

0.03492

22°

0.19438

40°

0.36397

58°

0.55431

76°

0.78129

5°

0.04366

23°

0.20345

41°

0.37388

59°

0.56577

77°

0.79544

6°

0.05241

24°

0.21256

42°

0.38386

60°

0.57735

78°

0.80978

7°

0.06116

25°

0.22169

43°

0.39391

61°

0.58905

79°

0.82434

8°

0.06993

26°

0.23087

44°

0.40403

62°

0.60086

80°

0.83910

9°

0.07870

27°

0.24008

45°

0.41421

63°

0.61280

81°

0.85408

10°

0.08749

28°

0.24933

46°

0.42447

64°

0.62487

82°

0.86929

11°

0.09629

29°

0.25862

47°

0.43481

65°

0.63707

83°

0.88473

12°

0.10510

30°

0.26795

48°

0.44523

66°

0.64941

84°

0.90040

13°

0.11394

31°

0.27732

49°

0.45573

67°

0.66189

85°

0.91633

14°

0.12278

32°

0.28675

50°

0.46631

68°

0.67451

86°

0.93252

15°

0.13165

33°

0.29621

51°

0.47698

69°

0.68728

87°

0.94896

16°

0.14054

34°

0.30573

52°

0.48773

70°

0.70021

88°

0.96569

17°

0.14945

35°

0.31530

53°

0.49858

71°

0.71329

89°

0.98270

18°

0.15838

36°

0.32492

54°

0.50953

72°

0.72654

90°

1.00000

Multiply factor M by the tool radius r to determine the offset dimension

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Machinery's Handbook, 31st Edition Reamers

908

REAMERS Hand Reamers.—Hand reamers are made with both straight and helical flutes. Helical flutes provide a shearing cut and are especially useful in reaming holes having keyways or grooves, as these are bridged over by the helical flutes, thus preventing binding or chatter­ ing. Hand reamers are made in both solid and expansion forms. The American standard dimensions for solid forms are given in the accompanying table. The expansion type is use­ful whenever, in connection with repair or other work, it is necessary to enlarge a reamed hole by a few thousandths of an inch. The expansion form is split through the fluted section and a slight amount of expansion is obtained by screwing in a tapering plug. The diameter increase may vary from 0.005 to 0.008 inch (0.127–0.2 mm) for reamers up to about 1 inch (25.4 mm) diameter and from 0.010 to 0.012 inch (0.25–0.3 mm) for diameters between 1 and 2 inches (25.4 and 50.8 mm). Hand reamers are tapered slightly on the end to facilitate starting them properly. The actual diameter of the shanks of commercial reamers may be from 0.002 to 0.005 inch (0.05–0.13 mm) under the reamer size. That part of the shank that is squared should be turned smaller in diameter than the shank itself, so that, when apply­ing a wrench, no burr may be raised that may mar the reamed hole if the reamer is passed clear through it. When fluting reamers, the cutter is so set with relation to the center of the reamer blank that the tooth gets a slight negative rake; that is, the cutter should be set ahead of the center, as shown in the illustration accompanying the table giving the amount to set the cutter ahead of the radial line. The amount is so selected that a tangent to the circumference of the reamer at the cutting point makes an angle of approximately 95 degrees with the front face of the cutting edge. Amount to Set Cutter Ahead of Radial Line to Obtain Negative Front Rake Fluting Cutter a B A Reamer Blank

95

C

Size of Reamer

a, Inches

1 ⁄4

0.011

Size of Reamer 7⁄8

a, Inches

Size of Reamer

a, Inches

0.038

2

0.087

3 ⁄8

0.016

1

0.044

21 ⁄4

0.098

1 ⁄2

0.022

11 ⁄4

0.055

21 ⁄2

0.109

5 ⁄8

0.027

11 ⁄2

0.066

23⁄4

0.120

3 ⁄4

0.033

13⁄4

0.076

3

0.131

When fluting reamers, it is necessary to “break up the flutes,” that is, to space the cutting edges unevenly around the reamer. The difference in spacing should be very slight and need not exceed two degrees one way or the other. The manner in which the breaking up of the flutes is usually done is to move the index head to which the reamer is fixed a certain amount more or less than it would be moved if the spacing were regular. A table is given showing the amount of this additional movement of the index crank for reamers with dif­ ferent numbers of flutes. When a reamer is provided with helical flutes, the angle of spiral should be such that the cutting edges make an angle of about 10 or at most 15 degrees with the axis of the reamer. The relief of the cutting edges should be comparatively slight. An eccentric relief, that is, one where the land back of the cutting edge is convex, rather than flat, is used by one or two manufacturers and is preferable for finishing reamers, as the reamer will hold its size lon­ger. When hand reamers are used merely for removing stock, or simply for enlarging holes, the flat relief is better because the reamer has a keener cutting edge. The width of the land of the cutting edges should be about 1 ⁄32 inch (0.79 mm) for a 1 ⁄4 -inch (6.35 mm), 1 ⁄16 inch (1.59 mm) for a 1-inch (25.4 mm), and 3⁄32 inch (2.38 mm) for a 3-inch (76.2 mm) reamer.

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Machinery's Handbook, 31st Edition Reamers

909

Irregular Spacing of Teeth in Reamers Number of Flutes in Reamer Index Circle to Use

4

6

39

39

8 less 4 more 6 less … … … … … … … … … … … …

4 less 5 more 7 less 6 more 5 less … … … … … … … … … …

Before Cutting 2d flute 3d flute 4th flute 5th flute 6th flute 7th flute 8th flute 9th flute 10th flute 11th flute 12th flute 13th flute 14th flute 15th flute 16th flute

14

16

39 39 39 49 Move Spindle the Number of Holes Below More or Less than for Regular Spacing

8

20

3 less 5 more 2 less 4 more 6 less 2 more 3 less … … … … … … … …

10

2 less 3 more 5 less 2 more 2 less 3 more 2 less 5 more 1 less … … … … … …

12

4 less 4 more 1 less 3 more 4 less 4 more 3 less 2 more 2 less 3 more 4 less … … … …

3 less 2 more 2 less 4 more 1 less 3 more 2 less 1 more 3 less 3 more 2 less 2 more 3 less … …

2 less 2 more 1 less 2 more 2 less 1 more 2 less 2 more 2 less 1 more 2 less 2 more 1 less 2 more 2 less

Threaded-End Hand Reamers.—Hand reamers are sometimes provided with a thread at the extreme point in order to give them a uniform feed when reaming. The diameter on the top of this thread at the point of the reamer is slightly smaller than the reamer itself, and the thread tapers upward until it reaches a dimension of from 0.003 to 0.008 inch (0.076– 0.2 mm), according to size, below the size of the reamer; at this point, the thread stops and a short neck about 1 ⁄16 -inch (1.59 mm) wide separates the threaded portion from the actual reamer, which is provided with a short taper from 3⁄16 to 7⁄16 inch (4.76–11.1 mm) long up to where the standard diameter is reached. The length of the threaded portion and the number of threads per inch for reamers of this kind are given in the accompanying table. The thread employed is a sharp V-thread. Dimensions for Threaded-End Hand Reamers Sizes of Reamers

1 ⁄8 -5 ⁄16

Length of Threaded Part

No. of Threads per Inch

3 ⁄8

32

11 ⁄32 -1 ⁄2

7⁄16

25 ⁄32 -1

9 ⁄16

17⁄32 -3 ⁄4

1 ⁄2

28 24 18

Dia. of Thread at Point of Reamer Full diameter -0.006 -0.006 -0.008 -0.008

Sizes of Reamers

11 ⁄32 -11 ⁄2

Length of Threaded Part

No. of Threads per Inch

9 ⁄16

18

117⁄32 -2

9 ⁄16

217⁄32 -3

9 ⁄16

21 ⁄32 -21 ⁄2

9 ⁄16

18 18 18

Dia. of Thread at Point of Reamer Full diameter -0.010 -0.012 -0.015 -0.020

Fluted Chucking Reamers.—Reamers of this type are used in turret lathes, screw machines, etc., for enlarging holes and finishing them smooth and to the required size. The best results are obtained with a floating type of holder that permits a reamer to align itself with the hole being reamed. These reamers are intended for removing a small amount of metal, 0.005 to 0.010 inch (0.127–0.25 mm) being common allowances. Fluted chucking reamers are provided either with a straight shank or a standard taper shank. (See table on page 910 for standard dimensions.)

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Machinery's Handbook, 31st Edition Reamers

910

Fluting Cutters for Reamers 55 30

D

85

85

A

A

C

B

C

B 15

70

D

Reamer Dia. 1 ⁄8

3 ⁄16 1 ⁄4

3 ⁄8

1 ⁄2 5 ⁄8

1

3 ⁄4

Fluting Cutter Dia. A 13⁄4

13⁄4 13⁄4 2   2   2   2   21 ⁄4

Radius between Cutting Faces D

Fluting Cutter Thickness B

Hole Dia. in Cutter C

3 ⁄16

3 ⁄4

nonea

3 ⁄16

3 ⁄4

1 ⁄64

3 ⁄16 1 ⁄4

3 ⁄4 3 ⁄4

5 ⁄16

3 ⁄4

7⁄16

3 ⁄4

3 ⁄8

1 ⁄2

a Sharp corner, no radius

3 ⁄4

1

nonea 1 ⁄64 1 ⁄32 1 ⁄32 3 ⁄64

3 ⁄64

Reamer Dia. 11 ⁄4 11 ⁄2 13⁄4 2  21 ⁄4 21 ⁄2 23⁄4 3 

Fluting Cutter Dia. A 21 ⁄4 21 ⁄4 21 ⁄4 21 ⁄2 21 ⁄2 21 ⁄2 21 ⁄2 21 ⁄2

Fluting Cutter Thickness B 9 ⁄16 5 ⁄8 5 ⁄8 3 ⁄4 3 ⁄4

7⁄8 7⁄8

 1

Hole Dia. in Cutter C 1 1 1 1 1 1 1 1

Radius between Cutting Faces D 1 ⁄16 1 ⁄16

5 ⁄64 5 ⁄64 5 ⁄64 3 ⁄16 3 ⁄16 3 ⁄16

Rose Chucking Reamers.—The rose type of reamer is used for enlarging cored or other holes. The cutting edges at the end are ground to a 45-degree bevel. This type of reamer will remove considerable metal in one cut. The cylindrical part of the reamer has no cutting edges but merely grooves cut for the full length of the reamer body, providing a way for the chips to escape and a channel for lubricant to reach the cutting edges. There is no relief on the cylindrical surface of the body part, but it is slightly back-tapered so that the diame­ter at the point with the beveled cutting edges is slightly larger than the diameter farther back. The back-taper should not exceed 0.001 inch per inch (or mm/mm). This form of reamer usually produces holes slightly larger than its size; it is, therefore, always made from 0.005 to 0.010 inch (0.127-0.25 mm) smaller than its nominal size, so that it may be followed by a fluted reamer for finishing. The grooves on the cylindrical portion are cut by a convex cutter having a width equal to from one-fifth to one-fourth the diameter of the rose reamer itself. The depth of the groove should be from one-eighth to one-sixth the diameter of the reamer. The teeth at the end of the reamer are milled with a 75-degree angu­lar cutter; the width of the land of the cutting edge should be about one-fifth the distance from tooth to tooth. If an angular cutter is preferred to a convex cutter for milling the grooves on the cylindrical portion because of the higher cutting speed possible when mill­ing, an 80-degree angular cutter slightly rounded at the point may be used. Cutters for Fluting Rose Chucking Reamers.—The cutters used for fluting rose chuck­ ing reamers on the end are 80-degree angular cutters for 1 ⁄4 - and 5⁄16 -inch diameter reamers; 75-degree angular cutters for 3⁄8 - and 7⁄16 -inch reamers; and 70-degree angular cutters for all larger sizes. The grooves on the cylindrical portion are milled with convex cutters of approximately the following sizes for given diameters of reamers: 5⁄32 -inch convex cutter for 1 ⁄2 -inch reamers; 5⁄16 -inch cutter for 1-inch reamers; 3⁄8 -inch cutter for 11 ⁄2 -inch reamers; 13⁄32 -inch cutters for 2-inch reamers; and 15⁄32 -inch cutters for 21 ⁄2 -inch reamers.

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Machinery's Handbook, 31st Edition Reamers

911

Dimensions of Formed Reamer Fluting Cutters

A B Dia. = D

C

C

The making and maintenance of cutters of the formed type involve greater expense than the use of angular cutters, dimensions of which are given on the previous page; but the form of flute produced by the formed type of cutter is preferred by many reamer users. The claims made for the formed type of flute are that the chips can be more read­ily removed from the reamer and that the reamer has greater strength and is less likely to crack or spring out of shape in hardening.

G E

H

F 6

Reamer Size 1 ⁄8 -3 ⁄16 1 ⁄4 -5 ⁄16 3 ⁄8 -7⁄16

1 ⁄2 -11 ⁄16 3 ⁄4 -1 

No. of Teeth in Reamer 6 6 6

6-8 8

11 ⁄16 -11 ⁄2

10

21 ⁄4 -3 

14

19⁄16 -21 ⁄8

12

Cutter Dia. D

Cutter Width A

Hole Dia. B

1 ⁄4

7⁄8

…

3 ⁄8

7⁄8

1 ⁄8

7⁄16

7⁄8

1 ⁄8

21 ⁄4

1 ⁄2

7⁄8

5 ⁄32

9 ⁄16

7⁄8

5 ⁄32

25⁄8

5 ⁄8

7⁄8

3 ⁄16

11 ⁄16

7⁄8

3 ⁄16

13⁄4 13⁄4

17⁄8 2

21 ⁄8

23⁄8

3 ⁄16

7⁄8

Bearing Width C …

Bevel Length E 0.125 0.152

0.178 0.205 0.232 0.258 0.285 0.312

Radius F 0.016 0.022

0.029 0.036 0.042 0.049 0.056 0.062

Radius F 7⁄32

9 ⁄32 1 ⁄2 9 ⁄16 11 ⁄16 3 ⁄4 27⁄32 7⁄8

Tooth Depth H 0.21

No. of Cutter Teeth 14

0.28

12

0.25 0.30 0.32 0.38 0.40 0.44

13 12 12 11 11

10

The smaller sizes of reamers, from 1 ⁄4 to 3⁄8 inch in diameter, are often milled, with regular double-angle reamer fluting cutters having a radius of 1 ⁄64 inch for 1 ⁄4 -inch reamer, and 1 ⁄32 inch for 5⁄16 - and 3⁄8 -inch sizes.

Reamer Terms and Definitions.—Reamer: A rotary cutting tool with one or more cut­ ting elements used for enlarging to size and contour a previously formed hole. Its principal support during the cutting action is obtained from the workpiece. (See Fig. 1.) Actual Size: The actual measured diameter of a reamer, usually slightly larger than the nominal size to allow for wear. Angle of Taper: The included angle of taper on a taper tool or taper shank. Arbor Hole: The central mounting hole in a shell reamer.

Axis: The imaginary straight line that forms the longitudinal centerline of a reamer, usually established by rotating the reamer between centers.

Back Taper: A slight decrease in diameter, from front to back, in the flute length of ream­ers. Bevel: An unrelieved angular surface of revolution (not to be confused with chamfer).

Body: The fluted full-diameter portion of a reamer, inclusive of the chamfer, starting taper, and bevel.

Chamfer: The angular cutting portion at the entering end of a reamer (see also Secondary Chamfer).

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Machinery's Handbook, 31st Edition Reamers

912

Vertical Adjustment of Tooth-Rest for Grinding Clearance on Reamers Hand Reamer for Steel. Cutting Clearance Land 0.006 inch Wide

Size of Reamer 1 ⁄2 5 ⁄8 3 ⁄4

7⁄8

1 11 ⁄8 11 ⁄4 13⁄8 11 ⁄2 15⁄8 13⁄4 17⁄8 2 21 ⁄8 21 ⁄4 23⁄8 21 ⁄2 25⁄8 23⁄4 27⁄8 3 31 ⁄8 31 ⁄4 33⁄8 31 ⁄2 35⁄8 33⁄4 37⁄8 4 41 ⁄8 41 ⁄4 43⁄8 41 ⁄2 45⁄8 43⁄4 47⁄8 5

Hand Reamer for Cast Iron and Bronze. Cutting Clearance Land 0.025 inch Wide

Chucking Reamer for Cast Iron and Bronze. Cutting Clearance Land 0.025 inch Wide

For Cutting Clearance

For Second Clearance

For Cutting Clearance

For Second Clearance

For Cutting Clearance

For Second Clearance

0.012 0.012 0.012 0.012 0.012 0.012 0.012 0.012 0.012 0.012 0.012 0.012 0.012 0.012 0.012 0.012 0.012 0.012 0.012 0.012 0.012 0.012 0.012 0.012 0.012 0.012 0.012 0.012 0.012 0.012 0.012 0.012 0.012 0.012 0.012 0.012 0.012

0.052 0.062 0.072 0.082 0.092 0.102 0.112 0.122 0.132 0.142 0.152 0.162 0.172 0.172 0.172 0.172 0.172 0.172 0.172 0.172 0.172 0.172 0.172 0.172 0.172 0.172 0.172 0.172 0.172 0.172 0.172 0.172 0.172 0.172 0.172 0.172 0.172

0.032 0.032 0.035 0.040 0.040 0.040 0.045 0.045 0.048 0.050 0.052 0.056 0.056 0.059 0.063 0.063 0.065 0.065 0.065 0.070 0.072 0.075 0.078 0.081 0.084 0.087 0.090 0.093 0.096 0.096 0.096 0.096 0.100 0.100 0.104 0.106 0.110

0.072 0.072 0.095 0.120 0.120 0.120 0.145 0.145 0.168 0.170 0.192 0.196 0.216 0.219 0.223 0.223 0.225 0.225 0.225 0.230 0.232 0.235 0.238 0.241 0.244 0.247 0.250 0.253 0.256 0.256 0.256 0.256 0.260 0.260 0.264 0.266 0.270

0.040 0.040 0.040 0.045 0.045 0.045 0.050 0.050 0.055 0.060 0.060 0.060 0.064 0.064 0.064 0.068 0.072 0.075 0.077 0.080 0.080 0.083 0.083 0.087 0.090 0.093 0.097 0.100 0.104 0.104 0.106 0.108 0.108 0.110 0.114 0.116 0.118

0.080 0.090 0.100 0.125 0.125 0.125 0.160 0.160 0.175 0.200 0.200 0.200 0.224 0.224 0.224 0.228 0.232 0.235 0.237 0.240 0.240 0.240 0.243 0.247 0.250 0.253 0.257 0.260 0.264 0.264 0.266 0.268 0.268 0.270 0.274 0.276 0.278

Rose Chucking Reamers for Steel For Cutting Clearance on Angular Edge at End 0.080 0.090 0.100 0.125 0.125 0.125 0.160 0.175 0.175 0.200 0.200 0.200 0.225 0.225 0.225 0.230 0.230 0.235 0.240 0.240 0.240 0.240 0.245 0.245 0.250 0.250 0.255 0.255 0.260 0.260 0.265 0.265 0.265 0.270 0.275 0.275 0.275

Chamfer Angle: The angle between the axis and the cutting edge of the chamfer mea­ sured in an axial plane at the cutting edge. Chamfer Length: The length of the chamfer measured parallel to the axis at the cutting edge. Chamfer Relief Angle: See under Relief. Chamfer Relief: See under Relief. Chipbreakers: Notches or grooves in the cutting edges of some taper reamers designed to break the continuity of the chips. Circular Land: See preferred term Margin. Clearance: The space created by the relief behind the cutting edge or margin of a reamer.

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Machinery's Handbook, 31st Edition Reamers

913

Illustration of Terms Applying to Reamers Land Width

Relieved Land Relief Angle

Margin Cutting Edge

Starting Taper

Heel Cutting Face

Flute

Actual Size Land

Helical Flutes Left-Hand Helix Shown

Actual Size

Core Diameter

Bevel

Chamfer Length Chamfer Relief Margin

Chamfer Angle

Chamfer Relief Angle

Straight Flutes Shown

Radial Zero Degrees Rake Angle and Right-Hand Rotation Shown

Flute

Land

Land Width Radial Rake Angle

Core Diameter Positive Radial Rake Angle and Right-Hand Rotation Shown

Machine Reamer

Hand Reamer Neck Shank Length Guide

Starting Taper Neck

Overall Length Flute Length

Pilot Axis

Cutter Sweep

Squared Shank

Straight Flutes

Actual Size

Cutter Sweep

Hand Reamer, Pilot and Guide Shank Length Tang

Overall Length

Taper Shank

Helix Angle

Flute Length Chamfer Angle

Chamfer Length

Helical Flutes R.H. Helix Shown

Body

Straight Shank Shank Length

Actual Size

Chucking Reamer, Straight and Taper Shank

Core: The central portion of a reamer that joins the lands below the flutes. Core Diameter: The diameter at a given point along the axis of the largest circle which does not project into the flutes. Cutter Sweep: The section removed by the milling cutter or grinding wheel in entering or leaving a flute. Cutting Edge: The leading edge of the relieved land in the direction of rotation for cut­ting. Cutting Face: The leading side of the relieved land in the direction of rotation for cutting on which the chip impinges. External Center: The pointed end of a reamer. The included angle varies with manufac­ turing practice. Flutes: Longitudinal channels formed in the body of the reamer to provide cutting edges, permit passage of chips, and allow cutting fluid to reach the cutting edges. Angular Flute: A flute that forms a cutting face lying in a plane intersecting the reamer axis at an angle. It is unlike a helical flute in that it forms a cutting face that lies in a single plane. Helical Flute: Sometimes called a spiral flute, a flute formed in a helical path around the axis of a reamer. Spiral flute: 1) On a taper reamer, a flute of constant lead; or 2) in reference to a straight reamer, see preferred term Helical Flute. Straight Flute: A flute that forms a cutting edge lying in an axial plane. Flute Length: The length of the flutes, not including the cutter sweep. Guide: A cylindrical portion following the flutes of a reamer to maintain alignment. Heel: The trailing edge of the land in the direction of rotation for cutting.

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914

Machinery's Handbook, 31st Edition Reamers

Helix Angle: The angle which a helical cutting edge at a given point makes with an axial plane through the same point. Hook: A concave condition of a cutting face. The rake of a hooked cutting face must be determined at a given point. Internal Center: A 60-degree countersink with clearance at the bottom, in one or both ends of a tool, which establishes the tool axis. Irregular Spacing: A deliberate variation from uniform spacing of the reamer cutting edges. Land: The section of the reamer between adjacent flutes. Land Width: The distance between the leading edge of the land and the heel, measured at a right angle to the leading edge. Lead of Flute: The axial advance of a helical or spiral cutting edge in one turn around the reamer axis. Length: The dimension of any reamer element measured parallel to the reamer axis. Limits: The maximum and minimum values designated for a specific element. Margin: The unrelieved part of the periphery of the land adjacent to the cutting edge. Margin Width: The distance between the cutting edge and the primary relief measured at a right angle to the cutting edge. Neck: The section of reduced diameter connecting shank to body, or connecting other portions of the reamer. Nominal Size: The designated basic size of a reamer overall length—the extreme length of the complete reamer from end to end, but not including external centers or expansion screws. Periphery: The outside circumference of a reamer. Pilot: A cylindrical portion preceding the entering end of the reamer body to maintain alignment. Rake: The angular relationship between the cutting face, or a tangent to the cutting face at a given point and a given reference plane or line. Axial Rake: Applies to angular (not helical or spiral) cutting faces. It is the angle between a plane containing the cutting face, or tangent to the cutting face at a given point, and the reamer axis. Helical Rake: Applies only to helical and spiral cutting faces (not angular). It is the angle between a plane, tangent to the cutting face at a given point on the cutting edge, and the reamer axis. Negative Rake: Describes a cutting face in rotation whose cutting edge lags the surface of the cutting face. Positive Rake: Describes a cutting face in rotation whose cutting edge leads the surface of the cutting face. Radial Rake Angle: The angle in a transverse plane between a straight cutting face and a radial line passing through the cutting edge. Relief: The result of the removal of tool material behind or adjacent to the cutting edge to provide clearance and prevent rubbing (heel drag). Axial Relief: The relief measured in the axial direction between a plane perpendicular to the axis and the relieved surface. It can be measured by the amount of indicator drop at a given radius in a given amount of angular rotation. Cam Relief : The relief from the cutting edge to the heel of the land produced by a cam action. Chamfer Relief Angle: The axial relief angle at the outer corner of the chamfer. It is measured by projection into a plane tangent to the periphery at the outer corner of the chamfer. Chamfer Relief: The axial relief on the chamfer of the reamer. Eccentric Relief: A convex relieved surface behind the cutting edge. Flat Relief: A relieved surface behind the cutting edge which is essentially flat.

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Machinery's Handbook, 31st Edition Reamers

915

Radial Relief: Relief in a radial direction measured in the plane of rotation. It can be measured by the amount of indicator drop at a given radius in a given amount of angu­lar rotation. Primary Relief: The relief immediately behind the cutting edge or margin. Properly called relief. Secondary Relief: An additional relief behind the primary relief. Relief Angle: The angle, measured in a transverse plane, between the relieved surface and a plane tangent to the periphery at the cutting edge. Secondary Chamfer: A slight relieved chamfer adjacent to and following the initial chamfer on a reamer. Shank: The portion of the reamer by which it is held and driven. Squared Shank: A cylindrical shank having a driving square on the back end. Starting Radius: A relieved radius at the entering end of a reamer in place of a chamfer. Starting Taper: A slight relieved taper on the front end of a reamer. Straight Shank: A cylindrical shank. Tang: The flatted end of a taper shank which fits a slot in the socket. Taper per Foot: The difference in diameter between two points 12 in. apart measured along the axis. Taper Shank: A shank made to fit a specific (conical) taper socket. Direction of Rotation and Helix.—The terms “right hand” and “left hand” are used to describe both direction of rotation and direction of flute helix or reamers. Hand of Rotation (or Hand of Cut): Right-hand Rotation (or Right-hand Cut): When viewed from the cutting end, the reamer must revolve counterclockwise to cut Left-hand Rotation (or Left-hand Cut): When viewed from the cutting end, the reamer must revolve clockwise to cut Hand of Flute Helix: Right-hand Helix: When the flutes twist away from the observer in a clockwise direction when viewed from either end of the reamer. Left-hand Helix: When the flutes twist away from the observer in a counterclockwise direction when viewed from either end of the reamer. The standard reamers on the tables that follow are all right-hand rotation. Dimensions of Centers for Reamers and Arbors

A

B 60

C

D

1 ⁄4

Large Center Dia. B 1 ⁄8

5 ⁄16

5 ⁄32

Arbor. Dia. A

3 ⁄8

3 ⁄16

7⁄16

7⁄32

1 ⁄2

1 ⁄4

9 ⁄16

9 ⁄32

5 ⁄8

5 ⁄16

11 ⁄16

11 ⁄32

Arbor Dia. A 3 ⁄4 13 ⁄16

13 ⁄32

7⁄8

7⁄16

15 ⁄16

15 ⁄32

1

11 ⁄8 Drill No. C

Hole Depth D

55

5 ⁄32

48

7⁄32

52 43 39

3 ⁄16 1 ⁄4 5 ⁄16

33

11 ⁄32

29

13 ⁄32

30

3 ⁄8

Large Center Dia. B 3 ⁄8

1 ⁄2 33 ⁄64

11 ⁄4

17⁄32

11 ⁄2 …

9 ⁄16

13⁄4

19 ⁄32

13⁄8

15⁄8 17⁄8 2

21 ⁄8 21 ⁄4

23⁄8

35 ⁄64

…

37⁄64 39 ⁄64 5 ⁄8 41 ⁄64 21 ⁄32 43 ⁄64

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Drill No. C 25

Hole Depth D 7⁄16

17

17⁄32

20 12 8 5 3 2 1

Letter A B

C E F

G H

1 ⁄2 9 ⁄16 19 ⁄32 5 ⁄8 21 ⁄32 21 ⁄32 11 ⁄16

…

23 ⁄32 23 ⁄32 3 ⁄4 3 ⁄4 25 ⁄32 13 ⁄16 27⁄32

Arbor Dia. A 21 ⁄2 25⁄8 23⁄4

27⁄8 3

Large Center Dia. B 11 ⁄16 45 ⁄64 23 ⁄32 47⁄64 3 ⁄4

31 ⁄8

49 ⁄64

33⁄8

51 ⁄64

35⁄8

53 ⁄64

37⁄8 4

55 ⁄64

41 ⁄2

15 ⁄16

31 ⁄4

31 ⁄2 33⁄4

41 ⁄4 43⁄4 5

25 ⁄32 13 ⁄16 27⁄32 7⁄8 29 ⁄32 31 ⁄32

1

Drill No. C J K L

M N N

O O

Hole Depth D 27⁄32 7⁄8 29 ⁄32 29 ⁄32 15 ⁄16 31 ⁄32 31 ⁄32

1

P

1

R

11 ⁄16

S

11 ⁄8

Q R

11 ⁄16

11 ⁄16

T

11 ⁄8

W

11 ⁄4

V

X

13⁄16 11 ⁄4

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Machinery's Handbook, 31st Edition Reamers

916

Straight-Shank Center Reamers and Machine Countersinks ANSI/ASME B94.2-1995 (R2015) D S

S

A

Center Reamers (Short Countersinks) Dia. of Cut 1 ⁄4 3 ⁄8 1 ⁄2 5 ⁄8 3 ⁄4

Approx. Length Overall, A 11 ⁄2 13⁄4 2

21 ⁄4

25⁄8

Length of Shank, S

Dia. of Shank, D

Dia. of Cut

3 ⁄4

3 ⁄16

1 ⁄2

1 ⁄4

5 ⁄8

3 ⁄8

3 ⁄4

3 ⁄8

7⁄8

1

11 ⁄4

A Machine Countersinks

7⁄8

1

D

37⁄8 4

41 ⁄8 41 ⁄4

1

1 ⁄2

Approx. Length Overall, A

43⁄8

Length of Shank, S 21 ⁄4

21 ⁄4 21 ⁄4 21 ⁄4 21 ⁄4

Dia. of Shank, D 1 ⁄2 1 ⁄2 1 ⁄2 1 ⁄2 1 ⁄2

All dimensions are given in inches. Material is high-speed steel. Reamers and countersinks have 3 or 4 flutes. Center reamers are standard with 60, 82, 90, or 100 degrees included angle. Machine countersinks are standard with either 60 or 82 degrees included angle. Tolerances: On overall length A, the tolerance is ±1 ⁄8 inch for center reamers in a size range of from 1 ⁄4 to 3⁄8 inch, incl., and machine countersinks in a size range of from 1 ⁄2 to 5⁄8 inch. incl.; 3 ± ⁄16 inch for center reamers, 1 ⁄2 to 3⁄4 inch, incl.; and machine countersinks, 3⁄4 to 1 inch, incl. On shank diameter D, the tolerance is -0.0005 to -0.002 inch. On shank length S, the tolerance is ±1 ⁄16 inch.

Calculating Countersink and Spot Drill Depths.—The following formulas are for countersinks and spot drills that have been sharpened to a point, allowances on tool depth will have to be made if the countersink or spot drill is not sharpened to a point.

D

Z

Fig. 1. For Countersinks and Spot Drills that have been Sharpened to a Point

α = countersink angle D = countersink diameter Z = depth of tool travel required from surface of part To find the tool travel “Z” required for any sharp-point countersink or spot drill angle, use either of the following formulas:

Z=

D 1 ∙ 2 tan α 2

or

Z=

D α ∙ cot 2 2

To find the tool travel “Z-a” required for any truncated countersink or spot drill angle, use either of the following formulas:

Z–a=

D–T D–T 1 ∙ cot α ∙ or Z – a = 2 2 2 tan α     2

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Machinery's Handbook, 31st Edition Reamers

917

D d T Z a Fig. 2. For Countersinks and Spot Drills that are a Truncated Cone

a = allowance for truncated countersink tool d = diameter of hole T = diameter of small end of truncated tool

The table below is a quick reference for some predefined angles on countersinks and spot drills that have been sharpened to a point. To calculate the tool travel “Z”, multiply the factor by the countersink diameter. Countersink Angle α, degrees 60 82 90 100

Factor (Imperial or Metric Units) 0.8660 0.5752 0.5000 0.4195

Countersink Angle α, degrees 118 120 135 142

Factor (Imperial or Metric Units) 0.3004 0.2887 0.2071 0.1722

Reamer Difficulties.—Certain frequently occurring problems in reaming require reme­ dial measures. These difficulties include the production of oversize holes, bellmouth holes, and holes with a poor finish. The following is taken from suggestions for correction of these difficulties by the National Twist Drill and Tool Co. and Winter Brothers Co.* Oversize Holes: The cutting of a hole oversize from the start of the reaming operations usually indicates a mechanical defect in the setup or reamer. Thus, the wrong reamer for the workpiece material may have been used or there may be inadequate workpiece support; inadequate or worn guide bushings; misalignment of the spindles, bushings, or work­piece; or runout of the spindle or reamer holder. The reamer itself may be defective due to chamfer runout or runout of the cutting end due to a bent or nonconcentric shank. When reamers gradually start to cut oversize, it is due to pickup or galling, principally on the reamer margins. This condition is partly due to the workpiece material. Mild steels, cer­tain cast irons, and some aluminum alloys are particularly troublesome in this respect. Corrective measures include reducing the reamer margin widths to about 0.005 to 0.010 inch (0.127-0.25 mm), use of hard case surface treatments on high-speed steel reamers, either alone or in combination with black oxide treatments, and the use of a highgrade fin­ish on the reamer faces, margins, and chamfer relief surfaces. Bellmouth Holes: The cutting of a hole that becomes oversize at the entry end with the oversize decreasing gradually along its length always reflects misalignment of the cutting portion of the reamer with respect to the hole. The obvious solution is to provide improved guiding of the reamer by the use of accurate bushings and pilot surfaces. If this solution is not feasible and the reamer is cutting in a vertical position, a flexible element may be employed to hold the reamer in such a way that it has both radial and axial float, with the hope that the reamer will follow the original hole and prevent the bellmouth condition. In horizontal setups where the reamer is held fixed and the workpiece rotated, any mis­ alignment exerts a sideways force on the reamer as it is fed to depth, resulting in the forma­ tion of a tapered hole. This type of bellmouthing can frequently be reduced by shortening * “Some Aspects of Reamer Design and Operation,” Metal Cuttings, April 1963.

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Machinery's Handbook, 31st Edition Reamers

918

the bearing length of the cutting portion of the reamer. One way to do this is to reduce the reamer diameter by 0.010 to 0.030 inch (0.25–0.76 mm), depending on size and length, behind a short full-diameter section, 1 ⁄8 to 1 ⁄2 inch (3.18 to 12.7 mm) long according to length and size, following the chamfer. The second method is to grind a high back taper, 0.008 to 0.015 inch per inch (or mm/mm), behind the short full-diameter section. These modifica­tions reduce the length of the reamer tooth that can cause the bellmouth condition. Poor Finish: The most obvious step toward producing a good finish is to reduce the reamer feed per revolution. Feeds as low as 0.0002 to 0.0005 inch (0.005 to 0.013 mm) per tooth have been used successfully. However, better reamer life results if the maximum fea­sible feed is used. The minimum practical amount of reaming stock allowance will often improve finish by reducing the volume of chips and the resulting heat generated on the cutting portion of the chamfer. Too little reamer stock, however, can be troublesome in that the reamer teeth may not cut freely but will deflect or push the work material out of the way. When this happens, excessive heat, poor finish, and rapid reamer wear can occur. Because of superior abrasion resistance, carbide reamers are often used when fine fin­ ishes are required. When properly conditioned, carbide reamers can produce a large num­ ber of good-quality holes. Careful honing of the carbide reamer edges is very important. Types and Sizes of Reamers American National Standard Fluted Taper Shank Chucking Reamers— Straight and Helical Flutes, Fractional Sizes ANSI/ASME B94.2-1995 (R2015)

B

A

Reamer Dia. 1 ⁄4 5 ⁄16 3 ⁄8 7⁄16 1 ⁄2 17⁄32 9 ⁄16 19 ⁄32 5 ⁄8 21 ⁄32 11 ⁄16 23 ⁄32 3 ⁄4 25 ⁄32 13 ⁄16

Length Overall A

Flute Length B

No. of Morse Taper Shanka

No. of Flutes

6

11 ⁄2

1

4 to 6

6 7 7 8 8 8 8 9

11 ⁄2 13⁄4 13⁄4 2 2 2 2

1 1 1

1 1 1 1

21 ⁄4

2

21 ⁄4

2

91 ⁄2

21 ⁄2

2

91 ⁄2

21 ⁄2

2

9 9 9 91 ⁄2

21 ⁄4

2

21 ⁄4

2

21 ⁄2

2

4 to 6 4 to 6 6 to 8

6 to 8 6 to 8 6 to 8 6 to 8

6 to 8 6 to 8 6 to 8 6 to 8 6 to 8

8 to 10 8 to 10

Reamer Dia. 27⁄32 7⁄8 29 ⁄32 15 ⁄16 31 ⁄32

1

B

Length Overall A

No. of Morse Taper Shanka 2

91 ⁄2

21 ⁄2

10

25⁄8

2

25⁄8

3

23⁄4

3

27⁄8

3

3

4

10 10 10

101 ⁄2

11 ⁄16

101 ⁄2

13⁄16

11

15⁄16

111 ⁄2

11 ⁄8

Flute Length B

11

25⁄8

2

25⁄8

3

23⁄4

3

27⁄8

3

3

4

No. of Flutes

8 to 10 8 to 10 8 to 10 8 to 10 8 to 10 8 to 12 8 to 12 8 to 12 8 to 12 8 to 12

1 1 ⁄4

11 1 ⁄2

13⁄8

12

31 ⁄4

4

10 to 12

121 ⁄2

31 ⁄2

4

10 to 12

17⁄16

12

…

…

11 ⁄2

4

31 ⁄4 …

…

8 to 12

10 to 12 …

a American National Standard self-holding tapers (see Table 7a on page 1020).

All dimensions are given in inches. Material is high-speed steel. Helical flute reamers with right-hand helical flutes are standard. Tolerances: On reamer diameter, 1 ⁄4 -inch size, +.0001 to +.0004 inch; over 1 ⁄4 - to 1-inch size, + .0001 to +.0005 inch; over 1-inch size, +.0002 to +.0006 inch. On length overall A and flute length B, 1 ⁄4 - to 1-inch size, incl., ±1 ⁄16 inch; 11 ⁄16 - to 11 ⁄2 -inch size, incl., 3⁄32 inch.

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Machinery's Handbook, 31st Edition Reamers

919

Expansion Chucking Reamers—Straight and Taper Shanks ANSI/ASME B94.2-1995 (R2015) D

B

A Dia of Reamer

Length, A

3 ⁄8

7

13 ⁄32 7⁄16

Flute Length, B

Shank Dia., D

Max.

Min.

3 ⁄4

0.3105

0.3095

7

3 ⁄4

0.3105

0.3095

7

7⁄8

0.3730

0.3720

15 ⁄32

7

7⁄8

0.3730

0.3720

1 ⁄2

8

1

0.4355

0.4345

17⁄32

8

1

0.4355

0.4345

9 ⁄16

8

0.4355

0.4345

19 ⁄32

8

11 ⁄8

0.4355

0.4345

5 ⁄8

9

0.5620

0.5605

21 ⁄32

9

0.5620

0.5605

11 ⁄16

9

0.5620

0.5605

23 ⁄32

9

0.5620

0.5605

3 ⁄4

91 ⁄2

13⁄8

0.6245

0.6230

25 ⁄32

91 ⁄2

13⁄8

0.6245

0.6230

13 ⁄16

91 ⁄2

13⁄8

0.6245

0.6230

27⁄32

91 ⁄2

13⁄8

0.6245

0.6230

0.7495

0.7480

0.7495

0.7480

0.7495

0.7480

0.7495

0.7480

0.8745

0.8730

0.8745

0.8730

0.8745

0.8730

7⁄8

10

29 ⁄32

10

15 ⁄16

10

31 ⁄32

10

11 ⁄8 11 ⁄4 11 ⁄4 11 ⁄4 11 ⁄4

11 ⁄2 11 ⁄2 11 ⁄2 11 ⁄2

1

101 ⁄2

15⁄8

11 ⁄32

101 ⁄2

15⁄8

11 ⁄16

101 ⁄2

15⁄8

Dia.of Reamer

Length, A

13⁄32

101 ⁄2

11 ⁄8 15⁄32 13⁄16

11

Flute Length, B 15⁄8 13⁄4

11

13⁄4

11

13⁄4

Shank Dia., D

Max.

Min.

0.8745

0.8730

0.8745

0.8730

0.8745

0.8730

0.9995

0.9980

0.9995

0.9980

0.9995

0.9980

17⁄32

11

11 ⁄4

111 ⁄2

17⁄8

15⁄16

111 ⁄2

17⁄8

0.9995

0.9980

2

0.9995

0.9980

17⁄16

12

2

1.2495

1.2480

11 ⁄2

121 ⁄2

21 ⁄8

1.2495

1.2480

19⁄16 a

121 ⁄2

21 ⁄8

1.2495

1.2480

1.2495

1.2480

1.2495

1.2480

1.2495

1.2480

1.4995

1.4980

1.4995

1.4980

1.4995

1.4980

1.4995

1.4980

…

…

13⁄8

15⁄8

13⁄4

12

13

21 ⁄4

111 ⁄16 a

13

13⁄4

131 ⁄2

23⁄8

113⁄16 a

131 ⁄2

23⁄8

17⁄8 115⁄16 a

21 ⁄4

14

21 ⁄2

14

21 ⁄2

2

14

21 ⁄8 b

141 ⁄2

23⁄4

21 ⁄4 b

141 ⁄2

23⁄4

…

…

3

…

…

15

3

…

…

…

…

…

…

23⁄8 b 21 ⁄2 b …

15

21 ⁄2

a Straight shank only. b Taper shank only.

All dimensions in inches. Material is high-speed steel. The number of flutes is as follows: 10; 13⁄4 - to to 14. The expansion feature of these reamers provides a means of adjustment that is important in reaming holes to close tolerances. When worn undersize, they may be expanded and reground to the original size. Tolerances: On reamer diameter, 3⁄8 - to 1-inch sizes, incl., +0.0001 to +0.0005 inch; over 1-inch size, + 0.0002 to + 0.0006 inch. On length A and flute length B, 3⁄8 - to 1-inch sizes, incl., ±1 ⁄16 inch; 1 1 ⁄32 - to 2-inch sizes, incl., ±3⁄32 inch; over 2-inch sizes, ±1 ⁄8 inch. Taper is Morse taper: No. 1 for sizes 3⁄8 to 19⁄32 inch, incl.; No. 2 for sizes 5⁄8 to 29⁄32 incl.; No. 3 for sizes 15⁄16 to 17⁄32 , incl.; No. 4 for sizes 11 ⁄4 to 15⁄8 , incl.; and No. 5 for sizes 13⁄4 to 21 ⁄2 , incl. For amount of taper, see Table on page 1013. 3 ⁄ - to 15 ⁄ -inch sizes, 4 to 6; 1 ⁄ - to 31 ⁄ -inch sizes, 6 to 8; 1- to 111 ⁄ -inch sizes, 8 to 8 32 2 32 16 115⁄16 -inch sizes, 8 to 12; 2- to 21 ⁄4 -inch sizes, 10 to 12; 23⁄8 - and 21 ⁄2 -inch sizes, 10

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Machinery's Handbook, 31st Edition Reamers

920

Hand Reamers—Straight and Helical Flutes ANSI/ASME B94.2-1995 (R2015) D

C

B

A

D

C

Straight Flutes 1 ⁄8 9 ⁄64 5 ⁄32 11 ⁄16 3 ⁄16 13 ⁄64 7 ⁄32 15 ⁄64 1 ⁄4 17 ⁄64 9 ⁄32 19 ⁄64 5 ⁄16 21 ⁄64 11 ⁄32 23 ⁄64 3 ⁄8 25 ⁄64 13 ⁄32 27 ⁄64 7 ⁄16 29 ⁄64 15 ⁄32 31 ⁄64 1 ⁄2 17 ⁄32 9 ⁄16 19 ⁄32 5 ⁄8 21 ⁄32 11 ⁄16 23 ⁄32 3 ⁄4 … 7 ⁄8 …

1 11 ⁄8 11 ⁄4 13 ⁄8 11 ⁄2

Reamer Diameter Helical Decimal Flutes Equivalent … 0.1250 … 0.1406 … 0.1562 … 0.1719 … 0.1875 … 0.2031 … 0.2188 … 0.2344 1 ⁄4 0.2500 … 0.2969 … 0.2812 … 0.2969 5 ⁄16 0.3125 … 0.3281 … 0.3438 … 0.3594 3 ⁄8 0.3750 … 0.3906 … 0.4062 … 0.4219 7 ⁄16 0.4375 … 0.4531 … 0.4688 … 0.4844 1 ⁄2 0.5000 0.5312 … 9 ⁄16 0.5625 0.5938 … 5 ⁄8 0.6250 0.6562 … 11 ⁄16 0.6875 0.7188 … 3 ⁄4 0.7500 13 ⁄16 0.8125 7 ⁄8 0.8750 15 ⁄16 0.9375 1.0000 1 1.1250 11 ⁄8 1.2500 11 ⁄4 1.3750 13 ⁄8 1.5000 11 ⁄2

B

A

Length Overall A 3 31 ⁄4 31 ⁄4 31 ⁄2 31 ⁄2 33 ⁄4 33 ⁄4 4 4 41 ⁄4 41 ⁄4 41 ⁄2 41 ⁄2 43 ⁄4 43 ⁄4 5 5 51 ⁄4 51 ⁄4 51 ⁄2 51 ⁄2 53 ⁄4 53 ⁄4 6 6 61 ⁄4 61 ⁄2 63 ⁄4 7 73 ⁄8 73 ⁄4 81 ⁄8 83 ⁄8 91 ⁄8 93 ⁄4 101 ⁄4 107 ⁄8 115 ⁄8 121 ⁄4 125 ⁄8 13

Flute Length B 1 1 ⁄2 15 ⁄8 15 ⁄8 13 ⁄4 13 ⁄4 17 ⁄8 17 ⁄8 2 2 21 ⁄8 21 ⁄8 21 ⁄4 21 ⁄4 23 ⁄8 23 ⁄8 2 1 ⁄2 21 ⁄2 25 ⁄8 25 ⁄8 23 ⁄4 23 ⁄4 27 ⁄8 27 ⁄8 3 3 31 ⁄8 31 ⁄4 33 ⁄8 31 ⁄2 311 ⁄16 37 ⁄8 41 ⁄16 43 ⁄16 49 ⁄16 47 ⁄8 51 ⁄8 57 ⁄16 513 ⁄16 61 ⁄8 65 ⁄16 61 ⁄2

Square Length C 5 ⁄32 5 ⁄32 7 ⁄32 7 ⁄32 7 ⁄32 1 ⁄4 1 ⁄4 1 ⁄4 1 ⁄4 1 ⁄4 1 ⁄4 5 ⁄16 5 ⁄16 5 ⁄16 5 ⁄16 3 ⁄8 3 ⁄8 3 ⁄8 3 ⁄8 7 ⁄16 7 ⁄16 7 ⁄16 7 ⁄16 1 ⁄2 1 ⁄2 1 ⁄2 9 ⁄16 9 ⁄16 5 ⁄8 5 ⁄8 11 ⁄16 11 ⁄16 3 ⁄4 13 ⁄16 7 ⁄8 15 ⁄16 1 1 1 1 11 ⁄8

Size of Square 0.095 0.105 0.115 0.130 0.140 0.150 0.165 0.175 0.185 0.200 0.210 0.220 0.235 0.245 0.255 0.270 0.280 0.290 0.305 0.315 0.330 0.340 0.350 0.365 0.375 0.400 0.420 0.445 0.470 0.490 0.515 0.540 0.560 0.610 0.655 0.705 0.750 0.845 0.935 1.030 1.125

No. of Flutes 4 to 6 4 to 6 4 to 6 4 to 6 4 to 6 4 to 6 4 to 6 4 to 6 4 to 6 4 to 6 4 to 6 4 to 6 4 to 6 4 to 6 4 to 6 4 to 6 4 to 6 6 to 8 6 to 8 6 to 8 6 to 8 6 to 8 6 to 8 6 to 8 6 to 8 6 to 8 6 to 8 6 to 8 6 to 8 6 to 8 6 to 8 6 to 8 6 to 8 8 to 10 8 to 10 8 to 10 8 to 10 8 to 10 8 to 12 10 to 12 10 to 14

All dimensions in inches. Material is high-speed steel. The nominal shank diameter D is the same as the reamer diameter. Helical-flute hand reamers with left-hand helical flutes are standard. Ream­ers are tapered slightly on the end to facilitate proper starting. Tolerances: On diameter of reamer, up to 1 ⁄4 -inch size, incl., + .0001 to + .0004 inch; over 1 ⁄4 to 1-inch size, incl., +.0001 to + .0005 inch; over 1-inch size, +.0002 to +.0006 inch. On length overall A and flute length B, 1 ⁄8 - to 1-inch size, incl., ± 1 ⁄16 inch; 11 ⁄8 - to 11 ⁄2 -inch size, incl., ±3⁄32 inch. On length of square C, 1 ⁄8 - to 1 inch size, incl., ±1 ⁄32 inch; 11 ⁄8 -to 11 ⁄2 -inch size, incl., ±1 ⁄16 inch. On shank diameter D, 1 ⁄8 - to 1-inch size, incl., -.001 to -.005 inch; 11 ⁄8 - to 11 ⁄2 -inch size, incl., -.0015 to -.006 inch. On size of square, 1 ⁄8 - to 1 ⁄2 -inch size, incl., -.004 inch; 17⁄32 - to 1-inch size, incl., -.006 inch; 11 ⁄8 - to 11 ⁄2 -inch size, incl., -.008 inch.

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Machinery's Handbook, 31st Edition Reamers

921

American National Standard Expansion Hand Reamers—Straight and Helical Flutes, Squared Shank ANSI/ASME B94.2-1995 (R2015) D C

B

A

D

C

Reamer Dia. 1 ⁄4

5 ⁄16 3 ⁄8

7⁄16 1 ⁄2

9 ⁄16 5 ⁄8

11 ⁄16 3 ⁄4

1

7⁄8

11 ⁄8 11 ⁄4 1 ⁄4

5 ⁄16 3 ⁄8

7⁄16 1 ⁄2 5 ⁄8 3 ⁄4

1

7⁄8

11 ⁄4

B

A

Length Overall A Max. Min. 43⁄8 43⁄8 53⁄8 53⁄8 61 ⁄2 61 ⁄2 7

75⁄8 8 9 10

101 ⁄2 11 43⁄8 43⁄8 61 ⁄8 61 ⁄4 61 ⁄2 8

85⁄8 93⁄8 101 ⁄4 113⁄8

33⁄4 4

41 ⁄4 41 ⁄2 5

53⁄8 53⁄4 61 ⁄4 61 ⁄2 71 ⁄2 83⁄8 9 93⁄4

37⁄8 4

41 ⁄4

41 ⁄2 5 6 61 ⁄2 71 ⁄2 83⁄8 93⁄4

Flute Length B Max. Min. 13⁄4 17⁄8 2 2

21 ⁄2 21 ⁄2 3 3 31 ⁄2 4

41 ⁄2 43⁄4 5 13⁄4 13⁄4 2 2

21 ⁄2 3 31 ⁄2 4 41 ⁄2 5

Length of Square C

Straight Flutes

1 ⁄4 11 ⁄2 5 ⁄16 11 ⁄2 3 ⁄8 13⁄4 7⁄16 13⁄4 1 ⁄2 13⁄4 9 ⁄16 17⁄8 5 ⁄8 21 ⁄4 11 ⁄16 21 ⁄2 3 ⁄4 25⁄8 7⁄8 31 ⁄8 1 31 ⁄8 1 31 ⁄2 1 41 ⁄4 Helical Flutes

11 ⁄2 11 ⁄2 13⁄4 13⁄4 13⁄4 21 ⁄4 25⁄8 31 ⁄8 31 ⁄8 41 ⁄4

Shank Dia. D 1 ⁄4

5 ⁄16 3 ⁄8

7⁄16

1 ⁄2

9 ⁄16

5 ⁄8

11 ⁄16

3 ⁄4

1

11 ⁄8 11 ⁄4

1 ⁄4

1 ⁄4

5 ⁄16

5 ⁄16

3 ⁄8

3 ⁄8

7⁄16

7⁄16

1 ⁄2

1 ⁄2

3 ⁄4

3 ⁄4

5 ⁄8

1 1

7⁄8

7⁄8

5 ⁄8

1

7⁄8

11 ⁄4

Size of Square

Number of Flutes

0.185 0.235 0.280 0.330 0.375 0.420 0.470 0.515 0.560 0.655 0.750 0.845 0.935

6 to 8 6 to 8 6 to 9 6 to 9 6 to 9 6 to 9 6 to 9 6 to 10 6 to 10 8 to 10 8 to 10 8 to 12 8 to 12

0.185 0.235 0.280 0.330 0.375 0.470 0.560 0.655 0.750 0.935

6 to 8 6 to 8 6 to 9 6 to 9 6 to 9 6 to 9 6 to 10 6 to 10 6 to 10 8 to 12

All dimensions are given in inches. Material is carbon steel. Reamers with helical flutes that are left hand are standard. Expansion hand reamers are primarily designed for work where it is necessary to enlarge reamed holes by a few thousandths. The pilots and guides on these reamers are ground under­size for clearance. The maximum expansion on these reamers is as follows: .006 inch for the 1 ⁄4 - to 7⁄16 -inch sizes. .010 inch for the 1 ⁄2 - to 7⁄8 -inch sizes and .012 inch for the 1to 11 ⁄4 -inch sizes. Tolerances: On length overall A and flute length B, ±1 ⁄16 inch for 1 ⁄4 - to 1-inch sizes, ± 3⁄32 inch for 11 ⁄8 -to 11 ⁄4 -inch sizes; on length of square C, ±1 ⁄32 inch for 1 ⁄4 - to 1-inch sizes, ± 1 ⁄16 inch for 1 1 ⁄8 -to 11 ⁄4 -inch sizes; on shank diameter D -.001 to -.005 inch for 1 ⁄4 - to 1-inch sizes, -.0015 to -­ .006 inch for 11 ⁄8 - to 11 ⁄4 -inch sizes; on size of square, -.004 inch for 1 ⁄4 - to 1 ⁄2 -inch sizes. -.006 inch for 9⁄16 - to 1-inch sizes, and -.008 inch for 11 ⁄8 - to 11 ⁄4 -inch sizes.

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Machinery's Handbook, 31st Edition Reamers

922

Taper Shank Jobbers Reamers—Straight Flutes ANSI/ASME B94.2-1995 (R2015)

B

A Reamer Diameter Fractional Dec. Equiv. 1 ⁄4 5 ⁄16 3 ⁄8 7⁄16 1 ⁄2 9 ⁄16 5 ⁄8 11 ⁄16 3 ⁄4 13 ⁄16 7⁄8

1

15 ⁄16

11 ⁄16 11 ⁄8 13⁄16 11 ⁄4 13⁄8 11 ⁄2

Length Overall A

0.2500

Length of Flute B 2

53⁄16

0.3125 0.3750 0.4375 0.5000 0.5625 0.6250 0.6875 0.7500 0.8125 0.8750 0.9375 1.0000 1.0625 1.1250 1.1875 1.2500 1.3750 1.5000

21 ⁄4 21 ⁄2 23⁄4 3

51 ⁄2 513⁄16 61 ⁄8 67⁄16 63⁄4 79⁄16 8

31 ⁄4 31 ⁄2 37⁄8 43⁄16 49⁄16 47⁄8 51 ⁄8 57⁄16 55⁄8 513⁄16 6

83⁄8 813⁄16 93⁄16 10

103⁄8 105⁄8 107⁄8 111 ⁄8 129⁄16 1213⁄16 131 ⁄8

61 ⁄8 65⁄16 61 ⁄2

No. of Morse Taper Shanka

No. of Flutes

1 1 1 1 1 2 2 2 2 2 3 3 3 3 3 4 4 4

6 to 8 6 to 8 6 to 8 6 to 8 6 to 8 6 to 8 8 to 10 8 to 10 8 to 10 8 to 10 8 to 10 8 to 10 8 to 10 8 to 10 8 to 12 8 to 12 10 to 12 10 to 12

1

6 to 8

a American National Standard self-holding tapers (Table 7a on page 1020.)

All dimensions in inches. Material is high-speed steel. Tolerances: On reamer diameter, 1 ⁄4 -inch size, +.0001 to +.0004 inch; over 1 ⁄4 - to 1-inch size, incl., +.0001 to +.0005 inch; over 1-inch size, +.0002 to +.0006 inch. On overall length A and length of flute B, 1 ⁄4 - to 1-inch size, incl., ±1 ⁄16 inch; and 11 ⁄16 - to 11 ⁄2 -inch size, incl., ±3⁄32 inch.

American National Standard Driving Slots and Lugs for Shell Reamers or Shell Reamer Arbors ANSI/ASME B94.2-1995 (R2015) L

W

J

M

Arbor Size No. 4 5 6 7 8 9

Fitting Reamer Sizes 3⁄ 4 13 ⁄ to 1 16 11 ⁄16 to 11 ⁄4 15⁄16 to 15⁄8 111 ⁄16 to 2 21 ⁄16 to 21 ⁄2

Driving Slot Width Depth W J 5⁄ 3⁄ 32 16 3⁄ 1⁄ 16 4 3⁄ 1⁄ 16 4 1⁄ 5⁄ 4 16 1⁄ 5⁄ 4 16 5⁄ 3⁄ 16 8

Lug on Arbor Width Depth L M 9⁄ 5⁄ 64 32 11 ⁄ 7⁄ 64 32 11 ⁄ 7⁄ 64 32 15 ⁄ 9⁄ 64 32 15 ⁄ 9⁄ 64 32 19 ⁄ 11 ⁄ 64 32

Reamer Hole Dia. at Large End 0.375 0.500 0.625 0.750 1.000 1.250

All dimension are given in inches. The hole in shell reamers has a taper of 1 ⁄8 inch per foot, with arbors tapered to correspond. Shell reamer arbor tapers are made to permit a driving fit with the reamer.

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Machinery's Handbook, 31st Edition Reamers

923

Straight-Shank Chucking Reamers—Straight Flutes, Wire Gage Sizes ANSI/ASME B94.2-1995 (R2015) D

B A Reamer Diameter Wire Gage

Inch

Lgth. Overall A

Shank Dia. D

Lgth. of Flute B

Max.

Min.

No. of Flutes

Reamer Diameter Wire Gage

Inch

Lgth. Overall A

Shank Dia. D

Lgth. of Flute B

Max.

Min.

No. of Flutes

60

.0400

21 ⁄2

1 ⁄2

.0390

.0380

4

49

.0730

3

3 ⁄4

.0660

.0650

4

59

.0410

21 ⁄2

1 ⁄2

.0390

.0380

4

48

.0760

3

3 ⁄4

.0720

.0710

4

58

.0420

21 ⁄2

1 ⁄2

.0390

.0380

4

47

.0785

3

3 ⁄4

.0720

.0710

4

57

.0430

21 ⁄2

1 ⁄2

.0390

.0380

4

46

.0810

3

3 ⁄4

.0771

.0701

4

56

.0465

21 ⁄2

1 ⁄2

.0455

.0445

4

45

.0820

3

3 ⁄4

.0771

.0761

4

55

.0520

21 ⁄2

1 ⁄2

.0510

.0500

4

44

.0860

3

3 ⁄4

.0810

.0800

4

54

.0550

21 ⁄2

1 ⁄2

.0510

.0500

4

43

.0890

3

3 ⁄4

.0810

.0800

4

53

.0595

21 ⁄2

1 ⁄2

.0585

.0575

4

42

.0935

3

3 ⁄4

.0880

.0870

4

52

.0635

21 ⁄2

1 ⁄2

.0585

.0575

4

41

.0960

31 ⁄2

7⁄8

.0928

.0918

4 to 6

51

.0670

3

3 ⁄4

.0660

.0650

4

40

.0980

31 ⁄2

7⁄8

.0928

.0918

4 to 6

50

.0700

3

3 ⁄4

.0660

.0650

4

39

.0995

31 ⁄2

7⁄8

.0928

.0918

4 to 6

38

.1015

31 ⁄2

7⁄8

.0950

.0940

4 to 6

19

.1660

41 ⁄2

11 ⁄8

.1595

.1585

4 to 6

37

.1040

31 ⁄2

7⁄8

.0950

.0940

4 to 6

18

.1695

41 ⁄2

11 ⁄8

.1595

.1585

4 to 6

36

.1065

31 ⁄2

7⁄8

.1030

.1020

4 to 6

17

.1730

41 ⁄2

11 ⁄8

.1645

.1635

4 to 6

35

.1100

31 ⁄2

7⁄8

.1030

.1020

4 to 6

16

.1770

41 ⁄2

11 ⁄8

.1704

.1694

4 to 6

34

.1110

31 ⁄2

7⁄8

.1055

.1045

4 to 6

15

.1800

41 ⁄2

11 ⁄8

.1755

.1745

4 to 6

33

.1130

31 ⁄2

7⁄8

.1055

.1045

4 to 6

14

.1820

41 ⁄2

11 ⁄8

.1755

.1745

4 to 6

32

.1160

31 ⁄2

7⁄8

.1120

.1110

4 to 6

13

.1850

41 ⁄2

11 ⁄8

.1805

.1795

4 to 6

31

.1200

31 ⁄2

7⁄8

.1120

.1110

4 to 6

12

.1890

41 ⁄2

11 ⁄8

.1805

.1795

4 to 6

30

.1285

31 ⁄2

7⁄8

.1190

.1180

4 to 6

11

.1910

5

11 ⁄4

.1860

.1850

4 to 6

29

.1360

4

1

.1275

.1265

4 to 6

10

.1935

5

11 ⁄4

.1860

.1850

4 to 6

28

.1405

4

1

.1350

.1340

4 to 6

9

.1960

5

11 ⁄4

.1895

.1885

4 to 6

27

.1440

4

1

.1350

.1340

4 to 6

8

.1990

5

11 ⁄4

.1895

.1885

4 to 6

26

.1470

4

1

.1430

.1420

4 to 6

7

.2010

5

11 ⁄4

.1945

.1935

4 to 6

25

.1495

4

1

.1430

.1420

4 to 6

6

.2040

5

11 ⁄4

.1945

.1935

4 to 6

24

.1520

4

1

.1460

.1450

4 to 6

5

.2055

5

11 ⁄4

.2016

.2006

4 to 6

23

.1540

4

1

.1460

.1450

4 to 6

4

.2090

5

11 ⁄4

.2016

.2006

4 to 6

22

.1570

4

1

.1510

.1500

4 to 6

3

.2130

5

11 ⁄4

.2075

.2065

4 to 6

21

.1590

41 ⁄2

11 ⁄8

.1530

.1520

4 to 6

2

2210

6

11 ⁄2

.2173

.2163

4 to 6

20

.1610

41 ⁄2

11 ⁄8

.1530

.1520

4 to 6

1

.2280

6

11 ⁄2

.2173

.2163

4 to 6

All dimensions in inches. Material is high-speed steel. Tolerances: On diameter of reamer, plus .0001 to plus .0004 inch. On overall length A, plus or minus 1 ⁄16 inch. On length of flute B, plus or minus 1 ⁄16 inch.

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Machinery's Handbook, 31st Edition Reamers

924

Straight-Shank Chucking Reamers—Straight Flutes, Letter Sizes ANSI/ASME B94.2-1995 (R2015) D B

A Reamer Diameter Letter Inch A 0.2340 B 0.2380 C 0.2420 D 0.2460 E 0.2500 F 0.2570 G 0.2610 H 0.2660 I 0.2720 J 0.2770 K 0.2810 L 0.2900 M 0.2950

Lgth. Over­ all A 6 6 6 6 6 6 6 6 6 6 6 6 6

Lgth. of Flute B 11 ⁄2 11 ⁄2 11 ⁄2 11 ⁄2 11 ⁄2 11 ⁄2 11 ⁄2 11 ⁄2 11 ⁄2 11 ⁄2 11 ⁄2 11 ⁄2 11 ⁄2

Shank Dia. D Max. Min. 0.2265 .2255 0.2329 .2319 0.2329 .2319 0.2329 .2319 0.2405 .2395 0.2485 .2475 0.2485 .2475 0.2485 .2475 0.2485 .2475 0.2485 .2475 0.2485 .2475 0.2792 .2782 0.2792 .2782

No. of Flutes 4 to 6 4 to 6 4 to 6 4 to 6 4 to 6 4 to 6 4 to 6 4 to 6 4 to 6 4 to 6 4 to 6 4 to 6 4 to 6

Reamer Diameter Letter Inch N 0.3020 O 0.3160 P 0.3230 Q 0.3320 R 0.3390 S 0.3480 T 0.3580 U 0.3680 V 0.3770 W 0.3860 X 0.3970 Y 0.4040 Z 0.4130

Lgth. Over­ all A 6 6 6 6 6 7 7 7 7 7 7 7 7

Lgth. of Flute B 11 ⁄2 11 ⁄2 11 ⁄2 11 ⁄2 11 ⁄2 13⁄4 13⁄4 13⁄4 13⁄4 13⁄4 13⁄4 13⁄4 13⁄4

Shank Dia. D Max. Min. 0.2792 0.2782 0.2792 0.2782 0.2792 0.2782 0.2792 0.2782 0.2792 0.2782 0.3105 0.3095 0.3105 0.3095 0.3105 0.3095 0.3105 0.3095 0.3105 0.3095 0.3105 0.3095 0.3105 0.3095 0.3730 0.3720

No. of Flutes 4 to 6 4 to 6 4 to 6 4 to 6 4 to 6 4 to 6 4 to 6 4 to 6 4 to 6 4 to 6 4 to 6 4 to 6 6 to 8

All dimensions in inches. Material is high-speed steel. Tolerances: On diameter of reamer, for sizes A to E, incl., plus .0001 to plus .0004 inch and for sizes F to Z, incl., plus .0001 to plus .0005 inch. On overall length A, plus or minus 1 ⁄16 inch. On length of flute B, plus or minus 1 ⁄16 inch.

Straight-Shank Chucking Reamers— Straight Flutes, Decimal Sizes ANSI/ASME B94.2-1995 (R2015) D B

A

Lgth. Reamer Overall Dia. A 0.1240 0.1260 0.1865 0.1885 0.2490 0.2510 0.3115

31 ⁄2 31 ⁄2 41 ⁄2 41 ⁄2 6 6 6

Lgth. of Flute B 7⁄8 7⁄8 11 ⁄8 11 ⁄8 11 ⁄2 11 ⁄2 11 ⁄2

Shank Diameter D Max. 0.1190 0.1190 0.1805 0.1805 0.2405 0.2405 0.2792

Min. 0.1180 0.1180 0.1795 0.1795 0.2395 0.2395 0.2782

No. of Flutes

Lgth. Reamer Overall Dia. A

4 to 6 4 to 6 4 to 6 4 to 6 4 to 6 4 to 6 4 to 6

0.3135 0.3740 0.3760 0.4365 0.4385 0.4990 0.5010

6 7 7 7 7 8 8

Lgth. of Flute B 11 ⁄2 13⁄4 13⁄4 13⁄4 13⁄4 2 2

Shank Diameter D Max. 0.2792 0.3105 0.3105 0.3730 0.3730 0.4355 0.4355

Min. 0.2782 0.3095 0.3095 0.3720 0.3720 0.4345 0.4345

No. of Flutes 4 to 6 6 to 8 6 to 8 6 to 8 6 to 8 6 to 8 6 to 8

All dimensions in inches. Material is high-speed steel. Tolerances: On diameter of reamer, for 0.124 to 0.249-inch sizes, plus .0001 to plus .0004 inch and for 0.251 to 0.501-inch sizes, plus .0001 to plus .0005 inch. On overall length A, plus or minus 1 ⁄ inch. On length of flute B, plus or minus 1 ⁄ inch. 16 16

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Machinery's Handbook, 31st Edition Reamers

925

American National Standard Straight-Shank Rose Chucking and Chucking Reamers— Straight and Helical Flutes, Fractional Sizes ANSI/ASME B94.2-1995 (R2015) D

Reamer Diameter Chucking Rose Chucking 3 ⁄64 a … 1 ⁄16 … 5 ⁄64 … 3 ⁄32 … 7⁄64 … 1 ⁄8 a 1 ⁄8 9 ⁄64 … 5 ⁄32 … 11 ⁄64 … 3 ⁄16 a 3 ⁄16 13 ⁄64 … 7⁄32 … 15 ⁄64 … 1 ⁄4 a 1 ⁄4 17⁄64 … 9 ⁄32 … 19 ⁄64 … 5 ⁄16 a 5 ⁄16 21 ⁄64 … 11 ⁄32 … 23 ⁄64 … 3 ⁄8 a 3 ⁄8 25 ⁄64 … 13 ⁄32 … 27⁄64 … 7⁄16 a 7⁄16 29 ⁄64 … 15 ⁄32 … 31 ⁄64 … 1 ⁄2 a 1 ⁄2 17⁄32 … 9 ⁄16 … 19 ⁄32 … 5 ⁄8 … 21 ⁄32 … 11 ⁄16 … 23 ⁄32 … 3 ⁄4 … 25 ⁄32 … 13 ⁄16 … 27⁄32 … 7⁄8 … 29 ⁄32 … 15 ⁄16 … 31 ⁄32 … 1 … 1 … 1 ⁄16 … 11 ⁄8 … 13⁄16 … 11 ⁄4 … 15⁄16 b … 13⁄8 … 17⁄16 b 1 … 1 ⁄2

Length Overall A 21 ⁄2 21 ⁄2 3 3 31 ⁄2 31 ⁄2 4 4 41 ⁄2 41 ⁄2 5 5 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 8 8 8 8 8 9 9 9 9 91 ⁄2 91 ⁄2 91 ⁄2 91 ⁄2 10 10 10 10 101 ⁄2 101 ⁄2 11 11 111 ⁄2 111 ⁄2 12 12 121 ⁄2

B

B

A

Flute Length B 1 ⁄2 1 ⁄2 3 ⁄4 3 ⁄4 7⁄8 7⁄8 1 1 11 ⁄8 11 ⁄8 11 ⁄4 11 ⁄4 11 ⁄2 11 ⁄2 11 ⁄2 11 ⁄2 11 ⁄2 11 ⁄2 11 ⁄2 11 ⁄2 13⁄4 13⁄4 13⁄4 13⁄4 13⁄4 13⁄4 13⁄4 13⁄4 2 2 2 2 2 21 ⁄4 21 ⁄4 21 ⁄4 21 ⁄4 21 ⁄2 21 ⁄2 21 ⁄2 21 ⁄2 25⁄8 25⁄8 25⁄8 25⁄8 23⁄4 23⁄4 27⁄8 27⁄8 3 3 31 ⁄4 31 ⁄4 31 ⁄2

Shank Dia. D Max. Min. 0.0455 0.0445 0.0585 0.0575 0.0720 0.0710 0.0880 0.0870 0.1030 0.1020 0.1190 0.1180 0.1350 0.1340 0.1510 0.1500 0.1645 0.1635 0.1805 0.1795 0.1945 0.1935 0.2075 0.2065 0.2265 0.2255 0.2405 0.2395 0.2485 0.2475 0.2485 0.2475 0.2792 0.2782 0.2792 0.2782 0.2792 0.2782 0.2792 0.2782 0.3105 0.3095 0.3105 0.3095 0.3105 0.3095 0.3105 0.3095 0.3730 0.3720 0.3730 0.3720 0.3730 0.3720 0.3730 0.3720 0.4355 0.4345 0.4355 0.4345 0.4355 0.4345 0.4355 0.4345 0.4355 0.4345 0.5620 0.5605 0.5620 0.5605 0.5620 0.5605 0.5620 0.5605 0.6245 0.6230 0.6245 0.6230 0.6245 0.6230 0.6245 0.6230 0.7495 0.7480 0.7495 0.7480 0.7495 0.7480 0.7495 0.7480 0.8745 0.8730 0.8745 0.8730 0.8745 0.8730 0.9995 0.9980 0.9995 0.9980 0.9995 0.9980 0.9995 0.9980 1.2495 1.2480 1.2495 1.2480

No. of Flutes 4 4 4 4 4 to 6 4 to 6 4 to 6 4 to 6 4 to 6 4 to 6 4 to 6 4 to 6 4 to 6 4 to 6 4 to 6 4 to 6 4 to 6 4 to 6 4 to 6 4 to 6 4 to 6 4 to 6 4 to 6 4 to 6 6 to 8 6 to 8 6 to 8 6 to 8 6 to 8 6 to 8 6 to 8 6 to 8 6 to 8 6 to 8 6 to 8 6 to 8 6 to 8 6 to 8 8 to 10 8 to 10 8 to 10 8 to 10 8 to 10 8 to 10 8 to 10 8 to 12 8 to 12 8 to 12 8 to 12 8 to 12 10 to 12 10 to 12 10 to 12 10 to 12

a Reamer with straight flutes is standard only. b Reamer with helical flutes is standard only.

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Machinery's Handbook, 31st Edition Reamers

926

All dimensions are given in inches. Material is high-speed steel. Chucking reamers are end cutting on the chamfer and the relief for the outside diameter is ground in back of the margin for the full length of land. Lands of rose chucking reamers are not relieved on the periphery but have a relatively large amount of back taper. Tolerances: On reamer diameter, up to 1 ⁄4 -inch size, incl., + .0001 to + .0004 inch; over 1 ⁄4 to 1-inch size, incl., + .0001 to + .0005 inch; over 1-inch size, + .0002 to + .0006 inch. On length overall A and flute length B, up to 1-inch size, incl., ±1 ⁄16 inch; 11 ⁄16 - to 11 ⁄2 -inch size, incl., ±3⁄32 inch. Helical flutes are right- or left-hand helix, right-hand cut, except sizes 11 ⁄16 through 11 ⁄2 inches, which are right-hand helix only.

Shell Reamers—Straight and Helical Flutes ANSI/ASME B94.2-1995 (R2015)

H

H

A Diameter of Reamer 3 ⁄4 7⁄8

1

15 ⁄16 a

11 ⁄16 11 ⁄8 13⁄16 11 ⁄4 15⁄16 13⁄8 17⁄16 11 ⁄2 19⁄16 15⁄8 111 ⁄16 13⁄4 113⁄16 17⁄8 115⁄16 2

21 ⁄16 a 21 ⁄8 23⁄16 a 21 ⁄4 23⁄8 a 21 ⁄2 a

Length Overall A 21 ⁄4 21 ⁄2 21 ⁄2 21 ⁄2 23⁄4 23⁄4 23⁄4 23⁄4 3 3 3 3 3 3 31 ⁄2 31 ⁄2 31 ⁄2 31 ⁄2 31 ⁄2 31 ⁄2 33⁄4 33⁄4 33⁄4 33⁄4 33⁄4 33⁄4

B

A Flute Length B

11 ⁄2 13⁄4 13⁄4 13⁄4 2 2 2 2 21 ⁄4 21 ⁄4 21 ⁄4 21 ⁄4 21 ⁄4 21 ⁄4 21 ⁄2 21 ⁄2 21 ⁄2 21 ⁄2 21 ⁄2 21 ⁄2 23⁄4 23⁄4 23⁄4 23⁄4 23⁄4 23⁄4

Hole Diameter Large End H 0.375 0.500 0.500 0.500 0.625 0.625 0.625 0.625 0.750 0.750 0.750 0.750 0.750 0.750 1.000 1.000 1.000 1.000 1.000 1.000 1.250 1.250 1.250 1.250 1.250 1.250

B Fitting Arbor No. 4 5 5 5 6 6 6 6 7 7 7 7 7 7 8 8 8 8 8 8 9 9 9 9 9 9

Number of Flutes 8 to 10 8 to 10 8 to 10 8 to 10 8 to 12 8 to 12 8 to 12 8 to 12 8 to 12 8 to 12 8 to 12 10 to 14 10 to 14 10 to 14 10 to 14 12 to 14 12 to 14 12 to 14 12 to 14 12 to 14 12 to 16 12 to 16 12 to 16 12 to 16 14 to 16 14 to 16

a Helical flutes only.

All dimensions are given in inches. Material is high-speed steel. Helical flute shell reamers with left-hand helical flutes are standard. Shell reamers are designed as a sizing or finishing reamer and are held on an arbor provided with driving lugs. The holes in these reamers are ground with a taper of 1 ⁄8 inch per foot. Tolerances: On diameter of reamer, 3⁄4 - to 1-inch size, incl., + .0001 to + .0005 inch; over 1-inch size, + .0002 to + .0006 inch. On length overall A and flute length B, 3⁄4 - to 1-inch size, incl., ± 1 ⁄16 inch; 11 ⁄16 - to 2-inch size, incl., ± 3⁄32 inch; 21 ⁄16 - to 21 ⁄2 -inch size, incl., ± 1 ⁄8 inch.

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Machinery's Handbook, 31st Edition Reamers

927

American National Standard Arbors for Shell Reamers— Straight and Taper Shanks ANSI/ASME B94.2-1995 (R2015) D B

A

L

A Arbor Size No. 4 5 6

Overall Length A 9

91 ⁄2 10

Approx. Length of Taper L 21 ⁄4 21 ⁄2 23⁄4

Reamer Size 3 ⁄4

to 1 11 ⁄16 to 11 ⁄4 13 ⁄16

Taper Shank No.a 2 2 3

Straight Shank Dia. D 1 ⁄2 5 ⁄8 3 ⁄4

Arbor Size No. 7 8 9

Overall Length A 11 12 13

Approx. Length of Taper L 3 31 ⁄2 33⁄4

Taper Shank No.a 3 4 4

Reamer Size

15⁄16 to 15⁄8 111 ⁄16 to 2 21 ⁄16 to 21 ⁄2

Straight Shank Dia. D 7⁄8 11 ⁄8 13⁄8

a American National Standard self-holding tapers (see Table 7a on page 1020).

All dimensions are given in inches. These arbors are designed to fit standard shell reamers (see table). End that fits reamer has taper of 1 ⁄8 inch per foot.

Stub Screw Machine Reamers—Helical Flutes ANSI/ASME B94.2-1995 (R2015)

D H 3 8

Series No. 00

0 1 2 3 4 5 6 7 8 9

10 11

Diameter Range

.0600-.066

.0661-.074 .0741-.084 .0841-.096 .0961-.126 .1261-.158 .1581-.188 .1881-.219 .2191-.251 .2511-.282

.2821-.313 .3131-.344 .3441-.376

Dia. of Length Length Over­all of Flute Shank A

13⁄4 13⁄4 13⁄4 13⁄4 2

21 ⁄4 21 ⁄4 21 ⁄4 21 ⁄4

21 ⁄4 21 ⁄4

21 ⁄2 21 ⁄2

B

D

A Size of Hole H

1 ⁄2

1 ⁄8

1 ⁄16

1 ⁄2

1 ⁄8

1 ⁄16

1 ⁄2

1 ⁄8

1 ⁄16

1 ⁄2

1 ⁄8

1 ⁄16

3 ⁄4

1 ⁄8

1 ⁄16

1 ⁄4

3 ⁄32

1 ⁄4

3 ⁄32

1 ⁄4

3 ⁄32

1 ⁄4

3 ⁄32

3 ⁄8

1 ⁄8

3 ⁄8

1 ⁄8

3 ⁄8

1 ⁄8

3 ⁄8

1 ⁄8

1 1 1 1 1 1

11 ⁄4 11 ⁄4

Flute No. 4 4 4 4 4 4 4 6 6 6 6 6 6

Series No. 12 13 14 15 16 17 18 19 20 21 22 23 …

B

Diameter Range

.3761- .407 .4071- .439 .4391- .470 .4701- .505 .5051- .567 .5671- .630 .6301- .692 .6921- .755 .7551- .817 .8171- .880 .8801- .942

.9421-1.010 …

Dia. of Length Length Over­all of Flute Shank

Size of Hole

11 ⁄4

1 ⁄2

3 ⁄16

1 ⁄2

3 ⁄16

1 ⁄2

3 ⁄16

1 ⁄2

3 ⁄16

5 ⁄8

1 ⁄4

5 ⁄8

1 ⁄4

5 ⁄8

1 ⁄4

3 ⁄4

5 ⁄16

3 ⁄4

5 ⁄16

3 ⁄4

5 ⁄16

3 ⁄4

5 ⁄16

3 ⁄4

5 ⁄16

A

21 ⁄2 21 ⁄2 21 ⁄2 21 ⁄2 3 3

B

11 ⁄4 11 ⁄4 11 ⁄4

11 ⁄2 11 ⁄2

3

11 ⁄2

3

11 ⁄2

3

11 ⁄2

3

11 ⁄2

3

3

…

11 ⁄2 11 ⁄2 …

D

…

H

…

Flute No. 6 6 6 6 6 6 6 8 8 8 8 8

…

All dimensions in inches. Material is high-speed steel.

These reamers are standard with right-hand cut and left-hand helical flutes within the size ranges shown.

Tolerances: On diameter of reamer, for sizes 00 to 7, incl., plus .0001 to plus .0004 inch and for sizes 8 to 23, incl., plus .0001 to plus .0005 inch. On overall length A, plus or minus 1 ⁄16 inch. On length of flute B, plus or minus 1 ⁄16 inch. On diameter of shank D, minus .0005 to minus .002 inch.

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Machinery's Handbook, 31st Edition Reamers

928

American National Standard Morse Taper Finishing Reamers ANSI/ASME B94.2-1995 (R2015) Chamber Optional D C

B

A

B

A

Taper No.a 0

Small End Dia. (Ref.) 0.2503

Large End Dia. (Ref.) 0.3674

2

0.5696

0.7444

1 3 4 5

Taper No.a 0

1

2

3 4 5

0.3674 0.7748 1.0167 1.4717

0.5170 0.9881 1.2893 1.8005

Straight Flutes and Squared Shank Length Flute Square Overall Length Length A B C 33⁄4 5 6

71 ⁄4

81 ⁄2 93⁄4

21 ⁄4 3

31 ⁄2 41 ⁄4 51 ⁄4 61 ⁄4

Straight and Spiral Flutes and Taper Shank Length Flute Large Small Overall Length End Dia. End Dia. A B (Ref.) (Ref.) 0.2503 0.3674 21 ⁄4 511 ⁄32 3 0.3674 0.5170 65⁄16 0.5696

0.7444

73⁄8

1.0167

1.2893

107⁄8

0.7748 1.4717

0.9881 1.8005

87⁄8

131 ⁄8

31 ⁄2 51 ⁄4

41 ⁄4 61 ⁄4

5 ⁄16

Shank Dia. D 5 ⁄16

Square Size 0.235 0.330

7⁄16

7⁄16

5 ⁄8

5 ⁄8

7⁄8

7⁄8

0.655

11 ⁄2

1.125

1

11 ⁄8

11 ⁄8 Taper Shank No.a 0

1

2

0.470 0.845

Squared and Taper Shank Number of Flutes 4 to 6 incl. 6 to 8 incl.

6 to 8 incl.

3

8 to 10 incl.

5

10 to 12 incl.

4

8 to 10 incl.

a Morse. For amount of taper see Table on page 1013.

All dimension are given in inches. Material is high-speed steel. The chamfer on the cutting end of the reamer is optional. Squared shank reamers are standard with straight flutes. Tapered shank ream­ers are standard with straight or spiral flutes. Spiral flute reamers are standard with left-hand spiral flutes. Tolerances: On overall length A and flute length B, in taper numbers 0 to 3, incl., ±1 ⁄16 inch, in taper numbers 4 and 5, ±3⁄32 inch. On length of square C, in taper numbers 0 to 3, incl., ±1 ⁄32 inch; in taper numbers 4 and 5, ±1 ⁄16 inch. On shank diameter D, -.0005 to -.002 inch. On size of square, in taper numbers 0 and 1, -.004 inch; in taper numbers 2 and 3, -.006 inch; in taper numbers 4 and 5, -.008 inch.

Center Reamers.—A “center reamer” is a reamer the teeth of which meet in a point. By their use, small conical holes may be reamed in the ends of parts to be machined as on lathe centers. When large holes—usually cored—must be center-reamed, a large reamer is ordi­narily used in which the teeth do not meet in a point, the reamer forming the frustum of a cone. Center reamers for such work are called “bull” or “pipe” center reamers. Bull Center Reamer: A conical reamer used for reaming the ends of large holes—usually cored—so that they will fit on a lathe center. The cutting part of the reamer is generally in the shape of a frustum of a cone. It is also known as a pipe center reamer.

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Machinery's Handbook, 31st Edition Reamers

929

Taper Pipe Reamers—Spiral Flutes ANSI/ASME B94.2-1995 (R2015) D C Nom. Size 1 ⁄8 1 ⁄4

3 ⁄8

1 ⁄2 3 ⁄4

1

11 ⁄4 11 ⁄2 2

Diameter Small Large End End 0.362 0.316 0.472 0.406 0.606 0.540 0.751 0.665 0.962 0.876 1.212 1.103 1.553 1.444 1.793 1.684 2.268 2.159

Length Overall A 21 ⁄8 27⁄16 29⁄16 31 ⁄8 31 ⁄4 33⁄4 4

Flute Length B 3 ⁄4 11 ⁄16 11 ⁄16 13⁄8 13⁄8 13⁄4 13⁄4 13⁄4 13⁄4

41 ⁄4 41 ⁄2

B

A Square Length C 3 ⁄8 7⁄16 1 ⁄2 5 ⁄8 11 ⁄16 13 ⁄16 15 ⁄16 1 11 ⁄8

Shank Diameter D 0.4375 0.5625 0.7000 0.6875 0.9063 1.1250 1.3125 1.5000 1.8750

Size of Square 0.328 0.421 0.531 0.515 0.679 0.843 0.984 1.125 1.406

No. of Flutes 4 to 6 4 to 6 4 to 6 4 to 6 6 to 10 6 to 10 6 to 10 6 to 10 8 to 12

All dimensions are given in inches. These reamers are tapered 3⁄4 inch per foot and are intended for reaming holes to be tapped with American National Standard Taper Pipe Thread taps. Material is high-speed steel. Reamers are standard with left-hand spiral flutes. Tolerances: On length overall A and flute length B, 1 ⁄8 - to 3⁄4 -inch size, incl., ±1 ⁄16 inch; 1- to 11 ⁄2 -inch size, incl., ±3⁄32 inch; 2-inch size, ±1 ⁄8 inch. On length of square C, 1 ⁄8 - to 3⁄4 -inch size, incl., ±1 ⁄32 inch; 1- to 2-inch size, incl., ±1 ⁄16 inch. On shank diameter D, 1 ⁄8 -inch size, - .0015 inch; 1 ⁄4 - to 1-inch size, incl., - .002 inch; 11 ⁄4 - to 2-inch size, incl., - .003 inch. On size of square, 1 ⁄8 -inch size, - .004 inch; 1 ⁄4 - to 3⁄4 -inch size, incl., - .006 inch; 1- to 2-inch size, incl., - .008 inch.

B & S Taper Reamers—Straight and Spiral Flutes, Squared Shank Taper No.a 1 2 3 4 5 6 7 8 9 10

Dia., Small End

0.1974 0.2474 0.3099 0.3474 0.4474 0.4974 0.5974 0.7474 0.8974 1.0420

Dia., Large End

0.3176 0.3781 0.4510 0.5017 0.6145 0.6808 0.8011 0.9770 1.1530 1.3376

Overall Length 43⁄4 51 ⁄8 51 ⁄2 57⁄8 63⁄8 67⁄8 71 ⁄2 81 ⁄8 87⁄8 93⁄4

Square Length 1 ⁄4

5 ⁄16 3 ⁄8

7⁄16 1 ⁄2

5 ⁄8

3 ⁄4

13 ⁄16

1

7⁄8

Flute Length 27⁄8 31 ⁄8 33⁄8 311 ⁄16 4 43⁄8 47⁄8 51 ⁄2 61 ⁄8 67⁄8

Dia. of Shank 9 ⁄32

11 ⁄32

13 ⁄32 7⁄16

9 ⁄16

5 ⁄8

3 ⁄4

13 ⁄16 1 11 ⁄8

Size of Square 0.210 0.255 0.305 0.330 0.420 0.470 0.560 0.610 0.750 0.845

No. of Flutes 4 to 6 4 to 6 4 to 6 4 to 6 4 to 6 4 to 6 6 to 8 6 to 8 6 to 8 6 to 8

a For taper per foot, see Table 10 on page 1023.

These reamers are no longer ANSI/ASME Standard. All dimensions are given in inches. Material is high-speed steel. The chamfer on the cutting end of the reamer is optional. All reamers are finishing reamers. Spiral flute reamers are standard with left-hand spiral flutes. (Tapered reamers, especially those with left-hand spirals, should not have circular lands because cutting must take place on the outer diameter of the tool.) B & S taper reamers are designed for use in reaming out Brown & Sharpe standard taper sockets. Tolerances: On length overall A and flute length B, taper nos. 1 to 7, incl., ±1 ⁄16 inch; taper nos. 8 to 10, incl., ±3⁄32 inch. On length of square C, taper nos. 1 to 9, incl., ±1 ⁄32 inch; taper no. 10, ±1 ⁄16 inch. On shank diameter D, - .0005 to - .002 inch. On size of square, taper nos. 1 to 3, incl., - .004 inch; taper nos. 4 to 9, incl., - .006 inch; taper no. 10, - .008 inch.

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Machinery's Handbook, 31st Edition Reamers

930

American National Standard Die-Makers’ Reamers ANSI/ASME B94.2-1995 (R2015) A Letter Size AAA AA A B C D E F

Diameter Small Large End End 0.055 0.070 0.065 0.080 0.075 0.090 0.085 0.103 0.095 0.113 0.105 0.126 0.115 0.136 0.125 0.148

Length A

B

Letter Size

21 ⁄4 21 ⁄4 21 ⁄4 23⁄8 21 ⁄2 25⁄8 23⁄4 3

11 ⁄8 11 ⁄8 11 ⁄8 13⁄8 13⁄8 15⁄8 15⁄8 13⁄4

G H I J K L M N

B

Diameter Small Large End End 0.135 0.158 0.145 0.169 0.160 0.184 0.175 0.199 0.190 0.219 0.205 0.234 0.220 0.252 0.235 0.274

Length A 3

31 ⁄4 31 ⁄4 31 ⁄4 31 ⁄2 31 ⁄2 4 41 ⁄2

B

Letter Size

13⁄4 17⁄8 17⁄8 17⁄8 21 ⁄4 21 ⁄4 21 ⁄2 3

O P Q R S T U …

Diameter Small Large End End 0.250 0.296 0.275 0.327 0.300 0.358 0.335 0.397 0.370 0.435 0.405 0.473 0.440 0.511 … …

Length A 5

51 ⁄2 6 61 ⁄2 63⁄4 7 71 ⁄4 …

B 31 ⁄2 4 41 ⁄2 43⁄4 5 51 ⁄4 51 ⁄2 …

All dimensions in inches. Material is high-speed steel. These reamers are designed for use in die­making, have a taper of 3⁄4 degree included angle or 0.013 inch per inch, and have 2 or 3 flutes. Ream­ers are standard with left-hand spiral flutes. Tip of reamer may have conical end. Tolerances: On length overall A and flute length B, ±1 ⁄16 inch.

Taper Pin Reamers — Straight and Left-Hand Spiral Flutes, Squared Shank; and Left-Hand High-Spiral Flutes, Round Shank ANSI/ASME B94.2-1995 (R2015) D C

No. of Taper Pin Reamer 8 ⁄ 0b 7 ⁄ 0 6 ⁄ 0 5 ⁄ 0 4 ⁄ 0 3 ⁄ 0 2 ⁄ 0 0 1 2 3 4 5 6 7 8 9 10

Diameter at Large End of Reamer (Ref.) 0.0514 0.0666 0.0806 0.0966 0.1142 0.1302 0.1462 0.1638 0.1798 0.2008 0.2294 0.2604 0.2994 0.3540 0.4220 0.5050 0.6066 0.7216

Diameter at Small End of Reamer (Ref.) 0.0351 0.0497 0.0611 0.0719 0.0869 0.1029 0.1137 0.1287 0.1447 0.1605 0.1813 0.2071 0.2409 0.2773 0.3297 0.3971 0.4805 0.5799

A

Overall Length of Reamer A 15⁄8 13 1 ⁄16 115⁄16 23⁄16 25⁄16 25⁄16 29⁄16 215⁄16 215⁄16 33⁄16 311 ⁄16 41 ⁄16 45⁄16 57⁄16 65⁄16 73⁄16 85⁄16 95⁄16

a Not applicable to high-spiral flute reamers.

B

Length of Flute B 25 ⁄32 13 ⁄16 15 ⁄16 13⁄16 15⁄16 15⁄16 19⁄16 111 ⁄16 111 ⁄16 115⁄16 25⁄16 29⁄16 213⁄16 311 ⁄16 47⁄16 53⁄16 61 ⁄16 613⁄16

Length of Square Ca … 5 ⁄32

5 ⁄32 5 ⁄32

5 ⁄32 5 ⁄32

7⁄32 7⁄32 7⁄32 1 ⁄4 1 ⁄4

1 ⁄4

5 ⁄16

3 ⁄8 3 ⁄8

7⁄16

9 ⁄16 5 ⁄8

Diameter of Shank D 1 ⁄16 5 ⁄64 3 ⁄32 7⁄64 1 ⁄8 9 ⁄64 5 ⁄32 11 ⁄64 3 ⁄16 13 ⁄64 15 ⁄64 17⁄64 5 ⁄16 23 ⁄64 13 ⁄32 7⁄16 9 ⁄16 5 ⁄8

Size of Squarea … 0.060 0.070 0.080 0.095 0.105 0.115 0.130 0.140 0.150 0.175 0.200 0.235 0.270 0.305 0.330 0.420 0.470

b Not applicable to straight and left-hand spiral-fluted, squared-shank reamers.

All dimensions in inches. Reamers have a taper of 1 ⁄4 inch per foot and are made of high-speed steel. Straight flute reamers of carbon steel are also standard. The number of flutes is as follows; 3 or 4, for 7 ⁄ 0 to 4 ⁄ 0 sizes; 4 to 6, for 3 ⁄ 0 to 0 sizes; 5 or 6, for 1 to 5 sizes; 6 to 8, for 6 to 9 sizes; 7 or 8, for the 10 size in the case of straight- and spiral-flute reamers; and 2 or 3, for 8 ⁄ 0 to 8 sizes; 2 to 4, for the 9 and 10 sizes in the case of high-spiral flute reamers. Tolerances: On length overall A and flute length B, ±1 ⁄16 inch. On length of square C, ±1 ⁄32 inch. On shank diameter D, -.001 to -.005 inch for straight- and spiral-flute reamers and -.0005 to -.002 inch for high-spiral flute reamers. On size of square, -.004 inch for 7 ⁄ 0 to 7 sizes and -.006 inch for 8 to 10 sizes.

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Machinery's Handbook, 31st Edition Twist Drills

931

TWIST DRILLS AND COUNTERBORES Twist drills are rotary end-cutting tools having one or more cutting lips and one or more straight or helical flutes for the passage of chips and cutting fluids. Twist drills are made with straight or tapered shanks, but most have straight shanks. All but the smaller sizes are ground with “back taper,” reducing the diameter from the point toward the shank, to pre­vent binding in the hole when the drill is worn. Straight-Shank Drills: Straight-shank drills have cylindrical shanks that may be of the same or of a different diameter than the body diameter of the drill and may be made with or without driving flats, tang, or grooves. Taper-Shank Drills: Taper-shank drills are preferable to the straight-shank type for drill­ing medium- and large-size holes. The taper on the shank conforms to one of the tapers in the American Standard (Morse) Series. American National Standard.—American National Standard B94.11M-1993 covers nomenclature, definitions, sizes and tolerances for High-Speed Steel Straight- and TaperShank Drills and Combined Drills and Countersinks, Plain and Bell types. It covers both inch and metric sizes. Dimensional tables from the Standard will be found on the following pages. Definitions of Twist Drill Terms.—The following definitions are included in the Stan­dard. Axis: The imaginary straight line that forms the longitudinal center of the drill. Back Taper: A slight decrease in diameter from point to back in the body of the drill. Body: The portion of the drill extending from the shank or neck to the outer corners of the cutting lips. Body Diameter Clearance: That portion of the land that has been cut away so it will not rub against the wall of the hole. Chisel Edge: The edge at the ends of the web that connects the cutting lips. Chisel Edge Angle: The angle included between the chisel edge and the cutting lip as viewed from the end of the drill. Clearance Diameter: The diameter over the cutaway portion of the drill lands. Drill Diameter: The diameter over the margins of the drill measured at the point. Flutes: Helical or straight grooves cut or formed in the body of the drill to provide cut­ting lips, to permit removal of chips, and to allow cutting fluid to reach the cutting lips. Helix Angle: The angle made by the leading edge of the land with a plane containing the axis of the drill. Land: The peripheral portion of the drill body between adjacent flutes. Land Width: The distance between the leading edge and the heel of the land measured at a right angle to the leading edge. Lips—Two-Flute Drill: The cutting edges extending from the chisel edge to the periph­ery. Lips—Three- or Four-Flute Drill (Core Drill): The cutting edges extending from the bot­ tom of the chamfer to the periphery. Lip Relief: The axial relief on the drill point. Lip Relief Angle: The axial relief angle at the outer corner of the lip. It is measured by projection into a plane tangent to the periphery at the outer corner of the lip. (Lip relief angle is usually measured across the margin of the twist drill.) Margin: The cylindrical portion of the land that is not cut away to provide clearance. Neck: The section of reduced diameter between the body and the shank of a drill. Overall Length: The length from the extreme end of the shank to the outer corners of the cutting lips. It does not include the conical-shank end often used on straight-shank drills, nor does it include the conical cutting point used on both straight- and taper-shank drills. (For core drills with an external center on the cutting end, the overall length is the same as for two-flute drills. For core drills with an internal center on the cutting end, the overall length is to the extreme ends of the tool.)

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Machinery's Handbook, 31st Edition Twist Drills

932

Point: The cutting end of a drill made up of the ends of the lands, the web, and the lips. In form, it resembles a cone but departs from a true cone to furnish clearance behind the cut­ting lips. Point Angle: The angle included between the lips projected upon a plane parallel to the drill axis and parallel to the cutting lips. Shank: The part of the drill by which it is held and driven. Tang: The flattened end of a taper shank, intended to fit into a driving slot in the socket. Tang Drive: Two opposite parallel driving flats on the end of a straight shank. Web: The central portion of the body that joins the end of the lands. The end of the web forms the chisel edge on a two-flute drill. Web Thickness: The thickness of the web at the point unless another specific location is indicated. Web Thinning: The operation of reducing the web thickness at the point to reduce drill­ ing thrust. Neck Diam. Tang

Point Angle

Taper Shank

Axis

Neck Length

Shank Diam.

Shank Length

Straight Shank

Shank Length

Straight Shank

Lip Relief Angle Rake or Helix Angle

Flutes Flute Length Body Length Overall Length

Drill Diam.

Clearance Diam. Body Diam. Clearance Chisel Edge Angle

Margin Lip Web Chisel Edge

Flute Length

Land

ANSI Standard Twist Drill Nomenclature

Types of Drills.—Drills may be classified based on the type of shank, number of flutes or hand of cut. Straight-Shank Drills: Those having cylindrical shanks that may be the same or a differ­ ent diameter than the body of the drill. The shank may be with or without driving flats, tang, grooves, or threads. Taper-Shank Drills: Those having conical shanks suitable for direct fitting into tapered holes in machine spindles, driving sleeves, or sockets. Tapered shanks generally have a driving tang. Two-Flute Drills: The conventional type of drill used for originating holes. Three-Flute Drills (Core Drills): Drills commonly used for enlarging and finishing drilled, cast or punched holes. They will not produce original holes. Four-Flute Drills (Core Drills): Used interchangeably with three-flute drills. They are of similar construction except for the number of flutes. Right-Hand Cut: When viewed from the cutting point, the counterclockwise rotation of a drill in order to cut. Left-Hand Cut: When viewed from the cutting point, the clockwise rotation of a drill in order to cut. Teat Drill: The cutting edges of a teat drill are at right angles to the axis, and in the center there is a small teat of pyramid shape which leads the drill and holds it in position. This form is used for squaring the bottoms of holes made by ordinary twist drills or for drilling the entire hole, especially if it is not very deep and a square bottom is required. For instance, when drilling holes to form clearance spaces at the end of a keyseat, preparatory to cutting it out by planing or chipping, the teat drill is commonly used.

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Machinery's Handbook, 31st Edition Twist Drills L

933

F

D

Conical Point Optional with Manufacturer

Table 1. ANSI Straight-Shank Twist Drills — Jobbers Length through 17.5 mm, Taper Length through 12.7 mm, and Screw Machine Length through 25.4 mm Diameter ANSI/ASME B94.11M-1993

Frac­ tion No. or Ltr. 97

96 95 94 93 92 91 90 89 88 87 86 85

84 83 82 81 80 79

1 ⁄64

78

77

76 75 74

Drill Diameter, Da

Jobbers Length

Equivalent

mm 0.15

0.16 0.17 0.18 0.19 0.20 0.22

0.25

0.28 0.30

0.32

0.35 0.38 0.40 0.42 0.45 0.48 0.50

0.55 0.60

Decimal Inch 0.0059

0.0063 0.0067 0.0071 0.0075 0.0079

0.0083

0.0087

0.0091 0.0095

0.0098

0.0100 0.0105 0.0110

0.0115

0.0118

0.0120 0.0125

0.0126

0.0130 0.0135

0.0138

0.0145

0.0150

0.0156

0.0157

0.0160

0.0165 0.0177

0.0180

0.0189 0.0197

0.0200 0.0210

0.0217 0.0225 0.0236

mm 0.150

0.160 0.170 0.180 0.190 0.200

0.211

0.221

0.231 0.241

0.250

0.254 0.267 0.280

0.292

0.300

0.305 0.318

0.320

0.330 0.343

0.350

0.368

0.380

0.396

0.400

0.406

0.420 0.450

0.457

0.480 0.500

0.508 0.533

0.550 0.572 0.600

Flute Inch 1 ⁄16 1 ⁄16 1 ⁄16 1 ⁄16 1 ⁄16 1 ⁄16 5 ⁄64 5 ⁄64 5 ⁄64 5 ⁄64 5 ⁄64 5 ⁄64 3 ⁄32 3 ⁄32 3 ⁄32 3 ⁄32 3 ⁄32 3 ⁄32 3 ⁄32 3 ⁄32 1 ⁄8 1 ⁄8 1 ⁄8 3 ⁄16 3 ⁄16 3 ⁄16 3 ⁄16 3 ⁄16 3 ⁄16 3 ⁄16 3 ⁄16 3 ⁄16 3 ⁄16 1 ⁄4 1 ⁄4 1 ⁄4 5 ⁄16

F

Overall

mm

Inch

1.6

3 ⁄4

1.6 1.6 1.6 1.6 1.6 2.0 2.0 2.0 2.0

2.0 2.0 2.4 2.4 2.4 2.4 2.4 2.4 2.4 2.4 3 3 3 5 5 5 5 5 5 5

5

5 5

3 ⁄4 3 ⁄4 3 ⁄4 3 ⁄4 3 ⁄4 3 ⁄4 3 ⁄4 3 ⁄4 3 ⁄4 3 ⁄4 3 ⁄4 3 ⁄4 3 ⁄4 3 ⁄4 3 ⁄4 3 ⁄4 3 ⁄4 3 ⁄4 3 ⁄4 3 ⁄4 3 ⁄4 3 ⁄4 3 ⁄4 3 ⁄4 3 ⁄4 7⁄8 7⁄8 7⁄8 7⁄8 7⁄8 7⁄8

Inch

19 19

19 19 19 19 19 19 19 19 19 19 19

11 ⁄8

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Inch

…

…

…

…

… … … … … … … … … … …

… … … … … … … … … … …

Flute

mm

Inch

…

…

…

…

… … … … … … … … … … …

… … … … … … … … … … …

F

Overall L

mm

Inch

…

…

…

…

…

…

…

…

… … … … … … … … … … …

… … … … … … … … … … …

… … … … … … … … … … …

mm … … … … … … … … … … …

…

…

…

…

…

…

…

19

…

…

…

…

…

…

…

…

19 19 19 19 19 19 19 19 19 19 19 22 22 22 22 22

… … … … … … … … … … … … … … … …

… … … … … … … … … … … … … … … …

… … … … … … … … … … … … … … … …

… … … … … … … … … … … … … … … …

… … … … … … … … … … … … … … … …

… … … … … … … … … … … … … … … …

… … … … … … … … … … … … … … … …

… … … … … … … … … … … … … … … …

22

…

…

…

…

…

…

…

…

25

…

…

…

…

…

…

…

…

25

1

mm

L

Screw Machine Length

…

1

6

F

Overall

19

22

1

8

mm

7⁄8

6 6

L

Taper Length

Flute

25

29

…

… …

…

…

… …

…

…

… …

…

…

… …

…

…

… …

…

…

… …

…

…

… …

…

…

… …

…

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Machinery's Handbook, 31st Edition Twist Drills

934

Table Table 1. (Continued) Straight-Shank — Jobbers Length through 1. ANSI ANSI Straight-Shank Twist Twist Drills Drills — Jobbers Length through 17.5 mm, Taper Length through 12.7 mm, and Screw Machine Length through 25.4 mm Diameter ANSI/ASME B94.11M-1993 Frac­ tion No. or Ltr.

Drill Diameter, Da

mm

73

72 71 70 69

68 1 ⁄32

67 66 65 64 63 62 61 60 59 58 57

56 3 ⁄64

54

0.65 0.70

0.75

0.80

0.85 0.90

0.95

1.00

1.05

1.10 1.15

1.20 1.30 1.35 1.40 1.45

53 1 ⁄16

52

Decimal Inch 0.0240

1.25 55

Jobbers Length

Equivalent

1.50 1.55 1.60 1.65

0.0250

0.0256

0.0260

0.0276

0.0280 0.0292

0.0295

0.0310

0.0312

0.0315

0.0320 0.0330

0.0335

0.0350

0.0354

0.0360 0.0370

0.0374

0.0380 0.0390

0.0394

0.0400 0.0410

0.0413

0.0420 0.0430

0.0433 0.0453

0.0465

0.0469

0.0472 0.0492 0.0512

0.0520

0.0531

0.0550

0.0551 0.0571 0.0591

0.0595

0.0610

0.0625

0.0630

0.0635

0.0650

mm 0.610

0.635

0.650

0.660

0.700

0.711

0.742

0.750

0.787

0.792

0.800

0.813 0.838

0.850

0.889

0.899

0.914 0.940

0.950

0.965 0.991

1.000

1.016 1.041

1.050

1.067 1.092

1.100 1.150

1.181

1.191

1.200 1.250 1.300

1.321

1.350

1.397

1.400 1.450 1.500 1.511

1.550

1.588

1.600

1.613

1.650

Flute Inch 5 ⁄16 5 ⁄16 3 ⁄8 3 ⁄8 3 ⁄8 3 ⁄8 1 ⁄2 1 ⁄2 1 ⁄2 1 ⁄2 1 ⁄2 1 ⁄2 1 ⁄2 5 ⁄8 5 ⁄8 5 ⁄8 5 ⁄8 5 ⁄8 5 ⁄8 5 ⁄8 11 ⁄16 11 ⁄16 11 ⁄16 11 ⁄16 11 ⁄16 11 ⁄16 3 ⁄4 3 ⁄4 3 ⁄4 3 ⁄4 3 ⁄4 7⁄8 7⁄8 7⁄8 7⁄8 7⁄8 7⁄8 7⁄8 7⁄8 7⁄8 7⁄8 7⁄8 7⁄8 7⁄8 7⁄8

1

F

Taper Length

Overall

mm

Inch

8 10

8

10 10 10

L

Flute

mm

Inch

11 ⁄8

29

11 ⁄4

32

11 ⁄8 11 ⁄4 11 ⁄4 11 ⁄4

29 32 32 32

F

mm

Inch

…

…

…

…

… … … …

… … … …

Screw Machine Length

Overall L

Flute

mm

Inch

…

…

…

…

… … … …

… … … …

F

Overall L

mm

Inch

…

…

…

…

…

…

…

…

… … … …

… … … …

… … … …

mm … … … …

13

13⁄8

35

…

…

…

…

…

…

…

…

13

13⁄8

35

…

…

…

…

…

…

…

…

13 13 13 13 13

13⁄8 13⁄8 13⁄8 13⁄8 13⁄8

16

11 ⁄2

16

11 ⁄2

16 16 16 16 16 17

11 ⁄2 11 ⁄2 11 ⁄2 11 ⁄2 11 ⁄2 15⁄8

35 35 35 35 35 38 38 38 38 38 38 38 41

… … … … … … … … … … … … …

… … … … … … … … … … … … …

… … … … … … … … … … … … …

… … … … … … … … … … … … …

17

15⁄8

41

11 ⁄8

29

21 ⁄4

57

17

15⁄8

41

11 ⁄8

29

21 ⁄4

57

17 17 17 19 19 19 19 19 22 22 22 22 22 22 22 22 22 22 22 22 22 22

25

15⁄8 15⁄8 15⁄8 13⁄4 13⁄4 13⁄4 13⁄4 13⁄4

17⁄8 17⁄8 17⁄8 17⁄8 17⁄8 17⁄8 17⁄8 17⁄8 17⁄8 17⁄8 17⁄8 17⁄8 17⁄8 17⁄8 2

Copyright 2020, Industrial Press, Inc.

41 41 41 44 44 44 44 44 48 48 48 48 48 48 48 48 48 48 48 48 48 48

51

11 ⁄8 11 ⁄8 11 ⁄8 11 ⁄8 11 ⁄8 11 ⁄8 11 ⁄8 11 ⁄8 13⁄4 13⁄4 13⁄4 13⁄4 13⁄4 13⁄4 13⁄4 13⁄4 13⁄4 13⁄4 13⁄4 13⁄4 2 2

2

29 29 29 29 29 29 29 29 44 44 44 44 44 44 44 44 44 44 44 44

51 51

51

21 ⁄4 21 ⁄4 21 ⁄4 21 ⁄4 21 ⁄4 21 ⁄4 21 ⁄4 21 ⁄4 3 3 3 3 3 3 3 3 3 3 3 3

33⁄4 33⁄4 33⁄4

57 57 57 57 57 57 57 57

76 76 76 76 76 76 76 76 76 76 76 76

95 95 95

… … … … … … … … … … … … … 1 ⁄2 1 ⁄2 1 ⁄2 1 ⁄2 1 ⁄2 1 ⁄2 1 ⁄2 1 ⁄2 1 ⁄2 1 ⁄2 5 ⁄8 5 ⁄8 5 ⁄8 5 ⁄8 5 ⁄8 5 ⁄8 5 ⁄8 5 ⁄8 5 ⁄8 5 ⁄8 5 ⁄8 5 ⁄8 11 ⁄16 11 ⁄16 11 ⁄16

… … … … … … … … … … … … …

… … … … … … … … … … … … …

… … … … … … … … … … … … …

13

13⁄8

35

13

13⁄8

35

13 13 13 13 13 13 13 13 16 16 16 16 16 16 16 16 16 16 16 16 17 17 17

13⁄8 13⁄8 13⁄8 13⁄8 13⁄8 13⁄8 13⁄8 13⁄8 15⁄8 15⁄8 15⁄8 15⁄8 15⁄8 15⁄8 15⁄8 15⁄8 15⁄8 15⁄8 15⁄8 15⁄8

111 ⁄16 111 ⁄16 111 ⁄16

35 35 35 35 35 35 35 35 41 41 41 41 41 41 41 41 41 41 41 41 43 43 43

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Machinery's Handbook, 31st Edition Twist Drills

935

Table Table 1. (Continued) Straight-Shank — Jobbers Length through 1. ANSI ANSI Straight-Shank Twist Twist Drills Drills — Jobbers Length through 17.5 mm, Taper Length through 12.7 mm, and Screw Machine Length through 25.4 mm Diameter ANSI/ASME B94.11M-1993 Frac­ tion No. or Ltr. 51 50

49 48

Drill Diameter, Da

mm 1.70 1.75 1.80 1.85 1.90 1.95

5 ⁄64

47

46 45

44

43

42 3 ⁄32

41 40 39 38 37 36

2.00 2.05

2.10 2.15 2.20 2.25 2.30 2.35

2.40 2.46 2.50

2.60 2.70

34 33 32 31

Decimal Inch 0.0669

0.0670

0.0689

0.0700

0.0709 0.0728

0.0730

0.0748

0.0760

0.0768

0.0781

0.0785

0.0787 0.0807

0.0810 0.0820

0.0827 0.0846

0.0860

0.0866 0.0886

0.0890

0.0906 0.0925

0.0935

0.0938

0.0945

0.0960

0.0965

0.0980

0.0984

0.0995 0.1015

0.1024

0.1040

0.1063

0.1065

0.1094

7⁄64

35

Jobbers Length

Equivalent

2.80

2.90 3.00

0.1100

0.1102 0.1110

0.1130

0.1142

0.1160

0.1181

0.1200

mm 1.700

1.702

1.750

1.778

1.800 1.850

1.854

1.900

1.930

1.950

1.984

1.994

2.000 2.050

2.057 2.083

2.100 2.150

2.184

2.200 2.250

2.261

2.300 2.350

2.375

2.383

2.400

2.438

2.450

2.489

2.500

2.527

Flute Inch 1

1

1

1

1 1

1

1

1

1

1

1

1

11 ⁄8 11 ⁄8 11 ⁄8

11 ⁄8 11 ⁄8 11 ⁄8 11 ⁄4 11 ⁄4 11 ⁄4

11 ⁄4 11 ⁄4 11 ⁄4

11 ⁄4

13⁄8 13⁄8 13⁄8 13⁄8 13⁄8 13⁄8

2.578 17⁄16

2.600 17⁄16 2.642 17⁄16

2.700 17⁄16 2.705 17⁄16

2.779

2.794

2.800

2.819

2.870

2.900

2.946

3.000

3.048

11 ⁄2 11 ⁄2

11 ⁄2 11 ⁄2 11 ⁄2 15⁄8 15⁄8

15⁄8 15⁄8

F

mm

Inch

25

2

25

25

25

25 25

25

25

25

25

25

25

25

29 29 29 29 29 29 32 32 32 32 32 32 32 35 35 35 35 35 35 37 37 37 37 37 38 38 38 38 38 41 41 41 41

Taper Length

Overall

2

2

2

2 2

2

2

2

2

2

2

2

21 ⁄8 21 ⁄8 21 ⁄8 21 ⁄8 21 ⁄8 21 ⁄8 21 ⁄4 21 ⁄4 21 ⁄4 21 ⁄4 21 ⁄4 21 ⁄4 21 ⁄4

23⁄8 23⁄8 23⁄8 23⁄8 23⁄8 23⁄8

21 ⁄2 21 ⁄2 21 ⁄2 21 ⁄2 21 ⁄2 25⁄8 25⁄8 25⁄8 25⁄8 25⁄8 23⁄4 23⁄4 23⁄4 23⁄4

Copyright 2020, Industrial Press, Inc.

L

Flute

mm

Inch

51

2

51

51

51

51 51

51

51

51

51

51

51

51

54 54 54 54 54 54 57 57 57 57 57 57 57 60 60 60 60 60 60 64 64 64 64 64 67 67 67 67 67 70 70 70 70

2

2

2

2 2

2

2

2

2

2

21 ⁄4

21 ⁄4

21 ⁄4 21 ⁄4 21 ⁄4 21 ⁄4 21 ⁄4 21 ⁄4 21 ⁄4 21 ⁄4 21 ⁄4 21 ⁄4 21 ⁄4 21 ⁄4 21 ⁄4

21 ⁄2 21 ⁄2 21 ⁄2 21 ⁄2 21 ⁄2 21 ⁄2 21 ⁄2 21 ⁄2 21 ⁄2 21 ⁄2 21 ⁄2 21 ⁄2 23⁄4 23⁄4 23⁄4 23⁄4 23⁄4 23⁄4 23⁄4 23⁄4

F

mm

Inch

51

33⁄4

51

51

51

51 51

51

51

51

51

51

57 57 57 57 57 57 57 57 57 57 57 57 57 57 57 64 64 64 64 64 64 64 64 64 64 64 64 70 70 70 70 70 70 70 70

Screw Machine Length

Overall

33⁄4

33⁄4 33⁄4

33⁄4 33⁄4 33⁄4

33⁄4 33⁄4

33⁄4

33⁄4

41 ⁄4 41 ⁄4 41 ⁄4 41 ⁄4 41 ⁄4 41 ⁄4 41 ⁄4 41 ⁄4 41 ⁄4 41 ⁄4 41 ⁄4 41 ⁄4 41 ⁄4 41 ⁄4 41 ⁄4

45⁄8 45⁄8 45⁄8 45⁄8 45⁄8 45⁄8 45⁄8 45⁄8 45⁄8 45⁄8 45⁄8 45⁄8

51 ⁄8 51 ⁄8 51 ⁄8 51 ⁄8 51 ⁄8 51 ⁄8 51 ⁄8 51 ⁄8

L

Flute

mm

Inch

95

11 ⁄16

95 95 95 95 95 95 95 95 95 95

108 108 108 108 108 108 108 108 108

108 108 108 108 108 108 117 117 117 117 117 117 117 117 117 117 117 117

130 130 130 130 130 130 130 130

11 ⁄16 11 ⁄16 11 ⁄16 11 ⁄16 11 ⁄16 11 ⁄16 11 ⁄16 11 ⁄16 11 ⁄16 11 ⁄16 11 ⁄16 11 ⁄16 3 ⁄4 3 ⁄4 3 ⁄4 3 ⁄4 3 ⁄4 3 ⁄4 3 ⁄4 3 ⁄4 3 ⁄4 3 ⁄4 3 ⁄4 3 ⁄4 3 ⁄4 13 ⁄16 13 ⁄16 13 ⁄16 13 ⁄16 13 ⁄16 13 ⁄16 13 ⁄16 13 ⁄16 13 ⁄16 13 ⁄16 13 ⁄16 13 ⁄16 7⁄8 7⁄8 7⁄8 7⁄8 7⁄8 7⁄8 7⁄8 7⁄8

F

Overall L

mm

Inch

17

111 ⁄16

43

17

111 ⁄16

43

17 17 17 17 17 17 17 17 17 17 17 19 19 19 19 19 19 19 19 19 19 19 19 19 21 21 21 21 21 21 21 21 21 21 21 21 22 22 22 22 22 22 22 22

111 ⁄16 111 ⁄16 111 ⁄16 111 ⁄16 111 ⁄16 111 ⁄16 111 ⁄16 111 ⁄16 111 ⁄16 111 ⁄16 111 ⁄16 13⁄4 13⁄4 13⁄4 13⁄4 13⁄4 13⁄4 13⁄4 13⁄4 13⁄4 13⁄4 13⁄4 13⁄4 13⁄4

113⁄16 113⁄16 113⁄16 113⁄16 113⁄16 113⁄16 113⁄16 113⁄16 113⁄16 113⁄16 113⁄16 113⁄16 17⁄8 17⁄8 17⁄8 17⁄8 17⁄8 17⁄8 17⁄8 17⁄8

mm 43 43 43 43 43 43 43 43 43 43 43 44 44 44 44 44 44 44 44 44 44 44 44 44 46 46 46 46 46 46 46 46 46 46 46 46 48 48 48 48 48 48 48 48

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Machinery's Handbook, 31st Edition Twist Drills

936

Table Table 1. (Continued) Straight-Shank — Jobbers Length through 1. ANSI ANSI Straight-Shank Twist Twist Drills Drills — Jobbers Length through 17.5 mm, Taper Length through 12.7 mm, and Screw Machine Length through 25.4 mm Diameter ANSI/ASME B94.11M-1993 Frac­ tion No. or Ltr.

Drill Diameter, Da

mm 3.10

1 ⁄8

30

29 28 9 ⁄64

27 26 25 24 23

3.20 3.30 3.40 3.50

3.60 3.70

21 20

19 18

16 15 14 13 3 ⁄16

12 11

10 9 8

0.1220

0.1250

0.1260

0.1285

0.1299 0.1339

0.1360

0.1378

0.1405

mm

Inch

3.100

15⁄8

3.175

3.200

3.264

3.300 3.400

3.454

3.500

3.569

0.1406

3.571

0.1440

3.658

0.1417

15⁄8 15⁄8 15⁄8 13⁄4 13⁄4 13⁄4

13⁄4 13⁄4

13⁄4

3.600

17⁄8

0.1457

3.700

17⁄8

0.1495

3.797

0.1470

3.734

17⁄8 17⁄8 17⁄8

F

mm

Inch

41

23⁄4

41 41 41 44 44 44 44 44 44 48 48 48 48 48 48

23⁄4 23⁄4 23⁄4

27⁄8 27⁄8 27⁄8 27⁄8 27⁄8 27⁄8 3 3 3 3 3 3

3.800

3.90

0.1535

3.900

2

51

31 ⁄8

0.1562

3.967

2

51

31 ⁄8

0.1575

4.000

21 ⁄8

54

31 ⁄4

0.1610

4.089

21 ⁄8

4.00

4.10 4.20 4.30

0.1520 0.1540 0.1570 0.1590

3.861 3.912 3.988 4.039

17⁄8 2 2 2

21 ⁄8

0.1614

4.100

21 ⁄8

0.1660

4.216

21 ⁄8

0.1654

4.200

21 ⁄8

0.1693

4.300

21 ⁄8

0.1719

4.366

21 ⁄8

4.40

0.1732

4.400 23⁄16

4.50

0.1772

4.500 23⁄16

4.60

0.1811

4.600 23⁄16

0.1850

4.700 25⁄16

4.70 4.80 4.90

5.00

0.1695 0.1730 0.1770 0.1800 0.1820

4.305

21 ⁄8

4.394 23⁄16 4.496 23⁄16 4.572 23⁄16 4.623 23⁄16

0.1875

4.762 25⁄16

0.1910

4.851

25⁄16

4.915

27⁄16

0.1890

4.800 25⁄16

0.1929

4.900

0.1960

4.978

27⁄16

5.054

27⁄16

0.1935

0.1969

0.1990

5.000

27⁄16

27⁄16

51 51 51 54 54 54 54 54 54 54 54

56

62 62 62

70 73 73 73 73 73 73

76 76 76 76 76 76

3 3 3 3 3 3 3 3

3 3 3 3 3 3

70

76 76 76 76 76 76 76 76

76 76 76 76 76 76

51 ⁄8

53⁄8 53⁄8 53⁄8 53⁄8 53⁄8 53⁄8 53⁄8 53⁄8 53⁄8 53⁄8 53⁄8 53⁄8 53⁄8 53⁄8

3

76

53⁄8

31 ⁄8 31 ⁄4 31 ⁄4 31 ⁄4 31 ⁄4 31 ⁄4 31 ⁄4 31 ⁄4 31 ⁄4

33⁄8 33⁄8 33⁄8 33⁄8 33⁄8

31 ⁄2

62

70

23⁄4

51 ⁄8

79

59 62

70

70

31 ⁄8

31 ⁄2

59

23⁄4

Inch

53⁄8

59 59

70

mm

76

33⁄8

56

Inch

3

56 56

mm

F

31 ⁄2 31 ⁄2 35⁄8 35⁄8 35⁄8 35⁄8 35⁄8

Copyright 2020, Industrial Press, Inc.

79 79 79 83 83 83 83 83 83 83 83 83 86 86 86 86 86 86 86 89 89 89 89 92 92 92 92 92

3 3

33⁄8 33⁄8 33⁄8 33⁄8 33⁄8 33⁄8 33⁄8 33⁄8 33⁄8 33⁄8 33⁄8 33⁄8 33⁄8 33⁄8 33⁄8 33⁄8 33⁄8 33⁄8 33⁄8 35⁄8 35⁄8 35⁄8 35⁄8 35⁄8 35⁄8 35⁄8

76 76

86 86 86 86 86 86 86 86 86 86 86 86 86 86 86 86 86 86 86 92 92 92 92 92 92 92

Screw Machine Length

Overall

79

33⁄8

56

L

Flute

31 ⁄8

56 56

Taper Length

Overall

0.1496

11 ⁄64

17

Decimal Inch

Flute

3.80

5 ⁄32

22

Jobbers Length

Equivalent

53⁄8 53⁄8 53⁄4 53⁄4 53⁄4 53⁄4 53⁄4 53⁄4 53⁄4 53⁄4 53⁄4 53⁄4 53⁄4 53⁄4 53⁄4 53⁄4 53⁄4 53⁄4 53⁄4 53⁄4 53⁄4

L

Flute

mm

Inch

130

7⁄8

130 137 137 137 137 137 137 137 137

7⁄8 15 ⁄16 15 ⁄16 15 ⁄16 15 ⁄16 15 ⁄16 15 ⁄16 15 ⁄16 15 ⁄16

24

115⁄16

49

24 24 24 24 24 24 24

17⁄8

115⁄16 115⁄16 115⁄16 115⁄16 115⁄16 115⁄16 115⁄16

25

21 ⁄16

137 137 137 137 137 137 137 146 146 146 146 146 146 146 146 146 146 146 146 146 146 146 146 146 146 146

1 1 1 1 1 1 1

11 ⁄16 11 ⁄16 11 ⁄16 11 ⁄16 11 ⁄16 11 ⁄16 11 ⁄16 11 ⁄16 11 ⁄16 11 ⁄16 11 ⁄8 11 ⁄8 11 ⁄8 11 ⁄8 11 ⁄8 11 ⁄8 11 ⁄8 11 ⁄8 11 ⁄8

6

152

6

48

22

1

1

152 152 152 152 152

13⁄16 13⁄16 13⁄16 13⁄16 13⁄16

mm

17⁄8

137

137

L

22

21 ⁄16

13⁄16

6

Inch

25

13⁄16

6

mm

1

152

6

Overall

137

6 6

F

25 25 25 25 25 25 25 25

27 27 27 27 27 27 27 27 27 27 29 29 29 29 29 29 29 29 29 30 30 30 30 30 30 30

21 ⁄16 21 ⁄16 21 ⁄16 21 ⁄16 21 ⁄16 21 ⁄16 21 ⁄16 21 ⁄16 21 ⁄8 21 ⁄8 21 ⁄8 21 ⁄8 21 ⁄8 21 ⁄8 21 ⁄8 21 ⁄8 21 ⁄8 21 ⁄8

23⁄16 23⁄16 23⁄16 23⁄16 23⁄16 23⁄16 23⁄16 23⁄16 23⁄16 21 ⁄4 21 ⁄4 21 ⁄4 21 ⁄4 21 ⁄4 21 ⁄4 21 ⁄4

48 49 49 49 49 49 49 49 52 52 52 52 52 52 52 52 52 52 54 54 54 54 54 54 54 54 54 54 56 56 56 56 56 56 56 56 56 57 57 57 57 57 57 57

ebooks.industrialpress.com

Machinery's Handbook, 31st Edition Twist Drills

937

Table Table 1. (Continued) Straight-Shank — Jobbers Length through 1. ANSI ANSI Straight-Shank Twist Twist Drills Drills — Jobbers Length through 17.5 mm, Taper Length through 12.7 mm, and Screw Machine Length through 25.4 mm Diameter ANSI/ASME B94.11M-1993 Frac­ tion No. or Ltr. 7

Drill Diameter, Da

Equivalent

mm

5 4 3 7⁄32

2 1

A 15 ⁄64

B C D E, 1 ⁄4

F G

I J K 9 ⁄32

Inch

mm

Inch

L

Flute

mm

Inch

F

Overall

mm

Inch

L

Screw Machine Length Flute

mm

Inch

F

Overall

mm

Inch

L

mm

5.100 27⁄16

62

35⁄8

92

35⁄8

92

6

152

13⁄16

30

21 ⁄4

57

0.2031

5.159 27⁄16

62

35⁄8

92

35⁄8

92

6

152

13⁄16

30

21 ⁄4

57

5.20

0.2047

5.200

5.30

0.2087

5.300

5.40

0.2126

5.400

5.50

0.2165

5.500

5.60

0.2010 0.2040 0.2055 0.2090 0.2130

5.105 27⁄16 5.182

21 ⁄2

5.220

21 ⁄2

5.309 5.410

0.2340

5.944

0.2280

0.2323

5.791

5.900

25⁄8 25⁄8 25⁄8 25⁄8 25⁄8 25⁄8

0.2344

5.954

25⁄8

0.2380

6.045

23⁄4

0.2362

6.000

6.10

0.2402

6.100

6.20

0.2441

6.200

6.30

0.2480

6.300

0.2420 0.2460

6.147 6.248

0.2500

6.350

0.2559

6.500

0.2520

23⁄4 23⁄4 23⁄4 23⁄4 23⁄4 23⁄4

23⁄4

6.400

27⁄8

6.528

27⁄8

6.629

27⁄8

0.2656

6.746

27⁄8

0.2677

6.800

27⁄8

0.2720

6.909

0.2570

6.60

0.2598

6.600

6.70

0.2638

6.700

6.90

21 ⁄2

25⁄8

5.800

6.80

21 ⁄2

5.613

5.600

0.2283

6.50

21 ⁄2

0.2210

0.2205

5.80

6.40

21 ⁄2

21 ⁄2

5.700

6.00

21 ⁄2

5.558

0.2244

5.90

21 ⁄2

0.2188

5.70

17⁄64

H

mm

F

Taper Length

Overall

0.2008

5.10

13 ⁄64

6

Decimal Inch

Jobbers Length

Flute

0.2610

0.2660

0.2717

6.756

6.900

27⁄8 27⁄8 27⁄8

27⁄8 27⁄8

27⁄8

7.00

0.2756

7.000

7.10

0.2795

7.100

215⁄16

0.2812

7.142

215⁄16

0.2874

7.300

215⁄16

7.20 7.30

0.2770 0.2810

0.2835

7.036 7.137

7.200

27⁄8

27⁄8

215⁄16 215⁄16

62 64 64 64 64 64 64 64 64 64

35⁄8 33⁄4 33⁄4 33⁄4 33⁄4 33⁄4 33⁄4 33⁄4 33⁄4 33⁄4

67

37⁄8

67

37⁄8

67 67 67 67 67 67 70 70 70 70 70 70 70 70

37⁄8 37⁄8 37⁄8 37⁄8 37⁄8 37⁄8

92 95 95 95 95 95 95 95 95 95 98 98 98 98 98 98 98

35⁄8 35⁄8 35⁄8 35⁄8 35⁄8 35⁄8 35⁄8 35⁄8 35⁄8 35⁄8 33⁄4 33⁄4 33⁄4 33⁄4 33⁄4 33⁄4

…

73 73 73 75 75 75 75 75

92 92 95 95 95 95 95 95

…

6 6 6

61 ⁄8 61 ⁄8 61 ⁄8 61 ⁄8 61 ⁄8 61 ⁄8 …

152 152 152 152

156 156 156 156 156 156 …

11 ⁄4 11 ⁄4 11 ⁄4 11 ⁄4 11 ⁄4

15⁄16 15⁄16 15⁄16 15⁄16 15⁄16 15⁄16 15⁄16

102

…

…

…

…

13⁄8

102

4

102

33⁄4

95

61 ⁄8

156

13⁄8

102

33⁄4

95

61 ⁄8

156

13⁄8

102

33⁄4

95

61 ⁄8

156

13⁄8

98

61 ⁄4

159

17⁄16

4 4 4 4 4 4

102 102 102

33⁄4

… …

33⁄4

…

73

92

6

152

11 ⁄4

4

105

73

92

6

152

11 ⁄4

15⁄16

41 ⁄8

73

92

6

152

11 ⁄4

156

73

73

92

6

152

11 ⁄4

61 ⁄8

37⁄8

73

92

6

152

13⁄16

95

105

73

92

6

152

33⁄4

41 ⁄8

73

92

6

98

73 73

92

41 ⁄8 41 ⁄8 41 ⁄8 41 ⁄8 41 ⁄8 41 ⁄8 41 ⁄8 41 ⁄8 41 ⁄8 41 ⁄8 41 ⁄8 41 ⁄4 41 ⁄4 41 ⁄4 41 ⁄4 41 ⁄4

Copyright 2020, Industrial Press, Inc.

105 105 105 105

37⁄8

… … …

95

… …

95 98

… … … …

61 ⁄8

… …

61 ⁄8 61 ⁄4

… … … …

156

… …

156 159 … … … …

13⁄8

13⁄8 13⁄8

13⁄8

17⁄16 17⁄16 17⁄16 17⁄16 17⁄16

105

37⁄8

98

61 ⁄4

159

17⁄16

105

37⁄8

98

61 ⁄4

159

11 ⁄2

105

…

…

…

…

11 ⁄2

105 105

…

…

… …

…

…

… …

11 ⁄2 11 ⁄2

105

37⁄8

98

61 ⁄4

159

11 ⁄2

108

…

…

…

…

11 ⁄2

105 108

… …

108

37⁄8

108

…

108

4

… …

98

102 …

… …

… …

11 ⁄2 11 ⁄2

61 ⁄4

159

11 ⁄2

…

…

19⁄16

63⁄8

162

19⁄16

30 32 32 32 32 32 32 32 32 32 33 33 33 33 33 33 33 33 35 35 35 35 35 35 35 35 37 37 37 37 37 37 37 38 38 38 38 38 38 38 38 38 40 40

21 ⁄4 23⁄8 23⁄8 23⁄8 23⁄8 23⁄8 23⁄8 23⁄8 23⁄8 23⁄8

27⁄16 27⁄16 27⁄16 27⁄16 27⁄16 27⁄16 27⁄16 27⁄16 21 ⁄2 21 ⁄2 21 ⁄2 21 ⁄2 21 ⁄2 21 ⁄2 21 ⁄2 21 ⁄2 25⁄8 25⁄8 25⁄8 25⁄8 25⁄8 25⁄8 25⁄8

211 ⁄16 211 ⁄16 211 ⁄16

57 60 60 60 60 60 60 60 60 60 62 62 62 62 62 62 62 62 64 64 64 64 64 64 64 64 67 67 67 67 67 67 67 68 68 68

211 ⁄16

68

211 ⁄16

68

211 ⁄16 211 ⁄16 211 ⁄16 211 ⁄16 23⁄4 23⁄4

68 68 68 68 70 70

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Machinery's Handbook, 31st Edition Twist Drills

938

Table Table 1. (Continued) Straight-Shank — Jobbers Length through 1. ANSI ANSI Straight-Shank Twist Twist Drills Drills — Jobbers Length through 17.5 mm, Taper Length through 12.7 mm, and Screw Machine Length through 25.4 mm Diameter ANSI/ASME B94.11M-1993 Frac­ tion No. or Ltr. L M 19 ⁄64

N

Drill Diameter, Da

Equivalent

mm 7.40 7.50 7.60

O

P 21 ⁄64

Q

R 11 ⁄32

S

T 23 ⁄64

U

7.493 31 ⁄16

78

0.2969

7.541 31 ⁄16

0.3020

7.671 31 ⁄16

0.2953 0.2992

7.400 31 ⁄16 7.500 31 ⁄16 7.600 31 ⁄16

7.900 33⁄16

8.20 8.30 8.40 8.50 8.60

0.3071

7.800 33⁄16

0.3125

7.938 33⁄16

0.3160

8.026 33⁄16

0.3150

8.000 33⁄16

0.3189

8.100 35⁄16

0.3230

8.204 35⁄16

0.3228 0.3268

0.3281

0.3307

0.3320

8.200 35⁄16 8.300 35⁄16

8.334 35⁄16 8.400 37⁄16

8.433 37⁄16

0.3346

8.500 37⁄16

0.3390

8.611

0.3386

8.600 37⁄16 37⁄16

8.70

0.3425

8.700 37⁄16

8.80

0.3465

8.800

31 ⁄2

8.90

0.3504

8.900

31 ⁄2

0.3580

9.093

9.00 9.10 9.20 9.30 9.40

0.3438 0.3480

0.3543 0.3583

8.733 37⁄16 8.839

9.000 9.100

31 ⁄2 31 ⁄2 31 ⁄2 31 ⁄2

0.3594

9.129

31 ⁄2

0.3661

9.300

35⁄8

0.3622 0.3680

9.200 9.347

35⁄8 35⁄8

0.3701

9.400

0.3750

9.525

35⁄8

9.60

0.3780

9.600

33⁄4

9.80

0.3858

9.800

9.70

25 ⁄64

0.2950

0.2913

0.3110

8.10

9.90 10.00

Inch

75

0.3740 0.3770

0.3819

9.500 9.576

9.700

35⁄8 35⁄8 35⁄8 33⁄4 33⁄4

0.3860

9.804

33⁄4

0.3906

9.921

33⁄4

0.3898 0.3937

9.900

10.000

33⁄4 33⁄4

Inch

41 ⁄4

108

…

43⁄8

111

43⁄8

78

43⁄8

78 78

L

43⁄8 43⁄8 43⁄8

111 111 111 111 111

81

41 ⁄2

114

81

41 ⁄2

114

81 81 81 81 84 84 84 84 84 87 87 87 87 87 87 87

41 ⁄2 41 ⁄2 41 ⁄2 41 ⁄2 45⁄8 45⁄8 45⁄8 45⁄8 45⁄8

89 92 92 92 92 92 92 92

4

… …

…

…

…

40

23⁄4

70

19⁄16

…

…

…

15⁄8

102 …

…

63⁄8

…

…

…

162 …

…

19⁄16 15⁄8

15⁄8

114

4

102

63⁄8

162

15⁄8

114

4

102

63⁄8

162

15⁄8

114 114 117

…

…

…

…

…

…

…

…

111 ⁄16

111 ⁄16

117

41 ⁄8

105

61 ⁄2

165

111 ⁄16

117

…

…

…

…

111 ⁄16

117 117

…

41 ⁄8

105

61 ⁄2

165

111 ⁄16

165

111 ⁄16

121

…

…

…

…

111 ⁄16

43⁄4 43⁄4 43⁄4 43⁄4

47⁄8 47⁄8 47⁄8 47⁄8 47⁄8 5 5 5 5 5 5 5

121 121

… …

… …

165

111 ⁄16

124

…

…

…

…

13⁄4

124 124

41 ⁄4

…

108 …

…

171 …

13⁄4 13⁄4

41 ⁄4

108

63⁄4

171

13⁄4

124

…

…

…

…

13⁄4

124

…

124

41 ⁄4

127

…

127 127 127

…

108

…

63⁄4

…

171

13⁄4

13⁄4

41 ⁄4

108

63⁄4

171

113⁄16

…

…

…

…

113⁄16

…

… …

… …

… …

113⁄16 113⁄16

127

41 ⁄4

108

63⁄4

171

113⁄16

127

…

…

…

…

17⁄8

127

41 ⁄4

108

130

…

…

Copyright 2020, Industrial Press, Inc.

63⁄4

124

111

51 ⁄8

111 ⁄16

61 ⁄2

130

51 ⁄8

…

111 ⁄16

105

95

51 ⁄8

…

…

111 ⁄16

41 ⁄8

43⁄8

51 ⁄8

…

…

111 ⁄16

121

51 ⁄8

51 ⁄8

…

…

111 ⁄16

61 ⁄2

…

…

…

105

…

…

…

41 ⁄8

121

…

…

130 130 130 130

… …

43⁄8 43⁄8

… …

111 111

63⁄4

… …

7

… …

7 7

171

113⁄16

…

17⁄8

178

17⁄8

…

17⁄8

… …

178 178

41 41 41

213⁄16 213⁄16 213⁄16 213⁄16

215⁄16

111 ⁄16

…

41

23⁄4

43

165

…

40

23⁄4

213⁄16

61 ⁄2

…

40

23⁄4

41

105

…

40

Inch

15⁄8

41 ⁄8

…

95

19⁄16

162

…

95

70

63⁄8

130

95

23⁄4

102

51 ⁄8

95

40

19⁄16

95 95

19⁄16

…

mm

L

…

… …

Overall

Inch

… …

F

mm

121

47⁄8

89

4

Inch

Flute

43⁄4

43⁄4

89 89

…

mm

L

Screw Machine Length

121

47⁄8

89

…

F

Overall

43⁄4

89 89

Flute

mm

78

78

Taper Length

Overall

mm

7.366 215⁄16

7.90 8.00

F

0.2900

7.700 33⁄16

3 ⁄8

W

Inch

0.3031

9.50 V

mm

7.70 7.80

5 ⁄16

Decimal Inch

Jobbers Length

Flute

17⁄8 17⁄8

17⁄8

115⁄16

41 43 43 43 43 43 43 43 43 43 43 43 43 43 44 44 44 44 44 44 44 46 46 46 46 46 46 48 48 48 48 48 48 48 49

213⁄16

215⁄16 215⁄16 215⁄16 215⁄16 215⁄16 215⁄16

mm 70 70 70 71 71 71 71 71 71 75 75 75 75 75 75 75

3

76

3

76

3 3 3 3 3

31 ⁄16 31 ⁄16 31 ⁄16 31 ⁄16 31 ⁄16 31 ⁄16 31 ⁄16 31 ⁄8 31 ⁄8 31 ⁄8 31 ⁄8 31 ⁄8 31 ⁄8 31 ⁄4 31 ⁄4 31 ⁄4 31 ⁄4 31 ⁄4 31 ⁄4 31 ⁄4

35⁄16

76 76 76 76 76

78 78 78 78 78 78 78 79 79 79 79 79 79 83 83 83 83 83 83 83 84

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Machinery's Handbook, 31st Edition Twist Drills

939

Table Table 1. (Continued) Straight-Shank — Jobbers Length through 1. ANSI ANSI Straight-Shank Twist Twist Drills Drills — Jobbers Length through 17.5 mm, Taper Length through 12.7 mm, and Screw Machine Length through 25.4 mm Diameter ANSI/ASME B94.11M-1993 Frac­ tion No. or Ltr. X Y

Drill Diameter, Da

Equivalent

mm 10.20

13 ⁄32

Z

27⁄64

7⁄16

10.50

15 ⁄32

1 ⁄2

17⁄32

37⁄64

39 ⁄64

41 ⁄64

21 ⁄32

10.262

37⁄8

98

37⁄8

0.4062

10.317

37⁄8

0.4134

10.500

37⁄8

0.4130

10.490

37⁄8

98 98 98 98

103

11.20

0.4409

11.200 43⁄16

0.4531

11.509 43⁄16

0.4252

0.4375

0.4528

10.800 41 ⁄16 11.112 41 ⁄16

11.500 43⁄16

11.80

0.4646

11.800 45⁄16

12.00

0.4724

12.000

43⁄8

0.4844

12.304

43⁄8

0.5000

12.700

41 ⁄2

0.5118

13.000

41 ⁄2

12.50 12.80

13.20 13.50

14.00

0.4688

0.4803 0.4921 0.5039

11.908 45⁄16

12.200 12.500 12.800

43⁄8

Inch

51 ⁄8

130

…

51 ⁄4

133

51 ⁄4 51 ⁄4 51 ⁄4 51 ⁄4

53⁄8

103

51 ⁄2

103

51 ⁄2

106 106 106 110 110

51 ⁄2 55⁄8 55⁄8 55⁄8 53⁄4 53⁄4

111

57⁄8

111

57⁄8

111

41 ⁄2

114

41 ⁄2

114

114 114

57⁄8 6 6 6 6

Inch

…

…

Flute

115⁄16

49

35⁄16

84

115⁄16

49

35⁄16

84

43⁄8

111

7

178

115⁄16

133

43⁄8

111

7

178

115⁄16

133

45⁄8

117

71 ⁄4

184

2

51

33⁄8

140

45⁄8

117

71 ⁄4

184

21 ⁄16

52

37⁄16

133 137 140 140 143 143 143 146 146 149 149 149

152 152 152 152

… …

45⁄8 45⁄8 45⁄8

… …

117 117 117

… …

71 ⁄4 71 ⁄4 71 ⁄4

43⁄4

121

71 ⁄2

43⁄4

121

71 ⁄2

43⁄4 43⁄4 43⁄4 43⁄4 43⁄4 43⁄4 43⁄4 43⁄4 … …

121 121 121 121 121 121 121 121 … …

71 ⁄2 71 ⁄2 71 ⁄2 73⁄4 73⁄4 73⁄4 73⁄4 73⁄4 … …

… …

184 184 184 190 190 190 190 190 197 197 197 197 197 … …

2

2

21 ⁄16 21 ⁄16 21 ⁄8 21 ⁄8 21 ⁄8 21 ⁄8 21 ⁄8

23⁄16 23⁄16 23⁄16 21 ⁄4 21 ⁄4

23⁄8 23⁄8

…

…

…

23⁄8

0.5312

13.492 413⁄16

122

65⁄8

168

…

…

…

…

23⁄8

0.5433

13.800 413⁄16

13.500 413⁄16

0.5469

13.891 413⁄16

0.5610

14.250 413⁄16

0.5512

0.5625

14.000 413⁄16

14.288 413⁄16

122 122 122 122 122 122

65⁄8 65⁄8 65⁄8 65⁄8 65⁄8

0.5807

14.750 53⁄16

132

71 ⁄8

0.5938

15.083 53⁄16

132

71 ⁄8

0.6094

15.479 53⁄16

0.6201

15.750 53⁄16

0.5781

0.5906 0.6004

0.6102

14.684 413⁄16 15.000 53⁄16 15.250 53⁄16

15.500 53⁄16

0.6250

15.875

53⁄16

16.25

0.6398

16.250

53⁄16

16.50

0.6496

16.500

53⁄16

0.6299

0.6406

0.6562

16.000

16.271

53⁄16

53⁄16

16.669 53⁄16

122

65⁄8

122

132 132 132 132 132 132 132 132 132 132 132

65⁄8

168 168 168 168 168 168 168 168

… … … … … … … …

… … … … … … … …

… … … … … … … …

… … … … … … … …

23⁄8 23⁄8

21 ⁄2 21 ⁄2 21 ⁄2

181

…

…

…

…

25⁄8

71 ⁄8 71 ⁄8 71 ⁄8 71 ⁄8 71 ⁄8 71 ⁄8 71 ⁄8 71 ⁄8 71 ⁄8 71 ⁄8

Copyright 2020, Industrial Press, Inc.

181 181 181 181 181 181 181 181 181 181

… … … … … … … … … …

… … … … … … … … … …

… … … … … … … … … … …

… … … … … … … … … … …

54 54 54 56 56 56 57 57 60 60 60 60 60 60 64 64 64

39⁄16 39⁄16 35⁄8 35⁄8

311 ⁄16 311 ⁄16 311 ⁄16 33⁄4 33⁄4

37⁄8

87 87 87 90 90 90 92 92 94 94 94 95 95 98

37⁄8

98

37⁄8

98

37⁄8 37⁄8 37⁄8 4 4 4

98 98 98

102 102 102

105

71 ⁄8

…

54

39⁄16

86

41 ⁄8

25⁄8

…

54

37⁄16

86

67

…

181

52

37⁄16

86

25⁄8

21 ⁄2

…

…

52

33⁄8

84

102

…

…

51

33⁄8

4

…

…

51

35⁄16

84

64

168

…

49

35⁄16

mm

21 ⁄2

65⁄8

181

49

Inch

133

…

0.5315

mm

L

…

168

65⁄8

Overall

Inch

65⁄8

122

F

mm

122

13.200 413⁄16

14.75

16.00

mm

L

13.096 413⁄16

0.5197

14.500 413⁄16

15.50

F

Screw Machine Length

Overall

0.5156

0.5709

15.25

L

Taper Length

Flute

mm

14.50

15.75 5 ⁄8

95

11.000 41 ⁄16

15.00 19 ⁄32

0.4040

10.200

33⁄4

0.4331

14.25 9 ⁄16

10.084

0.4016

Inch

11.00

10.80

13.80 35 ⁄64

0.3970

Overall

mm

100

13.00 33 ⁄64

Inch

10.716 315⁄16

12.20 31 ⁄64

mm

F

0.4219

11.50 29 ⁄64

Decimal Inch

Jobbers Length

Flute

25⁄8 25⁄8 23⁄4 23⁄4 23⁄4 23⁄4 23⁄4

27⁄8 27⁄8 27⁄8 27⁄8 27⁄8

64 67 67 67 67 70 70 70 70 70 73 73 73 73

73

4

41 ⁄8 41 ⁄8 41 ⁄8 41 ⁄8 41 ⁄4

102

105 105 105 105 108

41 ⁄4

108

41 ⁄4

108

41 ⁄4 41 ⁄4

41 ⁄2 41 ⁄2

108 108 114 114

41 ⁄2

144

41 ⁄2

114

41 ⁄2

114

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Machinery's Handbook, 31st Edition Twist Drills

940

Table Table 1. (Continued) Straight-Shank — Jobbers Length through 1. ANSI ANSI Straight-Shank Twist Twist Drills Drills — Jobbers Length through 17.5 mm, Taper Length through 12.7 mm, and Screw Machine Length through 25.4 mm Diameter ANSI/ASME B94.11M-1993 Frac­ tion No. or Ltr.

Drill Diameter, Da

Equivalent

mm

Decimal Inch

16.75

0.6594

17.00

0.6693

Jobbers Length

Flute

mm

Inch

16.750 17.000

F

Overall

mm

Inch

55⁄8

143

55⁄8

143

L

Taper Length

Flute

mm

Inch

75⁄8

194

75⁄8

194

… …

F

mm

Inch

…

…

…

Screw Machine Length

Overall

…

L

Flute F

Overall

mm

Inch

…

27⁄8

73

27⁄8

73

…

mm

Inch 41 ⁄2 41 ⁄2

L

mm 114 114

0.6719

17.066

55⁄8

143

75⁄8

194

…

…

…

…

27⁄8

73

41 ⁄2

114

17.25

0.6791

17.250

55⁄8

143

75⁄8

194

…

…

…

…

27⁄8

73

41 ⁄2

114

0.6875

17.462

55⁄8

143

75⁄8

194

…

…

…

…

27⁄8

73

41 ⁄2

114

17.50

0.6890

17.500

55⁄8

143

75⁄8

194

…

…

…

…

3

76

43⁄4

121

0.7031

17.859

…

…

…

…

…

…

…

…

3

76

43⁄4

121

0.7087

18.000

…

…

…

…

…

…

…

…

3

76

43⁄4

121

0.7188

18.258

…

…

…

…

…

…

…

…

3

76

43⁄4

121

0.7283

18.500

…

…

…

…

…

…

…

…

31 ⁄8

79

5

127

0.7344

18.654

…

…

…

…

…

…

…

…

31 ⁄8

79

5

127

0.7480

19.000

…

…

…

…

…

…

…

…

31 ⁄8

79

5

127

3 ⁄4

0.7500

19.050

…

…

…

…

…

…

…

…

31 ⁄8

79

5

127

49 ⁄64

0.7656

19.446

…

…

…

…

…

…

…

…

31 ⁄4

83

51 ⁄8

130

0.7677

19.500

…

…

…

…

…

…

…

…

31 ⁄4

83

51 ⁄8

130

0.7812

19.845

…

…

…

…

…

…

…

…

31 ⁄4

83

51 ⁄8

130

0.7879

20.000

…

…

…

…

…

…

…

…

33⁄8

86

51 ⁄4

133

0.7969

20.241

…

…

…

…

…

…

…

…

33⁄8

86

51 ⁄4

133

0.8071

20.500

…

…

…

…

…

…

…

…

33⁄8

86

51 ⁄4

133

0.8125

20.638

…

…

…

…

…

…

…

…

33⁄8

86

51 ⁄4

133

0.8268

21.000

…

…

…

…

…

…

…

…

31 ⁄2

89

53⁄8

137

53 ⁄64

0.8281

21.034

…

…

…

…

…

…

…

…

31 ⁄2

89

53⁄8

137

27⁄32

0.8438

21.433

…

…

…

…

…

…

…

…

31 ⁄2

89

53⁄8

137

0.8465

21.500

…

…

…

…

…

…

…

…

31 ⁄2

89

53⁄8

137

0.8594

21.829

…

…

…

…

…

…

…

…

31 ⁄2

89

53⁄8

137

0.8661

22.000

…

…

…

…

…

…

…

…

31 ⁄2

89

53⁄8

137

0.8750

22.225

…

…

…

…

…

…

…

…

31 ⁄2

89

53⁄8

137

43 ⁄64

11 ⁄16

45 ⁄64

18.00 23 ⁄32

18.50 47⁄64

19.00

19.50 25 ⁄32

20.00 51 ⁄64

20.50 13 ⁄16

21.00

21.50 55 ⁄64

22.00 7⁄8

0.8858

22.500

…

…

…

…

…

…

…

…

35⁄8

92

55⁄8

143

0.8906

22.621

…

…

…

…

…

…

…

…

35⁄8

92

55⁄8

143

0.9055

23.000

…

…

…

…

…

…

…

…

35⁄8

92

55⁄8

143

29 ⁄32

0.9062

23.017

…

…

…

…

…

…

…

…

35⁄8

92

55⁄8

143

59 ⁄64

0.9219

23.416

…

…

…

…

…

…

…

…

33⁄4

95

53⁄4

146

0.9252

23.500

…

…

…

…

…

…

…

…

33⁄4

95

53⁄4

146

0.9375

23.812

…

…

…

…

…

…

…

…

33⁄4

95

53⁄4

146

0.9449

24.000

…

…

…

…

…

…

…

…

37⁄8

98

57⁄8

149

0.9531

24.209

…

…

…

…

…

…

…

…

37⁄8

98

57⁄8

149

0.9646

24.500

…

…

…

…

…

…

…

…

37⁄8

98

57⁄8

149

0.9688

24.608

…

…

…

…

…

…

…

…

37⁄8

98

57⁄8

149

0.9843

25.000

…

…

…

…

…

…

…

…

4

102

6

152

63 ⁄64

0.9844

25.004

…

…

…

…

…

…

…

…

4

102

6

152

1

1.0000

25.400

…

…

…

…

…

…

…

…

4

102

6

152

22.50 57⁄64

23.00

23.50 15 ⁄16

24.00 61 ⁄64

24.50 31 ⁄32

25.00

a Fractional inch, number, letter, and metric sizes.

Copyright 2020, Industrial Press, Inc.

ebooks.industrialpress.com

Machinery's Handbook, 31st Edition Twist Drills B

N

S

941

D

F

L

Nominal Shank Size is Same as Nominal Drill Size

Table 2. ANSI Straight-Shank Twist Drills — Taper Length — Over 1 ⁄2 in. (12.7 mm) Dia., Fractional and Metric Sizes ANSI/ASME B94.11M-1993 Frac.

33 ⁄64 17⁄32

35 ⁄64

9 ⁄16 37⁄64

19 ⁄32 39 ⁄64

5 ⁄8

41 ⁄64 21 ⁄32

43 ⁄64 11 ⁄16 45 ⁄64 23 ⁄32 47⁄64 3 ⁄4 49 ⁄64 25 ⁄32

D

Diameter of Drill mm 12.80 13.00 13.20 13.50 13.80 14.00 14.25 14.50 14.75 15.00 15.25 15.50 15.75 16.00 16.25 16.50 16.75 17.00 17.25 17.50 18.00 18.50 19.00

19.50

Decimal Inch Equiv.

Millimeter Equiv.

