Dudley's Handbook of Practical Gear Design and Manufacture [4 ed.] 0367649020, 9780367649029

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DUDLEY’S HANDBOOK OF PRACTICAL GEAR DESIGN AND MANUFACTURE

DUDLEY’S HANDBOOK OF PRACTICAL GEAR DESIGN AND MANUFACTURE Fourth Edition

Edited by

Stephen P. Radzevich

Fourth edition published 2022 by CRC Press 6000 Broken Sound Parkway NW, Suite 300, Boca Raton, FL 33487-2742 and by CRC Press 2 Park Square, Milton Park, Abingdon, Oxon, OX14 4RN © 2022 selection and editorial matter, Stephen P. Radzevich; individual chapters, the contributors First edition published by McGraw-Hill Book Company 1984 Third edition published by CRC Press 2016 CRC Press is an imprint of Taylor & Francis Group, LLC Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, access www.copyright.com or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. For works that are not available on CCC please contact [email protected] Trademark notice: Product or corporate names may be trademarks or registered trademarks and are used only for identification and explanation without intent to infringe. ISBN: 978-0-367-64902-9 (hbk) ISBN: 978-0-367-64906-7 (pbk) ISBN: 978-1-003-12688-1 (ebk) Typeset in Times by MPS Limited, Dehradun Access the Support Materials: https://www.routledge.com/9780367649029.

Contents Preface...................................................................................................................................................... xix Acknowledgments.................................................................................................................................... xxi Author .................................................................................................................................................... xxiii Contributors ............................................................................................................................................ xxv

Introduction............................................................................................................................................... 1 Chapter 1

Foundations of Advanced Gear Systems............................................................................. 7 1.1 1.2

The Law of Contact: The First Fundamental Law of Gearing................................ 9 The Conjugate Action Law: The Second Fundamental Law of Gearing.............. 12 1.2.1 Conjugate Action Law in Parallel-Axes Gearing ......................................12 1.2.2 Conjugate Action Law in Intersected-Axes Gearing, and in Crossed-Axes Gearing.................................................................................14 1.2.3 Examples of Violation of the Conjugate Action Law ...............................19 1.3 The Law of Equal Base Pitches: The Third Fundamental Law of Gearing..................................................................................................................... 22 Conclusion .......................................................................................................................... 28 References........................................................................................................................... 28 Bibliography ....................................................................................................................... 29 Chapter 2

Gear-Design Trends............................................................................................................ 31 2.1 2.2 2.3

Manufacturing Trends ............................................................................................. 31 Features of Gears of Different Kinds ..................................................................... 31 Selection of the Right Kind of Gear....................................................................... 42 2.3.1 External Spur Gears ....................................................................................43 2.3.2 External Helical Gears ................................................................................45 2.3.3 Internal Gears ..............................................................................................47 2.3.4 Straight Bevel Gears....................................................................................48 2.3.5 Zerol Bevel Gears........................................................................................50 2.3.6 Spiral Bevel Gears.......................................................................................51 2.3.7 Hypoid Gears ...............................................................................................52 2.3.8 Face Gears ...................................................................................................54 2.3.9 Crossed-Helical Gears (Non-Enveloping Worm Gears) ............................55 2.3.10 Single-Enveloping Worm Gears .................................................................57 2.3.11 Double-Enveloping Worm Gears................................................................59 2.3.12 Spiroid Gears ...............................................................................................62 Bibliography ....................................................................................................................... 64 Chapter 3

Gear Types and Nomenclature .......................................................................................... 65 3.1

Types 3.1.1 3.1.2 3.1.3

of Gears......................................................................................................... 65 Classifications ..............................................................................................65 Parallel-Axes Gears.....................................................................................68 Nonparallel, Coplanar Gears (Intersecting-Axes) ......................................73 v

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3.1.4 Nonparallel, Non-Coplanar Gears (Non-Intersecting Axes)......................79 3.1.5 Nonconjugate Gears ....................................................................................88 3.1.6 Special Gear Types .....................................................................................88 3.2 Nomenclature of Gears............................................................................................ 92 3.2.1 Spur Gear Nomenclature and Basic Formulas...........................................94 3.2.2 Helical Gear Nomenclature and Basic Formulas.......................................94 3.2.3 Internal Gear Nomenclature and Formulas ................................................96 3.2.4 Crossed Helical Gear Nomenclature and Formulas...................................96 3.2.5 Bevel Gear Nomenclature and Formulas ...................................................96 3.2.6 Worm Gear Nomenclature and Formulas ................................................102 3.2.7 Face Gears .................................................................................................104 3.2.8 Spiroid Gear Nomenclature and Formulas...............................................104 3.2.9 Helicon Gears ............................................................................................105 3.3 An Advanced Set of Terms and Definitions for Design Parameters in Gearing................................................................................................................... 106 References......................................................................................................................... 121 Bibliography ..................................................................................................................... 122 Chapter 4

Gear Tooth Design ........................................................................................................... 123 4.1

4.2

4.3

4.4

Basic Requirements of Gear Teeth ....................................................................... 123 4.1.1 Definition of Gear Tooth Elements ..........................................................123 4.1.2 Basic Considerations for Gear Tooth Design ..........................................126 4.1.3 Long- and Short-Addendum Gear Design................................................137 4.1.4 Special Design Considerations..................................................................143 Standard Systems of Gear Tooth Proportions ...................................................... 150 4.2.1 Standard Systems for Spur Gears.............................................................150 4.2.2 System for Helical Gears ..........................................................................161 4.2.3 System for Internal Gears .........................................................................166 4.2.4 Standard Systems for Bevel Gears ...........................................................169 4.2.5 Standard Systems for Worm Gears ..........................................................174 4.2.6 Standard System for Face Gears ..............................................................180 4.2.7 System for Spiroid and Helicon Gears.....................................................182 General Equations Relating to Center-Distance ................................................... 187 4.3.1 Center-Distance Equations ........................................................................188 4.3.2 Standard Center-Distance ..........................................................................190 4.3.3 Standard Pitch Diameters ..........................................................................191 4.3.4 Operating Pitch Diameters ........................................................................191 4.3.5 Operating Pressure Angle..........................................................................192 4.3.6 Operating Center-Distance ........................................................................193 4.3.7 Center-Distance for Gears Operating on Nonparallel Nonintersecting Shafts...............................................................................193 4.3.8 Center Distance for Worm Gearing..........................................................193 4.3.9 Reasons for Nonstandard Center-Distances..............................................194 4.3.10 Nonstandard Center-Distances ..................................................................194 Elements of Center-Distance................................................................................. 195 4.4.1 Effects of Tolerances on Center-Distance................................................195 4.4.2 Machine Elements That Require Consideration in Critical Center-Distance Applications....................................................................199 4.4.3 Control of Backlash ..................................................................................202

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4.4.4 Effects of Temperature on Center-Distance...........................................203 4.4.5 Mounting Distance ..................................................................................205 Bibliography ..................................................................................................................... 206 Chapter 5

Preliminary Design Considerations ................................................................................. 207 5.1

Stress Formulas...................................................................................................... 207 5.1.1 Calculated Stresses..................................................................................207 5.1.2 Gear-Design Limits.................................................................................211 5.1.3 Gear-Strength Calculations .....................................................................211 5.1.4 Gear Surface-Durability Calculations.....................................................216 5.1.5 Gear Scoring ...........................................................................................224 5.1.6 Thermal Limits........................................................................................230 5.2 Stress Formulas...................................................................................................... 231 5.2.1 Gear Specifications .................................................................................231 5.2.2 Size of Spur and Helical Gears by Q-Factor Method ...........................232 5.2.3 Indexes of Tooth Loading ......................................................................236 5.2.4 Estimating Spur- and Helical-Gear Size by K-Factor ...........................239 5.2.5 Estimating Bevel-Gear Size....................................................................239 5.2.6 Estimating Worm-Gear Size...................................................................243 5.2.7 Estimating Spiroid Gear Size .................................................................245 5.3 Data Needed for Gear Drawings........................................................................... 245 5.3.1 Gear Dimensional Data...........................................................................246 5.3.2 Gear-Tooth Tolerances ...........................................................................248 5.3.3 Gear Material and Heat-Treatment Data................................................250 5.3.4 Enclosed-Gear-Unit Requirements .........................................................255 Bibliography ..................................................................................................................... 256

Chapter 6

Design Formulas............................................................................................................... 257 6.1

Calculation of Gear-Tooth Data............................................................................ 257 6.1.1 Number of Pinion Teeth .........................................................................257 6.1.2 Hunting Teeth..........................................................................................259 6.1.3 Spur-Gear-Tooth Proportions..................................................................262 6.1.4 Root Fillet Radii of Curvature................................................................265 6.1.5 Long-Addendum Pinions ........................................................................266 6.1.6 Tooth Thickness ......................................................................................270 6.1.7 Chordal Dimensions ................................................................................272 6.1.8 Degrees Roll and Limit Diameter ..........................................................274 6.1.9 Form Diameter and Contact Ratio..........................................................278 6.1.10 Spur-Gear Dimension Sheet....................................................................281 6.1.11 Internal-Gear Dimension Sheet...............................................................283 6.1.12 Helical-Gear Tooth Proportions..............................................................287 6.1.13 Helical-Gear Dimension Sheet................................................................291 6.1.14 Bevel-Gear Tooth Proportions ................................................................293 6.1.15 Straight-Bevel-Gear Dimension Sheet....................................................295 6.1.16 Spiral-Bevel-Gear Dimension Sheet.......................................................300 6.1.17 Zerol-Bevel-Gear Dimension Sheet........................................................302 6.1.18 Hypoid-Gear Calculations.......................................................................304 6.1.19 Face-Gear Calculations ...........................................................................304 6.1.20 Crossed-Helical-Gear Proportions ..........................................................307

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6.1.21 Single-Enveloping-Worm-Gear Proportions...........................................308 6.1.22 Single-Enveloping Worm Gears .............................................................309 6.1.23 Double-Enveloping Worm Gears............................................................312 6.2 Gear-Rating Practice.............................................................................................. 316 6.2.1 General Considerations in Rating Calculations......................................316 6.2.2 General Formulas for Tooth Bending Strength and Tooth Surface Durability.................................................................................................319 6.2.3 Geometry Factors for Strength ...............................................................328 6.2.4 Overall Derating Factor for Strength......................................................330 6.2.5 Geometry Factors for Durability ............................................................344 6.2.6 Overall Derating Factor for Surface Durability .....................................346 6.2.7 Load Rating of Worm Gearing...............................................................349 6.2.8 Design Formulas for Scoring..................................................................360 6.2.9 Trade Standards for Rating Gears ..........................................................371 6.2.10 Vehicle-Gear-Rating Practice..................................................................372 6.2.11 Marine-Gear-Rating Practice ..................................................................374 6.2.12 Oil and Gas Industry Gear Rating..........................................................375 6.2.13 Aerospace-Gear-Rating Practice .............................................................376 References......................................................................................................................... 378 Chapter 7

Gear Reactions and Mountings........................................................................................ 379 7.1

7.2

7.3

7.4

7.5

Mechanics of Gear Reactions ............................................................................. 379 7.1.1 Summation of Forces and Moments.......................................................379 7.1.2 Application to Gearing............................................................................382 Basic Gear Reactions, Bearing Loads, and Mounting Types ............................ 383 7.2.1 The Main Source of Load.......................................................................383 7.2.2 Gear Reactions to Bearing ......................................................................384 7.2.3 Directions of Loads.................................................................................385 7.2.4 Additional Considerations.......................................................................385 7.2.5 Types of Mountings ................................................................................386 7.2.6 Efficiencies ..............................................................................................388 Basic Mounting Arrangements and Recommendations ..................................... 388 7.3.1 Bearing and Shaft Alignment .................................................................389 7.3.2 Bearings ...................................................................................................389 7.3.3 Mounting Gears to Shaft.........................................................................390 7.3.4 Housing....................................................................................................390 7.3.5 Inspection Hole........................................................................................391 7.3.6 Break-In ...................................................................................................391 Bearing Load Calculations for Spur Gears......................................................... 391 7.4.1 Spur Gears ...............................................................................................391 7.4.2 Helical Gears ...........................................................................................392 7.4.3 Gears in Trains ........................................................................................392 7.4.4 Idlers ........................................................................................................392 7.4.5 Intermediate Gears ..................................................................................392 7.4.6 Planetary Gears........................................................................................394 Bearing-Load Calculations for Helicals.............................................................. 397 7.5.1 Single-Helical Gears................................................................................397 7.5.2 Double-Helical Gears ..............................................................................399 7.5.3 Skewed or Crossed Helical Gears ..........................................................399

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ix

7.6

Mounting Practice for Bevel and Hypoid Gears ................................................ 401 7.6.1 Analysis of Forces...................................................................................402 7.6.2 Rigid Mountings......................................................................................402 7.6.3 Maximum Displacements........................................................................402 7.6.4 Rolling-Element Bearings .......................................................................403 7.6.5 Straddle Mounting...................................................................................405 7.6.6 Overhung Mounting ................................................................................406 7.6.7 Gear Blank Design ..................................................................................407 7.6.8 Gear and Pinion Adjustments .................................................................409 7.6.9 Assembly Procedure................................................................................410 7.7 Calculation of Bevel and Hypoid Bearing Loads .............................................. 412 7.7.1 Hand of Spiral .........................................................................................412 7.7.2 Spiral Angle.............................................................................................415 7.7.3 Tangential Load.......................................................................................415 7.7.4 Axial Thrust.............................................................................................417 7.7.5 Radial Load .............................................................................................417 7.7.6 Required Data for Bearing Load Calculations.......................................426 7.8 Bearing Load Calculations for Worms ............................................................... 428 7.8.1 Calculation of Forces in Worm Gears....................................................428 7.8.2 Mounting Tolerances...............................................................................430 7.8.3 Worm Gear Blank Considerations..........................................................432 7.8.4 Run-In of Worm Gears ...........................................................................433 7.9 Bearing Load Calculations for Spiroid Gearing................................................. 433 7.10 Bearing Load Calculations for Other Gear Types.............................................. 436 7.11 Design of the Body of the Gear ......................................................................... 437 References......................................................................................................................... 439 Chapter 8

Compensation of Shaft Deflections through Gear Micro-Geometry Modifications .................................................................................................................... 441 8.1 8.2

Introduction ............................................................................................................ 441 Determination of Errors of Alignment due to Shaft Deflections......................... 441 8.2.1 Transmissions with Parallel Shafts...........................................................442 8.2.2 Transmissions with Intersecting Shafts ....................................................447 8.2.3 Transmissions with Crossing Shafts.........................................................450 8.3 Compensation of Errors of Alignment during Gear Generation.......................... 451 8.4 Numerical Examples.............................................................................................. 453 8.4.1 Spur Gearset ..............................................................................................454 8.4.2 Spiral Bevel Gearset..................................................................................458 8.4.3 Face Gearset ..............................................................................................463 8.4.3.1 Alternative Methods of Compensating Shaft Deflections................................................................................. 473 References......................................................................................................................... 479 Chapter 9

Special Design Problems in Gear Drives ........................................................................ 481 9.1

Special 9.1.1 9.1.2 9.1.3 9.1.4

Calculations of Involute Gear Geometry................................................. 481 Main Symbols and Definitions .................................................................481 Tooth Undercutting in External Gears .....................................................483 Tooth Tip Thickness and Tooth Pointing in External Gears ..................485 Interference of Profiles in External Gearing ............................................488

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9.1.5 9.1.6

Interference of Profiles in Internal Gearing .............................................489 Measurements of Tooth Thickness...........................................................493 9.1.6.1 Direct Measurements of Tooth Thickness ............................... 494 9.1.6.2 Measurement of Base Tangent Length .................................... 497 9.1.6.3 Measurements Using Pins or Balls .......................................... 500 9.1.6.4 Measurements of Teeth with Profile Modifications ................ 503 9.1.7 Profile Modifications .................................................................................505 9.1.7.1 Basics ........................................................................................ 505 9.1.7.2 Parameters of Linear Tip Relief............................................... 507 9.2 Examples of Calculation of Involute Gear Pairs Geometry ................................ 520 9.3 Analysis of Motion and Power Transmission in Complex Cylindrical Gear Drives ..................................................................................................................... 526 9.3.1 Definition of Gear Ratio ...........................................................................526 9.3.2 Basic Kinematic Diagrams and Gear Ratios of Planetary Gear Drives................................................................................................527 9.3.3 MN Method of Kinematical Analysis ......................................................537 9.4 Efficiency of Cylindrical Involute Gear Drives and Complex Driving Systems .................................................................................................................. 544 9.4.1 Efficiency of a Single Gear Pair...............................................................544 9.4.2 Efficiency of Planetary Gear Drives.........................................................546 9.4.2.1 Efficiency of Planetary Gear Drives 1CG ............................... 546 9.4.2.2 Efficiency of Planetary Gear Drives 2CG Type A.................. 547 9.4.2.3 Efficiency of Planetary Gear Drives 2CG Type B.................. 548 9.4.2.4 Efficiency of Planetary Gear Drives 2CG Type C.................. 548 9.4.2.5 Efficiency of Planetary Gear Drives 2CG Type D.................. 550 9.4.2.6 Efficiency of Planetary Gear Drives 3CG ............................... 551 9.4.3 Efficiency of Planetary and Complex Gear Drives Built Up of Two or More Stages..................................................................................553 9.5 Lubrication and Cooling of Gear Drives .............................................................. 565 9.5.1 Lubrication.................................................................................................565 9.5.2 Cooling ......................................................................................................573 9.6 Design of Spur and Helical Gears ........................................................................ 577 9.6.1 Pinions .......................................................................................................578 9.6.2 Gears ..........................................................................................................579 9.6.2.1 Industrial Gears ......................................................................... 579 9.6.2.2 Light-Weight Gears .................................................................. 581 9.6.2.3 Large-Diameter Gears............................................................... 584 9.6.3 Arrangement of Gear Supports.................................................................590 References......................................................................................................................... 594 Chapter 10 Gear Materials .................................................................................................................. 595 10.1

10.2

Steels for Gears ................................................................................................... 595 10.1.1 Mechanical Properties ...........................................................................595 10.1.2 Heat-Treating Techniques .....................................................................598 10.1.3 Heat-Treating Data ................................................................................599 10.1.4 Hardness Tests.......................................................................................602 Localized Hardening of Gear Teeth.................................................................... 603 10.2.1 Carburizing ............................................................................................604

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xi

10.2.2

Nitriding.................................................................................................612 10.2.2.1 Features of Nitriding Process ............................................. 612 10.2.2.2 Nitride Case Depth ............................................................. 615 10.2.3 Induction Hardening of Steel................................................................616 10.2.3.1 Induction-Hardening by Scanning...................................... 618 10.2.3.2 Load-Carrying Capacity of Induction-Hardened Gear Teeth........................................................................... 622 10.2.4 Flame Hardening of Steel .....................................................................623 10.2.5 Combined Heat Treatments ..................................................................624 10.2.6 Metallurgical Quality of Steel Gears....................................................624 10.2.6.1 Geometric Accuracy ........................................................... 625 10.2.6.2 Material Quality .................................................................. 625 10.2.6.3 Quality Items for Carburized Steel Gears.......................... 625 10.2.6.4 Quality Items for Nitrided Gears ....................................... 625 10.2.6.5 Procedure to Get Grade 2 Quality ..................................... 626 10.3 Cast Irons for Gears ............................................................................................ 628 10.3.1 Gray Cast Iron.......................................................................................629 10.3.2 Ductile Iron............................................................................................631 10.4 Nonferrous Gear Metals ...................................................................................... 632 10.4.1 Kinds of Bronze ....................................................................................632 10.4.2 Standard Gear Bronzes..........................................................................634 10.5 Nonmetallic Gears ............................................................................................... 634 10.5.1 Thermosetting Laminates ......................................................................635 10.5.2 Nylon Gears...........................................................................................636 References......................................................................................................................... 637 Chapter 11 Load Carrying Capacities, Strength Numbers, and Main Influence Parameters for Different Gear Materials and Heat Treatment Processes.......................................... 639 11.1 11.2 11.3 11.4 11.5 11.6 11.7

11.8

Introduction .......................................................................................................... 639 Fundamentals of Gear Stresses and the Determination of the Load Carrying Capacity ................................................................................................ 640 Overview of Typical Gear Failure Modes.......................................................... 642 Requirements on the Properties of Gear Steels.................................................. 644 Steels for Quenching and Tempering ................................................................. 645 Steels for Surface Hardening .............................................................................. 645 Steels for Nitriding .............................................................................................. 651 11.7.1 Tooth Root Bending Strength...............................................................652 11.7.2 Tooth Flank Load Carrying Capacity...................................................652 11.7.3 Micropitting and Wear Performance ....................................................655 Steels for Case-Hardening and Carbonitriding................................................... 657 11.8.1 Influence of Gear Size ..........................................................................659 11.8.2 Influence of Case-Hardening Depth .....................................................660 11.8.3 Influence of Retained Austenite............................................................661 11.8.4 Influence of Cryogenic Treatment ........................................................663 11.8.5 Influence of Residual Stress Condition ................................................667 11.8.6 Influence on Tooth Root Load Carrying Capacity ..............................668 11.8.7 Change in the Fracture Mode—Unpenned vs Shot-Peened Condition ...............................................................................................670 11.8.8 Stepwise S–N Curve .............................................................................672

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11.8.9 Increase of the Tooth Flank Load Carrying Capacity (Pitting) ..........672 11.9 Summary and Outlook......................................................................................... 675 Acknowledgment .............................................................................................................. 678 References......................................................................................................................... 678 Chapter 12 Gear Load Capacity Calculation: Based on ISO 6336 ................................................... 685 12.1

12.2

12.3

12.4

Introduction and History of ISO 6336................................................................ 685 12.1.1 Introduction: Parts and Document Types.............................................685 12.1.2 History ...................................................................................................686 12.1.3 Overview and Structure of ISO 6336 Documents ...............................687 Calculation of Surface Durability—ISO 6336-2:2019 ....................................... 688 12.2.1 Description of the Failure Mode Pitting ..............................................688 12.2.2 Basic Calculation Principles .................................................................690 12.2.2.1 Strength Analysis and Safety Factor.................................. 690 12.2.2.2 Contact Stress σH ................................................................ 691 12.2.2.3 Pitting Stress Limit σHG ..................................................... 695 12.2.3 New Aspects and Updates of the Standard..........................................699 12.2.3.1 Contact Factor ZB,D—Factor fZCa ...................................... 700 12.2.3.2 Work Hardening Factor ZW ............................................... 700 12.2.4 Calculation Example .............................................................................700 12.2.5 Summary................................................................................................702 12.2.6 Outlook ..................................................................................................703 Calculation of Tooth Bending Strength—ISO 6336-3:2019.............................. 704 12.3.1 Description of the Failure Mode Tooth Root Breakage ......................704 12.3.2 Basic Calculation Principles .................................................................706 12.3.2.1 Strength Analysis and Safety Factor.................................. 706 12.3.2.2 Tooth Root Stress σF .......................................................... 707 12.3.2.3 Bending Stress Limit σFG ................................................... 710 12.3.3 New Aspects and Updates of the Standard..........................................715 12.3.3.1 Form Factor YF—Load Distribution Influence Factor fε ............................................................................... 715 12.3.3.2 Form Factor YF—Tooth Root Geometry of Internal Gears.................................................................................... 716 12.3.3.3 Helix Angle Factor Yβ........................................................ 717 12.3.3.4 Relative Notch Sensitivity Factor YδrelT.......................... 717 12.3.4 Calculation Example .............................................................................718 12.3.5 Summary................................................................................................720 12.3.6 Outlook ..................................................................................................720 Calculation of Micropitting Load Capacity—ISO/TS 6336-22:2018 ................ 721 12.4.1 Description of the Failure Mode Micropitting.....................................721 12.4.2 Basic Calculation Principles .................................................................723 12.4.2.1 Specific Lubricant Film Thickness λGF,Y ........................... 724 12.4.2.2 Permissible Specific Lubricant Film Thickness (according to the FZG Micropitting Test according to FVA 54/7)....................................................................... 727 12.4.2.3 Limits of the Calculation Method ...................................... 728 12.4.3 Micropitting Test Procedures................................................................729 12.4.3.1 FZG Micropitting Test according to FVA-Information Sheet 54/7............................................................................ 729

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12.4.3.2 DIN 3990-16:2020 .............................................................. 732 12.4.4 Calculation Example .............................................................................732 12.4.5 Summary................................................................................................735 12.4.6 Outlook ..................................................................................................736 12.5 Calculation of Tooth Flank Fracture Load Capacity—ISO/TS 6336-4:2019 ......................................................................................................... 737 12.5.1 Description of the Failure ModeTooth Flank Fracture........................737 12.5.2 Basic Calculation Principles .................................................................638 12.5.3 Influences on Tooth Flank Fracture......................................................745 12.5.4 Calculation Example .............................................................................746 12.5.5 Summary................................................................................................749 12.5.6 Outlook ..................................................................................................749 References......................................................................................................................... 750 Chapter 13 Potential and Challenges of High-Performance Plastic Gears........................................ 755 13.1 13.2

Introduction .......................................................................................................... 755 State of the Art and Application of Plastic Gears ............................................. 755 13.2.1 Materials and Properties........................................................................755 13.2.2 Manufacturing........................................................................................756 13.2.3 Design ....................................................................................................758 13.2.4 Fields of Application.............................................................................758 13.3 Design and Calculation Methods for Plastic Gear Applications ....................... 758 13.3.1 Tooth Temperature ................................................................................758 13.3.2 Tooth Load Carrying Capacity Acc. to VDI 2736 ..............................761 13.3.2.1 Tooth Root Load Carrying Capacity.................................. 761 13.3.2.2 Tooth Flank Load Carrying Capacity ................................ 762 13.3.2.3 Frictional Wear Load Carrying Capacity........................... 763 13.4 Recent Research Results ..................................................................................... 763 13.4.1 Thermal Behavior..................................................................................763 13.4.2 Low Loss Plastic Gears ........................................................................767 13.4.3 Tooth Root Load Carrying Capacity ....................................................771 13.4.4 Flank Load Carrying Capacity..............................................................776 13.4.5 Tribology ...............................................................................................780 13.5 Challenges for the Future Application of Plastic Gears .................................... 784 13.6 Conclusion ........................................................................................................... 787 Nomenclature.................................................................................................................... 788 References......................................................................................................................... 789 Chapter 14 The Kinds and Causes of Gear Failures.......................................................................... 793 14.1

14.2

Analysis 14.1.1 14.1.2 14.1.3 14.1.4 Analysis 14.2.1 14.2.2 14.2.3 14.2.4

of Gear-System Problems .................................................................... 793 Determining the Problem ......................................................................793 Possible Causes of Gear-System Failures ............................................794 Incompatibility in Gear Systems ..........................................................797 Investigation of Gear Systems ..............................................................797 of Tooth Failures and Gear-Bearing Failures ..................................... 799 Nomenclature of Gear Failure ..............................................................799 Tooth Breakage .....................................................................................800 Pitting of Gear Teeth ............................................................................806 Scoring Failures.....................................................................................808

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14.3

14.2.5 Wear Failures ........................................................................................810 14.2.6 Gearbox Bearings ..................................................................................813 14.2.7 Rolling-Element Bearings .....................................................................813 14.2.8 Sliding-Element Bearings......................................................................815 Some Causes of Gear Failure Other than Excess Transmission Load .............. 818 14.3.1 Overload Gear Failures .........................................................................818 14.3.2 Gear-Casing Problems...........................................................................819 14.3.3 Lubrication Failures ..............................................................................820 14.3.4 Thermal Problems in Fast-Running Gears ...........................................826

Chapter 15 Load Rating of Gears....................................................................................................... 829 15.1 15.2 15.3 15.4 15.5

15.6

15.7

15.8

15.9 15.10 15.11

Main Nomenclature........................................................................................... 829 Coplanar Gears (Involute Parallel Gears and Bevel Gears) ............................ 831 Coplanar Gears: Simplified Estimates and Design Criteria............................. 835 Coplanar Gears: Detailed Analysis, Conventional Fatigue Limits, and Service Factors .................................................................................................. 838 RH—Conventional Fatigue Limit of Factor K ................................................ 844 15.5.1 RH—Preliminary Geometric Calculations ........................................ 844 15.5.2 Adaption for Bevel Gears .................................................................. 849 15.5.3 RH—Unified Geometry Factor GH ................................................... 850 15.5.4 RH—Comments and Comparisons on the Unified Geometry Factor GH ............................................................................................ 851 15.5.5 RH—Elastic Coefficient Dp and Conventional Fatigue Limit sc lim of the Hertzian Pressure ................................................... 851 15.5.6 RH—Adaptation Factor, AH .............................................................. 853 15.5.7 RH—Hertzian Pressure...................................................................... 858 15.5.8 RH—Service Factor, CSF (Only for One Loading Level)................ 858 15.5.9 Power Capacity Tables....................................................................... 858 RF—Conventional Fatigue Limit of Factor UL ............................................... 859 15.6.1 RF—Geometry Factor, Jn .................................................................. 861 15.6.2 RF—Adaptation Factor, AF ............................................................... 865 15.6.3 RF—Size Factor, Ks ........................................................................... 868 15.6.4 RF—Conventional Fatigue Limit of the Fillet Stress, stlim ............... 868 15.6.5 RF—Tooth Root Stress at Fillet, st ................................................... 869 15.6.6 RF—Service Factor, KSF (Only for One Loading Level) ................ 869 Coplanar Gears: Detailed Life Curves and Yielding....................................... 869 15.7.1 Definition of the Life Curves and Gear Life Ratings for One Loading Level..................................................................................... 870 15.7.2 Yielding .............................................................................................. 873 15.7.3 Tooth Damage and Cumulative Gear Life........................................ 874 15.7.4 Reliability ........................................................................................... 875 Coplanar Gears: Prevention of Tooth Wear and Scoring................................ 876 15.8.1 Progressive Tooth Wear..................................................................... 876 15.8.2 Scoring and Scuffing.......................................................................... 876 Crossed Helical Gears....................................................................................... 877 Hypoid Gears..................................................................................................... 877 Worm Gearing................................................................................................... 878

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Chapter 16 Gear-Manufacturing Methods .......................................................................................... 881 16.1

Gear-Tooth Cutting.............................................................................................. 881 16.1.1 Gear Hobbing ........................................................................................881 16.1.2 Shaping—Pinion Cutter ........................................................................888 16.1.3 Shaping—Rack Cutter...........................................................................894 16.1.4 Cutting Bevel Gears ..............................................................................898 16.1.5 Gear Milling ..........................................................................................904 16.1.6 Broaching Gears ....................................................................................906 16.1.7 Punching Gears......................................................................................908 16.1.8 G-TRAC Generating .............................................................................909 16.2 Gear Grinding ...................................................................................................... 910 16.2.1 Form Grinding.......................................................................................912 16.2.2 Generating Grinding—Disk Wheel ......................................................918 16.2.3 Generating Grinding—Bevel Gears......................................................922 16.2.4 Generating Grinding—Threaded Wheel...............................................925 16.2.5 Thread Grinding ....................................................................................927 16.3 Gear Shaving, Rolling, and Honing.................................................................... 930 16.3.1 Rotary Shaving ......................................................................................931 16.3.2 Rack Shaving.........................................................................................935 16.3.3 Gear Rolling ..........................................................................................936 16.3.4 Gear Honing ..........................................................................................939 16.4 Gear Measurement............................................................................................... 940 16.4.1 Gear Accuracy Limits ...........................................................................940 16.4.2 Machines to Measure Gears..................................................................947 16.5 Gear Casting and Forming .................................................................................. 950 16.5.1 Cast and Molded Gears.........................................................................950 16.5.2 Sintered Gears .......................................................................................953 16.5.3 Cold-Drawn Gears and Rolled Worm Threads....................................954 Reference .......................................................................................................................... 955 Chapter 17 Design of Tools to Make Gear Teeth.............................................................................. 957 17.1 Shaper Cutters...................................................................................................... 957 17.2 Gear Hobs ............................................................................................................ 966 17.3 Spur-Gear Milling Cutters................................................................................... 978 17.4 Worm Milling Cutters and Grinding Wheels ..................................................... 982 17.5 Gear-Shaving Cutters........................................................................................... 988 17.6 Punching Tools .................................................................................................... 990 17.7 Sintering Tools..................................................................................................... 991 References......................................................................................................................... 993 Chapter 18 Dynamic Model of Technological System for Gear Finishing ...................................... 995 18.1 Introduction .......................................................................................................... 995 18.2 Development of a Generalized Dynamic Model ................................................ 995 18.3 Determining the Model Parameters .................................................................. 1001 18.4 Objective Functions of the Dynamic Model .................................................... 1012 18.5 Synthesis of Tools and Parameters of the Technological Systems ................. 1016 References....................................................................................................................... 1017

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Chapter 19 Powder Metal Gears....................................................................................................... 1019 19.1 19.2

Introduction ........................................................................................................ 1019 PM Materials for Gears..................................................................................... 1019 19.2.1 As Sintered (S) ....................................................................................1020 19.2.2 Sinter Hardened (SH)..........................................................................1020 19.2.3 Quench and Temper (QT)...................................................................1020 19.2.4 Induction Hardening (IH)....................................................................1021 19.2.5 Case Carburizing and Tempering (CQT) ...........................................1021 19.3 Manufacturing .................................................................................................... 1022 19.3.1 Compaction..........................................................................................1023 19.3.2 Sintering...............................................................................................1026 19.3.3 Post Processing....................................................................................1028 19.3.3.1 Roll Densification ............................................................. 1028 19.3.3.2 Hard Finishing .................................................................. 1028 19.3.3.3 Peening and Shot Blasting................................................ 1028 19.3.3.4 Hardening .......................................................................... 1029 19.3.3.5 Wire EDM......................................................................... 1031 19.3.3.6 Welding ............................................................................. 1031 19.4 Tolerances .......................................................................................................... 1031 19.5 Productivity ........................................................................................................ 1032 19.6 Powder Metal Gear Macro Design ................................................................... 1034 19.7 Powder Metal Micro Design ............................................................................. 1034 19.7.1 Root Optimization ...............................................................................1037 19.8 Stress Calculations of PM Gears ...................................................................... 1037 19.9 PM Specific Standards....................................................................................... 1038 19.9.1 MPIF 35...............................................................................................1038 19.9.2 AGMA 6008........................................................................................1039 19.9.3 AGMA 944..........................................................................................1039 19.9.4 AGMA 942..........................................................................................1039 19.9.5 AGMA 930..........................................................................................1039 References....................................................................................................................... 1039 Chapter 20 3D Printed Gears ............................................................................................................ 1041 20.1 20.2

Introduction ........................................................................................................ 1041 Plastic Gears ...................................................................................................... 1041 20.2.1 Fused Filament Deposition (FFD) ......................................................1041 20.2.2 Stereo Lithography (SL) .....................................................................1042 20.2.3 Selective Laser Sintering (SLS)..........................................................1042 20.3 Steel Gears ......................................................................................................... 1042 20.3.1 Fused Filament Deposition (FFD) ......................................................1042 20.3.2 Laser and Electron Beam Methods (LPBF, EB-PBF) .......................1044 20.3.3 Binder Jet (BJ) ....................................................................................1046 20.4 Steels for Powder Bed Printing......................................................................... 1047 20.5 Post Processing of AM Steel Gears.................................................................. 1048 20.6 Final Remarks .................................................................................................... 1049 References....................................................................................................................... 1050

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Chapter 21 Gear Noise and Vibration (NVH).................................................................................. 1051 21.1

Fundamentals of Gear Noise............................................................................. 1051 21.1.1 Transfer Path of Vibration ..................................................................1051 21.1.2 Gear Noise Excitation .........................................................................1052 21.1.3 Eigenfrequency and Resonance ..........................................................1055 21.2 Calculation Methods to Evaluate Gear Noise Excitation................................. 1056 21.2.1 Differential Equation for Vibration Phenomena ................................1057 21.2.2 Transmission Error by Quasi-Static Approach ..................................1057 21.2.3 Tooth Force Excitation by Quasi-Static Approach ............................1058 21.2.4 Dynamic Approach..............................................................................1058 21.2.5 Evaluation by Characteristic Values...................................................1060 21.3 Measurement of Vibration................................................................................. 1061 21.3.1 Sensors and Measurement Results......................................................1061 21.3.2 Positioning of Sensors and Operating Range.....................................1066 21.3.3 Sources of Possible Errors ..................................................................1067 21.4 Condition Monitoring ........................................................................................ 1068 References....................................................................................................................... 1069 Chapter 22 Planetary Gear Trains..................................................................................................... 1071 22.1 Introduction ...................................................................................................... 1071 22.2 Types of Simple Planetary Gear Trains ......................................................... 1071 22.3 Specific Conditions of Planetary Gear Trains ................................................ 1073 22.4 Meshing Geometry of Planetary Gear Trains................................................. 1077 22.5 Torque Method for Kinematic and Power Analysis of Planetary Gear Trains...1077 22.6 Type of Powers, Losses, and Basic Efficiency of Planetary Gear Trains..... 1080 22.7 Efficiency of Planetary Gear Trains ............................................................... 1083 22.8 Load Capacity of Gears of Planetary Gear Train .......................................... 1084 22.9 Load Distribution between the Planets, Its Unevenness, and Equalization .. 1086 22.10 Types of Compound Planetary Gear Trains ................................................... 1090 22.11 Two-Carrier Compound Planetary Gear Trains.............................................. 1092 22.12 Three-Carrier Compound Planetary Gear Trains............................................ 1101 22.13 Four-Carrier Compound Planetary Gear Trains ............................................. 1104 22.14 Wolfrom Planetary Gear Train ........................................................................ 1106 22.15 Ravigneaux Planetary Gear Train ................................................................... 1110 22.16 Warning—Planetary Gear Trains .................................................................... 1111 References....................................................................................................................... 1113 Index..................................................................................................................................................... 1117

Preface This publication is dedicated to engineers who work in the field of gear design and gear production. Gearing is a specific area of mechanical engineering. Gearing encompasses the gear design, production, inspection, and implementation of gears, as well as some other supplementary subjects. This book is primarily focused on the gear design and gear production. Most machine designers do not have the time to keep up with all the developments in the field of gear design. This makes it hard for them to quickly design gears which will be competitive with the best that are being used in their field. There is a great need for practical gear-design information. Even though there is a wealth of published information on gears, gear designers often find it hard to locate the information they need quickly. This book is written to help gear designers get the vital information they need as easily as possible. The book is a revised1, updated and expanded version of the earlier published book by Darle W. Dudley titled “Handbook of Practical Gear Design”. The book has been published by McGraw-Hill Book Company in 1984, and later by CRC Press ten years later in 1994. Both the editions are originated from the earlier book by that same author entitled “Practical Gear Design”, which was published by McGrawHill Book Company in 1954. Last edition 1984/1994 of the Gear Handbook is highly recognized by gear community all around the world. As time is passing, new knowledge is accumulated by the gear industry along with that that some of experience summarized in the Dudley’s Handbook is getting obsolete. The obsolete data should be removed from the book, while the novel data in the field should be added. Keeping in mind the importance of the Dudley’s Handbook for practical gear engineers and practitioners, I appreciated and supported the attempt undertaken by CRC Press to revise, to update, and to expand the 2016 edition of the Handbook: Radzevich, S.P., Dudley’s Handbook of Practical Gear Design and Manufacture, 3rd edition, CRC Press, Boca Raton, FL, 2016, 655 pages, 573 B/W Illustrations. One of the critical problems arose when working on the new edition of the Dudley’s Handbook was that: • The new edition of the Dudley’s Handbook should fulfill nowadays demands of the gear industry. This requires in significant revision of the original text • At that same time, much effort was undertaken to preserve the original text of the Dudley’s Handbook, and in that way retaining atmosphere of the Dudley’s text This contradiction between a desired maximum revision to get the book up to date, and between desirable minimum revision those allow the book to be titled as Dudley’s Handbook of Practical Gear Design and Manufacture required in tradeoff. There were two options to follow with. First, to make significant changes to the original text of the Dudley’s Handbook. Following this way, the charm and the philosophy of the Dudley’s Handbook would be lost (or it will be ruined in much). In this case the Dudley’s Handbook will no longer be the Dudley’s book. It will be another book on gearing having no the word “Dudley’s” in the title.

1

The first revision of the Gear Handbook was published by CRC Press in 2012 (See Radzevich, S.P., Dudley’s Handbook of Practical Gear Design and Manufacture, 2nd edition, CRC Press, 2012, Boca Raton, FL, 878 pages). The second revision of the Gear Handbook was published by CRC Press in 2016 (See Radzevich, S.P., Dudley’s Handbook of Practical Gear Design and Manufacture, 3rd edition, CRC Press, 2016, Boca Raton, FL, 629 pages).

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Preface

Second, to keep with the original text of the Dudley’s Handbook as close as possible. In this case the Dudley’s philosophy and his vision/understanding of gearing can be preserved. Both mentioned above ways have a right to live, and it was hard to make a decision which way to follow. Ultimately, the decision to make reasonable/limited changes to original text of the Dudley’s book (where appropriate) had been made. With that said, a limited number of repetitions is getting inevitable. The best was done in order to minimize total number of the repetitions. Prior to begin revising the Gear Handbook, all the earlier editions of the Dudley’s Handbook [starting from 1954 McGraw-Hill edition and ending with the latest 2016 CRC Press 3rd edition (including 1962 and 1984 McGraw-Hill editions)] were carefully studied. Obsolete material was removed from the text body where appropriate. Several new chapters, as well as new appendices are added. This makes possible to get the book updated. An international team of well-recognized gear experts all around the world actively participated on this 4th edition of Dudley’s Handbook. Stephen P. Radzevich Sterling Heights, Michigan January 15, 2021

Acknowledgments The author would like to share the credit for any success of the Gear Handbook to plenty discussions on the subject with numerous representatives of the gear community both domestic and international. The contribution of many friends and colleagues in overwhelming numbers cannot be acknowledged individually, and as much as our benefactors have contributed, even though their kindness and help must go unrecorded. Many thanks to Ms. Nicola Sharpe, editor, Mechanical Engineering, for the idea to revise the 2016 edition of the Dudley’s Gear Handbook, and in this way to make the revised, updated and enlarged edition available to the gear community all around the globe. My special thanks to my wife, Natasha, for her tolerance, support, encouragement, endless patience and love.

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Author Stephen P. Radzevich is a professor of mechanical engineering and a professor of manufacturing engineering. He earned an MSc in 1976, a PhD in 1982, and a Dr(Eng)Sc in 1991, all in mechanical engineering. Dr. Radzevich has extensive industrial experience in gear design and gear manufacture. He has developed numerous software packages dealing with computer-aided design (CAD) and computer-aided machining (CAM) of precise gear finishing for a variety of industrial sponsors. His main research interest is the field of kinematics and geometry of gearing, particularly with a focus on precision gear design, high-power-density gear trains, torque share in split-power-transmission systems (SPTS), design of special purpose gear cutting/finishing tools, and design and machine (finish) of precision gears for low-noise and noiseless transmissions of cars, light trucks and so on. Dr. Radzevich has spent about 45 years developing software, hardware and other processes for gear design and optimization. Besides his work for industry, he trains engineering students at universities and gear engineers in companies. He has authored and coauthored over 40 monographs, handbooks, and textbooks. The monographs Generation of Surfaces (RASTAN, 2001), Kinematic Geometry of Surface Machining (CRC Press, 2007, 2nd edition 2014), CAD/CAM of Sculptured Surfaces on Multi-Axis NC Machine: The DG/K-Based Approach (M&C Publishers, 2008), Gear Cutting Tools: Fundamentals of Design and Computation (CRC Press, 2010, 2nd edition, 2017), Precision Gear Shaving (Nova Science Publishers, 2010), Dudley’s Handbook of Practical Gear Design and Manufacture (CRC Press, 2nd edition, 2012, 3rd edition 2016), High-Conformal Gearing: Kinematics and Geometry (CRC Press, 2016; 2nd edition, Elsevier, 2020), Theory of Gearing: Kinematics, Geometry, and Synthesis (CRC Press, 2012, 2nd edition, 2018), Geometry of Surfaces: A Practical Guide for Mechanical Engineers (Wiley, 2013, Springer, 2019, 2nd edition), and Recent Advances in Gearing – Scientific Theory and Applications (Springer, 2021) are among his recently published books. He also authored or coauthored about 650 scientific papers and holds over 260 patents on inventions in the field (U.S.A., Japan, Russia, Europe, Canada, Soviet Union, South Korea, Mexico, and others).

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Contributors Kiril Arnaudov (Chapter 22) National Academy of Sciences Sofia, Bulgaria Niklas Blech (Chapter 11) FZG, Technical University of Munich Munich, Germany Holger Cermak (Chapter 11) FZG, Technical University of Munich Munich, Germany Anders Flodin (Chapters 19, 20) Höganäs AB, Höganäs Sweden Daniel Fuchs (Chapter 11) FZG, Technical University of Munich Munich, Germany Alfonso Fuentes-Aznar (Chapter 8) Rochester Institute of Technology, Rochester New York, U.S.A. Joshua Götz (Chapter 21) FZG, Technical University of Munich Munich, Germany Christopher M. Illenberger (Chapter 13) FZG, Technical University of Munich Munich, Germany Dimitar P. Karaivanov (Chapter 22) National Academy of Sciences Sofia, Bulgaria

Boris M. Klebanov (Chapter 9) Israel Aerospace Industry U.S.A./Israel

Daniel Müller (Chapter 12) FZG, Technical University of Munich Munich, Germany Michael Otto (Chapter 21) FZG, Technical University of Munich Munich, Germany Stephen P. Radzevich (Chapter 1) Eaton Corp. (retired), Southfield Michigan, U.S.A. Nadine Sagraloff (Chapter 12) FZG, Technical University of Munich Munich, Germany Stefan Sendlbeck (Chapter 12) FZG, Technical University of Munich Munich, Germany Sebastian Sepp (Chapter 21) FZG, Technical University of Munich Munich, Germany André Sitzmann (Chapter 11) FZG, Technical University of Munich Munich, Germany Adrian Sorg (Chapter 11) FZG, Technical University of Munich Munich, Germany

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Karsten Stahl (Chapters 11, 12, 13, 21) FZG, Technical University of Munich Munich, Germany Michael Storchak (Chapter 18) Institute for Machine Tools Stuttgart University Stuttgart, Germany Thomas Tobie (Chapters 11, 12) Technical University of Munich Munich, Germany

Contributors

Karl Jakob Winkler (Chapter 12) FZG, Technical University of Munich Munich, Germany The rest of the chapters in this edition of Dudley’s Gear Handbook are updated and expanded by Stephen P. Radzevich.

Introduction This book (further Dudley’s Gear Handbook) deals with gears and gearing in both aspects, namely in gear design, as well as in gear production. Gear industry Gear science

NO, thank you !!

We are too busy !!

Gears are produced in enormous amounts—billions of gears are produced by the industry every year. While the automotive industry ranks as the primary consumer of gears, numerous other industries also require huge number of gears: aerospace (helicopter transmission, and so forth), construction machinery, agricultural machinery, to name a few. If we place cost savings in the production of every gear in the range of just ten cents, the total cost savings could reach hundreds of millions of dollars. This makes it clear that the gear design and the gear production processes require in careful handling. There is much room for gear engineers and for gear practitioners for further improvements in cost saving processes. The Dudley’s Handbook can be helpful in solving many of the gear design, as well as of the gear production problems.

UNIQUENESS OF THIS BOOK Many books on gears and gearing, both in design and in manufacture areas are published to this end. This fourth edition of the Dudley’s Handbook is a unique one because of many reasons. First, the Dudley’s Handbook is based on practical experience of designing and production of gears. The collection of data in the Dudley’s Handbook is invaluable for practitioners in both areas, namely in the gear design, as well as in the gear production areas. Second, this fourth edition of the Dudley’s Handbook is significantly expanded in comparison to the earlier editions. New chapters are added, that makes the Dudley’s Handbook a comprehensive source of information for those who are involved in design and production of gears. Third, uniqueness of the Dudley’s Handbook is proven by several earlier editions, starting from the 1954 edition. For decades the earlier editions of the book were successfully used by gear experts all around the world. It is anticipated that this new revised, updated and enlarged edition of the Dudley’s Handbook will be acknowledged by the reader.

INTENDED AUDIENCE Many readers will benefit from the Dudley’s Handbook: mechanical and manufacturing engineers continuously involved in design and manufacture process improvement, those who are active or intend to 1

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Dudley’s Handbook of Practical Gear Design and Manufacture

become active in the field. Senior undergraduate and graduate university students of mechanical and of manufacturing engineering are among them. The Dudley’s Handbook in intended to be used as a reference book, as well as invaluable source of practical data for gear design engineers, and for gear manufacturers.

THE ORGANIZATION OF THIS BOOK The Dudley’s Handbook is organized in twenty-two chapters followed by two appendices. Chapter 1 is contributed by Prof. Stephen P. Radzevich. In this chapter, the key elements of advanced gear systems are briefly outlined. Three fundamental laws of gearing are covered in this section of the Dudley’s Handbook. They are as follows: • The Law of Contact: The First Fundamental Law of Gearing • The Conjugate Action Law: The Second Fundamental Law of Gearing • The Law of Equal Base Pitches: The Third Fundamental Law of Gearing The conjugate action law considered here more in detail, that is, this law of gearing is discussed separately for the cases of (a) parallel-axes gearing, (b) intersected-axes gearing, and (c) crossed-axes gearing. Examples of violation of the conjugate action law are discussed. Accurate gears can be designed and produced if and only if all three fundamental laws of gearing are fulfilled. Gear design trends are considered in Chapter 2. Both manufacturing trends, as well as selection of the right kind of gear are disclosed. The sub-section Manufacturing Trends encompasses small, low-cost gears for toys, gadgets, and mechanisms. Then application gears, control gears, vehicle gears, marine gears, aerospace gear, and others are discussed. Kinds of external spur and helical gears, straight bevel gears, spiral bevel gears, hypoid and face gears are covered in the sub-section Selection of the Right Kind of Gear. Gear types and gear nomenclature are discussed in Chapter 3. The discussion begins with classification of gears, which includes parallel axis gearing, nonparallel coplanar gears operating on intersecting axes, nonparallel noncoplanar gears operating on nonintersecting axes, and special gear types. The discussion is followed by nomenclature of all practical kinds of gears, which includes, but not limited to nomenclature of spur and helical gears, both external and internal, crossed helical gear nomenclature and formulas, bevel gear nomenclature and formulas, worm gear nomenclature and formulas, nomenclature of face gears, beveloid gears etc. Advanced terminology in gear design is briefly discussed at the end of the section. Gear tooth design is disclosed in Chapter 4. Basic requirements of gear teeth, standard systems of gear tooth proportions, general equations relating to center distance along with elements of center distance are covered in this section of the book. Special attention is given to standard systems of gear tooth proportions, including, but not limited to standard systems for spur, helical, and internal gears. Standard systems for bevel gearing, worm gearing, face gearing, Spiroid and Helicon gears are discussed as well. Preliminary design considerations are summarized in Chapter 5. Here, stress formula is discussed in detail. This consideration is followed by data needed for gear drawings. Design formulas are discussed in Chapter 6. Calculation of gear-tooth data and gear-rating practice are covered in this chapter. The discussion is illustrated by vehicle-gear-rating practice, marine-gearrating practice, oil and gas industry gear rating, and by aerospace-gear-rating practice. Gear reactions and mountings are outlined in Chapter 7. The discussion begins with an analysis of mechanics of gear reactions, which then follows by consideration of basic reactions, bearing loads and mounting types. After that basic mounting arrangements are considered and corresponding recommendations are given. This sub-section of the book is followed by an analysis of bearing load calculations for spur and for helical gears. Then mounting practice for bevel and hypoid gears along with

Introduction

3

calculation of bevel and hypoid bearing loads is considered. This section of the book is ended by an analysis of bearing load calculations for worms, for Spiroid and other gear types, as well as of design of the body of the gear. Chapter 8 is contributed by Prof. A. Fuentes-Aznar.1 Gear misalignments due to shaft deflections for the transmitted nominal torque can be predicted and consequently compensated by introducing surface modifications during the process of gear manufacturing. This way, the gear tooth surfaces will be prepared to provide the best conditions of meshing and contact for that transmitted torque. In this chapter, the determination of errors of alignment due to shaft deflections based on the application of the finite element method for gear transmissions with parallel, intersecting, and crossing shafts will be presented. Finite element models include the gears, their supporting shafts, and the position and rigidity of the bearings. Several numerical examples will show the advantages of compensation of errors of alignment in spur, spiral bevel, and face gear transmissions. Contributed by Dr. B.M. Klebanov2 is devoted to special design problems and consists of six sections. In Section 1 special problems of gear geometry are considered, such as undercutting and sharpening of teeth, interference of their profiles, measurements of tooth thickness, and the problem of tooth profile modification. In Section 2 examples of calculation of center distance, transversal contact ratio, and determination of the location of pitch point relative to the beginning and the end of the contact are presented. There is explained the difference between addendum and dedendum in contact strength of material. In Section 3 method of Mechanical Nodes (MN method) is offered for calculation of gear ratio of complex driving systems. Section 4 provides method of calculation efficiency of planetary and complex gear drives affiliated with the MN method of calculation of their gear ratios. Section 5 is all about lubrication and cooling of the gear drives. The natural and forced convection cooling are considered in detail with recommendation for calculation. Section 6 is dedicated to design of diverse gears: small and large, industrial and light-weight, gorged, welded and built-up. Also discussed the arrangement of bearings with respect to reduction of misalignments in the mesh of gears and in the bearings. In Chapter 10 gear materials are considered. Steel for gears, localized hardening of gear teeth, cast irons for gears, as well as nonferrous gear metals and nonmetallic hears are considered in this section of the book. Chapter 11 is contributed by a team of gear experts of Gear Research Center (FZG) at the Technical University of Munich3, Germany. Components of the power train are usually subject to various stresses. Increasing power densities of modern gear transmissions require a high functional safety and efficiency of the power transmitting parts. Especially the design and the material with its microstructural properties and parameters effect the performance characteristics of gears. Those properties can be specifically influenced by the choice of the material and the heat treatment. An essential requirement for choosing a suitable steel and heat treatment is the understanding of the considered stress condition. Hence, in this chapter, main principles of component stresses for gears are described and corresponding criteria for the selection of steel and heat treatment are derived. Chapter 12 is contributed by a team of gear experts of Gear Research Center (FZG) at the Technical University of Munich4, Germany. This chapter describes gear load capacity calculations and gear rating methods based on the current version of the international standard ISO 6336 [21]. The series of ISO 6336 documents contains full standards (ISO), technical specifications (ISO/TS) and technical reports (ISO/TR). All these documents serve the purpose of calculating the load capacity of spur and helical gears. The parts 1, 2, 3, and 6 were first published in May 1996. A first revision of these parts was published in September 2006 and the third and most recent revision was in November 2019. This chapter describes the newly

1 2 3

4

Rochester Institute of Technology (RIT), Rochester, New York, USA. Israel Aerospace Industry, Dallas, Texas, USA/Israel. Chapter 11 is authored by Niklas Blech, Holger Cermak, André Sitzmann, Adrian Sorg, Daniel Fuchs, Thomas Tobie, and Karsten Stahl. Chapter 12 is authored by Daniel Müller, Nadine Sagraloff, Stefan Sendlbeck, Karl Jakob Winkler, Thomas Tobie, and Karsten Stahl.

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Dudley’s Handbook of Practical Gear Design and Manufacture

revised ISO 6336-2 [30] on surface durability against pitting damage as well as ISO 6336-3 [31] on tooth root bending strength against tooth root breakage and focuses on the new aspects of the current publication. Besides the revised standards, there were two recent publications of technical specifications in the ISO 6336 series, which are also presented in this chapter. ISO/TS 6336-4 [28] was published in January 2019 and concerns the calculation of the load capacity against tooth flank fracture and ISO/TS 6336-22 [26] was published in August 2018 and concerns the calculation of the load capacity against micropitting. For these four parts, a description of the relevant failure mode, basic calculations principles, new aspects and updates of the document, a calculation example, a summary and an outlook are presented in the following. Chapter 13 is contributed by a team of gear experts of Gear Research Center (FZG) at the Technical University of Munich5, Germany. Plastic gears have been used for decades in a wide variety of applications such as consumer articles or electromechanical actuators in the automotive sector. Plastic specific material properties such as low density and high damping characteristics as well as the possibility of mass production through injection molding are advantageous and contribute to the increasing application of plastic gears. However, the comparatively large differences in material properties compared to steel result in plastic gears mostly being used in low-power drives. In particular, the high temperature dependency of the material properties and lower strength numbers often represent a challenge for the application of plastic gears. Current research provides new knowledge on the operating behavior and the load carrying capacity of modern thermoplastic materials and contributes to the optimized design of plastic gears. The following chapter presents an overview of the state of the art and application of plastic gears, introduces existing design and calculation methods for plastic gears and summarizes some main results of actual research work performed at FZG institute. Chapter 14 is devoted to consideration of the kinds and causes of gear failures. This includes an analysis of gear-system problems, an analysis of tooth failures and gear-bearing failures, along with the consideration of some causes of gear failure other than excess transmission load. Load rating of gears is discussed in Chapter 15. The discussion includes but is not limited to main nomenclature, coplanar gears, RH—conventional fatigue limit of factor K, RF—Conventional fatigue limit of factor UL, crossed helical gears, hypoid gears, and worm gearing. The section ends with analysis of the state of the standards for load rating of gears. In parallel to gear design, gear manufacturing methods are considered in the book. Chapter 16 is devoted to in detail consideration of various methods for producing the gears. This includes gear-tooth cutting by hobbing, shaping, milling, broaching, and others. Gear grinding methods, such as form grinding, generating grinding and thread grinding are discussed. This sub-section of the chapter is followed by the discussion of gear shaving, rolling, and honing methods. The consideration of the gear manufacturing methods is ended by a discussion of gear casting and forming methods. Design of tools to make gear teeth is disclosed in Chapter 17. Design of shaper cutters, gear hobs, spur-gear milling cutter, gear shaving cutter, punch tools, and sintering tools are considered in this section of the book. In Chapter 18, by Dr. Michael Storchak6, the interaction of the tool and the workpiece in the real cutting process, and therefore the processing results, is significantly affected by the dynamic state of the technological system. To take such an effect on process quality, productivity and tool life into account, the interaction between tool and workpiece should be considered based on the dynamic model of the process. For this, a generalized dynamic interaction model of the technological system links for the gears finishing has been developed. A methodology for determining the dynamic system parameters is proposed and implemented. The objective functions of the dynamic model that provide the optimization choice of design parameters are determined. An analysis of the normal vibrations spectrum and forced oscillations of the technological system has been carried out. Ways to reduce the vibrational activity of the gears finishing process are proposed as a means of increasing productivity and processing accuracy,

5 6

Chapter 13 is authored by C.M. Illenberger, Thomas Tobie, and Karsten Stahl. Institute for Machine Tools, Stuttgart University, Germany.

Introduction

5

tool life. Designs of rolling gear tools for finishing of machined gears are also proposed. These tool designs provide a significant reduction in the vibration activity of the machining process both during free rolling and when the tool is rigidly connected to the gear machined. Chapter 19 on powder metal gears is contributed by Dr. Anders Flodin of Höganäs AB, Sweden. This chapter discusses the materials, the processes and the design of powder metal gears to enable the reader to take full advantage of the possibilities inherent in Powder Metal Gearing and not just make a copy of an existing wrought steel design. There chapter starts with a review of materials and how the materials relate to the thermal processes that follows after compaction, the two are tied together. There are many processes being used today to make PM gears, the reader will learn about the most common heat treatment processes and post processes used typically for PM gears and also what can be expected in terms of performance and when a specific process in favor of another process. The reader will also get advice on post processes such as welding, peening and wire cutting to avoid common mistakes leading to problems. There is also a section on compaction tooling for gears, to enable the reader to understand how helical gears are made in a tool. There is also a discussion around how to treat the material as a continuum when doing stress analysis as well as where to find more information on plasticity models for Powder Metal Gear Several standards are listed at the end of the chapter that are Powder Metal Gear specific and is a good reference source. Chapter 20 on additive manufacturing of gears is contributed by Dr. Anders Flodin of Höganäs AB, Sweden. Additive manufacturing is a relatively new technology in gearing. 3D printed plastic gears is considered an established and developed technology and is in use both commercially and in the maker community. Steel printed gears is less established, there are very few commercial gears made in steel using additive technologies at the time of writing. In this chapter the most common print methods are discussed and explained as well as some difficulties related to printing gears in terms of tolerances, anisotropy and design. Printable steel materials are discussed and exemplified together with post processes to reach the normal tolerance classes when 3D printing steel gears. Chapter 21 is contributed by a team of gear experts of Gear Research Center (FZG) at the Technical University of Munich7, Germany. This chapter covers the noise, vibration and harshness (NVH) behavior of geared transmissions as a major field in gear design and analysis. Gear noise is dependent on transmitted load and running speed of the gears and is characterized by distinct frequencies that may be clearly audible. In most applications gear noise is unwanted and has to be prevented. Major efforts in engineering are concentrated on low noise design on one hand and on gear noise analysis in operation on the other hand. In gear design a low noise generation is valued as a major quality criterion. Some design rules concerning gear main geometry or simple shapes of recommended micro geometry are available for the engineer. Nevertheless, a significant portion of developing silent gears is an iterative trial and error activity. That makes sound understanding of the underlying basics mandatory for today’s gear engineer. The chapter illustrates different mechanical approaches to gear mesh NVH. General basics of gear noise excitation are covered, in particular the influence of main geometry and of micro geometry. Evaluation of different design criteria (TE, etc.) will be explained. Widely used is the calculation of transmission error but also established are more complex methods like tooth force level or application force level. Gear noise analysis is a second relevant aspect. Based on general acoustics an evaluation of airborne sound and noise of transmissions will be described. More detailed methods of analysis techniques of transmissions and gear meshes will be documented which require greater effort. These include measurement of structure borne noise and measurement of transmission error in special test rig environment. Contributed by Dr. Kiril Arnaudov8 and Dr. Dimitar P. KaraivanovChapter 22 deals with the basic information about planetary gear trains (PGTs). Without considering the great variety of different types, the focus is on the most commonly used simple PGT ( AI PGT with one external mesh and one internal mesh), the specific requirements, and the possibilities for its combination in compound multi-carrier PGTs. Using

7 8

Chapter 21 is authored by Joshua Götz, Sebastian Sepp, Michael Otto, and Karsten Stahl. Both (Drs. Kiril Arnaudov and Dimitar P. Karaivanov) are of National Academy of Sciences, Sofia, Bulgaria.

6

Dudley’s Handbook of Practical Gear Design and Manufacture

the torque method, the basics of kinematic analysis, the determination of power flows and efficiency of simple and compound PGTs are given. The specific features of load capacity calculations of AI PGT are mentioned. Particular attention is paid to one of the most important problems of these gear trains—the load distribution between the planets and its equalization. Special attention is paid to the reduced PGTs—Wolfrom and Ravigneaux— compound PGTs with one carrier, and in one of the variants of Wolfrom gear train— without carrier. At the end of the chapter some traps in the design of PGTs are shown. Two appendices are attached at the very end of the book. Appendix A: Complementary Material. Appendix B: Numerical Data Tables All comments and constructive suggestions on content of the book should be sent to the Publisher, CRC Press, Taylor & Francis Group at http://www.crcpress.com. Stephen P. Radzevich Sterling Heights, Michigan, USA January 15, 2021

1

Foundations of Advanced Gear Systems Stephen P. Radzevich EATON Corp. (retired), Southfield, MI, USA

Gears have been extensively used by humans for centuries, and are still vital for modern industry. An example of the application of gears in the design of a transaxle is shown in Figure 1.1. Much work is undertaken to improve the quality of gears and gearing for advanced gear systems. High durability and reliability of gearsets, low vibration generation and noise excitation in gear meshes, and the highest possible power density, are among the main goals facing a gear designer. There are two main purposes for application of gears: 1. Transmitting a rotary motion from a source (driving shaft) to an application (driven shaft) is the first purpose for the application of gears. In this scenario, the magnitude of the output rotation can be either altered or not: The input rotation can be either reduced (as in gear reduction), or in­ creased (as in gear increase), or, ultimately, it can be of the same magnitude. 2. Altering an initial configuration of an output shaft in relation to an input shaft is the second purpose for the application of gears. In this scenario, the output shaft can either intersect the input shaft at an angle, or it can cross the input shaft. The shaft angle in these cases can be either acute, obtuse, or right. When transmitting and transforming a rotary motion by means of gear teeth, it is strongly desirable to design the gearset so as to keep the angular velocity ratio of the gearset of a constant value—no variation of the angular velocity ratio is permissible as this inevitably entails excessive noise excitation and vibration generation, reducing gearset efficiency and durability. To attain the highest possible power density of a gearset is another important task when designing gears. The power density of a gearset is the ratio of the power being transmitted through the gearset to its weight. Occasionally, the term “power-to-weight ratio” is used instead of the term “power density” in a gear pair. In certain applications, use of the ratio of the power being transmitted to a volume of space occupied by the gearset is preferred. This makes sense when gearsets of different design are compared, for example, the difference in weight of an external and an internal gear pair of equal power density is not significant. However, the space occupied by gear pairs of these two kinds is significantly different. With that said, it becomes clear that designing a gear pair is a sophisticated problem that can be suc­ cessfully solved only by extensive use of scientific methods. Common sense and previously accumulated experience are also valuable. A great depth of science is involved when we design and manufacture gearsets with the highest possible power density, the lowest possible noise excitation, and the longest possible durability. The scientific theory of gearing is a separate science that aims to solve problems of this sort. The scientific theory of gearing is a self-consistent system of interconnected knowledge, scientific methods and means, capable of, on the one hand, describing all known facts and phenomena, and on the other hand, predicting yet unknown facts and phenomena covered by the theory; the scientific classi­ fication of the facts and phenomena covered by the theory is an integral part of the scientific theory. The necessity of a scientific foundation for gear design and manufacture was realized by gear practitioners long ago. Gears that operate on parallel axes of rotation of a gear and its mating pinion were 7

8

Dudley’s Handbook of Practical Gear Design and Manufacture

FIGURE 1.1

Gears in the design of a five-speed transaxle.

the first to attract the attention of mathematicians and gear theoreticians. L. Euler proved in the middle of the eighteenth century that an involute tooth profile best fits gears that operate on parallel axes of rotation. This scientific accomplishment is recognized as the origin of the scientific theory of gearing. Three fundamental laws of gearing form a foundation of current scientific theory. The design, produc­ tion, and application of advanced gear systems are based on these three laws of gearing. The three fundamental laws of gearing are as follows: • The Law of Contact: The First Fundamental Law of Gearing • The Conjugate Action Law: The Second Fundamental Law of Gearing • The Law of Equal Base Pitches: The Third Fundamental Law of Gearing This set of three fundamental laws of gearing1 is briefly discussed immediately below. 1

Taking into account that: • •

If the second fundamental law of gearing is fulfilled, the first fundamental law of gearing is always fulfilled (and not vice versa), and If third fundamental law of gearing is fulfilled, the second fundamental law of gearing is always fulfilled (and not vice versa), the set of three fundamental laws of gearing can be substituted with one fundamental law of gearing, namely, with the third fundamental law of gearing (equality of base pitches). In such a scenario, the first and the second fundamental laws of gearing are considered as a precondition to the third fundamental law of gearing.

Foundations of Advanced Gear Systems

1.1

9

THE LAW OF CONTACT: THE FIRST FUNDAMENTAL LAW OF GEARING

When gears operate, a rotary motion and a torque are transmitted from a driving shaft to a driven shaft by means of gear teeth. In gearing, the tooth flank of a gear, G , interacts with a tooth flank, P , of its mating pinion. The tooth flanks, G and P , make contact either at a point, or along a line2. In the first case contact occurs at “contact point, K”. In the second case the tooth flanks, G and P , interact with one another along a “line of contact, LC.” As considered below, the first fundamental law of gearing must be fulfilled either at a single contact point, K, or at every point of the line of contact, LC. Consider a relative motion of the tooth flanks, G and P , of a gear and a mating pinion as illustrated in Figure 1.2. For simplicity, but without loss of generality, normal sections through the contact point of the gear and the mating pinion are depicted there. It is also assumed that the gear tooth flank, G , is mo­ tionless and the pinion tooth flank, P , travels in relation to the gear tooth surface, G . An instant linear velocity vector of this motion is designated as V . Three different scenarios can be distinguished when the pinion tooth flank, P , travels in relation to the gear tooth flank, G . First, an instantaneous motion of point Ka over the pinion tooth flank, P , is specified by an in­ stantaneous linear velocity vector Va (Figure 1.2a). At point, A (in Figure 1.2a, point A coincides with contact point, Ka), the projection, Pr n Va , of the linear velocity vector, Va , onto the unit normal vector, nap, to the traveling surface, P , at contact point, Ka, is pointed to the interior of the motionless gear tooth flank, G , that is, the inequality Pr n Va > 0 is valid. This results in the traveling pinion tooth flank, P , penetrating the motionless gear tooth flank, G , in the differential vicinity of the contact point Ka. A relative motion of this kind is not permissible for the interacting tooth flanks, G and P , of a gear and a pinion. Second, an instantaneous motion of point, Kb, over the pinion tooth flank, P , is specified by an instantaneous linear velocity vector, Vb (Figure 1.2b). At point, Kb (in Figure 1.2b, point B coincides with contact point, Kb), the projection, Pr n Vb , of the linear velocity vector, Vb , onto the unit normal vector, nap , to the traveling tooth flank, P , at contact point, Kb, is equal to zero (that is, the equality Pr n Vb = 0 is valid). This results in the traveling pinion tooth flank, P , not penetrating the motionless gear tooth flank, G , in the differential vicinity of the contact point, Kb. Instead, the pinion tooth flank, P , rolls and slides in relation to the gear tooth flank, G . In a particular case, either the rolling com­ ponent or the sliding component of the resultant relative motion of this kind can be equal to zero. Relative motion of this particular kind is permissible for the interacting tooth flanks, G and P . Transmitting a motion from a driving shaft to a driven shaft is possible if and only if a relative motion of this particular kind is occurred. Third, an instantaneous motion of point, Kc, over the pinion tooth flank, P , is specified by an in­ stantaneous linear velocity vector, Vc (Figure 1.2c). At point, Kc (in Figure 1.2c, point C coincides with contact point, Kc), the projection Pr n Vc of the linear velocity vector, Vc , onto the unit normal vector, ncp , to the traveling pinion tooth flank, P , at contact point Kc is pointed outward to the motionless gear tooth flank, G (that is, the inequality Pr n Vc < 0 is valid). This results in the traveling pinion tooth flank, P , departing from the motionless gear tooth flank, G , in the differential vicinity of point, Kc. A relative motion of this kind is not permissible for the interacting tooth flanks, G and P , of a gear and a pinion. The analysis of the depicted in Figure 1.2 schematics illustrate the necessity of a proper configuration of the linear velocity vector, V , of the resultant relative motion of the tooth flanks, G and P , to the common tangent plane to the tooth flanks, G and P , at every point of their contact. This requirement is referred to as the “first fundamental law of gearing”:

2

“Surface-to-surface” contact between a gear and its mating pinion tooth flanks is not common in gearing. It is likely gear coupling is the only example of “surface-to-surface” contact between the tooth flanks. However, gear coupling can be viewed as a degenerate case of gearing that features no plane of action, no pressure angle, and so forth. The interested reader may wish to go to the book by Prof. Stephen P. Radzevich [1], as well as to other advance sources for details on gear couplings.

P

VaΣ

Prn VaΣ > 0

nap

Ka

nap .VaΣ > 0

G

VbΣ Prn VbΣ = 0

(b)

nbp P

Kb

nbp .VbΣ = 0

G

(c)

P

VcΣ

ncp

Kc

Prn VcΣ < 0

ncp .VcΣ < 0

G

FIGURE 1.2 Possible configurations of the linear velocity vector of resultant relative motion, V , of two interacting tooth flanks, G and P, to the common tangent: (a) penetration of the tooth flanks, G and P, (b) properly contacted tooth flanks, G and P, (c) separation of the tooth flanks, G and P.

(a)

10 Dudley’s Handbook of Practical Gear Design and Manufacture

Foundations of Advanced Gear Systems

11

The first fundamental law of gearing3: “At every point of contact of tooth flanks of a gear and its mating pinion the vector of their instantaneous relative motion has to be pointed perpendicular to the common perpendicular at every instant of time”. It is common practice to express the law of contact of two interacting tooth flanks of the mating gear teeth in the form of dot product: ng V = 0

(1.1)

where ng is the unit vector of common perpendicular through contact point of the tooth flanks, G and P , of a gear and its mating pinion, and V is the linear velocity vector the resultant instantaneous relative motion of the tooth flanks, G and P , at contact point K. Analysis of Eq. (1.1) reveals that a component of the linear velocity vector, V , along the common perpendicular, ng , is equal to zero. Otherwise, either separation, or interference of the tooth flanks, G and P , in the gear pair is observed. Neither separation, nor interference of the tooth flanks, G and P , of a gear and a mating pinion is permissible. Therefore, the linear velocity vector, V , is either located in a common tangent plane, or it is of a zero value. Equation (1.1) was proposed by Prof. V.A. Shishkov by 19484. Because of this, the equation of contact in the form of dot product [see Eq. (1.1)] is commonly referred to as the “Shishkov equation of contact,ng V = 0 ”5. The equation of contact in the form ng V = 0 is practical in cases when the interacting surfaces feature simple geometry, and when the resultant relative motion is also simple. The first makes it possible to determine the unit normal vector, ng , without the derivation of the expressions for the de­ rivatives of the equations of the contacting surface with respect to the surface parameters. The second allows the determination of the linear velocity vector, V , without derivation of the equation of the traveling surface with respect to the parameter of motion. Use of the “Shishkov equation of contact” in the form ng V = 0 simplifies the solution to gear problems in this particular case. In cases when de­ rivation of the equations of the derivatives for the purposes of determination of the vectors, ng and V , cannot be avoided, use of the “Shishkov equation of contact” in the form ng V = 0 is less beneficial. The law of contact between a gear tooth flank, G , and a mating pinion tooth flank, P , is violated in misaligned parallel-axes gear pairs, as schematically illustrated in Figure 1.3. In this particular case, strongly unfavorable edge contact between the teeth flanks, G and P , is observed. The discussion can be summarized as follows. Permissible instantaneous relative motions of tooth flanks, G and P , in a gear pair are illustrated in Figure 1.4. No relative motion of the tooth flanks, G and P , along the common perpendicular, ng , is permissible; and relative motion is allowed in any direction within the common tangent plane through contact point, K. It should be pointed out here that a swivel relative motion, ± n , of the tooth flanks, G and P , around the axis along the common perpendicular, ng , also meets the requirement specified by Eq. (1.1). Not all kinds of the swivel motion of the tooth flanks in Figure 1.4 are permissible. For example, no swivel relative motion is permissible about an axis that either intersects, or crosses a straight line along the common perpendicular, ng, as the condition of contact, ng V = 0, in these cases is violated. The swivel motion, ± n , of the tooth flanks is not necessary to

3 4

5

Occasionally, the first fundamental law of gearing is referred to as the “condition of contact of interacting tooth flanks”. For details, see two publications by Prof. V.A. Shishkov: “Elements of Kinematics of Generating and Conjugating in Gearing,” in: Theory and Computation of Gears, Vol. 6, Leningrad, LONITOMASH, 1948, page 123; and Generation of Surfaces in Continuous-Indexing Methods of Surface Machining, Moscow, Mashgiz, 1951, 152 pages. For details on why Prof. V.A. Shishkov is credited with the discovery of the equation of contact in the form ng V = 0 , the interested reader may wish to go to the 2010 paper by Prof. Stephen P. Radzevich, “Concisely on Kinematic Method and about the History of the Equation of Contact in the Form ng V = 0 ”, Theory of Mechanisms and Machines, No. 1 (15), 8, 2010, pp. 42–51. https://tmm.spbstu.ru

12

Dudley’s Handbook of Practical Gear Design and Manufacture

Gear tooth

P

θ

G FIGURE 1.3

Example of violation of the first funda­ mental law of gearing in a misaligned parallel-axes gear pair.

Pinion tooth

ng

VΣ

P

K G

±un

FIGURE 1.4 Permissible instantaneous relative motions in geometrically accurate gearing (in “perfect gearing”).

transmit a rotary motion from a driving shaft to a driven shaft. However, motion of this nature can be observed in crossed-axes gearing (in C a gearing, for simplicity).

1.2

THE CONJUGATE ACTION LAW: THE SECOND FUNDAMENTAL LAW OF GEARING

In order to transmit smoothly a uniform rotation from a driving shaft to a driven shaft by means of gear teeth, at every instant of rotation of the gears, the law of conjugacy of the interacting tooth flanks, G and P , has to be fulfilled6. As the conjugate action law with respect to parallel-axes gearing is more or less known to gear practitioners, it is convenient to begin this discussion from the case of fulfillment/violation of the conjugate action law in parallel-axes gears. Later on, this discussion can be evolved to more general cases of intersected-axes gearing, as well as crossed-axes gearing. These two latter cases are almost unknown to the majority of gear practitioners.

1.2.1 CONJUGATE ACTION LAW

IN

PARALLEL-AXES GEARING

In today’s practice, the analysis of the gear kinematics and geometry is limited to the fulfillment only of the “condition of contact” (or, in other words, of the “enveloping condition”) of the interacting tooth flanks of a gear, G , and its mating pinion, P . It is loosely assumed that once the condition of contact, 6

In parallel-axes gearing, the condition of conjugacy (the conjugate action law) was known to L. Euler. In the time of Euler and later, only verbal formulation of the conjugate action law was known; no analytical expression for this important law of gearing has been derived until the beginning of third millennium.

Foundations of Advanced Gear Systems

13

ng V = 0, between the interacting tooth flanks of a gear and a mating pinion is fulfilled, then the tooth flanks, G and P , are conjugate. The later statement is incorrect: Non-conjugate tooth flanks of a gear and a mating pinion also commonly fulfill the condition of contact, ng V = 0 . To be conjugate, common perpendiculars at every point of the line of contact, LC , between the tooth flanks, G and P , have to pass through the axis of instant rotation of the interacting surfaces at every instant of time, that is, for any and all possible configurations of the surfaces, G and P , relative to each other. The conjugate action law in parallel-axes gearing was discovered by L. Euler, and (later and in­ dependently, by Felix Savary) in the mid-18th century7. In Europe, the conjugate action law for parallelaxes gearing is referred to as the “Willis theorem” due to the input of Robert Willis of the United Kingdom, who introduced this theorem to the public in 18418. The conjugate action law in parallel-axes gearing (according to Willis): The angular velocities of the two pieces are to each other inversely as the segments into which the “line of action” divides the line of centers, or inversely as the perpendiculars from centers of motion upon the line of action. According to the “conjugate action law,” the center-distance, C , is divided by the pitch point, P , onto two segments, Og P = rg and Op P = rp , so that a proportion: Og P Op P

=

rg rp

=

p g

=u

(1.2)

is valid. The conjugate action law is illustrated in Figure 1.5. At the beginning, it is necessary to stress here on the difference between the line of action, LA, and the path of contact, Pc . In geometrically accurate parallel-axes (involute) gearing, the path of contact is a straight-line segment. This straight-line segment is aligned with the line of action, LA. Despite the line of action, LA, and the path of contact, Pc , aligning with each other, it is important to realize that they are two different lines. These two lines align to one another in parallel-axes involute gearing and do not align to one another in intersected-axes gearing, as well as in crossed-axes gearing. The second fundamental law of gearing (in parallel-axes gearing) is fulfilled9 in 1956 Novikov (conformal) gearing (see Pat. USSR No. 109,113), and is violated10 in 1923 Wildhaber helical gearing (US Pat. No. 1,601,750). In today’s practice, fulfillment of the condition of conjugacy of the interacting tooth flanks of a gear and its mating pinion is commonly limited to the analysis of the simplest case of perfect (with zero axes misalignment) parallel-axes gears, that is, to the case of perfect involute gearing. The condition of conjugacy of the tooth flanks for gears with another tooth flank geometry is not discussed at all11. Examples can be readily found for gears that feature a cycloidal tooth profile, a circular-arc tooth profile, 7

8

9

10 11

A lot of efforts on investigation of the conjugate action law in gearing were undertaken by Charles Camus. Unfortunately, Camus failed to discover/finalize the conjugate action law; however, his input to the subject is important. With that said, it is clear now why the conjugate action law for parallel-axes gearing is referred to as “Camus-Euler-Savary main theorem of gearing” (or, just “ng V = 0 main theorem of gearing”, for simplicity). For details, see: Willis, R., Principles of Mechanisms, Designed for the Use of Students in the Universities and for Engineering Students Generally, London, John W. Parker, West Stand, Cambridge: J. & J.J. Deighton, 1841, 446 pages. Pat. USSR, No. 109,113, Gear Pairs and Cam Mechanisms Having Point System of Meshing. /M.L. Novikov, National Classification 47h, 6; Filed: April 19, 1956, published in Bull. of Inventions No.10, 1957. Pat. USA, No. 1,601,750, Helical Gearing, /E. Wildhaber, Patented: October 5, 1926, Filed: November 2, 1923. Wrong practice is commonly adopted when analyzing whether the law of conjugacy of the tooth flanks, ng V = 0 and ng V = 0 , is fulfilled, or it is not. “Shishkov equation of contact, ng V = 0 ”, is loosely used for this purpose (here, in the equation, ng V = 0 is the unit normal vector to the tooth flanks ng V = 0 and ng V = 0 at contact point, ng V = 0 , and ng V = 0 is the velocity vector of the resultant relative motion of the surfaces ng V = 0 and ng V = 0 at the contact point, ng V = 0 ). The law of contact of the tooth flanks ng V = 0 and ng V = 0 , and not the law of conjugacy of the interacting tooth flanks of a gear and its mating pinion is analytically described by “Shishkov equation of contact”. Unfortunately, often this difference is not recognized at all.

14

Dudley’s Handbook of Practical Gear Design and Manufacture

Line of action, LA

φt

Pinion

K Og

Op

Gear

Path of contact, Pc

FIGURE 1.5 Conjugate action law in geometrically accurate parallel-axes involute gearing (the background in this illustration is adapted from: https://www.tec-science.com/mechanical-power-transmission/involute-gear/meshing-lineaction-contact-pitch-circle-law/).

as well as tooth profiles of other geometries different from the involute of a circle. Moreover, no analysis of the condition of conjugacy of the interacting tooth flanks of a gear and its mating pinion for gears that operate on intersected axes of rotation, as well as for gears that operate on crossed axes of rotation, has ever been performed. This is all that is known to this end on the conjugate action law. There is no evidence of understanding of the conjugate action law (a) in gearing with modified tooth flanks, (b) in intersected-axes gearing, and (c) in crossed-axes gearing. It is important to bridge this gap in modern gear theory and practice, as the conjugate action of a gear and a mating pinion tooth flanks is vital in both gear design and gear production.

1.2.2 CONJUGATE ACTION LAW

IN INTERSECTED-AXES

GEARING,

AND IN

CROSSED-AXES GEARING

The second fundamental law of gearing is often referred to as the “conjugate action law.” When the first fundament al law of gearing is fulfilled, the conjugate action law can either be fulfilled, or it can be violated. In Figure 1.6, an example of interaction of local patches of two interacting tooth flanks, G and P , is schematically illustrated. The tooth flanks, G and P , make contact at point, K . The radii of curvature of the interacting tooth flanks at contact point, K , equal to Rg and Rp , correspondingly (see Figure 1.6a). The centers of curvature of the tooth profiles, G and P , are denoted by Og and Op , correspondingly. In the instantaneous motion of the pinion, P , in relation to the gear, G , the pinion performs an instantaneous rotation, p / g, about the point Og . The radius of curvature of the generated actual gear tooth flank, G act , equals to Rp / g Rg . In this scenario, the second fundamental law of gearing is fulfilled, and the actual tooth flank, Gact , is identical to the desirable gear tooth flank, G , as shown in Figure 1.6a. Note: Here, in Figure 1.6, a rotation, n , of the pinion, P , in relation to the gear, G , about the contact perpendicular, ng , is not prohibited by the second fundamental law of gearing. If the instantaneous rotation is performed either about the center Op / g (when Rp / g < Rg , see Figure 1.6b), or about the center Op / g (when Rp / g > Rg , see Figure 1.6c), the second fundamental law of gear is violated, and the actual tooth flank, Gact , differs from the desirable gear tooth flank, G .

ωn

ng

G ≡ Gact

K

Og

Op

Gear

VΣ

Pinion

ω p/g

P

Rp

(b)

ng ⋅ V Σ = 0

Rg

Gact

ωn

ng

K

Og

Gear

VΣ

Pinion

ωp / g

Rp / g < R g

Op / g

Op

G P

Rp

(c)

n g ⋅ VΣ = 0

Rg

Gact

ωn

ng

K

Gear

Op / g

Og

VΣ

Pinion

ωp / g

Rp / g > R g

Op

G

FIGURE 1.6 Examples of (a) fulfillment, and violation; (b) interference; and (c) separation, of a gear, G, and its mating pinion, P, tooth flanks: the “Shishkov equation of contact, ng V = 0 ” is fulfilled in all three cases (a) through (c).

ng ⋅ VΣ = 0

Rg = Rp/g

P

Rp

(a)

Foundations of Advanced Gear Systems 15

16

Dudley’s Handbook of Practical Gear Design and Manufacture

Reminder: When the law of contact is fulfilled, the conjugate action law can be either fulfilled, or it can be violated, as illustrated in Figure 1.6. The first fundamental law of gearing (that is analytically described by the “Shishkov equation of contact, ng V = 0 ”) is fulfilled in all three cases shown in Figure 1.6, while the second fundamental law of gearing (the conjugate action law) is fulfilled only in the first case illustrated in Figure 1.6a. The schematic shown in Figure 1.6 is helpful for understanding the difference between the first and the second fundamental laws of gearing, and prevents from making wrong conclusions in this regard. Important: When the second fundamental law of gearing is fulfilled, the first fundamental law of gearing is always fulfilled, and not vice versa. Prior to discussing how the conjugate action law works in intersected-axes gearing and in crossedaxes gearing, it is convenient to consider more in detail the specific features of the conjugate action law in parallel-axes gearing. Referring to Figure 1.7, consider a schematic of meshing in geometrically accurate parallel-axes involute gearing. With gearing of this particular design, the tooth flanks, G and P , are shaped so as to fulfill both the law of contact and the law of conjugacy. Therefore, at every instant of meshing of the gears, the line of action, LA, intersects the center-line, ℄, at the stationary pitch point, P. Due to this, the law of contact (“Shishkov equation of contact”): ng V = 0

(1.3)

ωp rg = Og P r p = Op P

db.p

Op

u=

VKla

φt sl

P

VKg VKp

VKsl

Pi

Ng

ng VKsl

ng ⋅VK = 0

φt

V pa

K

P

Np

φ ti.

= Const

VKla

Pc ≡ LA

Pl

rp

rg + rp = C

pln ×V la ⋅n = 0 K g

LA

rg

LAi

C L

db.g

Pr C L

Pitch point, P , travel distance

rg.i = Og Pi rp.i = Op Pi

uu =

rg.i rp.i

Og

ωg

= var

FIGURE 1.7 An example of fulfillment of the condition of contact, ng VslK = 0 , and the law of conjugacy, p ln × Vla K ng = 0 , of the tooth flanks, G and P, in parallel-axes involute gear pair (here p ln is the unit vector along the axis of instantaneous rotation, Pln ; the axis of instantaneous rotation, P ln , is a straight line though the pitch point, P, perpendicular to the plane of drawing).

Foundations of Advanced Gear Systems

17

and the law of conjugacy (2017): p ln × Vm ng = 0

(1.4)

of the tooth flanks, G and P , are fulfilled every time. Here is designated: p ln – is the unit vector pointed along the axis, Pln , of instantaneous rotation of a gear and a mating pinion ng – is the unit vector of a perpendicular to the gear tooth flank, G (to the pinion tooth flank, P ) Vm – is the vector of linear velocity of a certain gear/pinion tooth flank point, m , together with the plane of action, PA (that is, Vm = Vpa )

The “equation of conjugacy” [see (1.4)] is derived based on the following analysis. The plane of action, PA (as well as the unit normal vector, npa , to the plane of action, PA: npa = p ln × Vm ), can be specified in terms of two vectors: p ln and Vm . The unit normal vectors, ng and npa , have to be pointed perpendicular to one another. Ther efore, the equality ng npa = 0 is valid. This expression can be represented in the form of (1.4). In the event that either a gear tooth flank, G , or a mating pinion tooth flank, P , or both are designed improperly, an instant pitch point migrates along the center-line, ℄, from a left extreme position, Pl , to a right extreme position, Pr . An arbitrary position of the pitch point, P , in such a reciprocation is labeled as Pi . Commonly, the current value of the transverse pressure angle, t . i , differs from it nominal value, t (that is, an inequality t . i t is observed). Ultimately, the reciprocation of the pitch point, P , and the variation of the transverse pressure angle, t . i , cause an excessive unfavorable noise excitation and vibration generation in gearing. Once the importance of fulfillment of the law of conjugacy of the interacting tooth flanks in parallelaxes gearing is realized, then one can proceed with an analysis of the fulfillment of the law of conjugacy of the interacting tooth flanks, G and P , in intersected-axes gearing, and in crossed-axes gearing. The provided verbal description of the conjugate action law of the interacting tooth flanks, G and P , of a gear and a mating pinion can be complemented with an analytical description of this law of gearing. To meet the requirements imposed by the equation of conjugacy [see (1.4)], namely, in order to have a gear tooth flank, G , and a mating pinion tooth flank, P , conjugate, the unit normal vector, ng , has to be entirely located within the plane of action, PA. A line of action of the unit normal vector, ng , intersects the axis of instantaneous rotation, Pln , if the condition of conjugacy, p ln × Vm ng = 0 , is fulfilled. An example of fulfillment of the condition of conjugacy of gear, G , and mating pinion, P , tooth flanks in crossed-axes gearing is illustrated in Figure 1.8 (in a case of intersected-axes gearing, the lengths of the distances Ag Apa = Ap Apa = 0, and all three apexes, Ag , Ap, and Apa , are snapped together: Ag Ap Apa ). A desirable line of contact, LCdes , between the tooth flanks, G and P , is initially specified in a reference system, Xpa Ypa Z pa , associated with the plane of action, PA. The desirable line of contact, LCdes, is entirely within the plane of action, PA, and travels with the PA as the gears rotate. The unit normal vector, ng , to the gear tooth flank, G , always intersects the axis of instant rotation along the unit vector, p ln (the pitch-line, Pln ). This immediately results in the condition of conjugacy of the interacting tooth flanks, G and P , in intersected-axes gearing, and in crossed-axes gearing is fulfilled. Generated this way, the tooth flank, G , of a geometrically accurate straight bevel gear for an intersected-axes gearset is shown in Figure 1.9. Depending on the gear tooth count and the actual values of the rest of the design parameters, different portions of the generated gear tooth surface are used to shape the bevel gear teeth. The concept of the geometrically accurate gears for intersected-axes gearing

18

Dudley’s Handbook of Practical Gear Design and Manufacture PA

C

n pl

ϕ pa

Ap

pln

A pa

Pln

LAinst

rl. pa

Y pa

Y pa

P

Pln

rm

ng

Vm

P

m

ng

ω inst

G

m

Ag

X pa

Z pa

LC des

r o. pa

ω pa FIGURE 1.8 Fulfillment of the law of conjugacy of a gear, G, and a mating pinion, P, tooth flanks in crossed-axes gearing (in a case of intersected-axes gearing, Ag Apa = Ap Apa = 0 , and all three apexes, Ag , Ap , and Apa , are snapped together: Ag Ap Apa ).

Spherical involute

Gear tooth flank,G

Opa

Ag Apa Plane of action, PA

Desirable line of contact, LCdes Og

Base cone: gear

FIGURE 1.9

Tooth flank, G, of a geometrically accurate straight bevel gear.

Foundations of Advanced Gear Systems

19

can be traced back to 1887 when American George Grant filed the patent named as “Machine for Planing Gear Teeth.”12 Constructed for an intersected-axes gearset, a schematic similar to that shown in Figure 1.9 also can be constructed for a crossed-axes gearset with any desirable geometry of the line of contact, LCdes . The proposed (~2008) by Dr. S. Radzevich geometrically accurate crossed-axes gearing with true-line contact between the tooth flanks, G and P , is referred to as “R gearing.” It is important to stress here that “R gearing” is the only possible kind of geometrically accurate crossed-axes gearing with true line contact between the inter­ acting tooth flanks G and P (similar to involute gearing is the only possible kind of geometrically accurate parallel-axes gearing with true line contact between the interacting tooth flanks G and P ). A consideration similar to that above is valid with respect to the pinion tooth flank, P (not illustrated in Figure 1.8). When the gears rotate, the desirable line of contact, LCdes , generates the gear and the pinion tooth flanks. The generation of the gear tooth flank, G , is considered in a reference system, Xg Yg Zg , associated with the gear, and the generation of the pinion tooth flank, P , is considered in a reference system, Xp Yp Z p , associated with the pinion. The second fundamental law of gearing (the conjugate action law) also is helpful when approximate gearing is analyzed. A point of a gear tooth flank, G , can be a contact point of the tooth flanks, G and P , when the condition of conjugacy is met. For a corresponding angular configuration of a gear and a mating pinion point of a pinion tooth flank, P , that is anticipated to be in contact with the gear tooth flank, G , can actually be a contact point only when the condition of conjugacy is met. A possibility of contact of the teeth flanks, G and P , in approximate gearing can be verified by means of comparison of coordinates of the anticipated “contact” point, and the actual contact point on the gear tooth flank, G , and that on the pinion tooth flank, P . If the coordinates of the points are identical to one another, then the anticipated contact point really is a contact point (the unit normal vectors, npl and nm , must be aligned with each other). Otherwise, contact of the teeth flanks, G and P , in these points is not possible. Precision gears for intersected-axes gear pairs and crossed-axes gear pairs have to be designed so as to fulfill the conjugate action law as discussed in [1]—this is a must. Moreover, gears for intersected-axes gear pairs and for crossed-axes gear pairs have to be finish-cut or ground/honed so as to fulfill the conjugate action law in the gear generating process [2]. Currently, this is the only way to eliminate the gear lapping process, which is required to finish poorly designed and poorly finish-cut gears for intersected-axes, as well as for crossed gear pairs. In today’s industry, none of the gear companies is concerned with the fulfillment of the conjugate action law when designing and finishing gears for intersected-axes gear pairs and for crossed-axes gear pairs. It is proven [1] that the contact pattern of a favorable geometry and location can be ensured only in conjugate intersected-axes and crossed-axes gear pairs.13 An in-depth understanding of conjugate action law in cases of Ia gearing, and Ca gearing is vital for the development of software for gear-measuring machines (GMMs). Accurate and reliable gear in­ spection is not possible in cases when the conjugate action law (in cases of Ia gearing, and Ca gearing) is ignored.

1.2.3 EXAMPLES

OF

VIOLATION

OF THE

CONJUGATE ACTION LAW

There are not many examples of fulfillment of the second fundamental law of gearing. The 1887 G. Grant method of cutting bevel gears for intersected-axes gearsets (U.S. Pat. No. 407.437) is among only a few available in the public domain. Many more illustrations show the second fundamental law of gearing being actually violated in both gear design and gear machining.

12

13

U.S. Pat. No. 407.437. Machine for Planing Gear Teeth. /G.B. Grant, Filed: January 14, 1887 (serial No. 224,382), Patent issued: July 23, 1889. For details, see also: Radzevich, S.P., et al, “Preliminary Results of Testing of Low-Tooth-Count Bevel Gears of a Novel Design”, Gear Solutions, Part 1, October 2014, pp. 25–26; Part 2, December 2014, pp. 20–21; Part 3, January 2015, pp. 20–23.

20

Dudley’s Handbook of Practical Gear Design and Manufacture

FIGURE 1.10 Wrong practice that is extensively used in modern industry for finishing accurate: (a) straight bevel gears, and (b) spiral bevel gears (both the images are known from many sources, and are available in the public domain). After: Radzevich, S.P., “Knowledge (of Gear Theory) Is Power in the Design, Production, and Application of Gears”, Gear Solutions magazine, August 2020, pages 38–44.

A few examples of violation of the second fundamental law of gearing are briefly outlined im­ mediately below. For finishing gears for geometrically accurate intersected-axes and crossed-axes gearing, appropriate methods of finishing gear tooth flanks are outlined in [2]. Unfortunately, in today’s industry, when finishing gears for Ia gearing and Ca gearing, the applied methods and means are based on application of the crown-generating rack with straight-sided teeth. Two conventional schematics of the crowngenerating rack are shown in Figure 1.10. Straight bevel gears are commonly finish-cut using methods and means based on the application of the crown generating rack shown in Figure 1.10a, which features a straight-sided tooth profile (trapezoidal tooth profile). Spiral bevel gears also are finish-cut/ground using methods and means based on the application of the crown generating rack shown in Figure 1.10b, which that also features a straight-sided tooth profile. No accurate gears for intersected-axes gearing, as well as for crossed-axes gearing, can be generated by straight-sided crown racks (see Figure 1.10), as none of these racks is capable of generating conjugate tooth flanks, G and P , of two mating gears. Straight bevel gears: The crown generating rack (see Figure 1.10a) is extensively used for generation of tooth flanks of straight bevel gears. In the gear-planing operation, the tooth flank of the crowngenerating rack is reproduced by means of the straight cutting edges of the two cutters that reciprocate toward the axis of rotation of the gear to be machined. A close-up of the gear planing operation is shown in Figure 1.11a. Milling straight bevel gears by means of the dual interlocking circular cutters is another method of finish-cut gears for intersected-axes gearsets. In the gear-milling operation, the tooth flank of the crown generating rack is reproduced by means of straight-cutting edges of two mill cutters that rotate about their axis of rotation. Cutting edges of the mill cutters are perpendicular to the axes of their rotation. In some applications, the straight cutting edges of the mill cutters slightly deviate from the orthogonal position. In this case, the gear-tooth flanks are generated by means of an internal cone reproduced by the cutting edges. A closeup of the gear milling operation is shown in Figure 1.11b. Only approximate gears can be finish-cut in both cases shown in Figure 1.10, as well as in similar cases where a crown generating with straight-sided teeth are used.

(b)

FIGURE 1.11 Generation of tooth flanks of approximate straight bevel gears in (a) the gear planing operation, and (b) the gear milling operation.

(a)

Foundations of Advanced Gear Systems 21

22

Dudley’s Handbook of Practical Gear Design and Manufacture

FIGURE 1.12

Generation of tooth flanks of approximate spiral bevel gears in the gear face-milling operation.

Spiral bevel gears: The straight-sided crown-generating rack (see Figure 1.10b) is extensively used for the generation of tooth flanks of spiral bevel gears. In the gear-machining operation, the tooth flank of the crowngenerating rack is reproduced by means of the straight cutting edges of multiple cutters that rotate about the axis of rotation of the cutter-head. A close up of the face-milling operation is depicted in Figure 1.12. Only approximate gears can be face-milled in the face-milling operation shown in Figure 1.12, where a straight-sided crown generating is used. Spiral bevel gear grinding operation: The straight-sided crown generating rack (see Figure 1.10b) is used for generating tooth flanks of spiral bevel gears. In the gear-grinding operation, the tooth flank of the crown-generating rack is reproduced by means of the abrasive grains of the abrasive wheel that rotates about the axis of rotation of the grinding wheel. A close up of the gear grinding operation is shown in Figure 1.13. Only approximate gears can be finished up in the gear grinding operation shown in Figure 1.13, where a crown generating with straight-sided teeth is used. Machining/finishing of spur/helical gears: When skiving spur gears (see Figure 1.14a), honing internal helical gears (see Figure 1.14b), as well as in numerous other gear-finishing operations, the conjugate action law is violated, and, thus, no geometrically accurate gears can be produced. An in-depth analysis of gearsets of various gear systems reveals that today, chiefly approximate gears are designed and manufactured. Double-enveloping gearing (cone drive) is one of the examples in this regard. Known for more than a century, the so-called “octoid gears” is another example of approximate gearing in which the conjugate action law is violated.

1.3

THE LAW OF EQUAL BASE PITCHES: THE THIRD FUNDAMENTAL LAW OF GEARING

The last (but not the least) is related to base pitches14 of a gear and its mating pinion. As adopted in common practice, equality of base pitches of a gear and its mating pinion is considered only in a simplest case of geometrically accurate (with zero axes misalignment) parallel-axes gearsets, 14

Reminder: base pitch can be specified only in geometrically accurate gears; no base pitch can be specified in approximate gears, as well as in non-involute gears of all kinds.

Foundations of Advanced Gear Systems

FIGURE 1.13

23

Generation of tooth flanks of approximate spiral bevel gears in the gear grinding operation.

that is, in the case of perfect involute gearing. The necessity of equality of base pitches of the interacting tooth flanks G and P has received no attention with respect to gears that operate on intersected axes of rotation, as well as to gears that operate on crossed axes of rotation. Gear teeth can be viewed as series of cam surfaces that act on similar surfaces of the mating gear to impart a driving motion. An additional requirement caused by multiple interacting tooth surfaces in gears has to be required to be fulfilled in this regard. Equality of base pitches of a gear and a mating pinion is required in order to fulfill the “third fun­ damental law of gearing.” At every instant of rotation of the gears, the gear angular base pitch, b . g , has to equal the operating base pitch of the gear pair, b . op (this means that an equality, b . g = b . op , has to be valid), and the pinion angular base pitch, b . p , has to be equal to the operating base pitch of the gear pair, b . op (this means that an equality, b . p = b . op , has to be valid). As the concept of equal base pitches of a gear and a mating pinion has been developed in a more general sense only in the recent years, here the discussion begins with geometrically accurate parallel-axes gearing, and a more general case of this concept is considered below. In Figure 1.15, local patches of the adjacent gear tooth flanks, G , and the mating pinion tooth flanks, P , are shown. The gear and the pinion tooth flanks in whole are not defined yet. Therefore, only small portions of the tooth flanks, G , are depicted in Figure 1.15a. All these portions of the tooth flanks of a gear are located in the differential vicinity of points of intersection, Kg , of the tooth flanks, G , by the line of action, LA. Each tooth flank, G , is perpendicular to the line of action, LA, at points Kg . All the points Kg are evenly spaced along the line of action. The distance between each pair of adjacent points Kg is said to be the gear base pitch, pb . g. A similar analysis can be performed with respect to the pinion tooth flank, P , shown in Figure 1.15b. All these portions of the tooth flanks of the pinion are located in the differential vicinity of points of intersection, Kp , of the tooth flanks, P , by the line of action, LA Pc . Each tooth flank P is perpendicular

(b)

FIGURE 1.14 Generation of tooth flanks of approximate spur and helical gears in (a) the gear skiving operation, and (b) the gear honing operation.

(a)

24 Dudley’s Handbook of Practical Gear Design and Manufacture

Foundations of Advanced Gear Systems

25

(a)

G G G

Kg

G

Kg

G

Kg Kg Kg

Pc , LA

pb.g pb.g pb.g pb.g

(b)

pb.p pb.p pb.p pb.p Pc , LA Kp Kp Kp

P

Kp

P

Kp

P P P

(c)

pb.p pb.p pb.p pb.p K K K K K

Pc , LA

pb.g pb.g pb.g pb.g

FIGURE 1.15 On the concept of equal base pitches of a gear and a mating pinion: (a) base pitch in a gear, pb . g , (b) base pitch in a pinion, pb . p , and (c) equal base pitches in a gear pair, pb . g pb . p .

26

Dudley’s Handbook of Practical Gear Design and Manufacture

to the line of action, LA, at points Kp . All the points Kp are evenly spaced along the line of action. The distance between each pair of neighboring points Kp is said to be the pinion base pitch, pb . p. When the gear base pitch, pb . g, and the mating pinion base pitch, pb . p, are equal to one another (that is, when the identity pb . g pb . p is fulfilled), then the gear and the pinion can be engaged in mesh as illustrated in Figure 1.15c. Each gear point, Kg , coincides with corresponding pinion point, Kp . Because of this, the gear and the pinion points, Kg and Kp , further are designated as contact point, K . As it follows from the analysis in Figure 1.15c, it is not a must to keep all the base pitches, pb . g, of a gear equal to one another, as well as to keep all the base pitches of a mating pinion, pb . p , also equal to one another. It is critical to keep equality of a gear base pitch to a corresponding pinion base pitch for each pair of teeth engaged in mesh. Physically this is possible, but it limits the gear ratio of a gear pair to an integer number, namely, to 1, 2, 3, and so forth. Such a design of gearing is impractical and is not considered in this book. When base pitches of a gear and a mating pinion are not equal to one another ( pb . g pb . p ), for example, the gear base pitch, pb . g, is smaller compared to the mating pinion base pitch, pb . p , and an inequality pb . g < pb . p is valid (as shown in Figure 1.16a), only one pair of teeth is engaged in mesh. A gap between the remaining pairs of teeth of the gear, G , and the pinion, P , is observed. No gaps of this sort are permissible in geometrically accurate (in perfect) parallel-axes gearing. In another example illustrated in Figure 1.16b, the gear base pitch, pb . g , is greater compared to the mating pinion base pitch, pb . p, and the inequality pb . g > pb . p is valid (as shown in Figure 1.16b). Again, only one pair of teeth is engaged in mesh in this scenario. A gap between the remaining pairs of teeth of the gear, G , and the pinion, P , is observed. No gaps of this sort are permissible in perfect parallel-axes gearing. In this second example (see Figure 1.16b), the distribution of the gaps is inverse to that shown in Figure 1.16a. This is because the gear and the mating pinion are rigid bodies that physically cannot interfere into one another. With that said, the “third fundamental law of gearing” can be formulated in the following manner15: The third fundamental law of (parallel-axes) gearing: “In parallel-axes gearing, in order to transmit a uniform rotary motion from a driving shaft to a driven shaft by means of gear teeth, base pitch of a gear and that of a mating pinion must be equal to one another at every instant of time.” Here, base pitches in the transverse section of the gear teeth are considered. If a discussion is limited just to parallel-axes gearing, the concept of “base pitch” is applicable only to gears with involute tooth profile, that is, for spur, helical, herringbone, double-helical, and so forth involute gears. No “base pitch” can be specified for gears with cycloidal as well as any other noninvolute tooth profile. Therefore, when base pitches in a gear and a mating pinion are equal ( pb . g pb . p ), the “third fundamental law of gearing” is always fulfilled. When equality of the angular base pitches (of the b . g , b . p , and b . op ) is discussed, it is assumed by default that tooth flanks, G and P , of a gear, and of a mating pinion, both are generated by that same desirable line of contact, LCdes , and with the identical configuration of the LCdes in relation to the plane of action, PA, for the gear and for the pinion. If the equalities b . g = b . op and b . p = b . op are valid, for example, for a helical gear and for a gear having a circular-arc profile in the lengthwise direction, these gears cannot be engaged in mesh with one another, as the gear tooth flank, G , and the mating pinion tooth flank, P , are generated by different desirable lines of contact, LCdes . A spur gear and a helical gear with 15

The condition of equal base pitches of interacting tooth flanks of a gearand a mating pinion is known at least since the beginning of the twentieth century (or even earlier). Unfortunately, the author failed trying to identify the name of the gear researcher to be credited with this fundamental discovery in the scientific theory of gearing. Later on (2008), the concept of “operating base pitch in a gear pair”, and the necessity of equality of three base pitches (namely, ng V = 0 and ng V = 0 ) was proposed by Prof. Stephen P. Radzevich.

Foundations of Advanced Gear Systems

27

(a)

pb.p

pb.g < pb.p

pb.p pb.p

pb.p Kg Kg Kg Kg LA ≡ Pc

Kg

Gaps pb.g pb.g pb.g pb.g

(b)

pb.g > pb.p

pb.p pb.p pb.p

pb.p Kg Kg Kg Kg LA ≡ Pc

Kg pb.g pb.g pb.g

Gaps

pb.g

FIGURE 1.16 Examples of violation of the condition of equal base pitches: (a) case when the base pitch of a gear, pb . g , is smaller compared to the base pitch, pb . p , of a mating pinion ( pb . g < pb . p ), and (b) case when the base pitch of a gear, pb . g, is larger compared to the base pitch, pb . p , of a mating pinion ( pb . g > pb . p); in both cases the gaps are observed.

the same value of the transverse base pitch, and two helical gears with different helix angle, are perfect examples in this regard. The considered equality of base pitches of a gear and a mating pinion tooth flanks, G and P , is said to be a reduced case of the third fundamental law of gearing, which all perfect gearsets have to fulfil. In a more general case, equality of base pitches of a gear and its mating pinion to the operating base pitch of the gear pair, b . op , is a necessary and sufficient condition for designing geometrically accurate gears.

28

Dudley’s Handbook of Practical Gear Design and Manufacture

CONCLUSION This chapter of the book deals with geometrically accurate gears with parallel-axes gearsets (Pa gearing), intersected-axes gearsets (Ia gearing), and crossed-axes gearsets (Ca gearing). The first fundamental law of gearing (along with the “Shishkov equation of contact, ng V = 0” that is extensively used to analytically describe this law of gearing) is briefly outlined here. The second fundamental law of gearing (or, in other words, the “conjugate action law” of two in­ teracting tooth flanks of a gear and a mating pinion) is discussed in more detail mainly with a focus on intersected-axes gearing, as well as on crossed-axes gearing. Examples of applying the results of the analysis with respect to approximate gearing are provided. The reader's attention should focus on the violation of the “conjugate action law” in today’s design practice and the production of gears for Ia gearing and Ca gearing. Interchangeable bevel gears can be produced if the conjugate action law is fulfilled when the gears for Ia gearing and Ca gearing are designed and finish-cut/ground. The inter­ changeable gears for Ia gearing and Ca gearing do not need to be paired, and the broken gears can be replaced individually and not as a gearset. Gears for intersected-axes gear pairs and crossed-axes gear pairs have to be designed to fulfill the conjugate action law—this is a must. Moreover, gears for intersected-axes gear pairs and crossed-axes gear pairs have to be finish-cut or ground/honed to fulfill the conjugate action in the gear generating process. Currently, this is the only way to eliminate the gear-lapping process, which is required to finish poorly designed and poorly finish-cut gears for intersected-axes, as well as for crossed gear pairs. No lapping is required when the gears are finish-cut/ground geometrically accurate (in accordance to the conjugate action law). It is shown that the contact pattern of a favorable geometry and location can be ensured only in conjugate intersected-axes and crossed-axes gear pairs. An in-depth understanding of the conjugate action law in cases of Ia gearing and Ca gearing is vital for the development of software for gear-measuring machines (GMMs). No accurate and reliable gear inspection is possible in cases when the conjugate action law (in cases of Ia gearing, and Ca gearing) is ignored. In summary, it becomes evident that the following statement is valid: Properly engineered gears for Ia , and Ca gearsets fulfill the conjugate action law. When the condition of conjugacy is violated, the gears cannot be used in the design of high-powerdensity gear transmissions, in gear transmissions with high-rotated gears, in the design of low-toothcount gears, and so forth. The third fundamental law of gearing requires that base pitch of a gear and that of a mating pinion, both, equal to operating base pitch of the gear pair. This requirement is a must for gears of all designs. This is because gears are machine elements that feature a plurality of functional surfaces (tooth flanks) that simultameously interact with the plurality of functional surfaces (tooth flanks) of the mating pinion. No evidence of understanding of the fundamentals of the Scientific Theory of Gearing can be found in [3–6], as well as in other recently published sources on gearing and gear production.

REFERENCES [1] Radzevich, S. P. (2018). Theory of gearing: Kinematics, geometry, and synthesis (2nd ed., revised and expanded, 934 pages). Boca Raton, FL: CRC Press. [2] Radzevich, S. P. (2017). Gear cutting tools: Science and engineering (2nd ed., 606 pages). Boca Raton, FL: CRC Press. [3] Stadtfeld, H. J. (2014). Gleason bevel gear technology: Basics of gear engineering and modern manu­ facturing methods for angular transmissions (503 pages). Rochester, NY: The Gleason Works. ISBN 0615964923, 9780615964928. [4] Stadtfeld, H. J. (2014). Gleason bevel gear technology: The science of gear engineering and modern manufacturing methods for angular transmissions (491 p.). Rochester, NY: The Gleason Works.

Foundations of Advanced Gear Systems

29

[5] Stadtfeld, H. J. (2013). Gleason Kegelradtechnologie: Ingenieurwissenschaftliche Grundlagen und modernste Herstellunsverfahren für Winkelgetribe (491 p.). Renningen: Expert-Verlag. [6] Stadtfeld, H. J. (2019). Practical gear engineering: Answers to common gear manufacturing questions (395 pages). Rochester NY: Gleason Corporation.

BIBLIOGRAPHY Klingelnberg, J. (Ed.). (2016). Bevel gear: Fundamentals and applications (328 pages). Springer-Verlag GmbH Berlin Heidelberg. Klingelnberg, J. (Ed.). (2008, July 22). Kegelräder: Grundlagen, Anwendungen (2008 ed., 381 pages). Springer. Radzevich, S. P. (2018). An examination of high-conformal gearing. Gear Solutions, (February), 31–39. Radzevich, S. P. (2014). Generation of surfaces: Kinematic geometry of surface machining (2nd ed., 747 pages). Boca Raton, Florida: CRC Press. Radzevich, S. P. (2019) Geometry of surfaces: A practical guide for mechanical engineers (2nd ed., 329 pages). Springer International Publishing. 182 illustrations. [ISBN-10: 3030221830, ISBN-13: 978-3030221836]. Radzevich, S. P. (2020). High-conformal gearing: Kinematics and geometry (2nd ed., revised and extended, 506 pages). Amsterdam: Elsevier. (1st ed.: Radzevich, S. P. (2015). High-conformal gearing: Kinematics and geometry (332 pages). Boca Raton, Florida: CRC Press. ISBN 9781498739184. August 1, 2017 – 256 pages. ISBN 9781138749528). Radzevich, S. P. (2010). On fundamental principles of gear cutting tool design: The DG/K-based approach. In Proceedings of International Conference on Gears, October 4–6, Munich, Germany.

2

Gear-Design Trends Stephen P. Radzevich

Gears are the means by which power is transferred from source to application. Gears have been in use for over three thousand years,1 and they are an important element in all manner of machinery used in current times. An example of old-style gears made out of wood are illustrated in Figure 2.1. Gearing and geared transmissions drive the machines of modern industry. Gears move the wheels and propellers that transport us over the sea, on the land, and in the air. Transmission and transformation of rotation from an input shaft to an output shaft is the main purpose of gearing of all kinds. A sizable section of industry and commerce in today’s world depends on gearing for its economy, production, and livelihood. No doubt gearing of all kinds will be extensively used in the future. Gear design is a highly complicated art. The constant pressure to build less expensive, quieter run­ ning, lighter weight, and more powerful machinery has resulted in a steady change in gear designs. At present much is known about gear load-carrying capacity, and many complicated processes for making gears are available. Those who are just starting to learn about gears need to start by learning some basic words that have special meaning in the gear field. The glossary in Table 2.1 is intended to give simple definitions of these terms as they are understood by gear people. Table 2.2 shows the metric and English gear symbols for the terms which are used in Chapter 2. The Chapter 6 glossary (see Table 6.1) defines gearmanufacturing terms. See AGMA standards2 for English and metric gear nomenclature.

2.1

MANUFACTURING TRENDS

Before plunging into the formulas for calculating gear dimensions, it is desirable to make a brief survey of how gears are presently being made and used in different applications. The methods used to manufacture gears depend on design requirements, machine tools available, quantity required, cost of materials, and tradition. In each particular field of gear work, certain methods have become established as the standard way of making gears. These methods tend to change from time to time, but the tradition of the industry tends to act as a brake to restrain any abrupt changes that result from technological developments. The gear designer, studying gears as a whole, can get a good per­ spective of gear work by reviewing the methods of manufacture in each field.

2.2

FEATURES OF GEARS OF DIFFERENT KINDS

The variety of kinds of gears and gearsets depends mainly on application of gears, including, but not limited to an actual configuration of the axes of rotation of the gears. Gears operate on:

1

2

The 1969 book by D.W. Dudley [Dudley, D.W. (1969). The evolution of the gear art. Washington, DC: AGMA, 93p.] gives a brief review of the history of gears through the ages. The 2020 book by V. Vullo [Vullo, V. (2020). Gears: A concise history, Series: Springer Series in Solid Structural Mechanics, Vol. 12, Springer International Publishing, XVI, p. 246] is another valuable source of information on the evolution of the gear art. AGMA stands for the American Gear Manufacturers Association.

31

32

FIGURE 2.1

Dudley’s Handbook of Practical Gear Design and Manufacture

Old-style gears made out of wood.

TABLE 2.1 Glossary of Gear Nomenclature, Chapter 2 Term

Definition

Gear

A geometric shape that has teeth uniformly spaced around the circumference. In general, a gear is made to mesh its teeth with another gear. (A sprocket looks like a gear but is intended to drive a chain instead of another gear.) When two gears mesh together, the smaller of the two is called the pinion. The larger is called the gear.

Pinion Ratio

Ratio is an abbreviation for gear-tooth ratio, which is the number of teeth on the gear divided by the number of teeth on its mating pinion.

Module

A measure of tooth size in the metric system. In units, it is millimeters of pitch diameter per tooth. As the tooth size increases, the module also increases. Modules usually range from 1 to 25. A measure of tooth size in the English system. In units, it is the number of teeth per inch of pitch diameter. As the tooth size increases, the diametral pitch decreases. Diametral pitches usually range from 25 to 1. The circular distance from a point on one gear tooth to a like point on the next tooth, taken along the pitch circle. Two gears must have the same circular pitch to mesh with each other. As they mesh, their circles will be tangent to one another.

Diametral pitch

Circular pitch

Pitch diameter

The diameter of the pitch circle of a gear.

Addendum Dedendum

The radial height of a gear tooth above the pitch circle. The radial height of a gear tooth below the pitch circle.

Whole depth

The total radial height of a gear tooth. (Whole depth = addendum + dedendum.)

Pressure angle

The slope of the gear tooth at the pitch-circle position. (If the pressure angle were 0º, the tooth is parallel to the axis of the gear—and is really a spur-gear tooth.) The inclination of the tooth in a lengthwise direction. (If the helix angle is 0º, the tooth is parallel to the axis of the gear—and is really a spur-gear tooth.)

Helix angle Spur gears

Gears with teeth straight and parallel to the axis of rotation.

Helical gears External gears

Gears with teeth that spiral around the body of the gear. Gears with teeth on the outside of a cylinder.

Internal gears

Gears with teeth on the inside of a hollow cylinder. (The mating gear for an internal gear must be an external gear.)

Bevel gears

Gears with teeth on the outside of a conical-shaped body (normally used on 90º axes.)

(Continued)

Gear-Design Trends

33

TABLE 2.1 (Continued) Glossary of Gear Nomenclature, Chapter 2 Term

Definition

Worm gears

Gear sets in which one member of the pair has teeth wrapped around a cylindrical body like screw threads. (Normally this gear, called the worm, has its axis at 90º to the worm-gear axis.)

Face gears Spiroid gears

Gears with teeth on the end of the cylinder. A family of gears in which the tooth design is in an intermediate zone between bevel-, worm-, and facegear design. The Spiroid design is patented by the Spiroid Division of Illinois Tool Works, Chicago, Illinois.

Transverse section

A section through a gear perpendicular to the axis of the gear.

Axial section Normal section

A section through a gear in a lengthwise direction that contains the axis of the gear. A section through the gear that is perpendicular to the tooth at the pitch circle. (For spur gears, a normal section is also a transverse section.)

Notes: 1. For terms relating to gear materials, see Chapter 5. 2. For terms relating to gear manufacture, see Chapter 6. 3. For terms relating to the specifics of gear design and rating, see Chapters 3 and 4. 4. For a simple introduction to gears, see Chapter 14.

TABLE 2.2 Gear Terms, Symbols, and Units, Chapter 2 Term

Module

Metric

English

Symbol

Units*

Symbol

m

mm

–

Pressure angle

deg

Number of teeth or threads

z

–

Number of teeth, pinion

z1

–

Number of teeth, gear

z2

–

Ratio (gear or tooth ratio)

u

Diametral pitch

–

Units* – deg

Eq. (2.1) Figures 2.4, 2.19

N Np or n

– –

Eq. (2.10)

–

Eq. (2.10)

–

NG or N mG

–

u = z 2 /z 1 (m G = NG / Np )

–

Pd or P

in.−1

–

Pi

First reference or definition

–

Eq. (2.1)

3.1415927

Pitch diameter, pinion

dp1

mm

d

in.

Figure 2.4, Eq. (2.5), (2.6)

Pitch diameter, gear

dp2

mm

D

in.

Figure 2.4, Eq. (2.5), (2.6)

Base (circle) diameter, pinion

db1

mm

db

in.

Figure 2.4

Base (circle) diameter, gear

db2

mm

Db

in.

Figure 2.4

Outside diameter, pinion

da1

mm

do

in.

Figure 2.4 (abbrev. O.D.)

Outside diameter, gear

da2

mm

Do

in.

Figure 2.4

Form diameter

d

mm

df

in.

Figure 2.4

in.

Figure 2.4, Eq. (2.2)

in.

Figure 2.4

f

Circular pitch

p

mm

Addendum

ha

mm

p a

Dedendum

hf

mm

b

in.

Figure 2.4

Face width

b

mm

F

in.

Figure 2.4

Whole depth

h

mm

ht

in.

Figure 2.4

Working depth

h

mm

hk

in.

Figure 2.4

(Continued)

34

Dudley’s Handbook of Practical Gear Design and Manufacture

TABLE 2.2 (Continued) Gear Terms, Symbols, and Units, Chapter 2 Term

Metric

English

Symbol

Units*

Clearance

c

Chordal thickness

s¯ h¯a

s

Chordal addendum

Symbol

Units*

mm

c

in.

Figure 2.4

mm

tc ac

in.

Figure 2.4

in.

Figure 2.4

t

in. in.

Figure 2.4 Eq. (2.7)

in.

Eq. (2.22), Eq. (2.8)

mm

Tooth thickness Center distance

a

mm mm

Circular pitch, normal

pn

mm

C pn

Circular pitch, transverse

pt

mm

pt

Lead angle

deg

Helix angle Pitch, base or normal

deg

pb or

First reference or definition

mm

pbn

pb or

in.

Eq. (2.22), Eq. (2.8)

deg

Eq. (2.22), Eq. (2.36)

deg

Eq. (2.22), Eq. (2.11)

in.

Eq. (2.22)

pN

Pn px

in.−1

Eq. (2.13)

in.

Eq. (2.14)

mm

Di

in.

Figure 2.6 (abbrev. I.D.)

mm

dR

in.

Figure 2.15

df 2

mm

DR

in.

Figure 2.6

Pitch angle, pinion

1

deg

deg

Figure 2.7, Eq. (2.19)

Pitch angle, gear

2

deg

deg

Figure 2.7, Eq. (2.19)

Root angle, pinion

f1

deg

R

deg

Figure 2.7

Root angle, gear

f2

deg

R

deg

Figure 2.7

a1

deg

o

deg

Figure 2.7

a2

deg

o

deg

Figure 2.7

f

deg

deg

Figure 2.7

deg

deg

Figure 2.7, Eq. (2.21)

deg

Figure 2.7

B

in. in.

Figure 2.7 Figure 2.7

Diametral, base or normal

– px

mm

Inside (internal) diameter

di

Root diameter, pinion

df 1

Root diameter, gear

Axial pitch

Face angle, pinion Face angle, pinion Dedendum angle Shaft angle Cone distance Circular (tooth) thickness Backlash

R s

–

deg

A t

j

mm mm

Throat diameter of worm

dt1

mm

dt

in.

Figure 2.7

Throat diameter of worm gear

dt2

mm

Dt

in.

Figure 2.15

Lead of worm

pzW

mm

LW

in.

Eq. (2.35)

Notes * Abbreviation for units: mm = millimeters, in. = inches, deg = degrees.

Parallel axes of rotation – gears of this kind are referred to as “parallel-axes gearing,” or just “Pa gearing,” for simplicity Intersected axes of rotation – gears of this kind are referred to as “intersected-axes gearing,” or just “Ia gearing,” for simplicity Crossing axes of rotation – gears of this kind are referred to as “crossed-axes gearing,” or just “Ca gearing,” for simplicity.

Gear-Design Trends

35

Small, low-cost gears for toys, gadgets, and mechanisms. There is a large field of gear work in which tooth stress is of almost no consequence. Speeds are slow, and life requirements do not amount to much. Almost any type of “cog” wheel that could transmit rotary motion might be used. In this sort of situation, the main thing the designer must look for is low cost and high volume of production. The simplest type of gear drive—such as those used in toys—frequently uses punched gears. Pinions with a small number of teeth may be die-cast or extruded. If loads are light enough and quietness of operation is desired, injection-molded gears and pinions may be used. Samples of plastic gears for: (a) parallel-axes gearing (Pa gearing), (b) intersected-axes gearing (Ia gearing), and (c) crossed-axes gearing (Ca gearing) are depicted in Figure 2.2. Things like film projectors, oscillating fans, cameras, cash registers, and calculators frequently need quiet-running gears to transmit insignificant amounts of power. Molded-plastic gears are widely used in such instances. It should be said, though, that the devices just mentioned often need precision-cut gears where loads and speeds become appreciable. Die-cast gears on zinc alloy, brass, or aluminum are often used to make small, low-cost gears. This process is particularly favored where the gear wheel is integrally attached to some other element, such as a sheave, cam, or clutch member. The gear teeth and the special contours of whatever is attached to the gear may all be finished to close accuracy merely by die-casting the part in a precision metal mold. Many low-cost gadgets on the market today would be very much more expensive if it were necessary to machine all the complicated gear elements that are in them. Metal forming is more and more widely used as a means of making small gear parts. Pinions and gears with small numbers of teeth may be cut from rod stock with cold-drawn or extruded teeth already formed in the rod. Small worms may have cold-rolled threads. The forming operations tend to produce parts with very smooth, work-hardened surfaces. This feature is important in many devices where the friction losses in the gearing tend to be the main factor in the power consumption of the device. Appliance gears. Home appliances like washing machines, food mixers, and fans use large numbers of small gears. Because of competition, these gears must be made for only a relatively few cents apiece. Yet they must be quiet enough to suit a discriminating homemaker and must be able to endure for many years with little or no more lubrication than that given to them at the factory. Medium-carbon-steel gears finished by conventional cutting used to be the standard in this filed. Cut gears are still in widespread use, but the cutting is often done by high-speed automatic machinery. The worker on the cutting machine does little more than bring up trays of blanks and take away trays of finished parts. Modern appliances are making more and more use of gears other than cut steel gears. Sintered-iron gears are very in expensive (in large quantities), run quietly, and frequently wear less than comparable cut gears. The sintered metal is porous and may be impregnated with a lubricant. It may also be im­ pregnated with copper to improve its strength. Gear teeth and complicated gear-blank shapes may all be completely finished in the sintering process. The tools need to make a sintered gear may cost as much as $50,000, but this does not amount to much if 100,000 or more gears are to be made on semiautomatic machines. Laminated gears using phenolic resins and cloth or paper have proved very good where noise re­ duction is a problem. The laminates in general have much more load-carrying capacity than moldedplastic gears. Nonmetallic gears with cut teeth do not suffer nearly so much from tooth inaccuracy as do metal gears. Under the same load, a laminated phenolic-resin gear tooth will bend about thirty times as much as a steel gear tooth. It has often been possible to take a set of steel gears which were wearing excessively because of tooth-error effects, replace one member with a nonmetallic gear, and have the set stand up satisfactorily. Nylon gear parts have worked very well in situations in which wear resulting from high sliding velocity is a problem. The nylon material seems to have some of the characteristics of a solid lubricant. Nylon gearing has been used in some processing equipment where the use of a regular lu­ bricant would pollute the material being processed.

(b)

(c)

FIGURE 2.2 Plastic gears for: (a) parallel-axes gearing (Pa gearing), (b) intersected-axes gearing (Ia gearing), and (c) crossed-axes gearing (Ca gearing).

(a)

36 Dudley’s Handbook of Practical Gear Design and Manufacture

Gear-Design Trends

37

Machine-tool gearing. Accuracy and power-transmitting capacity are quite important in machinetool gearing. Metal gears are usually used. The teeth are finished by some precise metal-cutting process. The machine tool is often literally full of gears. Speed-change gears of the spur or helical type are used to control feed rate and work rotation. Index drives to work or table may be worm or bevel. Sometimes they are spur or helical. Many bevel gears are used at the right-angle intersections between shafts in bases and shafts in columns. Worm gears and spiral gears are also commonly used at these places. Machine-tool gearing is often finish-machined in a medium-hard condition (250 to 300 HV or 25 to 30 HRC). Mild-alloy steels are frequently used because of their better machinability and physical properties. Cast iron is often favored for change gears because of the ease in casting the gear blank to shape, its excellent machinability, and its ability to get along with scanty lubrication. The higher cutting speeds involved in cutting with tungsten carbide tools have forced many machinetool builders to put in harder and more accurate gears. Shaving and grinding are commonly used to finish machine-tool gears to high tooth accuracy. In few cases, machine-tool gears are being made with such top-quality features as full hardness (700 HV or 60 HRC), profile modification, and surface finish of about 0.5 m or 20 in . The machine-tool designer has a hard time calculating gear sizes. Loads vary widely depending on feeds, speeds, size of work, and material being cut. It is anybody’s guess what the user will do with the machine. Because machine tools are quite competitive in price, the over-designed machine may by too expensive to sell. In general, the designer is faced with the necessity of putting in gears which have more capacity than the average load, knowing that the machine tool is apt to be neglected or overloaded on occasions. Control gears. The guns or ships, helicopters, and tanks are controlled by gear trains with the backlash held to the lowest possible limits. The primary job of these gears is to transmit motion. What power they may transmit is secondary to their job of precise control of angular motion. In power gearing, a worn-out gear is one with broken teeth or bad tooth-surface wear. In the control-gear field, a worn-out gear may be one whose thickness has been reduced by as small an amount as 0.01 mm (0.0004 in.)! Some of the most spectacular control gears are those used to drive radio telescopes and satellite tracking antennas. These gears are so large that the only practical way to make them is to cut rack sectors and then bend each section into an ark of a circle. The teeth on the rack sections are cut so that they will have correct tooth dimensions when they are bent to form part of the circular gear. The radar units on an aircraft carrier use medium-pitch gears of fairly large size. Radar-unit gearing is generally critical on backlash, must handle rather high momentary loads, and must last for many years with somewhat marginal lubrication. (Radar gears are often somewhat in the open, and therefore can only use grease lubrication). Control gears are usually spur, bevel, or worm. Helical gears are used to a limited extent. Control gears are often in the fine-pitch range—1.25 module (20 diametral pitch) or finer. In few cases, control gears become quite large. The gears which train a main battery must be very rugged. The reaction on the gears when a main battery is fired can be terrific. Control gears are usually made of medium-alloy, medium-carbon steels. In many cases, they are hardened to a medium hardness before final machining. In other cases, they are hardened to a moderately high hardness after final machining. These gears need hardness mainly to limit wear. Any hardening done after final finishing of the teeth must be done in such a way as to give only negligible dimensional change or distortion. To eliminate backlash, it is necessary to size the gear teeth almost perfectly (or use special “anti-backlash” gear arrangements). Shaving and/or grinding are used to control tooth thickness to the very close limits needed in control gearing. The inspection of control gears is usually based on checking machines which measure the variation in center distance when a master pinion or rack is rolled through mesh with the gear being checked. A spring constantly holds the master and the gear being checked in tight mesh. The chart obtained from such a checking machine gives a very clear picture of both the tooth thickness of the gear

38

Dudley’s Handbook of Practical Gear Design and Manufacture

being checked and the variations in backlash as the gear is rolled through the mesh. If the backlash variation can be held to acceptable limits in control-gear sets, there is usually no need to know the exact involute and spacing accuracy. In some types of radar and rocket-tracking equipment, control-gear teeth must be spaced very accurately with regard to accumulate error. For instance, it may be necessary to have every tooth all the way around a gear wheel within its true position within about 10 seconds of arc. On a 400-mm (15.7-in.) wheel, this would mean that every tooth had to be correctly spaced with respect to every other tooth within 0.01 mm (0.0004 in.). This kind of accuracy can be achieved only by special gear-cutting techniques. Inspection of such gears generally requires the use of gear-checking machines that are equipped to measure accumulated spacing error. The equipment commonly used will measure any angle to within 1 second of arc or better. Vehicle gears. The automobile normally uses spur and helical gears in the transmission and bevel gears in the rear end. If the car is a front-end drive, bevel gears may still be used or helical gears may be used. Automatic transmissions are now widely used. This does not eliminate gears, however. Most automatic transmissions have more gears than manual transmissions. Automotive gears are usually cut from low-alloy-steel forgings. At the time of tooth cutting, the material is not very hard. After tooth cutting, the gears are case-carburized and quenched. Quenching dies are frequently used to minimize distortion. The composition of hearts (batches) of steel is watched carefully, and all steps in manufacture are closely controlled with the aim of having each gear in a lot behave in the same way when it is carburized and quenched. Even if there is some distortion, it can be compensated for in machining, provided that each gear in a lot distorts uniformly and by the same amount. Since most automotive-gear teeth are not ground or machined after final hardening, it is essential that the teeth be quite accurate in the as-quenched condition. The only work that is done after hardening is the grinding of journal surfaces and sometimes a small amount of lapping. Finished automotive gears usually have a surface hardness of about 700 HV or 60 HRC and a core hardness of 300 HV or 30 HRC. A variety of machines are used to cut automotive-gear teeth. In the past, shapers, hobbers, and bevelgear generators were conventional machines. New types of these machines are presently favored. For instance, multi-station shapers, hobbers, and shaving machines are used. A blank is loaded on the ma­ chine at one station. While the worker is loading other stations, the piece is finished. This means that the worker spends no time waiting for work to be finished. There is a wide variety of special design machines to hob, shape, shave, or grind automotive gears. The high production of only one (or two or three) gear design makes it possible to simplify a general-purpose machine tool and then build a kind of processing center where these functions are performed: • • • • •

Incoming blank is checked for correct size Incoming blank is automatically loaded into the machine Teeth are cut (or finished) very rapidly Finished parts are checked for accuracy and sorted into categories of accept, rework, or reject Outgoing parts are loaded onto conveyor belts

Sometimes the processing center may be developed to the point where cutting, heat treating, and fin­ ishing area all done in one processing center. The gears for the smaller trucks and tractors are made somewhat like automotive gears, but the volume is not quite so great and sizes are large. Large tractors, large trucks, and “off-the-road” earthmoving vehicles use much larger gears and have much lower volume than automotive gears. More conventional machine tools are used. Special-purpose processing centers are generally not used. Vehicle gears are heavily loaded for their size. Fortunately, their heaviest loads are of short duration. This makes it possible to design the gears for limited life at maximum motor torque and still have a gear that will last many years under average driving torque.

Gear-Design Trends

39

Although carburizing has been widely used as a means of hardening automotive gears, other heat treatments are being used on an increased scale. Combinations of carburizing and nitriding are used. Processes of this type produce a shallower case for the same length of furnace time, but tend to make the gear harder and the distortion less. Induction hardening is being used on some flywheel-starter gears as well as some other gears. Flame hardening is also used to a limited extent. Transportation gears. Buses, subways, mine cars, and railroad diesels all use large quantities of spur and helical gears. The gears range up to 0.75 meter (30 in.) or more in diameter. Teeth are sometimes as coarse as 20 module (1.25 diametral pitch). Plain carbon and low-alloy steels are usually used. Much of the gearing is case-carburized and ground. Through-hardening is also used extensively. A limited use is being made of induction-hardened gears. Transportation gears are heavily loaded. Frequently their heaviest loads last for a long period of time. Diesels that pull trains over high mountains ranges have long periods of operation at maximum torque. In some applications, severe but infrequent shock loads are encountered. Shallow-hardening, mediumcarbon steels seem to resist shock better than gears with a fully hardened carburized case. Both, furnace and induction-hardening techniques are used to produce shallow-hardened teeth with high shock resistance. Gear cutting is done mostly by conventional hobbing or shaping machines. Some gears are shaved and then heat-treated, while others are heat-treated after cutting and then ground. In this field of work, the volume of production is much lower and the size of parts is much larger than in the vehicle-gear field. Both these conditions make it harder to keep heat-treat distortion so well under control that the teeth may be finished before hardening. The machine tools used to make transportation gears are quite conventional in design. The volume of production in this field is not large enough to warrant the use of the faster and more elaborate types of machine tools used in the vehicle-gear field. Marine gears. Powerful, high-speed, large-sized gears power merchant marine and navy fighting ships. Propeller drives on cargo ships use bull gears up to 5 meters (200 in.) in diameter. First reduction pitch-line speeds on some ships go up to 100 m/s (meters per second) or 20,000 fpm (feet per minute). Single propeller drives in a navy capital ship go up to 40,000 or more kW (kilowatts) of power. Some of the new cargo ships now in service have 30,000 kW [40,000 hp (horsepower)] per screw. Marine gears are almost all made by finishing the teeth after hardening. Extreme accuracy in tooth spacing is required to enable the gearing to run satisfactorily at high speed. As many as 6000 pair of teeth may go through one gear mesh in a second’s time! There is an increased use of carburized and ground gears in the marine field. The fully hard gear is smaller and lighter. This lowers pitch-line velocity and helps to keep the engine room reasonably small. Hard gears resist pitting and wear better than medium-hard gears (through-hardened gears). Single helical gears have long been used for electric-power-generating equipment. Double helical gears have generally been used for the main propulsion drive of large ships. Single helical gears—with special thrust runners on the gears themselves—are coming fairly common use even on large ships. Spur and bevel gear drives are often used on small ships, but they are seldom used on large turbinedrive ships. (Some slower-speed diesel-drive ships use spur gearing.) Marine gears generally use a medium-alloy carbon steel. It is difficult or impossible to heat-treat plain carbon and low-alloy carbon steels in large sizes and get hardnesses over about 250 HV (250 HB) and a satisfactory metallurgical structure. Special welding techniques for large gear wheels make it possible to use medium-alloy steels and get good through-hardened gears up to 5 meters (200 in.) with gear tooth hardness at 320 HV (300 HB) or higher. The smaller pinions are not welded and can usually be made up to 375 HV (350 HB) or higher in sizes up to 0.75 meter (30 in.). There is a growing use of carburized and ground gears for ship propulsion. Some of these gears are now being made as large as 2 meters (80 in.) in size. In a very large marine drive, it is quite common to have fully hard, carburized gears in the first reduction and medium-hard, through-hardened gears I the second reduction.

40

Dudley’s Handbook of Practical Gear Design and Manufacture

The designer of marine gearing has to worry about both noise and load-carrying capacity. Although the tooth loads are not high compared with those on aircraft or transportation gearing, the capacity of the medium-hardness gearing to carry load is not high either. Considering that during its lifetime a highspeed pinion on a cargo ship may make 10 to 11 billion cycles of operation at full-rated torque, it can be seen that load-carrying capacity is very important. Gear noise is a more or less critical problem on all ship gearing. The auxiliary gears that drive generators are frequently located quite close to passenger quarters. The peculiar high-pitched whine of a high-speed gear has a damaging effect on either an engine-room operator, or a passenger who may be quartered near the engine room. On fighting ships, there is the added problem of keeping enemy submarines from picking up water-borne noises and of operating one's own ship quietly enough to be able to hear water-borne noises from an enemy. Quietness is achieved on marine gears by making the gears with large numbers of teeth and cutting the teeth with extreme accuracy. A typical marine pinion may have around 60 teeth and a tooth-to-tooth spacing accuracy of 5 μm (0.0002 in.) maximum. A pinion of the same diameter used in a railroad-gear application would have about 15 teeth and a tooth-spacing accuracy of 12 μm (0.0005 in.). In the com­ parison just made, the marine-gear teeth would be only one-quarter the size of the railroad-gear teeth. Aerospace gears. Gears are used in a wide variety of applications on aircraft. Propellers are usually driven by single- or double-reduction gear trains. Accessories such as generators, pumps, hydraulic regulators, and tachometers are gear-driven. Many gears are required to drive these kinds of accessories even on jet engines—which have no propellers. Additional gears are used to raise landing wheels, open bomb-bay doors, control guns, operate computers for gun- or bomb-sighting devices, and control the pitch of propellers. Helicopters have a considerable amount of gearing to drive main rotors and tail rotors. Space vehicles often use power gears between the turbine and the booster fuel pumps. The most distinctive types of aircraft gears are the power gears for propellers, accessories, and he­ licopter rotors. The control and actuating types of gears are not too different from what would be used for ground applications of a similar nature (except that they are often highly loaded and made of extra hard, high-quality steel). Aircraft power gears are usually housed in aluminum or magnesium casings. The gears have thin webs and light cross sections in the rim or hub. Accessory gears are frequently made integral with an internally splined hub. Spur or bevel gears are usually used for accessory drives. Envelope clearance required to mount the accessory driven by the gear often make it necessary to use large center distance but narrow face width. This fact plus the thrust problem tends to rule out helical accessory gears. Propeller-drive gears have wider face widths. Spur, bevel, and helical gear drives are all currently in use. To get maximum power capacity with small size and lightweight gearboxes, it is often desirable to use an epicyclic gear train. In this kind of arrangement, there is only one output gear, but several pinions drive against it. Aerospace power gears are usually made of high-alloy steel and fully hardened (on the tooth surface) by either case carburizing or nitriding. In some designs it is feasible to cut, shave, case-carburize, and grind only the journals. Many designs have such thin, nonsymmetrical webs as to require grinding after hardening. In general, piston-engine gearing runs more slowly than gas-turbine gearing. This makes some difference in the required accuracy. So far, many more piston-engine gears have been successfully finished before hardening than have gas-turbine gears. The tooth loads and speeds are both very high on modern aircraft gears. The designer must achieve high tooth strength and high wear resistance. In addition, the thin oils used for low-temperature starting of military aircraft make the scoring type of lubrication failure a critical problem. Several special things are done to meet the demands of aircraft-gear service. Pinions are often made long addendum and gears short addendum to adjust the tip sliding velocities and to strengthen the pinion. Pinion tooth thicknesses are often increased at the expense of the gear to strengthen the pinion. High pressure angles such as 22.5°, 25°, and 27.5° are often used to reduce the base. Involute-profile modifications are generally used to compensate for bending and to keep the tips from cutting the mating part.

Gear-Design Trends

41

The most highly developed aerospace gears are those used in rocket engines. The American projects Vanguard, Mercury, Gemini, and Apollo succeeded in boosting heavy payloads into orbit and eventually putting men on the moon. Unusual aerospace-gear capability was developed to meet the special re­ quirements of power gears and control gears in space vehicles. Materials and dimensional tolerances must be held under close control. A gear failure can frequently result in the loss of human life. The gear designer and builder both have a grave responsibility to furnish gears that are always sure to work satisfactorily. Extensive ground and flight testing are required to prove new designs. The machine tools used to make aircraft gears are conventional hobbers, shapers, bevel-gear generators, shavers, and gear grinders. For the close tolerances, the machinery must be in the very best condition and precision tooling must be used. A complete line of checking line equipment is needed so that involute profile, tooth spacing, helix angle, concentricity, and surface finish can be precisely measured. Industrial gearing. A wide range of types and sizes of gears that are used in homes, factories, and offices come under the “industrial” category. In general, these gears involve electric power from a motor used to drive something. The driven device may be a pump, conveyor, or liquid stirring unit. It may also be a garage door opener, an air compressor for office refrigeration, a hoist, a winch, or a drive to mix concrete on a truck hauling the concrete to the job. Industrial gearing is relatively low speed and low horsepower. Typical pitch-line speeds range from about 0.5 m/s to somewhat over 20 m/s (100 fpm to 4000 fpm). The types of gears may be spur, helical, bevel, worm, or Spiroid3. The power may range from less than 1 kW up to a few hundred kW. Typical input speeds are those of the electric motor, like 1800, 1500, 1200, and 100 rpm (revolutions per minute). The industrial field also includes drives with hydraulic motors. The field is characterized more by relatively now pitch-line speeds and power inputs than by the means of making or using the power. Much of the gearing used in industrial work is made with through-hardened steel used as cut. There is, though, growing use of fully hardened gears where the size of the gearing or the life of the gearing is critical. In the past, industrial gearing has not generally required long life or high reliability. The trend now—in the more important factory installations—is to obtain gears for moderately long life and reliability. For instance, a pump drive with an 80% probability of running OK for 1000 hours might have been quite acceptable in the 1960s. The pump buyer in the 1980s may be more concerned with the cost of downtime and parts replacement, and may want to get gears good enough to have a 95% probability of lasting for 10,000 hours at rated load. Gears in the oil and gas industry. The production of petroleum products for the energy needs of the world requires a considerable amount of high-power, high-speed gearing. Gear units are used on oil platforms, pumping stations, drilling sites, refineries, and power stations. Usually the drive is a turbine, but it may be a large diesel engine. The power range goes from about 750 kW to over 50,000 kW. Pitchline speeds range from 20 to 200 m/s (4000 to 40,000 fpm). Bevel gears are used to limited extent. Sometimes a stage of bevel gearing is needed to make a 90° turn in a power drive. (As an example, a horizontal-axis turbine may drive a vertical-axis compressor.) Hardened and ground gears are widely used. With better facilities to grind and measure large gears and better equipment to case-harden large gears has come a strong tendency to design the powerful gears for turbine and diesel applications with fully hardened teeth. This reduces weight and size considerably. Pitch-line speeds become lower. Less space and less frame structure are required in a power package with the higher-capacity, fully hardened gears. Mill gears. Large mills make cement, grind iron ore, make rubber, roll steel, or do some other basic functions. It is common to have a few thousand kilowatts of power going through two or more gear stages to drive some massive processing drum or rolling device.

3

Spiroid is a registered trademark of the Spiroid Division of the Illinois Tool Works, Chicago, IL, U.S.A.

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Dudley’s Handbook of Practical Gear Design and Manufacture

FIGURE 2.3

Jumbo-size two-segment drive gear for a large application.

The mill is usually powered by electric motors, but diesel engines or turbines may be used. The characteristic of mill drives is high power (and frequently unusually high torques). A process mill will often run continuously for months at a time. Downtime is critical because the output ceases completely when a mill unit is shut down. Spur and helical gears are generally used. Pitch-line speeds are usually quite low. A first stage may be doing around 20 m/s (4000 fpm), but the final stage in a mill may run as slowly as 0.1 to 1.0 m/s (20 to 200 fpm). Some bevel gears are also used—where axes must be at 90°. The larger mill gears are generally made medium hard, but they may even be of low hardness. The very large sizes involved often make it impractical to use the harder gears. There is an increasing use, though, of fully hardened mill gears up to gear pitch diameters of about 2 meters. Mill gears are commonly made in sizes up to 11 meters. Such giant gears have to be made in two or more segments. Figure 2.3 shows a jumbo-size two-segment drive gear for a large application. The segments are bolted together for cutting and then unbolted for shipping. (It is impractical to ship round pieces of metal over about 5 meters in diameter).

2.3

SELECTION OF THE RIGHT KIND OF GEAR

The preceding section gave some general information on how gears are made and used in different fields. In this part of the chapter, we shall concentrate on the problem of selecting the right kind of gear. The first step in designing a set of gears is to pick the right kind. In many cases, the geometric arrangement of the apparatus which needs a gear drive will considerably affect the selection. If the gears must be on parallel axes, then spur or helical gears are the ones to use. Bevel and worm gears can be used if the axes are at right angles, but they are not feasible with parallel axes. If the axes are nonintersecting and nonparallel, then crossed-helical gears, hypoid gears, worm gears, or Spiroid gears may be used. Worm gears, though, are seldom used if the axes are not at right angles to each other. Table 2.3 shows in more detail the principal kinds of gears and how they are mounted. There are no dogmatic rules that tell the designer just which gear to use. Frequently the choice is made after weighing the advantages and disadvantages of two or three kinds of gears. Some generalizations, though, can be made about gear selection. In general, external helical gears are used when both high speeds and high horsepower are involved. External helical gears have been built to carry as much as 45,000 kW of power on a single pinion and gear. And this is not the limit for designing helical gears—bigger ones could be built if

Gear-Design Trends

43

TABLE 2.3 Kinds of Gears in Common Use Parallel Axes

Intersecting Axes

Nonintersecting Nonparallel Axes

Spur external

Straight bevel

Crossed-helical

Spur internal

Zerol bevel

Single-enveloping worm

Helical external Helical internal

Spiral bevel Face gear

Double-enveloping worm Hypoid Spiroid

anyone needed them. It is doubtful if any other kind of gear could be built and used successfully to carry this much power on a single mesh. Bevel gears are ordinarily used on right-angle drives when high efficiency in needed. These gears can usually be designed to operate with 98% or better efficiency. Worm gears often have a hard time getting above 90% efficiency. Hypoid gears do not have as good efficiency as bevel gears, but they make up for this by being able to carry more power in the same space—provided the speeds are not too high. Worm gears are ordinarily used on right-angle drives when very high ratios are needed. They are also widely used in low to medium ratios as packaged speed reducers. Single-thread worms and worm gears are used to provide the indexing accuracy on many machine tools. The critical job of indexing hobbing machines and gear shapers is nearly always done by a worm-gear drive. Spur gears are ordinarily thought of a slow-speed gears, while helical gears are thought of as highspeed gears. If noise is not a serious design problem, spur gears can be used at almost any speeds that can be handled by other types of gears. Aircraft gas-turbine spur gears sometimes run at pitch-line speeds above 50 m/s (10,000 fpm). In general, though spur gears are not used much above 20 m/s (4000 fpm).

2.3.1 EXTERNAL SPUR GEARS Spur gears are used to transmit power between parallel shafts. They impose only radial loads on their bearing. The tooth profiles are ordinarily curved in the shape of an involute. Variations in center distance do not affect the trueness of the gear action unless the change is so great as to either jam the teeth into the root fillets of the mating member, or withdraw the teeth almost out of action. Spur-gear teeth may be hobbed, shaped, milled, stamped, drawn, sintered, cast, or shear-cut. They may be given a finishing operation such as grinding, shaving, lapping, rolling, or burnishing. Speaking generally, there are more kinds of machine tools and processes available to make spur gears than to make any other gear type. This favorable situation often makes spur gears the choice where cost of manu­ facture is a major factor in the gear design. The standard measure of spur-gear tooth size in the metric system is the module. In the English system (or, in other words, in Imperial system), the standard measure of tooth size is diametral pitch. There meanings are as follows: Module is millimeters of pitch diameter per tooth. Diametral pitch is number of teeth per inch of pitch diameter (a reciprocal function). Mathematically, Module =

25.400 diametral pitch

(2.1)

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Dudley’s Handbook of Practical Gear Design and Manufacture

or Diametral pitch =

25.400 module

(2.2)

Curiously, module and diametral pitch are size dimensions that cannot be directly measured on a gear. They are really reference values used to calculate other size dimensions which are measurable. Gears can be made to any desired module or diametral pitch, provided that cutting tools are available for that tooth size. To avoid purchasing cutting tools for too many different tooth sizes, it is desirable to pick a progression of modules and design to these except where design requirements force the use of special sizes. The following commonly used modules are recommended as a start for a design series: 25, 20, 15, 12, 10, 8, 6, 5, 4, 3, 2.5, 2.0, 1.5, 1, 0.8, 0.5. Many shops are equipped with English-system 1 1 3 1 diametral pitches series: 1, 1 4 , 1 2 , 1 4 , 2, 2 2 , 3, 4, 5, 6, 8, 10 , 12 , 16, 20 , 24, 32, 48, 64, 128. Considering the trend toward international trade, it is desirable to purchase new gear tools in standard metric sizes, so that they will be handy for gear work going to any part of the world. (The standard measuring system of the world is the metric system.) Most designers prefer a 20° pressure angle for spur gears. In the past the 14.5° pressure angle was widely used. It is not popular today because it gets into trouble with undercutting much more quickly than the 20° tooth when small numbers of pinion teeth are needed. Also, it does not have the loadcarrying capacity of the 20° tooth. A pressure angle of 22.5° or 25° is often used. Pressure angles above 20° give higher load capacity but may not run quite as smoothly or quietly. Figure 2.4 shows the terminology used with a spur gear, or a spur rack (a rack is a section of a spur gear with an infinitely large pitch diameter). Pressure angle Face width

Circular pitch Dedendum Addendum

Addendum of mating gear Tooth fillet Working depth Whole depth

Pitch diameter

Clearance

Chordal thickness

Base circle diameter

Outside diameter

Form diameter

Spur gear

Addendum

Chordal addendum

Arc thickness Rise of arc

Tooth thickness Spur rack FIGURE 2.4

Spur-gear and rack terminology.

Circular pitch

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45

The following formulas apply to spur gears in all cases: Circular pitch = pi × module

= pi ÷ diametral pitch

metric

English

Pitch diameter = no. of teeth × modulemetric = no. of teeth ÷ diametral pitch

English

(2.3) (2.4) (2.5) (2.6)

The nominal center distance is equal to the sum of the pitch diameter of the pinion and the pitch diameter of the gear divided by 2: Center distance =

pinion pitch dia. + gear pitch dia. 2

(2.7)

Since the center distance is a machined dimension, it may not come out to be exactly what the design calls for. In addition, it is common practice to use a slightly larger center distance to increase the operation pressure angle. For instance, if the actual center distance is made 1.7116% larger, gears cut with 20° hobs or shaper-cutters will run at 22.5° pressure angle. (See Chapter 17 for methods of design for special center distance.) For the reasons just mentioned, it is possible to have two center distances, a nominal center distance and an operating center distance. Likewise, there are two pitch diameters. The pitch diameter for the tooth-cutting operation is the nominal pitch diameter and is given by Eq. (2.5) and Eq. (2.6). The op­ erating pitch diameters are: Pitch dia. (operating) of pinion =

2 × operating cent. dist. ratio + 1

Pitch dia. (operating) of gear = ratio × pitch dia. (operating) of pinion

(2.8) (2.9)

were: Gear (tooth) ratio =

no. gear teeth no. pinion teeth

(2.10)

2.3.2 EXTERNAL HELICAL GEARS Helical gears are used to transmit power or motion between parallel shafts. The helix angle must by the same in degrees on each member, but the hand of the helix on the pinion is opposite to that on the gear. (A RH pinion meshes with LH gear, and LH pinion meshes with a RH gear.) Single helical gears impose both thrust and radial load on their bearings. Double helical gears develop equal and opposite thrust reactions, which have the effect of canceling out the thrust load. Usually double helical gears have a gap between helices to permit a runout clearance for the hob, grinding wheel, or other cutting tool. One kind of gear shaper has been developed that permits double helical teeth to be made continuous (no gap between helices).

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Dudley’s Handbook of Practical Gear Design and Manufacture

Helical-gear teeth are usually made with an involute profile in the transverse section (the transverse section is a cross section perpendicular to the gear axis). Small changes in center distance do not affect the action of helical gears. Helical-gear teeth may be made by hobbing, shaping, milling, or casting. Sintering has been used with limited success. Helical teeth may be finished by grinding, shaving, rolling, lapping, or burnishing. The size of helical-gear teeth is specified by module for the metric system and by diametral pitch for the English system. The helical tooth will frequently have some of its dimensions given in the normal section and others given in transverse section. Thus, standard cutting tools could be specified for either section—but not for both sections. If the helical gear is small (less than 1 meter pitch diameter), most designers will use the same pressure angle and standard tooth size in the normal section of the helical gear as they would use for spur gears. This makes it possible to hob helical gears with a standard spur gear hob. (It is not possible, though, to cut helical gears with standard spur-gear shaper-cutters.) Helical gears often use 20° as the standard pressure angle in the normal section. However, higher pressure angles, like 22.5° or 25°, may be used to get extra load-carrying capacity. Figure 2.5 shows terminology of a helical gear and a helical rack. In the transverse plane, the elements of a helical gear are the same as those of a spur gear. Eqs. (2.1) to (2.10) apply just as well to transverse plane of a helical gear as they do to a spur gear. Additional general formulas for helical gears are the following: Normal circ. pitch = circ. pitch × cosine helix angle

(2.11)

Normal module = transverse module × cosine helix angle

(2.12)

Axis of gear

Normal circular pitch

Transverse circular pitch Lead angle Base pitch (or normal pitch) Whole depth

Top-land Addendum Form diameter

Dedendum

Base circle diameter

Helix angle Face width

Outside diameter Pitch diameter

Helical gear

Normal plane

Normal pitch line

Lead angle Front plane Circular pitch

FIGURE 2.5

Helical-gear and rack terminology.

Helix angle

Gear-Design Trends

47

Normal diam. pitch = trans. diam. pitch × cosine helix angle

(2.13)

Axial pitch = circ. pitch ÷ tangent helix angle

(2.14)

Axial pitch = norm. circ. pitch ÷ sine helix angle

(2.15)

2.3.3 INTERNAL GEARS Two internal gears will not mesh with each other, but an external gear may be meshed with an internal gear. The external gear must not be larger than about two-thirds the pitch diameter of the internal gear when full-depth 20° pressure angle teeth are used. The axes on which the gears are mounted must be parallel. Internal gears may be either spur or helical. Even double helical internal gears are used occasionally. An internal gear is a necessary in an epicyclic type of gear arrangement. The short center distance of an internal gear set makes it desirable in some applications where space is very limited. The shape of an internal gear forms are natural guard over the meshing gear teeth. This is very advantageous for some types of machinery. Internal gears have the disadvantage that fewer types of machine tool can produce them. Internal gears cannot be hobbed4. They can be shaped, milled, or cast. In small sizes they can be broached. Both helical and spur internals can be finished by shaving, grinding, lapping, or burnishing. An internal gear has the same helix angle in degrees and the same hand as its mating pinion (righthand pinion meshes with right-hand gear and vice versa). Figure 2.6 shows the terminology used for a spur internal gear. All the previously given formulas apply to internal gearing except those involving center distance [Eqs. (2.7), (2.8), and (2.9) do not hold for internals]. Formulas for internal-gear center distance are as follows: Circular pitch Addendum

Clearance Working depth

Dedendum Addendum of mating pinion

Whole depth

Chordal thickness Arc thickness

Root diameter Pitch diameter

Base circle diameter

Addendum Rise of arc

Inside diameter Width of face

Cutter clearance Center distance

FIGURE 2.6

Internal-gear terminology.

Chordal addendum

48

Dudley’s Handbook of Practical Gear Design and Manufacture

Center distance =

pitch dia. of gear 2

Pitch dia. (operating) of pinion =

Pitch dia. (operating) of gear =

pitch dia. of pinion

2 × op. cent. dist. ratio 1

2 × op. cent.dist. ×ratio ratio 1

(2.16)

(2.17)

(2.18)

2.3.4 STRAIGHT BEVEL GEARS Bevel gear blanks are conical in shape. The teeth are tapered in both tooth thickness and tooth height. At one end the tooth is larger, while at the other end it is small. The tooth dimensions are usually specified for the large end of the tooth. However, in calculating bearing loads, the central-section dimensions and forces are use. The simplest type of bevel gear is the straight bevel gear. These gears are commonly used for transmitting power between intersecting shafts. Usually the shaft angle is 90°, but it may be almost any angle. The gears impose both radial and thrust load on their bearing. Bevel gears must be mounted on axes whose shaft angle is almost exactly the same as the design shaft angle. Also, the axes on which they are mounted must either intersect or come very close to intersecting. In addition to the accuracy required of the axes, bevel gears must be mounted at the right distance from the cone center. The complications involved in mounting bevel gears make it difficult to use sleeve bearings with large clearances (which is often done on high-speed, high-power spur and helical gears). Ball and roller bearings are the kinds commonly used for bevel gears. The limitations of these bearings’ speed and load-carrying capacity indirectly limit the capacity of bevel gears in some high-speed applications. Straight bevel teeth are usually cut on bevel-gear generators. In some cases, where accuracy is not too important, bevel-gear teeth are milled. Bevel teeth may also be cast. Lapping is the process often used to finish straight bevel teeth. Shaving is not practical for straight bevel gears, but straight bevels may be ground. The size of bevel-gear teeth is defined in module for the metric system and in diametral pitch for the English system. The specified size dimensions are given for the large end of the tooth. A bevel gear tooth that is 12 module at the large end may be only around 10 module at the small end. The commonly used modules (or diametral pitches) are the same as those used for spur gears. There is no particular advantage to using standard tooth sizes for bevel gears. A set of cutting tools will cut more than a single pitch. The two views of bevel gears in Figure 2.7 show bevel-gear terminology. Bevel-gear teeth have profiles that closely resemble an involute curve. The shape of a straight bevel-gear tooth (in a section normal to the tooth) closely approximates that of an involute spur gear with a larger number of teeth. This larger number of teeth, called the virtual number of teeth, is equal to the actual number of teeth divided by the cosine of the pitch angle. Straight bevel-gear teeth have been commonly made with 14.5°, 17.5°, and 20° pressure angles. The 20° design is the most popular. The pitch angle of a bevel gear is the angle of the pitch cone. It is a measure of the amount of taper in the gear. For instance, as the taper is reduced, the pitch angle approaches zero and the bevel gear approaches a spur gear.

4

Some very special hobs and hobbing machines have been used – to a rather limited extent – to hob internal gears.

Gear

A

Outside diameter

Pitch diameter

Front angle

Pitch angle Uniform clearance

Shaft angle

Pitch apex

Root angle

Dedendum angle

Back cone

Face angle

FIGURE 2.7 Bevel-gear terminology.

Back angle

Face width

Cone distance

Crown

A

Top land

Crown to back

Base cone distance

Pinion

Pitch apex to crown

Pitch apex to back

Dedendum

Tooth profile

Circular pitch

Tooth fillet

Addendum

Clearance

Backlash

Working depth

Base cone distance

Circular thickness Chordal thickness

Working position

Bottom land

Whole depth

A-A

Gear-Design Trends 49

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Dudley’s Handbook of Practical Gear Design and Manufacture

The pitch angles in a set of bevel gears are defined by lines meeting at the cone center. The root and face angles are defined by lines which do not hit the cone center (or apex). In old style designs these angles did meet at the apex, but modern designs make the outside cone of one gear parallel to the root cone of its mate. This gives a constant clearance and permits a better cutting-tool design and gear-tooth design than the old-style design with its tapering clearance. The circular pitch and the pitch diameters of bevel gears are calculated the same as for spur gears. [See Eqs. (2.1) to (2.3).] The pitch-cone angles may be calculated by one of the following sets of equations: Tan pitch angle, pinion =

Tan pitch angle, gear =

no. teeth in pinion no. teeth in gear no. teeth in gear no. teeth in pinion

(2.19)

(2.20)

When the shaft angle is less than 90°, sin shaft angle ratio + cos shaft angle

(2.21)

sin shaft angle 1/ratio + cos shaft angle

(2.22)

Tan pitch angle, pinion =

Tan pitch angle, gear =

When the shaft angle is over than 90°, sin(180° shaft angle) ratio cos(180° shaft angle)

(2.23)

sin(180° shaft angle) 1/ratio cos(180° shaft angle)

(2.24)

Tan pitch angle, pinion =

Tan pitch angle, gear =

In all the above cases, Pitch angle, pinion + pitch angle, gear = shaft angle

(2.25)

2.3.5 ZEROL BEVEL GEARS Zerol5 bevel gears are similar to straight bevel gears except that they have a curved tooth in the lengthwise direction. Zerol bevel gears (see Figure 2.8) have 0° spiral angle at the middle of the tooth face. They are made in a different kind of machine from that used to make straight bevel gears. The straight-bevel-gear-generating machine has a cutting tool that moves back and forth in a straight line. The Zerol is generated by a rotary cutter that is like a face mill. It is the curvature of this cutter that makes the lengthwise curvature in the Zerol tooth.

5

Zerol is a registered trademark of the Gleason Works, Rochester, NY, U.S.A.

Gear-Design Trends

FIGURE 2.8

51

A pair of Zerol bevel gears (courtesy of the Gleason Works, Rochester, NY, U.S.A.).

The Zerol gear has a profile that somewhat resembles an involute curve. The pressure angle of the tooth varies slightly in going across the face width. This is cause by the lengthwise curvature of the tooth. Zerol gear teeth may be finished by grinding or lapping. Since the Zerol gear can be ground, it is favored over straight bevel gears in applications in which both high accuracy and full hardness are required. Even in applications in which cut gears of machinable hardness can be used, the Zerol may be the best choice if speeds are high. Because of its lengthwise curvature, the Zerol tooth has s slight overlapping action. This tends to make it run more smoothly than the straight-bevel-gear tooth. A cut Zerol bevel gear is usually more accurate than a straight bevel gear. In making a set of Zerol gears, one member is made first, using theoretical machine settings. Then a second gear is finished in such a way that its profile and lengthwise curvature will give satisfactory contact with the first gear. Several trial cuts and adjustments to machine settings may be required to develop a set of gears that will conjugate properly. If a number of identical sets of gears is required, a matching set of test gears is made. Then each production gear is machined so that it will mesh satisfactorily with one or the other of the test gears. In this way a number of sets of interchangeable gears may be made. Zerol gears are usually made to a 20° pressure angle. In a few ratios where pinion and gear have small numbers of teeth, 22.5° or 25° is used. The calculations for pitch diameter and pitch-cone angle are the same for Zerol bevel gears as for straight bevel gears.

2.3.6 SPIRAL BEVEL GEARS Spiral bevel gears have a lengthwise curvature like Zerol gears. However, they differ from Zerol gears in that they have an appreciable angle with the axis of the gear. See Figure 2.9. Although spiral bevel teeth do not have a true helical spiral, a spiral bevel gear looks somewhat like a helical bevel gear. Spiral bevel gears are generated by the same machines that cut or grind Zerol gears. The only dif­ ference is that the cutting tool is set at an angle to the axis of the gear instead of being set essentially parallel to the gear axis. Spiral bevel gears are made in matched sets like Zerol bevel gears. Different sets of the same design are not interchangeable unless they have been purposely built to match a common set of test gears.

52

FIGURE 2.9

Dudley’s Handbook of Practical Gear Design and Manufacture

A pair of spiral bevel gears (courtesy of the Gleason Works, Rochester, NY, U.S.A.).

Generating types of machines are ordinarily used to cut or grind spiral-bevel-gear teeth. In some highproduction jobs, a special kind of machine is used that cuts the teeth without going through a generating motion. Spiral-bevel-gear teeth are frequently given a lapping operation to finish the teeth and obtain the desired tooth bearing. In high-speed gear work, the spiral bevel is preferred over the Zerol bevel because its spiral angle tends to give the teeth a considerable amount of overlap. This makes the gear run more smoothly, and the load is distributed over more tooth surface. However, the spiral bevel gear imposes much more thrust load on its bearings than does a Zerol bevel gear. Spiral bevel gears are commonly made of 16°, 17.5°, 20°, and 22.5° pressure angles. The 20° angle has become the most popular. It is the only angle used on aircraft and instrument gears. The most common spiral angle is 35°.

2.3.7 HYPOID GEARS Hypoid gears resemble bevel gears in some respects. They are used on crossed-axes shafts, and there is a tendency for the parts to taper as do bevel gears. They differ from true bevel gears in that their axes do not intersect. The distance between a hypoid pinion axis and the axis of a hypoid gear is called offset. This distance is measured along the perpendicular common to the two axes. If a set of hypoid gears had no offset, they would simply be spiral bevel gears. See Figure 2.10 for offset and other terms. Hypoid pinions may have as few as five teeth in a high ratio. Since the various kinds of bevel gears do not often go below 100 teeth in a pinion, it can be seen that it is easier to get high ratios with hypoid gears. Contrary to the general rule with spur, helical, and bevel gears, hypoid pinions and gears do not have pitch diameters which are in proportion to their numbers of teeth. This makes it possible to use a large and strong pinion even with a high ratio and only a few pinion teeth. See Figure 2.11. Hypoid teeth have unequal pressure angles and unequal profile curvatures on the two sides of the teeth. This results from the unusual geometry of the hypoid gear rather than from a nonsymmetrical cutting tool.

Gear-Design Trends

53

Hypoid offset

Pinion mounting distance

Pinion mounting distance Pinion mounting distance

Crossing point

Gear mounting distance Gear mounting distance

FIGURE 2.10

Hypoid-gear arrangement.

FIGURE 2.11

A pair of hypoid gears (courtesy of the Gleason Works, Rochester, NY, U.S.A.).

Hypoid gears are matched to run together, just as Zerol or spiral gear-sets are matched. Interchangeability is obtained by making production gears fit with test masters. Hypoid gears and pinions are usually cut on a generating type of machine. They may be finished by either grinding or lapping.

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Dudley’s Handbook of Practical Gear Design and Manufacture

The hypoid gears for passenger cars and for industrial drives usually have a basic pressure angle of 21°15 . For tractors and trucks the average pressure angle is 22° 30 . Pinions are frequently made with a spiral angle of 45° or 50°. In hypoid gearing, module and diametral pitch are used for the gear only. Likewise, the pitch diameter and the pitch angle are figured for the gear only. If a pitch were used for the pinion, it would be smaller than that of the gear. The size of a hypoid pinion is established by its outside diameter and its number of teeth. The geometry of hypoid teeth is defined by the various dimensions used to set up the machines to cut the teeth.

2.3.8 FACE GEARS Face gears have teeth cut on the end face of a gear, just as the name “face” implies. They are not ordinarily thought of as bevel gears, but functionally they are more akin to bevel gears than to any other kind. A spur pinion and a face gear are mounted—like bevel gears—on shafts that intersect and have a shaft angle (usually 90°). The pinion bearing carries mostly radial load, while the gear bearing has both thrust and radial load. The mounting distance of the pinion from the pitch-cone apex is not critical, as it is in bevel or hypoid gears. Figure 2.12 shows the terminology used with face gears. The pinion that goes with a face gear is usually made spur, but it can be made helical if necessary. The formulas for determining the dimensions of a pinion to run with a face gear are no different from those for the dimensions of a pinions to run with a mating gear on parallel axes. The pressure angles and pitches used are similar to spur-gear (or helical gear) practice. The pinion may be finished or cut by all the methods previously mentioned for spur and helical pinions. The gear, however, must be finished with a shaper-cutter which is almost the same size as the pinion. Equipment to grind face gears is not available. The teeth can be lapped, and they might be shaved without too much difficulty, although ordinarily they are not shaved. Pitch cone apex Pinion

Gear Pitch diameter Outside diameter Gear axis

FIGURE 2.12

Face-gear terminology.

Pitch diameter

Gear-Design Trends

55

The face-gear tooth changes shape from one end of the tooth to the other. The face width of the gear is limited at the outside end by the radius at which the tooth becomes pointed. At the inside end, the limit is the radius at which undercut becomes excessive. Practical considerations usually make it desirable to make the face width somewhat short of these limits. The pinion to go with a face gear is usually made with 20° pressure angle.

2.3.9 CROSSED-HELICAL GEARS (NON-ENVELOPING WORM GEARS) The word “spiral” is rather loosely used in the gear trade. The word may be applied to both helical and bevel gears. In this section we shall consider the special kind of worm gear that is often called a “spiral gear.” More correctly, though, it is a crossed-helical gear. Crossed-helical gears are essentially non-enveloping worm gears. Both members are cylin­ drically shaped. (See Figure 2.13). In comparison, the single-enveloping worm gear set has a

FIGURE 2.13

Crossed-helical-gear drive.

56

Dudley’s Handbook of Practical Gear Design and Manufacture

cylindrical worm, but the gear is throated so that is tends to wrap around the worm. The doubleenveloping worm gear set goes still further; both members are throated, and both members wrap around each other. Crossed-helical gears are mounted on axes that do not intersect and that are at an angle to each other. Frequently the angle between the axes is 90°. The bearings for crossed-helical gears have both thrust and radial load. A point contact is made between two spiral gear teeth in mesh with each other. As the gears revolve, this point travels across the tooth in a sloping line. After the gears have worn in for period of time, a shallow, sloping line of contact is worn into each member. This makes the original point contact increase to a line as long as the width of the sloping band of contact. The load-carrying capacity of crossed-helical gears is quite small when they are new, but if they are worn in carefully, it increases quite appreciably. Crossed-helical gears are able to stand small changes in center distance and small changes in shaft angle without any impairment in the accuracy with which the set transmits motion. This fact, and the fact that shifting either member endwise makes no difference in the amount of contact obtained, makes this the easiest of all gears to mount. There is no need to get close accuracy in center distance, shaft alignment, or axial position—provided the teeth are cut with reasonably generous face width and backlash. Crossed-helical gears may be made by any of the processes used to make single helical gear of the same hand. Up to the point of mounting the gear in a gearbox, there is no difference between a crossedhelical gear and a helical gear. Usually, a crossed-helical gear of one hand is meshed with a crossed-helical gear of the same hand. It is not necessary, though, to do this. If the shaft angle is properly set, it is possible to mesh opposite hands together. Thus, the range of possibilities is as follows: • • • •

RH drive with RH drive LH drive with LH drive RH drive with LH drive LH drive with RH drive

The pitch diameters of crossed-helical gears—like those of hypoid gears—are not in proportion to the tooth ratio. This makes the use of the word “pinion” for smaller member of the pair inappropriate. In a crossed-helical gearset, the small pinion might easily have more teeth than the gear! The same helix angle in degrees does not have to be used for each member. Whenever different helix angles are used, the module (or diametral pitch) for two crossed-helical gears that mesh with each other is not the same. The thing that is the same in all cases is the normal module (and the normal circular pitch). This makes the normal module (or normal diametral pitch) the most appropriate measure of tooth size. Designers of crossed-helical gears usually get the best results when there is a contact ratio in the normal section of at least 2. This means that in all positions of tooth engagement, the load will be shared by at least two pair of teeth. To get this high contact ratio, a low normal pressure angle and a deep tooth depth are needed. When the helix is 45°, a normal pressure angle of 14.5° gives good results. Some of the basic formulas for crossed-helical gears are as follows: Shaft angle = helix angle of driver ± helix angle of driven

(2.26)

Normal module = normal circ. pitch ÷ pi

(2.27)

pi normal circ. pitch

(2.28)

Normal diam. pitch =

Gear-Design Trends

57

Pitch dia. =

Center distance =

no. of teeth × normal module cosine of helix angle

(2.29)

pitch dia. driver + pitch dia. driven 2

(2.30)

no. teeth × normal circ. pitch pi × pitch dia.

(2.31)

Cosine of helix angle =

2.3.10 SINGLE-ENVELOPING WORM GEARS Figure 2.14 shows a single-enveloping worm gear. Worm gears are characterized by one member having a screw thread. Frequently, the thread angle (lead angle) is only a few degrees. The worm in this case has the outward appearance of the thread on a bolt, greatly enlarged. When a worm has multiple threads and a lead angle approaching 45°, it may be (if it has an involute profile) geometrically just the same as a helical pinion of the same lead angle. In this case the only difference between a worm and a helical pinion would be in their usage. Worm gears are usually mounted on nonintersecting shafts which are at a 90° shaft angle. Worm bearing usually have a high thrust load. The worm-gear bearing has a high radial load and a low thrust load (unless the lead angle is high). The single-enveloping worm gear has a line contact which extends either across the face width, or across the part of the tooth that is in the zone of action. As the gear revolves, this line sweeps across the whole width and height of the tooth. The meshing action is quite similar to that of helical gears on parallel shafts, except that much higher sliding velocity is obtained for the same pitch-line velocity. In a helical gearset, the sliding velocity at the tooth tips is usually mot more than about one-fourth the pitch-line velocity. In a high-ratio worm gear set, the sliding velocity is greater than the pitch-line velocity of the worm. Worm gear sets have considerably more load-carrying capacity than crossed-helical gear sets. This results from the fact that they have line contact in stead of point contact. Worm gear sets must be mounted on shafts that are very close to being correctly aligned and at the correct center distance. The

FIGURE 2.14 Single-enveloping worm gear set (courtesy of Transamerica Delaval, Delroyd Worm Gear Div., Trenton, NJ, U.S.A.).

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Dudley’s Handbook of Practical Gear Design and Manufacture

axial position of a single-enveloping worm is not critical, but the worm gear must be in just the right axial position so that it can wrap around the worm properly. Several different kinds of worm-thread shapes are in common use. These are: • • • •

Worm Worm Worm Worm

thread thread thread thread

produced by straight-sided conical milling or grinding wheel straight-sided in the axial section straight-sided in the normal section an involute helicoid shape

The shape of the worm thread defines the worm-gear tooth shape. The worm gear is simply a gear element formed to be “conjugate” to a specified worm thread. A worm and a worm gear have the same hand of helix. A RH worm, for example, meshes with a RH gear. The helix angles are usually very different for a worm and a worm gear. Usually, the worm has a more than 45° helix angle, and the worm gear has a less than 45° helix angle. Customarily the lead angle—which is the complement of the helix angle—is used to specify the angle of the worm thread. See Figure 2.15 for a diagram of the worm gear and its terminology. When the worm gear set has a 90° shaft angle, the worm lead angle is numerically equal to the worm-gear helix angle. The axial pitch is the dimension that is used to specify the size of worm treads. It is the distance from thread to thread measured in an axial plane. When the shaft angle is 90°, the axial pitch of the worm is numerically equal to the worm-gear circular pitch. In the metric system, popular axial pitch values are 5, 7.5, 10 , 15, 20 , 30 and 40 mm . In the English system, commonly used values have been 0.250 , 0.375, 0.500 , 0.750 , 1.000 , 1.250 and 1.500 in. Fine pitch worm gears are often designed to standard lead- and pitch diameter values so as to obtain even lead-angle values. Worm threads are usually milled or cut with a single-point lathe tool. In fine pitches, some designs can be formed by rolling. Grinding is often employed as a finishing process for high-hardness worms. In fine pitches, worm treads are sometimes ground from the solid. Worm-gear teeth are usually hobbed. The cutting tool is essentially a duplicate of the mating worm in size and thread design. New worm designs should be based on available hobs wherever possible to avoid the need for procuring a special hob for each worm design. A variety of pressure angles are used for worms. Single-thread worms used for indexing purposes frequently have low axial pressure angles, like 14.5°. Multiple-threaded worms with high lead angles like 30 or 40° are often designed with about 30° axial pressure angles. The following formulas apply to worm gears that are designed to run on 90° axes: Axial pitch of worm = circ. pitch of worm gear

(2.32)

no. of teeth × circ. pitch pi

(2.33)

Pitch dia. of gear =

Pitch dia. of worm = 2 × center distance

pitch dia. of gear

Lead of worm = axial pitch × number of threads Tan lead angle =

lead of worm pitch dia. ×pi

Lead angle of worm = helix angle of gear

(2.34) (2.35) (2.36)

(2.37)

Gear-Design Trends

59 Outside diameter Addendum

Root diameter

Whole depth Dedendum

Worm

Worm wheel

Lead angle

Linear pitch

Circular pitch

FIGURE 2.15

Worm gear terminology (linear pitch = axial pitch).

2.3.11 DOUBLE-ENVELOPING WORM GEARS The double-enveloping worm gear is like the single-enveloping gear except that the worm envelopes the worm gear. Thus, both members are throated. See Figure 2.16. Double-enveloping worm gears are used to transmit power between non-intersecting shafts, usually those at a 90° angle. Double-enveloping worm gears load their bearings with thrust and radial loads the same as single-enveloping worm gears do. Double-enveloping worm gears should be accurately located on all mounting dimensions. Shafts should be at the right shaft angle and at the right center distance. The double-enveloping type of worm gear has more tooth surface in contact than a single-enveloping worm gear. Instead of line contact, it has area contact at any one instant of time. The larger contact area of the double-enveloping worm gear increases the load-carrying capacity. On most double-enveloping

60

FIGURE 2.16 MI, U.S.A.).

Dudley’s Handbook of Practical Gear Design and Manufacture

Double-enveloping worm gear set (courtesy of Ex-Cell-O Corp., Cone Drive Div., Traverse City,

worm gear sets, the worm rubbing speed is below 10 m/s (2000 fpm ). Above 10 m/s, it is possible to get good results with oil lubrication, using a circulating system and coolers. The lubrication system must be good enough to prevent scoring and overheating. The only double-enveloping worm gear that is in widespread use today is the Cone-Drive6 design. Figure 2.17 shows the terminology used with Cone-Drive worm gears. The Cone-Drive worm has a straight-sided profile in the axial section, but this profile changes its inclination as you move along thread. At any one position, this slope is determined by a line that is tangent to the base cylinder of the gear. The base cylinder of a Cone-Drive gear is like an involute base circle in that it is an imaginary circle used to define a profile. Geometrically, though, the base circle of a Cone-Drive gear is not used in the same way as the base circle of an involute gear. The size of Cone-Drive gear teeth is measured by the circular pitch of the gear. The normal pressure angle is ordinarily 20° to 22°. The Cone-Drive worm and gear diameters are not in proportion to the ratio. With low ratios, it is possible (although not recommended) to have a worm that is larger than the gear! Equations (2.33) and (2.34) apply to Cone-Drive gears as well as to regular worm gears. The other formulas for single-enveloping worm gears apply only to the center of the Cone-Drive worm, since it does not have a fixed axial pitch and lead like a cylindrical worm. In both single-enveloping worm gears and Cone-Drive gears, it is generally recommended that the worm or pinion diameter be made a function of the center distance. Thus Pitch dia. of worm =

(center distance)0.875 2.2

(2.38)

Following the recommendation of Eq. (2.38) is, of course, not necessary. This formula merely re­ commends a good proportion of worm to gear diameter for best power capacity. In instrumental and control work, the designer may not be interested in power transmission at all. In such cases, it may be 6

Cone-Drive is a registered trademark of the Ex-Cell-O Corp., Cone-Drive Dif., Traverse City, Michigan, U.S.A.

Begin relief

Relief 90° end

Pinion face

1 pinion face 2 Thrust shoulder

Pinion throat diameter 1 Gear width 2 at O.D.

Gear throat radius

Pinion axis

Pinion face angle

Center distance

Gear axis

Base circle diameter

FIGURE 2.17 Terminology on Cone-Drive worm gears. Max.: maximum; OD: outside diameter; rad: radius.

Angular end

Pinion O.D. Maximum hob radius

CL

Gear face

Gear width at O.D.

Maximum hob radius

Gear face angle

1 Gear face 2

Gear-Design Trends 61

62

FIGURE 2.18

Dudley’s Handbook of Practical Gear Design and Manufacture

Spiroid gear pair (courtesy of Spiroid Div. of Illinois Tool Works, Inc., Chicago, IL, U.S.A.).

desirable to depart considerably from Eq. (2.38) in picking the size of a worm or pinion. In fact, the source shows a whole series of worm diameters for fine-pitch work that do not agree with Eq. (2.38). When a worm diameter is picked in accordance with Eq. (2.38), the gear diameter and the circular pitch may be obtained by working backward through Eqs. (2.34), Eq. (2.33), and Eq. (2.22). The helix angle of a worm gear or a Cone-Drive gear may be obtained from the following general formula: Tan center helix angle of gear =

pitch dia. of gear pitch dia. of worm × ratio

(2.39)

2.3.12 SPIROID GEARS Discussed here are Spiroid and Helicon brand gearing. Suitable for right-angle power transmission in applications with high power density requirements, these skew-axes gear forms operate on nonintersecting and non-parallel axes. The Spiroid family has Helicon7 and Planoid types as well as the Spiroid type. The Helicon is essentially a Spiroid with no taper in the pinion. The Planoid is used for lower ratios than the Spiroid, and its offset is lower—more in the range of the hypoid gear. The most famous family member is called Spiroid. It involves a tapered pinion that somewhat resembles a worm (see Figure 2.18). The gear member is a face gear with teeth curved in a lengthwise direction; the inclination to the tooth is like a helix angle—but not a true helical spiral. Figure 2.19 shows the terminology used with Spiroid gears.

7

Helicon and Planoid are registered trademarks of Illinois Tool Works Inc., Chicago, IL., U.S.A., as is the term Spiroid.

Gear O.D. Gear I.D.

Pinion length

Thread angle

Pinion mountig distance

FIGURE 2.19 Spiroid-gear terminology.

(a)

Gear taper angle

Center distance Pinion O.D.

Pinion taper angle

Pinion taper angle

(b)

Low-side normal pressure angle

Root flat

Normal tooth thickness

Tip width

High-side normal pressure angle

Full depth of tooth

Addendum

Gear-Design Trends 63

64

Dudley’s Handbook of Practical Gear Design and Manufacture

The Spiroid pinions may be made by hobbing, milling, rolling, or thread chasing. Spiroid gears are typically made by hobbing, using a specially built (or modified) hobbing machine and special hobs. The gear may be made with molded or sintered gear teeth using tools (dies or punches) that have teeth resembling hobbed gears. Shaping and milling are not practical to use in making Spiroid gears. The Spiroid gears may be lapped as a final finishing process. Special “shaving” type hobs may also be used in a finishing operation. Compared to traditional right-angle bevel and worm gearing, the gear-centerline offset in Spiroid and Helicon branded gearing allows for more tooth-surface contact and results in higher contact ratios. This boosts torque capacity and smoothness of motion transmission. Spiroid brand gears use advanced software and tooling to make the proprietary gearing fit specific application requirements. The gearsets are quiet, stiff, and compact, delivering ratios from 3:1 to 300:1 and beyond. Spiroid gears are used in a wide variety of applications, ranging from aerospace actuators to auto­ motive and appliance use. The combination of a high ratio in compact arrangements, low cost when mass-produced, and good load-carrying capacity makes the Spiroid-type gear attractive in many situa­ tions. The fact that the gearing can be made with lower-cost machine tools and manufacturing processes is also an important consideration.

BIBLIOGRAPHY Radzevich, S.P. (2014). Generation of surfaces: Kinematic geometry of surface machining, 2nd Edition. Boca Raton, Florida: CRC Press. Radzevich, S.P. (2017). Gear cutting tools: Science and engineering, 2nd Edition. Boca Raton, FL: CRC Press. Radzevich, S.P. (2018). Theory of gearing: Kinematics, geometry, and synthesis, 2nd Edition, revised and expanded. Boca Raton, FL: CRC Press. Radzevich, S.P. (2019). Geometry of surfaces: A practical guide for mechanical engineers, 2nd Edition. Springer International Publishing. [ISBN-10: 3030221830, ISBN-13: 978-3030221836]. Radzevich, S.P. (2020). High-conformal gearing: Kinematics and geometry, 2nd edition, revised and extended, Amsterdam: Elsevier, 2020. (1st edition. Boca Raton, Florida: CRC Press. ISBN 9781138749528).

3

Gear Types and Nomenclature Stephen P. Radzevich

Before we can discuss the different types of gears, we would do well to define just what we mean by a “gear.” Perhaps the most sufficient definition ever heard was provided by a loading dock worker who asked where he should deliver the “wheels with notches in them.” In more technically correct terms, a gear is a toothed wheel that is usually, but not necessarily, round. The teeth may have any of an almost infinite variety of profiles. The purpose of gearing is to transmit motion and/or power from one shaft to anther. This motion transfer may or may not be uniform, and may also be accompanied by changes in direction, speed, and shaft torque. Ultimately, even among gear specialists there is some ambiguity in the terms used to describe gears and gear-related parameters. In our treatment herein, to the extent possible, we follow the conventions recommended by the American Gear Manufacturers Association.

3.1

TYPES OF GEARS

The first part of this chapter covers the types of gears in common use. The basic nomenclature and formulas for each type of gear are covered in the second part of this chapter. The text material for each type of gear gives some comments on where the gears are used, how the gears may be made, the possible efficiency, and occasional comments on lubrication of the gears. These comments should all be considered introductory. Much more specific information is given in the fol­ lowing chapters of this book. Also, the references and bibliography given in this section of the book, show the reader where more comprehensive data can be obtained.

3.1.1 CLASSIFICATIONS In general, gears may be divided into several broad classifications based on the arrangement of the axes of the gear pair. The most general type of gearing consists of a gear pair whose axes are neither per­ pendicular nor parallel and do not intersect (i.e., they do not lie in the same plane). All other types are special cases of this basic form. In the ensuing sections we discuss most of the major gear types, generally in order of increasing complexity. We limit our discussion to specific gear types and do not include arrangements of these types. Our discussions consider, qualitatively, the various aspects of many different types of gears. In order to provide an easy basis for comparing these many and varied gear types, Table 3.1 provides a broadbased comparison. It is by no means an exhaustive comparison, but it serves to provide a reasonable basis for preliminary design. In using this table, the reader must keep clearly in mind the fact that the data are “typical” or “nominal.” Surely, any experienced gear technologist can point out specific cases that vary substantially in virtually every category. Within each type of gear classification there exist many variations in actual tooth form. The involute or involute-based form is, at least for parallel-axes gears, the most common tooth form; however, many other special forms exist. We do not discuss tooth forms, per se, in this section.

65

1/3 cone distance 28% of cone distance 1/3 cone

97–99.5

97–99.5

97–99.5

95–99.5

95–99.5

50–95

50–90

Spiral bevel gear

Face gear

Beveloid

Crossed helical

Cylindrical worm

F1 = 4pn sin 1 F2 = 4pn sin 2

From 0.2 d at low ratio to d at a high ratio 5/ pn

F = 2d

97–99.5

Radial and thrust Radial and thrust

Radial and thrust Radial and thrust Radial and thrust Radial and thrust

Radial

Directed by mating gear Directed by mating gear

97–99.5

97–99.5

Radial and thrust Radial and thrust Radial

F=d

97–99.5

External helical gears Internal helical gears Internal herringbone or double helical External herringbone or double helical Intersectingaxes Straight bevel gear Zerol bevel gear

Radial

Type of Load Imposed on Support Bearings

Radial

F=d

Max*. Width Generally Used**

Directed by mating gear

97–99.5

97–99.5

Approx. Range of Efficiency

Internal spur gears

Parallel-axes External spur gears

Type of Gear

TABLE 3.1 General Comparison of Gear Types

3:1 – 100:1

1:1 – 100:1

1:1 – 8:1

3:1 – 8:1

1:1 – 8:1

1:1 – 8:1

1:1 – 8:1

1:1 – 20:1

2:1 – 20:1

5:1 – 10:1

1:1 – 10:1

1:5 – 7:1

1:1 – 5:1

Nominal range of reduction ratio***

10,000

10,000

5,000

5,000

25,000

10,000

10,000

40,000

20,000

20,000

40,000

20,000

20,000

Nominal Max. Pitch Line Velocity, Fpm ( 5.08 10 3m/s ) Aircraft and High Precision

5,000

4,000

4,000

4,000

4,000

1,000

1,000

4,000

4,000

4,000

4,000

4,000

4,000

Commercial

Pinion

Both

Gear Both

Pinion

Both Gear

Both Gear

Both

Both

Both

Both

Member of Pair

Mill, hob

Hob, shape, mill

Generating forming Generating forming Generating forming Same as external spur gear Shape Hob generating

Hob, shape, mill

Shape

Shape, mill

Hob, shape, mill, broach, stamp, sinter Shape, mill, broach, stamp, sinter Hob, shape, mill

Methods of Manufacture

(Continued)

Grind

Grind, lap, shave

Lap, grind, shave, hone Shape, lap Lap, grind

Grind, lap

Grind

Grind

Grind, shave, lap, hone

Grind, shave, lap (crossed axis only), hone Grind, shave, lap (crossed axis only), hone Grind, shave, lap, hone Shave, grind, lap, hone Lap, shave, hone

Methods of Refining

66 Dudley’s Handbook of Practical Gear Design and Manufacture

Fp = 0.24 D Fg = 0.14 D F = 1/3 cone distance Fp = 0.21 D Fg = 0.12 D From 0.2 D at low ratio to D at a high ratio 5/ pn

50–97

90–98

50–98

95–99.5

50–95

High-reduction hypoid Spiroid

Planoid

Helicon

Face gear

Beveloid Radial and thrust

Radial and thrust 1:1 – 100:1

3:1 – 8:1

3:1 – 100:1

1.5:1 – 10:1

9:1 – 100:1

10:1 – 50:1

1:1 – 10:1

3:1 – 100:1

Nominal range of reduction ratio***

10,000

10,000

10,000

10,000

10,000

10,000

10,000

10,000

Nominal Max. Pitch Line Velocity, Fpm ( 5.08 10 3m/s) Aircraft and High Precision

4,000

4,000

6,000

4,000

6,000

4,000

4,000

4,000

Commercial

Gear Both

Pinion Gear Pinion Gear Pinion Gear Pinion Gear Pinion

Both

Worm Wheel

Gear

Member of Pair

* Face width given is only a “nominal” maximum. Consult other parts of the Gear Handbook for detail limitations on face width. ** d= pinion pitch diameter D= gear pitch diameter = lead angle = helix angle pn = normal circular pitch *** The gear types showing an upper ratio of 100:1 can be built to 300:1 or higher. The ratio of 10:1 is shown as a normal maximum limit

50–90

Radial and thrust Radial and thrust

Radial and thrust Radial and thrust Radial and thrust

Fg = 1/3 cone distance Fg = 0.15 D

90–98

50–98

Radial and thrust Radial and thrust

Fw = 5pn cos Fg = 0.67 d Fw = 0.9 D Fg = 0.9 d

Doubleenveloping worm Hypoid

Type of Load Imposed on Support Bearings

Approx. Range of Efficiency

Type of Gear

Max*. Width Generally Used**

TABLE 3.1 (Continued) General Comparison of Gear Types

Generating Forming Generating Forming Mill, hob Hob, mold Hob Mill, broach Mill, hob Hob, mold Same as external spur Shape Generating, hob

Shape, hob Hob, mill

Hob

Methods of Manufacture

Same as external spur Lap Grind, lap

Lap, grind Grind Grind, chase

Grind, lap Grind, lap Grind, chase

Grind, lap

Lap, grind

Lap

Methods of Refining

Gear Types and Nomenclature 67

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Dudley’s Handbook of Practical Gear Design and Manufacture

3.1.2 PARALLEL-AXES GEARS The simplest types of gears are those that connect parallel shafts. They are generally relatively easy to manufacture and are capable of transmitting large amounts of power with high efficiency. Parallel-axes gears transmit power with greater efficiency than any other type or form of gearing. Spur gears. The spur gear has teeth on the outside of a cylinder and the teeth are parallel to the axis of the cylinder. This simple type of gear is the most common type. Its volume of usage is the largest of all types. The shape of the tooth is that of an involute form. There are, however, some notable exceptions. Precision mechanical clocks very often use cycloidal teeth since they have lower separating loads and generally operate more smoothly than involute gears and have fewer tendencies to bind. The cycloidal form is not used for power gearing because such gears are difficult to manufacture, sensitive to small changes in center distance, and not as strong or as durable as their involute brothers. Figure 3.1 shows a close-up view of the teeth on a set of spur gears having about a 5:1 ratio. The teeth are 20 involute tooth form. The pinion is made with more than a standard addendum and the gear addendum is shorter than standard. The whole depth is standard for high-strength gears. Note the large radius of curvature in the root fillet region. This reduces the bending stress due to a lower stress con­ centration factor. Also note that the pinion teeth look very sturdy. This is an effect of the long- and shortaddendum design just mentioned. (Details of specific design strategy are covered in later chapters of this book.) In a case of non-reversible gear trains, as well as in cases of significant difference in loading of gearsets when rotating in different directions, spur gears can be designed so as to have an asymmetric tooth profile. Such a gear design is illustrated in Figure 3.2 [1]. The involute tooth profile of a gear can be truncated so as to remain only single point of the tooth profile—the rest of the tooth profile that does not interact with the tooth profile of a mating gear can be, in certain sense, of arbitrary geometry. In such a scenario, the remaining point of the tooth profile is referred to as the “involute tooth point.” This is a way in which high-conformal gearing (Novikov gearing) can be designed. See Figure 3.3. where a pair of high-conformal gears is depicted [2], [3], [4]. Important notice: It is a widely spread mistake to refer to Novikov gearing (as well as to highconformal gearing) as to “Wildhaber-Novikov gearing,” or just to “W-N gearing,” for simplicity. Novikov gearing meets all three fundamental laws of gearing (see Chapter 1), and, thus, a uniform

FIGURE 3.1 Sixteen-tooth spur pinion driving a gear with about 75 teeth. This is a special design of 20 involute teeth for high load-carrying capacity.

Gear Types and Nomenclature

69

FIGURE 3.2 Spur gear with asymmetric tooth profile (Adopted from: Kapelevich, A.L. ]2018]. Asymmetric gearing. Boca Raton, FL: CRC Press).

FIGURE 3.3

High-conformal gearing (Novikov gearing).

rotation can be smoothly transmitted from a driving shaft to a driven shaft by gearing of this design. In “Wildhaber-Novikov gearing” (or in “W-N gearing”) the second and the third fundamental laws of gearing are voilated. Therefore, gearing of this design is not capable of transmitting a steady rotation smoothly. Only poorly experienced and poorly educated gear engineers use the incorrect terms “Wildhaber-Novikov gearing” and “W-N gearing.” The most common pressure angles used for spur gears are 14.5°, 20°, and 25°. In general, the 14.5 pressure angle is not used for new designs (and has, in fact, been withdrawn as an AGMA standard tooth form); however, it is used for special designs and for some replacement gears. Lower pressure angles have the advantage of smoother and quieter tooth action because of the large profile contact ratio. In addition, lower loads are imposed on the support bearings because of a decreased radial load component;

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Dudley’s Handbook of Practical Gear Design and Manufacture

however, the tangential load component remains unchanged with pressure angle. The problem of un­ dercutting associated with small numbers of pinion teeth is more severe with the lower pressure angle. Lower pressure angle gears also have lower bending strength and surface durability ratings and operate with higher sliding velocities (which contribute to their relatively poor scoring and wear performance characteristics) than their higher pressure angle counterparts. Higher pressure angles have the advantage of better load-carrying capacity, with respect to both strength and durability, and lower sliding velocities (thus, better scoring and wear performance char­ acteristics). In some cases, very high pressure angles, 28 , 30 , and, in a few cases, as high as 45 , are employed in some special slow-speed gears for very high load capacity where noise is not the pre­ dominant consideration. Minimal equipment is required to produce this type of gear; thus, it is usually the least expensive of all forms of gearing. While the most common tooth form for spur gears is the involute, other tooth forms are possible as long as they provide conjugate motion. Helical gears. When gear teeth are cut on a spiral that wraps around a cylinder, they are called helical. Helical teeth enter the meshing zone progressively and, therefore, have a smoother action than spur gear teeth and tend to be quieter. In addition, the load transmitted may be somewhat larger, or the life of the gears may be greater for the same loading, than with an equivalent pair of spur gears. Conversely, in some cases smaller-size helical gears (compared with spur gears) may be used to transmit the same loading. Helical gears produce an end thrust along the axis of the shafts in addition to separating and tangential (driving) loads of spur gears. Where suitable means can be provided to take this thrust, such as thrust collars, or ball, or tapered roller bearings, it is no great disadvantage. The efficiency of a helical gear set, which is dependent on the total normal tooth load (as well as the sliding velocity and friction coefficient, and so forth), will usually be slightly lower than for an equivalent spur gear set. Conceptually, helical gears may be thought of as stepped spur gears in which the size of the step becomes infinitely smaller. For external parallel-axes helical gears to mesh, they must have the same helix angle but be of different hand. Just the opposite is true for an internal helical mesh; that is, the external pinion and the internal gear must have the same hand of helix. For crossed-axes helical gears (discussed later in this section), the helix angles on the pinion and gear may or may not be of the same hand, depending on the direction of rotation desired and the magnitude of their individual helix angles. Involute profiles are usually employed for helical gears and the same comments made earlier about spur gears hold for helical gears. In order to provide significant improvement in noise over spur gears, the face overlap must be at least unity. That is, one end of the gear face must be advanced at least one circular pitch from the other. If the face overlap (also called face contact ratio) is less than unity, the helical gear, for most analytical purposes (i.e., stress calculations) is treated as if it were a spur gear with no advantage taken of the fact that it is helical. Helix angles from only a few degrees up to about 45 are practical. As the helix angle increases from zero, in general, the noise level is reduced and the load capacity is increased. At angles much above 15 to 20 , however, the tooth bending capacity generally begins to drop off. This is due to the fact that the transverse tooth thickness decreases rapidly. Double-helical or herringbone gears are frequently used to obtain the noise benefits of single-helical gears without the disadvantage of thrust loading. The common types of helical gears are shown in Figure 3.4. The terms double-helical and herringbone are sometimes used interchangeably. Actually, herringbone is more correctly used to describe double-helical teeth that are cut continuously into a solid blank, as shown in Figure 3.4,c. Double-helical gears may be cut integral with the blank, either inline or staggered, or two separate parts may be assembled to form the double. While double-helical or herringbone gears do eliminate the net thrust load on the shaft, it is important to note that the two halves of the gear must internally react to full thrust load. This being the case, the design of the gear blank for one-piece gears, or the construction of the assembly, must be carefully

Gear Types and Nomenclature

(a)

Center distance

71

(b)

Center distance

(c)

Conventional

FIGURE 3.4

Staggered

Continuous (herringbone)

Types of helical gear teeth. (a) Single helical; (b) Double-helical; and (c) Types of double-helical.

evaluated to ensure that it is strong enough to take the thrust loads and to provide enough rigidity so that the deflections are not excessive. The circular-arc gear teeth in the length-wise direction (see Figure 3.5) is the next step of evolution of helical gears. The efficiency of helical gears is quite good though not quite equal to that of simple spurs. This is due to the fact that all other things being equal, the normal (or total) tooth load on a helical gear is higher than that or a spur for an equivalent tangential load. Internal gears. The internal gear has teeth on the inside of a cylinder. The teeth may be made either spur or helical. The teeth of an involute form internal gear have a concave shape rather than a convex shape. Internal gears are generally more efficient since the sliding velocity along the profile is lower than for an equivalent external set. Because of the concave nature of the internal tooth profile, its base is thicker than an equivalent external gear tooth (either spur or helical). The tooth strength of an internal gear is greater than that of an equivalent external gear. The internal gear has other advantages. It operates at a closer center distance with its mating pinion than do external gears of the same size. This permits a more compact design. The internal gear eliminates the use of an idler gear when it is necessary to have two parallel shafts rotate in the same direction. The internal gear forms its own guard over the meshing gear teeth. This is highly desirable for preventing accidents in some kinds of machines. Internal gears cannot be used where the number of teeth in the pinion is almost the same as that of the gear. When this occurs, the tips of the pinion teeth interfere with the tips of the gear teeth. While a good guide is to maintain a ratio of 2:1 between the number of teeth on the internal gear and its mating pinion, lower ratios may be practical in some cases, particularly if the tooth utilizes shifted involute design characteristics. For full-depth 20 pressure-angle teeth, for example, the internal gear pitch diameter must be at least one and one-half times that of the external gear; smaller ratios require considerable

72

FIGURE 3.5

Dudley’s Handbook of Practical Gear Design and Manufacture

Parallel-axes gearset with circular-arc teeth in the lengthwise direction.

modification of the tooth shape to prevent interference as the teeth enter and leave the mating gear. Internal gears also have the disadvantage that few machine tools can produce them. It may be more difficult to support a pinion in mesh with an internal gear because the pinion must usually be overhung mounted. Internal gears tend to find their greatest application in various forms of epicyclic gear systems and in large slewing-type drives, generally with integral bearings. In such con­ figurations, the internal gear is frequently much larger than its mating external pinion, and thus, it is usually possible to build a carrier structure inside the internal gear so that the pinion can be straddle mounted. A typical internal gear arrangement is shown in Figure 3.6, while Figure 3.7 shows a large internal spur gear. In the case of an internal helical gear, its hand of helix must be the same as its mating pinion; that is, both the external helical pinion and its mating helical internal gear must be either right- or left-handed.

Width of face

Cutter clearance

Center distance

FIGURE 3.6

Typical Internal-Gear Arrangement.

Gear Types and Nomenclature

FIGURE 3.7

73

A Large Internal Spur Gear.

3.1.3 NONPARALLEL, COPLANAR GEARS (INTERSECTING-AXES) While the involute is the tooth form of almost universal choice for parallel-axes gears, most gears that operate on nonparallel, coplanar axes do not employ involute profiles. A wide variety of gear types fall into this category. The most obvious use for gears of this type is the reduction of power flow around a corner, such as might be required, for example, when connecting a horizontally mounted turbine engine to the vertically mounted rotor shaft on a helicopter. In fact, while power flow is a frequent reason for choosing one of these gear types, other features unique to a particular type of gear, such as the ability of straight bevel gears to be used in a differential arrangement for a rear-wheel-drive automobile, might well be the determining factor in making the decision to use one of these gear types. Description of the major, and some interesting minor, ones follow. Bevel gears. There are four basic types of bevel gears: straight, Zerol1, spiral, and skew tooth. In addition, there are three different manufacturing methods (face milling, face hobbing, and tapered hobbing) that are employed to produce true curved tooth bevel gears, each of which produces different tooth geometry. In our discussion herein we address the overall generic attributes of the different types of bevel gears without addressing the manufacturing variations. All bevel gears impose both thrust and radial loads in addition to the transmitted tangential forces on their support bearings. The thrust is a result of the tapered nature of the bear blank regardless of the tooth form employed (i.e., even straight bevel gears produce an axial thrust). The diametral pitch for all bevel gears is conventionally measured at the heel of the tooth, while the pressure and spiral angles are conventionally measured at the mean section (tooth mid-face). In all cases, the pitch apex of the pinion and that of its mating gear must intersect at the point at which the axes of the pinion and gear shafts intersect. Figure 3.8 shows the basic bevel gear configuration that is common to all types of bevel gears.

1

Zerol is a registered trademark of the Gleason Works, Rochester, New York.

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Dudley’s Handbook of Practical Gear Design and Manufacture

FIGURE 3.8 Basic Bevel-Gear Configuration. Teeth May Be Tapered in Depth or They May Be Made with a Parallel Depth, Depending on Manufacturing System Used.

By convention, the direction of rotation of a bevel gear is specified as clockwise or counterclockwise as viewed from the heel end of the gear looking toward pitch apex. Bevel gears: Straight. The simplest form of the bevel gear. Except for the thrust load produced by the blank taper, straight bevel gear teeth are a three-dimensional analogy to spur gear teeth. One very common use for straight bevel gears is in a bevel gear differential. Extensions of the straight teeth intersect at the axis of the gear. The tooth profile in a section normal to the tooth approximates that of an involute spur gear having a number of teeth equal to the actual number of teeth in the bevel gear divided by the cosine of the pitch angle (equivalent number of teeth). The teeth are always tapered in thickness and may have either constant or tapering height. The outer or heel part of the tooth is larger than the inner part called the toe. See Figure 3.9. Also, straight bevel gears have instantaneous line contact they are often manufactured to have localized contact to permit more tolerance in mounting. One method of achieving this type of contact is the Coniflex2 system, employed by Gleason Works. Bevel gears: Zerol. An improvement over straight bevel gears in terms of contact conditions, noise level, and power capacity is the Zerol bevel gear (see Figure 3.10). This gear is similar to a straight bevel gear except that the teeth are curved along their axis; however, the mean spiral angle is zero3, thus, the bearing reaction loads (particularly the thrust loads) are the same as for straight bevels. Zerol gears are manufactured with the same type of cutters and on the same machines as spiral bevel gears. This is economically important as it eliminates the need for more than one type of bevel gear cutting equipment. Localized tooth bearing of Zerol gears is accomplished by lengthwise mismatch (face curvature mis­ match) and profile modification. The face width of a Zerol bevel gear is restricted to specific values due to cuter limitations. Zerol bevel gears are very frequently used in turbine engine, helicopter, and other high-speed devices such as accessory drives. Zerol bevel gears may be compared to double-helical or herringbone parallel-axes gear teeth in that they have no more thrust load than their straight counterparts but provide advantages related to the improved contact ratio due to their lengthwise curvature. A pressure angle of 20 is the most common; however, Zerol gears may be manufactured with 14.5 , 22.5 , or 25 as well. Because Zerol gears are not generally used in very-high-load applications, the use of high pressure angles, which tend to reduce contact ratio and thus increase noise, should be carefully considered. Bevel gears: Spiral. The most complex form of bevel gear is the spiral bevel. It is commonly used in applications that require high load capacity at higher operating speeds than are typically possible with

2 3

Coniflex is a registered trademark of the Gleason Works, Rochester, New York. Actually, the term Zerol applies to any “spiral” bevel gear with a spiral angle of 10 or less; but most often Zerol bevels have a zero mean spiral angle.

Gear Types and Nomenclature

FIGURE 3.9

Straight-Bevel Gears (Courtesy of the Gleason Works, Rochester, N.Y.).

FIGURE 3.10

Zerol-Bevel Gears. (Courtesy of the Gleason Works, Rochester, N.Y.).

75

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Dudley’s Handbook of Practical Gear Design and Manufacture

FIGURE 3.11

Spiral-Bevel Gears. (Courtesy of the Gleason Works, Rochester, N.Y.).

either straight or Zerol bevel gears. The relationship between spiral and straight bevel gears is com­ parable to that between helical and spur gears. By convention, the hand of spiral is defined as either leftor right-hand as determined by viewing the gear from the apex end looking toward the heel end. The teeth of spiral bevel gears are curved and oblique (see Figure 3.11), and as a result they have a considerable amount of overlap. This means more than one tooth in contact at all times and results in gradual engagement and disengagement with continuous multiple-tooth contact. This results in higher overlap contact ratios than are possible with either straight or Zerol gears. Because of this improved contact ratio and the resultant load sharing, spiral bevel gears have better load carrying capacity and run more smoothly and quietly than either straight or Zerol bevel gears of the same size. The improved load capacity of spiral bevel gears also allows them to be smaller in size for a given load capacity than an equivalent straight or Zerol bevel gear. Spiral bevel gears impose significantly more thrust load on their support bearings than do either Zerol or straight bevel gears. While plain thrust bearings have been used successfully, a rolling element thrust bearing is usually required with spiral bevel gears. The more common pressure angles for spiral bevel gears are 16 and 20 , with the latter now being almost standard. The most common spiral angle in use is 35 . In more highly loaded applications, particularly those that operate at high speeds as well, such as aircraft or helicopter transmissions, a higher pressure angle (typically 20.5 to 25 ) and lower spiral angles (typically 15 to 25 ) are more often used. The profile contact ratio on all bevel gears is usually substantially lower than equivalent spur gears; thus, they tend to be somewhat noisier. In addition, due to their higher total load for a given tangential load and the higher sliding that accompanies the localized contact, they are less efficient than either spur or helical gears. Still, the efficiency of a well-made set of bevel gears is generally better than 98%, and that of hardened and ground sets is usually above 99.2%. Spiral bevel gears can be made by any of three generic methods. Each of these three methods pro­ duces a unique lengthwise tooth curvature and profile. In general, while gears within one system may be interchangeable, and gear sets made by each system are set-wise interchangeable, pinions and gears are not interchangeable between systems, despite the fact that the basic configuration (i.e., diametral pitch, spiral angle, and pressure angle) may be identical. 1. Spiral bevel gears: Circular lengthwise tooth curvature: For this system, developed in the United States, face-milled cutters with multiple blades are used. While several methods of implementing this process exist, they are all somewhat similar in their basic motions. The blank executes a rolling motion relative to the cutter, which simulates one tooth of the imaginary crown gear. This

Gear Types and Nomenclature

77

is repeated for every tooth space, the cutter being withdrawn and turned back into starting position every time while the job is indexed for the next gap. This process is known as single indexing. All tooth spaces have first to undergo a roughing operation and are finished by a subsequent operation. In general, the root lines of the teeth are not parallel to the pitch line, but set at a specific root angle. The cutters must be set parallel to the root lines while producing the gears; that is, they must be tilted relative to the pitch line. This causes the normal pressure angle to change, so that the pressure angles of the cutters must be corrected accordingly. This correction depends on the root angle and on the spiral angle and is different for either tooth side. In operation, the concave tooth surfaces of one member are always in mesh with the convex tooth side of the other. This system is used by Gleason machines and is the only one for which grinding machines have been developed. 2. Spiral bevel gears: Involute lengthwise tooth curvature: This system was originally developed in Germany. The tooth is a tapered hob, usually single-thread, with a constant pitch determined in accordance with the normal diametral pitch. The machining process is by continuous indexing: Tool and hob rotate continuously and uniformly, the relationship between the two rotary motions being established by change gears as a function of the number of teeth being cut. The job is adjusted with its cone surface tangential to the surface of the imaginary flat gear. This system is utilized by some Klingelnberg machines but is rapidly being replaced by the more versatile face-milling and face-hobbing processes. 3. Spiral bevel gears: Epicycloidal lengthwise tooth curvature: This system, originally evolved in Italy and England but still not free from some shortcomings, has been further developed in Switzerland and Germany. The machines available for producing this type of spiral bevel gear, are also known as cyclo-palloid tooth form bevel gears, are similar to the one for the production of circular tooth length curves in that the tool is also a face-type milling cutter with inserted blades, which can be of the inside-cutting, outside-cutting, and gashing type, alternatively. The indexing is now, however, not intermittent but continuous. The cutter and the blank rotate continuously and uniformly, so that not only is one tooth space generated at a time, but all tooth spaces con­ secutively. Accordingly, the blades are not evenly distributed over the periphery of the cutter, but combined in groups (that is, the cutter has multiple starts). One group generally comprises one roughing, one outside-cutting, and one inside-cutting blade, and allows simultaneous machining of the tooth bottom and both tooth sides. The number of blade groups depends on size of the cutter and usually varies from 3 to 11. This basic system is utilized by both Oerlikon and Klingelnberg, with some variation on each. Bevel gears: Skew. The final type of bevel gear is the skew tooth. This is similar to a spiral bevel gear and is quite often also referred to (incorrectly) as a spiral bevel gear. Actually, the skew tooth has no lengthwise curvature; rather the teeth are simply cut straight but at an angle to the shaft centerline (as shown in Figure 3.12) so that some face overlap may occur. This provides an improvement in load capacity (see Figure 3.13) when compared with a straight bevel gear, but not to the level possible with a true spiral (curved tooth) bevel gear. Skew tooth gears are used primarily in large [over 30-in. (762 mm) gear pitch diameter] sizes only. They are produced on planing generator machines. Until the late 1980s, because of limitations in the size of the face-milling machines available, most “spiral” bevel gears over 30 to 40 in. (762 mm to 1016 mm) in diameter were planed; thus, they were actually skew tooth, not true spiral bevel gears. Since pinion and gear must be conjugate, the pinions that mate with these large gears must also be planed. Relatively recent advances in spiral bevel gear man­ ufacturing have led to the development of large machines of both the face-milling and face-hobbing type that can cut true spiral bevel gear teeth on gears over 100 in. (2540 mm) in pitch diameter.

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Dudley’s Handbook of Practical Gear Design and Manufacture

FIGURE 3.12

Skew Bevel Gearset.

Skew tooth

Spiral bevel

Straight bevel

The gear pitch diameter, inches.

FIGURE 3.13 Comparison of Large-Straight Skew and SpiralBevel-Gear Capacities (Data supplied by Philadelphia Gear Corp., King of Prussia, PA).

Face gears (On-Center). A face gear set is actually composed of a spur or helical pinion that is in mesh with a “face” gear. The pinions are really not any different from their parallel-axes counterparts except for the fact that they are in mesh with a face gear. Face gears have teeth cut into the blank such that the axis of the teeth lies in a plane that is perpendicular to the shaft axis. The mating pinion is either a spur or a helical gear. The pinion and gear axes are coplanar for on-center face gears. The pinion and face gear most often form a 90 shaft angle; however, small deviations from 90 are possible. In operation, this type of gear is similar to an equivalent set of straight bevel gears.

Gear Types and Nomenclature

79

Beveloid angles

FIGURE 3.14

Beveloid-Gear Arrangement.

Face gear tooth changes shape form one end of the tooth to the other. The load capacity of face gears, compared with that of bevel gears, is rather small; thus, they are used mostly for motion transmission rather than as power gears. Face gears are, relative to bevel gears, easy to make and somewhat less expensive as well. A typical arrangement is shown in Figure 3.12 (see Chapter 2). Toys and games make extensive use of face gears. Conical involute gearing. Conical involute gearing is a completely generalized form of involute gear. It is an involute gear (as shown in Figure 3.14) with tapered tooth thickness, root, and outside diameter. Commonly known as Beveloid gears, they are useful primarily for precision instrument drives where the combination of high precision and limited load-carrying ability fits the application. Like other involute gears, Beveloid gears are relatively insensitive to positional errors although shaft angles must be quite accurate. They will mate conjugately with all other involute gears (spur, worms, helical gears, and racks). Like crossed-axes helical gears (see later discussion), however, the area of contact these gears and Beveloid gears is limited; thus, load capacity is low. For small tooth numbers and large cone angles, undercutting is quite severe. This type of gear is not in widespread use at this time, but may be used to advantage in some circumstances, especially for precision motion-transmission applications in which the gear shafts must be located at raying orientations in space.

3.1.4 NONPARALLEL, NON-COPLANAR GEARS (NON-INTERSECTING AXES) Gears in this classification are generally the most complex, both in terms of geometry and manufacture. The simpler types, discussed first in the following discussions, are, however, quite easy to manufacture and are reasonably inexpensive, but they do not carry large loads. The more complex types are generally more expensive but provide better load capacity and other features that make them especially suited for a wide variety of special applications. Crossed-axes helical gears. Crossed helical gears are satisfactory for the normal range of ratios used for single reduction helical gears. They provide both speed reductions and extreme versatility of shaft positioning at a relatively low initial cost. At higher ratios or for anything above moderate loads, however, worm, Spiroid, or Helicon gears are generally preferable. [See Figure 3.13 (see Chapter 2) for typical crossed helical gears.] Crossed-axes helical gears are usually cut in the same manner as conventional helical gears using identical tooling, since they are, until mounted on their crossed axes, actually nothing more than

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Dudley’s Handbook of Practical Gear Design and Manufacture

Center distance

Gear axis 1

Pinion axis 2

FIGURE 3.15

Crossed-Axes-Helical-Gear Arrangement.

conventional helical gears! Since crossed helical gears have a great deal of sliding action, special at­ tention must be given to the selection of gear materials and their lubricants to reduce friction to a minimum and eliminate any possibility of seizing between mating gears. Experience has indicated that iron functions satisfactorily. However, a hardened steel pinion driving a bronze gear will also work quite well. Figure 3.15 shows some of special relations that must exist for this type of gearing to function properly. It is interesting to note that, while mating parallel-axes external helical gears must have op­ posite hands of helix, crossed-axes helical gears can have either the same or opposite hands of helix, depending on the shaft angle and the relative directions of rotation of the driver and driven gears. Additionally, the reduction ratio between the pinion and the gear is a function of their relative numbers of teeth but not directly of their pitch diameters. This provides a great deal of flexibility in choosing the ratio, center distance, and diameters of the gears. Finally, because the relative contact between the mating gears is the theoretical intersection of two cylinders, the load capacity of the gears is not directly influenced by the face width; that is, once the face width is extended such that it covers the full cylindrical intersection, extending either or both face widths farther is of no practical value since the extra face width will not be in contact. Cylindrical worm gearing. The most basic form of worm gearing is a straight cylindrical worm in mesh with a simple helical gear. In reality, this is really an extreme case of crossed-axes helical gearing. Such gears can provide considerably higher reduction ratios than simple crossed-axes gear sets, but their load capacity is low, and the wear rate is high. For light loads, however, this configuration can be an economical alternative. Single-enveloping worm gearing. Better load capacity can be achieved if the simple helical gear is modified such that it is throated to allow the worm to fit down farther into the gear to achieve greater tooth contact area and thus, smoother operation and improved load capacity. This common worm gear set is often referred to as single-enveloping since the gear envelopes the worm but the worm remains straight. While many variations of this configuration are used, in the most common case the worm is essentially straight-sided while the gear or wheel is generally involute form. The contact point is the­ oretically a line varying in length up to full face width of the gear with different tooth designs. Under load, this line becomes a thin elliptical band of contact. See Figure 3.16 for a two-stage worm gear drive. Basic arrangement views are shown in Figure 3.17. The hand of helix for both members is the same. While the worm is generally made of steel, the wheel is usually made of either cast iron or, more frequently, one of several types of bronze. The worm teeth may be through hardened, but they are often, especially in higher-load and higher-speed applications, case hardened and ground. In some cases, however, to improve accuracy and efficiency, the worm teeth are ground even though they are not

FIGURE 3.16 Single Stage Cylindrical-Worm Gearset (Courtesy of Hamilton Gear and Machine Co., Ltd., Hamilton, Canada).

Gear Types and Nomenclature 81

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Dudley’s Handbook of Practical Gear Design and Manufacture

Linear pitch

Circular pitch FIGURE 3.17

Schematic of Cylindrical-Worm Gears.

hardened after cutting. The wheel, due to the high sliding conditions that exist in this type of gearing, is usually made of cast iron or bronze, as noted in the foregoing; however, in some high-temperature environments it is necessary to make the wheel from steel as well. By virtue of their inherent high contact ratio, the mechanical power rating of worm gears is quite high; however, in practice, their actual continuous-duty rating is substantially lower. This is due to high heat generation that can raise the lubricant temperature to unacceptable levels when the box is operated continuously. Fan-cooled worm boxes are quite common, and higher-power-capacity worm housings are almost always fanned to aid heat dissipation. This large difference between the thermal and mechanical ratings of the typical worm gear drives rise to a peculiar property associated with worm drives: that is, their apparent ability to sustain relatively high short-term overloads without experiencing any damage. In reality, worm gears do not have a particularly good overload capacity; rather, their thermal limitations cause them to be operated at loads below their mechanical limits. When operated for short periods of time at “overload,” they are actually operating above their continuous-duty thermal limits but below their mechanical (stressed-related) ratings; thus, since it takes an appreciable period of time for the tem­ perature to rise, they sustain these short-term “overloads” quite well. The advent of synthetic fluids has been a boon for worm drives of all types for two reasons. First, synthetic fluids have the ability to operate at higher bulk temperatures than the compounded mineralbased fluids that are commonly used for worm gears. Second, the friction coefficient associated with the use of synthetic fluids tends to be somewhat lower than that associated with compounded worm gear oils; thus, less heating is produced. These factors combine to decrease the margin between the thermal and mechanical limits of newer work gear sets designed and rated to run with synthetic fluids; thus, their apparent overload capability is reduced. Worm gear efficiency is quite dependent on operating speed. The same set may show an efficiency of, say, 75% at a low speed and 85% at a higher speed. Ratio, material, accuracy, and geometric design all affect worm gear efficiency. Typical efficiencies run from 35% to 90%, with higher or lower values occurring in special cases.