0.5039 0.5117 0.5156 0.5197 0.5312 0.5315 0.5433 0.5419 0.5512 0.5610 0.5625 0.5709 0.5781 0.5807 0.5906 0.5938 0.6004 0.6094 0.6102 0.6201 0.6250 0.6299 0.6398 0.6406 0.6496 0.6562 0.6594 0.6693 0.6719 0.6791 0.6875 0.6890 0.7031 0.7087 0.7188 0.7283 0.7344 0.7480 0.7500 0.7656 0.7677 0.7812

12.800 13.000 13.096 13.200 13.492 13.500 13.800 13.891 14.000 14.250 14.288 14.500 14.684 14.750 15.000 15.083 15.250 15.479 15.500 15.750 15.875 16.000 16.250 16.271 16.500 16.667 16.750 17.000 17.066 17.250 17.462 17.500 17.859 18.000 18.258 18.500 18.654 19.000 19.050 19.446 19.500 19.842

Flute Length F Inch mm 43⁄4 43⁄4 43⁄4 43⁄4 43⁄4 43⁄4 47⁄8 47⁄8 47⁄8 47⁄8 47⁄8 47⁄8 47⁄8 47⁄8 47⁄8 47⁄8 47⁄8 47⁄8 47⁄8 47⁄8 47⁄8 51 ⁄8 51 ⁄8 51 ⁄8 51 ⁄8 51 ⁄8 53⁄8 53⁄8 53⁄8 53⁄8 53⁄8 55⁄8 55⁄8 55⁄8 55⁄8 57⁄8 57⁄8 57⁄8 57⁄8 6 6 6

121 121 121 121 121 121 124 124 124 124 124 124 124 124 124 124 124 124 124 124 124 130 130 130 130 130 137 137 137 137 137 143 143 143 143 149 149 149 149 152 152 152

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Overall Length L Inch mm 8 8 8 8 8 8

81 ⁄4 81 ⁄4 81 ⁄4 81 ⁄4 81 ⁄4 83⁄4 83⁄4 83⁄4 83⁄4 83⁄4 83⁄4 83⁄4 83⁄4 83⁄4 83⁄4 9 9 9 9 9

91 ⁄4 91 ⁄4 91 ⁄4 91 ⁄4 91 ⁄4 91 ⁄2 91 ⁄2 91 ⁄2 91 ⁄2 93⁄4 93⁄4 93⁄4 93⁄4 97⁄8 97⁄8 97⁄8

203 203 203 203 203 203 210 210 210 210 210 222 222 222 222 222 222 222 222 222 222 228 228 228 228 228 235 235 235 235 235 241 241 241 241 247 247 247 247 251 251 251

Length of Body B Inch mm 47⁄8 47⁄8 47⁄8 47⁄8 47⁄8 47⁄8 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5

51 ⁄4 51 ⁄4 51 ⁄4 51 ⁄4 51 ⁄4 51 ⁄2 51 ⁄2 51 ⁄2 51 ⁄2 51 ⁄2 53⁄4 53⁄4 53⁄4 53⁄4 6 6 6 6 61 ⁄8 61 ⁄8 61 ⁄8

124 124 124 124 124 124 127 127 127 127 127 127 127 127 127 127 127 127 127 127 127 133 133 133 133 133 140 140 140 140 140 146 146 146 146 152 152 152 152 156 156 156

Minimum Length of Shk. S Inch mm 25⁄8 25⁄8 25⁄8 25⁄8 25⁄8 25⁄8 23⁄4 23⁄4 23⁄4 23⁄4 23⁄4 31 ⁄8 31 ⁄8 31 ⁄8 31 ⁄8 31 ⁄8 31 ⁄8 31 ⁄8 31 ⁄8 31 ⁄8 31 ⁄8 31 ⁄8 31 ⁄8 31 ⁄8 31 ⁄8 31 ⁄8 31 ⁄8 31 ⁄8 31 ⁄8 31 ⁄8 31 ⁄8 31 ⁄8 31 ⁄8 31 ⁄8 31 ⁄8 31 ⁄8 31 ⁄8 31 ⁄8 31 ⁄8 31 ⁄8 31 ⁄8 31 ⁄8

66 66 66 66 66 66 70 70 70 70 70 79 79 79 79 79 79 79 79 79 79 79 79 79 79 79 79 79 79 79 79 79 79 79 79 79 79 79 79 79 79 79

Maximum Length of Neck N Inch mm 1 ⁄2 1 ⁄2 1 ⁄2 1 ⁄2 1 ⁄2 1 ⁄2 1 ⁄2 1 ⁄2 1 ⁄2 1 ⁄2 1 ⁄2 5 ⁄8 5 ⁄8 5 ⁄8 5 ⁄8 5 ⁄8 5 ⁄8 5 ⁄8 5 ⁄8 5 ⁄8 5 ⁄8 5 ⁄8 5 ⁄8 5 ⁄8 5 ⁄8 5 ⁄8 5 ⁄8 5 ⁄8 5 ⁄8 5 ⁄8 5 ⁄8 5 ⁄8 5 ⁄8 5 ⁄8 5 ⁄8 5 ⁄8 5 ⁄8 5 ⁄8 5 ⁄8 5 ⁄8 5 ⁄8 5 ⁄8

13 13 13 13 13 13 13 13 13 13 13 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16

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Machinery's Handbook, 31st Edition Twist Drills

942

Table Table 2. (Continued) 2. ANSI ANSI Straight-Shank Straight-Shank Twist Twist Drills Drills — Taper — Taper Length Length — — Over 1 ⁄2 in. (12.7 mm) Dia., Fractional and Metric Sizes ANSI/ASME B94.11M-1993 Frac. 51 ⁄64 13 ⁄16 53 ⁄64 27⁄32 55 ⁄64 7⁄8 57⁄64 29 ⁄32 59 ⁄64 15 ⁄16 61 ⁄64 31 ⁄32 63 ⁄64

1

11 ⁄64 11 ⁄32 13⁄64 11 ⁄16 15⁄64 13⁄32 17⁄64 11 ⁄8 19⁄64 15⁄32 111 ⁄64 13⁄16 113⁄64 17⁄32

115⁄64

D

Diameter of Drill mm 20.00 20.50 21.00

21.50 22.00 22.50 23.00

23.50 24.00 24.50 25.00

25.50 26.00 26.50

27.00 27.50 28.00 28.50

29.00 29.50 30.00 30.50

31.00 31.50

Decimal Inch Equiv.

Millimeter Equiv.

0.7874 0.7969 0.8071 0.8125 0.8268 0.8281 0.8438 0.8465 0.8594 0.8661 0.8750 0.8858 0.8906 0.9055 0.9062 0.9219 0.9252 0.9375 0.9449 0.9531 0.9646 0.9688 0.9843 0.9844 1.0000 1.0039 1.0156 1.0236 1.0312 1.0433 1.0469 1.0625 1.0630 1.0781 1.0827 1.0938 1.1024 1.1094 1.1220 1.1250 1.1406 1.1417 1.1562 1.1614 1.1719 1.1811 1.1875 1.2008 1.2031 1.2188 1.2205 1.2344 1.2402

20.000 20.241 20.500 20.638 21.000 21.034 21.433 21.500 21.829 22.000 22.225 22.500 22.621 23.000 23.017 23.416 23.500 23.812 24.000 24.209 24.500 24.608 25.000 25.004 25.400 25.500 25.796 26.000 26.192 26.560 26.591 26.988 27.000 27.384 27.500 27.783 28.000 28.179 28.500 28.575 28.971 29.000 29.367 29.500 29.766 30.000 30.162 30.500 30.559 30.958 31.000 31.354 31.500

Flute Length F Inch mm 61 ⁄8 61 ⁄8 61 ⁄8 61 ⁄8 61 ⁄8 61 ⁄8 61 ⁄8 61 ⁄8 61 ⁄8 61 ⁄8 61 ⁄8 61 ⁄8 61 ⁄8 61 ⁄8 61 ⁄8 61 ⁄8 61 ⁄8 61 ⁄8 63⁄8 63⁄8 63⁄8 63⁄8 63⁄8 63⁄8 63⁄8 61 ⁄2 61 ⁄2 61 ⁄2 61 ⁄2 65⁄8 65⁄8 65⁄8 65⁄8 67⁄8 67⁄8 67⁄8 71 ⁄8 71 ⁄8 71 ⁄8 71 ⁄8 71 ⁄4 71 ⁄4 71 ⁄4 73⁄8 73⁄8 73⁄8 73⁄8 71 ⁄2 71 ⁄2 71 ⁄2 77⁄8 77⁄8 77⁄8

156 156 156 156 156 156 156 156 156 156 156 156 156 156 156 156 156 156 162 162 162 162 162 162 162 165 165 165 165 168 168 168 168 175 175 175 181 181 181 181 184 184 184 187 187 187 187 190 190 190 200 200 200

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Overall Length L Inch mm

Length of Body B Inch mm

10 10 10 10 10 10 10 10 10 10 10 10 10 10 10

61 ⁄4 61 ⁄4 61 ⁄4 61 ⁄4 61 ⁄4 61 ⁄4 61 ⁄4 61 ⁄4 61 ⁄4 61 ⁄4 61 ⁄4 61 ⁄4 61 ⁄4 61 ⁄4 61 ⁄4 61 ⁄4 61 ⁄4 61 ⁄4 61 ⁄2 61 ⁄2 61 ⁄2 61 ⁄2 61 ⁄2 61 ⁄2 61 ⁄2 65⁄8 65⁄8 65⁄8 65⁄8 63⁄4 63⁄4 63⁄4 63⁄4 7 7 7

103⁄4 103⁄4 103⁄4 11 11 11 11 11 11 11

111 ⁄8 111 ⁄8 111 ⁄8 111 ⁄8 111 ⁄4 111 ⁄4 111 ⁄4 111 ⁄4 111 ⁄2 111 ⁄2 111 ⁄2 113⁄4 113⁄4 113⁄4 113⁄4 117⁄8 117⁄8 117⁄8 12 12 12 12

121 ⁄8 121 ⁄8 121 ⁄8 121 ⁄2 121 ⁄2 121 ⁄2

254 254 254 254 254 254 254 254 254 254 254 254 254 254 254 273 273 273 279 279 279 279 279 279 279 282 282 282 282 286 286 286 286 292 292 292 298 298 298 298 301 301 301 305 305 305 305 308 308 308 317 317 317

71 ⁄4 71 ⁄4 71 ⁄4 71 ⁄4 73⁄8 73⁄8 73⁄8 71 ⁄2 71 ⁄2 71 ⁄2 71 ⁄2 75⁄8 75⁄8 75⁄8 8 8 8

159 159 159 159 159 159 159 159 159 159 159 159 159 159 159 159 159 159 165 165 165 165 165 165 165 168 168 168 168 172 172 172 172 178 178 178 184 184 184 184 187 187 187 191 191 191 191 194 194 194 203 203 203

Minimum Length of Shk. S Inch mm 31 ⁄8 31 ⁄8 31 ⁄8 31 ⁄8 31 ⁄8 31 ⁄8 31 ⁄8 31 ⁄8 31 ⁄8 31 ⁄8 31 ⁄8 31 ⁄8 31 ⁄8 31 ⁄8 31 ⁄8 37⁄8 37⁄8 37⁄8 37⁄8 37⁄8 37⁄8 37⁄8 37⁄8 37⁄8 37⁄8 37⁄8 37⁄8 37⁄8 37⁄8 37⁄8 37⁄8 37⁄8 37⁄8 37⁄8 37⁄8 37⁄8 37⁄8 37⁄8 37⁄8 37⁄8 37⁄8 37⁄8 37⁄8 37⁄8 37⁄8 37⁄8 37⁄8 37⁄8 37⁄8 37⁄8 37⁄8 37⁄8 37⁄8

79 79 79 79 79 79 79 79 79 79 79 79 79 79 79 98 98 98 98 98 98 98 98 98 98 98 98 98 98 98 98 98 98 98 98 98 98 98 98 98 98 98 98 98 98 98 98 98 98 98 98 98 98

Maximum Length of Neck N Inch mm 5 ⁄8 5 ⁄8 5 ⁄8 5 ⁄8 5 ⁄8 5 ⁄8 5 ⁄8 5 ⁄8 5 ⁄8 5 ⁄8 5 ⁄8 5 ⁄8 5 ⁄8 5 ⁄8 5 ⁄8 5 ⁄8 5 ⁄8 5 ⁄8 5 ⁄8 5 ⁄8 5 ⁄8 5 ⁄8 5 ⁄8 5 ⁄8 5 ⁄8 5 ⁄8 5 ⁄8 5 ⁄8 5 ⁄8 5 ⁄8 5 ⁄8 5 ⁄8 5 ⁄8 5 ⁄8 5 ⁄8 5 ⁄8 5 ⁄8 5 ⁄8 5 ⁄8 5 ⁄8 5 ⁄8 5 ⁄8 5 ⁄8 5 ⁄8 5 ⁄8 5 ⁄8 5 ⁄8 5 ⁄8 5 ⁄8 5 ⁄8 5 ⁄8 5 ⁄8 5 ⁄8

16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16

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Machinery's Handbook, 31st Edition Twist Drills

943

Table Table 2. (Continued) Straight-Shank — Taper Length 2. ANSI ANSI Straight-Shank Twist Twist Drills Drills — Taper Length — — Over 1 ⁄2 in. (12.7 mm) Dia., Fractional and Metric Sizes ANSI/ASME B94.11M-1993 Frac. 11 ⁄4

mm

Decimal Inch Equiv.

Millimeter Equiv.

1.2500 1.2598 1.2795 1.2812 1.2992 1.3125 1.3189 1.3386 1.3438 1.3583 1.3750 1.3780 1.3976 1.4062 1.4173 1.4370 1.4375 1.4567 1.4688 1.4764 1.4961 1.5000 1.5625 1.6250 1.7500

31.750 32.000 32.500 32.542 33.000 33.338 33.500 34.000 34.133 34.500 34.925 35.000 35.500 35.717 36.000 36.500 36.512 37.000 37.308 37.500 38.000 38.100 39.688 41.275 44.450

32.00 32.50

19⁄32

33.00

15⁄16

111 ⁄32 13⁄8

113⁄32

17⁄16 115⁄32

11 ⁄2 19⁄16 15⁄8 13⁄4

Diameter of Drill

D

33.50 34.00 34.50 35.00 35.50 36.00 36.50 37.00 37.50 38.00

Flute Length F Inch mm 77⁄8 81 ⁄2 81 ⁄2 81 ⁄2 85⁄8 85⁄8 83⁄4 83⁄4 83⁄4 87⁄8 87⁄8 9 9 9

91 ⁄8 91 ⁄8 91 ⁄8 91 ⁄4 91 ⁄4 93⁄8 93⁄8 93⁄8 95⁄8 97⁄8 101 ⁄2

200 216 216 216 219 219 222 222 222 225 225 229 229 229 232 232 232 235 235 238 238 238 244 251 267

Overall Length L Inch mm 317 359 359 359 362 362 365 365 365 368 368 372 372 372 375 375 375 378 378 381 381 381 387 397 413

121 ⁄2 141 ⁄8 141 ⁄8 141 ⁄8 141 ⁄4 141 ⁄4 143⁄8 143⁄8 143⁄8 141 ⁄2 141 ⁄2 145⁄8 145⁄8 145⁄8 143⁄4 143⁄4 143⁄4 147⁄8 147⁄8 15 15 15

151 ⁄4

155⁄8 161 ⁄4

Length of Body B Inch mm 8

85⁄8 85⁄8 85⁄8 83⁄4 83⁄4 87⁄8 87⁄8 87⁄8 9 9

91 ⁄8

91 ⁄8 91 ⁄8 91 ⁄4 91 ⁄4 91 ⁄4 93⁄8 93⁄8 91 ⁄2 91 ⁄2 91 ⁄2 93⁄4 10

105⁄8

203 219 219 219 222 222 225 225 225 229 229 232 232 232 235 235 235 238 238 241 241 241 247 254 270

Minimum Length of Shk. S Inch mm 37⁄8 47⁄8 47⁄8 47⁄8 47⁄8 47⁄8 47⁄8 47⁄8 47⁄8 47⁄8 47⁄8 47⁄8 47⁄8 47⁄8 47⁄8 47⁄8 47⁄8 47⁄8 47⁄8 47⁄8 47⁄8 47⁄8 47⁄8 47⁄8 47⁄8

Maximum Length of Neck N Inch mm

98 124 124 124 124 124 124 124 124 124 124 124 124 124 124 124 124 124 124 124 124 124 124 124 124

5 ⁄8 5 ⁄8 5 ⁄8 5 ⁄8 5 ⁄8 5 ⁄8 5 ⁄8 5 ⁄8 5 ⁄8 5 ⁄8 5 ⁄8 5 ⁄8 5 ⁄8 5 ⁄8 5 ⁄8 5 ⁄8 5 ⁄8 5 ⁄8 5 ⁄8 5 ⁄8 5 ⁄8 5 ⁄8 5 ⁄8 3 ⁄4 3 ⁄4

16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 19 19

K

J

A

Table 3. American National Standard Tangs for Straight-Shank Drills ANSI/ASME B94.11M-1993 Nominal Diameter of Drill Shank, A

Inches 1 ⁄8

thru 3⁄16

over 3⁄16 thru 1 ⁄4 over 1 ⁄4 thru 5⁄16

over 5⁄16 thru 3⁄8

Millimeters

3.18 thru 4.76

over 4.76 thru 6.35 over 6.35 thru 7.94 over 7.94 thru 9.53

over 3⁄8 thru 15⁄32

over 9.53 thru 11.91

over 9⁄16 thru 21 ⁄32

over 14.29 thru 16.67

over 15⁄32 thru 9⁄16 over 21 ⁄32 thru 3⁄4 over 3⁄4 thru 7⁄8 over 7⁄8 thru 1

over 1 thru 13⁄16

over 13⁄16 thru 13⁄8

over 11.91 thru 14.29

over 16.67 thru 19.05 over 19.05 thru 22.23

over 22.23 thru 25.40

over 25.40 thru 30.16 over 30.16 thru 34.93

Thickness of Tang, J

Max.

Inches

0.094

0.122 0.162 0.203

0.243

0.303

0.373 0.443 0.514

0.609

0.700 0.817

Min.

0.090 0.118

0.158 0.199

0.239

0.297

0.367 0.437 0.508

0.601

0.692 0.809

Millimeters

Max.

2.39

3.10 4.11

5.16

6.17

7.70

9.47

11.25

13.05

15.47

17.78 20.75

Min.

2.29

3.00 4.01

5.06

6.07

7.55

9.32

11.10

12.90

15.27

17.58 20.55

Length of Tang, K

Inches 9 ⁄32 5 ⁄16 11 ⁄32 3 ⁄8 7⁄16 1 ⁄2 9 ⁄16 5 ⁄8 11 ⁄16 3 ⁄4 13 ⁄16 7⁄8

Millimeters 7.0 8.0 8.5 9.5

11.0

12.5 14.5 16.0 17.5 19.0 20.5 22.0

To fit split-sleeve collet-type drill drivers. See page 955.

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Machinery's Handbook, 31st Edition Twist Drills

944

Table 4. American National Standard Straight-Shank Twist Drills — Screw Machine Length — Over 1 in. (25.4 mm) Dia. ANSI/ASME B94.11M-1993 D

F A

L

mm

Decimal Inch Equivalent

Millimeter Equivalent

25.50

1.0039

26.00

D Frac.

Flute Length

Diameter of Drill

Overall Length

F

Shank Diameter

L

A

Inch

mm

Inch

mm

Inch

mm

25.500

4

102

6

152

0.9843

25.00

1.0236

26.000

4

102

6

152

0.9843

25.00

1.0625

26.988

4

102

6

152

1.0000

25.40

1.1024

28.000

4

102

6

152

0.9843

25.00

1.1250

28.575

4

102

6

152

1.0000

25.40

1.1811

30.000

41 ⁄4

108

65⁄8

168

0.9843

25.00

13⁄16

1.1875

30.162

41 ⁄4

108

65⁄8

168

1.0000

25.40

11 ⁄4

1.2500

31.750

43⁄8

111

63⁄4

171

1.0000

25.40

1.2598

32.000

43⁄8

111

7

178

1.2402

31.50

1.3125

33.338

43⁄8

111

7

178

1.2500

31.75

34.00

1.3386

34.000

41 ⁄2

114

71 ⁄8

181

1.2402

31.50

1.3750

34.925

41 ⁄2

114

71 ⁄8

181

1.2500

31.75

36.00

1.4173

36.000

43⁄4

121

73⁄8

187

1.2402

31.50

1.4375

36.512

43⁄4

121

73⁄8

187

1.2500

31.75

1.4961

38.000

47⁄8

124

71 ⁄2

190

1.2402

31.50

11 ⁄2

1.5000

38.100

47⁄8

124

71 ⁄2

190

1.2500

31.75

19⁄16

1.5625

39.688

47⁄8

124

73⁄4

197

1.5000

38.10

1.5748

40.000

47⁄8

124

73⁄4

197

1.4961

38.00

1.6250

41.275

47⁄8

124

73⁄4

197

1.5000

38.10

1.6535

42.000

51 ⁄8

130

8

203

1.4961

38.00

1.6875

42.862

51 ⁄8

130

8

203

1.5000

38.10

1.7323

44.000

51 ⁄8

130

8

203

1.4961

38.00

1.7500

44.450

51 ⁄8

130

8

203

1.5000

38.10

1.8110

46.000

53⁄8

137

81 ⁄4

210

1.4961

38.00

113⁄16

1.8125

46.038

53⁄8

137

81 ⁄4

210

1.5000

38.10

17⁄8

1.8750

47.625

53⁄8

137

81 ⁄4

210

1.5000

38.10

1.8898

48.000

55⁄8

143

81 ⁄2

216

1.4961

38.00

1.9375

49.212

55⁄8

143

81 ⁄2

216

1.5000

38.10

1.9685

50.000

55⁄8

143

81 ⁄2

216

1.4961

38.00

2.0000

50.800

55⁄8

143

81 ⁄2

216

1.5000

38.10

11 ⁄16 28.00 11 ⁄8 30.00

32.00 15⁄16

13⁄8

17⁄16 38.00

40.00 15⁄8 42.00 111 ⁄16 44.00 13⁄4 46.00

48.00 115⁄16 50.00 2

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Machinery's Handbook, 31st Edition Twist Drills L

945

F

D

Table 5. American National Taper-Shank Twist Drills Fractional and Metric Sizes ANSI/ASME B94.11M-1993

Fraction 1 ⁄8

9 ⁄64 5 ⁄32

11 ⁄64 3 ⁄16

13 ⁄64

7⁄32 15 ⁄64

1 ⁄4 17⁄64

9 ⁄32

19 ⁄64 5 ⁄16

21 ⁄64 11 ⁄32

23 ⁄64

3 ⁄8 25 ⁄64

Drill Diameter, D Equivalent mm 3.00 3.20 3.50 3.80 4.00 4.20 4.50 4.80 5.00 5.20 5.50 5.80 6.00 6.20 6.50 6.80 7.00 7.20 7.50 7.80 8.00 8.20 8.50 8.80 9.00 9.20 9.50 9.80 10.00

Decimal Inch

mm

Morse Taper No.

0.1181 0.1250 0.1260 0.1378 0.1406 0.1496 0.1562 0.1575 0.1654 0.1719 0.1772 0.1875 0.1890 0.1969 0.2031 0.2047 0.2165 0.2183 0.2223 0.2344 0.2362 0.2441 0.2500 0.2559 0.2656 0.2677 0.2756 0.2812 0.2835 0.2953 0.2969 0.3071 0.3125 0.3150 0.3228 0.3281 0.3346 0.3438 0.3465 0.3543 0.3594 0.3622 0.3740 0.3750 0.3858 0.3906 0.3937

3.000 3.175 3.200 3.500 3.571 3.800 3.967 4.000 4.200 4.366 4.500 4.762 4.800 5.000 5.159 5.200 5.500 5.558 5.800 5.954 6.000 6.200 6.350 6.500 6.746 6.800 7.000 7.142 7.200 7.500 7.541 7.800 7.938 8.000 8.200 8.334 8.500 8.733 8.800 9.000 9.129 9.200 9.500 9.525 9.800 9.921 10.000

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

Regular Shank Flute Length Overall Length F L Inch mm Inch mm 17⁄8 17⁄8 21 ⁄8 21 ⁄8 21 ⁄8 21 ⁄8 21 ⁄8 21 ⁄2 21 ⁄2 21 ⁄2 21 ⁄2 21 ⁄2 23⁄4 23⁄4 23⁄4 23⁄4 23⁄4 23⁄4 27⁄8 27⁄8 27⁄8 27⁄8 27⁄8 3 3 3 3 3 31 ⁄8 31 ⁄8 31 ⁄8 31 ⁄8 31 ⁄8 31 ⁄4 31 ⁄4 31 ⁄4 31 ⁄4 31 ⁄4 31 ⁄2 31 ⁄2 31 ⁄2 31 ⁄2 31 ⁄2 31 ⁄2 35⁄8 35⁄8 35⁄8

48 48 54 54 54 54 54 64 64 64 64 64 70 70 70 70 70 70 73 73 73 73 73 76 76 76 76 76 79 79 79 79 79 83 83 83 83 83 89 89 89 89 89 89 92 92 92

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51 ⁄8 51 ⁄8 53⁄8 53⁄8 53⁄8 53⁄8 53⁄8 53⁄4 53⁄4 53⁄4 53⁄4 53⁄4 6 6 6 6 6 6 61 ⁄8 61 ⁄8 61 ⁄8 61 ⁄8 61 ⁄8 61 ⁄4 61 ⁄4 61 ⁄4 61 ⁄4 61 ⁄4 63⁄8 63⁄8 63⁄8 63⁄8 63⁄8 61 ⁄2 61 ⁄2 61 ⁄2 61 ⁄2 61 ⁄2 63⁄4 63⁄4 63⁄4 63⁄4 63⁄4 63⁄4 7 7 7

130 130 137 137 137 137 137 146 146 146 146 146 152 152 152 152 152 152 156 156 156 156 156 159 159 159 159 159 162 162 162 162 162 165 165 165 165 165 171 171 171 171 171 171 178 178 178

Morse Taper No. … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … 2 … 2 …

Larger or Smaller Shanka Flute Length Overall Length F L Inch mm Inch mm … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … 31 ⁄2 … 35⁄8 …

… … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … 89 … 92 …

… … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … 73⁄8 … 71 ⁄2 …

… … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … 187 … 190 …

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Machinery's Handbook, 31st Edition Twist Drills

946

Table Table 5. (Continued) American National Taper-Shank 5. American National Taper-Shank Twist Twist Drills Drills Fractional and Metric Sizes ANSI/ASME B94.11M-1993

Fraction 13 ⁄32 27⁄64

7⁄16

29 ⁄64 15 ⁄32

31 ⁄64 1 ⁄2

33 ⁄64 17⁄32

35 ⁄64

9 ⁄16 37⁄64

19 ⁄32 39 ⁄64

5 ⁄8

41 ⁄64 21 ⁄32

43 ⁄64 11 ⁄16 45 ⁄64 23 ⁄32 47⁄64

Drill Diameter, D Equivalent mm 10.20 10.50 10.80 11.00 11.20 11.50 11.80 12.00 12.20 12.50 12.80 13.00 13.20 13.50 13.80 14.00 14.25 14.50 14.75 15.00 15.25 15.50 15.75 16.00 16.25 16.50 16.75 17.00 17.25 17.50 18.00 18.50

Decimal Inch 0.4016 0.4062 0.4134 0.4219 0.4252 0.4331 0.4375 0.4409 0.4528 0.4531 0.4646 0.4688 0.4724 0.4803 0.4844 0.4921 0.5000 0.5034 0.5118 0.5156 0.5197 0.5312 0.5315 0.5433 0.5469 0.5572 0.5610 0.5625 0.5709 0.5781 0.5807 0.5906 0.5938 0.6004 0.6094 0.6102 0.6201 0.6250 0.6299 0.6398 0.6406 0.6496 0.6562 0.6594 0.6693 0.6719 0.6791 0.6875 0.6880 0.7031 0.7087 0.7188 0.7283 0.7344

mm 10.200

10.320 10.500 10.716 10.800 11.000 11.112 11.200 11.500 11.509 11.800 11.906 12.000 12.200 12.304 12.500 12.700 12.800 13.000 13.096 13.200 13.492 13.500 13.800 13.891 14.000 14.250 14.288 14.500 14.684 14.750 15.000 15.083 15.250 15.479 15.500 15.750 15.875 16.000 16.250 16.271 16.500 16.667 16.750 17.000 17.066 17.250 17.462 17.500 17.859 18.000 18.258 18.500 18.654

Morse Taper No. 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

Regular Shank Flute Length Overall Length F L Inch mm Inch mm 92 7 178 35⁄8 35⁄8 37⁄8 37⁄8 37⁄8 37⁄8 37⁄8 41 ⁄8 41 ⁄8 41 ⁄8 41 ⁄8 41 ⁄8 43⁄8 43⁄8 43⁄8 43⁄8 43⁄8 45⁄8 45⁄8 45⁄8 45⁄8 45⁄8 45⁄8 47⁄8 47⁄8 47⁄8 47⁄8 47⁄8 47⁄8 47⁄8 47⁄8 47⁄8 47⁄8 47⁄8 47⁄8 47⁄8 47⁄8 47⁄8 51 ⁄8 51 ⁄8 51 ⁄8 51 ⁄8 51 ⁄8 53⁄8 53⁄8 53⁄8 53⁄8 53⁄8 55⁄8 55⁄8 55⁄8 55⁄8 57⁄8 57⁄8

92 98 98 98 98 98 105 105 105 105 105 111 111 111 111 111 117 117 117 117 117 117 124 124 124 124 124 124 124 124 124 124 124 124 124 124 124 130 130 130 130 130 137 137 137 137 137 143 143 143 143 149 149

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7 71 ⁄4 71 ⁄4 71 ⁄4 71 ⁄4 71 ⁄4 71 ⁄2 71 ⁄2 71 ⁄2 71 ⁄2 71 ⁄2 81 ⁄4 81 ⁄4 81 ⁄4 81 ⁄4 81 ⁄4 81 ⁄2 81 ⁄2 81 ⁄2 81 ⁄2 81 ⁄2 81 ⁄2 83⁄4 83⁄4 83⁄4 83⁄4 83⁄4 83⁄4 83⁄4 83⁄4 83⁄4 83⁄4 83⁄4 83⁄4 83⁄4 83⁄4 83⁄4 9 9 9 9 9 91 ⁄4 91 ⁄4 91 ⁄4 91 ⁄4 91 ⁄4 91 ⁄2 91 ⁄2 91 ⁄2 91 ⁄2 93⁄4 93⁄4

178 184 184 184 184 184 190 190 190 190 190 210 210 210 210 210 216 216 216 216 216 216 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 229 229 229 229 229 235 235 235 235 235 241 241 241 241 248 248

Morse Taper No. … 2 … 2 … … 2 … … 2 … 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 … … … … … … … … … … … … 3 … 3 … … 3 … 3 … 3 … 3 … 3

Larger or Smaller Shanka Flute Length Overall Length F L Inch mm Inch mm … … … … 35⁄8 … 37⁄8 … … 37⁄8 … … 41 ⁄8 … 41 ⁄8 43⁄8 43⁄8 43⁄8 43⁄8 43⁄8 45⁄8 45⁄8 45⁄8 45⁄8 45⁄8 45⁄8 47⁄8 47⁄8 47⁄8 47⁄8 47⁄8 … … … … … … … … … … … … 51 ⁄8 … 51 ⁄8 … … 53⁄8 … 53⁄8 … 55⁄8 … 55⁄8 … 57⁄8

92 … 98 … … 98 … … 105 … 105 111 111 111 111 111 117 117 117 117 117 117 124 124 124 124 124 … … … … … … … … … … … … 130 … 130 … … 137 … 137 … 143 … 143 … 149

71 ⁄2 … 73⁄4 … … 73⁄4 … … 8 … 8 73⁄4 73⁄4 73⁄4 73⁄4 73⁄4 8 8 8 8 8 8 81 ⁄4 81 ⁄4 81 ⁄4 81 ⁄4 81 ⁄4 … … … … … … … … … … … … 93⁄4 … 93⁄4 … … 10 … 10 … 101 ⁄4 … 101 ⁄4 … 101 ⁄2

190 … 197 … … 197 … … 203 … 203 197 197 197 197 197 203 203 203 203 203 203 210 210 210 210 210 … … … … … … … … … … … … 248 … 248 … … 254 … 254 … 260 … 260 … 267

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Table Table 5. (Continued) American National Taper-Shank 5. American National Taper-Shank Twist Twist Drills Drills Fractional and Metric Sizes ANSI/ASME B94.11M-1993

Fraction

Drill Diameter, D Equivalent mm 19.00

3 ⁄4 49 ⁄64 25 ⁄32 51 ⁄64 13 ⁄16 53 ⁄64 27⁄32 55 ⁄64 7⁄8 57⁄64 29 ⁄32 59 ⁄64 15 ⁄16 61 ⁄64 31 ⁄32 63 ⁄64

1

11 ⁄64 11 ⁄32 13⁄64 11 ⁄16 15⁄64 13⁄32 17⁄64 11 ⁄8 19⁄64 15⁄32 111 ⁄64 13⁄16 113⁄64

19.50 20.00 20.50 21.00

21.50 22.00 22.50 23.00

23.50 24.00 24.50 25.00

25.50 26.00 26.50

27.00 27.50 28.00 28.50

29.00 29.50 30.00 30.50

Decimal Inch 0.7480 0.7500 0.7656 0.7677 0.7812 0.7821 0.7969 0.8071 0.8125 0.8268 0.8281 0.8438 0.8465 0.8594 0.8661 0.8750 0.8858 0.8906 0.9055 0.9062 0.9219 0.9252 0.9375 0.9449 0.9531 0.9646 0.9688 0.9843 0.9844 1.0000 1.0039 1.0156 1.0236 1.0312 1.0433 1.0469 1.0625 1.0630 1.0781 1.0827 1.0938 1.1024 1.1094 1.1220 1.1250 1.1406 1.1417 1.1562 1.1614 1.1719 1.1811 1.1875 1.2008 1.2031

Morse Taper mm No. 19.000 2

19.050 19.446 19.500 19.843 20.000 20.241 20.500 20.638 21.000 21.034 21.433 21.500 21.829 22.000 22.225 22.500 22.621 23.000 23.017 23.416 23.500 23.813 24.000 24.209 24.500 24.608 25.000 25.004 25.400 25.500 25.796 26.000 26.192 26.500 26.591 26.988 27.000 27.384 27.500 27.783 28.000 28.179 28.500 28.575 28.971 29.000 29.367 29.500 29.797 30.000 30.162 30.500 30.559

2 2 2 2 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4

Regular Shank Flute Length Overall Length F L Inch mm Inch mm 149 248 93⁄4 57⁄8 57⁄8 6 6 6 61 ⁄8 61 ⁄8 61 ⁄8 61 ⁄8 61 ⁄8 61 ⁄8 61 ⁄8 61 ⁄8 61 ⁄8 61 ⁄8 61 ⁄8 61 ⁄8 61 ⁄8 61 ⁄8 61 ⁄8 61 ⁄8 61 ⁄8 61 ⁄8 63⁄8 63⁄8 63⁄8 63⁄8 63⁄8 63⁄8 63⁄8 61 ⁄2 61 ⁄2 61 ⁄2 61 ⁄2 65⁄8 65⁄8 65⁄8 65⁄8 67⁄8 67⁄8 67⁄8 71 ⁄8 71 ⁄8 71 ⁄8 71 ⁄8 71 ⁄4 71 ⁄4 71 ⁄4 73⁄8 73⁄8 73⁄8 73⁄8 71 ⁄2 71 ⁄2

149 152 152 152 156 156 156 156 156 156 156 156 156 156 156 156 156 156 156 156 156 156 162 162 162 162 162 162 162 165 165 165 165 168 168 168 168 175 175 175 181 181 181 181 184 184 184 187 187 187 187 190 190

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93⁄4 97⁄8 97⁄8 97⁄8 103⁄4 103⁄4 103⁄4 103⁄4 103⁄4 103⁄4 103⁄4 103⁄4 103⁄4 103⁄4 103⁄4 103⁄4 103⁄4 103⁄4 103⁄4 103⁄4 103⁄4 103⁄4 11 11 11 11 11 11 11 111 ⁄8 111 ⁄8 111 ⁄8 111 ⁄8 111 ⁄4 111 ⁄4 111 ⁄4 111 ⁄4 121 ⁄2 121 ⁄2 121 ⁄2 123⁄4 123⁄4 123⁄4 123⁄4 127⁄8 127⁄8 127⁄8 13 13 13 13 131 ⁄8 131 ⁄8

248 251 251 251 273 273 273 273 273 273 273 273 273 273 273 273 273 273 273 273 273 273 279 279 279 279 279 279 279 283 283 283 283 286 286 286 286 318 318 318 324 324 324 324 327 327 327 330 330 330 330 333 333

Morse Taper No. … 3 3 … 3 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 … … … … … … … … … 4 … … … 4 … … 4 … 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3

Larger or Smaller Shanka Flute Length Overall Length F L Inch mm Inch mm … … … … 57⁄8 6 … 6 61 ⁄8 61 ⁄8 61 ⁄8 61 ⁄8 61 ⁄8 61 ⁄8 61 ⁄8 61 ⁄8 61 ⁄8 61 ⁄8 61 ⁄8 61 ⁄8 61 ⁄8 61 ⁄8 61 ⁄8 … … … … … … … … … 63⁄8 … … … 61 ⁄2 … … 65⁄8 … 67⁄8 67⁄8 67⁄8 71 ⁄8 71 ⁄8 71 ⁄8 71 ⁄8 71 ⁄4 71 ⁄4 71 ⁄4 73⁄8 73⁄8 73⁄8 73⁄8 71 ⁄2 71 ⁄2

149 152 … 152 156 156 156 156 156 156 156 156 156 156 156 156 156 156 156 … … … … … … … … … 162 … … … 165 … … 168 … 175 175 175 181 181 181 181 184 184 184 187 187 187 187 190 190

101 ⁄2 105⁄8 … 105⁄8 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 … … … … … … … … … 12 … … … 121 ⁄8 … … 121 ⁄4 … 111 ⁄2 111 ⁄2 111 ⁄2 113⁄4 113⁄4 113⁄4 113⁄4 117⁄8 117⁄8 117⁄8 12 12 12 12 121 ⁄8 121 ⁄8

267 270 … 270 254 254 254 254 254 254 254 254 254 254 254 254 254 254 254 … … … … … … … … … 305 … … … 308 … … 311 … 292 292 292 298 298 298 298 302 302 302 305 305 305 305 308 308

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Table Table 5. (Continued) American National Taper-Shank 5. American National Taper-Shank Twist Twist Drills Drills Fractional and Metric Sizes ANSI/ASME B94.11M-1993

Fraction 17⁄32 115⁄64 11 ⁄4 117⁄64 19⁄32 119⁄64 15⁄16 121 ⁄64 111 ⁄32 123⁄64 13⁄8 125⁄64 113⁄32 127⁄64 17⁄16 129⁄64 115⁄32 131 ⁄64 11 ⁄2 133⁄64 117⁄32 135⁄64 19⁄16 137⁄64 119⁄32 139⁄64 15⁄8 141 ⁄64 121 ⁄32 143⁄64 111 ⁄16 145⁄64 123⁄32

Drill Diameter, D Equivalent mm 31.00 31.50 32.00 32.50

33.00 33.50 34.00 34.50

35.00 35.50 36.00 36.50

37.00 37.50 38.00

39.00

40.00

41.00

42.00

43.00

44.00

Decimal Inch 1.2188 1.2205 1.2344 1.2402 1.2500 1.2598 1.2656 1.2795 1.2812 1.2969 1.2992 1.3125 1.3189 1.3281 1.3386 1.3438 1.3583 1.3594 1.3750 1.3780 1.3906 1.3976 1.4062 1.4173 1.4219 1.4370 1.4375 1.4531 1.4567 1.4688 1.4764 1.4844 1.4961 1.5000 1.5156 1.5312 1.5354 1.5469 1.5625 1.5748 1.5781 1.5938 1.6094 1.6142 1.6250 1.6406 1.6535 1.6562 1.6719 1.6875 1.6929 1.7031 1.7188 1.7323

Morse Taper mm No. 30.958 4

31.000 31.354 31.500 31.750 32.000 32.146 32.500 32.542 32.941 33.000 33.338 33.500 33.734 34.000 34.133 34.500 34.529 34.925 35.000 35.321 35.500 35.717 36.000 36.116 36.500 36.512 36.909 37.000 37.308 37.500 37.704 38.000 38.100 38.496 38.892 39.000 39.291 39.688 40.000 40.084 40.483 40.879 41.000 41.275 41.671 42.000 42.067 42.466 42.862 43.000 43.259 43.658 44.000

4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 … 5 5 … 5 5 … 5 … 5 5 … 5 5 … 5 5 … 5 5

Regular Shank Flute Length Overall Length F L Inch mm Inch mm 190 333 71 ⁄2 131 ⁄8 77⁄8 77⁄8 77⁄8 77⁄8 81 ⁄2 81 ⁄2 81 ⁄2 81 ⁄2 85⁄8 85⁄8 85⁄8 83⁄4 83⁄4 83⁄4 83⁄4 87⁄8 87⁄8 87⁄8 9 9 9 9 91 ⁄8 91 ⁄8 91 ⁄8 91 ⁄8 91 ⁄4 91 ⁄4 91 ⁄4 93⁄8 93⁄8 93⁄8 93⁄8 … 93⁄8 95⁄8 … 95⁄8 97⁄8 … 97⁄8 … 10 10 … 101 ⁄8 101 ⁄8 … 101 ⁄8 101 ⁄8 … 101 ⁄8 101 ⁄8

200 200 200 200 216 216 216 216 219 219 219 222 222 222 222 225 225 225 229 229 229 229 232 232 232 232 235 235 235 238 238 238 238 … 238 244 … 244 251 … 251 … 254 254 … 257 257 … 257 257 … 257 257

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131 ⁄2 131 ⁄2 131 ⁄2 131 ⁄2 141 ⁄8 141 ⁄8 141 ⁄8 141 ⁄8 141 ⁄4 141 ⁄4 141 ⁄4 143⁄8 143⁄8 143⁄8 143⁄8 141 ⁄2 141 ⁄2 141 ⁄2 145⁄8 145⁄8 145⁄8 145⁄8 143⁄4 143⁄4 143⁄4 143⁄4 147⁄8 147⁄8 147⁄8 15 15 15 15 … 163⁄8 165⁄8 … 165⁄8 167⁄8 … 167⁄8 … 17 17 … 171 ⁄8 171 ⁄8 … 171 ⁄8 171 ⁄8 … 171 ⁄8 171 ⁄8

343 343 343 343 359 359 359 359 362 362 362 365 365 365 365 368 368 368 371 371 371 371 375 375 375 375 378 378 378 381 381 381 381 … 416 422 … 422 429 … 429 … 432 432 … 435 435 … 435 435 … 435 435

Morse Taper No. 3 3 3 3 3 … … … … … … … … … … … … … … … … … … … … … … … … … … … … … 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4

Larger or Smaller Shanka Flute Length Overall Length F L Inch mm Inch mm 190 308 71 ⁄2 121 ⁄8 77⁄8 77⁄8 77⁄8 77⁄8 … … … … … … … … … … … … … … … … … … … … … … … … … … … … … 93⁄4 93⁄8 95⁄8 95⁄8 95⁄8 97⁄8 97⁄8 97⁄8 10 10 10 101 ⁄8 101 ⁄8 101 ⁄8 101 ⁄8 101 ⁄8 101 ⁄8 101 ⁄8 101 ⁄8 103⁄8

200 200 200 200 … … … … … … … … … … … … … … … … … … … … … … … … … … … … … 238 238 244 244 244 251 251 251 254 254 254 257 257 257 257 257 257 257 257 264

121 ⁄2 121 ⁄2 121 ⁄2 121 ⁄2 … … … … … … … … … … … … … … … … … … … … … … … … … … … … … 15 15 151 ⁄4 151 ⁄4 151 ⁄4 151 ⁄2 151 ⁄2 151 ⁄2 155⁄8 155⁄8 155⁄8 153⁄4 153⁄4 153⁄4 153⁄4 153⁄4 153⁄4 153⁄4 153⁄4 161 ⁄4

318 318 318 318 … … … … … … … … … … … … … … … … … … … … … … … … … … … … … 381 381 387 387 387 394 394 394 397 397 397 400 400 400 400 400 400 400 400 413

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Table Table 5. (Continued) American National Taper-Shank 5. American National Taper-Shank Twist Twist Drills Drills Fractional and Metric Sizes ANSI/ASME B94.11M-1993

Fraction 147⁄64 13⁄4

125⁄32 113⁄16 127⁄32 17⁄8 129⁄32 115⁄16 131 ⁄32 2 21 ⁄32 21 ⁄16 23⁄32 21 ⁄8 25⁄32 23⁄16 27⁄32 21 ⁄4 25⁄16

23⁄8 27⁄16

21 ⁄2

29⁄16 25⁄8

211 ⁄16 23⁄4

213⁄16

Drill Diameter, D Equivalent mm

45.00 46.00

47.00 48.00 49.00 50.00

51.00 52.00 53.00

54.00 55.00 56.00 57.00 58.00 59.00 60.00 61.00 62.00 63.00 64.00 65.00 66.00 67.00 68.00 69.00 70.00 71.00

Decimal Inch 1.7344 1.7500 1.7717 1.7812 1.8110 1.8125 1.8438 1.8504 1.8750 1.8898 1.9062 1.9291 1.9375 1.9625 1.9688 2.0000 2.0079 2.0312 2.0472 2.0625 2.0866 2.0938 2.1250 2.1260 2.1562 2.1654 2.1875 2.2000 2.2188 2.2441 2.2500 2.2835 2.3125 2.3228 2.3622 2.3750 2.4016 2.4375 2.4409 2.4803 2.5000 2.5197 2.5591 2.5625 2.5984 2.6250 2.6378 2.6772 2.6875 2.7165 2.7500 2.7559 2.7953 2.8125

Morse Taper mm No. 44.054 …

44.450 45.000 45.242 46.000 46.038 46.833 47.000 47.625 48.000 48.417 49.000 49.212 50.000 50.008 50.800 51.000 51.592 52.000 52.388 53.000 53.183 53.975 54.000 54.767 55.000 55.563 56.000 56.358 57.000 57.150 58.000 58.738 59.000 60.000 60.325 61.000 61.912 62.000 63.000 63.500 64.000 65.000 65.088 66.000 66.675 67.000 68.000 68.262 69.000 69.850 70.000 71.000 71.438

5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5

Regular Shank Flute Length Overall Length F L Inch mm Inch mm … … … … 101 ⁄8 101 ⁄8 101 ⁄8 101 ⁄8 101 ⁄8 101 ⁄8 103⁄8 103⁄8 103⁄8 103⁄8 103⁄8 103⁄8 103⁄8 103⁄8 103⁄8 103⁄8 103⁄8 101 ⁄4 101 ⁄4 101 ⁄4 101 ⁄4 101 ⁄4 101 ⁄4 101 ⁄4 101 ⁄4 101 ⁄4 101 ⁄8 101 ⁄8 101 ⁄8 101 ⁄8 101 ⁄8 101 ⁄8 101 ⁄8 101 ⁄8 101 ⁄8 111 ⁄4 111 ⁄4 111 ⁄4 111 ⁄4 111 ⁄4 117⁄8 117⁄8 117⁄8 117⁄8 117⁄8 123⁄4 123⁄4 123⁄4 123⁄4 123⁄4 133⁄8 133⁄8 133⁄8

257 257 257 257 257 257 264 264 264 264 264 264 264 264 264 264 264 260 260 260 260 260 260 260 260 260 257 257 257 257 257 257 257 257 257 286 286 286 286 286 302 302 302 302 302 324 324 324 324 324 340 340 340

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171 ⁄8 171 ⁄8 171 ⁄8 171 ⁄8 171 ⁄8 171 ⁄8 173⁄8 173⁄8 173⁄8 173⁄8 173⁄8 173⁄8 173⁄8 173⁄8 173⁄8 173⁄8 173⁄8 173⁄8 173⁄8 173⁄8 173⁄8 173⁄8 173⁄8 173⁄8 173⁄8 173⁄4 173⁄8 173⁄8 173⁄8 173⁄8 173⁄8 173⁄8 173⁄8 173⁄8 173⁄8 183⁄4 183⁄4 183⁄4 183⁄4 183⁄4 191 ⁄2 191 ⁄2 191 ⁄2 191 ⁄2 191 ⁄2 203⁄8 203⁄8 203⁄8 203⁄8 203⁄8 211 ⁄8 211 ⁄8 211 ⁄8

435 435 435 435 435 435 441 441 441 441 441 441 441 441 441 441 441 441 441 441 441 441 441 441 441 441 441 441 441 441 441 441 441 441 441 476 476 476 476 476 495 495 495 495 495 518 518 518 518 518 537 537 537

Morse Taper No. 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … …

Larger or Smaller Shanka Flute Length Overall Length F L Inch mm Inch mm 264 413 103⁄8 161 ⁄4 103⁄4 103⁄8 103⁄8 103⁄8 103⁄8 103⁄8 101 ⁄2 101 ⁄2 101 ⁄2 101 ⁄2 105⁄8 105⁄8 105⁄8 105⁄8 105⁄8 … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … …

264 264 264 264 264 264 267 267 267 267 270 270 270 270 270 … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … …

161 ⁄4 161 ⁄4 161 ⁄4 161 ⁄4 161 ⁄4 161 ⁄4 161 ⁄2 161 ⁄2 161 ⁄2 161 ⁄2 165⁄8 165⁄8 165⁄8 165⁄8 165⁄8 … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … …

413 413 413 413 413 413 419 419 419 419 422 422 422 422 422 … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … …

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Table Table 5. (Continued) American National Taper-Shank 5. American National Taper-Shank Twist Twist Drills Drills Fractional and Metric Sizes ANSI/ASME B94.11M-1993

Drill Diameter, D Equivalent

Fraction

mm 72.00 73.00

27⁄8

74.00

215⁄16

75.00 76.00

3

77.00 78.00

31 ⁄8

31 ⁄4 31 ⁄2

Decimal Inch 2.8346 2.8740 2.8750 2.9134 2.9375 2.9528 2.9921 3.0000 3.0315 3.0709 3.1250 3.2500 3.5000

mm 72.000 73.000 73.025 74.000 74.612 75.000 76.000 76.200 77.000 78.000 79.375 82.550 88.900

Morse Taper No. 5 5 5 5 5 5 5 5 6 6 6 6 …

Regular Shank Flute Length Overall Length F L Inch mm Inch mm 3 1 340 537 21 ⁄8 13 ⁄8 340 537 133⁄8 211 ⁄8 340 537 133⁄8 211 ⁄8 14 356 552 213⁄4 14 356 552 213⁄4 3 552 14 356 21 ⁄4 14 356 552 213⁄4 14 356 552 213⁄4 371 622 145⁄8 241 ⁄2 5 1 371 622 14 ⁄8 24 ⁄2 5 1 371 622 14 ⁄8 24 ⁄2 394 648 151 ⁄2 251 ⁄2 … … … …

Morse Taper No. … … … … … … … … 5 5 5 5 5

a Larger or smaller than regular shank.

Larger or Smaller Shanka Flute Length Overall Length F L Inch mm Inch mm … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … 362 22 559 141 ⁄4 1 362 22 559 14 ⁄4 362 22 559 141 ⁄4 387 23 584 151 ⁄4 413 24 610 161 ⁄4

Table 6. American National Standard Combined Drills and Countersinks — Plain and Bell Types ANSI/ASME B94.11M-1993 PLAIN TYPE

60°

A C

L Size Designation 00 0 1 2 3 4 5 6 7 8

Size Designation

Body Diameter A Millimeters 1 ⁄8 3.18 1 ⁄8 3.18 1 ⁄8 3.18 3 ⁄16 4.76 1 ⁄4 6.35 5 ⁄16 7.94 7⁄16 11.11 1 ⁄2 12.70 5 ⁄8 15.88 3 ⁄4 19.05

11

12 13

14

15 16

17

18

Inches 1 ⁄8 3 ⁄16 1 ⁄4 5 ⁄16 7⁄16 1 ⁄2 5 ⁄8 3 ⁄4

A

mm

3.18 4.76

Inches 3 ⁄64 1 ⁄16 3 ⁄32

11.11

5 ⁄32

12.70 15.88 19.05

Plain Type Drill Diameter D Inches Millimeters .025 0.64 1 ⁄32 0.79 3 ⁄64 1.19 5 ⁄64 1.98 7⁄64 2.78 1 ⁄8 3.18 3 ⁄16 4.76 7⁄32 5.56 1 ⁄4 6.35 5 ⁄16 7.94

Drill Diameter

6.35 7.94

60°

C

Inches

Body Diameter

BELL TYPE

120° D

D

7⁄64

3 ⁄16 7⁄32 1 ⁄4

D

mm

1.19 1.59 2.38 2.78 3.97 4.76 5.56 6.35

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0.10 0.15 0.20 0.25 0.35

3 ⁄64 5 ⁄64 7⁄64 1 ⁄8 3 ⁄16 7⁄32 1 ⁄4 5 ⁄16

mm 2.5 3.8

Drill Length Inches 3 ⁄64 1 ⁄16

5.1

3 ⁄32

8.9

5 ⁄32

6.4

0.40

10.2

0.60

15.2

0.50

Drill Length C Millimeters 0.76 0.97 1.19 1.98 2.78 3.18 4.76 5.56 6.35 7.94

Inches .030 .038

Bell Diameter

Inches

E

L

Bell Type

E

A

12.7

7⁄64

3 ⁄16 7⁄32 1 ⁄4

C

Overall Length L Millimeters 29 11 ⁄8 29 11 ⁄8 32 11 ⁄4 7 48 1 ⁄8 2 51 54 21 ⁄8 70 23⁄4 3 76 1 83 3 ⁄4 89 31 ⁄2

Inches

Overall Length

mm

Inches

1.59

17⁄8 2

1.19 2.38 2.78 3.97

11 ⁄4

21 ⁄8

4.76

23⁄4 3

6.35

31 ⁄2

5.56

31 ⁄4

L

mm 32 48

51

54 70

76

83 89

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118° A

A

D

F L Drill Diameter 11/32” (8.737 mm and Smaller 118° Chamfer

D 60% of Drill Diameter

F L Drill Diameter 3/8” (9.525 mm) and Larger

Table 7. American National Standard Three- and Four-Flute TaperShank Core Drills — Fractional Sizes Only ANSI/ASME B94.11M-1993

Drill Diameter, D

Equivalent

Inch

Decimal Inch

mm

Morse Taper No. A

1 ⁄4

0.2500

6.350

1

0.3175

7.938

1

9 ⁄32 5 ⁄16 11 ⁄32 3 ⁄8 13 ⁄32 7⁄16 15 ⁄32 1 ⁄2 17⁄32 9 ⁄16 19 ⁄32 5 ⁄8 21 ⁄32 11 ⁄16 23 ⁄32 3 ⁄4 25 ⁄32 13 ⁄16 27⁄32 7⁄8 29 ⁄32 15 ⁄16 31 ⁄32

0.2812 0.3438 0.3750

7.142 8.733 9.525

0.4062

10.319

0.4688

11.908

0.4375

11.112

Three-Flute Drills Overall Length

Flute Length F Inch mm

1

31 ⁄4

83

61 ⁄2

1 1

1 1

31 ⁄2 35⁄8 37⁄8

41 ⁄8

47⁄8

124

47⁄8

124

0.6250 0.6562 0.6875 0.7188 0.7500 0.7812 0.8125 0.8438 0.8750 0.9062 0.9375 0.9688

15.815 16.668 17.462 18.258 19.050 19.842 20.638 21.433 22.225 23.019 23.812 24.608

2 2 2 2 2 2

45⁄8 47⁄8

51 ⁄8 53⁄8 55⁄8

3 3

61 ⁄8 63⁄8

1.0625

26.988

3

65⁄8

17⁄32 11 ⁄4

19⁄32

1.0938 1.1250 1.1562 1.1875 1.2188 1.2500 1.2812

26.192 27.783 28.575 29.367 30.162 30.958 31.750 32.542

3 4 4 4 4 4 4

…

61 ⁄2 67⁄8

71 ⁄8 71 ⁄4

73⁄8

71 ⁄2

77⁄8 …

…

…

…

…

…

…

…

…

… … …

… … …

…

81 ⁄4

83⁄4

184 210 216 222 222 222

…

…

43⁄8

111

2

47⁄8 47⁄8

2 2 2

45⁄8 47⁄8

130

229

2

51 ⁄8

143

91 ⁄2

241

2

55⁄8

91 ⁄4

156

93⁄4

103⁄4 103⁄4 103⁄4

235 248 251 273 273 273 273 273

2 2 2

53⁄8

61 ⁄8

3 3

61 ⁄8

162

279

3

165

111 ⁄8

283

3

61 ⁄2

4

67⁄8

168 175 181 184 187 190 200

…

Copyright 2020, Industrial Press, Inc.

11

111 ⁄4

121 ⁄2 123⁄4

127⁄8 13

131 ⁄8

131 ⁄2 …

279

286 318 324

3

3

61 ⁄8 63⁄8 63⁄8 65⁄8

4

71 ⁄8

330

4

73⁄8

343

4

327 333 …

4

71 ⁄4

4

71 ⁄2

4

81 ⁄2

77⁄8

… … …

83⁄4

222

124 130

83⁄4 9

229

143

91 ⁄2

241

117

124

137 149

156

3

3

… …

124

61 ⁄8

3

…

210

152

61 ⁄8

…

mm

81 ⁄4

57⁄8 6

103⁄4 11

162

…

2

83⁄4 9

156

…

…

83⁄4

156

…

…

124

156

…

…

81 ⁄2

149

…

…

190

117

137

…

71 ⁄2

71 ⁄4

156

61 ⁄8

…

…

3 3

…

…

103⁄4

61 ⁄8

171

Inch

…

61 ⁄8

3

165

mm

…

152

3

159

L

Inch

178

2

61 ⁄8

L

Four-Flute Drills Overall Length

Flute Length F

7

97⁄8

11 ⁄16

1.0312

111

63⁄4

57⁄8 6

63⁄8

13⁄16

98

105

2

2

3

15⁄32

92

14.288

25.400

11 ⁄8

89

0.5625

1.0000

13⁄32

79

43⁄8

15.083

162

61 ⁄4

2

0.5938

63⁄8

76

31 ⁄8

12.700 13.492

156

1

0.5000 0.5312

61 ⁄8

Inch

27⁄8 3

1

11 ⁄32

73

mm

Morse Taper No. A

81 ⁄2

83⁄4

91 ⁄4 93⁄4

97⁄8

156

103⁄4

156

103⁄4

156 156

103⁄4 103⁄4

216 222 222 235 248 251 273 273 273 273 273

162

103⁄4 11

279

165

111 ⁄8

283

162 168 175 181 184

11

111 ⁄4

121 ⁄2 123⁄4

279

286 318 324 327

187

127⁄8 13

330

200

131 ⁄2

343

190 216

131 ⁄8 141 ⁄8

333 359

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Machinery's Handbook, 31st Edition Twist Drills

952

Table Table 7. (Continued) American National Standard and Four-Flute 7. American National Standard Three-Threeand Four-Flute Taper-TaperShank Shank Core Core Drills Drills — — Fractional Fractional Sizes Sizes Only Only ANSI/ASME ANSI/ASME B94.11M-1993 B94.11M-1993 Drill Diameter, D Equivalent

Inch

Decimal Inch

111 ⁄32

1.3438

15⁄16

13⁄8

Morse Taper No. A

mm

1.3125

33.338

1.3750

34.925

…

34.133

… …

Three-Flute Drills Flute Length Overall Length F L Inch mm Inch mm …

…

…

…

…

…

…

…

…

…

…

…

Morse Taper No. A 4

85⁄8

219

141 ⁄4

362

4

87⁄8

225

141 ⁄2

368

9

229

145⁄8

371

91 ⁄4

235 238

147⁄8 15

378

244

165⁄8

4

113⁄32

1.4062

35.717

…

…

…

…

…

4

115⁄32

1.4688

37.306

…

…

…

…

…

4

17⁄16 11 ⁄2

117⁄32 19⁄16

119⁄32 15⁄8

121 ⁄32 111 ⁄16

123⁄32 13⁄4

125⁄32 113⁄16

127⁄32 17⁄8

129⁄32 115⁄16

1.4375 1.5000 1.5312 1.5675 1.5938 1.6250 1.6562 1.6875 1.7188 1.7500 1.7812 1.8125 1.8438 1.8750 1.9062 1.9375 1.9688

36.512

…

38.100

…

38.892

…

39.688

…

40.483

…

41.275

…

42.067

…

42.862

…

43.658

…

44.450

…

45.244

…

46.038

…

46.833

…

47.625

…

48.417

…

49.212

…

50.008

…

… … … … … … … … … … … … … … … … …

… … … … … … … … … … … … … … … … …

…

…

…

…

…

…

… …

…

… …

…

…

…

…

… … …

5

5

…

5

…

101 ⁄8 101 ⁄8

5

…

97⁄8 10

101 ⁄8

5

…

93⁄8

5

5

…

…

95⁄8

5

…

…

5

5

…

…

93⁄8

5

…

…

4

5

…

…

91 ⁄8

5

…

83⁄4

4

5

…

Four-Flute Drills Flute Length Overall Length F L Inch mm Inch mm

5

101 ⁄8 101 ⁄8 101 ⁄8 101 ⁄8 103⁄8 103⁄8 103⁄8 103⁄8

131 ⁄32 2

2.0000

50.800

…

…

…

…

…

5

103⁄8

21 ⁄4

2.2500

57.150

…

…

…

…

…

5

101 ⁄8

5

111 ⁄4

21 ⁄8 23⁄8

21 ⁄2

2.1250 2.3750 2.5000

53.975

…

60.325

…

63.500

…

… … …

… … …

…

…

…

5

…

…

5

…

101 ⁄4

101 ⁄8

222

232

238 251

143⁄8

143⁄4

163⁄8

365

375 381 416 422 429

254

167⁄8 17

432

257

171 ⁄8

435

257 257 257 257 257 257 264 264 264 264 264 260 257 257 286

171 ⁄8 171 ⁄8 171 ⁄8 171 ⁄8 171 ⁄8 171 ⁄8 173⁄8 173⁄8 173⁄8 173⁄8 173⁄8 173⁄8 173⁄8 173⁄8 183⁄4

435 435 435 435 435 435 441 441 441 441 441 441 441 441 476

Table 8. American National Standard Drill Drivers — Split-Sleeve, Collet-Type ANSI/ASME B94.35-1972 (R2015) B

D

A

E J – Taper per Foot K G

H L

Taper Number

G Overall Length

H Diameter at Gage Line

J Taper per Foota

K Length to Gage Line

L Driver Projection

1

2.62

0.475

0.59858

2.44

0.19

0b

2

2.38

3.19

0.356

0.700

0.62460

0.59941

2.22

2.94

0.16

0.25

a Taper rate in accordance with ANSI/ASME B5.10-1994 (R2013), Machine Tapers.

b Size 0 is not an American National Standard but is included here to meet special needs.

All dimensions are in inches.

Copyright 2020, Industrial Press, Inc.

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Machinery's Handbook, 31st Edition Twist Drills

953

Table 9. ANSI Three- and Four-Flute Straight-Shank Core Drills — Fractional Sizes Only ANSI/ASME B94.11M-1993 D

118° F L Drill Diameter 11/32” (8.733 mm) and Smaller 118° Chamfer

D 60% of Drill Diameter

F L Drill Diameter 3/8” (9.525 mm) and Smaller

Nominal Shank Size is Same Size as Nominal Drill Size Drill Diameter, D

Equivalent

Three-Flute Drills

Flute Length

Inch

Decimal Inch

mm

Inch

1 ⁄4

0.2500

6.350

0.3125

7.938

9 ⁄32 5 ⁄16 11 ⁄32 3 ⁄8 13 ⁄32 7⁄16 15 ⁄32 1 ⁄2 17⁄32 9 ⁄16 19 ⁄32 5 ⁄8 21 ⁄32 11 ⁄16 23 ⁄32 3 ⁄4 25 ⁄32 13 ⁄16 27⁄32 7⁄8 29 ⁄32 15 ⁄16 31 ⁄32

1

0.2812 0.3438 0.3750 0.4062 0.4375 0.4688

mm

Inch

33⁄4

95

4

102

43⁄8

111

7

43⁄4

121

71 ⁄2

43⁄4

121

8

203

83⁄4

222

11.112

11.908

45⁄8

15.875

47⁄8

0.7188 0.7500 0.7812 0.8125

0.8438 0.8750 0.9062

0.9375 0.9688

1.0000

162

10.317

47⁄8

0.6875

63⁄8

61 ⁄2

14.288

0.6562

156

105

0.5625 0.6250

61 ⁄8

41 ⁄8

12.700

0.5938

Inch

8.733

41 ⁄8

13.492

15.083

43⁄4

47⁄8

16.667

51 ⁄8

18.258

…

17.462

53⁄8

105 117

121

61 ⁄4

63⁄4

171

178

Inch

…

…

…

…

…

…

…

…

… … …

…

…

…

43⁄4

121

73⁄4

197

210

47⁄8

124

81 ⁄4

222

47⁄8

124

83⁄4

124

…

43⁄4

47⁄8

9

229

51 ⁄8

…

…

55⁄8

137

91 ⁄4

235

53⁄8

…

121 124

130

…

…

8

203

83⁄4

222

210 222

9

229

143

91 ⁄2

241

152

97⁄8

137

235

57⁄8

149

93⁄4

248

20.638

…

…

…

…

61 ⁄8

156

10

254

…

61 ⁄8

156

10

254

19.842 21.433 22.225 23.017

23.812 24.608

25.400

…

… … …

… …

…

…

… … …

… …

…

…

… … …

… …

…

…

… …

… …

…

57⁄8 6

61 ⁄8

61 ⁄8 61 ⁄8 63⁄8 63⁄8

…

61 ⁄2

13⁄32

1.0938

27.783

…

…

…

…

67⁄8

26.988 28.575

31.750

… …

…

… …

…

Copyright 2020, Industrial Press, Inc.

… …

…

… …

…

65⁄8

71 ⁄8 77⁄8

149

91 ⁄4

19.050

…

1.2500

…

…

197

190

…

11 ⁄4

…

…

73⁄4

…

1.1250

…

…

…

…

26.192

11 ⁄8

…

…

…

1.0312 1.0625

…

mm

…

11 ⁄32

11 ⁄16

L

mm

…

83⁄4

…

165

Overall Length

184

81 ⁄4

130

159

F

71 ⁄4

124 124

Four-Flute Drills

Flute Length

mm

37⁄8

9.525

98

L

7.142

0.5000 0.5312

F

Overall Length

156 156

93⁄4

10 10

156

103⁄4

162

11

162

11

248 251

254 254

273

279 279

165

111 ⁄8

283

175

111 ⁄2

292

200

121 ⁄2

168 181

111 ⁄4 113⁄4

286 298 318

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Size of Drill

Decimal Equivalent

Length of Point when Included Angle = 90°

Length of Point when Included Angle = 118°

Size of Drill

Decimal Equivalent

Length of Point when Included Angle = 90°

Length of Point when Included Angle = 118°

Size or Dia. of Drill

Decimal Equivalent

Length of Point when Included Angle = 90°

Length of Point when Included Angle = 118°

954

Dia. of Drill

Decimal Equivalent

Length of Point when Included Angle = 90°

Length of Point when Included Angle = 118°

60

0.0400

0.020

0.012

37

0.1040

0.052

0.031

14

0.1820

0.091

0.055

3 ⁄8

0.3750

0.188

0.113

59

0.0410

0.021

0.012

36

0.1065

0.054

0.032

13

0.1850

0.093

0.056

25 ⁄64

0.3906

0.195

0.117

58

0.0420

0.021

0.013

35

0.1100

0.055

0.033

12

0.1890

0.095

0.057

13 ⁄32

0.4063

0.203

0.122

57

0.0430

0.022

0.013

34

0.1110

0.056

0.033

11

0.1910

0.096

0.057

27⁄64

0.4219

0.211

0.127

56

0.0465

0.023

0.014

33

0.1130

0.057

0.034

10

0.1935

0.097

0.058

7⁄16

0.4375

0.219

0.131

55

0.0520

0.026

0.016

32

0.1160

0.058

0.035

9

0.1960

0.098

0.059

29 ⁄64

0.4531

0.227

0.136

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54

0.0550

0.028

0.017

31

0.1200

0.060

0.036

8

0.1990

0.100

0.060

15 ⁄32

0.4688

0.234

0.141

53

0.0595

0.030

0.018

30

0.1285

0.065

0.039

7

0.2010

0.101

0.060

31 ⁄64

0.4844

0.242

0.145

52

0.0635

0.032

0.019

29

0.1360

0.068

0.041

6

0.2040

0.102

0.061

1 ⁄2

0.5000

0.250

0.150

51

0.0670

0.034

0.020

28

0.1405

0.070

0.042

5

0.2055

0.103

0.062

33 ⁄64

0.5156

0.258

0.155

50

0.0700

0.035

0.021

27

0.1440

0.072

0.043

4

0.2090

0.105

0.063

17⁄32

0.5313

0.266

0.159

49

0.0730

0.037

0.022

26

0.1470

0.074

0.044

3

0.2130

0.107

0.064

35 ⁄64

0.5469

0.273

0.164

48

0.0760

0.038

0.023

25

0.1495

0.075

0.045

2

0.2210

0.111

0.067

9 ⁄16

0.5625

0.281

0.169

47

0.0785

0.040

0.024

24

0.1520

0.076

0.046

1

0.2280

0.114

0.068

37⁄64

0.5781

0.289

0.173

46

0.0810

0.041

0.024

23

0.1540

0.077

0.046

15 ⁄64

0.2344

0.117

0.070

19 ⁄32

0.5938

0.297

0.178

45

0.0820

0.041

0.025

22

0.1570

0.079

0.047

1 ⁄4

0.2500

0.125

0.075

39 ⁄64

0.6094

0.305

0.183

44

0.0860

0.043

0.026

21

0.1590

0.080

0.048

17⁄64

0.2656

0.133

0.080

5 ⁄8

0.6250

0.313

0.188

43

0.0890

0.045

0.027

20

0.1610

0.081

0.048

9 ⁄32

0.2813

0.141

0.084

41 ⁄64

0.6406

0.320

0.192

42

0.0935

0.047

0.028

19

0.1660

0.083

0.050

19 ⁄64

0.2969

0.148

0.089

21 ⁄32

0.6563

0.328

0.197

41

0.0960

0.048

0.029

18

0.1695

0.085

0.051

5 ⁄16

0.3125

0.156

0.094

43 ⁄64

0.6719

0.336

0.202

40

0.0980

0.049

0.029

17

0.1730

0.087

0.052

21 ⁄64

0.3281

0.164

0.098

11 ⁄16

0.6875

0.344

0.206

39

0.0995

0.050

0.030

16

0.1770

0.089

0.053

11 ⁄32

0.3438

0.171

0.103

23 ⁄32

0.7188

0.359

0.216

38

0.1015

0.051

0.030

15

0.1800

0.090

0.054

23 ⁄64

0.3594

0.180

0.108

3 ⁄4

0.7500

0.375

0.225

Machinery's Handbook, 31st Edition Twist Drills

Copyright 2020, Industrial Press, Inc.

Table 10. Length of Point on Twist Drills and Centering Tools

Machinery's Handbook, 31st Edition Twist Drills

955

British Standard Combined Drills and Countersinks (Center Drills).—BS 328: Part 2: 1972 (1990) provides dimensions of combined drills and countersinks for center holes. Three types of drill and countersink combinations are shown in this standard but are not given here. These three types will produce center holes without protecting chamfers, with protecting chamfers, and with protecting chamfers of radius form. Drill Drivers—Split-Sleeve, Collet-Type.—American National Standard ANSI/ASME B94.35-1972 (R2015) covers split-sleeve, collet-type drivers for driving straight shank drills, reamers, and similar tools, without tangs from 0.0390-inch through 0.1220-inch diameter, and with tangs from 0.1250-inch through 0.7500-inch diameter, including metric sizes. For sizes 0.0390 through 0.0595 inch, the standard taper number is 1, and the optional taper number is 0. For sizes 0.0610 through 0.1875 inch, the standard taper number is 1, first optional taper number is 0, and second optional taper number is 2. For sizes 0.1890 through 0.2520 inch, the standard taper number is 1, first optional taper number is 2, and second optional taper number is 0. For sizes 0.2570 through 0.3750 inch, the standard taper number is 1 and the optional taper number is 2. For sizes 0.3860 through 0.5625 inch, the standard taper number is 2 and the optional taper number is 3. For sizes 0.5781 through 0.7500 inch, the standard taper number is 3 and the optional taper number is 4. The depth B that the drill enters the driver is 0.44 inch for sizes 0.0390 through 0.0781 inch; 0.50 inch for sizes 0.0785 through 0.0938 inch; 0.56 inch for sizes 0.0960 through 0.1094 inch; 0.62 inch for sizes 0.1100 through 0.1220 inch; 0.75 inch for sizes 0.1250 through 0.1875 inch; 0.88 inch for sizes 0.1890 through 0.2500 inch; 1.00 inch for sizes 0.2520 through 0.3125 inch; 1.12 inches for sizes 0.3160 through 0.3750 inch; 1.25 inches for sizes 0.3860 through 0.4688 inch; 1.31 inches for sizes 0.4844 through 0.5625 inch; 1.47 inches for sizes 0.5781 through 0.6562 inch; and 1.62 inches for sizes 0.6719 through 0.7500 inch. British Standard Metric Twist Drills.—BS 328: Part 1:1959 (incorporating amend­ ments issued March 1960 and March 1964) covers twist drills made to inch and metric dimensions that are intended for general engineering purposes. ISO recommendations are taken into account. The accompanying tables give the standard metric sizes of Morse taper-shank twist drills and core drills, parallel-shank jobbing and long-series drills, and stub drills. All drills are right-hand cutting unless otherwise specified, and normal, slow, or quick helix angles may be provided. A “back-taper” is ground on the diameter from point to shank to provide longitudinal clearance. Core drills may have three or four flutes and are intended for opening up cast holes or enlarging machined holes, for example. The parallel-shank jobber, long-series drills, and stub drills are made without driving tenons. Morse taper shank drills with oversize dimensions are also listed, and Table 11 shows metric drill sizes superseding gage and letter-size drills, which are now obsolete in Britain. To meet special requirements, the Standard lists nonstandard sizes for the various types of drills. The limits of tolerance on cutting diameters, as measured across the lands at the outer corners of a drill, shall be h8, in accordance with BS 1916, Limits and Fits for Engineering (Part I, Limits and Tolerances); Table 14 shows the values common to the different types of drills mentioned before. The drills shall be permanently and legibly marked whenever possible, preferably by rolling, showing the size, and the manufacturer’s name or trademark. If they are made from high-speed steel, they shall be marked with the letters H.S. where practicable. Drill Elements: The following definitions of drill elements are given. Axis: The longitudinal center line. Body: That portion of the drill extending from the extreme cutting end to the commence­ ment of the shank.

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Machinery's Handbook, 31st Edition Twist Drills

956

Shank: That portion of the drill by which it is held and driven. Flutes: The grooves in the body of the drill that provide lips, permit the removal of chips and allow cutting fluid to reach the lips. Web (Core): The central portion of the drill situated between the roots of the flutes and extending from the point end toward the shank; the point end of the web or core forms the chisel edge. Lands: The cylindrical-ground surfaces on the leading edges of the drill flutes. The width of the land is measured at right angles to the flute helix. Body Clearance: The portion of the body surface that is reduced in diameter to provide diametral clearance. Heel: The edge formed by the intersection of the flute surface and the body clearance. Point: The sharpened end of the drill, consisting of all that part of the drill that is shaped to produce lips, faces, flanks, and chisel edge. Face: That portion of the flute surface adjacent to the lip on which the chip impinges as it is cut from the work. Flank: The surface on a drill point that extends behind the lip to the following flute. Lip (Cutting Edge): The edge formed by the intersection of the flank and face. Relative Lip Height: The relative position of the lips measured at the outer corners in a direction parallel to the drill axis. Outer Corner: The corner formed by the intersection of the lip and the leading edge of the land. Chisel Edge: The edge formed by the intersection of the flanks. Chisel Edge Corner: The corner formed by the intersection of a lip and the chisel edge. Table 11. British Standard Drills — Metric Sizes Superseding Gauge and Letter Sizes BS 328: Part 1:1959, Appendix B

Obsolete Drill Size 80 79 78 77 76 75 74 73 72 71 70 69 68 67 66 65 64 63 62 61 60 59

Recom­ mended Metric Size (mm) 0.35 0.38 0.40 0.45 0.50 0.52 0.58 0.60 0.65 0.65 0.70 0.75 1 ⁄32 in. 0.82 0.85 0.90 0.92 0.95 0.98 1.00 1.00 1.05

Obsolete Drill Size 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37

Recom­ mended Metric Size (mm) 1.05 1.10 3 ⁄64 in. 1.30 1.40 1.50 1.60 1.70 1.80 1.85 1.95 2.00 2.05 2.10 2.20 2.25 3 ⁄32 in. 2.45 2.50 2.55 2.60 2.65

Obsolete Drill Size 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15

Recom­ mended Metric Size (mm) 2.70 2.80 2.80 2.85 2.95 3.00 3.30 3.50 9 ⁄64 in. 3.70 3.70 3.80 3.90 3.90 4.00 4.00 4.10 4.20 4.30 4.40 4.50 4.60

Obsolete Drill Size 14 13 12 11 10 9 8 7 6 5 4 3 2 1 A B C D E F G H

Recom­ mended Metric Size (mm) 4.60 4.70 4.80 4.90 4.90 5.00 5.10 5.10 5.20 5.20 5.30 5.40 5.60 5.80

in. 6.00 6.10 6.20 1 ⁄4 in. 6.50 6.60 17⁄64 in.

15 ⁄64

Obsolete Drill Size I J K L M N O P Q R S T U V W X Y Z … … … …

Recom­ mended Metric Size (mm) 6.90 7.00 in. 7.40 7.50 7.70 8.00 8.20 8.40 8.60 8.80 9.10 9.30

9 ⁄32

in. 9.80 10.10 10.30 10.50 … … … … 3 ⁄8

Gauge and letter-size drills are now obsolete in the United Kingdom and should not be used in the production of new designs. The table is given to assist users in changing over to the recommended standard sizes.

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Machinery's Handbook, 31st Edition Twist Drills

957

Table 12. British Standard Morse Taper-Shank Twist Drills and Core Drills — Standard Metric Sizes BS 328: Part 1:1959 Diameter 3.00 3.20 3.50 3.80 4.00 4.20 4.50 4.80 5.00 5.20 5.50 5.80 6.00 6.20 6.50 6.80 7.00 7.20 7.50 7.80 8.00 8.20 8.50 8.80 9.00 9.20 9.50 9.80 10.00 10.20 10.50 10.80 11.00 11.20 11.50 11.80 12.00 12.20 12.50 12.80 13.00 13.20 13.50 13.80 14.00 14.25 14.50 14.75 15.00 15.25 15.50 15.75 16.00 16.25 16.50

Flute Length 33 36 39

Overall Length 114 117 120

43

123

47

128

52

133

57

138

63

144

69

150

75

156

81

87

94

162

168

175

101

182

108

189

114

212

120

218

125

223

Diameter 16.75 17.00 17.25 17.50 17.75 18.00 18.25 18.50 18.75 19.00 19.25 19.50 19.75 20.00 20.25 20.50 20.75 21.00 21.25 21.50 21.75 22.00 22.25 22.50 22.75 23.00 23.25 23.50 23.75 24.00 24.25 24.50 24.75 25.00 25.25 25.50 25.75 26.00 26.25 26.50 26.75 27.00 27.25 27.50 27.75 28.00 28.25 28.50 28.75 29.00 29.25

29.50 29.75 30.00

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Flute Length

Overall Length

125

223

130

228

135

233

140

238

145

243

150

248

155

253

155

276

160

281

165

286

170

291

175

296

175

296

Diameter 30.25 30.50 30.75 31.00 31.25 31.50 31.75 32.00 32.50 33.00 33.50 34.00 34.50 35.00 35.50 36.00 36.50 37.00 37.50 38.00 38.50 39.00 39.50 40.00 40.50 41.00 41.50 42.00 42.50 43.00 43.50 44.00 44.50 45.00 45.50 46.00 46.50 47.00 47.50 48.00 48.50 49.00 49.50 50.00 50.50 51.00 52.00 53.00 54.00 55.00 56.00 57.00 58.00 59.00 60.00

Flute Length

Overall Length

180

301

185

306

185

334

190

339

195

344

200

349

205

354

210

359

215

364

220

369

225

374

225

412

230

417

235

422

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Machinery's Handbook, 31st Edition Twist Drills

958

Table Table 12. (Continued) British Standard Taper-Shank 12. British Standard MorseMorse Taper-Shank Twist Twist DrillsDrills and Core Drills — Standard Metric Sizes BS 328: Part 1:1959 Diameter 61.00 62.00 63.00 64.00 65.00 66.00 67.00 68.00 69.00 70.00 71.00 72.00 73.00 74.00 75.00

Flute Length

Overall Length

240

427

245

432

250

437

250

437

255

442

Diameter 76.00 77.00 78.00 79.00 80.00 81.00 82.00 83.00 84.00 85.00 86.00 87.00 88.00 89.00 90.00

Flute Length

Overall Length

260

514

265

519

270

524

260

477

Diameter 91.00 92.00 93.00 94.00 95.00 96.00 97.00 98.00 99.00 100.00

Flute Length

Overall Length

275

529

280

534

All dimensions are in millimeters. Tolerances on diameters are given in the table below. Table 13 shows twist drills that may be supplied with the shank and length oversize, but they should be regarded as nonpreferred. The Morse taper shanks of these twist and core drills are as follows: 3.00 to 14.00 mm diameter, M.T. No. 1; 14.25 to 23.00 mm diameter, M.T. No. 2; 23.25 to 31.50 mm diameter, M.T. No. 3; 31.75 to 50.50 mm diameter, M.T. No. 4; 51.00 to 76.00 mm diameter, M.T. No. 5; 77.00 to 100.00 mm diameter, M.T. No. 6.

Table 13. British Standard Morse Taper-Shank Twist Drills — Metric Oversize Shank and Length Series BS 328: Part 1:1959 Dia. Range

Overall Length

M. T. No.

Dia. Range

Overall Length

M. T. No.

Dia. Range

Overall Length

M. T. No.

12.00 to 13.20

199

2

22.50 to 23.00

276

3

45.50 to 47.50

402

5

18.25 to 19.00

256

3

29.00 to 30.00

324

4

50.50

412

5

13.50 to 14.00

19.25 to 20.00

20.25 to 21.00

21.25 to 22.25

206

251

266

271

2

3

3

3

26.75 to 28.00

30.25 to 31.50

40.50 to 42.50

43.00 to 45.00

319

329

392

397

4

48.00 to 50.00

4

64.00 to 67.00

5

72.00 to 75.00

5

68.00 to 71.00

407

499

504

509

5

6

6

6

Diameters and lengths are given in millimeters. For the individual sizes within the diameter ranges given, see Table 12. This series of drills should be regarded as non-preferred.

Table 14. British Standard Limits of Tolerance on Diameter for Twist Drills and Core Drills — Metric Series BS 328: Part 1:1959 Drill Size (Diameter measured across lands at outer corners)

Tolerance (h8)

0 to 1 inclusive

Plus 0.000 to Minus 0.014

Over 3 to 6 inclusive

Plus 0.000 to Minus 0.018

Over 1 to 3 inclusive Over 6 to 10 inclusive

Over 10 to 18 inclusive Over 18 to 30 inclusive Over 30 to 50 inclusive Over 50 to 80 inclusive

Over 80 to 120 inclusive

All dimensions are given in millimeters.

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Plus 0.000 to Minus 0.014 Plus 0.000 to Minus 0.022 Plus 0.000 to Minus 0.027 Plus 0.000 to Minus 0.033 Plus 0.000 to Minus 0.039 Plus 0.000 to Minus 0.046 Plus 0.000 to Minus 0.054

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Machinery's Handbook, 31st Edition Twist Drills

959

6

22

7

24

8

26

9

28

10

30

11

32

12

34

14

36

16

38

1.35 1.40 1.45 1.50

18

40

1.55 1.60 1.65 1.70

20

43

27

53

30

57

33

61

36

65

39

70

43

75

47

52

80

86

9.60 9.70 9.80 9.90 10.00 10.10

63

101

69

109

75

81

87

117

125

133

Overall Length

20

49

93

Flute Length

5

24

57

Diameter

19

5.40 5.50 5.60 5.70 5.80 5.90 6.00 6.10 6.20 6.30 6.40 6.50 6.60 6.70 6.80 6.90 7.00 7.10 7.20 7.30 7.40 7.50 7.60 7.70 7.80 7.90 8.00 8.10 8.20 8.30 8.40 8.50 8.60 8.70 8.80 8.90 9.00 9.10 9.20 9.30 9.40 9.50

Overall Length

4

46

Flute Length

19

22

Diameter

3.0

1.75 1.80 1.85 1.90 1.95 2.00 2.05 2.10 2.15 2.20 2.25 2.30 2.35 2.40 2.45 2.50 2.55 2.60 2.65 2.70 2.75 2.80 2.85 2.90 2.95 3.00 3.10 3.20 3.30 3.40 3.50 3.60 3.70 3.80 3.90 4.00 4.10 4.20 4.30 4.40 4.50 4.60 4.70 4.80 4.90 5.00 5.10 5.20 5.30

Overall Length

19

Flute Length

2.5

Diameter

Overall Length

0.20 0.22 0.25 0.28 0.30 0.32 0.35 0.38 0.40 0.42 0.45 0.48 0.50 0.52 0.55 0.58 0.60 0.62 0.65 0.68 0.70 0.72 0.75 0.78 0.80 0.82 0.85 0.88 0.90 0.92 0.95 0.98 1.00 1.05 1.10 1.15 1.20 1.25 1.30

Flute Length

Diameter

Table 15. British Standard Parallel Shank Jobber Series Twist Drills — Standard Metric Sizes BS 328: Part 1:1959

87

133

94

142

101

151

108

160

14.25 14.50 14.75 15.00

114

169

15.25 15.50 15.75 16.00

120

178

10.20 10.30 10.40 10.50 10.60 10.70 10.80 10.90 11.00 11.10 11.20 11.30 11.40 11.50 11.60 11.70 11.80 11.90 12.00 12.10 12.20 12.30 12.40 12.50 12.60 12.70 12.80 12.90 13.00 13.10 13.20 13.30 13.40 13.50 13.60 13.70 13.80 13.90 14.00

All dimensions are in millimeters. Tolerances on diameters are given in Table 14.

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Machinery's Handbook, 31st Edition Twist Drills

960

Table 16. British Standard Parallel Shank Long Series Twist Drills — Standard Metric Sizes BS 328: Part 1:1959 Diameter 2.00 2.05 2.10 2.15 2.20 2.25 2.30 2.35 2.40 2.45 2.50 2.55 2.60 2.65 2.70 2.75 2.80 2.85 2.90 2.95 3.00 3.10 3.20 3.30 3.40 3.50 3.60 3.70 3.80 3.90 4.00 4.10 4.20 4.30 4.40 4.50 4.60 4.70 4.80 4.90 5.00 5.10 5.20 5.30 5.40 5.50 5.60 5.70 5.80 5.90 6.00 6.10 6.20 6.30 6.40 6.50 6.60 6.70

Flute Length

Overall Length

56

85

59

90

62

95

66

69

73

78

100

106

112

119

82

126

87

132

91

97

139

148

Diameter 6.80 6.90 7.00 7.10 7.20 7.30 7.40 7.50 7.60 7.70 7.80 7.90 8.00 8.10 8.20 8.30 8.40 8.50 8.60 8.70 8.80 8.90 9.00 9.10 9.20 9.30 9.40 9.50 9.60 9.70 9.80 9.90 10.00 10.10 10.20 10.30 10.40 10.50 10.60 10.70 10.80 10.90 11.00 11.10 11.20 11.30 11.40 11.50 11.60 11.70 11.80 11.90 12.00 12.10 12.20 12.30 12.40 12.50 12.60

Flute Length

Overall Length

102

156

109

115

121

128

134

165

175

184

195

205

Diameter 12.70 12.80 12.90 13.00 13.10 13.20 13.30 13.40 13.50 13.60 13.70 13.80 13.90 14.00 14.25 14.50 14.75 15.00 15.25 15.50 15.75 16.00 16.25 16.50 16.75 17.00 17.25 17.50 17.75 18.00 18.25 18.50 18.75 19.00 19.25 19.50 19.75 20.00 20.25 20.50 20.75 21.00 21.25 21.50 21.75 22.00 22.25 22.50 22.75 23.00 23.25 23.50 23.75 24.00 24.25 24.50 24.75 25.00

Flute Length

Overall Length

134

205

140

214

144

220

149

227

154

235

158

241

162

247

166

254

171

261

176

268

180

275

185

282

All dimensions are in millimeters. Tolerances on diameters are given in Table 14.

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Machinery's Handbook, 31st Edition Twist Drills

961

3.80 4.00 4.20 4.50 4.80

22

55

24 26

58 62

66

31

70

34

74

37

79

40

84

Overall Length

49 52

28

Flute Length

46

18 20

62

Diameter

16

6.20 6.50 6.80 7.00 7.20 7.50 7.80 8.00 8.20 8.50 8.80 9.00 9.20

26

Overall Length

5.00 5.20 5.50 5.80 6.00

Flute Length

Diameter

20 24 26 30 32 36 38 40 43

Diameter

Overall Length

3 5 6 8 9 11 12 13 14

Overall Length

Flute Length

0.50 0.80 1.00 1.20 1.50 1.80 2.00 2.20 2.50 2.80 3.00 3.20 3.50

Flute Length

Diameter

Table 17. British Standard Stub Drills — Metric Sizes BS 328: Part 1:1959

9.50 9.80 10.00 10.20 10.50

40

84

14.00 14.50 15.00 15.50 16.00

54

107

10.80 11.00 11.20 11.50 11.80 12.00 12.20 12.50 12.80 13.00 13.20 13.50 13.80

43

47

89

95

51

102

54

107

16.50 17.00 17.50 18.00 18.50 19.00 19.50 20.00 21.00 22.00 23.00 24.00 25.00

56

111

58

115

60

119

62

123

64

127

66

131

68 70 72

136 141 146

75

151

All dimensions are given in millimeters. Tolerances on diameters are given in Table 14.

Steels for Twist Drills.—Twist drill steels need good toughness, abrasion resistance, and ability to resist softening due to heat generated by cutting. The amount of heat generated indicates the type of steel that should be used. Carbon Tool Steel may be used where little heat is generated during drilling. High-Speed Steel is preferred because of its combination of red hardness and wear resis­ tance, which permits higher operating speeds and increased productivity. Optimum proper­ties can be obtained by selection of alloy analysis and heat treatment. Cobalt High-Speed Steel alloys have higher red hardness than standard high-speed steels, permitting drilling of materials such as heat-resistant alloys and materials with hardness greater than 38 RC (Rockwell C scale). These high-speed drills can withstand cutting speeds beyond the range of conventional high-speed steel drills and have superior resistance to abrasion but are not equal to tungsten carbide-tipped tools. Accuracy of Drilled Holes.—Normally, the diameter of drilled holes is not given a toler­ ance; the size of the hole is expected to be as close to the drill size as can be obtained. The accuracy of holes drilled with a two-fluted twist drill is influenced by many factors, including accuracy of the drill point; size of the drill; length and shape of the chisel edge; whether or not a bushing is used to guide the drill; work material; length of the drill; runout of the spindle and the chuck; rigidity of the machine tool, workpiece, and the setup; and cutting fluid used, if any. The diameter of the drilled holes will be oversize in most materials. The table Oversize Diameters in Drilling on page 962 provides the results of tests reported by The United States Cutting Tool Institute in which the diameters of over 2800 holes drilled in steel and cast iron were measured. The values in this table indicate what might be expected under average shop conditions; however, when the drill point is accurately ground and the other machining conditions are correct, the resulting hole size is more likely to be between the mean and average minimum values given in this table. If the drill is ground and used incor­rectly, holes that are even larger than the average maximum values can result.

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Machinery's Handbook, 31st Edition twist drills

962

Oversize Diameters in Drilling Drill Dia., Inch 1 ⁄16 1 ⁄8 1 ⁄4

Amount Oversize, Inch Average Max. Mean Average Min. 0.002 0.0015 0.001 0.0045 0.003 0.001 0.0065 0.004 0.0025

Drill Dia., Inch 1 ⁄2 3 ⁄4

1

Amount Oversize, Inch Average Max. Mean Average Min. 0.008 0.005 0.003 0.008 0.005 0.003 0.009 0.007 0.004

Courtesy of The United States Cutting Tool Institute

Some conditions will cause the drilled hole to be undersize. For example, holes drilled in light metals and in other materials having a high coefficient of thermal expansion such as plastics may contract to a size that is smaller than the diameter of the drill as the material surrounding the hole is cooled after having been heated by the drilling. The elastic action of the material surrounding the hole may also cause the drilled hole to be undersize when drilling high-strength materials with a drill that is dull at its outer corner. The accuracy of the drill point has a great effect on the accuracy of the drilled hole. An inaccurately ground twist drill will produce holes that are excessively oversize. The drill point must be symmetrical; i.e., the point angles must be equal, as well as the lip lengths and the axial height of the lips. Any alterations to the lips or to the chisel edge, such as thin­ ning the web, must be done carefully to preserve the symmetry of the drill point. Adequate relief should be provided behind the chisel edge to prevent heel drag. On conventionally ground drill points, this relief can be estimated by the chisel edge angle. When drilling a hole, as the drill point starts to enter the workpiece, the drill will be unsta­ble and will tend to wander. Then, as the body of the drill enters the hole, the drill will tend to stabilize. The result of this action is a tendency to drill a bellmouth shape in the hole at the entrance and perhaps beyond. Factors contributing to bellmouthing are: an unsymmet­r ically ground drill point; a large chisel edge length; inadequate relief behind the chisel edge; runout of the spindle and the chuck; using a slender drill that will bend easily; and lack of rigidity of the machine tool, workpiece, or the setup. Correcting these conditions as required will reduce the tendency for bellmouthing to occur and improve the accuracy of the hole diameter and its straightness. Starting the hole with a short stiff drill, such as a cen­ter drill, will quickly stabilize the drill that follows and reduce or eliminate bellmouthing; this procedure should always be used when drilling in a lathe, where the work is rotating. Bellmouthing can also be eliminated almost entirely and the accuracy of the hole improved by using a close-fitting drill jig bushing placed close to the workpiece. Although specific recommendations cannot be made, many cutting fluids will help to increase the accuracy of the diameters of drilled holes. Double-margin twist drills, available in the smaller sizes, will drill a more accurate hole than conventional twist drills having only a single margin at the leading edge of the land. The second land, located on the trailing edge of each land, pro­vides greater stability in the drill bushing and in the hole. These drills are especially useful in drilling intersecting off-center holes. Single and double-margin step drills, also available in the smaller sizes, will produce very accurate drilled holes, which are usually less than 0.002 inch (0.051 mm) larger than the drill size. Counterboring.—Counterboring (called spot-facing if the depth is shallow) is the enlargement of a previously formed hole. Counterbores for screw holes are generally made in sets. Each set contains three counterbores: one with the body of the size of the screw head and the pilot the size of the hole to admit the body of the screw; one with the body the size of the head of the screw and the pilot the size of the tap drill; and the third with the body the size of the body of the screw and the pilot the size of the tap drill. Counterbores are usually provided with helical flutes to provide positive effective rake on the cutting edges. The four flutes are so positioned that the end teeth cut ahead of center to provide a shearing action and eliminate chatter in the cut. Three designs are most common: solid, two-piece, and three-piece. Solid designs have the body, cutter, and pilot all in one piece. Two-piece designs have an integral shank and counterbore cutter, with an interchangeable pilot, and provide true concentricity of the cutter diameter with the shank while

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Machinery's Handbook, 31st Edition counterbores

963

allowing use of various pilot diameters. Three-piece counterbores have separate holder, counterbore cutter, and pilot, so that a holder will take any size of counterbore cutter. Each counterbore cutter, in turn, can be fitted with any suitable size diameter of pilot. Counterbores for brass are fluted straight. Counterbores with Interchangeable Cutters and Guides

F

B

A

D

E

L No. of Holder

No. of Morse Taper Shank

2

2 or 3

4

4 or 5

1

3

1 or 2

Range of Cutter Diameters, A

Range of Pilot Diameters, B

Total Length, C

11 ⁄8 -19⁄16

11 ⁄16 -11 ⁄8

91 ⁄2

21 ⁄8 -31 ⁄2

1-21 ⁄8

3 ⁄4 -11 ⁄16

3 or 4

15⁄8 -21 ⁄16

1 ⁄2 -3 ⁄4

7⁄8 -15 ⁄8

71 ⁄4

121 ⁄2 15

Length of Cutter Body, D 1

13⁄8 13⁄4

21 ⁄4

Length of Pilot, E

Dia. of Shank, F

5 ⁄8

3 ⁄4

7⁄8

11 ⁄8

11 ⁄8

15⁄8

13⁄8

21 ⁄8

Small counterbores are often made with three flutes, but should then have the size plainly stamped on them before fluting, as they cannot afterwards be conveniently measured. The flutes should be deep enough to come below the surface of the pilot. The counterbore should be relieved on the end of the body only, and not on the cylindrical surface. To facil­itate the relieving process, a small neck is turned between the guide and the body for clear­ance. The amount of clearance on the cutting edges is, for general work, from 4 to 5 degrees. The accompanying table gives dimensions for straight shank counterbores. Solid Counterbores with Integral Pilot Pilot Diameters Counterbore Diameters 0.110

0.133 0.155 0.176 0.198 0.220 0.241 0.285 0.327 0.372

Nominal 0.060

0.073 0.086 0.099 0.112

0.125 0.138 0.164 0.190 0.216

13 ⁄32

1 ⁄4

+1 ⁄64

0.076

0.089 0.102 0.115

0.128

0.141 0.154 0.180 0.206 0.232 17⁄64

+1 ⁄32 …

… …

… …

… … … … …

9 ⁄32

Straight Shank Diameter 7⁄64 1 ⁄8 5 ⁄32 11 ⁄64 3 ⁄16 3 ⁄16 7⁄32 1 ⁄4 9 ⁄32 5 ⁄16 3 ⁄8

1 ⁄2

5 ⁄16

21 ⁄64

11 ⁄32

3 ⁄8

19 ⁄32

3 ⁄8

25 ⁄64

13 ⁄32

1 ⁄2

11 ⁄16

7⁄16

29 ⁄64

15 ⁄32

1 ⁄2

25 ⁄32

1 ⁄2

33 ⁄64

17⁄32

1 ⁄2

Overall Length Short

Long

21 ⁄2

…

21 ⁄2

…

21 ⁄2

…

21 ⁄2

…

21 ⁄2

…

21 ⁄2

…

21 ⁄2

…

21 ⁄2

…

23⁄4

…

31 ⁄2

51 ⁄2

4

6

23⁄4

31 ⁄2 4 5

… 51 ⁄2 6 7

All dimensions are in inches.

Three-Piece Counterbores.—Data shown for the first two styles of counterbores are for straight-shank designs. These tools are also available with taper shanks in most sizes. Sizes of taper shanks for cutter diameters of 1 ⁄4 to 9⁄16 in. are No. 1; for 19⁄32 to 7⁄8 in., No. 2; for 15⁄16 to 13⁄8 in., No. 3; for 11 ⁄2 to 2 in., No. 4; and for 21 ⁄8 to 21 ⁄2 in., No. 5.

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Machinery's Handbook, 31st Edition COUNTERBORES AND CARBIDE BORING TOOLS

964

Counterbore Sizes for Hex-Head Bolts and Nuts.—Table 3a, page 1689, shows the maximum socket wrench dimensions for standard 1 ⁄4 -, 1 ⁄2 - and 3⁄4 -inch drive socket sets. For a given socket size (nominal size equals the maximum width across the flats of nut or bolt head), the dimension K given in the table is the minimum counterbore diameter required to provide socket wrench clearance for access to the bolt or nut. Sintered Carbide Boring Tools.—Industrial experience has shown that the shapes of tools used for boring operations need to be different from those of single-point tools ordi­narily used for general applications such as lathe work. Accordingly, Section 5 of Ameri­can National Standard ANSI B212.1-2002 gives standard sizes, styles and designations for four basic types of sintered carbide boring tools, namely: solid carbide square; carbide-tipped square; solid carbide round; and carbide-tipped round boring tools. In addition to these ready-to-use standard boring tools, solid carbide round and square unsharpened bor­ing tool bits are provided.

Style Designations for Carbide Boring Tools: Table 1 shows designations used to spec­ ify the styles of American Standard sintered carbide boring tools. The first letter denotes solid (S) or tipped (T). The second letter denotes square (S) or round (R). The side cutting edge angle is denoted by a third letter (A through H) to complete the style designation. Solid square and round bits with the mounting surfaces ground but the cutting edges unsharpened (Table 3) are designated using the same system except that the third letter indicating the side cutting edge angle is omitted. Table 1. American National Standard Sintered Carbide Boring Tools — Style Designations ANSI B212.1-2002 (R2007) Side Cutting Edge Angle E Degrees

Designation

0 10 30 40 45 55 90 (0° Rake) 90 (10° Rake)

A B C D E F G H

Boring Tool Styles Solid Square (SS)

SSC SSE

Tipped Square (TS) TSA TSB TSC TSD TSE TSF

Solid Round (SR)

Tipped Round (TR)

SRC

TRC

SRE

TRE TRG TRH

Size Designation of Carbide Boring Tools: Specific sizes of boring tools are identified by the addition of numbers after the style designation. The first number denotes the diam­eter or square size in number of 1 ⁄32 nds for types SS and SR and in number of 1 ⁄16 ths for types TS and TR. The second number denotes length in number of 1 ⁄ 8ths for types SS and SR. For styles TRG and TRH, a letter “U” after the number denotes a semi-finished tool (cut­ ting edges unsharpened). Complete designations for the various standard sizes of carbide boring tools are given in Table 2 through Table 7. In the diagrams in the tables, angles shown without tolerance are ± 1°.

Examples of Tool Designation: The designation TSC-8 indicates a carbide-tipped tool (T); square cross section (S); 30-degree side cutting edge angle (C); and 8 ⁄16 or 1 ⁄2 inch square size (8). The designation SRE-66 indicates a solid carbide tool (S); round cross section (R); 45-degree side cutting edge angle (E); 6 ⁄32 or 3⁄16 inch diameter (6); and 6 ⁄8 or 3⁄4 inch long (6).

The designation SS-610 indicates a solid carbide tool (S); square cross section (S); 6 ⁄32 or 3⁄16 inch square size (6); 10 ⁄8 or 11 ⁄4 inches long (10). It should be noted in this last example that the absence of a third letter (from A to H) indi­cates that the tool has its mounting surfaces ground but that the cutting edges are unsharp­ened.

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Machinery's Handbook, 31st Edition Carbide Boring Tools

965

Table 2. ANSI Carbide-Tipped Round General-Purpose Square-End Boring Tools Style TRG with 0° Rake and Style TRH with 10° Rake ANSI B212.1-2002 (R2007) Tool Designation and Carbide Grade

90° ± 1° D + 0.0005 + 0.0015

W C±

30° REF

1 16

B

Rake Angle ± 1°

T

M

H ± 0.010 6° ± 1°

L

Tool Designation Finished

Shank Dimensions, Inches

Semifinisheda

TRG-5

TRG-5U

TRH-5

TRH-5U

TRG-6

TRG-6U

TRH-6

TRH-6U

TRG-7

TRG-7U

TRH-7

TRH-7U

TRG-8

TRG-8U

TRH-8

TRH-8U

Dia. D

Length C

Dim.Over Flat B

Nose Height H

Set­back M (Min)

19 ⁄64

±.005

3 ⁄16

3 ⁄16

7⁄32

3 ⁄16

11 ⁄32

7⁄32

11 ⁄2

5 ⁄16

13⁄4

3 ⁄8

±.010 ±.010

9 ⁄32

±.010

0 10 0 10 0

1 ⁄4

11 ⁄32

0

10

3 ⁄16

5 ⁄16

15 ⁄32

21 ⁄2

1 ⁄2

1 ⁄4

Tip Dimensions, Inches

Rake Angle Deg.

3 ⁄16

1 ⁄4

13 ⁄32

21 ⁄2

7⁄16

Optional Design

10° ± 2°

10

Tip No.

T

W

L

1025

1 ⁄16

1 ⁄4

1 ⁄4

1030

1 ⁄16

5 ⁄16

1 ⁄4

1080

3 ⁄32

5 ⁄16

3 ⁄8

1090

3 ⁄32

3 ⁄8

3 ⁄8

a Semifinished tool will be without Flat (B) and carbide unground on the end.

Table 3. Solid Carbide Square and Round Boring Tool Bits +0 B –.002 +0 A –.002

C

Square Bits A

B

C

SS-58

5 ⁄32

5 ⁄32

1

SS-610

3 ⁄16

3 ⁄16

SS-810

11 ⁄4

1 ⁄4

1 ⁄4

SS-1012

11 ⁄4

5 ⁄16

5 ⁄16

SS-1214

11 ⁄2

3 ⁄8

3 ⁄8

13⁄4

Tool Designation

C

10°

Solid Carbide Square Boring Tool Bits Style SS

Tool Designation

+.0005 –.0015

D

Solid Carbide Square Boring Tool Bits Style SR Round Bits

D

C

Tool Designation

SR-33

3 ⁄32

3 ⁄8

SR-55

5 ⁄32

SR-34

3 ⁄32

1 ⁄2

SR-64

3 ⁄16

1 ⁄2

SR-810

1 ⁄4

SR-44

1 ⁄2

SR-66

3 ⁄16

3 ⁄4

SR-1010

11 ⁄4

1 ⁄8

5 ⁄16

SR-46

3 ⁄4

SR-69

3 ⁄16

…

SR-48

1 ⁄8

1

SR-77

11 ⁄8

…

11 ⁄4

1 ⁄8

7⁄32

7⁄8

…

…

C

D

C

5 ⁄8

SR-88

1 ⁄4

1

All dimensions are in inches. Tolerance on Length: Through 1 inch, + 1 ⁄32 , - 0; over 1 inch, +1 ⁄16 , -0.

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Tool Designation

D

… …

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6° ± 1°

Tool Designation and Carbide Grade

Optional Design

60

SSE-58

45

SSC-610

60

SSE-610

45

SSC-810

60

SSE-810

45

SSC-1012

60

SSE-1012

45

Height B

Length C

5 ⁄32

5 ⁄32

1

3 ⁄16

3 ⁄16

11 ⁄4

1 ⁄4

1 ⁄4

11 ⁄4

5 ⁄16

5 ⁄16

11 ⁄2

Bor. Bar Angle from Axis, Deg.

SSC-58

Width A

Tool Designation

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Boring Bar Angle, Deg. from Axis

Shank Dimensions, Inches

45

TRC-5

60

TRE-5

45

60

TRC-6

60

53

45

TRE-6

45

30

38

60

TRC-7

60

45

53

45

TRE-7

45

30

38

60

TRC-8

60

45

53

45

TRE-8

45

Side Cutting Edge Angle E,Deg.

End Cutting Edge Angle G,Deg.

Shoulder Angle F,Deg.

30

38

60

45

53

30

38

45

L

12° ± 2° Along angle “G”

Shank Dimensions, Inches D

C

5 ⁄16

11 ⁄2

3 ⁄8

13⁄4

7⁄16

21 ⁄2

1 ⁄2

21 ⁄2

B 19 ⁄64

±.005 11 ⁄32

±.010 13 ⁄32

±.010 15 ⁄32

±.010

H ± 0.010 6° ± 1° Along angle “G”

H 7⁄32

9 ⁄32

5 ⁄16

3 ⁄8

R 1 ⁄64

Shoulder Angle F, Deg.

6° ± 1° Along Angle “G”

8° ± 2° 6° ± 1°

6° ± 1°

16

+ 0.000 B – 0.002

Tool Designation

1 C±

End Cut. Edge Angle G, Deg.

Tool Designation and Carbide Grade

T

1 to sharp corner D/2 ± 64

F ± 1°

B

A ± 0.005 to Sharp Corner 2

1 64 12° ± 1°

W

Side Cut. Edge Angle E, Deg.

C±

R

30

38

60

Tip Dimensions, Inches Tip No.

T

W

L

2020

1 ⁄16

3 ⁄16

1 ⁄4

±.005

45

53

45

1 ⁄64

30

38

60

2040

3 ⁄32

3 ⁄16

5 ⁄16

±.005

45

53

45

2020

1 ⁄16

3 ⁄16

1 ⁄4

60

1 ⁄32

30

38

±.010

45

53

45

2060

3 ⁄32

1 ⁄4

3 ⁄8

1 ⁄32

30

38

60

2060

3 ⁄32

1 ⁄4

3 ⁄8

±.010

45

53

45

2080

3 ⁄32

5 ⁄16

3 ⁄8

Machinery's Handbook, 31st Edition Carbide Boring Tools

0.010 R ± 0.003 E ± 1°

6° ± 1°

G ± 1°

F Ref

D +0.0005 –0.0015

6° ± 1°

G ± 1°

F REF + 0.000 A – 0.002

Table 5. ANSI Carbide-Tipped Round Boring Tools Style TRC for 60° Boring Bar and Style TRE for 45° Boring Bar ANSI B212.1-2002 (R2007)

966

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Table 4. ANSI Solid Carbide Square Boring Tools Style SSC for 60° Boring Bar and Style SSE for 45° Boring Bar ANSI B212.1-2002 (R2007)

Machinery's Handbook, 31st Edition Carbide Boring Tools

967

Table 6. ANSI Carbide-Tipped Square Boring Tools — ANSI B212.1-2002 (R2007) Styles TSA and TSB for 90° Boring Bar, Styles TSC and TSD for 60° Boring Bar, and Styles TSE and TSF for 45° Boring Bar

G ± 1°

Shoulder angle Ref F

W

R Ref to Sharp Corner

A +0.000 –0.010 1 C±

T

E ± 1°

L

16

B 5 ⁄16

5 ⁄16

5 ⁄16

5 ⁄16 5 ⁄16 5 ⁄16 5 ⁄16 3 ⁄8 3 ⁄8 3 ⁄8 3 ⁄8 3 ⁄8 3 ⁄8

5 ⁄16 5 ⁄16 5 ⁄16 5 ⁄16 3 ⁄8 3 ⁄8 3 ⁄8 3 ⁄8 3 ⁄8 3 ⁄8

7⁄16

7⁄16

7⁄16

7⁄16

7⁄16 7⁄16 7⁄16 7⁄16 1 ⁄2 1 ⁄2 1 ⁄2 1 ⁄2 1 ⁄2 1 ⁄2 5 ⁄8 5 ⁄8 5 ⁄8 5 ⁄8 5 ⁄8 5 ⁄8 3 ⁄4 3 ⁄4 3 ⁄4 3 ⁄4 3 ⁄4 3 ⁄4

7⁄16 7⁄16 7⁄16 7⁄16 1 ⁄2 1 ⁄2 1 ⁄2 1 ⁄2 1 ⁄2 1 ⁄2 5 ⁄8 5 ⁄8 5 ⁄8 5 ⁄8 5 ⁄8 5 ⁄8 3 ⁄4 3 ⁄4 3 ⁄4 3 ⁄4 3 ⁄4 3 ⁄4

C 11 ⁄2 11 ⁄2 11 ⁄2 11 ⁄2 11 ⁄2 11 ⁄2 13⁄4 13⁄4 13⁄4 13⁄4 13⁄4 13⁄4 21 ⁄2 21 ⁄2 21 ⁄2 21 ⁄2 21 ⁄2 21 ⁄2 21 ⁄2 21 ⁄2 21 ⁄2 21 ⁄2 21 ⁄2 21 ⁄2 3 3 3 3 3 3 31 ⁄2 31 ⁄2 31 ⁄2 31 ⁄2 31 ⁄2 31 ⁄2

R

JK 1 NO KK 64 OO KK ! OO KK OO L0.005P

KJK 1 ONO KK 32 OO KK ! OO K O L0.010P

JK 1 NO KK 32 OO KK O KK ! OOO 0.010 L P

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Shoulder Angle F, Deg.

A 5 ⁄16

B +0.000 –0.010 End Cut. Edge Angle G, Deg.

Bor. Bar Angle from Axis, Deg. 90 90 60 60 45 45 90 90 60 60 45 45 90 90 60 60 45 45 90 90 60 60 45 45 90 90 60 60 45 45 90 90 60 60 45 45

Shank Dimensions, Inches

0° ± 1° Along angle “G” 10° ± 2° Along angle “G”

SideCut. Edge Angle E, Deg.

Tool Designation

12° ± 1° Tool Designation and Carbide Grade

TSA-5 TSB-5 TSC-5 TSD-5 TSE-5 TSF-5 TSA-6 TSB-6 TSC-6 TSD-6 TSE-6 TSF-6 TSA-7 TSB-7 TSC-7 TSD-7 TSE-7 TSF-7 TSA-8 TSB-8 TSC-8 TSD-8 TSE-8 TSF-8 TSA-10 TSB-10 TSC-10 TSD-10 TSE-10 TSF-10 TSA-12 TSB-12 TSC-12 TSD-12 TSE-12 TSF-12

10° ± 1° 7° ± 1° 6° ± 1°

0 10 30 40 45 55 0 10 30 40 45 55 0 10 30 40 45 55 0 10 30 40 45 55 0 10 30 40 45 55 0 10 30 40 45 55

8 8 38 38 53 53 8 8 38 38 53 53 8 8 38 38 53 53 8 8 38 38 53 53 8 8 38 38 53 53 8 8 38 38 53 53

90 90 60 60 45 45 90 90 60 60 45 45 90 90 60 60 45 45 90 90 60 60 45 45 90 90 60 60 45 45 90 90 60 60 45 45

Tip Dimensions, Inches Tip No.

2040 2040 2040 2040 2040 2040 2040 2040 2040 2040 2040 2040 2060 2060 2060 2060 2060 2060 2150 2150 2150 2150 2150 2150 2220 2220 2220 2220 2220 2220 2300 2300 2300 2300 2300 2300

T

W

L

3 ⁄32

3 ⁄16

5 ⁄16

3 ⁄32

3 ⁄16

5 ⁄16

3 ⁄32 3 ⁄32 3 ⁄32 3 ⁄32 3 ⁄32 3 ⁄32 3 ⁄32 3 ⁄32 3 ⁄32 3 ⁄32 3 ⁄32 3 ⁄32 3 ⁄32 3 ⁄32 3 ⁄32 3 ⁄32

3 ⁄16 3 ⁄16 3 ⁄16 3 ⁄16 3 ⁄16 3 ⁄16 3 ⁄16 3 ⁄16 3 ⁄16 3 ⁄16 1 ⁄4 1 ⁄4 1 ⁄4 1 ⁄4 1 ⁄4 1 ⁄4

5 ⁄16 5 ⁄16 5 ⁄16 5 ⁄16 5 ⁄16 5 ⁄16 5 ⁄16 5 ⁄16 5 ⁄16 5 ⁄16 3 ⁄8 3 ⁄8 3 ⁄8 3 ⁄8 3 ⁄8 3 ⁄8

1 ⁄8

5 ⁄16

7⁄16

1 ⁄8

5 ⁄16

7⁄16

1 ⁄8 1 ⁄8 1 ⁄8 1 ⁄8

5 ⁄32 5 ⁄32 5 ⁄32 5 ⁄32 5 ⁄32 5 ⁄32

5 ⁄16 5 ⁄16 5 ⁄16 5 ⁄16

9 ⁄16

3 ⁄8 3 ⁄8 3 ⁄8

7⁄16

3 ⁄16

7⁄16

3 ⁄8

3 ⁄8

3 ⁄16 3 ⁄16

7⁄16 9 ⁄16

7⁄16

3 ⁄16

7⁄16

3 ⁄8

3 ⁄16 3 ⁄16

7⁄16

7⁄16 7⁄16 7⁄16 7⁄16

9 ⁄16 9 ⁄16 9 ⁄16 9 ⁄16 5 ⁄8 5 ⁄8 5 ⁄8 5 ⁄8 5 ⁄8 5 ⁄8

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Machinery's Handbook, 31st Edition Carbide Boring Tools

968

Table 7. ANSI Solid Carbide Round Boring Tools — ANSI B212.1-2002 (R2007) Style SRC for 60° Boring Bar and Style SRE for 45° Boring Bar 6° ± 1° Tool Designation and Carbide Grade

G ± 1°

F Ref

0.010 R ± 0.003

D +0.0005 –0.0015 B +0.000 –0.005 1

6° ± 1°

D ±0.005 to sharp corner 2

E ± 1°

C ± 64

H 6° ± 1° Along angle “G”

Bor. Bar Angle Tool from Axis, Designation Deg.

Dia. D

Shank Dimensions, Inches Dim. Nose Length Over Height Flat B C H

+ 0.000 ; E − 0.005

SRC-33

60

3 ⁄32

3 ⁄8

0.088

0.070

SRE-33

45

3 ⁄32

3 ⁄8

0.088

0.070

SRC-44

60

1 ⁄8

1 ⁄2

0.118

0.094

SRE-44

45

1 ⁄8

1 ⁄2

0.118

0.094

+ 0.000 ; E − 0.005

SRC-55

60

5 ⁄32

5 ⁄8

0.149

0.117

5 ⁄32

5 ⁄8

3 ⁄16

3 ⁄4

0.177

3 ⁄16

3 ⁄4

0.177

1

0.240

SRE-55

SRC-66 SRE-66

SRC-88 SRE-88

SRC-1010

SRE-1010

45 60 45 60

45 60

45

1 ⁄4 1 ⁄4 5 ⁄16 5 ⁄16

1

11 ⁄4 11 ⁄4

0.149

0.240

0.300 0.300

Side Cut. Edge Angle E, Deg.

End Cut. Edge Angle G, Deg.

Shoulder Angle F, Deg.

30

38

60

45

53

45

30

38

60

45

53

45

±0.005

30

38

60

0.140

±0.005

30

38

60

0.187

±0.005

0.117

0.140

0.187

0.235 0.235

±0.005

±0.005

±0.005

±0.005 ±0.005

45 45

30 45

30 45

53 53

38 53

38 53

45 45

60 45

60 45

Boring Machines, Origin.—The first boring machine was built by John Wilkinson in 1775. John Smeaton had built one in 1769 which had a large rotary head, with inserted cutters, carried on the end of a light, overhanging shaft. The cylinder to be bored was fed forward against the cutter on a rude carriage, running on a track laid in the floor. The cutter head followed the inaccuracies of the bore, doing little more than smooth out local roughness of the surface. Watt’s first steam cylinders were bored on this machine and he complained that one, 18 inches in diameter, was 3⁄8 inch out of true. Wilkinson thought of the expedient, which had escaped both Smeaton and Watt, of extending the boring-bar completely through the cylinder and giving it an out-board bearing, at the same time making it much larger and stiffer. With this machine, cylinders 57 inches in diameter were bored which were within 1 ⁄16 inch of true. Its importance can hardly be overestimated as it insured the commercial success of Watt’s steam engine, which, up to that time, had not passed the experimental stage.

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Machinery's Handbook, 31st Edition Taps

969

TAPS A tap is a mechanical device applied to make a standard thread on a hole. A range of tap pitch diameter (PD) limits, from which the user may select to suit local conditions, is avail­ able. Taps included in the ANSI/ASME B94.9 standard are categorized according to type, style, size and chamfer, and blank design. General dimensions and tap markings are given in the standard ANSI/ASME B94.9 Taps: Ground Thread With Cut Thread Appendix (Inch and Metric Sizes) for straight-fluted taps, spiral-pointed taps, spiral-pointed-only taps, spiral-fluted taps, fast spiral-fluted taps, thread-forming taps, pulley taps, nut taps, and pipe taps. The standard also gives the thread limits for taps with cut threads and ground threads. The tap thread limits and toler­ances are given in Table 2 to Table 4, tap dimensions for cut thread and ground thread are given in Table 5a through Table 10. Pulley tap dimensions and tolerances are given in Table 12, straight and taper pipe thread tap dimensions and tolerances are given on Table 13a and Table 13b, and thread limits for cut thread and ground thread taps are given in Table 15 through Table 26a. Thread Form, Styles, and Types

Thread Form.—The basic angle of thread between the flanks of thread measured in an axial plane is 60 degrees. The line bisecting this 60° angle is perpendicular to the axis of the screw thread. The symmetrical height of the thread form, h, is found as follows: 0.64951905 h = 0.64951905P = (1) n

The basic pitch diameter (PD) is obtained by subtracting the symmetrical single thread height, h, from the basic major diameter as follows: Basic Pitch Diameter = Dbsc − h (2)

Dbsc = basic major diameter P = pitch of thread h = symmetrical height of thread n = number of threads per inch Types and Styles of Taps.—Tap type is based on general dimensions such as standard straight thread, taper and straight pipe, pulley, etc., or is based on purpose, such as thread forming and screw thread inserts (STI). Tap style is based on flute construction for cutting taps, such as straight, spiral, or spiral point, and on lobe style and construction for forming taps, such as straight or spiral. Straight-Flute Taps: These taps have straight flutes of a number specified as either stan­dard or optional, and are for general purpose applications. This standard applies to machine screw, fractional, metric, and STI sizes in high-speed steel ground thread, and to machine screw and fractional sizes in high speed and carbon steel cut thread, with taper, plug, semibottom, and bottom chamfer.

BLANK Design 1

BLANK Design 3

BLANK Design 2

Spiral-Pointed Taps: These taps have straight flutes, and the cutting face of the first few threads is ground at an angle to force the chips ahead and prevent clogging in the flutes. This standard applies to machine screw, fractional, metric, and STI sizes in high-speed

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970

Machinery's Handbook, 31st Edition Taps

steel ground thread, and to cut thread in machine screw and fractional sizes with plug, semibottom, and bottom chamfer.

Blank Design 1

Blank Design 2

Blank Design 3

Spiral-Pointed-Only Taps: These taps are made with the spiral point feature only, without longitudinal flutes. These taps are especially suitable for tapping thin materials. This stan­dard applies to machine screw and fractional sizes in high-speed steel, ground thread, with plug chamfer.

Blank Design 1

Blank Design 2

Blank Design 3

Spiral-Fluted Taps: These taps have right-hand helical flutes with a helix angle of 25 to 35 degrees. These features are designed to help draw chips from the hole or to bridge a key­way. This standard applies to machine screw, fractional, metric, and STI sizes in highspeed steel and to ground thread with plug, semibottom, and bottom chamfer.

Blank Design 2

Blank Design 1

Blank Design 3

Fast Spiral-Fluted Taps: These taps are similar to spiral-fluted taps, except the helix angle is from 45 to 60 degrees.This standard applies to machine screw, fractional, metric, and STI sizes in high-speed steel with plug, semibottom, and bottom chamfer.

Blank Design 1

Blank Design 2

Blank Design 3

Thread-Forming Taps: These taps are fluteless except as optionally designed with one or more lubricating grooves. The thread form on the tap is lobed, so that there are a finite num­ber of points contacting the work thread form. The tap does not cut, but forms the thread by extrusion. This standard applies to machine screw, fractional, and metric sizes, in high-speed steel, ground thread form, with plug, semibottom, and bottom entry taper.

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Machinery's Handbook, 31st Edition Taps

971

Blank Design 1

Blank Design 2

Blank Design 3

Pulley Taps: These taps were originally designed for tapping line-shaft pulleys by hand. Today, these taps have shanks that are extended in length by a standard amount for use where added reach is required. The shank is the same nominal diameter as the thread. This standard applies to fractional size and ground thread with plug and bottom chamfer.

Pipe Taps: These taps are used to produce standard straight or tapered pipe threads. This standard applies to fractional size in high-speed steel, ground thread, to high-speed steel and carbon steel in cut thread, and to straight pipe taps having plug chamfers and taper pipe taps.

Standard System of Tap Marking.—Ground thread taps specified in the US custom­a ry system are marked with the nominal size, number of threads per inch, the proper sym­ bol to identify the thread form, “HS” for high-speed steel, “G” for ground thread, and designators for tap pitch diameter and special features, such as left-hand and multi-start threads. Cut thread taps specified in the US customary system are marked with the nominal size, number of threads per inch, and the proper symbol to identify the thread form. High-speed steel taps are marked “HS,” but carbon steel taps need not be marked. Ground thread taps made with metric screw threads (M profile) are marked with “M,” followed by the nominal size and pitch in millimeters, separated by “X.” Marking also includes “HS” for high-speed steel, “G” for ground thread, designators for tap pitch diam­ eter and special features, such as left-hand and multi-start threads. Thread symbol designators are listed in the accompanying table. Tap pitch diameter des­ignators, systems of limits, special features, and examples for ground threads are given in the following section. Standard System of Tap Thread Limits and Identification for Unified Inch Screw Threads, Ground Thread.—H or L Limits: For Unified inch screw threads, when the maximum tap pitch diameter is over basic pitch diameter by an even multiple of 0.0005 inches, or the minimum tap pitch diameter limit is under basic pitch diameter by an even multiple of 0.0005 inches, the taps are marked “H” or “L,” respectively, followed by a limit number, determined as follows: Tap PD − Basic PD H Limit number = 0.0005

L Limit number =

Basic PD − Tap PD 0.0005

The tap PD tolerances for ground threads are given in Table 2, column D; PD tolerances for cut threads are given in Table 3, column D. For standard taps, the PD limits for various H limit numbers are given in Table 20. The minimum tap PD equals the basic PD minus the

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972

Machinery's Handbook, 31st Edition Taps

number of half-thousandths (0.0005 in.) represented by the limit number. The maximum tap PD equals the minimum PD plus the PD tolerance given in Table 20. Tap Marking with H or L Limit Numbers Example 1: 3⁄8 -16  NC  HS  H1

Maximum tap PD = Basic PD + 0.0005 3 1 = 8 − a 0.64951904 # 16 k + 0.0005 = 0.3344 + 0.0005 = 0.3349 Minimum tap PD = Maximum tap PD − 0.0005 = 0.3349 − 0.0005 = 0.3344

Example 2: 3⁄8 -16  NC  HS  G  L2

Minimum tap PD = Basic PD − 0.0010 3 1 = 8 − a 0.64951904 # 16 k − 0.0010 = 0.3344 − 0.0010 = 0.3334

Maximum tap PD = Minimum tap PD + 0.0005 = 0.3334 + 0.0005 = 0.3339

Oversize or Undersize: When the maximum tap PD over basic PD or the minimum tap PD under basic PD is not an even multiple of 0.0005, the tap PD is usually designated as an amount oversize or undersize. The amount oversize is added to the basic PD to establish the minimum tap PD. The amount undersize is subtracted from the basic PD to establish the minimum tap PD. The PD tolerance from Table 2 is added to the minimum tap PD to estab­lish the maximum tap PD for both. Example: 7⁄16 -14 NC plus 0.0017  HS  G Min. tap PD = Basic PD + 0.0017 in. Max. tap PD = Min. tap PD + 0.0005 in. Whenever possible for oversize or other special tap PD requirements, the maximum and minimum tap PD requirements should be specified. Special Tap Pitch Diameter: Taps not made to H or L limit numbers, to the specifications in, or to the formula for oversize or undersize taps may be marked with the letter “S” enclosed by a circle or by some other special identifier. Example: 1 ⁄2 -16 NC HS G. Left-Hand Taps: Taps with left-hand threads are marked “LEFT HAND” or “LH.” Example: 3⁄8 -16  NC  LH  HS  G  H3.

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Machinery's Handbook, 31st Edition Taps Table 1. Thread Series Designations Standard Tap Marking M

Product Thread Designation M

M

MJ

NC

NC5IF

NC

NC5INF

Third Series

Metric Screw Threads—M Profile, with basic ISO 68 profile

Metric Screw Threads: MJ Profile, with rounded root of radius 0.15011P to 0.18042P (external thread only) Entire ferrous material range

American Standard straight pipe threads in pipe couplings

NPSH

NPSH

American Standard straight hose coupling threads for joining to American Standard taper pipe threads

NPSI

NPSI

NPSL

NPSL

NPSM

NPSM

ANPT

ANPT

NPTF

NPTF

NPT

NPTR

PTF

PTF-SPL

STI

N

NPT

NPTR

PTF

PTF-SPL

STI

UN

Dryseal American Standard fuel internal straight pipe threads

Dryseal American Standard intermediate internal straight pipe threads

American Standard straight pipe threads for loose-fitting mechanical joints with locknuts American Standard straight pipe threads for free-fitting mechanical joints for fixtures Pipe threads, taper, aeronautical, national form

American Standard taper pipe threads for general use

Dryseal American Standard taper pipe threads

NF

UNJF

NEF

UNJEF

N

UNR

NC

UNRC

NF

UNRF

NEF

UNREF

NS

UNS

B1.20.1 AS-71051 B1.20.1

B1.20.3

B18.29.1

Unified Inch Screw Thread

Constant-pitch series

Extra-fine pitch series

UNJC

B1.20.1

B1.20.3

Helical coil screw thread insertsfree running and screw locking (inch series)

UNEF

NC

B1.20.7 B1.20.3

B1.20.3

Dryseal American Standard pipe threads

NEF

UNJ

B1.20.3

Dryseal American Standard pipe threads

Coarse pitch series

N

B1.20.1

B1.20.1

UNC UNF

B1.12

American Standard taper pipe threads for railing joints

NC NF

B1.13M

B1.12

Entire nonferrous material range

NPSC NPSF

American National Standard References

Class 5 interference-fit thread

NPS

NPSF

973

B1.1 B1.1

Fine pitch series

B1.1

Constant-pitch series, with rounded root of radius 0.15011P to 0.18042P (external thread only)

Coarse pitch series, with rounded root of radius 0.15011P to 0.18042 P (external thread only) Fine pitch series, with rounded root of radius 0.15011P to 0.18042P (external thread only)

Extra-fine pitch series, with rounded root of radius 0.15011P to 0.18042P (external thread only) Constant-pitch series, with rounded root of radius not less than 0.108P (external thread only) Coarse thread series, with rounded root of radius not less than 0.108P (external thread only) Fine pitch series, with rounded root of radius not less than 0.108P (external thread only)

Extra-fine pitch series, with rounded root of radius not less than 0.108P (external thread only) Special diameter pitch, or length of engagement

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B1.1

AS-8879 B1.15 AS-8879

B1.15 AS-8879

B1.15 AS-8879 B1.1 B1.1 B1.1 B1.1 B1.1

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Machinery's Handbook, 31st Edition Taps

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Table 2. Tap Thread Limits and Tolerances ANSI/ASME B94.9-2008 (R2018) Formulas for Unified Inch Screw Threads (Ground Thread) Max. Major Diameter = Basic Diameter + A Min. Major Diameter = Max. Maj. Dia. − B

Max. Pitch Diameter = Basic Diameter + C

Min. Pitch Diameter = Max. Pitch Dia. − D

A = Constant to add = 0.130P for all pitches B = Major diameter tolerance= 0.087P for 48 to 80 TPI; 0.076P for 36 to 47 TPI; 0.065P for 4 to 35 TPI C = Amount over basic for maximum pitch diameter D = Pitch Diameter Tolerance

Threads per Inch

C A

B

0 to 5⁄8

D

5⁄8 to 1

1 to 11 ⁄2

11 ⁄2 to 21 ⁄2

Over 21 ⁄2 0.0030

0.0020

0.0020

0 to 5⁄8

0.0005

5⁄8 to 1

0.0005

1 to 11 ⁄2 11 ⁄2 to 21 ⁄2 Over 21 ⁄2

80

0.0016 0.0011

0.0010

0.0015

0.0010

0.0010

0.0015

72

0.0018 0.0012

0.0010

0.0015

0.0020

0.0020

0.0030

0.0005

0.0005

0.0010

0.0010

0.0015

64

0.0020 0.0014

0.0010

0.0015

0.0020

0.0020

0.0030

0.0005

0.0005

0.0010

0.0010

0.0015

56

0.0023 0.0016

0.0010

0.0015

0.0020

0.0020

0.0030

0.0005

0.0005

0.0010

0.0010

0.0015

48

0.0027 0.0018

0.0010

0.0015

0.0020

0.0020

0.0030

0.0005

0.0005

0.0010

0.0010

0.0015

44

0.0030 0.0017

0.0010

0.0015

0.0020

0.0020

0.0030

0.0005

0.0005

0.0010

0.0010

0.0015

40

0.0032 0.0019

0.0010

0.0015

0.0020

0.0020

0.0030

0.0005

0.0005

0.0010

0.0010

0.0015

36

0.0036 0.0021

0.0010

0.0015

0.0020

0.0020

0.0030

0.0005

0.0005

0.0010

0.0010

0.0015

32

0.0041 0.0020

0.0015

0.0015

0.0020

0.0020

0.0030

0.0005

0.0005

0.0010

0.0010

0.0015

28

0.0046 0.0023

0.0015

0.0015

0.0020

0.0020

0.0030

0.0005

0.0005

0.0010

0.0010

0.0015

24

0.0054 0.0027

0.0015

0.0015

0.0020

0.0025

0.0030

0.0005

0.0005

0.0010

0.0015

0.0015

20

0.0065 0.0032

0.0015

0.0015

0.0020

0.0025

0.0030

0.0005

0.0005

0.0010

0.0015

0.0015

18

0.0072 0.0036

0.0015

0.0015

0.0020

0.0025

0.0030

0.0005

0.0005

0.0010

0.0015

0.0015

16

0.0081 0.0041

0.0015

0.0015

0.0020

0.0025

0.0035

0.0005

0.0005

0.0010

0.0015

0.0020

14

0.0093 0.0046

0.0015

0.0020

0.0025

0.0030

0.0035

0.0005

0.0005

0.0010

0.0015

0.0020

13

0.0100 0.0050

0.0015

0.0020

0.0025

0.0030

0.0035

0.0005

0.0005

0.0010

0.0015

0.0020

12

0.0108 0.0054

0.0015

0.0020

0.0025

0.0030

0.0035

0.0005

0.0005

0.0010

0.0015

0.0020

11

0.0118 0.0059

0.0015

0.0020

0.0025

0.0030

0.0040

0.0005

0.0005

0.0010

0.0015

0.0020

10

0.0130 0.0065

…

0.0020

0.0025

0.0030

0.0040

…

0.0005

0.0010

0.0015

0.0020

9

0.0144 0.0072

…

0.0020

0.0025

0.0030

0.0040

…

0.0005

0.0010

0.0015

0.0020

8

0.0162 0.0081

…

0.0020

0.0025

0.0030

0.0040

…

0.0005

0.0010

0.0015

0.0020

7

0.0186 0.0093

…

0.0025

0.0025

0.0035

0.0045

…

0.0010

0.0010

0.0020

0.0025

6

0.0217 0.0108

…

0.0025

0.0025

0.0035

0.0045

…

0.0010

0.0010

0.0020

0.0025

51 ⁄2

0.0236 0.0118

…

0.0025

0.0030

0.0035

0.0045

…

0.0010

0.0015

0.0020

0.0025

5

0.0260 0.0130

…

0.0025

0.0030

0.0035

0.0045

…

0.0010

0.0015

0.0020

0.0025

41 ⁄2

0.0289 0.0144

…

0.0025

0.0030

0.0035

0.0045

…

0.0010

0.0015

0.0020

0.0025

4

0.0325 0.0162

…

0.0025

0.0030

0.0035

0.0045

…

0.0010

0.0015

0.0020

0.0025

Dimensions are given in inches. The tables and formulas are used in determining the limits and tolerances for ground thread taps having a thread lead angle not in excess of 5°, unless otherwise specified. The tap major diameter must be determined from a specified tap pitch diameter: the minimum major diameter equals the maximum specified tap pitch diameter minus constant C, plus 0.64951904P plus constant A.

Minimum Major Diameter = Tap Pitch Diameter − C + 0.64951904P + A

For intermediate pitches, use value of next coarser pitch; for C and D use formulas for A and B. Lead Tolerance:± 0.0005 inch within any two threads not farther apart than 1 inch. Angle Tolerance: ± 20′ in half angle for 4 to 51 ⁄2 pitch; ± 25′ in half angle for 6 to 9 pitch; and ± 30′ in half angle for 10 to 80 pitch.

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Table 3. Tap Thread Limits and Tolerances ANSI/ASME B94.9-2008 (R2018) Formulas for Unified Inch Screw Threads (Cut Thread) Min. Major Diameter = Basic Diameter + B + C

Max. Major Diameter = Min. Maj. Dia. + A

Min. Pitch Diameter = Basic Diameter + B

Max. Pitch Diameter = Min. Pitch Dia. + D

A = Major diameter tolerance B = Amount over basic for minimum pitch diameter  constant to add for major diameter: 20% of theoretical truncation for 2 to 5.5 threads per inch C=A and 25% for 6 to 80 threads per inch D = Pitch diameter tolerance B D Diameter of Coarser than Tap (Inch) A 36 or more TPI 34 or less TPI N.F. N.F. and Finera 0 to 0.099 0.0015 0.0002 0.0005 0.0010 0.0010 0.10 to 0.249 0.0020 0.0002 0.0005 0.0015 0.0015 1 ⁄ to 3 ⁄ 0.0025 0.0005 0.0005 0.0020 0.0015 4 8 3 ⁄ to 5 ⁄ 0.0030 0.0005 0.0005 0.0025 0.0020 8 8 5 ⁄ to 3 ⁄ 0.0040 0.0005 0.0005 0.0030 0.0025 8 4 3 ⁄ to 1 0.0040 0.0010 0.0010 0.0030 0.0025 4 0.0045 0.0010 0.0010 0.0035 0.0030 1 to 11 ⁄2 0.0055 0.0015 0.0015 0.0040 0.0030 11 ⁄2 to 2 0.0060 0.0015 0.0015 0.0045 0.0035 2 to 21 ⁄4 0.0060 0.0020 0.0020 0.0045 0.0035 21 ⁄4 to 21 ⁄2 0.0070 0.0020 0.0020 0.0050 0.0035 21 ⁄2 to 3 0.0070 0.0025 0.0025 0.0055 0.0045 over 3 a Taps over 11 ⁄ inches with 10 or more threads per inch have tolerances for N.F. and finer. 2

Threads per Inch 2

21 ⁄2 3 31 ⁄2 4

41 ⁄2 5 51 ⁄2 6

Threads per Inch

4 and coarser 41 ⁄2 to 51 ⁄2 6 to 9

C

0.0217 0.0173 0.0144 0.0124 0.0108 0.0096 0.0087 0.0079 0.0078

Threads per Inch 7 8 9 10 11 12 13 14 16

Deviation in Half angle ± 30′ ± 35′ ± 40′

C

0.0077 0.0068 0.0060 0.0054 0.0049 0.0045 0.0042 0.0039 0.0034

Threads per Inch 18 20 22 24 26 27 28 30 32

Angle Tolerance Deviation in Threads per Half angle Inch ± 45′ ± 53′ ± 60′

10 to 28 30 and finer

C

0.0030 0.0027 0.0025 0.0023 0.0021 0.0020 0.0019 0.0018 0.0017

Threads per Inch 36 40 48 50 56 60 64 72 80

Deviation in Half angle ± 45′ ± 60′

C

0.0015 0.0014 0.0011 0.0011 0.0010 0.0009 0.0008 0.0008 0.0007 Deviation in Half angle ± 68′ ± 90′

Dimensions are given in inches. The tables and formulas are used in determining the limits and tolerances for cut thread metric taps having special diameter, special pitch, or both. For intermediate pitches, use value of next coarser pitch. Lead Tolerance: ± 0.003 inch within any two threads not farther apart than 1 inch. Taps over 11 ⁄2 in. with 10 or more threads per inch have tolerances for N.F. and finer.

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Machinery's Handbook, 31st Edition Taps

976

Standard System of Ground Thread Tap Limits and Identification for Metric Screw Threads, M Profile.—All calculations for metric taps use millimeter values. When US customary values are needed, they are translated from the three-place millimeter tap diam­eters only after the calculations are completed. Table 4. Tap Thread Limits and Tolerances ANSI/ASME B94.9-2008 (R2018) Formulas for Metric Thread (Ground Thread)

Min. major diameter = Basic diameter + W

Max. major diameter = Min. maj. dia. + X

Max. pitch diameter = Basic diameter + Y Min. pitch diameter = Max. pitch dia. + Z

W = Constant to add with basic major diameter (W = 0.08P) X = Major diameter tolerance Y = Amount over basic for maximum pitch diameter Z = Pitch diameter tolerance

P Pitch (mm) 0.30 0.35 0.40 0.45 0.50 0.60 0.70 0.75 0.80 0.90 1.00 1.25 1.50 1.75 2.00 2.50 3.00 3.50 4.00 4.50 5.00 5.50 6.00

W (0.08P) 0.024 0.028 0.032 0.036 0.040 0.048 0.056 0.060 0.064 0.072 0.080 0.100 0.120 0.140 0.160 0.200 0.240 0.280 0.320 0.360 0.400 0.440 0.480

X 0.025 0.025 0.025 0.025 0.025 0.025 0.041 0.041 0.041 0.041 0.041 0.064 0.064 0.064 0.064 0.063 0.100 0.100 0.100 0.100 0.100 0.100 0.100

M1.6 to M6.3 0.039 0.039 0.039 0.039 0.039 0.052 0.052 0.052 0.052 0.052 0.065 0.065 0.065 … … … … … … … … … …

Over M6.3 to to M25 0.039 0.039 0.052 0.052 0.052 0.052 0.052 0.065 0.065 0.065 0.065 0.065 0.078 0.078 0.091 0.091 0.104 0.104 0.104 … … … …

Y

Over M25 to M90 0.052 0.052 0.052 0.052 0.052 0.065 0.065 0.065 0.065 0.065 0.078 0.078 0.078 0.091 0.091 0.104 0.104 0.117 0.117 0.130 0.130 0.143 0.143

Over M90 0.052 0.052 0.052 0.052 0.065 0.065 0.065 0.078 0.078 0.078 0.091 0.091 0.091 0.104 0.104 0.117 0.130 0.130 0.143 0.143 0.156 0.156 0.156

M1.6 to M6.3 0.015 0.015 0.015 0.015 0.015 0.020 0.020 0.020 0.020 0.020 0.025 0.025 0.025 … … … … … … … … … …

Over M6.3 to to M25 0.015 0.015 0.015 0.020 0.020 0.020 0.020 0.025 0.025 0.025 0.025 0.031 0.031 0.031 0.041 0.041 0.041 0.041 0.052 0.052 … … …

Z

Over M25 to M90 0.020 0.020 0.020 0.020 0.025 0.025 0.025 0.025 0.025 0.025 0.031 0.031 0.031 0.041 0.041 0.041 0.052 0.052 0.052 0.052 0.064 0.064 0.064

Over M90 0.020 0.020 0.025 0.025 0.025 0.025 0.025 0.031 0.031 0.031 0.031 0.041 0.041 0.041 0.041 0.052 0.052 0.052 0.064 0.064 0.064 0.064 0.064

Dimensions are given in millimeters. The tables and formulas are used in determining the limits and tolerances for ground thread metric taps having a thread lead angle not in excess of 5°, unless otherwise specified. They apply only to metric thread having a 60° form with a P/8 flat at the major diameter of the basic thread form. All calculations for metric taps are done using millimeter values as shown. When inch values are needed, they are translated from the three-place millimeter tap diameters only after calculations are performed. The tap major diameter must be determined from a specified tap pitch diameter: the mini­ mum major diameter equals the maximum specified tap pitch diameter minus constant Y, plus 0.64951905P plus constant W. Minimum major diameter = Max. tap pitch diameter − Y + 0.64951904P + W For intermediate pitches, use value of next coarser pitch. Lead Tolerance:± 0.013 mm within any two threads not farther apart than 25 mm. Angle Tolerance: ± 30′ in half angle for 0.25 to 2.5 pitch; ± 25′ in half angle for 2.5 to 4 pitch; and ± 20′ in half angle for 4 to 6 pitch.

D or DU Limits: When the maximum tap pitch diameter is over basic pitch diameter by an even multiple of 0.013 mm (0.000512 in. reference), or the minimum tap pitch diameter limit is under basic pitch diameter by an even multiple of 0.013 mm, the taps are marked

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Machinery's Handbook, 31st Edition Taps

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with the letters “D” or “DU,” respectively, followed by a limit number. The limit number is determined as follows: Tap PD − Basic PD D Limit number = 0.0013

DU Limit number =

Basic PD − Tap PD 0.0013

Example: M1.6 × 0.35  HS  G  D3 Maximum tap PD = Basic PD + 0.0039 = 1.6 − ^0.64951904 # 0.35h + 0.0039

= 1.3727 + 0.039 = 1.412 Minimum tap PD = Maximum tap PD − 0.015 = 1.412 − 0.015 = 1.397

M6 × 1  HS  G  DU4 Minimum tap PD = Basic PD − 0.052 = 6 − ^0.64951904 # 1.0h − 0.052

= 5.350 − 0.052 = 5.298

Maximum tap PD = Minimum tap PD + 0.025 = 5.298 + 0.025 = 5.323

Definitions of Tap Terms.—The definitions that follow are taken from ANSI/ASME B94.9 but include only the more important terms. Some tap terms are the same as screw thread terms; therefore, see Definitions of Screw Threads starting on page 1942. Actual size: The measured size of an element on an individual part. Allowance: A prescribed difference between the maximum material limits of mating parts. It is the minimum clearance or maximum interference between such parts. Basic Size: The size from which the limits are derived by application of allowance and tolerance. Bottom Top: A tap having a chamfer length of 1 to 2 pitches. Chamfer: Tapering of the threads at the front end of each land or chaser of a tap by cut­ ting away and relieving the crest of the first few teeth to distribute the cutting action over several teeth. Chamfer Angle: Angle formed between the chamfer and the axis of the tap measured in an axial plane at the cutting edge. Chamfer Relief: The gradual degrees in land height from cutting edge to heel on the chamfered portion of the land to provide radial clearance for the cutting edge. Chamfer Relief Angle: Complement of the angle formed between a tangent to the relieved surface at the cutting edge and a radial line to the same point on the cutting edge. Classes of Thread: Designation of the class that determines the specification of the size, allowance, and tolerance to which a given threaded product is to be manufactured. It is not applicable to the tools used for threading. Concentric: Having a common center. Crest: The surface of the thread that joins the flanks of the thread and is farthest from the cylinder or cone from which the thread projects.

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Machinery's Handbook, 31st Edition Taps

Cutter Sweep: The section removed by the milling cutter or the grinding wheel in enter­ ing or leaving a flute. Cutting Edge: The intersection of cutting edge and the major diameter in the direction of rotation for cutting, which does the actual cutting. Core Diameter: The diameter of a circle that is tangent to the bottom of the flutes at a given point on the axis. Diameter, Major: The major cylinder on a straight thread. Diameter, Minor: The minor cylinder on a straight thread. Dryseal: A thread system used for both external and internal pipe threads applications designed for use where the assembled product must withstand high fluid or gas pressure without the use of sealing compound. Eccentric: Not having a common center. Eccentricity: One half of the total indicator variation (TIV) with respect to the tool axis. Entry Taper: The portion of the thread forming, where the thread forming is tapered toward the front to allow entry into the hole to be tapped. External Center: The pointed end on a tap. On bottom-chamfered taps, the point on the front end may be removed. Flank: The flank of a thread is the surface connecting the crest with the root. Flank Angle: Angle between the individual flank and the perpendicular to the axis of the thread, measured in an axial plane. A flank angle of a symmetrical thread is commonly termed the “half angle of thread.” Flank, Leading: 1) Flank of a thread facing toward the chamfered end of a threading tool; and 2) The leading flank of a thread is the one which, when the thread is about to be assembled with a mating thread, faces the mating thread. Flank, Trailing: The trailing flank of a thread is the one opposite the leading flank. Flutes: Longitudinal channels formed in a tap to create cutting edges on the thread pro­ file and to provide chip spaces and cutting fluid passages. On a parallel or straight-thread tap, they may be straight, angular or helical; on a taper-thread tap, they may be straight, angular or spiral. Flute Lead Angle: Angle at which a helical- or spiral-cutting edge at a given point makes with an axial plane through the same point. Flute, Spiral: A flute with uniform axial lead in a spiral path around the axis of a coni­ cal tap. Flute, Straight: A flute that forms a cutting edge lying in an axial plane. Flute, Tapered: A flute lying in a plane intersecting the tool axis at an angle. Full Indicator Movement (FIM): The total movement of an indicator where appropri­ ately applied to a surface to measure its surface. Functional Size: The functional diameter of an external or internal thread is the PD of the enveloping thread of perfect pitch, lead, and flank angles, having full depth of engagement but clear at crests and roots, and of a specified length of engagement. Heel: Edge of the land opposite the cutting edge. Height of Thread: The height of a thread is the distance, measured radially, between the major and minor cylinders or cones, respectively. Holes, Blind: A hole that does not pass through the work piece and is not threaded to its full depth. Holes, Bottom: A blind hole that is threaded close to the bottom. Hook Angle: Inclination of a concave cutting face, usually specified either as Chordal Hook or Tangential Hook. Hook, Chordal Angle: Angle between the chord passing through the root and crest of a thread form at the cutting face, and a radial line through the crest at the cutting edge. Hook, Tangential Angle: Angle between a line tangent to a hook cutting face at the cut­ ting edge and a radial line to the same point. Internal Center: A countersink with clearance at the bottom, in one or both ends of a tool, which establishes the tool axis.

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Machinery's Handbook, 31st Edition Taps

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Interrupted Thread Tap: A tap having an odd number of lands with alternate teeth in the thread helix removed. In some designs, alternate teeth are removed only for a portion of the thread length. Land: One of the threaded sections between the flutes of a tap. Lead: Distance a screw thread advances axially in one complete turn. Lead Error: Deviation from prescribed limits. Lead Deviation: Deviation from the basic nominal lead. Progressive Lead Deviation: (1) On a straight thread, the deviation from a true helix where the thread helix advances uniformly. (2) On a taper thread, the deviation from a true spiral where the thread spiral advances uniformly. Tap Terms

Max. Tap Major Dia.

Min. Tap Major Dia.

Basic Major Dia.

Basic Height of Thread

No Relief

Cutting Face

Relieved to Cutting Edge

Heel

Eccentric

Concentric

Tap Crest Basic Crest Angle of Thread Flank

Basic Pitch Dia.

Cutting Edge

Pitch

Basic Minor Dia. Base of Thread Basic Root Concentric Margin Eccentric Relief

Con-Eccentric Relief

Land

Negative Rake Angle

Zero Rake

Positive Rake Angle

Negative Rake

Radial

Positive Rake

Positive Hook

0 Deg. Hook

Negative Hook

Fig. 3. Tap Terms

Left Hand Cut: Rotation in a clockwise direction from cutting when viewed from the chamfered end of a tap. Length of Engagement: The length of engagement of two mating threads is the axial dis­ tance over which two mating threads are designed to contact. Length of Thread: The length of the thread of the tap includes the chamfered threads and the full threads, but does not include an external center. It is indicated by the letter “B” in the illustrations at the heads of the tables. Limits: The limits of size are the applicable maximum and minimum sizes. Major Diameter: On a straight thread, the major diameter is that of the major cylinder. On a taper thread, the major diameter at a given position on the thread axis is that of the major cone at that position.

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Machinery's Handbook, 31st Edition Taps

980

Tap Terms Overall Length, L Shank Length I

Thread Length I

Core Dia.

4

External Center

I

Land Width Flute

Driving Square Length

2

Truncated Center Optional Transitional Optional with Manufacturer BLANK Design 1 Overall Length, L

Thread Length I

Shank Length I4

Neck Length I1

External Center Neck Diameter d2

I

2

Driving Square Length

Truncated Neck to Shank Optional with Manufacturer

BLANK Design 2 with Optional Neck Overall Length, L Shank Length I

Thread Length I

4

I

2

Driving Square Length

Truncated Center Optional

External Center

Transitional Optional with Manufacturer BLANK Design 2 (without Optional Neck) Overall Length Shank Thread Length Length I I4 Chamfer I5 Length

Point Dia.

Driving Square Length, I 2

Size of Square across flats

d3 Internal Center

Chamfer Angle

90°

Shank Dia. d 1

Thread Lead Angle

BLANK Design 3

Fig. 4. Taps Terms

Minor Diameter: On a straight thread, the minor diameter is that of the minor cylinder. On a taper thread, the minor diameter at a given position on the thread axis is that of the minor cone at that position. Neck: A section of reduced diameter between two adjacent portions of a tool. Pitch: The distance from any point on a screw thread to a corresponding point in the next thread, measured parallel to the axis and on the same side of the axis.

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Machinery's Handbook, 31st Edition Taps

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Pitch Diameter (Simple Effective Diameter): On a straight thread, the pitch diameter is the diameter of the imaginary coaxial cylinder, the surface of which would pass through the thread profiles at such points as to make the width of the groove equal to one-half the basic pitch. On a perfect thread, this coincidence occurs at the point where the widths of the thread and groove are equal. On a taper thread, the pitch diameter at a given position on the thread axis is the diameter of the pitch cone at that position. Point Diameter: Diameter at the cutting edge of the leading end of the chamfered sec­tion. Plug Tap: A tap having a chamfer length of 3 to 5 pitches. Rake: Angular relationship of the straight cutting face of a tooth with respect to a radial line through the crest of the tooth at the cutting edge. Positive rake means that the crest of the cutting face is angularly ahead of the balance of the cutting face of the tooth. Negative rake means that the crest of the cutting face is angularly behind the balance of the cutting face of the tooth. Zero rake means that the cutting face is directly on a radial line. Relief: Removal of metal behind the cutting edge to provide clearance between the part being threaded and the threaded land. Relief, Center: Clearance produced on a portion of the tap land by reducing the diameter of the entire thread form between cutting edge and heel. Relief, Chamfer: Gradual decrease in land height from cutting edge to heel on the cham­ fered portion of the land on a tap to provide radial clearance for the cutting edge. Relief, Con-Eccentric Thread: Radial relief in the thread form starting back of a concen­ tric margin. Relief, Double Eccentric Thread: Combination of a slight radial relief in the thread form starting at the cutting edge and continuing for a portion of the land width, and a greater radial relief for the balance of the land. Relief, Eccentric Thread: Radial relief in the thread form starting at the cutting edge and continuing to the heel. Relief, Flatted Land: Clearance produced on a portion of the tap land by truncating the thread between cutting edge and heel. Relief, Grooved Land: Clearance produced on a tap land by forming a longitudinal groove in the center of the land. Relief, Radial: Clearance produced by removal of metal from behind the cutting edge. Taps should have the chamfer relieved and should have back taper, but may or may not have relief in the angle and on the major diameter of the threads. When the thread angle is relieved, starting at the cutting edge and continuing to the heel, the tap is said to have “eccentric” relief. If the thread angle is relieved back of a concentric margin (usually onethird of land width), the tap is said to have “con-eccentric” relief. Right Hand Cut: Rotation in clockwise direction for cutting when viewed from the chamfered end of a tap or die. Roots: The surface of the thread that joins the flanks of adjacent thread forms and is iden­ tical to cone from which the thread projects. Screw Thread: A uniform section produced by forming a groove in the form of helix on the external or the internal surface of a cylinder. Screw Thread Inserts (STI): Screw thread bushing coiled from diamond-shape crosssection wire. They are screwed into oversized tapped holes to size nominal-size internal threads. Screw Thread Insert (STI) Taps: These taps are over the nominal size to the extent that the internal thread they produce will accommodate a helical coil screw insert, which at final assembly will accept a screw thread of the nominal size and pitch.

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Machinery's Handbook, 31st Edition Taps

982

Shank: The portion of the tool body by which it is held and driven. Shaving: The excessive removal of material from the product thread profile by the tool thread flanks caused by an axial advance per revolution less than or more than the actual lead in the tool. Size, Actual: Measured size of an element on an individual part. Size, Basic: That size from which the limits of size are derived by the application of allowances and tolerances. Size, Functional: The functional diameter of an external or internal thread is the pitch diameter of the enveloping thread of perfect pitch, lead and flank angles, having full depth of engagement but clear at crests and roots, and of a specified length of engagement. It may be derived by adding to the pitch diameter in an external thread, or subtracting from the pitch diameter in an internal thread, the cumulative effects of deviations from specified profile, including variations in lead and flank angle over a specified length of engagement. The effects of taper, out-of-roundness, and surface defects may be positive or negative on either external or internal threads. Size, Nominal: Designation used for the purpose of general identification. Spiral Flute: See Flutes. Spiral Point: Angular fluting in the cutting face of the land at the chamfered end. It is formed at an angle with respect to the tap axis of opposite hand to that of rotation. Its length is usually greater than the chamfer length and its angle with respect to the tap axis is usually made great enough to direct the chips ahead of the tap. The tap may or may not have longi­tudinal flutes. Taper, Back: A gradual decrease in the diameter of the thread form on a tap from the chamfered end of the land towards the back, which creates a slight radial relief in the threads. Taper per Inch: The difference in diameter in one inch measured parallel to the axis. Taper Tap: A tap having a chamfer length of 7 to 10 pitches. Taper Thread Tap: A tap with tapered threads for producing a tapered internal thread. Thread, Angle of: The angle between the flanks of the thread measured in an axial plane. Thread Lead Angle: On a straight thread, the lead angle is the angle made by the helix of the thread at the pitch line with a plane perpendicular to the axis. On a taper thread, the lead angle at a given axial position is the angle made by the conical spiral of the thread, with the plane perpendicular to the axis, at the pitch line. Thread per Inch: The number of thread pitches in one inch of thread length. Tolerance: The total permissible variation of size or difference between limits of size. Total Indicator Variation (TIV): The difference between maximum and minimum indi­ cator readings during a checking cycle. L

L I

I

I2

I2

d1

d1 BLANK Design 2

BLANK Design 1 L I

I2

a d1

BLANK Design 3

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Table 5a. Standard Tap Dimensions (Ground and Cut Thread) ANSI/ASME B94.9-2008 (R2018) Nominal Diameter, inch

Nominal Metric Diameter

Tap Dimensions, inch

Over

To

Machine Screw Size No. and Fractional Sizes

0.052

0.065

0

(0.0600)

M1.6

0.0630

1

0.078

0.091

2

(0.0860)

M2.0

0.0787

1

0.065

0.078

1

Decimal Equiv. (0.0730)

0.091

0.104

3

(0.0990)

0.117

0.130

5

(0.1250)

0.145

0.171

0.104 0.130

0.171 0.197

0.223

0.117

0.145

0.197 0.223

4

6

8

10 12

(0.1120)

(0.1380)

(0.1640)

(0.1900) (0.2160)

mm

M1.8

M2.2 M2.5 …

M3.0

inch

Blank Design No.

0.0709

1

0.0866

Shank Diameter d1

Size of Square a

1.69

0.38

0.19

0.141

0.110

1.63

1.75

0.31

0.44

0.19

0.19

0.141

0.141

0.110

0.110

1.81

0.50

0.19

0.141

0.110

0.1182

1

1.94

0.63

0.19

0.141

0.110

M4.5

0.1772

…

Square Length I2

1

…

0.1378

M5

Thread Length I

0.0984

M3.5

M4.0

Overall Length L

0.1575

0.1969 …

1

1

1

1 1

1.88

2.00

2.13

2.38 2.38

0.56

0.69

0.75

0.88 0.94

0.19

0.19

0.25

0.25 0.28

0.141

0.141

0.168

0.194 0.220

0.110

0.110

0.131

0.152 0.165

1 ⁄4

(0.2500)

M6

0.2363

2

2.50

1.00

0.31

0.255

0.191

0.260

0.323

5 ⁄16

(0.3125)

M7

0.2756

2

2.72

1.13

0.38

0.318

0.238

0.323

0.395

3 ⁄8

(0.3750)

M10

0.3937

2

2.94

1.25

0.44

0.381

0.286

0.395

0.448

7⁄16

(0.4375)

…

…

3

3.16

1.44

0.41

0.323

0.242

0.448

0.510

1 ⁄2

(0.5000)

M12

0.4724

3

3.38

1.66

0.44

0.367

0.275

0.510

0.573

9 ⁄16

0.573

0.635

5 ⁄8

0.635

0.709

11 ⁄16

0.709

0.760

3 ⁄4

0.760

0.823

13 ⁄16

(0.8125)

0.823

0.885

7⁄8

(0.8750)

M8

0.3150

(0.5625)

M14

0.5512

3

3.59

1.66

0.50

0.429

0.322

  (0.6250)

M16

0.6299

3

3.81

1.81

0.56

0.480

0.360

(0.6875)

M18

0.7087

3

4.03

1.81

0.63

0.542

0.406

(0.7500)

…

…

3

4.25

2.00

0.69

0.590

0.442

M20

0.7874

3

4.47

2.00

0.69

0.652

0.489

M22

0.8661

3

4.69

2.22

0.75

0.697

0.523

983

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0.260

Machinery's Handbook, 31st Edition Taps

Copyright 2020, Industrial Press, Inc.

Nominal Diameter Range, inch

Over

To

0.885

0.948

1.010

1.073

0.948

1.010

Nominal Diameter, inch

Machine Screw Size No. and Fractional Sizes

Nominal Metric Diameter

984

Tap Dimensions, inch

Overall Length L

Thread Length I

Square Length I2

Shank Diameter d1

5.13

2.50

0.81

0.800

Size of Square a

Decimal Equiv.

mm

inch

Blank Design No.

(0.9375)

M24

0.9449

3

11 ⁄16

(1.0625)

M27

1.0630

3

5.13

2.50

0.88

0.896

0.672 0.672

1

15 ⁄16

(1.0000)

M25

1.073

1.135

11 ⁄8

(1.1250)

…

1.135

1.198

13⁄16

(1.1875)

M30

1.198

1.260

11 ⁄4

(1.2500)

…

1.260

1.323

15⁄16

(1.3125)

M33

1.323

1.385

13⁄8

(1.3750)

…

1.358

1.448

17⁄16

(1.4375)

M36

1.448

1.510

11 ⁄2

  (1.5000)

…

0.9843

3

4.91

2.22

0.75

0.760

0.570

0.600

…

3

5.44

2.56

0.88

0.896

1.1811

3

5.44

2.56

1.00

1.021

0.766

…

3

5.75

2.56

1.00

1.021

0.766

1.2992

3

5.75

2.56

1.06

1.108

0.831

…

3

6.06

3.00

1.06

1.108

0.831

1.4173

3

6.06

3.00

1.13

1.233

0.925

…

3

6.38

3.00

1.13

1.233

0.925 0.979

1.510

1.635

15⁄8

(1.6250)

M39

1.5353

3

6.69

3.19

1.13

1.305

1.635

1.760

13⁄4

(1.7500)

M42

1.6535

3

7.00

3.19

1.25

1.430

1.072

1.760

1.885

17⁄8

(1.8750)

…

…

3

7.31

3.56

1.25

1.519

1.139

1.885

2.010

2

(2.0000)

M48

1.8898

3

7.63

3.56

1.38

1.644

1.233

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Special taps greater than 1.010 inch to 1.510 inch in diameter inclusive, having 14 or more threads per inch or 1.75- mm pitch and finer, and sizes over 1.510 inch in diameter with 10 or more threads per inch or 2.5- mm pitch and finer are made to general dimensions shown in Table 10. For standard ground thread tap limits see Table 20, and Table 21 for inch and Table 16 for metric. For cut thread tap limits see Table 22 and Table 23. Special ground thread tap limits are determined by using the formulas shown in Table 2 for unified inch screw threads and Table 4 for metric M profile screw threads. Tap sizes 0.395 inch and smaller have an external center on the thread end (may be removed on bottom taps). Sizes 0.223 inch and smaller have an external center on the shank end. Sizes 0.224 inch through 0.395 inch have truncated partial cone centers on the shank end (of diameter of shank). Sizes greater than 0.395 inch have internal centers on both the thread and shank ends. For standard thread limits and tolerances see Table 17 for unified inch screw threads and Table 19 for metric threads. For runout tolerances of tap elements see Table 14. For number of flutes see Table 11.

Machinery's Handbook, 31st Edition Taps

Copyright 2020, Industrial Press, Inc.

Table Table 5a. (Continued) Standard Tap Dimensions (Ground andThread) Cut Thread) ANSI/ASME B94.9-2008 (R2018) 5a. Standard Tap Dimensions (Ground and Cut ANSI/ASME B94.9-2008 (R2018)

Nominal Diameter Range, inch

Machinery's Handbook, 31st Edition Taps

985

Table 5b. Standard Tap Dimensions Tolerances (Ground and Cut Thread) ANSI/ASME B94.9-2008 (R2018)

Tolerance, inch

Nominal Diameter Range, inch

Element

Length overall, L

Over

To (inclusive)

Direction

Ground Thread

Cut Thread

1.0100

2.0000

±

0.0600

0.0600

0.5200

Length of thread, I

1.0100

0.0520

0.2230

0.2230

0.5100

1.5100

0.0520

Diameter of shank, d1

0.6350

±

-

1.5100

-

0.0520

0.0020

0.0020

0.0030

-

0.5100

0.0040

-

1.0100

1.0100

0.0015

0.0015

-

2.0000

0.5100

0.0300

0.0600

-

0.6350

1.0100

1.5100

0.0900

0.1300

±

0.2230

1.0100

Size of square, a

±

1.0100

0.0520

0.0600

±

2.0000

0.2230

0.0500

±

2.0000

1.0100

0.0300

±

0.5100

1.5100 Length of thread, I2

±

0.0060

-

2.0000

0.0080

-

0.0300

0.0500

0.0600

0.0900

0.1300

0.0300

0.0600

0.0040

0.0050

0.0050

0.0070

0.0070

0.0040

0.0060

0.0080

Entry Taper Length.—Entry taper length is measured on the full diameter of the thread-forming lobes and is the axial distance from the entry diameter position to the theoretical intersection of tap major diameter and entry taper angle. Beveled end threads provided on taps having internal center or incomplete threads retained when external center is removed. Whenever entry taper length is specified in terms of number of threads, this length is measured in number of pitches, P.

1 Bottom length = 1 + 2 2 pitches Plug length = 3 + 5 pitches

Entry diameter measured at the thread crest nearest the front of the tap is an appropriate amount smaller than the diameter of the hole drilled for tapping. L I

I1

I4

L I

2

I2

I

a +0.000

d 1 -0.032 BLANK Design 2 with Optional Neck

BLANK Design 3

Optional Neck and Optional Shortened Thread Length, Ground and Cut Thread (Table 6)

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Nominal Diameter, inch Over

0.260 … 0.323 0.395

0.448

0.510

0.573

0.635

0.709

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0.760

0.823

0.885 0.948

0.117 0.130 0.145 0.171 0.197 … 0.223 0.260

0.323 … 0.395 0.448

4 5 6 8 10 … 12 1 ⁄4

5 ⁄16 … 3 ⁄8 7⁄16

0.510

1 ⁄2

0.573

9 ⁄16

0.635

5 ⁄8

0.709

0.760

0.823

11 ⁄16 3 ⁄4 13 ⁄16

0.885

0.948 1.010

7⁄8

1

15 ⁄16

Nominal Metric Diameter

Blank Design No.

Thread Length I

Neck Length I1

Square Length I2

Shank Diameter d1

Size of Square a

2 … 2

2.72 … 2.94

0.69 … 0.75

0.44 … 0.50

0.38 … 0.44

0.318 … 0.381

0.238 … 0.286

Decimal Equiv.

mm

inch

(0.2160) (0.2500)

M3.0 M3.5 M4.0 M4.5 M5.0 … M6.0

0.1181 0.1378 0.1575 0.1772 0.1969 … 0.2362

1 1 1 1 1 … 1 2

…

3

(0.1120) (0.1250) (0.1380) (0.1640) (0.1900)

(0.3125) … (0.3750)

M7.0 M8.0 M10.0

(0.5000)

M12.0

(0.4375)

(0.5625)

…

M14.0

0.2756 0.3150 0.3937

0.4724

0.5512

3

3

1.88 1.94 2.00 2.13 2.38 … 2.38 2.50

3.16

3.38

3.59

0.31 0.31 0.38 0.38 0.50 … 0.50 0.63

0.88

0.94

1.00

0.50 …

…

0.19 0.19 0.19 0.25 0.25 … 0.28 0.31

0.41

0.44

0.50

0.141 0.141 0.141 0.168 0.194 … 0.220 0.255

0.323

0.367

0.429

0.110 0.110 0.110 0.131 0.152 … 0.165 0.191

0.242

0.275

0.322

M16.0

0.6299

3

3.81

1.09

…

0.56

0.480

0.360

(0.7500)

…

…

3

4.25

1.22

…

0.69

0.590

0.442

0.8661

3

4.69

1.34

…

0.75

0.697

0.523

(0.6875)

M18.0

0.7087

(0.8125)

M20.0

0.7874

(0.9375) (1.0000)

M24.0 M25.0

0.9449 0.9843

(0.8750)

M22.0

3

3

3 3

4.03

4.47

4.91 5.13

1.09

1.22

1.34 1.50

Thread length, I, is a minimum value and has no tolerance.

When thread length, I, is added to neck length, I1, the total shall be no less than the minimum thread length, I.

Unless otherwise specified, all tolerances are in accordance with Table 5b. For number of flutes see Table 11.

0.25 0.31 0.31 0.38 0.38 … 0.44 0.38

(0.6250)

Thread length, I, is based on a length of 12 pitches of the UNC thread series.

For runout tolerances, see Table 14.

Tap Dimensions, inch

Overall Length L

…

…

… …

0.63

0.69

0.75 0.75

0.542

0.652

0.760 0.800

0.406

0.489

0.570 0.600

Machinery's Handbook, 31st Edition Taps

0.104 0.117 0.130 0.145 0.171 … 0.197 0.223

To (inclusive)

Nominal Diameter, inch

Machine Screw Size No. and Fractional Sizes

986

Copyright 2020, Industrial Press, Inc.

Table 6. Optional Neck and Optional Shortened Thread Length (Tap Dimensions, Ground and Cut Thread) ANSI/ASME B94.9-2008 (R2018)

Machinery's Handbook, 31st Edition Taps

987

Table 7. Machine Screw and Fractional Size Ground Thread Dimensions for Screw Thread Insert (STI) Taps ANSI/ASME B94.9-2008 (R2018) Threads per inch

Nominal Size (STI) 1 2 3 4 5 6

NC

NF

24

32

2

8

64 56 48 40 40 32 … 32

12

24

10 1 ⁄4

5 ⁄16 3 ⁄8

7⁄16

1 ⁄2

9 ⁄16

5 ⁄8

3 ⁄4

20

18

16

1 11 ⁄8 11 ⁄4 13⁄8 11 ⁄2

… 64 56 48 … … 40 36

…

28

24

…

1 1 1 1 1 1 1 1

2

2

2

3

Table 5a Blank Equivalent (Reference)

Overall length, L

Thread Length, I

Square Length, I2

Shank Diameter, d1

Size of Square, a

2.50

1.00

0.31

0.255

0.191

No. 3 No. 4 No. 5 No. 6 No. 8 No. 10 No. 8 No. 12 1 ⁄4

0.238

5 ⁄16

0.275

1 ⁄2

0.322

9 ⁄16

0.360

5 ⁄8

0.406

11 ⁄16

0.442

3 ⁄4

7⁄8

1.81 1.88 1.94 2.00 2.13 2.38 2.13 2.38

2.72

2.72

2.94

3.38

0.50 0.56 0.63 0.69 0.75 0.88 0.75 0.94

1.13

1.13

1.25

1.66

0.19 0.19 0.19 0.19 0.25 0.25 0.25 0.28

0.38

0.38

0.44

0.44

0.141 0.141 0.141 0.141 0.168 0.194 0.168 0.220

0.318

0.318

0.381

0.367

0.110 0.110 0.110 0.110 0.131 0.152 0.131 0.165

0.238

5 ⁄16

0.286

3 ⁄8

…

24

3

3.16

1.44

0.41

0.323

0.242

7⁄16

…

20

3

3.38

1.66

0.44

0.367

0.275

1 ⁄2

…

20

3

3.59

1.66

0.50

0.429

0.322

9 ⁄16

…

18

3

3.81

1.81

0.56

0.480

0.360

5 ⁄8

…

18

3

4.03

1.81

0.63

0.542

0.406

11 ⁄16

14 13

12 11

… …

… …

3 3

3 3

3.59 3.81

4.03 4.25

1.66 1.81

1.81 2.00

0.50 0.56

0.63 0.69

0.429 0.480

0.542 0.590

10

…

3

4.69

2.22

0.75

0.697

0.523

9

14

3

5.13

2.50

0.81

0.800

0.600

…

7⁄8

Tap Dimensions, inch

Blank Design No.

8

16

…

3

3

4.47

5.75

2.00

2.56

0.69

1.00

0.652

1.021

0.489

0.766

13 ⁄16

1

11 ⁄4

…

12, 14 NS

3

5.44

2.56

0.88

0.896

0.672

11 ⁄8

…

12

3

5.75

2.56

1.00

1.021

0.766

11 ⁄4

…

12

3

6.06

3.00

1.06

1.108

0.831

13⁄8

…

12

3

6.38

3.00

1.13

1.233

0.925

11 ⁄2

…

12

0.979

15⁄8

7 7 6

6

… … …

…

3 3 3 3

3

6.06 6.38 6.69 7.00

6.69

3.00 3.00 3.19 3.19

3.19

1.06 1.13 1.13 1.25

1.13

1.108 1.233 1.305 1.430

1.305

0.831 0.925 0.979 1.072

13⁄8

11 ⁄2 15⁄8 13⁄4

These threads are larger than nominal size to the extent that the internal thread they produce will accommodate a helical coil screw insert, which at final assembly will accept a screw thread of the nominal size and pitch. For optional necks, refer to Table 6 using dimensions for equivalent blank sizes. Ground Thread Taps: STI sizes 5⁄16 inch and smaller have external center on thread end (may be removed on bottom taps); sizes 10 through 5⁄16 inch will have an external partial cone center on shank end, with the length of the cone center approximately 1 ⁄4 of the diameter of shank; sizes larger than 5⁄16 inch may have internal centers on both the thread and shank ends.

For runout tolerances of tap elements, refer to Table 14 using dimensions for equivalent blank sizes. For number of flutes, refer to Table 11 using dimensions for equivalent blank sizes. For general dimension tolerances, refer to Table 5b using Table 5a equivalent blank size.

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Machinery's Handbook, 31st Edition Taps

988

Table 8. Standard Metric Size Tap Dimensions for Screw Thread Insert (STI) Taps ANSI/ASME B94.9-2008 (R2018) Tap Dimensions, inch

Nominal Size (STI) M2.2

M2.5

M3

M3.5

Thread Pitch, mm Coarse   Fine 0.45

0.45

0.50

0.60

…

…

Blank Design No.

Overall length, L

Thread Length, I

Square Length, I2

Shank Diameter, d1

Size of square, a

1

1.88

0.56

0.19

0.141

0.110

1

…

1

…

1

1.94

2.00

2.13

0.63

0.69

0.75

0.19

0.19

0.25

0.141

0.141

0.168

0.110

0.110

0.131

Blank Diameter No.4

No.5

No.6

No.8

M4

0.70

…

1

2.38

0.88

0.25

0.194

0.152

No.10

M5

0.80

…

2

2.50

1.00

0.31

0.255

0.191

1 ⁄4

M6

1

…

2

2.72

1.13

0.38

0.318

0.238

5 ⁄16

M7

1

…

2

2.94

1.25

0.44

0.381

0.286

3 ⁄8

M8

1.25

1

2

2.94

1.25

0.44

0.381

0.286

3 ⁄8

M10

1.5

1.25

3

3.38

1.66

0.44

0.367

0.275

1 ⁄2

…

…

3

3.16

1.44

0.41

0.323

0.242

7⁄16

M12

1.75

1.5

3

3.59

1.66

0.50

0.429

0.322

9 ⁄16

M14

2

…

3

4.03

1.81

0.63

0.542

0.406

11 ⁄16

…

1.5

3

3.81

1.81

0.56

0.480

0.360

5 ⁄8

2

…

3

4.25

2.00

0.69

0.590

0.442

3 ⁄4

…

1.5

3

4.03

1.81

0.63

0.542

0.406

11 ⁄16

2.5

…

3

4.69

2.22

0.75

0.697

0.523

7⁄8

…

2.0

3

4.47

2.00

0.69

0.652

0.489

13 ⁄16

2.5

2.0

3

4.91

2.22

0.75

0.760

0.570

15 ⁄16

3

4.69

2.22

0.75

0.697

0.523

M16 M18 M20

…

1.25

1.25 1.25

7⁄8

1

M22

2.5

2.0

3

5.13

2.50

0.81

0.800

0.600

…

1.5

3

4.91

2.22

0.75

0.760

0.570

M24

3

…

3

5.44

2.56

0.88

0.896

0.672

…

2

3

5.13

2.50

0.88

0.896

0.672

11 ⁄16

M27

3

…

3

5.75

2.56

1.00

1.021

0.766

11 ⁄4

…

2

3

5.44

2.56

0.88

0.896

0.672

11 ⁄8

M30

3.5

…

3

6.06

3.00

1.06

1.108

0.831

13⁄8

…

2

3

5.75

2.56

1.00

1.021

0.766

11 ⁄4

M33

3.5

…

3

6.38

3.00

1.13

1.233

0.925

11 ⁄2

…

2

3

6.06

3.00

1.06

1.108

0.831

13⁄8

M36

4

3

2

3

6.69

3.19

1.13

1.305

0.979

15⁄8

M39

4

3

2

3

7.00

3.19

1.25

1.430

1.072

13⁄4

15 ⁄16

11 ⁄8

These taps are larger than nominal size to the extent that the internal thread they produce will accommodate a helical coil screw insert, which at final assembly will accept a screw thread of the nominal size and pitch. For optional necks, use Table 6 and dimensions for equivalent blank sizes. Ground Thread Taps: STI sizes M8 and smaller have external center on thread end (may be removed on bottom taps); STI sizes M5 through M10 will have an external partial cone center on shank end, with the length of the cone center approximately 1 ⁄4 of the diameter of shank; STI sizes larger than M10 inch may have internal centers on both the thread and shank ends. For runout tolerances of tap elements, refer to Table 14 using dimensions for equivalent blank sizes. For number of flutes, refer to Table 11 using dimensions for equivalent blank sizes. For general dimension tolerances, refer to Table 5b using Table 5a equivalent blank sizes.

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Table 9. Special Extension Taps ANSI/ASME B94.9-2008 (R2018), Appendix (Tap Dimensions, Ground and Cut Threads) L

I4

I

I2

d1+0.003 Nominal Tap Size Fractional

Machine Screw

… …

Nominal Tap Size Fractional

Machine Screw

Pipe

Shank Length I4

0.88

11 ⁄2

…

…

3.00

1.00

15⁄8

…

3

3.13

…

1.13

13⁄4

…

…

3.13

…

1.25

17⁄8

…

…

3.25

1.38

2

…

…

3.25

1.50

21 ⁄8

…

…

3.38

Pipe

Shank Length I4

0-3

…

4

…

…

5-6

…

8

…

10-12

1 ⁄4

14

1 ⁄16

d1

to 1 ⁄4 incl. …

5 ⁄16

…

…

1.56

21 ⁄4

…

…

3.38

3 ⁄8

…

…

1.63

23⁄8

…

…

3.50

7⁄16

…

1.69

21 ⁄2

…

…

3.50

1 ⁄2

…

…

1.69

25⁄8

…

…

3.63

9 ⁄16

…

3 ⁄4

1.88

23⁄4

…

…

3.63

5 ⁄8

…

1

2.00

27⁄8

…

…

3.75

11 ⁄16

…

…

2.13

3

…

…

3.75

3 ⁄4

…

11 ⁄4

2.25

31 ⁄8

…

…

3.88

13 ⁄16

…

11 ⁄2

2.38

31 ⁄4

…

…

3.88

7⁄8

…

…

2.50

33⁄8

…

4

4.00

15 ⁄16

…

…

2.63

31 ⁄2

…

…

4.00

1

…

…

2.63

35⁄8

…

…

4.13

11 ⁄8

…

2

2.75

33⁄4

…

…

4.13

11 ⁄4

…

21 ⁄2

2.88

37⁄8

…

…

4.25

13⁄8

…

…

3.00

4

…

…

4.25

3 ⁄8

to 1 ⁄2 incl.

Tolerances For shank diameter, d1 for I4 length Fractional, Inch 1 ⁄4 11 ⁄16

to 5⁄8 incl. to 11 ⁄2 incl.

15⁄8 to 4 incl.

Machine Screw 0 to 14 incl. … …

Pipe, Inch

Tolerances

to 1 ⁄8 incl.

-0.003

to 1 incl.

-0.004

11 ⁄4 to 4 incl.

-0.006

1 ⁄16 1 ⁄4

Unless otherwise specified, special extension taps will be furnished with dimensions and toler­ ances as shown for machine screw and fractional taps in Table 5a, Table 5b, and Table 6, and for pipe taps in Table 13a. Exceptions are as follows: Types of centers are optional with manufacturer. Tolerances on shank diameter d1 and I4 length as shown on the above Table 9. Shank runout tolerance applies only to the I4 length shown on the above Table 9.

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990

Table 10. Special Fine Pitch Taps, Short Series ANSI/ASME B94.9-2008 (R2018), Appendix (Taps Dimensions, Ground and Cut Threads) L I4

I

I2

a

d1 Nominal Diameter Range, inch

Nominal Fractional Diameter

Nominal Metric Diameter

Taps Dimensions, inches Overall Length

Thread Length

Square Length

Shank Diameter

Size of Square

Over

To

inch

mm

L

I

I2

d1

a

1.070

1.073

11 ⁄16

M27

4.00

1.50

0.88

0.8960

0.672 0.672

1.073

1.135

11 ⁄8

…

4.00

1.50

0.88

0.8960

1.135

1.198

13⁄16

M30

4.00

1.50

1.00

1.0210

0.766

1.198

1.260

11 ⁄4

…

4.00

1.50

1.00

1.0210

0.766

1.260

1.323

15⁄16

M33

4.00

1.50

1.00

1.1080

0.831

1.323

1.385

13⁄8

…

4.00

1.50

1.00

1.1080

0.831

1.385

1.448

17⁄16

M36

4.00

1.50

1.00

1.2330

0.925

1.448

1.510

11 ⁄2

…

4.00

1.50

1.00

1.2330

0.925

1.510

1.635

15⁄8

M39

5.00

2.00

1.13

1.3050

0.979

1.635

1.760

13⁄4

M42

5.00

2.00

1.25

1.4300

1.072

1.760

1.885

17⁄8

…

5.00

2.00

1.25

1.5190

1.139

2.010

1.885

2.135

M48 …

5.00

2.00

1.38

1.6440

21 ⁄8

2.135

2.260

21 ⁄4

M56

5.25

2.00

1.44

1.8940

1.420

2.260

2.385

23⁄8

…

5.25

2.00

1.50

2.0190

1.514

2.385

2.510

21 ⁄2

…

5.25

2.00

1.50

2.1000

1.575

2.510

2.635

25⁄8

M64

5.50

2.00

1.50

2.1000

1.575

2.635

2.760

23⁄4

…

5.50

2.00

1.50

2.1000

1.575

2.760

2.885

27⁄8

M72

3.010

3.135

3.135

3.260

3.260

2.885

2.010

5.50

2.00

2.00

1.44

1.50

1.7690

2.1000

1.233

1.327

1.575

2.00

1.50

2.1000

31 ⁄8

…

5.50

31 ⁄4

M80

5.75

2.00

1.50

2.1000

1.575

3.385

33⁄8

…

5.75

2.00

1.50

2.1000

1.575

3.385

3.510

31 ⁄2

…

5.75

2.00

1.50

2.1000

1.575

3.510

3.635

35⁄8

M90

6.00

2.00

1.75

2.1000

1.575

3.635

3.760

33⁄4

…

6.00

2.00

1.75

2.1000

1.575

3.760

3.885

37⁄8

4.010

3

5.25

…

3.885

3.010

2

4

…

M100

5.75

6.00

6.00

2.00

2.00

2.00

1.50

1.75

1.75

2.1000

2.1000

2.1000

1.575

1.575

1.575

1.575

Unless otherwise specified, special taps 1.010 inches to 1.510 inches in diameter, inclusive, have 14 or more threads per inch or 1.75 mm pitch and finer. Sizes greater than 1.510 inch in diameter with 10 or more threads per inch or 2.5 mm pitch and finer will be made to the general dimensions shown above. For tolerances, see Table 5b. For runout tolerances of tap elements, see Table 14.

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(0.0600) (0.0730) (0.0860) (0.0990) (0.1120) (0.1250) (0.1380) (0.1640) (0.1900)

13⁄4 2

(1.7500) (2.0000)

(0.2160) (0.2500) (0.3125) (0.3750) (0.4375) (0.5000) (0.5625) (0.6250) (0.7500) (0.8750) (1.0000) (1.1250) (1.2500) (1.3750) (1.5000)

mm

M1.6 … M2.0 M 2.5 … M3.0 M3.5 M4.0 M4.5 M5 … M6 M7 M8 M10 … M12 M14 M16 … M20 … M24 … … M30 … … … … M36 … … … …

inch

0.0630 … 0.0787 0.0984 … 0.1181 0.1378 0.1575 0.1772 0.1969 … 0.2362 0.2756 0.3150 0.3937 … 0.4724 0.5512 0.6299 … 0.7874 … 0.9449 … … 1.1811 … … … … 1.4173 … … … …

TPI/Pitch

UNC NC … 64 56 48 40 40 32 32 24 … 24 20 18 18 16 14 13 12 11 10 … 9 … 8 7 … 7 … 6 … … 6 … 5 41 ⁄2

UNF NF 80 72 64 56 48 44 40 36 32 … 28 28 24 24 24 20 20 18 18 16 … 14 … 12 12 … … 12 … 12 … … 12 … …

Straight Flutes

mm

0.35 … 0.40 0.45 0.50 0.60 0.70 0.75 0.80 … 1.00 1.00 1.25 1.50 … 1.75 2.00 2.00 … 2.5 … 3.00 … 4.00 3.50 … … … … 4.00 … … … …

Standard 2 2 3 3 3 3 3 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 … 4 4 6 6 6 4 4 6 6 6

Spiral Point

Optional … … 2 2 2 2 2 2 ⁄3 2 ⁄3 2 ⁄3 2 ⁄3 2 ⁄3 2 ⁄2 2 ⁄3 3 3 3 … … … … … … … … … … … … … … … … … …

Standard 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 … … … … … … … … … … … … … … …

Optional … … … … … … … … … … … 3 3 3 … … … … … … … … … … … … … … … … … … … … …

Spiral Point Only … … … … 2 2 2 2 2 2 2 2 2 2 3 3 … … … … … … … … … … … … … … … … … … …

Reg. Spiral Flute

… … … … 2 2 2 2 2 2 2 3 (optional) 3 3 3 3 … … … … … … … … … … … … … … … … … … …

Fast Spiral Flute … … … 2 2 2 2 3 3 3 3 3 3 3 3 3 3 … … … … … … … … … … … … … … … … … …

For pulley taps, see Table 12. For taper pipe, see Table 13a. For straight pipe taps, see Table 13a. For STI taps, use number of flutes for blank size equivalent on Table 5a. For optional flutes, see Table 6.

991

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 0  1  2  3  4  5  6  8 10 … 12 1 ⁄4 … 5 ⁄16 3 ⁄8 7⁄16 1 ⁄2 9 ⁄16 5 ⁄8 3 ⁄4 … 7⁄8 … 1 11 ⁄8 … 11 ⁄4 … 13⁄8 … … 11 ⁄2

Table 11. Standard Number of Flutes (Ground and Cut Thread) ANSI/ASME B94.9-2008 (R2018)

Nominal Metric Dia.

Machinery's Handbook, 31st Edition Taps

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Machine Screw Size, Nom. Fractional Dia. inch

Machinery's Handbook, 31st Edition Taps

992

Table 12. Pulley Taps, Fractional Size (High-Speed Steel, Ground Thread) ANSI/ASME B94.9-2008 (R2018) L I

I4

I1

I2

a d1

+0.03

d 1 -0.03

Threads per Dia. Inch Number of NC of Tap UNC Flutes 1⁄ 20 4 4 5⁄ 18 4 16 3⁄ 8

7⁄ 16

16

1⁄ 2

13

3⁄ 4

10

5⁄ 8

Length Overall, L 6, 8 6, 8

4

14

6, 8, 10

4

11

Length  d Dia. Size  b of Shank Thread Neck Square of of Close Length, Length, Length Tolerance, Shank, Square, I I1 I2 d1 a I4 1.00 0.38 1.50 0.255 0.191 0.31 1.13 0.38 1.56 0.318 0.238 0.38 1.25

6, 8

4

6, 8, 10, 12

4

10,12

4

1.44 1.66

6, 8,10,12

1.81

2.00

0.38

0.44

0.44

1.63

0.381

0.286

0.56

1.69

0.507

0.380

0.75

2.25

0.759

0.569

1.69

0.50

0.50 0.63

2.00

0.69

0.75

Tolerances for General Dimensions

Element Overall length, L

Thread length, I

Square length, I2 Neck length, I1

Diameter Range 1⁄ 4

to 3⁄4

0.633

Diameter Range

0.333

0.475

Tolerance

Element

±0.06

Shank Diameter, d1a

1⁄ 4

to 1 ⁄2

-0.005

Size of Square, ab

1 ⁄ to 1 ⁄ 4 2 5 ⁄ to 3 ⁄ 8 4

-0.004 -0.006

to 3⁄4

±0.06

1⁄ 4

to 3⁄4

±0.03

1⁄ 4

to 3⁄4

c

1⁄ 4

0.444

Length of close tolerance shank, I4

1⁄ 4

Tolerance

to 3⁄4

d

a Shank diameter, d

1, is approximately the same as the maximum major diameter for that size. b Size of square, a, is equal to 0.75d to the nearest 0.001 in. 1 c Neck length, I , is optional with manufacturer. 1 d Length of close tolerance shank, I , is a min. length that is held to runout tolerances per Table 14. 4

These taps are standard with plug chamfer in H3 limit only. All dimensions are given in inches. These taps have an internal center in thread end. For standard thread limits, see Table 20. For runout tolerances of tap elements, see Table 14.

L

L I

I

I2

I1

I2 d1

d1

L I

I2 d1

Straight and Taper Pipe Tap Dimensions, Ground and Cut Thread (Table 13a and Table 13b)

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Nominal Size, Inch  a

Threads per Inch

1⁄ 16

27

1⁄ 8

27

3⁄ 8

18

3⁄ 4

14

1⁄ 8

27

1⁄ 4

18

1⁄ 2

14

1

11 1 ⁄2

1 1 ⁄2

11 1 ⁄2

1 1 ⁄4 2

11 1 ⁄2 8

8

4

…

4

5

4 4 4 4 5

5 5 7 7

8

8

5 5 5 5 5

5 5 7 7

…

…

Ground Thread NPSC, NPT, NPTF, NPSM, ANPT NPSF

Length Overall, L

Thread Length, I

Square Length, I2

Shank Diameter, d1

Size of Square, a

Length Optional Neck, I1

2.13

0.75

0.38

0.3125

0.234

…

b, c

0.375

2.13 2.13 2.44 2.56 3.13 3.25

3.75 4.00 4.25 4.25

5.50

6.00

0.69 0.75 1.06 1.06 1.38 1.38

1.75 1.75 1.75 1.75

2.56

2.63

0.38 0.38 0.44 0.50 0.63 0.69

0.81 0.94 1.00 1.13

1.25

1.38

0.3125 0.4375 0.5625 0.7000 0.6875 0.9063

1.1250 1.3125 1.5000 1.8750

2.2500

2.6250

0.234

0.375

Cut Thread Only NPT

b­

… d, e

f, g, h

…

NPSC, NPSM …

…

0.328

0.375

b, c

d, e

f, g, h

b, c

d, e

f, g, h

a

0.531

0.375

b, c

d, e

f, g, h

a

…

b, c

d, e

f, g, h

a

…

b, c

d

f, g, h

a

…

b, c

d

f, g, h

a

…

b, c

f, g, h

a

…

b, i

… …

f, h f, h

…

…

…

…

…

h

…

h

0.421 0.515 0.679

0.843 0.984 1.125 1.406

1.687

1.968

a Pipe taps 1 ⁄ inch are furnished with large-size shanks unless the small shank is specified. 8 b High-speed ground thread 1 ⁄ to 2 inches including noninterrupted (NPT, NPTF, and ANPT). 16 c High-speed ground thread 1 ⁄ to 11 ⁄ inch including interrupted (NPT, NPTF, and ANPT). 8 4 d High-speed ground thread 1 ⁄ to 1 inch including noninterrupted (NPSC and NPSM). 8 e High-speed cut thread 1 ⁄ to 1 inch including noninterrupted (NPSC and NPSM). 8 f High-speed cut thread 1 ⁄ to 1 inch including noninterrupted (NPT). 8 g High-speed cut thread 1 ⁄ to 11 ⁄ inch including interrupted (NPT). 8 4 h Carbon cut thread 1 ⁄ to 11 ⁄ inch including interrupted (NPT). 8 4 i High-speed ground thread 11 ⁄ to 2 inches including interrupted (NPT). 2

…

…

b, i

…

a

… … …

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2 1 ⁄2 3

11 1 ⁄2

Number of Flutes Inter­ Regular rupted Thread Thread

Machinery's Handbook, 31st Edition Taps

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Table 13a. Straight and Taper Pipe Tap Dimensions (Ground and Cut Thread) ANSI/ASME B94.9-2008 (R2018)

Machinery's Handbook, 31st Edition Taps

994

Table 13b. Straight and Taper Pipe Taps Tolerances (Ground and Cut Thread) ANSI/ASME B94.9-2008 (R2018) Ground Thread Nominal Diameter Range, inch To Over (inclusive)

Element Length overall, L

3 ⁄4

11 ⁄2

11 ⁄4 2

1 1 ⁄16 1

Length of thread, I

2 3 ⁄4

1 ⁄16

Length of square, I2 Diameter of shank, d1

Size of square, a

1 ⁄16

3 ⁄4

1 1 ⁄16

2 1 ⁄8 1

1 ⁄4

2

11 ⁄4

1 ⁄16

1 ⁄8

3 ⁄4

1 ⁄4

1

2

Tolerances, inch ±0.031 ±0.063 ±0.063

Cut Thread Nominal Diameter Range, inch To Over (inclusive)

Element

Length of thread, I

±0.031 ±0.063

Length of square, I2

-0.002 -0.002 -0.002 -0.004 -0.006 -0.008

3 ⁄4

11 ⁄2

11 ⁄4 3

1 1 ⁄8 1

±0.094

±0.125

1 ⁄8

Length overall, L

1 ⁄8

Diameter of shank, d1 Size of square, a

1 1 ⁄8 3 ⁄4

3 3 ⁄4

3 ⁄4

3 1 ⁄2 3

1 ⁄8

…

1 ⁄4

3 ⁄4

1

3

Tolerances, inch ±0.031 ±0.063 ±0.063 ±0.094

±0.125

±0.031 ±0.063 - 0.007 - 0.009 - 0.004 - 0.006 - 0.008

All dimensions are given in inches. The first few threads on interrupted thread pipe taps are left full. These taps have inter­nal centers. For runout tolerances of tap elements, see Table 14. Taps marked NPS are suitable for NPSC and NPSM. These taps have 2 to 31 ⁄2 threads chamfer, see Table 5a. Optional neck is for man­ufacturing use only. For taper pipe thread limit, see Table 24a. For straight pipe thread limits, see Table 23a, Table 23b, and Table 23d.

Table 14. Runout and Locational Tolerance of Tap Elements ANSI/ASME B94.9-2008 (R2018) a, A-B

,

, d1, A-B

A

B , c, A-B Chamfer

, d2, A-B Pitch Diameter

, da, A-B Major Diameter

Range Sizes (Inclusive) Machine Screw Shank, d1 Major diameter, da Pitch Diameter, d2 Chamfer a , c

Square, a

Metric

#0 to 5⁄16

M1.6 to M8

#0 to 5⁄16

M1.6 to M8

#0 to 5⁄16

M1.6 to M8

#0 to 1 ⁄2

M1.6 to M12

to 4

M14 to M100

#0 to 1 ⁄2

M1.6 to M12

to 4

M14 to M100

11 ⁄32 11 ⁄32 11 ⁄32

17⁄32

17⁄32

to 4 to 4 to 4

M10 to M100 M10 to M100 M10 to M100

Pipe, Inch 1 ⁄16 1 ⁄8

to 4

1 ⁄16 1 ⁄8

to 4

1 ⁄16 1 ⁄8

to 4

1 ⁄16 to 1 ⁄8 1 ⁄8

to 4

1 ⁄16 to 1 ⁄8 1 ⁄8

to 4

Total Runout FIM, Inch Ground Cut Thread Thread

Location, Inch

0.0060

0.0010

…

0.0050

0.0010

…

0.0080 0.0080 0.0050 0.0080

0.0016 0.0016 0.0010 0.0016

… … … …

0.0040

0.0020

0.0060

0.0030

… …

…

…

0.0060

…

…

0.0080

Chamfer should preferably be inspected by light projection to avoid errors due to indicator contact points dropping into the thread groove. a

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Machinery's Handbook, 31st Edition Taps

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Table 15. Tap Thread Limits: Metric Sizes, Ground Thread (M Profile Standard Thread Limits in Inches) ANSI/ASME B94.9-2008 (R2018) Nom. Dia mm

1.6 2 2.5 3 3.5 4 4.5 5 6 7 8 10 12 14 14 16 18 20 24 30 36 42 48

Major Diameter (Inches)

Pitch, mm 0.35 0.4 0.45 0.5 0.6 0.7 0.75 0.8 1 1 1.25 1.5 1.75 2 1.25 2 1.5 2.5 3 3.5 4 4.5 5

Basic

0.06299 0.07874 0.09843 0.11811 0.13780 0.15748 0.17717 0.19685 0.23622 0.27559 0.31496 0.39370 0.47244 0.55118 0.55118 0.62992 0.70870 0.78740 0.94488 1.18110 1.41732 1.65354 1.88976

Min.

0.06409 0.08000 0.09984 0.11969 0.13969 0.15969 0.17953 0.19937 0.23937 0.27874 0.31890 0.39843 0.47795 0.55748 0.55500 0.63622 0.71350 0.79528 0.95433 1.19213 1.42992 1.66772 1.90552

Max.

0.06508 0.08098 0.10083 0.12067 0.14067 0.16130 0.18114 0.20098 0.24098 0.28035 0.32142 0.40094 0.48047 0.56000 0.55600 0.63874 0.71450 0.79780 0.95827 1.19606 1.43386 1.71102 1.98819

Basic

Limit # D

0.05406 0.06850 0.08693 0.10531 0.12244 0.13957 0.15799 0.17638 0.21063 0.25000 0.28299 0.35535 0.42768 0.50004 0.51920 0.57878 0.67030 0.72346 0.86815 1.09161 1.31504 1.53846 1.76189

3

4

5 6 7 4 7 4 7 8 9 10

Pitch Diameter (Inches)

D # Limit

Min.

0.05500 0.06945 0.08787 0.10626 0.12370 0.14083 0.15925 0.17764 0.21220 0.25157 0.28433 0.35720 0.42953 0.50201 0.52070f 0.58075 0.67180f 0.72543 0.87063 1.0942 1.3176 1.5415 1.7649

Max.

0.05559 0.07004 0.08846 0.10685 0.12449 0.14161 0.16004 0.17843 0.21319 0.25256 0.28555 0.35843 0.43075 0.50362 0.52171f 0.58236 0.67230f 0.72705 0.8722 1.0962 1.3197 1.5436 1.7670

Limit # D

… … … 5 … 6 … 7 8 … 9 10 11 … … … 7 … … … … … …

D # Limit

Min.

Max.

… … … 0.10278a,b

… … … 0.10787a, b

0.14185a, b … 0.17917b, c 0.21374b, c

0.14264a, b … 0.17996b, c 0.2147b, c

0.2864b, d 0.3593b, e 0.43209e … … … 0.58075 … … … … … …

0.2875b, d 0.3605b, e 0.43331e … … … 0.58236 … … … … … …

a Minimum and maximum major diameters are 0.00102 larger than shown. b Standard D limit for thread-forming taps.

c Minimum and maximum major diameters are 0.00154 larger than shown.

d Minimum and maximum major diameters are 0.00205 larger than shown. e Minimum and maximum major diameters are 0.00256 larger than shown.

f These sizes are intended for spark plug applications; use tolerances from Table 2, column D.

All dimensions are given in inches. Not all styles of taps are available with all limits listed. For cal­culation of limits other than those listed, see formulas in Table 4.

Table 16. Tap Thread Limits: Metric Sizes, Ground Thread (M Profile Standard Thread Limits in Millimeters) ANSI/ASME B94.9-2008 (R2018) Size mm 1.6 2 2.5 3 3.5 4 4.5 5 6 7 8 10 12 14 14 16 18

Pitch 0.35 0.4 0.45 0.5 0.6 0.7 0.75 0.8 1.00 1.00 1.25 1.50 1.75 2.00 1.25 2.00 1.50

Major Diameter Basic Min. Max.

1.60 2.00 2.50 3.00 3.50 4.00 4.50 5.00 6.00 7.00 8.00 10.0 12.0 14.0 14.0 16.0 18.0

1.628 2.032 2.536 3.040 3.548 4.056 4.560 5.064 6.121 7.121 8.10 10.12 12.14 14.01 14.16 16.16 18.12

1.653 2.057 2.561 3.065 3.573 4.097 4.601 5.105 5.351 6.351 8.164 10.184 12.204 14.164 14.224 16.224 18.184

Basic

1.373 1.740 2.208 2.675 3.110 3.545 4.013 4.480 5.391 6.391 7.188 9.026 10.863 13.188 12.701 14.701 17.026

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D# 3

4

5 6 4 7 4

Pitch Diameter D # Limit Min. Max. D#

1.397 1.764 2.232 2.699 3.142 3.577 4.045 4.512 5.391 6.391 7.222 9.073 10.910 7.222f 12.751 14.751 17.063f

1.412 1.779 2.247 2.714 3.162 3.597 4.065 4.532 5.416 6.416 7.253 9.104 10.941 7.253f 12.792 14.792 17.076f

… … … 5 … 6 … 7 8 … 9 10 11 … … … …

D # Limit Min. Max.

… … … 2.725a,b … 3.603a,b … 4.551b,c 5.429b,c … 7.274b,d 9.125b,d 10.975b,e … … … …

… … … 2.740a,b … 3.623a,b … 4.571b,c 5.454b,c … 7.305b,d 9.156b,d 11.006b,e … … … …

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Machinery's Handbook, 31st Edition Taps

996

Table Table 16. (Continued) Tap Thread Limits: Ground Thread 16. Tap Thread Limits: MetricMetric Sizes, Sizes, Ground Thread (M Profile Standard Thread Limits in Millimeters) ANSI/ASME B94.9-2008 (R2018) Size mm 20 24 30 36 42 48

Pitch 2.50 3.00 3.50 4.00 4.50 5.00

Major Diameter Basic Min. Max. 20.0 20.20 20.263 24.0 24.24 24.34 30.0 30.28 30.38 36.0 36.32 36.42 42.0 42.36 42.46 48.0 48.48 48.58

Basic 18.376 22.051 27.727 33.402 39.077 44.103

D# 7 8 9 10

Pitch Diameter D # Limit D# Min. Max. 18.426 18.467 … 22.114 22.155 … 27.792 27.844 … 33.467 33.519 … 39.155 39.207 … 44.182 44.246 …

a Minimum and maximum major diameters are 0.026 larger than shown.

D # Limit Min. Max. … … … … … … … … … … … …

b Standard D limit for thread-forming taps.

c Minimum and major diameters are 0.039 larger than shown.

d Minimum and major diameters are 0.052 larger than shown.

e Minimum and major diameters are 0.065 larger than shown.

f These sizes are intended for spark plug applications; use tolerances from Table 2, column D.

Notes for Table 16: Inch translations are listed in Table 15. Limits listed in Table 16 are the most commonly used in industry. Not all styles of taps are available with all limits listed. For calculations of limits other than listed, see formulas in Table 4.

Table 17. Tap Size Recommendations for Class 6H Metric Screw Threads

Nominal Diameter, mm 1.6 2 2.5 3 3.5 4 4.5 5 6 7 8 10 12 14 16 20 24 30 36

Pitch, mm 0.35 0.4 0.45 0.5 0.6 0.7 0.75 0.8 1 1 1.25 1.5 1.75 2 2 2.5 3 3.5 4

Recommended Thread Limit Number D3 D3 D3 D3 D4 D4 D4 D4 D5 D5 D5 D6 D6 D7 D7 D7 D8 D9 D9

Internal Threads, Pitch Diameter

Min. (mm) 1.373 1.740 2.208 2.675 3.110 3.545 4.013 4.480 5.350 6.350 7.188 9.206 10.863 12.701 14.701 18.376 22.051 27.727 33.402

Max. (mm) 1.458 1.830 2.303 2.775 3.222 3.663 4.131 4.605 5.500 6.500 7.348 9.206 11.063 12.913 14.913 18.600 22.316 28.007 33.702

Min. (inch)

Max. (inch)

0.05406 0.06850 0.08693 0.10537 0.12244 0.13957 0.15789 0.17638 0.201063 0.2500 0.28299 0.35535 0.42768 0.50004 0.57878 0.72346 0.86815 1.09161 1.31504

0.05740 0.07250 0.09067 0.10925 0.12685 0.14421 0.16264 0.18130 0.21654 0.25591 0.28929 0.36244 0.43555 0.50839 0.58713 0.73228 0.87858 1.10264 1.32685

The above recommended taps normally produce the class of thread indicated in average materials when used with reasonable care. However, if the tap specified does not give a satisfactory gage fit in the work, a choice of some other limit tap will be necessary.

Table 18. Standard Chamfers for Thread Cutting Taps ANSI/ASME B94.9-2008 (R2018)

Type of tap Straight threads taps

Chamfer length

Bottom Semibottom Plug Taper

Min. 1P 2P 3P 7P

Max. 2P 3P 5P 10P

Type of tap Taper pipe taps

Chamfer length

Min.

Max.

2P

31 ⁄2  P

P = pitch. The chamfered length is measured at the cutting edge and is the axial length from the point diame­ter to the theoretical intersection of the major diameter and the chamfer angle. Whenever chamfer length is specified in terms of threads, this length is measured in number of pitches as shown. The point diameter is approximately equal to the basic thread minor diameter.

Copyright 2020, Industrial Press, Inc.

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Machinery's Handbook, 31st Edition Taps

997

Table 19. Taps Sizes for Classes 2B and 3B Unified Screw Threads Machine Screw, Numbered, and Fractional Sizes ANSI/ASME B94.9-2008 (R2018) Size

Threads per Inch NC NF UNC UNF

0 1 1 2 2 3 3 4 4 5 5 6 6 8 8 10 10 12 12

… 64 … 56 … 48 … 40 … 40 … 32 … 32 … 24 … 24 …

80 … 72 … 64 … 56 … 48 … 44 … 40 … 36 … 32 … 28

1 ⁄4

20 … 18 … 16 … 14 … 13 … 12 … 11 … 10 … 9 … 8 … 14NS 7 … 7 … 6 … 6 …

… 28 … 24 … 24 … 20 … 20 … 18 … 18 … 16 … 14 … 12 14NS … 12 … 12 … 12 … 12

1 ⁄4

5 ⁄16 5 ⁄16 3 ⁄8 3 ⁄8

7⁄16 7⁄16 1 ⁄2 1 ⁄2

9 ⁄16 9 ⁄16 5 ⁄8 5 ⁄8 3 ⁄4 3 ⁄4

7⁄8 7⁄8

1 1 1

11 ⁄8 11 ⁄8 11 ⁄4 11 ⁄4 13⁄8 13⁄8 11 ⁄2 11 ⁄2

Recommended Tap For Class of Threada

Pitch Diameter Limits For Class of Thread Min., All Max. Max. Class 2Bb Class 3Bc Classes (Basic) Class 2B Class 3B Machine Screw Numbered-Size Taps G H2 G H2 G H2 G H2 G H2 G H2 G H2 G H2 G H2 G H2 G H2 G H3 G H2 G H3 G H2 G H3 G H3 G H3 G H3 G H5 G H4 G H5 G H4 G H5 G H4 G H5 G H5 G H5 G H5 G H5 G H5 G H5 G H5 G H5 G H5 G H6 G H6 G H6 G H6 G H6 G H8 G H6 G H8 G H6 G H8 G H6 G H8 G H6

G H1 G H1 G H1 G H1 G H1 G H1 G H1 G H2 G H1 G H2 G H1 G H2 G H2 G H2 G H2 G H3 G H2 G H3 G H3 Fractional-Size Taps G H3 G H3 G H3 G H3 G H3 G H3 G H3 G H3 G H3 G H3 G H3 G H3 G H3 G H3 G H5 G H3 G H4 G H4 G H4 G H4 G H4 G H4 G H4 G H4 G H4 G H4 G H4 G H4 G H4

0.0519 0.0629 0.0640 0.0744 0.0759 0.0855 0.0874 0.0958 0.0985 0.1088 0.1102 0.1177 0.1218 0.1437 0.1460 0.1629 0.1697 0.1889 0.1928

0.0542 0.0655 0.0665 0.0772 0.0786 0.0885 0.0902 0.0991 0.1016 0.1121 0.1134 0.1214 0.1252 0.1475 0.1496 0.1672 0.1736 0.1933 0.1970

0.0536 0.0648 0.0659 0.0765 0.0779 0.0877 0.0895 0.0982 0.1008 0.1113 0.1126 0.1204 0.1243 0.1465 0.1487 0.1661 0.1726 0.1922 0.1959

0.2175 0.2268 0.2764 0.2854 0.3344 0.3479 0.3911 0.4050 0.4500 0.4675 0.5084 0.5264 0.5660 0.5889 0.6850 0.7094 0.8028 0.8286 0.9188 0.9459 0.9536 1.0322 1.0709 1.1572 1.1959 1.2667 1.3209 1.3917 1.4459

0.2224 0.2311 0.2817 0.2902 0.3401 0.3528 0.3972 0.4104 0.4565 0.4731 0.5152 0.5323 0.5732 0.5949 0.6927 0.7159 0.8110 0.8356 0.9276 0.9535 0.9609 1.0416 1.0787 1.1668 1.2039 1.2771 1.3291 1.4022 1.4542

0.2211 0.2300 0.2803 0.2890 0.3387 0.3516 0.3957 0.4091 0.4548 0.4717 0.5135 0.5308 0.5714 0.5934 0.6907 0.7143 0.8089 0.8339 0.9254 0.9516 0.9590 1.0393 1.0768 1.1644 1.2019 1.2745 1.3270 1.3996 1.4522

a Recommended taps are for cutting threads only and are not for roll-form threads. b Cut-thread taps in sizes #3 to 11 ⁄ in. NC and NF, inclusive, may be used under all normal 2 conditions and in average materials for producing Class 2B tapped holes. c Taps suited for class 3B are satisfactory for class 2B threads.

All dimensions are given in inches. The above recommended taps normally produce the class of thread indicated in average materials when used with reasonable care. However, if the tap specified does not give a satisfactory gage fit in the work, a choice of some other limit tap will be necessary.

Copyright 2020, Industrial Press, Inc.

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Pitch Diameter

Threads per Inch

Size

4 4 5

5 6

6

8

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8 10 10 12 12

… 64 … 56

NS

48 … 40

56 …

… … … … … … … …

…

48

…

… 40

… 32

…

32

… 24 … 24 …

80 … 72 … 64

… …

44 …

40

…

36 … 32 … 28

36 …

… …

…

…

… … … … …

Basic

Min.

Max.

Basic

Min.

Max.

H2 limit Min.

Min.

Max.

Min.

Max.

0.0950

…

…

…

…

0.0605 0.0736 0.0736 0.0866 0.0866 0.0999 0.0997 0.1134

0.0616 0.0750 0.0748 0.0883 0.0880 0.1017 0.1013 0.1153

0.0519 0.0629 0.0640 0.0744 0.0759 0.0855 0.0874 0.0958

0.0519 0.0629 0.064 0.0744 … … 0.0874 0.0958

0.0524 0.0634 0.0645 0.0749 … … 0.0879 0.0963

0.0524 0.0634 0.0645 0.0749 0.0764 0.086 0.0879 0.0963

0.0529 0.0639 0.0650 0.0754 0.0769 0.0865 0.0884 0.0968

0.1120

0.1129

0.1147

0.0985

0.0985

0.0990

0.0990

0.0995

0.1250

0.1250 0.1380

0.1380

0.1640

0.1640 0.1900 0.1900 0.2160 0.2160

0.1135

0.1264

0.1262 0.1400

0.1394

0.1660

0.1655 0.1927 0.1920 0.2187 0.2183

0.1156

0.1283

0.1280 0.1421

0.1413

0.1681

0.1676 0.1954 0.1941 0.2214 0.2206

0.0940

0.1088 0.1102 0.1177

0.1218

0.1437

0.1460 0.1629 0.1697 0.1889 0.1928

0.094

0.1088 …

0.1177

0.1218

0.1437

… 0.1629 0.1697 … …

0.0945

0.1093 …

0.1182

0.1223

0.1442

… 0.1634 0.1702 … …

0.0945

0.1093 0.1107

0.1182

0.1223

0.1442

0.1465 0.1634 0.1702 … …

H4 limit

Max.

0.0600 0.0730 0.0730 0.0860 0.0860 0.0990 0.0990 0.1120 0.1120

H3 limit

0.1098 0.1112

… … … … … … … …

…

…

…

… … … … … … … …

…

…

…

… … … … … … … …

…

…

…

… … … … … … … 0.0983d

…

0.1005

d

0.1010d

…

…

…

…

0.1447

0.1447 0.1452

…

…

…

…

…

a Minimum and maximum major diameters are 0.0010 larger than shown.

b Minimum and maximum major diameters are 0.0020 larger than shown. c Minimum and maximum major diameters are 0.0035 larger than shown.

d Minimum and maximum major diameters are 0.0015 larger than shown.

General notes: Limits listed in above table are the most commonly used in the industry. Not all styles of taps are available with all limits listed. For calculation of limits other than those listed, see formulas and Table 2.

Max.

… … … … … … … d 0.0978

0.1187 0.1192

0.1470 … … … 0.1639 0.1639 0.1644 0.1644 0.1707 0.1707 0.1712 0.1712 … 0.1899 0.1904 0.1904 … 0.1938 0.1943 0.1943

Min.

… … … … … … … …

0.1187

0.1228

H6 limita

H5 limit

…

0.0960d 0.1108

d d

0.1122

0.1197

a

0.1238

a

0.1457a

… 0.1480a 0.1649 … 0.1717 … 0.1909 … 0.1948 …

0.0965d 0.1113

d

0.1127

d a

0.1202

a

0.1243

0.1462a 0.1485a … … … …

H7 limitb

H8 limitc

Min.

Max.

Min.

Max.

Min.

Max.

…

…

…

…

…

…

… … … … … … … … …

… … … … …

… 0.1654 0.1722 0.1914 0.1953

… … … … … … … … …

… …

… … … … … … … … …

… …

… … … … … … … … …

… …

… … … … … … … … …

… …

… … … … … … … … …

… …

…

0.1207 0.1212 0.1222 0.1227

…

0.1467 0.1472 0.1482 0.1487

…

…

…

…

…

… … … … … 0.1659 0.1659 0.1664 … … 0.1727 0.1727 0.1732 0.1742 0.1747 0.1919 … … … … 0.1958 … … … …

Machinery's Handbook, 31st Edition Taps

0 1 1 2 2 3 3 4

NC NF UNF UNF

H1 limit

Major Diameter

998

Copyright 2020, Industrial Press, Inc.

Table 20. Tap Thread Limits: Machine Screw Sizes, Ground Thread ANSI/ASME B94.9-2008 (R2018) (Unified and American National Thread Forms, Standard Thread Limits)

Pitch Diameter

Major Diameter Size NC NF inch UNC UNF NS

H1 limit Basic

Min.

Max.

Basic

Min.

Max.

H2 limit Min.

Max.

H3 limit Min.

Max.

H4 limit Max.

Min.

Max.

Min.

Max.

Min.

Max.

Min.

Max.

…

…

0.2195a

0.2200a

…

…

…

…

…

…

…

…

…

…

…

…

…

…

0.2784a

0.2789a

…

…

0.2794c

0.2799c

…

…

20

…

…

0.2500 0.2532 0.2565 0.2175 0.2175 0.2180 0.2180 0.2185 0.2185 0.2190

1 ⁄4

…

28

…

0.2500 0.2523 0.2546 0.2268 0.2268 0.2273 0.2273 0.2278 0.2278 0.2283 0.2283 0.2288

5 ⁄16

18

…

…

0.3125 0.3161 0.3197 0.2764 0.2764 0.2769 0.2769 02774 0.2774 0.2779

5 ⁄16

…

24

…

0.3125 0.3152 0.3179 0.2854 0.2854 0.2859 0.2859 0.2864 0.2864 0.2869 0.2869 0.2874

3 ⁄8

16

…

…

0.3750 0.3790 0.3831 0.3344 0.3344 0.3349 0.3349 0.3354 0.3354 0.3359

3 ⁄8

…

24

…

0.3750 0.3777 0.3804 0.3479 0.3479 0.3484 0.3484 0.3489 0.3489 0.3494 0.3494 0.3499

7⁄16

14

…

…

0.4375 0.4422 0.4468 0.3911

…

…

7⁄16

…

20

…

0.4375 0.4407 0.4440 0.4050

…

…

…

…

…

…

…

0.2884c

0.2889c

…

…

0.3364a

0.3369a

…

…

0.3374c

0.3379c

…

…

…

…

…

…

0.3509c

0.3514c

…

…

…

…

0.3931a

0.3936a

…

…

…

…

0.3946 0.3951

…

…

0.4070a

0.4075a

a

…

…

…

0.4085 0.4090

…

0.5000 0.5050 0.5100 0.4500 0.4500 0.4505 0.4505 0.4510 0.4510 0.4515

…

…

0.4520

0.4525a

…

…

…

…

0.4535 0.4540

1 ⁄2

…

20

…

0.5000 0.5032 0.5065 0.4675 0.4675 0.4680 0.4680 0.4685 0.4685 0.4690

…

…

0.4695a

0.4700a

…

…

…

…

0.4710 0.4715

9 ⁄16

12

…

…

0.5625 0.5679 0.5733 0.5084

…

…

0.5094 0.5099

…

…

0.5104a

0.5109a

…

…

0.5114c

0.5119c

…

…

9 ⁄16

…

18

…

0.5625 0.5661 0.5697 0.5264

…

…

0.5269 0.5274 0.5274 0.5279

…

…

0.5284a

0.5289a

…

…

0.5294c

0.5299c

…

…

5 ⁄8

11

…

…

0.6250 0.6309 0.6368

0.566

…

…

0.5665

0.5675

…

…

0.5680a

0.5685a

…

…

0.5690c

0.5695c

…

…

5 ⁄8

…

18

…

0.6250 0.6286 0.6322 0.5889

…

…

0.5894 0.5899 0.5899 0.5904

…

…

0.5909a

0.5914a

…

…

0.5919c

0.5924c

…

…

11 ⁄16

…

…

11

0.6875 0.6934 0.6993 0.6285

…

…

…

…

0.6295 0.6300

…

…

…

…

…

…

…

…

…

…

11 ⁄16

…

…

16

0.6875 0.6915 0.6956 0.6469

…

…

…

…

0.6479 0.6484

…

…

…

…

…

…

…

…

…

…

3 ⁄4

10

…

…

0.7500 0.7565 0.7630 0.6850

…

…

0.6855 0.6860 0.6860 0.6865

…

…

0.6870

0.6875

…

…

0.6880d

0.6885d

…

…

3 ⁄4

…

16

…

0.7500 0.7540 0.7581 0.7094 0.7094 0.7099 0.7099 0.7104 0.7104 0.7109

…

…

0.7114a

0.1119a

…

…

0.7124d

0.4129d

…

…

7⁄8

9

…

…

0.8750 0.8822 0.8894 0.8028

…

…

0.8053

0.8058

…

…

…

…

…

…

0.567

…

0.567

…

…

0.8043 0.8048

999

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…

…

…

0.4060 0.4065

…

…

13

…

…

…

1 ⁄2

…

…

H8 limitb

H7 limit

Min.

1 ⁄4

0.3916 0.3921 0.3921 0.3926

H6 limita

H5 limit

Machinery's Handbook, 31st Edition Taps

Copyright 2020, Industrial Press, Inc.

Table 21. Tap Thread Limits: Fractional Sizes, Ground Thread ANSI/ASME B94.9-2008 (R2018) (Unified and American National Thread Forms, Standard Thread Limits)

Pitch Diameter

Major Diameter Size NC NF inch UNC UNF NS

H1 limit Basic

Min.

Max.

…

…

1

8

…

…

1.0000 1.0082 1.0163 0.9188

…

…

…

1

…

12

…

1.0000 1.0054 1.0108 0.9459

…

…

1

…

…

…

1.0000 1.0047 1.0093 0.9536

…

11 ⁄8

7

…

1.1250 1.1343 1.1436 1.0322

… …

12

… …

1.3750 1.3859 1.3967 1.2667

…

…

…

…

…

…

1.2682 1.2687

…

…

…

…

…

…

…

…

12

…

1.3750 1.3804 1.3858 1.3209

…

…

…

…

…

…

1.3224 1.3229

…

…

…

…

…

…

…

…

…

1.5000 1.5109 1.5217 1.3917

…

…

…

…

…

…

1.3932 1.3937

…

…

…

…

…

…

…

…

12

…

1.5000 1.5054 1.5108 1.4459

…

…

…

…

…

…

1.4474 1.4479

…

…

…

…

…

…

…

…

13⁄4

5

…

1.7500 1.7630 1.7760 1.6201

…

…

…

…

…

…

1.6216 1.6221

…

…

…

…

…

…

…

…

2

4.5

…

2.0000 2.0145 2.0289 1.8557

…

…

…

…

…

…

1.8572 1.8577

…

…

…

…

…

…

…

…

13⁄8

6

13⁄8 11 ⁄2 11 ⁄2

6

Min.

Max.

Min.

Max.

Min.

Max.

Min.

Max.

Min.

Max.

…

…

0.8301 0.8306

…

…

…

…

…

…

…

…

…

…

…

0.9203 0.9208

…

…

0.9213

0.9218

…

…

…

…

…

…

…

…

0.9474 0.9479

…

…

…

…

…

…

…

…

…

…

…

…

…

0.9551 0.9556

…

…

…

…

…

…

…

…

…

…

…

…

…

…

1.0337 1.0342

…

…

…

…

…

…

…

…

1.1250 1.1304 1.1358 1.0709

…

…

…

…

…

…

1.0724 1.0729

…

…

…

…

…

…

…

…

1.2500 1.2593 1.2686 1.1572

…

…

…

…

…

…

1.1587 1.1592

…

…

…

…

…

…

…

…

1.2500 1.2554 1.2608 1.1959

…

…

…

…

…

…

1.1974 1.1979

…

…

…

…

…

…

…

…

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a Minimum and maximum major diameters are 0.0010 larger than shown.

b Minimum and maximum major diameters are 0.0035 larger than shown.

c Minimum and maximum major diameters are 0.0020 larger than shown. d Minimum and maximum major diameters are 0.0015 larger than shown.

General notes: Limits listed in Table 21 are the most commonly used in the industry. Not all styles of taps are available with all limits listed. For calculation of limits other than those listed, see formulas and Table 2.

Machinery's Handbook, 31st Edition Taps

Min.

11 ⁄4

Max.

H8 limitb

H7 limit

0.8750 0.8797 0.8843 0.8286

0.8291 0.8296

Min.

H6 limita

H5 limit

…

12

Max.

H4 limit

14

7

Min.

H3 limit

…

11 ⁄4

Basic

H2 limit

7⁄8

11 ⁄8

Max.

1000

Copyright 2020, Industrial Press, Inc.

Table Table 21.  (Continued) Tap Thread Limits: Fractional Ground Thread ANSI/ASME B94.9-2008 (R2018) 21. Tap Thread Limits: Fractional Sizes, Sizes, Ground Thread ANSI/ASME B94.9-2008 (R2018) (Unified and American National Thread Forms, Standard Thread Limits)

Machinery's Handbook, 31st Edition Taps

1001

Table 22. Tap Thread Limits: Machine Screw Sizes, Cut Thread ANSI/ASME B94.9-2008 (R2018) Unified and American National Thread Forms, Standard Thread Limits Threads per Inch

Major Diameter

Pitch Diameter

Size

NC UNC

NF UNF

NS UNS

Basic

Min.

Max.

Basic

Min.

Max.

0

…

80

…

0.0600

0.0609

0.0624

0.0519

0.0521

0.0531

1

64

…

…

0.0730

0.0739

0.0754

0.0629

0.0631

0.0641

1

…

72

…

0.0730

0.0740

0.0755

0.0640

0.0642

0.0652

2

56

…

…

0.0860

0.0872

0.0887

0.0744

0.0746

0.0756

2

…

64

…

0.0860

0.0870

0.0885

0.0759

0.0761

0.0771

3

48

…

…

0.0990

0.1003

0.1018

0.0855

0.0857

0.0867

3

…

56

…

0.0990

0.1002

0.1017

0.0874

0.0876

0.0886

4

…

…

36

0.1120

0.1137

0.1157

0.0940

0.0942

0.0957

4

40

…

…

0.1120

0.1136

0.1156

0.0958

0.0960

0.0975 0.1002

4

…

48

…

0.1120

0.1133

0.1153

0.0985

0.0987

5

40

…

…

0.1250

0.1266

0.1286

0.1088

0.1090

0.1105

6

32

…

…

0.1380

0.1402

0.1422

0.1177

0.1182

0.1197

6

…

…

36

0.1380

0.1397

0.1417

0.1200

0.1202

0.1217

6

…

40

…

0.1380

0.1396

0.1416

0.1218

0.1220

0.1235

8

32

…

…

0.1640

0.1662

0.1682

0.1437

0.1442

0.1457

8

…

36

…

0.1640

0.1657

0.1677

0.1460

0.1462

0.1477

8

…

…

40

0.1640

0.1656

0.1676

0.1478

0.1480

0.1495

10

24

…

…

0.1900

0.1928

0.1948

0.1629

0.1634

0.1649

10

…

32

…

0.1900

0.1922

0.1942

0.1697

0.1702

0.1717

12

24

…

…

0.2160

0.2188

0.2208

0.1889

0.1894

0.1909

12

…

28

…

0.2160

0.2184

0.2204

0.1928

0.1933

0.1948

14

…

…

24

0.2420

0.2448

0.2473

0.2149

0.2154

0.2174

Angle Tolerance Threads per Inch

Half Angle

20 to 28

±0°45′

Full Angle ±0°65′

30 and finer

±0°60′

±0°90′

A maximum lead error of ±0.003 inch in 1 inch of thread is permitted. All dimensions are given in inches. Thread limits are computed from Table 3.

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Machinery's Handbook, 31st Edition Taps

1002

Table 23. Tap Thread Limits: Fractional Sizes, Cut Thread ANSI/ASME B94.9-2008 (R2018) (Unified and American National Thread Forms) Size 1⁄ 8 5⁄ 32 3⁄ 16 3⁄ 16 1⁄ 4 1⁄ 4 5⁄ 16 5⁄ 16 3⁄ 8 3⁄ 8 7⁄ 16 7⁄ 16 1⁄ 2 1⁄ 2 9⁄ 16 9⁄ 16 5⁄ 8 5⁄ 8 3⁄ 4 3⁄ 4 7⁄ 8 7⁄ 8

1 1 1 11 ⁄8 11 ⁄8 11 ⁄4 11 ⁄4 13⁄8 13⁄8 11 ⁄2 11 ⁄2 13⁄4 2

Threads per Inch NC NF NS UNC UNF UNS … … … … 20 … 18 … 16 … 14 … 13 … 12 … 11 … 10 … 9 … 8 … … 7 … 7 … 6 … 6 … 5 4.5

… … … … … 28 … 24 … 24 … 20 … 20 … 18 … 18 … 16 … 14 … 12 … … 12 … 12 … 12 … 12 … …

40 32 24 32 … … … … … … … … … … … … … … … … … … … … 14 … … … … … … … … … …

Threads per Inch 41 ⁄2 to 51 ⁄2 6 to 9 10 to 28 30 to 64

Major Diameter

Basic

Min.

0.1250 0.1563 0.1875 0.1875 0.2500 0.2500 0.3125 0.3125 0.3750 0.3750 0.4375 0.4375 0.5000 0.5000 0.5625 0.5625 0.6250 0.6250 0.7500 0.7500 0.8750 0.8750 1.0000 1.0000 1.0000 1.1250 1.1250 1.2500 1.2500 1.3750 1.3750 1.5000 1.5000 1.7500 2.0000

0.1266 0.1585 0.1903 0.1897 0.2532 0.2524 0.3160 0.3153 0.3789 0.3778 0.4419 0.4407 0.5047 0.5032 0.5675 0.5660 0.6304 0.6285 0.7559 0.7539 0.8820 0.8799 1.0078 1.0055 1.0049 1.1337 1.1305 1.2587 1.2555 1.3850 1.3805 1.5100 1.5055 1.7602 2.0111 Half Angle ±0° 35′ ±0° 40′ ±0° 45′ ±0° 60′

Pitch Diameter

Max.

0.1286 0.1605 0.1923 0.1917 0.2557 0.2549 0.3185 0.3178 0.3814 0.3803 0.4449 0.4437 0.5077 0.5062 0.5705 0.5690 0.6334 0.6315 0.7599 0.7579 0.8860 0.8839 1.0118 1.0095 1.0089 1.1382 1.1350 1.2632 1.2600 1.3895 1.3850 1.5145 1.5100 1.7657 2.0166

Basic

0.1088 0.13595 0.1604 0.1672 0.2175 0.2268 0.2764 0.2854 0.3344 0.3479 0.3911 0.4050 0.4500 0.4675 0.5084 0.5264 0.5660 0.5889 0.6850 0.7094 0.8028 0.8286 0.9188 0.9459 0.9536 1.0322 1.0709 1.1572 1.1959 1.2667 1.3209 1.3917 1.4459 1.6201 1.8557

Min.

0.1090 0.13645 0.1609 0.1677 0.2180 0.2273 0.2769 0.2859 0.3349 0.3484 0.3916 0.4055 0.4505 0.4680 0.5089 0.5269 0.5665 0.5894 0.6855 0.7099 0.8038 0.8296 0.9198 0.9469 0.9546 1.0332 1.0719 1.1582 1.1969 1.2677 1.3219 1.3927 1.4469 1.6216 1.8572

Max.

0.1105 0.1380 0.1624 0.1692 0.2200 0.2288 0.2789 0.2874 0.3369 0.3499 0.3941 0.4075 0.4530 0.4700 0.5114 0.5289 0.5690 0.5914 0.6885 0.7124 0.8068 0.8321 0.9228 0.9494 0.9571 1.0367 1.0749 1.1617 1.1999 1.2712 1.3249 1.3962 1.4499 1.6256 1.8612

Full Angle ±0° 53′ ±0° 60′ ±0° 68′ ±0° 90′

A maximum lead error of ±0.003 inch in 1 inch of thread is permitted. All dimensions are given in inches. Thread limits are computed from Table 3.

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Machinery's Handbook, 31st Edition Taps

1003

Table 23a. Straight Pipe Thread Limits: NPS, Ground Thread ANSI Straight Pipe Thread Form (NPSC, NPSM) ANSI/ASME B94.9-2008 (R2018) Major Diameter

1⁄ 8

Threads per Inch, NPS, NPSC, NPSM 27

Plug at Gaging Notch 0.3983

3⁄ 8

18

0.6640

0.6701

3⁄ 4

14

1.0364

1.0447

Nominal Size, Inches 1⁄ 4

1⁄ 2

1

18

Min. G 0.4022

0.5286

14

0.8260

1.2966

111 ⁄2

Pitch Diameter

Max. H 0.4032

Plug at Gaging Notch E 0.3736

0.6711

0.6270

0.6287

0.6292

1.0457

0.9889

0.9906

0.9916

0.5347

0.5357

0.8347

0.8357

1.3062

1.3077

0.4916

0.7784

1.2386

Min. K 0.3746

0.4933

0.7806

1.2402

Max. L 0.3751

0.4938 0.7811

1.2412

Formulas for NPS Ground Thread Tapsa Nominal Size 1⁄ 4

Min. G

Major Diameter

Max. H

Minor Dia. Max.

1⁄ 8

H - 0.0010

(K + A) - 0.0010

M-B

1

H - 0.0015

(K + A) - 0.0021

M-B

to 3⁄4

H - 0.0010

(K + A) - 0.0020

M-B

Threads per Inch

A

B

27 18

0.0296 0.0444

0.0257 0.0401

111 ⁄2

0.0696

0.0647

14

0.0571

0.0525

a In the formulas, M equals the actual measured pitch diameter.

All dimensions are given in inches. Maximum pitch diameter of tap is based upon an allowance deducted from the maximum product pitch diameter of NPSC or NPSM, whichever is smaller. Minimum pitch diameter of tap is derived by subtracting the ground thread pitch diameter tolerance for actual equivalent size. Lead tolerance: A maximum lead deviation pf ± 0.0005 inch within any two threads not farther apart than one inch. Angle Tolerance: 111 ⁄2 to 27 threads per inch, plus or minus 30 min. in half angle. Taps made to the specifications in Table 23a are to be marked NPS and used for NPSC and NPSM.

Table 23b. Straight Pipe Thread Limits: NPSF Ground Thread ANSI Standard Straight Pipe Thread Form (NPSF) ANSI/ASME B94.9-2008 (R2018) Major Diameter

Pitch Diameter

Min. K 0.2772

Max. L 0.2777

Minora Dia. Flat, Max. 0.004

1⁄ 16

Threads per Inch 27

Min. G 0.3008

Max. H 0.3018

Plug at Gaging Notch E 0.2812

1⁄ 4

18

0.5239

0.5249

0.4916

0.4859

0.4864

0.005

1⁄ 2

14

0.8230

0.8240

0.7784

0.7712

0.7717

0.005

Nominal Size, Inches 1⁄ 8 3⁄ 8 3⁄ 4

27

18

14

0.3932 0.6593

1.0335

0.3942 0.6603

1.0345

0.3736

0.6270

0.9889

0.3696 0.6213

0.9817

0.3701 0.6218

0.9822

0.004

0.005

0.005

a As specified or sharper.

All dimensions are given in inches.

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Machinery's Handbook, 31st Edition Taps

1004

Table 23c. ASME Standard Straight Pipe Thread Limits: NPSF Ground Thread Dryseal ANSI Standard Straight Pipe Thread Form (NPSF) ANSI/ASME B94.9-2008 (R2018) Formulas For American Dryseal (NPSF) Ground Thread Taps

Nominal Size, Inches

Min. G

Major Diameter

Pitch Diameter

Max. H

Min. K

Max. Minor Dia.

Max. L

1⁄ 16

H - 0.0010

K + Q - 0.0005

L - 0.0005

E-F

M-Q

1⁄ 8

H - 0.0010

K + Q - 0.0005

L - 0.0005

E-F

M-Q

1⁄ 4

H - 0.0010

K + Q - 0.0005

L - 0.0005

E-F

M-Q

3⁄ 8

H - 0.0010

K + Q - 0.0005

L - 0.0005

E-F

M-Q

1⁄ 2

H - 0.0010

K + Q - 0.0005

L - 0.0005

E-F

M-Q

3⁄ 4

H - 0.0010

K + Q - 0.0005

L - 0.0005

E-F

M-Q

Threads per Inch 27 18 14

E

Values to Use in Formulas F 0.0035 0.0052 0.0067

Pitch diameter of plug at gaging notch

M

Q 0.0251 0.0395 0.0533

Actual measured pitch diameter

All dimensions are given in inches. Lead Tolerance: A maximum lead deviation of ±0.0005 inch within any two threads not farther apart than one inch. Angle Tolerance: Plus or minus 30 min. in half angle for 14 to 27 threads per inch, inclusive.

Table 23d. ANSI Standard Straight Pipe Tap Limits: (NPS) Cut Thread ANSI Straight Pipe Thread Form (NPSC) ANSI/ASME B94.9-2008 (R2018)

Nominal Size 1 ⁄8

Threads per Inch, NPS, NPSC

Size at Gaging Notch

Pitch Diameter

Min.

Max.

A

B

C

27

0.3736

0.3721

0.3751

0.0267

0.0296

0.0257

0.0408

0.0444

0.0401

0.0535

0.0571

0.0525

0.0658

0.0696

0.0647

1 ⁄4

18

0.4916

0.4908

0.4938

3 ⁄8

18

0.6270

0.6257

0.6292

1 ⁄2

14

0.7784

0.7776

0.7811

3 ⁄4

14

0.9889

0.9876

0.9916

111 ⁄2

1.2386

1.2372

1.2412

1

Values to Use in Formulas

The following are approximate formulas, in which M = measured pitch diameter in inches: Major dia., min. = M + A Major dia., max. = M + B

Minor dia., max. = M - C

Maximum pitch diameter of tap is based on an allowance deducted from the maximum product pitch diameter of NPSC. Minimum pitch diameter of tap equals maximum pitch diameter minus the tolerance. All dimensions are given in inches. Lead Tolerance: ± 0.003 inch per inch of thread. Angle Tolerance: For all pitches, tolerance will be ± 45″ for half angle and ± 68″ for full angle. Taps made to these specifications are to be marked NPS and used for NPSC thread form. Taps made to the specifications in Table 23a are to be marked NPS and used for NPSC. As the American National Standard straight pipe thread form is to be maintained, the major and minor diameters vary with the pitch diameter. Either a flat or rounded form is allowable at both the crest and the root.

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Machinery's Handbook, 31st Edition Taps

1005

Table 24a. Taper Pipe Thread Limits (Ground and Cut Thread: Ground Thread For NPS, NPTF, and ANPT; Cut Thread for NPT only) ANSI/ASME B94.9-2008 (R2018)

Nominal Size 1 ⁄16 1 ⁄8 1 ⁄4 3 ⁄8 1 ⁄2 3 ⁄4 1 11 ⁄4

11 ⁄2 2 21 ⁄2 3

Threads per Inch 27 27 18 18 14 11.5 11.5 11.5 11.5 8 8 20

Gage Measurement Tolerance ± Cut Ground Projection Thread Thread Inchb 0.312 0.312 0.459 0.454 0.579 0.565 0.678 0.686 0.699 0.667 0.925 0.925

0.0625 0.0625 0.0625 0.0625 0.0625 0.0625 0.0937 0.0937 0.0937 0.0937 0.0937 0.0937

0.0625 0.0625 0.0625 0.0625 0.0625 0.0625 0.0937 0.0937 0.0937 0.0937 0.0937 0.0937

Reference Dimensions

Taper per Inch on Diametera Cut Thread Ground Thread Min.

0.0599 0.0599 0.0599 0.0599 0.0599 0.0599 0.0599 0.0599 0.0599 0.0599 0.0612 0.0612

Max.

0.0703 0.0703 0.0703 0.0703 0.0677 0.0677 0.0677 0.0677 0.0677 0.0677 0.0664 0.0664

Min.

0.0599 0.0599 0.0599 0.0599 0.0599 0.0599 0.0599 0.0599 0.0599 0.0599 0.0612 0.0612

Max.

0.0651 0.0651 0.0651 0.0651 0.0651 0.0651 0.0651 0.0651 0.0651 0.0651 0.0651 0.0651

L1 , Lengthc 0.1600 0.1615 0.2278 0.2400 0.3200 0.3390 0.4000 0.4200 0.4200 0.4360 0.6820 0.7660

Tap Drill Size NPT, ANPT, NPTF  d C Q

7⁄16

9 ⁄16

45 ⁄64

29 ⁄32 19⁄64 131 ⁄64 123⁄32 23⁄16 239⁄64 315⁄16

a Taper is 0.0625 inch per 1.000 inch on diameter (1:16) (3 ⁄ inch per 12 inches). 4 b Distance, small end of tap projects through L taper ring gage. 1 c Dimension, L

1, thickness on thin ring gage; see ANSI/ASME B1.20.1 and B1.20.5.

Given sizes permit direct tapping without reaming the hole, but only give full threads for approximate L1 distance. d

All dimensions are given in inches.

Lead Tolerance: ± 0.003 inch per inch on cut thread, and ± 0.0005 inch per inch on ground thread.

Angle Tolerance: ± 40 min. in half angle and 60 min. in full angle for 8 cut threads per inch; ± 45 min. in half angle and 68 min. in full angle for 111 ⁄2 to 27 cut threads per inch; ±25 min. in half angle for 8 ground threads per inch; and ±30 min. in half angle for 111 ⁄2 to 27 ground threads per inch.

Table 24b. Taper Pipe Thread — Widths of Flats at Tap Crests and Roots for Cut Thread NPT and Ground Thread NPT, ANPT, and NPTF ANSI/ASME B94.9-2008 (R2018) Column I

Threads per Inch 27 18 14 111 ⁄2 8

Tap Flat Width at

Major diameter Minor diameter Major diameter Minor diameter Major diameter Minor diameter Major diameter Minor diameter Major diameter Minor diameter

Column II

NPT—Cut and Ground Thread a ANPT—Ground Thread a Min.b

0.0014 … 0.0021 … 0.0027 … 0.0033 … 0.0048 …

Max.

0.0041 0.0041 0.0057 0.0057 0.0064 0.0064 0.0073 0.0073 0.0090 0.0090

NPTF

Ground Thread  a

0.0040 … 0.0050 … 0.0050 … 0.0060 … 0.0080 …

0.0055 0.0040 0.0065 0.0050 0.0065 0.0050 0.0083 0.0060 0.0103 0.0080

Min. b

Max.

Cut thread taps made to Column I are marked NPT but are not recommended for ANPT applications. Ground thread taps made to Column I are marked NPT and may be used for NPT and ANPT applications. Ground thread taps made to Column II are marked NPTF and used for dryseal application. a

b Minimum minor diameter flats are not specified and may be as sharp as practicable.

All dimensions are given in inches.

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Machinery's Handbook, 31st Edition Taps

1006

Table 25. Tap Thread Limits for Screw Thread Inserts (STI), Ground Thread, Machine Screw, and Fractional Size ANSI/ASME B94.9-2008 (R2018) Nominal Screw Size STI 1 2

Fractional Size STI

12

… … … … … … … … … … … … … … …

…

1 ⁄4 …

3 4 5 6 8 10

…

Threads Per Inch

NC

NF

20

…

64 56 … 48 … 40 … 40 32 … 32 … 24 … 24 …

… … 64 … 56 … 48 … … 40 … 36 … 32 … 28

Tap Major Diameter

Min.

Max.

H limit

0.3177

0.3187

H3

0.0948 0.1107 0.1088 0.1289 0.1237 0.1463 0.1409 0.1593 0.1807 0.1723 0.2067 0.2022 0.2465 0.2327 0.2725 0.2985

0.0958 0.1117 0.1088 0.1289 0.1247 0.1473 0.1419 0.1603 0.1817 0.1733 0.2077 0.2032 0.2475 0.2337 0.2735 0.2995

H2 H2 H2 H2 H2 H2 H2 H2 H3 H2 H3 H2 H3 H3 H3 H3

2B

Pitch Diameter Limits

3B

Max.

H limit

0.2835

0.2840

H2

0.2830

0.2835

Min.

0.0837 0.0981 0.0967 0.1131 0.1111 0.1288 0.1261 0.1418 0.1593 0.1548 0.1853 0.1826 0.2180 0.2113 0.2440 0.2742

0.0842 0.0986 0.0972 0.1136 0.1116 0.1293 0.1266 0.1423 0.1598 0.1553 0.1858 0.1831 0.2185 0.2118 0.2445 0.2747

H1 H1 H1 H1 H1 H1 H1 H1 H2 H1 H2 H1 H2 H2 H2 H2

Min.

0.0832 0.0976 0.0962 0.1126 0.1106 0.1283 0.1256 0.1413 0.1588 0.1543 0.1848 0.1821 0.2175 0.2108 0.2435 0.2737

Max.

0.0837 0.0981 0.0967 0.1131 0.1111 0.1288 0.1261 0.1418 0.1593 0.1548 0.1853 0.1826 0.2180 0.2113 0.2440 0.2742

…

5 ⁄16

18

…

0.3874

0.3884

H4

0.3501

0.3506

H3

0.3496

0.3501

…

3 ⁄8

16

…

0.4592

0.4602

H4

0.4171

0.4176

H3

0.4166

0.4171

…

7⁄16

14

…

0.5333

0.5343

H4

0.4854

0.4859

H3

0.4849

0.4854

…

1 ⁄2

13

…

0.6032

0.6042

H4

0.5514

0.5519

H3

0.5509

0.5514

…

9 ⁄16

12

…

0.6741

0.6751

H4

0.6182

0.6187

H3

0.6117

0.6182

…

5 ⁄8

11

…

0.7467

0.7477

H4

0.6856

0.6861

H3

0.6851

0.6856

3 ⁄4

10

…

0.8835

0.8850

H5

0.8169

0.8174

H3

0.8159

0.8164

7⁄8

9

…

1.0232

1.0247

H5

0.9491

0.9496

H3

0.9481

0.9486

…

1.3151

1.3171

H6

1.2198

1.2208

H4

1.2188

1.2198

7

…

1.4401

1.4421

H6

1.3448

1.3458

H4

1.3438

1.3448

6

…

1.5962

1.5982

H8

1.4862

1.4872

H6

1.4852

1.4862

6

…

1.7212

1.7232

H8

1.6112

1.6122

H6

1.6102

… … … … … …

…

…

…

… … … …

… … … … …

… … … 1 … …

…

11 ⁄8

…

11 ⁄4 …

… …

…

…

…

…

13⁄8 …

11 ⁄2 …

… … … … …

…

… … 8 … … 7

… … …

24 24 20 20 18

18

18

14 … 12 14 NS 12 12 12

12

0.3690 0.4315 0.5052 0.5677 06374

0.6999

0.8342 0.9708 1.1666 1.1116 1.0958

1.2366 1.3616 1.4866 1.6116

0.3700 0.4325 0.5062 0.5687 0.6384

0.7009

0.8352 0.9718 1.1681 1.1126 1.0968

1.2376 1.3626 1.4876

1.6126

H3 H3 H4 H4 H4

H4

H4 H4 H6 H6 H6

H6 H6 H6

H6

0.3405 0.4030 0.4715 0.5340 0.6001

0.6626

0.7921 0.9234 1.0832 1.0562 1.0484

1.1812 1.3062 1.4312

1.5562

0.3410 0.4035 0.4720 0.5345 0.6006

0.6631

0.7926 0.9239 1.0842 1.0572 1.0494

1.1822 1.3072 1.4322

1.5572

H2 H2 H3 H3 H3

H3

H3 H3 H4 H4 H4

H4 H4 H4

H4

0.3400 0.4025 0.4710 0.5335 0.5996

0.6621

0.7916 0.9224 1.0822 1.0552 1.0474

1.1802 1.3052 1.4302

1.5552

0.3405 0.4030 0.4715 0.5340 0.6001

0.6626

0.7921 0.9229 1.0832 1.0562 1.0484

1.1812 1.3062 1.4312 1.6112

1.5562

These taps are over the nominal size to the extent that the internal thread they produce will accom­modate a helical coil screw insert, which at final assembly will accept a screw thread of the normal size and pitch.

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Machinery's Handbook, 31st Edition Taps

1007

Table 26a. Tap Thread Limits ANSI/ASME B94.9-2008 (R2018) for Screw Thread Inserts (STI), Ground Thread, Metric Size (Inch) Metric Size STI M2.5 M3 M3.5 M4 M5 M6 M7 M8 M10

M12

M14 M16 M18

M20

M22

M24 M27 M30 M33 M36

M39

Pitch, mm 0.45 0.5 0.6 0.7 0.8 1 1 1 1.25 1 1 1.25

1.25 1.5 1.75 1.5 2 1.5 2 1.5 2 2.5 1.5 2

2.5 1.5 2

2.5 2 3 2

3 2 3.5 2 3 2 3 4 2 3 4

Tap Major Diameter, inch

Min.

0.1239 0.1463 0.1714 0.1971 0.2418 0.2922 0.3316 0.3710 0.3853 0.4497 0.4641 0.4776 0.5428 0.5564 0.5700 0.6351 0.6623 0.7139 0.7410 0.7926 0.8198 0.8470 0.8713 0.8985 0.9257 0.9500 0.9773 1.0044 1.0559 1.1117 1.1741 1.2298 1.2922 1.3750 1.4103 1.4931 1.5284 1.5841 1.6384 1.6465 1.7022 1.7565

Max.

0.1229 0.1453 0.1704 0.1955 0.2403 0.2906 0.3300 0.3694 0.3828 0.4481 0.4616 0.4751 0.5403 0.5539 0.5675 0.6326 0.6598 0.7114 0.7385 0.7901 0.8173 0.8445 0.8688 0.8960 0.9232 0.9475 0.9748 1.0019 1.0534 1.1078 1.1716 1.2259 1.2897 1.3711 1.4078 1.4892 1.5259 1.5802 1.6345 1.6440 1.6983 1.7516

Tap Pitch Diameter Limits, inch

Tolerance Class 4H

H limit 1 1 1 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 6 6 4 6 6

Min.

0.1105 0.1314 0.1537 0.1764 0.2184 0.2629 0.3022 0.3416 0.3480 0.4203 0.4267 0.4336 0.5059 0.5123 0.5187 0.5911 0.6039 0.6698 0.6826 0.7485 0.7613 0.7741 0.8273 0.8401 0.8529 0.9060 0.9188 0.9316 0.9981 1.0236 1.1162 1.1417 1.2343 1.2726 1.3525 1.3907 1.4706 1.4971 1.5226 1.5887 1.6152 1.6407

Max.

0.1100 0.1309 0.1532 0.1759 0.2179 0.2624 0.3017 0.3411 0.3475 0.4198 0.4262 0.4331 0.5054 0.5118 0.5182 0.5906 0.6034 0.6693 0.6821 0.7480 0.7608 0.7736 0.8268 0.8396 0.8524 0.9055 0.9183 0.9311 0.9971 1.0226 1.1152 1.1407 1.2333 1.2716 1.3515 1.3797 1.4696 1.4961 1.5216 1.5877 1.6142 1.6397

Tolerance Classes 5H and 6H

H limit 2 2 2 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 5 5 6 6 6 6 6 6 6 6 6 8 8 6 8 8

Min.

0.1110 0.1319 0.1542 0.1769 0.2187 0.2634 0.3027 0.3421 0.3485 0.4208 0.4272 0.4341 0.5064 0.5128 0.5192 0.5916 0.6049 0.6703 0.6836 0.7490 0.7623 0.7751 0.8278 0.8411 0.8539 0.9065 0.9198 0.9326 0.9991 1.0246 1.1172 1.1427 1.2353 1.2736 1.3535 1.3917 1.4716 1.4981 1.5236 1.5897 1.6162 1.6417

Max.

0.1105 0.1314 0.1537 0.1764 0.2184 0.2629 0.3022 0.3416 0.3480 0.4203 0.4267 0.4336 0.5059 0.5123 0.5187 0.5911 0.6044 0.6698 0.6831 0.7485 0.7618 0.7748 0.8273 0.8406 0.8534 0.9060 0.9193 0.9321 0.9981 1.0236 1.1162 1.1417 1.2343 1.2726 1.3525 1.3907 1.4706 1.4971 1.5226 1.5887 1.6152 1.6407

These taps are over the nominal size to the extent that the internal thread they produce will accom­modate a helical coil screw insert, which at final assembly will accept a screw thread of the normal size and pitch. STI basic thread dimensions are determined by adding twice the single thread height (2 × 0.64952P) to the basic dimensions of the nominal thread size. Formulas for major and pitch diameters are presented in MIL-T-21309E.

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Machinery's Handbook, 31st Edition Taps

1008

Table 26b. Tap Thread Limits ANSI/ASME B94.9-2008 (R2018) for Screw Thread Inserts (STI), Ground Thread, Metric Size (mm) Metric Size STI M2.5 M3 M3.5 M4 M5 M6 M7 M8 M10

M12

M14 M16 M18

M20

M22

M24 M27 M30 M33 M36

M39

Pitch, mm 0.45 0.5 0.6 0.7 0.8 1 1 1 1.25 1 1 1.25 1.25 1.5 1.75 1.5 2 1.5 2 1.5 2 2.5 1.5 2 2.5 1.5 2 2.5 2 3 2 3 2 3.5 2 3 2 3 4 2 3 4

Tap Major Diameter, mm

Min.

3.147 3.716 4.354 5.006 6.142 7.422 8.423 9.423 9.787 11.422 11.788 12.131 13.787 14.133 14.478 16.132 16.822 18.133 18.821 20.132 20.823 21.514 22.131 22.822 23.513 24.130 24.823 25.512 26.820 28.237 29.822 31.237 32.822 34.925 35.822 37.925 38.821 40.236 41.615 41.821 43.236 44.615

Max.

3.122 3.691 4.328 4.966 6.104 7.381 8.382 9.383 9.723 11.382 11.725 12.068 13.724 14.069 14.415 16.068 16.759 18.070 18.758 20.069 20.759 21.450 22.068 22.758 23.449 24.067 24.760 25.448 26.756 28.132 29.759 31.138 32.758 34.826 35.758 37.826 38.758 40.137 41.516 41.758 43.137 44.516

Tap Pitch Diameter Limits, mm

Tolerance Class 4H

H limit 1 1 1 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 6 6 4 6 6

Min.

2.807 3.338 3.904 4.481 5.547 6.678 7.676 8.677 8.839 10.676 10.838 11.013 12.850 13.012 13.175 15.014 15.339 17.013 17.338 19.012 19.337 19.662 21.013 21.339 21.664 23.012 23.338 23.663 25.352 25.999 28.351 28.999 31.351 32.324 34.354 35.324 37.353 38.026 38.674 40.353 41.026 41.674

Max.

2.794 3.325 3.891 4.468 5.535 6.665 7.663 8.664 8.827 10.663 10.825 11.001 12.837 13.000 13.162 15.001 15.326 17.000 17.325 18.999 19.324 19.649 21.001 21.326 21.651 23.000 23.325 23.650 25.352 25.974 28.326 28.974 31.326 32.299 34.324 35.298 37.328 38.001 38.649 40.328 4 1.001 41.648

Tolerance Classes 5H and 6H

H limit 2 2 2 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 5 5 6 6 6 6 6 6 6 6 6 8 8 6 8 8

Min.

2.819 3.350 3.917 4.493 5.555 6.690 7.689 8.689 8.852 10.688 10.851 11.026 12.863 13.025 13.188 15.027 15.364 17.026 17.363 19.025 19.362 19.688 21.026 21.364 21.689 23.025 23.363 23.688 25.377 26.025 28.377 29.025 31.377 32.349 34.379 35.349 37.379 38.052 38.699 40.378 41.051 41.699

Max.

2.807 3.338 3.904 4.481 5.547 6.678 7.676 8.677 8.839 10.676 10.838 11.013 12.850 13.012 13.175 15.014 15.352 17.013 17.351 19.012 19.350 19.675 21.013 21.351 21.676 23.012 23.350 23.675 25.352 25.999 28.351 28.999 31.351 32.324 34.354 35.324 37.353 38.026 37.674 40.353 41.026 41.674

These taps are over the nominal size to the extent that the internal thread they produce will accom­modate a helical coil screw insert, which at final assembly will accept a screw thread of the normal size and pitch. STI basic thread dimensions are determined by adding twice the single thread height (2 × 0.64952P) to the basic dimensions of the nominal thread size. Formulas for major and pitch diameters are presented in MIL-T-21309E.

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Machinery's Handbook, 31st Edition Acme and Square-Threaded Taps

1009

Acme and Square-Threaded Taps These taps are usually made in sets, three taps in a set being the most common. For very fine pitches, two taps in a set will be found sufficient, whereas as many as five taps in a set are used for coarse pitches. The table on the next page gives dimensions for proportioning both Acme and square-threaded taps when made in sets. In cutting the threads of squarethreaded taps, one leading tap maker uses the following rules: The width of the groove between two threads is made equal to one-half the pitch of the thread, less 0.004 inch (0.102 mm), making the width of the thread itself equal to one-half of the pitch, plus 0.004 inch (0.102 mm). The depth of the thread is made equal to 0.45 times the pitch, plus 0.0025 inch (0.064 mm). This latter rule produces a thread that for all the ordinarily used pitches for square-threaded taps has a depth less than the generally accepted standard depth, this latter depth being equal to one-half the pitch. The object of this shallow thread is to ensure that if the hole to be threaded by the tap is not bored out so as to provide clearance at the bottom of the thread, the tap will cut its own clearance. The hole should, however, always be drilled out large enough so that the cutting of the clearance is not required of the tap. The table, Dimensions of Acme Threads Taps in Sets of Three Taps, may also be used for the length dimensions for Acme taps. The dimensions in this table apply to single-threaded taps. For multiple-threaded taps or taps with very coarse pitch relative to the diameter, the length of the chamfered part of the thread may be increased. Square-threaded taps are made to the same table as Acme taps, with the exception of the figures in column K, which for square-threaded taps should be equal to the nominal diameter of the tap, no oversize allow­ ance being customary in these taps. The first tap in a set of Acme taps (not square-threaded taps) should be turned to a taper at the bottom of the thread for a distance of about one-quar­ ter of the length of the threaded part. The taper should be so selected that the root diameter is about 1 ⁄32 inch (0.794 mm) smaller at the point than the proper root diameter of the tap. The first tap should preferably be provided with a short pilot at the point. For very coarse pitches, the first tap may be provided with spiral flutes at right angles to the angle of the thread. Acme and square-threaded taps should be relieved or backed off on the top of the thread of the chamfered portion on all the taps in the set. When the taps are used as machine taps, rather than as hand taps, they should be relieved in the angle of the thread, as well as on the top, for the whole length of the chamfered portion. Acme taps should also always be relieved on the front side of the thread to within 1 ⁄32 inch (0.794 mm) of the cutting edge. Adjustable Taps.—Many adjustable taps are now used, especially for accurate work. Some taps of this class are made of a solid piece of tool steel that is split and provided with means of expanding sufficiently to compensate for wear. Most of the larger adjustable taps have inserted blades or chasers that are held rigidly but are capable of radial adjustment. The use of taps of this general class enables standard sizes to be maintained readily. Drill Hole Sizes for Acme Threads.—Many tap and die manufacturers and vendors make available to their customers computer programs designed to calculate drill hole sizes for all the Acme threads in their ranges from the basic dimensions. The large variety and combination of dimensions for such tools prevent inclusion of a complete set of tables of tap drills for Acme taps in this Handbook. The following formulas (dimensions in inches) for calculating drill hole sizes for Acme threads are derived from the American National Standard, ANSI/ASME B1.5-1997 (R2014), Acme Screw Threads. To select a tap drill size for an Acme thread, first calculate the maximum and minimum internal product minor diameters for the thread to be produced. (Dimensions for general purpose, centralizing, and stub Acme screw threads are given in the Threads and Thread­ ing section, starting on page 2054.) Then select a drill that will yield a finished hole some­ where between the established maximum and minimum product minor diameters. To reduce the amount of material to be removed when cutting the thread, consider staying close to the maximum product limit in selecting the hole size. If there is no standard drill size that matches the hole diameter selected, it may be necessary to drill and ream, or bore, the hole to size, to achieve the required hole diameter.

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Machinery's Handbook, 31st Edition Acme and Square-Threaded Taps

1010

Table 27. Dimensions of Acme Threads Taps in Sets of Three Taps A

B 1ST TAP IN SET

ROOT DIA. – 0.010"

C

D

2ND TAP IN SET

F

ROOT DIA. – 0.010"

A 41 ⁄4

B 17⁄8

11 ⁄ 16

51 ⁄2 6

23⁄8

13 ⁄ 16

67⁄8

15 ⁄ 16

79⁄16

9⁄ 16 5⁄ 8 3⁄ 4

7⁄ 8

1

11 ⁄8

11 ⁄4

13⁄8

11 ⁄2 15⁄8

47⁄8

61 ⁄2

211 ⁄16

71 ⁄4 77⁄8

31 ⁄4

81 ⁄2 9 91 ⁄2 10

13⁄4

21 ⁄4

121 ⁄2

23⁄4 3

15

21 ⁄2

21 ⁄2

213⁄16 3

101 ⁄2 11

17⁄8 2

21 ⁄8

113⁄8 113⁄4

131 ⁄4 14

E 17⁄8

5⁄ 8

31 ⁄8

5⁄ 8

21 ⁄2

7⁄ 8

313⁄16

11 ⁄ 16

31 ⁄8

41 ⁄4

3⁄ 4

31 ⁄2

41 ⁄16 47⁄16

39⁄16

415⁄16

41 ⁄4

41 ⁄2 43⁄4

45⁄8 51 ⁄4

51 ⁄2

53⁄4 6

61 ⁄4

47⁄8 5

63⁄4

51 ⁄2

73⁄4

51 ⁄4 53⁄4

61 ⁄4

I

1⁄ 2

31 ⁄8

33⁄4 4

H

C 23⁄8

23⁄4

61 ⁄2

D

G

K

FINISHING TAP Nomi­nal Dia. 1⁄ 2

E

9⁄ 16

313⁄16

3⁄ 4

213⁄16 35⁄16

3⁄ 4

G 13⁄4 2

15 ⁄ 16

29⁄16

1

11 ⁄16

11 ⁄2

41 ⁄16 41 ⁄4

17⁄8 2

41 ⁄2

21 ⁄8

1

43⁄4 5

11 ⁄16

53⁄16

11 ⁄8

55⁄8

11 ⁄4 11 ⁄4

23⁄4

37⁄8

17⁄16

81 ⁄4

0.770

17⁄16

13⁄8

41 ⁄2

13⁄4

27⁄16

11 ⁄4

15⁄8

45⁄16

11 ⁄8

13⁄8

33⁄8

15 ⁄ 16

11 ⁄16

213⁄16 3

11 ⁄4

15⁄16

1

0.645

313⁄16 41 ⁄16

57⁄16

11 ⁄2

19⁄16 19⁄16 15⁄8

31 ⁄4

35⁄8

13⁄4

21 ⁄8

2 5⁄8 27⁄8 3

0.707

0.832

0.895

0.957

1.020

33⁄16

1.145

31 ⁄2

1.395

33⁄8

35⁄8

1.270

1.520

1.645

411 ⁄16

21 ⁄4

51 ⁄8

23⁄8

43⁄8

2.020

57⁄8

25⁄8

51 ⁄8

2.520

53⁄4

3.020

415⁄16

13⁄16

51 ⁄2

71 ⁄2

2

63⁄4

17⁄8 2

19⁄16

21 ⁄4

37⁄8 4

61 ⁄8

69⁄16 7

0.582

13⁄4 2

11 ⁄2

13⁄16

K 0.520

11 ⁄8

31 ⁄8

7⁄ 8

1

1

I 11 ⁄2

21 ⁄4

11 ⁄8

35⁄8

13 ⁄ 16

H

7⁄ 8

31 ⁄2

13 ⁄ 16

71 ⁄4

83⁄4

23⁄16

F

61 ⁄4

21 ⁄4

21 ⁄2

23⁄4 3

41 ⁄4 43⁄4

51 ⁄2

1.770

1.895

2.270 2.770

Diameters of General-Purpose Acme Screw Threads of Classes 2G, 3G, and 4G may be calculated from pitch = 1/number of threads per inch, and: minimum diameter = basic major diameter – pitch maximum diameter = minimum minor diameter + 0.05 × pitch

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Machinery's Handbook, 31st Edition Acme and Square-Threaded Taps

1011

Table 28. Proportions of Acme and Square-Threaded Taps Made in Sets R –0.010"

B

A C

R = root diameter of thread T = double depth of full thread

Types of Tap

No. of Taps in Set 2 3

Acme Thread Taps

4

Order of Tap in Set

R + 0.65T

1st

R + 0.45T

2d 2d

SquareThreaded Taps

4

R + 0.010

3d

R + 0.90T

D

A on 2d tap - 0.005

R + 0.70T

A on 1st tap - 0.005

D

A on 3d tap - 0.005

R + 0.63T

A on 1st tap - 0.005

4th

R + 0.94T

A on 3d tap - 0.005

1st

R + 0.67T

2d

2d

3d

2d

1st

R + 0.37T

R + 0.82T D

D

R + 0.41T

2d

R + 0.080T

1st

R + 0.32T

3d

R + 0.90T

3d 2d

4th

1st 5

1⁄ 8

R + 0.40T

5th

3

R + 0.010

A on 1st tap - 0.005

1st

3d

2d

3d

4th

5th

D

A on 2d tap - 0.005 R + 0.010

A on 2d tap - 0.005

A on 4th tap - 0.005 R A on 1st tap - 0.005 R

A on 1st tap - 0.005 A on 2d tap - 0.005 R

R + 0.62T

A on 1st tap - 0.005

D

A on 3d tap - 0.005 R

R + 0.26T

A on 2d tap - 0.005

R + 0.50T

A on 1st tap - 0.005

R + 0.92T

A on 3d tap - 0.005

R + 0.72T D

C 1⁄ 8

A on 1st tap - 0.005

1st

2

D

B

R + 0.010

R + 0.80T

4th

5

A

1st

L D = full diameter of tap

A on 2d tap - 0.005

A on 4th tap - 0.005

L to 1 ⁄6 L

1⁄ 4

L to 1 ⁄3 L

1⁄ 6

L to 1 ⁄4 L

L to 1 ⁄6 L

1⁄ 4

L to 1 ⁄3 L

1⁄ 6

L

1⁄ 4

L to 1 ⁄3 L

1⁄ 6

L

1⁄ 8 1⁄ 5 1⁄ 8 1⁄ 5 1⁄ 5

L

L

L

L

L to 1 ⁄4 L

1⁄ 4

L to 1 ⁄3 L

1⁄ 4

L to 1 ⁄3 L

1⁄ 6

L to 1 ⁄4 L

1⁄ 8 1⁄ 8

L to 1 ⁄6 L L to 1 ⁄6 L

1⁄ 4

L to 1 ⁄3 L

1⁄ 6

L

1⁄ 4

L to 1 ⁄3 L

1⁄ 6

L

1⁄ 8 1⁄ 5 1⁄ 8 1⁄ 5 1⁄ 5 1⁄ 4

L

L

L

L

L to 1 ⁄4 L

L to 1 ⁄3 L

Example: 1 ⁄2 -10 Acme 2G, pitch = 1 ⁄ 10 = 0.1 minimum diameter = 0.5- 0.1 = 0.4

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1012

Machinery's Handbook, 31st Edition Acme and Square-Threaded Taps

maximum diameter = 0.4 + (0.05 × 0.1) = 0.405 drill selected = letter X or 0.3970 + 0.0046 (probable oversize) = 0.4016 Diameters of Acme Centralizing Screw Threads of Classes 2C, 3C, and 4C may be cal­ culated from pitch = 1/number of threads per inch, and: minimum diameter = basic major diameter - 0.9 × pitch maximum diameter = minimum minor diameter + 0.05 × pitch Example: 1 ⁄2 -10 Acme 2C, pitch = 1 ⁄ 10 = 0.1 minimum diameter = 0.5- (0.9 × 0.1) = 0.41 maximum diameter = 0.41 + (0.05 × 0.1) = 0.415 drill selected = 13⁄32 or 0.4062 + 0.0046 (probable oversize) = 0.4108. Diameters for Acme Centralizing Screw Threads of Classes 5C and 6C: These classes are not recommended for new designs, but may be calculated from: minimum diameter = [basic major diameter - (0.025 √ basic major dia.)] - 0.9 × pitch maximum diameter = minimum minor diameter + 0.05 × pitch pitch = 1/number of threads per inch Example: 1 ⁄2 -10 Acme 5C, pitch = 1 ⁄ 10 = 0.1 minimum diameter = [0.5- (0.025 √ 0.5)] - (0.9 × 0.1) = 0.3923 maximum diameter = 0.3923 + (0.05 × 0.1) = 0.3973 drill selected = 25⁄64 or 0.3906 + 0.0046 (probable oversize) = 0.3952 Tapping Square Threads.—If it is necessary to tap square threads, this should be done by using a set of taps that will form the thread by a progressive cutting action, the taps vary­ ing in size in order to distribute the work, especially for threads of comparatively coarse pitch. From three to five taps may be required in a set, depending upon the pitch. Each tap should have a pilot to steady it. The pilot of the first tap has a smooth cylindrical end from 0.003 to 0.005 inch (0.076-0.127 mm) smaller than the hole, and the pilots of following taps should have teeth. Collapsible Taps.—The collapsing tap shown in the accompanying illustration is one of many different designs that are manufactured. These taps are often used in turret lathe practice in place of solid taps. When using this particular style of collapsing tap, the adjust­able gage A is set for the length of thread required. When the tap has been fed to this depth, the gage comes into contact with the end of the work, which causes the chasers to collapse automatically. The tool is then withdrawn, after which the chasers are again expanded and locked in position by the handle seen at the side of the holder.

A

Collapsing Tap

Collapsible taps do not need to be backed out of the hole at the completion of the thread, reducing the tapping time and increasing production rates.

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Machinery's Handbook, 31st Edition Standard Tapers

1013

STANDARD TAPERS Certain types of small tools and machine parts, such as twist drills, end mills, arbors, lathe centers, etc., are provided with taper shanks which fit into spindles or sockets of cor­ responding taper, providing not only accurate alignment between the tool or other part and its supporting member, but also more or less frictional resistance for driving the tool. There are several standards for “self-holding” tapers, but the American National, the Morse, and the Brown & Sharpe are the standards most widely used by American manufacturers. The name self-holding has been applied to the smaller tapers—like the Morse and the Brown & Sharpe—because, where the angle of the taper is only 2 or 3 degrees, the shank of a tool is so firmly seated in its socket that there is considerable frictional resistance to any force tending to turn or rotate the tool relative to the socket. The term “self-holding” is used to distinguish relatively small tapers from the larger or self-releasing type. A milling machine spindle having a taper of 31 ⁄2 inches per foot is an example of a self-releasing taper. The included angle in this case is over 16 degrees and the tool or arbor requires a pos­itive locking device to prevent slipping, but the shank may be released or removed more readily than one having a smaller taper of the self-holding type. Tapers for Machine Tool Spindles.—Various standard tapers have been used for the taper holes in the spindles of machine tools, such as drilling machines, lathes, milling machines, or other types requiring a taper hole for receiving either the shank of a cutter, an arbor, a center, or any tool or accessory requiring a tapering seat. The Morse taper rep­resents a generally accepted standard for drilling machines. See more on this subject, page 1024. The headstock and tailstock spindles of lathes also have the Morse taper in most cases; but the Jarno, the Reed (which is the short Jarno), and the Brown & Sharpe have also been used. Milling machine spindles formerly had Brown & Sharpe tapers in most cases. In 1927, the milling machine manufacturers of the National Machine Tool Builders’ Association adopted a standard taper of 31 ⁄2 inches per foot. This comparatively steep taper has the advantage of insuring instant release of arbors or adapters. National Machine Tool Builders’ Association Tapers

Taper Numbera 30

40

Large End Diameter 11 ⁄4 13⁄4

a Standard taper of 31 ⁄ inches per foot 2

Taper Numbera

Large End Diameter

60

41 ⁄4

50

23⁄4

The British Standard for milling machine spindles is also 31 ⁄2 inches taper per foot and includes these large end diameters: 13⁄8 inches, 13⁄4 inches, 23⁄4 inches, and 31 ⁄4 inches. Morse Taper.—Dimensions relating to Morse standard taper shanks and sockets may be found in an accompanying table. The taper for different numbers of Morse tapers is slightly different, but it is approximately 5⁄8 inch per foot in most cases. The table gives the actual tapers, accurate to five decimal places. Morse taper shanks are used on a variety of tools, and exclusively on the shanks of twist drills. Dimensions for Morse Stub Taper Shanks are given in Table 1a, and for Morse Standard Taper Shanks in Table 1b. Also see Table 8 and Table 9 on page 1022. Brown & Sharpe Taper.—This standard taper is used for taper shanks on tools such as end mills and reamers, the taper being approximately 1 ⁄2 inch per foot for all sizes except for taper No. 10, where the taper is 0.5161 inch per foot. Brown & Sharpe taper sockets are used for many arbors, collets, and machine tool spindles, especially milling machines and grinding machines. In many cases there are a number of different lengths of sockets corre­ sponding to the same number of taper; all these tapers, however, are of the same diameter at the small end. See Table 10, page 1023.

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Machinery's Handbook, 31st Edition Standard Tapers

1014

Table 1a. Morse Stub Taper Shanks

Y

A

REAMER

X

P

SHANK

PLUG GAGE

B

D

M

RAD G

F

SHANK

O

E N

C H

F

TAPER 1 3/4´´ PER FOOT No. of Taper

Taper per Foota

Taper per Inchb

Small End of Plug, b D

Dia. End of Socket, a A

Shank

Total Length, B

Tang

Depth, C

Thickness, E

Length, F 5 ⁄16

1

0.59858

0.049882

0.4314

0.475

0.59941

0.049951

0.6469

0.700

15⁄16

11 ⁄8

13 ⁄64

2

111 ⁄16

17⁄16

19 ⁄64

7⁄16

3

0.60235

0.050196

0.8753

0.938

2

13⁄4

25 ⁄64

9 ⁄16

4

0.62326

0.051938

1.1563

1.231

23⁄8

21 ⁄16

33 ⁄64

11 ⁄16

5

0.63151

0.052626

1.6526

1.748

3

211 ⁄16

3 ⁄4

15 ⁄16

Socket End to Tang Slot, M

Width, N

25 ⁄32

7⁄32

15 ⁄16

5 ⁄16

No. of Taper

Tang

Socket

Min. Depth of Tapered Hole Drilled Reamed X Y

Radius of Mill, G

Diameter, H

3 ⁄16

13 ⁄32

7⁄8

2

7⁄32

39 ⁄64

11 ⁄16

15⁄32

17⁄64

3

9 ⁄32

13 ⁄16

11 ⁄4

13⁄8

15⁄16

11 ⁄16

1

Plug Depth, P

15 ⁄16

29 ⁄32

Tang Slot Length, O 23 ⁄32 15 ⁄16

13 ⁄32

11 ⁄8

4

3 ⁄8

13⁄32

17⁄16

19⁄16

11 ⁄2

13⁄16

17⁄32

13⁄8

5

9 ⁄16

119⁄32

113⁄16

115⁄16

17⁄8

17⁄16

25 ⁄32

13⁄4

All dimensions in inches. Radius J is 3⁄64 , 1 ⁄16 , 5⁄64 , 3⁄32 , and 1 ⁄8 inch respectively for Nos. 1, 2, 3, 4, and 5 tapers. a These are basic dimensions. b These dimensions are calculated for reference only.

Jarno Taper.—The Jarno taper was originally proposed by Oscar J. Beale of the Brown & Sharpe Mfg. Co. This taper is based on such simple formulas that practically no calcula­ tions are required when the number of taper is known. The taper per foot of all Jarno taper sizes is 0.600 inch on the diameter. The diameter at the large end is as many eighths, the diameter at the small end is as many tenths, and the length as many half inches as are indi­ cated by the number of the taper. For example, a No. 7 Jarno taper is 7⁄8 inch in diameter at the large end; 7⁄10, or 0.700 inch at the small end; and 7⁄2 , or 31 ⁄2 inches long; hence, diameter at large end = No. of taper ÷ 8; diameter at small end = No. of taper ÷ 10; length of taper = No. of taper ÷ 2. The Jarno taper is used on various machine tools, especially profiling machines and die-sinking machines. It has also been used for the headstock and tailstock spindles of some lathes.

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Machinery's Handbook, 31st Edition Standard Tapers

1015

Table 1b. Morse Standard Taper Shanks

H

Plug P

Reamer

A K

B D

S

d

R t

X

W

T

L

8°19’

Angle of Key, Taper, 1.75 in 12 No. of Taper 0

Taper per Foot 0.62460

Taper per Inch 0.05205

Small End of Plug D

Diameter End of Socket A

0.252

0.3561

Shank Length B 211 ⁄32

1

0.59858

0.04988

0.369

0.475

29⁄16

3

0.60235

0.05019

0.778

0.938

37⁄8

2 4 5 6 7

Plug Depth P 2 21 ⁄8

0.59941 0.62326 0.63151 0.62565 0.62400 Thickness t 0.1562

0.05262 0.05213

1.020 1.475 2.116

0.05200 2.750 Tang or Tongue Length Radius T R

0.700 1.231 1.748 2.494 3.270 Dia. d 0.235

1 ⁄4

5 ⁄32

3 ⁄8

3 ⁄16

0.343

0.3125

7⁄16

1 ⁄4

17⁄32

9 ⁄16

9 ⁄32

23 ⁄32

5 ⁄8

5 ⁄16

3 ⁄4

3 ⁄8

0.2500

41 ⁄16

0.4687

53⁄16

0.05193

0.572

0.2031

29⁄16

33⁄16

0.04995

0.6250

31 ⁄32

113⁄32 2

31 ⁄8 47⁄8

Depth S

Depth of Hole H

27⁄32

21 ⁄32

27⁄16

25⁄32

215⁄16

239⁄64

45⁄8

41 ⁄8

311 ⁄16

31 ⁄4

57⁄8

51 ⁄4

115⁄8

111 ⁄4

105⁄64

11 ⁄64 0.218

9 ⁄16

0.328

13⁄16

0.656

11 ⁄2

61 ⁄8

89⁄16

81 ⁄4

Keyway Width Length W L

0.266

0.484

3 ⁄4 7⁄8

11 ⁄4

721 ⁄64

Keyway to End K 115⁄16

21 ⁄16 21 ⁄2

31 ⁄16

37⁄8

415⁄16 7

1 ⁄2 0.7500 0.781 71 ⁄4 11 ⁄8 13⁄4 3 ⁄4 1.156 10 1.1250 25⁄8 25⁄8 91 ⁄2 13⁄8 Tolerances on rate of taper: all sizes 0.002 in. per foot. This tolerance may be applied on shanks only in the direction that increases the rate of taper, and on sockets only in the direction that decreases the rate of taper.

American National Standard Machine Tapers.—This standard includes a self-holding series (Table 2, Table 3, Table 4, Table 5 and Table 7a) and a steep taper series, Table 6. The self-holding taper series consists of 22 sizes, which are listed in Table 7a. The reference gage for the self-hold­ing tapers is a plug gage. Table 7b gives the dimensions and tolerances for both plug and ring gages applying to this series. Table 2 through Table 5 inclusive give the dimensions for self-holding taper shanks and sockets, which are classified as to (1) means of transmitting torque from spindle to the tool shank, and (2) means of retaining the shank in the socket. The steep machine tapers consist of a preferred series (bold-face type, Table 6) and an intermediate series (light-face type). A self-holding taper is defined as “a taper with an angle small enough to hold a shank in place ordinarily by friction without holding means. (Sometimes referred to as slow taper.)” A steep taper is defined as “a taper having an angle sufficiently large to insure the easy or self-releasing feature.” The term “gage line” indi­cates the basic diameter at or near the large end of the taper.

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Machinery's Handbook, 31st Edition Standard Tapers

1016

Table 2. American National Standard Taper Drive with Tang, Self-Holding Tapers ANSI/ASME B5.10-1994 (R2013) J

Gage Line

N E

Key Taper 1 3/4 in. per Ft.

K

A

G C B

No. of Taper

Diameter at Gage Line (1) A

No. of Taper 0.239 0.299 0.375 1 2 3 4 41 ⁄2 5 6

Radius J 0.03 0.03 0.05 0.05 0.06 0.08 0.09 0.13 0.13 0.16

0.239 0.299 0.375 1 2 3 4 41 ⁄2 5 6

0.23922 0.29968 0.37525 0.47500 0.70000 0.93800 1.23100 1.50000 1.74800 2.49400

Optional

J P

F O

H Gage Line M

Shank

Gage Line to End of Shank C 1.19 1.50 1.88 2.44 2.94 3.69 4.63 5.13 5.88 8.25 Socket Min. Depth of Hole K

Total Length of Shank B 1.28 1.59 1.97 2.56 3.13 3.88 4.88 5.38 6.12 8.25

Drilled 1.06 1.31 1.63 2.19 2.66 3.31 4.19 4.62 5.31 7.41

Reamed 1.00 1.25 1.56 2.16 2.61 3.25 4.13 4.56 5.25 7.33

Tang Thickness E

Length F

Radius of Mill G

Diameter H

Gage Line to Tang Slot M 0.94 1.17 1.47 2.06 2.50 3.06 3.88 4.31 4.94 7.00

Width N 0.141 0.172 0.203 0.218 0.266 0.328 0.484 0.578 0.656 0.781

Length O 0.38 0.50 0.63 0.75 0.88 1.19 1.25 1.38 1.50 1.75

Shank End to Back of Tang Slot P 0.13 0.17 0.22 0.38 0.44 0.56 0.50 0.56 0.56 0.50

0.125 0.156 0.188 0.203 0.250 0.312 0.469 0.562 0.625 0.750

0.19 0.25 0.31 0.38 0.44 0.56 0.63 0.69 0.75 1.13

0.19 0.19 0.19 0.19 0.25 0.22 0.31 0.38 0.38 0.50 Tang Slot

0.18 0.22 0.28 0.34 0.53 0.72 0.97 1.20 1.41 2.00

All dimensions are in inches. (1) See Table 7b for plug and ring gage dimensions. Tolerances: For shank diameter A at gage line, +0.002, -0.000; for hole diameter A, +0.000, -0.002. For tang thickness E up to No. 5 inclusive, +0.000, -0.006; No. 6, +0.000, -0.008. For width N of tang slot up to No. 5 inclusive, +0.006, -0.000; No. 6, +0.008, -0.000. For centrality of tang E with center line of taper, 0.0025 (0.005 total indicator variation). These centrality tolerances also apply to the tang slot N. On rate of taper, all sizes 0.002 per foot. This tolerance may be applied on shanks only in the direction that increases the rate of taper and on sockets only in the direction that decreases the rate of taper. Tolerances for two-decimal dimensions are plus or minus 0.010, unless otherwise specified.

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Machinery's Handbook, 31st Edition Standard Tapers

1017

Table 3. American National Standard Taper Drive with Keeper Key Slot, Self-Holding Tapers ANSI/ASME B5.10-1994 (R2013) P

N

O

E

G

J F

B C

K

N”

Taper 1 3/4 in. per Ft.

H

MX

Z

Y”

Y

A Gage Line

Gage Line

Shank

Tang

No. of Taper

Dia. at Gage Line (1) A

Total Length B

Gage Line to End C

3

0.938

3.88

4

1.231

41 ⁄2

Socket Min. Depth of Hole K

Gage Line to Tang Slot M

Thickness E

Length F

Radius of Mill G

3.69

0.312

0.56

0.28

0.78

0.08

3.31

3.25

4.88

4.63

0.469

0.63

0.31

0.97

0.09

4.19

4.13

3.88

1.500

5.38

5.13

0.562

0.69

0.38

1.20

0.13

4.63

4.56

4.32

5

1.748

6.13

5.88

0.625

0.75

0.38

1.41

0.13

5.31

5.25

4.94

6

2.494

8.56

8.25

0.750

1.13

0.50

2.00

0.16

7.41

7.33

7.00

7

3.270

11.63

11.25

1.125

1.38

0.75

2.63

0.19

10.16

10.08

9.50

Tang Slot

Diameter H

Radius J

Keeper Slot in Shank

0.266

1.13

1.19

0.266

0.391

1.50

1.25

0.391

0.453

1.81

1.38

0.453

1.38

0.516

2.13

1.50

0.516

1.63

0.641

2.25

1.75

0.641

1.69

0.766

2.63

1.81

0.766

1.19

0.56

1.03

1.13

0.484

1.25

0.50

1.41

1.19

41 ⁄2

0.578

1.38

0.56

1.72

1.25

5

0.656

1.50

0.56

2.00

6

0.781

1.75

0.50

2.13

7

1.156

2.63

0.88

2.50

4

Keeper Slot in Socket

Width N ′

0.328

3

3.06

Length Z

Length O

Gage Line to Bottom of Slot Y ′

Width N

Ream

Gage Line to Front of Slot Y

Shank End to Back of Slot P

No. of Taper

Drill

Length X

Width N ′

All dimensions are in inches. (1) See Table 7b for plug and ring gage dimensions. Tolerances: For shank diameter A at gage line, +0.002, -0; for hole diameter A, +0, -0.002. For tang thickness E up to No. 5 inclusive, +0, -0.006; larger than No. 5, +0, -0.008. For width of slots N and N ′ up to No. 5 inclusive, +0.006, -0; larger than No. 5, +0.008, -0. For centrality of tang E with center line of taper 0.0025 (0.005 total indicator variation). These centrality tolerances also apply to slots N and N ′. On rate of taper, see footnote in Table 2. Tolerances for two-decimal dimensions are ±0.010 unless otherwise specified.

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Machinery's Handbook, 31st Edition Standard Tapers

1018

Table 4. American National Standard Nose Key Drive with Keeper Key Slot, Self-Holding Tapers ANSI/ASME B5.10-1994 (R2013) V

T N” R”

U

R” D”

A O P

N

S” S” Y

Z M

D

D

Q I I”

B”

C A

N”

S Gage Line

W X

Gage Line

AE AE

Taper 1 3/4 in. per Ft. Taper

A(1)

200 250 300 350 400 450 500 600 800 1000 1200 Taper

2.000 2.500 3.000 3.500 4.000 4.500 5.000 6.000 8.000 10.000 12.000 D

5.13 5.88 6.63 7.44 8.19 9.00 9.75 11.31 14.38 17.44 20.50 D ′a

200 250 300 350 400 450 500 600 800 1000 1200

1.81 2.25 2.75 3.19 3.63 4.19 4.63 5.50 7.38 9.19 11.00

1.00 1.00 1.00 1.25 1.25 1.50 1.50 1.75 2.00 2.50 3.00

200 250 300 350 400 450 500 600 800 1000 1200 Taper

1.41 1.66 2.25 2.50 2.75 3.00 3.25 3.75 4.75 … … U

B ′

0.375 0.375 0.375 0.375 0.375 0.500 0.500 0.500 0.500 … … V

C

R/2 R

Q

I ′

I

R

S

W

0.25 0.25 0.25 0.31 0.31 0.38 0.38 0.44 0.50 0.63 0.75 X

1.38 1.38 1.63 2.00 2.13 2.38 2.50 3.00 3.50 4.50 5.38 N ′

1.63 2.06 2.50 2.94 3.31 3.81 4.25 5.19 7.00 8.75 10.50 R ′

1.010 1.010 2.010 2.010 2.010 3.010 3.010 3.010 4.010 4.010 4.010 S ′

0.562 0.562 0.562 0.562 0.562 0.812 0.812 0.812 1.062 1.062 1.062 T

4.50 5.19 5.94 6.75 7.50 8.00 8.75 10.13 12.88 15.75 18.50

0.656 0.781 1.031 1.031 1.031 1.031 1.031 1.281 1.781 2.031 2.531

0.94 1.25 1.50 1.50 1.50 1.75 1.75 2.06 2.75 3.31 4.00

2.00 2.25 2.63 3.00 3.25 3.63 4.00 4.63 5.75 7.00 8.25

1.69 1.69 1.69 2.13 2.38 2.56 2.75 3.25 4.25 5.00 6.00

Min. 0.003 Max. 0.035 for all sizes

3.44 3.69 4.06 4.88 5.31 5.88 6.44 7.44 9.56 11.50 13.75 M

1.56 1.56 1.56 2.00 2.25 2.44 2.63 3.00 4.00 4.75 5.75 N

0.656 0.781 1.031 1.031 1.031 1.031 1.031 1.281 1.781 2.031 2.031 O 1.56 1.94 2.19 2.19 2.19 2.75 2.75 3.25 4.25 5.00 6.00

1.000 1.000 2.000 2.000 2.000 3.000 3.000 3.000 4.000 4.000 4.000 P

0.50 0.50 0.50 0.50 0.50 0.75 0.75 0.75 1.00 1.00 1.00 Y

4.75 5.50 6.25 6.94 7.69 8.38 9.13 10.56 13.50 16.31 19.00 Z

a Thread is UNF-2B for hole; UNF-2A for screw. (1) See Table 7b for plug and ring gage dimensions. All dimensions are in inches. AE is 0.005 greater than one-half of A. Width of drive key R″ is 0.001 less than width R″ of keyway. Tolerances: For diameter A of hole at gage line, +0, -0.002; for diameter A of shank at gage line, +0.002, -0; for width of slots N and N ′, +0.008, -0; for width of drive keyway R ′ in socket, +0, -0.001; for width of drive keyway R in shank, 0.010, -0; for centrality of slots N and N ′ with center line of spindle, 0.007; for centrality of keyway with spindle center line: for R, 0.004 and for R ′, 0.002 T.I.V. On rate of taper, see footnote in Table 2. Two-decimal dimensions, ±0.010 unless otherwise specified.

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Machinery's Handbook, 31st Edition Standard Tapers

1019

Table 5. American National Standard Nose Key Drive with Drawbolt, Self-Holding Tapers ANSI/ASME B5.10-1994 (R2013) V U

Gage Line

T

A d

R”

B”

D’

Gage Line D AL AO R” D 60° Countersink

S” S”

G J

No. of Taper

200 250 300 350 400 450 500 600 800 1000 1200

Dia. at Gage Line Aa

2.000 2.500 3.000 3.500 4.000 4.500 5.000 6.000 8.000 10.000 12.000

Sockets Drive Keyway

Drive Key

Screw Holes Center Line UNF 2B to Center Hole UNF of Screw 2A Screw D D ′ 1.41 1.66 2.25 2.50 2.75 3.00 3.25 3.75 4.75 … …

0.38 0.38 0.38 0.38 0.38 0.50 0.50 0.50 0.50 … …

Width R″ 0.999 0.999 1.999 1.999 1.999 2.999 2.999 2.999 3.999 3.999 3.999

Width R ′ 1.000 1.000 2.000 2.000 2.000 3.000 3.000 3.000 4.000 4.000 4.000

a See Table 7b for plug and ring gage dimensions.

Depth S ′ 0.50 0.50 0.50 0.50 0.50 0.75 0.75 0.75 1.00 1.00 1.00

C A S

Gage Line to Front of Relief T 4.75 5.50 6.25 6.94 7.69 8.38 9.13 10.56 13.50 16.31 19.00

Dia. of Relief U

1.81 2.25 2.75 3.19 3.63 4.19 4.63 5.50 7.38 9.19 11.00

Shanks Drawbar Hole

No. of Taper 200

Length from Gage Line B ¢ 5.13

250 300 350 400

5.88 6.63 7.44 8.19

500

9.75

450 600

9.00

11.31

Dia. UNC-2B AL

Depth of Drilled Hole E

Depth of Thread AP

Dia. of Counter Bore G

7⁄8 -9

2.44

1.75

0.91

1.00 1.00 1.00 1.25 1.25 1.50 1.50 1.75 2.00 2.50 3.00

Drive Keyway Gage Line to First Thread AO 4.78

Dia. of Draw Bolt Hole d 1.00 1.00 1.13 1.13 1.63 1.63 1.63 2.25 2.25 2.25 2.25

Depth of 60° Chamfer J

Width R

Depth S

Center Line to Bottom of Keyway AE

0.13

1.010

0.562

1.005

1-8 1-8 11 ⁄2 -6

1.75 2.00 2.00 3.00

0.91 1.03 1.03 1.53

5.53 6.19 7.00 7.50

0.13 0.19 0.19 0.31

1.010 2.010 2.010 2.010

0.562 0.562 0.562 0.562

1.255 1.505 1.755 2.005

11 ⁄2 -6

4.00

3.00

1.53

9.06

0.31

3.010

0.812

2.505

11 ⁄2 -6 2-41 ⁄2

14.38

2-41 ⁄2

1200

20.50

2-41 ⁄2

17.44

Depth of Relief V

2.44 2.75 2.75 4.00

7⁄8 -9

800

1000

R AE AE

AP

E

R/2

2-41 ⁄2

4.00

5.31 5.31 5.31 5.31

3.00

4.00 4.00 4.00 4.00

1.53

8.31

2.03

10.38

2.03

16.50

2.03 2.03

13.44 19.56

0.31

0.50 0.50 0.50 0.50

3.010

3.010 4.010 4.010 4.010

0.812

0.812 1.062 1.062 1.062

2.255

3.005 4.005 5.005 6.005

All dimensions in inches. Exposed length C is 0.003 minimum and 0.035 maximum for all sizes. Drive Key D ′ screw sizes are 3⁄8 -24 UNF-2A up to taper No. 400 inclusive and 1 ⁄2 -20 UNF-2A for larger tapers. Tolerances: For diameter A of hole at gage line, +0.000, -0.002 for all sizes; for diameter A of shank at gage line, +0.002, -0.000; for all sizes; for width of drive keyway R ′ in socket, +0.000, -0.001; for width of drive keyway R in shank, +0.010, -0.000; for centrality of drive keyway R ′, with center line of shank, 0.004 total indicator variation, and for drive keyway R ′, with center line of spin­dle, 0.002. On rate of taper, see footnote in Table 2. Tolerances for two-decimal dimensions are ±0.010 unless otherwise specified.

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Machinery's Handbook, 31st Edition Standard Tapers

1020

Table 6. ANSI Standard Steep Machine Tapers ANSI/ASME B5.10-1994 (R2013) No. of Taper 5 10 15 20 25 30

Taper per Foota 3.500 3.500 3.500 3.500 3.500 3.500

Dia. at Gage Lineb

0.500 0.625 0.750 0.875 1.000 1.250

Length Along Axis 0.6875 0.8750 1.0625 1.3125 1.5625 1.8750

No. of Taper 35 40 45 50 55 60

Taper per Foota 3.500 3.500 3.500 3.500 3.500 3.500

Dia.at Gage Lineb

Length Along Axis

1.500 1.750 2.250 2.750 3.500 4.250

2.2500 2.5625 3.3125 4.0000 5.1875 6.3750

a This taper corresponds to an included angle of 16°, 35′, 39.4″. b The basic diameter at gage line is at large end of taper.

All dimensions given in inches. The tapers numbered 10, 20, 30, 40, 50, and 60 that are printed in heavy-faced type are designated as the “Preferred Series.” The tapers numbered 5, 15, 25, 35, 45, and 55 that are printed in lightfaced type are designated as the “Intermediate Series.”

Table 7a. American National Standard Self-holding Tapers — Basic Dimensions ANSI/ASME B5.10-1994 (R2013) No. of Taper .239 .299 .375 1 2 3 4

41 ⁄2

5 6 7 200 250 300 350 400 450 500 600 800 1000 1200

Taper per Foot

0.50200 0.50200 0.50200 0.59858 0.59941 0.60235 0.62326 0.62400

Dia. at Gage Line a A

0.23922 0.29968 0.37525 0.47500 0.70000 0.93800 1.23100 1.50000

0.63151 1.74800 0.62565 2.49400 0.62400 3.27000 0.750 2.000 0.750 2.500 0.750 3.000 0.750 3.500 0.750 4.000 0.750 4.500 0.750 5.000 0.750 6.000 0.750 8.000 0.750 10.000 0.750 12.000

Means of Driving and Holdinga   Tang Drive With Shank Held in by Friction (See Table 2)

   Tang Drive With Shank Held in by Key (See Table 3)

  Key Drive With Shank Held in by Key     (See Table 4)   Key Drive With Shank Held in by Draw-bolt     (See Table 5)

Origin of Series Brown & Sharpe Taper Series

Morse Taper Series

Inch per Foot Taper Series

3⁄ 4

a See illustrations above Table 2 through Table 5.

All dimensions given in inches.

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Machinery's Handbook, 31st Edition Standard Tapers

1021

Table 7b. American National Standard Plug and Ring Gages for the Self-Holding Taper Series ANSI/ASME B5.10-1994 (R2013) L

A”

A Gage Line

L A”

No. of Taper

0.239 0.299 0.375 1 2 3 4 41 ⁄2 5 6 7 200 250 300 350 400 450 500 600 800 1000 1200

Tapera per Foot

0.50200 0.50200 0.50200 0.59858 0.59941 0.60235 0.62326 0.62400 0.63151 0.62565 0.62400 0.75000 0.75000 0.75000 0.75000 0.75000 0.75000 0.75000 0.75000 0.75000 0.75000 0.75000

L”

Gage Line Diametera at Gage Line A

0.23922 0.29968 0.37525 0.47500 0.70000 0.93800 1.23100 1.50000 1.74800 2.49400 3.27000 2.00000 2.50000 3.00000 3.50000 4.00000 4.50000 5.00000 6.00000 8.00000 10.00000 12.00000

A

Tolerances for Diameter Ab Class X Gage

0.00004 0.00004 0.00004 0.00004 0.00004 0.00006 0.00006 0.00006 0.00008 0.00008 0.00010 0.00008 0.00008 0.00010 0.00010 0.00010 0.00010 0.00013 0.00013 0.00016 0.00020 0.00020

Class Y Gage

0.00007 0.00007 0.00007 0.00007 0.00007 0.00009 0.00009 0.00009 0.00012 0.00012 0.00015 0.00012 0.00012 0.00015 0.00015 0.00015 0.00015 0.00019 0.00019 0.00024 0.00030 0.00030

Class Z Gage

0.00010 0.00010 0.00010 0.00010 0.00010 0.00012 0.00012 0.00012 0.00016 0.00016 0.00020 0.00016 0.00016 0.00020 0.00020 0.00020 0.00020 0.00025 0.00025 0.00032 0.00040 0.00040

Diameter at Small End A ′

0.20000 0.25000 0.31250 0.36900 0.57200 0.77800 1.02000 1.26600 1.47500 2.11600 2.75000 1.703 2.156 2.609 3.063 3.516 3.969 4.422 5.328 7.141 8.953 10.766

Length Gage Line to End L 0.94 1.19 1.50 2.13 2.56 3.19 4.06 4.50 5.19 7.25 10.00 4.75 5.50 6.25 7.00 7.75 8.50 9.25 10.75 13.75 16.75 19.75

Depth of Gaging­ Notch, Plug Gage L ′ 0.048 0.048 0.048 0.040 0.040 0.040 0.038 0.038 0.038 0.038 0.038 0.032 0.032 0.032 0.032 0.032 0.032 0.032 0.032 0.032 0.032 0.032

a The taper per foot and diameter A at gage line are basic dimensions. Dimensions in Column A′ are calculated for reference only. b Tolerances for diameter A are plus for plug gages and minus for ring gages. All dimensions are in inches. The amount of taper deviation for Class X, Class Y, and Class Z gages are the same, respectively, as the amounts shown for tolerances on diameter A. Taper deviation is the permissible allowance from true taper at any point of diameter in the length of the gage. On taper plug gages, this deviation may be applied only in the direction that decreases the rate of taper. On taper ring gages, this devi­ ation may be applied only in the direction that increases the rate of taper. Tolerances on two-deci­ mal dimensions are ±0.010.

British Standard Tapers.—British Standard 1660: 1972, “Machine Tapers, Reduction Sleeves, and Extension Sockets,” contains dimensions for self-holding and self-releasing tapers, reduction sleeves, extension sockets, and turret sockets for tools having Morse and metric 5 percent taper shanks. Adapters for use with 7⁄24 tapers and dimensions for spindle noses and tool shanks with self-release tapers and cotter slots are included in this Standard.

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Machinery's Handbook, 31st Edition Standard Tapers

1022

Table 8. Dimensions of Morse Taper Sleeves A = No. Morse Taper Outside L

K

G M

F

B = No. Morse Taper Inside D

A

B

C

2

1

39⁄16

3

1

3

2

4 4

D

E

E

H

C F

G

I

H

I

K

L

M

0.700

5 ⁄8

1 ⁄4

7⁄16

23⁄16

0.475

21 ⁄16

3 ⁄4

0.213

315⁄16

0.938

1 ⁄4

5 ⁄16

9 ⁄16

23⁄16

0.475

21 ⁄16

3 ⁄4

0.213

47⁄16

0.938

3 ⁄4

5 ⁄16

9 ⁄16

25⁄8

0.700

21 ⁄2

7⁄8

0.260

1

47⁄8

1.231

1 ⁄4

15 ⁄32

5 ⁄8

23⁄16

0.475

21 ⁄16

3 ⁄4

0.213

2

47⁄8

1.231

1 ⁄4

15 ⁄32

5 ⁄8

25⁄8

0.700

21 ⁄2

7⁄8

0.260

4

3

53⁄8

1.231

3 ⁄4

15 ⁄32

5 ⁄8

31 ⁄4

0.938

31 ⁄16

13⁄16

0.322

5

1

61 ⁄8

1.748

1 ⁄4

5 ⁄8

3 ⁄4

23⁄16

0.475

21 ⁄16

3 ⁄4

0.213

5

2

61 ⁄8

1.748

1 ⁄4

5 ⁄8

3 ⁄4

25⁄8

0.700

21 ⁄2

7⁄8

0.260

5

3

61 ⁄8

1.748

1 ⁄4

5 ⁄8

3 ⁄4

31 ⁄4

0.938

31 ⁄16

13⁄16

0.322

5

4

65⁄8

1.748

3 ⁄4

5 ⁄8

3 ⁄4

41 ⁄8

1.231

37⁄8

11 ⁄4

0.478

6

1

85⁄8

2.494

3 ⁄8

3 ⁄4

11 ⁄8

23⁄16

0.475

21 ⁄16

3 ⁄4

0.213

6

2

85⁄8

2.494

3 ⁄8

3 ⁄4

11 ⁄8

25⁄8

0.700

21 ⁄2

7⁄8

0.260

6

3

85⁄8

2.494

3 ⁄8

3 ⁄4

11 ⁄8

31 ⁄4

0.938

31 ⁄16

13⁄16

0.322

6

4

85⁄8

2.494

3 ⁄8

3 ⁄4

11 ⁄8

41 ⁄8

1.231

37⁄8

11 ⁄4

0.478

6

5

85⁄8

2.494

3 ⁄8

3 ⁄4

11 ⁄8

51 ⁄4

1.748

415⁄16

11 ⁄2

0.635 0.322

7

3

115⁄8

3.270

3 ⁄8

11 ⁄8

13⁄8

31 ⁄4

0.938

31 ⁄16

13⁄16

7

4

115⁄8

3.270

3 ⁄8

11 ⁄8

13⁄8

41 ⁄8

1.231

37⁄8

11 ⁄4

0.478

7

5

115⁄8

3.270

3 ⁄8

11 ⁄8

13⁄8

51 ⁄4

1.748

415⁄16

11 ⁄2

0.635

7

6

121 ⁄2

3.270

11 ⁄4

11 ⁄8

13⁄8

73⁄8

2.494

7

13⁄4

0.760

Table 9. Morse Taper Sockets — Hole and Shank Sizes

Morse Taper

Morse Taper

Morse Taper

Size

Hole

Shank

Size

Hole

Shank

Size

Hole

1 by 2

No. 1

No. 2

2 by 5

No. 2

No. 5

4 by 4

No. 4

1 by 3

1 by 4

1 by 5

2 by 3

2 by 4

No. 1

No. 1

No. 1

No. 2

No. 2

No. 3

No. 4

No. 5

No. 3

No. 4

3 by 2

3 by 3

3 by 4

3 by 5

4 by 3

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No. 3

No. 3

No. 3

No. 3

No. 4

No. 2

No. 3

No. 4

No. 5

No. 3

4 by 5

4 by 6

5 by 4

5 by 5

5 by 6

No. 4

No. 4

No. 5

No. 5

No. 5

Shank No. 4

No. 5

No. 6

No. 4

No. 5

No. 6

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Machinery's Handbook, 31st Edition Standard Tapers

1023

Table 10. Brown & Sharpe Taper Shanks

18

Drill P

D

Reamer

K

S

1 16

”

18

Num­ ber of Taper

Taper per Foot (inch)

1c

.50200

2c 3c 4 5

.50200 .50200 .50240 .50160

Dia. of Plug at Small End D

Mill. Mach. Stan­ dard

Miscell.

.20000

15 ⁄16

…

…

.31250 .35000 .45000

11 ⁄2 … …

…

111 ⁄16

…

… 21 ⁄8

6

.50329

.50000

23⁄8

7

.50147

.60000

27⁄8

8

.50100

.75000

9

.50085

.90010

10 11 12 13

.51612 .50100 .49973 .50020

1.04465 1.24995 1.50010 1.75005

14

.50000

2.00000

16

.50000

2.50000

15 17

18

.5000

.50000

.50000

2.25000 2.75000

3.00000

…

…

39⁄16 …

41 ⁄4

5

… …

515⁄16 …

71 ⁄8 …

73⁄4

…

…

… …

11 ⁄4 …

13⁄4 …

…

…

…

…

3

…

4

…

…

511 ⁄16 …

…

63⁄4

71 ⁄8

…

…

…

…

13⁄4 2

…

…

… 2

…

… 21 ⁄2

…

…

…

…

…

Keyway from End of Spindle K

15 ⁄16

111 ⁄64 115⁄32

123⁄32 131 ⁄32

115⁄16

21 ⁄16

61 ⁄4 …

229⁄32

317⁄32

41 ⁄8

41 ⁄8

47⁄8

37⁄8

517⁄32

525⁄32

619⁄32

615⁄16 …

79⁄16

…

…

9

101 ⁄4

…

…

…

31 ⁄32

329⁄64

91 ⁄4

…

29⁄16

313⁄32

81 ⁄32

93⁄4

27⁄16

225⁄32

213⁄32

…

…

23⁄32

27⁄8

81 ⁄4 …

23⁄8

219⁄64

61 ⁄16

…

21 ⁄8

23⁄16

67⁄32 …

11 ⁄2

17⁄8

111 ⁄16

141 ⁄64

427⁄32

…

S

13⁄16

121 ⁄32

…

…

Shank Depth

113⁄64

81 ⁄4 83⁄4

Arbors

Collets

Taper 1 3 4 ” per Ft.

Plug Depth, P

13⁄16

d

L

B & Sb Stan­ dard

.25000

Plug Depth (Hole)

Plug Depth

T t W

”

”

817⁄32 …

…

45⁄8

523⁄32

613⁄32 615⁄16

621 ⁄32

715⁄32

Length of Key­ waya L

3 ⁄8

5 ⁄8

…

…

t

1 ⁄8

.197

5 ⁄16

.228

11 ⁄16

.228

11 ⁄16

.260

3 ⁄4

.260

3 ⁄4

.260

3 ⁄4

.291

7⁄8 15 ⁄16 15 ⁄16 15 ⁄16

1

.322 .322 .322

.353

11 ⁄8

11 ⁄8

.385 .385

15⁄16

.447

15⁄16

.447

15⁄16

15⁄16 15⁄16

11 ⁄2

921 ⁄32

d

.170

.197

5 ⁄8

89⁄16

101 ⁄4

T

3 ⁄16

.197

5 ⁄8

11 ⁄2

95⁄32

W

…

Length Diame­ Thick­ of ter of ness of Arbor Arbor Arbor Tongue Tongue Tongue

.135 .166

1 ⁄2

715⁄16 …

Width of Key­ way

.447 .447 .447

.510

1 ⁄4

5 ⁄16 5 ⁄16 11 ⁄32 11 ⁄32 3 ⁄8 3 ⁄8 3 ⁄8 7⁄16 15 ⁄32 15 ⁄32 15 ⁄32 1 ⁄2 9 ⁄16 9 ⁄16 21 ⁄32 21 ⁄32 21 ⁄32 21 ⁄32 21 ⁄32 3 ⁄4

…

…

111 ⁄16

.572

27⁄32

17⁄8

.635

15 ⁄16

…

…

111 ⁄16 …

…

.510

.572 …

3 ⁄4

27⁄32

…

.220

.282

5 ⁄32 3 ⁄16

.282

3 ⁄16

.320

7⁄32

.282

.320

.420

3 ⁄16

7⁄32 1 ⁄4

.420

1 ⁄4

.460

9 ⁄32

.420 .560

1 ⁄4

5 ⁄16

.560

5 ⁄16

.710

11 ⁄32

.560

5 ⁄16

.860

3 ⁄8

1.010

7⁄16

.860

1.010 1.010 1.210

3 ⁄8

7⁄16 7⁄16 7⁄16

1.210

7⁄16

…

…

1.960

9 ⁄16

1.460 1.710

1 ⁄2

1 ⁄2

2.210

9 ⁄16

…

…

2.450

…

5 ⁄8

…

a Special lengths of keyway are used instead of standard lengths in some places. Standard lengths

need not be used when keyway is for driving only and not for admitting key to force out tool. b “B & S Standard” Plug Depths are not used in all cases. c Adopted by American Standards Association.

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Machinery's Handbook, 31st Edition Standard Tapers

1024

Table 11. Jarno Taper Shanks

D

D= Number of Taper 2

3

4

5

6

7

8

9

10 11

12

13

14

15

16

17

18

19

20

C

A

No. of Taper 8

Length A

C=

Length B

11 ⁄8

1

23⁄16

2

33⁄16

3

43⁄16

4

51 ⁄4

5

B

No. of Taper 10 Diameter C 0.20

B=

No. of Taper 2 Diameter D 0.250

Taper per foot 0.600

11 ⁄2

0.30

0.375

0.600

21 ⁄2

0.50

0.625

0.600

31 ⁄2

0.70

0.875

0.600

411 ⁄16

41 ⁄2

0.90

1.125

0.600

53⁄4

51 ⁄2

1.10

1.375

0.600

61 ⁄2

1.30

1.625

0.600

73⁄4

71 ⁄2

1.50

1.875

0.600

813⁄16

81 ⁄2

1.70

2.125

0.600

913⁄16

91 ⁄2

1.90

2.375

0.600

15⁄8

211 ⁄16 311 ⁄16

61 ⁄4

6

71 ⁄4

7

63⁄4

85⁄16 95⁄16 105⁄16

8 9

10

0.40 0.60 0.80 1.00 1.20 1.40 1.60 1.80

2.00

0.500 0.750 1.000 1.250 1.500 1.750 2.000 2.250

2.500

0.600 0.600 0.600 0.600 0.600 0.600 0.600 0.600

0.600

Tapers for Machine Tool Spindles.—Most lathe spindles have Morse tapers, most mill­ ing machine spindles have American Standard tapers, almost all smaller milling machine spindles have R8 tapers, page 1033, and large vertical milling machine spindles have Amer­ican Standard tapers. The spindles of drilling machines and the taper shanks of twist drills are made to fit the Morse taper. For lathes, the Morse taper is generally used, but lathes may have the Jarno, the Brown & Sharpe, or a special taper. Of 33 lathe manufacturers, 20 use the Morse taper; 5, the Jarno; 3 use special tapers of their own; 2 use modified Morse (longer than the standard but the same taper); 2 use Reed (which is a short Jarno); 1 uses the Brown & Sharpe standard. For grinding machine centers, Jarno, Morse, and Brown & Sharpe tapers are used. Of ten grinding machine manufacturers, 3 use Brown & Sharpe; 3 use Morse; and 4 use Jarno. The Brown & Sharpe taper is used extensively for milling machine and dividing head spindles. The standard milling machine spindle adopted in 1927 by the milling machine manufacturers of the National Machine Tool Builders’ Association (now The Association for Manufacturing Technology [AMT]) has a taper of 31 ⁄2 inches per foot. This comparatively steep taper was adopted to ensure easy release of arbors.