Gear Types and Nomenclature

83

A worm set can be used where irreversibility is desired, since, if the lead angle is less than the friction angle, the wheel cannot drive the worm. Usually, worms with lead angles less than 5 are self-locking. Care should be exercised when designing self-locking worms, since this feature is a static one. Vibration can cause the set to slip under dynamic conditions (i.e., during a cutoff of power under load, the wheel, due to the inertia of the driving load, may overdrive the worm for a considerable time). Similarly, a worm set that, when stationary, cannot be driven through the gear shaft, may well begin to rotate if the unit is subjected to vibrations. The property of self-locking or irreversibility is better thought of as anti-backdriving rather than positive irreversibility. In critical applications, a brake on the worm should also be provided to ensure that the system will positively not back-drive. When the gears are assembled into their housing, the position of the gear along its axis must be adjusted such that an acceptable contact pattern is obtained on the bench. This contact pattern should favor the leaving side of the mesh so that a wedge is formed at the entering side of the mesh. This wedge at the entering side will cause lubricant to be drawn in the contact zone and thus, minimize wear. Double-enveloping worm gearing. The capacity of a single-enveloping worm gear set as described in the foregoing discussion is improved by allowing the gear or wheel to envelope the worm. A further improvement in capacity may be achieved by allowing the worm to envelope the wheel as well. Such drives are known as double-enveloping. Double-enveloping worm gear drives, by getting more teeth into contact, tend to provide higher load capacity than do cylindrical or single-enveloping worm sets. This is accomplished by changing the shape of the worm (as shown in Figure 3.18) from a cylinder to an hourglass. All of the comments made earlier pertaining to cylindrical worms also apply to double-enveloping worms. Because of the shape of the worm, clearly shown in Figure 3.19, this type of worm gearing is more expensive to produce; but where weight or size are considerations, the cost differential is relatively small.

Circular pitch: Worm

Circular pitch: Gear FIGURE 3.18

Schematic of Double-Enveloping-Worm Gears.

84

FIGURE 3.19

Dudley’s Handbook of Practical Gear Design and Manufacture

Double-Enveloping-Worm-Gear Set.

The forms of the worm and the gear in the double-enveloping drive design are regenerative; that is, the worm and gear tend to reproduce each other in use. This condition aids proper break-in and contact pattern development. Since the worm and wheel envelope each other, assembly is not as straightforward with double-enveloping gear sets as it is with a single-enveloping set. In general, the worm and wheel must be assembled obliquely, and thus, provisions must be made in the design of the housing to ac­ commodate this requirement. This can often be accomplished by splitting the housing such that the worm is one part and the wheel is the other half. In addition, the position of the worm along its axis and the position of the gear along its axis must be adjusted simultaneously such that an acceptable pattern is obtained. This bench patterning procedure is in contrast to the simpler procedure required of a singleenveloping set in which only the position of the gear along its axis need be adjusted at assembly to obtain an acceptable contact pattern at assembly. Hypoid gears. Hypoid gears (see Figure 3.20) resemble spiral bevel gears except that the teeth are asymmetrical; that is, the pressure angle on each side of the tooth is different. Many of the same ma­ chines used to manufacture spiral bevel gears can also be used to manufacture hypoid gears. The pitch surfaces of hypoid gears are hyperboloids of revolution. The teeth in mesh have line contact; however, under load, these lines spread to become elliptical regions of contact inclined across the face width of the teeth. One condition that must exist if a hypoid gear set is to have conjugate action is that the normal pitch of both members must be the same. The number of teeth in a gear and pinion are not, however, directly proportional to the ratio of their pitch diameters. This makes it possible to make large pinions while mini­ mizing the size of the driven gear. This one of the most attractive features of hypoid gearing. High reduction ratios with small offsets must usually utilize an overhung-mounted pinion. Frequently, however, the pinion must be straddle mounted (see Figure 3.21) if sufficient offset exists. Aside from space considerations, the tradeoff usually centers on efficiency, since efficiency generally decreases with increasing offset. In operation, hypoid gears are usually smoother and quieter than spiral bevel gears due to their inherent higher total contact ratio. However, as in all cases of nonintersecting gear sets, high sliding takes place across the face of the teeth. The efficiency of hypoid gears is thus, much less than that or a similar set of spiral bevel gears, typically 90 to 95% as compared with over 99% for many spiral bevel gears. Hypoid gears do, however, generally have greater tolerance to shock loading and can frequently be used at much higher single-stage ratios than spiral bevel gears.

(b)

(2)

(4)

(1)

(3)

FIGURE 3.20 Hypoid-Gear Data. (1) Hypoid Tooth Profile Showing Unequal Pressure Angles and Unequal Profile Curvatures on the Two Sides of the Tooth. (2) Hypoid Gears and Pinions (a) and (b) Are Referenced to Having an Offset below Center, While Those in (c) and (d) Have an Offset above Center. In Determining the Direction of Offset, It Is Customary to Look at the Gear with the Pinion at the Right. For Below-Center Offset, the Pinion Has a Left-Hand Spiral, and for AboveCenter Offset the Pinion Has a Right-Hand Spiral.

(a)

Gear Types and Nomenclature 85

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Dudley’s Handbook of Practical Gear Design and Manufacture

FIGURE 3.21 Hypoid-Gear Set. Both Pinion and Gear Are Straddle Mounted (Courtesy of the Gleason Works, Rochester, N.Y.).

Spiroid and Helicon gearing.4 Like most other skew axis concepts, Spiroid and Helicon are primarily screw action gears while, by comparison, spur, helical, and bevel gears are primarily rolling action gears. The gear member of either a Spiroid (see Figure 3.22), or Helicon (see Figure 3.23), set resembles a high-pitch-angle bevel gear, while the pinions are more akin to worms. Both forms have more teeth in contact than an equivalent size worm set. The primarily difference between Spiroid and Helicon gears is their minimum ratio capabilities [about 0:1 for Spiroid and 4:1 for Helicon (although lower ratios are practical if powdered metal fabrication is employed)] and the range of acceptable offsets. Spiroid pinions (see Figure 3.22) are typically tapered by about 5 or 10 on a side, while Helicon pinions (see Figure 3.23) are cylindrical. Pinions of both types, used in lower ratio designs (i.e., less than 30:1), typically utilize multiple threads so that the number of teeth in the gear may be maintained at least at 30. This type of gearing fits in between bevel and worm gears (see Figure 3.24) in terms of ratio, and generally provides performance than do worms. They can, like worms, be designed to be anti-backdriving, but again like worms, they cannot be designed to entirely self-locking under all operating conditions, particularly where significant vibrations are present. They can also incorporate accurate control of backlash since the contact pattern is largely controlled by adjusting the pinion position, while the backlash is largely controlled by adjusting the position of the gear along its axis at assembly. These features make them a good choice for positioning drives such as radar or another antenna. These gear types are relatively in sensitive to small position shifts; thus, in precision position-control applications, they are usually shimmed into intimate double flank contact to eliminate backlash. The contact conditions are line, again like worm and hypoid gears, but the contact line is essentially radial; thus, the flow of oil onto the contact zone is good, and efficiency is improved over worms—usually about as good, as hypoid gearing. As with worms, the efficiency of Spiroid or Helicon sets varies with speed, but not to the large extent that occurs with worms.

4

Spiroid and Helicon are registered trademarks of Illinois Tool Works, Chicago, Illinois.

Gear Types and Nomenclature

FIGURE 3.22

Spiroid-Gear Set (Courtesy of Illinois Tool Works, Spiroid Division, Chicago, Ill.).

FIGURE 3.23

Helicon-Gear Set (Courtesy of Illinois Tool Works, Spiroid Division, Chicago, Ill.).

87

One feature of Spiroid and Helicon gears that makes them both quiet and capable of great positional accuracy is the relatively high number of teeth in contact. The multiplicity of teeth in contact (for a typical design perhaps 10% of the gear teeth are theoretical contact) provides an error averaging function so that individual tooth errors are not significant in terms of gear position. Although generally less efficient than spur or helical gears, Spiroid or Helicon gears are usually superior to worm drives particularly at ratios less than 40:1, when compared on the basis of constant pinion size. The offset required for a typical Spiroid gear set is about one-third the gear diameter, but small variations can also be accommodated. Similarly, the shaft angle is usually limited to 80 to 100 , with 90 being the standard in the vast majority of cases. In some special applications, shaft angle as low as 70 or as high as 120 are possible, but each case is a special design.

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Dudley’s Handbook of Practical Gear Design and Manufacture

Worm

Spiroid

Hypoid

Spiral bevel FIGURE 3.24 Bevel Gears.

Schematic Comparison of Worm Gears, Spiroid Gears, Helicon Gears, Hypoid Gears, and Spiral-

In general, the gears may be cut on standard hobbing machines. For a given configuration, a Spiroid gear set will have a greater load capacity than a Helicon set, but the Helicon position is some what easier to manufacture due to the cylindrical nature of its pinion as compared with the tapered shape of the Spiroid pinion. Face gears (off-center). If the pinion of a face gear set is offset such that its axis and that of the gear do not intersect, it is termed off-center. The discussion provided earlier for on-center face gears applies equally well here. Such gear sets are not ordinarily used to transmit significant amounts of power; rather they are generally used in motion-transmission applications where uniformity of motion is not a critical factor.

3.1.5 NONCONJUGATE GEARS Along with gears that feature conjugate tooth flanks of the interacting teeth gears having non-conjugate tooth flanks are also used in the nowadays industry. Rotors of oil pump. An example of gearing of such a design is illustrated in Figure 3.25, where rotors in the design of an oil pump are not conjugate to one another. Because of this, an additional conjugate gear pair is required to drive the rotors. Violation of the second and the third fundamental laws of gearing (see Chapter 1) is the root cause for this violation. Roller worm gearset. A design of worm gearset with rollers makes possible a significant reduction of friction between the interacting surfaces of the worm and of the worm gear (see Figure 3.26). However, no smooth transmission of an input rotary motion is possible by means of worm gearing of this particular design, as the worm threads and the rollers are not conjugate to one another. Violation of the second and the third fundamental laws of gearing (see Chapter 1) is the root cause for this violation.

3.1.6 SPECIAL GEAR TYPES Our discussion to this point would lead the reader to believe that all gears are in the form of surfaces of revolution (i.e., cylinders and cones). Such is not the case. There are many applications in which gears

Gear Types and Nomenclature

FIGURE 3.25

Non-Conjugate Profiles in the Design of Rotors in the Design of an Oil Pump.

FIGURE 3.26

Roller Worm Gearset.

89

are used to transmit motions that are intentionally non-uniform. There are few areas in which man’s mechanical ingenuity can be employed more effectively than in the design of special-motion gears. The variety is almost endless, limited only by application and imagination. In order to provide some insight into the subject, we discuss some representative cases. Square or rectangular gears. Consider the case in which a constant input speed is to be converted into varying output speed. Gears that are square or rectangular, depending on actual output required, can be used to produce such motion (as shown in Figure 3.27). While the gears are in the position shown, the greatest radius of the driver mates with the smallest radius of the driven gear; the speed of the driven gear is, therefore, then in its maximum. As the gears revolve in the directions indicated by the arrows, the

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Dudley’s Handbook of Practical Gear Design and Manufacture c

c′

b b′ a a′

Driver

Driven One driver revolution

FIGURE 3.27

Square-Gear Characteristics.

radius of the driver gradually decreases, and that of the driven gear gradually increases, until the points b and b are in contact. The speed of the driven gear, therefore, gradually decreases during this eighth of a revolution. From the moment of contact between the points b and b , the reverse action takes place, and the speed of the driven gear gradually increases until the points c and c are in contact. Thus, during the entire revolution, the driven gear continues to alternate from a gradually decreasing to a gradually increasing speed, and vice versa. For rectangular gears to work properly together, it is necessary, first, that the pitch peripheries of the two gears be equal in length, and, second, that the sum of the radii of each pair of points (points that come into contact with each other) on the two pitch peripheries be equal to the distance between the centers of the gears. Triangular gears. Figure 3.28 represents a pair of triangular gears, the object of which is to obtain an alternating, varying speed from the uniformly rotating driver, as in rectangular gears. Triangular gears give fewer changers of speed per revolution than do rectangular gears. In Figure 3.28, the speed of the driven gear is at its minimum when the gears are in the position shown in the figure. Since, from the positions, the radius of the driver gradually increases, and that of the driven gear decreases, as far as the points b and b , the speed of the driven gear will gradually increase until the points b and b are in contact—or for one-sixth of an entire revolution. The reverse action will then take place until the points c and c are in contact, and so on. Thus, while in rectangular gears each gradually increasing or decreasing period takes place during one-eighth of revolution, in triangular gears, there are eight alternatively increasing and decreasing periods in one entire revolution of the driven gear, and in triangular gearing there are but six. b b′

c a

Driver

FIGURE 3.28

c′

a′

Driven

Triangular-Gear Characteristics.

One driver revolution

Gear Types and Nomenclature

91 b′

b

c

a

c′ a′

Driver Driven

FIGURE 3.29

One driver revolution

Elliptical-Gear Characteristics.

Elliptical gears. With elliptical gears (see Figure 3.29), we have still another means of obtaining the same result, with the difference that in elliptical gears each period of gradually increasing and decreasing speed takes place during one-fourth of a revolution. In other words, there are but four periods of in­ creasing and decreasing speed during an entire revolution of the driver. Scroll gears. Figure 3.30 shows a pair of scroll gears. From the positions shown in the figure (in which the greatest radius of the driver meshes with the smallest radius of the driven gear), as the gears revolve in the directions indicated by the arrows, the radius of the driver gradually and uniformly decreases, while that of the driven gear gradually and uniformly increases. The speed of the driven gear is, therefore, at its maximum when the gears are in the position shown, and gradually and uniformly decreases during the entire revolution. The moment before the position shown in the figure is reached, the smallest radius of the driver gear with the greatest radius of the driven gear, the speed of the latter is at its minimum and suddenly (as the gears assume the positions in the figure) changes to its maximum. Multiple-sector gears. The mechanism represented in Figure 3.31 is a pair of multiple-sector gears, and the object is to obtain a series of discrete, different uniform speeds. In the figure, as long as the arcs ab and a b are in mesh, the speed of the driven gear is the same. When the arcs cd and c d come into mesh, the speed of the driven gear becomes slower, but remains the same throughout the meshing of these two arcs. Similarly, when the arcs ef and e f come into mesh, the speed of the driven gear becomes

Driver

One driver revolution

FIGURE 3.30

Scroll-Gear Characteristics.

Driven

One driver revolution

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Dudley’s Handbook of Practical Gear Design and Manufacture b

c

f

b′

a

c′

Driver

f ′ a′ d

e

Driven

d′

e′

One driver revolution

FIGURE 3.31

Multiple-Sector-Gear Characteristics.

still slower, but remains uniform during the meshing of these arcs. Thus, during each revolution, the driven gear has three periods of uniform speed, each differing from the others. For sector gears to work properly together, it is necessary that the arcs that mesh together be equal in length (ab = a b , cd = c d , etc.) and that the sum of the arc lengths on one gear be equal to the sum of the arc lengths on the other (ab + cd + ef = a b + c d + e f ). Also, the sum of the radii of each two arcs together must be equal to the distance between the centers of the gears. Sector gears are somewhat difficult to construct because considerable care must be taken to ensure that no two sectors of the driver gear mesh at the same time with the driven gear. To illustrate, suppose that the arcs ab and a b mesh together at the same time as do the arcs ef and e f ; that is, that the last few teeth of ab mesh with the driven gear at the moment when the first few teeth of cf do the same. The driven gear will then strive to drive the driven gear at its maximum and minimum speeds at the same time, an attempt that must obviously result in a fracture. In the figure, the arc cf ceases to mesh with the driven gear at the moment when the arc ab begins to gear. Thus, each arc of the driver must escape engagement just in time for its successor to begin engagement, and yet leave between these events no appreciable interval to disturb the uniformity of motion. In actuality, a discontinuous condition will exist for all sector-type gears at the time when different sectors exchange control of the speed of the driven member. These seemingly instantaneous periods of acceleration and deceleration limit the application of such gears to very low speeds lest the acceleration becomes exceedingly large.

3.2

NOMENCLATURE OF GEARS

In this part of Chapter 3, the technical nomenclature of the various types of gears is given. Table 3.2 shows metric system symbols and English system (Imperial system) symbols. The following tables (due to space limitations) show nomenclature tables in the English system only.

Gear Types and Nomenclature

93

TABLE 3.2 Gear Terms, Symbols, and Units Used in the Calculation of Gear Dimensional Data Term

Metric Symbol

English Units

Symbol

Units

Number of teeth, pinion Number of teeth, gear

z1 z2

– –

NP or n NG or N

– –

Number of threads, worm

–

NW

–

–

Nc

–

Tooth ratio

z1 z u

–

mG

–

Addendum, pinion

ha1

mm

aP

in.

Addendum, gear

ha2

mm

aG

in.

Addendum, chordal

ha

mm

ac

in.

Rise of arc

–

mm

–

in.

Dedendum

hf

mm

b

in.

Working depth

h

mm

hk

in.

Whole depth

h c

mm

ht c

in.

s

mm mm

in. in.

mm

t tP tG tc

mm

B

in.

Number of crown teeth

Clearance Tooth thickness Arc tooth thickness, pinion

mm

in.

Backlash, transverse

s1 s2 s¯ j

Backlash, normal

jn

mm

Bn

in.

Pitch diameter, pinion

dp1

mm

d

in.

Pitch diameter, gear

dp2

mm

D

in.

Pitch diameter, cutter

dp0

mm

dc

in.

Base diameter, pinion

db1

mm

db

in.

Base diameter, gear

db2

mm

Db

in.

Outside diameter, pinion

da1

mm

do

in.

Outside diameter, gear

da2

mm

Do

in.

Inside diameter, face gear

di 2

mm

Di

in.

Root diameter, pinion or worm

df 1

mm

dR

in.

Root diameter, gear

df 2

mm

DR

in.

Form diameter

d

f

mm

df

in.

Limit diameter

dl

mm

dl

in.

dl

mm

in.

a

– mm

dl m

Face width

b

mm

C F

Net face width

b

mm

Fe

in.

m or m t mn

mm

–

–

mm

–

–

Diametral pitch, transverse

–

–

in.−1

Diametral pitch, normal

–

–

Pd or Pt Pn

Circular pitch

p

mm

p

in.

Arc tooth thickness, gear Tooth thickness, chordal

Excess involute allowance Ratio of diameters Center distance

Module, transverse Module, normal

mm

in. in.

– in. in.

in.−1

(Continued)

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Dudley’s Handbook of Practical Gear Design and Manufacture

TABLE 3.2 (Continued) Gear Terms, Symbols, and Units Used in the Calculation of Gear Dimensional Data Term

Metric Symbol

English Units

Symbol

Units

Circular pitch, transverse

pt

mm

pt

in.

Circular pitch, normal

pn

mm

pn

in.

Base pitch

pb

mm

pb

in.

Axial pitch

px

mm

px

in.

Lead (length)

pz

mm

L

in.

Pressure angle

or

deg

or

deg

t

t

Pressure angle, normal

n

deg

n

deg

Pressure angle, axial

x

deg

x

deg

Pressure angle of cutter

0

deg

c

deg

Helix angle

deg

deg

Lead angle

deg

deg

Shaft angle

deg

Roll angle

deg

r

deg r

deg

Pitch angle, pinion

1

deg

deg

Pitch angle, gear

2

deg

deg

Pi

–

–

a

–

mp

–

Zone of action

ga

mm

Z

in.

Edge radius, tool

rao

mm

rT

in.

f

mm

f

in.

Contact ratio

Radius of curvature, root fillet Circular thickness factor

k

–

k

–

Cone distance

R

mm

in.

Outer cone distance

Ra

mm

A Ao

Mean cone distance

Rm Rf

mm

Am

in.

mm

Ai

in.

Inner cone distance

in.

Note: Abbreviations for units: mm = millimeters, in. = inches, deg = degrees.

3.2.1 SPUR GEAR NOMENCLATURE

AND

BASIC FORMULAS

Figure 3.32 shows tooth elements that apply directly to spur gears. The same elements apply to other gear types. Table 3.3 shows spur gear formulas. Note that the value of may need to be used as 3.14159265 when very precise calculations are needed. Also, the determination of tooth thickness has complications of what backlash is needed and effects of long or short addendum. (These matters are covered later in this book.)

3.2.2 HELICAL GEAR NOMENCLATURE

AND

BASIC FORMULAS

Helical gear teeth spiral a “base” cylinder. The angle of this spiral with respect to the axis is the helix angle. The advance of the spiral in going 360 around the base cylinder is a fixed quantity called the lead.

Gear Types and Nomenclature

95

Whole depth

Tooth fillet

Working depth

Bottom land

Top land Chordal addendum Circular thickness

Tooth profile Backlash Clearance

Chordal thickness Base circle thickness

Addendum Dedendum

FIGURE 3.32

Spur Gear Nomenclature.

TABLE 3.3 Spur Gear Formulas To Find

Having

Diametral pitch

Number of teeth and pitch diameter

Diametral pitch

Circular pitch

Pitch diameter

Number of teeth and diametral pitch

Formula

P= P=

N D

3.1416 p

D=

N P

Do = D + 2a

Outside diameter

Pitch diameter and addendum

Root diameter

Outside diameter and whole depth

DR = Do

Root diameter

Pitch diameter and dedendum

Number of teeth Base-circle diameter

Pitch diameter and diametral pitch Pitch diameter and pressure angle

DR = D 2b N=D×P

Circular pitch

Pitch diameter and number of teeth

p=

Circular pitch

Diametral pitch

p=

Center distance

Number of gear teeth and number of pinion teeth and diametral pitch

Approximate thickness of tooth

Diametral pitch

*

Exact value for

is 3.14159265…

2ht

Db = D × cos

C=

3.1416D N 3.1416 P

NG + NP 2P

tt

=

DG + DP 2

1.5708 P

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Dudley’s Handbook of Practical Gear Design and Manufacture

The spur gear tooth is parallel to the axis. This means that a spur gear can be thought of a special case of a helical gear in which the numerical value of the lead is infinity! Figure 3.33 shows the nomenclature of helical gear teeth. View (a) shows a section through the teeth that is perpendicular to the gear axis. This is known as the transverse section. View (b) shows how a “normal” section data can be taken through a tooth perpendicular to a point on the pitch helix. Both the transverse section and the normal section data must be used in helical gear calculations. Table 3.4 shows the formulas needed to calculate date for both of these sections. A rack is a gear part in which the number of teeth in a 360 circle is infinite. Circular gears, of course, become straight when the radius of curvature is infinite. Involute teeth become straight-sided for the rack tooth form. Figure 3.34 shows rack tooth parts for spur, helical, and straight bevel teeth. The helical rack shows very clearly the difference between the normal plane and the front plane (which corresponds to the transverse plane of circular gears).

3.2.3 INTERNAL GEAR NOMENCLATURE

AND

FORMULAS

The internal gear has teeth on the inside of a ring rather than on the outside. An internal gear must mesh with a pinion having external teeth. (Two internal gears cannot mesh with each other.) The nomenclature for the transverse section of an internal gear drive is shown in Figure 3.35. Note that the internal tooth tends to have concave curves rather than convex curves. Also note that the pips of the teeth tend almost to hit each other as they enter and leave the meshing zone. Formulas for internal gear dimensions are given in Table 3.5. Note that some of these formulas are similar to formulas in Table 3.3 except for the minus sign. Internal gears may be either spur or helical. It the teeth are helical, data for the teeth in the normal section may be determined by formulas in Table 3.4.

3.2.4 CROSSED HELICAL GEAR NOMENCLATURE

AND

FORMULAS

Since the axes of crossed helical gears are not parallel to each other and do not intersect, the two meshing gears tend to have different helix angles. (If the numerical amount of each helix angle is the same, then the two gears cannot have opposite helix angles—because this would make the axes parallel instead of crossed.) Figure 3.36 shows standard nomenclature for the crossed-axes situation with helical gears. The basic formulas are given in Table 3.6.

3.2.5 BEVEL GEAR NOMENCLATURE

AND

FORMULAS

Section 3.1.3 gave general information on gear types in bevel gear family of gears. Nomenclature and formulas will now be given for three of these types that are in rather common usage. Straight. Figure 3.37 shows section through of a pair of straight bevel gears in mesh. Note how the teeth are on the outside of cones. Dimensions are measured to a “crown point” that exists in space but may not exist in metal! (Usually, this sharp corner is rounded off in the process of actual manufacture.) The formulas for the straight bevel are given in Table 3.7. By tradition, the pitch diameters are taken at the large end of the tooth. This end also goes by names such as the back face or the heel end. (The other end of the tooth is termed the front face or toe end.) Spiral. The spiral bevel gear has a curved shape (lengthwise). This curved shape is positioned at an angle to a pitch cone element. The angle at the center of the face width (not the large end of the tooth where the pitch diameter is specified). Figure 3.38 shows the spiral angle. The formulas for conventional spiral bevel gears are given in Table 3.8. Spiral bevel gears are often made to pressure angles other than 20 . They can also be made for shaft angles other than 90 . Beyond this, there is more than one system for the design and manufacture of bevel gears. Considerably more bevel gear data can be found out in later chapters of the book.

Gear axis

Base circle

Normal circular pitch

Pitch circle

Normal helix

Pitch helix

Gear axis

Root radius Base radius

Whole depth

Addendum Dedendum

Outside helix angle Pitch helix angle Base helix angle

(d)

Base circle

Pitch circle

Pitch helix

Normal plane

(b)

Normal helix

Normal pressure angle

Gear axis

Base circle

Normal plane

Gear axis

Pitch circle

Normal helix

Perpendicular to gear axis

Transverse pressure angle

FIGURE 3.33 Helical-Gear Nomenclature: (a) Radial Dimensions of a Gear, (b) Pressure Angles of a Gear, (c) Gear Pitches, and (d) Helix Angles of a Gear.

Base pitch

Normal plane

Form radius Pitch radius Outside radius

Transverse circular pitch

(c)

(a)

Gear Types and Nomenclature 97

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Dudley’s Handbook of Practical Gear Design and Manufacture

TABLE 3.4 Helical Gear Formulas To Find

Having

Formula

Normal diametral pitch

Number of teeth, pitch diameter, and helix angle

Pn =

N D cos

Normal diametral pitch

Normal circular pitch

Pn =

3.1416 pn

Normal diametral pitch

Transverse diametral pitch and helix angle

Pn =

P cos

Normal circular pitch

Normal diametral pitch

pn =

3.1416 Pn

Normal circular pitch

Transverse circular pitch

pn = p cos

Pitch diameter

Number of teeth, normal diametral pitch, and helix angle

D=

N Pn cos

Center distance

Pinion and gear pitch diameter

C=

DP + DG 2

Outside diameter

Pitch diameter and addendum

Do = D + 2a

Approximate normal tooth thickness

Normal diametral pitch

tn

1.571 Pn

Transverse tooth thickness

Normal tooth thickness and helix angle

tt =

tn cos

Normal pressure angle

Transverse pressure angle and helix angle

tan

Circular pitch

Tooth thickness Spur rack N ormal plane Normal pitch line

Lead angle Helix angle

Circular pitch Helical rack Gear axis

Straight bevel rack FIGURE 3.34

Spur-, Helical- and Bevel-Gear Racks.

Front plane

n

= tan cos

Gear Types and Nomenclature

99 Point at which contact starts

Point at which conact finishes Common tangent Line of action (or contact) Pressure angle Pitch point

Base diameter Working depth Whole depth Root diameter

Addendum

Clearance

Arc thickness

Dedendum

Chordal thickness

Chordal addendum

Addendum Rise of arc

FIGURE 3.35

Internal-Gear and Pinion Nomenclature.

Zerol. The Zerol bevel gear tooth has lengthwise curvature like the spiral bevel gear tooth. At the center of the face width, though the spiral angle is 0 . Figure 3.39 shows how the hand of Zerol bevel gear is designated. Note that a right-hand spiral bevel pinion has to have left-hand spiral bevel gear to mesh with it, and vice versa. This same rule applies to spiral bevel gears. Table 3.9 shows formulas for Zerol bevel gears. Both Zerol and spiral bevel gears have the pitch diameters specified at the large end of the tooth.

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Dudley’s Handbook of Practical Gear Design and Manufacture

TABLE 3.5 Internal Spur Gear Formulas To Find

Having

Formula

Pitch diameter of gear

Number of teeth in gear and diametral pitch

D=

NG P

Pitch diameter of pinion

Number of teeth in pinion and diametral pitch

d=

NP P

Internal diameter, gear

Pitch diameter and addendum

Center distance

Number of teeth in gear and pinion; diametral pitch

Center distance

Pitch diameter of gear and pinion

Di = D

2a

NG NP 2P

C= C=

D

d 2

Driven

Shaft angle, Σ

ψ 1 , helix angel of driver ψ 2 , helix angle of driven

Axis of driven Driver

Axis of driver

FIGURE 3.36

Crossed-Helical-Shaft Angle and Helix.

TABLE 3.6 Crossed Helical Gear (Nonparallel Shaft, Helicals) To Find

Having

Formula

=

+

Shaft angle

Helix angle of pinion and helix angle of gear

Pitch diameter of pinion

Number of teeth in pinion, normal diametral pitch, and helix angle of pinion

DP =

NP Pn × cos P

Pitch diameter of gear

Number of teeth in gear, normal diametral pitch, and helix angle of gear

DG =

NG Pn × cos G

Center distance

Pitch diameter of pinion, and pitch diameter of gear

Speed ratio

Number of teeth in gear, and number of teeth in pinion

C=

P

G

DP + Dg

mG =

2 NG NP

Gear Types and Nomenclature

101 Pitch apex to back Pitch apex to crown Crown

Pitch apex

Crown to back

Pinion

Face angle Root angle

Pitch angle

Face width

Uniform clearance Shaft angle

Front angle

Gear

Back angle Pitch diameter Outside diameter Back cone

FIGURE 3.37

Bevel-Gear Nomenclature.

TABLE 3.7 Straight Bevel Gear Formulas (20∘ Pressure Angle, 90∘ Shaft Angle) To Find

Having

Formula

Pitch diameter of pinion

Number of pinion teeth and diametral pitch

d=

NP Pd

Pitch diameter of gear

Number of gear teeth and diametral pitch

D=

NG Pd

Pitch angle of pinion

Number of pinion teeth and number of gear teeth

Pitch angle of gear

Pitch angle of pinion

Outer cone distance of pinion and gear Circular pitch of pinion and gear

Gear pitch diameter and pitch angle of gear

Ao =

D 2 sin

Diametral pitch

p=

3.1416 Pd

Dedendum angle of pinion

Dedendum of pinion and outer cone distance

Dedendum angle of gear

Dedendum of gear and outer cone distance

Face angle of pinion blank

Pinion pitch angle and dedendum angle of gear

o

=

+

G

Face angle of gear blank

Gear pitch angle and dedendum angle of pinion

o

=

+

P

Root angle of pinion

Pitch angle of pinion and dedendum angle of pinion

R

=

P

Root angle of gear

Pitch angle of gear and dedendum angle of gear

R

=

G

Outside diameter of pinion

Pinion pitch diameter, pinion addendum and pitch angle of pinion Pitch diameter of gear, gear addendum and pitch angle of gear

Outside diameter of gear Pitch apex to crown of pinion

Pitch diameter of gear, addendum, and pitch angle of pinion

Pitch apex to crown of gear

Pitch diameter of pinion, addendum, and pitch angle of gear

( )

1 NP NG

= tan = 90

( ) ( )

P

= tan

1 boP Ao

G

= tan

1 boG Ao

do = d + 2aoP cos Do = D + 2aoG cos xo =

D 2

aoP sin

Xo =

d 2

aoG sin

(Continued)

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Dudley’s Handbook of Practical Gear Design and Manufacture

TABLE 3.7 (Continued) Straight Bevel Gear Formulas (20∘ Pressure Angle, 90∘ Shaft Angle) To Find

Having

Formula

t=p

Circular tooth thickness of pinion

Circular pitch and gear circular tooth thickness

Chordal thickness of pinion

Circular tooth thickness, pitch diameter of pinion and backlash Circular tooth thickness, pitch diameter of gear and backlash

Chordal thickness of gear Chordal addendum of pinion Chordal addendum of gear

tc = t Tc = T

Addendum angle, circular tooth thickness, pitch diameter and pitch angle of pinion Addendum angle, circular tooth thickness, pitch diameter and pitch angle of gear

Tooth angle of pinion

Outer cone distance, tooth thickness, dedendum of pinion, and pressure angle

3.438 Ao

Tooth angle of gear

Outer cone distance, tooth thickness, dedendum of gear, and pressure angle

3.438 Ao

Note: tan

1

T t3 6d 2

B 2

T3 6D 2

B 2

acP = aoP +

t 2 cos 4d

acG = aoG +

T 2 cos 4D

( (

t 2

+ boP tan

T 2

+ boG tan

), minutes ), minutes

means “the angle whose tangent is”

Tooth spiral Circular pitch

Cone distance Face advance

Spiral angle

Face width FIGURE 3.38

Spiral-Bevel-Gear Nomenclature.

3.2.6 WORM GEAR NOMENCLATURE

AND

FORMULAS

When the word worm is used in connection with gear types, the implication is that there is some “en­ veloping.” A single-enveloping set of worm gears has a cylindrical worm in mesh with a gear that is throated to tend to wrap around the worm. Figure 3.40 shows a typical arrangement. When both the worm and the gear wrap around each other, the combination is designated as doubleenveloping. Figure 3.40 shows a typical arrangement of this kind. Cylindrical worm gears. The worm thread (or tooth) may be dimensioned in both a normal section and an axial section. The traverse section is normally not used. Figure 3.40 shows the nomenclature that is used. The special formulas for worm gearing are given in Table 3.10. The pitch diameter of the throated gear changes as you go across the face width. Standard practice is to take the pitch diameter at the center of the throat. Note the details of how this is done in Figure 3.40.

Gear Types and Nomenclature

103

TABLE 3.8 Spiral Bevel Gear Formulas (20∘ Pressure Angle, 90∘ Shaft Angle) To Find

Having

Formula

Pitch diameter of pinion

Number of pinion teeth and diametral pitch

d=

NP Pd

Pitch diameter of gear

Number of gear teeth and diametral pitch

D=

NG Pd

Pitch angle of pinion

Number of pinion teeth and number of gear teeth

Pitch angle of gear

Pitch angle of pinion

Outer cone distance of pinion and gear

Pitch diameter of gear and pitch angle of gear

Ao =

D 2 sin

Circular pitch of pinion and gear

Diametral pitch

p=

3.1416 Pd

Dedendum angle of pinion

Dedendum of pinion and outer cone distance

Dedendum angle of gear

Dedendum of gear and outer cone distance

Face angle of pinion blank

Pitch angle of pinion and dedendum angle of gear

o

=

+

G

Face angle of gear blank

Pitch angle of gear and dedendum angle of pinion

o

=

+

P

Root angle of pinion

Pitch angle of pinion and dedendum angle of pinion

R

=

P

Root angle of gear

Pitch angle of gear and dedendum angle of gear

R

=

G

Outside diameter of pinion

Pitch diameter, addendum, and pitch angle of pinion

do = d + 2aoP cos

Outside diameter of gear

Pitch diameter, addendum, and pitch angle of gear

Do = D + 2aoG cos

Pitch apex to crown of pinion

Pitch diameter of gear, pitch angle, and addendum of pinion

Pitch apex to crown of gear

Pitch diameter of gear, pitch angle, and addendum of gear

Circular tooth thickness of pinion

Circular pitch of pinion and circular pitch of gear

= 90

Zerol bevel gears

Right-hand

Left-hand

Left-hand

Right-hand

Zerol bevel pinions

FIGURE 3.39

( )

1 NP NG

= tan

Zerol-Bevel-Gear Nomenclature.

P G

( ) ( )

= tan

1 boP Ao

= tan

1 boG Ao

xo =

D 2

aoP sin

Xo =

d 2

aoG sin

t=p

T

104

Dudley’s Handbook of Practical Gear Design and Manufacture

TABLE 3.9 Zero Bevel Gear Formulas (20∘ Pressure Angle, 90∘ Shaft Angle) To Find

Having

Formula

Pitch diameter of pinion

Number of pinion teeth and diametral pitch

d=

NP Pd

Pitch diameter of gear

Number of gear teeth and diametral pitch

D=

NG Pd

Pitch angle of pinion

Number of pinion teeth and number of gear teeth

Pitch angle of gear

Pitch angle of pinion

Outer cone distance of pinion and gear Circular pitch of pinion and gear

Pitch diameter of gear and pitch angle of gear

Ao =

D 2 sin

Diametral pitch

p=

3.1416 Pd

Face angle of pinion blank

Pitch angle of pinion, and dedendum angle of gear

o

=

+

G

Face angle of gear blank

Pitch angle of gear, and dedendum angle of pinion

o

=

+

P

Root angle of pinion

Pitch angle of pinion and dedendum angle of pinion

R

=

P

Root angle of gear

Pitch angle of gear and dedendum angle of gear

R

=

G

Outside diameter of pinion

Pitch diameter, pitch angle, and dedendum of pinion

Outside diameter of gear

Pitch diameter, pitch angle, and dedendum of gear

Pitch apex to crown of pinion

Pitch diameter of gear, pitch angle, and addendum of pinion

Pitch apex to crown of gear

Pitch diameter of pinion, pitch angle, and addendum of gear

Circular tooth thickness of pinion

Circular pitch of pinion and circular thickness of gear

Circular tooth thickness of gear

Circular pitch, pressure angle, and addendum of pinion and gear

( )

1 NP NG

= tan = 90

do = d + 2aP cos

Do = D + 2aG cos xo =

D 2

aP sin

Xo =

d 2

aG sin

t=p

T

T=

p 2

(aP

aG )tan

Double-enveloping worm gears. The typical style of design and the normal nomenclature for this kind of gearing is shown in Figure 3.41. The pitch diameters are specified at the center of the throat on the worm and the center of the throat of the gear. The formulas for double-enveloping worm gears are given in Table 3.11.

3.2.7 FACE GEARS The gear member of a face gearset has teeth cut on the end of a cylindrical-shaped blank. Figure 3.42 shows a typical face gear arrangement and the nomenclature used (Table 3.12). The formulas for an on-center 90 shaft-angle set of face gears are given in Table 3.13. Additional design data (for a standard design) are given in Tables 3.13 and Table 3.14. There are other possible face gear designs with shaft angles that are not 90 , or conditions, or not being on-center. These can be thought of as special designs beyond the scope of this book.

3.2.8 SPIROID GEAR NOMENCLATURE

AND

FORMULAS

There is a family of gears that is generally described by the name Spiroid and Helicon. The gear member of each of these kinds of gear sets has teeth cut on the end of a blank. The pinion member does not mesh on center but instead meshes with a considerable offset from the central position. Figure 3.43 shows a standard Spiroid arrangement. The dimension A in Table 3.15 is the offset.

Gear Types and Nomenclature

105 Outside diameter Root diameter

Addendum

Dedendum

Worm

Pitch diameter

Center distance

Worm wheel

Pitch diameter

Whole depth

Throated diameter Maximum diameter

Normal pressure angle Lead angle Addendum

Axial pressure angle

Dedendum A A

Pitch diameter

Normal circular pitch Axial pitch

FIGURE 3.40

Cylindrical-Worm-Gear Nomenclature.

The Spiroid pinion is tapered. The teeth spiral around the pinion somewhat like threads on a worm, but they are on the surface of a cone rather than on a cylinder. The Helicon pinion differs from the Spiroid pinion by being cylindrical. This makes it quite comparable to a cylindrical worm. Table 3.16 gives some formulas for Beveloid gear sets. Other dimensions needed to make a standard design are given as part of Figure 3.43. Specific data for Beveloid gear sets are not given. The Spiroid Division of the Illinois Tool Works in Chicago, Illinois should be consulted for additional data.

3.2.9 HELICON GEARS This special type of gear is patented by the Vinco Corporation.

106

Dudley’s Handbook of Practical Gear Design and Manufacture

TABLE 3.10 Worm Gear Formulas To Find

Having

Formula

Worm:

L = NW × px

Lead

Number of threads in worm and axial pitch

Pitch diameter

Center distance

Rood diameter

Outside diameter of pinion and whole depth of tooth

Outside diameter

Pitch diameter and addendum

Minimum face

Throat diameter, pitch diameter, and addendum

Lead angle

Lead and pitch diameter

tan =

Normal pitch

Axial pitch and lead angle

pn = p × cos

Nominal pitch diameter

Gear: Number of teeth in gear and axial pitch

Throat diameter

Nominal pitch diameter and addendum

Effective face

Pitch diameter and working depth

Center distance

d=

C 0.875 C 0.875 to 3 1.7

dR = do

2ht

do = d + 2a

( ) ( Dt 2 2

f=2

D=

D 2

a

)

2

L 3.1416 × d

NG × px 3.1416

Dt = D + 2a Fe =

Pitch diameter of gear and pitch diameter of worm

(d + hk )2 C=

d2

D+d 2

Figure 3.23 shows a Helicon gear and its features. Helicon gears can be engaged with spur gears, helical gears, cylindrical worms, racks, and other Helicon gears. They can be used with intersecting, parallel, or skew shafts. They are relatively insensitive to mounting errors.

3.3

AN ADVANCED SET OF TERMS AND DEFINITIONS FOR DESIGN PARAMETERS IN GEARING5

Almost all necessary design parameters in a gear pair are covered by a conventional set of terms and definitions known from many sources, that is, from national, as well as international standards on gearing, handbooks, manuals, and so forth. As the theory of gearing evolves, new design parameters and definitions are introduced targeting more in-depth description of the gear kinematics of gearing, the tooth flank geometry, operating of a gearset, and others. Below, a short list of newly introduced along with some known terms and definitions that pertain to gearing is provided. The terms and definitions are listed in the order starting from the most fundamental. Axis of rotation – is a straight line associated with a gear, Og , (with a pinion, Op ) that remains motionless when the gear (the pinion) rotates (see Figure 3.44). Rotation of a gear – is an angular motion with an angular velocity, g, of a rotated gear. The similar is valid with respect to a rotated pinion, p . Rotation vector of a gear – is a vector6, g , that is pointed along the axis of rotation of the gear, Og . The magnitude of the vector, g , equals the angular velocity, g , of a rotated gear. The direction of the

5 6

The beginning of wisdom is to call things by their right names – An old Chinese proverb. It should be stressed here that a rotation is not a vector in nature. However, rotations can be treated as vectors if special care is taken, that is, when coordinate system transformations are applied, the order of multipliers becomes critical and it can not be altered.

Begin relief

Thrust shoulder

1 worm face 2

Worm face angle Worm throat diameter

FIGURE 3.41 Double-Enveloping-Worm-Gear Nomenclature.

Angular end Relief 90° end

Maximum hob radius

C L

Gear hob radius

1 gear width at O.D. 2

Worm axis

Gear axis

Gear face

Gear width at O.D.

Maximum hob radius

Gear face angle

1 gear face 2

Gear Types and Nomenclature 107

108

Dudley’s Handbook of Practical Gear Design and Manufacture

TABLE 3.11 Double-Enveloping Worm Gear Formulas To Find

Having

Worm root diameter*

Pitch diameter of pinion and dedendum of gear

Worm root diameter*

Center distance *

Center distance

Gear pitch diameter

Center distance and worm pitch diameter

Axial circle pitch

Gear pitch diameter and number of teeth in gear

Normal circular pitch

Axial circular pitch and pitch cone angle of pinion

Whole depth of tooth*

Normal circular pitch

Working depth of tooth

Whole depth of tooth

Dedendum*

Working depth of tooth

Normal pressure angle*

dR = d dR =

Worm pitch diameter

*

Formula

C 0.875

d=

2bG

C 0.875 (approx. 3 2.2

)

(approx. )

D=C px =

2d D NG

pn = px cos ht =

pn 2

hk = 0.9ht bG = 0.611hk –

Axial pressure angle

Normal pressure angle and lead angle at center of worm

Lead angle at center of worm

Pitch diameter, gear ratio, and worm pitch diameter

Lead angle average*

Pitch diameter, gear ratio, and worm pitch diameter

n

= 20

= tan

x c

= tan

= tan

( )

1 tan n cos c

( ) ( ) 1

D mg d

1 0.87D mg d

* Proportions given in these formulas represent recommendations of Cone-drive gears. Cone-drive is a registered trademark of Cone-Drive Textron, Traverse City, Michigan.

rotation vector g is specified by the right-hand rule. The rotation vector, g , is a type of sliding vector. This means that the vector, g , can be applied at any point within the axis of rotation of the gear, Og . This is similarly valid with respect to a rotation vector of a pinion, p . Center distance – is the closest distance of approach, C , between a gear, Og , and a mating pinion, Op , axes of rotation (see Figure 3.45). Center-line – is the straight line, ℄, along which the center distance, C , is measured. Crossed-axes angle (also theshaft angle) – is the angle, , formed by the rotation vectors of a gear, = ( g , p). Often, the crossed-axes angle is specified as , g and a mating pinion, p , that is, = 180 ( g , p) which is incorrect. Gear apex – is point of intersection, A g , of a gear axis of rotation, Og , by the center-line, ℄. Pinion apex – is point of intersection, A p , of a pinion axis of rotation, Op , by the center-line, ℄. Vector of instantaneous rotation – is a vector, pl , of instant rotation of a pinion in relation to the mating gear. Commonly, this vector is specified as pl = p g. Gear cone angle – is the angle, g , that the rotation vector of a gear, g , forms with the vector of instantaneous rotation, pl , that is, g = ( pl , g). The gear cone angle, g , is a signed value. Pinion cone angle – is the angle, p , that the rotation vector of a pinion, p , forms with the vector of instantaneous rotation, pl , that is, p = ( pl , p). The pinion cone angle, p , is a signed value. Axis of instantaneous rotation (also thepitch line) – is a straight line, Pln , along the vector, pl , of instantaneous rotation of a pinion in relation to the mating gear.

FIGURE 3.42 On-Center-Face-Gear Nomenclature.

Face to registering surface Mounting surface

Inner diameter

Outer diameter

Face to mounting surface Registering surface

Face width

Mounting distance

Gear axis Pitch cone

Pinion axis

Face gear

Involute spur pinion

Gear Types and Nomenclature 109

110

Dudley’s Handbook of Practical Gear Design and Manufacture

TABLE 3.12 Face Gears—On-Center 90∘ Shaft Angle To Find Gear ratio

Having

Formula

Number of gear and pinion teeth

Inside diameter

Diametral pitch, number of gear and pinion teeth, pressure angle

Outside diameter (pointed teeth)

Diametral pitch, number of gear teeth, pressure angle, and tangency pressure angle

Maximal face width

Face-gear inside and outside diameters

Face to pinion axis

Pinion outside diameter and working depth of tooth

Face to mounting surface

Mounting distance and face to pinion axis

Minimal number of teeth

Di = DoG =

(from 1.5 to 12.5)

8NP + NG cos2

NG cos sec o (see P Do

Table 2.13)

Di 2

Face to pinion axis =

do 2

hk

Face to mounting surface= = mounting distance face to pinion axis (see Table 2.14)

TABLE 3.13 Tangency Pressure Angle at Face Gear Outside Diameter o

1 P

Fg =

–

NP

NG NP

mG =

sec

o

12 13

40.765 39.669

1.32-31 1.29913

14

38.692

1.28120

15 16

37.815 37.022

1.26583 1.25249

17

36.300

1.24081

18 19

36.012 35.778

1.23626 1.23261

20

35.557

1.22919

21 22

35.346 35.145

1.22598 1.22295

23

34.954

1.22009

24 25

34.771 34.596

1.21738 1.21481

26

34.429

1.21237

27 28

34.268 34.113

1.21005 1.20783

29

33.965

1.20572

30 31

33.821 33.683

1.20369 1.20176

32

33.550

1.19990

(Continued)

Gear Types and Nomenclature

111

TABLE 3.13 (Continued) Tangency Pressure Angle at Face Gear Outside Diameter NP

sec

o

o

33

33.421

1.19812

34

33.297

1.19640

35 36

33.176 33.059

1.19475 1.19317

37

32.946

1.19164

38 39

32.837 32.730

1.19016 1.18874

40

32.627

1.18736

41 42

32.525 32.427

1.18602 1.18473

43

32.332

1.18352

44 45

32.240 32.150

1.18228 1.18111

46

32.061

1.17996

47 48

31.976 31.893

1.17887 1.17781

49

31.810

1.17675

50

31.734

1.17578

TABLE 3.14 Minimum Numbers of Teeth in Pinion and Face Gear Diametral Pitch Range

Minimum Number of Teeth Pinion

Gear

20–48

12

18

49–52 53–56

13 14

20 21

57–60

15

23

61–64 65–68

16 17

24 26

69–72

18

27

73–76 77–80

19 20

29 30

81–83

21

32

85–88 89–92

22 23

33 35

93–96

24

36

97–100

25

38

Tapered root

Base cylinder

FIGURE 3.43 Beveloid-Tooth and Gear Nomenclature.

Dished (or flat) front face

Tapered outside cone

Working position

112 Dudley’s Handbook of Practical Gear Design and Manufacture

Gear Types and Nomenclature

113

TABLE 3.15 Spiroid Gear Dimensions

A

B

C

D

E

F

G

H

K

0.500 0.750

0.625 1.000

1.500 2,250

0.129 0.176

0.596 0.894

0.2969 0.4219

0.365 0.548

1.101 1.651

0.2031 0.2812

1.000

1.375

3.000

0.248

1.192

0.5938

-.731

2.202

0.3750

1.250 1.500

1,625 2.000

3.750 4.500

0.295 0.338

1.490 1.788

0.7031 0.8281

0.914 1.096

2.752 3.303

0.4531 0.5312

1.875

2.625

5.625

0.402

2.236

1.0156

1.370

4.129

0.6406

2.250 2.750

3.187 4.000

6.750 8.250

0.461 0.536

2.683 3.279

1.1562 1.3594

1.644 2.010

4.954 6.055

0.7500 0.8750

3.250

4.750

9.750

0.608

3.875

1.5625

2.375

7.156

1.0156

3.750 4.375

5.625 6.750

11.250 13,125

0.677 0.757

4.471 5.216

1.7656 2.000

2.741 3.197

8.257 9.633

1.1406 1.2969

5.125

8.000

15.375

0.851

6.111

2.2812

3.745

11.285

1.4844

TABLE 3.16 Spiroid Gear Formulas To Find

Having

Formula NG NP

Ratio

Number of teeth in gear, number of threads in pinion

Pinion spiral angle

Theoretical lead, pinion OD cone angle, pinion pitch radius

Gear spiral angle

Ratio, pinion pitch radius, gear pitch radius, pinion spiral angle

sin

Pinion pitch point

Length of pinion primary pitch cone

= 3 axial length primary

mG =

tan G

P

=

= mG

L sec 2 r

( ) sin r RG

P

1

pitch cone from small end

114

Dudley’s Handbook of Practical Gear Design and Manufacture

ωg Og

ωg

FIGURE 3.44

Axis of Rotation of a Gear (After Dr. S.P. Radzevich, ~2008).

Og

Ag

ωg

ωg

Apa

ωpl

Pln

Ap Σ C L

Z

FIGURE 3.45

Op

ωp

C

Y

ωp

X

Elements of the Kinematics of a Gear Pair (After Dr. S.P. Radzevich, ~2008).

Gear Types and Nomenclature

115 PA − plane Cln − plane Ap

Nln − plane

ωp

ωpl

Op Apa

φt.ω

C

Pln Ag ωg Pln − plane

Og

C L

FIGURE 3.46 Principal Planes: the Pitch-Line Plane (Pln plane), the Center-Line Plane (Cln plane), and the Normal Plane (Nln plane), along with the Plane of Action, PA, Associated with a Gear Pair (After Dr. S.P. Radzevich, ~2008).

Plane-of-action apex – is the point of intersection, Apa , of the pinion axis of instant rotation, Pln , by the center-line, ℄. Pitch-line plane – is the plane through the pitch line, Pln , and the center line, ℄, of the gear pair (see Figure 3.46). In a case of I a gearing, the Pln plane is the plane through a gear and a mating pinion axes of rotation, Og and Op . In a case of Pa gearing, the Pln plane is the plane through the pitch line, Pln , and the center line, ℄. Center-line plane – is the plane through the center line, ℄, of the gear pair perpendicular to the pitch line, Pln . It is revealed later that the center-line plane is congruent with the pitch plane of the gear pair. In a case of I a gearing, the Cln plane is the plane through plane-of-action apex, A pa , perpendicular to the pitch line, Pln . In a case of Pa gearing, the Cln plane is a plane perpendicular to the gear and pinion axes of rotation, Og and Op . Normal plane – is the plane through the plane-of-action apex, Apa , perpendicular to the center line, ℄, of the gear pair. In a case of I a gearing, the Nln plane is a plane perpendicular to the gear and pinion axes of rotation, Og and Op . In a case of Pa gearing, the Nln plane is a plane perpendicular to the gear and pinion axes of rotation, Og and Op . Main reference system – The origin of the main reference system, Xln Yln Zln , is coincident with the plane-of-action apex, Apa . The axes Xln , Yln , and Zln are pointed along the lines of intersection of the principal planes as illustrated in (see Figure 3.47): the axis Zln is along the vector of instant rotation, pl ; the axis Yln is along the center-line, ℄; and the Xln axis complements the axes Yln and Zln to a left-handoriented Cartesian coordinate system Xln Yln Zln . Motionless gear reference system – The origin of the reference system, Xg.m Yg.m Zg.m , is coincident with the gear apex, Ag , as shown in Figure 3.48,a. The axis Zg.m is along the rotation vector of the gear, g ; the axis Yg.m is along the center-line, ℄; and the Xg.m axis complements the axes Yg.m and Zg.m to a left-hand-oriented Cartesian coordinate system Xg.m Yg.m Zg.m . Actually, the motionless gear reference system, Xg.m Yg.m Zg.m , is turned through the gear cone angle, g , about Zln axis of the reference system Xln Yln Zln associated with a gear pair as illustrated in Figure 3.48,b. Gear main reference system – The gear main reference system, Xg Yg Zg , is rigidly associated with the gear, and is rotated together with the gear about its axis of rotation, Og . As illustrated in Figure 3.48,c,

116

Dudley’s Handbook of Practical Gear Design and Manufacture C L Ap

ωp

Apa

Xln −ωg

ωg

ωpl

C

Ag

Zln Yln

FIGURE 3.47 Main Reference System Xln Yln Zln Associated with a Gear Pair (After Dr. S.P. Radzevich, ~2008).

the reference systems Xg Yg Zg and Xg.m Yg.m Zg.m share the common axis Zg Zg.m , and is turned about the Z m.g axis through a gear rotation angle, g . A gear rotation angle, g , and a pinion rotation angle, p , correlate to one another in the following manner: p = u g , where u is the gear ratio of the gear pair. Motionless pinion reference system – The origin of the reference system, Xp.m Yp.m Z p.m , is coincident with the pinion apex, Ap, as shown in Figure 3.48,d. The axis Z p.m is pointed along the rotation vector of the pinion, p ; the axis Yp.m is pointed along the center-line, ℄; and the Xp.m axis complements the axes Yp.m and Z p.m to a left-hand-oriented Cartesian coordinate system Xp.m Yp.m Z p.m . Actually, the motionless gear reference system, Xp.m Yp.m Z p.m , is turned through the pinion cone angle, p , about Zln axis of the main reference system Xln Yln Zln associated with a gear pair as illustrated in Figure 3.48,e. Pinion reference system – The pinion reference system, Xp Yp Z p , is rigidly associated with the pi­ nion, and is rotated together with the pinion about its axis of rotation, Op . As illustrated in Figure 3.48,f, the reference systems Xp Yp Z p and Xp.m Yp.m Z p.m share the common axis Z p Z p.m , and is turned about the Zg.m axis through a pinion rotation angle, p . Principal reference systems – The reference systems Xln Yln Zln , Xg.m Yg.m Zg.m , Xg Yg Zg , Xp.m Yp.m Z p.m , and Xp Yp Z p are commonly referred to as the principal reference systems associated with a gear pair. Plane of action – is the plain, PA, within which the tooth flanks of the gear and the pinion, G and P , interact with one another. In a case of crossed-axis gearing (Ca gearing, for simplicity), the plane of action, PA, is a plane through the pitch line, Pln , and the center line, ℄, of the gear pair (see Figure 3.46). This definition is identical to that given earlier to the pitch-line plane, Pln . Therefore, in a case of gearing, the plane of action is congruent to the Pln plane. In a case of intersected-axis gearing (Ia gearing, for simplicity), the plane of action, PA, is a plane through the pitch line, Pln , that forms the transverse pressure angle, t. , with the perpendicular to the plane through the axes of rotation of the gear and the pinion. In a case of parallel-axis gearing (Pa gearing, for simplicity), the plane of action, PA, is a plane through the pitch line, Pln , that forms the transverse pressure angle, t. , with the perpendicular to the plane through the axes of rotation of the gear and the pinion. Transverse pressure angle – is the angle, t. , formed by the pitch plane, PP , and the plane of action, PA, as shown in Figure 3.49.

Gear Types and Nomenclature

(a)

117

(b)

Ap

Zg.m ωg

ωp Σ

Apa

Σg

Ag , Apa

C Zg.m

Xg.m

Xln

ωp rw.g

ωg

ω pl

Xg.m

Ag

Zln

Xln ωpl

Zln

Yln , Yg.m

(c)

(d) Xp.m Yg.m

Yp.m Ap

Yg

rw.p

ωp Zp.m

Xg

Apa C Xln

ϕg

ωg

Xg.m

Zg ≡ Zg.m ωpl

Ag

Zln

(e)

(f)

Xp.m

Yp.m

ωg Σ

ϕp Ap , Apa

Σp

Yln

Yp

ωp Xln Zp.m

Xp.m ωpl Zp ≡ Zm.p Zln

Xp

FIGURE 3.48

The Rest of the Main Reference Systems (After Dr. S.P. Radzevich, ~2008).

118

Dudley’s Handbook of Practical Gear Design and Manufacture rp

Base cone: gear

C

Op

rg

ωp

Ap Apa

ωpa

ωp

ωpl

Ag

ωg ωg

PA

Base cone: pinion PP

φt.ω

CL Op

ωp

Ap

φt

ωpl Apa

PP C Pln − plane

Cln − plane Og

ωg Ag

FIGURE 3.49 Plane of Action, PA, and Base Cones Associated with a Gear Pair Shown in the Center-Line Plane (Cln plane) (After Dr. S.P. Radzevich, ~2008).

Base cone, gear – The gear base cone is a cone with apex coincident with the gear apex, Ag , the axis of the base cone is aligned with the gear axis of rotation, Og , and the gear base cone is tangent to the plane of action, PA. Base cone, pinion – The pinion base cone is a cone with apex coincident with the pinion apex, Ap, axis of the base cone is aligned with the pinion axis of rotation, Op , and the pinion base cone is tangent to the plane of action, PA.

Gear Types and Nomenclature

119

ωpa

LCdes ry.pa Φpa

Pln ϕb.op

ωpl

Ag

Apa

PA Ap

FIGURE 3.50

Operating Base Pitch,

b . op ,

of a Gear Pair (After Dr. S.P. Radzevich, ~2008).

Operating base pitch – The operating base pitch of a gear pair,

b . op , is the angular distance between i ±1 i and LCdes (see Figure 3.50). contact, LCdes

corresponding points of each two adjacent desired lines of The distance, b . op , can be measured on a circular arc of an arbitrary radius, ry . pa . The operating base pitch, b . op , is a calculated parameter, that is, it can not be measured directly in a gear pair. The operating base pitch, b . op , is specified in the plane of action, PA. The vertex of the operating base pitch angle, b . op , is coincident with the plane-of-action apex, Apa . The accuracy of assembly of a gear and a mating pinion in the gear housing can be expressed in terms of the base pitch of the gear pair, b . op , that is, in ac terms of the deviation, b . op , of the actual operating base pitch of the gear pair, b . op , from its nominal ac value, b . op . The deviation b . op is a signed value, and equals to b . op = b . op b . op . Base pitch, gear – The base pitch of a gear, b . g , is the angular distance between corresponding points

of each two adjacent lines of intersection of the gear tooth flanks, G i and G i±1, by the plane of action, PA. The gear base pitch, b . g , can be measured directly in a gear. The gear base pitch, , is measured within the plane of action, PA, at an arbitrary angular configuration of the gear in relation to the mating pinion. The vertex of the gear base pitch angle, b . g , is coincident with the plane-of-action apex, Apa . The accuracy of the machined gear can be expressed in terms of the base pitch of the gear, b . g , that is, in terms of the ac deviation, b . g , of the actual base pitch of the gear, b . g , from its nominal value, b . g . The deviation is a signed value, and equals to b.g = b.g Base pitch, pinion – The base pitch of a pinion,

ac b.g.

b.g

b.p,

is the angular distance between corresponding

points in each two adjacent lines of intersection of the pinion tooth flanks, P i and P i±1, by the plane of action, PA. The pinion base pitch, b . p , can be measured directly in a pinion. The pinion base pitch, b . p , is measured in the plane of action, PA, at an arbitrary angular configuration of the pinion in relation to the gear. The vertex of the pinion base pitch angle, b . p , is coincident with the plane-of-action apex, A pa . The accuracy of the machined pinion can be expressed in terms of the base pitch of the pinion, b . p , that is, in terms of the deviation, b . p , of the actual base pitch of the pinion, bac. p , from its nominal value, b . p . The deviation

b.p

is a signed value, and equals to

b.p

=

b.p

ac b. p.

120

FIGURE 3.51

Dudley’s Handbook of Practical Gear Design and Manufacture

Hand of a Helical Gear.

The first fundamental law of gearing – is the so-called, the law of contact. The law of contact can be analytically described by the Shishkov equation of contact: At every point of contact, K , of tooth flanks of a gear, G , and a mating pinion, P , the vector of the relative motion, V , of the tooth flanks is always perpendicular to the unit vector of the common perpendicular, n , to the contacting surfaces G and P , that is, n V = 0. The second fundamental law of gearing – is also referred to as the conjugate action law of the interacting tooth flanks, G and P , of the gear and the pinion (Radzevich, ~2008). According to this law, the instant line of action, LAinst , that is, a straight line along the common perpendicular, n , is always located within the plane of action, PA, and intersects the pitch line, Pln . Therefore, the path of contact, Pc , is a planar curve that is entirely located within the plane of action, PA. The conjugate action law of the interacting tooth flanks, G and P , of a gear and a mating pinion is more robust than the condition of their contact. This means that if conjugate action law is met, then there is no necessity to verify whether or not the law of contact is met: the law of contact is met for sure (but vice versa). The third fundamental law of gearing – This law of gearing requires equality of the base pitches (Radzevich, ~2008). In order to transmit a rotary motion smoothly by means of gear teeth, first, the base pitch of a gear, b . g , must be equal to the operating base pitch, b . op , of the gear pair, that is, the equality b . g = b . op must be observed; second, the base pitch of the pinion, b . p , must be equal to the operating base pitch, b . op , of the gear pair, that is, the equality b . p = b . op must be observed. Two equalities b . g = b . op and b . p = b . op can be combined into a generalized equality: b . g = b . p = b . op . All geometrically-accurate gearsets fulfill this equality. It should be stressed here that the two equalities, b . g = b . op and b . p = b . op , as well as the combined equality b . g = b . p = b . op , are more robust compared to the conjugate action law in the tooth flanks G and P , of the gear and the pinion. This means that in a case when base pitches of the gear and the pinion are equal to the operating base pitch of the gear pair, there is no need to verify whether or not the conjugate action law, as well as the law contact of the tooth flanks G and P , are fulfilled: in this case both the laws are satisfied for sure (but not vice versa). Ultimately, one can conclude that the equality of the base pitches of a gear and a mating pinion to the operating base pitch of the gear pair ( b . g = b . p = b . op ) is the most robust law all the perfect gear pairs must be met7 (Radzevich, ~2008). If the conditions b . g = b . op and b . p = b . op (or the condition:

7

In a particular case of geometrically-accurate (ideal or perfect Pa– gearing), that is, in a case of Pa– gearing with zero axis misalignment, the angular base pitches φb.g, φb.p, and φb.op reduce to the corresponding linear base pitches pb.g, pb.p, and pb.op correspondingly. In reduced case, the fundamental law of gearing is represented either by two equations pb.g = pb.op and pb.p = pb.op, or by a generalized equation pb.g = pb.p = pb.op.

Gear Types and Nomenclature

121 Crest

Tip Dedendum flank

Addendum flank Root curve

Flank (general) Tip circle

Fillet curve

Root

Pitch circle

Root circle

FIGURE 3.52

Tooth Profile Elements.

Face Width (dimension) Face (descriptive)

Pitch line (descriptive)

Tooth chamfer

End radius Tooth rounding Square face

FIGURE 3.53

Lengthwise Tooth Elements of a Spur Gear.

b . g = b . p = b . op ) are not fulfilled, then the gear pair is not capable of transmitting a rotary motion smoothly, and an excessive vibration generation and noise excitation become inevitable. Hand of helix – The hand of helix in helical gearing can be determined by means of the rule illu­ strated in (see Figure 3.51). This approach can be enhanced to the cases of gears for Ia axis, as well as Ca axis gearings. Conventional design parameters and terms those used in gearing are summarized and depicted in (see Figure 3.52) and (see Figure 3.53).

REFERENCES [1] Kapelevich, A.L. (2018). Asymmetric gearing. Boca Raton, FL: CRC Press. [2] Radzevich, S.P. (Editor). (2019). Advances in gear design and manufacture. Boca Raton, Florida: CRC Press. [3] Radzevich, S.P. (2018). An examination of high-conformal gearing. Gear Solutions magazine, February 2018, 31–39. [4] Radzevich, S.P. (2020). High-conformal gearing: Kinematics and geometry, 2nd edition. Amsterdam: Elsevier. ISBN-13: 978-0128212240, ISBN-10: 0128212241.

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Dudley’s Handbook of Practical Gear Design and Manufacture

BIBLIOGRAPHY Radzevich, S.P. (2014). Generation of surfaces: Kinematic geometry of surface machining, 2nd Edition. Boca Raton, Florida: CRC Press. Radzevich, S.P. (2017). Gear cutting tools: Science and engineering, 2nd Edition. Boca Raton, FL: CRC Press. Radzevich, S.P. (2018). Theory of gearing: Kinematics, geometry, and synthesis, 2nd Edition, revised and expanded, Boca Raton, FL: CRC Press. Radzevich, S.P. (2019). Geometry of surfaces: A practical guide for mechanical engineers, 2nd edition. Springer International Publishing. [ISBN-10: 3030221830, ISBN-13: 978-3030221836]. Radzevich, S.P. (2020). High-conformal gearing: Kinematics and geometry, 2nd edition, revised and extended, Elsevier, Amsterdam. (1st edition, 2015. Boca Raton, Florida: CRC Press. ISBN 9781498739184.)

4

Gear Tooth Design Stephen P. Radzevich

In this chapter the subject of the design of the gear tooth shape is covered. To the causal observer, some gear teeth appear tall and slim while others appear short and fat. Gear specialists talk about things like the “pressure angle”, long and short addendum, root fillet design, and the like. Obviously, there is some well-developed logic in the gear trade with regard to how to choose and specify factors related to the gear tooth shape. This chapter presents the basic data that are needed to exercise good judgment in gear tooth design.

4.1

BASIC REQUIREMENTS OF GEAR TEETH

Gear teeth mesh with each other and thereby transmit non-slip motion from one shaft to another. Those, making and using gear teeth may expect the teeth to conform to some standard design system. If this is case, the gear maker may be able to use some standard cutting tools that are already on hand. If a standard design is used that is already familiar to the gear user, the functional characteristics of the gears, such as relative load-carrying capacity, efficiency, quietness of operation, and the like, can be expected to be similar for new gear drives to those of gear drives already in service. It should be kept in mind that the gear art has progressed to a point where there is much more versatility than previously. Machine tools are computer-controlled and can be programmed to cover much more variations than were possible in the past. In the gear design office, computers can find what may be believed to be truly optimum design for some important application. Frequently, the design chose does not agree with a design that might have been considered standard in earlier years. What this means is that a great variety of gear tooth designs are being used. This trend in gear engineering can be a good one—providing the gear designer has an in-depth knowledge of all the things that need to be considered. Serious mistakes, though, can be made when a decision is made by computer data and the computer program failed to consider some critical constraints in the application.

4.1.1 DEFINITION

OF

GEAR TOOTH ELEMENTS

Many features of gear teeth need to be recognized and specified with appropriate dimensions (either directly or indirectly). Spur and helical gears are usually made with an involute tooth form. If a section through the gear tooth is taken perpendicular to the axis of the part, the features shown in Figures 4.1 and are revealed. Note the nomenclature used. Outside diameter (dashed line in Figure 4.1) is the maximum diameter of the gear blank for spur, helical, worms, or worm gears. All tooth elements lie inside this circle. A tolerance on this diameter should always be negative. Modification diameter (dashed line in Figure 4.1) is the diameter at which any tip modification is to begin. It is a reference dimension and may be given in terms of degrees of roll. Pitch diameter (dashed line in Figures 4.1 and 4.2) is the theoretical diameter established by dividing the number of teeth in the gear by the diametral pitch of the cutter to be used to produce the gear. This diameter can have no tolerance. 123

124

Dudley’s Handbook of Practical Gear Design and Manufacture Name of circle

e Pr ofile

Outside diameter Effective outside diameter Start of modification diameter

Act iv

Pitch diameter Limit diameter Form diameter Undercut diameter Base-circle diameter Working-depth diameter Working position

FIGURE 4.1

Nomenclature of Gear Circles.

Limit diameter (dashed line in Figure 4.1) is the lowest portion of a tooth that can actually come in contact with the teeth of a mating gear. It is a calculated value and is not to be confused with form diameter. It is the boundary between the active profile and the fillet area of the tooth. Form diameter (dashed line in Figure 4.1) is a specified diameter on the gear above which the transverse profile is to be in accordance with drawing specification on profile. It is an inspection di­ mension and should be placed at a somewhat smaller radius than the limit diameter to allow for shop tolerances. Undercut diameter (dashed line in Figure 4.1) is the diameter at which the trochoid producing un­ dercut in a gear tooth intersects the involute profile.

Tip round Tooth thickness Active profile

arc

chordal

Space width

Circular pitch Outside circle of mating tooth

Tip of mating tooth

Fillet Origin of involute Form-circle Pitch-circle diameter diameter

FIGURE 4.2

Gear Tooth Nomenclature.

Root-circle diameter

Clearance Base-circle diameter

Outside-circle diameter

Gear Tooth Design

125

Base-circle diameter (dashed line in Figures 4.1 and 4.2) is the diameter established by multiplying the pitch diameter (see above) by the cosine of the pressure angle of the cutter to be used to cut the gear. It is a basic dimension of the gear. [Note: In low-tooth-count gears, the base-circle diameter is greater compared to its root diameter, and is smaller in gears with regular tooth count (for involute gears with standard tooth profile, gears with 41 tooth and fewer, the base-circle diameter is greater compared to the root diameter, and for gears with 42 and more tooth it is smaller compared to the root diameter)]. Root diameter (dashed line in Figures 4.1 and 4.2) is the diameter of the circle that establishes the root lands of the teeth. All tooth elements should lie outside this circle. The tolerance should be negative. Figure 4.3 shows a side view of a spur gear tooth to further depict the features and nomenclature of gear teeth. Active profile (shaded area in Figure 4.3,a) is a surface and is that portion of the surface of the gear tooth which at some phase of the meshing cycle contacts the active profile of the mating gear tooth. It extends from the limit diameter (see Figure 4.1) near the root of the tooth to the tip round (see Figure 4.2) at the tip of the tooth and, unless the mating gear is narrower, extends from one side of the gear or edge round (see Figure 4.3,k) at one end of the tooth to the other side of the gear or edge round. Top land (shaded area in Figure 4.3,b) is a surface bounded by the sides of the gear (see Figure 4.3,d) and active profiles; or if the tooth has been given end and tip rounds (see Figure 4.3,h and 4.3,i), the top land is bounded by these curved surfaces. The top land forms the outside diameter of the gear. Fillet. The fillet of a tooth (shaded area in Figure 4.3,c) is a surface that is bounded by the form diameter (see Figure 4.1) and the root land (if present) (see Figure 4.3,e) and by the ends of the teeth. In full-fillet teeth, the fillet of one tooth is considered to extend from the center line of the space to the form diameter. Sides of gear. The sides of gear (shaded area in Figure 4.3,d) are surfaces and are the ends of the teeth in spur and helical gears. (a)

(b)

Active profile

(e)

(c)

(d)

Top land

(f)

Fillet

Sides of gear

(g)

(h) R

Root land

Transverce profile

(i)

(j)

Normal section

(k)

R

End round FIGURE 4.3

Fillet radius

Nomenclature of Gear Tooth Details.

Edge round

Tip round

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Dudley’s Handbook of Practical Gear Design and Manufacture

Root land (also known as bottom land). The root land (shaded area in Figure 4.3,e) is a surface bounded by fillets (see Figure 4.3,c) of the adjacent teeth and sides of the gear blank. Transverse profile (heavy line in Figure 4.3,f) is the shape of the gear tooth as seen in a plane perpendicular to the axis of rotation of the gear. Axial profile (heavy line in Figure 4.3,g) is the shape of the gear tooth as seen in a plane tangent to the pitch cylinder at the surface of the tooth. In the case of helical gears, it is the shape of a tooth as seen on a pitch cylinder and may be developed to be shown in a plane. Tip round (shaded area in Figure 4.3,h) is a surface that separates the active profile and the top land. It is sometimes applied to gear teeth either to remove the burrs, or to lessen the chance of chipping, particularly in the case of hardened teeth. It may also be added as a very mild and crude form of profile modification. End round (shaded area in Figure 4.3,i) is a surface that separates the sides and the top land of the tooth. It is sometimes applied to gear teeth to reduce the chance of chipping, particularly in the case of hardened teeth. Fillet radius (shaded area in Figure 4.3,j) is the minimum radius that a gear tooth may have. End round (shaded area in Figure 4.3,k) is the surface that separates the active profiles of the teeth from the sides of the gear. These edges are of importance in the cases of helical gears, spiral bevel gears, and worms, since they become very sharp on the leading edge.

4.1.2 BASIC CONSIDERATIONS

FOR

GEAR TOOTH DESIGN

Gear teeth are a series of cam surfaces that contact similar surfaces on a mating gear in an orderly fashion. In order to drive in a given direction and to transmit power or motion smoothly and with a minimum loss of energy, the contacting cam surface on mating gears must have the following properties: • The height and the lengthwise shape of the active profiles of the teeth (cam surfaces) must be such that, before one pair of teeth goes out of contact during mesh, a second pair will have picked up its share of the load. This is called continuity of action. • The shape of the contacting surfaces of the teeth (active profiles) must be such that the angular velocity of the driving member of the pair is smoothly imparted to the driven member in the proper ratio. The most widely used shape for active profiles of spur gears and helical gears that meets these requirements is the involute curve. There are many other specialized curves, each with specific advantages in certain applications. This subject is developed further in the section, “conjugate action”. • The spacing between the successive teeth must be such that a second pair of tooth-contacting surfaces (active profiles) is in the proper position to receive the load before the first leave mesh. Continuity of action and conjugate action are achieved by proper selection of the gear tooth proportions. Manufacturing tolerances on the gears govern the spacing accuracies of the teeth. Thus, to achieve a satisfactory design, it is necessary to specify correct toot proportions, and, in addition, the tolerances on the tooth elements must be properly specified. As a general rule, gearing designed in accordance with the standard systems will not have problems of continuity of action or conjugate action. In those cases where it is necessary to depart from the tooth proportions given in the standard systems, the designer should check both the continuity of action and the conjugate action of the resulting gear design. Continuity of action. As discussed above, all gear tooth contact must take place along the “line of action”. The shape of this line of action is controlled by the shape of the active profile of the gear teeth, and the length of lines of action is controlled by the outside diameters of the gears (see Figure 4.4). In order to provide a smooth continuous flow of power, at least one pair of teeth must be in contact at all

Gear Tooth Design

127 Effective length of line of action

Outside-diameter circles

Base pitch ( pb)

a pb

b

pb

FIGURE 4.4

Zone of Tooth Action.

times. This means that during a part of the meshing cycle, two pairs of teeth will be sharing the load. The second pair of teeth must be designed such that they will pick up their share of the load and be prepared to assume the full load before the first pair of teeth goes out of action. The base pitch p b is defined as follows: pb = p cos

=

cos Pd

(4.1)

where:1 p– is the circular pitch of gear – is the pressure angle of gear Pd – is the diametral pitch

Thus, either the outside-diameter circles, the operating pressure angle, or the base pitch must be adjusted so that ab exceeds the base pitch p b by from 20 to 40%. The most general way of checking continuity of action is by calculating the contact ratio. A numerical index of the existence and degree of continuity of action is obtained by dividing the length of the line of action by the base pitch of the teeth (see Figure 4.4). This is called contact ratio, m p : mp =

La pb

1.2

(4.2)

A spur gear mesh has only a transverse contact ratio, m p , whereas a helical gear mesh has a transverse contact ratio, m p , an axial contact ratio, mF , (face contact ratio), and a total contact ratio, m t . 1

Equations (4.1) through (4.6) given dimensionless. For English-system calculations, use inches for all dimensions. For metricsystem calculations, use millimeters for all dimensions.

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Dudley’s Handbook of Practical Gear Design and Manufacture

Equations for contact ratio are as follows: • Spur gears and helical gears

mp =

( ) ( ) do 2 2

db 2 2

+

( ) ( ) Do 2 2

Db 2 2

C sin

(4.3)

pb

• Internal gears, spur, and helical gears

mp =

( ) ( ) do 2 2

db 2 2

+

( ) ( ) Di 2 2

Dbi 2 2

C sin

pb

(4.4)

• Helical gears, axial contact ratio mF =

F tan p

(4.5)

• Helical gears, total contact ratio mt = m p + mF

(4.6)

where: d o – is the outside diameter (effective) of pinion D o – is the effective outside diameter of gear (diameter to intersection of tip round and active profile); see the following discussion D i – is the inside diameter (effective), internal gear db – is the base diameter of pinion Db – is the base diameter of gear Dbi – is the base diameter of internal gear C – is the operating center-distance of pair – is the operating pressure angle p– is the circular pitch (in plane of rotation) pb – is the base pitch F – is the length of tooth, axial, the face width of the gear – is the helix angle of helical gears

Notes: To achieve correct answers on contact ratio, the following points should be observed. • The effective outside diameter d o or D o is actually the diameter to the beginning of the tip round, usually D o = Do 2 (edge round specification) (max). This value rather than the drawing outside diameter should be used since in many cases manufacturing practices for removing burrs produce a large radius, particularly in fine-pitch gears. Thus, a considerable percent of the addendum may not be effective. If the teeth are given a very heavy profile modification, consideration should be given to performing the calculation under (a) full load, assuming contact to the tip of the active profile, and (b) light load, assuming contact near the start of modification. In this case d o or D o is selected

Gear Tooth Design

129

to have a value close to the diameter at the start of modification. This will give an index to the smoothness of operation at these conditions. • This equation also assumes that the form diameter is a larger value than the undercut diameter. If not, use the value of undercut diameter. • On occasion, the outside diameter of one or both members are so large relative to the centerdistance and operating pressure angle that the tip extends below the base circle when it is tangent to the lien of action. Since no involute action can take place below the base circle, the value C sin (the total length of the line of action) should be substituted in the equation in place of the value (d o /2)2 (db /2)2 or (D o /2)2 (Db /2)2 in the case that one or both become larger than the value of C sin . Conjugate action. Gear teeth are a series of cam surfaces that act on similar surfaces of the mating gear to impart a driving motion. Curves that act on each other with a resulting smooth driving action and with a constant driving ratio are called conjugate curves. The fundamental requirements governing the shapes that any pair of these curves must have are summarized in Willis’ “basic law of gearing” (1841), which states: Basic law of gearing: Normals to the profiles of mating teeth must, at all points of contact, pass through a fixed point located on the line of centers.2 (This law of gearing was known to L. Euler, and to F. Savary. Nowadays, this law of gearing is commonly referred to as “Camus-Euler-Savary fundamental law of parallel-axes gearing”, or as “CES–fundamental law of parallel-axes gearing”, for simplicity). In the case of spur- and helical-type gears, the curves used almost exclusively are those of the involute family. In this type of curves, the fixed point mentioned in the basic law is the pitch point. Since all contact takes place along the line of action, and since the line of action is normal to both the driving and driven involutes at all possible points of contact, and, lastly, since the line of action passes through the pitch point, it can be seen that the involute satisfies all requirements of the basic law of gearing. Worm gearing, like bevel gearing, is non-involute. The tooth form of worm gearing is usually based on the shape of the worm; that is, the teeth of the worm gear are made conjugate to the worm. In general, worms can be chased, as on a lathe, or cut by such process as milling or hobbing, or can be ground. Each process, however, produces a different shape of worm thread and generally requires a different shape of worm gear tooth in order to run properly. In the case of face gearing, the pinion member is a spur or helical gear of involute form, but the gear tooth is a special profile conjugate only to the specific pinion. Thus, pinions having a number of teeth larger or smaller than the number for which the face gear was designed will not run properly with the face gear. Pitch diameter. Although pitch circles are not the fundamental circles on gears, they are traditionally the starting point on most tooth designs. The pitch circle is related to the base circle, which is the fundamental circle, by the relationships that follow. Some authorities list as many as nine distinct de­ finitions of pitch circles. The following definitions cover the pitch circles considered in this chapter. Standard pitch diameter. The diameters of the circle on a gear determined by dividing the number of teeth in the gear by the diametral pitch. The diametral pitch is that of the basic rack defining the pitch and pressure angle of the gear: D=

2

N Pd

(4.7)

See: Willis, R. (1841). Principles of mechanisms, Designed for the use of students in the universities and for engineering students generally. Cambridge: J. & J.J. Deighton.

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Dudley’s Handbook of Practical Gear Design and Manufacture

where: D – is the diameter of standard pitch circle Pd – is the diametral pitch of basic rack N – is the number of teeth in gear

Operating pitch diameter. The diameter of the circle on a gear, which is proportional to the gear ratio, and to the actual center-distance, at which the gear pair will operate. A gear does not have an operating pitch diameter until it is meshed with a mating gear. The equations for operating pitch diameter are: • External spur and helical gearing3 d =

2 C mg + 1

2 C mg

D =

mg + 1

pinion member

(4.8)

gear member

(4.9)

pinion member

(4.10)

gear member

(4.11)

• Internal spur and helical gearing d =

2 C mg 1

D =

2 C mg mg

1

where: d – is the operating pitch diameter of pinion D – is the operating pitch diameter of gear C – is the operating center-distance of mesh number of teeth in gear of teeth in pinion

mg – is the gear ratio = number

The operating and the standard pitch circles will be the same for gears operated on center-distances that are exactly standard. The distinction to be made usually involves tolerances on the gear centerdistance. Most practical gear designs involve center-distance tolerances that are the accumulated effects of machining tolerances on the center bores and tolerances in bearings (clearances, runout of outer races, and so forth). Thus, gears all operate with maximum and minimum operating pitch diameters. In Figure 4.5, a pair of gears designed to operate on enlarged center-distances is shown. Both the standard and operating pitch circles are shown. It will be noted that the pitch circles are related to the base circle as follows: Db D= standard pitch diameter (4.12) cos

3

Eq. (4.8) through (4.13) should use inches for English calculations, and millimeters for metric calculations.

Gear Tooth Design

131

Gear Standard pressure angle Operating pressure angle

CL

Root diameter circle, gear Base diameter circle, gear Form diameter circle, gear Standard diameter circle, gear Operating diameter circle, gear Outside diameter circles Operating pitch diameter circle, pinion Standard pitch diameter circle, pinion

Pinion

FIGURE 4.5 Distance.

Standard pressure angle Operating pressure angle

Form diameter circle, pinion Base diameter circle, pinion Root diameter circle, pinion

Nomenclature for Gear Cut to Standard Pressure Angle and Operated at an Increased Center-

D =

Db cos

operating pitch diameter

(4.13)

It will also be noted that the standard pitch circles do not contact each other by the amount that the center-distance has been increased from standard. In worm gearing, it is convenient to use pitch circles. In this case, however, it is common practice to make the pitch circle of the gear go through the teeth at a diameter at which the tooth thickness is equal to the space width. In the case of the worm, the pitch circle also defines a cylinder at which the width of the threads and spaces are equal. It is also good practice to modify the worm teeth slightly to achieve the required backlash. When this is done, the space widths are greater than the thread thicknesses, as measured on the “standard” pitch cylinder, by the amount of backlash introduced. The pitch cylinder also defines the diameter at which the lead angle, as well as the pressure angle, is to be measured. Zones in which involute gear teeth exist. Although many gear designs utilize “standard” or “equaldedendum” tooth proportions, it is not always necessary or even desirable to use these proportions. One of the outstanding features of the involute tooth profile is the opportunity it affords for the use of different amounts of addendum and tooth thicknesses on gears of any given pitch and numbers of teeth. These variations can be produced with standard gear tooth generating tools. It is not necessary to buy different cutting or checking tooling for each new value of tooth thicknesses or addendum in the proper tooth thicknesses-addendum relationship in maintained in the design.

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Dudley’s Handbook of Practical Gear Design and Manufacture

As discussed in other chapters of this handbook, the limits of true involute action are established by the length of the line of action. It has been shown that the end limits defining the maximum usable portion of the line of action are fixed by the points at which this line becomes tangent to the basic circles. These limits, a and b, are shown in Figure 4.6,a. The largest pinion or gear that will have correct gear tooth action is defined by circles that pass through points a and b (see Figure 4.6,b). Therefore, any gear teeth that lie fully within the crosshatched area on Figure 4.6,b will have correct involute action on any portions of the teeth that are not undercut. Undercut limitations are discussed more fully under “un­ dercut” below. One should not infer that the largest usable outside diameters will be used simultaneously on both members on a given gear design. The equations for maximum usable outside diameter are:4 DoG = 2 (C sin

)2 +

DG cos 2

2

DoP = 2 (C sin

)2 +

DP cos 2

2

max

(4.14)

min

(4.15)

where: C – is the operating center-distance – is the cutting or standard pressure angle – is the operating pressure angle DG – is the pitch diameter of gear DP – is the pitch diameter of pinion

Operating pitch circles for the gear ratio (2:1) and center-distance chosen for this illustration are shown in place in Figure 4.6,c. If this center-distance is “standard” for the number of teeth and pitch, the operating pitch circles shown also be the “standard” pitch circles. To transmit uniform angular motion, a series of equally spaced involute curves are arranged to act on each other. These are shown in place in Figure 4.6,d. On both the pinion and the gear member, the involute curves originate at the base circles and theoretically can go on forever. The more practical lengths are suggested by solid lines. The spacing of the involutes of both members measured on the base circles must be equal, and the interval chosen is called the base pitch. The base pitch of the gear or pinion times the number of teeth in the member must exactly equal the circumference of the base circles. If similar involute curves of opposite hand are drawn for both base circles, the familiar gear teeth are achieved. It is customary to measure the distance from one involute curve to the next along the standard pitch circle. This distance is called the circular pitch. It is also customary to make “standard” tooth proportions with tooth thicknesses equal to one-half the circular pitch. The standard addendum used for the gearing system shown in this chapter is equal to the circular pitch divided by . Figure 4.6,e shows “standard” teeth developed on the base circles of Figure 4.6,a through Figure 4.6,d. It will be noted that the addendum of the 12-tooth pinion is equal to that of the gear and that the outside diameters Thus, established do not reach out to the maximum values established by the lineof-action limits. In modern gear-cutting practice, the tooth thickness of gears as measured on the “standard” pitch circle is established by the depth to which the generating-type cutter (usually hob or shaper cutter) is fed, relative to the “standard” pitch diameter. In order to obtain a correct whole depth of tooth, the outside

4

Equations (4.14) through (4.18) should use inches for English calculations, and millimeters for metric calculations.

a

Dbp 2

b

2

Dbg

Pinion

Gear

(f )

Maximum usable outside diameter of pinion

Pinion Dbp(max) 2 a

Gear

Standard outside diameter crcle

Working outside diameter circle

Base circle, gear

Base circle, pinion

(b)

∆d

Dbg(max) 2

b Gear

Pinion Dp 2

Gear

Pinion "Pointed tooth" outside diameter

Maximum usable outside diameter of gear

c

(g)

a

(c)

b Base circle, gear Gear

Pitch circle, gear

Undercut diameter

Gear

pb

pb

(h)

pb

pb

Pitch circle, Pinion pinion

Theoretical outside Pinion diameter (for desired tooth thickness)

Dg 2

Base circle pinion

(d)

pb

pb

Base circle, gear

Base circle, pinion

FIGURE 4.6 Study of Involute Tooth Development: (a) Maximum Usable Length of Line of Action; (b) Maximum Zone in Which Conjugate Action Can Take Place; (c) Operating Pitch Circles; (d) Development of Involute Profiles; (e) Standard Addendum Tooth Proportions; (f) Short- (Pinion) Addendum Tooth Proportions; (g) Long- (Pinion) Addendum (Abnormal) Tooth Proportions; (h) Long- (Pinion) Addendum (Normal) Tooth Proportions.

Gear

Pinion

(e)

Gear

Pinion

(a)

Gear Tooth Design 133

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Dudley’s Handbook of Practical Gear Design and Manufacture

diameters of the gear blanks are made larger or smaller than “standard” by twice the amount that the cutter will be fed in or held out relative to the normal or standard amount that would be used for standard gears. If the cutter is to be held out a distance c , the outside diameter of the blank is made 2 c larger than standard and the resultant gear is said to be long addendum. Figure 4.6,f shows an example of a gear in which the cutter was held out sufficiently so that the outside diameter of the gear was equal to the diameter of the maximum useable outside-diameter circle (see also Figure 4.6,b). This is called a long addendum gear. The cutter was fed into the pinion an equal amount, making it short addendum. Reasons for making pinions short addendum are discussed under “Speed increasing drive,” and the effects of the undercutting produced are discussed under “Undercut.” In order to avoid undercut, or to achieve a more equal balance in tooth strength, it is customary to make the pinion addendum long and that of the gear short. If an attempt is made to design a pinion with the maximum usable outside diameter as established by the line of action (see Figure 4.6,b), difficulties may arise. In the example shown having 12 and 24 teeth, it is not possible to generate a pinion having such an outside diameter. In Figure 4.6,g is shown the tooth form resulting from such an attempt. The pinion blank was turned to the maximum usable outside diameter, and the cutter was held out an equivalent distance c . As a result of the generating action, the sides of the teeth are involute curves “crossed over” at a diameter smaller than the desired outside diameter. Thus, the teeth are pointed at the outside diameter and also the whole depth is less than anticipated. This effect is less serious in gears having large number of teeth. The gear tooth that results from this extreme modification is badly undercut. This effect would not have been so great if the gear had more teeth. The amount of long and short addendum that may be applied to each member of a gear mesh is limited by the three following considerations: • The length of the usable portion of the line of action will form a maximum limit on the outside diameter of a gear. Diameters in excess of this will not provide additional tooth contact area, since there is no involute portion on the mating gear that can contact this area • The diameter at which the teeth become pointed limits the actual or effective outside diameter • Undercutting may limit the short-addendum gear. The undercut diameter should be always less than the form diameter. These three considerations show the extreme limits that bound gear tooth modifications. The designer should not infer that it is necessary to approach these limits in any given gear design. In the treatment of gear tooth modifications, some reasons for making given tooth modifications are considered. In each case, however, the designer must be sure that the amount actually used does not exceed the limits discussed here. Pointed teeth. The previous sections have shown how gear teeth generated by a specific basic rack can have different tooth thicknesses, and that the outside diameters of such gears are altered from standard as a function of the change in tooth thickness. In practice, these tooth profiles are achieved by “feeding in” or “holding out” the gear-generating tools. It is customary to feed a specific cutter to a definite depth into the gear blank which has a greater than standard outside diameter. The cutter, when working at its full cutting depth, will still be held out from the standard N / Pd pitch circle. Teeth generated thicker than standard will have tips of a width less than standard, since the cutter must be “held out”. For any given number of teeth, the tooth thickness can be increased such that the tip will become pointed at the outside-diameter circle. In Figure 4.7, four gears all having the same number of teeth, diametral pitch, and pressure angle are shown. In Figure 4.7a, the cutter shown by a rack has been fed to the standard depth. The outside diameter of this gear is standard, (N + 2) Pd . Note the tooth thickness at the tip. In Figure 4.7b, the outside diameter was somewhat enlarger and the cutter fed in the standard whole-depth distance starting from the enlarged outside diameter. This results in the thicker tooth (which would operate correctly with a standard gear on an enlarged center distance) and a

Gear Tooth Design

135

(a) Standard pitch line of cutter

(b) Outside diameter circle Standard pitch circle of gear

Standard pitch line of cutter Amount cutter is "held out"

Outside diameter circle

Standard pitch circle of gear

(c)

(d) Standard pitch line of cutter

Outside diameter circle

Standard pitch circle of gear

Tooth thickness

Amount cutter is "held out"

Outside diameter circle for full depth teeth Resultant outside diameter circle

Standard pitch line of cutter

Circular pitch Amount cutter

Tooth thickness

Standard pitch Tooth thickness circle of gear

is "held out"

FIGURE 4.7 Cutting Long-Addendum Teeth: (a) Standard Tooth; (b) Long-Addendum Tooth; (c) LongAddendum (Pointed) Tooth; (d) Long-Addendum (Over-Enlarged) Tooth.

somewhat thinner tip. In Figure 4.7c, the outside diameter has been established at a maximum value at which it is still possible to achieve a standard whole-depth tooth; the tooth Thus, generated is pointed. Figure 4.7d shows what happens if the maximum outside diameter and tooth thickness are exceeded. The resulting tooth does not have the correct whole depth because the involute curves cross over below the expected outside diameter. This tooth is similar to the one shown on the pinion member in Figure 4.6g. The amount that the outside diameter of a gear is to be modified is usually a function of the tooth thickness desired. (4.18) below gives the relationship usually employed. The maximum amount that the tooth thickness of a gear can be increased over standard to just achieve a pointed tooth can be calculated from the simultaneous5 solution of (4.16) and (4.17):

Domax = N

5

cos cos

1

2

(4.16)

Equations (4.16) and (4.17) are solved by assuming a series of values for . Curves are plotted for vs. . The point where the curves cross indicates a simultaneous solution for the two equations. For instance, a 20-tooth pinion of 1 pitch having a pressure angle and cut with a standard cutter has a crossing point for of and of 2.44 in. It should be kept in mind that these equations solve problems for one pitch only (divided by pitch to adjust answer for other pitches) and make no allowance for backlash. In the metric system the solution is in millimeters and for one module. (For other modules, multiply the answer by the module).

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Dudley’s Handbook of Practical Gear Design and Manufacture

Domax =

N (inv

inv

)

2

tan

(4.17)

The equivalent amount that the tooth thickness must be increased above the standard to achieve this increase in outside diameter is given by (4.18): T = d (inv

inv )

T

(4.18)

where Domax is the maximum outside diameter, at which a tooth having full working depth will come to a point, as follows: Domax =

N+2 + P

Do

(4.19)

where: T – is the tooth thickness at standard pitch diameter – is the standard pressure angle of hob or rack – is the pressure angle at tip of tooth

Undercut. An undercut tooth is one in which a portion of the profile in the active zone has been removed by secondary cutting action. Under certain conditions, the path swept out by the tip of a generating-type cutter will intersect the involute active profile at a diameter greater than the limit dia­ meter. Such a tooth is said to be undercut since no contact with the mating gear can take place between the undercut and the limit circles. The amount of undercut will depend on the type of tool used to cut the gears. Form-type cutters do not normally produce undercut. In calculating the undercut diameter, it is customary to use an equation based on the type of tooling that will produce the greatest undercut, usually a hob. This is done in order to give the shop the greatest freedom of choice in selecting tools. If the most adverse choice will produce a satisfactory tooth profile, then all other types of cutter that might be chosen by the shop would prove to be satisfactory, except in those cases where undercut is intended to provide clearance for the tips of the mating teeth. In general, spur gears of 20˚ pressure angle having 18 or more teeth and made to standard- or longaddendum tooth proportions will not be undercut. In each of the “standard systems” for different types of gear, the minimum number of teeth recommended for each pressure angle is usually based on undercut. Helical gears can usually be made with fewer tooth than can spur gears without getting into problems of undercut. Figure 4.8 shows three gears having the same number of teeth, each produced by generating-type cutters. Figure 4.8a is the tooth form of a gear having standard addendum. Figure 4.8b shows the same shape of the tooth that results when the tip of the rack-type cutter is operating at a depth exactly passing through the point of intersection of the line of action and the base circle. Figure 4.8c shows the shape of a tooth with results when the cutter works at a depth somewhat below the intersection of the line of action and the base circle. Undercutting is product of generating-type cutters. Gears cut by forming-type cutters do not usually have undercut. In certain cases, undercut may ensure tip clearance in mating gears. Racks operating with gears having small number of teeth may show tip interference unless the pinions were generated with special shaper-type cutters or hobs.

Gear Tooth Design

137

(a) Standard pitch line of cutter

Outside diameter circle Standard pitch circle of gear

(b) Standard pitch line of cutter

Outside diameter circle Amount cutter is "fed in"

Standard pitch circle of gear

(c) Standard pitch line of cutter

Outside diameter circle Amount cutter is "fed in"

Standard pitch circle of gear FIGURE 4.8 Cutting Short-Addendum Teeth: (a) Standard Tooth; (b) Short-Addendum Tooth; (c) ShortAddendum (Abnormal) Tooth.

4.1.3 LONG-

AND

SHORT-ADDENDUM GEAR DESIGN

The previous article shows the geometric limitations on the amount that gears can be made long or short addendum. This section indicates cases in which long and short addendum should be considered. Modification of the addendum of the pinion, and in most cases the gear number, is recommended for gears serving the following applications: • Meshes in which the pinion has a small number of teeth • Meshes operating on non-standard center-distances because of limitations on ratio or centerdistances • Meshes of speed-increasing drives • Meshes designed to carry maximum power for the given weight allowance. (This type of gearing is usually designed to achieve the best balance in strength, wear, specific sliding, pitting, or scoring) • Meshes in which an absolute minimum of energy loss through friction is to be achieved Addendum modification for gears having small number of teeth. Undercutting is one of the most serious problems occurring in gearing having small numbers of teeth. The amount that gears with small numbers of teeth should be enlarged (made long addendum) to avoid undercut, has been standardized.

138

Dudley’s Handbook of Practical Gear Design and Manufacture

The values of modification are based on the use of a hob or rack-type cutter and as a result are more than adequate for gears cut with circular, shaper-type cutters. The values of addendum modification re­ commended for each number of teeth are shown in Table 4.1. When two gears, each containing a small number of teeth, must be operated together, it may be necessary to make both members long addendum to avoid undercut. Such teeth will have a tooth thickness which is larger than standard and which will necessitate the use of greater-than-standard center-distance for the pair. Modifications to avoid undercut fall into three categories: • If both members have fewer teeth than the number critical to avoid undercut, increase the centerdistance so that the operating pressure angle is increased. Then get appropriate values of the addendum and whole depth for the pinion and for the gear • If the pinion member has fewer teeth than the critical number, and the mating gear has con­ siderably more, the usual practice is to decrease the addendum of the gear by the amount pro­ portional to the amount that the pinion is increased. This results in a pair of gears free from undercut that will operate on a standard center-distance • If the pinion member has fewer teeth than the critical number, and the gear just slightly more, a combination of the first two practices above may be employed An alternate is to increase the pinion addendum by the required amount and increase the center-distance by an equivalent amount to make it possible to use a standard gear. Speed-increasing drives. Most gear trains are speed reducing (torque increasing), and most data on gear tooth proportions are based on the requirements of this type of gear application. The kinematics of speed-increasing drives is somewhat different, and as a result, special tooth proportions should be considered for this type of gear application. As in the case of conventional drives, the problems to be discussed here are most serious in meshes involving small number of teeth. The first of these problems involves the tendency of the tip edge of the pinion tooth to gouge into the flank of the driving gear tooth. This gouging can come about as a result of spacing errors in the teeth of either member which allow the flank of the gear tooth to arrive at the theoretical contact point on the line of action before the pinion does. The pinion tooth has to deflect to get into the right position or it will gouge off a sliver of gear tooth side. If the gears are highly loaded, the unloaded pinion tooth entering the mesh will be out of position (lagging) since it is not deflected. The gear tooth is in effect slightly ahead of where it should be. The result is the same as if the gear tooth had an angular position error. The bearing and lubricating problems at the beginning point of contact are particularly bad. The edge of the pinion tooth tends to act as a scrapper and remove any lubricating film that may be present, for some distance along the flank of the tooth. One possible solution to this problem is to give the tip of the pinion tooth a moderate amount of tip relief. This provides a sort of sled-runner condition which is easier to lubricate and which helps the pinion tooth find the proper position relative to the gear tooth with less impact. A better solution, which may also be combined with the tip modification, is to modify the tooth thickness and addendums so as to get as much of the gear tooth contact zone in the arc of recess as possible. Power drives (optimal design). As shown in other chapters, gears fail in one or more of the following ways: actual breaking of the teeth, pitting, scoring, or by wear. In the case of drives with gears of standard tooth proportions and similar metallurgy, the weakest member is the pinion, and if tooth breakage does occur, it is generally in the pinion. This is a result of the weaker shape of the pinion tooth, as well as the larger number of fatigue cycles that it accumulates. This problem can be relieved to a considerable degree by making the pinion somewhat longer and, in so doing, increasing the thickness of the teeth and also improving their shapes. If a standard center-distance is to be maintained, the gear addendum is reduced a proportional amount. If the proper values are chosen, the pinion tooth strength

1.468 1.409

1.351

1.292 1.234

1.175

1.117 1.058

1.000

10 11

12

13 14

15

16 17

18

…

… …

Gear

1.000

0.883 0.942

0.825

0.708 0.766

0.649

0.532 0.591

G

…

19 18

20

22 21

23

25 24

…

… …

Recommended Min. No. of Teeth P

1.000

1.000 1.000

1.000

1.000 1.000

1.000

1.184 1.095

…

… …

pinion G

1.000

1.000 1.000

1.000

1.000 1.000

1.000

0.816 0.905

…

… …

Gear

12

12 12

12

12 12

12

15 14

…

… …

Recommended Min. No. of Teeth

Coarse-Pitch Teeth (1 through 19 Pd ), 25˚

1.0000

1.0642 1.0057

1.1227

1.2397 1.1812

1.2982

1.4151 1.3566

1.0000

0.9358 0.9943

0.8774

0.7604 0.8189

0.7019

0.5849 0.6434

0.5264

Gear

1.4190*

G

0.4094 0.4679

Pinion

1.4143* 1.4369*

P

…

19 18

21

25 23

27

33 30

36

42 39

Recommended Min. No. of Teeth

Fine-Pitch Teeth (20 Pd and finer), 20˚

Note: The values in this table are for gears of 1 diametral pitch. For other sizes divide by the required diametral pitch. The values of addendum shown are the minimum increase necessary to avoid undercut. Additional addendum can be provided for special applications to balance strength. See Table 4.2. * These values are less than the proportional amount that the tooth thickness is increased (see Table 4.15) in order to provide a reasonable top land.

…

9

Pinion

… …

P

Coarse-Pitch Teeth (1 through 19 Pd ), 20˚

7 8

No. of Teeth in Pinion

TABLE 4.1 Values of Addendum

Gear Tooth Design 139

140

Dudley’s Handbook of Practical Gear Design and Manufacture

will be increased and the gear tooth strength somewhat reduced, which will result in almost equal gear and pinion tooth strength. This will result in an overall increase in the strength of the gear pair. Several authorities have suggested addendum modifications which will balance scoring, specific sliding, and tooth strength. Unfortunately, each balance results in different tooth proportions so that the designer has to use proportions that will balance only one feature or else proportions that are a com­ promise. Table 4.2 gives values that are such a compromise. Experimental data seem to indicate that a pair that is corrected to the degree that seems to be indicated by tooth layouts or by calculation for balanced tooth strength will usually result in an overcorrection to the pinion member. The gear is not as strong as form factors seem to indicate. Notch sensitivity in higher hardness ranges seems to be a problem, especially if the gear is to experience a great number of cycles of loading. In small numbers of teeth, the correction required to avoid undercut on gears operated on standard center-distances is excessive, in many cases, in respect to equal tooth strength. An overcorrection in pinion tooth thickness can lead to an excessive tendency to score. As a result, the values of addendum recommended in Table 4.2 represent a compromise among balanced strength, sliding, and scoring.

TABLE 4.2 Values of Addendum for Balance Strength m G , (Gear Ratio) NG /NP

a (Addendum)

m G , (Gear Ratio) NG /NP

a (Addendum)

From

To

Pinion, aP

Gear, a G

From

To

Pinion, aP

Gear, a G

1.000 1.001

1.000 1.020

1.000 1.010

1.000 0.990

1.421 1.451

1.450 1.480

1.240 1.250

0.760 0.750

1.021

1.030

1.020

0.980

1.481

1.520

1.260

0.0740

1.031 1.041

1.040 1.050

1.030 1.040

0.970 0.960

1.521 1.561

1.560 1.600

1.270 1.280

0.730 0.720

1.051

1.060

1.050

0.950

1.601

1.650

1.290

0.710

1.061 1.081

1.080 1.090

1.060 1.070

0.940 0.930

1.651 1.701

1.700 1.760

1.300 1.310

0.700 0.690

1.091

1.110

1.080

0.920

1.761

1.820

1.320

0.680

1.111 1.121

1.120 1.140

1.090 1.100

0.910 0.900

1.821 1.891

1.890 1.970

1.330 1.340

0.670 0.660

1.141

1.150

1.110

0.890

1.971

2.060

1.350

0.650

1.150 1.170

1.170 1.190

1.120 1.130

0.880 0.870

2.061 2.161

2.160 2.270

1.360 1.370

0.640 0.630

1.190

1.210

1.140

0.860

2.271

2.410

1.380

0.620

1.210 1.231

1.230 1.250

1.150 1.160

0.850 0.840

2.411 2.581

2.580 2.780

1.390 1.400

0.610 0.600

1.251

1.270

1.170

0.830

2.781

3.050

1.410

0.590

1.271 1.291

1.290 1.310

1.180 1.190

0.820 0.810

3.051 3.411

3.410 3.940

1.420 1.430

0.580 0.570

1.311

1.330

1.200

0.800

3.941

4.820

1.440

0.560

1.331 1.361

1.360 1.390

1.210 1.220

0.790 0.780

4.821 6.811

6.810

1.450 1.460

0.550 0.540

1.391

1.420

1.230

0.770

–

–

–

–

Note: Do not select values from this table for the pinion member that are smaller than those given in Table 4.1.

Gear Tooth Design

141

Gears with teeth finer than about 20 diametral pitch (generally) cannot score, since the tooth is not strong enough to support a scoring load; therefore, the values for addendum increase in fine-pitch gears are somewhat larger than the values for coarse-pitch power gearing. Low-friction gearing. In cases where a speed-increasing gear train is to transmit power or motion with the least possible loss of energy, the selection of the tooth proportions is of considerable im­ portance. The sliding should be kept as low as possible, and as much of the tooth action should be put into the arc of recess as possible. Figure 4.9 shows two involute curves (tooth profiles) in contact at two different points along the line of action. The direction in which the driven pinion tooth slides along the driving gear tooth is shown by the arrows. This example is a speed-increasing drive which is the most sensitive to friction between the teeth. At the pitch point (where the line of centers crosses the line of action) there is no sliding, and it is at this point that the direction of relative sliding of one tooth on the other changes. The forces shown in Figure 4.9 are those acting on the driven pinion. The subscripts “a” are the values considered in the arc of approach and “r” are those considered in the arc of recess. The normal driving force WN is the force that occurs at the pitch point, if there were no friction at the point of gear tooth contact, would be the force at all other points of contact along the line of action. Since there is friction, the friction vectors fa and fr oppose the sliding of the gear teeth in the arcs of approach and recess. Note the change in direction due to the change in direction of sliding. The angle of friction is and is assumed R f Pa

Rf Pr

RNP

fr Wfr

Wn

Φ

Wn

fa Wfa

Φ

R f Gr RNG

Rf Ga

TDR FIGURE 4.9

Effect of Friction on Tooth Reactions.

142

Dudley’s Handbook of Practical Gear Design and Manufacture

to be the same in both cases. The torque exerted by the shaft driving the driving gear TDR manifests itself in arc of approach as: TDRa = WN × RNG TDRa = W fa × RfGa

if no friction

(4.20)

if friction is assumed

(4.21)

and in the arc of approach as: TDRa = WN × RNG TDRr = W fr × RfGr

if no friction

(4.22)

if friction is assumed

(4.23)

The resisting moments are, in the arc of approach: TDNa = WN × RNP

TDNa = W fa × Rf

Pa

if no friction

(4.24)

if friction is assumed

(4.25)

and the corresponding moments are, in the arc of recess: TDNr = WN × RNP TDNr = W fr × Rf

Pr

if no friction

(4.26)

if friction is assumed

(4.27)

Note that, in all cases above, single tooth contact is assumed. Efficiency is output divided by input and in this case is the torque that would appear on the driven shaft when friction losses are considered, compared with the torque that would result if no losses occurred. (4.28) below shows the efficiency of the mesh (single tooth contact) for the contacts occurring in the arc of approach, and (4.29) shows the efficiency in the arc of recess:6 Eapproach =

E recess =

Rf

Pa

Rf

Ga

Rf

Pr

Rf

Gr

RNG RNP RNG RNP

(4.28)

(4.29)

In the case of speed-increasing drives, the increase efficiency can have considerable significance; cases have occurred in which, for every high ratios and poor lubrication, the speed increaser actually became self-locking.

6

Equations (4.28) and (4.29) can be used in the metric system by using newtons for force and millimeters for distance. This makes the units of torque .

Gear Tooth Design

143

4.1.4 SPECIAL DESIGN CONSIDERATIONS Several special considerations should be kept in mind during the evolution of a gear design. Interchangeability. Two types of interchangeability are related to gearing. The first, and most generally recognized, is part interchangeability. This means that, if a part made to a specific drawing is damaged, a similar part made to the same drawing can be put in its place and can be expected to perform exactly the same quality of service. In order to achieve this kind of interchange­ ability, all parts must be held to carefully selected tolerances during manufacture, and the geometry of the parts must be clearly defined. The second kind is engineering interchangeability. This type provides the means whereby a large variety of different sizes or number of teeth can be produced with a very limited number of standard tools. Parts so made may or man not be designed to have part interchangeability. The several systems of gear tooth proportions are based on engineering interchangeability. By careful design and skill in the application of each system, it is possible to design gears that will perform in almost any application and that can be made with standard tools. Only in the most exceptional cases will special tools be required. The advantages of gearing designed with engineering inter­ changeability are the lower tool costs and the ease with which replacement or alternate gears can be designed. Gears procured through catalogue source are good examples of gears having both part and en­ gineering interchangeability. Tooth thickness. The thickness of gear teeth determines the center-distance at which they will operate, the backlash that they will have, and, as discussed in previous sections, their basic shape. One of the important calculations made during the design of gear teeth is establishment of tooth thickness. It is essential to specify the distance from the gear axis at which the desired tooth thickness is to exist. The usual convention is to use the distance that is established by the theoretical pitch circle N / P . Thus, if no other distance is shown, the specified tooth thickness is assumed to lie on the standard pitch circle. In certain cases, the designer may wish to specify a thickness, such as the chordal tooth thickness, at a diameter other than the standard pitch diameter. This specified diameter should be clearly defined on the gear drawings. The actual calculation of tooth thickness is usually accomplished by the following procedure: 1. Theoretical tooth thickness is established. a. If the gears are of conventional design and are to be operate on standard center-distance, the tooth thickness used is one-half the circular pitch. b. If special center-distances are to be accommodated, the below (4.30) and (4.31) may be used: cos

2

=

C cos C2

(4.30)

where: – is the operating pressure angle C – is the standard center-distance – is the standard pressure angle C2 – is the operating center-distance 2

TP + TG =

+ (inv

2

inv

) (NP + NG )

(4.31)

144

Dudley’s Handbook of Practical Gear Design and Manufacture

where:7 TP – is the tooth thickness of pinion member (at 1 diametral pitch) TG – is the tooth thickness of gear member (at 1 diametral pitch) NP – is the number of teeth in pinion NG – is the number of teeth in gear

c. If special tooth thicknesses for tooth strength considerations are required, the total thickness (sum of pinion and gear) must satisfy (4.31). If standard centers are used the sum of the modified thickness must equal 1 circular pitch. 2. After the theoretical tooth thickness is established, the allowance for backlash is made. a. Backlash allowance may be shared equally by pinion and gear. In this case the theoretical tooth thickness of each member is reduced by one-half the backlash allowance. b. In case of pinions with small numbers of teeth which have been enlarged to avoid problems of undercut, it is customary to take all the backlash allowance on the gear member. This avoids the absurdity of increasing the tooth thickness to avoid undercut and then thinning the teeth to introduce backlash. In case (a): B 2

(4.32)

= Ttheoretical

(4.33)

Tactual = Ttheoretical

where B is backlash allowance. In case (b): Tactual,

pinion

Tactual,gear = Ttheoretical

B

(4.34)

3. The teeth are lastly given an allowance for machining tolerance. This tolerance gives the machine operator a size or processing tolerance. Usually this is a unilateral tolerance. It is beyond the scope of this general-purpose book on gear technology to go deep into gear design details. Tooth profile modifications. Errors of manufacture, deflections of mountings under load, and de­ flections of the teeth under load all combine to prevent the attainment of true involute contact in gear meshes. As a result, the teeth do not perform as they are assumed to by theory. Premature contact at the tips or excessive contact pressures at the ends of the teeth give rise to noise and/or gear failures. In order to reduce these causes of excessive tooth loads, profile modification is a usually practice. Remember: Only base pitch preserving modification of the tooth profile is allowed. Transverse of profile modification. It is customary to consider the shape of the individual tooth profile. Actually, an operating gear presents to the mating gear tooth profiles whose shapes are distorted by errors in the profile shape (involute) and in tooth spacing, as well as by deflections due to contact loads acting at various places along the tooth. Ideally, teeth under load should appear to the mating gear 7

The best way to use this equation is to start with C being the center-distance for 1 diametral pitch (or for 1 module in metric dimensions). The C2 is computed by Eq. (4.30). The tooth thicknesses determined are not at the operating pitch diameters but are at the 1 diametral pitch center-distance of C.

Gear Tooth Design

145

to have perfect involute profiles and to have perfect spacing. This would require a tooth profile when subjected to a load. Since loads vary, this is not practical. Usual practice is to give tip or root profile modifications or otherwise correct involute profiles that will be distorted by contact loads. Tip modification usually takes the form of a sliding thinning of the tip of the tooth, starting at a point about halfway up the addendum. The amount of this modification is based on the probable accumulated effects of the following. Allowances for errors of gear manufacture. The correct involute profile may not be achieved as a result of manufacturing tolerances in the cutter used to cut the gear teeth or in the machine guiding the cutter. Errors in spacing may also be introduced by the cutter or machine. The result is that the tips of the teeth will attempt to contact the mating gear too soon or too late. Tip modification produces a sort of sled-runner shape to help guide the teeth into full contact with the least impact. Allowances for deflection under load. Although the teeth of a gear may have a correct profile under static conditions, the loads imposed may deflect the teeth in engagement to such a degree that the teeth that are just entering mesh will not be in relative positions that are correct for smooth engagement. These tooth deflections cause two errors: The actual tooth profiles are not truly conjugate under load and therefore do not transmit uniform angular motion, and the spacing, or relative angular placement of the driving teeth relative to the driven teeth is such that smooth tooth engagement cannot take place. Several methods may be used to compensate for these effects. The most usual is to provide tip relief as discussed above. The flanks of the teeth may also be relieved. Tip and flank relief assume that the normal tooth would tend to contact too soon. If the tips of the driven gear are made slightly thin, the tooth will be able to get into a position on the line of contact before contact between that tooth and its driving mate actually occurs. The sled-runner effect of the tip modification will allow the tooth to assume full contact load gradually. In the case of spur gearing, the amount of tip relief should be based on the sum of the allowance for probable tooth-to-tooth spacing errors and for assumed deflection of the teeth already in mesh. (4.35) and (4.36) below give good first approximations of the amount of tip relief required. The values obtained by these equations should be modified by experience for the best overall performance. Modification at first point of contact: Modification =

driving load(lb) × 3.5 × 10 face width (in.)

7

driving load(lb) × 2.0 × 10 face width (in.)

7

(4.35)

Remove stock from tip of driven gear. Modification at last point of contact: Modification =

(4.36)

Remove stock from tip of driven gear. In general, gear teeth that carry a load in excess of 2000 lb per in. of face width for more than 1,000,000 cycles should have modification. Those under 1000 lb per in. of face width do not generally require modification. Bevel gear teeth are often modified to accommodate mounting misalignments, tooth errors, and de­ flections due to load. The geometry of bevel gear teeth allows profile modifications both along the length of the teeth and form root to tip. In the case spiral and Zerol8 bevel gears cut with face mill-type cutters,

8

Zerol is a trademark registered by the Gleason Works, Rochester, NY.

146

Dudley’s Handbook of Practical Gear Design and Manufacture

the contacting faces of the gear and pinion teeth can be easily made to a slightly different radius of curvature on each member. This is equivalent to crowning of spur gears and is called mismatch. Tips of the teeth may be given relief, and in some cases the root of the flank is relieved (undercut). This is most commonly done on spiral bevel gears and hypoid gears that are to be lapped. It is done by cutters having a special protuberance called Toprem9cutters. Mismatch is calculated into the machine settings and is therefore beyond the scope of this chapter. Axial modifications. In general, it is expected that a gear tooth will carry its driving load across the full face width (if a spur gear). Because of deflections in shafts, bearings, or mountings, when under load, or because of errors in the manufacturing of these parts or in the gears, the teeth may not be quite parallel, and end loading may result. Since a heavy load on the end of a gear tooth will often cause it to break off, attempts to avoid loading by relieving the ends of the teeth (crowning) are often made. A gear tooth that has been crowned is slightly thicker at the center section of the tooth than at the ends when measured on the pitch cylinder. In effect, crowning allows a rocking-chair-like action between the teeth when the shafts deflect into in­ creasingly nonparallel positions. Heavy concentrations of load at the ends of the teeth are avoided. The fact that the whole face cannot act when the shafts are parallel requires that the load imposed upon a set of crowned gears be less than could be carried on a similar non-crowned pair if parallel teeth could be maintained. In general, the ends of crowned gears are made from 0.0005 to 0.002 in. thinner at the ends as compared with the middle, when measured on the circular-arc tooth thickness. Spur and helical gearing are crowned by means of special attachments on the gear tooth finishing machines. Bevel gearing is crowned, in effect, by the shape of the cutting tools and the way in which they are driven in relation to the teeth. Coniflex10 teeth are straight bevel teeth cut on special generating machines which produce lengthwise crowning. Spiral and Zerol bevel gears, and hypoid gears are given the effect of crowing by the use of a different radius of lengthwise curvature on the convex and the concave sides of the teeth. Throated worm gears can be given the effect of crowning by the use of slightly oversize hobs. Face gears are given the effect of crowning by the selection of a cutter having one or more teeth than the mating gear. Root fillets. The shape and minimum radius of curvature that the root fillet of a tooth will have depends on the type and design of the cutting tool used to produce the gear tooth. The shape of the root fillet, as well as its radius, and the smoothness with which it blends into the root land and active profile of a gear tooth can have a profound effect on the fatigue strength of the finished gear. The radius of the fillet at the critical cross section of the tooth is controlled on drawings by specifying the minimum acceptable fillet radius. The point of minimum radius occurs almost adjacent to the root land in the case of gears cut by hobs. An equation giving a reasonable evaluation or the minimum radius for teeth cut by hobs is:11

r f = 0.7 rT +

(ht d 2 cos2

a + ht

rT )2 (a + rT )

where: r f – is the minimum calculated fillet radius produced by hobbing or generating grinding rT – is the edge radius of the generating rack, hob, or grinding wheel

9 10 11

Toprem is a trademark registered by the Gleason Works, Rochester, NY. Coniflex is a trademark registered by the Gleason Works, Rochester, NY. (4.37) is valid for the metric system by using millimeters instead of inches.

(4.37)

Gear Tooth Design

147

a– is the addendum of gear h t – is the whole depth of gear d – is the pitch diameter of gear – is the helix angle (use 0˚ for spur gears)

The edge radius specified for the generating tool will, in general, depend on the service the gear is to perform or on special manufacturing considerations. Table 4.3 shows suggested values of edge radius for various gear applications. Other types of manufacturing tools can be designed to produce the minimum fillet radius as obtained from (4.37). The shape of the fillet will be somewhat different, however. A common method of checking the minimum radius of the coarser-pitch gears is to lay a pin in the fillet zone and note that the contact is along a single line. The constant 0.7 is to allow a reasonable working tolerance to the manufacturer of the tools. Edge breakdown of hobs and cutters will tend to increase the radius produced. This is particularly true of hobs and cutters for very-fine-pitch gears. Effective outside diameter. It is customary to consider the outside diameter of a gear as the outer boundary of the active profile of the tooth. In several cases this approximation is not good enough. • In very-fine-pitch gears that have been burr brushed, the tip round may be quite large in proportion to the size of the teeth even though it is only a few thousands in radius by actual measurement. Since no part of this radius can properly contact the mating tooth, the outside diameter is, from the standpoint of conjugate action, limited to the diameter where the tip radius starts. Effective outside diameter should be used instead of outside diameter in calculations of contact ratio. • In some gear meshes in which the pinion member contains a small number of teeth, the tips of gear teeth may be found to be extending into the pinion spaces to a depth greater than that bounded by the line of action. In Figure 4.6b the dimension DoP /2 is the maximum effective diameter to the pinion. The actual outside diameter of the pinion may exceed this value, however. In calculating contact ratio, the effective diameter as limited by the pinion base circle and center-distance should be used. Width of tip of tooth. The tooth thickness at the tip of the tooth is a convenient index of the quality of a gear design. For most power gearing applications, the thickness of the gear tooth should not be more than 1½ to 2 times that of the pinion tooth at the tip. If the tooth thickness (arc), at the pitch diameter (standard) is known, the following equation will give its thickness at the tip: to = Do

t + inv D

cos

o

=

inv

o

D cos Do

where: t o – is the tooth thickness at outside diameter Do Do – is the outside diameter of gear, or diameter where tooth thickness is wanted t – is the arc tooth thickness at D , or at known diameter D – is the standard pitch diameter, or diameter where t is known – is the standard pressure angel, or pressure angle where tooth thickness t is known o – is the pressure angle at outside diameter or at diameter where tooth thickness is wanted

(4.38)

(4.39)

Circular

1.03528

1.08360

1.08360

1.15470

1.22077

1.41421

1

1

1

1

1

1

1

1

1

1

1

1

1

1.15470

1.10338

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

1

1

…

…

…

…

…

…

…

…

…

…

…

…

…

1

1

1

1

1

1

2.84725

2.72070

…

…

…

…

…

…

…

…

…

…

…

…

…

2.22144

2.57340

2.72070

2.89185

2.89190

3.03454

3.14259

3.14259

…

…

…

…

…

…

…

…

…

…

…

…

…

3.14259

3.14259

3.14259

3.14259

3.14259

3.14259

Normal Transverse Normal Transverse p Pd pn Pnd

Diametral

Pitch of Teeth

1

Tooth Form No.

TABLE 4.3 Basic Tooth Proportions for Helical Gears

1.8400 1.7400 1.6400 1.4200

… 2.0000 2.0000 1.7400 1.5000

20 20 20 20 20 … … … … … … … … … … …

20 20 25 33 41

27 13 35

18 31 22 18 31 22

17 29 43 16 36 06 14 25 58

23 14 07 23 14 07

21 58 50 21 58 50 21 58 50 21 58 50 18 31 35 24 39 57 24 51 32 24 54 59 15 00 00 … …

22 30 00 25 00 00

6.73717

5.44140

…

…

…

…

…

…

…

…

…

…

…

…

…

3.14259

4.48666

5.44140

7.40113

…

…

1.8280

1.6300

2.0000

2.0000

2.0000

2.0000

2.0000

2.0000

2.0000

7.40113

2.0000

hk

Transverse Working

20

n

Normal

1.7500

2.0500

2.200+0.002Pd

2.3500

2.6150

2.1550

2.1570

2.1170

2.0180

2.4500

2.4000

2.3500

2.2700

2.3300

2.2500

1.7000

1.9500

2.0500

2.2000

2.3500

2.3500

ht

Whole

Depth of Teeth

11.72456 19 22 12

pn

Axial

Pressure Angle

1.0000 0.8700 0.7500

30

25

1.0000

1.1580

0.9230

0.9260

0.9140

0.8150

1.0000

1.0000

1.0000

1.0000

1.0000

1.0000

0.7100

0.8200

0.8700

0.9200

1.0000

1.0000

a

…

…

…

…

…

…

…

…

…

…

…

…

…

…

…

1.1107

1.2867

1.3604

1.4460

1.4460

1.5710

t

0.200

0.250

…

0.350

0.410

0.264

0.220

0.158

0.204

0.275

0.290

0.300

0.310

0.270

0.240

0.250

0.300

0.300

0.350

0.350

0.350

rT

Addendum Tooth Thick- Edge Radius of ness (arc) Generating Rack

…

…

…

…

…

…

…

…

…

…

…

…

…

45

35

30

23

23

15

Helix Angle

148 Dudley’s Handbook of Practical Gear Design and Manufacture

Gear Tooth Design

149

Pointed tooth diameter. An independent method of checking the quality of a long- and shortaddendum gear design is to calculate the diameter at which the teeth would come to a point. If this value is smaller than the value of outside diameter chosen by other means, the design should be recalculated. Equation (4.40) and (4.41) provide one method of calculating pointed tooth diameters: inv

oP

=

DoP =

t + inv D D cos cos oP

(4.40)

(4.41)

where oP equals to pressure angle at pointed tooth diameter, and DoP is the pointed tooth diameter. See (4.39) for other symbols. Purpose for backlash. In general, backlash is the lost motion between mating gear teeth. It may be measured along the line of action or on the pitch cylinder of the gears (transverse backlash) and, in the case of helical gears, normal to the teeth. In a set of meshing gears, the backlash that exists is the result of the actual center- distance at which the gears operate, and the thickness of the teeth. Changes in temperature, which may cause differential expansion of the gears and mountings, can produce appreciable changes in backlash. When establishing the backlash that a set of gears will require, the following should be considered: • The minimum and maximum center-distance. These values are the result of tolerance buildings in the distance between the bores supporting the bearings on which the gears are mounted as well as the basic or design values. Antifriction bearings, for example, have runout between the bore and the inner ball path. As the shaft and the inner race of the bearing rotate, and the outer race creeps in the housing, the center-distance will vary by the amount of the eccentricities of these bearing elements. • The thickness of the teeth, as measured at a fixed distance from the center on which the gear rotates, will vary because of gear runout. Also, the workman cutting the gear is given a tooth thickness tolerance to work to, which introduces tooth thickness variations from one gear to the next. The minimum backlash will occur when all the tolerances react all at the same time to give the shortest center-distance and the thickest teeth with the high points of gear runout. The maximum backlash will occur when all the tolerances move in the opposite directions. Design backlash is incorporated into the mesh to ensure that contact will not occur on the non-driving sides of the gear teeth. Although backlash may be introduced by increasing the center-distance, it is usually introduced by thinning the teeth. The minimum value should be at least sufficient to accom­ modate for a lubricating film on the teeth. Sometimes a statistical approach is used, since there are a sufficient number of tolerances involved; thus, the design backlash introduced may not be numerically as large as the possible adverse buildup of tolerances. This approach is particularly handy in instrument gearing, since the maximum backlash allowable usually has to be held to a minimum. By definition, backlash cannot exist in a single gear. Backlash is a function of the actual centerdistance on which the gears are operated, and the actual thicknesses of the teeth of each gear. It is customary to use generally recognized values of center-distance tolerance and gear tooth tol­ erances for power gearing. If this is done, and the calculated tooth thicknesses are reduced by the amounts of design backlash as shown in Table 4.4, satisfactory gears should result. In the case of instrument gearing, either the values in Table 4.4 may be used or the tooth thickness actually required and the resulting maximum backlash may be calculated.

150

Dudley’s Handbook of Practical Gear Design and Manufacture

TABLE 4.4 Recommended Backlash Allowance for Power Gearing Normal Diametral Pitch, Pnd

Center-Distance, in. ( 3.93701 10

2

mm )

0 – 5”

5 – 10”

10 – 20”

20 – 30”

30 – 50”

50 – 80”

80 – 120”

½ 1

… …

… …

… …

… 0.035

0.045 0.040

0.060 0.050

0.080 0.060

2

…

…

0.025

0.030

0.035

0.045

0.055

3 4

… …

0.018 0.016

0.022 0.020

0.027 0.025

0.033 0.030

0.042 0.040

… …

6

0.008

0.010

0.015

0.020

0.025

…

…

8

0.006

0.008

0.012

0.017

…

…

…

10 12

0.005 0.004

0.007 0.006

0.010 …

… …

… …

… …

… …

16

0.004

0.005

…

…

…

…

…

20 32

0.004 0.003

… …

… …

… …

… …

… …

… …

64

0.002

…

…

…

…

…

…

The backlash that is measured in gears under actual operation will in all probability be considerably larger than the values given in Table 4.4, since these values are for design and do not include a correction for normal machining tolerances. Backlash: Recommended values. These dimensions, shown in Figure 4.2, are the distances from the axis of spur, helical, and internal gears at which the active profiles of the teeth begin. Form diameter is the lowest point (spur and helical gearing) at which the mating tooth can contact the active profile (see Figure 4.5).

4.2

STANDARD SYSTEMS OF GEAR TOOTH PROPORTIONS

A standard system of gear tooth proportions provides a means of achieving engineering interchange­ ability for gears of all numbers of teeth of a given pitch and pressure angle. Because of the large variety of tooth proportions that are possible, it has been found desirable to standardize on a limited number of tooth systems. These systems specify the various relationships among tooth thicknesses, addendum, working depth, and pressure angle. The following sections provide the basic data covered by most of the systems. In each of the fol­ lowing systems, the tooth proportions are shown in terms of the basic rack of that system. In each case, all gears designed to the basic tooth proportions will have engineering interchangeability.

4.2.1 STANDARD SYSTEMS

FOR

SPUR GEARS

The following data are based on the information contained in standards for “20-Degree Involute FinePitch System for Spur and Helical Gears”, and for “Tooth proportions for Coarse-Pitch Involute Spur Gears”. Limitations in use of standard tables. Caution should be exercised in using the data contained in Table 4.5. The items shown apply only to gears that meet the following requirements:

Gear Tooth Design

151

TABLE 4.5 Basic Tooth Proportions of Spur Gears Symbol

a

Item

Coarse Pitch (Coarser than 20P), Full Depth

Fine Pitch (20P and Finer), Full Depth

Explanation No.

Pressure angle

20

25

20

1

Addendum (basic*)

1.000 P

1.000 P

1.000 P

2

+ 0.002

3

2.000 P

4

+ 0.002

5

b

Dedendum (min.) (basic )

1.250 P

1.250 P

hk

Working depth

2.000 P

2.000 P

ht

Whole depth (min.) (basic**)

2.250 P

2.250 P

t

2P

2P

1.5708 P

6

rf

Circular tooth thickness (basic*) Fillet radius (in basic rack)

0.300 P

0.300 P

Not standardized

7

c

Clearance (min.) (basic*)

0.250 P

0.250 P

0.200 P

+ 0.002

8

c

Clearance, shaved or ground teeth

0.350 P

0.350 P

0.350 P

+ 0.002

9

**

N

to

P

1.200 P

2.200 P

Min. numbers of teeth*: Pinion

18

12

18

10

Pair

36

24

–

11

0.250 P

0.250 P

Not standardized

12

Min. width of top land

* These values are basic, for equal-addendum gearing. When the gearing is made long- and short-addendum these values will be altered. ** These values are minimum. Shaved or ground teeth should be given proportions suitable to these processes.

• Standard-addendum gears must exceed the minimum numbers of teeth shown in Table 4.6. • Long- and short-addendum designs, as derived from Tables 4.5, 4.7, and 4.8, are to be used for speed-reducing drives only. The tooth proportions that result from the data shown in this section or from the application of the original standards will be suitable for most speed-reducing (torque-increasing) applications. All gears, including pinions with small numbers of teeth, designed in accordance with the procedure shown will be free from undercut. In order to avoid the problems of undercut in pinions having fewer than the minimum standard number of teeth, each system shows proportions for long- and short-addendum teeth. Gears designed with long- and short-addendum teeth cannot be operated interchangeable on standard centers. In general, such gears should be designed to operate only as pairs. These gears can be cut with the same generating-type cutters and checked with the same equipment as standard-addendum gears. The pro­ portions of long and short addendum shown in Table 4.1 are based on avoiding undercut. Such teeth will not have the optimum balance of strength and wear. Slightly different proportions can be used to achieve equal sliding balanced strength or a reduction in the tendency to score. Table 4.2 shows tooth proportions that have a good balance of strength and a minimum tendency to score. Standard tooth forms that have become obsolete. Because industry design standards are continuously reviewed by the sponsoring organizations to ensure that the standards embody the most modern

152

Dudley’s Handbook of Practical Gear Design and Manufacture

TABLE 4.6 Minimum Number of Pinion Teeth vs. Pressure Angle and Helix Angle Having No Undercut Helix Angle, Deg.

Minimum Number of Teeth to Avoid Undercut When Normal Pressure Angle,

n,

Deg.

14 ½

20

22 ½

25

0 (spur gear)

32

17

14

12

5

32

17

14

12

10

31

17

14

12

15

29

16

13

11

20

27

15

12

10

23

25

14

11

10

25

24

13

11

9

30

21

12

10

8

35

18

10

8

7

40

15

8

7

6

45

12

7

5

5

Note: Addendum 1/Pd; whole depth 2.25/Pd.

TABLE 4.7 General Recommendations on Numbers of Pinion Teeth for Spur and Helical Power Gearing (The maximum number of teeth in the range is based on providing a suitable balance between strength and wear capacity. The minimum number of teeth is intended to require either no en­ largement to avoid undercutting or no more enlargement than necessary) Ratio, mG

Diametral Pitch, Pd

19 – 60

1 – 1.9

1 – 19.9

200 – 240 BHN

19 – 50

2 – 3.9

19 – 45 19 – 45

4–8 1 – 1.9

1 – 19.9

Rockwell C 33 – 38

19 – 38

2 – 3.9

19 – 35 19 – 30

4–8 1 – 1.9

1 – 19.9

Rockwell C 38 – 63

17 – 26

2 – 3.9

15 – 24

4–8

Range of No. of Pinion Teeth

Hardness

Notes: Small numbers of teeth in the pinion require special tooth proportions to avoid undercut. Gears containing prime numbers of teeth of 101 and over may be difficult to cut on the gear-manufacturing equipment available. In general, prime numbers over 101 and all numbers over 200 should be checked with the shop to be sure the teeth can be made.

Gear Tooth Design

153

TABLE 4.8 Equations and References for Addendum Calculations, Spur Gears Type of Tooth Design

Operating Condition

Pinion

1.00 ; Pd

Gear

1.00 ; Pd

Standard or equal addendum

Numbers of teeth in gear and pinion greater than minimum numbers shown Tables 4.6 and 4.11

(a)

Long and short addendum

Numbers of teeth n pinion less than minimum shown in Table 4.6 and numbers of teeth in gear more than minimums shown in Table 4.11

(c)

Table 4.1 Pd

(d)

Table 4.2 Pd

Numbers of teeth in pinion less than minimum numbers shown in Table 4.11 and numbers of teeth in gear less than minimum numbers of teeth shown in Table 4.11

Increase center-distance sufficiently to avoid undercut, in pinion and gear (e)

Table 4.1 Pd

See Section 4.3

Gearing designed for balanced strength

(f)

Table 4.2 Pd

Table 4.2 Pd

see Table 4.5

(b) Value from Table 4.13 or if not less than (c)

Note: If value is less than given by Table 4.1 check for undercut Designed for nonstandard center-distance

(a)

see Table 4.5

(b) Value from Table 4.13 (c)

Table 4.1 Pd

(d)

Table 4.2 Pd

or

Note: If value is less than given by Table 4.1 check for undercut

(g) Calculate tooth thickness to meet required center-distance from Eq. (4.75).

Notes: The values in this table are for gears of 1 diametral pitch. For other sized divide by the required diametral pitch. The values of addendum shown are the minimum increase necessary to avoid undercut. Additional addendum can be provided for special applications to balance strength. See Table 4.2.

technology, there now exists a group of obsolete tooth form standards. On occasion, a designer is confronted with a situation in which a replacement gear must be made to mesh in a gear train conforming to one of these earlier standards. Table 4.9 shows basic data for some of the obsolete standards. The use of this data for new designs is not recommended. Brown and Sharp system. This system, see Table 4.9, was developed by the Brown and Sharp Company to replace the cycloidal tooth system. It was therefore given similar tooth proportions. It was intended to be cut by form-milling cutters. The departure from the true involute curve in this system is made to avoid the problems of undercut in pinions having small numbers of teeth. Backlash is achieved by feeding the cutter deeper than standard. AGMA 14.5˚ composite system. This system, is interchangeable with the B and S system. Replacement parts may be designed from the data in Table 4.9. Fellows 20˚ stub tooth system. In order to achieve a stronger tooth form for special drives, the Fellows Gear Shaper Company developed a stub tooth system in 1898. This system avoided the problem of tooth interference by the combined means of higher pressure angle and smaller values of addendum and dedendum. The pitch was specified as a combination of two standard diametral pitches; Thus, 10/12 (read ten-twelve). The circular pitch, pitch diameter, and tooth thickness are based on the first number, which is, 10, and are the same as for a standard 10-diametral-pitch gear. The addendum, dedendum, and clearance, however, are based on the second number (12) and are the same as for a 12-diametralpitch gear.

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TABLE 4.9 Tooth Proportions of Spur Gears, Obsolete System* (Tooth proportions for fully interchangeable gears operate on standard center distances) Symbol

Item

Brown & Sharp system

AGMA 14.5˚ composite system

Fellows 20˚ stub system

AGMA full-depth composite system

Pressure angle, deg.

14 ½

14 ½

20

14 ½

a

Addendum (basic**)

1.000 P

1.000 P

0.800 P

1.000 P

b

Dedendum (basic***)

1.157 P

1.157 P

1.000 P

1.157 P

hk

Working depth

2.000 P

2.000 P

1.600 P

2.000 P

ht

Whole depth (min.) (basic***)

2.157 P

2.157 P

1.800 P

2.157 P

t

Circular tooth thickness (basic**)

2P

2P

2P

2P

rf

Fillet radius (in basic rack) Clearance (min.) (basic**)

0.157 P

0.157 P

Not standardized

1.33 × clearance

0.157 P

0.157 P

0.200 P

0.157 P

32 64

– –

14 –

32 64

c c N

Clearance, shaved or ground teeth P

Min. numbers of teeth** Pinion Pair

* See Table 4.5. ** These values are basic for equal-addendum gearing. When the gearing is made long- and short-addendum these values will be altered. *** These values are minimum. Shaved or ground teeth should be given proportions suitable to these processes.

AGMA 14.5˚ full-depth system. In this system, the tooth proportions are identical with those of the 14.5 composite system. The sides of the rack teeth are straight lines and therefore produce involute gear tooth profiles in the generating process. The minimum number of teeth to obtain full tooth action is 31 unless the teeth are modified. Cycloidal tooth profiles. The cycloidal tooth profile is no longer used for any types of gears except clockwork and certain types of timer gears. A combination of involute and cycloidal tooth profile is found in the now-obsolete “composite system”. Clockwork gearing is based on the cycloid but has been greatly modified for practical reasons. The cycloidal tooth is derived from the trace of a point on a circle (called the describing circle) rolling without slippage on the pitch circle of the proposed gear. The addendum portions of the tooth are the trace of a point on the describing circle rolling on the outside of the pitch circle. This trace is an epicycloid. The dedendum portion of the gear tooth is formed by a trace of a point on the describing circle rolling on the inside of the pitch circle. This trace is a hypocycloid. Interchangeable systems of gears must have describing circles that are identical in diameter, and teeth that have the same circular pitch. The faces and flanks of the teeth must be generated by describing circles of the same size. The tooth proportions for cycloidal teeth were made to the same proportions as were adopted by the B and S system for gears.

Gear Tooth Design

155

The cycloidal system does not have a standard pressure angle. The operating pressure angle varies from zero at the pitch line to a maximum at the tips of the teeth. In order to achieve correct meshing, the gears must be operated on centers that will maintain the theoretical pitch circles in exact contact. This was one of the major disadvantages of the cycloidal tooth form. If the diameter of the describing circles were made equal to the radius of the pitch circle of the smallest pinion to be used in the system, the flanks of the teeth would be radial. This member, in effect, establishes the describing circle diameters of each system. In general, the systems for industrial gears were based on pinions of 12 to 15 teeth. Clockwork and timer tooth profiles. The tooth form of most clockwork and timer gearing involving some type of escapement differs from other types of gearing because of the peculiar requirements im­ posed by the operating conditions. Two requirements of this type of gearing are outstanding: • The gearing must be of the highest possible efficiency • The gearing is speed increasing, that is, larger gears driving smaller pinions, with high rations (between 6:1 and 12:1) due to the need for minimizing the number of gear wheels and economize on space. As is discussed in Section 4.1.3 under “Low friction gearing”, the most efficient mode of tooth en­ gagement is found in the arc of recess. In the case of speed-increasing involute profile drives, this means short-addendum pinions. This requirement may be in direct conflict with the need to make the pinion long addendum to avoid undercut. Clock gearing also does not have the requirement that it transmit smooth angular motion. A typical watch train will come to a complete stop five times a second. Furthermore, the motion and force are in one direction, minimizing the need for accurate control of backlash. The modified cycloidal tooth form is the most commonly used form for the going train of clocks, timers, and watches. Experience has shown that the various modifications to this basic tooth have little effect on the performance because of the scale effect of tolerances required. Specific spur gear calculation procedure. The following directions give a step-by-step procedure for determining spur gear proportions: 1. The application requirements should give the requirements of ratio, input speed, and kind of duty to be performed. The duty may be one of power transmission or of motion transmission 2. Based on application requirements, decide what pressure angle to use and plan to use a standard system (if possible) (see Table 4.5) 3. Pick approximate number of pinion teeth a. Consult Table 4.10 for general information b. Check Tables 4.6 and 4.11 for undercut conditions c. If power gearing, check Table 4.7 for data on balancing strength and wear capacity 4. Having approximate number of pinion teeth, determine approximate center distance and face width 5. Based on number of pinion teeth and center-distance, determine approximate pitch. Check Table 4.12 for standard pitches, and, if possible, use a standard pitch. Readjust pinion tooth numbers, center-distance, and ratio to agree with pitch chosen. See Chapter 2 for basic relations of pitch, center-distance, and ratio. Also use Eq. (4.8), Eq. (4.9), Eq. (4.10), and Eq. (4.11) if special “operating” center-distance is used 6. Determine whole depth of pinion and gear. Use Tables 4.5 and 4.13. Divide tabular value for 1 diametral pitch by actual diametral pitch to get design whole depth

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Dudley’s Handbook of Practical Gear Design and Manufacture

TABLE 4.10 Numbers of Pinion Teeth, Spur Gearing No. of Teeth

Remarks

7–9 a. b. c. d. e. f.

Requires long addendum to avoid undercut on all pressure angles If 20˚, outside diameter should be reduced in proportion to tooth thickness to avoid pointed teeth Should be made with 25˚ pressure angle if feasible May result in poor contact ratio in very fine diametral pitches because of accumulation of tolerances See Table 4.11 for minimum number of teeth in mating gear Subject to high specific sliding and usually have poor wear characteristics

10

Smallest practical number with 20˚ pressure angle

12

a. Requires long addendum to avoid undercut if 20˚ pressure angle or less b. Contact ratio may be critical in very fine pitches c. See Table 4.11 for minimum number of teeth in mating gear Smallest practical number for power gearing of pitches coarser than 16 diametral pitch a. Requires long addendum to avoid undercut if 20˚ pressure angle or less b. Smallest number of teeth that can be made “standard” if 25˚ pressure angle c. About the minimum number of teeth for any good fractional-horsepower gear design where long life is important d. See Table 4.11 for minimum number of teeth in mating gear

15

Used where strength is more important than wear a. Requires long addendum to avoid undercut if 20˚ pressure angle or less b. See Table 4.11 for minimum number of teeth in mating gear

19

Can be made standard addendum if 20˚ pressure angle or greater

25

Allows good balance between strength and wear for hard steels. Contact (contact diameter) is well away from critical base-circle region.

35

If made of hard steels, strength may be more critical than wear. If made of medium-hard (Rockwell C 30) steels strength and wear about equal Excellent wear resistance. Favored for high-speed gearing because of quietness. Critical on strength on all but low-hardness pinions

50

7. Determine addendum of pinion and gear. Consult Tables 4.1, 4.2, 4.5, and 4.8. Follow these rules: a. Use enough long addendum to avoid undercut (see Table 4.1) b. If a critical power job—speed decreasing—balance addendum for strength (see Table 4.2) c. If no undercut or power problems, use standard addendum (see Table 4.5) 8. Determine operating circular pitch. If standard pitch is used, consult Table 4.14. If enlarged center-distance is used, determine operating circular pitch form operating pitch diameters. [Eq. (4.8) to Eq. (4.11)]. 9. Determine design tooth thickness. Decide first on how much to thin teeth for backlash. Table 4.4 gives recommended amounts for power gearing having in mind normal accuracy of centerdistance and normal operating temperature variations between gear wheels and casings. If special designs with essentially “no backlash” or unusual materials and accuracy are involved, consult the last part of this chapter, titled “Center-Distance”, for special calculations. If the pinion and gear have equal addendum, make the theoretical tooth thicknesses equal, and do this by dividing the circular pitch by 2. If a long-addendum pinion is used with a short-addendum gear, adjust tooth thicknesses by the following:

Gear Tooth Design

157

TABLE 4.11 Numbers of Teeth in Pinion and Gear vs. Pressure Angle and Center-Distance No. of Teeth in Pinion

No. of Teeth in Gear and When Pressure Angle Is Equal To 14 ½ Coarse Pitch*

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

52 51 50 49 48 47 46 45 44 43 42 41 439 38 37 36 35 34 33

20 Coarse Pitch**

20 Fine Pitch**

25 24 23 22 21 20 19 18

42*** 39*** 36*** 33 30 27 25 23 21 19 18

25 Coarse Pitch**

15 14 12

Note: Pinions having fewer than 10 teeth are not recommended. Gears having fewer teeth than shown for any given pinion-gear combination will require an enlarged center-distance for proper operation.

* Gears of this pressure angle not recommended for new designs. ** Gears with these numbers of teeth can be made standard addendum and operated on standard center-distances. *** Not recommended; but if essential, use these values.

TABLE 4.12 Recommended Diametral Pitches Coarse Pitch

Fine Pitch

2

4

12

20

48

120

2.25 2.5

6 8

16

24 32

64 80

150 200

3

10

40

96

Note: These diametral pitches are suggested as a means of reducing the great amount of gear-cutting tooling that would have to be inventoried if all possible diametral pitches were to be specified.

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Dudley’s Handbook of Practical Gear Design and Manufacture

TABLE 4.13 Equations and References for Whole-Depth Calculations Pitch Range

Equation

Remarks

2.25 Pd

General application

AGMA standard

2.35 Pd

Gears to be given finishing operations such as shave or grind

AGMA standard

2.40 Pd

Maximum strength for full-radius (tip) hobs

AGMA standard

2.20 Pd

+ 0.002

Standard whole depth for general applications

AGMA standard

2.35 Pd

+ 0.002

Whole depth for pre-shave cutters for shaving or grinding

AGMA standard

Coarse pitch up to 19.99

Fine pitch* 20 Pd and finer

Application

* Does not apply to pinions containing nine teeth or less.

t = a 2 tan

n

(4.42)

If the pinion only is enlarged and the center-distance is enlarged to accommodate a standard gear, enlarge the pinion tooth thickness only (see Table 4.14). After the theoretical tooth thicknesses are obtained, subtract one-half the amount the teeth are to be thinned for backlash from the pinion and gear theoretical tooth thicknesses. This is the maximum design tooth thickness. Obtain the minimum design tooth thickness by subtracting a reasonable tolerance for machining from the maximum design tooth thickness. 10. Recheck load capacity using design proportions just obtained. If not within allowable limits, change design 11. If the gear design is for critical power gears, additional items will need calculation: a. Root fillet radius [(4.37)] b. Form diameter c. Modification of profile [(4.35), (4.36)] d. Diameter over pins (see corresponding section below) 12. Certain general dimensions must be calculated and toleranced: a. Outside diameter, D + 2 (a ± a) b. Root diameter, Do 2 h t c. Face width d. Chordal addendum and chordal thickness 13. It may be necessary to specify: a. Tip round (at outside diameter) (see Table 4.14) b. Edge round (see Table 4.14) c. Roll angle d. Base radius 14. In some unusual designs, it may be necessary to check the following to assure a sound design: • Diameter at which teeth become pointed [(4.41)] • Width of top land [(4.38)] • Effective contact ratio [Eq. (4.3) and Eq. (4.4)] • Undercut diameter

Gear Tooth Design

159

TABLE 4.14 Standard Tooth Parts Diametral Pitch, Pd , (1/in.)

Circular Pitch, p , in.

Module 1/ Pd Addendum a , in.

Whole Depth* Shave or Grind, h t , in.

Cut Teeth, h t , in.

Tooth Thicknesses ( p/2 ), t , in.

Double Teeth Cut Teeth 2 h t , in.

Ground Teeth 2 h t , in.

1 1¼

3.1415927 2.5132741

1.0000000 0.8000000

2.35000 1.88000

2.25000 1.80000

1.5707963 1.2566371

4.5000 3.6000

4.7000 3.7600

1½

2.0943951

0.6666667

1.56667

1.50000

1.0471980

3.0000

3.1333

1¾ 2

1.7951958 1.5707963

0.5714286 0.5000000

1.34286 1.17500

1.28571 1.12500

0.8975979 0.7853982

2.5714 2.2500

2.6857 2.3500

2½

1.2566371

0.4000000

0.94000

0.90000

0.6283185

1.8000

1.8800

3 3½

1.0471976 0.8975979

0.3333333 0.2857143

0.78333 0.67143

0.75000 0.64286

0.5235988 0.4487990

1.5000 1.2857

1.5667 1.3429

4

0.7853982

0.2500000

0.58750

0.56250

0.3926991

1.1250

1.1750

5 6

0.6283185 0.5235988

0.2000000 0.1666667

0.47000 0.39167

0.45000 0.37500

0.3141593 0.2617994

0.9000 0.7500

0.9400 0.7833

8

0.3926991

0.1250000

0.29375

0.28125

0.1963495

0.5625

0.5875

10 12

0.3141593 0.2617994

0.1000000 0.0833333

0.23500 0.19583

0.22500 -.18750

0.1570796 0.1308997

0.4500 0.3750

0.4700 0.3917

16

0.1963495

0.0625000

0.14688

0.14063

0.0981748

0.2813

0.2938

20 24

0.1570796 0.1308997

0.0500000 0.0416667

0.11950 0.09992

0.11200 0.09367

0.0785398 0.0654499

0.2240 0.1873

0.2390 0.1998

32

0.0981748

0.0312500

0.07544

0.07075

0.0490873

0.1415

0.1509

40 48

0.0785398 0.0654498

0.0250000 0.0208333

0.06075 0.05096

0.05700 0.04780

0.0392699 0.0327249

0.1140 0.0956

0.1215 0.1019

64

0.0490874

0.0156250

0.03872

0.03638

0.0245437

0.0728

0.0774

80 96

0.0392699 0.0327249

0.0125000 0.0104167

0.03138 0.02648

0.02950 0.02492

0.0196350 0.0163625

0.0590 0.0498

0.0628 0.0530

120

0.0261799

0.0083333

0.02158

0.02033

0.0130900

0.0407

0.0432

150

0.0209440

0.0066667

0.01767

0.01667

0.0104720

0.0333

0.0353

* h t for fine pitch (20 Pd ) is 2.2/ Pd + 0.002 .

Explanation and discussion of items in Table 4.5. Table 4.1 shows the “basic” or standard tooth proportions. 1. Pressure angle. The pressure angle and numbers of teeth in a given pair of gears are related in that low pressure angles (14 ½) should be avoided for low numbers of teeth (see Tables 4.6 and 4.11). 2. Addendum. For general applications, use equal-addendum teeth when the numbers of teeth in the pinion and in the pair exceed the values shown in items 10 and 11. Use long addendums for pinions having numbers of teeth, pressure angles, and diametral pitches as shown in Table 4.1. The amount of decrease of addendum in the mating gear is to be limited to values that will not produce undercut. The values of long addendum for pinions are limited to speed-decreasing drives. Data on speedincreasing drives are given in Section 4.1.3.

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Dudley’s Handbook of Practical Gear Design and Manufacture

3. Dedendum. The values shown in the table are the minimum standard values. Shaved or ground teeth should be given a greater dedendum (whole depth and clearance values); see items 5 and 9. A constant value, 0.002 in., is added to the dedendum of fine-pitch gears, which allows space for the accumulation of foreign matter at the bottoms of spaces. This provision is particularly im­ portant in the case of very fine diametral pitches. 4. Working depth. The working depth customarily determines the type of tooth; that is, a tooth with a 1.60/P working depth is called a full-depth tooth. 5. Whole depth. The value shown is the standard minimum whole depth. It will increase in pro­ portion to the increase in backlash cut into the teeth (unless the outside diameter is also corre­ spondingly adjusted). It will also increase slightly if long- and short-addendum teeth are generated with pinion-shaped cutters. See also items 3, 8, and 9 for increase due to manufacturing processing requirements. 6. Circular tooth thickness, basic. This is the basic circular tooth thickness on the standard pitch circle. These values will be slightly altered if backlash is introduced into the gears to allow them to mesh at a standard center-distance. These values will be drastically altered in the case of long- and short-addendum designs. Table 4.15 gives values of tooth thickness corresponding to standard values of long and short addendum for small numbers of teeth. 7. Fillet radius, basic rack. The fillet radius shown is that in the basic rack. The tooth form standard directs that the edge radius on hobs and rack-type shaper cutters should be equal to the fillet radius in the basic rack. It also directs that pinion-shaped cutters should be designed using the basic rack as a guide so that the gear teeth generated by these cutters will have a fillet radius approximating those produce by hobs and rack-type shaper cutters. It can be calculated approximately by the data shown in (4.42).

TABLE 4.15 Values of Tooth Thicknesses for Pinions, tP , and Gears, tG , with Small Numbers of Teeth (Long and Short Addendums) No. of Teeth in Pinion

Coarse-Pitch System 20˚ Pressure Angle Pinion

Gear

Fine Pitch System

25˚ Pressure Angle Pinion

Gear

7 8 9

20˚ Pressure Angle Pinion

Gear

2.0007 1.9581

1.1409 1.1835

1.9155

1.2261

10 11

1.9120 1.8680

1.2300 1.2730

1.7420 1.6590

1.3990 1.4820

1.8730 1.8304

1.2686 1.3112

12

1.8260

1.3150

1.5708

1.5708

1.7878

1.3538

13 14

1.7830 1.7410

1.3580 1.4000

1.5708 1.5708

1.5708 1.5708

1.7452 1.7027

1.3964 1.4389

15

1.6980

1.4430

1.5708

1.5708

1.6601

1.4815

16 17

1.6560 1.6130

1.4860 1.5290

1.5708 1.5708

1.5708 1.5708

1.6175 1.5749

1.5241 1.5667

18

1.5708

1.5708

1.5708

1.5708

1.5708

1.5708

19

1.5708

1.5708

1.5708

1.5708

1.5708

1.5708

Note: These tooth thicknesses go with Table 4.1 addendums. The above values are for 1 diametral pitch. These basic tooth thicknesses do not include an allowance for backlash.

Gear Tooth Design

161

In the case of 25˚– pressure-angle teeth, the fillet radius shown must be reduced for teeth having a clearance of 0.250/P. This is discussed in Section 4.1.4. In the case of fine-pitch teeth, the fillet radius usually will be larger than the clearance customarily given as c = 0.157/ P because of edge breakdown in the cutter. The effects of tool wear on fine-pitch gears are proportionally larger than the effects produced by coarse-pitch tools. 8. Clearance. The value shown is the minimum standard. Greater clearance is usually required for teeth that are finished by grinding or shaving. In general, the value shown in item 9 will be suitable for these processes. See also items 3 and 5. 9. Clearance for shaved or ground teeth. This is the recommended clearance for teeth to be fin­ ished by shaving or grinding. In the case of 25˚–pressure-angle teeth, the fillet radius shown must be reduced for teeth having a clearance of 0.250/P. 10. Minimum number of teeth in pinion. These are the lowest numbers of teeth that can be gen­ erated in pinions having standard addendums and tooth thicknesses that will not be undercut. Pinions having fewer teeth should be made long addendum in accordance with Table 4.1. 11. Minimum numbers of teeth in pair. This is the smallest number of teeth in pinion and gear that can be meshed on a standard center-distance without one member’s being undercut. For pairs with fewer teeth the members will have to be meshed on a nonstandard (enlarged) centerdistance. 12. Minimum width of top land. This is the approximate minimum width of top land allowable in standard long-addendum pinions. Increases in addendum that cause the tops of the teeth to have less than this value should be generally avoided.

4.2.2 SYSTEM

FOR

HELICAL GEARS

The following data are based on the information contained in AGMA standard for “20-Degree Involute Fine-Pitch System, for Spur and Helical Gears”. In addition, tooth proportions are shown that are used by several of the larger gear manufacturers for the design of helical gears. These tooth proportions are shown since there are no AGMA standards for coarse-pitch helical gears. In general, helical tooth proportions are based on either getting the most out of a helical gear design or on using existing tooling. The tooth proportions referred to above have been found to yield very good gears. In some cases, for less critical applications, tools on hand may be used. Very often hobs for spur gears are available, and gears for general service can be made with these tools. Since such hobs have no taper, they are not well suited to cut helix angles much over 30˚. In helical gear calculations, care should be taken to avoid confusion as to which plane the various tooth proportions are measured in. In some cases, the transverse plane is used. This plane is perpen­ dicular to the axis of the helical gear blank. In some cases, it is desirable to work in the normal plane. If a spur gear hob is used to cut helical teeth, the relationship between the transverse and normal pressure angles and the transverse and normal pitches as well as the base pitches should be established: cos pn

Pd =

tan

=

tan

(4.43)

n

cos

pN = pn cos

n

(4.44)

(4.45)

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Dudley’s Handbook of Practical Gear Design and Manufacture

where: Pd – is the diametral pitch (in transverse plane) – is the helix angle pn – is the circular pitch (in normal plane) n – is the pressure angle (in normal plane) pN – is the normal base pitch (normal to surface)

When gears mesh together or with hobs, racks, shaper, or shaving cutters, their normal base pitches must be equal. When a gear shaper is to be used to cut helical teeth, special guides are installed in the machine. These impart the twist into the cutter spindle, which produces the proper helix angle and lead. The cutter used has the same lead ground into the teeth as is produced by the guides. The cutter and guides in turn produce a given lead angle to the gear being cut. Practical considerations limit the lead angle on the guides to roughly 35˚. In order to control costs, helical gears that are to be shaped should be designed either to a standard series of values of leads, or to values of lead available in existing guides. Equation (4.46) shows the relationship of lead, number of teeth, diametral pitch, and helix angle in a given cutter, or helical gear: tan

=

N Pd L

(4.46)

where: – is the helix angle N – is the number of teeth, cutter or gear Pd – is the diametral pitch L – is the lead of cutter

Selection of tooth form. The tooth forms shown in Table 4.3 are used for ranges of applications as follows (see Table 4.16). The foregoing types each require a separate set of tooling. They have the advantage of getting the most out of good helical gear design. When an existing spur gear hob is to be used to produce the gear, use types 18 and 19 tooth forms. The data in Table 4.17 will be found helpful in making calculations for gears having types 18 and 19 tooth forms. Type 19 tooth form is similar to type 18 tooth form except that its proportions are based on the use of hobs designed to cut fine-pitch gears (20 diametral pitch or finer). Selection of helix angle. Single helical gears usually are given lower helices angles than double helical gears in order to limit thrust loads. Typical helix angles 15˚ and 23˚. Low helix angles do not provide so many axial crossovers as can be achieved on high-helix-angle gears of a given face width. Double-helix gears have helix angles that typically range from 30˚ through 45˚. Although higher helix angles provide smoother operation, the tooth strength is lower. In order to get the quietest gears and at the same time achieve good tooth strength, special cutting tools for each helix angle should be provided. Table 4.3 shows typical tooth proportions. Tooth types 1 through 5 require special cutters, and the helix angle shown should be used. These teeth are stubbed more and more as the helix angle is higher. Tooth types 6 through 12 can be made with any helix angle desired. Types 13 and 14 can also be made with almost any helix angle. These tooth proportions are based on the use of standard spur gear hobs. Since such hobs are not usually tapered, they do not do as god a job is cutting as do the specially designed helical hobs, and, as a result, they should be limited in use to helix angles less than 25˚.

Gear Tooth Design

163

TABLE 4.16 Value for Tip Round and Edge Round and End Round

Diametral Pitch

20 and finer 16

Edge Round

Tip Round and End Round

General Applications

Medium Strength

High Strength

Burr-brush edges Burr-brush edges

0.001 – 0.005 0.003 – 0.015

0.005 – 0.010 0.010 – 0.025

0.001 – 0.005 0.003 – 0.010

12

Burr-brush edges

0.005 – 0.020

0.012 – 0.030

0.005 – 0.015

10 8

Burr-brush edges Burr-brush edges

0.010 – 0.025 0.010 – 0.025

0.015 – 0.035 0.020 – 0.045

0.005 – 0.015 0.010 – 0.030

5

Burr-brush edges

0.010 – 0.025

0.025 – 0.060

0.010 – 0.030

3 2

Burr-brush edges Burr-brush edges

0.015 – 0.035 0.015 – 0.035

0.040 – 0.090 0.060 – 0.125

0.010 – 0.050 0.010 – 0.050

TABLE 4.17 Ranges of Gear Tooth Forms Applications Type of Gear

Suggested Tooth Form

Single helix, low allowable thrust reaction

Type 1

Single helix, moderate allowable thrust reaction

Types 2 and 3

Double helix, general purpose Double helix, minimum noise

Types 4 and 5 Type 6

Double helix, high load

Types 20 and 21

Face width. In the design of helical gears, the face width is usually based on that needed to achieve the required load-carrying capacity. In addition, the face width and lead are interrelated in that it is necessary to obtain at least two axial pitches of face width (F = 2 px ) to get reasonable benefit from the helical action, and four or more if high speeds, noise, or critical designs are confronted. Long- and short-addendum designs are not as common among helical gears as among spur gears. This is because a much lower number of teeth can be cu in helical gears without undercut. In low-hardness helical gearing, pitting, which is little affected by changes in addendum, is usually the limiting feature. In high-hardness designs, the same proportions as those used for spur gear addendums should be employed. Specific calculation procedure for helical gears. The procedure for helical gears is very similar to that given in Section 4.2.1 for spur gears: 1. Determine application requirements: ratio, power, speed, and so forth 2. Decide on basis tooth proportions and helix angle (see Table 4.3)

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Dudley’s Handbook of Practical Gear Design and Manufacture

TABLE 4.18 Tooth Proportions for Helical Gears Helix Angle, Deg

Diametral Pitch, Pd

Circular Pitch, Pt

Axial Pitch, px

Pressure Angle*

Working Depth, hk

Whole Depth, h t **

0

1.000000

3.14159

5

0.996195

20 00 00

2.000

2.250

3.15359

36.04560

20 4 13. 1

2.000

8

2.250

0.990268

3.17247

22.57327

20 10 50. 6

2.000

2.250

10

0.984808

3.19006

18.09171

20 17 00. 7

2.000

2.250

12

0.978148

3.21178

15.11019

20 24 37. 1

2.000

2.250

15

0.965926

3.25242

12.13817

20 38 48. 8

2.000

2.250

18

0.951057

3.30326

10.16640

20 56 30. 7

2.000

2.250

20

0.939693

3.34321

9.18540

21 10 22. 0

2.000

2.250

21

0.933580

3.36510

8.76638

21 17 56. 4

2.000

2.250

22

0.927184

3.38832

8.38636

21 25 57. 7

2.000

2.250

23

0.920505

3.41290

8.04029

21 34 26. 3

2.000

2.250

24

0.913545

3.43890

7.72389

21 43 22. 9

2.000

2.250

25

0.906308

3.46636

7.43364

21 52 58. 7

2.000

2.250

26

0.898794

3.49534

7.16651

22 02 44. 2

2.000

2.250

27

0.891007

3.52589

6.91994

22 13 10. 6

2.000

2.250

28

0.882948

3.55807

6.69175

22 24 09. 0

2.000

2.250

29

0.874620

3.59195

6.48004

22 35 40. 0

2.000

2.250

30

0.866025

3.62760

6.28318

22 47 45. 1

2.000

2.250

* Pressure angle based on 20 normal pressure angle. ** The values shown for whole depth are for coarse-pitch gears. If the gears are to be shaved or ground, use h t = 2.35. For fine-pitch gears, use 2.2/Pnd + 0.002 for general purpose gearing and 2.35/Pnd + 0.002 for gear to be shaved or ground.

3. 4. 5. 6. 7.

8. 9.

10. 11.

Pick appropriate number of pinion teeth (see Tables 4.6, 4.7, and 4.10) Determine approximate center-distance and face width Determine pitch of teeth Determine whole depth Determine addendum of pinion and gear. Use equal addendum for pinion and gear except in special cases where addendum must be increased to avoid undercut or special consideration must be given to increasing the pinion strength. (see Table 4.19) Determine operating circular pitch, operating helix angle, and operating normal pitch if special center-distance was used Determine design tooth thickness. If long and short addendum, adjust theoretical tooth thicknesses accordingly. Thin teeth for backlash. Consult Table 4.4 for power gearing backlash allowance Recheck load capacity. Check number of axial crossovers [(4.5)]. Check ability of thrust bear­ ings to handle thrust reactions. If results of these checks are unsatisfactory, change proportions If gear design is for critical power gears, additional items may need calculation a. Root fillet radius [(4.37)] b. Form diameter

Long- and shortaddendum

types 1 Pd

6

Designed for nonstandard center- distance

Increase addendum sufficiently to avoid undercut aG = a + K n / Pn

Decrease addendum by amount pinion addendum is increased. If undercut, increase centerdistance.See item below

If standard center-distance and values of aP as shown at (a) left.If nonstandard centerdistance, see item below

Gear

Increase center-distance by amount sufficient to accommodate increased addendum of booth pinion and gear. Calculate tooth thickness to meet required center-distance from Eq. (4.75), then calculate required addendum from (4.18).

Numbers of teeth in pinion less Increase addendum sufficiently to avoid undercut than minimum numbers aP = a + K n / Pn shown in Table 4.6, and numbers of teeth in gear less than minimum numbers of teeth shown in Figure 4.3

Value from special tools to be used Numbers of teeth in pinion less Increase addendum sufficiently to avoid undercut than minimum numbers aP = a + K n / Pn shown in Table 4.6, and See Figure 4.10 for K n numbers of teeth in gear more than minimums shown in Figure 4.3

a=

Numbers of teeth in gear and a = value from Table 4.3 types 1 6 pinion greater than a= Pd minimum numbers shown in types 7 19 ,a = , Figure 4.10 Pnd

Standard or equal addendum

Pinion

Operating Conditions

Type of Tooth Design

TABLE 4.19 Equations and References for Addendum Calculations, Helical Gears

Gear Tooth Design 165

166

Dudley’s Handbook of Practical Gear Design and Manufacture 50 40 30

7 8

20

11

14 10 0

16

13

10

9

12

15

17 0 K'n

FIGURE 4.10

Addendum Increment K

n

for 1 Diametral Pitch.

c. Modification of profile [(4.35), (4.36)] d. Diameter over balls 12. General dimensions to be determined and toleranced a. Outside diameter b. Root diameter c. Face width d. Chordal addendum and chordal tooth thickness 13. It may be necessary to specify a. Tip round or chamber (see Table 4.16) b. Edge round or end bevel (see Table 4.16) c. Roll angle d. Base radius 14. Generally helical gears are not designed for such critical applications as to be in danger of pointed teeth or undercut.

4.2.3 SYSTEM

FOR INTERNAL

GEARS

In general, both spur internal meshes and helical internal meshes may be calculated by the same methods as external meshes. However, in internal gear meshes, there are several problems unique to this type of gearing. The first of these is tip interference. In this type of interference, the pinion member cannot be as­ sembled radially with the gear. Only axial assembly is possible, and it should be provided for in the design. If a shaper cutter having a number of teeth equal to or greater than the pinion is used to cut the internal gear, it will cut its way into mesh but in so doing will remove some material from the flanks of a few of the teeth that should have been left in place in order to have good tooth operation. This cutting action is also known as trimming. Such teeth will have poor contact and will tend to be noisy. If the proper shaper cutter or a broach is used, this problem will not occur. The second problem is sometimes known as fouling. In this case, the internal gear teeth interfere with the flanks of the external tooth pinion if there is too small a difference in numbers of teeth between the pinion and the gear members.

Gear Tooth Design

167

Both these problems can be avoided in most gear designs by reducing the addendum of the internal gear (increasing its inside diameter). Tables 4.20 and 4.21 show a group of tooth proportions those will avoid these problems. A rather complicated graphical layout or involved calculations may be required to determine the exact proportions to avoid these problems. Since such calculations are beyond the scope of this chapter, Tables 4.20 and 4.21 are offered as a guide. Special calculations. In general, the tip interference problem can be limited by providing more than a 17-tooth difference between gear and pinion. The internal gear teeth have an addendum that extends toward the inside from the pitch circle. The critical dimension of addendum height is maintained by holding inside diameter.

TABLE 4.20 Addendum Proportions and Limiting Numbers of Teeth for Internal Spur Gears of 20˚ Pressure Angle No. of Pinion Teeth

Pinion Addendum

Min. No. of Gear Teeth

Gear Addendum

Axial Assembly

Radial Assembly

Minimum Ratio

Ratio of 2 Ratio of 4 Ratio of 8

12

1.350

19

26

0.472

0.510

0.582

0.616

13

1.510 1.290

19 20

26 27

0.390 0.507

0.412 0.556

0.451 0.635

0.471 0.673

1.470

20

27

0.419

0.445

0.488

0.509

14

1.230 1.430

21 21

28 28

0.543 0.447

0.601 0.479

0.688 0.525

0.729 0.548

15

1.180

22

30

0.574

0.642

0.733

0.777

16

1.400 1.120

22 23

30 32

0.470 0.608

0.506 0.688

0.554 0.786

0.577 0.834

1.380

23

32

0.487

0.526

0.574

0.597

17

1.060 1.360

24 24

33 33

0.642 0.505

0.734 0.546

0.839 0.594

0.890 0.617

18

1.000

25

34

0.676

0.779

0.892

0.947

19

1.350 1.000

25 27

34 35

0.516 0.702

0.558 0.792

0.605 0.808

0.628 0.950

1.330

27

35

0.539

0.578

0.625

0.648

20

1.000 1.320

28 28

36 36

0.713 0.550

0.802 0.590

0.903 0.636

0.952 0.658

22

1.000

30

39

0.733

0.821

0.912

0.957

24

1.290 1.000

30 32

39 41

0.577 0.750

0.621 0.836

0.666 0.920

0.688 0.960

1.270

32

41

0.599

0.644

0.687

0.709

26

1.000 1.250

34 34

43 43

0.766 0.620

0.849 0.666

0.926 0.709

0.963 0.729

30

1.000

38

47

0.792

0.870

0.936

0.968

40

1.220 1.000

38 48

47 57

0.654 0.836

0.702 0.903

0.741 0.952

0.761 0.976

1.170

48

57

0.718

0.764

0.797

0.814

*

Minimum ratio of number of gear teeth divided by number of pinion teeth for axial assembly.

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Dudley’s Handbook of Practical Gear Design and Manufacture

TABLE 4.21 Addendum Proportions and Limiting Numbers of Teeth for Internal Spur Gears of 25˚ Pressure Angle No. of Pinion Teeth

Pinion Addendum

Min. No. of Gear Teeth

Gear Addendum

Axial Assembly

Radial Assembly

Minimum Ratio

Ratio of 2 Ratio of 4 Ratio of 8

12

1.000 1.220

17 17

20 21

0.699 0.601

0.793 0.656

0.900 0.720

0.934 0.740

13

1.000

18

22

0.718

0.810

0.908

0.940

14

1.200 1.000

18 19

22 24

0.622 0.734

0.680 0.824

0.742 0.915

0.761 0.944

1.180

19

24

0.644

0.703

0.763

0.782

15

1.000 1.170

20 20

25 25

0.748 0.659

0.837 0.719

0.921 0.776

0.948 0.794

16

1.000

21

26

0.761

0.847

0.926

0.951

17

1.150 1.000

21 22

26 27

0.679 0.773

0.741 0.857

0.797 0.930

0.815 0.954

1.140

22

27

0.694

0.755

0.809

0.826

18

1.000 1.130

23 23

28 28

0.783 0.708

0.865 0.769

0.934 0.820

0.957 0.837

19

1.000

24

29

0.793

0.873

0.938

0.959

20

1.120 1.000

24 25

29 30

0.721 0.802

0.782 0.879

0.832 0.941

0.848 0.961

1.110

26

29

0.741

0.795

0.843

0.859

22

1.000 1.090

28 28

32 32

0.824 0.765

0.891 0.820

0.947 0.866

0.965 0.881

24

1.000

30

34

0.837

0.900

0.951

0.968

26

1.080 1.000

30 32

34 37

0.782 0.847

0.836 0.908

0.879 0.955

0.893 0.970

1.060

32

37

0.806

0.859

0.900

0.914

30

1.000 1.050

36 36

40 40

0.865 0.829

0.921 0.879

0.961 0.915

0.974 0.927

40

1.000

46

51

0.896

0.941

0.971

0.981

1.020

46

51

0.880

0.923

0.952

0.961

*

Minimum ratio of number of gear teeth divided by number of pinion teeth for axial assembly.

In general, the addenda of internal gears are made considerably shorter than those for equivalent external gears to avoid interference. The effect of the gear wrapping around the pinion tends to increase contact ratio, and also the chances for pinion-fillet interference. Internal gear-sets may be designed so that the pinion can be introduced into mesh at assembly by a radial movement. For any given number of teeth in the pinion, there must be a number of teeth in the gear that is somewhat greater than required if axial assembly were to be allowed. If a minimum difference in numbers of pinion and gear teeth is desired, the designer must design the gear casings and bearings so that the pinion can be introduced into mesh with the gear in an axial direction.

Gear Tooth Design

169

Tables 4.20 and 4.21 give the minimum numbers of teeth that a gear may have for any given pinion to achieve either radial or axial assembly. The “hand” of helical internal gears is determined by the direction in which the teeth move, right or left, as the teeth recede from an observer looking along the gear axis. Whereas two external helical gears must be of opposite hand to mesh on parallel axes, an internal helical gear must be of the same hand as its mating pinion. Specific calculation procedure for internal gears. Generally speaking, internal gears may be de­ signed by the same procedure as outlined in Section 4.2.1 for spur gears, or Section 4.2.2 for helical gears. However, there are some special considerations: 1. Number of teeth. The number of teeth in the pinion is based on the ratio required and also on tooth strength and wear considerations. In addition, there must be sufficiently large difference between the numbers of teeth in the pinion and gear to avoid problems of tip interference. Tables 4.20 and 4.21 give the minimum numbers of teeth in gear and pinion that can be specified and still be able to achieve either axial or radial assembly. 2. Number of teeth, gear. The minimum number of teeth in the gear member that can be used without getting into problems of assembly techniques or tip interference problems is shown in Tables 4.20 and 4.21. The manufacturing organization that will produce the gears should be checked to de­ termine the availability of suitable equipment when the number of teeth in the gear exceeds 200, or when prime numbers over 100 are used. If the internal gear is to be broached, consideration should be given to minimizing the number of teeth in the gear so as to keep broach cost down. 3. Helix angle. Since shaper guides are usually required for the cutting of the internal gear, first choice in helix angles should be based on existing guides. Because of the kinematics of the gear shaper, helix angle above about 30˚ should be avoided. The helix angle of the pinion member is of the same hand as that of the gear. 4. Diametral pitch. Since special tools are usually required to produce the internal gear, thought should be given to standardizing on ranges of numbers of teeth and diametral pitches for internal gearing. Table 4.12 shows suggested diametral pitches. Internal gearing can be made long and short addendum so that the need of nonstandard pitches to meet special center-distances is vir­ tually nonexistent. 5. Normal circular pitch. In helical internal gears the normal circular pitch may be specified on the internal gear cutter in the case of existing cutters.

4.2.4 STANDARD SYSTEMS

FOR

BEVEL GEARS

It is well established practice to follow the latest standards on gearing. If manufacturing equipment built in German, Swiss, or Japanese companies is in use, appropriate standard data should be obtained from the manufacturer. For the general guidance of the reader, the principal characteristics of systems now in use are discussed. Discussion of 20˚ straight bevel gear system. The tooth form of the gears in this system is based on a symmetrical rack. In order to avoid undercut and to achieve approximately equal strength, a different value of addendum is employed for each ratio. If these gears are cut on modern bevel gear generators, they will have a localized tooth bearing. The selection of addendum ratios and outside diameter is limited to a 1:1.5 ratio in width of top lands of pinion and gear. The face cone of the gear and pinion blanks is made parallel to the root cone elements to provide parallel clearance. This permits the use of larger edge radii on the generating tools with the attendant greater fatigue strength. Figure 4.11 shows data on the relation of dedendum angle to undercut for straight bevel gears. Discussion of spiral bevel gear system. Tooth thicknesses are proportioned so that the stresses in the gear and pinion will be approximately equal with the left-hand pinion driving clockwise or a left-hand

170

Dudley’s Handbook of Practical Gear Design and Manufacture

Relation between pressure angle, dedendum angle, and pitch angle for undercut in straight bevel gears tan =

(1+4tan2 sin2 cos2 )–1

2tan cos2 - is the maximum dedendum angle for undercut - is the angle of straight bevel gear - is the pressure angle

22

1° 2 20° 17

1° 2 14

1° 2

FIGURE 4.11 Relation between the Dedendum Angle and the Pitch Angle at Which Undercut Begins to Occur in Generating Straight Bevel Gears Using Sharp-Cornered Tools.

pinion driving counterclockwise. The values shown apply for gears operated below their endurance limits. For gears operated above their endurance limits, special proportions will be required. Special proportions will also be required for reversible drives on which optimum load capacity is desired. The method of establishing these balances is beyond the scope of this chapter but may be found in the Gleason Works publication “Strength of Bevel and Hypoid Gears”. The tooth proportions are based on the 35˚ spiral angle. A smaller angle may result in undercut and a reduction in contact ratio (see Figure 4.12). Discussion of Zerol bevel gear system. The considerations of tooth proportions to avoid undercut and loss of contact ratio as well as achieve optimum balance of strength are similar in this system to those of the straight and spiral systems. This system is based on the duplex cutting method in which the root cone elements do not pass through the pitch cone apex. The face cone of the mating member is made parallel to the root cone to produce a uniform clearance. Special tooth forms. The teeth of bevel gears can be given special form to reduce manufacturing costs or for engineering reasons.

Gear Tooth Design

171

22

1° 2 20°

16 14

1° 2

FIGURE 4.12 Relation between the Dedendum Angle and the Pitch Angle at Which Undercut Begins to Occur in Generating Spiral Bevel Gears at 35˚ Spiral Angle Using Sharp-Cornered Tools.

Because of the flexibility of the machining system used to cut most bevel gears, many forms are possible. The Formate12 tooth form is often used for gearing of high ratios because of its manufacturing economies. The teeth of the gear member are cut without generation, Thus, saving time, and the extra generation required to produce a conjugate pair is taken on the pinion. Since there are fewer pinion teeth, less time is spent in generation than if both pinion and gear teeth were generated. Another tooth form suitable for high-speed manufacture is the Revacile13 straight bevel gear. This special tooth form is generated by a large disc cutter, one space being completed with each cutter revolution. Both these forms are beyond the scope of this chapter. Limitations in 20 straight bevel gear system. The data contained in Table 4.22 apply only to gears that meet the following requirements: • The standard pressure angle is 20 ; in certain cases, depending on numbers of teeth, other pressure angles may be used (see Table 4.23). 12 13

Registered trademark of the Gleason Works, Rochester, NY. Registered trademark of the Gleason Works, Rochester, NY.

172

Dudley’s Handbook of Practical Gear Design and Manufacture

TABLE 4.22 Tooth Proportions of Standard Bevel Gears Item

Type of Tooth Straight Teeth

Pressure angle, deg. Working depth Clearance Face width Spiral angle Min. No. of teeth in system Whole depth Diametral range AGMA ref.

Spiral Teeth

Zerol Teeth

20 standard

20 standard

2.000/Pd 0.188/Pd + 0.002 F A o /3 or F 10/ Pd do not exceed smaller valueb – see Table 3.22

1.700/ Pd 0.188/Pd F A o /3 or F 10/ Pd whichever is smallerb 35 e 12

20 basic, 22 ½ or 25 where needed 2.000/ Pd 0.188/Pd + 0.002 F 0.25A o or F 10/ Pd whichever is smaller g F < 1 on duplex Zerols 0 13

2.18 Pd

2.18 f Pd

2.18 Pd

–

12 and coarser

3 and finer

208.02

209.02

202.02

a

+ 0.002

d

+ 0.002

Notes a 20 is standard pressure angle for straight-tooth bevel gears. Table 4.23 shows ratios that may be cut with 14 ½ degree pressure angle teeth. b It the face width exceeds one-third the outer cone distance, the tooth is in danger of breakage in the event that tooth contact shifts to the small end of the tooth. c This is the minimum number of teeth in the basic system (see Table 4.23 for equivalent minimum number of teeth in the gear member). d 20 is standard pressure angle for spiral bevel gears. Table 4.23 shows ratios that may be cut with a 16 degree pressure angle teeth. e 35 is standard pressure angle. If smaller spiral angles are used, undercut may occur and the contact ratio may be less. f For gears of 10 diametral pitch and coarser, the teeth are often rough-cut 0.005 deeper to avoid having the finish blades cut on their ends. g On duplex Zero bevel gears 1 in. is the maximum face width in all cases.

TABLE 4.23 Pressure Angle and Ratio, Minimum Number of Teeth in Gear and Pinion That Can Be Used with Any Given Pressure Angle Pressure Angel, Deg

Type of Bevel Gear Straight Tooth Pinion

20 (standard)

Spiral Tooth

Gear

Pinion

Zerol Tooth

Gear

Pinion

Gear

16

16

17

17

17

17

15

17

16

18

16

20

14 13

20 30

15 14

19 20

15 –

25 –

(Continued)

Gear Tooth Design

173

TABLE 4.23 (Continued) Pressure Angle and Ratio, Minimum Number of Teeth in Gear and Pinion That Can Be Used with Any Given Pressure Angle Pressure Angel, Deg

Type of Bevel Gear Straight Tooth

14 ½

25

Zerol Tooth

Pinion

Gear

Pinion

Gear

Pinion

Gear

– –

– –

13 12

22 26

– –

– –

29 28

29 29

28 27

28 29

27

31

26

30

26 25

35 40

25 24

32 33

24

57

23

36

– –

– –

22 21

40 42

–

–

20

50

–

–

19 24

70 24

23

25

22 21

26 27

20

29

19 18

31 36

17

45 59 16

16

22 ½

Spiral Tooth

Not used

Not used

Not used

13

13

16 16

14

14

–

–

16

17

13

15

– –

– –

16 16

18 19

– –

– –

–

–

15

15

–

–

– –

– –

15 15

16 17

– –

– –

–

–

15

18

–

–

– –

– –

15 15

19 20

– –

– –

–

–

15

21

–

–

– –

– –

15 15

22 23

– –

– –

–

–

15

24

–

–

– –

– –

14 13

14 15

– –

– –

12

12

13

13

13

13

– –

– –

13 12

14 12

– –

– –

174

Dudley’s Handbook of Practical Gear Design and Manufacture

• In all cases, full-depth teeth are used. Stub teeth are avoided because of the reduction in contact ratio, which may increase noise, and the reduction in wear resistance. • Long- and short-addendum teeth are used throughout the system (except for 1:1 ratios) to avoid undercut and to increase the strength of the pinion. • The face width is limited to between one-fourth and one-third cone distance. The use of greater face widths results in an excessively small tooth size at the inner ends of the teeth. Limitations in spiral bevel gear system. This system is more limited in its application than the straight tooth system. The data in this system do not apply to the following: • Automotive rear-axle drives • Formate pairs • Gears and pinions of 12 diametral pitch and finer, which are usually cut with one of the duplex spread-blade methods • Gear cut spread-blade and pinion cut single-side, with a spiral angle of less than 20 • Ratios with fewer teeth than those listed in Table 4.23 • Larger spiral bevel gears cut on the planning-type generators where the spiral angles should not exceed 30 Limitations in Zerol bevel gear system. The data contained in Table 4.22 apply only to Zerol bevel gears that meet the following requirements: • The standard pressure angle (basic) is 20 . Where needed to avoid undercut, 22 ½ and 25 pressure angles are standard (see Table 4.23). • The face width is limited to 25% of the cone distance since the small-end tooth depth decreases even more rapidly as the face width increases because of the duplex taper. In duplex Zerol gears, 1 in. face is the maximum value in any case. General comments. Table 4.7 will be found helpful in selecting the proper numbers of teeth for most power gearing applications. Table 4.23 gives the minimum numbers of teeth in the gear for each number of pinion teeth and pressure angle. In general, the more teeth that are in the pinion, the more quietly it will run and the greater will be its resistance to wear. The equipment used to produce bevel gearing imposes upper limits on numbers of teeth. In general, if the gear is to contain over 120 teeth if an even number or above 97 if a prime number, the manufacturing organization that is to produce the gears should be checked for capacity. Table 4.23 gives the minimum numbers of teeth in the pinion for each number of teeth in the gear for each pressure angle. The minimum number of pinion teeth is based on consideration of undercut. Table 4.24 shows addendum values that have been used in the past. These values are at the large end of the tooth (outer cone distance). These values show what was usually used in older designs that are in production. New standards that are now being developed may not agree with these values. (If a new standard is used, all values should be taken from the new standard—including addendum and backlash). Table 4.25 shows typical backlash values for bevel gears. In the field of bevel gears, mounting distance settings can change backlash and change the tooth contact pattern. Usually, the desired contact is achieved when a bevel gear set is mounted at the specified mounting distances. Figure 4.13 shows a typical mounting surface and body dimensions for a bevel gear.

4.2.5 STANDARD SYSTEMS

FOR

WORM GEARS

For general reference, Table 4.26 presents the tooth proportions for single-enveloping worm gears and for double-enveloping worm gears. These proportions are typical of past design practice.

Gear Tooth Design

175

TABLE 4.24 Bevel Gear Addendum (for 1 Diametral Pitch) Ratio (m G ) From

To

Ratio (m G )

Addendum Straight and Zerol

Spiral

From

To

Addendum Straight and Zerol

Spiral

1.00

1.00

1.000

0.850

1.52

1.56

0.730

0.620

1.01

1.02

0.990

0.840

1.56

1.57

0.720

0.620

1.02 1.03

1.03 1.04

0.980 0.970

0.830 0.820

1.57 1.60

1.60 1.63

0.720 0.710

0.610 0.610

1.04

1.05

0.960

0.820

1.63

1.65

0.710

0.600

1.05 1.06

1.06 1.08

0.950 0.940

0.810 0.800

1.65 1.68

1.68 1.70

0.700 0.700

0.600 0.590

1.08

1.09

0.930

0.790

1.70

1.75

0.690

0.590

1.09 1.11

1.11 1.12

0.920 0.910

0.780 0.770

1.75 1.76

1.76 1.82

0.690 0.680

0.580 0.580

1.12

1.13

0.900

0.770

1.82

1.89

0.670

0.570

1.13 1.14

1.14 1.15

0.900 0.890

0.760 0.760

1.89 1.90

1.90 1.97

0.660 0.660

0.570 0.560

1.15

1.17

0.880

0.750

1.97

1.99

0.650

0.560

1.17 1.19

1.19 1.21

0.870 0.860

0.740 0.730

1.99 2.06

2.06 2.10

0.650 0.640

0.550 0.550

1.21

1.23

0.850

0.720

2.10

2.16

0.640

0.540

1.23 1.25

1.25 1.26

0.840 0.830

0.710 0.710

2.16 2.23

2.23 2.27

0.630 0.630

0.540 0.530

1.26

1.27

0.830

0.700

2.27

2.38

0.620

0.530

1.27 1.28

1.28 1.29

0.820 0.820

0.700 0.690

2.38 2.41

2.41 2.58

0.620 0.610

0.520 0.520

1.29

1.31

0.810

0.690

2.58

2.78

0.600

0.510

1.31 1.33

1.33 1.34

0.800 0.790

0.680 0.680

2.78 2.82

2.82 3.05

0.590 0.590

0.510 0.500

1.34

1.36

0.790

0.670

3.05

3.17

0.580

0.500

1.36 1.37

1.37 1.39

0.780 0.780

0.670 0.660

3.17 3.41

3.41 3.67

0.580 0.570

0.490 0.490

1.39

1.41

0.770

0.66

3.67

3.94

0.570

0.480

1.41 1.42

1.42 1.44

0.770 0.760

0.650 0.650

3.94 4.56

4.56 4.82

0.560 0.560

0.480 0.470

1.44

1.45

0.760

0.640

4.82

6.81

0.550

0.470

1.45 1.48

1.48 1.52

0.750 0.740

0.640 0.630

6.81 7.00

7.00 8.00

0.540 0.540

0.470 0.460

Note: In the earlier standards and handbooks, the equations for addendum for all types of bevel gears wear aoG = Table 4.24/P . The values used in this table should be used only when checking calculations based on earlier standards.

• General practice. The following rules apply to conventional single-enveloping worm gears: • The worm axis is at right angle to the worm gear axis (90 ) • The worm gear is hobbed. Except of a small amount of oversize, the worm gear hob has the same number of threads, the same tooth profile, and the same lead as that of the mating worm. (A slight change in lead may be made to compensate for oversize effects)

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Dudley’s Handbook of Practical Gear Design and Manufacture

TABLE 4.25 Design Backlash for Bevel Gearing, in Diametral Pitch

Range of Design Backlash

Diametral Pitch

Range of Design Backlash

1.00 – 1.25

0.020 – 0.030

3.50 – 4.00

0.007 – 0.009

1.25 – 1.50

0.018 – 0.026

4.00 – 5.00

0.006 – 0.008

1.50 – 1.75 1.75 – 2.00

0.016 – 0.022 0.014 – 0.018

5.00 – 6.00 6.00 – 8.00

0.005 – 0.007 0.004 – 0.006

2.00 – 2.50

0.012 – 0.016

8.00 – 10.00

0.002 – 0.004

2.50 – 3.00 3.00 – 3.50

0.010 – 0.013 0.008 – 0.011

10.00 – 12.00 12 and finer

0.001 – 0.003

Note: In the earlier standards and handbooks, the equations for addendum for all types of bevel gears wear aoG = Table 4.24/P . The values used in this table should be used only when checking calculations based on earlier standards.

e ngl k a ce c a B stan di

Pitch cone apex Face cone apex

Face angle distance

Face width Crown to back Outside diameter FIGURE 4.13

Mounting surface

Bevel Gear Dimensions.

• Fine-pitch worms are usually milled or ground with a double-conical cutter or grinding wheel with an inclined angle of 40 (tool pressure angle is 20 ) • Coarser pitch worms are often made with a straight-sided milling or grinding tool—in American practice. Some European countries favor making the worm member in involute helicoid. Functionally, either type will work well in practice providing that the accuracy and fit of mating parts is of equal quality For double-enveloping worm gears, the defined profile is that of the worm gear. The worm is made with a special tool that has a shape similar to that of the worm gear. For the double-enveloping worm gear, the usual shape is straight-sided in the axial section and tangent to defined basic circle. Basic tooth forms for worm gearing. The tooth forms of worm gearing have not been standardized to the same degree as tooth forms for spur gearing, for example. Since a special hob has to be made to cut the gears that will mesh with each design of worm, and since the elements that constitute good worm gearing design are less well understood, there has been little incentive to standardize. The most general practice in worm gear design and manufacture is to establish the shape of the worm thread and then to design a hob that will generate teeth on the gear that are conjugate to those on the

Gear Tooth Design

177

TABLE 4.26 Typical Tooth Proportions for Worms and Worm Gears Application

No. of Worm Threads

Cutter Pressure Angle

Addendum

Working Depth

Whole Depth

Tooth Proportions for Single-Enveloping Worms and Worm Gears Index or holding mechanism Power gearing

1 or 2

14.5

1.000

2.000

2.250

1 or 2

20

1.000

2.000

2.250

25

0.900

1.800

2.050

20

1.000

2.000

20

1.000

2.000

2.200+0.002 in. English 2.200+0.002 mm Metric

3 or more Fine pitch (instrument)

1 – 10 1 – 10

Tooth Proportions for Double-Enveloping Worms and Worm Gears Item Power gearing

No. of worm threads 1 – 10

Pressure angle

Addendum

Working depth

20

0.700

1.400

Whole depth 1.600

Notes: The addendum, working-depth, and whole-depth values are for 1 normal diametral pitch. (The normal diametral pitch of the worm is equal to 3.141593 divided by the axial pitch of the worm and then the result divided by cosine of the worm lead angle). In the English system, for normal diametral pitches other than 1, divided by the normal diametral pitch. In the metric system the values are for 1 normal module. For modules other than 1, multiply by the normal module.

worm. This practice is not followed in the case of double-enveloping gearing since special tooling based on the shape of the gear member is required to generate the worm. The shape of the teeth of the worm member is dependent on the size of the tool and the method used to cut the threads. These threads may be cut with a straight-sided V-shaped tool in a lathe or milled with double conical milling cutters of up to 6-in. diameter in a thread mill or hobbed with a special hob, or ground in a thread-grinding machine with a grinding wheel having a diameter up to 20 in. In each case the worm thread profile will be noticeably different, especially in the case of the higher lead angles. Worms can also be rolled, in which case the thread profile will be still different. It is essential, therefore, for the designer of worm gearing either to specify the manufacturing method to be used to make the worm or to specify the coordinates of the worm profile. Specific calculations for worm gears. In designing worm gear units and making calculations several considerations should be kept in mind. • The ratio is the number of worm gear teeth divided by the number of worm threads. (The pitch diameter of the gear divided by the pitch diameter of the worm is almost never equal to the ratio!) • Due to the tendency of the worm gear to wear-in to best fit the worm, common factors between the number of worm threads and the number of worm gear teeth should be avoided • The normal circular pitch of the worm and the worm gear must be the same. Likewise, the normal pressure angle of worm and the normal pressure angle of the worm gear must be the same • The axial pitch of the worm and transverse circular pitch of the worm gear must be the same. In a like manner, the axial pressure angle of the worm must be the same as the transverse pressure angle of the worm gear • The lead of the worm equals the worm axial pitch multiplied by the number of threads. The lead can be thought of as the axial advance of a worm thread in one turn (360 ) of the worm

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Dudley’s Handbook of Practical Gear Design and Manufacture

• It has been customary to thin the worm threads to provide backlash but not thin the worm gear teeth • There is a small difference between the pressure angle of a straight-sided tool used to mill or grind worm threads and the normal pressure angle of the worm thread Table 4.26 shows typical tooth proportions for single-enveloping worm gear sets and for doubleenveloping worm gear-sets. This table shows the addendum is one-half the working depth. In some cases, it may be described to make the worm addendum larger than the addendum of the worm gear. Table 4.27 shows recommended minimum numbers of teeth for single-enveloping worm gears. Since the pressure angle changes going across the face width of the worm gear, larger numbers of teeth are needed to avoid undercut problems in worm gears than in spur gears. Table 4.28 shows that lead angle needs to increase with the numbers of threads. If, for instance, a lead angle of 20 is desired to get good efficiency, the designer should plan to use 4 or 5 threads on the worm. Worm gears are normally sized by picking the number of threads and teeth and then choosing a normal circular pitch that will give enough center-distance to have the necessary load-carrying ability. Table 4.29 shows suggested normal circular pitches. Table 4.30 shows nominal backlash values for worm gear sets. These values represent thinning of worm threads and tolerances on worm thread thickness and gear tooth thickness. Table 4.31 shows recommended cutter or grinding wheel outside diameters for the making of singleenveloping worms. Further details in worm gear design and rating are given below in consequent chapters of the book.

TABLE 4.27 Minimum Number of Worm Gear Teeth for Standard Addendum Pressure Angle, Deg

Min. No. of Teeth

14½

40

20

25

25 30

20 15

TABLE 4.28 Suggested Limits on Lead Angle No. Worm Threads 1 2

Lead Angle, Deg Less than 6 3 to 12

3

6 to 18

4 5

12 to 24 15 to 30

6 7 and higher

18 to 36 (not over 6˚ per thread)

Gear Tooth Design

179

TABLE 4.29 Some Recommended Normal Circular Pitches Fine Pitch English, in. 0.030 0.040

Coarse Pitch Metric, mm

English, in.

Metric, mm

1.000 1.250

0.200 0.250

5.000 6.500

0.050

1.500

0.300

8.000

0.065 0.080

2.000 2.500

0.400 0.500

10.000 12.000

0.100

3.000

0.625

15.000

0.130 0.160

3.500 4.000

0.750 1.000

20.000 25.000

1.250

30.000

1.500

40.000

Note: The normal circular pitch of the worm gear is equal to the axial pitch of the worm multiplied by the cosine of the lead angle.

TABLE 4.30 Recommended Values of Backlash for Single-Enveloping Worm Gearing and DoubleEnveloping Worm Gearing Center-Distance

Backlash (amount worm should be reduced)

2

0.003

0.008

6

0.006

0.012

12 24

0.012 0.018

0.020 0.030

TABLE 4.31 Recommended Values of Cutter Diameter (Single-Enveloping Gear Sets) Type of Worm

Suggested Cutter Diameter, Dc , in.

Process

Low-speed power

4

Milled

High-speed power

20

Ground

Fine-pitch: Commercial quality

3

Milled

Precision quality

20

Ground

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Dudley’s Handbook of Practical Gear Design and Manufacture

4.2.6 STANDARD SYSTEM

FOR

FACE GEARS

Fine pitch (20 diametral pitch and finer) face gears have been standardized. Coarse pitch face gears have not been standardized. The following data are arranged to provide a logical method of calculating the tooth proportions of face gears. Such gears will be suitable for most applications. Table 4.32 shows basic tooth proportions for face gear sets. Caution should be exercised in using Table 4.32. The items shown apply only to gears that meet the following requirements: • The axes of the gear and the pinion must intersect at an angle of 90 • The gear must be generated by means of a reciprocating pinion-shaped cutter having the same diametral pitch and pressure angle as the mating pinion and must be of substantially the same size • The pinion should be sized to meet requirements on load-carrying capacity • The minimum number of teeth in the pinion in this system is 12, and the minimum cutter pitch diameter is 0.250 in. • The minimum gear ratio is 1.5:1 and the maximum ratio is 12.5:1 • The long- and short-addendum designs for the pinion with less than 18 teeth should not generally be used on speed-increasing drives Pinion design. The pinion member in case of numbers of teeth less than 18 has an enlarged tooth thickness to avoid undercut when cut with a standard cutter (Tables 4.33 and 4.34). Compared with the pinions designed in accordance with data in Section 4.2.1, the outside diameter is somewhat less. This is to avoid the necessity of cutting the gear with shaper cutters that have excessively pointed teeth. The limit set is top land with 0.40/ Pd . Table 4.35 gives the tooth proportions for face gear pinions. Face gear design. The face-gear member is generated by a cutter having proportions based on the pinion with which the face gear will operate. The most important specification for the shape of the teeth of a face gear is a complete specification of the cutter to be used to cut it or, next best, a detailed specification of the mating pinion. The face gear has two dimensions unique to face gears which control the face width of the teeth: the outer and inner diameter of the face gear.

TABLE 4.32 Tooth Proportions for Pinions Meshing with Face Gears Coarse Pitch, Pd

Item No. of teeth in pinion, NP

NP < 18

20

Fine Pitch, Pd > 20

18

NP

NP < 18

18

NP

20

20

20

20

See Table 4.34

1 Pd

See Table 4.34

1 Pd

N Pd

N Pd

N Pd

N Pd

Working depth, h k

–

2.0 Pd

See Table 4.35

2.0 Pd

Whole depth, h t

–

2.25 2 Pd

See Table 4.35

Pressure angle, , deg Addendum, a Standard pitch diameter, Dp

Circular tooth thickness , t

–

Clearance, c

–

*

*

2 Pd 0.25 Pd

2.20 2 Pd

See Table 4.35 0.2 Pd

Thin teeth for backlash. See Section 4.5 for general information on spur gear backlash

+ 0.002

+ 0.002 2 Pd

0.2 Pd

+ 0.002

Gear Tooth Design

181

TABLE 4.33 Minimum Numbers of Teeth in Pinion and Face Gear Min.* Numbers of Teeth

Diametral Pitch Range Pinion

Gear

20 – 48 49 – 52

12 13

18 20

53 – 56

14

21

57 – 60 61 – 64

15 16

23 24

65 – 68

17

26

69 – 72 73 – 76

18 19

27 29

77 – 80

20

30

81 – 84 85 – 88

21 22

32 33

89 – 92

23

35

93 – 96 97 – 100

24 25

36 38

*

The minimum numbers of teeth in the pinion are limited by the design requirements of the gear cutter. These requirements, in addition to the minimum-gear-ratio limitation, limit the numbers of teeth in the gear.

TABLE 4.34 Addendum of Face Gears and Pinions No. of Teeth in Pinion

12

Coarse Pitch

Fine Pitch Pinion Addendum

Pinion Addendum

Gear Addendum

1.120

0.700

1.1215 Pd

0.002 0.002

13

1.100

0.760

1.1050 Pd

14

1.080

0.820

1.0865 Pd

0.002 0.002

15

1.060

0.880

1.0650 Pd

16

1.040

0.940

1.0420 Pd

0.002 0.002

0.002

17

1.020

0.980

1.0175 Pd

18 and 19

1.000 1.250

1.000 0.750

1.0000 Pd

20 and 21

1.000

1.000

22 through 29

1.250 1.000

0.750 1.000

(Continued)

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TABLE 4.34 (Continued) Addendum of Face Gears and Pinions No. of Teeth in Pinion

Coarse Pitch

Fine Pitch Pinion Addendum

Pinion Addendum

Gear Addendum

1.200 1.000

0.800 1.000

1.150

0.850

1.000 1.100

1.000 0.900

30 through 40 41 and higher

TABLE 4.35 Tooth Proportions for Fine-Pitch Face Pinions No. of Teeth Tooth Thickness at Standard Pitch Diameter

Whole Depth

Working Depth*

12

1.7878

2.0234

1.8234

13 14

1.7452 1.7027

2.0654 2.1054

1.8604 1.9054

15

1.6601

2.1424

1.9424

16 17

1.6175 1.5749

2.1778 2.2118

1.9778 2.0118

18

1.5708

2.2000

2.0000

19 and up

1.5708

2.2000

2.0000

*

Divide by diametral pitch and subtract 0.002 .

The maximum useable face width may be estimated from Table 4.36. The outside diameter of the gear should not exceed mo D. The inside diameter should not be less than m i D. Thus: Outer diameter = mo D = Do

(4.47)

Inner diameter = mi D = Di

(4.48)

Pitch diameter = D

(4.49)

Face width =

4.2.7 SYSTEM

FOR

SPIROID

AND

Do

Di 2

(4.50)

HELICON GEARS

At the present time there are no standards covering Spiroid, Helicon, or Planoid14 gears. The following material is based on information covered in “Spiroid Gearing”, paper 57-A-162 of the ASME, and on additional information supplied by its author, W.D. Nelson.

14

Spiroid, Helicon, and Planoid are trademarks registered by the Illinois Tool Works, Chicago, IL.

Gear Tooth Design

183

TABLE 4.36 Tooth Proportions and Diameter Constants for 1 Diametral Pitch Face Gears, 20˚– Pressure Angle No. of Pinion Teeth, NP

Gear Diameter Constraints

mg = 2

mg = 1.5

mg = 4

mg = 8

mo

mi

mo

mi

mo

mi

mo

mi

12 13

… 1.202

… 1.064

1.221 1.202

1.020 1.015

1.221 1.202

0.960 0.959

1.221 1.202

0.945 0.945

14

1.187

1.062

1.187

1.011

1.187

0.958

1.187

0.944

15 16

1.174 1.161

1.052 1.051

1.174 1.161

1.007 1.004

1.174 1.161

0.957 0.956

1.174 1.161

0.944 0.944

17

1.156

1.041

1.156

1.000

1.156

0.955

1.156

0.944

18

1.150 1.176

1.039 1.042

1.150 1.176

0.997 0.999

1.150 1.176

0.954 0.955

1.150 1.176

0.943 0.943

20

1.144

1.030

1.144

0.991

1.144

0.953

1.144

0.943

22

1.166 1.140

1.032 1.022

1.166 1.140

0.993 0.987

1.166 1.140

0.953 0.952

1.166 1.140

0.943 0.943

1.156

1.024

1.156

0.988

1.156

0.952

1.156

0.943

24

1.133 1.150

1.015 1.017

1.133 1.150

0.983 0.984

1.133 1.150

0.951 0.951

1.133 1.150

0.943 0.943

30

1.121

1.001

1.121

0.975

1.121

0.949

1.121

0.942

40

1.131 1.109

1.001 0.986

1.131 1.109

0.975 0.966

1.131 1.109

0.949 0.946

1.131 1.109

0.942 0.941

1.113

0.986

1.113

0.966

1.113

0.946

1.113

0.941

Although specialized designs can be developed that will best suit a given application, the standardized procedure presented here will yield designs that will meet the needs of most Spiroid and Helicon gear applications. Planoid gears are often associated with Spiroid and Helicon gearing but are an entirely different form of gearing, used for ratios generally under 10:1 where maximum strength and efficiency are required. Their design is beyond the scope of this section. The basic approach to the establishment of the Spiroid or Helicon tooth form is to establish the pinion tooth form and then develop a gear tooth form that is conjugate to it. The tooth form of the Spiroid gear is compromised of teeth having different pressure angles on each side of the teeth: a “low”-pressure-angle side and a “high”-pressure-angle side. The choice of pressure angle will control the extent of the fields of conjugate action of the teeth. The values of pinion taper angle have been standardized by the Illinois Tool Works in order to achieve the best general-purpose gearing. These standard values are shown in Table 4.37. The design of Helicon gearing is closely related to Spiroid gearing. Table 4.38 shows the general equations used in the design of both kinds of gearing. The general Spiroid formulas become Helicon formulas when the pinion taper angle is set equal to zero. Figure 4.14 shows a Spiroid gear set. A Helicon pinion and gear are similar except that the pinion taper angle and the gear face angle are both zero in Helicon gearing. Spiroid gearing. Spiroid gearing is suitable for gear ratios of 10:1 or higher. The numbers of teeth in the gear can range from 30 to 300.

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Dudley’s Handbook of Practical Gear Design and Manufacture

TABLE 4.37 Standard Tooth and Gear Blank Relationships, Spiroid and Helicon Gearing Spiroid gearing: 1. “Sigma” angle,

P

2. Pinion taper angle

= 40 (standard) vs. gear face angle

Pinion taper angle, , deg

Gear face angle*, , deg

5 7

8 preferred 11

10

14

3. Gear ratio m G = 10: 1 higher 4. Number of teeth in gear: Varies with center distance (see Table 4.18) Hunting ratios are desirable with all multiple-thread pinions 5. Tentative pressure angle selection Ratio

16: 1

15

35

m G > 16: 1

10

30

mG

*

Pressure angle selection Low side High side

Approximate values, for

P

= 40 .

TABLE 4.38 General Equations for Spiroid and Helicon Gearing Spiroid Gearing:

sin RP =

tan tan P

=

P

C R sin P + G R P

(4.52) cos P

RG R RP P

RG =

(4.53)

2 RG cos P

L=

RG sin P cos P RP

mG

xP = RG sin

ro = RP DW =

P

(4.55) (4.56)

L sec NP sin 1 sin 2 + cos( 1 + ) cos( 2 )

0.6

DN = DW cos L

P

= tan

RP xP

1

tan 2

m

=

(4.54)

xP tan

CLR = 0.07 N + 0.002 2L

(4.51)

+

= 2L

C zP RP (k yP xP )

(4.57)

(4.58) (4.59) (4.60)

L sec 2 rm

(4.61)

+5

(4.62)

(Continued)

Gear Tooth Design

185

TABLE 4.38 (Continued) General Equations for Spiroid and Helicon Gearing Spiroid Gearing: Helicon gearing:

RP =

mG

2L

Center distance, C

r, Outside diamete gear Point diam e gear ter,

Pinion shaft diameter, small end Taper angle,τ

(4.64)

RG sin P RP

0.6

DW =

L n

(4.65)

tan 1 + tan 2

= tan

= tan 2

(4.63)

2 RG cos P

L=

2L

C RG cos P RP

=

1

2L

1 RP xP

(4.66)

1

(4.67)

RG sin P RP

+7

(4.68)

Gear point to pinion axis Outside diameter, pinion Pinion shaft dia., large end

r0

γ

Face width, pinion

Mounting surface to pinion point

Gear axis to pinion point Mounting distance, pinion

FIGURE 4.14

γ

Gear face angle

Mounting distance, gear

Sprioid Gear Set—Mounting and Gear Blank Nomenclature.

The design procedure for Spiroid gearing differs somewhat from other types of gearing discussed in this chapter. The following are the basic steps: • Center-distance is the basic starting point for the calculation of Spiroid tooth proportions • Lead and pinion “zero” plane radius are then calculated • The tooth proportions and the blank proportions are based on the pinion zero plane radius and are then calculated From this procedure it can be seen that the pitch of the teeth is an end result of the calculating procedure for Spiroid gearing, rather than the beginning s n other types of gearing. Center-distance governs the horsepower capacity of the gear set.

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Table 4.37 shows the basic tooth proportions for Spiroid gearing. Table 4.38 gives the general equations used to determine Spiroid tooth proportions. If optimum efficiency is desired, the size of the pinion relative to the gear should be kept small by using values of R G / RP which are larger than those shown in Figure 4.15. This has the effect of producing an increased lead angle for a given center- distance and gear ratio. Figures 4.16 and 4.17 show re­ commended data for Helicon and Spiroid calculations. If control gear applications being considered, in which runout is critical, the driving side should be the low-pressure-angle side of the teeth. This minimizes the effects of runout in both members. A 10 taper angle may be used, since it requires only one-half the axial movement of the pinion to produce a given change in backlash. For gear sets of the highest strength, the number of teeth in the gear should be kept low (30 to 40 for lower ratios) and a stub tooth form employed. Such designs are beyond the scope of this section.

High point

DPG

σ

2

C DW

Gear

DOG 2

τ

r0

r Z Pinion

γ y

FIGURE 4.15

Working root cone

Desired high-point value

X (trial)

Gear Tooth High Point—Spiroid Gearing.

Read Sigma angle, p

Read

RG RP

RG RP

Reduction on ratio mG FIGURE 4.16

Dop

Sigma Angle for Different Ratios—Helicon Gearing.

Gear Tooth Design

187

RG RP

Center diatance, inches FIGURE 4.17

R G /RP vs. Center-Distance and Number of Teeth—Spiroid Gears.

Straddle-mounted pinions may require a shaft of larger size (greater stiffness) than would result from the data shown in Figure 4.14. In such cases, the values of R G , RP are decreased until a shaft of the desired size can be achieved. There is a definite limit to the maximum size of pinion for a given pressure angle. The size of the pinion may be increased until a “limiting pressure” condition is reached. Limiting pressure angle can be calculated by means of Eq. (4.62) Helicon gearing. Helicon gearing is suitable for gear ratios from 4:1 to 400:1 in light- to mediumloaded applications. The design procedure for Helicon gears is somewhat similar to that for Spiroid gears but starts with the gear outside diameter as the basis of load capacity instead of the center-distance as is used in Spiroid gearing. The following are the basic steps: • The gear ratio m G , pinion r/min, and horsepower are used to establish the necessary gear outside diameter • Pinion pitch radius, center-distance, lead, and working depth are next established • From these values, the gear and pinion blank dimensions are established

4.3

GENERAL EQUATIONS RELATING TO CENTER-DISTANCE

This section of Chapter 4 deals with the distance between the shafts of meshing gears. This distance is called center-distance in the case of gearing operating on nonintersecting shafts. Spur and helical gearing, both external and internal, worm gearing, hypoid gearing,15 spiral gearing, Spiroid16 gearing, and Helicon17 gearing must all operate at specific center-distances. Certain types of gears which operate on intersecting shafts, such as bevel gearing, do not have a center-distance dimension. However, their pitch surfaces must be maintained in the correct relationship; hence, the axial position of these gears along their shafts is critical. Such gears, therefore, have a mounting distance that defines the axial position of the gears.

15 16 17

In the case of hypoid gearing, offset is the correct term for the distance between gear and pinion shafts. Spiroid is a trademark of the Illinois Tool Works, Chicago, IL. Helicon is a trademark of the Illinois Tool Works, Chicago, IL.

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Certain types of gears, such as hypoid gears, Spiroid gears, throated worms and worm gears, face gears, Helicon gears, and Planoid18 gears, have both a center distance and a mounting distance. When establishing the center-distance for a set of gears, it is customary first to determine the theo­ retical center-distance for the gears. The actual operating center- distance is next determined. This center-distance includes considerations of tolerance and the interchangeability of the various parts that may be included in the final assembly such as bearing, mounting brackets, and housing. In spur and helical gears, it may be necessary or desirable to operate the gears on an appreciably larger center-distance than theoretical to get improved load-carrying capacity. For instance, gears can be de­ signed so that 20 hobs cut the tooth; but the “spread” center is calculated so as to make the gears operate at a 25 pressure angle. See Sections 4.3.4, 4.3.5, and 4.3.6. The next section covers the calculation of the theoretical center-distance at which various types of gearing will operate. The most generally used equations are summarized in Table 4.39. In the Sections that follow, the special consideration on which the equations are based are covered in detail.

4.3.1 CENTER-DISTANCE EQUATIONS Table 4.39 is a summary of equations convenient to use to obtain the values of center-distance for the various types of gearing shown in Figure 4.18. Center-distance and tooth thickness are inseparable. In most case, however, standard tooth propor­ tions are used, and the simplified equations of center-distance [Eq. (4.69) through Eq. (4.72)] are ade­ quate. Standard tooth proportions are defined in Section 4.3.2. When the sum of the tooth thickness of pinion and gear does not equal the circular pitch, nonstandard center-distance, as obtained from equa­ tions such as Eq. (4.73) and Eq. (4.76), will be required. Most problems involving center distance and tooth thickness fall into one of the following categories: • The thickness of the teeth of the gear and of the pinion is fixed. The center- distance at which the gears will mesh properly is to be established a. The sum of the tooth thicknesses of both members plus the design backlash is equal to the circular pitch. In this case, standard center-distance is correct. Use Eq. (4.69) and Eq. (4.72) b. The sum of the tooth thickness of both members plus the design backlash is not equal to the circular pitch. In this case, a nonstandard center-distance is required. Use Eq. (4.73) and Eq. (4.76). This problem is covered in greater detail in Section 4.3.9. • The center-distance is fixed. The tooth thicknesses for gears that will operate on the given centerdistance are to be established. If the center-distance for the given diametral pitch and number of teeth is different from that obtained from Eq. (4.70), then Eq. (4.73) must be solved and the operating pressure angle Thus, found used in Eq. (4.76), which is then solved for the term (tP + tG + B). In Section 4.1.4, the ways in which this total tooth thickness can be divided between gear and pinion are discussed • Neither the tooth thickness nor the center-distance is fixed; the best values for both are to be established. In this happy case, the tooth thickness is usually established first on the basis of strength or some other appropriate consideration and the center distance required is then based on Eq. (4.76) and Eq. (4.73) • Both the center-distance and the tooth thickness are fixed. The amount of backlash or the degree of tooth interference is to be established. This is a frequent “check problem” of an existing gear design. Equations (4.73) and (4.76) are used and solved for B. Minus values of B indicate tooth interference

18

Planoid is a trademark of the Illinois Tool Works, Chicago, IL.

Gear Tooth Design

189

TABLE 4.39 Center-Distance Equations Standard Center-Distance, C :

C=

d+D 2

(4.69)

C=

n+N 2P

(4.70)

C=

D

d

(4.71)

C=

N n 2P

(4.72)

Spur, helical, and worm gears

C =

C cos cos

(4.73)

Spur helical

C =

d +D 2

(4.74)

Internal

C =

D

(4.75)

Spur, helical, and worm gears

Internal gears

2

Operating center-distance, C :

d 2

General equations relating tooth thickness and center-distance for parallel axis gearing: External spur and helical gears

inv

=

Internal gears

inv

=

n (tP + tG + B ) (n + N ) d d

d

+ inv

(4.76)

n (tP + tG + B ) (N n ) d

+ inv

(4.77)

General equations relating tooth thickness and center-distance for nonparallel nonintersecting axis gearing

Ka = Kb = inv

G

n N

+ Ka inv Kd =

1 mG

=

(4.78) (pn

P

tnP tnG ) N pn

inv G inv P

(4.80)

Kc = Ka + Kd

inv inv

G

= Kb

K K

d

P

d

=

(4.81)

Kb Kc

=

Ka inv

(4.79)

(4.82) P

inv G inv P

= Kd (check equation)

(4.83) (4.84) (4.85)

DP =

DbP cos P

(4.86)

DG =

DbG cos G

(4.87)

Use values from Eqs. (4.84) and Eq. (4.85) in Eq. (4.74), where

Kc =

sin G sin P

=

sin G sin P

(4.88)

P – transverse diametral pitch C – “standard” center-distance [defined by Eq. (4.69) and Eq. (4.70)]. See also Section 4.13 C – operating center-distance d – “standard” (n / P ) pitch diameter of pinion. See also Section 4.14 d – operating pitch diameter of pinion. See also Section 4.15 D – “standard” (N / P ) pitch diameter of gear. See also Section 4.14

D – operating pitch diameter of gear. See also Section 4.15 n – number of teeth in pinion

N – number of teeth in gear – cutting pressure angle

B – design backlash, the total amount the teeth in both members are thinned for backlash considerations

(Continued)

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Dudley’s Handbook of Practical Gear Design and Manufacture

TABLE 4.39 (Continued) Center-Distance Equations Standard Center-Distance, C : – operating pressure angle. See also Section 4.16 tp – circular (transverse) thickness of pinion tooth tG – circular (transverse) thickness of gear tooth tnP – circular (normal) thickness of pinion tooth tnG – circular (normal) thickness of gear tooth P

– helix angle of pinion

G

– helix angle of gear

P

– rolling pressure angle of pinion – rolling pressure angle of gear

G *

Does not apply to worm gearing

(a)

(b) C

(c) C

(d)

(e)

(f )

C Offset

C C

C

FIGURE 4.18 Center-Distance, Shown for Various Kinds of Gears, Is the Distance between the Axes of the Shafts Measured on a Common Normal. (a) Spur and Helical Gearing—External; (b) Spur and Helical Gearing—Internal; (c) Worm Gearing—Single- and Double-Enveloping; (d) Spiral Gearing; (e) Spiroid, Planoid and Helicon Gearing; (f) Hypoid Gearing.

4.3.2 STANDARD CENTER-DISTANCE Most gear designs are based on standard tooth proportions. Such gears are intended to mesh on standard center-distances. The equations that establish standard center-distances are based on the following assumptions: 1. The sum of the circular tooth thicknesses (effective) equals to the circular pitch minus the backlash: tP + tG = p t

B

(4.89)

This rule covers the case of long and short addendums in which the gear teeth are corrected for such things as undercut or balanced strength. In such cases, the tooth thicknesses of both members are altered sufficiently to permit to mesh on standard center-distances. 2. The tooth thickness of an individual gear is one-half the circular pitch (transverse) minus one-half the backlash: t=

p

B 2

(4.90)

This covers the more common case. It represents the approach used by most makers of “catalogue”-type gears. Such gears are all expected to operate on standard center-distances regardless of the number of

Gear Tooth Design

191

teeth in the pinion. The center-distances on which such gears will operate may be calculated by means of Eq. (4.69) and Eq. (4.70). The tolerance on standard center-distance can be bilateral (Thus, 5.000 in. ± 0.002 in.) if the value of B was chosen sufficiently large. This is the most convenient way from the standpoint of manufacturer. If B is not large enough, a unilateral tolerance (Thus, 5.000 in. + 0.004 in. − 000 in.) may be used.

4.3.3 STANDARD PITCH DIAMETERS The standard pitch diameter of a gear is a dimension of a theoretical circle. It is given for each type of gear by the following relations: Spur gearing: N Pd

(4.91)

N Pt

(4.92)

N Pd

(4.93)

where Pd is the diametral pitch. Helical gearing:

where Pt is the transverse diametral pitch. Bevel gearing:

Worm gearing: NG p

(4.94)

where NG equals the number of worm gear teeth, and p equals the circular pitch of the worm gear. Since the diametral pitch of a gear is fixed by the tool that is used to cut it (hob, shaper cutter, shaving cutter, and so forth), and since the number of teeth in the gear is a whole number, the standard pitch diameter is an imaginary circle. It can have no tolerance, regardless of the variations in tooth thickness or in the center-distance on which it is to operate.

4.3.4 OPERATING PITCH DIAMETERS As shown elsewhere, it is entirely practical to operate involute gears of specific diametral pitch and numbers of teeth at various center-distances. It is convenient in such cases to calculate operating pitch diameters for such gears. See (4.95) through (4.98). External spur and helical pinions: d =

2C mG + 1

(4.95)

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Dudley’s Handbook of Practical Gear Design and Manufacture

Internal spur and helical pinions: d =

2C mG 1

(4.96)

D =

2C mG + 1

(4.97)

D =

2C mG 1

(4.98)

External spur and helical gears

Internal spur and helical gears:

where: C – is the operating center-distance mG – is the gear ratio,

N n

d – is the operating pitch diameter of pinion D – is the operating pitch diameter of gear

These equations define the operating pitch diameters as being proportional to the transmitted ratio and the instantaneous center-distance. Since gears with thicker-than-standard teeth must operate on enlarged center- distances, they will run on operating pitch diameters that are larger than their standard pitch diameters. The operating pitch diameter should not be specified as a drawing dimension on the detail drawing of the gear since it will vary for ever different gear and center-distance. The best place to show the operating center-distance is on an assembly drawing that shows both the pinion and gear that mesh together.

4.3.5 OPERATING PRESSURE ANGLE The pressure angle of an individual gear is based on the diameter of the base circle of the gear and on the identification of the specific radius at which the pressure angle is to be considered. In standard gears, this radius is customarily the standard pitch radius. It is convenient to consider the pressure angle of gears operating on nonstandard center-distances at the point of intersection of the line of action and line of centers. This is the definition of the operating pressure angle which may be calculated by means of (4.99): cos

where: – is the operating pressure angle C – is the standard center-distance [see Eq. (4.70)] C – is the operating center-distance – is the standard pressure angle

=

C cos C

(4.99)

Gear Tooth Design

193

It can be seen fro the foregoing that a gear can be cut with a cutter of one pressure angle and operate at a different pressure angle. This flexibility causes much of the confusion in gear design. It is necessary to define accurately each of the standard elements of a nonstandard gear—pitch and pressure angle. A specification of base-circle diameter is highly desirable.

4.3.6 OPERATING CENTER-DISTANCE The actual center-distance at which a gear will operate will have a large influence on the way in which the gear will perform in service. The actual operating center distance is made up of the combined effects of manufacturing tolerances, the basic center distance, differential expansions between the gears and their mountings, and deflections in the mountings due to service loads. The items that should be considered when determining the minimum and maximum operating centerdistance for any given gear design are discussed in detail in Section 4.3.9 and Section 4.4.1. In any critical evaluation of a gear design, particularly in the field of control gearing, the minimum and the maximum operating center-distance should be used in equations covering backlash, contact ratio, tooth tip clearance, and so forth. In this chapter the concept operating center-distance is used in two ways. In one case, it is considered to be the center-distance that results fro the buildup of all tolerances that influence center-distance in any one case. This concept is illustrated in case II in Section 4.4.1. Thus, it is the largest or smallest actual center-distance that could be encountered in a given design. In the other case, the one covered by Eq. (4.73) and Eq. (4.76). For example, it is the center-distance at which gears of a specified tooth thickness and backlash will operate. Depending on the problem encountered, the correct concept to use will have to be selected by the designer.

4.3.7 CENTER-DISTANCE

FOR

GEARS OPERATING

ON

NONPARALLEL NONINTERSECTING SHAFTS

The most frequently encountered example of this type of gear operation is in the shaving or lapping of gears. Here, a cutter, usually a helical-toothed member, is run in tight mesh with a spur or helical gear. The operating center-distance is dependent on the tooth thickness of each member. Equations (4.78) through Eq. (4.87) should be solved in sequence and the result used in Eq. (4.73) to obtain the operating center- distance. Equation (4.88) gives the ratio constant for the transverse pressure angles for crossed axis gearing.

4.3.8 CENTER-DISTANCE

FOR

WORM GEARING

The center-distance for worm gearing is based on the sum of the standard pitch diameters of the worm and worm gear [(4.100)]. The standard pitch diameter of the worm is obtained from (4.101). This circle has no kinematic significance but is the basis for worm tooth proportions:

where: C – is the center-distance d w – is the pitch diameter of worm D – is the pitch diameter of gear

C=

dw + D 2

(4.100)

dw =

L tan

(4.101)

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Dudley’s Handbook of Practical Gear Design and Manufacture

L – is the lead – is the lead angle

4.3.9 REASONS

FOR

NONSTANDARD CENTER-DISTANCES

On some occasions a set of gears must operate on a center-distance that is not one-half the sum of the standard pitch diameter of the meshing gears. The designer is confronted with nonstandard centerdistance in several situations, the more important of which are: • Gear trains in which the teeth are made to standard tooth thickness and backlash is introduced by increasing the standard center-distance slightly • Gear trains in which the number of teeth and pressure angle relationship requires a long addendum (enlarged tooth thickness design) to avoid undercut, and yet the number of teeth in the gear is so small that to make the gear sufficiently short addendum to compensate for the pinion enlargement would cause undercut. In such cases, an enlarged center-distance is usually indicated • Gear trains in which the sum of the tooth thicknesses of pinion and gear is not equal to the circular pitch for reasons of tooth strength, wear, or scoring • Gear trains in which a minor change in ratio (total number of teeth in mesh) has been made without a change in center-distance In each of these cases, the calculation of operating center-distance is performed using Eq. (4.73) and Eq. (4.76), or Eq. (4.77).

4.3.10 NONSTANDARD CENTER-DISTANCES By proper adjustments of the thickness of the teeth on each gear of a meshing pair, it is possible to achieve gear designs that will meet most nonstandard center-distances. In Section 4.1.3, the limitations governing tooth thickness are outlined. In cases where maximum strength is not critical, it will be found that gears not exceeding these limitations will satisfy most nonstandard center-distance problems en­ countered. This is discussed more fully in method 3 below. Sometimes designers who do not realize the possibilities of gear design will select such things as fractional diametral pitches (10.9652, etc.) in order to make gears fit a nonstandard center-distance. This is poor practice is that it necessitates special tooling, which may be costly. It is better to attempt to utilize standard-pitch tooling and, by proper adjustments of tooth thickness, to establish gear designs that will meet the nonstandard distances. In cases wherein the center-distance is nonstandard, the designer has four possible methods of de­ signing gears to meet the given center-distance: 1. A number of teeth in gear and pinion different from the numbers originally selected may be found which will more nearly meet the given center-distance. A Brocot table is very useful in finding the numbers of teeth that will give different gear ratios (1/ mG ). In this case, it is assumed that any of several ratios may be “good enough”. If not, the following alternatives, which all assume that a specific ratio must be obtained, may be considered 2. Equation (4.69), Eq. (4.70), Eq. (4.71), or Eq. (4.72) may be rearranged to solve for Pd . Thus, a diametral pitch that will meet the gear ratio and center-distance is found. This method will yield nonstandard diametral pitches for spur gears, which will almost always lead to large manu­ facturing costs because of the need for nonstandard tooling, cutters, and master gears. This method should be used only if methods 2, 3, and 4 do not meet the needs of application

Gear Tooth Design

195

3. The gear and pinion can be made with teeth that are thicker or thinner than standard. The easiest approach when using this method is to follow method 2 above, and then use the results as the basis for a selection of the standard diametral pitch nearest to the one found. Equation (4.73) is then used, which will yield the operating pressure angle for the gears based on the new selection of diametral pitch and the desired center-distance C . The value of C in this equation is determined from Eq. (4.69) or Eq. (4.70). Equation (4.76) is then used and solved for the term (tP + tG ). The selection of values for each term, tP and tG , should be based somewhat on the ratios of their number of teeth. In general, pinions cannot be changed from standard as much as gears with large numbers of teeth. The selected value of tooth thickness should be checked for undercut in the event that tP is less than p/2. If the thickness is appreciably greater than p/2 the pinion should be checked for pointed teeth. The same should be done for the gear, especially if the changed tooth thickness was not based on the gear ratio. Once the values of C , P , n, N , tP , and tG have been established, the remainder of the tooth proportions can be calculated from these values. If the addendum and whole depth are properly adjusted from the tooth thickness, standard tools can be used to cut and to inspect these gears 4. If helical gears can be used, the helix angle can be adjusted to obtain the required tooth thicknesscenter distance relationship. In effect, the designer is following method 3 above but is adjusting the transverse tooth thickness by the selection of the required helix angle. The procedure in this case is to use Eq. (4.70) or Eq. (4.72) to establish the diametral pitch required in the transverse plane P to suit given center-distance. The nearest finer (smaller) diametral pitch for which tools are available is selected (Pn ), and these values are used in (4.102) to establish the required helix angle: cos

=

P Pn

(4.102)

Although not an essential part of this problem, somewhat smoother-running gears will result if the designer can manage to pick diametral pitches that will result in a helical overlap of a least 2. After the helix angle has been found, the designer can determine the remaining tooth proportions from data found in this chapter. This method can be used economically if the hobbing process of cutting gears is to be used. Special guides and shaper cutters are required for every different lead of helical gear for gears cut by the shaping process. This then fixes limits on the values of helix angle that any given shop could cut.

4.4

ELEMENTS OF CENTER-DISTANCE

Up to this point, this chapter has considered only the means of determining the theoretical centerdistance that is required by a set of gears. The theoretical center- distance is either the basic centerdistance, as established by Eq. (4.69), Eq. (4.70), Eq. (4.71), or Eq. (4.72), or the theoretical tight-mesh center-distance, as established by Eq. (4.73) through Eq. (4.88). In an actual gear set, however, the gears will be force to operate on a real center-distance that may be larger or smaller than the theoretical centerdistance by amounts that are based on the way that the part tolerances (bearings, casing, bores, and so forth) will add up.

4.4.1 EFFECTS

OF

TOLERANCES

ON

CENTER-DISTANCE

In all but the very simplest gear casing designs, there are many tolerances that will govern the operating position of the gear and pinion shafts. Usually, power gearing is not designed to small values of backlash. It is customary, therefore, to apply a generous amount of backlash to the gear teeth, a value that has been found by experience sufficient to prevent binding of the teeth. In control gearing, however, particularly in

196

Dudley’s Handbook of Practical Gear Design and Manufacture

those applications that are operated in both directions, and are intended to have a minimum of lost motion, a critical study of the effects of part tolerances is usually required. The tooth thickness can then be made as large as possible, ensuring a minimum of backlash with little risk of binding. The following two examples indicate a very simple case and a more complex case of center-distance calculation. Case I. In Figure 4.19 is shown a spur gear pair mounted on a cast-iron gear case. The loads and speeds are such that the bearings are simply smooth bores in the cast iron with shafts running in them. Figure 4.20 is a vector diagram showing the forces produced on the bearings as a result of the gear tooth loads. Figure 4.21 shows the displacements of the shafts in the bearing resulting from these loads. In this example, it is desired to establish the largest and smallest center-distance that the gears will ever experience in order to determine the range of backlash existing in the gear set. Table 4.40 outlines the calculations made to establish the various center-distances at which the gears may be expected to operate in service. Part A of Table 4.40 shows the calculation of the maximum distance that the axis of either shaft can move away from the axis of its bore. This is given as the eccentricity of the shaft within the bearing. It is assumed that the bearing is dry. For a very rough approximation of maximum and minimum possible center-distances, these values of eccentricity can be added to and subtracted from the boring center-distance plus its tolerance. This is shown in part B of Table 4.40. This assumes the forces acting on the shafts are tooth-load reactions that act along line of centers. In the case of journal bearings that are properly lubricated, the calculated value of eccentricity of operation can be used. This calculation is discussed in texts on journal bearings design. In most cases if the designer can select the tolerances for the gear tooth thickness, and also the tolerances on the various parts of the casing and bearings that control the center- distance, such that the backlash is not excessive when the calculations shown thus far

Bore diameter

Shaft diameter

+0.0010

−0.0003

1.000 −0.0000

1.000 −0.0008

5.000 −0.001 (center distance)

+0.0010

1.500 −0.0000

Bore diameter

+0.0003

1.500 −0.0008

Shaft diameter

FIGURE 4.19 Spur or Helical Gears Operated in Bearings Bored in the Gear Casing Exhibit the Simplest Case of Tolerance Buildup on Center-Distance.

Gear Tooth Design

197

Reaction of bearings on pinion Driver Tangential component of driving force Radial (separating) component of driving force Ft

Line of action

Fn Normal force normal to tooth profiles at pitch line

Fr

φ′

Operating pressure angle

Driven

Reaction of bearings on gear

FIGURE 4.20 The Reactions Forces Developed by the Bearings to Overcome the Gear Teeth Driving Forces are Parallel to the Line of Action.

are carried out, no further calculations need be made. However, when the design is to achieve an absolute minimum of lost motion, the more accurate evaluation of center- distance should be carried out. Part C shows a calculation assuming that the gear tooth reactions will control the position of the journal in the bearing. The minimum center-distance will occur when the minimum boring centerdistance is achieved and the minimum bearing clearance occurs. The tooth loads tend to increase the center-distance in this example so that less than the minimum boring center-distance cannot occur. In actual service, the maximum and minimum operating center-distances will rarely reach these limits. In the case of maximum center-distance, a properly lubricated shaft will develop an oil wedge that keeps the journal away from the walls of the bore. Second, if the only separating loads on the shafts are those produced by gear tooth reactions, the radial movement of the shaft along the line of centers will be equal e (sin ). The proper evaluation of operating center-distance will depend on whether the appli­ cation is intermittent in rotation and on the direction of rotation. Case II. A more complex case occurs when the pinion and gear shafts are mounted in separate units. The following example illustrates a case in which a motor having a pinion cut integral with the shaft is mounted by means of a rabbet in a gear case. This type of construction is often found in power hand tools and in aircraft actuators. This example also illustrates a critical evaluation of antifriction bearings. It is desired to determine the largest and smallest center-distance that can occur in order to establish the maximum and minimum tooth thickness and the amounts of backlash that result (see Figure 4.22). Table 4.41, which summarizes the calculations, shows a total of 16 different elements that combine to control the maximum and minimum actual center-distance. Since there are so many tolerances, the use of the root mean square of these values is suggested as a reasonable approximation to operating conditions,

198

Dudley’s Handbook of Practical Gear Design and Manufacture If friction or lubricating oil wedge ia assumed, journal center wil lie in this zone. See texts on journal bearings.

b ∆C Reaction of gear tooth load on bearing

e

∆C – is increase in center distance due to movement of journal in bearing; e –is eccentricity of bearing bore diameter – journal diameter 2 FIGURE 4.21 The Position Occupied by the Journal in a Bearing Is a Function of the Gear Tooth Loads along the Shaft and the Effects of Friction or of the Oil Wedge in the Bearing.

TABLE 4.40 Summary of Calculations for Case I A. Calculation of bearing clearance and shaft eccentricity. See Figure 4.19 for drawing dimensions Bore for pinion shaft

Max. 1.0010

Min. 1.0000

Pinion-shaft diameter Clearance in bearing (c )

Min. 0.9992 Max. 0.0018

Max. 0.9997 Min. 0.0003

Max. 0.0009

Min. 0.00015

Eccentricity of shaft position in bearing (c/2) Bore for gear shaft

Max. 1.5010

Min. 1.5000

Gear shaft diameter Clearance in bearing (c )

Min. 1.4992 Max. 0.0018

Max. 1.4997 Min. 0.0003

Max. 0.0009

Min. 0.00015

Eccentricity of shaft position in bearing (c/2)

B. Calculation of theoretical maximum and minimum center-distance (assuming reaction forces acting along line of centers; values from above) Center-distance for bores (machining tolerance) Max. 5.0010 Min. 5.0000 Eccentricity of pinion bearing Eccentricity of gear-bearing center-distance

0.0009

−0.0009

0.0009 Max. 5.0028

−0.0009 Min. 4.9982

C. Calculation of theoretical maximum and minimum center-distance (assuming reaction forces acting along line of action; values from above, part B) Center-distance of bores (machining tolerance)

Max. 5.0010

Min. 5.0000

(Continued)

Gear Tooth Design

199

TABLE 4.40 (Continued) Summary of Calculations for Case I Eccentricity, radial component of pinion bearing (0.0009) sin 20

0.0003

–

–

0.0005

(0.00015) sin 20 Eccentricity, radial component of gear bearing (0.0009) sin 20

0.0003

–

(0.00015) sin 20

–

0.0005

Max. 5.0016

Min. 5.0001

Motor end shield

Actuator end plate

Pinion

Gear

FIGURE 4.22 Cross Section of the First Stage of Gearing in an Aircraft-Type Actuator. The Members Refer to the Surfaces Discussed in Table 4.41.

and it is used here to establish the maximum and minimum probable center-distance. This method gives over 90% assurance that the values shown will not be exceeded.

4.4.2 MACHINE ELEMENTS THAT REQUIRE CONSIDERATION APPLICATIONS

IN

CRITICAL CENTER-DISTANCE

As illustrated in cases I and II, the operating center-distance for a pair of gears is made up of several elements, each of which contributes to overall center-distance. The accumulation of tolerances on each of these elements must be considered in application in which a minimum of backlash is to be established. The following is a consideration of the elements that have the largest contribution to variations in center distance: • Rolling-element bearings. Ball-and-roller bearings consist of three major elements that contribute to backlash and to changes in center-distance. Some of these are illustrated in case II. The outer race has machining tolerances that cause eccentricity of the axes of the inner raceway and the outer

(3–4) (4)

Eccentricity of outer ring of bearingd Fit of outer race of bearing into motor end shielde

Concentricity of axis of bore with axis of rabbet on end shieldf

Clearance between rabbet of motor end shield and bore of actuator end plateg Concentricity of motor bore in actuator end plate and rabbeth

4. 5.

6.

7.

Distance between axis of gear shaft bore and rabbet in actuator housingi

Clearance between bore for bearing and bearing outer racee

9.

10.

8.

(2–3)

Radial clearance in bearingc

3.

(7)

(6–7)

(5–6)

(5)

(4–5)

(1) (1–2)

Fit of motor shaft in inner race of ball bearinga Eccentricity of inner ring bearingb

1. 2.

Surface Designation (see Figure 4.22) (1)

Item Discussed in Footnotes

No.

TABLE 4.41 Summary of Calculations for Case II

0.0008

0.0020

0.0010

0.0025

0.0020

0.0004 0.0008–0.0000

0.0004–0.0000

0.0000 0.0002

Tolerance (Clearance), in. (2)

0.0004

0.0010

0.0005

0.00125

0.0010

0.0002 0.0004–0.0000

0.0002

0.0000 0.0001

Equivalent Change in Center Distance (3)

–0.16

–1.00

–0.25

–1.56

–1.00

–0.04 –0.16

–0.04

–0.00 –0.01

(4a)

–0.16

–1.00

–0.25

–1.56

–1.00

–0.04 –0.16

+0.04

–0.00 –0.01

(4b)

Min. CenterDistance Col. (3)c×10-6

+0.16

+1.00

+0.25

+1.56

+1.00

+0.04 +0.16

–0.04

+0.00 +0.01

(5a)

(Continued)

+0.16

+1.00

+0.25

+1.56

+1.00

+0.04 +0.16

+0.04

+0.00 +0.01

(5b)

Max. CenterDistance Col. (3)c × 10-6

200 Dudley’s Handbook of Practical Gear Design and Manufacture

(8–9) (9–10) (10)

Radial clearance in bearingc Eccentricity, inner race of bearingb

Fit of gear shaft in inner race of bearinga

12. 13.

14. 0.0000

0.0004–0.0000 0.0002

0.0004

Tolerance (Clearance), in. (2)

0.0000

0.0002–0.0000 0.0001

0.0002

Equivalent Change in Center Distance (3)

–0.00

–0.04 –0.01

–0.04

–0.00

–0.04 –0.01

–0.04

Min. CenterDistance Col. (3)c×10-6 (4a) (4b)

+0.00

+0.04 +0.01

+0.04

+0.00

+0.04 +0.01

+0.04

Max. CenterDistance Col. (3)c × 10-6 (5a) (5b)

h i

g

f

e

d

c

b

a

Most bearing catalogues indicate bearing and shaft tolerances with will produce an interference fit. This fit cannot, therefore, contribute to the maximum center-distance at which the gears will operate. Most bearing catalogues indicate a tolerance on runout between the bore and the raceway of the inner ring. This eccentricity contributes to the instantaneous center- distance when the inner face is the moving portion of the bearing. Most bearing catalogues include a certain radial clearance between the inner and the outer race of a bearing. This clearance usually acts to allow an increase in center- distance due to the reaction between the two meshing gears. Most bearing catalogues indicate a tolerance on runout between the raceway and the outer diameter of the outer ring. This eccentricity contributes to the center- distance at which the gears will operate. This value will change during running since the outer race is often permits to creep. Most bearing catalogues indicate a bore tolerance on the hole that the bearing is fitted in. Usually the tolerance will provide clearance between the bearing and housing. When clearance exists, the gear reaction acts to increase the center-distance to the limit allowed by the clearance. Many manufacturers of pilot-mounted motors specify a value of runout between the motor shaft and the pilot surface. It is customary to measure this by holding the motor shaft fixed (as between centers) and to rotate the motor about its shaft. A dial indicator running on this surface will show runout (twice eccentricity). A clearance fit is usually specified on the hole into which the pilot of the motor is to fit. This clearance allows the motor to be shifted somewhat and, as a result, has a direct effect on center-distance. The value shown assumes a tolerance of + 0.0005 in. on the diameter of the pilot on the motor. 0.0005 in. In an assembly, such as shown in this example, a tolerance must be placed on the location of the bore to receive the motor relative to the rabbet on the end plate. This, in effect, is the same type of tolerance as discussed under “g” above.

Note: The maximum values of center-distance (Columns 4b and 5b) are based on the assumption that the separating forces between pinion and gear act to hold the shafts at their maximum separation within the limits of bearing clearance. The minimum values of center-distance (Columns 4a and 5a) are based on the assumption that the external forces on gear and pinion act to hold the shafts at their minimum separation within the limits of bearing clearance. The minimum center-distance (Column 4) is based on the assumption that all parts, made to their maximum clearances, are assembled to achieve the minimum possible center-distance. The maximum center-distance (Column 5) is base on the assumption that all the parts, made to their maximum clearances, are assembled to achieve the maximum possible center distance. Column (1) shows the surface designated on Figure 4.22 by a number in a circle; (2) shows the tolerance (usually on a diameter); (3) shows the amount this tolerance can contribute to center-distance (±); (4) shows the square of the tolerance contributing to the minimum center-distance; and (5) shows the square of the tolerance contributing to the maximum center-distance. The values in columns (4) and (5) are added algebraically and their square roots are shown at the foot of the column.

(7–8)

Eccentricity, outer race of bearingd

11.

Surface Designation (see Figure 4.22) (1)

Item Discussed in Footnotes

No.

TABLE 4.41 (Continued) Summary of Calculations for Case II

Gear Tooth Design 201

202

Dudley’s Handbook of Practical Gear Design and Manufacture

bore. Depending on how this member is installed, the center-distance established by the bores in which this element is fitted will be increased or decreased. It is customary to let this element creep; Thus, the eccentricity will go through all positions. There is also a tolerance on the outer diameter of this member. This tolerance, plus the tolerance on the bore in which this element is fitted, can cause varying degrees of looseness and therefore changes in operating center-distance. The inner race has machining tolerances that cause eccentricity in the same manner, as in the outer race member. In cases where the shaft rotates, it is customary to use a light interference fit between the inner race and the shaft. In such cases, the center of the gear will move in a path about the center of the ball or roller path of the bearing. The eccentricity of the gear will be that of the inner race of the bearing. This will cause a once-per-revolution change in center-distance. The clearance in the bearing is controlled by the selection of the diameter of the balls. This clearance allows greater or lesser changes in center-distance of the gears supported by the bearings • Journal bearings. Journal bearings can change the center-distance at which the gear was intended to operate by an amount that is a function of the clearance designed into the bearing, its direction of rotation, the speed of rotation, and the lubricant used. The change from the distance actually established by the distance between the centers of the bores is due to the oil wedge developed. Handbooks give convenient methods of evaluating the eccentricity that can be expected of a given bearing design. When necessary, these refinements can be introduced into the calculations illu­ strated by cases I and II • Sleeve bearings. The term sleeve bearings is used here to identify the journal-type bearing in which a sleeve of some bearing material is pushed into the gear casing or frames. The tolerance of the eccentricity of the outside diameter and the inside bearing surface is the element to consider in this type of bearing. In some cases, the clearance between the sleeve and the shaft is affected because of the interference fit between the sleeve and the bore in the casing • Casing bores. The distance between bearing bores in a gear casing can be made up of several elements. In the simplest case, a single frame or casing has two holes drilled at a specified centerdistance. The tolerance is selected to give the machine operator the necessary working allowance (see Figure 4.19). This tolerance has a direct effect on the operating center-distance of the gears. In some designs, the casing is made up of several parts bolted together. One part may carry the bore for the gear shaft, and another may carry the bore for the pinion shaft. The center-distance is then made up of a series of parts each having dimensions and tolerances that can add up in various ways to give maximum and minimum center- distance. Case II illustrates this type of gearing.

4.4.3 CONTROL

OF

BACKLASH

In cases where it is necessary to control the amount of backlash introduced by the mountings, two courses of action may be employed. In the first, the center-distance may be made adjustable. That is, provisions may be made so that, at assembly, the centers may be moved until the desired mesh is obtained. This entails a method of moving the parts through very small distances, and then being able to fix them securely when the desired distance has been reached. Provision must also be made for as­ semblers to see what they are doing when adjusting the mesh. In this approach, the difference between the smallest and the largest backlash in the adjusted mesh is only the effect of total composite error (runout) in both meshing parts. Size tolerance on the teeth can be generous since this is one of the tolerances adjusted out of the mesh by assembler. After the center-distance adjustment is completed, the parts may be drilled and doweled, although this causes difficulties if replacement gears are ever to be used on these centers.

Gear Tooth Design

203

In the second approach, very close tolerances are held both in the mountings (boring center-distance, bearings, etc.) and on the tooth thickness of the gear. This approach is used whenever interchangeability is specified and in mass production in which assembly costs is to be minimized.

4.4.4 EFFECTS

OF

TEMPERATURE

ON

CENTER-DISTANCE

Many gear designs consist of gears made of one material operating in mountings made of another material having a markedly different coefficient of expansion. Since it is customary to manufacture and assemble gears at room temperature of about 68 F , an analytical check of the effective center-distance present in the gears at their extremes of operating temperature is desirable. Power gears may heat up because of the frictional losses in the mesh and may achieve an operating temperature above that of their casing. If made of the same materials as their mountings, such gears would have a larger apparent pitch diameter19 relative to the center-distance and would therefore have a smaller effective center- distance at their running temperature. Some gears, particularly those in military applications, may be subjected to extreme cold for periods of time. Often, these are steel gears operating in aluminum or magnesium casings. In this case, the center- distance will shrink to a greater degree than the apparent pitch diameter of the gears, and the resulting backlash will be less at room temperature. If evaluating the effective center-distance of a mesh at an operating condition other than room temperature, the temperature of the gear blanks, as well as that of the mountings, which may be con­ siderably different, must be considered, as well as the coefficient of expansion of the gear and mounting materials. There are six operating conditions in which temperature can have an important effect on gear performance: 1. Gears operating at temperatures higher than their mountings: a. Operating temperatures lower than assembly-room temperature b. Operating temperatures higher than assembly-room temperature 2. Gears operating at temperatures lower than their mountings: a. Operating temperatures lower than assembly-room temperature b. Operating temperatures higher than assembly-room temperature 3. Gears operating at temperatures essentially the same as their mountings: a. Operating temperatures lower than assembly-room temperature b. Operating temperatures higher than assembly-room temperature (4.103) and (4.104) evaluate each of these possibilities and establish the most critical extremes at which minimum or maximum effective center-distance will occur. The significance of these possibilities can be appreciated by considering the following service con­ ditions. The first case is typical of power gearing under steady-state conditions. Condition (1.a) is a possibility at start-up under a cold operating environment. The case is usually a transitory condition where the gears may have to perform adjacent to an external source of heat. The third condition is most typical of control gears that do not transmit enough energy to be at a temperature greatly different from their mountings. The procedure recommended here is to calculate a minimum and a maximum effective centerdistance. These are based on the extremes of operating temperatures. From these values, the designer can calculate either a value of tooth thickness that will give the necessary operating backlash under the tightest mesh conditions, or the amounts of backlash that will be found in a given gear-set. (4.103) and (4.104) give minimum and maximum effective center-distances:

19

Apparent pitch diameter is the diameter at which a given value of tooth thickness is found.

204

Dudley’s Handbook of Practical Gear Design and Manufacture

Ce min = Cd + C t = CTT

(4.103)

Ce max = Cd + C

(4.104)

t

= CTL

where: Ce min – is the minimum effective center-distance that occurs under temperature extreme Cd – is the basic or nominal center-distance C t – is the minimum tolerance on center-distance CTT – is the change in center-distance; see note following (4.105) and (4.106) Ce max – is the maximum effective center-distance that occurs under temperature extreme C t – is the maximum tolerance on center-distance CTL – is the change in center-distance; see note following (4.105) and Eq. (4.106)

(4.105) and (4.106) indicate the amount that the center-distance (effective) will change because the gears and their mountings shrink or expand at different rates: C

C

T

T

=

=

D ( T 2

D ( T 2

M

KM

T

M

KM

T

G

G

KG ) +

d ( T 2

KG ) +

d ( T 2

M

KM

M

KM

T

G

T

KP )

G

KP )

(4.105)

(4.106)

Note: Compare C T and C T . Assign the smallest positive number or the largest negative number to CTT . Assign the largest positive number or the smallest negative number to CTL . If signs are opposite, use this negative value for CTT , and use the positive value for CTL . These values are used in (4.103) and (4.104) to obtain effective center-distance: T T

T T

M

= TM

M

=T

G

= TG

G

=T

M

G

TR TR

TR TR

where: KP – is the coefficient of expansion of pinion material, in. per in. per F KG – is the coefficient of expansion of gear material KM – is the coefficient of expansion of mounting material TM – is the minimum mounting temperature (operating) TR – is the assembly room temperature (68 F ) T M – is the maximum mounting temperature (operating) TG – is the minimum gearing temperature (operating) T G – is the maximum gearing temperature (operating)

(4.107) (4.108) (4.109) (4.110)

Gear Tooth Design

205 Axis of gear

Mounting distance, pinion

Axis of pinion

Mounting surface

Bevel pinion

Mounting distance, gear

Bevel gear

Mounting surface Mounting distance, gear

C

Mounting distance, worm Worm gear

Pitch cone apex Pinion

Face gear

Mounting distance, gear

Mounting surface FIGURE 4.23 Mounting Distance, Shown for Various Kinds of Gears, Is the Distance from the Axis of a Gear to the Mounting Surface of Its Nating Gear. This Distance Is Measured Parallel to the Shaft of the Subject Gear.

4.4.5 MOUNTING DISTANCE20 Bevel gearing, worm gearing, face gearing, and Spiroid gearing must be given close control on axial positioning (mounting distance) if good performance is to be achieved (see Figure 4.23). Most of the elements that must be given control to achieve proper center-distance must also be given close control to achieve proper mounting distance. If any of the above types of gearing can move along the axis of its shaft relative to the mating gear, the teeth will either bind or in the opposite case, have excessive backlash. Straight-toothed bevel gears are critical in respect to axial position, since backlash and tooth bearing pattern are affected by changes in axial position of the pinion and gear. Spiral bevel gears, hypoid gears, and Spiroid gears are also critical in that they tend to “screw” into mesh in one direction of rotation, which can cause binding, and will “unscrew” in the other direction of rotation, causing backlash, unless the mounting system is stiff enough to prevent axial shift. All properly designed mountings for gears of this type have bearings capable of withstanding all axial thrust loads imposed both by the gears and by possible external loads. Occasionally a design is attempted in which the designer relies on the separating

20

See Chapter 15 for general information on gear mounting tolerances and practices.

206

Dudley’s Handbook of Practical Gear Design and Manufacture

forces produce in the mesh to keep the gears away from a binding condition. Only a few such designs are truly successful. Worm gearing must also be accurately positioned if of the single- or double-enveloping design. If a throated member is permitted to move axially, the frictional force developed in the mesh will tend to move the member, and the geometry of the throated design will cause a reduction in backlash, sometimes to the point of binding. Mounting distance is usually specified for each gear member as the distance from a specific mounting surface on the gear blank, a face of a hub, for example, to the axis of the mating gear. Bevel gears are usually stamped with the correct mounting distance for the pair. This distance was established in a bevel gear test fixture at the time of manufacture. The design of mountings for gearing requiring a mounting distance should include provisions for shim­ ming or otherwise adjusting the position of the members at assembly. The calculation of mounting distance is accomplished in two steps. The design of the gear teeth (earlier in this chapter) includes a distance from the axes or pitch cone apex to a given surface of the gear. To this distance is added the distance to the closest bearing face that is used to provide the axial location of the gear and its shaft. Provision should be made in the design to secure the bearing seats that provide axial location at the specified mounting distances. In the previous chapter, radial clearance in rolling-element bearing was considered. In gear designs requiring a mounting distance, the axial clearance in one bearing on each shaft must be controlled. Reference to a bearing catalogue or handbook is recommended.

BIBLIOGRAPHY Radzevich, S.P. (2014). Generation of surfaces: Kinematic geometry of surface machining, 2nd Edition. Boca Raton, Florida: CRC Press. Radzevich, S.P. (2017). Gear cutting tools: Science and engineering, 2nd Edition. Boca Raton, FL: CRC Press. Radzevich, S.P. (2018). An examination of high-conformal gearing. Gear Solutions magazine, February, 31–39. Radzevich, S.P. (2018). Theory of gearing: Kinematics, geometry, and synthesis, 2nd Edition, revised and expanded. Boca Raton, FL: CRC Press. Radzevich, S.P. (Ed.). (2019). Advances in gear design and manufacture. Boca Raton, Florida: CRC Press. Radzevich, S.P. (2019). Geometry of surfaces: A practical guide for mechanical engineers, 2nd Edition. Springer International Publishing. Radzevich, S.P. (2019). Understanding the mounting distance: Intersected-axes gearing (bevel gearing). Gear Solutions magazine, December, 42–47. Radzevich, S.P. (2020). High-conformal gearing: Kinematics and geometry, 2nd edition, revised and extended. Amsterdam: Elsevier. Radzevich, S.P. (2020). Understanding the mounting distance: Crossed-axes gearing (hypoid gearing). Gear Solutions magazine, February, 38–43.

5

Preliminary Design Considerations Stephen P. Radzevich

In this chapter we shall begin to get into the detail work of designing a gear. Usually, the gear designer roughs out a preliminary design before doing all the work involved in a final design. The preliminary design stage involves consideration of the kinds of stress in the gears, an estimate of the approximate size, and consideration of the kind of data that will be required on the gear drawings. Each of these preliminary design considerations will be discussed in this chapter. Tables 5.1 and 5.2 are included for easy reference to the important gear nomenclature and symbols in Chapter 5.

5.1

STRESS FORMULAS

The gear designer’s first problem is to find a design that will be able to carry the power required. The gears must be big enough, hard enough, and accurate enough to do the job required. There are several kinds of stresses present in loaded and rotating gear teeth. The designer must consider all the possibilities so that the gears are proportioned to keep all the stresses within design limits.

5.1.1 CALCULATED STRESSES The stresses calculated in gear-design formulas are not necessarily true stresses. For instance, the tensile stress at the root of a gear tooth might be calculated at 275 N/mm2 (40,000 psi) using the formula for a cantilever beam. If the tooth was very hard and there were a large number of cycles, there might be an effective stress-concentration factor of about 2 to 1. This would tend to raise the stress to an effective value of perhaps 550 N/mm2 (80,000 psi) in this example. If the part was case-hardened, there might be a residual compressive stress in the outer fiber of the root fillet of as much as 140 N/mm2 (20,000 psi). If the root fillet was large and well polished and the number of cycles was low, the effective stress con­ centration might be as low as 1.0. In this case, a calculated stress of 275 N/mm2 (40,000 psi) might be effectively reduced to as little as 140 N/mm2 (20,000 psi)! It can be seen from the example just considered that things like stress concentration and residual stress can make it difficult to get correct answers about gear-tooth stresses. Other things also make it hard to get correct calculated stresses. The load that the gear teeth are transmitting may be known. However, whether this load is uniformly distributed (see Figure 5.1) across the face width and whether this load is properly shared by the two or more pairs of teeth that are in mesh at the same time may not be known. Errors in tooth spacing not only disrupt the sharing of tooth loads but may cause accelerations and decelerations, which will cause a dynamic overload. The masses of the rotating gears and connected apparatus resist velocity changes. Tachometer gears made of hardened steel have been known to pit in high-speed aircraft applications even when the transmitted load was negligible. Several assumptions have to be made to permit the calculation of stresses in gear teeth. It can be seen from the previous discussion that it is difficult to make assumptions that will properly allow for such things as stress concentration, residual stress, misalignment, and tooth errors. This means that the stress calculated is probably not true stress. 207

208

Dudley’s Handbook of Practical Gear Design and Manufacture

TABLE 5.1 Glossary of Gear Nomenclature, Chapter 5 Term

Definition

Backlash

The amount by which the sum of the circular tooth thicknesses of two gears in mesh is less than the circular pitch. Normally, backlash is thought of as the freedom of one gear to move while the mating gear is held stationary.

Bottom land

The surface at the bottom of the space between adjacent teeth.

Crown

A modification that results in the flank of each gear tooth having a slight outward bulge in its center area. A crowned tooth becomes gradually thinner toward each end. A fully crowned tooth has a little extra material removed at the tip and root areas also. The purpose of crowning is to ensure that the center of the flank carries its full share of the load even if the gears are slightly misaligned or distorted.

Face width

The length of the gear teeth as measured along a line parallel to the gear axis. (A gap between the helices of double helical gears excluded unless “total face width” is specified.) The rounded portion at the base of the gear tooth between the tooth flank and the bottom of the land.

Fillet Flank

The working, or contacting, side of the gear tooth. The flank of a spur gear usually has an involute profile in a transverse section.

Form diameter

The diameter set by the gear designer as the limit for contact with a mating part. Gears are manufactured so that the tooth profile is suitable to transmit load between the top of the tooth and the form diameter. A mathematical curve, which is commonly specified for gear-tooth profiles.

Involute curve Journal surfaces K factor Overhung

The finished surfaces of the part of a gear shaft, which has been prepared to fit inside a sleeve bearing or ball bearing. An index of the intensity of tooth load from the standpoint of surface durability. A gear with two bearings (for support) at one end of the face width and no bearing at the other end.

Pi

A dimensionless constant which is the ratio of the circumference of a circle to its diameter. Pi is denoted by the Greek latter and is approximately equal to 3.14159265.

Pitch-line velocity

The linear speed of a point on the pitch circle of a gear as it rotates. Pitch-line velocity is equal to rotational speed multiplied by the pitch-circle circumference.

Q factor Sliding velocity

A quantity factor from the weight of a gear set, defined by Eq. (5.48).

Straddle mount Throated

A method of gear mounting in which the gear has a supporting bearing at each end of the face width. A gear is throated when the gear blank has a smaller diameter in the center than at the ends of the “cylinder.” This concave shape causes the gear to partially envelope its mate and thus increases the area of contact between them. This design is often used to increase the load-carrying capacity of worm gearsets.

To land

The top surface of a gear tooth.

Undercut

When part of the involute profile of a gear tooth is cut away near its base, the tooth is said to be undercut. Undercutting becomes a problem when the numbers of teeth are small. Also, pre-shave or pre-grind may be designed to produce undercut, even when no undercut would be present because of the small number of teeth.

The linear velocity of the sliding component of the interaction between two gear teeth in mesh. The rate of sliding changes constantly; it is zero at the pitch line, and it increases as the contact point travels away from the pitch line in either direction.

Once a “stress” is calculated, there is usually no sure way of knowing how this stress is related to the physical properties of material. Ordinarily, the only properties of gear material known with some certainty are the ultimate strength and the yield point. Endurance-limit values may be available, but these will usually be taken from reverse-bending tests of small bars. The gear tooth is essentially a cantilever beam1 that is bent in one direction only. This is not as hard on the tooth as fully reversed bending would be. 1

The assumption that “tooth is essentially a cantilever beam” is not valid for gear teeth, as the Saint-Venant principle is inevitably violated under such an assumption [for details, see Section 24.2 (pages 657–662) in: Radzevich, S.P. (2018). Theory of gearing: Kinematics, geometry, and synthesis, 2nd Edition, revised and expanded. Boca Raton, FL: CRC Press].

Preliminary Design Considerations

209

TABLE 5.2 Gear Terms, Symbols, and Units, Chapter 5 Term

Metric

English

Reference or Comment

Symbol

Units

Symbol

Units

Module, transverse

mt

mm

–

–

Module, normal

mn

mm

–

–

Diametral pitch, transverse

–

–

Pt of Pd

in.

Diametral pitch, normal

–

–

Pnd

in.

m t = 25.4/ Pt Pt = 25.4/ m t

p

mm

p

in.

Pitch diameter, pinion

dp1

mm

d

in.

Pitch diameter, gear

dp2

mm

D

in.

Pitch radius, pinion

rp1

mm

r

in.

Figure 5.14

Pitch radius, gear

rp2

mm

R

in.

Figure 5.14

Outside radius, pinion

ra1

mm

ro

in.

Figure 5.14

Outside radius, gear

ra2

mm

in.

Figure 5.14

Face width Tooth ratio (gear ratio)

b u

mm

Ro F

–

mG

–

(No. gear teeth)/(No. pinion teeth)

Center distance

a

mm

C

in.

Figure 5.14 Figure 5.10

Circular pitch

in.

Pressure angle

deg

deg

Helix angle (or spiral)

deg

deg

Pitch angle, pinion

deg

Figure 5.21

Outer cone distance

Ra

mm

Ao

in.

Figure 5.21

Bending stress

st

N/mm2

st

psi

Section 5.3

Contact stress (Hertz)

sc xE

N/mm2

sc E

psi

Compressive stress also, Section 5.1.4

Modulus of elasticity Poisson’s ratio Load Tangential load (force) Power Torque

deg

W Wt

P T

Radius of curvature

N/mm2 – N N kW N*m

W Wt

P T

f

Roll angle (involute)

mm

– lb

Section 5.3

lb

Figure 5.4

hp in.*lb

Eq. (5.60), Eq. (5.61)

in.

Eq. (5.15), Eq. (5.37)

f

in.

Figure 5.6

mm

Radius of curvature, root fillet

psi

deg

r

deg

Figure 5.10

Zone of action

g

mm

Z

in.

Figure 5.14, Eq. (5.39)

Rotational speed, pinion

n1 n2 vt vs nc

rpm

rpm

m/s

nP nG vt

m/s

vs

fpm

Section 5.1.5

–

nc

–

Section 5.1.4

Rotational speed, gear Pitch-line velocity Sliding velocity Number of cycles

r

rpm

rpm fpm

(Rotational speed)*(pitch circumference)

Size factor for gearbox

Q

–

Q

–

Section 5.2.2

Loading index for strength

N/mm2

Section 5.2.3

N/mm2

Ul K

psi

Loading index for surface durability

Ul K

psi

Section 5.2.3

Scoring criterion number

Zc

C

Zc

F

Eq. (5.49)

–

mp

–

Eq. (5.44)

– –

B

in. –

Figure 5.11, Eq. (5.10) Eq. (5.36)

Contact ratio (profile or transverse) Band of contact width (between two cylinders) Specific film thickness

– –

(Continued)

210

Dudley’s Handbook of Practical Gear Design and Manufacture

TABLE 5.2 (Continued) Gear Terms, Symbols, and Units, Chapter 5 Term

Metric Symbol

Oil-film thickness (EHD) Surface roughness

English

Units

Symbol

Reference or Comment

Units

h min

m

h min

in.

Eq. (5.36), Figure 5.11

S

m

S

in.

Eq. (5.36)

Note: Abbreviations for units are as follows: Metric

English

mm

millimeters

in.

inches

deg

degrees

deg

degrees

N N*m

newtons newton-meters

lb in.*lb

pounds inch-pounds

N/mm2

Newtons per square millimeter

psi

pounds per square inch

kW rpm

kilowatts revolutions per minute

hp rpm

horsepower revolutions per minute

m/s

meters per second

fps

feet per second

C

degrees Celsius

F

degrees Fahrenheit

m

Micrometers (10−6 meters)

in.

microinches (in./106)

FIGURE 5.1 Cohecksntact checks at full torquefull torque for two setstwo sets of gears. left-hand view shows misalignment resultinggears. Left-Hand View Shows Misalignment Resulting In Overloadoverload At Endend Of Tooth More Than Twice What It Should Be. Right-hand View Shows Essentially Uniform Load Distributiontooth More Than Twice What It Should Be. Right-Hand View Shows Essentially Uniform Load Distribution across the Face Width.

Preliminary Design Considerations

211

Laboratory fatigue tests are made on small test specimens with a uniform and smoothly polished test section. In contrast, the actual gear tooth is usually a part of a large piece of metal of more or less nonuniform structure. The critical section of the gear tooth may have tear marks, rough finish, and possibly corrosion on the surface. Its failure is usually a fatigue failure. All these things make it hard for the gear designer to take ordinary handbook data on the strength of materials and design gears. The best way to find out how much load gears will carry is to build and test gears. Then the designer can work backward and calculate what “stress” was present when the gear worked properly.

5.1.2 GEAR-DESIGN LIMITS In spite of the difficulty of calculating of real stresses, gear-stress formulas are a valuable and necessary design tool. When (a) materials, (b) quality of manufacture, and (c) kind of design stay quite constant, then formulas can be used quite successfully to determine the proper size of new designs. Essentially the formula is used as a yardstick to make a new design a scale model of an old design that was known to be able to carry a certain load successfully. It is particularly important to base new designs on old designs that have been successful. If the designer has an application that is not similar to anything built in the past, it is important to realize that a new design cannot be rated with certainty. Formulas may be used to estimate how much the new gear will do. However, it should be kept in mind that one can only estimate the performance of a new gear design. Field experience will be required to give the final answer about what the gears will do. In many cases, it has been possible to raise the power rating of gearsets after a sufficient amount of field ex­ perience was obtained. Several aircraft engines today have gears in them that are running at 25 to 50 percent more power than the gears were rated at when they were first designed. The design engineer is obligated to design a gear unit so that all criteria are met in a reasonable fashion. Besides stress limits, there are temperature limits and oil-film-thickness limits. In addition to these first-order requirements, there are secondary considerations like vibration, noise, and environment. The general procedure in design is to first make the gears large enough to keep the tooth-bending stresses and surface compressive stresses within allowable limits. Further calculations are made to check the risks of scoring and over-temperature. Frequently, refinements in details of tooth design, kind of lubricant, tem­ perature of lubricant, and accuracy of the gear tooth are required to meet all the design limits.

5.1.3 GEAR-STRENGTH CALCULATIONS A gear tooth is essentially a stubby cantilever beam. At the base of the beam, there is tensile stress on the loaded side and compressive stress on the opposite side. When gear teeth break, they usually fail by a crack at the base of the tooth on the tensile-stress side. The ability of gear teeth to resist tooth breakage is usually referred to as their beam strength or their flexural strength. The flexural strength of gear teeth was first calculated to a close degree of accuracy by Wilfred Lewis in 1893. He conceived the idea of inscribing a parabola of uniform strength inside a gear tooth. It happens that when a parabola is made into a cantilever beam, the stress is constant along the surface of the parabola. By inscribing the largest parabola that will fit into a gear-tooth shape, one immediately locates the most critically stressed position on the gear tooth. This position is at the point at which the parabola of uniform strength becomes tangent to the surface of the gear tooth. The Lewis formula can be derived quite simply from the usual textbook formula for the stress at the root of a cantilever beam. Figure 5.2 shows a rectangular cantilever beam. The tensile stress at the root of this beam is:

st =

6W l F t2

(5.1)

212

Dudley’s Handbook of Practical Gear Design and Manufacture

t W l F

FIGURE 5.2

A loaded cantilever beam.

c Note: An easy way to find (a) is to lay in a bc straight line so that ab = bc . See left b side of the tooth. ab

b

a

a C L

Parabola

FIGURE 5.3 When gear tooth is loaded at point b , point a is the most critically stressed point. inscribed parabola is tangent to root fillet at point a and has its origin where load vector cuts the center line.

If we substitute a gear tooth for the rectangular beam, we can find the critical point in the root fillet of the gear by inscribing a parabola. This is point a in Figure 5.3. The next step is to draw some construction lines to get a dimension x . By similar triangles in Figure 5.4, it is apparent that: t2 4l

(5.2)

W l F (2x /3)

(5.3)

x=

By substituting x into (5.1), we get: st = Wn Wr l

Wt (or "W ")

C L 90°

a

x t

FIGURE 5.4 layout.

Determination of x dimension from a gear-tooth

Preliminary Design Considerations

213

The circular pitch p may be entered into both the numerator and the denominator without changing the value of the equation. This gives: st =

W lp F (2x /3) p

(5.4)

The term “2x /3p” was called “y” by Lewis. This was a factor that could be determined by a layout of the gear tooth. Since the factor was dimensionless, it could be tabulated and used for any pitch. Using y, the Lewis formula may be written as: st =

W F py

(5.5)

or: W = st p F y

(5.6)

In the present-day work, most engineers prefer to use module or diametral pitch instead of circular pitch in making stress calculations. This can be done by substituting Y for y (Y = y ), and Pd for p (Pd = /p ). The Lewis formula becomes: st =

W Pd , psi FY

(5.7)

The original Lewis formula was worked out for the transverse component of the applied load. In Figure 5.4, it is that the load normal to the tooth surface has a tangential and a radial component. The radial component produces a small compressive stress across the root of the gear tooth. When this component is considered, the tensile stress is reduced by a small amount, and the compressive stress on the opposite side of the tooth is increased by a slight amount. Seemingly this would indicate that the tooth would be most critically stressed on the compression side. This is not the case, however. In most materials, a tensile stress is more damaging than a somewhat higher compressive stress. Lewis took the application of load to be at the tip of the tooth. In his day, even the best gears were not very accurate. This meant that the load was usually carried by a single tooth instead of being shared between two teeth on a gear. If a single tooth carried full load, it is obvious that the greatest stress would occur when the tooth had rolled to a point at which the tip was carrying the load. Worst load. When gears are made accurate enough for the teeth to share load, the tip-load condition is not the most critical. In nearly all gear designs, the contact ratio is high enough to put a second pair of teeth in contact when one pair has reached the tip-load condition on one member. The worst-load condition occurs when a single pair of teeth carry full load and the contact has rolled to a point at which a second pair of teeth is just ready to come into contact. Figure 5.5 shows how to locate the worst-load condition on a spur pinion. Note that the contact point has advanced to where it is just one base pitch away from the first point of contact. In spur gears, the worst-load condition can be used in the Lewis formula by simply making the tooth layout with the load applied at the worst point instead of at the tip. Table 5.3 shows how this affects the Y factor for a few 20 spur pinions made to a standard depth of 2.250 in. for 1 pitch. The addendum is 1.000 in. for 1 pitch. In helical and spiral bevel gears, the teeth are usually accurate enough to share load. The geometry of the teeth, however, is such as to make any position of load that can accurately be labeled the “worst application” of load impossible to find. In all positions of contact, the design usually makes it possible

214

Dudley’s Handbook of Practical Gear Design and Manufacture C L Highest points of single tooth loading are on this circle

φ

Outside diameter of driven gear

Last point of contact

Direction of rotation

FIGURE 5.5 Worst application of beam loading on precision spur gear is one base pitch above the first point of contact.

TABLE 5.3 Y–Factors for Standard 20° Spur Pinions No. Pinion Teeth

No. Gear Teeth

Y–Factor Tip Load

Worst Load

20

20

0.287

0.527

20 20

60 120

0.287 0.287

0.527 0.600

25

25

0.310

0.583

25 25

60 120

0.310 0.310

0.657 0.693

30

30

0.332

0.640

30 30

60 120

0.332 0.332

0.673 0.740

for two or more pair of teeth to share the load. This situation is handled by calculating a geometry factor for strength. See Section 5.2.3 for further details. Stress concentration. In Lewis’ time, stress-concentration factors were not used in engineering calcu­ lations. In present gear work, almost all designers realize the necessity of using a stress-concentration factor. Figure 5.6 shows the dimensions used to calculate stress concentration. The equations established from the 1942 experimental work of Dolan and Broghamer are: Kr = 0.18 +

t

0.15

f

t h

0.45

t h

0.40

(5.8)

for 20 involute spur teeth, and Kr = 0.22 +

t f

for 14.5 involute spur teeth.

0.20

(5.9)

Preliminary Design Considerations

215

C L

h a ρf

ρf

t

FIGURE 5.6 Dimensions Used to Calculate Stress Concentration.

In the above equations, Kr is the stress-concentration factor. The radius f is the radius of curvature of the root fillet at the point at which the fillet joins the root diameter. Tests of actual metal gear teeth show that the real stress-concentration factor is not the same as the photo-elastic value determined on models made of plastic material. If the root fillet has deep scratches and tool-marks, it may be higher. Some materials are more brittle than others. In general, high-hardness steels show more stress-concentration effects than low-hardness steels. Some of the case-hardened and induction-hardened steels are an exception of this, however. When a high residual compressive stress is obtained in the surface layer of the material, stress-concentration effects are reduced. Stress concentration is also influenced by the number of cycles. Some of the low-hardness steels show little or no effect of stress concentration when loaded to destruction statically. Yet when fatigued for a million or more cycles, they show a definite reduction in strength due to stress concentration. Load distribution. The face width used by Lewis was the full face width. Actual gears are seldom loaded uniformly across the face width. Errors in shaft alignment and helix angles tend to increase the load on one end of the tooth or the other. Present practice is to use a load-distribution factor to account for the extra load imposed somewhere in the face width as a result of non-uniform load distribution. Dynamic load. The load applied to the gear tooth is greater than the transmitted load based on horsepower. In general, the faster the gears are running, the more shock due to tooth errors and the more dynamic effects due to imbalance and torque variations in the driving and driven apparatus. In design work, the overload on gear teeth is often handled by a dynamic factor. This factor is used as a multiplier of the transmitted load. The dynamic overload is really an incremental amount of load that is added to the transmitted load rather than being a multiplier of the transmitted load. For instance, low-hardness gears idling at little or no transmitted load may be inaccurate enough to develop such serious dynamic loads that they fail prematurely! Using the dynamic overload as an adder is somewhat unhandy in a rating equation. The usual rating formula is worked out for full rated power, and the dynamic factor—used as a multiplier—adds on what is considered to be the extra amount for dynamic load. Such rating formulas, of course, give the wrong answer at light loads, but they can be quite accurate at full load. Finite element. The German practice in calculating tooth strength has developed into taking the tangency point of a 30 angle as the critical stress point on the root fillet. Stress-concentration effects are determined by the finite element method of stress analysis instead of by photo-elastic results. The method of gear rating for tooth strength proposed by International Standards Organization (ISO) uses the 30 angle and equations for stress concentration based on finite element studies of gear teeth. German gear people feel that the 30 angle method is simpler than the inscribed parabola method and that the results are close enough to be suitable for any normal design work. The finite element method of getting stress-concentration values, though, gives results that are often considerably different from those found using the Dolan/Broghamer method. Figure 5.7 shows a comparison of the 30 angle method and the inscribed parabola method.

216

Dudley’s Handbook of Practical Gear Design and Manufacture

Wn

(1) (2)

30°

30°

(1) Most critically stressed point by inscribing parabola metnod (2) Most critically stressed point by 30° angle method

FIGURE 5.7 Comparison of 30 Method of Finding Most Critically Stressed Point with Inscribing Parabola Method.

5.1.4 GEAR SURFACE-DURABILITY CALCULATIONS Gears fail by pitting and wear as well as by tooth breakage. Frequently gears will wear to the point where they begin to run roughly. Then the increased dynamic load plus the stress-concentration effects of the worn tooth surface both cause the teeth ultimately to fail by breakage. Figure 5.8 shows the kinds of stress that are present in the region of the contact band. In the center of the band, there is a point of maximum compressive stress. Directly underneath this point, there is a maximum subsurface shear stress. The depth to the point of maximum shear stress is a little more than one-third the width of the band of contact. The gear-tooth surfaces move across each other with a combination of rolling and sliding motion. The sliding motion plus the coefficient of friction tends to cause additional surface stresses. Just behind the band of contact, there is a narrow region of tensile stress. A bit of metal on the surface of a gear tooth goes through a cycle of compression and tension each time a mating gear tooth passes over it. If the tooth is loaded heavily enough, there will usually be evidence of both surface cracks and plastic flow on the contacting surface. There may also be a rupturing of the metal as a result of subsurface shear stresses. Hertz derivations. The stresses on the surface of gear teeth are usually determined by formulas derived from the work of H. Hertz (of Germany). Frequently, these stresses are called Hertz stresses. Hertz determined the width of the contact band and the stress pattern when various geometric shapes were loaded against each other. Of particular interest to gear designers is the case of two cylinders with parallel axes loaded against each other. Figure 5.9 shows the case of two cylinders with parallel axes. The applied force is F pounds and the length of the cylinders is L inches. The width of the band of contact is B inches.

Contact band moving in this direction

Subsurfase compression Maximum subsurface compression Subsurfase tension Direction of sliding velocities

Subsurfase tension Band of contact Maximum subsurface shear Subsurfase compression Direction of rotation

FIGURE 5.8

Stress in Region of Tooth Contact.

Preliminary Design Considerations

217

F

R1

B

L R2

F

FIGURE 5.9

Parallel Cylinders in Contact and Heavily Loaded.

The Hertz formula for the width of the band of contact is: 16 F (K1 + K2 ) R 1 R2 L (R 1 + R2 )

B=

where: K1 =

2 1

1 E1

, and K2 =

2 2

1 E2

(5.10)

.

In the above equations, is Poisson’s ratio, and E is the modulus of elasticity. The maximum compressive stress is sc =

4F L B

(5.11)

The maximum shear stress is ss = 0.295 sc

(5.12)

Z = 0.393 B

(5.13)

The depth to the point of maximum shear is

Equation (5.10) and Eq. (5.11) may be combined and simplified to give the following when Poisson’s ratio is taken as 0.3: sc =

0.35

F (1/ R1 + 1/ R2 ) L (1/ E1 + 1/E2 )

(5.14)

218

Dudley’s Handbook of Practical Gear Design and Manufacture

The Hertz formulas can be applied to spur gears quite easily by considering the contact conditions for gears to be equivalent to those of cylinders that have the same radius of curvature at the point of contact as the gears have. This is an approximation because the radius of curvature of an involute tooth will change while going across the width of the band of contact. The change is not too great when contact is in the region of the pitch line. However, when contact is near the base circle, the change is rapid, and contact stresses calculated by the Hertz method for cylinders are not very accurate. Figure 5.10 shows how the radius of curvature may be determined at any point of an involute curve. At the pitch line, the radius of curvature is: =

d sin 2

(5.15)

At any other diameter, such as d 1, the radius of curvature is: 1

The angle

1

=

d1 sin 2

1

(5.16)

can be found out by the relation:

1

= cos

1

d cos d1

(5.17)

The compressive stress at the pitch line of a pair of spur gears can be obtained by substituting the following into Eq. (5.14)

INVOLUTE RELATIONS: db = d cos φ ε r (in radians) = tan φ Involute φ = εr −φ = tan φ − φ (in radians) db ε ρ= φ 2 r

db = d1 cos φ1 d ρ1 = b ε r1 2 φ = pressure angle Involute curve

ρ ρ1

inv φ

εr

εr1 φ1 φ

d1

d db

FIGURE 5.10

Radius of Curvature of an Involute and Other Basic Relations of the Involute Curve.

Preliminary Design Considerations

219

Wt cos

(5.18)

L=F

(5.19)

F=

R1 =

d sin 2

(5.20)

R 2 = mG R 1

(5.21)

Note: At this point, we are switching from the symbols of conventional mechanics to the symbols used by gear engineers. This can be confusing, unfortunately, because F is applied force in Eq. (5.10) to Eq. (5.18), but F is face width in Eq. (5.19) and following equations. In Eq. (5.18) Wt is the tangential driving pressure in pounds. It can be obtained by dividing the pinion torque by the pitch radius of the pinion. In Eq. (5.20), F is the face width in inches, and d is the pitch diameter of the pinion. The ratio of gear teeth to pinion teeth is mG . Combining Eq. (5.14), Eq. (5.18), and Eq. (5.20) gives: sc =

0.70 (1/ E1 + 1/E2 )sin

Wt mG + 1 Fd mG

cos

(5.22)

K factor derivations. For a steel spur pinion meshing with a steel gear, we may substitute the nominal value of 30,000,000 for E . Then:

sc = 5715

Wt mG + 1 Fd mG

(5.23)

when the teeth have a 20 pressure angle. Many gear designers have found it convenient to call the term under the square-root sign in Eq. (5.23) the K factor. This makes it possible to reduce Eq. (5.23) to: sc = 5715

K

(5.24)

where: K=

Wt mG + 1 Fd mG

(5.25)

The compressive stress on helical-gear teeth can be obtained by finding the radius of curvature of the teeth in a section normal to the pitch helix. This section has a pitch ellipse instead of pitch circle. Using an equation for an ellipse, we can bet values to use for the R values in Eq. (5.14). These are: R1 =

d sin 2 cos2

n

(5.26)

220

Dudley’s Handbook of Practical Gear Design and Manufacture

d sin

R 2 = mG

n

(5.27)

2 cos2

In accurate helical gears, the load is shared by several pair of teeth. The average length of tooth working is equal to the face width multiplied by the contact ratio and divided by the cosine of the helix angle. The normal load (load in normal section) applied is equal to the tangential load in the transverse plane divided by the cosine of the helix and divided by the cosine of the normal pressure angle. Substituting these values into Eq. (5.22) gives a basic equation for the compressive stress in helical-gear teeth: sc =

0.70 cos2 (1/ E 1 + 1/ E 2 ) m p sin

n

cos

n

Wt mG + 1 Fd mG

(5.28)

where m p is profile contact ratio (profile contact ratio in a transverse section). By introducing the K factor and a constant Ck , most of the terms in the equation can be eliminated. This gives us a simplified equation: K psi mp

sc = Ck

(5.29)

Table 5.4 gives some values of Ck and m p for typical gears made to full-depth proportions in the normal section. The profile-contact-ratio values were calculated for a 25-tooth pinion meshing with a 100-tooth gear. For other numbers of teeth, the contact ratio will change by a small amount; however, it makes very little difference in Eq. (5.29), since the contact-ratio term is under the square-root sign. Bevel and worm gears can be handled in a somewhat similar manner. Worst-load position. In most cases, the compressive stress on spur-gear teeth is calculated at the lowest point on a pinion tooth at which full load is carried by a single pair of teeth. This situation cannot occur in wide-face helicals, but it can occur in narrow helicals, bevels, and spur gears. Theoretically, if one pair of teeth carries full load and the position of loading is at the lowest possible position on the pinion, there is a worst-load condition for Hertz stress corresponding to the worst-load position for strength described in the preceding section. Different formulas for contact stress may include factors to account for the increase in load due to velocity and tooth accuracy. The same questions of dynamic load and misalignment across the face width

TABLE 5.4 Values of Ck and mp for Helical Gears Normal Pressure Angle,

15

Spur

n

Helix

30

45

Helix

Helix

Ck

mp

Ck

mp

Ck

mp

Ck

mp

14.5

6581

2.10

6357

2.01

5699

1.71

4653

1.26

17.5

6050

1.88

5844

1.79

5240

1.53

4278

1.13

20

5715

1.73

5520

1.65

4949

1.41

4041

1.05

25

5235

1.52

5057

1.45

4534

1.25

3702

0.949

Note: The above values are illustrative of the development of gear-rating equations, but should not be used for design purposes. Chapter 8 gives different values that are appropriate for actual design work.

Preliminary Design Considerations

221

are present in calculating contact stress as in calculating the root stress. Service factors allow for torque pulsations and the length of service required from the gears. Endurance limit. The tendency of gear teeth to pit has traditionally been thought of as a surface fatigue problem in which the prime variables were the compressive stress at the surface, the number of repetitions of the load, and the endurance strength of the gear material. In steel gears, the surface endurance strength is quite closely related to hardness, and so stress, cycles, and hardness then become the key item. It was also believed that there was an endurance limit for surface durability at about ten million cycles (107 cycles). For case-carburized steel gears, fully hardened, typical design values listed in Table 5.5. Gear work led to two very important conclusions: 1. Pitting is very much affected by the lubrication conditions. 2. There is no endurance limit against pitting. Work on the theory of elastohydrodynamic lubrication (EHD) showed that gears and rolling-element bearings often developed a very thin oil film that tended to separate the two contacting surfaces so that there was little or no metallic contact. When this favorable situation was obtained, the gear or the bearing could either carry more load without pitting or run for a longer time without pitting at a given load. Real gears in service frequently run for several thousand hours before pitting starts—or becomes serious. A gear can often run for up to a billion (109) cycles with little or no pitting, but after 2 or 3 billion (2 or 3 × 109) cycles, pitting—and the wear resulting from pitting—makes the gears unfit for further service. Regimes of lubrication. To handle the problem of EHD lubrication effects, the designer needs to think of three regimes of lubrication: Regime I: No appreciable EHD oil film (boundary) Regime II: Partial EHD oil film (mixed) Regime III: Full EHD oil film (full film)

In regime I, the gears may be thought of as running wet with oil, but the thickness of the EHD oil film developed is quite small compared with the surface roughness. Essentially full metal-to-metal contact is obtained in the hertzian contact band area. Regime I is typical of slow-speed, high-load gears running with a rough surface finish. Hand-operated gears in winches, food presses, and jacking devices are typical of slow-speed Regime I gears. Regime II is characterized by partial metal-to-metal contact. The asperities of the tooth surfaces hit each other, but substantial areas are separated by a thin film. Regime II is typical of medium-speed gears,

TABLE 5.5 Typical Design Values for Case-Carburized Steel Gears Metric*

Item

Maximum allowable stress Number of cycles Surface hardness *

English

Symbol

Value

Symbol

Value

sc nc HV

1724 N/mm2

sc

250,000 psi

107

nc HRC

60 min.

700 min.

107

In Chaper 5, most of the data will be given in English units to agree with previously published derivations. From Chap. 6 onward, the data will be primarily given in the metric system. For the convenience, equivalent English values will often be given as a second set of values.

222

Dudley’s Handbook of Practical Gear Design and Manufacture

highly loaded, running with a relatively thick oil and fairly good surface finish. Most vehicle gears are in Regime II. (Tractors, trucks, automobiles, and off-road equipment are vehicle gear applications.) In Regime III the EHD oil film is thick enough to essentially avoid metal-to-metal contact. Even the asperities miss each other. The well-designed and well-built high-speed gear is generally in Regime III. Turbine-gear applications in ship drives, electric generators, and compressors are good examples of highspeed gears. In the aerospace gearing field, turboprop drives are high-speed and in Regime III. Helicopter main rotor gears are in the high-speed gear region, except for some final-stage gears that may be slow enough to be out of the high-speed domain and into medium speed with Regime II conditions. Figure 5.11 shows a schematic representation of the three regimes of lubrication. Note the details of the hertzian contact band for each regime. Figure 5.12 shows the “nominal” zones for the three regimes. The quality of the surface finish of the new gear, the degree of finish improvement achieved in breaking in new gears, the thickness and kind of lubricant, the operating temperature, the pitch-line velocity, and the load intensity all enter into the determination of which regime the gear pair operates in. Figure 5.12 should be used only as a rough guide. The surface durability in the different regimes of lubrication varies considerably. Figure 5.13 shows the general trend2 of surface contact stress capacity at different numbers of cycles for each of the three regimes of lubrication. The three curves have these equations at 107 cycles:

Lb Ka = La Kb

3.2

Regime I

(5.30)

Gear profile (enlarged)

hmin

B Pinion profile Regime I

hmin (oil film thickness)

B

Regime II

B (contact band width)

FIGURE 5.11

hmin Regime III

Schematic Representation of the Three Regimes of Gear-Tooth Lubrication.

Preliminary Design Considerations

223 Pitch line velocity, fpm

Regime I II

III

Pitch line velocity, m/s

FIGURE 5.12

Location of Regimes of Lubrication for Average Gears.

sc = 362,500 psi sc = 290,000 psi 217,500

II I

145,000

Contact cycles

FIGURE 5.13

Trend of Contact Stress for Regimes I, II, and III. Carburized Gears.

Lb Ka = La Kb

5.3

Lb Ka = La Kb

8.4

Regime II

(5.31)

Regime III

(5.32)

Lb – is the cycles (any number from 106 to 108) L a – is the 107 cycles at 10 cycles Kb – is the allowable load intensity at Lb cycles

where:

Ka – is the allowable load intensity

7

The contact stress, sc , is proportional to the square root of the load intensity. For simplified gear rating, sc = Ck

K Cd

(5.33)

where Ck is the tooth geometry constant for the particular design.

2

Figure 5.13 is a nominal curve intended to show average conditions for reasonably good steel and nominal quality. These curves should not be used for final design.

224

Dudley’s Handbook of Practical Gear Design and Manufacture

K factor can be calculated from the equations K=

Wt mG + 1 Fd mG

external teeth

(5.34)

K=

Wt mG 1 Fd mG

internal teeth

(5.35)

2

Tangential driving force, Wt , is equal to Wt = torque × d , and Cd is overall derating factor for gear and gearbox design imperfections. The data shown in Figure 5.13 can be interpreted to show the results of Table 5.6. Table 5.6 shows the substantial difference in load-carrying capacity of the different regimes. Regime I lose over 50 percent of its capacity every time there is a tenfold increase in life (number of cycles). Regime II loses about 35 percent of its load capacity for a tenfold increase. In comparison, Regime III loses only about 24 percent of its capacity with a tenfold increase.

5.1.5 GEAR SCORING When excessive compressive stresses are carried on a gear tooth for a long period of time, pits will develop. If the gear is kept in operation after pitting starts, the whole surface of the tooth will eventually be worn away. Severe pitting leads to rough running and “hammering” of the gear teeth. This in turn leads to other types of surface wear, such as scoring, swaging, and abrasion. Scoring is characterized by radial scratch lines, swaging is an upsetting process similar to cold rolling, while abrasion is the tearing away of small particles when rough surfaces are rubbed across each other. Gear teeth may score when no pitting has taken place. This may occur when the gears are first put into operation. Sometimes the cause of scoring—when gears have not pitted first—is simply that the accuracy is not sufficient. The teeth do not have a good enough surface finish or good enough spacing and profile accuracy. Hot and cold scoring. The scoring problem is not limited to high-speed gears. Slow-speed gears may also score. The score marks on slow-speed gears look somewhat different. The failure mechanism seems to be more one of filing or abrading away the metal rather than welding and tearing. European gear people—with some reason—call the slow-speed scoring cold scuffing3 and the high-speed scoring hot scuffing. Cold scoring is basically a problem of gears that are running in Regime I or Regime II conditions with oil that does not have enough chemical additives to protect the surface. The key variable is specific film thickness , which is defined as:

TABLE 5.6 Nominal Comparison of Load-Carrying Capacity of the Regimes of Lubrication No. Cycles

Load Capacity for Different Regimes I

II

III

105

100%

100%

100%

106

48.7%

64.8%

76.0%

107 108

23.7% –

41.9% 27.2%

57.8% 44.9%

109

–

–

33.4%

1010

–

–

25.4%

Preliminary Design Considerations

225

=

where: S =

h min – is the EHD oil-film thickness S 12

+

S 22

h min S

(5.36)

S – is the composite surface roughness of the gear pair,

S 1– is the surface roughness of pinion

S 2 – is the surface roughness of gear

The surface roughness values used for S 1 and S 2 are normally the arithmetic-average (AA) values. The surface roughness that is used may be different from the value measured on new gears just finished in the gear shop. Most gears will wear in and improve their finish in the first 100 or so hours of operation. This process is helped by not loading the gears too heavily or running them at maximum temperature until they are well broken in. A special lubricant with extra additives or more viscosity may also be used to help lessen the danger of scoring when new gears are first put into service. Assuming that a favorable break-in will be achieved, in Eq. (5.36) the designer can use the surface roughness values after break-in rather than those from the gear shop. For instance, a precision-ground gear with a 20 AA surface roughness may wear into a 15 AA finish. It is necessary to calculate a scoring factor that will evaluate the combined effects of surface pressure, sliding velocity, coefficient of friction, kind of metals, and kind of oil from the standpoint of scoring. The gear trade has used several methods of calculating scoring risk. No method has yet proved to be entirely reliable. The remainder of this section will cover the basics of PVT, flash temperature, ad scoring index. PVT formula. A PVT formula was used with considerable success in designing small aircraft gears to be built by automotive-gear manufacturers. During World War II, a number of automotive-gear plants made large numbers of aircraft gears. The PVT formula developed as a result of this wartime experience. The factors in PVT are as follows: P – is the Hertz contact pressure. This is usually figured for the tip of the pinion and figured again for the root of the pinion (tip of the gear). The applied load is divided by the profile contact ratio to approximate the way load is shared by two pairs of teeth. V – is the sliding velocity in feet per second at the point at which P is figured. T – is the distance along the line of action from the pitch point to the point at which P is calculated.

The quantity PVT is always zero at the pitch point on spur or helical gears. It increases steadily as contact moves away from the pitch point, reaching a maximum at the point at which contact is at the tip of the tooth. For this reason, PVT is ordinarily calculated only for the tips of the teeth. The equations and nomenclature used in the following equations follow the procedure customarily used by automotive-gear designers. The reader will recognize that general equations like Eq. (5.14) and Eq. (5.16) could have been used as well. The calculation of PVT involves solving a series of equations. First, the radius of curvature at the pinion tip is calculated (see Figure 5.14): P

=

r 20

(r cos

t)

2

(5.37)

Similarly, the radius of curvature at the gear tip is: G

=

R 20

(R cos

t)

2

(5.38)

The length of the line of action is: 3

The terms “scuffing” and “scoring” are used somewhat interchangeably. Common practice recommends the word “scoring.” Other engineering groups in the U.S.A. and abroad often prefer “scuffing” over “scoring.”

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Dudley’s Handbook of Practical Gear Design and Manufacture

ro r

ρP

φt

ρG

Z C

Ro

φt

R

CL FIGURE 5.14

Dimensions Used in Scoring-Factor Calculation.

Z=

P

+

C sin

G

(5.39)

t

The hertz compressive stress for the tip of the pinion is: PP = 5740 TP – is the pinion torque, in.-lb center distance

where:

TP FZNP

C sin P (C

sin

n P)

t

NP – is the number of pinion teeth

F – is the face width, in.

(5.40) C – is the

Similarly, the stress at the gear tip is: PG = 5740

TG FZNG

C sin G (C sin

n t

G)

(5.41)

The scoring factor for the pinion tip is: PVTP =

nP N 1+ P ( 360 NG

P

r sin

t)

2P P

(5.42)

Preliminary Design Considerations

227

where nP is the pinion speed, rpm. At the gear tip, the scoring factor is: PVTG =

nP N 1+ P ( 360 NG

R sin

G

t)

2P G

(5.43)

The equations will work for either spur or helical gears. In the case of spur gears, there is only one pressure angle. This makes n = t . The profile contact ratio can be obtained easily from the Z value in Eq. (5.39). Since these values is needed frequently, we shall give an equation for it here: mp =

Z NP 2 r cos

(5.44) t

Equation (5.44) holds for either spur or helical gears. Literally the contact ratio represents the length4 of the line of action divided by the base pitch. It is the average number of teeth that are in contact in the transverse plane. When the contact ratio comes out to some number like 1.70, it does not actually mean that 1.70 teeth are working. If the ratio is between 1 and 2, there are alternately one pair and two pair of teeth working. From a time standpoint, though, it would work out that on the average there were 1.70 pair of teeth working. A design limit of 1,500,000 has been used frequently for a safe limit on PVT . This value works reasonably well with case-hardened gears that are of good accuracy and are lubricated with mediumweight petroleum oil. Pitch-line velocity should be above 2000 fpm. Flash temperature. The concept of flash temperature was first presented by Professor Herman Block. The basics of this formula are: Tf = Tb +

cf f Wt (v1 Fe (

v1 +

v2 )

v2 ) B /2 cos

(5.45) t

Tf – is the flash temperature, F Tb – is the temperature of blank surface in contact zone (often taken as inlet f – is the coefficient oil temperature), F cf – is the material constant for conductivity, density, and specific heat of friction Wt – is the tangential driving load, lb v1– is the rolling velocity of pinion at point of contact, fps v2 – is the rolling velocity of gear at point of contact, fps vs = (v1 v2 )– is the sliding velocity, fps Fe – is the face width in contact, in. B – is the width of band of contact, in. t – is the transverse pressure angle, degrees

where:

The rolling velocity may be obtained from the rpm of the pinion or gear by: v1 =

v2 =

nP

1

(5.46)

360 nP

2

(5.47)

360

Originally flash temperatures were calculated for the pinion tip and the gear tip. In this case, Pinion tip: 1

4

=

P

[see Eq. (4.36)],

2

=

G

Z (see Figure 5.14)

Length between the first point of contact and the last point of contact is the length used here.

(5.48)

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Dudley’s Handbook of Practical Gear Design and Manufacture

Gear tip: 1

=

G

[see Eq. (4.37)],

2

=

Z (see Figure 5.14)

P

(5.49)

Later development of the flash-temperature formula led to scoring being considered most apt to occur at the lowest point of single tooth contact on the pinion or the highest point of single tooth contact on the pinion. This situation is most apt to occur when very accurate aircraft spur gears have a small amount of profile modification at the tip to relieve tip loading. Figure 5.15 shows the location of the highest and lowest points of single tooth contact for the pinion. Normally, scoring calculations are made for the pinion only. The gear is handled indirectly, since the pinion can score only when in contact with the gear. (If the pinion is OK, the gear should be OK.) Table 5.7 shows how to calculate the radii of curvature of the highest and lowest points of single tooth contact. The material constant cf was taken as 0.0528 for straight petroleum oils. The coefficient of friction was taken as 0.06. The width of the band of contact for steel gears was obtained from: Wt

B = 0.00054

Fe (

1

1

+

2

2 )cos

(5.50) t

Scoring criterion. Some gears have a high risk of hot scoring, while others have little or no risk of hot scoring. The formula for this index number is: Wt Fe

Scring criterion number =

3/4 4

(nP )2 Pd

(5.51)

Table 5.8 shows scoring-criterion numbers. If the calculated scoring value exceeds those in Table 5.8, there is a risk of scoring, and more complex calculations should be made. If the value is below those in the table, the risk of scoring is low, but it may still exist as a result of such things as poor finish, appreciable overload due to inaccuracy, and inadequate lubricant proportions. The scoring-criterion number is a useful guide for spur, helical, and spiral bevel gears. The scoring criterion was derived from the flash-temperature equation, Eq. (5.45). By mathematical manipulation it is possible to write the flash-temperature equation in the form: Wt Fe

Tf = Tb + Z t

3/4 4

(nP )2 Pd

(5.52)

where: Z t = 0.0175

( [

1 2 /( 1

2 / mG ) )]1/4 (cos 4

1

+

2

Pd t)

3/4

(5.53)

The constant Z t is dimensionless when used with the scoring-criterion number. This makes it possible to tabulate Z t values for different proportions and styles of tooth design. The term Z t can be thought of as a tooth geometry factor for scoring.

Preliminary Design Considerations

229 C L 3 1

5

4

6

2

7 8

FIGURE 5.15

Layout of Teeth to Locate Critical Points for Scoring Calculations.

TABLE 5.7 Radii of Curvature of Involute Spur or Helical Teeth Contact Position

Pinion Curvature,

Lowest single tooth contact

r 2o

Highest single tooth contact

(r cos

t)

C sin

t

2

Gear Curvature,

1

p cos

C sin

t

R 2o

2

(R cos

2

1

t

t)

2

p cos

Note: In this table, the outside radius of the pinion is r o and the pitch radius is r . Radii of the gear are capitalized. The circular pitch is p.

TABLE 5.8 Critical Scoring-Criterion Numbers Critical Scoring-Index Numbers at Blank Temperatures, F

Kind of Oil

100

150

200

250

300

AGMA 1

9,000

6,000

3,000

–

–

AGMA 3 AGMA 5

11,000 13,000

8,000 10,000

5,000 7,000

2,000 4,000

– –

AGMA 7

15,000

12,000

9,000

6,000

–

AGMA 8A Grade 1065, Mil-O-6082B

17,000 15,000

14,000 12,000

11,000 9,000

8,000 6,000

– –

Grade 1010, Mil-O-6082B

12,000

9,000

6,000

2,000

–

Synthetic (Turbo 35) Synthetic Mil-L-7808D

17,000 15,000

14,000 12,000

11,000 9,000

8,000 6,000

5,000 3,000

t

230

Dudley’s Handbook of Practical Gear Design and Manufacture

5.1.6 THERMAL LIMITS The design of gear drives involves more than just making the gear teeth able to carry bending stresses and contact stresses and to resist scoring. Along with such things as bearing capacity, shaft design, and spline capacity, the designer must consider the thermal limits. Thermal limits at regular speed. Small gear drives (particularly those under 200 hp) are often splash lubricated by a quantity of liquid oil in the gearbox. Without a pumped oil system and oil coolers, the gearbox is cooled by the surrounding air. Such a gear drive will have a thermal rating as well as a mechanical rating. The application of the gear unit to a job should be such that neither the thermal rating nor the mechanical rating is exceeded. The customary procedure in calculating thermal ratings is based on finding the maximum horsepower that a unit can carry for 3 hours without the sump temperature exceeding 93 C (200 F ) when the ambient air temperature is not over 38 C (100 F ). Equations and calculation procedures for setting thermal ratings will not be given in this book. Instead, Table 5.9 is a rough guide to what is involved in thermal rating. The prime variables in determining thermal capacity are the gearbox size, the input pinion speed, and the ratio. Table 5.9 shows that if a gear unit is built twice as large, then the thermal rating increases about 3 to 1. The mechanical rating increases almost 8 to 1 when the size doubles, so this predicts that large gear units will be short of thermal capacity. Table 5.9 shows that for the same size unit, the thermal rating drops quite rapidly as the speed is increased. In contrast, the mechanical rating increases somewhat in proportion to an increase in speed. There is an influence from the ratio, but it is quite mild. From Table 5.9, at 400 rpm it might be possible to design medium-hard gears up to 600 mm (24 in.) center distance and still have enough thermal rating to match the mechanical rating. At 1200 rpm, though, it would probably not be possible to exceed 200 mm (8 in.) center distance and still have enough thermal capacity to match the mechanical capacity. When the normal thermal capacity is exceeded, the thermal capacity may be improved by using one or more fans mounted on input shafts. A favorable fan arrangement can as much as double the normal thermal capacity of a gear unit. For the large and more powerful gear units, the thermal capacity is completely inadequate. Pumpedoil lubrication systems with oil coolers must be used. With a pumped-oil system, calculations must be made to assure that the bearings and gear teeth are fed enough oil to adequately cool and lubricate all parts. The design of gear-lubrication systems will not be covered here.

TABLE 5.9 Some Approximate Values of Thermal Rating in Horsepower Pinion Speed, rpm

Center Distance, in. 4

8

16

24

Ratio, 2 to 1 400

80

240

720

1200

1200

40

110

315

500

2400

12

30 Ratio, 4 to 1

80

55

400

80

240

720

1200

1200 2400

42 15

115 40

325 110

525 85

Preliminary Design Considerations

231

Thermal limits at high speed. When spur-gear teeth run faster than 10 m/s (2000 fpm) or helical teeth run faster than 100 m/s (20,000 fpm), there may be problems with the trapping of air and oil in the gear mesh. Special design features have allowed small spur teeth in aerospace applications to run up to around 100 m/s reasonably successfully. Likewise, special designs of helical gears have permitted successful operation up to 200 m/s.

5.2

STRESS FORMULAS

The first part of the chapter has shown the general nature of the various kinds of stress formulas that may be used in designing a gearset. This material will probably help the reader to understand the problem of designing a gearset, but it does not give much help in knowing what to do first in designing a set after choosing the kind of gear. There are many ways to start the design of a set. Perhaps the easiest way is to skip all the detail design work and immediately estimate how big the gears must be. If a size that is close to being right can be chosen in the beginning, the designer can work through the dimensional calculations with the prospect of only minor adjustments after the design is checked with appropriate rating formulas. In this case, part of the chapter we shall take up the problem of estimating gear size.

5.2.1 GEAR SPECIFICATIONS Gears are used to transmit power from one shaft to another and to change rotational speed. The designer needs to have specifications for: • • • •

The The The The

amount of power to be transmitted pinion speed (or gear speed) required ratio of input and output speeds length of time the gears must operate

Frequently it is hard to find out how much power a gearset must carry. Take the example of a gear driven by a 10-hp motor. In some applications, the motor might be expected to run every day of the year at 10 hp. In other cases, the motor might run only intermittently at powers well below 10 hp. In still others the motor might be started every day and have to pull up to 20 hp for a short period of time while the driven machinery was warming up. The example just cited shows that the gear designer’s first problem may be to establish a proper power specification. In doing this, it is necessary to check both the driving and driven apparatus. It may be that the driven apparatus tends to stall or suffer severe shock at infrequent intervals. This might give rise to peak torques as much as five or ten times the full-load torque capacity of the driving motor. Although the gearset does not have to be the strongest of the three connected pieces of apparatus, neither should it be the weakest—unless there is good reason. Where power and speeds may be quite variable, the designer should strive to reduce the specifications to simple conditions. First, the maximum continuous load that the gear might be expected to handle should be established. Next, the maximum torque should be determined. This load will probably last for only a short period of time. In many designs, it is necessary to make stress calculations for only these two conditions. In some cases, though, there may be a high intermediate load which is less than the maximum but does not last so long as the continuous load. Here it is necessary to calculate stresses for the in­ termediate load also. In complex design situations, it is necessary to construct a gear histogram. The best procedure is to calculate the pinion torque for each design condition. The results are plotted on log-log graph paper, with the highest loads being put at the lowest number of cycles. Figure 5.16 shows two examples of load histograms. For the vehicle gear shown, it turns out that the critical design condition is low gear and

232

Dudley’s Handbook of Practical Gear Design and Manufacture

maximum torque. In comparison, the turbine application has its most critical situation at the maximum continuous torque, not the highest torque seen under starting conditions.

5.2.2 SIZE

OF

SPUR

AND

HELICAL GEARS

BY

Q-FACTOR METHOD

It is fairly easy to estimate the sizes of spur and helical gears. After the proper specifications have been established, a tentative gear size can be obtained by the Q factor method. This method was originally developed to estimate gear weights. It is very handy, though, in estimating center distance and face width. The Q factor method of estimating gear size is simply a method whereby the power, speed, and ratio of a gearset are all reduced to a single number. This number, called Q for quantity, is a measure of the size of the job the gear has to do. The value of the Q factor as an index can be demonstrated quite readily from the weight curves shown in Figures 5.17 and 5.18. These show how the average weight of complete single-reduction gearsets compares with calculated Q factors: kW power (u + 1)3 × metric pinion rpm u

(5.54)

horsepower (mG + 1)3 × English pinion rpm mG

(5.55)

Q=

Q=

Starting overloads 30

4350

20

2900 Example of a typical vehicle gistogram

15

2175 Low load

10

1450

2nd gear

8

Starting overloads

1160

3rd gear

5 Short-duration overload

4 3

4th gear Max. continuous rating

Example of a typical turbine/generator histogram

870 725 580 435

5th gear

2

290 Light load cruising

1.5

218

1

145

1

10

100

1000

104

105

106

107

108

109

1010

Cycles FIGURE 5.16 Histogram of load intensity (K factor) plotted against contact cycles for a vehicle gear mesh and a turbine gear mesh. Note that the vehicle is critical for the low gear rating and the turbine is critical for its maximum continuous rating.

Preliminary Design Considerations

233

The ratio factor in Eq. (5.54) and Eq. (5.55) has been tabulated in Table 5.10 for ratios from 1 to 10. The required center distance and face width can be obtained from the Q factor as soon as the designer decides how heavily it is safe to load the gears. For spur and helical gears, the K factor is a convenient index for measuring the intensity of tooth loads. Equation (5.68) through Eq. (5.71) define the K factor, and Table 5.11 gives some values of K factor that can be used for preliminary estimates of gearset sizes. As soon as the amount of K factor has been decided on, the following equation can be solved: a2b =

4, 774, 650 Q metric K

(5.56)

31, 500 Q English K

(5.57)

C 2F =

Face-width considerations. In spur or single-helical gears, using a face width that exceeds the pinion pitch diameter is often not advisable. If the face width is wider, torsional twist concentrates the load quite heavily on one end. In many applications, it is not even possible to effectively use a face width as wide as the pinion diameter. Errors in tooth alignment and shaft alignment may make it impossible to get tooth contact across this much face width. When a face width equal to the pinion diameter is intended, Eq. (5.56) and Eq. (5.57) may be modified to the following: a3 =

10,000

2, 387, 325 Q (u + 1) metric K

(5.58)

Single pinion and gear average values for full weitht design

1000

100

10 0.1

1.0

10

100

Q factor FIGURE 5.17 Gear Weights Plotted against Q Factor for Different Intensities of Tooth Loading. (English Units.)

234

Dudley’s Handbook of Practical Gear Design and Manufacture

Curves drawn for 500 K factor load intensity

10,000

1000

100

10

1

10

100

1000

Q factor

FIGURE 5.18

Comparison of Gear Weights for Different Designs.

C 3 = 15

31, 750 Q (mG + 1) English K

(5.59)

In double helical gears, the face width may be as wide as 1.75 times the pinion pitch diameter before the problems of torsional twist and beam bending get too serious. This results from the fact that double

TABLE 5.10 Ratio Factors for Single-Reduction Gears Speed Ratio, u

Ratio Factor (u + 1)3 /u

Speed Ratio, u

Ratio Factor (u + 1)3 /u

1.00

8.000

3.00

21.333

1.20

8.873

3.50

26.036

1.40 1.60

9.874 10.985

4.00 4.50

31.250 36.972

1.80

12.195

5.00

43.200

2.00 2.20

13.500 14.895

6.00 7.00

57.167 73.143

2.40

16.377

8.00

91.125

2.60 2.80

17.945 19.597

9.00 10.00

111.11 133.10

Preliminary Design Considerations

235

TABLE 5.11 Guide for Choice of Face Width in Spur or Helical Gears Aspect Ratio, ma , b / dp1 (metric) F / d

Situation

(English)

Numerical Constant for Center Distance, Eq. (5.58) and Eq. (5.59) Metric

English

Lower-accuracy gears, appreciable error in mounting dimensions. May need crowning, but helix modification not needed Medium accuracy, good mounting accuracy. Helix modification generally not needed. May need crowning.

59,683,125

39,375

2,387,325

15,750

1.5

High-accuracy parts and mounting. Helix modification generally not needed.

1,591,550

10,500

1.75

Very high accuracy. Probably need helix modification for single helical gears. May not need for double helix.

1,364,186

9,000

2.00

Very high accuracy. Will need helix modification in single-helix designs, and will probably need helix modification in double-helix designs. Very high accuracy. May not be practical to use, due to criticalness of helix modification required.

1,193,662

7,875

1,061,003

7,000

0.4

1.0

2.25

helical pinions shift axially under load to equalize the loading on each helix. This shifting compensates for most of the torsional twist. With both single- and double-helix gears, the face width can be made relatively wide, provided proper helix modification is made to compensate for the deflections involved. Table 5.11 shows guideline information on how to make an initial choice of face width. A constant is also given to permit the center distance to be immediately obtained from the Q factor for the chosen aspect ratio. Use this constant in Eq. (5.56) and Eq. (5.57) in place of the numerical constant shown. After the center distance is obtained, the pitch diameters are obtained by Eq. (5.5) and Eq. (5.6). Weight from volume. When other than simple gear pairs or simple planetary units are involved, the Q factor method of weight estimating rather becomes impractical. A simple alternative based on a summation of the face widths multiplied by the pitch diameters squared works quite well. In a 1963 paper by R.J. Willis it is shown how to pick gear ratios and gear arrangements for the lightest weight. His work was based on this concept. The basic weight equation is: bdP2 × weight constant = weight, kg metric 36,050

(5.60)

(Fd 2 × weight constant) = weight, kg English

(5.61)

The weight constant to be used is the same for metric or English units and is given in Table 5.12. Figure 5.19 shows in schematic form a few of the many possible gear arrangements. To get a rough approximation of the weight, pitch diameters and face widths are obtained for all gear parts. Then, Eq. (5.56) and Eq. (5.57) are used to get the summation of the face widths times the pitch diameters squared.

236

Dudley’s Handbook of Practical Gear Design and Manufacture

Table 5.12 shows some typical values of the weight constant needed for Eq. (5.56) and Eq. (5.57). The data in this table are based on general experience in gear design. Special situations, of course, may make the real weight of a well-designed unit quite different from the first estimate. For instance, a gearbox may support the motor or engine and need extra weight for the frame structure. Extra weight may be needed for oil-pump equipment or other accessory parts added to the preliminary gear unit. When weight is critical, special design efforts that use lightweight bearings, hollow shafts, thin casing walls, and gears with all excess material removed from the gear bodies may achieve surprisingly low weights.

5.2.3 INDEXES

OF

TOOTH LOADING

There are two indexes of tooth loading that are very important in gear design. Unit load is an index of tooth loading from the standpoint of tooth strength. The K factor is an index of tooth loading from the standpoint of tooth surface durability. To say it another way, the higher the unit load, the more risk of tooth breakage, and the higher the K factor, the more risk of tooth pitting. Both of these index numbers are calculated from the transmitted power. The normal calculations method for spur or helical gears is: Torque TP =

P × 9549.3 N m metric n1

(5.62)

where P is power in kilowatts. Torque TP =

P × 63, 025 in. nP

lb English

(5.63)

where P is power in horsepower. Tangential driving load: Wt = TP ×

Wt = TP ×

2000 N metric dp1

(5.64)

2 lb English d

(5.65)

where dp1 is pinion pitch diameter, mm, and d is pinion pitch diameter, in. The unit load index is derived from the Lewis formula for tooth strength. [See Eq. (5.7)]. It is

TABLE 5.12 Some Approximate Values of Thermal Rating in Horsepower Application

Factor

Typical Conditions

Aircraft

0.25 to 0.30

Magnesium or aluminum casings; limited-life design; high-stress levels; rigid weight control

Hydrofoil

0.30 to 0.35

Lightweight steel casings; relatively high stress levels; limited-life design; rigidity desired

Commercial

0.600 to 0.625

Cast or fabricated steel casings; relatively low stress levels; unlimited-life design; solid rotors and shafts

Note: Use these factors in Eq. (5.60) and Eq. (5.61).

Offset

(f)

(b)

Double-reduction four-branch

Offset with idler

FIGURE 5.19 Eight Kinds of Gear Arrangements for Spur or Helical Gears.

Double-reduction double-branch

(e)

(a)

(g)

(c)

Planetary

Offset with two idlers

(h)

(d)

Star

Double-reduction

Preliminary Design Considerations 237

238

Dudley’s Handbook of Practical Gear Design and Manufacture

Ul =

Wt × mn N/mm2 metric b

(5.66)

Wt × Pnd psi English F

(5.67)

Ul =

The K factor is based on the Hertz stress formula. It was first shown in Eq. (5.25). It will now be given in both metric and English forms and for both external and internal gears. External: K=

K=

Wt u + 1 N/mm2 metric dp1 b u

(5.68)

Wt mG + 1 psi English dF mG

(5.69)

Wt u 1 N/mm2 metric dp1 b u

(5.70)

Wt mG 1 psi English dF mG

(5.71)

Internal: K=

K=

The load indexes show the intensity of loading that the teeth are trying to carry. They are based on real quantities, and they measure what the user of gearing is getting out of the mesh. For instance, a K factor of 5 means that a gear mesh is carrying 5 Newtons of tooth load for each millimeter of pinion pitch diameter and each millimeter of face width in contact, with an appropriate adjustment for the relative size of the gear that is in mesh with the pinion. The gear user can understand unit load and K factor as measures of how much load is being carried per unit of size in the gear mesh, from bending strength and surface loading viewpoints. In comparison, the stress formulas have factors in them that are related to the quality and geometry of the application. The calculated stress number is, of course, very useful, but it does not directly tell the user the relative intensity of loading. For instance, a high stress may occur in a situation in which moderate load intensity is coupled with a poor geometric design and low quality. Obviously less gear-unit size is needed if acceptable stress levels can be achieved when the design has a relatively high intensity of loading and good enough quality and geometric design of teeth to keep these factors favorable. The general procedure in preliminary design is to size the gears based on the K factor. Then the tooth size in module (or pitch) is determined by figuring the unit load and making the teeth large enough to get an acceptable unit-load value. For instance, a spur pinion with 36 teeth might be OK on K factor but too high on unit load. If 18 teeth were put on the same pitch diameter (tooth size twice as great), the unit load is reduced 2 to 1 (50 percent as much unit load).

Preliminary Design Considerations

5.2.4 ESTIMATING SPUR-

AND

239

HELICAL-GEAR SIZE

BY

K-FACTOR

Since the K factor is so important in determining gear size, it is necessary to know how much K factor can be carried in different applications. The teeth of low- and medium-hard-steel gears usually have more strength than they have capacity to resist pitting. Hence the index of surface durability becomes the limiting factor in determining the loadcarrying capacity of the gears. If very thin oil is used or there is inadequate provision to cool the gearset, scoring might be a limiting condition. Generally, though, the designer will use heavy enough oil and provide enough cooling to get all the capacity out of the gears that is in the metal. Oil and oil-cooling systems are usually cheaper than larger-sized gears. In fully hardened gearing, the strength of the teeth may become as important as, or even more important than, surface durability. Even in this case, the gear designer can often so proportion the design that there will be enough strength available to match the capacity of the teeth to withstand pitting. If strength is more limiting than wear, a proper reduction in K factor can still aid in obtaining teeth of adequate strength. With this in mind, it is possible to use approximate K factors as a measure of the amount of load that can be carried on many types of spur and helical gears. Table 5.13 shows a study of K factor and unit-load values that are typical of nominal design practice for a variety of gear applications. These values are intended for use in the preliminary sizing of a gearset. Figure 5.20 shows in pictorial fashion the nominal design range for vehicle, aerospace, and turbine gears. After the preliminary sizing of gears, detail design work must be done to establish the pressure angle, helix angle, tooth addenda, tooth whole depth, and tooth thickness. Then, when complete tooth geometry has been picked, the designer should calculate tooth stresses and compare them with allowable values for the material being used and the degree of reliability required. Chapter 6 covers the determination of geartooth geometry and the calculation of load-carrying capacity to meet approximate gear-rating criteria. In this chapter, the data on how to use index values to size gears are intended only as a means of estab­ lishing the first approximation of an appropriate gear size.

5.2.5 ESTIMATING BEVEL-GEAR SIZE It is possible to estimate bevel-gear sizes by the Q factor method. So far, this method has had limited usage. It appears that the method will work fairly well for bevel gears but will not give quite as good results as for spur and helical gears. The geometry of bevel gears is more complicated than that of spur or helical gears. Under load, bevel gears tend to shift position more than gears on parallel shafts. It is frequently necessary to design a “mismatch” into the teeth. The mismatch concentrates the tooth load in the center of the face width and allows some shifting of shaft alignment to occur before the load is concentrated too heavily at one end of the tooth. These things make it hard for a simple formula to estimate capacity correctly. Some changes are required before the formulas in Section 5.2.2 can be applied to bevel gears. Bevel gears do not have any center-distance dimensions. This means that center distance must be removed from Eq. (5.56) and Eq. (5.57). This can be done with the help of Eq. (2.5). The result is: 2 dp1 b=

1.91 × 107Q metric (u + 1)2K

(5.72)

d 2F =

126, 000 Q English (mG + 1)2 K

(5.73)

Equation (5.72) and Eq. (5.73) can be used for both bevel and spur gears. It is handy to use when the design requirements have been reduced to a Q factor, but the designer has not yet decided whether spur,

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Dudley’s Handbook of Practical Gear Design and Manufacture

TABLE 5.13 Indexes of Tooth Loading for Preliminary Design Calculations Application

Minimum Hardness of Steel Gears

No. Pinion Cycles

Accuracy

K Factor

Unit Load

N/mm2

psi

N/mm2

psi

High precision High precision

0.69

100

45

6,500

1.04

150

59

8,500

1010

High precision

2.76

400

83

12,000

210 HB

109

0.48

70

31

4,500

335 HB

300 HB

109

0.76

110

38

5,500

58 HRC

58 HRC

109

High precision High precision High precision

2.07

300

55

8,000

225 HB

210 HB

108

Medium high precision

1.38

200

38

5,500

335 HB

300 HB

108

2.07

300

48

7,000

58 HRC

58 HRC

108

Medium high precision Medium high precision

5.52

800

69

10,000

225 HB

210 HB

108

Medium precision

0.83

120

24

3,500

335 HB

300 HB

108

1.24

180

31

4,500

58 HRC

58 HRC

108

Medium precision Medium precision

3.45

500

41

6,000

Aerospace, helical (single pair)

60 HRC

60 HRC

109

High precision

5.86

850

117

17,000

Aerospace, spur (epicyclic)

60 HRC

60 HRC

109

4.14

600

76

11,000

Vehicle transmission, helical

59 HRC

59 HRC

4 × 107

6.20

900

124

18,000

Vehicle transmission, spur

59 HRC

59 HRC

4 × 106

High precision Medium high precision Medium high precision

8.96

1300

124

18,000

Small commercial (pitch-line speed less than 5 m/s)

335 HB

Phenolic/ laminate

Medium precision

0.34

50

–

–

335 HB

Nylon

0.24

35

–

–

200 HB

Zinc alloy

106

Medium precision Medium precision

0.10

15

–

–

200 HB

Brass or aluminum

106

Medium precision

0.10

15

–

–

Turbine driving a generator

Internal combustion engine driving a compressor

General-purpose industrial drives, helical (relatively uniform torque for both driving and driven units)

Large industrial drives, spur – hoists, kilns, mills (moderate shock in driven units)

Small gadget (pitch-line speed less than 2.5 m/s)

Pinion

Gear

225 HB

210 HB

1010

335 HB

300 HB

1010

59 HRC

58 HRC

225 HB

Notes: 1. The above indexes of tooth loading assume average conditions. With a special design and favorable application, it may be possible to go higher. With an unfavorable application and/or a design that is not close to optimum, the indexes of tooth loading shown will be too high for good practice. 2. The table assumes that the controlling load must be carried for the pinion cycles shown.

Preliminary Design Considerations

241

FIGURE 5.20 Rough comparison of design K factors for carburized gears used in vehicle, aerospace, and turbine applications. The bandwidths result from quality of gearing, tooth design, application, level of risk, etc.

helical, or bevel gears are to be used. These equations can help the designer estimate the sizes of all three kinds. The calculation of unit load and K factor for bevel gears is somewhat different than for spur and helical gears. Figure 5.21 shows how these are calculated for the bevel gearset. The K factor in Eq. (5.72) and Eq. (5.73) is the same value that was used previously. However, it is sometimes necessary to use lower values than those shown in Table 5.13 to compensate for some of the special problems in bevel gears. For instance, if mountings do not maintain good tooth contact at full load, the K factor should be reduced. Bevel-gear designers have tended to reduce capacity for both size of pinion and increase in pitch-line velocity. Table 5.14 shows a study of bevel-gear K factors based on load-rating formulas in current use. These values represent averages of different rating designs for a uniform power source and a mild shock powerabsorbing device (like an electric motor driving a well-designed pump). If the designer has already decided to use a bevel gearset, it is not necessary to go to the trouble of calculating a Q factor. The equation can be simplified to: 1.910 107P (u + 1) metric K n1 u

(5.74)

126,000 P (mG + 1) English K nP mG

(5.75)

2 dp1 b=

d 2F =

Both Eq. (5.74) and Eq. (5.75) determine the quantity pitch diameter squared times face width. To get a complete solution, it is necessary to know the relation of pitch diameter to face width. It is recommended that the face width of straight and spiral bevel gears not exceed 0.3 times the cone distance or 10 in. divided by the diametral pitch. For Zerol gears, the only difference in the limits is that the cone-distance constant is 0.25 instead of 0.30.

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Dudley’s Handbook of Practical Gear Design and Manufacture

TP (in. lb) A0 (in.) mG

2sinpitch angle Wt

dF

FIGURE 5.21 Shaft Angle.

mG

The Method for Calculating Indexes of Tooth Loading for Straight or Spiral Bevel Gears at a 90

Table 5.15 shows how these limits work out in terms of pitch diameter or the pinion for different ratios. A shaft angle of 90 is assumed. Table 5.15 shows that the face width for a low ratio like 1 to 1 will probably be limited by the cone distance. For a high ratio like 5 to 1, the pitch will probably limit the face width. An appropriate value of face width can be picked from Table 5.15. Then this value can be used in Eq. (5.74) and Eq. (5.75).

Preliminary Design Considerations

243

TABLE 5.14 Approximate K Factor Values for Bevel Gears Pinion Pitch Diameter

K Factor Values for Pitch-Line Speed

Gear Ratio 5 m/s

1000 fpm

20 m/s

4000 fpm

N/mm2

psi

N/mm2

psi

570 490

3.45 2.96

500 430

3.27

475

2.90

420

2.79 2.72

405 395

2.48 2.41

360 350

2.38

345

2.10

305

Case 2. Industrial spiral bevel gears, pinion BHN 245, gear BHN 210 2.0 1 1.83 265 1.62

235

z2 / z1 (NG /NP )

mm

in.

50

2.0

1 3

3.93 3.38

100

5.0

1

200

10.0

3 1 3

50

Case 1. Industrial spiral bevel gears at HRC 58

3

1.65

239

1.46

212

100

5.0

1

1.41

205

1.24

180

200

10.0

3 1

1.23 1.12

179 163

1.08 0.99

157 143

3

0.94

137

0.83

120

50

Case 3. Industrial straight bevel gears, pinion BHN 245, gear BHN 210 2.0 1 0.86 125 –

(see Note 1)

3

0.77

112

–

(see Note 1)

100

5.0

1 3

0.83 0.76

120 110

– –

– –

200

10.0

1

0.79

115

–

–

3

0.72

105

–

–

Notes: 1. Straight bevel gears are usually not used above 10 m/s (2000 fpm). At 20 m/s (4000 fpm), spiral bevel gears are the normal choice. 2. The above K factor values assume average conditions. With a special design and a favorable application and/or design that is not close to being optimum, the K factor values shown will be too high for good practice.

Hypoid gears are hard to estimate. As a general rule, a hypoid pinion will carry about the same power as a bevel pinion. Since the hypoid pinion is bigger—for the same ratio—than a bevel pinion, the hypoid set will carry more power as a set. Face gears may be handled somewhat similar to straight bevel gearsets. Generally, it will be necessary to use les face width for the face gear than would be allowed as a maximum for the same ratio of bevel gears.

5.2.6 ESTIMATING WORM-GEAR SIZE It is much harder to estimate the capacity of worm gears. In spur, helical, and bevel gears, the magnitude of the surface compressive stress is the major thing that determines capacity. The limits of strength and scoring seldom determine the gear size—unless the set is poorly proportioned. In worm gearing, the tendency to score is often as important as the tendency to pit in determining capacity. Since scoring depends on both compressive stress and rubbing (or sliding) velocity, a rating formula has to be based on more than a K factor. Worm-gear sizes can be estimated reasonably well from a table of power vs. center distance for a series of worm speeds. Table 5.16 shows nominal capacity of a range of sizes of single-enveloping worm

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Dudley’s Handbook of Practical Gear Design and Manufacture

TABLE 5.15 Ratio of Maximum Face Width to Bevel-Pinion Pitch Ratio

Face Width Based on 0.3 Cone Distance

Face Width Based on 10 in. per Diametral Pitch 15 Teeth

20 Teeth

25 Teeth

1

0.212d

0.667d

0.500d

0.400d

1.5

0.270d

0.667d

0.500d

0.400d

2

0.335d

0.667d

0.500d

0.400d

3

0.474d

0.667d

0.500d

0.400d

4

0.618d

0.667d

0.500d

0.400d

5

0.765d

0.667d

0.500d

0.400d

6

0.912d

0.667d

0.500d

0.400d

7

1.061d

0.667d

0.500d

0.400d

d is the pinion diameter

*

gears. It is assumed that the worm is case-hardened and ground to good accuracy and finish. To meet the table specifications, the worm gear should be made of a good grade of chill-cast phosphor bronze and cut to give a good bearing with the worm. The ratings in the table should be regarded as nominal. Several manufacturers have been able to carry up to 100 percent more load than that shown in the table by the use of special materials and by obtaining a very high degree of precision in the worm and gear. Conversely, when accuracy has not been the best and when operating conditions have not been good (when such conditions as shock loads, vibration, or overheating have been present), it has been necessary to rate worm gearsets substantially lower than the values shown in Table 5.16. The nominal capacity of double-enveloping worm gears of the Cone-Drive design is shown in Table 5.17. Tables 5.16 and 5.17 are intended for worm gears subject to shock-free loading and in service for not more than 10 hours per day. If service is 24 hours per day with some shock loading, the table ratings should be reduced to about 75 percent of the values shown. The horsepower ratings shown in Tables 5.16 and 5.17 are mechanical ratings. The mechanical rating is the amount of power that the set is expected to carry without excessive wear or tooth breakage when the set is kept reasonably cool. In many cases, worm gearsets will overheat because there is not enough cooling of the gear case or the oil supply to remove the heat generated by the set. This makes it necessary to calculate a thermal rating. The thermal rating is the maximum amount of power that the set can carry before a dangerous operating temperature is reached. Quite obviously, the thermal rating depends as much on casing design and lubrication system as it does on the size of the gears themselves. In many cases, a worm-gear design will not carry so much thermal rating as mechanical rating. However, if adequate oil pumps, heat exchangers, and oil jets are used, it should be possible to operate any worm gearset up to its full mechanical rating. In each of the designs shown in the tables, an arbitrary size of worm was used. The size chosen represents good design practice. In many instances, though, it will be necessary to use different-sized worms. Often a worm is made to a “shell” design to slip over a large shaft. Large turbine shafts may have large worms mounted on them to drive small worm gears attached to oil pumps or governors. Good designs of this type can be made, but they are not as efficient as worm gearsets in which the worm and gear sizes can be more properly proportioned.

Preliminary Design Considerations

245

TABLE 5.16 Nominal Capacity of Cylindrical Worm Gearing Ratio, mG Center Distance, C , in.

Worm Pitch Effective Face Lead Angle, Diameter, d , in. Width, Fe , in. , Degrees

Output hp at Different rpm of Worm 100

5 8

2 2

15 25 50

0.825

720

1750

3600

10,000

0.46875 0.50000

37.6 25.7

0.19 0.15

1.20 0.90

2.10 1.70

2.90 2.30

4.40 3.60

2

0.46875

14.4

0.08

0.51

0.95

1.30

2.10

2 2

0.46875 0.46875

8.75 4.40

0.05 0.02

0.31 0.15

0.59 0.28

0.83 0.39

1.30 0.62

5

4

0.87500

40.3

1.60

8.00

12.00

17.00

22.00

8 15

4 4

0.93750 0.93750

28.0 15.8

1.20 0.72

6.40 3.90

9.90 6.10

14.00 8.70

19.00 12.00

25

4

0.93750

9.64

0.44

2.40

3.80

5.40

7.40

50 5

4 8

0.93750 1.68750

4.85 43.3

0.21 11.00

1.20 39.00

1.80 60.00

2.60 75.00

3.60 –

8

8

1.81250

30.5

8.10

33.00

50.00

64.00

–

15 25

8 8

1.81250 1.81250

17.4 10.7

4.80 3.00

21.00 13.00

31.00 19.00

41.00 26.00

– –

50

8

5 8

16 16

15 25 50

1.525

2.800

1.81250

5.39

1.40

6.20

9.30

12.00

–

3.12500 3.37500

46.5 33.4

63.00 50.00

186.00 156.00

254.00 221.00

– 263.00

– –

16

3.37500

19.4

30.00

98.00

143.00

169.00

–

16 16

3.37500 3.37500

11.9 6.02

19.00 8.90

62.00 30.00

90.00 43.00

107.00 52.00

– –

5.100

Notes: 1. Service factor Ks has a value of 1.0 for this table (See Chapter 6 for service factors.) 2. Sliding velocity is not over 6000 fpm.

5.2.7 ESTIMATING SPIROID GEAR SIZE Spiroid gears have less sliding than worm gears but much more sliding than spur or helical gears. It is not practical to estimate their size by a Q factor method. In general, both members of the Spiroid set are carburized. Final finish is done by grinding or lapping. Normally, the lubrication is handled by extreme pressure (EP) lubricants. Table 5.18 shows the nominal capacity of a range of Spiroid gearset sizes. This table can be used to make a preliminary estimate of the size needed for Spiroid gearset.

5.3

DATA NEEDED FOR GEAR DRAWINGS

After the size of a gear train has been determined, the designer must work out all the dimensional specifications and tolerances that are necessary to define exactly the gears required. Data on the gear material required and the heat treatment must also be given. Many years ago, it was customary to specify only a few major dimensions, such as pitch, number of teeth, pressure angle, and face width. It was assumed that the skilled mechanic in the shop would know what tooth thickness, whole depth, addendum, and degree of accuracy were required. Little or no consideration was given to such things as root fillet radius, surface finish, and profile modification.

246

Dudley’s Handbook of Practical Gear Design and Manufacture

TABLE 5.17 Nominal Capacity of Double-Enveloping Worm Gearing Ratio, mG

Center Distance, C , in.

Worm Pitch Diameter, d , in.

Output hp at Different rpm of Worm 100

5 15

720

2 2

0.830 0.830

0.40 0.17

2.19 0.99

50

2

0.850

0.05

0.31

5 15

4 4

1.730 1.550

3.65 1.65

16.90 8.15

50

4

1.660

0.52

2.56

5 15

8 8

3.450 2.940

25.90 11.80

95.20 48.10

1750 3.82 1.79

3600

10,000

5.53 2.61

7.93 3.83

0.56

0.82

1.22

26.60 13.10

34.60 17.30

– 22.90

4.11

5.47

7.28

135.00 70.10

163.00 90.90

– –

50

8

2.900

3.67

15.20

22.50

28.90

–

5 15

16 16

5.143 5.143

180.00 83.20

513.00 273.00

678.00 369.00

– 439.00

– –

50

16

5.143

26.10

87.00

119.00

140.00

–

5 15

24 24

7.333 7.333

473.00 227.00

1194.00 636.00

1488.00 829.00

– –

– –

50

24

7.333

71.70

204.00

270.00

–

–

Notes: 1. Service factor Ks is 1.0 for this table (See Chapter 6 for service factors.) 2. Sliding velocity is not over 6000 fpm.

Today the designer of gears for a highly developed piece of machinery such as an airplane engine, an ocean-going ship, or an automobile finds it necessary to specify dimensions and tolerances covering all features of the gear in close detail. There is often the risk that a buyer of gears will purchase thousands of dollars worth of gears and find that their quality is unsuitable for the application.

5.3.1 GEAR DIMENSIONAL DATA The dimensional data that may be required to make a gear drawing can be broken down into blank dimensions and tooth data. The blank dimensions are usually shown in cross-sectional views. The tooth data are either tabulated or shown directly on an enlarged view of one or more teeth. Some of the common blank dimensions are: • • • • • • • • • • •

Outside diameter Face width Outside cone angle (bevel gears) Back cone angle (bevel gears) Throat diameter (worm gears) Throat radius (worm gears) Root diameter Bore diameter (internal gears) Mounting distance (bevel gears) Inside rim diameter Web thickness

Preliminary Design Considerations

247

TABLE 5.18 Nominal Capacity of Spiroid Gears (Pinion and Gear Case-Hardened, 60 Rockwell C min.) Ratio, m G

10.250 14.667

Center Distance, C

Pinion O.D., do

0.500

0.437 0.421

Gear O.D., Do

1.500

Hp at Different rpm of Pinion 100

720

1750

3600

10,000

0.0142 0.0120

0.0697 0.0589

0.129 0.109

0.1948 0.1645

0.3354 0.2834

25.500

0.423

0.0088

0.0432

0.080

0.1208

0.2080

47.000 10.250

0.427 0.853

0.0064 0.0937

0.0313 0.4601

0.058 0.852

0.0876 1.2870

0.1508 –

1.000

3.000

14.667

0.821

0.0759

0.3726

0.690

1.0420

25.500 47.000

0.827 0.837

0.0550 0.0377

0.2700 0.1852

0.500 0.343

0.7550 0.5179

0.0299

0.1469

0.272

0.4107

0.4961 0.4125

2.4350 2.0250

4.510 3.750

6.8100 5.6630

71.000

0.761

10.250 14.667

1.875

1.507 1.463

5.625

25.500

1.435

0.2915

1.4310

2.650

4.0020

47.000 71.000

1.448 1.308

0.1804 0.1441

0.8856 0.7074

1.640 1.310

2.4760 1.9780

106.000 10.250 14.667

1.215 3.250

2.395 2.465

9.750

0.1188

0.5832

1.080

1.6310

2.1010 1.760

10.3100 8.6400

19.000 16.000

–

–

–

–

25.500

2.319

1.2100

5.9400

11.000

47.000 71.000

2.344 2.090

0.7205 0.5544

3.5370 2.7220

6.550 5.040

106.000 10.250 14.667

1.933 5.125

3.342 3.342

15.375

0.4455

1.1870

4.050

7.0070 5.9180

34.4000 29.0500

63.700 53.800

25.500

3.092

3.9490

19.3900

35.900

47.000 71.000

2.910 3.097

2.2550 1.7270

11.0700 8.4780

20.500 15.700

106.000

2.841

1.3860

6.8040

12.600

Notes: 1. Class 1 AGMA service. 2. Pitch-line speed not over 1700 fpm. 3. No allowance for shock loads, Ks = 1.0 . 4. Based on tooth proportions recommended by Spiroid Division of Illinois Tool Works, Chicago, IL, USA.

• Journal diameter The tabulated gear-tooth data will cover such items as: • • • • •

Number of teeth Module (or diametral pitch) Pitch diameter Circular pitch Linear pitch (worm gears)

–

248

• • • • • • • • • • • • •

Dudley’s Handbook of Practical Gear Design and Manufacture

Pressure angle Normal pressure angle (helical gears) Normal circular pitch (helical gears) Normal module or pitch (helical gears) Addendum Whole depth Helix angle Hand of helix Lead angle (worm gears) Pitch cone angle (bevel gears) Root cone angle (bevel gears) Tooth thickness Lead (worm gears)

Special views of gear teeth may be used to show things lime minimum root fillet radius, form diameter, tip radius, end radius, and surface finish. Notes may be added to the drawing to specify the heat treatment and to define the accuracy limits for checking the gear teeth. Further notes often define the reference axis used in checking and refer to drawings of cutting tools or processing procedures needed to make the gear come out right. The description just given of what may go on a gear drawing may leave the impression that gear drawings have to be very complicated. This is not necessarily the case. The designer should consider the gear quality required to meet design requirements, together with the responsibility that the shop making the gear will assume for making gears that will work satisfactorily. If the gear shop will assume re­ sponsibility for making a gear that will operate satisfactorily, and they understand the design require­ ments, a very simple drawing may be sufficient. Many fine gears are made from drawings that are very simple. The problem of the gear designer is to determine just how detailed the drawing must be to give the gear maker the responsibility for producing a gear that will do the job. Those involved in the details of gear and spline5 drafting should study carefully all the special data and instructions given in ANSI standards. After years of work, a general agreement has been reached on an international practice on defining and depicting the whole family of gears and splines.

5.3.2 GEAR-TOOTH TOLERANCES This is a difficult and controversial subject. Much has been written about gear tolerances, and yet there are no clear-cut answers. Space will not permit detailed study of tolerances in this book. From a general standpoint, tolerances on a gear must meet two kinds of requirements: 1. The tolerances must be broad enough to be met by the method of manufacture and the crafts­ manship of the plant undertaking the manufacture. 2. Tolerances must be close enough so that the gears will carry the required loads for a sufficient length of time and without objectionable noise or vibration. In timing and control applications, suitable accuracy of motion and freedom from “lost motion” on reversal must be obtained. The designer has several sources of information on tolerances. If the product is in productions, the accuracy already achieved and the performance in the field can be studied. This should give an answer to the questions of what can be done and what is needed. On entirely new products, studies of dynamic loads, effects of

5

The spline with gear teeth is closely related to gearing in its design and manufacture. Functionally, though, the spline is a joint connection rather than a gear mesh.

Preliminary Design Considerations

249

misalignment, and effects of surface finish may be required. Some of the considerations that will help a designer to estimate accuracy requirements are discussed in consequent sections of this book. Trade standards are very helpful in showing trade practices in regard to tolerances and inspection of different kinds of gears. Machine-tool builders can usually give good data as to the accuracy that their products can produce. The quality of gear teeth cannot be completely controlled until appropriate tolerances are specified on the following items: • • • • • •

Tooth spacing Tooth profile Concentricity of teeth with axis Tooth alignment (or lead, or helix) Tooth thickness (or backlash) Surface finish of flank and fillet

The best tooth-to-tooth spacing accuracy obtainable is about 2.5 μm (0.0001 in.). Very careful grinding or very good cutting and shaving is required to get this extreme degree of accuracy. Only a few of the best-designed machine tools are capable of this kind of work. This kind of accuracy is needed in a few very high-speed gears for marine and industrial uses. Tooth-to-tooth accuracy of most precision gear applications is of the order of 5 μm (0.0002 in.) or 8 μm (0.0003 in.). Good commercial gears range from 12 μm (0.0005 in.) to 40 μm (0.0016 in.) in tooth-to-tooth accuracy. In few cases, the involute profile is held to 5 μm (0.0002 in.) variation. Most precision gears have their involutes true within 12 μm (0.0005 in.). Good general-purpose gears range from 25 μm (0.001 in.) to 12 μm (0.0005 in.) on involute. Concentricity of control gears of small size is held to 12 μm for some applications. Most precision gears are in the range of 25 μm to 50 μm. Good commercial gears range up to about 120 μm. Concentricity is ordinarily measured as twice the eccentricity (full indicator reading used). Tooth alignment in very critical applications is held to as little as 15 μm and 10 μm over 250 mm to 500 mm. Extraordinary equipment and effort are required to hold a tight control over wide-face-width gears like the 500-mm-per-helix gears used in ship propulsion. Many commercial gears are held to limits like 12 μm to 20 μm for 25-mm face width. Power gears allow from 50 μm to 120 μm (0.0020 to 0.0050 in.) tooth-thickness variation. This variation, of course, is not within one gear but from one gear to another. All teeth of one gear will be very close to the same size if tooth spacing is close. Some control gears are ground or shave to as little as 5 μm tooth-thickness tolerance. This is very hard to do. Tooth-surface finish can be held within 0.4 μm (16 μin.) Many precision gears are held to about 0.8 μm AA. Commercial gears are apt to be around 1.2 μm to 2 μm or more. There is a general trend to use limits-of-accuracy sheets to define tolerances. This saves writing out all the tolerances on each drawing. All that has to be put on the drawing is the limits column that the gear is to be checked to. When a company issues its own engineering limits for gear accuracy, several things can be covered: • Several classes or grades of accuracy can be set. High accuracy grades can be set for long-life, high-speed gears. Lower accuracy grades will cover medium- or low-speed gears. • Things not covered in trade standard on accuracy can be covered. These may be things like root fillet radii tolerances, allowable mismatch between side of tooth profile and root fillet, or allowable waviness or irregularity in profile and helix. • An engineering limits document can define checking machine procedures to get tooth-to-tooth spacing, accumulated spacing, and runout readings. Procedures can be defined to handle allowed error in master gears used to check production gears. Stylus size, stylus pressure, and other

250

Dudley’s Handbook of Practical Gear Design and Manufacture

mechanics of checking can be defined to fit the checking machinery intended to be used in manufacture. Many major companies that build a goodly volume of important geared machinery have found a welldeveloped engineering document on gear checking essential. Trade standards are valuable guidelines for setting company engineering standards for gears. And, of course, it is quite appropriate to work directly to trade standards on gearing without special extra data that may be peculiar to an individual company. Table 5.19 shows typical gear tolerances for quality numbers 9 to 13. In comparison, Table 5.20 shows DIN tolerances from 10 to 3. Those designing and building gears are strongly advised to get the latest issues of the standards and keep abreast of the best current practices in gear inspection. In many cases, gear quality can be measured by simple functional tests. In some applications, the ability to run quietly in a noise-testing machine is a good check. Some power gearing is checked by running at full load and full speed for a period of time. If the gear runs smoothly and the surface polishes up without wear, it can be assumed that the gear will do its job in service. In control gearing, the measurement of backlash in different positions may serve to control quality. Since it costs a lot to give gears detailed checks, the designer should choose the least expensive checking system that will ensure proper quality. Then the gear drawing should be toleranced in keeping with the inspection that is planned.

5.3.3 GEAR MATERIAL

AND

HEAT-TREATMENT DATA

Historically, the geometric quality of gears (tooth tolerances) received much attention between 1940 and 1980. The metallurgical quality of gears has not (so far) received equal attention. The best gear people around the world are now coming to realize that metallurgical quality is just as important as geometric quality. A gear of good accuracy made from material that has a poor hardness pattern or substantial metallurgical flaws will not last as long as it should or will be unable to carry full load without serious distress. Take the case-hardened gear, for example. If the case is too thin, the tooth strength and wear re­ sistance will be unsatisfactory. If the case is too deep, the tooth is apt to be too brittle and subject to high internal stresses (the case will be tending to break away from the core material). If the carburizing gas is too rich, the outer case will contain too much carbon and is pat to spall under heavy load. If the gas is too lean, the tooth surface will not develop full hardness and wear resistance. If the case-carburized gear is unprotected just prior to quench, it may “out of gas” carbon and have deficient hardness right at the surface. If the quench is too slow for the alloy content and the size of the part, the core may lack strength and hardness and be unable to support the case under severe load conditions. The raw steel used to make forgings for carburized gears may be dirty or non-homogeneous. The best carburizing possible will not make a good gear if the machined forging has serious internal flaws before the final heat-treating operation. Those making drawings for carburized gears need to cover items like these: • Steel composition (maximum and minimum limits for all elements and impurities) • Cleanness of steel (vacuum-arc-remelt steel is often used where even the best air-melt steel is not clean enough) • Case depth (control is needed at tip, flank, and root) • Case and core hardness • Test for grinding burns • Metallurgical structure (proper martensitic quality is needed in both case and core)

Preliminary Design Considerations

251

TABLE 5.19 Typical Standard Gear Tolerances for Quality Number 9 to 13 AGMA Quality Number

Normal Diametral Pitch

Runout Tolerance for Pitch Diameter (Inches)

¾ 9

10

11

12

1½

½ 1

6

12

25

50

100

200

400

63.5

104.7 124.7 147.0 173.4 204.5 241.2 74.8 89.1 105.1 124.0 146.2 172.4

¾

1½

3

6

12

10.2

13.4 11.5

2

38.5 45.4

53.5

63.7

75.2

88.6

104.5 123.3

4 8

23.3 27.5 32.4 14.1 16.7 19.7 23.2

38.3 27.4

45.6 32.6

53.7 38.4

63.4 45.3

74.7 53.4

88.1 63.0

4.4

12

11.6 13.7 16.2 19.1

22.5

26.8

31.6

37.2

43.9

51.8

4.0

4.6

5.1

5.8

6.6

20 ½

9.1

10.7 12.6 14.9

17.6 74.8

20.9 89.0

24.7 29.1 34.3 40.4 3.6 105.0 123.8 146.1 172.3

4.1

4.6

5.2

5.9 9.4

1

45.3

53.5

63.7

75.1

88.5

104.4 123.2

7.2

8.1

2 4

27.5 32.4 16.7 19.6 23.2

38.2 27.3

45.5 32.5

53.7 38.4

63.3 45.3

74.7 53.4

88.1 63.0

4.1

5.4 4.6

6.1 5.2

6.9 5.9

8

10.1 11.9 14.0 16.6

19.5

23.3

27.4

32.4

38.2

45.0

3.1

3.5

4.0

4.5

5.1

12 20

8.3 6.5

11.5 13.6 9.0 10.6

16.1 12.5

19.1 14.9

22.6 17.6

26.6 20.8

31.4 24.5

37.0 28.9

2.8 2.5

3.2 2.9

3.6 3.2

4.1 3.7

4.6 4.1

53.4

63.6

75.0

88.5

104.3 123.0

1 2

32.4 19.6 23.1

38.2 27.3

45.5 32.5

53.6 38.3

63.2 45.2

74.6 53.3

88.0 62.9

3.8

5.0 4.3

5.7 4.9

4

11.9 14.0 16.6

19.5

23.2

27.4

32.3

38.1

45.0

14.0 11.5

16.6 13.7

19.6 16.1

23.1 19.0

27.3 22.4

32.2 26.4

9.8 7.6

½

8 12

7.2 5.9

8.5 7.0

10.0 11.8 8.2 9.7

20

4.6

5.5

6.4

½ 1 2

13

3

Pitch Tolerance for Pitch Diameter (Inches)

7.6

9.0

10.7

12.6

14.8

17.5

20.6

23.1

38.1 27.3

45.4 32.5

53.6 38.3

63.2 45.2

74.5 53.3

87.9 62.8

5.8 5.0

7.7

8.7

9.8

6.6 5.6

7.4 6.4

8.4 7.2

6.6

2.9

3.3

3.7

4.2

2.2 2.0

2.5 2.3

2.8 2.6

3.2 2.9

3.6 3.3

1.8

2.0

2.3

2.6

2.9

3.5

4.7 4.0

14.0 16.5

19.5

23.2

27.4

32.3

38.1

44.9

5.2

8.5 6.1

10.0 11.8 7.2 8.5

13.9 10.0

16.6 11.9

19.6 14.0

23.1 16.5

27.2 19.5

32.1 23.0

1.5

4.2

5.0

5.9

6.9

8.2

9.8

11.5

13.6

16.0

18.9

1.4

1.6

1.8

2.0

2.3

3.3

3.9

4.6

5.4

6.4 27.2

7.6 32.4

9.0 38.3

10.6 45.1

12.5 53.2

14.7 62.8

1.3

1.4

1.6

1.8

2.0 3.3

1

16.5

19.5

23.2

27.4

32.3

38.1

44.9

2.5

2.8

2 4

13.9 10.0

16.6 11.9

19.6 14.0

23.1 16.5

27.2 19.5

32.1 22.9

1.4

1.9 1.6

2.1 1.8

2.4 2.1

4 8 12 20 ½

2.0 1.7

2.7

3.0

3.4

2.3 2.0

2.6 2.2

2.9 2.5

6.1

10.0 11.8 7.2 8.4

8

3.7

4.3

5.1

6.0

7.1

8.5

10.0

11.8

13.9

16.4

1.1

1.3

1.4

1.6

1.8

12 20

3.0 2.4

3.6 2.8

4.2 3.3

5.0 3.9

5.9 4.6

7.0 5.4

8.2 6.4

9.7 7.6

11.4 8.9

13.5 10.5

1.0 0.9

1.1 1.0

1.3 1.1

1.4 1.3

1.6 1.4

Note: Tolerance values are in ten-thousands of an inch.

Through-hardened gears have their own metallurgical problems. If the quench is not fast enough for the size and alloy content of the gear part, there is apt to be a lack of hardness in the tooth root area. The structure may also be improper.

4

5

6

7

8

9

10

DIN Grade No.

10 – 16 6 – 10 3.55 – 6 2 – 3.55 10 – 16 6 – 10 3.55 – 6 2 – 3.55 10 – 16 6 – 10 3.55 – 6 2 – 3.55 10 – 16 6 – 10 3.55 – 6 2 – 3.55 10 – 16 6 – 10 3.55 – 6 2 – 3.55 10 – 16 6 – 10 3.55 – 6 2 – 3.55 10 – 16 6 – 10 3.55 – 6 2 – 3.55

Module

1.6 – 2.5 2.6 – 4.4 4.2 – 7.15 7.15 – 12.7 1.6 – 2.5 2.6 – 4.4 4.2 – 7.15 7.15 – 12.7 1.6 – 2.5 2.6 – 4.4 4.2 – 7.15 7.15 – 12.7 1.6 – 2.5 2.6 – 4.4 4.2 – 7.15 7.15 – 12.7 1.6 – 2.5 2.6 – 4.4 4.2 – 7.15 7.15 – 12.7 1.6 – 2.5 2.6 – 4.4 4.2 – 7.15 7.15 – 12.7 1.6 – 2.5 2.6 – 4.4 4.2 – 7.15 7.15 – 12.7

Diam. Pitch

Normal Tooth Size

110 100 90 80 80 71 63 56 56 50 45 40 40 36 32 28 28 25 22 20 20 18 16 14 14 12 11 10

μm

Runout

43 39 35 31 31 28 25 22 22 20 18 16 16 14 13 11 11 10 9 8 8 7 6 5.5 5.5 5 4 4

10−4 in.

TABLE 5.20 Typical DIN Gear Tolerances for Grades 10 to 3

63 56 50 40 40 36 32 25 32 25 20 18 22 18 16 12 16 12 11 9 11 9 8 6 8 6 5 4.5

t-to-t μm

25 22 20 16 16 14 13 10 13 10 8 7 9 7 6 5 6 5 4 3.5 4 3.5 3 2 3 2 2 2

10−4 in.

140 140 125 125 90 90 80 71 63 63 56 50 45 45 40 36 32 32 28 28 25 22 20 20 18 16 16 14

cum. μm

Spacing

Over 50 to 125 (Over 2 to 4.9)

125 110 100 90 90 80 71 63 63 56 50 45 45 40 36 32 32 28 25 22 22 20 18 16 16 14 12 11

μm

(Continued)

55 55 49 49 35 35 31 28 25 25 22 20 18 18 16 14 13 13 11 11 10 9 8 8 7 6 6 5.5

10−4 in.

49 43 39 35 35 31 28 25 25 22 20 18 18 16 14 13 13 11 10 9 9 8 7 6 6 5.5 5 4

10−4 in.

Runout

71 56 50 45 45 36 32 28 32 25 22 20 22 18 16 14 16 14 11 10 11 10 9 8 8 7 5.5 5

28 22 20 18 18 14 13 11 13 10 9 8 9 7 6 5.5 6 5.5 4 4 4 4 3.5 3 3 3 2 2

180 160 140 140 110 100 90 90 80 71 71 63 56 56 45 45 40 36 36 32 28 25 25 22 20 20 18 16

71 63 55 55 43 39 35 35 31 28 28 25 22 22 18 18 16 14 14 13 11 10 10 9 8 8 7 6

t-to-t cum. μm 10−4 μm 10−4 in. in.

Spacing

Over 125 to 280 (Over 4.9 to 11)

Pitch Diameter

140 125 110 100 100 90 80 71 71 63 56 50 50 45 40 36 36 32 28 25 25 22 20 18 18 16 14 12

55 49 43 39 39 35 31 28 28 25 22 20 20 18 16 14 14 13 11 10 10 9 8 7 7 6 5.5 5

71 63 56 45 45 40 36 28 36 28 25 20 25 20 18 16 18 14 12 10 12 10 9 8 9 8 6 5

200 180 180 160 125 110 110 100 90 80 80 71 63 63 56 50 45 40 40 36 32 28 28 25 22 22 20 18

79 71 71 63 49 43 43 39 35 31 31 28 25 25 22 20 18 16 16 14 13 11 11 10 9 9 8 7

(Continued)

28 25 22 18 18 16 14 11 14 11 10 8 10 8 7 6 7 5.5 5 4 5 4 3.5 3 3.5 3 2 2

cum. μm 10−4 in.

Spacing

t-to-t μm 10−4 μm 10−4 in. in.

Runout

Over 280 to 560 (Over 11 to 22)

252 Dudley’s Handbook of Practical Gear Design and Manufacture

Runout

t-to-t

Spacing

10 9 8 7

μm

Runout

cum.

4 3.5 3 3

10−4 in.

5.5 4.5 4 3

t-to-t μm

2 2 1.5 1

10−4 in.

Runout

t-to-t

cum.

Spacing

63

55

49 43

43

39 35

31

31 28

25

22 22

160

140

125 110

110

100 90

80

80 71

63

56 56

22 25

25

36 28

32

40 36

50

56 50

71

80

9 10

10

14 11

13

16 14

20

22 20

28

31

80 71

90

100 90

110

125 125

140

200 180

200

220

31 28

35

39 35

43

49 49

55

79 71

79

87

63 63

71

90 80

90

110 100

125

140 125

160

180

25 25

28

35 31

35

43 39

49

55 49

63

71

25 28

28

36 32

36

45 40

50

63 56

71

80

10 11

11

14 13

14

18 16

20

25 22

28

31

90 80

100

110 100

125

140 140

160

220 200

220

250

35 31

39

43 39

49

55 55

63

87 79

87

98

5 4 4 4

10−4 in.

20 28

25

40 32

28

45 36

56

56 45

71

90

4 4 3.5 3

10−4 in.

8 11

10

16 13

11

18 14

22

22 18

28

35

15

20

32

50

6

8

13

20

2 2 1.5 1

14 14 12 12

5.5 5.5 5 5

12 11 10 9

5 4 4 3.5

18

25

40

63

μm

7

10

16

25

10−4 in.

22

32

50

80

μm

40 – 100 mm (1.6 – 3.9 in.)

9

13

20

31

10−4 in.

2 2 2 1

6 6 5.5 5.5

9

13

20

31

10−4 in.

160 + mm (6.3 + in.)

16 16 14 14

(Continued)

22

32

50

80

μm

100 – 160 mm (3.9 – 6.3 in.)

6 5 4.5 3.5

cum. μm 10−4 in.

Spacing

t-to-t μm 10−4 μm 10−4 in. in.

Runout

Over 280 to 560 (Over 11 to 22)

Helix Slope for Face Width

5.5 5 4 3.5

t-to-t cum. μm 10−4 μm 10−4 in. in.

Spacing

20 – 40 mm (0.8 – 1.6 in.)

11 10 9 8

μm

Runout

Over 125 to 280 (Over 4.9 to 11)

Pitch Diameter

10−4 in. μm 10−4 in.

Profile Slope (Total)

12 11 10 10

cum. μm

Spacing

Over 50 to 125 (Over 2 to 4.9)

Over 1000 to 1600 (over 39 to 63)

Pitch Diameter

1.6 – 2.5 2.6 – 4.4 4.2 – 7.15 7.15 – 12.7

Diam. Pitch

Over 560 to 1000 (over 22 to 39)

10 – 16 6 – 10 3.55 – 6 2 – 3.55

Module

Normal Tooth Size

μm 10−4 in. μm 10−4 in. μm 10−4 in. μm 10−4 in. μm 10−4 in. μm 10−4 in. μm

3

DIN Grade No.

TABLE 5.20 (Continued) Typical DIN Gear Tolerances for Grades 10 to 3

Preliminary Design Considerations 253

t-to-t

Spacing

cum.

Runout

t-to-t

cum.

Spacing

Over 1000 to 1600 (over 39 to 63)

8

10 9

25 22

5

4 4

14 14

12

11 10

5 4.5

5.5

5.5 6

7

9 8

12

14 11

14

16 18

2 2

2

2 2

3

3.5 3

3

4 4

5

5.5 4

5.5

6 7

8

8

16 16

18

20 18

22

25 25

28

32 32

36

45 40

45

56 50

63

71

6 6

7

8 7

9

10 10

11

13 13

14

18 16

18

22 20

25

28

12 11

14

16 16

18

22 20

22

28 25

32

36 32

40

45 45

50

56

5 4

5.5

6 6

7

9 8

9

11 10

13

14 13

16

18 18

20

22

5 5

6

7 7

8

10 9

9

11 10

14

16 12

16

18 20

20

22

2 2

2

3 3

3

4 3.5

3.5

4 4

5.5

6 5

6

7 8

8

9

18 18

18

22 20

25

28 28

32

36 36

40

50 45

50

63 56

71

71

7 7

7

9 8

10

11 11

13

14 14

16

20 18

20

25 22

28

28

5 4

6

5 8

7

11 8

7

12 9

16

12 10

16

14 22

18

22

2 1.5

2

2 3

3

4 3

3

5 3.5

6

5 4

6

5.5 9

7

9

5

6

8

10

2

2

3

4

10−4 in.

6

8

10

12

μm

20 – 40 mm (0.8 – 1.6 in.)

2

3

4

5

10−4 in.

8

10

12

16

μm

40 – 100 mm (1.6 – 3.9 in.)

3

4

5

6

10−4 in.

8

10

12

16

μm

100 – 160 mm (3.9 – 6.3 in.)

Helix Slope for Face Width

Notes: 1. Tolerance values above are given in both microns (μm) and ten-thousands of an inch (10−4 in.). 2. For definition of tolerances, see Chapter 16. 3. t-to-t stands for tooth-to-tooth spacing tolerance; cum. stands for cumulative spacing tolerance. 4. Runout values are about 10% lower than concentricity values determined with a master gear. 5. Tolerance values in this table were extracted from DIN Standard 3962 (1978), Beuth Verlag GmbH, Berlin 30 und Koen 1, German Federal Republic.

6

5.5 5.5

16

8

11

28

8 7

13 11

32 28

20

14

36

20 18

11 10

16 16

40 40

20

18

45

20

20

50

10−4 in. μm

Profile Slope (Total)

μm 10−4 in. μm 10−4 in. μm 10−4 in. μm 10−4 in. μm 10−4 in. μm 10−4 in. μm

Runout

Over 560 to 1000 (over 22 to 39)

Pitch Diameter

TABLE 5.20 (Continued) Typical DIN Gear Tolerances for Grades 10 to 3

3

4

5

6

10−4 in.

160 + mm (6.3 + in.)

254 Dudley’s Handbook of Practical Gear Design and Manufacture

Preliminary Design Considerations

255

5.3.4 ENCLOSED-GEAR-UNIT REQUIREMENTS The gear designer has much more to do than just design gear teeth and gear parts. The gear unit will have gear casings, shafts, bearings, seals, and a lubrication system. All these things are part of the gear product and therefore become the responsibility of the gear designer. Those in gear-design work soon find that they have to become highly skilled in bearing selection, bearing design, and the fit of bearings on shafts and into casings. The load on the bearings from a gear mesh is not a steady load like a weight load, but a somewhat pulsating load because of tooth-error effects and tooth-stiffness effects as pairs of teeth roll through the meshing zone. Wear particles from contacting gear teeth tend to get into the lubricant and go through the bearings. Temperature changes resulting from changing gear loads or changing ambient conditions cause somewhat non-uniform expansions and contractions that disturb the rather critical gear-bearing fits. All these things make the design of gear bearings more critical than the design of bearings for a machinery application usually is. The lubrication system for a gear unit needs to keep all the tooth surfaces wet with lubricant and reasonably cool. In high-horsepower, high-speed gear units, oil-jet design, oil cooling by heat ex­ changers, oil-pump design, and oil-filter provision to remove dirt and wear particles all become very critical. Even in slow-speed designs with heavy oil or grease lubrication, where a pumped system is not needed, keeping the teeth wet and providing a lubricant that will work over a wide temperature range (from start-up on a cold day to maximum continuous load on a hot day) may be critical. The gear casing must support the gears and provide accurate alignment. For face widths up to about 10 mm, it is usually possible to make the gears and casing accurate enough to achieve satisfactory alignment with parts made to normal gear trade tolerances. Above 100 mm this becomes difficult, and at 250 mm face width it is usually impossible to be assured of a satisfactory tooth fit just by making all gears, bearings, and casings to close tolerances. The solution to this problem is to assemble a gear unit and check contact across the face width. In a critical application, contact checks may be needed at no load, ¼ load, ½ load, ¾ load, and full load. (When allowance for deflections under load is made by changing the helix angle, full contact under light load is not wanted, since the parts are meant to deflect and shift contact to achieve the desired load distribution as torque is increased.) When contact checks show an unsatisfactory condition under torque, the pinion or gear may be re-cut to better fit its mate. Sometimes casing bores are scrapped or re-machined to correct small errors in the original casing boring. In some situations, there may be several gear casings and several sets of gears available on the assembly floor. If the first gearset in a casing does not fit quite right, a second or third set may be tried. With selective assembly, it is often possible to fit up most of the units without doing any corrective work on gear parts or casings. In addition to fitting all right under torque loads, a critical gear unit may need to be tested at full speed to make sure that it runs without undue vibration and that all bearing temperatures and gear-part tem­ peratures are OK. In addition, the run test will show whether or not the gear teeth are tending to score or have some serious local misfit condition. (Generally, after full-speed power testing, the unit is dis­ assembled and all gear parts and bearings are inspected carefully by someone experienced with gears to see if there is any evidence of serious distress—that requires correction—in either the gear teeth or the bearings.) Those designing gear units have the responsibility of determining what testing may be needed to get satisfactorily under power. (Of course, if the gear unit is small and does not run too fast, the risk of an assembled unit not being OK when all parts are appropriately checked to part drawings my be relatively negligible.) To summarize this section, the gear unit designer has these major tasks:

256

Dudley’s Handbook of Practical Gear Design and Manufacture

• • • • • • • •

Design of gear parts Design of casing structures Design of bearings Design of lubrication system Design of seals, bolts, dowell pins, etc. Specification of assembly procedures Specification of gear running test procedure Definition of acceptable or unacceptable results from run tests and inspection or parts after testing.

More detailed information on the subject can be found in advanced sources on gear design and production.

BIBLIOGRAPHY Radzevich, S.P. (Ed.). (2019). Advances in Gear Design and manufacture. Boca Raton, Florida: CRC Press. Radzevich, S.P. (2018). Theory of Gearing: Kinematics, Geometry, and Synthesis, 2nd Edition, revised and ex­ panded. Boca Raton, FL: CRC Press.

6

Design Formulas Stephen P. Radzevich

The material presented so far has shown how to choose a kind of gear and how to make a preliminary estimate to the gear size required. The general nature of the detailed information required on gear drawings has also been discussed. In this chapter we shall take up the calculation of the detailed dimensions and the checking of the design against gear-rating formulas. The designer following this procedure should get almost the right size design on the first try. If the load-rating calculations show that the design is not quite right, it will usually be possible to make only one or two changes, such as a change in pitch or a change in face width, to adjust the capacity within the proper limits. It should not be necessary for the designer to scrap the whole design and start over from scratch after checking the load rating.

6.1

CALCULATION OF GEAR-TOOTH DATA

In this part of the chapter, we shall go over all the numbers and dimensions that will be needed on the drawings of the various kinds of gears. Designers who are not familiar with the limitations on gear size and shape that different materials or different manufacturing equipment may impose should consider the design worked out in this chapter as tentative until they have considered the things brought up in the later chapters of the book. Table 6.1 is a glossary of gear terms used in Chapter 6. Table 6.2 shows the metric and English symbols for the principal terms used in calculating gear data. (Many secondary terms are defined in figures and tables used for specific calculations.)

6.1.1 NUMBER

OF

PINION TEETH

In general, the more teeth a pinion has, the more quietly it will run and the better its resistance to wear will be. On the other hand, a smaller number of pinion teeth will give increased tooth strength, lower cutting costs, and larger tooth dimensions. In spur work, for instance, a seven-tooth pinion of 64 dia­ metral pitch has been found quite useful in some fractional-horsepower applications. Spur pinions used on railroad traction motors usually have from 12 to 20 teeth. High-load, high-speed aircraft pinions run from about 18 to 30 teeth. High-speed helical pinions for ship propulsion frequently have from 35 to 60 teeth. In general, low-ratio sets can stand more teeth than high-ratio sets. A 1 to 1 ratio set with 35 teeth will have about the same tooth strength as a 5 to 1 ratio set with 24 pinion teeth. Table 6.3 may be used as a general guide for numbers of teeth for spur and helical gears. Table 6.4 is worked out to give an approximate balance between gear-surface-durability capacity and gear-tooth-strength capacity. To understand this table, let’s consider a high-speed gear drive at 3 to 1 ratio with case-hardened and ground teeth at a surface hardness of 60 HRC (equivalent to about 600 HB or about 725 HV). The table says that a 25/75 tooth ratio ought to be OK. This means that when a large enough center distance and face width are picked based on surface-durability calculations, further cal­ culations on tooth strength out to be OK if the module (or pitch) is chosen to get 25 pinion teeth. Gear people would say that it was a “balanced design between durability and strength.” 257

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Dudley’s Handbook of Practical Gear Design and Manufacture

TABLE 6.1 Glossary of Gear Nomenclature, Chapter 6 Term

Definition

Approach action

Involute action before the point of contact between meshing gears has reached the pitch point. (A driving pinion has approach action on its dedendum.)

Breakage

A gear tooth or portion of a tooth breaking off. Usually the failure is from fatigue; pitting, scoring, and wear may weaken a tooth so that it breaks even though the stresses on the tooth were low enough to present no danger of tooth breakage when it was new. Sometimes a relatively new tooth will break as a result of a severe overload or a serious defect in the tooth structure.

Derating

Reducing the power rating of a gearset to compensate for tooth errors and for irregularities in the power transmission into and out of the gear drive. (When de-rated 2 to 1, a gear unit is rated to transmit only one-half the power that it might have transmitted under perfect conditions.)

Edge radius End easement

A radius of curvature at the end corner or top corner of a gear tooth. A tapering relief made at each end of a gear tooth while the middle portion of the tooth length is made to the true helix angle. The relief stops at some specified distance from the tooth end, such as one-eighth to one-sixth of the face width. End easement protects the teeth from misalignment and from stress concentrations peculiar to tooth ends. End easement is used on wide-face-width gears, while crowning may be used on narrow gears.

Flash temperature

The temperature at which a gear-tooth surface is calculated to be hot enough to destroy the oil film and allow instantaneous welding at the contact point.

Full-depth teeth

Gear teeth with a working depth of 2.0 times the normal module (or 2.0 divided by the normal pitch). When pinions are wide in face width for their diameter, there is appreciable bending and twisting of the pinion in mesh. A small change in helix angle (called helix correction or helix modification) may be used to compensate for the bending and twisting. This correction tends to give better load distribution across the face width. A ratio of numbers of gear and pinion teeth, which ensures that each tooth in the pinion will contact every tooth in the gear before it contacts any gear tooth second time. (13 to 48 is a hunting ratio; 12 to 48 is not a hunting ratio.)

Helix correction or Helix modification

Hunting ratio

Lead

The axial advance of a thread or a helical spiral in 360° (one turn about the shaft axis).

Lead angle Limit diameter

The inclination of a thread at the pitch line from a line at 90° to the shaft axis. The diameter at which the outside diameter of the mating gear crosses the line of action. The limit diameter can be thought of as a theoretical form diameter. See Figure 6.10.

Pitch point

The point on a gear-tooth profile that lies on the pitch circle of that gear. At the moment that the pitch point of one gear contacts its mating gear, the contact occurs at the pitch point of the mating gear, and this common pitch point lies on a line connecting the two gear centers.

Pitting

A fatigue failure of a contacting tooth surface, which is characterized by little bits of metal breaking out of the surface. Changing a part of the involute profile to reduce the load in that area. Appropriate profile modifications help gears to run more quietly and better resisting scoring, pitting, and tooth breakage.

Profile modification

Recess action

Involute action after the point of contact between meshing gears has passed the pitch point. (A driving pinion has recess action on its addendum.)

Runout

A measure of eccentricity relative to the axis of rotation. Runout is measured in a radial direction, and the amount is the difference between the highest and the lowest reading in 360° (one turn). For gear teeth, runout is usually checked by either putting pins between the teeth or using a master gear. Cylindrical surfaces are checked for runout by a measuring probe that reads in a radial direction as the part turned on its specified axis.

(Continued)

Design Formulas

259

TABLE 6.1 (Continued) Glossary of Gear Nomenclature, Chapter 6 Term

Definition

Scoring

A failure of a tooth surface in which the asperities tend to weld together and than tear, leaving radial scratch lines. Scoring failures come quickly and are thought of as lubrication failures rather than metal fatigue failures.

Zone of action

The distance, on the line of action between two gears, from the start of contact to the end of contact for each tooth.

For our 3 to 1 example with fully hard gears, the balance would be reasonably good in short-life vehicle gears or somewhat longer-life aircraft gears. For turbine power gears that would run for many more hours and would need a capability of around 1010 or more pinion cycles, the durability calculations would considerably reduce the intensity of tooth loading. A balanced design might then be achieved at higher numbers of teeth, like 30 to 38 pinion teeth. In high-speed turbine gearing, scoring hazards and noise and vibration considerations make it desirable to use as small a tooth size as possible. The designer might get a good durability/strength balance at something like 35 teeth for the 3 to 1 example, but decide to go up to 40 pinion teeth and put a little more face width and/ or center distance in to keep the rating within acceptable limits. This design would not have the lightest weight possible, but would compromise enough to avoid scoring and noise hazards reasonably well. In contrast to the example of high-speed turbine gear just cited, a vehicle designer with hard gears at a 3 to 1 ratio might to as low as 20/60 teeth rather than 25/76. The much shorter life, slow-speed vehicle gears can stand quite a little surface damage, but a broken tooth puts the gear drive out of action immediately. The designer of vehicle gears needs extra strength and can get it by using fewer and larger teeth. There is often a scoring hazard in vehicle gears, but this is handled more by using special extreme pressure (EP) lubricants than by using smaller teeth. To sum it up, Table 6.4 is a general guide to where to start on tooth numbers, but a complete design study may show that it is desirable to use pinion tooth numbers that differ from the table by a modest amount.

6.1.2 HUNTING TEETH The numbers of pinion and gear teeth must be whole numbers. It is generally desirable—particularly with low-hardness parts—to obtain hunting ratio between gear and pinion teeth. With a hunting ratio, any tooth on one member will—in time—contact all the teeth on the mating part. This tends to equalize wear and improve spacing accuracy. To illustrate this point, let us consider a tooth ratio of 21 to 76. The factors of 21 are 3 and 7. The factors of 76 are 2, 2, and 19. This ratio will hunt because the parts have no common factor. The gear should not be cut with a double-thread hob. A shaving cutter with 57 teeth would be a poor choice for either part. The cutter has factors of both 3 and 19. As a general rule, tooth numbers should be selected so that there is no common factor between the number of teeth of a pinion and a gear that mesh together and there should be no common factor between the number of teeth of a gear and of a cutting tool that has a gear-like meshing action with the part being cut. In certain cases, improved gear load-carrying capacity can be obtained with a non-hunting ratio. This is proven by tests made of spur gears having a hunting ratio of 25/27 and an integer ratio of 26/26. Pitchline speed was about 7 m/s (1400 fpm). At a low hardness of 185 HV (about 185 HB), it has been found that the 26/26 ratio would carry a little over 1000 N/mm2 surface compressive stress before pitting, while the 25/27 ratio would only carry

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Dudley’s Handbook of Practical Gear Design and Manufacture

TABLE 6.2 Gear Terms, Symbols, and Units, Chapter 6 Term

Metric

English

Reference or Formula

Symbol

Units

Symbol

Units

Number of teeth, pinion Number of teeth, gear

z1 z2

– –

NP or n NG or N

– –

Section 6.1 Section 6.1, 6.2

Number of threads, worm

–

NW

–

Table 6.29

–

Nc

–

Equation (6.46) and following

Tooth ratio

z1 z u

–

mG

–

z2 / z 1 (or NG /NP )

Addendum, pinion

h a1

mm

aP

in.

Section 6.3, 6.5

Addendum, gear

h a2 h¯ a

mm

aG

in.

Sections 6.3, 6.5

mm

ac

in.

Figure 6.6, Eq. (6.4)

Number of crow teeth

Addendum, chordal Rise of arc

–

mm

–

in.

Figure 6.6

Dedendum

hf

mm

b

in.

Equation (6.1)

Working depth

h

mm

hk

in.

2.0 × module (for full depth teeth)

Whole depth

h c

mm

ht c

in.

Section 6.3, Eq. (6.33)

in.

Equation (6.33), Section 6.32

s

mm mm

t tP tG tc

in. in.

Section 6.6 Sections 6.6, 5.11

in.

Sections 6.6, 5.14, Eq. (6.23)

in.

Figure 6.6, Eq. (6.5) Section 6.15, Eq. (6.24)

Clearance Tooth thickness Arc tooth thickness, pinion Arc tooth thickness, gear Tooth thickness, chordal

mm

s1 s2 s¯

mm mm

Backlash, transverse

j

mm

B

in.

Section 6.6, Table 6.21

Backlash, normal

jn

mm

Bn

in.

Section 6.6

dp1

mm

d

in.

Table 6.11

Pitch diameter, gear

dp2

mm

D

in.

Table 6.11

Pitch diameter, cutter

dp0

mm

dc

in.

Equation (6.36)

Base diameter, pinion

db1

mm

db

in.

Table 6.11

Base diameter, gear

db2

mm

Db

in.

Table 6.11

Outside diameter, pinion

da1

mm

do

in.

Pitch diameter +

Outside diameter, gear

da2

mm

Do

in.

Pitch diameter +

Inside diameter, face gear

di2

mm

Di

in.

Equation (6.31)

Root diameter, pinion or worm

df 1

mm

dR

in.

Equation (6.39)

Root diameter, gear

df 2

mm

DR

in.

Equation (6.39)

Form diameter

d

f

mm

df

in.

Sections 6.9, 6.11

Limit diameter

dl

mm

dl

in.

Figure 6.10, Section 6.8

dl

mm

in.

Allows for runout,

Ratio of diameters Center distance

a

– mm

dl m

Ratio, any diameter to pitch diameter Sections 5.7, 5.13

Face width

b

mm

C F

– in. in.

Section 5.8

b m or m t mn

mm

Fe

in.

Meshing face width

mm

–

–

mm of pitch diameter per tooth

mm

Module in normal section

–

–

in.−1

Teeth per in. of pitch diameter

Diametral pitch, normal

–

–

– Pd or Pt Pn

–

Diametral pitch, transverse

in.−1

Diametral pitch in normal section

Circular pitch

p

mm

p

in.

Pitch diameter, pinion

Excess involute allowance

Net face width Module, transverse Module, normal

center distance, Sections 6.9, 6.11

Pitch-circle arc length per tooth

(Continued)

Design Formulas

261

TABLE 6.2 (Continued) Gear Terms, Symbols, and Units, Chapter 6 Term

Metric

English

Reference or Formula

Symbol

Units

Symbol

Units

Circular pitch, transverse

pt

mm

pt

in.

Circular pitch, normal

pn

mm

pn

in.

Base pitch

pb

mm

pb

in.

Equation (6.17)

Axial pitch

px

mm

px

in.

pn /sin of helix angle

Lead (length)

pz

mm

L

in.

px × no. of threads

Pressure angle

or

deg

or

deg

Section 6.3

t

t

Pressure angle, normal

n

deg

n

deg

Section 6.12

Pressure angle, axial

x

deg

x

deg

Equation (6.40)

Pressure angle of cutter

0

deg

c

deg

Section 6.22

Helix angle

deg

deg

Section 6.12

Lead angle

deg

deg

Complement of helix angle

Shaft angle

deg

deg

Angle between gear shaft and pinion shaft

deg

Involute roll angle Equation (6.27)

Roll angle

r

deg

r

Pitch angle, pinion

1

deg

deg

Pitch angle, gear

2

deg

deg

–

–

Constant, about 3.14159265…

Pi Contact ratio Zone of action

g

Edge radius, tool

r a0

Radius of curvature, root fillet Circular thickness factor

f

Equation (6.26)

–

mp

–

Section 6.9

mm

Z

in.

Figure 6.11

mm

rT

in.

Generating tool, Eqs. (6.1), (6.22)

mm

f

in.

Generated root radius, Eqs. (6.1), (6.22)

k

–

k

–

(Bevel gears), Section 6.14

R

mm

(Bevel gears), Table 6.23

mm

A Ao

in. in.

Table 6.23

Mean cone distance

Ra Rm

mm

Am

in.

R a + Rf

Inner cone distance

Rf

mm

Ai

in.

Ra

Cone distance Outer cone distance

2

(or

A o + Ai ) 2

b (or A o

F)

Notes: 1. Abbreviations for units: mm = millimeters, in. = inches, deg = degrees. 2. See Table 6.33 for terms, symbols, and units in load rating of gears.

about 800 N/mm2 before pitting. The pitting limit was defined as the highest loading the gear pair would carry for 107 cycles without pitting. The numbers just quoted are based on a part being considered “pitted” when 1% of the contact area is pitted. To say it another way, the tests showed that the integer ratio could carry about 25% more Hertz stress. Since Hertz stress is proportional to the square root of K–factor, the integer ratio carried about 60% more K–factor. At a higher hardness of 300 HB for one part and over 600 HB for the other part, the hunting ratio carried about 6% more stress or about 12% more K–factor. The explanation seems to be that a single pair of teeth meshing with each other,—wear down as­ perities (or “coin” a fit between the surfaces) quite quickly. In the hunting ratio, a tooth on one part has to get worn and wear all the teeth on the other part into a fit with itself. Thus, a full fit cannot occur until all pinion teeth are worn alike, all gear teeth are worn alike, and the pinion-worn profile is a very close surface fit to the gear-worn profile.

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Dudley’s Handbook of Practical Gear Design and Manufacture

TABLE 6.3 General Guide to Selection of Number of Pinion Teeth No. Pnion Teeth

Design Considerations

7

Requires at least 25° pressure angle and special design to avoid undercutting. Poor contact ratio. Use only in fine pitches.

10

Smallest practical number with 20° teeth. Takes about 145% long addendum to avoid undercut. Poor wear characteristics.

15 19

Use where strength is more important than wear. Requires long addendum. No undercutting with 20° standard-addendum design.

25

Good balance between strength and wear for hard steels. Contact kept away from critical base-circle region.

35 50

Strength may be more critical than wear on hard steels – about even on medium-hard steels. Probably critical on strength on all but low-hardness pinions. Excellent wear resistance. Favored in highspeed work for quietness.

Note: The data given in Table 6.3 are rather general in nature. They are intended to give the reader a general view of the considerations involved in picking numbers of teeth. For somewhat more specific guidance in designing spur or helical gears, Table 6.4 shows how the choice tends to shift as hardness and ratio are varied.

TABLE 6.4 Guide to Selecting Number of Pinion Teeth z1 (or NP) for Good Durability and Adequate Strength Ratio, u (mG)

Long-Life, High-Speed Gears Having Brinell Hardness

Vehicle Gears, Short Life at Maximum Torque Having Brinell Hardness

200

300

400

600

200

300

400

600

1

80

50

39

35

50

37

29

26

1.5

67

45

32

30

45

30

24

22

2 3

60 53

42 37

28 25

27 25

42 37

27 24

21 18

20 18

4

49

34

24

24

34

23

17

17

5 7

47 45

32 32

23 22

23 22

32 31

22 21

17 16

17 16

10

43

30

21

21

30

20

16

16

Notes: 1. Typical high-speed applications, that fie this table, are turbine-driven helicopter gears. For turbine-driven industrial gears, more pinion teeth can be used because of lower allowable surface loading. (About 25% more is typical.) 2. Typical vehicle gears are spur and helical gears used in final wheel drives. (Considerably fewer pinion teeth are used in hypoid and spiral bevel final drives.)

The decision on whether or not to use hunting ratios becomes much more complex. Several pros and cons need to be considered. Table 6.5 shows the range of considerations.

6.1.3 SPUR-GEAR-TOOTH PROPORTIONS The proportions of addendum equal to 1.000mn and whole depth equal to 2.250mn have been used for many years. This design allows only a very small root fillet radius of curvature, and it is a hard design to work with when designing shaper cutters, pre-shave hobs, or shaving cutters.

Design Formulas

263

TABLE 6.5 Hunting Tooth Considerations Approximate Hardness, HB Pinion

Lubrication Regime

Gear

I

II

III

200 300

200 300

#1 #4

#2 #5

#3 #6

600

300

#7

#8

#9

600

600

# 10 Situation

# 11

# 12

#1, 2, 4, 5

Substantial gain in load capacity with integer ratio. After serious pitting, failure may be hastened as a result of growth of spacing errors and rough running of integer ratio.

# 3, 6

Hunting ratio probably best if parts kept in service after some teeth pitted more than 1%. No data available to prove gain in load carrying for integer ratio parts to be taken out of service when a “worst” tooth pits more than 5%. Substantial gain in load capacity with integer ratio. After serious pitting, failure may be quick as a result of spacing error effects.

# 7, 8 #9

Hunting ratio probably best. Worst error spots on lower-hardness gear will have fewer cycles and more chance to be worked into a fit by the hard pinion teeth.

# 10, 11

Possibly a small gain in load capacity with integer ratio. Hinting ratio probably best. Wear-in effects will be quite small. Worst tooth pair (with highest stresses resulting) will contact much less frequently.

# 12

TABLE 6.6 Spur-Gear Proportions Use

Pressure Angle, ( )

Working Depth, h (hk )

Whole Depth, h (h t )

Edge Radius of Generating Rack, r a0 (rT )

General purpose

20

2.000

2.250

0.300

Extra depth for shaving Aircraft, full fillet, high fatigue strength

20 20

2.000 2.000

2.350 2.400

0.350 0.380

Alternative high-strength design

25

2.000

2.250

0.300

Fine-pitch gears

20

2.000

2.200 + * (constant)

–

Notes: For metric design, the depth values and tool radius are in millimeters and for 1 module. (For other modules, multiply by module.) For English design, the depth values and tool radius are in inches and for 1 diametral pitch. (For other diametral pitches, divide by pitch.) * The extra amount of whole depth is 0.05 mm or 0.002 in. This constant is added to the calculated whole depth for small teeth 1.27 module or smaller. (In the English system fine pitch is 20 pitch and larger pitch numbers.)

Table 6.6 shows the more popular spur-gear-tooth proportions. Note that teeth finished by shaving or grinding require more whole depth than gears finished by cutting only (hobbing, shaping, or milling). When maximum load capacity is desired, the teeth need as large a root fillet radius as possible. This in turn leads to a need for more whole depth and a relatively large corner radius on the hob, grinding wheel, or other tool used to make the root area of the tooth.

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Dudley’s Handbook of Practical Gear Design and Manufacture

The most common pressure angle now in use for spur gears is 20°. Case-hardened aircraft or vehicle gears very often use a 25° pressure angle. The 25° form makes the teeth thicker at the base, and this improves the bending strength. In addition, the 25° teeth have larger radii of curvature at the pitch line, and this enables more load to be carried before the contact stress exceeds allowable limits. The 25° teeth generally run with more noise, as a result of the lower contact ratio. The tips of the teeth are thinner, and this may lead to fracturing of the tip when case-hardened teeth are cased too deep or the metallurgical structure is faulty. When made just right, and when the application can stand somewhat rougher running, the 25° tooth form will carry about 20% more torque (or power in kilowatts) than a 20° tooth form. A good compromise design that is easier to make successfully than 25° and has more capacity than 20° is a 22.5° tooth form. This design has about 11% more capacity than a 20° design. Figure 6.1 shows some comparison of tooth forms when they are cut with a rack (tooth number = ∞). In the gear trade, there was considerable use of stub teeth in early days. A typical stub tooth would have a working depth of around 1.60 (1.60 × m). When teeth are quite inaccurate, load sharing between two pair of teeth in the meshing zone may not exist. This means that one pair has to carry full load right

FIGURE 6.1

Comparison of tooth forms.

Design Formulas

265

out to the very tip of the pinion or gear. In such a situation, it is obvious that a shorter and stubbier tooth can take more load without breakage than taller, full-depth tooth. The use of stub teeth has declined to where they are seldom used. In general, gear teeth are designed and built accurately enough to share load when they should. Test data and field experience show that fulldepth teeth, made accurately, will carry more load and/or last longer than stub teeth. This leads to the consideration of extra-depth teeth. (If accurate full-depth teeth are better than stub teeth, why not go on to even deeper teeth than full-depth teeth?) Some gears are in production and doing very well with a whole depth great as 2.30. At this much depth, a 25° pressure angle is impossible and 22.5° is difficult. Pressure angles in the 17.5° to 20° range are reasonably practical with 2.30 working depth. The extra-depth teeth have problems with thin tips and smaller root fillet radii. Also, the effect on radii of curvature of the lower pressure angle somewhat offsets the gain from the higher contact ratio of the extra depth. In addition, very-high-speed extra-depth teeth are more sensitive to scoring troubles. In some situations, a less than 20° pressure angle may be used to get smoother and quieter running. If the pinion has more than 30 teeth, fairly good load capacity can be obtained with a pressure angle as low as 14.5°. Also, in instrument and control gears, center-distance changes have less effect on backlash at 14.5° than at 20°. Stub teeth, extra-depth teeth, and low-pressure-angle teeth all have possible advantages for limited applications. For general use, though, the full-depth 20° design presents the best compromise.

6.1.4 ROOT FILLET RADII

OF

CURVATURE

To get adequate tooth strength, it is desirable to cut or grind gear teeth to get either a full-radius root fillet or one that is almost full radius. Figure 6.2 shows an example of a 25-tooth pinion with a 22.5° pressure angle as it would be cut with a hob having a 0.36 edge radius (0.36 × module). The pinion has a whole depth of 2.35 and a long addendum of 1.2. Note the shape of the trochoidal root fillet and the path of the hot tip. Note also that the generated fillet of this type has its smallest radius of curvature near the root diameter. For design purposes, a 22.5° spur tooth would not be made exactly as shown in Figure 6.2. A slightly smaller whole depth of 2.3 and a pre-shave or pre-grind edge radius of 0.35 would be used. These changes facilitate the overall tool design and make it possible to use normal tolerances. The fillet ob­ tained is still very good and has a relatively low stress-concentration factor. The formula for calculating the minimum radius of curvature produced by hobbing or generating grinding is

FIGURE 6.2

25-tooth pinion with 22.5° pressure angle.

266

Dudley’s Handbook of Practical Gear Design and Manufacture

f

= ra0 +

f

= rT +

(h f

r a 0 )2

0.5 dp1 + (h f

r a0 )

(b rT )2 0.5 d + (b rT )

metric

English

(6.1)

(6.2)

The term f is the minimum radius of curvature in the generated trochoidal fillet. The edge radius of the hob or grinding tool is ra0 (rT ). The dedendum of the part is hf (or b). For a list of gear symbols and units used in this chapter, see Tables 6.2 and 6.3. For general purpose work, the value of the minimum root fillet radius on the drawing should not be more than about 70% of the calculated minimum. This allows the toolmaker for a reasonable margin of error in not obtaining the design tool radius. When gears are shaped, form ground, or milled, it is possible to get about the same minimum radius as would be obtained in hobbing. The shape of the fillet will be slightly different, though. From a practical standpoint, the designer can design to the method shown above and be assured that the gears can be produced by any of the cutting methods if the tools are properly designed. Figure 6.3 shows curve sheet based on Eqs. (6.1) and (6.2) for teeth with 20° and 25° pressure angles. Values for 22.5° are about midway between the values for 20° and 25°. Figure 6.3 shows an example of special curve sheet paper used to plot gear data. The abscissa distance is proportional to the reciprocal of the tooth number. Note that the distance from 12 to ∞ is just twice the distance the distance from 24 to ∞. The effect1 of tooth numbers on many variables having to do with involute curvature or trochoidal curvature is somewhat proportional to the reciprocal of the tooth number. Hence this graph paper results in relatively straight-line curves for many tooth variables.

6.1.5 LONG-ADDENDUM PINIONS A long-addendum pinion is one that has an addendum that is longer than that of the mating gear. Since the working depth is 2.000m for full-depth teeth, long-addendum pinions—in a full-depth design—have an addendum greater than 1.000m. Long-addendum pinions mate with “short-addendum” gears. The standard design practice is to make the gear addendum short by the same amount that the pinion addendum is made long. The most compelling reason to use long-addendum pinions is to avoid undercut2. An undercut design is bad for several reasons. From a load carrying standpoint, the undercut pinion is low in strength and wears easily at the point at which the undercut ends. In addition, there is the danger of interference. If the cutting tool does not make a big enough undercut, the mating gear may try to enlarge the undercut. Since the mating gear is not a cutting tool, it does a poor job of removing metal. It tends to bind in the undercut. This creates an interference condition, which is very detrimental. The more teeth on the cutting tool, the greater the undercut in the gear. Since a hob corresponds to a rack (a rack is a section of a gear with an infinite number of teeth), the hob produces the most undercut. As a general rule, it can be stated that hobbed teeth will not have interference due to lack of undercut. Besides avoiding undercut with low numbers of teeth, the long-addendum design, together with a proportional increase in tooth thickness at the pitch line [see Eqs. (6.5) and (6.6) below], makes the

1

2

The reader should note from this graph paper that a change of one tooth at the 12-tooth point means a change of more than 20 teeth at the 80-tooth point! See Figure 21.1 for an example of a 16-tooth standard-addendum pinion that is undercut.

Design Formulas

267

FIGURE. 6.3 Minimum root fillet radius graphs.

pinion tooth stronger and the gear tooth weaker. Thus, a long addendum can be used to balance strength. Figure 6.4 shows the remarkable difference that a 35% long addendum makes on a 12-tooth pinion. The highest sliding velocity and the greatest compressive stress occur at the bottom of the pinion tooth. This condition can be helped by using a long-addendum design to get the start of the active involute farther away from the pinion base circle. Figure 6.5 shows a curve sheet giving amounts of addendum for 20° standard teeth. The dashed curve on the left-hand side shows the bare minimum that is necessary to avoid undercut. Not gear or pinion should have less addendum than that given by this curve. The solid lines in Figure 6.5 show the addendum for both pinion and gear. Suppose a 20-tooth pinion was meshing with a 50-tooth gear. The curve would be read for 20 meshing with 50, and then again for 50 meshing with 20. The answers would be 1.16m for the pinion and 0.84m for the gear. If the dashed curve is read for pinion addendum, there is no place to read the gear addendum. In this case, the designer should subtract from the gear addendum the same amount is added to the pinion addendum. The sum of the pinion addendum and the gear addendum must always equal the working depth. The curve sheet of Figure 6.5 was drawn to give an approximate balance between the strength of the pinion and the strength of the gear. It also took care of the problem of pinion undercut.

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Dudley’s Handbook of Practical Gear Design and Manufacture

FIGURE 6.4 How long addendum eliminates undercut. Pinion at left, with an addendum of 0.100-in., is badly undercut, but pinion at right, with 0.135-in. addendum, has no undercut. Both of these 10-pitch, 12-tooth pinions were cut with the same hob and shaving cutter to the same whole depth.

FIGURE 6.5 Recommended addendum constant for pinions and gears of 20° pressure angle. The addendum constant is for 1 module or for 1 diametral pitch.

The problem of undercut is not as critical with 22.5° teeth, and it almost ceases to exist with 25° teeth. The general formula to find the minimum number of teeth needed to avoid undercut in spur gears is: z=

where hfx = hf – ra0 (1 – sin α).

2 hf

x

m sin2

metric

(6.3)

Design Formulas

269

N=

2X Pt sin2

English

(6.4)

where X = b – rT (1 – sin ϕ). When the pinion has a standard addendum of 1.000m and the spur teeth are designed to the proportions shown in Table 6.6, the minimum number of teeth that can be used without undercut is: 20° pressure angle 22.5° pressure angle 25° pressure angle

19 teeth 15 teeth 13 teeth

When using a long addendum to avoid undercut, the amounts of long addendum shown in Table 6.7 are needed. When the pinion has enough teeth to avoid undercut, the question of how much long addendum to use becomes more complex. A modest amount of long addendum will tend to balance strength between the pinion and the gear. Some benefit also results from the standpoint of surface durability and scoring. If a large amount of long addendum is used, the pinion is apt to be substantially stronger than the gear, and the tendency to score at the pinion tip becomes much greater. Some gear designers have used and advocated a 200% pinion addendum and a 0% gear addendum. This kind of design has all recess action when the pinion drives and no approach action. If gears are somewhat accurate, they will run more smoothly and quietly as the arc of recess is increased and the arc of approach decreased. There are special cases in which the design of a gearset justifies a very long pinion addendum and a very short gear addendum. For most designs of power gears, only a modest amount of long addendum is recommended. This somewhat balances pinion and gear strength and gives favorable results from the standpoint of surface durability and scoring. In service, the dedendum of the pinion and gear may pit extensively (see Chapter 14). If the gear dedendum is not unduly large, the gear can survive heavy pitting fairly well. Table 6.8 shows some guideline values for amounts of long addendum. These relatively low amounts of long addendum will give good results in rating calculations. From a practical standpoint, these gears will run backwards (gear driving rather than pinion) relatively well. Most gear drives have a power reversal under “coast” conditions, so the ability to run in reverse is usually a design requirement. In addition, the gear dedendum is not large enough to be a major problem when the gear member happens to have premature pitting. In speed-increasing drives, the gear is the driver. A long pinion addendum (in this case) makes the drive run more roughly and noisily. As a general rule, only enough long addendum is used to avoid

TABLE 6.7 Amount of Long Addendum to Avoid Undercut with Hobbed Teeth Pressure Angle

At 10 Teeth

At 12 Teeth

At 15 Teeth

20°

1.50

1.40

1.27

22.5° 25°

1.37 1.20

1.22 1.03

1.00 OK 1.00 OK

Note: For metric design the addendum of the pinion is the long addendum given above multiplied by the module. The gear addendum = 2.00m—pinion addendum. For English design, divide the number given by the diametral pitch for the pinion addendum. For gear addendum, divide 2.00 by diametral pitch and subtract pinion addendum.

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Dudley’s Handbook of Practical Gear Design and Manufacture

TABLE 6.8 Long and Short Addendum for Speed-Reducing Spur Gears Tooth Ratio, z 1/z 2 (NP /NG )

= 20 ( = 20 ) Pinion Addendum, h a1 (aP )

= 25 ( = 25 )

Gear Addendum, h a2 (a G )

Pinion Addendum, h a1 (aP )

Gear Addendum, h a2 (a G )

12/35

1.24

0.76

1.16

0.84

12/50

1.32

0.68

1.22

0.78

12/75 12/125

1.38 1.44

0.62 0.56

1.25* 1.25*

0.75 0.75

12/∞

1.48

0.52

1.25*

0.75

16/35 16/50

1.115 1.22

0.85 0.78

1.10 1.15

0.90 0.85

16/75

1.27

0.73

1.18

0.82

16/125 16/∞

1.32 1.36

0.68 0.64

1.21 1.24

0.79 0.76

24/35

1.06

0.94

1.04

0.96

24/50 24/75

1.11 1.15

0.89 0.85

1.08 1.10

0.92 0.90

24/125

1.20

0.80

1.13

0.87

24/∞

1.24

0.76

1.16

0.84

Note: Data in millimeters for 1-module teeth. (For 1-pitch teeth, read addendum data in inches.) * It is not practical to use more addendum here because the tip of the tooth is as thin as good practice will allow.

undercut. Generally, it is possible—and also advisable—to design the speed-increasing drive with en­ ough pinion teeth to be out of the undercut problem area. This means that speed-increasing gear drives will usually have the same addendum for the pinion and the gear.

6.1.6 TOOTH THICKNESS The tooth thickness of standard-addendum pinions and gears can be obtained by subtracting the minimum backlash from the circular pitch and dividing by 2. When long- and short-addendum pro­ portions are used, it is necessary to adjust the tooth thicknesses. Usually, this adjustment is made with a formula that will permit a shaper cutter to cut long and short proportions just as well as it cuts standard proportions. However, if the tooth thicknesses are adjusted for hobbing, it is usually possible to shape them with standard shaping tools and have only a small error in whole depth. With this thought in mind, the following formula for hobbed teeth is a good one to use to design all kinds of gears3:

s=

3

pt

j 2

+ ha (2 tan )

metric

(6.5)

Unfortunately, a small “s” is used for both tooth thickness and stress in the metric system. The reader needs to be aware that some symbols can have double meanings. For involute helical teeth, use normal module, normal circular pitch, and normal pressure angle. The tooth thickness obtained is in the normal section. (For English calculation, use similar normal-section values and the result will be normal tooth thickness.) The normal circular pitch pn = pt cosψ for English.

Design Formulas

271

where pt = m , backlash is denoted as j, and t=

pt

B 2

h a = ha

0.5 h .

+ a (2 tan )

English

(6.6)

where pt = / P d , backlash is denoted as B, and a = a 0.5 hk . The tooth-thickness values obtained from Eqs. (6.5) and (6.6) are arc tooth thickness. Backlash. The design amount of backlash j (or B) should be chosen to meet the requirements of the application. In power gearing, it is usually a good policy to use relatively generous backlash. The amount should be at least enough to let the gears turn freely when they are mounted on the shortest center distance and are subject to the worst condition of temperature and tooth error. In control gearing, the design usually can have but very little backlash. Some binding of the teeth as a result of eccentricity and tooth error may be preferable to designing so that there is appreciable lost motion due to backlash. Table 6.9 shows amounts of backlash as a guide for gear design with spur, helical, bevel, and spiral bevel gears. When the gears are not running fast and there are relatively small temperature changes, the table is useful for determining the minimum amount of backlash to use—assuming that extra backlash represents lost motion and may be somewhat undesirable from a performance standpoint. (A reversing gear drive, for instance, should not have any more backlash than necessary.) In epicyclic gears, it may be necessary to use a low backlash to keep “floating” suns or ring gears from floating too far and causing high vibration at light load. Table 6.9 is again useful as guide. Although Table 6.9 is based on general practice, it should be kept in mind that the values are sug­ gested values and must be used with discretion. Where gears have large center distance or where the casing material has a different expansion from that of the gears, some of the values shown are risky. For instance, 2.5-module high-speed gears used for either marine or aircraft applications will need a minimum backlash of the order of 0.15 or 0.20 mm (0.006 or 0.008 in.) to operate satisfactorily under all temperature conditions. Tolerances and tooth thickness The variation in backlash will depend considerably on the tooththickness tolerance. In hobbing or shaping teeth, a tolerance of 0.05 mm (0.002 in.) on thickness represents close work. Adding tolerances for the pinion and gear, a backlash variation of 0.10 mm (0.004 in.) is obtained from cutting alone! Obviously, it would not be good policy to design gear teeth so that the

TABLE 6.9 Suggested Backlash When Assembled Metric Module

English Backlash, mm

Diametral Pitch

Backlash, Inches

25

0.63 – 1.02

1

0.025 – 0.040

18 12

0.46 – 0.69 0.35 – 0.51

1½ 2

0.018 – 0.027 0.014 – 0.020

10

0.28 – 0.41

2½

0.011 – 0.016

8 6

0.23 – 0.36 0.18 – 0.28

3 4

0.009 – 0.014 0.007 – 0.011

5

0.15 – 0.23

5

0.006 – 0.009

4 3

0.13 – 0.20 0.10 – 0.15

6 8 and 9

0.005 – 0.008 0.004 – 0.006

2

0.08 – 0.13

10 – 13

0.003 – 0.005

1

0.05 – 0.10

14 – 32

0.002 – 0.004

272

Dudley’s Handbook of Practical Gear Design and Manufacture

backlash tolerances made the tooth-cutting costs prohibitive. Table 6.10 shows a study of tooth-thickness tolerances for different cutting methods for teeth in the 2- to 5-module range. These values should be considered when setting ranges of backlash and tooth-thickness tolerances.

6.1.7 CHORDAL DIMENSIONS The tooth thickness of a spur or helical gear is often measured with calipers. This instrument is set for a depth corresponding to the chordal addendum, and it measures a width corresponding to the chordal tooth thickness. Figure 6.6 defines the chordal dimensions. The chordal addendum is obtained by adding the rise of arc to the addendum. The equations generally use are s 2 cos2 h¯a = ha + 4d p

ac = a +

t 2 cos2 4d

metric

(6.7)

English

(6.8)

TABLE 6.10 Tolerances on Tooth Thickness Method of Cutting

Degree of Care Very Best

Close Work

Easy to Meet

mm

in.

mm

in.

mm

in.

Grinding

0.005

0.0002

0.013

0.0005

0.05

0.002

Shaving Hobbing

0.005 0.013

0.0002 0.0005

0.013 0.050

0.0005 0.0020

0.05 0.10

0.002 0.004

Shaping

0.018

0.0007

0.050

0.0020

0.10

0.004

FIGURE 6.6

Chordal dimensions of a gear or pinion tooth.

Design Formulas

273

Equations (6.7) and (6.8) work for both helical and spur gears. In helical gears, the tooth thickness used should be the normal tooth thickness. Figure 6.7 shows the rise of arc for a range of diameters and tooth thicknesses. For helical gears, curve readings are multiplied by the cosine squared of the helix angle. Chordal tooth thicknesses are obtained by subtracting a small amount from the arc tooth thickness. This arc-to-chord correction is so small that it can be neglected in may gear designs. The calculation should always be made, though, for coarse-pitch pinions, master gears, and gear-tooth gages. The ap­ proximate equations that are usually used are:

FIGURE 6.7

s¯ = s

s 3 cos4 6 dp2

metric

(6.9)

tc = t

t 3 cos4 6 d2

English

(6.10)

Rise of arc graph. Read the rise of arc as shown, and then correct for helix angle.

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Dudley’s Handbook of Practical Gear Design and Manufacture

For helical gears, the normal tooth thickness should be used in Eqs. (6.9) and (6.10). The arc-to-chord correction sc is plotted in Figure 6.8 for a range of diameters and tooth thicknesses. Multiplying constants are shown to take care of helical gears.

6.1.8 DEGREES ROLL

AND

LIMIT DIAMETER

The checking machines used to measure the involute profile of spur and helical gears usually record error in involute against degrees of roll. This makes it necessary in many cases to determine the degrees roll at the start of active profile and at the end of active profile. The start of active profile—so far as checking goes—is at the form diameter. The end of active profile is at the outside diameter. Figure 6.10 showed the basic involute relations in the English system. Figure 6.9 is a similar diagram showing basic involute relations in the metric system. In Figure 4.10 the roll angle to the pitch circle (or to the pitch point) is the angle designated r . The roll angle to “any point” on the involute curve is r1.

FIGURE 6.8

Arc-to-chord correction graph. Read the arc-to-chord correction as shown, and then correct for helix angle.

Design Formulas

FIGURE 6.9

275

Roll angles, which define the active profile of the gear.

The solution of the roll angle problem lies in the fact that the length of the line unwrapped from the base cylinder is equal to the ark length from the origin of the involute to the tangency pint of the involute action line to the base cylinder. An angle in radians is equal to the arc length of the angle divided by the radius. These relations plus other obvious triangular relations lead to the basic relations of the involute shown in Figure 6.9. The degrees roll at the outside diameter may be calculated by solving the following equations:

Metric a

=

cos ra

da dp a

=

English

mo = =

cos t a

180 tan a

cos ro

=

o

=

do d cos

(6.11) t

(6.12)

o

(6.13)

mo

180 tan

276

Dudley’s Handbook of Practical Gear Design and Manufacture

FIGURE 6.10 Limit diameter and form diameter. The allowance for extra involute should be large enough to allow for center-distance variations and variations in outside diameter of the mating gear.

The angle ra ( ro ) is the degrees roll at the outside diameter. The roll angle at the limit diameter is calculated in a similar manner. The limit diameter is the diameter at which the outside diameter of the mating gear crosses the line of action. See Figure 6.10. Roll angles for limit diameter or form diameter can be obtained by:

Metric

=

dl dp

cos

l

l

rl f

cos rf

=

cos t l

cos rl

l

=

df

=

=

cos t f

180 tan f

cos rf

(6.14)

cos

t

(6.15)

l

(6.16)

ml

df d

(6.17)

cos t mf

(6.18)

180 tan f

(6.19)

f

=

dl d

180 tan

mf =

dp

f

=

ml =

180 tan l

=

=

English

=

Design Formulas

277

Calculation Table 6.11 gives the step-by-step procedure for calculating all the dimensions in the zone of action. The following items are of particular interest: • Line of action for the addendum (18) • Zone of action (19) • Line of action, total (21)

TABLE 6.11 Calculation Sheet for the Meshing-Zone Dimensions for a Pair of Involute Gear Teeth (See Figure 6.11 for Identification of Items in This Table) Basic Data 1. 2.

Pressure angle Cosine, pressure angle

20 0.939693

3.

Tangent, pressure angle

0.363970

4. 5.

Circular pitch Base pitch, (2) × (4)

9.424778 8.856394 Pinion

Gear

6. 7.

Number of teeth Addendum

25 3.54

96 2.46

8.

Pitch diameter, (4) × (6) ÷ π

Calculations

75.00

288.00

9. 10.

Outside diameter, (8) + 2 × (7) Base diameter, (8) × (2)

82.08 70.47695

292.92 270.63148

11.

(9) ÷ (10)

1.164636

1.082357

12. 13.

(11) × (11) [(12) – 1.000]0.5

1.356377 0.596973

1.171498 0.414123

14.

(13) – (3)

0.233003

0.050153

15. 16.

(6) ÷ 6.283185 Contact ratio, addendum, (14) × (15)

3.978873 0.92709

15.27887 0.76628

17.

Contact ratio, pair, (16)1 + (16)2

1.69337

18. 19.

Line of action, addendum, (16) × (5) Zone of action, (18)1 + (18)2

8.21067 14.99715

6.78647

20.

0.5 × (10) × (3)

12.82575

49.25090

21. 22.

Line of action, total, (18) + (20) (21) – (19)

21.03642 6.03927

56.03737 41.04022

23.

[2 × (22)]2

145.89113

6737.1986

24. 25.

Limit diameter, [(23) + (10)2]0.5 (10) ÷ (9)

71.50449 0.858637

282.80487 0.923909

26.

Pressure angle, OD, arcos (25)

30.836078

22.495559

27. 28.

Roll angle, OD, 57.29578 × tan (26) (10) ÷ (24)

34.204057 0.985630

23.727487 0.956955

29.

Pressure angle, LD, arcos (28)

9.725034

16.872143

30.

Roll angle, LD, 57.29578 × tan (29)

9.819514

17.377365

Notes: 1. This table can be calculated in the metric system by using millimeters for all dimensions. Columns 1 and 2 show a metric example of 25 teeth pinion meshing with 96 teeth gear. 2. This table can be calculated in the English system by using inches for all dimensions. 3. All angles are in degrees.

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Dudley’s Handbook of Practical Gear Design and Manufacture

• Limit diameter (24) • Roll angle, outside diameter (27) • Roll angle, limit diameter (30) This calculation sheet is illustrated by an example worked out for 25 pinion teeth meshing with 96 gear teeth with a pressure angle of 20°. The teeth are 3 module and spur. The constant 57.29578 is 180° divided by pi (3.14159265). Figure 6.11 illustrates the dimensions that are calculated in Table 6.11. This table works for either metric or English dimensions.

6.1.9 FORM DIAMETER

AND

CONTACT RATIO

Form diameter is an important design parameter of a gear and of a pinion. Torque capacity of a gear pair, as well as other output capabilities of it depend on contact ratio. Actual value of contact ratio is tightly connected with actual value of from diameter of a gear and of a mating pinion. Form diameter. The form diameter of a gear is that diameter that represents the design limit of involute action. A theoretical limit diameter dl is first obtained, and then practical value is obtained by subtracting an allowance dl for outside-diameter runout and center-distance tolerance. Limit diameter may be calculated graphically or by solving a series of equations. The preceding section gave the necessary equations, and Table 6.11 showed a sample calculation. After dl is obtained, the form diameter is: d

f

= dl

dl

(6.20)

The allowance dl should be large enough to accommodate for the effect of various tooth errors in extending the contact below the theoretical point plus the effect of tooth bending under load in extending the contact deeper. For general applications the value of 0.05 mt has worked out quite well. If the pinion has a small number of teeth, so that the limit diameter is close to the base circle, a form diameter that is below the base circle should never be specified. With a standard-addendum pinion of 20 pressure angle, an allowance for excess involute as large as 0.05 mt should be used only when there are 25 or more teeth in the pinion. The value of 0.05 mt (transverse module is denoted by mt ) has an English equivalent of 0.050 in. / Pd . Some sample values are:

Tooth size

Δdl

1 module

0.05 mm

5 module

0.25 mm

20 module 20 pitch

1.00 mm 0.0025 in.

5 pitch

0.01000 in.

1 pitch

0.0500 in.

For calculation purposes, it is handy to make the excess involute a function of the circular pitch (instead of a function of the module or an inverse function of the diametral pitch). On this basis, it works out that: dl = 0.016 × circular pitch = 0.016 ×

mt

metric

(6.21)

Design Formulas

279

FIGURE 6.11 Identification of dimensions used in calculations for the meshing or a pair of involute gear teeth. The numbers correspond to the line number in Table 6.11.

dl = 0.016 × circular pitch = 0.016 ×

Pt

English

(6.22)

Contact ratio. The contact ratio of a pair of spur or helical gears represents the length of the zone of action divided by the base pitch. The basic equation is:

Contact ratio =

zone of action base pitch

(6.23)

280

Dudley’s Handbook of Practical Gear Design and Manufacture

=

g pb

metric

(6.24)

mp =

Z pb

English

(6.25)

where gα (or Z) designates zone of action [Item (19), Table 6.11]; pb = ( mt )cos

pb =

Pt

cos

t

t

for metric;

for English.

Figure 6.12 shows plotted contact-ratio values for involute spur-gear teeth of 20 , 22.5 , and 25 pressure angles. Note that the 25% long-addendum teeth have somewhat lower contact ratios than standardaddendum teeth. Also note that the 25 pressure-angle teeth have a much lower contact ratio than 20 teeth. The term contact ratio should be thought of as “average number of teeth in contact.” As spur gears roll through mesh, there are either two pair of teeth in mesh or one pair. (A 20 pair with a 1.7 contact ratio never really has 1.7 pair of teeth in contact!)

FIGURE 6.12 Transverse contact ratios for standard addendum and 25% long addendum spur-gear teeth at dif­ ferent pressure angles.

Design Formulas

281

6.1.10 SPUR-GEAR DIMENSION SHEET In calculating gear dimensions, it is helpful to use a dimension sheet that lists all the items to be cal­ culated. This ensures that all necessary values are calculated, and it is a convenient sheet to file in a design book as a record of different jobs. Table 6.12 shows a recommended dimension sheet for spur gears. The sheet shows how to obtain design values for each item. To illustrate the use of the sheet, numerical values are given for a sample problem. Several of the dimensions in Table 6.12 require tolerances. In power gearing of small size, a centerdistance tolerance of 0.05 mm (0.002 in.) is often used. This affects the backlash about 0.038 mm (0.0015 in.). On control gearing, a closer value would be needed. The face width of the pinion is usually made a little wider than the gear. It is easier to provide the extra width for error in axial positioning on the pinion than on the gear.

TABLE 6.12 Spur-Gear Dimensions Item

Reference

Metric

English

1.

Center distance

Sections 5.2.2 to 5.2.4

181.50

7.145655

2.

Circular pitch

Equation (2.2)

9.42478

0.37105

3. 4.

Pressure angle Working depth

Section 6.3 Table 6.6

20 6.00

20 0.2362

5.

Module, transverse

Eqs. (2.5) to (2.6)

3

–

6.

Diametral pitch, transverse

Eqs. (2.5) to (2.6)

–

8.66667

7.

Number of teeth

Sections 6.1, 6.2

8. 9.

Pitch diameter Addendum

Equation (2.7) Section 6.5

10.

Whole depth

Table 6.6

11. 12.

Outside diameter Root diameter

(8) + [2 × (9)] (11) – [2 × (10)]

13.

Arc tooth thickness

Section 6.6

14. 15.

Chordal tooth thickness Chordal addendum

Section 6.7 Section 6.7

Pinion

Gear

Pinion

25

96

25

Gear 96

75.00 3.54

288.00 2.46

2.9527 0.1394

11.3386 0.0968

7.05

7.05

0.2775

0.2775

82.08 67.98

292.92 278.82

3.2315 2.6765

11.5322 10.9772

4.991

4.204

0.1965

0.1655

5.026 3.62

4.243 2.47

0.1979 0.1397

0.1671 0.0969

16.

Pin size

Table A.1

5.334

5.334

0.210

0.210

17. 18.

Diameter over pins Face width

Table A.2 Section 5.8

83.1905 37.0

304.2113 35.0

3.2752 1.45

11.9768 1.38

19.

Form diameter

Table A.3

71.35

282.65

2.8090

11.1279

20. 21.

Roll angle, O.D. Roll angle, F.D.

Table A.3 Table A.3

34.20 9.07

23.73 12.27

34.20 9.07

23.73 17.27

70.477

270.631

2.77468

10.6548

22.

Base circle, diameter

Table 6.1

23. 24.

Minimum root radius Accuracy

Section 6.4

0.84

0.79 0.033 See Note 4

0.031

Notes: 1. See Chapter 10 for gear materials and specifications. 2. See Chapter 17 for an example of spread center design. (This table is for standard center design.) 3. Metric dimensions are in millimeters; English dimensions are in inches. Angles are in degrees (and decimals of degree). 4. After the designer ahs done the load retain and studied Section 16.4.1, accuracy needs may be satisfied by an ISO or AGMA quality level, or the designer may need to write a special recipe for accuracy limits.

282

Dudley’s Handbook of Practical Gear Design and Manufacture

The tolerance on outside diameter is based on whatever tolerance is reasonable to hold in lathe or cylindrical grinding operations. This dimension does not need to be held too close unless accuracy is needed in order to set calipers for close tooth-thickness checking. The tolerance on root diameter should allow for variation in tool thickness as well as for the worker’s error in setting the gear-cutting machine. Tooth thickness tolerances should be governed by backlash requirements of the application as well as by Table 6.10 considerations. Tolerances on tooth accuracy and data on material and heat treatment will be needed (see Section 12.2.4). Usually, these data are not shown on the specification sheet unless they can be given by referring to a standard specification. Frequently, they are shown only on the part drawing. In many cases, it is desired to make layouts of pinion or gear teeth to see what they will look like. This can be done by calculating the involute part of the tooth profile as a first step, then calculating the root fillet trochoid as a second step. The 25-tooth pinion of Table 6.12 is as an example. Figure 6.13 shows how this tooth was plotted.

FIGURE 6.13

Oversize layout of 25-tooth pinion.

Design Formulas

283

The 25-tooth pinion in Figure 6.13 looks good. The tip of the tooth is generously wide (should be no trouble in carburizing). The root fillet is generous in curvature, and the base of the tooth is wide. This tooth can be expected to have good beam strength.

6.1.11 INTERNAL-GEAR DIMENSION SHEET The dimensions for internal gears may be calculated in a manner quite similar to that just described for spur gears. However, internal gears are subject to two kinds of trouble that do not affect external gears. If the internal gearset is designed for full working depth, the gear may contact the pinion at a lower point on the tooth flank than the cutting tool generated on the involute profile. When internal gearsets have too little difference between the number of teeth in the pinion and the number of teeth in the gear, there may be interference between the tips of the teeth. The interference is most apt to occur as the pinion is moved radially into mesh with the gear. It is possible to get around the radial-interference difficulty by assembling the set by an axial movement of the pinion. The difference between numbers of teeth can be less when this method of assembly is used, but there will still be interference if the difference is too small. Figures 6.14 and 6.15 show radial interference, and axial assembly of 19/26 teeth gearset having 20° pressure angle. The calculations (or layout procedure) that avoid these troubles with internal gears are rather com­ plicated. Instead of presenting the method here, Tables 6.13 and 6.14 will be given to show what di­ mensions can be safely used with internal gears. In Tables 6.13 and 6.14, the addendum of the internal gear has been shortened so that the internal gear will contact the pinion no deeper than it would be contacted by a rack tooth with an addendum equal to 2.0m ha1 (or 2.0/Pd aP ). This makes it possible to manufacture the pinion of an internal gearset in the same way that a pinion would be manufactured to go with an external gear with a large number of teeth. The minimum numbers of teeth for the different methods of assembly are calculated so that the tips clear each other by at least 0.02m millimeters (or 0.020/Pd inches). Tables 6.13 and 6.14 are worked out so as to get the most that is possible out of internal gears. Many designers have used internal-gear data that did not use quite so much working depth and quite so close

FIGURE 6.14 Internal gear of 26 teeth will not assemble radially with a 19-tooth pinion. Both parts cut to Table 6.13 proportions.

284

Dudley’s Handbook of Practical Gear Design and Manufacture

FIGURE 6.15 Internal gear of 26 teeth will assemble axially with a 19-tooth pinion. Both parts cut to Table 6.13 proportions.

tooth numbers as are shown in this table. If the design is not critical and center-distance accuracy is not close, it may be desirable not to use quite so low a ratio as would be permitted by the table. It will be noted that the tables show two addendum values for each number of teeth. The first one is the minimum addendum that might be used for the pinion. This addendum has been lengthened only in those cases in which there is danger of undercut. The second addendum has been lengthened to balance tooth strength. The lengthening is not quite so much as layouts of the teeth might indicate, but it is about the right amount to use considering the fact that the quality of material in the gear may not be quite as good as that in the pinion. Also, the stress concentration at the root of the internal-gear tooth is apt to be high. In general, the minimum addendum should be used when the gear drives the pinion, and the maximum addendum should be used when the pinion drives the gear. The internal gear usually cannot have a clearance of much over 0.250m (or 0.259/Pd) because of the effect of the concave (flaring) sides of the teeth. The “outside diameter” of an internal gear is really the inside diameter. The inside diameter is ob­ tained by subtracting twice the gear addendum from the pitch diameter. The tooth thickness of the pinion is usually adjusted for the amount of long addendum that the pinion has. The gear-tooth thickness is obtained by subtracting the pinion thickness and backlash from the circular pitch. No consideration is given to gear addendum in figuring gear-tooth thickness. This is a result of the fact that the gear addendum has been arbitrarily stubbed to avoid tip interference. The pinion thickness for a full-depth pinion is: s1 =

tp =

p

j 2

p

B 2

+ (ha1

1.000m ) (2 tan ) metric

(6.26)

+ aP

1.000 (2 tan ) Pd

(6.27)

English

Design Formulas

285

TABLE 6.13 Addendum Proportions and Limiting Numbers of Teeth for Internal Ppur Gears of 20° Pressure Angle No. of Pinion Teeth z1 (NP )

12

Pinion Addendum ha1 (aP )

Min. No. of Gear Teeth

Gear Addendum

Axial Assembly, z2 (NG )

Radial Assembly, z2 (NG )

u = min* (mG = min ), h a2 (a G )

u=2 (mG = min ), h a2 (a G )

u=4 (mG = min ), h a2 (a G )

u=8 (mG = min ), h a2 (a G )

1.350

19

26

0.472

0.510

0.582

1.510

19

26

0.390

0.412

0.451

0.471

1.290 1.470

20 20

27 27

0.507 0.419

0.556 0.445

0.635 0.488

0.673 0.509

14

1.230

21

28

0.543

0.601

0.688

0.729

15

1.430 1.180

21 22

28 30

0.447 0.574

0.479 0.642

0.525 0.733

0.548 0.777

1.400

22

30

0.470

0.506

0.554

0.577

16

1.120

23

32

0.608

0.688

0.786

0.834

17

1.380 1.060

23 24

32 33

0.487 0.642

0.526 0.734

0.574 0.839

0.597 0.890

13

0.616

1.360

24

33

0.505

0.546

0.594

0.617

1.000 1.350

25 25

34 34

0.676 0.516

0.779 0.558

0.892 0.605

0.947 0.628

19

1.000

27

35

0.702

0.792

0.808

0.950

20

1.330 1.000

27 28

35 36

0.539 0.713

0.578 0.802

0.625 0.903

0.648 0.952

18

1.320

28

36

0.550

0.590

0.636

0.658

1.000 1.290

30 30

39 39

0.733 0.577

0.821 0.621

0.912 0.666

0.957 0.688

24

1.000

32

41

0.750

0.836

0.920

0.960

26

1.270 1.000

32 34

41 43

0.599 0.766

0.644 0.849

0.687 0.926

0.709 0.963

22

30 40

*

1.250

34

43

0.620

0.666

0.709

0.729

1.000 1.220

38 38

47 47

0.792 0.654

0.870 0.702

0.936 0.741

0.968 0.761

1.000

48

57

0.836

0.903

0.952

0.976

1.170

48

57

0.718

0.764

0.797

0.814

Minimum u = z2/z1 is minimum for axial assembly.

The chordal addendum of the gear is not ordinarily needed unless the gear has a large enough inside diameter to permit tooth caliper to be used. It takes at least a 130-mm (5-in.) inside diameter to allow a person’s hand inside the gear to measure tooth thickness. Usually, small internal gears are checked for thickness by using measuring pins between the teeth. In an internal gear, the rise of arc must be sub­ tracted from the addendum. Because of the concave nature of the gear tip, only about two-thirds of the pitch-line rise of arc is effective. The form diameter of the gear is obtained by adding the allowance for excess involute instead of subtracting it. Thus:

286

Dudley’s Handbook of Practical Gear Design and Manufacture

TABLE 6.14 Addendum Proportions and Limiting Numbers of Teeth for Internal Spur Gears of 25° Pressure Angle No. of Pinion Teeth z1 (NP )

12

Pinion Addendum ha1 (aP )

Min. No. of Gear Teeth

Gear Addendum

Axial Assembly, z2 (NG )

Radial Assembly, z2 (NG )

u = min* (mG = min ), h a2 (a G )

u=2 (mG = min ), h a2 (a G )

u=4 (mG = min ), h a2 (a G )

u=8 (mG = min ), h a2 (a G )

1.000

17

20

0.699

0.793

0.900

1.220

17

21

0.601

0.656

0.720

0.740

1.000 1.200

18 18

22 22

0.718 0.622

0.810 0.680

0.908 0.742

0.940 0.761

14

1.000

19

24

0.734

0.824

0.915

0.944

15

1.180 1.000

19 20

24 25

0.644 0.748

0.703 0.837

0.763 0.921

0.782 0.948

1.170

20

25

0.659

0.719

0.776

0.794

16

1.000

21

26

0.761

0.847

0.926

0.951

17

1.150 1.000

21 22

26 27

0.679 0.773

0.741 0.857

0.797 0.930

0.815 0.954

13

0.934

1.140

22

27

0.694

0.755

0.809

0.826

1.000 1.130

23 23

28 28

0.783 0.708

0.865 0.769

0.934 0.820

0.957 0.837

19

1.000

24

29

0.793

0.873

0.938

0.959

20

1.120 1.000

24 25

29 30

0.721 0.802

0.782 0.879

0.832 0.941

0.848 0.961

18

1.110

26

29

0.741

0.795

0.843

0.859

1.000 1.090

28 28

32 32

0.824 0.765

0.891 0.820

0.947 0.866

0.965 0.881

24

1.000

30

34

0.837

0.900

0.951

0.968

26

1.080 1.000

30 32

34 37

0.782 0.847

0.836 0.908

0.879 0.955

0.893 0.970

22

30 40

*

1.060

32

37

0.806

0.859

0.900

0.914

1.000 1.050

36 36

40 40

0.865 0.829

0.921 0.879

0.961 0.915

0.974 0.927

1.000

46

51

0.896

0.941

0.971

0.981

1.020

46

51

0.880

0.923

0.952

0.961

Minimum u = z2/z1 is minimum for axial assembly.

d

f2

= dl + dl

Df = Dl + Dl

metric

(6.28)

English

(6.29)

The limit diameter for internal sets is figured by the same general procedure as that shown for external gears in Table 6.11. However, it is necessary to add instead of subtract. For the internal gear, the following equations should be used:

Design Formulas

where

r

=

180 tan

287 rl

=

=

r

r

+(

r)

ra

z1 z2

metric

(6.30)

English

(6.31)

, rl

+(

ro

r)

NP NG

180 tan

where r = . Once the roll angle at the limit diameter is known, the limit diameter is easily calculated by known method. The cutter that generates an internal gear will usually make a fillet that is almost the same radius of curvature as itself. Generally, a cutter for an internal gear will have only about two-thirds of the radius given in Table 6.6. Table 6.15 shows a dimension sheet for internal gears. Data for a sample problem of an 18-tooth pinion driving a 28-tooth gear is also shown. The calculation of the diameter between pins for the internal gear is somewhat different from the calculation of diameter over pins for an external gear.

6.1.12 HELICAL-GEAR TOOTH PROPORTIONS A wide variety of tooth proportions have been used for helical gears. In spite of efforts to standardize helical-gear-tooth proportions, there is still no recognized standard. In this section we will consider typical kinds of helical-gear designs and discuss why each kind is used. In most cases, a gear maker will have gear-cutting tools, on hand, and so a new design is very often based on existing tools. Hobs can be used to cut different helix angles merely by changing settings and gear ratios in the hobbing machine. If the gear teeth are to be shaped, a shaper cutter is generally needed to each helix angle, and so spur-gear shaper cutters are not usable for cutting helical gears. The guides used on a gear shaping machine cut a constant lead rather than cutting a constant helix angle. As tooth numbers change, the helix angle that is produced by a given guide will change. For designer of helical gears generally needs certain kinds of tooth proportions to make the gear unit function properly, and this is usually the chief reason for picking certain proportions. Consideration of available cutting tools then becomes secondary. The designer, of course, would hope that a gear shop cutting the design would have tools on hand, but the most important thing would be to make a good helical gear design regardless of the availability of cutting tools. Table 6.16 shows five kinds of helical-gear teeth, with examples shown for 10°, 15°, 30°, and 35° helix angles. When hobs are available, the normal-section data of the helical gear must agree with those of the hob. After a helix angle is picked, the hob can be put on a hobbing machine to cut gears at this helix angle, and the transverse-section data are obtained. In a helical-gear tooth, the transverse-section data must correlate, of course, with the normal-section data, regardless of how the teeth are made. When the helical-gear teeth are to be shaped, the shaping-tool design is primarily based on the transverse section. The normal-section data for shaped teeth are calculated the same way as for hobbed teeth. In Table 6.16, the general-purpose design is typical of relatively standard 20°—pressure-angle gear hobs being used to cut helical gears. The practice of using a gear hob for helical gears is OK when the helical gears are small and narrow in face width. Large, wide-face helical gears need a special tapered hob (see Section 19.2). The spur-gear hob without taper would break down too quickly. The extra-load-capacity tooth proportions are based on 22.5° normal pressure angle and standard teeth. These teeth are stronger from a beam-strength standpoint and also from the standpoint of surface

288

Dudley’s Handbook of Practical Gear Design and Manufacture

TABLE 6.15 Internal-Gear Dimensions Item

Reference

Metric

English

1.

Center distance

Section 2.14

15.000

0.590550

2.

Circular pitch

(8) × (7) ÷ π

9.42478

0.371054

3. 4.

Pressure angle Working depth

Section 6.1.3 Section 6.1.11

25 5.51

25 0.217

5.

Module, transverse

(8) ÷ (7)

3

–

6.

Diametral pitch, transverse

(7) ÷ (8)

–

8.466667 Pinion Gear

7.

Number of teeth

Tables 6.13, 6.14

8. 9.

Pitch diameter Addendum

Section 1.14 Section 6.1.11, Table 6.13, Table 6.14

Pinion

Gear

18

28

18

28

54.00 3.39

84.00 2.12

2.12598 0.133

3.30708 0.083

10.

Whole depth

Section 6.1.11

7.05

6.26

0.2775

0.2465

11. 12.

Outside diameter Inside diameter

(8) + 2 × (9) (8) – 2 × (9)

60.78 –

– 79.76

2.392 –

– 3.141

13.

Root diameter

(12) + 2 (10), gear

46.68

92.28

1.837

3.634

14. 15.

Arc tooth thickness Chordal tooth thickness

Section 6.1.11 Equation (6.9), (6.10)

4.980 4.973

4.240 4.238

0.1960 0.1958

0.1670 0.1669

16.

Chordal addendum

Section 6.1.11

17. 18.

Pin size Diameter over pins

Table B.1 Table B.2

19.

Diameter between pins

Table B.6

20. 21.

Face width Form diameter

Section 5.2.2 Equations (9.28), (9.29)

22.

Roll angle, O.D.

Table B.7

23. 24.

Roll angle, F.D. Base circle diameter

Table B.7 (8) × cos (3)

25.

Minimum root radius

Section 9.1.11

26.

Accuracy

3.34

2.12

0.1315

0.083

5.3340 61.912

5.3340 –

0.2100 2.4375

0.2100 –

–

77.092

–

3.0351

27.0 50.035

25.0 90.536

1.06 1.970

1.00 3.564

42.195

17.904

42.195

17.904

12.185 48.941

36.877 76.130

12.185 1.9268

36.877 2.9972

0.90

0.77

0.035

0.030

See Note 3

Notes: 1. Metric dimensions are in millimeters, English dimensions are in inches, and angles are in degrees. 2. This sheet shows a standard center-distance design. 3. After the designer has done the load rating and studied Section 16.4.1, accuracy needs may be satisfied by an ISO or AGMA quality level, or the designer may need to write a special recipe for accuracy limits.

durability. The contact ratio is still relatively good, and they will tend to run quite smoothly, although not quite as smoothly as the general-purpose 20°—normal-pressure-angle design. The high-load-capacity helical-gear tooth, 25° normal pressure angle, tends to have a maximum loadcarrying capacity. These teeth do not run quite as smoothly as the designs just mentioned because of the lower contact ratio. However, if tooth profile modifications are properly made, they will run quite well at the design load (teeth modified for a heavy load may run roughly and noisily at light loads). In high-horsepower designs running at high speed, it is often desirable to use a special tooth design to get quiet operation. (A good example would be 10,000 kW going through a helical gearset at a pitch-line speed of 100 m/s or higher.) The four grouping in Table 6.16 shows examples of special helical-gear-tooth proportions for low noise.

2.000 2.000 2.200 2.200 2.200 2.200

16.1599

15.3275 17.2500

16.9384

15.2727 14.4817

2.000 2.000 2.000

25

18.2377 17.9105

2.000

2.000 2.000 2.000

22.50

22.50 22.50 2.000 2.000

2.000 2.000

20 22.50

25 25

2.000

20

25

2.000 2.000

Working Depth, h (hk )

2.575 2.575

2.575

2.400 2.575

2.400

2.400 2.400

2.250

2.250 2.250

2.250

2.300 2.300

2.300

2.350 2.300

2.350

2.350 2.350

Whole Depth, h (h t )

Normal Section Data

20 20

Pressure Angle, n ( n)

0.375 0.375

0.375

0.400 0.375

0.400

0.400 0.400

0.300

0.300 0.300

0.300

0.325 0.325

0.325

0.350 0.325

0.350

0.350 0.350

Edge Radius, r a0 (rT )

30 35

15

35 10

30

10 15

35

15 30

10

30 35

15

35 10

30

10 15

Helix Angle, ( )

17.5000 17.5000

17.5000

18.5000 17.5000

18.5000

18.5000 18.5000

29.6510

25.7693 28.3000

25.3376

25.5614 26.8240

23.2109

23.9568 22.8118

22.7959

20.2836 20.6469

Pressure Angle, t ( t)

3.6276 3.8352

3.2524

3.8352 3.1901

3.6276

3.1901 3.2524

3.8352

3.2524 3.6276

3.1901

3.6276 3.8352

3.2524

3.8352 3.1901

3.6276

3.1901 3.2524

Circular Pitch, p t ( pt )

Transverse Section Data

6.283 5.477

12.138

5.477 18.092

6.283

18.092 12.138

5.477

12.138 6.283

18.092

6.283 5.477

12.138

5.477 18.092

6.283

18.092 12.138

Axial Pitch, px ( px )

1.154700 1.220776

1.035276

1.220776 1.015427

1.154700

1.015427 1.035276

1.220776

1.035276 1.154700

1.015427

1.154700 1.220776

1.035276

1.220776 1.015427

1.154700

1.015427 1.035276

Module, Transverse mt

Pitch

0.866025 0.819152

0.965926

0.819152 0.984808

0.866025

0.984808 0.965926

0.819152

0.965926 0.866025

0.984808

0.866025 0.819152

0.965926

0.819152 0.984808

0.866025

0.984808 0.965926

Diametral Pitch, Pt

Notes: All angles are in degrees. For the metric system, all dimensions are in millimeters. Multiply by normal module for dimensions of other than 1 module. For English system, all dimensions are in inches. Divide by normal diametral pitch for dimensions of other than 1 normal diametral pitch. Module in normal section is 1.000 for metric system. Diametral pitch is 1.000 in normal section for English system.

Special for very low noise

Special for low noise

High load capacity

Extra load capacity

General purpose

Typical Use

TABLE 6.16 Helical-Gear Basic Tooth Data

Design Formulas 289

290

Dudley’s Handbook of Practical Gear Design and Manufacture

In a few cases the helical-gear application may be so critical that a very special design is justified. The fifth grouping in Table 6.16 shows some examples in which a relatively low-pressure-angle is used and the teeth are made deeper than standard depth (the standard working depth for spur or helical gears is usually thought of as 2.000 times the module in the normal section). Helical-gear teeth made to the fifth grouping will not have maximum load-carrying capacity, but they will still have relatively good load-carrying capacity. The design, of course, is biased toward low noise rather than toward maximum torque capacity for the size of the gears. The designs at 10° and 15° helix angles are preferred for single helical gears because the thrust load is relatively low. It is usually possible to use a face width that is at least as much as the axial pitch. (To get the full benefit of helical tooth action, the face width should equal at least two axial pitches.) When the gears are made double helical, there is no thrust load, because the thrust of one helix is opposed by an equal and opposite thrust from the other helix. Double-helix gears should be made with at least 30° helix angle. At helix angles this high, it is possible to get several axial pitches within the face width of each helix. This is beneficial for quiet running. The full benefit of helical-tooth action can be obtained quite readily with double-helix gears. Note the double helical design illustrated in Figure 6.16. There are over five axial pitches per helix. In a double-helix-gear design, the usual design practice is to position the gear axially with thrust bearings, then permit the pinion to float axially so that the load is divided evenly between the two helices. There may be resistance to axial float resulting from friction in coupling devices. If there is enough

FIGURE 6.16 Helical gearset used to drive a 1750-kW generator at 1200 rpm. Turbine speed is 10,638 rpm. Source: (Courtesy of Transamerica DeLaval, Trenton, NJ, U.S.A.)

Design Formulas

291

resistance to axial movement, the division of load in a double helical gear may be impaired. A 30° helix angle, or higher, is needed in the double helical design to ensure that resistance to axial movement from coupling can be quite readily overcome (it would be a design mistake—in most cases—to use 15° helix angle for a double helical set). The relation between the normal and transverse pressure angles of helical gears is given by the following equation: tan

tan

t

=

t

=

tan cos tan cos

n

metric

(6.32)

n

English

(6.33)

6.1.13 HELICAL-GEAR DIMENSION SHEET Table 6.17 shows a helical-gear dimension sheet. All the tooth dimensions that will ordinarily be needed are shown on this sheet. In some low-cost applications, items like roll angles, form diameter, and axial pitch may not be required. The face-width dimensions should be based on the load-carrying requirements. Also, it should be remembered that the wider the face width, the harder it is to secure accuracy enough to make the whole face width carry load uniformly. In addition to these considerations, the designer should consider the fact that a certain amount of face width is required to get the benefit of helical-gear action. General ex­ perience indicates that it takes at least two axial pitches of face width to get full benefit from the overlapping action of helical teeth. In critical high-speed gears, where noise is the problem, the designer should aim to get four or more axial pitches in the face width. If the face width is less than one axial pitch, the tooth action will be in between that of a spur and a true helical gear. Helical pinions should have enough addendum to avoid undercut. Equations (6.3) and (6.4) can be used to calculate the number of teeth at which undercut just starts on a helical pinion. Two pressure angles have to be used, though, to get the right answer. In calculating the hf x dimension, the normal pressure angle n is used. In calculating z , the transverse pressure angle t is used. Helical pinions can go to lower numbers of teeth without undercut than spur pinions. It is seldom that helical gearing requires long and short addendum to avoid undercut. In general, there is not so much need to use long and short addendum to balance strength between pinion and gear in helical gearing as in spur gearing. Low-hardness helical gearing usually has strength to spare and is in no danger of scoring trouble. Wear in the form of pitting is the main thing that limits the design. In this situation, a long-addendum pinion meshing with a short-addendum gear offers only a slight advantage over equal addendum on both members. In high-hardness helical gearing where the pinion has a low number of teeth and the ratio is high, it may be desirable to use about the same ratio between pinion addendum and gear addendum as would be used for spur gears. In this case, Figure 6.5 may be used as a guide. The minimum root fillet radius of a helical gear is not usually specified on the drawing. This is a hard dimension to check because you cannot get a good projection view of the fillet without cutting out and mounting a section4 of the gear. If beam strength is not critical, then it is not necessary to hold a close control over root-fillet curvature. It is good judgment, though, to require that all cutting tools have as much radius as possible.

4

The section cut is normal to the helical tooth. The minimum radius specified is, of course, in the normal section.

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Dudley’s Handbook of Practical Gear Design and Manufacture

TABLE 6.17 Helical-Gear Dimensions Item

Reference

Metric

1.

Center distance

Sections 5.2.2, 5.2.4

2.

Normal circular pitch

Equation (2.11)

3. 4.

Normal pressure angle Working depth

Section 6.1.12, Table 6.16 Table 6.16

5.

Helix angle

Section 6.1.12

6. 7.

Transverse pressure angle Module, normal

Equation (6.32), (6.33) Equation (2.12)

8.

Normal diametral pitch

Equation (2.13)

9.

Number of teeth

English

187.9026

7.39774

9.4248

0.371054

20 6.00

20 0.236

15

15

20.64689 3

20.64689 –

–

8.466667

Section 6.1.13

Pinion 25

Gear 96

Pinion 25

Gear 96

77.6457

298.1595

3.05692

11.73856

3.00 7.05

3.00 7.05

0.118 0.2775

0.118 0.2775

10.

Pitch diameter

Equations (2.7), (2.8)

11. 12.

Addendum Whole depth

Section 6.1.13 Table 6.16

13.