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Machinery's Handbook, 31st Edition Standard Tapers

1025

Table 12. American National Standard Plug and Ring Gages for Steep Machine Tapers ANSI/ASME B5.10-1994 (R2013) L

L

A”

A

C”

A

Gage Line

Gage Line

Class X Type Gage

Class Y & Z Type Gage B L

L A

A”

A

A”

Gage Line

Gage Line

Class X Type Gage No. of Taper

Taper per Foota (Basic)

Diameter at Gage Linea A

Class Y & Z Type Gage

Tolerances for Diameter Ab Class X Gage

Class Y Gage

Class Z Gage

Diame­ ter at Small Enda A ′

Length Gage Line to Small End L

Overall Length Dia. of Gage of Body Opening B C

5

3.500

0.500

0.00004

0.00007

0.00010

0.2995

0.6875

0.81

0.30

10

3.500

0.625

0.00004

0.00007

0.00010

0.3698

0.8750

1.00

0.36

15

3.500

0.750

0.00004

0.00007

0.00010

0.4401

1.0625

1.25

0.44

20

3.500

0.875

0.00006

0.00009

0.00012

0.4922

1.3125

1.50

0.48

25

3.500

1.000

0.00006

0.00009

0.00012

0.5443

1.5625

1.75

0.53

30

3.500

1.250

0.00006

0.00009

0.00012

0.7031

1.8750

2.06

0.70

35

3.500

1.500

0.00006

0.00009

0.00012

0.8438

2.2500

2.44

0.84

40

3.500

1.750

0.00008

0.00012

0.00016

1.0026

2.5625

2.75

1.00

45

3.500

2.250

0.00008

0.00012

0.00016

1.2839

3.3125

3.50

1.00

50

3.500

2.750

0.00010

0.00015

0.00020

1.5833

4.0000

4.25

1.00

55

3.500

3.500

0.00010

0.00015

0.00020

1.9870

5.1875

5.50

1.00

60

3.500

4.250

0.00010

0.00015

0.00020

2.3906

6.3750

6.75

2.00

a The taper per foot and diameter A at gage line are basic dimensions. Dimensions in Column A′ are calculated for reference only. b Tolerances for diameter A are plus for plug gages and minus for ring gages. All dimensions are in inches. The amounts of taper deviation for Class X, Class Y, and Class Z gages are the same, respectively, as the amounts shown for tolerances on diameter A. Taper deviation is the permissible allowance from true taper at any point of diameter in the length of the gage. On taper plug gages, this deviation may be applied only in the direction that decreases the rate of taper. On taper ring gages, this devi­ ation may be applied only in the direction that increases the rate of taper. Tolerances on two-deci­ mal dimensions are ±0.010.

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Machinery's Handbook, 31st Edition Standard Tapers

1026

Table 13. Jacobs Tapers and Threads for Drill Chucks and Spindles

A

B

F

E D

G H

C

American Standard Thread Form Taper Series

No. 0 No. 1 No. 2 No. 2a No. 3

A

B

0.22844 0.33341 0.48764 0.48764 0.74610

0.2500 0.3840 0.5590 0.5488 0.8110

C

0.43750 0.65625 0.87500 0.75000 1.21875

Taper per Ft.

0.59145 0.92508 0.97861 0.97861 0.63898

Taper Series

a These dimensions are for the No. 2 “short” taper.

Thread Size 5 ⁄16 -24 5 ⁄16 -24 3 ⁄8 -24

1 ⁄2 -20 5 ⁄8 -11

5 ⁄8 -16

45 ⁄64 -16 3 ⁄4 -16

1-8 1-10

11 ⁄2 -8

Threada Size 5 ⁄16 -24 3 ⁄8 -24

1 ⁄2 -20 5 ⁄8 -11

5 ⁄8 -16

45 ⁄64 -16 3 ⁄4 -16

1-8 1-10 11 ⁄2 -8

Max.

Diameter D

0.531 0.633 0.633 0.860 1.125 1.125 1.250 1.250 1.437 1.437 1.871 Max. 0.3114 0.3739 0.4987 0.6234 0.6236 0.7016 0.7485 1.000 1.000 1.500

G

Min.

Max.

0.516 0.618 0.618 0.845 1.110 1.110 1.235 1.235 1.422 1.422 1.851

0.3245 0.3245 0.385 0.510 0.635 0.635 0.713 0.760 1.036 1.036 1.536

Min.

Hb

0.3042 0.3667 0.4906 0.6113 0.6142 0.6922 0.7391 0.9848 0.9872 1.4848

0.437c 0.562d 0.562 0.687 0.687 0.687 0.687 1.000 1.000 1.000

A

No. 4 No. 5 No. 6 No. 33 …

1.1240 1.4130 0.6760 0.6240 …

Diameter E

Max.

0.3195 0.3195 0.380 0.505 0.630 0.630 0.708 0.755 1.026 1.026 1.526

0.135 0.135 0.135 0.135 0.166 0.166 0.166 0.166 0.281 0.281 0.343

0.2902 0.3528 0.4731 0.5732 0.5906 0.6687 0.7159 0.9242 0.9395 1.4242

Taper per Ft.

C

1.6563 1.8750 1.0000 1.0000 …

Min.

Plug Gage Pitch Dia. Go Not Go 0.2854 0.3479 0.4675 0.5660 0.5844 0.6625 0.7094 0.9188 0.9350 1.4188

B

1.0372 1.3161 0.6241 0.5605 …

0.62886 0.62010 0.62292 0.76194 …

Dimension F

Min.

0.115 0.115 0.115 0.115 0.146 0.146 0.146 0.146 0.250 0.250 0.312

Ring Gage Pitch Dia. Go Not Go 0.2843 0.3468 0.4662 0.5644 0.5830 0.6610 0.7079 0.9188 0.9350 1.4188

0.2806 0.3430 0.4619 0.5589 0.5782 0.6561 0.7029 0.9134 0.9305 1.4134

a Except for 1-8, 1-10, 11 ⁄ -8, all threads are now manufactured to the American National 2 Standard Unified Screw Thread System, Internal Class 2B, External Class 2A. Effective date 1976. b Tolerances for dimension H are as follows: 0.030 inch for thread sizes 5 ⁄ -24 to 3 ⁄ -16, inclusive, 16 4 and 0.125 inch for thread sizes 1-8 to 11 ⁄2 -8, inclusive. c Length for Jacobs 0B5⁄16 chuck is 0.375 inch, length for 1B5⁄16 chuck is 0.437 inch. d Length for Jacobs No. 1BS chuck is 0.437 inch. Usual Chuck Capacities for Different Taper Series Numbers: No. 0 taper, drill diameters, 0-5⁄32 inch; No. 1, 0-1 ⁄4 inch; No. 2, 0-1 ⁄2 inch; No. 2 “Short,” 0-5⁄16 inch; No. 3, 0-1 ⁄2 , 1 ⁄8 -5⁄8 , 3⁄16 -3⁄4 , or 1 ⁄4 13 ⁄ inch; No. 4, 1 ⁄ -3 ⁄ inch; No. 5, 3 ⁄ -1; No. 6, 0-1 ⁄ inch; No. 33, 0-1 ⁄ inch. 16 8 4 8 2 2 Usual Chuck Capacities for Different Thread Sizes: Size 5⁄16 -24, drill diameters 0-1 ⁄4 inch; size 3 ⁄ -24, drill diameters 0-3 ⁄ , 1 ⁄ -3 ⁄ , or 5 ⁄ -1 ⁄ inch; size 1 ⁄ -20, drill diameters 0-1 ⁄ , 1 ⁄ -3 ⁄ , or 5 ⁄ -1 ⁄ 8 8 16 8 64 2 2 2 16 8 64 2 inch; size 5⁄8 -11, drill diameters 0-1 ⁄2 inch; size 5⁄8 -16, drill diameters 0-1 ⁄2 , 1 ⁄8- -5⁄8 , or 3⁄16 -3⁄4 inch; size 45⁄64 -16, drill diameters 0-1 ⁄2 inch; size 3⁄4 -16, drill diameters 0-1 ⁄2 or 3⁄16 -3⁄4 .

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X See Note 3 Max variation from gage line

D min

C

.015 .015

H

J

B

A gage

H

F G

X .0004 See note 4

Z F′ F G

Keyseat Key tight fit in slot when insert key is used G′

Optional Key Construction

Preferred Key Construction

1027

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–X– L min section Z-Z

Machinery's Handbook, 31st Edition Standard Tapers

Copyright 2020, Industrial Press, Inc.

Table 1. Essential Dimensions of American National Standard Spindle Noses for Milling Machines ANSI/ASME B5.18-1972 (R2014) E Slot and key location Face of column min X .002 total M M Usable 4 threads 5° 45° Z Standard steep machine taper 3.500 inch per ft K

Size No.

Gage Dia.of Taper A

30

Clearance Hole for Draw-in Bolt Min. D

Minimum Dimension Spindle End to Column E

Width of Driving Key F

Width of Keyseat F ′

Maximum Height of Driving Key G

Minimum Depth of Keyseat G ′

Distance from Center to Driving Keys H

Radius of Bolt Hole Circle J

Size of Threads for Bolt Holes UNC-2B K

Full Depth of Arbor Hole in Spindle Min. L

Depth of Usable Thread for Bolt Hole M

Pilot Dia. C

1.250

2.7493 2.7488

0.692 0.685

0.66

0.50

0.6255 0.6252

0.624 0.625

0.31

0.31

0.660 0.654

1.0625 (Note 1)

0.375-16

2.88

0.62

40

1.750

3.4993 3.4988

1.005 0.997

0.66

0.62

0.6255 0.6252

0.624 0.625

0.31

0.31

0.910 0.904

1.3125 (Note 1)

0.500-13

3.88

0.81

45

2.250

3.9993 3.9988

1.286 1.278

0.78

0.62

0.7505 0.7502

0.749 0.750

0.38

0.38

1.160 1.154

1.500 (Note 1)

0.500-13

4.75

0.81

50

2.750

5.0618 5.0613

1.568 1.559

1.06

0.75

1.0006 1.0002

0.999 1.000

0.50

0.50

1.410 1.404

2.000(Note 2)

0.625-11

5.50

1.00

60

4.250

8.7180 8.7175

2.381 2.371

1.38

1.50

1.0006 1.0002

0.999 1.000

0.50

0.50

2.420 2.414

3.500 (Note 2)

0.750-10

8.62

1.25

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All dimensions are given in inches. Tolerances: Two-digit decimal dimensions ± 0.010 unless otherwise specified. A—Taper: Tolerance on rate of taper to be 0.001 inch per foot applied only in direction that decreases rate of taper. F ′—Centrality of keyway with axis of taper 0.002 total at maximum material condition. (0.002 Total indicator variation) F—Centrality of solid key with axis of taper 0.002 total at maximum material condition. (0.002 Total indicator variation) Note 1: Holes spaced as shown and located within 0.006 inch diameter of true position. Note 2: Holes spaced as shown and located within 0.010 inch diameter of true position. Note 3: Maximum turnout on test plug:    0.0004 at 1-inch projection from gage line.    0.0010 at 12-inch projection from gage line. Note 4: Squareness of mounting face measured near mounting bolt-hole circle.

Machinery's Handbook, 31st Edition Standard Tapers

Dia.of Spindle B

1028

Copyright 2020, Industrial Press, Inc.

Table 1. (Continued) Essential Dimensions of American National Standard Spindle for Milling Machines ANSI/ASME B5.18-1972 (R2014) Table 1. Essential Dimensions of American National Standard Spindle NosesNoses for Milling Machines ANSI/ASME B5.18-1972 (R2014)

Machinery's Handbook, 31st Edition Standard Tapers

1029

Table 2. Essential Dimensions of American National Standard Tool Shanks for Milling Machines ANSI/ASME B5.18-1972 (R2014)

American Standard Taper 3.500 Inch per Foot

Maximum Variation from Gage Line

.015 .015

End of Spindle

K X P

R L M S

T

U

O

Z V

Dia.of Neck P

1.250

0.422 0.432

0.66 0.65

1.750

0.531 0.541

0.94 0.93

45

2.250

0.656 0.666

50

2.750

60

4.250

Size. No.

Distance from Rear of Flange to End of Arbor V

30 40

Gage Dia.of Taper N

30 40

X

Y

W

Tap Drill Size for Draw-in Thread O

Size No.

N Gage

Size of Thread for Draw-in Bolt UNC-2B M

CL

2 Slots B .004 Total M

Pilot Dia. R

Length of Pilot S

Minimum Length of Usable Thread T

Minimum Depth of Clearance Hole U

0.500-13

0.675 0.670

0.81

1.00

2.00

0.625-11

0.987 0.980

1.00

1.12

2.25

1.19 1.18

0.750-10

1.268 1.260

1.00

1.50

2.75

0.875 0.885

1.50 1.49

1.000-8

1.550 1.540

1.00

1.75

3.50

1.109 1.119

2.28 2.27

1.250-7

2.360 2.350

1.75

2.25

4.25

Clearance of Flange from Gage Diameter W

Tool Shank Centerline to Driving Slot X

Width of Driving Slot Y

2.75

0.045 0.075

0.640 0.625

0.635 0.645

3.75

0.045 0.075

0.890 0.875

0.635 0.645

45

4.38

0.105 0.135

1.140 1.125

0.760 0.770

50

5.12

0.105 0.135

1.390 1.375

60

8.25

0.105 0.135

2.400 2.385

Distance from Gage Line to Bottom of C’bore Z

Depth of 60° Center K

Diameter of C’bore L

2.50

0.05 0.07

0.525 0.530

3.50

0.05 0.07

0.650 0.655

4.06

0.05 0.07

0.775 0.780

1.010 1.020

4.75

0.05 0.12

1.025 1.030

1.010 1.020

7.81

0.05 0.12

1.307 1.312

All dimensions are given in inches. Tolerances: Two-digit decimal dimensions ± 0.010 inch unless otherwise specified. M—Permissible for Class 2B “NoGo” gage to enter five threads before interference. N—Taper tolerance on rate of taper to be 0.001 inch per foot applied only in direction that increases rate of taper. Y—Centrality of drive slot with axis of taper shank 0.004 inch at maximum material condition (0.004 inch total indicator variation).

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Machinery's Handbook, 31st Edition Standard Tapers

1030

Table 3. American National Standard Draw-In Bolt Ends ANSI/ASME B5.18-1972 (R2014) M

D B C

A

Size No.

Length of Small End A

Length of Usable Thread at Small End B

Length of Usable Thread on Large Diam­eter C

Size of Thread for Large End UNC-2A M

Size of Thread for Small End UNC-2A D

30

1.06

0.75

0.75

0.500-13

0.375-16

40

1.25

1.00

1.12

0.625-11

0.500-13

45

1.50

1.12

1.25

0.750-10

0.625-11

50

1.50

1.25

1.38

1.000-8

0.625-11

60

1.75

1.37

2.00

1.250-7

1.000-8

All dimensions are given in inches.

Table 4. American National Standard Pilot Lead on Centering Plugs for Flatback Milling Cutters ANSI/ASME B5.18-1972 (R2014) T .0002 .0250 .0625 30° Lead Diameter

–T–

.03125 American Standard Taper 3.500 Inch per Ft.

Plot Diameter

Face of Spindle Max Lead Diameter = Max Pilot Diameter – .003 Min Lead Diameter = Min Pilot Diameter – .006

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Machinery's Handbook, 31st Edition Standard Tapers

1031

Table 5. Essential Dimensions for American National Standard Spindle Nose with Large Flange ANSI/ASME B5.18-1972 (R2014) M2

M1

X See Note 1 C Max Variation from Gage Line E Min L Min

Gage Dia. of Taper A

Dia. of Spindle Flange B

Pilot Dia. C

50A

2.750

8.7180 8.7175

1.568 1.559

Size No.

Distance from Center to Driving Keys Second Position

H2

H2

1st Position Z

B

Keyseat Key Tight Fit in Slot G

Drive Key

Clearance Hole for Draw-in Bolt Min. D

Min. Dim. Spindle End to Column E

1.06

0.75

Width of Driving Key F 1.0006 1.0002

H2

J1

J2

K1

K2

2.420 2.410

2.000

3.500

0.625-11

0.750-10

5.50

Size of Threads for Bolt Holes UNC-2B

Outer

H1 H1

Full Depth of Arbor Hole in Spindle Min. L

Inner

Z

J2

J1

F1 F

Radius of Bolt Hole Circles (See Note 3)

45°

2nd Position

X .0004 See Note 2 Face of Column

G1

Size No.

X .002 Total M

45°

Usable Threads K1 .015 .015 A Gage

Section Z-Z

50A

–X–

K2

American Standard Taper 3.500 Inch per Ft.

D Min

Slot and Key Location

Usable Threads

Height of Driving Key Max. G

Depth of Keyseat Min. G1

0.50

0.50 Depth of Usable Thread for Bolt Holes

M1

M2

1.00

1.25

Distance from Center to Driving Keys First Position H1 1.410 1.404

Width of Keyseat F1 0.999 1.000

All dimensions are given in inches.

Tolerances: Two-digit decimal dimensions ± 0.010 unless otherwise specified.

A—Tolerance on rate of taper to be 0.001 inch per foot applied only in direction that decreases rate of taper.

F—Centrality of solid key with axis of taper 0.002 inch total at maximum material condition. (0.002 inch Total indicator variation)

F1—Centrality of keyseat with axis of taper 0.002 inch total at maximum material condition. (0.002 inch Total indicator variation) Note 1: Maximum runout on test plug:   0.0004 at 1-inch projection from gage line.   0.0010 at 12-inch projection from gage line.

Note 2: Squareness of mounting face measured near mounting bolt-hole circle.

Note 3: Holes located as shown and within 0.010 inch diameter of true position.

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Machinery's Handbook, 31st Edition Standard Tapers

1032

V-Flange Tool Shanks and Retention Knobs.—Dimensions of ANSI/ASME B5.18-1972 (R2014) standard tool shanks and corresponding spindle noses are detailed on pages 1027 through 1030, and are suitable for spindles used in milling and associated machines. Corre­sponding equipment for higher-precision numerically controlled machines, using reten­tion knobs instead of drawbars, is usually made to the ANSI/ASME B5.502015 standard. Essential Dimensions of V-Flange Tool Shanks ANSI/ASME B5.50-2015 C 0.002

A

S

E

B

D

No Alterations to M

P

0.002 A

T

M

V

G

30 deg N rad 0.002

F

A

H

0.375 dia. 0.015 DP.

–A–

7:24

A

60 deg. ± 15 deg. U

L Both Sides 0.005

J

A Size 30 40 45 50 60

Size 30 40 45 50 60

R over 0.2813

Z

K

X W Y

0.005

A pins

B

B

C

D

G

H

±0.010

Min.

UNC 2B

±0.010

±0.002

Gage Dia. 1.250 1.750 2.250 2.750

+ 0.015 - 0.000

F

±0.005

+0.000 - 0.015

+0.000 - 0.015

1.875 2.687 3.250 4.000

0.188 0.188 0.188 0.250

1.00 1.12 1.50 1.75

0.516 0.641 0.766 1.031

0.500-13 0.625-11 0.750-10 1.000-8

1.531 2.219 2.969 3.594

1.812 2.500 3.250 3.875

0.735 0.985 1.235 1.485

0.640 0.890 1.140 1.390

A

L

M

N

P

R

S

T

Tolerance

±0.001

Max.

±0.010

Min.

±0.002

±0.010

Min.

Gage Dia. 1.250 1.750 2.250 2.750

+0.000 - 0.005

0.645 0.645 0.770 1.020

1.812 2.500 3.250 3.875

0.020 0.040 0.040 0.040 0.040

1.38 1.38 1.38 1.38

2.176 2.863 3.613 4.238

0.590 0.720 0.850 1.125

0.652 0.880 1.233 1.427

1.250 1.750 2.250 2.750

4.250

6.375

1.020

0.312

5.500

2.25

1.281

1.250-7

1.500

5.219

5.863

5.500

1.375

J

A

Tolerance

4.250

E

0.005 0.002

0.000 R 0.020 + – 0.005

A

–B–

2.235

2.309

K

2.140 Z

4.250

Notes: Taper tolerance to be 0.001 in. in 12 in. applied in direction that increases rate of taper. Geo­metric dimensions symbols are to ASME Y14.5-2018. Dimensions are in inches. Deburr all sharp edges. Unspecified fillets and radii to be 0.03 ± 0.010R, or 0.03 ± 0.010 × 45 degrees. Data for size 60 are not part of Standard. For all sizes, the values for dimensions U (tol. ± 0.005) are 0.579: for V (tol. ± 0.010), 0.440; for W (tol. ± 0.002), 0.625; for X (tol. ± 0.005), 0.152; and for Y (tol. ± 0.002), 0.750.

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Machinery's Handbook, 31st Edition Standard Tapers

1033

Essential Dimensions of V-Flange Tool Shank Retention Knobs ANSI/ASME B5.50-2015 D 0.002 A 0.002 A –B–

C

H

M

E F

K B

A –A– Pitch Dia.

Size

30 deg.

0.500-13

40

0.625-11

45

0.750-10

50

1.000-8

60

1.250-7

Tolerances

UNC- 2A G

J

45 deg.

35 deg.

A

30

Size

L

G

R rad.

B

0.002 A

C

0.520

0.006 B

D

E

F

0.385

1.10

0.460

0.320

0.740

0.490

1.50

0.640

0.440

0.940

0.605

1.80

0.820

0.580

1.140

0.820

2.30

1.000

0.700

1.460

1.045

3.20

1.500

1.080

±0.005

±0.005

±0.005 H

±0.005 J

±0.040 K

L

M

R

30

0.04

0.10

0.187

0.65 0.64

0.53

0.19

0.094

40

0.06

0.12

0.281

0.94 0.92

0.75

0.22

0.094

45

0.08

0.16

0.375

1.20 1.18

1.00

0.22

0.094

0.468

1.44 1.42

1.25

0.25

0.125

0.500

2.14 2.06

1.50

0.31

50 60 Tolerances

0.10 0.14 ±0.010

0.20 0.30 ±0.010

±0.010

+0.000 -0.010

±0.040

0.125 +0.010 -0.005

Notes: Dimensions are in inches. Material: low-carbon steel. Heat treatment: carburize and harden to 0.016 to 0.028 in. effective case depth. Hardness of noted surfaces to be 56–60 RC (Rockwell C scale); core hardness 35–45 RC. Hole J shall not be carburized. Surfaces C and R to be free from tool marks. Deburr all sharp edges. Geometric dimension symbols are to ASME Y14.5-2018. Data for size 60 are not part of Standard.

Collets

R8 Collet.—The dimensions in this figure are believed reliable. However, there are varia­ tions among manufacturers of R8 collets, especially regarding the width and depth of the keyway. Some sources do not agree with all dimensions in this figure. R8 collets are not always interchangeable.

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Machinery's Handbook, 31st Edition Collets

1034

5 3 Keyway − ------ Wide × ------ Deep

7 ------ – 20 UNF Thread 16

32

32

0.9375

1.2500

16° 51'

0.9495 0.9490 1.25

0.9375

3.0625 4.00

0.125

All dimensions in inches.

Bridgeport R8 Collet Dimensions

Collet Styles for Lathes, Mills, Grinders, and Fixtures AC

A C

A C B

1

B

2

B

3

A A

AC

C

B

4

B

5

A C

6

A C

A C B

7

B

8

A B

B

9

A C

10

B

A C B

11

B

12

Collet Styles

Collets for Lathes, Mills, Grinders, and Fixtures Collet 1A 1AM 1B 1C 1J 1K 2A 2AB 2AM

Style 1 1 2 1 1 3 1 2 1

Bearing Dia., A 0.650 1.125 0.437 0.335 1.250 1.250 0.860 0.750 0.629

Dimensions

Length, B 2.563 3.906 1.750 1.438 3.000 2.813 3.313 2.563 3.188

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Thread, C 0.640 × 26 RH

1.118 × 24 RH 0.312 × 30 RH 0.322 × 40 RH 1.238 × 20 RH None

0.850 × 20 RH 0.500 × 20 RH 0.622 × 24 RH

Max. Capacity (inches)

Round 0.500 1.000 0.313 0.250 1.063 1.000 0.688 0.625 0.500

Hex

0.438 0.875 0.219 0.219 0.875 0.875 0.594 0.484 0.438

Square 0.344 0.719 0.188 0.172 0.750 0.719 0.469 0.391 0.344

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Machinery's Handbook, 31st Edition Collets

1035

ColletsCollets for Lathes, Mills, Mills, Grinders, and Fixtures (Continued) for Lathes, Grinders, and Fixtures Collet 2B 2C 2H 2J 2L 2M 2NS 2OS 2S 2VB 3AM 3AT 3B 3C 3H 3J 3NS 3OS 3PN 3PO 3S 3SC 3SS 4C 4NS 4OS 4PN 4S 5C

5M 5NS 5OS 5P 5PN 5SC 5ST 5V 6H 6K 6L 6NS 6R 7B 7 B&S 7P 7R 8H 8ST 8WN 9B 10L 10P 16C 20W 22J 32S

Style 2 1 1 1 1 4 1 1 1 2 1 1 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 5 1 1 1 1 1 1 1 1 1 1 1 1 4 4 1 6 1 1 1 4 1 1 1 1 1 1

Bearing Dia., A 0.590 0.450 0.826 1.625 0.950 2 Morse 0.324 0.299 0.750 0.595 0.750 0.687 0.875 0.650 1.125 2.000 0.687 0.589 0.650 0.599 1.000 0.350 0.589 0.950 0.826 0.750 1.000 0.998 1.250 1.438 1.062 3.500 0.812 1.312 0.600 1.250 0.850 1.375 0.842 1.250 1.312 1.375 7 B&S 7 B&S 1.125 1.062 1.500 2.375 1.250 9 B&S 1.562 1.500 1.889 0.787 2.562 0.703

Dimensions

Length, B 2.031 1.812 4.250 3.250 3.000 2.875 1.562 1.250 3.234 2.438 3.188 2.313 3.438 2.688 4.438 3.750 2.875 2.094 2.063 2.063 4.594 1.578 2.125 3.000 3.500 2.781 2.906 3.250 3.281 3.438 4.219 3.406 3.687 3.406 2.438 3.281 3.875 4.750 3.000 4.438 5.906 4.938 3.125 2.875 4.750 3.500 4.750 5.906 3.875 4.125 5.500 4.750 4.516 2.719 4.000 2.563

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Max. Capacity (inches)

Thread, C 0.437 × 26 RH 0.442 × 30 RH 0.799 × 20 RH 1.611 × 18 RH 0.938 × 20 RH 0.375 × 16 RH 0.318 × 40 RH 0.263 × 40 RH 0.745 × 18 RH 0.437 × 26 RH 0.742 × 24 RH 0.637 × 26 RH 0.625 × 16 RH 0.640 × 26 RH 1.050 × 20 RH 1.988 × 20 RH 0.647 × 20 RH 0.518 × 26 RH 0.645 × 24 RH 0.500 × 24 RH 0.995 × 20 RH 0.293 × 36 RH 0.515 × 26 RH 0.938 × 20 RH 0.800 × 20 RH 0.660 × 20 RH 0.995 × 16 RH 0.982 × 20 RH 1.238 × 20 RHa 1.238 × 20 RH 1.050 × 20 RH 0.937 × 18 RH 0.807 × 24 RH 1.307 × 16 RH 0.500 × 26 RH 1.238 × 20 RH 0.775 × 18 RH 1.300 × 10 RH 0.762 × 26 RH 1.178 × 20 RH 1.234 × 14 RH 1.300 × 20 RH 0.375 × 16 RH 0.375 × 16 RH 1.120 × 20 RH None

1.425 × 20 RH 2.354 × 12 RH 1.245 × 16 RH 0.500 × 13 RH 1.490 × 18 RH 1.495 × 20 RH 1.875 × 1.75 mm RHb 0.775 × 6-1 cm 2.550 × 18 RH 0.690 × 24 RH

Round 0.500 0.344 0.625 1.375 0.750 0.500 0.250 0.188 0.563 0.500 0.625 0.500 0.750 0.500 0.875 1.750 0.500 0.375 0.500 0.375 0.750 0.188 0.375 0.750 0.625 0.500 0.750 0.750 1.063 0.875 0.875 0.750 0.625 1.000 0.375 1.063 0.563 1.125 0.625 1.000 1.000 1.125 0.500 0.500 0.875 0.875 1.250 2.125 1.000 0.750 1.250 1.250 1.625 0.563 2.250 0.500

Hex 0.438 0.594 0.531 1.188 0.656 0.438 0.203 0.156 0.484 0.438 0.531 0.438 0.641 0.438 0.750 1.500 0.438 0.313 0.438 0.313 0.656 0.156 0.313 0.656 0.531 0.438 0.656 0.656 0.906

Square 0.344 0.234 1.000 0.438 1.000 0.344 0.172 0.125 0.391 0.344 0.438 0.344 0.531 0.344 0.625 1.250 0.344 0.266 0.344 0.266 0.531 0.125 0.266 0.531 0.438 0.344 0.531 0.531 0.750

0.484 1.938 0.438

0.391 1.563 0.344

0.750 0.750 0.641 0.531 0.875 0.328 0.906 0.484 0.969 0.531 0.875 0.859 0.969 0.406 0.406 0.750 0.750 1.063 1.844 0.875 0.641 1.063 1.063 1.406

0.625 0.625 0.516 0.438 0.719 0.266 0.750 0.391 0.797 0.438 0.719 0.703 0.781 0.344 0.344 0.625 0.625 0.875 1.500 0.719 0.531 0.875 0.875 1.141

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Machinery's Handbook, 31st Edition Collets

1036

ColletsCollets for Lathes, Mills, Mills, Grinders, and Fixtures (Continued) for Lathes, Grinders, and Fixtures Collet 35J 42S 50V 52SC 115 215 315 B3 D5 GTM J&L JC LB RO RO RO RO R8

Style 1 1 8 1 1 1 1 7 7 7 9 8 10 11 12 12 11 7

Bearing Dia., A 3.875 1.250 1.250 0.800 1.344 2.030 3.687 0.650 0.780 0.625 0.999 1.360 0.687 1.250 1.250 1.250 1.250 0.950

Dimensions

Length, B 5.000 3.688 4.000 3.688 3.500 4.750 5.500 3.031 3.031 2.437 4.375 4.000 2.000 2.938 4.437 4.437 2.938 4.000

Max. Capacity (inches)

Thread, C

Round 3.500 1.000 0.938 0.625 1.125 1.750 3.250 0.500 0.625 0.500 0.750 1.188 0.500 1.125 0.800 1.125 0.800 0.750

3.861 × 18 RH 1.236 × 20 RH 1.125 × 24 RH 0.795 × 20 RH 1.307 × 20 LH 1.990 × 18 LH 3.622 × 16 LH 0.437 × 20 RH 0.500 × 20 RH 0.437 × 20 RH None None None

0.875 × 16 RH 0.875 × 16 RH 0.875 × 16 RH 0.875 × 16 RH 0.437 × 20 RH

Hex 3.000 0.875 0.813 0.531 0.969 1.500 2.813 0.438 0.531 0.438 0.641 1.000 0.438 0.969 0.688 0.969 0.688 0.641

Square 2.438 0.719 0.656 0.438 0.797 1.219 2.250 0.344 0.438 0.344 0.516 0.813 0.344 0.781 0.563 0.781 0.563 0.531

a Internal stop thread is 1.041 × 24 RH.

b Internal stop thread is 1.687 × 20 RH.

Dimensions in inches unless otherwise noted. Courtesy of Hardinge Brothers, Inc. Additional dimensions of the R8 collet are given on page 1033.

DIN 6388, Type B, and DIN 6499, ER Type Collets 30 C A B

Collet Standard Type B,   DIN 6388

ER Type,   DIN 6499

A B

L

L

ER Type

Type B

Type

16 20 25 32 ERA8 ERA11 ERA16 ERA20 ERA25 ERA32 ERA40 ERA50

B (mm) 25.50 29.80 35.05 43.70 8.50 11.50 17 21 26 33 41 41 52

Copyright 2020, Industrial Press, Inc.

L (mm) 40 45 52 60 13.5 18 27 31 35 40 46 39 60

Dimensions

A (mm) 4.5-16 5.5-20 5.5-25 9.5-32 0.5-5 0.5-7 0.5-10 0.5-13 0.5-16 2-20 3-26 26-30 5-34

C

… … … … 8° 8° 8° 8° 8° 8° 8° 8° 8°

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Machinery's Handbook, 31st Edition Portable TOOL SPINDLES

1037

ARBORS, CHUCKS, AND SPINDLES Portable Tool Spindles Circular Saw Arbors.—ASME B107.4-2005 (R2011) “Driving and Spindle Ends for Portable Hand, Air, and Air Electric Tools” calls for a round arbor of 5⁄8 -inch diameter for nominal saw blade diameters of 6 to 8.5 inches, inclusive, and a 3⁄4 -inch diameter round arbor for saw blade diameters of 9 to 12 inches, inclusive.

Spindles for Geared Chucks.—Recommended threaded and tapered spindles for porta­ ble tool geared chucks of various sizes are as given in the following table: Recommended Spindle Sizes Recommended Spindles

Chuck Sizes, Inch 3⁄ 16

and 1 ⁄4 Light

1⁄ 4

and 5⁄16 Medium

3⁄ 8

Medium

3⁄ 8 1⁄ 2 1⁄ 2 5⁄ 8

Light

Light

Medium

and 3⁄4 Medium

Threaded

Tapera 1

3 ⁄ -24 8

3 ⁄ -24 or 1 ⁄ -20 8 2

2 Short

1 ⁄ -20 or 5 ⁄ -16 2 8

2

2

3 ⁄ -24 or 1 ⁄ -20 8 2 1 ⁄ -20 or 5 ⁄ -16 2 8

33

5 ⁄ -16 or 3 ⁄ -16 8 4

3

6

5 ⁄ -16 or 3 ⁄ -16 8 4

a Jacobs number.

Vertical and Angle Portable Tool Grinder Spindles.—The 5⁄8 -11 spindle with a length of 11 ⁄8 inches shown on page 1040 is designed to permit the use of a jam nut with threaded cup wheels. When a revolving guard is used, the length of the spindle is measured from the wheel bearing surface of the guard. For unthreaded wheels with a 7⁄8 -inch hole, a safety sleeve nut is recommended. The unthreaded wheel with 5⁄8 -inch hole is not recommended because a jam nut alone may not resist the inertia effect when motor power is cut off.

Straight Grinding Wheel Spindles for Portable Tools.—Portable grinders with pneu­ matic or induction electric motors should be designed for the use of organic bond wheels rated 9500 ft per min (48.25 m/s). Light-duty electric grinders may be designed for vitri­ fied wheels rated 6500 ft per min (33.0 m/s). Recommended maximum sizes of wheels of both types are as given in the following table: Recommended Maximum Grinding Wheel Sizes for Portable Tools

Spindle Size 3 ⁄ -24 × 11 ⁄ 8 8 1 ⁄ -13 × 13 ⁄ 2 4 5 ⁄ -11 × 21 ⁄ 8 8 5 ⁄ -11 × 31 ⁄ 8 8 5 ⁄ -11 × 31 ⁄ 8 8 3 ⁄ -10 × 31 ⁄ 4 4

Maximum Wheel Dimensions 9500 fpm 6500 fpm Diameter Thickness Diameter Thickness T D T D 21 ⁄2 4

1⁄ 2 3⁄ 4

6

2

8 8 8

1

11 ⁄2 2

4 5

1⁄ 2 3⁄ 4

…

…

…

…

8

…

1

…

Minimum T with the first three spindles is about 1 ⁄8 inch to accommodate cutting off wheels. Flanges are assumed to be according to ANSI B7.1 and threads to ANSI/ASME B1.1.

Copyright 2020, Industrial Press, Inc.

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Machinery's Handbook, 31st Edition Portable Tool Spindles

1038

American Standard Square Drives for Portable Air and Electric Tools ASA B5.38-1958 FM Diam. Plunger

BM

EM Diam. Thru Hole

DM

RM CM

AM

Chamfer or Radius Optional

DESIGN A Drive Size

Design.

Max.

1 ⁄4

A A

A

3 ⁄8 1 ⁄2

1

11 ⁄2

BM Max.

Max.

0.252

0.247

0.330

0.502

0.497

0.665

0.627

B B B

3 ⁄4

Min.

0.377

A

5 ⁄8

AM

0.752 1.002 1.503

0.372

0.500

0.622

0.834

0.747 0.997 1.498

1.000 1.340 1.968

CM

Design

Max.

1 ⁄4

A A

1 11 ⁄2

0.153

…

0.078

0.015

0.321

0.309

…

0.187

0.031

0.625

0.531

0.656

0.938 1.125 1.625

0.406 0.594

0.750 1.000 1.562

0.227

0.215

0.321

0.309

0.415 0.602 0.653

0.403 0.590 0.641

…

A A B B B

AF

Female End

0.216 0.234 0.310

AF

0.315

0.303

0.201

…

0.503

0.670

0.753 1.004 1.505

1.005 1.350 1.983

DF

…

0.508

0.839

DESIGN B

0.090

0.335

0.758 1.009 1.510

BF Min.

0.147

0.253

0.628

0.047 0.063 0.094

0.159

0.258

0.505

0.047

RF Max.

Max.

0.633

… … …

EF Min.

BF Min.

0.378

0.187

0.031

Min.

Min.

0.383

…

0.156

90° 90° Optional Groove Designs

Optional CF Min.

Drive Size

3 ⁄4

0.165

0.265

DESIGN A

5 ⁄8

RM Max.

RF Alternate Corner

Must Accept CM Max.

1 ⁄2

FM Max.

0.312

BF Min.

3 ⁄8

EM Min.

Max.

0.438

DM

Min.

Min.

Chamfer Optional See Note AF

DF

EF Min.

DESIGN B

Male End

0.221 0.315

0.409 0.596 0.647

0.209 0.303

0.397 0.584 0.635

0.170 0.201

0.216 0.234 0.310

… …

0.047 0.062 0.125

All dimensions in inches. Incorporating fillet radius (R M ) at shoulder of male tang precludes use of minimum diameter cross-hole in socket (EF), unless female drive end is chamfered (shown as optional). If female drive end is not chamfered, socket cross-hole diameter (EF) is increased to compensate for fillet radius R M , max. Minimum clearance across flats male to female is 0.001 inch through 3⁄4 -inch size; 0.002 inch in 1- and 11 ⁄2 -inch sizes. For impact wrenches, A M should be held as close to maximum as practical. CF, min., for both designs A and B should be equal to CM , max.

Copyright 2020, Industrial Press, Inc.

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Machinery's Handbook, 31st Edition Portable Tool Spindles

1039

American Standard Threaded and Tapered Spindles for Portable Air and Electric Tools ASA B5.38-1958 R DM

Threaded Spindle

Taper Spindle (Jacobs)

Pitch Dia.

Max.

Min.

R

L

3 ⁄8 -24

0.3479

0.3455

1 ⁄16

9 ⁄16 c

1 ⁄2 -20

0.4675

0.4649

1 ⁄16

9 ⁄16

5 ⁄8 -16

0.5844

0.5812

3 ⁄32

11 ⁄16

3 ⁄4 -16

0.7094

0.7062

3 ⁄32

11 ⁄16

a Jacobs taper number.

No.a 1

2Sd 2 33 6 3

DG LG

LM

L Nom. Dia. and Thd.

EG

Master Plug Gage

DM

LM

0.335-0.333

0.490-0.488 0.490-0.488 0.563-0.561 0.626-0.624 0.748-0.746

0.656

0.750 0.875 1.000 1.000 1.219

EG

DG

Taper per Footb

LG

0.38400 0.33341 0.65625

0.54880 0.55900 0.62401 0.67600 0.81100

0.48764 0.48764 0.56051 0.62409 0.74610

0.7500 0.87500 1.000 1.000 1.21875

0.92508

0.97861 0.97861 0.76194 0.62292 0.63898

b Calculated from E

G , D G , L G for the master plug gage. c Also 7⁄ inch. 16 d 2S stands for 2 Short.

All dimensions in inches. Threads are per inch and right-hand. Tolerances: On R, plus or minus

1 ⁄ inch; on L, plus 0.000, minus 0.030 inch. 64

American Standard Abrasion Tool Spindles for Portable Air and Electric Tools ASA B5.38-1958 Sanders and Polishers

Max. 3/32 + 0.000 15/16 – 1/16 5/8-11 UNC-2A Vertical and Angle Grinders

Max. 3/32

Max. 3/32 Guard 1-1/8 5/8-11 UNC-2A With Revolving Cup Guard

Copyright 2020, Industrial Press, Inc.

1-1/8 5/8-11 UNC-2A Stationary Guard

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Machinery's Handbook, 31st Edition Portable Tool Spindles

1040

American Standard Abrasion Tool Spindles for Portable Air and Electric Tools ASA (Continued) Portable Air and Electric ToolsB5.38-1958 ASA B5.38-1958

Straight Wheel Grinders

Cone Wheel Grinders

Max. 3/32

R H

L

L D H 3 ⁄8 -24 UNF-2A

R 1 ⁄4

1 ⁄2 -13 UNC-2A

L 11 ⁄8 13⁄4

3 ⁄8

5 ⁄8 -11 UNC-2A

1 ⁄2

3 ⁄4 -10 UNC-2A

1

L

9 ⁄16

1 ⁄2 -13 UNC-2A

21 ⁄8

1

5 ⁄8 -11 UNC-2A

D 3 ⁄8 -24 UNF-2A

11 ⁄16

5 ⁄8 -11 UNC-2A

31 ⁄8

15 ⁄16

31 ⁄4

All dimensions in inches. Threads are right-hand.

American Standard Hexagonal Chucks and Shanks for Portable Air and Electric Tools ASA B5.38-1958 B

H

1 ⁄4

0.253

0.314 0.442

7⁄16

B

L Max.

Nominal Hexagon

Min.

0.255

3 ⁄8

15 ⁄16

5 ⁄8

0.630

0.316 0.444

11 32

To End of Chuck 3 1 R 32

1 16

1 4

11 ⁄8

17⁄64

1 4

3 16

3 11 R 72 16

3 64

5 16

0.755

3 ⁄4

Shanks

…

…

0.4375 0.4350

0.312 0.310

0.250 0.248

3 16

1

13 ⁄64

To End of Chuck

5 ⁄16

H

Max.

1 2

11 32

1 4

7 11 R 4 64

1 32

7 16

Max.

B

L Max.

0.632

11 ⁄32

15⁄8

0.758 …

23 32 5 16

17 32

5 R 32 13 4

1 16

5 8

17⁄8

11 ⁄32

…

…

0.750 0.747

0.625 0.622 To End of Chuck

Min.

To End of Chuck

Nominal Hexagon

55 64 5 16

21 32

To End of Chuck

L H

5 R 32

1 16

2

3 4

All dimensions in inches. Tolerance on B is plus or minus 0.005 inch.

Copyright 2020, Industrial Press, Inc.

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Machinery's Handbook, 31st Edition Mounted Wheels and Points

1041

Mounted Wheels and Mounted Points These wheels and points are used in places that are hard to access and are available with a vitri­fied bond. The wheels are available with aluminum oxide or silicon carbide abrasive grains. The aluminum oxide wheels are used to grind tough and tempered die steels and the silicon carbide wheels, cast iron, chilled iron, bronze, and other nonferrous metals. The illustrations on pages 1041 and 1042 give the standard shapes of mounted wheels and points as published by the Grinding Wheel Institute. A note about the maximum operating speed for these wheels is given at the bottom of the first page of illustrations. Metric sizes are given on page 1043. 3” 4

5” 8

1” 2

5” 8

B-41

B-42 3” 8

1” 2

B-62

1” 2

B-131

7” 32

B-44

1” 8

3” 16

5” 8

3” 4

B-81

5” 16

B-104 1” 2

1” 4

3” 8

3” 16

B-103

3” 8

B-43

B-71

5” 8

5” 16

1” 2

3” 8

B-132

7” 16

3” 8

5” 16

B-52

1” 2

B-91

B-53

3” 4

B-61 3” 8

1” 4

1” 4

1” 4

1” 8

1” 8

B-96

B-97

B-92

5” 16

11” 16 5” 8

B-101

1” 2

1” 2

3” 8

B-112 1” 2

3” 8

B-133

5” 8

5” 8

7” 16

3” 8

3” 4

B-51

11” 16

B-111

3” 4

1” 4

B-135

B-121

3” 16

3” 8

B-122

B-123

1” 8

B-124

T

T

T

T

D

D

D

D

Group W

Fig. 1a. Standard Shapes and Sizes of Mounted Wheels and Points ANSI B74.2-2003 See Table 1 for inch sizes of Group W shapes and for metric sizes for all shapes.

The maximum speeds of mounted vitrified wheels and points of average grade range from about 38,000 to 152,000 rpm for diameters of 1 inch down to 1 ⁄4 inch. However, the safe operating speed usually is limited by the critical speed (speed at which vibration or whip tends to become excessive), which varies according to wheel or point dimensions, spindle diameter, and overhang.

Copyright 2020, Industrial Press, Inc.

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Machinery's Handbook, 31st Edition Mounted Wheels and Points

1042

2

2

1” 2

3” 4

1

3” 4

1”

A-1

1

A-3

1

1” 4

1” 8

1

A-12

1

1” 4

A-13

1

11” 16

A-14

1” 16

1” 4

A-15

5” 8

A-25

1

A-26

1”

A-35

1

3” 8

1”

3” 4

A-23

1” 4

A-24

3” 8

1

A-32

1” 4

5” 8

1

A-37

3” 4

1”

3” 8

A-36

1”

5” 8

A-31

3” 8

A-11

1”

A-21

1”

1”

7” 8

A-5

7” 8

1” 8

2”

1” 8 3” 4

A-4

1

11” 16

1” 4

1” 2

A-34

1”

1” 4

3” 4 3” 4

1”

A-38

A-39

Fig. 1b. Standard Shapes and Sizes of Mounted Wheels and Points ANSI B74.2-2003

Copyright 2020, Industrial Press, Inc.

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Machinery's Handbook, 31st Edition Mounted Wheels and Points

1043

Table 1. Shapes and Sizes of Mounted Wheels and Points ANSI B74.2-2003 Abrasive Shape Size Diameter Thickness mm mm 20 65 22 70 30 30 20 28 21 45 18 30 25 25 18 22 6 25 25 25 20 25 16 16 13 20 6 8 5.6 10 11 20 10 20 8 16 20 8 13 10 16 3 20 5 13 16 6 6 3 6

Abrasive Shape No.a A1 A3 A4 A5 A 11 A 12 A 13 A 14 A 15 A 21 A 23 B 41 B 42 B 43 B 44 B 51 B 52 B 53 B 61 B 62 B 71 B 81 B 91 B 92 B 96

Abrasive Shape No.a W 144

D mm 3

W 146

3

W 145

3

W 152

5

W 154

5

W 153 W 158 W 160

W 162

5 6 6

6

Abrasive Shape Size T D mm inch 6 1 ⁄8 10 13 6

10 13 3 6

10

W 163

6

13

W 174

10

6

W 164

W 175 W 176 W 177

6

10 10 10

W 178

10

W 181

13

W 179 W 182 W 183 W 184 W 185

W 186

W 187 W 188

W 189

W 195

10 13 13 13 13

13

13 13

13

16

20

10 13 20

25

30

1.5 3 6

10 13

20

25 40

50

20

T inch 1 ⁄4

1 ⁄8

3 ⁄8

1 ⁄8

1 ⁄2

3 ⁄16

1 ⁄4

3 ⁄16

3 ⁄8

3 ⁄16

1 ⁄2

1 ⁄4

1 ⁄8

1 ⁄4

1 ⁄4

1 ⁄4

3 ⁄8

1 ⁄4

1 ⁄2

1 ⁄4

3 ⁄4

3 ⁄8

1 ⁄4

3 ⁄8

3 ⁄8

3 ⁄8

1 ⁄2

3 ⁄8

3 ⁄4

3 ⁄8

1

3 ⁄8

11 ⁄4

1 ⁄2

1 ⁄16

1 ⁄2

1 ⁄8

1 ⁄2

1 ⁄4

1 ⁄2

3 ⁄8

1 ⁄2

1 ⁄2

1 ⁄2

3 ⁄4

1 ⁄2 1 ⁄2

1

11 ⁄2

1 ⁄2

2

5 ⁄8

3 ⁄4

A 24 A 25 A 26 A 31 A 32 A 34 A 35 A 36 A 37 A 38 A 39 B 97 B 101 B 103 B 104 B 111 B 112 B 121 B 122 B 123 B 124 B 131 B 132 B 133 B 135

Abrasive Shape No.a W 196

D mm 16

W 200

20

W 197

16

W 201

20

W 202

20

W 203

20

W 204

20

W 205

20

W 207

20

W 208

20

W 215

25

W 216

25

W 217

25

W 218

25

W 220

25

W 221

25

W 222

25

W 225

30

W 226 W 228 W 232

3 6

10 13 20 25 40 50 3 6

10 13 25

40

50 6

30

30

40

W 236

50

10

30

W 235

Abrasive Shape Size T D mm inch 26 5 ⁄8

30 30

W 230

40

20 50 6

13

W 237

40

25

W 242

50

25

W 238

40

a See shape diagrams in Fig. 1a and Fig. 1b on pages 1041 and 1042.

Copyright 2020, Industrial Press, Inc.

Abrasive Shape Size Diameter Thickness mm mm 6 20 25 … … 16 35 26 25 20 38 10 25 10 40 10 30 6 25 25 20 20 3 10 16 18 16 5 8 10 11 18 10 13 13 … 10 … 5 … 3 … 13 13 10 13 10 10 6 13

Abrasive Shape No.a

40

T inch 1

5 ⁄8

2

3 ⁄4

1 ⁄8

3 ⁄4

1 ⁄4

3 ⁄4

3 ⁄8

3 ⁄4

1 ⁄2

3 ⁄4

3 ⁄4

3 ⁄4 3 ⁄4 3 ⁄4

1 1 1

1

11 ⁄2 2 1 ⁄8 1 ⁄4 3 ⁄8

1

1 ⁄2

1

1

11 ⁄2 2

11 ⁄4

3 ⁄8

11 ⁄4 11 ⁄4

11 ⁄4 2

11 ⁄2

1 ⁄2

11 ⁄2 2

11 ⁄2 1

1

11 ⁄4 11 ⁄4

11 ⁄2 11 ⁄2

1

1 ⁄4 3 ⁄4

1 ⁄4

1

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1044

Machinery's Handbook, 31st Edition Broaches and Broaching BROACHES AND BROACHING The Broaching Process

The broaching process may be applied in machining holes or other internal surfaces and also to many flat or other external surfaces. Internal broaching is applied in forming either symmetrical or irregular holes, grooves, or slots in machine parts, especially when the size or shape of the opening, or its length in proportion to diameter or width, makes other machining processes impracticable. Broaching originally was utilized for such work as cutting keyways; machining round holes into square, hexagonal, or other shapes; forming splined holes; and for a large variety of other internal operations. The development of broaching machines and broaches finally resulted in extensive application of the process to external, flat, and other surfaces. Most external or surface broaching is done on machines of vertical design, but horizontal machines are also used for some classes of work. The broaching process is very rapid and accurate, and it leaves a finish of good quality. It is employed extensively in automotive and other plants where duplicate parts must be pro­duced in large quantities and for dimensions within small tolerances. Types of Broaches.—A number of typical broaches and the operations for which they are intended are shown by the diagrams in Fig. 1. Broach A produces a round-cornered, square hole. Prior to broaching square holes, it is usually the practice to drill a round hole having a diameter d somewhat larger than the width of the square. Hence, the sides are not com­pletely finished, but this unfinished part is not objectionable in most cases. In fact, this clearance space is an advantage during the broaching operation in that it serves as a chan­nel for the broaching lubricant; moreover, the broach has less metal to remove. Broach B is for finishing round holes. Broaching is superior to reaming for some classes of work because the broach will hold its size for a much longer period, thus insuring greater accu­racy. Broaches C and D are for cutting single and double keyways, respectively. Broach C is of rectangular section and, when in use, slides through a guiding bushing which is inserted in the hole. Broach E is for forming four integral splines in a hub. The broach at F is for producing hexagonal holes. Rectangular holes are finished by broach G. The teeth on the sides of this broach are inclined in opposite directions, which has the following advan­tages: The broach is stronger than it would be if the teeth were opposite and parallel to each other; thin work cannot drop between the inclined teeth, as it tends to do when the teeth are at right angles, because at least two teeth are always cutting; the inclination in opposite directions neutralizes the lateral thrust. The teeth on the edges are staggered, the teeth on one side being midway between the teeth on the other edge, as shown by the dotted line. A double-cut broach is shown at H. This type is for finishing, simultaneously, both sides f of a slot, and for similar work. Broach I is the style used for forming the teeth in internal gears. It is practically a series of gearshaped cutters, the outside diameters of which gradually increase toward the finishing end of the broach. Broach J is for round holes but differs from style B in that it has a continuous helical cutting edge. Some prefer this form because it gives a shearing cut. Broach K is for cutting a series of helical grooves in a hub or bushing. In helical broaching, either the work or the broach is rotated to form the helical grooves as the broach is pulled through. In addition to the typical broaches shown in Fig. 1, many special designs are now in use for performing more complex operations. Two surfaces on opposite sides of a casting or forging are sometimes machined simultaneously by twin broaches and, in other cases, three or four broaches are drawn through a part at the same time, for finishing as many duplicate holes or surfaces. Notable developments have been made in the design of broaches for external or “surface” broaching.

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Machinery's Handbook, 31st Edition Broaches and Broaching

1045

d

Square A

Round B

Single Keyway C

Double Keyway D

4-Spline E

Hexagon F

Round J

Helical Groove K

f f

Rectangular G

Double-Cut H

Internal Gear I

Fig. 1. Types of Broaches

Burnishing Broach: This is a broach having teeth or projections that are rounded on the top instead of being provided with a cutting edge, as in the ordinary type of broach. The teeth are highly polished, the tool being used for broaching bearings and for operations on other classes of work where the metal is relatively soft. The tool compresses the metal, thus making the surface hard and smooth. The amount of metal that can be displaced by a smooth-toothed burnishing broach is about the same as that removed by reaming. Such broaches are primarily intended for use on babbitt, white metal, and brass, but may also be satisfactorily used for producing a glazed surface on cast iron. This type of broach is also used when it is only required to accurately size a hole. Pitch of Broach Teeth.—The pitch of broach teeth depends upon the depth of cut or chip thickness, the length of cut, the cutting force required and the power of the broaching machine. In the pitch formulas that follow L = length, in inches (mm), of layer to be removed by broaching d = depth of cut per tooth as shown by Table 1 (For internal broaches, d = depth of cut as measured on one side of broach or one-half difference in diameters of successive teeth in the case of a round broach.) F = a factor (For brittle types of material, F = 3 or 4 for roughing teeth and 6 for finishing teeth. For ductile types of material, F = 4 to 7 for roughing teeth and 8 for finishing teeth.) b = width in inches (mm) of layer to be removed by broaching P = pressure required in tons per square inch (MPa) of an area equal to depth of cut times width of cut, in inches (mm) (Table 2) T = usable capacity, in tons (metric tons), of broaching machine = 70 percent of maxi­mum tonnage

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Machinery's Handbook, 31st Edition Broaches and Broaching

1046

Table 1. Designing Data for Surface Broaches Depth of Cut per Tooth Material to be Broached

Steel, High Tensile Strength Steel, Med.Tensile Strength

inch

0.0015–0.002

Cast Steel

0.0025–0.005 0.0025–0.005

Cast Iron, Soft

0.006–0.010

Malleable Iron

0.0025–0.005

Cast Iron, Hard

0.003–0.005

Zinc Die Castings

0.005–0.010

Cast Bronze

0.010–0.025

Wrought Aluminum Alloys

0.005–0.010

Cast Aluminum Alloys

0.005–0.010

Magnesium Die Castings

0.010–0.015

Face Angle or Rake, Degrees

Rough

Finish

0.013

14–18 10

1.5–3 1.53

0.5–1 0.5

0.013

10–15

0.025

12b

0.025

15b

0.025

20b

Finishing

Roughinga mm

inch

0.04–0.05

0.06–0.13

0.0005 0.0005

0.15–0.25

0.0005

0.13–0.25

0.0010

0.13–0.25

0.0010

0.25–0.38

0.0010

0.06–0.13

0.06–0.13

0.013

0.013

0.0005

0.08–0.13

0.0005

0.25–0.64

0.0005

0.13–0.25

mm

0.0005

0.013 0.013 0.013

0.0010

0.025

Clearance Angle, Degrees

10–12

1.5–3

7

0.5–1

1.5–3

0.5

1.5–3

5

0.5

1.5–3

0.5

5

2

8

0

0

12b

3

1

3

1

3

1

a The lower depth-of-cut values for roughing are recommended when work is not very rigid, the

tolerance is small, a good finish is required, or length of cut is comparatively short. b In broaching these materials, smooth surfaces for tooth and chip spaces are especially recommended.

Table 2. Broaching Pressure P for Use in Pitch Formulas (2a) and (2b) Depth d of cut per tooth, inch (mm) 0.01 (0.25)

0.004 (0.10) 0.002 (0.05)

0.001 (0.025)

…

…

…

…

…

…

…

…

…

…

…

…

…

…

…

…

…

…

…

35

483

…

…

Malleable Iron

Cast Iron

Cast Brass

Brass, Hot-pressed Zinc Die Castings

Cast Bronze

Wrought Aluminum Cast Aluminum

Magnesium Alloy

…

…

35

483

115 1586 50 85 70

35

70 85

35

689

1172 965

483

965

1172 483

158 128 108 115

2179 1765 1489

250

3447

312

4302

200 2758 0.004 0.10

185 158 128

2179 1765

243

3351

…

…

…

…

mm

143 1972 0.006 0.15

115 1586 0.006 0.15

100 1379 0.006 0.15

1972

…

…

…

…

…

…

…

…

…

…

…

…

…

…

…

…

…

965

…

…

…

…

…

…

…

…

…

…

…

…

…

…

…

…

…

…

70

965

…

…

…

…

…

…

…

…

85

70

689

1172

143

2551

inch

…

50

1586

Ton/in2

…

Steel, Med. Tensile Strength

Cast Steel

MPa

…

Ton/in2

…

Cut, d

MPa

MPa

…

Side-cutting Broaches Pressure, P

Ton/in2

Ton/in2

…

MPa

MPa

Steel, High Tensile Strength

Ton/in2

Material to be Broached

Ton/in2

Pressure, P

MPa

0.024 (0.60)

115 1586 0.020 0.51

85

1172

…

…

…

…

…

…

…

…

…

…

…

…

…

…

…

…

…

…

The minimum pitch shown by Formula (1) is based upon the receiving capacity of the chip space. The minimum pitch should not be less than 0.2 inch (5.0 mm) unless a smaller pitch is required for extremely short cuts to provide at least two teeth in contact simultane­ously with the part being broached. A reduction below 0.2 inch (5.0 mm) is seldom required in surface broaching but may be necessary in connection with internal broaching. Minimum pitch = 3 LdF (1)

Whether the minimum pitch may be used or not depends upon the power of the available machine. The factor F in the formula provides for the increase in volume as the material is broached into chips. If a broach has adjustable inserts for the finishing teeth, the pitch of the finishing teeth may be smaller than the pitch of the roughing teeth because of the smaller depth d of the cut. The higher value of F for finishing teeth prevents the pitch from becom­ing too small, so that the spirally curled chips will not be crowded into too small a

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Machinery's Handbook, 31st Edition Broaches and Broaching

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space. The pitch of the roughing and finishing teeth should be equal for broaches without separate inserts (notwithstanding the different values of d and F) so that, after wear makes this necessary, some of the finishing teeth may be ground into roughing teeth. US Units

Allowable pitch =

dLbP T inch (2a)

Metric Units

dLbP Allowable pitch = 9810T mm (2b)

If the pitch obtained by Formula (2a), or Formula (2b) in metric calculations, is larger than the minimum obtained by Formula (1), this larger value should be used because it is based upon the usable power of the machine. As the notation indicates, 70 percent of the maximum tonnage T is taken as the usable capacity. The 30 percent reduction is to provide a margin for the increase in broaching load resulting from the gradual dulling of the cutting edges. The procedure in calculating both minimum and allowable pitches will be illus­trated by an example. Example: Determine pitch of broach for cast iron if L = 220 mm; d = 0.1 mm; and F = 4.

Minimum pitch = 3 220 × 0.1 × 4 = 28.14 mm

Next, apply Formula (2b). Assume that b = 75 mm and T = 8 metric ton; for cast iron and depth d of 0.1 mm, P = 1586 MPa (Table 2). Then,

Allowable pitch =

0.1 × 220 × 75 = 1586 = 33.34 mm 9810 ^8h

This pitch is safely above the minimum. If in this case the usable tonnage of an available machine were, say, 7 metric tons instead of 8 metric tons, the pitch as shown by Formula (2b) might be increased to about 38.1 mm, thus reducing the number of teeth cutting simul­ taneously and, consequently, the load on the machine; or the cut per tooth might be reduced instead of increasing the pitch, especially if only a few teeth are in cutting contact, as might be the case with a short length of cut. If the usable tonnage in the preceding example were, say, 10 metric tons, then a pitch of 26.68 mm would be obtained by Formula (2b); hence, the pitch in this case should not be less than the minimum of approximately 28.14 mm obtained from Formula (1).

Depth of Cut per Tooth.—The term “depth of cut” as applied to surface or external broaches means the difference in the heights of successive teeth. This term, as applied to internal broaches for round, hexagonal or other holes, may indicate the total increase in the diameter of successive teeth; however, to avoid confusion, the term as here used means in all cases and regardless of the type of broach, the depth of cut as measured on one side. In broaching free-cutting steel, the Broaching Tool Institute recommends 0.003 to 0.006 inch (0.076–0.15 mm) depth of cut for surface broaching; 0.002 to 0.003 inch (0.05–0.076 mm) for multispline broaching; and 0.0007 to 0.0015 inch (0.018–0.038 mm) for round hole broaching. The accompanying table contains data from a German source and applies specifically to surface broaches. All data relating to depth of cut are intended as a general guide only. While depth of cut is based primarily upon the machinability of the material, some reduction from the depth thus established may be required, particularly when the work-supporting fixture in surface broaching is not sufficiently rigid to resist the thrust from the broaching operation. In some cases, the pitch and cutting length may be increased to reduce the thrust force. Another possible remedy in surface broaching certain classes of work is to use a side-cutting broach instead of the ordinary depth-cutting type. A broach designed for side-cutting takes relatively deep narrow cuts that extend nearly to the full depth required. The side-cutting section is followed by teeth arranged for depth cutting to obtain the required size and surface finish on the work. In general, small tolerances in sur­face broaching require a reduced cut per tooth to minimize

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1048

work deflection resulting from the pressure of the cut. See Cutting Speed for Broaching starting on page 1142 for broach­ing speeds. Face Angle

Pitch

Clearance Angle

Land

Depth Radius Terms Commonly Used in Broach Design

Face Angle or Rake.—The face angle (see diagram) of broach teeth affects the chip flow and varies considerably for different materials. While there are some variations in practice, even for the same material, the angles given in the accompanying table are believed to rep­resent commonly used values. Some broach designers increase the rake angle for finishing teeth in order to improve the finish on the work.

Clearance Angle.—The clearance angle (see illustration) for roughing steel varies from 1.5 to 3 degrees and for finishing steel from 0.5 to 1 degree. Some recommend the same clearance angles for cast iron, and others larger clearance angles varying from 2 to 4 or 5 degrees. Additional data will be found in Table 1.

Land Width.—The width of the land usually is about 0.25 × pitch. It varies, however, from about one-fourth to one-third of the pitch. The land width is selected so as to obtain the proper balance between tooth strength and chip space.

Depth of Broach Teeth.—The tooth depth as established experimentally and on the basis of experience usually varies from about 0.37 to 0.40 of the pitch. This depth is measured radially from the cutting edge to the bottom of the tooth fillet.

Radius of Tooth Fillet.—The “gullet” or bottom of the chip space between the teeth should have a rounded fillet to strengthen the broach, facilitate curling of the chips, and safeguard against cracking in connection with the hardening operation. One rule is to make the radius equal to one-fourth the pitch. Another is to make it equal 0.4 to 0.6 the tooth depth. A third method preferred by some broach designers is to make the radius equal one-third of the sum obtained by adding together the land width, one-half the tooth depth, and one-fourth of the pitch. Total Length of Broach.—After the depth of cut per tooth has been determined, the total amount of material to be removed by a broach is divided by this decimal to ascertain the number of cutting teeth required. This number of teeth multiplied by the pitch gives the length of the active portion of the broach. By adding to this dimension the distance over three or four straight teeth, the length of a pilot to be provided at the finishing end of the broach, and the length of a shank which must project through the work and the faceplate of the machine to the draw-head, the overall length of the broach is found. This calculated length is often greater than the stroke of the machine, or greater than is practical for a broach of the diameter required. In such cases, a set of broaches must be used.

Chipbreakers.—The teeth of broaches frequently have rounded chipbreaking grooves located at intervals along the cutting edges. These grooves break up wide curling chips and prevent them from clogging the chip spaces, thus reducing the cutting pressure and strain on the broach. These chipbreaking grooves are on the roughing teeth only. They are stag­gered and applied to both round and flat or surface broaches. The grooves are formed by a round-edged grinding wheel and usually vary in width from about 1 ⁄32 to 3 ⁄32 inch (0.79 to 2.38 mm) depending upon the size of broach. The more ductile the material, the wider the chipbreaker grooves should be and the smaller the distance between

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them. Narrow slotting broaches may have the right- and left-hand corners of alternate teeth beveled to obtain chipbreaking action. Shear Angle.—The teeth of surface broaches ordinarily are inclined so they are not at right angles to the broaching movement. The object of this inclination is to obtain a shear­ ing cut, which results in smoother cutting action and an improvement in surface finish. The shearing cut also tends to eliminate troublesome vibration. Shear angles for surface broaches are not suitable for broaching slots or any profiles that resist the outward move­ ment of the chips. When the teeth are inclined, the fixture should be designed to resist the resulting thrusts unless it is practicable to incline the teeth of right- and left-hand sections in opposite directions to neutralize the thrust. The shear angle usually varies from 10 to 25 degrees. Types of Broaching Machines.—Broaching machines may be divided into horizontal and vertical designs, and classified further according to the method of opera­tion, as, for example, whether a broach in a vertical machine is pulled up or pulled down in forcing it through the work. Horizontal machines usually pull the broach through the work in internal broaching, but short rigid broaches may be pushed through. External surface broaching is also done on some machines of horizontal design, but usually vertical machines are employed for flat or other external broaching. Although parts usually are broached by traversing the broach itself, some machines are designed to hold the broach or broaches stationary during the actual broaching operation. This principle has been applied both to internal and surface broaching. Vertical Duplex Type: The vertical duplex type of surface-broaching machine has two slides or rams that move in opposite directions and operate alternately. While the broach connected to one slide is moving downward on the cutting stroke, the other broach and slide is returning to the starting position, and this returning time is utilized for reloading the fixture on that side; consequently, the broaching operation is practically continuous. Each ram or slide may be equipped to perform a separate operation on the same part when two operations are required. Pull-up Type: Vertical hydraulically-operated machines that pull the broach or broaches up through the work are used for internal broaching of holes of various shapes and for broaching bushings, splined holes, small internal gears, etc. A typical machine of this kind is so designed that all broach handling is done automatically. Pull-down Type: The various movements in the operating cycle of a hydraulic pull-down type of machine equipped with an automatic broach-handling slide are the reverse of the pull-up type. The broaches for a pull-down type of machine have shanks on each end, an upper one for the broach-handling slide and a lower one for pulling through the work. Hydraulic Operation: Modern broaching machines, as a general rule, are operated hydraulically rather than by mechanical means. Hydraulic operation is efficient, flexible in the matter of speed adjustments, low in maintenance cost, and the “smooth” action required for fine precision finishing may be obtained. The hydraulic pressures required, which frequently are 800 to 1000 pounds per square inch (5.5 to 6.9 MPa), are obtained from a motor-driven pump forming part of the machine. The cutting speeds of broaching machines frequently are between 20 and 30 feet per minute (6.1 to 9.1 m/ min), and the return speeds often are double the cutting speed, or higher, to reduce the idle period. Ball-Broaching.—Ball-broaching is a method of securing bushings, gears, or other com­ponents without the need for keys, pins, or splines. A series of axial grooves, separated by ridges, is formed in the bore of the workpiece by cold plastic deformation of the metal when a tool, having a row of three rotating balls around its periphery, is pressed through the parts. When the bushing is pressed into a broached bore, the ridges displace the softer material of the bushing into the grooves, thus securing the assembly. The balls can be made of high-carbon chromium steel or carbide, depending on the hardness of the component.

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Broaching Difficulties.—The accompanying table has been compiled from information supplied by the National Broach and Machine Co. and presents some of the common broaching difficulties, their causes and means of correction. Causes of Broaching Difficulties

Broaching Difficulty

Possible Causes

Stuck broach

Insufficient machine capacity; dulled teeth; clogged chip gullets; failure of power during cutting stroke. To remove a stuck broach, remove workpiece and broach from the machine as a unit; never try to back out broach by reversing machine. If broach does not loosen by tapping workpiece lightly and trying to slide it off its starting end, mount workpiece and broach in a lathe and turn down work­piece to the tool surface. Workpiece may be sawed longitudinally into sev­eral sections in order to free the broach. Check broach design; perhaps tooth relief (back off) angle is too small or depth of cut per tooth is too great.

Galling and pickup

Lack of homogeneity of material being broached—uneven hardness, porosity; improper or insufficient coolant; poor broach design, mutilated broach; dull broach; improperly sharpened broach; improperly designed or outworn fixtures. Good broach design will do away with possible chip build-up on tooth faces and excessive heating. Grinding of teeth should be accurate so that the correct gullet contour is maintained. Contour should be fair and smooth.

Broach breakage

Overloading; broach dullness; improper sharpening; interrupted cutting stroke; backing up broach with workpiece in fixture; allowing broach to pass entirely through guide hole; ill fitting and/or sharp-edged key; crooked holes; untrue locating surface; excessive hardness of workpiece; insufficient clearance angle; sharp corners on pull end of broach. When grinding bevels on pull end of broach, use wheel that is not too pointed.

Chatter

Too few teeth in cutting contact simultaneously; excessive hardness of material being broached; loose or poorly constructed tooling; surging of ram due to load variations. Chatter can be alleviated by changing the broaching speed, by using shear cutting teeth instead of right-angle teeth, and by changing the coolant and the face and relief angles of the teeth.

Drifting or misalignment of tool during cutting stroke

Lack of proper alignment when broach is sharpened in grinding machine, which may be caused by dirt in the female center of the broach; inadequate support of broach during the cutting stroke, on a horizontal machine espe­cially; body diameter too small; cutting resistance variable around I.D. due to lack of symmetry of surfaces to be cut; variations in hardness around I.D.; too few teeth in cutting contact.

Streaks in broached surface

Lands too wide; presence of forging, casting or annealing scale; metal pickup; presence of grinding burrs and grinding and cleaning abrasives.

Rings in the broached hole

Surging resulting from uniform pitch of teeth; presence of sharpen­ing burrs on broach; tooth clearance angle too large; locating face not smooth or square; broach not supported for all cutting teeth passing through the work. The use of differential tooth spacing or shear cutting teeth helps in preventing surging. Sharpening burrs on a broach may be removed with a wood block.

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Machinery's Handbook, 31st Edition Files and Burs

1051

FILES AND BURS Files

Definitions of File Terms.—The following file terms apply to hand files but not to rotary files and burs. Axis: Imaginary line extending the entire length of a file equidistant from faces and edges. Back: The convex side of a file having the same or similar cross section as a half-round file. Bastard Cut: A grade of file coarseness between coarse and second cut of American pat­tern files and rasps. Blank: A file in any process of manufacture before being cut. Blunt: A file whose cross-sectional dimensions from point to tang remain unchanged. Coarse Cut: The coarsest of all American pattern file and rasp cuts. Coarseness: Term describing the relative number of teeth per unit length, the coarsest having the least number of file teeth per unit length; the smoothest, the most. American pattern files and rasps have four degrees of coarseness: coarse, bastard, second and smooth. Swiss pattern files usually have seven degrees of coarseness: 00, 0, 1, 2, 3, 4, 6 (from coarsest to smoothest). Curved-tooth files have three degrees of coarseness: stan­ dard, fine and smooth. Curved Cut: File teeth that are made in curved contour across the file blank. Cut: Term used to describe file teeth with respect to their coarseness or their character (single, double, rasp, curved, special). Double Cut: A file tooth arrangement formed by two series of cuts, namely the overcut followed, at an angle, by the upcut. Edge: Surface joining faces of a file. May have teeth or be smooth. Face: Widest cutting surface or surfaces that are used for filing. Heel or Shoulder: That portion of a file that abuts the tang. Hopped: A term used among file makers to represent a very wide skip or spacing between file teeth. Length: The distance from the heel to the point. Overcut: The first series of teeth put on a double-cut file. Point: The front end of a file; the end opposite the tang. Rasp Cut: A file tooth arrangement of round-topped teeth, usually not connected, that are formed individually by means of a narrow, punch-like tool. Re-cut: A worn-out file that has been re-cut and re-hardened after annealing and grinding off the old teeth. Safe Edge: An edge of a file that is made smooth or uncut, so that it will not injure that portion or surface of the workplace with which it may come in contact during filing. Second Cut: A grade of file coarseness between bastard and smooth of American pattern files and rasps. Set: To blunt the sharp edges or corners of file blanks before and after the overcut is made, in order to prevent weakness and breakage of the teeth along such edges or corners when the file is put to use. Shoulder or Heel: See Heel or Shoulder. Single Cut: A file tooth arrangement where the file teeth are composed of single unbro­ ken rows of parallel teeth formed by a single series of cuts. Smooth Cut: An American pattern file and rasp cut that is smoother than second cut. Tang: The narrowed portion of a file which engages the handle. Upcut: The series of teeth superimposed on the overcut, and at an angle to it, on a doublecut file.

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Machinery's Handbook, 31st Edition Files and Burs

File Characteristics.—Files are classified according to their shape or cross section and according to the pitch or spacing of their teeth and the nature of the cut. Cross Section and Outline: The cross section may be quadrangular, circular, triangular, or some special shape. The outline or contour may be tapered or blunt. In the former, the point is more or less reduced in width and thickness by a gradually narrowing section that extends for one-half to two-thirds of the length. In the latter the cross section remains uni­form from tang to point. Cut: The character of the teeth is designated as single, double, rasp or curved. The singlecut file (or float as the coarser cuts are sometimes called) has a single series of parallel teeth extending across the face of the file at an angle of from 45 to 85 degrees with the axis of the file. This angle depends upon the form of the file and the nature of the work for which it is intended. The single-cut file is customarily used with a light pressure to produce a smooth finish. The double-cut file has a multiplicity of small pointed teeth inclining toward the point of the file arranged in two series of diagonal rows that cross each other. For general work, the angle of the first series of rows is from 40 to 45 degrees and of the second from 70 to 80 degrees. For double-cut finishing files, the first series has an angle of about 30 degrees, and the second from 80 to 87 degrees. The second, or upcut, is almost always deeper than the first or overcut. Double-cut files are usually employed, under heavier pressure, for fast metal removal and where a rougher finish is permissible. The rasp is formed by raising a series of individual rounded teeth from the surface of the file blank with a sharp, narrow, punch-like cutting tool and is used with a relatively heavy pressure on soft substances for fast removal of material. The curved-tooth file has teeth that are in the form of parallel arcs extending across the face of the file, the middle portion of each arc being closest to the point of the file. The teeth are usually single cut and are relatively coarse. They may be formed by steel displacement but are more commonly formed by milling. With reference to coarseness of cut, the terms coarse, bastard, second and smooth cuts are used, the coarse or bastard files for the heavier classes of work and the second- or smooth-cut files for the finishing or more exacting work. These degrees of coarseness are only comparable when files of the same length are compared, as the number of teeth per inch of length decreases as the length of the file increases. The number of teeth per inch varies considerably for different sizes and shapes and for files of different makes. The coarseness range for the curved-tooth files is given as standard, fine and smooth. In the case of Swiss pattern files, a series of numbers is used to designate coarseness instead of names; Nos. 00, 0, 1, 2, 3, 4 and 6 being the most common with No. 00 the coarsest and No. 6 the finest. Classes of Files.—There are five main classes of files: mill or saw files; machinists’ files; curved-tooth files; Swiss pattern files; and rasps. The first two classes are commonly referred to as American pattern files. Mill or Saw Files: These are used for sharpening mill or circular saws, and large crosscut saws; for lathe work; for draw filing; for filing brass and bronze; and for smooth filing gen­erally. The number identifying the following files refers to the illustration in Fig. 1. 1) Cantsaw files have an obtuse isosceles triangular section, a blunt outline, are single cut and are used for sharpening saws having M-shaped teeth and teeth of less than 60-degree angle; 2) Crosscut files have a narrow triangular section with the short side rounded and a blunt outline, are single cut and are used to sharpen crosscut saws. The rounded portion is used to deepen the gullets of saw teeth, and the sides are used to sharpen the teeth themselves; 3) Double ender files have a triangular section, are tapered from the middle to both ends, are tangless and single cut, and are used reversibly for sharpening saws; 4) The mill file itself is usually single cut, tapered in width, and often has two square cutting edges in addi­tion to the cutting sides. Either or both edges may be rounded, however, for filing the gul­lets of saw teeth. The blunt mill file has a uniform rectangular cross section from tip to tang; 5) The triangular saw files or taper saw files have an equilateral

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Machinery's Handbook, 31st Edition Files and Burs

1053

triangular section, are tapered, are single cut and are used for filing saws with 60-degree angle teeth. They come in taper, slim taper, extra-slim taper and double-extra-slim taper thicknesses. Blunt tri­angular and blunt hand-saw files are without taper; and 6) Web saw files have a diamond-shaped section, a blunt outline, are single cut and are used for sharpening pulpwood or web saws.

Machinists’ Files: These files are used throughout industry where metal must be removed rapidly and finish is of secondary importance. Except for certain exceptions in the round and half-round shapes, all are double cut. 7) Flat files have a rectangular section, are tapered in width and thickness, are cut on both sides and edges, and are used for general utility work; 8) Half-round files have a circular segmental section, are tapered in width and thickness, have their flat side double cut, their rounded side mostly double but sometimes single cut, and are used to file rounded holes, concave corners, etc., in general filing work; 9) Hand files are similar to flat files but taper in thickness only. One edge is uncut or “safe.”; and 10) Knife files have a “knife-blade” section, are tapered in width only, are double cut, and are used by tool and die makers on work having acute angles.

Machinist’s general purpose files have a rectangular section, are tapered and have single-cut teeth divided by angular serrations that produce short cutting edges. These edges help stock removal but still leave a smooth finish and are suitable for use on various mate­r ials, including aluminum, bronze, cast iron, malleable iron, mild steels and annealed tool steels.

11) Pillar files are similar to hand files but are thicker and not as wide; 12) Round files have a circular section, are tapered, single cut, and are generally used to file circular open­ings or curved surfaces; 13) Square files have a square section, are tapered, and are used for filing slots and keyways, and for general surface filing where a heavier section is preferred; 14) Three-square files have an equilateral triangular section and are tapered on all sides. They are double cut and have sharp corners, as contrasted with taper triangular files, which are single cut and have somewhat rounded corners. They are used for filing accurate inter­nal angles, for clearing out square corners, and for filing taps and cutters; and 15) Ward­ing files have a rectangular section, and taper in width to a narrow point. They are used for general narrow-space filing.

Wood files are made in the same sections as flat and half round files but with coarser teeth especially suited for working on wood.

1

2

9

4

3

10

11

6

5

12

7

14

13

8

15

Fig. 1. Styles of Mill or Saw Files

Curved-Tooth Files: Regular curved-tooth files are made in both rigid and flexible forms. The rigid type has either a tang for a conventional handle or is made plain with a hole at each end for mounting in a special holder. The flexible type is furnished for use in special holders only. Curved-tooth files come in standard fine and smooth cuts and in

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1054

Machinery's Handbook, 31st Edition Files and Burs

parallel flat, square, pillar, pillar narrow, half-round and shell types. A special curvedtooth file is available with teeth divided by long angular serrations. The teeth are cut in an “off center” arc. When moved across the work toward one edge of the file a fast cutting action is pro­vided; when moved toward the other edge, a smoothing action; thus the file is made to serve a dual purpose. Swiss Pattern Files: These are used by tool and die makers, model makers and delicate instrument parts finishers. They are made to closer tolerances than the conventional Amer­ican pattern files although with similar cross sections. The points of the Swiss pattern files are smaller, the tapers are longer, and they are available in much finer cuts. They are primar­ily finishing tools for removing burrs left from previous finishing operations truing up nar­row grooves, notches and keyways, cleaning out corners and smoothing small parts. For very fine work, round- and square-handled needle files, available in numerous cross-sec­tional shapes in overall lengths from 4 to 7 3⁄4 inches, are used. Die sinkers use die sinkers’ files and die sinkers’ rifflers. The files, also made in many different cross-sectional shapes, are 31 ⁄2 inches in length and are available in the cut Nos. 0, 1, 2, and 4. The rifflers are from 51 ⁄2 to 63⁄4 inches long, have cutting surfaces on either end, and come in numerous cross-sec­tional shapes in cut Nos. 0, 2, 3, 4 and 6. These rifflers are used by die makers for getting into corners, crevices, holes and contours of intricate dies and molds. Used in the same fashion as die sinkers’ rifflers, silversmiths’ rifflers, that have a much heavier cross section, are available in lengths from 6 7⁄8 to 8 inches and in cut Nos. 0, 1, 2, and 3. Blunt machine files in cut Nos. 00, 0, and 2 for use in ordinary and bench filing machines are available in many different cross-sectional shapes, in lengths from 3 to 8 inches. Rasps: Rasps are employed for work on relatively soft substances such as wood, leather, and lead where fast removal or material is required. They come in rectangular and half round cross sections, the latter with and without a sharp edge. Special Purpose Files: Falling under one of the preceding five classes of files, but modi­ fied to meet the requirements of some particular function, are a number of special-purpose files. The long-angle lathe file is used for filing work that is rotating in a lathe. The long tooth angle provides a clean shear, eliminates drag or tear and is self-clearing. This file has safe or uncut edges to protect shoulders of the work which are not to be filed. The foundry file has especially sturdy teeth with heavy-set edges for the snagging of castings—the removing of fins, sprues, and other projections. The die-casting file has extra-strong teeth on corners and edges as well as sides for working on die castings of magnesium, zinc, or aluminum alloys. A special file for stainless steel is designed to stand up under the abrasive action of stainless steel alloys. Aluminum rasps and files are designed to eliminate clog­ ging. A special tooth construction is used in one type of aluminum tile which breaks up the filings, allows the file to clear itself and overcomes chatter. A brass file is designed so that with a little pressure the sharp, high-cut teeth bite deep, while, with less pressure, their short uncut angle produces a smoothing effect. The lead float has coarse, single-cut teeth at almost right angles to the file axis. These shear away the metal under ordinary pressure and produce a smoothing effect under light pressure. The shear-tooth file has a coarse single cut with a long angle for soft metals or alloys, plastics, hard rubber and wood. Chainsaw files are designed to sharpen all types of chainsaw teeth. These files come in round, rectangular, square and diamond-shaped sections. The round- and square-sectioned files have either double- or single-cut teeth, the rectangular files have single-cut teeth, and the diamond-shaped files have double-cut teeth. Effectiveness of Rotary Files and Burs.—There it very little difference in the efficiency of rotary files or burs when used in electric tools and when used in air tools, provided the speeds have been reasonably well selected. Flexible-shaft and other machines used as a source of power for these tools have a limited number of speeds which govern the revolu­ tions per minute at which the tools can be operated.

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Machinery's Handbook, 31st Edition Files and Burs

1055

The carbide bur may be used on hard or soft materials with equally good results. The principle difference in construction of the carbide bur is that its teeth or flutes are provided with a negative rather than a radial rake. Carbide burs are relatively brittle and must be treated more carefully than ordinary burs. They should be kept cutting freely, in order to prevent too much pressure, which might result in crumbling of the cutting epics. At the same speeds, both high-speed steel and carbide burs remove approximately the same amount of metal. However, when carbide burs are used at their most efficient speeds, the rate of stock removal may be as much as four times that of ordinary burs. In certain cases, speeds much higher than those shown in the table can be used. It has been demon­ strated that a carbide bur will last up to 100 times as long as a high-speed steel bur of corre­sponding size and shape. Approximate Speeds of Rotary Files and Burs

Tool Diameter inches mm 1⁄ 8 3.2 1⁄ 4 6.4 3⁄ 8 9.5 1⁄ 2 12.7 5⁄ 8 15.9 3⁄ 4 19.1 7⁄ 8 22.2 1 25.4 1 1 ⁄8 28.6 11 ⁄4 31.8

Medium Cut, High-Speed Steel Bur or File Mild Steel 4600 3450 2750 2300 2000 1900 1700 1600 1500 1400

Cast Iron Bronze Aluminum Speed, Revolutions per Minute 7000 15,000 20,000 5250 11,250 15,000 4200 9000 12,000 3500 7500 10,000 3100 6650 8900 2900 6200 8300 2600 5600 7500 2400 5150 6850 2300 4850 6500 2100 4500 6000

Carbide Bur Medium Fine Magnesium Cut Cut Any Material 30,000 45,000 30,000 22,500 30,000 20,000 18,000 24,000 16,000 15,000 20,000 13,350 13,350 18,000 12,000 12,400 16,000 10,650 11,250 14,500 9650 10,300 13,000 8650 9750 … … 9000 … …

As recommended by the Nicholson File Company.

Steel Wool.—Steel wool is made by shaving thin layers of steel from wire. The wire is pulled, by special machinery built for the purpose, past cutting tools or through cutting dies that shave off chips from the outside. Steel wool consists of long, relatively strong, and resilient steel shavings having sharp edges. This characteristic renders it an excellent abra­sive. The fact that the cutting characteristics of steel wool vary with the size of the fiber, which is readily controlled in manufacture, has adapted it to many applications. Metals other than steel have been made into wool by the same processes as steel, and when so manufactured have the same general characteristics. Thus wool has been made from copper, lead, aluminum, bronze, brass, monel metal, and nickel. The wire from which steel wool is made may be produced by either the Bessemer, or the basic or acid open-hearth processes. It should contain from 0.10 to 0.20 percent carbon; from 0.50 to 1.00 percent manganese; from 0.020 to 0.090 percent sulphur; from 0.050 to 0.120 percent phos­phorus; and from 0.001 to 0.010 percent silicon. When drawn on a standard tensilestrength testing machine, a sample of the steel should show an ultimate strength of not less than 120,000 pounds per square inch (828 MPa). Steel Wool Grades

Description

Super Fine Extra Fine Very Fine Fine

Grade 0000 000 00 0

Fiber Thickness Inch Millimeter

0.001 0.0015 0.0018 0.002

0.025 0.035 0.04 0.05

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Description

Medium Medium Coarse Coarse Extra Coarse

Grade 1 2 3 4

Fiber Thickness Inch Millimeter

0.0025 0.003 0.0035 0.004

0.06 0.075 0.09 0.10

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Machinery's Handbook, 31st Edition Knurls and Knurling

1056

KNURLS AND KNURLING Standard Knurls and Knurling.—The ANSI/ASME Standard B94.6-1984 (R2014) cov­ ers knurling tools with standardized diametral pitches and their dimensional relations with respect to the work in the production of straight, diagonal, and diamond knurling on cylin­drical surfaces having teeth of uniform pitch parallel to the cylinder axis or at a helix angle not exceeding 45 degrees with the work axis. These knurling tools and the recommendations for their use are equally applicable to general purpose and precision knurling. The advantage of this ANSI/ASME Standard system is the provision by which good tracking (the ability of teeth to mesh as the tool penetrates the work blank in successive revolutions) is obtained by tools designed on the basis of diame­tral pitch instead of TPI (teeth per inch) when used with work blank diameters that are mul­tiples of 1 ⁄64 inch for 64 and 128 diametral pitch or 1 ⁄32 inch for 96 and 160 diametral pitch. The use of knurls and work blank diameters that will permit good tracking should improve the uniformity and appearance of knurling, eliminate costly trial and error methods, reduce the failure of knurling tools and the production of defective work, and decrease the number of tools required. Preferred sizes for cylindrical knurls are given in Table 1, and detailed specifications appear in Table 2. Table 1. ANSI Standard Preferred Sizes for Cylindrical Type Knurls ANSI/ASME B94.6-1984 (R2014)

Nominal Outside Diameter Dnt

Width of Face F

Diameter of Hole A

1 ⁄2

3 ⁄16

3 ⁄16

3 ⁄4

3 ⁄8

1 ⁄4

5 ⁄8

5 ⁄16

7⁄32

3 ⁄4

5 ⁄8

1 ⁄4

1 ⁄4

5 ⁄8

3 ⁄8

7⁄8

1

64

1 ⁄4

Standard Diametral Pitches, P 96 128

Number of Teeth, Nt , for Standard Pitches

32 40 48 56

48 60 72 84

64 80 96 112

80 100 120 140

40

60

80

100

1 ⁄4 Additional Sizes for Bench and Engine Lathe Tool Holders

48 64

5 ⁄16

3 ⁄8

160

72 96

96 128

120 160

The 96 diametral pitch knurl should be given preference in the interest of tool simplification. Dimensions D nt , F, and A are in inches.

Table 2. ANSI Standard Specifications for Cylindrical Knurls with Straight or Diagonal Teeth ANSI/ASME B94.6-1984 (R2014) Diame­ tral Pitch P

Nominal Diameter, Dnt 1 ⁄2

5 ⁄8

3 ⁄4

7⁄8

1

Tracking Correction Factor Q

Major Diameter of Knurl, Dot , +0.0000, -0.0015

Tooth Depth, h, + 0.0015, - 0.0000 Straight

Diagonal

64

0.4932

0.6165

0.7398

0.8631

0.9864

0.0006676

0.024

0.021

96

0.4960

0.6200

0.7440

0.8680

0.9920

0.0002618

0.016

0.014

128

0.4972

0.6215

0.7458

0.8701

0.9944

0.0001374

0.012

0.010

160

0.4976

0.6220

0.7464

0.8708

0.9952

0.00009425

0.009

0.008

Radius at Root R 0.0070 0.0050 0.0060 0.0040 0.0045 0.0030 0.0040 0.0025

All dimensions except diametral pitch are in inches. Approximate angle of space between sides of adjacent teeth for both straight and diagonal teeth is 80 degrees. The permissible eccentricity of teeth for all knurls is 0.002 inch maximum (total indica­tor reading). Number of teeth in a knurl equals diametral pitch multiplied by nominal diameter. Diagonal teeth have 30-degree helix angle, y.

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Machinery's Handbook, 31st Edition Knurls and Knurling

1057

The term Diametral Pitch applies to the quotient obtained by dividing the total number of teeth in the circumference of the work by the basic blank diameter; in the case of the knurl­ing tool, the diametrical pitch would be the total number of teeth in the circumference divided by the nominal diameter. In the Standard, the diametral pitch and number of teeth are always measured in a transverse plane that is perpendicular to the axis of rotation for diagonal as well as straight knurls and knurling.

Cylindrical Knurling Tools.—The cylindrical type of knurling tool is comprised of a tool holder and one or more knurls. The knurl has a centrally located mounting hole and is pro­ vided with straight or diagonal teeth on its periphery. The knurl is used to reproduce this tooth pattern on the work blank as the knurl and work blank rotate together. *Formulas for Cylindrical Knurls (1)

P = diametral pitch of knurl = Nt ÷ Dnt

Dnt = nominal diameter of knurl = Nt ÷ P

(2)

(3)

Nt = no. of teeth on knurl = P × Dnt

*Pnt = circular pitch on nominal diameter = π ÷ P

(4)

(5)

*Pot = circular pitch on major diameter = πDot ÷ Nt Dot = major diameter of knurl = Dnt − (Nt Q ÷ π)

(6)

Q = Pnt − Pot = tracking correction factor in Formula

(7)

Tracking Correction Factor Q: Use of the preferred pitches for cylindrical knurls, Table 2, results in good tracking on all fractional work-blank diameters that are multiples of 1 ⁄64 inch for 64 and 128 diametral pitch, and 1 ⁄32 inch for 96 and 160 diametral pitch; an indica­tion of good tracking is evenness of marking on the work surface during the first revolution of the work. The many variables involved in knurling practice require that an empirical correction method be used to determine what actual circular pitch is needed at the major diameter of the knurl to produce good tracking and the required circular pitch on the workpiece. The empirical tracking correction factor, Q, in Table 2 is used in the calculation of the major diameter of the knurl, Formula (6). Pot h R F

Dot

Ψ

Dnt A

Straight

Diagonal

45° Bevel to Depth of Teeth Optional Cylindrical Knurl * Note: For diagonal knurls, Pnt and Pot are the transverse circular pitches that are measured in the plane perpendicular to the axis of rotation.

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Machinery's Handbook, 31st Edition Knurls and Knurling

1058

Flat Knurling Tools.—The flat type of tool is a knurling die, commonly used in recipro­ cating types of rolling machines. Dies may be made with either single or duplex faces hav­ing either straight or diagonal teeth. No preferred sizes are established for flat dies. Flat Knurling Die with Straight Teeth: Linear Pitch = Pl Radius at Root = R



Tooth Depth = h

R = radius at root

(8)

P = diametral pitch = Nw ÷ Dw

(9) (10)

Dw = work blank (pitch) diameter = Nw ÷ P

Nw = number of teeth on work = P × Dw

h = tooth depth Q = tracking correction factor (see Table 2) Pl = linear pitch on die

= circular pitch on work pitch diameter = P − Q

Table 3. ANSI Standard Specifications for Flat Knurling Dies ANSI/ASME B94.6-1984 (R2014)

(11)

Tooth Depth, Tooth Depth, Radius Radius h h at Diametral Linear at Diametral Linear Root, Pitch, Pitch,a Root, Pitch, Pitch,a Pl Pl Straight Diagonal R P Straight Diagonal R P 64

0.0484

0.024

0.021

0.0070 0.0050

128

0.0244

0.012

0.010

0.0045 0.0030

96

0.0325

0.016

0.014

0.0060 0.0040

160

0.0195

0.009

0.008

0.0040 0.0025

a The linear pitches are theoretical. The exact linear pitch produced by a flat knurling die may vary slightly from those shown, depending upon the rolling condition and the material being rolled. All dimensions except diametral pitch are in inches.

Addendum = a

Work Blank Diameter = Dw

Tooth Depth = h

Knurled Diameter = Dow

Teeth on Knurled Work

Formulas Applicable to Knurled Work.—The following formulas are applicable to knurled work with straight, diagonal, and diamond knurling.

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Machinery's Handbook, 31st Edition Knurls and Knurling

1059

Formulas for Straight or Diagonal Knurling with Straight- or Diagonal-Tooth Cylindri­ cal Knurling Tools Set with Knurl Axis Parallel with Work Axis: (12) P = diametral pitch = N ÷ D w

w

Dw = work blank diameter = Nw ÷ P

(13)

(14)

Nw = no. of teeth on work = P × Dw

a = “addendum” of tooth on work = (Dow − Dw) ÷ 2

(15)

h = tooth depth (see Table 2)

Dow = knurled diameter (outside diameter after knurling) = Dw + 2a

(16)

Formulas for Diagonal and Diamond Knurling with Straight-Tooth Knurling Tools Set at an Angle to the Work Axis: y = angle between tool axis and work axis If, P = diametral pitch on tool Py = diametral pitch produced on work blank (as measured in the transverse plane) by setting tool axis at an angle y with respect to work blank axis D w = diameter of work blank; and Nw = number of teeth produced on work blank (as measured in the transverse plane) then, Pψ = P cos ψ (17)

and,

N = Dw P cos ψ (18) For example, if 30-degree diagonal knurling were to be produced on 1-inch diameter stock with a 160 pitch straight knurl: Nw = Dw Pcos30° = 1.000 # 160 # 0.86603 = 138.56 teeth

Good tracking is theoretically possible by changing the helix angle as follows to corre­ spond to a whole number of teeth (138): cosψ = Nw ' Dw P = 138 ' ^1 # 160h = 0.8625 1 ψ = 30 2 degrees, approximately Whenever it is more practical to machine the stock, good tracking can be obtained by reducing the work blank diameter as follows to correspond to a whole number of teeth (138): Nw 138 Dw = = = 0.996 inch Pcosψ 160 # 0.866 Table 4. ANSI Standard Recommended Tolerances on Knurled Diameters ANSI/ASME B94.6-1984 (R2014) Toler­ ance Class I II III

Diametral Pitch 128 160 64 96 128 96 Tolerance on Work-Blank Tolerance on Knurled Outside Diameter Diameter Before Knurling + 0.005 + 0.004 + 0.003 + 0.002 ± 0.0015 ± 0.0010 ± 0.0007 - 0.012 - 0.010 - 0.008 - 0.006 + 0.000 + 0.000 + 0.000 + 0.000 ± 0.0015 ± 0.0010 ± 0.0007 - 0.010 - 0.009 - 0.008 - 0.006 + 0.000 + 0.000 + 0.0000 + 0.000 + 0.000 + 0.000 + 0.000 - 0.006 - 0.005 - 0.004 - 0.003 - 0.0015 - 0.0010 - 0.0007 64

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160

± 0.0005 ± 0.0005 + 0.0000 - 0.0005

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Machinery's Handbook, 31st Edition Knurls and Knurling

1060

Recommended Tolerances on Knurled Outside Diameters.—The recommended applications of the tolerance classes shown in Table 4 are as follows: Class I: Tolerances in this classification may be applied to straight, diagonal and raised diamond knurling where the knurled outside diameter of the work need not be held to close dimensional tolerances. Such applications include knurling for decorative effect, grip on thumb screws, and inserts for moldings and castings. Class II: Tolerances in this classification may be applied to straight knurling only and are recommended for applications requiring closer dimensional control of the knurled outside diameter than provided for by Class I tolerances. Class III: Tolerances in this classification may be applied to straight knurling only and are recommended for applications requiring closest possible dimensional control of the knurled outside diameter. Such applications include knurling for close fits. Note: The width of the knurling should not exceed the diameter of the blank, and knurl­ ing wider than the knurling tool cannot be produced unless the knurl starts at the end of the work. Marking on Knurls and Dies.—Each knurl and die should be marked as follows: a. when straight to indicate its diametral pitch; b. when diagonal, to indicate its diametral pitch, helix angle, and hand of angle. Concave Knurls.—The radius of a concave knurl should not be the same as the radius of the piece to be knurled. If the knurl and the work are of the same radius, the material com­pressed by the knurl will be forced down on the shoulder D and spoil the appearance of the work. A design of concave knurl is shown in the accompanying illustration, and all the important dimensions are designated by letters. To find these dimensions, the pitch of the knurl required must be known, and also, approximately, the throat diameter B. This diam­eter must suit the knurl holder used and be such that the circumference contains an even number of teeth with the required pitch. When these dimensions have been decided upon, all the other unknown factors can be found by the following formulas: Let R = radius of piece to be knurled; r = radius of concave part of knurl; C = radius of cutter or hob for cut­ting the teeth in the knurl; B = diameter over concave part of knurl (throat diameter); A = outside diameter of knurl; d = depth of tooth in knurl; P = pitch of knurl (number of teeth per inch circumference); p = circular pitch of knurl; then r = R + 1 ⁄2 d; C = r + d; A = B + 2r - (3d + 0.010 inch); and d = 0.5 × p × cot a/2, where a is the included angle of the teeth. C As the depth of the tooth is usually very slight,

A

d

B

D

the throat diameter B will be accurate enough for all practical purposes for calculat­ing the pitch, and it is not necessary to take into consideration the pitch circle. For exam­ple, assume that the pitch of a knurl is 32, that the throat diameter B is 0.5561 inch, that the radius R of the piece to be knurled is 1 ⁄16 inch, and that the angle r of the teeth is 90 degrees; find the dimensions of the knurl. Using the notation given:

1 1 p = P = 32 = 0.03125 inch d = 0.5 # 0.03125 # cot45 ° = 0.0156 inch 1 0.0156 r = 16 + = 0.0703 inch C = 0.0703 + 0.0156 = 0.0859 inch 2 A = 0.5561 + 0.1406 − ^0.0468 + 0.010h = 0.6399 inch

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Machinery's Handbook, 31st Edition Tool Wear

1061

TOOL WEAR AND SHARPENING Metal cutting tools wear constantly when they are being used. A normal amount of wear should not be a cause for concern until the size of the worn region indicates that the tool should be replaced. Normal wear cannot be avoided and should be differen­tiated from abnormal tool breakage or excessively fast wear. Tool breakage and an exces­sive rate of wear indicate that the tool is not operating correctly, and steps should be taken to correct this situation. There are several basic mechanisms that cause tool wear. It is generally understood that tools wear as a result of abrasion caused by hard particles of work material plow­ing over the surface of the tool. Wear is also caused by diffusion or alloying between the work and tool material. In regions where the conditions of contact are favor­able, work material reacts with tool material, causing an attrition of the tool material. The rate of this attrition is dependent upon the temperature in the region of contact and the reactivity of the tool and work materials with each other. Diffusion or alloying also occurs where particles of the work material are welded to the surface of the tool. These welded deposits are often quite visible in the form of a built-up edge, as particles or a layer of work material inside a crater, or as small mounds attached to the face of the tool. The dif­fusion or alloying occurring between these deposits and the tool weakens the tool material below the weld. Frequently these deposits are again rejoined to the chip by welding, or they are simply broken away by the force of collision with the passing chip. When this happens, a small amount of the tool material may remain attached to the deposit and be plucked from the surface of the tool, to be carried away with the chip. This mechanism can cause chips to be broken from the cutting edge and cause the formation of small craters, or pull-outs, on the tool face. It can also contribute to the enlargement of the larger crater that sometimes forms behind the cutting edge. Among the other mechanisms that can cause tool wear are severe thermal gradients and thermal shocks, which cause cracks to form near the cutting edge, ultimately leading to tool failure. This condition can be caused by improper tool grinding procedures, by heavy interrupted cuts, or by the improper application of cutting flu­ids when machining at high cutting speeds. Chemical reactions between the active constit­uents in some cutting fluids sometimes accelerate the rate of tool wear. Oxidation of the heated metal near the cutting edge also contributes to tool wear, particularly when fast cut­ting speeds and high cutting temperatures are encountered. Breakage of the cutting edge caused by overloading, heavy shock loads, or improper tool design is not normal wear and should be corrected. The wear mechanisms described above bring about visible manifestations of wear on the tool, which should be understood so that the proper corrective measures can be taken, when required. These visible signs of wear are described in the following paragraphs, and the cor­rective measures that might be required are given in the table Tool Troubleshooting Check List on page 1083. The best procedure when troubleshooting is to try to correct only one condition at a time. When a correction has been made, it should be checked. After one con­dition has been corrected, work can then be started to correct the next condition. Flank Wear.—Tool wear occurring on the flank of the tool below the cutting edge is called flank wear. Flank wear always takes place and cannot be avoided. It should not give rise to concern unless the rate of flank wear is too fast or the flank wear land becomes too large in size. The size of the flank wear can be measured as the distance between the top of the cutting edge and the bottom of the flank wear land. In practice, a visual estimate is usu­ally made instead of a precise measurement, although in many instances flank wear is ignored and the tool wear is “measured” by the loss of size on the part. The best measure of tool wear, however, is flank wear. When it becomes too large, the rubbing action of the wear land against the workpiece increases and the cutting edge must be replaced. Because conditions vary, it is not possible to give an exact amount of flank wear at which the tool should be replaced. Although there are many exceptions, as a rough estimate, high-speed steel tools should be replaced when the width of the flank wear land reaches 0.005 to

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1062

Machinery's Handbook, 31st Edition TOOL WEAR AND SHARPENING

0.010 inch (0.13–0.25 mm) for finish turning and 0.030 to 0.060 inch (0.76–1.52 mm) for rough turning; and for cemented carbides 0.005 to 0.010 inch (0.13–0.25 mm) for finish turning and 0.020 to 0.040 inch (0.51–1.02 mm) for rough turning. Under ideal conditions, which, surprisingly, occur quite frequently, the width of the flank wear land will be very uniform along its entire length. When the depth of cut is uneven, such as when turning out-of-round stock, the bottom edge of the wear land may become somewhat slanted, the wear land being wider toward the nose. A jagged-appearing wear land usually is evidence of chipping at the cutting edge. Sometimes, only one or two sharp depressions of the lower edge of the wear land will appear to indicate that the cutting edge has chipped above these depressions. A deep notch will sometimes occur at the “depth of cut line,” or that part of the cutting opposite the original surface of the work. This can be caused by a hard surface scale on the work, by a work-hardened surface layer on the work, or when machining high-temperature alloys. Often the size of the wear land is enlarged at the nose of the tool. This can be a sign of crater breakthrough near the nose or of chipping in this region. Under certain conditions, when machining with carbides, it can be an indica­tion of deformation of the cutting edge in the region of the nose. When a sharp tool is first used, the initial amount of flank wear is quite large in relation to the subsequent total amount. Under normal operating conditions, the width of the flank wear land will increase at a uniform rate until it reaches a critical size, after which the cut­ ting edge breaks down completely. This is called catastrophic failure, and the cutting edge should be replaced before this occurs. When cutting at slow speeds with high-speed steel tools, there may be long periods when no increase in the flank wear can be observed. For a given work material and tool material, the rate of flank wear is primarily dependent on the cutting speed and then on the feed rate. Cratering.—A deep crater will sometimes form on the face of the tool, which is easily recognizable. The crater forms at a short distance behind the side cutting edge, leaving a small shelf between the cutting edge and the edge of the crater. This shelf is sometimes covered with the built-up edge and at other times it is uncovered. Often the bottom of the crater is obscured with work material that is welded to the tool in this region. Under normal operating conditions, the crater will gradually enlarge until it breaks through a part of the cutting edge. Usually this occurs on the end cutting edge just behind the nose. When this takes place, the flank wear at the nose increases rapidly and complete tool failure follows shortly. Sometimes cratering cannot be avoided, and a slow increase in the size of the crater is considered normal. However, if the rate of crater growth is rapid, leading to a short tool life, corrective measures must be taken. Cutting Edge Chipping.—Small chips are sometimes broken from the cutting edge, which accelerates tool wear but does not necessarily cause immediate tool failure. Chip­ ping can be recognized by the appearance of the cutting edge and the flank wear land. A sharp depression in the lower edge of the wear land is a sign of chipping, and, if this edge of the wear land has a jagged appearance, it indicates that a large amount of chipping has taken place. Often the vacancy or cleft in the cutting edge that results from chipping is filled up with work material that is tightly welded in place. This occurs very rapidly when chipping is caused by a built-up edge on the face of the tool. In this manner, the damage to the cutting edge is healed; however, the width of the wear land below the chip is usually increased and the tool life is shortened. Deformation.—Deformation occurs on carbide cutting tools when taking a very heavy cut using a slow cutting speed and a high feed rate. A large section of the cutting edge then becomes very hot, and the heavy cutting pressure compresses the nose of the cutting edge, thereby lowering the face of the tool in the area of the nose. This reduces the relief under the nose, increases the width of the wear land in this region, and shortens tool life.

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Machinery's Handbook, 31st Edition TOOL WEAR AND SHARPENING

1063

Surface Finish.—The finish on the machined surface does not necessarily indicate poor cutting tool performance unless there is a rapid deterioration. A good surface finish is, however, sometimes a requirement. The principal cause of a poor surface finish is the built-up edge that forms along the edge of the cutting tool. The elimination of the built-up edge will always result in an improvement of the surface finish. The most effective way to eliminate the built-up edge is to increase cutting speed. When cutting speed is increased beyond a certain critical limit, there will be a rather sudden and large improvement in the surface finish. Cemented carbide tools can operate successfully at higher cutting speeds, where the built-up edge does not occur and where a good surface fin­ish is obtained. Whenever possible, cemented carbide tools should be operated at cutting speeds where a good surface finish will result. There are times when such speeds are not possible. Also, highspeed tools cannot be operated at the speed where the built-up edge does not form. In these conditions, the most effective method of obtaining a good surface finish is to employ a cutting fluid that has active sulphur or chlorine additives. Cutting tool materials that do not alloy readily with the work material are also effective in obtaining an improved surface finish. Straight titanium carbide and diamond are the two principal tool materials that fall into this category. The presence of feed marks can mar an otherwise good surface finish, and attention must be paid to the feed rate and the nose radius of the tool if a good surface finish is desired. Changes in tool geometry can also be helpful. A small “flat,” or secondary cutting edge, ground on the end cutting edge behind the nose will sometimes provide the desired surface finish. When the tool is in operation, the flank wear should not be allowed to become too large, particularly in the region of the nose where the finished surface is produced. Sharpening Twist Drills.—Twist drills are cutting tools designed to perform concur­ rently several functions, such as penetrating directly into solid material, ejecting the removed chips outside the cutting area, maintaining the essentially straight direction of the advance movement and controlling the size of the drilled hole. The geometry needed for these multiple functions is incorporated into the design of the twist drill in such a manner that it can be retained even after repeated sharpening operations. Twist drills are resharp­ ened many times during their service life, with the practically complete restitution of their original operational characteristics. However, in order to obtain all the benefits which the design of the twist drill is capable of providing, the surfaces generated in the sharpening process must agree with the original form of the tool’s operating surfaces, unless a change of shape is required for use on a different work material. The principal elements of tool geometry essential for the adequate cutting performance of twist drills are shown in Fig. 1. The generally used values for these dimen­sions are the following: Point angle: Commonly 118°, except for high strength steels, 118° to 135°; aluminum alloys, 90° to 140°; and magnesium alloys, 70° to 118°. Helix angle: Commonly 24° to 32°, except for magnesium and copper alloys, 10° to 30°. Lip relief angle: Commonly 10° to 15°, except for high strength or tough steels, 7° to 12°. The lower values of these angle ranges are used for drills of larger diameter, the higher values for the smaller diameters. For drills of diameters less than 1 ⁄4 inch (6.35 mm), the lip relief angles are increased beyond the listed maximum values up to 24°. For soft and free machining materials, 12° to 18° except for diameters less than 1 ⁄4 inch (6.35 mm), 20° to 26°. Relief Grinding of the Tool Flanks.—In sharpening twist drills the tool flanks contain­ ing the two cutting edges are ground. Each flank consists of a curved surface, which pro­vides the relief needed for the easy penetration and free cutting of the tool edges. In grinding the flanks, Fig. 2, the drill is swung around the axis A of an imaginary cone while resting in a support that holds the drill at one-half the point angle B with respect to the face of the grinding wheel. Feed f for stock removal is in the direction of the drill axis.

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The relief angle is usually measured at the periphery of the twist drill and is also specified by that value. It is not a constant but should increase toward the center of the drill. The relief grinding of the flank surfaces will generate the chisel angle on the web of the twist drill. The value of that angle, typically 55°, which can be measured, for example, with the protractor of an optical projector, is indicative of the correctness of the relief grinding. Margin

Chisel Edge Angle Point Angle

Land Lip Relief Angle

Web Thickness Helix Angle

Standard Point Fig. 1. The principal elements of tool geometry on twist drills.

C

55°

T

Fig. 3. The chisel edge C after thinning the web by grinding off area T.

f /2 A Fig. 2. In grinding the face of the twist drill, the tool is swung around the axis A of an imaginary cone, while resting in a support tilted by half of the point angle b with respect to the face of the grinding wheel. Feed f for stock removal is in the direction of the drill axis.

Fig. 4. Split-point or “crankshaft” type web thinning.

Drill Point Thinning.—The chisel edge is the least efficient operating surface element of the twist drill because it does not cut, but actually squeezes or extrudes, the work material. To improve the inefficient cutting conditions caused by the chisel edge, the point width is often reduced in a drill-point thinning operation, resulting in a condition such as that shown in Fig. 3. Point thinning is particularly desirable on larger size drills and also on those which become shorter in usage, because the thickness of the web increases toward the shaft of the twist drill, thereby adding to the length of the chisel edge. The extent of point thin­ning is limited by the minimum strength of the web needed to avoid splitting of the drill point under the influence of cutting forces.

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1065

Both sharpening operations—the relieved face grinding and the point thinning—should be carried out in special drill-grinding machines or with twist drill-grinding fixtures mounted on general-purpose tool-grinding machines designed to assure the essential accu­racy of the required tool geometry. Off-hand grinding may be used for the important web thinning when a special machine is not available; however, such operations require skill and experience. Improperly sharpened twist drills, e.g. those with unequal edge length or asymmetrical point angle, will tend to produce holes with poor diameter and directional control. For deep holes and for drilling into stainless steel, titanium alloys, high temperature alloys, nickel alloys, very high-strength materials and in some cases tool steels, splitpoint grinding, resulting in a “crankshaft” type drill point, is recommended. In this type of point­ing, see Fig. 4, the chisel edge is entirely eliminated, extending the positive rake cutting edges to the center of the drill, thereby greatly reducing the required thrust in drilling. Points on modified-point drills must be restored after sharpening to maintain their increased drilling efficiency. Sharpening Carbide Tools.—Cemented carbide indexable inserts are usually not resharpened, but sometimes they require a special grind in order to form a contour on the cutting edge to suit a special purpose. Brazed-type carbide cutting tools are resharpened after the cutting edge has become worn. On brazed carbide tools, the cutting-edge wear should not be allowed to become excessive before the tool is re-sharpened. One method of determining when brazed carbide tools need resharpening is by periodic inspection of the flank wear and the condition of the face. Another method is to determine the amount of production normally obtained before excessive wear has taken place, or to deter­m ine the equivalent period of time. One disadvantage of this method is that slight varia­tions in the work material will often cause the wear rate not to be uniform and the number of parts machined before regrinding will not be the same each time. Usually, sharpening should not require removal of more than 0.005 to 0.010 inch (0.13−0.25 mm) of carbide. General Procedure in Carbide-Tool Grinding: The general procedure depends upon the kind of grinding operation required. If the operation is to resharpen a dull tool, a diamond wheel of 100- to 120-grain size is recommended, although a finer wheel—up to 150-grain size—is sometimes used to obtain a better finish. If the tool is new or is a “standard” design and changes in shape are necessary, a 100-grit diamond wheel is recommended for rough­ing and a finer grit diamond wheel can be used for finishing. Some shops prefer to rough-grind the carbide with a vitrified silicon carbide wheel, the finish grinding being done with a diamond wheel. A final operation commonly designated as lapping may or may not be employed for obtaining an extra-fine finish. Wheel Speeds: The speed of silicon carbide wheels usually is about 5000 feet per minute (25.4 m/s). The speeds of diamond wheels generally range from 5000−6000 fpm (25.4−30.5 m/s); yet lower speeds (550−3000 fpm or 2.8−15.2 m/s) can be effective. Offhand Grinding: In grinding single-point tools (excepting chipbreakers), the common practice is to hold the tool by hand, press it against the wheel face and traverse it continu­ ously across the wheel face while the tool is supported on the machine rest or table, which is adjusted to the required angle. This is known as “offhand grinding” to distinguish it from the machine grinding of cutters as in regular cutter-grinding practice. The selection of wheels adapted to carbide-tool grinding is very important. Silicon Carbide Wheels.—The green-colored silicon carbide wheels generally are pre­ ferred to the dark gray or gray-black variety, although the latter are sometimes used. Grain or Grit Sizes: For roughing, a grain size of 60 is very generally used. For finish grinding with silicon carbide wheels, a finer grain size of 100 or 120 is common. A silicon carbide wheel such as C60-I-7V may be used for grinding both the steel shank and carbide tip. However, for under-cutting steel shanks up to the carbide tip, it may be advantageous to use an aluminum oxide wheel suitable for grinding softer, carbon steel.

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Machinery's Handbook, 31st Edition TOOL WEAR AND SHARPENING

Grade: According to the standard system of marking, different grades from soft to hard are indicated by letters from A to Z. For carbide-tool grinding fairly soft grades such as G, H, I, and J are used. The usual grades for roughing are I or J and for finishing H, I, and J. The grade should be such that a sharp free-cutting wheel will be maintained without excessive grinding pressure. Harder grades than those indicated tend to overheat and crack the car­bide. Structure: The common structure numbers for carbide-tool grinding are 7 and 8. The larger cup-wheels (10 to 14 inches or 254−356 mm) may be of the porous type and be des­ignated as 12P. The standard structure numbers range from 1 to 15 with progressively higher numbers indicating less density and more open wheel structure.

Diamond Wheels.—Wheels with diamond-impregnated grinding faces are fast and cool cutting and have a very low rate of wear. They are used extensively both for resharpening and for finish grinding of carbide tools when preliminary roughing is required. Diamond wheels are also adapted for sharpening multi-tooth cutters such as milling cutters, reamers, etc., which are ground in a cutter-grinding machine. Resinoid bonded wheels are commonly used for grinding chipbreakers, milling cutters, reamers or other multi-tooth cutters. They are also applicable to precision grinding of car­ bide dies, gages, and various external, internal and surface grinding operations. Fast, cool cutting action is characteristic of these wheels.

Metal bonded wheels are often used for offhand grinding of single-point tools especially when durability or long life and resistance to grooving of the cutting face are considered more important than the rate of cutting. Vitrified bonded wheels are used both for roughing of chipped or very dull tools and for ordinary resharpening and finishing. They provide rigidity for precision grinding, a porous structure for fast cool cutting, sharp cutting action and durability.

Diamond Wheel Grit Sizes.—For roughing with diamond wheels, a grit size of 100 is the most common both for offhand and machine grinding. Grit sizes of 120 and 150 are frequently used in offhand grinding of single-point tools 1) for resharpening; 2) for a combination roughing and finishing wheel; and 3) for chipbreaker grinding. Grit sizes of 220 or 240 are used for ordinary finish grinding all types of tools (offhand and machine) and also for cylindrical, internal and surface finish grinding. Grits of 320 and 400 are used for “lapping” to obtain very fine finishes, and for hand hones. A grit of 500 is for lapping to a mirror finish on such work as carbide gages and boring or other tools for exceptionally fine finishes. Diamond Wheel Grades.—Diamond wheels are made in several different grades to bet­ ter adapt them to different classes of work. The grades vary for different types and shapes of wheels. Standard Norton grades are H, J, and L for resinoid-bonded wheels, grade N for metal-bonded wheels and grades J, L, N, and P for vitrified wheels. Harder and softer grades than standard may at times be used to advantage.

Diamond Concentration.—The relative amount (by carat weight) of diamond in the dia­mond section of the wheel is known as the “diamond concentration.” Concentrations of 100 (high), 50 (medium) and 25 (low) ordinarily are supplied. A concentration of 50 rep­resents one-half the diamond content of 100 (if the depth of the diamond is the same in each case) and 25 equals one-fourth the content of 100 or one-half the content of 50 concentra­tion. 100 Concentration: Generally interpreted to mean 72 carats of diamond/in3 of abrasive section. (A 75 concentration indicates 54 carats/in3.) Recommended (especially in grit sizes up to about 220) for general machine grinding of carbides, and for grinding cutters and chipbreakers. Vitrified and metal-bonded wheels usually have 100 concentration.

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50 Concentration: In the finer grit sizes of 220, 240, 320, 400, and 500, a 50 concentra­ tion is recommended for offhand grinding with resinoid-bonded cup-wheels. 25 Concentration: A low concentration of 25 is recommended for offhand grinding with resinoid-bonded cup-wheels with grit sizes of 100, 120 and 150. Depth of Diamond Section: The radial depth of the diamond section usually varies from ⁄16 to 1 ⁄4 inch (1.6 to 6.4 mm). The depth varies somewhat according to the wheel size and type of bond. 1

Dry Versus Wet Grinding of Carbide Tools.—In using silicon carbide wheels, grinding should be done either absolutely dry or with enough coolant to flood the wheel and tool. Satisfactory results may be obtained either by the wet or dry method. However, dry grind­ing is the most prevalent usually because, in wet grinding, operators tend to use an inade­quate supply of coolant to obtain better visibility of the grinding operation and avoid getting wet; hence checking or cracking in many cases is more likely to occur in wet grind­ ing than in dry grinding. Wet Grinding with Silicon Carbide Wheels: One advantage commonly cited in connec­ tion with wet grinding is that an ample supply of coolant permits using wheels about one grade harder than in dry grinding, thus increasing the wheel life. Plenty of coolant also pre­vents thermal stresses and the resulting cracks, and there is less tendency for the wheel to load. A dust exhaust system also is unnecessary. Wet Grinding with Diamond Wheels: In grinding with diamond wheels, the general prac­ tice is to use a coolant to keep the wheel face clean and promote free cutting. The amount of coolant may vary from a small stream to a coating applied to the wheel face by a felt pad. Coolants for Carbide Tool Grinding.—In grinding either with silicon carbide or dia­ mond wheels, a coolant that is used extensively consists of water plus a small amount either of soluble oil, sal soda, or soda ash to prevent corrosion. One prominent manufacturer rec­ommends for silicon carbide wheels about 1 ounce of soda ash per gallon of water, and for diamond wheels kerosene. The use of kerosene is quite general for diamond wheels and is usually applied to the wheel face by a felt pad. Another coolant recommended for dia­mond wheels consists of 80 percent water and 20 percent soluble oil. Peripheral Versus Flat Side Grinding.—In grinding single-point carbide tools with sili­ con carbide wheels, the roughing preparatory to finishing with diamond wheels may be done either by using the flat face of a cup-shaped wheel (side grinding) or the periphery of a “straight” or disk-shaped wheel. Even where side grinding is preferred, the periphery of a straight wheel may be used for heavy roughing as in grinding back-chipped or broken tools (see left-hand diagram). Reasons for preferring peripheral grinding include faster cutting with less danger of localized heating and checking especially in grinding broad sur­faces. The advantages usually claimed for side grinding are that proper rake or relief angles are easier to obtain and the relief or land is ground flat. The diamond wheels used for tool sharpening are designed for side grinding. (See right-hand diagram.)

18” or 20” Diam.

Tungsten Carbide Tip 15°

Cut Steel Back to 15° Angle

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Grind Carbide Only Steel Cleared 15°

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1068

Machinery's Handbook, 31st Edition TOOL WEAR AND SHARPENING

Lapping Carbide Tools.—Carbide tools may be finished by lapping, especially if an exceptionally fine finish is required on the work as, for example, with tools used for precision boring or turning nonferrous metals. If the finishing is done by using a diamond wheel of very fine grit (such as 240, 320, or 400), the operation is often called “lapping.” A second lapping method is by means of a power-driven lapping disk charged with diamond dust, Norbide powder, or silicon carbide finishing compound. A third method is by using a hand lap or hone usually of 320 or 400 grit. In many plants, the finishes obtained with carbide tools meet requirements without a special lapping operation. In all cases, any feather edge which may be left on tools should be removed, and it is good practice to bevel the edges of roughing tools at 45 degrees to leave a chamfer 0.005 to 0.010 inch wide (0.127−­ 0.254 mm). This is done by hand honing, and the object is to prevent crumbling or flaking off at the edges when hard scale or heavy chip pressure is encountered. Hand Honing: The cutting edge of carbide tools, and tools made from other tool materi­ als, is sometimes hand honed before it is used in order to strengthen the cutting edge. When interrupted cuts or heavy roughing cuts are to be taken, or when the grade of carbide is slightly too hard, hand honing is beneficial because it will prevent chipping or even possi­bly, breakage of the cutting edge. Whenever chipping is encountered, hand honing the cut­ting edge before use will be helpful. It is important, however, to hone the edge lightly and only when necessary. Heavy honing will always cause a reduction in tool life. Normally, removing 0.002 to 0.004 inch (0.051−0.102 mm) from the cutting edge is sufficient. When indexable inserts are used, the use of pre-honed inserts is preferred to hand honing, although sometimes an additional amount of honing is required. Hand honing of carbide tools in between cuts is sometimes done to defer grinding or to increase the life of a cutting edge on an indexable insert. If correctly done, so as not to change the relief angle, this pro­cedure is sometimes helpful. If improperly done, it can result in a reduction in tool life. Chipbreaker Grinding.—For this operation a straight diamond wheel is used on a uni­ versal tool and cutter grinder, a small surface grinder, or a special chipbreaker grinder. A resinoid-bonded wheel of grade J or N commonly is used, and the tool is held rigidly in an adjustable holder or vise. The width of the diamond wheel usually varies from 1 ⁄8 to 1 ⁄4 inch (3.2−6.4 mm). A vitrified bond may be used for wheels as thick as 1 ⁄4 inch (6.35 mm), and a resinoid bond for relatively narrow wheels. Summary of Miscellaneous Points.—In grinding a single-point carbide tool, traverse it across the wheel face continuously to avoid localized heating. This traverse movement should be quite rapid in using silicon carbide wheels and comparatively slow with dia­ mond wheels. A hand traversing and feeding movement, whenever practicable, is gener­ ally recommended because of greater sensitivity. In grinding, maintain a constant, moderate pressure. Manipulating the tool so as to keep the contact area with the wheel as small as possible will reduce heating and increase the rate of stock removal. Never cool a hot tool by dipping it in a liquid, as this may crack the tip. Wheel rotation should preferably be against the cutting edge or from the front face toward the back. If the grinder is driven by a reversing motor, opposite sides of a cup wheel can be used for grinding right-and left-hand tools and with rotation against the cutting edge. If it is necessary to grind the top face of a single-point tool, this should precede the grinding of the side and front relief, and top-face grinding should be minimized to maintain the tip thickness. In machine grinding with a diamond wheel, limit the feed per traverse to 0.001 inch (0.025 mm) for 100 to 120 grit; 0.0005 inch (0.013 mm) for 150 to 240 grit; and 0.0002 inch (0.005 mm) for 320 grit and finer. Meshes, Sieves, and Screens

Sieving or screening is a method of categorizing powder particle size by running the powder through a specific size screen or screens. Powder can be separated into two or more size fractions by stacking the screens (Fig. 1), thereby determining the powder

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Machinery's Handbook, 31st Edition Meshes, Sieves, and Screens

1069

particle size distribution. Sieves and screens are usually used for larger particle sized materials, greater than 44 micron (325 Mesh). U S S TA N D A R D S I E V E S E R I E S A.S.T.M. MICRONS 425 E-11

SIEVE NO. 40 OPENING

0.0165

IN

SPECIFICATIONS

OPENING

0.425

MM

U S S TA N D A R D S I E V E S E R I E S SIEVE NO. A.S.T.M. MICRONS 100 150 E-11 OPENING

0.0059

IN

SPECIFICATIONS

OPENING

0.150

MM

Fig. 1.

Two scales are frequently used to classify particle sizes; US Sieve Series and Tyler Mesh Size, also called Tyler Standard Sieve Series. The larger the mesh number the smaller the particle size of the powder. The Tyler mesh number indicates the number of openings in the screen per linear inch. •  A -6 mesh powder contains particles that pass through a 6 mesh screen. A -6 mesh powder has particles that measure less than 3360 microns. •  A powder that is -325 mesh has particles that measure less than 44 micron. Market Grade sieves use thicker wire than other commercial grades, and are commonly used for applications where mesh strength (and therefore screen life) is important. Mill Grade sieves use a thinner wire diameter, which provide more open area for a given mesh size. Therefore, Mill Grade sieves are used when throughput is more important than durability. Tensile Bolting Cloth uses very fine wire diameters, and thus provides the highest frac­tion of open area of all sieve types; it is often used for fine sifting and screen printing. A comparison of standard screen and sieve sizes are indicated in Table 2 and Table 5. Example 1: A tantalum powder has the following description: Tantalum Powder, -325 Mesh, 3N8 Purity The minus (-) sign in the designation -325 indicates that particle sizes of less than 44 micron (µm) (Table 1) are able to pass through a 325 mesh screen. Much of the powder may be considerably smaller and still qualify as a -325 mesh powder, but the designation −325 mesh simply means that the powder will pass through a 325 mesh screen. The 3N8 Purity indicates a purity of 99.98% relative to other rare earth powders; similar­ ily, 2N5 indicates 99.5% purity, 5N8 indicates 99.9998%, and 6N indicates 99.9999%, etc. Example 2: illustrates two stacked screens. If a powder is required to fall within a specified max/min range of sizes, two mesh sizes maybe specified such as in the following: Tantalum Powder, −40+100 Mesh, 3N8 Purity The -40 designation indicates that the powder will pass through an 40 mesh screen, and the designation +100 indicates that particles are trapped (do not pass through) by a 100 mesh screen. Thus -40+100 indicates a range of particle sizes small enough to pass the 40 mesh and too large to pass the 100 mesh screen.

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Machinery's Handbook, 31st Edition Meshes, Sieves, and Screens

1070 Sieve size (mm)

11.2 6.35 5.6 4.75 4 3.36 2.8 2.36 2 1.85 1.7 1.4 1.18 1.04 1 0.841 0.787 0.71 0.681 0.63 0.595 0.541 0.5 0.47 0.465 0.437 0.4 0.389 0.368 0.355 0.33 0.323 0.31 0.3 0.282 0.27 0.26 0.25 0.241 0.231 0.224 0.21 0.2 0.193 0.177 0.165 0.149 0.125 0.105 0.088 0.074 0.063 0.053 0.044 0.037

Opening (in.)

(µm)

0.438 11200 0.25 6350 0.223 5600 0.187 – 0.157 – 0.132 – 0.11 – 0.0937 – 0.0787 – – – 0.0661 – 0.0555 – 0.0469 – – – 0.0394 – 0.0331 841 – – 0.0278 – – – – – 0.0232 595 – – 0.0197 – – – – – – – 0.0165 400 – – – – 0.0139 – – – – – – – 0.0117 – – – – – – – 0.0098 250 – – – – – – 0.0083 210 – – – – 0.007 177 – – 0.0059 149 0.0049 125 0.0041 105 0.0035 88 0.0029 74 0.0024 63 0.0021 53 0.0017 44 0.0015 37

Table 1. Commercial Sieve Mesh Dimensions Standard Mesh US

7/16 1/4 3.5 4 5 6 7 8 10 – 12 14 16 – 18 20 – 25 – – 30 – 35 – – – 40 – – 45 – – – 50 – – – 60 – – – 70 – – 80 – 100 120 140 170 200 230 270 325 400

Tyler – – 3.5 4 5 6 7 8 9 – 10 12 14 – 16 20 – 24 – – 28 – 32 – – – 35 – – 42 – – – 48 – – – 60 – – – 65 – – 80 – 100 115 150 170 200 250 270 325 400

Tensile Bolting Cloth

Mesh Opening – – – – – – – – – – 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54 58 60 62 64 70 72 74 76 78 80 84 88 90 94 105 120 145 165 200 230 – 300 – –

– – – – – – – – – – 0.062 0.0535 0.0466 0.041 0.038 0.0342 0.031 0.0282 0.0268 0.0248 0.0229 0.0213 0.0198 0.0185 0.0183 0.0172 0.0162 0.0153 0.0145 0.0137 0.013 0.0127 0.0122 0.0116 0.0111 0.0106 0.0102 0.0098 0.0095 0.0091 0.0088 0.0084 0.0079 0.0076 0.0071 0.0065 0.0058 0.0047 0.0042 0.0034 0.0029 – 0.0021 – –

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Wire

– – – – – – – – – – 0.009 0.009 0.009 0.009 0.0075 0.0075 0.0075 0.0075 0.0065 0.0065 0.0065 0.0065 0.0065 0.0065 0.0055 0.0055 0.0055 0.0055 0.0055 0.0055 0.0055 0.0045 0.0045 0.0045 0.0045 0.0037 0.0037 0.0037 0.0037 0.0037 0.0037 0.0035 0.0035 0.0035 0.0035 0.003 0.0025 0.0022 0.0019 0.0016 0.0014 – 0.0012 – –

Mill Grade

Mesh Opening 2 3 4 – 5 6 7 8 9 10 12 14 16 – 18 20 22 24 26 28 30 32 34 36 38 – 40 – – 45 – – 50 55 – – – 60 – – – – – – – – – – – – – – – – –

0.466 0.292 0.215 – 0.168 0.139 0.115 0.1 0.088 0.08 0.065 0.054 0.0465 – 0.0406 0.036 0.032 0.0287 0.0275 0.0275 0.0238 0.0223 0.0204 0.0188 0.0178 – 0.0165 – – 0.0142 – – 0.0125 0.0112 – – – 0.0102 – – – – – – – – – – – – – – – – –

Wire

0.054 0.041 0.035 – 0.032 0.028 0.028 0.025 0.023 0.02 0.018 0.017 0.016 – 0.015 0.014 0.0135 0.013 0.011 0.01 0.0095 0.009 0.009 0.009 0.0085 – 0.0085 – – 0.008 – – 0.0075 0.007 – – – 0.0065 – – – – – – – – – – – – – – – – –

Market Grade Mesh Opening 2 3 4 4 5 6 7 8 10 11 12 14 16 – 18 20 – 24 – – – – 30 – – 35 – 40 – – – – – – 50 – – – – 60 – – – – 80 – 100 120 150 180 200 250 270 325 400

0.437 0.279 0.2023 0.187 0.159 0.132 0.108 0.0964 0.0742 0.073 0.0603 0.051 0.0445 – 0.0386 0.034 – 0.0277 – – – – 0.0203 – – 0.0176 – 0.015 – – – – – – 0.011 – – – – 0.0092 – – – – 0.007 – 0.0055 0.0046 0.0041 0.0033 0.0029 0.0024 0.0021 0.0017 0.0015

Wire

0.063 0.054 0.0475 0.063 0.041 0.0348 0.035 0.0286 0.0258 0.018 0.023 0.0204 0.0181 – 0.0173 0.0162 – 0.014 – – – – 0.0128 – – 0.0118 – 0.0104 – – – – – – 0.009 – – – – 0.0075 – – – – 0.0055 – 0.0045 0.0037 0.0026 0.0023 0.0021 0.0016 0.0016 0.0014 0.001

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Machinery's Handbook, 31st Edition Meshes, Sieves, and Screens

1071

As indicated by the fragment below from Table 1, the specified particles are therefore smaller than 40 mesh and larger than 100 mesh. Particles trapped above the 100 mesh screen range in size from greater than 149 µm to 400 µm. Particles smaller than 149 micron are passed through the 100 mesh screen. Approximately 90 percent of the particles trapped will fit within the specified range. Standard Mesh

Opening (in.) (µm)

Sieve size (mm) … 0.4 0.149

… 0.0165 0.0059

US … 40 100

… 400 149

Tyler … 35 100

Sieves and screens are stacked with the largest opening (smallest mesh size) above, and the smallest opening (largest mesh size) below, as in Fig. 2. U S S TA N D A R D S I E V E S E R I E S A.S.T.M. MICRONS 425 E-11

SIEVE NO. 40 OPENING

0.0165

IN

OPENING

SPECIFICATIONS

0.425

MM

U S S TA N D A R D S I E V E S E R I E S SIEVE NO. A.S.T.M. MICRONS 100 150 E-11 OPENING

0.0059

IN

SPECIFICATIONS

OPENING

0.150

MM

Fig. 2.

Mesh Size µm 5 10 15 20 22 25 28 32 36 38 40 45 50 53 56 63 71

Table 2. Standard Sieves and Mesh Sizes

TYLER Mesh 2500 1250 800 625 … 500 … … … 400 … 325 … 270 … 250 …

ASTM E-11 No. … … … … … … … … … 400 … 325 … 270 … 230 …

BS 410 Mesh 2500 1250 800 625 … 500 … … … 400 … 350 … 300 … 240 …

DIN 4188 mm

0.005 0.01 0.015 0.02 0.022 0.025 0.028 0.032 0.036 … 0.04 0.045 0.05 … 0.056 0.063 0.071

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Mesh Size µm 106 112 125 140 150 160 180 200 212 250 280 300 315 355 400 425 450

TYLER Mesh 150 … 115 … 100 … 80 … 65 60 … 48 … 42 … 35 …

ASTM E-11 No. 140 … 120 … 100 … 80 … 70 60 … 50 … 45 … 40 …

BS 410 Mesh 150 … 120 … 100 … 85 … 72 60 … 52 … 44 … 36 …

DIN 4188 mm

… 0.112 0.125 0.14 … 0.16 0.18 0.2 … 0.25 0.28 … 0.315 0.355 0.4 … 0.45

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Machinery's Handbook, 31st Edition Meshes, Sieves, and Screens

1072 Mesh Size µm 75 80 90 100

Table Table 2. (Continued) Standard and Mesh 2. Standard SievesSieves and Mesh Sizes Sizes TYLER Mesh 200 … 170 …

Sieve size (mm) 4.75 3.35 2.81 2.38 2 1.68 1.4 1.2 1 0.853 0.71 0.599 0.5 0.422

ASTM E-11 No. 200 … 170 …

BS 410 Mesh 200 … 170 …

DIN 4188 mm … 0.08 0.09 0.1

Mesh Size µm 500 560 600 …

TYLER Mesh 32 … 28 …

ASTM E-11 No. 35 … 30 …

Table 3. Typical Openings in Laboratory Sieve Series BSS 5 6 7 8 10 12 14 16 18 22 25 30 36

Tyler (approx) 4 6 7 8 9 10 12 14 16 20 24 28 32 35

US (approx) 4 6 7 8 10 12 14 16 18 20 25 30 35 40

Sieve size (mm) 0.354 0.297 0.251 0.211 0.178 0.152 0.125 0.104 0.089 0.075 0.066 0.053 0.044 0.037

BSS 44 52 60 72 85 100 120 150 170 200 240 300 350 440

BS 410 Mesh 30 … 25 …

Tyler (approx) 42 48 60 65 80 100 115 150 170 200 250 270 325 400

DIN 4188 mm 0.5 0.56 … … US (approx) 45 50 60 70 80 100 120 140 170 200 230 270 325 400

BS 812-103.1:1985 Withdrawn replaced by BS EN 933-1:2011. BS EN 933-1:2011 describes two methods for the determination of the particle size distribution of samples of aggregates and fillers by sieving.

Abrasive Grit Number.—Standard abrasive grain sizes are designated by numbers. These numbers range from number 8, which is the coarsest, to number 240, which is the finest. The allowable limits for the sizing of aluminum-oxide and silicon-carbide abrasives for grinding wheel manufacture are given in US Simplified Practice Recommendation 118. These numbers in most cases equal approximately the number of sieve openings per inch in the United States Standard Fine Sieve series. For example, a number 30 sieve has 0.0232-inch openings and a sieve wire diameter of 0.0130 inch, making the pitch equal to 0.0362 inch; hence there are 27.6 meshes per inch. The United States Standard Fine Sieve series ranges from number 31 ⁄2 to number 400. Grading Abrasives: In the actual grading of abrasives, several standard sieves are used. To illustrate, take grit No. 10. All material must pass through the coarsest sieve—in this case the No. 7. Through the next to the coarsest sieve, termed the “control sieve”—in this case the No. 8—all material may pass, but not more than 15 percent may be retained on it. At least 45 percent must pass through No. 8, and be retained on No. 10 sieve, but it is per­ missible to have 100 percent pass through No. 8, and remain on No. 10 sieve, the require­ ment being that the grain passing through No. 8, and retained on No. 10 and No. 12 must add to at least 80 percent; consequently, if 45 percent passed through No. 8 sieve and was retained on No. 10 sieve, then at least 35 percent must be retained on the No. 12 sieve. Not more than 3 percent is permitted to pass through the No. 14 sieve.

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Machinery's Handbook, 31st Edition TABLE OF CONTENTS

MACHINING OPERATIONS CUTTING SPEEDS AND FEEDS 1077 Work Materials 1077 Cutting Tool Materials 1082 Cutting Speed, Feed, Depth of Cut, Tool Wear, and Tool Life 1082 Selecting Cutting Conditions 1083 Tool Troubleshooting 1084 Cutting Speed Formulas 1085 Cutting Speeds and Equivalent RPM for Drills of Number and Letter Sizes 1086 RPM for Various Cutting Speeds and Diameters SPEEDS AND FEEDS TABLES 1090 How to Use the Speeds and Feeds Tables 1094 Speeds and Feeds Tables for Turning 1095 Plain Carbon and Alloy Steels 1099 Tool Steels 1100 Stainless Steels 1101 Ferrous Cast Metals 1103 Turning Speed Adjustments 1103 Tool Life Factors 1105 Copper Alloys 1106 Titanium and Titanium Alloys 1107 Superalloys 1108 Speeds and Feeds Tables for Milling 1111 Slit Milling 1112 Aluminum Alloys 1113 Plain Carbon and Alloy Steels 1117 Tool Steels 1118 Stainless Steels 1120 Ferrous Cast Metals 1122 High-Speed Steel Cutters 1124 Speed Adjustment Factors 1125 Radial Depth of Cut Adjustments 1127 Tool Life Adjustments 1128 Using the Speeds and Feeds Tables for Drilling, Reaming, and Threading 1129 Plain Carbon and Alloy Steels 1134 Tool Steels 1135 Stainless Steels 1136 Ferrous Cast Metals 1138 Light Metals 1139 Adjustment Factors for HSS 1140 Copper Alloys 1140 Tapping and Threading 1142 Cutting Speed for Broaching

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ESTIMATING SPEEDS AND MACHINING POWER 1143 1143 1143 1143 1143 1143 1145 1145 1148 1148 1149 1151 1151 1152 1153 1153 1154 1154 1155 1155 1155

Estimating Planer Cutting Speeds Cutting Speed for Planing and Shaping Cutting Time for Turning, Boring, and Facing Planing Time Speeds for Metal-Cutting Saws Speeds for Turning Unusual Material Estimating Machining Power Power Constants Feed Factors Tool Wear Factors Metal Removal Rates Estimating Drilling Thrust, Torque, and Power Work Material Factor Chisel Edge Factors Feed Factors Drill Diameter Factors Boring Cutting Force Moduli of Elasticity Moment of Inertia Deflection of the Boring Bar MICROMACHINING

1156 1156 1157 1157 1158 1160 1161 1164 1166 1170 1172 1172 1173 1173 1174 1175 1176 1176 1177 1177

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Introduction Machine Tool Requirements Microcutting Tools Tool Stiffness Tool Sharpness Tool Materials Tool Offset and Positioning Tool Damage Tool Life Workpiece Materials Ductile Regime Micromachining Crystallographic Directions and Planes Introduction Theory Case Study Cutting Fluids in Micromachining Safety Benefits Selection of Cutting Fluid Drop Size Measurement

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Machinery's Handbook, 31st Edition TABLE OF CONTENTS

MACHINING OPERATIONS

(Continued)

MICROMACHINING

SCREW MACHINES, BAND SAWS, CUTTING FLUIDS

1179 Contact Angle Measurement 1182 Dynamics of Microdroplets 1184 Case Studies 1186 Microfabrication Processes and Parameters 1186 Micromilling 1189 Microdrilling 1191 Microturning 1192 Speeds and Feeds MACHINING ECONOMETRICS 1196 Tool Wear and Tool Life 1196 Equivalent Chip Thickness (ECT) 1197 Tool Life Relationships 1198 Taylor’s Equation 1201 G- and H-Curves 1202 V-ECT-T Graph and the Tool Life Envelope 1205 Forces and Tool Life 1207 Surface Finish and Tool Life 1209 Shape of Tool Life Relationships 1210 Minimum Cost 1211 Production Rate 1211 The Cost Function 1212 Global Optimum 1213 Optimization Models, Economic Tool Life when Feed is Constant 1216 Determination of Machine Settings and Calculation of Costs 1217 Formulas Valid for All Operation Types Including Grinding 1217 Calculation of Cutting Time and Feed Rate 1217 Scheduling of Tool Changes 1218 Calculation of Cost of Cutting and Grinding Operations 1221 Variation of Tooling and Total Cost with the Selection of Speeds and Feeds 1222 Selection of Optimized Data 1225 High-Speed Machining Econometrics 1226 Chip Geometry in Milling 1228 Mean Chip Thickness 1228 Formulas 1230 Forces and Tool Life 1231 High-Speed Milling 1232 Econometrics Comparison

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1234 Automatic Screw Machine Tool Operations 1234 Knurling 1234 Revolutions for Top Knurling 1234 Cams for Threading 1235 Cutting Speeds and Feeds 1237 Spindle Revolutions 1238 Practical Points on Cam and Tool Design 1239 Stock for Screw Machine Products 1241 Band Saw Blade Selection 1242 Tooth Forms 1242 Types of Blades 1243 Band Saw Speed and Feed Rate 1244 Bimetal Band Saw Speeds 1245 Band Saw Blade Break-In 1246 Cutting Fluids for Machining 1247 Cutting and Grinding Fluids 1247 Cutting Oils 1247 Water-Miscible Fluids 1248 Selection of Cutting Fluids 1249 Turning, Milling, Drilling and Tapping 1250 Machining Operations 1251 Machining Magnesium 1252 Metalworking Fluids 1252 Classes of Metalworking Fluids 1252 Occupational Exposures 1253 Fluid Selection, Use, and Application 1254 Fluid Maintenance 1255 Respiratory Protection MACHINING NONFERROUS METALS AND NONMETALLIC MATERIALS 1256 Machining Nonferrous Metals 1256 Aluminum 1257 Magnesium 1258 Zinc Alloy Die Castings 1258 Monel and Nickel Alloys 1259 Copper Alloys 1259 Machining Non-Metals 1259 Hard Rubber 1259 Formica 1260 Micarta 1260 Ultrasonic Machining

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MACHINING OPERATIONS GRINDING FEEDS AND SPEEDS 1261 Basic Rules 1261 Wheel Life and Grinding Ratio 1262 ECT in Grinding 1263 Optimum Grinding Data 1265 Surface Finish, Ra 1266 Spark-Out Time 1267 Grinding Cutting Forces 1268 Grinding Data Selection 1269 Grindability Groups Side Feed, Roughing and Finishing 1269 1270 Relative Grindability 1271 Grindability Overview 1271 Procedure to Determine Data 1277 Calibration of Recommendations 1279 Optimization GRINDING AND OTHER ABRASIVE PROCESSES 1280 Grinding Wheels 1280 Abrasive Materials 1281 Bond Properties 1281 Structure 1282 ANSI Markings 1282 Sequence of Markings 1283 Standard Shapes and Sizes 1283 Selection of Grinding Wheels 1291 Grinding Wheel Faces 1293 Hardened Tool Steels 1297 Constructional Steels 1298 Cubic Boron Nitride 1299 Dressing and Truing Dressing and Truing with Single 1301 Point Diamonds 1303 Size Selection Guide for Diamonds 1304 Diamond Wheels 1304 Shapes 1305 Core Shapes and Designations 1305 Cross Sections and Designations 1306 Designations for Location 1307 Composition 1308 Designation Letters 1309 Selection of Diamond Wheels 1309 Abrasive Specification 1310 Handling and Operation 1310 Speeds and Feeds 1311 Grinding Wheel Safety 1311 Safety in Operating 1311 Handling, Storage and Inspection 1311 Machine Conditions

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GRINDING AND OTHER ABRASIVE PROCESSES

(Continued)

1312 Grinding Wheel Mounting 1312 Safe Operating Speeds 1313 Portable Grinders 1315 Cylindrical Grinding 1315 Plain, Universal, and LimitedPurpose Machines 1315 Traverse or Plunge Grinding 1315 Work Holding on Machines 1316 Work-Holding Methods 1316 Selection of Grinding Wheels 1317 Wheel Recommendations 1317 Operational Data 1318 Basic Process Data 1318 High-Speed Cylindrical Grinding 1319 Areas and Degrees of Automation 1319 Troubles and Correction 1323 Centerless Grinding 1324 Through-Feed Method of Grinding 1324 In-Feed Method 1324 End-Feed Method 1324 Automatic Centerless Method 1324 Internal Centerless Grinding 1324 Troubleshooting 1325 Surface Grinding 1325 Advantages 1326 Principal Systems 1328 Grinding Wheel Recommendations 1329 Process Data for Surface Grinding 1330 Faults and Possible Causes 1330 Vitrified Grinding Wheels 1330 Silicate Bonding Process 1330 Oilstones 1332 Offhand Grinding 1332 Floor- and Bench-Stand Grinding 1332 Portable Grinding 1332 Swing-Frame Grinding 1333 Abrasive Belt Grinding 1333 Abrasive Cutting 1334 Application of Abrasive Belts 1335 Selection of Contact Wheels 1336 Cutting-Off Difficulties 1336 Honing Process 1336 Rate of Stock Removal 1337 Formulas for Rotative Speeds 1337 Abrasive Stones for Honing 1338 Eliminating Undesirable Honing Conditions 1338 Tolerances

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Machinery's Handbook, 31st Edition TABLE OF CONTENTS

MACHINING OPERATIONS GRINDING AND OTHER ABRASIVE PROCESSES

CNC NUMERICAL CONTROL PROGRAMMING

(Continued)

(Continued)

1361 1361 1361 1361 1362 1362 1363

1338 Laps and Lapping 1338 Material for Laps 1339 Laps for Flat Surfaces 1339 Grading Abrasives 1339 Charging Laps 1340 Rotary Diamond Lap 1340 Grading Diamond Dust 1340 Cutting Properties 1341 Cutting Qualities 1341 Wear of Laps 1341 Lapping Abrasives 1341 Effect of Lapping Lubricants 1342 Lapping Pressures 1342 Wet and Dry Lapping 1342 Summary of Lapping Tests

1364 1364 1364 1365 1366 1366 1371 1372 1374 1374 1375 1375 1375 1376 1376 1376 1377 1377

NONTRADITIONAL MACHINING AND CUTTING 1342 Introduction 1344 Mechanical Erosion Processes 1344 Water Jet Machining 1345 Ultrasonic Machining 1345 Abrasive Flow Machining 1345 Magnetic Abrasive Finishing 1345 Electro-Thermal Processes 1345 Electrical Discharge Machining 1345 Laser Beam Machining 1348 Plasma Arc Machining 1350 Electron Beam Machining

1377 1378

CNC NUMERICAL CONTROL PROGRAMMING 1350 1351 1352 1352 1353 1354 1355 1355 1358 1359 1359

Introduction CNC Coordinate Geometry CNC Programming Process Word Address Format Program Development Control System CNC Program Data Program Structure Miscellaneous Function (M-Codes) Unit of Measurement (G20, G21) Absolute and Incremental Programming (G90, G91) 1359 Spindle Function (S-Address) 1360 Feed Rate Function (F-Address) 1360 Inverse Time Feed Rate

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1378 1379 1379 1379 1380 1380 1381 1381 1382 1384 1385 1386 1390 1390 1390 1392 1393 1396

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Feed Rate Override Tool Function (T-Address) Tool Nose Radius Compensation Rapid Motion (G00) Linear Interpolation (G01) Circular Interpolation (G02, G03) Helical and Other Interpolation Methods Offsets for Milling Work Offset (G54 though G59) Tool Length Offset (G43, G44) Cutter Radius Offset (G41, G42) Machining Holes Fixed Cycles Contouring Turning and Boring Thread Cutting on CNC Lathes Depth of Thread Calculations Infeed Methods Radial Infeed Compound Infeed Threading Operations Threading Cycle (G32) Threading Cycle (G76) Multi-Start Threads Subprograms, Macros and Parametric Programming Subprograms Macros and Parametric Programming Basic Macro Skills Confirming Macro Capability Common Features and Applications Macro Structure Macro Definition and Call Variable Definition (G65) Types of Variables Variable Declarations and Expressions Macro Functions Branching and Looping Macro Example Axis Nomenclature Total Indicator Reading CAD/CAM Types of CAD Drawings Drawing Projections Drawing Tips and Traps Drawing Exchange Standards

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Machinery's Handbook, 31st Edition MACHINING OPERATIONS

1077

CUTTING SPEEDS AND FEEDS Work Materials.—The large number of work materials that are commonly machined vary greatly in their basic structure and the ease with which they can be machined. Yet it is possible to group together certain materials having similar machining characteristics, for the purpose of recommending the cutting speed at which they can be cut. Most materials that are machined are metals and it has been found that the most important single factor influencing the ease with which a metal can be cut is its microstructure, followed by any cold-work that may have been done to the metal, which increases its hardness. Metals that have a similar, but not necessarily the same microstructure, will tend to have similar machining characteristics. Thus, the grouping of the metals in the accompanying tables has been done on the basis of their microstructure. Except for a few soft and gummy metals, experience indicates that harder metals are more difficult to cut than softer metals. Also, any given metal is more difficult to cut when it is in a harder form than when it is softer. It is more difficult to penetrate the harder metal and more power is required. These factors in turn will generate a higher cutting tempera­ture at any given cutting speed, thereby making it necessary to use a slower speed, for the cutting temperature must always be kept within the limits that can be sustained by the cut­ting tool without failure. Hardness, then, is an important property that must be considered when machining a given metal. Hardness alone, however, cannot be used as a measure of cutting speed. For example, if pieces of AISI 11L17 and AISI 1117 steel both have a hard­ness of 150 BHN (Brinell Hardness Number), their recommended cutting speeds for high-speed steel tools may be 140 fpm (0.71 m/s) and 130 fpm (0.66 m/s), respectively. In some metals, two entirely different microstructures can produce the same hardness. As an example, a fine pearlite microstruc­t ure and a tempered martensite microstructure can result in the same hardness in a steel. These microstructures will not machine alike. For practical purposes, however, informa­tion on hardness is usually easier to obtain than information on microstructure; thus, hard­ness alone is usually used to differentiate between different cutting speeds for machining a metal. In some situations, the hardness of a metal to be machined is not known. When the hardness is not known, the material condition can be used as a guide. The surface of ferrous metal castings has a scale that is more difficult to machine than the metal below. Some scale is more difficult to machine than others, depending on the foundry sand used, the casting process, the method of cleaning the casting, and the type of metal cast. Special electrochemical treatments sometimes can be used that almost entirely eliminate the effect of the scale on machining, although castings so treated are not fre­quently encountered. Usually, when casting scale is encountered, the cutting speed is reduced approximately 5 or 10 percent. Difficult-to-machine surface scale can also be encountered when machining hot-rolled or forged steel bars. Metallurgical differences that affect machining characteristics are often found within a single piece of metal. The occurrence of hard spots in castings is an example. Different microstructures and hardness levels may occur within a casting as a result of variations in the cooling rate in different parts of the casting. Such variations are less severe in castings that have been heat treated. Steel bar stock is usually harder toward the outside than toward the center of the bar. Sometimes there are slight metallurgical differences along the length of a bar that can affect its cutting characteristics. Cutting Tool Materials.—The recommended cutting speeds and feeds in the accompa­ nying tables are given for high-speed steel, coated and uncoated carbides, ceramics, cer­mets, and polycrystalline diamonds. More data are available for HSS and carbides because these materials are the most commonly used. Other materials that are used to make cutting tools are cemented oxides or ceramics, cermets, cast nonferrous alloys (Stellite), single-crystal diamonds, polycrystalline diamonds, and cubic boron nitride.

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Machinery's Handbook, 31st Edition SPEEDS AND FEEDS

Carbon Tool Steel: It is used primarily to make the less expensive drills, taps, and ream­ ers. It is seldom used to make single-point cutting tools. Hardening in carbon steels is very shallow, although some have a small amount of vanadium and chromium added to improve their hardening quality. The cutting speed to use for plain carbon tool steel should be approximately one-half of the recommended speed for high-speed steel.

High-Speed Steel: This designates a number of steels having several properties that enhance their value as cutting tool material. They can be hardened to a high initial or room-temperature hardness ranging from 63 to 65 RC (Rockwell C scale) for ordinary high-speed steels and up to 70 RC for the so-called super high-speed steels. They retain sufficient hardness at tempera­tures up to 1,000 or 1,100°F (573 or 593°C) to enable them to cut at cutting speeds that will generate these tool temperatures, and return to their original hardness when cooled to room temperature. They harden very deeply, enabling high-speed steels to be ground to the tool shape from solid stock and to be reground many times without sacrificing hardness at the cutting edge. High-speed steels can be made soft by annealing so that they can be machined into complex cutting tools such as drills, reamers, and milling cutters and then hardened. The principal alloying elements of high-speed steels are tungsten (W), molybdenum (Mo), chromium (Cr), vanadium (V), together with carbon (C). There are a number of grades of high-speed steel that are divided into two types: tungsten high-speed steels and molybdenum high-speed steels. Tungsten high-speed steels are designated by the prefix T before the number that designates the grade. Molybdenum high-speed steels are desig­ nated by the prefix letter M. There is little performance difference between comparable grades of tungsten or molybdenum high-speed steel. The addition of 5 to 12 percent cobalt to high-speed steel increases its hardness at the temperatures encountered in cutting, thereby improving its wear resistance and cutting efficiency. Cobalt slightly increases the brittleness of high-speed steel, making it suscepti­ ble to chipping at the cutting edge. For this reason, cobalt high-speed steels are primarily made into single-point cutting tools that are used to take heavy roughing cuts in abrasive materials and through rough abrasive surface scales. The M40 series and T15 are a group of high-hardness or so-called super high-speed steels that can be hardened to 70 RC; however, they tend to be brittle and difficult to grind. For cutting applications, they are usually heat treated to 67–68 RC to reduce their brittleness and tendency to chip. The M40 series is appreciably easier to grind than T15. They are recom­mended for machining tough die steels and other difficult-to-cut materials; they are not recommended for applications where conventional high-speed steels perform well. High-speed steels made by the powder-metallurgy process are tougher and have an improved grindability when compared with similar grades made by the customary process. Tools made of these steels can be hardened about 1 RC higher than comparable high-speed steels made by the customary process without a sacrifice in toughness. They are particularly use­ful in applications involving intermittent cutting and where tool life is limited by chipping. All these steels augment rather than replace the conventional high-speed steels. Cemented Carbides are also called sintered carbides or simply carbides. They are harder than high-speed steels and have excellent wear resistance. Information on these and other hard metal tools is included in the section CEMENTED CARBIDES starting on page 860. Cemented carbides retain a very high degree of hardness at temperatures up to 1400°F (760°C) and even higher; therefore, very fast cutting speeds can be used. When used at fast cutting speeds, they produce good surface finishes on the workpiece. Carbides are more brittle than high-speed steel and, therefore, must be used with more care.