Outside diameter

(10) + 2 × (11)

83.65

304.16

3.293

11.974

14. 15.

Root diameter Normal arc tooth thickness

(13) – 2 × (12) Section 6.1.6

69.55 4.617

290.06 4.617

2.738 0.1818

11.419 0.1818

16.

Chordal tooth thickness

Equations (6.9), (6.10)

4.594

4.609

0.1809

0.1815

17. 18.

Chordal addendum Ball size

Equations (6.7), (6.8) Table A.1

3.047 5.33401

3.014 5.33401

0.1200 0.210

0.1185 0.210

19.

Diameter over balls

Table A.9

20. 21.

Face width Form diameter

Sections 6.1.12, 6.1.13 Table A.3

78 73.393

77 293.338

3.07 2.8894

3.03 11.5487

22.

Roll angle, O.D.

Table A.3

32.68

24.87

3268

24.87

23. 24.

Roll angle, F.D. Base circle diameter

Table A.3 (10) × cos (6)

8.17 72.6586

18.60 279.0091

8.17 2.86057

18.60 10.98458

25.

Hand of helix

Section 2.13

26. 27.

Lead Minimum root radius

(2) × (9) ÷ sin (5) Section 6.1.13

28.

Accuracy

Section 16.4.1

RH

LH

RH

LH

910.364 0.89

3495.796 0.78

35.8410 0.035

137.625 0.030

See Note 3

Notes: 1. Metric dimensions are in millimeters, English dimensions are in inches, and angles are in degrees. 2. This sheet shows a standard center-distance design. See Section 17.1 for an example of a special “spread center” design. 3. After the designer has done the load rating and studied Section 15.4.1, accuracy needs may be satisfied by an ISO or AGMA quality level, or the designer may need to write a special recipe for accuracy limits.

The equations for the minimum radius of curvature in the root fillet of a helical gear is: =

f

where hf = h

r a20

(h f

r a 0 )2

dp /(2 cos2 ) + (hf

r a0 )

metric

(6.34)

English

(6.35)

ha , and r a0 is edge radius of generating rack;

f

=

r T2 (b rT )2 d /(2 cos2 ) + (b rT )

Design Formulas

293

where b = ht − a, and rT is edge radius of generating rack. Although Eqs. (6.34) and (6.35) are based on the generating action of a rack, the designer can use it for any type of helical-gear manufacture on the assumption that other methods of tooth cutting can meet the same minimum radius value. The shape of helical teeth in the normal section is almost exactly the same as that of a spur gear with a larger number of teeth. For instance, the 25-tooth helical pinion on a 77.6457-mm pitch diameter shown on the sample dimension sheet would be matched in tooth contour by a spur pinion of 27.74 teeth on an 83.220-mm pitch diameter with a 20° pressure angle. The matching number of teeth in a spur gear is called the virtual number of teeth. The virtual number of spur teeth is equal to the number of helical teeth divided by the cube of the cosine of the helix angle.

6.1.14 BEVEL-GEAR TOOTH PROPORTIONS The proportions of straight bevel, Zerol5 bevel, and spiral bevel gears can quite logically be considered together. The Gleason Works of Rochester, N.Y., has done an excellent job of standardizing the designs of these kinds of gears. Their work is generally accepted by the gear industry. The material in this section and the next three sections is taken from the revised editions of Gleason bevel-gear systems for straight, spiral, and Zerol bevel gears. The pressure angles and tooth depths generally recommended for bevel gears are given in Table 6.18. The whole depth specified is sometimes slightly exceeded by the practice of rough-cutting some pitches a small amount deeper than the calculated depth to save wear on the finishing cutters. The various systems have the amount of addendum for the gear and the pinion worked out so as to avoid undercut with low number of teeth and balance the strength of gear and pinion teeth. In each case, though, there is a limit to how far the system will go. Table 6.19 shows the minimum numbers or teeth that can be used in different combinations. In spur- and helical-gear work, the amount of addendum for the pinion is first determined. Then what is left of the working depth is used for the gear addendum. In bevel-gear practice, just the opposite procedure is used. The gear addendum is determined first. Table 6.20 shows the amount of gear addendum recommended for bevel gears. Since bevel gears and bevel pinions usually have different addendums, it is necessary to use different tooth thicknesses for the two members. The standard systems adjust the tooth thicknesses so that ap­ proximately equal strength is obtained for each member. The adjustment of the tooth thickness is ac­ complished by a factor called k . This k should not be confused with the K factor used as an index of tooth load. The k values can be read from Figures 6.17 to 6.20. The thicknesses of bevel-gear teeth are calculated for the large ends of the teeth. It is customary to calculate the circular thicknesses of the teeth without allowing any backlash. In straight bevel gearing, chordal tooth thicknesses are calculated with backlash. The chordal thicknesses of a straight-bevel-gear tooth can be measured with tooth calipers. The formula for the gear circular tooth thickness of any kind of bevel gear is:

5

s2 =

pt 2

(ha1

ha 2 )

tG =

pt 2

(aP

aG )

tan cos

tan cos

t

t

k mt

k Pd

metric

(6.36)

English

(6.37)

Trademark registered in the U.S. Patent Office by the Gleason Works of Rochester, NY, U.S.A.

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Dudley’s Handbook of Practical Gear Design and Manufacture

TABLE 6.18 Bevel-Gear Proportions Kind of Bevel Gear

Size Range Module, m t Diametral Pitch, Pt

Pressure Angle, (or )

Working Depth, h (or h k )

Straight

25.4–0.40

1–64

20°

2.000

Spiral

25.4–1.27

1–20

20°

1.700

25.4–1.27

1–20

16°

1.700

5.00–0.40 5.00–0.80

5–64 5–32

20° 22.5°

2.000 2.000

5.00–1.27

5–20

25°

2.000

Zerol

Whole Depth, h (or h t )

2.188 +

0.005 mt

or 0.002Pt

1.888 1.888

2.188 +

0.05 mt

or 0.002Pt

Notes: 1. For metric design, the values are multiplied by the module to get the working depth and the whole depth. 2. For English design, the values are divided by the diametral pitch to get the whole working depth and the whole depth. 3. To protect fine pitches, a constant value is added to the whole depth. This is done for all pitches, but it is not very significant on large pitches. The value is 0.05 mm or 0.002 in. 4. Examples for straight bevel: 5 module

1 module

Working depth

10.00 mm

2.000 mm

Whole depth

10.99 mm

2.398 mm

TABLE 6.19 Minimum Tooth Numbers for Bevel Gears Kind of Gear

Pressure Angle

Number of Pinion Teeth

Min. Number of Gear Teeth

Straight bevel

20°

13

31

14 15

20 17

16

16

12 13

26 22

14

20

15 16

19 18

17

17

15 16

25 20

Spiral bevel

Zerol bevel

20°

20°

17

17

22.5°

13 14

15 14

25°

13

14

13

13

Design Formulas

295

TABLE 6.20 Gear Addendum for Bevel Gears Kind of Bevel Gear

Metric

English

Straight or Zerol

0.540m t +

0.460m t u2

0.540 Pd

+

0.460 2 PdmG

Spiral bevel

0.540m t +

0.390m t u2

0.540 Pd

+

0.390 2 PdmG

Note: The pinion addendum is obtained by subtracting the gear addendum from the working depth.

FIGURE 6.17

Circular thickness factor for straight bevel gears with 20° pressure angle.

where k is the circular tooth thickness factor given by the appropriate curve for straight, Zerol, or spiral bevel gears. See Figures 6.17 to 6.20. The recommended amount of backlash for bevel gearsets when they are assembled ready to run is given in Table 6.21. In instrument and control gearing, it may be desirable to use values even lower than those shown. Conversely, some high-speed gears and gears mounted in casing with a material different from that of the gear may require more backlash.

6.1.15 STRAIGHT-BEVEL-GEAR DIMENSION SHEET A dimension sheet for calculating straight-bevel-gear-tooth data is given in Table 6.22. This sheet gives several formulas not previously discussed. These can be tried out quite readily by working through numerical calculations for the sample design that is shown in the table.

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Dudley’s Handbook of Practical Gear Design and Manufacture

FIGURE 6.18 Circular thickness factor for spiral bevel gears with 20° pressure angle and 35° spiral angle. LH pinion driving clockwise or RH pinion driving counterclockwise.

The backlash allowance in item (20) of the table is taken from Table 6.21. The designer should consult the text in Section 6.1.6 and then decide whether or not special requirements of the application might take it desirable to depart from standard backlash. The corrections for chordal tooth thickness and chordal addendum are made by these equations: Chordal tooth thickness s¯2 = s2

(s2 )3 6 (d p 2 ) 2

tcG = tG

(tG )3 6 D2

s¯ 1 = s 1

(s 1)3 6 (dp1)2

j 2

B 2 j 2

metric

(6.38)

English

(6.39)

metric

(6.40)

Design Formulas

297

FIGURE 6.19 Circular thickness factor for Zerol bevel gears with 20° pressure angle operating at 90° shaft angle.

FIGURE 6.20 Circular thickness factor for Zerol bevel gears with 25° pressure angle operating at 90° shaft angle.

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Dudley’s Handbook of Practical Gear Design and Manufacture

TABLE 6.21 Nominal Backlash for Bevel Gears at Tightest Point of Mesh Tooth Size Range Module, mt

Backlash for Low Accuracy

Backlash for High Accuracy

Diametral Pitch, Pd

Millimeters

Inches

Millimeters

Inches

25–20 20–17

1.00–1.25 1.25–1.50

1.14–1.65 0.89–1.40

0.045–0.065 0.035–0.055

0.51–0.76 0.46–0.66

0.020–0.030 0.018–0.026

17–14

1.50–1.75

0.63– .10

0.025–0.045

0.41–0.56

0.016–0.022

14–12 12–10

1.75–2.00 2.00–2.50

0.51–1.00 0.46–0.76

0.020–0.040 0.018–0.030

0.36–0.46 0.30–0.41

0.014–0.018 0.012–0.016

10–8.5

2.50–3.00

0.38–0.63

0.015–0.025

0.25–0.33

0.010–0.013

8.5–7.2 7.2–6.3

3.00–3.50 3.50–4.00

0.30–0.56 0.25–0.51

0.012–0.022 0.010–0.020

0.20–0.28 0.18–0.22

0.008–0.011 0.007–0.009

6.3–5.1

4.00–5.00

0.20–0.41

0.008–0.016

0.15–0.20

0.006–0.008

5.1–4.2 4.2–3.1

5.00–6.00 6.00–8.00

0.15–0.33 0.13–0.25

0.006–0.013 0.005–0.010

0.13–0.18 0.10–0.15

0.005–0.007 0.004–0.006

3.1–2.5

8.00–10.00

0.10–0.20

0.004–0.008

0.076–0.127

0.003–0.005

2.6–1.6 1.6–1.2

10.00–16.00 16.00–20.00

0.076–0.127 0.051–0.102

0.003–0.005 0.002–0.004

0.051–0.102 0.025–0.076

0.002–0.004 0.001–0.003

tcP = tP

(tP )3 6 d2

B 2

English

(6.41)

Chordal addendum (s )2 cos h¯a2 = ha2 + 2 4 d p2 acG = aG +

(tG )2 cos 4D

(s )2 cos h¯a1 = ha1 + 1 4 dp1 acP = aP +

2

(tP )2 cos 4d

1

metric

(6.42)

English

(6.43)

metric

(6.44)

English

(6.45)

where backlash (in millimeters for metric) is designated as j, and backlash (in inches for English) is denoted be B. The tooth angle is a machine setting used on a Gleason two-tool straight-bevel-gear generator. See Table 6.2 for the definition of this angle. The limit-point width is the maximum width of the point of a straight-sided V-tool that will touch the sides and bottom of the tooth at the small end. The tool actually used must not be larger than this dimension, but it may be a small amount less.

Design Formulas

299

TABLE 6.22 Straight-Bevel-Gear Dimensions Item

Reference

Metric

English 0.61842

1.

Circular pitch

(8) × π ÷ (7)

15.7079

2.

Pressure angle

Section 2.15

20

20

3. 4.

Working depth Shaft angle

Table 6.18 Assumed to be 90°

20.00 90

0.3937 90

5.

Module

(8) ÷ (7)

5.000

–

6.

Diametral pitch

(7) ÷ (8)

–

5.080

7.

Number of teeth

Table 6.19

8. 9.

Pitch diameter Pitch angle

10. 11. 12.

Pinion

Gear

Pinion

16

49

16

Gear 49

Section 5.2.5 Section 5.15, Table 6.23

80.00 18.08

245.00 71.92

3.1496 18.08

9.6457 71.92

Outer cone distance

Table 6.23

128.87

128.87

5.074

5.074

Face width Addendum

Section 5.2.5, 6.1.15 Table 6.20

40.00 7.05

40.00 2.95

1.575 0.278

1.575 0.116

13.

Whole depth

Table 6.18

10.99

10.99

0.433

0.433

14. 15.

Dedendum angle Face angle

Table 6.23 Table 6.23

1.73 21.63

3.55 73.65

1.73 21.63

3.55 73.65

16.

Root angle

Table 6.23

16.35

68.37

16.35

68.37

17. 18.

Outside diameter Pitch apex to crown

Table 6.23 Table 6.23

93.41 120.31

246.83 37.20

3.678 4.737

9.718 1.465

19.

Circular tooth thickness

Equations (6.36), (6.37)

9.514

6.194

0.3745

0.2438

20. 21.

Backlash Chordal tooth thickness

Table 6.21, Section 6.1.6 Equations (6.38), (6.41)

0.13–0.18 9.43

0.13–0.18 6.13

0.005–0.007 0.3712

0.005–0.007 0.2413

22.

Chordal addendum

Equations (6.42) through (6.45)

7.32

2.96

0.2882

0.1165

23. 245.

Tooth angle Limit point width: large end

Table 6.23 Section 6.1.15, Table A.12

2.75 3.30

2.67 3.63

2.75 0.130

2.67 0.143

25.

Limit point width: small end

Section 6.1.15, Table A.12

2.31

2.54

0.091

0.100

26. 27.

Tool point width Tool edge radius

Section 6.1.15, Table A.12 Section 6.1.15, Table A.12

1.65 0.63

1.78 0.63

0.065 0.025

0.070 0.025

Note: Metric dimensions are in millimeters, English dimensions are in inches, and angles are in degrees.

The tool advance corresponds to the 0.05 mm or 0.002 in. in the whole depth formula (see Table 6.18). This dimension sets the tool deeper and increases the clearance along the length of the tooth. Table 6.23 gives the calculations for bevel gears mounted on a 90 shaft angle only. If bevel gears are to be mounted on some other angle, special calculations have to be made. See Table 6.24 for a complete dimension sheet for the bevel gears. The face width of straight bevel gears should not exceed 32% of the outer cone distance nor exceed 3.18 times the circular pitch. These rules may be rounded off to 30% of efface width and three times the circular pitch for convenience. If either rule is exceeded by much, there is apt to be trouble in making and fitting the teeth properly. Real load-carrying capacity will probably be lost as a result of poor running fits when bevel teeth gears are made wider in face width than the rules just mentioned will allow.

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Dudley’s Handbook of Practical Gear Design and Manufacture

TABLE 6.23 Calculation of Bevel-Gear Body Dimensions

1.

No. pinion teeth

16

22.

Pinion RA, (10) – (16)

16.3322

2.

No. gear teeth

19

23.

Gear RA, (11) – (19)

68.3464

3. 4.

Pinion, pitch diameter Gear, pitch diameter

80.00 245.00

24. 25.

2.0 (5) × cos (10) Pinion OD, (3) + (24)

13.4035 93.4035

5.

Pinion addendum

7.05

26.

2.0 (6) × cos (11)

6. 7.

Gear addendum Working depth

2.95 10.00

27. 28.

Gear OD, (4) + (26) 0.50 × (4) – (5) sin (10)

8.

Whole depth

9. 10.

(1) ÷ (2) Pinion PA, arctan (9)

10.99

29.

Pinion pitch apex to crown

0.326531 18.0834

30. 31.

0.50 × (3) – (6) sin (11) Gear pitch apex to crown

11.

Gear PA, 90° – (10)

71.9165

12. 13.

2 sin (10) Outer cone distance, (3) ÷ (12)

0.620804 128.8652

32. 33.

Pinion circular thickness Pressure angle

14.

(8) – (5)

15. 16.

(14) ÷ (13) Pinion ded. angle, arctan (15)

17.

(8) – (6)

18. 19.

(17) ÷ (13) Gear ded. angle, arctan (18)

20.

Pinion FA, (19) + (10)

21.6535

21.

Gear FA, (16) + (11)

73.6677

1.8314 246.8314 120.3117 (28) 37.1957 (30)

For straight bevels only: 9.514 20°

3.94

34.

0.50 × (32) + (14) tan (33)

6.19104

0.030575 1.7512

35. 36.

57.30 ÷ (13) Pinion tooth angle, (34) × (35)

0.44465 2.7528

8.04

37.

Gear circular thickness

0.062391 3.5701

38. 39.

0.50 × (37) + (17) tan (33) Gear tooth angle, (35) × (38)

6.194 6.02332 2.6782

Notes: 1. These calculations are for straight, spiral, or Zerol bevel gears on a 90° shaft angle. 2. The abbreviations used are: PA = pitch angle; FA = face angle; RA = root angle; OD = outside diameter.

6.1.16 SPIRAL-BEVEL-GEAR DIMENSION SHEET Table 6.24 shows the relatively simple design sheet that is used for spiral bevel gears. Control of the spiral bevel gear’s geometry is obtained by controlling the machine settings of the machine used to make the gears and by testing each gear with standard test gears.

Design Formulas

301

TABLE 6.24 Spiral-Bevel-Gear Dimensions Item

Reference

Metric

English

1.

Circular pitch

(9) × π ÷ (8)

15.7079

0.61842

2.

Pressure angle

Section 1.17

20

20

3. 4.

Spiral angle Working depth

Section 5.1.16 Tables 5.18

35 8.30

35 0.327

90

90

5.00 –

– 5.080

5.

Shaft angle

Assumed to be 90°

6. 7.

Module Diametral pitch

(9) ÷ (8) (8) ÷ (9)

8. 9.

Number of teeth Pitch diameter

Table 4.19 Section 5.2.5

10.

Pitch angle

Table 6.23

11. 12.

Outer cone distance Face width

Table 6.23 Section 6.1.16

13.

Addendum

Table 6.20

14. 15.

Whole depth Dedendum angle

Table 6.18 Table 6.23

16.

Face angle

17. 18.

Root angle Outside diameter

19.

Pitch apex to crown

Table 6.23

20. 21.

Mean circular tooth thickness Backlash

Section 6.1.16 Section 6.1.16, Table 6.21

22.

Hand of spiral

One LH, other RH

LH

RH

LH

RH

23. 24.

Function Direction of rotation

Design choice Design choice

Driver CCW

Driven CW

Driver CCW

Driven CW

25.

Cutter diameter

Table A.13

190.5

190.5

7.50

7.50

26.

Cutter edge radius

Table A.13

0.63

0.63

0.025

0.025

Pinion

Gear

Pinion

Gear

16 80.00

49 245.00

16 3.1496

49 9.6457

18.08

71.92

18.08

71.92

128.87 38

128.87 38

5.074 1.496

5.074 1.496

5.91

2.38

0.233

0.094

9.22 1.13

9.22 2.83

0.363 1.13

0.363 2.83

Table 6.23

20.92

73.05

20.92

73.05

Table 6.23 Table 6.23

16.95 91.26

69.08 246.48

16.95 3.593

69.08 9.704

120.67

37.74

4.751

1.485

8.13 0.13–0.18

5.04 0.13–0.18

0.320 0.005–9.007

1.1984 0.005–9.007

Note: 1. Metric dimensions are in millimeters, English dimensions are in inches, and angles are in degrees.

Table 6.21 is issued as a design guide for backlash of spiral bevel gears. In general, all the backlash is subtracted from the pinion thickness. Table 6.24 should be used only for spiral bevel gears on 90° shaft angle. The standard spiral angle is 35°. The mean circular thickness shown in item (20) is the tooth thickness at the midsection (midway from the face width). The dimensions in the midsection are all smaller (due to the conical shape of a bevel gear) than they are at the large end. The relation of midsection to large end is Midsection 2.0 × outer cone distance face width = Large end 2.0 × outer cone distance

(6.46)

If the face width is 30% of the cone distance, this ratio will come out to be 0.850. This means the mean tooth thicknesses will be about 85% of the thicknesses at the large end.

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The tooth thicknesses shown in Table 6.24 for the 16/49 spiral bevel set have been reduced from the theoretical to allow for a minimum backlash for precision spiral bevel teeth. The sum of the tooth thicknesses is less than the mean circular pitch by the backlash. (The mean circular pitch for the example is 13.2700 mm or 0.5224 in.) With somewhat more complex calculations—handled easily in a large computer program—the data shown in Tables 6.22 and 6.25 can be modified to give best results for the planned method of manu­ facture. A slightly tilted root-line taper, for instance, is often used to allow for maximum tooling-point widths. This achieves the best tool life and the largest fillet radii (for decreased bending stresses). These dimension sheets show considerably more data than the basic engineering data. Small varia­ tions to favor a particular job can be made very readily. Spiral bevel gears should not have a face width exceeding 30% of the outer cone distance.

6.1.17 ZEROL-BEVEL-GEAR DIMENSION SHEET The Zerol bevel gear has curved teeth like the spiral bevel gear, but its spiral angle is 0 . The dimension sheet (see Table 6.25) is similar in form to the one used for spiral bevel gears, but the dimensions have numerical values more like those used for straight bevel teeth.

TABLE 6.25 Zerol-Bevel-Gear Dimensions Item

Reference

Metric

English

1.

Circular pitch

(8) × π ÷ (7)

7.8540

0.30921

2. 3.

Pressure angle Working depth

Section 2.16 Table 6.18

20 5.00

20 0.197

4.

Shaft angle

Assumed to be 90°

90

90

5. 6.

Module Diametral pitch

(8) ÷ (7) (7) ÷ (8)

2.500 –

– 10.160

7. 8.

Number of teeth Pitch diameter

Table 6.19 Section 5.2.5

Pinion

Gear

Pinion

Gear

32 80

98 245.00

32 3.1496

98 9.6456

9.

Pitch angle

Table 6.23

18.08

71.92

18.08

71.92

10. 11.

Outer cone distance Face width

Table 6.23 Section 6.1.17

128.87 32

128.87 32

5.074 1.260

5.074 1.260

12.

Addendum

Table 6.20

3.53

1.47

1.1390

0.0579

13. 14.

Whole depth Dedendum angle

Table 6.18 Table 6.23

5.47 1.37

5.47 3.28

0.2154 1.37

0.2154 3.28

15.

Face angle

Table 6.23

21.37

73.28

21.37

73.28

16. 17.

Root angle Outside diameter

Table 6.23 Table 6.23

16.72 86.71

68.63 245.91

16.72 3.414

68.63 9.681

121.41

38.60

4.780

1.520

3.92 0.05–0.10

2.86 0.05–0.10

0.1543 0.002–004

0.1126 0.002–004

152.4

152.4

6.00

6.00

18.

Pitch apex to crown

Table 6.23

19. 20.

Mean circular tooth thickness Backlash

Section 6.1.17 Section 6.1.17, Tables 5.21

21.

Cutter diameter

Table A.14

Note: 1. Metric dimensions are in millimeters, English dimensions are in inches, and angles are in degrees.

Design Formulas

303

The mean tooth-thickness values shown in Table 6.25 are similar to those for spiral bevel gears. They are for the middle of the face width, and they have been reduced to allow for an amount of backlash for precision gears. The dedendum angle for Zerol bevel gears has a small amount added to the angle that would be obtained using the method shown in Table 6.23 for bevel-gear-body dimensions. This added amount, in degrees, can be obtained from the following: dedendum angle = A

B

C

(6.47)

where A is:

Pressure Angle

Value of A Metric

English

20° 22.5°

111.13 ÷ z 81.13 ÷ z

111.13 ÷ Nc 81.13 ÷ Nc

25°

56.87 ÷ z

56.87 ÷ Nc

Number of crown teeth: z a = 2.0 (Ra ÷ m t )

Nc = 2.0Pd Ao

metric

English

(6.48) (6.49)

The value B is the same regardless of pressure angle: B=

25.2

B=

dp1 sin

2

zb 5

d sin Nc F

metric

English

(6.50)

(6.51)

The value C is also the same regardless of pressure angle: C = 5.918 ÷ (z × m t )

metric

(6.52)

C = (0.2333 Pd ) ÷ Nc

English

(6.53)

Zerol bevel gears are quite frequently used in critical instrument work. For this kind of work, it is frequently necessary to use almost no backlash. If the gears are to be used for general-purpose work, Table 6.21 may be used as a design guide for backlash. The dedendum angles shown in Table 6.25 are made to suit the Duplex method of cutting Zerol bevel gears. This is a rapid method of cutting, which allows both pinion and gear to be cut spread-blade (both sides of a tooth space are finished simultaneously). Present machine capacity limits the use of Duplex method to cut gears approximately 2.5 module (10 pitch) and finer and to ground gears approximately 4 module (6 pitch) and finer. The minimum number of pinion teeth is 13.

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Dudley’s Handbook of Practical Gear Design and Manufacture

Table 6.25 is for use only when the shaft angle is 90°. Angular Zerol bevel gears require special calculations. The face width of Zerol bevel gears should not exceed 25% of the outer cone distance.

6.1.18 HYPOID-GEAR CALCULATIONS The hypoid gear quite closely resembles a spiral bevel gear in appearance. The major difference is that the pinion axis is offset above or below the gear axis. In the regular bevel-gear family, the axis of the pinion and that of the gear always intersect. The hypoid type of gear does not have intersecting axes. Since this kind of gear is basically different from the bevel gear, hypoid gears are not ordinarily called hypoid bevel gears. Instead, they are just called hypoid gears. From a manufacturing standpoint, though, they are cut or ground with the same kinds of machinery that are used to make spiral bevel gears. Hypoid gears can be more readily designed for low numbers of pinion teeth and high ratios than can spiral bevel gears. The Gleason Works recommends the following minimum numbers of teeth for Formate6 pinions:

Ratio

Minimum Number of Pinion Teeth

2½

15

3

12

4 5

9 7

6

6

10

5

The calculations involved in designing a set of hypoid gears are quite long. A calculation sheet of about 150 items must be calculated to get all the answers needed. About 45 items on the sheet must be worked through on a trial basis, then repeated two or three times until assumed and calculated values check. With the advent of computers, hypoid-gear calculations became much easier, since they could be programmed. In view of the large amount of material needed to explain hypoid-gear calculations, it is not possible to present the method here.

6.1.19 FACE-GEAR CALCULATIONS The pinion of a face gearset can be designed in just the same manner as a spur or helical pinion would be designed. The pinion should have enough addendum to avoid being undercut. Equations (6.3) and (6.4) may be used to check a design to see if it is in danger of undercutting. The meshing action of the face gear with the pinion is somewhat similar to that of a rack meshing with the pinion. The involute profile of the pinion should be finished accurately to a deep enough depth to permit the pinion to mesh with a rack. One of the principal design problems in face gears is to calculate the face width. Since the face-gear teeth run across the end of a cylindrical blank instead of across the outside diameter, the length of the tooth sets an inside diameter and an outside diameter. The face-gear tooth changes its shape as you move lengthwise along the tooth. The minimum inside diameter is determined by the point at which the un­ dercut portion of the gear profile extends to about the middle of the tooth height. The maximum outside

6

Formate pinions are pinions that are matched to run with formate gears. Formate gears are cut without any generating action.

Design Formulas

305

diameter is established by the point at which the top land of the tooth narrows to a knife-edge. In general, it is good design practice to make the face width of the gear somewhat shorter than these two extremes. Table 6.26 shows some recommended proportions for spur face gears of 20° pressure angle. A ratio of less than 1.5 is not generally recommended. The diameter constants are used to calculate the limiting outside and inside diameters of the gear as follows: da 2 =

o z2 m t

metric

(6.54)

English

(6.55)

metric

(6.56)

English

(6.57)

where εo = m0; Do = di 2 =

mo NG Pd 1 z2 m t

where ε1 = mi; Di =

m i NG Pd

TABLE 6.26 Tooth Proportions and Diameter Constants for Face Gears, 20° Pressure Angle No. of Pinion Teeth z 1 (NP )

Pinion Addendum h a1 (aP )

Pinion Tooth Thickness s 1 (tP )

Gear Addendum h a2 (aG )

Gear Diameter Constants for Gear Ratios

o

Ratio = 1.5 (mo ) i (m i )

o

Ratio = 2.0 (mo ) i (m i )

o

Ratio = 4.0 (mo ) i (m i )

o

Ratio = 8.0 (mo ) i (m i )

12

1.120

1.790

0.700

…

…

1.221

1.020

1.221

0.960

1.221

0.945

13 14

1.100 1.080

1.745 1.700

0.760 0.820

1.202 1.187

1.064 1.062

1.202 1.187

1.015 1.011

1.202 1.187

0.959 0.958

1.202 1.187

0.945 0.944

15

1.060

1.660

0.880

1.174

1.052

1.174

1.007

1.174

0.957

1.174

0.944

16 17

1.040 1.020

1.620 1.580

0.940 0.980

1.161 1.156

1.051 1.041

1.161 1.156

1.004 1.000

1.161 1.156

0.956 0.955

1.161 1.156

0.944 0.944

18

1.000

1.570

1.000

1.150

1.039

1.150

0.997

1.150

0.954

1.150

0.943

20

1.250 1.000

1.752 1.570

0.750 1.000

1.176 1.144

1.042 1.030

1.176 1.144

0.999 0.991

1.176 1.144

0.955 0.953

1.176 1.144

0.943 0.943

1.250

1.752

0.750

1.166

1.032

1.166

0.993

1.166

0.953

1.166

0.943

22

1.000 1.200

1.570 1.715

1.000 0.800

1.140 1.156

1.022 1.024

1.140 1.156

0.987 0.988

1.140 1.156

0.952 0.952

1.140 1.156

0.943 0.943

24

1.000

1.570

1.000

1.133

1.015

1.133

0.983

1.133

0.951

1.133

0.942

30

1.200 1.000

1.715 1.570

0.800 1.000

1.150 1.121

1.017 1.001

1.150 1.121

0.984 0.975

1.150 1.121

0.951 0.949

1.150 1.121

0.943 0.942

1.150

1.680

0.850

1.131

1.001

1.131

0.975

1.131

0.949

1.131

0.942

1.000 1.100

1.570 1.640

1.000 0.900

1.109 1.113

0.986 0.986

1.109 1.113

0.966 0.966

1.109 1.113

0.946 0.946

1.109 1.113

0.941 0.941

40

Notes: For metric design, the dimensions are in millimeters and are for 1 module. For English design, the dimensions are in inches and are for 1 diametral pitch.

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Dudley’s Handbook of Practical Gear Design and Manufacture

In Eqs. (6.54) through (6.57), z 2 and NG are the number of face-gear teeth. The constants m o and m i may be read from Table 6.26. The maximum usable face width of the gear is: b2 =

FG =

da 2

di 2

metric

(6.58)

English

(6.59)

2 Do

Di 2

The face width of the pinion is usually made a little greater than that of the gear to allow for error in axial positioning of the pinion. The face gear is cut with a pinion-shaped cutter that has either the same number of teeth as the pinion or just slightly more. If the cutter has slightly more teeth, the gear tooth will be “crowned” slightly, and contact will be heavy in the center of the face width, with little or no contact at the ends of the teeth. The face gear is never cut with a cutter that has a smaller number of teeth than the pinion. If it were, contact would be heavy at the ends and hollow in the center—a very unsatisfactory condition. When the pinion of a face gearset has less than 17 teeth, it is necessary to reduce the cutter outside diameter to keep the top land of the cutter from becoming too narrow. If the land is too narrow, the cutter will wear too fast at the tip. It should be noted that the combinations shown in Table 6.26 have a working depth of less than 2.000 when the pinion has less than 17 teeth. Also, the pinion tooth thickness is increased more than might be expected for the amount of long addendum. These things are done to make it possible to design cutters for the gear with a reasonable top land. The whole depth of the face gear is: h = ha1 + ha2 + c

metric

(6.60)

ht = aP + ag + c

English

(6.61)

where clearance is designated by c. A reasonable value for gear clearance is 0.25 m t (or 0.25/Pd ). The pinion may be made to the same whole depth as that given in Table 6.26, or the depth may be adjusted to agree with the gear. The choice usually depends on what tools are on hand to cut the pinion. The face-gear-tooth thickness can be obtained by subtracting the pinion thickness plus backlash from the circular pitch. Since the gear tooth is tapered, this value does not mean much. The size of face-gear teeth is usually controlled by measuring backlash when the face gear is assembled in a test fixture with a master pinion. The shape of the tooth can be checked by contacting the tooth with the tooth of a mating pinion. The shape of the pinion tooth can be checked in an involute machine. In figuring the roll angle of the pinion at the form diameter, Table 6.11 does not apply. Instead, the roll angle at the limit diameter is: rl

=

rl

=

r1

rP

360 ha2 dp1 sin t cos 360 aG d sin t cos

, degrees

metric

(6.62)

English

(6.63)

t

, degrees t

After the pinion roll angle of the limit diameter is obtained, the pinion form diameter and roll angle of the form diameter may be obtained in the usual manner.

Design Formulas

307

The top and bottom of the face-gear tooth are defined by axial dimensions. The addendum and wholedepth dimensions of the gear are used in figuring these axial dimensions.

6.1.20 CROSSED-HELICAL-GEAR PROPORTIONS The elements of a crossed-helical gear are the same as those of a parallel-helical gear. The bigger difference in the calculations is that the crossed-helical gears meshing together may not have the same pitch, the same pressure angle, or the same helix angle. The only dimensions that are necessarily the same for both members are those in the normal section. By comparison, a mating parallel-helical gear and pinion have the same dimensions in both the transverse and normal planes. Most designers favor making crossed-helical gears with deeper teeth and lower pressure angles than parallel-helical gears. This is done to get a contact ratio equal to or greater than 2. Since the crossedhelical gear has only point contact instead of contact across a face width, its load-carrying capacity and its smoothness of running are much improved by having a design, which gives a contact ratio of 2 or better. When the ratio is this high, there will be either two or three pair of teeth in contact at every instant of time as the teeth roll through mesh. If the contact ratio were less than 2, there would be one interval of time when only a single pair of teeth was carrying the load. There is no trade standard for crossed-helical-gear-tooth proportions. Many manufacturers use crosshelical gears only occasionally. To save tool cost, they frequently design and cut crossed-helical gears with the same hobs or other tools used for parallel-helical gears. This practice will usually give a workable design, but it will not have the load-carrying capacity and smoothness of running that can be obtained with tooth proportions designed for crossed-helical gears. Table 6.27 shows a recommended set of proportions for the design of crossed-helical gears. These proportions have been used with good results in many high-speed applications. Since the crossed-helical gear is not critical on center-distance accuracy or on axial position, it is very handy to use when the power is low and costs are critical. The proportions shown in Table 6.27 will give a contact ratio greater than 2 for all combinations of teeth where undercut is not present. The minimum number of driven teeth needed to avoid undercut is 20. This same number holds for all four combinations shown in the table. The minimum number of driver teeth needed to avoid undercut is

TABLE 6.27 Tooth Proportions for Crossed-Helical Gears Normal module mn = 1 Normal circular pitch pn = 3.14159 mm (Normal diametral pitch Pnd = 1 Normal circular pitch pn = 3.14159 in.) Helix Angle Driver, ( 1)

1

Driven, (

2

Normal Pressure Angle, ( n)

n

Addendum, ha (a )

Working Depth h (h k )

Whole Depth h (h t )

2)

45

45

14 30

1.200

2.400

2.650

60

30

17 30

1.200

2.400

2.650

75

15

19 30

1.200

2.400

2.650

86

4

20

1.200

2.400

2.650

Note: The addendum, working depth, and whole depth values are for 1 normal module (metric) in millimeters or for 1 diametral pitch (English) in inches.

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Helix Angle of Driver

Minimum Number of Teeth

45°

20

60° 75°

9 4

86°

1

One of the first two combinations, shown in Table 6.27, should be used when both members are to be either shaved or hobbed. Both these operations are difficult to perform when the helix angle is above 60°. When very high ratios are needed or there is need for a large-diameter driver, the last two combinations become attractive. The whole depth shown does not allow for a very large root fillet radius. In crossed-helical gears, root stresses are not high, and so a full fillet radius is not needed. Long and short addendum is sometimes used in crossed-helical gear design, but there is not much need for it. Since the strength of the teeth is not a problem in most cases, there is no need to juggle addendums either to balance strength between two members or to avoid undercut and permit the use of a smaller number of driver teeth and a resultant coarser pitch. Table 6.28 shows a dimension sheet and a sample problem for calculating crossed-helical gears. It is based on a 90° shaft angle, and helix angles of like hand for each member. The exact face width needed on crossed-helical gears depends on how far the point of contact moves as the teeth roll through mesh. When the set is new, the arc of action will be a little longer than two normal circular pitches. After the set has worn in, there will be meshing action over about three or slightly more normal circular pitches. Table 6.28 shows the pinion face width to be equivalent to a projection of three normal circular pitches plus an increment. The added increment keeps contact away from the ends of the tooth and allows for error in axial positioning and variation in the amount of wear. For 2.5 normal module (10 pitch), this increment should be at least 6.3 mm (¼ in.).

6.1.21 SINGLE-ENVELOPING-WORM-GEAR PROPORTIONS Like crossed-helical gearing, worm gearing seldom has two mating members with the same module (diametral pitch). The axial pitch of the worm equals the circular pitch of the gear. Also, both members have the same normal circular pitch. In the past, it has been quite common practice to make the tooth proportions a function of the circular pitch (or axial pitch). An addendum of one m n (Pd )—which is standard in many kinds of gears—is equivalent to 0.3183 pn , where pn is the normal circular pitch in either millimeters or inches. With low lead angles and only one or two worm threads, it has been quite customary to use an addendum of 0.3183 pn , a working depth of 0.6366 pn , and a whole depth equal to the working depth plus 0.050 pn . With multiple threads and high lead angles, it is necessary to use quite high pressure angles. It is necessary to shorten the addendum and working depth to avoid getting sharp-pointed teeth and to keep out of undercut trouble. If the tooth proportions are based on the circular pitch of the gear, a high lead angle like 45 may require that the addendum and working depth be cut back to about 70% of the values mentioned above. However, if the tooth proportions are based on the normal circular pitch, the ad­ dendum will stay in about the right proportion to the lead angle. For instance, 0.3183 pn at a 45 lead angle is equal to 0.2250 p. In the fine-pitch field, fine-pitch worm gearing is discussed in considerable detail in the corresponding standards. The standards base the proportions of fine-pitch worm gears on the normal circular pitch.

Design Formulas

309

TABLE 6.28 Crossed-Helical-Gear Dimensions Item

Reference

Metric

English

1.

Center distance

[(9)driver + (9)driven]/2

100.725

3.9655

2.

Normal circular pitch

Design choice (Note 4)

7.853982

0.309212

3. 4.

Normal pressure angle Working depth

Table 6.27 Table 6.27

17.50 6.00

17.50 0.236

5.

Module, normal

(2) ÷ 3.14159265

2.500

–

6.

Normal diametral pitch

3.14159265 ÷ (2)

–

10.160

7.

Number of teeth

Section 6.1.20

8. 9.

Helix angle Pitch diameter

Eqs. (2.26), (2.31) (7) × (2) ÷ [π × cos (8)]

Driver

Driven

Driver

12

49

12

Driven 49

60° RH 60.00

30° RH 141.451

60° RH 2.3622

30° RH 5.5689

10.

Addendum

Table 6.27

3.00

3.00

0.118

0.118

11. 12.

Whole depth Outside diameter

Table 6.27 (9) + 2.0 × (10)

6.63 66.00

6.63 147.45

0.261 2.598

0.261 5.805

13.

Root diameter

(12) – 2.0 × (11)

52.74

134.19

2.076

5.283

14. 15.

Normal tooth thickness Chordal tooth thickness

Section 6.1.6 Eqs. (6.9), (6.10)

3.88 3.88

3.88 3.88

0.1525 0.1525

0.1525 0.1525

16.

Chordal addendum

Eqs. (6.7), (6.8)

3.02

3.02

0.1188

0.1188

17. 18.

Face width Lead

Section 6.1.20 (2) × (7) ÷ sin (8)

27 108.828

19 769.690

1.0625 4.2846

0.75 30.3028

19.

Base circle diameter

See Note 2

48.0508

130.4024

1.9098

5.1339

20.

Accuracy

Section 6.1.18

See Note 5

Notes: 1. Metric dimensions are in millimeters, English dimensions are in inches, and angles are in degrees. 2. Base circle diameter = cos (transverse pressure angle) × (9), and tan (transverse pressure angle) = tan (3) ÷ cos (8). 3. The shaft angle is assumed to be 90° for this table, and the helix angles are assumed to be of the same hand for both members of the set. 4. After a preliminary size is chosen, a rating estimate for the set may be made with the help of Section 6.2.7 in this chapter 5. After the design has done the load rating and studied Section 16.4.1, accuracy needs may be satisfied by an ISO or AGMA quality level, or the designer may need to write a special recipe for accuracy limits.

There are no generally accepted trade standards for the proportions of medium-pitch worm gears. From a practical standpoint, it is best to base the proportions of all sizes of worm gears on the normal circular pitch. Table 6.29 shows recommended proportions for three general kinds of applications. The design for index and holding mechanisms represents average shop practice.

6.1.22 SINGLE-ENVELOPING WORM GEARS Low-lead-angle worm gearsets are frequently self-locking. This means that the worm cannot be driven by the gear. The set will “hold” when the worm has no power applied to it. The exact lead angle at which a worm will be self-locking depends on variables like the surface finish, the kind of lubrications, and the amount of vibration where the drive is installed. Generally speaking, though, self-locking occurs if the lead angle is below 6 , and it may occur with as much as 10 lead angle. The pressure angle of the tool shown in Table 6.29 is the pressure angle of a straight-sided conical milling or grinding wheel used to finish the worm threads. The normal pressure angle of the worm is a small amount less than this value. The equations for the normal pressure angle are:

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Dudley’s Handbook of Practical Gear Design and Manufacture

TABLE 6.29 Tooth Proportions for Single-Enveloping Worm Gears No. of Worm Threads, z 1 (Nw )

Cutter Pressure Angle, 0 ( c )

Addendum, h a (a )

Index or holding mechanism

1 or 2

14 30

0.3183 pn

0.6366 pn

0.7000 pn

Power gearing

1 or 2

20

0.3183 pn

0.6366 pn

0.7000 pn

3 or more

25

0.2860 pn

0.5720 pn

0.6350 pn

1 – 10

20

0.3183 pn

0.6366 pn

0.7003 pn + 0.05*

Item

Fine-pitch (instrument)

Working Depth, Whole Depth, h (h t ) h (h k )

Note: The addendum, working depth, and whole depth values are for 1 normal module (metric) in millimeters or for 1 normal diametral pitch (English) in inches. * 0.05 is for metric system. Add 0.002 in. for English system.

n

=

n

=

0

c

metric

(6.64)

English

(6.65)

where: =

=

90 dp1 sin3 z 1 (dp0 cos2 + dp1) 90 d sin3 Nw (dc cos2 + d )

(6.66)

(6.67)

where: γ (λ) – is the lead angle dp1 (d) – is the worm pitch diameter dp0 (dc) – is the cutter pitch diameter

The angle ( ) is in degrees. Table 6.29 allows more clearance than the old figure of 0.050 p . This is in line with the practice of putting more generous tip radii on tools. The pitch diameter of the worm usually represents a compromise among several considerations. If the worm is small compared with the gear, the lead angle will be high and efficiency will be good. However, the face width of the gear will be small, and there may be trouble getting the bearing close enough together to prevent bending of the small worm. A large worm compared with the gear gives lower efficiency, but it may be possible to use a large enough bore in the worm to permit keying it on its shaft instead of making it integral with the shaft. An approximate value for the worm mean-diameter is: dp1 =

a0.875 2.2

metric

(6.68)

Design Formulas

311

d=

C 0.875 2.2

English

(6.69)

It is believed that this value gives a good practical size for the worm, considering all the factors men­ tioned. Of course, if the designer is not primarily interested in the efficient transmission of power, it is possible to depart widely from Eqs. (6.68) to (6.69), and still get satisfactory operation. The worm pitch diameter may range from a0.875 /1.7 to a0.875 /3.0 (C 0.875 /1.7 to C 0.875 /3.0 ) without substantial effect on the power capacity. If the worm addendum equals the gear addendum, the mean worm diameter is actually the pitch diameter. In the fine-pitch field, the efficiency and power-transmitting ability of a given size set are usually not too important. For this reason, no particular effort has been made to proportion the worm pitch diameter to the equations like Eqs. (6.68) and (6.69). A series of standard axial pitches, and a series of standard lead angles, are listed in gear standards. Indirectly, this results in a series of standard pitch diameters. The axial pitches range from 0.75 to 4.0 mm (0.03 to 0.16 in.), and the lead angles range from 30 to 30 . A considerable amount of tabulated data is given in the standard for each of the combinations shown. The lead angle of the worm is equal to the helix angle of the worm gear when the shaft angle is 90 . The lead angle may be calculated from any of the following relations: tan

=

tan

=

sin

dp2

metric

(6.70)

px z1 dp1

metric

(6.71)

=

pn z 1 dp1

metric

(6.72)

tan

=

D mG d

English

(6.73)

tan

=

px NW d

English

(6.74)

sin

=

pn NW d

English

(6.75)

u dp1

In the Eqs. (6.70) through (6.75), px is the axial pitch and u (mG ) is the number of gear teeth divided by the number of worm threads. It is wise to not use more than 6 lead angle per thread. For instance, if the lead angle is 30 , there should be at least five threads of the worm. If too few threads are used, the problem of designing tools and producing accurate curvatures on the worm threads and on the gear teeth becomes too critical for good manufacturing practice. In general, the number of teeth on the gear should not be less than 29. As an exception, the number can be reduced to 20 when the cutter pressure angle is 25 and the lead angle does not exceed 15 . The number of teeth on the gear and the number of threads on the worm should be picked to get a hunting ratio. This is particularly important in worm gearing, because the hob for generating the gear should have the same number of threads as the worm.

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Different designers use several different means to get the worm-gear face width and the worm-gear outside diameter. A simple method, which is always safe and which will utilize practically all the wormgear tooth that is worth using, is to make the face width equal to (or just slightly greater than) the length of a tangent to the worm pitch circle between the points at which it is intersected by the worm outside diameter. The worm-gear maximum diameter is made just large enough to have about 60% of the face width throated. This design is shown as design A in Figure 6.21. Design B shows an alternative that has been widely used for power gearing. Design C shows a non-throated design that may be used for instrument or other applications in which maximum power transmission is not an important design consideration. The profile of the worm will have a slight convex curvature when it is produced by a straight-sided milling cutter or grinding wheel. Section 17.4 shows how to calculate the normal-section curvature of the worm. In worm gearing, it has been customary to obtain backlash by thinning the worm threads only. The worm-gear teeth are given a design thickness equal to half the normal circular pitch (when equal ad­ dendums are used for the worm and the worm gear). Generally speaking, worm gears require more backlash than do spur or helical gears. Quite often a steel worm and a bronze gear are housed in a castiron casing. The temperature of the set may change quite considerably during operation because of the high sliding velocity of the worm threads. Different expansions can easily cause an appreciable change in backlash. Except for slow-speed control gears, the design should provide enough backlash to keep the gears from binding at any speed or temperature condition under which the set may have to operate. Table 6.30 shows a dimension sheet for single-enveloping worm gears and the solution of a sample problem.

6.1.23 DOUBLE-ENVELOPING WORM GEARS Although several types of double-enveloping worm gears are mentioned in gear literature, one type is at present much more widely used in industry than any other type. This is the Cone-Drive gear. Other types of double-enveloping worm gears are in use, but the Cone-Drive type is produced in such large quantities in the United States that it seems most appropriate to cover this type only in this book. Table 6.31 shows the proportions recommended for double-enveloping worm gears.

FIGURE 6.21

Worm-gear design examples. (Dimensions shown have English system symbols.)

Design Formulas

313

TABLE 6.30 Single-Enveloping-Worm-Gear Dimensions Item

Reference

1.

Center distance

Section 5.2.6

2.

Axial pitch of worm

π × (10)G ÷ (9)G

3. 4.

Cutter pressure angle Worm lead angle

Table 6.29 Equations (6.70) to (6.75)

5.

Working depth

Table 6.29

6. 7.

Module, transverse Diametral pitch, transverse

(2) ÷ 3.141593 3.141593 ÷ (2)

8.

Normal circular pitch

(2) × cos (4)

9.

Number of teeth

Metric

English

249.565

9.82537

25.00

0.98425

25° 25°

25° 25°

14.42

0.568

7.95775 –

– 3.191858

22.6577

0.892035

Section 6.1.22

Worm 5

Gear 52

Worm 5

Gear 52

10.

Pitch diameter

Section 6.1.22

85.327

413.803

3.3593

16.2914

11. 12.

Addendum Whole depth

Table 6.29 Table 6.29

7.21 15.86

7.21 15.86

0.284 0.625

0.284 0.625

13.

Outside diameter

(10) + 2.0 × (11)

99.75

428.22

3.927

16.859

14. 15.

Maximum outside diameter, gear Root diameter

Section 6.1.22 (13) – 2.0 × (12)

– 68.03

435.35 396.50

– 2.677

17.140 15.609

16.

Normal tooth thickness

Section 6.1.22

10.88

11.32

0.428

0.446

17. 18.

Chordal addendum Face width

Equations (6.7), (6.8) Section 6.1.22

7.27 115.0

7.27 54.20

0.2865 4.52

0.2865 2.13

Equations (6.68), (6.67)

0.5559

19.

Pressure angle change

–

0.5559

–

20. 21.

Normal pressure angle Equations (6.64), (6.65) 24.4441 – Tool to produce the worm. Straight-sided milling cutter. 150 mm (5.905 in.) diameter

24.4441

–

22.

Accuracy

See Note 3

Notes: 1. The helix angle of the worm gear is the same as the lead angle of the worm when the shaft angle is 90°. 2. If the worm is to be hardened and ground, then the tool definition in (21) should state the diameter of the grinding wheel and the form of grinding wheel. 3. There are some individual company standards for the accuracy of worm gearing, but there are no generally recognized trade standards. 4. Metric data is in millimeters, English data is in inches, and all angles are in degrees.

TABLE 6.31 Tooth Proportions for Double-Enveloping Worm Gears Normal Pressure Angle 20°

n

(

n)

Addendum ha (a)

Working Depth h (h k )

Whole Depth h (h t )

0.225 pn

0.450 pn

0.500 pn

The pitch diameter of the worm should be approximately a0.875 /2.2 (C 0.875 /2.2 ). This is the same value as that given by Eqs. (6.68) and (6.69). The worm root diameter is:

df 1 = dp1 + 2 ha1

2 h1

metric

(6.76)

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Dudley’s Handbook of Practical Gear Design and Manufacture

dR = d + 2 aP

2 h tP

English

(6.77)

If this value comes out less than a0.875 /3 (C 0.875 /3), the worm pitch diameter should be increased so that the root diameter is not less than this value. In picking numbers of threads, the designer should not have more than about 6 per thread lead angle for sets of 125 mm (5 in.) or less center distance. The number of gear teeth should be picked so as to give a hunting ratio if possible. This is not quite so important as with single-enveloping worm gears. It is further recommended that the relation of the number of gear teeth to the center distance be as follows:

Center Distance

Number of Gear Teeth

mm

in.

50

2

24–40

150 300

6 12

30–50 40–60

600

24

60–80

The axial pressure angle of any kind or worm is determined by the relation tan

tan

=

x

x

=

tan cos tan

n

metric

(6.78)

n

English

(6.79)

cos

For double-enveloping worm gears, the lead angle that is used is at the center of the worm. The base-circle diameter of the double-enveloping worm gear can be determined by layout, or it can be calculated by: db2 = dp2 sin(

x

+

Db = D sin(

x

+ )

)

metric English

(6.80) (6.81)

= px /2dp2 (metric) or sin = px /2D (English). where sin The active part of the worm face width should almost equal the base circle diameter. Usually, the face width of the worm is made a little shorter. Thus: b1 = db2 Fw = Db

0.03 a

metric

(6.82)

0.03 C

English

(6.83)

In general, the face width of the gear is made slightly less than the root diameter of the worm. The thickness of the worm threads and gear teeth are controlled by side-feeding operations. It is customary to make the worm thread equal to 45% of the axial circular pitch and the gear tooth equal to 55% of the normal circular pitch, provided that the tool design will permit and gear strength is critical.

Design Formulas

315

Backlash is subtracted from the worm thread thickness. For average applications, the following amounts of backlash are reasonable:

Center Distance

Backlash

mm

in.

mm

in.

50 150

2 6

0.08–0.20 0.15–0.30

0.003–0.008 0.006–0.012

300

12

0.30–0.50

0.012–0.020

600

24

0.45–0.75

0.018–0.030

Table 6.32 is a dimension sheet for calculating double-enveloping worm gears. The solution of a sample problem is shown on the dimension sheet.

TABLE 6.32 Double-Enveloping-Worm-Gear Dimension Item

Reference

Metric

English

253.558

9.9826

25.40 20

1.000 20

1.

Center distance

Section 5.2.6

2. 3.

Axial pitch of worm* Normal pressure angle

π × (10)G ÷ (9)G Table 6.31

4.

Worm lead angle*

Eqs. (6.70) to (6.75)

5. 6.

Working depth Module, transverse

Table 6.31 (2) ÷ π

7.

Diametral pitch, transverse

π ÷ (2)

8.

Normal circular pitch

(2) × cos (4)

9.

Number of teeth

Section 6.1.23

5

52

5

52

10. 11.

Pitch diameter Addendum

Section 6.1.23 Table 6.31

86.69 5.18

420.42 5.18

3.413 0.203

16.552 0.203

25

25

10.36 8.0851

0.406 –

–

3.141593

23.020

0.906308 Worm Gear

Worm

Gear

12.

Whole depth

Table 6.31

11.51

11.51

0.453

0.453

13. 14.

Outside diameter Throat diameter

From layout (10) + 2 × (11)

115.57 97.05

445.64 430.78

4.550 3.819

17.545 16.958

15.

Root diameter

Equations (6.76), (6.77)

74.03

–

2.913

–

16. 17.

Normal tooth thickness Axial pressure angle

Section 6.1.23 Equations (6.78), (6.79)

8.99 21.88020

13.82 –

0.354 21.88020

0.544 –

18.

Base circle diameter

Equations (6.80), (6.81)

–

168.39

–

6.63

19. 20.

Face width Face angle

Section 6.1.23 From layout

160.3 45

66.7 87.5

6.3125 45

2.625 87.5

21.

Accuracy

See Note 3

Notes: 1. The helix angle of the worm gear is the same as the lead angle of the worm when the shaft angle is 90°. 2. Metric data are in millimeters, English data are in inches, and all angles are in degrees. 3. There are no trade standards for accuracy of double-enveloping worm gears. * At the center of the worm.

316

6.2

Dudley’s Handbook of Practical Gear Design and Manufacture

GEAR-RATING PRACTICE

After the gear-tooth data have been calculated, it is necessary to calculate the capacity of the gearset. Since the design was started from an estimate (see Chapter 5), it may happen that the first design that is worked out in detail is too small or too large. Once all the gear-tooth data have been calculated, it is possible to use design formulas to determine a rated capacity of the gearset. This rated capacity should be larger than the actual load that will be applied to the gearset. In some field of gearing, there are well-established trade standards in regard to the rating of gearsets. Sometimes these are quite conservative. In a few unusual cases, a trade standard may rate a gearset as able to carry more load than it really will carry. It must be recognized that a trade standard is based on the general level of quality that representative members of the industry can produce. This quality includes the accuracy of the gear teeth, the accuracy with which the gears are positioned in their casings, and the quality of the materials from which the gears are made. Manufacturers who cannot live up to the normal level of quality for a particular kind of gears cannot expect their gears to safely carry the full ratings allowed by a trade standard. Likewise, manufacturers who can build gearing of appreciably better quality than the general level of the trade might expect that their gears would be able to carry somewhat more load than that allowed by a trade standard. As a general rule, in checking gear capacity, it is wise to first check the design using general formulas for tooth strength and durability. Then, if the application falls into the field of a particular trade standard, it should be checked against the trade standard. If the design meets both tests, it is probably all right. In this part of the chapter, we shall consider both general formulas and trade standards for rating gears.

6.2.1 GENERAL CONSIDERATIONS

IN

RATING CALCULATIONS

Gear designers make rating calculations to establish that a given gear design is suitable in size and in quality to meet the specified requirements of a gear application. (The buyer of a gear unit or the user of a gear unit may also make rating calculations to check the design being bought or used.) Rating calculations are concerned with more than direct load-carrying capacity. All three of these somewhat different concerns tend to apply: • Has the rating been calculated correctly with a good formula, and have all the right assumptions and decisions been made in regard to materials, material quality, and geometric quality? • Does the design calculate a suitable rated power, length of life, and degree of reliability to satisfy a specified formula in the business contract or one established and recognized in normal trade practice? There is a legal obligation in many projects, both in the United States and worldwide7, to meet all applicable AGMA standards. • Will the gear unit meet its normal rating capacity in the power package using the gears? In a power system there are often situations in which gears are prematurely damaged by undefined overloads, misalignments, unexpected temperature excursions, or contamination of the lubricant by some foreign material (water, ash, chemical vapors at the site, and so forth). The designer needs to rate the gears to meet the application. In a new application with unknown hazards, an appropriate gear rating may not be possible until several prototype units have been in service for a suitable period of time. A reliable gear rating may not be practical until enough development work has

7

In an international project, the gears might be built in Europe and installed in South America. If the project was insured or financed by U.S. interests, there would probably be a contract requirement that the gears meet AGMA standards, even though they were not built in the United States.

Design Formulas

317

been done and enough field experience acquired to prove out a gear design suitable for the hazards of the application. (See Chapter 14 for a discussion of things that may be problems in power breakages.) Somewhat aside from the mechanics of making gear-rating calculations and the concerns of contract requirements and application hazards is the fact that there is a shortage of good technical data and specifications to closely control all the things that to into a rating. In earlier times, the mechanical designer tried to calculate a safe stress. It was thought that all parts with less than the safe stress limit would perform without failure. Gears, bearings, and many other mechanical components have a probability of failure. A safe stress is in reality only a stress at which the probability of failure is low. Theoretical gear design has now shifted to using stress values based primarily on some level of probability of failure. Unfortunately, the data available to set probability of failure are still rather limited. This problem is compounded by the fact that a system of probability of failure versus stress needs to be tied into a system of material quality grades. Material quality grades are still not very clearly defined. Two grades of quality for aircraft gears were established in the 1960s. In the 1970s, two grades were established for vehicle gears. Work is under way to establish grades for high-speed gearing used with marine and land turbine units. Generally speaking, there is about a 20 to 25% change in load-carrying capacity as you go from Grade 1 to Grade 2 material. In a fully developed gear grading system, there should probably be more than two grades of material (at least three), and the material specifications of each grade should be more closely defined than is now the case. The designer who is going to shift a rating up or down 20% for material grade certainly needs some rather clear-cut definitions of what must be achieved to qualify the material for its designated grade. (See Section 10.2.6 for detailed information on material quality.) In earlier times, failure by tooth breakage meant a broken tooth, and failure by pitting meant a substantial number of “destructive” pits. Recent work shows that both of these criteria of failure are inadequate. An aircraft gear with some small cracks in the root found at overhaul has usually “failed.” It is not safe to use the part if the cracks are above a certain size in a critical location, so the gear is scrapped—even if it might run another 1000 hours. At the other extreme, an industrial gear may have a whole layer of material pitted away from the lower flank and still be able to run for several more years. In this case, failure by pitting only comes when so much metal has worn away and load sharing between teeth has become so bad that the teeth break. A further design complication comes from the fact that gear teeth may work-harden in service and may polish up rough tooth surfaces. The designer used to design gears based on the hardness of the part and the finish and fit of the part when it left the gear shop. Now the final finishing of the gear teeth and the final metallurgical character of the tooth surface are often established in service. The data on how to allow for improvements in finish and fit and for work hardening is still rather limited. Technical papers have shown quantitative data that indicate a 2 to 1 improvement in finish and a change in hardness of 100 points Vickers or 100 BHN are not uncommon in lower-hardness gears running at medium to low pitch-line velocities. Calculation procedure. The calculation begins with a requirement that a gearset must handle a specified power at a given input speed. In a simple rating calculation, the preliminary sizing of the unit has established these things: • • • • • • •

Power to be transmitted, in kilowatts (kW) or horsepower (hp) Rotational speed of pinion Number of pinion teeth and number of gear teeth Face width Pitch diameters of pinion and gear Pressure angle (normal pressure angle for helical gears) Size of teeth by module or diametral pitch

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Dudley’s Handbook of Practical Gear Design and Manufacture

• Helix angle • Hours of life needed at rated load The first calculation step is to get the tangential driving load. The pinion torque is calculated first, then the torque is converted to a tangential force acting at the pitch diameter. The torque is: Pinion torque =

T1 =

TP =

power × constant pinion rpm

(6.84)

P × 9549.3 n1

metric

(6.85)

P × 63, 025 nP

English

(6.86)

In metric equation, the power P is in kilowatts, and the calculated torque will be in Newton-meters (N m ). For English equation, the power is in horsepower, and the answer will be in inch-pounds (in. lb). The tangential driving force is: Wt =

T1 × 2000 , N metric dp1

(6.87)

Wt =

TP × 2.0 , lb English d

(6.88)

The geometry factor for strength is usually read from a table or a curve sheet. The overall derating factor may be read from a table or calculated. The bending and contact stresses can now be calculated. Before checking whether or not these stresses are permissible, the designer must make some further determinations. The life cycles required for the pinion and for the gear need to be known, and an estimate of which quality grade of material will be used needs to be made. Also, the level of reliability needed for the application should be determined. The loading cycles are: n c1 = pinion rpm × hours life × 60 × no. contacts

(6.89)

n c2 = gear rpm × hours life × 60 × no. contacts

(6.90)

In the above equations, a pinion driving three gears would have three contacts per revolution. Likewise, if five planet pinions drive a single gear, there would be five contacts per revolution of the gear. This kind of situation is common in epicyclic gears. A single pinion driving a single gear has one contact per revolution. (The gear also has one contact for each revolution it makes.) Grades of material quality. In general, two quality grades can be considered possible. These are: Grade 2 The best quality obtainable with an approximate choice of material composition and processing that is close to optimum. Usually there is extra cost for the best quality.

Design Formulas

319

Grade 1 A quality of material that is good, but not optimum. This could be considered a typical quality from gear makers doing good industrial work at competitive costs. For highly critical work in space vehicles, or highly sensitive applications, a Grade 3 quality can be considered. This might be thought of as essentially perfect material made under such rigid controls that there is relatively absolute assurance that the highest possible perfection is obtained. Obviously, the very high cost Grade 3 gears will carry more stress—or will have higher reliability at the same stress—than Grade 2 gears. Grade 3 gear data will not be given in this book. It is too complex and limited in usage. Reliability of gears In regard to reliability, these general concepts should be kept in mind: Reliability level

L.1 L1

Fewer than one failure in 1000 Fewer than one failure in 100

Seldom used Typical gear design

L10

Fewer than one failure in 10

May be used in vehicle gears

L20

Fewer than one failure in 5

Expendable gearing

L30

Fewer than one failure in 2

Highly expendable gearing

The L.1 level has been used in some highly critical aerospace work (particularly space vehicles). L1 is typical of industrial turbine work, helicopter work, and high-grade electric motor gearing. Vehicle gears for land use have tended to be in the L1 to L10 range. Home tools, toys, gadgets, etc., may be in the L20 area, or even up to L50. There is a limited amount of laboratory test data, and an average curve is drawn through failure points, the curve will be around L50. (It takes much test data and field experience to determine where a stress level that is equivalent to an L1 level of reliability can be located.)

6.2.2 GENERAL FORMULAS

FOR

TOOTH BENDING STRENGTH

AND

TOOTH SURFACE DURABILITY

The rating formulas for gear-tooth strength and gear surface durability have become very long and very difficult to handle in many of the rating standards and other trade groups. In this work, short and simple formulas will be used. The short formulas group all variables into essentially three factors: • An index of load intensity, with dimensions • A geometry factor, dimensionless • A derating factor, dimensionless The load-intensity evaluations are based on all real numbers that relate the size of the gears to the power being carried by the gears. The geometry factor evaluates the shape of the tooth. This involves pressure angle, helix angle, depth of tooth, root fillet radius, and proportion of addendum to dedendum. The derating factor handles all the things that tend to reduce the load-carrying capacity. (Concentration of too much load at spots on the surfaces of some teeth reduces the average load al­ lowable, since failure can be avoided only when the most overloaded spots are still within safe limits.) The derating factor handles non-uniform load distribution across the face width and in a cir­ cumferential direction. It also handles dynamic overloads due to spacing error and the masses of the pinion and gear meshing together, and other things—quality related—such as surface finish effects,

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Dudley’s Handbook of Practical Gear Design and Manufacture

overload effects due to non-steady power, and variations in metal quality between very large gears and small gears. Strength formula. The simplified general formula for tooth bending stress of spur, helical, and bevel8 gears is: st = K t Ul Kd

(6.91)

The bending stress st is measured in “N/mm2” when metric system is applied, and in “psi” for English system. In Eq. (6.91), the geometry factor for bending strength is designated as K t (see Section 6.2.3): Kt =

constant × cos (helix angle) J factor

(6.92)

Unit load, index for tooth breakage, is denoted by Ul : Ul =

Wt N/mm2 metric b mn

(6.93)

Wt Pnd psi English F

(6.94)

Ul =

Overall derating for bending strength is designated as Kd (see Section 6.2.4): Kd =

Ka Km Ks Kv

metric or English

(6.95)

Individual items in the above equations are defined in Table 6.33, along with many other used in loadrating equations. Durability formula. The simplified general formula for tooth-surface durability of spur, helical, and bevel9 gears give the contact stress: sc = Ck

K Cd

(6.96)

where geometry factor for durability Ck is equal (see Section 6.2.5): Ck = constant ×

1 ratio I (ratio + 1)

(6.97)

K factor (index for pitting) is calculated from: K=

8 9

Wt u + 1 b dp1 u

See the subsection “Rating bevel gears” at the end of this section. See the subsection “Rating Bevel Gears” at the end of this section.

metric

(6.98)

Design Formulas

321

TABLE 6.33 Gear Terms, Symbols, and Units Used in Load Rating of Gears Term

Metric

English

Symbol Units

Symbol

Reference or Formula

Units

m or m t mn

mm

–

–

Metric tooth size

mm

–

–

m n = 25.5/ Pnd

Diametral pitch, transverse

–

–

in.−1

Diametral pitch, normal

– a

–

Pd or Pt Pnd

mm

in.

Figure 5.14

Face width

b

mm

C F

in.

Figure 2.18

Face width, effective

b

mm

Fe

in.

Equations (6.123), (6.124)

Pitch diameter

dp

mm

d

in.

Figure 5.10

Of pinion

dp1

mm

d

in.

Equations (2.5), (2.6)

Of gear

dp2

mm

D

in.

Equations (2.7), (2.16)

Of driver (crossed-helical)

dp1

mm

D1

in.

Equation (6.118)

Of driven (crossed-helical)

dp2

mm

D2

in.

dt1 u

mm

d1

in.

–

mG

–

Pressure angle, normal

n

deg

Pressure angle, transverse

t

deg

Module, transverse Module, normal

Center distance

Throat diameter, worm Ratio (tooth)

in.−1

t

deg

Figure 2.21 Figure 2.22

deg

deg

deg

Rotational speed, gear Sliding velocity Power Torque Torque on pinion Tangential load or force Dynamic load In normal plane Unit load Number of load cycles For pinion For gear Reliability level K factor, pitting index Stress on tooth surface

rpm m/s kW Nm Nm N N N N/mm2 –

No. gear teeth ÷ No. pinion teeth

n

deg rpm

Equations (6.134), (6.135) Equation (6.118)

Lead angle

n1 n2 vs P T T1 Wt Wd Wn Ul nc nc1 nc2 L K s

Pnd = 25.5/ m n

deg

Helix or spiral angle Rotational speed, pinion or worm

English tooth size

nP or n W rpm nG rpm vs fpm, fps P hp T in.-lb TP in.-lb Wt lb Wd lb Wn lb Ul psi nc –

Figure 2.22 Equations (6.84) to (6.86) Equations (6.127), (6.128) Equations (6.122), (6.125), (6.126), (6.133), (6.134) Equations (6.84) to (6.85) Equations (6.84) to (6.86) Equations (6.84) to (6.86) Equations (6.87), (6.88), (6.125), (6.126) Equation (6.113) Equation (6.121) Equations (6.93), (6.94), (6.129), (6.130)

–

ncP

–

Equation (6.89)

–

–

Equation (6.90)

– psi

End of Section 6.2.1 Equations (6.96), (6.98), (6.99)

N/mm2

ncG L K s

– N/mm2

psi

Equation (6.66)

Bending stress (strength)

st

N/mm2

st

psi

Equation (6.91)

Contact stress (surface)

sc

N/mm2

sc

psi

Equation (6.96)

For bending strength

Kd

–

Kd

–

Section 6.2.2, Equation (6.105)

For surface durability

Cd

–

Cd

–

Section 6.2.6, Equation (6.116)

Kt Ck

–

Kt Ck

–

Sections 6.2.2, 6.2.3

–

Sections 6.2.2, 6.2.5

Overall derating factor:

Geometry factor: For bending strength For surface durability

–

(Continued)

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Dudley’s Handbook of Practical Gear Design and Manufacture

TABLE 6.33 (Continued) Gear Terms, Symbols, and Units Used in Load Rating of Gears Term

Metric

English

Symbol Units

Symbol

Reference or Formula

Units

Application factor: For bending strength

Ka

–

Ka

–

Section 6.2.4

For surface durability

Ca

–

Ca

–

Section 6.2.6

Size factor: For bending strength

Ks

–

Ks

–

Section 6.2.4

Cs

–

Cs

–

Section 6.2.6

For bending strength

Kv

–

Kv

–

Section 6.2.4, 6.2.7

For surface durability

Cv

–

Cv

–

Section 6.2.4, 6.2.6

Km Kmf

–

Section 6.2.4

–

Km Kmf

–

(face width effects)

–

Equations (6.106), (6.107), (6.108), (6.109), (6.110)

(transverse effects)

Kmt

–

Kmt

–

Equation (6.106)

For surface durability

Cm Cmf

–

Cm Cmf

–

Section 6.2.6

–

Section 6.2.4

E ew

psi in.

Equation (6.119) Table 6.39 Equations (6.107), (6.108)

For surface durability Dynamic load factor:

Load-distribution factor: For bending strength

(face width effects)

xE

Modulus of elasticity Wear-in amount

ew

Mismatch error Material and type constant Wear load Aspect ratio J factor (geometric strength) I factor (geometric pitting)

– N/mm mm

2

et Cp

mm

et Cp

in.

–

–

Equation (6.114), (6.115)

– ma

–

Ww

lb

Equation (6.117)

–

ma

–

Section 6.2.4

–

–

Alt. to Kt , Section 6.2.3

–

Alt. to Ck , Section 6.2.5

–

J I mp

–

Section 6.2.5

J I

Contact ratio

–

Flash temperature index

Tf

C

Tf

F

(6.136)

Gear body temperature

Tb

C

Tb

F

(6.136)

Geometry constant

–

(6.136)

–

(6.136)

C

Zt Zs Zc

–

Scoring-criterion number

Zt Zs Zc

°F*

(6.136)

Oil-film thickness (EHD)

h min

μm

h min

μin.

(6.144)

S

μm

S

μin.

(6.149)

–

(6.150)

Surface finish constant

Effective surface finish Lambda ratio

–

–

Notes: 1. Abbreviations for units are as follows: 2. See Table 6.2 for terms, symbols, and units in the calculation of gear dimensional data. * Zc is not a temperature, but becomes a change of temperature when multiplied by Zt and Zs. Metric System

English System

mm deg

millimeters degrees

in. deg.

inches degrees

rpm

revolutions per minute

rpm

revolutions per minute

m/s

meters per second

fpm fps

feet per minute feet per second

(Continued)

Design Formulas

323

TABLE 6.33 (Continued) Gear Terms, Symbols, and Units Used in Load Rating of Gears Metric System

English System

kW

kilowatts

hp

horsepower

N Nm

Newtons Newton-meters

lb in.-lb

pounds inches-pounds

N/mm2

Newtons per square millimeter

psi

pounds per square inch

°C μm

degrees Celsius micrometers (10-6 m)

°F μin.

degrees Fahrenheit microinches (10-6 in.)

K=

Wt m G + 1 Fd mG

English

(6.99)

and, finally, overall derating for durability Cd is calculated by (see Section 6.2.6): Cd =

Ca Cm Cs Cv

(6.100)

In the metric system the contact stress sc [see Eq. (6.96)] is measured in N/mm2, and English system the contact stress sc is measured in psi. In most long-life power gearing, the size of the gear drive (center distance and face width) is de­ termined by the durability formula. The calculated contact stress must not exceed the allowable stress for the number of cycles of life required and the quality (grade) of material used. The allowed stress is adjusted for the permissible risk of failure (L1, L10, etc.). The gear-strength calculations are used primarily to determine what size of tooth (module or pitch) is needed to keep the bending stress within allowable limits for the number of cycles of life required, the quality of the material, and the permissible risk of failure. An exception to this occurs when gears with a very high torque rating are needed for a low number of cycles. (Final drive vehicle gears are often in this situation.) The gear size may beset primarily by gear-strength considerations. The gears may suffer some degree of surface failure and still run satisfactorily for the need life. Generally speaking, gear teeth that suffer surface damage in less than 1 million (106) cycles of high torque cannot keep on running for more than about 100 million (108) cycles, even if the continuous torque is rather light. Turbine gearing, for instance, generally has to run more than a billion (109) cycles. Turbine gears cannot have surface damage due to a high torque at low cycles and be expected to last for many years of operation. In writing this section on rating formulas and the next five sections on factors in the rating formulas, the following strategy has been accepted: • The general formulas and their principal terms will be given in a manner that should remain unchanged for many years. • Some items that are being changed—and perhaps will continue to change for many years—will be given in numerical values that are conservative for the gearing design practice. The reader will be given practical guidance in what to use in rating gears, but obviously may be able to find better numerical values to use as worldwide gear research and field experience give more information about how to evaluate all of the many things that enter into gear rating.

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Dudley’s Handbook of Practical Gear Design and Manufacture

• This book will endeavor to give the reader a means of determining the approximate size of gear teeth needed to handle a desired rating10. Whenever recognized trade standards or contract spe­ cifications are involved, the final sizing and design of the gears should be adjusted to meet these obligations. (In general, gears sized by the principles in this book should not need much adjustment to fit contract obligations. If a contract would allow significantly smaller or weaker gears, the designer should review the field experience available and decide whether or not going beyond the recommendations given here is reasonable.) • In many areas, not enough is known to rate the gears with any great confidence that the rating is really right. This is particularly true in Regime I lubrication conditions. (See Section 5.1.4 for a decision of lubrication regimes.) If extensive field experience is not available to verify a gearrating procedure, it is generally necessary to do initial factory and field testing of prototype units to prove their specified ratings before going into production of the units. Rating curves of stress versus cycles. The following rating curves are presented as general guides to the gear designer: Figure Figure Figure Figure *

6.22 6.23 6.24 6.25

Covers Covers Covers Covers

allowable bending stress* allowable bending stress* contact stress for Regime contact stress for Regime

for short cycles. for long cycles. II lubrication and short cycles. III lubrication and long cycles.

The bending stress that is considered in Figures 6.22 and 6.23 is a unidirectional stress for normal gear action. When gear teeth are subject to loading in two directions, such as a planet in an epicyclic set that is loaded on both sides of the teeth, the rating must be reduced for reverse bending. An approximately correct rating can be obtained by multiplying the allowable bending stress by 0.70 before comparing it to the calculated stress.

These curves are somewhat different from the presently published curves used in gear standards. They reflect the concept that high loads in fewer than 107 cycles can cause microscopic metal damage that makes the gear tooth unable to run for 109 cycles or longer. They also take into account regimes of lubrication and the change in slope of the stress versus cycles curves for durability. There are not enough good data in the gear trade to plot a set of curves like those in Figures 6.22 to 6.25 with a high degree of confidence. The data shown are intended to represent as good a judgment as can be determined, given the present state of the gear art. Hopefully, more research work and field experience will make it possible to have better data. These curves are intended to be used with the load histogram manner of design and an obligation to control the scoring problem. See Section 6.2.8 for more on the load histogram method, and for s dis­ cussion of how to handle the scoring hazard. Rating bevel gears. When the intensities of tooth loading for bevel gears (K−factor and unit load) are calculated by a known method, the values obtained are comparable to those of spur and helical gears. This is very helpful in preliminary design work when a decision as to whether the body shape of the gears should be cylindrical or conical has not yet been made. As a general rule, a reasonable K−factor on a bevel gear tends to be about equal to that on a matching spur or helical gear with the same hardness, quality, and pitch-line speed. The unit load may be somewhat less, since it is harder to get a large root fillet radius on a bevel gear than on a matching spur or helical gear. Size is another factor. Large bevel gears need more derating for size than cylindrical gears. With these factors in mind, this book was written with the plan that the basic calculations for surface compressive stress and bending stress of bevel gears would be handled the same way and use the same allowable stress as those for spur and helical gears.

10

See Chapter 14 for a discussion of gear ratings and of possible factors other than the basic rating that may cause failures.

Design Formulas

FIGURE 6.22

Bending stress for a short-life design.

FIGURE 6.23

Bending stress for a long-life design.

325

For a long while, bevel-gear-rating equations were put together differently than those for spur and helical gears. Furthermore, the allowable stress values were different. The trend now is to use similar formulas and to bring the stress values together. (Seemingly the stress values would be the same if they were true stresses. Of course, as Section 5.1 points out, the stresses used in gear rating are not really true values. They are “stress numbers” that have been established by experience in developing gear ratings.)

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Dudley’s Handbook of Practical Gear Design and Manufacture

FIGURE 6.24

Contact stress for Regime I and a short-life design.

FIGURE 6.25

Contact stress for Regime III and a long-life design.

The geometry factors and the derating factors for bevel gears in this book follow quite closely the general practice developed in the United States over last years. Using Eq. (6.91) for strength and Eq. (6.96) for surface durability and the definitions of all the factors in Sections 6.2.2 through 6.2.6 results in calculation stress for bevel gears that generally come out high compared with the allowable stresses in Figures 6.22 through 6.26. Sample calculations show that for bevel gears known to be satisfactory in service, calculated stresses predict quick failure. This is a serious problem. How should bevel gears be treated?

Design Formulas

FIGURE 6.26 (HPSTC).

327

Geometry factors for strength of spur gears based on the highest point of single tooth loading

328

Dudley’s Handbook of Practical Gear Design and Manufacture

In time, there will be adjustment in the geometry and the derating factors for both bevel and cy­ lindrical gears. Quite likely, these changes, plus some adjustments in the stress curves, will make it possible to use the same stresses for both bevel and cylindrical gears. It would seem, though, that it will take certain time for this process to run its course. For those using this book, the best plan is the following: • Establish initial designs by the methods shown here, but cross-check these with applicable AGMA standards, proven experience in bevel-gear work, and recommendations by the Gleason Machine Division in Rochester, New York, U.S.A. • Keep in mind that this book does not cover ratings of bevel gears for aerospace work or for vehicle gears. Only industrial bevel gears are covered. • Strength of bevel gears is more critical than that of cylindrical gears. Lower numbers of pinion teeth are advisable, particularly on fully hardened gears. The strength rating is often the controlling rating for bevel gears, whereas the durability rating is generally controlling for cylindrical gears. • The bending stress of a bevel gear calculated by Eq. (6.91) can generally be divided by an ad­ justment factor of 1.4 before it is compared with the strength rating curves of Figure 6.23. • The surface compressive stress of a bevel gear calculated by Eq. (6.96) can generally be divided by an adjustment factor of 1.25 before it is compared with the durability rating curves of Figure 6.25.

6.2.3 GEOMETRY FACTORS

FOR

STRENGTH

The geometry factors for strength is a dimensionless factor that evaluates the shape of the tooth, the amount of load sharing between teeth, and the stress concentration in the root area. There is a standard procedure for handling this factor. The ISO standards now being developed have somewhat different procedures. Chapter 5, Section 5.3, gave a general background on how gear-tooth strength has been handled in the past. The geometry factors for strength presented in this section are based on AGMA standards. The standards show the procedures for determining J factors and, in addition, shows man graphs of J factor values for different gear designs. Although the ISO is developing a somewhat different method for determining geometry factors for strength, the AGMA system has worked quite well in practice and is well accepted in the gear trade. In this book simplified calculation methods are used, and a K t factor—rather than a J factor—is used in the stress formulas. The relation of K t to J is: Kt =

1.0 J

spur gears

Kt =

cos (helix angle) J

Kt =

1.0 J

Kt =

(6.101)

helical gears

(6.102)

straight Zerol bevel gears

(6.103)

cos (spiral angle) J

spiral bevel gears

(6.104)

Since the strength factor is dimensionless, it can be used equally well in metric and English calculations.

Design Formulas

329

Lack of load sharing. Normally the strength of spur-gear teeth is calculated on the basis of the teeth sharing load at the first point of contact and at last point of contact. This is why the critical load is taken at the highest point of single-tooth contact. If the teeth are not cut accurately enough to share load, they may still wear in enough so that load sharing exists before there have been many stress cycles. In some cases, the accuracy may be poor enough or the metal hard enough so that essentially no useful load sharing is achieved. If this happens, then the geometry factor for strength should be determined with full load taken at the tip of the tooth. Table 6.34 gives some typical strength factors for full load taken at the tip. Table 6.35 gives some guideline information as to how much error it takes to cause a failure of load sharing. Calculations for helical gears and bevel gears are not often made for the condition where load sharing does not exist. In general, these gears are either made accurately enough to share load or lapped until a satisfactory contact pattern is achieved. (If they can’t be fitted so that they contact properly, they may be rejected.) Inaccurate low-hardness gears may wear and cold-flow enough to develop relatively good contact patterns—and good load sharing—early in their service life. Helical gears with narrow face width. A good helical gear should have enough face width so that contact ratio in the axial plane is at least 2. If the face width is narrow relative to the pitch diameter of the pinion or if the helix angle is quite low (like 8 or less), the axial contact ratio may be less than 1. How should the strength of a helical pinion or gear be calculated when the axial contact ratio is only 0.50? When the axial contact ratio is less than 1.0, the helical gear can best be thought of as in an inter­ mediate zone between a helical gear and a spur gear. At an axial contact ratio of 0.50, a close ap­ proximation of the geometry factor for strength can be taken midway between the value for a spur gear and the value for a helical gear. It is shown in AGMA standards how to derive the J factor for helical gears, either wide face width or narrow face width. This standard can be used to get data beyond those given in this book. Geometry factors for strength for some standard designs. The geometry factors for strength have been calculated and plotted on curve sheets for several standard designs of spur and helical gears. See Figures 6.26 and 6.27. The method used was that proposed by AGMA. The data shown agree with tabulated data AGMA.

TABLE 6.34 Geometry Factors Kt for Strength of Spur Gears Loaded at the Tip No. Teeth

20° Pressure Angle 1.25 Addendum .00 Addendum

25° Pressure Angle 0.75 Addendum 1.25 Addendum 1.00 Addendum 0.75 Addendum

12

3.95

–

–

3.10

3.70

–

15 18

3.77 3.66

4.50 4.29

– –

2.97 2.88

3.51 3.34

– –

25

3.52

3.97

–

2.77

3.09

–

35 50

3.40 –

3.70 3.57

4.12 3.85

2.67 –

2.92 2.79

3.20 2.99

100

–

3.39

3.51

–

2.66

2.75

275

–

3.26

3.29

–

2.58

2.61

Notes: 1. It is assumed that a pinion with 1.25 basic addendum meshes with a gear of 0.75 addendum and that the tooth thicknesses are adjusted for the change in addendum. 2. It is assumed that a 1.00-addendum pinion meshes with a 1.00-addendum gear and that the tooth thicknesses are standard. 3. These data are for extra-depth teeth cut with a relatively full radius fillet (whole depth 2.35 to 2.40).

330

Dudley’s Handbook of Practical Gear Design and Manufacture

TABLE 6.35 Limiting Error in Action for Steel Spur Gears Load Intensity per Unit of Face Width

Dimensional Error on Line of Action between Contact Points Teeth Share Load

Metric, Wt/b, N/mm

Teeth Failed to Share Load

English, Wt/F, lb/in.

Metric, mm

English, in.

Metric, mm

English, in.

100

571

0.005

0.0002

0.013

0.0005

200 300

1142 1713

0.008 0.010

0.0003 0.0004

0.020 0.030

0.0008 0.0012

400

2284

0.013

0.0005

0.041

0.0016

600 1000

3426 5710

0.020 0.033

0.0008 0.0013

0.061 0.076

0.0024 0.0030

1600

9140

0.050

0.0020

0.122

0.0048

Notes: 1. The error values above are useful only as rough guides. Perfect load sharing, of course, can only occur with perfect accuracy. The meaning of “teeth share load” is that reasonably good load sharing can be expected. The meaning of “teeth fail to share load” is that very little useful load sharing can be expected unless the teeth wear enough to remove most of the error. 2. The “error on the line of action” may be either profile error or spacing error, or some combination of the two. Adjacent spacing error is usually the most troublesome.

Although much design practice is based on AGMA standards, it is not known with certainty that the geometry factors are just right. The ISO method does not agree with the AGMA method. In particular, the relation of spur teeth to helical teeth is different. Figures 6.26 and 6.27 give results that are reliable as long as gear rims are reasonably thick and centrifugal forces are low. When gear research work provides better values, they will probably not be very different from those shown. Geometry factors for the strength of bevel gears are given in Table 6.36. These are used when the unit load is calculated for the middle of the bevel tooth.

6.2.4 OVERALL DERATING FACTOR

FOR

STRENGTH

The overall derating factor evaluates all the things that tend to make the load higher than it should be for the load being transmitted. Specifically, the factor is: Kd =

Ka Km Ks Kv

(6.105)

where: Ka– Km– Ks– Kv–

is is is is

the the the the

application factor load-distribution factor size factor dynamic factor

Each of these factors is explained in the following subsections. The numerical values given should be considered as somewhat typical of what is being used, but not necessarily precise values for any given situation. Most of them are based on experience as much as on theoretical logic. Experience, of course, tends to be an average of known data. Anything that is an average represents the mean of a range of

Design Formulas

331

FIGURE 6.27 Geometry factors for strength of helical gears based on the highest point of single tooth loading and sharing of load between teeth in the zone of contact.

332

Dudley’s Handbook of Practical Gear Design and Manufacture

TABLE 6.36 Geometry Factors Kt for the Strength of Bevel Gears Number of Pinion Teeth

15

20

25

35

50

Number of Gear Teeth

20 35

Straight Bevel Gears

Spiral Bevel Gears

Pinion

Gear

Pinion

Gear

5.00 4.48

5.75 5.56

4.93 3.79

4.93 3.79

50

4.35

5.49

3.21

3.21

100 20

3.79 5.04

4.90 5.04

2.57 4.45

2.57 4.45

35

4.25

5.00

3.62

3.62

50 100

4.15 3.57

4.90 4.35

3.05 2.48

3.05 2.48

20

4.61

4.91

3.80

3.80

35 50

4.11 3.95

4.67 4.61

3.48 2.94

3.48 2.94

100

3.42

4.00

2.42

2.42

35 50

4.14 3.75

4.14 4.20

3.13 2.78

3.13 2.78

100

3.16

3.66

2.35

2.35

50 100

3.71 2.95

3.71 3.33

2.64 2.29

2.64 2.29

Notes: 1. The values in this table are subject to change as refinements are made in the procedure of determining stress intensity due to bending load on gear teeth. 2. This table is based on the normal design and manufacturing procedure recommended by the Gleason Works. In many cases, there will be reason to modify the actual design, and this will tend to cause changed in the geometry factors from the “guideline” values given. 3. The straight bevel data are from Coniflex bevel gears having 20° pressure angle. 4. The geometry factors for Zerol bevel teeth tend to be the same as those for straight bevels.

actual values. Also, the known data may be somewhat limited compared with the unknown data—which would have given a different value, had they been known. Another problem is that much data has been accumulated in the past on gears that were not too high in power and had low-hardness gear teeth. These parts were not too large in size, and the lower hardness was helpful in letting the teeth wear in to accommodate manufacturing errors. As gears get larger in size, they are more difficult to make with good quality; if they are much harder, the teeth have very little tendency to wear in during service. A good example of this kind of changes has occurred in mill gearing. Certain mill gearing formerly transmitted around 1000 kW and used gearing around 300 HV (280 HB). In recent years, the power has gone up to 3000 kW and much gearing is being built with surface hardness over 600 HV (550 HB). In this case, past derating practice need modification to handle the new situation, but the bulk of the experience in the gear trade was accumulated in earlier years on the smaller, lower-hardness gears. How can the gear trade quickly obtain the experience needed to design new products out of the range of older products? The derating values given in this section must be considered as general guides. In a particular contract job, the buyer may specify that the gearing must meet certain design standards. Since derating is so critical and so potentially variable, the gear designer should endeavor to meet these objectives: • Any contract design specifications should either be met or variations negotiated.

Design Formulas

333

• The design should look reasonable by the logic of this book and other books applicable to the gear job at hand. • If the organization building the gears has no good depth of experience in making gears of the size, the hardness, or the kind required, the job should be considered developmental until adequate experience is obtained. This involves things like bench testing of components, factory testing of whole units at full load and full speed for some millions of contact cycles, and then a field eva­ luation of a few prototype production units under actual field service conditions. (A design standard is useful, but it does not remove the need for gear builders to obtain a good depth of experience in the product they are making.) Application factor Ka. This factor evaluates external factors that tend to apply more load to the gear teeth than the applied load Wt . A rough-running prime mover and/or a rough-running piece of driven equipment can seriously increase the effect of the applied load. On an instantaneous basis, the torque being transmitted may be fluctuating considerably. The transmitted torque is then just the mean value of the torque fluctuations. Other things like accelerations and decelerations, rudder turns or a ship, engine misfiring, powersystem vibrations, etc., enter into the application factor. In the past, the service factor was somewhat similar to the application factor. The service-factor concept, though, involved life and reliability as well as overloads. The best present practice treats life on the basis of cycles and reliability on a probability of failure basis. The application factor is used, and service factor is not used. The application factor is determined by experience. An application factor of 1.0 is best thought of as a perfectly smooth turbine driving a perfectly smooth generator at an always constant load and speed. If another application has to have the load reduced 2 to 1 so that the same gears will last as long as in the ideal turbine/generator application, then the application factor is 2.0. Table 6.37 gives application factors for a range of gear applications. Load distribution factor Km. This factor evaluates non-uniform load distribution across the face width and non-uniform load distribution in the meshing direction (transverse plane). The general formula is:

Km = Kmf Kmt

(6.106)

If the gear teeth fit reasonably well, the effect of Km is relatively small and will be taken care of by making Kmf appropriately large. The data given in this book anticipated that a value of 1.0 will be used for Kmt . The factor Kmf should evaluate all the effects that may be due to the following: • Helical spiral of pinion does not match helical spiral of mating gear (helix error effect). • The pinion body bends and twists under load so that there is a mismatch between the pinion and the gear teeth (deflection effects). • The pinion axis, under load, is not parallel to the gear axis. Or, in bevel gears, the pinion axis is not at a 90° axis angle (position error effects, under load). • Centrifugal forces distort the shape of the pinion or gear and mismatch the teeth (centrifugal effects). • Thermal gradients distort the shape of the pinion or gear and mismatch the teeth (thermal effect). • Deliberate design modifications, such as crowning, end easement or helix correction, concentrate the load in one area and relieve the load in another area. This is usually done to lessen the effect of one of the preceding items, but it is an effect in itself (design effects). The listing just given shows why load distribution is one of the most complex subjects in gear design. The later proposed standard method of handling all these variables had about 150 pages—and it was found that it did not adequately cover all the possible situations!

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Dudley’s Handbook of Practical Gear Design and Manufacture

TABLE 6.37 Typical Application Factors Ka for Power Gearing Prime Mover Turbine

Driven Equipment

Motor

Internal Combustion Engine

1.3

1.3

1.7

1.1

1.1

1.3

Generators and exciters: Based load or continuous Pack duty cycle Compressors: Centrifugal

1.7

1.5

1.8

1.7

1.5

1.8

Axial

1.8 2.2

1.7 2.0

2.0 2.5

Rotary lobe (radial, axial, screw, and so forth) Reciprocating

1.5 2.0

1.3 1.7

1.7 –

2.0

1.7

–

1.7 1.5

1.5 1.5

2.0 1.8

2.0

2.0

2.3

Pumps: Centrifugal (all service except as listed below) Centrifugal – boiler feed High-speed centrifugal (over 3600 rpm) Centrifugal – water supply Rotary – axial flow – all types Reciprocating Blowers: Centrifugal

1.7

1.5

1.8

1.7 1.7

1.4 1.4

1.8 1.8

2.0

1.7

2.2

Induced draft Paper industry: Jordan or refiner

Fans: Centrifugal Forced draft

1.5

1.5

–

1.3

1.3

–

–

1.5

–

Pulp beater Sugar industry:

1.5

1.5

1.8

Cane knife

1.7 1.7

1.7 1.7

2.0 2.0

Centrifugal Mill

– –

1.75 1.75

– –

–

1.75

–

Paper machine, line shaft

Processing mills: Autogenous, ball Pulverizers Cement mills

–

1.4

–

Metal rolling or drawing: Rod mills

–

2.0

–

Plate mills, roughing

–

2.75

–

Hot blooming or slabbing

Notes: 1. The values given are illustrative. As more experience is gained, new applications factors will be established in the gear trade. 2. The values given may vary in a multistage drive. Experience and study will often show that the first stage needs a different application factor than that needed for the last stage. 3. The power rating and the kind of gear arrangement affect the application factor. The values given here represent somewhat average situations. (Be wary of new gear designs of high power. The old experience on application factors may be wrong for the new situation.)

Design Formulas

335

Effect of helix error and shaft misalignment. In this section, some limited data will be given on misalignment effects that are due to error in matching the helical spirals and on deflection errors that are due to the aspect ratio of the pinion. Table 6.38 gives some target values that the designer should try to meet. (Instead of figuring how high the Km is when all tolerances and deflections are allowed for, figure close the tolerances and the deflections must be held to get a reasonable value of Km .) Figure 6.28 shows the approximate face-load-distribution factor as a function of mismatch error et and intensity of load. The curves were plotted up to Kmf = 2.0 from the relations: Kmf =

Kmf =

10, 000b × et + 1 metric Wt

1, 450, 000 F × et + 1 Wt

(6.107)

English

(6.108)

In Equations (6.107) and (7.107), Wt is in Newtons for the metric calculation and in pounds for the English calculation. The face width and error are in millimeters for the metric calculation and in inches for the English calculation. The answer is dimensionless. The basics for Equations (6.107) and 6.108) is that the total mesh deflection is: Mesh deflection =

load stiffnes constant × face width

TABLE 6.38 Some Target Values of Km for Spur and Helical Gears Hardness of Gear Set and Load per Unit of Face Width

Target Values of Km for Gears with Face Width 50 mm (2 in.) 100 mm (4 in.) 250 mm 750 mm (10 in.) (30 in.)

High hardness (675 HV):

Wt / b = 100 (Wt / F = 571)

1.7

1.9

–

–

Wt / b = 300 (Wt / F = 1713)

1.4

1.6

1.8

–

Wt / b = 800 (Wt / F = 4568)

1.2

1.3

1.6

1.8

Medium hardness (300 HV):

1.8

Wt / b = 100 (Wt / F = 571)

1.5

1.6

1.8

–

Wt / b = 300 (Wt / F = 1713)

1.2

1.3

1.6

1.8

Wt / b = 800 (Wt / F = 4568)

1.1

1.2

1.4

1.5

Wt / b = 100 (Wt / F = 571)

1.3

1.4

1.5

1.6

Wt / b = 300 (Wt / F = 1713)

1.1

1.2

1.4

1.5

Wt / b = 800 (Wt / F = 4568)

1.0

1.1

1.2

1.3

Low hardness (6210 HV):

Notes: 1. These values assume that the accuracy is adjusted, with high-hardness gears being more accurate than low-hardness gear. 2. For gears over 100 mm face width, it is assumed that each set is matched or fitted to get an acceptable contact pattern at full load. 3. It is assumed that the low-hardness gears wear in (or cold-flow the metal) to improve the contact. This happens to a lesser extent with medium hard gears, and almost no useful wear-in occurs with high-hardness gears.

336

Dudley’s Handbook of Practical Gear Design and Manufacture

FIGURE 6.28 Approximate load-distribution factors Kmf from combined effects of helix mismatch and shaft misalignment.

=

Wt 20,000 b =

metric

Wt 2,900,000 F

(6.109)

(6.110)

When Kmf is over 2.0, the error is great enough to make the face width in contact less than the total face width. The equation for the face-load-distribution factor then becomes: Kmf =

Kmf =

40,000 b et Wt 5,800,000 F et Wt

metric

English

(6.111)

(6.112)

The choice of stiffness constant of 20,000 N/mm2 (2,900,000 psi) deserves some comment. Tests of gear teeth show that this is good average value for typical gear design. If it is an error, it is more apt to be a little low than a little high. In a limited amount of testing, it is found that 23,000 N/mm2 to be typical for high-strength tooth design. In earlier years, values like 2,000,000 psi and even 1,000,000 psi have been used (2,000,000 psi is equivalent to about 14,000 N/mm2). These values seemed to give reasonable correlation with results observed in practice. The reason they did was that either the teeth tend to wear in to develop a good fit or the gears were flexible enough in their mountings to shift so that the contact was better than it should have been, based on errors and true stiffness of teeth. The problem with this design approach is that large gears and very hard gears shift scarcely at all, and they don’t wear in, either. Therefore, it is best to make a relatively true calculation of Kmf . If there is

Design Formulas

337

wear-in or gear-body shifting, this can more appropriately be allowed for by determining the compen­ sating amounts directly and then subtracting them from the helix error. Table 6.39 shows amounts of wear-in ew that might be expected. There is relatively little good data on wear-in of gears, and so this table should be considered a good opinion on what is likely to happen rather than the results of any study in depth. Aspect-ratio effects. The relative slenderness of the pinion is called the aspect ratio. The aspect ratio ma is the contacting face width of the pinion divided by the pitch diameter. For double helical pinions, the best practice is to use the total face width (the width of the two helices plus the gap between the helices). When the aspect ratio approaches 2.0, both single helical and double helical pinions bend and twist enough to tend to develop relatively high Kmf values. Figure 6.29 shows a plot of Kmf against the aspect ratio for both single enveloping and double enveloping pinions. This curve sheet was drawn to fit these assumptions: • The teeth are cut to true and exactly matching helix angles. • The pinions and gears are straddle-mounted on bearings, and the bearings are a reasonable distance from the tooth ends. • There is no wear-in to compensate for deflection. • An approximation formula was used to get the deflection. The deflection was taken as an error and converted to Kmf by the formulas just discussed. Since deflection is proportional to load intensity, the answer in Kmf is the same regardless of the load intensity.

TABLE 6.39 Approximate Wear-in Amounts ew That May Be Distribution Factors for Bevel Gears, Realized with Approximate Lubricants and a Good Break-in Procedure Material Hardness HV

HB

210 320

200 300

Initial Misfit in Helix Width 0.023 mm (0.0009 in.) mm in.

0.038 mm (0.0015 in.) mm in.

0.064 mm (0.0025 in.) mm in.

0.010 mm (0.0040 in.) mm in.

Regime I—less than 1 m/s (200 fpm) pitch-line velocity 0.0150 0.0125

0.0006 0.0005

0.0230 0.0200

0.0009 0.0008

0.0380 0.0300

0.0015 0.0012

0.0500 0.0380

0.0020 0.0015

415

400

0.0100

0.0004

0.0180

0.0007

0.0230

0.0009

0.0300

0.0012

530 675

500 600

0.0075 0.0050

0.0003 0.0002

0.0125 0.0075

0.0005 0.0003

0.0150 0.0125

0.0006 0.0005

0.0250 0.0200

0.0010 0.0008

210 320

200 300

0.0125 0.0100

0.0300 0.0250

0.0012 0.0010

Regime II—less than 5 m/s (1000 fpm) pitch-line velocity 0.0005 0.0004

0.0180 0.0150

0.0007 0.0006

0.0250 0.0200

0.0010 0.0008

415

400

0.0075

0.0003

0.0125

0.0005

0.0150

0.0006

0.0200

0.0008

530 675

500 600

0.0500 0.0025

0.0002 0.0001

0.0075 0.0050

0.0003 0.0002

0.0125 0.0100

0.0005 0.0004

0.0150 0.0125

0.0006 0.0005

210 320

200 300

0.0075 0.0050

0.0150 0.0125

0.0006 0.0005

Regime III—less than 20 m/s (4000 fpm) pitch-line velocity 0.0003 0.0002

0.0100 0.0075

0.0004 0.0003

0.0125 0.0100

0.0005 0.0004

415

400

0.0050

0.0002

0.0050

0.0002

0.0075

0.0003

0.0100

0.0004

530 675

500 600

0.0025 0.0025

0.0001 0.0001

0.0025 0.0025

0.0001 0.0001

0.0050 0.0025

0.0002 0.0001

0.0075 0.0050

0.0003 0.0002

338

FIGURE 6.29

Dudley’s Handbook of Practical Gear Design and Manufacture

Effect of pinion bending and twisting on load distribution across the face width.

Figure 6.29 makes it look like the double helix does better than the single helix. This is really not so, because of the gap. When a single-helix pinion and a double-helix pinion have the same working face width, the Kmf value tends to be about the same. The high Kmf value for an aspect ratio around 2.0 can be reduced considerably by helix modification. In general, helix modification (also called helix correction) should be given strong consideration when the aspect ratio of single helical pinions exceeds 1.15 or that of double helical pinions exceeds 1.60. The problem of Kmf begins to look rather formidable from all the foregoing. Some things can be done, though, to improve the situation for spur and helical gears. (Bevel gears are discussed next.) • If the aspect ratio is under 1.15 and the pinion and gear are straddle-mounted in their bearings, there should be no serious effect from bending and twisting. Of course, if the pinion is overhung, special calculations will be needed to find the overhung deflection, even with an aspect ratio of 1.0. The main problem when the aspect ratio is under 1.15 is manufacturing error. • If the aspect ratio is over 1.5, the bending and twisting effects need to be calculated. Helix cor­ rection is apt to be needed. When helix correction is made, the pinions are generally fitted to their gears so that under no load (or very light load), the contact pattern is open where the teeth have been relieved. Using special measuring techniques, the amount the teeth are open can be measured. If the pinion does not match its gear properly, it can be re-shaved or re-ground. (In some cases, bearings may be shifted slightly to fix the fit.) When the fitting job is done, the desired mismatch to compensate for deflection has been achieved, and manufacturing errors in cutting teeth or boring cases have been taken care of. • When the aspect ratio is quite low—like 0.25 or less—there is often a tendency for the gear (or both pinion and gear) to deflect or shift so as to distribute the load relatively evenly, even when an appreciable helix mismatch exists. Tests and observations may reveal that a pair of gears that would have a Kmf of around 2.5 may actually run with a real Kmf of only 1.4! It is beyond the scope of this book to represent engineering analysis data on how gears can deflect sideways or shift on their mountings to achieve a substantial reduction in Kmf (or Cmf). The designer should be alert for this possibility and test new designs thoroughly as that any favorable reduction in Kmf can be noted and used.

Design Formulas

339

As a matter of interest, it is quite common for vehicle transmission gears to have a low aspect ratio and a favorable tendency to run at much lower Kmf values than a simple calculation based on helix errors and tooth load per unit of face width would indicate. • Helical gears are loaded on an inclined line that runs from the top of the tooth to the limit diameter. Under misaligned conditions, the area of high loading is localized, and the developed field of stress in the tooth root is not as severe as it would be for a spur gear. It is often practical to use a somewhat lower Kmf for helical gears than for spur gears when the aspect ratio is low. For high-aspect-ratio gears, Kmf is generally considered to be equal to Cmf, the durability factor for misalignment effects. Load distribution factor Kmf for bevel gears. Bevel gears are generally made with the following design controls: • The face width is 0.30 or 0.25 of the outer cone distances (an aspect ratio control). • The teeth are cut or ground so that the tooth contact is localized. A slight crown is introduced to compensate for small errors in tooth making and in mounting the bevel gears on their axes. • The bevel gears are tested in special contact-checking machines. If the fit is not right, the bevel gears are either rejected or lightly lapped to make the fit acceptable. If the misfit is more than what lapping can correct, the pinion or gear may be re-cut or re-ground to bring the tooth fit on the tester under control. The results of these special practices and controls is to make it much easier to establish reasonable loaddistribution factors for bevel gears than for spur or helical gears. Table 6.40 shows the general pattern of load-distribution factors for bevel gears. The same loaddistribution factor is generally used for both strength and durability rating calculations (Km = Cm ). Industrial bevel gears are often made in large sizes. In general, both members are straddle-mounted. Figure 6.30 shows the general trend for the increase in mounting factor as the parts get larger and there is more difficulty in controlling the accuracy. The mounting factors for bevel gears are right for the rated load. The reason for this is that the localized contact is developed to suit the rated torque. If the bevels are operated at lower than rated torque, the mounting factors tend to increase because the fit is not right. This is not too critical, though, because the torque generally decreases faster than the load-distribution factor increases. A serious problem can occur when a bevel set has to handle very high overload torques for a small number of cycles (like 105) and then must run for a large number of cycles (like 109) at rated torque. At

TABLE 6.40 Load-Distribution Factors for Bevel Gears, Km and Cm. Application

Both Members Straddle-Mounted

One Member Straddle-Mounted

Neither Member Straddle-Mounted

General industrial Vehicle

1.00–1.10 1.00–1.10

1.10–1.25 1.10–1.25

1.25–1.40 –

Aerospace

1.00– .25

1.10–1.40

1.25–1.50

Notes: 1. These values are based on setting the position on the cone element very close to the right position. For instance, if a bevel gear with 25 mm (1 in.) face width was out of position about 0.15 mm (0.006 in.), a Km value of 1.05 would increase to about 1.8! 2. These values are based on the face width not exceeding 25 mm. Industrial bevel gears are made in rather large sizes. Figure 6.30 shows how the load-distribution factor tends to increase as the face width increases.

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Dudley’s Handbook of Practical Gear Design and Manufacture

FIGURE 6.30 Load-distribution factor for bevel gears, Km and Cm . This curve is applicable for straddle mounting only and industrial units.

rated torque, the load distribution will not be as good as it should be, and a higher-than-normal loaddistribution factor will be needed. Size factor Ks. The size factor de-rates the gear design for the adverse effect of size on material properties. Large gears may have the same hardness as small gears, but the material may not be as strong or fatigue-resistant. Inclusions or other flaws in the steel tend to be more numerous in a large stressed area than in a small stressed area. The making of gears involves pouring ingots, forging ingots, quenching and normalizing of forgings, hardening and tempering of gears, etc. All these operations can be done with better control on small part than on very large parts. The size factor tends to go up to around 2 in going from rather small gears to very large gears—when all the adverse effects of size are considered. Fortunately, some things can be done to decrease the effect of size. If the steel used is chosen to have enough alloy content, a much better heat-treating response can be obtained. Also, if temperatures or other variables are somewhat out of control, the richer-alloy steels are more forgiving of less than optimum conditions. Figure 6.31 shows size factors for the strength of spur, helical, and bevel gears. The bevel curve is not the one normally used in older standards. The old curve had a unity value for a very large tooth and a 0.5 value for a small tooth. This was compensated for by using about one-half the allowed stress in bending for bevel teeth as for spur and helical teeth. The new trend is to use the same allowed stresses for spur, helical, and bevel teeth. This is done by making the size factor greater than unity as size becomes detrimental. Dynamic load factors Kv and Cv. The dynamic factor makes allowance for overload effects generated by a pair of meshing gears. If both members of the gear pair were perfect, a uniform angular rotation of one member would result in a uniform angular rotation of the other member. The ratio of the number of gear teeth to the number of pinion teeth would be exact ratio of the two angular velocities. Gear teeth are never perfect, although a high-precision pair of gears is much more perfect than a lowprecision or “commercial” accuracy pair. The tooth error results in a transmission error, which makes the ratio of input speed to output speed tends to fluctuate. On an instantaneous basis, each member of the gear pair is constantly going through slight accelerations and decelerations. This results, in turn, in dynamic forces being developed because of the mass of the pinion and its shaft and the gear and its shaft.

Design Formulas

FIGURE 6.31

341

Size factor for strength of spur, helical, and bevel gears.

Tooth errors in spacing, runout, and profile cause transmission error. Helix errors influence trans­ mission error because of their indirect effect on load sharing between teeth and their effect on the stiffness constant for the mesh. The dynamic factor is used as a de-rating factor to compensate for the adverse effect of the dynamic overloads caused by the driving or driven machinery. The application factors Kv and Cv are used to handle dynamic loads unrelated to gear-tooth accuracy. The equation for dynamic factor is: Kv = Cv =

Wt Wt + Wd

metric or English

(6.113)

where Cv– is the dynamic factor for durability Wt– is the transmitted load, newtons or pounds Wd– is the dynamic load, newtons or pounds

Equation (6.113) makes the dynamic factor for strength the same as the dynamic factor for durability. The definition used for dynamic load is unrelated to the kind of trouble that the load can cause when it is too great. Excessive dynamic loads are a hazard to gear-tooth strength, gear-tooth surface durability, and gear-tooth scoring resistance. For AGMA calculations of bending stress or contact stress, it has been customary to put the dynamic factor in the denominator, whereas the other derating factors are put in the numerator. This makes a high dynamic factor come out to some value like 0.50 instead of 2.0. The ISO work puts the dynamic factor in the numerator. In this book, AGMA practice will be followed. Figure 6.32 shows dynamic factors for spur and helical gears. A range of values is shown for four general levels of gear accuracy. These levels can be described as follows:

342

FIGURE 6.32

Dudley’s Handbook of Practical Gear Design and Manufacture

Dynamic factor for strength and durability for spur and helical gears.

A. High accuracy Generally ground gears. May be gears hobbed or shaped to very close limits and then precisionshaved. The tooth design provides good profile modification and helix modification (across the face width). Tooth spacing (both tooth-to-tooth and accumulated), profile (both slope and mod­ ification), helix (crown, end easement, and/or allowance for bending and twisting), gear runout, and gear surface finish (smoothness and lack of waviness) are held to very close limits. The modifications in profile and helix are based on a single load that is reasoned to be the significant load determining the rating. B. Medium high accuracy Ground or shaved gears, but quality is a significant step lower than that outlined above. C. Medium accuracy Gears are precision-finished by hobbing or shaping. Modifications in profile or helix are either not made or made without close control. This is the best that skilled gear people can do by taking extra time for finishing and using the most suitable machine tools. D. Low accuracy Gears are hobbed or shaped to normal practice. Profile modifications are either not made or not made with close control. Machine tools used are good, but may not have the accuracy capability of the newest types. These levels of accuracy cannot be tied closely to the limits recommended by AGMA. Some requisite quality items are not specified in AGMA standards, and some are not set to match the latest machine tool capability. As a rough guide, though, high accuracy is like AGMA quality level 12 to 13, and medium accuracy is like AGMA quality level 6. DIN quality grade 5 is in the high-accuracy range. The range shown in Figure 6.32 is typical of the practice that has been followed successfully for many years. A gear designer can judge how closely every accuracy item is apt to be held in gears that will be made to a given design. If the gears are going to be extra good for the way they are made, then there is reason to go to the top of the range. Figure 6.32 stops low-accuracy gears at a pitch-line velocity of 15 m/s (3000 fpm) and medium-highaccuracy gears at 35 m/s (7000 fpm). Generally speaking, these kinds of gears are not apt to be used at

Design Formulas

343

higher speeds, although they might give satisfactory service. (If a low-accuracy gear is run too fast, there is apt to be trouble with noise and vibration, besides the uncertainty as to whether or not the gear unit will perform as well as its rating would anticipate.) High-accuracy gears (and super-high-accuracy gears) are being used with relatively good success at pitch-line velocities up to 200 m/s (40,000 fpm). What dynamic load is appropriate for high-precision gears in the 40- to 200 m/s speed range? High-speed gears must be made very precisely. In general, the dynamic load factor will be quite low – from 0.85 to 0.96. To get the best results, the teeth need to be helical, with enough face width and axial contact ratio to get two axial crossovers. The following values show what may be expected in high-speed gearing: • Helical, axial contact ratio over 2.0, truly high precision in all details Kv = Cv = 0.95 • Helical, axial contact ratio of 1.0, truly precision Kv = Cv = 0.90 • Helical, high precision, but profile and helix corrections not made Kv = Cv = 0.85 • Spur truly high precision in all details. (The high-speed spur gear needs to be narrow in face width to run above 40 m/s. Generally, the upper limit for spur gears is around 20 m/s) Kv = Cv = 0.85 The dynamic load factors for bevel gears are shown in Figure 6.33. The curves represent the practices that were followed successfully for a long while.

FIGURE 6.33

Dynamic factor for strength and durability for bevel gears.

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Dudley’s Handbook of Practical Gear Design and Manufacture

Straight bevel gears are not often used above 10 m/s (2000 fpm) pitch-line velocity. If they are not cut to precision accuracy and fitted for the best contact pattern, the dynamic factor should be read from the second or third curve. Spiral bevel gears that are finished by cutting and fitted for a good contact pattern should use the dynamic factor given by the second curve.

6.2.5 GEOMETRY FACTORS

FOR

DURABILITY

The geometry factors for durability evaluate the shape of the tooth and the amount of load sharing between teeth. The I factors shown in AGMA standard are geometry factors, but not the same kind as those in this book, with one exception: The Ck factor given by AGMA for spur and helical vehicle gears is the same kind. The relation of Ck to I is: Ck =

Ck =

Cp 12.043 Cp 1.00

1 u I u+1 1 mG I mG + 1

metric

(6.114)

English

(6.115)

where: Cp – is the constant for material and gear type (Cp = 2300 for steel spur or helical gears, and Cp = 2800 for steel straight, spiral, or Zerol bevel gears) I – is the dimensionless constant defined by AGMA standards u – is the metric symbol for tooth ratio mG – is the English symbol for tooth ratio

Figure 6.34 shows geometry factors for the durability of spur gears of standard-addendum and 25% longaddendum pinion designs for both 20 and 25 pressure angles. These geometry values are based on the most critical stress being taken at the lowest point of single tooth contact (LPSTC). The choice of the lowest point is felt to be more conservative than determining the stress at the pitch line. If the number of pinion teeth is over 25 and the contact ratio is 1.7 or higher, there is not much practical difference between stress taken at the lowest point of single tooth contact and those taken at the pitch line. Figure 6.35 shows geometry factors for helical-gear teeth of 15 and 30 helix angles and normal pressure angles of 20 and 25 . The critical stress is determined by an AGMA method that allows for load sharing in the zone of action. It is assumed that the axial contact ratio is 2.0 or more (minimum of two axial crossovers). If the axial contact ratio is only 1.0, the geometry factors will increase by a small amount. AGMA standards have provision for calculating this special case. If the axial contact ratio is less than 1.0, the helical gear approaches the spur gear. (At an axial contact ratio of 0.5, the character of the gearset is about halfway between the character of a spur gear and that of a helical gear.) Table 6.41 shows geometry factors for the durability of bevel gears. It seems likely that gear research will develop values for the durability geometry factors that are somewhat more complex and therefore more accurate than those given in Figures 6.34 and 6.35 and in Table 6.41. There is not much change, though, that they will change by any large amount. Experience in the gear trade shows that the geometry constants for surface durability are quite good.

Design Formulas

345

FIGURE 6.34 Geometry factors for durability of spur gears based on the lowest point of single tooth contact (LPSTC).

346

Dudley’s Handbook of Practical Gear Design and Manufacture

FIGURE 6.35 Geometry factors for durability of helical gears based on tip loading and sharing of load between teeth in the zone of contact.

6.2.6 OVERALL DERATING FACTOR

FOR

SURFACE DURABILITY

The overall derating factor evaluates all the things that tend to make the load higher than it should be for the torque being transmitted. The factor is: Cd =

Ca Cm Cs Cv

metric or English

(6.116)

Design Formulas

347

TABLE 6.41 Geometry Factors Ck for Durability of Bevel Gears No. Pinion Teeth 15

20

25

No. Gear Teeth

Straight Bevel Gears

Spiral Bevel Gears

20

8433

7780

35

8731

7281

50 100

8967 8752

6837 6260

20

7952

7537

35 50

8102 8315

7285 6803

100

8144

6283

20 35

7766 7808

7379 7251

50

7935

6742

35

100 35

7766 7565

6271 6956

50

7593

6627

50

100 50

7317 7353

6222 6458

100

6989

6121

Notes: 1. These values are subject to change as refinements are made in the procedure to determine stress intensity due to bending load on gear teeth. 2. This table is based on the normal design and manufacturing process recommended by the Gleason Works. In many cases, there will be reason to modify the actual design, and this will tend to cause changes in the geometry factors from the “guideline” values given here. 3. The straight bevel data are for Coniflex bevel gears with a 20° pressure angle. 4. The spiral bevel data are for 20° pressure angle, 35° spiral angle bevel gears. 5. The geometry factors for Zerol bevel teeth tend to be the same as those for straight bevel gears.

where: Ca – is the application factor. Generally, Ca is taken to be the same as Ka . See Section 6.2.4 Cm – is the load distribution factor. This factor is generally taken to be the same as Km . See Section 6.2.4. In some special cases, Km may justifiably be lower than Cm . Cs – is the size factor. This factor is not the same as Ks . Cv – is the dynamic factor. This factor is the same as Kv . See Section 6.2.4

Since the overall derating for surface durability is rather close to being the same as the overall derating factor for strength, Section 6.2.4 should be read for general information and for specific data on the factors that are the same in derating for strength and for durability. Size factor Cs. From a surface durability standpoint, face width is probably the best way to evaluate the detrimental effect of size. (For tooth breakage, tooth size seems to be the best measure of the size effect.) The size factor for durability (like that for strength, discussed in Section 6.2.4) is intended primarily for derating gears for the fact that large pieces of steel tend to have more flaws and are more difficult to forge and heat-treat for ideal metallurgical properties than are small pieces of steel. AGMA standards have acknowledged the need for a size factor in the rating formula for durability of spur and helical gears. However, no numerical values have yet been agreed upon and written into standards. It is left to each designer to assign an appropriate value when large gears cannot be made with

348

Dudley’s Handbook of Practical Gear Design and Manufacture

metal quality comparable with that of the small gears that were used to set basic allowable stresses in the standards. For bevel gears, a size factor Cs , with numerical values is set. This standard, though, deviates from spur and helical practice in that the detrimental effect of size is not handled by a factor greater than unity. Instead, a factor near unity is given to a large and a factor of 0.5 to a small gear. This, in effect, says that the large gear is the standard of reference and gives the small gear credit for being better than the large gear. Large gears are then derated for the fact that their size tends to give imperfection that reduces the amount of stress that can be carried. AGMA standards issued to cover spur, helical, and spiral bevel gears. These standards show a minimum Cs value of 1.0 for spur and helical gears, but it does show Cs values less than 1.0 for spiral bevel gears. Figure 6.36 shows a plot of size factors for durability versus face width. The spiral bevel curve is worked out to place the unity value on the small gear and de-rate the large gear (rather than up-rate the small gear). This change was made for this book by shifting the allowable contact stress so that the value used is right for the small gear at Cs = 1.0. The size factor—by AGMA definition—does include “area of stress pattern.” This variable shifts more in going from small to large bevel gears than it does in spur or helical gears. This has to do with the way the teeth are made and fitted. (It must be kept in mind that bevel teeth are on the surface of a cone, while spur and helical teeth are on a cylinder.) The size factor for spiral bevel gears changes about 2 to 1 in going from a small gear to a large gear. The dashed curve for spur and helical gears changes about 1.5 to 1.0 in going from a small gear to a very large gear. The relative difference is essentially due to the geometric difference between bevel gears and helical gears. For spur and helical gears, the curve for Cs in Figure 6.36 is shown dashed. The curve represents what can normally be expected in good industrial gear manufacture. Complementary considerations. Here are the considerations those relate to recommended value of size factor Cs for gears of various applications: • Up to 125 mm (5 in.) face width, it should be possible to pick steel and process it well enough to have essentially no size effect. Cs = 1.00. • Above 125 mm face width, it becomes increasingly difficult to avoid size effects. A gearset with about 400 mm (16 in.) face width is apt to have a pinion around 400 mm pitch diameter mating

FIGURE 6.36

Size factor for durability of spur, helical, and spiral bevel gears.

Design Formulas

349

with a gear in the range of 1600 mm (64 in.) pitch diameter. Metallurgical studies11 of gears of this size generally show quality degradations ranging from small to very serious. A size derating of 1.3 for a gear with 400 mm face width seems to be about average for industrial gear manufacture around the world by those skilled in the gear art. (Those unskilled may, of course, do much worse.) • In the aerospace field, great skill and effort are used to keep gear metallurgical quality under close control. Aerospace people could probably make gears up to 250 mm face width without any noticeable size effects. (Most aerospace gears are less than 125 mm face width.) • Marine gears are made in face widths up to 1000 mm (40 in.). Marine practice for ocean-going ships is highly specialized. An appropriate size derating for a 1000-mm marine drive might be around Cs = 1.20.

6.2.7 LOAD RATING

OF

WORM GEARING

The complete worm-gear family comprises: • Crossed-helical gears (non-enveloping) • Cylindrical worm gearing (single-enveloping) • Double-enveloping worm gears This section will cover methods for estimating the load-carrying capacity of each of the three family members. There are no trade standards on crossed-helical gears. The synopsis given here will be in English symbols and units only. (Readers can probably best check the reference by sticking to English units.) The rating for cylindrical worm gearing will follow AGMA standards. This is a good guide developed over many years. Not much change can be anticipated in the immediate future. The rating for double-enveloping worm gears will follow AGMA standards. This again gives good guidance and will probably not change much in the immediate future. The rating of all kinds of worm gearing is primarily based on surface durability. Tooth strength is usually not much of risk—unless abnormally small threads or teeth are sued. For this reason, strength in handled on a very approximate basis. The set is sized from the durability estimates. Crossed-helical-gear durability. Crossed-helical gears are generally rated only by a surface-durability formula. With point contact, not enough load can be carried to cause much danger of tooth breakage. The discussed method is a handy general formula for crossed-helical gears, which gives very reasonable results. The first step is to calculate a wear load Ww: Ww = A6 B3K Q

(6.117)

The factors A and B depend on the ratio of the radii of curvature of the profiles of the driver and driven. The radius of curvature of the driver is: R c1 =

D 1 sin 2 cos2

n

(6.118)

1

The radius of curvature of the driven member is determined by a similar equation. The pitch diameter of the driver is D 1, and the normal pressure angle is n . The helix angle is 1.

11

See Section 10.2.4.

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Dudley’s Handbook of Practical Gear Design and Manufacture

Factors for rating crossed-helical gears. Table 6.42 shows the constants A6 and B3. The constant K is not the same as the K factor discussed in Chapter 5. This K is: K = 29.7662 s 3

1 1 + E1 E2

2

(6.119)

The value s is the stress on the tooth surface, and E is the modulus of elasticity. The value Q is a ratio factor: Q=

R c 1 Rc 2 Rc1 + Rc2

2

(6.120)

The allowable K for crossed-helical gears that Buckingham recommends is given in Table 6.43. It should be noted that Table 6.43 shows quite a difference in capacity with running in. A little wear at light load tends to broaden the point of contact considerably. The calculated wear load in Eq. (6.117) should be compared with the dynamic load in the normal plane. If the torque on the driver is T1, the transmitted load in the normal plane is: Wn =

2 T1 D 1 cos 1 cos

(6.121) n

If high degree of accuracy is obtained, it is usually possible to keep the dynamic load on crossed-helical gears within about 150% of the transmitted load in the normal plane. The calculated wear load should be enough less than the transmitted load to allow for dynamic-load effects and to allow for any application factor that may be appropriate for the job. There is little conclusive information to go on to judge the limiting rubbing velocity that can be handled by spiral gears of different materials. Hardened steel on bronze will handle the most speed. With ordinary materials and good commercial accuracy, the rubbing speed should be probably not exceed 30 m/s (6000 fpm). Use of special bronze material in combination with a case-hardened and ground driver makes it possible to handle rubbing speed up to 50 m/s (10,000 fpm) in aircraft and steam-turbine applications. The other material combinations shown in Table 6.43 should probably not be used above a rubbing speed of 20 m/s (4000 fpm) in ordinary applications.

TABLE 6.42 Factors for Rating Crossed-Helical Gears A6

B3

A6B3

1.000

0.560

1.0000

0.560

1.500

1.302

0.4490

0.583

2.000 3.000

2.411 6.053

0.2520 0.1120

0.609 0.678

4.000

11.620

0.0640

0.744

6.000 10.000

30.437 106.069

0.0292 0.0108

0.889 1.141

Curvature Ratio, Rc2/Rc1

Design Formulas

351

TABLE 6.43 Load-Stress Factors for Crossed-Helical Gears Pinion (Driver)

Gear (Driven)

s, psi

K, lb

Hardened steel

Hardened steel

Hardened steel Cast iron

150,000

446

Bronze, phosphor Bronze, phosphor

83,000 83,000

170 302

Cast iron

90,000

385

–

446

With initial point contact

Cast iron

With short running-in period Hardened steel

Hardened steel Hardened steel

Bronze, phosphor

–

230

Cast iron Cast iron

Bronze, phosphor Cast iron

– –

600 770

– –

446 300

With extensive running in Hardened steel Hardened steel

Hardened steel Bronze, phosphor

Cast iron

Bronze, phosphor

–

1200

Cast iron

Cast iron

–

1500

The sliding or rubbing velocity may be calculated by the formula: vs = 0.262

n 1D 1 sin 1

fpm

(6.122)

where n1 is the rotational speed of the driver in rpm. A special word of caution should be added. The crosswise rubbing in crossed-helical sets tends to destroy the EHD oil film. This means that these gears can get into more trouble than other worm gears, bevel gears, or regular helical gears, when the oil is thin or surface finishes are poor. Those using crossed-helical gears need past experience or test-stands results to make sure that the choice of lubricant and the quality of the tooth-surface finishes are appropriate for the application. Cylindrical-worm-gear durability. The load-carrying capacity of worm gearsets may be estimated from the general formula given below. Since this formula has been developed more by experience than by rational derivation, it is more reasonable to calculate tangential load capacity than it is to calculate stress and compare the calculated stress to allowable stress. The general formula for cylindrical worm gearsets is: Wt =

Ks dp0.8 .2 b Km Kv 75.948

Wt = Ks D 0.8 Fe Km Kv

N metric

(6.123)

lb English

(6.124)

where: Ks – is the materials factor (see Table 6.44; this is different from the Ks size factor). d2p or D – is the worm-gear pitch diameter, mm or in.

352

Dudley’s Handbook of Practical Gear Design and Manufacture

TABLE 6.44 Material Factor Ks for Cylindrical Worm Gears For Units of 75 mm (3 in.) Center Distance to About 1 Meter (40 in.) Center Distance Gear Pitch Diameter Sand Cast Static Chill Cast mm

in.

65 75

2.5 3

1000 960

Centrifugal Cast

– –

– –

100

4

900

–

–

125 150

5 6

855 820

– –

– –

175

7

790

–

–

200 250

8 10

760 715

1000 955

– –

375

15

630

875

–

505 635

20 25

570 525

815 770

– 1000

760

30

–

740

985

1015 1270

40 50

– –

680 635

960 945

1775

70

–

570

920

For Units with Less than 75 mm (3 in.) Center Distance Maximum Ks Value

Center Distance mm

in.

12 25

0.5 1.0

725 735

38

1.5

760

50 63

2.0 2.5

800 880

75

3.0

1000

Notes: 1. For bronze worm gear and steel worm with at least HV 655 (HRC 58) surface hardness. 2. See Chapter 10 for worm-gear-material data. 3. Sliding velocity not to exceed 30 m/s (6000 fpm); worm speed not more than 3600 rpm.

b or Fe – is the effective face width, mm or in. (not exceed 2/3 of the worm pitch diameter). Km – is the ratio correction factor, dimensionless (see Table 6.45). Kv – is the dynamic factor, dimensionless (see Table 6.46).

The sliding velocity is: vs =

dp1 n 1 19, 098 cos

vs = 0.2618

where:

dn w cos

m/s

fpm

metric

(6.125)

English

(6.126)

Design Formulas

353

TABLE 6.45 Ratio Correction Factor Km for Cylindrical Worm Gears Ratio, u (mG)

Km

Ratio, u (mG)

Km

3.0

0.500

14.0

0.799

3.5

0.554

16.0

0.809

4.0 4.5

0.593 0.620

20.0 30.0

0.820 0.825

5.0

0.645

40.0

0.815

6.0 7.0

0.679 0.706

50.0 60.0

0.785 0.745

8.0

0.724

70.0

0.687

9.0 10.0

0.744 0.760

80.0 90.0

0.622 0.555

12.0

0.783

100.0

0.490

TABLE 6.46 Velocity Factor Kv for Cylindrical Worm Sliding Velocity m/s

Velocity Factor, Kv

fpm

Sliding Velocity m/s

fpm

Velocity Factor, Kv

0.005 0.025

1 5

0.649 0.647

3.0 3.5

600 700

0.340 0.310

0.050

10

0.644

4.0

800

0.289

0.100 0.150

20 30

0.638 0.631

4.5 5.0

900 1000

0.272 0.258

0.200

40

0.625

6.0

1200

0.235

0.300 0.400

60 80

0.613 0.600

7.0 8.0

1400 1600

0.216 0.200

0.500

100

0.588

9.0

1800

0.187

0.750 1.000

150 200

0.588 0.588

10.0 11.0

2000 2200

0.175 0.165

1.250

250

0.500

12.0

2400

0.156

1.500 1.750

300 350

0.472 0.446

13.0 14.0

2600 2800

0.148 0.140

2.000

400

0.421

15.0

3000

0.134

2.250 2.500

450 500

0.398 0.378

20.0 25.0

4000 5000

0.106 0.089

2.750

550

0.358

30.0

6000

0.079

354

Dudley’s Handbook of Practical Gear Design and Manufacture n 1 or n w – is the rotational speed of worm, rpm or – is the lead angle of threads at mean worm diameter, degrees

After the allowable value of Wt is determined, an output power capacity (at an application factor of 1) can be determined from: P=

Wd t p2n2 19, 090, 800

P=

kW metric

(6.127)

WDn t G hp English 126, 800

(6.128)

where n2 (or nG) is gear speed, rpm. The available output power from the worm gearset alone has to be divided by a service factor to make allowance for the smoothness or roughness of the driving and driven equipment. In worm-gear practice, this factor is still a service factor, since it also allows for the amount of time the unit is to be operated. As was explained in Section 6.2.4, the latest practice for spur and helical gears is to use an application factor instead of a service factor. The application factor considers roughness of connected equipment but does not evaluate life cycles or hours of operation. Table 6.47 shows typical service factors.

TABLE 6.47 Service Factors for Cylindrical Worm-Gear Units Prime Mover

Duration of Service per Day

Electric motor

Multi-cylinder internal combustion engine

Single-cylinder internal combustion engine

Electric motor

Driven Machine Load Classification Uniform

Moderate Shock

Heavy Shock

Occasional ½ hour Intermittent 2 hours

0.80 0.90

0.90 1.0

1.00 1.25

10 hours

1.00

1.25

1.50

24 hours Occasional ½ hour

1.25 0.90

1.50 1.00

1.75 1.25

Intermittent 2 hours

1.00

1.25

1.50

10 hours 24 hours

1.25 1.50

1.50 1.75

1.75 2.00

Occasional ½ hour

1.00

1.25

1.50

Intermittent 2 hours 10 hours

1.25 1.50

1.50 1.75

1.75 2.00

24 hours

1.75

2.00

2.25

Following service factors apply for applications involving frequent starts and stops Occasional ½ hour 0.90 1.00

1.25

Intermittent 2 hours

1.00

1.25

1.50

10 hours 24 hours

1.25 1.50

1.50 1.75

1.75 2.00

Notes: 1. Time specified for intermittent and occasional service refers to total operating time per day. 2. Term “frequent starts and stops” refers to more than 10 starts per hour.

Design Formulas

355

If input power is needed, the output power must have added to it the losses in the gearbox. These losses come from friction on the gear teeth, bearing losses, seal losses, and losses due to oil churning. The relation is: Input power = power used by driven machine + sum of all losses

The calculation of worm-gear efficiencies is too complex to be included in this book. Worm gears are used with materials other than a bronze gear and a steel worm. There are no trade standards for these other materials. Some rough guidance as to what these other materials might be expected to do is given in Table 6.48. The strength of worm-gear teeth is not well established. Since strength is usually not critical, there has not been any extensive research activity to establish calculation means and allowable bending stress. (Some worm-gearing builders have done a considerable amount of testing to establish data for their own products.) A unit load can be calculated for worm gears: Ul =

Ul =

Wt b mn

N/mm2 metric

(6.129)

psi English

(6.130)

Wt Pnd Fe

where: Wt – is the tangential load, N (metric) or lb (English). b – is the effective face width, mm (metric). Fe – is the effective face width, in. (English). px cos 3.14159 3.14159 Pnd = p cos x

mn =

(metric)

English

Table 6.49 gives some rather approximate values for unit load. Generally, there is no problem if the unit load is less than the values shown (providing no unusual shock loads are present). Double-enveloping-worm-gear durability. The load carrying capacity of double-enveloping worm gears is calculated as the input horsepower by a simplified empirical formula:

TABLE 6.48 Guide to Approximate Material Capacity for a Variety of Worm-Gear Material Combinations Material Worm

Durability Constant, Ks

Speed Range, Rubbing Velocity

Up to 30 m/s (6000 fpm) Up to 10 m/s (2000 fpm)

Worm Gear

Steel, 53 HRC minimum Steel, 35 HRC minimum

Bronze, phosphor Bronze, phosphor

600 600

Steel, 53 HRC minimum

Bronze, super manganese

1000

Up to 2 m/s (400 fpm)

Steel, 53 HRC minimum Cast iron

Bronze, forged manganese Cast iron

700 700

Up to 10 m/s (2000 fpm) Up to 2 m/s (400 fpm)

Cast iron

Bronze, phosphor

600

Up to 10 m/s (2000 fpm)

356

Dudley’s Handbook of Practical Gear Design and Manufacture

TABLE 6.49 Approximate Values of Unit Load for Cylindrical Worm Gears Material Combination

Ul, psi

Worm

Worm Gear

Steel, hardened Steel, hardened

Bronze, phosphor Bronze, super manganese

Running

Static

1350 3200

5,500 12,500

Steel, hardened

Bronze, forged manganese

1800

7,000

Cast iron Cast iron

Cast iron Bronze, phosphor

1000 1350

5,500 5,500

Notes: 1. The hardened steel worm will usually be HV 655 or HRC 58 to get surface durability. Some slow-speed worm gears may use through-hardened steel worms. These have less surface durability. It is assumed that the hardened steel worm will at least have HV 354 or HRC 35 hardness. 2. These values assume an average contact ratio of only 1.5. Many worm-gear designs will have a higher contact ratio and therefore somewhat higher unit load capability.

P=

0.7457 n 1 Ks Km Ka Kv u

P=

nW Ks Km Ka Kv mG

kW

metric

hp English

(6.131)

(6.132)

where: n 1 or n W – is the worm speed, rpm. u or m G – is the tooth ratio (= no . gear teeth ). no . worm threads Ks – Km – Ka – Kv –

is is is is

the the the the

pressure constant based on center distance (see Figure 6.37). ratio correction factor (see Table 6.50). face width and materials factor based on center distance (see Table 6.51). velocity factor based on rubbing or sliding speed (see Figure 6.38).

The sliding speed of double-enveloping worm gear is: vs =

vs =

dt1 n 1 19,089 cos 0.2618 dt n W cos

m/s metric

(6.133)

fpm

(6.134)

English

where: dt1 or dt – is the worm throat diameter (Figure 2.17). n 1 or n W – is the worm rpm. or – is the worm-gear helix angle.

The output power can be obtained by subtracting all the losses from the input power. Thus:

Design Formulas

357

FIGURE 6.37 Basic pressure constant based on center distance for standard-design double-enveloping worm gears.

Output power = input power

sum of all losses

(6.135)

The standard method of calculating the rating for double-enveloping worm gears is shown in AGMA standards. In this standard, overall efficiency values are given for a wide range of standard designs. The rating data shown above are extracted from this standard. Those designing double-enveloping worm gears should use this standard to obtain more data than can be covered in this book. In double-enveloping-worm-gear practice, the service factors are based on the shock in the system. (There is no particular difference between an electric motor drive and a turbine.) The useable power is the power obtained from Eqs. (6.131) and (6.132) divided by the service factor. Table 6.52 shows service factors extracted from AGMA. Comparison of double-enveloping and cylindrical worm-gear rating procedures. There are several significant differences in practice. A wider variety of design practices are used for cylindrical worm gears. This has led to the use of more general-purpose formulas rather than highly specialized formulas. To help keep the reader from getting confused over the data presented, these comparisons are worth noting:

358

Dudley’s Handbook of Practical Gear Design and Manufacture

TABLE 6.50 Ratio Correction Factor Km for Double-Enveloping Worm Gears Ratio, u (m G )

Km

Ratio, u (m G )

Km

3.0

0.380

14.0

0.720

3.5 4.0

0.435 0.490

16.0 20.0

0.727 0.737

4.5

0.520

30.0

0.746

5.0 6.0

0.550 0.604

40.0 50.0

0.748 0.750

7.0

0.632

60.0

0.751

8.0 9.0

0.665 0.675

70.0 80.0

0.752 0.752

10.0

0.690

90.0

0.753

12.0

0.706

100.0

0.753

TABLE 6.51 Face Width and Materials Factors for Standard Design Double-Enveloping Worm Gears Center Distance

Materials Factor, Ka

mm

in.

50.08

2.000

0.620

63.5

2.500

0.684

76.2 88.9

3.000 3.500

0.755 0.780

101.9

4.000

0.855

127.0 152.4

5.000 6.000

0.934 1.014

177.8

7.000

1.073

203.2 254.0

8.000 10.000

1.113 1.175

304.8

12.000

1.250

381.0 457.2

15.000 18.000

1.281 1.328

533.4

21.000

1.368

609.6 660.4

24.000 26.000

1.398 1.411

711.2

28.000

1.425

762.0 812.8

30.000 32.000

1.438 1.445

863.6

34.000

1.453

914.4 965.2

36.000 38.000

1.460 1.469

1016.0

40.000

1.476

Design Formulas

FIGURE 6.38

359

Velocity factor for double-enveloping worm gears.

TABLE 6.52 Service Factors for Double-Enveloping Worm Gears Hours/Day ½ 1

Uniform

Moderate Shock

Heavy Shock

Extreme Shock

0.6 0.7

0.8 0.9

0.9 1.0

1.1 1.2

2

0.9

1.0

1.2

1.3

10 24

1.0 1.2

1.2 1.3

1.3 1.5

1.5 1.75

Item

Comparison

Power

Cylindrical ratings are based on output power. Double-enveloping ratings are based on input power.

Size

The worm-gear pitch diameter is the primary size quantity for cylindrical worm-gear units. The doubleenveloping units use center distance as the primary size quantity.

Materials

Several materials are in general use for the worms or worm gears of cylindrical worm units. Double-enveloping worm-gear units usually use through-hardened steel worms and chill-cast or centrifugally cast bronze gears. For each center distance and speed, there is essentially only one choice of material. (This explains why the “material factor” is handled indirectly as a function of center distance and rubbing speed.) Cylindrical worm-gear practice allows some flexibility in regard to worm pitch diameter for a given center distance and some flexibility on other design variables. For double-enveloping worm gears, all variables are tied quite closely to the center distance and the desired ratio.

Design flexibility

360

Dudley’s Handbook of Practical Gear Design and Manufacture

6.2.8 DESIGN FORMULAS

FOR

SCORING

The problem of designing gears to resist scoring is still not completely solved. Section 5.1.5 gives the historical background of PVT and flash-temperature calculations. Much work has been done on elastohydrodynamic (EHD) calculations of oil-film thickness and the possible relation of gear-tooth surface roughness to EHD film. The best that can be done is to estimate the hazard of scoring. There is no positive assurance that gears that are calculated to be quite good will not score, and gears that are calculated to be somewhat poor may still perform satisfactorily. The best procedure is to calculate the scoring risk and then plan to handle it by either design changes to lessen the scoring risk or special development of the gearset and its lubrication system to handle a latent scoring hazard. This section will give two design approaches. One is based on flash temperature and hot scoring; the other is based on oil-film thickness and cold scoring. Hot scoring A general design formula for spur and helical gears is: Tf = Tb + ZtZsZc

(6.136)

where: Tf – is the flash temperature index, °C or °F. Tb – is the gear body temperature, °C or °F. Zt – is the geometry constant, dimensionless. Zs – is the surface finish constant, dimensionless. Zc – is the scoring criterion number, °C factor or °F factor.

The body temperature is hard to measure, but it can be measured with thermocouples and a means of getting the reading out of the rotating part (by slip rings or by miniature radio). In well designed gears with oil nozzles delivering enough oil, the temperature rises of the gear body (over the incoming temperature) ought not exceed the following values: • 25°C (45°F) for aerospace gears • 15°C (27°F) for turbine gears The geometry constant Zt was defined in Section 5.1.5, Eq. (5.52). If the teeth have no tip relief, Zt should be calculated for the tips of the pinion and the gear, since the hazard of scoring will be highest at the tips. Table 6.53 gives some representative values of Zt for tip conditions. High performance gears with teeth 2.5 module (10 diametral pitch) or larger are generally given a standard profile modification. With profile modification, the most critical point for scoring is generally at the start of modification. Table 6.54 shows a design guide for profile modification worked out for general-purpose gearing with moderately heavy loads. (This guide may not be appropriate for highly critical aerospace power gears or the most critical vehicle gears—more on this see Section 17.2.) The values of Zt that match the modification depth shown in Table 6.54 are plotted in Figures 6.39 to 6.42. A standard amount of modification for general design can be set: Gear tip modification =

6.5Cm Wt 105b

mm

Gear tip modification =

4.5Cm Wt 107F

in.

metric

(6.137)

English

(6.138)

Design Formulas

361

TABLE 6.53 Geometry Constant for Scoring Zt at the Tip of the Tooth Pressure Angle, t or

20°

25°

t

No. Pinion Teeth, z 1 or NP

No. Gear Teeth, z 2 or NG

Pinion Addendum, h a1 or aP (for m = 1.0 or Pd = 1.0 )

Zt Gear Addendum, h a2 or aG (for m = 1.0 or At At Pd = 1.0 ) Pinion Tip Gear Tip

18

25

1.00

1.00

0.0184

–0.0278

18 18

35 85

1.00 1.00

1.00 1.00

0.0139 0.0092

–0.0281 –0.0307

25

25

1.00

1.00

0.0200

–0.0200

25 25

35 85

1.00 1.00

1.00 1.00

0.0144 0.0088

–0.0187 –0.0167

12

35

1.25

0.75

0.0161

–0.0402

18 25

85 85

1.25 1.25

0.75 0.75

0.0107 0.0104

–0.0161 –0.0112

35

85

1.25

0.75

0.0101

–0.0087

35 18

275 25

1.25 1.00

0.75 1.00

0.0070 0.0135

–0.0072 –0.0169

18

35

1.00

1.00

0.0107

–0.0168

18 25

85 25

1.00 1.00

1.00 1.00

0.0074 0.0141

–0.0141 –0.0141

25

35

1.00

1.00

0.0107

–0.0126

25 12

85 35

1.00 1.25

1.00 0.75

0.0069 0.0328

–0.0103 –0.0160

12

85

1.25

0.75

0.0500

–0.0151

18 25

85 85

1.25 1.25

0.75 0.75

0.0056 0.0082

–0.0095 –0.0073

35

85

1.25

0.75

0.0078

–0.0060

35

275

1.25

0.75

0.0056

–0.0048

Note: When proper profile modification is made, the risk of scoring is probably more critical at the start of modification than at the tip of the tooth. Values of Zt at the approximate start of modification are given in Figures 6.39 to 6.42.

Pinion

tip modification =

Pinion tip modification =

4.1Cm Wt 105b 2.8Cm Wt 107F

mm metric

in.

English

(6.139)

(6.140)

where Cm is the load-distribution factor for surface durability. The factor Zs allows for surface finish conditions. This is a most difficult variable to determine. With much test-stand experience, a manufacturer can determine reasonable values for calculation purposes. On a new product, one must consider the finish achieved in the shop, the initial wear-in (which may considerably improve the effective finish, from a scoring standpoint), and the beneficial effects of lu­ bricant additives. (Special oils and additives may be used to condition the surfaces of gears critical scoring.)

362

Dudley’s Handbook of Practical Gear Design and Manufacture

TABLE 6.54 Depth to Start of Profile Modification for General-Purpose Gears

Schematic Diagram to Show Locations of Amounts of Profile Modification and Depth of Profile Modification on a Pinion and on a Gear Normal Pressure Angle

Pinion (Driver)

Gear (Driven)

20° 22.5°

0.400 0.365

0.450 0.415

25°

0.325

0.375

Notes: 1. The depth shown is in millimeters for 1 module, normal, or in inches for 1 normal diametral pitch. For other size teeth, multiply depth by normal module for metric, or divide depth by normal diametral pitch for English. 2. In some cases, manufacturing considerations lead to putting all the modification on the pinion. If this is done, the modification that might have been put at the gear tip is put at the pinion form diameter. In this case, a diameter for start modification can be calculated for the gear. Then as a next step, Table A.16 formulas can be used to find the diameter on the pinion that matches this diameter. This matching pinion diameter is then used as a start of modification diameter for the lower flank of the pinion.

The following values represent a rough guide for Zs: Initial Finish, AA

Zs

Comment

0.3 μm (12 μin.) 0.5 μm (20 μin.)

1.2 1.5

Usually honed, after finish ground Fine finish; some break-in needed

0.75 μm (30 μin.)

1.7

Good finish; special break-in needed (for Zs to equal 1.7)

1 μm (40 μin.) 1.5 μm (60 μin.)

2.0 2.5

Nominal finish; extensive break-in needed (for Zs to equal 2.0) Poor finish; special break-in procedure should be used (then Zs = 2.5 is possible)

The factor Zc is the scoring criterion number. This index of scoring risk was presented in Section 5.1.5, Eq. (5.51). This factor will be given in somewhat more detail here:

where:

Zc =

Wte b

0.75

Zc =

Wte F

0.75

(m t )0.25 n1 1.094 1 nP (Pt )0.25

C factor

F factor

metric

(6.141)

English

(6.142)

Design Formulas

363

FIGURE 6.39 Geometry factor for scoring, Zt, at the start of standard profile modification. Full depth, 20° pressure angle spur gears. Pinion addendum 1.00, gear addendum 1.00.

Wte – is the tangential load that is applied to a point on the profile in danger of scoring, newtons (metric) or pounds (English). b and F – is the face width, mm or in. n 1 and n P – is the pinion rpm. m t – is the module, transverse. Pt – is the transverse diametral pitch.

The load applied at the start of profile modification should generally be the full load. For spur gears with modification, the design is generally worked out so that the start of modification becomes a real highest point of single-tooth loading. For helical gears, the design may have a high enough transverse contact ratio to keep the maximum effective load somewhat below the full load. (Usually doing an extensive analysis of load sharing to find out if the helical tooth merited less than full load at the start of mod­ ification is not worth the trouble.) If the teeth are not modified, close to full load may be applied at the tooth tips. The spur gear, unmodified, would only get about 50% load at the tip—since two pair of teeth are sharing the load. However, scoring is a hazard at the position of the worst tooth-to-tooth spacing error. At this position, the tooth tip may get full load because the spacing error prevented load sharing. Then the few worst teeth

364

Dudley’s Handbook of Practical Gear Design and Manufacture

FIGURE 6.40 Geometry factor for scoring, Zt, at the start of standard profile modification. Full depth, 20° pressure angle spur gears. Pinion addendum 1.5, gear addendum 0.76.

may score, and the scoring may wear away enough metal to restore load sharing. If the scoring is not too bad, it may heal up, and then the unit may run without further scoring. For both spur and helical gears that are unmodified, a simple and conservative design practice is to take full load at the tooth tips when figuring the scoring risk. The load distribution across the face width also enters into the choice of load. If Cm = 1.5, then the load Wte should be 50% higher. Scoring, of course, is most apt to happen at the end of the face width that is overloaded. To sum it up: Wte = Wt × Cm × percent load for profile position

(6.143)

Some values of the flash temperature and a rough guess as to the probability of scoring are given in Table 6.55. Cold scoring. Cold scoring occurs when EHD oil film is small compared with the surface roughness, and the lubricant does not have enough additives to prevent scoring as the asperities on the contacting gear-tooth surfaces abrasively wear. The first design step is to calculate an approximate EDD oil-film thickness, hmin. A relatively exact calculation of hmin is quite complicated and requires special data about the oil. (Paraffinic oils have somewhat different data than naphthenic oils.)

Design Formulas

365

FIGURE 6.41 Geometry factor for scoring, Zt, at the start of standard profile modification. Full depth, 25° pressure angle spur gears. Pinion addendum 1.00, gear addendum 1.00.

A simple, but approximate, calculation for the minimum oil film at the pitch line will be given. The calculation will be given in English units only. The answer, of course, can easily be changed into metric units: h min =

44.6re (lubricant factor) (velocity factor) (loading factor)

(6.144)

where effective radius of curvature at pitch diameter, re, is: re =

C sin cos2

n

mG (m G + 1)2

Lubricant factor = ( E )0.54

where: ∝ – is the lubricant pressure-viscosity coefficient, in.2/lb (see Table 6.56).

(6.145)

(6.146)

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Dudley’s Handbook of Practical Gear Design and Manufacture

FIGURE 6.42 Geometry factor for scoring, Zt, at the start of standard profile modification. Full depth, 25° pressure angle spur gears. Pinion addendum 1.25, gear addendum 0.76.

TABLE 6.55 Flash Temperature Limits Tf and Scoring Probability Kind of Lubricant

Risk of Scoring Low

High

°C

°F

°C

°F

135 150

275 300

175 190

350 375

Mil-O-6081. grade 1005 Mil-L-6086, grade medium

65 160

150 325

120 200

250 400

SAE 50 motor oil with mild EP

200

400

260

500

260

500

315

600

Synthetic oil Mil-L-7808 Mil-L-23699 Mineral oil

Mil-L-2105, grade 90 (SAE 90 gear oil)

Design Formulas

367

TABLE 6.56 Nominal Lubricant Properties Kind of Oil

Temperature

Viscosity, cP

Lubricant Pressure-Viscosity Coefficient, in.2/lb

°C

°F

Mil-L-7808

100 71

212 160

3.6 min. 5.5 min.

0.000105 0.000115

Mil-L-23699

100

212

5.0 min.

0.000105

Mil-O-6081, grade 1005

71 100

160 212

8.0 min. 5.0 min.

0.000115 0.000075

Mil-L-6086, grade medium

100

212

8.0 min.

0.000096

SAE 30 motor oil

71 100

160 212

17.0 min. 11.5 min.

0.000110 0.000096

SAE 50 motor oil

100

212

17.0 min.

0.000096

SAE 90 gear oil

100 71

212 160

16.5 min. 45.0 min.

0.000096 0.000110

SAE 140 gear oil

100

212

35.0 min.

0.000096

E effective elastic modulus for a steel gearset:

(E =

E 2 (1

)2

= 51.7 106 psi, since E = 30, 000, 000 psi, and Poisson s ratio = 0.3)

Velocity factor =

0u

0.7

(6.147)

E re

where: 0 – is the lubricant viscosity at operating temperatures, cP (centipoises); see Table 6.56 u – is the rolling velocity (at the pitch line), in./second (u = nP d sin t ). 60

d – is the pinion pitch diameter. nP – is the pinion rpm. t – is the transverse pressure angle.

Loading factor =

Wt F E re

0.13

(6.148)

where: Wt– is the tangential load, lb. F – is the face width, in.

The answer, h min , comes out in micro-inches (μ in.). This value can be changed to micrometers (μm) by dividing by 39.37. Figure 6.43 shows some plotted values of h min for a heavy vehicle oil, SAE 90 gear oil, and for a light synthetic oil, Mil-L-23699, that is used in the aerospace field. Note that the cold synthetic oil gives less oil-film thickness than the hot vehicle oil.

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Dudley’s Handbook of Practical Gear Design and Manufacture

FIGURE 6.43 Approximate EHD oil-film thickness for a heavy mineral oil used in vehicles and a light synthetic oil used in aircraft. Note size of gearset.

Table 6.57 shows a general study of how oil-film thickness changes with gear size, gear-tooth load, and pitch-line velocity. Note that pitch-line velocity is by far the most important variable. This table was made for an oil that might be used with medium-speed industrial gears. After the minimum EHD oil-film thickness is determined, it can be compared with the surface finish of the gear teeth by Figure 6.44. The surface finish to be used is not the finish before the gears have operated, but the effective finish after they have been given whatever break-in or wear-in treatment is intended. The effective finish for a pair of gears running together is: S12 + S22

S =

(6.149)

where: S1– is the finish of one gear after break-in, arithmetic average. S2– is the finish of mating gear after break-in, arithmetic average.

The ratio of film thickness to surface finish is: =

h min S

(6.150)

This ratio, which can be called the lambda ratio, is usually over 1.0 for Regime III conditions. In Regime II, it will tend to be less than 1.0, with usual values around 0.4 to 0.6. In Regime I, the lambda ratio is often around 0.1 to 0.3.

Design Formulas

369

TABLE 6.57 Approximate EHD Minimum Oil-Film Thickness for Mil-L-6086 Oil Gear Mesh Temperature °C

hmin, μ m for Pitch-Line Speed, m/s

hmin, μ in. for Pitch-Line Speed, fpm

K factor*, N/mm2

°F

0.5

2.5

10

50

100

500

2000

10,000

1.33 1.15

2.08 1.80

6.41 5.56

17.50 15.17

52.40 45.40

Small gear unit** 60

140

80

176

100

212

60

80

100

1.38 4.14

0.053 0.045

0.163 0.141

0.44 0.38

13.80

0.039

0.120

0.33

0.98

1.54

4.75

12.97

38.80

1.38 4.14

0.031 0.027

0.097 0.084

0.26 0.23

0.79 0.69

1.24 1.08

3.82 3.31

10.42 9.03

31.20 27.10

13.80

0.023

0.072

0.20

0.58

0.92

2.83

7.72

23.10

1.38 4.14

0.021 0.018

0.065 0.056

0.18 0.15

0.53 0.46

0.83 0.72

2.56 2.22

6.98 6.05

20.90 18.10

13.80

0.015

0.048

0.13

0.39

0.61

1.89

5.17

15.40

1.38

0.086

Large gear unit*** 0.265 0.72 2.15

3.40

10.43

28.40

85.10

4.14

0.073

0.229

0.62

1.87

2.90

9.04

24.60

73.80

176

13.80 1.38

0.063 0.051

0.196 0.157

0.53 0.43

1.60 1.28

2.50 2.01

7.73 6.20

21.00 16.90

63.00 50.61

4.14

0.044

0.136

0.37

1.11

1.74

5.38

14.60

43.89

212

13.80 1.38

0.038 0.034

0.116 0.105

0.32 0.29

0.95 0.86

1.49 1.35

4.59 4.16

12.50 11.30

37.50 33.97

140

4.14

0.029

0.092

0.25

0.75

1.117

3.61

9.80

29.50

13.80

0.025

0.078

0.21

0.64

1.00

3.08

8.37

25.18

*1.38 N/mm2 = 200 psi; 4.14 N/mm2 = 600 psi; 13.8 N/mm2 = 2000 psi. **re = 8.39 mm = 0.3304 in. ***re = 41.96 mm = 1.652 in.

Since lambda ratio varies considerably, the best way to estimate probable gear running conditions is to use a diagram like Figure 6.44. (Gear builders can plot their own Figure 6.44 once they gain considerable experience in the application of their gears.) The secret, of course, to successful gear application over all regimes of lubrication is to make up for small lambda ratio values by appropriately strong additives in the oil. In general, these conclusions hold:

Regime

Conclusion

I

High hazard of cold scoring. Need a strong EP oil.

II III

Moderate hazard of cold scoring. Need an anti-wear oil or an EP oil. A mild anti-wear oil or a straight mineral oil will probably OK. No serious hazard of cold scoring.

370

FIGURE 6.44 finish.

Dudley’s Handbook of Practical Gear Design and Manufacture

Regimes of lubrication are determined by minimum EHD oil-film thickness and effective surface

Design practice to handle scoring. In most cases, gears are sized to meet durability requirements. Then the tooth size is made large enough to meet tooth-strength requirements. Gear sizing is seldom done to meet scoring requirements. The usual procedure is to establish the gear design and then check the design for scoring hazard. What can the designer do if: 1. There is a serious risk of hot scoring? 2. There is a serious risk of cold scoring? In the case of hot scoring, the most effective things to do are to improve the surface finish and to use a more score-resistant oil. Using smaller gear teeth and changing the addendum proportions may be of some help. (Note in Figures 6.39 to 6.42 how the Zt value changes as more or less long addendum is put on the pinion.) A higher-pressure angle or a special profile modification can also help, as can better tooth accuracy (note data on tooth accuracy in Section 16.4.1). If a design is marginal and all the obvious things have been done, it may be necessary to copper-plate or silver-plate the teeth. A very thin deposit of copper or silver is quite effective in preventing scoring. Large turbine gears, running at very high speeds, are routinely plated with copper or silver to control the scoring hazard. Some gear drives for large propeller-driven aircraft have had to use silver-plated teeth. If cold scoring is the hazard and the gears are in Regime I or in the transition zone to Regime II, the situation may be rather critical. The most obvious thing to do is to use a stronger EP oil. (The chemical additives in the extreme pressure oil tend to make a chemical film that will substitute for the EHD film that is missing.) In Regime I, and to a lesser extent in Regime II, the gears tend to machine themselves in service. A gear pair with a 2-μm finish when it left the factory, may, in a few days of running, wear in a new surface that is as smooth as 0.5 μm (in the direction of sliding).

Design Formulas

371

This means that it is critical to use an extra-strong EP oil during the break-in period. Also, the gears should not be operated at maximum torque or maximum temperature while they are wearing in. If the wear-in is accomplished without serious damage to the profiles, then the gears can often run for years without the need for a special EP oil. When cold scoring is hazard, it is helpful to use as heavy an oil as possible and to use the right amount of profile modification. In conclusion to this section, it should be said that experience is most important in handling scoring problems. If gears score on the test stand or in the field, it is usually possible to try different im­ provements in lubrication and in the way the gear teeth are finished. Sometimes the pattern of grinding or cutting marks can be changed, with much improvement in scoring resistance. Sometimes an oil specialist can recommend a change in lubricant or a lubricant additive that will clear up the scoring problem. Sometimes a change in the heat-treatment procedure used in making the gear teeth will result in a more score-resistant tooth surface. Each builder of gears for applications with high scoring hazard needs to learn a “recipe” for making gears that will survive without undue scoring trouble.

6.2.9 TRADE STANDARDS

FOR

RATING GEARS

Sections 6.2.1 to 6.2.7 gave general methods for determining gear load-carrying capacity. The reader was probably surprised to note that the load-carrying capacity is quite elastic. If one assumes that a high degree of accuracy is obtained in the gearing and that the driving and driven apparatuses do not impose shock, one obtains a large amount of capacity for a given size of gearset. On the other hand, if poor accuracy and rough-running apparatus are assumed, the calculated capacity becomes quite low. In a sense, the general rating formulas require the designer to use considerable judgment in evaluating all the factors that affect the capacity of a given design. In contrast, standard rating formulas are usually not elastic. They give a definite answer. Standards are established by a technical group that meets to discuss the intensity of loading that their field experience has shown to be safe and conservative. In some cases, the degree of quality on which a standard is based is spelled out in the standard. In many cases, though, not all the things that contribute to the quality of a gear are precisely defined. A newcomer in a given gear field may have to scout around to find out just how good the gears that are being made by established manufacturers in the field are. The standards for gear design or for gear rating may be written to cover a broad field, or they may be product standards for a limited area of usage. For instance, a “mother” standard on gear rating will tend to cover the principles and practice for rating the surface durability and strength of all kinds of spur and helical gears. This kind of standard will often give more than one method for determining a variable. A “product” standard will cover only a limited field of usage. The product standard will be quite specific on how to rate the gears. The principle is that the product standard is derived from the mother standard, and so it can be much shorter and more specific. In a limited field, those who are building and using the gears should have rather good experience to judge what is correct for good technical practice. Product standards on gear units cover a number of things beyond the capacity of the gear teeth. Frequently the capacity of a set has to be limited by the capacity of the casing to radiate and conduct heat away. The kind of lubrication, the kind of material, the kind of bearings and the amount of the service factor are some of the other things that may be specified in gear standards. The material in this book covers many of the things that are in trade standards. However, a designer who wishes to design a gear in accordance with a particular standard should get a copy of the standard and study every detail of it. Standards are like legal documents. All the fine print must be read and followed before one can claim to be acting in accordance with the standard. Purchase contracts frequently specify standards or other details of gear unit design. A builder of gears under contract is, of course, obligated to meet the contract. What should be done if some requirement relating to the gears seems unreasonable or impractical, or if it allows gears to be built that have too

372

Dudley’s Handbook of Practical Gear Design and Manufacture

much risk of failure? (This book is intended to be conservative and to alert the reader to risks in gear making—hence a gear contract may allow things that do not agree with this book.) If this book is more conservative than some gear standard or some contract provision, the gear builder or gear buyer would be well advised to consider whether or not there is enough proven experience to justify going beyond the recommendations given here. In the last analysis, it is proven experience that sets the design. Certainly, progress is being made in all aspects of the gear art. It is to be expected that there will be many situations in which there is enough experience and technical know-how to go beyond things in this book.

6.2.10 VEHICLE-GEAR-RATING PRACTICE Vehicle-gear designers are always under great pressure to make their gears both small and inexpensive. A gear failure is not too serious, because there is not much risk to the safety of the vehicle driver. Expensive equipment is not tied up by a failure, as it is when gears in a power plant or a ship fail. However, failure cannot be allowed to happen very often, or the vehicle will get a reputation for being poorly design. Also, the customer-complaint charges may become serious. The vehicle designer needs to know quite precisely where the danger point is in loading the gears. There are two good sources of information. Since vehicles are built in very large quantities, a large amount of statistical data is available. These can be collected and plotted to give design curves taht are quite reliable for the limited field of gear work for which they apply. The second source of information is the product experience of individual manufacturers. After many thousands of transmissions and rear ends have been built and put into service, the builder finds out which parts are weak and which parts are stronger than they need to be. If the weak spots are doctored, the rating of the gears can usually be increased. In many cases, automotive manufacturers have been able to improve their gear drives at about the same rate as engine horsepower has been increased. Present-day vehicle engines are much more powerful than they were 20 years ago. In general, though, the gear drives are not larger. They are just made with better accuracy and better materials. The AGMA standard for spur and helical vehicle gears defines two quality grades, Grade 1 and Grade 2. Grade 2 steel is specified to be cleaner, harder, and having better metallurgical structure than Grade 1. The standard shows composition of low-alloy, medium-alloy, and high-alloy steel. The text explains that that the higher-alloy steels are required as the gears become larger and the vehicle is subjected to more severe duty requirements. The standard gives typical de-rating factors for different vehicle applications. Table 6.58 shows a sampling of these factors. The design stress limits are adjusted for Grade 2 or Grade 1 material, and also adjusted for a probability of failure of L10 or L1. An L10 designation means that 10 out of 100 gears might fail prematurely by the mode of failure for which the stress level is given. Table 6.59 shows the design stress limits in tabular form. (The standard presents the values in curve form.) Vehicle-gear design is characterized by two things: • Very severe loads are permitted at low numbers of cycles (less than 107). This means that the teeth are apt to have some surface or subsurface damage from the high loading. The damage may take the form of micro-cracking, surface cold work, or small amounts of pitting. This kind of damage is not expected to result in failure before 108 cycles, provided the gear meets the design limits. At 108 cycles the gear may be definitely damaged, and it may be quite unfit to run for 109 or 1010 cycles. • In general, vehicle gears operate under Regime II conditions. To meet their design ratings, they must have appropriate lubricants that have the right viscosity and appropriate additives. Because of Regime II conditions, the load that can be carried on the surface of the tooth drops rather rapidly as the number of cycles increases.

Design Formulas

373

TABLE 6.58 A Sampling of Overall Derating Factors Application

Overall Derating Factor

Final drive

Medium Precision

Medium-High Precision

1.2 1.4

1.0 1.2

Off-highway Truck

Shifting transmission

Epicyclic

Off-highway

1.5

1.3

Truck Passenger car

1.6 1.7

1.4 1.5

Off-highway

1.7

1.6

Passenger car

–

2.0

TABLE 6.59 Design Stress Limits No. Stress Cycles

Bending Stress Grade 2 L10

Contact Stress Grade 1

L1

L10

Grade 2 L1

L10

Grade 1 L1

L10

L1

10

1270

1170

Metric, N/mm2 980 880

–

–

–

–

104

960

880

750

690

–

3270

3450

2860

105 106

760 580

690 520

570 450

520 410

3100 2480

2620 2140

2690 2140

2240 1790

2 × 106

530

480

410

370

–

–

–

–

107 108

530 530

480 480

410 410

370 370

2000 1650

1720 1380

1690 1310

1400 1100

103 104

185 140

170 128

142 109

128 100

– –

– 475

– 500

– 415

105

110

100

83

76

450

380

390

325

106 2 × 106

84 77

75 70

65 60

59 54

360 –

310 –

310 –

260 –

107

77

70

60

54

290

250

245

203

108

77

70

60

54

240

200

190

160

3

English, psi (multiply by 1000)

Notes: 1. These limits are for short life (2 × 108 maximum). 2. Some micro damage can be tolerated at cycles below 106. 3. These limits are for operation in Regime II. * The bending stress data show no slope after 2 × 106 cycles. However, the standard concedes that a shallow slope may be needed and suggests a drop of 34 N/mm2 (5000 psi) in going from 2 × 106 cycles to 108 cycles.

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Dudley’s Handbook of Practical Gear Design and Manufacture

In general, vehicles are built in large quantities. This makes it practical (on new design) to build a few gear drives that are made just as small as possible by vehicle criteria. The first few drives are then extensively tested. The test results will probably show some failures. In general, design modifications are made to improve the design without changing the overall size. With improvements, the design will pass the test satisfactorily, and it is then read to go into initial production. After a unit has been in production for a period of time, field results may show a need for further design improvements. The design improvements might include any one or all of the following things: • Change to a higher-alloy steel or improve the heat-treating practice to get better hardness and metallurgy. • Change the profile modification or the helix modification to get better fit. • Change the pitch of the teeth, the pressure angle, or the tooth proportions to get better geometric conditions. • Improve the accuracy of the teeth, or improve the bearing and casing structures that support the gears. • Use a better lubricant and/or a better means of cooling the gears when they are running under the worst heavy-loading, hot-had conditions. More could be added.

6.2.11 MARINE-GEAR-RATING PRACTICE The gears used to drive large ships are almost all helical. The gears are so large that it is difficult to caseharden them and retain enough dimensional accuracy to permit finish grinding. The high speed and longlife requirements of marine gears make it necessary to grind the gears after case-carburizing. In some cases, the smaller marine gears (epicyclic marine gears, for instance) are cut, shaved, or ground before nitriding, then nitrided with such overall skill and control of the manufacturing process that they can be put into service after final hardening without any grinding in the final hard condition. There has been considerable use of through-hardened gears for marine service that are precision-cut and shave or ground. These gears have no case and are essentially uniform in hardness throughout the tooth area. The gear drives on large ship is a very important piece of machinery. Ship owners want the gearing to be good enough to stay in service for something like 20 to 30 years. The gears run at fairly high speeds, so the lifetime cycles may get up to 1010 or even higher. If a gear drive on a ship were to fail, there would be a chance that a storm might be in progress, and the ship might be blown onto the rocks and wrecked. There is also the risk that a ship with disabled gears might be so long in getting into port that a perishable cargo would be lost. The design of gearing for ships tends to be very conservative. Those who purchase gear drives frequently control the conservatism of the design by specifying maximum K factor and unit-load values. As an example, a large ship using medium-hard gears might have gears designed to meet values like the following: First stage High-speed pinions Hardness 262–311 HB High-speed gears Hardness 241–285 HB K– factor not to exceed 0.86 N/mm2 (125 psi) Second stage Low-speed pinions Hardness 241–285 HB Low-speed gears Hardness 223–269 HB K– factor not to exceed 0.69 N/mm2 (100 psi)

Design Formulas

375

Those who operate large ships almost always insure the ship. Those who write insurance for ships or for machinery on the ship need independent certification of the capability and condition of the ship or the piece of equipment under consideration. This kind of work is ordinarily handled by organizations that are commonly called classification societies. These groups are concerned with the power rating of gears, the quality of gears, and the character of the drive system associated with the gears. This latter concern involves things like propeller shaft bearings, mountings for turbines, and hull deflections under different sea conditions. The classification societies are involved in approving new designs, as well as evaluating machinery in service. Each of them tends to have rules pertaining to gear rating. A marine-gear designer will often find that a design is required by contract to meet the rules of a designated classification society.

6.2.12 OIL

AND

GAS INDUSTRY GEAR RATING

The gears used in the oil and gas industry need high reliability. Power gearing often handles power from 1000 kW to as much as 30,000 kW. A compressor or generator drive system will often be required to run 20,000 to 30,000 hours before overhauling, and have a total life requirement of 50,000 to 100,000 hours. At an overhaul, seals and couples, and sometimes bearings, might be replaced. It is generally expected that the gears themselves will last for the total design life. (Sometimes a gear or pinion will last for the total design life, but some rework of teeth or journal bearing surfaces may be required during the life.) Very often, fast-running turbines drive the gear units in this field. The total life cycles of gear parts are often in the area of 1010 to 1011 cycles. Table 6.60 shows some material extracted from API Standard 613. The “material index number” is really the K factor when the application factor is equal to 1. In most cases, thought, API recommends an application factor greater than 1, and so the real K factor is obtained by dividing the material index number by the application factor. In some cases, large power gears used in the oil and gas industry heat to the point where the fit of the teeth is considerably affected by thermal distortions and the body temperature of the pinion may be over 50 degrees hotter than the incoming oil. Thermal problems are being handled, but no trade standards have yet been developed. Section 15.3.4 of this book has further material on this subject.

TABLE 6.60 Some Design Limits Kind of Material

Material Index Number

Bending Stress Number

N/mm2

psi

N/mm2

psi

3.10 2.83

450 410

269 248

39,000 36,000

1.38 0.90

200 130

179 145

26,000 21,000

Carburized HRC 59 HRC 55 Through-hardened HB 300 HB 200

Notes: 1. When the material index is divided by the application factor, it becomes a design K− factor. 2. The bending stress equation used by API has the application factor in it and an overall derating factor of 1.8. 3. API uses the term “service factor.” The values used, though, are appropriate for the new AGMA definition of application factor. (API sets their limits low enough to handle about 1010 cycles, and so their so-called SF values are really application factors.) 4. Extracted from API 613, “Special-Purpose Units for Refinery Services.”

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Dudley’s Handbook of Practical Gear Design and Manufacture

6.2.13 AEROSPACE-GEAR-RATING PRACTICE Aerospace gears must be very light in weight for the loads carried, and yet they must be very reliable. Many of the gears are so important to the operation of the aircraft that a broken tooth would cause an aircraft to crash. The life of an aircraft gear at full torque may vary anywhere from about 106 to 109 cycles. Laboratory and bench tests are used extensively to provide design data for aircraft gears. New aircraft engines are so expensive to design and build that it is worth doing a lot of work to develop component parts before they are put into an engine. The stakes are high. A small reduction in the volume and weight of a gearbox could result in one manufacturer’s engine being accepted for production contracts and another’s being rejected. Gear failures in the field can cause all aircraft using the particular design to be grounded until the questionable gearing is replaced. Under laboratory conditions, gears will perform best of all. Usually, the quality is better than that obtained in large-quantity production. Test stands usually provide a gear with better lubrication and alignment than the worst of the engines that the gears will have to run in. When the load-carrying capacity of a gear has been determined in a test stand, it is necessary to reduce that capacity for use in an actual engine. Allowances must be made so that even the poorest gear operating in the worst engine will not fail. New engines are given extensive testing in engine test cells to demonstrate that they will meet design requirements. Then, if they appear to be satisfactory, they are flight-tested. After this the engine may be approved for use in commercial or military aircraft. Aircraft-gear designers are usually free to use any design formula that looks good to them, but their gears—to be successful—must go through the de­ velopment and testing procedure just described. In propeller aircraft, the gear unit is generally considered to be a part of the engine package. Many small propeller-driven aircraft are used for pleasure, but relatively few are now used to carry passengers. Jet-engine aircraft usually use no main power gearing, but have a considerable amount of gearing in accessory drives. The smaller military cargo planes do use propellers and propulsion gears. A large new use of aerospace gearing is in helicopters. Helicopters are used for commercial purposes, and they have a major use in the Army. A typical Army helicopter will have one or two main rotor drives, and each main rotor drive will have about three stages of gearing. The first stage of gearing is often built into the turbine engine package and furnished with the engine. The second and third stages are generally in a gear unit furnished with the helicopter. The helicopter gears work rather hard most of the time. Even when the helicopter is loitering, the helicopter rotor must provide lift to support the weight of the helicopter. The widely used design guide for aerospace gearing recognizes two grades of material and lists derating factors for several kinds of applications. Stress levels are given for both grades of material. The Grade 2 material is a high-quality steel of the AISI 9310 type. It is specified to have very good cleanness, high hardness, and a very good metallurgical structure. The Grade 1 steel may be carburized 9310, or it may be a somewhat similar steel if appropriate hardness and quality can be obtained. It may be air-melt rather than vacuum-melt. Not quite so high hardness is specified. Nitriding steel of the Nitralloy 135 type is permitted for Grade 1, provided that the teeth are not larger than 2.5 module (10 pitch). The nitride case is thin, even with rather long nitriding time. The standard reflects concern that the case depth may not be adequate for Grade 1 levels of tooth loading if the teeth are larger than 2.5 module. Table 6.61 shows some of the overall de-rating factors. Table 6.62 shows the design stress limits for different numbers of cycles. These stress limits are intended to go with an L1 probability of failure. The aerospace field is somewhat like the vehicle field. The industry standards, in both cases, are really intended as design guides rather than rigid rules. They show what is being achieved by those who are relatively skilled in the gear art and serve as a guide for new designs. The new design, though, is usually made as small as the designer dares to make it.

Design Formulas

377

TABLE 6.61 A Sampling of Overall Derating Factors Application

Overall Derating Factor Medium-High Precision

High Precision

1.8 1.5

1.2 1.0

Power take-off accessory gears

2.1

1.5

Auxiliary power units

2.1

1.5

Main propulsion drive gears

Continuous Take-off and early climb

TABLE 6.62 Design Stress Limits No. Stress Cycles

Bending Stress Grade 2

Contact Stress Grade 1

Grade 2

Grade 1

Metric, N/mm2 1 104

1103 827

965 710

2296 2296

2041 2041

105

627

538

2034

1793

106 107

489 448

414 379

1779 1551

1586 1379

108

414

352

1358

1207

109 1010

379 345

324 303

1193 1041

1069 938

1 104

160,000 120,000

140,000 103,000

333,000 333,000

296,000 296,000

105

91,000

78,000

295,000

260,000

106 107

71,000 65,000

60,000 55,000

258,000 225,000

230,000 200,000

108

60,000

51,000

197,000

175,000

109 1010

55,000 50,000

47,000 44,000

173,000 151,000

155,000 136,000

English, psi

Notes: 1. These stress limits are based on a probability of failure of 1 I 100 (LI). 2. These limits are intended for long life; no micro damage is expected at cycles below 106. 3. These limits are for operating in Regime III.

As was said earlier in this section, the new design has to be proven by considerable amount of testing on the ground and in the air. Frequently, design modifications are made to improve the fit of the gears or the quality of the material in the gears. Profile modifications are very important, as is surface finish. The lubrication system must work well. Generally, the oil is specified, and it is not possible to solve gear problems by using a heavy oil or one with more additives. Aircraft are apt to travel all over the world, and so they have to use the lubricants stocked at airfields. Also, an aircraft may start in the Arctic and a

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few hours later be operating under hot desert conditions. Usually, the oil for the gears has to be the same oil used by the turbine engine. These oils stand hot temperatures quite well, and are thin enough for the equipment to operate in most arctic conditions. The additives in the oil are mild, but they are sufficient for Regime III operation under favorable conditions.

REFERENCES Radzevich, Radzevich, Radzevich, CRC

S. P. (Ed.). (2019). Advances in gear design and manufacture, CRC Press, 549p. S. P. (2017). Gear Cutting tools: Science and engineering (2nd ed.). CRC Press, 606p. S. P. (2018). Theory of gearing: Kinematics, geometry, and synthesis (2nd ed.). revised and expanded. Press, 934p.

7

Gear Reactions and Mountings Stephen P. Radzevich

It is often assumed that gear engineers are primarily concerned with toothed wheels that mesh with each other. The gear engineer, of course, has to understand and deal with the design application of gear teeth. In a broad sense, though, the gear engineer has to be concerned with the whole gearbox. For instance, the bearings that support the toothed gear wheels and the mountings of gears on shafts is a very important part of the technology of gear units. This chapter covers how to determine the reaction forces that come from typical loaded gears, and it also covers typical gear mounting arrangements.

7.1

MECHANICS OF GEAR REACTIONS

The main function of a gear is to transmit motion and/or power. The main function of the supporting body is to neutralize or create a state of equilibrium. Since a gear is a rotating or moving body, a state of dynamic equilibrium must be obtained. To be in dynamic equilibrium, all the reactions from the rotating gear must be neutralized by equal and opposite forces supporting the gear shaft. All couples and moments from the gear or power source must also be contained. In essence, the total work forces and moments must equal the total forces and work out. Only the most basic form of dynamics is considered. It is realized that shaft unbalance, accelerating and decel­ erating changes, and various other dynamic forces exist. Normally these are not an important factor. Gears generally rotate about a near-uniform center axis, which always tends to pass through its center of gravity and percussion; gears and the connecting bodies reach a state of relative constant velocity. Only short time periods of extreme acceleration or deceleration exist. However, these conditions must be considered and taken into account when they form the major loading conditions. They will not be considered further here. In this section, a brief review is made to the basic mechanics used in gear reaction calculations. The units to be used may be either English units or metric units. Table 7.1 shows the units normally used. To have dynamic equilibrium, all the laws of dynamics must be obeyed and all forces must have direction, magnitude, and a point of application. This is shown by a graphical or pictorial representation by means of a line with an arrowhead. The arrow denotes direction; the length of the line, magnitude; and the tail, the point of origin.

7.1.1 SUMMATION

OF

FORCES

AND

MOMENTS

Any number of lines of force in three planes can be resolved into one result. Figure 7.1 shows two forces in a XY plane and their resultant: L= tan

X2 + Y 2

x

=

Y X

(7.1) (7.2)

Conversely, X = L cos

x

(7.3) 379

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TABLE 7.1 Units Used for Gear Reactions Item

Metric

English

Force

Newtons

Distance

Meters, or millimeters

Pounds Inches or feet

Time Work

Minutes or seconds Newton·meters

Minutes or seconds Inch*pounds

Power

Kilowatts

Horsepower

Velocity

Meters per second

Feet per minute

FIGURE 7.1 Vector diagram.

Y = L sin

(7.4)

x

A moment of a force is equivalent to the sum of the individual moments of its components. This is true in either coplanar or three planes. In Figure 7.2 a moment is shown in the XY plane: M=Ld

(7.5)

M = X dy + Y dx

(7.6)

These same laws hold for three planes, and for dynamic equilibrium their total sum must equal zero except for the power transmitted or lost. In Figure 7.3 for forces:

FIGURE 7.2

Moment diagram (M = L·d = X·dy + Y·dx).

Gear Reactions and Mountings

FIGURE 7.3

381

Three-dimensional vector diagram.

L=

X2 + Y 2 + Z 2

(7.7)

where: X = L cos Y = L cos Z = L cos

Summation of moments is shown in Figure 7.4:

FIGURE 7.4

Three-dimensional moment diagram.

x y z

(7.8)

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Dudley’s Handbook of Practical Gear Design and Manufacture

M = Mx + My + Mz

(7.9)

or: M=

Mx2 + My2 + Mz2 cos

x

=

cos

y

=

cos

z

=

Mx M My

(7.10)

(7.11)

M Mz M

It is also important to know two other facts pertaining to couples and moments: 1. Figure 7.5a shows a couple that has no resultant; the two forces are equal but act in opposite directions. This produces rotation, and the moment is the product of one force multiplied by the entire distance between them. Only an equal and opposite couple can balance another couple. It cannot be balanced or held stationary by a single force or reaction (see Figure 7.5b) 2. All gears rotate about an axis, and it is here that the application of moments in space can be simplified. No force has a moment about a parallel axis.

7.1.2 APPLICATION

TO

GEARING

Figure 7.6a and b show how the moments in space are made coplanar and the X−axis is a point. Therefore, the moment of force about the X−axis is due only to the sum of the Y− and Z− forces and arms. To have equilibrium, X = 0,

Y = 0,

FIGURE 7.5

Z=0

(7.12)

A couple: (a) couple, (b) couple in equilibrium.

FIGURE 7.6 Force vectors at a point in space.

Gear Reactions and Mountings

383

Mx = 0,

My = 0,

Mz = 0

(7.13)

Since this shows only three independent variables, there can exist a maximum of three unknown values. To have dynamic equilibrium, there cannot be more than six unknown quantities to be solved, and both sets of Eqs. (7.12) and (7.13) must be completely satisfied. These rules always hold true for gears rotating about any axis. To reduce gear reactions and their resultant calculations to the simplest form, no matter how many moments or forces there are on a gear, always resolve them into two basic bearing loads, one of which is axial and the other radial, as shown in Figure 7.7. The one basic rule that must be applied to all gear mountings and mechanics analysis is, the sum­ mation of all forces must equal zero and the summation of all moments must equal zero. This rule is valid for gears, those rotating steadily, with no acceleration/deceleration.

7.2

BASIC GEAR REACTIONS, BEARING LOADS, AND MOUNTING TYPES

There are various sources of gear reactions and bearing loads. These are further complicated by the manner in which the gear and its shaft are mounted.

7.2.1 THE MAIN SOURCE

OF

LOAD

The main sources of loads are torque, reactions, weight, centrifugal forces, and vibrations. In most cases the gear torque is the main applied load and is usually caused by power input and work being done at the output. To determine torque or twisting moment and loads from horsepower, the amount of horsepower, the revolutions per minute, the pitch diameters of gear and pinion, and the reduction ratio must be known: T1 =

P × 9549.3 n1

(7.14)

TP =

Ph × 63.025 nP

(7.15)

where: P ( Ph ) – T1 (TP )– n 1 (nP )– d1 (d )–

is is is is

the the the the

power [metric (English)], kW (hp). pinion torque, N m(in. lb). speed, r/min. pitch diameter, mm (in.).

FIGURE 7.7 Load reactions at a bearing: W—combined load, Wr—radial load at 90° to axis of reaction, Wx—thrust load directly along axis of rotation.

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Dudley’s Handbook of Practical Gear Design and Manufacture

The gear torque is: T2 =

T1 × number of gear teeth number of pinion teeth

(7.16)

TG =

TP × number of gear teeth number of pinion teeth

(7.17)

The tangential load at the pinion pitch diameter is designated for both metric and English units: Wt =

7.2.2 GEAR REACTIONS

TO

2 × pinion torque pitch diameter of pinion

(7.18)

BEARING

With Wt known it is possible to calculate the separating force W r and the axial force Wx. Once these are determined, the values may be added vectorially so that Wt and W r determine the total radial load Wr on the gear and bearing and the axial force Wx. As stated in Section 7.1 and shown in Figure 7.8, the total radial (Wr) and axial load (Wx) determined in space are all that is necessary to determine total loads W on the gear teeth and reactions on bearings: Wr = W r + Wt

FIGURE 7.8

Tooth reactions on a gear loaded in three planes.

(7.19)

Gear Reactions and Mountings

385

W = Wr + Wx

(7.20)

Here the following equalities take place: Wr = |Wr|, |W′r| = W′r, |Wt| = Wt, and |Wx| = Wx.

7.2.3 DIRECTIONS

OF

LOADS

The method of finding the calculating W′r, Wr, Wx is found later on and is shown for spur and helical gears. There are also general rules that hold true for directions of loads on driving and driven members: • The radial load on the bearing supporting a gear tooth that is driving is always opposite to its direction of rotation. • The radial load on the bearing supporting a gear tooth that is being driven is always the same as its direction of rotation. The rules for determining axial loads tend to be more complicated: • When viewed from the axis of rotation and when the hand of spiral is left on a counter-clockwiserotating driving gear, its axial bearing thrust is away from the viewer. • When viewed from the axis of rotation and when the hand of spiral is right on a counter-clockwiserotating driving gear, its axial bearing thrust is toward the viewer. • When viewed from the axis of rotation and when hand of spiral is left on a clockwise-rotating driving gear, its axial bearing thrust is toward the viewer. • When viewed from the axis of rotation and when hand of spiral is right on a clockwise-rotating driving gear, the direction of axial bearing thrust is away from the viewer. • The direction of the driven gears’ axial thrust is always directly opposite the axial thrust of the driving gear. This can be best shown in tabular form in Table 7.2. It should always be kept in mind that the tangential, radial, axial, and total vectorial sums of any of these same components on driving gear are always directly opposite to those on the driven gear.

7.2.4 ADDITIONAL CONSIDERATIONS Consideration must be given, not only for the axial and radial bearing loads generated from the geometry of the gears, but for the other external forces that must be supported by the shaft and bearings. For example, spur gears should not normally generate any axial thrust, but, on a common shaft, a pump, fan,

TABLE 7.2 Direction of Axial Thrusts on Driving and Driven Gears Hand of Spiral

Direction of Rotation

Driving

Driven

Left

Clockwise

Toward viewer

Away from viewer

Right

Counterclockwise Clockwise

Away from viewer Away from viewer

Toward viewer Toward viewer

Counterclockwise

Toward viewer

Away from viewer

*

For thrust directions on bevel and hypoid gears, see Section 7.7.

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Dudley’s Handbook of Practical Gear Design and Manufacture

or propeller could be attached, generating substantial axial or centrifugal forces; and this would have to be restrained as well as the gear forces. Transfer of power in any mechanism always results in losses due to inefficiency. In the calculations presented here, 100% efficiency is assumed. In types of gearing with high rubbing rates, such as worms, the output power should be reduced because of the higher losses incurred. Weights of actual gears are usually small that they are of no consequence. Direction of gravity and weight must be considered when large gears or when large masses are attached to the gear shaft. Care must also be taken in allowing for the effect of “g,” or gravity loadings. In some cases, values as high as 15g s are encountered. Forces caused by the rotation of a mass around a center different from its own center of gravity are called centrifugal forces. Parts that are out of balance, have large external masses, or rotate in a planetary field, have centrifugal forces that must be considered. The most common formula for calculations is: Wcf =

r¯ wn2 36,128

(7.21)

where: Wcf – is the centrifugal force, lb. r¯ – is the radius from rotational center to center of gravity, in. n– is the rotation per minute. w – is the weight, lb.

or, where in oz of unbalance are known, Wcf = 1.73

n 1000

× in. oz of unbalance

(7.22)

For those needing an answer in kilograms of force, the simple English formula shown above can be used, and the answer in pounds can be converted to kg f by dividing by 2.20462. All gear reactions can be resolved into tangential force, separating force, and axial thrust. The relative values of these varies in intensity depending on the type of gears; and the loads applied to the bearings are extremely dependent on the type of mounting used.

7.2.5 TYPES

OF

MOUNTINGS

There are really two basic types of mountings—straddle and overhung. In straddle mountings, as shown in Figure 7.9, the radial load is divided in inverse proportion to the distance from point of application to the total distance between shaft supports. The bearing reactions act opposite to the direction of the load from the gearing. In overhung mountings as in Figure 7.10, the load is applied outside the two supports. This produces a greater reaction or load on the bearing nearest the applied load. The load on the bearing farther away is smaller. The bearing loads are not in the same directions. The bearing reaction nearest the applied load is opposite to the direction of the load. The reaction on the other bearing, farthest away, is in the same directions as the applied load. Since, in an overhung-mounted gear, the largest bearing load is nearest the mesh, this assumes greater importance in heavily loaded optimum-designed bearings mountings. To compensate for this, generally the bearing nearest the mesh is of type that has the greatest radial load-carrying ability. The bearing farthest away is often a ball or axial resistance bearing and constrains both the axial and the smaller radial loads. This method is used when there are no great temperature or deflection problems. In some

Gear Reactions and Mountings

387

FIGURE 7.9 Example of straddle mounting.

FIGURE 7.10

Example of overhung mounting.

cases, two different-sized bearings are used, with the greater-capacity bearing nearest the overhung member. It is wise to locate the axial-thrust retaining bearing next to the gear when large temperature variations or deflections exist. Change of rotation and potential thrust-direction changes must be taken into account. While it is necessary to balance out reactions loads, from now on whenever a load is calculated, it will be the actual load applied to the bearing or support in the correct direction. It will not be the bearing reaction, which opposes the load and is the opposite direction of the bearing load. There are relative merits between the overhung- and straddle-mounted gears and shafts. The most commonly used and preferred method is the straddle mounting. It balances loads, more efficiently uti­ lizes the bearings, and in most cases reduces misalignment and deflection problems.

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Dudley’s Handbook of Practical Gear Design and Manufacture

When there are extreme or unusual axial-thrust loads or special mounting problems, the overhung mounting has its merits.

7.2.6 EFFICIENCIES Gear efficiencies and resultant power losses can cause great variation in the actual power delivered and the load transmitted. These loss values vary from ½% per mesh to values as high as 80% per mesh, depending on the type of gear, lubrication, bearings, and degree of accuracy of manufacturing. It is a poor gear design or application that has efficiency less than 50%. It is generally a non-planar application that has great losses. Some designers allow a set value for each mesh. While 100% efficiency is assumed in all these calculations, it is always necessary at least to consider the effect of efficiency. If the power loss is low, generally a few percentage points or less, it is best to ignore this effect on bearings and gears. Coplanar gears can be from 97 to 80% efficient. When critical, experience or actual test data should be used to determine losses. As a quick guide for efficiencies, or if useful data are not available, the values in Table 7.3 are suggested for good conditions.

7.3

BASIC MOUNTING ARRANGEMENTS AND RECOMMENDATIONS

Once the bearing loads are calculated for any gear, experience, art, and common sense give some pointed recommendations and rules. These rules apply to all simply mounted gears no matter what type or shape they take. Various other rules for each specific type of gear are shown later under their appropriate heading. Keep bearing mountings for gears as close as possible to their faces, allowing reasonable space for lubrication and arrangements. This eliminates large moments and reduces vibration problems. Whenever possible, use only two bearings for each gear shaft. When making a center-distance layout for gear proportions, also consider shaft, spline, and bearing sizes and load-carrying abilities. These often are limiting conditions. Whenever possible, straddle mounts both members of a mesh. For straddle-mounted gears a spread between bearings of approximately 70% of the pitch diameter is a minimum. When only one gear can be straddle-mounted, it should be the gear with the highest radial load.

TABLE 7.3 Approximate Gear Losses for Various Types Different Types of Gears

Total Range of Losses per Mesh, %

Spur and helical external

½–3 Range of Losses per Mesh

Spur and helical internal

½–3

Bevel gears Worms

½–3 2–50

Spiroids*

2–50

Hypoids Crossed helicals

2–50 2–50

Note: All non-planar gears have great variation due to ratios, sizes, materials, lubricants, and relative sliding. * Registered trademark of Spiroid Division, Illinois Tool Works, Chicago, Illinois.

Gear Reactions and Mountings

389

Overhung gears should have a spread between bearings of 70% of the pitch diameter, and the spread between bearings should be at least twice the overhang. The shaft supporting the overhung gears should be greater in diameter than the overhang (see Figure 7.14 below). Idlers can often be positioned to offset and minimize bearing loads. Consideration must be given to the potential for the two meshes on the gear train setting up undesirable torsional vibration and overloads. In general, for all types of gears, the tangential forces applied to the bearing are opposite to the direction of rotation of the driving member. The separating forces are away from the tooth surfaces. The driven gear has its tangential forces and separating forces always opposite to the driving pinion or driving force. Reverse direction of rotation, and there is a reversal of tangential driving loads, but the separating tends to remain in the same direction away from the teeth (toward center). Axial thrusts vary for all types of gears, but they also reverse direction with reversal of rotation except in the case of Zerol and straight bevel gears. If the total force on any pinion or any gear of a single mesh is found, the force on the mating gear of that mesh is directly opposite to it. When calculations are laborious, this fact can be used to establish the direction and magnitude of the forces on the mating gear for that individual meshing point in space. All shifting should be checked for torsional and moment load-carrying ability. The distance between supports and shaft diameter should preferably be designed so that operating speed is below the first critical speed. The shaft should not operate near any critical vibration frequency.

7.3.1 BEARING

AND

SHAFT ALIGNMENT

It is extremely important to minimize gear tooth mounting displacements. The bearing sizes and shaft dimensions are determined only in part by strength and life considerations. The above-mentioned rules are good examples that should be followed in addition to utilizing strength and stress knowledge. The bearing spacing and shaft stiffness, as well as the housing and methods of attachments between gear, shaft, bearing, and housing, must be considered. In most applications, the deflections should not exceed 0.001 at the working surfaces between the gear and pinion. Total amount of allowable misalignment between gear surfaces should be further investigated and calculated. The method of calculating increase stresses in spur and helical gears due to misalignment should be investigated. The total contributing factors to misalignment should not exceed the calculated or known allowable value.

7.3.2 BEARINGS From the tentative gear layout, calculate the bearing loads and establish the preferred types and sizes of bearings. Actual calculations should be based on data that follow in detail each type of gearing and mounting that is involved. It is an accepted practice to preload the bearings carrying the thrust loads in order to obtain mountings with minimum total axial displacements under operating loads. The amount of preload depends on the mounting, load, and operating speed and should be established in collaboration with the bearing manufacturer. At this time the design of the gearbox with complete sped, load, and operating data should be submitted to the bearing manufacturer for approval of bearing sizes, types, bearing mountings, lubrication, and bearing life. The calculation of the bearing load capacities and life is not included here, but such data and information are readily available for any first-rate bearing manu­ facturer. Many bearing manufacturers have engineering design books that are given to users of their products. Selection of dimensional fits can also be learned. Generally, the most important rule is for the tight or interference fit to be between the inner race and shaft, and a loose or controlled slippage to be between the outer race and the housing or bearing liner.

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7.3.3 MOUNTING GEARS

Dudley’s Handbook of Practical Gear Design and Manufacture TO

SHAFT

Whenever possible, the gear should be made integral with the shaft. However, because of complex shapes and bearing and mounting arrangements, and with several gears and components mounted on a single shaft, this is not always possible. When gears must be assembled onto the shaft, involute splines are often used. In mounting gears with splined bores, a cylindrical centering fit is recommended to avoid excessive eccentricity in the final mounting. The most satisfactory method is to provide a suitable length of cylindrical fit at one or both ends of the gear hub for centering purposes. The splines are then used for driving only. This method is particularly applicable in the case of involute side-fit splines that cannot be adequately centered from the major or minor diameters (see Figure 7.35 below). When space or length allowable is critical, the major-diameter-fit involute spline is often used. The internal spline is often broached and the major diameter of the external member is ground to fit the broached part. When used this way, the spline is stressed from both tangential driving torques and possible tip loading from in­ terference fits caused by assembly or temperature contractional effects. Present practice is to avoid the use of minor-diameter-fit involute splines whenever possible. In many applications, straight-sided splines and serrations are used. Serrations and straight-sided splines are generally treated in the same manner and their rules of usage are interchangeable. Hardened gears with straight-sided splines in the bore should be centered in assembly by the bore or minor diameter of the splines, which, after hardening, should be ground concentric with the teeth. Unhardened gears with straight-sided splines should be centered in cutting and in assembly by the major diameter of the splines. Since heat treatment may introduce distortion in all splines, it is important that the splines be of no greater length than is actually required for load transmission1. In blanks with long hubs, the splines should be located as nearly as possible to the gear teeth and, if possible, directly under the gear teeth. Keys, pins, and press-fit diameters are used in many gear applications. In data gearing, loads normally are light and the mounting method is picked to give the least amount of running-index error. The integral gearand-shaft is preferred with the cylinder fit of gear to shaft next. When heavy loads and extreme accuracy requirements are needed, ingenious combinations of various basic mounting methods must be employed.

7.3.4 HOUSING The function of the housing is to form a strong base in which to mount the bearings and support the gears and shafts and to create an environment and space where a satisfactory lubricant may be introduced to lubricate and cool the gears. The housing is also used to mount and support various other components such as accessories and parts near or common to the gearing. In many illustrations, the lubrication system is contained entirely with the housing; in some oil systems, lines into and out of the housing must be supplied. The actual design and calculation of most gearboxes can be so complicated that a general practice is to employ experience or test data or actually to submit the gear housing to loads and measure the results by dial or strain gauges. In simple-type shapes, basic stress analysis is used. With the bearings selected and the general arrangement established, it is possible to proceed with the design of the supporting structure. The design should ensure, whenever possible, that loads from external sources such as belts, propellers, and pumps are carried entirely independently of the gear mountings; otherwise, the design must be made sufficiently rigid to carry these external loads without affecting the operating position of the gears. Flexible-type couplings are preferred to connect the assembly to the input and output loads in order to protect the gears from outside forces resulting from shaft deflections or misalignments. The housing and supports should be of adequate section and of suitable shape, with ribbing to support the bearings rigidly against the forces, as indicated by the analysis of their magnitude and directions. The choice of material for the housing depends to a great extent on the application. If weight is not a factor,

1

Generally, the length of a spline is equal to or less than the spline pitch diameter.

Gear Reactions and Mountings

391

ferrous castings or a combination of ferrous castings with weldments may be used. Where weight is very important, aluminum or magnesium castings may be used for the housing. When light metal housings are used, special design considerations must be given to minimize the effect of temperature changes on the mountings and to provide adequate rigidity under all operating conditions. Best practice calls for mounting the bearings in steel liners, either fitted and pinned in the housing bores, or cast in and machined in place. Also, the thrust bearings that control the position of the gears on their axes should be located close to the gear teeth. Two opposed bearings should be arranged so that temperature changes will not seriously affect preloading. It should be noted that in order to obtain equal rigidity in a light metal casting it may be necessary to use sections of two to four times the thickness of those used in steel or iron.

7.3.5 INSPECTION HOLE As an aid to assembly and for periodic inspection in service, an opening should be provided in the housing to permit the inspection of the tooth surfaces of at least one member of the pair without dis­ assembly. This is especially desirous in non-planar axis gears or in bevel and hypoid gears.

7.3.6 BREAK-IN While gears and bearings are designed, mounted, and manufactured to carry their rated loads without trouble, the initial period of operation is most critical, and preferably a “break-in” run at lighter loads or slower speed should be make. The load should gradually be increased until full operating load and speed are reached to check the complete gearbox.

7.4

BEARING LOAD CALCULATIONS FOR SPUR GEARS

The most common gear used is the spur-type. This is true not only because of the simplicity of stress calculations, design, and manufacturing but largely because of the ease of mounting, calculations, and retention of bearing loads. Spur gears are almost always devoid of any large self-generated axial thrusts. Their only components are the tangential driving load and its separating components, which results in a radial load only, which is the vectorial sum of these two forces [see Eq. (7.19)].

7.4.1 SPUR GEARS In a simple spur gear, only a radial load is present and Wr can be used interchangeably with the total load W. Because this fact assumes greater importance in other types of gears such as helical, Wr is used here. The tangential driving load can be calculated by Eq. (7.18). The separating load is designated as W r . For universal usage, the separating load is a function of the transverse pressure angle or that pressure angle formed by a plane slicing through the gear tooth at 90 to the axis of rotation. In a spur gear, the normal pressure angle, , and the transverse pressure angle, t , are one and the same. To simplify matters later on, the transverse pressure angle, t , will always be used in separating and radial-load calculations. (For metric calculations, use tan t and use Newtons for Wt instead of pounds): W

r

= Wt tan

t

(7.23)

2 r

(7.24)

Let Wr designates the total radial load: Wr =

Wt2 + W

If Wt is designated in the X plane and Wr in the Y plane, (7.24) agrees exactly with Eq. (7.1), which is for forces in one plane.

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The values Wr , Wt , and W r will all be considered as originating from the pitch diameter of the pinion or gear and in the center of its contact face, as will W and Wx later on. Figure 7.11 shows a perspective of two spur gears and their respective driving or driven load. The pressure angle of the gear controls the effect of the separating component. The smaller pressure angle gives lesser separating and total force, as shown in Figure 7.12. (7.24) is plotted against the tangent of the pressure angle to calculate Wr proportions shown in Figure 7.13. A pressure angle of 0° to 45° represents one unit of tangential driving load in pounds. Figures 7.9 and 7.10 and sample problems, show how to calculate overhung- and straddle-mounted gear resultant bearing reactions. Spur gears have no calculable axial thrust; however, they tend to walk or to be displaced by torque and other components. It is wise to allow for small amounts of axial movements or to restrain the gear. In heavily loaded spurs, or when spurs are common to shafts carrying other axial loads, it is a necessity to restrain or position them.

7.4.2 HELICAL GEARS All data shown so far have applied to external gears. However, internal gears are used in many appli­ cations. In external gears, the direction of rotation is changed in the mesh. In internal gear, the rotation between the two gears is in the same direction. In most cases one or both of the gears are overhung. While it is possible to arrange straddle mounting, normally space and length do not permit it. Internal gears offer the advantage of larger reduction or step-up changes in smaller center distances, but at the expense or allowable space for bearing mounting arrangements. Procedure for calculating bearing loads are shown in Figure 7.14; W r , Wt , and Wr are calculated before.

7.4.3 GEARS

IN

TRAINS

Many times, gears are in trains, either in one continuous plane with idlers or in offset panes on inter­ mediate shafts and in planetary drives. Planetary gears are considered as a separate type of gear train.

7.4.4 IDLERS The idlers may be used to change direction of rotation, to gain additional distance between centers with smaller-sized and greater quantity of parts, or to provide additional mounting pads at various r/min for driven components, shafts, or accessories. If only the input and output shafts are used and a constant horsepower is driven through the train, the load on all the gear teeth in the train is the same. When the idlers are all arranged in one straight line, the two separating components W r cancel output, but the tangential driving loads are added directly. Under the aforementioned conditions, the resultant bearing load is twice Wt . When the gear train is offset at an angle between centers, the total forces tend to combine or cancel out. For accuracy of results, a graphical or vector analysis should be made. Figure 7.15 shows a vector analysis of idler loads. If in Figure 7.15 the angle on BC in relation to AB have been acute instead of obtuse, the reaction WB would have been much smaller in magnitude. If it is possible, proper selection of rotation and angles of intersection can be adjusted to reduce the total bearing loads generated in an idler gear train.

7.4.5 INTERMEDIATE GEARS The other most common gear train in spur gears is the intermediate shaft arrangement. Its main ad­ vantage is the combination of greater gear ratio reduction and a gain in center distances for mounting various combinations.

Gear Reactions and Mountings

FIGURE 7.11

Spur gear bearing loads.

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Dudley’s Handbook of Practical Gear Design and Manufacture

FIGURE 7.12

Chart of relation W

FIGURE 7.13

r

for Wt=1.0 .

Chart of total radial load versus

t.

There are basically two gear meshes but only three shafts instead of four. Also, the planes of mesh are parallel but separated (see Figure 7.16). The total loads that are applied to and must be reacted against by bearing D are found by summation of moments about bearing C . Summation of forces, however, is taken about bearing D. When contact is at an angle or several meshes are concerned, both vertical and horizontal planes of moments must be taken. The bearings must resist making the total moment equal zero. The proportions of spacing between bearings are very critical. The distances or moment arms should be selected to favor the F bearing in Figure 7.16 would, in most gearboxes, be the smaller because of lack of space. Therefore, the distance g on the y gear was intentionally made longer to reduce the bearing load on F and put the greater load on E , as it should have more space available for a larger bearing with greater load-carrying capacity. Also, whenever possible, the distance between gears and bearings is as small as is feasible to eliminate large moment loads in the shafts. The shaft on x n gears will have a coupling tendency to cock or walk, and consideration should be given to axial-thrust bearings to resist the coupling effect. To aid the positioning of planes of intersection on the gear teeth, avoid large axial moments and overhangs.

7.4.6 PLANETARY GEARS Planetary gears are used most often for maximum horsepower transmitted in the smallest space or when large changes in speed ratios are needed. The increased carrying capacity is generally possible because the gear reactions tend to equalize each other, reducing bearing loads, size required, and total weight. Helical, bevel, and spur gears are used in planetary drives, but spurs are often preferred because of ease of manufacturing and elimination or reduction of axial-thrust loads. For simplicity’s sake, spur gears have been assumed in these discussions, but helical and other types of gears can just as easily be used. In a simple planetary arrangement, there is generally a sun gear, a planet gear, an internal ring gear, and a cage or arm (see Figure 7.17). One planet. If the planetary drive has one planet gear, load reactions are different from those in the case where there are two or more planet gears. The one-planet-gear calculations are basically the same as the combination of an external idler and an internal gear and external pinion. For example, the sun-andone-mesh planetary drive is the same as two external gears meshing. The reaction on the internal gear is

Gear Reactions and Mountings

FIGURE 7.14

395

Internal spur gear bearing loads.

FIGURE 7.15

Vector analysis of idler loads.

396

FIGURE 7.16

Dudley’s Handbook of Practical Gear Design and Manufacture

Intermediate gear trains.

the same as in a simple one meshing with an external pinion. However, on the planet gear, consideration must be given for the reactions on its center of rotation from the cage or arm. Several planets. If two or more planets are used, preferably three or more, the reactions are different because three planets mate with one sun gear and one ring gear; and the bearing reactions tend to balance out. There are basically zero resultant separating bearing forces on the sun pinion and the internal ring gear. The planet gear bearing has the loads from the two meshes and the cage driving arm, plus the weight and centrifugal force from the planet and its bearings. When there are high-speed planets, the effect of centrifugal force must be considered. If speeds and weights are low, centrifugal force may be ignored. If they are high, compute them by means of (7.21) or (7.22). Centrifugal force may become the prime factor in the final decision of bearing design or type of planetary gear drive used. When the total gear reaction W or Wcf (centrifugal force) are found, they are added vectorially or computed as follows: L=

Wcf2 + W 2

(7.25)

L – is the total load of all forces. Wcf – is the centrifugal force. W – is the total gear reaction.

However, helix forces, unbalance, and driven propeller or fan reactions must be considered when they are tied into the planetary drive system. When two or three planets are used, the loads should be

Gear Reactions and Mountings

397

FIGURE 7.17

Simple planetary arrangement.

considered as shared equally by all planets. When there are more than three planets, it is not always possible to share the load equally, and some allowance should be made for unequal loading of all the individual planets. The following special recommendations apply to idlers. Preferably use two or more planets to carry the load. When possible, use three planets. If more than three are needed, may designers feel it is wise to go to five or more because of a lack of load-sharing tendencies. Always mount the entire system as flexibly as possible. For example, always mount the planets on spherical floating bearings or mounts. Make all mountings or actual holding surfaces as light in weight as can be utilized to reduce centrifugal forces.

7.5

BEARING-LOAD CALCULATIONS FOR HELICALS

So far, all data have been based on spur gears and all radial values calculated are directly applicable to helical gears if the transverse pressure angle is considered.

7.5.1 SINGLE-HELICAL GEARS Helical gears have thrust and reactions in all three planes. Besides the tangential driving and separating forces, there is also an axial-thrust load. This thrust is due to the helix angle and is normally considered emanating from the tooth contact at the pitch diameter. Since thrust is applied at the distance of the pitch radius from the axis of rotation a radial moment is produced. However, in helicals the moment is produced within the gear. This information on helicals can be transcribed into the laws of mechanics. Three forces are shown in Figure 7.18: Wt – tangential driving load W r – separating force Wx – axial thrust

FIGURE 7.18 Helical gear-face advance. Face advance = QF + F tan ψ. Contact ratio of faces QF/Pt ≥ 1.

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Dudley’s Handbook of Practical Gear Design and Manufacture

W – total load Ma – moment produced by axial thrust Wt and W r are found as before. It is important, though, to consider a new effect on the pressure angle: tan

tan

t

t

=

=

tan cos tan

n

n

cos

metric

(7.26)

English

(7.27)

where: t ( t )– n ( n )–

is the transverse pressure angle. is the normal pressure angle.

( )– is the helix angle.

The separating load is a function of transverse pressure angle, which in a helical gear is always greater than n , the normal pressure angle. Therefore, W r will always be a slight amount larger than in a spur gear but is calculated as before in (7.23). The helical angle assumes great importance in calculating the axial and separating load, and in the equations presented, the helix angle will be used for all calculations. tan

=

× pitch diameter lead

(7.28)

Lead is the distance a point would advance axially across the face of the gear for one revolution measured at the pitch diameter. A small helix angle has a high lead and a large helix angle has a low lead. Wx increases with higher helix angle and, whenever possible, should be kept to the smallest angle. However, for a helix angle to be of any value, it should have a face overlap of a little over 1. This means the contacting distance of face advance along the pitch-diameter periphery of a helical tooth is equal to or greater than the transverse circular pitch (see Figure 7.18). Normally, for high-speed gears, a face advance of over 2 is preferred. Face contact ratio =

Face advance Circular pitch

(7.29)

As gears become more heavily loaded with higher speeds, and quietness is a requirement, a greater overlap ratio is needed. Therefore, the selection of the helix angle becomes a direct function of face advance. tan helix angle =

Face advance Face width

(7.30)

With the selection of t helix angle, the helical gear forces can be determined as shown in Figure 7.8. In its simplest form: Wx = Wt × tan helix angle

(7.31)

Gear Reactions and Mountings

399

The total load W is the vector summation of Wt , Wx and W r : W=

Wt2 + W

2 r

+ Wx2

(7.32)

for X , Y , and Z axis. More consideration must now be given to summation of forces in various planes. In some cases, values are additive and in other cases they are subtractive. The moment produced from the helix is now calculated: Ma = Wx × pitch radius of gear

(7.33)

where Ma is the axial moment, and Wx is the axial thrust. Ma produces an additional moment and, in turn, an additional radial load, which must be contained by the bearings and supports. Also notice that, if the vectors representing Wt , W r , and Wx are assumed to be respectively, the X , Y , and Z axis; (7.32) conforms to Eq. (7.7). From the standpoint of load produced axially, it is always desired to keep it to the absolute minimum value. Since a helical gear always requires a bearing to resist an axial force, this can be a disadvantage in some installations. The forces can become as large as to be determined in large helix angles. See Figure 7.19 and Table 7.4 for details of bearing load calculations.

7.5.2 DOUBLE-HELICAL GEARS To overcome axial thrust, the forces are sometimes counteracted or nullified by double-helical gears. The total face is made into two equal halves, both with the same helix angle but opposite hands. The axial thrusts oppose one another and the forces are contained in the gear and not transmitted to the bearing. The double-helical gears are often made of two separate halves of helical gears bolted or joined together, or else a space between the gears is allowed in the middle for tool clearances. There are some gear machine tools that permit a continuous herringbone when face width is limited. Only one gear in a double-helical train is positioned or restrained axially by a bearing, and the rest of the gears then get their location from this one part. For speed-increasing gears it is often wise to use or consider either helical or double-helical gears. As axial leads and helix angles increase, so also do the merits of double-helical gears increase.

7.5.3 SKEWED

OR

CROSSED HELICAL GEARS

Besides helical gears operate on parallel axes, it is possible to have the axes of rotation skewed or at an angle with each other. The latter gears are called crossed helical or skewed helical gears. When the individual gear reactions are found, the same methods are used. In helical gears, the total reaction for the pinion is equal and opposite to the mating gear. In a pair of crossed helicals, each gear and pinion must be considered and calculated separately. The direction of Wx is generally obvious. In crossed helical gears, one of the mating parts could be a helical or spur gear and it would mate with a helical gear. All spiral helicals tend to have only point contact, are used to transmit light loads, and have small bearing reactions. Allowance must be made for the helix angle in relation to the axis of rotation of each gear. In external helical gears, the hands of the mating parts are equal and opposite, but in crossed helical gears all types of hands and angles of helix can and must be considered. Calculations of the bearing loads from the gear reactions are the same as for helical gears. More care and consideration must be given to the angles of helix in relation to the shaft angles. The hand of the helices and shaft

r

WrA =

Wx r 1 a+b

Wra a+b

Wt a a+b

Wr 3)2 Wr B =

Wx r 1 a+b

= Wr 3

Wrb a+b

Wt b a+b

Wr24 + (Wr 5 + Wr 6 )2

Wr 6 =

Wr 5 =

Wr 4 =

Bearing B

Wx (may be applied to either bearing A or B)

Wr21 + (Wr 2

Wr 3 =

Wr 2 =

Wr 1 =

Bearing A

Gear Bearing Loads

Note: See Figure 7.19 for definition of terms and bearing locations.

Total thrust

Total radial load

Thrust Wx

Separating force W

Tan force Wt

Force

TABLE 7.4 Equations for Helical Gear Bearing Loads with Straddle Mounting

Wr C =

Wx r 2 c+d

= Wr 3

Wrc c+d

Wt c c+d

Wr D =

Wx r 2 c+d

= Wr 9

Wrd c+d

Wt d c+d

Wr210 + (Wr 11

Wr 12 =

Wr 11 =

Wr 10 =

Bearing D

Wx (may be applied to either bearing C or D)

Wr27 + (Wr 8 + Wr 9 )2

Wr 9 =

Wr 8 =

Wr 7 =

Bearing C

Pinion Bearing Loads

Wr 12 )2

400 Dudley’s Handbook of Practical Gear Design and Manufacture

Gear Reactions and Mountings

FIGURE 7.19

401

Helical gear bearing loads.

angles can be varied so that direction of rotation is reversed. Once the true helix angle and the actual direction of rotation between the driver and driven gears are determined, the reactions are easily found. Figure 7.20 shows a pair of crossed helical gears. The light lines on the driving gear show the teeth as they would appear to the viewer. The meshing point considered in the diagram is underneath the driver. After allowing for the direction of rotation, shaft angle, and hand of spirals, the direction of the forces can be found using Table 7.2. The angles P and G can be found, and, knowing Wt , the axial thrust for the gear and pinion are determined as in any helical gear. See Figure 7.19 for details on bearing load calculations. In the metric system, the symbol for pinion helix angle is 1 instead of P . For the gear the symbol is 2 instead of G .

7.6

MOUNTING PRACTICE FOR BEVEL AND HYPOID GEARS

All gears operate best when their axes are maintained in correct alignment, and, although bevel and hypoid gears have the ability to absorb reasonable displacements without detriment to the tooth action, excessive misalignments reduce the load capacity and complicate the manufacture and assembly of gears. Accurate alignment of the gears under all operating conditions may be accomplished only by good design, accompanied by accurate manufacture and assembly of both mountings and gears. The following recommendations are given to aid in the design of bevel and hypoid gearbox assemblies in order to obtain the best performance of the gears.

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Dudley’s Handbook of Practical Gear Design and Manufacture

FIGURE 7.20 Bearing loads on crossed helical gears. Regardless of shaft angle between crossed helical pinion and crossed helical gear, the directions of the pinion tangential force Wt and the pinion thrust force Wx are as shown above for the mesh point.

7.6.1 ANALYSIS

OF

FORCES

In the design of the mountings for bevel and hypoid gears, the first step is to establish the magnitude and directions of the axial and separating forces, from Figures 7.21 and 7.22, and draw a diagram of the resultant force on the gears as shown in Figures 7.23 to 7.26. These provide the basis for the design of the gear blank and the mountings and for the selection and arrangement of the bearings. In case the gear must operate in both directions of rotation or if there is reversal of torque, the force diagram should include both conditions of operation. Clockwise and counterclockwise rotations are as viewed from the back of the gear looking toward cone center. Values shown are for shaft angles at right angles, but values and determination of directions also apply to shaft angles other than 90° (Figures 7.24 and 7.25).

7.6.2 RIGID MOUNTINGS Gear mountings should be designed for maximum rigidity since the problems involved in producing satisfactory gears multiply rapidly when deflections in the mountings cause excessive gear displace­ ments. It is necessary to modify the standard cutting in order to narrow and shorten the tooth contacts to suit the flexible mounting. The decrease in bearing area raises the unit tooth pressure and reduces the number of teeth in contact, which results in problems of noise and increases the danger of surface failure and tooth breakage.

7.6.3 MAXIMUM DISPLACEMENTS For gears from 6- to 15 in. diameter (150 to 375 mm), the desirable limits of gear and pinion dis­ placements at maximum load are listed as follows: • • • •

The pinion offset should not exceed ±0.003 in. from correct position (±0.076 mm). The pinion should not yield axially more than 0.003 in. in either direction (±0.076 mm). The gear offset should not exceed ±0.003 in. from correct position (±0.076 mm). The gear should not yield axially more than 0.003 in. in either direction (±0.076 mm) in either direction on miters or near miters, or more than 0.010 in. (0.25 mm) away from the pinion on higher ratios.

Gear Reactions and Mountings

FIGURE 7.21

403

Design chart for axial thrust of bevel gears.

The aforementioned limits are for average applications in which the gears operate over a range of loads and at maximum load approximately 10% of the time. When maximum load occurs for longer periods, reduce the limits accordingly. Somewhat wider tolerances are allowable for large gears.

7.6.4 ROLLING-ELEMENT BEARINGS While plain bearings may be used in bevel or hypoid gear mountings, it is usually much easier to maintain the gears in satisfactory alignment with rolling-element bearings. Either ball or roller bearings are satisfactory. However, it is important that the type and size of bearings be carefully selected to suit the particular application, considering especially the operating speeds, loads, the desired life of the bearings, and the rigidity required for the best operation of the gears.

404

FIGURE 7.22

Dudley’s Handbook of Practical Gear Design and Manufacture

Design chart for separating component of radial load for bevel gears.

FIGURE 7.23 Resultant force in axial plane due to right-hand gear being driven counterclockwise or driving clockwise; also, lefthand gear being driven clockwise or driving counterclockwise.

Gear Reactions and Mountings

405

FIGURE 7.24 Resultant force in axial plane due to right-hand gear being driven clockwise or driving counterclockwise; also, lefthand gear being driven counterclockwise or driving clockwise.

FIGURE 7.25 Resultant force in axial plane due to left-hand pinion driving clockwise or being driven counterclockwise; also, right-hand pinion driving counterclockwise or being driven clockwise.

FIGURE 7.26 Resultant force in axial plane due to left-hand pinion driving counterclockwise or being driven clockwise; also, right-hand pinion driving clockwise or being driven counterclockwise.

7.6.5 STRADDLE MOUNTING The preferred design of a bevel or hypoid gearbox provides straddle mountings for both gear and pinion, and this arrangement is generally used for industrial and other heavily loaded applications. Usually, the desired bearing life and mounting rigidity may be obtained more economically with a straddle mounting. When it is not feasible to use this arrangement for both members of a pair, the gear member that usually has the higher radial load should be straddle-mounted.

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Dudley’s Handbook of Practical Gear Design and Manufacture

FIGURE 7.27 Webless-type gear member, straddle-mounted using two opposed tapered roller bearings widely spaced to provide good control of gear.

In straddle mounting the gear or larger member of a pair, usually two tapered roller bearings or angular-contact ball bearings are mounted opposed with sufficient spacing between them to provide control of the gear (see Figure 7.27). A second straddle-mounting arrangement (see Figure 7.28), used mostly for pinions, provides an inboard or pilot bearing for pure radial support of one end of the pinion shaft; and, on the opposite end, bearings suitable for carrying both radial and thrust loads. In this location several different types of bearings are used: a single double-row, deep-groove angular-contact ball bearings; or two single-row angular-contact bearings mounted DB or DF; or two tapered roller bearings, either direct- or indirect-mounted. A similar arrangement is used also for straddle mounting the gear or larger member of a pair in large applications, and in an aluminum or magnesium housing, where temperature changes would seriously affect the preloading or thrust bearings if spaced on either end of the gear shaft (see Figure 7.29).

7.6.6 OVERHUNG MOUNTING When the pinion is mounted overhung on tapered roller bearings, the indirect mounting should be used (see Figure 7.30). This arrangement provides greater stability to the mounting for a given spacing of the

Gear Reactions and Mountings

FIGURE 7.28

407

Straddle pinion mounting for short shafts, showing use of combined thrust and radial bearings.

bearings, and the thrust load for normal operation is thus carried by the bearing adjacent to the pinion, adding further to its stability. The bearing adjacent to the pinion should be located as closely as possible to the pinion teeth to reduce the overhang (see Figure 7.31). All spiral bevel and hypoid gears should be located against thrust in both directions. Provision against in ward thrust on straight and Zerol bevel gears is often omitted, provided the conditions of operating are such that inward thrust cannot occur. One of the advantages of Zerol and straight bevel gears is that change of rotation does not change direction of axial thrust. Since the contact pattern of spiral bevel and hypoid gears is controlled by their axial position, special attention must be given to the selection of the type and to the preloading of the bearings.

7.6.7 GEAR BLANK DESIGN The gear blank should be designed to avoid excessive localized stresses and serious deflections within it. The direction of the resultant force should be considered in the design of the blank and in the method employed in mounting it on its shaft or centering hub (Figures 7.32 to 7.37)). Ring gears are made in three general types: the webless ring, web-type ring, and counterbored type. Of these, the bolted-on webless ring design shown in Figure 7.27 is best for hardened gears larger than 7 in. (180 mm) in diameter. The fit of the gear on its centering hub should be either size-to-size or slight

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Dudley’s Handbook of Practical Gear Design and Manufacture

FIGURE 7.29 Straddle mounting for both members of a spiral bevel pair, using a straight roller radial bearing on one end of the shaft and combined thrust and radial bearing on another end. Shim-type adjustments for positioning both gear and pinion.

interference. These gears should be clamped to the centering hub with fine-thread cap screws as shown in Figures 7.27, 7.33, and 7.37, or with bolts. Several methods of locking screws and nuts in place are indicated in the illustrations. Self-locking-type nuts and cap screws are widely used for this purpose. For severe operating conditions, cap screws should be locked more positively than is possible by wiring. The method of centering the gear on the counterbore, shown in Figure 7.37, is recommended for large gears, especially near miters. Figure 7.33 illustrates mounting for a gear that operates with an inward thrust. Designs in which screws or bolts are subjected to added tension from gear forces should be avoided. On severe reversing or vibration installations, separate dowel drives may be used with the designs or splines as in Figures 7.35 and 7.38. The use of dowels or dowel-fit bolts has been found unnecessary in most automotive and industrial drives because, with bolts or cap screws drawn tightly, the friction of the ring gear in its hub prevents shearing of the screws. Hardened gears smaller than 7 in. (less than 180 mm) in diameter may best be designed with integral hubs. A front hub on a Zerol bevel, spiral bevel, or hypoid gear should in no case intersect the extended root line so as to interfere with the path of the cutter toward the root cone apex as shown in Figure 7.39.

Gear Reactions and Mountings

409

FIGURE 7.30 Typical overhung pinion mounting with shim adjustment for positioning, and selected spacer for preloading bearings.

On hub-type gears, the length of the hub should be at least equal to the bore diameter, and the end of the hub should be securely clamped against a shoulder on the shaft. Whether the gear is mounted on a flanged hub or is made integral with the hub, the supporting flange should be made sufficiently rigid to prevent serious deflections in the direction of the gear axis at the mesh point. For larger gears the web preferably should made conical and without ribbing to permit machining for balance, to eliminate oil churning when dip lubrication is used and to lessen the danger of stress concentration being set up with the castings. The gear hub and bearing arrangement should be designed with sufficient rigidity that the use of a thrust button behind the gear mesh in unnecessary. The area of a thrust button is too small to provide durable support. For hardened gears, the blanks should be designed with sections and shape suitable to hardening with minimum distortion.

7.6.8 GEAR

AND

PINION ADJUSTMENTS

Provision should be made for adjusting both gear and pinion axially at assembly. Shim-type adjustments as shown in Figures 7.28 and 7.29 or threads and nuts with locks as shown in Figure 7.40 may be used. Shims less than 0.015 in. (0.38 mm) thick may “pound out” in service. When used adjacent to a bearing, the shims should be placed next to the stationary member. If an assortment of shims varying in thickness in steps of 0.003 in. (0.076 mm) is used [such as 0.020 in. (0.50 mm), 0.023 in. (0.60 mm), 0.026 in. (0.65 mm)], the member to be adjusted may be positioned within ± 0.0015 in. (± 0.04 mm).

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FIGURE 7.31 Typical overhung pinion mounting and straddle gear mounting with shim adjustment for positioning both gear and pinion.

FIGURE 7.32

Method of mounting gear to absorb the thrust with minimum amount of deflection.

7.6.9 ASSEMBLY PROCEDURE The following is a recommended procedure for the assembly of bevel and hypoid gearboxes: • Thoroughly clean housing and parts, following the bearing manufacturer’s instructions in handling the rolling-element bearings. • Keep lapped or matched gears in their original sets. • Assemble even-ratio pairs with the teeth mating as lapped. Tooth and mating spaces are usually designated with X markings. • Set pinion on its correct axial position by measurement. • Adjust gear along its axis until the specified backlash is obtained.

Gear Reactions and Mountings

411

FIGURE 7.33

Method of mounting gear when thrust is inward.

FIGURE 7.34 Blank sections beneath the teeth should be sufficiently rigid in the direction of thrust so that deflections will be minimized.

• Securely lock gears and thrust bearings in this position. Note, for spiral bevel and hypoid gears, both members should be locked against thrust in both directions. • As a final check, paint the teeth with marking compound and observe the tooth contact pattern under light load. This may indicate the need for further adjustment. Check to ensure that there is an adequate supply of the specified lubricant to the mesh and to the bearings.

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Dudley’s Handbook of Practical Gear Design and Manufacture

FIGURE 7.35

Gear blank designed to take care of inward thrust. Also, a preferred mounting with spline drive.

FIGURE 7.36

Method of mounting gear to hub to absorb high outward thrust.

7.7

CALCULATION OF BEVEL AND HYPOID BEARING LOADS

The normal load on the tooth surfaces of bevel or hypoid gears may be resolved into two components: one in the direction along the axis of the gear, and the other perpendicular to the axis. The axial force produces an axial thrust on the bearings, while the force perpendicular to the axis produces a radial load on the bearings. The direction and magnitude of the normal load depends on the ratio, pressure angle, spiral angle, hand of spiral, direction of rotation, and whether the gear is the driving or driven members.

7.7.1 HAND

OF

SPIRAL

The hand of spiral-on-spiral bevel and hypoid gears is donated by the direction in which the teeth curve; that is, left-hand teeth incline away from the axis in the counterclockwise direction when an observer

Gear Reactions and Mountings

FIGURE 7.37

Webless-type miter gear—counter-bored type.

FIGURE 7.38

Positive drive for severe reversing or shock loads.

413

414

FIGURE 7.39 shown.

Dudley’s Handbook of Practical Gear Design and Manufacture

When using Zerol bevel, spiral bevel, and hypoid gears, clearance must be provided for the cutter as

looks at the face of the gear, and right-hand teeth incline away from the axis in the clockwise direction. The hand of spiral of one member of a pair is always opposite to that of its mate. It is customary to use the hand of spiral of the pinion to identify the combination; that is, a left-hand combination is one with the left-hand spiral on the pinion and a right-hand spiral on the gear. The hand of spiral has no effect on the smoothness and quietness of operation or on the efficiency. Attention, however, is called to the difference in the effect of the thrust loads as stated in the following paragraph. A left-hand spiral pinion driving clockwise (viewed from the back) tends to move axially away from the cone center, while a right-hand pinion tends to move toward the center because of the oblique direction on the curved teeth. If there is excessive end play in the pinion shaft because of faulty assembly or bearing failure, the movement of a right-hand pinion driving clockwise will take up the backlash under load, and the teeth of the gear and pinion may wedge together, while a left-hand spiral pinion under the same conditions would back away and introduce additional backlash between the teeth, a condition that would not prevent the gears from fluctuating. When the ratio, pressure angle, and spiral angle are such that doing so is possible, the hand of spiral should be selected to give an axial thrust that tends to move both the gear and the pinion out of mesh. Otherwise, the hand of spiral should be selected to give an axial thrust that tends to move the pinion out of mesh. Often the mounting conditions dictate the hand of spiral to be selected. In a reversible drive there is, of course, no choice unless the pair performs a heavier duty in one direction a greater part of the time. In the calculation of reactions, Zerol and straight bevel gears may be treated as special case of spiral bevel gears in which the spiral angle is zero degrees. In these, the direction of the axial thrust is always outward on both members regardless of the direction of rotation and whether the pinion is the driving or driven member. On hypoids, when the pinion is below center and to the right when facing the front of the gear, the pinion hand of spiral should always be left. With the pinion above center and to the right, the pinion hand should always be right (see Figure 7.41). Specification of hand of Zerol gears is illustrated in Figure 7.42.

Gear Reactions and Mountings

FIGURE 7.40

415

Truck axle—straddle-mounted pinion and with threaded-nut adjustments for the gear bearing.

7.7.2 SPIRAL ANGLE The spiral angle, if possible, should be selected to give a face contact ratio of at least 1.25. For maximum smoothness, the face contact ratio should be between 1.50 and 2.0. On straight or Zerol bevel gears the spiral angle will, of course, be zero.

7.7.3 TANGENTIAL LOAD The tangential load on a bevel or hypoid gear is given2 by:

Wt =

2

19, 098, 600P d2m × n2

In all calculations herein, the tangential load is calculated at the mean pitch radius.

(7.34)

416

Dudley’s Handbook of Practical Gear Design and Manufacture

FIGURE 7.41 Both pinions shown in (a) and (b) are referred to as having an offset “below center,” while those in (c) and (d) have an offset “above center.” In determining the direction of offset, it is customary to look at the face of the gear with the pinion at the right.

FIGURE 7.42 Diagram illustrating hand of Zerol. The upper drawing shows a left-hand, and the lower, the right-hand Zerol bevel gear combination.

Wt =

126, 050Ph dmG × nG

where: Wt (Wt )– is the tangential load [metric (English)], N (lb). P (Ph )– is the power, kW (hp).

(7.35)

Gear Reactions and Mountings

417

d2m (d mG )– is the mean pitch diameter of gear, mm (in.). n 2 , (nG )– is the speed, r/min.

The mean pitch diameter of a bevel gear may be determined by subtracting the term (bevel gear face width times the sine of the bevel gear pitch cone angle) from the bevel gear pitch diameter at the large end of the bevel gear. The tangential load on the mating bevel pinion will be equal to the tangential load on the gear. The tangential load on the mating hypoid pinion, however, must be determined as follows: WtP = WtG

cos spiral angle of pinion cos spiral angle of gear

(7.36)

7.7.4 AXIAL THRUST The value of the axial thrust is dependent on the tangential tooth load, spiral angle, pressure angle, and pitch angle. This may be determined with aid of Figure 7.21. Figure 7.21 is symmetrical about a hor­ izontal center line; therefore, there are two points of intersection of the pitch-angle and the spiral-angle curves. Selection of the proper point is dependent on the hand of spiral, the direction of rotation, and whether the member is the driving or driven one. This is given in Figure 7.21. Note that the intersection points for the pinion and its mating gear are always on opposite sides of the horizontal center line. The pressure angle is given on the scale at the right. A straight line connecting the pressure angle with the intersection point of the spiral-angle and pitch-angle curves can now be extended to the scale on the left. This scale gives the axial thrust Wx in percent of the tangential tooth load Wt . The spiral angle on straight and Zerol gears is zero. For determining the axial thrust in hypoid gears, the pressure angle of the driving side should be used. Instead of using the pitch angles, the face angle of the pinion and the root angle of the gear should be used. Note: The spiral angle, pitch angle, and pressure angle for the corresponding member must be selected in each case. The actual axial-thrust load may be determined by multiplying the tangential load in pounds for the corresponding member by the percent given in Figure 7.21: Wx = (percent from Fig. 11.21) × Wt

(7.37)

Figure 7.43 illustrates the direction of rotation and direction of thrust.

7.7.5 RADIAL LOAD The radial load caused by the separating force between the two members may be determined in a similar manner to that used for the axial thrust with the aid of Figure 7.7. The same procedure is followed here as outlined in the foregoing section, “Axial thrust.” The scale on the right gives the component part of the radial load Wr in percent of the tangential tooth load Wt . This component of the actual radial load may be determined by multiplying the tangential load in pounds for the corresponding member by the percent given in Figure 7.22: W

r

= (percent from Fig. 11.22) × Wt

(7.38)

In addition to the foregoing element, there is a radial component caused by the tangential load itself, and a third component caused by the couple that the axial thrust on the given member produces.

418

Dudley’s Handbook of Practical Gear Design and Manufacture

FIGURE 7.43 Diagram illustrating hand of spiral and its effect on the direction of axial thrust in spiral bevel gears. The upper drawing shows a left-hand; and the lower, a right-hand spiral bevel combination.

The total radial load produced on the bearings will therefore depend on the resultant of these three components. For the overhung mounting the radial components of bearing A will be (see Figure 7.44): Wr1 = W

r

Wr 2 = Wt

where W

r

L+M 2 L+M 2

(7.39)

(7.40)

is the separating component from Figure 7.23, and Wt is the tangential load.

Wr 3 = Wx

dm 2M

(7.41)

where: L – is the distance along the axis from center of the face width to center of bearing A [for hypoid gearing see Eq. (7.58). M – is the spacing between centers of bearings A and B. Wx – is the axial-thrust load.

Gear Reactions and Mountings

FIGURE 7.44

419

Diagram showing load components on a pinion and bearings—overhung mounting.

dm – is the mean pitch diameter (dm = d − F sin γ). [for a hypoid pair use (7.55) for dm and N by directions given for (7.58)]. d – is the pitch diameter. F – is the face width. – is the pitch angle.

The total radial component on bearing A will be: Wr A =

Wr22 + (Wr1

Wr 3)2

(7.42)

In like manner, the radial components on bearing B will be: Wr 5 = Wt

L M

(7.44)

Wr 6 = Wx

dm 2M

(7.45)

The total radial component on bearing B will be: Wr B =

Wr25 + (Wr 4

Wr 6 )2

(7.46)

For the straddle mounting shown in Figure 7.45 the radial components on bearing C will be: Wr 7 = W

r

K N+K

(7.47)

420

FIGURE 7.45

Dudley’s Handbook of Practical Gear Design and Manufacture

Diagram showing load components on a pinion and bearings—straddle mounting.

K N+K

(7.48)

dm 2(N + K )

(7.49)

Wr8 = Wt

Wr 9 = Wx

where N is the distance3 along the axis from center of face width to center of bearing C , and K is the distance4 along the axis from center of face width to center of bearing D. The total radial component on bearing C will be: Wr C =

Wr28 + (Wr 7

Wr 8)2

(7.50)

In like manner, the radial components on bearing D will be: Wr10 = W

r

3 4

(7.51)

K N+K

(7.52)

dm 2(N + K )

(7.53)

Wr11 = Wt

Wr12 = Wx

K N+K

For hypoid pair use Eq. (7.55) for dm and find N by directions given for Eq. (7.58). For hypoid pair find K by Eq. (7.57), see Figure 7.45.

Gear Reactions and Mountings

421

The total radial component on bearing D will be: Wr D =

Wr211 + (Wr10 + Wr12 )2

(7.54)

For hypoid gear pair, change the preceding equations as follows: dmG = gear pitch diameter

dmP = dmG ×

face width × sin gear root angle

(7.55)

number of pinion teeth cos gear spiral angle × number of gear teeth cos pinion spiral angle

(7.56)

JP

K=G

J

(7.57)

L=H

J

(7.58)

dmG cos for hypoid pinion 2

(7.59)

dmP cos for hypoid gear 2

(7.60)

JG

where: G – is the distance along axis from center line of mate to center of bearing D. H – is the distance along axis from center line of mate to center of bearing A. ε – is the arctan [tan(spiral angle pinion—spiral angle gear) × sin gear root angle]. M – is the spacing between centers of bearings A and B. N – is the spacing between centers of bearings C and D minus dimension K.

See Figures 7.44 and 7.45. Example of Spiral-Bevel-Gear Bearing Load Calculation. A sample problem for a 16-toth pinion driving a 49-tooth gear demonstrates how the equations given in this section are used. Given: a 16 combination, 5 diametral pitch, 1.500-in. face width, 20° pressure angle, 35° spiral angle. 49

(See Table 7.5 for further data relative to this problem.) Load data: 71 hp at 1800 r/min or pinion; left-hand pinion driving clockwise; gear is straddle mounted; pinion, overhung. The tangential load on this spiral bevel gear will be: WtG =

126, 050Ph 126, 050 × 71 = = 1818lb dmG n 8.374 × 588

(7.61)

16

where Ph = 74 hp, dmG = 8.374 in., and n = 49 × 1800 = 588 r/min of gear . The tangential load on the aforementioned spiral bevel pinion will be equal to the tangential load on the gear: WtP = Wt G = 1818 lb

(7.62)

422

Dudley’s Handbook of Practical Gear Design and Manufacture

TABLE 7.5 Spiral Bevel Gear Sample Data and Bearing Arrangement (English Units) Item

Pinion Symbol

English Units

d

Pitch diameter

Gear Symbol

D

3.200

9.800

18 50

Pitch angle

sin Mean pitch diameter

71 55

sin

0.3104

dmP = d

F sin

2.734

English Units

dmG = D

0.9506

F sin

8.374

Overhung mounting:

L

1.500

–

–

M

5.000

–

–

Straddle mounting:

K

–

–

2.500

N

–

–

3.500

The axial-thrust loads for both gear and pinion are obtained from Figure 7.22: Wx = (percent from Fig. 22.22) × Wt

(7.63)

WxP = +0.80 × 1818 = 1454 lb(outward)

(7.64)

WxG = +0.20 × 1818 = 364 lb(outward)

(7.65)

The separating force for both gear and pinion is obtained from Figure 7.23:

W W

W

r

= (percent from Fig. 22.23) × Wt

(7.66)

rP

= +0.20 × 1818 = 364 lb(separating)

(7.67)

= +0.80 × 1818 = 1454 lb(separating)

(7.68)

rG

Table 7.6 tabulates the formulas and sample calculations for the resultant bearing loads on the afore­ mentioned pair of gears. Note: Since an overhung mounting is used on the pinion, the formulas for bearings A and B are used. If both members had been overhung-mounted (see Figure 7.46), the formulas for bearings A and B would have been used for both gear and pinion; or if both members had been straddle-mounted (see Figure 7.47), the formulas for C and D would have been used for both gear and pinion. In any case, the dimensions and loads for the corresponding member must be used. Example of Hypoid Gear Bearing Load Calculation. A sample problem of a 14-tooth hypoid pinion driving a 47-tooth hypoid gear is given in the following: Given: a 14 combination, 5.013 gear diametral pitch, 1.438 in. face width, 1.750-in. pinion offset 47

below center. Pressure angles 15 20 on forward drive; 27 10 on reverse drive. Spiral angles 21 28 on

Total radial load

126, 050P dmG n

Wr =

Wx

(loads)2

dm 2

Wx = Wt × thrust factor See Figure 7.21 W r = Wt × separating factor See Figure 7.22

Wt =

Load Calculations

– L+M M

d

= 473

r

363

Wr22 + (Wr1

WrA = 2.3532 + (473 = 2, 364

WrA = 398)2

Wr 3)2

Wr3 = 1, 454 25.0 = 398

2.734

Wr3 = Wx 2Mm

1.5 + 5.0 5.0

Wr1 = W Wr1 = 364

Wr2 = 1,

Wr 2 = Wt

L+M M 1.5 + 5.0 818 5.0 = 2,

Bearing A L

= 545

Wr25 + (Wr 4

WrB = 5452 + (109 = 617

WrB = 398)2

Wr 6 )2

Wr6 = 1, 454 25.0 = 398

2.734

d

Wr 6 = Wx 2Mm

K

–

8.374

Wr28 + (Wr 7 WrC = 7582 + (606 = 836

WrC =

254)2

Wr 9)2

Wr9 = 364 2(3.5 + 2.5) = 254

d

Wr 9 = Wx 2(N m+ K )

2.5

Wr7 = 1, 454 3.5 + 2.5 = 606

1, 060

8.374

Wr211 + (Wr10 + Wr12 )2 WrD = 1, 060 2 + (848 + 254)2 = 1, 536

WrD =

Wr12 = 364 2(3.5 + 2.5) = 254

d

Wr12 = Wx 2(N m+ K )

3.5

N r N+K

Wr10 = 1, 454 3.5 + 2.5 = 848

Wr10 = W

Wx = thrust factor × Wt Wx = +0.20 × 1, 818 = 364

Wr11 = 1,

3.5 818 3.5 + 2.5 =

N

Bearing D

Wr11 = Wt N + K

Straddle Mounting

758

K r N+K

2.5 818 3.5 + 2.5 =

Wr8 = Wt N + K

Bearing C

Wr 7 = W

Wr8 = 1,

Bearing Loads

Wx = thrustfactor × Wt Wx = +0.80 × 1, 818 = 1, 454 L Wr 4 = W r M 1.5 Wr4 = 364 5.0 = 109

1.5 818 5.0

Wr 5 = Wt M

Bearing B

Wr5 = 1,

Overhung Mounting

Wt – tangential tooth load, lb P – power transmitted, hp dmG = D F sin mean pitch diameter of gear, in. D – gear pitch diameter F – gear face width – gear pitch angle n – speed of gear W r – separating component Wx – axial-thrust component L – axial distance from center of face width to center line of bearing A M – spacing between center line of bearings A and B N – axial distance from center of face width to center line of bearing C K – axial distance from center of face width to center line of bearing D dm = d F sin d – pitch diameter, in. – pitch angle

Thrust couple

Separating, Wr

Thrust, Wx

Tangential, Wt

Forces

TABLE 7.6 Solution to Sample Problem on Spiral Bevel Gear Bearing Loads (English Units)

Gear Reactions and Mountings 423

424

Dudley’s Handbook of Practical Gear Design and Manufacture

FIGURE 7.46

Overhung mounting.

FIGURE 7.47

Straddle mounting.

Gear Reactions and Mountings

425

TABLE 7.7 Hypoid Gear Sample Data and Bearing Arrangement (English Units) Item

Pinion

Gear

Symbol

English Units

Symbol

Pitch diameter

D=

Pitch angle

21 47

o

sin

0.37110

o

Mean pitch diameter

dmP =

n cos dmG N cos G P

3.161

Distance from center line of mate to center of bearing

H (from layout)

5.615 3.740

JP =

dmG 2

cos

Overhung mounting (see Figures 7.44 and 7.46): L=H

JP

M

dmg

N diam . pitch

English Units

9.375

ΓR

66 27

sin ΓR

0.91671

= D – F sin ΓR

G (from layout)

JG =

8.57

1.581

dmP 2

–

1.875

–

–

4.750

–

Straddle mounting (see Figures 7.45 and 7.47):

K=G

JG

N

–

–

3.000

–

–

6.250

the gear; 45 3 on the pinion. Pinion face angle is equal to 21 47 . Gear root angle is equal to 66 27 . (See Table 7.7 for further data relative to this problem.) Load data: 60 hp at 1500 rpm of pinion; left-hand pinion driving clockwise; gear is straddle-mounted; pinion, overhung. The tangential load on the aforementioned hypoid gear will be: WtG =

126, 050PH 126, 050 × 60 = = 2100 lb dmG h 8.057 × 447

(7.69)

14

where PH = 60 hp, dmG = 8057, n = 47 × 1500 = 447 r/min, gear . The tangential load on the hypoid pinion will be: WtP = WtG

P

G

cos

P

cos

G

= 2100 ×

0.70649 = 1594 lb 0.93060

= 45 3 ,

cos

P

= 21 28 ,

cos

G

(7.70)

= 0.70649

(7.71)

= 0.93060

(7.72)

The axial-thrust loads for both gear and pinion for forward drive are obtained from Figure 7.22: Wx = (percent from Fig. 22.22) × Wt

(7.73)

WxG = +0.11 × 2100 = 231lb(outward)

(7.74)

426

Dudley’s Handbook of Practical Gear Design and Manufacture

WxP = +1.08 × 1594 = 1722lb(outward)

(7.75)

The separating forces for both gear and pinion are obtained from Figure 7.23: W W

W

rG

rP

= (percent from Fig. 22.23) × Wt

(7.76)

= +0.48 × 2100 = 1008 lb(separating)

(7.77)

=

(7.78)

r

0.01 × 1594 =

16 lb(attracting)

Table 7.8 tabulates the formulas and sample calculations for the resultant bearing loads on the hypoid pair.

7.7.6 REQUIRED DATA

FOR

BEARING LOAD CALCULATIONS

To calculate the bearing loads, the following data must be available:

Bevel Gears Gear ratio (number of teeth in gear and pinion) Gear pitch diameter or diametral pitch Pitch angles (both gear and pinion) Face width Pressure angle Spiral angle Horsepower or gear torque to be transmitted Gear r/min Hand of spiral Direction of rotation (viewed from the back) Sketch showing method of mounting and bearing spacing Shaft angle

Hypoid Gears Gear ratio (number of teeth in gear and pinion) Gear pitch diameter or diametral pitch Gear root angle Pinion face angle Gear face width Pressure angles (both sides of tooth) Spiral angles (both gear and pinion) Horsepower or gear torque to be transmitted Gear r/min Hand of spiral Direction of rotation (viewed from the back) Sketch showing method of mounting and bearing spacing Shaft angle

Total radial load

126, 050P dm n

Wr =

(loads)2

d Wx 2m

Wx = Wt × thrust factor See Figure 7.21 W r = Wt × separating factor See Figure 7.22

Wt =

Load Calculations

= 2, 223

Wr22 + (Wr1

r

Wr 3

WrA = 2.2232 + ( 22 = 2, 301

WrA =

Wr3 =

Wr1 =

Wr1 = W

573)2

)2

L+M M 1.875 + 4.750 16 4.750 = 22 dm Wr3 = Wx 2M 3.161 1, 722 2 × 4.750 = 573

Wx = thrust factor × Wt Wx = +1.08 × 1, 594 = 1, 722

Wr2 = 1,

Wr 2 = Wt

L+M M 1.875 + 4.750 594 4.750

Bearing A

mean pitch diameter of hypoid pinion, in.

D – gear pitch diameter, in. F – gear face width, in. R – gear root angle n – number of teeth in hypoid pinion N – number of teeth in hypoid gear – gear spiral angle G – pinion spiral angle P

P

n cos G

= d mG N cos

6

L r M

Wr25 + (Wr 4 573)2

)2

573

Wr 6

d Wr 6 = Wx 2Mm 3.161 1, 722 2 × 4.750 =

16 4.750 =

1.875

Wr 4 = W

–

WrB = 6292 + ( 6 = 855

WrB =

Wr6 =

Wr4 =

= 681

Wr11 = 2,

+ (Wr 7

WrC = 6812 + (327 = 718

WrC =

Wr28

101)2

)2

101

327

Wr 9

K r N+K 3.00 Wr7 = 1, 008 6.25 + 3.00 = d Wr 9 = Wx 2(N m+ K ) 8.057 Wr9 = 231 2(6.25 + 8.00) =

Wr 7 = W

–

Wr211 + (Wr10 + Wr12 )2

WrD = 1, 4192 + (681 + 101)2 = 1, 620

WrD =

Wr12 =

Wr10 =

N r N+K 6.25 1, 008 6.25 + 3.00 = 681 d Wr12 = Wx 2(N m+ K ) 8.057 231 2(6.25 + 3.00) = 101

Wr10 = W

Wx = thrust factor × Wt Wx = +0.11 × 2, 100 = 231

= 1, 419

6.25 100 6.25 + 3.00

Wr8 = 2,

Wr11 = Wt N + K

3.00 100 6.25 + 3.00

N

Bearing D

Wr8 = Wt N + K

K

Bearing C

Straddle Mounting

1.875 594 4.750

= 629

Bearing Loads

Wr 5 = Wt M

L

Bearing B

Wr5 = 1,

Overhung Mounting

Wt – tangential tooth load, lb P – power transmitted, hp n – speed, rpm (values for corresponding member) W r – separating component Wx – axial-thrust component L – axial distance from center of face width to center line of bearing A M – spacing between center line of bearings A and B N – axial distance from center of face width to center line of bearing C K – axial distance from center of face width to center line of bearing D dm = D F sin R mean pitch diameter of hypoid gear, in.

Thrust couple

Separating, Wr

Thrust, Wx

Tangential, Wt

Forces

TABLE 7.8 Solution to Sample Problem on Hypoid Gear Bearing Loads (English Units)

Gear Reactions and Mountings 427

428

7.8

Dudley’s Handbook of Practical Gear Design and Manufacture

BEARING LOAD CALCULATIONS FOR WORMS

Basically, the worm-and-gear bearing load calculations are similar to helical gear reactions except for three main differences. All the members of the worm gear family run on crossed axes. Generally, the axes are crossed at 90°. Besides the shaft arrangement, worm gears have “overlap.” There may be no overlap, overlap in one plane, or overlap in two planes. Overlap is that characteristic whereby one tries to envelope the mating part by being so curved as to tend to wrap around the mating part. Overlap gives a grater area of tooth engagement. It also makes the mounting more sensitive. See Figure 7.48 for worm types. The third unusual feature of worm gears is that of ratio. The ratio is defined as the number of gear teeth divided by the number of worm threads (or worm “starts”). A single-start worm may have several turns of thread; but, if it meshes with a gear with 100 teeth, the ratio is 100:1. If the worm had two starts and the gear had 100 teeth, the ratio would be 50:1.

7.8.1 CALCULATION

OF

FORCES

IN

WORM GEARS

Figure 7.3 showed the basics of a force that acted in three planes. This concept is helpful in under­ standing how the force at the worm gear mesh acts. The efficiency of worm gears is quite variable. A single-tread worm may have efficiency as low as 50%. A multiple-thread worm with at least 15 lead angle and a reasonable rubbing speed will usually

FIGURE 7.48 Types of worm gears: (a) non-throated crossed helical (point contact), (b) single-throated cylindrical worm gear (line contact), (c) double-throated, double-enveloping gear (line contact or area contact).

Gear Reactions and Mountings

429

have efficiency in the range of 90 to 95%. Since worm gear efficiency is so variable, it cannot be neglected in calculating load reactions. The worm gear overlap gives a relatively wide zone of contact. The pressure angle and the thread angle vary quite appreciably throughout this area. It is customary to calculate the worm gear load reactions if all the contact were at the theoretical pitch point. This permits use of the theo­ retical pressure angle, the theoretical lead angle, etc., as variables in the equations. Although this method is not rally exact, it is good enough to get load reactions to size bearings and design casing structures.

FIGURE 7.49

Worm gear bearing loads.

430

Dudley’s Handbook of Practical Gear Design and Manufacture

Figure 7.49 shows the case of a worm and gear mounted on 90° axes and each straddle-mounted. Load reactions at the mesh and at each of the bearings are shown. The tangential force on the worm may be calculated by: Wt =

126,050Ph nw d

English

(7.80)

metric

(7.81)

English

(7.82)

where: Wt (Wt )– P (Ph )– n 1 (n w )– dp1 (d )–

is is is is

the the the the

tan driving force [metric (English)], N (lb). input power, kW (hp). rotations per minute of worm. pitch diameter of worm.

The separating force is: W

W

r

r

Wt tan tan

=

=

Wt tan

n

n

tan

where: W r (W r )– is the separating force [metric (English)], N (lb). ( )– is the worm lead angle. ( n n )– is the normal pressure angle.

The worm thrust force is: Wx =

Wt tan(lead angle)

(7.83)

The details for worm gear bearing load calculation are shown in Table 7.9. Figure 7.49 illustrates the bearing loads in a worm gear set. The method of worm gear bearing load calculations just described may be used for crossed helical gears on right-angle axes, for cylindrical worm gears, and for double-enveloping worm gears.

7.8.2 MOUNTING TOLERANCES The worm gear types vary all the way from the non-enveloping crossed helical to the double-enveloping type. In the crossed helical, alignment is not critical in any direction. Axial movement merely shifts the contact area along the length of the part. Center-distance error changes backlash and the amount of tooth contact, but neither of these things is highly critical in mot applications. The cylindrical worm gearset is critical on center distance, shaft angle, and axial position of the gear. Slight errors in any of these three things will tend to change a full-face-width contact pattern to only a small amount of contact at the end of the tooth. The contact area in a worm gearset may drop from 100 to 15% with an error in alignment of as little as 0.010 in. in a foot. The double-enveloping worm gearset is critical in the three directions of the cylindrical set plus a fourth direction of the axial position of the worm.

Note: See Figure 7.49 for definition of terms and bearing locations.

Wx may be applied to either bearing A or B

WrB =

Total thrust

Wr 3)2

Wr21 + (Wr 2

b r a+b

Wr24 + (Wr 5 + Wr 6)2

r

Wr 5 = W Wr 6 = Wx a +1b = Wr 3

WrA =

r

a r a+b

Total radial load

Wr 2 = W Wr3 = Wx a +1b

r

b

Wr 4 = Wt a + b

a

Wr1 = Wt a + b

Bearing B

Bearing A

Worm Bearing Loads

Thrust, Wx

Separating, W

Worm tangential force, Wt

Forces

TABLE 7.9 Equations for Worm Gear Bearing Loads with Straddle Mounting

Wr29 + (Wr 7

Wr 8)2

WrD =

Wr212 + (Wr10 + Wr11)2

c

c r c+d

Wr12 = Wx c + d

Wr11 = W

r

Wr10 = Wt c +2d = Wr 7

Bearing D

WxG = Wt (may be applied to either bearing C or D )

WrC =

d

d r c+d

Wr 9 = Wx c + d

Wr8 = W

r

Wr 7 = Wt c +2d

Bearing C

Worm-Gear Bearing Loads

Gear Reactions and Mountings 431

432

Dudley’s Handbook of Practical Gear Design and Manufacture

TABLE 7.10 Typical Mounting Tolerances for High-Capacity Worm Gearsets Center Distance

Tolerance on Center Distance

Tolerance on Axial Position

Tolerance on Alignment

Metric values, mm 0–75

± 0.013

± 0.0025

± 0.650 per m

75–150 150–375

± 0.025 ± 0.050

± 0.050 ± 0.075

± 0.400 per m ± 0.225 per m

375 up

± 0.075

± 0.100

± 0.150 per m

English values, in. 0–3

± 0.0005

± 0.001

± 0.008 per ft

3–6

± 0.0010

± 0.002

± 0.005 per ft

6–15 15 up

± 0.0020 ± 0.0030

± 0.003 ± 0.004

± 0.003 per ft ± 0.002 per ft

Table 7.10 shows some typical mounting tolerances for a high-capacity worm gearset. The most precise worm gears used in timing devices, radar work, etc., are mounted with about half the error shown in Table 7.10. On the other hand, worm gearsets for commercial service and relatively light load service are mounted with about twice the error shown in Table 7.10.

7.8.3 WORM GEAR BLANK CONSIDERATIONS The worm member is usually made of steel. Generally, this member offers no problem in design. In a few cases the worm may be long and slender with a wide span between its bearings. In this case there may be danger of serious “bowing” of the worm between supports. Calculations should be made in such a case to

FIGURE 7.50

Worm gear blank proportions.

Gear Reactions and Mountings

433

make sure that the worm shaft is not too highly stressed. Also, the deflection should be calculated. This deflection acts as a change in center distance and should be judged on this basis. The worm gear is generally made of bronze in the toothed area. Quite often the hub is made of steel or case iron. The hub should have a large enough hole in it to assure a strong shaft. The length of the hub should be about one-third the gear pitch diameter to assure a stable mount. Figure 7.50 shows a typical worm gear with a bronze ring bolted to a steel hub.

7.8.4 RUN-IN

OF

WORM GEARS

All types of worm gears benefit by a “run-in.” This is particularly true when the gear member is made of bronze. The as-cut worm gear generally has generating flats on its surface, and the surface itself is somewhat imperfect because of “oversize” effects of the hob. As the gear is broken in, its surface wears a slight amount to conform to that of the worm. In addition, the surface of the gear becomes smooth and polished. Tests on bronze gears have shown that the amount of load that will cause seizure of the surface may go up as much as tenfold after a careful break-in! In breaking in worm gears, the load is gradually increased and the speed is kept low. The surfaces are watched carefully. If small amounts of bronze start adhering to the worm surface, it is necessary to stop the break-in and dress down the worm threads with fine abrasive paper to remove the bronze. The breakin may then continue. After the break-in is complete and the tooth surfaces polished, and the contact pattern has been obtained to a satisfactory level of quality, the gears are stamped as matched set. Exact axial-distance settings used in the break-in are stamped on the set to aid in field installation.

7.9

BEARING LOAD CALCULATIONS FOR SPIROID GEARING

The Spiroid gear is a crossed-axes-type gear somewhat intermediate between a worm gear and a hypoid gear. Its originators, the Illinois Tool Works of Chicago, Illinois, considers it to be a screw-type gear. The pinion of the Spiroid set is like a tapered worm. The pinion meshes on the side of t gear rather than the outside diameter of the gear. The gear does not wrap around the worm as a conventional worm gear does. The Spiroid gear is coned. Another member of the Spiroid family is the Helicon gear. A Helicon pinion is cylindrical rather than coned.

FIGURE 7.51

High and low side of Spiroid gear.

434

FIGURE 7.52

Dudley’s Handbook of Practical Gear Design and Manufacture

Examples of mounting positions for left-hand and right-hand Helicon gearsets.

The Spiroid pinion has two effective pressure angles, which are called the “high” pressure angle and the ”low” pressure angle. Figure 7.51 shows the high side and the low side of a typical Spiroid pinion. The low side of the Spiroid pinion is preferred for driving because it has the largest amount of tooth contact area and it exerts the least separating force on the gear.

FIGURE 7.53

Spiroid pinion lead angle.

Gear Reactions and Mountings

435

Spiroid and Helicon gears may be designed for a right-hand or left-hand system. The choice of hand on the pinion determines the direction of rotation of the gear. The position of the pinion (normally in one of four quadrants) and the offset between the axes of the pinion and the gear have much to do with the direction of bearing reactions. Figure 7.52 illustrates the configurations just described for Helicon gear sets. Before bearing load calculations can be made for a Spiroid set, it is necessary to establish design details like ratio, cone angle, offset, and pressure angle. The pinion has a constant lead; but, because of the taper, the lead angle is variable. Figure 7.53 shows how the lead angle varies along the length of the cone. Bearing load calculations are based on the lead angle, the pressure angles, the pitch diameters, and the center distance at the mean length of the pinion. The exact calculation of Spiroid bearing loads is a complicated process. Generally, it is satisfactory to calculate approximate bearing loads using special constants. The error is usually less than 10%. If exact load data are needed, the Illinois Tool Works is willing to furnish assistance on a specific request. The approximate tangential driving force may be obtained from the output horsepower or output torque as follows: Wt =

Wt =

19, 098, 600P n 2 dp2 126, 050Ph nG D

metric

English

(7.84)

(7.85)

where: Wt (Wt )– P (Ph )– n 2 (n G )– dp2 (D)–

is is is is

the the the the

tangential driving force [metric (English)], N (lb). output power, kW (hp). speed of gear, r/min. gear pitch diameter, mm (in.).

An approximate radial force on the gear is: WrG = 0.91Wt

(7.86)

The radial force on the pinion is: WrP =

Wt

(7.87)

= 0.60 for 35° pressure angle on high side = 0.52 for 30° pressure angle on high side = 0.35 for 15° pressure angle on low side = 0.21 for 10° pressure angle on low side The thrust force on the pinion is equal to the radial force of the gear. Likewise, the thrust force on the gear is equal to the radial force on the pinion. Thus:

where

WxP = WrG

(7.88)

WxG = WrP

(7.89)

436

FIGURE 7.54

Dudley’s Handbook of Practical Gear Design and Manufacture

Spiroid gear bearing loads.

The point of application of these forces should be taken at the mid-face of the gear and the mid-depth of the tooth. For straddle-mounted Spiroid set, the bearing loads may be determined by a procedure similar to that shown for worm gears in Table 7.8. Substitute WrP for the worm tangential force Wt . Assume zero for the separating force. Substitute WxP for the worm thrust Wx in the Spiroid gear calculation, use zero for the separating force and substitute Wr G for the Wx term in the worm gear bearing load part of the table. The radii r 1 and r 2 should be taken to the midpoint of the Spiroid mesh. Figure 7.54 shows the Spiroid gear meshing point and the direction of the tangential force. The Spiroid Division of the Illinois Tool Works in Chicago, Illinois has established computer pro­ grams to determine bearing reactions for the commonly used Spiroid and Helicon arrangements.

7.10 BEARING LOAD CALCULATIONS FOR OTHER GEAR TYPES Space does not permit detail bearing load equations for several additional types of gears. The Planoid5, Helicon, face gear, Beveloid6, and others are not covered. However, the principles used in this chapter make it relatively easy for designers to set up equations for any special types. The first step in solving the problem is to find the radial, separating, and thrust forces at some center point of the contact area. Then calculations are made to determine the final radial and axial reaction at each bearing location. If there is more than one load point on the same shaft, proper vector addition must be made to get the overall resultant. In solving complex bearing load problems, the following rules should be kept in mind: • When all the gearing forces have been transferred to a given mount, the final result is one resultant radial load and (depending on the bearing type) one resultant thrust (or axial) load. • The summation of all forces must be equal zero. • The summation of all moments must equal zero.

5 6

Registered trademark of the Spiroid Division of Illinois Tool Works, Chicago, Illinois. Registered trademark of Vinco Corp., Detroit, Michigan.

Gear Reactions and Mountings

437

In the case of new gear types, the manufacturer of the type or the inventor of the type is usually glad to furnish potential customers with detail calculations for the forces at the meshing point.

7.11 DESIGN OF THE BODY OF THE GEAR To conclude this chapter, some comments should be made relative to the structure of the body that supports the gear teeth. It sometime happens that a gear designer does an excellent job of designing gear teeth and then making bearing selections and calculations to adequately support the gear, but does almost no engineering work relative to the gear body. In a gearset, the pinion is often small and essentially amounts to teeth cut on a section of shafting. The gear member of the pair, though, tends to be much larger in diameter and therefore is apt to involve a hub, a web, and a rim. Each of the three things just mentioned need to be so designed as to have adequate strength, adequate rigidity, and be manageable from the standpoint of resonance and vibration. The prime usage of a gearset may be the transmission of power, or it may be the transmission of motion. The following kinds of gear applications need to be recognized before design work is done to settle details of a rim, a web, or a hub: • Data gears: These gears are generally sized to fit an arrangement scheme and a need to bridge the distance from one shaft to another shaft. Some examples are tachometer drives, governor drives, or instrument drives. The amount of power transmitted though the gearing is usually relatively in­ significant for the size of gears chosen to meet arrangement requirements. • Power gears: These gears are primarily used to transmit power and change speed. They may be either speed-reducing or speed-increasing. The size of the gears is usually determined by making the gear large enough to carry the required power. For weight reasons and cost reasons, the ten­ dency is to design them as small as possible but still have adequate power capacity and reliability. • Accessory gears: In a power-producing engine or motor there may be a train of power gearing that takes the main power from the prime mover to the driven device. This might be a turbine driving gear connected to a pump, a compressor, or a propeller. In addition to the main power drive, there is often a need to have a package of gears driving accessories. These accessories can involve oil pumps, fuel pumps, pumps for hydraulic power, and so forth. The arrangement of the accessory package frequently requires gears larger in diameter and wider in face width than what is needed for the transmitted power. This means that the body structure of accessory gears can often be less rugged than that needed for main drive power gears. A further consideration in gear body structures has to do with the kind of application. For instance, rocket engines and aircraft engines have to be very lightweight. This means the use of very-high-strength ma­ terials and extra cost in manufacture to achieve the ultimate capacity in power per kilogram of metal or per pound of metal. At the other end of the spectrum, industrial gearing used in mills and in processing plants can be much heavier in weight, but there is much concern to keep the cost relatively low. Relatively heavy structures are used. The allowable stresses are held to much lower values than in aerospace work. Table 7.11 has been put together to give some general guidance relative to rim thickness, web thickness, holes in the web, and hub thickness. Those using this table, should consider the table as primarily historical with regard to what is commonly done rather than as positive design values that must be used. For instance, a well-designed aircraft gear may be acceptable with a rim thickness underneath the gear teeth that is only 1.25 times the gear tooth height. If the face width happened to be rather wide and the rim underneath the gear lacked stiffness due to a thin web or the tendency to have “spokes” connecting to the hub, it may be necessary to increase the rim thickness to two or three times the tooth height to get adequate stiffness in the gear body. The tendency to have a spoked condition is quite common in marine applications and gearing for sta­ tionary power. Gears of this type often have a rim thickness that is as much as four times the tooth height.

438

Dudley’s Handbook of Practical Gear Design and Manufacture

TABLE 7.11 Approximate Ratio of Certain Gear Body Dimensions to the Whole Depth of the Gear Tooth Gear Drive Application

Rim Thickness

Web Thickness

Lightening Holes Recommended?

Hub Thickness

0.5 to 0.7

Normally if space available

0.6 to 1.0 (0.07 in. min)

Not recommended

1.5–2.0

Recommended Odd number of holes

1.5–2.5 1.5–2.5

Data gears: Instrument or Data accessory Power gears: Rocket engine

0.6 to 0.7 (0.07 in. min) 1.0

1.0

(0.125 in. min)

(0.125 in. min)

0.7 1.0 to 1.24

0.5 0.7 to 1.0

(0.125 in. min)

(0.125 in. min)

1.25–1.50

1.00–1.35

Normally gears only

–

1.5 avg. 1.25

1.35 1.00

Gear only Gear only

– –

Commercial hypoid

2.0

2.0

Yes

8

Commercial Spiroid Marine, submarine

1.0–2.5 3 to 4

2.0 3 to 4

Yes Holes required for welding, cleaning etc.

11 –

Stationary power

4 to 6

Depends on design

–

–

main power Accessory rocket Accessory Aircraft Aircraft and helicopter main drives Planets Accessory

Notes: Anti-backlash: Use values appropriate for data gears or for power gears. Ratio of 1.0 means value equals whole depth; 2.0 is twice whole depth.

At times, rather thin webs are used that tend to go beyond Table 7.11 recommendations. This helps save weight, but there may be failures of the web de to resonant vibrations. This risk can be handled by using damper rings under the rim of the gear to stop vibration. If stresses induced by vibrations are not too high, the shot-peening of a thin web in critical areas may give the added capacity that is needed to handle stresses from vibrations. Another general area of concern is resonance in vibration. A webbed gear tends to have up to 3 or more critical speeds. If the gear is not running too fast and the web is fairly heavy, the resonant fre­ quencies tend to be very high compared with the frequencies that the teeth are meshing. In aerospace applications, though, if often happens that a critical frequency can be rather close to the operating frequency. This is a bad situation and should be avoided. Sometimes the first critical comes at about twothirds of the operating frequency and the second critical is well above the operating frequency. This situation may be tolerable if the gear unit tends to run at a constant speed and the gear unit comes up to full speed rather quickly so that there is almost no time that the gear is operating at the first critical. To sum things up, the design of the body structure of the gear should be handled just as thoroughly as the design of the gear teeth. It is beyond the scope of this book to present many pages of engineering data on the design of structures such as gear bodies, turbine wheel bodies, and so forth. The main point is that structure design is just as important in gearing as it is in the design of other types of machinery.

Gear Reactions and Mountings

439

REFERENCES [1] Radzevich, S. P. (2020). Conjugate action law in intersected-axes gear pairs and in crossed-axes gear pairs. Gear Solutions Magazine, June, pp. 42–48. [2] Radzevich, S. P. (2014). Principal design parameters of gearing with non-parallel axes of rotations. Gear Solutions, August, pp. 65–73. [3] Radzevich, S. P. (2020). Understanding the mounting distance: Crossed-axes gearing (hypoid gearing). Gear Solutions Magazine, February, pp. 38–43. [4] Radzevich, S. P. (2019). Understanding the mounting distance: Intersected-axes gearing (bevel gearing). Gear Solutions Magazine, December, pp. 42–47.

8

Compensation of Shaft Deflections through Gear Micro-Geometry Modifications Alfonso Fuentes-Aznar

8.1

INTRODUCTION

One of the main problems that affects the mechanical performance of gear drives is the appearance of edge contacts on the gear tooth surfaces. These are typically caused by misalignments due to assembly errors and the deflection of the shafts supporting the gears. An edge contact results on a non-favorable condition of the bearing contact, which yields high levels of contact and bending stresses. These stresses may lead to premature failure of the gears. Gear tooth surface modifications can be designed to accommodate the expected errors of alignment of the gears due to shaft deflections, positioning errors of the bearing supports, and misalignments due to assembly errors. These cause larger levels of contact and bending stresses, and therefore, limit the maximum torque that the gearset is able to transmit. In order to compensate the misalignments and localize the bearing contact, larger surface modifications on the gear tooth surfaces are needed. Misalignments due to shaft deflections can be predicted and consequently compensated by introducing surface modifications during the manufacturing process of the gear tooth surfaces. This way, the gear tooth surfaces will be prepared to achieve the best conditions of meshing and contact under the expected errors of alignment. Additional tooth surface modifications will still be needed to account for other errors of align­ ment. They will provide smaller surface modifications that allow a larger portion of the gear tooth surfaces to be used for the bearing contact, significantly reducing the maximum contact and bending stresses.

8.2

DETERMINATION OF ERRORS OF ALIGNMENT DUE TO SHAFT DEFLECTIONS

The determination of errors of alignment presented in this chapter is based on the application of the finite element method to determine the deformation of the shafts during the transmission of load and its effect on the gear tooth surfaces. The theory presented here can be also applied with alternative analytical or nu­ merical methods for the determination of the slope and deflections of the shafts at each gear position. However, the application of the finite element method allows the analysis of the actual geometry of the gears and the consideration of the position of the contact pattern for each position of meshing and contact. Finite element models applied here include the gears, their supporting shafts, and the position and rigidity of the bearings. These finite element models will be created following the ideas presented in [1,2]. In [3], the comparison of using three-dimensional solid element type C3D8I [4] and linear threedimensional beam element type B31 [4] was carried out. Element type C3D8I represents hexahedral first order elements enhanced by incompatible deformation modes in order to improve their bending behavior. Element type B31 is based on Timoshenko's theory of beams, and, consequently, transverse shear de­ formation is allowed. It was concluded that linear three-dimensional beam elements provide similar results to those obtained from the model of the shaft meshed with three-dimensional solid elements but with much lower computation cost. 441

442

Dudley’s Handbook of Practical Gear Design and Manufacture

ABAQUS® software [4] has been used for finite element analysis of the gears with their supporting shafts. Gear teeth and rim volumes are meshed using three-dimensional solid elements (C3D8I). Supporting shafts are meshed with linear three-dimensional beam elements (B31). The active tooth surfaces of the wheel are considered master surfaces, whereas those corresponding to the pinion are considered slave surfaces. Steel has been considered for all solid parts, including supporting shafts, with general properties of Young's module of 210 GPa and Poisson's ratio 0.30. The determination of the errors of alignment depicted in the following subsections depends on the type of transmission. The following three general types of gear transmissions are being considered to illustrate the process of determination of errors of alignment due to shaft deflections: (i) transmissions with parallel shafts, (ii) transmissions with intersecting shafts, and (iii) transmissions with crossed shafts.

8.2.1 TRANSMISSIONS

WITH

PARALLEL SHAFTS

The approach presented in this section can be applied to any type of transmission with parallel shafts. Figure 8.1 shows the finite element model of a spur gear drive considering the shafts, the gears, and the location of the supports. This model will serve as a reference for the determination of errors of alignment due to shaft deflections in this type of transmission. The shaft of the pinion is supported at points A1 and A2, located at both ends. The shaft of the wheel is also supported at both ends where points B1 and B2 are located. A global coordinate system Sf will be considered with origin at point Pr having its axis zf directed toward point A2 and axis yf pointing toward the shaft of the wheel. Displacements in x f , yf , and zf directions will be constrained at points A1 and B1, whereas displacements in x f and yf directions will be constrained at A2 and B2 .

FIGURE 8.1

Finite element model of a spur gear drive with supporting shafts.

Compensation of Shaft Deflections

443

The finite element model of pinion and wheel implements five pairs of contacting teeth. The nodes on the sides and bottom part of the rim of the pinion conform a rigid surface connected to reference point Pr on the pinion shaft. Points P1 and P2 are located at both sides of the pinion web. Beam elements from P1 to Pr and from Pr to P2 will take into account the flexural rigidity of the pinion web. Similarly, nodes on the sides and bottom part of the rim of the wheel conform a rigid surface connected to reference point Wr on the wheel shaft and points W1 and W2 are located at both sides of the web. Beam elements from W1 to Wr and from Wr to W2 will also consider the flexural rigidity of the wheel web. The transmitted torque is applied at point A2 of the pinion shaft in a counter clockwise direction and the rotation of point B2 on the wheel shaft is constrained (Figure 8.1). For parallel shaft transmissions, the following relative errors of alignment between pinion and wheel are considered: (i) axial displacement of the wheel with respect to the pinion, A2 , (ii) crossing shaft error, c , (iii) intersecting shaft error, i , and (iv) center distance error, C . Figure 8.2 shows the positive directions considered for each of those relative errors of alignment. Figure 8.3 shows the supporting shafts of pinion and wheel in both undeformed and deformed po­ sitions as well as the coordinate systems and points used to calculate the errors of alignment due to shaft deflections. As mentioned above, points Pr and Wr are the reference points of the pinion and the wheel, respectively. Under load, those reference points will take positions represented by points Pr and Wr . The relative errors of alignment between pinion and wheel are determined from the displacements (Uxji, Uyji , Uzji ) and rotations (Rxji , Ryji , Rzji ) that reference nodes Pr and Wr of pinion (i = 1) and wheel (i = 2) experiment under the applied torque T at each contact position j analyzed. The mentioned dis­ placements (Uxji, Uyji , Uzji ) and rotations (Rxji , Ryji , Rzji ) are obtained from the results of the finite element analysis for each contact position j . A certain number of contact positions are distributed uniformly along two angular pitches of rotation of the pinion and the average value of the displacements and rotations obtained. The averaged displacements are denoted here as (Ux1, Uy1, Uz1) for the pinion and

FIGURE 8.2 Relative errors of alignment in spur gear transmissions: (a) axial displacement of the wheel with respect to the pinion, A2 , (b) crossing shaft error, c , (c) intersecting shaft error, i , and (d) center distance error, C.

444

FIGURE 8.3

Dudley’s Handbook of Practical Gear Design and Manufacture

Supporting shafts of pinion and wheel in aligned and deformed positions.

(Ux2, Uy2, Uz2 ) for the wheel. Similarly, the averaged rotations of the reference points of pinion and wheel

are denoted as (Rx1, Ry1, Rz1) for the pinion and (Rx2 , Ry2 , Rz2 ), respectively. Under no load, the axis of rotation of the pinion z1 in its aligned position coincides with axis zf of the global coordinate system. Under load, the axis of rotation of the pinion is directed as z1 as shown in Figure 8.3. The matrix of transformation from z1 to z1 in the fixed coordinate system Sf is given as a function of the rotations Rx1 and Ry1 of the reference point of the pinion model and expressed as follows: 1 0 1 La1 = 0 cos Rx 1 0 sin Rx cos Ry1 Lb1 =

(8.1)

cos Rx1 0 sin Ry1

0 1 0 sin Ry1 0 cos Ry1

cos Ry1 Lz1 z1 = L b1La1 =

0 sin Rx1

sin Rx1 sin Ry1 cos Rx1 sin Ry1 cos Rx1

0 sin

(8.2)

Ry1

cos

Ry1

sin

sin Rx1 Rx1

cos

Rx1

cos

(8.3) Ry1

Similarly, the matrix of coordinate transformation from z2 to z2 in the global coordinate system Sf is given as a function of the rotations Rx2 and Ry2 of the reference point of the wheel model as follows:

Compensation of Shaft Deflections

445

cos Ry2 Lz2 z2 =

sin Rx2 sin Ry2 cos Rx2 sin Ry2 cos Rx2

0 sin

Ry2

cos

Ry2

sin

sin Rx2 Rx2

Rx2

cos

cos

(8.4)

Ry2

Considering that the unit vectors of the axes of rotation of pinion and wheel in coordinate system Sf are: 0 e(fz1) = e(fz 2) = 0 1

(8.5)

and by applying the transformations above, the unit vector along the axis of rotation of the pinion under load is: cos Rx1 sin Ry1 e(fz1 ) = Lz1 z1 e(fz1) =

sin Rx1 cos

Rx1

cos

(8.6) Ry1

and the unit vector along the axis of rotation of the wheel under load: cos Rx2 sin Ry2 e(fz2 ) = Lz2 z2 e(fz2) =

sin Rx2 cos

Rx2

cos

(8.7) Ry2

The cross product of the unit vectors of the axes of rotation of pinion and wheel in their deformed position under load allows the determination of the crossing angle error, c , and the intersecting angle error, i , as follows: cos Rx2 cos Ry2 sin Rx1 + cos Rx1 cos Ry1 sin Rx2 e(fz1 ) × e(fz2 ) =

cos Rx1 cos Rx2 cos Ry2 sin Ry1 + cos Rx1 cos Ry1 cos Rx2 sin Ry2 cos

The intersecting angle error, i

The crossing angle error, c

=

i,

sin

Ry2

sin

Rx2

+ cos

Rx2

sin

Rx1

sin

(8.8)

Ry2

is given by the x f -component of the cross product e(fz1 ) × e(fz2 ) as: cos Rx2 cos Ry2 sin Rx1 + cos Rx1 cos Ry1 sin Rx2

= c,

Rx1

(8.9)

is given by the yf -component of the cross product e(fz1 ) × e(fz2 ) as:

cos Rx1 cos Rx2 cos Ry2 sin Ry1 + cos Rx1 cos Ry1 cos Rx2 sin Ry2

(8.10)

The center distance error is determined as: C=

(x 2

x1 )2 + (y2

y1 )2

(x2

x1)2 + (y2

y1)2

(8.11)

446

Dudley’s Handbook of Practical Gear Design and Manufacture

and the axial displacement of the wheel with respect to the pinion as: A2 = z 2

z1

(8.12)

where x1 , y1 , z1 are the coordinates of the reference point of the pinion in its deformed position Pr :

x1 = x1 + Ux1 y1 = y1 + Uy1 z1 = z1 +

(8.13)

Uz1

and x 2 , y2 , z 2 the coordinates of the reference point of the wheel in its deformed position Wr : x 2 = x2 + Ux2 y2 = y2 + Uy2

(8.14)

z 2 = z2 + Uz2

In Eqs. (8.13) and (8.14), x1, y1, z1 and x2 , y2 , z2 are the coordinates of the reference points Pr and Wr , respectively, in their undeformed positions (see Figure 8.3). An alternative approach for the determination of errors of alignment is based on the consideration of the loaded and unloaded positions of points P1 and P2 at both sides of the pinion web, and that of the points W1 and W2 at both sides of the wheel web. The deformed position of points P1, P2 , W1, and W2 of the shafts due to the applied nominal torque are determined by finite element analysis and represented in the global coordinate system, Sf , as follows: r (fP1 ) = [ xf(P1 ) yf(P1 ) z (f P1 ) ]T r (fP2 ) = [ xf(P2) yf(P2) z (f P2 ) ]T r (fW1 ) = [ xf(W1 ) yf(W1 ) z (f W1 ) ]T

(8.15)

r (fW2 ) = [ xf(W2) yf(W2) z (f W2 ) ]T

Assuming that the shafts of pinion and wheel are rigid between points P1 and P2 , and between points W1 and W2, the unit vectors along the deflected axes of pinion and wheel can be determined as follows: e(fz1 ) =

e(fz2 ) =

r (fP2)

r (fP1 )

|r (fP2 )

r (fP1 ) |

r (fW2 )

r (fW1 )

|r (fW2 )

r(fW1 ) |

(8.16)

(8.17)

Based on them, Eqs. (8.8), (8.9), and (8.10) can be applied to determine the intersecting shaft error, i and the crossing shaft error, c , respectively. For determination of the center distance error, C , and the axial displacement of the wheel with respect to the pinion, A2 , the loaded position of reference points Pr and Wr should be evaluated.

Compensation of Shaft Deflections

447

Assuming that the distance from P1 to Pr is equal to half the face width of the pinion, and knowing the unit vector e(fz1 ) from P1 to P2 , the loaded position of reference point Pr can be determined as:

r(fPr ) = r(fP1 ) + (

Fw1 (z1 ) )e f 2

(8.18)

Similarly, the loaded position of reference point Wr will be determined as: r(fWr ) = r(fW1 ) + (

Fw2 (z 2 ) )ef 2

(8.19)

Based on the coordinates of position vectors r(fPr ) and r(fWr ), Eqs. (8.11) and (8.12) can be applied to determine the center distance error, C , and the axial displacement of the wheel with respect to the pinion, A2 .

8.2.2 TRANSMISSIONS

WITH INTERSECTING

SHAFTS

The approach presented in this section for the determination of the relative errors of alignment between the pinion and the wheel can be applied to any transmission with intersecting shafts. This method will be based on the deformed coordinates of points P1 and P2 located on the pinion shaft at both sides of its web, and points W1 and W2 located on the wheel shaft at both sides of its web. The deformed coordinates of mentioned points will be determined by finite element analysis according to the model shown in Figure 8.4. Given the deformed coordinates of the reference points located on the axes of rotation of pinion and wheel for the considered load, this approach is valid for any loading condition applied to the pinion shaft. If they are determined as proposed here, it is assumed that the performed elastic finite element analysis is accurately describing the behavior of the material. Figure 8.4 shows the position and orientation of the global coordinate system Sf , the unloaded po­ sition of reference points P1 and P2 located on the pinion shaft, and points W1 and W2 located on the wheel shaft. The pinion shaft is supported at points A1 and A2. Their radial displacements will be constrained as well as the axial displacement of A1. The wheel shaft is supported at points B1 and B2 . Their radial displacement will also be constrained. Support B1 is considered as fixed and therefore its axial dis­ placement will be constrained. The torque is applied at point A2 whereas the rotation of the wheel shaft is constrained at point B2 . Figure 8.5 shows the positive direction of the relative errors of alignment commonly considered for intersecting shaft transmissions: (i) shortest distance error, E , (ii) shaft angle error, , (iii) axial displacement of the pinion, A1, and (iv) axial displacement of the wheel, A2 . Figure 8.6 shows the deformed position of reference points P1 and P2 on the pinion shaft, and points W1 and W2 on the wheel shaft under load. The position vectors of those points in the global coordinate system are given by Eq. (8.15) The unit vector along the deflected axis of the pinion, e(fz1 ), and that of the wheel, e(fz2 ) will therefore be given by Eqs. (8.16) and (8.17). The unit vectors along the deflected shafts of the pinion and the wheel, and the position of points Ok and Ol limiting the shortest distance between the axes are also shown in Figure 8.6. The relative errors of alignment due to shaft deflections in intersecting shaft transmissions will be determined as follows: (1) Shaft angle error,

. The shaft angle error is determined by the following function of e(fz1 ), and e(fz2 ):

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Dudley’s Handbook of Practical Gear Design and Manufacture

FIGURE 8.4 Position of reference points P1 and P2 on the pinion shaft, and points W1 and W2 on the wheel shaft under no load (undeformed position).

(

= arccos e(fz1 ) e(fz2 )

)

(8.20)

Here, is the shaft angle. A negative value in the shaft angle error represents a decrease in the shaft angle. (2) Shortest distance error, ΔE. The shortest distance between shafts is determined along the normal to both shafts given by: n(12) = e(fz1 ) × e(fz2 ) f

(8.21)

The vector that goes from point W1 in the axis of rotation of the wheel to point P1 in the axis of rotation of the pinion is given by: r(fP1 / W1 ) = r (fP1 )

r(fW1 )

(8.22)

Then, the shortest distance between the two shafts will be the projection of r(fP1 / W1 ) on the common normal, n(12) f , to both shafts determined as:

Compensation of Shaft Deflections

449

FIGURE 8.5 Relative errors of alignment for a spiral bevel gearset: (a) shortest distance error, E, (b) shaft angle error, , (c) axial displacement of the pinion, A1, (d) axial displacement of the wheel, A2 .

E = Ol ¯Ok =

(r(fP /W ) n(12) f ) 1

1

(8.23)

The parametric coordinates of the points of intersection of line Ol ¯Ok with the axes of rotation of the pinion, s0 , and the wheel, t0 , are given by:

(

)

(8.24)

(

)

(8.25)

s0 = n(12) e(fz2 ) × r (fP1 / W1 ) f

t0 = n(12) e(fz1 ) × r (fP1 / W1 ) f

Therefore, the position vectors of points Ok and Ol can be determined by: r(fOk ) = r (fP1 ) + s0 e(fz1 )

(8.26)

r(fOl ) = r (fW1 ) + t0 e(fz2 )

(8.27)

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Dudley’s Handbook of Practical Gear Design and Manufacture

FIGURE 8.6 Position of reference points P1 and P2 on the pinion shaft, and points W1 and W2 on the wheel shaft under load (deformed position).

Here, s0 and t0 will take negative values, as points Ok and Ol are located toward the negative direction of vectors e(fz1 ) and e(fz2 ). The absolute value of the minimum distance between the pinion and wheel shafts can also be determined as: | E | = |r (fOk )

r (fOl ) |

(8.28)

(3) Axial displacement of pinion, A1. The axial displacement of the pinion is given by: A1 = Ok¯P1

Of¯P1

(8.29)

where P1 corresponds to the unloaded position of point P1 in coordinate system Sf. (4) Axial displacement of the wheel, A2 . The axial displacement of the wheel is given by: A2 = Ol ¯W1 Of ¯W1 where W1 corresponds to the unloaded position of point W1 in coordinate system Sf .

8.2.3 TRANSMISSIONS

WITH

(8.30)

CROSSING SHAFTS

The determination of errors of alignment due to shaft deflections in gear transmissions with crossing shafts is similar to that described for gear drives with intersecting shafts. Below is a summary of the

Compensation of Shaft Deflections

451

equations to be applied to determine the relative errors of alignment in transmissions with crossing shafts: (1) Shaft angle error, Δγ:

(

= arccos e(fz1 ) e(fz2 )

)

(8.31)

(2) Shortest distance error, E : E = Ol ¯Ok =

(r(fP /W ) n(12) f ) 1

1

E

(8.32)

Here, E is the offset distance of the gearset. (3) Axial displacement of pinion, ΔA1: A1 = Ok¯P1

Of¯P1

(8.33)

Of ¯W1

(8.34)

(4) Axial displacement of the wheel, A2 : A2 = Ol ¯W1

8.3

COMPENSATION OF ERRORS OF ALIGNMENT DURING GEAR GENERATION

Compensation of errors of alignment due to shaft deflections during the process of pinion or wheel generation is necessary in order to achieve the best conditions of meshing and contact for the transmitted nominal torque in gear drives. The process of compensation of errors of alignment due to shaft deflection will be based on the following general steps: • Shaft deflections due to the transmitted nominal torque have to be determined. Finite element models comprising the gear models, shafts, and bearing locations are applied here. Alternative analytical methods to determine the deflection and slope of the shafts due to the transmitted forces on the gear tooth surfaces can also be applied. As a result, the relative errors of alignment due to shaft deflections will be known. • The previous step may involve several iterations of analysis because each iteration of compen­ sation will change the position of the bearing contact, and therefore the deflection and slope of the shafts at the gear positions. Usually, this process needs three or four iterations to converge. Convergence is achieved when the errors of alignment determined after one iteration are equal to or within an acceptable tolerance from those compensated in previous iterations. • Depending on the type of gearset, the pinion, the wheel, or both will be generated considering their computed misaligned position under the evaluated conditions. • The gear tooth surfaces obtained before and after compensation of errors of alignment will be finally compared to get the type and amount of compensation needed on the gear tooth surfaces. The process of compensation of errors of alignments due to shaft deflections in parallel shaft trans­ missions will be presented below. Although it is presented for a spur pinion, it can be applied to any type of parallel shaft transmission. Regarding other types of gear transmissions, details of the process of compensation of errors of alignment caused by shaft deflections in spiral bevel gear drives can be found

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in [5]. In addition, the derivation and compensation of the errors of alignment caused by shaft deflections in face-gear drives generated by shaper cutters is provided in [6]. Figure 8.7 shows the coordinate systems applied to generate a spur pinion in misaligned position, as well as the errors of alignment to be compensated. The errors shown in Figure 8.7 are all considered positive. Coordinate systems Sc and S1 are considered rigidly connected to the generating cutter and the to-begenerated pinion with errors of alignment compensated, respectively. Transformation from coordinate system Sc to coordinate system S1 takes into account the errors of alignment to be compensated. Consider that the generating surfaces of the rack-cutter are known and given as rc (u, ). Here, u and are the surface parameters along the profile and lead directions of the generating rack-cutter tooth surfaces. During generation, the rack cutter is translated in the negative direction of axis xc a distancerp1 1, while the beinggenerated pinion is rotated an angle 1. Here, rp1 is the pitch radius of the pinion. The family of rack-cutter surfaces c is represented in coordinate system S1, fixed to the beinggenerated pinion, by: (8.35) r1(u, , 1) = M1n ( 1) Mnm Mmk Mkc rc (u, ) where: 1 Mkc = 0 0 0

Mmk =

Mnm

cos 0 sin 0

0 1 0 0

0 rp1 1 0 r1 + C 1 A2 0 1 0 sin 1 0 0 cos 0 0

c

1 0 0 cos = 0 sin 0 0

c

i i

0 sin cos 0

(8.36)

c

c

i i

0 0 0 1

(8.37)

0 0 0 1

(8.38)

cos 1 sin 1 0 0 sin 1 cos 1 0 0 (8.39) M1n = 0 0 1 0 0 0 0 1 Here, r1 = rp1 + mx1 is the shifted radius of the pinion, which considers the generating profile shift coefficient for the pinion, x1. The equation of meshing is determined as:

fc1 (u, ,

1)

=

r1 r × 1 u

r1

=0

(8.40)

1

The surface of the pinion with errors of alignment compensated during generation is determined by vector equation r1(u, , 1) and the equation of meshing fc1 (u, , 1) = 0.

Compensation of Shaft Deflections

453

FIGURE 8.7 Toward generation of the pinion of a spur gearset with errors of alignment due to shaft deflections compensated.

When pinion tooth surfaces include compensation of errors of alignment due to shaft deflections for the nominal torque as shown above, the conditions of meshing and contact under such a torque are similar to those achieved with the theoretical geometries under the presence of no errors of alignment. It is true that for a lower torque than that used for compensation of errors of alignment, the conditions of meshing and contact will not be as favorable as those obtained for the nominal torque. However, the application of a lower torque will contribute to yield lower contact and bending stresses and will limit the risk of damaging the gear tooth surfaces. Next sections show several numerical examples of compensation of errors of alignment in spur, spiral bevel, and face gear drives.

8.4

NUMERICAL EXAMPLES

In this section, examples of compensation of errors of alignment due to shaft deflections will be pre­ sented for a spur gearset, a spiral bevel gearset, and a face gearset.

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Dudley’s Handbook of Practical Gear Design and Manufacture

8.4.1 SPUR GEARSET The basic geometry of the spur gearset used as example of compensation of errors of alignment caused by shafts deflection is shown in Table 8.1. The gearset includes a pinion of 21 teeth and a wheel of 34 teeth with 3 mm of module, pressure angle of 20 degrees, and operating center distance of 82.5 mm. Figure 8.8 provides the dimensions of the shafts supporting the pinion and the gear of the drive. Points A1 and A2 represent the location of the bearings on the pinion shaft, and points B1 and B2 represent the location of the bearings on the wheel shaft. The diameters of the pinion and wheel shafts are 30 mm and 40 mm, respectively. A torque T of 340 Nm is applied to the pinion shaft at point A2 in counter clockwise direction as shown in Figure 8.8. The wheel shaft will be considered as fixed at end B2. Both gears have a face width of 30 mm. Figure 8.9 shows the evolution of the maximum contact stress by means of the von Mises equivalent stress on the active tooth surfaces of the pinion (Figure 8.9a) and the wheel (Figure 8.9b) when a nominal torque of 340 Nm is applied to the pinion with and without the consideration of the shafts supporting the gears. If the gear supporting shafts are not considered in the finite element model, the maximum value of the contact stress is 1064 MPa at contact position 21. However, when the gear supporting shafts are considered, the maximum contact stresses reach a peak of almost 2800 MPa due to the appearance of severe edge contacts that will lead to the premature failure of the gears, or the need to limit substantially the maximum torque that the gearset can transmit. The wheel experiences a similar behavior as shown in Figure 8.9b. Figure 8.10 shows the evolution of the maximum contact stress on the active tooth surfaces of the pinion (Figure 8.10a) and the wheel (Figure 8.10b) as a function of the applied torque of the pinion when it varies from 90 Nm to 340 Nm. By assuming a limiting contact stress of 1200 MPa, the allowable transmitted torque should not be larger than 140 Nm for this gear drive. Figure 8.11 shows the contact patterns on the pinion tooth surface when considering the relative errors of alignment caused by shaft deflections when a torque varying from 0 Nm (no load and perfectly aligned gear drive) to a maximum of 340 Nm (nominal torque) is applied to the pinion shaft. As shown in Figure 8.11a, a perfectly distributed contact pattern is obtained on the pinion tooth surface for no load and therefore a perfectly alignment gearset due to shaft deflections. As the torque increases, shaft

TABLE 8.1 Basic Geometry of a Spur Gearset Design Parameter Tooth number

Units

Pinion

[-]

21

Wheel 34

Module

[mm]

3

Pressure angle Operating center distance

[deg] [mm]

20 82.5

Addendum diameter

[mm]

70.098

106.920

Pitch diameter Root diameter

[mm] [mm]

63.000 56.598

102.000 93.402

Base diameter

[mm]

59.201

95.849

Root form diameter Hob tip radius

[mm] [mm]

59.639 1.140

96.885 1.140

Circular tooth thickness

[mm]

5.112

4.313

Face width Nominal torque

[mm] [Nm]

30.000 340

30.000 -

Compensation of Shaft Deflections

FIGURE 8.8

455

Geometric model of a spur gearset with supporting shafts.

deflection increases and the contact pattern is shifted toward the back section of the pinion tooth surface. Notice that as the torque increases, the bearing contact on the tooth surfaces decreases, yielding the high levels of contact stresses shown in Figure 8.10. Table 8.2 shows the relative errors of alignment due to shaft deflections for each iteration of com­ pensation. The first iteration yields errors of alignment that provide more than 85% of the total com­ pensation. The iterative process is needed because for the first iteration the path of contact will be located in one of the edges of the surfaces. As the errors of alignment are being compensated, the bearing contact is extended over a larger portion on the gear tooth surfaces and therefore the corresponding shaft deflections change. After four iterations, the errors of alignment obtained due to shaft deflections do not change much with respect to the previous iteration and the process of compensation can be considered converged.

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Dudley’s Handbook of Practical Gear Design and Manufacture

FIGURE 8.9 Evolution of the maximum contact stress on the tooth surfaces of (a) the pinion and (b) the wheel with and without the consideration of the shafts.

The errors of alignment due to shaft deflections listed in Table 8.2 are obtained for a nominal torque of 340 Nm applied to the pinion shaft and will be compensated during the process of generation of the pinion. Table 8.3 shows the relative errors of alignment when the torque varies from 90 Nm to 340 Nm for the geometry with errors of alignment compensated. For lower values of the torque, because the relative errors of alignment are different from those compensated, the contact pattern will be shifted toward the front sides of the tooth surfaces but considering the lower value of the torque, the stresses on the surfaces will also be limited. Figure 8.12 shows the contact patterns on the pinion tooth surface when the applied torque to the pinion varies from 90 Nm to 340 Nm and the errors of alignment caused by shaft deflections are compensated for a nominal torque of 340 Nm. When a torque of 90 Nm is applied, the contact pattern is shifted toward the front section of the pinion. When the torque increases, the contact pattern increases and takes a larger area on the pinion tooth surface. The contact pattern continues increasing with the torque until it reaches the value of 340 Nm. For the nominal torque of 340 Nm, the contact pattern is distributed perfectly on the pinion tooth surfaces, yielding the lower values of the maximum contact stress.

FIGURE 8.10 Evolution of the maximum contact stress on the tooth surfaces of (a) the pinion and (b) the wheel as a function of the applied torque to the pinion.

Compensation of Shaft Deflections

457

FIGURE 8.11 Contact patterns on the pinion tooth surface when the applied torque to the pinion varies from 0 Nm to 340 Nm before compensation of errors of alignment caused by shaft deflections.

TABLE 8.2 Relative Errors of Alignment due to Shaft Deflections for Each Iteration of Compensation Errors of Alignment

[Units]

Iteration 1

Iteration 2

Iteration 3

Iteration 4

Axial displacement of wheel, ΔA2 Center distance error, ΔC

[mm] [mm]

0.0003 0.0970

0.0003 0.1022

0.0003 0.1033

0.0003 0.1035

Intersecting shaft angle error, Δγi

[deg]

-0.0503

-0.0573

-0.0587

-0.0590

Crossing angle error, Δγc

[deg]

-0.1041

-0.1189

-0.1216

-0.1222

Figure 8.13 shows the evolution of the maximum contact stress on the active tooth surfaces of the pinion (Figure 8.13a) and the wheel (Figure 8.13b) with errors of alignment caused by shaft deflection compensated as a function of the applied torque of the pinion when it varies from 90 Nm to 340 Nm. When errors of alignment are compensated and the torque increases, the maximum contact stresses decrease because of a more favorable contact pattern that extends over a larger portion of the gear tooth surfaces. When the errors of alignment caused by shaft deflections are compensated, the gearset is able to transmit a larger torque with lower contact stresses than for the case when no errors of alignment are compensated. For the case of design shown in Table 8.1 and shaft dimensions shown in Figure 8.8, the standard geometry of the spur gear and the geometry obtained after compensation of errors of alignment caused by shaft deflections are compared. The maximum deviation required on the left side of the pinion tooth surface is of 70.7 m as shown in Figure 8.14. The modification of the pinion tooth surfaces needed to compensate errors of alignment caused by shaft deflections for this case of design is compatible with a helix deviation determined as: = arctan

Max deviation 0.0707 = arctan = 0.315° Face width 30

The helix deviation for the left side of the tooth should be right-handed.

(8.41)

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Dudley’s Handbook of Practical Gear Design and Manufacture

TABLE 8.3 Relative Errors of Alignment as a Function of Applied Torque Errors of Alignment

[Units]

90 Nm

140 Nm

190 Nm

240 Nm

290 Nm

340 Nm 0.0003

Axial displacement of wheel, ΔA2

[mm]

0.0000

0.0001

0.0001

0.0002

0.0002

Center distance error, ΔC

[mm]

0.0299

0.0459

0.0612

0.0755

0.0896

0.1035

Intersecting shaft angle error, Δγi Crossing angle error, Δγc

[deg] [deg]

-0.0187 -0.0391

-0.0283 -0.0591

-0.0370 -0.0773

-0.0446 -0.0931

-0.0518 -0.1078

-0.0590 -0.1222

FIGURE 8.12 Contact patterns on the pinion tooth surface when the applied torque to the pinion varies from 90 Nm to 340 Nm and the errors of alignment caused by shaft deflections compensated.

Additional micro-geometry modifications including an additional lead crowning are recommended to account for other sources of errors of alignment. However, those additional micro-geometry modifica­ tions should be lower than those needed if errors caused by shaft deflections were not compensated as proposed here.

8.4.2 SPIRAL BEVEL GEARSET In this section, an example of compensation of errors of alignment caused by shaft deflections for a facemilled generated spiral bevel gear drive is presented. The basic design parameters of the spiral bevel gear drive are listed in Table 8.4. Geometric data (diameters and lengths) of pinion and wheel supporting shafts with definition of the bearing location are shown in Figure 8.15. Table 8.5 represents the machinetool settings for pinion and wheel generation of the spiral bevel gearset, which are considered as known. The process of compensation of errors of alignment caused by shaft deflections in spiral bevel gears yields new machine-tool settings for the pinion tooth surfaces, whereas the wheel will be kept un­ modified. The process of compensation is carried out by the application of the local synthesis method [1]. Details of this process can be found in [5].

Compensation of Shaft Deflections

459

FIGURE 8.13 Evolution of the maximum contact stress on the tooth surfaces of (a) the pinion and (b) the wheel as a function of the applied torque to the pinion and errors of alignment caused by shaft deflections compensated.

FIGURE 8.14 Comparison of the standard geometry of a spur gear and the geometry obtained after compensation of errors of alignment caused by shaft deflections.

Figure 8.16 shows the evolution of the maximum contact stress on the active tooth surfaces of pinion and wheel of the considered spiral bevel gear drive under a nominal torque of 500 Nm when (i) no shafts are considered and (ii) the pinion and wheel supporting shafts are considered. Shaft deflections cause severe edge contacts on the spiral bevel gear tooth surfaces and may lead to a premature failure of the gear drive. In this analysis, the convex side of the pinion tooth surfaces has been considered as driving and the concave side of the wheel tooth surfaces as driven.

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Dudley’s Handbook of Practical Gear Design and Manufacture

Figure 8.17 shows the evolution of the maximum contact stress on the pinion and wheel active tooth surfaces considering the gear supporting shafts and a torque applied to the pinion shaft varying from 100 Nm to 500 Nm. As expected, the higher the applied torque, the higher the contact stresses. It can be observed that, for a torque between 200 Nm and 300 Nm, edge contacts occur yielding high values of contact stresses at some contact positions along the cycle of meshing. Figure 8.18 shows the contact patterns when the torque applied to the pinion shaft varies from 0 Nm (perfectly aligned gear drive and no load) to 500 Nm. Those results reveal that, firstly, under no load, a longitudinally oriented bearing contact is obtained, and secondly, as the transmitted torque increases, the contact pattern is shifted toward the heel of the pinion and wheel tooth surfaces, causing severe edge contacts on the heel side of gear tooth surfaces. Table 8.6 shows the relative errors of alignment for each iteration of design that has been compen­ sated throughout the redesign of, in each case, the pinion member of the spiral bevel gearset. A long­ itudinally oriented path of contact is always considered. Table 8.7 shows the new machine-tool settings for the pinion with compensated geometry. Notice the change of machine-tool settings for the convex side of the pinion. Figure 8.19 shows the contact patterns for the spiral bevel gearset with compensated geometry and applied torque to the pinion varying from 0 Nm to 500 Nm. For low torque transmission, the contact patterns are now shifted toward the toe edges of the active gear tooth surfaces. However, when the torque is increasing to its nominal value, contact patterns are shifted toward the center of the active pinion and wheel tooth surfaces. Therefore, the spiral bevel gearset will perform under the desired conditions of meshing and contact at the nominal torque. Figure 8.20 shows the evolution of the maximum contact stress on the pinion and wheel active tooth surfaces of the spiral bevel gear drive with compensated geometry and torque applied to the pinion varying from 100 Nm to 500 Nm. Now, the higher the applied torque, the lower the maximum contact stresses. Notice the smooth evolution of maximum contact stresses for the nominal torque of 500 Nm. Finally, Figure 8.21 shows the field of von Mises stresses for the existing geometry of the pinion (Figure 8.21a) and for the geometry compensated for shaft deflections caused by the nominal torque of

TABLE 8.4 Basic Design Parameters of a Face-Milled Spiral Bevel Gear Drive Design Parameter

Units

Pinion

Tooth number Outer transverse module

[-] [mm]

16

Shaft angle

[deg]

29.0000

Mean spiral angle Face width

[deg] [mm]

35.0000 31.0000

Hand of spiral Outer whole depth Outer addendum

Wheel 40 4.9485

[-]

Left

Right

[mm] [mm]

8.9868 6.0389

8.9868 2.0820

Outer dedendum

[mm]

2.9480

6.9049

Outer working depth Pitch angle

[mm] [deg]

8.1208 21.8014

8.1208 68.1986

Root cone angle

[deg]

20.6720

64.9232

Face cone angle

[deg]

25.0768

69.3280

Compensation of Shaft Deflections

FIGURE 8.15

461

Pinion and wheel supporting shaft dimensions and bearing locations.

TABLE 8.5 Basic Machine-Tool Settings and Cutter Geometry of Existing Geometry of a Spiral Bevel Gearset Pinion [Units]

Concave

Wheel Convex

Concave

Convex

Machine center to back

[mm]

-4.9200

3.3222

0.0000

Sliding base

[mm]

0.8904

-2.0192

-0.8034

Blank offset Radial distance

[mm] [mm]

5.9734 67.7858

-6.5572 87.3045

0.0000 78.2788

Cradle angle

[deg]

51.3459

51.3792

52.8824

Machine root angle Velocity ratio

[deg] [-]

20.6720 2.3541

20.6720 2.9623

64.9232 1.0753

Modified roll coefficient C

[-]

-0.0226

0.0173

0.0000

Modified roll coefficient D Mean cutter radius

[-] [mm]

0.0733 N/A

-0.0658 N/A

0.0000 76.2000

Cutter tip point radius

[mm]

70.5501

79.5542

N/A

Point width Blade pressure angle

[mm] [deg]

N/A 21.8410

N/A 18.0818

2.5000 21.8410

Blade edge radius

[mm]

1.1906

18.0818

1.2371

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Dudley’s Handbook of Practical Gear Design and Manufacture

FIGURE 8.16 Evolution of maximum contact stress on (a) the pinion tooth surfaces and (b) the wheel tooth surfaces with and without consideration of the gear supporting shafts.

FIGURE 8.17 Evolution of the maximum contact stress on (a) the pinion and (b) the wheel of the given spiral bevel gearset for a torque variation from 100 Nm to 500 Nm applied to the pinion shaft.

FIGURE 8.18

Contact pattern as a function of the applied torque to the pinion varying from 0 Nm to 500 Nm.

Compensation of Shaft Deflections

463

TABLE 8.6 Evolution of Relative Errors of Alignment Compensated in Each Iteration of Design Errors of Alignment

[Units]

Iteration 1

Iteration 2

Iteration 3

Iteration 4

Shortest distance error, ΔE

[mm]

0.533493

0.619400

0.642909

0.649654

Axial displacement of pinion, ΔA1

[mm]

-0.009778

-0.009462

-0.009403

-0.009386

Axial displacement of wheel, ΔA2 Shaft angle error, Δγ

[mm] [deg]

0.655969 -0.239113

0.677513 -0.250556

0.684088 -0.253888

0.686003 -0.254862

500 Nm (Figure 8.21b). Figure 8.22 shows the contact stresses on the wheel of the spiral gearset for both the existing geometry (Figure 8.22a) and the compensated geometry (Figure 8.22b).

8.4.3 FACE GEARSET The third and last example of compensation of errors of alignment caused by shaft deflections considers a face-gear drive with basic geometric parameters listed in Table 8.8. A tip relief of 15 m of parabolic progression along the tooth profile for a radial length of 1.3 mm is applied to the pinion tooth surfaces. A root relief of 18 m of parabolic progression is also applied in profile direction for a radial length of 1.7 mm starting at the form diameter of the pinion. The top and bottom reliefs applied to the pinion intend to avoid edge contacts on the tooth surfaces of the pinion and the face gear. Figure 8.23a shows the deviations applied to the pinion tooth surfaces as mentioned above. They are measured with respect to the standard involute profile of the spur pinion. Notice that the face gear is generated by a shaper of 26 teeth, whereas the pinion of the gear drive has 25 teeth. This tooth difference of one tooth between the pinion and the shaper contributes to provide a

TABLE 8.7 Machine-Tool Settings and Cutter Geometry of the Spiral Bevel Gearset with Compensated Geometry Pinion [Units]

Concave

Wheel Convex

Concave

Convex

Machine center to back

[mm]

-4.9200

3.1599

0.0000

Sliding base Blank offset

[mm] [mm]

0.8904 5.9734

-1.9619 -7.9184

-0.8034 0.0000

Radial distance

[mm]

67.7858

87.7117

78.2788

Cradle angle Machine root angle

[deg] [deg]

51.3459 20.6720

51.7059 20.6720

52.8824 64.9232

Velocity ratio

[-]

2.3541

2.9760

1.0753

Modified roll coefficient C Modified roll coefficient D

[-] [-]

-0.0226 0.0733

0.0175 -0.0706

0.0000 0.0000

Mean cutter radius

[mm]

N/A

N/A

76.2000

Cutter tip point radius Point width

[mm] [mm]

70.5501 N/A

79.5542 N/A

N/A 2.5000

Blade pressure angle

[deg]

21.8410

Blade edge radius

[mm]

18.0818 1.1906

18.0818

21.8410 1.2371

464

Dudley’s Handbook of Practical Gear Design and Manufacture

FIGURE 8.19 Contact patterns for the spiral gearset with compensated geometry and applied torque to the pinion varying from 0 Nm to 500 Nm.

FIGURE 8.20 Evolution of the maximum contact stress on (a) the pinion and (b) the wheel of the spiral bevel gearset with compensated geometry and torque applied to the pinion shaft varying from 100 Nm to 500 Nm.

small crowning to the surface of the face gear and with it the localization of the contact pattern on the pinion and face gear tooth surfaces [1]. The amount of crowning applied to the face gear by using a shaper of 26 teeth for the gear drive with design parameters as provided in Table 8.8 is of about 30 m, as shown in Figure 8.23b. The provided longitudinal crowning applied to the face gear makes the gear drive less sensitive to errors of alignment and makes this study to be aligned with actual designs of facegear transmissions. Table 8.9 shows the design parameters of the pinion and face gear shafts. Hollow shafts have been considered for both the pinion and the face gear. Both gear elements are overhung mounted outside the bearing span as shown in Figure 8.24. When the shafts supporting the gears are not taken into account, the conditions of meshing and contact can be considered theoretical or ideal. In this situation, the contact patterns on the gear tooth

Compensation of Shaft Deflections

465

FIGURE 8.21 Contact stresses on the pinion tooth surfaces for (a) existing geometry and (b) compensated geometry for shaft deflections.

FIGURE 8.22 Contact stresses on the wheel tooth surfaces for (a) existing geometry and (b) compensated geometry for shaft deflections.

TABLE 8.8 Design Parameters of a Face-Gear Drive Design Parameter Tooth number of the pinion Tooth number of the shaper Tooth number of the face gear Pressure angle Module

Units

Value

[-] [-]

25 26

[-]

160

[deg] [mm]

25 6.35

Shaft angle

[deg]

90

Inner radius of the face gear Outer radius of the face gear

[mm] [mm]

471 559

Pitch radius of the pinion

[mm]

79.375

Pitch radius of the face gear Face width of the pinion

[mm] [mm]

508 88

466

Dudley’s Handbook of Practical Gear Design and Manufacture

FIGURE 8.23 Deviations of (a) the pinion and (b) the face-gear tooth surfaces from their respective standard geometries.

surfaces are said to correspond to the unloaded (or lightly loaded) gear drive. However, when the supporting shafts are considered for stress analysis, real conditions of meshing and contact are simulated and the deflections of shafts under the transmitted torque will influence the contact pattern location and the level of stresses on the gear tooth surfaces. A nominal torque of 2500 Nm will be applied to the pinion shaft on the opposite side of the pinion location in the shaft. Figure 8.25 shows the evolution of the maximum contact stress on the tooth surfaces of the pinion (Figure 8.25a) and the face gear (Figure 8.25b) when (i) no shafts are considered (ideal conditions of meshing with no errors of alignment) and (ii) the actual supporting shafts of the pinion and the face gear are considered. When shafts are not considered, areas of high contact stresses appear on the tooth surfaces of the pinion and the face gear. Those areas are located in the upper corner of the inner section of the active tooth surface of the face gear and the corresponding area in the pinion tooth surface. However, when shafts are considered into the finite element model, the contact pattern is shifted toward the outer radius of the active tooth surfaces of the face gear and those areas of high contact stresses appearing in the inner radius are avoided, as well as the maximum contact stress reduced. Figure 8.26 shows the field of von Mises stress for the case when supporting shafts are not considered (Figure 8.26a) and for the case when elastic shafts are considered (Figure 8.26b). The results shown in Figure 8.26 suggest that the active tooth surfaces of the face gear should be trimmed at the inner and outer upper corners of the surface to avoid those areas of high contact stresses. Figure 8.27a shows the theoretical boundaries of the active tooth surfaces of the face gear considered in the stress analysis shown above and Figure 8.27b shows the corresponding trimmed active tooth

TABLE 8.9 Design Parameters of the Pinion and Face-Gear Shafts Design Parameter

Units

Value

Inner diameter of the pinion shaft

[mm]

80

Outer diameter of the pinion shaft

[mm]

100

Inner diameter of the face gear shaft Outer diameter of the face gear shaft

[mm] [mm]

160 200

Compensation of Shaft Deflections

FIGURE 8.24

467

Shaft dimensions and 3D representation of the face-gear drive.

FIGURE 8.25 Evolution of the maximum contact stress on (a) the pinion and (b) the face-gear tooth surfaces with and without the consideration of the gear supporting shafts.

surfaces that are being considered to avoid those edge contacts that occurred in the previous analysis. The inner and outer upper corners of the active surface of the face gear have been removed. The finite element analysis is repeated to study the variation of contact and bending stresses for the new redesigned face gearset. Figure 8.28 shows the variation of the maximum contact stress on the pinion (Figure 8.28a) and the face gear tooth surfaces (Figure 8.28b) upon application of flank edge coordinates as shown in Figure 8.27b for the cases with and without consideration of the corresponding supporting shafts. The nominal torque of 2500 Nm is being considered here. Notice that the case without consideration of supporting shafts now yields a smooth variation of low values of contact stresses that reaches a max­ imum of 665 MPa for the pinion and 545 MPa for the face gear. However, when shafts are considered, the maximum contact stress is considerably increased all over the cycle of meshing. Figure 8.29 shows the evolution of the maximum bending stress in terms of the maximum principal stress for the pinion (Figure 8.29a) and the face gear (Figure 8.29b) when flank edge coordinates are applied to limit the boundaries of the active surfaces and for the cases with and without consideration of the gear supporting shafts. The maximum bending stress experiences an important increment when the supporting shafts are considered with respect to the case of ideal conditions of meshing and contact (no deflections or misalignments).

468

Dudley’s Handbook of Practical Gear Design and Manufacture

FIGURE 8.26 Field of equivalent von Mises stresses on the face-gear tooth surfaces (a) without considering supporting shafts and (b) with supporting shafts being considered.

Figure 8.30 shows the contact patterns when considering the relative errors of alignment caused by shaft deflections for different torques applied to the pinion shaft varying between 0 Nm (perfectly aligned gear drive) and 2500 Nm. The results obtained reveal that, firstly, under no load, a centered bearing contact is achieved, and secondly, as the torque increases, the contact pattern is shifted toward the outer radius of the tooth surface of the face gear. When the torque is larger than 1500 Nm, edge contacts will occur on the outer section of the active surface of the face gear causing high contact stresses. Figure 8.31 shows the evolution of the maximum contact stress on the pinion (Figure 8.31a) and the face gear tooth surfaces (Figure 8.31b) considering the supporting shafts and a torque applied to the pinion varying from 500 Nm to 2500 Nm. When the input torque is larger than 1000 Nm, edge contacts start to have an effect on the maximum contact stresses along the cycle of meshing. The deflections of supporting shafts of pinion and face gear are obtained along 41 contact positions by using finite element analysis. Then, the relative errors of alignment for each contact position are de­ termined according to Eqs. (8.20) to (8.30) and averaged along the 41 contact positions. According to [7,8] errors of alignment , E , and A2 have an influence on the contact pattern location. If the pinion tooth surfaces are longitudinally crowned, then the error A1 will also influence the contact pattern location.

Compensation of Shaft Deflections

469

FIGURE 8.27 Application of flank edge coordinates to the active surfaces of the face gear to trim the corners where edge contact and areas of high contact stresses occur. All dimensions are given in mm.

FIGURE 8.28 Evolution of maximum contact stress on (a) the pinion and (b) the face-gear tooth surfaces upon appli­ cation of flank edge coordinates for the cases with and without consideration of the corresponding supporting shafts.

In order to achieve the needed micro-geometry modifications of the tooth surfaces of the face gear to compensate errors of alignment caused by shaft deflections, the generation of the face gear by adjusting the position of the shaper cutter to replicate the relative position of the pinion in misaligned condition, will be considered. The computation of errors of alignment caused by shaft deflections and their com­ pensation is repeated through two or more iterations until the obtained errors of alignment to be

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Dudley’s Handbook of Practical Gear Design and Manufacture

FIGURE 8.29 Evolution of the maximum bending stress for (a) the pinion and (b) the face-gear fillet tooth surfaces upon application of flank edge coordinates for the cases with and without consideration of the corre­ sponding supporting shafts.

compensated converge within an acceptable precision. Table 8.10 shows the averaged relative errors of alignment due to shaft deflections under the nominal torque of 2500 Nm for three iterations of com­ pensation of errors of alignment. Notice that after the first iteration, the errors of alignment approach in more than 80% to the values obtained after iteration 3, and usually compensation of those errors of alignment in the first iteration gives more than 90% of the expected improvement on the maximum contact and bending stress.

FIGURE 8.30 Contact patterns on the face-gear tooth surface considering shaft deflections under a torque varying from 0 Nm to 2500 Nm applied to the pinion shaft.

Compensation of Shaft Deflections

471

FIGURE 8.31 Evolution of the maximum contact stress on (a) the pinion and (b) the face-gear tooth surface for an input torque applied to the pinion shaft varying from 500 Nm to 2500 Nm.

Figure 8.32 shows the evolution of the maximum contact stress for the pinion (Figure 8.32a) and the face gear tooth surfaces (Figure 8.32b) after each iteration of compensation of errors of alignment due to shaft deflections as listed in Table 8.10. Notice that after the first iteration of compensation of errors of alignment due to shaft deflections, the maximum contact stress is reduced more than 25% for the pinion and 50% for the face gear. Successive iterations do not show much improvement on the contact stresses for both pinion and face gear with respect to the improvement achieved after the first iteration. This is mainly due to the slight crowning provided to the tooth surfaces of the face gear of the initial design by considering a shaper with 26 teeth that allows absorbing small values of misalignments. If we want to achieve a contact closely to line contact after compensation of errors of alignment caused by shaft deflections, the application of a shaper cutter with the same number of teeth of the pinion is needed. However, in that case, the gear drive would be very sensitive to errors of alignment caused of other sources and therefore, keeping a small crowning on the gear tooth surfaces is necessary as con­ sidered here. Figure 8.33 shows the variation of the maximum contact stress on the pinion (Figure 8.33a) and the active tooth surfaces of the face gear (Figure 8.33b) before and after the compensation of errors of alignment due to shaft deflections. The case with no shafts is also included as a reference. The maximum

TABLE 8.10 Relative Errors of Alignment due to Shaft Deflections for Each Iteration of Compensation Errors of Alignment

[Units]

Iteration 1

Iteration 2

Iteration 3

Shortest distance error, ΔE

[mm]

1.0577

1.2286

1.2923

Axial displacement of pinion, ΔA1 Axial displacement of face gear, ΔA2

[mm] [mm]

-0.0223 0.5198

-0.0198 0.5041

-0.0188 0.4917

Shaft angle error, Δγ

[deg]

-0.0317

-0.0311

-0.0304

472

Dudley’s Handbook of Practical Gear Design and Manufacture

FIGURE 8.32 Evolution of the maximum contact stress for (a) the pinion and (b) the face gear after each iteration of compensation of errors of alignment due to shaft deflections.

contact stress on both the pinion and the active tooth surfaces of the face gear are considerably reduced after the compensation of errors of alignment due to shaft deflections. Figure 8.34 shows the contact patterns when the tooth surfaces of the face gear include the needed micro-geometry modifications to absorb the errors of alignment caused by shaft deflections. For a low value of the transmitted torque, the contact pattern is shifted toward the inner radius of the tooth surfaces of the face gear. However, when the torque is increasing to its nominal value of 2500 Nm, the contact pattern is shifted toward the center of the tooth surfaces as desired, reducing in this way the maximum contact stress with the nominal torque applied. Figure 8.35 shows the evolution of the maximum contact stress on the pinion and face gear tooth surfaces for a torque applied to the pinion shaft varying from 500 Nm to 2500 Nm. The variation of the maximum contact stress corresponding to each level of torque variation is represented by curves that are very close to each other. Even for low values of the torque when the contact pattern is shifted toward the inner radius of the tooth surfaces of the face gear, contact stresses are still low because the torque is small. When torque is increased, the contact pattern is shifted toward the center of the face gear tooth

FIGURE 8.33 Evolution of the maximum contact stress for (a) the pinion and (b) the face-gear tooth surfaces before and after the process of compensation of errors of alignment due to shaft deflections.

Compensation of Shaft Deflections

473

FIGURE 8.34 Contact patterns on the face-gear tooth surface after compensation of errors of alignment caused by shaft deflections when a torque from 0 Nm to 2500 Nm is applied to the pinion shaft.

surfaces (Figure 8.34) and therefore the increment of torque is compensated mostly with a better location of the contact pattern. Finally, Figure 8.36 shows the field of von Mises equivalent stress on the tooth surfaces of the face gear when errors of alignment due to shaft deflections have been compensated for different values of applied torque at the same contact position along the cycle of meshing. Notice that the contact ellipse moves toward the center of the surface when the torque increases. 8.4.3.1 Alternative Methods of Compensating Shaft Deflections Here, alternative methods to prepare the tooth surfaces of the face gear to absorb the errors of alignment due to shaft deflections are presented. The first one is based on application of an offset between the axis of rotation of the shaper and the face gear, which is equivalent to the compensation of an error E , the

FIGURE 8.35 Evolution of maximum contact stresses on (a) the pinion and (b) the face-gear tooth surfaces for a torque varying from 500 Nm to 2500 Nm applied to the pinion shaft after compensation of errors of alignment due to shaft deflections.

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Dudley’s Handbook of Practical Gear Design and Manufacture

minimum distance between shafts. The second one is based on the application of a shaper with 28 teeth to increase the longitudinal crowning applied to the tooth surface of the face gear and therefore to make the face gearset less sensitive to errors of alignment. 8.4.3.1.1 Alternative Method 1: Application of Offset between the Axis of the Shaper and the Face Gear Figure 8.37 shows the comparison of the compensated geometry of the tooth surface of the face gear after the third iteration of compensation (see Table 8.10) with respect to the standard geometry achieved with the generation of the face gear by a shaper with 26 teeth and no errors of alignment compensated. Notice that the deviations obtained after compensation of errors of alignment caused by shaft deflections show a longitudinal deviation pattern with a maximum of 140 m at the inner radius of the tooth surface of the face gear. It is equivalent to the application of a small left hand helix angle to the tooth surfaces of the face gear. Figure 8.38 shows the schematic representation of a deviated shaper for generation of the face gear in order to achieve the desired deviation max on the tooth surfaces of the face gear. This deviation is equivalent to the compensation of only an error E of minimum distance between the shafts of the shaper and the face gear. If the face width of the face gear is denoted by Fw2 and the required deviation of the tooth surface of the face-gear is max at the inner radius (point A2 in Figure 8.38), the shaft of the shaper has to be rotated around point A1 of an angle given by: = arctan(

max

Fw2

)

(8.42)

Point A1 is located at the outer radius of the face gear denoted by L 2. Therefore, the applied error E , as shown in Figure 8.38, can be obtained as: E = L 2 tan

= L2 (

max

Fw2

)

(8.43)

Considering the data shown in Table 8.8 for the face gear and a desired deviation at the inner radius of the face gear of 140 m as shown in Figure 8.37, the error E to be compensated is of 0.89 mm. For the convenience of identifying different designs, the reference design with all errors compensated, that is to say the face gear with compensated geometry for E = 1.2923 mm, A1 = 0.0188 mm, A2 = 0.4917 = 0.0304°, will be referred to as Case 0. The design in which only an error E = 0.89 mm mm, and is compensated will be referred to as Case 1. Later, the case of design in which the face gear is generated by a shaper with 28 teeth and without errors of alignment compensated during generation will be referred to as Case 2. After compensating an error E = 0.89 mm during generation of the face gear, the deviations ob­ tained with respect to the standard geometry of the face gear achieved with the generation by a shaper with 26 teeth and no errors of alignment compensated are shown in Figure 8.39. The maximum deviation obtained at the inner radius is of 0.140 mm as desired. Figure 8.40 shows the contact patterns on the tooth surfaces of the face gear for Case 1 with different torques applied to the pinion shaft varying between 0 Nm and 2500 Nm. The shift of the contact pattern for Case 1 is similar to that of Case 0 shown in Figure 8.34 and goes from the inner radius to the outer radius as the torque increases. In this case, we can observe that the final contact

Compensation of Shaft Deflections

475

FIGURE 8.36 Field of von Mises equivalent stress on the face gear after compensation of errors of alignment caused by shaft deflection for different values of torque applied to the pinion shaft.

476

FIGURE 8.37 with 26 teeth.

Dudley’s Handbook of Practical Gear Design and Manufacture

Comparison of compensated geometry of face gear and standard geometry generated by a shaper

FIGURE 8.38 Schematic representation of the deviation of the shaper axis during generation of the face gear to achieve the desired deviation max on the face-gear tooth surface.

Compensation of Shaft Deflections

477

pattern is extended all over the tooth surface of the face gear and with it, a reduction of contact and bending stresses is expected. Figures 8.41 and 8.42 show the variation of the maximum contact and bending stresses, re­ spectively, on the pinion (Figures 8.41a and 8.42a) and face gear tooth surfaces (Figures 8.41b and 8.42b) considering a torque of 2500 Nm applied to the pinion shaft. We recall that the pinion for all cases of design have the same geometry and micro-geometry modifications. Case 1 reduces the maximum contact stress on the pinion tooth surfaces from 672 MPa (Case 0) to 572 MPa (-14.9%) and from 592 MPa (Case 0) to 548 MPa on the face gear tooth surfaces (-7.4%). Regarding bending stresses, a reduction from 130 MPa to 114 MPa was achieved for the face gear. The obtained results show that compensation of only an error E to achieve the desired deviations on the face-gear tooth surfaces is a very effective way to prepare face-gear drives to absorb errors of alignment caused by shaft deflections. 8.4.3.1.2 Alternative Method 2: Application of a Shaper with 28 Teeth and No Compensations of Errors of Alignment Here, the effect of applying a shaper with 28 teeth (three more than the mating pinion) on the capacity of the face gear to absorb errors of alignment caused by shaft deflections is studied. In this case of design, none other compensation for errors of alignment is applied. As mentioned above, this case of design will be referred to as Case 2. Figure 8.43 shows the contact patterns on the face-gear tooth surfaces for this case of design. Because of the larger crowning of the face-gear tooth surfaces, the contact patterns are centered on the face-gear tooth surfaces for low values of the torque and shift toward the outer radius when the torque is increased. Due to the higher values of crowning applied to the face gear tooth surfaces, the contact patterns extend over a smaller area of the surface than those achieved after

FIGURE 8.39 Comparison of geometry of the face gear with an error E = 0.89 mm compensated during gen­ eration and the standard geometry generated by a shaper with 26 teeth and no errors of alignment compensated.

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Dudley’s Handbook of Practical Gear Design and Manufacture

FIGURE 8.40 Contact patterns on the face-gear tooth surfaces with compensation for E = 0.89 mm for a torque applied to the pinion shaft varying from 0 Nm to 2500 Nm.

FIGURE 8.41 Evolution of the maximum contact stress on (a) the pinion and (b) the face-gear tooth surfaces when the face gear is generated in different ways and a torque of 2500 Nm is applied to the pinion shaft.

compensation of errors of alignment (Figures 8.34 and 8.40). In terms of stresses, Figures 8.41 and 8.42 also show the variations of the maximum contact and bending stresses, respectively, on the pinion (left) and face gear tooth surfaces (right) for this case of design (Case 2) considering a torque of 2500 Nm applied to the pinion shaft. The maximum contact stress experiences an increment from 672 MPa (Case 0) to 763 MPa for the pinion and from 592 MPa (Case 0) to 813 MPa for the face gear. The maximum bending stress increases from 147 MPa (Case 0) to 255 MPa for the pinion and from 130 MPa (Case 0) to 186 MPa for the face gear. Therefore, increasing the crowning of the face gear to make the gearset less

Compensation of Shaft Deflections

479

FIGURE 8.42 Variation of maximum bending stress for (a) the pinion fillet surfaces and (b) the face-gear fillet surfaces when the face gear is generated in different ways and a torque of 2500 Nm is applied to the pinion shaft.

FIGURE 8.43 Contact patterns on the face gear tooth surfaces generated by a shaper with 28 teeth when a torque from 0 Nm to 2500 Nm is applied to the pinion shaft.

sensitive to errors of alignment, which is the current practice of design, although effective, yields larger values of contact and bending stresses and therefore limits the maximum torque that the face gearset can safely transmit.

REFERENCES [1] Litvin F. L., & Fuentes-Aznar A. (2004). Gear geometry and applied theory (2nd Ed.). Cambridge University Press. [2] Argyris J., Fuentes-Aznar A., & Litvin F. L. (2002). Computerized integrated approach for design and stress analysis of spiral bevel gears. Computer Methods in Applied Mechanics and Engineering, 191(11), 1057–1095.

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Dudley’s Handbook of Practical Gear Design and Manufacture

[3] Roda-Casanova V., Iserte-Vilar J. L., Sanchez-Marin F., Fuentes-Aznar A., & Gonzalez-Perez I. (2011). Development and comparison of shaft-gear models for the computation of gear misalignments due to power transmission. Vol. 8 of International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, 11th International Power Transmission and Gearing Conference, pp. 279–287. [4] ABAQUS/Standard User's Manual. Providence, Rhode Island 02909-2499 (US), 2019. [5] Fuentes-Aznar A., Ruiz-Orzaez R., & Gonzalez-Perez I. (2018). Compensation of errors of alignment caused by shaft deflections in spiral bevel gear drives. (pp 301–319). Springer International Publishing. [6] Guo, H., & Fuentes-Aznar A. (2020). Compensation of errors of alignment caused by shaft deflections in face-gear drives generated by shaper cutters. Mechanism and Machine Theory, 144, 103667. [7] Zanzi C. & Pedrero J. I.(2005). Application of modified geometry of face gear drive. Computer Methods in Applied Mechanics and Engineering, 194,3047– 3066. [8] Litvin F. L., Fuentes-Aznar A., Zanzi C., Pontiggia M., & Handschuh R. F.(2002). Face-gear drive with spur involute pinion: Geometry, generation by a worm, stress analysis. Computer Methods in Applied Mechanics and Engineering, 191(25), 2785–2813.

9

Special Design Problems in Gear Drives Boris M. Klebanov

9.1

SPECIAL CALCULATIONS OF INVOLUTE GEAR GEOMETRY

9.1.1 MAIN SYMBOLS

AND

DEFINITIONS

mn = normal module (mm); z1, z2 = numbers of teeth of the pinion and the gear, respectively; β = helix angle on the reference diameter d; d = reference diameter (mm); d=

mn z ; cos

da = tip diameter (mm); αP = pressure angle of the standard basic rack tooth profile; x = profile shift coefficient; Notes: 1. x∙mn is the radial displacement of the basic rack from the position where its reference line is tangent to the reference circle d. According to ISO 21771, this displacement is given sign “+” when it increases the tooth thickness on the reference diameter. That means, for external teeth x is positive when the basic rack (that is, the hob or shaper cutter) is displaced outward from the gear center, and for internal teeth x is positive when the cutter is displaced inward, toward the gear center. The displacements in opposite directions have sign “–”. 2. According to ISO 21771, for internal gearings the gear tooth number z2 has sign “–”. b = face width of a gear (mm). Derived quantities: mt = transverse module (mm):

mt =

mn cos

;

αt = transverse pressure angle at the point of intersection of the involute profile with the reference circle d: tan

t

=

tan cos

P

;

db = base circle diameter (mm): db = mt z cos

t

= d cos t ; 481

482

Dudley’s Handbook of Practical Gear Design and Manufacture

pbt = transverse base pitch: pbt =

db ; z

αn = normal pressure angle at the reference cylinder: tan

n

= tan

t

cos = tan P; hence,

n

=

P;

αyn = normal pressure angle at the Y-cylinder (whose diameter equals dy): tan

yn

= tan

yt

cos y ;

αyt = transverse pressure angle at the Y-cylinder: cos

yt

=

db d = cos t ; dy dy

βb = helix angle on the base cylinder: tan

b

=

db tan = cos d

t

tan .

The tooth profile in involute gears is the involute of the base circle. Figure 9.1, where 1 is the base circle and 2 is its involute, demonstrates the features of the involute. The most important feature is that in each point of the involute, the normal to it is tangent to the base circle. The length of arc ab of the base circle equals the length of line segment bc. Angle θ is denoted “invα” and called involute function. It is one of the parameters used often in the gear geometry calculations. From Figure 9.1 we can see that θ = φ – α. Since: =

ab bc = = tan , rad , 0.5 db 0.5 db

c 2 1

a

b

θ

db

o

FIGURE 9.1 Involute of a circle.

Special Design Problems in Gear Drives

483

0.20

inv =tan -

0.15

0.10

0.05

0 o 15

FIGURE 9.2

20o

25o

30o

35o

40o

45o

Involutes.

= inv

= tan

, rad .

Angle \α for any point c on the involute is given by: cos =

0.5 db . oc

If angle α is known, involute function θ is easily calculated, though it can be found using program “Involute Function Calculator” on the Internet. Vice versa, if the θ value is known, angle α can be determined by iterations taking the initial value from Figure 9.2. Much quicker the exact α value can be found using program “Inverse of Involute Function Calculator” from the Internet, if it is available. The length L of involute from start point a to any point c is given by: L=

rb 2

2

=

rb (tan )2 , 2

(9.2)

where rb = 0.5db.

9.1.2 TOOTH UNDERCUTTING

IN

EXTERNAL GEARS

The involute profiles of hobbed gear teeth are formed in the generating process by the linear cutting profiles of the hob teeth. However, the involute cannot be generated inside its base circle. Figure 9.3a presents the meshing of a hob with the cut gear in the position of minimum profile shift with no tooth undercutting. Here line hh that passes through the ends of the straight-line segments of the hob cutting profiles passes through point a. Line ab is the line of action in the engagement of the cutting rack with the cut gear. This line is normal to the rack linear profile and tangent to the base circle db of the gear. Point a is the point of tangency of line ab with the base circle. If the hob enters deeper so that line hh passes below point a as shown in Figure 9.3b, the teeth become undercut. The undercutting makes the teeth thinner in the root (because fillet f increases) and also cuts away part of the involute profile denoted “ip.” Therefore, if the undercutting cannot be avoided, the location of the end point of the involute profile u, i.e., diameter du, see Figure 9.3b, should be determined

484

(a)

Dudley’s Handbook of Practical Gear Design and Manufacture rack or hob

reference line

90°

b h

haP

a

P

f

t

da

h

d

ip

db O

(b)

reference line

b h

h

a

P

u f

t

da

db du

ip

FIGURE 9.3

Undercutting of hobbed teeth.

to calculate the contact ratio εα. If diameter du is less than the Start of Active Profile (SAP) diameter dNf, the contact ratio is not influenced by the undercutting. Diameter du, as well as the shape of the tooth root, is obtained usually by graphical simulation of the generating process in the computer. Tooth undercutting is not something definitely forbidden. Sometimes the designer is forced to decrease the gear diameter, so he uses negative profile shift and accept the undercutting provided that both the contact ratio and strength of the teeth meet the requirements. Sometimes undercutting is used to provide an outlet for the grinding disc. However, the designers mostly try to avoid undercutting. From Figure 9.3a, the minimal number of teeth that has no undercutting without the profile shift (x = 0) can be calculated. Taking into account that at x = 0 the reference line of the rack is tangent to the reference diameter d of the gear, we obtain the following relations for a spur gear from ΔOaP: OP =

d m z d m z = n ; Oa = b = n cos P; 2 2 2 2

OP = Oa cos

P

=

mn z 2 sin 2

P

= haP ;

From here: z=

2haP m n sin2

. P

At haP = mn and αP = 200, z = 17.1 ≈ 17. This is the minimal tooth number of a spur gear with no profile shift that can be hobbed virtually without undercutting. The undercutting can be avoided by increasing the distance between the hob and the gear axis, i.e., by a positive profile shift, to move the hh line above point a. The minimal profile shift coefficient xmin required to avoid the tooth undercutting in spur and helical gears equals:

Special Design Problems in Gear Drives

485

1.0 0.5 0 -0.5

0o

-1.0

12o 20o

-1.5

30o 35o

-2.0

40o 45o

xmin

-2.5

5

10

15

20

25

30

35

40

z

FIGURE 9.4

Minimal profile shift coefficients without undercutting.

xmin =

haP cos mn

0.5 zsin2 t .

For example, if z = 10, β = 120, haP = mn, αP = 200, then: tan

t

=

tan20 o = 0.372102; cos12o

xmin = 1 cos 12o

t

= 20.41o ;

0.5 10 sin2 20.41o = 0.37.

Diagrams in Figure 9.4 enable the quick estimation of the xmin value. The undercutting of teeth generated by a shaper cutter occurs if point a on the base diameter db of the gear is located within the tip circle da0 of the cutter (Figure 9.5a). This effect can be avoided by a positive profile shift (Figure 9.5b) or using a cutter with a smaller number of teeth, i.e., a smaller tip diameter. (Notes: 1. In Figure 9.5, diameters related to the cutter are provided with subscript “0.” 2. Lines of action aa0 are tangent to the base circles of both the cutter and the cut gear.)

9.1.3 TOOTH TIP THICKNESS

AND

TOOTH POINTING

IN

EXTERNAL GEARS

Tooth pointing (Figure 9.6) is the critical thinning of the tooth top land when it becomes sharp. It is usually unacceptable. For strength and wear reasons, the normal thickness on the tooth tip san should not be usually less than (0.3–0.4)mn. The tooth thickness in the transversal plane sat on the tip diameter da equals: sat = da

where αat is given by:

2z

+

2x tan z

n

+ inv

t

inv

at

(9.3)

486

Dudley’s Handbook of Practical Gear Design and Manufacture

(a)

(b) da0 db0

shaper cutter

shaper cutter

da0

db0 a0

a0

u a

da

t

t db

db du

gear

FIGURE 9.5

a

gear

da

Tooth undercutting with a shaper cutter. Sat Sdt

dsharp

p shar

da d db

FIGURE 9.6 Tooth thickness and tooth pointing.

cos

at

=

db . da

The true thickness san (in normal section of the tooth) equals: san = sat cos

a,

where βa = helix angle on the outer diameter da: tan

a

=

da tan . d

If the sat value is negative, that means that the right and the left profiles of the tooth intersect before they reach the given diameter da. The diameter where the profiles intersect can be obtained from formula: dsharp =

db cos

sharp

(9.4)

Special Design Problems in Gear Drives

487

where the pressure angle on the pointed tip αsharp is obtained from Eq. (9.3) assumed that the bracketed expression equals 0: inv

sharp

=

2z

+

2 x tan z

n

+ inv

t.

(9.5)

For the first approximation, tables of involutes or the diagram in Figure 9.2 can be used, and then the higher accuracy of the αsharp value can be achieved by iterations. Certainly, the computer program “Inverse of Involute Function Calculator” is the best tool if available. Example 9.1: Helical gear z = 10, β = 120, mn = 5 mm, αP = 200, x = 0.37, da = 64.5 mm. What is the tip thickness of the tooth? tan

mt =

t

=

tan 20 o = 0.372102; cos 12o

= 20.4103350 = 0.356227 rad .

t

inv

t

= 0.372102

0.356227 = 0.015875 rad .

5 = 5.111703; d = mt z = 51.117030 mm; cos 12o

db = mt z cos

t

= 5.111703 10 cos 20.410335o = 47.907857 mm.

cos

at

at

inv

at

=

db 47.907857 = = 0.742757; da 64.5

= 42.033198o = 0.733618 rad ;

= 0.901454

0.733618 = 0.167836 rad .

Now, from Eq. (9.3) we obtain the tooth tip thickness in transversal plane: sat = 64.5

2 10 tan

+

a

2 0.37 0.363970 + 0.015875 10

da 64.5 = tan 12o = 0.268206; d 51.117030

= tan

a

sn = sat cos

0.167836 = 2.067 mm ;

a

= 15.0137270 ;

= 2.067 cos15.013727o = 1.996 mm .

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Dudley’s Handbook of Practical Gear Design and Manufacture

Note: The tooth thickness sat is calculated along the arc of the tip circle but it is measured over a chord. This refinement is usually needless because the difference in numerical values is negligible. In this example, sat = 2.067 mm along the da circle, and its chord dimension is shorter by 0.3 μm only. Let us calculate the diameter where the teeth become sharp. From Eq. (9.5): inv

sharp

=

2 10

+

2 0.37 0.363970 + 0.015875 = 0.199888. 10

As a first approximation, from Figure 9.2, αsharp ≈ 44.10. Using “Inverse of Involute Function Calculator” we obtain αsharp ≈ 44.1311300. Then, from Eq. (9.4): dsharp =

9.1.4 INTERFERENCE

OF

PROFILES

IN

47.907857 = 66.747 mm. cos 44.131130 o

EXTERNAL GEARING

The tip-to-root interference may occur between the tooth tip of one gear and the root fillet of the mating gear tooth. The idea is illustrated in Figure 9.7a where the main elements of external mesh geometry are represented: base circles db1 and db2, tip circles da1 and da2, and line of action a1a2 that is tangent to the base circles. Since the teeth can interact properly with their involute profiles only, the manufacturing process must ensure that the SAP (Start of Active Profile) diameters in the engagement with the cutter

(a)

(b)

O1

rack or hob b2 bc

h

db1

da1

dNf1 p

a1

bt

b2

dNfC

da2

aw

h

a2

b1

db2

tC

a2

(c) da2

dNf2

db0 ac b 2

wt

db2

da0 shaper cutter bc a2 dNfC

da2 O2

FIGURE 9.7

Determination of the required SAP diameters in the external gearing.

tC

db2

Special Design Problems in Gear Drives

489

(dNfC) are not bigger than SAP diameters dNf required by the gearing geometry, i.e., dNfC1 ≤ dNf1 = 2O1b2 and dNfC2 ≤ dNf2 = 2O2b1. In other words, the SAP point of one tooth must be unreachable for the mating tooth because it is located beyond the tip circle of the mating tooth. Figure 9.7b and c show the formation of the involute profiles in the generating process of hobbed and shaped gears, respectively. (Here letter “C” is added to the values related to meshing with the cutter.) Often the designer includes the required dNf value in the gear drawing for the profile control. Diameters dNf1 and dNf2 are easy to calculate from Figure 9.7a: a1 b2 = a1 a2

a2 b2; a2 b1 = a1 a2

a1 b1;

dNf 1 = 2O1 b 2 =

db21 + 4a1 b22 ;

dNf 2 = 2O2 b1 =

db22 + 4 a2 b12 .

Here: a1 a2 = a1 b1 = 0.5 da21

a w2

0.25(db1 + db2 )2 ;

db21 ;

a2 b 2 = 0.5 da22

b1 b2 = a1 b1 + a2 b2

db22 ;

a1 a2.

Distance b1b2 where the mating teeth can be in contact is called length of action. Dividing it by transverse base pitch pbt = π∙mt∙cosαt, we obtain the transverse contact ratio:

=

b1 b2 . pbt

For this case, the tip diameters da1 and da2 shall be reduced by the amount of tooth edge rounding or chamfering.

9.1.5 INTERFERENCE

OF

PROFILES

IN INTERNAL

GEARING

The situation in the internal gearing is similar, see Figure 9.8. Here are shown: base circles db1 and db2, tip circles da1 and da2, and line of action a1a2 that is tangent to both base circles. The contact of teeth occurs within segment b1b2 bordered by the tip circles. Points b1 and b2 are the points of intersection of circles da1 and da2 with the extension of line a1a2. Distance b1b2 is the length of action where the teeth may contact properly. Circle da2 must not intersect with the line of action between points a1 and a2; i.e., da2 must be anyway greater than 2O2a1, where O2 a1 =

(0.5db2 )2 + (a w sin

wt )

2

.

However, to avoid the interference of the gear tooth tip with the non-involute part of the pinion tooth profile, the following condition must be fulfilled:

490

Dudley’s Handbook of Practical Gear Design and Manufacture b1 b2 dNf2 dNf1

a2

a1

db1 O1

db2

da1

d a2

wt

aw

O2 FIGURE 9.8

Determination of the required SAP diameters in the internal gearing. 2

da 2

2 (0.5db2 )2 + a w sin

wt

+

min

.

Here ρmin = minimal radius of curvature of the involute profile of the pinion tooth. To prevent the interference of the pinion tooth tip into the root fillet of the gear, the SAP diameter of the internally toothed gear (dNf02) obtained in the gear cutting process must not be less than dNf2 = O2b1. The SAP diameter of the pinion dNf01 obtained in the tooth cutting process must be no bigger than dNf1 = O1b2. The values of O1b2 and O2b1can be calculated on the basis of geometry presented in Figure 9.8. The interference of tips of the teeth may occur in the internal gearings with a small difference z2 – z1 (see Figure 9.9). The minimal values of z2 – z1 for spur gears with standard basic rack (αP = 200, haP = 1) and x1 = x2 = 0 are given in Figure 9.10 [1]. The difference z2 – z1 can be reduced by using profile shift x

d1

interference d2 z1=50 z2=56

FIGURE 9.9

Tip-to-tip interference.

Special Design Problems in Gear Drives

491

7

(z2

z1)min

6 5 4 3 2 40

FIGURE 9.10

60

80

100

120 z2

140

160

180

200

Minimum difference (z2 – z1) which prevents the interference of tips in internal gearing [ 1].

directed outward from the gear center, increased pressure angles αP of the basic rack, or reduced ad­ dendum height of the teeth. For such cases, the analytical detection of whether or not the tip-to-tip interference exists can be made on the following considerations, see Figure 9.11 [2]. As the pinion turns by angle φ1, its tooth 1 comes to an intersection with the tip circle of the ring gear (position 1*). At the same time, the ring gear turns by angle φ2, and its tooth 2 takes up position 2*. If tooth 2 goes ahead of tooth 1 as shown in the draft, the interference of tips cannot occur. The mathematical expression of this dependence is given by the following Equation: 2

(

1

+ inv

at1

inv

wt )

z1 + inv z2

wt

inv

at 2,

rad .

(9.6)

Angles Φ1, Φ2, αat1, αat2 and αwt are determined from the following equations:

2*

2

1*

cos

1 =

cos

2

1 a1 1 2

=

a w2

(0.5da2 )2

(0.5da1)2 da1 a w

(0.5da2 )2

(0.5da1)2 + a w2 ; da 2 a w

;

da1 a2

2 wt

1

db1 da2

O1

aw

O2

db2

FIGURE 9.11 The conditions for the interference of tips [ 2].

492

Dudley’s Handbook of Practical Gear Design and Manufacture

cos

at1

=

cos

db1 ; cos da1

wt

=

at 2

=

db 2 ; da 2

db2 db1 . 2 aw

Here αwt = working pressure angle of a gear pair; aw = center distance (mm); z1 and z2 = numbers of teeth of the pinion and the ring gear, respectively (positive numbers); db1 and db2 = base diameters of the gears (mm); da1 and da2 = tip diameters of the gears (mm). Example 9.2: z1 = 116, z2 = 120, the basic profile is characterized by angle αP = 200 and addendum height haP = mn; the profile shifts x1 = x2 = 0, β = 0 (spur gears). Is this mesh problemfree? Since the value of the module makes no difference, we take mn = 1 mm, then the dimensions of gears are as follows: da2 = 1 (120

2) = 118 mm;

da1 = 1 (116 + 2) = 118 mm ; db2 = 120 cos 20 o = 112.763 mm; db1 = 116 cos 20 o = 109.004 mm; aw =

cos

at 2

116 2

=

= 2 mm;

112.763 = 0.955619; 118

= 17.133875o = 0.299043 rad;

cos

at1

at2

120

at1

=

109.004 = 0.923763; 118

= 22.517441o = 0.393003 rad ;

inv

at2

= tan 0.299043

0.299042 = 0.009244;

inv

at1

= tan 0.393003

0.393003 = 0.021567;

inv

cos

2

=

P

= tan20 o

(0.5 118)2

20

180

= 0.014904;

(0.5 118)2 + 22 = 0.016949; 118 2

Special Design Problems in Gear Drives 2

cos

1

=

Since there is no profile shift,

493

= 89.028847o = 1.553847 rad ;

0.016949;

wt

=

P

1

= 90.971153o = 1.587746 rad .

= 20 o; inv 200 = 0.014904 rad .

Now we revert to Eq. (9.4): (1.587746 + 0.021567

0.014904)

116 + 0.014904 120

0.009244 = 1.546922
da, we have to increase the pin diameter, for instance, to DM = 8 mm. From Eq. (9.13), with αt = αP = 200, we have: inv

Kt

=

8 + 0.014904 + 51 5 cos 20 o Kt

+ 4 0.2 tan20 o = 0.015173; 2 51

= 20.115281o ;

From Eq. (9.12): dK = 255

From Eq. 9.15:

cos 20 o = 255.187395 mm ; cos 20.115281o

Special Design Problems in Gear Drives

503

MdK = 255.187395 cos

90 o 51

8 = 247.066365 mm.

The pins project beyond the tip diameter which is 249 mm. To determine the location of contact point P (see Figure 9.16b) on the tooth profile we shall determine radius RP: From the triangle acO: ac = 0.5

dK2

db2 = 0.5

255.1872

239.6222 = 43.880 mm;

aP = ac + Pc = 43.880 + 4 = 47.880 mm;

From the triangle aPO: RP =

(0.5 db )2 + aP2 =

119.8112 + 47.880 2 = 127.594 mm .

Thus, the height hP of point P from the tooth top equals: hP = 0.5 249

127.594 = 3.09 mm.

9.1.6.4 Measurements of Teeth with Profile Modifications The formulae given above for the tooth thickness measurements are valid on condition that the mea­ suring tool contacts with the non-modified involute profile of the tooth. It is important, then, to know the location of start point St of the profile modification and take into account that in reality this location is not exact and may be somewhat different on different teeth of the same gear if the profile modification is machined (ground) separately of the main profile. Therefore, for the sake of definiteness, the point of contact (POC) with the measuring tool should be chosen far enough from the theoretical St location. When the tooth thickness is measured by a vernier tooth caliper or using pins (balls), the location of POC can always be chosen as required by using a pin (ball) of proper diameter. However, when the base tangent length is measured, we can only change the number of spanned teeth by a whole number. In this case, the measurement over the modified profiles may be unavoidable, especially when the modification is long and covers about а half of the tooth height. The most widespread tooth profile modification is the linear tip relief that has an involute shape. Figure 9.17a presents the main involute profile 1 of the tooth with its base circle 2. Involute 3 of smaller base circle 4 is so chosen as to intersect with the main profile at point St on a specified radius rs and remove from the main profile a thin wedge-shaped layer of maximum thickness Cαa at the tooth edge. The modification involute is characterized by pressure angle of its basic rack P which is slightly greater than αP—usually by not more than 10…1020'. Figure 9.17b shows the profile of a hob intended for cutting teeth with profile modification. Here 5 is the standard profile of the basic rack, and 6 is the modification. Dimension A is obtained from geometry calculations of the meshing of the gear with the hob: A is the distance between the point of intersection of circle rst with the line of action in this meshing and the hob reference line. As soon as the angle of modification involute P and the tooth tip thickness after modification sat (see Figure 9.17a) are determined, the base tangent length measured over the modified parts of tooth flanks can be calculated as shown in Figure 9.18, where the teeth are drawn as if their profiles are cut entirely with a hob with pressure angle P . Here: db = mt z cos

t

;

504

Dudley’s Handbook of Practical Gear Design and Manufacture

(a)

(b)

S*at C

1 3

*

a

reference line

P

St

Sat

6 P

A

rst 2 4

FIGURE 9.17

nmn/2

5

Formation of a linear tip relief.

Wk*

* Sat

da

d*b

B

A 1

a

b

*at

2 3

FIGURE 9.18

Calculation of the span length measured over the modified parts of tooth profiles.

mn ; tan cos

mt =

t

=

tan cos

P

.

The base tangent length Wkt in the transversal plane equals the length of arc ab that, in turn, equals: Wkt = AB = ab = 0.5 db

As one can see from Figure 9.18, =

1

=

1

+ 2(

2 (k z

2

+

1), rad

where k = number of spanned teeth. 2

=

3);

sat , rad . da

.

(9.16)

Special Design Problems in Gear Drives

3

505

= inv

at ;

cos

at

=

db . da

After the Wkt value is found from Eq. (9.16), it shall be multiplied by cos tangent length Wk that is measured in the direction normal to the teeth. tan

b

= tan

b

to obtain the real base

db . d

9.1.7 PROFILE MODIFICATION 9.1.7.1 Basics Figure 9.19a presents a gear pair in a position where teeth 1 and 1* transmit the torque between the gears, while teeth 2 and 2* are about to touch each other and start their participation in the load transmission. This ideal situation is possible if the base pitches of gears are equal. However, what if the pitches are slightly different? Imagine that the base pitch of driving teeth 1-2 is greater than that of driven teeth 1*-2*. In this case, there is formed a gap between teeth 2 and 2*, and they, in fact, do not participate in transmitting load until pair 1-1* disengages as the gears rotate. From this point on, teeth 2 and 2* close the gap with an impact under the influence of torques applied to the shafts of the gears. This is called mid-impact. If the base pitch of driving teeth 1-2 is smaller than that of driven teeth 1*-2*, then the profiles of teeth 2 and 2* intersect as shown in Figure 9.19b. In reality, the profiles cannot intersect, just teeth 2 and 2* come into contact prematurely, before they reach the line of action, with an impact (called edge impact) that is particularly dangerous because it is applied to the very tip of the driven tooth 2* and causes a sharp peak of bending stress. Since this tooth enters into contact with its edge (corner contact), this may cause local plastic deformation and scuffing. Therefore, it is important to round-off and polish the sharp edges, but first to avoid the edge impact. The situation when the driving gear pitch is smaller than that of the driven gear can be caused not only by a manufacturing error that can be kept very small, but also due to elastic deflections of the teeth under load that are unavoidable. Figures 9.20a and b demonstrate the deflection of the teeth (the initial shape of

(a) driving 2

1

line of action 2*

1*

driven

(b) 2 1 1*

line of action 2*

FIGURE 9.19

Teeth engagement.

506

Dudley’s Handbook of Practical Gear Design and Manufacture

(a) driver 2

1

(b) 1*

2* driven

FIGURE 9.20

Elastic deformation of teeth under load.

the loaded tooth is shown with dashed lines). It is obvious that under load the distance between teeth 1 and 2 shortens, and between teeth 1* and 2* grows. Thus, the deformations of teeth change their pitch: under load, the pitch of the teeth to come into contact decreases on the driving gear and rises on the driven gear. This difference in tooth spacing between the engaging teeth disturbs the engagement and requires tooth profile modifications to avoid the edge impacts of teeth. To prevent the interference of teeth 2 and 2* shown in Figure 9.19b, part of their profile in this place must be removed. It is easier to remove material from the upper side of the driven tooth, and this profile modification is called tip relief. Another kind of corner contact occurs when the loaded tooth pair disengages (see Figure 9.21). During the disengagement of pair 1-1*, the growing load and deformation of the following pair 2-2* increases the distance between teeth 1 and 2, while the distance between teeth 1* and 2* shortens. Therefore, pair 1-1* does not really disengage at point b2 where the tip diameter of the driving gear crosses the line of action, but continues being in corner contact beyond this point. To avoid this corner contact, the tip relief of the driving tooth is also necessary, though in lesser amount. It is obvious that the amount of tip relief should be proportional to the deflection of the teeth under load, which equals tooth force Fn (N per mm of face width) divided by tooth stiffness cʹ.(N/(mm∙μm). It is recommended [1] for the gears with thick rim (say, SR > 4mn) to take the stiffness of a gear pair cʹ = 12.5 N/(mm∙μm) on the ends of contact and 18 N/(mm∙μm) in the middle of contact. Experimentally, the stiffness of a standard tooth pair with m = 4.5 mm, z1/z2 = 16/23 and x1/x2 = 0.44/0.51 was found changing nearly parabolically between about 12.5 N/(mm·µm) on the ends of contact and 17 N/(mm·µm) in the middle of contact [3]. The stiffness of high contact ratio (HCR) tooth pair with m = 3.738 mm, z1 = z2 = 54, α = 180 and addendum height ha = 1.4 m was also found changing parabolically between 8.7 N/ (mm·µm) on the ends of contact and 14.3 N/(mm·µm) in the middle of contact [4]. Detailed re­ commendations on calculation of tooth pair stiffness are given in ISO 6336-1: 2006, Section 9. The most exact results can be obtained by using FEM. driving 2

1 1* b2 line of action

2* driven FIGURE 9.21 Disengagement of loaded teeth.

Special Design Problems in Gear Drives

507

Gears with small unit loading and running-in ability may work well without profile modification if the accuracy of the teeth geometry is high enough. For instance, if a gear with tooth module 3 mm is made of tempered steel and loaded with a force of 50 N per mm of face width, the deflection of a tooth pair, with the stiffness cʹ ≈ 17 N/(mm·µm), equals 50/17 = 2.9 µm. Natural tip relief due to the running-in wear can compensate for this deformation. However, if a gear with case hardened teeth of module 10 mm is loaded by a force of 1000 N/mm, then the deflection of a tooth pair equals 58.8 µm, and no running-in wear can be expected there. In this case, the tip relief shall be made by machining. Usually the profile correction is made on the tooth addendums of both gears. The geometry of tip relief, as it looks on the profile control chart, is usually parabolic or linear, however, the last one is mostly used because it is the simplest in calculation and in production. The formation of linear tip relief is described in Section 9.1.6.4 and Figure 9.17. 9.1.7.2 Parameters of Linear Tip Relief Tip relief is characterized by amount Cαa and location of starting point st (Hr or Lr in Figure 9.22a). There is a kind of consensus concerning the tip relief magnitude Cαa: it should equal to the tooth pair deflection f under load plus the base pitch errors of both gears [5]. About the location of the starting point the offers are different. The problem is that, in principle, the ideal tip relief can be designed for a certain torque only. For the lower or higher torque, the tip relief would be too big or too small, and in both cases, it may lead to increased vibrations. The oldest offer valid for the normal contact ratio gears (NCR, with εα = 1.5 – 1.7) is called short tip relief, see Figure 9.22b. Here the non-modified parts of involute profiles provide contact ratio εα = 1, so that even at low load the gears rotate smoothly. The long tip relief [3] starts at each gear at the upper point of single tooth contact (STC) area (Figure 9.22c). This kind of profile modification gives the minimal transmission error (TE) under design load, and the authors present the results of experiments that approve high effectiveness of long relief when the drive works under this load. However, they warn about increased noise when the load is smaller or greater than the design load, because at a smaller load the profile relief appears too big, so that part of the relieved surface does not participate in the load transmission, and at a greater load the tip relief becomes insufficient. Therefore, the designers often prefer using the short relief, though its ad­ vantage over the long relief at low loads becomes apparent only at loads below 10–15% of the design load. Experiments with tip reliefs of the same amount and different extents showed [6,7] that the best results over the range of loads was achieved using tip relief of intermediate extent, see Figure 9.22d. Here the length of action b1b2 is divided into three equal parts, and the midsection belongs to unmodified C a

gear 1 b2

Lr

s1

line of action

Hr

s2

b1

pbt

st b

gear 2

s1

s2

short relief` pbt LCa1

c pbt gear 1 tip relief a

d

LCa2

long relief " " " s1 s2 intermediate relief

C a gear 2 tip relief

FIGURE 9.22 Extension of linear tip relief. b, c, d—profile control charts for different extensions of tip reliefs; slanting lines on the left and on the right sides of the charts are the tip reliefs for gear 1 and gear 2, respectively.

508

Dudley’s Handbook of Practical Gear Design and Manufacture

involute profiles, while the other two parts are associated with relieved tips. Close to the intermediate relief is the recommendation to make the length of the unmodified part between points s1 and s2 equal to 2/3pbt symmetrically relative to the pitch point [5]. For smooth run of the gears it is important to make the gradient of the tip relief very small, the angle of the “wedge” should not exceed 10-1020'. Hence, there is dependence between the amount of tip relief Cαa and the length Lr: Ca Lr

0.01745 … 0.02327 rad .

(9.17)

The dependence between length Lr on the tooth and roll length LCа on the profile chart can be obtained from Figure 9.23 as follows: • The length of involute between points b and st equals (see Eq. (9.2)): Lr =

rb [(tan a )2 2

tan

Here

a=

st

=

(tan

st )

2 ] mm .

2L r . rb

(tan a )2

(9.18)

arcos(rb/ra) = pressure angle on the end of the effective tooth profile. rst =

rb cos

.

(9.19)

st

b Lr

LCa

of line n i t ac o

s

st

a 90° rb

ra

rst

a

O

FIGURE 9.23

Extension of tip relief: on the tooth (Lr) and on the line of action (roll length LCa).

Special Design Problems in Gear Drives

509

LCa = bs = ab

as ;

(9.20)

ab =

ra2

rb2 ;

(9.21)

as =

rst2

rb2 .

(9.22)

• From here we obtain: • Now we can determine radius rst: • From Figure 9.23 we can see that The real angle γ between the main and the relief involutes at their intersection point st is calculated as follows (see Figure 9.24): =

,

(9.23)

= arc sin

rb ; rst

(9.24)

= arc sin

rb ; rst

(9.25)

where:

where: rb = base radius of the main involute; rb = base radius of the tip relief involute; rst = radius of location of point st. Notes. 1. The unmodified parts of profiles in normal-contact-ratio (NCR) gearing with long relief (Figure 9.22c) provide εαu = (εα – 1), i.e., εαu < 1. It does not mean that the gears disconnect like in the case of εα < 1, because the deformation of the teeth under load brings the relieved surfaces into contact. However, if the load is low, the tooth deflection becomes insufficient, the teeth lose their contact in some parts of the relieved areas, the transmission error (TE) grows, and this is evidenced in an increased noise.

St 1 2

* rst

O

rb r*b

FIGURE 9.24 Calculation of the angle between the main involute and tip relief involute at the point of their intersection. 1—base circle of the main involute, 2—base circle of the tip relief involute.

510

Dudley’s Handbook of Practical Gear Design and Manufacture e

=

KF ,

(9.26)

where: KF = (1.03 ÷ 1.05) + 4.8 10

4

Fu.

(9.27)

For example, if εα = 1.85 and Fu = 100 N/mm2, then: 2. Computer simulations of non-modified gear pairs with contact ratios εα = 1.19…2.15 showed [8] that under load, due to elastic deformations of the teeth, the contact ratio grows significantly. The value of the effective contact ratio εαe in relation to unit load Fu = Fn/m (N/mm2) can be ap­ proximated for the interval of Fu = 30…100 N/mm2 (i.e., N per mm of face width and per mm of module) as follows: e

= 1.85[(1.03 ÷ 1.05) + 4.8 10

4

100] = 1.99 ÷ 2.03

2.

The tip reliefs, if their amount and extent are optimal for a certain load, return the contact ratio at this load approximately back to the theoretical εα value. Example 9.9: Design tip relief for NCR spur gear pair with z1 = 31, z2 = 65, mn = 10 mm, αP = 200, addendum height ha = mn, profile shift coefficients x1 = x2 = 0. The geometrics of the mesh (see Figure 9.25): tip diameters: of the pinion 330 mm and of the gear wheel—670 mm. Tooth edge rounding radius r = 0.5 mm, so the effective tip radii equal ra1 = 164.5 mm and ra2 = 334.5 mm. Base radii: rb1 = 0.5·10·31·cos200 = 145.652 mm, rb2 = 0.5·10·65·cos200 = 305.400 mm. Center distance aw = 0.5·10(31 + 65) = 480 mm. Base pitch ptb = π·10·cos200 = 29.521 mm.

3. Symmetrical models of tip relief presented in Figures 9.22b–d are not always optimal because the problem of impacts in the mesh is associated mainly with the moment when the teeth come into driving O1 gear

d1

ra1 rb1

1

a1 b2 p

2

b1 a2

wt

driven gear

ra2 rb2

O2

d2

FIGURE 9.25

Involute gearing geometry.

Special Design Problems in Gear Drives

511

contact. The moment of disengagement is less dangerous. Therefore, it is recommended [9] to make the Cαa value at the last point of contact about 60% of that at the first point.

a1 a2 =

a1 b1 =

ra21

rb21 =

164.52

145.652 = 75.81 mm;

a2 b2 =

ra22

rb22 =

334.52

305.40 2 = 136.46 mm;

a w2

(rb1 + rb2 )2 =

480 2

b1 b2 = 75.81 + 136.46 =

(145.65 + 305.40)2 = 164.18 mm;

164.18 = 48.09 mm;

48.09 = 1.629. 29.521

The teeth are case hardened and ground, the base pitch error not more than 11 μm, tooth force Fn = 1000 N/mm. Calculation of tooth pair deformation: Fn 1000 = c 17

f=

59

m.

The required amount of tip relief for the driven gear: C

a2

= 59 + 2 11 = 81

m.

1. Design of the tip relief for the driven gear. To make the transition between the relief and the main involute surfaces not too sharp, let us decide that the angle γ between these surfaces shall not exceed 10 = 0.0175 rad. Hence, for the driven gear: Lr 2 =

cos

a2

=

C a2 81 = = 4642 m = 4.642 mm. sin 0.01745 rb2 305.400 = = 0.913004; ra2 334.5 tan

a2

a2

= 24.0761600 .

= 0.446822.

From Eq. (9.20): tan

0.4468222

2 4.642 = 0.411401. 305.4

st2

=

st 2

= 22.362315o ; cos

st 2

= 0.924796;

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Dudley’s Handbook of Practical Gear Design and Manufacture

From Eq. (9.21): rst2 =

305.4 = 330.235 mm. 0.924796

Now we use Eqs. (9.22), (9.23), and (9.24): a2 b 2 = 136.46 mm(see above); 330.2352

a2 s2 =

305.42 = 125.64 mm;

LCa2 = 136.46 C L r1 =

cos

a1

=

a1

125.64 = 10.82 mm .

= 0.6C

a2

= 0.6 81

49

Ca 49 = = 2808 sin 0.01745

m = 2.808 mm .

r b2 145.652 = = 0.885422; ra2 164.5 tan

a1

m;

a1

= 27.6965020 .

= 0.524934.

2. Design of the tip relief for the driving gear. From Eq. (9.20): tan

0.5249342

2 2.808 = 0.486824. 145.652

st1

=

st1

= 25.957931o ; cos

st1

= 0.899116;

From Eq. (9.21): rst1 =

145.652 = 161.995 mm. 0.899116

Now we use Eqs. (9.22), (9.23), and (9.24): a1 b1 = 75.81 mm (see above); a1 s1 =

161.9952

LCa1 = 75.81

s1 s2 = b1 b2

145.6522 = 70.91 mm; 70.91 = 4.90 mm.

(LCa1 + L Ca2 ) = 48.09

(4.90 + 10.82) = 32.37 mm.

Special Design Problems in Gear Drives

513

The transversal contact ratio provided by the unmodified part of main involute profile equals: u

=

s1 s2 32.37 = = 1.097. pbt 29.521

3. Determination of the parameters of base circle 6 of involute 5 (see Figure 9.17b) that describes the relief profile. The radius of the base circle for the relief involute rb equals: rb = 0.5m n z cos ,

where mn = 10 mm = module of the gear (given); z = tooth number of the calculated gear (given); α* = pressure angle of the relief involute (to be found). The sequence of steps in determining the pressure angle of the relief involute: • Calculate the half-thickness of tooth on the effective tip radius (ra) and on the radius of the relief start point (rst) for the main tooth profile; • Choose preliminarily the value of angle α* and calculate radius rb*; • Calculate the half-thickness of tooth on radius ra and on radius rst for the tooth profile formed by the relief involute; • Determine the differences of the half-thicknesses on radius ra and on radius rst; • Rotate the relief involute about the gear center until it intersects with the main involute at the st point; • Calculate the remaining difference between the tooth half-thicknesses at radius ra and multiply it by cosαa. This is the tip relief amount Cαa for the chosen angle α*. If it is less than required, angle α* should be increased, and vice versa. Note: For each gear, points s and st are characterized by the same radius rst, see Figure 9.23. The Equation for the tooth half-thickness SR on radius R (at x = 0): SR = R

2z

+ inv

inv

R

(9.28)

where α = pressure angle related to the base circle (for the mane involute α = αP = 200; for the relief involute α = α*); αR = arcos(rb/R) for the main involute and αR = arcos(rb*/R) for the relief involute. (Note: if the profile shift coefficient x ≠ 0 or the gear is helical, i.e. β ≠ 0, Eq. (9.7) should be used.) 4. Calculation of the tooth half-thicknesses for the main profile (α = 200) of the driven gear (z2= 65, ra2= 334.5 mm, rst2= 330.235 mm; rb2= 305.40 mm): inv 200 = 0.014904;

• Determine the tooth half-thickness on radius ra2: a2

= arcos

rb 2 305.40 = arcos = 24.0761600 ; ra2 334.5

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Dudley’s Handbook of Practical Gear Design and Manufacture

inv Sa2 = 334.5

2 65

= 0.026614;

ra2

+ 0.014904

0.026614 = 4.166 mm.

• Determine the tooth half-thickness on radius rst2: st 2

= arcos

rb2 305.40 = arcos = 22.3623860 ; rst 2 330.235 inv

Sst2 = 330.235

2 65

st2

= 0.021105;

+ 0.014904

0.021105 = 5.933 mm.

Now we choose for the first approximation the pressure angle for the relief involute α* = 210. 5. Calculation of the tooth half-thicknesses for the relief profile (α* = 210) of the driven gear (z2= 65, ra2= 334.5 mm, rst2= 330.235 mm; rb* = 0.5·10·65·cos210= 303.414 mm): = inv 210 = 0.017345;

inv

• Determine the tooth half-thickness on radius ra2: a2

= arcos

rb2 303.414 = arcos = 24.8968720 ; ra2 334.5 inv

Sa2 = 334.5

2 65

ra

= 0.029586;

+ 0.017345

0.029586 = 3.989 mm .

• Determine the tooth thickness on radius rst2: st 2

= arcos

rb2 303.414 = arcos = 23.251337o ; rst 2 330.235 inv

Sst2 = 330.235

2 65

st2

= 0.023849;

+ 0.017345

0.023849 = 5.833 mm.

• Difference of the half-thicknesses on ra: a2

= Sa2

Sa2 = 4.166

3.989 = 0.177 mm.

Special Design Problems in Gear Drives

515

• Difference of the half-thickness on rst2 st 2

= Sst 2

Sst 2 = 5.933

5.833 = 0.100 mm .

• To obtain the real difference of the half-thicknesses on radius ra when the half-thicknesses on radius rs are equal, the 210-involute shall be rotated around the gear center until it in­ tersects with the 200-involute on radius rs, then the difference on radius ra becomes smaller: real a2

=

a2

st 2

ra2 = 0.177 rst 2

0.100

334.5 = 0.076 mm. 330.235

The real amount of relief equals: C

a2

=

real a2

cos

a2

= 0.076 cos 24.0760 = 0.069 mm.

The required relief amount Cαa2 = 0.081 mm, so the angle α* of the relief involute must be increased. The calculations show that at α* = 21.20, Cαa2 = 0.082 mm. Taking into consideration inaccuracy of our data concerning the tooth stiffness and load level, it is not worth making additional calculations for the sake of couple of microns. Thus, we accept the pressure angle of the tip relief involute α2* = 21.20. The height Hr from the outer diameter to point st equals: Hr = ra

rst cos ,

S

where = arcsin 2rst . st Radius ra shall be taken here without deduction of rounding or chamfering of the tooth edges. 6. Determination of the real angle γ between the main and the relief involutes at their intersection point st. From Figure 9.24 one can see that: =

,

where: = arcsin

rb ; rst

= arcsin

rb ; rst

rb = 0.5·m·z·cos200 = base circle radius of the main involute; rb*= 0.5·m·z·cosα* = base circle radius of the relief involute. In this case, rb = 0.5·10·65·cos200 = 305.4 mm; rb* = 0.5·10·65·cos 21.20 = 303.005 mm; rst = 330.235 mm. = arcsin

305.4 = 67.6380 ; 330.235

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Dudley’s Handbook of Practical Gear Design and Manufacture

= arcsin

= 67.638

303.005 = 66.5700 ; 330.235

66.570 = 1.0680 = 10 4 .

It is less than the limit recommended above. 7. Similar calculations made for the driving gear (with z1 = 31) give the following results: 1

= 21.40 ; C

a1

= 0.050 mm(0.049 mm required), and

= 10 3. 6 .

The recommendations provided above for NCR gears are not valid for high-contact-ratio (HCR) gears, with εα ≥ 2. Figure 9.26 shows the position when pair 3-3* enters into contact. At the same moment, pair 2-2* contacts in the middle of the teeth, and pair 1-1* is about to disengage. Apparently, at εα ≈ 2 (as shown), if we want the teeth to engage and disengage with zero load, teeth 1 and 3* shall have the same depth of modification Cαa = Fn/cʹ, where cʹ is the stiffness of pair 2-2* which takes the full load. However, the modification must have a certain extension, and it changes part of double-tooth-contact (DTC) area into single-tooth-contact (STC). This is undesirable because the idea of HCR gears is to provide double-tooth contact in any angular position of the gears. Keeping this in mind, the HCR gears are usually designed with theoretical contact ratio εα > 2, and the tip reliefs are made shorter to retain the εαu ≈ 1.85…1.90. Higher contact ratio is achieved by reducing the pressure angle αP to, for example, 180, but mainly by increasing the tooth height. The undercutting and sharpening of the teeth must be checked. Figure 9.27 presents the correlation between ha/m and Sa/m for 200-gears with x = 0. The best shape of tip reliefs for the HCR gears are parabolas. Nevertheless, also linear tip relief can be successfully used here. Example 9.10: Design profile modification for HCR gears with the following parameters: z1 = 31, z2 = 91, αP = 200, mn = 5 mm, addendum height ha = 1.35mn, profile shift coefficients x1 = x2 = 0, Fn = 500 N/mm. The geometrics of the mesh (see Figure 9.25): tip diameters: of the pinion 168.5 mm and of the gear wheel – 468.5 mm. Tooth edge rounding radius r = 0.25 mm, so the effective tip radii equal ra1 = 84 mm and ra2 = 234 mm. Base radii: rb1 = 0.5·5·31·cos200 = 72.826 mm, rb2 = 0.5·5·91·cos200 = 213.780 mm. Center distance aw = 0.5·5(31+91) = 305 mm. Base pitch ptb = π·5·cos200 = 14.761 mm, base pitch error not more than 5 μm. a1 b1 = a2 b2 =

driving

1*

of line n o i act b1

2* 1

2342

72.8262 = 41.861 mm; 213.780 2 = 95.153 mm;

b2

p Sа

842

3*

3

2

driven

FIGURE 9.26

HCR meshing.

Special Design Problems in Gear Drives 1.6

517

40 35 30

1.5

ha/m

z=25

1.4

1.3 0.1

FIGURE 9.27

0.2

0.3 Sa/m

0.4

0.5

Tooth tip thickness Sa depending on the addendum height ha at x = 0.

a1 a2 =

3052

(72.826 + 213.780)2 = 104.317 mm;

b1 b2 = 41.861 + 95.153 =

104.317 = 32.697 mm;

32.697 = 2.215. 14.761

1. Design of the tip reliefs for the driven and driver gears. Let us assume that the stiffness of two tooth pairs, one of whom contacts in the middle and the other in the end of the length of action, equals: cʹ ≈ 14.3 + 8.7 = 23 N/mm·µm (see above). The deformation of the two engaged tooth pairs under load equals: f=

Fn 500 = = 21.7 c 23

m.

The required depth of the driven gear tip relief: C

a2

= 21.7 + 2 5 = 31, 7

m.

To shorten the relief length, let us make the transition angle γ between the relief and the main involute surfaces equal the maximum recommended: γ = 1020ʹ = 0.02327 rad. Then the relief length equals: Lr 2 =

cos

a2

=

C a2 31.7 = = 1362 m = 1.362 mm. sin 0.02327 rb2 213.78 = = 0.913590; ra2 234

a2

= 23.9937250 .

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Dudley’s Handbook of Practical Gear Design and Manufacture

From Eq. (9.20): tan

(tan23.993725o )2

=

st 2

2 1.362 = 0.430546. 213.78

= 23.294101o

st 2

From Eq. (9.21): rst 2 =

213.78 = 232.752 mm; cos 23.294101o 2342

a2 b2 =

213.782 = 95.153 mm ;

232.7522

a2 s2 =

LCa2 = 95.153

213.782 = 92.041 mm; 92.041 = 3.112 mm .

Let us take the modification depth for the second point (the pinion tooth tip relief) equal to 60% of the depth in the first point, as recommended, i.e.,: C

a1

= 31.7 0.6 = 19

m.

Then we obtain the following: L r1 =

cos

a1

=

19 = 817 m = 0.817 mm. 0.02327

rb1 72.826 = = 0.866976; ra1 84

a1

= 29.8908900 .

From Eq. (9.20): tan

st1

=

(tan29.890890 o)2

st1

2 0.817 = 0.554954. 72.826

= 29.028260 o

From Eq. (9.21): rst1 =

72.826 = 83.289 mm; cos 29.028260 o

Special Design Problems in Gear Drives

519

a1 b1 = 41.861 mm; a1 s1 =

83.2892

72.8262 = 40.416 mm;

LCa1 = 41.861 s1 s2 = b1 b2

40.416 = 1.445 mm.

L Ca2 = 32.697

L Ca1

u

=

1.445

3.112 = 28.14 mm.

s1 s2 28.14 = = 1.906. pbt 14.761

Let us use Eqs. (9.26) and (9.27) to determine the effective contact ratio provided by the non-modified parts of profiles under load: ue

=

KF ,

where: KF = (1.03 ÷ 1.05) + 4.8 10 ue

4

100 = 1.078 ÷ 1.098.

= 1.906 (1.078 ÷ 1.098) = 2.055 ÷ 2.093.

The process of determination of the relief involute parameters is identical to that demonstrated in Example 9.9. The investigators [10] concluded that the NCR gears with profile modification reach similar or even lower level of vibrations than the unmodified HCR gears. The lowest level of vibrations is achieved on modified HCR gears. In low-contact-ratio (LCR) gears, with εα = 1.1…1.2, the entire profile can be modified, and the parabolic shape of modification is used for such gears. The different options of profile modification of spur gears demonstrated above provide improvement of the gears' functioning. However, this way the designer cannot be sure that the chosen parameters of modification are the best possible, though any modification is better than no modification at all. In principle, the aim of profile modification is to reduce transmission error (TE) in the situation of load transfer. It is experimentally shown that the minimum TE leads also to minimum vibrations and to minimum stresses. Therefore, solving for the optimum in this case consists in analysis of modifications with different parameters of profile relief for TE until the modification with minimum TE is found. Modern sophisticated methods use computer programs in which the computer calculates the tooth stiffness and tooth deflections using FEM, and the transmission error is determined by “rotation” of the computer models of gears in small intervals. The computer program changes the parameters of profile modification and finds the optimum with minimal TE. More detailed and complicated computer programs provide the comprehensive analysis of the gears as part of the entire transmission, together with its shafts, bearings and connections. These programs perform calculations for strength, determine the load distribution across the face width, calculate the thickness of oil film between the teeth and the instantaneous temperature in the contact, torsional and linear vibrations etc. In addition, these programs find the optimal tooth profile and tooth direction modifications to achieve more uniform load distribution across the face width and minimum stresses and vibrations taking into consideration all the levels of speed and load.

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Dudley’s Handbook of Practical Gear Design and Manufacture

9.2

EXAMPLES OF CALCULATION OF INVOLUTE GEAR PAIRS GEOMETRY

Figure 9.28 provides an extremely simplified presentation of a gear pair. Here are shown only the base diameters db, tip diameters da, and the center distance aw, however, this gives us a lot of knowledge about this gear pair. First, we know the gear ratio: u=

db2 . db1

(“Minus” because the gears rotate in opposite directions.) Second: passing a straight-line tangent to circles db1 and db2, we obtain segment a1a2 called line of action. The teeth can interact correctly only within this segment. Tip circles da1 and da2 intersecting with line a1a2 in points b1 and b2, restrict the zone of real contact of teeth to segment b1b2 which is called length of action. The rest of segment a1a2 is unachievable for teeth of one of the gears. Dividing the length of segment b1b2 by transverse pitch on the base cylinder pbt, we obtain the transverse contact ratio εα:

=

where pbt =

db1 z1

=

db2 z2

b1 b2 , pbt

.

The length b1b2 is obtained as follows: • in ΔO1BO2: O1B = a1a2; O2B = 0.5(db1+db2); O1O2 = aw; hence, a1 a2 =

a w2

[0.5(db1 + db2 )]2 ,

1

driving gear O1

db1

da1 b 2 p

B

a1

b1 aw

wt

a2 wt

da2 db2 driven gear

O2

FIGURE 9.28 Basic lines of involute gearing geometry.

Special Design Problems in Gear Drives

521

• passing line O1b1 we obtain ΔO1a1b1 from which da21

a1 b1 = 0.5

db21 ;

• passing line O2b2 we obtain ΔO2a2b2 from which da22

a2 b2 = 0.5

db22 ;

and, finally, b1 b2 = a1 b1 + a2 b2 a1 a2. Point p where line a1a2 intersects with center line O1O2 is called pitch point. When the teeth interact at this point, there is no sliding between their profiles. This point is the border between addendum and dedendum of the tooth. This division is not formal. When the gears rotate, the profiles of teeth roll and slide relative to each other, and in this motion the addendum surface move faster. Therefore, the ad­ dendum material comes to the contact area compressed by the friction forces while the dedendum material comes tensioned. This explains the phenomenon of higher resistance of addendum surfaces to pitting as compared with that of the dedendum surfaces. When the contact of teeth comes to point p, the speed of the contacting surfaces becomes equal, the sliding speed equals zero, and the direction of sliding changes. The direction of friction forces changes as well, and this is one of the factors exciting vibrations. With distance from point p, the sliding speed grows linearly, therefore, to reduce the maximum sliding speed and the susceptibility of teeth to scoring, it is desirable to have point p placed approxi­ mately in the middle of segment b1b2. This can be controlled by choosing profile shift coefficients x1 and x2. Keeping the sum x1 + x2 = const, we can move segment b1b2 along the line of action closer to the pinion or to the gear. Besides the mentioned above, we have from this picture the working transverse pressure angle αwt which is important for the following calculations of the gear pair geometry. It is given by: cos

wt

db1 + db2 . 2 aw

=

(9.29)

Several examples of calculation of gear pairs are given below. The main symbols and definitions pre­ sented in Section 9.1.1 are valid here. Example 9.11: Spur gears, z1 = 25, x1 = 0.4, z2 = 57, x2 = 0.3, m = 3 mm, αP = 200, β = 0. What is the center distance? inv

wt

= 2 tan

n

x1 + x2 + inv z1 + z2

t.

Here, because the gear is spur: αn = αt = αP = 200; inv 200 = 0.014904. inv

wt

= 2 tan 20 o

0.4 + 0.3 + 0.014904 = 0.021118; 25 + 57 wt

Center distance modification coefficient [11]:

= 22.3667o .

(9.30)

522

Dudley’s Handbook of Practical Gear Design and Manufacture

y=

z1 + z2 2 cos

aw =

cos cos

t

1 =

wt

cos 20 o cos 22.3667o

25 + 57 2

1 = 0.6617;

z1 + z2 25 + 57 + y mn = + 0.6617 3 = 124.985 mm. 2 cos 2

This value can be rounded off to 125 mm. Let us calculate the tip dimeters using formula [11]: da = d + 2ha ,

where d = reference diameter (see Section 9.1.1), h a1 = (1 + y–x2)mn; h a2 = (1 + y–x1)mn.

Thus, ha1 = (1 + 0.6617

0.3)3 = 4.085 mm;

da1 = 3 25 + 2 4.085 = 83.17 ha2 = (1 + 0.6617

83.2 mm;

0.4)3 = 3.785 mm;

da2 = 3 57 + 2 3.785 = 178.57

178.6 mm.

Now we can use Figure 9.28 and the related equations to determine the transverse contact ratio εα and the position of point p within segment b1b2. Notes: 1. Here and in the examples that follow, the profile shift coefficients x are nominal, i.e., related to a zero-backlash mesh. 2. If the tip edges of the teeth are chamfered or rounded off, the tip diameters da used for the calculation of the contact ratio shall be reduced by the doubled size of the chamfer or rounding radius. Example 9.12: Helical gears, z1 = 25, x1 = 0.4, z2 = 57, x2 = 0.3, mn = 3 mm, β=150, αP = 200. What is the center distance? inv

wt

x1 + x2 + inv z1 + z2

= 2 tan

n

tan cos

tan 20 o = 0.376810; cos 15o

t

.

Angle αt is given by (see Section 9.1): tan

t

t

=

P

=

= 20.646911o ; inv

t

= 0.016453 rad .

Special Design Problems in Gear Drives

inv

wt

523

= 2 tan 20 o

0.4 + 0.3 + 0.016453 = 0.022667 ; 25 + 57 wt

y=

z1 + z2 2 cos

aw =

cos cos

t

1 =

wt

= 22.877775o . cos 20.646911o cos 22.877775o

25 + 57 2 cos15o

1 = 0.6650;

z1 + z2 25 + 57 + y mn = + 0.6650 3 = 129.334 mm. 2 cos 2 cos15o

Example 9.13: Center distance aw = 125 mm. It is required to calculate a gear drive with gear ratio z2/z1 = 2.5. First, let us determine approximately the reference diameters of the gears: d1

2 aw 2 125 = = 71.4 mm; d2 = 2.5 d1 1+u 1 + 2.5

178.6 mm .

Because the gear ratio must be 2.5 precisely, the number of teeth of the gear must be divisible by 2.5. Considering several standard modules (mn = 2.5, 3 and 4 mm), we obtain the approximate numbers of teeth of the gear (z2 ≈ 71, 60 and 45). The two last options meet our requirements, and we choose mn = 3 mm, z2 = 60, z1 = 24. Now we have the module, numbers of teeth, and we can calculate the reference and base diameters of the gears: d1 = m n z1 = 3 24 = 72 mm; db1 = d1 cos

t

= 72 cos20 o = 67.657869 mm;

d2 = mn z2 = 3 60 = 180 mm . db2 = 180 cos 20 o = 169.144672 mm .

From Eq. (9.29): cos wt

wt

=

67.657869 + 169.144672 2 125

= 18.700050 o ; inv

wt

= 0.947210;

= 0.012105;

From Eq. (9.30): x1 + x2 = (inv

wt

inv t )

z1 + z2 . 2 tan n

524

Dudley’s Handbook of Practical Gear Design and Manufacture

Since the gears are spur, αt = 200; inv200 = 0.014904; x1 + x2 = (0.012105

0.014904)

24 + 60 = 2 tan20 o

0.323.

The negative profile shift reduces the tooth thickness in the root. Because the pinion teeth are thinner and have more cycles of loading, let us make x1 = 0 and x2 = – 0.323. In this example, no requirements about the load capacity of this gear drive are mentioned. If the load is high enough, other combinations of module and numbers of teeth can be considered, as well as the use of helical gears and different kinds of heat treatment and surface hardening. Example 9.14: Spur gear drive z1 = 16, x1 = 0, z2 = 62, x2 = 0, mn = 5 mm, aw = 195 mm, αP = 200. In service, many cases of pitting and tooth breakage had happened with the pinion, and it was decided to replace the pinion by a stronger one while leaving the gear and the housing untouched. The strengthening of the pinion teeth is achieved due to reducing its number of teeth (z1 = 15 instead of 16) with the positive profile shift required for the normal mesh with the gear. The base diameters of the new pinion (db1) and the old gear (db2) equal: db1 = mn z1 cos

t

= 5 15 cos 20 o = 70.476946 mm;

db2 = mn z2 cos

t

= 5 62 cos 20 o = 291.304712 mm .

From Eq. (9.29): cos

wt

70.476946 + 291.304712 = 0.927645; 2 195

=

wt

= 21.929360 o ; inv

wt

= 0.019853.

From Eq. (9.30): x1 + x2 = (inv

wt

inv t )

z1 + z2 = (0.019853 2 tan n

0.014904)

15 + 62 = 0.5235. 2 tan20 o

Since x2 = 0, x1 = 0.5235. The tip diameter of the pinion can be chosen equal to that of the old pinion, with z1 = 16 and x1 = 0, which was determined as follows: da1 = d1 + 2m n = 5 16 + 2 5 = 90 mm.

Bending strength of the new pinion teeth is 20% higher than that of the original pinion because of thicker tooth root. Example 9.15: Center distance aw = 300 mm. It is required to design double-helical gear pair with gear ratio |u| = 2.8-2.9 and helix angle β ≈ 350—for better load distribution between the left and right helices.

Special Design Problems in Gear Drives

525

Let us determine approximately the reference diameter of the pinion: 2a w 2 300 = = 158 ÷ 154 u +1 (2.8 ÷ 2.9) + 1

d1

156 mm .

Since the gears are of medium hardness, the number of teeth of the pinion shall be chosen between 30 and 35. Because d1 = mt∙z1, the transversal module equals: d1 156 = = 5.2 ÷ 4.46 mm. z1 30 ÷ 35

mt =

m n = mt cos = 5.2cos30 o ÷ 4.46cos35o = 4.5 ÷ 3.65 mm .

Let us take mn = 4 mm and β = 350. In this case: mn 4 = = 4.883098 mm ; cos cos35o

mt =

2 aw 2 300 = = 122.87 mt 4.883098

z1 + z2 =

z1 =

123 ;

z1 + z2 123 = = 32.4 ÷ 31.5 u +1 (2.8 ÷ 2.9) + 1

32 .

Now we can choose between two options: z2 = 91 with u = 91/32 = 2.84 and a small negative profile shift, and z2 = 90 with u = 90/32 = 2.81and a small positive profile shift. Let us calculate the first option: tan

t

db1 = mt z1 cos

t

t

=

tan cos

P

=

tan20 o = 0.444326; cos35o

= 23.956822o ; inv

t

= 0.026201;

= 4.883098 32 cos 23.956822o = 142.797679 mm;

db2 = 4.883098 91 cos 23.956822o = 406.080901 mm;

From Eq. (9.29): cos

From Eq. (9.30):

wt

=

142.797679 + 406.080901 = 0.914798; 2 300

wt

= 23.822943o ; inv

wt

= 0.025742.

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Dudley’s Handbook of Practical Gear Design and Manufacture

x1 + x2 = (inv

wt

inv t )

z1 + z2 = (0.025742 2 tan n

0.026201)

32 + 91 = 2 tan20 o

0.0776.

Finally, the parameters of the double-helical gear pair are as follows: mn = 4 mm, z1 = 32, x1 = 0, z2 = 91, x2 = –0.0776,

9.3

= 350 .

ANALYSIS OF MOTION AND POWER TRANSMISSION IN COMPLEX CYLINDRICAL GEAR DRIVES

9.3.1 DEFINITION

OF

GEAR RATIO

One of the main features of a gear drive is its gear ratio u that equals the ratio of the angular velocity of one shaft to that of the other shaft: u1

2

1

=

2

; u2

1

=

2

;

1

where 1 and 2 are the symbols of the shafts. Note: the lower index at the symbol u indicates the “direction” of the gear ratio: from – to. It is also important to know the direction of rotation of one shaft depending on that of the other shaft. In the simplest cases like that shown in Figure 9.29a the gear ratio and direction of rotation are clear: u1

2

=

1

=

2

z2 . z1

Note: 1. Since the meshing gears rotate in opposite directions, the minus sign is added to the gear ratio. If there are several gears in a gear train (Figure 9.29b), it is very easy to know the gear ratio, though it may be confusing for a beginner. Here idling gears – z4, z5, and z6 – do not influence the modulus of gear ratio u3-7 but influence its sign. In this case, u3

7

=

3 7

=

z7 . z3

(a)

z2

z1

1

2

(b)

z3 z5 z7

z6

z4 FIGURE 9.29

Gear ratio.

Special Design Problems in Gear Drives

527

The direction of rotation of gears z3 and z7 is the same. However, in complex gear systems this parameter, as well as the gear ratio, is often a matter of kinematic analysis.

9.3.2 BASIC KINEMATIC DIAGRAMS

AND

GEAR RATIOS

OF

PLANETARY GEAR DRIVES

Gear drives with movable axes are called planetary because some of the gears (called planet gears) are not coaxial with the central axis and rotate around it resembling the rotation of planets around the sun. The planet gears are assembled in a planet carrier that rotates about the central axis. There are also central gears (coaxial with the central axis) in a planetary gear drive. Depending on the number of central gears, the planetary gear drives can be divided in three groups with different design and kinematic features: • drives with only one central gear (1CG); • drives with two central gears (2CG); and • drives with three central gears (3CG). The typical elements of a planetary mechanism are (see Figure 9.30): • ring gear or annulus: central gear with internal teeth, denoted by a; • sun gear: central gear with external teeth, denoted by s; • planet gear: off-center gear with external teeth, denoted by p; it is mounted rotatable in a planet carrier; the planet gear rotates around its axis and, along with the planet carrier, around the central axis. The number of planet gears can be (and usually is) more than one, depending on the space available to place them; • planet carrier: denoted by c; contains the axles and bearings of the planet gears. Physically, it is just an arm or a crank that rotates about the central axis of the drive. The gear ratio of a planetary gear drive can be determined using two methods. We demonstrate them in Figure 9.30 for a 2CG drive with the fixed ring gear (ωa = 0). The first method is based on the velocity vector diagram. All the gears are replaced by cylinders with their radii proportional to their tooth numbers, i.e., rs = k·zs, rp = k·zp, and ra = k·za, where zs, zp, and za are the numbers of teeth of the sun gear, planet gear and ring gear, respectively. It is supposed that the cylinders roll with no slippage relative to each other. If the angular speed of sun roller s equals ωs, its peripheral speed equals:

Vs = rp

s rs.

A p Vp

c

Vs

rc

c

s

rs ra

s a FIGURE 9.30 Kinematic diagram of a planetary gear drive.

528

Dudley’s Handbook of Practical Gear Design and Manufacture

Point A is the instantaneous center of rotation of planet roller p because this point belongs also to fixed roller a, thus the speed of this point equals zero. Therefore, the speed in the center of roller p equals: Vp =

Vs = 2

s rs

2

.

The center of the planet gear belongs also to planet carrier c, thus the angular speed of the planet carrier equals: c

=

Vp rc

; rc = rs + rp = rs +

c

2 Vp

=

rs + ra

=

ra

rs 2

=

rs + ra ; 2

ws rs . rs + ra

From these relations we obtain the gear ratio from sun roller s to planet carrier c at roller a fixed: usa

c

s

=

rs + ra r z = 1 + a = 1 + a. rs rs zs

=

c

(9.31)

(Notice: the gear ratio symbol u is provided, in addition to lower index, with an upper index that indicates the fixed member.) The idea of the second method (called Willys’ method) is to consider first the planetary gear drive as a regular gear drive with the planet carrier fixed. So, we assume that carrier c in Figure 9.30 is fixed (ωc = 0), and the sun gear rotates with angular speed ωs, for example, counter clockwise. In this case, ring gear a rotates clockwise with the angular speed: =

a

s

zs . za

But in reality the ring gear is fixed. To restore the real situation, we have to rotate the entire gear drive in the counter clockwise direction with the –ωa speed; in other words, we have to add the –ωa speed to the angular speeds obtained above for the gear drive with the fixed planet carrier. The new angular speeds we denote with the additional sign “r” which means “real”: sr

cr

=

=0

s

a

=

a

s

=

zs ; za

s

1+

zs ; za

ar

=

a

a

= 0.

Now, the gear ratio of a planetary gear drive with the ring gear fixed equals: usa

c

=

sr cr

=

1 + zs /z a z = 1 + a. zs /z a zs

Special Design Problems in Gear Drives

p

529

4

3

c 2

1 a FIGURE 9.31 1CG planetary gear drive.

This formula is identical with Eq. (9.31) developed using the velocity vector diagram. Both these methods can be applied to all types of planetary gear drives. Let us examine the different types of planetary gear drives. Figure 9.31 presents a kinematic diagram of 1CG drive (with one central gear). Here ring gear a is stationary and connected to housing 1. Planet carrier c (just a crank in this case) is a part of input shaft 2 of the drive. Planet gear p is mounted on the eccentric pin of the crank and supported by bearings. Planet gear p is engaged with ring gear a. Cardan shaft 3 transmits rotation from planet gear p to shaft 4 of the driven machine. The gear ratio of 1CG gear drive with fixed ring gear a equals: uca

p

=

2 4

=

zp za

zp

,

(9.32)

where za = number of teeth of the ring gear; zp = number of teeth of the planet gear. The designer is often interested in a greater gear ratio. That means the difference between za and zp is desired to be as small as possible. But if this difference is too small, there may be interference between the teeth tips of the planet gear and ring gear (look for details in Section 9.1.5). Reduction of the difference za – zp can be achieved by a certain reduction of the tooth height (re­ ducing the tip diameter of the planet gear dap and increasing the tip diameter of the ring gear daa) provided that the transverse contact ratio of the teeth, with the tip edges rounded off, is not less than 1.1–1.2. (Take notice: decreasing the contact ratio leads to reduced contact strength of the teeth flanks.) Outward profile shift is also helpful. By using a basic rack profile with αP = 30° and reduced tooth height (haP = 0.75 mn), the difference (za – zp) can be decreased to 3, 2, or even 1. Often cycloidal gears are used for the 1CG planetary gears. In Figure 9.32 planet carrier c is mounted on input shaft 1, and two cycloidal planet gears (p1 and p2) are mounted on the plane’s carrier on two bearings with opposite eccentricity. The planet gears are engaged with a stationary lantern gear a that is formed by pins 2. The pins are installed in housing frame 3. The torque transmission from planet gears to output shaft 4 is performed through pins 5 fixed rigidly in the disc of shaft 4 and circular holes 6 in the planet gears. The pins enter holes 6, and each pin revolves around the hole center in the relative motion (so called W-mechanism). 2CG planetary gear drives (with two central gears) are the most widespread. There are several dif­ ferent configurations of such drives. Type A (Figure 9.33) contains sun gear s, ring gear a, and planet gears p. (The number of planet gears depends on their diameter: the less the difference between za and zs, the more planet gears can be placed in the ring-shaped space between them. For example, the Wright

530

Dudley’s Handbook of Practical Gear Design and Manufacture

A

section A-A lantern ring gear, planet gear and pins are shown

3 a p2

5

6

5

6 p1

1 c 4 2

A

FIGURE 9.32 1CG planetary pin gear drive.

(a)

FIGURE 9.33

u = 3...9 0.99 - 0.97

(b)

(c)

-2...-8 0.98...0.96

p

p

p

s

s

s

c

c

c

a

a

a

1.13...1.50 0.99

2CG planetary gear drives type A.

“Cyclone” aviation planetary gear drive contained 20 planetary gears [12].) The gear ratio depends on which of the three elements is stationary. For the gear drives type A, the gear ratios are as follows: • ring gear a is stationary (Figure 9.33a): usa

c

=

• planet carrier c is stationary (Figure 9.33b):

s c

=1+

za , zs

(9.33)

Special Design Problems in Gear Drives

531

usc

s

=

a

za , zs

=

a

(9.34)

(Strictly speaking, gear drive with a stationary planet carrier is not planetary because the axes of all gears are fixed; it is called star gear drive.) • sun gear s is stationary (Figure 9.33c): uas

a

=

c

=1+

c

zs , za

(9.35)

Planetary gear drives 2CG type B (Figure 9.34) are distinguished by double-toothed planet gears. They are more complicated in production but allow greater gear ratio. For those gears: Figure 9.34a usa

c

s

=

z p1 z a

=1+

z s z p2

c

,

(9.36)

Figure 9.34b usc

a

s

=

=

a

z p1 z a z s z p2

,

(9.37)

Figure 9.34c uas

(a) u = 7...16

c

=

(b)

0.99 - 0.97

p1

FIGURE 9.34

p2

p1

a

=1+

c

z s z p2 zp1 z a

,

(9.38)

(c)

-6...-15 0.98 - 0.96

p2

p1

s

s

s

c

c

c

a

a

a

2CG planetary gear drives type B.

1.04...1.10 0.99

p2

532

Dudley’s Handbook of Practical Gear Design and Manufacture

(a)

(b) u = 8...30

(c)

0.80...0.75

p1

p1

p2

(d)

25...300 0.90...0.40

p1

p2

p2 p1 c

c

s1

s2

s

c

c

a1

FIGURE 9.35

p2

a1

a2

a2

a

2CG planetary gear drive type C.

In gear drives 2CG type C illustrated in Figure 9.35a–c the planet gears are double-toothed, and the central gears have the same type of toothing: both external or both internal. These drives allow having very high gear ratio but with a low efficiency. Since in the internal gearing the energy losses are less than in the external gearing (see Section 9.4.1), the design with internal gearings is preferable. Minimization of the energy loss requires the increase of the planet gear diameter to a maximum, but only one planet gear is available in this case (see Figure 9.35c). Figure 9.35d shows a peculiar design of 2CG planetary gear drive. Here the single planet gear is provided with internal toothing p1 engaged with sun gear s and external toothing p2 engaged with ring gear a. Formulae for the gear ratios are as follows: Figure 9.35a (sun gear s1 is stationary): uss21

c

=

s2

=1

usc2

s1

=1

=1

uac2

a1

=1

=1

usc

c

z p 2 z s1 z s2 zp1

,

(9.39)

Figure 9.35b-c (ring gear a1 is stationary): uaa21

c

=

a2 c

z p2 z a1 z a2 z p1

,

(9.40)

Figure 9.35d (ring gear a is stationary): usa

Note: In all cases, uXZ

Y

=

1 , uYZ X

fixed member). For example:

c

=

s c

a

=1

zp1 z a z s z p2

,

(9.41)

where X, Y, and Z are the symbols of the links of the mechanism (Z is the

Special Design Problems in Gear Drives

533

usa

c

=

1 uca

. s

Planetary gear drives 2CG type D (Figure 9.36) are characterized by pairs of planet gears p1 and p2 in mesh with each other and, in addition, they may be in mesh with sun gears (s1, s2) or ring gears (a1, a2), see Figure 9.36a. (Planet carrier c is mostly round, but here it is pictured as a polygon to distinguish it among the many circles symbolizing gears.) There are three versions of such gear drives: with two sun gears (Figure 9.36b), with one sun gear and one ring gear (Figure 9.36c), and with two ring gears (Figure 9.36d). The formulae for their gear ratios are as follows: Figure 9.36b (sun gear s2 is stationary): uss12

c

=

usa

c

=

uaa12

c

=

s1

=1+

z s2 , z s1

(9.42)

=1

za , zs

(9.43)

=1+

z a2 , z a1

(9.44)

c

Figure 9.36c (ring gear a is stationary): s c

Figure 9.36d (ring gear a2 is stationary): a1 c

Notice: here the numbers of teeth of the planet gears do not influence the gear ratio. Planetary gear drives 3CG (Figure 9.37) are configured of a 2CG gear drive shown in Figure 9.35b (with two ring gears a1 and a2) with the addition of sun gear s. This addition greatly increases its efficiency and, consequently, the reasonable gear ratio as compared with the gear drive presented in Figure 9.35b. The formulae for gear ratio are as follows: Figure 9.37a and b (ring gear a1 is stationary): usa1a2 =

s

=

a2

1 + z a1/ z s , (z a1 z p2 )/(zp1 z a2 )

1

(9.45)

Figure 9.37c and d (ring gear a1 is stationary): usa1a2 =

s a2

=

1 + (z a1 z p2 )/(zp1 z s ) 1

(z a1 z p2 )/(zp1 z a2 )

,

(9.46)

For technological simplification, 3CG gear drives with a common planet gear are often used (Figure 9.37e). Here planet gear p with doubled face width is engaged with two ring gears a1 and a2. The ring gears have different numbers of teeth (za1 and za2), however the tip and root diameters of their toothing are approximately the same. It is achieved by appropriate profile shift. Formulae 9.45 and 9.46 are valid also here, but with zp1 = zp2 they look as follows: usa1a2 =

s a2

=

1 + z a1/ z s . 1 z a1/ z a2

a1

c s2

p2 s1

FIGURE 9.36 2CG planetary gear drives type D.

a2

p1

(a)

c

s1

p1

(b) u = 2...3 0.97...0.95

s2

p2

c

s

p1

(c)

za > 2zs

-2...-7 0.98...0.96 a p2

a1

p1

(d)

about 2 0.99...0.96

a2

c

p2

534 Dudley’s Handbook of Practical Gear Design and Manufacture

FIGURE 9.37 3CG planetary gear drives.

a1

a1

a2

c

c

p1

s

p2

(b)

s

p1

0.90...0.80

(a) u = 20...500

a2

p2

a1

c

s

p1

(c)

a2

p2

a1

a1

s

p

c

a2

p2

0.84...0.70

(e) u = 60...500

c

s

p1

(d)

a2

Special Design Problems in Gear Drives 535

536

Dudley’s Handbook of Practical Gear Design and Manufacture

(a)

(b)

(c)

a1

a2

a1

a2

a1

p1

p2

p1

p2

p1

p2

s1

s2

s1

s2

s1

s2

c1

c2

(d)

c1

c2

(e)

c1

a2

c2

(f )

a1

a2

a1

a2

a1

a2

p1

p2

p1

p2

p1

p2

s1

s2

s1

s2

s1

s2

c1

FIGURE 9.38

c2

c1

c2

c1

c2

Kinematic diagrams of two-stage planetary gear drives.

In Figures 9.32–9.37 some of the mostly used kinematic diagrams are provided with approximations about their reasonable gear ratio u and efficiency η [2,13]. Planetary gear drives with two or more stages are made up usually of units 2CG type A because they are both more efficient and more producible as compared with other types of planetary drives. Figure 9.38 presents six two-stage planetary gear drives where each stage is just a 2CG type A unit, but they are connected with each other in different way. In the first case, shown in Figure 9.38a, the planetary stages are connected in series, so that their common gear ratio u is determined easily: u = u sa11

c1

u sa22

c2 ,

and the members of this equation are calculated using Eqs. (9.33). It is also easy to calculate the common gear ratio of the gear drive shown in Figure 9.38f: u = u sc11

a1

u sa22

c2 ,

and the members of this equation are determined using Eqs. (9.34) and (9.33). The calculations of gear ratios of other four types of drives presented in Figure 9.38, where one of the external shafts is connected with two drive elements, require more sophisticated methods.

Special Design Problems in Gear Drives

537

(a)

(b) Tin

Tout

FIGURE 9.39

Tc

Th

Ts

Ta

Mechanical nodes (MNs).

9.2.3 MN METHOD

OF

KINEMATICAL ANALYSIS

One of the simplest is the method of Mechanical Nodes (MN) [14] which is applicable to all gear drives with parallel shafts. Following this method, each gear unit, independently on its type, possesses three elements that connect it with the outer world: input shaft, output shaft and a fixed (nonrotating) member, let’s call it “housing.” Therefore, the MN is depicted as a circle provided with three “legs.” Figure 9.39a presents a Mechanical Node for any gear drive, and here Tin, Tout and Th are the input torque, output torque, and the moment applied to the housing, respectively. Figure 9.39b presents the MN for a pla­ netary drive 2CG type A, and here Ta, Tc , and Ts are torques applied to the ring gear (annulus), planet carrier, and the sun gear, respectively. Interdependence of the torques applied to these legs is determined by the design of the gear unit, with the understanding that the sum of these three torques must equal zero according to the Newton’s first law. As soon as we know the torque on one “leg,” we immediately can know the torques on two other “legs” from the kinematic of the mechanism. Notes: 1. The outward shafts and the housing can be loaded not only by torques, but also by radial and axial forces and bending moments. However, our goal is to determine the gear ratio of a complex gear drive. It is clear that no other loads but the torques are related to the gear ratio. Therefore, only the torques applied to the outward shafts and the housing of each gear unit should be taken into account in this case. 2. Since the gear ratio depends on the ratio of dimensions only, like in the lever theory, the energy losses must be neglected. The direction of rotation is of fundamental importance at the determination of the gear ratio. Therefore, the gear ratios, as well as the torques applied to the shafts, should be given signs “plus” or “minus” basing on the following reasons: • The direction of torque applied to a driving shaft is always the same as the direction of its rotation. • The direction of torque applied to a driven shaft is always opposite to the direction of its rotation. • Hence, if the input and output shafts rotate in opposite directions (i.e., the gear ratio is negative, as in Figure 9.29a), the input and output torques have the same direction and, consequently, the same sign. And vice versa, if the input and output shafts rotate in the same direction like gears z3 and z7 in Figure 9.29b, the input and output torques are directed oppositely, and their signs are opposite. Thus, the relation between the input and output torques of a gear drive (neglecting the energy losses) is given by the following equations:

538

Dudley’s Handbook of Practical Gear Design and Manufacture

Tout =

Tout , Tin

Tin u; u =

(9.47)

where u = ωin /ωout = gear ratio with its sign; Tin and Tout = input and output torques with their signs. Figures 9.40a and 9.41a present kinematical diagrams and MN-diagrams of a single-stage reduction gear. It consists of input shaft 1, output shaft 2, pinion 3, gear 4 and housing 5. The gear ratio u = – z4/z3, and the output torque, according to Eqs. (9.47), equals: (a)

(b)

1

6 8 10

9

3 4

7

2

5 11 12

FIGURE 9.40

Kinematical diagrams of single- and double-stage reduction gear drives.

(a)

(b) Th

Th h

Tin z 3

h

z4 Tout

Tin

z11

z8

Tout

(c) Th2

Th1

Tin1

z8

h

h

1

2 z9

Tout1

z10

z11

Tin2 FIGURE 9.41

MN-diagrams of reduction gear drives presented in Figure 9.40.

Tout2

Special Design Problems in Gear Drives

539

Tout = Tin

z4 . z3

Since, for the static balance, the sum of all torques applied to the unit must equal zero, the following expressions give us the Th value: Th =

(Tin + Tout ) =

z4 . z3

Tin 1 +

Two-stage reduction gear shown in Figure 9.40b has input shaft 6, output shaft 7, and two stages of speed reduction: high speed stage with pinion 8 and gear 9 and low speed stage with pinion 10 and gear 11. All the shafts and gears are mounted into common housing 12. The MN-diagram of this reduction gear can be made as shown in Figure 9.41b where it is considered as one unit, or as shown in Figure 9.41c where it is considered as two gear units connected with each other in a certain way. In the first case, the total gear ratio u equals: z9 z8

u = uHS uLS =

z11 z z = 9 11 . z10 z8 z10

Here uHS and uLS = gear ratios of highs speed and low speed stages, respectively. Thus, for the MNdiagram pictured in Figure 9.41b the output torque, taking into consideration Eqs. (9.47), equals: Tout =

Tin u =

Tin

z 9 z11 . z 8 z10

Let us consider now the option of MN-diagram shown in Figure 9.41c. Here the two-stage gear drive is considered as two separate drives: high speed (it is given number 1) and low speed (number 2). Their housings are stationary (does not matter together or separately), and the output shaft of drive 1 is connected to the input shaft of drive 2. (Note: These two drives may be designed as absolutely separate units. In this case, their housings would be connected through a common foundation, and the output shaft of unit 1 would be connected to input shaft of unit 2 through a coupling. Such a big change of the gear design would make no change in the MN-diagram.) Now we go back to Figure 9.41c and determine the torques: Tout1 = Tz9 =

Tin1 u1 =

Tin1

z9 z = Tin1 9 ; z8 z8

Since gears z9 and z10 are connected by a piece of shaft, the torques applied to the ends of the shaft are equal in modulus and have opposite directions, i.e., opposite signs: Tin2 = Tz10 =

Tout 2 = Tz11 =

Tout1 =

Tin2 u2 = Tin1

z9 z8

Tin1

z9 ; z8

z11 = z10

Tin1

z9 z11 . z8 z10

540

Dudley’s Handbook of Practical Gear Design and Manufacture

The total gear ratio of these two gears equals: u=

Tout 2 z z = 9 11 . Tin1 z 8 z10

As expected, the result is the same. This simple example was to demonstrate the procedure of operations with MN-diagrams. Note: The use of MN-diagrams for analysis of kinematic diagrams of drives with non-parallel shafts is not considered here because the issue of the torque directions needs in this case some additional detailed elaboration to avoid mistakes. Let’s create the MN-diagram for planetary gear drive 2CG type A presented in Figure 9.33. The “legs” of MN that represents a planetary gear drive should be titled a, s and c (meaning ring gear, sun gear and planet carrier, see Figure 9.39b). The relations between torques applied to this drive can be derived from the equations for the gear ratio. From Eqs. (9.33) and (9.47) we obtain: Ts usa

Tc =

c

=

Ts 1 +

za , zs

where Ts = torque applied to sun gear s. The driving member s and driven member c rotate in the same direction; hence, the gear ratio is a positive number here, while the output torque sign is opposite to that of the input torque. From Equation 9.34 we can write: Ta =

Ts usc

a

= Ts

za , zs

It is convenient to denote ratio za/zsas k, i.e., za/zs= k. In this case: Tc =

Ts (1 + k ); Ta = Ts k ;

(9.48)

As dictated by the static equilibrium requirement, the sum of all three torques (Ta, Ts, and Tc) must equal zero. From Eq. (9.48) also follows: Ts =

Tc

1 ; Ta = 1+k

Tc

k ; 1+k

(9.49)

As was previously noted, each Mechanical Node represents a gear drive or a set of gear drives with known relations between the torques applied to its three external connections (“legs”). If a complex gear drive consists of several MNs, their “legs” should be connected according to the mechanical diagram, and then, from the static equilibrium considerations, the dependence between the input and output torques that represents the total gear ratio can be established. Figure 9.38 presents six variants of connections of two planetary gear drives 2CG type A, with each design having its own formula for the gear ratio. Only for two of them (Figures 9.38a and f) the gear ratio can be determined in the simplest way: by determining the gear ratio of each planetary stage separately and then taking their product. The other options included in Figure 9.38 have one common feature: one of their outward shafts is connected to two drive members. For instance, in Figure 9.38b the right shaft is connected to two planet carriers (c1 and c2); in Figure 9.38c, the right shaft is connected to ring gear a1 and planet carrier c2, and so on. For such gear drives, the MN method of kinematic analysis proved to be very effective.

Special Design Problems in Gear Drives

541 12 10 13

14

15 6

3

7 1 2

4

5 11

FIGURE 9.42

8

9

7

Two-stage planetary aircraft speed reducer.

Example 9.16: Figure 9.42 [15] represents a two-stage planetary speed reducer of a turboprop aero-engine with a gear ratio of u = 11.5. Before starting with the MN-diagram, it is essential to understand the design of this complicated drive. Here sun gear 1 of the first stage is connected to the turbine by torsion shaft 2 (input shaft). Spline connections on both ends of this shaft make the sun gear floating and self-aligning relative to planet gears 3. The stem of planet carrier 4 of the first stage is connected by spline 5 with output shaft 6. Ring gear 7 of the first stage is connected with sun gear 8 of the second stage by cone disc 9 that is provided with splines both on the inner and the outer diameters. This enables the motion of disc 9 and the self-alignment of ring gear 7 and sun gear 8 relative to planet gears 3 of the first stage and planet gears 10 of the second stage. Planet carrier 11 of the second stage is connected by spline with stationary disc 12 and does not rotate. Hinge 13 that stops disc 12 against rotation is connected to a hydraulic torquemeter. Ring gear 14 of the second stage is connected through splined cone disc 15 with output shaft 6. Proceeding to the kinematical diagram language, we have here a gear drive combined of two planetary stages, 1 and 2, where the input shaft is connected to s1; planet carrier c1 and ring gear a2 are connected with the output shaft; ring gear a1 is connected with sun gear s2, and planet carrier c2 is fixed against rotation. Such a design is presented in Figure 9.38d, and its MN-diagram—in Figure 9.43. Now the torques applied to all elements of the gear drive can be determined. The input torque Tin = Ts1. Using Eqs. (9.48) we obtain the following: Tc1 =

Ts1 (1 + k1) ; Ta1 = Ts1 k1;

From the static balance of the member that connects a1 and s2: Ta1 + Ts2 = 0; Ts2 =

Ta1 =

Ts1 k1;

542

Dudley’s Handbook of Practical Gear Design and Manufacture Tout a2

c1 s1

s2

a1

c2 FIGURE 9.43 Figure 9.42.

Tin

MN-diagram of planetary gear drive shown in

From Eq. (9.48) we have the following for the second stage: Ta2 = Ts2 k2 =

Ts1 k1 k2;

The output torque Tout equals: Tout = Tc1 + Ta2 =

Ts1 (1 + k1) u=

Ts1 k1 k2 =

Ts1 (1 + k1 + k1 k2 );

Tout = 1 + k1 (1 + k2 ). Tin

Thus, we have obtained the gear ratio formula for this type of gear drive. Torques Tin and Tout have opposite signs, that means the input and output shafts rotate in the same direction. The same meaning has the positive sign of the gear ratio (because k1 and k2 are essentially positive numbers). The MN-diagrams of all the combinations of two-stage planetary gear drives presented in Figure 9.38 are given in Figure 9.44. The formulae for gear ratio obtained using the method demonstrated above are as follows: (a)

(d) s2

c1 s1

a1

c2

a2

(b)

a1

s2

c2

(e)

s1

a1

a2

s2

a2

a1

(c)

c1

s1

c2

s2

(f) c2

a1

FIGURE 9.44

s1

c2

c1

s1

a2

c1

c1

s2

s2

a1 a2

s1

c1

a2

c2

MN diagrams for kinematical diagrams of planetary gear drives shown in Figure 9.38.

Special Design Problems in Gear Drives

a. b. c. d. e. f.

u u u u u u

= = = = = =

543

(1 + k1)(1 + k2 ); 1 k1 k2; 1 + k2 (1 + k1); 1 + k1 (1 + k2 ); k2 k1 (1 + k2 ); k1 (1 + k2 ).

Notice: k1 = za1/zs1 and k2 = za2/zs2. Many other examples of kinematical analysis of complex planetary gear drives using MN method can be found in [14]. Let us consider a gear drive where the planetary stage is combined with regular drives.

Example 9.17: Figure 9.45 presents a two-path gear drive where pinions A and B transmit the power together from the input shaft to the output gear C. The tooth numbers of the pinions are equal, i.e., zA = zB. To ensure the equal load distribution between pinions A and B, mechanism called torque distributor is used. It consists of planetary gear drive 1 and regular drive 2 (gears E and D) that connects ring gear a with gear A. Gear B is connected to planet carrier c by a shaft. Let us determine the relations between the torques: • From Eqs. (9.48) for the planetary drive (k = za/zs): Tc =

Ts (1 + k ); Ta = Ts k ;

• For transmission between planet carrier c and gear B: TB =

Tc = Ts (1 + k );

• For transmission between ring gear a and gear A: TD =

TE = TD

output shaft

A

E

Ta =

zE = zD

Ts k ;

Ts k

zE ; zD

a p

E

C TC s

B D

FIGURE 9.45

c

input shaft

a s

Ts TC

D

c

A C

Ts

B

Kinematic and MN diagrams of a two-path gear drive with a torque distributor.

544

Dudley’s Handbook of Practical Gear Design and Manufacture

TA =

TE = Ts k

zE ; zD

The equal load distribution between pinions A and B requires the equality of their torques: TA = TB. Using this and the expressions above, we obtain the required relation between the tooth numbers: zE 1+k = . zD k

The output torque: TC = (TA + TB )

zC z z z = 2Ts k E C = 2Ts (1 + k ) C . zA zD z A zA

Accordingly, the total gear ratio of the entire gear drive equals: us

9.4

C

=

TC = Ts

2(1 + k )

zC . zA

EFFICIENCY OF CYLINDRICAL INVOLUTE GEAR DRIVES AND COMPLEX DRIVING SYSTEMS

Estimation of gear drive efficiency is essential because in many cases the efficiency is the governing factor in choosing the type of drive. Also, it is necessary for design of cooling systems.

9.4.1 EFFICIENCY

OF A

SINGLE GEAR PAIR

The power losses in gear drives are mainly caused by friction between the meshing teeth and in the bearings. Smaller part of the energy is spent for the oil agitation (churning losses), windage, and friction in the contact seals. This smaller part may be considerable in high-speed drives, especially at partial load when the oil temperature sinks and the churning losses grow significantly. But in the high-speed drives the friction losses in the teeth contact are less because of thicker oil film and lesser coefficient of friction, so the efficiency, i.e., the total power lost relative to the power transmitted, remains nearly the same and is usually calculated using simplified formulae or on the basis of testing. Detailed recommendations for estimation of power losses can be found in technical literature [16]. The designer can minimize the power losses by certain means: for example, by making smoother the working flanks of the teeth to lower the friction coefficient, avoiding ribs on the rotating parts to decrease the windage and churn losses, choosing a more effective lubrication system to diminish the effect of the oil jamming between the teeth, increasing the pressure angle in the mesh to reduce the slip velocity and increase the rolling speed in the tooth contact, and so on. After all, the gear drives are studied quite well, and their features, including the limits of efficiency variation, are known. Figure 9.46 presents the efficiency values achieved in different kinds of gear drives (data provided in Reference [3] are used). Here numbers 1 to 3 are related to high-power gear drives with helical gears, plain bearings and cir­ culating lubrication system (1—teeth hobbed and lapped; 2—teeth case hardened and ground; 3—planetary gear drive with floating sun gear, teeth shaved and nitrided). Numbers 4 to 6 are related to industrial gear drives with rolling bearings and submerged lubrication (4—cylindrical gearing, teeth hobbed; 5—cylindrical gearing, teeth case hardened and ground; 6—spiral bevel gearing, teeth hardened

Special Design Problems in Gear Drives

545

3.0

2.5

2.0 % 1.5

1.0

0.5

0 1

2

3

4

mesh power loss

5

6

7

8

total power loss

FIGURE 9.46 stage).

Power loss ψ in different gear drives (% per

and lapped). Numbers 7 and 8 are related to vehicular transmissions (7—cylindrical gearing, teeth shaved and hardened; 8—spiral bevel gearing, teeth hardened and lapped). We can see that the mesh power loss in high-power drives is relatively high and ranges from 1% to 1.5% though the friction coefficient between the teeth is low due to liquid friction in contact. This can be explained by increased churning losses in the mesh. The total power losses are also quite high because the system of lubrication and heat removal in such drives consumes remarkable amount of power. Industrial gear drives, with lower speed and splash lubrication, have lesser power losses both in mesh and total. Bevel gear drives have lesser efficiency than the cylindrical ones. The efficiency of a single-stage gear drive equals: =1

,

where η = efficiency; ψ = power loss ratio: =

Pout ; Pin

=

Plost P Pout = in ; Pin Pin

where Pout = output power; Pin = input power; Plost = lost power. The goal of this section is the estimation of efficiency of complicated multi-stage gear drives provided that the rate of mesh power loss ψ in every single stage is known. Therefore, hereafter the simplest formula for the mesh power loss offered in [2] will be used: 2.3 f

1 1 ± z1 z2

(9.50)

where f = friction coefficient; z1 = pinion teeth number; z2 = gear teeth number. Sign “+” shall be used for external engagement, sign “–” for internal engagement. Equation (9.50) is an approximation. First of all, factor 2.3 is most adequate for gears where the pitch point is located approximately in the middle of the length of action. When it is not so, this

546

Dudley’s Handbook of Practical Gear Design and Manufacture

factor may have to be corrected. Secondly, the friction coefficient f is variable. It depends on numerous factors, such as the smoothness of the teeth flanks, the contact stress magnitude, the oil viscosity and rotational speed. Experiments have shown that in typical gear drives f ≈ 0.05…0.1. For example, if both gears have external toothing and z1 = 20, z 2 = 80, and f = 0.08, then the mesh power loss equals: = 2.3 0.08

1 1 + 20 80

= 0.0115 = 1.15%.

This is approximately in the range of the mesh power loss shown in Figure 9.46. To take into account also the energy loss in the rolling bearings, we suggest to take for the efficiency calculations f = 0.10. More exact determination of the gear drive efficiency, if needed, should be made using more detailed methods of calculation or experimentally. If the gear drive contains more than one gear pair (for example, a pinion, an idler and a gear), or it consists of two or more stages connected in series, its efficiency equals the product of efficiencies of all gear pairs transmitting the power. If, for instance, a gear drive is composed of three stages with their efficiencies η1, η2 and η3, its total efficiency η equals: =

9.4.2 EFFICIENCY

OF

1 2 3

PLANETARY GEAR DRIVES

The formulae for the efficiency of planetary gear drives are taken from sources [2] and [13]. The power loss ratio at fixed planet carrier ( c ) is calculated using Eq. (9.50) which, in essence, allows for the mesh losses only. The energy losses in the bearings, churning losses, windage and others, can be calculated separately and added to the mesh losses or taken into account by increasing the friction coefficient f “by eye,” based on the experience: for example, to f = 0.12. The efficiency is different at different loads because different kinds of energy losses depend on the load level differently. Therefore, in important cases more detailed calculations of the energy losses must be undertaken and then verified by experi­ mental investigations. 9.4.2.1 Efficiency of Planetary Gear Drives 1CG (Figure 9.31) With ring gear a fixed and planet carrier c driving, a

=

c 1 a 1 + |uc p|

c

,

where: |uca p| =

zp za

zp

,

(see Eq. (9.32) and Figure 9.31); ψc = power-loss ratio at a fixed planet carrier c is obtained from Eq. (9.50):

Special Design Problems in Gear Drives

547 c

1 zp

= 2.3 f

1 . za

9.4.2.2 Efficiency of Planetary Gear Drives 2CG Type A (Figure 9.33) The power-loss ratio ψc at the planet carrier c fixed (“star gear,” Figure 9.33b) is obtained from Eq. (9.50): c

= 2.3 f

1 2 + zs zp

1 za

(9.51)

The efficiency of gear drive with fixed planet carrier (Figure 9.33b) equals: c s a

=

c a s

c.

=1

Note: Eq. (9.51) is not exact. This type of a planetary gear drive, when the planet carrier is fixed, is just a regular gear drive with a pinion (sun gear s), idler (planet gear p) and gear (ring gear a). For the power loss calculation, this drive should be considered as two single-pair gear drives: gear pair s – p and gear pair p – a. Their power loss ratios and efficiencies are as follows:

s p

= 2.3 f

1 1 + ; zs zp

s p

=1

p a

= 2.3 f

1 zp

p a

=1

1 ; za

s p;

p a;

The total efficiency of these two drives equals the product of the partial efficiencies: s a

=

s p p a

= (1

s p )(1

p a)

=1

(

s p

+

p a)

+

s p p a

1

(

s p

+

p a ).

Member ψs-p∙ψp-a is neglected. It is very small because the partial ψ values are as small as 0.01…0.02. If ring gear a is fixed (Figure 9.33a), a s c

=

a c s

=1

usc a 1 + usc

c

=1

a

k 1+k

c

(9.52)

z

Here usc a = za = k , see Eq. (9.33). s If sun gear s is fixed (Figure 9.33c), s a c

=

s c a

=1

c

1+

usc a

=1

c

1+k

(9.53)

548

Dudley’s Handbook of Practical Gear Design and Manufacture

For these drives, the efficiency is independent on whether it is used as a speed reducer or a speed increasing drive. 9.4.2.3 Efficiency of Planetary Gear Drives 2CG Type B (Figure 9.34) In this case, the power-loss rate at fixed carrier c (Figure 9.34b) equals: c

1 1 1 + + zs z p1 z p2

= 2.3 f

1 za

and c s a

c a s

=

=1

c.

If the ring gear a is fixed (Figure 9.34a), a s c

=

a c s

=1

usc a 1 + usc

c; a

If the sun gear s is fixed (Figure 9.34c), s a c

Here usc

a

=

z a z p1 z p2 z s

=

s c a

=1

c

1 + usc

; a

, see Eq. (9.37).

9.4.2.4 Efficiency of Planetary Gear Drives 2CG Type C (Figure 9.35) The 2CG gear drives types A and B considered above have the gear ratio at fixed planet carrier uc < 0, and this determines their relatively high efficiency. Gear drives 2CG type C, with uc > 0, on the contrary, are characterized by a low efficiency. However, they have the important advantage that a very big gear ratio, such as several hundreds or even thousands, can be achieved in one stage, though with low efficiency. The power-loss ratio with carrier c fixed ψc equals: • for Figure 9.35a: c

= 2.3 f

1 1 1 1 + + + ; z s1 zs2 zp1 z p2

c

= 2.3 f

1 1 + zp1 zp2

• for Figure 9.35b and c:

• for Figure 9.35d:

1 z a1

1 z a2

Special Design Problems in Gear Drives c

549

1 1 + zs z p2

= 2.3 f

1 zp1

1 . za

Correspondingly, at planet carrier fixed, the efficiency equals: c

c.

=1

With other fixed members, the efficiencies are as follows: • for Figure 9.35a with fixed gear s1:

1 uss21

Here ucs1 s2 =

c

, uss21

c

1

s1 c s2

=

a1 s2 c

=1

1+

;

|ucs1 s2

1|

c

uca1 s2

1

c;

uca1 a2

1

c

uca1 a2

1

c;

is given in Eq. (9.39).

• for Figure 9.35b and c with ring gear a1 fixed:

Here uca1 a2 =

1 uaa21

c

, uaa21

c

1

a1 c a2

=

a1 a2 c

=1

1+

;

is given in Eq. (9.40).

• for Figure 9.35d with ring gear a fixed:

Here uca s =

1 , usa c

usa

c

1

a c s

=

a s c

=1

1+

;

|uca s

1|

c

uca

1

c;

s

is given in Eq. (9.41).

Example 9.18: Planetary gear drive 2CG type C according to Figure 9.35c with numbers of teeth as follows: za1 = 150, zp1 = 120, za2 = 147, zp2 = 117. Ring gear a1 is stationary. What are the gear ratio and efficiency if the driving members are planet carrier c or ring gear a2? From Eq. (9.40), if ring gear a2 is driving: uaa21

c

=

a2 c

=1

z p2 z a1 z a2 z p1

=1

117 150 = 0.00510204; 147 120

550

Dudley’s Handbook of Practical Gear Design and Manufacture

If the planet carrier is driving, then: c

uca1 a2 =

=

a2

1

= 196.

uaa21 c

From the formulae above, the efficiency in the case of planet carrier c driving equals: a1 c a2

c

= 2.3 f

1 1 + zp1 zp2

1 z a1

=

1 1+

|uca1 a2

c

1|

;

1 1 1 = 2.3 0.1 + z a2 120 117

a1 c a2

=

1 1 7.85 10

1 + 196

4

1 150

1 147

= 7.85 10 4;

= 0.867;

The efficiency in the case of ring gear a2 driving equals: a1 a2 c

=1

uca1 a2

1

c

=1

196

1 7.85 10

4

= 0.847.

9.4.2.5 Efficiency of Planetary Gear Drives 2CG Type D (Figure 9.36) For Figure 9.36b: s2 c s1

where

c

=

s2 s1 c

= 2.3 f

usc1

=1

usc1 usc1

s2

c,

1

s2

1 2 2 1 + + + ; z s1 z p1 z p2 z s2

s2

=

zs2 . z s1

Note: This kind of gear drive with the planet carrier fixed should be considered as three single-pair gear drives connected in series. But also, here the note to Eq. (9.51) remains valid, that is why the simplified formula for ψc is used). For Figure 9.36c: a c s

where

c

=

a s c

= 2.3 f

=1

usc usc

a

a

1

1 2 2 + + zs z p1 z p2

c,

1 ; za

Special Design Problems in Gear Drives

551

usc

=

a

za . zs

For Figure 9.36d: a2 a1 c

c

where

a2 c a1

=

uac1

=1

uac1

a2

c,

1

a2

1 2 2 + + z a1 z p1 z p2

= 2.3 f

uac1

a2

=

1 ; z a2

z a2 . z a1

Example 9.19: Planetary gear drive 2CG type D according to Figure 9.36c with numbers of teeth as follows: za = 81, zp1 = 26, zp2 = 25, zs = 27. Ring gear a is stationary. What are the gear ratio and efficiency if the driving member is sun gear s? From Eq. (9.37), the gear ratio equals: usa

c

a s c

c

a c s

=

= 2.3 0.1

=

a c s

2; uca usc

=1

usc

a

=

1

s

=

0.5;

c;

1 = 0.04177; 81

za 81 = = 3; zs 27 3

=1

a

a

1 2 2 + + 27 26 25 usc

a s c

81 = 27

za =1 zs

=1

3

1

0.04177 = 0.937.

9.4.2.6 Efficiency of Planetary Gear Drives 3CG, Figure 9.37 (the Mesh Loss Only) (da1> da2): a1 s a2

1+

(

0.98 usa1a2 1 + z a1 / z s

1

)

c a 2 a1

;

552

Dudley’s Handbook of Practical Gear Design and Manufacture a1 a2 s

usa1a2 1 + z a1/z s

0.98 1

c a1 a 2

.

(da1< da2): 0.98

a1 s a2

a1 a2 s

usa1a2

1+

1 + z a1 / z s

; c a1 a 2

usa1a2 +1 1 + z a1/ z s

0.98 1

c a 2 a1

.

Here: c a2 a1

=

c a1 a2

1 1 + zp1 z p2

= 2.3 f

1 z a1

1 ; z a2

usa1a2 – see Eq. (9.45) for Figures 9.37a and b, and Eq. (9.46) for Figures 9.37c and d. Figure 9.37e: the formulae for efficiency are the same as for Figure 9.37a, but: c a1 a 2

=

c a 2 a1

= 2.3 f

1 2 + z a1 zp

1 , z a2

and the friction coefficient f shall be increased because of increased profile shift in the meshing of ring gears with the planet gear. Example 9.20: Planetary gear drive 3CG according to Figure 9.37a with the numbers of teeth as follows: zs = 12, za1 = 84, za2 = 75, zp1 = 36, zp2 = 27. Ring gear a1 is stationary. What are the gear ratio and efficiency if the driving member is sun gear s or ring gear a2? From Eq. (9.45), usa1a2 =

c a1 a 2

=

c a 2 a1

a1 s a2

a1 a2 s

s

=

a2

1

= 2.3 0.1

1+

(

1 + 84/12 = 50; (84 27)/(36 75)

1 1 + 36 27 0.98

50 1 + 84 / 12

0.98 1

)

1 84

1 0.0091

1 = 0.0091; 75 = 0.935.

50 0.0091 = 0.916. 1 + 84/12

Special Design Problems in Gear Drives

553

1.0

0.9

0.8

0.7

zp

2

=6

0

0.6 35

0.5

45

27

0.4 20

15

0.3

0.2

0 50 100 200 300

400 500 600 u

700 800

900 1000

FIGURE 9.47 Efficiency of planetary gear drives 3CG with the allowance for energy losses in rolling bearings (sun gear driving) [ 2].

This calculation takes into account the mesh losses only. The efficiency of 3CG planetary gear drives with allowance made for the energy losses in meshes and rolling bearings are shown in Figure 9.47 [2]. For this example, the efficiency obtained from Figure 9.47 equals sa1a2 ≈ 0.87.

9.4.3 EFFICIENCY

OF

PLANETARY

AND

COMPLEX GEAR DRIVES BUILT UP

OF

TWO

OR

MORE STAGES

If the stages are connected in series, so that the output shaft of the previous stage is coupled to the input shaft of the following stage and each of the two outward shafts is connected to only one gear element, the total efficiency can easily be obtained as the product of the efficiencies of all stages. Example 9.21: We shall calculate the efficiency of a two-stage gear drive shown in Figure 9.38f. It consists of two drives: star gear drive (first stage, Figure 9.33b) and planetary gear drive 2CG type A (Figure 9.33a). The tooth numbers are as follows: zs1 = zs2 = 18; zp1 = zp2 = 21; za1 = za2 = 60. From Eq. (9.50), for the first stage (star gear drive): c 1

= 2.3 0.1

1 1 + 18 21

1 = 0.031. 60

Since the first stage has a fixed planet carrier, its efficiency equals:

554

Dudley’s Handbook of Practical Gear Design and Manufacture 1

c 1

=1

=1

0.031 = 0.969.

For the second stage, since the numbers of teeth are identical, c 2

c 1

=

= 0.031.

However, in this stage the ring gear is fixed, thus from Eq. (9.36) a 2

=1

1

c 2 z + z s2 a2

=1

0.031 1+

18 60

= 0.976.

And finally, the efficiency of the entire drive: =

c a 1 2

= 0.969 0.976 = 0.946

However, if one of the outward shafts is connected to two gear elements (similar to the drives in Figure 9.38b–e), the easiest way to calculate the overall efficiency is to make use of the same MN diagrams that have been already used for the gear ratio determination. Below we explain the algorithm of this calculation by presenting several examples, but first some general rules should be stated [2,14]: 1. The combined drive as a whole has two outward shafts, driving (input) and driven (output), and one fixed member. (The last can be just the common foundation of the drive members.) One of the planetary stages that contains the outward shaft with only one connection to a drive element should be regarded as the main stage, and its three elements (“legs” in the MN terminology) should be denoted by X, Y, and Z. 2. The members of the main planetary stage should be denoted as follows: • the outward shaft of this stage (remember: it is connected to only one drive element!) shall always be denoted by Z; • if shaft Z is connected to planet carrier c, sun gear s should be denoted by X, and ring gear a by Y; • if shaft Z is connected to ring gear a, sun gear s should be denoted by X, and planet carrier c by Y; • if shaft Z is connected to sun gear s, ring gear a should be denoted by X, and planet carrier c by Y. 3. The other outward shaft of the complex drive that is connected to two drive elements shall be denoted by W. Thus, if shaft Z is driving, shaft W is driven; and vice versa: if shaft Z is driven, shaft W is driving. 4. The efficiency of this type of gear drives is obtained from the following formulae: • If shaft Z is driving and connected to planet carrier c of the main stage,

Z W

=1

uW

Z

uZY

W

uZY

X uZ W

c m

+ uZY

W

X

+ uZX

• If shaft Z is driven and connected to planet carrier c of the main stage,

W

Y

;

(9.54)

Special Design Problems in Gear Drives

W Z

555

1

= 1 + uW

uZY W

Z

;

uZY X uZ W

c m

+

uZY W

X

+

uZX W

(9.55)

Y

• If shaft Z is driving and not connected to planet carrier c of the main stage:

Z W

=1

uW

Z

uZ

W

uY

c m

W

+ uZY

W

X

+ uZX

W

Y

;

(9.56)

• If shaft Z is driven and not connected to planet carrier c of the main stage, W Z

=

1 1 + uW

Z

( uZ

W

uY

W

c m

+ uZY

W

X

+ uZX

W

Y)

;

(9.57)

Following is the physical interpretation of the members of Eqs. (9.54 to 9.57): 1

T

• uZ W = Z = TW ; uW Z = u = gear ratios between shafts Z and W and between shafts W and W Z Z W Z, correspondingly; • uZY W = gear ratio between shafts Z and W with member Y fixed; • uZX W = gear ratio between shafts Z and W with member X fixed; • mc = power loss ratio of the main planetary gear with planet carrier fixed, see Eq. (9.50) and all developments; • X = power loss ratio of the transmission between member X and shaft W; • Y = power loss ratio of the transmission between member Y and shaft W; 1 • uZY X = 1 + k , where k1 = za1/zs1 – kinematic parameter of the main planetary gear, used in Eqs. 1 (9.54) and (9.55) only; • uY W = gear ratio of the transmission between member Y and shaft W, used in Eqs. (9.56) and (9.57) only. Now, following the rules presented, we can proceed with the examples.

Example 9.22: It is required to calculate the efficiency of a two-stage planetary gear drive presented in Figure 9.48 with the following teeth numbers: zs1 = 16, zp1 = 20; za1 = 56; zs2 = 18; zp2 = 27; za2 = 72. As previously stated, planetary gear drive 1 is selected as the main one because its outward shaft is connected to only one drive member (c1). This shaft we denote by Z, member s1 by X and member a1 by Y. Shaft W receives torques from two members: s1 and s2, we denote those torques by TW1 and TW2 respectively. Now the torques on all the gear members can be determined starting from shaft Z with TZ = 1 and using Eqs. (9.48) and (9.49). k1 =

z a1 56 z 72 = = 3.5; k2 = a2 = = 4; z s1 16 z s2 18

TZ = Tc1 = 1; TS1 = TW1 =

Tc1

1 = 1 + k1

0.222;

556

Dudley’s Handbook of Practical Gear Design and Manufacture

a2

a1

p2

p1 a1

a2 s2

s1

c2

c1

c2

s2

s1 TW2

TW1

c1 TZ

TW

FIGURE 9.48

Kinematic and MN diagrams to Example 9.22.

Ta1 =

Tc1

k1 = 1 + k1

0.778;

(For chekup: Tc1 + Ts1 + Ta1 = 1

0.222

Ta2 =

1 0.778 = = 0.194. k2 4

Ta1 = 0.778; Ts2 = TW 2 = Ta2

TW = TW1 + TW 2 =

0.222 + 0.194 =

0.028; uW

0.778 = 0.)

Z

=

TZ = 35.7. TW

Notes: 1. If we take more decimal digits in our calculation, we will obtain more exact value of the gear ratio: in this case, uW-Z = 36. 2. The fact that torques TW1 and TW2 have opposite signs is noteworthy. Only a part of the larger torque TW1 that equals (TW1 – TW2) is transmitted to shaft Z, while the lesser of the two torques (TW2) circulates in the mechanism. The circulating power may exceed vastly the transmitted power and cause great energy losses. The ratio between the parasitic torque TW2 and the useful torque equals: TW 2 0.194 = = 6.93. TW1 + TW 2 0.028

That means, the parasitic torque is about 7 times larger than the useful output torque from the gear drive; consequently, the efficiency of this gear cannot be high. To calculate it, we shall use Eq. (9.54) if shaft Z is driving and Eq. (9.55) if shaft Z is driven. Here are the members of those equations: uW

Z

= 36; uZ

W

=

1 = 0.0278; uZY 36

X

=

TX = 0.222; TZ

Special Design Problems in Gear Drives

uZX

c m

W

557

TW 2 = TZ

=

= 2.3 0.1

0.194; uZY

1 1 + 16 20

W

TW1 = 0.222; TZ

=

1 = 0.0333; 56

X

= 0.