There are four distinct types of carbides:1) straight tungsten carbides; 2) crater-resistant carbides; 3) titanium carbides; and 4) coated carbides.

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Machinery's Handbook, 31st Edition SPEEDS AND FEEDS

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Straight Tungsten Carbide: This is the most abrasion-resistant cemented carbide and is used to machine gray cast iron, most nonferrous metals, and nonmetallic materials, where abrasion resistance is the primary criterion. Straight tungsten carbide will rapidly form a crater on the tool face when used to machine steel, which reduces the life of the tool. Tita­ nium carbide is added to tungsten carbide in order to counteract the rapid formation of the crater. In addition, tantalum carbide is usually added to prevent the cutting edge from deforming when subjected to the intense heat and pressure generated in taking heavy cuts. Crater-Resistant Carbides: These carbides, containing titanium and tantalum carbides in addition to tungsten carbide, are used to cut steels, alloy cast irons, and other materials that have a strong tendency to form a crater. Titanium Carbides: These carbides are made entirely from titanium carbide and small amounts of nickel and molybdenum. They have an excellent resistance to cratering and to heat. Their high hot hardness enables them to operate at higher cutting speeds, but they are more brittle and less resistant to mechanical and thermal shock. Therefore, they are not rec­ommended for taking heavy or interrupted cuts. Titanium carbides are less abrasion-resis­tant and not recommended for cutting through scale or oxide films on steel. Although the resistance to cratering of titanium carbides is excellent, failure caused by crater formation can sometimes occur because the chip tends to curl very close to the cutting edge, thereby forming a small crater in this region that may break through.

Coated Carbides: These are available only as indexable inserts because the coating would be removed by grinding. The principal coating materials are titanium carbide (TiC), titanium nitride (TiN), and aluminum oxide (Al2O3). A very thin layer approximately 0.0002 inch (5.08 mm) of coating material is deposited over a cemented carbide insert; the material below the coating is the substrate. The overall performance of the coated carbide is limited by the substrate, which provides the required toughness, resistance to deforma­ tion, and thermal shock. With an equal tool life, coated carbides can operate at higher cut­ ting speeds than uncoated carbides. The increase may be 20 to 30 percent and sometimes up to 50 percent faster. Titanium carbide and titanium nitride coated carbides usually oper­ ate in the medium (200–800 fpm, 1.0–4.1 m/s) cutting speed range, and aluminum oxide coated carbides are used in the higher (800–1600 fpm, 4.1–8.1 m/s) cutting speed range.

Carbide Grade Selection: The selection of the best grade of carbide for a particular application is very important. An improper grade of carbide will result in a poor perfor­ mance—it may even cause the cutting edge to fail before any significant amount of cutting has been done. Because of the many grades and the many variables that are involved, the carbide producers should be consulted to obtain recommendations for the application of their grades of carbide. A few general guidelines can be given that are useful to form an orientation. Metal cutting carbides usually range in hardness from about 89.5 to 93 RA (Rockwell A scale) with the exception of titanium carbide, which has a hardness range of 90.5 to 93.5 RA. Generally, the harder carbides are more wear-resistant and more brittle, whereas the softer carbides are less wear-resistant but tougher. A choice of hard­ness must be made to suit the given application. The very hard carbides are generally used for taking light finishing cuts. For other applications, select the carbide that has the highest hardness with sufficient strength to prevent chipping or breaking. Straight tungsten car­bide grades should always be used unless cratering is encountered. Straight tungsten car­bides are used to machine gray cast iron, ferritic malleable iron, austenitic stainless steel, hightemperature alloys, copper, brass, bronze, aluminum alloys, zinc alloy die castings, and plastics. Crater-resistant carbides should be used to machine plain carbon steel, alloy steel, tool steel, pearlitic malleable iron, nodular iron, other highly alloyed cast irons, fer­r itic stainless steel, martensitic stainless steel, and certain high-temperature alloys. Tita­nium carbides are recommended for taking high-speed finishing and semifinishing cuts on steel, especially the low-carbon, low-alloy steels, which are less abrasive and have a strong tendency to form a crater. They are also used to take light cuts on alloy cast iron and on some high-nickel alloys. Nonferrous materials, such as some aluminum alloys

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Machinery's Handbook, 31st Edition SPEEDS AND FEEDS

and brass, that are essentially nonabrasive may also be machined with titanium carbides. Abrasive materials and others that should not be machined with titanium carbides include gray cast iron, titanium alloys, cobalt- and nickel-base superalloys, stainless steel, bronze, many aluminum alloys, fiberglass, plastics, and graphite. The feed used should not exceed about 0.020 inch/rev (0.51 mm/rev). Coated carbides can be used to take cuts ranging from light finishing to heavy roughing on most materials that can be cut with these carbides. The coated carbides are recom­ mended for machining all free-machining steels, all plain carbon and alloy steels, tool steels, martensitic and ferritic stainless steels, precipitation-hardening stainless steels, alloy cast iron, pearlitic and martensitic malleable iron, and nodular iron. They are also rec­ommended for taking light finishing and roughing cuts on austenitic stainless steels. Coated carbides should not be used to machine nickel- and cobalt-base superalloys, tita­ nium and titanium alloys, brass, bronze, aluminum alloys, pure metals, refractory metals, and nonmetals such as fiberglass, graphite, and plastics.

Ceramic Cutting Tool Materials: These are made from finely powdered aluminum oxide particles sintered into a hard dense structure without a binder material. Aluminum oxide is also combined with titanium carbide to form a composite, which is called a cermet. These materials have a very high hot hardness enabling very high cutting speeds to be used. For example, ceramic cutting tools have been used to cut AISI 1040 steel at a cutting speed of 18,000 fpm (91.4 m/s) with a satisfactory tool life. However, much lower cutting speeds, in the range of 1000–4000 fpm (5.1–20.3 m/s) and lower, are more common because of lim­itations placed by the machine tool, cutters, and chucks. Although most applications of ceramic and cermet cutting tool materials are for turning, they have also been used success­fully for milling. Ceramics and cermets are relatively brittle and a special cutting edge preparation is required to prevent chipping or edge breakage. This preparation consists of honing or grinding a narrow flat land, 0.002 to 0.006 inch (50.8– 152.4 mm) wide, on the cutting edge that is made about 30 degrees with respect to the tool face. For some heavy-duty applications, a wider land is used. The setup should be as rigid as possible and the feed rate should not normally exceed 0.020 inch (508 mm), although 0.030 inch (762 mm) has been used successfully. Ceramics and cermets are recommended for roughing and finish­ing operations on all cast irons, plain carbon and alloy steels, and stainless steels. Materials up to a hardness of 60 RC (Rockwell C scale) can be cut with ceramic and cermet cutting tools. These tools should not be used to machine aluminum and aluminum alloys, magnesium alloys, titanium, and titanium alloys. Cast Nonferrous Alloy: Cutting tools of this alloy are made from tungsten, tantalum, chromium, and cobalt plus carbon. Other alloying elements are also used to produce mate­ rials with high temperature and wear resistance. These alloys cannot be softened by heat treatment and must be cast and ground to shape. The room-temperature hardness of cast nonferrous alloys is lower than for high-speed steel, but the hardness and wear resistance is retained to a higher temperature. The alloys are generally marketed under trade names such as Stellite, Crobalt, and Tantung. The initial cutting speed for cast nonferrous tools can be 20 to 50 percent greater than the recommended cutting speed for high-speed steel.

Diamond Cutting Tools are available in three forms: single-crystal natural diamonds shaped to a cutting edge and mounted on a tool holder on a boring bar; polycrystalline dia­ mond indexable inserts made from synthetic or natural diamond powders that have been compacted and sintered into a solid mass, and chemically vapor-deposited diamond. Sin­ gle-crystal and polycrystalline diamond cutting tools are very wear-resistant, and recom­ mended for machining abrasive materials that cause other cutting tool materials to wear rapidly. Typical of abrasive materials machined with single-crystal and polycrystalline diamond tools and cutting speeds used are the following: fiberglass, 300 to 1000 fpm (1.5 to 5.1 m/s); fused silica, 900 to 950 fpm (4.6 to 4.8 m/s); reinforced melamine plastics, 350 to 1000 fpm (1.8 to 5.1 m/s); reinforced phenolic plastics, 350 to 1000 fpm (1.8 to 5.1 m/s); thermosetting plastics, 300 to 2000 fpm (1.5 to 10.2 m/s); Teflon, 600 fpm

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(3.0 m/s); nylon, 200 to 300 fpm (1.0 to 1.5 m/s); mica, 300 to 1000 fpm (1.5 to 5.1 m/s); graphite, 200 to 2000 fpm (1.0 to 10.1 m/s); babbitt bearing metal, 700 fpm (3.6 m/s); and aluminum-silicon alloys, 1000 to 2000 fpm (5.1 to 10.2 m/s). Another important application of diamond cutting tools is to produce fine surface fin­ ishes on soft nonferrous metals that are difficult to finish by other methods. Surface fin­ ishes of 1 to 2 microinches (0.025 to 0.051 mm) can be readily obtained with single-crystal diamond tools, and finishes down to 10 microinches (0.25 mm) can be obtained with poly­ crystalline diamond tools. In addition to babbitt and the aluminum-silicon alloys, other metals finished with diamond tools include: soft aluminum, 1000 to 2000 fpm (5.1 to 10.2 m/s); all wrought and cast aluminum alloys, 600 to 1500 fpm (3.0 to 7.6 m/s); copper, 1000 fpm (5.1 m/s); brass, 500 to 1000 fpm (2.5 to 5.1 m/s); bronze, 300 to 600 fpm (1.5 to 3.0 m/s); oilite bearing metal, 500 fpm (2.5 m/s); silver, gold, and platinum, 300 to 2500 fpm (1.5 to 12.7 m/s); and zinc, 1000 fpm (5.1 m/s). Ferrous alloys, such as cast iron and steel, should not be machined with diamond cutting tools because the high cutting temperatures generated will cause the diamond to transform into carbon.

Chemically Vapor-Deposited (CVD) Diamond: This tool material offers performance charac­teris­tics well suited to highly abrasive or corrosive materials, and hard-tomachine composites. CVD diamond is available in two forms: thick-film tools, which are fabricated by brazing CVD diamond tips, approximately 0.020 inch (0.51 mm) thick, to carbide sub­strates; and thin-film tools, having a pure diamond coating over the rake and flank surfaces of a ceramic or carbide substrate. CVD is pure diamond, made at low temperatures and pressures, with no metallic binder phase. This diamond purity gives CVD diamond tools extreme hardness, high abrasion resistance, low friction, high thermal conductivity, and chemical inertness. CVD tools are generally used as direct replacements for PCD (polycrystalline diamond) tools, primarily in finishing, semifinishing, and continuous turning applications of extremely wear-inten­ sive materials. The small grain size of CVD diamond (ranging from less than 1 μm to 50 μm) yields superior surface finishes compared with PCD, and the higher thermal conduc­tivity and better thermal and chemical stability of pure diamond allow CVD tools to oper­ate at faster speeds without generating harmful levels of heat. The extreme hardness of CVD tools may also result in significantly longer tool life. CVD diamond cutting tools are recommended for the following materials:aluminum and other ductile; nonferrous alloys such as copper, brass, and bronze; and highly abra­sive composite materials such as graphite, carbon-carbon, carbon-filled phenolic, fiber­glass, and honeycomb materials.

Cubic Boron Nitride (CBN): Next to diamond, CBN is the hardest known material. It will retain its hardness at a temperature of 1800°F and higher, making it an ideal cutting tool material for machining very hard and tough materials at cutting speeds beyond those possible with other cutting tool materials. Indexable inserts and cutting tool blanks made from this material consist of a layer, approximately 0.020 inch thick, of polycrystalline cubic boron nitride firmly bonded to the top of a cemented carbide substrate. Cubic boron nitride is recommended for rough and finish turning hardened plain carbon and alloy steels, hardened tool steels, hard cast irons, all hardness grades of gray cast iron, and super­ alloys. As a class, the superalloys are not as hard as hardened steel; however, their combi­ nation of high strength and tendency to deform plastically under the pressure of the cut, or gumminess, places them in the class of hard-to-machine materials. Conventional materials that can be readily machined with other cutting tool materials should not be machined with cubic boron nitride. Round indexable CBN inserts are recommended when taking severe cuts in order to provide maximum strength to the insert. When using square or triangular inserts, a large lead angle should be used, normally 15°, and whenever possible, 45°. A negative rake angle should always be used, which for most applications is negative 5°. The relief angle should be 5° to 9°. Although cubic boron nitride cutting tools can be used with­ out a coolant, flooding the tool with a water-soluble type coolant is recommended.

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Machinery's Handbook, 31st Edition SPEEDS AND FEEDS

Cutting Speed, Feed, Depth of Cut, Tool Wear, and Tool Life.—The cutting condi­ tions that determine the rate of metal removal are the cutting speed, the feed rate, and the depth of cut. These cutting conditions and the nature of the material to be cut determine the power required to take the cut. The cutting conditions must be adjusted to stay within the power available on the machine tool to be used. Power requirements are discussed in ESTIMATING SPEEDS AND MACHINING POWER starting on page 1143. Cutting conditions must also be considered in relation to the tool life. Tool life is defined as the cutting time to reach a predetermined amount of wear, usually flank wear. Tool life is determined by assessing the time—the tool life—at which a given predetermined flank wear is reached, 0.01 in. (0.25 mm), 0.015 in. (0.38 mm), 0.025 in. (0.64 mm), 0.03 in. (0.76 mm), for example. This amount of wear is called the tool wear criterion, and its size depends on the tool grade used. Usually, a tougher grade can be used with a bigger flank wear, but for finishing operations, where close tolerances are required, the wear criterion is relatively small. Other wear criteria are a predetermined value of the machined surface roughness and the depth of the crater that develops on the rake face of the tool. ANSI/ASME B94.55M, specification for tool life testing with single-point tools, defines the end of tool life as a given amount of wear on the flank of a tool. This standard is followed when making scientific machinability tests with single-point cutting tools in order to achieve uniformity in testing procedures so that results from different machinability laboratories can be readily compared. It is not practicable or necessary to follow this standard in the shop; however, it should be understood that the cutting conditions and tool life are related. Tool life is influenced most by cutting speed, then by feed rate, and least by depth of cut. When depth of cut is increased to about 10 times greater than the feed, a further increase in the depth of cut will have no significant effect on tool life. This characteristic of the cutting tool performance is very important in determining the operating or cutting conditions for machining metals. Conversely, if the cutting speed or feed is decreased, the increase in the tool life will be proportionately greater than the decrease in the cutting speed or the feed. Tool life is reduced when either feed or cutting speed is increased. For example, the cut­ ting speed and the feed may be increased if a shorter tool life is accepted; furthermore, the reduction in the tool life will be proportionately greater than the increase in the cutting speed or the feed. However, it is less well understood that a higher feed rate (feed/rev 3 speed) may result in a longer tool life if a higher feed/rev is used in combination with a lower cutting speed. This principle is well illustrated in the speed tables of this section, where two sets of speed and feed data are given (labeled optimum and average) that result in the same tool life. The optimum set results in a greater feed rate (i.e., increased produc­ tivity) although the feed/rev is higher and cutting speed lower than the average set. Com­ plete instructions for using the speed tables and for estimating tool life are given in How to Use the Tables starting on page 1090. Selecting Cutting Conditions.—The first step in establishing cutting conditions is to select depth of cut. The depth of cut will be limited by the amount of metal to be machined from the workpiece, by the power available on the machine tool, by the rigidity of the workpiece and cutting tool, and by the rigidity of the setup. Depth of cut has the least effect upon tool life, so the heaviest possible depth of cut should always be used. The second step is to select the feed (feed/rev for turning, drilling, and reaming, or feed/ tooth for milling). The available power must be sufficient to make the required depth of cut at the selected feed. The maximum feed possible that will produce an acceptable sur­face finish should be selected. The third step is to select the cutting speed. Although the accompanying tables provide recommended cutting speeds and feeds for many materials, experience in machining a cer­tain material may form the best basis for adjusting given cutting speeds to a particular job. In general, depth of cut should be selected first, followed by feed, and last cutting speed.

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Machinery's Handbook, 31st Edition SPEEDS AND FEEDS

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Table 1. Tool Troubleshooting Check List

Problem Excessive flank wear—tool life too short

Tool Material Carbide

HSS

Excessive cratering

Carbide

HSS

Cutting edge chipping

Carbide

HSS

Carbide and HSS Cutting edge deformation

Carbide

Poor surface finish

Carbide

Remedy 1. Change to harder, more wear-resistant grade 2. Reduce the cutting speed 3. Reduce the cutting speed and increase the feed to maintain produc­tion 4. Reduce the feed 5. For work-hardenable materials—increase the feed 6. Increase the lead angle 7. Increase the relief angles 1. Use a coolant 2. Reduce the cutting speed 3. Reduce the cutting speed and increase the feed to maintain produc­tion 4. Reduce the feed 5. For work-hardenable materials—increase the feed 6. Increase the lead angle 7. Increase the relief angle 1. Use a crater-resistant grade 2. Use a harder, more wear-resistant grade 3. Reduce the cutting speed 4. Reduce the feed 5. Widen the chip breaker groove 1. Use a coolant 2. Reduce the cutting speed 3. Reduce the feed 4. Widen the chip breaker groove 1. Increase the cutting speed 2. Lightly hone the cutting edge 3. Change to a tougher grade 4. Use negative-rake tools 5. Increase the lead angle 6. Reduce the feed 7. Reduce the depth of cut 8. Reduce the relief angles 9. If low cutting speed must be used, use a high-additive EP cutting fluid 1. Use a high additive EP cutting fluid 2. Lightly hone the cutting edge before using 3. Increase the lead angle 4. Reduce the feed 5. Reduce the depth of cut 6. Use a negative rake angle 7. Reduce the relief angles 1. Check the setup for cause if chatter occurs 2. Check the grinding procedure for tool overheating 3. Reduce the tool overhang 1. Change to a grade containing more tantalum 2. Reduce the cutting speed 3. Reduce the feed 1. Increase the cutting speed 2. If low cutting speed must be used, use a high additive EP cutting fluid 4. For light cuts, use straight titanium carbide grade 5. Increase the nose radius 6. Reduce the feed 7. Increase the relief angles 8. Use positive rake tools

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Machinery's Handbook, 31st Edition SPEEDS AND FEEDS

1084

Table Table 1. (Continued) Tool Troubleshooting 1. Tool Troubleshooting CheckCheck List List Tool Material HSS

Problem Poor surface finish (Continued)

Notching at the depth of cut line

Remedy

Diamond Carbide and HSS

1. Use a high additive EP cutting fluid 2. Increase the nose radius 3. Reduce the feed 4. Increase the relief angles 5. Increase the rake angles 1. Use diamond tool for soft materials 1. Increase the lead angle 2. Reduce the feed

Cutting Speed Formulas Most machining operations are conducted on machine tools having a rotating spindle. Cut­ting speeds are usually given in feet or meters per minute and these speeds must be con­verted to spindle speeds, in revolutions per minute, to operate the machine. Conversion is accomplished by use of the following formulas: US Units: 12V 12 # 252 N = π D = π # 8 = 120 rpm

N=

Metric Units: 1000V V π D = 318. 3 D rpm

where N is the spindle speed in revolutions per minute (rpm); V is the cutting speed in feet per minute (fpm) for US units and meters per minute (m/min) for metric units. In turning, D is the diameter of the workpiece; in milling, drilling, reaming, and other operations that use a rotating tool, D is the cutter diameter in inches for US units and in millimeters for metric units. π = 3.1416. Example: The cutting speed for turning a 4-inch (101.6-mm) diameter bar has been found to be 575 fpm (175.3 m/min). Using both the inch and metric formulas, calculate the lathe spindle speed.

Metric Units:

US Units:

12V 12 × 575 N = π D = 3.1416 × 4 = 549 rpm

1000V 1000 × 175. 3 N = π D = 3.1416 × 101.6 = 549 rpm

When the cutting tool or workpiece diameter and the spindle speed in rpm are known, it is often necessary to calculate the cutting speed in feet or meters per minute. In this event, the following formulas are used. US Units: π DN V = 12 fpm

Metric Units: π DN V = 1000 m/min

As in the previous formulas, N is the rpm and D is the diameter in inches for the US Units formula and in millimeters for the metric formula. Example: Calculate the cutting speed in feet per minute and in meters per minute if the spindle speed of a 3 ⁄4 -inch (19.05-mm) drill is 400 rpm. US Units: Metric Units:

π DN π × 0.75 × 400 = 78.5 fpm 12 = 12 π DN π × 19. 05 × 400 V = 1000 = = 24.9 m/min 1000

V=

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Machinery's Handbook, 31st Edition SPEEDS AND FEEDS

1085

Cutting Speeds and Equivalent RPM for Drills of Number and Letter Sizes Size No.

1 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54 Size A B C D E F G H I J K L M N O P Q R S T U V W X Y Z

30'

40'

50'

503 518 548 562 576 592 606 630 647 678 712 730 754 779 816 892 988 1032 1076 1129 1169 1226 1333 1415 1508 1637 1805 2084

670 691 731 749 768 790 808 840 863 904 949 973 1005 1039 1088 1189 1317 1376 1435 1505 1559 1634 1777 1886 2010 2183 2406 2778

838 864 914 936 960 987 1010 1050 1079 1130 1186 1217 1257 1299 1360 1487 1647 1721 1794 1882 1949 2043 2221 2358 2513 2729 3008 3473

491 482 473 467 458 446 440 430 421 414 408 395 389 380 363 355 345 338 329 320 311 304 297 289 284 277

654 642 631 622 611 594 585 574 562 552 544 527 518 506 484 473 460 451 439 426 415 405 396 385 378 370

818 803 789 778 764 743 732 718 702 690 680 659 648 633 605 592 575 564 549 533 519 507 495 481 473 462

Cutting Speed, Feet per Minute 60' 70' 80' 90' 100' Revolutions per Minute for Number Sizes

1005 1173 1340 1508 1675 1037 1210 1382 1555 1728 1097 1280 1462 1645 1828 1123 1310 1498 1685 1872 1151 1343 1535 1727 1919 1184 1382 1579 1777 1974 1213 1415 1617 1819 2021 1259 1469 1679 1889 2099 1295 1511 1726 1942 2158 1356 1582 1808 2034 2260 1423 1660 1898 2135 2372 1460 1703 1946 2190 2433 1508 1759 2010 2262 2513 1559 1819 2078 2338 2598 1631 1903 2175 2447 2719 1784 2081 2378 2676 2973 1976 2305 2634 2964 3293 2065 2409 2753 3097 3442 2152 2511 2870 3228 3587 2258 2634 3010 3387 3763 2339 2729 3118 3508 3898 2451 2860 3268 3677 4085 2665 3109 3554 3999 4442 2830 3301 3773 4244 4716 3016 3518 4021 4523 5026 3274 3820 4366 4911 5457 3609 4211 4812 5414 6015 4167 4862 5556 6251 6945 Revolutions per Minute for Letter Sizes 982 963 947 934 917 892 878 862 842 827 815 790 777 759 725 710 690 676 659 640 623 608 594 576 567 555

1145 1124 1105 1089 1070 1040 1024 1005 983 965 951 922 907 886 846 828 805 789 769 746 727 709 693 672 662 647

For fractional drill sizes, use the following table.

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1309 1284 1262 1245 1222 1189 1170 1149 1123 1103 1087 1054 1036 1012 967 946 920 902 878 853 830 810 792 769 756 740

1472 1445 1420 1400 1375 1337 1317 1292 1264 1241 1223 1185 1166 1139 1088 1065 1035 1014 988 959 934 912 891 865 851 832

1636 1605 1578 1556 1528 1486 1463 1436 1404 1379 1359 1317 1295 1265 1209 1183 1150 1127 1098 1066 1038 1013 989 962 945 925

110'

130'

150'

1843 1901 2010 2060 2111 2171 2223 2309 2374 2479 2610 2676 2764 2858 2990 3270 3622 3785 3945 4140 4287 4494 4886 5187 5528 6002 6619 7639

2179 2247 2376 2434 2495 2566 2627 2728 2806 2930 3084 3164 3267 3378 3534 3864 4281 4474 4663 4892 5067 5311 5774 6130 6534 7094 7820 9028

2513 2593 2741 2809 2879 2961 3032 3148 3237 3380 3559 3649 3769 3898 4078 4459 4939 5162 5380 5645 5846 6128 6662 7074 7539 8185 9023 10417

1796 1765 1736 1708 1681 1635 1610 1580 1545 1517 1495 1449 1424 1391 1330 1301 1266 1239 1207 1173 1142 1114 1088 1058 1040 1017

2122 2086 2052 2018 1968 1932 1903 1867 1826 1793 1767 1712 1683 1644 1571 1537 1496 1465 1427 1387 1349 1317 1286 1251 1229 1202

2448 2407 2368 2329 2292 2229 2195 2154 2106 2068 2039 1976 1942 1897 1813 1774 1726 1690 1646 1600 1557 1520 1484 1443 1418 1387

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Machinery's Handbook, 31st Edition RPM FOR VARIOUS SPEEDS AND DIAMETERS

1086

Revolutions per Minute for Various Cutting Speeds and Diameters Dia., Inches 1 ∕4 5 ∕16 3 ∕8

7∕16 1 ∕2

9 ∕16 5 ∕8

11 ∕16 3 ∕4

13 ∕16 7∕8

15 ∕16 1 11 ∕16 11 ∕8 13∕16 11 ∕4 15∕16 13∕8 17∕16 11 ∕2 19∕16 15∕8 111 ∕16 13∕4 17∕8 2 21 ∕8 21 ∕4 23∕8 21 ∕2 25∕8 23∕4 27∕8 3 31 ∕8 31 ∕4 33∕8 31 ∕2 35∕8 33∕4 37∕8 4 41 ∕4 41 ∕2 43∕4 5 51 ∕4 51 ∕2 53∕4 6 61 ∕4 61 ∕2 63∕4 7 71 ∕4 71 ∕2 73∕4 8

40

50

60

70

611

764

917

1070

489 408 349 306 272 245 222 203 190 175 163 153 144 136 129 123 116 111 106 102 97.6 93.9 90.4 87.3 81.5 76.4 72.0 68.0 64.4 61.2 58.0 55.6 52.8 51.0 48.8 46.8 45.2 43.6 42.0 40.8 39.4 38.2 35.9 34.0 32.2 30.6 29.1 27.8 26.6 25.5 24.4 23.5 22.6 21.8 21.1 20.4 19.7 19.1

611 509 437 382 340 306 273 254 237 219 204 191 180 170 161 153 146 139 133 127 122 117 113 109 102 95.5 90.0 85.5 80.5 76.3 72.5 69.5 66.0 63.7 61.0 58.5 56.5 54.5 52.5 51.0 49.3 47.8 44.9 42.4 40.2 38.2 36.4 34.7 33.2 31.8 30.6 29.4 28.3 27.3 26.4 25.4 24.6 23.9

733 611 524 459 407 367 333 306 284 262 244 229 215 204 193 183 175 167 159 153 146 141 136 131 122 115 108 102 96.6 91.7 87.0 83.4 79.2 76.4 73.2 70.2 67.8 65.5 63.0 61.2 59.1 57.3 53.9 51.0 48.2 45.9 43.6 41.7 39.8 38.2 36.7 35.2 34.0 32.7 31.6 30.5 29.5 28.7

856 713 611 535 475 428 389 357 332 306 285 267 251 238 225 214 204 195 186 178 171 165 158 153 143 134 126 119 113 107 102 97.2 92.4 89.1 85.4 81.9 79.1 76.4 73.5 71.4 69.0 66.9 62.9 59.4 56.3 53.5 50.9 48.6 46.5 44.6 42.8 41.1 39.6 38.2 36.9 35.6 34.4 33.4

Cutting Speed, Feet per Minute 80 90 100 120 Revolutions per Minute

1222

978 815 699 611 543 489 444 408 379 349 326 306 287 272 258 245 233 222 212 204 195 188 181 175 163 153 144 136 129 122 116 111 106 102 97.6 93.6 90.4 87.4 84.0 81.6 78.8 76.4 71.8 67.9 64.3 61.1 58.2 55.6 53.1 51.0 48.9 47.0 45.3 43.7 42.2 40.7 39.4 38.2

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1376

1100 916 786 688 611 552 500 458 427 392 366 344 323 306 290 274 262 250 239 230 220 212 203 196 184 172 162 153 145 138 131 125 119 114 110 105 102 98.1 94.5 91.8 88.6 86.0 80.8 76.3 72.4 68.8 65.4 62.5 59.8 57.2 55.0 52.8 50.9 49.1 47.4 45.8 44.3 43.0

1528

1222 1018 874 764 679 612 555 508 474 438 407 382 359 340 322 306 291 278 265 254 244 234 226 218 204 191 180 170 161 153 145 139 132 127 122 117 113 109 105 102 98.5 95.6 89.8 84.8 80.4 76.4 72.7 69.4 66.4 63.6 61.1 58.7 56.6 54.6 52.7 50.9 49.2 47.8

1834

1466 1222 1049 917 813 736 666 610 569 526 488 458 431 408 386 367 349 334 318 305 293 281 271 262 244 229 216 204 193 184 174 167 158 152 146 140 136 131 126 122 118 115 108 102 96.9 91.7 87.2 83.3 80.0 76.3 73.3 70.4 67.9 65.5 63.2 61.1 59.0 57.4

140 2139

1711 1425 1224 1070 951 857 770 711 664 613 570 535 503 476 451 428 407 389 371 356 342 328 316 305 286 267 252 238 225 213 203 195 185 178 171 164 158 153 147 143 138 134 126 119 113 107 102 97.2 93.0 89.0 85.5 82.2 79.2 76.4 73.8 71.0 68.9 66.9

160 2445

1955 1629 1398 1222 1086 979 888 813 758 701 651 611 575 544 515 490 466 445 424 406 390 374 362 349 326 306 288 272 258 245 232 222 211 203 195 188 181 174 168 163 158 153 144 136 129 122 116 111 106 102 97.7 93.9 90.6 87.4 84.3 81.4 78.7 76.5

180 2750

2200 1832 1573 1375 1222 1102 999 914 853 788 733 688 646 612 580 551 524 500 477 457 439 421 407 392 367 344 324 306 290 275 261 250 238 228 219 211 203 196 189 184 177 172 162 153 145 138 131 125 120 114 110 106 102 98.3 94.9 91.6 88.6 86.0

200 3056

2444 2036 1748 1528 1358 1224 1101 1016 948 876 814 764 718 680 644 612 582 556 530 508 488 468 452 436 408 382 360 340 322 306 290 278 264 254 244 234 226 218 210 205 197 191 180 170 161 153 145 139 133 127 122 117 113 109 105 102 98.4 95.6

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Machinery's Handbook, 31st Edition RPM FOR VARIOUS SPEEDS AND DIAMETERS

1087

Revolutions per Minute for Various Cutting Speeds and Diameters Dia., Inches 1 ∕4 5 ∕16 3 ∕8

7∕16 1 ∕2

9 ∕16 5 ∕8

11 ∕16 3 ∕4

13 ∕16

7∕8 15 ∕16 1 11 ∕16 11 ∕8 13∕16 11 ∕4 15∕16 13∕8 17∕16 11 ∕2 19∕16 15∕8 111 ∕16 13∕4 113∕16 17∕8 115∕16 2 21 ∕8 21 ∕4 23∕8 21 ∕2 25∕8 23∕4 27∕8 3 31 ∕8 31 ∕4 33∕8 31 ∕2 35∕8 33∕4 37∕8 4 41 ∕4 41 ∕2 43∕4 5 51 ∕4 51 ∕2 53∕4 6 61 ∕4 61 ∕2 63∕4 7 71 ∕4 71 ∕2 73∕4 8

225

250

275

300

3438

3820

4202

4584

2750 2292 1964 1719 1528 1375 1250 1146 1058 982 917 859 809 764 724 687 654 625 598 573 550 528 509 491 474 458 443 429 404 382 362 343 327 312 299 286 274 264 254 245 237 229 221 214 202 191 180 171 163 156 149 143 137 132 127 122 118 114 111 107

3056 2546 2182 1910 1698 1528 1389 1273 1175 1091 1019 955 899 849 804 764 727 694 664 636 611 587 566 545 527 509 493 477 449 424 402 382 363 347 332 318 305 293 283 272 263 254 246 238 224 212 201 191 181 173 166 159 152 146 141 136 131 127 123 119

3362 2801 2401 2101 1868 1681 1528 1401 1293 1200 1120 1050 988 933 884 840 800 764 730 700 672 646 622 600 579 560 542 525 494 468 442 420 400 381 365 350 336 323 311 300 289 280 271 262 247 233 221 210 199 190 182 174 168 161 155 149 144 139 135 131

3667 3056 2619 2292 2037 1834 1667 1528 1410 1310 1222 1146 1078 1018 965 917 873 833 797 764 733 705 679 654 632 611 591 573 539 509 482 458 436 416 398 381 366 352 339 327 316 305 295 286 269 254 241 229 218 208 199 190 183 176 169 163 158 152 148 143

Cutting Speed, Feet per Minute 325 350 375 400 Revolutions per Minute

4966

3973 3310 2837 2483 2207 1987 1806 1655 1528 1419 1324 1241 1168 1103 1045 993 946 903 863 827 794 764 735 709 685 662 640 620 584 551 522 496 472 451 431 413 397 381 367 354 342 331 320 310 292 275 261 248 236 225 215 206 198 190 183 177 171 165 160 155

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5348

4278 3565 3056 2675 2377 2139 1941 1783 1646 1528 1426 1337 1258 1188 1126 1069 1018 972 930 891 855 822 792 764 737 713 690 668 629 594 563 534 509 486 465 445 427 411 396 381 368 356 345 334 314 297 281 267 254 242 232 222 213 205 198 190 184 178 172 167

5730

4584 3820 3274 2866 2547 2292 2084 1910 1763 1637 1528 1432 1348 1273 1206 1146 1091 1042 996 955 916 881 849 818 790 764 739 716 674 636 603 573 545 520 498 477 458 440 424 409 395 382 369 358 337 318 301 286 272 260 249 238 229 220 212 204 197 190 185 179

6112

4889 4074 3492 3057 2717 2445 2223 2038 1881 1746 1630 1528 1438 1358 1287 1222 1164 1111 1063 1018 978 940 905 873 843 815 788 764 719 679 643 611 582 555 531 509 488 470 452 436 421 407 394 382 359 339 321 305 290 277 265 254 244 234 226 218 210 203 197 191

425

450

500

550

6493

6875

7639

8403

5195 4329 3710 3248 2887 2598 2362 2165 1998 1855 1732 1623 1528 1443 1367 1299 1237 1181 1129 1082 1039 999 962 927 895 866 838 811 764 721 683 649 618 590 564 541 519 499 481 463 447 433 419 405 383 360 341 324 308 294 282 270 259 249 240 231 223 216 209 203

5501 4584 3929 3439 3056 2751 2501 2292 2116 1965 1834 1719 1618 1528 1448 1375 1309 1250 1196 1146 1100 1057 1018 982 948 917 887 859 809 764 724 687 654 625 598 572 549 528 509 490 474 458 443 429 404 382 361 343 327 312 298 286 274 264 254 245 237 229 222 215

6112 5093 4365 3821 3396 3057 2779 2547 2351 2183 2038 1910 1798 1698 1609 1528 1455 1389 1329 1273 1222 1175 1132 1091 1054 1019 986 955 899 849 804 764 727 694 664 636 611 587 566 545 527 509 493 477 449 424 402 382 363 347 332 318 305 293 283 272 263 254 246 238

6723 5602 4802 4203 3736 3362 3056 2802 2586 2401 2241 2101 1977 1867 1769 1681 1601 1528 1461 1400 1344 1293 1245 1200 1159 1120 1084 1050 988 933 884 840 800 763 730 700 672 646 622 600 579 560 542 525 494 466 442 420 399 381 365 349 336 322 311 299 289 279 271 262

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Machinery's Handbook, 31st Edition RPM FOR VARIOUS SPEEDS AND DIAMETERS

1088

Revolutions per Minute for Various Cutting Speeds and Diameters (Metric Units) Dia., mm

Cutting Speed, Meters per Minute 5

6

8

10

12

16

20

25

30

35

40

45

Revolutions per Minute

5

318

382

509

637

764

1019

1273

1592

1910

2228

2546

2865

6

265

318

424

530

637

849

1061

1326

1592

1857

2122

2387

8

199

239

318

398

477

637

796

995

1194

1393

1592

1790

10

159

191

255

318

382

509

637

796

955

1114

1273

1432

12

133

159

212

265

318

424

531

663

796

928

1061

1194

119

159

199

239

318

398

497

597

696

796

895

95.5

127

159

191

255

318

398

477

557

637

716

102

16

99.5

20

79.6

25

63.7

76.4

30

53.1

63.7

84.9

127

153

204

255

318

382

446

509

573

106

127

170

212

265

318

371

424

477

35

45.5

54.6

72.8

90.9

40

39.8

47.7

63.7

79.6

109 95.5

145

182

227

273

318

364

409

127

159

199

239

279

318

358

45

35.4

42.4

56.6

70.7

84.9

113

141

177

212

248

283

318

50

31.8

38.2

51

63.7

76.4

102

127

159

191

223

255

286

55

28.9

34.7

46.3

57.9

69.4

92.6

116

145

174

203

231

260

60

26.6

31.8

42.4

53.1

63.7

84.9

106

133

159

186

212

239

65

24.5

29.4

39.2

49

58.8

78.4

98

122

147

171

196

220

70

22.7

27.3

36.4

45.5

54.6

72.8

90.9

114

136

159

182

205

75

21.2

25.5

34

42.4

51

68

84.9

106

127

149

170

191

80

19.9

23.9

31.8

39.8

47.7

63.7

79.6

99.5

119

139

159

179

106

90

17.7

21.2

28.3

35.4

42.4

56.6

70.7

88.4

100

15.9

19.1

25.5

31.8

38.2

51

63.7

79.6

124

141

159

95.5

111

127

143

101

116

130

110

14.5

17.4

23.1

28.9

34.7

46.2

57.9

72.3

86.8

120

13.3

15.9

21.2

26.5

31.8

42.4

53.1

66.3

79.6

92.8

130

12.2

14.7

19.6

24.5

29.4

39.2

49

61.2

73.4

85.7

106

119

97.9

110 102

140

11.4

13.6

18.2

22.7

27.3

36.4

45.5

56.8

68.2

79.6

90.9

150

10.6

12.7

17

21.2

25.5

34

42.4

53.1

63.7

74.3

84.9

95.5

160

9.9

11.9

15.9

19.9

23.9

31.8

39.8

49.7

59.7

69.6

79.6

89.5

170

9.4

11.2

15

18.7

22.5

30

37.4

46.8

56.2

65.5

74.9

84.2

180

8.8

10.6

14.1

17.7

21.2

28.3

35.4

44.2

53.1

61.9

70.7

79.6

190

8.3

10

13.4

16.8

20.1

26.8

33.5

41.9

50.3

58.6

67

75.4

200

8

39.5

12.7

15.9

19.1

25.5

31.8

39.8

47.7

55.7

63.7

71.6

220

7.2

8.7

11.6

14.5

17.4

23.1

28.9

36.2

43.4

50.6

57.9

65.1

240

6.6

8

10.6

13.3

15.9

21.2

26.5

33.2

39.8

46.4

53.1

59.7

260

6.1

7.3

9.8

12.2

14.7

19.6

24.5

30.6

36.7

42.8

49

55.1

280

5.7

6.8

9.1

11.4

13.6

18.2

22.7

28.4

34.1

39.8

45.5

51.1

300

5.3

6.4

8.5

10.6

12.7

17

21.2

26.5

31.8

37.1

42.4

47.7

350

4.5

5.4

7.3

9.1

10.9

14.6

18.2

22.7

27.3

31.8

36.4

40.9

400

4

4.8

6.4

8

9.5

12.7

15.9

19.9

23.9

27.9

31.8

35.8

450

3.5

4.2

5.7

7.1

8.5

11.3

14.1

17.7

21.2

24.8

28.3

31.8

500

3.2

3.8

5.1

6.4

7.6

10.2

12.7

15.9

19.1

22.3

25.5

28.6

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Machinery's Handbook, 31st Edition RPM FOR VARIOUS SPEEDS AND DIAMETERS

1089

Revolutions per Minute for Various Cutting Speeds and Diameters (Metric Units) Dia., mm

Cutting Speed, Meters per Minute 50

55

60

65

70

75

80

85

90

95

100

200

Revolutions per Minute

5

3183

3501

3820

4138

4456

4775

5093

5411

5730

6048

6366

12,732

6

2653

2918

3183

3448

3714

3979

4244

4509

4775

5039

5305

10,610

8

1989

2188

2387

2586

2785

2984

3183

3382

3581

3780

3979

7958

10

1592

1751

1910

2069

2228

2387

2546

2706

2865

3024

3183

6366

12

1326

1459

1592

1724

1857

1989

2122

2255

2387

2520

2653

5305

16

995

1094

1194

1293

1393

1492

1591

1691

1790

1890

1989

3979

20

796

875

955

1034

1114

1194

1273

1353

1432

1512

1592

3183

25

637

700

764

828

891

955

1019

1082

1146

1210

1273

2546

30

530

584

637

690

743

796

849

902

955

1008

1061

2122

35

455

500

546

591

637

682

728

773

819

864

909

1818

40

398

438

477

517

557

597

637

676

716

756

796

1592

45

354

389

424

460

495

531

566

601

637

672

707

1415 1273

50

318

350

382

414

446

477

509

541

573

605

637

55

289

318

347

376

405

434

463

492

521

550

579

1157

60

265

292

318

345

371

398

424

451

477

504

530

1061

65

245

269

294

318

343

367

392

416

441

465

490

979

70

227

250

273

296

318

341

364

387

409

432

455

909

75

212

233

255

276

297

318

340

361

382

403

424

849

80

199

219

239

259

279

298

318

338

358

378

398

796

90

177

195

212

230

248

265

283

301

318

336

354

707

100

159

175

191

207

223

239

255

271

286

302

318

637

110

145

159

174

188

203

217

231

246

260

275

289

579

120

133

146

159

172

186

199

212

225

239

252

265

530

130

122

135

147

159

171

184

196

208

220

233

245

490

140

114

125

136

148

159

171

182

193

205

216

227

455

150

106

117

127

138

149

159

170

180

191

202

212

424

160

99.5

109

119

129

139

149

159

169

179

189

199

398

170

93.6

103

112

122

131

140

150

159

169

178

187

374

180

88.4

97.3

106

115

124

133

141

150

159

168

177

354

190

83.8

92.1

101

109

117

126

134

142

151

159

167

335

200

79.6

87.5

95.5

103

111

119

127

135

143

151

159

318

220

72.3

79.6

86.8

94

101

109

116

123

130

137

145

289

240

66.3

72.9

79.6

86.2

92.8

99.5

106

113

119

126

132

265

260

61.2

67.3

73.4

79.6

85.7

91.8

97.9

104

110

116

122

245

280

56.8

62.5

68.2

73.9

79.6

85.3

90.9

96.6

102

108

114

227

300

53.1

58.3

63.7

69

74.3

79.6

84.9

90.2

95.5

101

106

212

350

45.5

50

54.6

59.1

63.7

68.2

72.8

77.3

81.8

99.1

91

182

400

39.8

43.8

47.7

51.7

55.7

59.7

63.7

67.6

71.6

75.6

79.6

159

450

35.4

38.9

42.4

46

49.5

53.1

56.6

60.1

63.6

67.2

70.7

141

500

31.8

35

38.2

41.4

44.6

47.7

50.9

54.1

57.3

60.5

63.6

127

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1090

Machinery's Handbook, 31st Edition SPEEDS and FEEDS SPEEDS AND FEEDS TABLES How to Use the Tables

The principal tables of speed and feed values are listed in the table below. In this section, Table 1 through Table 9 give data for turning, Table 10 through Table 15e give data for milling, and Table 17 through Table 23 give data for ream­ing, drilling, threading.

The materials in these tables are categorized by description, and Brinell Hardness Num­ ber (BHN) range or material condition. So far as possible, work materials are grouped by similar machining characteristics. The types of cutting tools (HSS end mill, for example) are identified in one or more rows across the tops of the tables. Other important details con­cerning the use of the tables are contained in the footnotes to Table 1, Table 10 and Table 17. Informa­tion concerning specific cutting tool grades is given in notes at the end of each table. Principal Speeds and Feeds Tables Speeds and Feeds for Turning

Table 1. Cutting Speeds and Feeds for Turning Plain Carbon and Alloy Steels Table 2. Cutting Speeds and Feeds for Turning Tool Steels Table 3. Cutting Speeds and Feeds for Turning Stainless Steels Table 4a. Cutting Speeds and Feeds for Turning Ferrous Cast Metals Table 4b. Cutting Speeds and Feeds for Turning Ferrous Cast Metals Table 5a. Turning-Speed Adjustment Factors for Feed, Depth of Cut, and Lead Angle Table 5b. Tool Life Factors for Turning with Carbides, Ceramics, Cermets, CBN, and Polycrystalline Diamond Table 5c. Cutting-Speed Adjustment Factors for Turning with HSS Tools Table 6. Cutting Speeds and Feeds for Turning Copper Alloys Table 7. Cutting Speeds and Feeds for Turning Titanium and Titanium Alloys Table 8. Cutting Speeds and Feeds for Turning Light Metals Table 9. Cutting Speeds and Feeds for Turning Superalloys

Speeds and Feeds for Milling

Table 10. Cutting Speeds and Feeds for Milling Aluminum Alloys Table 11. Cutting Speeds and Feeds for Milling Plain Carbon and Alloy Steels Table 12. Cutting Speeds and Feeds for Milling Tool Steels Table 13. Cutting Speeds and Feeds for Milling Stainless Steels Table 14. Cutting Speeds and Feeds for Milling Ferrous Cast Metals Table 15a. Recommended Feed in Inches per Tooth (ft) for Milling with High-Speed Steel Cutters Table 15b. End Milling (Full Slot) Speed Adjustment Factors for Feed, Depth of Cut, and Lead Angle Table 15c. End, Slit, and Side Milling Speed Adjustment Factors for Radial Depth of Cut Table 15d. Face Milling Speed Adjustment Factors for Feed, Depth of Cut, and Lead Angle Table 15e. Tool Life Adjustment Factors for Face Milling, End Milling, Drilling, and Reaming Table 16. Cutting Tool Grade Descriptions and Common Vendor Equivalents Table 17. Table 18. Table 19. Table 20. Table 21. Table 22. Table 23.

Speeds and Feeds for Drilling, Reaming, and Threading

Speeds and Feeds for Drilling, Reaming, and Threading Plain Carbon and Alloy Steels Speeds and Feeds for Drilling, Reaming, and Threading Tool Steels Speeds and Feeds for Drilling, Reaming, and Threading Stainless Steels Speeds and Feeds for Drilling, Reaming, and Threading Ferrous Cast Metals Speeds and Feeds for Drilling, Reaming, and Threading Light Metals Diameter Speed and Feed Adjustment Factors for HSS Twist Drills and Reamers Speeds and Feeds for Drilling and Reaming Copper Alloys

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Machinery's Handbook, 31st Edition USING THE SPEEDS AND FEEDS TABLES

1091

Each of the cutting speed tables in this section contains two distinct types of cutting speed data. The speed columns at the left of each table contain traditional Handbook cutting speeds for use with high-speed steel (HSS) tools. For many years, this extensive collection of cutting data has been used successfully as starting speed values for turning, milling, drilling, and reaming operations. Instructions and adjustment factors for use with these speeds are given in Table 5c (feed and depth-of-cut factors) for turning, and in Table 15a (feed, depth of cut, and cutter diameter) for milling. Feeds for drilling and reaming are dis­cussed in Using the Speeds and Feeds Tables for Drilling, Reaming, and Threading on page 1128. With traditional speeds and feeds, tool life may vary greatly from material to material, making it very difficult to plan efficient cutting operations, in particular for setting up unattended jobs on CNC equipment where the tool life must exceed cutting time, or at least be predict­able so that tool changes can be scheduled. This limitation is reduced by using the com­bined feed/speed data contained in the remaining columns of the speed tables.

The combined feed/speed portion of the speed tables gives two sets of speed and feed data for each material represented. These feed/speed pairs are the optimum and average data (identified by Opt. and Avg.); the optimum set is always on the left side of the column and the average set is on the right. The optimum feed/speed data are approximate values of speed and feed that achieve minimum-cost machining by combining a high productivity rate with low tooling cost at a fixed tool life. The average feed/speed data are expected to achieve approximately the same tool life and tooling costs, but productivity is usually lower, so machining costs are higher. The data in this portion of the tables are given in the form of two numbers, of which the first is the feed in thousandths of an inch per revolution (or per tooth, for milling) and the second is the cutting speed in feet per minute. For exam­ ple, the feed/speed set 15∕215 represents a feed of 0.015 in/rev (0.38 mm/rev) at a speed of 215 fpm (65.6 m/min). Blank cells in the data tables indicate that feed/speed data for these materials were not available at the time of publication. Generally, the feed given in the optimum set should be interpreted as the maximum safe feed for the given work material and cutting tool grade, and the use of a greater feed may result in premature tool wear or tool failure before the end of the expected tool life. The primary exception to this rule occurs in milling, where the feed may be greater than the optimum feed if the radial depth of cut is less than the value established in the table foot­ note; this topic is covered later in the milling examples. Thus, except for milling, the speed and tool life adjustment tables, to be discussed later, do not permit feeds that are greater than the optimum feed. On the other hand, the speed and tool life adjustment factors often result in cutting speeds that are well outside the given optimum to average speed range.

The combined feed/speed data in this section were contributed by Dr. Colding of Colding International Corp., Ann Arbor, MI. The speed, feed, and tool life calculations were made by means of a special computer program and a large database of cutting speed and tool life testing data. The COMP computer program uses tool life equations that are extensions of the F. W. Taylor tool life equation, first proposed in the early 1900s. The Colding tool life equations use a concept called equivalent chip thickness (ECT), which simplifies cutting speed and tool life predictions, and the calculation of cutting forces, torque, and power requirements. ECT is a basic metal cutting parameter that combines the four basic turning variables (depth of cut, lead angle, nose radius, and feed per revolution) into one basic parameter. For other metal cutting operations (milling, drilling, and grinding, for exam­ple), ECT also includes additional variables such as the number of teeth, width of cut, and cutter diameter. The ECT concept was first presented in 1931 by Prof. R. Woxen, who showed that equivalent chip thickness is a basic metal cutting parameter for high-speed cutting tools. Dr. Colding later extended the theory to include other tool materials and metal cutting operations, including grinding. The equivalent chip thickness is defined by ECT = A/CEL, where A is the cross-sectional area of the cut (approximately equal to the feed times the depth of cut), and CEL is the cut­ ting edge length or tool contact rubbing length. ECT and several other terms related to tool

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Machinery's Handbook, 31st Edition USING THE SPEEDS AND FEEDS TABLES

1092

geometry are illustrated in Fig. 1 and Fig. 2. Many combinations of feed, lead angle, nose radius and cutter diameter, axial and radial depth of cut, and numbers of teeth can give the same value of ECT. However, for a constant cutting speed, no matter how the depth of cut, feed, or lead angle, etc., are varied, if a constant value of ECT is maintained, the tool life will also remain constant. A constant value of ECT means that a constant cutting speed gives a constant tool life and an increase in speed results in a reduced tool life. Likewise, if ECT were increased and cutting speed were held constant, as illustrated in the generalized cutting speed versus ECT graph that follows, tool life would be reduced. EC

CE L

T

CELe

a

r

A'

A f

LA (ISO)

a = depth of cut A = A'   = chip cross-sectional area CEL = CELe = engaged cutting edge length ECT = equivalent chip thickness = A'/CEL f = feed/rev r = nose radius LA = lead angle (US) L A(ISO) = 90−LA

LA (US)

Fig. 1. Cutting Geometry, Equivalent Chip Thickness, and Cutting Edge Length

CEL

A

LA (ISO)

A– A Rake Angle

A

LA (US)

Fig. 2. Cutting Geometry for Turning

In the tables, the optimum feed/speed data have been calculated by COMP to achieve a fixed tool life based on the maximum ECT that will result in successful cutting, without premature tool wear or early tool failure. The same tool life is used to calculate the average feed/speed data, but these values are based on one-half of the maximum ECT. Because the data are not linear except over a small range of values, both optimum and average sets are required to adjust speeds for feed, lead angle, depth of cut, and other factors.

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Machinery's Handbook, 31st Edition USING THE SPEEDS AND FEEDS TABLES

1093

Tool life is the most important factor in a machining system, so speeds and feeds cannot be selected as simple numbers, but must be considered with respect to the many parameters that influence tool life. The accuracy of the combined feed/speed data presented is believed to be very high. However, machining is a variable and complicated process and use of the speeds and feeds tables requires the user to follow the instructions carefully to achieve good predictability. The results achieved, therefore, may vary due to material con­dition, tool material, machine setup, and other factors, and cannot be guaranteed. The feed values given in the tables are valid for the standard tool geometries and fixed depths of cut that are identified in the table footnotes. If the cutting parameters and tool geometry established in the table footnotes are maintained, turning operations using either the optimum or average feed/speed data (Table 1 through Table 9) should achieve a constant tool life of approximately 15 minutes; tool life for milling, drilling, reaming, and threading data (Table 10 through Table 14 and Table 17 through Table 22) should be approximately 45 minutes. The reason for the different economic tool lives is the higher tooling cost associated with milling-drilling operations than for turning. If the cutting parameters or tool geometry are different from those established in the table footnotes, the same tool life (15 or 45 minutes) still may be maintained by applying the appropriate speed adjustment factors, or tool life may be increased or decreased using tool life adjustment factors. The use of the speed and tool life adjustment factors is described in the examples that follow. Both the optimum and average feed/speed data given are reasonable values for effective cutting. However, the optimum set with its higher feed and lower speed (always the left entry in each table cell) will usually achieve greater productivity. In Table 1, for example, the two entries for turning 1212 free-machining plain carbon steel with uncoated carbide are 17∕805 and 8∕1075. These values indicate that a feed of 0.017 in/rev and a speed of 805 ft/min, or a feed of 0.008 in/rev and a speed of 1075 ft/min can be used for this material. The tool life, in each case, will be approximately 15 minutes. If one of these speed and feed pairs is assigned an arbitrary cutting time of 1 minute, then the relative cutting time of the second pair to the first is equal to the ratio of their respective feed 3 speed products. Here, the same amount of material that can be cut in 1 minute, at the higher feed and lower speed (17∕805), will require 1.6 minutes at the lower feed and higher speed (8∕1075) because 17 3 805/(8 3 1075) = 1.6 minutes.

V = Cutting Speed (m/min)

1000

Tool Life, T (min)

100

T=5 T = 15 T = 45 T = 120

10 0.01

0.1

Equivalent Chip Thickness, ECT (mm)

1

Cutting Speed versus Equivalent Chip Thickness with Tool Life as a Parameter

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1094

Machinery's Handbook, 31st Edition SPEEDS AND FEEDS FOR TURNING

Speeds and Feeds Tables for Turning.—Speeds for HSS (high-speed steel) tools are based on a feed of 0.012 inch/rev and a depth of cut of 0.125 inch; use Table 5c to adjust the given speeds for other feeds and depths of cut. The combined feed/speed data in the remaining columns are based on a depth of cut of 0.1 inch, lead angle of 15 degrees, and nose radius of 3∕64 inch. Use Table 5a to adjust given speeds for other feeds, depths of cut, and lead angles; use Table 5b to adjust given speeds for increased tool life up to 180 min­ utes. Examples are given in the text. Examples Using the Speeds and Feeds Tables for Turning: The examples that follow give instructions for determining cutting speeds for turning. In general, the same methods are also used to find cutting speeds for milling, drilling, reaming, and threading, so reading through these examples may bring some additional insight to those other metalworking processes as well. The first step in determining cutting speeds is to locate the work material in the left column of the appropriate table for turning, milling, or drilling, reaming, and threading.

Example 1, Turning: Find the cutting speed for turning SAE 1074 plain carbon steel of 225 to 275 BHN (Brinell Hardness Number), using an uncoated carbide insert, a feed of 0.015 in/rev, and a depth of cut of 0.1 inch.

In Table 1, speed and feed data for two types of uncoated carbide tools are given, one for hard tool grades, the other for tough tool grades. In general, use the speed data from the tool category that most closely matches the tool to be used because there are often significant differences in the speeds and feeds for different tool grades. From the uncoated carbide hard grade values, the optimum and average feed/speed data given in Table 1 are 17∕615 and 8∕815, or 0.017 in/rev at 615 ft/min and 0.008 in/rev at 815 ft/min. Because the selected feed (0.015 in/rev) is different from either of the feeds given in the table, the cut­ting speed must be adjusted to match the feed. The other cutting parameters to be used must also be compared with the general tool and cutting parameters given in the speed tables to determine if adjustments need to be made for these parameters as well. The general tool and cutting parameters for turning, given in the footnote to Table 1, are depth of cut = 0.1 inch, lead angle = 15°, and tool nose radius = 3∕64 inch.

Table 5a is used to adjust the cutting speeds for turning (from Table 1 through Table 9) for changes in feed, depth of cut, and lead angle. The new cutting speed V is found from V = Vopt 3 Ff 3 Fd , where Vopt is the optimum speed from the table (always the lower of the two speeds given), and Ff and Fd are the adjustment factors from Table 5a for feed and depth of cut, respectively.

To determine the two factors Ff and Fd , calculate the ratio of the selected feed to the opti­mum feed, 0.015∕0.017 = 0.9, and the ratio of the two given speeds Vavg and Vopt , 815∕  615 = 1.35 (approximately). The feed factor Ff = 1.07 is found in Table 5a at the intersection of the feed ratio row and the speed ratio column. The depth-of-cut factor Fd = 1.0 is found in the same row as the feed factor in the column for depth of cut = 0.1 inch and lead angle = 15°, or for a tool with a 45° lead angle, Fd = 1.18. The final cutting speed for a 15° lead angle is V = Vopt 3 Ff 3 Fd = 615 3 1.07 3 1.0 = 658 fpm. Notice that increasing the lead angle tends to permit higher cutting speeds; such an increase is also the general effect of increasing the tool nose radius, although nose radius correction factors are not included in this table. Increasing lead angle also increases the radial pressure exerted by the cutting tool on the workpiece, which may cause unfavorable results on long, slender workpieces. Example 2, Turning: For the same material and feed as the previous example, what is the cutting speed for a 0.4-inch depth of cut and a 45° lead angle?

As before, the feed is 0.015 in/rev, so Ff is 1.07, but Fd = 1.03 for depth of cut equal to 0.4 inch and a 45° lead angle. Therefore, V = 615 3 1.07 3 1.03 = 676 fpm. Increasing the lead angle from 15° to 45° permits a much greater (four times) depth of cut, at the same feed and nearly constant speed. Tool life remains constant at 15 minutes.  (Continued on page 1104)

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Material AISI/SAE Designation Free-machining plain carbon steels (resulfurized): 1212, 1213, 1215

1108, 1109, 1115, 1117, 1118, 1120, 1126, 1211

1132, 1137, 1139, 1140, 1144, 1146, 1151

{

{

{

Plain carbon steels: 1006, 1008, 1009, 1010, 1012, 1015, 1016, 1017, 1018, 1019, 1020, 1021, 1022, 1023, 1024, 1025, 1026, 1513, 1514

HSS

Brinell Hardness Number  a

Speed (fpm)

100-150

150

Tool Material Coated Carbide Hard Tough

Hard

Ceramic

Tough

Cermet

f s f s

f = feed (0.001 in/rev), s = speed (ft/min)  Metric Units: f 3 25.4 = mm/rev, s 3 0.3048 = m/min Opt. Avg. Opt. Avg. Opt. Avg. Opt. Avg. Opt. Avg. Opt. Avg. Opt. 17 8 36 17 17 8 28 13 15 8 15 8 7 805 1075 405 555 1165 1295 850 1200 3340 4985 1670 2500 1610 17 8 36 17 28 13 28 13 15 8 15 8 7 745 935 345 470 915 1130 785 1110 1795 2680 1485 2215 1490

Avg. 3 2055 3 1815

7 1355

3 1695

150-200

160

100-150

130

150-200

120

f s

17 730

8 990

36 300

17 430

17 1090

8 1410

28 780

13 1105

15 1610

8 2780

15 1345

8 2005

175-225

120

275-325

75

f s

17 615

8 815

36 300

17 405

17 865

8 960

28 755

13 960

13 1400

7 1965

13 1170

7 1640

325-375

50

375-425

40

f s

17 515

8 685

36 235

17 340

17 720

8 805

28 650

13 810

10 1430

5 1745

10 1070

5 1305

100-150

140

150-200

145

f s

17 745

8 935

36 345

17 470

28 915

13 1130

28 785

13 1110

15 1795

8 2680

15 1485

8 2215

7 1490

3 1815

200-250

110

100-125

120

f s f s

125-175

110

175-225

90

36 300 36 405 36 345 36 300

17 405 17 555 17 470 17 405

17 865 17 1165 28 915 17 865

8 960 8 1295 13 1130 8 960

28 755 28 850 28 785 28 755

13 960 13 1200 13 1110 13 960

13 1400 15 3340 15 1795 13 1400

7 1965 8 4985 8 2680 7 1965

13 1170 15 1670 15 1485 13 1170

7 1640 8 2500 8 2215 7 1640

3 2055 3 1815

70

8 815 8 1075 8 935 8 815

7 1610 7 1490

225-275

17 615 17 805 17 745 17 615

f s

f s

1095

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(Leaded): 11L17, 11L18, 12L13, 12L14

{

Uncoated Carbide Hard Tough

Machinery's Handbook, 31st Edition SPEEDS AND FEEDS FOR TURNING

Copyright 2020, Industrial Press, Inc.

Table 1. Cutting Speeds and Feeds for Turning Plain Carbon and Alloy Steels

Material AISI/SAE Designation

Speed (fpm)

125-175

100

225-275

70

275-325

60

325-375

40

375-425

30

125-175

100

175-225

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Free-machining alloy steels, (resulfurized): 4140, 4150

Tool Material Coated Carbide Hard Tough

Hard

Ceramic

Tough

Cermet

f s

f = feed (0.001 in/rev), s = speed (ft/min)  Metric Units: f 3 25.4 = mm/rev, s 3 0.3048 = m/min Opt. Avg. Opt. Avg. Opt. Avg. Opt. Avg. Opt. Avg. Opt. Avg. Opt. 17 8 36 17 28 13 28 13 15 8 15 8 7 745 935 345 470 915 1130 785 1110 1795 2680 1485 2215 1490

f s

17 615

8 815

36 300

17 405

17 865

8 960

28 755

13 960

13 1400

7 1965

13 1170

7 1640

f s

17 515

8 685

36 235

17 340

17 720

8 805

28 650

13 810

10 1430

5 1745

10 1070

5 1305

80

f s

17 730

8 990

36 300

17 430

17 8 1090 1410

28 780

13 1105

15 1610

8 2780

15 1345

8 2005

7 1355

3 1695

225-275

65

f s

17 615

8 815

36 300

17 405

17 865

8 960

28 755

13 960

13 1400

7 1965

13 1170

7 1640

7 1365

3 1695

325-375

35

375-425

30

f s

17 515

8 685

36 235

17 340

17 720

8 805

28 650

13 810

10 1430

5 1745

10 1070

5 1305

175-200

110

200-250

90

17 525

8 705

36 235

17 320

17 505

8 525

28 685

13 960

15 1490

8 2220

15 1190

8 1780

7 1040

3 1310

17 355

8 445

36 140

17 200

17 630

8 850

28 455

13 650

10 1230

5 1510

10 990

5 1210

7 715

3 915

17 330

8 440

36 125

17 175

17 585

8 790

28 125

13 220

8 1200

4 1320

8 960

4 1060

7 575

3 740

275-325

250-300

85

50

65

300-375

50

375-425

40

f s f s f s

Avg. 3 1815

Machinery's Handbook, 31st Edition SPEEDS AND FEEDS FOR TURNING

Plain carbon steels (continued): 1055, 1060, 1064, 1065, 1070, 1074, 1078, 1080, 1084, 1086, 1090, 1095, 1548, 1551, 1552, 1561, 1566

HSS

Brinell Hardness Number  a 175-225

Plain carbon steels (continued): 1027, 1030, 1033, 1035, 1036, 1037, 1038, 1039, 1040, 1041, 1042, 1043, 1045, 1046, 1048, 1049, 1050, 1052, 1524, 1526, 1527, 1541

Uncoated Carbide Hard Tough

1096

Copyright 2020, Industrial Press, Inc.

Table Table 1. (Continued) Cutting Speeds and Feeds for Turning Plain Carbon and Alloy 1. Cutting Speeds and Feeds for Turning Plain Carbon and Alloy SteelsSteels

Material AISI/SAE Designation

Free-machining alloy steels: (leaded): 41L30, 41L40, 41L47, 41L50, 43L47, 51L32, 52L100, 86L20, 86L40

Alloy steels: 4012, 4023, 4024, 4028, 4118, 4320, 4419, 4422, 4427, 4615, 4620, 4621, 4626, 4718, 4720, 4815, 4817, 4820, 5015, 5117, 5120, 6118, 8115, 8615, 8617, 8620, 8622, 8625, 8627, 8720, 8822, 94B17

HSS

Brinell Hardness Number  a

Speed (fpm)

150-200

120

200-250

100

300-375

55

375-425

50

125-175

100

175-225

90

225-275

70

275-325

60

f s f s f s

375-425

30 (20)

175-225

85 (70)

225-275

70 (65)

275-325

375-425

250-300

325-35

325-375

Tool Material Coated Carbide Hard Tough

Hard

Ceramic

Tough

Cermet

f s f s

f = feed (0.001 in/rev), s = speed (ft/min)  Opt. Avg. Opt. Avg. Opt. Avg. 17 8 36 17 17 8 730 990 300 430 1090 1410 17 8 36 17 17 8 615 815 300 405 865 960

Metric Units: f 3 25.4 = mm/rev, s 3 0.3048 = m/min Opt. Avg. Opt. Avg. Opt. Avg. Opt. 28 13 15 8 15 8 7 780 1105 1610 2780 1345 2005 1355 28 13 13 7 13 7 7 755 960 1400 1965 1170 1640 1355

f s

17 515

8 685

36 235

17 340

17 720

8 805

28 650

13 810

10 1430

5 1745

10 1070

5 1305

17 525

8 705

36 235

17 320

17 505

8 525

28 685

13 960

15 1490

8 2220

15 1190

8 1780

7 1040

3 1310

17 355 17 330

8 445 8 440

36 140 36 135

17 200 17 190

17 630 17 585

8 850 8 790

28 455 28 240

13 650 13 350

10 1230 9 1230

5 1510 5 1430

10 990 8 990

5 1210 5 1150

7 715 7 655

3 915 3 840

f s

17 330

8 440

36 125

17 175

17 585

8 790

28 125

13 220

8 1200

4 1320

8 960

4 1060

7 575

3 740

60 (50)

f s f s f s

17 525 17 355 17 330

8 705 8 445 8 440

36 235 36 140 36 135

17 320 17 200 17 190

17 505 17 630 17 585

8 525 8 850 8 790

28 685 28 455 28 240

13 960 13 650 13 350

15 1490 10 1230 9 1230

8 2220 5 1510 5 1430

15 1190 10 990 8 990

8 1780 5 1210 5 1150

7 1020 7 715 7 655

3 1310 3 915 3 840

30 (20)

f s

17 330

8 440

36 125

17 175

17 585

8 790

28 125

13 220

8 1200

4 1320

8 960

4 1060

7 575

3 740

75

50

40 (30)

Avg. 3 1695 3 1695

1097

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Alloy steels: 1330, 1335, 1340, 1345, 4032, 4037, 4042, 4047, 4130, 4135, 4137, 4140, 4142, 4145, 4147, 4150, 4161, 4337, 4340, 50B44, 50B46, 50B50, 50B60, 5130, 5132, 5140, 5145, 5147, 5150, 5160, 51B60, 6150, 81B45, 8630, 8635, 8637, 8640, 8642, 8645, 8650, 8655, 8660, 8740, 9254, 9255, 9260, 9262, 94B30 E51100, E52100 use (HSS Speeds)

Uncoated Carbide Hard Tough

Machinery's Handbook, 31st Edition SPEEDS AND FEEDS FOR TURNING

Copyright 2020, Industrial Press, Inc.

TableTable 1. (Continued) Cutting and Feeds for Turning Plain Carbon andSteels Alloy Steels 1. Cutting SpeedsSpeeds and Feeds for Turning Plain Carbon and Alloy

Material AISI/SAE Designation

Maraging steels (not AISI): 18% Ni, Grades 200, 250, 300, and 350

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Nitriding steels (not AISI): Nitralloy 125, 135, 135 Mod., 225, and 230, Nitralloy N, Nitralloy EZ, Nitrex 1

HSS

Brinell Hardness Number  a 220-300

Speed (fpm) 65

300-350

50

Tool Material Coated Carbide Hard Tough

Hard

Ceramic

Tough

Cermet

f = feed (0.001 in/rev), s = speed (ft/min)  Metric Units: f 3 25.4 = mm/rev, s 3 0.3048 = m/min Opt. Avg. Opt. Avg. Opt. Avg. Opt. Avg. Opt. Avg. Opt. Avg. Opt.

Avg.

f s

17 220

8 295

36 100

17 150

20 355

10 525

28 600

13 865

10 660

5 810

7 570

3 740

f s

17 165

8 185

36 55†

17 105

17 325

8 350

28 175

13 260

8 660

4 730

7 445

3 560

17 55†

8 90

36 100

17 150

7

3

17 55

8 90

350-400

35

43-48 RC

25

48-52 RC

10

250-325

60

f s

50-52 RC

10

f s

200-250

70

f s

17 525

8 705

36 235

17 320

17 505

8 525

28 685

300-350

30

f s

17 330

8 440

36 125

17 175

17 585

8 790

28 125

f s 17 220

8 295

20 355

10 525

28 600

7 385

3 645

10 270

5 500

660

810

10 570

5 740

7 385‡

3 645

10 270

5 500

13 960

15 1490

8 2220

15 1190

8 1780

7 1040

3 1310

13 220

8 1200

4 1320

8 960

4 1060

7 575

3 740

13 865

a Brinell Hardness Number given unless otherwise indicated by RC for hardness on the Rockwell C scale. Speeds for HSS (high-speed steel) tools are based on a feed of 0.012 inch/rev and a depth of cut of 0.125 inch; use Table 5c to adjust the given speeds for other feeds and depths of cut. The combined feed/speed data in the remaining columns are based on a depth of cut of 0.1 inch, lead angle of 15 degrees, and nose radius of 3 ∕ inch. Use Table 5a to adjust given speeds for other feeds, depths of cut, and lead angles; use Table 5b to adjust given speeds for increased tool life up to 180 minutes. 64 Examples are given in the text. The combined feed/speed data in this table are based on tool grades (identified in Table 16) as follows: uncoated carbides, hard = 17, tough = 19, † = 15; coated carbides, hard = 11, tough = 14; ceramics, hard = 2, tough = 3, ‡ = 4; cermet = 7 .

Machinery's Handbook, 31st Edition SPEEDS AND FEEDS FOR TURNING

Ultra-high-strength steels (not ASI): AMS alloys 6421 (98B37 Mod.), 6422 (98BV40), 6424, 6427, 6428, 6430, 6432, 6433, 6434, 6436, and 6442; 300M and D6ac

Uncoated Carbide Hard Tough

1098

Copyright 2020, Industrial Press, Inc.

Table Table 1. (Continued) Cutting Speeds and Feeds for Turning Plain Carbon and Alloy 1. Cutting Speeds and Feeds for Turning Plain Carbon and Alloy SteelsSteels

Material AISI Designation

Water hardening: W1, W2, W5 Shock resisting: S1, S2, S5, S6, S7 Cold-work, oil hardening: O1, O2, O6, O7 Cold-work, high carbon, high chromium: D2, D3, D4, D5, D7 Cold-work, air hardening: A2, A3, A8, A9, A10 A4, A6 A7 Hot-work, chromium type: H10, H11, H12, H13, H14, H19

Hot-work, tungsten type: H21, H22, H23, H24, H25, H26 Hot-work, molybdenum type: H41, H42, H43

Brinell Hardness Number  a

Uncoated HSS Speed (fpm)

200-250

45

150-200 175-225 175-225

100 70 70

200-250

70

200-250 225-275 150-200 200-250

55 45 80 65

325-375

50

48-50 RC 50-52 RC 52-56 RC 150-200 200-250 150-200 200-250

20 10 — 60 50 55 45

150-200

75

Mold: P2, P3, P4, P5, P6, P26, P21

100-150 150-200

90 80

200-250

65

High-speed steel: M1, M2, M6, M10, T1, T2,T6 M3-1, M4 M7, M30, M33, M34, M36, M41, M42, M43, M44, M46, M47, T5, T8 T15, M3-2

225-275

55

225-275

45

f s

17 455

8 610

36 210

17 270

17 830

8 1110

28 575

13 805

13 935

7 1310

13 790

7 1110

7 915

3 1150

f s

17 445

8 490

36 170

17 235

17 705

8 940

28 515

13 770

13 660

7 925

13 750

7 1210

7 1150

3 1510

f s

17 165

8 185

36 55

17 105

17 325

8 350

28 175

13 260

8 660

4 730

7 445

3 560

17 55†

8 90

f s

7 385‡

3 645

10 270

5 500

f s

17 445

8 490

36 170

17 235

17 705

8 940

28 515

13 770

13 660

7 925

13 750

7 1210

7 1150

3 1510

f s f s

17 445 17 445

8 610 8 610

36 210 36 210

17 270 17 270

17 830 17 830

8 1110 8 1110

28 575 28 575

13 805 13 805

13 935 13 935

7 1310 7 1310

13 790 13 790

7 1110 7 1110

7 915 7 915

3 1150 3 1150

f s

17 445

8 490

36 170

17 235

17 705

8 940

28 515

13 770

13 660

7 925

13 750

7 1210

7 1150

3 1510

a Brinell Hardness Number given unless otherwise indicated by RC for hardness on the Rockwell C scale. Speeds for HSS (high-speed steel) tools are based on a feed of 0.012 inch/rev and a depth of cut of 0.125 inch; use Table 5c to adjust the given speeds for other feeds and depths of cut. The combined feed/speed data in the remaining columns are based on a depth of cut of 0.1 inch, lead angle of 15 degrees, and nose radius of 3 ∕ inch. Use Table 5a to adjust given speeds for other feeds, depths of cut, and lead angles; use Table 5b to adjust given speeds for increased tool life up to 64 180 minutes. Examples are given in the text. The combined feed/speed data in this table are based on tool grades (identified in Table 16) as follows: uncoated carbides, hard = 17, tough = 19, † = 15; coated carbides, hard = 11, tough = 14; ceramics, hard = 2, tough = 3, ‡ = 4; cermet = 7.

1099

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Special purpose, low alloy: L2, L3, L6

Tool Material Uncoated Carbide Coated Carbide Ceramic Hard Tough Hard Tough Hard Tough Cermet f = feed (0.001 in/rev), s = speed (ft/min)  Metric Units: f 3 25.4 = mm/rev, s 3 0.3048 = m/min Opt. Avg. Opt. Avg. Opt. Avg. Opt. Avg. Opt. Avg. Opt. Avg. Opt. Avg.

Machinery's Handbook, 31st Edition SPEEDS AND FEEDS FOR TURNING

Copyright 2020, Industrial Press, Inc.

Table 2. Cutting Speeds and Feeds for Turning Tool Steels

Uncoated

(Martensitic): 416, 416Se, 416 Plus X, 420F, 420FSe, 440F, 440FSe Stainless steels (Ferritic): 405, 409 429, 430, 434, 436, 442, 446, 502 (Austenitic): 201, 202, 301, 302, 304, 304L, 305, 308, 321, 347, 348 (Austenitic): 302B, 309, 309S, 310, 310S, 314, 316, 316L, 317, 330

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(Martensitic): 403, 410, 420, 501

(Martensitic): 414, 431, Greek Ascoloy, 440A, 440B, 440C (Precipitation hardening):15-5PH, 17-4PH, 17-7PH, AF-71, 17-14CuMo, AFC-77, AM-350, AM-355, AM-362, Custom 455, HNM, PH13-8, PH14-8Mo, PH15-7Mo, Stainless W

HSS

Brinell Hardness Number

Speed (fpm)

135-185

110

135-185 225-275 135-185 185-240 275-325 375-425

100 80 110 100 60 30

135-185

90

135-185 225-275

75 65

135-185

70

135-175 175-225 275-325 375-425 225-275 275-325 375-425 150-200 275-325 325-375 375-450

95 85 55 35 55-60 45-50 30 60 50 40 25

Hard

f s

Tool Material Tough

Hard

20 480

10 660

36 370

17 395

7 640

36 310

13 520

7 640

36 310

f s f s

13 210

7 260

36 85

17 135

f s

13 520

7 640

36 310

f s

13 210

7 260

f s

13 520

f s

13 195

f s

Cermet

Tough

f = feed (0.001 in/rev), s = speed (ft/min)  Metric Units: f 3 25.4 = mm/rev, s 3 0.3048 = m/min Opt. Avg. Opt. Avg. Opt. Avg. Opt. Avg. Opt. Avg.

13 520

f s

Coated Carbide

20 480

8 945

28 640

13 810

7 790

3 995

28 625

13 815

7 695

3 875

28 625

13 815

7 695

3 875

28 130

13 165

13 810

7 790

3 995

17 345

28 625

13 815

7 695

3 875

36 85

17 135

28 130

13 165

13 200†

7 230

7 640

36 310

17 345

28 625

13 815

13 695

7 875

7 240

36 85

17 155

10 660

36 370

17 345

17 755

17 395

17 755

8 945

28 640

See footnote to Table 1 for more information. The combined feed/speed data in this table are based on tool grades (identified in Table 16) as follows: uncoated car­bides, hard = 17, tough = 19; coated carbides, hard = 11, tough = 14; cermet = 7, † = 18.

Machinery's Handbook, 31st Edition SPEEDS AND FEEDS FOR TURNING

Material Free-machining stainless steel (Ferritic): 430F, 430FSe (Austenitic): 203EZ, 303, 303Se, 303MA, 303Pb, 303Cu, 303 Plus X

Uncoated Carbide

1100

Copyright 2020, Industrial Press, Inc.

Table 3. Cutting Speeds and Feeds for Turning Stainless Steels

Tool Material Uncoated Carbide HSS

Brinell Hardness Number

Speed (fpm)

ASTM Class 20

120-150

120

ASTM Class 30, 35, and 40

190-220

80

Material

ASTM Class 25

ASTM Class 45 and 50

ASTM Class 55 and 60

ASTM Type 1, 1b, 5 (Ni resist) ASTM Type 2, 3, 6 (Ni resist) ASTM Type 2b, 4 (Ni resist)

160-200 220-260

250-320 100-215

120-175 150-250

CBN

Avg.

Opt.

Gray Cast Iron

Avg.

Opt.

Avg.

Opt.

Avg.

Opt.

Avg.

Opt.

Avg.

Opt.

Avg.

15 1490

8 2220

15 1180

8 1880

8 395

4 510

24 8490

11 36380

f s

28 160

13 245

28 400

13 630

28 360

13 580

11 1440

6 1880

11 1200

6 1570

8 335

4 420

24 1590

11 2200

65

f s

28 110

13 175

28 410

13 575

15 1060

8 1590

15 885

8 1320

8 260

4 325

f s

28 180

13 280

28 730

13 940

28 660

13 885

15 1640

8 2450

15 1410

8 2110

f s

28 125

13 200

28 335

13 505

28 340

13 510

13 1640

7 2310

13 1400

7 1970

f s

28 100

13 120

28 205

13 250

11 1720

6 2240

11 1460

6 1910

60

35 70

50

200-255

70

75 60 50

Malleable Iron

30

Speeds for HSS (high-speed steel) tools are based on a feed of 0.012 inch/rev and a depth of cut of 0.125 inch; use Table 5c to adjust the given speeds for other feeds and depths of cut. The combined feed/speed data in the remaining columns are based on a depth of cut of 0.1 inch, lead angle of 15 degrees, and nose radius of 3∕6 4 inch. Use Table 5a to adjust the given speeds for other feeds, depths of cut, and lead angles; use Table 5b to adjust given speeds for increased tool life up to 180 minutes. Examples are given in the text. The combined feed/speed data in this table are based on tool grades (identified in Table 16) as follows: uncoated carbides, tough = 15; Coated carbides, hard = 11, tough = 14; ceramics, hard = 2, tough = 3; cermet = 7; CBN = 1.

1101

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13 945

(Martensitic): 53004, 60003, 60004

250-320

Cermet

28 585

(Pearlitic): 40010, 43010, 45006, 45008, 48005, 50005

(Martensitic): 90001

Tough

f = feed (0.001 in/rev), s = speed (ft/min)  Metric Units: f 3 25.4 = mm/rev, s 3 0.3048 = m/min

13 1040

95

240-280

Hard

28 665

160-200

220-260

Ceramic

Tough

13 365

130

(Martensitic): 80002

Opt.

Hard

28 240

110-160

(Martensitic): 70002, 70003

Tough

f s

90

(Ferritic): 32510, 35018

200-240

Coated Carbide

Machinery's Handbook, 31st Edition SPEEDS AND FEEDS FOR TURNING

Copyright 2020, Industrial Press, Inc.

Table 4a. Cutting Speeds and Feeds for Turning Ferrous Cast Metals

1102

Tool Material

Brinell Hardness Number

Material (Ferritic): 60-40-18, 65-45-12 (Ferritic-Pearlitic): 80-55-06

{

(Pearlitic-Martensitic): 100-70-03 (Martensitic): 120-90-02

{

Uncoated Carbide

Uncoated HSS

Hard

Coated Carbide

Tough

Hard

Ceramic

Tough

Hard

Tough

Cermet

f = feed (0.001 in/rev), s = speed (ft/min)  Metric Units: f 3 25.4 = mm/rev, s 3 0.3048 = m/min Speed (fpm)

Avg.

Opt.

Avg.

Opt.

Avg.

Opt.

Avg.

Opt.

Avg.

Opt.

Avg.

28 200

13 325

28 490

13 700

28 435

13 665

15 970

8 1450

15 845

8 1260

8 365

4 480

f s

28 130

13 210

28 355

13 510

28 310

13 460

11 765

6 995

11 1260

6 1640

8 355

4 445

30

f s

28 40

13 65

28 145

13 175

10 615

5 750

10 500

5 615

8 120

4 145

110

f s

17 370

8 490

36 230

17 285

17 665

8 815

28 495

13 675

15 2090

8 3120

7 625

3 790

f s

17 370

8 490

36 150

17 200

17 595

8 815

28 410

13 590

15 1460

8 2170

7 625

3 790

80

70

f s

17 310

8 415

36 115

17 150

17 555

8 760

15 830

8 1240

300-350

55

45

f s

28 70†

13 145

1544 5

8 665

350-400

30

f s

28 115†

13 355

100

190-225

80

240-300

45

300-400

15

225-260 270-330

65

f s

Avg.

Nodular (Ductile) Iron

Cast Steels 100-150

(Low-carbon): 1010, 1020 (Medium-carbon): 1030, 1040, 1050

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Opt.

140-190

Opt.

(Low-carbon alloy): 1320, 2315, 2320, 4110, 4120, 4320, 8020, 8620

125-175

{ {

175-225

225-300

90

250-300

60

200-250

225-250

{

9070

150-200

175-225 (Medium-carbon alloy): 1330, 1340, 2325, 2330, 4125, 4130, 4140, 4330, 4340, 8030, 80B30, 8040, 8430, 8440, 8630, 8640, 9525, 9530, 9535

100

250-300

80

28 335

13 345

15 955

8 1430

The combined feed/speed data in this table are based on tool grades (identified in Table 16) as shown: uncoated carbides, hard = 17; tough = 19, † = 15; coated car­ bides, hard = 11; tough = 14; ceramics, hard = 2; tough = 3; cermet = 7. Also, see footnote to Table 4a.

Machinery's Handbook, 31st Edition SPEEDS AND FEEDS FOR TURNING

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Table 4b. Cutting Speeds and Feeds for Turning Ferrous Cast Metals

1.00 0.90 0.80 0.70 0.60 0.50 0.40 0.30 0.20 0.10

Ratio of the two cutting speeds given in the tables 1.00

1.10

Vavg /Vopt 1.25 1.35 1.50

1.0 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00

1.0 1.02 1.03 1.05 1.08 1.10 1.09 1.06 1.00 0.80

1.0 1.05 1.09 1.13 1.20 1.25 1.28 1.32 1.34 1.20

Feed Factor, Ff 1.0 1.07 1.10 1.22 1.25 1.35 1.44 1.52 1.60 1.55

1.0 1.09 1.15 1.22 1.35 1.50 1.66 1.85 2.07 2.24

1.75

2.00

1.0 1.10 1.20 1.32 1.50 1.75 2.03 2.42 2.96 3.74

1.0 1.12 1.25 1.43 1.66 2.00 2.43 3.05 4.03 5.84

1 in. (25.4 mm) 15°

45°

0.4 in. (10.2 mm)

0.74 0.75 0.77 0.77 0.78 0.78 0.78 0.81 0.84 0.88

1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0

0.79 0.80 0.81 0.82 0.82 0.82 0.84 0.85 0.89 0.91

15°

Depth of Cut and Lead Angle 0.2 in. (5.1 mm) 0.1 in. (2.5 mm)

45° 15° 45° 15° Depth of Cut and Lead Angle Factor, Fd

1.03 1.03 1.03 1.03 1.03 1.03 1.03 1.02 1.02 1.01

0.85 0.86 0.87 0.87 0.88 0.88 0.89 0.90 0.91 0.92

1.08 1.08 1.07 1.08 1.07 1.07 1.06 1.06 1.05 1.03

1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0

0.04 in. (1.0 mm)

45°

15°

45°

1.18 1.17 1.15 1.15 1.14 1.14 1.13 1.12 1.10 1.06

1.29 1.27 1.25 1.24 1.23 1.23 1.21 1.18 1.15 1.10

1.35 1.34 1.31 1.30 1.29 1.28 1.26 1.23 1.19 1.12

Use with Table 1 through Table 9. Not for HSS tools. Table 1 through Table 9 data, except for HSS tools, are based on depth of cut = 0.1 inch, lead angle = 15 degrees, and tool life = 15 minutes. For other depths of cut, lead angles, or feeds, use the two feed/speed pairs from the tables and calculate the ratio of desired (new) feed to optimum feed (largest of the two feeds given in the tables), and the ratio of the two cutting speeds (Vavg /Vopt). Use the value of these ratios to find the feed factor Ff at the intersection of the feed ratio row and the speed ratio column in the left half of the table. The depth-of-cut factor Fd is found in the same row as the feed factor in the right half of the table under the column corresponding to the depth of cut and lead angle. The adjusted cutting speed can be calculated from V = Vopt 3 Ff 3 Fd , where Vopt is the smaller (optimum) of the two speeds from the speed table (from the left side of the column containing the two feed/speed pairs). See the text for examples.

Tool Life, T (minutes) 15 45 90 180

Turning with Carbides: Workpiece < 300 BHN fs 1.0 0.86 0.78 0.71

fm 1.0 0.81 0.71 0.63

Turning with Carbides: Workpiece > 300 BHN; Turning with Ceramics: Any Hardness fl 1.0 0.76 0.64 0.54

fs 1.0 0.80 0.70 0.61

fm 1.0 0.75 0.63 0.53

fl 1.0 0.70 0.56 0.45

Turning with Mixed Ceramics: Any Workpiece Hardness fs 1.0 0.89 0.82 0.76

fm 1.0 0.87 0.79 0.72

fl 1.0 0.84 0.75 0.67

Except for HSS speed tools, speeds and feeds given in Table 1 through Table 9 are based on 15-minute tool life. To adjust speeds for another tool life, multiply the cutting speed for 15-minute tool life V15 by the tool life factor from this table according to the following rules: for small feeds where feed ≤ 1 ∕2 fopt , the cutting speed for desired tool life is VT = fs 3 V15; for medium feeds where 1 ∕2 fopt < feed < 3∕4 fopt , VT = fm 3 V15; and for larger feeds where 3∕4 fopt ≤ feed ≤ fopt , VT = f l 3 V15. Here, fopt is the largest (optimum) feed of the two feed/speed values given in the speed tables.

1103

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Table 5b. Tool Life Factors for Turning with Carbides, Ceramics, Cermets, CBN, and Polycrystalline Diamond

Machinery's Handbook, 31st Edition TURNING-SPEED ADJUSTMENT FACTORS

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Table 5a. Turning-Speed Adjustment Factors for Feed, Depth of Cut, and Lead Angle Ratio of Chosen Feed to Optimum Feed

Machinery's Handbook, 31st Edition TURNING-SPEED ADJUSTMENT FACTORS

1104

Table 5c. Cutting-Speed Adjustment Factors for Turning with HSS Tools in. 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.010 0.011 0.012 0.013 0.014 0.015 0.016 0.018 0.020 0.022 0.025 0.028 0.030 0.032 0.035 0.040 0.045 0.050 0.060

Feed

mm 0.05 0.08 0.10 0.13 0.15 0.18 0.20 0.23 0.25 0.28 0.30 0.33 0.36 0.38 0.41 0.46 0.51 0.56 0.64 0.71 0.76 0.81 0.89 1.02 1.14 1.27 1.52

Feed Factor Ff 1.50 1.50 1.50 1.44 1.34 1.25 1.18 1.12 1.08 1.04 1.00 0.97 0.94 0.91 0.88 0.84 0.80 0.77 0.73 0.70 0.68 0.66 0.64 0.60 0.57 0.55 0.50

in. 0.005 0.010 0.016 0.031 0.047 0.062 0.078 0.094 0.100 0.125 0.150 0.188 0.200 0.250 0.312 0.375 0.438 0.500 0.625 0.688 0.750 0.812 0.938 1.000 1.250 1.250 1.375

Depth of Cut

mm 0.13 0.25 0.41 0.79 1.19 1.57 1.98 2.39 2.54 3.18 3.81 4.78 5.08 6.35 7.92 9.53 11.13 12.70 15.88 17.48 19.05 20.62 23.83 25.40 31.75 31.75 34.93

Depth-of-Cut Factor Fd 1.50 1.42 1.33 1.21 1.15 1.10 1.07 1.04 1.03 1.00 0.97 0.94 0.93 0.91 0.88 0.86 0.84 0.82 0.80 0.78 0.77 0.76 0.75 0.74 0.73 0.72 0.71

For use with HSS tool data only from Table 1 through Table 9. Adjusted cutting speed V = VHSS 3 Ff 3 Fd , where VHSS is the tabular speed for turning with high-speed tools.

Example 3, Turning: Determine the cutting speed for turning 1055 steel of 175 to 225 BHN (Brinell Hardness Number) using a hard ceramic insert, a 15° lead angle, a 0.04-inch depth of cut and 0.0075 in/rev feed. The two feed/speed combinations given in Table 5a for 1055 steel are 15∕1610 and 8∕2780, corresponding to 0.015 in/rev at 1610 fpm and 0.008 in/rev at 2780 fpm, respec­tively. In Table 5a, the feed factor Ff = 1.75 is found at the intersection of the row corre­sponding to feed/fopt = 7.5∕15 = 0.5 and the column corresponding to Vavg /Vopt = 2780∕1610 = 1.75 (approximately). The depth-of-cut factor Fd = 1.23 is found in the same row, under the column heading for a depth of cut = 0.04 inch and lead angle = 15°. The adjusted cutting speed is V = 1610 3 1.75 3 1.23 = 3466 fpm. Example 4, Turning: The cutting speed for 1055 steel calculated in Example 3 represents the speed required to obtain a 15-minute tool life. Estimate the cutting speed needed to obtain a tool life of 45, 90, and 180 minutes using the results of Example 3. To estimate the cutting speed corresponding to another tool life, multiply the cutting speed for 15-minute tool life V15 by the adjustment factor from the Table 5b, Tool Life Fac­tors for Turning. This table gives three factors for adjusting tool life based on the feed used, fs for feeds less than or equal to 1 ∕2 fopt , fm for midrange feeds between 1 ∕2 and 3∕4 fopt and f l for large feeds greater than or equal to 3∕4 fopt and less than fopt . In Example 3, fopt is 0.015 in/rev and the selected feed is 0.0075 in/rev = 1 ∕2 fopt . The new cutting speeds for the various tool lives are obtained by multiplying the cutting speed for 15-minute tool life V15 by the factor for small feeds fs from the column for turning with ceramics in Table 5b. These calcula­tions, using the cutting speed obtained in Example 3, follow.

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Machinery's Handbook, 31st Edition SPEEDS AND FEEDS FOR TURNING Tool Life 15 min 45 min 90 min 180 min

1105

Cutting Speed V15 = 3466 fpm

V45 = V15 3 0.80 = 2773 fpm V90 = V15 3 0.70 = 2426 fpm V180 = V15 3 0.61 = 2114 fpm

Depth of cut, feed, and lead angle remain the same as in Example 3. Notice, increasing the tool life from 15 to 180 minutes, a factor of 12, reduces the cutting speed by only about one-third of the V15  speed. Table 6. Cutting Speeds and Feeds for Turning Copper Alloys Group 1 Architectural bronze (C38500); Extra-high-headed brass (C35600); Forging brass (C37700); Free-cutting phosphor bronze, B2 (C54400); Free-cutting brass (C36000); Free-cutting Muntz metal (C37000); High-leaded brass (C33200; C34200); High-leaded brass tube (C35300); Leaded com­mercial bronze (C31400); Leaded naval brass (C48500); Medium-leaded brass (C34000) Group 2

Aluminum brass, arsenical (C68700); Cartridge brass, 70% (C26000); High-silicon bronze, B (C65500); Admiralty brass (inhibited) (C44300, C44500); Jewelry bronze, 87.5% (C22600); Leaded Muntz metal (C36500, C36800); Leaded nickel silver (C79600); Low brass, 80% (C24000); Low-leaded brass (C33500); Low-silicon bronze, B (C65100); Manganese bronze, A (C67500); Muntz metal, 60% (C28000); Nickel silver, 55-18 (C77000); Red brass, 85% (C23000); Yellow brass (C26800) Group 3

Aluminum bronze, D (C61400); Beryllium copper (C17000, C17200, C17500); Commercial­bronze, 90% (C22000); Copper nickel, 10% (C70600); Copper nickel, 30% (C71500); Electrolytic tough pitch copper (C11000); Guilding, 95% (C21000); Nickel silver, 65-10 (C74500); Nickel sil­ver, 65-12 (C75700); Nickel silver, 65-15 (C75400); Nickel silver, 65-18 (C75200); Oxygen-free copper (C10200) ; Phosphor bronze, 1.25% (C50200); Phosphor bronze, 10% D (C52400) Phos­phor bronze, 5% A (C51000); Phosphor bronze, 8% C (C52100); Phosphorus deoxidized copper (C12200) Uncoated Carbide

HSS

Wrought Alloys Description and UNS Alloy Numbers

Group 1 Group 2 Group 3

Material Condition

Speed (fpm)

A CD

300 350

A CD

A CD

200 250

100 110

Polycrystalline Diamond

f = feed (0.001 in/rev), s = speed (ft/min) Metric Units: f 3 25.4 = mm/rev, s 3 0.3048 = m/min

f s

f s

f s

Opt.

Avg.

28 1170

13 1680

28 715

28 440

13 900

13 610

Opt.

Avg.

7 1780

13 2080

Abbreviations designate: A, annealed; CD, cold drawn. The combined feed/speed data in this table are based on tool grades (identified in Table 16) as fol­lows: uncoated carbide, 15; diamond, 9. See the footnote to Table 7.

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1106

Table 7. Cutting Speeds and Feeds for Turning Titanium and Titanium Alloys Brinell Hardness Number

Material

Tool Material Uncoated Carbide (Tough)

HSS

f = feed (0.001 in/rev), s = speed (ft/min)       Metric Units: f 3 25.4 = mm/rev, s 3 0.3048 = m/min Opt. Avg.

Speed (fpm)

Commercially Pure and Low Alloyed 99.5Ti, 99.5Ti-0.15Pd

110-150

99.1Ti, 99.2Ti, 99.2Ti-0.15Pd,  98.9Ti-0.8Ni-0.3Mo

180-240

99.0 Ti

250-275

f 100-105 s f 85-90 s f 70 s

28 55 28 50 20 75

13 190 13 170 10 210

17 95

8 250

17 55

8 150

Alpha Alloys and Alpha-Beta Alloys 5Al-2.5Sn, 8Mn, 2Al-11Sn-5Zr  1Mo, 4Al-3Mo-1V, 5Al-6Sn-2Zr  1Mo, 6Al-2Sn-4Zr-2Mo, 6Al-2Sn  4Zr-6Mo, 6Al-2Sn-4Zr-2Mo-0.25Si

6Al-4V 6Al-6V-2Sn, Al-4Mo, 8V-5Fe-IAl

6Al-4V, 6Al-2Sn-4Zr-2Mo,  6Al-2Sn-4Zr-6Mo,  6Al-2Sn-4Zr-2Mo-0.25Si

4Al-3Mo-1V, 6Al-6V-2Sn, 7Al-4Mo I Al-8V-5Fe 13V-11Cr-3Al, 8Mo-8V-2Fe-3Al,  3Al-8V-6Cr-4Mo-4Zr,  11.5Mo-6Zr-4.5Sn

{

300-350

50

310-350 320-370 320-380

40 30 20

320-380

40

375-420 375-440

20 20

275-350

375-440

f s

Beta Alloys 25 f 20 s

The speed recommendations for turning with HSS (high-speed steel) tools may be used as starting speeds for milling titanium alloys, using Table 15a to estimate the feed required. Speeds for HSS (high-speed steel) tools are based on a feed of 0.012 inch/rev and a depth of cut of 0.125 inch; use Table 5c to adjust the given speeds for other feeds and depths of cut. The combined feed/ speed data in the remaining columns are based on a depth of cut of 0.1 inch, lead angle of 15 degrees, and nose radius of 3∕6 4 inch. Use Table 5a to adjust given speeds for other feeds, depths of cut, and lead angles; use Table 5b to adjust given speeds for increased tool life up to 180 minutes. Examples are given in the text. The combined feed/speed data in this table are based on tool grades (identified in Table 16) as follows: uncoated carbide, 15.

Table 8. Cutting Speeds and Feeds for Turning Light Metals HSS

Material Description

All wrought and cast magnesium alloys All wrought aluminum alloys, including 6061-T651, 5000, 6000, and 7000 series All aluminum sand and permanent mold casting alloys Alloys 308.0 and 319.0 Alloys 390.0 and 392.0 Alloy 413 All other aluminum die-casting alloys including alloys 360.0 and 380.0

Material Condition

A, CD, ST, and A CD ST and A AC ST and A

Speed (fpm) 800 600 500 750 600

Tool Material

Uncoated Carbide (Tough) Polycrystalline Diamond f = feed (0.001 in/rev), s = speed (ft/min) Metric: f 3 25.4 = mm/rev, s 3 0.3048 = m/min

f s

Aluminum Die-Casting Alloys —

—

AC ST and A — ST and A

80 60 — 100

AC

125

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f s

f s

f s f s

Opt.

Avg.

36 2820

17 4570

36 865

17 1280

24 2010

32 430 36 630

11 2760

15 720 17 1060

Opt.

Avg.

11 5890a

8 8270

8 4765

10 5085 11 7560

4 5755

5 6570 6 9930

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Machinery's Handbook, 31st Edition SPEEDS AND FEEDS FOR TURNING

1107

a The speeds and feeds for turning Al alloys 308.0 and 319.0 with (polycrystalline) diamond tooling represent an expected tool life T = 960 minutes = 16 hours; corresponding speeds and feeds for 15-minute tool life are 11⁄28600 and 6⁄37500.

Abbreviations for material condition: A, annealed; AC, as cast; CD, cold drawn; and ST and A, solution treated and aged, respectively. Speeds for HSS (high-speed steel) tools are based on a feed of 0.012 inch/rev and a depth of cut of 0.125 inch; use Table 5c to adjust the HSS speeds for other feeds and depths of cut. The combined feed/speed data are based on a depth of cut of 0.1 inch, lead angle of 15 degrees, and nose radius of 3∕6 4 inch. Use Table 5a to adjust given speeds for other feeds, depths of cut, and lead angles; use Table 5b to adjust given speeds for increased tool life up to 180 minutes. The data are based on tool grades (identified in Table 16) as follows: uncoated carbide, 15; diamond, 9.

Table 9. Cutting Speeds and Feeds for Turning Superalloys Uncoated Carbide Tough

HSS Turning Rough Finish Material Description T-D Nickel Discalloy 19-9DL, W-545 16-25-6, A-286, Incoloy 800, 801, and 802, V-57 Refractaloy 26 J1300 Inconel 700 and 702, Nimonic 90 and 95 S-816, V-36 S-590 Udimet 630 N-155 Air Resist 213; Hastelloy B, C, G and X (wrought); Haynes 25 and 188; J1570; M252 (wrought); Mar-M905 and M918; Nimonic 75 and 80 CW-12M; Hastelloy B and C (cast); N-12M Rene 95 (Hot Isostatic Pressed) HS 6, 21, 2, 31 (X 40), 36, and 151; Haynes 36 and 151; Mar-M302, M322, and M509, WI-52 Rene 41 Incoloy 901 Waspaloy Inconel 625, 702, 706, 718 (wrought), 721, 722, X750, 751, 901, 600, and 604 AF2-1DA, Unitemp 1753 Colmonoy, Inconel 600, 718, K-Monel, Stellite Air Resist 13 and 215, FSH-H14, Nasa C-W-Re, X-45 Udimet 500, 700, and 710 Astroloy

Speed (fpm) 70-80 80-100 15-35 35-40 25-35 30-40 {

{

30-35

35-40

15-20 15-25

20-25 20-30

10-12

12-15

10-15

10-20

15-20 15-30 20-25 15-25

{

15-20

20-25

{

8-12

10-15

{

10-12

10-15

10-15 10-20 10-30

12-20 20-35 25-35

15-20

20-35

8-10

10-15

{

{

{

Mar-M200, M246, M421, and Rene 77, 80, and 95 (forged) B-1900, GMR-235 and 235D, IN 100 and 738, Inconel 713C and 718 { (cast), M252 (cast)

—

—

10-12

10-15

8-10

Ceramic Hard Tough CBN f = feed (0.001 in/rev), s = speed (ft/min) Metric Units: f 3 25.4 = mm/rev, s 3 0.3048 = m/min

Opt.

Avg.

Opt.

Avg.

Opt.

Avg.

Opt.

Avg.

f s

24 90

11 170

20 365

10 630

f s

20 75

10 135

20 245

10 420

f s

20 75

10 125

11 1170

6 2590

11 405

6 900

20 230

10 400

f s

28 20

13 40

11 895

6 2230

10 345

5 815

20 185

10 315

f s

28 15

13 15

11 615

6 1720

10 290

5 700

20 165

10 280

—

— 10-15 5-10

Tool Material

12-20 5-15 10-12 10-15 8-10

The speed recommendations for rough turning may be used as starting values for milling and drill­ing with HSS tools. The combined feed/speed data in this table are based on tool grades (identified in Table 16) as follows: uncoated carbide = 15; ceramic, hard = 4, tough = 3; CBN = 1.

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1108

Machinery's Handbook, 31st Edition SPEEDS AND FEEDS FOR MILLING

Speeds for HSS (high-speed steel) tools are based on a feed of 0.012 inch/rev and a depth of cut of 0.125 inch; use Table 5c to adjust the given speeds for other feeds and depths of cut. The combined feed/speed data in the remaining columns are based on a depth of cut of 0.1 inch, lead angle of 15 degrees, and nose radius of 3∕6 4 inch. Use Table 5a to adjust given speeds for other feeds, depths of cut, and lead angles; use Table 5b to adjust given speeds for increased tool life up to 180 minutes. Exam­ples are given in the text.

Speeds and Feeds Tables for Milling.—Table 10 through Table 14 give speeds and feeds for milling. The data in the first speed column can be used with high-speed steel tools using the feeds given in Table 15a; these are the same speeds contained in previous editions of the Handbook. The remaining data in Table 10 through Table 14 are combined speeds and feeds for end, face, and slit, slot, and side milling that use the speed adjustment factors given in Table 15b, Table 15c, and Table 15d. Tool life for the combined feed/speed data can also be adjusted using the factors in Table 15e. Table 16 lists cutting tool grades and vendor equivalents. End Milling: Table data for end milling are based on a 3-tooth, 20-degree helix angle tool with a diameter of 1.0 inch, an axial depth of cut of 0.2 inch, and a radial depth of cut of 1 inch (full slot). Use Table 15b to adjust speeds for other feeds and axial depths of cut, and Table 15c to adjust speeds if the radial depth of cut is less than the tool diameter. Speeds are valid for all tool diameters. Face Milling: Table data for face milling are based on a 10-tooth, 8-inch diameter face mill, operating with a 15-degree lead angle, 3∕64 -inch nose radius, axial depth of cut = 0.1 inch, and radial depth (width) of cut = 6 inches (i.e., width of cut to cutter diameter ratio = 3∕4). These speeds are valid if the cutter axis is above or close to the center line of the work­piece (eccentricity is small). Under these conditions, use Table 15d to adjust speeds for other feeds and axial and radial depths of cut. For larger eccentricity (i.e., when the cutter axis to workpiece center line offset is one half the cutter diameter or more), use the end and side milling adjustment factors (Table 15b and Table 15c) instead of the face milling factors. Slit and Slot Milling: Table data for slit milling are based on an 8-tooth, 10-degree helix angle tool with a cutter width of 0.4 inch, diameter D of 4.0 inch, and a depth of cut of 0.6 inch. Speeds are valid for all tool diameters and widths. See the examples in the text for adjustments to the given speeds for other feeds and depths of cut. Tool life for all tabulated values is approximately 45 minutes; use Table 15e to adjust tool life from 15 to 180 minutes. Using the Speeds and Feeds Tables for Milling: The basic feed for milling cutters is the feed per tooth (f), which is expressed in inches per tooth. There are many factors to con­ sider in selecting the feed per tooth and no formula is available to resolve these factors. Among the factors to consider are the cutting tool material; the work material and its hard­ ness; the width and the depth of the cut to be taken; the type of milling cutter to be used and its size; the surface finish to be produced; the power available on the milling machine; and the rigidity of the milling machine, the workpiece, the workpiece setup, the milling cutter, and the cutter mounting. The cardinal principle is to always use the maximum feed that conditions will permit. Avoid, if possible, using a feed that is less than 0.001 inch per tooth because such low feeds reduce the tool life of the cutter. When milling hard materials with small-diameter end mills, such small feeds may be necessary, but otherwise use as much feed as possible. Harder materials in general will require lower feeds than softer materials. The width and the depth of cut also affect the feeds. Wider and deeper cuts must be fed somewhat more slowly than narrow and shallow cuts. A slower feed rate will result in a better surface fin­ ish; however, always use the heaviest feed that will produce the surface finish desired. Fine chips produced by fine feeds are dangerous when milling magnesium because sponta­ neous combustion can occur. Thus, when milling magnesium, a fast feed that will produce a relatively thick chip should be used. Cutting stainless steel produces a work-hardened layer on the surface that has been cut. Thus, when milling this material, the feed should

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Machinery's Handbook, 31st Edition SPEEDS AND FEEDS FOR MILLING

1109

be large enough to allow each cutting edge on the cutter to penetrate below the workhardened layer produced by the previous cutting edge. The heavy feeds recommended for face mill­ing cutters are to be used primarily with larger cutters on milling machines having an ade­quate amount of power. For smaller face milling cutters, start with smaller feeds and increase as indicated by the performance of the cutter and the machine.

When planning a milling operation that requires a high cutting speed and a fast feed, always check to determine if the power required to take the cut is within the capacity of the milling machine. Excessive power requirements are often encountered when milling with cemented carbide cutters. The large metal removal rates that can be attained require a high horsepower output. An example of this type of calculation is given in the section on Machining Power that follows this section. If the size of the cut must be reduced in order to stay within the power capacity of the machine, start by reducing the cutting speed rather than the feed in inches per tooth. The formula for calculating the table feed rate, when the feed in inches per tooth is known, is as follows:

fm = ft nt N where fm =  milling machine table feed rate in inches per minute (ipm)

ft =  feed in inch per tooth (ipt)

nt =  number of teeth in the milling cutter

N =  spindle speed of the milling machine in revolutions per minute (rpm) Example: Calculate the feed rate for milling a piece of AISI 1040 steel having a hardness of 180 BHN. The cutter is a 3-inch diameter high-speed steel plain or slab milling cutter with 8 teeth. The width of the cut is 2 inches, the depth of cut is 0.062 inch, and the cutting speed from Table 11 is 85 fpm. From Table 15a, the feed rate selected is 0.008 inch per tooth.

12V 12 # 85 N = π D = 3.14 # 3 = 108 rpm fm = ft nt N = 0.008 # 8 # 108 = 7 ipm (approximately)

Example 1, Face Milling: Determine the cutting speed and machine operating speed for face milling an aluminum die casting (alloy 413) using a 4-inch polycrystalline diamond cutter, a 3-inch width of cut, a 0.10-inch depth of cut, and a feed of 0.006 inch/tooth.

Table 10 gives the speeds and feeds for milling aluminum alloys. The feed/speed pairs for face milling die-cast alloy 413 with polycrystalline diamond (PCD) are 8∕2320 (0.008 in/tooth feed at 2320 fpm) and 4∕4755 (0.004 in/tooth feed at 4755 fpm). These speeds are based on an axial depth of cut of 0.10 inch, an 8-inch cutter diameter D, a 6-inch radial depth (width) of cut ar, with the cutter approximately centered above the workpiece, i.e., eccentricity is low, as shown in Fig. 3. If the preceding conditions apply, the given speeds and feeds can be used without adjustment for a 45-minute tool life. The given speeds are valid for all cutter diameters if a radial depth of cut to cutter diameter ratio (ar/D) of 3∕4 is maintained (i.e., 6 ∕8 = 3∕4). However, if a different feed or axial depth of cut is required, or if the ar/D ratio is not equal to 3∕4 , the cutting speed must be adjusted for the conditions. The adjusted cutting speed V is calculated from V = Vopt 3 Ff 3 Fd 3 Far, where Vopt is the lower of the two speeds given in the speed table, and Ff, Fd , and Far are adjustment factors for feed, axial depth of cut, and radial depth of cut, respectively, obtained from Table 15d(face milling); except, when cutting near the end or edge of the workpiece as in Fig. 4, Table 15c (side milling) is used to obtain Ff.

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Machinery's Handbook, 31st Edition SPEEDS AND FEEDS FOR MILLING

1110 Work ar

Work Feed ar Feed

D

Cutter

D Cutter

e

Fig. 3.

Fig. 4.

In this example, the cutting conditions match the standard conditions specified in the speed table for radial depth of cut to cutter diameter (3 in ⁄4 in), and depth of cut (0.01 in), but the desired feed of 0.006 in/tooth does not match either of the feeds given in the speed table (0.004 or 0.008). Therefore, the cutting speed must be adjusted for this feed. As with turn­ing, the feed factor Ff is determined by calculating the ratio of the desired feed f to maxi­mum feed fopt from the speed table, and from the ratio Vavg /Vopt of the two speeds given in the speed table. The feed factor is found at the intersection of the feed ratio row and the speed ratio column in Table 15d. The speed is then obtained using the following equation:

f Chosen feed 0.006 Optimum feed = fopt = 0.008 = 0.75

Vavg 4755 Average speed Optimum speed = Vopt = 2320 . 2.0

Ff = ^1.25 + 1.43h ⁄ 2 = 1.3

Fd = 1.0

Far = 1.0

V = 2320 # 1.34 # 1.0 # 1.0 = 3109 fpm , and 3.82 # 3109 ⁄ 4 = 2970 rpm Example 2, End Milling: What cutting speed should be used for cutting a full slot (i.e., a slot cut from the solid, in one pass, that is the same width as the cutter) in 5140 steel with hardness of 300 BHN using a 1-inch diameter coated carbide (insert) 0° lead angle end mill, a feed of 0.003 in/tooth, and a 0.2-inch axial depth of cut? The speed and feed data for end milling 5140 steel, Brinell Hardness Number = 275– 325, with a coated carbide tool are given in Table 11 as 15∕80 and 8∕240 for optimum and average sets, respectively. The speed adjustment factors for feed and depth of cut for full slot (end mill­ing) are obtained from Table 15b. The calculations are the same as in the previous exam­ples: f/fopt = 3∕15 = 0.2 and Vavg /Vopt = 240∕80 = 3.0, therefore, Ff = 6.86 and Fd = 1.0. The cutting speed for a 45-minute tool life is V = 80 3 6.86 3 1.0 = 548.8, approximately 550 ft/min. Example 3, End Milling: What cutting speed should be used in Example 2 if the radial depth of cut ar is 0.02 inch and axial depth of cut is 1 inch? In end milling, when the radial depth of cut is less than the cutter diameter (as in Fig. 4), first obtain the feed factor Ff from Table 15c, then the axial depth of cut and lead angle fac­tor Fd from Table 15b The radial depth of cut to cutter diameter ratio ar/D is used in Table 15c to determine the maximum and minimum feeds that guard against tool failure at high feeds and against premature tool wear caused by the tool rubbing against the work at very low feeds. The feed used should be selected so that it falls within the minimum to maxi­mum feed range, and then the feed factor Ff can be determined from the feed factors at min­imum and maximum feeds, Ff 1 and Ff 2 as explained below.

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Machinery's Handbook, 31st Edition SPEEDS AND FEEDS FOR MILLING

1111

The maximum feed fmax is found in Table 15c by multiplying the optimum feed from the speed table by the maximum feed factor that corresponds to the ar/D ratio, which in this instance is 0.02∕1 = 0.02; the minimum feed fmin is found by multiplying the optimum feed by the minimum feed factor. Thus, fmax = 4.5 3 0.015 = 0.0675 in/tooth and fmin = 3.1 3 0.015 = 0.0465 in/tooth. If a feed between these maximum and minimum values is selected, 0.050 in/tooth for example, then for ar/D = 0.02 and Vavg /Vopt = 3.0, the feed fac­tors at maximum and minimum feeds are Ff 1 = 7.90 and Ff 2 = 7.01, respectively, and by interpolation, Ff = 7.90 + (0.050 − 0.0465)/(0.0675 − 0.0465) 3 (7.01 − 7.90) = 7.75.

The depth of cut factor Fd is obtained from Table 15b, using fmax from Table 15c instead of the optimum feed fopt for calculating the feed ratio (chosen feed/optimum feed). In this example, the feed ratio = chosen feed/fmax = 0.050∕0.0675 = 0.74, so the feed factor is Fd = 0.93 for a depth of cut = 1.0 inch and 0° lead angle. Therefore, the final cutting speed is 80 3 7.75 3 0.93 = 577 ft/min. Notice that fmax obtained from Table 15c was used instead of the optimum feed from the speed table, in determining the feed ratio needed to find Fd . Slit Milling.—The tabular data for slit milling is based on an 8-tooth, 10-degree helix angle cutter with a width of 0.4 inch, a diameter D of 4.0 inch, and a depth of cut of 0.6 inch. The given speeds and feeds are valid for any diameters and tool widths, as long as suffi­cient machine power is available. Adjustments to cutting speeds for other feeds and depths of cut are made using Table 15c or Table 15d, depending on the orientation of the cutter to the work, as illustrated in Case 1 and Case 2 of Fig. 5. The situation illustrated in Case 1 is approximately equivalent to that illustrated in Fig. 3, and Case 2 is approxi­ mately equiva­lent to that shown in Fig. 4. Case 1: If the cutter is fed directly into the workpiece, i.e., the feed is perpendicular to the surface of the workpiece, as in cutting off, then Table 15d (face milling) is used to adjust speeds for other feeds. The depth of cut portion of Table 15d is not used in this case (Fd = 1.0), so the adjusted cutting speed V = Vopt 3 Ff 3 Far. In determining the factor Far from Table 15d, the radial depth of cut ar is the length of cut created by the portion of the cutter engaged in the work. Case 2: If the cutter feed is parallel to the surface of the workpiece, as in slotting or side milling, then Table 15c (side milling) is used to adjust the given speeds for other feeds. In Table 15c, the cutting depth (slot depth, for example) is the radial depth of cut ar that is used to determine maximum and minimum allowable feed/tooth and the feed factor Ff. These minimum and maximum feeds are determined in the manner described previously, however, the axial depth of cut factor Fd is not required. The adjusted cutting speed, valid for cutters of any thickness (width), is given by V = Vopt 3 Ff. Slit Mill

f Case 1 ar

Chip Thickness

Work Case 2

ar

f feed/rev, f

Fig. 5. Determination of Radial Depth of Cut or in Slit Milling

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End Milling

Material Condi­ tion*

Material

CD ST and A CD ST and A

Alloys 308.0 and 319.0

—

Alloys 360.0 and 380.0

—

Alloys 390.0 and 392.0

—

Alloy 413 {

— ST and A AC

Face Milling

Indexable Insert Uncoated Carbide

Slit Milling

Polycrystalline Diamond

HSS

f = feed (0.001 in/tooth), s = speed (ft/min)  Metric Units: f 3 25.4 = mm/rev, s 3 0.3048 = m/min

Indexable Insert Uncoated Carbide

Opt.

Avg. Opt.

Avg. Opt.

Avg. Opt.

Avg. Opt.

Avg. Opt.

Avg.

f s

15 165

8 15 850 620

8 39 2020 755

20 8 1720 3750

4 16 8430 1600

8 39 4680 840

20 2390

f s f s f s

15 30 15 30

Aluminum Die-Casting Alloys 8 15 8 39 100 620 2020 755 8 15 8 39 90 485 1905 555 39 220

20 1720 20 8 1380 3105 20 370

16 160 4 16 7845 145

8 375 8 355

39 840 39 690

20 2390 20 2320

f s

15 30

8 90

20 8 1380 3105

39 500

20 1680

f s

15 355

15 485

8 39 1385 405

8 39 1905 555

20 665

8 2320

4 4755

4 16 7845 145

8 335

39 690

20 2320

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Abbreviations designate: A, annealed; AC, as cast; CD, cold drawn; and ST and A, solution treated and aged, respectively. End Milling: Table data for end milling are based on a 3-tooth, 20-degree helix angle tool with a diameter of 1.0 inch, an axial depth of cut of 0.2 inch, and a radial depth of cut of 1 inch (full slot). Use Table 15b to adjust speeds for other feeds and axial depths of cut, and Table 15c to adjust speeds if the radial depth of cut is less than the tool diameter. Speeds are valid for all tool diameters. Face Milling: Table data for face milling are based on a 10-tooth, 8-inch diameter face mill, operating with a 15-degree lead angle, 3∕6 4 -inch nose radius, axial depth of cut = 0.1 inch, and radial depth (width) of cut = 6 inches (i.e., width of cut to cutter diameter ratio = 3∕4). These speeds are valid if the cutter axis is above or close to the center line of the workpiece (eccentricity is small). Under these conditions, use Table 15d to adjust speeds for other feeds and axial and radial depths of cut. For larger eccentricity (i.e., when the cutter axis to workpiece center line offset is one half the cutter diameter or more), use the end and side milling adjustment factors (Table 15b and Table 15c) instead of the face milling factors. Slit and Slot Milling: Table data for slit milling are based on an 8-tooth, 10-degree helix angle tool with a cutter width of 0.4 inch, diameter D of 4.0 inch, and a depth of cut of 0.6 inch. Speeds are valid for all tool diameters and widths. See the examples in the text for adjustments to the given speeds for other feeds and depths of cut. Tool life for all tabulated values is approximately 45 minutes; use Table 15e to adjust tool life from 15 to 180 minutes. The combined feed/speed data in this table are based on tool grades (identified in Table 16) as follows: uncoated carbide = 15; diamond = 9.

Machinery's Handbook, 31st Edition SPEEDS AND FEEDS FOR MILLING

All wrought aluminum alloys, 6061-T651, 5000, 6000, 7000 series All aluminum sand and permanent mold casting alloys

All other aluminum die-casting alloys

HSS

Indexable Insert Uncoated Carbide

1112

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Table 10. Cutting Speeds and Feeds for Milling Aluminum Alloys

End Milling

Material Free-machining plain carbon steels (resulfurized): 1212, 1213, 1215

{

(Resulfurized): 1108, 1109, 1115, { 1117, 1118, 1120, 1126, 1211

(Resulfurized): 1132, 1137, 1139, 1140, 1144, 1146, 1151

Plain carbon steels: 1006, 1008, 1009, 1010, 1012, 1015, 1016, 1017, 1018, 1019, 1020, 1021, 1022, 1023, 1024, 1025, 1026, 1513, 1514

{

Brinell Hardness Number  a

Speed (fpm)

100-150

140

150-200

130

100-150

130

150-200

115

175-225

115

275-325

70

325-375

45

HSS

f s

Avg. Opt.

7 45

4 125

7 35

f s

730

f s

7 30

4 85

f s

7 25

4 70

f s

7 35

4 100

7 30

4 85

f s

7 35

4 100

f s

7 30

4 85

375-425

35 140

150-200

130

200-250

110

f s

100-125

110

f s

125-175

110

175-225

90

225-275

65

7 45

Slit Milling

f = feed (0.001 in/tooth), s = speed (ft/min)  Metric Units: f 3 25.4 = mm/rev, s 3 0.3048 = m/min

Opt.

f s

100-150

Face Milling

Uncoated Carbide Coated Carbide Uncoated Carbide Coated Carbide Uncoated Carbide Coated Carbide

4 100

7 465

Avg. Opt.

4 735

7 800

Avg. Opt.

4 39 1050 225

Avg. Opt.

20 335

39 415

39 215

Avg. Opt.

20 685

20 405

39 265

Avg. Opt.

20 495

39 525

Avg. 20 830

4

7

4

7

4

39

20

39

20

39

20

39

20

85

325

565

465

720

140

220

195

365

170

350

245

495

39 185

20 350

39 90

20 235

39 135

20 325

39 265

20 495

39 525

20 830

4 125

7 210

7 465

4 435

4 735

7 300

7 800

4 560

39 90

4 39 1050 225

20 170

20 335

39 175

20 330

39 215

20 405

39 185

20 350

39 415

20 685

39 185

20 350

39 215

20 405

1113

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(Leaded): 11L17, 11L18, 12L13, 12L14

{

HSS

Machinery's Handbook, 31st Edition SPEEDS AND FEEDS FOR MILLING

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Table 11. Cutting Speeds and Feeds for Milling Plain Carbon and Alloy Steels

End Milling

Material

Plain carbon steels: 1055, 1060, 1064, 1065, 1070, 1074, 1078, 1080, 1084, 1086, 1090, 1095, 1548, 1551, 1552, 1561, 1566

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Free-machining alloy steels (Resul­furized): 4140, 4150

Brinell Hardness Number  a

Speed (fpm)

125-175

100

175-225

85

225-275

70

275-325

55

325-375

35

375-425

25

125-175

90

175-225

75

225-275

60

275-325

45

325-375

30

HSS Opt.

Face Milling

f = feed (0.001 in/tooth), s = speed (ft/min)  Metric Units: f 3 25.4 = mm/rev, s 3 0.3048 = m/min

Avg. Opt.

Avg. Opt.

Avg. Opt.

f s

7 35

4 100

f s

7 30

4 85

f s

7 25

4 70

7 210

4 435

7 300

4 560

39 90

7 30

4 85

7 325

4 565

7 465

4 720

39 140

f s

7 30

4 85

f s

7 25

4 70

7 210

4 435

7 300

4 560

15 7

8 30

15 105

8 270

15 270

f s

Avg. Opt.

Avg. Opt.

39 185

20 350

20 170

39 175

20 330

39 90

20 235

39 135

20 325

20 220

39 195

20 365

39 170

20 350

39 245

20 495

39 185

20 350

39 175

20 330

39 90

20 235

39 135

20 325

8 450

39 295

20 475

39 135

20 305

7 25

4 70

39 70

20 210

7 25

4 70

39 90

20 170

15 100

200-250

90

f s

250-300

60

f s

15 6

8 25

15 50

8 175

15 85

8 255

39 200

20 320

f s

15 5

8 20

15 40

8 155

15 75

8 225

39 175

20 280

45 35

Avg.

20 405

375-425

300-375

Avg. Opt.

39 215

175-200

375-425

Slit Milling

Uncoated Carbide Coated Carbide Uncoated Carbide Coated Carbide Uncoated Carbide Coated Carbide

Machinery's Handbook, 31st Edition SPEEDS AND FEEDS FOR MILLING

Plain carbon steels: 1027, 1030, 1033, 1035, 1036, 1037, 1038, 1039, 1040, 1041, 1042, 1043, 1045, 1046, 1048, 1049, 1050, 1052, 1524, 1526, 1527, 1541

HSS

1114

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Table Table 11. (Continued) Cutting Speeds and Feeds for Milling Plain Carbon and Alloy 11. Cutting Speeds and Feeds for Milling Plain Carbon and Alloy SteelsSteels

End Milling

Material

Free-machining alloy steels (Leaded): 41L30, 41L40, 41L47, 41L50, 43L47, 51L32, 52L100, 86L20, 86L40

Brinell Hardness Number  a

Speed (fpm)

150-200

115

200-250

95

250-300

50

125-175

100

225-275

60

175-225

40

90

275-325

50

325-375

40

375-425

25

175-225

75 (65)

225-275

60

275-325

50 (40)

325-375

35 (30)

375-425

20

HSS

f s

Opt.

7 30

Face Milling

Slit Milling

Uncoated Carbide Coated Carbide Uncoated Carbide Coated Carbide Uncoated Carbide Coated Carbide f = feed (0.001 in/tooth), s = speed (ft/min)  Metric Units: f 3 25.4 = mm/rev, s 3 0.3048 = m/min

Avg. Opt.

Avg. Opt.

Avg. Opt.

Avg. Opt.

4 85

7 325

4 565

7 465

4 720

39 140

20 220

39 90

20 170

39 195

Avg. Opt.

Avg. Opt.

Avg.

39 185

20 365

20 350

39 170

20 350

39 245

20 495

39 175

20 330

39 90

20 235

39 135

20 325

f s

7 30

4 85

f s

7 25

4 70

7 210

4 435

7 300

4 560

f s

15 7

8 30

15 105

8 270

15 220

8 450

39 295

20 475

39 135

20 305

39 265

20 495

f s

15 6

8 25

15 50

8 175

15 85

8 255

39 200

20 320

39 70

20 210

39 115

20 290

39 135

20 305

39 265

20 495

f s

15 5

8 20

15 45

8 170

15 80

8 240

39 190

20 305

f s

15 5

8 20

15 40

8 155

15 75

8 225

39 175

20 280

f s

15 5

8 30

15 105

8 270

15 220

8 450

39 295

20 475

f s

15 5

8 25

15 50

8 175

8 170

15 80

8 240

39 190

20 305

f s

15 5

8 20

15 40

8 155

15 75

8 225

39 175

20 280

f s

15 5

8 25

15 45

15 85

8 255

39 200

20 320

39 70

20 210

39 115

20 290

1115

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Alloy steels: 1330, 1335, 1340, 1345, 4032, 4037, 4042, 4047, 4130, 4135, 4137, 4140, 4142, 4145, 4147, 4150, 4161, 4337, 4340, 50B44, 50B46, 50B50, 50B60, 5130, 5132, 5140, 5145, 5147, 5150, 5160, 51B60, 6150, 81B45, 8630, 8635, 8637, 8640, 8642, 8645, 8650, 8655, 8660, 8740, 9254, 9255, 9260, 9262, 94B30 E51100, E52100: use (HSS speeds)

70

300-375 375-425

Alloy steels: 4012, 4023, 4024, 4028, 4118, 4320, 4419, 4422, 4427, 4615, 4620, 4621, 4626, 4718, 4720, 4815, 4817, 4820, 5015, 5117, 5120, 6118, 8115, 8615, 8617, 8620, 8622, 8625, 8627, 8720, 8822, 94B17

HSS

Machinery's Handbook, 31st Edition SPEEDS AND FEEDS FOR MILLING

Copyright 2020, Industrial Press, Inc.

Table Table 11. (Continued) Cutting Speeds and Feeds for Milling Plain Carbon and Alloy 11. Cutting Speeds and Feeds for Milling Plain Carbon and Alloy SteelsSteels

End Milling

Material

Maraging steels (not AISI): 18% Ni Grades 200, 250, 300, and 350 Nitriding steels (not AISI): Nitralloy 125, 135, 135 Mod., 225, and 230, Nitralloy N, Nitralloy EZ, Nitrex 1

Brinell Hardness Number  a

Speed (fpm)

220-300

60

300-350

45

350-400

20

43-52 RC

—

250-325

50

50-52 RC

—

200-250

60

300-350

25

HSS Opt.

f s

8 15

4 45

f s

f s

f s

f s

15 7

15 5

Slit Milling

f = feed (0.001 in/tooth), s = speed (ft/min)  Metric Units: f 3 25.4 = mm/rev, s 3 0.3048 = m/min

Avg. Opt.

f s f s

Face Milling

Uncoated Carbide Coated Carbide Uncoated Carbide Coated Carbide Uncoated Carbide Coated Carbide

8 30

8 20

Avg. Opt.

8 165

4 355

8 150

4 320

8 165

4 355

5 20†

5 20†

15 105

15 40

3 55

3 55

8 270

8 155

8 300

Avg. Opt.

Avg. Opt.

39 130

8 300

15 220

15 75

Avg. Opt.

Avg. Opt.

Avg.

4 480 20 235

4 480

8 450

8 225

a Brinell Hardness Number given unless otherwise indicated by RC for hardness on the Rockwell C scale.

39 295

39 175

20 475

20 280

39 75

20 175

39 5

20 15

39 5

39 135

20 15

20 305

39 265

20 495

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For HSS (high-speed steel) tools in the first speed column only, use Table 15a for recommended feed in inches per tooth and depth of cut. End Milling: Table data for end milling are based on a 3-tooth, 20-degree helix angle tool with a diameter of 1.0 inch, an axial depth of cut of 0.2 inch, and a radial depth of cut of 1 inch (full slot). Use Table 15b to adjust speeds for other feeds and axial depths of cut, and Table 15c to adjust speeds if the radial depth of cut is less than the tool diameter. Speeds are valid for all tool diameters. Face Milling: Table data for face milling are based on a 10-tooth, 8-inch diameter face mill, operating with a 15-degree lead angle, 3 ∕64 -inch nose radius, axial depth of cut = 0.1 inch, and radial depth (width) of cut = 6 inches (i.e., width of cut to cutter diameter ratio = 3 ∕4). These speeds are valid if the cutter axis is above or close to the center line of the workpiece (eccentricity is small). Under these conditions, use Table 15d to adjust speeds for other feeds and axial and radial depths of cut. For larger eccentricity (i.e., when the cutter axis to workpiece center line offset is one half the cutter diameter or more), use the end and side milling adjustment factors (Table 15b and Table 15c) instead of the face milling factors. Slit and Slot Milling: Table data for slit milling are based on an 8-tooth, 10-degree helix angle tool with a cutter width of 0.4 inch, diameter D of 4.0 inches, and a depth of cut of 0.6 inch. Speeds are valid for all tool diameters and widths. See the examples in the text for adjustments to the given speeds for other feeds and depths of cut. Tool life for all tabulated values is approximately 45 minutes; use Table 15e to adjust tool life from 15 to 180 minutes. The combined feed/speed data in this table are based on tool grades (identified in Table 16) as follows: end and slit milling uncoated carbide = 20 except † = 15; face milling uncoated carbide = 19; end, face, and slit milling coated carbide = 10.

Machinery's Handbook, 31st Edition SPEEDS AND FEEDS FOR MILLING

Ultra-high-strength steels (not AISI): AMS 6421 (98B37 Mod.), 6422 (98BV40), 6424, 6427, 6428, 6430, 6432, 6433, 6434, 6436, and 6442; 300M, D6ac

HSS

1116

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Table Table 11. (Continued) Cutting Speeds and Feeds for Milling Plain Carbon and Alloy 11. Cutting Speeds and Feeds for Milling Plain Carbon and Alloy SteelsSteels

End Milling

Material

Water hardening: W1, W2, W5 Shock resisting: S1, S2, S5, S6, S7 Cold-work, oil hardening: O1, O2, O6, O7 Cold-work, high carbon, high chro­mium: D2, D3, D4, D5, D7 Cold-work, air hardening: A2, { A3, A8, A9, A10 A4, A6 A7

Hot-work, chromium type: H10, H11, H12, H13, H14, H19

Mold: P2, P3, P4, P5, P6 P20, P21 High-speed steel: M1, M2, M6, M10, T1, T2, T6 M3-1, M4, M7, M30, M33, M34, M36, M41, M42, M43, M44, M46, M47, T5, T8 T15, M3-2

{

Speed (fpm)

175-225

50

200-250

40

200-250

50

200-250 225-275 150-200 200-250

45 40 60 50

325-375

30

48-50 RC 50-52 RC 52-56 RC 150-200

— — — 55

200-250

45

150-200

65

100-150 150-200

75 60

200-250

50

225-275

40

225-275

30

150-200 175-225

85 55

HSS Opt. f s

8 25

8 15

4 70

4 45

f s

8 235

4 8 455 405

8 150

4 320

5 20†

3 55

f s f s

f s

8 25

Face Milling

Uncoated Carbide

Slit Milling

Uncoated Carbide

CBN

Coated Carbide

f = feed (0.001 in/tooth), s = speed (ft/min)  Metric Units: f 3 25.4 = mm/rev, s 3 0.3048 = m/min Avg. Opt. Avg. Opt. Avg. Opt. Avg. Opt. Avg. Opt. Avg. Opt.

f s f s

Coated Carbide

4 70

8 235

4 8 455 405

4 39 635 235

20 385

39 255

20 385

39 130

20 235 39 50

39 255

20 385

4 39 635 235

20 385

39 255

20 385

a Brinell Hardness Number given unless otherwise indicated by RC for hardness on the Rockwell C scale.

39 115

20 39 265 245

39 75

20 175

20 39 135 5†

39 115

Avg. 20 445

20 15

20 39 265 245

20 445

1117

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Hot-work, tungsten and molybde­num types: H21, H22, H23, H24, H25, H26, H41, H42, H43 Special-purpose, low alloy: L2, L3, L6

HSS

Brinell Hardness Number  a

Uncoated Carbide

Machinery's Handbook, 31st Edition SPEEDS AND FEEDS FOR MILLING

Copyright 2020, Industrial Press, Inc.

Table 12. Cutting Speeds and Feeds for Milling Tool Steels

End Milling

Material

Brinell Hardness Number

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Free-machining stainless steels (Ferritic): 430F, 135-185 430FSe (Austenitic): 203EZ, 303, 303Se, 303MA, 135-185 { 225-275 303Pb, 303Cu, 303 Plus X 135-185 (Martensitic): 416, 416Se, 416 Plus X, 420F, 185-240 { 275-325 420FSe, 440F, 440FSe 375-425 Stainless steels (Ferritic): 405, 409, 429, 430, 135-185 434, 436, 442, 446, 502 (Austenitic): 201, 202, 301, 302, 304, 304L, 305, 308, 321, 347, 348 (Austenitic): 302B, 309, 309S, 310, 310S, 314, 316, 316L, 317, 330

(Martensitic): 403, 410, 420, 501

135-185 { 225-275

HSS Speed (fpm) 110 100 80 110 100 60 30 90 75 65

135-185

70

135-175 { 175-225 275-325 375-425

95 85 55 35

Uncoated Carbide

HSS

f s

Face Milling Coated Carbide

Slit Milling

Uncoated Carbide

Coated Carbide

f = feed (0.001 in/tooth), s = speed (ft/min)  Metric Units: f 3 25.4 = mm/rev, s 3 0.3048 = m/min Opt. Avg. Opt. Avg. Opt. Avg. Opt. Avg. Opt. Avg. Opt. 7 30

4 80

7 305

4 780

7 20

4 55

7 210

4 585

f s

7 30

4 80

7 305

4 780

f s

7 20

4 55

7 210

4 585

f s

Coated Carbide

7 420

7 420

4 1240

4 1240

39 210

39 210

20 385

20 385

39 120

20 345

39 75

20 240

39 120

20 345

39 75

20 240

39 155

39 155

Avg.

20 475

20 475

Machinery's Handbook, 31st Edition SPEEDS AND FEEDS FOR MILLING

Table 13. Cutting Speeds and Feeds for Milling Stainless Steels

1118

Copyright 2020, Industrial Press, Inc.

For HSS (high-speed steel) tools in the first speed column only, use Table 15a for recommended feed in inches per tooth and depth of cut. End Milling: Table data for end milling are based on a 3-tooth, 20-degree helix angle tool with a diameter of 1.0 inch, an axial depth of cut of 0.2 inch, and a radial depth of cut of 1 inch (full slot). Use Table 15b to adjust speeds for other feeds and axial depths of cut, and Table 15c to adjust speeds if the radial depth of cut is less than the tool diameter. Speeds are valid for all tool diameters. Face Milling: Table data for face milling are based on a 10-tooth, 8-inch diameter face mill, operating with a 15-degree lead angle, 3∕6 4 -inch nose radius, axial depth of cut = 0.1 inch, and radial depth (width) of cut = 6 inches (i.e., width of cut to cutter diameter ratio = 3∕4). These speeds are valid if the cutter axis is above or close to the center line of the workpiece (eccentricity is small). Under these conditions, use Table 15d to adjust speeds for other feeds and axial and radial depths of cut. For larger eccentricity (i.e., when the cutter axis to workpiece center line offset is one half the cutter diameter or more), use the end and side milling adjustment factors (Table 15b and Table 15c) instead of the face milling factors. Slit and Slot Milling: Table data for slit milling are based on an 8-tooth, 10-degree helix angle tool with a cutter width of 0.4 inch, diameter D of 4.0 inches, and a depth of cut of 0.6 inch. Speeds are valid for all tool diameters and widths. See the examples in the text for adjustments to the given speeds for other feeds and depths of cut. Tool life for all tabulated values is approximately 45 minutes; use Table 15e to adjust tool life from 15 to 180 minutes. The combined feed/speed data in this table are based on tool grades (identified in Table 16) as follows: uncoated carbide = 20, † = 15; coated carbide = 10; CBN = 1.

End Milling

Material Stainless Steels (Martensitic): 414, 431, Greek Ascoloy, 440A, 440B, 440C

(Precipitation hardening): 15-5PH, 17-4PH, 17-7PH, AF-71, 17-14CuMo, AFC-77, AM-350, AM-355, AM-362, Custom 455, HNM, PH13-8, PH14-8Mo, PH15-7Mo, Stainless W

{

HSS

Brinell Hardness Number

Speed (fpm)

225-275

55-60

275-325

45-50

375-425

30

150-200

60

275-325

50

325-375

40

375-450

25

Uncoated Carbide

HSS

Coated Carbide

Face Milling Coated Carbide

Slit Milling

Uncoated Carbide

Coated Carbide

f = feed (0.001 in/tooth), s = speed (ft/min)  Metric Units: f 3 25.4 = mm/rev, s 3 0.3048 = m/min Opt. Avg. Opt. Avg. Opt. Avg. Opt. Avg. Opt. Avg. Opt.

f s

7 20

4 55

7 210

4 585

39 75

Avg.

20 240

1119

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For HSS (high-speed steel) tools in the first speed column only, use Table 15a for recommended feed in inches per tooth and depth of cut. End Milling: Table data for end milling are based on a 3-tooth, 20-degree helix angle tool with a diameter of 1.0 inch, an axial depth of cut of 0.2 inch, and a radial depth of cut of 1 inch (full slot). Use Table 15b to adjust speeds for other feeds and axial depths of cut, and Table 15c to adjust speeds if the radial depth of cut is less than the tool diameter. Speeds are valid for all tool diameters. Face Milling: Table data for face milling are based on a 10-tooth, 8-inch diameter face mill, operating with a 15-degree lead angle, 3∕6 4 -inch nose radius, axial depth of cut = 0.1 inch, and radial depth (width) of cut = 6 inches (i.e., width of cut to cutter diameter ratio = 3∕4). These speeds are valid if the cutter axis is above or close to the center line of the workpiece (eccentricity is small). Under these conditions, use Table 15dto adjust speeds for other feeds and axial and radial depths of cut. For larger eccentricity (i.e., when the cutter axis to workpiece center line offset is one half the cutter diameter or more), use the end and side milling adjustment factors (Table 15b and Table 15c) instead of the face milling factors. Slit and Slot Milling: Table data for slit milling are based on an 8-tooth, 10-degree helix angle tool with a cutter width of 0.4 inch, diameter D of 4.0 inch, and a depth of cut of 0.6 inch. Speeds are valid for all tool diameters and widths. See the examples in the text for adjustments to the given speeds for other feeds and depths of cut. Tool life for all tabulated values is approximately 45 minutes; use Table 15e to adjust tool life from 15 to 180 minutes. The combined feed/speed data in this table are based on tool grades (identified in Table 16) as follows: uncoated carbide = 20; coated carbide = 10.

Machinery's Handbook, 31st Edition SPEEDS AND FEEDS FOR MILLING

Copyright 2020, Industrial Press, Inc.

Table Table 13. (Continued) Cutting Speeds and Feeds for Milling Stainless 13. Cutting Speeds and Feeds for Milling Stainless SteelsSteels

End Milling

HSS

Brinell Hardness Speed Number (fpm)

Material ASTM Class 20

100

190-220

70

160-200

ASTM Class 45 and 50

220-260

ASTM Class 30, 35, and 40 ASTM Class 55 and 60

250-320

ASTM Type 1, 1b, 5 (Ni resist)

100-215

ASTM Type 2, 3, 6 (Ni resist)

120-175

ASTM Type 2b, 4 (Ni resist)

150-250

80 50 30

(Pearlitic): 40010, 43010, 45006, 45008, 48005, 50005

160-200

80

(Martensitic): 53004, 60003, 60004

200-255

55

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220-260 240-280

(Martensitic): 90001

250-320

(Ferritic): 60-40-18, 65-45-12 (Ferritic-Pearlitic): 80-55-06 (Pearlitic-Martensitic): 100-70-03 (Martensitic): 120-90-02

{

65

Ceramic

CBN

Gray Cast Iron

f 5 s 35

3 90

5 520

3 855

f 5 s 30

3 70

5 515

3 1100

f 5 s 30

3 70

5 180

f 5 s 25

3 65

5 150

f 7 s 15

4 35

7 125

f 7 s 10

4 30

7 90

39 140

20 225

39 285

20 535

39 1130

20 39 1630 200

20 39 530 205 20 39 400 145

20 420

39 95

20 39 160 185

20 395

39 845

20 39 1220 150

20 380

3 250

39 120

20 39 195 225

20 520

39 490

20 925

39 85

20 150

3 215

39 90

20 39 150 210

20 400

39 295

20 645

39 70

20 125

4 240

39 100

20 39 155 120

20 255

39 580

20 920

39 60

20 135

4 210

39 95

20 39 145 150

20 275

39 170

20 415

39 40

20 100

Malleable Iron

50 45 25

140-190

75

190-225

60

225-260

50

270-330

25

240-300

Slit Milling

Uncoated Carbide

f = feed (0.001 in/tooth), s = speed (ft/min)  Metric Units: f 3 25.4 = mm/rev, s 3 0.3048 = m/min Opt. Avg. Opt. Avg. Opt. Avg. Opt. Avg. Opt. Avg. Opt. Avg. Opt. Avg. Opt. Avg.

30 110

(Martensitic): 80002

Face Milling

Coated Carbide

40

110-160

(Martensitic): 70002, 70003

Uncoated Carbide

50

(Ferritic): 32510, 35018

200-240

HSS

Coated Carbide

40

Nodular (Ductile) Iron

Coated Carbide

Opt.

Avg.

Machinery's Handbook, 31st Edition SPEEDS AND FEEDS FOR MILLING

120-150

ASTM Class 25

Uncoated Carbide

1120

Copyright 2020, Industrial Press, Inc.

Table 14. Cutting Speeds and Feeds for Milling Ferrous Cast Metals

End Milling

HSS Brinell Hardness Speed Number (fpm)

Material (Low carbon): 1010, 1020

(Medium carbon): 1030, 1040 1050

100-150

{

(Medium-carbon alloy): 1330, 1340, 2325, 2330, 4125, 4130, 4140, 4330, { 4340, 8030, 80B30, 8040, 8430, 8440, 8630, 8640, 9525, 9530, 9535

Uncoated Carbide

Face Milling

Coated Carbide

Ceramic

Slit Milling

CBN

Uncoated Carbide

f = feed (0.001 in/tooth), s = speed (ft/min)  Metric Units: f 3 25.4 = mm/rev, s 3 0.3048 = m/min Opt. Avg. Opt. Avg. Opt. Avg. Opt. Avg. Opt. Avg. Opt. Avg. Opt. Avg. Opt. Avg. Cast Steels

Coated Carbide

Opt.

Avg.

4 7 70 245†

4 410

7 420

4 650

39 265‡

20 430

39 135†

20 39 260 245

20 450

85

f 7 s 20

4 7 55 160†

4 400

7 345

4 560

39 205‡

20 340

39 65†

20 39 180 180

20 370

225-250

65

f 7 s 15

4 7 45 120†

4 310

39 45†

20 135

300-350

30

f s

95

175-225

80

225-300

60

200-250 250-300 175-225

75 50 70

150-200 (Low-carbon alloy): 1320, 2315, 2320, 4110, 4120, 4320, 8020, 8620

Coated Carbide

f 7 s 25

125-175 {

100

HSS

Uncoated Carbide

250-300

50

39 25

20 40

1121

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For HSS (high-speed steel) tools in the first speed column only, use Table 15a for recommended feed in inches per tooth and depth of cut. End Milling: Table data for end milling are based on a 3-tooth, 20-degree helix angle tool with a diameter of 1.0 inch, an axial depth of cut of 0.2 inch, and a radial depth of cut of 1 inch (full slot). Use Table 15b to adjust speeds for other feeds and axial depths of cut, and Table 15c to adjust speeds if the radial depth of cut is less than the tool diameter. Speeds are valid for all tool diameters. Face Milling: Table data for face milling are based on a 10-tooth, 8-inch diameter face mill, operating with a 15-degree lead angle, 3∕6 4 -inch nose radius, axial depth of cut = 0.1 inch, and radial depth (width) of cut = 6 inches (i.e., width of cut to cutter diameter ratio = 3∕4). These speeds are valid if the cutter axis is above or close to the center line of the workpiece (eccentricity is small). Under these conditions, use Table 15d to adjust speeds for other feeds and axial and radial depths of cut. For larger eccentricity (i.e., when the cutter axis to workpiece center line offset is one half the cutter diameter or more), use the end and side milling adjustment factors (Table 15b and Table 15c) instead of the face milling factors. Slit and Slot Milling: Table data for slit milling are based on an 8-tooth, 10-degree helix angle tool with a cutter width of 0.4 inch, diameter D of 4.0 inches, and a depth of cut of 0.6 inch. Speeds are valid for all tool diameters and widths. See the examples in the text for adjustments to the given speeds for other feeds and depths of cut. Tool life for all tabulated values is approximately 45 minutes; use Table 15e to adjust tool life from 15 to 180 minutes. The combined feed/speed data in this table are based on tool grades (identified in Table 16) as follows: uncoated carbide = 15 except † = 20; end and slit milling coated carbide = 10; face milling coated carbide = 11 except ‡ = 10. ceramic = 6; CBN = 1.

Machinery's Handbook, 31st Edition SPEEDS AND FEEDS FOR MILLING

Copyright 2020, Industrial Press, Inc.

Table Table 14. (Continued) Cutting Speeds and Feeds for Milling Ferrous Cast Metals 14. Cutting Speeds and Feeds for Milling Ferrous Cast Metals

1122

End Mills

Depth of Cut, .250 inch (6.35 mm)

Depth of Cut, .050 inch (1.27 mm)

Cutter Diameter, inch (mm) Brinell Hardness Number

Material

inch (25.4 mm)

1 inch and up (19.05 mm)

inch (6.35 mm)

3 ∕4

Plain or Slab Mills

Cutter Diameter, inch (mm)

inch (12.7 mm) 1 ∕2

1 ∕4

inch (12.7 mm) 1 ∕2

inch (19.05 mm) 3 ∕4

1 inch and up (25.4 mm)

Form Relieved Cutters

Face Mills and Shell End Mills

Slotting and Side Mills

ft = feed per tooth, inch;  Metric Units: ft 3 25.4 = mm

Free-machining plain carbon steels

100-185

.001

.003

.004

.001

.002

.003

.004

.003-.008

.005

.004-.012

.002-.008

Plain carbon steels, AISI 1006 to 1030; 1513 to 1522

100-150

.001

.003

.003

.001

.002

.003

.004

.003-.008

.004

.004-.012

.002-.008

AISI 1033 to 1095; 1524 to 1566

{

{

Alloy steels having less than 3% carbon. Typical examples: AISI 4012, 4023, 4027, 4118, 4320 4422, 4427, 4615, 4620, 4626, 4720, 4820, 5015, 5120, 6118, 8115, 8620 8627, 8720, 8820, 8822, 9310, 93B17

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Alloy steels having 3% carbon or more. Typical examples: AISI 1330, 1340, 4032, 4037, 4130, 4140, 4150, 4340, 50B40, 50B60, 5130, 51B60, 6150, 81B45, 8630, 8640, 86B45, 8660, 8740, 94B30

Tool steel

Gray cast iron Free malleable iron

150-200

.001

.002

.003

.001

.002

.002

.003

.003-.008

.004

.003-.012

.002-.008

120-180

.001

.003

.003

.001

.002

.003

.004

.003-.008

.004

.004-.012

.002-.008

180-220

.001

.002

.003

.001

.002

.002

.003

.003-.008

.004

.003-.012

.002-.008

220-300

.001

.002

.002

.001

.001

.002

.003

.002-.006

.003

.002-.008

.002-.006

125-175

.001

.003

.003

.001

.002

.003

.004

.003-.008

.004

.004-.012

.002-.008

175-225

.001

.002

.003

.001

.002

.003

.003

.003-.008

.004

.003-.012

.002-.008

225-275

.001

.002

.003

.001

.001

.002

.003

.002-.006

.003

.003-.008

.002-.006

275-325

.001

.002

.002

.001

.001

.002

.002

.002-.005

.003

.002-.008

.002-.005

175-225

.001

.002

.003

.001

.002

.003

.004

.003-.008

.004

.003-.012

.002-.008

225-275

.001

.002

.003

.001

.001

.002

.003

.002-.006

.003

.003-.010

.002-.006

275-325

.001

.002

.002

.001

.001

.002

.003

.002-.005

.003

.002-.008

.002-.005

325-375

.001

.002

.002

.001

.001

.002

.002

.002-.004

.002

.002-.008

.002-.005

150-200

.001

.002

.002

.001

.002

.003

.003

.003-.008

.004

.003-.010

.002-.006

200-250

.001

.002

.002

.001

.002

.002

.003

.002-.006

.003

.003-.008

.002-.005

120-180

.001

.003

.004

.002

.003

.004

.004

.004-.012

.005

.005-.016

.002-.010

180-225

.001

.002

.003

.001

.002

.003

.003

.003-.010

.004

.004-.012

.002-.008

225-300

.001

.002

.002

.001

.001

.002

.002

.002-.006

.003

.002-.008

.002-.005

110-160

.001

.003

.004

.002

.003

.004

.004

.003-.010

.005

.005-.016

.002-.010

Machinery's Handbook, 31st Edition SPEEDS AND FEEDS FOR MILLING

Copyright 2020, Industrial Press, Inc.

Table 15a. Recommended Feed in Inches per Tooth (ft) for Milling with High-Speed Steel Cutters

End Mills

Depth of Cut, .250 inch (6.35 mm)

Depth of Cut, .050 inch (1.27 mm)

Cutter Diameter, inch (mm)

Material Pearlitic-Martensitic malleable iron

Cast steel Zinc alloys (die castings) Copper alloys (brasses & bronzes) Free cutting brasses & bronzes

Plain or Slab Mills

Cutter Diameter, inch (mm) inch (6.35 mm)

160-200

.001

.003

.004

.001

.002

.003

.004

.003-.010

.004

.004-.012

.002-.018

200-240

.001

.002

.003

.001

.002

.003

.003

.003-.007

.004

.003-.010

.002-.006

Brinell Hardness Number

1 ∕2

3 ∕4

1 inch and up (25.4 mm)

Slotting and Side Mills

1 inch and up (19.05 mm)

1 ∕4

inch (19.05 mm)

Face Mills and Shell End Mills

inch (25.4 mm) 3 ∕4

inch (12.7 mm)

Form Relieved Cutters

inch (12.7 mm) 1 ∕2

ft = feed per tooth, inch;  Metric Units: ft 3 25.4 = mm

240-300

.001

.002

.002

.001

.001

.002

.002

.002-.006

.003

.002-.008

.002-.005

100-180

.001

.003

.003

.001

.002

.003

.004

.003-.008

.004

.003-.012

.002-.008

180-240

.001

.002

.003

.001

.002

.003

.003

.003-.008

.004

.003-.010

.002-.006

240-300

.001

.002

.002

.005

.002

.002

.002

.002-.006

.003

.003-.008

.002-.005

…

.002

.003

.004

.001

.003

.004

.006

.003-.010

.005

.004-.015

.002-.012

100-150

.002

.004

.005

.002

.003

.005

.006

.003-.015

.004

.004-.020

.002-.010

150-250

.002

.003

.004

.001

.003

.004

.005

.003-.015

.004

.003-.012

.002-.008

80-100

.002

.004

.005

.002

.003

.005

.006

.003-.015

.004

.004-.015

.002-.010

Cast aluminum alloys—as cast

…

.003

.004

.005

.002

.004

.005

.006

.005-.016

.006

.005-.020

.004-.012

Cast aluminum alloys—hardened

…

.003

.004

.005

.002

.003

.004

.005

.004-.012

.005

.005-.020

.004-.012

…

.003

.004

.005

.002

.003

.004

.005

.004-.014

.005

.005-.020

.004-.012

…

.002

.003

.004

.001

.002

.003

.004

.003-.012

.004

.005-.020

.004-.012

Magnesium alloys Ferritic stainless steel Austenitic stainless steel

…

.003

.004

.005

.003

.004

.005

.007

.005-.016

.006

.008-.020

.005-.012

135-185

.001

.002

.003

.001

.002

.003

.003

.002-.006

.004

.004-.008

.002-.007

135-185

.001

.002

.003

.001

.002

.003

.003

.003-.007

.004

.005-.008

.002-.007

185-275

.001

.002

.003

.001

.002

.002

.002

.003-.006

.003

.004-.006

.002-.007

135-185

.001

.002

.002

.001

.002

.003

.003

.003-.006

.004

.004-.010

.002-.007

Martensitic stainless steel

185-225

.001

.002

.002

.001

.002

.002

.003

.003-.006

.004

.003-.008

.002-.007

225-300

.0005

.002

.002

.0005

.001

.002

.002

.002-.005

.003

.002-.006

.002-.005

Monel

100-160

.001

.003

.004

.001

.002

.003

.004

.002-.006

.004

.002-.008

.002-.006

1123

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Wrought aluminum alloys— cold drawn Wrought aluminum alloys—hardened

Machinery's Handbook, 31st Edition SPEEDS AND FEEDS FOR MILLING

Copyright 2020, Industrial Press, Inc.