(ψX = 0 because the transmission between member X and shaft W is just a shaft with no losses of energy.) As we can see from the MN diagram in Figure 9.48, the power is transmitted between member Y and shaft W through planetary gear 2. Since its carrier c2 is fixed, Y

Z W

=1

W Z

=

=

c 2

= 2.3 0.1

1 1 + 18 27

1 = 0.0266; 72

36( 0.222

0.222 0.0278 0.0333 + 0.222 0 + 0.194 0.0266) = 0.555;

1 + 36( 0.222

1 = 0.692. 0.222 0.0278 0.0333 + 0.222 0 + 0.194 0.0266)

Thus, if shaft W is driving, the gear ratio uW-Z= 36 and the efficiency ηW-Z= 0.692. If shaft Z is driving, uZ-W= 0.0278 and ηZ-W= 0.555. Example 9.23: The two-stage planetary gear drive shown in Figure 9.39 has the following tooth numbers: zs1 = 16, zp1 = 20; za1 = 56; zs2 = 18; zp2 = 27; za2 = 72. Drive 1 should be selected as the main one because its outward shaft is connected to one drive member (s1) only. This shaft we shall denote by Z, member a1 by X and member c1 by Y. Shaft W receives torques from two members: c1 and a2, we denote them TW1 and TW2 respectively. Now the torques on all the gear members can be determined starting from shaft Z and using Eqs. (9.48) and (9.49). k1 =

z a1 56 z 72 = = 3.5; k2 = a2 = = 4; z s1 16 zz 2 18

TZ = Ts1 = 1; Tc1 = TW1 =

Ts1 (1 + k1) =

4.5;

Ta1 = Ts1 k1 = 3.5; (For checkup: Ts1 + Tc1 + Ta1 = 1 Ts2 =

Ta1 =

TW = TW1 + TW 2 =

4.5 + 3.5 = 0.)

3.5; Ta2 = TW 2 = Ts2 k2 = 4.5

14 =

18.5; uZ

3.5 4 =

W

=

14;

TW = 18.5. TZ

Torques TW1 and TW2 have identical signs, thus the efficiency of this gear drive is expected to be high.

558

Dudley’s Handbook of Practical Gear Design and Manufacture

To calculate the efficiency, we should use Eqs. (9.56) and (9.57) because shaft Z is connected to the sun gear, not to the planet carrier. Below are the members of those equations: uZ

W

= 18.5; uW

uZX

c m

W

=

= 2.3 0.1

=

Z

1 = 0.054; uY uZ W

TW1 = 4.5; uZY TZ 1 1 + 16 20

W

W

= 1;

TW 2 = 14; TZ

=

1 = 0.0333; 56

Y

= 0.

(ψY = 0 because the transmission between member Y and shaft W is just a shaft.) As we can see from the MN diagram in Figure 9.49, the power transmission between member X and shaft W is performed by planetary gear drive 2. Since its carrier c2 is fixed, X

Z W

=1

W Z

=

=

c 2

1 1 + 18 27

= 2.3 0.1

1 = 0.0266; 72

0.054 ( 18.5

1 0.0333 + 14 0.0266 + 4.5 0) = 0.948;

1 + 0.054( 18.5

1 = 0.951. 1 0.0333 + 14 0.0266 + 4.5 0)

Thus, if shaft Z is driving, the gear ratio uZ-W= 18.5 and the efficiency ηZ-W= 0.948. If shaft W is driving, the gear ratio uW-Z= 0.054 and ηW-Z= 0.951. Example 9.24: Let us calculate the efficiency of a two-stage planetary gear drive presented in Figure 9.50 with the following tooth numbers: zs1 = 22, z p1 = 22; za1 = 65; z s2 = 21; z p2 = 21; za2 = 62.

a1

a2

p1

p2 a1

s1 c1

s2 c2

s1 TZ

s2 c1 TW1

a2 TW2 TW

FIGURE 9.49

Kinematic and MN diagrams to Example 9.23.

c2

Special Design Problems in Gear Drives a1

559

a2

p1

p2 c1

s1

s2 c1

a1

c2

c2 s1

s2 TW2

TW1

TZ

a2

TW

FIGURE 9.50

Kinematic and MN diagrams to Example 9.24.

As established above, we select drive 1 as the main one because its outward shaft is connected to one drive member (a1) only. This shaft we denote by Z, member s1 by X and member c1 by Y. Shaft W receives torques from two members: s1 and s2, we denote those by TW1 and TW2 respectively. Now the torques on all the gear members can be determined starting from shaft Z and using Eqs. (9.48) and (9.49). k1 =

z a1 65 z 62 = ; k2 = a2 = ; z s1 22 zs2 21

TZ = Ta1 = 1; Ts1 = TW1 = Ta1

Tc1 =

Ta1

1 + k1 = k1

Ts2 = TW 2 =

Tc2

1.338461538; Tc2 =

1 = 1 + k2

W

=

Tc1 = 1.338461538;

1.338461538 = 1 + 62/21

TW = TW1 + TW 2 = 0.338461538

uZ

1 22 = = 0.338461538; k1 65

0.338646895;

0.338646895 = 0.000185357;

TW = 0.000185357; uW TZ

Z

=

TZ = 5395 ; TW

Note: the high precision of calculation of TW1 and TW2 (with many decimal digits) is required because their difference is very small, and it reflects the gear ratio of the drive. Torques TW1 and TW2 have opposite sings, so a considerable power loss is expected. TW 2 0.338646895 = = 1827. TW1 + TW 2 0.000185357

Thus, the circulating torque is 1827 times greater than the useful torque; consequently, the efficiency must be very low. To calculate the efficiency, we should use Eqs. (9.56) and (9.57), because shaft Z is connected to the ring gear. Below are the members of those equations:

560

Dudley’s Handbook of Practical Gear Design and Manufacture

uZ

W

uY

uZX

W

= 0.000185357; uW

W

TW 2 = TY

=

c m

= 2.3 0.1

1 2 + 22 22

= 5395;

0.253012048;

TW 2 = 0.338646895; uZY TZ

=

Z

W

=

TW 1 = TZ

1 = 0.0278; 65

0.338461538;

X

= 0.

(ψX = 0 because the transmission between member X and shaft W is a piece of shaft.) As shown in the MN diagram, the transmission between member Y and shaft W is performed by planetary gear drive 2. As its ring gear a2 is fixed, Eqs. (9.35) and (9.36) should be used to determine the ψY value: c 2

a 2

=1

1 2 + 21 21

= 2.3 0.1 c 2

1 + z s 2 / z a2 Y

=1

=1 a 2

1 = 0.0291; 62 0.0291 = 0.9783; 1 + 21/62

= 0.0217.

Now, we can calculate the efficiencies: Z W

=1

W Z

=

5395( 0.000185357 + 0.253012048 0.0278 + 0.338646895 0.0217) =

76.6;

1 = 0.0127. 1 + 5395( 0.000185357 + 0.253012048 0.0278 + 0.338646895 0.0217)

Notes: 1. The input shaft here can be shaft W only, so the direction of arrows in the MN diagram chosen in the beginning was incorrect. However, it does not play any role. 2. In the last two equations we omitted members uZY W X because X = 0. In the previous examples, these members were still kept in order to follow strictly the algorithm. Thus, if shaft Z is driving, the gear drive is self-locking because its efficiency ηZ–Wis negative. If shaft W is driving, the gear ratio uW-Z = 5395 and ηW-Z = 0.0127 = 1.27%. This efficiency looks very low. How can we check the correctness of this result? In this relatively simple case, we can easily calculate the power losses because from the calculation using MN diagram we know the torques applied to the drive members. Since we know also the rotational speeds, we can determine the useful power PU and the lost powers PL1 and PL2 in planetary stages 1 and 2, respectively. Let us begin:

Special Design Problems in Gear Drives

561

1. Useful power equals: PU = TZ

Z,

where TZ = 1 = torque applied to output shaft Z. So PU = ωZ. PL 2 = Ts2

s 2 (1

2 ).

2. Lost power in stage 2 equals: From the MN diagram we have already got Ts2 ≈ 0.3386, ωs2 = 5395ωZ and loss in this stage equals: PL 2 = 0.3386 5395

Z (1

PL1 = Ts1

s1 (1

0.9783) = 39.64

a 2

= 0.9783, so the power

Z.

1).

3. Lost power in gear drive 1 equals: From the MN diagram we know Ts1 ≈ 0.3385 and ωs1 = 5395ωZ. Concerning the power lost in this stage, we have (in principle) to take into consideration that ring gear a2 rotates, and this influences the relative speed of the gears and the amount of the lost power. However, in this case rotation of the ring gear is very slow, and it can be neglected. Considering this stage as 2CG type A gear drive with ring gear fixed, we use Eq. (9.52). The rate of power loss with planet carrier fixed was already calculated and equals ψmc = 0.0278. Thus, a1 s1 c1

=1

c m

1 + z s1/z a1

=1

0.0278 = 0.9792. 1 + 22/65

Hence, PL1 = 0.3385 5395

Z

(1

0.9792) = 37.99

Z.

4. The efficiency of the entire gear drive equals: =

PU Z = PU + PL1 + PL 2 (1 + 39.64 + 37.99)

= 0.0127. Z

Thus, the calculation of efficiency was correct.

Example 9.25: Figure 9.51 presents a continuously variable transmission (CVT) which consists of planetary gear drive 1, V-belt variator 2 and a pair of gears A and B. The following numbers of teeth are given for gears in this device: zA = 19; zB = 95; for the planetary stage: za = 74; zs = 16, zp = 29. It is required to calculate how does the total ratio uΣ = ωW/ωZ and efficiency change when the V-belt variator ratio changes from 1:3 to 3:1.

562

Dudley’s Handbook of Practical Gear Design and Manufacture

A 2 1 Z

W input shaft

output shaft

s

B

c a FIGURE 9.51

Continuously variable transmission with a belt variator ( Example 9.25).

The kinematic diagram of this transmission is presented in Figure 9.52. The following designations are used here: • h = housing of a component drive (along with its index); • W = input shaft of the entire transmission; • Z = output shaft of the entire transmission. The MN diagram of this device is given in Figure 9.53. Here the planetary stage is denoted by index P, gear pair A-B by index 1, and the belt variator—by index V. To determine the torques and the gear ratio of the entire transmission, we start as always from the outward shaft that is connected to only one member of the drive: with output torque TZ = Tc = 1. Then, the torques applied to members a and s of the planetary stage can be obtained from Eq. (9.49): Ts =

variator C

A

Tc

1 ; Ta = 1+k

k z 74 ;k= a = = 4.625. 1+k zs 16

p

Z

W TW

Tc

D

s X B

TZ

c a Y FIGURE 9.52

Kinematic diagram to Example 9.25.

Special Design Problems in Gear Drives

1

563

h1

A (A-B) B Y a

C

T c Z Z (planetary) s X T

hV V (variator) D TW2

P

W1

W TW

FIGURE 9.53 MN diagram to Example 9.25.

Thus, Ts = TW 1 =

1 = 1 + 4.625

0.1778; Ta =

4.625 = 1 + 4.625

0.8222;

(Here TW1 = part of the input torque TW that comes from sun gear s.) Further, we consider the shafts one after another: TB =

Ta = 0.8222; TA =

TB uB

TC =

A

TA =

= TB

zA 19 = 0.8222 = 0.1644. zB 95

0.1644.

From this point on, we should make our calculations separately for the different transmission ratios of the variator. • Option 1 (uV = ωC/ωD = 1:3): TD =

TC

C

= 0.1644

D

TW = TW1 + TW 2 =

1 = 0.0548 = TW 2; 3

0.1778 + 0.0548 =

0.123;

TZ 1 = = 8.13. TW 0.123

u =

Shafts W and Z rotate in the same direction. • Option 2 (uV = 3:1): TD =

TC

C D

= 0.1644 3 = 0.4392 = TW 2;

564

Dudley’s Handbook of Practical Gear Design and Manufacture

TW = TW1 + TW 2 =

u =

0.1778 + 0.4392 = 0.3154;

TZ = TW

1 = 0.3154

3.17.

Shafts W and Z rotate in the opposite directions. The diagram of 1/uΣ versus uV is given in Figure 9.54. (The1/uΣ value is presented instead of uΣ because in some interval uΣ grows immensely and, when the output shaft stops before its direction of rotation is changed, uΣ = ∞.) The expression for the transmission ratio is given by: u =

1 . 0.1778 0.1644uV

At uV = 1.0815, the denominator of this formula becomes zero, and the output shaft does not rotate. Let’s calculate the efficiency of this transmission at uV = 1:3 (Option 1). Following the re­ commendations stated above, we have defined in Figure 9.53 the output shaft connected with the planet carrier c as Z, sun wheel s as X, ring gear a as Y, and the input shaft as W. Then, we calculate the efficiency using Eq. (9.55): 1

=

W Z

1 + uW

Z

uZY

W

uZY

X uZ

;

W

c m

+ uZY

W

X

+ uZX

W

Y

Using the results of the above calculation of the transmission ratio, we obtain the following members of Eq. (9.55):

0.1

- 0.1

Z

W

1/u

0

- 0.2

- 0.3 0

1.0 C

FIGURE 9.54

Diagram of gear ratios to Example 9.25.

D

2.0 uV

3.0

Special Design Problems in Gear Drives

uW

Z

= u = 8.13; uZ

uZY

uZX

W

c m

=

565

=

X

TW 2 = TZ

= 2.3 0.1

TX = TZ

W

1 uW

= 0.123; Z

Ts = 0.1778; Tc

0.0548; uZY

1 2 + 16 29

=

W

=

TW 1 = 0.1778; TZ

1 = 0.027; 74

X

= 0.

Member Y (ring gear a) and shaft W are connected through gear pair A-B and the variator. Let us set the energy losses in the gear mesh as 2% and in the variator as 5%. Thus, Y

= 0.02 + 0.05 = 0.07.

Now, as we have calculated all the terms of Eq. (9.55), the efficiency can be determined: W Z

=

1 + 8.13( 0.1778

1 = 0.939. 0.1778 0.123 0.027 + 0.0548 0.07)

Many more examples of calculation of the gear ratio and efficiency of complex gear drives using MNdiagrams are given in [14].

9.5

LUBRICATION AND COOLING OF GEAR DRIVES

9.5.1 LUBRICATION The system of lubrication is intended to provide the rubbing surfaces with an oil film and take away the heat and wear debris. The kind of lubricant and method of gear lubrication depend in the first place on the pitch line velocity of a gear. Low speed gears, particularly those which are open (that means, they don’t have a sealed housing), are often greased. The grease is applied to the teeth by brush once a day or once a week depending on the need. Because the grease is unable to take off the heat from the teeth, this type of lubrication is used only for light-duty applications, where the small amount of heat may be successfully taken off from the gears by convective heat transfer. The recommended speed limit for the grease-lubricated open gears is up to 4 m/s. At a higher speed or continuous duty, the periodic manual greasing becomes too laborious and is replaced by electrically driven grease applicators or by dipping the gears into a pan filled with thick sticky oil. Gears which have a sealed housing are usually lubricated with oil. As compared with grease, oil is more heat resistant, long lasting, and easily replaceable. It washes away the wear debris from the wearing surfaces and, by filtering of the oil, the mechanism can be kept clean during long-term usage. The metallic wear debris can be continuously collected by a permanent magnet. It is particularly important for bearings, and especially for plain bearings, to be lubricated with clean oil. But the other wearing parts also have sufficiently lesser wear rate when the wear debris is perma­ nently removed from the friction surfaces. Oil transfers the heat from the teeth to the housing walls, and then it is removed into the outside ambient.

566

Dudley’s Handbook of Practical Gear Design and Manufacture oil level

(a)

(b)

1 2 3 4

(c)

(d)

FIGURE 9.55 Oil bath lubrication of gear drives.

The simplest method of gear lubrication is oil-bath lubrication, Figure 9.55. To diminish hydraulic losses, the gears should not be immersed too deeply, but not less than the tooth height. Therefore, the oil level in the housing should be checked while the gear is running because the oil splashed on the walls of the housing flows down slowly, and the oil level in operation may become considerably lower than it was in the non-operated gear drive. The author has observed recession of oil level in a reduction gear, after the machine was started up, by as much as 25 mm. If the oil gauge is placed far from the oil-immersed gear wheel, even a small inclination of the gear housing may result in lack of lubrication. That is why, the gear drive should be mounted level on its foundation, and the housing should be provided with special flat surfaces for the level gauge. To guarantee reliable bath lubrication in a multistage gearset, one gear of each stage should deep into the oil. For this purpose, in the low-speed stages the oil level can be increased as shown in Figure 9.55a. At higher speed, the oil level should be lowered to avoid excessive churning losses. The smaller gear wheel can be lubricated with an additional idler gear 1 (so called “rotaprint lubrication,” Figure 9.55b) made of gray cast iron or plastics which possess good wettability in oil. It rotates freely on an axle and supplies oil to the power train. The face width of the idler should be not less than 0.7-0.8 of the lubricated gear widths. If the width of the idler is relatively small, the load concentrates in the lubricated area, because this part of a tooth “protrudes” due to thicker oil film. This may result in severe pitting in the area of flooded lubrication. Sometimes the housing is designed with an inclined joint surface that enables equal dipping of the gears of all stages (Figure 9.55c). The diagram of a two-stage gear drive with gears located in vertical plane is presented in Figure 9.55d. Here the high-speed gear wheel 2 is lubricated by idler gear 3 which is mounted on the shaft of output gear 4 on a rolling or plane bearings. In this case, the rolling bearings of the upper and intermediate shafts should be grease lubricated. Rolling bearings are usually packed with grease while assembly in such a way as to be sure that all the voids are filled with grease. These bearings typically operate in enclosed volumes where the grease forced out of the bearing would be contained. About one half of this volume should also be filled with grease. If the operating life of the grease is shorter than the interval between the overhauls, certain means should be provided in the design for proper re-greasing. Figure 9.56a demonstrates design of a grease lubricated end bearing of a shaft. Here, ball bearing 1 is fixed on the shaft by retaining ring 2. In housing 3, it is fixed between partition 4 and ring 5. The partition seals the bearing chamber, and it is used also for delivery of grease through nipple 6 and channels in the partition. To prevent grease flow out of the right side, felt seal 7 is installed. On the left side, the bearing housing is closed by thin cover 8 attached by bolts to ring 5. When the bearing is re-greased, cover 8 should be removed, and the maintenance worker should ensure that the old grease is forced out of the bearing. The design with an overhanging shaft end is presented in Figure 9.56b. The space of bearing 9 is closed on the right side by partition 10 and two-stage labyrinth seal 11, and on the left side by cover 12

Special Design Problems in Gear Drives

(a)

567 6 3 4 7

8 1

(b) 12 9

11

2 5

10 14 13

FIGURE 9.56

Grease lubricated bearings [ 14].

incorporating a labyrinth seal. The cover is provided with an opening closed by lid 13. During regreasing, the lid should be removed to allow the old grease to flow out. Note: If the rolling bearing doesn’t require re-greasing (for example, because of short period of use of the drive, or because the bearing is provided with sealing shields and greased for life), the design of the bearing assembly can be simplified. If the axes of rolling bearings are placed in a horizontal plane, they are usually splash oil fed. It is believed that at oil viscosity less than 250 cSt, reliable splash-oil-feed is achievable when the centrifugal force of the oil drops is greater than their weight [9]. That means: mV 2 > mg; V > R

gR ,

where: V = linear speed of the mesh (m/s); R = radius of the gear; g = 9.81 m/s2 = acceleration of gravity. Example 9.26: When R = 0.2 m, the linear speed, for satisfactory oil sprinkling, should be not less than: V=

n=

9.81 0.2 = 1.4

m ; s

60 V 60 1.4 = = 67 rpm . D 0.2 2

If the oil is sprinkled on the walls of the housing, it doesn’t mean that it gets into the bearings. Investigation of splash oil feed for bearings has been performed on a two-stage reduction gear with a transparent housing made of acrylic plastic [17]. No special grooves were made to lead the oil to the bearings, but they were completely open and accessible for the air and oil splashes from within. Before each experiment, all the bearings were dismounted from the shafts and degreased. Then the experimental unit was assembled and filled up with oil of needed viscosity, so that the intermediate gear wheel was immersed into the oil as deep as two heights of the teeth (as shown in Figure 9.55a). Then the gear was

568

Dudley’s Handbook of Practical Gear Design and Manufacture section A-A lower part of housing

section B-B

gear face

upper part of housing

1

2

B A B A

1 FIGURE 9.57

2

1

gear face

Lubrication grooves and scrapers.

rotated for 5 min with a certain rotational speed in a certain direction. After that the gear was dismantled, the bearings dismounted from the shafts, and the presence of oil on the races and rolling elements of the bearings was checked by white filter paper. We have found that in many cases the bearings remained dry although inside the housing there was a real oil storm, particularly at higher speeds. The experiments have shown that without appropriate facilities, splash lubrication of the bearings cannot be guaranteed. There were cases in which (at a high speed) the oil streamed abundantly down the walls, but tore off the edge of the bearing seat and didn’t enter the bearing. When the gear was stopped, the oil flow was getting less intensive, and some small amount of oil was entering the bearing. But such a lubrication can be considered neither reliable nor satisfactory. Reliable oil supply to the bearings is achieved by oil-catching grooves 1 (Figure 9.57) made on the joint surfaces of the housing. In middle-sized gear drives, the grooves are 5–10 mm in depth and 7–15 mm in width. The grooves catch the oil that flows down the walls. Another kind of oil catchers are dripping pans presented in Figure 9.58; they collect the splashed oil that flows down the walls and lead it to the bearings. The design shown in Figure 9.58a looks simple, however, the casting of such housing and cleaning of the inner surfaces after casting are complicated. Design shown in Figure 9.58b is handier in this respect. Here plate 1 is attached with screws to the inner face of the bearing seat and creates the oil (a)

(b)

1

(c)

2

FIGURE 9.58

Oil collectors and lubrication holes.

Special Design Problems in Gear Drives

569

collector. If the housing is welded (Figure 9.58c), the oil collector 3 is machined in the bearing seat before welding. If the gear drive runs slowly and the sprinkling is insufficient, the oil can be taken off the oilimmersed gear face by a scraper (2 in Figure 9.57). The tip of the scraper can be installed at a distance of about 0.5 mm from the gear face. If the tip is made of hard rubber or plastic, it may touch the gear. The scraper shown in Figure 9.57 is designed so that it can take oil off the gear face and lead it to the oil groove independently of the gear’s direction of rotation. The oil grooves and ducts are configured so that the oil can flow to the oil reservoir only through the bearings. This way, bearing lubrication is assured. Gear drives designed to operate at a wide range of working conditions, such as rotational speed, direction of rotation, load, temperature etc., should have a lubrication system that assures the proper functioning at any specified condition. For small drives, it may be increasing the oil level until all the gears and bearings are partly or completely immersed. In these cases, the seals should be of good quality and reliability. For larger drives, a force-feed lubrication system (with an oil pump) is commonly used. The capacity of oil reservoir should be big enough to reduce the rate of oil decomposition and give the wear debris the place to settle. Depending on the power transmitted, the volume of oil is recommended in the range of 3.5–10 liter per kW of lost power. For instance, if a 2-stage gear drive transmits power of 100 kW, and its efficiency is 97%, the recommended oil content equals 100∙0.03(3.5 to 10) = 10.5 to 30 l. In practice, it is not always achievable. At smaller oil volume, the intervals between oil replacement is shorter, however, it depends on duty. In these cases, field experience is the best adviser. The oil level is mostly monitored by a dip stick. The oil surface in operation changes sufficiently, and it is neither plane nor level because the oil-immersed gears displace the upper layers of oil in the direction of their rotation. Therefore, it is important to place the dip stick closer to the immersed gear. Figure 9.59 presents a design of a dip stick that allows measuring the oil level while the gear drive is running. Pipe 1 protects the stick from the oil splashes. Rubber knob 2 of the stick serves as a plug. Hole 3 in the upper part of the pipe lets the air in or out of the pipe during the changing of the oil level, and thus enables the correct measuring. The forced-feed lubrication system is unavoidable if design or service conditions don’t allow sa­ tisfactory lubrication by immersion or splashing; for example, when the gear shafts are vertical, or when the speed is too high or too slow, or when the lubrication system contains oil filters, heat exchangers, and other devices that need oil circulation. Sliding bearings, usually, also require forced-feed lubrication to provide required oil flow through the bearing for satisfactory heat removal. Oil circulation is commonly provided by a gear-type pump or plunger pump, attached to the gear housing and driven by one of the gear shafts. The gear-type pump is preferable in reversible version, with appropriate system of valves, so that the lubrication system and the gear drive would not be damaged if the gear drive is accidentally reversed.

2

3 1 FIGURE 9.59

Dip stick.

570

Dudley’s Handbook of Practical Gear Design and Manufacture

4

6

3 5 7 8

oil level

1

2

FIGURE 9.60 Three-stage reduction gear drive with vertical shafts.

In middle-sized gear drives, the pump usually takes oil from the drive housing and supplies it through oil ducts to the gear mesh and the bearings. The capacity of the oil reservoir is usually chosen as large as the pump can supply in 0.5–2 min. depending on the design features. Example of forced-feed lubrication system for a reduction gear drive with vertical shafts is presented in Figure 9.60. Pump plunger 1 is set in motion by cam 2 mounted on the end of the intermediate shaft. The oil comes through ducts 3 and 4 into the upper housing, and there it is distributed among the bearings and gears. The low-speed stage is lubricated by sprayer 5. A small part of the oil flows through oil indicator 6, so that the oil flow can be seen inside a glass pipe. Rotating collar 7 press-fitted to the output shaft prevents the access of oil to the labyrinth sealing of the output shaft 8. In bigger gear drives, which transmit great powers, the lubrication system may include, besides the associated pumps, the following: • An electric pump for oil circulation before setting the machine in operation; • An oil tank with a capacity that equals the pump supply during 3–4 min. or more; • An oil cooler and, possibly, also a heater—to heat the oil before starting, lest the oil pressure in the system be too high in the beginning because of increased flow resistance; • Oil filters; • Indicator of metal particles in the oil; • Safety valves; • Manometers, thermometers, and alarm devices that alert the personnel when the pressure and/or the temperature of the oil deviate from the permissible values, and other devices. The recommended quantity of oil to be sprinkled on the gear teeth is about 5–8 l/min per kilowatt of mesh power loss, which averages between 1% and 1.5% of the power transmitted per stage. At this quantity, the oil warms up by 8–5°C, correspondingly. With forced-feed lubrication, uniform distribution of oil across the face width of the gears is achieved using tubular sprayers with several holes (Figure 9.61a). To avoid their clogging, the diameter of the holes should be not less than 0.8 mm, usually 1.2–3.0 mm. The oil flow through a single hole depends on its diameter, oil pressure, and oil viscosity. At the viscosity of 110 cSt, the oil flow Q equals approximately:

Special Design Problems in Gear Drives

571

(a)

(d)

A

(b)

A section A-A

(c)

FIGURE 9.61

Oil sprayers.

Q = 1.6 d 2 p l / min

where: d = diameter of the hole, mm; p = oil pressure, MPa (N/mm2). For example, if d = 2 mm and p = 0.3 MPa, Q = 3.5 l/min. Narrow gears may be lubricated by sprayers shown in Figure 9.61b made from a pipe with de­ formed end. In high speed gear drives, there may occur erosion of metal in places where the teeth hit the oil jet. In these cases, a sprayer as shown in Figure 9.61c may be useful. It fragments the oil jet and creates some mixture of air and small particles of oil. Flat and wide (up to 150–200 mm) oil jet is achieved using the sprayer presented in Figure 9.61d [17]. The most effective cooling of teeth is achieved when the oil is directed at the beginning of the teeth engagement as shown in Figure 9.62a. This method may be recommended up to speed 20 m/s for spur gears and up to 50 m/s for helical gears. For high-speed gears, particularly spur gears, this method is unsuitable, because when the oil is forced out of the tooth spaces in a very short time, there may be created a very high hydrodynamic oil pressure. This pressure occurs cyclically with the teeth engagement frequency and may lead to great radial loads, severe vibrations and failure of gear drive elements, such as bearings, shafts, housing, etc. In these cases, the oil should be sprinkled as shown in Figure 9.62b. If the oil pics up insufficient heat, additional oil sprayers can be installed to cool the teeth after their disengagement, see Figure 9.62c. In some cases, where it is required, the gear rim may be additionally chilled by sprinkling oil on its inner surface. The sprayers and all elements of the piping should be rigid and strong lest they be broken from vibrations. The natural frequencies of the elements of lubrication system should not be close to the teeth engagement frequency. It is desirable to locate the bolt connections so that undone nut, a piece of a broken cotter pin, or lock wire can’t fall and get into the gear mesh. Among the locking elements for the threaded joints, tab washers and locking plates are the best.

572

Dudley’s Handbook of Practical Gear Design and Manufacture

(a)

(b)

(c)

0.5

FIGURE 9.62

Directions of oil spray on gears.

When the prototype gear drive is assembled, the lubrication system should be pumped with oil to check whether the jets get exactly to the needed places. This is particularly important when the sprayers are distant from the lubricated surfaces. If the rotational speed of the gear changes in the course of operation, the pump delivery changes as well. In this case, the pump should be designed for excessive oil supply at high speed and provided with a bypass valve to keep constant oil pressure in the system. Modern gear drives, medium to large, which transmit high powers mostly employ forced-feed lu­ bricating systems. Those systems provide an accurate and controlled flowrate of lubricant to each running component of the drive. A typical forced-feed lubrication system for a gearbox is presented in Figure 9.63. Oil is drawn from the oil sump by main pump 1 through suction strainer 2. Generally, the main pump is driven by the gear drive input shaft. If the drive speed changes, the pump delivery changes proportionally, and at low speed the oil supply can be insufficient to sustain the minimal required flowrate in the lubrication system. Where such a problem exists, an auxiliary electric pump 3 can be installed in parallel to the main pump. This auxiliary pump should support and provide the required oil flowrate for the gearbox lubrication in case of extremely slow rotational speed of the input shaft or failure of the main pump. It also serves for pumping oil through the lubrication system before starting the machine, to warm the oil and to avoid the dry run in the first moments of the machine operation.

8

E

3 2

9

7

6

5

4

1

FIGURE 9.63 Diagram of a forced-feed lubrication system [ 14].

Special Design Problems in Gear Drives

573

The role of suction strainer 2 is to protect the pumps from the penetration of large foreign objects which could cause catastrophic damage and immediate failure of the pump. External piping or internal oil pas­ sages built in the gear drive casing connect the discharge of both pumps through check valves 4 to oil filter 5. In small systems, the replaceable filters of a paper type are used, while in larger systems, duplex filters with cleanable metal mesh elements are preferred. The oil is then directed through a heat exchanger 6 where it exchanges heat with the cooling media, typically coolant liquid or air. Relief/pressure regulating valve 7 controls and limits the oil pressure in the system. The oil is then discharged to the main oil gallery. External piping and internal oil passages connect to the oil gallery and direct oil for the lubrication of the different consumers: gears, bearings, couplings and clutches. The flowrate for each consumer is regulated by fixed or adjustable orifices 8. The oil lubricates and removes heat from the consumers and returns back to the sump by gravity. Sensors, such as flow, pressure and temperature transducers 9, are installed in critical locations and provide their output signals into the machine or the plant monitoring and control computer. Logic diagrams coded in the computer software react to the signals’ values and provide warnings, alarms, limitation of the operation or emergency shutdown in an attempt to protect the machine in case of a serious deviation from the nominal system operation values. In modern systems, novel sensors for monitoring of the oil condition were introduced. Those perform on-line analysis of the oil to detect metallic chips, debris, oil oxidation, changes in the oil acidity or additives depletion.

9.5.2 COOLING Almost the entire power lost in mechanisms turns into heat. This heat energy created inside the gear drive warms all the parts inside, the oil, the housing, and then it is dissipated into the air. The hotter the gear drive, the greater the amount of heat removed from it into the ambient per unit of time, and the drive temperature grows until the balance between the heat input and the heat removal is achieved. The task of the designer is to provide an effective cooling system which ensures the input-removal heat balance at acceptable temperature of oil in the drive. There are several ways of heat removal: • natural convection (always exists but not always adequate); • forced convection—due to use of fans, shaft-mounted or electric (electric fan that potentially can provide more effective cooling may be mounted on the gear drive housing); • use of pipes with running water installed in the oil sump (the source of water is required); • use of external oil-to-air heat exchangers (oil circulation system and a fan are required); • use of external oil-to-water heat exchangers (oil circulation system and a water source are required). Here only the first two ways of heat removal are discussed in detail. The calculation of cooling begins with the determination of heat input. The efficiency η of a gear drive is tentatively taken relying on the information on performance of similar devices, but finally, it should be determined experimentally. It is assumed that all the power loss turns into heat, so the heat input Pin is given by: Pin = P (1

) 103 W

(9.58)

Here, P = power on the input shaft of a gear drive, kW. The heat energy Pout removed from the gear drive due to convective cooling depends on the area A (m2) of the heat-exchange surfaces, the temperature difference Δt (°C) between the gear housing (tg) and the surrounding medium (ts), and also on convective heat transfer factor k W/(m2∙ °C) that shows how much heat energy can be removed from 1 m2 of a surface if the temperature difference Δt = 1°C:

574

Dudley’s Handbook of Practical Gear Design and Manufacture

Pout = A (tg

ts ) k = A t k W

(9.59)

The numerical value of factor k depends on the material of the housing, on how it is painted if at all, on how it is cleaned from dust and oil, on the condition of surrounding air and its speed relative to the heatexchange surface, and others. The rotational speed of the shafts affects the heat exchange very much: the higher the speed, the more intensive and uniform the heat transfer is from the oil to the housing walls and the greater the fanning effect of the rotating couplings. Besides, the k value obtained experimentally depends on the surface taken into account. For instance, the bottom of the housing is usually ineffective unless it is provided with enough space for the air exchange. Fanning and finning (in the direction of the air motion) can make the bottom surface extremely effective in heat dissipation, see Figure 9.64. The fins are less effective than the wall itself because their temperature decreases gradually with the distance from the wall. It is recommended to take into account only one half of the fin surface—say, only one side of a fin. The temperature of the housing is also not even: it is higher in the zone of the oil sump. Thus, the k value is an approximate magnitude obtained in experiments with real gear drives. According to experiments with worm drives [18], the heat transfer coefficient k is in the range from 10 to 25 W/(m2∙°C). The smaller values are recommended for painted casting of steel and gray iron, low rotational speed and dusty air. The greater values are suitable for aluminum non-painted housings, rotational speed of input shaft of 1500 rpm and more, and clear air. For blower cooling, k = 7 + 12 v W /(m2 °C ),

(9.60)

where v = air speed, m/s. For example, if v = 5 m/s, k = 34 W/(m2∙ °C). Example 9.27: The reduction gear shown in Figure 9.57 transmits continuous load of 30 kW. The efficiency of this gear drive η = 0.96, it works indoor, protected from solar heating, and the air temperature in summer ts = 30°C. The outer area of the housing A = 0.8 m2 (without the bottom). What can be the oil temperature without additional cooling means? From Eq. (9.58), the input heat energy is: Pin = 30 (1

0.96) 103 = 1200 W .

From Eq. (9.59), taking k = 18 W/(m2∙℃) (average between 10 and 25 W/(m2∙℃), see above), we determine the output energy: Pout = 0.8 18 t = 14.4 t W .

Equating Pin and Pout, we obtain the temperature difference between the housing and the air: t=

1200 = 83.3 °C . 14.4

Because the air temperature ts = 30°C, the oil sump temperature is: tg = 30 + 83.3 = 113.3 °C .

Such a high temperature is unacceptable, and the gear drive must be provided with some additional cooling device. Let’s mount a fan on the input shaft to blow on the side walls and the upper wall of the

FIGURE 9.64 Worm gear drive with bottom cooling fins and a fan.

Special Design Problems in Gear Drives 575

576

Dudley’s Handbook of Practical Gear Design and Manufacture

housing. The area A remains approximately the same. If the air speed equals 5 m/s, from Eq. (9.60) we obtain factor k = 34 W/(m2∙℃). Then the dissipated heat energy equals: Pout = 0.8 34 t = 27 t W . t=

1200 = 44.4 °C . 27

The oil temperature tg = 44.4 + 30 = 74.4°C. We can see that the fan cooling is very effective. The shaft end shall be made somewhat longer to ma “But different folk have different views.” (R. Kipling.) Different experts have different experience and, therefore, somewhat different recommendations concerning the convective heat transfer factor k. For instance, the German source [3] recommends to take for a calm air and unimpeded convection k = 15–25 W/(m2∙℃). The lower limit should be used for dirty housing walls, low linear speed of the mesh, and nonuniform distribution of temperature over the housing surfaces—for example, in big-sized gear drives. For such drives, more exact equation for factor k is recommended: k = fk [10 + 0.07(tg

ts )]

1 h

0.15

W /(m2°C ).

(9.61)

Here h is the gear housing height (m), and factor fk which depends on the air speed shall be taken from Figure 9.65. When the gear drive is installed in a big room, fk = 1 shall be substituted in Eq. (9.61). If a fan is mounted on the high-speed shaft, it is recommended: • for cylindrical gearings with one fan: fk = 1.4 (because the fan blows on one side wall only), with two fans: fk = 2.5 (both sides are aired); • for bevel and bevel-cylindrical gearings with one fan on the bevel pinion shaft: fk = 2.0 (because it blows on both sides); the same may be assumed for worm gears as well. Note: Factor k depends not on the presence or absence of a fan, but on the air speed which, in turn, depends on the rotational speed and characteristics of the fan. Thus, the last recommendations shall be considered taking into account Figure 9.65 and Eq. (9.60).

7 6 5 fk 4 3 2 1 1 2 3

5

7

10 vm/s

15

20

FIGURE 9.65 Factor fk as a function of air speed V [ 3].

Special Design Problems in Gear Drives

577

1.2 1.1 1.0 Bref 0.9 0.8 0.7 0.6 10

20 30 40 air temperature toC

50

FIGURE 9.66 t0C [ 19].

Factor Bref as a function of air temperature

FIGURE 9.67

Factor BA as a function of altitude A [ 19].

1.0 0.9 BA 0.8 0.7 0.6 0

1

2 3 4 altitude A, km

5

6

The common principle of all calculations for cooling is the equalization of the input (Pin) and output (Pout) heat energy. So, Eqs. (9.58) and (9.59) are valid also here. And finally, the standard AGMA ISO 14179-1 “Gear Reducers – Thermal Capacity” must be men­ tioned [19]. This document offers detailed recommendations for calculation of both energy losses and convection cooling of gear drives of different kinds. Here we present from this standard only three coefficients (“modifiers”): • Bref that takes account of the influence of the ambient temperature (i.e., of the air density) on the effectiveness of cooling, see Figure 9.66; • BA that takes account of the influence of altitude (i.e., the air density again) on the effectiveness of cooling, see Figure 9.67, and • BD that takes account for the duty cycle (% of operation time per each hour), see Figure 9.68. The convective heat transfer factor k obtained as recommended above should be multiplied by factors Bref, BA, and BD which are applicable to any method of convective cooling, natural or fan enhanced.

9.6

DESIGN OF SPUR AND HELICAL GEARS

Modern gear drives are highly stressed, their parts are made of high-strength steel, and the teeth of gears are usually surface-hardened. This enables increasing of the allowable load of the gears by a factor of 3 to 5 as compared with the gears with non-hardened teeth. However, the modulus of elasticity of steel remains approximately the same independently of its strength, so the deformations of the teeth and the gear bodies, as well as the shafts, bearing, and other elements of power transmission, including the housing, grow in direct proportion to the growing load. These deformations cannot be neglected, and the designer is to give considerable attention to their calculation and minimization of their harmful effect.

578

Dudley’s Handbook of Practical Gear Design and Manufacture 2.0 1.8 1.6

BD 1.4 1.2 1.0 100%

80

60 top

40

20%

FIGURE 9.68 Factor BD as a function of duty [ 19]. top = operation time per each hour, %.

The deformations of teeth change their pitch: under load, the pitch of the teeth to come into contact decreases on the driving gear and rises on the driven gear. This difference in tooth spacing between the engaging teeth disturbs the engagement and requires tooth profile modifications to avoid the edge im­ pacts of teeth, see Section 9.1.7. The bending and torsional strain of the pinion, as well as the deformations of the gear body caused by the tooth forces, centrifugal forces, and temperature expansion, account for the misalignment of teeth and uneven load distribution over the face width. The same effect causes the deformation of bearings under load. Therefore, the considerations of reduction of elastic deformations under load or, in some cases, purposeful increase of flexibility, are one of the main considerations in the design of gear drives.

9.6.1 PINIONS As a rule, pinions are made integral with the shafts. Being relatively small in diameter, especially at greater gear ratios, the pinions are quite flexible both in bending and twisting. Their deformations under load cause nonuniform load distribution across the face width. To reduce the pinion deformations, it is common to limit the gear ratio. For example, in gear drives with surface hardened teeth, the gear ratio in a single pair usually does not exceed 6. The ratio of active face width bw to the pinion’s diameter d1 does not exceed 0.8-1.0. Figure 9.69 demonstrates the design of a midsized industrial reduction gear drive produced in unified housing 1 with gear ratios from 7 to 36. At smaller gear ratios (Figure 9.69a, u = 7.6) the pinions are very rigid. At higher gear ratios (Figure 9.69b, u = 36.2) the pinions are small in diameter, and the diameters of their necks intended for bearings are reduced to provide free entry and exit of grinding discs. It is not recommended to mount bearings on necks slotted by the hob if the neck diameter exceeds the root diameter of the pinion teeth. Small overcutting on a neck due to entrance or exit of the hob can be accepted if it takes, say, no more than 10-15% of the bearing width though it is undesirable. In Figure 9.69b, size of the bearings of pinion 2 is reduced as compared with pinion 3 in Figure 9.69a to fit the smaller diameters of necks. Bearings of pinion 4 are not changed, however, the right bearing is mounted on sleeve 5 press-fitted to the pinion neck. Probably, to meet the close fitting tolerances re­ quired for the rolling bearings, the outer diameter of sleeve 5 shall be finished after mounting it on the pinion neck. In Figure 9.70, the housing is provided with an additional support 1, so that input pinion 2 becomes much shorter and more rigid in bending. Bearing 3 is mounted on sleeve 4 press-fitted to the neck of pinion 2 and fixes the pinion in axial direction. The left bearing is floating. The housing with support 1 remains unified and can be used for the entire range of the gear ratios and design solutions presented in Figure 9.69.

Special Design Problems in Gear Drives

(a)

FIGURE 9.69 1

3

1

579

(b)

2

4

5

Design of pinions and gears for an industrial double-stage reduction gear drive. 2

4

3

FIGURE 9.70 Reduction of bending deformation of the pinion.

9.6.2 GEARS 9.6.2.1 Industrial Gears It is well known that helical gearing is characterized by sufficiently higher load capacity and lesser vibration than spur gearing. The main disadvantage of the helical gearing is the creation of an axial force in the mesh which bends the gear body, influences adversely the bearing system, and requires sturdier housing walls which eventually take the axial force. All these factors increase the weight, but the in­ dustrial gear drives don’t have tight restrictions on weight. Thus, the design of helical gears with casehardened teeth is very simple, see Figure 9.71a. This design is the best from almost all points of view: the forging is the cheapest, the stiffness of the gear body is the greatest, the warpage during carburization and hardening is minimal, and the connection with the shaft through a press fit with a key is the cheapest, reliable and provides exact positioning of the gear on the shaft. The only disadvantage of this design is the increased weight of the gear. However, for the industrial gear drives it is not critical, especially in comparison with the entire drive which contents many weighty elements, such as a housing made of grey cast iron or cast steel, solid shafts and pinions, massive couplings etc. Helical gears with non-hardened teeth, low-loaded and capable of running-in due to wear and plastic deformation of the overloaded areas of the teeth flanks, can be made less massive, with connection between the rim and the hub in the form of a web (disc). Several steel models of a webbed gear with D/d = 2.5 were tested for rigidity [17]. (Here D and d are the outer and the inner diameters of the web.) Axial force Fa was applied to the rim, and the angle φ was measured, see Figure 9.71b. According to this research, angle φ is given by:

580

Dudley’s Handbook of Practical Gear Design and Manufacture

(a)

(b)

(c)

Fa

Fa SR

r

bs b

bs b

FIGURE 9.71 Design of industrial gears.

= C 10

3

Fa bs D2 b

1.5

rad .

(9.62)

Here: φ = local angular deflection of the rim in the area of application of force Fa; Fa = axial force (N); D = web outer diameter = rim inner diameter (mm);bs = web thickness (mm); b = rim width (mm); C = experimental factor, see Figure 9.72. This deformation causes misalignment of contacting teeth, and the angle of misalignment γ is given by: =

sin

wt ,

where αwt is the working transverse pressure angle of the gear pair. Equation 9.62 gives the deformation magnitude for the gear proportions similar to that of the ex­ amined models. Exact magnitudes of the bending deformation of real gears under force Fa can be obtained using FEM. 2.1 1.9 1.7

C

1.5 1.3 1.1 0.9 0.7 0.5

FIGURE 9.72

0°

Factor C to Eq. (9.62).

10°

20°

30°

Special Design Problems in Gear Drives

581

0.7

bs/bw

0.6

o

30

0.5

o

15 0.4

o

8 0.3 0.2 200

300

400

tooth hardness, HB

FIGURE 9.73 steel.

Optimal thickness of a flat web for helical gears of tempered

With growing web thickness, grow both the weight and the load capacity of the gears. Figure 9.73 shows the optimal relation bs/bw of a flat web at which the ratio of drive unit weight to load capacity is minimum [17]. The running-in ability of the teeth and the weight of the housing and the pinion were taken account of in the calculations. The thickness of conical webs with angle α ≥ 200 can be halved versus the flat webs, but should not be less than 0.15mn + 0.1bw [3]. Spur gears can be designed with a thin flat web; its thickness bs is limited by stresses in the area where the rim connects to the web. Greater fillet radius r enables thinner web, but usually bs = (0.1-0.2)bw. The thickness of the rim SR ≈ 2mn. Compliance of the thin web contributes to the ability of self-alignment of toothed rim relative to the mating gear and provides partial compensation for the tooth misalignment caused by bending of the shafts, displacements in the bearings etc. 9.6.2.2 Light-Weight Gears Where the weight of a gear drive is critical, the design options presented in Figures 9.69 to 9.71 are unacceptable because both the gears and their connections with the shafts are too heavy. First of all, helical gears are rarely used in light-weight drives because they require very rigid gear body, robust housing walls subjected to the axial forces, and bearings to react to those forces. Figure 9.74a shows an example of a toothed rim and web of a heavy loaded light-weight spur gear. The thinner the rim and the web, the less the gear weight. It is recommended, however, to make the rim thickness SR ≥ 1.6m (mostly SR ≈ 2m) because at thinner rim the fatigue crack, starting from the tooth fillet, may propagate into the rim and result in an immediate and very grave failure. The strength balance

(a)

(b)

(c)

bw

SR

br

r

1

bs FIGURE 9.74

Design of light-weight spur gears and load distribution across the face width.

582

Dudley’s Handbook of Practical Gear Design and Manufacture

of the teeth and the disc can be achieved at bs/b = 0.08-0.12 if the teeth are not hardened and at bs/b = 0.12-0.20 if the teeth are case hardened. The transition area between the disc and the rim should be made with reasonably big fillet radii r to thicken this area and reduce the stress concentration. In Figure 9.74a, the rim thickness SR = 1.6 m, web thickness bs = 0.125 b, r = 1.5 bs. Such a small rim thickness increases significantly the bending deformation of the tooth under load except for the web area where the local tooth stiffness remains high. The nonuniform tooth stiffness leads to uneven load dis­ tribution, and the tooth in the web zone becomes overloaded. To make the tooth rigidity and load distribution across the face width more even, the rim is provided with ring ribs 1. These ribs add only a little to the gear weight, but they increase the bending rigidity of the rim and reduce the stresses in the transition area between the rim and the web. At the same time, the bending deformation of the tooth is also reduced. According to ISO 6336-1: 2006, the average tooth bending stiffness is proportional to factor CR that allows for the thickness of the rim and the web and is determined as follows: CR = 1 +

ln (bS / b ) . 5 e SR /(5mn)

For solid gear (that is, at bs/b = 1) CR = 1. For the gear design presented in Figure 9.74a, with big fillets r and two ribs 1 with br = 0.7bs, we can suppose that the equivalent web thickness and bs ≈ 0.2b. Then we obtain: CR = 1 +

ln (0.2) = 0.766. 5 e1.6/5

That means that the bending deformation of teeth in this case is greater by a factor of 1:0.766 = 1.3 than that of the solid gear shown in Figure 9.71a. This must be taken into consideration while choosing dimensions for the profile modification. The load distribution across the face width depends on the design of both meshing gears. The design shown in Figure 9.74b is the worst one because the load is concentrated in the zone of maximal tooth pair stiffness, and the unit load here exceeds the average by about 30% at SR = 2 m. Figure 9.74c presents a better design option with а more uniform distribution of load. The keyed press-fit connection of gears with their shafts is usually impracticable in light-weight drives because it hardly can be realized with thin-walled parts. The minimum weight is achieved when the gears are made integral with their shafts as shown in Figure 9.75a. Distance A allows grinding of the smaller gear teeth. The additional weight reduction is achieved here due to use of the end journals as the inner races of roller bearings. If the gears must be made separately from the shaft for technological or other reasons, spline con­ nections are used as a rule. The double gear shown in Figure 9.75b is made separable because otherwise a rolling bearing cannot be mounted on neck 1. Bevel gear 2 is centered on cylindrical spigots 3 and 4 with a light interference fit and clamped by nut 5. The torque is transmitted through splines 6. The second bearing of this assembly is mounted on neck 7. Such a connection is very common. The centering on cylindrical surfaces is more exact than on the splines. At the same time, the centering surfaces unload the splines from radial forces, and this reduces sharply the movements, wear and heat generation in the spline connection. The other widespread and reliable method of fastening a gear to the shaft is flange connection, see Figure 9.75c. It is essential to fit the parts tightly on the centering diameter d, but not too tight because the flanges must be strongly pressed against each other. In this case, the centering surfaces take the radial force, and the bolts transmit the circumferential forces only. The average force Fav applied to a single bolt is given by:

Special Design Problems in Gear Drives

583

(a)

(b) 5 4

7

6

2

1

3

(d)

(c) W

(e) section A-A lk

A

S bs

10 15 13 14

A

d1

dk

12

d 9

db d

FIGURE 9.75

11

8

Hub-shaft connections in light-weight gear drives.

Fav =

2T , N db n

Here, T = torque, N∙mm; db = diameter of the bolt circle, mm; n = number of bolts. The load distribution between the bolts can be determined by numerical methods only. The maximal force: Fmax = K Fav.

(9.63)

The values of factor K calculated for bs/w = 0.15–0.25 are given in Figure 9.76. Note: Fitted bolts used as a rule in flange connections in light-weight machinery, and the bores for them should be made within very close tolerances, often the bolts are selected for certain bores to provide very small or zero interference fit. Excessive interference may create great friction forces between the bolts and their bores and reduce too much the essential compressive force between the flanges. The same is true in regard to the fit of the gear to the centering diameter d. The fitted surfaces, as well as the bolt

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Dudley’s Handbook of Practical Gear Design and Manufacture

1.6 1.4 K 1.4

n=1

2

1.3

n=6 1.2 1.6

1.9

2.2 d1/d

2.5

2.8

FIGURE 9.76 Factor K to Eq. (9.63).

and nut threads and their bearing surfaces, should be oiled during assembly to reduce the friction. If the torque is irreversible and the level of vibrations is low, transition fits are mostly used for the fitted bolts. Thin-walled parts can be connected using round keys (items 8 in Figure 9.75d). These keys and the mating parts are checked for bearing stress: b

=K

16 T , MPa d dk lk n

where T = torque, N∙mm; d = diameter of connection, mm; dk = key diameter, mm; lk = key length, mm; n = number of keys. Factor K that accounts for nonuniformity of load distribution between the keys can be determined only using FEM. Between the round keys, screws 9 are installed to prevent axial displacement of the connected parts. Both the keys and the screws are secured by punching against falling out. Design option presented in Figure 9.75e is a variant of the idea of transmitting the torque by splines while centering on additional surfaces. Here gear 10 is centered relative to shaft 11 using split cones 12 and 13. To make dismantling easier, the angle of the cones is made larger than the coefficient of friction. Usually, the vertex angle is 300 and 600 for cones 12 and 13, respectively. The cones may be made of bronze or brass to prevent fretting in the contact with the shaft and the gear. Nut 14 tightens the cones and the gear on the shaft and creates a very strong connection. The torque is transmitted through splines 15. 9.6.2.3 Large-Diameter Gears Gears of 1200 mm and more in diameter, but often starting from diameters 700–800 mm, are mostly combined of a relatively thin forged rim made of a high-strength steel and a center joined with the rim by welded or mechanical connections. If the toothed rim is relatively narrow, it is connected with the hub by a single disc, see Figure 9.77. Fully-welded gear is shown in Figure 9.77a. The weldment consists of rim 1, web 2, and hub 3. Ribs 4 are required for helical gears only. The thickness of ribs can be about 0.6–0.7 of the disc thickness. The height and number of ribs can be chosen based on FEM calculation of the gear deformation under force Fa. The problem here is that the weldability of steel is inversely proportional to its hardenability. Alloyed and mild-carbon steels used as the rim materials have limited or poor weld­ ability, and their welding to discs requires special processing procedure and qualified operators to avoid welding cracks. Where it is not handy, mechanical connections are used. In Figure 9.77b, gear rim 5 is fastened by bolts to center 6 cast of steel or iron. It is preferable to locate the flanges of bolted connection so that axial force Fa would help the bolts to clamp the joint. Ribs 7 stiffen the web of the center. It is recommended to place the ribs on the compressed side of the web loaded by the axial force in bending. This is because the bending stresses are maximal on the top of ribs.

Special Design Problems in Gear Drives

(a)

585

(b)

(c)

(d)

Fa

Fa

Fa

1 9 13

5 8 6 7

4 2

11 10

12

3

FIGURE 9.77

Large-diameter gears with relatively narrow toothed rim.

The rim is fitted to spigot 8 of the center with a light interference (of about 0.1 mm per m of the spigot diameter). In this case, the vector of bolt load does not rotate around the bolt, but keeps tangential direction, and only its magnitude changes. Figure 9.77c shows a similar design of a bevel gear. In Figure 9.77d, gear rim 9 is riveted to welded center that consists of three parts: hub 10, disc 11 and ribs 12 required for a helical gear only. Recess 13 is made to reduce stress concentration on the end of welded seam. It is especially important if the stiffeners are welded on the stretched side of the disc. Investigation of load distribution between the bolts was performed using FEM [17]. The dimensions of FE models are shown in Figure 9.78a. Calculations were made for bs/b = 0.1…1.0; h/δ = 1, 2, 4; δ/R = 0.02, 0.06 (a)

(b) b h

0.8 a 0.7

h/ =1

/R = 0.02

h/ =2

bs

h/ =4

0.6

R

Rb r

/R = 0.06 0.5 h/ =0.5

0.4 0.3

=1 h/ h/ = 2 h/ = 3

h/ =1 h/ =2

0.2 /R = 0.10 0.1 0.1

0.2

0.3 bS/b

FIGURE 9.78

Diagrams of exponent “a” to Eq. (9.64).

0.4

0.5

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Dudley’s Handbook of Practical Gear Design and Manufacture

and 0.10; r/R = 1/6 and 1/2 (the influence of r/R was found negligible); number of bolts n = 2, 4, 6, 8, 12 and 24. Here, R is the outer radius that for the geared rim can be taken approximately equal to the bottom radius of the teeth + 1m; Rb is the joint radius of the rim and the center in the FE model (radius of location of the centers of bolts). The results of calculations were approximated by the following formulas: Fmax = Fav na ; Fmin

Fav . na

(9.64)

Here, Fav = average load of the bolt: Fav = T/(Rb·n), N; T = torque (N·mm); Rb = radius of location of the bolts or rivets (mm); n = number of bolts; a = exponent, see Figure 9.78b. One can see that the thinner the disc as compared with the rim width, the less the a value. Imagine, if the disc were very compliant (for example, it were made of rubber), the load would be distributed between the bolts evenly. That means, in this case Fmax = Fmin = Fav, and a = 0. If a = 1, the entire load is transmitted through only one bolt. Stiffening of disc by radial ribs influences very little its tangential compliance and, hence, the load distribution between the bolts. Note: if the gears revolve fast, the ribs should be avoided or shielded to reduce the churning and windage losses. Large-diameter gears with wide toothed rims (say, above 250 mm) require two or even three discs to provide reasonably strong connection between the rim and the hub. This can be made on the basis of bolted connections, however mostly such gears are fully welded. Figure 9.79а presents a typical rigid structure of a welded gear. Here rim 1 is connected with hub 2 through two flat discs 3. Ribs 4 welded between the discs increase their bending stiffness, so that this structure is able to withstand the axial forces developed in the helical mesh. Bores 5 are intended to let the oil out from the inner space of the gear. Double-helical gear, Figure 9.79b, does not require stiffening ribs because there is no axial force transmitted from the gear rim to the hub. On the contrary, pliable discs enable easier self-adjustment of the rim in axial direction to achieve more even load distribution between the gear helices. High-speed gears have one more feature that is worthy of notice: the rim increases in diameter due to centrifugal forces, and this increase is greater at a distance from the disc than in the disc area. This causes misalignment of the teeth. The third disc in the middle (Figure 9.79c), in addition to stronger connection between the rim and the hub, helps to solve this problem. To taste the practical importance of this problem, let’s calculate the radial deformation of a gear rim spinning with a peripheral speed of V = 100 m/s. (a)

(b)

1

4

2 3 5

FIGURE 9.79

Large-diameter welded gears with wide toothed rim.

(c)

Special Design Problems in Gear Drives

587

The tensile stress excited in the rim (without the teeth) due to centrifugal forces equals: t

=

g

V 2 MPa.

Here, γ = 7.7∙10 – 5 N/mm3 = specific weight of steel; g = 9.81∙103 mm/s2 = acceleration of gravity; V = rim speed (mm/s). Thus, in this case, t

=

7.7 10 5 (100 103)2 = 78.5 MPa. 9.81 103

The increase of the rim radius ΔR is given by: R=

t

E

R,

where, E = 2.06∙105 MPa = modulus of elasticity of steel. R=

78.5 1000 = 0.381 mm. 2.06 105

The influence of the teeth is considerable; it may be as great as 20 to 40%. Thus, the radial deformation of the ring-shaped toothed rim may amount up to 0.5 mm. The discs block most part of the radial deformation, but between the discs it remains big enough and may cause considerable misalignment of teeth. Connection of the toothed rim with the center using shrink fit (Figure 9.80a) is an old-fashioned design because such gears are characterized by a great increase in weight. Besides, the shrink fit induces significant tensile stress into the rim that reduces the bending strength of the teeth. However, such gears are still in use. They consist of bandage 1 (it is the toothed rim) and cast or welded center 2. The rim thickness SR and the center barrel thickness SC must be big enough to provide the required surface pressure p on joint surface 3 and, thereby, the friction forces between the rim and the center which are able to transmit the torque. There are two concomitants there: a) the tension stress in the rim caused by the interference fit must not be too high because it influences the bending strength of the teeth; b) the interference of the fit must not be too high to enable mounting of the rim on the center by moderate temperatures of heating. If the teeth are case hardened, heating above 120–150°C is unacceptable be­ cause it reduces the hardness and residual compressive stresses in the hardened layer and is harmful for the load capacity of the gear. In this case, heating of the rim can be combined with cooling of the center, or the thicknesses SR and SC can be increased to reduce the required amount of interference and, in this way, the necessary heating temperature. Torque Tc at which the friction connection starts creeping equals: Tc = 0.5 dc2 bc pfc , N mm;

(9.65)

where, dc = diameter of connection (mm) bc = width (length) of the connection (mm); p = surface pressure in the connection (MPa); fc = friction coefficient in the connection. Equation (9.65) is written on the assumption that the entire surface of the connection participates in transmitting load simultaneously. In the case of a built-up gear, this is not so. The load is taken by only a small part of the connection near the loaded tooth (see Figure 9.81a), and the thinner the rim, the smaller

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Dudley’s Handbook of Practical Gear Design and Manufacture

(a)

(b)

Fa

p

b SR

1

SC

2

dc df

3 5

4 bc

FIGURE 9.80

(a)

Gears with shrink-fitted rim.

Fn

SR SC

(b)

Fn

1

FIGURE 9.81

Local slippage (a) and separation (b) of bandage.

the local area that assists in the load transmission. We are unable now to describe this complicated process quantitatively, that is why Equation (9.65) is still used for the calculations, however, the value of the conventional friction coefficient fc is determined experimentally or based on operating experience. Experiments [20] showed that under static load, when the entire rim slips, fc ≈ 0.28, but under dynamic load, when the gear rotates, the creeping of the rim relative to the center begins at much lesser torques. This effect is caused by local microslip near the loaded tooth due to local lowering of the friction coefficient because of vibrations in the area of working teeth and local tensile strain of the rim. This microslip moves along with the tooth force around the gear and accumulates with each turn as the gear rotates. It was found that at the rim thickness SR = 1m, 2m, 4m, and 5m, the no-creep condition is achieved at torques that match the friction coefficients fc = 0.005, 0.017, 0.024, and 0.028 respectively

Special Design Problems in Gear Drives

589

(in Eq. (9.65)). This result correlates well with the operational data [21] that gears with a shrink-fitted rim make no troubles in service if their maximal working torque does not exceed the Tc value obtained using Eq. (9.65) at fc = 0.016…0.033. The lowering of pressure in the joint is caused by the bending deformation of the bandage and may lead to a complete local separation in the joint (item 1 in Figure 9.81b). This, in turn, may result in fretting and wear of the joint surfaces and even in the oil ingress between the joint surfaces. It is recommended [21] to determine the maximal torque Tsep that doesn’t cause separation as follows: • for the rim and center made of steel: Tsep =

d p SR b 0.44 +

9m n SR

, N mm;

• for the steel rim and gray cast iron center: Tsep =

d p SR b 0.2 +

6.6 mn SR

, N mm.

Here, d = pitch diameter of the gear, mm; p = pressure in the joint, MPa (N/mm2); SR = rim thickness, mm; b = face width of the rim, mm; mn = module, mm. In helical gears, the center barrel is provided with shoulder 4 (Figure 9.80a), and the helix direction shall be so chosen as to direct the axial force Fa toward the shoulder to prevent the rim from lateral sliding off the center. If the load is reversal, screws 5 are installed at the interface between the rim and the center. Diameter of screws ds ≈ (0.4…0.5)SR, length ls ≈ 3ds, number of screws 6…8. After assembly, the edges of the threaded holes shall be plastically deformed (for instance, by punching) to prevent the screws from falling out. It was mentioned above that tension of the shrink-fitted rim caused by the pressure between the rim and the center influences the bending strength of the teeth. Experiments [3] with spur gears showed that at tensile stress σt = 100, 150, and 200 MPa, the bending strength of tempered teeth with hardness HV230 and HV303 was reduced by 10…11%, 12…16% and 14…18% respectively. The bending strength of case-hardened teeth, at the same magnitudes of σt, was reduced by 17%, 25.5%, and 31.5% respectively. Since the tension of the rim may have such a sizeable effect on the bending strength of teeth, the calculation of pressure p on the contact surfaces and, then, determination of tensile stresses in the rim, - shall be made thoroughly, using FEM, to see the increase of stresses in the area of discs. (See the joint pressure distribution across the face width shown in Figure 9.80b.) The outer diameter do of the toothed rim for such calculations can be determined from formula [22]: d 0 = d f + 2 mn (0.22 + 0.009 ) mm,

where df = tooth root diameter (mm);mn = normal module (mm);β = helix angle (degrees). It is desirable not to exceed maximum tensile stress σt = 150 MPa for tempered teeth and 100 MPa for surface hardened teeth. This aim can be achieved by increasing the thicknesses SR and SC. Mostly, SR ≈ (4–5)mn, and SC is greater by 15–20% depending on the strength calculations. Large-diameter gears are more pliable as compared with smaller gears, and their elastic deformation, whether caused by operational loads or by weight loads, are much more sizeable. Therefore, the weight loads of such gears should never be forgotten when their mounting conditions are considered. For ex­ ample, the big-sized gears are mounted on a gear-cutting machine with the gear axis vertical. In this position, the weights of the rim and the center shall be supported, otherwise the flat discs would be

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Dudley’s Handbook of Practical Gear Design and Manufacture

elastically deformed by their weights and deflect the adjacent portions of the rim in radial direction towards the center. In the operating position, the gear axis is horizontal, the deformed parts return to their initial shape, and the teeth in the disc areas would become protruding, thus the load would concentrate on these places. Another example: the weight of a large-diameter gear with its shaft and coupling shall be taken into account when the reaction of supports is calculated.

9.6.3 ARRANGEMENT

OF

GEAR SUPPORTS

Discussing the design of gears with regard to deformations under load and their influence on the mis­ alignment of teeth, we have to give due attention to the influence of bearings. If the bearings were zeroclearance and absolutely rigid, we would place them just close to the gear in order to reduce the bending deformation of the shaft to a minimum. However, the bearings do have clearances and elastic de­ formations under load, and the part of the shaft connected with the bearing deflects from the center of the bearing sit in the housing by a certain amount Δ. If one bearing deflects by Δ1 and the second by Δ2 in the same direction (Figure 9.82a), then the angle of misalignment of the shaft γ equals: =

1

2

L

rad .

(Here, Δ1, Δ2, and L must be substituted in the same units, for example, in mm.) If Δ1 = Δ2, then γ = 0. If the forces applied to the gear unit are so directed that the bearings deflect from the center in opposite directions (Figures 9.82b and c), the angle of misalignment increases drastically: =

1

+ L

2

rad .

L bearing sits center line

F1

1

2 shaft center line

(a)

shaft center line

F1

1

F2

F1

(b)

C

bearing sits center line

L

(c) FIGURE 9.82

Deflection of bearings and misalignment of shafts.

2

Special Design Problems in Gear Drives

591

From the last Equations we can see that the greater the distance L between the bearings, the less the influence of the radial displacement of bearings on the shaft misalignment. The designer can increase the distance between bearings only within reasonable limits because increasing this results in increased bending strain and stress of the shaft, not to mention the increase in dimensions of the mechanism. For example, for the overhung mounted gear (Figure 9.82c) it is common to make L = (2-2.5)C, and this is the relation that may help to minimize the summary harmful effect of the bearings deflections and the shaft bending strain on the misalignment of the gear. However, the best solution for heavily loaded gears is the straddle mounting. The shafts with two gears deserve special consideration of the direction of applied forces. Figure 9.83 presents two diagrams of two-stage reduction gear drives, and Figure 9.84 shows the direction of forces applied to intermediate shafts 1 and 2. From Figure 9.84 we can see that in the coaxial design the intermediate shaft is skewed by the applied forces, and the bearings of this shaft must be rigid (for example, roller bearings, not ball bearings) and have minimal clearances. Increased distance between bearings of intermediate shaft 2 caused by the placement of additional bearings 3 reduces the shaft skewing in supports, nevertheless, the requirements to the stiffness of supports must be increased as compared with the common design.

(a)

(b) 1 2

FIGURE 9.83 Double-stage reduction gear drives: a—common design, b—coaxial design.

3

(b)

(a)

F1

G1

F2 F2

G2

F1 1

G1

F1

F1 F2

L

F2

G2 2

FIGURE 9.84 Direction of forces applied to intermediate shafts in Figure 9.83: a—for common design, b—for coaxial design.

592

Dudley’s Handbook of Practical Gear Design and Manufacture

(a)

(c)

L

(b)

(d)

L

FIGURE 9.85

L

L

Supports of double-toothed planet gears.

The planetary drives with double-toothed planet gears may have the same problem. In drives 2CG type B (Figure 9.34) the direction forces applied to the planet gear is favorable, such as shown in Figure 9.84a. However, in drives 2CG type C (Figure 9.35) and 3CG (Figure 9.37) the tooth forces are directed as shown in Figure 9.84b. The design of supports for such planet gears deserves discussion. In Figure 9.85a, the roller bearings are installed inside the planet gear, and each bearing is placed under a toothed rim. This is a false idea. There is a one-piece cluster gear, not two separate gears, and the short distance between bearings leads to great misalignment of the teeth and in the bearings and quick failure. Increase of the bearing spacing within the planet gear width (Figure 9.85b) helps very much and can be satisfactory if the bearings are mounted with a certain preload. However, the use of preload in such a design is very costly and is used in very special cases only. It is preferable to place the bearings in the planet carrier (Figure 9.85c). This increase in spacing of the bearings enables to obviate the need for their preload. Design with spherical bearings presented in Figure 9.85d rules out the problem of misalignment in bearings that is no less important than the misalignment of gear teeth. In general, the main actions the designer can take to reduce the misalignment of gears and bearings due to deflections of shafts in their supports are as follows: • to avoid where it is possible situations where the working forces deflect the bearings in opposite directions (like in Figure 9.82b and c); • to reduce the internal clearances in bearings through the choice of clearance group of the bearing and the fit tightness; • to avoid clearance fits of the bearing rings and the intermediate parts between the bearing ring and the housing; • to choose roller bearings instead of ball bearings which are more pliable; • to use preload of rolling bearings where it is unavoidable. In big-sized gear drives, where the plain bearings are used, the recommendation to avoid design shown in Figure 9.82b and c becomes imperative. Figure 9.86 demonstrates the special design of double-gear idler

Special Design Problems in Gear Drives

(a)

593

F1 F2 1

(b)

2 e F1 F2

(c)

F1

e

FIGURE 9.86 Improvement of alignment of double-toothed idlers on plane bearings.

F2

from Figure 9.82b. The idea is that the double-toothed idler with two bearings (Figure 9.86a) is replaced by two separated gears connected by a torque transmitting element. Each of the two gears is supported separately. The bearings are placed symmetrically relative to the toothed rim, and the loads of the bearings are equal. Thus, the displacements in the bearings are equal, and the misalignment angle γ = 0. That means, that the axis of the gear remains parallel to the bearing sits of the housing. There are two options: • the bearings of the separated gears are coaxial (Figure 9.86b) and the gears are out-of-line by amount e that depends on the clearances in the bearings and the positions of the shafts inside the bearings in operation; • the bearings of one gear are displaced in radial direction by amount e relative to the bearings of the other gear (Figure 9.86c), so that the gears in operation become coaxial. The commonly used implementation of the option diagrammed in Figure 9.86b is demonstrated in Figure 9.87. Here double helical gears 1 and 2 are connected through torsion bar 3 with gear-type couplings 4 and 5 on its ends. Gears 1 and 2 are radially displaced relative to each other under load because of different directions of their tooth forces. The gear-type couplings compensate for this mis­ alignment. In this design, due to symmetry of load applied to each gear, the axes of gears 1 and 2 remain parallel to the axes of their bearings, - on the understanding than the clearances in the bearings of one gear are equal. Thus, the clearances in the bearings do not impair neither the alignment of the mating gears, nor the alignment of the gears with their bearings.

4

FIGURE 9.87

1

3

Implementation of diagram presented in Figure 9.86b.

2

5

594

Dudley’s Handbook of Practical Gear Design and Manufacture

Implementation of diagram shown in Figure 9.86c requires very detailed calculation of the plain bearings in order to know the exact location of the gears relative to their bearings. In this case, it is possible to achieve an exact coaxiality of the gears and connect them by a short spline coupling.

REFERENCES [1] Kudryavtsev V. N. (1957). Gear drives. MASHGIZMoscow – Leningrad. (In Russian). [2] Kudryavtsev V. N. (1966). Planetary gear drives. Mashinostroenie, Moscow-Leningrad. (In Russian). [3] Niemann G., Maschinenelemente W. H., & Getriebe Allgemein B.II. (1985). Zahnradgetriebe – Grundlagen, Stirnradgetriebe. Zweite Auflage. Springer-Verlag. [4] Munro R. G., Palmer D., & Morrish L. (2001). An experimental method to measure gear stiffness throughout and beyond the path of contact. proc. of the institution of mechanical engineers, part C. Journal of Mechanical Engineering Science, 215(7), 793–803. [5] Smith J. D. (2003). Gear noise and vibration. (2nd ed.). Cambridge University, Marcel Dekker Inc. [6] Palmer D., & Fish M. (January-February 2012). Evaluation of methods for calculating effects of tip relief on transmission error, noise and stress in loaded spur gears. Gear Technology, 56–67. [7] Yildirim N., & Munro L. G. (1999). Systematic approach to profile relief design of low and high contact ratio spur gears. Journal of Mechanical Engineering Science, 213(C6), 551–562. [8] Kapelevich A. L. (2018). Asymmetric gearing. CRC Press. [9] Dudley D. W. (Ed.) (1962). Gear handbook. McGRAW-HILL Book Company. [10] Utakapan T., Kohn B., Fromberger M., Heider M., Otto M., Hӧhn B.-R., & Stahl K. (2016). Measurements of gear noise behavior for different microgeometries. Gears Research Center (FZG). Technical University in Munich. [11] KHK Stock Gears. (2015). Calculation of gear dimensions. Kohara Gear Industry Co., Ltd. [12] Dollezhal V. A. (1945). Speed reducers of aviation engines. Oborongiz. (In Russian). [13] Kudryavtsev, V. N. & Kirdyashev Y. N. (Ed.). (1972). Planetary Gear Drives Handbook. Mashinostroenie. (In Russian). [14] Klebanov B. M., & Groper M. (2016). Power mechanisms of rotational and cyclic motions. CRC Press. [15] Nikitin Y. M. (1968). Design of elements and units of aircraft engines. Masninostroenie, Moscow. (In Russian) [16] AGMA ISO 14179-1. Gear reducers – thermal capacity based on ISO/TR 14179-1. American Gear Manufacturers Association. [17] Klebanov B. M., Barlam D. M., Nystrom F. E. (2008). Machine elements. Life and design. CRC Press. [18] Levitan Y. V., Obmornov V. P., & Vasilyev V. I . Worm gears. Handbook. Mashinostroyenie, Leningrad. (In Russian). [19] AGMA ISO 14179-1. (2004). Gear reducers – Thermal capacity based on ISO/TR 14179-1. American Gear Manufacturers Association. [20] Arai N., Oeda M., & Aida T. (1981). Study of the stress at the root fillet and safety for slipping of forcefitted helical gears. Bulletin JSME, 24(198), 2203–2209. [21] Safronov Y. V. (1980). Up-to-date calculation of the rim thickness and the interference fit of large spur gears. Transactions of the Rostov-na-Donu Agriculture University. (In Russian). [22] Dinovich M. Y., & Shelomov N. M. (1977). Stiffness of the rim of helical gears. Vestnik Mashinostroeniya, 7. (In Russian), 25–27.

10

Gear Materials

A wide variety of steels, cast irons, bronzes, and phenolic resins have been used for gears. As their outstanding characteristics, the steels have the greatest strength per unit volume and the lowest cost per pound. In many fields of gear work, some kind of steel is the only material to consider. The cast irons have long been popular because of their good wearing characteristics, their excellent machinability, and the ease with which complicated shapes may be produced by the casting method. The bronzes are very important in worm-gear work because of their ability to withstand high sliding velocity and their ability to wear in to fit hardened-steel worms. They are also very useful in applications in which corrosion is a problem. The ease with which bronze can be worked makes it a good choice where small gear teeth are produced by stamping or by drawing rod through dies. The phenolic resins are used in various combinations to produce laminated gears with remarkably good load-carrying capacity in spite of the low physical strength of the material. In general, these materials are about thirty times as elastic as steel. Their “rubbery” nature makes them operate with less shock due to tooth errors and allows their teeth to bend enough to make more teeth contact and share load than in metal designs. Some of these materials can stand quite high sliding velocities. In this chapter we shall study all the kinds of gear materials just mentioned. Condensed information will be given to define the composition, heat treatment, and mechanical properties of these materials. To help the reader with the language of materials, a short glossary is given in Table 10.1.

10.1 STEELS FOR GEARS There are a number of steels used for gears, ranging from plain carbon steels through the highly alloyed steels and from low to high carbon contents. The choice will depend upon a number of factors, including size, service, and design. The following discussions should serve as a guide in selecting steels.

10.1.1 MECHANICAL PROPERTIES In designing gears, the mechanical properties of the material are of some interest to the designer from a comparative standpoint, but they cannot be used directly in calculating the load-carrying capacity. As was pointed out in Section 5.1.1, the calculated stresses on gear teeth are not necessarily the true stresses in the material. Moreover, tensile properties—the most commonly published mechanical properties of a material—are determined by loading small bars in simple tension. The gear tooth is a different geometric shape with a more complex stress pattern, and therefore the actual properties of the material as a gear can best be determined by testing the material in the shape of a gear tooth. The allowable stresses for rating gears (see Sections 6.2.1 and 6.2.2) are based on tests and field experience with gears rather than on the mechanical properties of the material as determined by routine laboratory tests. The mechanical prop­ erties are valuable indirectly in that they indicate how gears made of a particular material might be expected to perform. Figure 10.1 is a rough guide to the tensile properties of steel. In gear terminology, some steels are considered to be alloy steels and some are considered to be plain carbon steels. The steels used for gears tend to vary from those with a small number of alloys to steels rich in alloys. From a practical standpoint, the steels that are considered to be plain carbon usually have some alloy content, like manganese and silicon. The steels that are normally thought of as alloy steels usually have chromium, nickel, and molybdenum as well as manganese and silicon.

595

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Dudley’s Handbook of Practical Gear Design and Manufacture

TABLE 10.1 Glossary of Metallurgical and Heat-treating Nomenclature Term

Definition

Aging

A change in an alloy by which the structure recovers from an unstable or meta-stable condition produced by quenching or by cold working. The degree of stable equilibrium obtained for any given grade of steel is a function of time and temperature. The change in structure consists of precipitation and is marked by change in physical and mechanical properties. Aging that takes place slowly at room temperature may be accelerated by an in crease in temperature. A broad term used to describe the heating and cooling of steel in the solid form. The term annealing usually implies relatively slow cooling. In annealing, the size, shape, and composition of steel produced and the purpose of the treatment determine the temperature of the operation, the rate of temperature change, and the time at heat. Annealing is used to induce softness, to remove stress, to alter physical and mechanical properties, to remove gases, to change the crystalline structure, and to produce a desired microstructure.

Annealing

Austenite

In steels, the gamma form of iron with carbon in solid solution. Austenite is tough and nonmagnetic and tends to harden rapidly when worked below the critical temperature.

Austenitic steels Brinell hardness

Steels that are austenitic at room temperature. A hardness number determined by applying a known load to the surface of the material to be tested through a hardened-steel ball of known diameter. The diameter of the resulting permanent indentation is measured. This method is unsuitable for measuring the hardness of sheet or strip metal.

Carburizing

Diffusing carbon into the surface of iron-base alloys by heating such alloys in the presence of carbonaceous materials at high temperatures. Such treatments followed by appropriate quenching and tempering, harden the surface of the metal to a depth proportional to the time of carburizing.

Case hardening

Hardening the outer layer of an iron-base alloy by a process that changes the surface chemical composition, followed by an appropriate thermal treatment. Carburizing and nitriding are typical case-hardening techniques. Induction heating and the hardening of the outer layer of metal is another form of case hardening.

Cold working Ductility

Permanent deformation of a metal below its re-crystallization temperature. Some metals can be hotworked at room temperature, while others can be cold-worked at temperatures in excess of 1000°F. The property of metal that allows it to be permanently deformed before final rupture. It is commonly evaluated by tensile testing in which the amount of elongation or reduction of area of the broken test specimen, as compared with the original, is measured.

Elastic limit

Maximum stress to which a metal can be subjected without permanent deformation.

Endurance limit Ferrite

The maximum stress to which material may be subjected an infinite number of times without failure. Iron in the alpha from in which alloying constituents may be dissolved. Ferrite is magnetic, soft, and acts as a solvent for manganese, nickel, and silicon.

Flame hardening

Hardening an iron-based alloy by using a high-temperature flame to heat the surface layer above the transformation temperature range at which austenite begins to form, then cooling the surface quickly by quenching. This process is usually followed by tempering. An oxyacetylene torch is often used in flame hardening.

Grain size

The grain-size number is determined by a count of a definite microscopic area, usually at 100× magnification. The larger the grain-size number, the smaller the grains.

Grains Hardenability

Individual crystals in metals. The property that determines the depth and distribution of hardness induced by quenching. The higher the Hardenability value, the greater the depth to which the material can be hardened and the slower the quench that can be used.

Hardness

The property of materials that is measured by resistance to indentation.

Heat treatment

A general term that refers to operations involving the heating and cooling of a metal in the solid state for the purpose of obtaining certain desired conditions or properties. Heating and cooling for the

(Continued)

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597

TABLE 10.1 (Continued) Glossary of Metallurgical and Heat-treating Nomenclature Term

Inclusions

Definition prime purpose of mechanical working are not included in the meaning of the definition. Gear metal heat treatments include heating in a furnace followed by quenching, flame hardening, and induction hardening. Particles of nonmetallic materials, usually silicates, oxides, or sulfides, which are mechanically entrapped or are formed during solidification or by subsequent reaction within the solid metal. Impurities in metals.

Induction hardening

Hardening of steel by using an alternating current to induce heating, followed by quenching. This process can be used to hardened only the surface by using high-frequency current, or it can throughharden the steel if low frequency is used. Induction hardening is a fast process that can be controlled to produce the desired depth and hardness in a localized area and with low distortion.

Jominy test

The Jominy test is used to determine the end-quench hardenability of steel. It consists of waterquenching one end of a 1-in.-diameter bar under closely controlled conditions and measuring the degree of hardness at regular intervals along the side of the bar from the quenched end up. A microconstituent in quenched steel characterized by an acicular, or needle-type, structure. It has the maximum hardness of any structure obtained from the decomposition of austenite.

Martensite Modulus of elasticity

The ratio, within the elastic limit, of the stress to the corresponding strain. The stress in pounds per square inch is divided by the elongation of the original gage length of the specimen in inches per inch. Also known as Young’s modulus.

Nitriding

Adding nitrogen to solid iron-base alloys by heating the steel in contact with ammonia gas or other suitable nitrogenous material. This process is used to harden the surface of gears. A process in which steel is heated to a temperature above the transformation range and subsequently cooled in still air to room temperature.

Normalizing Pearlite

The lamellar aggregate of ferrite and cementite resulting from the direct transformation of austenite at the lower critical point.

Quenching Residual stress

Rapid cooling by contact with liquids, gases, or solids, such as oil, air, water, brine, or molten salt. Stresses remaining in a part after the completion of working, heat treating, welding, etc., due to phase changes, expansion, contraction, and other phenomena.

Rockwell hardness

A hardness number determined by a Rockwell hardness tester, a direct-reading machine that may use a steel ball or diamond penetrator.

Secondary hardening

An increase in hardness developed by tempering high-alloy steels after quenching, usually associated with precipitation reactions.

Stress relief

A process of reducing internal residual stresses in a metal part by heating the part to a suitable temperature and holding it at that temperature for a proper time. Stress relieving may be applied to parts that have been welded, machined, cast, heat-treated, or worked. A process that reduces brittleness and internal strains by the reheating of quench-hardened or normalized steels to a temperature below the transformation range. A draw treatment is a tempering treatment that is hot enough to reduce hardness.

Tempering

Tensile strength

The maximum load per unit of original cross-sectional area carried by a material during a tensile test.

Through hardening

Increasing the hardness of a metallic part by a process that hardens the core material as well as the surface layers, with the hardness uniform throughout the whole gear tooth and the metal immediately adjacent to the gear tooth. The temperature range within which austenite forms in ferrous alloys.

Transformation range Yield point

The load per unit of original cross section at which a marked increase in deformation occurs without any increase in load.

598

FIGURE 10.1

Dudley’s Handbook of Practical Gear Design and Manufacture

Approximate tensile properties of steel.

There is a general feeling in the gear trade that alloy steels are inherently stronger and more fatigueresistant than the so-called plain carbon steels. The real situation is rather complex. If two steels have equal hardness and each has the same tempered martensitic structure, then the ultimate strength, the yield strength, and the endurance strength will be essentially the same—in a small piece of the metal. The alloy content helps out in several very important ways: • The cooling rate in quenching can be considerably slower. This makes it possible to get a good metallurgical structure in the large gears. • Gears with high alloy content can be carburized with too rich or too lean a carburizing atmosphere and still come out fairly good. Nickel in particular makes heat-treating operations less sensitive to precise control. • Certain alloy combinations are helpful in developing fracture toughness. (With these combinations, a small crack grows very slowly, or perhaps ceases to grow.) • In general, the impact properties are considerably improved by alloy content. Nickel and mo­ lybdenum are particularly valuable for impact strength. The overall situation is that the composition of the steel used in gears is very important. A poor choice of alloy content had often led to early failure in gears built with good precision and sized large enough to meet design standards.

10.1.2 HEAT-TREATING TECHNIQUES Gear steels are heat-treated for two general purposes. First, they must be put in condition for proper machinability. Second, the necessary hardness, strength, and wear resistance for the intended use must be

Gear Materials

599

developed. Steel in the as-rolled or as-forged condition may be coarser-grained in structure and nonuniform in hardness as a result of uncontrolled cooling after the forging or rolling operations. Therefore, a heat treatment followed by controlled cooling is used to develop the type of metallurgical structure most suited to the subsequent machining operations. Gear forgings of carburizing steels are usually normalized or normalized and tempered to develop a uniform microstructure and reduce their tendency to distort during later hardening operations. Normalizing consists of heating the steel to well above the critical temperature and somewhat above the final hardening temperature, followed by an air cool. Another treatment similar to normalizing is annealing. The part to be annealed is heated above the critical range, just as in normalizing, but it is cooled at a slow rate either by controlling the furnace cooling rate or by allowing the furnace and load to cool off together with the doors closed. Slow cooling in the furnace usually produces a pearlitic or lamellar structure, which provides good surface finishes after machining. An interrupted or cycle annealing procedures a spheroidized structure, which often has maximum machinability. Steel parts are hardened by quenching and tempering to develop the combination of strength, toughness, hardness, and wear resistance that may be needed to make the part function properly in use. Proper hardening consists of cooling the steel from above the critical temperature quickly enough to form a fully hardened structure and to prevent the formation of undesired structures, which could occur at intermediate temperatures if the cooling rate is too slow. Table 10.2 shows the important temperatures that must be observed with different kinds of steel. Table 10.3 shows the critical cooling rates for steels made to the minimum composition range for each kind of steel. The size of round that will through-harden in a mild quench and the hardness obtained with 60% Martensite formation are also shown in Table 10.3. Table 10.4 shows the approximate cooling rates needed to get a high level of metallurgical quality in gears. The steels listed in Table 10.2 and Table 10.3 are those commonly used for gears. The compositions of these steels that have been established by the American Iron and Steel Institute are given in Table 10.5. The exact composition of steel may vary somewhat from one steel company to another or from steels used in vehicles to those used in aircraft. The aircraft composition tends to have closer ranges for each alloy element, and there is more control of trace elements. The hardening temperature of a steel must be slightly above the critical temperature. The critical temperature is the temperature at which the steel is completely austenized and ready to undergo the hardening reaction upon quenching. The critical cooling rate is the rate of cooling that is just fast enough to produce a fully hardened structure in a particular steel composition. The Ms temperature is the temperature at which the fully hard structure (martensite) first starts to form during the quench. If the cooling rate is slower than the critical rate, other structures (ferrite, Pearlite, and upper bainite) may form at intermediate temperatures between the critical temperature and the Ms temperature. After quenching, steels are usually tempered or stress-relieved. The tempering operation may be used to reduce the hardness of a part and increase the toughness. By properly adjusting the tempering tem­ perature, a wide range of hardness may be obtained. Even when no reduction in hardness is desired, a low-temperature (250 to 350°F) tempering operation is desirable to reduce stresses in the steel and produce a kind of martensite that is tougher than the kind produced immediately upon quenching.

10.1.3 HEAT-TREATING DATA The gear designer should remember that, in most cases, the strength and hardness of the steel gear will depend primarily upon the skill and intelligence with which the steel has been heat-treated. Improper quenching or tempering at the wrong temperature will completely defeat the designer’s selection of the best steel for the job. Heat treating is still a skilled art. The best gears are made by those who study carefully the microstructure of their product and alter their heat-treating technique to get the best possible results.

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Dudley’s Handbook of Practical Gear Design and Manufacture

TABLE 10.2 Heat-Treating Data for Typical Gear Steels AISI No.

Normalizing Temperature, °F

Annealing Hardening Carburizing Reheat Temperature, °F Temperature, °F Temperature, °F Temperature, °F

Ms Temperatures*, °F

1015

1700

1600

–

1650–1700

1025

1650–1750

1600

1575–1650

1500–1650

1040 1045

1650–1750 1600–1700

1450 1450

1525–1575 1450–1550

1060

1550–1650

1400–1500

1450–1550

–

–

555

1118 1320

1700 1600–1650

1450 1500–1700

– –

1650–1700 1650–1700

1650–1700 1450–1500

740

1335

1600–1700

1500–1600

1500–1550

–

–

640

2317 2340

1650–1750 1600–1700

1575 1400–1500

– 1425–1475

1650–1700 –

1450–1500 –

725 555

3310

1650–1750

1575

–

1650–1700

1450–1500

655

3140 4028

1600–1700 1600–1700

1450–1550 1525–1575

1500–1550 –

– 1600–1700

– 1450–1500

590 750

4047

1550–1750

1525–1575

1475–1550

4130 4140

1600–1700 1600–1700

1450–1550 1450–1550

1550–1650 1525–1625

– –

– –

685 595

4320

1600–1800

1575

–

1650–1700

1425–1475

720

4340 4620

1600–1700 1700–1800

1100–1225 1575

1475–1525 –

– 1650–1700

– 1475–1525

545 555

4640

1600–1700

1450–1550

1450–1550

–

–

605

4820 5145

1650–1750 1600–1700

1575 1450–1550

– 1475–1525

1650–1700

1450–1500

685

5210

–

1350–1450

1425–1600

–

–

485

6120 6150

1700–1800 1650–1750

1600 1550–1650

– 1550–1650

1700 –

1475–1550 –

760 545

8620

1600–1800

1575

–

1700

1425–1550

745

9310

1650–1750

1575

–

1650–1700

1425–1550

650

EX 24 EX 29

1650–1750 1650–1750

1600 1600

1660 1600

1650–1700 1650–1700

1500–1550 1500–1550

830 830

EX 30

1650–1750

1550

1600

1650–1700

1500–1550

830

EX 55

1650–1750

1525

1600

1650–1700

1500–1550

790

Notes * The Ms temperature is the temperature at which martensite forms.

Table 6.2 gave heat-treating data for some typical gear steels. The values shown should be considered to be nominal and should be varied in actual practice as experience dictates. Figure 10.2 shows the way in which structures form during the cooling of 4340 steel. Transformation diagrams such as this one can be obtained from several companies that are involved in steel making. These diagrams are very helpful in developing a suitable heat-treating procedure for a particular steel. The carbon content of a steel establishes the maximum hardness that can be reached in the fully hardened condition, while the alloying elements determine the critical cooling rate necessary for full hardening and therefore the section thickness that will harden with the quench that is available. For

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601

TABLE 10.3 Hardenability Data for Typical Gear Steels. (Data Based on Minimum Composition Range Except Where Noted) AISI No.

Hardness of 60% Martensite, HRC

Critical Cooling rate, °F per Second at 1300°F

Size of Round That Will Through-harden, in.

Mildly Agitated Quenching Medium

Through-hardening steels 1045 1060

50.5 54

400* 125*

0.50** 1.20**

Water Water

1335

46

195

1.00

Water

2340 3140

49 49

125 125

0.60 0.60

Oil Oil

4047

52

195

1.00

Water Oil Water

4130

44

305

0.40 0.70

4140

49

56

1.00

Oil

4340 5145

49 51

10 125

2.80 0.60

Oil Oil

5210

60

30

1.30

Oil

53 Minimum core hardness

77 Critical cooling rate, °F per second at 1300°F

0.80 Size of round that will through-harden, in.

Oil Mildly agitated quenching medium

6150 AISI No.

Data for core of case-carburizing steels 1015 1025

15 18

400** 400**

0.50** 0.50**

Water Water

1118

20

400**

0.50**

Water

1320 3310

20 25

305 3

0.70 3.00

Water Oil

4028

24

300*

0.80**

Water

4320 4620

32 30

195 305

0.40 0.20

Oil Oil

0.70

Water

4820 8620

35 25

77 250*

0.80 0.80

Oil Water

0.30

Oil

9310 EX 24

25 26

21 195*

1.70 0.80

Oil Water

EX 29

32

175*

1.00**

Water

35 40

*

1.20 2.00

Water Oil

EX 30 EX 55

150 42**

Notes * Estimated values. ** Obtained from nominal composition rather than from minimum of specification range.

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Dudley’s Handbook of Practical Gear Design and Manufacture

TABLE 10.4 Approximate Cooling Rates Needed to Achieve Appropriate Metallurgical Quality for Long-life, Highly Loaded Gears Carburizing Steel

Time*, s

Core Hardness HV10

to 600°C (1112°F)

to 400°C (752°F)

to 200°C (392°F)

339** 339

3 5

8 12

10 28

4028 8620 EX 24

339

6

20

35

4620 4320

339 339

3 8

8 22

10 60

EX 29

339

8

22

60

9310 4815

339 339

10 6

35 21

90 58

EX 30

339

6

21

58

EX 35

390

13

50

125

Notes * Time is from start of quench to the temperature listed. ** 339 HV10 is approximately 34 HRC. 390 HV10 is approximately 40 HRC.

example, plain carbon steels have such high critical cooling rates that they must be water- or brinequenched to be fully hardened even when the section is relatively small. Alloy steels transform more slowly and can be hardened with an oil quench. In the case of some high-alloy tool and die steels, even cooling in still air will develop high hardness! End-quench-hardenability curves demonstrate how alloys slow down the reaction rates of steels and give the heat treater time enough to develop full hardness with a mild quench. With large parts made of plain carbon or low-alloy steels, there is a distinct danger of cracking the parts if they are quenched drastically enough to harden completely. On the other hand, a milder quench may not develop structures with the required strength. In extreme cases, the part may be so large that even with considerable alloy content, it is not practical to use a fast-enough quench to develop full hardness. In this case, the designer has to recognize that compromise is necessary, and design with low enough stresses to get by with whatever properties can be obtained in steel. Figure 10.3 shows end-quench-hardenability curves for several kinds of gear steels. These data, to­ gether with those shown in Table 10.3 and Table 10.4, will help the designer choose a steel that will have sufficient hardenability for a specific gear. The designer should also note material quality grades in gearrating specifications (see Sections 6.2.1, 6.2.10, and 6.2.13).

10.1.4 HARDNESS TESTS The easiest way to determine the approximate tensile strength of a piece of steel is to check its hardness. Gear parts do not always have the same hardness in the teeth as in the rim, web, or hub. One of the best ways for the designer to control the final condition of the heat-treated gear is to specify the hardness of the teeth. In some cases, it is desirable to check both the gear teeth and the gear blank for hardness. Figure 10.4 shows how this is done on small aircraft gears. Table 10.6 shows some of the commonly used hardness checks and the kind of gear for which they are most suited. The amount of load used and the kind of ball or point used to indent the piece being tested are also shown.

Gear Materials

603

TABLE 10.5 Composition of Typical Gear Steels AISI No.

1015 1025

Chemical Composition Limits, % C

Mn

P

0.13/0.18 0.22/0.28

0.30/0.60 0.30/0.60

0.040 0.040

S 0.050 0.050

Si – –

Ni – –

Cr – –

Mo – –

1045

0.43/0.50

0.60/0.90

0.040

0.050

–

–

–

–

1060 1118

0.55/0.65 0.14/0.20

0.60/0.90 1.30/1.60

0.040 0.045

0.050 0.080/0.13

– –

– –

– –

– –

1320

0.18/0.23

1.60/1.90

0.040

0.040

0.20/0.35

–

–

–

1335 3140

0.33/0.38 0.38/0.43

1.60/1.90 0.70/0.90

0.040 0.040

0.040 0.040

0.20/0.35 0.20/0.35

– 1.10/1.40

– 0.55/0.75

– –

3310

0.08/0.13

0.45/0.60

0.025

0.025

0.20/0.35

3.25/3.75

1.40/1.75

–

4028 4047

0.25/0.30 0.45/0.50

0.70/0.90 0.70/0.90

0.040 0.040

0.040 0.040

0.20/0.35 0.20/0.35

– –

– –

0.20/0.30 0.20/0.30

4130

0.28/0.33

0.40/0.60

0.040

0.040

0.20/0.35

–

0.80/1.10

0.15/0.25

4140 4320

0.38/0.43 0.17/0.22

0.75/1.00 0.45/0.65

0.040 0.040

0.040 0.040

0.20/0.35 0.20/0.35

– 1.65/2/00

0.80/1.10 0.40/0.60

0.15/0.25 0.20/0.30

4340

0.38/0.43

0.60/0.80

0.040

0.040

0.20/0.35

1.65/2/00

0.70/0.90

0.20/0.30

4620 4640

0.17/0.22 0.38/0.43

0.45/0.65 0.60/0.80

0.040 0.040

0.040 0.040

0.20/0.35 0.20/0.35

1.65/2/00 1.65/2/00

– –

0.20/0.30 0.20/0.30

4820

0.18/023

0.50/0.70

0.040

0.040

0.20/0.35

3.25/3.75

–

0.20/0.30

5145 5210

0.43/048 0.95/1.10

0.70/0.90 0.25/0.45

0.040 0.025

0.040 0.025

0.20/0.35 0.20/0.35

– –

0.70/0.90 1.30/1.60

– –

6120

0.17/0.22

0.70/0.90

0.040

0.040

0.20/0.35

–

0.70/0.90

V 0.10 min

6150 8620

0.48/053 0.18/0.23

0.70/0.90 0.70/0.90

0.040 0.040

0.040 0.040

0.20/0.35 0.20/0.35

– 0.40/0.70

0.80/1.10 0.40/0.60

V 0.15 min Mo 0.15/0.25

9310

0.08/0.13

0.45/0.65

0.025

0.025

0.20/0.35

3.00/3.50

1.00/1.40

0.08/0.15

EX 24 EX 29

0.18/0.23 0.18/0.23

0.75/1.00 0.75/1.00

0.040 0.040

0.040 0.040

0.20/0.35 0.20/0.35

– 0.40/0.70

0.45/0.65 0.45/0.65

0.20/0.30 0.30/0.40

EX 30

0.13/0.18

0.70/0.90

0.040

0.040

0.20/0.35

0.70/1.00

0.45/0.65

0.45/0.60

EX 35

0.15/0.20

0.20/1.00

0.040

0.040

0.20/0.35

1.65/2.00

0.45/0.65

0.65/0.80

The approximate relation between the various hardness-test scales is shown in Table 10.7. The values shown are averages of tests on carbon and alloy steels. Because of difference in structure, cold-working tendencies, and other factors, there is no exact mathematical conversion between the scales. For this reason, hardness requirements on a gear drawing should be given only in the scale by which they will be checked.

10.2 LOCALIZED HARDENING OF GEAR TEETH Several methods are used either to case-harden only the gear teeth or to harden the surfaces of gear teeth and leave the inside part of the tooth at an intermediate hardness. Carburizing, nitriding, induction hardening, and flame hardening can all be used to produce gear teeth that are much harder than the gear blank that supports the teeth.

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Dudley’s Handbook of Practical Gear Design and Manufacture

FIGURE 10.2 Transformation diagram of AISI 4340. (Courtesy of International Nickel Co., New York, NY, U.S.A.)

FIGURE 10.3

Some typical Jominy curves showing end-quench hardenability.

10.2.1 CARBURIZING Carburizing is the oldest and probably the most widely used process for hardening gear teeth. It consists of heating a 0.10 to 0.25% carbon steel at a temperature above its critical range in a gaseous, solid, or liquid medium capable of giving up carbon to the steel. The surface layer becomes enriched in carbon content and therefore is capable of developing a high degree of hardness after quenching.

Gear Materials

605

TABLE 10.6 Hardness-Testing Apparatus and Application for Gears Instrument

Shape and Type of Indenter

Loading

Recommended Use

Brinell

10-mm steel or tungsten carbide ball

3000 kg

Rockwell C

Diamond Brale penetrator (120° diamond cone)

150 kg

For large gears and shafts in range of hardness from 100 to 400 Brinell. (If gears have a hard case, the case depth must be sufficient and the core strength adequate to support the area under test. Rockwell C tests may be used to reveal an error from such irregular conditions.) Brinell tests, when fairly representative of the general hardness, are good measure of the ultimate tensile strength of the material. For gears medium to large in size. Range approximately 25 to 68 Rockwell C.

Rockwell A

Diamond Brale penetrator (120° diamond cone)

60 kg

For small gears and tips of large gear teeth. Range Rockwell A 62 to 85.

Rockwell 30-N

Diamond Brale penetrator (120° diamond cone, special indenter) Diamond Brale penetrator (120° diamond cone, special indenter)

30 kg

For small parts and shallow case-hardened parts.

15 kg

Lightest Rockwell load. For testing very small parts and checking the working side of teeth. Check for very thinly case-hardened parts. Rockwell 15-N is used 72-93.

Rockwell 15-N

Vickers, pyramid

136° diamond pyramid

50 kg

All applications where piece will not be too heavy for machine. For use in testing hardness of shallow cases, etc.

Scleroscope

Does not indent surface. Diamond-tipped tup bounced on specimen

–

For applications permitting no damage or indentation to surfaces (results not always comparable with indentation hardness tests).

Tukon

Knoop indenter

500–1000 g for practical use

A laboratory instrument used only for finding the hardness of material on pieces of cross sections, e.g., hardness from surface inward, every 0.05 mm (0.0002 in.), or hardness of individual microconstituents. All test areas must have flat mirrorpolished surfaces. An extremely delicate and precise test. Used for any hardness.

The concentration of carbon in the surface layer is determined by many factors. For best strength and toughness, though, the concentration should be kept under 1.0%, preferably around 0.80 to 0.90%. Control of the richness of the carbon case is obtained by controlling the richness of the carburizing atmosphere. The development of the case depends upon the diffusion of carbon into the steel; therefore, time and temperature are the main factors that control the case depth. Steel composition has no great influence on the rate of carbon penetration. Figure 10.5 shows the approximate effect of temperature and time on the depth of case obtained in a gaseous carburizing operation. Carburized parts may be heat-treated in several ways to achieve a variety of case and core properties. Several of the most important treatments are illustrated in Figure 10.6 and explained in the table ac­ companying the chart. For heavy-duty gearing, treatment C generally provides both a good case and a strong core. With only one quench, distortion is minimized. In some high-production gear jobs, it has

606

FIGURE 10.4

Dudley’s Handbook of Practical Gear Design and Manufacture

Hardness-checking an aircraft gear near the pitch line of the tooth.

been possible to get satisfactory gears by directly quenching from the carburizing temperature. The designer should hesitate to use this short-cut method unless there is a steel and a procedure that will produce gears that are known (by proper testing) to be good enough for the job. In the vehicle field, the direct quench method is normally used for carburized gears. The gears are somewhat small, the volume of the production is high, and the facilities used are picked to be just right for the part. All these things tend to make it practical to use direct quench. In the turbine field, most carburized gears are given a reheat quench. Here the gear parts tend to be fairly large, general-purpose heat-treat equipment is used, and the volume is low. The reheat method seems to give better metallurgical structure and more assurance that the gears can run satisfactorily for 109 or 1010 cycles. (In the vehicle field, gears seldom run at high loads for more than 108 cycles.) Commonly used carburized steels are shown in Table 10.3. For best load-carrying capacity, the case should be up to 700 HV (60 HRC), and the core should be in the range of 340 to 415 HV (35 to 42 HRC). Too hard a core promotes brittleness and cuts down the compressive stress that is developed by the slight difference in volume between the case material and the core material. Too soft a core does not provide strength enough to support the high loads that the case can carry. For aircraft work, most designers favor a steel of the AISI 9310 type. The AMS 62601 specification quite closely follows the specification of AISI 9310 (AMS means “aircraft material specification”). This is nickel–chromium–molybdenum type

1

Most of the critical aircraft gears are made from AMS 6265 steel. This is a vacuum-arc remelt steel of unusual cleanness. Both AMS 6260 and 6265 are 9310 types of steel. (AMS 6260 is an air-melt quality of steel.)

Gear Materials

607

TABLE 10.7 Approximate Relation between Hardness-Test Scales Brinell 3000 kg, 10 mm

614 587

Rockwell

Vickers Pyramid

Scleroscope (Shore)

Tukon (Knoop)

C

A

30-N

15-N

70 65

86.5 84.0

86.0 82.0

94.0 92.0

1076 820

90

840

63

83.0

80.0

91.5

763

87

790

60 58

81.0 80.0

77.5 75.5

90.0 89.3

695 655

81 79

725 680

547

55

78.5

73.0

88.0

598

74

620

522 484

53 50

77.5 76.0

71.0 68.5

87.0 85.5

562 513

71 67

580 530

460

48

74.5

66.5

84.5

485

64

500

426 393

45 42

73.0 71.5

64.0 61.5

83.0 81.5

446 413

61 56

460 425

352

38

69.5

57.5

79.5

373

51

390

301 250

33 24

67.0 62.5

53.0 45.0

76.5 71.5

323 257

45 37.5

355

230

20

60.5

41.5

69.5

236

34

B

Rockwell 30-T

15-T

200

93

78.0

91.0

210

30

180 150

89 80

75.5 70.0

89.5 86.5

189 158

28 24

100

56

54.0

79.0

105

80* 70*

47 34

47.7 38.5

75.7 71.5

Note: * Based on 500-kg load and 10-mm ball.

of steel. Other types of steels used for making heavy-duty gears for marine, tractor, railroad, and other applications are the nickel, nickel–chromium, and nickel–molybdenum types. In a few gear applications (usually smaller-sized parts), good load-carrying capacity has been obtained using steels with a low alloy content. On a new job, the designer should do some development work to determine the most suitable type of steel for a given application. The carburizing cycle brings gears up to a red-hot temperature. When the red-hot gear is quenched, it cools rapidly, but somewhat unevenly. The outer surface, of course, cools the fastest because it is in contact with the quenching medium. As the steel cools, the structure changes and grows stronger. The unequal cooling rates in different parts of the gear body tend to make the gear change size very slightly and to distort. In addition, the carburized case tends to be slightly larger in volume than the core material. This also contributes to the size change and distortion tendency. The overall result of this is that the pressure angle of the tooth tends to increase slightly, and the helix angle tends to unwind. There is also a tendency for the bore to shrink, for the part to develop axial and radial runout, and for the outside diameter to become slightly coned. Because of these things, it is necessary either to make allowances for dimensional changes, or to grind the part after carburizing. In

608

FIGURE 10.5

Dudley’s Handbook of Practical Gear Design and Manufacture

Nominal time and temperature requirements for different case depths.

either event, quenching dies may be helpful in cutting down distortion and reducing the amount of change that must be either allowed for or ground off. Carburized gears have to have enough surface hardness to resist surface-initiated pitting. The al­ lowable contact stress is based on the surface hardness. Small gears, with about 2.5-module (10-pitch) teeth and pitch diameters in the range of about 40 to 300 mm, can be carburized to achieve a surface hardness in the range of 700 to 759 HV (60 to 63 HRC). Larger gears, with 5-module (5-pitch) teeth and diameters going up to 1 meter (40 in.), are more difficult to carburize with optimum results in hardness and metallurgical structure. The design hardness of the medium-large gear should generally be in the range of 675 to 725 HV (58 to 62 HRC). If there are problems in processing carburized gears, the achievable minimum surface hardness may be only 600 HV (55 HRC). The problems may come from the parts being rather small or very large. Parts as small as 1-module (25 pitch), or as large as 25-module (1 pitch), are carburized. Either of these sizes is difficult to handle, and the surface hardness may be low. Besides surface hardness, the carburized pinion or gear needs an adequate case depth. There are subsurface stresses that are strong enough to cause cracks in the region of case-to-core interface if the case is too thin. The carburized case needs to be deep enough to give adequate bending strength, and deep enough to resist case crushing or case spalling. The depth needed for bending strength is a function of the normal module (or the normal diametral pitch), but the depth needed to resist subsurface stresses due to contact loads is a function of the pitch diameter of the pinion, the ratio, and the pressure angle (primarily). Figure 10.7 shows an example of a carburized tooth sectioned to study case and core structure—before finish grinding. The minimum effective case depth in the root fillet for tooth bending strength may be estimated by this relation:

Gear Materials

609

FIGURE 10.6 Diagrammatic representation of different treatments subsequent to carburization and the case and core characteristics obtained. (Courtesy of International Nickel Co., NY, U.S.A.)

het = 0.16 mn het =

0.16 Pnd

mm metric in.

(10.1)

English

(10.2)

where: mn– is the normal module. Pnd – is the normal diametral pitch.

The minimum effective case depth on the flank of the tooth for surface durability may be estimated by this relation: hec =

hec =

sc dp1 sin

t

48, 250 cos sc d sin

7×

10 6

cos

b

t b

u u+1 mG mG + 1

mm metric

(10.3)

in.

(10.4)

English

610

FIGURE 10.7

Dudley’s Handbook of Practical Gear Design and Manufacture

Carburized gear tooth. (a) Section about 5 ×. (b) Case structure about 1000 ×. (c) Core about 250 ×.

Gear Materials

611

where: sc dp1 d

t,

– – – t–

is is is is

the the the the

maximum contact stress in the region of 106 to 107 cycles (N/mm2 for metric, psi for English). pinion pitch diameter, mm (metric). pinion pitch diameter, in. (English). pressure angle, transverse, metric, English.

b , b – is the base helix angle, metric, English. u , mG – is the tooth ratio, metric, English.

Figure 10.8 shows a schematic view of the case on a gear tooth. The minimum thickness needed at region A is determined by Eqs. (10.3) and (10.4). The minimum thickness needed at region B is determined by Eqs. (10.1) and (10.2). If the case is too deep, the depth at region C becomes too great. A deep case at region C is a hazard because the whole top of the tooth may break off. Generally speaking, the depth at region C should not be greater than: hem = 0.40 mn hem =

0.40 Pnd

mm in.

metric English

(10.5) (10.6)

In carburizing gears, the case depth is customarily specified for region A, with maximum and minimum limits of effective case depth. The effective case depth for parts with a minimum surface hardness of at least 675 HV (58 HRC) is taken at the point at which the case is at least HV 510 (50 HRC). For parts with less surface hardness, it is advisable to put the limit point for effective case depth at about 7 HRC points lower than the minimum surface hardness. Thus, the effective case depth for 600 HV (55 HRC) might be taken at HV 485 (48 HRC). In specifying case depth, a set of values like this needs to be used: • Effective case depth, flank, 0.50 to 0.75 mm (0.020 to 0.030 in.) • Minimum effective case depth, root, 0.40 mm (0.016 in.) • Maximum effective case depth, tip, 1.00 mm (0.040 in.) The above might be about right for a 2.5-module spur pinion, heavily loaded.

FIGURE 10.8 Carburized case pattern. Shaded area is all 510 HV (50 HRC) or higher in hardness. Unshaded area is less than 510 HV (50 HRC).

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Dudley’s Handbook of Practical Gear Design and Manufacture

The amount of case depth needed for surface durability is illustrated in Figure 10.9. This figure shows an example of spur gears with 22.5 pressure angle and a tooth contact stress of 1800 N/mm2 (261,000 psi) for 5 × 106 cycles. This is typical of a very good vehicle gear, with very high torque at relatively slow pitch-line speed. The faster-running turbine gears, for instance, would have lower design stresses in the region of 107 cycles and still lower stress levels at a rated load in the region of 109 or 1010 cycles.

10.2.2 NITRIDING Nitriding is a case-hardening process in which the hardening agents are nitrides formed in the surface layers of steel through the absorption of nitrogen from a nitrogenous medium, usually dissociated am­ monia gas. 10.2.2.1 Features of Nitriding Process Almost any steel composition will absorb nitrogen, but useful cases can be obtained only on steels that contain appreciable amounts of aluminum, chromium, or molybdenum. Other elements, such as nickel and vanadium, may be needed for their special effects on the properties of the nitrided steel. The gear steels most commonly nitrided are shown in Table 10.8. The temperatures used and the hardness ob­ tained are shown in Table 10.9. A nitride case does not form as fast as a carbon case. Figure 10.10 shows the nominal relation between nitriding time and case depth obtained. Nitriding, like carburizing, is a difficult process, but because the rate of penetration is slower, the time cycles are quire long. Nitriding is conducted in sealed retorts in an atmosphere of dissociated ammonia at temperatures between 930 and 1000°F. The modern practice is to start the process with 30% ammonia dissociation for the first several hours, then to allow dissociation to increase to 85%. At the lower dissociation rate, a weak and brittle layer of overly rich nitrides is formed at a surface layer about 0.05 mm (0.002 in.) deep.

FIGURE 10.9 Minimum effective case depth for carburized spur gears of 22.5 pressure angle and a contact stress of 1800 N/mm2 (261,000 psi) for not over 107 cycles.

Gear Materials

613

TABLE 10.8 Nitriding Gear Steels Steel

Carbon

Manganese

Silicon

Chromium

Aluminum

Molybdenum

Nickel

Nitralloy 135

0.35

0.55

0.30

1.20

1.00

0.20

–

Nitralloy 135 (modified)

0.41

0.55

0.30

1.60

1.00

0.35

–

Nitralloy N AISI 4340

0.23 0.40

0.55 0.70

0.30 0.30

1.15 0.80

1.00 –

0.25 0.25

3 1

AISI 4140

0.40

0.90

0.30

0.95

–

0.20

–

31 Cr Mo V 9

0.30

0.55

0.30

2.50

–

0.20

–

TABLE 10.9 Nominal Temperatures Used in Nitriding and Hardnesses Obtained Steel

Temperature before Nitriding, F

Nitriding, F

Hardness, Rockwell C Case

Core

Nitralloy 135*

1150

975

62 – 65

30 – 35

Nitralloy 135 modified Nitralloy N

1150 1000

975 975

62 – 65 62 – 65

32 – 36 40 – 44

AISI 4340

1100

975

48 – 53

27 – 35

AISI 4140 31 Cr Mo V 9

1100 1100

975 975

49 – 54 58 – 62

27 – 35 27 – 33

Note: * Nitralloy is trademark of the Nitralloy Corp., New York, N.Y.

This is called a white layer because it etches out white in a micrograph. The higher dissociation rate that is used at the end of the cycle allows most of the excess nitrides to diffuse into the metal, leaving only traces of q white layer. Figure 10.11 shows examples of nitride cases on spline and gear teeth. For best results, parts to be case-hardened by nitriding, should be rough-machined and then quenched and tempered. The tempering temperature is usually between 1000 and 1150°F. After heat treatment, the part is finish-machined, stress-relieved at about 1100°F, and then nitrided. Since the nitriding temperature is lower than the original tempering temperature of the steel, hard­ ening occurs with a minimum of distortion. The method is therefore suitable for complex parts such as gears that can be machined while in the medium-hardness region and then hardened without enough distortion to require grinding. Parts that are too flimsy to stand a quench and draw without serious distortion can often be successfully brought up to full hardness by nitriding. The formation of nitrides in the case causes the steel to expand, and, as in carburized cases, a fa­ vorable compressive stress is developed. This also results in a slight overall growth of the part. In spur gears this is manifested by an increase in outside diameter and in the diameter over pins. As an example, a 2-module (12-pitch) spur gear with a lightweight web showed an average increase in outside diameter of 0.13 mm (0.005 in.) and a 0.20-mm (0.008-in.) growth in diameter over pins. The pressure angle tends to decrease slightly. (This is just the opposite to the change in profile caused by case-carburizing.)

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Dudley’s Handbook of Practical Gear Design and Manufacture

FIGURE 10.10 Nominal time required for different nitride case depths. Note: Under the most favorable condi­ tions, about 40% more case depth is possible.

In general, there is no distortion of the gear blank, provided that the blank was thoroughly tempered and stabilized before nitriding. If the blank has residual stresses before nitriding, the many hours at nitriding temperature will relieve these stresses and cause the part to warp. The tendency for the nitride case to grow produces a “cornice” effect, or an overhang of quite brittle material on sharp corners and edges of parts, unless precautions have been taken to break such corners with a small radius before nitriding. If the controlled dissociation method of nitriding is not used, it is usually advisable to grind or lap off the white layer [0.08 mm (0.003 in.) nominally] after nitriding. It is also advisable to take precautions to avoid decarburization of areas to be nitrided. A very weak and brittle case is produced on decarburized steels or on steels that have not been properly quenched and tempered prior to nitriding. If the steel contains aluminum, a nitrided case is relatively harder and inherently more brittle than a carburized case. Under high pressures, it may crack and spall because of either too little case depth or too weak a core material under the case. In general, nitrided gears need almost as much case as shown in Figure 10.9, but with the coarser pitches, it is not possible to get the required depth. In severe load applications, some help can be obtained by using the Nitralloy N type material. This material goes through a precipitation hardening of the core during the nitriding cycle. This gives the case added support and may, to some extent, compensate for the case being too thin on coarser-pitch gears. The nitrided gear—because of its high hardness—resists scoring and abrasion better than other types of gears. Tests of 2.5-module (10-pitch) and finer nitrided gears have shown that with proper nitriding technique, just as heavy loads could be carried on a nitrided tooth2 as on a good case-carburized tooth. In cases where shock loading is present, or where the pitch is medium to coarse, most designers have found that other types of hardened gears would stand more loading than the nitrided gear.

2

This statement assumes that the nitrided case is the same hardness as the carburized case and that the governing load was one for 107 cycles or more (not a very high initial load for less than 106 cycles).

Gear Materials

615

FIGURE 10.11 Examples of nitrided gear tooth structures: (a) small spline, Nitralloy 135M. White layer 0.03 mm (0.0012 in). Case depth 0.41 mm (0.016 in.); (b) medium size, internal spline 4340. White layer 0.01 mm (0.0004 in.). Case depth 0.38 mm (0.015 in.); (c) part of 5-module gear tooth, 31 Cr Mo V 9. White layer 0.016 mm (0.0006 in.). Case depth 0.65 mm (0.026 in.).

10.2.2.2 Nitride Case Depth From the standpoint of tooth breakage, the nitride case does not generally give a high degree of strength. For spur teeth in the range of 4-module (6-pitch) to 10-module (2.5-pitch), nitriding has some value in increasing the surface load-carrying capacity, but the beam strength has to be based almost entirely on the core hardness of the tooth. Equations (10.1) and (10.2) are a good approximation of the case depth

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Dudley’s Handbook of Practical Gear Design and Manufacture

needed if the nitrided case—instead of the core—is to govern the beam strength of the nitrided tooth. It can be seen from Figure 10.10 that it takes about fifty-five hours nitriding time to get a case depth of 0.48 mm (0.019 in.). Equations (10.1) and (10.2) show that a 3-module spur tooth needs a case depth of about 0.48 mm if the case is to control the beam strength. This means that a very long nitriding time is needed if teeth larger than 3-module are going to get much beam-strength help from the nitride case. From the standpoint of surface durability, the situation is also rather critical on nitride case depth. A pinion of 150-mm (6-in.) pitch diameter running with a gear of 450 mm (18 in.) needs about 1.5 mm (0.060 in.) case depth when carburized and designed to carry a load of 1800 N/mm2 (261,000 psi). There is some limited experience that says nitride case depth does not need to be quite as deep as carburized depth to handle the same load. If this is true, the nitride case depth would need to be at least 85% of the values given by Eqs. (10.3) and (10.4). The net result is that only small pinions (less than 100 mm) will be able to have enough nitride case depth to carry as much load as their surface hardness would indicate. Many large pinions (and their mating gears) are nitrided and used successfully in turbine-gear ap­ plications. How do they get by with a very thin case for their size? The answer seems to lie in using a surface loading that is lower than what the surface hardness could handle, but with subsurface stresses that are within the capability of the core material. If the nitrided gear has a core hardness of at least HV 335 (34 HRC), the subsurface capability of the material underneath the case should be able to handle a surface loading up to about 1000 N/mm2 (145,000 psi). In actual practice, nitrided turbine gears tend to be designed for surface loading in the range of 900 to 1000 N/mm2 (130,000 to 145,000 psi). This means that they can get by with a much thinner case than Eqs. (10.3) and (10.4) would indicate—providing they have good core hardness. For nitrided gears, the effective3 case depth based on HV 395 (40 HRC) seems to work the best. The low-chromium nitriding steels develop a surface hardness of only about HV 515 (50 HRC). They need a much lower-case determination point than do carburized steels. The aluminum types of nitriding steels and the high-chromium nitriding steels develop hardness comparable to carburized steels. The shape of their case hardness curves, though, is such that most of the hard metal is above HRC 40. Figure 10.12 shows two actual examples. Note how well the effective case is defined by the HRC 40 point.

10.2.3 INDUCTION HARDENING OF STEEL High-frequency alternating currents can be used to heat locally the surface layers of steel gear teeth. Since the gear body remains relatively cold during the induction hardening, it serves as a fixture to maintain the dimensional accuracy of the induction-hardened teeth. The surface layers of metal undergo some plastic upsetting as they are heated because of the restraint of the cold blank. Upon hardening, the surface layers of metal undergo some expansion due to the volume increase of the hardened metal. There are thus two conflicting tendencies in induction hardening. One is a tendency for the part to shrink, and the other is a tendency for the part to expand. If the right technique is used with the right steel on the right pitch of teeth, good residual compression can be obtained. If the cycle is not just right, damaging residual tension stresses can result. Some gear manufacturers have developed proprietary processes for induction hardening of certain kinds of gears, which enable them to build induction-hardened gears similar in strength to case-carburized or nitrided teeth. Many manufacturers have not been able to build inductionhardened gears with as much capacity as case-hardened gears. Designers of induction-hardened gears should plan on running enough experiments to develop just the right kind of cycle for the particular pitch and size of blank they are concerned with.

3

If the nitrided gear has a core hardness approaching HV 395 (40 HRC), then a higher value, such as 446 HV (45 HRC), must be used for effective case depth.

Gear Materials

617

FIGURE 10.12 Examples of two steels nitrided for a total time of forty-five hours. 4340 is a low-chromium steel. 135M is high-chromium and aluminum steel.

Induction-hardened gears have some tendency to warp. Usually there is no change in tooth profile, but there may be a “coning” of outside diameter, axial runout, and radial bumps or hollows near holes in the web or near spokes in the wheel. Fortunately, much can be done to control induction-hardened-gear distortion by changes in blank design and changes in the induction-heating technique. Both plain carbon and alloy steels are used for induction-hardened gears. Carbon content is usually either 0.40 or 0.50%. If a very fast cycle of heating is used, the choice of alloy will depend on the time that is required for the steel to austenitize. A small gear may be brought to the upper critical temperature in as little as four seconds. The hardness pattern obtained with induction hardening will depend on the alloy used, the amount of power per square inch of gear surface, the heating time, the frequency, and the pitch of the tooth. Figure 10.13 shows some examples of hardness pattern obtained in very-high-capacity gear teeth. The induction-hardening current may be obtained from motor-generator sets, spark-gap oscillators, or vacuum-tube oscillators. The power and frequency available from these sources are as follows:

Source

Power, kW

Frequency, cycles per second

Motor-generator Spark-gap oscillator

5–1000 2–15

5000–12,000 10,000–300,000

Vacuum-tube oscillator

2–100

300,000–1,000,000

618

Dudley’s Handbook of Practical Gear Design and Manufacture

FIGURE 10.13 Hardness patterns obtained with induction-hardened gear teeth. (a) 2-module (12-pitch) pinion. (b) 10-module (2.5-pitch) pinion.

Table 10.10 shows that quite large amounts of power are required to induction-harden gears when a coil is wrapped around the whole gear. Many designs cannot be induction-hardened simply because equipment with power enough to handle then is not available. Fine-pitch gears require very high frequency for best results, while coarse-pitch gears require low frequency. If the wrong frequency is used, the heat may be developed below the teeth (fine pitch) or only in the tooth tips (coarse pitch). Table 10.11 shows the frequencies generally recommended. 10.2.3.1 Induction-Hardening by Scanning During the 1960th and 1970th, methods were developed to induction-harden gear teeth by moving an inductor across the face width of the gear to harden either a single tooth or the space between two teeth. In this way, a large gear could be progressively hardened with a relatively small amount of power. If a gear had 100 teeth, it would take 100 passes of the inductor across the face width to harden the whole gear. Although this is somewhat time-consuming, the method was quite practical. A carburized gear might require five to ten hours of furnace time, while a nitrided gear might require fifty hours of furnace time. If it took three hours to harden a gear by the scanning method, this was not bad. Of course, if the gear could have been induction-hardened by a coil wrapped around the gear, it might have been done in something like five minutes. The scanning method offered the ability to achieve quite close control over the contour that was hardened. Figure 10.14 shows some examples of tests to establish a desired case depth. In the illustration, the S values are rates of coil traverse, and the P values are power settings. Note the deep case at the left with S 10 and P 4.0, and the thinner case at the right with S 17. 5 and P 5.5. These tests were done with an inductor between the teeth. The scanning method will do large gears with large teeth. It will do internal gears as well as external gears. The inductor can follow a helical spiral to do helical gears. Figure 10.15 shows a close-up of the

Gear Materials

619

TABLE 10.10 Power and Time Required to Induction-Harden Spur Gears Power, kW

Gear Size Diameter

Tooth Size

Approximate Heating Time, Seconds

Face Width in.

m

Pd

25

12

0.5

12

0.50

1.25

20

5

25 25

50 50

2.0 2.0

6 25

0.25 1.00

2.50 2.50

10 10

10 50

mm

in.

mm

50

25

1.0

6

0.25

1.25

20

4

50 50

25 125

1.0 5.0

18 6

0.75 0.25

1.25 2.50

20 10

15 5

50

125

5.0

18

0.75

2.50

10

7

100 100

150 250

6.0 10.0

125 25

5.00 1.00

8.00 2.50

3 10

90 12

700

150

6.0

50

2.00

3.00

8

6

700 700

150 750

6.0 30.0

125 125

5.00 5.00

8.00 8.00

3 3

20 130

Note: This table is for hardening the whole gear in one operation.

TABLE 10.11 Frequencies Generally Recommended for Induction Hardening Module

Diametral Pitch

Frequency, Cycles per Second

32 10

500,000 – 1,000,000 300,000 – 500,000

5

5

10,000 – 300,000

10

2.5

6000 – 10,000

0.7 2.5

FIGURE 10.14 Induction-hardened test piece for case depth. This work was done with a patented submerged inductor by National Automatic Tool Co., Inc., Richmond, IN, U.S.A.

620

Dudley’s Handbook of Practical Gear Design and Manufacture

FIGURE 10.15 Close-up view of scanning mechanism on a large induction-hardening machine. Note round guide pin, inductor, electric power cables, coolant lines, and traversing ram. (Courtesy of National Automatic Tool Co., Inc., IN, U.S.A.)

mechanism that scans the teeth. Note in this view the guide pin, the inductor, the power lines, and the lines to pass coolant through the inductor. In some equipment, the gear and the inductor are submerged in coolant while the inductor hardens the teeth. In other equipment, the gear is kept cool by jets of coolant going along behind the inductor; the inductor is not submerged. In some cases, the setup is quite flexible and can be used in a variety of ways to best suit the part being done. The basics of measuring case depth on teeth hardened by the scanning method are shown in Figure 10.16. As with the carburized tooth shown in Figure 10.8, there are three places to check the case and one location for measuring core hardness. An example of the hardness pattern that may be obtained with the scanning method is shown in Figure 10.17. Note the rather uniform hardness across the case and then the abrupt drop to core hardness. The effective case depth can be taken at 40 HRC like nitrided gears. (It wouldn’t make much difference if it were 45 HRC). The scanning method uses frequency values that range from 10 to 300 kHz (a kilohertz is a thousand cycles per second; 10 kHz = 10,000 cycles per second). Table 10.12 shows the nominal range of case depths that can be obtained for different tooth sizes and different frequency values. At a given frequency, the case depth is adjusted by the rate of scanning and the power setting. (Figure 10.14 shows test samples done at 200 kHz on 10-module teeth.) The case depth needed with induction-hardened teeth is somewhat greater than that needed for car­ burized teeth. Since the transition from case to core is not gradual, the case/core interface may not be able to carry as much load as its hardness would indicate. This makes it desirable to get the interface zone deeper into the tooth, where the subsurface stresses are lower.

Gear Materials

621

FIGURE 10.16 Induction-hardened case pattern by the scan­ ning method of tooth heating.

FIGURE 10.17

Example of 4340 gear induction-hardened by the scanning method.

622

Dudley’s Handbook of Practical Gear Design and Manufacture

TABLE 10.12 Scanning Induction Heating Frequency and Case Depth Ranges That Are Practical for Different Sizes of Gear Teeth Tooth Size

Frequency, kHz

Module

Diametral Pitch

2.5

10

300 0.5–1.0

150

50

10

–

–

–

(0.02–0.04) 3

8

0.75–1.25 (0.03–0.05)

–

–

–

4

6

–

0.75–1.50

0.75–1.50

–

–

(0.03–0.06) 0.89–1.65

(0.03–0.06) 0.89–1.75

1.75–2.50

(0.035–0.065)

(0.035–0.070)

(0.07–0.10)

5

5

6

4

–

1.00–1.75 (0.04–0.07)

1.00–2.00 (0.04–0.08)

2.00–2.75 (0.08–0.11)

12

2

–

1.50–2.25

1.50–2.50

2.50–3.25

–

(0.06–0.09) –

(0.06–0.10) –

(0.10–0.13) 2.75–3.50

25

1

(0.11–0.14) Notes: 1. The flank case depths shown are ranges that are practical for tooth size and frequency. The tolerance on case depth should be about ± 15%. (For 1-mm case, use 0.85–1.15 mm.) 2. The case in the root should be about 70% of the case on the flank. 3. The case depths shown above are in millimeters, with equivalent depth in inches shown in parentheses.

10.2.3.2 Load-Carrying Capacity of Induction-Hardened Gear Teeth The induction-hardened tooth may have about the same load-carrying capacity as a carburized tooth. If the surface hardness is up to about 58 HRC and the pattern of the case is good, a high load-carrying capacity may be achieved by getting an appropriate metallurgical structure and an appropriate residual stress pattern. The problem, though, is that the fast heating and cooling times involved tend to cause considerable variation in both metallurgical structure and the residual stress pattern. The hardness and depth of the case may seem to be right for the job, but the gears may fail prematurely in service because their structure and/or residual stress pattern did not come out right. (“Right” means able to carry the load—not necessarily right from a theory of metals standpoint.) Pitting is another problem. An induction-hardened tooth may pit, and then a tooth fracture may start at a pit. In many cases, induction-hardened gears have shown less capability to survive under moderately serious pitting than have case-carburized teeth. In view of the above, the builder of induction-hardened gears needs to develop each design to meet its job requirements. This development involves: • Metallurgical study of sample teeth to verify that the alloy chosen and the induction-hardening cycle will give satisfactory case depth, case pattern, case hardness, and metallurgical structure • Study of process variable to control uniformity of case from tooth to tooth, from end to end, and from top of tooth to bottom of tooth • Control study of the induction-hardening machine to maintain its power setting, maintain inductor positioning, avoid over- and under-temperature conditions due to electrical malfunction, etc. (Of

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particular concern is how to handle a machine stoppage when a gear is partly done without putting a weak spot in the circumference of the gear.) • Verification that the expected load-carrying capacity is achieved by appropriate full-load testing in the factory, and then follow-up observation of gears in service in the field in order to verify that the expected life and reliability is being achieved. (Of particular concern is field damage to the teeth by pitting, foreign material going through the mesh, overloads in transient operating conditions, etc.) Many experienced builders are producing induction-hardened gears that are very satisfactory for their application. There have been quite a few cases in which inexperienced builders have not understood the complexity of the induction-hardening process and have built gears that have failed prematurely under load ratings that seemed reasonable. With all this in mind, it seems most prudent not to try to establish an industry load rating for induction-hardened gears. Instead, a load rating should become established for a given gearset as the builder acquires enough experience with it to check the manufacturing “recipe” and to acquire adequate knowledge of how the gears stand up in actual service.

10.2.4 FLAME HARDENING

OF

STEEL

Flame hardening is similar to induction hardening in both results obtained and kinds of steel used. It differs from induction hardening in that the heat is applied to the surface by oxyacetylene flames in stead of being generated electrically in a layer extending from the surface to a small distance below the surface. In some cases, it is difficult—or impossible—to get the same hardness pattern and fatigue strength by flame as can be obtained by induction hardening. Special burners have been developed to impinge gas flames simultaneously on all tooth surfaces. Some types of flame-hardening apparatus use electronic control to turn off the gas at precisely the time when the temperature of the part just exceeds the critical temperature. The flame-hardening process can be used quite handily either to harden the whole tooth or to harden just the working part of the tooth. Where beam strength is critical, the designer can usually improve the tooth strength by hardening the root fillet as well as the working part of the tooth. If wear is the only consideration, hardening the tooth to the form diameter will do the job and reduces the risk of distorting the gear blank. Figure 10.18 shows some examples of flame-hardened teeth.

FIGURE 10.18 Sections through flame-hardened gear teeth. (a) 1.3-module (20-pitch) teeth, 5 ×. (b) 4-module (6pitch) teeth, 5 ×.

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10.2.5 COMBINED HEAT TREATMENTS A number of combination heat treatments have been developed. The skilled gear designer can often make use of such heat treatments to get results that are impossible with one procedure alone. For instance, a double-cycle heat treatment of carburized gears produces a strong, long-wearing case. However, both distortion and time consumed in heat treating may be more than is desired. In some cases, it is practical to carburize, quench from the carburizing heat, and draw the gear to medium hardness. Then the teeth can be finished by shaving or light grinding and, as a last step, induction-hardened. This procedure saves furnace time, since induction heating is very rapid. Also, the distortion is appreciably reduced because the whole gear blank does not undergo a high heat and quench after the teeth are finished. Induction hardening can be done using a furnace preheat and then a period of both high-frequency and low-frequency heating. The carburizing and nitriding treatments may be combined by using molten salts (or gases) that liberate both available carbon and nitrogen. The heating temperature does not need to be as high as that required for carburizing, and the case formed requires less time than nitriding alone would require. The case is harder, generally, than that obtained by carburizing alone but softer than nitrided cases on steels containing aluminum. In general, this type of treatment does not achieve quite so high a fatigue strength and quite so much pitting resistance as can be obtained with straight carburizing. In automotive gear work, where the cycles at maximum load are not too great, many types of gears have been produced with a cyanide hardening process, and they appear to perform just as well as carburized gears. In aircraft work, where the cycles at maximum load are usually quite large, very little use has been made so far of gears heat-treated by cyaniding or carbo-nitriding. The gear designer who wants to know what the best hardening process for a particular gear is should plane on experimenting with the processes just described. Things like pitch, blank design, steel used, and application requirements all enter into why one heat treatment may work better than others in a parti­ cular job. Hot quenching of gears and other parts may be used to minimize distortions. One such treatment is called austempering. It consists of quenching and holding the austenitized gear in a salt bath heated to some temperature below A c14 but below the Ms temperature. A range of hardnesses between 400 and 650 HV (400 and 575 HB) can be developed, depending upon the temperature that is used. An extended time in the quench is required, particularly with the more highly alloyed steels, to permit full transformation. The hardness results are similar to those obtained by a quench and medium-temperature draw, but the structures developed are quite different and generally have better toughness at a given hardness. Another hot-quench type of treatment is martempering. The part is quenched in a bath, which is either just above or just below the Ms temperature for a time that is just long enough for all sections of the part to cool down to the bath temperature. Then the part is removed and air-cooled to room temperature. This treatment produces full hardness. After martempering, the part should be given a conventional tempering treatment to the desired hardness. Even if no reduction in hardness is desired, tempering is needed to remove stresses and improve the structure. Austempered pieces, however, do not require further tempering. The two treatments just described are successful only on parts with small sections or with parts of large sections made of highly alloyed steels. Both are relatively slow quenching processes.

10.2.6 METALLURGICAL QUALITY

OF

STEEL GEARS

For reliable performance at the design load rating, the gears in a gear unit must have two kinds of quality that are suitable for the application: 4

A c1 is the critical temperature of the case.

Gear Materials

625

10.2.6.1 Geometric Accuracy This involves profile, spacing, helix, runout, finish, balance, etc. 10.2.6.2 Material Quality This involves hardness, hardness pattern, grain structure, inclusions, surface defects, residual stress pattern, internal seams or voids, etc. The possible gradations in gear-material quality have been thought of in these terms: Grade 0 Grade 1 Grade 2 Grade 3

Ordinary quality. No gross defects, but no close control of quality items Good quality. Modest level of control on the most important quality items. (The normal practice of experienced gear and metals people doing good work.) Premium quality. Close control on essential all critical items. (Some extra material expense to achieve better load-carrying capacity and/or improved reliability.) Super quality. Essentially absolute control of all critical items. (Much extra expense to achieve the ultimate. Has been used in space vehicle work. The need for this level of quality will be rather rare.)

10.2.6.3 Quality Items for Carburized Steel Gears The principal items to control for material quality of carburized gears are: • • • • • • • • •

Surface hardness Core hardness Case structure Core structure Steel cleanness Surface condition, flanks Surface condition, root fillet Grain size Non-uniformity in hardness or structure

The appropriate controls on these items will not be the same for aerospace gears and vehicle gears. For instance, some retained austenite is generally permissible (or even desired) in vehicle gears. In long-life aerospace or turbine gears, almost no retained austenite is permitted. Cleanness is another item. Certain kinds of dirt are tolerable in vehicle gears when the very high stress is applied for less than 107 cycles. Aerospace gears with a heavy design load for over 109 cycles are often made with vacuum-arc remelt steel (VAR) to remove essentially all the dirt particles in the steel. Table 10.13 shows some typical data for long-life, high-speed gears illustrating the possible differ­ ence between Grade 1 and Grade 2 gears. Figure 10.19 shows some metallurgical illustrations of Grade 1 and Grade 2 permissible structures. As it was said earlier, metallurgical quality grades are just being established in the gear industry. Table 10.13 and Figure 10.19 (and Table 10.14 and Figure 10.20) should be considered as examples of what is involved. Much study and work are needed to fully establish material grades in several important fields of gear work. 10.2.6.4 Quality Items for Nitrided Gears The same list of items for carburized gears applies to nitrided gears. The nitrided gear tends to have a “white layer” and may have internal cracks as a result of improper nitriding. These things make the actual control of nitrided-gear quality somewhat different from that of carburized gears. Table 10.14 and Figure 10.20 show concepts of quality grade considerations for nitrided gears.

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TABLE 10.13 General Comparison of Carburized Steel Gear Metallurgical Quality. (For Turbine Gears but Not for Vehicle Gears) Quality Item

Grade 1

Grade 2

Metallurgy of case

Tempered martensite. Retained austenite 20% maximum. Minor carbide network permitted. Some transformation products permitted. Acceptable and unacceptable conditions defined by 1000 × microphotos.

Tempered martensite. Retained austenite 10% maximum. No carbide networks. Almost no transformation products such as bainite, pearlite, proeutectoid ferrite, or cementite permitted. Acceptable and unacceptable conditions defined by 1000 × microphotos.

Metallurgy of core (at tooth root diameter)

Low-carbon martensite. Some transformation products permitted. Acceptable and unacceptable conditions defined by 250 × microphotos.

Low-carbon martensite. Almost no transformation products permitted. Acceptable and unacceptable conditions defined by 250 × microphotos.

Material cleanness

Air-melt (aircraft quality). AMS 2301 E

Magnetic indications of cracks or flaws on gear teeth

No indications parallel to axis. Not more than 4 surface indications, nonparallel to axis, per tooth, maximum length 4 mm. Not more than 6 subsurface indications, nonparallel to axis, maximum length 5 mm.

Vacuum-arc remelt (VAR) or vacuum slag process. AMS 2300 No indications parallel to axis. Not more than 2 surface indications, nonparallel to axis, per tooth, maximum length 2 mm. Not more than 3 subsurface indications, nonparallel to axis, maximum length 3 mm.

Forging grain flow

Forgings required for parts over 200 mm (8 in.) diameter. Close grain definition on forging drawing.

Forgings required for part over 125 mm (5 in.) in diameter. Very close grain flow requirements on forging drawing.

Surface defects after carburizing

Surface oxidation not to exceed 0.012 mm (0.0005 in.). Decarburization effects not to exceed 60 HKN (3 HRC). Minimum hardness of case at surface

Surface oxidation not to exceed 0.0008 mm (0.0003 in.). Decarburization effects not to exceed 40 HKN (2 HRC).

Flank (A)

685 HKN (57.8 HRC)

725 HKN (59.6 HRC)

Root (B) Minimum hardness of core at rood diameter (D)

660 HKN (56.5 HRC) 30 HRC

695 HKN (58 HRC) 34 HRC

Notes: 1. This table shows a partial amount of the magnetic particle specifications that are needed. 2. Details of acceptable and unacceptable decarburization can be shown by examples of plots of case hardness versus case depth. 3. Maximum case hardness is generally 800 HKN (63 HRC), but it may be specified lower to ensure that the draw treatment after final hardening lowers the hardness slightly and increases the impact strength. 4. The maximum core hardness is generally HRC 41, but it may be made slightly lower to better control the residual stress pattern.

10.2.6.5 Procedure to Get Grade 2 Quality The premium Grade 2 quality requires extra effort and expense. The principal steps that are generally needed are: 1. Choose a steel with enough alloy in it to respond well in the heat-treat procedures planned. 2. Check incoming raw material for cleanness. (A certification from the supplier is usually not en­ ough. Take samples and run laboratory tests.) 3. Prove out the heat-treat cycle planned by running one or more test gears. Set time, temperature, location in the furnace, etc., so that sure results can be obtained on the gears.

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FIGURE 10.19 Metallurgical examples, carburized gears. (a) Case structure, acceptable for Grade 2 (and Grade 1). (b) Core structure, acceptable for Grade 2. (c) Core structure, not acceptable for Grade 1. (d) Carbide network at tooth tip. Acceptable for Grade 1 but not for Grade 2. (e) Inter-granular oxide layer, acceptable for Grade 1 but not for Grade 2. (f) Subsurface crack. Not acceptable for any grade.

4. Use a portion of a toothed gear (or a whole gear) as a heat-treat sample, when a batch of gears is done. 5. Do laboratory work on the furnace samples after carburizing or nitriding. 6. Inspect finished gears for cracks and possible improper surface condition by nondestructive methods.

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TABLE 10.14 General Comparison of Nitrided Steel Gear Metallurgical Quality. (For Small Gears Heavily Loaded or Medium-Large Gears with Low Enough Loading to Permit Nitriding) Quality Item

Grade 1

Grade 2

Metallurgy of case

White layer permitted up to 0.02 mm (0.0008 in.) maximum. Complete grain boundary network acceptable. No micro-cracks, soft spots, or gross nitrogen penetration areas permitted. Acceptable and unacceptable conditions defined in 500 × micro-photos.

No white layer permitted on working surfaces of the tooth, other surfaces 0.02 mm (0.0008 in.) maximum. No micro-cracks, soft spots, or heavy nitrogen penetration along grain boundaries permitted. Continuous iron nitride network not permitted. Acceptable and unacceptable conditions defined in 500 × micro-photos.

Metallurgy of core (at tooth root diameter)

Medium-carbon tempered martensite. Some transformation products permitted. Acceptable and unacceptable conditions defined in 250 × micro-photos.

Medium-carbon tempered martensite. Almost no transformation products (bainite and pearlite) or ferrite permitted. Acceptable and unacceptable conditions defined in 250 × micro-photos.

Material cleanness Air-melt (turbine quality). Minimum hardness of case at surface (A, B):

Air-melt (aircraft quality).AMS 2301 E

Nitralloy 135, 135 M, and N

730 HKN (60 HRC)

740 HKN (60.5 HRC)

31 Cr Mo V 9 AISI 4140, 4340

700 HKN (58.5 HRC) 520 HKN (48.6 HRC)

725 HKN (59.6 HRC) 545 HKN (50.2 HRC)

Minimum hardness of core at root (D): Nitralloy 135, 135 M Nitralloy N

31 HRC

34 HRC

37 HRC

40 HRC

31 Cr Mo V 9

28 HRC

32 HRC

AISI 4140, 4340

30 HRC

34 HRC

Notes: 1. This table shows a partial amount of the magnetic particle specifications that are needed. 2. Details of acceptable and unacceptable decarburization can be shown by examples of plots of case hardness versus case depth. 3. Maximum case hardness is generally 800 HKN (63 HRC), but it may be specified lower to ensure that the draw treatment after final hardening lowers the hardness slightly and increases the impact strength. 4. The maximum core hardness is generally HRC 41, but it may be made slightly lower to better control the residual stress pattern.

10.3 CAST IRONS FOR GEARS Cast iron has long been used as a gear material. It has special merit, and some disadvantages, derived from its inherent structure and resulting properties. The gray cast irons, especially the alloyed types commonly used for gears, range in strength up to that of low-hardness steel. A cast iron with spheroidal graphite extends well into the strength range of heat-treated steels. Cast irons differ from steel in both composition and structure. Their carbon content usually ranges from 2.5 to 4%, whereas gear steels contain substantially less than 1% carbon. The resulting structural differences, however, are most important. In cast irons as a class, the carbon is predominantly in the free, or graphitic, state, with only a small proportion in the combined, or pearlitic, form. Steels normally contain only the combined form of carbon. It is the amount, form, and size of the graphite in cast iron that are responsible for is characteristic com­ bination of properties. From the standpoint of mechanical properties, graphite, being weak in itself, reduces ductility, strength, elasticity, and impact resistance; but, on the other hand, it increases the ability of cast iron to damp out vibrations and noise. The graphite also helps cast-iron gear teeth to operate with scanty lubrication.

Gear Materials

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FIGURE 10.20 Metallurgical examples, nitrided gears. (a) Case structure, thin white layer. Acceptable for Grade 2 by honing off white layer. (b) Case structure, thicker white layer but acceptable for Grade 1 without removal. (c) Continuous grain boundary nitrides. Not acceptable for Grade 1. (d) 4340 core structure, acceptable for Grade 2 (or Grade 1). (e) Nitralloy core structure, acceptable for Grade 1 but not for Grade 2. (f) 4340 core structure, not acceptable for Grade 1 or 2.

10.3.1 GRAY CAST IRON The gray cast irons, either plain or alloyed, are characterized by graphite in the flake form. These flakes act as a source of stress concentration, breaking up the continuity of the steel-like matrix. Figure 10.21 shows the characteristic structure of gray cast iron in comparison with other cast irons.

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FIGURE 10.21 Comparison of structures (all un-etched, 250 ×). Dark particles are graphite. (a) Gray iron. (b) Malleable iron. (c) Ductile iron.

As cast iron solidifies and cools in the mold, metallurgical changes occur that are similar in many respects to those discussed in the heat treatment of steel. In fact, the mold cooling conditions should be considered a heat treatment—the only one that many gray irons ever receive. The size and distribution of flake graphite are controlled by the relationship between composition and section size, and by several phases of foundry practice. The matrix behaves like steel, and therefore the composition defines the extent of hardening under given cooling conditions. Light sections that cool rapidly are inclined to freeze “white,” with the carbon in the form of iron carbide (cementite). Higher carbon and silicon inoculation, and nickel and copper additions, reduce the chilling power and produce graphitic and machinable

Gear Materials

631

structures in light sections. High carbon and silicon alone, however, are likely to produce large flakes and weak structures in heavy sections, and so castings with heavy sections are usually made from controlled and inoculated nickel irons. A pearlitic matrix is highly desirable for a high-strength, wear-resistant iron, and the usual alloying elements—nickel, molybdenum, and copper—are used to increase the ability of gray-iron matrix to harden to this structure either in the mold or by subsequent heat treatment. Gray iron has a low modulus of elasticity, varying with the strength level, that ranges from one-third to two-thirds that of steel. For gearing this is an advantage, since it tends to reduce the compressive stress (Hertz stress) developed by a given tooth load. It also tends to reduce slightly the beam strength, because the teeth bend more and there is more sharing of load. Gray iron is so brittle, though, that it should not be considered in gears subject to serve shock.

10.3.2 DUCTILE

IRON

For a long while, malleable iron was the only type of cast iron available that would provide a useful degree of toughness or ductility. In malleable iron, the free graphite forms from white iron during annealing in clusters of tiny flakes. Malleable iron, as compared with gray iron, has ductility and better yield and elastic properties. Malleable iron has not been used widely for gears, however, because of its poor wear resistance and limitation to parts of relative light section size. The iron with true spheroidal graphite is variously referred to as ductile iron, spheroidal-graphite iron, or nodular iron. Figure 10.21, which shows flake graphite in gray iron and spheroidal graphite in ductile iron, clearly demonstrates that the term nodular iron is not a completely accurate term to describe the new material. The spheroidal form of graphite, furthermore, provides a material with a different combination of properties than is possible with malleable iron. Ductile iron may be made by treating a low-phosphorus gray-iron base composition with a magnesium or equivalent additive. It has tensile strength ranging from 60,000 to 180,000 psi, yield strength from 45,000 to about 150,000, and elongation up to 25%. The modulus of elasticity is about 24 × 106 psi. The standard grades now listed are shown in Table 10.15. Ductile iron can be made with a high carbon content and still have low carbide content. It is, therefore, not surprising that both its machinability and its wear resistance are excellent. The combination of (a) strength, (b) toughness, (c) wear and fatigue resistance, and (d) susceptibility to heat treatment offered by ductile iron, has been of considerable interest to gear designers. This material has been used to advantage in applications formerly filled by carbon and alloy through-hardened steels, flame-hardened steel, gray cast iron, and gear bronze.

TABLE 10.15 Typical Properties of the Different Grades of Ductile Iron Commercial Designation

60-40-18 65-45-12

Recommended Heat Treatment

Annealed As cast or annealed

Brinell Hardness Range 140–180 150–200

Tensile Strength, psi, min. 60,000 65,000

Yield Strength, psi, min.

40,000 45,000

Elongation in 2 in., %, min.

18.0 12.0

80-55-06

Quenched and tempered

180–250

80,000

55,000

6.0

100-70-03 120-90-02

Quenched and tempered Quenched and tempered

230–285 270–330

100,000 120,000

70,000 90,000

3.0 2.0

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Dudley’s Handbook of Practical Gear Design and Manufacture

10.4 NONFERROUS GEAR METALS The metals copper, zinc, tin, aluminum, and manganese are use in various combinations as gear ma­ terials. The most important, perhaps, is the alloy called bronze. The bronzes have become a widely used family of materials for gears, largely because of their ability to withstand heavy sliding loads. Like cast iron, the bronzes are easy to cast into complex shapes, while certain types are available in wrought forms. Because bronze casting as a rule will be more expensive than cast iron, the designer will select the material primarily for its ability to meet special service requirements. Most gear designers are of the opinion that bronze will withstand high-speed rubbing—such as in a worm gearset—better than cast iron. When the best tin bronzes are compared with the best gray cast irons, the bronzes indeed seem to be superior as a “bearing” material, although individual tests have been recorded in which certain bronzes were found inferior to some cast irons. Even in low-speed applications, bronze may be the most desirable material. Although the tables of hobbing machines do not turn very fast, experience indicates that higher table speeds can be used without danger of scoring when the hobbers have bronze index wheels than when they have cast-iron wheels. A number of special alloys are used for die-casting gears. In this type of work, a low-melting-point material with good casting characteristics is essential. Zinc-base alloys such as SAE alloy 903 are very popular. This alloy contains about 4% aluminum and about 0.04% magnesium, with the remainder zinc. It melts at about 700 F and has a tensile strength of around 40,000 psi. Aluminum-base die castings have about the same tensile strength, but they weigh only about 40% as much as the zinc castings. Their melting point is around 1100 F . A good composition for gear work is the aluminum alloy with 10% silicon and 0.5% magnesium.

10.4.1 KINDS

OF

BRONZE

Like the terms “steel” and “cast iron,” “bronze” is really the name of a family of materials that may vary over a wide range of composition and properties. A “bronze” family is an alloy of copper and tin, as compared with “brass,” which is an alloy of copper and zinc. In practice, however, certain bronzes contain both tin and zinc, and the aluminum and manganese bronzes, for example, may contain very little or no tin. It might have been less confusing if these had been called brasses. The basic bronze is an alloy of 90% copper and 10% tin that exhibits the desired two-phase structure. The best strength and bearing properties are developed if the hard and soft constituents are finely and intimately dispersed. This is usually accomplished by carefully chilling the casting. See Figure 10.22, a for the “delta” constituent. Zinc is added to copper-tin bronze to increase strength, but at some sacrifice in bearing properties. Nickel also increases strength and has a further beneficial effect on hardness and uniformity of the structure. Lead added to bronze does not combine with the base metal but remains distributed throughout the casting to act as a solid lubricant, much as graphite does in cast iron. See Figure 10.22b for lead particles in bronze. Lead is weak; therefore, it softens and weakens the bronze but improves machinability and allows quicker wearing in with a mating part. If bronze is not melted with the proper techniques, the tin may oxidize, and very hard crystals of tin may be formed. These may be as destructive to the bearing surface as emery added to the lubricant! Proper de-oxidation will prevent this danger. The tin bronzes deoxidized with phosphor are usually the best for gear applications. The term phosphor bronze means literally a bronze deoxidized with phosphor. Aluminum bronzes are more complex than the copper-tin grades and may not have such good bearing properties. These alloys have a high strength as cast, however, and the strength can be increased by heat treatment. Aluminum bronzes have been used in many gear applications where sliding velocity is not high. Manganese bronzes have high zinc content along with smaller amounts of manganese, iron, and aluminum. Their strength far exceeds that of the tin bronzes. Super-manganese bronze has a tensile strength above 100,000 psi. With steel worms of 300 or more Brinell hardness, it makes about the

Gear Materials

633

FIGURE 10.22 Two typical tin-base worm-gear bronzes (both 250 ×). (a) Etched phosphor bronze. Light islands are “delta” constituent, the hard particles in the structure. (b) Un-etched lead bronze.

strongest worm-gear combination that can be obtained for slow-speed applications, outside of using steel on steel. Manganese bronzes are also available as forgings or as cold-drawn, or extruded bars. The wroughtmanganese bronze is lower in alloy content and strength than the super-manganese bronze, but has surprisingly good bearing properties. Some worm-gear applications with rubbing speeds of up to 50 m/s (10,000 fpm) have been handled with wrought-manganese bronze. It is also suitable for many low-speed, heavy-load applications. Silicon-alloy bronzes are available in both, cast and wrought forms. Silicon increases strength only moderately in the cast grades, while the wrought forms can be further strengthened and hardened by cold drawing. There has been only limited use of silicon bronze for gears.

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Table 10.16 shows some of the nominal compositions, tensile properties, and uses of the several kinds of bronze. It should be understood that there are a host of other bronzes, which differ somewhat in composition from the typical kinds shown in the table. The strength and hardness values shown are average values. There is a considerable spread in properties for each composition, depending upon the size of the part, the rate of cooling of the casting, and the quality of the bronze.

10.4.2 STANDARD GEAR BRONZES At the present time, three kinds of copper-tin bronzes are specified as standard materials for use in wormgear applications. Besides copper-tin bronzes, aluminum bronzes and manganese bronzes are shown. The compositions are not given, but expected physical properties are. Many organizations have set up standards on bronze. The situation is somewhat confused, because there are no standards that are accepted by all bronze users. Frequently contracts will specify that bronze parts are to be made of some kind of bronze that may not be appropriate for the kind of gearing that is required. In such cases, the gear designer should obtain approval to substitute an appropriate gear bronze.

10.5 NONMETALLIC GEARS In early times, gears were often made of wood. At present, few wooden gears are made. A large quantity of “nonmetallic” gears is used, however.

TABLE 10.16 Types of Gear Bronzes and Typical Examples of Each Type Type

Composition, %

Strength*, psi

HB (500 kg)

Uses

Bronze (zinc-deoxidized)

Cu 88, Sn 10, Zn 2

ult. 46,000y.s. 19,000e.l. 12,000

65

Spur, bevel, worm gears

Phosphor bronze (chill-cast) Nickel-phosphor bronze (chill-cast)

Cu 88, Sn 10, Pb 0.25

ult. 50,000y.s. 22,000

85

Cu 88, Sn 10.5, Ni 1.5, Ph 0.2

ult. 55,000y.s. 28,000

90

Medium-speed wormgears Medium-speed wormgears

Lead phosphor bronze (sand-cast)

Cu 87.5, Sn 11, Pb 1.5

ult. 50,000y.s. 22,000

75

High-speed worm-gears

Aluminum bronze (sand-cast) Aluminum bronze

Cu 89, Al 10, Fe 1

120

Same as above but heat-treated

ult. 65,000y.s. 27,000e.l. 20,000 ult. 95,000y.s. 60,000e.l. 50,000

Spur, bevel, low-speed worm gears Heavy-duty low-speed gears

Super-manganese bronze (sand-cast)

Cu 64, Zn 23, Fe 2.75, Mn 3.75, Al 6.75

ult. 110,000y.s. 70,000e.l. 52,000

236

Heavy-duty low-speed gears

Manganese bronze (forged)

Cu 58, Zn 37.5, Al 1.5, Mn 2.5, Si 0.5

ult. 75,000y.s. 40,000

135

Moderate-load gears. Small high-speed gears

Silicon bronze (sand-cast)

Cu 95, Si 4, Mn 1

ult. 45,000y.s. 20,000

80

Low-speed gears or moderate load

Notes * ult. – ultimate strength y.s. – yield strengths e.l. – elastic limit.

210

Gear Materials

635

The cotton-phenolic type of gear belongs to a general class of material called thermosetting laminates. In addition to the laminates, several other plastic materials are used for gears. Bakelite and similar plastics are used to injection-mold complete toothed gears. These materials do not have the strength of the laminates, but they are inexpensive and easy to manufacture, and so suitable when large quantities of small, light-duty gears are needed. Later, several remarkable new plastics have been appeared on the market.

10.5.1 THERMOSETTING LAMINATES A wide variety of sheet materials, such as paper, asbestos, cotton fabric or mat, wood veneer, nylon fabric, and glass fabric, may be used in making laminates. The binders may be phenolic resins, melamine resin, or silicones. The laminates used for gears are generally of cotton fabrics, although paper is used occasionally. Laminates are made by coating the sheet material with the liquid binder. After drying, the sheets are cut, stacked between metal plates, and bonded to form a board under a pressure of 1000 to 2500 psi at a temperature between 270 and 350 F . Such a board will show directional properties, because the com­ ponents differ in strength from one direction to another. The teeth of gears cut from these pre-laminated boards will differ in strength, depending upon the position of the teeth with respect to the grain of the board. When the highest strength is needed, it is possible to prepare special gear blanks with the fibers for the laminate running in all possible directions. Then all teeth will have equal strength. The laminates are sold under a variety of trademarks. Compositions and properties vary considerably. Phenolic-laminated material has several characteristics that make it attractive for gearing. Being nonmetallic, it runs quietly even when meshed with a steel pinion. Nonmetallic gears show very little tendency to vibrate or respond to vibrations. Nonmetallic gears weigh only about one-fifth to one-sixth as much as steel gears of the same size. Even though their strength is only from one-tenth to one thirteenth that of steel, it is possible, in some applications, to have nonmetallic gears carry about the same load as a cast-iron gear or low-hardness steel gear of the same size. If both the steel gear and the nonmetallic gear are made to commercial accuracy limits, the effect of tooth errors on the steel gearset is much greater because the teeth deform less under load, and so small errors create severe dynamic overloads in the steel gear teeth. The non­ metallic gear deforms about thirty times more than steel. This ensures good sharing of load between teeth and a low Hertz stress as a result of the wide contact band. Laminated nonmetallic gears have been run together in some applications with only water used as a lubricant. A steel pinion and a laminated gear will often run with less lubrication than a pair of steel gears. In designing nonmetallic gears, the relatively large tooth deflection creates a problem. The driven gear—because of deflection—has a tendency to gouge the driver at the first point of contact. If the driver is a steel or cast-iron gear, this problem is not serious because the nonmetallic driven gear does not have enough hardness to gouge the hard metal. If the driven gear is steel, the gouging tendency can be controlled fairly well by making the driver with a very long addendum and the driven gear with little or no addendum. For highest load capacity, nonmetallic gears are mated with steel or cast-iron gears. The best wear resistance is obtained when the metal gear of the set has a hardness of 300 Brinell (or more). In selecting a phenolic laminate, the pitch should be considered. When the teeth are 1.5 module (16 pitch) or coarser, a 15 oz/yd2 tightly woven duck is about right. Teeth of 1 module (24 pitch) require a fabric base of about 6.5-oz duck. Finer-pitch gears use about 3-oz fine cambric. Laminated nonmetallic gears have been used in a wide variety of applications. A few typical ap­ plications are air compressors, automotive timing gears, shoemaking machinery, electric clocks, and household appliances, bottling machinery, and calculating machinery. When mated with a good steel gear, sliding velocities up to about 15 m/s (3000 fpm) can be handled. Figure 10.23 shows a gear after testing at 40 m/s (8000 fpm).

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FIGURE 10.23 Phenolic laminated gear with a steel hub, after testing at a pitch-line velocity of 40 m/s (8000 fpm). (Courtesy of General Electric Co., Lynn, MA, U.S.A.)

10.5.2 NYLON GEARS Nylon has surprisingly high strength for a molded plastic. It has good bearing properties and con­ siderable toughness. Nylon may be injection-molded to form a complete toothed gear, or it may be molded in the form of rods or sheets from which gears may be machined. The best accuracy is obtained when the teeth are cut. The lowest cost, of course, is obtained when the complete part is molded. A commonly used nylon for gears is the du Pont material FM-10001. Some of the properties of this material are shown in Table 10.17.

TABLE 10.17 Some Properties of FM-10001 Nylon Tensile Strength:

70 F At 77 F

At

15,700

At 170 F

10,000 7600

Modulus of elasticity, at 77 F

400,000

Rockwell hardness (R) Specific gravity

118 1.14

Mold shrinkage, in. per in.

0.015

Gear Materials

637

FIGURE 10.24 Solid nylon gear after testing at a pitch-line velocity of 40 m/s (8000 fpm) and overload rating. (Courtesy of General Electric Co., Lynn, MA, U.S.A.)

The values in Table 10.17 indicate that nylon has almost as much strength as the best laminates. Its weight is slightly less than that of the laminates. Nylon will deform under load about 75 times as much as steel! One of the attractive features of nylon gears is their ability to operate with marginal lubrication. Small nylon gears have been able to run at high speed under light load with no lubricant at all. This is important in some applications, e.g., where a lubricant would soil yarn or film handled by a machine. Nylon gears have worked well in many gear applications, such as in movie cameras, textile machines, food mixers, and timing devices. In some cases, it has been reported that nylon did not stand up well. It is proven by tests that nylon gears are capable of running at 40 m/s (8000 fpm) pitch-line velocity and tooth loads up to those carried by low-hardness steel gears. This indicates that nylon—when properly processed into a well-designed gear—is a surprisingly good gear material (see Figure 10.24).

REFERENCES Davis, J. R. (Ed.) (2005). Gear materials, properties, and manufacture. ASM Press, 339p. ISBN 978-0-87170-815-1. Linke, H., Börner, J., & Heß, R. (September 5, 2016). Cylindrical gears: Calculation – Materials – Manufacturing. Hanser, Har/Psc Edition, 848p.

11

Load Carrying Capacities, Strength Numbers, and Main Influence Parameters for Different Gear Materials and Heat Treatment Processes Niklas Blech, Holger Cermak, André Sitzmann, Adrian Sorg, Daniel Fuchs, Thomas Tobie, and Karsten Stahl

11.1 INTRODUCTION1 Components of the power train are usually subjected to various stresses. Increasing power densities of modern gear transmissions require a high safety and efficiency of the power transmitting parts. Especially the design and the material, with its microstructural properties and parameters, affect the performance characteristics of gears. Those properties can be specifically influenced by the choice of the material and the heat treatment. An essential requirement for choosing a suitable material and heat treatment is the understanding of the considered stress condition. Hence, in the following, main prin­ ciples of component stresses for gears are described and corresponding criteria for the selection of material and heat treatment are derived. Gears represent a complex machine element in terms of geometry and stresses. For the selection of material and heat treatment, numerous aspects must be taken into account, see Figure 11.1. A large number of different materials are available for use of gears in gearboxes. For applications with low power transmission requirements, plastics, non-ferrous metals, sintered and cast-iron materials, or structural steels are often used. Gears made of these materials are usually manufactured in lower gear quality and in higher quantities and thus at lower unit costs. However, for gears with high and the highest performance requirements, alloyed steels suitable for heat treatment are mainly used. For example, by means of a case-hardening of corresponding steels, high load carrying capacities are achievable. Figure 11.2 shows the influence of the material and the heat treatment process on the size (center distance “a” in mm), overall weight, and costs of a gearbox based on an exemplary single-stage industrial gearbox. Due to the use of case-hardened gears, instead of e.g., quenched and tempered gears, the gearbox can be built smaller, lighter, and relatively more cheaply, while still meeting the requirements. In the same way, the load carrying capacity of the gearing is influenced with regard to the various types of damage and the operating parameters such as power loss, oil temperature, or required lubricant. In this chapter, the focus is on gears with high performance requirements and thus on steels suitable for heat treatment. This chapter gives a brief overview of load carrying capacities, strength numbers and main influence parameters for different gear materials and heat treatment processes of selected current research projects

1

Parts of the subchapter are mainly based on [17, 32, 35, 39, 43–46, 63, 75, 79, 88, 97].

639

640

FIGURE 11.1

Dudley’s Handbook of Practical Gear Design and Manufacture

Influences on the choice of suitable (gear) material.

and does not claim to be exhaustive. For further information, please keep up to date on future research projects at FZG.

11.2 FUNDAMENTALS OF GEAR STRESSES AND THE DETERMINATION OF THE LOAD CARRYING CAPACITY Loaded gears show a complex stress state at the surface as well as below the surface in greater material depths. Therefore, a distinction must be made between the stress in the tooth root and on the active tooth flank of a gear tooth (see Figure 11.3). While the stress profile in the tooth root is mainly determined by the bending stresses from the acting tooth force, the tooth flank is subjected to rolling and sliding stresses, characterized by the Hertzian pressure, and the kinematics of the gearing. Taking into account the presence of a lubricant, the stress profile of the tooth flank can be char­ acterized by the gradient of the maximum shear stresses (e.g., main shear stress or alternating shear stress) according to the Hertzian equations and the theory of elastohydrodynamics (EHL) and differs significantly from the conditions in the tooth root. The endurance life of the tooth flank depends, among other aspects, on the lubrication conditions, whereby typical occurring types of damage are pitting, micropitting, flank fracture, wear, and scuffing. In the tooth root, the failure mode of tooth root breakage is decisive. The determination of the tooth root stress is generally based on the model of a one-sided clamped bending beam. The stress gradient results in consequence of the notch effect and the section modulus against bending in the area of the tooth root fillet. The decisive bending stress is additionally superimposed by compressive and shear stresses. The basis for the load capacity calculation of involute gears form the calculation methods of ISO 6336 [3], DIN 3990 [4], or AGMA 2001-D04 [5] as well as extended calculation approaches derived from extensive theoretical and experimental investigations. The basic ideas of the general strength calculation also apply specifically to the component “gear.” The material must have adequate strength to ensure the required endurance life or to prevent a failure. For this purpose, for each gear failure mode, the relevant stress and the material strength has to be compared. According to ISO 6336 [3], the load carrying capacity is determined on the basis of the nominal tangential force of the deviation-free gearing acting on a gear tooth. The proper operating conditions in the gearbox are taken into account by a number of additional factors. Characteristic material strength values for the tooth flank and tooth root, for different gear materials and heat treatment processes, are documented in ISO 6336-5 [6,7] together with re­ quirements for the given material quality groups (ML, MQ, and ME). The strength values are based on extensive experimental investigations on so-called standard reference test gears under standard test conditions, further information see ISO 6336 [3]. By means of different factors, a conversion from these

FIGURE 11.2 Influence of gear material on size, weight and costs of a single-stage gearbox (example) with identical torque to be transmitted acc. to [ 1].

Load Carrying Capacities 641

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Dudley’s Handbook of Practical Gear Design and Manufacture

FIGURE 11.3 General stress state in a gear tooth (F—normal tooth force) acc. to [ 2].

strength numbers of a standard reference test gear to the proper gear geometry and operating conditions of the particular application is possible.

11.3 OVERVIEW OF TYPICAL GEAR FAILURE MODES Different types of damage, depending on the gear geometry (e.g.,: pressure angle [8]; symmetric – asymmetric [9]; internal gear [10]; gear size [11,13,14]), material properties (e.g.,: plastic [15–18]), operating conditions (e.g.,: [19–24]) and lubricant characteristics (e.g.,: [25–27]), influence the gear load carrying capacity. In addition, each type of damage is influenced by different physical parameters in different ways and is subjected to different mechanisms. Therefore, a comprehensive understanding of the underlying mechanisms as well as the corresponding load and stress conditions is necessary for a suitable gear design. This enables the selection of a suitable material with optimum properties with regard to the required high load carrying capacities. Gear damages can basically be characterized as failures due to material fatigue or as failures without previous material fatigue, the latter being primarily caused by tribological effects in the lubricated contact zone. In addition, a differentiation can be made depending on the origin of the gear damage. On the one hand, this can be related to the position in the gear area—on the tooth flank or in the tooth root—and on the other hand to the damage origin in the material volume - on the surface or below the surface in a greater material depth. This can lead to different requirements on the material properties in different areas of the gear. Figure 11.4 summarizes the most significant types of gear damages to be considered in gear design. Pitting and tooth root fracture are the classic types of damage due to material fatigue. Both types of damage usually occur at or near the surface and are characterized by a crack that propagates further in the material. While the load carrying capacity regarding pitting is strongly influenced by the Hertzian flank

FIGURE 11.4

Main gear failure modes.

Load Carrying Capacities

643

pressure with additional consideration of EHL-theory in the tooth contact zone, the tooth root load carrying capacity depends on the bending stresses in the tooth root fillet. Differences between the flank contact pressure and bending stress and the resulting stress states can lead to different requirements of material properties in the relevant areas. In addition, micropitting can (negatively) influence the performance of a gear. Under unfavorable lubrication conditions, this kind of damage is observed directly on the surface of the loaded tooth flank. Micropitting is seen as fatigue damage, where crack propagation is limited to the boundary area close to the surface. Consequently, this type of damage is influenced by the lubricant, the lubrication condition and the material properties at and near the surface. The contact load on the tooth flank also induces stresses at greater material depths below the surface. If these stresses exceed the local strength of the material, further damage with crack initiation below the surface can occur. Such damage is referred to in the literature as TFF (tooth flank fracture) or as TIFF (tooth internal flank fracture) or fatigue below the surface [28,29]. Since the load-induced stresses in greater material depth rise with increasing gear size, the strength properties of the material in greater material depths become more important for gears of large size. In addition to fatigue failure modes, other types of damage to the tooth flank are known, but these are subjected to different mechanisms. Scuffing is a spontaneously occurring, short-term local welding of the tooth flanks of pinion and wheel due to an insufficiently separating lubricating film in the tooth mesh. The temperature in contact is the decisive parameter for determining the risk of scuffing. The oil tem­ perature, the load, the friction coefficient in contact and the speed conditions influence this temperature in the contact. A further typical non-fatigue gear damage is wear (slow-running wear). This type of damage can occur at very slow circumferential speeds and the associated low lubricant film thickness in the tooth contact. Wear is a continuous, linear material removal that occurs as soon as wear-critical conditions in tooth contact are reached or go below the wear limit. The gear geometry, the manufacturing route and operating conditions in the gearbox, but especially the material used and the corresponding heat treatment ultimately define which types of damage occur and have to be considered for the practical application. Figure 11.5 shows a comparison of the main load carrying capacity limits for gears made of quenched and tempered (left) and case-hardened steel (right).

FIGURE 11.5 Main load carrying capacity limits for gears made of quenched and tempered steel (left) and casehardened steel (right) acc. to [ 1].

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Dudley’s Handbook of Practical Gear Design and Manufacture

It can be seen that the load carrying capacity limits for case-hardened gears are generally significantly higher than those of quenched and tempered gears. In addition, the individual damage limits in relation to each other can be influenced in significantly different ways by material and heat treatment. The ratios given are to be understood as examples and can be modified by a number of parameters that partly influence each other.

11.4 REQUIREMENTS ON THE PROPERTIES OF GEAR STEELS The stress condition of an operating gear demands different requirements on the local material properties of the tooth flank and in the tooth root, both at and below the surface in greater material depth. The local strength properties are influenced by the complex interaction of different variables, such as chemical composition, steel production process, heat treatment or gear production route (see Figure 11.6). In general, highly loaded gears should have an adequate strength and thus hardness in the area close to the surface, up to a certain distance below the surface, which leads to a sufficiently high fatigue and wear durability. Ideally, the core area should be ductile to avoid brittle fracture under high impact loads (DIN EN 10084 [30]). Consequently, various alloy concepts as well as thermal and thermo­ chemical heat treatment processes have been developed to achieve this combination of properties. Alloy concepts for different gear sizes differ considerably in different markets due to historical cir­ cumstances (e.g., automotive engineering, mechanical engineering, military), experience and local availability of alloy elements. The essential requirements (see Figure 11.7) are explained below using case-hardening steels as an example, but can also be applied to other materials for heat treatment. Case-hardening steels have to meet the following essential requirements with regard to component properties and durability: • • • • • •

Chemical composition and hardenability Homogeneity (microscopic and macroscopic) Cleanliness Mechanical properties (tensile strength, fatigue strength and durability) Wear durability, rolling durability, bending strength High and uniform form stability (DIN 3990 [4])

The technical delivery conditions for case-hardening steels are specified in common standards, e.g., DIN EN 10084 [30] and ISO 683-3 [31]. In addition to the classification and labeling of steel grades, pro­ duction processes and property requirements (e.g., hardenability), testing and inspection procedures are

FIGURE 11.6

Influencing parameters on (local) material properties.

Load Carrying Capacities

FIGURE 11.7

645

Overview of common gear materials and common use of the respective heat treatment processes.

also specified. Besides these general standards, many manufacturers have published internal delivery specifications, which describe in detail special requirements (e.g., grain size, chemical composition etc.). This is justified through the numerous possible process routes for the production of gears. Depending on the requirements, different sequences of annealing, hardening, and machining are used. If, for ex­ ample, high form stability of the gear is required, pre-tempering is carried out before and stress-relief heat treatment after machining. To optimize the material, it is important to consider the entire process chain. For the design of gears, the steel grades are often selected according to the requirements of ISO 6336-5 [6].

11.5 STEELS FOR QUENCHING AND TEMPERING Tempering is a thermal process: the gear is usually hardened at temperatures above 550 °C, quenched and afterwards tempered. Common steels are 30CrNiMo8, 36NiCrMo16 or 42CrMo4. Quenching and tempering is usually used as intermediate process before a heat treatment or as last step of the heat treatment process, especially for large-size gears. Figure 11.8 exemplary shows the influences of the carbon content and of alloy elements on the mechanical properties of steels for quenching and tempering. The load carrying capacity of quenched and tempered gears is substantially lower in comparison to case-hardened, nitrided, or surface hardened gears. The benefits of quenched and tempered gears are the relatively cheap and easy production, a well controllable heat treatment process, good running-in properties and a high safety against tooth root breakage. Quenched and tempered gears are mainly used in larger applications (e.g., mills) as well as in combination with hardened gears (e.g., as ring gears in planetary gearboxes). When paired with a hardened partner a higher risk of wear needs to be considered.

11.6 STEELS FOR SURFACE HARDENING Surface Hardening is a thermal hardening process that limits the heat input and therefore the resulting microstructural transformations to the surface area. As a benefit, not only reduced energy to harden gears is needed, but also the distortion is usually reduced compared to hardening processes for which the complete gear has to be heated. Surface hardening is often the only usable hardening treatment when it comes to gears of larger size as technological boundaries of other hardening methods are reached, e.g., hardening oven size. The aim of surface hardening is to achieve a high surface hardness combined with an as smooth as possible hardness transition to the (tempered) core hardness. Common materials for surface hardening of gears are 51CrV4, 34CrNiMo6 or 42CrMo4, in special applications also cast steel

Elongation at break

FIGURE 11.8 tempering.

Tensilestrength, yieldstrength

hardened

annealed

Brinell hardness

Carbon content

Elongation at break

Yield strength

Tensilestrength

Brinell hardness

Influence of carbon content

Rockwell hardness C Distance from the quenched and tempered surface of the sample®

Influence of alloy composition

Influence of carbon content (left acc. [ 32]) and alloys (right, acc. DIN EN 10083 [ 33]) on the mechanical properties of steels for quenching and

Brinell hardness

646 Dudley’s Handbook of Practical Gear Design and Manufacture

Load Carrying Capacities

647

TABLE 11.1 Overview of Usual Surface Hardening Processes for Gears Depending on Quantities and Size Processes Spin hardening

Tooth-by-tooth hardening

Quantities

Diameter in mm Module in mm

Hardness

Flame hardening

Series, Batch Up to 200Up to 1500

≤ 6≤ 18

Induction hardening: single or dual frequency

Series

Up to 200

≤6

Tip/Flank: 50…56 HRCTooth root: 36… 56 HRC

Tip-by-tip hardening (flame)

Batch

No limit

>8

50…56 HRC

Tip-by-tip hardening (induction) Gap-by-gap hardening (flame)

Batch

-

6…40

-

-

-

10…40

47…52 HRC

-

-

5…40

-

Gap-by-gap hardening (induction)

or even cast iron. Usual processes are flame-, induction-, laser- or electron beam hardening, with flame and induction hardening being the most common ones, see Table 11.1. Induction hardening in comparison to the other mentioned surface hardening processes has the ad­ vantage, that the heat is generated inside the gear by eddy current. This helps achieving shorter heating times and greater surface hardening depths compared to other processes where the material needs to be heated from the surface and heating time and depth are limited to heat conduction within the gear material. The short heating time of induction hardening allows for better implementation in the process chain of gear manufacturing. Depending on the hardening process either one tooth at the time (tooth-bytooth-hardening) or all teeth simultaneously (spin hardening) are hardened. Another great benefit, especially for smaller gear sizes, that comes with modern induction hardening technologies such as dual frequency induction hardening, is the relatively large freedom in design of the hardening pattern of gears. Figure 11.9 shows the range of the achievable hardening contours from through hardened (contour on the left: contour 1) to contour hardened (contour on the right: contour 3) teeth. The result can be an adapted contour, with an optimized hardening pattern on the tooth flank and in the tooth root. However, this freedom inherits a great complexity. The hardening pattern not only determines the surface hardening depth in different parts of the tooth but also the resulting residual stresses (see Figures 11.10 and 11.11) and therefore the achievable load carrying capacity of the gear. In [36] different variants of induction hardened gears with different gear sizes (normal module 2 and 4 mm) were investigated. Table 11.2 shows selected variants. The tooth root bending strength was in­ vestigated for all variants listed in Table 11.2. The main focus of the investigations was on the contour hardened variants. Variant 0 represents the reference, which was through hardened on the tooth flank. The pitting load carrying capacity was investigated only for the variants A, D, and E. The determined allowable stress numbers of all variants are plotted in Figure 11.12 with the specification for the al­ lowable bending stress and contact stress numbers for induction hardened gears acc. to ISO 6336-5 [6]. With regard to the bending strength number σF lim, it can be seen that variant A ranges between the specified numbers for material quality MQ and ME. Variants 0 and C are located between the specified numbers for material qualities ML and MQ. The result of variant B, which is characterized by a very small surface hardening depth, show a determined allowable stress number for bending that is sig­ nificantly lower than the specified numbers for material quality ML. Variant D is located in the upper

FIGURE 11.9 3) [ 34,35].

contour 1

(b)

contour 2

(c)

contour 3

Range of achievable hardening contours of induction hardened gears from through hardened (left, contour 1) to contour hardened (right, contour

(a)

648 Dudley’s Handbook of Practical Gear Design and Manufacture

FIGURE 11.10

Surface residual stresses in MPa

-0,5

0,0

0,5

1,0

Distance from 30°-tangent in mm

1,5

V3 - Contour 3

V1 - Contour 1

2,0

Near surface residual stresses for hardening contour on the left and on the right in Figure 11.9 (gear module of 4 mm) ([ 34], acc. [ 35]).

-1000 -1,0

-800

-600

-400

-200

0

200

400

600

Residual stresses along the surface profile

Load Carrying Capacities 649

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Dudley’s Handbook of Practical Gear Design and Manufacture

FIGURE 11.11 Residual stress depth profile for different hardening contours in Figure 11.9 in the tooth root fillet for gears with a normal module of 4 mm ([ 34], acc. [ 35]).

TABLE 11.2 Investigated Variants (* Variants 0 and C Were through Hardened on the Tooth Flank) [36] Variant

0*

A

B

C*

D

E

Module

4 mm

4 mm

4 mm

4 mm

2 mm

2 mm

Material Hardening profile

42CrMo4 ill1

42CrMo4 ill2

42CrMo4 ill3

44MnSiVS6 ill4

42CrMo4 ill5

C56E ill6

FIGURE 11.12 Classification of the determined allowable bending stress (left) and contact stress (right) acc. to ISO 6336-5 [ 6] for induction hardened gears, additionally the specified strength numbers for case-hardened gears are presented acc. [ 36].

Load Carrying Capacities

(a)

(b)

Metallographic cross section of a compound layer (white) and case structure close to the surface

FIGURE 11.13

651

Metallographic cross section of a diffusion layer (dark) and core structure

Exemplary metallographic cross section of a nitrided gear [ 39].

range of the specification acc. to ISO 6336-5 [6] and shows the highest allowable stress number of all investigated variants, whereas the value of variant E is nearly identical with the allowable stress number of variant A. Furthermore, the specifications for the allowable stress number for case-hardened gears are additionally plotted. The contour hardened variants A and E are located in the area between material qualities ML and MQ and take a typical value for unpenned case-hardened gears. The through hardened variants 0 and C reached a bending strength number in the range of the material quality ML of casehardened gears while variant D reached a strength number that complies with the specifications of material quality MQ of case-hardened gears. All in all, the allowable bending strength numbers of the variants with the small normal module of 2 mm (variant D and E) are equal or even higher compared to variants 0, A, B, and C (normal module 4 mm). For selected variants, the flank load carrying capacity (pitting) was additionally investigated. This includes herein the variants A, D, and E. The tooth flanks of all test gears were grinded. The tooth flanks are characterized by a low surface roughness (Roughness: Ra ≈ 0.2 μm). It can be seen that all variants exceed the specified strength numbers of the standard for induction hardening. It can also be established that the allowable contact stresses of the variants D and E (normal module of 2 mm) are higher compared to the value of variant A (normal module 4 mm). Furthermore, the specifications for the allowable contact stress numbers for case-hardened gears are also plotted. All induction hardened variants are characterized by values in the typical range of case-hardened gears. This includes the surface hardness as well as the allowable contact stress.

11.7 STEELS FOR NITRIDING2 Nitriding is, next to case-hardening, a common thermochemical surface treatment for highly loaded gears. In contrast to case-hardening, the nitriding treatment is carried out at lower process temperatures and therefore creates lower distortion. As a result, grinding after nitriding is usually not necessary. Nitrided gears are usually characterized by a thin but very hard compound layer of iron and alloy element nitrides directly on the surface with a thickness of a few microns and the subsequent diffusion layer reaching more deeply into the material, as shown in Figure 11.13. The selection of the adequate nitriding depth is linked to the module of the gear and should increase with increasing module. However, it should be noted that due to technological restrictions (diffusion rate of nitrogen, temperature during heat

2

Subchapter is mainly based on [47, 42].

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treatment, heat treatment duration) the nitriding depth is generally limited to approx. 0.6 to 0.8 mm. Using a modern nitriding heat treatment, even nitriding depths of up to 1 mm can be achieved [40–42]. Common nitriding alloy compositions for gears are e.g., 31CrMoV9, 34CrNiMo6 or 42CrMo4. Usual processes are gas, bath or plasma nitriding. The achievable strength numbers, especially in the area of static strength, are mostly below case-hardened gears. However, there is a possible load carrying ca­ pacity potential, especially for gears of smaller sizes. For larger gears, nitriding is mainly used for reasons of wear protection. The micropitting and wear performance is primarily influenced by the properties of the compound layer: closed compound layer, sufficient thickness of the compound layer and composition of the compound layer. With regard to fatigue related damages (e.g., pitting), the properties of the diffusion layer are important, too [47]. Compared to case-hardening, the decrease in hardness from surface to core hardness is generally much more significant. In contrast to case-hardened gears, nitrided gears show usually a higher overload sensitivity. Nitrided gears are particularly resistant to scuffing, wear and corrosion. Therefore, nitriding provides an alternative to case-hardening for specific application and offers potential for cost savings in the production of distortion-sensitive highly loaded gears [42]. In order to evaluate the load carrying capacity of nitrided gears, experimental investigations were carried out as part of the research projects [41,48,49] and [50]. For the investigations, several heat treatable and nitriding steels such as 31CrMoV9, 30CrNiMo8, 15CrMoV5-9, and 32CDV13 were used. Both the flank load carrying capacity and the tooth root bending strength were investigated.

11.7.1 TOOTH ROOT BENDING STRENGTH As part of [48,49] and [41], the tooth root bending strength of nitrided gears has been investigated intensively on a pulsator test rig. The investigations were limited to gears that did not receive any further machining after nitriding, so the compound layer at the surface was intact. Figure 11.14 shows the experimentally determined tooth root bending strength of the nitrided gears, which were investigated in the research projects [48,49]. The determined tooth root bending strength for the nitrided gears was in the area of the material quality MQ to ME acc. to ISO 6336-5 [6] and thereby much better than that of only quenched and tempered gears [48]. The effects of an increased nitriding hardening depth (NHD) were investigated in [41]. Here, the calculated tooth root bending strength of the nitrided gears was even higher as specified in ISO 6336-5 [6] and comparable to the tooth root bending strength of case-hardened and shot-blasted gears of the same size. For the review of the determined tooth root bending strength, it must be taken into account, that in all research projects the variation of the results was relatively high and a nameable number of gears failed by fracture of the tooth root after few load cycles. These gears were further investigated in [48]. The cracks started at the 30° tangent of the tooth root fillet so it was concluded that the maximum tensile stresses were decisive for the fracture. The tooth root fractures in [41] occurred at relatively low load cycles, too. This indicates also a high sensitivity of the nitrided gears for loads exceeding the endurance limit [47].

11.7.2

TOOTH FLANK LOAD CARRYING CAPACITY

The flank load carrying capacity has been investigated on a FZG back-to-back test rig on as-nitrided gears with intact compound layer as well as on gears, where the compound layer was removed after nitriding by grinding. For both the gears with and without the compound layer, respectively, a pitting endurance within the expectations of ISO 6336-5 [6] has been determined. The observed damage characteristics, however, were different between the variants. The passed specimen of the gears with an intact compound layer were free of micropitting. The failed gears however, showed severe micropitting on the tooth flanks before the gears eventually failed due to pitting. In contrast to that, some of the gears without compound layer, which failed due to pitting, did not show any micropitting before. If no mi­ cropitting has been observed, the overall pitting endurance of the gears with compound layer was higher,

30CrNiMo8 (T = 530 °C, KN = 0.5, NHD ≈ 1.0 mm) [FVA15a]

31CrMoV9 (T = 530 °C, KN = 0.5, NHD ≈ 1.0 mm) [FVA15a]

Source: FVA05 = [49] FVA12a =[48] FVA 15a = [41]

Surface Hardnessin HV

32CDV13 (T = 570 °C, KN = 0.5, NHD ≈ 1.0 mm) [FVA15a]

15CrMoV5-9 (T = 520 °C, KN = 1.0, NHD ≈ 0.4 mm) [FVA12a]

15CrMoV5-9 (T = 510 °C, KN = 0.8, NHD ≈ 0.5 mm) [FVA05]

15CrMoV5-9 (T = 510 °C, KN = 0.4, NHD ≈ 0.5 mm) [FVA05]

31CrMoV9 (T = 520 °C, KN = 1.0, NHD ≈ 0.6 mm) [FVA12a]

31CrMoV9 (T = 510 °C, KN = 0.8, NHD ≈ 0.6 mm) [FVA05]

FIGURE 11.14 Test results for the tooth root bending strength of nitrided gears acc. to [ ,48,49] and [ 41] and comparison with the specified strength numbers for nitrided gears acc. to ISO 6336 [ 3] acc. to [ 47] (T: nitriding temperature, KN: nitriding index, NHD: nitriding hardening depth acc. to ISO 6336 [ 3]).

σFlimin N/mm2

31CrMoV9 (T = 510 °C, KN = 0.4, NHD ≈ 0.6 mm) [FVA05]

Load Carrying Capacities 653

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Dudley’s Handbook of Practical Gear Design and Manufacture

compared to the variants that were ground after nitriding. If micropitting occurred, both variants showed a comparable pitting performance [42,47,48]. As part of the project [41], gears with increased nitriding depth of up to 1 mm (deep nitriding) have been investigated. The gears investigated in [41] were ground before the nitriding process. The increased NHD values have been achieved by increasing the nitriding temperature and the nitriding time in a twostage nitriding process. Details of the heat treatment can be found in [40]. Due to the increased nitriding temperature and time, the negative effects such as edge bulges, increased gear roughness and reduced gear quality were slightly increased compared to a state-of-the-art nitriding process. Also, the core hardness is slightly reduced due to tempering effects [51]. The results of the experiments on the FZG back-to-back test rig showed a dependence on whether the compound layer stayed intact on the tooth flank during the test run or whether it started to delaminate. In Figure 11.15, the tooth flank picture of a highly loaded flank is shown after 30 million load cycles. On the surface, no defects can be observed optically. A metallographic cross section of the compound layer of the deep nitrided material 32CDV13 is shown in Figure 11.16. The comparison of the as-nitrided state and the state after 30 million load cycles shows, that for this material and heat treatment, the compound layer stayed intact during the test run. As a result, the gear passed the running test at a high load and the tooth flank does not show any wear, micropitting or pitting.

FIGURE 11.15 Tooth flank picture of the deep nitrided gear (end relief: amount CI(II) ≈ 40 µm, width bI(II) ≈ 3,5 mm), material 32CDV13, after 30 million cycles (σH0 = 2050 N/mm2, passed specimen) [ 41] acc. to [ 47].

(a)

(b)

25 µm as-nitrided in new condition

25 µm after 30 million load cycles

FIGURE 11.16 Compound layer of a deep nitrided gear, material 32CDV13 (σH0 = 2050 N/mm2, passed spe­ cimen) [ 41] acc. to [ 47].

Load Carrying Capacities

655

FIGURE 11.17 Tooth flank picture of the deep nitrided gear (end relief: amount CI(II) ≈ 40 µm, width bI(II) ≈ 3,5 mm), material 30CrNiMo8, after 6 million cycles (σH0 = 1450 N/mm2, pitting failure) [ 41] acc. to [ 47].

(a)

(b)

25 µm as-nitrided in new condition

25 µm after 6 million load cycles

FIGURE 11.18 Compound layer of a deep nitrided gear, material 30CrNiMo8 (σH0 = 1450 N/mm2, pitting failure) [ 41] acc. to [ 47].

In Figure 11.17, a tooth flank of a gear made from another nitriding steel is shown. The deep nitriding treatment was carried out with different heat treatment parameters due to the lower temperature re­ sistance of the material 30CrNiMo8 at long nitriding times [51]. On the tooth flank, a combination of micropitting and pitting is visible after only 6 million load cycles on a load level that was much lower, compared to the variant shown in Figure 11.15. Figure 11.18 shows the cross section of the compound layer. Due to the lower nitriding temperature, the compound layer in the as-nitrided state is much thinner than that of the material 32CDV13 shown in Figure 11.16. After the running test, many cracks in the thin compound layer could be observed. One of them is exemplarily shown in Figure 11.18. As a conclusion, it can be stated, that as long as the compound layer remains intact, a high load carrying capacity of the tooth flank can be achieved. However, if the compound layer is damaged or starts to delaminate, this significantly reduces the flank load carrying capacity.

11.7.3 MICROPITTING

AND

WEAR PERFORMANCE

In the research project [50], the micropitting and wear performance of nitrided gears has been in­ vestigated. The results were compared to [52], where similar investigations have been done by means of

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Dudley’s Handbook of Practical Gear Design and Manufacture

case-hardened gears. The gears, which were compared to case-hardened gears, were nitrided in an in­ dustrial gaseous nitriding process (510°C; 30 hours). After nitriding, there was no further machining applied [47]. The wear performance has been tested acc. to DGMK 377 [53]. The gears were lubricated with the FVA reference oil 3 + 4% Anglamol99 (FVA 3A, mineral oil of ISO VG 100 with sulfur-phosphorus additives) [54]. In comparison to the case-hardened test variant, the nitrided gears in general showed a much better wear performance [47]. In further tests, the influence of the nitrided layer, i.e., compound layer thickness (CLT), composition of the compound layer, thickness of the porous layer (CLTP) as well as the influence of the nitriding hardness depth (NHD) acc. to ISO 6336 [3], on the micropitting and wear performance was investigated. By a variation of the nitriding parameters, different nitrided layers could be created on the material 31CrMoV9. The variations of the nitrided layer, which were tested in this research and additionally the corresponding core hardnesses (CH), are shown in Table 11.3. 47 [47]. The investigations showed that the composition of the compound layer is the main influencing factor on the wear performance. The nitriding depth shows less influence, if a certain minimum thickness of the nitrided layer is ensured. Almost no influence has been determined for the type of material and the type of nitriding process, as long as the properties of the compound layer are similar. Figures 11.19 and 11.20 show exemplary results of the wear tests on a case-hardened and a nitrided tooth flank, respectively. Both gears were tested with comparable conditions [47]. The micropitting performance has been investigated acc. to the test method FZG-micro-pitting test FVA 54 [55]. Again, reference oil FVA 3A (mineral oil of ISO VG 100 with sulfur-phosphorus ad­ ditives) has been used. Similar to the wear tests, the nitrided gears also performed significantly better in the micropitting test, compared to the case-hardened gears. During the tests, the porous zone has been

TABLE 11.3 Variations of NHD and/or Nitrided Layer for the Micropitting and Wear Tests acc. to [47] Variant

NHD in mm

CH in HV10

CLT in µm

CLTp in µm

Nitride Phases

R (industrial reference)

0.38

276

11.4

3.7

mainly ε

PN (plasma nitriding) NHD1

0.48 0.12

337 326

7.9 0.7

2.2 0

γ‘ γ‘

NHD2

0.22

322

2.6

0

γ‘

CL1 CL2

0.43 0.53

326 327

4.1 12.6

0.5 4.5

γ‘ mainly γ‘

NC (nitrocarburising)

0.43

321

17.6

4.3

mixture of γ‘/ε

FIGURE 11.19 Case-hardened toot flank after wear test acc. to DGMK 377 [ 52,53] acc. to [ 47].

Load Carrying Capacities

657

FIGURE 11.20 Nitrided tooth flank (end relief: amount CI(II) ≈ 12 µm, width bI(II) ≈ 2,5 mm) after wear test acc. to DGMK 377 [ 50,53] acc. to [ 47].

FIGURE 11.21 Case-hardened tooth flank after micropitting test FVA 54 [ 52,55] acc. to [ 47].

FIGURE 11.22 Nitrided tooth flank (end relief: amount CI(II) ≈ 12 µm, width bI(II) ≈ 2,5 mm) after micropitting test FVA 54 [ 50,55] acc. to [ 47].

identified as one of the main influencing factors on the micropitting performance. As long as a sufficient thickness of the porous zone at the tooth flank can be ensured, no micropitting has been observed. However, it must be noted that the porous zone has decreased during the micropitting test. Consequently, this will lead to micropitting if no porous zone is left on the tooth flank. Figures 11.21 and 11.22 show exemplary results of the micropitting tests on a case-hardened and a nitrided tooth flank, respec­ tively [47]. Based on the results it can be concluded that in general the same principles as for case-hardened gears on the formation of micropitting and/or wear apply [56]. However, for nitrided gears, the additional influence of the compound layer must be taken into account [47].

11.8 STEELS FOR CASE-HARDENING AND CARBONITRIDING For highly stressed transmissions, usually case-hardened gears are used. During the case-hardening process, the surface-near case layer is enriched with carbon by a thermochemical treatment, to create a stress-adapted strength profile with a high surface hardness and a smooth strength transition to the core

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Dudley’s Handbook of Practical Gear Design and Manufacture

Hardness

structure. Case-hardening depths (CHD550 HV at 550 HV acc. to ISO 6336-5 [6]) of approx. 0.2…5.0 mm can usually be achieved. The case-hardened layer usually has a microstructure consisting of martensite and retained austenite. Figure 11.23 shows an example of a schematic hardness depth curve of a casehardened gear and a corresponding hardness contour. For the use in gearboxes, a wide range of case-hardening steels is available. An overview of common, internationally established case-hardening steels of different alloying concepts is given in Table. 11.4. In general, the following essential material parameters influence the load carrying capacity of casehardened gears:

550 HV

CHD550 HV

Distance from surface

FIGURE 11.23

Hardness profile and cross section of a case-hardened gear tooth (following [ 57]).

TABLE 11.4 Chemical Composition of Selected Internationally Used Case-Hardening Steels for MediumSized and Large-Sized Gears, Specifications in Mass Percent acc. to [39] Name

Standard

Alloy Components

C

Si

Mn

EN 10084 0

min

18CrNiMo7 6

EN 10084 0

min min

0.17

-

1.10

0.22 0.15

0.40 -

min

0.21

15CrNi6

EN 10084 0

min min

0.14 0.19

17NiCrMo6 4

EN 10084 0

min

SAE 8620

SAE J1249 0

min min

SAE 9310

SAE J1249 0

20CrMnTi

GB T 3077 0

20CrMnMo

GB T 3077 0

SCM420

JIS G4105 0

20MnCr5

P

S

1.40 0.50

0.035 -

0.035 -

0.40

0.90

0.025

0.40

0.40 0.60

0.035

0.14

-

0.60

0.20 0.18

0.40 0.15

0.90 0.70

0.025 -

0.035 -

min

0.23

0.35

0.90

0.030

min min

0.08 0.13

0.15 0.35

0.45 0.65

0.025

min

0.17

0.17

0.80

min min

0.23 0.17

0.37 0.17

1.10 0.90

0.035 -

0.035 -

min

0.23

0.37

1.20

0.025

min min

0.18 0.23

0.15 0.35

0.60 0.85

0.030

-

-

-

Cr

Mo

Ni

1.00

-

-

1.30 1.50

0.25

1.40

0.035

1.80

0.35

1.70

0.035

1.40 1.70

-

1.40 1.70

0.80

0.15

1.20

1.10 0.40

0.25 0.15

1.50 0.40

0.040

0.60

0.25

0.70

0.040

1.00 1.40

0.08 0.15

3.00 3.50

1.00

0.00

-

1.30 1.10

0.15 0.20

0.30 -

0.035

1.40

0.30

0.30

0.030

0.90 1.20

0.15 0.30

-

-

-

-

Load Carrying Capacities

• • • • • • • • • •

659

Case-hardening depth Surface hardness Core strength Microstructure and grain size Retained austenite content Degree of cleanliness Degree of deformation Homogeneity of the material structure Internal oxidation Residual stress condition

Some of these parameters are discussed more in detail in the following sub-sections from various points of view, as there are often several and sometimes complex interactions between the individual influence factors.

11.8.1 INFLUENCE

OF

GEAR SIZE

The influence of the gear size is considered by the size factor YX in the calculation acc. to ISO 6336-3 [58]. A significant influence of the material (and thus the hardenability) on the size factor can be de­ termined, see Figure 11.24. Materials with a high hardenability (17NiCrMo14 and 18CrNiMo7-6) show significant advantages in load carrying capacity compared to a material with a low hardenability (16MnCr5).

(b)

m = 20 mm

m = 15 mm (c)

m = 10 mm 18CrNiMo7-6 ISO 6336

50 mm

(a)

FIGURE 11.24 Influence of the gear size on the tooth root bending strength for case-hardening steels with dif­ ferent hardenabilities acc. to [ 57].

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Dudley’s Handbook of Practical Gear Design and Manufacture

TABLE 11.5 Overview of Typical Case-Hardening Processes for Gears and Their Area of Application, Possible Advantages and Disadvantages [39] Procedure

Application

Advantages

Disadvantages

Direct hardening

Smaller gears, lower CHD

Simple and cheap hardening procedure, energy- and time-saving, highest core hardness

Coarse-grained martensite, highest retained austenite contents, danger of grain growth

Single hardening Double hardening

Larger gears, higher CHD Distortion-resistant gears

Fine acicular martensite, reduced retained austenite content Highest core hardness, fine acicular martensite, low retained austenite content

Energy- and time-intensive, distortion Distortion, highest effort on energy and time

Further investigations prove an increase of the load carrying capacity for gear sizes with mn < 5 mm [59–63]. This influence of the gear size is not yet considered in ISO 6336 [3]. Likewise, the casehardening process has to be applied to the gear size. The different hardening processes have advantages and disadvantages. An overview of the most common case-hardening processes under indication of advantages and disadvantages and possible areas of application is given in Table 11.5. Through special measures (e.g., shot-blasting or shot-peening, optimized material composition or heat treatment) it is possible to achieve higher experimentally verified strength values in individual cases. Acc. to ISO 6336-5 [6], both the requirements to the material as well as to the tests to ensure the material quality increase significantly from material quality grade ML to material quality grade ME.

11.8.2 INFLUENCE

OF

CASE-HARDENING DEPTH

In former research projects [64,65], the influence of the case-hardening depth on pitting and tooth root load capacity was investigated. The results from [65] also served as a basis for the work of TOBIE [66] and were published internationally in [67–69]. The test gears investigated comprise several gear sizes (normal moduli 2.3; 3; 5; 8 and 10 mm) and were manufactured from the materials 16MnCr5 and 20MnCr5, respectively, which have a low to medium hardenability. Figure 11.25 summarizes the results on the influence of case depth on the tooth root endurance limit. The results prove that the highest tooth root load carrying capacity is achieved at a case-hardening depth of 0.1…0.2 ∙ normal module. This range thus represents the optimum case-hardening depth in the tooth root for MnCr-alloyed steels with regard to the tooth root endurance strength. Outside this range a reduction of the tooth root load capacity is to be expected (see Figure 11.26). If the CHD is too low, this is due to the steep hardness or strength gradients and the associated low supporting effect of the surface layer. In case of higher CHD values, this reduction is attributed to an increase in the internal oxidation depth, possibly grain coarsening and low residual compressive stresses. This observation is also confirmed in [1,70–72] and [73]. Figure 11.27 shows the influence of the case-hardening depth by the factor YCHD. Additionally, also gear running tests with gears of different CHD were performed in order to de­ termine the influence of CHD on the pitting load carrying capacity. The results prove that there is also an optimal CHD necessary to achieve maximum pitting strength. CHDopt for pitting depends on the radii of curvature of the gears and can be calculated as follows: CHDopt,pitting = ( C + 10)/25 ± 0, 15 mm . Depending on the gear design for a specific application, the requirements regarding tooth root bending strength and pitting strength may differ. Therefore, a reasonable compromise for choosing suitable CHD often is necessary. Recommended values are given in ISO 6336-5 [6].

Normalized tooth root endurance

σFlim, Eht σFlim, max

Load Carrying Capacities

661

100 [%] 80 Tooth root: ~ module of the gears Zahnfuß: Eht CHD ~ Modul der Verzahnung

60

40 CHDA : mn= 8 mm, z = 24 CHDB : mn= 3 mm, z = 67 CHDC : mn= 3 mm, z = 29

20 0

0.1 mn

0.2 mn

FVA -Nr. 8:

0.3 mn

0.4 mn

mn = 2,3 mm mn = 3 mm mn = 5 mm mn = 10 mm 0.5 mn

0.6 mn

Normalized case hardening depth FIGURE 11.25 Tooth root bending strength depending on case-hardening depth (CHD) acc. to [ 64] and [ 65] (materials 16MnCr5 or 20MnCr5).

ZCHD,exp

Area acc. to[Nie03]

CHD/CHDth FIGURE 11.26 Pitting strength depending on case hardening depth (CHD) acc. to [ 65] (CHDth: threshold value of case hardening depth).

11.8.3 INFLUENCE

OF

RETAINED AUSTENITE3

In the following section, the impact of retained austenite on the strength of gears is described in more detail. The microstructure of martensite and finely dispersed retained austenite is considered the standard

3

The subchapter is mainly based on [74], further information are published in [37] and [76].

Dudley’s Handbook of Practical Gear Design and Manufacture

YCHD

662

Optimal value

Normalized case hardening depth CHD FIGURE 11.27 YCHD factor for the influence of the case-hardening depth on tooth root load carrying capacity of case-hardened gears acc. to [ 65].

for high strength case layers (acc. to ISO 6336-5 [6]). Investigations on test gears made of 20MnCr5 or 18CrNiMo7-6 have shown that a carbonitrided variant with a high content of retained austenite increases the tooth flank carrying capacity, while negative influences on the tooth root bending strength were not determined. Gas carburized variants with a high content of retained austenite showed similar tendencies regarding the gear load carrying capacity. Figure 11.28 shows the hardness depth profiles for a variant with high content of retained austenite made of the material 20MnCr5. The determination of the hardness depth gradients was carried out at the left and right side of the flank and root of an unloaded, not grinded and shot-blasted pulsator wheel tooth. The hardness depth profiles of the material 18CrNiMo7-6 were comparable to the profiles of the material 20MnCr5. The plotted profiles show a decrease of the hardness values toward the surface, which cor­ relates with high retained austenite contents in these areas. The minimal required hardness at the surface

Left flank 800,0

Right flank Left root

700,0 Hardness in HV1

Right root 600,0 500,0 400,0 300,0 200,0 0,00

0,50

1,00

1,50 2,00 2,50 Distance from surface in mm

3,00

3,50

FIGURE 11.28 Hardness profiles of 20MnCr5 gear with high content of retained austenite (maximum about 50%, average about 35%, measured by X-ray diffraction) acc. to [ 74].

Load Carrying Capacities

663

of 600 HV1 or rather 660 HV1 acc. to ISO 6336-5 [6] is not fulfilled but the CHD is still appropriate for the investigated gear size mn = 5 mm. Figure 11.29 shows the nominal tooth root bending strength for a failure probability of 50% (σF0∞,50%) of several variants with alternative carburizing or carbonitriding heat treatments in a bar chart and compares the results with the case-hardened reference R. In addition, the scatter regarding the determined strength values of the individual variants as well as a scattering range of ± 5% around the nominal tooth root bending strength of the case-hardened reference made of the material 20MnCr5 is depicted. It shows that the results of all variants are classified within the illustrated scatter range. Furthermore, all variants show a test scattering, which is still common for case-hardened gears. Overall, it can be said that the different heat treatment processes respectively alternative microstructures do not have a significant negative influence on the tooth root bending strength concerning endurance life for the investigated shot-blasted condition. In the present case, a few variants made of the material 18CrNiMo76 tend to show higher scatters than the variants made of the material 20MnCr5. The higher scatters of the variants made of the material 18CrNiMo7-6 are presumably due to inhomogeneities in the basic material. For all results it must be taken into account, that the test gears were mechanically cleaned by shotblasting and that the shot-blasting treatment may influence or cover up certain other effects such as different surface hardness or internal oxidation. In Figure 11.31, the determined tooth root bending strength values of the individual variants are entered in the standard strength diagram for case-hardened gears acc. to ISO 6336-5 [6]. The variants marked with * are estimated strength values based on a limited number of data points. Additionally, strength values for case-carburized and shot-blasted gears made of the material 18CrNiMo7-6 and casecarburized and shot-blasted gears made of the material 16MnCr5 are given from the literature. As stated, the tooth root bending strength values are all within a scatter range, which would still be common for case-hardened gears (reference variant R). Within this scatter range, the variant with an optimized thermal post-treatment at 280°C TPT280 made of 20MnCr5 shows the highest and the variant with 50% retained austenite and gas carbonitriding has the lowest tooth root bending strength. Based on these results, proving a suitable tooth root bending strength, benefits of an increased amount of retained austenite resp. different heat treatment parameters regarding the tooth flank load carrying capacity—pitting and micro-pitting—can be used (see Figure 11.30). The micro-pitting and wear load carrying capacity are influenced in a positive way by carbonitriding. A higher sensitivity against impact loads has to be considered [77].

11.8.4 INFLUENCE

OF

CRYOGENIC TREATMENT4

Another possibility to affect the load carrying capacity of gears is a cryogenic treatment. In Figure 11.32, it can be seen that the tooth root bending strength for shot-blasted gears that have been exposed to different low temperatures before the test does not show a significant deviation and especially no re­ duction for any low temperature treatment, while the unpenned batches and the samples with higher retained austenite suffer a reduction in the tooth root bending strength. The shot-blasting seems to oppose possible low temperature-induced effects on the tooth root bending strength of gears with a retained austenite content acc. to the industrial practice. Regarding the investigated unpenned gears and those with increased retained austenite content, a decrease in tooth root bending strength is visible irrespective of the applied low temperature treatment. For the gears with intentionally increased retained austenite content, the shot-blasting leads to a higher tooth root bending strength compared to the respective variants without a shot-blasting.

4

The subchapter is mainly based on [78]. Further information are published in [12].

FIGURE 11.29

Case carburized (reference variant) 50 % retained austenite gas carbonitrided 50 % retained austenite low pressure carbonitrided 10 % bainite 50 % retained austenite and optimized thermal post treatment at 150 °C 50 % retained austenite and optimized thermal post treatment at 280 °C Grain boundary carbides

R

RA50GCN

RA50LPCN

B10

TPT150

TPT280

C

Tooth root bending strength of different variants with alternative heat treatments compared to a conventional case-hardening process acc. to [ 74].

Variant

Abbreviation

664 Dudley’s Handbook of Practical Gear Design and Manufacture

Case carburized 50 % retained austenite and optimized thermal post treatment at 280 °C 30 % bainite 50 % retained austenite gas carburized

EH

TN280

BN30

RA50GAK

FIGURE 11.30 Comparison of the individual running test results (number of load cycles and average number of load cycles until pitting failure) for 20MnCr5 variants (following [ 76]).

Variant

Variant

LC

Test run

Test run

Test run

Test run

Abbreviation

Number of load cycles

Area of endurance

Load Carrying Capacities 665

666

FIGURE 11.31 acc. to [ 74].

Dudley’s Handbook of Practical Gear Design and Manufacture

Standard strength diagram acc. to ISO 6336-5 [ 6] for variants with alternative heat treatments

FIGURE 11.32 Endurable nominal tooth root bending stress (50% failure probability) of several case-hardened variants under application of low temperature treatments (HCT: low temperature treatment before tempering; H+CT: low temperature treatment after tempering (and shot-blasting)) [ 78].

Based on the performed pulsator tests to obtain the tooth root bending strength and extensive in­ vestigations of material properties to characterize the consequences of different low temperature treat­ ments on case-hardened gears, the following observations and tendencies can be derived: • Gears that were low temperature treated at –60°C or lower before tempering, reveal an increased surface hardness in comparison to the corresponding reference batch. In contrast, gears that were low temperature treated after tempering and additional mechanically cleaning by shot-blasting (H +TC) showed almost no change in surface hardness. • For the batches, which were low temperature treated before tempering (HCT), a corresponding reduction of the retained austenite content near the surface from partly over 20% to less than 10%

Load Carrying Capacities

667

was observed. Gears that were low temperature treated after shot-peening, in contrast, reveal si­ milar retained austenite contents irrespective of the applied treatment temperature. • A low temperature treatment before tempering results in increased or steady compressive residual stresses for unpenned gears, while shot-blasted ones show an increase at a treatment temperature of –196°C. • The shot-blasted H+TC-treated gears reveal no significant influence of the low temperatures on the tooth root bending strength. • The unpenned gears and the ones with increased retained austenite (unpenned and shot-blasted) in contrast show a clearly reduced tooth root bending strength after applying a low temperature treatment before tempering. Overall, low temperature treatments cause different effects on gears’ material properties, depending on the treatment sequence and low temperature magnitude. For unpenned gears, low temperatures may result into lower tooth root bending strength and therefore should be considered in the gear design. A mechanically cleaning by a shot-blasting treatment seems to possibly counteract all or at least some of the performance reducing effects of the low temperature treatments. Regarding the influence of cryogenic treatments on the flank load carrying capacity, differing results and statements can be found in the literature. In some investigations a decrease of the pitting strength after cryogenic treatments is stated [80], investigations in contrast show increased pitting strength after cryogenic treatments [81,82]. Systematic investigations at FZG are performed at the moment.

11.8.5 INFLUENCE

OF

RESIDUAL STRESS CONDITION5

Residual stresses are stresses that occur in a component that is not loaded by any force or torque. There are tensile and compressive residual stresses. Both kinds of residual stresses are balanced within a component. Compressive residual stresses in the surface layer typically have a positive influence on the load carrying capacity, whereas tensile residual stresses in the surface layer can significantly reduce the load carrying capacity. The residual stress state is influenced by the manufacturing process, which in­ cludes soft machining, heat treatment, and finishing. The heat treatment process “case-hardening” usually leads to compressive residual stresses in the surface layer and tensile residual stresses in the core. The residual stresses take only small compressive stress values in a range of about −200 up to −400 N/mm2 after typical case-hardening treatment. During the quenching, the component does not cool down uniformly, so a volume difference occurs whereby residual stresses arise. Furthermore, austenite transforms into martensite. Both microstructures have different specific volumes, which result in addi­ tional residual stresses. Overall, the residual stress state results from a combination of quenching and volume change. In many cases, the gears are subjected to a mechanical cleaning by shot-blasting or shot-peening process after case-hardening. Thereby, there is a distinction between mechanical cleaning by shotblasting and shot-peening. The aim of the mechanical cleaning process by shot-blasting (see Figures 11.33 and 11.34) is to remove the scale layer and clean the component after heat treatment. By this process, compressive residual stresses are induced, which have a positive effect on the load carrying capacity. Furthermore, according to ISO 6336-5 [6], the bending stress numbers for case-hardened gears of material quality MQ are purposefully achievable with shot-blasting. Shot-peening differs from shotblasting. In opposite to mechanically cleaning by shot-blasting, several parameters such as blasting

5

Contents of this subchapter were previously presented at the 2017 American Gear Manufactures Association’s Fall Technical Meeting as 17FTM20 [38] []. Printed here with permission of the copyright holder, the American Gear Manufacturers Association, 1001 N. Fairfax Street, 5th Floor, Alexandria, Virginia 22314.

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FIGURE 11.33 Schematic presentation of the mechanical cleaning by shot-blasting by an impeller acc. to [ 83].

FIGURE 11.34 Schematic presentation of the shot-peening by jet nozzle acc. to [ 83].

material, hardness of the blasting material, size of the material, degree of coverage and intensity are defined and are monitored [84–86]. In both cases, the generation of the residual stresses is based on the model representation of Wohlfahrt [87]. For case-hardened steels (e.g., 16MnCr5 or 18CrNiMo7-6), the residual stresses are generated by the elastic-plastic deformation of the surface. Due to the local stresses, which exceed the yield point of the material, compressive residual stresses arise. Furthermore, the retained austenite transforms into martensite during the shot-peening process. Due to different specific volumes of the microstructures, compressive residual stresses arise [2,88]. During the finishing process “grinding,” the residual stress state is also changed. The surface layer is mechanically and thermally influenced [89]. Due to the mechanical influence by the grinding wheel, compressive or tensile residual stresses may be generated. Excessive heat exposure during the grinding process results in tensile residual stresses. The residual stress condition after grinding is a result of a superimposition of both influences. The process parameters have a significant influence on the residual stress state. In the surface layer, tensile as well as compressive residual stresses may arise. In Figure 11.35, typical residual stress profiles in the surface layer are plotted for case-hardened gears in the unpenned, mechanical cleaned by shot-blasted, and shot-peened condition. The influence of in­ creased residual stresses due to shot-blasting or shot-peening is limited to only a depth of about 0.1… 0.15 mm. The highest compressive residual stresses are achievable with shot-peening.

11.8.6 INFLUENCE

ON

TOOTH ROOT LOAD CARRYING CAPACITY

Due to the increase of compressive residual stresses by shot-blasting or shot-peening on and near the surface, the tooth root bending strength of case-hardened gears can be increased. The investigations concerning the tooth root bending strength of [2,90–92], and [93] were performed on a pulsator test rig.

Load Carrying Capacities

669

Residual stress in N/mm2

Depthinmm

Unpeened Mechanicalcleaned Shotpeened

FIGURE 11.35 Comparison of the residual stresses in the unpenned, mechanically cleaned by shot-blasted, and shotpeened condition—surface layer of case-hardened gears (measurement results, exemplary presentation) acc. to [ 2].

Investigations done by Weigand [93] or Stenico [2] dealt with the influence of shot-blasting and shotpeening using case-hardened gears made out of the materials 16MnCr5 and 18CrNiMo7-6. Thereby, different gear sizes were used including module 1.75…8 mm. Weigand [93] identified an increase of the tooth root bending strength of 30% (concerning material 16MnCr5, module 5 mm) and 15% (concerning material 18CrNiMo7-6, module 5 mm) between case-hardened gears in the unpenned condition and in the shot-blasted condition. By using two different shot-peening processes, the increase in the tooth root bending strength was of 35 to 45% (material 16MnCr5) and 30 to 50% (material 18CrNiMo7-6) compared to the unpenned condition. Inoue et al. [94] dealt with the effect of shot-peening on the bending strength of carburized gears. In this research work, gears (module 5 mm) made out of the material JIS SCM 415 and JIS SCM 420H were shotpeened with different process parameters and different resulting residual stress states. The results show that an increase of the tooth root bending strength of up to 30% compared to the unpenned condition is possible. Stenico [2] investigated the influence of different shot-peening parameters on the tooth root bending strength of case-hardened gears with different materials and gear sizes. The investigations showed an increase of the tooth root bending strength of about 27 to 43% for material 16MnCr5 (gear sizes/normal module: 1.75; 3 and 5 mm) between the unpenned condition and after shot-blasting. This increase can be further enhanced by shot-peening. In the case of shot-peening, the tooth root bending strength was about 42 to 66% higher than in the unpenned condition. The material 18CrNiMo7-6 showed increases of about 38 to 50% between the unpenned and shot-peened condition for the gear size module 5 mm. In Figure 11.36, the determined S–N-curves for the tooth root bending strength material 16MnCr5 (normal module 3 mm) are plotted in the unpenned, shot-blasted, and shot-peened condition (failure probability 50%). The results prove that by shot-blasting or shot-peening, the tooth root bending strength can be increased significantly. A consideration of the low cycle fatigue indicates that the slope of the low cycle fatigue curve gradient is different for the three variants. The unpenned variant has the highest value and the shot-peened variant the lowest one. This means that shot-blasting as well as shot-peening influences especially the endurance limit, but not the static tooth root bending strength. The consequence is that shot-blasted, but especially shot-peened, gears show higher overload sensitivities than gears in the un­ penned condition. Furthermore, for gears, especially in the shot-peened condition, failures due to sub­ surface cracks were detected.

Dudley’s Handbook of Practical Gear Design and Manufacture 3000 case carburized 2500 m = 3 mm n 2000

50% failure probability

35 30 25 20

1500 15 1000 10

500

1

5

shot peened mechanically cleaned unpeened 103

104

105 Number of load cycles N

106

Pulsating load FPn In kN

Tooth root stress σF0 in N/mm2

670

107

FIGURE 11.36 Exemplary presentation of the influence of mechanical cleaning by shot-blasting and shot-peening on the tooth root bending strength of a carburized gear (material 16MnCr5, normal module mn 3 mm) acc. to [ 2].

FIGURE 11.37

Tooth root bending test results of a shot-peened variant at a cycle limit of 100 ∙ 106 [ 90].

Figure 11.37 is taken from [90] and presents the results of the tooth root bending strength of the material 16MnCr5 (gear size module 5 mm) in the shot-peened condition. The test runs were terminated after 100 ∙ 106 load cycles. The test runs that fail in the range lower than 106 load cycles are characterized by surface-initiated tooth root fracture damages. In the range of higher load cycles, all breakages ori­ ginate from the subsurface. Furthermore, the results show that the nominal tooth root stresses σF0 of the failures in the range of more than 106 load cycles are lower than for breakages from the surface.

11.8.7 CHANGE

IN THE

FRACTURE MODE—UNPENNED

VS.

SHOT-PEENED CONDITION

Case-hardened gears in the unpenned condition and most of the gears in the shot-blasted condition fail due to tooth root breakages with a crack initiation at the surface. In Figure 11.38 a typical tooth root fracture surface is shown. Typical load cycles for such damages are in the range of 105…106 load cycles.

Load Carrying Capacities

671

FIGURE 11.38 Typical tooth root fracture surface, crack initiation on the surface acc. to [ 92].

FIGURE 11.39 Typical tooth root fracture surface, crack initiation subsurface on an inclusion (oxide) acc. to [ 92].

Tooth root breakages of gears in the shot-peened condition that fail in the region of endurance limit show mostly a different fracture surface compared to gears in the unpenned condition. The fracture surface contains a small round and bright spot. In the literature, this is often called “fish-eye” due to the characteristic appearance. Figure 11.39 represents such a typical fracture surface. The crack is initiated subsurface on an inclusion (oxide). More information about the crack area characteristics relevant for tooth root fracture damages of case carburized and shot-peened high strength gears of different sizes made of high-quality steels can be found in [95]. Shot-peened gears in the range of endurance life usually fail in the range of more than 106 load cycles. In Figure 11.40, the load stresses induced for different gear sizes are plotted as well as a typical hardness profile of case-hardened gears. Furthermore, typical residual stresses of unpenned and shotpeened gears are shown. The schematic distribution of the load-induced stresses presents that the highest load stresses occur on the surface of the tooth root. Furthermore, it is shown that the decrease of the load stresses depends on the gear size, whereas the compressive residual stresses do not depend on the gear size. Considering case-hardened gears in the unpenned condition, the crack initiation takes places on the surface where the highest load stresses occur. By shot-peening, the surface is strengthened, and a crack initiation on the surface is mostly prevented. The crack initiation takes place subsurface usually at an inclusion or other defect of the material. BRETL [90] developed a model that considers the local strength of the material. Thereby, the material strength is compared with the local stress situation. The material strength mainly depends on the hardness; the local stress situation is characterized by load-induced stresses and residual stresses. Compressive residual stresses have a positive effect on the tooth root bending fatigue strength and are taken into account in an appropriate way. Acc. to this model, a crack can be initiated when the local stress situation exceeds the local strength of the material. An inclusion or defect in the material causes a local increase of the stresses, which can lead to an excess of the local strength, and a subsurface crack can be initiated. With the calculation model of BRETL, it is possible to determine whether an inclusion can cause a crack that is growable. The model approach of BRETL was

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FIGURE 11.40 Schematic distributions of load stress, residual stress, and hardness in tooth root area of case-hardened gears depending on material depth acc. to [ 90,96].

further developed by SCHURER [92,96] (internationally presented in [98]) including in addition the work of STENICO [2]. PRASANNAVENKATESAN [99] found out that the subsurface crack initiation takes place in material depths in which the positive influence of the shot-peening is subsided. In Figure 11.41, the residual stress state is plotted and the depth of the crack initiation is determined. The subsurface crack initiation took place in a material depth of about 220 µm. The maximum of the compressive residual stresses is located in a depth of about 100 µm. In a material depth of more than 200 µm, the compressive residual stresses of the shotpeening are already subsided. This is also confirmed by extensive investigations of BRETL [90] and SCHURER [96]. The results show that the crack initiation takes place in a material depth in which the high compressive residual stresses due to the shot-peening are subsided but are still located within the case-hardened layer.

11.8.8 STEPWISE S–N CURVE All the results and theoretical considerations draw a conclusion that there is a kind of “double” S–Ncurve for the tooth root bending fatigue of shot-peened case-hardened gears. In Figure 11.42, a schematic illustration of the stepwise S–N curve is shown. In the literature, it is also often called “twofold” S–N curve. The “classical” S–N curve for tooth root bending strength is limited by crack initiation at the surface and is extensively validated for unpenned and shot-blasted gears. Besides this, for shot-peened gears, there is a second S–N curve, which is determined by subsurface crack initiation. This S–N curve determines the load carrying capacity in the high cycle fatigue. Due to the subsurface crack initiation, a further decrease of the tooth root bending strength has to be considered. As shown in the research works [90,91], such a kind of two-step S–N curves theoretically also exists for unpenned and shot-blasted gears. However, as the surface is not strengthened by increased com­ pressive residual stresses, the load-induced stresses exceed the fatigue strength at the surface before a critical stress condition below the surface can initiate subsurface cracks. Therefore, the crack initiation on the surface decisively determines the lifetime of unpenned and shot-blasted gears.

11.8.9 INCREASE

OF THE

TOOTH FLANK LOAD CARRYING CAPACITY (PITTING)

In [101], the influence of shot-peening on the tooth flank capacity (pitting) was investigated. Therefore, the gears were grinded (tooth flank) and after that shot-peened. The gears used in the investigation have a

FIGURE 11.41

Residual stress distribution (left) and subsurface crack initiation (right) acc. to [ 99].

Load Carrying Capacities 673

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FIGURE 11.42 Schematic illustration of the two-step S–N-curve acc. to [ 100].

gear size of normal module 5 mm and were made out of the materials 16MnCr5 and 18CrNiMo7-6 (former name: 17NiCrMo6). The results indicate that an increase of the tooth flank load carrying ca­ pacity up to 10% is possible. Furthermore, especially concerning the material 16MnCr5, a correlation of the load carrying numbers to the amount of retained austenite was detected. Regardless of the material, all shot-peened gears have a higher surface roughness after shot-peening compared to the grinded state. Consequently, in the experimental investigations, the shot-peened gears showed a stronger tendency toward micropitting. This damage type can be reduced by using an adjusted oil with a higher micro­ pitting load carrying capacity. TOWNSEND [102] investigated the influence of shot-peening on the tooth flank load carrying capacity of gears. Therefore, some gears were additionally shot-peened after grinding. The test runs were performed at a constant Hertzian stress. The gears that were shot-peened showed a pitting lifetime that is about 46% higher than without shot-peening. The increase of the load carrying capacity is assumed to go back to the residual stress state. In a further investigation, the influence of the intensity (a middle one and a high one) of the shot-peening process on the lifetime of case-hardened and shot-peened gears was determined. After the shot-peening, the tooth flanks were honed to improve the roughness con­ dition of the tooth flanks. The results show that the lifetime of the gears that were shot-peened with the high intensity was 1.7 times higher than the lifetime of the variant with the middle intensity (failure probability 50%). Due to the fact that a shot-peening process often leads to increased surface roughness values, a further processing of the shot-peened surface typically is necessary to use the potential of the flank load carrying capacity [83]. Therefore, the influence of a shot-peening treatment and a finishing process was focused on in a research project and [103] to further increase the flank load carrying capacity of gears. In experimental test runs, the flank load carrying capacity (pitting) of gears made of the material 16MnCr5 and a gear size of module 5 mm was determined. The finishing of the gears was varied, which included grinding, barrel finishing, and shot-peening in combination with an additional barrel finishing. The re­ sults plotted in Figure 11.43 show that the highest pitting load carrying numbers can be achieved by shotpeening after grinding and an additional barrel finishing. In this case, the increase of the nominal contact stress for endurance was about 20% compared to the variant that was grinded only. However, it should be noticed that with the gain in the pitting load carrying capacity, the risk of a tooth flank fracture damages rises. Tooth flank fractures are characterized by a crack initiation subsurface in the area of the active tooth flank. Further applications of shot-peening are repair measures for grinding burn (see [83,89,101,104,105]) and for avoiding facing edge tooth fracture (see [106,107]).

Load Carrying Capacities

675

c

1900 1800 b

1700

750 700 650 600 550 500

1600

450 a

1500

400

1400

350

1300

300

1200 1100

Verzahnung „Basis“ Basic data: z 1 /z 2 = 17/18, a = 91,5 mm z1/z2= 17/18,-1a = 91,5 mm n 1 = 3000 min = 3000 min-1 1 FVA 3 + 4 % A99 ϑ E = 60n°C; E=

106

60 °C; FVA 3 + 4% A99

a:a)R1/O1:geschliffen(Ra ground: Ra≈≈0,3 0,3µm) μm b: ≈ 0,1 µm) b)R2:gleitgeschliffen barrel finished: (Ra Ra ≈ 0,1 μm c: R3:festigkeitsgestrahlt + c) s.p. + barrel finished: Ra ≈ 0,1 μm gleitgeschliffen (Ra ≈ 0,1 µm) failure probability 50 %50% Ausfallwahrscheinlichkeit

107

Pinion torque T1 in Nm

Nominal flank pressure σH0 in N/mm2

2000

250

200

108

number of load cycles FIGURE 11.43

S–N-curve of the pitting load carrying capacity acc. to [ 83] and [ 103]; s.p. means shot-peening.

11.9 SUMMARY AND OUTLOOK6 The diverse and complex stress and strain conditions on gears in gearboxes require special carefulness in the selection of material and heat treatment processes, taking into account the manufacturing and operating conditions. Different gear failure mechanisms and a large number of parameters, some of which influence each other, require the most precise knowledge of the possible interactions and often necessitate com­ promises with regard to the desired material properties. Simplified, standardized calculation methods like AGMA 2001-D4 [5], DIN 3990 [4] or ISO 6336 [3] and design rules based on experience and test results enable the gear designer to make a quick and reliable choice of material and heat treatment processes that meet the requirements. Due to the characteristic stress profiles of gears over the material depth in the area of the tooth flank and the tooth root, steels with the possibility of heat treatment are of outstanding importance, especially when high performance is required. Only through the interaction of design and heat treatment technology, the existing knowledge can be used to optimize solutions for drive train components. Modern gear transmissions are constantly faced with new requirements such as higher power density, reduced costs, increased component reliability or further exploitation of material potentials. Figures 11.44 and 11.45 show as an example the expected tooth root respectively tooth flank pitting fatigue strength of different groups of steel alloy and heat treatment concepts acc. to ISO 6336-5 [6]. Within the strength ranges, usually three material quality levels can be distinguished: • Material grade ML for low requirements, • Material grade MQ for requirements that can be met by experienced manufacturers at reasonable cost • Material grade ME for applications with highest requirement to the gear load carrying capacity. It can be seen that in both cases the highest strength values are achieved by case-hardened gears of material quality ME. The diagrams relate an easily measurable property, such as surface hardness, to a complex system property, such as fatigue strength in the tooth root respectively the tooth flank. The fact that for a given surface hardness a quite broad level of strength can be achieved with respect to the tooth root and the 6

Parts of the subchapter are mainly based on [1, 7, 32, 35, 39, 43–46, 63, 75, 76, 79, 88, 97].

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FIGURE 11.44 Strength fields for tooth root fatigue strength for different heat treatment processes acc. To SO 6336-5 [ 6] acc. to [ 39] (translated into English).

FIGURE 11.45 Strength fields for pitting fatigue strength for different heat treatment processes acc. To ISO 63365 [ 6] acc. to [ 39] (translated into English).

tooth flank fatigue strength indicates that the alloy composition, the microstructure and the thermo-chemical treatment have a very large influence on the actual performance of the gear. Consequently, optimizations of the base material, heat treatment process and manufacturing route offer further potentials to increase the load carrying capacity respectively power density of gears in modern applications in the future. A profound understanding of the underlying stress conditions and failure mechanism, however, is required. Figure 11.46 summarizes finally advantages and disadvantages of selected heat treatment processes for gears, as well as typical applications.

FIGURE 11.46

Summary of advantages, disadvantages and typical applications of selected heat treatment processes for gears.

Load Carrying Capacities 677

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ACKNOWLEDGMENT Research projects with the cross reference [FVA…] were conducted with the kind support of the FVA (Forschungsvereinigung Antriebstechnik e.V.) research association. The projects were sponsored by the German Federal Ministry of Economics and Technology (BMWi) through the AiF (Arbeitsgemeinschaft industrieller Forschungsvereinigungen) in the course of a program for the support of collective industrial research (IGF) as a result of a decision by the German Bundestag or AVIF (Forschungsvereinigung der Arbeitsgemeinschaft der Eisen und Metall verarbeitenden Industrie e.V.).

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Dudley’s Handbook of Practical Gear Design and Manufacture Hoja, S., Schurer, S., Hoffmann, F., Tobie, T., Zoch, H.-W., & Stahl, K. (2015). Tiefnitrieren von Zahnrädern [engl.: Deep nitriding of gears], FVA-no. 615 II - issue 1147 - final report. Forschungsvereinigung Antriebstechnik e.V. (FVA). Sitzmann, A., Tobie, T., Stahl, K., & Schurer, S. Influence of the case properties after nitriding on the load carrying capacity of highly loaded gears. Volume 10: 2019 International Power 18.08.2019. SAE J1249. (2008). Former SAE standard and former sae ex-steels. Society of Automotive Engineers (SAE). Spitzer, H., Bleck, W., & Flesch, R. (1998). Einsatzstähle - Normung und Entwicklungstendenzen [eng.: Case hardening steels - Standardization and development trends] - ATTT/AWT - Conference Case Hardening, Conference Proceedings, pp. 11–20, Aachen. Stahl, K., Höhn, B.-R., Tobie, T. (2013). Tooth flank breakage: influences on subsurface initiated fatigue failures of case hardened gears. Proceedings of the ASME 2013 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, Portland, Oregon, USA, August 4-7 2013, Paper No. DETC2013-12183. GB/T 3077-2015. (2015). Alloy structural steels. Standardization Administration of the People’s Republic of China (SAC). König, J., Hoja, S., Tobie, T., Hoffmann, F., & Stahl, K. (2019). Increasing the load carrying capacity of highly loaded gears by nitriding. MATEC Web of Conferences 287(i), 2001. Bretl, N. T., Schurer, S., Tobie, T., Stahl, K., & Höhn, B.-R. (2012). Ergänzungsvorhaben Produktsicherheit nitrierter Zahnräder (engl.: Additional project product safety of nitrided gears), FVA-no. 386 II - issue 1025 final report. Forschungsvereinigung Antriebstechnik e.V. (FVA). Günther, D., Pouteau, P., Bruckmeier, S., Hoffmann, F., Tobie, T., Mayr, P., & Zoch, H.-W. (2005). Produktsicherheit nitrierter Zahnräder [engl.: Product safety of nitrided gears], FVA-no. 386 - issue 777 final report. Forschungsvereinigung Antriebstechnik e.V. (FVA). Zornek, B., Hoja, S., Tobie, T., Hoffmann, F., Stahl, K., & Zoch, H.-W. (2017). Tribologische Tragfähigkeit von nitrierten Innen- und Außenverzahnungen bei geringen Umfangsgeschwindigkeiten [engl.: Tribological load carrying capacity of nitrided internal and external gears at low circumferential speeds], FVA-no. 482 IV - issue 1206 - final report. Forschungsvereinigung Antriebstechnik e.V. (FVA). Hoja, S., Hoffmann, E., Steinbacher, M., & Zoch, H.-W. (2018). Investigation of the Tempering Effect during Nitriding - Untersuchung des Anlasseffektes beim Nitrieren. HTM Journal of Heat Treatment and Materials, 73, 335–343. Schudy, J., Tobie, T., & Höhn, B.-R. (2008). Flankentragfähigkeit von innen- und außenverzahnten Stirnrädern bei geringen Umfangsgeschwindigkeiten [eng.: Flank load carrying capacity of internal and external spur gears at low circumferential speeds], FVA-no. 482 I - issue 867 - final report. Forschungsvereinigung Antriebstechnik e.V. (FVA). Bayerdörfer, I., Michaelis, K., Höhn, B.-R., & Winter, H. (1997). Method to assess the wear character­ istics of lubricants - FZG test method C/0, 05/90:120/12, DGKM. Laukotka, E. M. (2007). Referenzölkatalog (engl.: Reference oil catalog), issue 660. Forschungsvereinigung Antriebstechnik e.V. (FVA). Emmert, S., Schönnenbeck, G., Rettig, H., Oster, P., Höhn, B.-R., & Winter, H. (1993). Testverfahren zur Untersuchung des Schmierstoffeinflusses auf die Entstehung von Grauflecken bei Zahnrädern [eng.: Test method for investigating the influence of lubricants on the development of micro-pitting on gears], FVA information sheet no. 54/7. Forschungsvereinigung Antriebstechnik e.V. (FVA). König, J., Felbermaier, M., Tobie, T., & Stahl, K. (2018). Influence of low circumferential speeds on the lubrication conditions and the damage characteristics of case-hardened gears. Minneapolis/Minnesota/USA. Tobie, T., Hippenstiel, F., & Mohrbacher, H. (2015). Optimized gear performance by alloy modification of carburizing steels for application in large gear boxes, São Paulo, 2015. International Symposium on Wear Resistant Alloys for the Mining and Processing Industry. ISO 6336-3. (2006). Calculation of load capacity of spur and helical gears- part 3: calculation of tooth bending strength. ISO International Organization for Standardization. Braykoff, C. (2007). Tragfähigkeit kleinmoduliger Zahnräder [eng.: Load carrying capacity of small module gears][Dissertation]. Technical University of Munich (TUM). Dobler, A., Hergesell, M., Tobie, T., & Stahl, K. (2016). Increased tooth bending strength and pitting load capacity of fine-module gears. Gear Technology, 34(1), 48–53. Hergesell, M. (2013). Grauflecken- und Grübchenbildung an einsatzgehärteten Zahnrädern mittlerer und kleiner Baugröße [eng.: Micro-pitting and pitting on case-hardened gears of medium and small size] [Dissertation]. Technical University of Munich (TUM).

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Hergesell, M., Tobie, T., Höhn, B.-R., & Stahl, K. (2011). Überprüfung der Grübchen- und Zahnfußtragfähigkeit gerad- und schrägverzahnter, kleinmoduliger Zahnräder und Zusammenfassung von Empfehlungen zum Erreichen optimaler Tragfähigkeit für Zahnräder mit Modul ≤ 1mm [eng.: Verification of pitting and tooth root load carrying capacity of spur and helical gears with small module and summary of recommendations to achieve optimal load carrying capacities for gears with module ≤ 1mm], FVA-no. 410 II - issue 986 - final report, Frankfurt am Main, Germany. Hippenstiel, F., Johann, K.-P., & Caspari, R. (2009). Tailor made carburizing steels for use in power generation plants. Paper presented at the European Conference on Heat Treatment, Strasbourg, France. Börnecke, K., Rösch, H., Käser, W., Bagh, P., Rettig, H., Hösel, T., Weck, M., & Winter, H. (1976). Grundlagenversuche zur Ermittlung der richtigen Härtetiefe bei Wälz- und Biegebeanspruchungen [eng.: Basic experimental investigations to determine the right hardness depth for rolling and bending stresses], FVA-no. 8 - issue 36 - final report. Forschungsvereinigung Antriebstechnik e.V. (FVA). Tobie, T., Oster, P., & Höhn, B.-R. (2001). Härtetiefe-Großzahnräder - Einfluss der Einsatzhärtetiefe auf die Grübchen und Zahnfußtragfähigkeit großer Zahnräder [eng.: Influence of the case-hardening depth on pitting and tooth root load carrying capacity of big gears], FVA-no. 271 - issue 622 - final report. FVA (Forschungsvereinigung Antriebstechnik e.V.). Tobie, T. (2001). Zur Grübchen- und Zahnfußtragfähigkeit einsatzgehärteter Zahnräder [eng.: To the pitting and tooth root load carrying capacity of case-hardened gears] [Dissertation]. Technical University of Munich (TUM). Tobie, T. (2000). Profondita di cementazione e Capacita di Carico [eng.: Case hardening depth and load carrying capacity]. Organi di Trasmissione, 82–92. Tobie, T., Oster, P., & Höhn, B.-R. (2002). Case depth and load capacity of case-carburized gears. Gear Technology March/April 2002. Tobie, T., Oster, P., & Höhn, B.-R. (2005). Systematic Investigations on the Influence of Case Depth on the Pitting and Bending Strength of Case Carburized Gears. Gear Technology July/August 2005, 40–48. Funatani, K. (1970). Einfluß von Einsatzhärtungstiefe und Kernhärte auf die Biegedauerfestigkeit von aufgekohlten Zahnrädern [eng.: Influence of case hardening depth and core hardness on bending en­ durance limit of carburized gears]. Härterei-Technische Mitteilungen, 25(2), 92–98. Mallener, H., & Schulz, M. Wärmebehandlung von Zahnrädern [eng.: Heat treatment of gears]. Berichtsband zur AWT-Tagung “Randschichtermüdung im Wälzkontakt,” pp. 93–110 (Suhl, 1992). Yoshida, A., Fujita, K., Kanehara, T., & Ota, K. (1986) Effect of case depth on fatigue strength of casehardened gears. Bulletin of JSME, 29(247). Pedersen, R., & Rice, L. (1961). Case crushing of carburized and hardened gears. Transactions SAE, 370–380. Winkler, K. J., Tobie, T., Stahl, K., Güntner, C., & Schurer, S. (2019). Material properties and tooth root bending strength of shot blasted, case carburized gears with alternative microstructures. AGMA Fall Technical Meeting 2019, Alexandria. Previously presented at the 2019 American Gear Manufactures Association’s Fall Technical Meeting as 19FTM16. Printed here with permission of the copyright holder, the American Gear Manufacturers Association, 1001 N. Fairfax Street, 5th Floor, Alexandria, Virginia 22314. Winter, H. (1981). 20 Jahre Antriebstechnik [eng.: 20 years of drive train technology] - special issue, 11–16. Schurer, S., Güntner, C., Tobie, T., Saddei, P., & Steinbacher, M. (2017). Alternative mehrphasige Randschichtgefüge beim Einsatzhärten zur Steigerung der Festigkeitseigenschaften von verzahnten Getriebebauteilen [eng.: Alternative multiphase case layer structures during case hardening to increase the strength properties of gear components], FVA-no. 513 III - issue 1248 - final report. Forschungsvereinigung Antriebstechnik e.V. (FVA). Lombardo, S., & Steinbacher, M. (2011). Carbonitrieren von verzahnten Getriebebauteilen [eng.: Carbonitriding of gear components], FVA-no. 513 I - issue 970 - final report. Forschungsvereinigung Antriebstechnik e.V. (FVA). Kratzer, D., Dobler, F., Tobie, T., Hoja, S., Steinbacher, M., & Stahl, K. (2019). Effects of low-temperature treatments on surface hardness, retained austenite content, residual stress condition and the resulting tooth root bending strength of case-hardened 18CrNiMo7-6 gears. Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science 64. Krause C., Biasutti, F., & Davis, M. (2011). Induction hardening of gears with superior quality and flexibility using Simultaneous Dual Frequency (SDF), Alexandria: AGMA Technical Paper. Cavallaro, G. P. (1995). Fatigue porperties of carburised gear steels [Dissertation]. University of South Australia.

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Dudley’s Handbook of Practical Gear Design and Manufacture Leube, H., Volger, J., & Weck, M. (1987). Einfluß des Randkohlenstoffgehaltes auf die Tragfähigkeit einsatzgehärteter Zylinderräder [eng.: Surface carbon content - Influence of the surface carbon content on the load carrying capacity of case-hardened cylindrical gears], FVA-no. 76 - issue 263 - final report part 1. Forschungsvereinigung Antriebstechnik e.V. (FVA). Wagar, R., Speer, G., Matlock, D., & Mendez, P. (2005). Examination of pitting fatigue in carburized steels with controlled retained austenite fractions. SAE Technical Papers. Koller, P., Tobie, T., Stahl, K., & Höhn, B.-R. (2012). Zahnflankentragfähigkeit unter besonderer Berücksichtigung einer zusätzlichen Oberflächenbearbeitung [eng.: Tooth flank load carrying capacity with special consideration of an additional surface treatment], FVA-no. 453 II - issue 1034 - final report. Forschungsvereinigung Antriebstechnik e.V. (FVA). SAE AMS 2432D. (2013). Shot peening, computer monitored. SAE International. SAE J 2441. (2015). Shot peening. SAE International. SAE AMS 2430U. (2018). Shot peening. SAE International. Wohlfahrt, H. (1984). The influence of peening conditions on the resulting distribution of residual stress. Conf Proc: ICSP-2. Chicago, IL, USA. Steutzger, M., Oster, P., & Höhn, B.-R. (1997). Größeneinfluß auf die Zahnfußtragfähigkeit [eng.: Size influence on the tooth root load carrying capacity], FVA-no. 162 - issue 529 - final report. Forschungsvereinigung Antriebstechnik e.V. (FVA). Schwienbacher, S., Wolter, B., Tobie, T., Höhn, B.-R., & Kröning, M. (2007). Ermittlung und Charakterisierung von Randzonen-Kennwerten und -eigenschaften und deren Einfluss auf die Flankentragfähigkeit einsatzgehärteter, geschliffener Zahnräder [eng.: Determination and characterization of edge zone parameters and properties and their influence on the flank load capacity of case-hardened, ground gears], FVA-no. 453 - issue 830 - final report. Forschungsvereinigung Antriebstechnik e.V. (FVA). Bretl, N. T. (2010). Einflüsse auf die Zahnfußtragfähigkeit einsatzgehärteter Zahnräder im Bereich hoher Lastspielzahlen [eng.: Influences on the tooth root load carrying capacity of case-hardened gears in the range of high load cycles], Munich, Germany [Dissertation]. Technical University of Munich (TUM). Bretl, N. T., Schurer, S., Tobie, T., Stahl, K., & Höhn, B.-R. (2013). Investigations on tooth root bending strength of case hardened gears in the range of high cycle fatigue. AGMA Technical Paper 13FTM09. Schurer, S., Tobie, T., & Stahl, K. (2015). Tragfähigkeitsgewinn im Zahnfuß durch hochreine Stähle [eng.: Load carrying capacity gain in the tooth root due to high-strength steels], FVA-no. 293 III - issue 1148 final report. Forschungsvereinigung Antriebstechnik e.V. (FVA). Weigand, U. (1999). Werkstoff- und Wärmebehandlungseinflüsse auf die Zahnfußtragfähigkeit [eng.: Material and heat treatment influences on the tooth root load carrying capacity], Munich, Germany [Dissertation]. Technical University of Munich (TUM). Inoue, K., Maehara, T., Yamanaka, M., & Kato, M. (1989). The effect of shot peening on the bending strength of carburized gear teeth. JSME International Journal. Ser. 3, Vibration, Control Engineering, Engineering For Industry, 32(3), 448–454. Fuchs, D., Schurer, S., Tobie, T., & Stahl, K. (2019). Investigations into non-metallic inclusion crack area characteristics relevant for tooth root fracture damages of case carburised and shot-peened high strength gears of different sizes made of high-quality steels. Forschung im Ingenieurwesen. Issue: 03/2019. Schurer, S. (2016). Einfluss nichtmetallischer Einschlüsse in hochreinen Werkstoffen auf die Zahnfußtragfähigkeit [eng.: Influence of non-metallic inclusions in high-strength materials on the tooth root load carrying capacity], [Dissertation]. Technical University of Munich (TUM). Fischer, U. (2002). Tabellenbuch Metall [eng.: Mechanical and metal trades handbook]. Verl. EuropaLehrmittel Nourney Vollmer, Haan-Gruiten, 42. Aufl., [Reprint]. Fuchs, D., Schurer, S., Tobie, T., & Stahl, K. (2019). A model approach for considering nonmetallic inclusions in the calculation of the local tooth root load-carrying capacity of high-strength gears made of high-quality steels. Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science661, 7309–7317. Prasannavenkatesan, R., Zhang, J., McDowell, D. L., Olson, G. B., & Jou, H.-J. (2009). 3D modeling of subsurface fatigue crack nucleation potency of primary inclusions in heat treated and shot peened mar­ tensitic gear steels. International Journal of Fatigue, 31(7), 1176–1189. Nishijima, S., & Kanazawa, K. (1999). Stepwise S-N curve and fisheye failure in gigacycle fatigue. Fatigue & Fracture of Engineering Materials & Structures, 22(7), 601–607. https://doi.org/10.1046/ j.1460-2695.1999.00206.x.

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Köcher, J., & Weck, M. (1995). Einfluss von Kugelstrahlen auf die Zahnflankentragfähigkeit einsatzgehärteter Zylinderräder [eng.: Influence of shot peening on the tooth flank load carrying capacity of case-hardened spur gears], FVA-no. 185 I - issue 449 - final report. Forschungsvereinigung Antriebstechnik e.V. (FVA). Townsend, D. P. (1993). Improvement in surface fatigue life of hardened gears by high-intensity shot peening. International Journal of Fatigue, 15(5), 439. König, J., Koller, P., Tobie, T., & Stahl, K. (2015). Correlation of relevant case properties and the flank load carrying capacity of case-hardened gears. ASME (Hg.) 2015 – ASME. International Design engineering. Koller, P., Schwienbacher, S., Tobie, T., Oster, P., & Höhn, B.-R. (2009). Indicating grinding burn on gears - comparison and evaluation of different testing methods. 7th International Conference on Barkhausen Noise and Micromagnetic Testing, pp. 43–52. Aachen, Germany. Koller, P., Schwienbacher, S., Tobie, T., Oster, P., Stahl, & K., Höhn, B.-R. (2013). Grinding burn on gears: Correlation between flank-load-carrying capacity and material characteristics. Dobre, G. Power Transmissions, Dordrecht. Kadach, D., Matt, P., Tobie, T., & Stahl, K. Influences of the facing edge condition on the flank load carrying capacity of helical gears. ASME 2015 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, Boston, Massachusetts, USA (08022015). Matt, P., Tobie, T., & Höhn, B.-R. (2012). Einfluss der Stirnkante auf die Tragfähigkeit von Zahnrädern unter Berücksichtigung des Schrägungswinkels [eng.: Influence of the end edge on the load capacity of gears under consideration of the helix angle], FVA-no. 284 IV - issue 1023 -final report. Forschungsvereinigung Antriebstechnik e.V. (FVA).

12

Gear Load Capacity Calculation Based on ISO 6336 Daniel Müller, Nadine Sagraloff, Stefan Sendlbeck, Karl Jakob Winkler, Thomas Tobie, and Karsten Stahl

12.1 INTRODUCTION AND HISTORY OF ISO 6336 The following subsections give information on the introduction and history of the ISO 6336 series [1]. The series of ISO 6336 documents describes the calculation of the load capacity of spur and helical gears. The different documents of the calculation method are outlined in the following subsections beginning with an introduction, continuing with the history and closing with an overview on the structure of ISO 6336 documents. The information provided in the following subsections is mainly based on the ISO 6336 series itself as well as standards referenced or related to the ISO 6336 series. The information of the ISO 6336 series is summarized for the reader as an easy and comprehensible approach to the calculation methods. Additional and supporting information has been added to ensure a good understanding for a reader with basic gear knowledge. The application of the calculation methods stated in the standards is for experts only and should not be used by individuals without appropriate knowledge and background. It has to be noted, that this chapter gives an introduction on the basic principles, influence factors and updates of the calculation method based on the ISO 6336 series. A complete and comprehensive cal­ culation of the load capacity based on this chapter cannot be achieved without the respective standards. The respective standards have to be obtained and applied for a full perception and valid calculation of the load capacity. Possible liabilities concerning this chapter and the described calculation method are hereby excluded.

12.1.1 INTRODUCTION: PARTS

AND

DOCUMENT TYPES

ISO 6336 [1] consists of different document types, which provide the user with calculation methods and examples to evaluate the load capacity of gears. ISO 6336 consists of the three following types of documents. • International Standards (ISO) contain calculation methods that are based on widely accepted practices and have been validated. International standards are systematically revised every five years. The result of a revision can be a withdrawal, a confirmation without changes or a revision with the publication of the standard as a new edition. • Technical Specifications (ISO/TS) contain calculation methods that are still subject to further development. Technical specifications are systematically revised every three years and can be confirmed as technical specification for a maximum of six years. If not before, it has to be decided

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after six years whether the technical specification is withdrawn, transformed to a standard without changes or transformed to a standard with revisions. • Technical Reports (ISO/TR) contain data that is informative, such as example calculations. Technical reports have no systematic reviews and are valid until otherwise decided. Each part of the ISO 6336 series is one of the three document types. The parts of ISO 6336 correspond to different topics, which are listed in the following: • Parts 1 to 19 of the ISO 6336 series cover calculation methods on the fatigue analyzes for gear rating. • Parts 20 to 29 of the ISO 6336 series cover calculation methods that are predominantly related to the tribological behavior of the lubricated flank surface contact. • Parts 30 to 39 of the ISO 6336 series include example calculations. Only a part of the numbers for are already reserved for existing documents, further numbers are still vacant. The vacant numbers of the ISO 6336 series allow the addition of new parts under appropriate numbers to reflect knowledge gained in the future. Table 12.1 gives an overview on the existing parts and document types of ISO 6336.

TABLE 12.1 Parts and Document Types of ISO 6336 Calculation of Load Capacity of Spur and Helical Gears

International Standard

Part 1: Basic principles, introduction and general influence factors Part 2: Calculation of surface durability (pitting)

X

Part 3: Calculation of tooth bending strength

X

Technical Report

X

Part 4: Calculation of tooth flank fracture load capacity Part 5: Strength and quality of materials

X

Part 6: Calculation of service life under variable load

X

Part 20: Calculation of scuffing load capacity—Flash temperature method Part 21: Calculation of scuffing load capacity—Integral temperature method Part 22: Calculation of micropitting load capacity

Technical Specification

X

X X X

Part 30: Calculation examples for the application of ISO 6336 parts 1, 2, 3, and 5

X

Part 31: Calculation examples of micropitting load capacity

X

Gear Load Capacity Calculation (ISO 6336)

687

12.1.2 HISTORY The history of the ISO 6336 series [1] starts with the publication of the parts 1 (Basic Principles), 2 (Surface Durability), 3 (Bending Strength), and 5 (Materials) in 1996. These parts as well as the following parts 6 (Service Life), 20 (Scuffing Flash Temperature), and 21 (Scuffing Integral Temperature) were among others based on calculation methods of parts of the German standard series DIN 3990 [2] as well as American standards such as ANSI/AGMA 2001-D04 [3]. The first edition of the DIN 3990 series was published in 1970, the second edition in 1987. During those years, the calculation methods of the DIN 3990 series were applied in the field and verified by the industry. When creating the first edition of ISO 6336, the calculation methods of DIN 3990 were eventually being overtaken by ISO 6336 with minor adjustments but no fundamental changes. For the publication of the first edition, the reference to the DIN 3990 series gave ISO 6336 the advantage to possess a verified basis as well as years of experience for the published calculation methods. The following list shows the publications of ISO 6336 in chronological order. Published corrections, so called technical corrigenda or corrected versions, are not listed. • In 1996 the first edition of the parts 1 (Basic Principles), 2 (Surface Durability), 3 (Bending Strength), and 5 (Materials) was published. • In 2003 the second edition of part 5 (Materials) was published. • In 2006 the revised second edition of the parts 1 (Basic Principles), 2 (Surface Durability), 3 (Bending Strength), and the first edition of part 6 (Service Life) was published. • In 2016 the revised third edition of part 5 (Materials) was published. • In 2017 the first edition of the parts 20 (Scuffing Flash Temperature, formerly ISO/TR 13989-1), 21 (Scuffing Integral Temperature, formerly ISO/TR 13989-2) and 30 (Examples Parts 1, 2, 3, and 5) were published. • In 2018 the first edition of the parts 22 (Micropitting, formerly ISO/TR 15144-1) and 30 (Examples Micropitting) was published. • In 2019 the first edition of part 4 (Tooth Flank Fracture), the revised third edition of the parts 1 (Basic Principles), 2 (Surface Durability), 3 (Bending Strength), and the revised second edition of part 6 (Service Life) was published. The chronological order of the publications shows that the parts of the ISO 6336 series have been growing strongly in the recent years. If mistakes are found long before the next revision, there is the possibility of publishing a corrected version or a technical corrigendum for the concerned part. A cor­ rected version is the publication of the entire document including the corrections whereas a technical corrigendum is a separate additional document, which contains only the corrections. If the mistakes are of editorial nature, a corrected version is preferred whereas a technical corrigendum is more likely when the mistakes are of technical nature. The user of standards should always make sure that the current version including possible corrections is used. The current version of a standard can be referenced with or without the year of publication, for instance 6336-1:2019 or 6336-1. If a certain edition from a certain year is referenced, the year must be included in the reference, for instance 6336-1:1996.

12.1.3 OVERVIEW

AND

STRUCTURE

OF

ISO 6336 DOCUMENTS

Each document of ISO 6336 [1] follows a basic structure. The major parts of the structure and their information are explained in the following. • The foreword provides general information on the International Organization for Standardization (ISO) as well as main changes of the content to the previous edition. The latter can be of interest, when comparing different editions of ISO 6336 documents and trying to find sources for deviating calculation results.

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• The introduction lists the different types of documents and parts of ISO 6336. The content of the introduction is comparable to Section 12.1.1. • The scope defines the range in which the calculation methods can be applied. It is important to know and to comply with the boundaries of application for the calculation methods. Unreasonable and incorrect calculation results can be caused by not complying with the boundaries defined in the scope. • The normative references provide information on other standards that are referenced in the document and thus give a good overview on connected and supporting standards. When looking up those standards, it is important to pay attention to the year of publication. If no year is given, the most recent publication of the referenced standard is meant. • The symbols and abbreviations are sorted in tables with descriptions of all symbols and abbreviations used in the document. The table can be helpful when comparing and understanding the numerous equations. • The annex is at the penultimate section of the document, not part of the official document and merely provides additional information for a better and further comprehension. • The bibliography is the last section of the document and provides publications with further in­ formation as well as sources to certain equations and number values of the calculation method. The main parts of the documents of ISO 6336 are different. Most sections begin with general in­ formation, influence factors and limits before going into more detailed explanations, the calculation methods and the formulae. There can exist up to three different methods to perform a calculation of a needed value. The different methods differ in their accuracy and thus reliability. Regarding accuracy and reliability, Method A is superior to Method B and Method B is superior to Method C. Depending on the calculation, not all methods have to be defined, often Method C is left out due to a lacking accuracy. The different methods originate from the times when computers were not easily accessible and rough but easy to calculate estimates were necessary for engineers. In the following, the different methods are specified in more detail. • Method A relies on full scale load tests, precise measurements, comprehensive mathematical analysis, simulations and further state of the art tools to calculate and derive the results. All gear and loading data has to be available. • Method B relies on mostly analytical formulae with sufficient accuracy for most applications. Assumptions involved in the method are listed and for each case, it is necessary to assess whether these assumptions apply to the existing conditions. • Method C relies on analytical formulae with simplified and approximated values, often taken from tables and graphs instead of being calculated. As for Method B, assumptions involved in the method are listed and for each case, it is necessary to assess whether these assumptions apply to the existing conditions.

12.2 CALCULATION OF SURFACE DURABILITY—ISO 6336-2:2019 The following subsections give information on the recently reviewed and published ISO 6336-2:2019 [4]. ISO 6336-2 describes the calculation of surface durability against pitting. The standard is presented in the following subsections by a description of pittings, basic calculation principles, new aspects and updates of the standard, a calculation example, a summary and an outlook on the future of ISO 6336-2:2019.

Gear Load Capacity Calculation (ISO 6336)

12.2.1 DESCRIPTION

OF THE

689

FAILURE MODE PITTING

Pittings can be described as shell shaped material breakouts from the loaded flank surface. Pittings are caused by fatigue failure due to the rolling-sliding contact. The fatigue cracks initiate either at the surface of the gear tooth or at a shallow depth below the surface. The orientation of the cracks initiated at the surface is against the direction of the sliding force. The crack usually propagates for a short distance in a direction roughly parallel to the tooth surface before turning or branching to the surface. When the cracks have grown long enough to separate a piece of the surface material, a pit is formed. The edges of a pit are usually sharp and angular. Figure 12.1 shows characteristics of pittings regarding the exemplary shape as well as crack initiation and orientation [5].

FIGURE 12.1

Characteristics of pittings—exemplary shape, crack initiation, and orientation.

Pittings can be categorized into the following two groups. • Nonprogressive pittings normally consist of small, shallow pits that occur in localized areas. They tend to redistribute the load by removing high asperities. When the load is more evenly distributed, the pitting stops. This is most often observed on gears that are not surface or case hardened. • Progressive pittings normally consist of pits that grow at an increasing rate until a significant portion of the tooth surface has pits of various shapes and sizes. The shape of pittings is to a certain degree dependent on the heat treatment of the gear tooth flank. Case carburized gears usually show singular pittings on a singular tooth flank in the initial stage. If the tooth flank with the pitting is continuously loaded, the singular pitting can lead to excessive local stresses and progressive pitting growth. Existing cracks propagate and new cracks initiate. Due to these additional cracks, secondary breakouts can occur and under continued load, the tooth flank will be eventually covered by a large pitting area. Additionally, pitting damages on the teeth may occur due to increased dynamical effects. Figure 12.2 shows different stages of pitting damages on a case carburized test gear [5]. Pittings on through hardened gears appear usually in large numbers on multiple or all tooth flanks. The shapes are similar to common pittings and they appear in different sizes. Nitrided gears show rather shallow pittings with a shape similar to a teardrop. It seems that when pitting occurs at nitrided gears, a delamination of the surface hardened layer, also called white layer flaking, is taking place before the pitting damage. Figure 12.3 shows exemplary pittings on a through hardened gear on the left side and a nitrided gear on the right side.

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FIGURE 12.2

Different stages of pitting damages on a case carburized test gear.

FIGURE 12.3

Exemplary pittings on a through hardened gear (left) and a nitrided gear (right).

Pittings can lead to increasing noise and vibrations. Progressive development of pittings can lead to high quantity breakouts, destruction of the tooth flank, and eventually tooth breakages. For applications with high impact failures, for instance danger to human lives, pittings are intolerable. To avoid failure by pittings, ISO 6336-2:2019 [4] can be used to calculate the surface durability of spur and helical gears.

12.2.2 BASIC CALCULATION PRINCIPLES The following subsections on the basic calculation principles of ISO 6336-2:2019 [4] give information on strength analysis and safety factor, contact stress and pitting stress limit. 12.2.2.1 Strength Analysis and Safety Factor The basic strength analysis with regard to pitting damage according to ISO 6336-2:2019 [4] is based on the comparison of the occurring contact stress and the permissible component strength (pitting stress limit) of the gear. The contact stress depends on loading, external influences (i.e., the application), influences from the Hertzian contact theory and material elasticity. Furthermore, it takes into account characteristics of the gear shape with regard to contact overlap and helical gearing effects on the load distribution. The pitting stress limit contains material and surface properties, lubrication and limited life influences. The ratio of pitting stress limit and contact stress leads to the safety factor for surface durability (see (12.1)). SH1,2 =

Pitting stress limit = Contact stress

HG1,2 H1,2

> SH min

(12.1)

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TABLE 12.2 Recommendations for Safety Factor SH Normal cases (industrial turbo gear units, plant gear units)

SH = 1.0

Maximum torque against endurance strength

SH = 0.5

1.2

0.7

High reliability and critical cases (ship and aircraft transmissions)

SH = 1.2

1.6

where: SH HG

H

SH min 1, 2

– – – – –

is is is is is

the the the the the

safety factor for surface durability. pitting stress limit, in N/mm2. contact stress, in N/mm2. minimum required safety factor for surface durability. index for pinion (1) or wheel (2).

The safety factor should be higher than the minimum required safety factor for the pinion and the wheel. ISO 6336-2:2019 [4] does not give explicit proposals regarding the minimum safety factor. However, recommendations for typical safety factors in practical applications are to be found in the literature, i.e., [6]. Recommendations are listed in Table 12.2. The strength analysis of gears with regard to surface durability is mainly based on the Hertzian contact theory. Other influences, e.g., properties according to the elastohydrodynamic lubrication theory, contact sliding properties, or frictional specific behavior are not directly part of the standard calculation, but are covered indirectly within the experimentally verified gear strength numbers. Various influence factors and material properties were developed and validated on numerous gear tests, which still allow the standard to cover a large part of influences and provide a well-founded strength analysis. 12.2.2.2 Contact Stress σH For strength analysis with regard to surface durability (pitting damage), the determinant stress is the occurring contact stress H . It is calculated based on the nominal contact stress at the pitch point H 0 , which in turn is determined on the basis of the nominal tangential load Ft . However, depending on the geometry of the spur or helical gears, the contact stress at the pitch point can differ from the determinant contact stress for the standard strength analysis. Therefore, various factors are part of the standard calculation to account for these and further influences. Exemplary influences are the type of material or the type of application and will be explained in the following. The resulting contact stress H is usually different for pinion and wheel. Hence, a separate strength analysis for both gears is necessary. As displayed in (12.2)–(12.4), the calculation of the contact stress H includes a contact factor specific to pinion or wheel ZB, D , the nominal contact stress at the pitch point H 0 , and so-called load influence factors KH 0 . ISO 6336-1:2019 [7] specifies these load influence factors, which, for example, account for external load influences, dynamic effects and influences of load distribution. H1,2

H0

= ZB, D

= ZH ZE Z Z

H0

KH 0 Ft u + 1 d1 b u

(12.2)

(12.3)

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KH 0 = KA K Kv KH KH

(12.4)

where: – ZB, D – H0 – KH 0 – 1, 2 – ZH – ZE – Z – Z –

is the contact stress, in N/mm2. is the contact factor of the pinion (B) or wheel (D). is the nominal contact stress at the pitch point, in N/mm2. are load influence factors. is the index for pinion (1) or wheel (2). is the zone factor. is the elasticity factor. is the contact ratio factor. is the helix angle factor.

Ft d1 b u KA K

is is is is is is

H

– – – – – –

the the the the the the

nominal tangential load, in N. reference diameter of the pinion, in mm. face width of the gear pairing, in mm. transmission ratio. application factor. mesh load factor.

Kv – is the dynamic factor. KH – is the face load factor KH – is the transverse load factor.

Table 12.3 shows a summary of the factors with specific influence on the contact stress H . Contact factor ZB, D . Depending on the geometry of the gears, it can be necessary to consider the stress not at the pitch point, but at another location at the tooth flank. The influence of the tooth flank curvatures on the contact stress is considered with the contact factor ZB, D in ISO 6336-2:2019 [4]. For example, for spur gears the contact stress is transformed to the corresponding contact stress at the inner point of the single pair tooth contact (B for pinion, D for wheel). Figure 12.4 shows the radii of curvature for special points in the line of action while referring to the driving gear 1 [8]. For helical gears, the relevant contact point is determined by interpolation, if the pitch point is not determinant. The basis for the calculation is the ratio of radii of curvature according to Eq. (12.5).

TABLE 12.3 Summary of Factors of Influence on the Contact Stress

H

Contact factor ZB, D

The contact factor accounts for the transformation of the contact stress at the pitch point to the determinant contact stress. In case of a spur gear, the stress is determined at the inner point of the single pair tooth contact (B for pinion, D for wheel). For helical gears, the relevant contact point is the basis for transformation from the pitch point.

Zone factor ZH

The zone factor considers the Hertzian contact theory and the tooth flank curvature. It transforms the load tangential to the reference cylinder into a normal load at the pitch cylinder.

Elasticity factor ZE

The elasticity factor accounts for the elasticity of the material by considering the modulus of elasticity and Poisson’s ratio. The contact ratio factor accounts for influences of the transverse contact ratio and the overlap ratio.

Contact ratio factor Z Helix angle factor Z

The helix angle factor takes into account the influence of helical gears, i.e., their load distribution with regard to the lines of contact.

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FIGURE 12.4 Radii of curvature for special points in the line of action, referring to driving gear 1: (a) start of active profile, contact point A, (b) inner point of single tooth contact, contact point B, (c) pitch point, contact point C, (d) outer point of single tooth contact, contact point D, (e) end of active profile, contact point E [ 8].

M1 =

C1

C2

B1

B2

;

M2 =

C1

C2

D1

D2

(12.5)

where: M1,2 – is the contact factor auxiliary quantity. – is the radius of curvature, in mm.

Depending on the auxiliary quantity M1,2 , the contact factor is determined differently for spur and helical gears: For spur gears with a transverse contact ratio of > 1 the inner point of the single pair tooth contact (B for pinion, D for wheel) is considered according to Eqs. (12.6) and (12.7): ZB = 1, if M1

1;

ZB = M1, if M1 > 1;

ZD = 1, if M2

1

ZD = M2, if M2 > 1

For helical gears with a transverse contact ratio of applies: ZB = ZD =

> 1 and an overlap ratio of

fZCa

(12.6) (12.7) 1, Eq. (12.8)

(12.8)

With the auxiliary factor fZCa flank modifications and their design methods are considered. The auxiliary factor varies from fZCa = 1.0 for flank modifications based on advanced 3D numerical computations to fZCa = 1.2 for no flank modifications applied (see Section 12.2.3.1. for more details).

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Equations (12.9) and (12.10) apply to helical gears with a transverse contact ratio of overlap ratio of < 1: ZB = 1 + ZB = M1 +

( fZCa ( fZCa

1), if M1

1;

ZD = 1 +

M1), if M1 > 1;

( fZCa

ZD = M2 +

( fZCa

1), if M2

> 1 and an

1

M2 ) , if M2 > 1

(12.9) (12.10)

where: ZB, D – is the contact factor of the pinion or wheel. fZCa – is the auxiliary factor for flank modifications. – is the contact ratio. B , D – is the index for pinion (B) or wheel (D).

The factor fZCa was newly introduced in ISO 6336-2:2019 and is not included in previous editions (see Section 12.2.3.1). It allows a more reliable consideration of the occurring stresses for helical gears with and without profile modifications and is based on extensive theoretical and experimental research work [9,10]. Zone factor ZH . Due to the loaded contact of two curved surfaces at the point of contact of the gears, the Hertzian contact theory applies and is used in the calculation according to ISO 6336-2:2019 [4]. The zone factor ZH considers the flank curvature influences on the Hertzian stress at the pitch (see Eq. (12.11)). This includes the transformation of the load tangential to the reference cylinder into a normal load at the pitch point. ZH =

2 cos

b

cos

cos2

t

sin

wt wt

(12.11)

where: ZH – is the zone factor. b – is the base helix angle, in °. t

– is the transverse pressure angle, in °. is the working transverse pressure angle at the pitch cylinder, in °.

wt –

Elasticity factor ZE . To account for the influences of material properties on the contact stress, the elasticity factor ZE considers the modulus of elasticity and Poisson’s ratio. It is calculated according to Eq. (12.12): ZE =

where: ZE – – E – 1, 2–

is is is is

the elasticity factor. Poisson’s ratio. the modulus of elasticity, in N/mm2. the index for pinion (1) or wheel (2).

(

1 2 1

1 E1

+

2 2

1 E2

)

(12.12)

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Contact ratio factor Z . Depending on the geometry of the mating gears, the transverse contact ratio and the overlap ratio can vary. This influences the strength analysis and is accounted for with the contact ratio factor Z (see Eqs. (12.13) and (12.14)). Z =

4

1

3

1

ZE =

+

,

,

SH min

(12.19)

where the symbols are described in (12.1). The pitting stress limit can be calculated according to Eq. (12.20). HG1,2

= ZNT ZL Z v ZR ZW ZX

(12.20)

H lim

where the symbols are described in Eqs. (12.16) and (12.17). The contact stress can be calculated according to Eq. (12.21). H1,2

=

KA K Kv KH KH ZB, D ZH ZE Z Z

Ft u + 1 d1 b u

(12.21)

where the symbols are described in Eqs. (12.2) to (12.4). The various Z- and K-factors have to be calculated in order to determine the contact stress and the pitting stress limit. Detailed information on the factors and their calculation can be found in Section 12.2.2. Table 12.7 shows the basic output data including the values for the Z- and K-factors, contact stress, pitting stress limit and the safety factor against pitting. The calculated safety factors for surface durability should be superior to the minimum safety factor, SH > SH min. Depending on the application, the experience and the consequences of a gear failure, lower or higher minimum safety factors can be chosen. It is recommended that the manufacturer and the customer agree on the values of the minimum safety factor for surface durability SH min. [12]

12.2.5 SUMMARY ISO 6336-2:2019 [4] is a standard that describes the calculation of surface durability against pitting. The following list shows the history of the standard. • In 1996 the first edition of ISO 6336-2 was published. • In 2006 the second edition of ISO 6336-2 was published. • In 2019 the third and currently valid edition of ISO 6336-2 was published. The surface durability is calculated against the failure mode pitting. Pittings can be characterized by the following properties. • Pittings can be described as shell shaped material breakouts from the loaded flank surface. • Pittings are caused by fatigue failure due to the rolling-sliding contact. • Pittings can lead to increasing noise and vibrations as well as destruction of the tooth flank with consequential tooth breakages. The basic calculation principles for the surface durability against pitting are according to the fol­ lowing bullet points.

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TABLE 12.7 Basic Output Data (Note: Values are calculated according to ISO 6336-2:2019 and may differ from ISO/TR 6336-30:2017.) Description

Unit

Symbol

Pinion 0.91

Wheel

Life factor

–

ZNT

Lubricant factor Velocity factor

– –

ZL Zv

1.05 0.97

0.96

Roughness factor

–

ZR

0.97

Work hardening factor Size factor

– –

ZW ZX

1.00 1.00

Allowable stress number (surface)

N/mm2

Mesh load factor Velocity factor

– –

Face load factor Transverse load factor Single pair tooth contact factor

σH

lim

1500

1500

Kγ Kv

1.00 1.00

–

KHβ

1.16

– –

KHα ZB,D

Zone factor

–

ZH

2.40

Elasticity factor Contact ratio factor

– –

ZE Zε

189.91 0.80

Helix angle factor Nominal tangential load Reference diameter Gear ratio Pitting stress limit Nominal contact stress Safety factor for surface durability

1.00 1.00

1.00

–

Zβ

1.02

N mm

Ft d

127352

–

u

N/mm2 N/mm2

σHG σH

1338 1301

1415 1301

–

SH

1.03

1.09

141.34

856.35 6.06

• The safety factor for surface durability against pitting is the ratio of the pitting stress limit and contact stress. • The pitting stress limit represents the load capacity depending on various factors such as the material, lubricant, velocity, roughness, and more (see Section 12.2.2.3). • The contact stress represents the load depending on various factors such as material, macro and micro-geometry of the gear, load influence factors and more (see Section 12.2.2.2). With the recent publication of the third edition of ISO 6336-2 in 2019, the following key aspects have changed with respect to the previous version. • The factor fZCa , which accounts for flank modifications and their design method, was introduced for the calculation of the contact factor ZB, D for pinion or wheel. • The work hardening factor ZW was extended to the combination of surface-hardened steel pinions with ductile iron gears. ISO 6336-2:2019 has been recently published and will be reviewed systematically after five years. The current standard and its calculation methods are applied by the industry. New topics and results of

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recent research projects are prepared, reviewed and discussed to be possibly added for the next edition. In the meantime, ISO 6336-2:2019 is the state-of-the-art standard to calculate the surface durability against pitting for spur and helical gears.

12.2.6 OUTLOOK The outlook for ISO 6336-2:2019 [4] describes how the standard for the calculation of the surface durability against pitting could be developed in the future. The next systematic revision of ISO 6336-2 will be five years from 2019, in the year 2024. During this time, the standard will be applied in the industry and the calculation methods of the standard will be further verified and validated in practice. The recent changes in the last edition of the standard will be used to calculate industrial applications. If necessary, the calculation methods shall be updated to include further experiences and observations in practice. Changes have to be thoroughly investigated and statistically validated in advance. Besides the revision of the existing calculation methods, new approaches and state of the art knowledge can be added for a further developed standard. New manufacturing processes, new materials and new calculation possibilities can lead to the necessity of new or advanced calculation methods. The following list gives exemplary topics that are possibly relevant in the future. The topics are all pre­ sumptions and do not claim to be complete or in any form binding. • New manufacturing processes could be additively manufactured gears. The research and experience regarding additively manufactured gears is still very young. New designs, which are not possible with conventional gear manufacturing processes, can be achieved by additive manufacturing. Cooling pipes within the gear teeth or light weight structures in the gear hub are possible applications. The existing calculation methods of the standards will have to be reviewed and verified for these new manu­ facturing processes. • New materials for gears could be the usage of plastic. Up to now, there is only limited knowledge and experience regarding the calculation of the load capacity of plastic gears, with limited national guidelines such as VDI 2736 [15]. The calculation methods for plastic gears should be further developed and could be included as extension in an existing standard or as a separate standard for plastic gears. • New calculation possibilities are achieved by advancements in computer technology as well as software applications. Detailed tooth contact analysis is possible for quite some time now and are used more and more often by the industry. Currently, comprehensive mathematical analysis and simulations are covered by Method A in ISO 6336 [1] with no further guidelines or support on how to achieve such analysis. The existing standard could be extended to further take into account such new calculation possibilities and give additional guidance to users. The above-mentioned topics for the further development of ISO 6336-2 are current assumptions and further topics will likely be included until the next revision. It can be assumed with a high degree of certainty, that the current version of ISO 6336-2 will be further developed and adapted to the current state of the art concerning the calculation of the surface durability against pitting.

12.3 CALCULATION OF TOOTH BENDING STRENGTH – ISO 6336-3:2019 The following subsections give information on the recently reviewed and published ISO 6336-3:2019 [16]. ISO 6336-3 describes the calculation of bending strength against tooth root breakage. The standard

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is presented in the following subsections by a description of tooth root breakages, basic calculation principles, new aspects and updates of the standard, a calculation example, a summary and an outlook on the future of ISO 6336-3:2019.

12.3.1 DESCRIPTION

OF THE

FAILURE MODE TOOTH ROOT BREAKAGE

Tooth root breakage can be described as the fracture of a large portion of a tooth from the main body of the gear. They are characterized by bending fatigue cracks initiated in the tooth root fillet area. Tooth root breakage involves the propagation of a crack during a large number of load cycles. If the load application is sufficient, the crack will propagate until the tooth finally breaks off the gear [5]. Figure 12.11 shows an exemplary tooth root breakage.

FIGURE 12.11

Exemplary tooth root breakage.

The fracture surface typically consists of two different zones, the fatigue and the final fracture surface. • The fatigue fracture surface is free from any visible sign of plastic deformation. The surface is smooth, matt in appearance, possibly traversed by arrest lines, and may exhibit steps between successive crack-propagation stages. • The final fracture surface shows strong plastic deformation. The surface shows macro- and microasperities, is rough and is clearly differentiated to the fatigue zone. The final fracture can be described as an overload fracture. With rising load, the fatigue fracture surface will be reduced and the final fracture surface will grow. Figure 12.12 shows exemplary fracture surfaces for gear teeth broken under different loads. For the fracture surface of the higher loaded gear tooth, additional asperities at the side of the tooth root where the crack initiated can be observed. Although bending fatigue cracks can occur elsewhere, they are usually initiated on the tensile side of the tooth root fillet. The crack origin may occur at the surface but also below the surface of the tooth root fillet. Sub-surface initiated bending fatigue cracks in the tooth root fillet are often associated with a nonmetallic inclusion. In some cases, predominantly for gears with uneven load distributions along the face width, the fracture is limited to one end of the tooth. Tooth root fractures can occur on several teeth or a single tooth. Figure 12.13 shows exemplary tooth root breakages to one end of several teeth for a bevel gear as well as a single broken tooth for a spur gear [5]. A tooth root breakage will most likely lead to a total failure of the gearset. Sometimes, the destruction of all gears in a transmission can be the consequence of the breakage of one tooth. Due to the high possibility of collateral damage, gears are usually designed with higher safety factors against tooth root

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FIGURE 12.12 Exemplary tooth root breakages broken under a lower load (left) and a higher load (right), crack initiation from the upper edge of the surface.

FIGURE 12.13 Exemplary tooth root breakages to one end of several teeth of a bevel gear and a single broken tooth of a spur gear [ 5].

breakages than against pittings. To avoid failure by tooth root breakages, ISO 6336-3:2019 [16] can be used to calculate the bending strength of spur and helical gears.

12.3.2 BASIC CALCULATION PRINCIPLES 12.3.2.1 Strength Analysis and Safety Factor The basic strength analysis with regard to tooth root breakage (tooth bending strength) according to ISO 6336-3:2019 [16] is based on the comparison of the permissible component strength (bending stress limit) and the occurring stress (tooth root stress) of the gear. The tooth root stress contains geometry influences with regard to tooth shape, helical gearing or tooth height. Moreover, taking into account the complex stress situation in the tooth root and influences of hollow gears with rims are also part of the tooth root stress in ISO 6336-3:2019 [16]. In comparison, the bending stress limit depends on material and surface properties including notch effects, limited life influences, as well as size effects. The ratio of bending stress limit and tooth root stress leads to the safety factor for tooth root stress (see Eq. (12.22)):

SF1,2 =

Bending stress limit = Tooth root stress

FG1,2 F1,2

> SF min

(12.22)

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where: SF FG

F

SF min 1, 2

– – – – –

is is is is is

the the the the the

safety factor for tooth root stress. tooth root stress limit, in N/mm2. tooth root stress, in N/mm2. minimum required safety factor for tooth root stress. index for pinion (1) or wheel (2).

The safety factor should be higher than the minimum required safety factor for each, the pinion and the gear wheel. ISO 6336-3:2019 [16] does not give explicit proposals regarding the minimum safety factor. However, recommendations for typical safety factors in practical applications are to be found in the literature, i.e., [6]. Recommendation are listed in Table 12.8:

TABLE 12.8 Recommendations for Safety Factor SF Normal cases (industrial turbo gear units, plant gear units)

SF = 1.2

1.5

Maximum torque against endurance strength

SF = 0.7

1.0

High reliability and critical cases (ship and aircraft transmissions)

SF = 1.4

2.0

The strength analysis with regard to tooth root strength of gears is mainly based on the tensile stress at the tooth root. Especially in case of unidirectional loading, conventionally shaped gear teeth tend to start cracking at the tension fillet, which leads to tooth root breakage. 12.3.2.2 Tooth Root Stress σF For strength analysis with regard to tooth root breakage, the nominal stress is the basis for the occurring tooth root stress F . For ISO 6336-3:2019 [16], the maximum tensile stress at the surface of the tooth root at the critical section is taken into account. Therefore, the loading at the outer point of the single pair tooth contact is relevant. In case of helical gears, a recalculation to a virtual spur gear is the basis for evaluation. As displayed in Eqs. (12.23)–(12.25), the calculation of the occurring tooth root stress includes load influence factors (see Section 12.2.2.2) and the nominal tooth root stress. Table 12.9 shows a summary of the factors with specific influence on the tooth root stress F : F1,2

F0

=

F0

KF 0

= YF YS Y YB YDT

(12.23) Ft b mn

KF 0 = KA K Kv KF KF

where: – is the tooth root stress, in N/mm2. 2 F0 – is the nominal tooth root stress, in N/mm . KF0 – are load influence factors. F

(12.24) (12.25)

708 1, 2– YF – YS – Y – YB – YDT – Ft – b – mn – KA – K – Kv – KF – KF –

Dudley’s Handbook of Practical Gear Design and Manufacture is is is is

the the the the

index for pinion (1) or wheel (2). form factor. stress correction factor. helix angle factor.

is is is is is is is

the the the the the the the

rim thickness factor. deep tooth factor. nominal tangential load, in N. face width of the gear pairing, in mm. normal module, in mm. application factor. mesh load factor.

is the dynamic factor. is the face load factor. is the transverse load factor.

TABLE 12.9 Summary of Factors of Influence on the Tooth Root Stress

F

Form factor YF

The form factor accounts for shape influences of external and internal gears on the tooth root stress. The tooth thickness at the root, the load bending moment arm, and the curvature of the critical tooth root part are the main factors of influence.

Stress correction factor YS

The stress correction factor considers the transformation from the nominal tooth root stress (bending stress) to the local tooth root stress. This includes the stress-increasing effect of the notch-like tooth root curvature. The helix angle factor takes into account the influence of helical gears on the tooth root stress, i.e., their load distribution properties.

Helix angle factor Y Rim thickness factor YB Deep tooth factor YDT

The rim thickness factor accounts for the changed stress condition in case of thin-rimmed gears. The deep tooth factor takes into account the stress-decreasing effect of precise manufactured gears with high contact ratios due to their load distribution properties.

Form factor YF . Different tooth shapes of internal or external gears influence the nominal tooth root stress (bending stress). In ISO 6336-3:2019 [16], the determinant load for tooth root strength is the load at the outer point of the single pair tooth contact. The normal chord between the touching points of the root fillet and the 30° tangent for external gears respectively the 60° tangent for internal gears defines the critical section for evaluation. Figure 12.14 shows the normal chord of the tooth root at the critical section for an external gearing [6]. The choice of the 30° tangent and the 60° tangent for internal and external gears, respectively, follow the assumption that they localize the point of highest stresses in the tooth root in a good manner. Overall, together with the point of load application, the critical section specifies the main geometrical factors of influence for the strength analysis with regard to tooth root breakage. That is the tooth thickness at the critical section, the load bending moment arm, and the curvature of the root fillet at the critical section. The shape influence is described with the form factor YF (see Eq. (12.26)). For high contact ratios n 2 the recently included, additional factor f takes into account load distribution properties of multi teeth meshing (see Section 12.3.3.1).

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FIGURE 12.14 Normal chord of the tooth root at the critical section for an external gearing [ 6].

YF =

6 hFe mn

cos

( ) sFn mn

2

Fen

cos

f

(12.26)

n

where: YF – is the hFe – is the mn – is the Fen – is the sFn – is the n – is the f – is the

form factor. bending moment arm for tooth root stress, in mm. normal module, in mm. load direction angle, in °. tooth root chord at the critical section, in mm. normal pressure angle, in °. load distribution influence factor.

In case of helical gears, the geometry of an adapted and recalculated virtual spur gear is used to determine the form factor. Additional influences of helical gears consider the helix angle factor Y . The determination of the shape specific properties like the tooth root thickness, the bending moment arm, or the radius of curvature at the tooth root in ISO 6336-3:2019 [16] are based on the gear tool properties and process movement. Available tools for calculation are a gear hob for external gears and a shaper cutter for both, external and internal gears. Stress correction factor YS . ISO 6336-3:2019 [16] is based on the tooth root bending stress. However, the actual local tooth root stress at the determined location of the tooth root fillet might differ from the pure tensile stresses due to bending only. The local stress state is more complex and results from a multi-axis stress condition. Furthermore, the notch-like effect of the tooth root curvature influences the stress at the tooth root. ISO 6336-3:2019 [16] provides an empirical formula, namely the stress correction factor YS , to ac­ count for these phenomena (see Eq. (12.27)).

s YS = 1.2 + 0.13 Fn qs hFe

1 1.21+2.3

hFe sFn

;

qs =

sFn 2 F

(12.27)

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where: YS – sFn – hFe – F –

is is is is

the the the the

stress correction factor. tooth root chord at the critical section, in mm. bending moment arm for tooth root stress, in mm. tooth root radius at the critical section, in mm.

Helix angle factor Yβ

Depending on the design and manufacturing of a gear, a so-called grinding notch in the tooth root can remain on the finished gear. Because it influences the occurring stresses in the tooth root, there are calculation specifications in the standard to account for that. Helix angle factor Y . In case of a helical gear, the helix angle influences the toot root stress, i.e., due to the resulting load distribution properties. An exemplary correlation between the reference helix angle and the helix angle factor for a gear with an overlap ratio of = 1 is presented in Figure 12.15. Helix angles of > 20° lead to helix angle factors of Y > 1.0 , which eventually leads to high tooth root stresses. Values of the helix factor for further overlap ratios and corresponding equations can be found in ISO 6336-3:2019 [16] or calculated by Eq. (12.36) and approximately range between Y = 0.95 1.5.

1.1

0.9 0

20

40

Reference helix angle β in°

FIGURE 12.15 Exemplary helix angle factor for a gear with an overlap ratio of corresponding equations in ISO 6336-3:2019 [ 16]).

= 1 (further overlap ratios and

Rim thickness factor YB. If the rim thickness gets to thin, the gears’ rim tends to break earlier than the gear teeth. To avoid that, the rim needs to have a certain thickness. The standard accounts for the strength-weakening effect of thin rims up to a specified minimum thickness for both, external and in­ ternal gears with the rim thickness factor YB , which is shown in Figure 12.16. Corresponding equations can be found in ISO 6336-3:2019 [16]. For even thinner rims an advanced analysis is recommended, which the ISO 6336-3:2019 [16] does not cover. Deep tooth factor YDT . Gear pairings with a high contact ratio of 2 n < 2.5 have on average more than one tooth pair in contact. However, this is only the case in practice, if the manufacturing precision is above a certain level, i.e., with minimized pitch error. Therefore, gears with an ISO tolerance class ≤ 4 are assumed to have a smoother load distribution with more than one tooth pair contact on average. This form of gear pair can bear more stress in total, which is accounted for by the deep tooth factor of YDT . For an ISO tolerance class ≤ 4 the values for the deep tooth factor are between YDT = 0.7 1.0 depending on the virtual contact ratio. Exact values and corresponding equations for the deep tooth contact can be found in ISO 6336-3:2019 [16]. 12.3.2.3 Bending Stress Limit σFG The bending strength of the gears builds on material gear tests. The bending stress limit FG (tooth root stress limit) results from the permissible bending stress FP including the minimum required safety factor SF min with regard to tooth root stress (see Eq. (12.28)).

Gear Load Capacity Calculation (ISO 6336)

FIGURE 12.16

711

Rim thickness factor according to ISO 6336-3:2019 [ 16]. FG

=

FP

SF min

(12.28)

where: FG

FP

SF min

– is the tooth root stress limit, in N/mm2. – is the permissible bending stress, in N/mm2. – is the minimum required safety factor for tooth root stress.

Permissible bending stress FP (Method B). Based on several tests with standard reference test gears, different influences like material and surface properties and phenomena, the notch sensitivity or the number of load cycles, determine the gears bending stress limit (see Eq. (12.29)). This test-based approach is called Method B and is primarily described in the ISO 6336-3:2019 [16]. FP

=

YST YNT Y SF min

F lim

rel T

YR rel T YX

(12.29)

where: FP F lim

YST YNT SF min Y relT YRrelT YX

– – – – – – – –

is is is is is is is is

the the the the the the the the

permissible bending stress, in N/mm2. nominal stress number (bending), in N/mm2. stress correction factor. life factor. minimum required safety factor for tooth root stress. relative notch sensitivity factor. relative surface factor. size factor.

The nominal stress number with regard to bending is based on numerous tests. Figure 12.17 shows exemplary nominal stress numbers F lim for case hardened wrought steel for the different material qualities ML, MQ, and ME. Depending on the core hardness and the Jominy hardenability, the material quality MQ is divided in three subsections. The specifications of the material qualities, allowable stress

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Nominal stress number σF lim (bending) in N/mm2

550 ME a b

MQ

c 400

ML

250 500

700

900

Surface hardness in HV

FIGURE 12.17 Exemplary nominal stress numbers (bending) F lim depending on the surface hardness HV for case hardened wrought steels with different material qualities ML, MQ, and ME (further materials, material qualities (e.g., MQ a, b, c) and corresponding equations in ISO 6336-5:2016 [ 11]).

numbers for further materials and corresponding equations can be found in ISO 6336-5:2016 [11]. Additional influences are considered with various factors, which are presented in the following and are summarized in Table 12.10. Stress correction factor YST . Also, in case of the standard reference test gear, the local tooth root stress is transformed into the bending stress at the tooth root, as described in Section 12.3.2.2. The stress correction factor equals YST = 2.0 for reference test gears in the standard. Life factor YNT . If the targeted number of load cycles is below a material specific value determined for reference gears, the fatigue limit, higher tooth root (bending) stresses could be tolerable due to limited life. The life factor YNT therefore tends to increase with a decreasing number of load cycles. Figure 12.18 shows the exemplary life factor for standard reference test gears made out of case

TABLE 12.10 Summary of Factors of Influence on the Permissible Tooth Root Stress Stress correction factor YST

Life factor YNT

FP

The stress correction factor considers the transformation from the nominal tooth root stress (bending stress) to the local tooth root stress for the standard reference test gears. This includes the stress-increasing effect of the notch-like tooth root curvature. The life factor accounts for higher allowable contact stresses in case of a limited target lifetime.

Relative notch sensitivity factor Y relT

The relative notch sensitivity factor accounts for the notch sensitivity of the material and its dependency on the stress gradient.

Relative surface factor YRrelT

The relative surface factor takes into account the surface properties of the tooth root and their influence on the relevant stress level for tooth breakage. The size factor considers size effects of the gears on the permissible stress level.

Size factor YX

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Casecarburized NL ≤ NL, static= 104: NL = NL, Ref = 3 106: NL = 1010:

Life factor YNT

YNT, static

ZNT, static = 2.5 ZNT, Ref = 1.0 ZNT, long life = 0.85 – 1.0

YNT, Ref

static NL, static

limited life

NL, Ref

long life

Number of load cycles NL

FIGURE 12.18 Exemplary life factor for standard reference test gears made out of case carburized material (further materials and corresponding equations in ISO 6336-3:2019 [ 16]).

Relative notch sensitivity factor Yδ rel T

carburized material. Values of the life factor for further materials as well as corresponding equations can be found in ISO 6336-3:2019 [16]. Relative notch sensitivity factor Y rel T . Notches generally increase stresses in close proximity. To account for this phenomenon, the relative notch sensitivity factor Y rel T (see Figure 12.19) considers the stress sensitivity of the material and it is based on the material properties and the stress gradient. The notch sensitivity normally differs for the static and the dynamic case. High notch parameters lead to high relative notch sensitivity factors, which eventually lead to high bending stress limits. Figure 12.19 shows the exemplary relative notch sensitivity factor for a gear made out of case carburized material. Values of the relative notch sensitivity factor for further materials and corresponding equations can be found in ISO 6336-3:2019 [16].

1.2

1

0.8 1 .E+00

1 .E+01

1 .E+02

Notch parameter qs= SFn/ 2ρF

FIGURE 12.19 Exemplary relative notch sensitivity factor for reference stress and a gear made out of case car­ burized material (further materials and corresponding equations in ISO 6336-3:2019 [ 16]).

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Relative surface factor YR rel T . Another influence on the stress condition in the tooth root is the root surface, which the ISO 6336-3:2019 [16] considers with the relative surface factor YR rel T (see Figure 12.20). Depending on the material properties and the surface roughness, the stresses vary. The surface roughness can be seen as micro-notches, which generally result in an increasing relative surface factor with decreasing surface roughness. High values for the surface roughness led to a low relative surface factor, which eventually leads to low bending stress limits. Figure 12.20 shows the exemplary relative surface factor for a gear made out of case carburized material. Values of the relative surface factor for further materials and corresponding equations can be found in ISO 6336-3:2019 [16]. Size factor YX . When increasing the size of a component, the strength of the component cannot be scaled up in the same way in general. Due to an increased number of impurities or defects in the material structure, bigger components are more likely to fail at lower stress levels. The size factor YX offers a way to consider this effect in the standard. Large sizes lead to low size factors, which eventually lead to low bending stress limits. Figure 12.21 shows the exemplary size factor for a gear made out of case car­ burized material. Values of the size factor for further materials and corresponding equations can be found in ISO 6336-3:2019 [16] and range between YX = 0.7 1. Permissible bending stress FP for limited and long life (Method B). The standard also offers another way to calculate the permissible bending stress for limited or long life. Based on a material specific S–N curve the stress depending on the number of load cycles can be determined. For example, for steel gears the permissible bending stress calculates as follows (see Eq. (12.30)).

Relative surface factor YR rel T

1.15

1

0.85 1 .E+00

1 .E+01

1 .E+02

Roughness Rz in lm

FIGURE 12.20 Exemplary relative surface factor for a gear made out of case carburized material (further materials and corresponding equations in ISO 6336-3:2019 [ 16]).

Size factor YX

1.2 1 0.8 0.6

0

17.5

35

Normal module mn in mm

FIGURE 12.21 Exemplary size factor for a gear made out of case carburized material (further materials and corresponding equations in ISO 6336-3:2019 [ 16]).

Gear Load Capacity Calculation (ISO 6336)

=

FP

715

3 10 6 NL

FPref

0.4037 log

FPstat FPref

(12.30)

where: FP FPref

– is the permissible bending stress, in N/mm2. – is the reference permissible bending stress, in N/mm2. is the static permissible bending stress, in N/mm2. – is the number of load cycles.

FPstat –

NL

12.3.3 NEW ASPECTS

AND

UPDATES

OF THE

STANDARD

The revision and update of ISO 6336-3:2006 [17] to the current ISO 6336-3:2019 [16] has changed the following key aspects with regard to tooth root bending strength. 12.3.3.1 Form Factor YF—Load Distribution Influence Factor fε To account for reduced loads due to high contact ratios, the load distribution factor f extends the calculation of the form factor YF (see Eq. (12.26)) in ISO 6336-3:2019 [16]. The load distribution factor f depends on the contact ratio of the virtual spur gear n as well as the overlap ratio and considers the meshing influences on the load distribution (see Eqs. (12.31)–(12.35)). In general, the load distribution 1. Hence, it tends to reduce the calculated nominal tooth root stress F . factor is f In case of f = 1, ISO 6336-3:2006 [17] and ISO 6336-3:2019 [16] correspond to each other with regard to the form factor YF . If more than two teeth of a spur gear are meshing with each other on average (contact ratio n > 2), the new load distribution factor reduces the form factor by thirty percent, resulting in a correspondingly reduced nominal tooth root stress. For helical gears, the form factor and therefore the nominal tooth root stress decreases gradually with increasing overlap ratio with a rapid decrease at a contact ratio of the virtual gear of n = 2. f = 1, if

= 0,

f = 0.7, if

= 0,

n

1

(12.38)

= 1, if

where: Yβ – is the helix angle factor. – is the helix angle, in °. – is the overlap ratio.

12.3.3.4 Relative Notch Sensitivity Factor YδrelT For steel with a well-defined yield point the relative notch sensitivity factor is updated according to Eq. (12.39).

Y relT =

where: Y relT – is the relative notch sensitivity factor. – is the yield stress, in N/mm2. S

1 + 0.93 (YS 1 + 0.93

1) 4

200 S

4

200 S

(12.39)

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12.3.4 CALCULATION EXAMPLE Calculation examples for the standard ISO 6336-3:2019 [16] can be found in technical report ISO/TR 6336-30 [14]. ISO/TR 6336-30 contains calculation examples for the application of ISO 6336 parts 1, 2, 3, and 5. The technical report was first published in the year 2017 and the values and results of the calculation examples are based on the calculation methods of ISO 6336-3:2006 [17]. Currently, ISO 6336-30:2017 is under revision to update the examples and the number values to the current parts 1, 2, and 3 of the ISO 6336 series [1] published in the year 2019. The following example is summarized by an input and output table and can be calculated according to the equations of Section 12.3.2. and ISO 6336-3:2019 [16]. The calculation example presented here corresponds to the first of eight calculation examples from ISO/TR 6336-30. Table 12.11 shows selected input data for the calculation example. The complete input data as well as a detailed step-by-step cal­ culation can be found in ISO/TR 6336-30. The standard ISO 6336-3:2019 enables the user to calculate the safety factor against tooth root breakage. The safety factor against tooth root breakage is the ratio of the tooth root stress limit and the tooth root stress according to Eq. (12.40). SF1,2 =

Tooth root stress limit = Tooth root stress

FG1,2 F1,2

> S F min

(12.40)

where the symbols are described in Eq. (12.19).

TABLE 12.11 Basic Input Data Description

Unit

Symbol

Pinion

Number of teeth Normal module

– mm

z mn

17

Wheel 103 8

º

αn

20

Helix angle Face width

º mm

β b

15.8

Center distance

mm

aw

Nominal addendum correction factor Span measurement

– mm

x Wk

Normal pressure angle

Number of teeth spanned

100

500 500

0.145 38.196

0 307.943

–

k

2

13

Outside diameter Basic rack dedendum

mm mm

da hfP

159.66

872.35

Basic rack fillet root radius

mm

ρfP

Material allowance for finishing ISO accuracy grade

mm –

q Q

Material type

–

–

Eh

Eh

Material quality Lubricant

– –

– –

MQ

MQ

Application factor

–

KA

Nm rpm

T n

Pinion torque Pinion speed

1.4 · mn = 11.2 0.39 · mn = 3.12 0 5

0 5

ISO-VG-320 1.0 9 000 360

(54529.4) (59.4)

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The tooth root stress limit can be calculated according to Eq. (12.41). FG1,2

= YST YNT Y relT YR

rel T

YX

(12.41)

F lim

where the symbols are described in Eqs. (12.28) and (12.29). The tooth root stress can be calculated according to Eq. (12.42). F1,2

= KA K Kv KF KF YF YS Y YB YDT

Ft b mn

(12.42)

where the symbols are described in Eqs. (12.23) to (12.25). The various Y- and K-factors have to be calculated in order to determine the tooth root stress and the tooth root stress limit. More detailed information on the factors can be found in Section 12.3.2. Table 12.12 shows the basic output data including the values for the Y- and K-factors, tooth root stress, tooth root stress limit and the safety factor against tooth root breakage. The calculated safety factors for bending strength should be superior to the minimum safety factor, SF > SF min. Depending on the application, the experience and the consequences of a gear failure, lower or higher minimum safety factors can be chosen. It is recommended that the manufacturer and the customer agree on the values of the minimum safety factor for bending strength SF min [16].

TABLE 12.12 Basic Output Data (Note: Values are calculated according to ISO 6336-3:2019 and may differ from ISO/TR 6336-30:2017.) Description

Unit

Symbol

Pinion

Wheel

Stress correction factor

–

YST

Life factor

–

YNT

0.89

0.92

Relative notch sensitivity factor Relative surface factor

– –

Yδ rel T YR rel T

0.99 0.96

1.00 0.96

–

YX

0.97

0.97

N/mm2 –

σF lim Kγ

500

Size factor Allowable stress number (bending) Mesh load factor

2.00

500 1.00

Velocity factor

–

Kv

1.00

Face load factor Transverse load factor

– –

KHβ KHα

1.16 1.00

Form factor

–

YF

1.26

Stress correction factor Helix angle factor

– –

YS Yβ

1.79 1.00

1.07 2.05 0.97

Rim thickness factor

–

YB

Deep tooth factor Nominal tangential load

– N

YDT Ft

Tooth root stress limit

N/mm2

σFG

824

860

Tooth root stress Safety factor for bending strength

N/mm2 –

σF SF

396 2.08

383 2.24

1.00 1.00 127352

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Dudley’s Handbook of Practical Gear Design and Manufacture

12.3.5 SUMMARY ISO 6336-3:2019 [16] is a standard that describes the calculation of bending strength against tooth root breakage. The following list shows the history of the standard: • In 1996 the first edition of ISO 6336-3 was published. • In 2006 the second edition of ISO 6336-3 was published. • In 2019 the third and currently valid edition of ISO 6336-3 was published. The bending strength is calculated against the failure mode tooth root breakage. Tooth root breakages can be described by the following properties. • Tooth root breakage can be described as the fracture of a large portion of a tooth from a gear. • Tooth root breakages are caused by bending fatigue cracks initiated in the tooth root fillet area. • Tooth root breakages will most likely lead to a total failure of the gearset. The basic calculation principles for the bending strength against tooth root breakage are according to the following principle. • The safety factor for bending strength against tooth root breakage is the ratio of the tooth root stress limit and the tooth root stress. • The tooth root stress limit represents the load capacity depending on various factors such as the material, surface roughness, gear size, and more (see Section 12.3.2.3). • The tooth root stress represents the load depending on various factors such as the geometry of the gear, rim thickness, load influence factors, and more (see Section 12.3.2.2.). With the recent publication of the third edition of ISO 6336-3 in 2019, the following key aspects have changed with respect to the previous version. • The load distribution influence factor f , which accounts for reduced stresses due to high contact ratios, was introduced for the calculation of the contact factor YF . • The gear geometry calculation for internal gears was refined and is now more precise. • The helix angle factor Y was extended by the additional term 1/cos3β, leading to higher stresses if the helix angle is increased. ISO 6336-3:2019 has been recently published and will be reviewed systematically after five years. The current standard and its calculation methods are applied by the industry. New topics and results of recent research projects are prepared, reviewed and discussed to be possibly added for the next edition. In the meantime, ISO 6336-3:2019 is the state-of-the-art standard to calculate the bending strength against tooth root breakage for spur and helical gears.

12.3.6 OUTLOOK The outlook for ISO 6336-3:2019 [16] describes how the standard for the calculation of the bending stress against tooth root breakage could be developed in the future. The next systematic revision of ISO 6336-3 will be five years from 2019, in the year 2024. During this time, the standard will be applied in the industry. If necessary, the calculation methods shall be updated to cover further experiences and observations in practice. Alternatively, the scope and the application limits need to be updated to exclude the applications in practice that cannot be covered by the existing calculation methods. In both cases, the

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further development of the calculation methods as well as the limitation of the scope, changes have to be thoroughly investigated and statistically validated in advance. The following topics have already been identified based on input from the gear industry for further development of ISO 6336-3. • The form factor YF shall be extended for the manufacturing process of form grinding. In the current standard, the form factor can be calculated for gears generated with a hob or a shaper cutter. Formulae shall be added to calculate the form factor for gear geometries that are defined by the finishing process form grinding. • The calculation for the tooth root notch factor shall be reviewed and if possible extended. The reference given in the standard is from 1972 and delivers results on the conservative side. Since then, further investigations and simulations have given additional information. Since the review of this topic is complex and wide-ranging, a thorough review will need its time. Besides the revision of the existing calculation methods, new approaches and state of the art knowledge can be added for a further developed standard. New manufacturing processes, new materials and new calculation possibilities can lead to the necessity of new or advanced calculation methods. A list of exemplary topics that are possibly relevant in the future can be found in Section 12.2.6. In the list of Section 12.2.6, the computer aided analysis of the tooth contact is to be substituted with the computer aided analysis of tooth root bending. The above-mentioned topics for the further development of the ISO 6336-3 are current assumptions and further topics will likely be included in the next revision. It can be assumed with a high degree of certainty, that the current version of ISO 6336-3 will be further developed and adapted to the current state of the art concerning the calculation of the bending strength against tooth root breakage.

12.4 CALCULATION OF MICROPITTING LOAD CAPACITY— ISO/TS 6336-22:2018 The following subsections give information on ISO/TS 6336-22:2018 [21]. ISO/TS 6336-22:2018 is a technical specification describing the calculation of the micropitting load capacity. The calculation method is outlined in the following subsections relating to the description of micropitting, the basic calculation principles, further standards and test procedures, a calculation example, a summary and an outlook on the future of ISO/TS 6336-22:2018 and the FZG micropitting test according to FVA 54/7 [22] respectively DIN 3990-16:2020 [23].

12.4.1 DESCRIPTION

OF THE

FAILURE MODE MICROPITTING

The phenomenon micropitting is usually observed on materials with a high surface hardness and under highly loaded conditions in Hertzian type of rolling and sliding contact under lubrication regimes in EHL-contacts, especially in the regime of mixed lubrication. It can be visualized as a coherent area, which appears as gray and dull on affected surfaces. Micropitting is a surface fatigue phenomenon and therefore the result of numerous very small surface cracks. Occurring cracks near the surface do not necessarily grow deeper into the material. They may terminate or grow back to the surface. This will lead to small outbreaks, which are typically 10 μm to 20 μm deep and change the profile form [21]. Figure 12.24 shows a typical micropitted area and the changed profile of the flank due to micropitting. Figure 12.25 shows a micropitted flank surface with a magnification of 250 times. Figure 12.26 shows a micropitted flank surface with a magnification of 950 times. The occurrence of micropitting is strongly influenced by the operating conditions, the used lubricant and the condition of the surface of the mating contact partners [26]. The operating conditions include the load, the operating temperature and the circumferential speed. The lubricant usually consists of a base oil and additives, which mainly affect the lubrication conditions and therefore the micropitting load carrying

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Micropitting

FIGURE 12.24

Micropitting on a case hardened gear flank and the changed profile of the flank [ 24].

FIGURE 12.25

250x magnification of a micropitted flank [ 25].

FIGURE 12.26

950x magnification of a micropitted flank [ 25].

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capacity. The viscosity of the lubricant strongly influences the lubricant film thickness and therefore the lubrication conditions as well. The viscosity and the effect of the additives depend on the operating temperature. Considering micropitting on gears, the condition of the tooth flank surface depends on roughness, gear geometry, material and manufacturing processes. Concerning the manufacturing process, the heat treatment, residual stresses and residual austenite are decisive for the tooth flank condition. The gear geometry refers to the tooth flank geometry, which depends on size, profile shift and profile modification [21]. Micropitting occurs under unfavorable lubrication conditions and preferably in the area of negative specific sliding. With increasing running time, it can spread over the entire tooth flank. Micropitting changes the profile form and as a result reduces the quality of the gearing. Additionally, dynamic forces and gear noise can increase when micropitting occurs [27]. Micropitting can affect the pitting load carrying capacity in a positive or negative way. The profile form derivations caused by micropitting can lead to a better load distribution along the tooth flank and therefore to an increased pitting load carrying capacity. Contrariwise, the small cracks can encourage the development of pitting damages. Often pitting damages occur on the edge of the micropitted area, since the contact stresses can increase on the transition from the micropitted area to the undamaged surface [25,28]. Figure 12.27 shows a pitting damage in the transition area of a micropitted area to an undamaged surface.

FIGURE 12.27

Pitting damage on the transition of a micro pitted area and undamaged surface [ 28].

Due to the various effects of micropitting, it is important to determine the load carrying capacity of micropitting for a given application. A safety factor against micropitting can be calculated according to ISO/TS 6336-22:2018 [21] for a given gear drive.

12.4.2 BASIC CALCULATION PRINCIPLES The calculation method according to ISO/TS 6336-22:2018 [21] aims at evaluating a micropitting safety factor Sλ for a given gear drive. Since micropitting on gear flanks occurs in a tribological system between the meshing tooth flanks and the lubricant in between, the lubricant film thickness is a main parameter regarding micropitting. Therefore, the safety factor against micropitting is expressed as the ratio of the minimum specific lubricant film thickness λGF,min in the actual contact area and the permissible specific

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lubricant film thickness λGFP. The safety factor against micropitting can be calculated according to Eq. (12.43).

S =

GF , min GFP

(12.43)

where: Sλ – is the safety factor against micropitting. λGF,min – is the minimum specific lubricant film thickness. λGFP – is the permissible specific lubricant film thickness.

The permissible specific lubricant film thickness λGFP for oils with additives cannot be calculated without any experimental investigation, since the effect of the additives is highly dependent on the lubricant temperature. In order to take the permissible specific lubricant film thickness λGFP into con­ sideration, two methods are applicable. One method is to determine the permissible specific lubricant film thickness λGFP by experimental investigations or service experience relating to micropitting on gears for industrial applications. For this method, the test gears need to have the same design as the actual gear pair and the gear manufacturing, gear accuracy, operating conditions, lubricant and operating tem­ perature have to be appropriate for the actual gear box. As a result, this method causes high costs. The other method is to use the test result of a standardized micropitting test like the widely used FZG micropitting test according to FVA 54/7 [22]. The FZG micropitting test investigates the micropitting performance of gear lubricants according to FVA 54/7 and gives a classification of the lubricant re­ garding the micropitting load capacity expressed as a failure load stage (SKS). By using the failure load stage, determined at the actual operating temperature, the permissible specific lubricant film thickness λGFP for the investigated specific lubricant can be calculated directly using the same approach as the occurring specific lubricant film thickness described in Section 12.4.2.1. 12.4.2.1 Specific Lubricant Film Thickness λGF,Y The specific lubricant film thickness λGF,Y can be calculated for every point in the field of contact, whereas the index Y indicates the considered calculation point on the path of contact. According to ISO/TS 6336-22:2018 two methods (A and B) are possible. For method A the specific lubricant film thickness λGF,Y is determined in the complete contact area by an appropriate gear computing program. The method A is the more accurate one, but needs more input values. For simplification, the calculation of the local specific lubricant film thickness according to Method B is limited to 7 points on the path of contact including the points A (start of active profile – SAP), B, C, D, and E (end of active profile—EAP) as well as AB and DE. The points AB and DE are placed midway between points A and B and D and E. Method B involves the assumption that micropitting mainly occurs in areas of negative specific sliding. These areas are found along the path of contact between points A and C on the driving gear and between points C and E on the driven gear. The specific lubricant film thickness λGF,Y can be calculated from the lubricant film thickness hY and the effective arithmetic mean roughness Ra of the surface, according to Eq. (12.44). GF , Y

where: λGF,Y – is the specific lubricant film thickness. hY – is the lubricant film thickness in µm. Ra – is the effective surface roughness in µm.

=

hY Ra

(12.44)

Gear Load Capacity Calculation (ISO 6336)

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The effective surface roughness Ra used in the calculation method ISO/TS 6336-22:2018 [21] is determined as the arithmetic mean value of the arithmetic mean roughness values of pinion Ra1 and wheel Ra2, as shown in Eq. (12.45) Ra = 0.5 (Ra1 + Ra2 )

(12.45)

where: Ra – is the effective surface roughness in µm. Ra1 – is the arithmetic mean roughness value of pinion in µm. Ra2 – is the arithmetic mean roughness value of wheel in µm.

Equation (12.46) shows that the lubricant film thickness hy is calculated in accordance to Dowson/ Higginson [29] with an additional factor: the local sliding parameter SGF,Y. As a result, the lubricant film thickness hY is influenced by the normal radius of relative curvature ρn,Y, the material parameter GM, the local velocity parameter UY, the local load parameter WY, and the local sliding parameter SGF,Y. hY = 1600

n, Y

0,22 GM0,6 UY0,7 WY 0,13 SGF ,Y

(12.46)

where: hY ρn,Y GM UY WY SGF,Y

– – – – – –

is is is is is is

the the the the the the

lubricant film thickness in µm. normal radius of relative curvature in mm. material parameter. local velocity parameter. local load parameter. local sliding parameter.

The normal radius of relative curvature ρn,Y can be calculated for each considered contact point from the gear geometry according to ISO 21771:2014 [30]. The material parameter GM is calculated according to Eq. (12.47) and is influenced by the reduced modulus of elasticity Er and the pressure-viscosity coefficient of the lubricant at bulk temperature αθM. GM = 10 6

M

Er

(12.47)

where: GM – is the material parameter. αϴM – is the pressure-viscosity coefficient of the lubricant at bulk temperature in m2/N. Er – is the reduced modulus of elasticity in N/mm2.

The local velocity parameter UY is dependent on the dynamic viscosity ηθM of the lubricant at bulk temperature, the sum of the tangential velocities vΣ,Y, the reduced modulus of elasticity Er, and the normal radius of relative curvature ρn,Y and can be calculated according to Eq. (12.48): UY =

,Y M

2000 Er

where: UY – is the local velocity parameter. ηϴM – is the dynamic viscosity at bulk temperature in N⸱s/m2.

(12.48) n, Y

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νΣ,Y – is the sum of the tangential velocities in m/s. Er – is the reduced modulus of elasticity in N/mm2. ρn,Y – is the normal radius of relative curvature in mm.

The local load parameter WY is shown in Eq. (12.49) and can be determined by using the local Hertzian contact stress pdyn,Y and the reduced modulus of elasticity Er. WY =

2 pdyn ,Y

2

2000 Er2

(12.49)

where: WY – is the local load parameter. pdyn,Y – is the local Hertzian contact stress in N/mm2. Er – is the reduced modulus of elasticity N/mm2.

The local sliding parameter SGF,Y describes the influence of local sliding on the calculation of the lubricant film thickness hY and as a consequence on the risk of micropitting. Commonly, the lubricant film thickness is smaller in areas of sliding than in rolling friction. This can be explained with the assumption that sliding is responsible for an additional local heating of the lubricant. As a result, a higher temperature in the contact zone leads to changing properties of the lubricant, i.e., that the viscosity of the lubricant drops with an increasing temperature leading to a smaller lubricant film thickness. The pressure-viscosity coefficient as well depends strongly on the temperature. Therefore the local sliding parameter SGF,Y accounts for a comparison of the pressure-viscosity coefficient at local contact tem­ perature αϴB,Y and at bulk temperature αϴM as well as a comparison of the dynamic viscosity at local contact temperature ηϴB,Y and at bulk temperature ηϴM, which can be seen in (12.50). SGF, Y =

B, Y M

B, Y

(12.50)

M

where: SGF,Y αϴB,Y ηϴB,Y αϴM ηϴM

– – – – –

is is is is is

the the the the the

local sliding parameter. pressure-viscosity coefficient at local contact temperature in m2/N. dynamic viscosity at local contact temperature in N⸱s/m2. pressure-viscosity coefficient of the lubricant at bulk temperature in m2/N. dynamic viscosity at bulk temperature in N⸱s/m2.

The local contact temperature θB,Y can be calculated as the sum of the local flash temperature θfl,Y and the bulk temperature θM, which can be seen in equation (12.51). B, Y

where: θB,Y – is the local contact temperature in °C. θfl,Y – is the local flash temperature in °C. θM – is the bulk temperature in °C.

=

fl , Y

+

M

(12.51)

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12.4.2.2

Permissible Specific Lubricant Film Thickness (According to the FZG Micropitting Test According to FVA 54/7) The permissible specific lubricant film thickness λGFP can be calculated based on the results of a mi­ cropitting test according to FVA 54/7 [22]. The material factor WW and the critical specific lubricant film thickness λGFT are taken into account according to Eq. (12.52). GFP

= 1.4 WW

(12.52)

GFT

where: λGFP – is the permissible specific lubricant film thickness. WW – is the material factor. λGFT – is the critical specific lubricant film thickness.

The material factor WW respects the load carrying behavior of different steels compared to the casehardened material 16MnCr5 used for the standard “FZG-C” test gears, which can be seen in Table 12.13.

TABLE 12.13 Material Factor WW Material

Material Factor, WW

Case carburized steel, with austenite content: Less than 25%

1.0

Greater than 25%

0.95

Gas nitrided steel (HV > 850) Induction, flame hardened steel

1.5 0.65

Through hardened steel

0.5

Basically, the critical specific lubricant film thickness λGFT can be determined in accordance to Eq. (12.44) of the specific lubricant film thickness λGF,Y, but with the gearing data for the FZG mi­ cropitting test according to FVA 54/7 (gear type “FZG-C”) and the specific test parameters of the given failure load stage (SKS). Table 12.14 shows basic information of the "FZG-C" test gears for the FZG micropitting test according to FVA 54/7. TABLE 12.14 Basic Information of “FZG-C” Test Gears for the FZG Micropitting Test according to FVA 54/ 7 [22] Dimension

Symbol

Numerical Value

Unit

Center distance Effective tooth width

a b

91.5 14.0

mm mm

Module

m z1 z2

4.5

mm

16 24

-

20.0

°

0

°

Number of teeth pinion Number of teeth wheel Pressure angle Helix angle Tooth correction

Without tip relief and root relief, no longitudinal crowning

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The calculation of the reference value λGFT can be done for only point A, since the minimum specific lubricant film thickness for the gear type “FZG-C” is at point A. The critical specific lubricant film thickness λGFT is calculated with the torque at the pinion for the given failure load stage. The torque for the load stages 5 to 10 is listed in Table 12.15.

TABLE 12.15 Failure Load Stage and Corresponding Torque on Pinion SKS

5

6

7

8

9

10

Torque

70 Nm

98.9 Nm

132.5 Nm

171.6 Nm

215.6 Nm

265.1 Nm

The failure load stage for the used lubricant has to be determined with the same oil temperature in the FZG micropitting test according to FVA 54/7 as in the considered application, especially for oils with complex additive packages. 12.4.2.3 Limits of the Calculation Method The method according to ISO 6336-22 [21] has been developed on the basis of testing and observation of oil-lubricated gear transmissions with modules between mn = 3 mm and mn = 11 mm and pitch line velocities of vt = 8 m/s to vt = 60 m/s. However, the procedure can be used for any gear pair with available suitable reference data, providing the criteria specified in ISO/TS 6336-22. At circumferential speeds of vt > 80 m/s the additional heating by churning of the oil in the mesh and windage losses can have a decisive influence on the operating temperature. Therefore, this additional heat input should be taken into account for calculating the mass temperature by using Method A. Method A can be used to calculate the micropitting load capacity for pitch line velocities of vt > 2 m/s [21]. For gears operating at small circumferential speed a more detailed method was established in the research project FVA 482 III [31], since an overlapping of micropitting and sliding wear (slow speed wear) may occur. While some operating conditions are clearly critical toward micropitting, others are critical toward sliding wear. The minimum specific lubricant film thickness in the gear contact λmin is the main characteristic parameter to evaluate whether micropitting or sliding wear will emerge and which of both will be the dominating damage mechanism. It can be observed that micropitting only occurs if sliding wear is not the dominant type of damage [32]. In [25,31,32] a limitation of the applicability of the existing calculation models for micropitting and sliding wear is proposed for case-hardened gears operated at low to medium circumferential speed. The risk zones, which are described in (1)–(4), depend on the minimum specific lubricant film thickness in the gear contact λmin for a given lubricant. The risk zones are visualized in Figure 12.28. 1. For λmin ≥ 2 · λGFP no micropitting and no sliding wear is expected, since the system operates at a sufficiently high specific film thickness. This means that the minimum specific lubricant film thickness in the gear contact λmin is more than double the permissible specific lubricant film thickness λGFP and the safety factor against micropitting according to ISO/TS 6336-22:2018 [21] is Sλ ≥ 2. 2. For λGFP ≤ λmin < 2∙λGFP a moderate risk of micropitting may be expected. The risk of micropitting rises when the value of the minimum specific lubricant film thickness λmin tends toward the permissible specific lubricant film thickness λGFP. However, the specific lubricant film thickness for sliding wear λC, wear according to Plewe [33,34] has to be checked in comparison with the permissible specific lubricant film thickness λGFP. The specific lubricant film thickness for sliding wear λC,wear determines whether a superposition of micropitting and wear occurs.

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FIGURE 12.28 Risk zones depending on the minimum specific lubricant film thickness in the gear contact λmin for a given lubricant (according to FVA 482 III [ 31]) [ 32].

3. For λC,wear < λmin < λGFP a significant risk of micropitting is to be expected. For this area the specific lubricant film thickness for sliding wear λC,wear, which determines whether a superposition of micropitting and wear occurs, must be checked in comparison to the permissible specific lu­ bricant film thickness λGFP. For the zones (2) and (3) the following considerations should be noted: For λmin > 2∙λC,wear the calculation of the safety factor against micropitting Sλ provides accurate results. If λmin < 2∙λC,wear is fulfilled, it can be assumed that a risk of micropitting exists, but the expected micropitting can be decisively influenced by sliding wear as main failure mechanism. As a result, the calculated safety factor against micropitting Sλ has a limited informational value in this case. The value of the permissible specific lubricant film thickness λGFP is depending on the micropitting performance of the lubricant in use, whereas the value of the specific lubricant film thickness for sliding wear λC,wear is depending on the wear performance of the lubricant in use. As a result, the relationship of both values is not a constant factor and absolute values as well as the ratio may be different for different lubricants and operating conditions. 4. For λmin ≤ λC,wear sliding wear can be expected as the dominant type of damage. If an application operates in such critical lubricating conditions, a calculation of a safety factor against micropitting Sλ is not recommended.

12.4.3 MICROPITTING TEST PROCEDURES The following sections give information on the FZG micropitting test according to FVA 54/7 according FVA 54/7 [22] and German standard DIN 3990-16:2020 [23], which describes the test procedure of the FZG micropitting test according to FVA 54/7 in detail. 12.4.3.1 FZG Micropitting Test According to FVA-Information Sheet 54/7 The FZG micropitting test according to FVA 54/7 [22] is carried out for a specific oil under defined conditions especially regarding oil temperature and circumferential speed. The FZG micropitting test according to FVA 54/7 consists of a load stage test and an endurance test. The load stage test is per­ formed for predefined load stages with increasing torque. After conducting the load stage test, a

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TABLE 12.16 Load Stages with the Defined Torque on the Pinion and Corresponding Hertzian Contact Pressure of the Fzg Micropitting Test according to Fva 54/7 running-in

5

6

7

8

9

10

Torque in Nm

Load stage

28.8

70

98.9

132.5

171.6

215.6

265.1

Hertzian contact pressure in N/mm2

510.0

795.1

945.1

1093.9

1244.9

1395.4

1547.3

classification of the lubricant regarding the micropitting load capacity expressed as a failure load stage (SKS) is available. The load stage, in which the defined failure criterion (profile form deviation f fm ) is exceeded, defines the failure load stage. After the load stage test, a subsequent endurance test usually is performed. This test part provides information about the progress of micropitting at high numbers of load cycles. Execution of the test method. The FZG micropitting test according to FVA 54/7 is conducted at the standard FZG back-to-back test rig with a center distance of a = 91.5 mm. The “FZG-C“ test gears are being used for the FZG micropitting test according to FVA 54/7. The test gears are designed as spur gears with 16 teeth on the pinion and 24 teeth on the gear, the normal module of 4.5 mm, no tooth correction, a pressure angle of 20° and the effective tooth width of 14 mm for both partners. The casecarburized test gears are made of 16MnCr5 with a surface hardness of 750 ± 30 HV1 in the area of the tooth flank. The grinding process of the test gears is Maag 0°-grinding, therefore a gear quality grade of ≤ 5 acc. to DIN 3962:1978 [34] can be reached. The arithmetic roughness of the tooth flanks is set to Ra = 0.5 µm ± 0.1 µm. Basic information of this type of gearing is summarized in Table 12.14. The load stage test starts with a running-in stage, which is followed by the subsequent load stages 5 to 10. During the load stage test, the torque on the pinion is increasing from 30 Nm (running-in) to 265 Nm (stage 10). At the same time the Hertzian contact pressure is increasing. The Hertzian contact pressure at pitch point pC can be seen in Table 12.16. In Table 12.17 the standard test conditions for the load stage test and the endurance test are listed. It is possible to change the speed and oil temperature in order to use test conditions that are closer to real application. Because the effect of the additives is strongly dependent on the temperature, changes in oil temperature may lead to significant changes in the obtained failure load stage (SKS). There it is strongly recommended to test the oil at or close to the typical temperature used in the application. The short description of the standard test is GF-C/8.3/90. The short description gives information of the gearing type (C-GF), the circumferential speed (vt = 8.3 m/s) and the oil temperature (ϑOil = 90°C). Determination of the micropitting load capacity. In order to determine the failure load stage, the profile shape of at least three teeth distributed evenly over the circumference is measured after every conducted load stage. After measuring the profile shape, the profile form deviation f f according to DIN 3960:1978 [34] is determined. For the reference line in the profile diagram, the actual involute of the new test gear is applied. The failure load stage is defined as load stage, when the mean value f fm of the measured values passes a fixed level, which is set to 7.5 µm, for the first time. If the lubricant passes the highest defined load stage 10, the failure load stage of the lubricant is set to SKS > 10. After conducting the load stage test, a subsequent endurance test is usually performed. The endurance test consists of eighty hours running time in the 8th load stage and then up to five times eighty hours in the 10th load stage on the same gear flanks. The tooth flanks are inspected every eighty hours. In this test part, a higher number of load cycles (N = 10.5 ⸱ 106 on the pinion) per inspection interval is realized and therefore this test part gives information about the progress of micropitting. If the mean profile deviation due to micropitting is bigger than 20 µm or the maximum running time of 400 hours in load stage 10 is reached, the test is finished.

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TABLE 12.17 Standard Test Conditions in the FZG Micropitting Test according to FVA 54/7 acc. to FVAInformation Sheet [22] Dimension

Symbol

Value

Unit

Pinion speed

n1 v

2250

rpm

8.3

m s

ϑOil

Pinion 90± 2

°C

Running time during running-in Running time per load stage

-

1 16

hours hours

Running time per inspection interval in endurance test

-

80

hours

Lubrication

-

Spray lubrication

-

Circumferential speed Driving gear Oil inlet temperature

-

Exemplary test results from the standard test (C/8.3/90) using lubricants with different levels of micropitting load capacity as well as the visualized failure criterions can be found in Figure 12.29. The maximum profile form deviation after every test interval is plotted over the running time of the test. Lubricants with a high micropitting load capacity show less profile form derivations than lubricants with a lower micropitting load capacity. It should be noticed, that the defined failure criteria of ffm = 7,5 μm respectively ffm = 20 μm are applicable for the test procedure only and are not intended to define a criterion when micropitting is considered to be damaging for industrial applications.

FIGURE 12.29 Test results from micropitting tests using lubricants with different levels of micropitting load capacity (example) [ 22].

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12.4.3.2 DIN 3990-16:2020 In May 2020 the German standard DIN 3390-16:2020 “Calculation of load capacity of cylindrical gears – Part 16: Determination of the micro-pitting load-carrying capacity of lubricants using FZG-test-method GT-C/8,3/90” was published. DIN 3390-16:2020 [23] describes the test procedure of the FZG micro­ pitting test according to FVA 54/7 [22] in detail and comprises clarifications and improvements on how to run and evaluate the test. Additionally, precise data for the test method based on round robin testing is included. By proceeding this test, the micropitting load carrying capacity (GFT) of gear lubricants can be evaluated. The new standard DIN 3390-16:2020 allows a comparison of different lubricants for a specific application. Additionally, the test result can be used for the calculation of the micropitting load capacity of a given gear drive for example according to ISO/TS 6336-22:2018 [21]. The execution of the test procedure, which is described in DIN 3990-16:2020, is in accordance with FVA 54/7 and even further specified in order to achieve more comparable results between different laboratories. Additionally, a new classification system was defined. According to the information sheet FVA 54/7 [22] the test is finished after conducting the load stage test and the endurance test. However, in order to ensure reliable results and scientific depth a second load stage test is commonly added as verification run for the determined failure load stage of the first test run. As a result, the determination of the failure load stage SKS according to DIN 3990-16:2020 [23] depends on the failure load stage of the first load stage test SKSP and the failure load stage of the second load stage SKSW. SKS corresponds to the failure load stage of the first load stage test SKSP, if the result of the repeat run confirms the result of the test run within the specified repeatability. In case of larger deviations in the load stage test from the test run and repeat run, it is recommended to repeat one load stage test to verify the results. Alternatively, the minimum achieved failure load stage from the test run and repeat run can be specified as the failure load stage of the micro-pitting test: If SKSP If SKSP

SKSW

1: SKS = SKSP

SKSW > 1: SKS = min(SKSP ; SKSW )

(12.11) (12.12)

where: SKSP – is the failure load stage of the first load stage test (reference run). SKSW – is the failure load stage of the second load stage (repetition run). SKS – is the failure load stage of the conducted FZG micropitting test according to FVA 54/7.

According to FVA 54/7, the micropitting load capacity of a lubricant is divided into the three classes (GF-classes) “GFT-low,” “GFT-medium,” and “GFT-high.” This can be seen in Figure 12.29. In DIN 3990-16:2020 [23] a new GF-class “GFT-very high” is introduced. Table 12.18 shows the classification according to DIN 3990-16:2020 for “GFT-very high.” For the new class “GFT-very high” only a limited experience is available and the defined limits for the GFT-class were not validated by round-robin testing so far. Thus, the evaluation of the class “GFT-very high” is described in an informative annex for the moment.

12.4.4 CALCULATION EXAMPLE For a better understanding, an example for the calculation of the micropitting load capacity according to ISO/TS 6336-22:2018 [21] is shown in the following. For this calculation, the micropitting test results according to FVA 54/7 [22] are used. More examples can be found in ISO/TS 6336-31 [35]. This example is part of the research of FVA 482 III [31] and was published as well in [36]. The gears showed micro pitted areas and had an averaged profile deviation of 14 μm after 20 million load cycles on the pinion.

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TABLE 12.18 GF-Class “GFT-Very High” according to DIN 3990-16:2020 GF-Class

1st Load Stage Test

2nd Load Stage Test

Endurance Test Requirements/Characteristics

GFTvery high

SKSP > 10 and ffm after LS9≤ 5,5 µm

SKSW > 10

Criteria

z10 ≥ 2: failure due to pitting or achieving maximum running time

z10 ffmz10

2 ≤9 µm

3 ≤11 µm

4 ≤13 µm

5 ≤14 µm

and: (ffmz10 – ffmafter LS 8) / z10 ≤ 1,5 μm Definitions: z10: number of executed intervals in the endurance test in LS 10 until end of endurance test without scuffing nor wear. The end of endurance test generally is reached, if: - the failure criterion considering micro-pitting (ffm > 20 µm) is exceeded or - pitting, scuffing and/or wear appears or - the maximum running time of 80 h in LS 8 and 400 h in LS 10 is reached. ffm: absolute value of the average profile deviation caused by micro-pitting ffmafter LS 8: ffm after 80 h running time in LS 8 in the endurance test ffmz10: ffm after z10 (correlates to the last executed interval in the endurance test without scuffing nor wear) SKS: failure load stage GFT: micro-pitting load carrying capacity.

new

after test run 50 40 30 20 10 0 Profile form deviation in µm FIGURE 12.30

Diameter

Diameter

The test was finished if the mean profile deviation due to micropitting was more than 20 µm or the maximum running time of 20 million load cycles was reached. As it can be seen in Figure 12.30, the profile form deviation regarding the pinion was located between point A and C for the pinion and for the wheel between point C and E of the line of action [31]. Figure 12.31 shows an exemplary picture of the gear flanks of pinion and wheel after the test run. Table 12.19 shows the data of this lubricant. In Table 12.20, the geometry data of the gear drive is given. For this example, a mineral oil, which reaches failure load stage (SKS) 9 at test temperature 60 °C acc. to FVA 54/7 [22] is used.

new

after test run 50 40 30 20 10 0 Profile form deviation in µm

Exemplary profile deviation of pinion (left side) and wheel (right side) [ 31].

734

FIGURE 12.31

Dudley’s Handbook of Practical Gear Design and Manufacture

Exemplary pictures of the gear flanks of pinion (left side) and wheel (right side) [ 31].

TABLE 12.19 Lubricant Data Dimension

Symbol

Numerical Value

Unit

Oil inlet temperature Kinematic viscosity at 40°C

ϑOil

60 90

°C

40

Kinematic viscosity at 100°C

100

11

mm2 s

Density of the lubricant at 15°C

12

880

mm2 s

kg m3

Oil type

-

Mineral oil

-

Failure load stage at test temperature (60°C) according to FVA 54/7 [22]

-

SKS 9

-

TABLE 12.20 Geometry Data of the Tested Gear Drive Dimension

Symbol

Numerical Value

Unit

Center distance

a

91.5

mm

Effective tooth width

b

14

mm

Module Number of teeth pinion

m z1

5 17

mm -

Number of teeth wheel

z2

Pressure angle Helix angle Tooth correction

18

-

20.0 0

° °

tip relief on pinion and wheel Ca = 22 µm, no longitudinal crowning

By using the SKS value of the lubricant, the permissible specific lubricant film thickness λGFP can be calculated according to ISO/TS 6336-22:2018 [21] to λGFP = 0.203. The determination of the permissible specific lubricant film thickness λGFP takes place on point A. This point represents the critical point on the line of action for the “FZG-C” test gears. On this point the minimum value of the specific lubricant

Gear Load Capacity Calculation (ISO 6336)

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TABLE 12.21 Results of the Calculation according to ISO/TS 6336-22:2018 Point

A

AB

B

C

D

DE

E

GF , Y

0.247

0.215

0.237

0.365

0.236

0.217

0.231

1.163

1.069

1.138

0.215

GF , min

0.203

GFP

S ,Y

1.118

S

1.059

1.167

1.798 1.059

film thickness is located. The determination of the minimum value of the specific lubricant film thickness λGF,min for an application depends according to ISO/TS 6336-22:2018 [21] generally on the geometry, material, application, load and data of the lubricant. The oil inlet temperature, which has to be the same or close to the test temperature, is 60°C. Table 12.21 shows the values of the specific lubricant film thickness λGF,Y for the points A, AB, B, C, DE, and E on the line of action of the tested gear drive. The safety factor against micropitting Sλ is expressed as the quotient of the minimum value of the specific lubricant film thickness and the permissible specific lubricant film thickness. The safety factor Sλ,Y of this calculation example is calculated for all seven points on the path of contact. The results are listed in Table 12.21. As it can be seen in Table 12.21 the minimum value of the specific lubricant film thickness is located in points AB and DE. These results appear realistic, because micropitting mainly appears in areas of negative specific sliding. These areas are found along the path of contact between points A and C on the driving gear and between points C and E on the driven gear. Additionally, this matches well with the profile from deviations shown in Figure 12.30. The resulting safety factor against micropitting for this gear drive is Sλ = 1.059. Considering Figure 12.28 this safety factor is not in the range of the highest probability of micropitting, since λGFP < λmin < 2∙λGFP. The results are clearly lower than Sλ ≥ 2, which represents the range where no micropitting is expected. This leads to the assumption, that micropitting may occur on the flank, but the defined failure criterion of ffm = 20 µm is not exceeded during the test run. Hence, the calculated safety factor against micropitting matches with the experimental findings and behavior. It should be noticed that a safety factor against micropitting of Sλ < 2 typically is not sufficient to avoid micropitting. Based on field experience and test data, safety factors of 1,0 < Sλ < 2,0 can be allowable if the input parameters for the calculation are carefully chosen and proven experience is available. For a safety factor against micropitting of Sλ > 2, no micropitting is expected.

12.4.5 SUMMARY Micropitting is a surface fatigue phenomenon, which is the result of numerous small surface cracks. It occurs in Hertzian rolling and sliding contacts under unfavorable lubrication regimes in EHL-contacts and usually on materials with a high surface hardness. It can be optically detected as a gray and dull area on affected surfaces. The main influence factors on micropitting are the lubricant, the flank surface roughness, the gear geometry and the operating conditions. The micropitting load carrying capacity can be calculated according to ISO/TS 6336-22:2018 [21] based on a comparison of the minimum specific lubricant film thickness λGF,min und the permissible specific lubricant film thickness λGFP. The ratio of these values results in a safety factor against mi­ cropitting Sλ. The permissible specific lubricant film thickness λGFP can be obtained with two methods. One method is to determine the permissible specific lubricant film thickness λGFP by experimental

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investigations or service experience relating to micropitting on real gears. The other method is to cal­ culate directly the permissible specific lubricant film thickness λGFP by using the test result from a standardized micropitting test such as the FZG micropitting test according to FVA 54/7 expressed as the SKS. The simpler, cheaper and most widely used method is certainly to use the result of a standardized micropitting test. In Figure 12.32, an overview of the calculation for the safety factor against micro­ pitting is shown by using the result of the micropitting test according FVA 54/7.

Calculation acc. ISO/TS 6336-22 Lubricant and operating conditions (esp. lubricant temperature) Given Gear drive (incl. Geometry, Roughness, Material and Load)

Circumferential speed Gear drive acc. to Lubricant temperature Micropittingtest FVA 54/7 SKS (Failure load stage)

λGF,min

λGFP

Sλ

Micropitting test acc. FVA 54/7

For example C/8.3/90

λGF,min λGFP

FIGURE 12.32 Overview of the calculation of the safety factor against micropitting using the result of the FZG micropitting test according to FVA 54/7 [ 22].

The FZG micropitting test according to FVA 54/7 aims to determine the micropitting load carrying capacity (GFT) of gear lubricants. This also allows a comparison of different lubricants for a specific application. The test procedure is described and specified within the German standard DIN 3990-16:2020 [23] and was published in May 2020. Within this new standard, a new classification system was pro­ posed, including the new GF-class “GFT-very high”.

12.4.6 OUTLOOK A high load carrying capacity regarding micropitting is requested in many modern gear applications. As result more research work is to be expected on this topic, which may lead to a further development of the calculation method according to ISO/TS 6336-22:2018 [21]. For instance, research projects on the in­ fluence of high-speed applications like e-mobility on micropitting are planned. The micropitting resistance in a gear-lubricant system is strongly influenced by the applied lubricant. In order to avoid the appearance of micropitting, a lubricant with a sufficient micropitting load carrying capacity should be used. The load carrying capacity of gear lubricants with additives cannot be determined theore­ tically, since the additives have a strong effect on the micropitting load carrying capacity depending on the operating temperature. As a result, the load carrying capacity of gear lubricants has to be determined by physical testing. The FZG micropitting test according to FVA 54/7 is a well-established test procedure to determine the load carrying capacity regarding micropitting and also the basis of the recently published German standard DIN 3990-16:2020 [23]. This test method was defined about 30 years ago. As a result, the standardized test gears and test conditions used during the testing can differ from commonly used gears in modern applications and the test conditions do not match with all conditions occurring in industrial

Gear Load Capacity Calculation (ISO 6336)

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applications. In the recent research project FVA 779 I [24] the suitability of the test results for modern practical gear applications and the correlation of the test results with different industrial applications were verified. The results have proven that the well-established test procedure delivers reliable results for modern practical gear applications. Possible modifications of the test procedure were proposed in order to simplify, standardize and reduce the measuring and evaluation effort of the test method and to further differentiate oils with a high load carrying capacity regarding micropitting. The research project was recently finished and the results will most likely be published in the near future.

12.5 CALCULATION OF TOOTH FLANK FRACTURE LOAD CAPACITY—ISO/TS 6336-4:2019 The following subsections give information on ISO/TS 6336-4:2019 [13]. ISO/TS 6336-4:2019 is a technical specification describing the calculation of the tooth flank fracture load capacity. The calculation method is outlined in the following subsections relating to the description of tooth flank fracture, the basic calculation principles, further influences on tooth flank fracture, calculation examples, a summary and an outlook on the future of ISO/TS 6336-4.

12.5.1 DESCRIPTION

OF THE

FAILURE MODE TOOTH FLANK FRACTURE

Tooth flank fracture (TFF) also referred to as tooth flank breakage, is for some applications, especially large gear sizes, one of the main fatigue failure modes of gears. Unlike the other fatigue failure modes at the gear flank, the crack start is located well below the active flank surface. The crack usually initiates in an area of reduced material strength and reduced stresses. Since the crack propagates below the surface, the failure is not visible from the outside until the fracture of the tooth flank takes place. Tooth flank fracture can occur on cylindrical spur and helical gears, bevel and hypoid gears as well as internal gears. The failure has occurred on gears with smaller module such as mn = 1.5 mm but more frequently on larger gears up to module mn = 30 mm. Often, gears with a relatively small module and a larger number of teeth are prone to tooth flank fracture if counter measures are not taken into account during the design of the gear. Tooth flank fracture leads to the breakage of a tooth and usually causes severe consequential damages, such as further tooth breakages. Tooth flank fracture can lead to the failure of the entire transmission. Tooth flank fracture can and should be considered in the design phase of gears and a sufficient and adequate safety factor against this damage should be ensured. For the assessment of tooth flank fracture risk, a practical calculation approach was included in ISO/TS 6336-4:2019 [13]. Further partly more complex calculation approaches are proposed [37–42]. Description of Tooth Flank Fracture. The fatigue failure of tooth flank fracture usually occurs after several million load cycles. The fracture characteristics of tooth flank fracture are shown in Figure 12.33. The figure shows a schematic section of a tooth, where the crack initiation point is marked. The arrows indicate the direction and velocity of the crack propagation. The primary crack is initiated in larger material depth below the active flank. Usually, the crack is initiated below the active flank near the pitch point C in material depths of y = 2-3 · CHD550HV where the material strength is reduced. However, the primary crack initiation point can be in larger material depths for gears with larger radius of relative curvature because the crack initiation depth depends not only on the hardness depth profile but also strongly on the radius of relative curvature. Further influences on the crack initiation depth are in­ homogeneities such as larger grains or nonmetallic inclusions, which can exist in the material. The inhomogeneities can displace the crack initiation point in larger or smaller material depths as well as along the tooth height. On gears where the crack initiation is located at a large inclusion, the position of the crack initiation point can vary significantly. After the crack initiation, the crack grows slowly in vacuum with every load cycle, which often leads to a clearly visible fisheye that can later observed in the fracture zone of the broken tooth. In Figure 12.33 also a fracture zone is shown, where a fisheye can be recognized. After varying loads, where the crack

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active flank primary crack and crack initiation

fisheye for constant loading view on fracture zone

fisheye for varying loads fisheye

secondary crack tertiary crack

FIGURE 12.33 Overview of fracture characteristics of tooth flank fracture. Left: crack initiation and propagation. Right: View on the fracture zone with a fisheye and examples for fisheyes after constant and varying loads.

grows discontinuous, beachmarks can be possibly visible in the fisheye, whereas at a constant torque a smooth fisheye forms. The crack growth rate in direction toward the active flank is slower in the case-hardened area with compressive residual stresses than toward the tooth root of the unloaded flank side. The lower material strength in the core area and the lower compressive residual stresses or possibly tensile residual stresses cause a faster crack propagation. The crack propagates further within approximately 105 load cycles [43]. At some point, the crack has advanced so far, that the tooth finally breaks with a brittle and sudden rupture. Until the tooth flank fracture takes place, the crack is not visible from the outside. The crack path of the primary crack mostly has a approx. ca. 45° angle to the active flank. At the unloaded flank side the primary crack path can end at the dedendum flank area but often runs below the tooth root fillet and eventually back up to end in the middle of the tooth root fillet as shown in Figure 12.33. The primary crack can be accompanied by secondary and tertiary cracks as shown in Figure 12.33 as well as further smaller brittle fractures, which presumably start at the active flank. The fracture often does not affect the whole face width of the gear, especially for gearings with larger face width or for helical gears. For such gears, mostly only a part of the face width breaks. Tooth Interior Fatigue Fracture. Tooth interior fatigue fracture (TIFF) is a similar failure mode but occurs usually on reverse loaded gears such as idler gears. The name “tooth interior fatigue fracture” was first defined by MackAldener [44] as a observed failure on thin idler gears. The crack initiation point is comparable to tooth flank fracture underneath the surface. However, the main crack propagation is in tangential direction in the lower strength core area of the tooth. The formation of a plateau is presumably caused by the alternating contact stresses on both flank sides in combination with the alternating sec­ ondary bending stresses [43]. Figure 12.34 shows the fracture characteristics of tooth interior fatigue failure with crack initiation and crack propagation. The mechanisms and influences on the failure mode tooth flank fracture and tooth interior fatigue failure are assumed comparable for crack initiation. The crack propagation differs significantly for these failure modes, which is presumably caused by secondary bending stresses on both flank sides.

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FIGURE 12.34 Overview of fracture characteristics of tooth interior fatigue fracture with crack initiation and propagation.

12.5.2 BASIC CALCULATION PRINCIPLES The calculation approach of ISO/TS 6336-4 [13] is based on a practical approach (WgFB) developed by FZG/Witzig [45]. This practical approach is derived from a complex material-physically based model, which was developed by FZG/Oster [46] and continuously extended at FZG. This physical model compares the local equivalent shear stress of the shear stress intensity hypothesis (SIH) [47] with the local material strength. The calculation result is a material exposure depth profile at a given contact point on the path of contact. This model was verified based on experimental investigations in test rigs as well as reported failures in the industry. An overview of this model is given in [48]. From this high-order model, the practical calculation approach “WgFB” was derived by Witzig [45], which is the basis of today’s ISO/TS 6336-4 calculation approach. To validate the calculation results of the simplified calculation approach, approx. 1.5 M calculations where performed and compared with the high-order model. The input factors were varied for case carburized involute spur and helical gears (as specified in ISO 53 [49]) within the following limits: • Hertzian stress: 500 N/mm² ≤ pH ≤ 3000 N/mm². • Radius of relative curvature: 5 mm ≤ ρred ≤ 3000 mm. • Case hardening depth at 550 HV in finished condition: 0.3 mm ≤ CHD ≤ 4.5 mm. After the validation, this practical approach was implemented in the detailed standard calculation ap­ proach of ISO/TS 6336-4 in 2018. Calculation approaches for hardness depth profiles and the calculation of the maximum contact stress with consideration of flank profile modifications were added. The calculation approach of ISO/TS 6336-4 is based on the following steps: • • • •

Calculation Calculation Calculation Calculation

of of of of

hardness and residual stresses depth profiles material strength depth profile the equivalent stress depth profiles material exposure depth profile from equivalent stress and strength depth profile

Unlike, other calculation approaches of ISO 6336 [1], the result is not a safety factor based on the occurring stresses and the permissible stress. The calculation of ISO/TS 6336-4 results in a material exposure depth profile AFF,CP(y) at a considered contact point on the tooth flank. With the calculated maximum material exposure in larger material depths, it is possible to derive statements about the tooth flank fracture load capacity or risk of tooth flank fracture. Only maximum material exposures in depths larger than the Hertzian half width allow a statement on the risk of tooth flank fracture. The Hertzian half width can be calculated according to Eq. (12.53)

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bH , CP = 4

pdyn, CP red , CP

Er

(12.53)

where: bH,CP pdyn,CP ρred,CP Er

– – – –

is is is is

half of the Hertzian contact width at the contact point CP. the local Hertzian contact stress at the contact point CP. the local normal radius of relative curvature at the contact point CP. the reduced modulus of elasticity.

A calculated global maximum material exposure in a depth of y ≤ bH indicates that a near surface failure such as pitting is more likely. At the same time, it does not allow any conclusions to be drawn about the risk of pitting damage because near surface influences are not considered in the calculation approach for tooth flank fracture. The surface roughness and the sliding velocity (rotational speed of the gear) do affect pitting failure but not subsurface failures like tooth flank fracture. If reliable data on the limiting shear strength and the residual stress state in bigger material depths is available, the material-physically based model is applicable to all steel materials and heat-treatment processes. However, the practical approach for the calculation of the load capacity against tooth flank fracture has so far been validated for case carburized gears only. Therefore, the calculation of the load capacity against tooth flank fracture according to ISO/TS 6336-4 is only valid for case carburized gears at the moment. Contact Point Definition. In the calculation approach of ISO/TS 6336-4, the location of the contact points, which should be investigated on the active flank are not specified. Usually, seven contact points on the contact path as shown in Figure 12.35 are investigated, although it is usually sufficient to calculate the material exposure depth profile at the pitch point and for the points B and D, which are located next to the pitch point. Using modern software tools, it is possible to analyze the complete area of meshing.

FIGURE 12.35

Contact points on the active flank.

The calculation of the material exposure depth profile for each considered contact point, which in­ cludes a flank of the pinion and the wheel. The depth is always assumed normal to the flank at the contact point. The calculation should be performed up to the middle of the chordal tooth thickness. Therefore, hardness and residual stress depth profiles should also be calculated up to the middle of the chordal tooth thickness. The input parameters in this calculation approach such as the contact pressure are identical at the pitch point for pinion and wheel, so that only one gear can be investigated provided both, pinion and wheel, have the same material and hardness profiles. Calculation of the Material Exposure Depth Profile. For the assessment of the tooth flank fracture load capacity the material exposure depth profile is calculated. In Table 12.22 an overview of main calculation methods for different parameters as specified in ISO/TS 6336-4 is given. The methods are further explained in the following sections.

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TABLE 12.22 Overview of Main Calculation Methods Specified in ISO/TS 6336-4 Calculation Methods

Symbol

Method

Hardness depth profile

HV (y)

A

Description Not yet specified

B

Approach of Thomas based on measured hardness values

C1 C2

Approach of Lang Approach of Thomas based on design parameters

Residual stresses depth profile

σRS (y)

A

Based on measurement

Local Hertzian contact stress

pdyn,CP

B A

Approach of Lang based on hardness depth profile Calculation based on maximum contact stress along face width determined from 3D-pressure distribution on the gear flank

B

Based on input torque with load sharing factor XCP

B B

Unmodified profiles Adequate profile modifications

Local load sharing factor

XCP

The material exposure depth profile at a contact point is calculated according to (12.54) based on a comparison of the occurring stresses including load induces stresses, compressive residual stresses and the material strength. AFF, CP (y ) =

eff , L, CP (y )

–

eff , L , RS , CP (y ) per , CP (y )

–

eff , RS (y )

+ c1

(12.54)

where: AFF,CP(y) y CP τper,CP(y) τeff,L,CP(y) Δτeff,L,RS,CP(y) τeff,RS(y) c1

– – – – – – – –

is is is is is is is is

the the the the the the the the

is the local material exposure in the material depth y for contact point CP. local material depth in the contact point CP. considered contact point. local material shear strength in the material depth y for contact point CP. local equivalent stress state without consideration of residual stresses. influence of the residual stresses on the local equivalent stress state. quasi-stationary residual stress state in the material depth y. material exposure calibration factor, which is 0.04 for case carburized steels.

Local occurring stresses τeff,L, Δτeff,L,RS, τeff,RS. The local occurring stresses consist of three equivalent stress state depth profiles: the local equivalent stress state without consideration of residual stresses τeff,L, the influence of the residual stresses on the local equivalent stress state Δτeff,L,RS and the quasi-stationary residual stress state τeff,RS. The occurring stresses are mainly influenced by the Hertzian contact pressure and the residual stresses depth profile. The Hertzian contact stress pdyn,CP is decisive for the calculated material exposure and considered in the calculation of the local occurring equivalent stress state. According to ISO/TS 6336-4 Method A, the local Hertzian contact stress at the contact point can be calculated based on a 3D elastic contact model. From the calculated contact pressure distribution the maximum contact stress over the face width at a contact point should be used as pH,CP,A. In addition, the application factor KA (KFFA-A)

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and the dynamic factor Kv are taken into account, which are defined in ISO 6336-1:2019. The local occurring equivalent stress for method A is calculated with Equation (12.55). pdyn, CP, A = pH , CP, A

KA KV

(12.55)

where: pdyn,CP,A pH,CP,A KA KV

– – – –

is is is is

the the the the

local Hertzian contact stress. local nominal Hertzian contact stress, calculated with a 3D load distribution program. application factor (according to ISO 6336-1). dynamic factor (according to ISO 6336-1).

If no 3D elastic contact model is available there is a sufficiently accurate Method B stated in ISO/TS 6336-4, which calculates the nominal contact stress pH,CP,B at all contact points based on the applied torque with consideration of possible profile modifications. Method B usually shows comparable results to a 3D elastic model for smaller as well as larger gears but is an approximation nonetheless. The load sharing factor XCP can be calculated for unmodified profiles and different adequate profile mod­ ifications. If no information about the applied profile modification on helical gears is available, the load sharing factor XCP should be calculated with the assumption of an adequate profile modification. The local nominal Hertzian contact stress according to Method B is calculated according to Eq. (12.56). pH , CP, B = ZE

b

Ft XCP red , CP cos

t

(12.56)

where: pH,CP,B XCP Ft ρred,CP b αt ZE

– – – – – – –

is is is is is is is

the the the the the the the

local nominal Hertzian contact stress (Method B). load sharing factor. transverse tangential load at reference cylinder. local normal radius of relative curvature. face width. transverse pressure angle. elasticity factor (according to ISO 6336-2).

The local occurring equivalent stress for Method B is calculated with Eq. (12.57). pdyn, CP, B = pH , CP, B

KA K KV KH KH

(12.57)

where: pdyn,CP,B pH,CP,B KA Kγ KV KHα KHβ

– – – – – – –

is is is is is is is

the the the the the the the

local Hertzian contact stress. local nominal Hertzian contact stress, calculated based on the applied torque. application factor (according to ISO 6336-1). mesh load factor (according to ISO 6336-1). dynamic factor (according to ISO 6336-1). transverse load factor (according to ISO 6336-1). mesh load factor (according to ISO 6336-1).

Based on the local Hertzian contact stress the local equivalent stress without considering of residual stresses can be calculated τeff,L,CP with Eq. (12.58). The resulting equivalent stress depth distribution is an approximation of the contact stresses without consideration of residual stresses.

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eff , L, CP (y )

=

y 2 Er

y Er

0.149 pdyn, CP +

4

red , CP

0.4

16

2 red . CP

y Er 4

red , CP

pdyn,CP

(

)+

pdyn, CP 2 Er

y 4

red , CP

+ 1.54

2

(12.58)

where: τeff,L,CP(y) CP y pdyn,CP Er ρred,CP

– – – – – –

is is is is is is

the the the the the the

local equivalent stress without considering of residual stresses. considered contact point. local material depth in the contact point CP. local Hertzian contact stress. is the reduced modulus of elasticity. local normal radius of relative curvature.

The influence of residual stresses in the local equivalent stress state is calculated with Eq. (12.59). In the residual stresses depth profile σRS(y) only compressive residual stresses can be considered yet, therefore the residual stresses depth profile σRS(y) must consist of only compressive or zero residual stresses: eff , L, RS , CP (y )

= K1

RS (y )

100

32 tanh(9 y1.1)

K2

(12.59)

where: ∆τeff,L,RS,CP(y) CP y σRS(y) K1, K2

– – – – –

is the influence of the residual stresses on the local equivalent stress. is the considered contact point. is the local material depth in the contact point CP. is the residual stresses depth profile. are adjustment factors, for which the equations are stated in ISO 6336-4.

The residual stress depth profile σRS(y) can be calculated based on the hardness according to Method B. For the calculation of the residual stress depth profile an improved approach of Lang [50] is stated in ISO/TS 6336-4. As described in method A, a measured residual stress depth profile can also be used. The residual stress depth profiles can be measured for example with an X-ray diffractometer and layer by layer removal of the flank surface by electrochemical erosion. The residual stresses have to be measured up to larger material depths (approx. y = 5 ∙ bH or 3 ∙ CHD550HV). In larger material depths (approx. y ≥ 0.5 mm), the effect of the electrochemical erosion on the residual stresses has to be considered with an appropriate correction. A near surface residual stress measurement is not sufficient for a correct cal­ culation of the material exposure in larger depths. The calculation approach does not yet consider tensile residual stresses, only negative (compressive) stresses can be considered. With the residual stress depth profile, the quasi-stationary assumed residual stress τeff,RS(y) can be calculated with Eq. (12.60). eff , RS (y )

where: τeff,RS – is the quasi-stationary residual stress state. y – is the material depth. σRS(y) – is the residual stresses depth profile.

=

2 15

RS (y )

(12.60)

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Dudley’s Handbook of Practical Gear Design and Manufacture

Material Strength. The local material strength τper,CP(y) of case carburized gears fulfilling the re­ quirements according to ISO/TS 6336-5 with a material quality of MQ is calculated with the hardness conversion factor Kτ,per and the material factor Kmaterial based on the hardness depth profile HV(y). Both factors are based on experimental investigations and available practical experience. The local material strength is calculated with (12.61). per , CP (y )

=K

, per

Kmaterial HV (y )

(12.61)

where: τper,CP(y) y Kτ,per Kmaterial HV(y)

– – – – –

is is is is is

the the the the the

local material shear strength in the material depth y for contact point CP. material depth. hardness conversion factor. material factor. hardness at the material depth y (hardness depth profile).

The hardness conversion factor is set to Kτ,per = 0.4 according to the calculation method. This is based on the calibration of the calculation approach for case carburized gears with usual material and mi­ crostructure properties. The material factor is in the range of Kmaterial = 0.7-1.13 depending on the material tensile strength and the normal chordal tooth thickness between the contact points B and D. This factor can be different for pinion and wheel. The factor Kmaterial takes into account the material properties and the size of the tooth (module of the gear). The size of the tooth has to be considered because of the technological size effect, which causes a strength reduction for larger components due to a higher risk of larger microstructure inhomogeneities. This factor does not consider unusual inhomogeneities in the microstructure or different sized nonmetallic inclusions. The technological size effect of the tooth face width is not yet considered. If there are more or larger known inhomogeneities in the material, a lower factor should be used to be on the safe side, especially for larger gears. The hardness depth profile HV(y) for the calculation of the material strength can be calculated with the calculation approaches stated in ISO 6336-4 Method C1 and C2 or the calculation approaches can be used to approximate the measured hardness depth profile (Method B). Methods C1 and C2 require only a limited number of hardness values as input data, e.g., surface hardness, CHD and core hardness. This data can be derived from hardness measurements or specified numbers are used in the design phase. Method B requires representative measured hardness depth profiles. The hardness depth profile HV(y) has significant influence on the calculated maximum material exposure. Often, the residual stresses are calculated according to Method B with an improved version of the approach of Lang [50,51] based on the hardness depth profile, which creates even further impact of the hardness on the material exposure. Therefore, the hardness profile should be as accurately as pos­ sible. It is best to be taken from hardness depth profile measurements at the flank point and then either interpolated and smoothed or approximated with a hardness calculation approach as stated in method B. A calculation of the material strength τper,CP(y) with the interpolated measured hardness depth profile values is not recommended. There are usually fluctuations and discontinuities in the measured hardness depth profile, which may have a strong effect on the calculated material exposure and can lead to unreasonable results. There are several calculation approaches for hardness depth profiles of case carburized gears available [52–54]. These approaches can be used to approximate the measured hardness depth profile. If no measurements are available, these calculation approaches can be used to calculate a hardness depth profile based on the surface hardness, core hardness and the case hardening depth for 550 HV. The

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calculation approach of Lang [53] as stated in ISO/TS 6336-4 is likely to fit well for smaller gears with smaller case hardening depths. The calculation approach of Thomas [54] is also stated in ISO/TS 6336-4 and was developed for larger gears with larger case hardening depths. The calculation approach of Thomas hardness depth profiles with a plateau near the surface, where the hardness is approximately constant can be approximated with the y-coordinate of the maximum hardness yHV,max. In Figure 12.36 a measured hardness depth profile is shown as well as two calculated hardness depth profiles. These hardness depth profiles are calculated based on the measured values: surface hardness, core hardness and case hardening depth CHD550HV. Both approaches lead to comparable hardness depth profiles, when compared to the measurements. However, the approach of Thomas gives a better ap­ proximation of the lower measured hardness values in larger material depths. The application of the approach of Lang in the example of Figure 12.36 may lead to overestimation of the hardness in larger material depths and therefore to underestimation of the risk of tooth flank fracture. In the design phase of the gear, in which no measured hardness profile is yet available, calculation studies with different calculated hardness profiles should be performed.

Approach of Lang Approach of Thomas Measurement

700

Hardness in HV

600

500

400

HVsurface = 700 HV, HVcore = 380 HV,

300

CHD = 1.00 mm, yHV,max = 0.20 mm

200 0

0.5

1

1.5

2

2.5

3

Depth in mn

FIGURE 12.36 values.

Exemplary application of hardness depth profile calculation approaches on measured hardness

12.5.3 INFLUENCES

ON

TOOTH FLANK FRACTURE

Tooth flank fracture was theoretically and experimentally investigated at FZG (Technical University of Munich) with test gearings for tooth flank fracture with a module of mn = 3 and 5 mm on back-to-back test rigs with center distance a = 200 mm [33,45,55]. Experimental investigations with a test gearing of module mn = 8 mm were carried out at the WZL (RWTH Aachen University) [40]. In addition, theo­ retical investigations and recalculations of tooth flank fractures in failed industry gearboxes exist [56,57]. The main influences on tooth flank fracture are the subsurface stress condition and the local strength in larger material depth. The subsurface stress condition is a multiaxial phase-shifted stress condition caused by the rolling-sliding contact and depends on the gear geometry and the load distribution on the path of contact [58]. The subsurface stresses consist mainly of the contact stresses (Hertzian Contact Stresses) and partly on the lower secondary stresses, such as normal and shear stresses caused by tooth

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bending. The subsurface contact stresses are influenced by the local contact pressure at the contact point, which results from the applied pressure distribution. The main shear stresses according to Hertz are assumed to be decisive for the crack initiation. The maximum main shear stress is located in the depth of approx. y = 0.8 bH and has a magnitude of approx. τmax = 0.3 pH. A larger radius of relative curvature places the maximum stress in lager material depths. However, tooth flank fracture is typically initiated at material depths further below the flank surface than the location of the maximum main shear stress according to Hertz. In large material depth, the shear stresses decrease but also the material strength of case carburized gears is reduced due to the decreasing hardness and reduced residual stresses. [48,59] The occurring secondary stresses are often significantly lower than the contact stresses. The normal stresses from bending with their maximum stress at the flank surfaces are presumably not decisive for the initiation of the subsurface crack. However, the bending stresses as well as the shear stresses due to tooth bending with their maximum stress in the middle of the chordal tooth thickness can presumably have an impact on the crack propagation as well as the crack initiation for some gear geometries [59]. The local material strength in larger material depths depends on the gear material. The hardness depth profile is assumed to be decisive on the load capacity concerning tooth flank fracture. Theoretical in­ vestigations suspect a larger case hardening depth CHD550HV leads to a higher tooth flank fracture load capacity [60,61]. An optimal CHD550HV, such as for tooth breakage and for pitting, has to be assumed but not yet defined. The hardness depth profile at the flank, especially in larger material depths can influence the risk of tooth flank fracture significantly. A higher hardness in larger material depths reduces the risk of tooth flank fracture. In addition, materials with higher hardenability showed higher tooth flank fracture load capacity for a comparable material quality [45]. The material quality and the micro-structure are sus­ pected to have significant impact on the tooth flank fracture load carrying capacity. Research projects such as FVA 848 I (ongoing) [62] and FVA 293 IV (ongoing) [63] show, that less or only small nonmetallic inclusion (NMI), a smaller grain size and homogenous micro-structure can increase the tooth flank fracture strength significantly. Another main influence on tooth flank fracture are the residual stresses. The residual stresses depth profile is currently often derived from the hardness depth profile according to the approach of Lang [50]. Compressive residual stresses near the flank surface prevent crack initiation in this area. However, in the transition area of the case-carburized layer to the core, the compressive residual stresses are close to zero and tensile residual stresses have to be assumed in even larger material depth of the core. Influences of the lubricant or surface roughness only affect the near surface area and so far have shown no impact on tooth flank fracture load capacity. In experimental investigations of the author, no influence of the rotational speed was observed. The driving direction could have an impact for some gear geometry’s but a general statement is not yet possible.

12.5.4 CALCULATION EXAMPLE Example calculations concerning the tooth flank fracture risk according to ISO/TS 6336-4 are described for two different gearsets in the following. The gearset with a normal module of mn = 3 mm gearing was experimentally investigated in a backto-back test rig and the endurance limit regarding tooth flank fracture was determined [45,48]. In con­ trast, the pinion of the module mn = 5 mm gearing failed in experimental investigations in a back-to-back test rig due to pitting damage only [64]. For each gearsets the material exposure depth profile is calculated at the pitch point C according to ISO/TS 6336-4. In Table 12.23 the input parameters for the calculation with ISO/TS 6336-4 are listed. First, based on the gear geometry the radius of relative curvature at the pitch point C is calculated. The local Hertzian contact stress is calculated based on the torque (Method B) for unmodified flanks with consideration of the K-factors such as the load distribution factors. Especially, for the calculation of

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TABLE 12.23 Calculation Input Parameters and Calculation Methods Symbol

Description

Unit

Pinion

Wheel

Combined Geometry

Material strength

Operational Data

z mn

Number of teeth Normal module

mm

αn

Normal pressure angle

β b

Helix angle Face width (common)

67

69

Pinion

Wheel

Combined 17

18

3

5

°

20

20

° mm

0 18

0 14

a

Center distance

mm

x da

Profile shift coefficient Tip diameter

– mm

ρred,C

Radius of rel. curvature at Pitch point

mm

A HVsurface

Tolerance class acc. to ISO 1328-1 Surface hardness

HV

HVcore

Core hardness

CHD HV(y)

CHD550 Hardness depth profile

200 −0.61 201.2

91.5

−0.6169 207.16

0.475 99.75

14.25 5

0.445 104.45 10.03

5

5

5

690

650

HV

410

450

mm –

0.48 Method B: Lang

0.9 Method B: Lang

σRS(y)

Residual stress depth profile

–

Method B: Lang

Method B: Lang

Kmaterial T

Material factor Torque on pinion

– Nm

1.0 1360

1.13 456

KA

Application factor

-

1

1

Kv KHα

Dynamic factor Transverse load factor

-

1.05 1

1.04 1

KHβ

Face load factor

-

1.05

1.03

Kγ pH,CP

Mesh load factor Pressure values

N/mm²

1 Method B

1 Method B

XCP

Load sharing factor (Unmodified profiles)

-

1

1

pdyn,C

Hertzian contact stress at pitch point

N/mm²

1445

1789

helical gears, it is recommended to calculate the maximum Hertzian contact stress with a mesh load distribution program (Method A) or a suitable profile modification should be considered with help of the load sharing factor XCP. For the recalculation of tests in a test rig with constant torque an application factor of KA = 1.0 is used. For applications with changing loads a sufficient application factor according to ISO 6336-1 should be considered. The depth profiles of the hardness and residual stresses are calculated based on the approach of Lang as stated in ISO/TS 6336-4. The material factors are calculated based on the tensile strength of the materials and the chordal tooth thickness between contact point B and D. In Figure 12.37, the calculated hardness and residual stress depth profiles are shown as well as the calculated material exposure depth profile at the pitch point C for the module mn = 3 mm gearing. This gearing failed due to tooth flank fracture in all test runs that were performed within the experimental investigations. In Figure 12.38 the material exposure depth profile at the pitch point C for the module mn = 5 mm gearing is shown. This gearing did not fail due to tooth flank fracture but due to pitting. Calculated material exposure depth profiles can usually be roughly divided into three different types.

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FIGURE 12.37 Example calculation at the pitch point C of a pinion failed due to tooth flank fracture in ex­ perimental investigations (example in Table 12.23).

FIGURE 12.38 Example calculation at the pitch point C of a pinion failed due to Pitting in experimental in­ vestigations (example in Table 12.23).

Material exposure depth profile type 1 shows a clear single maximum, which is located in the near surface area in depths of approx. y = 0-2∙bH. This is also the case for the example calculation of the module mn = 5 mm gearing in Figure 12.38, which failed due to pitting damage. The height of the near surface maximum does not allow any conclusions regarding pitting load carrying capacity because the surface related influences are not considered in the calculation. However, the near surface maximum indicates a risk regarding surface or near surface failure such as pitting instead of tooth flank fracture. Material exposure depth profile type 2 has a single flatter maximum in a depth of y > 1 ∙ bH. This is the case for the gearset shown in Figure 12.37. If the maximum material exposure is close to AFF,max = 0.8 or even higher, tooth flank fracture in experimental investigations and failure analysis was observed. A max­ imum material exposure of AFF,max = 0.8 corresponds to the fatigue strength with 50% failure probability determined with FZG back-to-back test rigs at constant input torque. This means AFF,max = 0.8 corresponds to a safety factor of STFF = 1.0 against tooth flank fracture. The location or depth of the calculated maximum usually matches the observed crack initiation point of the tooth flank fracture damage. The material exposure depth profile type 3 has two clear maxima, where the first maxima is in the near surface area and the second maximum is located in a larger material depth. In this case, the height of the second maxima in a larger material depth is used for the evaluation of the tooth flank fracture risk. However, it should be noted, that the gearing could fail due to surface as well as sub-surface-initiated failure. A maximum material exposure in larger material depths of AFF,max = 0.8 should also not be exceeded for type 3. The material properties in larger material depths are often not precisely predictable. If larger material inhomogeneities or larger inclusions are possible at this material depth, a significantly lower maximum material exposure should be specified for the design of the gearset. Nevertheless, for typical gear steels and material qualities, the defined limit of AFF,max = 0.8 has proven applicable in the past.

Gear Load Capacity Calculation (ISO 6336)

749

Nonetheless, for practical applications, a minimum safety factor against tooth flank fracture of STFF > 1.0 is strongly recommended.

12.5.5 SUMMARY Tooth flank fracture is a fatigue failure mode, which usually leads to breakage of the tooth and therefore most likely to the total failure of the gearset. Tooth flank fracture usually occurs after several million load cycles. The failure is mainly characterized by a crack initiation in larger material depths below the surface of the loaded gear flank. It occurs in the area of reduced contact stresses and reduced material strength in depths of approx. y = 2 – 3 ∙ CHD550HV. The main influencing factors are the contact stresses, the relative radius of curvature of the gear, the residual stresses and the material strength, which is calculated based on the hardness depth profile. These influencing factors are currently considered in the calculation approach of ISO/TS 6336-4:2019 [13]. With the approach of ISO/TS 6336-4, the tooth flank fracture load capacity can be calculated for cylindrical spur and helical gears with external teeth. Based on the gear properties, such as geometry and material properties as well as operational data, a material exposure depth profile is calculated. The calculated material exposure depth profile is then used for the assessment of the risk of tooth flank fracture. A material exposure maximum in larger material depths indicates a risk of tooth flank fracture. A maximum material exposure in larger depths of Amax(y > bH) = 0.8 corresponds to the endurance limit determined in test-rigs and also shows good correlation with tooth flank fractures observed in different applications. A minimum safety factor of STFF > 1.0 is strongly recommended for practical applications. However, the approach is yet only validated for case carburized gears with specifications within certain limits. These limits as well as the use or application of ISO/TS 6336-4 are subject to further in­ vestigations, review, and development.

12.5.6 OUTLOOK The calculation method in ISO/TS 6336-4 is relatively new compared to the other calculation methods of the main gear failure modes such as pitting in ISO 6336-2 or tooth root breakage in ISO 6336-3. The calculation method was based on systematic experimental investigations with test rigs and recalculations of failures in different applications in the industry. The technical specification was published in order to give broad access to the calculation method and for companies to give feedback on the accuracy, desired improvements, or extensions. The main influences are already taken into account, but a large number of further influencing factors are insufficiently or not fully taken into account due to a lack of systematic experimental investigations. One of the biggest challenges is on the side of the material strength, where the strong influence of different non-metallic inclusions should be considered properly. A first step was taken in the work of Wickborn [65], where a factor Kdefect was proposed to consider the influences of different sized nonmetallic inclusions in the calculation of the material strength. Materials with high purity showed a significant increase of tooth flank fracture load capacity in recent researches at FZG, where a first step was accomplished by developing a method to consider different non-metallic inclusions in the calcu­ lation of tooth root bending strength. The influence of the micro-structure, especially the grain size and distribution but also segregations, can have a significant effect on the tooth flank fracture load capacity, as a few fatigue tests showed in recent researches at FZG. Furthermore, the technological size effect of large gears can lead to a reduction of the tooth flank fracture load capacity and is not fully investigated yet. Another major task is to calculate residual stress depth profiles with consideration of tensile residual stresses in the core area of a case-hardened gear reliably and also to consider the tensile residual stresses in the calculation approach. Research is ongoing to develop a new calculation approach for residual

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stress depth profiles based on numerical studies with different case-carburizing processes and validation by experimental investigations. For the calculation of induction hardened, nitrided, or carbonitrided gears, the calculation approach is basically applicable. However, calculation methods for representative hardness and residual stress depth profiles are not available yet. For induction hardened gears, research is ongoing. The application factor KFFA for tooth flank fracture is described in ISO 6336-1 but not yet specified. It is already observed, that overloads can lead to a reduced tooth flank fracture endurance strength but not yet systematically investigated or quantified [45]. Until systematic research is done, the application factor of the pitting or bending strength calculation is recommended. The application of ISO/TS 6336-4 for internal gears should be possible with the current calculation approach, if the relative radius of curvature is used as input value. Nonetheless, systematic research regarding tooth flank fracture on internal gears is still outstanding. In summary, the presented calculation methods according to ISO/TS 6336-4 allow the calculation of the risk concerning tooth flank fracture for state-of-the-art gears. The results should be evaluated carefully and compared to field experience and regarding differences in the design. Further influences shall be included in the future to improve the reliability and applicability of the existing methods. Appropriate minimum safety factors against tooth flank fracture shall be chose to cover the simplifi­ cations of the current calculation methods.

REFERENCES [1] International Organization for Standardization, Since. (1996). ISO 6336 Series - Calculation of load ca­ pacity of spur and helical gears. Standard, Technical Specification, Technical Report ISO 6336 Series. Beuth Verlag GmbH, Berlin. [2] Deutsches Institut für Normung e.V. (DIN), Since. (1970). DIN 3990 Series - Tragfähigkeitsberechnung von Stirnrädern [Eng: load capacity calculation of cylindrical gears]. Standards DIN 3990 Series. Beuth Verlag GmbH, Berlin. [3] American National Standards Institute/ American Gear Manufacturing Association. (2016). Fundamental rating factors and calculation methods for involute spur and helical gear teeth. Standard ANSI/AGMA 2001-D04 (R2016). ANSI, USA. [4] International Organization for Standardization. (2019). Calculation of load capacity of spur and helical gears - Part 2: Calculation of surface durability (pitting). Standard ISO 6336-2:2019. Beuth Verlag GmbH, Berlin. [5] International Organization for Standardization. (2020). Gears -wear and damage to gear teeth Terminology. Standard ISO 10825:2020 (under revision). Beuth Verlag GmbH, Berlin. [6] Niemann, G., Winter, H., & Höhn, B.-R. (2003). Maschinenelemente. Band 2: Getriebe allgemein, Zahnradgetriebe - Grundlagen, Stirnradgetriebe; Machine Elements - Volume 2: General Transmissions, Gearboxes - Fundamentals, Spur Gears. Springer Berlin Heidelberg. [7] International Organization for Standardization. (2019). Calculation of load capacity of spur and helical gears - Part 1: Basic principles, introduction and general influence factors. Standard ISO 6336-1:2019. Beuth Verlag GmbH, Berlin. [8] Linke, H., Börner, J., & Heß, R. (2016). Cylindrical Gears. Calculation - Materials - Manufacturing. Hanser, Munich. [9] Stahl, K. (2001). Grübchentragfähigkeit einsatzgehärteter Gerad- und Schrägverzahnungen unter be­ sonderer Berücksichtigung der Pressungsverteilung [Eng: pitting load capacity of case-hardenes spur- and helical gears taking into account the Hertzian pressure distribution] [PhD Thesis], Gear Research Center (FZG), Technical University of Munich. [10] Steinberger, G. (2006). Optimale Grübchentragfähigkeit von Schrägverzahnungen [Eng: optimal pitting load capacity of helical gears] [Dissertation], Gear Research Center, Technical University of Munich. [11] International Organization for Standardization. (2016). Calculation of load capacity of spur and helical gears - Part 5: Strength and quality of materials. Standard ISO 6336-5:2016. Beuth Verlag GmbH, Berlin.. [12] International Organization for Standardization. (2006). Calculation of load capacity of spur and helical gears Part 2: Calculation of surface durability (pitting). Standard ISO 6336-2:2006. Beuth Verlag GmbH, Berlin. [13] International Organization for Standardization. (2019). Calculation of load capacity of spur and helical

Gear Load Capacity Calculation (ISO 6336)

[14]

[15]

[16] [17]

[18]

[19] [20]

[21] [22]

[23]

[24]

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[26]

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[29] [30]

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gears - Calculation of tooth flank fracture load capacity. Technical Specification ISO/TS 6336-4:2019. Beuth Verlag GmbH, Berlin. International Organization for Standardization. (2020). Calculation of load capacity of spur and helical gears - Part 30: Calculation examples for the application of ISO 6336 parts 1,2,3,5. Standard ISO/TR 6336-30:2020 (under revision). Beuth Verlag GmbH, Berlin.. Verein Deutscher Ingenieure. (2014). Thermoplastische Zahnräder - Stirnradgetriebe - Tragfähigkeitsberechnung [Eng: thermoplastic gears - cylindrical gears - load capacity calculation]. VDI-Guideline VDI 2736 Blatt 2. Beuth Verlag GmbH, Berlin.. International Organization for Standardization. (2019). Calculation of load capacity of spur and helical gears - Part 3: Calculation of tooth bending strength. Standard ISO 6336-3:2019. Beuth Verlag GmbH, Berlin.. International Organization for Standardization. (2006). Calculation of load capacity of spur and helical gears - Part 3: Calculation of tooth bending strength. Standard ISO 6336-3:2006. Beuth Verlag GmbH, Berlin.. Otto, M. (2009). Lastverteilung und Zahnradtragfähigkeit von schrägverzahnten Stirnrädern [Eng. load distribution and load capacity of helical gears] [PhD Thesis], Gear Research Center (FZG), Technical University of Munich. Schinagl, S. (2002). Zahnfußtragfähigkeit schrägverzahnter Stirnräder unter Berücksichtigung der Lastverteilung [PhD Thesis], Gear Research Center (FZG, Technical University of Munich. Verein Deutscher Ingenieure. (2016). VDI 2737 Berechnung der Zahnfußtragfähigkeit von Innenverzahnungen mit Zahnkranzeinfluss [Calculation of the load capacity of the tooth root in internal toothings with influence of the gear rim]. VDI-Guideline VDI 2737. Beuth Verlag GmbH, Berlin. International Organization for Standardization. (2018). Calculation of load capacity of spur and helical gears - Part 22: Calculation of micropitting load capacity. Technical Specification ISO/TS 6336-22:2018. Beuth Verlag GmbH, Berlin. Schönnenbeck, G. & Emmert, S. (1993). FVA-Nr. 54/I-IV - Mirco-Pitting Infoblatt englisch. Test pro­ cedure for the investigation of the micro-pitting capacity of gear lubricants. Forschungsvereinigung Antriebstechnik e.V., Frankfurt/Main. Deutsches Institut für Normung e.V. (DIN). (2020). Tragfähigkeitsberechnung von Stirnrädern - Teil 16: Bestimmung der Graufleckentragfähigkeit von Schmierstoffen im FZG-Prüfverfahren GT -C/8,3 /90 [engl.: Calculation of load capacity of cylindrical gears – Part 16: Determination of the micro-pitting loadcarrying capacity of lubricants using FZG-test-method GT-C/8,3/90]. DIN 3990-16.. Sagraloff, N., Tobie, T., & Stahl, K. (2020). “FVA 779 I - FVA-Heft 1408 - Graufleckentragfähigkeit praxisnaher Verzahnungen – Anwendbarkeit des FVA-Graufleckentests als praxisnaher Anwendungstest [engl.: Micro-pitting load carrying capacity of practice-oriented gears – Applicability of the test method FVA micro-pitting test for common industrial gear applications],” Forschungsvereinigung Antriebstechnik e.V., 2020 (2020). Felbermaier, M. (2018). Untersuchungen zur Graufleckenbildung und deren Einfluss auf die Grübchentragfähigkeit einsatzgehärteter Stirnräder [engl.: Investigations on micropitting and its influence on the pitting load capacity of case-hardened cylindrical gears] [PhD Thesis], Gear Research Center (FZG), Technical University of Munich. Schrade, U. (1999). Einfluß von Verzahnungsgeometrie und Betriebsbedingungen auf die Graufleckentragfähigkeit von Zahnradgetrieben [engl.: Influence of gear geometry and operating condi­ tions on the micropitting load carrying capacity of gears] [PhD Thesis], Gear Research Center (FZG), Technical University of Munich. Schönnenbeck, G. (1984). Einfluß der Schmierstoffe auf die Zahnflankenermüdung (Graufleckigkeit und Grübchenbildung) hauptsächlich im Umfangsgeschwindigkeitsbereich 1…9 m/s [engl.: Influence of lu­ bricants on tooth flank fatigue (micropitting and pitting) mainly in the circumferential speed range of 1…9 m/s] [PhD Thesis], Gear Research Center (FZG), Technical University of Munich. Hergesell, M. (2012). Grauflecken- und Grübchenbildung an einsatzgehärteten Zahnrädern mittlerer und kleiner Baugröße [engl.: Micropitting and pitting of case-hardened gears of medium and small size] [PhD Thesis], Gear Research Center (FZG), Technical University of Munich. Dowson, D. & Higginson, G. R. (1966). Elastohydrodynamic lubrication. Oxford: Pergamon Press. Deutsches Institut für Normung e.V. (DIN). (2014). Zahnräder – Zylinderräder und Zylinderradpaare mit Evolventenverzahnung – Begriffe und Geometrie [engl.: Gears - Cylindrical gears and pairs of cylindrical gears with involute gearing - Terms and geometry]. DIN ISO 21771.

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[31] Felbermaier, M., Tobie, T., & Stahl, K. (2015). FVA-Nr. 482 III - Heft 1122 - Langsamlauf-Graufleckigkeit Abschlussbericht [engl.: Slow speed-micropitting], Frankfurt/Main. [32] Koenig, J., Felbermaier, M., Tobie, T., & Stahl, K., (Eds.) (2018). Influence of low circumferential speeds on the lubrication conditions and the damage characteristics of case-hardened gears. STLE Annual Meeting & Exhibition, Minneapolis, Minnesota. [33] Plewe, H.-J. (1980). Untersuchungen über den Abriebverschleiß von geschmierten, langsam laufenden Zahnrädern (engl. Studies on the abrasion wear of lubricated, slow-running gears) PhD Thesis, Gear Research Center (FZG), Technical University of Munich. [34] Bruckmeier, S. (2006). Flankenbruch bei Stirnradgetrieben. [eng.: Tooth flank fracture in spur gears] [PhD Thesis]. Gear Research Center (FZG), Technical University of Munich. [35] Deutsches Institut für Normung e.V. (DIN). (1978). Toleranzen für Stirnradverzahnungen: Toleranzen für Abweichungen einzelner Bestimmungsgrößen [engl.: Tolerances for spur gears: Tolerances for deviations of individual determinants]. DIN 3962-1. [36] International Organization for Standardization. (2018). Calculation of load capacity of spur and helical gears - Part 31: Calculation examples of micropitting load capacity. ISO/TS 6336-31.. [37] VDI-Verein Deutscher Ingenieure (Ed.) (2017). Practical use of micropitting test results according to FVA 54/7 for calculation of micropitting load capacity acc. to ISO/TR 15144-1.. [38] DNVGL-CG-0036, Calculation of gear rating for marine transmissions. Classification. [39] Ghribi, D., & Octrue, M. (2015). Comparative study of the tooth flank fracture in cylindrical gears. Efficient method to assess the risk of the tooth flank fracture on the cylindrical gears. VDI (Hg.) 2015 – International Conference on Gears.. [40] Hein, M., Tobie, T., & Stahl, K. (2017). Calculation of tooth flank fracture load capacity – Practical applicability and main influence parameters. AGMA 2017 Fall Technical Meeting, Columbus, USA. . [41] Konowalczyk, P. (2018). Grübchen- und Zahnflankenbruchtragfähigkeit großmoduliger Stirnräder. Pitting and tooth flank fracture load capacity of large modul spur gears. Edition Wissenschaft Apprimus, Band 21/ 2018. Apprimus Verlag, Aachen. [42] Leimann, D.-O. (2019) Calculation of Tooth Flank Fracture Load Capacity acc. to the method of Leimann. Method for case carburized cylindrical helical and spur gears with external teeth and bevel gears with and without axial shift.. [43] Weber, R. (2015). Auslegungskonzept gegen Volumenversagen bei einsatzgehärteten Stirnrädern. [eng.: Design concept against volume failure in case-hardened cylindrical gears] [PhD Thesis], University Kassel. [44] MackAldener, M. & Olsson, M. (2002). Analysis of crack propagation during tooth interiorfatigue fracture. Engineering Fracture Mechanics, 69, 2147–2162. [45] MackAldener, M. & Olsson, M. (2000). Interior fatigue fracture of gear teeth. Fatigue & Fracture of Engineering Materials & Structures, 2000(23), 283–292. [46] Witzig, J. (2012). Flankenbruch - Eine Grenze der Zahnradtragfähigkeit in der Werkstofftiefe. [eng.: Tooth Flank Fracture - A Limit of Gear Load Capacity in Material Depth] [PhD Thesis], Gear Research Center (FZG), Technical University of Munich. [47] Oster, P. (1982). Beanspruchung der Zahnflanken unter Bedingungen der Elastohydrodynamik. [eng.: Stress on Tooth Flanks under Conditions of Elastohydrodynamics] [PhD Thesis], Gear Research Center (FZG), Technical University of Munich. [48] Zenner, H. & Richter, I. (1977). Festigkeitshypothese für die Dauerfestigkeit bei beliebigen Beanspruchungskombinationen. [eng.: Fatigue strength hypothesis for arbitrary stress combinations]. Konstruktion, 29(1), 11–18.. [49] Stahl, K., Höhn, B.-R., & Tobie, T. (2013). Tooth flank breakage - influences on subsurface initiated fatigue failures of case hardened gears; 889. ASME 2013 International Design Engineering Technical Conferences & Computers and Information in Engineering Conference (DETC2013), ASME, Portland, Oregon, USA. [50] International Organization for Standardization. (1998). Cylindrical gears for general and heavy en­ gineering — Standard basic rack tooth profile. Standard ISO 53:1998. Beuth Verlag GmbH, Berlin. [51] Lang, O. R. (1979). Dimensionierung komplizierter Bauteile aus Stahl im Bereich der Zeit- und Dauerfestigkeit. [eng.: Dimensioning of complicated components made of steel in the area of fatigue strength and fatigue resistance]. Zeitschrift für Werstofftechnik, 10, 10–29.. [52] Hertter, T. (2003). Rechnerischer Festigkeitsnachweis der Ermüdungstragfähigkeit vergüteter und

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einsatzgehärteter Stirnräder [eng.: Computational strength verification of the fatigue load capacity of quenched and tempered and case-hardened cylindrical gears] [PhD Thesis], Gear Research Center (FZG), Technical University of Munich. Beermann, S. (2015). Tooth flank fracture - Influence of macro and micro geometry. International Conference on Gears 2015 International Conference on High Performance Plastic Gears 2015 International Conference on Gear Production 2015. VDI-Berichte 2255, Garching.. Lang, O. R. (1988). Berechnung und Auslegung induktiv randschichtgehärteter Bauteile. [eng.: Calculation and design of inductive surface hardened components]. In: Kloos, K. H. & Grosch, J., Induktives Randschichthärten, Berichtsband, Tagung 23. bis 25. März 1988, München: Arbeitsgemeinschaft Wärmebehandlung und Werkstofftechnik (AWT), 1989, S. 332-348. Thomas, J. (1998). Flankentragfähigkeit und Laufverhalten von hart-feinbearbeiteten Kegelrädern. [eng.: Flank load carrying capacity and running behavior of hard-soft-finished bevel gears] [ PhD Thesis], Gear Research Center (FZG), Technical University of Munich. Tobie, T. (2001). Zur Grübchen- und Zahnfußtragfähigkeit einsatzgehärteter Zahnräder. [eng.: For the Pitting and Tooth Root Load Capacity of Case-Hardened Gears]. Einflüsse aus Einsatzhärtungstiefe, Wärmebehandlung und Fertigung bei unterschiedlicher Baugröße [PhD Thesis], Gear Research Center (FZG), Technical University of Munich. Bauer, E. & Böhl, A. (2011). Flank breakage on gears for energy systems. Gear Technology, 28(8), 36–42. Octrue, M., Ghribi, D., & Sainsot, P. (2018). A contribution to study the tooth flank fracture (TFF) in cylindrical gears. Procedia Engineering, 213, 215–226.. Zenner, H., Simbürger, A., & Liu, J. (2000). On the fatigue limit of ductile metals under complex multiaxial loading. International Journal of Fatigue, 22(2), 137–145.. Tobie, T., Stahl, K., & Höhn, B.-R. (2013). Tooth Flank Breakage - Influences on Subsurface Initiated Fatigue Failures of Case Hardened Gears, ASME 2013. Boiadjiev, I., Witzig, J., Tobie, T., & Stahl, K. (2014). Tooth flank fracture - basic principles and calcu­ lation model for a sub surface initiated fatigue failure mode of case hardened gears; 924. International Conference on Gears, Lyon Villeurbanne, France, pp. 670–680. Boiadjiev, I., Witzig, J., Tobie, T., & Stahl, K. (2015). Tooth flank fracture – basic principles and cal­ culation model for a sub-surface-initiated fatigue failure mode of case-hardend gears.. Fuchs, D. & Steinbacher, M. (ongoing), FVA 848 I: Einflüsse auf die Ausbildung und Wirkung von Grobkorn in Al-N stabilisierten Einsatzstählen [eng.: Influences on the formation and effect of larger grains in Al-N stabilized case-hardened steels]. Fuchs, D. & Tobie, T. (ongoing),Untersuchungen zum Fehlstellenversagen an Zahnrädern und deren Einfluss auf die Zahnradtragfähigkeit. [eng.: Investigations into the defect failure of gears and its influence on gear load capacity]. Kadach, D. (2016). Influences of load distribution on the pitting load carrying capacity of cylindrical gears. Power Transmissions Proceedings of the International Conference on Power Transmissions (ICPT).. Wickborn, C. (2017). Erweiterung der Flankentragfähigkeitsberechnung von Stirnrädern in der Werkstofftiefe- Einfluss von Werkstoffeigenschaften und Werkstoffdefekten. [eng.: Extension of the flank load capacity calculation of cylindrical gears in the material depth - influence of material properties and material defects] [PhD Thesis], Gear Research Center (FZG), Technical University of Munich.

13

Potential and Challenges of High-Performance Plastic Gears C. M. Illenberger, T. Tobie, and K. Stahl

13.1 INTRODUCTION Plastic gears are gaining more and more importance and are used in new applications. In the past, typical applications for plastic gears were motion transmissions in actuators, but current developments show an increasing trend toward applications with higher drive power. In the automotive sector for example, in addition to a large number of electric actuators driven by plastic gears, even safety-critical applications such as braking and steering systems are increasingly equipped with plastic gears. In the drive unit of ebikes, plastic gears are the current state of the art, and the use of plastic gears is also conceivable in the drive train of small urban electric vehicles. In order to dimension plastic gears in line with requirements and to fully exploit the potential of thermoplastic materials, solid knowledge of material behavior is required both during production and in subsequent operation. This chapter explains the current state of the art for plastic gears and offers comprehensive insights into current research projects.

13.2 STATE OF THE ART AND APPLICATION OF PLASTIC GEARS 13.2.1 MATERIALS

AND

PROPERTIES

The mechanical-thermal properties resulting from the molecular structure of different plastics allow a classification of the materials into different sub-categories: thermosetting plastics, elastomers, and thermoplastics. In the environment of drive technology, semi-crystalline thermoplastics are mainly used as gear materials. Among others acc. to [1], the following semi-crystalline thermoplastics are particularly relevant for the production of plastic gears: • • • • •

High molecular polyethylene of high density (PE-HD) Polyoxymethylene (POM) Polyamide (PA) Polybutylene terephthalate (PBT) Polyetheretherketone (PEEK)

To further increase the mechanical properties or to specifically adjust them, fiber reinforcements are often added to the respective base material. Glass or carbon fibers are usually applied to the matrix materials to increase the mechanical parameters such as Young's modulus and tensile strength. Further, the addition of aramid fibers is possible to improve the mechanical properties. In order to positively influence the frictional properties, especially in dry running, friction reducing additives such as graphite and polytetrafluoroethylene (PTFE) may be utilized. This enables on the one hand the reduction of the friction coefficient and the associated heat generation in the tooth contact, on the other hand efficiency, wear, and service life can be optimized. When selecting friction-reducing fillers, however, it should be noted that the fillers are primarily recommended for improving the tribological properties on the tooth

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TABLE 13.1 Mechanical Properties of Typically Applied Thermoplastic Gear Materials [ 2] Material

Density in kg/m3

Young´s Modulus at 23°C in N/mm2

Tensile Strength at 23°C in N/mm2

Max. Continuous Operating Temperature in °C

PA 46

1.20

3300

100

140

PA 66

1.15

3000

85

90

POM PEEK

1.41 1.28

2900 3600

65 100

90 250

flank, where they have a positive effect, while other mechanical properties such as strength can be negatively influenced [2]. The mechanical properties of thermoplastics differ significantly from those of metallic construction materials. For instance, the Young's modulus and tensile strength are not only several times lower than those of steel, but are also highly temperature-dependent. High loading speeds increase tensile strength and Young's modulus, while elongation at break decreases. In static loading, plastics are deformed by creep [2]. Table 13.1 shows the mechanical properties as well as the maximum continuous operating temperature of exemplary thermoplastic gear materials that are chosen typically when designing plastic gears. An increase in temperature reduces the fraction of semi-crystalline areas in the material, which results in a reduction of the Young's modulus and the strength properties. If the glass transition temperature is exceeded, a significant reduction of the mechanical properties is to be expected. Humidity is another influencing parameter on the properties of plastic gears. Thermoplastic materials absorb moisture from the environment to varying degrees. An increasing moisture content reduces the yield stress and the modulus of elasticity. Furthermore, the absorption of moisture leads to volume changes due to expansion [2]. Thermoplastic materials exhibit viscoelastic material behavior. This behavior is responsible for the material damping of thermoplastic materials. The damping coefficients differ depending on the type of material and temperature. The high damping characteristics compared to steel materials have a positive effect on the noise emissions of plastic gears. Another important property of thermoplastics is their resistance to gaseous and liquid chemicals. Table 13.2 shows the resistance of exemplary thermoplastics to chemicals acc. to [2]. Resistance to chemicals allows plastic gears to operate in environments where steel gears cannot be used due to corrosion.

13.2.2 MANUFACTURING The vast majority of plastic gears are produced by injection molding. Depending on the material used, the desired gear quality as well as the quantities produced and the size of the individual components, alternative manufacturing processes such as casting or machining can also be considered. For the pro­ duction of high volumes, the injection molding process is typically the most cost-effective manufacturing process in terms of individual unit costs. In this process, the thermoplastic granulate is melted and injected into a cavity, which provides the shape of the gear after solidification. In addition to the low unit costs, another major advantage of the injection molding process is the largely free design of the mold. In addition to conventional involute gears, alternative geometries and highly integrated components can also be manufactured which could only be machined to a limited extent by conventional hobbing. Highly integrated components. In order to achieve a sufficiently precise gearing quality, however, it is necessary

High-Performance Plastic Gears

757

TABLE 13.2 Resistance of Typically Applied Thermoplastics to Chemicals acc. to [ 2] (+ Resistant, ○ Limited Resistance, - Not Resistant) Chemica Alcohols

PA 46

PA 66

POM

PEEK

Methanol

+

○

+

+

Ethanol Propanol

+ +

○ ○

+ +

+ +

Water

Cold

+

+

+

+

Fuels

Hot Gasoline

○ +

○ +

○ +

+ +

Diesel

+

+

+

+

Acids

Hydrochloric acid Sulfuric acid

-

-

-

○ -

Acetic acid

-

-

-

+

Bases

Potassium hydroxide Sodium hydroxide

+ +

+ +

+ +

Ammonium hydroxide

o

+

+

to consider the shrinkage of the material during the cooling process. Components with large wall thicknesses tend to form voids and to shrink during cooling, which is why the production of injection molded gears is subject to production limitations. The addition of fiber reinforcements can reduce shrinkage. Depending on the flow direction, however, considerable differences in shrinkage behavior can occur, which can lead to distortion in the component [2]. A further manufacturing process is the unpressurized casting of larger components with part masses starting at approximately 1 kg. The raw component is machined after polymerization, since the materials used (PA6G, PA12G) lead to comparatively high shrinkage, which cannot be compensated by correc­ tions in the mold [2]. For small and medium series, machining processes are applied to a large extent. For this purpose, the same tools can be used like for the production of steel gears. The manufacturing accuracy and the resulting gearing quality are strongly dependent on the manu­ facturing method and the respective process conditions.

13.2.3 DESIGN Injection molding offers diverse design possibilities. In particular, it is possible to injection mold stepped gears and over-mold shafts and hubs as well as highly integrated components. Different wall thicknesses cause an unequal shrinkage behavior during cooling of the component and can lead to distortion and internal stresses. For the purpose of economic production, the gears can be equipped with cutouts. On the one hand, this reduces the amount of resources required during production and, on the other hand, reduces the weight of the component. It is advisable to ensure that the cutouts are symmetrically arranged to minimize distortion of the plastic gear. In order to increase the stiffness of the gear body ribs can be applied.

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Dudley’s Handbook of Practical Gear Design and Manufacture

If possible, the torque should be transmitted form-fittingly from the gear to the corresponding shaft. Over-molding of inserted shafts generally results in good centering of the gear teeth. When manu­ facturing large-size plastic gears, larger steel inserts can be over molded to reduce dimensional changes due to shrinkage [2].

13.2.4 FIELDS

OF

APPLICATION

Plastic gears are used within a wide range of power starting from less than 0.001 kW in small consumer goods to up to approximately 1 kW in high performance applications. A large share of the applications for plastic gears is in the household appliance sector. Here, plastic gears are used not only for drives in toys, but also, and in particular, in kitchen appliances and tools. In many cases, the plastic gears are operated dry or with starved lubrication by lifetime lubrication with grease. In applications in printing and food technology, the use of a lubricant is often completely avoided due to hygiene requirements. A growing market for plastic gear wheels can be found especially in the smart home sector, where new functions are realized by means of electric drives. Here, automatically controlled roller blinds, venetian blinds as well as automatic garage doors, roof windows and other applications are typical for the use of plastic gears. Plastic gears are also increasingly used in the automotive sector. The reason for this is the constant introduction of new functions such as electrically adjustable mirrors, belt tensioners, seat adjustment and many other applications, which increase comfort. But also, in the area of safetyrelevant functions plastic gears are increasingly used in the automotive sector. These include, for example, the parking brake and electric power steering, where plastic gears are used. In the pow­ ertrain segment, plastic gears are used primarily in the field of electric micro-mobility and are an essential part of the drive train of e-bikes. Plastic gears are not yet used for the currently growing market of electrified urban light vehicles but for drive powers of 0.7nrP . Of course, the actual situation may be far better, as the speed may be in the interval between two load peaks, but the as­ sessment of resonant or vibration conditions presents a number of uncertainties. The main problems are: 1. The real overall tooth stiffness 2. How masses influence actual resonance speed when they are not adjacent and rigidly connected with pinion or gear. For instance, the polar moment of inertia of the gear of the previous stage, adjacent to a pinion, certainly must be added to that of the pinion, but the effect of an external coupling is uncertain 3. External excitations can cause resonance. As mentioned above, a rational approach should sub­ stitute the dynamic analysis of the entire system for the product Ka Kv . Long and elastic external transmission shaft often exclude effects of masses that are far from gears, but in particular cases can themselves originate vibration conditions. Figure 15.6 refers to a starting process, that is, to a transient regime: The vibrations are much stronger in the case (b), and the only difference consists in a more elastic final transmission. If there is any doubt regarding the third problem—or any explicable failure has occurred—it is advisable to interpose elastic couplings or hydraulic couplings. Special design features must be adopted when calculations show that the speed approaches resonance. Gears can work in a resonance condition (it should be avoided if possible); a number of marine gears do, as their speed is variable and can incur resonance. But they must be built with very good precision and must be helical gears with higher overlap ratios. Note that when the calculation indicates a resonance risk, it may be easier to avoid it by diminishing rather than increasing the resonance speed. This has also the advantage that uncertainties of load peaks corresponding to resonance sub-multipliers are avoided. This can be accomplished by increasing either pinion mass or pinion tooth number or both, with the aim of operating above the resonance range. Higher

Load Rating of Gears

843

(a) 5

Torque/Regime torque

4

2

3

1

0 0

1

2

3

4

Time, (sec)

0

1

2

3

4

Time, (sec)

(b)

Torque/Regime torque

5

4

2

3

1

0

FIGURE 15.6

Gear vibration in transient condition depending on the stiffness of the external output shaft.

tooth numbers, that is, lower modules, also reduce the risk of scoring. On the other hand, this often makes RF more restrictive than RH and requires careful verifications. Gears that are to operate above the resonance speed should be precision gears. Unground tool-finished marine gears operated in the past above the resonance speed. In this case, medium precision may be insufficient as the gear pair may incur a multiple a multiple of the resonance speed. Then it will be

844

Dudley’s Handbook of Practical Gear Design and Manufacture

subjected to a load peak, even though it may be lower than the one corresponding to the main resonance condition. For gear speeds either in the resonance range or above it, high-precision gears with AGMA number not less than 11, and AGMA Kv factor rating with a Qv factor not greater than 10, are advisable as a summary general criterion. More optimistic estimates, as considered by AGMA for very accurate gearing, may be allowed if the system is simple enough and the calculation indicates that the speed is far enough from both the main resonance speed and its multiples. Spur gears should not operate in the resonance range, although they do in particular cases. Then specific tests are available. The procedure for the calculation of n r P cannot be considered valid for planetary gears, although the general concepts are the same. Load distribution factor, Km. For AGMA, Km = Cm = Cmf Cmt , respectively, for face and transverse load distribution factors, but usually Cmt = 1. ISO defines products K H K H for tooth surface and K F K F for tooth fillet where K F is somewhat less than K H . Both standards give simplified and analytical methods. Warning: the “empirical” or simplified methods of ISO and AGMA standards largely differ from the analytical ones, especially when the shaft of the wheel deforms elastically, as often happens in multistage gear units. However, the analytical methods require preliminary computation of the tooth misalignment. A preliminary indication only is given in the following section. Assume that Km varies from 1.1 to 1.8; greater values would imply bad general design or unacceptable manufacturing errors. Table 15.2 helps for the choice of Km . Carter deformations often are an important item for tooth alignment. They should be investigated by FEM, at least for large, important gear units or for mass production. This is not an easy task because of the complex form of the carters, usually obtained by the assembling of different parts. Experimental checks are useful when possible. Note on overload factors: The choice of the single overload factors or the direct assumption of their synthetic product V is within the competence of the gear designer, even if the indications and the formulas of the standards are helpful. Their assessment is generally more important for the final results than the geometry calculations that follow, even if the latter are also necessary. AGMA and ISO overload factors should be interchangeable, according to their definitions, and in fact it may be necessary to alternate them to make direct assumptions, for example, when the validity field of an equation is limited or when the standard indications are doubtful for a given application.

15.5 RH—CONVENTIONAL FATIGUE LIMIT OF FACTOR K A unified formula of Klim for both RHA and RHI is as follows: Klim = GH

sc lim CP

2

AH V

(15.37)

where all parameter are in consistent units: Klim and sc lim in [N/mm2], and CP in [N/mm2]0.5, or Klim and sc lim in [lb/in2], and c in [lb/in2]0.5. The overall overload factor V has been defined [Eq. (15.21)]. The other factors are given in the following.

15.5.1 RH—PRELIMINARY GEOMETRIC CALCULATIONS Pressure and helix angles:

Load Rating of Gears

845

TABLE 15.2 Guideline for Assumption of the Load Distribution Factor, Km * Item (a)**: Stiffness of the shafts of pinion and wheel Good: A=0 Average:

A=1

Bad:

A=2

Item (b)#: Gear helix precision and shaft parallelism in the housing Good: B=0 Average:

B=1

Bad:

B=2

Item (c)##: Choice of Km

A+B Km

0

1

2

3

4

1.1

1.25

1.4

1.6

1.8

*

Note: The Table is valid for gears designed and manufactured following normal criteria and for normal housing stiffness. A low housing stiffness may produce opposite effects: 1. It may compensate tooth misalignment for one stage gear pairs, or 2. For multistage gear trains, it may help some gear pairs, especially those under most load, and worsen the misalignment of others. ** Note to item (a): In normal multistage gear trains, deflection of the second pinion shaft (which carries the first wheel) cause misalignment of the first gear pair, and so on. # Notes to item (b): Helix precision: what is of most interest in the difference in pinion and wheel helices. Shaft eccentricity in journals is due to elastic deformations of the bearing and to clearances: Internal bearing clearance and possible clearances between bearing and housing, and between bearing and shaft. Remember that overhung gear mounting causes shaft eccentricity and gear misalignment even in the case of preloaded bearings, because of elastic deformations in the bearing, especially when an overhung pinion is the gear on a shaft under greatest load. This is typical on most bevel pinions. ## Notes to item (c): Lower Km of 10–20% (minimum: Km = 1.1) for appropriate helix correction or longitudinal tooth flank correction or crowning. (Warning: inappropriate corrections or crowning may cause worse misalignment!). Increase Km of 10–30% for bad tooth spacing or profile precision, especially in helical gears. Increase Km of 10–20% because of torsional pinion deflection for the first gear pair of multistage speed reducers if the gear ratio is high and the pinion is sited at the side of the power input.

B=

s

= tan

b

= sin

2C [mm] cos

s

m n [mm](NG + S2 NP )

t

= cos

1

1

=

1

tan

ns

cos

s

sin

s

cos

ns

(15.38)

(15.39)

2C [in. ] Pnd [in 1]cos NG + S2 NP

cos B

s

s

(15.40)

(15.41)

846

Dudley’s Handbook of Practical Gear Design and Manufacture 1

= tan

B tan

(15.42)

s

Base diameters: d bP = dP cos

t

(15.43)

d bG = dG cos

t

(15.44)

m p = m pA + m pE

(15.45)

Profile (transverse) contact ratio:

where m pA and m pE are the operating addendum contact ratios relating to the stretches AC , CE in Figures 15.7 and 15.8:

doG dbG

m pA =

2

doP dbP

m pE =

1

tan

t

2

1

tan

t

NP 2

(15.46)

S2 NG 2

(15.47)

pbt

E

ft

D A

dbP 2

C B

pbt

ft

dP 2

FIGURE 15.7 Involute meshing of an external gear pair in point B , Lower point of single contact of pinion or higher point of single contact of gear.

Load Rating of Gears

847 pbt

D C

E

ft

B

A pbt

dbP 2

ft dP 2

FIGURE 15.8

Involute meshing of an internal gear pair in point B , LPSC of pinion or HPSC of gear.

Outside and base diameters do and db can be given in any consistent units. If the contact does not extend as far as the tooth tip because of semi-topping or other reasons, then the diameters at the contact limit must be introduced instead of doP and doG . Design criteria of gear teeth with semi-topping can become an important item for the surface load capacity, as they can greatly affect the contact ratio. This involves a variation of the overall contact length of helical teeth. Curvature coefficient, XB , relating to the lower point of single contact (LPSC) of the pinion involutes, B in Figures 15.7 and 15.8: 2 ME =

XB = 1

1

m pE

tan

t

ME NP

(15.48)

1 + S2

ME NG

(15.49)

The XB formula is valid for usual gears with 1 < m p < 2. The LPSCB depicts a real step in tooth meshing for spur gears, whereas it works as a reference point for helical gears and enables the trend of the profile curvature to be followed. In usual design cases with a lower gear ratio and a larger path CE with regard to AC (see Figure 15.9), the relative curvature of the involutes may be worse in the higher point of single contact (HPSD) D. Then a coefficient Xd in every formula of the Geometry factor GH : 2 MA =

1

m PA

tan

t

(15.50)

848

Dudley’s Handbook of Practical Gear Design and Manufacture

pbt E

ft

D

B

C

A pbt

ft dbP 2

dP 2

FIGURE 15.9

Involute meshing of an external gear pair in point D , pinion HPSC.

XD = 1

S2

MA NG

1+

MA NP

(15.51)

Face contact ratio (overlap ratio): mF =

F [mm] sin m n [mm]

s

=

F [in.] Pnd [in 1]sin

s

(15.52)

A single face width must be considered here for double-helical gear pairs. The following contact line coefficient m1 and load-sharing ratio mN are for RHA only (as well as for RFA). For spur gears: m1 = 1

(15.53)

mN = 1

(15.54)

For helical gears, m pi , m Fi the integer, m pd , m Fd the decimal parts of m p , mF :

Load Rating of Gears

849

m1 =

m p m Fi + m pi m Fd + max (m Pd + m Fd

mN =

15.5.2 ADAPTION

FOR

1.0)

(15.55)

m p mF

cos

b

(15.56)

m1 mP

BEVEL GEARS

Pressure and helix angles: Equations (15.38) and (15.39) are valid. Standard and operating angles coincide for bevel gears, then t = s , = s . Addenda in the middle point of the face width, Figure 15.10: a Pm = a oP

(F /2)tan(

oP

P)

(15.57)

aGm = a oG

(F /2)tan(

oG

G)

(15.58)

Virtual outside diameters: doPv = dPmv + 2aPm

(15.59)

pbt

D C

E

B

A pbt

ft dbP 2

FIGURE 15.10

dP 2

Involute meshing of an internal gear pair in point D , pinion HPSC.

ft

850

Dudley’s Handbook of Practical Gear Design and Manufacture

doGv = dGmv + 2aGm

(15.60)

Virtual base diameters: dbPv = dPm v cos

t

(15.61)

dbGv = dGm v cos

t

(15.62)

Then Eqs. (15.46) and (15.47) can be applied by introducing all virtual data, virtual tooth number included, in while Eq. (15.52) the mean normal module m nm or the mean normal diametral pitch Pndm must be introduced. No variation effects Eqs. (15.45), (15.48), (15.49), (15.54), (15.55), and (15.56).

15.5.3 RH—UNIFIED GEOMETRY FACTOR GH RHA Spur gears: GH = XB

sin(2 t )

(15.63)

4

Helical gears: GH =

XB sin(2 t ) 0.9 4mN

(15.64)

Note: AGMA’s original geometry factor I is, as a concept: I=

GH

(15.65)

N

1 + S2 NP

G

Thus, Eqs. (15.63) and (15.65) give exactly the same I factor as the AGMA standards for spur gears. For helical gears, Eq. (15.64) is simplified with regard to the standards but maintains the trend de­ pending on the profile curvature, whereas the results are somewhat more conservative in most common cases, for example, for small tooth numbers of the pinion. No modification is adopted for the so-called LCR gears, as the mN ratio accounts for their load sharing. (LCR is an AGMA term that means low contact ratio—which is disputable, as it refers to overlap or face contact ratio and not to transverse contact ratio). RHI–Spur and Helical Gears GH =

sin(2 t ) XB 0.9 4 cos b cos s Z 2

(15.66)

where Z is a factor of the contact ratios according to ISO: Z2 =

4

mp 3

[1

min (mF , 1)] +

min (mF , 1) mp

(15.67)

Load Rating of Gears

851

Note: Equation (15.66) leads to the same final result as the original ISO factors, expected for the effect of the correction coefficient XB /0.9 that is introduced as a simple criterion for taking into account the trend of the variation of the relative profile curvature. The DIN method of 1987 is equivalent to introducing the factor XB only for spur gears and only when it is less than 1, without dividing it by 0.9. It is possible that the final draft of the ISO method coincides with DIN’s. See the following comparisons and discussion. In practice, the XB coefficient can be maintained as calculated from Eq. (15.49), also when m p > 2 as an estimation, for both RHA and RHI. Specific tests must be performed if one wants to fully exploit the performances of such an unusual kind of gears. Note that involute gear teeth with higher contact ratios can be obtained by adopting higher tooth numbers and longer teeth. Low tooth numbers are possible for the pinion if the tooth profiles differ from the involute, but this does not regard this chapter. Tables 15.3 and 15.4 indicate GH values for typical meshing cases of external and internal gear pairs with nominal addendum = mn or 1/ Pnd . The addendum modification coefficients xP and xG are obtained by dividing the modifications of the nominal addendum by m n or multiplying by Pnd before any further correction of the tooth outside diameters, that the gear designer may adopt for any reason. The coeffi­ cients are positive if they involve an increase of the diametral size (Note that the Germans adopt the opposite convention for internal gears).

15.5.4 RH—COMMENTS

AND

COMPARISONS

ON THE

UNIFIED GEOMETRY FACTOR GH

Important differences between the various methods depend on the choice of the point along the contact line where the relative profile curvature is rated. Figures 15.11 and 15.12 show the geometry factor as depending on the addendum modification coefficients (see above). When xG = xP , the center distance is unmodified; otherwise, it is modified. No addendum shortening has been considered as it is unnecessary in both cases. AGMA-V and ISO-V refer to the “variants” that are adopted in this chapter, Eqs. (15.64) and (15.66). Equation (15.63) for spur gears agrees with all the AGMA standards mentioned in the following. Both the variants for RHA and RHI show a very similar trend of AGMA standards with regard to the influence of xP and xG coefficients, but GH is far higher according to the AGMA standards for helical pinions with a low tooth numbers, Figure 15.12, that requested for surface-hardened gears if surface and root resistances have to be balanced. Of course, the success of gears such as these that have been designed according to AGMA standards depends on other items, for example, on a conservative choice of the allowable Hertzian pressure. But a more cautious choice of GH , as given by Eq. (15.64), becomes opportune especially if higher values of the Hertzian pressure are allowed according to ANSI/AGMA standards. On the whole, a far greater gap between the GH factors or helical and spur gears can be observed for AGMA ratings with regard to ISOs. The DIN method reduces GH as opposed to ISO for spur gears, when XB < 1. The field experience often disagrees with such a gap. Cases are known where helical gears, substituting for spur ones, did not avoid pitting. A lot of common industrial planetary units built with case-hardened spur gears, which have been working successfully for many years, would soon have failed if they behaved according to AGMA or even to DIN.

15.5.5 RH—ELASTIC COEFFICIENT DP HERTZIAN PRESSURE

AND

CONVENTIONAL FATIGUE LIMIT sc lim

OF THE

Table 15.5 gives the elastic coefficient CP (ISO symbol: ZE ) for gear pairs made from the same category of materials. Otherwise, CP must be calculated. For metal materials with Poisson’s ratio = 0.3: CP [(N/mm2)0.5] =

0.175Em [N/mm2]

(15.68)

852

Dudley’s Handbook of Practical Gear Design and Manufacture

TABLE 15.3 Geometry Factors for RHA and RHI, External Gear Pairs s

0

15

30

NP / NG

xP

xG

13/82 13/82 13/41 13/41 13/21 13.21 26/82 26/82 26/82 26/41 26/41 26/41 26/26 26/26 13/82 13/82 13/41 13/41 13/21 13/21 26/82 26/82 26/82 26/41 26/41 26/41 26/26 26/26 13/82 13/82 13/41 13/41 13/21 13/21 26/82 26/82 26/82 26/41 26/41 26/41 26/26 26/26

0.4 0.4 0.4 0.4 0.4 0.4 0.0 0.4 0.4 0.0 0.4 0.4 0.0 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.0 0.4 0.4 0.0 0.4 0.4 0.0 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.0 0.4 0.4 0.0 0.4 0.4 0.0 0.4

0 0.4 0 0.4 0 0.4 0 0 0.4 0 0 0.4 0 0.4 0 0.4 0 0.4 0 0.4 0 0 0.4 0 0 0.4 0 0.4 0 0.4 0 0.4 0 0.4 0 0 0.4 0 0 0.4 0 0.4

t

21.240 22.338 22.085 23.807 23.132 25.529 20 21.099 22.085 20 21.714 23.173 20 23.929 21.810 22.851 22.610 24.254 23.608 25.912 20.647 21.677 26.610 20.647 22.258 23.646 20.647 24.370 23.745 24.616 24.413 25.818 25.261 27/274 22.796 23.635 24.413 22.796 24.118 25.294 22.796 25.919

mpA

mpE

mp

GHA

GHI

0.592 0.634 0.587 0.650 0.573 0.657 0.914 0.629 0.701 0.859 0.636 0.736 0.810 0.746 0.561 0.602 0.559 0.620 0.550 0.631 0.866 0.597 0.666 0.818 0.606 0.703 0.774 0.717 0.475 0.510 0.480 0.534 0.480 0.554 0.731 0.505 0.566 0.699 0.520 0.607 0.669 0.627

0.893 0.847 0.858 0.784 0.813 0.709 0.810 0.986 0.904 0.810 0.935 0.811 0.810 0.746 0.861 0.817 0.828 0.757 0.785 0.684 0.774 0.946 0.867 0.774 0.897 0.779 0.774 0.716 0.766 0.728 0.737 0.675 0.700 0.610 0.669 0.827 0.760 0.699 0,785 0.682 0.669 0.627

1.485 1.481 1.444 1.434 1.386 1.366 1.725 1.615 1.605 1.670 1.570 1.547 1.621 1.493 1.423 1.419 1.387 1.377 1.334 1.315 1.641 1.542 1.533 1.592 1.503 1.482 1.548 1.432 1.241 1.238 1.217 1.209 1.180 1.163 1.400 1.332 1.325 1.368 1.305 1.290 1.338 1.254

0.149 0.148 0.152 0.152 0.161 0.162 0.146 0.167 0.167 0.152 0.169 0.173 0.158 0.182 0.228 0.226 0.228 0.225 0.223 0.230 0.263 0.281 0.280 0.267 0.280 0.280 0.272 0.286 0.212 0.209 0.215 0.211 0.225 0.220 0.257 0.272 0.271 0.265 0.276 0.275 0.275 0.284

0.198 0.196 0.199 0.197 0.205 0.205 0.214 0.233 0.233 0.217 0.232 0.234 0.222 0.242 0.249 0.246 0.248 0.245 0.254 0.251 0.287 0.307 0.305 0.291 0.305 0.305 0.297 0.312 0.257 0.254 0.261 0.257 0.273 0.268 0.312 0.331 0.329 0.322 0.336 0.334 0.334 0.346

General data: ns = 20 Nominal addendum = mn = 1/ Pnd x —is the addendum modification / mn = addendum modification Pnd ( xP , x G for pinion and gear, respectively), that is, do = Nmn / cos s + 2mn (1 + x ) Helical gear pairs: m1 = 0.95 for GHA , and mF 1 for GHI

Load Rating of Gears

853

TABLE 15.4 Geometry Factors Capacity of Case-Hardened RHA and RHI, Internal Spur Gear Pairs xG

xP

13/82

0.4

0

17.959

0.343

13/82 13/41

0.4 0.4

0.4 0.4

20.000 20.000

0.412 0.431

26/82

0.0

0

20.000

0.863

26/82 26/82

0.4 0.4

0 0.4

17.406 20.000

0.204 0.412

mpA

t

mp

GHA

GHI

1.027

1.369

0.153

0.194

0.944 0.944

1.356 1.376

0.147 0.145

0.185 0.185

0.810

1.673

0.135

0.193

1.285 1.077

1.490 1.489

0.186 0.172

0.247 0.228

mpE

NP / NG

General data: ns = 20 s = 0 Nominal addendum = mn = 1/ Pnd x —is the addendum modification / m n = addendum modification Pnd ( xP , xG for pinion and gear, respectively) Notes: Addendum modifications are assumed > 0 , for both pinion and internal gear, when they involve an increase of the diametral size. Therefore: For the pinions: do = Nmn + 2mn (1 + xP ) For the internal gears of the table the addenda are shortened by 0.2 mn = 0.2/ Pnd , thereby avoiding false contacts and widening the range of usable cutters, that is, do = Nmn 2mn (1 xG 0.2)

CP [(lb/in2)0.5] =

0.175Em [lb/in2]

(15.69)

where Em is the mean Young’s modulus of elasticity of pinion (EP ) and gear (EG ):

Em =

2EP EG EP + EG

(15.70)

Table 15.6 gives values near to the maximum indicated by the standards for sc lim (AGMA symbol: sac —ISO symbol: Hlim ). Warning: the notes to the table are essential!

15.5.6 RH—ADAPTATION FACTOR, AH The values are meant for gears similar to those of previous field experience or of laboratory tests. Adaptation factors aim to adjust the calculation to the manufacturing and operating peculiarities of the examined gear pair. In this item AGMA and ISO present great differences. RHA: AH =

CH2 Cs Cf CT2

(15.71)

854

Dudley’s Handbook of Practical Gear Design and Manufacture 0.35

0.35 RHI

GH Helical (Ψs =15º)

0.30

AGMA-V AGMA 211

0.25

GH

ISO-V ISO 6336/1983

0.30

0.25

ISO-V ISO 6336/1983

0.20 AGMA 218 AGMA 218

0.15

0 0

0.2 -0.2

0.4 -0.4

0.6 -0.6

XP XG

DIN 3990/1987

0.15

0 0

0.2 -0.2

0.10 0.6 -0.6

0.4 -0.4

FIGURE 15.11 Unified geometry factor for external teeth with unmodified center distance. NP /NG = 26/82 , mF = 1 for the helical gears. 0.35 RHI

AGMA 218

GH Helical (Ψs =15º)

GH 0.30

AGMA 211 0.25

0.30 ISO 6336/1983 0.25

ISO-V

AGMA-V ISO 6336/1983 0.20

Spur gears

AGMA unified geometry factor

= 20 ,

0.35 RHA

AGMA 218 0.15

0.10 0.4 -0.4

ns

0.20

ISO-V ISO 3990/1987

0.15

AGMA 210

0.4 -0.2

0.4 0

ISO unified geometry factor

0.10

0.20

Spur gears

AGMA unified geometry factor

AGMA 218

ISO unified geometry factor

RHA

0.4 0.2

0.4 0.4

XP XG

0.4 -0.4

0.4 -0.2

0.4 0

0.4 0.2

0.10 0.4 0.4

FIGURE 15.12 Unified geometry factor for external teeth with unmodified center distance and a low tooth number of the pinion. ns = 20 , NP /NG = 13/82 , mF = 1 for the helical gears.

Load Rating of Gears

855

TABLE 15.5 Elastic Coefficient CP for RH Pairing of Materials

Em, N/mm2 (lb/in2)

Em, N/mm2 (lb/in2)

Steel/steel Nodular or malleable iron/nodular or malleable iron

207,000 (30,000,000) 172,000 (25,000,000)

190 (2290) 173 (2090)

Cast iron/cast iron

138,000 (20,000,000)

155 (1870)

Steel/nodular or malleable iron Steel/cast iron

188,000 (27,000,000) 165,000 (24,000,000)

180 (2170) 170 (2050)

Nodular or malleable iron/cast iron

153,000 (22,000,000)

163 (1960)

TABLE 15.6 RH and RF Conventional Fatigue Limits Material

Flank and Root Hardness

sc lim , N/mm2 lb/in2 RHA

RHI

sc lim , N/mm2 lb/in2 RFA

RFI

Cast iron

175 HB

470 (68,000)

360 (52,000)

55 (8,000)

150 (22.000)

Nodular iron

200 HB 180 HB

530 (77,000) 560 (81,000)

400 (58,000) 500 (73,000)

80 (12,000) 190 (27,000)

165 (24,000) 370 (54,000)

240 HB

680 (98,000)

580 (84,000)

230 (33,000)

410 (59,000)

Malleable iron

180 HB 240 HB

510 (74,000) 620 (89,000)

480 (69,000) 550 (80,000)

80 (12,000) 130 (19,000)

370 (54,000) 410 (59,000)

Through-hardened and tempered steel

220 HB

700 (102,000)

690 (100,000)

240 (35,000)

550 (80,000)

260 HB 300 HB

795 (115,000) 890 (129,000)

745 (108,000) 800 (115,000)

265 (38,000) 290 (42,000)

580 (84,000) 610 (88,000)

Induction-hardened steel

~55 HRC

1,250 (180,000)

1,300 (190,000)

340 (50,000)

740 (105,000)

Carburized case-hardened steel

58-62 HRC

1,500 (215,000)

1,550 (225,000)

430 (63,000)

930 (135,000)

Case-nitrided steel, 2–3% Cr

700-750 HV

1,300 (190,000)

1,380 (200,000)

400 (58,000)

840 (120,000)

Notes: For cast steels ISO suggests lowering sc lim by 10% and st lim by 20%. Data for surface-hardened steels require proper hardened depth. The tabulated data are not applicable to large-sized gears, in­ duction or case hardened, or to medium- to large-sized gears, gas nitride. Core hardness is a determining factor. st lim values for induction hardening refer to root hardening. st lim must be reduced by 30% for alternating load. Intermediate values should be adopted for bidirectional loads according to the number of cycles in each direction: max reduction 30% for one load application. Tabulated data take no account of shock sensitivity. All mentioned steels are alloy steels. All materials are considered tested and in correct metallurgical condition. ANSI/AGMA standards allow as much as sc lim = 1910 N/mm2 (275,000 lb/in2) and st lim = 520 N/mm2 (75,000lb/in2) for car­ burized case-hardened steels with good metallurgical quality and certified cleanliness (AGMA Grade 3).

The factors at the denominator, that is, for size, flank finish, and temperature, are usually assumed equal to unity. They should be > 1 if anomalous conditions arise. The hardness factor CH equals 1 except in two cases for the gear.

856

Dudley’s Handbook of Practical Gear Design and Manufacture

Case 1. For non-surface-hardened gears, if the Brinell hardnesses HBP and HBG of pinion and gear are in the range: 1.2

HBP HBG

1.7

(15.72)

then:

CH

(1.08 =1+

)

HBP

1 (m G

HBG

120

1)

(15.73)

Case 2. If the pinion teeth are surface hardened with minimum HRCP = 48 Rockwell hardness, and the gear Brinell hardness is in the range 180 HBG 400 : CH = 1 +

450 1333

HBG 0.0125 fP e

(15.74)

where e = 2.718 (base of the natural logarithms), and fP is the flank of the pinion (roughness, arithmetic average, μin.). Note: The factor CH implies work hardening of the gear, but field experience shows that in case (2) the pinion teeth have a file effect rather than a work-hardening effect on the gear teeth if fP is higher, with disastrous wear. The pinion of such gear pairs should not be finished with fP > 40 in. or 1 m . RHA: AH = (ZX , ZW , ZL , Z v , ZR )

(15.75)

The size factor ZX is usually assumed equal to unity. The hardness factor ZW > 1 only if the pinion has surface-hardened teeth with a flank finish fP > 1 m or 40 in . (roughness, arithmetic average) and the gear is in the range 130 HBG 400 . Then: ZW = 1.2

HBG 130 1700

(15.76)

The factors ZL , Z v , and ZR for lubrication, velocity, and roughness are the most typical ISO factors. Simplified equations are given for them, and some modifications for ZR are suggested. RHI Lubrication Factor ZL =

v40 167

A

(15.77)

where v40 is the nominal kinematic velocity of the lubricant at 40 C(104 F ) in cSt (1 cSt = 10 6 m2/s). Assume A = 0.05 for surface-hardened, A = 0.1 for non-surface-hardened teeth (maximum ZL = 1.25). RHI Velocity Factor Zv =

vt [m/s] 10

B

(15.78)

Load Rating of Gears

857

Zv =

B

vt [ft/min] 1970

(15.79)

where B = 0.025 for surface-hardened teeth (minimum: Z v = 0.93), B = 0.05 for non-surface-hardened teeth (minimum: Z v = 0.85). RHI Roughness Factor. Tooth flank roughness is the mean value of some teeth of pinion fP and of fG . Then a reduced value fr is calculated, which relates to the size of test gears: fr [ m] =

fP [ m] + fG [ m] 200 3 2 dP [mm] + dG [mm]

(15.80)

fr [ in.] =

fP [ in. ] + fG [ in. ] 7.87 3 2 dP [in. ] + dG [in. ]

(15.81)

Consider fr as arithmetic average as above. Then: ZR =

0.5 fr [ m]

ZR =

20 fr [ in. ]

C

(15.82)

C

(15.83)

where C = 0.08 for surface-hardened, C = 0.15 for non-surface-hardened teeth. This agrees with the original ISO data with good approximation, but greater C exponents are advisable when tool-finished gears are subjected to a continuous and constant load, except in the case of case-hardened gears where C = 0.08 seems sufficiently de-rating. Note: The ISO surface factors have an experimental basis, but surface fatigue tests always give a lot of scattering and ISO explicitly states that such factors are only an estimation. However, they account for the roughness disturbances with ZR as well as for the lubrication regime, even if only summarily, by means of ZL and Z v . ZL is somehow disputable as it refers to a nominal viscosity, Z v apparently indicates a right trend as speedy gears in EHD conditions are favored whereas slow gears in boundary lubrication regime are routinely de-rated, as they will meet progressive wear even if not always pitting. There is actually a surface condition effect, which is sometimes more important than it is thought to be, which risks misleading one’s appreciation of the real nature of some field experience. For instance, a lot of big gears probably work in much more severe application conditions than those considered by K a and do not have pitting just because of a more effective work hardening than that allowed by CH . It would be risky both to rely on K a not directly checked and to assume optimistic CH especially if not proven for different kinds of materials. Sometimes the gear designer should take into account other favorable or unfavorable circumstances not considered by the standards. This task can accomplish by varying directly the AH factor. Shotpeening increases the pitting resistance. The type of lubrication apparatus as well as the lubrication and cooling conditions are also influential parameters, because the lubricant action and viscosity influence EHD conditions. Synthetic oils are generally favorable to pitting resistance according to their type and the gear material. Oil additives, even if conceived for other purposes, have sometimes a favorable in­ fluence, and any operating condition affecting tooth temperature can be indirectly important.

858

Dudley’s Handbook of Practical Gear Design and Manufacture

15.5.7 RH—HERTZIAN PRESSURE The value of the Hertzian pressure sc (ISO symbol: H ) does not directly enter the ratings based on the K factor and is given merely for information. AGMA’s and ISO’s equations differ for RHA and RHI. RHA VKCs Cf

sc = Cp

GH

(15.84)

RHI sc = Cp

VK GH

(15.85)

Both formulas are fictitious for various reasons, but this does not affect the reliability of the general calculations.

15.5.8 RH—SERVICE FACTOR, CSF (ONLY

FOR

ONE LOADING LEVEL)

Two cases are distinguished for the procedure. Case 1. Gear sizes and load are given, K and K lim have been calculated. Then such data are equivalent to a service factor: CSF =

Ka Klim K

(15.86)

which substitutes and summarizes application as well as life and reliability problems. Case 2. K lim has been rate for a new design with provisional data and a given CSF is required. Then the gear design must be newly sized by assuming:

K=

Ka Klim CSF

(15.87)

The service factor is often mistaken for the application factor. This does not supply any margin for reliability, unless a cautious sc lim value has been chosen, but an explicit margin CSF / Ka is preferable for the sake of clearness. The margin must be increased for a long gear life and may be reduced for a short one, even to the point where CSF itself may be < 1. In such cases the direct estimation of tooth damage and gear life is preferable. The CSF procedure normally excludes life analysis, and vice versa.

15.5.9 POWER CAPACITY TABLES Tables 15.7 through 15.12 give a general survey of performance of industrial case-hardened gears. As the present rating standards require detailed geometrical data, Tables 15.7 and 15.8 give them for gear pairs with parallel axes, which can be built by metric standardized tools. Of course, the perfor­ mances are not too different for gears built with standardized diametral pitches and general sizes and tooth numbers near to those of the tables. The addendum modifications are so chosen that the so-called Almen factors are balanced, that is, the product of theoretical Hertzian pressure multiplied by sliding

Load Rating of Gears

859

TABLE 15.7 Spur Gears: Data and General Assumptions for Capacity Table 15.8 Gear Ratio mG

Tooth Numbers NP /NG

Addendum Modif. Coefficients xP xG

Center Distance C, mm

Operating Pressure Angle t 50

100

200

300

Normal Module, mn, mm 1.0

25/25

0

0

20

2.00

4.0

8

1.6

22/35

0.1394

– 0.0673

20.3899

1.75

3.5

7

–

2.5

19/47

0.2849

0.0607

21.5190

1.50

3.0

6

12

4.0

16/64

0.2981

– 0.2981

20

1.25

2.5

5

10

6.3

14/86

0.3628

– 0.3628

20

1.00

2.0

4

8

–

Notes: External teeth: Normal standard pressure angle ns = 20 . Outside diameters do = Nmn + 2mn (1 + x ). Carburized case-hardened alloy steel, 58–62 HRC. Ground teeth. Service factor CSF = 1.5 including application, reliability, and life factors. Power-sharing factor Ksh = 1. Dynamic factors Kv rated according to RHA or RHI methods for AGMA quality number =11. Load distribution factor Km = 1.25. RHA: sc lim = 1500 N/mm2 ; AH = 1. RHI: sc lim = 1500 N/mm2 ; AH rated for f = 0.8 m (32 in.), v40 = 200 cSt , and pitch velocity (other influential factor =1).

velocity and access or recess length is the same at the extreme points of the contact path. This criterion is suitable for both speed-reducing and speed-increasing gear pairs. The addenda are adopted for bevel gears: Tables 15.11 and 15.12. Tables 15.8, 15.10, 15.11, and 15.12 refer to the surface, that is, to pitting and generic wear. The tooth numbers are chosen such that the tooth strength generally is no problem in the case of RFI rating and unidirectional loading. RFI rating and/or bidirectional loading may give more restrictive results. Higher power may require oil cooling especially in continuously transmitted, otherwise, the real power capacity will be reduced, apart from other drawbacks. Anomalous dynamic conditions, for ex­ ample, big masses rigidly connected to pinions or external causes of vibration excitation, may invalidate the data (see Section 15.4). Values in brackets are doubtful for scoring resistance, see Section 15.8. Otherwise, scoring should not be a problem, provided that good EP oils are used. If the scoring capacity of the lubricant is lower, problems may arise for the highest gear sizes and speeds indicated in the tables. Greater powers can be transmitted at higher speeds, but lower modulus, that is, greater tooth numbers are advisable, and this can make RF more restrictive than RH. Each case must be solved by itself according to the risk of operating in resonance ranges (see Section 15.4) and according to the loading conditions, the prescribed rating method, and the desired reliability against tooth breakage; no general tabulated indications can be given. Service and load distribution factors, as well as quality numbers, tooth roughness, and lubricant viscosity, as indicated in Tables 15.7, 15.9, 15.11, and 15.12, must be considered only as reference values.

15.6 RF—CONVENTIONAL FATIGUE LIMIT OF FACTOR UL A unified formula for both RFA and RFI can be given as follows:

860

Dudley’s Handbook of Practical Gear Design and Manufacture

TABLE 15.8 Surface Capacity of Case-Hardened Spur Gears Gear ratio mG

Pinion speed nP r/min

Center Distance C, mm/Net face width, mm 50/17.5

100/35

200/70

400/140

Power, kW (multiply by 1.341 for power in hp)

1.0

1.6

2.5

4.0

6.3

RHA

RHI

RHA

RHI

RHA

100

0.6

0.7

250

1.5

1.8

4.7

5.8

37

12.0

15.0

91

500 710

2.9 4.1

3.6 5.2

23.0 32.0

31.0 44.0

178 250

RHI

RHA

RHI

50

–

–

128

–

–

257 (364)

– –

– –

1000

5.7

7.4

45.0

62.0

–

–

–

–

1420 1700

8.0 9.5

11.0 13.0

63.0 74.0

88.0 104.0

– –

– –

– –

– –

100

0.4

0.5

3.4

4.4

27

35

–

–

250 500

1.1 2.1

1.3 2.6

8.3 16.0

11.0 22.0

66 129

91 185

– –

– –

710

2.9

3.7

23.0

32.0

182

264

–

–

1000 1420

4.1 5.8

5.3 7.6

32.0 45.0

45.0 64.0

253 354

370 (520)

– –

– –

1700

6.9

9.1

54.0

76.0

–

–

–

–

100 250

0.28 0.7

0.32 0.8

2.2 5.5

2.7 6.8

18 43

22 58

139 342

191 (492)

500

1.4

1.6

11.0

14.0

85

118

–

–

710 1000

1.9 2.7

2.4 3.4

15.0 21.0

20.0 29.0

120 167

170 240

– –

– –

1420

3.8

4.8

30.0

41.0

235

340

–

–

1700 100

4.6 0.15

5.8 0.17

36.0 1.2

49.0 1.4

279 9.2

405 12

– 73

– 99

250

0.36

0.42

2.9

3.5

23

30

179

256

500 710

0.7 1.0

0.8 1.2

5.7 8.0

7.2 10.0

45 63

62 89

352 (494)

(523) (746)

1000

1.4

1.7

11.0

15.0

88

126

–

–

1420 1700

2.0 2.4

2.5 3.0

16.0 19.0

21.0 26.0

123 147

180 215

– –

– –

100

0.08

0.09

0.6

0.7

4.8

6.0

38

50

250 500

0.19 0.38

0.22 0.44

1.5 3.0

1.8 3.7

12 24

15 32

95 186

131 268

710

0.50

0.60

4.2

5.3

33

45

262

385

1000 1420

0.70 1.10

0.90 1.30

5.9 8.3

7.6 11.0

47 66

65 93

365 (512)

(546) (778)

1700

1.30

1.50

9.9

13.0

78

111

–

–

Note: Gear and rating data: See Table 15.7. See the text for due reservations about table validity.

Load Rating of Gears

861

TABLE 15.9 Helical Gears: Data and General Assumptions for Capacity Table 15.10 Gear Ratio m G

Tooth Numbers NP /NG

Helix Angle

Addendum Modif. Coefficients

s

xP

Transverse Operating Pressure Angle t

Center Distance C, mm 50

100

200

300

Normal module, m n, mm

xG

1.0

23/23

21

0.1903

0.1903

23.3437

2

4

8

1.6

20/32

22

0.3036

0.2566

23.9929

1.75

3.5

7

–

2.5

18/45

18

0.2413

–0.0245

21.8755

1.5

3

6

12

4.0

15/60

19

0.3218

0.0270

22.2820

1.25

2.5

5

10

6.3

13/82

17

0.3736

–0.0366

21.8078

1

2

4

8

–

Notes: External teeth: Normal standard pressure angle ns = 20 . Outside diameters do = Nmn / cos s + 2mn (1 + x ). Carburized case-hardened alloy steel, 58–62 HRC. Ground teeth. Service factor CSF = 1.5 including application, reliability, and life factors. Power-sharing factor Ksh = 1. Dynamic factors Kv rated according to RHA or RHI methods for AGMA quality number =11. Load distribution factor Km = 1.25. RHA: contact line coefficient m 1 rated according to the actual tooth data; sc lim = 1500 N/mm2 ; AH = 1. RHI: sc lim = 1500 N/mm2 ; AH rated for f = 0.8 m (32 in.), v40 = 200 cSt , and pitch velocity (other influential factor =1).

ULlim =

Jn AF sc lim Ks V

(15.88)

where ULlim and sc lim are in consistent units, either N/mm2 or lb/in2. The overall overload factor V has been defined in the foregoing. See the following discussion for the other factors.

15.6.1 RF—GEOMETRY FACTOR, JN The geometry factor accounts for tooth form and stress concentration at fillet. The higher point of single contact, D for the pinions, Figures 15.9 and 15.10, and B for the gears, Figures 15.7, and 15.8, is taken as the rating point for spur gears in RFA and for both spur and helical gears in RFI, so-called method B, according to the procedure of Castellani and Parenti-Castelli that is followed entirely here. Tip meshing is considered in RFA normal method for helical gears instead. Both methods introduce adjusting factors. The detailed computation of the geometry factor requires programming. The equations can be de­ ducted from the original standards or from the cited references that defines the actual fillet form as generated by pinion cutter as well as by rack cutter or by hobs. Then, a formally unified geometry factor can be deduced from the original factors by means of the following equations. RFA: Jn =

J cos

s

cos

t

where J is the geometry factor. RFI: Jn =

(15.89)

YF YS Y cos

(15.90) s

862

Dudley’s Handbook of Practical Gear Design and Manufacture

TABLE 15.10 Surface Capacity of Case-Hardened Helical Gears Gear Ratio mG

Pinion Speed nP r/min

Center Distance C, mm/Net face width, mm 50/17.5

100/35

200/70

400/140

Power, kW (multiply by 1.341 for power in hp) RHA

1.0

1.6

2.5

4.0

6.3

RHI

RHA

RHI

RHA

RHI

RHA

RHI

100

1.1

1.0

8.5

8.1

67

69

–

–

250

2.6

2.5

21

21

165

180

–

–

500 710

5.2 4.9

5.1 5.1

41 58

43 62

(323) –

(367) –

– –

– –

1000

6.9

7.2

81

89

–

–

–

–

1420 1700

9.7 12.0

10.0 13.0

113 135

126 152

– –

– –

– –

– –

100

0.44

0.41

5.60

5.70

45

48

–

–

250 500

1.1 2.2

1.00 2.10

14.0 27.0

15.0 30.0

110 216

124 255

– –

– –

100

4.9

5.1

39

43

303

366

–

–

250 500

6.9 9.7

7.2 10.0

54 76

62 88

(422) –

(519) –

– –

– –

710

12.0

13.0

90

106

–

–

–

–

1000 1420

0.44 1.10

0.41 1.00

3.5 8.6

3.40 8.60

28 68

28 74

219 (536)

242 (6.29)

1700

2.20

2.10

17

18.0

134

151

–

–

100 250

3.0 4.3

3.0 4.3

24 34

26 37

188 263

218 310

– –

– –

500

6.0

6.2

47

52

368

443

–

–

710 1000

7.2 0.21

7.4 0.21

56 1.7

63 1.8

438 14

(531) 15

– 108

– 124

1420

0.53

0.53

4.2

4.5

34

38

265

323

1700 100

1.10 1.5

1.10 1.5

8.4 12

9.1 13

66 93

78 112

(520) –

(664) –

250

2.1

2.2

17

19

130

160

–

–

500 710

3.0 3.5

3.2 3.8

23 28

27 33

183 217

230 276

– –

– –

1000

0.10

0.10

0.8

0.8

6.6

7.0

52

58

1420 1700

0.26 0.51

0.26 0.51

2.1 4.1

2.1 4.3

16.0 32.0

18.0 37.0

129 254

153 315

710

0.7

0.7

5.8

6.2

46

53

358

453

1000 1420

1.0 1.4

1.0 1.5

8.1 11.0

8.9 13.0

64 89

76 109

(499) –

(646) –

1700

1.7

1.8

13.0

15.0

107

131

–

–

Note: Gear and rating data: See Table 15.9. See the text for due reservations about table validity.