Table Table 15a. (Continued) Recommended Feed in Inches per Tooth with High-Speed Steel Cutters 15a. Recommended Feed in Inches per Tooth (ft) for(fMilling with High-Speed Steel Cutters t) for Milling

1124

Cutting Speed, V = Vopt 3 Ff 3 Fd Ratio of Chosen Feed to Optimum Feed

Ratio of the two cutting speeds

Depth of Cut and Lead Angle

(average/optimum) given in the tables Vavg /Vopt 1.00

1.25

1.50

2.00

1 in 2.50

3.00

4.00

Feed Factor, Ff

0°

(25.4 mm) 45°

0.4 in 0°

(10.2 mm) 45°

0.2 in 0°

(5.1 mm) 45°

0.1 in 0°

Depth of Cut and Lead Angle Factor, Fd

(2.4 mm) 45°

0.04 in 0°

(1.0 mm) 45°

1.00

1.0

1.0

1.0

1.0

1.0

1.0

1.0

0.91

1.36

0.94

1.38

1.00

0.71

1.29

1.48

1.44

1.66

0.90

1.00

1.06

1.09

1.14

1.18

1.21

1.27

0.91

1.33

0.94

1.35

1.00

0.72

1.26

1.43

1.40

1.59

0.80

1.00

1.12

1.19

1.31

1.40

1.49

1.63

0.92

1.30

0.95

1.32

1.00

0.74

1.24

1.39

1.35

1.53

0.70

1.00

1.18

1.30

1.50

1.69

1.85

2.15

0.93

1.26

0.95

1.27

1.00

0.76

1.21

1.35

1.31

1.44

0.60

1.00

1.20

1.40

1.73

2.04

2.34

2.89

0.94

1.22

0.96

1.25

1.00

0.79

1.18

1.28

1.26

1.26

0.50

1.00

1.25

1.50

2.00

2.50

3.00

4.00

0.95

1.17

0.97

1.18

1.00

0.82

1.14

1.21

1.20

1.21

0.40

1.00

1.23

1.57

2.29

3.08

3.92

5.70

0.96

1.11

0.97

1.12

1.00

0.86

1.09

1.14

1.13

1.16

0.30

1.00

1.14

1.56

2.57

3.78

5.19

8.56

0.98

1.04

0.99

1.04

1.00

0.91

1.04

1.07

1.05

1.09

0.20

1.00

0.90

1.37

2.68

4.49

6.86

17.60

1.00

0.85

1.00

0.95

1.00

0.99

0.97

0.93

0.94

0.88

0.10

1.00

0.44

0.80

2.08

4.26

8.00

20.80

1.05

0.82

1.00

0.81

1.00

1.50

0.85

0.76

0.78

0.67

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For HSS (high-speed steel) tool speeds in the first speed column of Table 10 through Table 14, use Table 15a to determine appropriate feeds and depths of cut. Cutting speeds and feeds for end milling given in Table 11 through Table 14 (except those for high-speed steel in the first speed column) are based on milling a 0.20-inch deep full slot (i.e., radial depth of cut = end mill diameter) with a 1-inch diameter, 20-degree helix angle, 0-degree lead angle end mill. For other depths of cut (axial), lead angles, or feed, use the two feed/speed pairs from the tables and calculate the ratio of desired (new) feed to optimum feed (largest of the two feeds are given in the tables), and the ratio of the two cutting speeds (Vavg /Vopt). Find the feed factor Ff at the intersection of the feed ratio row and the speed ratio column in the left half of the Table. The depth of cut factor Fd is found in the same row as the feed factor, in the right half of the table under the column corresponding to the depth of cut and lead angle. The adjusted cutting speed can be calculated from V = Vopt 3 Ff 3 Fd , where Vopt is the smaller (optimum) of the two speeds from the speed table (from the left side of the column containing the two feed/speed pairs). See the text for examples. If the radial depth of cut is less than the cutter diameter (i.e., for cutting less than a full slot), the feed factor Ff in the previous equation and the maximum feed fmax must be obtained from Table 15c. The axial depth of cut factor Fd can then be obtained from this table using fmax in place of the optimum feed in the feed ratio. Also see the footnote to Table 15c.

Machinery's Handbook, 31st Edition MILLING SPEED ADJUSTMENT FACTORS

Copyright 2020, Industrial Press, Inc.

Table 15b. End Milling (Full Slot) Speed Adjustment Factors for Feed, Depth of Cut, and Lead Angle

Ratio of Radial Depth of Cut to Diameter 1.00

0.75

0.60

0.50

0.40

0.30

0.20

0.10

0.05

0.02

Cutting Speed, V = Vopt 3 Ff 3 Fd Maximum Feed/Tooth Factor 1.00

1.00

1.00

1.00

1.10

1.35

1.50

2.05

2.90

4.50

1.25

1.50

2.00

Vavg /Vopt

2.50

3.00

4.00

Feed Factor Ff at Maximum Feed per Tooth, Ff 1 1.00

1.15

1.23

1.25

1.25

1.20

1.14

0.92

0.68

0.38

1.00

1.24

1.40

1.50

1.55

1.57

1.56

1.39

1.12

0.71

1.00

1.46

1.73

2.00

2.17

2.28

2.57

2.68

2.50

1.93

1.00

1.54

2.04

2.50

2.83

3.05

3.78

4.46

4.66

4.19

1.00

1.66

2.34

3.00

3.51

3.86

5.19

6.77

7.75

7.90

1.00

1.87

2.89

4.00

4.94

5.62

8.56

13.10

17.30

21.50

Minimum Feed/Tooth Factor 0.70

0.70

0.70

0.70

0.77

0.88

1.05

1.44

2.00

3.10

1.25

1.50

Vavg /Vopt

2.00

2.50

3.00

4.00

Feed Factor Ff at Minimum Feed per Tooth, Ff 2 1.18

1.24

1.24

1.20

1.25

1.23

1.40

0.92

0.68

0.38

1.30

1.48

1.56

1.58

1.55

1.57

1.56

1.29

1.12

0.70

1.50

1.93

2.23

2.44

2.55

2.64

2.68

2.50

2.08

1.38

1.69

2.38

2.95

3.42

3.72

4.06

4.43

4.66

4.36

3.37

1.85

2.81

3.71

4.51

5.08

2.15

3.68

5.32

6.96

8.30

5.76

10.00

7.76

17.40

6.37

8.00

7.01

11.80

20.80

22.20

1125

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This table is for side milling, end milling when the radial depth of cut (width of cut) is less than the tool diameter (i.e., less than full slot milling), and slit milling when the feed is parallel to the work surface (slotting). The radial depth of cut to diameter ratio is used to determine the recommended maximum and minimum values of feed/tooth, which are found by multiplying the feed/tooth factor from the appropriate column above (maximum or minimum) by feed opt from the speed tables. For example, given two feed/speed pairs 7∕15 and 4 ∕4 5 for end milling cast, medium-carbon, alloy steel, and a radial depth of cut to diameter ratio ar/D of 0.10 (a 0.05-inch width of cut for a 1 ∕2 -inch diameter end mill, for example), the maximum feed fmax = 2.05 3 0.007 = 0.014 in/tooth and the minimum feed fmin = 1.44 3 0.007 = 0.010 in/tooth. The feed selected should fall in the range between fmin and fmax . The feed factor Fd is determined by interpolating between the feed factors Ff 1 and Ff 2 corresponding to the maximum and minimum feed per tooth, at the appropriate ar/D and speed ratio. In the example given, ar/D = 0.10 and Vavg /Vopt = 45∕15 = 3, so the feed factor Ff 1 at the maximum feed per tooth is 6.77, and the feed factor Ff 2 at the minimum feed per tooth is 7.76. If a working feed of 0.012 in/tooth is chosen, the feed factor Ff is half way between 6.77 and 7.76 or by formula, Ff = Ff 2 + (feed − fmin)/( fmax − fmin) 3 (Ff 1 − Ff 2 ) = 7.76 + (0.012 − 0.010)/ (0.014 − 0.010) 3 (6.77 − 7.76) = 7.27. The cutting speed is V = Vopt 3 Ff 3 Fd , where Fd is the depth of cut and lead angle factor from Table 15b that corresponds to the feed ratio (chosen feed)/fmax , not the ratio (chosen feed)/optimum feed. For a feed ratio = 0.012∕0.014 = 0.86 (chosen feed/fmax), depth of cut = 0.2 inch and lead angle = 45°, the depth of cut factor Fd in Table 15b is between 0.72 and 0.74. Therefore, the final cutting speed for this example is V = Vopt 3 Ff 3 Fd = 15 3 7.27 3 0.73 = 80 ft/min. Slit and Side Milling: This table only applies when feed is parallel to the work surface, as in slotting. If feed is perpendicular to the work surface, as in cutting off, obtain the required speed-correction factor from Table 15d (face milling). The minimum and maximum feeds/tooth for slit and side milling are determined in the man­ner described above, however, the axial depth of cut factor Fd is not required. The adjusted cutting speed, valid for cutters of any thickness (width), is given by V = Vopt 3 Ff. Examples are given in the text.

Machinery's Handbook, 31st Edition MILLING SPEED ADJUSTMENT FACTORS

Copyright 2020, Industrial Press, Inc.

Table 15c. End, Slit, and Side Milling Speed Adjustment Factors for Radial Depth of Cut

Cutting Speed V = Vopt 3 Ff 3 Fd 3 Far Depth of Cut, inch (mm), and Lead Angle

Ratio of the two cutting speeds (average/optimum) given in the tables Vavg /Vopt 1.00 1.0 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00

1.10 1.0 1.02 1.03 1.05 1.08 1.10 1.09 1.06 1.00 0.80

1.25 1.35 1.50 Feed Factor, Ff 1.0 1.0 1.0 1.05 1.07 1.09 1.09 1.10 1.15 1.13 1.22 1.22 1.20 1.25 1.35 1.25 1.35 1.50 1.28 1.44 1.66 1.32 1.52 1.85 1.34 1.60 2.07 1.20 1.55 2.24

1.75 1.0 1.10 1.20 1.32 1.50 1.75 2.03 2.42 2.96 3.74

2.00 1.0 1.12 1.25 1.43 1.66 2.00 2.43 3.05 4.03 5.84

1 in (25.4 mm) 15° 45° 0.78 0.78 0.80 0.81 0.81 0.81 0.82 0.84 0.86 0.90

1.11 1.10 1.10 1.09 1.09 1.09 1.08 1.07 1.06 1.04

0.4 in 0.2 in 0.1 in (10.2 mm) (5.1 mm) (2.4 mm) 15° 45° 15° 45° 15° 45° Depth of Cut Factor, Fd 0.94 1.16 0.90 1.10 1.00 1.29 0.94 1.16 0.90 1.09 1.00 1.27 0.94 1.14 0.91 1.08 1.00 1.25 0.95 1.14 0.91 1.08 1.00 1.24 0.95 1.13 0.92 1.08 1.00 1.23 0.95 1.13 0.92 1.08 1.00 1.23 0.95 1.12 0.92 1.07 1.00 1.21 0.96 1.11 0.93 1.06 1.00 1.18 0.96 1.09 0.94 1.05 1.00 1.15 0.97 1.06 0.96 1.04 1.00 1.10

Ratio of Radial Depth of Cut/Cutter Diameter, ar/D

0.04 in (1.0 mm) 15° 45°

1.00

1.47 1.45 1.40 1.39 1.38 1.37 1.34 1.30 1.24 1.15

0.72 0.73 0.75 0.75 0.76 0.76 0.78 0.80 0.82 0.87

1.66 1.58 1.52 1.50 1.48 1.47 1.43 1.37 1.29 1.18

0.75 0.50 0.40 0.30 0.20 Radial Depth of Cut Factor, Far 1.00 1.53 1.89 2.43 3.32 1.00 1.50 1.84 2.24 3.16 1.00 1.45 1.73 2.15 2.79 1.00 1.44 1.72 2.12 2.73 1.00 1.42 1.68 2.05 2.61 1.00 1.41 1.66 2.02 2.54 1.00 1.37 1.60 1.90 2.34 1.00 1.32 1.51 1.76 2.10 1.00 1.26 1.40 1.58 1.79 1.00 1.16 1.24 1.31 1.37

0.10 5.09 4.69 3.89 3.77 3.52 3.39 2.99 2.52 1.98 1.32

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For HSS (high-speed steel) tool speeds in the first speed column, use Table 15a to determine appropriate feeds and depths of cut. Tabular speeds and feeds data for face milling in Table 11 through Table 14 are based on a 10-tooth, 8-inch diameter face mill, operating with a 15-degree lead angle, 3∕6 4 -inch cutter insert nose radius, axial depth of cut = 0.1 inch, and radial depth (width) of cut = 6 inches (i.e., width of cut to cutter diameter ratio = 3∕4). For other depths of cut (radial or axial), lead angles, or feed, calculate the ratio of desired (new) feed to optimum feed (largest of the two feeds given in the speed table), and the ratio of the two cutting speeds (Vavg /Vopt). Use these ratios to find the feed factor Ff at the intersection of the feed ratio row and the speed ratio column in the left third of the table. The depth of cut factor Fd is found in the same row as the feed factor, in the center third of the table, in the column corresponding to the depth of cut and lead angle. The radial depth of cut factor Far is found in the same row as the feed factor, in the right third of the table, in the column corresponding to the radial depth of cut to cutter diameter ratio ar/D. The adjusted cutting speed can be calculated from V = Vopt 3 Ff 3 Fd 3 Far, where Vopt is the smaller (optimum) of the two speeds from the speed table (from the left side of the column containing the two feed/speed pairs). The cutting speeds as calculated above are valid if the cutter axis is centered above or close to the center line of the workpiece (eccentricity is small). For larger eccen­t ricity (i.e., the cutter axis is offset from the center line of the workpiece by about one-half the cutter diameter or more), use the adjustment factors from Table 15b and Table 15c (end and side milling) instead of the factors from this table. Use Table 15e to adjust end and face milling speeds for increased tool life up to 180 minutes. Slit and Slot Milling: Tabular speeds are valid for all tool diameters and widths. Adjustments to the given speeds for other feeds and depths of cut depend on the circumstances of the cut. Case 1: If the cutter is fed directly into the workpiece, i.e., the feed is perpendicular to the surface of the workpiece, as in cutting off, then this table (face milling) is used to adjust speeds for other feeds. The depth of cut factor is not used for slit milling (Fd = 1.0), so the adjusted cutting speed V = Vopt 3 Ff 3 Far. For determining the factor Far, the radial depth of cut ar is the length of cut created by the portion of the cutter engaged in the work. Case 2: If the cutter is fed parallel to the surface of the workpiece, as in slotting, then Table 15b and Table 15c are used to adjust the given speeds for other feeds. See Fig. 5.

Machinery's Handbook, 31st Edition MILLING SPEED ADJUSTMENT FACTORS

Ratio of Chosen Feed to Optimum Feed 1.00 0.90 0.80 0.70 0.60 0.50 0.40 0.30 0.20 0.10

1126

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Table 15d. Face Milling Speed Adjustment Factors for Feed, Depth of Cut, and Lead Angle

Machinery's Handbook, 31st Edition MILLING SPEED ADJUSTMENT FACTORS

1127

Table 15e. Tool Life Adjustment Factors for Face Milling, End Milling, Drilling, and Reaming Tool Life, T (minutes) 15 45 90 180

Face Milling with Carbides and Mixed Ceramics fs

1.69 1.00 0.72 0.51

fm

1.78 1.00 0.70 0.48

fl

1.87 1.00 0.67 0.45

End Milling with Carbides and HSS fs

1.10 1.00 0.94 0.69

fm

1.23 1.00 0.89 0.69

fl

1.35 1.00 0.83 0.69

Twist Drilling and Reaming with HSS fs

1.11 1.00 0.93 0.87

fm

1.21 1.00 0.89 0.80

fl

1.30 1.00 0.85 0.72

The speeds and feeds given in Table 11 through Table 14 and Table 17 through Table 23 (except for HSS speeds in the first speed column) are based on a 45-minute tool life. To adjust the given speeds to obtain another tool life, multiply the adjusted cutting speed for the 45-minute tool life V45 by the tool life factor from this table according to the following rules: for small feeds, where feed ≤ 1 ∕2 fopt , the cutting speed for the desired tool life T is VT = fs 3 V15; for medium feeds, where 1∕ f 3 3 2 opt < feed < ∕4 fopt , VT = fm 3 V15; and for larger feeds, where ∕4 fopt ≤ feed ≤ fopt , VT = f l 3 V15 . Here, fopt is the largest (opti­mum) feed of the two feed/speed values given in the speed tables or the maximum feed fmax obtained from Table 15c, if that table was used in calculating speed adjustment factors.

Table 16. Cutting Tool Grade Descriptions and Common Vendor Equivalents Grade Description

Cubic boron nitride Ceramics

Cermets Polycrystalline Coated carbides

Uncoated carbides

Tool Identification Code

1 2 3 4 (Whiskers) 5 (Sialon) 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Approximate Vendor Equivalents Sandvik Coromant

CB50 CC620 CC650 CC670 CC680 CC690 CT515 CT525 CD10 GC-A GC3015 GC235 GC4025 GC415 H13A S10T S1P S30T S6 SM30

Kennametal

KD050 K060 K090 KYON2500 KYON2000 KYON3000 KT125 KT150 KD100 — KC910 KC9045 KC9025 KC950 K8, K4H K420, K28 K45 — K21, K25 KC710

Seco

Valenite

CBN20 VC721 480 — 480 Q32 — — 480 — — Q6 CM VC605 CR VC610 PAX20 VC727 — — TP100 SV310 TP300 SV235 TP200 SV325 TP100 SV315 883 VC2 CP20 VC7 CP20 VC7 CP25 VC5 CP50 VC56 CP25 VC35M

See Table 2 on page 866 and the section Cemented Carbides and Other Hard Materials for more detailed information on cutting tool grades. The identification codes in column two correspond to the grade numbers given in the footnotes to Table 1 to Table 4b, Table 6 to Table 14, and Table 17 to Table 23.

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Machinery's Handbook, 31st Edition 1128 SPEEDS AND FEEDS FOR DRILLING, REAMING, THREADING Using the Speeds and Feeds Tables for Drilling, Reaming, and Threading.—The first two speed columns in Table 17 through Table 23 give traditional Handbook speeds for drilling and reaming. The following material can be used for selecting feeds for use with the tradi­tional speeds. The remaining columns in Table 17 through Table 23 contain combined feed/speed data for drilling, reaming, and threading, organized in the same manner as in the turning and mill­ing tables. Operating at the given speeds and feeds is expected to result in a tool life of approximately 45 minutes, except for indexable insert drills, which have an expected tool life of approximately 15 minutes per edge. Examples of using this data follow. Adjustments to HSS drilling speeds for feed and diameter are made using Table 22; Table 5a is used for adjustments to indexable insert drilling speeds, where one-half the drill diameter D is used for the depth of cut. Tool life for HSS drills, reamers, and thread chasers and taps may be adjusted using Table 15e and for indexable insert drills using Table 5b. The feed for drilling is governed primarily by the size of the drill and by the material to be drilled. Other factors that also affect selection of the feed are the workpiece configuration, the rigidity of the machine tool and the workpiece setup, and the length of the chisel edge. A chisel edge that is too long will result in a very significant increase in the thrust force, which may cause large deflections to occur on the machine tool and drill breakage. For ordinary twist drills, the feed rate used is given in the table that follows. For addi­ tional information also see the table Approximate Cutting Speeds and Feeds for Standard Automatic Screw Machine Tools—Brown and Sharpe on page 1235. Feet Rate for Twist Drills

Drill Size, inch (mm)

smaller than 1 ∕8 inch (3.175 mm)

Feed Rate, inch/rev (mm/rev)

0.001 to 0.003 in/rev (0.025–0.08 mm/rev)

from 1 ∕8 - to 1 ∕4 -inch (3.175–6.35 mm)

0.002 to 0.006 in/rev (0.05–0.15 mm/rev)

from 1 ∕2 - to 1-inch (12.7–25.4 mm) larger than 1 inch (25.4 mm)

0.007 to 0.015 in/rev (0.18–0.38 mm/rev) 0.010 to 0.025 in/rev (0.25–0.64 mm/rev)

from 1 ∕4 - to 1 ∕2 -inch (6.35–12.7 mm)

0.004 to 0.010 in/rev (0.10–0.25 mm/rev)

The lower values in the feed ranges should be used for hard materials such as tool steels, super­ alloys, and work-hardening stainless steels; the higher values in the feed ranges should be used to drill soft materials such as aluminum and brass.

Example 1, Drilling: Determine the cutting speed and feed for use with HSS drills in drilling 1120 steel.

Table 17 gives two sets of speed and feed parameters for drilling 1120 steel with HSS drills. These sets are 16∕50 and 8∕95, i.e., 0.016 in/rev feed at 50 ft/min and 0.008 in/rev at 95 fpm, respectively. These feed/speed sets are based on a 0.6-inch diameter drill. Tool life for either of the given feed/speed settings is expected to be approximately 45 minutes.

For different feeds or drill diameters, the cutting speeds must be adjusted and can be determined from V = Vopt 3 Ff 3 Fd , where Vopt is the minimum speed for this material given in the speed table (50 fpm in this example) and Ff and Fd are the adjustment factors for feed and diameter, respectively, found in Table 22.

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Drilling

Material {

(Resulfurized): 1108, 1109, 1115, 1117, 1118, 1120, 1126, 1211

{

(Resulfurized): 1132, 1137, 1139, 1140, 1144, 1146, 1151

{

(Leaded): 11L17, 11L18, 12L13, 12L14

{

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Plain carbon steels: 1006, 1008, 1009, 1010, 1012, 1015, 1016, 1017, 1018, 1019, 1020, 1021, 1022, 1023, 1024, 1025, 1026, 1513, 1514

Plain carbon steels: 1027, 1030, 1033, 1035, 1036, 1037, 1038, 1039, 1040, 1041, 1042, 1043, 1045, 1046, 1048, 1049, 1050, 1052, 1524, 1526, 1527, 1541

{

{

Drilling

HSS

Brinell Hardness Number

Free-machining plain carbon steels (Resulfurized): 1212, 1213, 1215

Reaming

HSS

Indexable Insert Coated Carbide

Reaming

Threading

HSS

HSS

f = feed (0.001 in/rev), s = speed (ft/min)  Metric Units: f 3 25.4 = mm/rev, s 3 0.3048 = m/min

Speed (fpm)

Opt.

100-150

120

80

150-200 100-150 150-200

125 110 120

80 75 80

175-225

100

65

275-325 325-375 375-425 100-150 150-200

70 45 35 130 120

45 30 20 85 80

200-250

90

60

100-125

100

65

125-175 175-225 225-275 125-175 175-225 225-275 275-325 325-375 375-425

90 70 60 90 75 60 50 35 25

60 45 40 60 50 40 30 20 15

f 21 s 55

11 125

8 310

4 620

Avg. Opt. 36 140

Avg. Opt. 18 83 185 140

20 185

f 16 s 50

8 95

8 370

4 740

27 105

14 83 115 90

20 115

8 365

4 735

36 140

18 83 185 140

20 185

f s

8 365

4 735

f s

8 365

4 735

f s

f s

f 21 s 55

Avg. Opt.

11 125

8 365

8 310

4 735

4 620

Avg.

Machinery's Handbook, 31st Edition SPEEDS AND FEEDS FOR DRILLING, REAMING, THREADING 1129

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Table 17. Speeds and Feeds for Drilling, Reaming, and Threading Plain Carbon and Alloy Steels

Drilling

Brinell Hardness Number 125-175 175-225

Material

Plain carbon steels (Continued): 1055, 1060, 1064, 1065, 1070, 1074, 1078, 1080, 1084, 1086, 1090, 1095, 1548, 1551, 1552, 1561, 1566

Free-machining alloy steels (Resulfurized): 4140, 4150

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(Leaded): 41L30, 41L40, 41L47, 41L50, 43L47, 51L32, 52L100, 86L20, 86L40

Alloy steels: 4012, 4023, 4024, 4028, 4118, 4320, 4419, 4422, 4427, 4615, 4620, 4621, 4626, 4718, 4720, 4815, 4817, 4820, 5015, 5117, 5120, 6118, 8115, 8615, 8617, 8620, 8622, 8625, 8627, 8720, 8822, 94B17

{

{

{

{

Reaming

Drilling

HSS

85 70

Speed (fpm)

HSS

Indexable Insert Coated Carbide

Reaming

Threading

HSS

HSS

f = feed (0.001 in/rev), s = speed (ft/min)  Metric Units: f 3 25.4 = mm/rev, s 3 0.3048 = m/min 55 45

225-275

50

30

275-325 325-375 375-425 175-200 200-250

40 30 15 90 80

25 20 10 60 50

250-300

55

30

300-375 375-425

40 30

25 15

150-200

100

65

200-250

90

60

250-300 300-375 375-425 125-175 175-225

65 45 30 85 70

40 30 15 55 45

225-275

55

35

275-325

50

30

325-375 375-425

35 25

25 15

Opt.

Avg. Opt.

Avg. Opt.

f 16 s 50

8 95

8 370

4 740

27 105

f 16 s 75 f s

8 140

8 410

4 685

8 310

4 525

f s

f s f 16 s 50 f s

8 95

f 16 s 75

8 140

f s f 11 s 50 f s

6 85

Avg. Opt.

Avg.

14 83 115 90

20 115

26 150

13 83 160 125

20 160

4 740 4 735

27 105

14 83 115 90

20 115

8 410

4 685

26 150

13 83 160 125

20 160

8 310

4 525

19 95

10 83 135 60

20 95

8 365

8 355 8 370 8 365

8 355 8 335

4 735

4 600

4 600 4 570

Machinery's Handbook, 31st Edition 1130 SPEEDS AND FEEDS FOR DRILLING, REAMING, THREADING

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TableTable 17. (Continued) and Feeds for Drilling, Reaming, and Threading Plain Carbon andSteels Alloy Steels 17. SpeedsSpeeds and Feeds for Drilling, Reaming, and Threading Plain Carbon and Alloy

Drilling

Material Alloy steels: 1330, 1335, 1340, 1345, 4032, 4037, 4042, 4047, 4130, 4135, 4137, 4140, 4142, 4145, 4147, 4150, 4161, 4337, 4340, 50B44, 50B46, 50B50, 50B60, 5130, 5132, 5140, 5145, 5147, 5150, { 5160, 51B60, 6150, 81B45, 8630, 8635, 8637, 8640, 8642, 8645, 8650, 8655, 8660, 8740, 9254, 9255, 9260, 9262, 94B30 E51100, E52100: use (HSS speeds) Ultra-high-strength steels (not AISI): AMS 6421 (98B37 Mod.), 6422 (98BV40), 6424, 6427, 6428, 6430, 6432, 6433, 6434, 6436, and 6442; 300M, D6ac Maraging steels (not AISI): 18% Ni Grade 200, 250, 300, and 350 Nitriding steels (not AISI): Nitralloy 125, 135, 135 Mod., 225, and 230, Nitralloy N, Nitralloy EZ, Nitrex I

Reaming

Drilling

HSS

Brinell Hardness Number

HSS

Indexable Insert Coated Carbide

Reaming

Threading

HSS

HSS

f = feed (0.001 in/rev), s = speed (ft/min)  Metric Units: f 3 25.4 = mm/rev, s 3 0.3048 = m/min

Speed (fpm)

175-225

75 (60)

50 (40)

225-275

60 (50)

40 (30)

275-325

45 (35)

30 (25)

325-375 375-425 220-300 300-350

30 (30) 20 (20) 50 35

15 (20) 15 (10) 30 20

350-400

20

10

250-325

50

30

200-250

60

40

300-350

35

20

f s f s f s

f s

Opt. 16 75 11 50

Avg. Opt. 8 8 140 410 8 355 6 8 85 335 8 310

f s

f s f s f 16 s 75 f s

8 325

8 140

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8 270 8 325 8 410 8 310

Avg. 4 685 4 600 4 570 4 525

Opt. 26 150

Avg. Opt. 13 83 160 125

Avg. 20 160

19 95

10 83 135 60

20 95

26 150

13 83 160 125

20 160

4 545

4 450 4 545 4 685 4 525

The two leftmost speed columns in this table contain traditional Handbook speeds for drilling and reaming with HSS steel tools. The section Feed Rates for Drilling and Reaming contains useful information concerning feeds to use in conjunction with these speeds. HSS Drilling and Reaming: The combined feed/speed data for drilling are based on a 0.60-inch diameter HSS drill with standard drill point geometry (2-flute with 118° tip angle). Speed adjustment factors in Table 22 are used to adjust drilling speeds for other feeds and drill diameters. Examples of using this data are given in the text. The given speeds and feeds for reaming are based on an 8-tooth, 25∕32 -inch diameter, 30° lead angle reamer, and a 0.008-inch radial depth of cut. For other feeds, the correct speed can be obtained by interpolation using the given speeds if the desired feed lies in the recommended range (between the given values of optimum and average feed). If a feed lower than the given average value is chosen, the speed should be maintained at the corresponding average speed (i.e., the highest of the two speed values given). The cutting speeds for reaming do not require adjustment for tool diameters for standard ratios of radial depth of cut to reamer diameter (i.e., fd = 1.00). Speed adjustment factors to modify tool life are found in Table 15e.

Machinery's Handbook, 31st Edition SPEEDS AND FEEDS FOR DRILLING, REAMING, THREADING 1131

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Table Table 17. (Continued) Speeds and Feeds for Drilling, Reaming, and Threading Plain Carbon and Alloy 17. Speeds and Feeds for Drilling, Reaming, and Threading Plain Carbon and Alloy SteelsSteels

Machinery's Handbook, 31st Edition 1132 SPEEDS AND FEEDS FOR DRILLING, REAMING, THREADING Indexable Insert Drilling: The feed/speed data for indexable insert drilling are based on a tool with two cutting edges, an insert nose radius of 3∕6 4 inch (1.2 mm), a 10-degree lead angle, and diameter D = 1 inch (2.54 mm). Adjustments to cutting speed for feed and depth of cut are made using Table 5a on page 1103 (Adjustment Factors) using a depth of cut of D⁄2, or one-half the drill diameter. Expected tool life at the given speeds and feeds is approximately 15 minutes for short hole drilling (i.e., where maximum hole depth is about 2D or less). Speed adjustment factors to increase tool life are found in Table 5b. Tapping and Threading: The data in this column are intended for use with thread chasers and for tapping. The feed used for tapping and threading must be equal to the lead (feed = lead = pitch) of the thread being cut. The two feed/speed pairs given for each material, therefore, are representative speeds for two thread pitches, 12 and 50 threads per inch (1∕0.083 = 12, and 1∕0.020 = 50). Tool life is expected to be approximately 45 minutes at the given speeds and feeds. When cutting fewer than 12 threads per inch (pitch ≥ 0.08 inch or 2.1 mm), use the lower (optimum) speed; for cutting more than 50 threads per inch (pitch ≤ 0.02 inch or 0.51 mm), use the larger (average) speed; and, in the intermediate range between 12 and 50 threads per inch, interpolate between the given average and optimum speeds. The combined feed/speed data in this table are based on tool grades (identified in Table 16) as fol­lows: coated carbide = 10.

Example 2, Drilling: If the 1120 steel of Example 1 is to be drilled with a 0.60-inch drill at a feed of 0.012 in/rev, what is the cutting speed in ft/min? Also, what spindle rpm of the drilling machine is required to obtain this cutting speed? To find the feed factor Fd in Table 22, calculate the ratio of the desired feed to the opti­ mum feed and the ratio of the two cutting speeds given in the speed tables. The desired feed is 0.012 in/rev and the optimum feed, as explained above is 0.016 in/rev, therefore, feed/fopt = 0.012∕0.016 = 0.75 and Vavg /Vopt = 95∕50 = 1.9, approximately 2.

The feed factor Ff is found at the intersection of the feed ratio row and the speed ratio col­umn. Ff = 1.40 corresponds to about halfway between 1.31 and 1.50, which are the feed factors that correspond to Vavg /Vopt = 2.0 and feed/fopt ratios of 0.7 and 0.8, respectively. Fd , the diameter factor, is found on the same row as the feed factor (halfway between the 0.7 and 0.8 rows, for this example) under the column for drill diameter = 0.60 inch. Because the speed table values are based on a 0.60-inch drill diameter, Fd = 1.0 for this example, and the cutting speed is V = Vopt 3 Ff 3 Fd = 50 3 1.4 3 1.0 = 70 ft/min. The spindle speed in rpm is N = 12 3 V/(π 3 D) = 12 3 70/(3.14 3 0.6) = 445 rpm.

Example 3, Drilling: Using the same material and feed as in the previous example, what cutting speeds are required for 0.079-inch and 4-inch diameter drills? What machine rpm is required for each?

Because the feed is the same as in the previous example, the feed factor is Ff = 1.40 and does not need to be recalculated. The diameter factors are found in Table 22 on the same row as the feed factor for the previous example (about halfway between the diameter fac­ tors corresponding to feed/fopt values of 0.7 and 0.8) in the column corresponding to drill diameters 0.079 and 4.0 inches, respectively. Results of the calculations are summarized below. Drill diameter = 0.079 inch

Drill diameter = 4.0 inches

Ff = 1.40

Ff = 1.40

Fd = (0.34 + 0.38)⁄2 = 0.36

Fd = (1.95 + 1.73)⁄2 = 1.85

V = 50 3 1.4 3 0.36 = 25.2 fpm

V = 50 3 1.4 3 1.85 = 129.5 fpm

12 3 25.2/(3.14 3 0.079) = 1219 rpm

12 3 129.5/(3.14 3 4) = 124 rpm

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Machinery's Handbook, 31st Edition SPEEDS AND FEEDS FOR DRILLING, REAMING, THREADING 1133 Drilling Difficulties: A drill split at the web is evidence of too much feed or insufficient lip clearance at the center due to improper grinding. Rapid wearing away of the extreme outer corners of the cutting edges indicates that the speed is too high. A drill chipping or breaking out at the cutting edges indicates that either the feed is too heavy or the drill has been ground with too much lip clearance. Nothing will “check” a high-speed steel drill quicker than to turn a stream of cold water on it after it has been heated while in use. It is equally bad to plunge it in cold water after the point has been heated in grinding. The small checks or cracks resulting from this practice will eventually chip out and cause rapid wear or breakage. Insufficient speed in drilling small holes with hand feed greatly increases the risk of breakage, especially at the moment the drill is breaking through the farther side of the work, due to the operator’s inability to gage the feed when the drill is running too slowly. Small drills have heavier webs and smaller flutes in proportion to their size than do larger drills, so breakage due to clogging of chips in the flutes is more likely to occur. When drill­ing holes deeper than three times the diameter of the drill, it is advisable to withdraw the drill (peck feed) at intervals to remove the chips and permit coolant to reach the tip of the drill. Drilling Holes in Glass: The simplest method of drilling holes in glass is to use a stan­ dard, tungsten carbide-tipped masonry drill of the appropriate diameter, in a gun-drill. The edges of the carbide in contact with the glass should be sharp. Kerosene or other liquid may be used as a lubricant, and a light force is maintained on the drill until just before the point breaks through. The hole should then be started from the other side if possible, or a very light force applied for the remainder of the operation, to prevent excessive breaking of material from the sides of the hole. As the hard particles of glass are abraded, they accumu­late and act to abrade the hole, so it may be advisable to use a slightly smaller drill than the required diameter of the finished hole. Alternatively, for holes of medium and large size, use brass or copper tubing, having an outside diameter equal to the size of hole required. Revolve the tube at a peripheral speed of about 100 feet per minute (30.5 m/min), and use carborundum (80 to 100 grit) and light machine oil between the end of the pipe and the glass. Insert the abrasive under the drill with a thin piece of soft wood, to avoid scratching the glass. The glass should be supported by a felt or rubber cushion, not much larger than the hole to be drilled. If practicable, it is advisable to drill about halfway through, then turn the glass over, and drill down to meet the first cut. Any fin that may be left in the hole can be removed with a round second-cut file wetted with turpentine. Smaller-diameter holes may also be drilled with triangular-shaped cemented carbide drills that can be purchased in standard sizes. The end of the drill is shaped into a long tapering triangular point. The other end of the cemented carbide bit is brazed onto a steel shank. A glass drill can be made to the same shape from hardened drill rod or an old threecornered file. The location at which the hole is to be drilled is marked on the workpiece. A dam of putty or glazing compound is built up on the work surface to contain the cutting fluid, which can be either kerosene or turpentine mixed with camphor. Chipping on the back edge of the hole can be prevented by placing a scrap plate of glass behind the area to be drilled and drilling into the backup glass. This procedure also provides additional sup­port to the workpiece and is essential for drilling very thin plates. The hole is usually drilled with an electric hand drill. When the hole is being produced, the drill should be given a small circular motion using the point as a fulcrum, thereby providing a clearance for the drill in the hole. Very small round or intricately shaped holes and narrow slots can be cut in glass by the ultrasonic machining process or by the abrasive jet cutting process.

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Drilling

Material

150-200

Shock resisting: S1, S2, S5, S6, S7

Cold-work (oil hardening): O1, O2, O6, O7

(High carbon, high chromium): D2, D3, D4, D5, D7 (Air hardening): A2, A3, A8, A9, A10

{

A4, A6 A7

Hot-work (chromium type): H10, H11, H12, H13, H14, H19

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{

(Molybdenum type): H41, H42, H43

{

Special-purpose, low alloy: L2, L3, L6

T15, M3-2

20

200-250

45

30

200-250

200-250

30 60 50

35

30 20

55

35

200-250

40

25

200-250

35

20

45 60 75 60

HSS

Avg. Opt.

Avg. Opt.

Avg. Opt.

Avg.

f 15 s 45

7 85

8 360

4 24 605 90

12 95

83 75

20 95

8 270

4 450

8 360

4 24 605 90

12 95

83 75

20 95

40

150-200

150-200

HSS

20

30

150-200

{

50

Threading

30

325-375

100-150

High-speed steel: M1, M2, M6, M10, T1, T2, T6

M3-1, M4, M7, M30, M33, M34, M36, M41, M42, M43, M44, M46, M47, T5, T8

30

150-200

Mold steel: P2, P3, P4, P5, P6P20, P21

Opt.

200-250

45

Reaming

55

35

150-200

(Tungsten type): H21, H22, H23, H24, H25, H26

HSS

50

225-275 {

85

Indexable Insert Uncoated Carbide

f = feed (0.001 in/rev), s = speed (ft/min)  Metric Units: f 3 25.4 = mm/rev, s 3 0.3048 = m/min

Speed (fpm)

175-225 175-225

Drilling

HSS

Brinell Hardness Number

Water hardening: W1, W2, W5

Reaming

f s

30 40 50 40

200-250

45

30

225-275

35

20

225-275

25

15

f 15 s 45

7 85

See the footnote to Table 17 for instructions concerning the use of this table. The combined feed/speed data in this table are based on tool grades (identified in Table 16) as follows: coated carbide = 10.

Machinery's Handbook, 31st Edition 1134 SPEEDS AND FEEDS FOR DRILLING, REAMING, THREADING

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Table 18. Speeds and Feeds for Drilling, Reaming, and Threading Tool Steels

Drilling Brinell Hardness Number

Material Free-machining stainless steels (Ferritic): 430F, 430FSe (Austenitic): 203EZ, 303, 303Se, 303MA, 303Pb, 303Cu, 303 Plus X

{

(Martensitic): 416, 416Se, 416 Plus X, 420F, 420FSe, 440F, 440FSe

{

Stainless steels (Ferritic): 405, 409, 429, 430, 434 (Austenitic): 201, 202, 301, 302, 304, 304L, 305, 308, 321, 347, 348 (Austenitic): 302B, 309, 309S, 310, 310S, 314, 316

{

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(Martensitic): 403, 410, 420, 501

{

(Martensitic): 414, 431, Greek Ascoloy

{

(Martensitic): 440A, 440B, 440C

{

(Precipitation hardening): 15-5PH, 17-4PH, 17-7PH, AF-71, 17-14CuMo, AFC-77, AM-350, AM-355, AM-362, Custom 455, HNM, PH13-8, PH14-8Mo, PH15-7Mo, Stainless W

{

Reaming

HSS

Drilling HSS

Indexable Insert Coated Carbide

Reaming

Threading

HSS

HSS

f = feed (0.001 in/rev), s = speed (ft/min)  Metric Units: f 3 25.4 = mm/rev, s 3 0.3048 = m/min Opt. Avg. Opt. Avg. Opt. Avg. Opt. Avg.

Speed (fpm)

135-185

90

60

135-185 225-275 135-185 185-240 275-325 375-425

85 70 90 70 40 20

55 45 60 45 25 10

135-185

65

45

135-185 225-275 135-185 135-175 175-225 275-325 375-425 225-275 275-325 375-425 225-275 275-325 375-425

55 50 50 75 65 40 25 50 40 25 45 40 20

35 30 30 50 45 25 15 30 25 15 30 25 10

150-200

50

30

275-325 325-375 375-450

45 35 20

25 20 10

f 15 s 25

7 45

8 320

4 24 540 50

12 50

83 40

20 51

f 15 s 20

7 40

8 250

4 24 425 40

12 40

83 35

20 45

f 15 s 25

7 45

8 320

4 24 540 50

12 50

83 40

20 51

f 15 s 20

7 40

8 250

4 24 425 40

12 40

83 35

20 45

f 15 s 20

7 40

8 250

4 24 425 40

12 40

83 35

20 45

See the footnote to Table 17 for instructions concerning the use of this table. The combined feed/speed data in this table are based on tool grades (identified in Table 16) as follows: coated carbide = 10.

Machinery's Handbook, 31st Edition SPEEDS AND FEEDS FOR DRILLING, REAMING, THREADING 1135

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Table 19. Speeds and Feeds for Drilling, Reaming, and Threading Stainless Steels

Drilling

Reaming

Drilling

Reaming

Threading

HSS

HSS

Indexable Carbide Insert

Material

HSS

Brinell Hardness Number

HSS

Uncoated

Coated

f = feed (0.001 in/rev), s = speed (ft/min)  Metric Units: f 3 25.4 = mm/rev, s 3 0.3048 = m/min

Speed (fpm)

Opt.

ASTM Class 20

120-150

100

65

ASTM Class 25

160-200

90

60

ASTM Class 30, 35, and 40

190-220

80

55

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ASTM Class 45 and 50

220-260

60

40

ASTM Class 55 and 60

250-320

30

20

ASTM Type 1, 1b, 5 (Ni resist)

100-215

50

30

ASTM Type 2, 3, 6 (Ni resist)

120-175

40

25

ASTM Type 2b, 4 (Ni resist)

150-250

30

20

(Ferritic): 32510, 35018

110-160

110

75

(Pearlitic): 40010, 43010, 45006, 45008, 48005, 50005

160-200

80

55

200-240

70

45

(Martensitic): 53004, 60003, 60004

200-255

55

35

(Martensitic): 70002, 70003

220-260

50

30

(Martensitic): 80002

240-280

45

30

(Martensitic): 90001

250-320

25

15

(Ferritic): 60-40-18, 65-45-12

140-190

100

65

f s f s

Avg. Opt.

Avg. Opt.

Avg. Opt.

Avg. Opt.

6 26 485 85

13 83 65 90

20 80

21 50

10 83 30 55

20 45

30 95

16 83 80 100

20 85

22 65

11 83 45 70

20 60

28 80

14 83 60 80

20 70

16 80

8 90

11 85

6 180

11 235

13 50

6 50

11 70

6 150

11 195

6 405

Avg.

Malleable Iron f s

19 80

10 100

f s

14 65

7 65

11 85

11 270 6 180 11 235

6 555 6 485

Nodular (Ductile) Iron f s

17 70

9 80

11 85

6 180

11 235

6 485

Machinery's Handbook, 31st Edition 1136 SPEEDS AND FEEDS FOR DRILLING, REAMING, THREADING

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Table 20. Speeds and Feeds for Drilling, Reaming, and Threading Ferrous Cast Metals

Drilling

Reaming

Drilling

Reaming

Threading

HSS

HSS

Indexable Carbide Insert

Material (Martensitic): 120-90-02

HSS

Brinell Hardness Number {

(Ferritic-Pearlitic): 80-55-06

270-330

HSS

25

Opt.

330-400

10

5

190-225

70

45

225-260

50

30

240-300

40

25

(Low carbon): 1010, 1020

100-150

100

65

125-175

90

60

175-225

70

45

225-300

55

35

150-200

75

50

200-250

65

40

250-300

50

30

175-225

70

45

225-250

60

35

250-300

45

30

300-350

30

20

350-400

20

10

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(Low-carbon alloy): 1320, 2315, 2320, 4110, 4120, 4320, 8020, 8620

{

{

(Medium-carbon alloy): 1330, 1340, 2325, 2330, 4125, 4130, 4140, 4330, 4340, { 8030, 80B30, 8040, 8430, 8440, 8630, 8640, 9525, 9530, 9535

Coated

Avg. Opt.

Avg. Opt.

Avg. Opt.

6 150

6 405

Avg. Opt.

Avg.

15

(Pearlitic-Martensitic): 100-70-03

(Medium carbon): 1030, 1040, 1050

Uncoated

f = feed (0.001 in/rev), s = speed (ft/min)  Metric Units: f 3 25.4 = mm/rev, s 3 0.3048 = m/min

Speed (fpm)

f s

13 60

f s

18 35

9 70

f s

15 35

7 60

6 60

11 70

11 195

21 55

11 83 40 60

20 55

29 75

15 83 85 65

20 85

24 65

12 83 70 55

20 70

Cast Steels

f s

8 195†

4 475

8 130†

4 315

See the footnote to Table 17 for instructions concerning the use of this table. The combined feed/speed data in this table are based on tool grades (identified in Table 16) as follows: uncoated = 15; coated carbide = 11, † = 10.

Machinery's Handbook, 31st Edition SPEEDS AND FEEDS FOR DRILLING, REAMING, THREADING 1137

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Table Table 20. (Continued) Speeds and Feeds for Drilling, Reaming, and Threading Ferrous Cast Metals 20. Speeds and Feeds for Drilling, Reaming, and Threading Ferrous Cast Metals

Drilling

Material

CD

All aluminum sand and permanent mold casting alloys

Alloys 308.0 and 319.0 Alloys 360.0 and 380.0 Alloys 390.0 and 392.0

{

Alloys 413

400

HSS Opt.

350

350

500

500

ST and A

350

—

—

HSS

Avg. Opt.

Avg. Opt.

Avg. Opt.

Avg.

f 31 s 390

16 580

11 3235

6 11370

52 610

26 615

20 565

83 635

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—

f 23 s 110

11 145

11 945

6 3325

38 145

19 130

83 145

20 130

f 27 s 90

14 125

11 855

6 3000

45 130

23 125

83 130

20 115

f 24 s 65

12 85

11 555

6 1955

40 85

20 80

83 85

20 80

f 27 s 90

14 125

11 855

6 3000

45 130

23 125

83 130

20 115

—

—

—

300

300 70 —

45

40

AC

125

100

All wrought magnesium alloys

A,CD,ST and A

500

500

All cast magnesium alloys

A,AC, ST and A

450

450

{

HSS

350

ST and A

All other aluminum die-casting alloys

Threading

Aluminum Die-Casting Alloys

AC

70

Reaming

400

AC

—

Indexable Insert Uncoated Carbide

f = feed (0.001 in/rev), s = speed (ft/min)  Metric Units: f 3 25.4 = mm/rev, s 3 0.3048 = m/min

Speed (fpm)

ST and A

ST and A

Drilling

HSS

Brinell Hardness Number

All wrought aluminum alloys, 6061-T651, 5000, 6000, 7000 series

Reaming

Magnesium Alloys

Abbreviations designate: A, annealed; AC, as cast; CD, cold drawn; and ST and A, solution treated and aged, respectively. See the footnote to Table 17 for instruc­ tions concerning the use of this table. The combined feed/speed data in this table are based on tool grades (identified in Table 16) as follows; uncoated carbide = 15.

Machinery's Handbook, 31st Edition 1138 SPEEDS AND FEEDS FOR DRILLING, REAMING, THREADING

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Table 21. Speeds and Feeds for Drilling, Reaming, and Threading Light Metals

Cutting Speed, V = Vopt 3 Ff 3 Fd Ratio of Chosen Feed to Optimum Feed

1.00

1.00

Ratio of the two cutting speeds (average/optimum) given in the tables Vavg /Vopt 1.25

1.50

2.00

1.00

1.00

1.00

1.00

0.90

1.00

1.06

1.09

0.80

1.00

1.12

0.70

1.00

0.60

2.50

Tool Diameter

0.08 in

0.15 in

0.25 in

0.40 in

0.60 in

1.00 in

2.00 in

3.00 in

4.00 in

(15 mm)

(25 mm)

(50 mm)

(75 mm)

(100 mm)

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3.00

4.00

(2 mm)

(4 mm)

(6 mm)

(10 mm)

1.00

1.00

1.00

0.30

0.44

0.56

0.78

1.00

1.32

1.81

2.11

2.29

1.14

1.18

1.21

1.27

0.32

0.46

0.59

0.79

1.00

1.30

1.72

1.97

2.10

1.19

1.31

1.40

1.49

1.63

0.34

0.48

0.61

0.80

1.00

1.27

1.64

1.89

1.95

1.15

1.30

1.50

1.69

1.85

2.15

0.38

0.52

0.64

0.82

1.00

1.25

1.52

1.67

1.73

1.00

1.23

1.40

1.73

2.04

2.34

2.89

0.42

0.55

0.67

0.84

1.00

1.20

1.46

1.51

1.54

0.50

1.00

1.25

1.50

2.00

2.50

3.00

5.00

0.47

0.60

0.71

0.87

1.00

1.15

1.30

1.34

1.94

0.40

1.00

1.23

1.57

2.29

3.08

3.92

5.70

0.53

0.67

0.77

0.90

1.00

1.10

1.17

1.16

1.12

0.30

1.00

1.14

1.56

2.57

3.78

5.19

8.56

0.64

0.76

0.84

0.94

1.00

1.04

1.02

0.96

0.90

0.20

1.00

0.90

1.37

2.68

4.49

6.86

17.60

0.83

0.92

0.96

1.00

1.00

0.96

0.81

0.73

0.66

0.10

1.00

1.44

0.80

2.08

4.36

8.00

20.80

1.29

1.26

1.21

1.11

1.00

0.84

0.60

0.46

0.38

Feed Factor, Ff

Diameter Factor, Fd

This table is specifically for use with the combined feed/speed data for HSS twist drills in Table 17 through Table 23; use Table 5a and Table 5b to adjust speed and tool life for indexable insert drilling with carbides. The combined feed/speed data for HSS twist drilling are based on a 0.60-inch diameter HSS drill with standard drill point geom­etry (2-flute with 118° tip angle). To adjust the given speeds for different feeds and drill diameters, use the two feed/speed pairs from the tables and calculate the ratio of desired (new) feed to optimum feed (largest of the two feeds from the speed table), and the ratio of the two cutting speeds Vavg /Vopt . Use the values of these ratios to find the feed factor Ff at the intersection of the feed ratio row and the speed ratio column in the left half of the table. The diameter factor Fd is found in the same row as the feed factor, in the right half of the table, under the column corresponding to the drill diameter. For diameters not given, interpolate between the nearest available sizes. The adjusted cutting speed can be calculated from V = Vopt 3 Ff 3 Fd , where Vopt is the smaller (optimum) of the two speeds from the speed table (from the left side of the column containing the two feed/speed pairs). Tool life using the selected feed and the adjusted speed should be approximately 45 minutes. Speed adjustment factors to modify tool life are found in Table 15e.

Machinery's Handbook, 31st Edition SPEEDS AND FEEDS FOR DRILLING, REAMING, THREADING 1139

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Table 22. Diameter Speed and Feed Adjustment Factors for HSS Twist Drills and Reamers

Machinery's Handbook, 31st Edition 1140 SPEEDS AND FEEDS FOR DRILLING, REAMING, THREADING Table 23. Speeds and Feeds for Drilling and Reaming Copper Alloys Group 1 Architectural bronze(C38500); Extra-high-leaded brass (C35600); Forging brass (C37700); Free-cutting phosphor bronze (B-2) (C54400); Free-cutting brass (C36000); Free-cutting Muntz metal (C37000); High-leaded brass (C33200, C34200); High-leaded brass tube (C35300); Leaded com­mercial bronze (C31400); Leaded naval brass (C48500); Medium-leaded brass (C34000) Group 2 Aluminum brass, arsenical (C68700); Cartridge brass, 70% (C26000); High-silicon bronze, B (C65500); Admiralty brass (inhibited) (C44300, C44500); Jewelry bronze, 87.5% (C22600); Leaded Muntz metal (C36500, C36800); Leaded nickel silver (C79600); Low brass, 80% (C24000); Low-leaded brass (C33500); Low-silicon bronze, B (C65100); Manganese bronze, A (C67500); Muntz metal, 60% (C28000); Nickel silver, 55-18 (C77000); Red brass, 85% (C23000); Yellow brass (C26800) Group 3 Aluminum bronze, D (C61400); Beryllium copper (C17000, C17200, C17500); Commercial bronze, 90% (C22000); Copper nickel, 10% (C70600); Copper nickel, 30% (C71500);Electro­lytic tough-pitch copper (C11000); Gilding, 95% (C21000); Nickel silver, 65-10 (C74500); Nickel silver, 65-12 (C75700); Nickel silver, 65-15 (C75400); Nickel silver, 65-18 (C75200); Oxygen-free copper (C10200); Phosphor bronze, 1.25% (C50200); Phosphor bronze, 10% D (C52400); Phosphor bronze, 5% A (C51000); Phosphor bronze, 8% C (C52100); Phosphorus deoxidized copper (C12200) Drilling Reaming Alloy Description and UNS Alloy Material Numbers Condition

HSS Speed (fpm)

Drilling Reaming Indexable Insert HSS Uncoated Carbide HSS f = feed (0.001 in/rev), s = speed (ft/min) Metric Units: f 3 25.4 = mm/rev, s 3 0.3048 = m/min Opt. Avg. Opt. Avg. Opt. Avg. Wrought Alloys

A 160 160 21 11 11 6 36 18 f CD 175 175 210 265 405 915 265 230 s A 120 110 24 12 11 6 40 20 f Group 2 CD 140 120 100 130 205 455 130 120 s A 60 50 23 11 11 6 38 19 f Group 3 CD 65 60 155 195 150 340 100 175 s Abbreviations designate: A, annealed; CD, cold drawn. The two leftmost speed columns in this table contain traditional Handbook speeds for HSS steel tools. The text contains information con­ cerning feeds to use in conjunction with these speeds. HSS Drilling and Reaming: The combined feed/speed data for drilling and Table 22 are used to adjust drilling speeds for other feeds and drill diameters. Examples are given in the text. The given speeds and feeds for reaming are based on an 8-tooth, 25∕32 -inch diameter, 30° lead angle reamer, and a 0.008-inch radial depth of cut. For other feeds, the correct speed can be obtained by interpolation using the given speeds if the desired feed lies in the recommended range (between the given values of optimum and average feed). The cutting speeds for reaming do not require adjustment for tool diameter as long as the radial depth of cut does not become too large. Speed adjustment factors to modify tool life are found in Table 15e. Indexable Insert Drilling: The feed/speed data for indexable insert drilling are based on a tool with two cutting edges, an insert nose radius of 3∕6 4 inch, a 10-degree lead angle, and diameter D of 1 inch. Adjustments for feed and depth of cut are made using Table 5a (Turning Speed Adjustment Factors) using a depth of cut of D⁄2, or one-half the drill diameter. Expected tool life at the given speeds and feeds is 15 minutes for short hole drilling (i.e., where hole depth is about 2D or less). Speed adjust­ment factors to increase tool life are found in Table 5b. The combined feed/speed data in this table are based on tool grades (identified in Table 16) as follows: uncoated carbide = 15.

Group 1

Using the Speeds and Feeds Tables for Tapping and Threading.—The feed used in tap­ping and threading is always equal to the pitch of the screw thread being formed. The threading data contained in the tables for drilling, reaming, and threading (Table 17

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Machinery's Handbook, 31st Edition SPEEDS AND FEEDS FOR TAPPING AND THREADING

1141

through Table 23) are primarily for tapping and thread chasing, and do not apply to thread cutting with single-point tools. The threading data in Table 17 through Table 23 give two sets of feed (pitch) and speed values, for 12 and 50 threads/inch, but these values can be used to obtain the cutting speed for any other thread pitches. If the desired pitch falls between the values given in the tables, i.e., between 0.020 inch (50 tpi) and 0.083 inch (12 tpi), the required cutting speed is obtained by interpolation between the given speeds. If the pitch is less than 0.020 inch (more than 50 tpi), use the average speed, i.e., the largest of the two given speeds. For pitches greater than 0.083 inch (fewer than 12 tpi), the optimum speed should be used. Tool life using the given feed/speed data is intended to be approximately 45 minutes, and should be about the same for threads between 12 and 50 threads per inch.

Example: Determine the cutting speed required for tapping 303 stainless steel with a ∕2 -20 coated HSS tap. The two feed/speed pairs for 303 stainless steel, in Table 19, are 83∕35 (0.083 in/rev at 35 fpm) and 20∕45 (0.020 in/rev at 45 fpm). The pitch of a 1 ∕2 -20 thread is 1∕20 = 0.05 inch, so the required feed is 0.05 in/rev. Because 0.05 is between the two given feeds (Table 19), the cutting speed can be obtained by interpolation between the two given speeds as fol­lows: 0.05 − 0.02 V = 35 + 0.083 − 0.02 ^45 − 35h = 40 fpm 1

The cutting speed for coarse-pitch taps must be lower than for fine-pitch taps with the same diameter. Usually, the difference in pitch becomes more pronounced as the diameter of the tap becomes larger and slight differences in the pitch of smaller-diameter taps have little significant effect on the cutting speed. Unlike all other cutting tools, the feed per rev­olution of a tap cannot be independently adjusted—it is always equal to the lead of the thread and is always greater for coarse pitches than for fine pitches. Furthermore, the thread form of a coarse-pitch thread is larger than that of a fine-pitch thread; therefore, it is necessary to remove more metal when cutting a coarse-pitch thread. Taps with a long chamfer, such as starting or tapper taps, can cut faster in a short hole than short chamfer taps, such as plug taps. In deep holes, however, short chamfer or plug taps can run faster than long chamfer taps. Bottoming taps must be run more slowly than either starting or plug taps. The chamfer helps to start the tap in the hole. It also functions to involve more threads, or thread form cutting edges, on the tap in cutting the thread in the hole, thus reducing the cutting load on any one set of thread form cutting edges. In so doing, more chips and thinner chips are produced that are difficult to remove from deeper holes. Shortening the chamfer length causes fewer thread form cutting edges to cut, thereby producing fewer and thicker chips that can easily be disposed of. Only one or two sets of thread form cutting edges are cut on bottoming taps, causing these cutting edges to assume a heavy cutting load and produce very thick chips. Spiral-pointed taps can operate at a faster cutting speed than taps with normal flutes. These taps are made with supplementary angular flutes on the end that push the chips ahead of the tap and prevent the tapped hole from becoming clogged with chips. They are used primarily to tap open or through holes although some are made with shorter supple­ mentary flutes for tapping blind holes. The tapping speed must be reduced as the percentage of full thread to be cut is increased. Experiments have shown that the torque required to cut a 100 percent thread form is more than twice that required to cut a 50 percent thread form. An increase in the percentage of full thread will also produce a greater volume of chips. The tapping speed must be lowered as the length of the hole to be tapped is increased. More friction must be overcome in turning the tap and more chips accumulate in the hole. It will be more difficult to apply the cutting fluid at the cutting edges and to lubricate the

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1142

Machinery's Handbook, 31st Edition SPEEDS AND FEEDS FOR TAPPING AND THREADING

tap to reduce friction. This problem becomes greater when the hole is being tapped in a hori­zontal position.

Cutting fluids have a very great effect on the cutting speed for tapping. Although other operating conditions when tapping frequently cannot be changed, a free selection of the cutting fluid usually can be made. When planning the tapping operation, the selection of a cutting fluid warrants a very careful consideration and perhaps an investigation.

Taper threaded taps, such as pipe taps, must be operated at a slower speed than straight thread taps with a comparable diameter. All the thread form cutting edges of a taper threaded tap that are engaged in the work cut and produce a chip, but only those cutting edges along the chamfer length cut on straight thread taps. Pipe taps often are required to cut the tapered thread from a straight hole, adding to the cutting burden.

The machine tool used for the tapping operation must be considered in selecting the tap­ping speed. Tapping machines and other machines that are able to feed the tap at a rate of advance equal to the lead of the tap, and that have provisions for quickly reversing the spin­dle, can be operated at high cutting speeds. On machines where the feed of the tap is con­trolled manually—such as on drill presses and turret lathes—the tapping speed must be reduced to allow the operator to maintain safe control of the operation. There are other special considerations in selecting the tapping speed. Very accurate threads are usually tapped more slowly than threads with a commercial grade of accuracy. Thread forms that require deep threads for which a large amount of metal must be removed, producing a large volume of chips, require special techniques and slower cutting speeds. Acme, buttress, and square threads, therefore, are generally cut at lower speeds.

Cutting Speed for Broaching.—Broaching offers many advantages in manufacturing metal parts, including high production rates, excellent surface finishes, and close dimen­ sional tolerances. These advantages are not derived from the use of high cutting speeds; they are derived from the large number of cutting teeth that can be applied consecutively in a given period of time, from their configuration and precise dimensions, and from the width or diameter of the surface that can be machined in a single stroke. Most broaching cutters are expensive in their initial cost and are expensive to sharpen. For these reasons, a long tool life is desirable, and to obtain a long tool life, relatively slow cutting speeds are used. In many instances, slower cutting speeds are used because of the limitations of the machine in accelerating and stopping heavy broaching cutters. At other times, the avail­able power on the machine places a limit on the cutting speed that can be used; i.e., the cubic inches of metal removed per minute must be within the power capacity of the machine.

The cutting speeds for high-speed steel broaches range from 3 to 50 feet per minute, although faster speeds have been used. In general, the harder and more difficult to machine materials are cut at a slower cutting speed and those that are easier to machine are cut at a faster speed. Some typical recommendations for high-speed steel broaches are: AISI 1040, 10 to 30 fpm; AISI 1060, 10 to 25 fpm; AISI 4140, 10 to 25 fpm; AISI 41L40, 20 to 30 fpm; 201 austenitic stainless steel, 10 to 20 fpm; Class 20 gray cast iron, 20 to 30 fpm; Class 40 gray cast iron, 15 to 25 fpm; aluminum and magnesium alloys, 30 to 50 fpm; copper alloys, 20 to 30 fpm; commercially pure titanium, 20 to 25 fpm; alpha and beta titanium alloys, 5 fpm; and the superalloys, 3 to 10 fpm. Surface broaching operations on gray iron castings have been conducted at a cutting speed of 150 fpm, using indexable insert cemented car­bide broaching cutters. In selecting the speed for broaching, the cardinal principle of the performance of all metal cutting tools should be kept in mind; i.e., increasing the cutting speed may result in a proportionately larger reduction in tool life, and reducing the cutting speed may result in a proportionately larger increase in the tool life. When broaching most materials, a suitable cutting fluid should be used to obtain a good surface finish and a better tool life. Gray cast iron can be broached without using a cutting fluid although some shops prefer to use a soluble oil.

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Machinery's Handbook, 31st Edition SPEEDS AND MACHINING POWER

1143

ESTIMATING SPEEDS AND MACHINING POWER Estimating Planer Cutting Speeds.—Whereas most planers of modern design have a means of indicating the speed at which the table is traveling, or cutting, many older planers do not. The following formulas are useful for planers that do not have a means of indicating the table or cutting speed. It is not practicable to provide a formula for calculating the exact cutting speed at which a planer is operating because the time to stop and start the table when reversing varies greatly. The formulas below will provide a reasonable estimate. Vc Vc ≈ Sc L and S c ≈ ----L where Vc  =  cutting speed; fpm or m/min Sc  =  number of cutting strokes per minute of planer table L  =  length of table cutting stroke; ft or m

Cutting Speed for Planing and Shaping.—The traditional HSS cutting tool speeds in Table 1 through Table 4b and Table 6 through Table 9, pages 1095 through 1107, can be used for planing and shaping. The feed and depth of cut factors in Table 5c should also be used, as explained previously. Very often, other factors relating to the machine or the setup will require a reduction in the cutting speed used on a specific job. Cutting Time for Turning, Boring, and Facing.—The time required to turn a length of metal can be determined by the following formula in which T = time in minutes, L = length of cut in inches (or mm), f = feed in inches per revolution (or mm/min), and N = lathe spin­ dle speed in revolutions per minute. L T = fN

When making job estimates, the time required to load and to unload the workpiece on the machine, and the machine handling time, must be added to the cutting time for each length cut to obtain the floor-to-floor time. Planing Time.—The approximate time required to plane a surface can be determined from the following formula in which T = time in minutes, L = length of stroke in feet (or meter), Vc = cutting speed in feet per minute (m/min), Vr = return speed in feet per minute (m/min); W = width of surface to be planed in inches (or mm), F = feed in inches (or mm), and 0.025 = approximate reversal time factor per stroke in minutes for most planers: 1 1 W T = F ;L # c + m + 0.025E Vc Vr

Speeds for Metal-Cutting Saws.—The table on page 1144 gives speeds and feeds for solid-tooth, high-speed steel, circular, metal-cutting saws as recommended by Saws International, Inc. (sfpm = surface feet per minute = 3.142 3 blade diameter in inches 3 rpm of saw shaft ÷ 12). Also see page 1245 for bandsaw blade speeds. Speeds for Turning Unusual Materials.—Slate, on account of its peculiarly stratified formation, is rather difficult to turn, but if handled carefully, can be machined in an ordi­ nary lathe. The cutting speed should be about the same as for cast iron. A sheet of fiber or pressed paper should be interposed between the chuck or steadyrest jaws and the slate, to protect the latter. Slate rolls must not be centered and run on the tailstock. A satisfactory method of supporting a slate roll having journals at the ends is to bore a piece of lignum vitae to receive the turned end of the roll, and center it for the tailstock spindle. Rubber can be turned at a peripheral speed of 200 feet per minute (61 m/min), although it is much easier to grind it with an abrasive wheel that is porous and soft. For cutting a rubber roll in two, the ordinary parting tool should not be used, but a tool shaped like a knife; such a tool severs the rubber without removing any material.

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

1144

First entry is cutting speed, second entry is feed α = Cutting angle (Front), degree β = Relief angle (Back), degree

 Materials Aluminum

Rake Angle α

β

24

12 10

Light Alloys with Cu, Mg, and Zn

22

Light Alloys with High Si

20

8

Copper

20

10

Bronze

15

8

Hard Bronze

10

8

Cu-Zn Brass

16

8

Gray Cast Iron

12

8

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Carbon Steel

20

8

Medium Hard Steel

18

8

Hard Steel

15

8

Stainless Steel

15

8

1 ∕4 –3 ∕4

6500 sfpm 100 in/min 3600 sfpm 70 in/min 650 sfpm 16 in/min 1300 sfpm 24 in/min 1300 sfpm 24 in/min 400 sfpm 6.3 in/min 2000 sfpm 43 in/min 82 sfpm 4 in/min 160 sfpm 6.3 in/min 100 sfpm 5.1 in/min 66 sfpm 4.3 in/min 66 sfpm 2 in/min

Stock Diameters, inches

3 ∕4 –11 ∕2

6200 sfpm 85 in/min 3300 sfpm 65 in/min 600 sfpm 16 in/min 1150 sfpm 24 in/min 1150 sfpm 24 in/min 360 sfpm 6 in/min 2000 sfpm 43 in/min 75 sfpm 4 in/min 150 sfpm 5.9 in/min 100 sfpm 4.7 in/min 66 sfpm 4.3 in/min 63 sfpm 1.75 in/min

11 ∕2 –21 ∕2

6000 sfpm 80 in/min 3000 sfpm, 63 in/min 550 sfpm 14 in/min 1000 sfpm 22 in/min 1000 sfpm 22 in/min 325 sfpm 5.5 in/min 1800 sfpm 39 in/min 72 sfpm 3.5 in/min 150 sfpm 5.5 in/min 80 sfpm 4.3 in/min 60 sfpm 4 in/min 60 sfpm 1.75 in/min

21 ∕2 –31 ∕2

5000 sfpm, 75 in/min 2600 sfpm 60 in/min 550 sfpm 12 in/min 800 sfpm 22 in/min 800 sfpm 20 in/min 300 sfpm 5.1 in/min 1800 sfpm 35 in/min 66 sfpm 3 in/min 130 sfpm 5.1 in/min 80 sfpm 4.3 in/min 57 sfpm 3.5 in/min 57 sfpm 1.5 in/min

6–19

1981 m/min 2540 mm/min 1097 m/min 1778 mm/min 198 m/min 406 mm/min 396 m/min 610 mm/min 396 m/min 610 mm/min 122 m/min 160 mm/min 610 m/min 1092 mm/min 25 m/min 102 mm/min 49 m/min 160 mm/min 30 m/min 130 mm/min 20 m/min 109 mm/min 20 m/min 51 mm/min

Stock Diameters, millimeters 19–38

1890 m/min 2159 mm/min 1006 m/min 1651 mm/min 183 m/min 406 mm/min 351 m/min 610 mm/min 351 m/min 610 mm/min 110 m/min 152 mm/min 610 m/min 192 mm/min 23 m/min 102 mm/min 46 m/min 150 mm/min 30 m/min 119 mm/min 20 m/min 109 mm/min 19 m/min 44 mm/min

38–63

1829 m/min 2159 mm/min 914 m/min 1600 mm/min 168 m/min 356 mm/min 305 m/min 559 mm/min 305 m/min 559 mm/min 99 m/min 140 mm/min 549 m/min 991 mm/min 22 m/min 89 mm/min 46 m/min 140 mm/min 24 m/min 109 mm/min 18 m/min 102 mm/min 18 m/min 44 mm/min

63–89

1524 m/min 2159 mm/min 792 m/min 1524 mm/min 168 m/min 305 mm/min 244 m/min 559 mm/min 244 m/min 508 mm/min 91 m/min 130 mm/min 549 m/min 889 mm/min 20 m/min 76 mm/min 40 m/min 130 mm/min 24 m/min 109 mm/min 17 m/min 89 mm/min 17 m/min 38 mm/min

Gutta percha can be turned as easily as wood, but the tools must be sharp and a good soap-and-water lubricant used. Copper can be turned easily at 200 feet per minute (61 m/min). See also Table 6 on page 1105. Limestone such as is used in the construction of pillars for balconies, etc., can be turned at 150 feet per minute (46 m/min), and the formation of ornamental contours is quite easy. Marble is a treacherous material to turn. It should be cut with a tool such as would be used for brass, but at a speed suitable for cast iron. It must be handled very carefully to prevent flaws in the surface.

Machinery's Handbook, 31st Edition SPEEDS AND MACHINING POWER

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Speeds, Feeds, and Tooth Angles for Sawing Various Material with Solid-Tooth, High-Speed Steel, Circular, Metal-cutting Saws

Machinery's Handbook, 31st Edition SPEEDS AND MACHINING POWER

1145

The foregoing speeds are for high-speed steel tools. Tools tipped with tungsten carbide are adapted for cutting various non-metallic products which cannot be machined readily with steel tools, such as slate, marble, synthetic plastic materials, etc. In drilling slate and marble, use flat drills; and for plastic materials, tungsten carbide-tipped twist drills. Cut­ting speeds ranging from 75 to 150 feet per minute (23–46 m/min) have been used for drill­ing slate (without coolant) and a feed of 0.025 inch per revolution (0.64 mm/rev) for drills 3∕4 and 1 inch (19.05 and 25.4 mm) in diameter. Estimating Machining Power Knowledge of the power required to perform machining operations is useful when plan­ ning new machining operations, for optimizing existing machining operations, and to develop specifications for new machine tools that are to be acquired. The available power on any machine tool places a limit on the size of the cut that it can take. When much metal must be removed from the workpiece it is advisable to estimate the cutting conditions that will utilize the maximum power on the machine. Many machining operations require only light cuts to be taken for which the machine obviously has ample power; in this event, esti­mating the power required is a wasteful effort. Conditions in different shops may vary and machine tools are not all designed alike, so some variations between the estimated results and those obtained on the job are to be expected. However, by using the methods provided in this section a reasonable estimate of the power required can be made, which will suffice in most practical situations. The measure of power in customary inch units is the horsepower; in SI metric units it is the kilowatt, which is used for both mechanical and electrical power. The power required to cut a material depends upon the rate at which the material is being cut and upon an exper­imentally determined power constant, Kp, which is also called the unit horsepower, unit power, or specific power consumption. The power constant is equal to the horsepower required to cut a material at a rate of one cubic inch per minute; in SI metric units the power constant is equal to the power in kilowatts required to cut a material at a rate of one cubic centimeter per second, or 1000 cubic millimeters per second (1 cm3 = 1000 mm3). Differ­ent values of the power constant are required for inch and for metric units, which are related as follows: to obtain the SI metric power constant, multiply the inch power constant by 2.73; to obtain the inch power constant, divide the SI metric power constant by 2.73. Values of the power constant in Table 1a, and Table 1b can be used for all machining operations except drilling and grinding. Values given are for sharp tools. Table 1a. Power Constants Kp Using Sharp Cutting Tools

Material

Brinell Kp Kp Hardness Inch Metric Number Units Units

Material

Gray Cast  Iron

100-120 120-140 140-160 160-180 180-200 200-220 220-240

0.28 0.35 0.38 0.52 0.60 0.71 0.91

Ferrous Cast Metals 0.76 Malleable Iron 0.96  Ferritic 1.04 1.42 1.64  Pearlitic 1.94 2.48

Alloy Cast  Iron

150-175 175-200 200-250

0.30 0.63 0.92

0.82 1.72 2.51

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Cast Steel …

Brinell Hardness Number

Kp Kp Inch Metric Units Units

150-175

0.42

1.15

175-200 200-250 250-300

0.57 0.82 1.18

1.56 2.24 3.22

150-175 175-200 200-250 …

0.62 0.78 0.86 …

1.69 2.13 2.35 …

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Machinery's Handbook, 31st Edition Machining Power

1146

Table Table 1a. (Continued) Constants Kp Using Cutting 1a. PowerPower Constants Kp Using SharpSharp Cutting Tools Tools Material

Brinell Kp Kp Hardness Inch Metric Number Units Units

Material

Brinell Hardness Number

Kp Kp Inch Metric Units Units

High-Temperature Alloys, Tool Steel, Stainless Steel, and Nonferrous Metals High-Temperature Alloys 150-175 0.60 0.82 2.24 Stainless Steel 175-200 0.72  A286 165  A286 285 0.93 2.54 200-250 0.88  Chromoloy 200 0.78 3.22 Zinc Die-Cast Alloys … 0.25  Chromoloy 310 1.18 3.00 Copper (pure) … 0.91   Inco 700 330 1.12 3.06 Brass   Inco 702 230 1.10 3.00  Hard … 0.83  Hastelloy-B 230 1.10 3.00  Medium … 0.50  M-252 230 1.10 3.00  Soft … 0.25  M-252 310 1.20 3.28  Leaded … 0.30  Ti-150A 340 0.65 1.77 1.10 3.00 Bronze  U-500 375 0.91  Hard … Monel Metal … 1.00 2.73  Medium … 0.50

Tool Steel

175-200 200-250 250-300 300-350 350-400

0.75 0.88 0.98 1.20 1.30

2.05 2.40 2.68 3.28 3.55

1.64 1.97 2.40 0.68 2.48 2.27 1.36 0.68 0.82

2.48 1.36

Aluminum  Cast   Rolled (hard)

… …

0.25 0.33

0.68 0.90

Magnesium Alloys

…

0.10

0.27

The value of the power constant is essentially unaffected by the cutting speed, the depth of cut, and the cutting tool material. Factors that do affect the value of the power constant, and thereby the power required to cut a material, include the hardness and microstructure of the work material, the feed rate, the rake angle of the cutting tool, and whether the cut­ting edge of the tool is sharp or dull. Values are given in the power constant tables for dif­ferent material hardness levels, whenever this information is available. Feed factors for the power constant are given in Table 2. All metal cutting tools wear but a worn cutting edge requires more power to cut than a sharp cutting edge. Factors to provide for tool wear are given in Table 3. In this table, the extra-heavy-duty category for milling and turning occurs only on operations where the tool is allowed to wear more than a normal amount before it is replaced, such as roll turning. The effect of the rake angle usually can be disregarded. The rake angle for which most of the data in the power constant tables are given is positive 14 degrees. Only when the deviation from this angle is large is it necessary to make an adjustment. Using a rake angle that is more positive reduces the power required approximately 1 percent per degree; using a rake angle that is more negative increases the power required; again approximately 1 percent per degree. Many indexable insert cutting tools are formed with an integral chip breaker or other cut­ting edge modifications, which have the effect of reducing the power required to cut a material. The extent of this effect cannot be predicted without a test of each design. Cutting fluids will also usually reduce the power required, when operating in the lower range of cutting speeds. Again, the extent of this effect cannot be predicted because each cutting fluid exhibits its own characteristics.

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Machinery's Handbook, 31st Edition Machining Power

1147

Table 1b. Power Constants Kp Using Sharp Cutting Tools Material

All Plain Carbon Steels

AISI 1108, 1109, 1110, 1115, 1116, 1117, 1118, 1119, 1120, 1125, 1126, 1132

AISI 4023, 4024, 4027, 4028, 4032, 4037, 4042, 4047, 4137, 4140, 4142, 4145, 4147, 4150, 4340, 4640, 4815, 4817, 4820, 5130, 5132, 5135, 5140, 5145, 5150, 6118, 6150, 8637, 8640, 8642, 8645, 8650, 8740 AISI 1330, 1335, 1340, E52100

Brinell Hardness Number

80-100 100-120 120-140 140-160 160-180 180-200 200-220

Kp Kp Inch Metric Units Units

Material

Wrought Steels

0.63 0.66 0.69 0.74 0.78 0.82 0.85

100-120 120-140 140-160 160-180 180-200

0.41 0.42 0.44 0.48 0.50

140-160 160-180 180-200 200-220 220-240 240-260 260-280 280-300 300-320 320-340 340-360 160-180 180-200 200-220

0.62 0.65 0.69 0.72 0.76 0.80 0.84 0.87 0.91 0.96 1.00 0.79 0.83 0.87

Plain Carbon Steels 1.72 1.80 1.88 2.02 2.13 2.24 2.32

All Plain Carbon Steels

1.12 1.15 1.20 1.31 1.36

AISI 1137, 1138, 1139, 1140, 1141, 1144, 1145, 1146, 1148, 1151

Free Machining Steels

Alloy Steels

1.69 1.77 1.88 1.97 2.07 2.18 2.29 2.38 2.48 2.62 2.73 2.16 2.27 2.38

AISI 4130, 4320, 4615, 4620, 4626, 5120, 8615, 8617, 8620, 8622, 8625, 8630, 8720

Brinell Hardness Number

Kp Inch Units

Kp SI Metric Units

220-240 240-260 260-280 280-300 300-320 320-340 340-360

0.89 0.92 0.95 1.00 1.03 1.06 1.14

2.43 2.51 2.59 2.73 2.81 2.89 3.11

180-200 200-220 220-240 240-260 …

0.51 0.55 0.57 0.62

1.39 1.50 1.56 1.69

140-160 160-180 180-200 200-220 220-240 240-260 260-280 280-300 300-320 320-340

0.56 0.59 0.62 0.65 0.70 0.74 0.77 0.80 0.83 0.89

1.53 1.61 1.69 1.77 1.91 2.02 2.10 2.18 2.27 2.43

… … … …

…

… … … …

…

… … … …

The machine tool transmits the power from the driving motor to the workpiece, where it is used to cut the material. The effectiveness of this transmission is measured by the machine tool efficiency factor, E. Average values of this factor are given in Table 4. For­ mulas for calculating the metal removal rate, Q, for different machining operations are given in Table 5. These formulas are used together with others given below. The following formulas can be used with either customary inch or with SI metric units.

Pc = Kp CQW Kp CQW Pc = Pm = E E where Pc  =  power at the cutting tool; hp, or kW

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Table 2. Feed Factors, C, for Power Constants

Inch Units Feed in.a 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.010 0.011 0.012 0.013

C 1.60 1.40 1.30 1.25 1.19 1.15 1.11 1.08 1.06 1.04 1.02 1.00 0.98

SI Metric Units

Feed in.a 0.014 0.015 0.016 0.018 0.020 0.022 0.025 0.028 0.030 0.032 0.035 0.040 0.060

C 0.97 0.96 0.94 0.92 0.90 0.88 0.86 0.84 0.83 0.82 0.80 0.78 0.72

Feed mmb 0.02 0.05 0.07 0.10 0.12 0.15 0.18 0.20 0.22 0.25 0.28 0.30 0.33

C 1.70 1.40 1.30 1.25 1.20 1.15 1.11 1.08 1.06 1.04 1.01 1.00 0.98

Feed mmb 0.35 0.38 0.40 0.45 0.50 0.55 0.60 0.70 0.75 0.80 0.90 1.00 1.50

a Turning, in/rev; milling, in/tooth; planing and shaping, in/stroke; broaching, in/tooth.

C 0.97 0.95 0.94 0.92 0.90 0.88 0.87 0.84 0.83 0.82 0.80 0.78 0.72

b Turning, mm/rev; milling, mm/tooth; planing and shaping, mm/stroke; broaching, mm/tooth.

Table 3. Tool Wear Factors, W

Type of Operation For all operations with sharp cutting tools Turning: Finish turning (light cuts) Normal rough and semifinish turning Extra-heavy-duty rough turning Milling: Slab milling End milling Light and medium face milling Extra-heavy-duty face milling Drilling: Normal drilling Drilling hard-to-machine materials and drilling with a very   dull drill Broaching: Normal broaching Heavy-duty surface broaching Planing and Use values given for turning Shaping

W 1.00 1.10 1.30 1.60-2.00 1.10 1.10 1.10-1.25 1.30-1.60 1.30 1.50 1.05-1.10 1.20-1.30

Pm =  power at the motor; hp, or kW Kp =  power constant (see Table 1a and Table 1b) Q =  metal removal rate; in3/min or cm3/s (see Table 5) C =  feed factor for power constant (see Table 2) W =  tool wear factor (see Table 3) E =  machine tool efficiency factor (see Table 4) V =  cutting speed, fpm, or m/min N =  cutting speed, rpm f =  feed rate for turning; in/rev or mm/rev f =  feed rate for planing and shaping; in/stroke, or mm/stroke

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ft =  feed per tooth; in/tooth, or mm/tooth fm =  feed rate; in/min or mm/min dt =  maximum depth of cut per tooth: inch, or mm d =  depth of cut; inch, or mm nt =  number of teeth on milling cutter nc =  number of teeth engaged in work w =  width of cut; inch, or mm Table 4. Machine Tool Efficiency Factors, E Type of Drive

Direct Belt Drive Back Gear Drive

Type of Drive

E

0.90

E

Geared Head Drive

0.75

0.70-0.80

Oil-Hydraulic Drive

0.60-0.90

Table 5. Formulas for Calculating the Metal Removal Rate, Q Metal Removal Rate

For Inch Units Only Q = in3/min

For SI Metric Units Only Q = cm3/s

Single-Point Tools (Turning, Planing, and Shaping)

12Vfd

V 60 fd

Milling

fm wd

Surface Broaching

12Vwnc dt

Operation

fm wd

60, 000

V 60 unc dt

Example: A 180-200 BHN AISI 4130 shaft is to be turned on a geared head lathe using a cutting speed of 350 fpm (107 m/min), a feed rate of 0.016 in/rev (0.40 mm/rev), and a depth of cut of 0.100 inch (2.54 mm). Estimate the power at the cutting tool and at the motor, using both the inch and metric data. Inch units: Kp =  0.62 (from Table 1b) C =  0.94 (from Table 2) W =  1.30 (from Table 3) E =  0.80 (from Table 4) Q =  12 Vfd = 12 3 350 3 0.016 3 0.100 (from Table 5) Q =  6.72 in3/min Pc =  Kp CQW = 0.62 # 0.94 # 6. 72 # 1. 30 = 5.1 hp

Pm =   Pc = 5 = 6.4 hp E 0.80 SI metric units: Kp =  1.69 (from Table 1b) C =  0.94 (from Table 2) W =  1.30 (from Table 3) E =  0.80 (from Table 4) 107 V Q =   60 fd = 60 × 0.40 × 2.54 = 1.81 cm3/ s (from Table 5)

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Machinery's Handbook, 31st Edition Machining Power



Pc =   Kp CQW = 1.69 × 0.94 × 1. 81 × 1. 30 = 3.74 kW



Pc Pm =   E = 3.74 = 4.677 kW 0.80

Whenever possible the maximum power available on a machine tool should be used when heavy cuts must be taken. The cutting conditions for utilizing the maximum power should be selected in the follow­ing order: 1) select the maximum depth of cut that can be used; 2) select the maximum feed rate that can be used; and 3) estimate the cutting speed that will utilize the maximum power available on the machine. This sequence is based on obtaining the longest tool life of the cutting tool and at the same time obtaining as much production as possible from the machine. The life of a cutting tool is most affected by cutting speed, then by feed rate, and least of all by depth of cut. The maximum metal removal rate that a given machine is capable of machining from a given material is used as the basis for estimating the cutting speed that will utilize all the power available on the machine. Example: A 160–180 BHN gray iron casting that is 6 inches wide is to have 1 ⁄8 inch stock removed on a 10 hp milling machine, using an 8 inch diameter, 10 tooth, indexable insert cemented carbide face milling cutter. The feed rate selected for this cutter is 0.012 in/ tooth, and all the stock (0.125 inch) will be removed in one cut. Estimate the cutting speed that will utilize the maximum power available on the machine. Kp =  0.52 (From Table 1a) C =  1.00 (From Table 2) W =  1.20 (From Table 3) E =  0.80 (From Table 4) K CQW Pm E 10 # 0.80 c Pm = p m Qmax = = 0.52 # 1.00 # 1.20 = 12.82 in 3/ min E K pCW

Qmax 12.82 fm = wd = 6 # 0.125 = 17.1 in / min f 17 N = max = 0.012 # 10 = 142.4 rpm ft nt π DN π # 8 # 142 V = 12 = = 298.3 fpm 12

^Q = fm wdh ^ fm = ft nt N h

a N = 12V k πD

Calculated cutting speed of 298.3 fpm is correct, but very low for available cutting tools. Today, milling of gray irons is performed, mostly, using cubic boron nitride indexable inserts. Typical range of the cutting speed for milling with large engagement (0.8) and 0.004–0.008 inch per tooth chip load is 2700–2000 sfm. Estimating maximum productivity of cutting should be based on the recommended cutting speeds versus work materials. Example: A 3.2 mm deep cut is to be taken on a 200–210 BHN AISI 1050 steel part using a 7.5 kW geared head lathe. The feed rate selected for this job is 0.45 mm/rev. Estimate the cutting speed that will utilize the maximum power available on the lathe. Kp =  2.32 (From Table 1b) C =  0.92 (From Table 2) W =  1.30 (From Table 3) E =  0.80 (From Table 4)

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Machinery's Handbook, 31st Edition Machining Power Qmax =

Pm E 7. 5 × 0. 80 = 2.32 × 0.92 × 1. 30 K p CW

1151 K pCQW m E

c Pm =

= 2.16cm3 / sec V=

60 Qmax 60 × 2.16 = 0.45 × 3. 2 fd

aQ =

V fdk 60

= 90.0m / min The calculated cutting speed of 90.0 m/min is correct, but too low for carbide cutting tools. Thirty years ago, recommended cutting speed for turning AISI 1050 steel, 200– 210 BHN, depth of cut 4.0 mm, was 160 m/min. Currently, using advanced coated carbide inserts, the cutting speed is increased up to 300 m/min for the depth of cut of 2.5–5.0 mm and a 0.38 mm/rev feed rate. Estimating Drilling Thrust, Torque, and Power.—Although the lips of a drill cut metal and produce a chip in the same manner as the cutting edges of other metal cutting tools, the chisel edge removes the metal by means of a very complex combination of extrusion and cutting. For this reason a separate method must be used to estimate the power required for drilling. Also, it is often desirable to know the magnitude of the thrust and the torque required to drill a hole. The formulas and tabular data provided in this section are based on information supplied by the National Twist Drill Division of Regal-Beloit Corp. The val­ues in Table 6 through Table 9 are for sharp drills and the tool wear factors are given in Table 3. For most ordinary drilling operations 1.30 can be used as the tool wear factor. When drill­ing most difficult-to-machine materials and when the drill is allowed to become very dull, 1.50 should be used as the value of this factor. It is usually more convenient to measure the web thickness at the drill point than the length of the chisel edge; for this reason, the approximate w/d ratio corresponding to each c/d ratio for a correctly ground drill is pro­vided in Table 7. For most standard twist drills the c/d ratio is 0.18, unless the drill has been ground short or the web has been thinned. The c/d ratio of split-point drills is 0.03. The for­mulas given below can be used for spade drills, as well as for twist drills. Separate formulas are required for use with customary inch units and for SI metric units. Table 6. Work Material Factor, Kd , for Drilling with a Sharp Drill Work Material AISI 1117 (Resulfurized   free machining mild steel) Steel, 200 BHN Steel, 300 BHN Steel, 400 BHN Cast Iron, 150 BHN Most Aluminum Alloys Most Magnesium Alloys Most Brasses Leaded Brass

Material Constant, Kd 12,000 24,000 31,000 34,000 14,000 7,000 4,000 14,000 7,000

Work Material Austenitic Stainless Steel   (Type 316) Titanium Alloy Ti6Al4V   40Rc René 41 

40Rc

Hastelloy-C

Material Constant, Kd 24,000a for Torque 18,000a for Torque 29,000a for Thrust 40,000ab min. 30,000a for Torque 37,000a for Thrust

a Values based upon a limited number of tests. b Will increase with rapid wear.

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Table 7. Chisel Edge Factors for Torque and Thrust c/d 0.03 0.05 0.08 0.10 0.13 0.15

Approx. w/d 0.025 0.045 0.070 0.085 0.110 0.130

Torque Factor A 1.000 1.005 1.015 1.020 1.040 1.080

Thrust Factor B 1.100 1.140 1.200 1.235 1.270 1.310

Thrust Factor J 0.001 0.003 0.006 0.010 0.017 0.022

c/d 0.18 0.20 0.25 0.30 0.35 0.40

Approx. w/d 0.155 0.175 0.220 0.260 0.300 0.350

Torque Factor A 1.085 1.105 1.155 1.235 1.310 1.395

For drills of standard design, use c/d = 0.18; for split-point drills, use c/d = 0.03 c/d = Length of Chisel Edge ÷ Drill Diameter. w/d = Web Thickness at Drill Point ÷ Drill Diameter.

Thrust Factor B 1.355 1.380 1.445 1.500 1.575 1.620

For inch units only: 2 T = 2Kd Ff FT BW + Kd D JW

(1) (2)

M = Kd Ff FM AW

Pc = MN ⁄ 63,025

(3)

For SI metric units only: T = 0.05 Kd Ff FT BW + 0.007 Kd D2JW

(4)

Kd Ff FM AW  = 0.000025 Kd Ff FM AW 40, 000 Pc = MN⁄ 9550

M=

Use with either inch or metric units:

Thrust Factor J 0.030 0.040 0.065 0.090 0.120 0.160

Pc Pm = E

(5) (6) (7)

where Pc = Power at the cutter; hp, or kW   Pm =  Power at the motor; hp, or kW M =  Torque; in. lb, or N.m T =  Thrust; lb, or N Kd =  Work material factor (See Table 6) Ff =  Feed factor (See Table 8) FT  =  Thrust factor for drill diameter (See Table 9) FM =  Torque factor for drill diameter (See Table 9) A =  Chisel edge factor for torque (See Table 7) B =  Chisel edge factor for thrust (See Table 7) J =  Chisel edge factor for thrust (See Table 7) W =  Tool wear factor (See Table 3) N =  Spindle speed; rpm E =  Machine tool efficiency factor (See Table 4) D =  Drill diameter; in., or mm c =  Chisel edge length; in., or mm (See Table 7) w =  Web thickness at drill point; in., or mm (See Table 7)

Example: A standard 7∕8 inch drill is to drill steel parts having a hardness of 200 BHN on a drilling machine having an efficiency of 0.80. The spindle speed to be used is 350 rpm and the feed rate will be 0.008 in/rev. Calculate the thrust, torque, and power required to drill these holes: Kd = 24,000 (From Table 6)  Ff =   0.021 (From Table 8) FT = 0.899 (From Table 9)  FM =   0.786 (From Table 9) A = 1.085 (From Table 7)  B = 1.355 (From Table 7) J =   0.030 (From Table 7)

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Table 8. Feed Factors Ff for Drilling

Inch Units

Feed, in/rev 0.0005 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.010

Ff 0.0023 0.004 0.007 0.010 0.012 0.014 0.017 0.019 0.021 0.023 0.025

Feed, in/rev 0.012 0.013 0.015 0.018 0.020 0.022 0.025 0.030 0.035 0.040 0.050

Ff 0.029 0.031 0.035 0.040 0.044 0.047 0.052 0.060 0.068 0.076 0.091

SI Metric Units

Feed, mm/rev 0.01 0.03 0.05 0.08 0.10 0.12 0.15 0.18 0.20 0.22 0.25

Ff 0.025 0.060 0.091 0.133 0.158 0.183 0.219 0.254 0.276 0.298 0.330

Feed, mm/rev 0.30 0.35 0.40 0.45 0.50 0.55 0.65 0.75 0.90 1.00 1.25

Ff 0.382 0.432 0.480 0.528 0.574 0.620 0.708 0.794 0.919 1.000 1.195

Table 9. Drill Diameter Factors: FT for Thrust, FM for Torque Drill Dia., in. 0.063 0.094 0.125 0.156 0.188 0.219 0.250 0.281 0.313 0.344 0.375 0.438 0.500 0.563 0.625 0.688 0.750 0.813

Inch Units

FT

FM

0.110 0.151 0.189 0.226 0.263 0.297 0.330 0.362 0.395 0.426 0.456 0.517 0.574 0.632 0.687 0.741 0.794 0.847

0.007 0.014 0.024 0.035 0.049 0.065 0.082 0.102 0.124 0.146 0.171 0.226 0.287 0.355 0.429 0.510 0.596 0.689

Drill Dia., in. 0.875 0.938 1.000 1.063 1.125 1.250 1.375 1.500 1.625 1.750 1.875 2.000 2.250 2.500 2.750 3.000 3.500 4.000

FT

FM

0.899 0.950 1.000 1.050 1.099 1.195 1.290 1.383 1.475 1.565 1.653 1.741 1.913 2.081 2.246 2.408 2.724 3.031

0.786 0.891 1.000 1.116 1.236 1.494 1.774 2.075 2.396 2.738 3.100 3.482 4.305 5.203 6.177 7.225 9.535 12.13

Drill Dia., mm 1.60 2.40 3.20 4.00 4.80 5.60 6.40 7.20 8.00 8.80 9.50 11.00 12.50 14.50 16.00 17.50 19.00 20.00

SI Metric Units

FT

FM

1.46 2.02 2.54 3.03 3.51 3.97 4.42 4.85 5.28 5.96 6.06 6.81 7.54 8.49 9.19 9.87 10.54 10.98

2.33 4.84 8.12 12.12 16.84 22.22 28.26 34.93 42.22 50.13 57.53 74.90 94.28 123.1 147.0 172.8 200.3 219.7

Drill Dia., mm 22.00 24.00 25.50 27.00 28.50 32.00 35.00 38.00 42.00 45.00 48.00 50.00 58.00 64.00 70.00 76.00 90.00 100.00

FT

FM

11.86 12.71 13.34 13.97 14.58 16.00 17.19 18.36 19.89 21.02 22.13 22.86 25.75 27.86 29.93 31.96 36.53 39.81

260.8 305.1 340.2 377.1 415.6 512.0 601.6 697.6 835.3 945.8 1062 1143 1493 1783 2095 2429 3293 3981

W =  1.30 (From Table 3) T =  2Kd Ff FT BW + Kd d 2 JW = 2 3 24,000 3 0.021 3 0.899 3 1.355 3 1.30 + 24,000 3 0.8752 3 0.030 3 1.30 = 2313 lb M =  Kd Ff FM AW = 24,000 3 0.021 3 0.786 3 1.085 3 1.30 = 559 in-lb P 3.1- = 3.9 hp MN = 559 × 350- = 3.1 hp ----------------------P m = -----c = --------P c = ---------------E 0.80 63 ,025 63 ,025 Twist drills are generally the most highly stressed of all metal cutting tools. They must not only resist the cutting forces on the lips, but also the drill torque resulting from these forces and the very large thrust force required to push the drill through the hole. Therefore, often when drilling smaller holes, the twist drill places a limit on the power used and for very large holes, the machine may limit the power.

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1154

Machinery's Handbook, 31st Edition Boring Boring

Boring is an internal turning operation performed with a boring bar to enlarge a previously drilled hole to form an internal shape of required dimensions. Boring operations range from semi-roughing to finishing. An indexable boring bar has three basic elements: indexable cutting insert, shank, and anchor. The designation system for indexable inserts is the same as for turning. The anchor is the clamping portion of the shank that is held in the tool block. The minimum clamping length is approximately three to four diameters of the shank. The distance the boring bar extends beyond the tool block is the overhang; it determines the cutting depth. The overhang is the unsupported portion of the boring bar. Long overhang causes excessive deflection of the shank and generates vibration (chatter), which worsens the surface finish of the bore. Eliminating chatter, especially in long-bore workpieces, is one of the greatest challenges faced by manufacturers and users of boring bars. Deflection of a boring bar depends on the mechanical properties of the shank material, the length of the overhang, and the cutting conditions. The following equation is used for calculating deflection (y) of a boring bar. (1) where F = the cutting force, lbf or N L = unsupported length of a boring bar (overhang), in. or mm E = the modulus of elasticity (in tension) of a boring bar material, psi or N/mm2 I = the moment of inertia of a boring bar cross-sectional area, in4 or mm4. Cutting Force.—Cutting force (F) is calculated by Formula (2) in customary US units. (2) where d = depth of cut, in. f = a feed rate, ipm Kp = a power constant, hp/in3/min. The power constant Kp data are given on pages 1145–1147 (Table 1a and 1b). The feed rate factors, C, for power constants are given on page 1148 (Table 2). 396,000 = a conversion factor that represents a unit of power equal to 550 in-lb/min Example 1(a): A workpiece is AISI 4140 chromium-molybdenum steel, 220 HB; depth of cut d = 0.080 in.; feed rate f = 0.008 ipr. Calculate the cutting force F in customary US units of measure. Power constant Kp = 0.72 hp/ in 3/min (from Table 1b, page 1147) Feed factor C = 1.08 (from Table 2, page 1148) Adjusted power constant, Kpa = Kp × C = 0.72 × 1.08 ≅ 0.78. Calculating: F = 396,000dfKpa = 396,000 × 0.08 × 0.008 × 0.78 = 197.7 lbf

Cutting force (F) is calculated by Formula (3) in metric units.

(3)

where d = depth of cut, mm f = feed rate, mm/min. Kp = is a power constant, kW/cm3/min. The power constant Kp data are given on pages 1145–1147 (Tables 1a and 1b). The feed rate factors, C, for power constants are given on page 1148 (Table 2). 60,000 = is a conversion factor that represents a unit of power equal to 1 kW × mm/min.

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Example 1(b): A workpiece is AISI 4140 chromium-molybdenum steel, 220 HB; depth of cut d = 2.03mm; feed rate f = 0.20 mm/rev. Calculate the cutting force F in metric units. Power constant Kp = 0.0328 kW/cm 3/min. (from Table 1b, page 1147) Feed factor C = 1.08 (from Table 2, page 1148) Adjusted power constant, K pa = K p × C = 0.0328 × 1.08 ≅ 0.0354 kW/cm 3/min. Calculating: F = 60,000dfK pa = 60,000 × 2.03 × 0.20 × 0.0354 = 863.6 N

Comparing the cutting forces calculations of Example 1(a) and 1(b), the cutting forces are approximately equivalent within about 2 percent.

Moduli of Elasticity (E) of Boring Bar Materials.—Boring bar shanks are made of steels, tungsten-base metals, or cemented carbides. Most commonly used boring bar materials are alloy steels. Some boring bar manufacturers use AISI 1144 free machining medium-carbon steel. Regardless of grades, all carbon and alloy steels have approximately the same modulus of elasticity: Customary US units, E = 30 × 106 psi Metric units, E = 20.6 × 104 N/mm2 A common mistake is to assume that a steel shank with a higher hardness, or one made from a higher quality of steel will deflect less. As can be seen from Equation (1), the material property that determines deflection is the modulus of elasticity. Hardness does not appear in this equation. Tungsten heavy alloys for boring bars, E = (45–48) × 106 psi (customary US units) and E = (31–33) × 104 N/mm 2 (metric units). Boring bars made of tungsten heavy alloys will deflect less than steel boring bars of the same diameter and overhang by 50 to 60 percent when cutting at the same depth of cut and feed rate. Cemented carbides for boring bars, E = (84–89) × 106 psi (customary US units) and E = (52–61) × 104 N/mm 2 (metric units). Boring bars made of cemented carbides provide minimum deflection because their moduli of elasticity are higher than those of steels and tungsten heavy alloys. Moment of Inertia (I) of a Boring Bar Cross-Sectional Area.—Moment of inertia is a property of areas. Since boring bars are available in a variety of diameters, it is important to calculate the moment of inertia of a bar cross section using appropriate formulas. Boring bars are usually round with a solid or tubular cross section. Moments of inertia of solid or tubular cross-sectional areas are calculated by: (4a)

(4b)

where D is the diameter of the bar in inches or mm, Do is the outside diameter of the bar and Di is the inside diameter, in inches or mm.

Example 2: A boring bar diameter is 1 inch. Moment of inertia I = π × (1)4/64 = 0.0491 in4. Moments of inertia of cross sections are given starting on page 241.

Deflection (y) of the Boring Bar.—To calculate the deflection of the boring bar, it is necessary to enter one more data point in Equation (1). It is the unsupported length of the boring bar (L).

Example  , Boring Bar Deflection: The unsupported length of a 1-inch diameter boring bar is 4 inches. Using F calculated in Example 1(a) and I calculated in Example 2:

y=

(197.7)43 FL3 = = 0.0028 inch 3 EI 3 (30 × 106) 0.0491

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1156

Machinery's Handbook, 31st Edition MICROMACHINING MICROMACHINING Introduction

Recent technological advancement and market need for product miniaturization demand three-dimensional (3D) microcomponents. Although microelectronic manufacturing techniques can produce pseudo 3D microdevices using silicon and other semiconducting materials, such materials are neither robust nor biocompatible for demanding applications in aerospace, medical, sensor, defense, petroleum, and transportation. Examples of robust applications include microdrilling holes for fuel or ink injection nozzles, electronic printed circuit boards, microfabrication of watch components, air bearings, cooling holes in tur­bo­machinery, high aspect ratio features on tool steel molds and dies, etc. There are alterna­tive nontraditional processes to produce microfeatures on robust engineering materials such as laser micromachining, electrical discharge microdrilling, electrochemical micro­machining, chemical etching, electron/ion beam machining; however, these processes are either cost prohibitive, limited to conductive materials, or inferior when comparing result­ing surface integrity, subsurface damage, high aspect ratio, or microfeature quality. Micro­fabrication with traditional processes such as micromilling, microdrilling, microturning… are still the preferred choice in most applications. There is no standard that defines micro­machining, but most researchers uses cutting tools to produce components with key dimensions less than 1 mm (0.040 inch) or when depth of cut is comparable to tool sharp­ness or tool grain size in their micromachining studies. Realizing the needs for traditional micromanufacturing, there are more commercially available machine tools and microtools in the market. However, costly equipment, lack of in-depth understanding of micromachining, and limited guidelines for effective use of microtools are still the bottleneck for full application of micromachining. Universities and research institutes worldwide have started theoretical investigation of micromachining and produced positive results from the academic point of view. Without practical guide­ lines on micromachining, technicians and machinists probably would make wrong and costly decisions when simply extending macroscale machining practices into microscale machining applications—a microtool simply breaks at even conservative macroscale parameters for speeds, feeds, and depth of cut.This section, while complementing other chapters in this Handbook, focuses on practicality, based on proven theories and published data, to help decision makers to understand the requirements for micromachining, and as a guide to people on the shop floor to quickly and confidently begin using the recommended parameters and techniques. Both US standard and SI metric units are included for conve­ nience. Examples of how to use the data and equations are given throughout this chapter. Machine Tool Requirements

To obtain the same surface speed as in macromachining, a machine tool must: a) Be capable of rotating a workpiece or tool at high speeds of 25,000 rpm or above b) Control spindle runout to submicron level c) Have a very robust mechanical and thermal structure that is not affected by vibration or thermal drift d) Have high resolution tool positioning and feeding mechanisms Success in micromachining depends on tool quality and precision of the machine tool. Machine spindle runout, tool concentricity and tool positioning accuracy must be in the neighborhood of 1⁄100th of tool diameter or less for successful operation. Tolerance stack up for spindle runout, tool eccentricity, and wandering of a microdrill causes cyclic bend­ ing of the tool that lead to catastrophic failure. At a low rotatational speeds, the displace­ ment of a spindle can be monitored with a sensitive mechanical indicator. However, this option is not applicable for machines that operates at or above a few thousands rpm. Other non-contact techniques using capacitance, magnetism, or light would be more appropriate. Fig. 1a shows an example of spindle runout measuring setup. A laser beam is

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Machinery's Handbook, 31st Edition MACHINE TOOL REQUIREMENTS

1157

pointed at a rotating precision plug gage. The spindle displacement is then recorded on a computer for further analysis and display in either frequency or time domain. Commercial laser systems can provide displacement readings to ±0.1 mm resolution. An example of spindle runout is shown in Fig. 1b; the spindle runout of a Haas OM2 machine was measured with a Keyence laser system to be ±1.25 mm. Care must be prac­ ticed to isolate vibration of the spindle or it would affect the sensor reading, and avoid direct eye contact with the reflected laser from the shiny plug gage. Machine tool spindle

Data acquisition system

Precision plug gage

Fig. 1a. Setup for Spindle Runout Measurement.

Runout ( m)

Laser displacement sensor

2.0

Laser on 3mm plug gage rotating at 10,000 rpm

1.0 0.0 –1.0 –2.0

0

2 4 6 8 10 Spindle Rotating Time (s)

Fig. 1b. Spindle Runout of Haas OM2 CNC Micromilling Machine.

Example 1, Spindle Speed for Macro versus Micro Machining: The speed and feed table on page 1118 recommends a milling speed of 585 ft/min (178 m/min) and feed of 0.004 in/tooth (0.1 mm/tooth) for end milling 316L stainless steel using an uncoated carbide tool. Macromachining: To have said surface speed for an Ø1⁄2 inch (Ø12.5 mm) end mill, the required spindle speed is 585 ^ft/min h V N = πD = # 12 ^in/ft h = 4, 469 rpm π ^rad/rev h # 0.5 ^in h Micromachining: To obtain the same surface speed for an Ø0.004 inch (Ø0.1 mm) micromill, the new spindle speed is 585 ^ft/min h V # 12 ^in/ft h = 558, 633 rpm N = πD = π ^rad/rev h # 0.004 ^in h To turn, face, or bore a stainless steel microshaft of Ø0.004 inch (Ø0.1 mm) at this cutting speed, a lathe spindle would need to rotate at 558,633 rpm too. A machine tool with spindle speed exceeding 500,000 rpm is rare or simply not commercially available at this time. Applying the recommended macro feed of 0.004 in/tooth (0.1 mm/tooth) for an 0.004 inch (0.1 mm) diameter micromill would break the tool because the feed/tooth is as large as the microtool diameter.

Microcutting Tools Tool Stiffness.—It is relatively easy to have a rigid turning or facing microtool, but it requires careful planning to maintain rigidity of a micromill or a microdrill. Geometries of macroscale and microscale drilling/milling tools are the same: tool diameter, number of cutting flutes, point included angle for microdrill, helix angle, web thickness, clearance angle, flute length, shank diameter, and overall length. A careful selection of microtools must consider the intended machined features and highest possible tool stiffness. The two most important geometries that affect the microtool stiffness are the tool diameter and flute length assuming the number of flutes has been chosen. It can be shown that the torsional stiffness of a mill/drill is proportional to (tool diameter)4 and (flute length) −2. For a specific mill/drill tool dimension, the milling/drilling strategy must be adjusted accordingly to avoid tool breakage.

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Example 2, Stiffness of Microtools: If a drill diameter of 0.8 mm is selected instead of 1.0 mm, then the 20% reduction of diameter will result in a reduction in torsional stiffness E of:

∆E =

^ D2h4 − ^ D1h4 0.84 − 1.04 = = − 59% ^ D1h4 1.0 4

Similarly, if a flute length of 1.2 mm is chosen instead of 1.0 mm, the 20% change in flute length will lead to a decrease in torsional stiffness E of:

∆E =

^ L2h−2 − ^L 1 h−2 1.2 −2 − 1.0−2 = = − 30% ^ L1h− 2 1.0− 2

Tool Sharpness.—The tool edge radius is critical in micromachining. Fig. 2a through Fig. 2d shows two scenarios for the same microcutting tools with edge radius r. The tool can be either a turning, facing, or boring microtool that linearly engages a workpiece material at a certain depth of cut. A similar tool can move in a circular path as a microdrill or micromill, and engage a workpiece at a certain chip load (feed per tooth). If the depth of cut (or chip load) is too shallow, the tool simply plows the material and pushes it away elastically. This elastic material layer just springs back after the tool passes by. If the depth of cut is substan­tial (recommended), then a chip is formed and a new machined surface is generated with negligible spring back. Chip load is commonly used interchangeably with feedrate for a cutting tool with multiple cutting edges (teeth) such as in milling or drilling. Chip load is defined as tool feed distance for each tooth and represents the chip size forming for each tooth. Chip load can also be interpreted as the radial depth of cut for each tooth in milling. The following equation converts chip load of a cutting edge to feedrate of a multiple-edge cutting tool:

f = cL nN

where f = feedrate of tool (mm/min, in/min) cL = chip load of a cutting edge (mm/tooth, in/tooth) n = number of cutting flutes or cutting edges (#teeth/rev) N = rotational speed (rpm) Example 3: A two-flute uncoated carbide end mill with diameter Ø1 mm (Ø0.040 in) is used for micromilling pure titanium. Table 13b suggests a chip load of 17 mm/tooth and cutting speed of 90 m/min. The rotational speed is computed as:

V N = πD =

The feedrate for this operation is:

90 ( m/min) = 28, 600 rpm rad π a rev k # 0.001 ( m )

µm µm teeth rev mm in f = cL nN = 17 ` tooth j # 2 a rev k # 28,600 ` min j = 972,400 min . 972 min . 38 min

Typical fine grain carbide tools are first sintered from submicron carbide particles in a cobalt matrix, and then ground and lapped to final geometry. Optimal edge radii of 1–4 mm (39–156 μinch) are typically designed for sintered tools to balance edge sharpness and edge strength. Only single crystalline diamond tools can be ground and lapped to edge radii within the nanometer range. The threshold for minimum depth of cut has been investigated theoretically and verified experimentally by many researchers. It varies from 5 to 40 percent of the tool edge radius depend­ing on the workpiece material and original rake angles. The threshold depth of cut or chip load, therefore, can be conservatively set to be 50 percent of the tool edge radius. When machin­ing below this threshold, a microtool just rubs and plows the surface with negative effec­tive rake angle and deforms it elastically during the first pass. This results in high cutting force, high specific energy, fast tool wear, rough surface finish, and significant burrs. In subsequent passes when the cumulative depth is greater than the critical depth of cut, then a tool can remove materials as chips and the cycle repeats.

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Effect of Depth of Cut (Chip Load) in Micromachining Negative rake angle Shallow depth

Positive rake angle

Microtool

Deep depth Chip

Spring back

r

r

Workpiece

Workpiece Fig. 2a. Microfacing, Depth of Cut < 0.5 r.

Fig. 2b. Microfacing, Depth of Cut > 0.5 r.

Negative rake angle

Positive rake angle

Microtool Spring back

Low chip load

Microtool

Microtool

High chip load r

r Workpiece

Workpiece

Fig. 2c. Micromilling, Chip Load < 0.5 r

Fig. 2d. Micromilling, Chip Load > 0.5 r.

Fig. 2a and Fig. 2c illustrate rubbing and plowing of material with negative effective rake angle at a shallow depth of cut. Fig. 2b and Fig. 2d illustrate chip removal from material with positive effective rake angle at a deep depth of cut.

It is crucial to verify the tool edge radius before deciding on cutting parameters. Measur­ ing of tool edge radius, however, is not trivial. A tool edge radius can be estimated from a scanning electron microscopic picture when the cutting edge is parallel to the electron beam (Fig. 6), or from a scanned image at the neighborhood of a cutting edge on an atomic force microscope (Fig. 3a and Fig. 3b), or by scanning an edge on an optical microscope pro­filer in different views to reconstruct a 3D image of the tool edge before finding its radius. Tool Edge Measurement by Atomic Force Microscopy Note the different vertical and horizontal scales.

nm

40 nm

500

20

200

400

600

800

nm

Fig. 3a. New Polycrystalline Diamond Tool with a 750 nm Edge Radius.

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50

100

150

200

nm

Fig. 3b. New Single Crystalline Diamond Tool with a 10 nm Edge Radius.

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Tool Materials.—Using the right microtool is essential for micromachining. A microtool that successfully drills through holes on a plastic printed circuit board is not necessarily able to drill deep blind holes on titanium alloys. Understanding the requirements and selecting the right microtool for each condition saves time, money, and frustration. It has been theoretically derived and experimentally proven that the smaller is the chip, then the higher is the stress to generate it. Microcutting tools, therefore, have to be designed for higher stress with extreme geometrical constraints. When depth of cut is smaller than the average grain size of a workpiece, each grain generates different stress on the cutting edge and eventually fatigues the tool. Microtools as small as 25 mm (0.001 inch) are commercially available. Common tool materials are high-speed steel (HSS), cermet, carbide, cubic boron nitride (CBN), poly­ crystalline diamond (PCD), and single crystalline diamond (SCD). HSS is commonly not used in micromachining of metal since it does not have required hardness and strength to resist plastic deformation. A SCD tool is available for microturning, but not for micro­ drilling or micromilling. Carbide and cermet, having properties between HSS and dia­ mond, are most suitable for microcutting tools. They are sintered from random abrasive grains in either cobalt or nickel binder with a small addition of molybdenum or chromium. A higher binder content increases the tool toughness and crack resistance, but reduces the tool bulk hardness. Using ultra fine grain (submicron size) abrasives in a lesser amount of binder is the optimal solution because a tool with submicron carbide grains can maintain a high hardness while improving its crack resistance against chattering, interrupted cuts, or cyclic deflection due to spindle runout. Microtool failure modes include shearing, chipping, and wear. To minimize shearing and catastrophic tool failure, a tool should be made from a high hardness substrate and with a geometry suitable for micromachining, i.e., large included angle and sharp cutting edge (Fig. 4). A tool with smaller than minimum included angle will be deformed and fractured in service. Rake angle Tool edge radius

Included angle

Tool

Uniform coating Nonuniform coating

r (uncoated) Relief angle (a)

(b)

Fig. 4. (a) Tool Geometry, and (b) Change of Tool Edge Radius Due to Coating.

11 10 0° 0°

Workpiece Strength (MPa)

2500 2000 1500 A

°

90

°

gle

80

C l too

°

70

e

lud

inc

n da

1000 500

5

B 10 15 Tool Hardness (HV, GPa)

20

Fig. 5. Microtool Minimum Included Angles.

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Example 4, Required Included Angle: Find the minimum included angle for an ultra-fine grain carbide tool of 12 GPa Vicker hardness to machine the F799 Co-Cr alloy of 1500 MPa (0.2 3 106 psi) tensile strength. Solution: Referring to Fig. 5, locate workpiece strength at point A on the vertical axis (1500 MPa). Locate point B for tool hardness on the horizontal axis (12 GPa). The intersection at C of the horizontal line from A and the vertical line from B indicates that the minimum included tool angle should be 75°.

Coating of microtools is still a technical challenge due to conflicting constraints for tool performance. Chemical or physical vapor deposition (CVD or PVD) techniques have been developed to coat cutting tools with mono/multiple layers of intermetallic or ceramic com­pounds (Table 1). Criteria for acceptable tool coating are numerous: uniformity, high hard­ness, high toughness, low friction, high wear resistance, surface smoothness, high chemical/diffusion resistance, and high temperature stability at a reasonable cost. Although a coating thickness of 2–4 mm (79–157 min) is acceptable for a macrotool, the coating thickness on a microtool should be thinner to minimize fracture and peeling of the coating. Both CVD and PVD processes not only add the coating thickness to the edge radius, but the extra coating also increases the radius at sharp corners (Fig. 4b). This is unfortunate since the thicker coating reduces the tool sharpness by enlarging the tool edge radius and causes an unfavorable plowing effect with negative effective rake angle. An uncoated microtool might perform satisfactorily, but the same machining parameters can be devastating to an over-coated microtool (Fig. 6). A thin coating of less than 1.5 mm fol­lowing by an edge sharpening process would improve the tool performance, however, at the expense of higher tool cost. Published data indicate that micrograin carbide tools with 1.5 mm TiN coating is the best for micromilling of H13 tool steel hardened to 45 RC. Table 1. Commercial Coatings for Microtools Coating

TiN TiCN TiAlCN TiAlN AlTiN ZrN CrN Diamond like AlTiN/Si3 N4 AlCrN/Si3 N4

Structure

monolayer gradient gradient multilayer gradient monolayer monolayer gradient nanocomposite nanocomposite

Hardness

GPa 24 37 28 28 38 20 18 20 45 42

106 psi 3.5 5.4 4.1 4.1 5.5 2.9 2.6 2.9 6.5 6.1

Coefficient of Friction 0.55 0.20 0.30 0.60 0.70 0.40 0.30 0.15 0.45 0.35

Coating Thickness

mm

1-5 1-4 1-4 1-4 1-3 1-4 1-4 0.5-1.5 1-4 1-5

minch

Maximum Temperature ºC

ºF

39-197 600 1110 39-157 400 750 39-157 500 930 39-157 700 1290 39-118 900 1650 39-157 550 1020 39-157 700 1290 20-59 400 750 39-157 1200 2190 39-197 1100 2010

Tool Offset and Positioning.—Tool offset and tool positioning are crucial in micromill­ ing and microdrilling because a tool is small and extremely fragile especially if it has a high aspect ratio (length to diameter ratio). Common shop practices to find tool offset and posi­tion often damage a tool or workpiece. Non-contact techniques using light, magnetism, capacitance, ultrasound, etc. are the preferred choice for precisely locating the relative position between tool and workpiece. Selection of a suitable sensor depends at least on fol­lowing criteria: •  Better resolution compared to that of the machine tool axis •  Small working zone to cover a microtool •  Fast sampling rate for intended tool speed

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b

a

Fig. 6. Peeling (a) and cracking (b) of 4mm-thick TiN coating layer on a WC micromill. Back scattered electron technique shows high contrast of a dark TiN coating layer against the bright WC/Co substrate in the background. Example 5, Sensor Requirement: Select a sensor for microdrilling using Microdrill: 100 mm diameter, 1 mm flute length (Ø0.004 inch, 0.040 inch flute length). Machine tool: 1 mm (40 minch) repeatability and 500,000 rpm capability.

Solution: A laser displacement sensor is selected to satisfy the following specifications: Resolution: 0.1 mm (4 minch) Spot size: 25–75 mm (0.001–0.003 minch). Although most drill shanks are Ø3.175 mm (Ø0.125 inch), the working zone should be as small as possible to detect the shank center. In order to make 6 measurements when a tool is rotating at 500,000 rpm, the time between measurements is:

1 rev 1 min 60 s t = a 6 measurements ka 500, 000 rev ka 1 min k = 2 # 10 − 5 s = 20µs

A laser with minimum 20 ms sampling rate (50 kHz) would be sufficient.

A mechanical edge finder is adequate for most macromachining setups, but it is not suit­able for micromachining especially with small and pliable parts. Fig. 7a shows a noncon­tact technique to detect part edge or find lateral tool offset. A rotating precision plug gage, mounted on a machine spindle, is positioned between a stationary laser sensor and the workpiece. The small laser beam is aimed at the plug gage center and on the part edge when the plug gage is withdrawn away from the beam path. These two laser sensor readings allow computing the tool center offset. A precision plug gage should be used instead of a cutting tool shank for better repeatability. Example 6, Lateral Tool Offset Calculation: Use a laser displacement sensor and a Ø3.175 mm (Ø1/8 inch) plug gage to detect the edge of a ground block. Solution: i) Mount the plug gage on the machine spindle and rotate it at 5000 rpm. ii) Scan a laser beam across the plug gage and stop when the distance from the laser source to the target is minimum, i.e., the beam is at the gage center. Read L1 = 35 mm. iii) Jog the plug gage away from the beam path, read distance to the part edge L2 = 55 mm. iv) The lateral offset from the spindle center to the workpiece edge is then: Lateral offset = L2 − L1 − D⁄2 = 55 − 35 − (3.175⁄2) = 18.412 mm

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z Precision plug gage Laser displacement sensor

x Workpiece

L1

D/2

L

L2

Fig. 7a. Setup for Lateral Edge Detection Using Laser Sensor.

Deviation from Mean (µm)

Fig. 7b compares the accuracy and repeatability of the non-contact method shown in Fig. 7a against those of a mechanical edge finder. 3 2 1 0 –1 –2 –3

Laser on 5/16 inch drill Mechanical edge drill Laser on 1/8 inch plug gage

0

10 20 Trial Number

30

Fig. 7b. Superior Accuracy and Repeatability of Laser Edge Detection Technique Compared to Mechanical Technique.

It is rather simple to find the lateral offset as illustrated in Example 6, but it is more diffi­cult to find the exact vertical offset for a slender microdrill or micromill without damaging it. One can attempt to use the common “paper technique” or take a risk with an available contact sensor for z-setting. In the paper technique, one would use a hand to slide a piece of paper on top of a workpiece while gradually lowering a tool. The tool stops when a resis­tance on the paper is felt. The paper technique is tedious, subjective, and tool dependent. Fig. 8b shows scattering of data up to ±5 mm when finding z-offset for a center drill, but it is ±15 mm for a milling cutter with 4 teeth. A commercial contact sensor requires a tool to move down and press against a solid sur­face. A pressure sensor then triggers an audible or visual signal to indicate a positive con­tact. The pressure level on such sensor is preset for macrotool setting and cannot be adjusted for a microtool. In both cases, the tool tip is one paper thickness or one contact sensor height above the workpiece—if the tool survives. A non-contact sensor is more practical and reliable. The same laser displacement sensor used for lateral tool offset can also be used for vertical tool offset. Fig. 8a shows a precision ring with secured circular plastic membrane that is used for indirect measurement. The membrane center is marked with a reference (e.g., crossing lines) at which the height can be measured with the laser displacement sensor. Upon placing the fixture on top of a work­ piece and then lowering a tool onto the reference mark, a slight contact of the tool and the flexible membrane is precisely detected with the laser beam pointing near the contact point. When this happens, the tool tip is at the same height as the membrane. The repeat­ ability of tool offset using this technique is well within the positioning repeatability 3 mm of the tested Haas OM2 machine tool (Fig. 8b).

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Microtool Laser displacement sensor

Plastic membrane with marked center // A 5µm

z

h

x

-A-

Workpiece

Deviation from Mean (µm)

Fig. 8a. Microtool Offset and Microtool Height Detection Using Laser. 15 10 5 0 –5 –10 –15

Paper & center drill Paper & 4-insert cutter Laser & membrane

0

10

20 30 Trial Number

40

Fig. 8b. Superior Accuracy and Repeatability of Laser Offset and Height Detection Technique Compared to Paper Technique.

When a part is small or does not have a large surface for the fixture to rest on, then an indi­ rect technique to find vertical tool offset for a microdrill or micromill is recommended. The following example illustrates this. Example 7, Vertical Tool Offset Calculation: A vice or collet is used to clamp a micropart for drilling. The micropart protrudes upward a distance h1 = 0.1000 inch. If the vice surface has been qualified as a reference, it can be used to find the vertical offset of a microdrill tip (Fig. 9). i) Measure the fixture height at the reference mark using the laser sensor, h2 = 0.3500 inch. ii) Position the fixture on top of the vice. iii) Lower the microdrill onto the reference mark of the membrane. Stop when the membrane is slightly deflected which can be detected easily with the laser sensor. iv) Calculate the required drill vertical offset: Vertical offset = h2 − h1 = 0.3500 − 0.1000 = 0.2500 inch

Tool Damage.—Tool damage can be categorized by the relative size of the damage, rang­ ing from submicron to hundreds of microns, as indicated in Table 2. The tool failure mech­ anisms include damages due to mechanical, thermal, and chemical effects, and adhesion. Examples of microtool damages are illustrated in Fig. 10a through Fig. 10d.

Mechanical effect is the most common source of tool damage. Abrasive wear is caused by sliding of hard particles from workpiece or tool against the cutting tool surface. Attri­tion wear is larger than abrasion wear; it occurs when one or a few grains of the tool are weakened at their grain boundaries and are dislodged from the tool. Microchipping and chipping are larger chunks of tool being removed due to mechanical or thermal shocks upon loading and unloading. Machining at optimal parameters and with a rigid setup will reduce vibration, shock, and mechanical damage to a microtool.

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Microtool Laser displacement sensor z x

Fixture

h2 h1 0

Part

Vice Fig. 9. Indirect Vertical Tool Offset Detection Technique.

Microtool Damages

2

10.0 μm

1

3 Fig. 10a. Abrasion with progressive wear from center to edge (label “1”) due to different cutting speed on a microdrill. 10.0 μm

50.0 μm

Fig. 10b. Built-up-edge at a cutting lip (label “2”) and side (label “3”) of a microdrill. 10.0 μm

4 5 Fig. 10c. Adhesion wear (label “4”) due to built-up-edge on a micromill (label “5”).

6 Fig. 10d. Chipping at the cutting edge of a carbide micromill due to local depletion of cobalt binder.

Thermal effect is the second cause of tool damage. A cutting tool edge is softened at high machining temperature, deformed plastically, and removed from the tool. Both high-speed steel tools and carbide tools with high cobalt content are vulnerable to thermal damage. High temperature also promotes diffusion, i.e., atoms from the tool and workpiece move

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mutually across their interfaces, therefore degrading their properties and causing diffusion wear. Diamond with a carbon-rich matrix cannot be used with low-carbon ferrous alloys like steels or stainless steels because diamond carbonizes at temperatures exceeding 1112ºF (600°C) and carbon diffuses to the steel due to its lower carbon content and high affinity to carbon. The useful life of a tool can be extended by proper application of coolant to reduce thermal damage, or by use of a protective coating that blocks undesirable thermal diffusion from/to a tool surface. Chemical damage of a tool is due to a chemical reaction between a tool material and its environment like air, cutting fluid, or workpiece material. Tool oxidation is common when cutting in air at high speed. An oxidation reaction is accelerated with temperature, but can be eliminated when inert gas is used to shield the cutting tool from surrounding oxygen. A chain reaction can also occur and further degrade a tool. For example, iron in steel is first oxidized at high cutting temperature to form iron oxide; the iron oxide then weakens the aluminum oxide coating of a tool and leads to peeling and chipping of the coating. Adhesion tool damage happens when a built-up-edge (BUE) welds strongly to a tool sur­ face and then breaks away with a minute amount of tool material. When machining soft materials, a chip tends to adhere to the tool and grow in size. When the BUE is large and becomes unstable, it is removed with the chip while also shearing off part of the cutting tool due to the higher adhesion strength between BUE and tool than the intergrain binding strength of the tool. Stainless steel, nickel and titanium alloys are known for causing adhe­ sion wear on carbide microtools. Adhesion damage can be reduced by using proper lubri­ cant to reduce friction between chip and tool, by coating the tool with a smooth and low friction layer, by reducing tool edge radius, or by increasing cutting speed to raise the tool surface temperature and soften the BUE while reducing its weldability to the tool surface. Microtool failures occur due to a combination of the above mechanisms. For example, peeling of tool coating might be due to coating defects, or to mechanical mechanisms when a large gradient of stress exists across a thick coating layer; the loosened coating particles then rub and cause mechanical abrasive wear on a tool. Thermal mechanisms may cause workpiece atoms to diffuse, weaken, and dislodge several tool grains as microchipping. Table 2. Categories of Tool Damage Microtool damage Abrasion Attrition Peeling Microchipping Chipping Fracture

Damage size mm μinch 100 > 3940

Mechanism Mechanical, thermal Mechanical, thermal Mechanical, chemical Mechanical, adhesion Mechanical Mechanical

Tool Life.—Tool life criteria in macromachining are documented in ANSI/ASME B94.55M-1985 (R2019), Tool Life Testing with Single-Point Turning Tools. This standard suggests an end of tool life when a tool exhibits: •  An average flank wear of 300 mm (0.0118 in), or •  Any maximum flank wear land of 600 mm (0.0236 in), or •  Any tool wear notch of 1000 mm (0.0394 in), or •  A crater wear of 100 mm (0.0039 in). It is obvious that such criteria for a macrotool cannot be applied to a microtool because (i) it would be cost prohibitive to continue testing until 300 mm flank wear, and (ii) the wear criteria are even larger than most tool dimensions. In the absence of a microtool standard, researchers have set their own criteria based on direct observation and/or indirect monitoring of microtool tool wear effects. Published data varies on microtool wear thresholds: 5 mm flank/nose wear on diamond tools, or

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50 mm flank/nose wear on carbide tools, or chipping dimensions relative to cutting tool grain size, or peeling of tool coating, etc. A variety of techniques have been suggested for tool monitoring; the direct techniques measure the tool conditions (e.g., flank wear, crater wear) while the indirect techniques measure the consequence of tool wear (e.g., burr size, change of microhole diameter). Table 3. Microtool Wear and Monitoring Techniques

Direct

Measurement Tool wear Tool edge conditions Wear particles Particle radioactivity Tool-workpiece junction resistance Workpiece features (hole diameter, slot depth…) Cutting force, torque, power

Sound emitted from tool-workpiece friction Indirect

Vibration Temperature Surface roughness Burr dimension

Metrology Equipment / Sensor

Microscope Spectrophotometer, scintillator Voltmeter Microscope Interferometer Dynamometer Strain gage, ampere meter Acoustic emission transducer Microphone Accelerometer Displacement sensor Thermocouple Pyrometer Profilometer, interferometer, optical profiler… Microscope, interferometer, optical profiler…

The importance of tool life monitoring and tool life prediction is presented in the section MACHINING ECONOMETRICS starting on page 1196. The following material expands from that and covers relevant information for tool life of microtools. The general Taylor equation that relates tool life and machining parameters also applies in micromachining: a b c

V fdT =g where V =   surface cutting speed (m/min, ft/min) f =   tool feed (mm/rev, in/rev) or chip load (mm/tooth, in/tooth) d =   depth of cut (mm, inch) T =   tool life (min) g, a, b, c =   constants

(1)

When thermal damage mechanism dominates then a >> b, c in Equation (1). The term a dominates mathematically and the effects of feed and depth of cut are insignificant com­ pared to speed. The general Taylor equation can be rewritten as: g a V T = b c (2) fd If n = 1/a, then this equation is the same as that in the Econometrics section: g n n (3) VT = c b c m = C fd When tool chipping occurs then both terms b, c >> a, therefore the feed and depth of cut are more important than surface speed. The general tool life reduces to

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Machinery's Handbook, 31st Edition Microcutting Tools f bd cT =

g = C′ Va

(4)

Chipping is generally not acceptable since a chipped tool generates excessive burr and a very rough surface. By reducing depth of cut and feed, then chipping should be eliminated assuming micromachining with a quality tool and machine tool. When stable parameters are applied, then the only damage mechanism is thermal and tool life can be predicted with Equation (3). It has been shown that flank wear due to abrasion is directly proportional to the magni­ tude of acoustic signal or feeding force. An increase of 300 percent in micromilling feeding force from an initial value was established as a threshold for reaching the tool life. A reduction in feeding force, however, might indicate gradual failure of a microtool due to fatigue crack propagation. Indirect monitoring of tool wear by monitoring feeding force for both micro­m illing and microdrilling would be a preferred technique since this does not interfere with the machining process and reduce productivity. In the absence of a sensitive commercial system that can reliably and accurately monitor tool force and tool life in micromachining, direct tool wear monitoring should still be a popular practice. Traditional tests using the Taylor approach would machine at the same cutting speed until reaching the predetermined tool failure criteria. Such tests can be time consuming if a chosen speed is too low, and only applicable to turning since a constant cutting speed is required. In reality, a part must be machined with the same tool in different directions and speeds to obtain the final profile and surface finish. Several techniques were developed to accelerate the testing method since turning tests alone are tedious, expensive, and do not reflect actual part machining. The cumulative wear technique, assuming that the abrasion wear mechanism is the same at different cutting speeds, is more flexible and can reduce the testing time and cost. The proposed cumulative tool life testing technique: •  Is flexible. If an initial speed is too slow, testing speed can be increased and the cumu­ lative time and tool wear recorded. •  Is simple. Manual machines can be used instead of CNC machines. The same rpm on a manual lathe can be used for the turning test until tool failure. Times and cutting speeds for all passes are used to calculate the equivalent time and speed. •  Is more cost-effective. Both turning and facing can be combined to completely con­ sume an expensive workpiece material. •  Is order independent. The level of cutting speed is not important if providing the same tool wear mechanism. Experimental data for macromachining shows no difference of tool life if changing cutting speeds from low to high, or in reverse order. Consider a tool that machines at cutting speed V and stops after machining time Δt before reaching its tool life T. The tool then cuts at different speeds and times until reaching the tool life criteria—for example, 50 mm flank wear on a carbide microendmill. The fraction of tool life when cutting at each speed and time is Δt/T, and the total tool life fraction is k ∆t ∆t ∆ t1 ∆ t2 (5) + +f+ k = / i = Q T1 T2 Tk = i 1 Ti The theoretical value of the total tool life fraction Q should be one. Experimental values for Q were found to be in the range 1.2–1.5. When combined with Taylor Equation (3), then Equation (5) becomes k

/ ∆ ti Vi1 ⁄ n = QC 1 ⁄ n i =1

(6)

After machining with a tool at different times and speeds in different conditions (e.g., dif­ferent tool coatings), it is necessary to compare the tool performance by calculating its

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equivalent tool life and equivalent tool speed. The equivalent tool life Te is just the sum of all machining time periods: k

/ ∆ti

Te =

(7)

i =1

The equivalent tool speed must produce the same tool damage as a tool after cumulative machining. The total tool damage is given in Equation (6) as: k

1 ∆ ti Vi 1 ⁄ n = C 1 ⁄ n = TeVe1 ⁄ n Q/ i =1

(8a)

Solving for the equivalent cutting speed Ve

When Q = 1 then, (Ve)

1/n

KJK K1 Ve = KKK Q KK L

k

Nn

/ ∆ tiVi 1 ⁄ nOOO i =1

k

/ ∆ ti i= 1

(8b)

OO OO O P

is the mathematical average of all (Vi)1/n terms, by definition.

Mathematical models for cumulative tool wear are now derived for most popular machining operations, namely turning, drilling, facing, and milling. •  For turning with different cutting speeds, Equation (6) is applied. If turning speeds are kept the same from one pass to another, substitute V = Vi into Equation (6) to obtain: k

(9)

V 1 ⁄ n / ∆ ti = QC 1 ⁄ n i =1

•  For drilling, tool wear would be most substantial at the cutting lip where cutting speed is at the highest. Since cutting speed is constant during drilling as in turning, the tool wear model for drilling is the same as in Equation (6) for variable speeds, and Equation (9) for constant speed. •  For facing, the cutting speed reduces linearly from the maximum Vi at the outermost radius to zero at the spindle center. It can be shown that the cumulative tool life model for facing is: k

/ ∆ tiVi 1 ⁄ n i =1

=

1 ⁄n n+1 n QC

(10)

•  For milling, the actual machining time is the time during which chips are produced. The chip generating time involves geometry of a tool and milling parameters. The cumulative tool life model for face milling is k

/ ∆ tiVi 1 ⁄ n =

i=1

1 QC 1 ⁄ n λ

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and

λ=

a M cos −1 ` 1 − 2 D j 360°

(11)

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Machinery's Handbook, 31st Edition Microcutting Tools

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where λ =  milling factor a = width of cut (radial depth) in milling M = number of teeth D = milling cutter diameter Example 8, Cumulative Tool Life: Turning Test: Dry turning a metal matrix composite rod (Ø18 mm, 100 mm long) at constant 256 rpm on a manual lathe, depth of cut 0.5 mm, feed 0.07 mm/rev. Carbide tool TNPR331M-H1, tool holder MTENN2020-33. This Al-SiC composite is very abrasive and is ideal for tool life model testing since abrasive wear is the main mechanism and flank wear is clearly seen and measured on a carbide tool. In this test, a tool is turned at constant rpm until reaching 300 mm flank wear. At least two data points are required to calculate the effect of speed, or the slope n in Taylor equation. From Table 4, the speed and tool life pairs are (14.48 m/min, 3.54 min) and (9.56 m/min, 5.58 min). The slope n derived from Equation (3) for these two data points is

n=

log (V2 ⁄ V1) log (9.56 ⁄ 14.48) = 0.91 = log (T1 ⁄ T2) log (3.54 ⁄ 5.58)

When considering many data points, the averaged value of n is 0.94. A spreadsheet such as Table 4 is a convenient way to tabulate cumulative values of each Δt and ΔtV1/n term and then use these to calculate the equivalent tool life Te with Equation (7), and the equivalent tool speed Ve with Equation (8b). The plot for all experimental data at constant cutting speeds and cumula­tive speeds is shown in Fig. 11a. Having all data points fitting on the same line indicates the validity of cumulative tool life models.

Facing Test: The same material and cutting tools are used in the facing test. Tool wear and tool life plots are shown in Fig. 11b and Fig. 11c. There is no difference in tool life when machining at low then high speed or the other way around.

Table 4. Spread Sheet for Example 8, Cumulative Tool Life in Turning Δ Length (mm) RPM 256 256 256 179 179 179

26.5 16.0 21.0 18.5 21.5 25.0

Diameter (mm) 18.0 18.0 18.0 17.0 17.0 17.0

Speed (m/min) 14.48 14.48 14.48 9.56 9.56 9.56

Cumulative

Δt (min) 1.48 0.89 1.17 1.48 1.72 2.00

ΔtV1/n

25.39 15.33 20.12 16.30 18.95 22.03 projected

ΔtV1/n 25.39 40.72 60.84 16.30 35.25 57.28 61.61

Flank Wear Feed Δt (min) (mm) (mm/rev) 1.48 2.37 3.54 1.48 3.19 5.19 5.58

199 242 300 160 233 289 300

0.07 0.07 0.07 0.07 0.07 0.07

Equivalent

Ve (m/min)

Te (min)

14.48

3.54

9.56

5.58

Workpiece Materials

Micromachining is often utilized to fabricate components for miniaturized sensors, med­ical, optical, and electronic devices, etc. Common engineering materials for these applica­tions include stainless steel, aluminum, titanium, copper, and tool steel for miniature molds and dies. Workpiece materials must meet certain conditions for successful micromachining. Un­ like macromachining, a micromachining tool is subjected to fluctuating cutting force when it encounters each grain since microtool size is comparable to material grain size. A microtool is more vulnerable to fatigue fracture and the resulting surface—if the tool sur­vives—would be rough due to different spring-back protrusion from each grain due to dif­ferent crystallographic orientations of the grains, and direction-dependent properties of the material. Homogenous workpiece materials with very fine and uniform grain sizes should be chosen for micromachining. Inclusions and large precipitates should be mini­ mized to avoid damage to a fragile tool edge.

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Cir. Speed (m/min)

102 5 4 3 2

Q = 1.25 n = 0.94 C = 40.79

101 5 4 3 2 100

Constant cir. speed High–low cir. speed Low–high cir. speed 3 4 56

2 3 4 56 Turning Time (min)

100

2

101

3

Fig. 11a. Tool life plot for turning tools. Circumference speed refers to the maximum cutting speed at the outer radius in turning. Cast A359/SiC⁄20p; tool H1 WC (-8,0,9,5,60,30,0.4mm); 0.5 mm depth; 0.07 mm/rev feed; dry. Tool wear criterion

Flank Wear (micron)

300 250

45 to 34 m/min 45 to 25 45 to 19 45 to 14 45 to 10 33 to 25 32 to 24 18 to 12

200 150

0

1 2 3 4 5 6 Accumulative Facing Time (min)

7

Fig. 11b. Cumulative flank wear of tool facing at high-to-low circumference speed. Cast A359/SiC⁄20p; tool H1 WC (-8,0,9,5,60,30,0.4mm); 0.5 mm depth; 0.07 mm/rev feed; dry.

Cir. Speed (m/min)

102 6 5 4 3 2

Q = 1.5

101 n = 0.8957

Constant cir. speed High–low cir. speed Low–high cir. speed

6 5 4 3 2

100

2

3 4 5 6 7 101 Facing Time (min)

2

3 4

Fig. 11c. Tool life plot for facing tools. Circumference speed refers to the maximum cutting speed at the outer radius in facing. Cast A359/SiC⁄20p; tool H1 WC (-8,0,9,5,60,30,0.4mm); 0.5 mm depth; 0.07 mm/rev feed; dry.

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Example 9, Grain Size Consideration: The speed and feed table on page 1118 recommends a chip load (feed) of 0.1 mm/tooth (0.004 in/tooth) for macro-scale end milling 316L stainless steel using an uncoated carbide tool. Assume the average material grain size is 15 mm. Macromilling: Using a Ø1⁄2 inch (Ø12.5 mm) end mill, the number of grains being cut by each tooth would be

100µ m 0.1mm chip load = 6.67 grains grain size = 15 µm ⁄ grain = 15 µm ⁄ grain

Micromilling: Selecting a Ø0.1 mm (Ø0.004 in) end mill, the recommended chip load would be 13 mm for stainless steel (see Table 13b, page 1195). The number of grains being cut by each tooth is

13µ m chip load = 0.87 grains grain size = 15 µm ⁄ grain

The cutting force on the macrotool and resulting surface finish are uniform due to the averag­ ing effect from seven grains. Because a microtool shears less than one single grain at a time, the micromachined surface is irregular due to different spring-back amounts of each individual grain, and the cutting force on the microtool fluctuates depending on each grain orientation.

Ductile Regime Micromachining

Crystallographic Directions and Planes.—When machining in micro or nano scale, workpiece atom orientation affects machining performance because material prop­erties change with crystalline orientation. Fig. 12a and Fig. 12b show blocks of the same material but with different surfaces. For example, the surface of the silicon block shown in Fig. 12a is harder, stiffer (higher elastic modulus), and is more difficult to machine than the same silicon block in Fig. 12b. Miller indices are commonly used to specify particular crystallo­graphic orientations of atoms.

(110)

(100)

Fig. 12a. Block of Material with (110) Surface. Fig. 12b. Block of Same Material with (100) Surface.

Consider a simple cubic system where atoms are located at corners (cubic as with manga­nese), at corners and inside (body centered cubic as with iron and chrominum), or at cor­ners and on the surfaces (face centered cubic as with aluminum and copper) systems. For convenience, we will set a coordinate system Oxyz, as shown in Fig. 13a, and the size of the cube is set at one atomic spacing unit (OA = OC = OD = 1). z D (001)

E

G F

O A x

z

z

[001] C

B Fig. 13a.

(110)

y x

(111)

y

y

x Fig. 13b.

Fig. 13c.

During a micro/nano machining process, a cutting tool pushes and shears a grain. The cutting action forces some atoms to slide in certain directions and planes to form chips

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on the tool rake face. These specific directions and planes are called slip systems. Soft materi­als such as copper and aluminum have more slip systems, therefore, are easier to be machined compared to harder materials such as steel with fewer slip systems. The directional Miller index is the coordinate of a vector representing the atom sliding direction. In Fig. 13a, if the atom slides from C to G (also the same as sliding from O to D), then this vector with corresponding coordinates are given by:

CG = OD = 6x D − xO, yD − yO, zD − z O@ = [0, 0, 1] or [001] direction Similarly, if an atom slides from B to A (or C to O) then the Miller direction is:

BA = CO = 6x O − xC, yO − yC, zO− zC@ = 60, −1, 0@ or 6010 @ direction

The Miller plane represents the planes that intersect with the coordinate system. The plane DEFG in Fig. 13a, congruent with OABC, intersects the z-axis while parallel with the x and y axes. The Miller index for this plane is represented by the inverse of the axis intersection:

1 1 1 1 1 1 Plane DEFG = a x - intercept , y - intercept , z - intercept k = a ∞ , ∞ , 1 k = ^0,0,1h or ^001h plane

The plane EGCA in Fig. 13b, congruent with DFBO, intersects the x and y axes while parallel with z axis. The Miller index for this plane is:

1 1 1 1 1 1 Plane EGCA = a x - intercept , y - intercept , z - intercept k = a 1 , 1 , ∞ k = ^1,1,0h or ^110h plane

The plane DCA in Fig. 13c, congruent with BEG, intersects all the x, y, and z axes. The Miller index for this plane is: 1 1 1 1 1 1 Plane DCA = a x - intercept , y - intercept , z - intercept k = a 1 , 1 , 1 k = ^1,1,1h or ^111h plane

Miller Index Nomenclature: In both direction and plane Miller indices, any minus sign is written on top of the number, for example [111], and all commas are omitted for simplicity. Square brackets “[ ]” are used to indicate a specific direction, and pointed brackets “< >” are used to indicate a family of directions with similar geometries. For example, the family has 12 directions similar to [100], [001], …, which are all the edges of the cube in Fig. 13a. Parentheses “( )” are used to indicate a specific plane, and curly brackets “{ }” are used to indicate a family of planes with similar geometries. For example, the {100} family has 6 planes similar to (100), (001), …, which are all the surfaces of the cube in Fig. 13a. Introduction.—The concept of ductile-regime machining has been investigated since the 1960s for amorphous brittle materials such as glasses. Silicon, germanium, and glasses have become strategic materials that are widely used to fabricate intricate components in microelectronics, optical, defense industries, and recently as micro opticalelectrical-mechanical systems. Silicon and other brittle materials are known for their low machin­ability unless they are machined in the ductile-regime conditions. When utilized at the optimal machining conditions, only minimum effort is required for the subsequent etch­ing, grinding, or polishing to remove the damaged subsurface. This section summarizes the theory and provides practical guidance for ductile regime machining. Theory.—The mechanism of ductile-regime machining has been studied by many researchers. Using a fracture mechanics approach, it can be shown that there is a threshold below which the ductile regime prevails:

dc =

plastic flow energy fracture energy

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E K 2 = Aa H ka Hc k

(12)

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1174

Machinery's Handbook, 31st Edition DUCTILE REGIME MICROMACHINING

where dc =  critical depth of cut (m, inch) A =  constant E =  Young’s modulus (Pa, psi) Kc =  surface fracture toughness (Pa·m0.5, psi·in0.5) H =  surface microhardness (Pa, psi) A shallow depth of cut, therefore, would energetically promote plastic flow rather than brittle fracture in the substrate and the chips. Table 5 tabulates properties of some brittle materials and their experimental critical depths of cut. Table 5. Selected Properties of Some Brittle Materials Materials

α-Al2 O3 SiC Si

Young modulus (GPa)

275–393 382–475 168

Fracture toughness (MPa·m0.5)

3.85–5.90 2.50–3.50 0.6

Knoop hardness (GPa)

19.6–20.1 24.5–25.0 10

Critical depth of cut (mm)

1.0 0.2 0.5

The constant A in Equation (12) varies in the range 0.1–0.6 due to measuring uncertainty of surface toughness Kc, elastic modulus E, and microhardness H in a testing environment. These properties depend on crystalline orientation of the materials, surface conditions, and tool geometry. •  The critical resolved shear stress, on a crystalline plane due to the cutting action, is directly proportional to the Schmid factor cosλcosφ, where φ and λ are the orientations of the slip plane and slip direction. An ideal ductile mode machining would happen when the cutting shear stress is parallel to both the slip plane and the slip direction, oth­erwise a pseudo ductile mode with micro cleavages occurs. True ductile-regime machining happens only along certain crystalline orientations, but brittle machining occurs at other crystalline orientations. This explains why micromachining a crystal­ line specimen at the same speed, depth of cut, and coolant produces ductile machined surfaces in one direction but brittle machined surfaces on others. •  Cutting fluid changes the surface properties of materials (Kc, E, and H) and affects conditions for ductile regime micromachining. When micromachining the (100) ger­manium using a single crystalline diamond tool, the critical depth of cut changes from 0.13 mm (5 min.) with distilled water as cutting fluid to 0.29 mm (11 min.) in dry machin­ing. •  Tool geometry also affects the results. Plowing and fracture of material occurs when depth of cut is less than approximately half of the tool cutting edge radius (see Microcutting Tools on page 1157). Tools with negative top rake angle are usually utilized because a negative rake causes a compressive zone in the workpiece ahead of and below the tool and suppresses microcrack formation. Example 10, Mirror-finish Micromachining: Diamond tools with sharp cutting edge radii are very effective for machining brittle or ductile material with the exception of ferrous alloys such as tool steels or stainless steels. The cutting speed has minimum effect on surface finish, but a reduction of the feedrate leads to improvement of surface finish. An optical quality surface of 1.4–1.9 nm R max was obtained when turning single crystalline quartz with a diamond tool (−20° rake, 0.8 mm nose radius) at < 0.3 mm depth of cut, 3 m/s speed, and 8.1 mm/rev feedrate.

Case Study.—A study used polished (001) p-type silicon wafers of Ø100 mm (Ø4 inch). Small grooves were faced at different constant depth of cut or gradually changing depth of cut to study the ductile behavior (Fig. 14). Single crystalline diamond tools with (001) rake surface, 10–40 nm edge sharpness, +5° rake angle, and 0.51 mm or 2.00 mm nose radii were used for a facing operation. The complete tool nomenclature follows the American Stan­dards Association (back rake angle, side rake angle, end relief angle, end clearance

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angle, side relief angle, side clearance angle, end cutting edge angle, side cutting edge angle, nose radius) with the addition of edge sharpness as (5°, 0°, 0°, 5°, 5°, 30°, 0°, 0.51– 2.00 mm, 10–40 nm). The ultraprecision machining process was performed on a rigid system that has 9 nm positioning accuracy. Compressed air was used to blow chips away from the finish machined area. Surface finish of a machined wafer was measured with an atomic force microscope (AFM) and a phase-shift interferometer (PSI). Surface finish measurements indicated ductile or brittle chip fracture on machined sur­ faces. As depth of cut reduced below 1 mm, the surface finish was also diminished due to a higher percentage of ductile machined surfaces. Perfect ductile regime machining was achieved when depth of cut was between 0.1–0.5 mm. A smaller depth of cut in the neigh­ borhood of 0.05 mm (50 nm), however, worsened the surface finish because machining at such shallow depth of cut (close to the cutting tool edge radius of 40 nm) would plow and fracture the material surface. At the same cutting parameters, micromachining along the silicon directions gave better surface finish while brittle chipping was seen when cutting along the silicon directions (Fig. 15, Fig. 16a, and Fig. 16b). 200 Surface Finish Ra (nm)

[100]

[110] Fig. 14. Machining plan on (001) wafers. A wafer was faced at a constant depth in different zones (left) or changing cutting depth in a taper cut (right).

400 nm 200

150 100 50 0 0.0

PSI in PSI in 0.2 0.4 0.6 0.8 1.0 Maximum Chip Thickness ( µ m)

Fig. 15. Surface finish as a function of the maximum chip thickness (depth of cut) and crystalline direction of the silicon wafer. The minimum surface finish is with ductile machined surface.

400 nm 200

6

nm

4 2

5

10

15

nm

Fig. 16a. Perfect ductile regime machining Fig. 16b. Mixed mode of ductile regime of (001) silicon along [110]. and pitting along [100]. Speed 75 m/min, feed 2.5mm/rev; depth 0.5mm; SCD tool (5°,0°,0°,5°,5°,30°, 0°, 0.5mm,10–40nm).

Cutting Fluids in Micromachining Micromilling and microdrilling, among the most versatile manufacturing processes, can be leveraged from existing technology to produce 3D microparts or microcavities in molds and dies for mass-replication. Although macro-scale milling and drilling technology is mature, micro-scale milling/drilling technology is yet to be fully developed. Extending common practices in macromachining to micromachining often ends up with failure. Very short tool life is experienced with micromachining, and flood cooling is

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not effective in microdrilling because coolant cannot flow into a partially drilled microhole. This section recommends how to select and apply a cutting fluid for effective micro milling/drilling. Micro milling/drilling requires high rotation speed exceeding 25,000 rpm of a small tool to achieve an acceptable surface cutting speed for material removal. When drilling steel, 50 percent of the heat generated conducts into the drill, but 80 percent of heat will go to the tool when drilling titanium. A microdrill with sharp cutting edges subjected to high temperature and high stress will fail easily if cutting fluid is not adequate. When rotating a microtool at a very high rotating speed, flood coolant is not effective since it does not have enough momentum to penetrate the boundary layer (fast moving air layer) around a fast rotating tool, or wet the bottom of a deep microhole. In addition, any unfiltered chip from recycled coolant can damage a microtool or fragile workpiece. Micromist (minimum quantity lubri­cation, MQL) has been studied by many researchers and is proven to provide proper cool­ing and lubricating in micromachining. In ideal conditions, a stream of micron-size lubricant particles in micromist: •  Does not contain any chip or solid contaminant •  Has enough momentum to penetrate the boundary layer of a fast rotating tool •  Adheres to the fast rotating tool despite high centrifugal force, and •  Wets the tool and workpiece to provide effective cooling and lubricating. The following section discusses safety, selection of cutting fluid, application method, and recommends optimal setup for micromachining. Safety.—The aspect of health and safety when using micromist is a concern. A mist does not only cause potential health issues for workers in the environment, but also contami­ nates other instruments and machines nearby. Biodegradable fluids must be used; polyol esters are superior to common vegetable oils because the former have higher biodegrad­ ability, are less “sticky” due to oxidizing, and increase in molecular weight with time and temperature. Due to the aerosol formation during mist flow at high pressure, an air purifi­cation unit or proper ventilating fan should be installed to minimize breathing of the aero­sol particles by operators, and prevent damage to adjacent equipment. Benefits.—Most conventional machining processes like turning, milling, drilling, and grind­ing can benefit from micromist lubrication when applied properly. Although applica­ tion of micromist is limited when the mist flow is obstructed — as in gun drilling — suc­cessful microdrilling has been reported for microholes with 10:1 aspect ratio (depth/diameter). At optimal conditions, micromist significantly minimizes built-up-edges, reduces burr size and cutting force, and therefore improves tool life for both coated and uncoated tools. Depending on which cutting fluid is used and how it is applied, researchers have found the effect of micromist ranges from “the same as flood cooling” to “extending tool life 3–10 times over flood cooling.” There is yet any published paper on inferior results of micromist over dry and flood cooling. Air Inlet Pulse Generator

Reservoir Manifold

Hose

Hypodermic Needle Tip

Metering Pump Fig. 17. Schematic of a Micromist System for Micromachining. Courtesy of Unist, Inc.

Systems that can generate micromist for minimum quantity lubrication machining are commercially available. A typical design (Fig. 17) includes a resevoir for biocompatible oil, feeding tubes, and an atomizing unit that mixes a compressed air flow with a controlled volume of oil. A needle is necessary to direct the mist to a predetermined location.

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Fig. 18a. Side View of a Pendant Drop Below a Stainless Steel Tube.

28

18

26

16

24

14

22

12

20

Pendant Area (x)

Surface Tension (mN/m2)

The resulting oil microdroplets—size and speed—should be adjustable to effectively penetrate and wet a tool/part interface. This can be done by adjusting the air pressure, type of oil, and volume of oil released into the air stream. Selection of Cutting Fluid.—A cutting fluid is selected for both cooling and lubricating purposes in micromachining. It should be environmentally friendly, should not interact chemically at high temperature with tool or workpiece, and can be cleaned and disinfected from the machined parts. It must have low surface energy relative to the surface energy of the cutting tool and workpiece material, high thermal diffusivity, and lubricity. For micro­m ist applications, a cutting fluid must be able to flow easily in a small tube (low viscosity) and to form microdroplets. Complete wetting is desirable for a cutting fluid because it covers large surface areas of tool and workpiece and effectively removes heat from the source. Its self-spreading capa­bility due to the differential surface energy allows cutting fluid to penetrate deep into the chip/tool interface to effectively lubricate and cool this zone. Wetting condition can be assessed by two methods: Pendant drop technique: A drop of liquid is formed and suspended vertically at the end of a solid tube. The side view of a drop is analyzed to compute the liquid surface tension using a tensiometer (Fig. 18a and Fig. 18b).

0

26 56 84 112 Time into Run (s)

10 140

Fig. 18b. Calculated Surface Tension from Starting to Full Forming of a Droplet.

Sessile drop technique: A drop of liquid is placed on a horizontal surface. The side view of the drop is analyzed to calculate the liquid contact angle or measure it with a goniometer (Fig. 19).

Complete wetting

Contact angle θ

Liquid

Partial wetting

Non-wetting

Solid

Fig. 19. Sessile Drop Technique to Assess Wetting of Cutting Fluid.

The following section presents a simpler approach to calculating the contact angle and drop size using a modified sessile drop technique. This technique uses a toolmaker’s microscope, available at most manufacturing shops, to measure the top view of a drop, instead of a goniometer to measure the side view. Drop Size Measurement.—A microdroplet must have sufficient momentum to penetrate the boundary air layer moving around a fast rotating microtool and to wet the tool after­ ward. Calculation of momentum and contact angle for wetting assessment requires the

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droplet dimension. Knowing the lubricant drop size allows proper calibration of a micro­ mist system to maintain the system effectiveness. Table 6 summarizes different techniques to measure the liquid drop size. The techniques are basically intrusive and nonintrusive methods to either collect the droplets for subsequent analysis, or for in situ imaging of the in-flight droplets.

Nonintrusive techniques use dedicated laboratory research instruments to provide accurate dimensions and comprehensive statistical information of microdroplets. The effect of variables like air pressure on drop size and speed can be automatically calculated and analyzed.

Intrusive techniques use less sophisticated instruments to collect and analyze drop­lets directly or indirectly. These simpler techniques, however, depend on operator skills for collecting reliable data. Table 6. Liquid Droplet Measurement Techniques Intrusive Slide: collect droplets on a slide for micro­scopic assessment. Solidification: transform droplets to solid for sieving or weighting. Momentum: analyze droplet impact. Heat Transfer: analyze cooling effect of drop­lets with a hot wire anemometer.

Nonintrusive Light shadowing: analyze shadows of in flight droplets. Laser Doppler Anemometry: analyze visibil­ity, intensity and phase shift of scattered laser from a small sample of droplets. Laser Diffractometry: scan and analyze a large group of droplets.

The slide technique is a simple way to study drop size and its wetting characteristic. The setup is shown in Fig. 20a, in which a mask and glass plate are quickly exposed to a steady stream of micromist droplets. Only a few droplets are able to pass through the mask opening and deposit on a clean glass plate behind it. P Micromist nozzle

Droplet

θ

h

R

Mask Glass plate

Fig. 20a. Setup for microdroplet collection.

Fig. 20b. Analysis of droplet geometry.

It is assumed that (i) droplet volume remains the same before and after touching the glass plate, (ii) gravity effect on a microscale droplet is negligible, and (iii) the droplet forms part of a sphere on the plate to minimize its total surface energy. Using Equations (13) and (14) that follow, the average volume of a single droplet can be calculated by measuring the average projected droplet diameter P and its height h on a toolmaker’s microscope:

V=

π h 2 a h P2 k π D3 2 3 + 4h = 6 1⁄3

6V D= a π k

2

(13)

h P = :3h 2 a 3 + 4h kD

1⁄3

(14)  

where V =  volume of microdroplet (mm3, in3)

P =  projected droplet diameter (mm, in) h =  height of a microdroplet (mm, in)

D =  air-borne diameter of microdroplet (mm, in)

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Drop size varies with air pressure and volume of oil for atomization. In general, higher air pressure and velocity give more uniform and smaller drop size. Table 7 lists re­sults of average drop sizes measured using this technique. Among different cutting flu­ids, the average diameter of in-flight droplets is approximately 1 mm for the CL2210EP lubricant, but it can be as large as 9 mm for other fluids. Table 7. Properties of Selected Lubricants

Lubricant

2210EP

Surface tension (mN/m) Droplet diameter (mm) Viscosity (Pa-s) @270s–1 shear rate Contact angle on 316L (°) Contact angle on WC (°)

2210

26 0.97 0.016 7 7

29 2.3 0.014 14 7

2200

34 6.7 0.023 10 10

2300HD 34 8.4 0.061 18 12

Example 11, Drop Size Calculation: Set the Unist system at 32 strokes/min, 3.6 mm stroke length, 3.78 bar pressure. Collect droplets of CL2100EP and measure with a toolmaker’s micro­scope. The average projected drop size is 2 mm and average drop height is 0.3 mm. The drop volume is calculated using Equation (13) to be

V=

2 ^2 µmh 2 π h 2 a h P2 k π ^0.3 µ m h c 0.3µ m 3 m 2 3 + 4h = 2 3 + 4 # 0.3 µm = 0.49 µm

Using Equation (14), the average air-borne diameter of a droplet is 1⁄3

6V D= a π k

3 1⁄3

6 # 0.49µ m k = a π

= 0.97µ m

The following practical guides will assist in obtaining reliable results using the slide technique. •  The solid surface (glass plate, workpiece, or cutting tool) should be as smooth as pos­ sible, flat and polished, clean and positioned horizontally. Any surface defect such as machining marks or burrs will distort the droplet profile. •  The solid must be cleaned thoroughly before testing to avoid contamination of the tested liquid and distorted data. Ultrasonic cleaning in alcohol or degreaser follow­ing by dry air blowing should be adequate. •  For meaningful information, about 10–20 droplets should be measured. Ignore very large drops that are coalesced from smaller droplets, and very small satellite drops that are splashed off upon impact of a droplet on the glass plate. •  A measurement should be as quick as possible, using minimum light since a tiny liq­uid droplet might evaporate or spread when heated in bright light.

Contact Angle Measurement.—Droplet volume is calculated in the previous section by the sessile drop technique, and the same volume can be used for calculating the droplet contact angle. Alternatively, a predetermined droplet volume can be set and dispensed on a solid surface using a micropipette. The contact angle of a sessile droplet on a flat surface can be computed from the following equation:

V

P

1⁄3

2 3⁄2 24 ( 1 − K cos i) F = 0.16 in. (> 4 mm)

0.50 to 0.40 0.45 to 0.35 0.35 to 0.25 0.60 to 0.50

The real force of the shearing machine is

FM = 1.3F

(16)

Shearing with Rotary Cutters: The rotary shearing operation is much like shearing with straight inclined blades because the straight blade may be thought of as a rotary cutter with an endless radius. It is possible to make straight line cuts as well as to produce circular blanks and irregular shapes by this method. Fig. 4 illustrates the conventional arrangement of the cutters in a rotary shearing machine for the production of a perpendic­ ular edge. Only the upper cutter is rotated by the power drive system. The upper cutter pinches the material and causes it to rotate between the two cutters.

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1408

Machinery's Handbook, 31st Edition Shearing Driven cutter Work material

c

T Shear line

Fig. 4. Schematic Illustration of Shearing with Rotary Cutters

Shearing force with rotary cutters can be calculated approximately as T2 F = n . k . UTS . λ (17) 2 tan φ µ + T where φ = acos :1 − D D; μ = lap of cutters (in.); T = thickness of material (in.); D = diameter of cutter (in.); n = 0.75 to 0.85 (for most materials); k = 0.7 to 0.8 (ratio UTS ⁄ τ); and, l = relative amount of penetration of the cutters (Table 3). The real force of a shearing machine with rotary cutter is

FM = 1.3F

(18)

Rotary shearing machines are equipped with special holding fixtures that rotate the work material to generate the desired circle. Clearance.—Clearance is defined as the space between the upper and lower blades. With­ out proper clearance, the cutting action no longer progresses. With too little clearance, a defect known as “secondary shear” is produced. If too much clearance is used, extreme plastic deformation will occur. Proper clearance may be defined as that clearance which causes no secondary shear and a minimum of plastic deformation. The clearance between straight blades (parallel and inclined) is: c = (0.02 to 0.05), mm. The clearance between rotary cutters with parallel inclined axes is

c = ^0.1 to 0. 2hT

(19)

where T = material thickness (in.). Cutoff and Parting Cutoff.—Cutoff is a shearing operation in which the shearing action must be along a line. The pieces of sheet metal cutoff are the blanks. Fig. 5 shows several types of cutoff opera­ tions. As seen in the illustration, a cutoff is made by one or more single line cuts. The line of cutting may be straight, curved, or angular. The blanks need to be nested on the strip in such a way that scrap is avoided. Some scrap may be produced at the start of a new strip or coil of sheet metal in certain cases. This small amount is usually negligible. The use of cutoff operations is limited by the shape of a blank. Only blanks that nest per­ fectly may be produced by this operation. Cutoff is performed in a die and therefore may be classified as a stamping operation. With each cut, a new part is produced. More blanks may be produced per stroke of the press ram by adding more single-line cutting edges.

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Machinery's Handbook, 31st Edition Cutoff and Parting Line of cutting

1409 Blank

Strip Line of cutting Scrap

Blank

Strip Blank

Line of cutting Strip Line of cutting Scrap

Blank

Strip

Fig. 5. Types of Cutoff

Parting.—Parting is a cutting operation of a sheet metal strip by a die with cutting edges on two opposite sides. During parting, some amount of scrap is produced, as shown in Fig. 6. This might be required when the blank outline is not a regular shape and is precluded from perfectly nesting on the strip. Thus, parting is not as efficient an operation as cutoff. Line of cutting

Blank

Strip Scrap

Blank

Line of cutting

Scrap

Strip

Fig. 6. Types of Parting

Blanking and Punching

Blanking and punching are fabricating processes used to cut materials into forms by the use of a die. Major variables in these processes are as follows: the punch force, the speed of the punch, the surface condition and materials of the punch and die, the condition of the blade edge of the punch and die, the lubricant, and the amount of clearance. In blanking, a workpiece is removed from the primary material strip or sheet when it is punched. The material that is removed is the new workpiece or blank. Punching is a fabri­ cating process that removes a scrap slug from the workpiece each time a punch enters the punching die. This process leaves a hole in the workpiece (Fig. 7). Workpiece

Blanking

Workpiece

Punching

Fig. 7. Blanking and Punching

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1410

Machinery's Handbook, 31st Edition Blanking and Punching

Characteristics of the blanking process include: 1) Ability to produce workpieces in both strip and sheet material during medium and mass production. 2) Removal of the workpiece from the primary material stock as a punch enters a die. 3) Control of the quality by the punch and die clearance. 4) Ability to produce holes of varying shapes quickly.

Characteristics of the punching process include: 1) Ability to produce holes in both strip and sheet material during medium and mass pro­duction. 2) Ability to produce holes of varying shapes quickly. There are three phases in the process of shearing during blanking and punching as illus­ trated in Fig. 8. Punch

Phase I

Phase II

Work material

Phase III

Die

Fig. 8. Phases in the Process of Shearing

In Phase I, work material is compressed across and slightly deformed between the punch and die, but the stress and deformation in the material does not exceed the plastic limit. This phase is known as the elastic phase. In Phase II, the work material is pushed farther into the die opening by the punch; at this point in the operation the material has been obviously deformed at the rim, between the cutting edges of the punch and the die. The concentration of outside forces causes plastic deformation at the rim of the material. At the end of this phase, the stress in the work mate­ rial close to the cutting edges reaches a value corresponding to the material shear strength, but the material resists fracture. This phase is called the plastic phase. During Phase III, the strain in the work material reaches the fracture limit, and microcracks appear, which turn into macro-cracks, followed by separation of the parts of the workpiece. The cracks in the material start at the cutting edge of the punch on the upper side of the work material, and at the die edge on the lower side of the material; the crack propagates along the slip planes until complete separation of the part from the sheet occurs. A slight burr is generally left at the bottom of the hole and at the top of the slug. The slug is then pushed farther into the die opening. The slug burnish zone expands and is held in the die opening. The whole burnish zone contracts and clings to the punch. Blanking and Punching Clearance.—Clearance, c, is the space (per side) between the Dd − dp punch and the die opening shown in Fig. 9, such that: c = 2 Ideally, proper clearance between the cutting edges enables the fractures to start at the cutting edge of the punch and the die. The fractures will proceed toward each other until they meet. The fractured portion of the sheared edge then has a clean appearance. For opti­mum finish of a cut edge, correct clearance is necessary. This clearance is a function of the type, thickness, and temper of the material. When clearance is not sufficient, additional layers of the material must be cut before complete separation is accomplished. With correct clearance, the angle of the fracture will permit a clean break below the burnish zone because the upper and lower fractures will extend toward one another. Excessive clearance will result in a tapered cut edge, because for any cutting operation, the opposite side of the material that the punch enters after cut­ting will be the same size as the die opening.

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Machinery's Handbook, 31st Edition Blanking and Punching

1411

dp c

Dd

Fig. 9. Punch and Die Clearance

Where Clearance is Applied: Whether clearance is deducted from the dimensions of the punch or added to the dimensions of the die opening depends upon the nature of the work­ piece. In the blanking process (a blank of given size is required), the die opening is made to that size and the punch is made smaller. Conversely, in the punching process (when holes of a given size are required), the punch is made to the dimensions and the die opening is made larger. Therefore, for blanking, the clearance is deducted from the size of the punch, and for piercing the clearance is added to the size of the die opening. Value for Clearance: Clearance is generally expressed as a percentage of the material thickness, although an absolute value is sometimes specified. Table 4 shows the value of the shear clearance in percentages, depending on the type and thickness of the material. Table 4. Values for Clearance as a Percentage of the Thickness of the Material Material Thickness, T

Material Low carbon steel Copper and soft brass Medium carbon steel 0.20% to 0.25% carbon Hard brass Hard steel, 0.40% to 0.60% carbon

< 0.040 in. (< 1.0 mm) 5.0 5.0

0.040-0.080 in. (1.0-2.0 mm) 6.0 6.0

0.082-0.118 in. (2.1- 3.0 mm) 7.0 7.0

0.122-0.197 in. (3.1-5.0 mm) 8.0 8.0

0.200-0.275 in. (5.1-7.0 mm) 9.0 9.0

6.0

7.0

8.0

9.0

10.0

6.0

7.0

8.0

9.0

10.0

7.0

8.0

9.0

10.0

12.0

Table 5 shows absolute values for the blanking and punching clearance for high-carbon steel (0.60% to 1.0% carbon) depending on the thickness of the work material. Table 5. Absolute Values of Clearance for Blanking and Punching High-Carbon Steel

Material Thickness, T (in.) (mm) 0.012 0.3048 0.020 0.5004 0.032 0.8001 0.040 1.0160 0.047 1.1938 0.060 1.5240 0.078 1.9812 0.098 2.4892 0.118 2.9972 0.138 3.5052

Clearance, c (in.) (mm) 0.00006 0.001524 0.0009 0.02286 0.0013 0.03302 0.0016 0.04064 0.0020 0.05080 0.0026 0.06604 0.0035 0.08890 0.0047 0.11938 0.0059 0.14986 0.0077 0.19558

Material Thickness, T (in.) (mm) 0.157 3.9878 0.177 4.4958 0.197 5.0038 0.236 5.9944 0.275 6.9850 0.315 8.0010 0.394 10.0076 0.472 11.9888 0.590 14.9860 0.748 18.9992

Clearance, c (in.) (mm) 0.0095 0.24130 0.0116 0.29464 0.0138 0.35052 0.0177 0.44958 0.0226 0.57404 0.0285 0.72390 0.0394 1.00076 0.0502 1.27508 0.0689 1.75006 0.0935 2.37490

Effect of Clearance: Manufacturers have performed many studies on the effect of clear­ ance on punching and blanking. Clearance affects not only the smoothness of the fracture, but also the deformation force and work. A tighter blanking and punching clearance generates more heat on the cutting edge and the bulging area tightens around the punch. These effects produce a faster breakdown of the cutting edge. If the clearance increases, the

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Machinery's Handbook, 31st Edition Blanking and Punching

1412

bulging area disappears and the roll-over surface is stretched and will retract after the slug breaks free. Less heat is generated with increases in the blanking and punch­ing clearance, and the edge breakdown rate is reduced. The deformation force is greatest when the punch diameter is small compared to the thickness of the work material. In one example, a punching force of about 142 kN was required to punch 19 mm holes into 8 mm mild steel when the clearance was about 10 percent. With a clearance of about 4.5 percent, the punching force increased to 147 kN, and a clearance of 2.75 percent resulted in a force of 153.5 kN. Die Opening Profile.—Die opening profiles depend on the purpose and required toler­ ance of the workpiece. Two opening profiles are shown in Fig. 10a and Fig. 10b.

Fig. 10a. Opening Profile for High Quality Part

Fig. 10b. Opening for Low Accuracy Part

The profile in Fig. 10a gives the highest quality workpiece. To allow a die block to be sharpened more times, the height h of the block needs to be greater than the thickness of the workpiece. The value of h is given in Table 6. The die opening profile in Fig. 10b is used for making a small part with low accuracy from very soft material, such as soft thin brass. The angle of the cone α = 15′ to 45′. Table 6. Value of Dimension h Based on Material Thickness < 0.04 in. ( < 1 mm) Height h

0.14 in (3.5 mm)

Work material thickness, T > 0.04 to 0.2 in. ( >1 to 5 mm )

0.26 in (6.5 mm)

> 0.2 to 0.4 in. ( > 5 to 10 mm) 0.45 in (11.5 mm)

Angle α = 3° to 5°

Deformation Force, Deformation Work, and Force of Press.—Deformation force F for punching and blanking with flat face of punch is defined by the following equation: (20) F = LT τm = 0.8L T ^UTS h where F =  deformation force, lb (N) L =  the total length of cutting, in. (mm) T =  thickness of the material, in. (mm) τm =  shear stress, lb/in2 (MPa) UTS =   the ultimate tensile strength of the work material, lb/in2 (MPa)

Force of Press: Such variables as unequal thickness of the material, friction between the punch and workpiece, or dull cutting edges, can increase the necessary force by up to 30 percent, so these variables must be considered in selecting the power requirements of the press. That is, the force requirement of the press, Fp is

(21) Fp = 1.3F The blanking and punching force can be reduced if the punch or die has bevel-cut edges. In blanking operations, bevel shear angles should be used on the die to ensure workpiece remains flat. In punching operations, bevel shear angles should be used on the punch. Deformation Work W for punching and blanking with flat face of punch is defined by the following equation: (22) W = kFT

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Machinery's Handbook, 31st Edition Blanking and Punching

1413

where k =  a coefficient that depends on the shear strength of the material and the thick­ness of the material F =  deformation force (lb) T = material thickness (in.) Table 7. Values for Coefficient k for Some Materials

Material Low carbon steel Medium carbon steel 0.20 to 0.25% carbon Hard steel 0.40 to 0.60% carbon Copper, annealed

Shear Strength lb/in2 (MPa)

35,000–50,000 (240–345) 50,000–70,000 (345–483) 70,000–95,000 (483–655) 21,000 (145)

< 0.040 in. < (1.0 mm)

Material Thickness, inch (mm) 0.078–0.157 in. 0.040−0.078 in. (2.0−4.0 mm) (1.0−2.0 mm)

> 0.157 in. > 4.0 mm

0.70–0.65

0.64−0.60

0.58–0.50

0.45–0.35

0.60–0.55

0.54–0.50

0.49–0.42

0.40–0.30

0.45–0.42

0.41−0.38

0.36–0.32

0.30–0.20

0.75–0.69

0.70–0.65

0.64–0.55

0.50–0.40

Stripper Force.—Elastic Stripper: When spring strippers are used, it is necessary to cal­culate the amount of force required to effect stripping. This force may be calculated by the following equation: 1 Fs = 0.00117 PT = 855PT

(23a)

where Fs = stripping force (lb)



P = sum of the perimeters of all the punch­ing or blanking faces (in.) T = thickness of material (in.)

Fs = 5.9PT

where Fs = stripping force (N)

(23b)

P = sum of the perimeters of all the punch­ing or blanking faces (mm) T = thickness of material (mm)

This formula has been used for many years by a number of manufacturers and has been found to be satisfactory for most punching and blanking operations. After the total stripping force has been determined, the stripping force per spring must be found in order to establish the number and dimensions of springs required. Maximum force per spring is usually listed in the manufacturer’s catalog. The correct determined force per spring must satisfy the following relationship: F (24) Fmax 2 Fso 2 ns where Fmax =  maximum force per spring, lb (newton) FSO =   stripping force per spring, lb (newton) Fs =  total stripping force, lb (newton) n =   number of springs Blanking Pressure.—When designing parts that are to be blanked in a press, it is often necessary to work out the pressure required. A press of sufficient tonnage, providing the necessary blanking force, must be used. Otherwise, the press is likely to stall. To calculate the pressure required in tons for blanking various materials, use the following formula: P=A×T×S where

P = pressure required (tons) A = perimeter of blank (in.) T = thickness of material (in.) S = shear strength of material (tons per in2)

Example: Calculate the pressure required in tons for the blank illustrated by Fig. 11. Soft sheet brass is to be used with a thickness of 0.125 inch and a shear strength of 17 tons per square inch.

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Machinery's Handbook, 31st Edition BLANKING PRESSURE

1414

4" 1" 0.8"

Ø0.8" 2 PLACES

2.45"

0.85" 0.8" 0.5" 3.5"

Fig. 11. Dimensioned Drawing of Part to be Blanked

Perimeter of blank = 4 + 2.45 + 3.5 + 0.8 + 0.5 + 0.85 + 1 + 0.8 = 13.9 in. Circumference of holes = 2(π ×  0.8) = 5.026 in. Total perimeter = 13.9 + 5.026 = 18.926 in. Pressure required = P = A  × T ×  S = 18.926  × 0.125 ×  17 = 40.22 tons When metric dimensions are used, the same formula applies: P = pressure required (kg) where A = perimeter of blank (mm) T = thickness of material (mm) S = shear strength of material (kg per mm 2) Note: The general rule for punches that have a shear, where the cutting faces are ground at an angle, is to halve the tonnage for material thicknesses of up to 0.25 inch or 6 mm. For material thicknesses over 0.25 inch or 6 mm, two-thirds of the tonnage should be used. Fine Blanking.—This process uses special presses and tooling to pro­duce flat components from sheet metal or plate, with high dimensional accuracy. Accord­ing to Hydrel A. G., Romanshorn, Switzerland, fine-blanking presses can be powered hydraulically or mechanically, or by a combination of these methods, but they must have three separate and distinct movements. These movements clamp the work mate­r ial, perform the blanking operation, and eject the finished part from the tool. Forces of 1.5–2.5 times those used in conventional stamping are needed for fine blanking, so machines and tools must be designed accordingly. In mechanical fine-blanking presses, clamping and ejection forces are exerted hydraulically. Such presses generally are of toggle-type design and are limited to total forces of up to about 280 tons. Higher forces usually require all-hydraulic designs. These presses are also suited to embossing, coining, and impact extrusion work. Cutting elements of tooling for fine blanking generally are made from 12 percent chro­ mium steel, although high-speed steel and tungsten carbide also are used for long runs or improved quality. Cutting clearances between the punch and die as a percentage of the thickness of material are given in Table 8. Table 8. Values for Clearances Used in Fine-Blanking Tools as a Percentage of the Thickness of the Material Material Thickness (in.) (mm) 3 to 6 > 6 to 10 > 10 to 18 > 18 to 30 > 30 to 50 > 50 to 80 > 80 to 120 > 120 to 180 > 180 to 250 > 250 to 315 > 315 to 400 > 400 to 500 > 500 to 630 > 630 to 800 > 800 to 1000

37.5

45

Standard Pressure Angle, in Degrees 30

37.5

d 0.035 0.052 0.069 0.087 0.113 0.139 0.173 0.208 0.251 0.294 0.329 0.364 0.398 0.450 0.502 0.554

0.026 0.039 0.052 0.065 0.085 0.104 0.130 0.156 0.189 0.222 0.248 0.274 0.300 0.339 0.378 0.417

45

30

Classes of Fit

37.5

e

0.020 0.030 0.040 0.050 0.065 0.080 0.100 0.120 0.145 0.170 0.190 0.210 0.230 0.260 0.290 0.320

0.024 0.035 0.043 0.055 0.069 0.087 0.104 0.125 0.147 0.173 0.191 0.217 0.234 0.251 0.277 0.294

es/tan αD in millimeters 0.018 0.026 0.033 0.042 0.052 0.065 0.078 0.094 0.111 0.130 0.143 0.163 0.176 0.189 0.209 0.222

0.014 0.020 0.025 0.032 0.040 0.050 0.060 0.072 0.085 0.100 0.110 0.125 0.135 0.145 0.160 0.170

45

f 0.010 0.017 0.023 0.028 0.035 0.043 0.052 0.062 0.074 0.087 0.097 0.107 0.118 0.132 0.139 0.149

0.008 0.013 0.017 0.021 0.026 0.033 0.039 0.047 0.056 0.065 0.073 0.081 0.089 0.099 0.104 0.112

All h

0.006 0.010 0.013 0.016 0.020 0.025 0.030 0.036 0.043 0.050 0.056 0.062 0.068 0.076 0.080 0.086

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

These values are used with the applicable formulas in Table 13.

cF 0.5m 0.75m Min hs = 0.6m Min cF = 0.1m

0.5πm Internal Spline

πm

0.5πm

rfi = 0.2m Min.* 0.75m

30°

External Spline Reference Line rfe = 0.2m Min.*

*If full radius is smaller use full radius

Fig. 6a. Profile of Basic Rack for 30° Flat Root Spline

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Machinery's Handbook, 31st Edition Metric Module Involute Splines cF 0.5m

0.5πm Internal Spline

πm

rfi = 0.4m Min.* 0.9m

30° External Spline Reference Line

0.9m Min hs = 0.6m Min hF = 0.1m

0.5πm

2357

rfe = 0.4m Min.*

*If full radius is smaller, use full radius

Fig. 6b. Profile of Basic Rack for 30° Fillet Root Spline

cF 0.45m 0.7m Min hs = 0.55m Min hF = 0.1m

0.5πm

πm

0.5πm

Internal Spline

rfi = 0.3m Min. 0.7m

37.5° External Spline Reference Line rfe = 0.3m Min.

Fig. 6c. Profile of Basic Rack for 37.5° Fillet Root Spline

cF 0.4m 0.6m Min hs = 0.5m cF = 0.1m

0.5πm Internal Spline

πm

0.5πm

rfi = 0.25m Min. 0.6m

45° External Spline Reference Line rfe = 0.25m Min. Fig. 6d. Profile of Basic Rack for 45° Fillet Root Spline

British Standard Straight Splines.—British Standard BS 2059:1953, “Straight-sided Splines and Serrations,” was introduced because of the widespread development and use of splines, and because of the increasing use of involute splines it was necessary to provide a separate standard for straight-sided splines. BS 2059 was prepared on the hole basis, the hole being the constant member, and provided for different fits to be obtained by varying the size of the splined or serrated shaft. Part 1 of the standard deals with 6 splines only, irre­spective of the shaft diameter, with two depths termed shallow and deep. The splines are bottom fitting with top clearance. The standard contains three different grades of fit, based on the principle of variations in the diameter of the shaft at the root of the splines, in conjunction with variations in the widths of the splines themselves. Fit 1 represents the condition of closest fit and is designed for minimum backlash. Fit 2 has a positive allowance and is designed for ease

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2358

Machinery's Handbook, 31st Edition Straight-Sided Splines

of assembly, and Fit 3 has a larger positive allowance for applications that can accept such clearances. All these splines allow for clearance on the sides of the splines (the widths), but in Fit 1, the minor diameters of the hole and the shaft may be of identical size. Assembly of a splined shaft and hole requires consideration of the designed profile of each member, and this consideration should concentrate on the maximum diameter of the shafts and the widths of external splines, in association with the minimum diameter of the hole and the widths of the internal splineways. In other words, both internal and external splines are in the maximum metal condition. The accuracy of spacing of the splines will affect the quality of the resultant fit. If angular positioning is inaccurate, or the splines are not parallel with the axis, there will be interference between the hole and the shaft. Part 2 of the Standard deals with straight-sided 90 serrations having nominal diameters from 0.25 to 6.0 inches. Provision is again made for three grades of fits, the basic constant being the serrated hole size. Variations in the fits of these serrations is obtained by varying the sizes of the serrations on the shaft, and the fits are related to flank bearing, the depth of engagement being constant for each size and allowing positive clearance at crest and root. Fit 1 is an interference fit intended for permanent or semi-permanent ass­emblies. Heating to expand the internally-serrated member is needed for assembly. Fit 2 is a transition fit intended for assemblies that require accurate location of the serrated members, but must allow disassembly. In maximum metal conditions, heating of the outside member may be needed for assembly. Fit. 3 is a clearance or sliding fit, intended for general applications. Maximum and minimum dimensions for the various features are shown in the Standard for each class of fit. Maximum metal conditions presupposes that there are no errors of form such as spacing, alignment, or roundness of hole or shaft. Any compensation needed for such errors may require reduction of a shaft diameter or enlargement of a serrated bore, but the measured effective size must fall within the specified limits. British Standard BS 3550:1963, “Involute Splines”, is complementary to BS 2059, and the basic dimensions of all the sizes of splines are the same as those in the ANSI B92.11996, for major diameter fit and side fit. The British Standard uses the same terms and symbols and provides data and guidance for design of straight involute splines of 30 pressure angle, with tables of limiting dimensions. The standard also deals with manufac­ turing errors and their effect on the fit between mating spline elements. The range of splines covered is: Side fit, flat root, 2.5/5.0 to 32/64 pitch, 6 to 60 splines. Major diameter, flat root, 3.0/6.0 to 16/32 pitch, 6 to 60 splines. Side fit, fillet root, 2.5/5.0 to 48/96 pitch, 6 to 60 splines.

British Standard BS 6186, Part 1:1981, “Involute Splines, Metric Module, Side Fit” is identical with sections 1 and 2 of ISO 4156 and with ANSI B92.2M-1980 (R1989) “Straight Cylindrical Involute Splines, Metric Module, Side Fit – Generalities, Dimen­ sions and Inspection”.

SAE Standard Spline Fittings.—The SAE spline fittings (Table 18 through Table 21 inclusive) have become an established standard for many applications in the agricultural, automotive, machine tool, and other industries. The dimensions given, in inches, apply only to soft broached holes. Dimensions are illustrated in Fig. 7a, Fig. 7b, and Fig. 7c. The toler­ances given may be readily maintained by usual broaching methods. The tolerances selected for the large and small diameters may depend upon whether the fit between the mating part, as finally made, is on the large or the small diameter. The other diameter, which is designed for clearance, may have a larger manufactured tolerance. If the final fit between the parts is on the sides of the spline only, larger tolerances are permissible for both the large and small diameters. The spline should not be more than 0.006 inch per foot out of parallel with respect to the shaft axis. No allowance is made for corner radii to obtain clearance. Radii at the corners of the spline should not exceed 0.015 inch.

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Machinery's Handbook, 31st Edition Straight-Sided Splines

2359

W

W

W

D

h

d

Fig. 7a. 4-Spline Fitting

h

D

h

D

d

d

Fig. 7b. 6-Spline Fitting

Fig. 7c. 10-Spline Fitting

Table 18. SAE Standard 4-Spline Fittings Nom. Diam 3 ∕4

D

For All Fits

W

d

4A—Permanent Fit h

4B—To Slide, No Load

Max.

Ta

0.749 0.750 0.179 0.181 0.636 0.637 0.055 0.056

78

Min.

Max.

Min.

Max.

Min.

Max.

Min.

Min.

d

Max.

Min.

h

Max.

Ta

0.561 0.562 0.093 0.094

123

1 1 ∕8

0.874 0.875 0.209 0.211 0.743 0.744 0.065 0.066 0.999 1.000 0.239 0.241 0.849 0.850 0.074 0.075 1.124 1.125 0.269 0.271 0.955 0.956 0.083 0.084

107 139 175

0.655 0.656 0.108 0.109 0.749 0.750 0.124 0.125 0.843 0.844 0.140 0.141

167 219 277

13∕8

1.374 1.375 0.329 0.331 1.168 1.169 0.102 0.103

262

1.030 1.031 0.171 0.172

414

7∕8

1

11 ∕4 11 ∕2 15∕8

13∕4 2

2 1 ∕4 2 1 ∕2 3

1.249 1.250 0.299 0.301 1.061 1.062 0.093 0.094 1.499 1.500 0.359 0.361 1.274 1.275 0.111 0.112

1.624 1.625 0.389 0.391 1.380 1.381 0.121 0.122

1.749 1.750 0.420 0.422 1.486 1.487 0.130 0.131 1.998 2.000 0.479 0.482 1.698 1.700 0.148 0.150 2.248 2.250 0.539 0.542 1.910 1.912 0.167 0.169

2.498 2.500 0.599 0.602 2.123 2.125 0.185 0.187 2.998 3.000 0.720 0.723 2.548 2.550 0.223 0.225

217 311

367

424 555 703

865 1249

0.936 0.937 0.155 0.156 1.124 1.125 0.186 0.187

1.218 1.219 0.202 0.203

1.311 1.312 0.218 0.219 1.498 1.500 0.248 0.250 1.685 1.687 0.279 0.281 1.873 1.875 0.310 0.312 2.248 2.250 0.373 0.375

341 491

577

670 875 1106

1365 1969

a See note at end of Table 21.

Table 19. SAE Standard 6-Spline Fittings

Nom. Diam. 3 ∕4

Min.

D

For All Fits Max.

Min.

W

Max.

0.749 0.750 0.186 0.188

0.674

0.675

1.249 1.250 0.311 0.313

1.124

1.125

11 ∕8

0.874 0.875 0.217 0.219 0.999 1.000 0.248 0.250 1.124 1.125 0.279 0.281

13∕8

1.374 1.375 0.342 0.344

7∕8

1

11 ∕4 11 ∕2 15∕8

13∕4 2

21 ∕4

21 ∕2 3

6A—Permanent Fit d Max. Ta

Min.

1.499 1.500 0.373 0.375 1.624 1.625 0.404 0.406

1.749 1.750 0.436 0.438 1.998 2.000 0.497 0.500 2.248 2.250 0.560 0.563 2.498 2.500 0.622 0.625 2.998 3.000 0.747 0.750

0.787 0.899 1.012 1.237 1.349 1.462

1.574 1.798 2.023 2.248 2.698

80

0.788 0.900 1.013

109 143 180

1.238

269

1.350 1.463

1.575 1.800 2.025 2.250 2.700

223 321 376

436 570 721

891 1283

6B—To Slide, No Load 6C—To Slide Under Load d d Ta Ta Min. Max. Min. Max. 0.637

0.638

1.062

1.063

0.743 0.849 0.955 1.168 1.274 1.380

1.487 1.698 1.911 2.123 2.548

117

0.599

0.600

152

325

0.999

1.000

421

0.744 0.850 0.956

159 208 263

1.169

393

1.275 1.381

1.488 1.700 1.913 2.125 2.550

468 550

637 833 1052

1300 1873

0.699 0.799 0.899 1.099 1.199 1.299

1.399 1.598 1.798 1.998 2.398

0.700 0.800 0.900 1.100 1.200 1.300

1.400 1.600 1.800 2.000 2.400

207 270 342 510 608 713

827 1080 1367 1688 2430

a See note at end of Table 21.

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Machinery's Handbook, 31st Edition Straight-Sided Splines

2360

Table 20. SAE Standard 10-Spline Fittings Nom. Diam. 3 ∕4 7∕8

1 11 ∕8 11 ∕4 13∕8 11 ∕2 1 5 ∕8 13∕4 2 21 ∕4 21 ∕2 3 31 ∕2 4 41 ∕2 5 51 ∕2 6

Min.

0.749 0.874 0.999 1.124 1.249 1.374 1.499 1.624 1.749 1.998 2.248 2.498 2.998 3.497 3.997 4.497 4.997 5.497 5.997

D

For All Fits Max.

0.750 0.875 1.000 1.125 1.250 1.375 1.500 1.625 1.750 2.000 2.250 2.500 3.000 3.500 4.000 4.500 5.000 5.500 6.000

Min.

W

0.115 0.135 0.154 0.174 0.193 0.213 0.232 0.252 0.271 0.309 0.348 0.387 0.465 0.543 0.621 0.699 0.777 0.855 0.933

10A—Permanent Fit Max.

0.117 0.137 0.156 0.176 0.195 0.215 0.234 0.254 0.273 0.312 0.351 0.390 0.468 0.546 0.624 0.702 0.780 0.858 0.936

a See note at end of Table 21.

Min.

d

0.682 0.795 0.909 1.023 1.137 1.250 1.364 1.478 1.592 1.818 2.046 2.273 2.728 3.182 3.637 4.092 4.547 5.002 5.457

Max.

0.683 0.796 0.910 1.024 1.138 1.251 1.365 1.479 1.593 1.820 2.048 2.275 2.730 3.185 3.640 4.095 4.550 5.005 5.460

Ta

120 165 215 271 336 406 483 566 658 860 1088 1343 1934 2632 3438 4351 5371 6500 7735

10B—To Slide, No Load 10C—To Slide Under Load Min.

d

0.644 0.752 0.859 0.967 1.074 1.182 1.289 1.397 1.504 1.718 1.933 2.148 2.578 3.007 3.437 3.867 4.297 4.727 5.157

Max.

0.645 0.753 0.860 0.968 1.075 1.183 1.290 1.398 1.505 1.720 1.935 2.150 2.580 3.010 3.440 3.870 4.300 4.730 5.160

Ta

183 248 326 412 508 614 732 860 997 1302 1647 2034 2929 3987 5208 6591 8137 9846 11718

Min.

d

0.607 0.708 0.809 0.910 1.012 1.113 1.214 1.315 1.417 1.618 1.821 2.023 2.428 2.832 3.237 3.642 4.047 4.452 4.857

Max.

0.608 0.709 0.810 0.911 1.013 1.114 1.215 1.316 1.418 1.620 1.823 2.025 2.430 2.835 3.240 3.645 4.050 4.455 4.860

Ta

241 329 430 545 672 813 967 1135 1316 1720 2176 2688 3869 5266 6878 8705 10746 13003 15475

Table 21. SAE Standard 16-Spline Fittings Nom. Diam. 2 21 ∕2 3 31 ∕2 4 41 ∕2 5 51 ∕2 6

Min.

1.997 2.497 2.997 3.497 3.997 4.497 4.997 5.497 5.997

D

For All Fits Max.

2.000 2.500 3.000 3.500 4.000 4.500 5.000 5.500 6.000

Min.

W

0.193 0.242 0.291 0.340 0.389 0.438 0.487 0.536 0.585

16A—Permanent Fit Max.

0.196 0.245 0.294 0.343 0.392 0.441 0.490 0.539 0.588

Min.

1.817 2.273 2.727 3.182 3.637 4.092 4.547 5.002 5.457

d

Max.

1.820 2.275 2.730 3.185 3.640 4.095 4.550 5.005 5.460

T

a

1375 2149 3094 4212 5501 6962 8595 10395 12377

16B—To Slide, No Load 16C—To Slide Under Load Min.

1.717 2.147 2.577 3.007 3.437 3.867 4.297 4.727 5.157

d

Max.

1.720 2.150 2.580 3.010 3.440 3.870 4.300 4.730 5.160

Ta

2083 3255 4687 6378 8333 10546 13020 15754 18749

Min.

1.617 2.022 2.427 2.832 3.237 3.642 4.047 4.452 4.857

d

Max.

1.620 2.025 2.430 2.835 3.240 3.645 4.050 4.455 4.860

Ta

2751 4299 6190 8426 11005 13928 17195 20806 24760

a Torque Capacity of Spline Fittings: The torque capacities of the different spline fittings are given in the columns headed “T.” The torque capacity, per inch of bearing length at 1000 pounds pressure per square inch on the sides of the spline, may be determined by the following formula, in which T = torque capacity in inch-pounds per inch of length, N = number of splines, R = mean radius or radial distance from center of hole to center of spline, h = depth of spline: T = 1000 NRh

Table 22. Formulas for Determining Dimensions of SAE Standard Splines No. of Splines Four Six Ten Sixteen

W For All Fits 0.241Da 0.250D 0.156D 0.098D

A Permanent Fit h d 0.075D 0.850D 0.050D 0.900D 0.045D 0.910D 0.045D 0.910D

B To Slide Without Load h d 0.125D 0.750D 0.075D 0.850D 0.070D 0.860D 0.070D 0.860D

C To Slide Under Load h d … … 0.100D 0.800D 0.095D 0.810D 0.095D 0.810D

a Four splines for fits A and B only.

The formulas in the table above give the maximum dimensions for W, h, and d, as listed in Table 18 through Table 21 inclusive.

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Machinery's Handbook, 31st Edition Polygon Shafts

2361

Polygon-Type Shaft Connections.—Involute-form and straight-sided splines are used for both fixed and sliding connections between machine members such as shafts and gears. Polygon-type connections, so called because they resemble regular polygons but with curved sides, may be used similarly. German DIN Standards 32711 and 32712 include data for three- and four-sided metric polygon connections. Data for 11 of the sizes shown in those Standards, but converted to inch dimensions by Stoffel Polygon Systems, are given in the accompanying table. Dimensions of Three- and Four-Sided Polygon-Type Shaft Connections Drawing for 3-sided Designs

Bmax

Drawing for 4-sided Designs

2e

DM

DI DA

Q

R

DI DA

Bmax 2e Three-Sided Designs

Nominal Sizes DA (in.) 0.530

DI (in.) 0.470

e (in.) 0.015

Four-Sided Designs

Design Data

Area (in.2) 0.194

ZP (in.3) 0.020

Nominal Sizes DA (in.) 0.500

DI (in.) 0.415

Design Data

e (in.) 0.075

Area (in.2) 0.155

ZP (in.3) 0.014

0.350

0.048

0.665

0.585

0.020

0.302

0.039

0.625

0.525

0.075

0.800

0.700

0.025

0.434

0.067

0.750

0.625

0.125

0.250

0.028

0.930

0.820

0.027

0.594

0.108

0.875

0.725

0.150

0.470

0.075

1.080

0.920

0.040

0.765

0.153

1.000

0.850

0.150

0.650

0.12

1.205

1.045

0.040

0.977

0.224

1.125

0.950

0.200

0.810

0.17

1.330

1.170

0.040

1.208

0.314

1.250

1.040

0.200

0.980

0.22

1.485

1.265

0.055

1.450

0.397

1.375

1.135

0.225

1.17

0.29

1.610

1.390

0.055

1.732

0.527

1.500

1.260

0.225

1.43

0.39

1.870

1.630

0.060

2.378

0.850

1.750

1.480

0.250

1.94

0.64

2.140

1.860

0.070

3.090

1.260

2.000

1.700

0.250

2.60

0.92

Dimensions Q and R shown on the diagrams are approximate and used only for drafting purposes: Q ≈ 7.5e; R ≈ D I /2 + 16e. Dimension D M = D I + 2e. Pressure angle Bmax is approximately 344e/D M degrees for three sides, and 299e/D M degrees for four sides. Tolerances: ISO H7 tolerances apply to bore dimensions. For shafts, g6 tolerances apply for sliding fits; k7 tolerances for tight fits.

Choosing Between Three- and Four-Sided Designs: Three-sided designs are best for applications in which no relative movement between mating components is allowed while torque is transmitted. If a hub is to slide on a shaft while under torque, four-sided designs, which have larger pressure angles Bmax than those of three-sided designs, are better suited to sliding even though the axial force needed to move the sliding member is approximately 50 percent greater than for comparable involute spline connections.

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2362

Machinery's Handbook, 31st Edition Polygon Shafts

Strength of Polygon Connections: In the formulas that follow, Hw = hub width, inches   Ht = hub wall thickness, inches Mb =  bending moment, lb-inch Mt =  torque, lb-inch Z =  section modulus, bending, in3 = 0.098DM 4/DA for three sides  = 0.15DI 3 for four sides Z P =  polar section modulus, torsion, in3 = 0.196DM 4/DA for three sides  = 0.196DI 3 for four sides DA and DM . See table footnotes. Sb =  bending stress, allowable, lb/in2 Ss =  shearing stress, allowable, lb/in2 St =  tensile stress, allowable, lb/in2 For shafts,

Mt (maximum) = Ss Z p

Mb (maximum) = Sb Z For bores,

Ht ^minimumh = K

Mt St Hw

in which K = 1.44 for three sides except that if D M is greater than 1.375 inches, then K = 1.2; K = 0.7 for four sides. Failure may occur in the hub of a polygon connection if the hoop stresses in the hub exceed the allowable tensile stress for the material used. The radial force tending to expand the rim and cause tensile stresses is calculated from 2Mt Radial Force, lb = D I n tan ^ Bmax + 11.3h This radial force acting at n points may be used to calculate the tensile stress in the hub wall using formulas from strength of materials. Manufacturing: Polygon shaft profiles may be produced using conventional machining processes such as hobbing, shaping, contour milling, copy turning, and numerically con­ trolled milling and grinding. Bores are produced using broaches, spark erosion, gear shapers with generating cutters of appropriate form, and, in some instances, internal grind­ers of special design. Regardless of the production methods used, points on both of the mating profiles may be calculated from the following equations: D X =  ------I + e cos α – e cos ( nα ) cos α – ne sin ( nα ) sin α 2 

D Y =  ------I + e sin α – e cos ( nα ) sin α + ne sin ( nα ) cos α 2  In these equations, α is the angle of rotation of the workpiece from any selected reference position; n is the number of polygon sides, either 3 or 4; D I is the diameter of the inscribed circle shown on the diagram in the table; and e is the dimension shown on the diagram in the table and may be used as a setting on special polygon grinding machines. The value of e determines the shape of the profile. A value of 0, for example, results in a circular shaft having a diameter of D I. The values of e in the table were selected arbitrarily to pro­vide suitable proportions for the sizes shown.

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Machinery's Handbook, 31st Edition CAMS AND CAM DESIGN

2363

CAMS AND CAM DESIGN Classes of Cams.—Cams may, in general, be divided into two classes: uniform motion cams and accelerated motion cams. The uniform motion cam moves the follower at the same rate of speed from the beginning to the end of the stroke; but as the movement is started from zero to the full speed of the uniform motion and stops in the same abrupt way, there is a distinct shock at the beginning and end of the stroke, if the movement is at all rapid. In machinery working at a high rate of speed, therefore, it is important that cams are so constructed that sudden shocks are avoided when starting the motion or when reversing the direction of motion of the follower. The uniformly accelerated motion cam is suitable for moderate speeds, but it has the dis­advantage of sudden changes in acceleration at the beginning, middle and end of the stroke. A cycloidal motion curve cam produces no abrupt changes in acceleration and is often used in high-speed machinery because it results in low noise, vibration and wear. The cycloidal motion displacement curve is so called because it can be generated from a * cycloid which is the locus of a point of a circle rolling on a straight line. Cam Follower Systems.—The three most used cam and follower systems are radial and offset translating roller follower, Fig. 1a and Fig. 1b; and the swinging roller follower, Fig. 1c. When the cam rotates, it imparts a translating motion to the roller followers in Fig. 1a and Fig. 1b and a swinging motion to the roller follower in Fig. 1c. The motion of the follower is, of course, dependent on the shape of the cam; and the following section on displacement dia­grams explains how a favorable motion is obtained so that the cam can rotate at high speed without shock.

M

Fig. 1a. Radial Translating Roller Follower

Fig. 1b. Offset Translating Roller Follower

Fig. 2a. Closed-Track Cam

Fig. 1c. Swinging Roller Fol­lower

Fig. 2b. Closed-Track Cam With Two Rollers

The arrangements in Fig. 1a, Fig. 1b, and Fig. 1c show open-track cams. In Fig. 2a and Fig. 2b the roller is forced to move in a closed track. Open-track cams build smaller than * Jensen, P. W., Cam Design and Manufacture, Industrial Press Inc.

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2364

Machinery's Handbook, 31st Edition CAMS AND CAM DESIGN

closed-track cams but, in general, springs are necessary to keep the roller in contact with the cam at all times. Closed-track cams do not require a spring and have the advantage of positive drive throughout the rise and return cycle. The positive drive is sometimes required as in the case where a broken spring would cause serious damage to a machine. Displacement Diagrams.—Design of a cam begins with the displacement diagram. A simple displacement diagram is shown in Fig. 3. One cycle means one whole revolution of the cam; i.e., one cycle represents 360°. The horizontal distances T1, T2 , T3, T4 are expressed in units of time (seconds); or radians or degrees. The vertical distance, h, rep­resents the maximum “rise” or stroke of the follower. Rise Interval T1

Dwell Interval T3

Return Interval T2

Dwell Interval T4

h= Follower Stroke One Cycle = 360° Fig. 3. A Simple Displacement Diagram

The displacement diagram of Fig. 3 is not a very favorable one because the motion from rest (the horizontal lines) to constant velocity takes place instantaneously and this means that accelerations become infinitely large at these transition points. Types of Cam Displacement Curves: A variety of cam curves are available for moving the follower. In the following sections only the rise portions of the total time-displacement diagram are studied. The return portions can be analyzed in a similar manner. Complex cams are frequently employed which may involve a number of rise-dwell-return intervals in which the rise and return aspects are quite different. To analyze the action of a cam it is necessary to study its time-displacement and associated velocity and acceleration curves. The latter are based on the first and second time-derivatives of the equation describing the time-displacement curve: y = displacement = f ^ t h y = f ^φ h or dy dy v = dt = velocity = ω dφ d 2y d 2y a = 2 = acceleration = ω 2 2 dt dφ

Meaning of Symbols and Equivalent Relations y =  displacement of follower, inch (m) h =  maximum displacement of follower, inch (m) t =  time for cam to rotate through angle φ, sec, = φ/ω, sec T =  time for cam to rotate through angle β, sec, = β/ω, or β/6N, sec φ =  cam angle rotation for follower displacement y, degrees β =  cam angle rotation for total rise h, degrees v =  velocity of follower, in/sec (m/s) a =  follower acceleration, in/sec2 (m/s2) t/T =  φ/β N =  cam speed, rpm ω =  angular velocity of cam, degrees/sec = β/T = φ/t = dφ/dt = 6N ωR =  angular velocity of cam, radians/sec = πω/180

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Machinery's Handbook, 31st Edition CAMS AND CAM DESIGN

2365

W =  effective weight, lbs (kg) g =  gravitational constant = 386 in/sec2 (9.81 m/s2) f(t) =   means a function of t f(φ) =   means a function of φ Rmin =  minimum radius to the cam pitch curve, inch (m) Rmax =  maximum radius to the cam pitch curve, inch (m) rf =  radius of cam follower roller, inch (m) ρ =  radius of curvature of cam pitch curve (path of center of roller follower), inch (m) Rc =  radius of curvature of actual cam surface, in., (m) = ρ − rf for convex surface; = ρ + rf for concave surface. B

Acceleration = ∞ Velocity

h

y

A

t

T Acceleration = ∞

Fig. 4. Cam Displacement, Velocity, and Acceleration Curves for Constant Velocity Motion

Four displacement curves are of the greatest utility in cam design. 1. Constant-Velocity Motion: (Fig. 4)

t y= hT

or y =

dy h v = dt = T a=

hφ β

or v =

d 2y = 0* dt 2

(1a)

hω β

(1b) }

0e Fr

Table Entering Factorsa

Contact Angle, α

Fa ≤e Fr

RADIAL CONTACT GROOVE BEARINGS

Fa /iZD2 Metric Inch Units Units

0.172 0.345 0.689 1.03 1.38 2.07 3.45 5.17 6.89

25 50 100 150 200 300 500 750 1000

Fa /ZD2 Metric Inch Units Units

0.014 0.172 0.028 0.345 0.056 0.689 0.085 1.03 0.11 1.38 0.17 2.07 0.28 3.45 0.42 5.17 0.56 6.89 0.014 0.172 0.029 0.345 0.057 0.689 0.086 1.03 0.11 1.38 0.17 2.07 0.29 3.45 0.43 5.17 0.57 6.89 0.015 0.172 0.029 0.345 0.058 0.689 0.087 1.03 0.12 1.38 0.17 2.07 0.29 3.45 0.44 5.17 0.58 6.89 … … … … … … … … … … Self-aligning Ball Bearings

25 50 100 150 200 300 500 750 1000 25 50 100 150 200 300 500 750 1000 25 50 100 150 200 300 500 750 1000 … … … … …

Fa >e Fr

e X Y X 0.19 2.30 0.22 1.99 0.26 1.71 0.28 1.56 0.30 1.45 0.56 1 0.34 1.31 0.38 1.15 0.42 1.04 0.44 1.00 ANGULAR CONTACT GROOVE BEARINGS e 0.23 0.26 0.30 0.34 0.36 0.40 0.45 0.50 0.52 0.29 0.32 0.36 0.38 0.40 0.44 0.49 0.54 0.54 0.38 0.40 0.43 0.46 0.47 0.50 0.55 0.56 0.56 0.57 0.68 0.80 0.95 1.14

X

Y

For this type use the X, Y, and e values applicable to single row radial contact bearings

1.5 tan α

0.46

0.44

0.43 0.41 0.39 0.37 0.35

0.40

1

1.88 1.71 1.52 1.41 1.34 1.23 1.10 1.01 1.00 1.47 1.40 1.30 1.23 1.19 1.12 1.02 1.00 1.00 1.00 0.87 0.76 0.66 0.57

0.4 cot α

a Symbol definitions are given on the following page.

X

1

1

1 1 1 1 1 1

Y

X

0

0.56

Y 2.78 2.40 2.07 1.87 1.75 1.58 1.39 1.26 1.21 2.18 1.98 1.76 1.63 1.55 1.42 1.27 1.17 1.16 1.65 1.57 1.46 1.38 1.34 1.26 1.14 1.12 1.12 1.09 0.92 0.78 0.66 0.55

X

0.42 cot α

0.78

0.75

0.72

0.70 0.67 0.63 0.60 0.57 0.65

Y 2.30 1.99 1.71 1.55 1.45 1.31 1.15 1.04 1.00

Y 3.74 3.23 2.78 2.52 2.36 2.13 1.87 1.69 1.63 3.06 2.78 2.47 2.20 2.18 2.00 1.79 1.64 1.63 2.39 2.28 2.11 2.00 1.93 1.82 1.66 1.63 1.63 1.63 1.41 1.24 1.07 0.98

0.65 cot α

b For single row bearings when F /F ≤ e, use X = 1, Y = 0. Two similar, single row, angular cona r tact ball bearings mounted face-to-face or back-to-back are considered as one double row, angular contact bearing.

Values of X, Y, and e for a load or contact angle other than shown are obtained by linear interpola­ tion.Values of X, Y, and e do not apply to filling slot bearings for applications in which ball-raceway contact areas project substantially into the filling slot under load. Symbol Definitions: Fa is the applied axial load in newtons (pounds); Co is the static load rating in newtons (pounds) of

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2486

Machinery's Handbook, 31st Edition Ball and Roller Bearings

the bearing under consideration and is found by Formula (20); i is the number of rows of balls in the bearing; Z is the number of balls per row in a radial or angular contact bearing or the number of balls in a single row, single direction thrust bearing; D is the ball diameter in millimeters (inches); and Fr is the applied radial load in newtons (pounds).

For radial and angular contact ball bearings with balls not larger than 25.4 mm (1 inch) in diameter, C is found by the formula:

C = fc ^i cos αh0.7Z 2 ⁄ 3 D1.8

(2)

and with balls larger than 25.4 mm (1 inch) in diameter C is found by the formula:

C = 3.647fc ^i cos αh0.7Z 2 ⁄ 3D1.4 (metric) C = fc ^i cos αh0.7Z 2 ⁄ 3 D1.4( inch )

(3a)

(3b) where fc =  a factor which depends on the geometry of the bearing components, the accu­ racy to which the various bearing parts are made and the material. Values of fc, are given in Table 27 i =  number of rows of balls in the bearing α =  nominal contact angle, degrees Z =  number of balls per row in a radial or angular contact bearing D =  ball diameter, mm (inches) The magnitude of the equivalent radial load, P, in newtons (pounds) for radial and angu­ lar contact ball bearings, under combined constant radial and constant thrust loads is given by the formula:

P = XFr + YFa

(4)

where Fr =  the applied radial load in newtons (pounds) Fa =  the applied axial load in newtons (pounds) X =  radial load factor as given in Table 30 Y =  axial load factor as given in Table 30 Thrust Ball Bearings: The magnitude of the Rating Life L10 in millions of revolutions for a thrust ball bearing application is given by the formula: C 3 L10 = c a m (5) Pa

where Ca =  the basic load rating, newtons (pounds). See Formulas (6) to (10) Pa =  equivalent thrust load, newtons (pounds). See Formula (11) For single row, single and double direction, thrust ball bearing with balls not larger than 25.4 mm (1 inch) in diameter, Ca is found by the formulas:

for α = 90 degrees, for α ! 90 degrees ,

Ca = fc Z

2⁄3

D

1.8

(6)

0.7 2 ⁄ 3 1.8

Ca = fc ^cos αh Z

D tan α

(7)

and with balls larger than 25.4 mm (1 inch) in diameter, Ca is found by the formulas:

for α = 90 degrees,

Ca = 3.647fc Z Ca = fc Z

for α ≠ 90 degrees,

2⁄3

2⁄3

1.4

D

(metric)

1. 4

D ( inch )

(8a) (8b)

Ca = 3.647fc ^cos αh0. 7Z 2 ⁄ 3D1. 4 tanα (metric)

(9a)

Ca = f c ^cos αh0. 7Z 2 ⁄ 3D1. 4 tan α ( inch )

(9b)

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Machinery's Handbook, 31st Edition Ball and Roller Bearings

2487

where fc =  a factor which depends on the geometry of the bearing components, the accu­ racy to which the various bearing parts are made, and the material. Values of fc are given in Table 29 Z =  number of balls per row in a single row, single direction thrust ball bearing D =  ball diameter, mm (inches) α =  nominal contact angle, degrees D dm 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.12 0.14 0.16 0.18 0.20 0.22 0.24 0.26 0.28 0.30 0.32 0.34

α = 90° Metrica 36.7 45.2 51.1 55.7 59.5 62.9 65.8 68.5 71.0 73.3 77.4 81.1 84.4 87.4 90.2 92.8 95.3 97.6 99.8 101.9 103.9 105.8

a Use to obtain C b Use to obtain C

Table 29. Values of fc for Thrust Ball Bearings

Inchb 2790 3430 3880 4230 4520 4780 5000 5210 5390 5570 5880 6160 6410 6640 6854 7060 7240 7410 7600 7750 7900 8050

α = 45°

Metrica 42.1 51.7 58.2 63.3 67.3 70.7 73.5 75.9 78.0 79.7 82.3 84.1 85.1 85.5 85.4 84.9 84.0 82.8 81.3 79.6 … …

D cos α 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.12 0.14 0.16 0.18 0.20 0.22 0.24 0.26 0.28 0.30 … …

Inchb 3200 3930 4430 4810 5110 5360 5580 5770 5920 6050 6260 6390 6470 6500 6490 6450 6380 6290 6180 6040 … …

a in newtons when D is given in mm.

α = 60°

Metrica 39.2 48.1 54.2 58.9 62.6 65.8 68.4 70.7 72.6 74.2 76.6 78.3 79.2 79.6 79.5 … … … … … … …

Inchb 2970 3650 4120 4470 4760 4990 5190 5360 5510 5630 5830 5950 6020 6050 6040 … … … … … … …

α = 75° Metrica 37.3 45.9 51.7 56.1 59.7 62.7 65.2 67.3 69.2 70.7 … … … … … … … … … … … …

Inchb 2840 3490 3930 4260 4540 4760 4950 5120 5250 5370 … … … … … … … … … … … …

a in pounds when D is given in inches.

For thrust ball bearings with two or more rows of similar balls carrying loads in the same direction, the basic load rating, Ca , in newtons (pounds) is found by the formula:

[

Z 1 10 ⁄ 3  Z 2  10 ⁄ 3 Z n 10 ⁄ 3 C a = ( Z 1 + Z 2 + … + Z n )  --------- + --------+ … +  ---------  C a1  C a2  C an

]

– 0.3

(10)

where Z1, Z2 , . . . , Z n =  number of balls in respective rows of a single-direction multi-row thrust ball bearing Ca1, Ca2 , . . . , Can =  basic load rating per row of a single-direction, multi-row thrust ball bear­ing, each calculated as a single-row bearing with Z1, Z2 , . . . , Zn balls, respec­tively The magnitude of the equivalent thrust load, Pa , in newtons (pounds) for thrust ball bear­ings with α ≠ 90 degrees under combined constant thrust and constant radial loads is found by the formula:

Pa = XFr + YFa where Fr = Fa = X = Y =

(11)

 the applied radial load in newtons (pounds)  the applied axial load in newtons (pounds)  radial load factor as given in Table 30  axial load factor as given in Table 30

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Machinery's Handbook, 31st Edition Ball and Roller Bearings Table 30. Values of X and Y for Computing Equivalent Thrust Load Pa for Thrust Ball Bearings Single Direction Bearings

Contact Angle α

e X 1.25 0.66 45° 2.17 0.92 60° 4.67 1.66 75° For α = 90°, Fr = 0 and Y = 1.

Fa 2e Fr

Y 1 1 1

X 1.18 1.90 3.89

Fa #e Fr

Double Direction Bearings

Y 0.59 0.54 0.52

X 0.66 0.92 1.66

Fa 2e Fr

Y 1 1 1

Roller Bearing Types Covered.—This standard* applies to cylindrical, tapered and selfaligning radial and thrust roller bearings and to needle roller bearings. These bearings are presumed to be within the size ranges shown in the AFBMA dimensional standards, of good quality and produced in accordance with good manufacturing practice. Roller bearings vary considerably in design and execution. Since small differences in rel­ative shape of contacting surfaces may account for distinct differences in loadcarrying ability, this standard does not attempt to cover all design variations, rather it applies to basic roller bearing designs. The following limitations apply: 1) Truncated contact area. This standard may not be safely applied to roller bearings sub­jected to application conditions which cause the contact area of the roller with the raceway to be severely truncated by the edge of the raceway or roller. 2) Stress concentrations. A cylindrical, tapered or self-aligning roller bearing must be expected to have a basic load rating less than that obtained using a value of fc taken from Table 31 and Table 32 if, under load, a stress concentration is present in some part of the roller-raceway contact. Such stress concentrations occur in the center of nominal point contacts, at the contact extremities for line contacts and at inadequately blended junctions of a roll­ing surface profile. Stress concentrations can also occur if the rollers are not accurately guided such as in bearings without cages and bearings not having rigid integral flanges. Values of fc given in Table 31 and Table 32 are based upon bearings manufactured to achieve optimized contact. For no bearing type or execution will the factor fc be greater than that obtained in Table 31 and Table 32. 3) Material. This standard applies only to roller bearings fabricated from hardened, good quality steel. 4) Lubrication. Rating Life calculated according to this standard is based on the assump­ tion that the bearing is adequately lubricated. Determination of adequate lubrication depends upon the bearing application. 5) Ring support and alignment. Rating Life calculated according to this standard assumes that the bearing inner and outer rings are rigidly supported, and that the inner and outer ring axes are properly aligned. 6) Internal clearance. Radial roller bearing Rating Life calculated according to this stan­dard is based on the assumption that only a nominal internal clearance occurs in the mounted bearing at operating speed, load, and temperature. 7) High-speed effects. The Rating Life calculated according to this standard does not account for high speed effects such as roller centrifugal forces and gyroscopic moments: These effects tend to diminish fatigue life. Analytical evaluation of these effects frequently requires the use of high-speed digital computation devices and hence, cannot be included. * All references to “standard” are to AFBMA and American National Standard “Load Ratings and Fatigue Life for Roller Bearings” ANSI/AFBMA Std 11-1990.

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2489

Table 31. Values of fc for Radial Roller Bearings D cos α dm

D cos α dm

fc

Metrica

0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.11 0.12 0.13 0.14 0.15 0.16 0.17

Inchb

52.1 60.8 66.5 70.7 74.1 76.9 79.2 81.2 82.8 84.2 85.4 86.4 87.1 87.7 88.2 88.5 88.7

a For α =

4680 5460 5970 6350 6660 6910 7120 7290 7440 7570 7670 7760 7830 7880 7920 7950 7970

0.18 0.19 0.20 0.21 0.22 0.23 0.24 0.25 0.26 0.27 0.28 0.29 0.30 0.31 0.32 0.33 0.34

0°, Fa = 0 and X = 1.

b Use to obtain C in pounds when l

D cos α dm

fc

Metrica

Inchb

88.8 88.8 88.7 88.5 88.2 87.9 87.5 87.0 86.4 85.8 85.2 84.5 83.8 83.0 82.2 81.3 80.4

7980 7980 7970 7950 7920 7890 7850 7810 7760 7710 7650 7590 7520 7450 7380 7300 7230

0.35 0.36 0.37 0.38 0.39 0.40 0.41 0.42 0.43 0.44 0.45 0.46 0.47 0.48 0.49 0.50 …

fc

Metrica

Inchb

79.5 78.6 77.6 76.7 75.7 74.6 73.6 72.5 71.4 70.3 69.2 68.1 67.0 65.8 64.6 63.5 …

7140 7060 6970 6890 6800 6700 6610 6510 6420 6320 6220 6120 6010 5910 5810 5700 …

ef f and D are given in inches.

Table 32. Values of fc for Thrust Roller Bearings D cos α dm 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.12 0.14 0.16 0.18 0.20 0.22 0.24 0.26 0.28 0.30

45° < α < 60°

Metrica

Inchb

109.7 127.8 139.5 148.3 155.2 160.9 165.6 169.5 172.8 175.5 179.7 182.3 183.7 184.1 183.7 182.6 180.9 178.7 … …

9840 11460 12510 13300 13920 14430 14850 15200 15500 15740 16120 16350 16480 16510 16480 16380 16230 16030 … …

a Use to obtain C b Use to obtain C

60° < α < 75°

Metrica 107.1 124.7 136.2 144.7 151.5 157.0 161.6 165.5 168.7 171.4 175.4 177.9 179.3 179.7 179.3 … … … … …

fc

Inchb

9610 11180 12220 12980 13590 14080 14490 14840 15130 15370 15730 15960 16080 16120 16080 … … … … …

75° ≤ α < 90°

Metrica 105.6 123.0 134.3 142.8 149.4 154.9 159.4 163.2 166.4 169.0 173.0 175.5 … … … … … … … …

Inchb

9470 11030 12050 12810 13400 13890 14300 14640 14930 15160 15520 15740 … … … … … … … …

D dm 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.12 0.14 0.16 0.18 0.20 0.22 0.24 0.26 0.28 0.30

α = 90°

Metrica 105.4 122.9 134.5 143.4 150.7 156.9 162.4 167.2 171.7 175.7 183.0 189.4 195.1 200.3 205.0 209.4 213.5 217.3 220.9 224.3

fc

Inchb

9500 11000 12100 12800 13200 14100 14500 15100 15400 15900 16300 17000 17500 18000 18500 18800 19100 19600 19900 20100

a in newtons when l eff and D are given in mm.

a in pounds when l ef f and D are given in inches.

Roller Bearing Rating Life.—The Rating Life L10 of a group of apparently identical roller bearings is the life in millions of revolutions that 90 percent of the group will com­ plete or exceed. For a single bearing, L10 also refers to the life associated with 90 percent reliability.

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2490

Radial Roller Bearings: The magnitude of the Rating Life, L10, in millions of revolu­ tions, for a radial roller bearing application is given by the formula:

C 10 ⁄ 3 L10 = a P k

(12)

where C = the basic load rating in newtons (pounds), see Formula (13); and, P =  equiva­lent radial load in newtons (pounds), see Formula (14). For radial roller bearings, C is found by the formula:

C = fc ^ileff cos αh 7 ⁄ 9 Z 3 ⁄ 4D 29 ⁄ 27

(13)

where fc =  a factor which depends on the geometry of the bearing components, the accu­ racy to which the various bearing parts are made, and the material. Maximum values of fc are given in Table 31 i =  number of rows of rollers in the bearing lef f =   effective length, mm (inches)   α =  nominal contact angle, degrees Z =  number of rollers per row in a radial roller bearing D =  roller diameter, mm (inches) (mean diameter for a tapered roller, major dia­m­ eter for a spherical roller) When rollers are longer than 2.5D, a reduction in the fc value must be anticipated. In this case, the bearing manufacturer may be expected to establish load ratings accordingly. In applications where rollers operate directly on a shaft surface or a housing surface, such a surface must be equivalent in all respects to the raceway it replaces to achieve the basic load rating of the bearing. When calculating the basic load rating for a unit consisting of two or more similar single-row bearings mounted “in tandem,” properly manufactured and mounted for equal load distribution, the rating of the combination is the number of bearings to the 7 ⁄ 9 power times the rating of a single-row bearing. If, for some technical reason, the unit may be treated as a number of individually interchangeable single-row bearings, this consideration does not apply. The magnitude of the equivalent radial load, P, in newtons (pounds), for radial roller bearings, under combined constant radial and constant thrust loads is given by the formula:

P = XFr + YFa where Fr = Fa = X = Y =

(14)

 the applied radial load in newtons (pounds)  the applied axial load in newtons (pounds)  radial load factor as given in Table 33  axial load factor as given in Table 33 Table 33. Values of X and Y for Computing Equivalent Radial Load P for Radial Roller Bearing

Fa #e Fr

Bearing Type

X

Self-Aligning and Tapered Roller Bearingsa α ≠ 0°

1

a For α =

0°, Fa = 0 and X = 1. e = 1.5 tan α

1

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Fa 2e Fr

Y X Single Row Bearings 0 0.4 Double Row Bearingsa 0.67 0.45 cot α

Y 0.4 cot α 0.67 cot α

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Machinery's Handbook, 31st Edition Ball and Roller Bearings

2491

Typical Bearing Life for Various Design Applications Uses

Agricultural equipment Aircraft equipment Automotive Race car Light motorcycle Heavy motorcycle Light cars Heavy cars Light trucks Heavy trucks Buses Electrical Household appliances

Motors ≤ 1 ⁄2 hp Motors ≤ 3 hp Motors, medium Motors, large Elevator cables sheaves Mine ventilation fans Propeller thrust bearings Propeller shaft bearings Gear drives Boat gearing units Gear drives Ship gear drives Machinery for 8 hour service which are not always fully utilized Machinery for 8 hour service which are fully uti­lized Machinery for continuous 24 hour service

Design life in hours

Uses

3000–6000 500–2000

Gearing units Automotive Multipurpose Machine tools Rail Vehicles Heavy rolling mill Machines Beater mills Briquette presses Grinding spindles Machine tools Mining machinery Paper machines Rolling mills Small cold mills Large multipurpose mills Rail vehicle axle Mining cars Motor rail cars Open-pit mining cars Streetcars Passenger cars Freight cars Locomotive outer bearings Locomotive inner bearings Machinery for short or intermittent operation where service interruption is of minor importance Machinery for intermittent service where reliable operation is of great importance Instruments and apparatus in frequent use

500–800 600–1200 1000–2000 1000–2000 1500–2500 1500–2500 2000–2500 2000–5000 1000–2000 1000–2000 8000–10000 10000–15000 20000–30000 40000–60000 40000–50000 15000–25000 > 80000 3000–5000 > 50000 20000–30000 14000–20000 20000–30000 50000–60000

Design life in hours 600–5000 8000–15000 20000 15000–25000 > 50000 20000–30000 20000–30000 1000–2000 10000–30000 4000–15000 50000–80000 5000–6000 8000–10000 5000 16000–20000 20000–25000 20000–25000 26000 35000 20000–25000 30000–40000 4000–8000 8000–14000 0–500

Roller bearings are generally designed to achieve optimized contact; however, they usu­ ally support loads other than the loading at which optimized contact is maintained. The 10 ⁄ 3 exponent in Rating Life Formulas (12) and (15) was selected to yield satisfactory Rat­ing Life estimates for a broad spectrum from light to heavy loading. When loading exceeds that which develops optimized contact, e.g., loading greater than C/4 to C/2 or Ca /4 to Ca /2, the user should consult the bearing manufacturer to establish the adequacy of the Rating Life formulas for the particular application. Thrust Roller Bearings: The magnitude of the Rating Life, L10, in millions of revolutions for a thrust roller bearing application is given by the formula:

L10 = c

Ca 10 ⁄ 3 m Pa

(15)

where Ca =  basic load rating, newtons (pounds). See Formulas (16) to (18) Pa =  equivalent thrust load, newtons (pounds). See Formula (19) For single row, single and double direction, thrust roller bearings, the magnitude of the basic load rating, Ca , in newtons (pounds), is found by the formulas:

for α = 90°,

Ca = fc l eff Z

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D

7⁄9 3⁄4

29 ⁄ 27

(16)

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Machinery's Handbook, 31st Edition Ball and Roller Bearings

2492

for α ! 90 °,

7 ⁄ 9 3 ⁄ 4 29 ⁄ 27 Ca = fc ^l eff cos αh Z D tan α

(17)

where fc =  a factor which depends on the geometry of the bearing components, the accu­ racy to which the various parts are made, and the material. Values of fc are given in Table 32 lef f =  effective length, mm (inches) Z =  number of rollers in a single row, single direction, thrust roller bearing D =  roller diameter, mm (inches) (mean diameter for a tapered roller, major diam­ eter for a spherical roller) α =  nominal contact angle, degrees For thrust roller bearings with two or more rows of rollers carrying loads in the same direction the magnitude of Ca is found by the formula:

 Z 1 l eff1 C a = ( Z 1 l eff1 + Z 2 l eff2 + … + Z n l effn )  --------------- C a1 Z n l effn --------------C an

9⁄2

Z 2 l eff2 + ---------------C a2

9 ⁄ 2 – 2 ⁄ 9

9⁄2

+…+ (18)

 

where Z1, Z 2 , …, Z n =  the number of rollers in respective rows of a single direction, multirow bearing Ca1, Ca2 , …, Can =  the basic load rating per row of a single direction, multi-row, thrust roller bearing, each calculated as a single row bearing with Z1, Z2 , … , Zn rollers respec­tively lef f 1, lef f 2 , …, lef f n =  effective length, mm (inches), or rollers in the respective rows In applications where rollers operate directly on a surface supplied by the user, such a surface must be equivalent in all respects to the washer raceway it replaces to achieve the basic load rating of the bearing. In case the bearing is so designed that several rollers are located on a common axis, these rollers are considered as one roller of a length equal to the total effective length of contact of the several rollers. Rollers as defined above, or portions thereof which contact the same washer-raceway area, belong to one row. When the ratio of the individual roller effective length to the pitch diameter (at which this roller operates) is too large, a reduction of the fc value must be anticipated due to excessive slip in the roller-raceway contact. When calculating the basic load rating for a unit consisting of two or more similar single row bearings mounted “in tandem,” properly manufactured and mounted for equal load distribution, the rating of the combination is defined by Formula (18). If, for some techni­ cal reason, the unit may be treated as a number of individually interchangeable single-row bearings, this consideration does not apply. The magnitude of the equivalent thrust load, Pa , in pounds, for thrust roller bearings with α not equal to 90 degrees under combined constant thrust and constant radial loads is given by the formula:

Pa = XFr + YFa

where Fr = Fa = X = Y =

(19)

 applied radial load, newtons (pounds)  applied axial load, newtons (pounds)  radial load factor as given in Table 34  axial load factor as given in Table 34

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Machinery's Handbook, 31st Edition Ball and Roller Bearings

2493

Table 34. Values of X and Y for Computing Equivalent Thrust Load Pa for Thrust Roller Bearings Single Direction Bearings

Bearing Type Self-Aligning Tapered Thrust Roller Bearingsa α≠0 a

Double Direction Bearings

Fa 2e Fr

Fa #e Fr

X

Y

tan α

1

Fa 2e Fr

X

Y

X

Y

1.5 tan α

0.67

tan α

1

For α = 90°, Fr = 0 and Y = 1.

e = 1.5 tan α

Life Adjustment Factors.—In certain applications of ball or roller bearings it is desirable to specify life for a reliability other than 90 percent. In other cases the bearings may be fab­r icated from special bearing steels such as vacuum-degassed and vacuum-melted steels, and improved processing techniques. Finally, application conditions may indicate other than normal lubrication, load distribution, or temperature. For such conditions a series of life adjustment factors may be applied to the fatigue life formula. This is fully explained in AFBMA and American National Standard “Load Ratings and Fatigue Life for Ball Bear­ings”ANSI/AFBMA Std 9-1990 and AFBMA and American National Standard “Load Ratings and Fatigue Life for Roller Bearings”ANSI/AFBMA Std 11-1990. In addition to consulting these standards it may be advantageous to also obtain information from the bearing manufacturer. Life Adjustment Factor for Reliability: For certain applications, it is desirable to specify life for a reliability greater than 90 percent which is the basis of the Rating Life.

To determine the bearing life of ball or roller bearings for reliability greater than 90 percent, the Rating Life must be adjusted by a factor a1 such that L n = a1 L10. For a reliability of 95 percent, designated as L 5, the life adjustment factor a1 is 0.62; for 96 percent, L 4 , a1 is 0.53; for 97 percent, L3, a1 is 0.44; for 98 percent, L2 , a1 is 0.33; and for 99 percent, L1, a1 is 0.21. Life Adjustment Factor for Material: For certain types of ball or roller bearings which incorporate improved materials and processing, the Rating Life can be adjusted by a factor a2 such that L10′ = a2 L10. Factor a2 depends upon steel analysis, metallurgical processes, forming methods, heat treatment, and manufacturing methods in general. Ball and roller bearings fabricated from consumable vacuum remelted steels and certain other special analysis steels, have demonstrated extraordinarily long endurance. These steels are of exceptionally high quality, and bearings fabricated from these are usually considered spe­cial manufacture. Generally, a2 values for such steels can be obtained from the bearing manufacturer. However, all of the specified limitations and qualifications for the applica­ tion of the Rating Life formulas still apply. Life Adjustment Factor for Application Condition: Application conditions which affect ball or roller bearing life include: 1) lubrication; 2) load distribution (including effects of clearance, misalignment, housing and shaft stiffness, type of loading, and thermal gradi­ ents); and 3) temperature. Items 2 and 3 require special analytical and experimental techniques, therefore the user should consult the bearing manufacturer for evaluations and recommendations.

Operating conditions where the factor a3 might be less than 1 include: a) exceptionally low values of Nd m (rpm times pitch diameter, in mm); e.g., Nd m < 10,000; b) lubricant vis­ cosity at less than 70 SSU for ball bearings and 100 SSU for roller bearings at operating temperature; and c) excessively high operating temperatures.

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2494

Machinery's Handbook, 31st Edition Ball and Roller Bearings

When a3 is less than 1 it may not be assumed that the deficiency in lubrication can be overcome by using an improved steel. When this factor is applied, L10′ = a3 L10. In most ball and roller bearing applications, lubrication is required to separate the rolling surfaces, i.e., rollers and raceways, to reduce the retainer-roller and retainer-land friction and sometimes to act as a coolant to remove heat generated by the bearing. Factor Combinations: A fatigue life formula embodying the foregoing life adjustment factors is L10′ = a1 a2 a3 L10. Indiscriminate application of the life adjustment factors in this formula may lead to serious overestimation of bearing endurance, since fatigue life is only one criterion for bearing selection. Care must be exercised to select bearings which are of sufficient size for the application. Ball Bearing Static Load Rating.—For ball bearings suitably manufactured from hard­ ened alloy steels, the static radial load rating is that uniformly distributed static radial bear­ing load which produces a maximum contact stress of 4,000 megapascals (580,000 pounds per square inch). In the case of a single row, angular contact ball bearing, the static radial load rating refers to the radial component of that load which causes a purely radial dis­placement of the bearing rings in relation to each other. The static axial load rating is that uniformly distributed static centric axial load which produces a maximum contact stress of 4,000 megapascals (580,000 pounds per square inch). Radial and Angular Contact Groove Ball Bearings: The magnitude of the static load rat­ ing Co in newtons (pounds) for radial ball bearings is found by the formula:

Co = fo iZD2 cos α

(20)

where fo =  a factor for different kinds of ball bearings given in Table 35 i =  number of rows of balls in bearing Z =  number of balls per row D =  ball diameter, mm (inches) α =  nominal contact angle, degrees This formula applies to bearings with a cross sectional raceway groove radius not larger than 0.52D in radial and angular contact groove ball bearing inner rings and 0.53D in radial and angular contact groove ball bearing outer rings and self-aligning ball bearing inner rings. The load-carrying ability of a ball bearing is not necessarily increased by the use of a smaller groove radius but is reduced by the use of a larger radius than those indicated above. Radial or Angular Contact Ball Bearing Combinations: The basic static load rating for two similar single row radial or angular contact ball bearings mounted side by side on the same shaft such that they operate as a unit (duplex mounting) in “back-to-back” or “faceto-face” arrangement is two times the rating of one single row bearing. The basic static radial load rating for two or more single row radial or angular contact ball bearings mounted side by side on the same shaft such that they operate as a unit (duplex or stack mounting) in “tandem” arrangement, properly manufactured and mounted for equal load distribution, is the number of bearings times the rating of one single row bearing. Thrust Ball Bearings: The magnitude of the static load rating Coa for thrust ball bearings is found by the formula:

Coa = fo ZD 2 cos α where fo = Z = D = α=

(21)

 a factor given in Table 35  number of balls carrying the load in one direction  ball diameter, mm (inches)  nominal contact angle, degrees

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Machinery's Handbook, 31st Edition Ball and Roller Bearings

2495

This formula applies to thrust ball bearings with a cross sectional raceway radius not larger than 0.54D. The load-carrying ability of a bearing is not necessarily increased by use of a smaller radius, but is reduced by use of a larger radius. Roller Bearing Static Load Rating: For roller bearings suitably manufactured from hard­ened alloy steels, the static radial load rating is that uniformly distributed static radial bear­ing load which produces a maximum contact stress of 4,000 megapascals (580,000 pounds per square inch) acting at the center of contact of the most heavily loaded rolling element. The static axial load rating is that uniformly distributed static centric axial load which pro­duces a maximum contact stress of 4,000 megapascals (580,000 pounds per square inch) acting at the center of contact of each rolling element.

D cos α dm 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.11 0.12 0.13 0.14 0.15 0.16 0.17 0.18 0.19 0.20 0.21 0.22 0.23 0.24 0.25 0.26 0.27 0.28 0.29 0.30 0.31 0.32 0.33 0.34 0.35 0.36 0.37 0.38 0.39 0.40 0.41 0.42 0.43 0.44 0.45

Table 35. fo for Calculating Static Load Rating for Ball Bearings Radial and Angular Contact Groove Type Inchb Metrica 12.7 1850 13.0 1880 13.2 1920 13.5 1960 13.7 1990 14.0 2030 14.3 2070 14.5 2100 14.7 2140 14.5 2110 14.3 2080 14.1 2050 13.9 2020 13.6 1980 13.4 1950 13.2 1920 13.0 1890 12.7 1850 12.5 1820 12.3 1790 12.1 1760 11.9 1730 11.6 1690 11.4 1660 11.2 1630 11.0 1600 10.8 1570 10.6 1540 10.4 1510 10.3 1490 10.1 1460 9.9 1440 9.7 1410 9.5 1380 9.3 1350 9.1 1320 8.9 1290 8.7 1260 8.5 1240 8.3 1210 8.1 1180 8.0 1160 7.8 1130 7.6 1100 7.4 1080 7.2 1050

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Radial Self-Aligning Metrica Inchb 1.3 187 1.3 191 1.3 195 1.4 198 1.4 202 1.4 206 1.5 210 1.5 214 1.5 218 1.5 222 1.6 226 1.6 231 1.6 235 1.7 239 1.7 243 1.7 247 1.7 252 1.8 256 1.8 261 1.8 265 1.9 269 1.9 274 1.9 278 2.0 283 2.0 288 2.0 293 2.1 297 2.1 302 2.1 307 2.1 311 2.2 316 2.2 321 2.3 326 2.3 331 2.3 336 2.4 341 2.4 346 2.4 351 2.5 356 2.5 361 2.5 367 2.6 372 2.6 377 2.6 383 2.7 388 2.7 393

Metrica 51.9 52.6 51.7 50.9 50.2 49.6 48.9 48.3 47.6 46.9 46.4 45.9 45.5 44.7 44.0 43.3 42.6 41.9 41.2 40.4 39.7 39.0 38.3 37.5 37.0 36.4 35.8 35.0 34.4 33.7 33.2 32.7 32.0 31.2 30.5 30.0 29.5 28.8 28.0 27.2 26.8 26.2 25.7 25.1 24.6 24.0

Thrust

Inchb 7730 7620 7500 7380 7280 7190 7090 7000 6900 6800 6730 6660 6590 6480 6380 6280 6180 6070 5970 5860 5760 5650 5550 5440 5360 5280 5190 5080 4980 4890 4810 4740 4640 4530 4420 4350 4270 4170 4060 3950 3880 3800 3720 3640 3560 3480

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Machinery's Handbook, 31st Edition Ball and Roller Bearings

2496

Table 35. fo for Calculating Static Static Load Rating for Ball Table 35. (Continued) fo for Calculating Load Rating forBearings Ball Bearings

D cos α dm 0.46 0.47 0.48 0.49 0.50

Radial and Angular Contact Groove Type Inchb Metrica 7.1 1030 6.9 1000 6.7 977 6.6 952 6.4 927

Radial Self-Aligning a Metric Inchb 2.8 399 2.8 404 2.8 410 2.9 415 2.9 421

a Use to obtain C

Metrica 23.5 22.9 22.4 21.8 21.2

Thrust

Inchb 3400 3320 3240 3160 3080

o or Coa in newtons when D is given in mm. o or Coa in pounds when D is given in inches. Note: Based on modulus of elasticity = 2.07 × 105 megapascals (30 × 106 pounds per square inch)

b Use to obtain C

and Poisson’s ratio = 0.3.

Radial Roller Bearings: The magnitude of the static load rating Co in newtons (pounds) for radial roller bearings is found by the formulas: D cos α m Co = 44 c 1 − iZleff D cos α ( metric) (22a) dm

Co = 6430 c 1 −

D cos α m iZleff D cos α ( inch) dm

(22b)

where D =  roller diameter, mm (inches); mean diameter for a tapered roller and major diameter for a spherical roller d m =  mean pitch diameter of the roller complement, mm (inches) i =  number of rows of rollers in bearing Z =  number of rollers per row lef f =  effective length, mm (inches); overall roller length minus roller chamfers or minus grinding undercuts at the ring where contact is shortest α =  nominal contact angle, degrees Radial Roller Bearing Combinations: The static load rating for two similar single row roller bearings mounted side by side on the same shaft such that they operate as a unit is two times the rating of one single row bearing. The static radial load rating for two or more similar single row roller bearings mounted side by side on the same shaft such that they operate as a unit (duplex or stack mounting) in “tandem” arrangement, properly manufactured and mounted for equal load distribution, is the number of bearings times the rating of one single row bearing. Thrust Roller Bearings: The magnitude of the static load rating Coa in newtons (pounds) for thrust roller bearings is found by the formulas: D cos α m Coa = 220 c 1 − Zleff D sin α ( metric) (23a) dm

Coa = 32150 c 1 −

D cos α m Zleff D sin α ( inch) dm

(23b)

where the symbol definitions are the same as for Formulas (22a) and (22b). Thrust Roller Bearing Combination: The static axial load rating for two or more similar single direction thrust roller bearings mounted side by side on the same shaft such that they operate as a unit (duplex or stack mounting) in “tandem” arrangement, properly manufac­tured and mounted for equal load distribution, is the number of bearings times the rating of one single direction bearing. The accuracy of this formula decreases in the case of single direction bearings when Fr > 0.44 Fa cot α, where Fr is the applied radial load in newtons (pounds) and Fa is the applied axial load in newtons (pounds).

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Machinery's Handbook, 31st Edition Ball and Roller Bearings

2497

Ball Bearing Static Equivalent Load.—For ball bearings the static equivalent radial load is that calculated static radial load which produces a maximum contact stress equal in magnitude to the maximum contact stress in the actual condition of loading. The static equivalent axial load is that calculated static centric axial load which produces a maximum contact stress equal in magnitude to the maximum contact stress in the actual condition of loading. Radial and Angular Contact Ball Bearings: The magnitude of the static equivalent radial load Po in newtons (pounds) for radial and angular contact ball bearings under com­bined thrust and radial loads is the greater of:

where Xo = Yo = Fr = Fa =

Po = Xo Fr + Yo Fa

(24)

Po = Fr

(25)

 radial load factor given in Table 36  axial load factor given in Table 36  applied radial load, newtons (pounds)  applied axial load, newtons (pounds) Table 36. Values of Xo and Yo for Computing Static Equivalent Radial Load Po of Ball Bearings

Contact Angle α = 0° α = 15° α = 20° α = 25° α = 30° α = 35° α = 40° …

Single Row Bearingsa

Double Row Bearings

Yo b

Xo

0.6

0.5

0.6

0.5

0.5 0.5 0.5 0.5 0.5 0.5

0.47 0.42 0.38 0.33 0.29 0.26

1 1 1 1 1 1

0.94 0.84 0.76 0.66 0.58 0.52

0.5

0.22 cot α

1

0.44 cot α

Xo

RADIAL CONTACT GROOVE BEARINGSc,a ANGULAR CONTACT GROOVE BEARINGS

SELF-ALIGNING BEARINGS

Yo b

a P is always ≥ F . o r b Values of Y for intermediate contact angles are obtained by linear interpolation. o c Permissible maximum value of F /C (where F is applied axial load and C is static radial load a o a o

rating) depends on the bearing design (groove depth and internal clearance).

Thrust Ball Bearings: The magnitude of the static equivalent axial load Poa in newtons (pounds) for thrust ball bearings with contact angle α ≠ 90° under combined radial and thrust loads is found by the formula: Poa = Fa + 2.3Fr tan α (26) where the symbol definitions are the same as for Formulas (24) and (25). This formula is valid for all load directions in the case of double direction ball bearings. For single direc­tion ball bearings, it is valid where Fr /Fa ≤ 0.44 cot α and gives a satisfactory but less con­servative value of Poa for Fr /Fa up to 0.67 cot α. Thrust ball bearings with α = 90° can support axial loads only. The static equivalent load for this type of bearing is Poa = Fa . Roller Bearing Static Equivalent Load.—The static equivalent radial load for roller bearings is that calculated, static radial load which produces a maximum contact stress act­ing at the center of contact of a uniformly loaded rolling element equal in magnitude to the maximum contact stress in the actual condition of loading. The static equivalent axial

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Machinery's Handbook, 31st Edition Ball and Roller Bearings

2498

load is that calculated, static centric axial load which produces a maximum contact stress acting at the center of contact of a uniformly loaded rolling element equal in magnitude to the maximum contact stress in the actual condition of loading. Radial Roller Bearings: The magnitude of the static equivalent radial load Po in newtons (pounds) for radial roller bearings under combined radial and thrust loads is the greater of:

where Xo = Yo = Fr = Fa =

Po = Xo Fr + Yo Fa

(27)

Po = Fr

(28)

 radial factor given in Table 37  axial factor given in Table 37  applied radial load, newtons (pounds)  applied axial load, newtons (pounds)

Table 37. Values of Xo and Yo for Computing Static Equivalent Radial Load Po for Self-Aligning and Tapered Roller Bearings Bearing Type Self-Aligning and Tapered α≠0

Xo

Single Rowa

0.5

Yo

Xo

0.22 cot α

1

Double Row

Yo

0.44 cot α

a P is always ≥ F . o r

The static equivalent radial load for radial roller bearings with α = 0° and subjected to radial load only is Por = Fr. Note: The ability of radial roller bearings with α = 0° to support axial loads varies considerably with bearing design and execution. The bearing user should therefore consult the bearing manufac­ turer for recommendations regarding the evaluation of equivalent load in cases where bearings with α = 0° are subjected to axial load.

Radial Roller Bearing Combinations: When calculating the static equivalent radial load for two similar single row angular contact roller bearings mounted side by side on the same shaft such that they operate as a unit (duplex mounting) in “back-to-back” or “faceto-face” arrangement, use the Xo and Yo values for a double row bearing and the Fr and Fa val­ues for the total loads on the arrangement. When calculating the static equivalent radial load for two or more similar single row angular contact roller bearings mounted side by side on the same shaft such that they oper­ ate as a unit (duplex or stack mounting) in “tandem” arrangement, use the Xo and Yo values for a single row bearing and the Fr and Fa values for the total loads on the arrangement. Thrust Roller Bearings: The magnitude of the static equivalent axial load Poa in newtons (pounds) for thrust roller bearings with contact angle α ≠ 90°, under combined radial and thrust loads is found by the formula:

Poa = Fa + 2.3Fr tan α

(29)

where Fa =  applied axial load, newtons (pounds) Fr =  applied radial load, newtons (pounds) α =  nominal contact angle, degrees The accuracy of this formula decreases for single direction thrust roller bearings when Fr > 0.44 Fa cot α. Thrust Roller Bearing Combinations: When calculating the static equivalent axial load for two or more thrust roller bearings mounted side by side on the same shaft such that they operate as a unit (duplex or stack mounting) in “tandem” arrangement, use the Fr and Fa values for the total loads acting on the arrangement.

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Machinery's Handbook, 31st Edition Lubrication Theory

2499

LUBRICATION Lubrication Theory

Whenever a solid surface moves over another, it must overcome a resistive, opposing force known as solid friction. The first stage of solid friction, known as static friction, is the frictional resistance that must be overcome to initiate movement of a body at rest. The sec­ond stage of frictional resistance, known as kinetic friction, is the resistive force of a body in motion as it slides or rolls over another solid body. It is usually smaller in magnitude than static friction. Although friction varies according to applied load and solid surface roughness, it is unaffected by speed of motion and apparent contact surface area. When viewed under a microscope a solid surface will appear rough with many asperities (peaks and valleys). When two solid surfaces interact without a lubricating medium, full metal-to-metal contact takes place in which the asperity peaks of one solid interferes with asperity peaks of the other solid. When any movement is initiated the asperities collide causing a rapid increase in heat and the metal peaks to adhere and weld to one another. If the force of motion is great enough the peaks will plow through each other’s surface and the welded areas will shear causing surface degradation, or wear. In extreme cases, the resistance of the welded solid surfaces could be greater than the motive force causing mechanical seizure to take place. Some mechanical systems designs, such as brakes, are designed to take advantage of fric­tion. For other systems, such as bearings, this metal-to-metal contact state and level of wear is usually undesirable. To combat this level of solid friction, heat, wear, and con­ sumed power, a suitable lubricating fluid or fluid film must be introduced as an intermedi­ ary between the two solid surfaces. Although lubricants themselves are not frictionless, the molecular resistive force of a gas or fluid in motion known as fluid friction is significantly less than solid friction. The level of fluid friction is dependent on the lubricant’s Viscosity (see page 2504.) Film Thickness Ratio, λ.—For all bearings, the working lubricant film thickness has a direct relationship to bearing life. The “working” or specific film thickness ratio lambda λ is defined by dividing the nominal film thickness by the surface roughness, as depicted in Fig. 1. T λ= R (1) where λ =  specific film thickness T =  nominal film thickness R =  surface roughness Bearing Nominal Lubricant Film Thickness (T)

Lubricant

Shaft

Maximum Surface Roughness (R)

Fig. 1. Determining the Working Film Thickness Ratio Lambda

The Lubrication Film.—Whenever a plain journal style bearing operates with a fluid film, the coefficient of friction μ or extent of friction reduction will depend on which one of three lubricant film conditions exists between the facing surfaces.

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Machinery's Handbook, 31st Edition Lubricating Films

2500

Full Film Hydrodynamic Lubrication (HDL): HDL is the desired lubrication condition for plain style bearings in which both surfaces are fully separated by a working or specific film thickness lambda, λ of more than 2 at the point of pressure distribution loading shown in Fig. 2. A fluid wedge is created in which the asperities will not collide. Both surfaces are said to be “metal-contact” free at all times. Bearing

Journal (Shaft)

Ro

HD

LL

Pre ubr

ssu

ica

re Z

nt

tation

– –

one

We dg

Full Film Magnified Thickness Surfaces

e

Fig. 2. HDL Hydrodynamic Lubrication of a Journal Bearing

As the shaft speed accelerates, rotation of the journal acts as a pump, forcing lubricant into the pressure distribution area. Providing the lubricant is of a high enough viscosity, the wedge-shaped lubricant channel will create a load-carrying pressure sufficient to com­pletely separate the two surfaces and support the moving journal. Full film thickness will vary between 5 and 200 microns depending on speed, load, and viscosity. As the speed increases so does the lubricating action and ability to carry heavier loads. Inversely, slow speeds do not allow the lubricant wedge to form, causing breakdown of the hydrodynamic action and an undesirable Boundary Layer Lubrication state to prevail. Boundary Layer Lubrication (BL): When a journal shaft is at rest in the bearing, any full film lubricant wedge has collapsed leaving a residual film of lubricant in its place, insuffi­cient to prevent metal-to-metal contact from occurring. Upon subsequent start up, the bear­ing surfaces partially collide and ride on the thin lubricant film (start up conditions promote heavy wear). When lubricant supply is inadequate, or heavy loads coupled with low shaft speeds is the only design possible, the boundary layer lubrication must rely heav­ily on the composition of the lubricant to provide specific anti-wear and extreme pressure sacrificial additives, designed to retard premature wear. These surface-active additives act to form a thin surface laminate that prevents metal adhesion. Boundary layer lubrication also occurs when a lubricant of too low a viscosity is chosen. Mixed Film Lubrication (MF): Mixed film lubrication state is generally encountered under shock load conditions when a minimum thickness hydrodynamic film momentarily breaks down or “thins out” into a boundary layer condition under severe shock load. Mixed film condition is also encountered as a shaft accelerates toward full speed and the film thickness transforms from boundary to full hydrodynamic condition. Choosing too light a viscosity lubricant can lead to momentary or full time mixed film condition. Mixed film condition is encountered when the specific film thickness lambda λ is between 1 and 2. Lubricating Film Transition: Boundary layer condition is encountered when the specific film thickness lambda λ ratio is less than 1, mixed-film when the lambda λ ratio is between 1 and 2, and hydrodynamic when the lambda λ ratio is more than 2. Once the lambda λ ratio surpasses 4, relative bearing life is increased four-fold as depicted in Fig. 3. Lambda λ < 1 Boundary Layer Film Lubrication

1 < Lambda λ < 2 Mixed Film Lubrication

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2 < Lambda λ < 4 Hydrodynamic Lubrication

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Machinery's Handbook, 31st Edition Lubricating Films Boundary BL

Mixed MF

2501

Hydrodynamic HDL

Relative Bearing Life

4x 3x

2x

1x 0.4

0.6 0.8 1.0

2.0

3.0

4.0

5.0 6.0 7.0

Lambda λ (Working Film Thickness)

10

Fig. 3. The Relationship between Film Thickness Ratio Lambda λ and Bearing Life

To achieve long life while supporting heavy loads, a plain bearing must successfully manage the relationship between load, speed, and lubricant viscosity. If the load and speed change, the lubricant viscosity must be able to compensate for the change. This relation­ship is shown in the Stribeck or ZNP curve illustrated in Fig. 4. Choosing the correct lubri­cant viscosity allows the bearing to run in the favored hydrodynamic range producing a low coefficient of friction ranging from 0.002 to 0.005. Mixed MF

Hydrodynamic HDL

Coefficient of Friction, µ

Boundary BL

ZN P

Fig. 4. Stribeck (ZNP) Curve

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2502

Machinery's Handbook, 31st Edition Lubricating Films

Rolling element bearings (point contact) and the rolling section of mating gear teeth (line contact) also favor full hydrodynamic lubrication film. They differ from sliding elements in that rolling elements require considerably less lubrication than their sliding counterparts and that the load is concentrated over a much smaller footprint on a non-conforming sur­ face—small diameter ball or roller “rolling” over or within a much larger diameter raceway. As the ball or roller “rolls” through the load zone, the point of contact experiences a rapid pressure rise causing momentary micro distortion of both the rolling element and race. This area of deformation is named the Hertzian Contact Area (Fig. 5) and is analogous to the contact patch of a properly inflated tire on a moving vehicle. As the loaded section of rolling element moves out of the Hertzian contact area the deformed surface elastically returns to its original shape. The lubricant trapped in the Hertzian contact area benefits greatly from a phenomenon in which a lubricant under pressure will experience a dramatic rise in viscosity and act as a solid lubricant, allowing small amounts of lubricant to provide full film separation under extreme loading conditions. Under these conditions the hydro­dynamic film is termed elastohydrodynamic lubrication (EHDL), and is unique to point/line contact situations typically found with rolling element bearings and mating gear teeth.

A

R o t a ti o n

Load

Trapped EHDL Lubricant Film

Hertzian Contact Area (Deformation Area of Ball and Race) Enlarged Section A.

Fig. 5. Hertzian Contact Area Found in Rolling Element Bearing Surfaces

The wheels of industry run literally on a microfilm of lubricant; practical examples of typical oil film thicknesses expressed in machine dynamic clearance are stated in the fol­lowing Table 1. Table 1. Typical Oil Film Thicknesses Expressed as Machine Dynamic Clearances Machine Component

Typical Clearance in Microns

Plain Journal Bearings Rolling Element Bearings Gears Hydraulic Spool to Sleeve Engine Piston Ring to Cylinder Engine Rod Bearing (Plain) Engine Main Bearing (Plain) Pump Piston to Bore

0.5–100 0.1–3 0.1–1 1–4 0.3–7 0.5–20 0.5–80 5–40

1 micron = 0.00003937 inch; 25.4 microns = 0.001 inch.

Lubricants A lubricant’s primary function is to reduce friction; in doing so it reduces wear and energy consumption. Secondary functions are to reduce temperature, impact shock, corro­ sion, and contamination.

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Machinery's Handbook, 31st Edition Lubricants

2503

A lubricant can be in liquid (oil), solid (grease), or gaseous (oil mist) form and can be for­mulated from animal, vegetable, hydrocarbon, or synthetic base oil stocks. Adding to each lubricant formulation numerous chemical thickeners, solids, and chemical additives, gives every single manufactured lubricant its own unique signature blend. Selection of lubricant style and type is arguably the most influential factor in assuring long bearing life. In the 1970’s, Dr. Ernest Rabinowicz of MIT performed a landmark study on the effects of lubrication on the Gross National Product (GNP) of the United States. The study con­ cluded that at that time, US manufacturing companies spent over $600 billion US annually to repair damage caused by friction-induced mechanical wear; more importantly, the study determined that over 70 percent of bearing loss of usefulness (failure) is directly attributable to surface degradation, a totally preventable condition. In his study, Rabinowicz determined there are four major contributors to surface degradation: Corrosive Wear: All metallic-bearing surfaces will corrode if left unprotected from con­tact with water and corrosive acids. Water is introduced into lubricated environments from outside sources penetrating the sealed reservoir or bearing (washout, product contamina­tion), or through condensation, causing ferrous metals to rust. Corrosive acids are pro­duced when the lubricant becomes oxidized and suffers loss or breakdown of its corrosion inhibitor additive packages. Specifying and using a lubricant with rust inhibitors and cor­rosion-inhibitor additives, and replacing the lubricant in a timely manner when additives are depleted from the oil, will prevent corrosion. Mechanical Wear by Adhesion: Adhesive wear occurs when a lubricant film separating two sliding surfaces fails to completely separate the two surfaces. Metal to metal contact occurs causing metal fragment transfer from one surface to the other. This transfer is com­ monly referred to as seizing, galling, scuffing, or scoring of surfaces. Correct lubricant vis­cosity and application frequency will significantly reduce or eliminate adhesive wear. Mechanical Wear by Abrasion : Abrasive wear, sometimes referred to as cutting wear, is the result of hard particles (wear particles or introduced contaminant particles) bridging two moving surfaces, scraping and cutting either one surface (two body abrasion) or both bearing surfaces (three body abrasion). Controlling abrasive wear requires reduction of adhesive wear combined with contamination control of lubricant transfer, application, and filtration processes. Mechanical Wear by Fatigue: Fatigue wear results when bridged wear particles cause small surface stress raisers (surface rippling) that eventually expand and break away from the parent metal as a spall (flake or splinter). Repeated cyclic stress at the damaged area accelerates the fatigue wear process. Correct lubricant viscosity choice and contamination control are essential to retard fatigue wear. In all four types of wear, the primary solution for wear retardation lies in the correct choice of lubricant. Lubricants are categorized into two specific families—oil, and grease. The choice to use either oil or grease will depend upon temperature range, speed of rota­tion, environment, budget, machine design, bearing and seal design, which operating con­ditions. Lubricating Oil.—For the majority of industrial applications requiring the separation of moving surfaces, the lubricant of choice continues to be petroleum based oil, also widely known as mineral oil. Although any liquid will provide a degree of lubrication, hydrocar­bon-based petroleum oils provide excellent surface wetting capabilities, water resistance, thermal stability, and sufficient fluid viscosity or “stiffness” to provide full film protection under load—all at an inexpensive price. By adding chemicals, metals, solids, and fillers, mineral base oil stock can be formulated into an infinite number of tailored lubricating products, including grease. These modified mineral oils can be formulated for virtually any industrial application and widen the lubricant’s specification and capabilities even fur­ther. The fundamental defining property for all lubricating oils is viscosity, and is the start­ing point for choosing one specific lubricant over another.

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2504

Machinery's Handbook, 31st Edition Viscosity

t io

no

o fR

ta t io

Bearing

n

Di

r ec

Viscosity: The viscosity of a fluid is measured as its resistance to flow and shear; resis­ tance caused by fluid friction is set up along the molecular shear planes of the lubricant as depicted in Fig. 6. Thin or light lubricants, such as machine and spindle oils shear at a faster rate than thick lubricants such as gear oils, and are said to be less viscous. Although lower viscosity oil is desirable for reducing energy (less drag), it likely would not be “stiff” or viscous enough to withstand the demands of a heavily loaded gearbox.

Shaft

Viscous Shear Planes

Fig. 6. Viscosity Shear Planes

Kinematic Viscosity: Oil viscosity is measured by a variety of classifications. The two generally accepted industrial standards are: Saybolt Universal Seconds or SUS (imperial measure), and ISO VG-centisokes-cSt (metric measure). These two standards rate oil by their kinematic viscosity values. The ratings, based on a fluid temperature of 100 °F (40 °C) and 212 °F (100 °C), relate the time taken for a fluid to flow through a viscosimeter capil­lary apparatus and directly measure oil’s resistance to flow and shear by the forces of grav­ity. Other common viscosity classifications and comparison equivalents are shown in Table 2. g ⁄ cc - at 60 °F cSt = -------------(2) η ( cP ) where h =  absolute or dynamic viscosity in centipoise g/cc =   lubricant density (specific gravity) cSt =  kinematic viscosity in centistokes

To convert cSt to SUS at 100 °F (40 °C), multiply by 4.632 To convert cSt to SUS at 210 °F (100 °C), multiply by 4.664 Absolute Viscosity: The absolute or dynamic viscosity is measured in poise (metric) or centipoise (cP) and reyn (imperial), where one reyn is equivalent to 68,950 poise. Onepoise is equivalent to a one-dyne force required to move a plane surface (shear plane) of unit area a distance of one centimeter with unit speed (one centimeter per second) over a second plane at a unit distance (one centimeter) from it. Absolute viscosity is calculated by multiplying the kinematic viscosity value by the density of the lubricant measured at the test temperature, and is the measure of oil’s resistance to flow and shear caused by internal friction. Absolute viscosity is the viscosity measured through oil analysis.

η ( cP ) = g ⁄ cc at 60 °F × cSt

(3)

where h =  absolute or dynamic viscosity in centipoise g/cc =  lubricant density (specific gravity) cSt =  kinematic viscosity in centistokes

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Machinery's Handbook, 31st Edition Viscosity

2505

Table 2. Viscosity Comparison Chart Centistokes 40 C 100 C 2000 1500

SAE Grades Engine Gear

70 1500

60 1000 800

250

50

200

140

30

6000 1000

8A

680

8

460

7

150 100 80 60

14

50

50

8 7

40

6

90

40 85W

12 10 9

5000 3000 2500 2000

60 20 18 16

8000

9

400 300

SUS 100 F 210 F

4000

40

600 500

ISO VG AGMA Grades Grades

30

320

6

220

5

150

4

1500 1000 800 600

100

3

68

2

20

46

1

300 250 200

75W

30

5

10W

20

4

5W

15 3 10 Viscosities based on 95VI single grades relate horizontally. SAE grades specified at 100 °C. SAE W grades are also specified at low temperatures. ISO and AGMA Grades specified at 40 °C.

32 22 15

300 250 200 180 160 140 120 100 90 80 70

500 400

80W

350

160 140 120 100

60 55 50 45

40

80 70 60

35

SAE - Society of Automotive Engineers (Automotive lubricants) AGMA - American Gear Manufacturers Assn. (Gear lubricants) ISO - International Standards Organization SUS - Saybolt Universal Seconds

Viscosity Index (VI): Viscosity is dependent on temperature. As oil heats up it becomes thinner or less viscous. Inversely, as oil cools down it becomes thicker or more viscous. This phenomenon dictates that all oils will change their physical properties once they have achieved their working environment temperature. Therefore, before a lubricant viscosity choice can be made, its expected working environment temperature must be known. To engineer for this phenomenon, oil is given a Viscosity Index, or VI rating, which defines the measure of a lubricant’s viscosity change due to temperature change. Higher VI ratings are more desirable, reflecting narrower viscosity change over a standard temperature range. To determine a specific oil’s VI rating, its kinematic viscosity is measured

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2506

Machinery's Handbook, 31st Edition VISCOSITY

at 100 °F (40 °C) and 212 °F (100 °C), then its results are compared with two or more series of oils. VI values once ranged between 0 and 100, but recent developments in lubricant technology and additives have allowed this index to raise its upper limit and include a Very High Vis­cosity Index (VHVI) group. Lubricants are generally classified in four basic VI groups depicted in Table 3. VI rating < 35 35–80 80–110 > 110

Table 3. Viscosity Index Rating

Viscosity Index Group Low (LVI) Medium (MVI) High (HVI) Very High (VHVI)

Composition of Oil.—Oil is composed of either a mineral (hydrocarbon based) or syn­ thetic oil base stock to which is added a variety of organic and inorganic compounds that are dissolved or suspended as solids in the formulated oil. Depending on the end use condi­tion the oil formulation is designed for, the additive package can make up from 1% up to 30% of the formulated oil volume.

Mineral-Based Oil.—Mineral oils are refined from crude oil stocks. Depending on where the crude stock is found in the world, the oil can be paraffinic or napthenic based. Paraffinic based stocks are generally found in the midcontinental United States, England’s North Sea, and the Middle East. They contain a 60/30/10 mix of paraffin/napthene/wax resulting in a very high VI rating up to 105. Because wax is present, they are known to have a wax pour point in which the oil’s flow is severely constricted or stopped by wax crystallization at lower temperatures. This type of base oil stock is preferred when blending high quality crankcase oils, hydraulic fluids. turbine oils, gear oils, and bearing oils. Napthenic based oil stocks are generally found in South America and the coastal regions of the United States. They contain a 25/75/trace mix of paraffin/napthene/wax, resulting in a less stable VI rating up to 70. Because only a trace of wax is present they are known as viscosity pour point oils in which oil flow is restricted by increases in the lubricant’s viscosity at low temperatures. Napthenic oils have lower pour points, higher flash points, and better addi­tive solvency than paraffinic oils. This type of base stock is preferred when blending loco­motive oils, refrigerant, and compressor oils.

Oil Additives.—When contact is likely between two bearing surfaces the lubricant should be designed to mitigate the friction through the addition of engineered additives to the base oil. Every manufactured lubricant on the market has its own unique formulation. In effect, it is an engineered liquid, custom built to perform a specific a job in a specific environment. All additives are sacrificial and therefore careful attention to additive package levels through the use of oil analysis will tell the user exactly when to change the oil to prevent damage to the bearing or contact parts. Typically oil additives as shown in Table 4 are used to enhance the existing base oil, add additional properties to the oil, and suppress any unde­sirable properties the base oil may have. Enhancement Additives Anti-oxidant Corrosion inhibitor Demulsifier Anti-foam

Table 4. Oil Additives New Property Additives EP Anti-wear Detergent Dispersant

Suppressant Additives Pour point depressant viscosity improver

The additive package formulation will depend on the end use. Table 5 references what oil type generally carries what additive package in its formulation.

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Machinery's Handbook, 31st Edition Oil Additives

2507

Table 5. Additive Package by Oil Type Guide Additive Anti-oxidant

Corrosion Inhibitor Demulsifier Anti-foam

Bear­ing Oil

Gear Oil

Turbine Oil

d

d

d

d

d

d

d

d

d

d

d

d

d

d

d

d

d

d

d

d

d

d

d

d

d

d

d

d

d

d

Extreme Pressure EP Anti-wear

Hydraulic Compres­sor Crankcase Grease Oil Oil Oil

d d

d

d

Detergent

Dispersant Pour Point

Viscosity Improver

d some

d some

d

d

d some

d some

d

d

Anti-oxidants: Oxygen attacks the base oil, especially at higher temperatures, leading to the formation of sludge, tars, varnish, and corrosive acids. Anti-oxidant additives can improve the oxidation stability of the oil by more than 10 times; lubricants designed for higher operating temperatures will contain higher levels of antioxidants. Corrosion Inhibitor or Antirust Agents : Used to form a protective shield against water on ferrous metals, and copper, tin, and lead-based bearing metals. They also act to neutral­ize any corrosive acids that may attack the bearing materials. Demulsifying Agents: Stop water from emulsifying with the oil. Antifoaming Agents: When oil is moved quickly, these agents, usually silicon-based compounds, act to retard the formation of air bubbles at the lubricant’s surface; air bubbles contain oxygen that will attack the base oil and cause cavitation in pumps. Extreme Pressure (EP) Additives: Additives such as sulphur, phosphorous, and chlorine are employed to “soften” bearing surfaces, allowing them to break away as small asperities without adhesive “tearing” when metal-to-metal contact is unavoidable. These additives can be detrimental to yellow metal bearing material. Anti-wear Agents: Solids such as molybdenum disulphide (moly), graphite, and PTFE, are employed to assist as additional sliding agents when metal-to-metal contact occurs under heavy loads. See Table 6. Detergents are organic metallic soaps of barium, calcium, and magnesium, acting as chemical cleaners to keep surfaces free from deposits and neutralize harmful combustion acids. Dispersants work in conjunction with detergents to chemically suspend the dirt particles in the oil and allow them to be extracted by the lubrication system filters. Pour Point Depressants prevent the formation of wax crystals in paraffinic-based min­ eral oil at low temperatures allowing it to be more fluid at colder temperatures. Viscosity Improvers : Sometimes a base oil of inferior quality will require thickeners to assist in achieving the specified viscosity levels over a varied temperature range. Viscosity improvers are also used to prevent the oil from thinning at higher temperatures allowing the manufacturer to build multi-grade lubricants that operate over wider temperature ranges. Viscosity improvers use long chain organic molecules such as polymethacrylates and ethylene propylene copolymers to retard the viscosity shearing and improve an oil’s viscosity performance.

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Machinery's Handbook, 31st Edition Oil Additives

2508

Table 6. Properties of Common Lubricant Solids Additives Solid Additive

Color

Load Capability

Molybdenum Disulphide

GrayBlack

Thermal Stability

Average Particle Size

Moisture Sensitivity

> 100,000 (psi)

< 750° F

< 1–6 micron

Detrimental

GrayBlack

< 50,000 (psi)

< 1200° F

2–10 micron

Necessary

White

< 6,000 (psi)

< 500° F

< 1 micron

No Effect

Graphite Polytetrafluoroethylene PTFE

1 micron = 0.00003937 inch; 1 psi = 6.8947 kPa; Temp. in °C = (°F – 32)/1.8

Solids additives shown in Table 6, can be added to both mineral and synthetic base oil stocks. In certain high temperature and high-pressure conditions, solids can be mixed with a mineral spirits carrier and applied directly to the bearing surfaces as a dry solid lubricant. The volatile carrier flashes off with the heat and leaves a dry solid film on the bearing sur­face. Synthetic-Based Oils.—Originally developed to cope with extreme high temperature sit­ uations encountered in early jet engines, synthetic based oil differs from mineral based oil in that its base stock is man-made. Using a polymerization similar to that used in plastics manufacturing, synthetic based oils are scientifically designed with identifiable molecular structures, resulting in fluids with highly predictable properties. Synthetic lubricants deliver many advantages; their stability under severe high and low temperature operating conditions enables equipment to operate in extreme conditions with a high degree of reliability. Although there are many different synthetic base stocks, indus­try is primarily served by the following five common synthetic lubricant types. Poly-Alph-Olefins (PAOs): PAOs, Table 7, are often described as man-made mineral oils (synthesized hydrocarbons) and were amongst the first developed synthetic lubricants for popular use in automotive crankcase oils. They are formulated in a similar molecular struc­ture to that of pure paraffin through the synthesis of ethylene gas molecules into a polym­erized uniform structure. They have a wide range of uses that include crankcase oil, gear oil, compressor oil, and turbine oils. Table 7. Poly-Alph-Olefins (PAOs)

Positive Features Low pour point (down to –90 °F or –68 °C) High viscosity index, VI > 140 High viscosity range Good mineral oil compatibility Good seal compatibility Excellent corrosion stability

Negative Features

Cost (4–8 × mineral oil cost) Poor additive solubility Poor biodegradability

Poly-alkylene Glycols (PAGs): PAGs, Table 8, are also known as organic chemical Ucon fluids that possess excellent lubricity and a unique property that causes decomposed or oxidized products to volatilize (clean burn) or become soluble, resulting in no sludge, var­nish, or damaging particles to be formed at high temperatures. PAG’s are polymers of alkylene oxides and are used for compressor oils, hydraulic oils (water glygols), and severe duty gear oils.

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Machinery's Handbook, 31st Edition SYNTHETIC-BASED OILS

2509

Table 8. Polyalkylene Glycols (PAGs)

Positive Features Low pour point (to –60 °F or –51 °C) High viscosity index, VI > 150 High viscosity range Fair seal compatibility Excellent biodegradability Do not produce sludge or varnish

Negative Features

Cost (4–8 × mineral oil cost) Poor mineral oil compatibility Poor PAO and synthetic ester based   oil compatibility

Di-Basic Acid Esters (Di-Esters): Due to their high shear VI stability under extreme tem­ perature, di-esters, Table 9, have become very popular in the aerospace industry. Formu­ lated from the reaction between alcohol and acid-laden oxygen, di-esters originally saw primary use in jet engine oils, but are now used mainly in high temperature compressor oils. Table 9. Di-Basic Acid Esters (Di-esters)

Positive Features Low pour point (to –80 °F or –62 °C) High viscosity index, VI > 150 High viscosity range Good mineral oil compatibility Good additive solvency

Negative Features

Cost (4–8 × mineral oil cost) Poor hydrolytic stability Poor seal compatibility Fair mineral oil compatibility Poor corrosion stability

Polyol-Esters: Due to their increased thermal stability over di-esters, polyol-esters, Table 10, have now taken over as the preferred oils for gas turbines, jet engines, and 2-cycle oil applications. Table 10. Polyol-Esters

Positive Features Low pour point (to –95 °F or –71 °C) High viscosity index, VI >160 High viscosity range Good oxidation stability Good mineral oil compatibility Good anti-wear properties Good additive solvency

Negative Features

Cost (10–15 × mineral oil cost) Poor hydrolytic stability Poor seal compatibility Fair mineral oil compatibility Poor corrosion stability

Silicones: Silicone lubricants, Table 11, are semi-inorganic compounds formulated to provide the stability of inorganic products, yet retain the versatility of organic products. Although they have poor lubricity, silicone lubricants find favor in lightly loaded instru­ ment bearings and oils and situations requiring high temperature change and compatibility with plastics. Additives are added to the base stocks to enhance lubricant performance, just as with mineral oils. Table 11. Silicone

Positive Features Low pour point (to –95 °F or –71 °C) High viscosity index, VI > 250 Very high Viscosity range Very high flash point Good seal compatibility

Negative Features High Cost (30–100 × mineral oil cost) Poor lubricity Poor seal compatibility Poor mineral oil compatibility Poor biodegradability Poor additive solvency

Temperature Effects on Oil.—Changes in temperature effect an oil’s viscosity and its ability to maintain a load carrying hydrodynamic film as depicted in Fig. 7. With the exception of silicone-based fluids, which unfortunately have poor lubricating qualities, most oils suffer a dramatic drop in viscosity once the temperature surpasses 100 °F (38 °C).

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Machinery's Handbook, 31st Edition TEMPERATURE EFFECTS

2510 10,000

Absolute Viscosity − cP

100 20 Silico

10

ne

50

Est

er

30 20 15

M

ine

ral

10

Oi

l

5 4 3

-30 -20 -10 0 10 20 30 40 50 60 70 80 90 10 11 12 13

Temperature °C Fig. 7. Temperature Effect on Viscosity for Different Oils

Temperature affects not only the viscosity of the oil, it affects the condition and life expectancy of the oil as shown in Fig. 8. For every 17° F (10° C) increase in temperature, oxidation rates double and effective oil life is halved. Operating temperature is the leading indicator in determining oil change out frequencies

Oil Working Temperature

C

300 250 Synthetic Oil (Esters and Silicones)

200 180

160

150

130 100

100 75

50 0

1

10

100

1000

Mineral Oil (Anti-oxidant Additive)

10,000

Oil Life (Hours) Fig. 8. Expected Oil Life at Varying Operating Temperatures.

Oxidation is the leading cause of lubricant failure. Fig. 9 shows typical upper and lower working limits for various lubricating oils. Lubricating Grease.—In situations where the containment and continued application of lubricating oil is not practical, lubricating grease is widely used—most specifically in roll­ing element bearings requiring only periodic lubrication, and slow-speed, high-load boundary lubrication applications. Easier to retain than oil, grease offers lower lubricant losses and good sealing qualities. When utilized in an automatic delivery system, grease can provide full film lubrication. Grease is a blended mix of the lubricating oil (mineral or synthetic—usually di-ester or silicone based), oil additive package, and fatty acids mixed with metallic alkaline soap

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Machinery's Handbook, 31st Edition GREASE

2511

Temperature ˚F -100

-50

0 32 100

200

300

400

500

600

Mineral Poly-alph-olefin (PAO) Poly Alkylene Glycol (PAG) Di Ester Polyol Ester

Normal Use Extended Range Due to Additve Package.

Silicone -75

-50 -25 0

100

200

300

Temperature °C Fig. 9. Temperature Limit Guidelines for Oil.

to form the thickening agent. Varying the oil, additive package, and soap blend produces many unique types of grease formulated for a variety of operating conditions. Greases are classified according to their soap base as depicted in Table 12.

Grease works in a similar way to a sponge; as the temperature of the grease rises, the oil bleeds from the soap filler and performs the lubricating function to the balls, raceways, and sliding surfaces. Inversely, once the grease cools down, the oil is soaked back up into the soap filler, which essentially acts as a semi-fluid container for the lubricating oil. An important step in selecting the correct grease is determining if the base oil viscosity is suit­able for the application. For example, grease designed for heavily loaded, high tempera­ture applications will probably use a heavy viscosity oil base, whereas generalpurpose grease is more likely to use a medium viscosity oil base. Table 12. Grease Types and Their Properties Type Calcium (Lime Soap) Sodium (Soda Soap) Calcium Complex Lithium Aluminum Complex Lithium Complex Barium Bentonite (non-soap) Urea

Appearance Buttery Fibrous Stringy Buttery Stringy Buttery Fibrous Buttery Buttery

Pumpability Fair Fair Fair Excellent Good Excellent Very Good Good Good

Heat Resistance Fair Very Good Good Good Excellent Excellent Excellent Excellent Excellent

Temperature Range 230 °F (110 °C) 250 °F (120 °C) 350 °F (175 °C) 350 °F (175 °C) 350 °F (175 °C) 375 °F (190 °C) 380 °F (193 °C) 500 °F (260 °C) > 500 °F (260 °C)

Water Compatibility Resistance with other greases Excellent Excellent Poor Good Very Good Fair Excellent Excellent Excellent Poor Very Good Excellent Excellent Fair Good Poor Excellent Excellent

Grease properties may change according to the additive package used

At sustained high temperatures, grease will soften substantially and could leak or drop from the bearing unless rated specifically for high temperature applications. High tem­ peratures rapidly oxidize the lubricant causing the soap to harden; higher temperatures require more frequent application of grease. Lower temperatures can be just as detrimental because the grease “stiffens” considerably as temperatures near –20 °F (–30 °C). At this temperature the rolling elements no longer rotate and they drag across the raceway. Under heavier loads this effect causes “smearing” of the bearing surfaces leading to premature bearing failure. Grease temperature guidelines by type are shown in Fig. 10.

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Machinery's Handbook, 31st Edition GREASE

2512

Temperature ˚F -100

-50

0 32 100

200

Calcium

300

400

500

600

Complex

Sodium Lithium Aluminum Complex Bentonite -75

-50 -25 0

100

200

300

Temperature ˚C Fig. 10. Temperature Limit Guidelines for Grease

Grease Classification: The National Lubricating Grease Institute—NLGI, classifies grease according to a rating standard that measures the consistency of the grease. Using a penetrometer apparatus under laboratory conditions, a conical weight is dropped from a known height into the grease sample, and its depth of penetration is measured after a 5 sec­ond time period. The Table 13 rating chart shows that stiffer greases are rated with a higher NLGI code than more fluid grease with higher levels of penetration. Grease consistency largely depends on the type and amount of soap thickener blended in the grease and the oil viscosity — NOT the base oil viscosity alone. Rolling element bearings will use grease in the NLGI 1 to 3 range. Centralized grease lubricating systems favor 0 to 2 NLGI rated grease. Table 13. NLGI Grease Consistency Rating Chart

NLGI Rating 6 5 4 3 2 1 0 00 000

Description Brick Hard Very Stiff Stiff Medium Medium Soft Soft Very Soft Semi Fluid Semi Fluid

Penetration Range (0.1mm@77° F) 85–115 130–160 175–205 220–250 265–295 310–340 355–385 400–430 445–475

Grease Additives: As with oil, grease will also contain solids additives such as graphite, molybdenum disulphide, and PTFE for use in extreme pressure and heavy wear condi­tions. Lubricant Application Selecting a Suitable Lubricant.—Selecting a suitable lubricant will depend on a number of factors such as type of operation (full film, boundary layer), temperature, speed, load, working environment, and machine design. Machine maintenance requirements and main­tenance schedules are not always taken into account in the equipment engineering design process of the lubrication system(s). Careful assessment of the conditions and consultation with a lubricant manufacturer/provider must take place to determine the

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Machinery's Handbook, 31st Edition Selecting Lubricants

2513

optimal lubricant choice for each specific application. Table 14 offers general guidelines for lubricant choice when operating conditions are known, and Fig. 11 offers lubricant viscosity guideline choices based on bearing speed in rpm. Once the initial lubricant choice is made, its vis­cosity must be checked against the specific operating temperature to ensure that the lubri­cant is suitable for speed, load, and temperature conditions. Table 14. General Guidelines for Choosing a Preferred Lubricant Type

Condition Clearances Designed for Oil

Oil

Grease

Solid

d

d

d

d

Clearances Designed for Grease

High Speed, Low Load

d

Low Speed, High Load

d

Low Speed, Oscillating Load

d

High Temperature

Full Film Applications

d

d a

d

d

Boundary Layer Applications

d

d

Contaminated Working Environment

d

d

Product Cannot Tolerate Leaks

d

Closed Gearbox

d

d

Isolated Bearings

d

a Automated delivery system.

Crancase Viscosity

SAE 50

ISO 320 ISO 220

SAE 40

ISO 100

Full-film Lubrication Heavy Load 250

SAE 30 SAE 20

ISO 68 ISO 46

ISO VG Viscocity

Boundary and Mixed-film Lubrication Heavy Load

Grease

SAE 10 ISO 32 ISO 15 ISO 10

SAE 5 10

20

40 60 100

200

40 60

100

Journal Speed, N (rpm)

2000 6000 4000 10,000

Fig. 11. Lubricant Viscosity Selection Guide based on Bearing Speed in rpm

Lubricant additives deliver different working characteristics to the lubricant. Knowing and documenting a machine or system’s lubricant application requirements will facilitate a consolidation of lubricant requirements and assist in determining the optimal lubricant additive package. Table 15 reviews typical lubricated components, and assigns priority guideline ratings against a number of important lubricant functional attributes. This infor­mation is a starting point when working with the lubricant manufacturer to enable consol­idation of lubricant needs and choose lubricants with suitable additives.

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Machinery's Handbook, 31st Edition Selecting Lubricants

2514

Table 15. Priority Guideline Ratings of Lubricant Functional Attributes for Different Lubricated Components Lubricant Attribute Friction Reduction Boundary Lubrication Cooling Ability Temperature Range Corrosion Protection Seal Out Contaminants 0 = Low Priority, 3 = High Priority

Sliding Bearing 1 1 2 1 1 0

Rolling Bearing 2 2 2 2 2 2

Wire Rope, Chain, Open Gears 1 2 0 1 2 1

Closed Gears 2 3 3 2 1 0

Oil Application.—Oil lubrication can be broken down into two major categories: termi­ nating (total loss), and recirculating oil systems. Terminating Oil Systems: Terminating oil systems are semi-automated and automated systems that dispense oil at a known rate to the bearing(s) and do not recover the oil. This system can be generally observed in use for lubricating plain bearings, gibs, and slide ways found in small to medium-sized machine tools. Reservoir oil is replenished with new oil on an “as used” basis. Recirculating Oil Systems: Recirculating oil systems pump oil through the bearing(s) on a semi-continuous or continuous cycle, using the oil to cool the bearing as it lubricates. Depending on the system design the oil can be filtered prior to the pump suction inlet, on the pump discharge, and again on the gravity return to the oil storage reservoir. Recirculating systems are used on every kind of small to very large equipment as long as the oil can be contained; reservoirs retain their original charge of oil, which is changed on a condition or time basis. A simple method used by lubrication delivery system manufacturers for determining the bearing oil requirements under normal load and speed conditions uses a volumetric requirement over a specified time period (See Table 16), designated by:

V = A#R

(4)

where V =   oil volume in cubic centimeters, (cc) A =   bearing contact surface area, (cm2) R =   film thickness replenishment, (mm) Table 16. Lubrication Film Replenishment Rate Guidelines for Oil and Grease Lubricant Delivery Method Automatic Terminating Oil Automatic Recirculating Oil

R - Film Thickness 0.025 mm (0.001 inch) 0.025 mm (0.001 inch)

Time 1 hour 1 minute

Other closed system oiling methods exist: Gearbox splash systems employ a simple recirculative pickup/transfer of oil by a submerged gear tooth from an oil reservoir bath. Constant level oilers maintain a constant level of oil in a specially designed oil bath bearing housing. Using air over oil technology, oil can be misted and “rifled” into the bearing allowing very high speeds of over 20,000 rpm at light loads. When replenishing oil reservoirs always use new clean oil of the exact same specifica­ tion, from the same manufacturer. Mixing different oils of similar specification can cause additive packages to react with one another causing detriment to the bearings. If changing to a new oil specification or manufacturer consult with the new manufacturer for the cor­ rect change-out procedure. Grease Application.—Because grease is easy to retain in a bearing housing over a long period of time and because it acts as a seal against contaminants, most rolling element bear­ings are grease lubricated. For most applications a NLGI 1 or 2 rating grease is used.

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Machinery's Handbook, 31st Edition GREASE Application

2515

The method of grease lubrication will depend on the greased bearing design; bearings can be hand-packed, manually-lubricated with a terminating style grease gun, or automatically greased. Open rolling bearings are received with a rust inhibiting compound from new and must be pre-packed on assembly—DO NOT remove bearings from their packaging until ready to use, and DO NOT spin dry bearings as this will significantly degrade the life of the bearing. Shielded or sealed bearings usually arrive pre-packed from the manufac­turer—always specify your preferred grease to your bearing supplier when ordering. The initial amount of grease is determined by adjusting the volume according to the known speed and load. For operating temperatures above 180°F (80°C) the bearing is packed to 25 percent of the full pack volume. For temperatures below 180°F (80°C), the guideline for pack volume is shown in Table 17 and is based on the bearing surface speed in operation calcu­lated as:

dn or Dn = SD # RPM where dn = Dn = SD = RPM =

(5)

 bearing surface speed factor, metric, mm  bearing surface speed factor, US customary, in.  shaft diameter of the bearing bore, mm or in.  velocity, rotations per minute at full speed

From 0 50,000 100,000 150,000 200,000+

dn (mm)

Table 17. Bearing Packing Guidelines To 50,000 100,000 150,000 200,000

From 0 2,000 4,000 6,000 8,000+

Dn (in.)

To 2,000 4,000 6,000 8,000

% Full Pack

100 75 50 33 25

For vibration applications, do not fill more than 60% of full pack

When hand packing, work the grease with fingers around all the rolling elements; the bearing can be dismantled to make this operation easier. The grease should fill the immediate bearing area. Before renewing grease in an existing bearing, the bearing must be removed and washed in kerosene or any suitable degreasing product. Once clean, the bearing is lightly coated in mineral oil, being careful not to spin the bearing at this point. Once filled with the appropriate amount of grease in the bearing area, the bearing can be hand spun to fling off grease excess, which is wiped away with a lint-free clean cloth. Free spacing in the housing should be filled from 30 to 50 percent. Overfilling bearings with grease is the leading cause of bearing lubrication-related failures. Over greasing causes the lubri­cant to “churn,” which in turn “spikes” the bearing internal temperature, significantly reducing the bearing life using considerably more energy to overcome fluid friction. Bear­ings designed to be lubricated by a grease handgun or automated delivery system will have a grease port built into the bearing housing and raceway. Grease lubrication intervals will depend on temperature and speed. Fig. 12 provides guidelines for renewing grease based on speed. Bearings running at high temperature extremes will require more frequent application based on the temperature, load, speed, and type of grease used. When replenishing grease always use new clean grease of the exact same specification, from the same manufacturer. Mixing different greases can lead to compatibility problems causing detriment to the bear­ings. If changing to a new grease specification or manufacturer, consult with the new man­ufacturer for the correct change out procedure. Lubrication Delivery Methods and Systems.—Numerous methods and systems are used to deliver oil and grease to bearing points. Automated centralized systems work on the premise of delivering a very small amount of lubrication on an almost continual basis,

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Machinery's Handbook, 31st Edition Lubrication Delivery Methods

2516

20,000 For bearings at 150 F. Half periods for every 25 F increase in temperature. Where grease nipples are provided, lubricate at each, one-fourth of renewal period. Open and Repack at full period.

10,000 900 8000 7000 6000

Note: One Working Year. 8 Hours/Day A) 2400 Hrs. 6 Day Week, 11 Holidays B) 2200 Hrs. 5.5 Day Week, 11 Holidays C) 2000 Hrs. 5 Day Week, 11 Holidays

Grease Renewal Period (Hours)

5000 4000 3000 2000

Ball Bearings

1000 900 800 700 600

Plain Roller Bearings

500 400 300

Taper and Spherical

200

100 0

40,000 80,000 120,00 160,00 180,00 200,00 240,00 280,00

Speed Factor dn, mm Fig. 12. Grease Renewal Period based on Running Time and Speed

providing optimal full film lubrication to the bearing. Although more expensive initially, automated centralized systems are credited with significant savings by extending bearing life up to three times longer than manually lubricated bearings. They also reduce downtime in changing out bearings, reduce lubricant consumption, and reduce energy consumption. Table 18 compares the different types of methods and delivery systems and some of their features, and can be used as a guide in determining a suitable lubrication delivery approach. Table 18. Lubrication Method and System Comparison Guide

Feature Oil Grease Continuous Delivery Cyclic Delivery Automatic Control Manual Control Positive Displacement Line Monitor Protection # Lube Points

Hand Pack d

Manual Gun d d

Single Point d d d

d d

Unlimited

Centralized Recirculating d

Self Contained Splash/Bath d

Gravity Fed d

d

d

d

d

d d d

d

d

d

d

d d

Centralized Total Loss d d

Unlimited

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d

d

20 Min.

200 Max.

d d

20 Max.

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Machinery's Handbook, 31st Edition Lubrication Delivery Methods

2517

Manual gun delivery systems are commonly known as grease guns and oil guns. These hand-dispensing devices are capable of delivering lubricant at pressures exceeding 15,000 psi (103 MPa), and must be used with extreme caution if the bearing seal is not to be com­promised—especially when lubricating from a remote located grease nipple. Bearings manually lubricated with grease and oil guns are lubricated with significantly more lubri­cant and less frequent applications than automatic centralized lubricated bearings. Manual lubrication results in a high degree of bearing fluid friction and a significant lower life expectancy.

Single point lubricators are self-contained automatic dispensing units that house a lubri­cant reservoir and can dispense oil or grease to a single bearing or a small number of bear­ings through a manifold system. Earlier versions of the grease units employed a spring-loaded follower plate that dispensed against a bearing back pressure through a controllable bleed valve; while oil units used gravity (also known as gravity units) to allow oil to drip through a bleed valve at a controlled rate onto a brush or wick device touching the moving shaft or part. Both unit types are refillable and are still available. Modern-day versions are mostly one-time-use units that employ programmable controlled battery operated positive displacement pumps, or electrochemical gas expandable bellows to move the lubricant to the bearing.

Centralized total loss systems employ a pump that can be automatically or manually acti­vated to pump oil (solid or mist) or grease to a series of metering valves mounted at the lubrication point, or in a manifold device piped to the bearing point. These systems are capable of delivering a metered amount of lubricant on a cyclic basis to many hundreds of lubricant points simultaneously. Because the lubricant is not reclaimed at the bearing point, the pump reservoir must be filled with lubricant on a regular basis. This lubrication system is the most common type of found on industrial equipment.

Centralized oil recirculating systems are designed to continually pump a metered amount of oil through each bearing point. The oil is channeled back to the reservoir through a filter system and pumped out again through the distribution system.

Self contained bath and splash installations are “pick-up” type systems that employ oil in a reservoir filled to an engineered level that covers the lowest submersed gear teeth. As the gear moves it picks up the oil and transfers lubricant as each gear engages and disengages. Higher rpm speed causes the lubricant to be splashed high into the gearbox cavity so that is is distributed to all the internal devices. Contamination Control

Before an oil lubricant gets to perform its lubrication function at the bearing point, it must often go through a torturous handling process where the oil must be transferred multiple times before it eventually resides in the final application reservoir. The lubricant is shipped from the refinery to the blending station, to the manufacturer’s bulk storage tank, to the supplier’s storage tank, to the barrel or pail, to the user’s storage facility, to the mainte­nance department, and finally to the machine’s reservoir. If the transfer equipment and storage tanks/devices are not dedicated to this exact lubricant type and scrupulously clean, and the oil is not filtered at every transfer point, the virgin oil will be contaminated when placed in the equipment reservoir. In a study performed by the National Research Council of Canada on bearing failure in primary industries it was found that 82 percent of wear-induced failure was particleinduced fail­ure from dirty lubricants, with the greatest wear caused by particles whose size equaled the oil film thickness. Perhaps the greatest contamination enemy for bearings is the ever pres­ent silt and its abrasive properties. Fig. 13 shows the Macpherson curve, which depicts the contaminant effect on roller bearing life based on contaminant micron size.

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Machinery's Handbook, 31st Edition Lubricant Contamination

2518

Millions of Cycles to Fatigue Failure

14 Points shown at 4.6 and 14 microns represent ISO cleanleaness Code 4406 (1999 sampling rates)

12 10 8 6 4 2 0

4

6

10

14

20

30

40

50

Contamination Size in Microns Fig. 13. Macpherson Contamination Effect Curve

The graph in Fig. 13 clearly shows the relationship between bearing life extension and contaminant size. By focusing on controlling contaminates less than 10 microns in size with quality filtration methods, expected bearing life is more than tripled. ISO Cleanliness Code.—When performing a solids lubricant analysis and cleanliness testing, the ISO Cleanliness Code ISO4406 (1999) is used as a guide. The number of 4-micron, 6-micron, and 14-micron diameter particles in a 1-ml lubricant sample are counted and compared to a particle concentration range (see Table 19), then assigned a cleanliness code number for each particle count size. Table 19. ISO Cleanliness Code 4406 (1999)

More Than 80,000 40,000 20,000 10,000 5,000 2,500 1,300 640 320 160 80 40

Particles per ml Up to and Including 160,000 80,000 40,000 20,000 10,000 5,000 2,600 1,280 640 320 160 80

Range Num­ber (R) 24 23 22 21 20 19 18 17 16 15 14 13

More Than 20 10 5 2.5 1.3 0.64 0.32 0.16 0.08 0.04 0.02 0.01

Particles per ml Up to and Including 40 20 10 5 2.6 1,28 0.64 0.32 0.16 0.08 0.04 0.02

Range Num­ber (R) 12 11 10 9 8 7 6 5 4 3 2 1

Example: An ISO code of 21/19/17 would represent findings of between 10,000 to 20,000 4-micron sized particles per ml, between 2,500 and 5,000 6-micron sized particles per ml, and between 640 and 1,280 14-micron sized particles per ml; this sample would be considered very dirty. Typical cleanliness targets for rolling element bearings would start at 16/14/12 or better, 17/15/12 or better for journal style bearings, 17/14/12 or better for industrial gearboxes, and 15/12 or better for hydraulic fluids.

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Machinery's Handbook, 31st Edition Lubricant Contamination

2519

Cleanliness Levels

A study conducted by the British Hydromechanics Research Association (BHRA) looked at the relationship between hydraulic fluid cleanliness and mean time between fail­ure (MTBF) of over 100 hydraulic systems in a variety of industries over a three year period. The results are seen in Fig. 14 and show that systems that were successful in filter­ ing out and excluding contaminants over 5 microns in size lasted tens of thousand of hours longer between system breakdowns.

< 5 Microns > 5 Microns > 15 Microns

100

10,000

1000

50,000

100,000

Mean Time Between Failures (MTBF) Fig. 14. MTBF vs. Cleanliness Levels

Percentage (%) Bearing Life Remaining

Solid particle ingression into a closed lubrication/hydraulic system can come from a variety of sources that include new oil, service and manufacturing debris, improper seals, vents/breathers, filter breakdown, and internal wear generation. For the most part, ingress prevention is all about filtration. Introducing filtered clean new oil into a system will sig­ nificantly retard the wear process and avoid clogging up breathers and in-line filter sys­tems. Water Contamination .—Water contamination is the other major lubricant contaminant that will significantly degrade the oil’s life (Fig. 15). Lubrication fluid typically saturates at 0.04% or 400 ppm, whereas hydraulic fluid (excluding water glycol fluids) saturates at an even lower 0.03% or 300 ppm. Typical water sources are found in the fluid storage areas when lubricants are stored outdoors and subjected to the elements, or stored in continually changing temperatures causing condensation and rust that can be transferred into the equipment’s lubrication system. 0.0025% = 25 parts per million 0.50 % = 5000 ppm

250 200 150 100 50 0 0.002 0.0

0.0

0.1

0.1

0.2

0.5

Percentage (%) of Water in Oil

Fig. 15. Effect of Water in Oil on Bearing Life

Copyright 2020, Industrial Press, Inc.

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Machinery's Handbook, 31st Edition Lubricant Filtration

2520

Filtration Systems.—Although contamination cannot be completely eliminated, with diligence and the use of effective filtration techniques and methods the effects of contami­ nation can be seriously mitigated.

Working on the understanding that fluid cleanliness is the basis for contamination con­ trol, primary filtration commences on the virgin stock oils prior to the lubricant being placed in the working reservoir or lubrication / hydraulic system.

Once the lubricant is in a working system it will immediately begin to attract contami­ nants already in the system, from the air, outside sources, manufacturing materials, and wear materials that must be filtered out as the lubricant moves through the system. Table 20 shows a typical pressure flow lubrication delivery, hydraulic system complete with a minimum filter media package. The function of these filters is to reduce operating costs and increase component life; therefore they must be properly sized for the system and be of the highest quality.

There are two basic types of filter design, surface filters and depth filters. Surface filters are the most common and use a screening material to trap debris. Depth filters are deep cleaning filtering devices that use multiple layers of dense materials to “polish” the lubri­ cant. Depth filters are usually set up in parallel with the basic filter system and allow a small percentage of the lubricant flow to bypass the pump and be depth cleaned. Table 20. Typical Minimum Filtration Requirements for a Closed Loop Lubrication Hydraulic System Delivery Section

Return Section

Pressured Gravity Lubrication Delivery System

B

H Pump

D

E

C Reservoir A F

G

B A Portable Filter Cart

Location

Type

Degree

Material

Purpose

Pump Suction

Surface

Medium

Gauze, Paper

Pump Protection

Surface

Fine

Return Line

Surface

Medium

D

Reservoir Vent

Surface

E

Reservoir Fill Port

Surface

F

Drain Plug

Surface

A B C

G H

Pump Delivery Header

Drain Valve

Delivery Bypass Filter

Felt, Paper, Cellulose, Sintered Metal

Primary System Protection

Course / Medium

Wire, Wool, Paper, Oil Bath Desiccant

Remove Airborne Contaminant and Condensation

Fine

Magnet

Capture Ferrite Wear Metals And Debris

Course

n/a

n/a

Depth

Very Fine

Copyright 2020, Industrial Press, Inc.

Felt, Paper, Cellulose Gauze, Paper n/a

Carbon, Cellulose, Diama­taceous Earth, Felt

Wear Products Protection

Prevent Course Solids Ingress Water Removal

Lubricant Deep Cleaning And Polishing

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Machinery's Handbook, 31st Edition Lubricant Filtration

2521

A diagram and tabulation of the filtration requirements of a closed loop hydraulic sys­ tem are shown in Table 20. Fluid in the reservoir is sucked up by the pump through the suc­tion filter (A) and pumped into the delivery header line under pressure. If a depth filter option is used, a small percentage, up to 15 percent, of the oil flow is diverted for deep cleaning through a depth filter (H) and sent back to the reservoir for recycling. The lubricant is then forced through the primary pressure filter and allowed to perform its work at the bearing point or hydraulic device before it eventually channels into the system return line under gravity to pass through a low pressure return line filter that takes out any wear materials gathered along the way. Once through the return filter the oil makes its way back into the reservoir. The reservoir is protected against airborne contaminants and condensation by the vent filter, and is protected against ingress of course solids by the fill neck screen filter. Because water is heavier than oil it will settle to the bottom of the tank where most of it can be drained off by opening the drain valve. Metallic debris also settle to the bottom and are captured by the magnetic drain plug at the bottom of the reservoir. As the lubricant oxi­dizes and breaks down, sludge will form on the bottom of the reservoir, which must be cleaned out periodically manually by removing the reservoir clean-out hatch. Filter Efficiency: Most surface filters are sold in either one-time-use, or cleanable-reus­ able forms. Depth filters are all one-time-use filters. All filters are performance rated according to the media’s particle removal efficiency, known as the filter’s filtration ratio, or beta ratio. Not all filters are made equal, and they are tested for dirt holding capacity, pressure differential capability, and filter efficiency, using an ISO 4572 Multipass Test Procedure in which fluid is circulated through a mock lube system in a controlled manner. Differential pressure across the test filter element is recorded as contamination is added into the lubricant upstream of the filter. Laser particle sensors determine contamination levels both upstream and downstream of the filter element and the beta ratio is determined using the following formula: # Upstream Particulate Bx = # Downstream Particulate (6) where B =  filter filtration ratio

x =  specific particle size

Example: If 100,000 particles, 10 microns and larger, are counted upstream of the test fil­ter, and 1000 particles are counted after or downstream of the test filter element, the beta ratio would equal: 100, 000 B10 = 1000 Efficiency is determined using the following equation: 1  × 100 Efficiency x =  1 – ---- B x (7) 1   Efficiency 10 = 1 – --------- × 100 = 99%  100 The higher the beta ratio, the better the capture efficiency of the filter, see Table 21. Beta Ratio at a Specific Particle Size 1.01 1.1 1.5 2

Filter Efficiency at Same Specific Particle Size 1% 9% 33% 50%

Table 21. Filter Efficiency Beta Ratio at a Specific Particle Size 5 10 20 75

Copyright 2020, Industrial Press, Inc.

Filter Efficiency at Same Specific Particle Size 80% 90% 95% 98.7%

Beta Ratio at a Specific Particle Size 100 200 1000 …

Filter Efficiency at Same Specific Particle Size 99% 99.5% 99.9% …

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Machinery's Handbook, 31st Edition COUPLINGS and Clutches

2522

COUPLINGS, CLUTCHES, BRAKES Connecting Shafts.—For couplings to transmit up to about 150 horsepower, simple flange-type couplings of appropriate size, as shown in the table, are commonly used. The design shown is known as a safety flange coupling because the bolt heads and nuts are shrouded by the flange, but such couplings today are normally shielded by a sheet metal or other cover. Safety Flange Couplings E

K

E F

D

H

C

G

A

J B A 1

B 13 ⁄4

C 21 ⁄4

11 ∕4

23∕16

13∕4 2

31 ∕16

33∕8 4

21 ∕4

315∕16

51 ∕8

23∕4 3

413∕16

11 ∕2

21 ∕2

25∕8

31 ∕2

43∕8

51 ∕4

31 ∕4

511 ∕16

33∕4 4

69∕16 7

31 ∕2

41 ∕2 5 51 ∕2 6 61 ∕2 7 71 ∕2 8 81 ∕2 9

91 ∕2 10 101 ∕2 11 111 ∕2 12

61 ∕8

23∕4

D 4 5 6

7

41 ∕2

8

55∕8

10

63∕4

12

61 ∕4 73∕8 8

81 ∕2 9

9

11

121 ∕4 131 ∕8 14

145∕8

153∕4

147∕8

167∕8 18

165∕8

201 ∕4

153∕4 171 ∕2

183∕8

191 ∕4

201 ∕8 21

20 22 24 26 28 28

30

31

213∕8

34

235∕8

243∕4 257∕8

11 ∕16

13∕16

1 ∕2 9 ∕16

15∕16

5 ∕8

19∕16

3 ∕4

17∕16

11 ∕16

21 ∕4

191 ∕8

221 ∕2

7∕16

16

15

20

131 ∕2

15 ∕16

15 ∕16

111 ∕4

113∕8

3 ∕8

115∕16

83∕4

123∕8

13 ∕16

14

13

18

101 ∕2

F 5 ⁄16

13 ∕16

101 ∕4 111 ∕4

E 11 ⁄16

111 ∕16

77∕8

83∕4

B

32 35 36 37 38

113∕16

7∕8

21 ∕16

1

21 ∕2

11 ∕4

23∕4 23∕4

215∕16 31 ∕8 31 ∕4

41 ∕4

47∕16 45∕8

413∕16

9 ∕3 2

21 ∕4

1 ∕4

9 ∕3 2

1 ∕4

1 ∕4

9 ∕3 2

1 ∕4

1 ∕4

9 ∕3 2

1 ∕4

1 ∕4

9 ∕3 2

5 ∕16

1 ∕4

9 ∕3 2

5 ∕16

41 ∕8

1 ∕4

9 ∕3 2

5 ∕16

1 ∕4

9 ∕3 2

5 ∕16

47∕8

1 ∕4

9 ∕3 2

3 ∕8

1 ∕4

9 ∕3 2

3 ∕8

1 ∕4

9 ∕3 2

3 ∕8

17∕8

25∕8 3

33∕8 33∕4

41 ∕2 51 ∕4

13∕8

1 ∕4

1 ∕4

9 ∕3 2

3 ∕8

9 ∕3 2

7∕16

71 ∕2

1 ∕4

9 ∕3 2

7∕16

1 ∕4

9 ∕3 2

7∕16

81 ∕4 9

1 ∕4

9 ∕3 2

7∕16

5 ∕16

11 ∕3 2

1 ∕2

5 ∕16

11 ∕3 2

1 ∕2

5 ∕16

11 ∕3 2

9 ∕16

107∕8

5 ∕16

11 ∕3 2

9 ∕16

5 ∕16

11 ∕3 2

5 ∕8

11 ∕3 2

5 ∕8

23∕8

115∕8 12

5 ∕16 5 ∕16

11 ∕3 2

11 ∕16

11 ∕3 2

11 ∕16

25∕8

123∕4

5 ∕16

131 ∕8

5 ∕16

11 ∕3 2

3 ∕4

131 ∕2

5 ∕16

11 ∕3 2

3 ∕4

5 ∕16

11 ∕3 2

7∕8

5 ∕16

11 ∕3 2

7∕8

5 ∕16

11 ∕3 2

13∕8

11 ∕2 15∕8

13∕4

21 ∕4

41 ∕8

K

1 ∕4

33∕4

315∕16

J

1 ∕4

55∕8 6

31 ∕2

311 ∕16

H

11 ∕2

11 ∕8

17∕8 2

37∕16

G

21 ∕8

21 ∕2 23∕4 27∕8 3

Copyright 2020, Industrial Press, Inc.

63∕4

71 ∕2

93∕4

101 ∕2 111 ∕4

137∕8

141 ∕4

1

No. 5 5 5 5 5 5 5 5 5 5

Bolts

Dia. 3 ∕8

7∕16 1 ∕2 9 ∕16 5 ∕8 11 ∕16 3 ∕4 13 ∕16 7∕8 15 ∕16

5

1

5

11 ∕8

5 5 5 5 5 5 6 6 7 7 8 8 8

10 10 10

10

11 ∕16 11 ∕4 13∕8 13∕8

17∕16 11 ∕2 11 ∕2

19∕16 11 ∕2

19∕16 11 ∕2

19∕16 15∕8 15∕8

111 ∕16 13∕4

113∕16

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Machinery's Handbook, 31st Edition COUPLINGS and Clutches

2523

For small sizes and low power applications, a setscrew may provide the connection between the hub and the shaft, but higher power usually requires a key and perhaps two set­screws, one of them above the key. A flat on the shaft and some means of locking the set­screw(s) in position are advisable. In the AGMA Class I and II fits, the shaft tolerances are −0.0005 inch from 1 ⁄2 to 11 ⁄2 inches diameter and −0.001 inch on larger diameters up to 7 inches. Class I coupling bore tolerances are + 0.001 inch up to 11 ⁄2 inches diameter, then + 0.0015 inch to 7 inches diameter. Class II coupling bore tolerances are + 0.002 inch on sizes up to 3 inches diameter, + 0.003 inch on sizes from 31 ⁄4 through 33 ⁄4 inches diameter, and + 0.004 inch on larger diameters up to 7 inches. Interference Fits.—Components of couplings transmitting over 150 horsepower often are made with an interference fit on the shafts, which may reduce fretting corrosion. These cou­plings may or may not use keys, depending on the degree of interference. Keys may range in size from 1 ⁄8 inch wide by 1 ⁄16 inch high for 1 ⁄2 -inch diameter shafts to 13 ⁄4 inches wide by 7 ⁄8 inch high for 7-inch diameter shafts. Couplings transmitting high torque or operating at high speeds or both may use two keys. Keys must be a good fit in their keyways to ensure good transmission of torque and prevent failure. AGMA standards provide recommenda­tions for square parallel, rectangular section, and plain tapered keys, for shafts of 5 ⁄16 through 7 inches diameter, in three classes designated commercial, precision, and fitted. These standards also cover keyway offset, lead, parallelism, finish and radii, and face keys and splines. (See also ANSI and other Standards in KEYS AND KEYSEATS starting on page 2539 of this Handbook.) Double-Cone Clamping Couplings.—As shown in the table, double-cone clamping cou­ plings are made in a range of sizes for shafts from 17 ⁄16 to 6 inches in diameter, and are easily assembled to shafts. These couplings provide an interference fit, but they usually cost more and have larger overall dimensions than regular flanged couplings. Double-cone Clamping Couplings F

M

G

G

K J

E

C

B

D H A 14

L A

17∕16 115∕16 27∕16 3 31 ∕2 4 41 ∕2 5 51 ∕2 6

B

51 ∕4 7

83∕4 101 ∕2 121 ∕4 14 151 ∕2 17 171 ∕2 18

C

23∕4 31 ∕2 45∕16 51 ∕2 7 7 8 9

91 ∕2 10

D

21 ∕8 27∕8 35∕8 43∕32 53∕8 51 ∕2 67∕8 71 ∕4 73∕4 81 ∕4

E

15∕8 21 ∕8 3 31 ∕2 43∕8 43∕4 51 ∕4 53∕4 61 ∕4 63∕4

14

F

5 ∕8 5 ∕8 3 ∕4 3 ∕4 7∕8 7∕8 7∕8 7∕8 1 1

”

” G

21 ∕8 23∕4 31 ∕2 43∕16 51 ∕16 51 ∕2 63∕4 7 7 7

Copyright 2020, Industrial Press, Inc.

H

43∕4 61 ∕4 713∕16 9 111 ∕4 12

131 ∕2 15

151 ∕2 16

J

11 ∕8 11 ∕8 17∕8 21 ∕4 25∕8 33∕4 33∕4 33∕4 33∕4 33∕4

K

1

13∕8 13∕4 2

21 ∕8 21 ∕2 23∕4 3 3 3

L

M

61 ∕4 77∕8 91 ∕2 111 ∕4 12

1 ∕2

5

141 ∕2 151 ∕4 151 ∕4 151 ∕4

1 ∕2 5 ∕8 5 ∕8 3 ∕4 3 ∕4 3 ∕4 3 ∕4 7∕8 7∕8

No. of Bolts 3 3 3 3 4 4 4 4 4 4

No. of Keys 1 1 1 1 1 1 1 1 1 2

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Machinery's Handbook, 31st Edition COUPLINGS and Clutches

2524

Flexible Couplings.—Shafts that are out of alignment laterally or angularly can be con­ nected by any of several designs of flexible couplings. Such couplings also permit some degree of axial movement in one or both shafts. Some couplings use disks or diaphragms to transmit the torque. Another simple form of flexible coupling consists of two flanges connected by links or endless belts made of leather or other strong, pliable material. Alter­natively, the flanges may have projections that engage spacers of molded rubber or other flexible materials that accommodate uneven motion between the shafts. More highly developed flexible couplings use toothed flanges engaged by correspondingly toothed ele­ments, permitting relative movement. These couplings require lubrication unless one or more of the elements is made of a self-lubricating material. Other couplings use dia­phragms or bellows that can flex to accommodate relative movement between the shafts. The Universal Joint.—This form of coupling, originally known as a Cardan or Hooke’s coupling, is used for connecting two shafts the axes of which are not in line with each other, but which merely intersect at a point. There are many different designs of universal joints or couplings, which are based on the principle embodied in the original design. One well-known type is shown by the accompanying diagram. As a rule, a universal joint does not work well if the angle α (see illustration) is more than 45 degrees, and the angle should preferably be limited to about 20 degrees or 25 degrees, excepting when the speed of rotation is slow and little power is transmitted. Variation in Angular Velocity of Driven Shaft: Owing to the angularity between two shafts connected by a universal joint, there is a variation in the angular velocity of one shaft during a single revolution, and because of this, the use of universal couplings is sometimes prohibited. Thus, the angular velocity of the driven shaft will not be the same at all points of the revolution as the angular velocity of the driving shaft. In other words, if the driving shaft moves with a uniform motion, then the driven shaft will have a variable motion and, therefore, the universal joint should not be used when absolute uniformity of motion is essential for the driven shaft. Determining Maximum and Minimum Velocities: If shaft A (see diagram) runs at a con­ stant speed, shaft B revolves at maximum speed when shaft A occupies the position shown in the illustration, and the minimum speed of shaft B occurs when the fork of the driving shaft A has turned 90 degrees from the position illustrated. The maximum speed of the driven shaft may be obtained by multiplying the speed of the driving shaft by the secant of angle α. The minimum speed of the driven shaft equals the speed of the driver multiplied by cosine α. Thus, if the driver rotates at a constant speed of 100 revolutions per minute and the shaft angle is 25 degrees, the maximum speed of the driven shaft is at a rate equal to 1.1034 × 100 = 110.34 rpm. The minimum speed rate equals 0.9063 × 100 = 90.63; hence, the extreme variation equals 110.34 − 90.63 = 19.71 rpm.

B



A

Copyright 2020, Industrial Press, Inc.

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Machinery's Handbook, 31st Edition COUPLINGS and Clutches

2525

Use of Intermediate Shaft between Two Universal Joints.—The lack of uniformity in the speed of the driven shaft resulting from the use of a universal coupling, as previously explained, is objectionable for some forms of mechanisms. This variation may be avoided if the two shafts are connected with an intermediate shaft and two universal joints, pro­ vided the latter are properly arranged or located. Two conditions are necessary to obtain a constant speed ratio between the driving and driven shafts. First, the shafts must make the same angle with the intermediate shaft; second, the universal joint forks (assuming that the fork design is employed) on the intermediate shaft must be placed relatively so that when the plane of the fork at the left end coincides with the center lines of the intermediate shaft and the shaft attached to the left-hand coupling, the plane of the right-hand fork must also coincide with the center lines of the intermediate shaft and the shaft attached to the right-hand coupling; therefore the driving and the driven shafts may be placed in a variety of positions. One of the most common arrangements is with the driving and driven shafts par­allel. The forks on the intermediate shafts should then be placed in the same plane. This intermediate connecting shaft is frequently made telescoping, and then the driving and driven shafts can be moved independently of each other within certain limits in longi­tudinal and lateral directions. The telescoping intermediate shaft consists of a rod which enters a sleeve and is provided with a suitable spline, to prevent rotation between the rod and sleeve and permit a sliding movement. This arrangement is applied to various machine tools. Knuckle Joints.—Movement at the joint between two rods may be provided by knuckle joints, for which typical proportions are seen in the table Proportions of Knuckle Joints on page 2526. Friction Clutches.—Clutches which transmit motion from the driving to the driven mem­ ber by the friction between the engaging surfaces are built in many different designs, although practically all of them can be classified under four general types, namely, conical clutches; radially expanding clutches; contracting-band clutches; and friction disk clutches in single and multiple types. There are many modifications of these general classes, some of which combine the features of different types. The proportions of various sizes of cone clutches are given in the table “Cast-iron Friction Clutches.” The multicone friction clutch is a further development of the cone clutch. Instead of having a single cone-shaped surface, there is a series of concentric conical rings which engage annular grooves formed by corresponding rings on the opposite clutch member. The internal-expanding type is provided with shoes which are forced outward against an enclosing drum by the action of levers connecting with a collar free to slide along the shaft. The engaging shoes are commonly lined with wood or other material to increase the coefficient of friction. Disk clutches are based on the principle of multiple-plane friction, and use alternating plates or disks so arranged that one set engages with an outside cylindrical case and the other set with the shaft. When these plates are pressed together by spring pressure, or by other means, motion is transmitted from the driving to the driven members connected to the clutch. Some disk clutches have a few rather heavy or thick plates and others a relatively large number of thinner plates. Clutches of the latter type are common in automobile trans­m issions. One set of disks may be of soft steel and the other set of phosphor-bronze, or some other combination may be employed. For instance, disks are sometimes provided with cork inserts, or one set or series of disks may be faced with a special friction material such as asbestos-wire fabric, as in “dry plate” clutches, the disks of which are not lubri­cated like the disks of a clutch having, for example, the steel and phosphor-bronze combi­nation. It is common practice to hold the driving and driven members of friction clutches in engagement by means of spring pressure, although pneumatic or hydraulic pressure may be employed.

Copyright 2020, Industrial Press, Inc.

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Machinery's Handbook, 31st Edition COUPLINGS and Clutches

2526

Proportions of Knuckle Joints

f a

D

For sizes not given below:

h g

a

e

b

D

f D

a = 1.2D b = 1.1D c = 1.2D e = 0.75D f = 0.6D g = 1.5D

i

e

c D

k

l

h = 2D i = 0.5D j = 0.25D k = 0.5D l = 1.5D

D

b j

D

a

b

c

e

f

g

1 ∕2

5 ∕8

9 ∕16

5 ∕8

3 ∕8

5 ∕16

3 ∕4

7∕8

9 ∕16

7∕16

3 ∕4

5 ∕8

15 ∕16

3 ∕4

3 ∕4

1

11 ∕4 11 ∕2 13∕4 2

21 ∕4

21 ∕2 23∕4 3

31 ∕4 31 ∕2 33∕4 4

7∕8

11 ∕4

11 ∕2 13∕4

21 ∕8

15∕8 2

11 ∕2 13∕4

21 ∕8

21 ∕4

23∕8

3

23∕4

3

23∕4 31 ∕4 35∕8

4

41 ∕4 41 ∕2 43∕4

51 ∕8

43∕4

53∕4

5

13∕8

11 ∕4

23∕8

41 ∕4

41 ∕2

3 ∕4

11 ∕8

51 ∕2 6

21 ∕2 3

31 ∕4 35∕8

37∕8

41 ∕8 43∕8 43∕4 5

51 ∕4 51 ∕2

23∕4 31 ∕4

35∕8 4

41 ∕4 41 ∕2 43∕4

11 ∕8

15∕16 11 ∕2

111 ∕16 17∕8

21 ∕16 21 ∕4

27∕16

25∕8

213∕16 3

7∕8

11 ∕16 13∕16 13∕8

11 ∕2 15∕8

113∕16 2

21 ∕8 21 ∕4 23∕8

11 ∕8 11 ∕2 17∕8

21 ∕4 25∕8 3

33∕8 33∕4

41 ∕8 41 ∕2 47∕8

51 ∕4

h 1 11 ∕2 2

21 ∕2 3

31 ∕2

i

j

k

l

1 ∕4

1 ∕8

1 ∕4

3 ∕4

3 ∕8

3 ∕16

3 ∕8

1 ∕2

1 ∕4

1 ∕2

5 ∕8

5 ∕16

5 ∕8

3 ∕4

3 ∕8

3 ∕4

7∕8

7∕16

7∕8

4

1

41 ∕2

11 ∕8

5

51 ∕2 6

61 ∕2 7

11 ∕4

13∕8

11 ∕2

15∕8

13∕4

55∕8

71 ∕2

17∕8

6

8

1 ∕2 9 ∕16 5 ∕8 11 ∕16 3 ∕4 13 ∕16 7∕8

11 ∕8 11 ∕2 17∕8

21 ∕4 25∕8

1

3

11 ∕8

33∕8

13∕8

41 ∕8

11 ∕4 11 ∕2 15∕8

13∕4

2

15 ∕16

1

17∕8 2

33∕4

41 ∕2 47∕8

51 ∕4 55∕8 6

51 ∕8

33∕16

29∕16

63∕8

81 ∕2

21 ∕8

11 ∕16

21 ∕8

63∕8

53∕4

39∕16

27∕8

71 ∕8

91 ∕2

23∕8

13∕16

23∕8

71 ∕8

51 ∕2

6

33∕8 33∕4

23∕4 3

63∕4

71 ∕2

9

10

21 ∕4 21 ∕2

11 ∕8 11 ∕4

21 ∕4 21 ∕2

63∕4

71 ∕2

Power Transmitting Capacity of Friction Clutches.—When selecting a clutch for a given class of service, it is advisable to consider any overloads that may be encountered and base the power transmitting capacity of the clutch upon such overloads. When the load varies or is subject to frequent release or engagement, the clutch capacity should be greater than the actual amount of power transmitted. If the power is derived from a gas or gasoline engine, the horsepower rating of the clutch should be 75 or 100 percent greater than that of the engine. Power Transmitted by Disk Clutches.—The approximate amount of power that a disk clutch will transmit may be determined from the following formula, in which HP = horse­ power transmitted by the clutch; μ = coefficient of friction; r = mean radius of engaging surfaces; F = axial force in pounds (spring pressure) holding disks in contact; N = number of frictional surfaces; S = speed of shaft in revolutions per minute:

µ rFNS HP = 63, 000

Copyright 2020, Industrial Press, Inc.

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Machinery's Handbook, 31st Edition COUPLINGS and Clutches

2527

Cast-iron Friction Clutches t

φ e D

a

e

D

1

11 ∕4 11 ∕2 13∕4 2

21 ∕4 21 ∕2 23∕4 3

31 ∕4 31 ∕2 33∕4 4

41 ∕4 41 ∕2 43∕4 5

51 ∕4 51 ∕2 53∕4 6

h

e D

c s

For sizes not given below: a  = 2 D b   =  4 to 8 D c  = 21 ∕4 D t  = 11 ∕2 D e   =  3∕8 D h   =  1 ∕2 D s   =  5∕16 D, nearly k   =  1 ∕4 D

k

b

a

Note: The angle φ of the cone may be from 4 to 10 degrees

e a

2

21 ∕2 3

31 ∕2 4 41 ∕2 5

51 ∕2 6 61 ∕2 7

71 ∕2 8 81 ∕2 9

91 ∕2 10

101 ∕2 11 111 ∕2 12

b

4-8

 5-10

 6-12

 7-14

 8-16  9-18 10-20

c

11 ∕2

33∕8 4

21 ∕4

27∕8

41 ∕2 5

16-32 18-36

101 ∕4

77∕8

91 ∕2

19-38

103∕4

21-42

113∕4

24-48

3∕ 4

33∕8

5∕ 8

7∕ 8

11 ∕8

81 ∕2 9

23-46

25∕8 3

5∕ 8

41 ∕2

15-30

22-44

1∕ 2

63∕4

73∕8

20-40

17∕8

1

13-26

17-34

3∕ 8

33∕4

61 ∕4

14-28

e

55∕8

11-22

12-24

t

21 ∕4

111 ∕4

123∕8 13

131 ∕2

41 ∕8 47∕8

51 ∕4 55∕8 6

1

85∕8 9

7∕ 8

1∕ 2

3∕ 8

5∕ 16

5∕ 8

7∕ 16

11 ∕8

5∕ 8

9∕ 16

13∕8

7∕ 8

1

11 ∕4 11 ∕2

5∕ 8

3∕ 4 7∕ 8

1∕ 2 5∕ 8

11 ∕ 16 3∕ 4

11 ∕2

11 ∕4

11 ∕4

1

13∕4

21 ∕4

13∕8

11 ∕8

13∕8

13∕4

81 ∕4

3∕ 8

1∕ 4

17∕8 2

13∕8

71 ∕8 77∕8

5∕ 8

3∕ 4

k

5∕ 16

1

15∕8

71 ∕2

s

1∕ 2

15∕8

11 ∕4

63∕8 63∕4

h

17∕8 2 2

21 ∕4

21 ∕4

13∕4

21 ∕8 23∕8

21 ∕2

25∕8

23∕4 27∕8 3

1

13∕8

11 ∕2 11 ∕2 15∕8

13∕4 13∕4

17∕8

13 ∕ 16 7∕ 8

15 ∕ 16

11 ∕16 13∕16 11 ∕4

15∕16 13∕8

17∕16 11 ∕2

Frictional Coefficients for Clutch Calculations.—While the frictional coefficients used by designers of clutches differ somewhat and depend upon variable factors, the fol­lowing values may be used in clutch calculations: For greasy leather on cast iron about 0.20 or 0.25, leather on metal that is quite oily 0.15; metal and cork on oily metal 0.32; the same on dry metal 0.35; metal on dry metal 0.15; disk clutches having lubricated surfaces 0.10. Formulas for Cone Clutches.—In cone clutch design, different formulas have been developed for determining the horsepower transmitted. These formulas, at first sight, do not seem to agree, there being a variation due to the fact that in some of the formulas the friction clutch surfaces are assumed to engage without slip, whereas, in others, some allowance is made for slip. The following formulas include both of these conditions:

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Machinery's Handbook, 31st Edition COUPLINGS and Clutches

2528

HP =  horsepower transmitted N =  revolutions per minute r =  mean radius of friction cone, in inches r1 =  large radius of friction cone, in inches r 2 =  small radius of friction cone, in inches R1 =  outside radius of leather band, in inches R2 =  inside radius of leather band, in inches V =  velocity of a point at distance r from the center, in feet per minute F =  tangential force acting at radius r, in pounds Pn =  total normal force between cone surfaces, in pounds Ps =  spring force, in pounds α =  angle of clutch surface with axis of shaft = 7 to 13 degrees β =  included angle of clutch leather, when developed, in degrees f =  coefficient of friction = 0.20 to 0.25 for greasy leather on iron p =  allowable pressure per square inch of leather band = 7 to 8 pounds W =  width of clutch leather, in inches W

Development of Clutch Leather

R2

β

α

R1

r1

r2

r R1 = sin1 α

r R 2 = sin2α r +r β = sin α # 360 r= 12 2 2 π rN V = 12 F=

HP # 33, 000 V

For engagement with some slip: P Pn = sins α For engagement without slip: P Pn = sin α +sf cos α

P W = 2 πnrp Ps =

Ps =

Pn frN HP = 63, 025

HP # 63, 025 sin α frN HP # 63, 025 ^sin α + f cos αh frN

Angle of Cone.—If the angle of the conical surface of the cone type of clutch is too small, it may be difficult to release the clutch on account of the wedging effect, whereas, if the angle is too large, excessive spring force will be required to prevent slipping. The mini­ mum angle for a leather-faced cone is about 8 or 9 degrees and the maximum angle about 13 degrees. An angle of 121 ⁄2 degrees appears to be the most common and is generally

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Machinery's Handbook, 31st Edition COUPLINGS and Clutches

2529

con­sidered good practice. These angles are given with relation to the clutch axis and are one-half the included angle. Magnetic Clutches.—Many disk and other clutches are operated electromagnetically with the magnetic force used only to move the friction disk(s) and the clutch disk(s) into or out of engagement against spring or other pressure. On the other hand, in a magnetic parti­ cle clutch, transmission of power is accomplished by magnetizing a quantity of metal par­ ticles enclosed between the driving and the driven components, forming a bond between them. Such clutches can be controlled to provide either a rigid coupling or uniform slip, useful in wire drawing and manufacture of cables. Another type of magnetic clutch uses eddy currents induced in the input member which interact with the field in the output rotor. Torque transmitted is proportional to the coil cur­rent, so precise control of torque is provided. A third type of magnetic clutch relies on the hysteresis loss between magnetic fields generated by a coil in an input drum and a close-fitting cup on the output shaft, to transmit torque. Torque transmitted with this type of clutch also is proportional to coil current, so close control is possible. Permanent-magnet types of clutches also are available, in which the engagement force is exerted by permanent magnets when the electrical supply to the disengagement coils is cut off. These types of clutches have capacities up to five times the torque-to-weight ratio of spring-operated clutches. In addition, if the controls are so arranged as to permit the coil polarity to be reversed instead of being cut off, the combined permanent magnet and elec­tromagnetic forces can transmit even greater torque. Centrifugal and Free-wheeling Clutches.—Centrifugal clutches have driving members that expand outward to engage a surrounding drum when speed is sufficient to generate centrifugal force. Free-wheeling clutches are made in many different designs and use balls, cams or sprags, ratchets, and fluids to transmit motion from one member to the other. These types of clutches are designed to transmit torque in only one direction and to take up the drive with various degrees of gradualness up to instantaneously. Slipping Clutch/Couplings.—Where high shock loads are likely to be experienced, a slipping clutch or coupling or both should be used. The most common design uses a clutch plate that is clamped between the driving and driven plates by spring pressure that can be adjusted. When excessive load causes the driven member to slow, the clutch plate surfaces slip, allowing reduction of the torque transmitted. When the overload is removed, the drive is taken up automatically. Switches can be provided to cut off current supply to the driving motor when the driven shaft slows to a preset limit or to signal a warning or both. The slip or overload torque is calculated by taking 150 percent of the normal running torque. Wrapped-Spring Clutches.—For certain applications, a simple steel spring sized so that its internal diameter is a snug fit on both driving and driven shafts will transmit adequate torque in one direction. The tightness of grip of the spring on the shafts increases as the torque transmitted increases. Disengagement can be effected by slight rotation of the spring, through a projecting tang, using electrical or mechanical means, to wind up the spring to a larger internal diameter, allowing one of the shafts to run free within the spring. Normal running torque Tr in lb-ft = (required horsepower × 5250) ÷ rpm. For heavy shock load applications, multiply by a 200 percent or greater overload factor. (See Motors, factors governing selection.) The clutch starting torque Tc, in lb-ft, required to accelerate a given inertia in a specific time is calculated from the formula:

Tc =

WR 2 # ∆ N 308t

where WR2 =   total inertia encountered by clutch in lb-ft2 (W = weight and R = radius of gyration of rotating part) ΔN =  final rpm − initial rpm

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Machinery's Handbook, 31st Edition COUPLINGS and Clutches

2530

308 =  constant (see Factors Governing Motor Selection on page 2652) t =  time to required speed in seconds

Example 1: If the inertia is 80 lb-ft2, and the speed of the driven shaft is to be increased from 0 to 1500 rpm in 3 seconds, the clutch starting torque in lb-ft is

80 # 1500 308 # 3 = 130 lb-ft The heat E, in BTU, generated in one engagement of a clutch can be calculated from the formula: T # W R 2 # ^N 21 − N 22 h E= c ^ Tc − T 1h # 4.7 # 10 6

Tc =

where WR2 =  total inertia encountered by clutch in lb-ft2 N1 = final rpm  N2 =    initial rpm Tc = clutch torque in lb-ft  T1 =   torque load in lb-ft

Example 2: Calculate the heat generated for each engagement under the conditions cited for Example 1. 130 # 80 # ^1500h2 E= = 41.5 BTU ^130 − 10h # 4.7 # 10 6

The preferred location for a clutch is on the high- rather than on the low-speed shaft because a smaller-capacity unit, of lower cost and with more rapid dissipation of heat, can be used. However, the heat generated may also be more because of the greater slippage at higher speeds, and the clutch may have a shorter life. For light-duty applications, such as to a machine tool, where cutting occurs after the spindle has reached operating speed, the cal­culated torque should be multiplied by a safety factor of 1.5 to arrive at the capacity of the clutch to be used. Heavy-duty applications such as frequent starting of a heavily loaded vibratory-finishing barrel require a safety factor of 3 or more. Positive Clutches.—When the driving and driven members of a clutch are connected by the engagement of interlocking teeth or projecting lugs, the clutch is said to be “positive” to distinguish it from the type in which the power is transmitted by frictional contact. The positive clutch is employed when a sudden starting action is not objectionable and when the inertia of the driven parts is relatively small. The various forms of positive clutches dif­fer merely in the angle or shape of the engaging surfaces. The least positive form is one having planes of engagement which incline backward, with respect to the direction of motion. The tendency of such a clutch is to disengage under load, in which case it must be held in position by axial pressure. e 8°

h θ A

θ

θ/2

B

C

e D

69° E

Fig. 1. Types of Clutch Teeth

This pressure may be regulated to perform normal duty, permitting the clutch to slip and disengage when over-loaded. Positive clutches, with the engaging planes parallel to the axis of rotation, are held together to obviate the tendency to jar out of engagement, but they provide no safety feature against over-load. So-called “under-cut” clutches engage more tightly the heavier the load, and are designed to be disengaged only when free from

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Machinery's Handbook, 31st Edition COUPLINGS and Clutches

2531

load. The teeth of positive clutches are made in a variety of forms, a few of the more common styles being shown in Fig. 1. Clutch A is a straight-toothed type, and B has angular or saw-shaped teeth. The driving member of the former can be rotated in either direction; the latter is adapted to the transmission of motion in one direction only, but is more readily engaged. The angle θ of the cutter for a saw-tooth clutch B is ordinarily 60 degrees. Clutch C is sim­ilar to A, except that the sides of the teeth are inclined to facilitate engagement and disen­gagement. Teeth of this shape are sometimes used when a clutch is required to run in either direction without backlash. Angle θ is varied to suit requirements and should not exceed 16 or 18 degrees. The straight-tooth clutch A is also modified to make the teeth engage more readily, by rounding the corners of the teeth at the top and bottom. Clutch D (commonly called a “spiral-jaw” clutch) differs from B in that the surfaces e are helicoidal. The driving member of this clutch can transmit motion in only one direction.

B

A

D

C

Fig. 2. Diagrammatic View Showing Method of Cutting Clutch Teeth

Clutch Blank

Cutter a d

Fig. 3.

Clutches of this type are known as right- and left-hand, the former driving when turning to the right, as indicated by the arrow in the illustration. Clutch E is the form used on the back-shaft of the Brown & Sharpe automatic screw machines. The faces of the teeth are radial and incline at an angle of 8 degrees with the axis, so that the clutch can readily be disen­gaged. This type of clutch is easily operated, with little jar or noise. The 2-inch (50.8 mm) diameter size has 10 teeth. Height of working face, 1 ⁄8 inch (3.175 mm). Cutting Clutch Teeth.—A common method of cutting a straight-tooth clutch is indicated by the diagrams A, B and C, Fig. 2, which show the first, second and third cuts required for forming the three teeth. The work is held in the chuck of a dividing-head, the latter being set at right angles to the table. A plain milling cutter may be used (unless the corners of the teeth are rounded), the side of the cutter being set to exactly coincide with the centerline. When the number of teeth in the clutch is odd, the cut can be taken clear across the blank as shown, thus finishing the sides of two teeth with one passage of the cutter. When the num­ber of teeth is even, as at D, it is necessary to mill all the teeth on one side and then set the cutter for finishing the opposite side. Therefore, clutches of this type commonly have an odd number of teeth. The maximum width of the cutter depends upon the width of the space at the narrow ends of the teeth. If the cutter must be quite narrow in order to pass the narrow ends, some stock may be left in the tooth spaces, which must be removed by a sep­ arate cut. If the tooth is of the modified form shown at C, Fig. 1, the cutter should be set as

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Machinery's Handbook, 31st Edition COUPLINGS and Clutches

2532

indicated in Fig. 3; that is, so that a point a on the cutter at a radial distance d equal to onehalf the depth of the clutch teeth lies in a radial plane. When it is important to eliminate all backlash, point a is sometimes located at a radial distance d equal to six-tenths of the depth of the tooth, in order to leave clearance spaces at the bottoms of the teeth; the two clutch members will then fit together tightly. Clutches of this type must be held in mesh. x



B Cutter Clutch Bank

A

C

D E

D1

E1

Fig. 4.

Angle of Dividing-Head for Milling V-Shaped Teeth with Single-Angle Cutter 360°/N

cos α =

tan (360° ⁄ N ) # cot θ 2

α is the angle shown in Fig. 4 and is the angle shown by the graduations on the dividing head. θ is the included angle of a single cutter, see Fig. 1. Angle of Single-angle Cutter, θ

No. of Teeth, N 5

60°

6

27° 60

8

73

7

9

10 11

12

13

14

15

16

17

68

75

77

79

80

81

82

82

83

83

70°

Dividing Head Angle, α 19.2' 55° 56.3' 74° 71 37.6 81

46.7

13.3

58.9

53.6

18.5

24.4

17.1

.536

36.9

7.95

34.7

76

79

81

82

83

83

84

84

85

85

85

48.5

30.9

13

24.1

17

58.1

31.1

58.3

21.2

40.6

57.4

83

84

85

86

86

87

87

87

87

87

88

80° 15.4' 13

39.2

56.5

45.4

19.6

45.1

4.94

20.9

34

45

54.4

2.56

Angle of Single-angle Cutter, θ

No. of Teeth, N 18

60°

19

83° 84

21

84

20

22

23

24

25

26

27

28

29

30

84

85

85

85

85

85

86

86

86

86

70°

Dividing Head Angle, α 58.1' 12.1' 88° 86° 18.8 86 25.1 88

37.1

53.5

8.26

21.6

33.8

45

55.2

4.61

13.3

21.4

28.9

86

86

86

87

87

87

87

87

87

87

87

36.6

46.9

56.2

4.63

12.3

19.3

25.7

31.7

37.2

42.3

47

88

88

88

88

88

88

88

88

88

88

88

80° 9.67' 15.9

21.5

26.5

31

35.1

38.8

42.2

45.3

48.2

50.8

53.3

55.6

Cutting Saw-Tooth Clutches: When milling clutches having angular teeth as shown at B, Fig. 1, the axis of the clutch blank should be inclined a certain angle α as shown at A in Fig. 4. If the teeth were milled with the blank vertical, the tops of the teeth would incline towards the center as at D, whereas, if the blank were set to such an angle that the tops of the teeth were square with the axis, the bottoms would incline upwards as at E. In either

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Machinery's Handbook, 31st Edition COUPLINGS and Clutches

2533

case, the two clutch members would not mesh completely: the engagement of the teeth cut as shown at D and E would be as indicated at D1 and E1 respectively. As will be seen, when the outer points of the teeth at D1 are at the bottom of the grooves in the opposite member, the inner ends are not together, the contact area being represented by the dotted lines. At E1 the inner ends of the teeth strike first and spaces are left between the teeth around the outside of the clutch. To overcome this objectionable feature, the clutch teeth should be cut as indi­cated at B, or so that the bottoms and tops of the teeth have the same inclination, converging at a central point x. The teeth of both members will then engage across the entire width as shown at C. The angle α required for cutting a clutch as at B can be determined by the fol­lowing formula in which α equals the required angle, N = number of teeth, θ = cutter angle, and 360°/N = angle between teeth:

tan (360° ⁄ N ) # cot θ 2 The angles α for various numbers of teeth and for 60-, 70- or 80-degree single-angle cut­ters are given in the table on page 2532. The following table is for double-angle cutters used to cut V-shaped teeth.

cos α =

Angle of Dividing-Head for Milling V-Shaped Teeth with Double-Angle Cutter 180°/N

cos α =

tan (180° ⁄ N ) # cot (θ ⁄ 2) 2

This is the angle (α, Fig. 4) shown by graduations on the dividing-head. θ is the included angle of a double-angle cutter, see Fig. 1. Included Angle of Cutter, θ No. of Teeth, N 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

60°

73° 75 76 77 78 79 80 80 81 81 82 82 82 83 83 83 83 84 84 84 84

90°

Dividing Head Angle, α 39.4' 16.1 34.9 40.5 36 23.6 4.83 41 13 41.5 6.97 30 50.8 9.82 27.2 43.1 57.8 11.4 24 35.7 46.7

80° 81 82 82 83 83 84 84 84 85 85 85 85 86 86 86 26 86 86 86 86

39' 33.5 18 55.3 26.8 54 17.5 38.2 56.5 12.8 27.5 40.7 52.6 3.56 13.5 22.7 31.2 39 46.2 53 59.3

Included Angle of Cutter, θ No. of Teeth, N 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51

60°

84° 85 85 85 85 85 85 85 85 86 86 86 86 86 86 86 86 86 86 86 86

90°

Dividing Head Angle, α 56.9' 6.42 15.4 23.8 31.8 39.3 46.4 53.1 59.5 5.51 11.3 16.7 22 26.9 31.7 36.2 40.6 44.8 48.8 52.6 56.3

87° 87 87 87 87 87 87 87 87 87 87 87 87 87 87 88 88 88 88 88 88

5.13' 10.6 15.8 20.7 25.2 29.6 33.7 37.5 41.2 44.7 48 51.2 54.2 57 59.8 2.4 4.91 7.32 9.63 11.8 14

The angles given in the table above are applicable to the milling of V-shaped grooves in brackets, etc., which must have toothed surfaces to prevent the two members from turning relative to each other, except when unclamped for angular adjustment

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Machinery's Handbook, 31st Edition Friction Brakes

2534

Friction Brakes Formulas for Band Brakes.—In any band brake, such as shown in Fig. 1, in the tabula­tion of formulas, where the brake wheel rotates in a clockwise direction, the tension in that part of the band marked x equals P

1 e µθ − 1

The tension in that part marked y equals P

e µθ . e µθ − 1

P =  tangential force in pounds at rim of brake wheel e =  base of natural logarithm = 2.71828 μ =  coefficient of friction between the brake band and the brake wheel θ =  angle of contact of the brake band with the brake wheel expressed in radians

180 deg deg 1 radian = π radians = 57. 296 radian See also Conversion Tables of Angular Measure starting on page 103. For simplicity in the formulas presented, the tensions at x and y (Fig. 1) are denoted by T1 and T2 respectively, for clockwise rotation. When the direction of the rotation is reversed, the tension in x equals T2 , and the tension in y equals T1, which is the reverse of the tension in the clockwise direction. The value of the expression eμθ in these formulas may be most easily found by using a hand-held calculator of the scientific type; that is, one capable of raising 2.71828 to the power μθ. The following example outlines the steps in the calculations.

Table of Values of eμθ Proportion of Contact to Whole Circumference,

θ 2π 0.1

0.2

0.3

0.4

0.425 0.45

0.475 0.5

0.525 0.55 0.6

0.7

0.8

0.9 1.0

Steel Band on Cast Iron, μ = 0.18

Wood

Leather Belt on

Cast Iron

1.12

Slightly Greasy; μ = 0.47 1.34

Very Greasy; μ = 0.12 1.08

Slightly Greasy; μ = 0.28 1.19

1.40

2.43

1.25

1.69

1.25

1.57

1.62

1.66

1.71

1.76

1.81

1.86

1.97

2.21

1.81

3.26

3.51

3.78 4.07

4.38 4.71

5.07

5.88

7.90

2.47

10.60

3.10

19.20

2.77

14.30

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1.16

1.35

1.38

1.42

2.02 2.11

1.40

2.21

1.46

2.41

1.43

1.49

1.51

1.57

1.66 1.83

1.97 2.12

2.31

2.52

2.63

2.81

3.43 4.09

4.87 5.81

Damp; μ = 0.38 1.27 1.61

2.05 2.60 2.76 2.93 3.11

3.30 3.50 3.72 4.19

5.32

6.75 8.57

10.90

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Machinery's Handbook, 31st Edition Friction Brakes

2535

Formulas for Simple and Differential Band Brakes

F = force in pounds at end of brake handle; P = tangential force in pounds at rim of brake wheel; e = base of natural logarithm = 2.71828; μ = coefficient of friction between the brake band and the brake wheel; θ = angle of contact of the brake band with the brake wheel, expressed in radians (1 radian = 57.296 degrees).

T1 = P

P

F x

e µθ eµθ − 1

For clockwise rotation:

bT2 Pb eµθ m F = a = a c µθ e −1

For counter clockwise rotation:

bT1 Pb 1 k F = a = a a µθ e −1

a

b

T2 = P

Simple Band Brake

P

y

1 eµθ − 1

Fig. 1.

For clockwise rotation:

bT1 Pb 1 k F = a = a a µθ e −1

For counter clockwise rotation:

a

b

bT Pb eµθ m F = a 2 = a c µθ e −1

F Fig. 2.

Differential Band Brake

F

For clockwise rotation:

F=

µθ b2 T2 − b1 T1 P b e − b1 m = a c 2 µθ a e −1

For counter clockwise rotation:

b1

a

F=

b2

b2 T1 − b1 T2 P b2 − b1e µθ m = a c µθ a e −1

In this case, if b2 is equal to, or less than, b1 eμθ, the force F will be 0 or negative and the band brake works automatically.

Fig. 3.

For clockwise rotation:

F=

F a b1

b2 Fig. 4.

b2 T2 + b1 T1 P b2 e µθ + b1 m = a c µθ a e −1

For counter clockwise rotation:

µθ b1 T2 + b2 T1 P b e + b2 m = a c 1 µθ a e −1 Pb1 eµθ + 1 m. If b2 = b1, both of the above formulas reduce to F = a c µθ e −1

F=

In this case, the same force F is required for rotation in either direction.

Example: In a band brake of the type in Fig. 1, dimension a = 24 inches, and b = 4 inches; force P = 100 pounds; coefficient μ = 0.2, and angle of contact = 240 degrees, or

240 θ = 180 # π = 4.18

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2536

Machinery's Handbook, 31st Edition Friction Brakes

The rotation is clockwise. Find force F required. Pb e µθ m F = a c µθ e −1 100 # 4 2.718280. 2# 4.18 k = 24 a 2.71828 0. 2# 4.18 − 1

. 400 2.71828 0836 2.31 = 24 # = 16.66 # 2.31 − 1 = 29.4 . −1 2.71828 0836

If a hand-held calculator is not used, determining the value of eμθ is rather tedious, and the table on page 2534 will save calculations. Coefficient of Friction in Brakes.—The coefficients of friction that may be assumed for friction brake calculations are as follows: Iron on iron, 0.25 to 0.3; leather on iron, 0.3; cork on iron, 0.35. Values somewhat lower than these should be assumed when the velocities exceed 400 feet per minute at the beginning of the braking operation. For brakes where wooden brake blocks are used on iron drums, poplar has proved the best brake block material. The best material for the brake drum is wrought iron. Poplar gives a high coefficient of friction, and is little affected by oil. The average coefficient of friction for poplar brake blocks and wrought-iron drums is 0.6; for poplar on cast iron, 0.35; for oak on wrought iron, 0.5; for oak on cast iron, 0.3; for beech on wrought iron, 0.5; for beech on cast iron, 0.3; for elm on wrought iron, 0.6; and for elm on cast iron, 0.35. The objection to elm is that the friction decreases rapidly if the friction surfaces are oily. The coefficient of friction for elm and wrought iron, if oily, is less than 0.4. Calculating Horsepower from Dynamometer Tests.—When a dynamometer is arranged for measuring the horsepower transmitted by a shaft, as indicated by the diagram­ matic view in Fig. 5, the horsepower may be obtained by the formula: 2π LPN HP = 33000 in which HP = horsepower transmitted; N = number of revolutions per minute; L = dis­ tance (as shown in illustration) from center of pulley to point of action of weight P, in feet; P = weight hung on brake arm or read on scale. L

P Fig. 5.

By adopting a length of brake arm equal to 5 feet 3 inches, the formula may be reduced to the simple form: NP HP = 1000

If a length of brake arm equal to 2 feet 71 ⁄2 inches is adopted as a standard, the formula takes the form: NP HP = 2000

The transmission type of dynamometer measures the power by transmitting it through the mechanism of the dynamometer from the apparatus in which it is generated, or to the

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Machinery's Handbook, 31st Edition Friction Brakes

2537

apparatus in which it is to be utilized. Dynamometers known as indicators operate by simultaneously measuring the pressure and volume of a confined fluid. This type may be used for the measurement of the power generated by steam or gas engines or absorbed by refrigerating machinery, air compressors, or pumps. An electrical dynamometer is for measuring the power of an electric current, based on the mutual action of currents flowing in two coils. It consists principally of one fixed and one movable coil, which, in the normal position, are at right angles to each other. Both coils are connected in series, and, when a current traverses the coils, the fields produced are at right angles; hence, the coils tend to take up a parallel position. The movable coil with an attached pointer will be deflected, the deflection measuring directly the electric current. Formulas for Block Brakes

F = force in pounds at end of brake handle; P = tangential force in pounds at rim of brake wheel; μ = coefficient of friction between the brake block and brake wheel.

F b

Block brake. For rotation in either direction: b 1 Pb 1 F = P a + b # µ = a + ba µ k

a

Fig. 1.

F c

b Fig. 2.

For counter clockwise rotation: Pb µ + Pc Pb 1 c F = a + b = a + ba µ + bk

a

F

c b

Fig. 3.



a

Block brake. For clockwise rotation: Pb µ − Pc Pb 1 c F = a + b = a + ba µ − bk

Block brake. For clockwise rotation: Pb µ + Pc Pb 1 c F = a + b = a + ba µ + bk For counter clockwise rotation: Pb µ − Pc Pb 1 c F = a + b = a + ba µ − bk The brake wheel and friction block of the block brake are often grooved as shown in Fig. 4. In this case, substitute for μ in the µ above equations the value where α is one-half sin α + µ cos α the angle included by the faces of the grooves.

Fig. 4.

Friction Wheels for Power Transmission

When a rotating member is driven intermittently and the rate of driving does not need to be positive, friction wheels are frequently used, especially when the amount of power to be transmitted is comparatively small. The driven wheels in a pair of friction disks should always be made of a harder material than the driving wheels, so that if the driven

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Machinery's Handbook, 31st Edition Friction Wheels

2538

wheel should be held stationary by the load, while the driving wheel revolves under its own pres­sure, a flat spot may not be rapidly worn on the driven wheel. The driven wheels, therefore, are usually made of iron, while the driving wheels are made of or covered with, rubber, paper, leather, wood or fiber. The safe working force per inch of face width of contact for various materials are as follows: Straw fiber, 150; leather fiber, 240; tarred fiber, 240; leather, 150; wood, 100 to 150; paper, 150. Coefficients of friction for different combina­tions of materials are given in the following table. Smaller values should be used for excep­tionally high speeds, or when the transmission must be started while under load. Horsepower of Friction Wheels.—Let D = diameter of friction wheel in inches; N = number of revolutions per minute; W = width of face in inches; f = coefficient of friction; P = force in pounds, per inch width of face. Then:

HP =

Assume

3.1416 # D # N # P # W # f 33,000# 12 3.1416 # P # f 33,000# 12 = C

then, for P = 100 and f = 0.20, C = 0.00016 for P =  150 and f = 0.20, C = 0.00024 for P =  200 and f = 0.20, C = 0.00032 Working Values of Coefficient of Friction Materials Straw fiber and cast iron Straw fiber and aluminum Leather fiber and cast iron Leather fiber and aluminum Tarred fiber and cast iron Paper and cast iron

Coefficient of Friction 0.26 0.27 0.31 0.30 0.15 0.20

Materials Tarred fiber and aluminum Leather and cast iron Leather and aluminum Leather and type metal Wood and metal

Coefficient of Friction 0.18 0.14 0.22 0.25 0.25

The horsepower transmitted is then:

HP = D # N # W # C

Example: Find the horsepower transmitted by a pair of friction wheels; the diameter of the driving wheel is 10 inches, and it revolves at 200 revolutions per minute. The width of the wheel is 2 inches. The force per inch width of face is 150 pounds, and the coefficient of friction 0.20.

HP = 10 # 200 # 2 # 0.00024 = 0.96 horsepower

Horsepower Which May Be Transmitted by Means of a Clean Paper Friction Wheel of One-inch Face When Run Under a Force of 150 Pounds (Rockwood Mfg. Co.) Dia. of Friction Wheel 4 6 8 10 14 16 18 24 30 36 42 48 50

25

0.023 0.035 0.047 0.059 0.083 0.095 0.107 0.142 0.178 0.214 0.249 0.285 0.297

50

0.047 0.071 0.095 0.119 0.166 0.190 0.214 0.285 0.357 0.428 0.499 0.571 0.595

75

0.071 0.107 0.142 0.178 0.249 0.285 0.321 0.428 0.535 0.642 0.749 0.856 0.892

100

0.095 0.142 0.190 0.238 0.333 0.380 0.428 0.571 0.714 0.856 0.999 1.142 1.190

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Revolutions per Minute 150 200 300

0.142 0.214 0.285 0.357 0.499 0.571 0.642 0.856 1.071 1.285 1.499 1.713 1.785

0.190 0.285 0.380 0.476 0.666 0.761 0.856 1.142 1.428 1.713 1.999 2.284 2.380

0.285 0.428 0.571 0.714 0.999 1.142 1.285 1.713 2.142 2.570 2.998 3.427 3.570

400

0.380 0.571 0.761 0.952 1.332 1.523 1.713 2.284 2.856 3.427 3.998 4.569 4.760

600

0.571 0.856 1.142 1.428 1.999 2.284 2.570 3.427 4.284 5.140 5.997 6.854 7.140

800

0.76 1.14 1.52 1.90 2.66 3.04 3.42 4.56 5.71 6.85 7.99 9.13 9.52

1000

0.95 1.42 1.90 2.38 3.33 3.80 4.28 5.71 7.14 8.56 9.99 11.42 11.90

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Machinery's Handbook, 31st Edition Metric Keys and Keyways

2539

KEYS AND KEYSEATS Metric Square and Rectangular Keys and Keyways The ANSI/ASME B18.25.1M (Withdrawn) standard covers requirements for square and rectangular parallel keys and keyways intended for both alignment of shafts and hubs, and transmitting torque between shafts and hubs. Keys covered by this standard have a relatively tight width toler­ance. The deviations are less than the basic size. Keys with greater width tolerance and with deviations greater than the basic size are covered by ANSI/ ASME B18.25.3M (Withdrawn). All dimen­sions in this standard are in millimeters (mm).

Comparison with ISO R773-1969 and 2491-1974.—This standard is based on ISO Standards R773­-1969, Rectangular or Square Parallel Keys and their corresponding key­ ways, and 2491-1974, Thin Parallel Keys and their corresponding keyways (dimensions in millimeters). Product manufactured to this standard will meet the ISO standards. Because of tighter width tolerances in this standard, products manufactured to the ISO standard may not meet the requirements of this standard. This standard differs from ISO in that it: a) does not restrict the corners of a key to be chamfered but allows either a chamfer or a radius on the key; and b) specifies a key mate­ rial hardness rather than a tensile property. Tolerances.—Many of the tolerances shown in Table 1 and Table 2 are from ANSI/ASME B4.2 (ISO 286-1 and ISO 286-2). As a result, in addition to plus-minus tolerances which are common in the United States, some are expressed as plus-plus or minus-minus deviations from the basic size. For further interpretation of these tolerances refer to ANSI/ ASME B4.2 or ISO 286.

Designation.—Keys conforming to this standard shall be designated by the following data, preferably in the sequence as follows: a) ASME document number; b) product name; c) nominal size, width (b) × height (h) × length; d) form; and e) hardness (if other than non-hardened). Example: ANSI/ASME B18.25.1M (Withdrawn) square key 3 × 3 × 15 form B. ANSI/ASME B18.25.1M (Withdrawn) rectangular key 10 × 6 × 20 form C hardened

Preferred Lengths and Tolerances.—Preferred lengths and tolerances of square and rectan­gular keys are shown below. Tolerances are JS16. To minimize problems due to lack of straightness, key length should be less than 10 times the key width. Length

± Tolerances

Length

0.38

90,  100, 110

12, 14, 16, 18

0.56

200,  220,  250

32,  36,  40, 45,  50

0.80

6

8, 10

20,  22,  25,  28 56,  63, 70,  80

0.45

0.65 0.95

± Tolerances 1.10

125, 140, 150, 180

1.25

280

1.60

320,  360,  400

1.45

1.80

Material Requirements.—Standard steel keys shall have a Vickers hardness of 183 HV mini­mum.HardenedkeysshallbealloysteelthroughhardenedtoaVickershardnessof390to 510 HV. When other materials and properties are required, these shall be as agreed upon by the supplier and customer. Dimensions and Tolerances.—Dimensions and tolerances for square and rectangular parallel keys are shown in Table 1. Recommended dimensions and tolerances for keyways are shown in Table 2.

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Machinery's Handbook, 31st Edition Metric Keys and Keyways

2540

Figures for Table 1 and Table 3 X

X X

h

h

y

L

b/2

b

b b/2

X y

y

y

Style A

Style B

s

45

X L

X L

y

y

Style C

y = sharp edges removed not to exceed smax

Table 1. Dimensions and Tolerances for Metric Square and Rectangular Parallel Keys ANSI/ASME B18.25.1M-1996 (Withdrawn)

Basic Size (mm)

Width, b

Tolerance, h8

Basic Size

2 3 4 5 6

0 −0.014

2 3 4 5 6

5

0 −0.018

6

0 −0.018

8

10

0 −0.022

16 18

4 5 7 6 8 6

12

14

3

8 0 −0.027

6 9 7 10 7 11

Thickness, h Tolerance, Square, h8 Rectangular, h11

Square Keys

0 −0.014 0 −0.018

Chamfer or Radius, s

Minimum 0.16

0.25

0.25

0.40

Rectangular Keys 0 −0.060 0 0.25 −0.075 0 −0.090 0 −0.075 0 −0.090 0 −0.075 0 −0.090 0.40 0 −0.075 0 −0.090 0 −0.110

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Maximum

0.40

0.60

0.60

Range of Lengths

From

Toa

6 6 8 10 14

20 36 45 56 70

10

56

14

70

18

90

22

110

28

110

36

160

45

180

50

200

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Machinery's Handbook, 31st Edition Metric Keys and Keyways

2541

Table 1. Dimensions and Tolerances for Metric Square and and Table 1. (Continued) Dimensions and Tolerances for Metric Square Rectangular Parallel Keys ANSI/ASME B18.25.1M-1996 (Withdrawn) Basic Size (mm)

Width, b

Tolerance, h8

8

20

22

12 6 14

0 −0.033

9

25

14 10

28

16 11 18 12 20 22 25 28 32 32 36 40 45 50

32 36 40 45 50 56 63 70 80 90 100

Basic Size

0 −0.039

0 −0.046 0 −0.054

Thickness, Chamfer or Radius, h s Tolerance, Square, h8 Rectangular, h11 Minimum Maximum Rectangular Keys 0 −0.090 0 −0.110 0 −0.075 0 −0.110 0.60 0.80 0 −0.090 0 −0.110 0 −0.090

Range of Lengths

From

Toa

56

220

63

260

70

280

80

320

90

360

100

400

0 −0.110 0 −0.110

0 −0.160

1.00

1.20

1.60

2.00

2.50

3.00

a See Preferred Lengths and Tolerances starting on page 2539 for preferred maximum length of key.

All dimensions in this standard are in millimeters (mm).

Figures for Table 2 and Table 4 x b

Hub

t2

t1

h Key

d

d+t2 d–t1

Shaft

x

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Section x-x

r

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Key size b×h Basic (mm) Size

Keyway Width. Tolerancea and Resulting Fitsb Normal Fit Close Fit Free Fit Shaft Hub Shaft and Hub Shaft Hub N9 Fit JS9 Fit P9 Fit H9 Fit D10 Fit

8×7

8

10 × 6

10

10 × 8

10

5

12 × 6

12

3.5

12 × 8

12

5

14 × 9 16 × 7 16 × 10 18 × 7 18 × 11

14 16 16 18 18

0.008L 0.031T

+0.025 0

0.039L +0.060 0T +0.020

0.074L 0.020L

 0 −0.030

0.018L +0.0150 0.030T −0.0150

0.033L 0.015T

−0.012 −0.042

0.006L 0.042T

+0.030  0

0.048L 0T

+0.078 +0.030

0.096L 0.030L

 0 −0.036

0.022L +0.0180 0.036T −0.0180

0.040L 0.018T

−0.015 −0.051

0.007L 0.051T

+0.036  0

0.058L 0T

+0.098 +0.040

0.120L 0.040L

 0 −0.043

0.027L +0.0215 0.0485L 0.043T −0.0215 0.0215T

−0.018 −0.061

0.009L 0.061T

+0.043  0

0.070L 0T

+0.120 +0.050

0.147L 0.050L

1.2 1.8 2.5 1.8 3 2.5 3.5 3

4 3.5

3.5 5.5 4 6 4 7

Tolerance

+0.1 0

+0.2 0 +0.1 0 +0.2 0 +0.1 0 +0.2 0 +0.1 0 +0.2  0

Hub, t2

Basic Size 1 1.4 1.8 1.4 2.8 1.8 2.8 2.8

3.3 2.8 3.3 2.8

Toler­ ance

+0.1 0

Min. Max. 0.08

0.16

0.16

0.25

0.25

0.4

+0.1 0 +0.1 0 +0.2 0 +0.1 0

3.3

+0.2

2.8

+0.1 0

3.8 3.3 4.3 3.3 4.4

Radius, r

+0.2  0

Machinery's Handbook, 31st Edition Metric Keys and Keyways

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2 3 4 5 6 6 6 8

14

−0.006 −0.031

Basic Size

2×2 3×3 4×4 5×3 5×5 6×4 6×6 8×5

14 × 6

−0.004 0.010L +0.0125 0.0265L −0.029 0.029T −0.0125 0.0125T

Depth Shaft, t1

2542

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Table 2. Keyway Dimensions and Tolerances for Metric Square and Rectangular Parallel Keys ANSI/ASME B18.25.1M-1996 (Withdrawn)

Keyway Width. Tolerancea and Resulting Fitsb Normal Fit Close Fit Free Fit Shaft Hub Shaft and Hub Shaft Hub N9 Fit JS9 Fit P9 Fit H9 Fit D10 Fit

0.011L 0.074T

+0.052  0

0.085L 0T

+0.149 +0.065

0.182L 0.065L

0.013L 0.088T

+0.062  0

0.101L 0T

+0.180 +0.080

0.219L 0.080L

0.014L 0.106T

+0.074  0

0.120L 0T

+0.220 +0.100

0.266L 0.100L

0.017L +0.087 0.1254T  0

0.139L 0T

+0 260 +0.120

0.314L 0.120L

Basic TolerSize ance 5 7.5 5.5 9 5.5 +0.2 9  0 6 10 7c 11c 7.5c 12 13 15 17 20 +0.3  0 20 22 25 28 31

Hub, t2

Basic Size 3.3 4.9 3.8 5.4 3.8 5.4 4.3 6.4 4.4 7.4 4.9 8.4 9.4 10.4 11.4 12.4 12.4 14.4 15.4 17.4 19.5

Toler­ ance

+0.2  0

+0.3  0

Radius, r Min. Max.

0.4

0.06

0.7

1.0

1.2

1.6

2.0

2.5

a Some of the tolerances are expressed as plus-plus. See Tolerances on page 2539 for more information.

b Resulting fits: L indicates a clearance between the key and keyway; T indicates an interference between the key and keyway. c This value differs from that given in ANSI/ASME B18.25.1M-1996 (Withdrawn), which is believed to be inaccurate.

2543

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Key size b×h Basic (mm) Size 20 20 × 8 20 20 × 12 22 22 × 9 22 22 × 14 0 0.033L +0.026 0.059L −0.022 25 −0.052 0.052T −0.026 0.026T −0.074 25 × 9 25 25 × 14 28 28 × 10 28 28 × 16 32 32 × 11 32 32 × 18 36 36 × 12 0 0.039L +0.031 0.070L −0.026 36 −0.062 0.062T −0.031 0.031T −0.088 36 × 20 40 40 × 22 45 45 × 25 50 50 × 28 56 56 × 32 63 63 × 32 0 0.046L +0.037 0.083L −0.032 70 −0.074 0.074T −0.037 0.037T −0.106 70 × 36 80 80 × 40 90 0.054L +0.0435 0.0975L −0.037 90 × 45 0 100 × 50 100 −0.087 0.87T −0.0435 0.0435T −0.1254

Depth Shaft, t1

Machinery's Handbook, 31st Edition Metric Keys and Keyways

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Table Keyway Dimensions and Tolerances for Metric and Rectangular Parallel ANSI/ASME B18.25.1M-1996 (Withdrawn) Table2.2.(Continued) Keyway Dimensions and Tolerances for Metric SquareSquare and Rectangular Parallel KeysKeys ANSI/ASME B18.25.1M-1996 (Withdrawn)

Machinery's Handbook, 31st Edition Metric Keys and Keyways

2544

Metric Square And Rectangular Keys and Keyways: Width Tolerances and Deviations Greater than Basic Size

This ANSI/ASME B18.25.3M standard covers requirements for square and rectangular parallel keys and keyways intended for both alignment of shafts and hubs, and transmitting torque between shafts and hubs. Keys covered by this standard have a relatively loose width toler­ance. All width tolerances are positive. Keys with minus width tolerances and a smaller tolerance range are covered by ANSI/ASME B18.25.1M-1996 (Withdrawn). Dimensions and tolerances for square and rectangular keys are shown in Table 3. Recommended dimensions and tolerances for keyways are shown in Table 4. All dimensions in this standard are in millimeters. Table 3. Dimensions and Tolerances for Metric Square and Rectangular Parallel Keys ANSI/ASME B18.25.3M-2003

Width, b

Width Tolerances and Deviations Greater than Basic Size Thickness, h

Basic Size

Tolerance

Basic Size

Tolerance

2 3 4 5 6

+0.040 −0.000

2 3 4 5 6

5 6

+0.045 −0.000

+0.045 −0.000

8

10

+0.050 −0.000

7 6

6 8 +0.075 −0.000

16 18

20

4 5

8

12

14

3

6 9 7 10 7 11

+0.050 −0.033

8 12

Chamfer or Radius, s Min.

Max.

+0.040 −0.000

0.16

0.25

+0.045 −0.000

0.25

0.40

Square Keys

Rectangular Keys +0.160 −0.000 +0.175 −0.000

+0.190 −0.000 +0.175 −0.000 +0.19 −0.000 +0.175 −0.000 +0.190 −0.000 +0.175 −0.000

0.25

0.40

0.40

0.40

0.60

0.60

+0.190 −0.000 +0.210 −0.000 +0.190 −0.000 +0.210 −0.000

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0.60

0.80

Range of Lengths From

Toa  

8 10 14

20 36 45 56 70

10

56

14

70

18

90

22

110

28

140

36

160

45

180

50

200

6

56 63

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Machinery's Handbook, 31st Edition Metric Keys and Keyways

2545

Table Table 3. (Continued) Dimensions and Tolerances for Metric Square 3. Dimensions and Tolerances for Metric Square and and Rectangular Rectangular Parallel Parallel Keys Keys ANSI/ASME ANSI/ASME B18.25.3M-2003 B18.25.3M-2003 Width, b

Basic Size

Width Tolerances Tolerances and and Deviations Deviations Greater Width Greater than than Basic Basic Size Size

Tolerance

Thickness, h

Basic Size 6

22

14 +0.050 −0.033

25

9 14 10

28 32 36

+0.090 −0.000

40 45 50 56 63 70 80 90 100

+0.125 −0.000 +0.135 −0.000

16 11 18 12 20 22 25 28 32 32 36 40 45 50

Chamfer or Radius, s

Tolerance +0.175 −0.000 +0.210 −0.000 +0.210 −0.000 +0.190 −0.000 +0.210 −0.000

Min.

0.60

+0.280 −0.000

+0.310 −0.000

Max.

Range of Lengths From

Toa  

70

280

80

320

90

360

100

400

0.80

1.00

1.20

1.60

2.00

2.50

3.00

a See Preferred Lengths and Tolerances on page 2545 for preferred maximum length of key except basic width of 2 mm.

Comparison with ISO R773-1969 and 2491-1974.—This standard has greater toler­ances than ISO Standards R773-1969 and 2491-1974. Product manufactured to this stan­dard is not interchangeable dimensionally with product manufactured to the ISO standards nor is product manufactured to the ISO standards dimensionally interchangeable with product manufactured to this standard. ISO standards do not include hardened keys. Preferred Lengths and Tolerances.—Preferred lengths and tolerances of square and rectangular keys are shown below. Tolerances are JS 16 from ANSI/ASME B4.2. To minimize problems due to lack of straightness, key length should be less than 10 times the key width. 6

Length

8, 10

12, 14, 16, 18

20, 22, 25, 28 32, 36, 45, 50 56, 63, 70, 80

±Tolerances

Length

±Tolerances

125, 140, 160, 180

1.25

0.375

90, 100, 110

0.55

200, 220, 250

0.45

0.65

0.80 0.95

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280

320, 360, 400

1.10

1.45

1.60

1.80

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Normal Fit

Key Size, b×h

2 3 4 5 5 6 6 8

Shaft Toler­ ance Fit

Hub Toler­ ance Fit

Close Fit Shaft and Hub Toler­ ance Fit

Keyway Depth

Shaft, t1

Free Fit

Shaft Toler­ ance Fit

Hub Toler­ ance Fit

+0.040 0.040L +0.050 0.050L +0.034 +0.010 0.030T +0.025 0.015T −0.008

0.034L +0.066 0.032T +0.040

0.066L +0.086 0T +0.060

0.086L 0.020L

+0.045 0.045L +0.060 0.060L +0.035 +0.015 0.030T +0.015 0.015T −0.005

0.035L +0.075 0.040T +0.045

0.075L +0.105 0T +0.075

0.105L 0.030L

+0.055 0.055L +0.075 0.075L +0.040 +0.015 0.035T +0.035 0.015T 0.000

0.040L 0.050T

0.090L +0.130 0T +0.090

0.130L 0.040L

Nominal 1.2 1.8 2.5 1.8 3 2.5 3.5 3

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8×7

8

10×6

10

10×8

10

5

12×6

12

3.5

12×8

12

14×6

14

14×9

14

+0.080 0.080L +0.095 0.095L +0.055 −0.030 0.045T +0.055 0.020T −0.015

0.055L 0.060T

+0.090 +0.050

+0.135 +0.075

0.135L +0.185 0T +0.125

0.185L 0.050L

4 3.5

5 3.5 5.5

Toler­ ance

+0.1 0

+0.2 0 +0.1 0 +0.2 0 +0.1 0 +0.2 0 +0.1 0 +0.2 0

Hub, t2 Nomi­ nal 1 1.4 1.8 1.4 2.8 1.8 2.8 2.8 3.3 2.8 3.3 2.8 3.3 2.8 3.8

Radius, r

Toler­ ance

Max. 0.16

+0.1 0 0.25 +0.2 0 +0.1 0 +0.2 0 +0.1 0 +0.2 0 +0.1 0 +0.2 0

0.6

Machinery's Handbook, 31st Edition Metric Keys and Keyways

2×2 3×3 4×4 5×3 5×5 6×4 6×6 8×5

Nominal

Keyway Width Tolerance and Resulting Fita

2546

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Table 4. Keyway Dimensions and Tolerances for Metric Square and Rectangular Parallel Keys ANSI/ASME B18.25.3M-2003 (R2008) Width Tolerances and Deviations Greater than Basic Size

Normal Fit

Keyway Width Tolerance and Resulting Fita Close Fit Shaft and Hub Toler­ ance Fit

Keyway Depth

Free Fit

Shaft, t1

Hub, t2

Radius, r

2547

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Shaft Hub Shaft Hub Key Size, Nom- Toler­ NomToler­ Nomi­ Toler­ Toler­ Toler­ Toler­ b×h inal inal ance nal ance Max. ance Fit ance Fit ance Fit ance Fit 16 4 3.3 16 × 7 16 6 4.3 16 × 10 +0.080 0.080L +0.095 0.095L +0.055 0.055L +0.135 0.135L +0.185 0.185L −0.030 0.045T +0.055 0.020T −0.015 0.060T +0.075 0T +0.125 0.050L 18 4 3.3 18 × 7 18 7 4.4 18 × 11 20 5 3.3 20 × 8 10 7.5 4.9 20 × 12 10 5.5 3.8 22 × 9 0.6 22 9 5.4 22 × 14 +0.085 0.085L +0.110 0.110L +0.050 0.050L +0.135 0.150L +0.200 0.200L −0.035 0.050T +0.060 0 025T −0.010 0.075T +0.085 0T +0.110 0.065L 25 5.5 3.8 25 × 9 25 9 5.4 25 × 14 +0.2 +0.2 28 6 4.3 28 × 10 0 0 28 10 6.4 28 × 16 32 7b 4.4 32 × 11 32 11b 7.4 32 × 18 36 7.5b 4.9 36 × 12 +0.110 0.110L +0.170 0.170L +0.090 0.090L +0.200 0.225L +0.300 0.300L 56 56 × 32 −0.050 0.075T +0.090 0.035T −0.020 0.105T +0.125 0T +0.225 0.100L 1.6 20 12.4 63 63 × 32 70 22 14.4 70 × 36 80 25 15.4 80 × 40 90 +0.130 0.130L +0.180 0.180L +0.095 0.095L +0.225 0.225L +0.340 0.340L 28 17.4 2.5 90 × 45 100 +0.255 0.120L 31 19.5 100 × 50 −0.050 0.085T +0.090 0.045T −0.015 0.120T +0.135 0T a In columns labeled “Fit,” L indicates the maximum clearance between the key and keyway; the T indicates the maximum interference between the key and keyway. b This value differs from that given in ANSI/ASME B18.25.3M, which is believed to be inaccurate.

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Table 4. KeywayKeyway Dimensions and Tolerances for Metric SquareSquare and Rectangular Parallel Keys Keys ANSI/ASME B18.25.3M-2003 (R2008) Table 4. (Continued) Dimensions and Tolerances for Metric and Rectangular Parallel ANSI/ASME B18.25.3M-2003 (2008) Width Tolerances and Deviations Greater than Basic Size

Machinery's Handbook, 31st Edition Metric Keys and Keyways

2548

Tolerances.—Many of the tolerances shown in Table 3 and Table 4 are from ANSI/ASME B4.2 (ISO 286-1 and ISO 286-2). As a result, in addition to plus-minus tolerances which are common in the United States, some are expressed as plus-plus deviations from the basic size. Designation.—Keys conforming to this standard shall be designed by the following data, preferably in the sequence shown: a) ASME document number; b) product name; c) nominal size, width (b) × height (h) × length; d) style; and e) hardness (if other than non-hardened). Optionally, a part identification number (PIN) per ANSI/ASME B18.24 may be used. Material Requirements.—Same as for ANSI/ASME B18.25.1M-1996 (Withdrawn). See page 2539. Nomimal Shaft Diameter, d Over 6 8 10 12 17

Metric Keyway Sizes According to Shaft Diameter Based on BS 4235:Part 1:1972 (1986)

Nominal Keyway Key Size, Up to Width, b b×h and Incl. Keyways for Square Parallel Keys

8 2 2×2 3 10 3×3 12 4 4×4 17 5 5×5 22 6 6×6 Keyways for Rectangular Parallel Keys 22 30 8 8×7 30 38 10 10 × 8 38 44 12 12 × 8 44 50 14 14 × 9 50 58 16 16 × 10 58 65 18 18 × 11 65 75 20 20 × 12 75 85 22 22 × 14

Nomimal Shaft Diameter, d

Nominal Keyway Key Size, Up to Width, b b×h and Incl. (Cont’d) Keyways for Rectangular Parallel Keys Over 85 95 110 130 150 170 200 230 260 290 330 380 440 …

95 110 130 150 170 200 230 260 290 330 380 440 500 …

25 × 14 28 × 16 32 × 18 36 × 20 40 × 22 45 × 25 50 × 28 56 × 32 63 × 32 70 × 36 80 × 40 90 × 45 100 × 50 …

25 28 32 36 40 45 50 56 63 70 80 90 100 …

Note: This table is NOT part of ANSI/ASME B18.25.1M-1996 (Withdrawn) or ANSI/ASME B18.25.3M-2003, and is included for refer­ence only. The selection of the proper size and type of key must rest with the design authority.

Metric Woodruff Keys and Keyways This ANSI/ASME B18.25.2M standard covers requirements for metric Woodruff keys and keyways intended for both alignment of shafts and hubs, and transmitting torque between shafts and hubs. All dimensions in this standard are in millimeters (mm). Dimensions and tolerances for Woodruff keys are shown in Table 5. Recommended dimensions and toler­ances for keyways are shown in Table 6. For inch series Woodruff keys and keyseats, see ANSI/ASME Standard Woodruff Keys and Keyseats starting on page 2556. Comparison with ISO 3912-1977.—This standard is based on ISO 3912-1977, Wood­ruff Keys and Keyways. However, to improve manufacturability, tolerances are decreased for the keyway width. The resulting fit is approximately the same. Keys manufactured to this standard are functionally interchangeable with keys manufactured to the ISO standard. Because of tighter width tolerances in this standard, products manufactured to the ISO standard may not meet the requirements of this standard.

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Machinery's Handbook, 31st Edition Metric Keys and Keyways

2549

ANSI/ASME B18.25.2M (Withdrawn) also differs from ISO 3912 in that it: a) does not restrict the corners of a key to be chamfered but allows either a chamfer or a radius on the key; b) specifies a key material hardness rather than a tensile property; and c) specifies h12 rather than h11 for the tolerance of the height of the keys. Tolerances.—Many of the tolerances shown in Table 5 and Table 6 are from ANSI/ASME B4.2, Pre­ferred Metric Limits and Fits (ISO 286-1 and ISO 286-2). As a result in addition to plus-minus tolerances which are common in the United States some are expressed as plus-plus devia­tions from the basic size. Table 5. Dimensions for Metric Woodruff Keys ANSI/ASME B18.25.2M-1996 (Withdrawn)

b

x

D

D

x

h1 s

b

x = Sharp edges removed not to exceed S max

Key Size b×h×D

b

Tolerance

h1

l × l.4 × 4

1

1.4

1.5 × 2.6 × 7 2 × 2.6 × 7 2 × 3.7 × 10 2.5 × 3.7 × 10 3 × 5 × 13 3 × 6.5 × 16 4 × 6.5 × 16

1.5 2 2 2.5 3 3 4

2.6 2.6 3.7 3.7 5.0 6.5 6.5

4 × 7.5 × 19

4

5 × 6.5 × 16

5

6.5

5 × 7.5 × 19 5 × 9 × 22 6 × 9 × 22 6 × 10 × 25 8 × 11 × 28

5 5 6 6 8

7.5 9.0 9.0 10.0 11.0

10 × 13 × 32

10

13.0

a Height h

 0 −0.025

b

x

h2

b

s Whitney Form (flat bottom type)

Normal Form (full radius type)

Width

x

h1

Chamfer Diameter, or Radius, Height D s Tolerance Tolerance Tolerance a h12 h2   h12 h12 Min. Max. D  0 −0.10  0 −0.12

7.5

2.1 2.1 3.0 3.0 4.0 5.2 5.2 6.0

 0 −0.15

 0 −0.18

4

1.1  0 −0.10

 0 −0.12

7 7 10 10 13 16 16 19

5.2

16

6.0 7.2 7.2 8.0 8.8

19 22 22 25 28

10.4

 0 −0.15  0 −0.18

32

 0 −0.120  0 −0.150

0.16

0.25

0.25

0.40

0.40

0.60

 0 −0.180  0 −0.210  0 −0.180  0 −0.210  0 −0.250

2 is based on 0.80 times height h1.

Designation.—Keys conforming to this standard shall be designated by the following data, preferably in the sequence as follows: a) ASME document number; b) product name; c) nominal size, width (b) × height (h) × length; d) form; and e) hardness (if other than non-hardened). Example: ANSI/ASME B18.25.2M (Withdrawn), Woodruff Key 6 × 10 × 25 normal hardened; ANSI/ASME B18.25.2M (Withdrawn), Woodruff Key 3 × 5 × 13 Whitney.

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Machinery's Handbook, 31st Edition Metric Keys and Keyways

2550

Material Requirements.—Same as for ANSI/ASME B18.25.2M-1996 (Withdrawn). See page 2539. Advantages of Woodruff Keys.—In the Woodruff key system, half-circular disks of steel are used as keys, the half-circular side of the key being inserted into the keyseat. Part of the key projects and enters into a keyway in the part to be keyed to the shaft in the ordi­nary way. The advantage of this type of key is that the keyway is easily milled by simply sinking a milling cutter, of the same diameter as the diameter of the stock from which the keys are made, into the shaft. The keys are also very cheaply made, as they are simply cut off from round bar stock and milled apart in the center. Examples of Woodruff keyseat cut­ters are shown on page 895. Table 6. Keyway Dimensions for Metric Woodruff Keys ANSI/ASME B18.25.2M-1996 (Withdrawn) A t2

b

Hub

t1

h1 z Key

d+t2 d–t1

d

s

Shaft Section A-A Width

Depth

Key Sizea   b × hl × D   l × l.4 × 4

Basic Size

Shaft N9

Hub S9

Close Fit Shaft & Hub P9

Free Fit Shaft H9

Hub D10

Shaft, t1 Basic Size

1

1.0

1.5

2.0

2 × 2.6 × 7

2

1.8

2 × 3.7 × 10

2

1.5 × 2.6 × 7

2.5 × 3.7 × 10

2.5

−0.004 −0.029

+0.0125 −0.0125

−0.006 −0.031

+0.025 +0.60  0 +0.20

2.9

Tolerance

Toleranceb Normal Fit

Detail z

r

Hub, t2 Basic Size

+0.1  0

2.7

1.0 1.0

3.8

1.4

3

5.3

1.4

4 × 6.5 × 16

4

5.0

1.8

4 × 7.5 × 19

4

6.0

5 × 6.5 × 16

5

4.5

2.3

5 × 7.5 × 19

5

5.5

2.3

5 × 9 × 22

5

7.0

2.3

6 × 9 × 22

6

+0.030 +0.078  0 +0.030

6.5 7.5

6 × 10 × 25 8 × 11 × 28

8

10 × 13 × 32

10

0 −0.036

+0.018 −0.018

−0.015 −0.051

+0.036 +0.098  0 +0.040

8.0 10.0

0.16

0.08

0.25

0.16

0.4

0.25

1.2

3

−0.012 −0.042

Max. Min.

0.8

3 × 6.5 × 16

+0.015 −0.015

Radius, R

0.6

3 × 5 × 13

−0.030   0

Tolerance

A

+0.2  0

+0.3  0

+0.1  0

1.8

2.8 2.8 3.3 3.3

+0.2  0

a The nominal key diameter is the minimum keyway diameter.

b Some of the tolerances are expressed as plus-plus or minus-minus. See Tolerances on page 2549 for more informations.

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Machinery's Handbook, 31st Edition INCH KEYS AND KEYSEATS

2551

ANSI Standard Inch Series Keys and Keyseats.—American National Standard, B17.1 Keys and Keyseats, based on current industry practice, was approved in 1967, and reaf­ firmed in 2013. This standard establishes a uniform relationship between shaft sizes and key sizes for parallel and taper keys as shown in Table 1. Other data in this standard are given in Table 2 and Table 3 through Table 7. The sizes and tolerances shown are for single key appli­cations only. The following definitions are given in the standard. Note: Inch dimensions converted to metric dimensions (enclosed in parentheses) are not included in the standard. Key: A demountable machinery part which, when assembled into keyseats, provides a positive means for transmitting torque between the shaft and hub. Keyseat: An axially located rectangular groove in a shaft or hub. This standard recognizes that there are two classes of stock for parallel keys used by industry. One is a close, plus toleranced key stock and the other is a broad, negative toler­ anced bar stock. Based on the use of two types of stock, two classes of fit are shown: Class 1: A clearance or metal-to-metal side fit obtained by using bar stock keys and key­ seat tolerances as given in Table 4. This is a relatively free fit and applies only to parallel keys. Class 2: A side fit, with possible interference or clearance, obtained by using key stock and keyseat tolerances as given in Table 4. This is a relatively tight fit. Class 3: This is an interference side fit and is not tabulated in Table 4 since the degree of interference has not been standardized. However, it is suggested that the top and bottom fit range given under Class 2 in Table 4, for parallel keys be used. Table 1. Key Size Versus Shaft Diameter ANSI/ASME B17.1-1967 (R2013) Nominal Shaft Diameter Over

7∕16

7∕16

9 ∕16

9 ∕16 7∕8

11 ∕4 13∕8 13∕4

21 ∕4 23∕4

31 ∕4 33∕4

41 ∕2 51 ∕2 61 ∕2 71 ∕2 9 a

To (Incl.)

5 ∕16

Width, W 3 ∕3 2 1 ∕8

7∕8

11 ∕4 13∕8 13∕4

21 ∕4 23∕4

31 ∕4 33∕4

41 ∕2 51 ∕2 61 ∕2

Nominal Key Size Height, H Square Rectangular 3 ∕3 2 …

1

1 ∕8

3 ∕3 2

Normal Keyseat Depth H/2 Square Rectangular 3 ∕6 4 … 1 ∕16

3 ∕6 4

3 ∕16

3 ∕16

1 ∕8

3 ∕3 2

1 ∕16

1 ∕4

1 ∕4

3 ∕16

1 ∕8

3 ∕3 2

5 ∕16

5 ∕16

1 ∕4

5 ∕3 2

1 ∕8 1 ∕8

3 ∕8

3 ∕8

1 ∕4

3 ∕16

1 ∕2

1 ∕2

3 ∕8

1 ∕4

3 ∕16

5 ∕8

5 ∕8

7∕16

5 ∕16

7∕3 2

3 ∕4

3 ∕4

1 ∕2

3 ∕8

1 ∕4

7∕8

7∕8

5 ∕8

7∕16

5 ∕16

11 ∕4

3 ∕4

1 ∕2

3 ∕8

7∕8

5 ∕8

7∕16

3 ∕4

1 ∕2

11 ∕4 11 ∕2

1

11 ∕2

1

Square Keys preferred for shaft diameters above this line; rectangular keys, below 7∕8 71 ∕2 13∕4 13∕4 11 ∕2 a 9 2 2 1 11 ∕2

11

21 ∕2

21 ∕2

13∕4

11 ∕4

3 ∕4 3 ∕4 7∕8

Some key standards show 1¼ inches; preferred height is 1½ inches.

All dimensions are given in inches. For larger shaft sizes, see ANSI/ASME Standard Woodruff Keys and Keyseats.

Key Size versus Shaft Diameter: Shaft diameters are listed in Table 1 for identification of various key sizes and are not intended to establish shaft dimensions, tolerances or selec­ tions. For a stepped shaft, the size of a key is determined by the diameter of the shaft at the point of location of the key. Up through 61 ⁄2 -inch (165.1 mm) diameter shafts square keys are preferred; rectangular keys are preferred for larger shafts.

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Machinery's Handbook, 31st Edition INCH KEYS AND KEYSEATS

2552

If special considerations dictate the use of a keyseat in the hub shallower than the pre­ ferred nominal depth shown, it is recommended that the tabulated preferred nominal stan­dard keyseat always be used in the shaft. Keyseat Alignment Tolerances: A tolerance of 0.010 inch (0.254 mm), max is provided for offset (due to parallel displacement of keyseat centerline from centerline of shaft or bore) of keyseats in shaft and bore. The following tolerances for maximum lead (due to angular displacement of keyseat centerline from centerline of shaft or bore and measured at right angles to the shaft or bore centerline) of keyseats in shaft and bore are specified: 0.002 inch (0.0508 mm) for keyseat length up to and including 4 inches (101.6 mm); 0.0005 inch per inch of length (0.0127 mm per mm) for keyseat lengths above 4 inches to and including 10 inches (254 mm); and 0.005 inch (0.127 mm) for keyseat lengths above 10 inches. For the effect of keyways on shaft strength, see Effect of Keyways on Shaft Strength on page 301.

T

S

Table 2. Depth Control Values S and T for Shaft and Hub ANSI/ASME B17.1-1967 (R2013) Nominal Shaft Diameter 1 ∕2

Shafts, Parallel and Taper

Hubs, Parallel

Hubs, Taper

Square

Rectangular

Square

Rectangular

Square

S

S

T

T

T

0.430

0.445

0.560

0.544

0.535

Rectangular T

0.519

7∕8 15 ∕16 1 11 ∕16 11 ∕8 13∕16 11 ∕4 15∕16 13∕8 17∕16 11 ∕2 19∕16 15∕8 111 ∕16 13∕4 113∕16 17∕8 115∕16 2 21 ∕16 21 ∕8 23∕16 21 ∕4 25∕16 23∕8

0.493 0.517 0.581 0.644 0.708 0.771 0.796 0.859 0.923 0.986 1.049 1.112 1.137 1.201 1.225 1.289 1.352 1.416 1.479 1.542 1.527 1.591 1.655 1.718 1.782 1.845 1.909 1.972 1.957 2.021

0.509 0.548 0.612 0.676 0.739 0.802 0.827 0.890 0.954 1.017 1.080 1.144 1.169 1.232 1.288 1.351 1.415 1.478 1.541 1.605 1.590 1.654 1.717 1.781 1.844 1.908 1.971 2.034 2.051 2.114

0.623 0.709 0.773 0.837 0.900 0.964 1.051 1.114 1.178 1.241 1.304 1.367 1.455 1.518 1.605 1.669 1.732 1.796 1.859 1.922 2.032 2.096 2.160 2.223 2.287 2.350 2.414 2.477 2.587 2.651

0.607 0.678 0.742 0.806 0.869 0.932 1.019 1.083 1.146 1.210 1.273 1.336 1.424 1.487 1.543 1.606 1.670 1.733 1.796 1.860 1.970 2.034 2.097 2.161 2.224 2.288 2.351 2.414 2.493 2.557

0.598 0.684 0.748 0.812 0.875 0.939 1.026 1.089 1.153 1.216 1.279 1.342 1.430 1.493 1.580 1.644 1.707 1.771 1.834 1.897 2.007 2.071 2.135 2.198 2.262 2.325 2.389 2.452 2.562 2.626

0.582 0.653 0.717 0.781 0.844 0.907 0.994 1.058 1.121 1.185 1.248 1.311 1.399 1.462 1.518 1.581 1.645 1.708 1.771 1.835 1.945 2.009 2.072 2.136 2.199 2.263 2.326 2.389 2.468 2.532

21 ∕2

2.148

2.242

2.778

2.684

2.753

2.659

9 ∕16 5 ∕8

11 ∕16 3 ∕4

13 ∕16

27∕16

2.084

2.178

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2.714

2.621

2.689

2.596

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Machinery's Handbook, 31st Edition INCH KEYS AND KEYSEATS

2553

Table Table 2. (Continued) Control T for Shaft and Hub 2. DepthDepth Control ValuesValues S and STand for Shaft and Hub ANSI/ASME B17.1-1967 (R2013) Nominal Shaft Diameter 29∕16 25∕8 211 ∕16 23∕4 213∕16 27∕8 215∕16 3 31 ∕16 31 ∕8 33∕16 31 ∕4 35∕16 33∕8 37∕16 31 ∕2 39∕16 35∕8 311 ∕16 33∕4 313∕16 37∕8 315∕16 4 43∕16 41 ∕4 43∕8 47∕16 41 ∕2 43∕4 47∕8 415∕16 5 53∕16 51 ∕4 57∕16 51 ∕2 53∕4 515∕16 6 61 ∕4 61 ∕2 63∕4 7 71 ∕4 71 ∕2 73∕4 8 9 10 11 12 13 14 15

Shafts, Parallel and Taper

Square S 2.211 2.275 2.338 2.402 2.387 2.450 2.514 2.577 2.641 2.704 2.768 2.831 2.816 2.880 2.943 3.007 3.070 3.134 3.197 3.261 3.246 3.309 3.373 3.436 3.627 3.690 3.817 3.880 3.944 4.041 4.169 4.232 4.296 4.486 4.550 4.740 4.803 4.900 5.091 5.155 5.409 5.662 5.760 6.014 6.268 6.521 6.619 6.873 7.887 8.591 9.606 10.309 11.325 12.028 13.043

Rectangular S 2.305 2.369 2.432 2.495 2.512 2.575 2.639 2.702 2.766 2.829 2.893 2.956 2.941 3.005 3.068 3.132 3.195 3.259 3.322 3.386 3.371 3.434 3.498 3.561 3.752 3.815 3.942 4.005 4.069 4.229 4.356 4.422 4.483 4.674 4.737 4.927 4.991 5.150 5.341 5.405 5.659 5.912 a 5.885 a 6.139 a 6.393 a 6.646 6.869 7.123 8.137 8.966 9.981 10.809 11.825 12.528 13.543

Hubs, Parallel

Square T 2.841 2.905 2.968 3.032 3.142 3.205 3.269 3.332 3.396 3.459 3.523 3.586 3.696 3.760 3.823 3.887 3.950 4.014 4.077 4.141 4.251 4.314 4.378 4.441 4.632 4.695 4.822 4.885 4.949 5.296 5.424 5.487 5.551 5.741 5.805 5.995 6.058 6.405 6.596 6.660 6.914 7.167 7.515 7.769 8.023 8.276 8.624 8.878 9.892 11.096 12.111 13.314 14.330 15.533 16.548

Hubs, Taper

Rectangular T 2.748 2.811 2.874 2.938 3.017 3.080 3.144 3.207 3.271 3.334 3.398 3.461 3.571 3.635 3.698 3.762 3.825 3.889 3.952 4.016 4.126 4.189 4.253 4.316 4.507 4.570 4.697 4.760 4.824 5.109 5.236 5.300 5.363 5.554 5.617 5.807 5.871 6.155 6.346 6.410 6.664 6.917 a 7.390 a 7.644 a 7.898 a 8.151 8.374 8.628 9.642 10.721 11.736 12.814 13.830 15.033 16.048

Square T 2.816 2.880 2.943 3.007 3.117 3.180 3.244 3.307 3.371 3.434 3.498 3.561 3.671 3.735 3.798 3.862 3.925 3.989 4.052 4.116 4.226 4.289 4.353 4.416 4.607 4.670 4.797 4.860 4.924 5.271 5.399 5.462 5.526 5.716 5.780 5.970 6.033 6.380 6.571 6.635 6.889 7.142 7.490 7.744 7.998 8.251 8.599 8.853 9.867 11.071 12.086 13.289 14.305 15.508 16.523

Rectangular T 2.723 2.786 2.849 2.913 2.992 3.055 3.119 3.182 3.246 3.309 3.373 3.436 3.546 3.610 3.673 3.737 3.800 3.864 3.927 3.991 4.101 4.164 4.228 4.291 4.482 4.545 4.672 4.735 4.799 5.084 5.211 5.275 5.338 5.529 5.592 5.782 5.846 6.130 6.321 6.385 6.639 6.892 a 7.365 a 7.619 a 7.873 a 8.126 8.349 8.603 9.617 10.696 11.711 12.789 13.805 15.008 16.023

a 1¾ × 1½ inch key. All dimensions are given in inches. See Table 4 for tolerances.

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Machinery's Handbook, 31st Edition INCH KEYS AND KEYSEATS

2554

Table 3. ANSI Standard Plain and Gib Head Keys ANSI/ASME B17.1-1967 (R2013)

Parallel

W

W

Hub length

…

Parallel

1 ∕8

3 ∕16

3 ∕16

H

3 ∕4

11 ∕2

+0.000

11 ∕4 3

+0.000 +0.000 +0.000 +0.000 +0.001 +0.002 +0.003

B

1 ∕4

5 ∕16

5 ∕16

1 ∕8

1 ∕4

1 ∕4

1 ∕4 7∕16

3 ∕8

3 ∕16

5 ∕16

5 ∕16

7∕16

1 ∕4

7∕16

3 ∕8

3 ∕8

3 ∕8

5 ∕8

1 ∕2

1 ∕4

7∕16

3 ∕8

1 ∕2

1 ∕2

7∕8

5 ∕8

3 ∕8

5 ∕8

1 ∕2

11 ∕4

3 ∕4

7∕16

3 ∕4

9 ∕16

7∕8

1 ∕2

7∕8

5 ∕8

7∕8

+0.000

A

1 ∕2

7∕8

+0.002 +0.003 +0.000

Gib Head Nominal Dimensions Rectangular Nominal Key Size H A B H Width, W 3 ∕3 2 3 ∕16 1 ∕8 1 1

1 ∕4

5 ∕8

+0.001

11 ∕2 3 4 6 7 11 ∕4 3 7

5 ∕16

3 ∕4

+0.000

11 ∕4 3 7 3 ∕4

1 ∕4

5 ∕8

+0.000

31 ∕2

5 ∕16

3 ∕4

+0.000

21 ∕2

3 ∕4

Square

Nominal Key Size Width, W 1 ∕8

+0.003

11 ∕2 3 4 6 …

Plain or Gib Head Square or Rectangular

+0.002

3 ∕4

11 ∕4 3 …

Bar Stock

+0.001

31 ∕2

21 ∕2 …

Keystock

Width, W

11 ∕4 3

11 ∕2

Rectangular

Taper

11 ∕4 3

…

Bar Stock

1

13∕8

1

5 ∕8

1

B

W*

W Alternate Plain Taper

Nominal Key Size Width W Over To (Incl.)

Square

A

H

Plain Taper

Keystock

W*

Gib Head Taper

H

Key

45°

H

Plain and gib head taper keys have a 1/8” taper in 12” W* Hub length

W

B/2 Approx

Hub length

H

3 ∕4

11 ∕4

11 ∕4

13∕4 2

13∕4 2

31 ∕2 …

31 ∕2 …

11 ∕2

21 ∕2 3

11 ∕2

21 ∕2 3

Tolerance

−0.000

Height, H

+0.001

−0.000

+0.003

−0.000

+0.002

−0.000 −0.000

+0.000

−0.002

−0.005 −0.006 −0.008 −0.013 −0.000

+0.005 +0.005

−0.000 −0.000 Square

…

−0.004

+0.000 +0.000 +0.000 +0.000 +0.005

−0.005 −0.006 −0.008 −0.013 −0.000

6

−0.005 −0.005 −0.003

+0.000

−0.004

5

−0.005

+0.005 +0.005 +0.000

−0.000 −0.000 −0.003

31 ∕2 4

−0.006

+0.005

−0.000

23∕4

−0.004

+0.000

−0.006

23∕8

−0.003

+0.000

−0.004

15∕8 2

−0.002

+0.000

−0.003

A

−0.000

−0.000 −0.000

Rectangular B

11 ∕8

17∕16

13∕4 2

21 ∕4 3 31 ∕2 4 …

H

A

B

13∕8

1

11 ∕2

23∕8

13∕4

13∕4 2

23∕4

13∕4 2

…

…

3 ∕4 7∕8

1

11 ∕2

21 ∕2 …

11 ∕4 15∕8

23∕8 31 ∕2 4

7∕8

11 ∕8

21 ∕4 3

All dimensions are given in inches. *For locating position of dimension H. Tolerance does not apply. For larger sizes the following relationships are suggested as guides for establishing A and B: A = 1.8H and B = 1.2H.

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Machinery's Handbook, 31st Edition INCH KEYS AND KEYSEATS

2555

Table 4. ASME Standard Fits for Parallel and Taper Keys ANSI/ASME B17.1-1967 (R2013)

Key Width Type of Key

Square

Over …

1 ∕2

1 ∕2

3 ∕4

3 ∕4

11 ∕2

11 ∕2

21 ∕2

21 ∕2

31 ∕2

…

1 ∕2

3 ∕4

Parallel Square

Parallel Rectan­ gular

Taper

1

1

1 ∕2

Rectan­ gular

To (Incl.)

3 ∕4

1

1

11 ∕2

11 ∕2

3

3

4

4

6

6

7

…

11 ∕4

11 ∕4

3

3

31 ∕2

…

11 ∕4

11 ∕4

3

3

7

…

11 ∕4

11 ∕4

3

3

b

Side Fit

Width Tolerance Key

Key-Seat

Top and Bottom Fit

Fit Rangea

Depth Tolerance

Class 1 Fit for Parallel Keys

Shaft Hub Key-Seat Key-Seat

Key

+0.000 +0.002 0.004 CL +0.000 0.000 −0.002 −0.000 −0.002 +0.000 +0.003 0.005 CL +0.000 0.000 −0.002 −0.000 −0.002 +0.000 +0.003 0.006 CL +0.000 0.000 −0.003 −0.000 −0.003 +0.000 +0.004 0.007 CL +0.000 0.000 −0.003 −0.000 −0.003 +0.000 +0.004 0.008 CL +0.000 0.000 −0.004 −0.000 −0.004 +0.000 +0.004 0.010 CL +0.000 0.000 −0.006 −0.000 −0.006 +0.000 +0.002 0.005 CL +0.000 0.000 −0.003 −0.000 −0.003 +0.000 +0.003 0.006 CL +0.000 0.000 −0.003 −0.000 −0.003 +0.000 +0.003 0.007 CL +0.000 0.000 −0.004 −0.000 −0.004 +0.000 +0.004 0.008 CL +0.000 0.000 −0.004 −0.000 −0.004 +0.000 +0.004 0.009 CL +0.000 0.000 −0.005 −0.000 −0.005 +0.000 +0.004 0.010 CL +0.000 0.000 −0.006 −0.000 −0.006 +0.000 +0.004 0.012 CL +0.000 0.000 −0.008 −0.000 −0.008 +0.000 +0.004 0.017 CL +0.000 0.000 −0.013 −0.000 −0.013 Class 2 Fit for Parallel and Taper Keys +0.001 −0.000 +0.002 −0.000 +0.003 −0.000 +0.001 −0.000 +0.002 −0.000 +0.003 −0.000 +0.001 −0.000 +0.002 −0.000 +0.003 −0.000

+0.002 −0.000 +0.002 −0.000 +0.002 −0.000 +0.002 −0.000 +0.002 −0.000 +0.002 −0.000 +0.002 −0.000 +0.002 −0.000 +0.002 −0.000

0.002 CL 0.001 INT 0.002 CL 0.002 INT 0.002 CL 0.003 INT 0.002 CL 0.001 INT 0.002 CL 0.002 INT 0.002 CL 0.003 INT 0.002 CL 0.001 INT 0.002 CL 0.002 INT 0.002 CL 0.003 INT

a Limits of variation. CL = Clearance; INT = Interference.

+0.001 −0.000 +0.002 −0.000 +0.003 −0.000 +0.005 −0.005 +0.005 −0.005 +0.005 −0.005 +0.005 −0.000 +0.005 −0.000 +0.005 −0.000

Fit Rangea

+0.000 −0.015 +0.000 −0.015 +0.000 −0.015 +0.000 −0.015 +0.000 −0.015 +0.000 −0.015 +0.000 −0.015 +0.000 −0.015 +0.000 −0.015 +0.000 −0.015 +0.000 −0.015 +0.000 −0.015 +0.000 −0.015 +0.000 −0.015

+0.010 −0.000 +0.010 −0.000 +0.010 −0.000 +0.010 −0.000 +0.010 −0.000 +0.010 −0.000 +0.010 −0.000 +0.010 −0.000 +0.010 −0.000 +0.010 −0.000 +0.010 −0.000 +0.010 −0.000 +0.010 −0.000 +0.010 −0.000

0.032 CL 0.005 CL 0.032 CL 0.005 CL 0.033 CL 0.005 CL 0.033 CL 0.005 CL 0.034 CL 0.005 CL 0.036 CL 0.005 CL 0.033 CL 0.005 CL 0.033 CL 0.005 CL 0.034 CL 0.005 CL 0.034 CL 0.005 CL 0.035 CL 0.005 CL 0.036 CL 0.005 CL 0.038 CL 0.005 CL 0.043 CL 0.005 CL

+0.000 −0.015 +0.000 −0.015 +0.000 −0.015 +0.000 −0.015 +0.000 −0.015 +0.000 −0.015 +0.000 −0.015 +0.000 −0.015 +0.000 −0.015

+0.010 −0.000 +0.010 −0.000 +0.010 −0.000 +0.010 −0.000 +0.010 −0.000 +0.010 −0.000 +0.010 −0.000 +0.010 −0.000 +0.010 −0.000

0.030 CL 0.004 CL 0.030 CL 0.003 CL 0.030 CL 0.002 CL 0.035 CL 0.000 CL 0.035 CL 0.000 CL 0.035 CL 0.000 CL 0.005 CL 0.025 INT 0.005 CL 0.025 INT 0.005 CL 0.025 INT

b To (Incl.) 3½-inch Square and 7-inch Rectangular key widths.

All dimensions are given in inches. See also text on page 2539.

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Machinery's Handbook, 31st Edition INCH KEYS AND KEYSEATS

2556

Table 5. Suggested Keyseat Fillet Radius and Key Chamfer ANSI/ASME B17.1-1967 (R2013) Keyseat Depth, H/2

Keyseat Depth, H/2

Over

To (Incl.)

Fillet Radius

45 deg. Chamfer

1∕ 8

1∕ 4

1∕ 32

3∕ 64

7∕ 8

11 ∕4

3∕ 16

7∕ 32

1∕ 4

1∕ 2

1∕ 16

5∕ 64

11 ∕4

13∕4

1∕ 4

9∕ 32

1∕ 2

7∕ 8

1∕ 8

5∕ 32

13∕4

21 ∕2

3∕ 8

13 ∕ 32

Over

To (Incl.)

Fillet Radius

45 deg. Chamfer

All dimensions are given in inches.

Table 6. Standard Keyseat Tolerances for Electric Motor and Generator Shaft Extensions ANSI/ASME B17.1-1967 (R2013) Keyseat Width Over

To (Incl.)

…

1∕ 4

1∕ 4

3∕ 4

Width Tolerance

Depth Tolerance

+0.001

+0.000

−0.001

−0.015

+0.000

+0.000

11 ∕4

3∕ 4

−0.002

−0.015

+0.000

+0.000

−0.003

−0.015

All dimensions are given in inches.

Table 7. Set Screws for Use Over Keys ANSI/ASME B17.1-1967 (R2013) Nom. Shaft Dia. Over

To (Incl.)

Nom. Key Width

Set Screw Dia.

Nom. Shaft Dia.

Nom. Key Width

Set Screw Dia. 1∕ 2

Over

To (Incl.)

5∕ 16

7∕ 16

3∕ 32

No. 10

21 ∕4

23∕4

5∕ 8

7∕ 16

9∕ 16

1∕ 8

No. 10

23∕4

31 ∕4

3∕ 4

5∕ 8

9∕ 16

7∕ 8

3∕ 16

1∕ 4

31 ∕4

33∕4

7∕ 8

3∕ 4

7∕ 8

11 ∕4

1∕ 4

5∕ 16

33∕4

41 ∕2

1

3∕ 4

11 ∕4

13∕8

5∕ 16

3∕ 8

41 ∕2

51 ∕2

11 ∕4

7∕ 8

13∕8

13∕4

3∕ 8

3∕ 8

51 ∕2

61 ∕2

11 ∕2

13∕4

21 ∕4

1∕ 2

1∕ 2

…

…

…

1 …

All dimensions are given in inches. These set screw diameter selections are offered as a guide but their use should be dependent upon design considerations.

ANSI/ASME Standard Woodruff Keys and Keyseats.—American National Standard B17.2 was approved in 1967, and reaffirmed in 2013. Data from this standard are shown in Table 8, Table 9, and Table 10.

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Machinery's Handbook, 31st Edition INCH KEYS AND KEYSEATS

2557

Table 8. ANSI Standard Woodruff Keys ANSI/ASME B17.2-1967 (R2013) +0.000 –0.010

+0.001 –0.000

E

W

Break corners 0.020 Max R Key No. 202 202.5 302.5 203 303 403 204 304 404 305 405 505 605 406 506 606 806 507 607 707 807 608 708 808 1008 1208 609 709 809 1009 610 710 810 1010 1210 811 1011 1211 812 1012 1212

C

+0.000 –0.010

B

B

F

F

Flat Bottom Type

Actual Length F +0.000 −0.010

Max.

1 ∕16 × 1 ∕4 1 ∕16 × 5 ∕16

0.248 0.311 0.311 0.374 0.374 0.374 0.491 0.491 0.491 0.612 0.612 0.612 0.612 0.740 0.740 0.740 0.740 0.866 0.866 0.866 0.866 0.992 0.992 0.992 0.992 0.992 1.114 1.114 1.114 1.114 1.240 1.240 1.240 1.240 1.240 1.362 1.362 1.362 1.484 1.484 1.484

0.109 0.140 0.140 0.172 0.172 0.172 0.203 0.203 0.203 0.250 0.250 0.250 0.250 0.313 0.313 0.313 0.313 0.375 0.375 0.375 0.375 0.438 0.438 0.438 0.438 0.438 0.484 0.484 0.484 0.484 0.547 0.547 0.547 0.547 0.547 0.594 0.594 0.594 0.641 0.641 0.641

3 ∕3 2 1 ∕16

× 5∕16

× 3∕8 3 ∕3 2 × 3 ∕8 1 ∕8 × 3 ∕8 1 ∕16 × 1 ∕2 3 ∕3 2 × 1 ∕2 1 ∕8 × 1 ∕2 3 ∕3 2 × 5 ∕8 1 ∕8 × 5 ∕8 5 ∕3 2 × 5 ∕8 3 ∕16 × 5 ∕8 1 ∕8 × 3 ∕4 5 ∕3 2 × 3 ∕4 3 ∕16 × 3 ∕4 1 ∕4 × 3 ∕4 5 ∕3 2 × 7∕8 3 ∕16 × 7∕8 7∕3 2 × 7∕8 1 ∕4 × 7∕8 3 ∕16 × 1 7∕3 2 × 1 1 ∕4 × 1 5 ∕16 × 1 3 ∕8 × 1 3 ∕16 × 11 ∕8 7∕3 2 × 11 ∕8 1 ∕4 × 11 ∕8 5 ∕16 × 11 ∕8 3 ∕16 × 11 ∕4 7∕3 2 × 11 ∕4 1 ∕4 × 11 ∕4 5 ∕16 × 11 ∕4 3 ∕8 × 11 ∕4 1 ∕4 × 13 ∕8 5 ∕16 × 13 ∕8 3 ∕8 × 13 ∕8 1 ∕4 × 11 ∕2 5 ∕16 × 11 ∕2 3 ∕8 × 11 ∕2

C

W

D

Full Radius Type

Nominal Key Size W×B

+0.001 –0.000

E

Break corners 0.020 Max R

Height of Key Min.

Max.

0.104 0.135 0.135 0.167 0.167 0.167 0.198 0.198 0.198 0.245 0.245 0.245 0.245 0.308 0.308 0.308 0.308 0.370 0.370 0.370 0.370 0.433 0.433 0.433 0.433 0.433 0.479 0.479 0.479 0.479 0.542 0.542 0.542 0.542 0.542 0.589 0.589 0.589 0.636 0.636 0.636

0.109 0.140 0.140 0.172 0.172 0.172 0.194 0.194 0.194 0.240 0.240 0.240 0.240 0.303 0.303 0.303 0.303 0.365 0.365 0.365 0.365 0.428 0.428 0.428 0.428 0.428 0.475 0.475 0.475 0.475 0.537 0.537 0.537 0.537 0.537 0.584 0.584 0.584 0.631 0.631 0.631

D

Min. 0.104 0.135 0.135 0.167 0.167 0.167 0.188 0.188 0.188 0.234 0.234 0.234 0.234 0.297 0.297 0.297 0.297 0.359 0.359 0.359 0.359 0.422 0.422 0.422 0.422 0.422 0.469 0.469 0.469 0.469 0.531 0.531 0.531 0.531 0.531 0.578 0.578 0.578 0.625 0.625 0.625

Distance Below Center E 1 ∕6 4 1 ∕6 4 1 ∕6 4 1 ∕6 4 1 ∕6 4 1 ∕6 4 3 ∕6 4 3 ∕6 4 3 ∕6 4 1 ∕16 1 ∕16 1 ∕16 1 ∕16 1 ∕16 1 ∕16 1 ∕16 1 ∕16 1 ∕16 1 ∕16 1 ∕16 1 ∕16 1 ∕16 1 ∕16 1 ∕16 1 ∕16 1 ∕16 5 ∕6 4 5 ∕6 4 5 ∕6 4 5 ∕6 4 5 ∕6 4 5 ∕6 4 5 ∕6 4 5 ∕6 4 5 ∕6 4 3 ∕3 2 3 ∕3 2 3 ∕3 2 7∕6 4 7∕6 4 7∕6 4

All dimensions are given in inches. The key numbers indicate normal key dimensions. The last two digits give the nominal diameter B in eighths of an inch and the digits preceding the last two give the nominal width W in thirtyseconds of an inch.

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Machinery's Handbook, 31st Edition INCH KEYS AND KEYSEATS

2558

Table 9. ANSI Standard Woodruff Keys ANSI/ASME B17.2-1967 (R2013) +0.001 –0.000

E

W

C Break corners 0.020 Max R

Key No. 617-1

817-1

1017-1 1217-1 617 817

1017 1217

822-1

1022-1 1222-1 1422-1 1622-1 822

1022 1222 1422 1622 1228 1428 1628 1828 2028 2228 2428

B +0.000 –0.010

B +0.000 –0.010

F

F

Full Radius Type

Flat Bottom Type

3∕ 16

× 21 ∕8

Actual Length F +0.000 −0.010 1.380

5∕ 16

× 21 ∕8

1.380

3∕ 16

× 21 ∕8

1.723

5∕ 16

× 21 ∕8

1.723

Nominal Key Size W×B 1∕ 4

× 21 ∕8

1.380

3∕ 8

× 21 ∕8

1.380

1∕ 4

× 21 ∕8

1.723

3∕ 8

× 21 ∕8

1.723

× 23∕4

2.000

3∕ 8

× 23∕4

2.000

1∕ 2

× 23∕4

2.000

1∕ 4

5∕ 16

× 23∕4

2.000

7∕ 16

× 23∕4

2.000

× 23∕4

2.317

3∕ 8

× 23∕4

2.317

1∕ 2

× 23∕4

2.317

1∕ 4

5∕ 16

× 23∕4

2.317

7∕ 16

× 23∕4

2.317

3∕ 8

× 31 ∕2

2.880

1∕ 2

× 31 ∕2

2.880

5∕ 8

× 31 ∕2

2.880

3∕ 4

× 31 ∕2

2.880

7∕ 16

× 31 ∕2

2.880

9∕ 16

× 31 ∕2

2.880

11 ∕ 16

× 31 ∕2

2.880

Max. 0.406 0.406

0.406 0.406 0.531 0.531 0.531 0.531 0.594 0.594 0.594 0.594 0.594 0.750 0.750 0.750 0.750 0.750 0.938 0.938 0.938 0.938 0.938 0.938 0.938

C

+0.001 – 0.000

E

W

D Break corners 0.020 Max R

Height of Key Min. 0.401

Max. 0.396

0.401

0.396

0.401 0.401 0.526 0.526 0.526 0.526 0.589 0.589 0.589 0.589 0.589 0.745 0.745 0.745 0.745 0.745 0.933 0.933 0.933 0.933 0.933 0.933 0.933

0.396 0.396 0.521 0.521 0.521 0.521 0.584 0.584 0.584 0.584 0.584 0.740 0.740 0.740 0.740 0.740 0.928 0.928 0.928 0.928 0.928 0.928 0.928

D

Min. 0.390 0.390 0.390 0.390 0.515 0.515 0.515 0.515 0.578 0.578 0.578 0.578 0.578 0.734 0.734 0.734 0.734 0.734 0.922 0.922 0.922 0.922 0.922 0.922 0.922

Distance Below Center E 21 ∕ 32 21 ∕ 32 21 ∕ 32 21 ∕ 32 17∕ 32 17∕ 32 17∕ 32 17∕ 32

25 ∕ 32 25 ∕ 32 25 ∕ 32 25 ∕ 32 25 ∕ 32

5∕ 8 5∕ 8 5∕ 8 5∕ 8 5∕ 8

13 ∕ 16 13 ∕ 16 13 ∕ 16 13 ∕ 16 13 ∕ 16 13 ∕ 16 13 ∕ 16

All dimensions are given in inches. The key numbers indicate nominal key dimensions. The last two digits give the nominal diameter B in eighths of an inch and the digits preceding the last two give the nominal width W in thirtysec­onds of an inch. The key numbers with the −1 designation, while representing the nominal key size have a shorter length F and due to a greater distance below center E are less in height than the keys of the same num­ber without the −1 designation.

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Machinery's Handbook, 31st Edition INCH KEYS AND KEYSEATS B

A

C

F

Key Above Shaft

Keyseat—Shaft

2559 E

D

Keyseat—Hub

Table 10. ANSI Keyseat Dimensions for Woodruff Keys ANSI/ASME B17.2-1967 (R2013) Key No.

202

202.5 302.5 203 303 403 204 304 404 305 405 505 605 406 506 606 806 507 607 707 807 608

708

808

1008

1208

609

709 809

1009

610

710 810

1010

1210

811

1011

Nominal Size Key 1 ∕16

× 1 ∕4

1 ∕16

× 5∕16

3 ∕3 2

× 5∕16

1 ∕16

× 3∕8

3 ∕3 2

× 3∕8

1 ∕8

× 3∕8

1 ∕16

× 1 ∕2

3 ∕3 2

× 1 ∕2

1 ∕8

× 1 ∕2

3 ∕3 2

× 5∕8

1 ∕8

× 5∕8

5 ∕3 2

× 5∕8

3 ∕16

× 5∕8

1 ∕8

× 3∕4

5 ∕3 2

× 3∕4

3 ∕16

× 3∕4

1 ∕4

× 3∕4

5 ∕3 2

× 7∕8

3 ∕16

× 7∕8

7∕3 2

× 7∕8

1 ∕4

× 7∕8

3 ∕16 7∕3 2 1 ∕4 5 ∕16

7∕3 2 1 ∕4 5 ∕16 3 ∕16 7∕3 2 1 ∕4 5 ∕16 3 ∕8 1 ∕4 5 ∕16

Width Aa

Depth B

Diameter F

Key Above Shaft

Keyseat—Hub

Height C

Width D

Depth E

Min.

Max.

+0.005 −0.000

Min.

Max.

+0.005 −0.005

0.0615

0.0630

0.0728

0.250

0.268

0.0312

0.0635

0.0372

0.0928

0.0943

0.0882

0.312

0.330

0.0469

0.0948

0.0529

0.0615 0.0615 0.0928 0.1240 0.0615 0.0928 0.1240 0.0928 0.1240 0.1553 0.1863 0.1240 0.1553 0.1863 0.2487 0.1553 0.1863 0.2175 0.2487

0.0630 0.0630 0.0943 0.1255 0.0630 0.0943

0.1038 0.1358 0.1202 0.1045 0.1668 0.1511

0.1255

0.1355

0.1255

0.1825

0.0943 0.1568 0.1880 0.1255 0.1568 0.1880 0.2505 0.1568 0.1880 0.2193 0.2505

0.1981 0.1669 0.1513 0.2455 0.2299 0.2143 0.1830 0.2919 0.2763 0.2607 0.2450

0.312 0.375 0.375 0.375 0.500 0.500 0.500 0.625 0.625 0.625 0.625 0.750 0.750 0.750 0.750 0.875 0.875 0.875 0.875

0.330 0.393 0.393 0.393 0.518 0.518 0.518 0.643 0.643 0.643 0.643 0.768 0.768 0.768 0.768 0.895 0.895 0.895 0.895

0.0312 0.0312 0.0469 0.0625 0.0312 0.0469 0.0625 0.0469 0.0625 0.0781 0.0937 0.0625 0.0781 0.0937 0.1250 0.0781 0.0937 0.1093 0.1250

+0.002 −0.000 0.0635 0.0635 0.0948 0.1260 0.0635 0.0948 0.1260 0.0948 0.1260 0.1573 0.1885 0.1260 0.1573 0.1885 0.2510 0.1573 0.1885 0.2198 0.2510

+0.005 −0.000 0.0372 0.0372 0.0529 0.0685 0.0372 0.0529 0.0685 0.0529 0.0685 0.0841 0.0997 0.0685 0.0841 0.0997 0.1310 0.0841 0.0997 0.1153

0.1310

×1

0.1863

0.1880

0.3393

1.000

1.020

0.0937

0.1885

0.0997

×1

0.2487

0.2505

0.3080

1.000

1.020

0.1250

0.2510

0.1310

0.3760

0.1935

×1 ×1

0.2175 0.3111

×1

0.3735

× 11 ∕8

0.2175

3 ∕8 3 ∕16

Keyseat—Shaft

× 11 ∕8 × 11 ∕8 × 11 ∕8

0.1863 0.2487

0.1880 0.2193 0.2505

0.2768 0.2455

0.3853 0.3697 0.3540

1.000 1.000

1.125 1.125 1.125

1.020 1.020 1.020

1.145 1.145 1.145

0.1093 0.1562 0.1875

0.0937 0.1093 0.1250

0.2198 0.3135 0.1885 0.2198 0.2510

0.3228

1.125

1.145

0.1562

0.3135

0.2175

0.2193

0.4327

1.250

1.273

0.1093

0.2198

× 11 ∕4

0.2487

0.1880 0.2505

0.4483 0.4170

1.250 1.250

1.273 1.273

0.0937 0.1250

0.1885 0.2510

0.3111

0.3130

0.3858

1.250

1.273

0.1562

0.3135

0.2487

0.2505

0.4640

1.375

1.398

0.1250

0.2510

× 11 ∕4

0.3735

× 13∕8

0.3111

× 13∕8

0.3755

1.000

0.3130

0.1863

× 11 ∕4

0.3130

0.3237

0.3111

× 11 ∕4 × 11 ∕4

0.2193

0.3755

0.3130

0.3545

0.4328

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1.250

1.375

1.273

1.398

0.1875

0.1562

0.3760

0.3135

0.1153

0.1622 0.0997 0.1153

0.1310

0.1622

0.0997 0.1153

0.1310

0.1622

0.1935 0.1310

0.1622

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Machinery's Handbook, 31st Edition INCH KEYS AND KEYSEATS

2560

10. ANSIANSI Keyseat Dimensions for Woodruff Keys Keys Table Table 10. (Continued) Keyseat Dimensions for Woodruff ANSI/ASME B17.2-1967 (R2013) Key No. 1211

812

1012

1212

617-1 817-1

Nominal Size Key 3 ∕8 1 ∕4 5 ∕16 3 ∕8 3 ∕16 1 ∕4

1017-1

5 ∕16

617

3 ∕16

1217-1 817

1017

1217

822-1

3 ∕8

1 ∕4 5 ∕16 3 ∕8 1 ∕4

1022-1

5 ∕16

1422-1

7∕16

1222-1

1622-1 822

3 ∕8

1 ∕2 1 ∕4

1022

5 ∕16

1422

7∕16

1222

1622 1228 1428 1628 1828 2028 2228 2428

3 ∕8

1 ∕2 3 ∕8 7∕16 1 ∕2 9 ∕16 5 ∕8

Width Aa

Depth B

Diameter F

Keyseat—Hub

Height C

Width D

Depth E

Min.

Max.

+0.005 −0.000

Min.

Max.

+0.005 −0.005

× 13∕8

0.3735

0.3755

0.4015

1.375

1.398

0.1875

0.3760

0.1935

× 11 ∕2

0.3111

0.3130

0.4798

1.500

1.523

0.1562

0.3135

0.1622

0.1863

0.1880

0.3073

2.125

2.160

0.0937

0.1885

0.0997

× 11 ∕2

0.2487

× 11 ∕2

0.3735

× 21 ∕8

0.2487

× 21 ∕8 × 21 ∕8

0.2505

0.4485 0.2760

1.500 1.500 2.125

1.523 1.523 2.160

0.1250 0.1875 0.1250

0.2510 0.3760 0.2510

0.1310 0.1935 0.1310

0.2448

2.125

2.160

0.1562

0.3135

0.1622

0.1863

0.1880

0.4323

2.125

2.160

0.0937

0.1885

0.0997

× 21 ∕8

0.2487

× 21 ∕8

0.3755

0.5110

+0.005 −0.000

0.3130

0.3735

× 21 ∕8

0.2505

+0.002 −0.000

0.3111

× 21 ∕8

0.3755 0.2505

0.2135 0.4010

2.125 2.125

2.160 2.160

0.1875 0.1250

0.3760 0.2510

0.3111

0.3130

0.3698

2.125

2.160

0.1562

0.3135

0.2487

0.2505

0.4640

2.750

2.785

0.1250

0.2510

× 23∕4

0.3111

0.3130

0.4328

2.750

2.785

0.1562

0.3135

0.1622

0.4360

0.4380

0.3703

2.750

2.785

0.2187

0.4385

0.2247

× 23∕4

0.3735

× 23∕4

0.4985

× 23∕4 × 23∕4 × 23∕4

0.2487

× 31 ∕2 × 31 ∕2

× 31 ∕2

× 31 ∕2

2.750

2.785 2.785

0.1875 0.2500 0.1250

0.3760

0.3760 0.5010 0.2510

0.1935 0.1310 0.1935 0.2560 0.1310

2.750

2.785

0.1562

0.3135

0.1622

0.4360

0.4380

0.5263

2.750

2.785

0.2187

0.4385

0.2247

0.4985

× 31 ∕2

0.6200

2.750

2.785

0.1875

0.5888

× 23∕4

× 31 ∕2

0.2505

0.3390

2.750

2.160

0.3130

0.3735

× 31 ∕2

0.5005

0.4015

2.125

0.3111

× 23∕4 × 23∕4

0.3755

0.3385

0.1310

0.1622

0.3735

× 23∕4

0.3755

0.1935

× 21 ∕8

11 ∕16 3 ∕4

Key Above Shaft

Keyseat—Shaft

0.3735 0.4360 0.4985 0.5610 0.6235 0.6860 0.7485

0.3755 0.5005 0.3755 0.4380 0.5005 0.5630 0.6255 0.6880 0.7505

0.5575 0.4950 0.7455 0.7143 0.6830 0.6518 0.6205 0.5893 0.5580

2.750 2.750 3.500 3.500 3.500 3.500 3.500 3.500 3.500

2.785 2.785 3.535 3.535 3.535 3.535 3.535 3.535 3.535

0.1875 0.2500 0.1875 0.2187 0.2500 0.2812 0.3125 0.3437 0.3750

0.3760 0.5010 0.3760 0.4385 0.5010 0.5635 0.6260 0.6885 0.7510

0.1935 0.2560 0.1935 0.2247 0.2560 0.2872 0.3185 0.3497 0.3810

a These Width A values were set with the maximum keyseat (shaft) width as that figure which will

receive a key with the greatest amount of looseness consistent with assuring the key’s sticking in the keyseat (shaft). Minimum keyseat width is that figure permitting the largest shaft distortion acceptable when assembling maximum key in minimum keyseat. Dimensions A, B, C, D are taken at side intersection. All dimensions are given in inches.

The following definitions are given in this standard: Woodruff Key: A demountable machinery part which, when assembled into key-seats, provides a positive means for transmitting torque between the shaft and hub. Woodruff Key Number: An identification number by which the size of key may be read­ily determined. Woodruff Keyseat—Shaft: The circular pocket in which the key is retained. Woodruff Keyseat—Hub: An axially located rectangular groove in a hub. (This has been referred to as a keyway.) Woodruff Keyseat Milling Cutter: An arbor type or shank type milling cutter normally used for milling Woodruff keyseats in shafts (see page 895).

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Machinery's Handbook, 31st Edition Keys and Keyseats

2561

Chamfered Keys and Filleted Keyseats.—In general practice, chamfered keys and fil­ leted keyseats are not used. However, it is recognized that fillets in keyseats decrease stress concentration at corners. When used, fillet radii should be as large as possible without causing excessive bearing stresses due to reduced contact area between the key and its mat­ing parts. Keys must be chamfered or rounded to clear fillet radii. Values in Table 5 assume general conditions and should be used only as a guide when critical stresses are encoun­tered. Depths for Milling Keyseats.—Table 11 on page 2562 has been compiled to facilitate the accurate milling of keyseats. This table gives the distance M (see illustration accompany­ ing table) between the top of the shaft and a line passing through the upper corners or 2 2h ^ edges of the keyseat. Dimension M is calculated by the formula: M = 1 2 S − S −E where S is diameter of shaft, and E is width of keyseat. A simple approximate formula that gives M to within 0.001 inch is M = E2 ÷ 4S.

Keyseating Machines.—The machines which are designed especially for cutting keyse­ ats or keyways in the hubs of pulleys, gears, etc., are generally known as keyseaters. Machines of this class usually have a base or frame which contains the mechanism for imparting a reciprocating motion to a cutter bar, which moves vertically for cutting a key­seat in the work. There are several types of machines which are used for internal keyseating operations in addition to the machines designed especially for this work. Broaching machines as well as slotters are commonly used, and keyseating is also done to some extent in shapers and planers.

Other Key Types.—The sunk key is the most common type and is of rectangular section that engages a groove or slot formed both in the shaft and hub of the gear or pulley. The width of an ordinary sunk key ordinarily is equal to about one-fourth of the shaft diameter and the thickness, when a flat key is preferred to the square form, is usually about one-sixth of the shaft diameter; these proportions are varied somewhat by different manufacturers. The flat key is a rectangular shape which bears upon a flat surface formed on one side of the shaft. The draw or gib key is a sunk key which has a head by means of which it can be removed. The round tapered key is simply a taper pin which is driven into a hole that is partly in the shaft and partly in the hub; this form is used for light work. The name feather or spline is applied to a key which is fixed to either a shaft or hub, as when a gear must be driven by a shaft, but at the same time be free to slide in a lengthwise direction. The taper of American Standard square and flat keys is 1.8 inch per foot. The saddle key does not enter a slot in the shaft. It has parallel sides and is curved on its under side to fit the shaft. It is slightly tapered on top so that, when it is driven tightly in place, the shaft is held by frictional resistance. This key should be fitted so that it bears lightly on the sides and heavily between the shaft and hub throughout its entire length. As the drive with this type of key is not positive, it is only used where there is little power to transmit. It is an inexpensive method of keying, as the shaft does is not machined.

Effect of Keyways on Shaft Strength.—See SHAFTS starting on page 295 and Effect of Keyways on Shaft Strength starting on page 301.

British Standard Keys and Keyways.—See Keys and Keyways in the ADDITIONAL material in the Machinery’s Handbook 31 Digital Edition.

Cotters.—A cotter is a form of key that is used to connect rods, etc., that are subjected either to tension or compression or both, the cotter being subjected to shearing stresses at two transverse cross sections. When taper cotters are used for drawing and holding parts together, if the cotter is held in place by the friction between the bearing surfaces, the taper should not be too great. Ordinarily a taper varying from 1 ⁄4 to 1 ⁄2 inch per foot is used for plain cotters. When a set screw or other device is used to prevent the cotter from backing out of its slot, the taper may vary from 11 ⁄2 to 2 inches per foot.

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Machinery's Handbook, 31st Edition Keys and Keyseats

2562

Table 11. Finding Depth of Keyseat and Distance from Top of Key to Bottom of Shaft M

C

D

E

J

S

For milling keyseats, the total depth to feed cutter in from outside of shaft to bottom of keyseat is M + D, where D is depth of keyseat. For checking an assembled key and shaft, caliper measurement J between top of key and bottom of shaft is used.

J = S − ^ M + Dh + C

where C is depth of key. For Woodruff keys, dimensions C and D can be found in Table 8 through Table 10. Assuming shaft diameter S is normal size, the tolerance on dimen­sion J for Woodruff keys in keyslots are + 0.000, −0.010 inch. Width of Keyseat, E

Dia. of Shaft, S Inches

1 ∕16

0.3125

.0032

0.3437

.0029 .0065

0.3750

3 ∕3 2

…

1 ∕8

5 ∕3 2

3 ∕16

7∕3 2

1 ∕4

5 ∕16

3 ∕8

Dimension M, Inch

7∕16

1 ∕2

9 ∕16

5 ∕8

11 ∕16

3 ∕4

…

…

…

…

…

…

…

…

…

…

…

…

…

…

…

…

…

…

…

…

…

…

…

…

…

…

.0026 .0060 .0107

…

…

…

…

…

…

…

…

…

…

…

…

0.4060

.0024 .0055 .0099

…

…

…

…

…

…

…

…

…

…

…

…

0.4375

.0022 .0051 .0091

…

…

…

…

…

…

…

…

…

…

…

…

0.4687

.0021 .0047 .0085 .0134

…

…

…

…

…

…

…

…

…

…

…

0.5000

.0020 .0044 .0079 .0125

…

…

…

…

…

…

…

…

…

…

…

.0161

…

…

…

…

…

…

…

…

…

…

…

…

…

…

…

…

…

…

…

…

…

…

…

…

…

…

…

0.5625

…

.0039 .0070

.0111

0.6250

…

.0035 .0063 .0099 .0144 .0198

0.6875

…

.0032 .0057 .0090 .0130 .0179 .0235

0.7500

…

.0029 .0052 .0082

.0119

.0163 .0214 .0341

…

…

…

…

…

…

…

0.8125

…

.0027 .0048 .0076

.0110

.0150 .0197 .0312

…

…

…

…

…

…

…

0.8750

…

.0025 .0045 .0070 .0102 .0139 .0182 .0288

…

…

…

…

…

…

…

0.9375

…

…

.0042 .0066 .0095 .0129 .0170 .0263 .0391

…

…

…

…

…

…

1.0000

…

…

.0039 .0061 .0089 .0121 .0159 .0250 .0365

…

…

…

…

…

…

1.0625

…

…

.0037 .0058 .0083

…

…

…

…

…

…

1.1250

…

…

.0035 .0055 .0079 .0107 .0141 .0221 .0322 .0443

…

…

…

…

…

1.1875

…

…

.0033 .0052 .0074 .0102 .0133 .0209 .0304 .0418

…

…

…

…

…

1.2500

…

…

.0031 .0049 .0071 .0097 .0126 .0198 .0288 .0395

…

…

…

…

…

1.3750

…

…

…

.0045 .0064 .0088

.0180 .0261 .0357 .0471

…

…

…

…

1.5000

…

…

…

.0041 .0059 .0080 .0105 .0165 .0238 .0326 .0429

…

…

…

…

1.6250

…

…

…

.0038 .0054 .0074 .0097 .0152 .0219 .0300 .0394 .0502

…

…

…

1.7500

…

…

…

…

.0050 .0069 .0090 .0141 .0203 .0278 .0365 .0464

…

…

…

1.8750

…

…

…

…

.0047 .0064 .0084 .0131 .0189 .0259 .0340 .0432 .0536

…

…

2.0000

…

…

…

…

.0044 .0060 .0078 .0123 .0177 .0242 .0318 .0404 .0501

…

…

2.1250

…

…

…

…

…

2.2500

…

…

…

…

…

…

2.3750

…

…

…

…

…

…

…

2.5000

…

…

…

…

…

…

…

…

.0141 .0193 .0253 .0321 .0397 .0482 .0576

2.6250

…

…

…

…

…

…

…

…

.0135 .0184 .0240 .0305 .0377 .0457 .0547

2.7500

…

…

…

…

…

…

…

…

…

.0175 .0229 .0291 .0360 .0437 .0521

2.8750

…

…

…

…

…

…

…

…

…

.0168 .0219 .0278 .0344 .0417 .0498

3.0000

…

…

…

…

…

…

…

…

…

.0114

.0149 .0235 .0342

.0115

.0056 .0074

.0116

.0167 .0228 .0298 .0379 .0470 .0572 .0684

.0070 .0109 .0157 .0215 .0281 .0357 .0443 .0538 .0643

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.0103 .0149 .0203 .0266 .0338 .0419 .0509 .0608

…

.0210 .0266 .0329 .0399 .0476

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Machinery's Handbook, 31st Edition CALCULATIONS FOR BELTS AND PULLEYS

2563

FLEXIBLE BELTS AND SHEAVES Flexible belt drives are used in industrial power transmission applications, especially when the speeds of the driver and driven shafts must be different or when shafts must be widely separated. The trend toward higher speed prime movers and the need to achieve a slower, useful driven speed are additional factors favoring the use of belts. Belts have numerous advantages over other means of power transmission; these advantages include overall economy, cleanliness, no need for lubrication, lower maintenance costs, easy installation, dampening of shock loads, and the abilities to be used for clutching and vari­ able speed power transmission between widely spaced shafts. Calculations for Belts and Pulleys Belt speed plays an important role in the amount of load a friction drive system can trans­ mit. Higher speeds will require higher preloads (increased belt tension) to compensate for the higher centrifugal force. In positive drive (toothed belt) systems, higher speeds gener­ ate dynamic forces caused by unavoidable tolerance errors that may result in increased tooth or pin stresses and shorter belt life. Pulley Diameters and Drive Ratios.—Minimum pulley diameters determined by belt manufacturers are based on the minimum radius that a belt can wrap around a pulley with­ out stressing the load-carrying members. For positive drive systems, minimum pulley diameters are also determined by the minimum number of teeth that must be engaged with the sprocket to guarantee the operating load. Diameters of driving and driven pulleys determine the velocity ratio of the input relative to the output shaft and are derived from the following formulas: for all belt systems, veloc­ity ratio V = Dpi /Dpo, and for positive (toothed) drive systems, velocity ratio V = Ni /No, where Dpi is the pitch diameter of the driving pulley, Dpo is the pitch diameter of the driven pulley, Ni is the number of teeth on the driving pulley, and No is the number of teeth on the driven pulley. For most drive systems, a velocity ratio of 8:1 is the largest that should be attempted with a single reduction drive, and 6:1 is a reasonable maximum. Wrap Angles and Center-to-Center Distances.—The radial distance for which the belt is in contact with the pulley surface, or the number of teeth in engagement for positive drive belts, is called the wrap angle. Belt and sprocket combinations should be chosen to ensure a wrap angle of about 120° around the smaller pulley. The wrap angle should not be less than 90°, especially with positive drive belts, because if too few teeth are in engage­ment, the belt may jump a tooth or pin and timing or synchronization may be lost. For flat belts, the minimum allowable center-to-center distance (CD) for any belt-andsprocket combination should be chosen to ensure a minimum wrap angle around the smaller pulley. For high-velocity systems, a good rule of thumb is a minimum CD equal to the sum of the pitch diameter of the larger sprocket and one-half the pitch diameter of the smaller sprocket. This formula ensures a minimum wrap angle of approximately 120°, which is generally sufficient for friction drives and will ensure that positive drive belts do not jump teeth. Pulley Center Distances and Belt Lengths.—Maximum center distances of pulleys should be about 15 to 20 times the pitch diameter of the smaller pulley. Greater spacing requires tight control of the belt tension because a small amount of stretch will cause a large drop in tension. Constant belt tension can be obtained by application of an adjustable ten­sioning pulley applied to the slack side of the belt. Friction drive systems using flat belts require much more tension than positive drive belt systems. Belt length can be calculated from: L = 2C + π(D 2 + D1)/2 + (D 2 − D1)2/4C for friction drives, and length L = 2C + π(D 2 + D1)/2 + (D 2 + D1)2/4C for crossed belt friction belt drives, where C is the center distance, D1 is the pitch diameter of the small pulley, and D 2

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Machinery's Handbook, 31st Edition Pulley Speeds and Belt Lengths

2564

is the pitch diameter of the large pulley. For serrated belt drives, the length determined by use of these equations should be divided by the serration pitch. The belt length must then be adjusted to provide a whole number of serrations. Pulley Diameters and Speeds.—If D = diameter of driving pulley, d = diameter of driven pulley, S = speed of driving pulley, and s = speed of driven pulley:

D=

d#s S ,

d=

D# S s ,

S=

d#s D ,

and

s= D# S d

Example 1: If diameter of the driving pulley D is 24 inches, its speed is 100 rpm, and the driven pulley runs at 600 rpm, diameter of the driven pulley, d = 24 × 100 ⁄ 600 = 4 inches. Example 2: If the diameter of the driving pulley D is 60 cm, its speed is 100 rpm, and the driven pulley is to run at 600 rpm, the diameter of the driven pulley, d = 60 × 100 ∕ 600 = 10 cm. Example 3: If the diameter of the driven pulley d is 36 inches, its required speed is to be 150 rpm, and the speed of the driving pulley is to be 600 rpm, the diameter of the driving pulley D = 36 × 150 ∕ 600 = 9 inches. Example 4: If diameter of driven pulley d is 4 inches, its required speed is 800 rpm, and diameter of driving pulley D is 26 inches, speed of driving pulley = 4 × 800 ∕ 26 = 123 rpm. Example 5: If diameter of driven pulley d is 10 cm, its required speed is 800 rpm, and diameter of driving pulley D is 25 cm, speed of the driving pulley = 10 × 800 ∕ 25 = 320 rpm. Example 6: If the diameter of the driving pulley D is 15 inches and its speed is 180 rpm, and the diameter of the driven pulley d is 9 inches, then the speed of the driven pulley = 15 × 180 ∕ 9 = 300 rpm. Pulley Diameters in Compound Drive.—If speeds of driving and driven pulleys, A, B, C, and D (see illustration) are known, the first step in finding their diameters is to form a fraction with the driving pulley speed as the numerator and the driven pulley speed as the, denominator, and then reduce this fraction to its lowest terms. Resolve the numerator and the denominator into two pairs of factors (a pair being one factor in the numerator and one in the denominator) and, if necessary, multiply each pair by a trial number that will give pulleys of suitable diameters. Example 7: If speed of pulley A is 260 rpm and the required speed of pulley D is 720 rpm, find the diameters of the four pulleys. Reduced to its lowest terms, the fraction 260 ∕ 720 = 13 ∕ 36, which represents the required speed ratio. Resolve this ratio 13 ∕ 36 into two factors:

13 1 # 13 36 = 2 # 18 Multiply by trial numbers 12 and 1 to get: ^1 # 12h # ^13 # 1 h 12 # 13 = ^2 # 12h # ^18 # 1 h 24 # 18

A

B

C D

Compound Drive with Four Pulleys

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Machinery's Handbook, 31st Edition Pulley Speeds and Belt Lengths

2565

The values 12 and 13 in the numerator represent the diameters of the driven pulleys, B and D, and the values 24 and 18 in the denominator represent the diameters of the driving pulleys, A and C, as shown in the illustration. Speed of Driven Pulley in Compound Drive.—If diameters of pulleys A, B, C, and D (see illustration above), and speed of pulley A are known, the speed of the driven pulley D is found from:

driving pulley diameter driven pulley diameter

driving p ulley diameter # driven pulley diameter

# speed of first driving pulley

Example 8: If the diameters of driving pulleys A and C are 18 and 24 inches, diameters of driven pulleys B and D are 12 and 13 inches, and the speed of driving pulley A is 260 rpm, speed of driven pulley

18 # 24 D = 12 # 13 # 260 = 720 rpm



Length of Belt Traversing Three Pulleys.—The length L of a belt traversing three pul­ leys, as shown in the diagram below, and touching them on one side only, can be found by the following formula. Pulley 2, Radius R2 = 2

C23 = 8

 2 = 36.87, or 0.6435 rad.

C12 = 10

 1 = 53.13, or 0.9273 rad.

3 = 90, or 1.5708 rad. Pulley 3, Radius R3 = 4

Pulley 1, Radius R1 = 1

C13 = 6

Flat Belt Traversing Three Pulleys

Referring to the diagram, R1, R2 , and R3 are the radii of the three pulleys; C12 , C13, and C23 are the center distances; and α1, α2 , and α3 are the angles, in radians, of the triangle formed by the center distances. Then: 2 ^ R − R 1h2 ^ R3 − R 2 h2 1 ^ R − R1 h F L = C 12+ C 13 + C 23+ 2 < 2 + 3 + C 12 C 13 C 23 + π ^ R1 + R 2 + R 3 h − ^α1 R1 + α 2 R2 + α 3R3 h Example 9: Assume R1 = 1, R2 = 2, R3 = 4, C12 = 10, C13 = 6, C23 = 8, α1 = 53.13 degrees or 0.9273 radian, α2 = 36.87 degrees or 0.6435 radian, and α3 = 90 degrees or 1.5708 radi­ ans. Then: 2 ^4 − 1 h2 ^4 − 2 h2 1 ^2 − 1h E + + L = 10 + 6 + 8 + 2 ; 10 6 8 + π ^1 + 2 + 4h + ^0.9273 # 1 + 0.6435 # 2 + 1.5708 # 4 h

= 24 + 1.05 + 21.9911 − 8.4975 = 38.5436

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2566

Machinery's Handbook, 31st Edition PULLEY SPEEDS AND BELT LENGTHS

Power Transmitted by Belts.—With belt drives, the force that produces work acts on the rim of a pulley or sheave and causes it to rotate. Since a belt on a drive must be tight enough to prevent slip, there is a belt pull on both sides of a driven wheel. When a drive is stationary or operating with no power transmitted, the pulls on both sides of the driven wheel are equal. When the drive is transmitting power, however, the pulls are not the same. There is a tight side tension TT and a slack side tension, TS . The difference between these two pulls (TT − TS) is called effective pull or net pull. This effective pull is applied at the rim of the pulley and is the force that produces work. Net pull equals horsepower (HP) × 33,000 ÷ belt speed (fpm). Belt speed in fpm can be set by changing the pulley, sprocket, or sheave diameter. The shaft speeds remain the same. Belt speed is directly related to pulley diameter. Double the diameter and the total belt pull is cut in half, reducing the load on the shafts and bearings. A belt experiences three types of tension as it rotates around a pulley: working tension (tight side − slack side), bending tension, and centrifugal tension. The tension ratio (R) equals tight side divided by slack side tension (measured in pounds). The larger R is, the closer a V-belt is to slipping—the belt is too loose. (Synchro­ nous belts do not slip, because they depend on the tooth grip principle.) In addition to working tension (tight side − slack side), two other tensions are developed in a belt when it is operating on a drive. Bending tension TB occurs when the belt bends around the pulley. One part of the belt is in tension and the other is in compression, so com­ pressive stresses also occur. The amount of tension depends on the belt’s construction and the pulley diameter. Centrifugal tension (TC) occurs as the belt rotates around the drive and is calculated by TC = MV2, where TC is centrifugal tension in pounds, M is a constant depen­dent on the belt’s weight, and V is the belt velocity in feet per minute. Neither the bending nor centrifugal tensions are imposed on the pulley, shaft, or bearing—only on the belt. Combining these three types of tension results in peak tension, which is important in determining the degree of performance or belt life: Tpeak = TT + TB + TC . Measuring the Effective Length.—The effective length of a V-belt is determined by placing the belt on a measuring device having two equal diameter sheaves with standard groove dimensions. The shaft of one of the sheaves is fixed. A specified measuring tension is applied to the housing for the shaft of the other sheave, moving it along a graduated scale. The belt is rotated around the sheaves at least two revolutions of the belt to seat it properly in the sheave grooves and to divide the total tension equally between the two strands of the belt. The effective length of the belt is obtained by adding the effective (outside) circumfer­ ence of one of the measuring sheaves to twice the center distance. Synchronous belts are measured in a similar manner. The following sections cover common belts used in industrial applications for power transmission and specified in Rubber Manufacturers Association (RMA), Mechanical Power Transmission Association (MPTA), and The Rubber Association of Canada (RAC) standards. The information presented does not apply to automotive or agricultural drives, for which other standards exist. The belts covered in this section are Narrow, Classical, Double, and Light-Duty V-Belts, V-Ribbed Belts, Variable-Speed Belts, 60 deg V-Belts, and Synchronous (Timing) Belts. Flat Belting

Flat belting was originally made from leather because it was the most durable material available and could easily be cut and joined to make a driving belt suitable for use with cylindrical or domed pulleys. This type of belting was popular because it could be used to transmit high torques over long distances and it was employed in factories to drive many

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Machinery's Handbook, 31st Edition Flat Belts and Pulleys

2567

small machines from a large common power source such as a steam engine. As electric motors became smaller, more efficient, and more powerful, and new types of belts and chains were made possible by modern materials and manufacturing processes, flat belts fell out of favor. Flat belts are still used for some drive purposes, but leather has been replaced by other natural and synthetic materials such as urethanes, which can be rein­forced by high-strength polyamide or steel fabrics to provide properties such as resistance to stretching. The high modulus of elasticity in these flat belts eliminates the need for peri­ odic retensioning that is usually necessary with V-belts. Driving belts can be given a coating of an elastomer with a high coefficient of friction, to enable belts to grip pulleys without the degree of tension common with earlier materials. Urethanes are commonly used for driving belts where high resistance to abrasion is required, and also resist attack by chemical solvents of most kinds. Flat belts having good resistance to high temperatures are also available. Typical properties of polyurethane belts include tensile strength up to 40,000 psi (276 MPa), depending on reinforcement type and Shore hardness of 85 to 95. Most polyurethane belts are installed under tension. The amount of tension varies with the belt cross section, being greater for belts of small sec­tion. Belt tension can be measured by marking lines 10 inches, or 100 mm, apart on an installed belt, then applying tension until the separation increases by the desired percent­age. For 2 percent tension, lines on the tensioned belt would be 10.2 inches, or 102 mm, apart. Mechanical failure may result when belt tensioning is excessive, and 2 to 2.5 percent elongation should be regarded as the limit. Flat belts offer high load capacities and are capable of transmitting power over long dis­ tances, maintaining relative rotational direction, can operate without lubricants, and are generally inexpensive to maintain or replace when worn. Flat belt systems will operate with little maintenance and only periodic adjustment. Because they transmit motion by friction, flat belts have the ability to slip under excessive loads, providing a fail-safe action to guard against malfunctions. This advantage is offset by the problem that friction drives can both slip and creep so that they do not offer exact, consistent velocity ratios nor preci­ sion timing between input and output shafts. Flat belts can be made to any desired length, being joined by reliable chemical bonding processes. Increasing centrifugal force has less effect on the load-carrying capacity of flat belts at high speeds than it has on V-belts, for instance. The low thickness of a flat belt, compared with a V-belt, places its center of gravity near the pulley surface. Flat belts therefore may be run at surface speeds of up to 16,000 or even 20,000 ft/min (81.28 to 101.6 m/s), although ideal speeds are in the range of 3,000 to 10,000 ft/min (15.25 to 50.8 m/s). Elasto­ meric drive surfaces on flat belts have eliminated the need for belt dressings that were often needed to keep leather belts in place. These surface coatings can also contain antistatic materials. Belt pulley wear and noise are low with flat belts, shock and vibration are damped, and efficiency is generally greater than 98 percent, compared to 96 percent for V-belts. Driving belt load capacities can be calculated from torque T = F(d/2) and horsepower HP = T × rpm/396,000, where T is the torque in in-lb, F is the force transmitted in lb, and d is the pulley diameter in inches. Pulley width is usually about 10 percent larger than the belt, and for good tracking, pulleys are often crowned by 0.012–0.10 inch (0.305– 2.54 mm) for diameters in the range of 1.5–80 inches (3.8–203 cm). Before a belt specification is written, the system should be checked for excessive startup and shut-down loads, which sometimes are more than 10 percent above operating condi­ tions. In overcoming such loads, the belt will transmit considerably more force than during normal operation. Large starting and stopping forces will also shorten belt life unless they are taken into account during the design stage.

Flat Belt Pulleys.—Flat belt pulleys are usually made of cast iron, fabricated steel, paper, fiber, or various kinds of wood. They may be solid or split and in either case the hub may be split for clamping to the shaft.

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Machinery's Handbook, 31st Edition Flat Belts and Pulleys

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Pulley face widths are nominally the same as the widths of the belts they are to carry. The pulley face should be approximately one inch more than the belt width for belts under 12 inches (30.5 cm) wide, 2 inches (5.1 cm) more for belts from 12 to 24 inches (30.5– 61 cm) wide, and 3 inches (7.6 cm) more for belts over 24 inches (61 cm) in width. Belts may be made to center themselves by the use of crowned pulleys. The usual amount of crowning is 1 ⁄8 inch per foot (10.4 mm per meter) of pulley width. Thus, the difference in maximum and minimum radii of a crowned 6-inch (152.4 mm) wide pulley would be 1 ⁄16 inch (1.59 mm). Crowned pulleys have a rim section either with a convex curve or a flat V form. Flanges on the sides of flat belt pulleys are in general undesirable as the belt tends to crawl against them. Too much crown is undesirable because of the tendency to “break the belt’s back.” This is particularly true for riding idlers close to driving pulleys where the cur­vature of the belt changes rapidly from one pulley to the other. Here, the idler should under no circumstances be crowned and the adjacent pulley should have very little crown. Pul­leys carrying shifting belts are not crowned. Open belt drives connecting pulleys on short centers with one pulley considerably larger than the other may be unsatisfactory due to the small angle of wrap on the smaller pulley. This angle may be increased by the use of idler pulleys on one or both sides of the belt. V-Belts Narrow V-Belts ANSI/RMA IP-22.—Narrow V-belts serve the same applications as multiple, classical V-belts, but allow for a lighter, more compact drive. Three basic cross sections—3V and 3VX, 5V and 5VX, and 8V—are provided, as shown in Fig. 1. The 3VX and 5VX are molded, notched V-belts that have greater power capacity than conventional belts. Narrow V-belts are specified by cross section and effective length and have top widths ranging from 3 ⁄8 to 1 inch (9.525–25.4 mm). Narrow V-belts usually provide substantial weight and space savings over classical belts. Some narrow belts can transmit up to three times the horsepower of conventional belts in the same drive space, or the same horsepower in one-third to one-half the space. These belts are designed to operate in multiples and are also available in the joined configuration. Belt Cross Sections: Nominal dimensions of the three cross sections are given in Fig. 1. Belt Size Designation: Narrow V-belt sizes are identified by a standard belt number. The first figure of this number followed by the letter V denotes the belt cross section. An X fol­ lowing the V indicates a notched cross section. The remaining figures show the effective belt length in tenths of an inch. For example, the number 5VX1400 designates a notched V-belt with a 5V cross section and an effective length of 140.0 in. Standard effective lengths of narrow V-belts are shown in Table 1. 1” 5/8”

5/8” 7/8”

3/8” 5/16” 3VX

17/32” 5VX

17/32” 5V

8V

Fig. 1. Nominal Narrow V-Belt Dimensions

Sheave Dimensions: Groove angles and dimensions for sheaves and face widths of sheaves for multiple belt drives are given in Table 2a and Table 2b, along with various tolerance values. Standard sheave outside diameters are given in Table 3.

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Standard Length Designationa

3V

25.0 26.5 28.0 30.0 31.5 33.5 35.5 37.5 40.0 42.5 45.0 47.5 50.0 53.0 56.0 60.0 63.0 67.0 71.0 75.0 80.0 85.0 90.0 95.0 100.0

5V

… … … … … … … … … … … … 50.0 53.0 56.0 60.0 63.0 67.0 71.0 75.0 80.0 85.0 90.0 95.0 100.0

8V

… … … … … … … … … … … … … … … … … … … … … … … … 100.0

±0.3 ±0.3 ±0.3 ±0.3 ±0.3 ±0.3 ±0.3 ±0.3 ±0.3 ±0.3 ±0.3 ±0.3 ±0.3 ±0.4 ±0.4 ±0.4 ±0.4 ±0.4 ±0.4 ±0.4 ±0.4 ±0.5 ±0.5 ±0.5 ±0.5

Matching Limits for One Set 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.30 0.30 0.30 0.30 0.30 0.30 0.30 0.30

Standard Length Designationa 1060 1120 1180 1250 1320 1400 1500 1600 1700 1800 1900 2000 2120 2240 2360 2500 2650 2800 3000 3150 3350 3550 3750 4000 4250

Standard Effective Outside Length

3V

Cross Section

106.0 112.0 118.0 125.0 132.0 140.0 … … … … … … … … … … … … … … … … … … …

a To specify belt size, use the Standard Length Designation prefixed by the cross section, for example, 5 V850. All dimensions in inches.

5V

106.0 112.0 118.0 125.0 132.0 140.0 150.0 160.0 170.0 180.0 190.0 200.0 212.0 224.0 236.0 250.0 265.0 280.0 300.0 315.0 335.0 355.0 … … …

8V

106.0 112.0 118.0 125.0 132.0 140.0 150.0 160.0 170.0 180.0 190.0 200.0 212.0 224.0 236.0 250.0 265.0 280.0 300.0 315.0 335.0 355.0 375.0 400.0 425.0

Permissible Deviation from Standard Length ±0.6 ±0.6 ±0.6 ±0.6 ±0.6 ±0.6 ±0.8 ±0.8 ±0.8 ±0.8 ±0.8 ±0.8 ±0.8 ±0.8 ±0.8 ±0.8 ±0.8 ±0.8 ±0.8 ±1.0 ±1.0 ±1.0 ±1.0 ±1.0 ±1.2

Matching Limits for One Set 0.30 0.30 0.30 0.30 0.30 0.30 0.30 0.45 0.45 0.45 0.45 0.45 0.45 0.45 0.45 0.45 0.60 0.60 0.60 0.60 0.60 0.60 0.60 0.75 0.75

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250 265 280 300 315 335 355 375 400 425 450 475 500 530 560 600 630 670 710 750 800 850 900 950 1000

Cross Section

Permissible Deviation from Standard Length

Machinery's Handbook, 31st Edition Narrow V-Belts

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Table 1. Narrow V-Belt Standard Effective Lengths ANSI/RMA IP-22 (1983)

Standard Effective Outside Length

a

dB

hg

Sg

RB dB

RB

File Break all Sharp Corners be

hg

Se

Se

Standard Groove Dimensions

Groove Angle α bg

Sg Deep Groove Dimensions

a

he

Face Width of Standard and Deep Groove Sheaves = sg (Ng – I) + 2Se, where Ng = Number of Grooves Standard Groove Dimensions Cross Section

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3V

5V

8V

Standard Groove Outside Diameter

Up through 3.49 Over 3.49 up to and including 6.00 Over 6.00 up to and including 12.00 Over 12.00 Up through 9.99 Over 9.99 up to and including 16.00 Over 16.00 Up through 15.99 Over 15.99 up to and including 22.40 Over 22.40

a See footnote b following Table 2b.

Groove Angle, α, ±0.25 deg

bg ±0.005

36

38 40

0.350

be (Ref)

hg (Min)

0.350

0.340

42 38 40

42

Sg ±0.015

0.3438

0.406

0.344 (+ 0.099, −0.031)

0.5938

0.688

0.500 +0.125, −0.047)

7.10

0.100

1.0000

1.125

0.750 (+0.250, −0.062)

12.50

0.200

0.181

0.183 0.186

0.600

0.600

0.590

0.332 0.336 0.575

1.000

1.000

0.990

0.580 0.585

Min Recommended OD

dB ±0.0005

0.188 0.329

42 38 40

RB (Min)

Design Factors a

Se

2.65

2a

0.050

Machinery's Handbook, 31st Edition Narrow V-Belts

Pitch Diameter

Diameter Effective and Outside

File Break all Sharp Corners be and bg

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Groove Angle α

Outside Diameter Effective and Outside Pitch Diameter

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Table 2a. Narrow V-Belt Standard Sheave and Groove Dimensions ANSI/RMA IP-22 (1983)

Deep Groove Dimensionsa Cross Section

3V

5V

Deep Groove Outside Diameter

Groove Angle, α, ±0.25 deg

bg ±0.005

Up through 3.71

36

0.421

Over 3.71 up to and including 6.22

38

0.425

Over 6.22 up to and including 12.22

40

0.429

Up through 10.31

Over 12.22

42

0.434

Over 10.31 up to and including 16.32

38

40

0.716

Over 16.32

42

0.723

Over 16.51 up to and including 22.92

40

1.191

42

1.201

Up through 16.51

8V

Over 22.92

38

be (Ref)

hg (Min)

RB (Min)

Design Factors b

Min Recommended OD

2a

2he

dB ±0.0005

Sg ±0.015

0.3438

0.500

0.375 (+0.094, −0.031)

2.87

0.050

0.218

0.5938

0.812

0.562 (+0.125, − 0.047)

7.42

0.100

0.320

1.0000

1.312

0.844 (+0.250, −0.062)

13.02

0.200

0.524

Se

0.070 0.350

0.449

0.073 0.076 0.078

0.710

0.168 0.600

0.750

0.172 0.175

1.180

0.312 1.000

1.252

0.316 0.321

Outside Diameter

Up through 8.0 in. outside diameter ±0.020 in. For each additional inch of outside diameter add ±0.0025 in. a Total indicator reading.

All dimensions in inches.

Other Sheave Tolerances

Radial Runouta 

Up through 10.0 in. outside diameter 0.010 in. For each additional inch of outside diameter add 0.0005 in.

Axial Runout a

Up through 5.0 in. outside diameter 0.005 in. For each additional inch of outside diameter add 0.001 in.

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a Deep groove sheaves are intended for drives with belt offset such as quarter-turn or vertical shaft drives. They may also be necessary where oscillations in the center distance may occur. Joined belts will not operate in deep groove sheaves. b Summation of the deviations from S for all grooves in any one sheave should not exceed ±0.031 in. The variations in pitch diameter between the grooves in any g one sheave must be within the following limits: Up through 19.9 in. outside diameter and up through 6 grooves—0.010 in. (add 0.0005 in. for each additional groove). 20.0 in. and over on outside diameter and up through 10 grooves—0.015 in. (add 0.0005 in. for each additional groove). This variation can be obtained by measuring the distance across two measuring balls or rods placed in the grooves diametrically opposite each other. Comparing this “diameter over balls or rods” measurement between grooves will give the variation in pitch diameter.

Machinery's Handbook, 31st Edition Narrow V-Belts

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Table 2b. Narrow V-Belt Standard Sheave and Groove Dimensions ANSI/RMA IP-22 (1983)

Machinery's Handbook, 31st Edition Narrow V-Belts

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Table 3. Standard Sheave Outside Diameters ANSI/RMA IP-22, 1983 Nom 2.65 2.80 3.00 3.15 3.35

3V Min 2.638 2.795 2.953 3.150 3.346

Max 2.680 2.840 3.000 3.200 3.400

Nom 7.10 7.50 8.00 8.50 9.00

5V Min 7.087 7.480 7.874 8.346 8.819

Max 7.200 7.600 8.000 8.480 8.960

Nom 12.50 13.20 14.00 15.00 16.00

8V Min 12.402 13.189 13.976 14.764 15.748

Max 12.600 13.400 14.200 15.000 16.000

3.55 3.65 4.00 4.12 4.50

3.543 3.642 3.937 4.055 4.409

3.600 3.700 4.000 4.120 4.480

9.25 9.75 10.00 10.30 10.60

9.291 9.567 9.843 10.157 10.433

9.440 9.720 10.000 10.320 10.600

17.00 18.00 19.00 20.00 21.20

16.732 17.717 18.701 19.685 20.866

17.000 18.000 19.000 20.000 21.200

4.75 5.00 5.30 5.60 6.00

4.646 4.921 5.197 5.512 5.906

4.720 5.000 5.280 5.600 6.000

10.90 11.20 11.80 12.50 13.20

10.709 11.024 11.811 12.402 13.189

10.880 11.200 12.000 12.600 13.400

22.40 23.60 24.80 30.00 31.50

22.047 23.622 24.803 29.528 31.496

22.400 24.000 25.200 30.000 32.000

6.30 6.50 6.90 8.00 10.00

6.299 6.496 6.890 7.874 9.843

6.400 6.600 7.000 8.000 10.000

14.00 15.00 16.00 18.70 20.00

13.976 14.764 15.748 18.701 19.685

14.200 15.000 16.000 19.000 20.000

35.50 40.00 44.50 50.00 52.00

35.433 39.370 44.094 49.213 51.969

36.000 40.000 44.800 50.000 52.800

10.60 12.50 14.00 16.00 19.00

10.433 12.402 13.976 15.748 18.701

10.600 12.600 14.200 16.000 19.000

21.20 23.60 25.00 28.00 31.50

20.866 23.622 24.803 27.953 31.496

21.200 24.000 25.200 28.400 32.000

63.00 71.00 79.00 99.00 …

62.992 70.866 78.740 98.425 …

64.000 72.000 80.000 100.000 …

20.00 25.00 31.50 33.50 … …

19.685 24.803 31.496 33.465 … …

20.000 25.200 32.000 34.000 … …

37.50 40.00 44.50 50.00 63.00 71.00

37.402 39.370 44.094 49.213 62.992 70.866

38.000 40.000 44.800 50.000 64.000 72.000

… … … … … …

… … … … … …

… … … … … …

All dimensions in inches. The nominal diameters were selected from R40 and R80 preferred num­bers (see page 687).

Minimum Sheave Size: The recommended minimum sheave size depends on the rpm of the faster shaft. Minimum sheave diameters for each belt cross section are listed in Table 3.

Cross Section Selection: The chart (Fig. 2, on page 2573) is a guide to the V-belt cross section to use for any combination of design horsepower and speed of the faster shaft. When the intersection of the design horsepower and speed of the faster shaft falls near a line between two areas on the chart, it is advisable to investigate the possibilities in both areas. Special circumstances (such as space limitations) may lead to a choice of belt cross section different from that indicated in the chart.

Horsepower Ratings: The horsepower ratings of narrow V-belts can be calculated using the following formula: K2 F + KSR r HP = dp r