Load Rating of Gears

863

TABLE 15.11 Surface Capacity of Case-Hardened Straight Bevel Gears Gear Ratio mG

Tooth Numbers NP /NG

Gear Pitch Diameter, dG , [mm] ([in. ], Approx.)

Pinion Speed nP [r/min]

50 (2)

100 (4)

200 (8)

400 (16)

Power, kW (Multiply by 1.341 for Power in hp.) RHA 1.0

1.6

2.5

4.0

6.3

25/25

18/29

15/37

14/56

13/82

RHI

RHA

RHI

RHA

RHI

RHA

RHI

100

0.32

0.34

2.5

2.8

19

24

–

–

500 1000

1.5 2.9

1.7 3.4

11 22

14 27

86 160

111 197

– –

– –

1420

3.9

4.7

30

36

–

–

–

–

1700 100

4.6 0.12

5.6 0.13

35 0.9

42 1.0

– 7.3

– 8.5

– –

– –

500

0.6

0.6

4.4

5.3

33

44

–

–

1000 1420

1.1 1.5

1.3 1.8

8.3 12

11 15

63 86

85 116

– –

– –

1700

1.8

2.2

14

18

101

135

–

–

100 500

0.05 0.25

0.05 0.25

0.39 1.9

0.43 2.2

3.1 14

3.5 18

24 108

29 151

1000

0.50

0.50

3.6

4.4

27

36

203

(293)

1420 1700

0.60 0.80

0.70 0.90

5.0 5.8

6.2 7.5

37 44

51 61

– –

– –

100

–

–

0.15

0.16

1.2

1.3

9.2

11

500 1000

0.09 0.18

0.10 0.19

0.7 1.4

0.8 1.6

5.5 11

6.8 14

42 80

58 115

1420

0.25

0.28

1.9

2.4

15

20

109

161

1700 100

0.30 –

0.34 –

2.3 0.06

2.8 0.06

17 0.47

24 0.53

128 3.7

190 4.4

500

–

–

0.28

0.32

2.2

2.7

17

23

1000

0.07

0.08

0.5

0.6

4.3

5.5

32

46

1420 1700

0.10 0.12

0.11 0.13

0.8 0.9

0.9 1.1

5.9 7.0

7.8 9.4

45 53

65 78

Notes: Shaft angle . Pressure angle ns = 20 . Net face width = 0.3 cone distance . Tooth proportions recommended by Gleason Works, Rochester, New York. Carburized and case-hardened alloy steel, 58–62 HRC. Tool finishing. Service factor CSF = 1.5 including application, reliability, and life factors. Ksh = 1, Kv for AGMA quality number = 8, Km = 1.4 . RHA: sc lim = 1500 N/mm2 ; AH = 1. RHA: sc lim = 1500 N/mm2 ; AH rated for f = 2 m(80 in.), v40 = 200 cSt , and actual pitch velocity. See the text for due reservations about table validity.

The RFI factors for tooth form (YF ), stress correction (YS = Kf ), and helix (Y ), are clarified in the cited references. The calculation of the helix factor Y is reported here because it shows how lower face contact ratio mF affects Jn according to ISO for helical gears:

864

Dudley’s Handbook of Practical Gear Design and Manufacture

TABLE 15.12 Surface Capacity of Case-Hardened Spiral Bevel Gears Gear Ratio mG

Tooth Numbers NP /NG

Gear Pitch Diameter, dG , [mm] ([in.] , Approx.)

Pinion Speed nP[r/min]

50 (2)

100 (4)

200 (8)

400 (16)

Power, kW (Multiply by 1.341 for Power in hp.) RHA 1.0

1.6

21/21

100

4.0

RHI

RHA

RHI

RHA

RHI

0.58

0.60

4.6

5.0

36

42

–

–

2.8 5.4

3.1 6.3

21.0 41.0

26.0 52.0

164 31-

217 423

– –

– –

1420

7.4

8.9

57.0

74.0

(425)

(583)

–

–

1700

8.8 0.21

11.0 0.21

67.0 1.7

87.0 1.8

– 1.3

– 15

– –

– –

500

1.0

1.1

7.9

9.2

61.0

78

–

–

1000 1420

2.0 2.7

2.2 3.2

15.0 21.0

19.0 27.0

116 160

158 224

– –

– –

16/25

100

3.3

3.9

25.0

32.0

188

267

–

–

500

0.07 0.36

0.07 0.38

0.58 2.8

0.64 3.3

4.6 22

5.3 28

36 166

45 236

1000

0.7

0.8

5.4

6.7

41

57

315

(477)

1420 1700

1.0 1.2

1.1 1.4

7.5 8.9

9.6 12.0

57 68

81 98

– –

– –

–

–

0.23

0.24

1.8

2.0

14

16

0.14 0.27

0.14 0.29

1.10 2.10

1.20 2.50

8.5 16

10 21

66 126

87 178

14/35

100

13/52

100 500 1000

6.3

RHA

500 1000

1700 2.5

RHI

1420

0.38

0.42

3.00

3.60

23

30

174

255

1700

0.45 –

0.50 –

3.50 0.09

4.30 0.09

27 0.7

36 0.8

205 5.5

305 6.5 34

12/75

100 500

–

–

0.43

0.47

3.3

4.0

26

1000

0.11

0.11

0.80

0.90

6.5

8.2

50

70

1420 1700

0.15 0.18

0.16 0.19

1.20 1.40

1.40 1.70

9.1 11.0

12.0 14.0

70 82

100 120

Notes: Shaft angle . Pressure angle ns = 20 . Spiral angle s = 35 . Net face width = 0.3 cone distance . Tooth proportions recommended by Gleason Works, Rochester, New York. Carburized and case-hardened alloy steel, 58–62 HRC. Lapped teeth. Service factor CSF = 1.5 including application, reliability, and life factors. Ksh = 1, Kv for AGMA quality number = 9 , Km = 1.4 . RHA: contact line coefficient m 1 rated according to the actual tooth data; sc lim = 1500 N/mm2 ; AH = 1. RHA: sc lim = 1500 N/mm2 ; AH rated for f = 2 m (80 in.), v40 = 200 cSt , and actual pitch velocity. See the text for due reservations about table validity.

Y =1

min(mF , 1) min( s , 30) 120

This must be remembered when consulting the following Jn tables that refer to mF

(15.91)

1 for RFI.

Load Rating of Gears

865

The operating angular parameters, instead of the standard ones, have been introduced by AGMA for both form and stress correction factors. However, the differences are small, especially for small centerdistance modifications. The tables of the cited references can be used for guidance, for both RFA and RFI. Warning: AGMA standards give an alternative procedure for the so-called LCR helical gears with mF 1, but they often give more optimistic results. Instead, the normal procedure is followed in this chapter, as based on the load-sharing ratio mN [see Eq. (15.56)] the same as for RH. Tables 15.13 and 15.14 are supplied here. The stress correction factors Kf are added, not only for comparison purposes, but also because they must be considered again for excluding the stress con­ centration effect in the yielding calculations RFA (which follow). Note that the DIN method coincides with ISO’s for spur gears, whereas in leads to more optimistic results for helical gears with regard to the RFI procedure considered here: With reference to Table 15.13, Jn increases by about 4 percent for s = 15 , and by as much as 17 to 21 percent for s = 30 . In Table 15.13, a European standard hob or rack cutter has been considered (though tip edge rounding is considered rather small for cautionary reasons). On the other hand, the results are not too different for standard AGMA tools with smaller addendum and smaller tip rounding, respectively, or with greater addendum and full tip rounding. In fact, the bending arm of the tooth load and the notch effect at the fillet are in some measure self-compensatory. Full rounding gives somewhat better Jn values; remember that the tooth root surface must be good to ensure a real improvement of tooth strength. The table can obviously serve only as a general guide. A programmed computation is necessary if specific manu­ facturing conditions must be taken into account. As regards internal teeth, some comparison with FEM analysis shows that the standard methods hardly agree with real fillet stresses. Thus, Table 15.14 serves more as pure information than for practical purposes. A 45 angle of the tangent to the fillet (instead of 30 ) has been considered for RFI. (The original ISO and DIN methods simplify the procedure by considering rack instead of internal gears, but they fail to give adequate estimations of the fillet radius). In the cases of Gleason bevel gears, if a computation is performed, the tooth thickness correction must be considered as independent of the addendum modification. When using tables, note that a slight in­ crease in Jn for the pinion and decrease for the gear should be considered if the well-known Gleason geometrical coefficient K has been introduced for the thickness correction and K > 0 .

15.6.2 RF—ADAPTATION FACTOR, AF A factor A F corrects the geometry factor by taking into account any peculiarity of geometry and material condition at fillet: A F > 1 for favorable, and A F < 1 for unfavorable circumstances as a whole. There is no similar overall factor in the standard methods, whereas there are some particular factors that may be thought of as a part of A F . Slim ring gears, both external and internal—the case is obviously more frequent internal ones—have as stress concentration increase at the tooth root. The ANSI/AGMA standards introduce a de-rating “rim thickness factor”, KB , and (“for informational purposes only”) suggests KB > 1 when the ratio of the rim thickness over the whole tooth depth is less than 1.2. This means that the adaptation factor AF should be divided by KB . On the other hand, the real root stress in such gears depends on many parameters, and direct analyzes should be made if their tooth strength is central to the overall performance of a gear pair. In the original AGMA method there is a factor KT 1 that de-rates strength for higher temperature. It may be included in AF inversely, that is, it may lower AF . There certainly is a de-rating temperature influence, but it is a problem to make a reliable assessment of it. As for ISO, there are two factors that can be thought of as included in AF : the notch sensitivity factor Y rel T , and the roundness factor, YRrel T . They are defined as relative factors, as they introduce modifications of the stress correction with regard to “test” gears.

866

Dudley’s Handbook of Practical Gear Design and Manufacture

TABLE 15.13 Geometry Factors for RFA and RFI, External Gear Pairs s

0

15

30

NP / NG

xP

xG

JnAP

KnAP

JnAG

KnAG

JnIP

KnIP

JnIG

KnIG

13/82 13/82

0.4 0.4

0.0 0.4

0.378 0.380

1.938 1.906

0.358 0.388

1.908 1.935

0.285 0.282

2.213 2.208

0.277 0.261

2.216 2.547

13/41

0.4

0.0

0.370

1.872

0.377

1.807

0.276

2.166

0.277

1.988

13/41 13/21

0.4 0.4

0.4 0.0

0.372 0.360

1.817 1.782

0.388 0.296

1.857 1.649

0.271 0.264

2.155 2.103

0.266 0.253

2.380 1.768

13/21

0.4

0.4

0.362

1.698

0.377

1.733

0.256

2.082

0.261

2.187

26/82 26/82

0.0 0.4

0.0 0.0

0.364 0.434

2.015 2.120

0.412 0.386

2.132 2.008

0.323 0.311

2.028 2.527

0.317 0.297

2.429 2.322

26/82

0.4

0.4

0.434

2.081

0.420

2.048

0.307

2.513

0.278

2.699

26/41 26/41

0.0 0.4

0.0 0.0

0.353 0.422

1.972 2.054

0.379 0.361

2.035 1.910

0.313 0.301

1.991 2.466

0.319 0.298

2.144 2.071

26/41

0.4

0.4

0.419

1.988

0.417

1.980

0.294

2.435

0.285

2.512

26/26 26/26

0.0 0.4

0.0 0.4

0.344 0.406

1.934 1.909

0.344 0.406

1.934 1.909

0.305 0.283

1.960 2.368

0.305 0.283

1.960 2.368

13/82

0.4

0.0

0.516

1.487

0.504

1.643

0.319

2.193

0.308

2.216

13/82 13/41

0.4 0.4

0.4 0.0

0.525 0.511

1.458 1.465

0.542 0.477

1.642 1.556

0.316 0.310

2.188 2.150

0.290 0.309

2.522 1.994

13/41

0.4

0.4

0.524

1.419

0.541

1.563

0.304

2.139

0.296

2.361

13/21 13/21

0.4 0.4

0.0 0.4

0.502 0.518

1.437 1.374

0.427 0.530

1.425 1.451

0.297 0.288

2.091 2.070

0.287 0.291

1.780 1.172

26/82

0.0

0.0

0.515

1.543

0.568

1.680

0.361

2.031

0.350

2.415

26/82 26/82

0.4 0.4

0.0 0.4

0.573 0.581

1.602 1.573

0.545 0.583

1.647 1.650

0.343 0.339

2.488 2.475

0.329 0.308

2.318 2.662

26/41

0.0

0.0

0.499

1.543

0.527

1.613

0.351

1.996

0.354

2.143

26/41 26/41

0.4 0.4

0.0 0.4

0.565 0.573

1.584 1.542

0.513 0.576

1.566 1.582

0.334 0.326

2.433 2.405

0.332 0.315

2.074 2.483

26/26

0.0

0.0

0.486

1.543

0.486

1.543

0.341

1.966

0.341

1.966

26/26 13/82

0.4 0.4

0.4 0.0

0.562 0.522

1.520 1.558

0.562 0.504

1.520 1.672

0.314 0.339

2.343 2.106

0.314 0.326

2.343 2.197

13/82

0.4

0.4

0.531

1.536

0.533

1.671

0.336

2.102

0.308

2.430

13/41 13/41

0.4 0.4

0.0 0.4

0.519 0.531

1.541 1.505

0.485 0.535

1.606 1.611

0.331 0.326

2.077 2.068

0.331 0.313

1.998 2.289

13/21

0.4

0.0

0.512

1.519

0.450

1.504

0.320

2.035

0.317

1.800

13/21 26/82

0.4 0.0

0.4 0.0

0.526 0.520

1.469 1.596

0.530 0.557

1.525 1.699

0.311 0.381

2.018 2.011

0.310 0.360

2.116 2.350

26/82

0.4

0.0

0.564

1.643

0.539

1.675

0.355

2.354

0.343

2.279

26/82 26/41

0.4 0.0

0.4 0.0

0.571 0.508

1.621 1.596

0.568 0.527

1.677 1.650

0.351 0.372

2.346 1.985

0.323 0.369

2.535 2.117

26/41

0.4

0.0

0.559

1.630

0.517

1.614

0.347

2.320

0.351

2.064

26/41 26/26

0.4 0.0

0.4 0.0

0.565 0.497

1.597 1.596

0.565 0.497

1.626 1.596

0.340 0.364

2.300 1.962

0.330 0.364

2.381 1.962

26/26

0.4

0.4

0.556

1.580

0.556

1.580

0.330

2.256

0.330

2.256

Notes: General data, t and contact ratios: same as for Table 15.3. Cutting data for m1 = Pnd = 1: • Hob or rack cutter without protuberance, addendum = 1.25, radius of tip edge rounding = 0.2 . • Reduction of normal base thickness of pinion and gear = 0.02 for tooth backlash.

Load Rating of Gears

867

TABLE 15.14 Geometry Factors for RFA and RFI, Internal Spur Gears KfAG

JnIG

KfIG

0.380

2.278

0.377

2.191

0.339 0.280

2/533 3.060

0.276 0.195

2.894 4.000

0.4

0.423

2.064

0.414

2.013

0.0 –0.4

0.406 0.394

2.115 2.177

0.334 0.276

2.411 2.854

0.0

0.0

0.301

2.854

0.197

4.000

0.2

0.0

0.398

2.155

0.298

2.667

0.0 –

0.4 0.0

0.420 0.375

2.305 2.577

0.362 0.256

2.461 3.392

–

–

–0.4

0.306

3.154

0.213

4.000

– –

0.2 –

0.4 0.0

0.469 0.455

2.076 2.124

0.407 0.330

2.204 2.653

–

–

–0.4

0.446

2.170

0.278

3.101

40 40

0.0 0.2

0.0 0.0

0.319 0.447

3.035 2.162

0.214 0.289

4.000 2.998

15

0.0

0.0

o.443

2.660

0.331

3.141

15 25

0.2 0.0

0.0 0.0

0.520 0.445

2.266 2.662

0.389 0.309

2.666 3.330

–

25

0.2

0.0

0.528

2.290

0.386

2.719

– –

40 40

0.0 0.2

0.0 0.0

0.397 0.515

2.934 2.313

0.253 0.339

4.000 3.045

0.4

0.0

25

0.0

0.0

0.365

2.654

0.290

3.039

– –

– –

25 40

0.2 0.0

0.0 0.0

0.440 0.323

2.219 2.982

0.355 0.217

2.513 4.000

NP / NG

xP

xG

ZC

aC

xC

JnAG

13/82

0.4

0.0

25

0.0

0.4

– –

– –

– –

– –

0.0 –0.4

–

–

–

0.2

– –

– –

– –

– –

–

–

40

–

–

40

0.4 –

0.4 –

25 –

–

–

– –

– –

–

–

– –

– –

13/41

0.4

0.4

26/82

– 0.0

– 0.0

– – –

13/82

26/82

–

–

40

0.2

0.0

0.431

2.251

0.315

2.792

0.4 –

0.4 –

25 25

0.0 0.2

0.0 0.0

0.417 0.508

2.688 2.237

0.269 0.353

3.644 2.814

–

–

40

0.0

0.0

0.354

3.144

0.242

4.000

–

–

40

0.2

0.0

0.498

2.264

0.307

3.203

Notes: General data, notes, t and contact ratios: the same as for Table 15.4. Cutting data for mn = 1 [mm] Pnd = 1 [in. 1 ]: • Shaper cutter with Zc tooth number, nominal addendum = 1.25, radius of tip edge rounding = relating to a given sharpening condition of the cutter. • Reduction of gear base thickness = 0.02 for tooth backlash. Angle of fillet tangent = 45 for RFI. The occasional value KfIG = 4 is arbitrary (ISO’s rating is out of range).

aC ,

addendum modification = xC

The relative variation of the notch sensitivity has no great influence on steel gear strength according to ISO. However, a reduced notch sensitivity can be favorable for bad fillets, unfavorable for good ones (for example, for fillets better than those considered in the table), that is, the computation of KfI gives too optimistic values in such a case.

868

Dudley’s Handbook of Practical Gear Design and Manufacture

A higher fillet roughness is unfavorable and tool scores are worse. It may be necessary to diminish AF by 5 to 10 percent. A factor AF = 0.95 to 0.90 can be suggested for teeth cut without tool protuberance and shaved, if no other peculiarity occurs. More severe restrictions are necessary for grinding steps that can be originated by many grinding conditions. AF = 0.95 to 0.75 in common cases. But AF should be as low as 0.3 in some extreme cases! It must be stressed that even teeth cut with tool protuberance may present fillet anomalies. The effect of short-peenting is doubtful for planet gears and in any case of bidirectional loading. Otherwise, it is generally favorable and suggests AF > 1, especially if applied on low-roughness fillets free from any irregularities. Specific tests are advisable for important cases or for mass production gears. Note: We may correct the precious indications for RFA ratings since the original AGMA values of fatigue limit stress are rather low (see the following). They have been deduced from field experience, contrary to ISO indications that originate from laboratory tests. Therefore, it may be supposed that AGMA’s fatigue limits already take into account the most common cases of bad fillet conditions and AF = 1 may be adopted in such cases. On the contrary, an accurate AF assessment is necessary for RFI ratings.

15.6.3 RF—SIZE FACTOR, KS AGMA does not give any numerical indication for Ks > 1. ISO prescribes Ks = 1 for any case if m n [mm] 5, and Ks = 1 for m n [mm] > 5. (Note that the original ISO factor is YX = 1/ Ks ). For RFI m n [mm] > 5Ks =

1

1 A (B

5)

(15.92)

where: For steel or nodular iron gears, non-surface-hardened: A = 0.006 B = mn for mn 30 B = 30 for m n > 30 For surface-hardened steel: A = 0.010 B = mn for mn 30 B = 30 for m n > 30 For cast iron: A = 0.015 B = mn for mn 25 B = 25 for m n > 25

Note: The meaning of the size factor Ks for ISO is as follows. It is known that equal materials with similar metallurgical structure and equal hardness have different fatigue resistance according to their size. Then Ks must affect the fatigue limit—as it does, see Eq. (15.88)—but not the stress value [Eq. (15.94)]. AGMA apparently considers it as a generic departing factor that takes into account any un­ determined cause of stress increasing for big gear sizes: In fact, Ks is included in the calculation of st according to AGMA [see the following discussion, Eq. (15.93)].

15.6.4 RF—CONVENTIONAL FATIGUE LIMIT

OF THE

FILLET STRESS, stlim

Table 15.6 reports values near the maximum indicated by the standards for stlim (AGMA symbol: sat . ISO and DIN symbol: FE ). Note that ISO intended a different parameter by the symbol F lim , which is a nominal stress and amounts to one-half of FE stress correction factor for test gears equals 2). The difference between AGMA and ISO indications is striking and is in part compensated by the difference of Kf and Jn factors. For instance, ISO indications for case-hardened gears lead to greater ULlim

Load Rating of Gears

869

so that Figure 15.4 generally gains in significance. However, the ISO RF assessments have been in­ dustrially tested. What is necessary in RFI ratings is a careful choice of the adaptation factors (see the foregoing), as well as a proper assumption of all factors influencing the final results. The indication given Table 15.6 about reducing stlim for bidirectional load, as applied for a number of cycles in every direction.

15.6.5 RF—TOOTH ROOT STRESS

AT

FILLET, st

The tooth root stress does not directly enter the ratings based on the UL factor, and is given only for information. Its value includes the stress correction factor Kf : Thus, it would mean the actual root stress, provided that no plastic phenomena occur (which is generally not true, at least for steel gears). RFA:

RFI:

st =

st =

VUL Ks Jn

(15.93)

VUL Jn

(15.94)

The difference between the two equations depends on the nature of the Ks factor (see the foregoing discussion).

15.6.6 RF—SERVICE

FACTOR,

KSF (ONLY

FOR

ONE LOADING LEVEL)

Once the detailed gear data have been established, the loading data are equivalent to the following service factor for RF: KSF =

Ka ULlim UL

(15.95)

If this way is adopted, the service factor summarizes application and reliability problems as well as those of gear life and excludes detailed ratings of the latter.

15.7 COPLANAR GEARS: DETAILED LIFE CURVES AND YIELDING In the following paragraphs equations and data are given for defining the life curves in a unified way for RH and RF. The simple approach may be disputed, especially for RH for the calculation of the cu­ mulative damage according to Miner’s rule, but it is usually accepted for industrial gears as best esti­ mated values and is preferable to no checking at all. The adopted procedure facilitates the assumption of life curves different from the indications of standards, if the designer prefers to follow the indications of specific gear design books or those of an available field experience. In fact, many factors can affect the real slope of the life curve and the real life for a given load. More appropriate approaches are advisable for high-performance gears in special fields, and require specific experimentation in conditions similar to the applications as regards sizes, materials, lubrication, loading, and velocity. The procedure is based on the definition of a loading factor QH or QF that coincides with the life factor of the standards KL for RF, or with the squared CL factor for RH, but works in an opposite way. The value of the life factor depends on the cycle number N and leads to determining a reliability factor CR or KR . Such a procedure cannot deal with the cases of more than one loading level. Instead, the QH or QF factor is calculated as depending directly on the given load, so that the cumulative damage and gear life are rated, and the reliability factor can be determined from the damage. Yielding ratings are facilitated.

870

Dudley’s Handbook of Practical Gear Design and Manufacture

Loading Factors QH , QF . For each loading level the following ratio is called loading factor:

15.7.1 DEFINITION

OF THE

RH:

QH =

K Klim

(15.96)

RF:

QF =

UL ULlim

(15.97)

LIFE CURVES

AND

GEAR LIFE RATINGS

FOR

ONE LOADING LEVEL

As the loading factor is proportional to the tangential load Wt , the life curve of Figure 15.1 can be redrawn with QH or QF as ordinates, see Figure 15.13. QW is the maximum initial value, QV corresponds to vertex V , and QS is a safety value below which some assume no failure will occur. The choice of Q S is up to the gear designer. It is advisable not to assume QHS greater than 0.8 or QFS greater than 0.7, with a possible exception: Higher QFS values might be allowed for RFA, especially if a proper adaptation factor AF is adopted, because a safety margin is implied in the fatigue limits. Lower values are necessary for an increased reliability, for example, when a gear failure may be dangerous for the operators (see Section 15.7.4 “Reliability”), for example, for hoists, marine applica­ tions, and so forth. The vertex value Q V equals unity with the exception of RFA, where QFV = 1.04.

QH , QF W

Qw

V

Qv

conventional fatigue limit S

Qs

N 103

104

105 NW

106

107

108

109

1010

NV

FIGURE 15.13 Unified definition of the characteristic points of the life curves for gear capacity ratings, RH curves (N , QH ) or RF curves (N , QF ).

Load Rating of Gears

871

The initial maximum value QW can be chosen by the designer, or it equals the maximum standard value of the life factor (squared for RH), CLW 2 for RH and KLW for RF, reported in Table 15.15. In the same table, the standard cycle numbers for points W and V are given, that is, NW and NV . AGMA does not consider any correction of such maximum. Thus, RHA:

QHW = CLW 2

(15.98)

RFA:

QFW = KLW

(15.99)

Note that the concept of Q W is “yielding” (for direct yielding assessments see the following). ISO suggest some corrections of the life curves that can be extended by assuming approximately that the adaptation factors, as well as size factor for RF, influence fatigue, but not yielding limits. Thus, such factors must be excluded, as they have been introduced in the calculation of the fatigue limit:

TABLE 15.15 Coordinates of Points Defining Life Curve Initial point W , Figure 15.13: number of cycles, NW maximum loading factor, QHW for RH, QFW for RF RHA:

NW = 10 4 QHW = CLW 2

RHI:

CLW 2 = 2.17

NW =

C

LW 2

AH

– all materials

CLW 2 = 1.69

– for cast iron and for gas-nitrided steels

CLW = 2.56

– for all other materials

2

103

QFW = KLW

– all steels

KLW = 2.7 KLW = 2.4 +

RFI:

– all steels

NW = 10 5

QHW = RFA:

– all steels

– for case-hardened steels HB 250 140

NW = 103 NW =

QFW =

HB

320

– for cast iron and case-hardened steels

10 4 KLW Ks AF

– for through-hardened steels in the range of 210

– for all other materials

KLW = 1.6

– for cast iron and gas-nitrided steels

KLW = 2.5

– for all other materials

Vertex V , Figure 15.13: number of cycles, NV conventional fatigue limit, QHV for RH, QFV for RF RHA:

NV = 107

QHV = 1

– all steels

RHI:

NV = 2 106

QHV = 1

– for cast iron and gas-nitrided steels

NV = 5 107

QHV = 1

– for all other materials

NV = 3

106

QHV = 1.040

– all steels

NV = 3

106

QHV = 1

– all materials

RFA: RFI:

The table refers to the following materials: through-hardened and tempered steel; carburized and case-hardened steel; gasnitrided steel; induction-hardened steel, with root hardening for RF. The RHI and RFI indications include cast iron, malleable iron (pearlitic), and nodular (ductile) iron.

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Dudley’s Handbook of Practical Gear Design and Manufacture

RHI:

RFI:

CLW 2 AH

(15.100)

KLW Ks AF

(15.101)

QHW =

QFW =

Thus, the estimations are adjusted with regard to the gears that have been tested in the laboratory. Note that the corrections do not mean that the yielding assessment is modified: on the contrary, they mean that the yielding limit is considered as independent of the cited factors that induce the fatigue limit. The ratio QW is adjusted for this purpose. (A single further adjustment will be added for RFI for the direct assessment of the margin against yielding, in the case of reverse of bidirectional load, see the following.) The sloped stretches WV and VS of the life curve follow the equations: RH:

NHf =

RF:

NFf =

NHV (QH ) A NFV

( )

QF A QFV

(15.102)

(15.103)

where: A = A1 for the first stretch; A = A2 for the second stretch; Nf – is the gear life (cycle number) until failure (pitting or tooth breakage). The curve stretch WV is fully defined by the previous choice of the coordinates of points W , V . For NW N NV :

( ) A=A = log ( ) log

1

NV NW

QW QV

(15.104)

The exponent A 2 for the stretch VS , that is, for N > NV and QH > QHS or > QFS , must be chosen by the designer. AGMA standards only give ranges that correspond: • For RHA, to A2min = A1, A2max = 22; • For RFA, to A2min = 31, 56 . ISO does not give any indication. Of course, A2 can not be less than A1, whose value is rather different from AGMA’s in its turn. Lower A2 leads to lower curve and more pessimistic results. There are a lot of influencing parameters, but for RH the velocity is probably the most important. A low velocity involves boundary lubrication and suggests a lower curve, that is, lower exponent A2, possibly A2 = A1; whereas a high velocity allows higher A2 values if it makes possible EHD lubrication of accurate gears with low surface roughness. For RF the choice is above all a matter of reliability from various points of view: A good and checked reliability of materials and manufacturing allows higher A2, while a desired better reliability of gear resistance suggests lower A2 values.

Load Rating of Gears

873

Different life curves can be established by adopting different coordinates of points W and V . Specific tests or indications of specific books for gear design can help. Any type of ordinates can be found there: pinion torque, tangential load, Hertzian pressure, root stress, and so forth. It is easy to translate them into QH and QF factors: A point that serves as the conventional fatigue limit should be chosen and a loading factor QV = 1 attributed. Remember that the Hertzian pressure must be squared when calculating the ratio QH . Finally, the exponent A 2 is chosen so that the desired curve is fully schematized. Now, the cycle number per minute nL may or may not be equal the gear revolutions per minute according to possible power sharing and uni- or bidirectional loading. If Nf is the cycle number until failure for a given loading level, as rated by Eqs. (15.102) or (15.103), the gear life until failure, in hours, is: Lf =

Nf 60 nL

(15.105)

Note: n L can be different for RH and RF. For example, consider a common planet pinion: Foe RF, it completes a load cycle for each revolution of the planet pinion, relative to the sun pinion and ring gear, and includes the load applications on both tooth sides. (Of course, its strength is de-rated because of the reverse bending load: This affects the preliminary assumption of stlim ). For RH, the surface K factor relating to the planet-ring mesh is usually non-damaging for the planet gear. One cycle per revolution must be considered if the load transmitted by the sun gear is unidirectional; but one-half cycle per revolution, as an average, if the load is bidirectional and its application is borne equally by the two tooth sides. Note: Uni- or bidirectional load is not the same as uni- or bidirectional motion. For instance, the speed reducer for a hoist has bidirectional motion but unidirectional loading, whereas the gears that control the traveling of a bridge crane have both bidirectional motion and load.

15.7.2 YIELDING As in the foregoing discussion, the Q W value of the loading factor should mean yielding. RH. AGMA does not say anything on this subject, whereas ISO explicitly attributes the meaning of surface yielding to the beginning of the life curve. No safety margin is defined in this regard. Field experience usually does not show any failures for occasional loads at this level, foe well-rated gears. In all probability, possible initial plastic deformations of the tooth surface turn into work hardening. Thus, a yielding loading factor Q Hy is simply given by Eq. (15.106): RHI QHy = QHW

(15.106)

Loading factors QH higher than QHW should not be allowed. Plastic destruction of the tooth surface is not uncommon if excessive loads are applied even for a short time. RF. AGMA gives an independent rating based on a yielding stress sty (original symbol: say ); Whereas ISO attributes the meaning of yielding to the initial maximum of the life curve. However, a safety margin is necessary for tooth root yielding and it is obtained by means of a factor defined by AGMA: Ky that must be < 1. The definition of the loading factor facilitates the RF yielding ratings as follows. It must be QF QFy , where QFy is a yielding loading factor that includes Ky . RFA QFy =

Ky sty Kf AF stlim

(15.107)

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Dudley’s Handbook of Practical Gear Design and Manufacture

The following equation corresponds approximately to AGMA’s indications for through-hardened gears and tempered steels in the range of 180 HB 410 : sty = A (HB

68)

(15.108)

where A is 3.3 for metric units [N/mm2], and A is 480 for English units [lb/in.2]. AGMA standards do not give any indications for surface-hardened steels. Here, the hardening depth is a determining factor. Besides, the yielding stress must not be thought of in general terms, as the AGMA geometry factor for bending strength refers to the tooth side where the load is applied. This has to do with normal kinds of fatigue failure, but the stress is higher at the opposite side because of compressive stress. The following data can be assumed as a guideline, provided that the proper hardening depth be adopted and that a suitable Ky coefficient be introduced (see the following discussion): Carburized case-hardened steels: sty = 1200 N/mm2 (175.000 lb/in2)

Tooth root induction hardened: sty = 950 N/mm2 (140.000 lb/in2)

Gas-nitrided steels: sty = 700 N/mm2 (100.000 lb/in2)

(Same general specifications as in Table 15.6). RFI QFy =

Ky QFV stlim T stlim

(15.109)

where stlim T is the tabulated or test stlim , while the adopted stlim can be lower because of reverse or bidirectional load or for any reason. Note that the introduction of the Ky factor can make QFy < QFW . In such cases QFW serves solely for determining the life curve, but the maximum load level must be kept lower than QFy . AGMA suggests Ky = 0.75 for general use, and Ky = 0.50 for conservative purposes. Lower values may be advisable in the ratings of cast iron and bronze if the indications of the AGMA standards are followed for sty . Cautious values may also be used for nitride and for induction-hardened gears. The introduction of Kf in Eq. (15.107) serves again to exclude the notch effect for yielding, but this can be more easily accepted for ductile materials. AGMA does not exclude Ks for yielding, whereas ISO does. This depends on the different concepts of the factor, but can be considered as an additional margin against yielding of big gears in the AGMA rating, provided that a Ks greater than 1 has been assumed.

15.7.3 TOOTH DAMAGE

AND

CUMULATIVE GEAR LIFE

The assumption of the safety loading level QS is equivalent to assuming that no damage Dg occurs if QH < QHS or QF < QFS : Dg = 0 , that is, an unlimited life is supposed for RH or RF, respectively, al­ though AGMA and other do not accept this view.

Load Rating of Gears

875

If, on the contrary, the loading level is

QS , the damage of the tooth flank or fillet is: Dg =

L Lf

(15.110)

where L is the desired life in hours for the loading level considered, and L f is the gear life until failure, as rated by Eq. (15.105). If x different damaging loads for RH or for RF are applied to the gear, a cumulative damage is rated: Dgc = Dg1 + Dg2 + ... + Dgx

(15.111)

The allowable cumulative gear life under damaging loads is, inversely: L fc =

Lc Dg

(15.112)

where Lc is the desired cumulative life for the considered damaging levels. If there are further, nondamaging loading levels, the total allowable gear life increases proportionally to the desired gear life.

15.7.4 RELIABILITY The reliability factors, CR for RH and KR for RF, can be easily calculated in accordance with the definitions of the standards if all cycle numbers of the damaging loads are included either in the first stretch WV of the curve (range: NW N NV ), or in the second stretch VS (range: N > NV , QH > QHS or QF > Q FS ). In this case: 1 Dgc

RH:

CR =

RF:

1 KR = Dgc

1/2A

(15.113)

1/ A

(15.114)

(ISO symbols: SH for CR , SF for KR ). If the cycle numbers correspond to different stretches of the curves, iterative calculations would be necessary. The assessment of reliability factors is not necessary in itself: It is given here just because the standards them, and indeed Eqs. (15.113) and (15.114) afford the calculations even in cases where the original methods do not, that is, when more than one loading level is considered. However, the damage assessment is sufficient in itself. A damage limitation is necessary to get reliability: Dgc < 0.3 is usually suitable for RH: lower values for RF, especially if a gear failure is dangerous for the operators or costly because it stops a plant far more important than the gears themselves. In such cases, a severe limitation of the RH damage is necessary, too, as extended wear may induce tooth breakage. QHS and QFS must be lowered accordingly when reliability must be increased.

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15.8 COPLANAR GEARS: PREVENTION OF TOOTH WEAR AND SCORING Lubrication regimes and the associated problems of tooth wear and cold and hot scoring are amply treated the book by Townsend. The reader is referred to it for a thorough examination of them and for a detailed calculation of the hot scoring risk. They involve not only ratings in themselves, but designer competence and suitable tooth profile definition. This chapter is limited to some first estimates.

15.8.1 PROGRESSIVE TOOTH WEAR Fast gears in EHD lubrication regime were examined after 40 years operation and had practically no flank wear. However, slow gears in boundary lubrication always wear. For instance, some very slow spur gears presented a large amount of metal removal in the single contact zone after 10 years—but they did operate for 10 years. In fact, wear was important if compared with cycle number, but the total cycle number was not great, due to the small number of revolutions per minute. The disturbance due to profile alteration was acceptable because of the slowness of motion. In practice, the same limitation of the tooth pressure that is considered as a pitting prevention serves as a wear prevention, especially if an adaptation factor for tooth velocity was introduced, like SSO’s Z v , that lowers the conventional fatigue limit for slow gears. If the desired cycle numbers involve the second sloped stretch of the life curve, then a low exponent of the life Eqs. (15.102) or (15.103) is necessary for slow gears. And a proper lubricant with sufficient density and viscosity is necessary. Lubricant additives are useful if they are appropriate: for example, additives that favor running-in and/or reduce the friction coefficient may threaten wear in the long term, besides their possible action in pitting prevention.

15.8.2 SCORING

AND

SCUFFING

Gears can score even at low speed (cold scoring) if inappropriate lubricants are used and if the gear design does not prevent contacts in high-sliding zones. Cold scoring is distinguished from progressive wear, not only because of a different tooth flank appearance, but because it can occur after a short operating time. “Scoring” usually means hot-scoring, that is, tooth surface destruction of fast loaded gears at high contact temperature. Many gear features can influence scoring. For instance, edge contact of gear teeth may be an important item for scoring risk, and a proper tip relief can prevent it. Another important item is the tooth surface finish. A detailed rating method is based on the concept of the “flash temperature”. Important indications on suitable profile features are also given there. Other detailed methods are given by the standards. Scoring prevention does not concern common industrial gear applications, since EP lubricants have become usual, but only fast, heavily loaded gears. This chapter gives only a simplified equation. For external gears with parallel shafts and ns = 20 , vt < 30 m/s, m n < 10 mm , m pA cos s , and m pE cos s : Wtscor =

dP 1 + NP / NG

3

0.48 F cos V vt 0.007mn2 +

s

(15.115) mn

where dP , F , and m n are in mm, vt is in m/s, and the scoring tangential load is determined in N. In the case of English units, it is better to translate then into metric units at first and apply the formula just as it is. The equation emphasizes the risk of scoring for higher loads. It usually gives a safety margin, as the best EP oil may allow loads even greater by two or three times to be applied without scoring. There are many circumstances in the risk of scoring, and for any doubtful case it is advisable that proper tooth

Load Rating of Gears

877

design and more detailed ratings be performed. For example, the basic gear temperature is an important item, and higher power losses mean higher gear temperature, if all other parameters are equal. Gear pairs with higher contact rations require special investigations, preferably experimental ones.

15.9 CROSSED HELICAL GEARS Crossed helical gears theoretically have point contact so that they can carry a very limited load for RH, while they present no problems for RF. Hertz theory can be applied to the specific case by means of tabulated factors. However, such kinds of gears are often employed as small accessories for mass production, for example, of combustion engines. Thus, it is both desired and possible to perform pre­ liminary tests. A summary review of performance of crossed helical gears, manufactured with typical materials, is given in Table 15.16. A variant can have case-hardened pinions with ground teeth and bath-nitrided gear with shaved teeth.

15.10 HYPOID GEARS In theory, the load capacity rating of hypoid gears might be approached in a way not too different from what is adopted for bevel gears, if one takes into account the real pinion size and corrects the curvature radii of the profiles for RH. In practice, a lot of conflicting observations can be made. Longitudinal sliding may increase the risk of scoring. However, the profile design of such gears can be very accurate at present. The tooth surface can be made less sensitive to axial displacements and misalignment due to mounting and to elastic deflections. They may then carry greater loads compared to bevel gears as designed by more common criteria. Finally, their load capacity must be fully exploited as the reduction of their overall size is important in their usual application fields, for example, for vehicle transmissions.

TABLE 15.16 Normal Capacity of Crossed Helical Gears (Service Factor =1) Pinion Pitch Diameter dP, mm (in.)

Gear Ratio mG

Gear Pitch Diameter dP, mm

Pinion Speed nP , r/min

Center Distance C, mm 500

1000

2000

3000

4000

Power, kW (multiply by 1.341 for hp) 25 (~1)

50 (~2)

75 (~3)

1

25

25

0.008

0.015

0.025

0.033

0.041

3

75

50

0.028

0.052

0.091

0.120

0.150

5 1

125 50

75 50

0.036 0.063

0.068 0.120

0.120 0.200

0.160 0.270

0.190 0.330

3

150

100

0.220

0.420

0.730

0.970

1.180

5 1

250 75

150 75

0.290 0.230

0.540 0.420

0.940 0.730

1.260 0.970

1.530 1.180

3

225

150

0.810

1.510

2.640

3.510

4.260

5

375

225

1.050

1.960

3.430

4.560

5.540

Gear data: n = 20 , = 45 . Pinion in alloy steel, carburized and case hardened, 58÷62 HRC, ground teeth. Gear is phosphorus bronze. Short running-in. Note: The table is deduced from the first edition of the Gear Handbook, but the helix angle is unified to 45 in order to enable grinding of the pinion teeth. The given power serves only as a rough indication. Higher performance can be achieved by special materials in tested operating conditions. Crossed helical gears should be avoided as far as possible in common industrial applications.

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All such observations lead to one conclusion: It is better to size hypoid gears according to previous field experience and test them for their specific application than to calculate their load capacity. An ap­ proximate idea of the performance of hypoid gears can be determined from the same Table 15.12 that deals with spiral bevel gears, corresponding to equal gear diameter.

15.11 WORM GEARING Double-enveloping worm gears are not considered here: In fact, either they are used as precision gearing where the transmitted power is very low and so they are oversized and a capacity rating is not worthwhile—or they require specific tests, as their operating behavior depends on a lot of specific manufacturing and application conditions. Cylindrical worm gearing. The existing standard methods for rating the capacity of worm gears are based on the Hertz theory for RH. They include some empirical factors or modifications that do not take into due consideration the surface finish of both worm threads and wheel teeth (which may be only partly improved by a running-in period) and the actual tooth bearing area. The hob that cuts the wheel teeth usually has larger diameter than the worm, and its parameters affect the contact conditions of the gear pair in an unforeseeable way, unless a specific software is used for investigating them. The lubrication conditions and the lubricant types greatly influence the real performance of the gear pair. Furthermore, the high sliding produces a gear amount of heat that must be dissipated through the housing. Therefore, the housing design and cooling are more important for the real performances of worm gears than for common coplanar gears, especially when a continuous load is transmitted. Thus, the ratings for worm gears are less reliable than those for coplanar gears (even if the latter are not completely reliable either). Table 15.17 gives approximate values of the allowable input power for speed-reducing worm gears manufactured by specialized firms. The values must be multiplied by the efficiency for obtaining the output power. The table refers to service factors equal to 1 for gear durability. Proper housing design and fan cooling ensure good heat dissipation for such speed reducers. Nevertheless, the power values must be thought of as mechanical rather than as thermal limits. The service factors usually account for the case of continuous and prolonged loading, as well as for appli­ cation conditions and required life as usual. The table relates to: • • • •

Worms: alloy-steel, case-hardened—ground threads with involute profile Wheels: centrifugally cast special bronze—hobbing with good accuracy and surface finish Suitable gear diameters for optimizing efficiency and durability Suitable diameter difference between the hob (generating the wheel) and the worm, and suitable bearing zone between thread and wheel tooth

Either super-finishing techniques are used for both worm and wheel, or some running-in period is required. There is a great difference between the allowable power transmission of such specialized speed reducers and that of worm gears applied, as a part of any machine, internally to a nonspecific housing. The same worm gears of said speed reducers are often employed in such case but must be essentially de-rated. The problem of designing and manufacturing a worm gear pair for a generic mechanical application is that only the best conditions of materials, manufacturing, lubrication, and housing cooling give sufficient reliability to load capacity assessments. Each one of such items, not only derates performance, but makes it more uncertain, if its condition is not the best. In industrial use the difference in types of thread profile, given equal surface finish and accuracy, are less important. Nevertheless, worms with so-called straight profiles may have a capacity reduction

Load Rating of Gears

879

TABLE 15.17 Nominal Capacity of Cylindrical Worm Gear Pairs for Standard Speed Reducers (Service Factor CSF = 1) Ratio

Worm Speed, r/min

Center Distance, mm (in.) 75

(3)

100

(4)

150

(6)

200

(8)

300

(12)

Input Power, kW (hp) 10

20

30

40

50

60

500

1.7

(2.3)

3.7

(4.9)

9.5

(13)

17

(23)

35

(47)

710

2.1

(2.8)

4.6

(6.1)

12.0

(16)

22

(29)

45

(60)

1000 1420

2.6 3.2

(3.5) (4.3)

5.7 7.1

(7.6) (9.6)

15.0 19.0

(20) (25)

27 35

(37) (47)

57 73

(77) (98)

1700

3.6

(4.8)

8.0

(11.0)

21.0

(29)

39

(53)

83

(111)

500 710

1.0 1.3

(1.4) (1.7)

2.3 2.9

(3.1) (3.9)

5.9 7.5

(8.0) (10.0)

11 14

(15) (19)

23 29

(30) (39)

1000

1.6

(2.1)

3.6

(4.8)

9.4

(13.0)

17

(23)

37

(49)

1420 1700

2.0 2.2

(2.7) (3.0)

4.5 5.1

(6.0) (6.8)

12.0 13.0

(16.0) (18.0)

22 25

(30) (33)

47 53

(63) (71)

500

0.8

(1.1)

1.7

(2.3)

4.4

(5.9)

8.1

(11)

17

(21)

710 1000

1.1 1.3

(1.4) (1.7)

2.2 2.7

(2.9) (3.6)

5.6 7.0

(7.5) (9.4)

10.0 13.0

(14) (17)

22 28

(29) (37)

1420

1.6

(2.2)

3.4

(4.5)

8.8

(12.0)

16.0

(22)

36

(48)

1700 500

1.8 0.7

(2.4) (1.0)

3.8 1.4

(5.1) (1.9)

9.9 3.6

(13.0) (4.8)

19.0 6.6

(25) (8.9)

40 14

(54) (19)

710

0.9

(1.2)

1.8

(2.4)

4.5

(6.1)

8.4

(11.0)

18

(24)

1000 1420

1.1 1.4

(1.5) (1.8)

2.2 2.8

(3.0) (3.7)

5.7 7.2

(7.6) (9.6)

11.0 13.0

(14.0) (18.0)

23 29

(31) (39)

1700

1.5

(2.0)

3.1

(4.1)

8.1

(11.0)

15.0

(20.0)

33

(44)

500

0.6

(0.8)

1.2

(1.5)

3.0

(4.1)

5.6

(7.5)

12

(16)

710 1000

0.7 0.9

(1.0) (1.2)

1.4 1.8

(1.9) (2.4)

3.8 4.8

(5.1) (6.4)

7.1 9.0

(9.6) (12.0)

15 19

(20) (26)

1420

1.1

(1.5)

2.3

(3.0)

6.0

(8.1)

11.0

(15.0)

25

(33)

1700 500

1.3 0.5

(1.7) (0.6)

2.5 1.1

(3.4) (1.3)

6.8 2.5

(9.1) (3.4)

13.0 4.7

(17.0) (6.4)

28 10

(38) (13)

710

0.6

(0.8)

1.2

(1.6)

3.2

(4.3)

6.0

(8.1)

13

(17)

1000 1420

0.7 0.9

(0.9) (1.2)

1.5 1.9

(2.0) (2.5)

4.0 5.0

(5.4) (6.8)

7.6 9.6

(10.0) (13.0)

16 21

(22) (28)

1700

1.0

(1.3)

2.1

(2.8)

5.7

(7.6)

11.0

(15.0)

23

(31)

depending on the diameter of the grinding wheel, as the ground profile usually is not really straight and differs from the profile of the hob that generates the worm wheel. Such a combination may in some cases obtain a favorable relief effect, but often reduces the tooth bearing area. A prolonged running-in can usually help, unless the profile combination is such as to produce a bad operating conjugation. Some tests show an advantage for special worms with concave thread flanks. The load capacity essentially decreases for worms with non-ground threads: In this case the usual assessments are absolutely unreliable.

16

Gear-Manufacturing Methods Stephen P. Radzevich

The many methods of making gear teeth must be considered by the gear designer. The size and geometric shape of the gear or pinion must be within the capacity range of some machine tool. If the lowest competitive cost is to be obtained, then the gear designer must make the gear of a size, shape, and material that will permit the most economical method of manufacture. The purpose of this chapter is to review all the commonly used methods of making gears. Illustrative data concerning the sizes and kinds of machine tools now available on the market will be given. Design limitations for each method of manufacture will be discussed so that the gear designer will have at least a general idea of what can be done by each method. Some data will be given on how fast gears can be made by each method. This is a controversial subject. The reader should use the data given with caution. They will not fit all situations. A subject like gear-manufacturing methods is broad enough to require several books [1]. In this chapter it will be possible to tell only a small part of the story. Further information may be found in the references at the end of the book. Gear-manufacturing terms used in this chapter are defined in the glossary Table 16.1. Figure 16.1 shows a broad outline of the methods used to make gear teeth. Methods which are geometrically similar are grouped together. This figure shows that the gear designer will have to evaluate a large number of methods to choose the most suitable method for each job.

16.1 GEAR-TOOTH CUTTING A wide variety of machines are used to cut gear teeth. As shown in Figure 16.1, there are four more or less distinct ways to cut material from a gear blank so as to leave a toothed wheel after cutting. The cutting tool may be threaded and gashed. If so, it is a hob, and the method of cutting is called hobbing. When the cutting tool is shaped like a pinion or a section of a rack, it will be used in a cutting method called shaping. In the milling process, the cutting tool is a toothed disk with a gear-tooth contour ground into the sides of the teeth. The fourth general method uses a tool, or series of tools, that wraps around the gear and cuts all teeth at the same time. Methods of this type are broaching, punching, and shear cutting.

16.1.1 GEAR HOBBING Spur, helical, crossed-helical (spiral), and worm gears can be produced by hobbing. All gears but worm gears are cut by feeding the hob across the face width of the gear. In the case of worm gears, the hob is fed either tangentially past the blank or radially into the blank. Figure 16.2 shows how a hob forms teeth on different kinds of gears. A wide variety of sizes and kinds of hobbing machines are used. Machines have been built to hob gears all the way from less than 2 mm to over 10 m (3/32 to 400 in.) in diameter. Large double-helical gears are frequently hobbed with double-stanchions machines. These hobbers use two cutting heads 180 apart to cut both helices at once. Figures 16.3 to 16.5 show several examples of hobbing. In designing gears to be hobbed, a number of things must be considered. A hob needs clearance to “run out” at the end of the cut. If the gear teeth come too close to a shoulder or other obstruction, it may 881

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Dudley’s Handbook of Practical Gear Design and Manufacture

TABLE 16.1 Glossary of Gear Manufacturing Nomenclature Term

Definition

Broaching

A machining operation which rapidly forms a desired contour in a working surface by moving a cutter, called a broach, entirely past the workpiece. The broach has a long series of cutting teeth that gradually increase in height. The broach can be made in many different shapes to produce a variety of contours. The last few teeth of the broach are designed to finish the cut rather than to remove considerably more metal. Broaches are often used to cut internal gear teeth, racks, and gear segments on small gears, and usually are designed to cut all teeth at the same time. A finishing operation which polishes a surface by rubbing.

Burnishing Casting

A process of pouring molten metal into a mold so that the metal hardens into desired shape. Casting is often used to make gear blanks that will have cut teeth. Small gears are frequently cast complete with teeth by the die-casting process, which uses a precision mold of tool steel and low-melting-point alloys for the gears.

Drawing

A metal forming process used to make gear teeth on a small-diameter rod by pulling the rod through a small gear-shaped hole. A process that uses extreme pressure to push solid metal through a die of the desired shape.

Extruding Generating

Any gear-cutting method in which the cutter rolls in mesh with the gear being cut. A generating method allows a straight-sided cutter to cut a curved involute profile in the gear blank.

Grinding

A process that shapes the surface by passes with a rotating abrasive wheel. Grinding is not practical way to remove large amounts of metal, so grinding is used to make very fine-pitch teeth, or to remove heat-treat distortion from large gears that have been cut and then fully hardened. Many different kinds of grinding operations are used in gear manufacture. A precise gear-tooth-cutting operation that uses a threaded and gashed cutting tool called a hob to remove the metal between the teeth. The rotating hob has a series of rack teeth arranged in a spiral around the outside of a cylinder (see Figure 11.9), so it cuts several gear teeth at one time.

Hobbing

Lapping

A polishing operation which uses an abrasive paste to finish the surfaces of gear teeth. Generally a toothed, cast-iron lap is rolled with the gear being finished.

Milling

A machining operation which removes the metal between two gear teeth by passing a rotating cutting wheel across the gear blanks.

Molding

A process like casting which involves filling a specially shaped container (the mold) with liquid plastic or metal, so that the material has the shape of the mold after it cools. Injection-molding machines use high pressure to force the hot plastic into steel gear molds. A fast, inexpensive method of producing small gears from thin sheets of metal. The metal is sheared by a punching die which stamps through the sheet stock into a mating hole.

Punching Rolling

A process which rapidly shapes fine gear teeth or worm threads by high pressure rolling with a toothed die.

Shaping

A gear-cutting method in which the cutting tool is shaped like a pinion. The shaper cuts while traversing across the face width and rolling with the gear blank at the same time. A finishing operation that uses a serrated gear-shaped or rack-shaped cutter to shave off small amounts of metal as the gear and cutter are meshed at an angle to one another. The crossed axes crate a sliding motion which enables the shaving cutter to cut.

Shaving

Shear cutting

A rapid gear-cutting process which cuts all gear teeth at the same time using a highly specialized machine. This method of cutting gear teeth is no longer used very frequently.

Sintering

A process for making small gears by pressing powder metal into a precision mold under great pressure and then baking the resulting gear-shaped briquette in an oven. Sintering is cost-effective only for quantity production because the molds and tools are very expensive.

Stamping

Another word for punching.

FIGURE 16.1 Outline of methods of making gear teeth.

Gear-Manufacturing Methods 883

884

FIGURE 16.2

Dudley’s Handbook of Practical Gear Design and Manufacture

Comparison of different kinds of hobbing.

FIGURE 16.3 A tapered hob ready to finish-cut a small helical gear. This picture shows the basics of gear hobbing.

be impossible to cut the part by hobbing. If the gear is double-helical, a gap must be left between the helices for hob runout. The method of calculating the width of this gap is given in Section 17.2. If it is not possible to use the narrowest possible gap, the values shown in Table 16.2 may be used. Some large hobbers do not have centers to mount the work. In such cases, the gear is normally clamped on its rim. This makes it necessary to provide rim surfaces that are true with the journals of the part. Most of the smaller hobbing machines mount the work on centers. If the part does not have shaft extensions with centers, it is necessary to provide tooling so that the part can be mounted on a cutting arbor with centers.

Gear-Manufacturing Methods

885

FIGURE 16.4 A large coupling hub with external spur teeth that has been precision hobbed on a medium-size hobbing machine.

The hobbing process is quite advantageous in cutting gears with very wide face widths or gears that have a toothed section which is integrated with a long shaft. A very high degree of tooth-spacing accuracy can be obtained with hobbing. High-speed marine and industrial gears with pitch-line speeds in the range of 15 to 100 m/s (3000 to 20,000 fpm) and diameters up to 5 m (200 in.) are very often cut by hobbing. A few large mill-gears up to 10 m (400 in.) in diameter are hobbed. (These are not high-speed gears.) Gears can be finished by setting the hob to full depth and making only one cut. Where highest accuracy is desired, it is customary to make a roughing and finishing cut. The roughing cut removes almost all the stock. The finishing cut may remove from 0.25 to 1.00 mm (0.010 to 0.040 in.) of tooth thickness, depending on the size of the tooth. The time required to make a hobbing cut can be calculated from the following formula: Hobbing time, min =

no. of gear teeth × (face + gap) no. of hob threads × feed × hob rpm

(16.1)

The reason for adding the gap to the face width is the fact that a hob has to travel a certain distance in going into the cut and coming out of the cut. This extra travel happens to equal the amount of space that is necessary for the gap between helices. Equation (16.1) works with either millimeters or inches for face and feed. In the past it was quite customary to use single-thread hobs for finish and single- or double-thread hobs for roughing. Single-thread hobs give the best surface finish and tooth accuracy. In many cases, though, multiple-thread hobs are used for both roughing and finishing. When gears are shaved or lapped

886

FIGURE 16.5

Dudley’s Handbook of Practical Gear Design and Manufacture

A large hobbing machine used to cut precision gears for large ships.

TABLE 16.2 Nominal Gap Width Tooth Size

Hob Diameter

Gap Width

15 Helix Module, normal

30 Helix

45 Helix

Normal diametral pitch

mm

in.

mm

in.

mm

in.

mm

in.

1.25 1.60

20 16

48 64

1.875 2.50

16 22

0.625 0.875

19 26

0.75 1.00

19 26

0.75 1.00

2.50

10

76

3.00

32

1.250

38

1.50

38

1.50

3.00 4.00

8 6

76 89

3.00 3.50

35 45

1.375 1.750

48 51

1.875 2.00

48 57

1.875 2.25

6.00

4

102

4.00

57

2.250

70

2.75

83

3.25

8.00

3

115

4.50

73

2.875

89

3.50

105

4.125

after hobbing, surface finish in hobbing is not quite so important. If the gear has enough teeth and they are not an even multiple of the number of the hob threads, hobs with as many as five or seven threads may be quite profitably used. For best results there should be about 30 gear teeth for each hob thread. This would mean that a five-thread hob would not be used to cut fewer than 151 gear teeth. For com­ mercial work of moderate accuracy, the number of gear teeth may be as low as 15 per hob thread. Hobbing feeds and speeds depend on how well the gear material cuts, the accuracy desired, the size of the gear, and the strength of the hobbing machine. Table 16.3 shows representative values for three

Gear-Manufacturing Methods

887

TABLE 16.3 Hobbing Feeds and Speeds Hob Diameter

Feed per Revolution Finishing, 1 Thread

mm

in.

mm

in.

Roughing, 1 Thread mm

in.

Roughing, 2 Thread mm

Roughing, 3 Thread

in.

mm

Speed, rpm

in.

Conservative

Normal

High

25

1

0.5

0.020

1.3

0.050

0.9

0.035

0.6

0.025

590

870

1150

47 75

1.875 3

0.7 1.1

0.030 0.045

1.5 2.2

0.060 0.085

1.1 1.7

0.045 0.065

0.7 1.0

0.030 0.040

270 145

370 210

490 270

100

4

1.4

0.055

2.5

0.100

1.9

0.075

1.3

0.050

105

150

200

125 150

5 6

1.5 1.7

0.060 0.065

2.5 2.8

0.100 0.110

1.9 2.0

0.075 0.080

1.3 1.4

0.050 0.055

80 65

115 95

150 125

200

8

1.8

0.070

3.0

0.120

2.3

0.090

1.5

0.060

50

70

95

Note: These values are based on AISI B1112 steel, 100% machinability rating.

classes of work. The conservative values show what would be used when best accuracy was needed. The high-speed values are about the highest that can be obtained when machine and hob design receive special attention, and accuracy is not the most important thing. The size of hob will depend on the pitch. Section 11.2 shows standard hob sizes for different pitches. Table 16.3 is based on a low-hardness steel of 100% machinability. For different hardnesses, the cutting speed should be reduced about as indicated in Table 16.4. The above table is based on hobbers of rugged design cutting parts well within the machine capacity. In many cases it is necessary to reduce feeds and speeds to about 60% of the values given to reduce wear on the hob caused by machine vibration. Many advancements were made in machine-tool design and cutting-tool design. The art of gear cutting made notable improvements in both the accuracy and the production rates that could be achieved. Also, the diversity of equipment and methods became much greater than it was in earlier years. It is not possible in this book to cover all the latest things in gear hobbing. To give the reader an appreciation of the state of the art in hobbing, Table 16.5 was prepared. This table shows nominal production time for a small gear-set and a large gear-set. It also shows fast pro­ duction time—what can be done using the most advanced hobbers, hobs, and hobbing techniques.

TABLE 16.4 The Recommended Values for Reduction of Cutting Speed Material

Steel

Cast iron

Hardness

Percentage of Table 15.3 Speed

HV

HB

370

350

32

320 205

300 200

40 56

180

175

85

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Dudley’s Handbook of Practical Gear Design and Manufacture

TABLE 16.5 Some Examples of Production Time for Hobbing or Milling Gear Teeth Examples of Gear-Sets

1

Number of Teeth

Module

Face Width, mm

Material and Hardness

25 102

2 2

32 30

Alloy steel, pre-grind cut at Vickers hardness of about 285

2

25

2

32

2 15

30 307

Alloy steel, pre-shave cut at Vickers hardness of about 375

3

102 25

4

102

15

300

25 102

15 15

307 300

Alloy steel, finish hob at Vickers hardness of about 325 Alloy steel, carburized and hardened to Vickers 700. Finish by skiving hobbing

Production Time per Piece N

F

5 min 11 min

1.5 min 3 min

15 min

3 min

30 min 6 hr

6 min 2.5 hr

22 hr

9 hr

4.5 hr 17 hr

1.7 hr 6.5 hr

Notes: 1. The above table is base on spur gears, or helical gears up to 15 helix angle, and 20 pressure angle. 2. The alloy steel would contain chromium, manganese, nickel, and molybdenum. Good examples are AISI 4320 and 4340. 3. Vickers 285 = approximately 270 Brinell hardness Vickers 375 = approximately 353 Brinell hardness Vickers 700 = approximately 58.5 Rockwell C 4. Production time cut: N Nominal job-shop work. Milling or hobbing teeth, single-start conventional hobs or cutters. Work-holding fixtures and hobbing machines somewhat old and not too well suited to the exact work being done. F A production hobbing facility doing fast repeat work. Multi-start and coated* hobs being used wherever possible. Work-holding fixtures of special design. Most modern, high-performance hobbing machines. * A popular hard coating for extra performance is titanium nitride (TIN).

The data shown in Table 16.5 are a composite of survey data collected. This means that the data are somewhat average. It is possible to do even better under the most favorable conditions. (Of course, under poor conditions, Table 16.5 times for either job-shop or fast production will not be achievable.) Table 16.5 brings out these important considerations: • Fast production work with the best equipment may be three to five times faster than job-shop work with more ordinary equipment. • Finish cutting of the harder steels may take two or three times longer than pre-grind cutting of lower-hardness steels. • Skive hobbing of fully hardened steels is as fast as or faster than finish hobbing medium-hard steels.

16.1.2 SHAPING—PINION CUTTER Spur, helical and face gears and worms can be cut with a pinion-type cutter. Either internal or external gears can be cut. Parts from less than 1 mm to over 3 m (1/16 to over 120 in.) may be shaped. Relatively wide face widths may be cut, but in certain cases shaping will not handle as much face width as the hobbing process. For example, a standard 1.25-m (50-in.) gear shaping machine will have a face-width capacity of 0.2 m (8 in.) regardless of helix angle, but a comparably sized hobbing machine would handle face width of 0.6 m (24 in.), depending on the helix angle. Figures 16.6 and 16.7 show examples of shaper cutting.

Gear-Manufacturing Methods

FIGURE 16.6

889

The basics of shaper cutting.

Shaper cutters need only a small amount of cutter-runout clearance at the end of the cut or stroke. Shaped teeth may be located close to shoulders. A cluster gear can be readily shaper-cut where it may be impossible to hob because of insufficient hob-runout clearance. One NC shaper, pictured in Figure 16.8, could be conceivably cut a cluster gear in one set-up, depending upon gear data. Double helical gears can be cut with very narrow gaps between helices. In fact, one design of shaper can actually produce a “continuous” double helical tooth. In designing gears to be shaped, it is necessary to machine a groove as deep as the gear tooth at the end of the face width for runout of the cutter. Normal values for the width of this groove are listed in Table 16.6. It is difficult to mount parts between centers while they are being shaper-cut, since the bottom portion must be driven. At least the one end must therefore be clamped to a fixture or gripped by a chuck. Gear parts which are integral with long shaft extensions may be supported at the upper end by a center or steady rest, and a long shaft extension downward may be accommodated by a hole in the base of the machine bed and even by a recessed portion of the machine foundation. Gear shaping is quite advantageous on parts with narrow face widths. In hobbing, it takes time for the hob to travel into and out of the cut. For helical gears, the hob travel must be increased proportionally. In shaping, there is a minimum of overtravel for spur gears, and this overtravel does not increase for helical gears. For instance, a 2.5-module (10-pitch) gear with 25-mm (1-in.) face width would have an extra travel in hobbing of about 32 mm (1¼ in.). In shaping, the extra travel would be only 5.56 mm (0.219 in.). In this example, the hob would cut across more than twice the face width that a shaper cutter would to do the same part. Furthermore, with the advent of the modern cutter spindle back-off type machines, it is not uncommon to see stroking rates of 1000 strokes per minute for face widths 25 mm (1 in.) or smaller. In fact, for narrow face widths of 6 mm (¼ in.), high-speed shapers capable of stroking rates of

890

Dudley’s Handbook of Practical Gear Design and Manufacture

FIGURE 16.7 A shaping machine that has just cut a medium-sized double helical gear. (Courtesy of Fellows Corp., Emhart Machinery Group, Springfield, VT, U.S.A.)

over 2000 strokes per minute are in use. Hydrostatically mounted guide and cutter spindle are necessary because of these high stroking rates. The time in minutes required to shaper-cut a gear with a disc-type cutter may be estimated by the following formula: Shaping time =

no. of gear teeth × strokes per rev. of cutter × no. of cuts no. of cutter teeth × no. of strokes per min

(16.2)

It should be noted here that a “stroke” is a cutting stroke and a return stroke. Thus, a stroke is really a double stroke. In shaping the coarser pitches, it is necessary to take several roughing and finishing cuts. Table 16.7 shows some nominal sizes of cutters and the number of cuts taken when shaping steel teeth of about 200 Brinell hardness. Table 16.8 gives some representative values for the rotary feed rate per double stroke. These re­ presents average values which might be used in cutting medium-precision gears. The number of double strokes per revolution of the cutter can be calculated by: Strokes per revolution =

cutter diameter × rotary feed rate

The number of double strokes per minute is calculated from the formula:

(16.3)

Gear-Manufacturing Methods

891

FIGURE 16.8 Schematic design of a numerically controlled (NC) gear-shaping machine. (Courtesy of Lorenz, a subsidiary of Maag Gear Wheel Co., Ettlingen, German Federal Republic.)

Strokes per minute =

1000 × max . cutting speed, m/ min metric stroke length, mm ×

Strokesperminute =

12 × max .cuttingspeed, ft/ min English strokelength, in. ×

(16.4)

(16.5)

892

Dudley’s Handbook of Practical Gear Design and Manufacture

TABLE 16.6 Normal Values of Width of the Runout Groove Tooth Size

Width of the Runout Groove

Module

23 Helix

15 Helix

Spur Diametral pitch

mm

in.

mm

in.

mm

in.

1.0

24

5.0

3/16

6.0

15/64

6.5

¼

1.8

14

5.0

3/16

6.5

¼

7.0

17/64

2.5 4.0

10 6

5.5 6.5

7/32 ¼

7.2 7.2

9/32 9/32

8.0 8.0

5/16 5/16

6.0

4

7.2

9/32

9.0

11/32

9.5

3/8

TABLE 16.7 Nominal Sizes of Shaper Cutters and Number of Shaping Cuts Tooth Size Module 1.25

Cutter Diameter

Number of Cuts

Diametral Pitch

mm

in.

Roughing

Finishing

20

50

2

1

1

75

3

1

75 100

3 4

1 1

2.5

10

1 1

4

6

100

4

1 or 2

1

6

4

115 100

4.5 4

1 or 2 2 or 3

1 1

125

5

2 or 3

1

12

2

150 175

6 7

2 or 3 2 or 3

1 1

The cutting speed depends on the face width of the gear, the hardness and machinability of the material, the degree of quality, the cutter material, and the desired cutter life (see Table 16.9). Just as considerable advancements have been made in hobbing, there have been notable advancements in shaper-cutting gears in the 1960th ant later. Perhaps the most notable are: • • • • •

Numerically controlled machines Cutter spindle back-off (instead of work table back-off) Cutter spindle moved hydraulically instead of by mechanical means New infeed methods Machine with column that moves and stationary work table (conventional shapers work the op­ posite way) • Better shaper cutter materials and special hard coatings.

These things have led to significant improvements in the precision of shaped gears and considerable improvement in gear shop production of shaped gears.

Gear-Manufacturing Methods

893

TABLE 16.8 Rotary Feed for Each Double Stroke of the Shaping Cutter Material

Brinell Hardness, HB

Machinability, %

Rotary Feed per Double Stroke 1.5 to 2.5 Module (10 to 17 DP)

2.5 to 4 Module (6 to 10 DP)

4 to 6Mmodule (4 to 6 DP)

6 to 9 Module (3 to 4 DP)

mm

in.

mm

in.

mm

in.

mm

in.

Steel to be casehardened

135

100

0.5

0.020

0.5

0.020

0.6

0.024

0.6

0.024

185 220

80 65

0.5 0.3

0.020 0.012

0.5 0.35

0.020 0.014

0.6 0.45

0.024 0.018

0.6 0.5

0.024 0.020

Throughhardened alloy steel

172

72

0.4

0.016

0.4

0.016

0.5

0.020

0.6

0.024

217 254

55 45

0.3 0.25

0.012 0.010

0.35 0.3

0.014 0.012

0.4 0.3

0.016 0.012

0.6 0.4

0.024 0.016

–

130

0.20

0.008

0.3

0.012

0.3

0.012

0.35

0.014

Plastics

Notes: 1. The above feed values are based on a roughing and a finishing cut for six module (four diametral pitch) and smaller teeth. For large teeth, up to ten module, two roughing cuts and one finishing cut are intended. 2. For finishing cuts of high-precision quality, the rotary feed will need to be reduced to get the required finish and the required spacing accuracy.

Figure 16.9 shows the Hydrostroke machine, developed by Fellows Corporation. The details of how the cutter spindle is stroked back and forth by pressurized oil are shown in Figure 16.10. The Hydrostroke machine is capable off doubling production rates under favorable conditions. It also has the potential to cut some gears so accurately that they do not need to be shaved (or ground) to get a relatively high precision. (The formulas just given for time required to shape gears do not apply to gears shaped in this machine.) Figure 16.11 shows an example of a high-production machine with a moveable column and a sta­ tionary work table. Note the special automation tooling. Figure 16.8 shows in a schematic fashion how a high-production, moving column type of machine works. Basic motions are shown in the upper left-hand corner. The drive system is shown in the upper right-hand corner. Note that the drives use dc motors and are infinitely variable. The lower left-hand corner shows a spiral infeed method. The machine just described has a cutter spindle back-off system and independent dc motor drives for stroking rates and for rotary and radial feed amounts. This machine provides a significant increase in

TABLE 16.9 Typical Cutting Speeds for Shaper Cutter Gear Face Width

Machinability 20%

mm

in.

m/min

200

8

100

4

50 30

40%

80%

100%

fpm

m/min

fpm

m/min

fpm

m/min

1.5

5

6.4

21

18.6

61

3.7

12

9.1

30

22.6

74

2 1.2

6.7 9.1

22 30

12.5 15.8

41 52

26.8 31.1

20

0.8

11.6

38

19.2

63

13

0.5

15.8

52

24.1

79

120%

130%

fpm

m/min

fpm

m/min

fpm

26.4

87

35.4

116

39.4

130

30.8

101

39.3

129

43.0

141

88 102

35.4 39.6

116 130

43.9 47.9

144 157

47.9 51.8

157 170

35.1

115

43.6

143

51.8

170

55.2

181

40.8

134

49.1

161

57.3

188

60.4

198

894

Dudley’s Handbook of Practical Gear Design and Manufacture

FIGURE 16.9 The Hydrostroke gear-shaping machine. (Courtesy of Fellows Corp., Emhart Machinery Group, Springfield, VT, U.S.A.)

productivity compared with the older-style table relief shapers that used feed cams and feed-change gears with a single motor. The special drives resulted in the developing of a new cutting method, sometimes referred to as spiral infeed. This cutting method tends to improve chip loading conditions and more evenly distributes tool wear around the cutter. Table 16.10 shows estimates of production time needed to shape a variety of parts. The time shown tends to be much less than would be obtained by using equations in this chapter. The equations, of course, are for job-shop rather than high-production conditions. The values in Table 16.10 may, of course, be better by even more advanced shaping methods—and with somewhat unfavorable conditions, it may not be possible to cut a part as quickly as the table shows.

16.1.3 SHAPING—RACK CUTTER Spur and helical gears as well as racks may be shaped with a rack cutter. Although machines designed to use rack cutters are mostly used for external gears, attachments for cutting internal gears with a pinionshaped cutter are available.

FIGURE 16.10 Schematic details showing the principle of the mechanism to reciprocate the cutter in a Hydrostroke gear shaper. (Courtesy of Fellows Corp., Emhart Machinery Group, Springfield, VT, U.S.A.)

Gear-Manufacturing Methods 895

896

Dudley’s Handbook of Practical Gear Design and Manufacture

FIGURE 16.11 High-production, numerically controlled (NC) gear-shaping machine. (Courtesy of Lorenz, a Subsidiary of Maag Gear Wheel Co., Ettlingen, German Federal Republic.)

The generating action is the same as rolling a gear along a mating rack. The rack tool, mounted in a clapper box, reciprocates while the gear rolls past its cutting tool. Cutting generally takes place during the down-stroke, and the clapper box clears the tool from the work on the upstroke (see Figure 16.12). Rack shapers cut only a few teeth in one generating cycle, then index to pick up the next teeth. The number of teeth per generation, the number of strokes per tooth (feed), and the stroking speed are all individually variable. The cutting ram is mounted in long guide-ways and is driven by a crank motion or, on heavy-duty versions, a multi-pitch screw. The rack tool can be likened to one flute of a hob. It is not as expensive as a hob, and it does not require as much runout clearance as a hob. Also, there is no diameter/tooth size restriction, so a singletooth rack tool can be made as large as the clapper box.

Gear-Manufacturing Methods

897

TABLE 16.10 Some Production Estimates of Shaping Time Using High-Production Equipment and Techniques Part

No. Teeth

Pitch Diameter

Face Width

Brinell Hardness

Cycle Time

165 160

27.5 sec. 1.6 min

mm

in.

mm

in.

10 21

25.4 67.3

1.00 2.65

15.9 17.1

0.625 0.673

Tractor transmission, helical gear, AISI 1045

27

119.6

4.71

44.5

1.750

140

4.7 min

Truck spur gear, SAE 5130 Truck, spur gear, malleable iron

55 47

232.9 223.8

9.17 8.81

19.1 15.9

0.750 0.625

225 255

9.3 min 1.67 min

Auto starter, spur pinion, AISI 4004 Auto transmission, spur gear, SAE 8620

Tractor, spur gear, AISI 4140

55

678.2

26.70

127.0

5.00

270

110 min

Mach., spur gear, AISI 1552 Mach., internal gear, ductile iron

252 94

1066.8 477.5

42.00 18.80

49.8 79.5

1.960 3.130

300 217

72 min 37.2 min

Mach., helical spline, 416 stainless

32

58.7

2.31

96.8

3.810

155

12 min

FIGURE 16.12

Rack shaping a helical gear. (Courtesy of Maag Gear Wheel Co., Zurich, Switzerland.)

898

Dudley’s Handbook of Practical Gear Design and Manufacture

The most popular use for the rack shaper is for coarse pitches, high-hardness material, and narrow-gap double helical gears. It is also ideal for cutting segments, since only the toothed portion must be rolled past the tool. The rack shaper can also machine two- and three-lobe rotors, as a result of the large tool size and large number of strokes per tooth. Rack shapers are commercially available in size that will handle gears all the way from a centimeter (0.4 in.) in diameter up to about 14 m (45 ft) in diameter. Face widths up to 1.55 m (61 in.) maximum may be cut (see Figure 16.13). Rack shapers are generally built to mount the work in a fixture instead of between centers. Gears with long shaft extensions are usually supported by a steady rest. The time in minutes required to cut a gear on a rack shaper may be calculated by the following formula: Shaping time = no. of gear teeth ×

strokes per tooth index time + × no. of cuts teeth per index strokes per min

(16.6)

The indexing time varies from about 0.08 minute on small machines to around 1.0 minute on large machines. When finishing high-precision gears, the machines are indexed once per tooth so that the same tool finishes all the gear teeth, thus preventing tool pitch errors or mounting errors from being transferred to the gear. In roughing, more teeth may be cut per index, depending on the gear diameter and tooth size. The number of strokes per tooth depends upon the number of teeth in the gear, the module (pitch), and the gear material. The fewer the strokes, the larger the generating marks on the given tooth. Gears of 100 to 200 teeth have almost straight-line profiles, while smaller numbers of teeth have more profile cur­ vature. For this reason, it is necessary to use more strokes per tooth when finishing gears with small numbers of teeth. In fine pitches, the roughing cuts do not remove too much metal. This makes it possible to use fewer strokes per tooth in roughing than in finishing. In coarser pitches, however, a lot of metal has to be removed in roughing. This makes it necessary to use more strokes per tooth in roughing that in finishing. But with heavy-duty machines, coarser pitch “gashing” is usually done with step-type plunge cutters (see Figure 16.14), which can remove a large volume of material with fewer strokes per tooth. Table 16.11 shows the average practice in strokes per tooth. More strokes may be needed in finishing if highest accuracy is desired, while fewer strokes may suffice in commercial work. The cutting speed, or number of strokes per minute, is determined by both the size of the machine and the face width. In cutting a very narrow face width, the cutting speed in meters per minute (feet per minute) may be limited by the strokes-per-minute capability. In general, small machines are built to reciprocate much faster than large machines. Rack-type tools are limited to about the same cutting speed in meters per minute as other tools (see Table 16.12), but heavier cuts are possible than with hobs or disctype shaping tools. Table 16.13 shows some typical values for number of cuts and strokes per minute for different kinds of work.

16.1.4 CUTTING BEVEL GEARS Straight bevel gears are produced by a generating machine which reciprocates a cutting tool in a motion somewhat like that of a shaper. The tools used do not resemble a pinion or a segment of a rack. The peculiar geometry of the gear makes the use of a special tool to cut each side of the bevel tooth desirable. These tools each have a single inclined cutting edge which generates the bevel gear tooth on one side or the other. The machines achieve a generating motion by rolling the work and the cutter head at a low rate, while the cutters reciprocate rapidly back and forth (see Figure 16.15). The latest models of straight-bevel-gear generators have a design feature which permits the tooth to be cut with a slight amount of crown. A cam can be set so that the cutting tool will remove a little extra

FIGURE 16.13 One of the largest gear-cutting machines in the world. A helical gear 11.9 m (469.9 in.) in diameter has just been rack-shaped. (Courtesy of Fuller Company, Allentown, PA, U.S.A.)

Gear-Manufacturing Methods 899

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Dudley’s Handbook of Practical Gear Design and Manufacture

FIGURE 16.14 Special rack cutters for the fast, rough cutting of large gears. (Courtesy of MAAG Gear Wheel Co., Zurich, Switzerland.)

TABLE 16.11 Average Strokes per Tooth for Rack Shaping No. of Teeth

Strokes per Tooth Roughing

2.5 m t (10 Pt )

4 m t (6 Pt )

Finishing

12 m t (2 Pt )

2.5 m t (10 Pt )

4 m t (6 Pt )

12 m t (2 Pt )

Std.

HD

Std.

HD

Std.

HD

Std.

HD

Std.

HD

Std.

HD

15 20

60 55

– –

103 90

– –

325 295

175 160

15 14

– –

22 16

– –

40 30

39 32

30

50

–

85

50

260

140

11

–

13

12

22

23

50 80

42 38

– –

75 65

44 38

230 200

130 110

7.5 7

– –

9 7

9 7

16 13

17 13

Note: Std.—standard machines, HD—heavy-duty machines, m t —transverse module, Pt —transverse diametral pitch.

metal at each end of the tooth. This makes it possible to secure a localized tooth bearing in the center of the face width. Bevel gears are overhung from the spindle of the generating machine during cutting. Those designing bevel gears should be careful to make the blank design suitable for mounting in the bevel-gear generator. Frequently it is necessary to design the bevel gear as a ring which is bolted onto its shaft after cutting. This gets the shaft out of the way during cutting. Bevel gears coarser than 1 module (25 pitch) are usually given a roughing and a finishing cut. Where production is high and parts are not over 350 mm (14 in.) in diameter, it is possible to use special highspeed roughing machines which do not have a generating motion. These machines rough faster than the generators, and they save the generators from the wear and tear of roughing. Teeth of 0.8 module (32

Gear-Manufacturing Methods

901

TABLE 16.12 Cutting Speeds for Different Materials Material

Hardness, HB

Cutting Speed Conservative

High-Speed

m/min

fpm

m/min

fpm

m/min

fpm

350

7.6

25

11

35

15

50

300 200

11 17

35 55

14 26

45 85

21 37

70 120

250

14

45

18

60

27

90

175 90 (500 kg)

23 46

75 150

30 85

100 280

46 122

150 400

–

107

350

168

550

213

700

Steel

Cast iron Bronze

Normal

Laminated plastic

TABLE 16.13 Number of Cuts and Strokes per Minute for Typical Rack Shaping Tooth Size

Pitch Diameter

Number of Cuts Roughing

Module

Diametral Pitch

mm

in.

1.6

16

100

4

2.5 2.5

10 10

100 200

4 8

4

6

200

4 12

6 2

1000 1000

12

2

2500

Finishing Commercial

Precision

1

1

1

1 1

1 1

1 1

8

2

1

2

40 40

2 3

1 1

2 2

100

3

1

2

200

Strokes per minute 80 55

32

Face width, mm (in.)

25 (1)

100 (4)

200 (8)

400 (16)

Cutting speed, m/min (fpm)

13 (42)

27 (90)

27 (90)

27 (90)

pitch) or finer are often cut in one cut. Special Duplete tools are used. These tools have two cutting edges in tandem – one edge roughs and one—finishes. The complications of bevel-gear cutting make it hard to write any general formulas for estimating cutting time. In view of this situation, the best way to give some general information on cutting time is to give a table with cutting time shown for a range of sizes of bevel gears. Table 16.14 shows the average cutting time for steel gears of about 250 Brinell hardness. Spiral and Zerol bevel gears are cut with a generating machine that uses a series of cutting blades mounted on a circular tool-holder. The tool-holder is rotated to cause a cutting action while the work slowly rotates with the tool-holder. The rotation of the work with respect to the tool-holder causes a generating action to occur. After one tooth space is finished, the machine goes through an indexing motion to bring the cutter into the next tooth slot (see Figure 16.16).

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Dudley’s Handbook of Practical Gear Design and Manufacture

FIGURE 16.15 This view shows the generation of a Coniflex straight bevel gear. Note the two reciprocating tools which travel in a curved path to produce a slight crowning on the teeth. (Courtesy of Gleason Works, Rochester, NY, U.S.A.)

TABLE 16.14 Average Time to Cut Spiral Bevel Gears No. of Teeth

Pitch-Cone Angle, Deg

Tooth Size

Face Width

Time, min

Module

Pitch

mm

in.

Low production

High production

12

8.5

1.1

24

9.5

0.375

2.4

1.2

20 30

22 45

1.1 1.1

24 24

6.4 3.2

0.250 0.125

4.0 5.0

2.0 2.5

50

68

1.1

24

6.4

0.250

10.0

5.0

80 12

81.5 8.5

1.1 4.2

24 6

9.5 41

0.375 1.625

16.0 14.6

8.0 4.9

20

22

4.2

6

38

1.500

23.7

7.9

30 50

45 68

4.2 4.2

6 6

32 38

1.250 1.500

27.6 34.2

9.2 11.4

80

81.5

4.2

6

41

1.625

44.8

14.9

12 20

8.5 22

8.5 8.5

3 3

83 83

3.250 3.250

34.2 57.0

11.4 19.0

30

45

8.5

3

83

3.250

78.0

26.0

50 80

68 81.5

8.5 8.5

3 3

83 83

3.250 3.250

130.0 150.7

43.3 50.2

Notes: 1. The 24-pitch gears are assumed to be cut on a Gleason No. 423 Hypoid Generator, the 6-pitch are cut on a No. 641 G-PLETE Generator, and the 3-pitch are cut on a No. 645 G-PLETE Generator. 2. For high production, the completing process is used to finish gear teeth in a single chucking from the solid blank. For lower production requirements, the 5-cut process can be used: rough and finish the gear member, rough the pinion member, then finish each side of the pinion teeth in a separate cutting operation.

Gear-Manufacturing Methods

903

FIGURE 16.16 Spiral gears up to approximately 1.25 m (50 in.) in diameter are generated on machines employing circular face-mill cutters. (Courtesy of Gleason Works, Rochester, NY, U.S.A.)

The line of machines using face-mill cutters will handle gears from 5 mm (3/16 in.) to 2.5 m (100 in.) diameter and teeth from 0.5 module (48 pitch) to 17 module (1.5 pitch). Large spiral gears up to about 2.5 m (100 in.) in diameter can also be cut on a planing type of generator which uses a single tool. The average time required to cut spiral or Zerol bevel gears may be estimated from Table 16.14. If face widths are less than those shown in the table, there may be some reduction in cutting time, but it will not be in proportion to the reduction of face width. If very high accuracy is required, the time may be longer. No allowance is made for setup time which may be required to adjust the machine to produce the desired tooth contour. The time shown in Table 16.14 is the time required per piece after the machine has produced a few satisfactory pieces. Loading and unloading time are not included. Figure 16.17 shows a hypoid gear being cut with a straddle type of cutting tool. Note that the face mill has two rows of cutting teeth. This method cuts gears faster, and it is believed that straddle-cut (or ground) gears have somewhat longer life.

FIGURE 16.17

Straddle-cutting a hypoid gear. (Courtesy of Gleason Works, Rochester, NY, U.S.A.)

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Dudley’s Handbook of Practical Gear Design and Manufacture

16.1.5 GEAR MILLING Worms, spur gears, and helical gears may be produced by the milling process. Bevel-gear teeth are sometimes produced by this method, but the geometric limitations of trying to produce accurate bevel teeth by this process greatly restrict its use except for roughing cuts. Conventional milling machines equipped with a dividing head may be used to mill gear teeth. A slot at a time is milled, and the machine is hand-indexed to the next slot. Several makes of special gearmilling machines, known in the trade as “gear cutting machines”, are on the market. These are designed for the sole purpose of cutting gear teeth or clutches and the like with milling cutters. They are usually equipped with automatic indexing equipment. The machines used to mill gears are a different type from those used to cut gear teeth. The wormmilling machine is essentially a thread-milling machine. Gear cutting machines cover the whole range of gear sizes up to more than 5 m (200 in.) in diameter. Pitches that are coarser than 34 module (3/4 pitch) are often cut with an end mill. Helical rolling-mill pinions in the circular-pitch range of 100 to 200 mm (4 to 8 in.) are often produced by end milling. Figure 16.18 shows a gear being milled. Figure 16.19 shows an end mill used to mill some large gear teeth. Gear-cutting machines and milling cutters are not so expensive as hobbing machines and hobs or gear shaper and shaper cutters. On the other hand, gear-cutting machines do not produce accuracy comparable with that produced by hobbing and shaping. Some high-precision gears are produced by rough milling followed by hardening and finish grinding. Parts to be milled must allow room for runout of the cutter at each end of the tooth. Some gear-cutting machines mount the work on centers, while others clamp the work on a fixture. Wide face widths can be milled, and it is usually possible to have fairly long shaft extensions on the gear. Gear teeth may be milled in one cut, or they may be given a rough cut almost to size and then finishmilled. The time required to make a milling cut can be calculated form the following formula: Milling time, min = no. of gear teeth × index time +

FIGURE 16.18

Milling a large gear.

face + overtravel feed per min

(16.7)

Gear-Manufacturing Methods

FIGURE 16.19

905

Coarse-pitch spur or helical gears can be cut on a hobbing machine with an end-mill type of cutter.

The index time on automatic machines runs from 0.04 to 0.08 minute per tooth. Hand indexing takes much longer. The minimum overtravel for milling a spur gear is: Overlap = 2 depth of cut × (cutter diameter

depth of cut)

(16.8)

The feed per minute ranges from about 12 to 500 mm/min (1/2 to 20 in./min). The feed rate may be calculated from: Feed per min =

rpm of cutter × no. of cutter teeth feed per tooth

(16.9)

The rpm of milling cutters is based on cutting speeds that high-speed steel tools have been able to stand when cutting different materials. Table 16.15 gives some representative values of cutting speeds for different materials and different degrees of care in cutting. The conservative values represent the con­ dition where long cutter life is desired, and best accuracy and finish are also sought. The high-speed values represent the condition where a rugged machine and cutter are being worked to the limit. In a few cases, cemented-carbide cutters have been used to make gears instead of high-speed steel cutters. Much higher cutting speeds can be used with carbides, provided that the machine is rugged enough and powerful enough to drive the carbide cutter. After the cutting speed is determined, the rpm of the cutter is: Rpm of cutter =

1000 × cutting speed, m/ min metric 0.262 × outside dia. of cutter, mm

Rpmofcutter =

12 × cuttingspeed, fpm English 0.262 × outsidedia. ofcutter, in.

(16.10)

(16.11)

The feed per tooth depends on the finish desired and on how well the material cuts. Some typical values are listed in Table 16.16.

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Dudley’s Handbook of Practical Gear Design and Manufacture

TABLE 16.15 Average Time to Cut Straight Bevel Gears No. of Teeth

Pitch-Cone Angle, Deg.

12 20

8.5 22

Tooth Size

Face Width

Time

Module

Pitch

mm

in.

Roughing

Finishing

1.1 1.1

24 24

9.5 6.4

0.375 0.250

1.1 1.7

1.1 1.4

30

45

1.1

24

3.2

0.129

2.6

2.1

50 80

68 81.5

1.1 1.1

24 24

6.4 9.5

0.250 0.375

4.3 3.4 DI*

3.5 6.8

12

8.5

2.5

10

25

1.000

3.1

3.1

20 30

22 45

2.5 2.5

10 10

19 9.5

0.750 0.375

4.5 5.9

4.5 6.8

50

68

2.5

10

19

0.750

11.3

11.3

80 12

81.5 8.5

2.5 6.4

10 4

25 64

1.000 2.500

10.3 DI* 14.4

20.7 10.0

20

22

6.4

4

51

2.000

20.0

14.0

30 50

45 68

6.4 6.4

4 4

38 51

1.500 2.000

21.0 50.0

16.1 35.0

80

81.5

6.4

4

64

2.500

96.0

66.6

Note * DI stands for “double index”.

TABLE 16.16 Some Typical Values of Feed per Tooth Feed per Tooth

Description

Steel Material

Cast Iron, Bronze, or Plastic

mm

in.

mm

in.

0.05 0.08

0.002 0.003

0.10 0.15

0.004 0.006

Conservative Normal

0.13

0.005

0.25

0.010

Fast-cutting

There have been important developments in milling just as in hobbing and shaping. Special cutters can be used for very rapid stock removal. With a very rugged machine, precision indexing, and precision cutters, very good accuracy can be achieved. The milling process is particularly suitable for making racks, especially large racks. Figure 16.20 shows the milling of a large rack. Note the special roughing cutter that leads the precision finishing cutter.

16.1.6 BROACHING GEARS Small internal gears can be cut in one pass of a broach provided that the gear designer does not put the teeth in a “blind” hole. Large internal gears can be made by using a surface type of broach to make several teeth at a pass. Indexing of the gear and repeated passes of the broach can make a complete gear. Gears as large as 1.5 m (60 in.) in diameter are made by this process.

Gear-Manufacturing Methods

907

FIGURE 16.20 Milling large rack segments for a radar antenna application. (Courtesy of Sewall Gear Mfg. Col, St.Paul, MN, U.S.A.)

Racks and gear segments are often made by broaching. The teeth formed by broaching may be either spur or helical. Present broaching machines and broach-making facilities impose quite definite limits to the size gear that can be made by a one-pass broach. The parts most commonly made range from about 6 to 75 mm (1/4 to 3 in.) in diameter. Parts up to 200 mm (8 in.) in diameter have been made in production by broaching with a one-pass broach. However, when broaches get to this size, they become very costly. It is very difficult to forge, heat-treat, and grind a piece of high-speed steel that is 200 mm (8 in.) in diameter and several feet long and get a hardness of over 62 Rockwell C and an accuracy within 0.005 mm (0.0002 in.). Figure 16.21 shows an example of a small broaching machine. Broaching is a rapid operation. If the teeth are not too deep or the face width too wide, the gear can be cut completely in one pass. The time in minutes for a one-pass broaching cut is: Broaching time =

length of stroke length of stroke + + handling time 12 × cutting speed 12 × return speed

(16.12)

The rate at which a broach can cut will depend on the material being cut and the quality desired. Table 16.17 shows some typical broaching speeds. On the return stroke, the broach is often moved as fast as the machine will operate. This may be on the order of 9 to 11 m/min (30 to 35 fpm). The handling time for loading and unloading the machine will run about 0.07 to 0.12 minute on a production setup. The length of the broach stroke will be longer than the cutting portion of the broach, but usually not as long as the overall length. The cutting length of the broach will depend on tooth depth, face width, and how easily the gear cuts. Table 16.18 shows some typical broaching lengths. A very important development in broaching was the pot-broaching technique. Conventional broaching pushes a long broach through the part. Pot broaching moves the part through the broach. Figure 16.22 shows a tool for push-up pot broaching. The parts are pushed through this tool. The tool is made up of a number of small parts with cutting teeth, instead of a very large part with cutting teeth. Figure 16.23 shows a diagram of a machine for pull-up pot broaching. When parts require a long cut, it is mechanically better to pull the parts through the pot broach. This avoids possible buckling of a long pushrod.

908

Dudley’s Handbook of Practical Gear Design and Manufacture

FIGURE 16.21 Broaching machine. (Courtesy of National Broach and Machine Co., a division of Lear Siegle, Detroit, MI, U.S.A.)

The pot-broaching method tends to make the production time much shorter, and the tool itself is much easier to make. Pot broaching is a high-production method that has come into very substantial use on high-volume jobs. Figure 16.24 shows a good example of pot-broach work.

16.1.7 PUNCHING GEARS Punching is undoubtedly the fastest and cheapest method of making gear teeth. Unfortunately, punched gears do not have the same degree of accuracy as many of the cut gears, and punching is limited to narrow face widths.

Gear-Manufacturing Methods

909

TABLE 16.17 Typical Speed of Broach on the Cutting Stroke Material

Hardness, HB

Broaching Speed Conservative m/min

Steel

Normal

fpm

m/min

High-Speed fpm

m/min

fpm

350

1.2

4

3.0

10

4.9

16

300 200

2.4 4.9

8 16

4.9 6.7

16 22

6.1 9.1

20 30

Cast iron

250

4.3

14

6.1

20

7.3

24

Bronze

175 90

5.5 5.5

18 18

7.3 7.3

24 24

9.1 9.1

30 30

TABLE 16.18 Typical Broach Lengths Gear Whole Depth, in.

Face Width, in.

Broach Length Free-Cutting Material Cutting Portion

Tough Material

Overall

m

in.

m

in.

Cutting Portion m

in.

Overall m

in.

0.250

2

2.2

85

2.5

100

–

–

–

–

0.200

1

0.9

35

1.3

50

1.9

75

2.3

90

0.100 0.050

1 0.5

0.5 0.15

20 6

0.9 0.25

35 10

0.8 0.25

30 10

1.1 0.5

45 20

The design of blanks for gears to be punched must suit the process. The punching machines can handle only thin sheet stock. The as-punched gear is a disk-like wafer. The shaft can be attached to the punching by pressing a piece of rod through the hole in the punching. Punched gears are often mated with pinions that have much wider face widths than the gear. This makes positioning the punched gear precisely at the right distance from the end of the shaft unnecessary. In some cases, though, punched gears must be accurately positioned. Both external and internal spur gears may be made by punching. Equipment currently in use will handle gears from 6 to 25 mm (1/4 to 1 in.). The face width of the punching should not be greater than two-thirds of the whole depth of the tooth. Only fairly soft materials can be punched. Brass is popular material for punching. Sheet bronze, aluminum, and steel are also used. One stroke of a punch can makes a punched gear. The punching rate depends on the thickness and hardness of the material as well as on the die life and gear quality desired. Table 16.19 gives some typical rates.

16.1.8 G-TRAC GENERATING One of the surprising developments in gear cutting is the G-TRAC generator, made by the Gleason Works. The machine uses rack-type cutting tools mounted on an endless chain. The gear being cut is rolling in mesh with the passing rack teeth in the chain. Figure 16.25 shows the whole machine. Figure 16.26 shows a stack of parts being cut so as to have helical gear teeth.

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Dudley’s Handbook of Practical Gear Design and Manufacture

FIGURE 16.22 Push-up pot-type broach tool. Note parts finished up at top and blanks starting at the bottom. (Courtesy of National Broach and Machine Co., a division of Lear Siegler, Detroit, MI, U.S.A.)

The G-TRAC machine is intended for high production, low cost, and high accuracy. It is possible, though, to use the machine effectively for small production. In this case, a single row of cutting tools is used, and one tooth slot is formed at a time. The simple rack tools can be made by gear shops that are using the G-TRAC. This feature should make it easy to change tooth module (pitch), tooth pressure angle, or tooth depth when a gear unit is under development. (Normally, gear-cutting tools have to be ordered from a tool-making company, and some month’s time is lost when new, nonstandard gear tools are needed.) The concept of G-TRAC machine opens several new possibilities in gear machine tools.

16.2 GEAR GRINDING In general, gear grinding is an operation that is performed after a gear has been cut and heat-treated to a high hardness and has had journals or other mounting surfaces finish-ground. Grinding is needed because it is very difficult to cut parts over 350 HB (38 HRC). Since fully hardened steels are needed in many applications and it is often difficult to keep the heat-treat distortion of a cut gear within acceptable limits, there is a large field of gear work in which the grinding process is needed. In a few cases, medium-hard gears that could be finished by cutting are ground. This may be done to save the cost of expensive cutting tools like hobs, shapers, or shaving cutters; or it may be done to get a desired surface finish on accuracy on a gear that is difficult to manufacture. In some of the fine pitches, gear teeth may be finish-ground from the solid. For instance, the whole volume of stock removed in making an 0.8-module (32-pitch) tooth less than the amount of stock

Gear-Manufacturing Methods

911

FIGURE 16.23 broaching.

Sketch of broaching machine for pull-up pot

FIGURE 16.24 Some small automotive pinions pot-broached at a rate of 195 parts per hour. Right-hand pinion is 2.5 module (10 pitch), and left-hand pinion is 3.17 module (8 pitch).

removed in finish-grinding a good 4-module (6-pitch) tooth, even when the 0.8-module and 4-module teeth have equal face widths. Figure 16.1 shows that most of the methods of cutting a gear tooth have a counterpart in a grinding method. Disc cutters are used to mill gear teeth, and disc grinding wheels grind gear teeth. Threaded hobs cut gear teeth, and threaded grinding wheels grind gear teeth. Rack shaping is matched by generating grinders that make a disc wheel go through the motion of a tooth on a rack. There is, however, no cutting counterpart to the dished wheel, base circle generating grinding method.

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Dudley’s Handbook of Practical Gear Design and Manufacture

TABLE 16.19 Typical Punching Rates Material

Hardness, HB

Thickness

Strokes per Min

mm

in.

0.5 2.0

0.020 0.080

300 250

Brass

60

Aluminum

–

0.5

0.020

300

150

2.0 0.5

0.080 0.020

250 150

2.0

0.080

100

Steel

FIGURE 16.25

G-TRAC generator. (Courtesy of Gleason Works, Rochester, NY, U.S.A.)

In the following sections, information is given on how to estimate grinding time. This is a controversial subject which is hard to handle. Things like the hardness of the grinding wheel, the accuracy required of the gear, and the toughness and hardness of the stock being ground all enter into the time required to grind a gear. In general, medium-hard gears which are made by through-hardening steel can be ground faster than carburized gears at full hardness. Also, the less stock left for grinding, the faster the gear can be ground. The grinding times and number of passes required show in the various tables should be considered as only nominal values, subject to considerable revision either upward or downward in individual cases.

16.2.1 FORM GRINDING Form grinders use a disc wheel to grind both sides of the space between two gear teeth. Their grinding action is very similar to the action of a machine for milling gear teeth. Form-grinding wheels have an

Gear-Manufacturing Methods

913

involute form dressed into the side of the wheel, while a generating grinding wheel is straight-sided. Figure 16.27 shows a comparison of the two kinds of wheels for the same pinion. The machines available to form-grind gears can handle external spur gears from about 10 mm (0.4 in.) to about 2 m (80 in.) in diameter. Internal gears from about 1 m (40 in.) major diameter to about 50 mm (2 in.) minor diameter can be ground. Some form grinders will grind both spur as well as helical gears, external or internal gear teeth. Figure 16.28 shows a general-purpose for grinder. Form-grinding machines usually mount the work on centers. The gear design must provide room for the wheel to run out at each end of the work. Pinions which require an undercut to allow them to mesh with a mating gear without interference cannot be ground with a single grinding wheel. However, pinions with undercut may be ground with two grinding wheels straddling one or more teeth. Ceramic form grinding. It is a common practice to grind gears with ceramic wheels such as alu­ minum oxide or silicon carbide.

FIGURE 16.26 Generating helical-gear teeth on a stack of gear blanks by the G-TRAC machine. (Courtesy of Gleason Works, Rochester, NY, U.S.A.)

914

FIGURE 16.27

Dudley’s Handbook of Practical Gear Design and Manufacture

Comparison of form grinding and generating grinding.

FIGURE 16.28 An automated general-purpose form—grinding machine capable of doing external or internal gears up to 600 mm (24 in.) pitch diameter and up to 45 helix angle. (Courtesy of National Broach and Machine Co., a division of Lear Siegler, Detroit, MI, U.S.A.)

The time required to grind a gear has three parts. These are: 1. time to rough-grind, 2. time to finish-grind, and 3. time to dress the grinding wheel. In addition, there are some handling times, such as time to load the work, time to centralize the grinding stock, time to unload the work, and time to check the work for size.

Gear-Manufacturing Methods

915

When ceramic grinding wheels are used, all but the smallest gears will require more than one wheel dressing per gear. The number of dressings required depends upon several factors, of which the principal ones are the number of gear teeth, the grinding-wheel diameter, and the face width. The hardness of the gear and the accuracy required also enter into the picture. Table 16.20 shows some typical data on number of wheel dressings required and time per dressing. It is assumed that the material is about 60 HRC and that moderate accuracy is required. Face width is assumed to be in these proportions (see Table 16.21). The time required for either rough grinding or finish grinding by the form-grinding method may be estimated by Eq. (16.13). In this equation, a cycle is the cutting action that occurs on a single tooth between successive feeds of the grinding wheel. The latest-model form grinder feed down at each end of the stroke while roughing. Thus, a cycle is only a stroke across the tooth. The older-style machines feed down only once per revolution of the work. On each tooth, the grinding wheel makes a stroke across and back; in this case, a cycle includes both strokes. Grinding time, min =

no. of gear teeth × no. of cuts cycles per minute

(16.13)

The number of cuts will depend upon the amount of stock left for grinding and the amount removed per cut. Enough stock must be removed to eliminate both the inaccuracy of the rough-cut gear teeth and the inaccuracy caused by neat-treat distortion. Large gears with thin webs tend to distort more than small gears of solid construction. Quenching dies can be used to limit heat-treat distortion considerably. With care and skill in heat treating, it is possible to hold heat-treat distortion to reasonably low limits. TABLE 16.20 Typical Data for Dressing Form-Grinding Wheels Size of Work No. of Teeth

Wheel Diameter

No. of Dressings per Gear

Minutes per Dressing

Module

Pitch

mm

in.

75 20

2.5 2.5

10 10

150 150

6 6

5 2

¾ ¾

75

4

6

300

12

6

1

20 75

4 8

6 3

300 300

12 12

2 8

1 1¾

20

8

3

300

12

3

1¾

TABLE 16.21 Tooth Size and Face Width Proportions Tooth Size Module

Face Width Pitch

mm

in.

2.5 4

10 6

40 50

1.5 2.0

8

3

100

4.0

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Dudley’s Handbook of Practical Gear Design and Manufacture

Table 16.22 shows the general range between low amounts of heat-treat distortion and high amounts. The stock shown on tooth thickness is the sum of the amounts left on the two sides of the tooth. The relation between diameter over pins and tooth thickness is only approximate. The worst condition that can occur before grinding is that the distortion is so bad that some tooth on the gear requires all the stock to be ground off one side. If the gear is worse than this, it will not clean up, and the gear might as well be scrapped instead of ground. Assuming that the gear distortion is within the stock allowed, the maximum number of cuts is: No. of cuts =

2 × stock left for grinding stock normally removed per cut

(16.14)

In Eq. (16.14), the stock left for grinding is in terms of tooth thickness. The normal amount of stock removed is the amount that would be removed from the tooth thickness if the grinding wheel were cutting on both sides. The minimum number of grinding cuts is just one-half of that given by Eq. (16.14). In an average case, the number of cuts might be expected to be somewhere about midway between the minimum and maximum number of cuts. Roughing cuts usually take as much stock per cut as can be removed without burning the work. Finishing cuts take only a small amount of stock, and the last cut may just “spark out”, taking almost no stock. Table 16.23 shows the normal amounts of stock removed in form grinding per cut. The rate at which a form-grinding machine strokes depends upon the face width being ground, the amount of overtravel, the kind of material being cut, and the range of speeds available on the machine. Some of the latest machines on the market can stroke up to 0.35 m/s (70 fpm). They rough out a tooth completely before indexing. This makes it possible to use a much smaller amount of overtravel than would be required if the wheel had to move clear of the work on each stroke to permit indexing. Table 16.24 shows some typical cutting rates for form grinders. The table is based on a 300-mm (12in.) grinding wheel for the multiple-cycle machine and 150-mm (6-in.) wheel for the single-cycle ma­ chine. The work is assumed to be 2.5 module (10 pitch). The coarser itches take slightly longer to grind because of more overtravel. The table is based on 0.25 m/s (50 fpm) for roughing and 0.15 m/s (30 fpm) for finishing for the multiple-cycle machine. For the single-cycle type of machine, roughing speed is assumed to be 0.15 m/s (30 fpm) and finishing speed 0.10 m/s (20 fpm). Borazon form grinding. Gear tooth grinding has long been done with ceramic wheels such as alu­ minum oxide or silicon carbide. These wheels have high abrasive performance.

TABLE 16.22 Amounts of Stock Normally Needed for Grinding Tooth Size

Stock Left for Grinding Low Heat-Treat Distortion Tooth Thickness

Module

High Heat-Treat Distortion

Diameter Over Pins

Tooth Thickness

Diameter Over Pins

Pitch

mm

in.

mm

in.

mm

in.

mm

in.

1.6

16

0.13

0.005

0.30

0.012

0.25

0.010

0.64

0.025

2.5

10

0.20

0.008

0.50

0.020

0.38

0.015

0.96

0.038

4.0 8.0

6 3

0.30 0.66

0.012 0.026

0.76 1.65

0.030 0.065

0.64 1.14

0.025 0.045

1.57 2.69

0.062 0.106

12.0

2

1.14

0.045

2.85

0.112

1.90

0.075

4.78

0.188

Gear-Manufacturing Methods

917

TABLE 16.23 Stock Removed per Form-Grinding Cut Tooth Size

Rough Grinding Tooth Flanks Only 45 HRC

Finish Grinding

Tooth Flanks and Root

60 HRC

45 HRC

60 HRC

Module

Pitch

mm

in.

mm

in.

mm

in.

mm

in.

mm

in.

2.4

10

0.05

0.0020

0.04

0.0015

0.04

0.0015

0.025

0.0010

0.012

0.0005

4

6

0.05

0.0020

0.04

0.0015

0.04

0.0015

0.025

0.0010

0.012

0.0005

8 12

3 2

0.06 0.08

0.0025 0.0030

0.05 0.06

0.0020 0.0025

0.05 0.06

0.0020 0.0025

0.040 0.040

0.0015 0.0015

0.012 0.020

0.0005 0.0008

TABLE 16.24 Typical Cutting Rates for Form Grinding Face Width of Gear

Multiple Cycles per Tooth Machine

Single Cycles per Tooth Machine

mm

in.

Roughing, Cycles per min

Finishing, Cycles per min

Roughing, Cycles per min

Finishing, Cycles per min

20

¾

125

18

26

19

50

2

100

16

23

16

100 150

4 6

75 60

14 12

18 15

13 10

200

8

50

10

13

9

Borazon is a kind of “super abrasive” material that is based on cubic boron nitride (CBN). This material began to be used to grind1 gear teeth at about 1980. The popular name for CBN grinding is Borazon grinding. Borazon CBN is a trademark of General Electric, U.S.A. The CBN grinding wheel is being made as a highly precise metal wheel with a single layer of CBN particles galvanically bonded onto the surface. A precisely calibrated grit is used. A 65-μm (0.0025-in.) grit is used for gear finishing, and 100- to 120-μm (0.005-in.) grit is used for gear roughing. The CBN grinding wheel actually works somewhat like a milling cutter. Each exposed crystal of CBN tends to cut very tiny metal chips. The chips produced by CBN grinding look quite different from the particles produced by a vitrified, ceramic grinding wheel. It is claimed that the CBN grinding wheel is about 3000 times more wear-resistant than a typical aluminum oxide, ceramic wheel. Borazon gear-tooth grinding is done quite differently from conventional form grinding with ceramic wheels: • No wheel dressing. (After a considerable amount of gear grinding, the CBN wheel is stripped and re-plated with a new layer of CBN particles.) • Slow feed. (The wheel progresses slowly across the face width, removing all the stock in one pass.) 1

Super abrasives had been used to sharpen hobs and cutters in the 1970th.

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Dudley’s Handbook of Practical Gear Design and Manufacture

• Cool grinding. (With plenty of coolant and a free cutting characteristic, the CBN wheel tends to remove stock in a very predictable and controllable manner. Gear-tooth random errors or variations in hardness do not seem to be serious obstacle to the very hard and strong CBN wheel. Local overheating is probably less likely to occur.) Since the CBN wheel is not dressed, a different wheel is needed for each pitch, each number of teeth, and each style of profile modification (or pressure angle variation). For this reason, the CBN process is less attractive for small-lot production than it is for volume production. Figure 16.29 shows a form grinder used for Borazon grinding. Figure 16.30 shows a close-up of the grinding wheel and work. Some of the many kinds of parts that can be efficiently ground by the Borazon grinding method are shown in Figure 16.31.

16.2.2 GENERATING GRINDING—DISK WHEEL There are two basic types of generating grinding with disc wheels. One type uses a single wheel which is dressed to the shape of the basic rack tooth. The workpiece is rolled past the grinding area while the wheel reciprocates past the face width. The other type uses two saucer-shaped wheels which are concave toward the tooth flank, so that only a narrow rim contacts the tooth while the workpiece is rolled past the grinding area. Since this latter type of grinder is now made only by the Maag Gear Wheel Company, it will be referred to here as the MAAG process, while the dressed-single-wheel type will be referred o as the conical-wheel process. Conical-wheel grinding machines can handle gears in the range of about 25 mm (1 in.) to 3.5 m (140 in.) diameter, while the largest Maag machines can handle gears of up to 4.7 m (184 in.) diameter. Both types must have room for the grinding wheel to run out at each end of the cut. Smaller machines usually mount the work between centers. Large machines usually mount the work vertically with a fixture. Since the grinding pressure is small, supports are not usually needed.

FIGURE 16.29 Form-grinding machine developed for Borazon grinding (Courtesy of KAPP Co., Coburg, German Federal Republic.)

Gear-Manufacturing Methods

FIGURE 16.30

919

Form grinding a helical-gear part. (Courtesy of KAPP Co., Coburg, German Federal Republic.)

The generating grinding machines can make both spur and helical external gears. Internal spur and helical gears can be made with one model, which can handle up to 0.85 m (33 ½ in.) base-circle diameter. The action of a conical-wheel grinding machine is similar to that of a rack shaper using a single point tool. The grinding wheel is dressed to the desired basic rack form and is reciprocated past the face width while the gear rolls past the grinding field on its pitch circle. Because the wheel must maintain a definite form, it must be of a relatively hard bond, and so it requires a coolant to minimize burning of the work. The MAAG process is somewhat similar for their larger machines, except that two saucer-shaped wheels form the rack “tooth”. The wheel contact area is limited to its outer edge, where wheel wear can be sensed and continuously compensated for by automatic wheel adjustment. Relatively soft bonded wheels are used, so that no coolant is needed. On the smaller MAAG machines, the process is somewhat different. The two grinding wheels are still saucer-shaped and contact the work only at their outer edges, so that wear compensation can be used for soft wheels without coolant. But with the smaller machines, up to 1 m (40 in.) diameter, the grinding wheels do not form a rack “tooth”, but are parallel, and always contact the workpiece involute at a point on a line tangent to the base circle. This is known as zero-degree grinding because the grinding wheels are set at a pressure angle of 0 (see Figure 16.32). The gear is rolled on the base-circle tangent plane Y Y . The grinding-wheel contact planes are shown as X X , and the contact points are shown as P . Longitudinal corrections are made on all disc-wheel generating grinders by moving the grinding wheel to cut deeper or shallower as it passes across the face. Profile modifications on conical-wheel machines are made by dressing the wheel. On MAAG machines, where there is no wheel “form” to dress, profile modifications are made by moving the wheels to cut deeper or shallower along the profile. Modification by means of wheel movement also allows “topological” modification, in which both profile and helix may be continuously varied across the face width. The time needed to make gears by generating grinding with disc wheels is made up of essentially the same elements as the time required to form-grinding a gear. The time required to dress a grinding wheel

920

Dudley’s Handbook of Practical Gear Design and Manufacture

FIGURE 16.31 Examples of gear parts ground by Borazon form grinding. (Courtesy of KAPP Co., Coburg, German Federal Republic.)

(when necessary) may be estimated from Table 16.25. The time in minutes required to do either rough grinding or finishing grinding may be estimated using the following formula: Grinding time = no. of gear teeth × no. of cuts × index time +

strokes per tooth strokes per minute

(16.15)

Equation (16.14) can be used to get the number of cuts for generating grinding as well as for form grinding. In solving this equation, though, Table 16.26 should be used to get the amount of stock that is normally removed per cut. Because the amount of stock to be removed does not depend on the method of grinding, this item may be obtained from Table 16.22. The time required for generating grinding machines to index varies from 0.04 to 0.30 min, depending on the make and size of machine. The strokes per tooth and the strokes per minute vary with machine designs and tooth numbers and face widths. When the workpiece is lightweight, the machine elements used to generate can also be relatively light. In these cases, the workpiece is rolled back and forth very rapidly in generation, while the stroking action is slow. The process lends itself to horizontal mounting of the work. Heavy work-pieces need heavy generating elements that cannot be moved rapidly. These parts are mounted vertically and are slowly generated, while the wheel stroking motion is rapid. The result is that vertical machines use many fast

Gear-Manufacturing Methods

921

FIGURE 16.32 Zero-degree pressure angle grinding with saucer wheels. (Courtesy of Maag Gear Wheel Col, Zurich, Switzerland.)

TABLE 16.25 Typical Frequency and Length of Time Required to Dress Conical Grinding Wheel Size of Work

Wheel Diameter

Number of Dressing per Gear

Minutes per Dressing

mm

in.

mm

in.

45 HRC

460HRC

750

30

500

20

4

5

2½

450 300

18 12

300 300

12 12

3 3

4 3

1 1

150

6

300

12

2

3

¾

50

2

300

12

2

2

¾

passes along the face for each tooth, while horizontal machines use many fast generating motions, but only one pass along the face for each tooth for each cut. Table 16.27 shows some typical values for vertical machines, and Table 16.28 shows some typical values for horizontal machines. Figure 16.33 shows a generating grinding machine of the type that uses a conical wheel. A close-up of the grinding wheel and a double helical gear being ground are shown in Figure 16.34. A grinding machine using double saucer wheels is shown in Figure 16.35. A close-up of the wheel and work is shown in Figure 16.36.

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Dudley’s Handbook of Practical Gear Design and Manufacture

TABLE 16.26 Typical Amount of Stock Removed per Cut When Generating Grinding Tooth Size

Roughing Grinding Tooth Flanks Only 45 HRC

Module 2.5

Finishing Grinding Flanks and Root

60 HRC

45 HRC

60 HRC

Pitch 10

mm 0.13

in. 0.005

mm 0.08

in. 0.003

mm 0.08

in. 0.003

mm 0.04

in. 0.0015

mm 0.013

in. 0.0005

4

6

0.15

0.006

0.08

0.003

0.08

0.003

0.04

0.0015

0.013

0.0005

12

2

0.23

0.009

0.10

0.004

0.10

0.004

0.06

0.0025

0.020

0.0008

TABLE 16.27 Typical Stroking Rates for Vertical Grinders Tooth Size

Strokes per Tooth Roughing Grinding

Module 2.5

Strokes per Minute

Finish Grinding

Face Width

Pitch 10

15 tooth 30

75 tooth 14

15 tooth 40

75 tooth 20

50 mm (2 in.) 200

100 mm (4 in.) 135

150 mm (6 in.) 90

4

6

35

17

55

28

200

135

90

8 12

3 2

40 45

20 25

65 70

35 50

150 60

100 45

65 30

TABLE 16.28 Typical Stroking Rates for Horizontal Grinders Tooth Size

Generating Strokes per Minute

Stroking Rate (Seconds per Tooth, per Cut), Rough Grinding* 15 Tooth—Face Width

Module Pitch 2.5 4

10 6

75 Tooth—Face Width

15 tooth

75 tooth

50 mm (2 in.)

100 mm (4 in.)

150 mm (6 in.)

50 mm (2 in.)

100 mm (4 in.)

150 mm (6 in.)

120 130

240 148

3.5 2.8

7.0 5.6

10.5 8.4

1.8 2.3

3.6 4.6

5.4 6.9

8

3

130

75

2.8

5.6

8.4

4.6

9.2

13.8

12

2

110

40

3.5

7.0

10.5

12.0

24.0

36.0

Note * For finish grinding, multiply seconds per tooth by a factor of 3.

16.2.3 GENERATING

GRINDING—BEVEL GEARS

Zerol, spiral, and hypoid gears can be finished by grinding. The machinery available to grind these gears uses a generating motion. The grinding wheel cuts somewhat like a face cutter. The wheel is shaped like a cup rather than like the disc wheels used to grind spur and helical gears. As the bevel gear is ground,

Gear-Manufacturing Methods

923

FIGURE 16.33 Generating grinding a double helical gear with a conical wheel. (Courtesy of BHS-Hofler, Ettlingen, German Federal Republic.)

FIGURE 16.34 Close-up view of conical grinding wheel and work. (Courtesy of BHS-Hofler, Ettlingen, Germa Federal Republic.)

the work and the grinding wheel roll through a generating motion. In principle, the grinding wheel acts like one tooth of a circular rack which is rolling through mesh with the gear being ground. Bevel-gear grinding machines index after each tooth space has been ground with the generating motion. The motions of the machine are so smooth that the process appears to be almost as continuous as a hobbing or shaping process.

924

Dudley’s Handbook of Practical Gear Design and Manufacture

FIGURE 16.35 Generating grinder using two saucer-shaped wheels on a single helical gear. (Courtesy of Maag Gear Wheel Co., Zurich, Switzerland.)

FIGURE 16.36

Close-up view of saucer-shaped grinding wheels and work.

Gear-Manufacturing Methods

925

No feed motion is needed to travel the grinding wheel across the face width of the work. Since the wheel is like a hoop laid across the face width, the whole face width is ground in the same operation. The fact tends to make bevel-gear grinding fast. With spur or helical gears, a feed motion is needed (unless the face width is very narrow). This requires time for both the grinding of the tooth profile and the advance of the wheel across the face width. In bevel gears, the face width is done just as soon as the tooth profile is done. At the present time, there are several sizes of bevel-gear grinders for general-purpose work on the market. These are more or less universal machines intended to take a range of pitches, ratios, and spiral angles. Many special bevel-gear grinders have been built for the high-production manufacture of a particular gear-set. The designer of bevel gears that require grinding should be careful to design something that is within the range of available machinery, unless the job is important enough to warrant the development of special machine tools. The capacity ranges of bevel-gear grinders are rather involved. The maximum gear diameter that a machine will do changes rather substantially as the gear ratio changes and as the spiral angle changes. In some cases, bevel gears up to 1 m (40 in.) can be ground on the larger bevel-gear grinders. Most bevel gears which are ground seem to be in the size range of about 75 mm (3 in.) to 500 mm (20 in.) in diameter. There are so many complications to bevel-gear grinding that it is desirable for most designers to check with the shop that is to build their gears as soon as they have laid out a preliminary design. The complications of bevel-gear grinding make it hard to write any formula for grinding time. The best way to give information on grinding time appears to be to give a table of approximate times for different combinations. Table 16.29 shows approximate times, assuming that distortion is not serious and that moderate precision is desired. The total time per piece shown in Table 16.29 includes the automatic wheel-dressing cycle. This is approximately 30 seconds per dress. Figure 16.37 shows a typical spiral-bevel-gear grinding machine.

16.2.4 GENERATING GRINDING—THREADED WHEEL Gears can be ground by a grinding machine that uses a threaded grinding wheel. The elements of this kind of machine are very similar to those of a hobbing machine. The process can grind either spur or helical gears, provided they are external.

TABLE 16.29 Average Time to Grind Spiral Bevel Gears No. of Teeth

Pitch-Cone Angle, Deg

Tooth Size Module Pitch

Face Width mm

No. of Passes

No. of Dresses

Seconds per Tooth

Total Time, min

in.

12

8.5

4

6

41

1.625

4

1

2.7

2.7 per side

20 30

22 45

4 4

6 6

38 32

1.500 1.250

6 7

2 2

2.7 2.7

6.4 per side 10.5

50

68

4

6

38

1.500

8

2

2.7

19.0

80 12

81.5 8.5

4 8

6 3

41 82

1.625 3.250

9 4

3 1

2.7 4.5

33.9 4.1 per side

20

22

8

3

82

3.250

6

2

4.5

10.0 per side

30 50

45 68

8 8

3 3

82 82

3.250 3.250

7 8

2 2

4.5 4.5

16.7 31.0

80

81.5

8

3

82

3.250

9

3

4.5

55.0

926

Dudley’s Handbook of Practical Gear Design and Manufacture

FIGURE 16.37

Grinding a spiral bevel gear. (Courtesy of Gleason Works, Rochester, NY, U.S.A.)

The threaded-wheel grinding machines are available up to 0.77 m (30 in.) outside diameter capacity. Either spur or helical, external gear teeth can be ground. Helix angles up to 45 are ground, but the maximum work diameter has to be considerably reduced. The sizes of teeth produced by this method range from about 0.2 module (120 pitch) to 8 module (3 pitch). Grinding machines of around 350 mm (13 ¾ in.) maximum work capacity use threaded grinding wheels about 350 in size, while the larger machines around 0.77-m capacity use wheel about 400 mm (15 ¾ in.) in size. The smaller machines (350 mm) are using grinding-wheel speeds up to 1900 rpm. Gears to be ground with a threaded wheel must allow room for the wheel to run out (see Table 16.30). Several roughing cuts are used to bring the gear down to size. One or more finishing cuts—depending on the finish accuracy required—are used to finish the gear. In most cases, it is not necessary to stop to dress the grinding wheel while a gear is being ground. As Table 16.30 shows, several gears can usually be done before the wheel needs dressing. This table shows some typical numbers of gears ground per wheel dressing and amounts of overtravel. It takes about 12 min to dress the grinding wheel. Diamond wheels are used to dress threads on the grinding wheel. The time required to grind a gear by the threaded-grinding-wheel process may be calculated in much the same way as hobbing time is calculated. The time in minutes is: Crinding time, min =

no. of gear teeth × (face + overtravel) × no. of cuts no. of threads × feed × rpm of wheel

(16.16)

The amount of stock removed with an average cut and the average rates of feed are shown in Table 16.31. The number of cuts may be figured from the stock left to grind, except that extra cuts may remove almost no stock.

Gear-Manufacturing Methods

927

TABLE 16.30 Number of Gears Ground per Wheel Dressing and Approximate Overtravel Tooth Size Module 1.6 2.5

Face Width

Helix Angle

Pitch

mm

in.

16

10 25

0.375 1.000

Spur

10

4.0

6

Gears per Dressing

Overtravel

25 Teeth

75 Teeth

mm

in.

15

200 100

100 50

1 6

0.06250 0.25000 0.09375

10

0.500

Spur

100

50

2

50

2.000

15

40

20

6

0.25000

19 100

0.750 4.000

Spur

20 5

10 2

3 9

0.12500 0.37500

15

TABLE 16.31 Typical Values for Feed and Stock Removal When Grinding with a Threaded Wheel Tooth Size

Feed per Revolution of Work Roughing

Module

Stock Removed per Cut, Roughing or Finishing

Finishing

Pitch

mm

in.

mm

in.

mm

in.

1.6

16

1.8

0.07

0.63

0.025

0.01

0.0004

2.5 4

10 6

1.8 1.8

0.07 0.07

0.63 0.63

0.025 0.025

0.01 0.01

0.0004 0.0004

Wheel wear is quite uniform with the threaded grinding wheel. When on spot gets worn, the wheel is shifted axially to let a new part of the wheel generate the work. This is one of the reasons for why a lot of grinding can be done between the wheel dressings. Gear teeth finer than 0.8 module (32 pitch) may be quite satisfactorily ground from the solid. Larger teeth are cut before grinding. The threaded-wheel grinder is used for high-precision gears. It is also coming into increased use for medium-precision gears which are hardened—and were formerly used without grinding after hardening. Closer quality control and more concern over gear noise make it harder to produce acceptable gears for lower-speed applications. Table 16.32 shows some comparisons of grinding time that may be expected. Note the substantial reduction in grinding time with lower accuracy. These less-accurate gears are adequate for vehicle gears and certain other industrial gears that do not run at high speeds. Figure 16.38 shows a threaded-wheel grinding machine being used to make high-speed helical pi­ nions. Figure 16.39 shows a close-up of a threaded wheel and a high-speed helical gear.

16.2.5 THREAD GRINDING Single-enveloping worms may be finished by grinding in a thread grinder. Ordinary worms are milled, hardened, and then ground. In fine pitches—5-mm (0.200-in.) linear pitch and less—it is possible to make the thread complete by grinding. Pitches as coarse as 40-mm (1.60-in.) linear pitch may be ground on available grinding machines. Some grinders limit the lead angle of the worm to 30 , while others will

928

Dudley’s Handbook of Practical Gear Design and Manufacture

TABLE 16.32 Comparison of Estimate Grinding Times Using the Threaded Grinding Wheel Method for HighPrecision and Medium-Precision Gears Tooth Size Module

No. of Teeth Pitch

Face Width mm

in.

Stock to Remove mm

Grinding Time, Min

in.

High-precision gears (turbine drives) 2.5

10

25

32

1.25

0.24

0.0095

6.3

2.5 2.5

10 10

102 102

32 76

1.25 3.0

0.24 0.24

0.0095 0.0095

20.6 49.3

5.0

5

25

76

3.0

0.30

0.0120

14.3

5.0 5.0

5 5

102 102

76 152

3.0 6.0

0;.38 0.46

0.0150 0.0180

65.4 150.0

2.5 2.5

10 10

25 102

32 32

1.25 1.25

0.24 0.24

0.0095 0.0095

3.2 8.8

2.5

10

102

76

3.0

0.24

0.0095

20.7

5.0 5.0

5 5

25 102

76 76

3.0 3.0

0.30 0.30

0.0120 0.0120

9.1 24.2

5.0

5

102

152

6.0

0.30

0.0120

48.1

Medium-precision gears (vehicle drives)

Notes: 1. The high-precision gears are ground with a single-thread wheel. 2. The medium-precision gears are ground with a double-threaded (two-start) wheel. 3. In all cases, the gears are rough-ground and then finish-ground.

FIGURE 16.38 Grinding helical pinions on a threaded-grinding-wheel machine. (Courtesy of Sier-Bath Gear Co., North Bergen, NJ, U.S.A.)

Gear-Manufacturing Methods

929

FIGURE 16.39 Threaded grinding wheel and helical gear. (Courtesy of Reishouer Corp., Elgin, IL, U.S.A., and Reishouer Ltd., Zurich, Switzerland.)

go up to as much as 50 . Worms up to 300 mm (12 in.) outside diameter may be ground with presently available equipment. See Figure 16.40 for an example of worm thread grinding. Worm designs to be ground must allow room for a relatively large grinding wheel to run out. A commonly used size of wheel is 500 mm (20 in.). When the worm grinding wheel has a straight-sided profile, it will produce a worm thread with a convex curvature. This curve is not an involute. The amount of this curvature may be calculated by the method shown in Sec. 17.4. To produce a straight-sided worm thread, a slight convex curvature in the grinding-wheel profile is required. The time required to grind worm threads may be estimated by the following formula:

FIGURE 16.40

Grinding a large precision index worm. (Courtesy of Jones and Lamson, Springfield, VT, U.S.A.)

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Grinding time, min =

thread length no. of threads × index time + × no. of cuts (16.17) threads per cut feed rate

The indexing time runs from about 0.1 to 0.5 min. The developed length of the thread is: Length of thread =

3.14 worm dia. face width × cos (lead angle) lead of worm

(16.18)

In fine-pitch worms, it is sometimes possible to use two or three ribs on the grinding wheel and grind all worm threads at once. This saves time. The number of cuts will depend on both the pitch and the amount of stock left for grinding. Table 16.33 shows the number of cuts and the feed rates normally used for different pitches of worms. It is assumed that a nominal amount of stock is left for finishing. This would be about 0.25 mm (0.010 in.) on tooth thickness for 12-mm (0.500-in.) linear pitch, and 0.65 mm (0.025 in.) for 32-mm (1.250-in.) linear pitch.

16.3 GEAR SHAVING, ROLLING, AND HONING There are three different ways of finishing involute gear teeth that involve a gear-like tool rolling with the work on a crossed axis: • Shaving • Rolling • Honing The shaving process finishes by cutting. The rolling process cold-forms the metal by very small amounts of cold flow on the gear teeth, produced by pressure from the hardened roll. The honing process abrasively removes very small amounts of metal from the gear-tooth surfaces. Gear shaving is strictly a finishing operation. Compared with grinding, shaving is generally a much faster process. Grinding is not limited to hardnesses. It is usually quite practical to shave gears up to 350 HB (38 HRC). Gears up to 450 HB (47 HRC) have been cut and shaved with fair results. At the higher hard­ nesses, tool wear is very fast, and special techniques and lubricants are required. Also, the shaving cutter needs to be extra hard.

TABLE 16.33 Number of Cuts and Feed Rates Normally Used in Worm Grinding Linear Pitch

No. of Cuts

Feed Rate, per Min 250-mm (10-in.) Wheel

mm

500-mm (20-in.) Wheel

in.

Roughing

Finishing

mm

in.

mm

in.

6

0.250

2

1

625

25

1000

40

12 19

0.500 0.750

3 4

1 1

500 375

20 15

875 750

35 30

25

1.000

5

2

300

12

625

25

31

1.250

7

2

250

10

500

20

Gear-Manufacturing Methods

931

Shaving is a corrective process. Most people in the trade express the view, “Shaving will not make a bad gear good, but it will make a good gear better”! Shaving readily improves surface finish and reduces gear runout. If the gear and the shaving cutter are properly designed, it is possible to improve profile accuracy considerably. Tooth-to-tooth spacing is improved by shaving, but accumulated spacing error is not changed by a large amount unless the accumulated error comes mostly from eccentricity effects. On narrow-face-width gears, helix errors can be controlled by shaving, but a narrow shaving cutter has little or no control on wide-face gears. Since shaved gears are not heat-treated between cutting and shaving there is no heat-treat distortion on clean up, and only a small amount of stock is needed for shaving. Shaved gears are often fully hardened after shaving. When this is done, heat-treat distortion is held to a minimum or controlled in such a manner that uniform distortion results, and this distortion is allowed for in the shaving process. There are two general methods of shaving, rotary shaving and rack shaving. The rotary method uses a pinion-like cutter with serrated teeth, while the rack method uses an actual rack with serrated teeth. Figure 16.41 shows rotary shaving, while Figure 16.42 below shows rack shaving.

16.3.1 ROTARY SHAVING The rotary shaving cutter has gear teeth ground with a profile that will conjugate with that of the part to be shaved. The cutter teeth are serrated, with many small rectangular notches. As the shaving cutter rotates with the gear, these notches scrape off little shavings of material; hence the name shaving. There is no index gearing in the shaving process. Small parts are driven by the cutter that is shaving them, while large parts drive the cutter instead. The shaving cutter is essentially a precision-ground gear made of tool steel. The reason that shaving can produce very high accuracy is that there is an averaging action as the cutter rolls with the work. This tends to mask the effect of any slight indexing errors that may have been ground into the cutter. The shaving cutter does not have the same helix angle in degrees as that of the part it is shaving. This makes the axis of the cutter sit at an angle with the axis of the work. The amount of this crossed-axis

FIGURE 16.41 VT, U.S.A.)

Shaving an instrument pinion. (Courtesy of Fellows Corp., Emhart Machinery Group, Springfield,

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FIGURE 16.42 Gear-shaving machine equipped with programmable controller. (Courtesy of National Broach and Machine Co., a division of Lear Siegler, Inc., Detroit, MI, U.S.A.)

angle governs the shaving action. The greater the angle, the more the cutter cuts. The crossed-axis angle causes a cutter tooth to slide sideways as it rolls with the gear tooth. It is this motion that makes the shaving cutter scrapes off metal. If shaving cutters were serrated in a radial direction and the crossed-axis angle were zero, a shaving cutter would not cut at all unless it was reciprocating axially. It so happens, that one design of shaving machine does use a rapid reciprocating motion to shave internal gears. In general, shaving cutters do not have any rapid reciprocating motion. Both external and internal gears may be shaved. The gears may be either spur or helical. Small external gears may be shaved by feeding the cutter either parallel to the gear axis or at some angle to the axis. If the cutter is fed at right angles to the gear axis, the cutter must be as wide as or slightly wider than the work. If the cutter is fed in a diagonal direction, there is a relation between the cutter minimum width and the gear face width. When the cutter is fed parallel to the gear axis, there is no geometric requirement on the cutter face width. With this direction of feed, satisfactory results have been achieved by shaving a gear as wide as 0.5 m (20 in.) face width with a 25-mm (1 in.) cutter.

Gear-Manufacturing Methods

933

Figure 16.42 shows a shaving machine for small gears that will shave with the feed at an angle to the axis. A vertical-axis shaving machine is shown in Figure 16.43. This type of machine is often used for large gears. Gears which are to be shaved should allow room for the cutter run out. Shaving-cutter runout is hard to figure, because the cutter sits at an angle with the work and contacts the work on an oblique line. Gears to be shaved are often cut (before shaving) with a cutting tool which has a slight protuberance in its tip. This is helpful because the shaving cutter is not intended to remove metal from the root fillet of the gear. The protuberance on the cutting tool produces a slight relief, or undercut. This undercut allows the tip of the shaving cutter to roll freely instead of hitting a shoulder where the shaving action stops at the bottom of the gear tooth. Gears to be shaved must have tooth designs that permit sufficient teeth to be in contact with the shaving cutter. Since the cutter either drives or is driven by work, the gear teeth must be capable of transmitting power smoothly. For instance, shaving standard spline teeth with a 30 pressure angle and a height which is stubbed to 50% of full proportions is usually impractical. These teeth do not have enough involute profile to transmit power smoothly when they roll. In general, the pressure angle should be in the range of 14.5 to 25 , and tooth height should be at least 75% of full depth to permit satisfactory rolling conditions. Clearance in the root fillet should be at least 0.3m n (0.3/ Pd ) to permit a suitable design of a pre-shaving type of cutting tool. Spur pinions with small numbers of teeth are somewhat more difficult to shave. There are problems with obtaining good involute when shaving pinions with 10 teeth and fewer. It is also difficult to design a suitable shaving cutter for internal gears with fewer than 40 teeth. When internal gears have 25 teeth, it is just about impossible to get a cutter inside that will shave on the crossed-axis principle. The time required to shave gears may be estimated by one of two formulas. If the feed is parallel to the gear axis, (16.19) should be used. If the feed is at an angle to the axis, (16.20) is the one to use. Feed parallel to axis:

FIGURE 16.43 A rotary gear shaver for finishing medium-sized internal and external gears. (Courtesy of National Broach and Machine Co., a division of Lear Siegler, Detroit, MI, U.S.A.)

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Shavingtime, min =

0.262 × pitch dia. ×(face width + overtravel) × no. of cuts shaving speed × feed

(16.19)

Feed at an angle to axis: Shaving time, sec = time per stroke × no. of cuts

(16.20)

The overtravel in shaving may be estimated at about 1.5 times per cutter width. The face width and pitch diameter in (16.19) are those of the gear being shaved. The number of cuts required will depend on the amount of stock left for shaving and the amount taken per cut. Sometimes extra cuts may be needed to secure helix-angle correction. The amount of stock that may be removed in shaving is fairly limited. It is practical to put in only a certain amount of undercut in the pre-shave cutting. If too much stock is removed, the undercut allowance will be exceeded, and there will be trouble from cutter “bottoming”. Table 16.34 shows the approximate amount of stock that may be left for shaving. The stock removed per cut and the rate of feed both depend somewhat on the size of the gear and considerably upon the degree of quality and surface finish desired. Table 16.35 gives some representative values that can be used for estimating purposes. The rolling velocity of a shaving cutter is generally described as shaving speed. The shaving speed, of course, varies considerably depending on the hardness and size of the gear parts being shaved. Some typical values of shaving speed are: Small steel parts, 250 BHN, 25 to 100 mm (1 to 4 in.) in diameter 120 to 150 m/min (400 to 500 fpm) Large gears 250 BHN, 1 to 2.5 m (40 to 100 in.) in diameter 90 to 120 m/min (300 to 400 fpm) Medium gears, 350 BHN, 100 mm to 1 m (4 to 40 in.) in diameter 75 to 90 m/min (250 to 300 fpm)

When gears are shaved with the cutter traveling at an angle to the axis, the cutter has to be wide enough to shave the whole gear face width in one stroke. This process will not handle as wide work as the parallel method. Typical machinery on the market can handle gears up to 0.6 m (24 in.) diameter and 100 mm (4 in.) face width by this process. By comparison, when the feed is parallel to the axis, available machinery will handle gears up to 5 m (200 in.) in diameter and 1.5 m (60 in.) face width. The amount of time required to make a stroke (or cut) when the feed is at an angle to the axis depends mostly on the gear diameter. Table 16.36 gives some representative values for different angles of feed.

TABLE 16.34 Amounts of Stock Left for Shaving Tooth Size

Stock Left on Tooth Thickness Minimum (High Accuracy)

Module

Pitch

mm

in.

Maximum (Medium Accuracy) mm

in.

1.6

16

0.025

0.0010

0.050

0.002

2.5

10

0.038

0.0015

0.075

0.003

4 12

6 2

0.050 0.075

0.0020 0.0030

0.100 0.150

0.004 0.006

Gear-Manufacturing Methods

935

TABLE 16.35 Shaving Stock Removal and Feed Rates Kind of Work Diameter mm

in.

150

6

Stock Removal per Cut (on TT)

Cross Feed Rate per Revolution

Accuracy Roughing, mm (in.)

Finishing, mm (in.)

Roughing, mm (in.)

Finishing, mm (in.)

High precision

0.018

0.008

0.38

0.15

(0.0007) 0.038

(0.0003) 0.018

(0.015) 0.50

(0.006) 0.50

(0.0015)

(0.0007)

(0.020)

(0.020)

0.018 (0.0007)

0.008 (0.0003)

0.38 (0.015)

0.20 (0.008)

150

6

Medium precision

600

24

High precision

600

24

Medium precision

0.038

0.018

0.63

0.63

(0.0015) 0.018

(0.0007) 0.008

(0.025) 0.38

(0.025) 0.25

(0.0007)

(0.0003)

(0.015)

(0.010)

0.038 (0.0015)

0.013 (0.0005)

0.63 (0.025)

0.38 (0.015)

2400

96

High precision

2400

96

Medium precision

TABLE 16.36 Seconds per Cycle When Shaving Cutter IIs Fed at an Angle to the Gear Axis Gear Diameter mm

Feed 90 to Gear Axis

Feed 60 to Gear Axis

Feed 30 to Gear Axis

in.

Commercial

Precision

Commercial

Precision

Commercial

Precision

75

3

28

40

32

45

36

50

150 300

6 12

45 57

65 85

50 63

70 90

54 68

75 95

450

18

70

100

76

105

82

120

600

24

85

125

91

130

97

149

16.3.2 RACK SHAVING In rack shaving, the gear to be shaved is rolled back and forth with a rack having serrated teeth. The rack is moved in a direction which is not perpendicular to the gear axis. This gives a crossed-axis effect which makes rack shaving follow the same kind of cutting action as rotary shaving. It is the crosswise sliding that makes the rack cut. As the work rolls with the reciprocating rack (see Figure 16.44), it is also moved across a portion of the rack so as to equalize the wear of the rack. After each stroke of the rack, the gear is fed in a slight amount towards the rack. External spur and helical gears can be shaved by the rack method. Presently available machines will handle gears up to 200 mm (8 in.) diameter and 50 mm (2 in.) face width. This method of shaving does not lend itself to shaving large parts. The racks used are quite expensive. The rack must be wider than the gear face width, and it must have a length (and stroke) longer than the gear circumference. The rack-

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Dudley’s Handbook of Practical Gear Design and Manufacture

FIGURE 16.44

The rack shaving method.

shaving process is very rapid, and a very large number of parts are obtained per sharpening of the rack. Rack shaving is quite economical on high-production jobs. The time required to finish steel gears by rack shaving may be estimated from Table 16.37. The table shows an average time based on a moderate amount of stock left for shaving and material with rea­ sonably good machinability. Special conditions or quality requirements would require proportionally more or less time.

16.3.3 GEAR ROLLING The gear-rolling processes finish the teeth by rolling the gear with a hardened tool that has very precisely ground teeth. This tool is called a die. The gear and die are pushed together with a high force. Figure 16.45 shows the schematic arrangement of single-die rolling. Note the roller steady rests used to transmit the force of the table feed to work. Figure 16.46 shows a two-die rolling machine. In this case two dies, 180 apart, apply force to the part being rolled. (The part does not need roller steady rests when two dies with equal and opposite forces do the rolling.)

TABLE 16.37 Average Amount of Time to Rack Shave Gears Gear Diameter mm

Time, Seconds

in.

1.6 module (16 pitch)

2 module (12 pitch)

2.5 module (10 pitch)

3 module (8 pitch)

4 module (6 pitch)

25 50

1 2

75 73

70 72

69 67

65 63

60 58

75

3

70

67

64

60

55

100 150

4 6

65 58

62 55

59 52

55 48

50 43

200

8

45

42

39

35

30

Gear-Manufacturing Methods

937

FIGURE 16.45 Operating principle of single-die gear-rolling machine. (Extracted with permission from Modern Methods of Gear Manufacture, published by National Broach and Machine Co., a division of Lear Siegler, Inc., Detroit, MI, U.S.A.)

FIGURE 16.46 Two-die gear-rolling machine. (Courtesy of National Broach and Machine Co., a division of Lear Siegler, Inc., Detroit, MI, U.S.A.)

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Small parts are generally done with the two-die rolling machine, while large parts are done with the single-die machine. Typical machines roll gears from about 25 mm (1 in.) to 150 mm (6 in.) using two dies. Single-die machines are being used to do gears up to 0.5 m (19 in.) diameter. The face widths are generally quite narrow [from about 20 mm (0.8 in.) to about 70 mm (2 ¾ in.)]. Gear rolling on high-production jobs is generally done with the axes parallel. It can, of course, be done with the axes crossed by a rather small angle. It takes less force to do the job with axes crossed, but there is less control of helix accuracy. Also, the rolling time is longer. In rolling teeth, there is some difficulty with involute accuracy and the quality of the rolled surface. If the gear is too hard or has too much stock left, there is a tendency for slivers of metal to roll over the edges and for there to be folds or flaws around the pitch line. The sliding velocities are different on the two sides of the gear being rolled. This fact tends to make the two sides different and to cause trouble at the pitch line, where the sliding velocity reverses. Figure 16.47 shows the characteristic conditions during gear rolling. Rolling is successful, though, in spite of the problems just mentioned. These factors will help ensure that the process will work: • • • • •

The amount of stock left for rolling is small (about one-half of that left for shaving). The teeth are cut before rolling with a protuberance cutter so that the die can roll clear of the root fillet. The teeth are usually chamfered at the outside diameter corner. The gear to be rolled should not be too hard. [A good hardness for rolling is 200 HB (210 HV).] Rolling jobs are “developed” by experimental rolling, involute checking, and then involute modification of the die to make the involute of the part come out as desired. • The rolling is often reversal in direction in order to make the two sides of the gear tooth come out alike.

Gear rolling is fast, and it produces a very smooth, burnished surface finish. In the production of small, high-volume vehicle gears, rolling gears that are going to be carburized and hardened before the

FIGURE 16.47

Different sliding directions as a die rolls with a gear.

Gear-Manufacturing Methods

939

carburizing process works out well. A low hardness is normal in parts to be carburized, since the final hardness of both case and core material will be established in the carburizing process. Small gears are being made to medium-high-precision accuracy by rolling before carburizing and no grinding after carburizing. (In some cases, such parts may be honed after carburizing.) To show the relative time involved in shaving, rolling, and honing, the following data from National Broach and Machine Co., Inc. are illustrative. These data are for a 25-tooth helical gear with 73.66 mm (2.90 in.) pitch diameter and 16 mm (5/8 in.) face width. Normal pressure angle was 20 , and helix angle was 32 : Gear shaving: Conventional 43 Diagonal 22 Gear rolling 10 Gear honing 21

seconds/piece seconds/piece seconds/piece seconds/piece

16.3.4 GEAR HONING In the gear honing process, an abrasive tool with gear teeth is rolled with the gear part on a crossed axis. The force holding the honing tool and work together is very light. The honing tool cuts because of abrasive particles in the composition of the hone or attached to the surface of the hone. The honing is done wet, with an appropriate fluid to serve as a lubricant, a coolant, and a medium to flush away the wear debris from honing. The gear hone may be a plastic resin material with abrasive grains of silicon carbide. This kind of hone is made by casting in a mold. The hone may be made to medium-precision accuracy and used on gears intended to have medium-high-precision accuracy. The basic accuracy is in the gear before honing. Honing averages the surface and takes off local bumps, scale, etc. Thus, the accuracy of the gear is not primarily derived from the hone. For high-precision gear work, the hone is made to a relatively high precision. It is often possible to get some improvement in involute, helix (lead), and concentricity by careful honing with a precision hone. Metal hones with bonded-on abrasives are used for fine-pitch gears and for certain medium-pitch gears where a plastic hone might tend to break. Vehicle gears are often carburized and then run without grinding. These gears are generally shaved or rolled to finish accuracy before carburizing. The final honing operation does these things: • Removes heat-treat scale and oxidation • Removes nicks and bumps from handling (the unhardened gear gets bruised very easily in handling) • Removes some heat-treat distortion. The honed vehicle gear generally runs more quietly. Its load-carrying capacity is higher because of more uniformity in accuracy and smoother tooth surface. High-speed gears used in helicopters and in certain other high-speed turbine applications are generally finish-ground after carburizing. Such gears may still need honing to get a very smooth surface finish, so that they will not fail due to scoring. Frequently, such gears need a surface finish better than the grinding machine will produce. The better the ground finish, the better the honed finish. Some guideline relations are: • Shave or grind to 0.7–0.9 μm (0.028–0.036 μin.) finish; hone to 0.4–0.5 μm (0.016–0.020 μin.). • Shave or grind to 0.4–0.6 μm (0.016–0.024 μin.) finish; hone to 0.25–0.35 μm (0.010–0.014 μin.). • Shave or grind to 0.25–0.35 μm (0.010–0.014 μin.) finish; hone to 0.15–0.2 μm (0.006–0.008 μin.).

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When the honed finish must be extra good, the final honing is often done with a rubber-like hone. The hone material is polyurethane. The abrasive is applied as a liquid compound during the honing. The abrasive particles lodge in pores on the hone tooth surface and thereby charge the hone. The type of honing just described removes almost no stock, and so it has little or no ability to correct involute or helix (lead). The primary purpose of this type of honing is to polish the tooth surface. (In some difficult gear jobs, two-stage honing is used. Normal hones clean up the surface and refine the accuracy. The final finish is obtained by the rubber-like polishing hone.) Gear-honing machines look like gear-shaving machines. They are special, though, in that they must handle fluids with abrasives, and they must be able to accurately apply very light forces to the hone. Figure 16.48 shows a close-up of a gear-honing operation. The top half of Figure 16.48 shows some typical hones made of steel and carbide coated. Gear-honing machines for external gears are readily available for gear sizes up to 0.6 m (24 in.) diameter. Machines for honing internal gears tend to be smaller in capacity, with 0.3 m (12 in.) being typical. The size of teeth commonly honed ranges from 1.2 to 12 module (2 to 20 pitch).

16.4 GEAR MEASUREMENT Those running gears shops find that they need a considerable capability to measure the geometric ac­ curacy of gear teeth. It is not enough to have machinery to cut, grind, and finish gear teeth. Additional machinery is needed to measure gear-tooth spacing, profile, helix, concentricity, and finish. The gears used in the aerospace field must be rigidly controlled for both geometrical and metallurgical quality. The high-speed gears used with turbine engines are almost as critical. A turbine, for instance, that runs at 20,000 rpm will usually be connected to a pinion. In the oil and gas industry, it is normal for the turbine gear drive to be designed for a life of 40,000 hours – at full-rated power. A pinion meshing with one gear makes 4.8×1010 cycles in 40,000 hours at 20,000 rpm. With a high pitch-line speed and full-rated load, it is obvious that a pinion like this will not last this long if its accuracy is deficient. Precise measurements are needed to verify that each and every item of geometric accuracy is within specification limits. Vehicle gears do not run as long—total life is often no more than 2×108 cycles. The vehicle gear, though, is much more heavily loaded than the turbine gear. In addition, noise and vibration requirements have become quite critical. The vehicle gear does not have to be as accurate as the turbine gears, because it does not run so fast or so long. But the vehicle gear has to meet its own level of accuracy. If id does not, vehicle drives that should last a few years will fail in a few months. In all fields of gearing, the control of gear accuracy is essential. Since this is the case, it is necessary to present some material on accuracy limits and on machines used to measure gear accuracy.

16.4.1 GEAR ACCURACY LIMITS The gear designer faces a dilemma when endeavoring to put accuracy limits on a gear drawing. To get high load-carrying capacity and reliability to meet the gear life requirements, each item must be specified very closely. (The load rating part of Chapter 6 showed the rather direct relation between accuracy and load-carrying capacity.) Besides load-carrying capacity, the designer needs to worry about two other things. First, the accuracy called for must be practical to meet in the gear shop (or shops) available to do the gear work. Specifying impossible accuracy limits is to no avail. Second, the gear parts need to be made at reasonable cost. In a competitive world, it is not the best gear that is needed. What is really needed is the lowest-cost gear that will adequately meet load, life, reliability, and quietness requirements. To solve the dilemma just given, it is advisable for the gear designer to first consider these things:

Gear-Manufacturing Methods

941

FIGURE 16.48 Gear hones with abrasives bonded to metal and a gear honing arrangement. (Courtesy of National Broach and Machine Co., a division of Lear Siegler, Inc., Detroit, MI, U.S.A.)

• Pitch-line velocity • Intensity of loading, length of life, and degree of reliability needed • Requirements as to noise, vibration, or need to run with hot, thin oils. The things just mentioned will guide the designer to the proper level of accuracy. Next, the designer needs to consider the machine tools and level of operator skill and discipline that are available to make the gears (good trade practice in the product area). After choosing an appropriate level of accuracy and reviewing manufacturing resources, the designer needs guidelines on the gear trade. Table 16.38 is a brief guide base on practical experience. The published standards are revised periodically to cover more quality items and to define items more clearly. Also, the mix may change. For instance, the helix accuracy specified for a high-precision gear may be too low to match the profile and spacing accuracy achievable with first-rate grinder. For longrange thinking, it is probably best to think in terms of the six levels described in Table 16.38, then try to find (or establish) a set of accuracy limits that is adequate for the job requirements and practical to meet in the gear shop making the parts.

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TABLE 16.38 Accuracy Levels for Gears Designation

AA Ultra-high accuracy

A High accuracy

B Medium-high accuracy

Description of Level

Highest possible accuracy. Achieved by special tool-room type methods. Used for master gears, unusually critical high-speed gears, or when both highest load and highest reliability are needed. High accuracy, achieved by grinding, shaving with first-rate machine tools, and skilled operators. Used extensively for turbine gearing and aerospace gearing. Sometimes used for critical industrial gears. A relatively high accuracy, achieved by grinding or shaving with emphasis on production rate rather than highest quality. May be achieved by hobbing or shaping with best equipment and favorable conditions. Used in medium-speed industrial gears and the more critical vehicle gears.

Approximate Relation to Trade Standards AGMA 390.03

DIN 3963

14 or 15

2 or 3

12 or 13

4 or 5

10 to 11

6 to 7

C Medium accuracy

A good accuracy, achieved by hobbing or shaping with first-rate machine tools and skilled operators. May be done by highproduction grinding or shaving. Typical use is for vehicle gears and electric motor industrial gearing running at slower speeds.

8 to 9

8 to 9

D Low accuracy

A nominal accuracy for hobbing or shaping. Can be achieved with older machine tools and less skilled operators. Typical use is for low-speed gears that wear into a reasonable fit. (Lower hardness helps to permit wear-in).

6 or 7

10 or 11

E Very low accuracy

An accuracy for gears used at slow speed and light load. Teeth may be cast or molded in small sizes. Typical use is in toys and gadgets. May be used for low-hardness power gears with limited life and reliability needs.

4 or 5

12

Many major companies2 in the U.S.A. have found it necessary to set up their own accuracy limits. In this way they can cover important items not covered in AGMA or DIN standards, and they can adjust the mix of limits to be right for their kind of gear work. Figure 16.49 illustrates the principal geometric quality items that need control in gearing. For each item, a method of specifying the item is shown. (Other methods are in use, but the ones shown are believed to be the most popular—and practical—for gear manufacturer). Table 16.39 shows examples of tolerances for three levels of accuracy and three sizes of gears. This table is based on a survey of key people in the gear trade. It does not agree exactly with any published (or unpublished) accuracy standard. It is hoped that these data will be useful to those studying and setting values for gear accuracy limits. As a last item, cost needs to be considered. The Fellows Corporation has studied this subject rather thoroughly. They emphasize that the achievement of high accuracy involves several important variables, including:

2

These are primarily companies making helicopter gears or high-speed turbine gears for the oil and gas industry.

Gear-Manufacturing Methods

FIGURE 16.49

Definitions of gear-tooth geometry tolerances.

943

Pitch cumulative

2.

Irregularities

5.

Slope

Crown

6.

7.

Helix

Modification

4.

Profile 3. Slope (total)

Pitch variation (t-to-t)

1.

Spacing

Quality Item

12 μm (0.0005 in.) – –

8 μm

(0.0003 in.) 10 μm

(0.0004 in.)

5 μm (0.0002 in.)

4 μm

13 μm (0.0005 in.)

(0.00016 in.)

10 μm (0.0004 in.)

9 μm (0.00035 in.)

7 μm

(0.0009 in.)

(0.0007 in.)

(0.0003 in.)

8 μm (0.0003 in.) 23 μm

5 μm

Medium Gear

(0.0002 in.) 17 μm

Small Gear

A. High Precision

–

(0.0005 in.) –

20 μm

(0.0003 in.)

7 μm

20 μm (0.0008 in.)

(0.0006 in.)

16 μm

(0.0020 in.)

(0.0004 in.) 50 μm

10 μm

Large Gear

(0.0007 in.)

(0.0005 in.) 18 μm

13 μm

(0.00024 in.)

6 μm

20 μm (0.0008 in.)

(0.0005 in.)

13 μm

(0.0012 in.)

(0.0004 in.) 30 μm

10 μm

Small Gear

–

(0.0008 in.) –

20 μm

(0.0003 in.)

8 μm

25 μm (0.0010 in.)

(0.0008 in.)

20 μm

(0.0019 in.)

(0.0005 in.) 48 μm

12 μm

Medium Gear

–

(0.0010 in.) –

25 μm

(0.0004 in.)

10 μm

36 μm (0.0014 in.)

(0.0010 in.)

25 μm

(0.0040 in.)

(0.0008 in.) 100 μm

20 μm

Large Gear

B. Medium-High Precision

TABLE 16.39 Examples of Tolerances for a Range of Gear Sizes and a Range of Quality Levels

(0.0013 in.)

(0.0010 in.) 33 μm

25 μm

(0.0005 in.)

13 μm

36 μm (0.0014 in.)

(0.0010 in.)

25 μm

(0.0020 in.)

(0.0008 in.) 50 μm

20 μm

Small Gear

–

(0.0016 in.) –

40 μm

(0.0008 in.)

20 μm

50 μm (0.0020 in.)

(0.0016 in.)

40 μm

(0.0036 in.)

(0.0010 in.) 90 μm

25 μm

Medium Gear

–

(0.0020 in.) –

50 μm

(0.0012 in.)

30 μm

75 μm (0.0030 in.)

(-.0024 in.)

60 μm

(0.0080 in.)

(0.0014 in.) 200 μm

35 μm

Large Gear

(Continued)

C. Medium Precision

944 Dudley’s Handbook of Practical Gear Design and Manufacture

Irregularities

9.

Composite, total

11.

Waviness

14.

1.8 μm (0.070 μin.)

1.5 μm

1.2 μm (0.028 μin.)

(0.060 μin.)

1.0 μm (0.040 μin.)

0.6 μm (0.024 μin.)

0.5 μm

(0.0008 in.)

(0.0006 in.)

(0.020 μin.)

9 μm (0.00035 in.) 20 μm

7 μm

(0.0003 in.) 15 μm

5 μm (0.0002 in.)

4 μm

12 μm (0.0005 in.)

Medium Gear

(0.00016 in.)

– –

Small Gear

A. High Precision

(0.080 μin.)

2.0 μm

1.6 μm (0.064 μin.)

(0.032 μin.)

0.8 μm

(0.0016 in.)

(0.0006 in.) 40 μm

15 μm

(0.0002 in.)

5 μm

12 μm (0.0005 in.)

Large Gear

(0.100 μin.)

2.5 μm

1.6 μm (0.064 μin.)

(0.032 μin.)

0.8 μm

(0.0012 in.)

(0.0006 in.) 30 μm

15 μm

(0.00024 in.)

6 μm

– –

Small Gear

(0.120 μin.)

3.0 μm

1.8 μm (0.070 μin.)

(0.036 μin.)

0.9 μm

(0.0020 in.)

(0.0008 in.) 50 μm

20 μm

(0.0003 in.)

8 μm

25 μm (0.0010 in.)

Medium Gear

(0.140 μin.)

3.5 μm

2.0 μm (0.080 μin.)

(0.040 μin.)

1.0 μm

(0.0032 in.)

(0.0011 in.) 80 μm

28 μm

(0.0004 in.)

10 μm

35 μm (0.0014 in.)

Large Gear

B. Medium-High Precision

Notes: 1. Small gear is 2.5 module (10 pitch), 50 mm (2 in.) face width, 64 mm (2.5 in.) pitch diameter. 2. Medium gear is 5 module (5 pitch), 125 mm (5 in.) face width, 250 mm (10 in.) pitch diameter. 3. Large gear is 12 module (2 pitch), 500 mm (20 in.) face width, 1250 mm (50 in.) pitch diameter.

Root fillet, AA

13.

Finish 12. Profile, AA

Composite (t-to-t)

10.

Concentricity

End casement

8.

Quality Item

TABLE 16.39 (Continued) Examples of Tolerances for a Range of Gear Sizes and a Range of Quality Levels

(0.200 μin.)

5.0 μm

3.2 μm (0.126 μin.)

(0.064 μin.)

1.6 μm

(0.0024 in.)

(0.0012 in.) 60 μm

30 μm

(0.0004 in.)

10 μm

– –

Small Gear

(0.280 μin.)

7.0 μm

4.0 μm (0.160 μin.)

(0.080 μin.)

2.0 μm

(0.0036 in.)

(0.0018 in.) 90 μm

45 μm

(0.0006 in.)

14 μm

50 μm (0.0020 in.)

Medium Gear

C. Medium Precision

(0.400 μin.)

10 μm

5.0 μm (0.200 μin.)

(0.100 μin.)

2.5 μm

(0.0050 in.)

(0.0024 in.) 130 μm

60 μm

(0.0008 in.)

20 μm

70 μm (0.0028 in.)

Large Gear

Gear-Manufacturing Methods 945

946

• • • • • • • • •

Dudley’s Handbook of Practical Gear Design and Manufacture

Machine operator’s skill Blank accuracy, material, and heat treatment Cutting or grinding tool accuracy Mounting of cutting tool or grinding wheel Work-holding fixture accuracy Accuracy in mounting of work-holding fixture Production method Distortion Inherent capability and condition of machine tool.

Fellows has released several charts and tables that show cost vs. accuracy trends for fine-pitch and medium-pitch gears and the relative cost of different methods of cutting or finishing gear teeth. Figure 16.50 shows the cost trend as involute accuracy is increased and the range of capability of several gear-tooth-making methods. Note that a change from an AGMA 8 quality number (DIN 9) to an AGMA 15 quality number (DIN 3) involves approximately a tenfold increase in cost. Figure 16.51 shows general cost trends for each method of gear-tooth making. Note that cutting (hobbing or shaping) is the lowest cost for AGMA 8 quality number (DIN 9). However, for AGMA 12 quality number (DIN 5), grinding or shaving is less expensive. (AGMA 12 is very difficult to achieve by cutting.)

FIGURE 16.50 Approximate change in cost of making teeth on a 3-module (8-pitch), 150 mm (6 in.) diameter gear for different degrees of involute accuracy. (Courtesy of Fellows Corp., Emhart Machinery Group, Springfield, VT, U.S.A.)

Gear-Manufacturing Methods

947

FIGURE 16.51 The general trend of the cost of making gear teeth for different manufacturing methods. (Courtesy of Fellows Corp., Emhart Machinery Group, Springfield, VT, U.S.A.)

16.4.2 MACHINES

TO

MEASURE GEARS

It is possible to make relatively high-accuracy gears without much special measuring equipment. The procedure is along these lines: • Make the gear teeth on good-quality machine tools • Set up the work accurately and cut or grind with precision cutters or grinding wheels • Check the runout of the finished gear by measuring runout over pins. (A precision cylindrical pin is put into each tooth space, and radial runout is measured with a high-accuracy indicator which reads to 0.0025 mm or 0.0001 in.) • Contact the mating gears and note how the involute profiles fit and how the helices fit across the face width • Observe the tooth finish and feel it with your fingernail The system just described was widely used in the past and it is still in use by those who make a limited quantity of gears for their own needs. It is possible for skilled mechanics with a general understanding of gear quality to handle things so that generally satisfactory results are obtained. In general, those who build gears for sale on the open market need a machine or machines to measure the prime variables of involute profile, helix across the face width, tooth spacing, tooth finish, and tooth action by meshing with a master gear. Gears cannot be put in quality grades without accurate mea­ surements. End easement, crown, and profile modification cannot be controlled by contact checks alone.

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Dudley’s Handbook of Practical Gear Design and Manufacture

(All these items can result in no contact in certain areas. The gap in a no-contact situation can be determined only by measurement. Use of CMM of modern designs is the best choice to do that.) In medium-production gear shops making gears up to 0.6 m (24 in.) in diameter, it is usually handiest to have separate machines to check each major variable—for instance, spacing checkers, involute checkers, helix (lead) checkers, master gear rolling checkers, and surface finish checkers. For large gears up to 2 m, it is common practice to have one or more general-purpose checking machines that will check involute, helix, spacing, and finish all on the same machine. It is desirable to use a checking machine of relatively complete capability or a large gear for these reasons: • The large gear is too heavy for one person to lift. It takes a relatively long time to hoist the gear onto the checking machine and get it in position so that it is turning on an exactly true axis. • Gears over 0.6 m are generally not mass-produced. With lower quantities, the economics of buying separate machines for each kind of check are generally not attractive; it is usually more costeffective to buy one machine with complete capability. Figure 16.52 shows a large gear being checked on a machine with complete capability. The equipment in the foreground does the integration to get cumulative tooth spacing from tooth-to-tooth spacing data. Tooth finish data are derived from local parts of involute and helix checks taken at high magnification.

FIGURE 16.52 Gear-checking machine for large gears. Spacing, involute, helix, and surface finish are all mea­ sured on machine of this type. (Courtesy of Maag Gear Wheel Co., Zurich, Switzerland.)

Gear-Manufacturing Methods

949

When gears are over 2 m (80 in.) in diameter, putting the gear on a checking machine becomes rather impractical. The solution to the problem is to take the checking machine to the gear. This may be done with portable checking machines that either mount on a machine tool or mount on the gear itself. Figure 16.53 shows an example of portable checking equipment temporarily mounted to the bed of a grinding machine. Figure 16.54 shows a portable involute checker mounted on a large marine gear. The checking of spacing, involute, helix, and finish does not necessarily determine whether or not a gear is a good gear. Normally, four teeth, 90 apart, are checked. The involute and spacing checks are taken in the center of the face width. The helix is checked at mid height. If just a few teeth are bad, the trouble may be in between the places checked. This possibility is covered by rolling the production gear with a master gear. If the master is as wide as the gear being checked, every bit of tooth surface will be checked. (If the master gear is narrower, the master-gear check can be taken at more than one location across the face width.) Mater-gear checks are composite checks. They reveal individual tooth action, and they show the total runout of the gear as a total composite reading (runout plus tooth-action effects). The machines used for rolling checks are of two types, double-flanks and single-flank. The double-flank machines force the master gear into tight mesh with the gear being checked. The machine reads center distance variation as the gear revolves. Since both flanks touch at all times, the readings show error effects of both sides of the teeth. This mixes up errors, and it becomes troublesome when gears are built to high accuracy on the drive side but are allowed to have considerably less accuracy on the essentially nonworking coast side. The single-flank rolling machines maintain a constant center distance. A small torque keeps the gear and master in contact on the side being measured. The machine measures the change in rotation of one part from a theoretical uniform rotation of both parts. The schematic design of one model of single-flank tester is shown in Figure 16.55 and 16.56. A single-flank checking machine is shown in Figure 16.57.

FIGURE 16.53 Portable profile-checking head mounted on a grinding machine. (Courtesy of Maag Gear Wheel Co., Zurich, Switzerland.)

950

Dudley’s Handbook of Practical Gear Design and Manufacture

FIGURE 16.54 Portable profile-checking head mounted on the rim of a large gear. (Courtesy of General Electric Co., Lynn, MA, U.S.A.)

The machines and methods just described apply to parallel-axis gears (involute-helical). Gears on nonparallel axes (bevel gears, worm gears, and Spiroid gears) do not have as extensive an array of checking machines, but the logic of using machines to check gears sold on the market is the same. A general review of gear-inspection machines and gear-inspection practice for all types of gears (any gear on any axis) can be found out from other sources.

16.5 GEAR CASTING AND FORMING In this part of the chapter, we shall consider some of the ways of making gear teeth other than by metal cutting or grinding.

16.5.1 CAST

AND

MOLDED GEARS

Although the casting process is most often used to make blanks for gears which will have cut teeth, there are several variations of the casting process that are used to make toothed gears in one operation. Many years ago, when gear-cutting machines were very limited, it was quite common practice to make a wooden pattern of the complete gear—teeth and all— and then cast the gear in a sand mold. A few old-times in the gear trade can still recall the days when a “precision” gear was one with cut teeth and an ordinary gear was one with teeth “as-cast”. In recent times there have been only very limited use of gears with teeth made by sand casting. In some instances, gears for farm machinery, stokers, and some

Gear-Manufacturing Methods

FIGURE 16.55

951

Principle of the Gleason/Goulder single-flank testing machine.

hand-operated devices have used cast teeth. The draft on the pattern and the distortion on cooling make it hard to obtain much accuracy in cast-iron or cast-steel gear teeth. Large quantities of small gear parts are made by die casting. In the die-casting process, a tool-steel mold is used. The cavity is filled with some low-melting material, such as alloys of zinc, aluminum, or copper. With a precision-machined mold and a blank design that is not vulnerable to irregular shrinkage (such as spoked wheel), accuracy comparable with that of commercial cutting can be obtained. Complicated gear shapes which would be quite costly to machine can be made quickly and at low cost by the die-casting process. The main disadvantage of the process is that the low-melting metals do not have enough hardness for high load-carrying capacity. In many applications, though, die-cast gears have sufficient capacity to do the job. Die-cast gears are usually less than 150 mm (6 in.) in diameter and from 2.5 module (10 pitch) to 0.5 module (48 pitch). There is no particular reason for why larger or smaller gears cannot be made, if dies and casting equipment of suitable size are provided.

952

FIGURE 16.56

Dudley’s Handbook of Practical Gear Design and Manufacture

Single-flank error graph.

A process somewhat similar to die casting used in the making of molded-plastic gears. These gears are made in one operation. The raw plastic material is heated in a cylinder to a temperature in the range of 400 to 600 F (depending on composition). It is forced into a steel mold under pressure as high as 140 N/mm2 (20,000 psi). Injection-molding machines now on the market range in size from 30 g to 10 kg per shot capacity. The cycle time in injection molding is very fast. Machines of 225-g capacity may make around 100 cycles per hour. One cycle fills the mold. The mold may contain a cavity for one gear, or a half dozen or more parts may be made by filling the cavities in one mold.

Gear-Manufacturing Methods

953

FIGURE 16.57 A single-flank testing machine that rolls a master with a gear. (Courtesy of Gleason/Goulder, Huddersfield, England, a subsidiary of Gleason Works, U.S.A.)

The accuracy of injection-molded gears ranges from good to fair. Some plastics are less subject to shrinkage than others. Some—like nylon—absorb water or oil and are subject to distortion through expansion. The investment-casting process is another method of casting gears. This process has often been called the”lost-wax” or the “precision” casting process. This process uses a master pattern or die. The master is filled with some low-melting-point metal like a lead-tin-bismuth alloy or some wax or plastic. After these cools, a pattern is formed which is a replica—except for shrinkage allowances—of the pattern to be made. The pattern is used to form a mold called an investment. The investment is made by covering the pattern with a couple layers of refractory material. The kind of refractory used depends mainly upon the temperature at which the investment will be heated when it is filled with the molten metal. The in­ vestment has to be heated to remove the pattern and leave a cavity for pouring. This heating, however, is not nearly so much as the heating which the investment gets when it is filled with the casting material. The investment process has had only limited use in gear making. Its most apparent value lies in the making of accurate gear teeth out of materials which are so hard that teeth cannot readily be produced by machining. The process can be used with a wide variety of steels, bronzes, and aluminum alloys. With machinable materials, the process would still be useful if the gear was integral with some complicated shape that was very difficult to produce by machining.

16.5.2 SINTERED GEARS Small spur gears may be made by sintering. Small helical gears of simple design may also be made by sintering, provided that the helix angle is not over about 15 . Machinery presently available will handle parts from about 5 mm (3/16 in.) diameter up to about 100 mm (4 in.) diameter. The sintering process consists of pouring a metal powder into a mold, compressing the powder into a gear-shaped briquette with a broach-like tool which fits the internal teeth of the mold, stripping the mold with another broach-like tool, and then baking the briquette in an oven. It takes about 415 N/mm2 (60,000 psi) pressure to briquette the larger gears. Presses of 300,000 kg (300 tons) capacity are used for these larger gears. Pitches in the range of 0.8 module (32 pitch) to 4 module (6 pitch) can be readily sintered. Face widths may range from about 2.5 mm (3/32 in.) to 38 mm (1.5 in.). Smaller face widths are difficult to

954

Dudley’s Handbook of Practical Gear Design and Manufacture

strip from the mold. Face widths of over 25 mm (1 in.) may give trouble because of loss briquetting pressure from excessive wall friction. The sintering process can be used to make a complete toothed gear in one operation. Splines, key­ ways, double-D bores, crank arms, and other projections may be made by the same sintering operation that forms the gear teeth. By using different powders in the mold, it is even possible to add a thin bronze clutch face integral with one side of the gear! Sintered gears compare favorably in accuracy with commercial cut gears. The surface finish of sin­ tered gears is usually much better than that of cut gears. The tools that are used to make the briquettes must be lapped to a mirror finish to minimize wall friction. In most cases, a sintered gear goes through the press only once. The pressing time per gear ranges from 2 to about 15 seconds (depending on the gear size and complexity). The sintering is done in conveyorized furnaces. Since the machinery is almost completely automatic, the operators have to do only such things as fill the powder hopper, transfer trays of green briquettes, gauge pieces for size, and remove finished gears. In spite of the fact that iron powder costs from about 30 cents to as much as $4.00 per pound, gears made by sintering are frequently cheaper than gears of comparable strength and quality made by other processes. The tools required sinter gears are very costly but are capable of making a large number of parts. Favorable costs for sintered gears are obtained only when there is enough production to liquidate tool costs adequately. The breakeven quantity may vary from about 20,000 to 50,000 pieces (depending on the complexity off the tools).

16.5.3 COLD-DRAWN GEARS

AND

ROLLED WORM THREADS

Spur pinion teeth may be formed by clod drawing, and worm threads may be rolled. Both these processes are limited in their application. When they can be used, it is possible to get very low costs and a highstrength part. Cold-drawn pinion stock comes in long rods. These rods have teeth formed in them for their entire length. A rod may be as long as 2.5 m (8 ft) with a diameter as small as 6 mm (¼ in.). The pinion rod is chucked in automatic or semiautomatic lathes. Short pinions are rapidly cut off the rod. The pinion may be formed with shaft extensions by turning off the teeth for a distance on each end, or the pinion may be drilled and cut off as a disc with a hole in it to permit mounting on a shaft. Cold-drawn pinions may be made of any material that has good cold-drawing properties. Carbon steel, stainless steel, phosphorous bronze, and many other metals can be used to cold-drawn pinions. The extruding operation of making pinion rod is done hot. Only soft, nonferrous metals like brass, aluminum, and bronze can be used for extruding. Both the cold drawing and the extruding processes use dies to form the pinion teeth. Pinions from about 3 mm (1/8 in.) to about 25 mm (1 in.) pitch diameter can be made by cold drawing or extruding. The number of teeth should not be les than about 15 nor more than about 24. Tooth size may range from 0.25 module (100 pitch) to 1 module (24 pitch). The teeth should be designed with at least 20 pressure angle and enough addendum to avoid undercut. Sharp corners have to be avoided. The tooth contour should have a full-radius root fillet and a full radius at the outside diameter. The time required making pinions by the cold-drawing or extrusion process is essentially the time required to cut off each pinion. This ranges from about 2 to 12 seconds, depending on the shaft ex­ tensions required and the diameter of the part. The time required to make the teeth is the time required to pull the rod stock through the dies. Worm threads may be made by cold rolling. The process is very rapid, and it produces a very smooth, work-hardened surface which is quite similar to that produced in cold-drawn pinions. Raw stock of 180 HB, for instance will have a surface hardness equivalent to about 260 HB after rolling. The surface finish may be in the range of 0.25 μm (0.010 μin.) to 0.75 μm (0.030 μin.). The rolling process exerts great pressure on the worm blank. To get a straight worm, it is necessary to use a tooth depth that is not greater than about one-sixth of the outside diameter. To get a satisfactory

Gear-Manufacturing Methods

955

TABLE 16.40 Typical Production Time to Roll Worm Threads Pitch Diameter

Linear Pitch

Threaded Length

Pieces per Minute

mm

in.

mm

in.

mm

in.

Hand-Fed

Automatic

6.5 9.5

0.250 0.375

1.2 2.0

0.050 0.075

9.5 12.5

0.375 0.500

48 48

100 130

12.5

0.500

2.5

0.100

19.0

0.750

48

150

profile, the worm should have a normal pressure angle of 20 or more. There should be a generous radius in the root fillet and on the thread crest. Lead angles should not exceed 25 . There is some spring-back of metal after rolling, and the die used will have a slight generating action because of its finite diameter. To get a symmetrical worm-thread profile of the desired shape, it is often necessary to make the shape of the die slightly different from the shape of the spaces between worm threads. The time required to roll threads may be estimated from Table 16.40.

REFERENCE [1] Radzevich, S. P. (2017). Gear cutting tools: Science and engineering. (2nd ed.). CRC Press, 606p. ISBN 9781138037069. [First edition: Radzevich, S. P. (2010). Gear cutting tools: Fundamentals of design and computation. CRC Press, 786p.]

17

Design of Tools to Make Gear Teeth Stephen P. Radzevich

The design of gear cutting tools is not necessarily a problem of the gear designer. Usually gearmanufacturing organizations and gear-tool companies will handle the design of tools to make gears. It often happens, though, that the gear designer has to help out on the tool design. Perhaps a special gear is needed which cannot be readily obtained with conventional tools. Perhaps some work on tool design will show that changes should be made in the gear design to facilitate the tooling. In many cases there is the problem of choosing the proper size or type of tool to best fulfill the requirements of the gear design. Still another problem is the case where tools are on hand from previous job and someone has to determine whether or not a new design can use these tools. In this chapter we shall take up some of the more common tool-design problems. Data and calculation methods will be shown so that—if need be—the gear engineer can calculate the dimensions of the cutting tool. The tool-design data shown in this chapter are of necessity limited. Many special problems involved in designing and manufacturing tools are not discussed here. The gear designer should, wherever possible, secure the services of competent tool designers who specialize in gear-cutting tools. The material in this chapter is intended only as an aid to the gear designer, not as a substitute for the services of a tool designer. The interested reader is referred to advanced sources for details (see [1], [2], and others).

17.1 SHAPER CUTTERS A variety of kinds and sizes of shaper cutters are available on the market. The rack-type shaper machines use rack cutters on external work and pinion cutters on internal work. The pinion-type shaper machines use pinion-shaped cutters on both external and internal gears. External spur gears usually use a disc type of pinion shaper cutter. Figure 17.1 shows a typical disctype cutter. For internal gears, it is frequently necessary to use such small cutters that the disc construction cannot be used. The smallest cutters are usually made shank type. Figure 17.1 shows some typical shaper cutters and their nomenclature. Table 17.1 shows the largest number of cutter teeth that can normally be used with different numbers of internal gear teeth. There are no official trade standards on shaper-cutter dimensions. Unofficially, though, all the manufacturers are able and willing to work to essentially the same dimensions and tolerances. Table 17.2 shows the most commonly used dimensions for disc and shank cutters. Shaper cutters frequently have small enough numbers of teeth to cause the base circle to come high on the tooth flank. The region below the base circle may be left as a simple radial flank, or it may be filled in. The filled-in design can be used to break the top corner of the gear tooth being cut. Shaper cutters can be made with a protuberance at the tip. The protuberance cuts an undercut at the root of the gear tooth. This provides a desirable relief for a shaving tool. The protuberance design is also used in some cases to permit the sides of the gear teeth to be ground without having to grind the root fillet. 957

958

Dudley’s Handbook of Practical Gear Design and Manufacture

FIGURE 17.1 Typical types of shaper cutters and their nomenclature. (Courtesy of Fellow Corp., Emhart Machinery Group, Springfield, VT, U.S.A.)

There are a variety of special features that can be provided—and are frequently needed—in shapercutter teeth. Figure 17.2 illustrates six different special features. Figure 17.3 shows some of the design details of a disc shaper cutter. Note that the face of the cutter is shown with a rake angle of 5 . Roughing cutters sometimes have a top rake angle1 as high as 10 . The sides of the cutter have a side clearance of about 2 . The top of the tooth also has a clearance angle. When the pressure angle is 20 , this angle is made about 1.5 times as much as the side clearance angle. The clearance angles and the top face angle are all necessary to make the shaper cutter an efficient metal-cutting tool.

1

Rake angle is sometimes loosely referred to as face angle, or as hook angle.

Design of Tools to Make Gear Teeth

959

TABLE 17.1 Maximum Number of Shaper-Cutter Teeth for Different Internal Gears No. of Internal Teeth

Maximum Number of Teeth in Cutter

14.5 PA , Full Depth

20 PA , Full Depth

20 PA , Stub, 25 PA , Full Depth

16

–

–

–

9

20

–

–

–

13

24 28

– –

– –

10 11

17 21

32

–

10

12

25

36 40

– 14

13 17

14 18

29 33

44

16

21

23

37

48 52

18 21

25 29

27 32

41 45

56

24

34

36

49

60 64

27 30

38 42

40 45

53 57

68

33

46

49

61

72 80

36 44

50 58

53 62

65 73

30 P A , Fillet Root Splines

Note: PA—is pressure angle.

TABLE 17.2 Typical Dimensions for Shaper Cutters Tooth Size

Approximate Pitch Diameter in.

Width mm

Bore

Module

Diametral Pitch

mm

in.

mm

9 to 10

2.5 to 2.75

150 to 175

6 to 7

32

1.25

2.5 to 6.5 2.5 to 6.5

4 to 10 4 to 10

100 100

4 4

22 22

0.875 0.875

2.5 to 5.0

5 to 10

75

3

22

0.875

1.7 to 5.0 1.0 to 1.6

5 to 14 16 to 24

75 75

3 3

17 22

0.6875 0.875

0.5 to 1.4

18 to 48

75

3

14

0.5625

Counterbore in.

mm

in.

70

2.75

105

4.125

45 32

1.75 1.25

65 65

2.5625 2.5625

32

1.25

52

2.0625

32 32

1.25 1.25

52 52

2.0625 2.0625

32

1.25

52

2.0625

Disk type

Shank type Shank diameter, large end mm

in.

2.5 to 3.5 1.6 to 2.3

7 to 10 11 to 16

38 30

1.5 1.1875

17 14

0.6875 0.5625

0.5 to 1.4

18 to 48

25

1.0

11

0.4375

960

Dudley’s Handbook of Practical Gear Design and Manufacture

FIGURE 17.2 Special features that can be provided in shaper-cutter teeth. (Courtesy of Fellows Corp., Emhart Machinery Group, Springfield, VT, U.S.A.)

Shaper cutters for finishing work are usually made to a very high degree of precision. Although there are no AGMA standard tolerances for shaper cutters, the standard tolerances of individual tool companies are generally close to being the same. Table 17.3 shows the tolerances that have been published by Barber Colman for five levels of shaper-cutter quality (Radial runout and profile tolerances, spur and helical). Table 17.4 shows the tolerances that have been published by Barber Colman for five levels of

Design of Tools to Make Gear Teeth

FIGURE 17.3

961

Design details of disc shaper cutter. (Courtesy of Illinois Tool Works, Chicago, IL, U.S.A.)

shaper-cutter quality (adjacent and nonadjacent indexing tolerances, spur and helical). Al values shown are in ten-thousands of an inch. Figure 17.4 shows how a shaper cutter cuts an internal gear. The cutter has external teeth and must be small enough to not destroy the corners of the internal teeth. See Table 17.1 for the maximum number of cutter teeth that can be used. Shaper cutters are normally made external. They can, however, be made internal. Figure 17.5 shows a comparison of an external and an internal shaper cutter. (The internal cutter is often called an enveloping cutter.) When the cutters are small in diameter for the tooth size, they are made integral with a shank. If the face width to be cut is wide, the shank has to be rather long and sturdy. Figure 17.6 shows some shank cutters of rugged design for wider face gears. Helical gears may be cut with shaper cutters provided that the cutter has an appropriate helix angle and the shaping machine has a helical guide of appropriate lead to twist the cutter as it strokes back and forth. Ordinarily, helical gears are cut with shaper cutters, which are sharpened normal to the helix of the cutter tooth. (See Figure 17.7. If the helix is low, the teeth may be sharpened in the transverse section. Herringbone gears of the continuous-tooth type must be cut with a pair of shaper cutters working together. To make the cutting match for both sides, it is necessary to use a cutter with the top face ground normal to the cutter axis. This makes the top face angle 0 , and it makes one side of the cutter tooth have an acute angle and the other side an obtuse angle. These features do not aid the cutting action of the tool, but they are necessary to produce the continuous tooth. Figure 17.8 shows an example of these cutters. Note that a special sharpening technique has produced a good cutting edge even on the obtuse-angle side. Helical shaper cutters must have the same normal pitch and the same normal pressure angle as the gear they are cutting. Since the axis of the gear and that of the cutter are parallel, the cutter transverse pitch and transverse pressure angle are also equal to those of the gear. The relation of the hand of helices is as follows:

1–1.999 2–2.999 3–4.999 5–7.999 8–13.999 14–19.999 1–1.999 2–2.999 3–4.999 5–7.999 8–13.999 14–19.999 1–1.999 2–2.999 3–4.999 5–7.999 8–13.999 14–19.999 1–1.999 2–2.999 3–4.999 5–7.999 8–13.999 14–19.999 1–1.999 2–2.999 3–4.999 5–7.999 8–13.999 14–19.999

Normal Diametral Pitch

4 3 2.5

5 4 3.5

7 6 5

10 8 6.5

13 12 10

3 2.5 2

4 3.5 2.5

6 4.5 3.5

8 6.5 5

11 9 7

4 3 2.5

5 4.5 4

7 6 5

11 8.5 7

15 14 11

3 2.5 2

4 3.5 2.5

6 5 4

8.5 7 5

11 9 7.5

P

R

R

P

2–2.999

0.5–1.999

4.5 4 3.5 3

6 5.5 4.5 4

8.5 7.5 6.5 5.5

12 11 9 7.5

18 16 15 13

R

3.5 3 2.5 2

5 4 3.5 3

7 6 5 4

10 8.5 7 5.5

15 12 10 7.5

P

3–3.999

4.5 4 3.5 3

6 5.5 5 4

9 8 7 5.5

13 11 9 8

19 18 16 14

R

4 3 2.5 2

5 4.5 4 3

7 6 5 4

10 8.5 7 5.5

15 12 10 7.5

P

4–4.999

7 4.5 4 3.5 3

9.5 6.5 6 5 4.5

13 9.5 8.5 7 6

19 13 12 10 8.5

27 20 19 17 14

R

5.5 4 3 2.5 2

7.5 5.5 4.5 4 3

10 7.5 6.5 5 4

15 10 8.5 7 5.5

21 15 12 10 8

P

5–5.999

Pitch Diameter, in.

11 7 5 4.5 3.5

14 10 7 6.5 5.5

19 13 10 8.5 7.5

27 19 14 13 11

38 29 22 20 18

R

8 5.5 4 3 2.5

11 8 5.5 4.5 4

16 10 7.5 6.5 5.5

22 15 11 9 7.5

31 22 15 13 10

P

6–7.999

12 7 5 4.5

15 10 7 6.5

20 14 11 9

29 20 15 13

40 30 24 22

R

8 5.5 4 3

12 8 5.5 5

16 11 7.5 6.5

23 16 11 9

31 23 15 13

P

8–9.999

13 7.5 5.5

16 11 7.5

21 15 11

30 22 16

41 32 25

R

8.5 6 4

12 8 5.5

16 11 8

23 16 11

32 23 16

P

10–11.9

140 87 65 60 58 33 75 43 35 27 17 12 44 32 22 16 13 11 26 17 13 11 8 6 23 15 7 6 5 4

2

91 58 43 40 38 22 50 29 23 18 12 8 29 21 15 11 9 7 17 12 9 8 6 4 16 10 5 4 3 3

3

Sharpening Tolerance Side Clearance

280 170 130 120 116 65 150 87 70 54 35 23 87 64 44 32 26 22 52 35 26 23 17 13 46 29 15 11 10 8

1

Notes: 1. This table was furnished by Barber Colman Co., Rockford, Illinois, U.S.A., as their recommendation for standard shaper cutter tolerances. 2. R is radial run-out, P is profile tolerance (denominator pitch). (All readings are in ten-thousandths of an inch.)

5

4

3

2

1

Quality Number

TABLE 17.3 Shaper-Cutter Table for Sizes and Tolerances (Radial Runout and Profile Tolerances, Spur and Helical)

962 Dudley’s Handbook of Practical Gear Design and Manufacture

Design of Tools to Make Gear Teeth

963

TABLE 17.4 Shaper-Cutter Table for Sizes and Tolerances (Adjacent and Nonadjacent Indexing Tolerances, Spur and Helical) Quality Number

Normal Diametral Pitch

Pitch Diameter, in. 0.5–1.999

N

A 1

2–2.999

A

N

N

A

N

A

N

8–9.999

A

N

10–11.9

A

N

6.5

16

7 6.5

21 17

7.5 6.5

22 17

7.5 7

22 19

5.5

15

5

12

5

12

5.5

13

5.5

14

5.5

14

9.5 9

5 4.5

11 9.5

5 4.5

11 10

5 4.5

12 10

5.5 5

12 11

5.5 5

13 12

5.5

13

14–19.999

4

7.5

4

8

4

9

4

9

4

9

4

9

4.5

11

5 4.5

15 11

5 4.5

16 12

5 4.5

16 13

4

9.5

1–1.999 2–2.999 3.5

8

3.5

8

4

8.5

4

9

4

9.5

5–7.999 8–13.999

3.5 3

7 6

3.5 3

7.5 6

3.5 3

7.5 6.5

3.5 3

7.5 6.5

4 3.5

8 7

4 3.5

8.5 7.5

4

9

14–19.999

2.5

5

2.5

5

3

5

3

6

3

6

3

7.5

3.5 3

11 7.5

3.5 3.5

11 8.5

4 3.5

11 9

3

7

1–1.999 2–2.999 3

6

3

6.5

3

6.5

3

6.5

3

7

5–7.999 8–13.999

3 2.5

5.5 5

3 2.5

5.5 5

3 2.5

5.5 5

3 2.5

6 5

3 2.5

6 5

3 2.5

6 5

3

6.5

14–19.999

2

4

2

4

2

4

2

4

2

4

2.5

6

2.5 2.5

7.5 6

3 2.5

8.5 6

3 2.5

9 6.5

2

4.5

1–1.999 2–2.999 3–4.999

5

A

6–7.999

5 4.5

3–4.999

4

N

5–5.999

5–7.999 8–13.999

3–4.999

3

A

4–4.999

1–1.999 2–2.999 3–4.999

2

3–3.999

2

4

2

4

2

4.5

2

4.5

2

4.5

5–7.999 8–13.999

2 1.5

4 3

2 1.5

4 3

2 1.5

4 3

2 1.5

4 3.5

2 1.5

4 3.5

2 2

4.5 4

2

4.5

14–19.999

1.5

3

1.5

3

1.5

3

1.5

3

1.5

3

2

4.5

2 2

6 4.5

2 2

6.5 4.5

2 2

6.5 4.5

1–1.999 2–2.999 3–4.999

1.5

3

1.5

3

1.5

3

1.5

3.5

1.5

3.5

1.5

3.5

5–7.999 8–13.999

1.5 1

3 2

1.5 1

3 2

1.5 1

3 2.5

1.5 1

3 2.5

1.5 1

3 2.5

1.5 1

3 2.5

1.5

3

1.5

3

14–19.999

1

2

1

2

1

2

1

2

1

2

Notes: 1. This table was furnished by Barber Colman Co., Rockford, Illinois, U.S.A., as their recommendation for standard shaper cutter tolerances. 2. A is adjacent indexing tolerance, N is nonadjacent indexing tolerance (exclusive of runout). (All readings are in ten-thousandths of an inch.)

External gear: RH cutter LH cutter Internal gear: RH cutter RH cutter

for LH gear for RH gear for RH gear for LH gear

964

Dudley’s Handbook of Practical Gear Design and Manufacture

FIGURE 17.4 How a shaper cutter cuts an internal gear. (Courtesy of Fellows Corp., Emhart Machinery Group, Springfield, VT, U.S.A.)

FIGURE 17.5 A comparison of a shaper cutter with external teeth and one with internal teeth. (Courtesy of Fellows Corp., Emhart Machinery Group, Springfield, VT, U.S.A.)

The helix angle that a shaper cutter produces depends on both the lead of the guide and the number of cutter teeth. The helix of the cutter must also agree with the helix angle being cut or serious “cutter rub” will occur. The formula for the relation of cutter teeth number to lead of guide is: no. of teeth in gear no. of teeth in cutter = Lead of guide lead of gear

(17.1)

Shaper cutters do not usually cut the same whole depth throughout their life. A shaper cutter can be designed with a front clearance angle that a certain number of teeth may be cut to an exact depth and thickness for the life of the cutter. However, if this cutter is used to cut gears of substantially larger or smaller numbers of teeth, the whole depth cut will vary slightly from the design value. If a shaper cutter designed to cut an external gear is used to cut an internal gear of the same tooth thickness, the discrepancy in whole depth may be quite ap­ preciable. In many cases, the designer of a shaper cutter does not know all the numbers of teeth that the cutter may have to cut during its life. This leads the designer to make the front clearance angle large enough so that the

Design of Tools to Make Gear Teeth

965

FIGURE 17.6 Shank-type cutters for relatively wide face gears. (Courtesy of Fellows Corp., Emhart Machinery Group, Springfield, VT, U.S.A.)

FIGURE 17.7 Helical-gear cutters for all but low helix angles are sharpened in the normal section. (Courtesy of Fellows Corp., Emhart machinery Group, Springfield, VT, U.S.A.)

966

Dudley’s Handbook of Practical Gear Design and Manufacture

FIGURE 17.8 Diagram of top half of a shaper cutter for continuous herringbone teeth. Note special sharpening method which changes original obtuse angle at the corner of the cutter to an angle that will cut.

cutter usually cuts a little extra on the whole depth. It is usually reasoned that the problem of a little extra depth is less than the problem of having the depth too shallow. In many high-production jobs, it is desirable to design the shaper cutter for a particular gear so that the cutter will do exactly what is wanted throughout its life. This is particularly true when cutters are used to pre-shave-cut a gear and leave an undercut. The position of the undercut must remain constant within close limits to tie in with the shaving-cutter design.

17.2 GEAR HOBS The gear hob is really a cylindrical worm converted into a cutting tool. Cutting edges are formed by “gashing” the worm with a number of slots. These slots are usually either parallel with the hob axis or perpendicular to the worm thread. The teeth of the hob are relieved back of the cutting edge to make an efficient cutting tool. The cutting face may be radial, or it may be given a slight “hook” (rake angle) to improve the cutting action. Usually finishing hobs are radial. Figure 17.9 shows the design details of a typical spur-gear hob. The involute generating portion of the hob-tooth profile is usually made straight in the normal section. Since the thread angle is usually low, this makes the profile come out essentially straight even if the hob is gashed axially instead of normally.

FIGURE 17.9

Typical shell-type hob. (Courtesy of Barber Colman Co., Rockford, IL, U.S.A.)

Design of Tools to Make Gear Teeth

967

Theoretically, the hob should have a slight curvature in its profile to cut a true involute gear. The curvature should correspond to that of an involute helicoid. Practically, though, the curvature required is so slight that it is disregarded. Only on multiple-thread hobs of coarse pitch does it become necessary to grind an involute curve. Hobs may be made with straight bores, tapered bores, or integral with hob arbors. The shell type of hob with a straight bore is the most commonly used type. Taper-bore hobs require more wall thickness than shell hobs. Some companies prefer the taper-bore hob because of the more rigid mounting which the taper provides. Integral-shank hobs are expensive. They are used when the hob diameter has to be made so small that a big enough hole can not be put through the hob. Very small hob diameters may be required when hob runout space or the gap between helices is limited. Worm-gear hobs sometimes have to be very small to match the diameter of small worms (see Figure 17.10). The nominal sizes of shell-type hobs are shown in Tables 17.5 and 17.6. Taper hobs generally require a smaller bore or a larger diameter than the values given in Tables 17.5 and 17.6. Where extra rigidity is required, it is often desirable to put in a larger-diameter hob and support the hob on a larger arbor. For instance, a 10-pitch hob ordinarily has q 1 ¼ -in. bore and a 3-in. outside diameter. In cutting high-speed gears to extreme precision, it is desirable to use a 4-in. hob with 1 ¾-in bore for 10 pitch. Tables 17.5 and 17.6 show that multiple-thread hobs require larger bores and diameters than single-thread hobs. Table 17.7 shows some of the commonly accepted tolerances for different classes of hobs. Although about 20 items may be specified in a hob tolerance sheet, the three shown in Table 17.7 are the most important. Lead variation in one turn and hob-profile error both directly affect the profile accuracy of the gear being cut. Other commonly used hob tolerances only indirectly affect the accuracy of the gear. Helical-gear hobs require a taper when the gear exceeds about 30 helix angle. Even below this angle, a taper is helpful if the gear ahs over 150 teeth. Figure 17.11 shows a typical hob for a helical gear of 250 teeth and 35 helix angle. The design of the gap between helices on a double helical gear and the bob design are tied to each other. The gap must be wide enough to accommodate for both the tapered can full parts of the hob. An exact calculation of gap width is quite difficult, but fortunately an approximate solution is usually close enough. The following formula is usually accurate within plus or minus 5%:

Min . gap =

Min . gap =

ho (do

ho ) cos

h (DH

h) cos

o

1

+

+

z o pn sin

o

cos

n1 pn sin cos

1

+

+

x sin tan

x sin tan

o

metric

(17.2)

1

English

(17.3)

n

n

FIGURE 17.10 Worm-gear hob with multiple threads and integral with shank. (Courtesy of Barber Colman Co., Rockford, IL, U.S.A.)

968

Dudley’s Handbook of Practical Gear Design and Manufacture

TABLE 17.5 Typical Dimensions of Shell Hobs, English Dimensions Diametral Pitch

No. of Threads

Hole, in.

1

1 or 2

2.50

10.75

15.00

5 8

×

5 16

2

1

1.50

5.75

8.00

3 8

×

3 16

2

1.50

6.50

8.00

3 8

×

3 16

1

1.25

4.00

4.00

1 4

×

1 8

2

1.50

5.00

4.00

3 8

×

3 16

3

1.50

5.50

4.00

3 8

×

3 16

1

1.25

3.50

3.50

1 4

×

1 8

2

1.50

4.50

3.50

3 8

×

3 16

3

1.50

5.00

3.50

3 8

×

3 16

1

1.25

3.50

3.50

1 4

×

1 8

2

1.50

4.50

3.50

3 8

×

3 16

3

1.50

5.00

3.50

3 8

×

3 16

1

1.25

3.00

3.00

1 4

×

1 8

2

1.25

3.75

3.00

1 4

×

1 8

3

1.25

4.00

3.00

1 4

×

1 8

1

1.25

3.00

3.00

1 4

×

1 8

2

1.25

3.50

3.00

1 4

×

1 8

3

1.25

3.75

3.00

1 4

×

1 8

12

1

1.25

2.75

2.75

1 4

×

1 8

16

1

1.25

2.50

2.50

1 4

×

1 8

20

1

0.75

1.875

1.875

1 8

×

1 16

32

1

0.75

1.50

1.125

1 8

×

1 16

100

1

0.75

1.375

0.625

4

5

6

8

10

Outside Diameter, in.

Length, in.

where: ho , h – – ht do , DH – –

is is is is

the the the the

depth of cut (finishing cut may be less depth than whole depth). total depth of cut. hob outside diameter (at point doing the cutting). helix angle of gear, degree, ( o = ).

– is the helix angle of gear, degree, ( ,

pn

1

=

– is the lead angle of hob, degrees sin , sin – is the normal circular pitch.

).

=

no . hob trheads × normal circular pitch . × hob pitch dieamter

Keyway, in

–

Design of Tools to Make Gear Teeth

969

TABLE 17.6 Typical Dimensions of Shell Hobs, German Dimensions Module, mm

Hole, mm

Outside Diameter, mm

Length, mm

1.00

22

50

44

1,25

22

50

44

1.50 1.75

22 22

56 56

51 51

2.00

27

63

60

2.25 2.50

27 27

70 70

70 70

2.75

27

70

70

3.00 3.25

32 32

80 80

85 85

3.50

32

80

85

3.75 4.00

32 32

90 90

94 94

4.50

32

90

94

5.00 5.50

32 32

100 100

104 104

6.00

40

110

126

6.50 7.00

40 40

110 110

126 126

8.00

40

125

156

9.00 10.00

40 40

125 140

156 188

11.00

50

160

200

12.00 13.00

50 50

170 180

215 230

14

50

190

245

15 16

60 60

200 210

258 271

18

60

230

293

20

60

250

319

z o , n1– is the number of pitches from hobbing center. – is the whichever is larger: gear addendum, or gear dedendum minus clearance. x n , n – is the normal profile angle, degrees.

All linear dimensions above are in millimeters for Eq. (17.2) or in inches for Eq. (17.3). Figure 17.12 shows schematically how this formula works. The hob must be fed out into the gap between helices until it stops cutting on the helix it was cutting. If it is a roughing cut, it is not necessary to completely stop cutting—the finish cut can take off a little extra at the tooth ends. Frequently large gears are rough-cut to full depth and then finish-cut with a smaller hob and a slightly shallower depth. As shown in the sketch, the generating zone of the hob must just clear the end of the helix. The minimum gap width is determined by the distance needed to keep either the first full tooth or any of the

970

Dudley’s Handbook of Practical Gear Design and Manufacture

TABLE 17.7 Summary of Hob Lead, Profile, and Tooth Thickness Tolerances Item

No. of Class Threads

Tooth Size 1 Module (25 Pitch) μm 10

Lead variation in one axial pitch of the hob

1

Multiple

Profile error in involutegenerating portion of hob tooth (tip relief modifica­ tion excluded)

1

2

3 or 4

Tooth thickness error (minus only)

1 or multiple

4

1.5 Module (16 Pitch)

in. μm 10

4

2.5 Module (10 Pitch)

in. μm 10

4

5 Module (5 Pitch)

in. μm 10

4

8 Module (3 Pitch)

in. μm 10

4

25 Module (1 Pitch)

in. μm 10

4

AA

5

2

5

2

8

3

10

4

20

8

–

–

A B

10 15

4 6

10 15

4 6

13 18

5 7

15 23

6 9

25 43

0 17

64 90

25 35

C

20

8

20

8

23

9

28

11

56

22

115

45

D A

40 10

10 4

46 10

18 4

50 13

20 5

64 15

25 6

100 25

40 10

150 64

60 25

B

18

7

18

7

20

8

25

10

43

17

90

35

C D

25 –

10 –

25 46

10 18

30 50

12 20

38 64

15 25

56 100

22 40

115 150

45 60

AA

4

1.7

4

1.7

4

1.7

4

1.7

5

2

–

–

A B

5 8

2 3

5 8

2 3

5 8

2 3

5 10

2 4

8 13

3 5

25 40

10 16

C

8

3

8

3

8

3

10

4

25

10

64

25

D A

13 5

5 2

15 5

6 2

20 5

8 2

30 8

12 3

75 13

30 5

200 30

80 12

B

8

3

8

3

8

3

13

5

18

7

46

18

C D

8 –

3 –

8 18

3 7

8 20

3 8

13 30

5 12

28 75

11 30

70 200

27 80

A

5

2

5

2

8

3

8

3

13

5

38

15

B C

8 8

3 3

8 8

3 3

10 10

4 4

13 13

5 5

18 28

7 11

50 70

20 27 80

D

–

–

18

7

20

8

30

12

75

30

200

AA

25

10

25

10

25

10

25

10

38

15

–

–

A B

25 25

10 10

25 25

10 10

25 25

10 10

25 25

10 10

38 38

15 15

75 75

30 30

C

38

15

38

15

38

15

38

18

50

20

90

35

D

50

20

50

20

50

20

50

20

75

30

100

40

in.

Notes: 1. 10 4 in. = ten-thousands of inch. 2. Hobs are classified by dimensional tolerances as follows: Classes AA and A—precision ground. Class B—commercial ground. Class C—accurate ground. Class D—commercial un-ground.

tapered teeth from hitting the opposite helix. Usually, the hob is centered 1.5 pitches back form the first full tooth. This makes n1 equal to 1.5 for the first solution. Since the tooth is a full tooth, h equals to ht , and DH is the full outside diameter. [This paragraph and Figure 17.12 are given in English symbols only. The metric equivalents are easily evident in Eqs. (17.2) and (17.3).] After the gap for the first full tooth has been determined, a check should be made to see if any of the tapered teeth require additional gap. As shown in Figure 17.12, the first tapered tooth has a value of

Design of Tools to Make Gear Teeth

FIGURE 17.11

971

Multiple-thread hob. (Courtesy of General Electric Co., Lynn, MA, U.S.A.)

n1 = 2.5, and h = ht a1. A series of calculations can be made for different tapered teeth to explore the cutting action of all the tapered teeth. Sometimes it is desirable to lay out the values calculated, as shown in Figure 17.12. In Eqs. (17.2) and (17.3), the curvature of the gear is not taken into account. An end view will show whether the point at which the hob tends to hit the opposite helix is off-center enough to permit the value of h to be reduced an appreciable amount to compensate for the curvature of the gear. In designing a tapered hob, it is usually desirable to use such an amount of taper that the tapered teeth cut a gap width that is either just equal to that of the first full tooth or slightly greater than that of the first full tooth. This ensures that the taper is really doing some work. If the tapered teeth are so short that they do not require as much gap as the first full tooth, they will still do some cutting on the sides of the gear teeth, but they will not be removing the amount of stock that they should. The end of the hob to be tapered for helical-gear cutting is opposite to that which is sometimes used on large spur gears. For helical gears, the rules are: Conventional hobbing: RH hob tapered on LH end, top coming LH hob tapered on RH end, top coming Climb hobbing: RH hob tapered on RH end, top coming LH hob tapered on LH end, top coming

Worm gears can be hobbed on the same machines used to hob spur and helical gears. Even the same kind of hobs can be used, provided that the work is of the same diameter and tooth profile as the hob.

972

FIGURE 17.12

Dudley’s Handbook of Practical Gear Design and Manufacture

Diagram of hob in gap between helices.

Usually, though, special hobs are required for worm gears. Worm gears which mesh with multiplethreaded worms need tangential feed hobbers to eliminate generating flats on the tooth profiles. The tangential hobber has a slow feed in a directions tangent to the gear been cut. The effect of this feed is to shift the center position of the hob continuously during the cut. This shifting makes the hob teeth move into different positions with respect to the gear. The hob cuts as if it had an almost infinite number of cutting edges. Figure 17.13 shows a worm-gear hob which has just finished cutting a single-enveloping worm-gear. Note the long taper on the hob. At the start of the cut, only the taper teeth engage the work. At the end of the cut, the gear is engaging only the full-depth hob teeth. This kind of hob is often called (for obvious reason) a “pineapple” hob. The end of the hob to be tapered depends on the direction of feed use on the hobber. For most hobbers, the direction is such as to require: RH hob to be tapered on RH end LH hob to be tapered on LH end

There is no handy way of figuring what amount of taper will give the best results on worm-gear hobs. A common practice is to make the length of taper about three times the whole depth and the depth of taper about three-fourths of the whole depth. Pineapple hobs are efficient cutting tools but rather expensive. Where production requirements are not great, a much simpler hob can be used if the tangential feed is slowed down. In fact, the tangential type of

Design of Tools to Make Gear Teeth

FIGURE 17.13

973

Worm-gear hob and worm gear. (Courtesy of General Electric Col, Lynn, MA, U.S.A.)

hobber can generate a complete worm gear with only a single cutting tooth. This scheme has been used quite successfully with cemented-carbide “fly” cutters. With high-speed tool-steel fly cutters, the single cutting edge often gets dull before it finishes a gear. Figure 17.14 shows such a hob. It has five teeth, corresponding to five worm threads (worm starts). This kind of hob has been called a “pancake” hob. Worm-gear hobs—no matter what kind—must have about the same diameter as the worm they imitate. Since the hob gets smaller in diameter as is sharpened, it is necessary to make the new wormgear hob slightly larger than the worm. The amount of oversize to use has not as yet been standardized. Since any oversize produces error in the gear, the amount of oversize is a function of the amount of error that may be tolerated. If no oversize is used, the hob might cut perfectly when new, but it would have to be scrapped after one sharpening unless the hobbing center distance was held constant and the hob was allowed to cut shallower and thicker teeth after each sharpening. The latter expedient is helpful in some jobs where worms and gears may be sized to fit each other, but it is an awkward way of making gears when the job requires all parts to be essentially the same size and quality. The amount of hob oversize boils down to a compromise between accuracy and hob life. The more oversize, the more hob sharpenings possible before the hob reaches its “spent” diameter. Usually, a hob is considered spent when its diameter is less than that of the worm. Undersize damages the accuracy of a hob much more than oversize. The exact calculation of the effects of hob oversize is too complicated problem to be treated in this book. For general applications where a quick yardstick is needed, the following formula re­ presents a good limit for hob oversize. High precision can usually be obtained when the hob oversize does not exceed the value given by the formula. If the formula is exceeded, the accuracy will probably by in the commercial class (good enough for many jobs, but not good enough for critical jobs). The formula is:

Hob oversize = d 0

dp1 = dp1 (0.030

0.028 tan )

15.24 metric px + 7.62

(17.4)

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Dudley’s Handbook of Practical Gear Design and Manufacture

FIGURE 17.14

Pancake worm-gear hob.

Hob oversize = dH

d = d (0.030

0.028 tan )

0.600 px + 0.300

English

(17.5)

where: d 0 , dH – is the hob pitch diameter in mm, (in.). d p1, d – is the worm pitch diameter in mm, (in.). ,

px

– is the worm lead angle in degrees. – is the axial pitch (of worm and hob are equal), mm, (in.).

The cutting edges of the worm-gear hob must have a curvature that will permit them to lie on an imaginary worm surface which has the same profile curvature as the worm. A curvature which corre­ sponds to that which is cut or ground into the worm must be ground into the hob. Since worms used in the U.S.A. usually do not have an involute helicoidal shape, calculations of hob profile must be based on the method used to make the worm (see Section 17.4). The details of designing a hob can best be understood by studying some design problems. Figure 17.15 shows the English design data for a pre-shave spur-gear hob designed for cutting 10diametral-pitch gears and pinions which will operate at pitch-line speeds up to 10,000 fpm and loads over 1000 lb per in. of face width. The hob has a full-radius tip. It cuts an extra deep tooth of 0.240 in. When cutting standard addendum gears, it produces a tear-tooth thickness of 0.156 in. This thickness and the depth of 0.240 in. remain constant throughout the life of the hob. It will be recalled that shaper cutters do not produce constant thickness and depth when they are used over a range of tooth numbers. The generating action of the hob is such that tooth numbers do not affect the thickness of the tooth it cuts. The hob has a protuberance of 0.0014 in. After the gear is shaved down to a design thickness, which is of 0.154 in to 0.153 in., the undercut caused by the protuberance blends smoothly into the contour of the gear tooth. The location of the protuberance is made high enough on the hob tooth so that only a small

Design of Tools to Make Gear Teeth

FIGURE 17.15

Gear hob, pre-shave.

975

976

FIGURE 17.16

Dudley’s Handbook of Practical Gear Design and Manufacture

Built-up hob with carbide blades. (Courtesy of Barber Colman Co., Rockford, IL, U.S.A.)

amount of shaving will be required to clean up the gear tooth down to the end of the active part of the involute. Note that the hob is straight for a distance of 0.095 in. above pitch line. Only the first 0.005 in. of the protuberance generates a possible part of the working involute on the gear. The rest of the protuberance cuts in the root fillet of the gear. The amount of protuberance is controlled mainly by changing the angle on the protuberance. Where 16 is used on this hob, about 18 would be used on a 4-pitch hob with a Hi-point2 of 0.002 in. There have been made several important developments in hobs during recent decades. Three de­ velopments that are quite important to the gear industry are these: • A built-up hob.

Special hob blades are attached to a body made of less costly material than the blade material. • Special roughing hobs. The special roughing hob is a very efficient tool to remove stock, but it does not produce gear teeth intended to run together. • Skiving hobs. This hob cans finish-cut fully hardened gear teeth. The built-up hobs may have blades brazed to an alloy steel body, or the blades may be mechanically attached. The blade material may be expensive high-speed steel, or it may even be a carbide material. Figure 17.16 shows an example of a built-up hob. The larger sizes of built-up hobs may be re-bladed. The built-up hob tends to be less expensive (in large sizes) because the body material is not nearly as costly as the blade material. Delivery time may be reduced because a large tool-steel forging is not needed. The special roughing hobs cut very efficiently. They remove larger chips, and the ships tend to cut and break away in an efficient manner. Figure 17.17 shows a patented K-Kut roughing hob developed by Barber Colman Co. Figure 17.18 shows the patented roughing hob developed by Azumi, called the Dragon hob. Figure 17.19 shows Dragon hob cutting a large gear. 2

Hi-point is the point on the protuberance or a hob that cuts the deepest undercut on the gear (to be shaved or ground).

Design of Tools to Make Gear Teeth

977

FIGURE 17.17 IL, U.S.A.)

K-Kut roughing hob, U.S. Patent No. 3,892,022. (Courtesy of Barber Colman Co., Rockford,

FIGURE 17.18

Dragon roughing hob. (Courtesy of Azumi Mfg. Co., Osaka, Japan).

The K-Kut hob and the Dragon hob were developed and patented at about the same time. One has a U.S. patent and the other a Japanese patent. It is known that Barber Colman Co. and Azumi Mfg. Co. were working independently and did not know details of each other’s design until their special roughing hobs were on the market. Another special patented hob that is very efficient at metal removal is the E-Z Cut hob. This hob will do both roughing and finishing. Figure 17.20 shows this hob and a detail of the design. This hob is intended to be used on very large gears where the cutting time is normally of several days’ duration. The skiving hob uses a special carbide blade and a negative rake angle of 30 . Figure 17.21 shows a skiving hob and details of the special rake angle. Figure 17.22 shows a skiving hob being set in position

978

FIGURE 17.19

Dudley’s Handbook of Practical Gear Design and Manufacture

Dragon hob roughing a large gear. (Courtesy of Azumi Mfg. Co., Osaka, Japan.)

to finish-cut a fully hardened gear that was finish-hobbed before hardening with a protuberance type of hob. (The pre-grinding type of hob can be used as a pre-skive hob.) It is a remarkable achievement, of course, to be able to finish a case-hardened gear by hobbing instead of grinding. In the case where very high accuracy and smooth finish are needed, the hard gear—finished by skiving—may be given a further honing operation or a very light final grinding. When a final grind is used, the skiving serves to remove the bulk of the heat-treat distortion, and to remove it in a quicker and more efficient manner than grinding.

17.3 SPUR-GEAR MILLING CUTTERS Spur gears can be formed by milling a slot at a time and indexing to the next slot. The milling cutter is made so that its contour has an involute curve matching that of the gear tooth. A crossing section through the cutter tooth is the same as that through the space between two gear teeth (see Figure 17.23). Standard involute gear cutters (see Figure 17.24) are designed to cut a range of gear-tooth numbers. Table 17.8 shows the tooth numbers that standard cutters will cut.

Design of Tools to Make Gear Teeth

979

FIGURE 17.20 Detail of E-Z Cut hob design, and the E-Z Cut hob. U.S. Patent No. 3.715,789. (Courtesy of Barber Colman Co., Rockford, IL, U.S.A.)

Spur gears cut by involute cutters of the same pitch are interchangeable. The form of the cutter is deigned to be correct for the lowest number of teeth in the range of the cutter. For instance, a No. 5 cutter has the involute curvature of 21 teeth. If it is used to cut 25 teeth in stead of 21 teeth, the curvature will be somewhat too great. However, this is not too serious. Teeth with too much curvature will run together better than teeth with too little curvature. If close accuracy is desired in form-milled teeth, it is necessary to get a cutter with a curvature which is right for the exact number of teeth. This can be done by purchasing a special single-purpose cutter. Standard gear cutters are usually not ground. The highest accuracy can be obtained by using both a single-purpose cutter and a ground cutter. Some companies build standard gear cutters in half numbers. For example, a No. 3 ½ cutter has a range of 30 to 34 teeth, whereas a No. 4 cutter has a range of 26 to 34. Gear designers can help themselves by choosing numbers of teeth that tie in with the design values for standard cutters. If a 4 to 1 ratio is desired, tooth numbers of 14 and 56 would be a good choice. A No. 7 cutter would be just right for the pinion, and a No. 2 cutter would be within one tooth of being just right for the gear. This would give close accuracy without paying extra for single-purpose cutters.

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Dudley’s Handbook of Practical Gear Design and Manufacture

FIGURE 17.21 Skiving hob and detail of negative rake angle. U.S. Patent No. 3,786,719. (Courtesy of Azumi Mfg. Co., Osaka, Japan.)

Where close accuracy is desired in form-milled gear teeth, it is often necessary to make precise layouts of gear and cutter teeth so that both the tool and the work may be checked. Points on the involute profile may be laid out by rectangular coordinates. The calculation of the cutter profile involves two steps: • The cutter is assumed to be like an internal gear tooth that completely fills the space between two external teeth. A series of arc tooth thicknesses are calculated for a series of assumed diameters. • The arc tooth thicknesses are converted to rectangular coordinates based on the point at which the gear pitch diameter intersects the center line of the cutter. These coordinates are plotted to give the cutter profile. Gear milling cutters range in size from 8.5 in. diameter and 2 in. bore for 1 diametral pitch to 1.75 in. diameter and 0.875 in. bore for 32 pitch. Several kinds of “stocking” cutters are commonly used to

Design of Tools to Make Gear Teeth

981

FIGURE 17.22 Skiving hob being set to finish a large case-hardened gear. Note undercut in gear teeth. Only the sides of gear teeth are skived, not the root fillet. (Courtesy of Azumi Mfg. Co., Osaka, Japan.)

FIGURE 17.23

Milling tool fits space between gear teeth exactly after gear is finished.

982

FIGURE 17.24

Dudley’s Handbook of Practical Gear Design and Manufacture

Gear milling cutter. (Courtesy of General Electric Co., Lynn, MA, U.S.A.)

TABLE 17.8 Standard Involute Gear Cutters No. 1 is used to cut

from 135 teeth to a rack

No. 2 is used to cut No. 3 is used to cut

from 55 teeth to 134 teeth from 35 teeth to 54 teeth

No. 4 is used to cut

from 26 teeth to 34 teeth

No. 5 is used to cut No. 6 is used to cut

from 21 teeth to 25 teeth from 17 teeth to 20 teeth

No. 7 is used to cut

from 14 teeth to 16 teeth

No. 8 is used to cut

from 12 teeth to 13 teeth

rough-cut teeth before finish cutting. Those planning to use gear milling cutters should consult tool vendor’s catalogs for further details on the kinds and types of cutters available.

17.4 WORM MILLING CUTTERS AND GRINDING WHEELS On casual observation, one might assume that a milling cutter would produce on a worm a normal section profile that is of the exact same curvature as the normal section of the cutter. However, there is a slight generating action between the cutter and the work. This is caused by the fact that the thread angle of the worm varies from the top to the bottom of the thread. The inclination of the cutter is set to correspond to the thread angle at the pitch line of the worm. This means that at the top or bottom of the thread, the cutter will not be tangent to the worm surface in the plane containing the cutter axis (normal

Design of Tools to Make Gear Teeth

983

section). Since the tangency points do not come in the plane of the normal section, more metal will be removed from the top and bottom of the worm thread than that corresponding to the cutter normal section profile. A straight-sided conical cutter will produce a worm thread with a convex curvature in either the normal or the axial sections of the worm. If a straight-sided worm profile is required, the cutter must be formed to a convex curvature which is conjugate with the straight worm. The amount of curvature produced by the generating action of a milling cutter is rather slight compared with that produced by a hob or shaper cutter. With low thread angles and fine pitches, the curvature is often almost negligible. With coarse pitches and high thread angles, the amount becomes quite significant. For example, a five-thread worm of 1.250 in. axial pitch and 30 thread angle has about 0.006 in. curvature when cut with a straight-sided cutter. A five-thread worm of 0.625-in. axial pitch has only about 0.001 in. curvature when the thread angle is 15 . Several kinds of worms have been in common use. When worm threads were made on a lathe, it was quite handy to make the worm profile either straight in the axial section, or straight in the normal section. This practice carried over into milled worms, with the result that many designs in current production call for a straight-sided worm. This practice puts a burden on the makers of a milling cutter (or the one who dresses a thread-grinding wheel). They must develop the required curvature in the cutting tool. Most gear engineers in the U.S.A. prefer to use a straight-sided milling cutter or grinding wheel. This puts all the curvature into the worm. This practice makes the job of producing a worm-gear hob somewhat more difficult, since it must have a curvature on the cutting edge corresponding to that of the worm thread. It is reasoned, though, that hob makes are better able to take care of this problem than are makers of milling cutters. Worm-gear hobs have helical gashes on most cases. The combination of gash angle and relief on the sides of the hob tooth would make it necessary to curve the cutting edge of the hob tooth even if the hob had to match a straight-sided worm. Since hob makers cannot escape curvature problems, it is perhaps reasonable to give them the whole problem. In England and some other countries, worms of involute helicoidal shape are popular. These worms have an involute curve in a transverse section. This design is not handy to work with unless involute generating equipment is available to make worm threads. Such equipment is not generally available in the U.S.A. Worm-gear designs frequently need to calculate the curvature produced by milling a worm. If a straight-sided cutter is used, the worm thread profile must be calculated before precise data can be put on the worm drawing to check the thread profile. Also, these data are needed to check the worm-gear hob. For a sample problem, we shall take a five-threaded worm with 25.00 mm axial pitch. We shall adjust the pitch diameter to give a 25 lead angle. This makes 5 per thread and is about as high a lead angle as should be used with five threads. Using Eqs. (5.69) through (5.73), the pitch diameter works out to 85.327 mm. Using the proportions of Table 6.29, we get an addendum of 7.21 mm. Cutter pressure angle is 25 . Normal circular pitch is 22.6577 mm. The worm thread thickness will be 10.88 mm. The worm thread cutter will have an outside diameter of 150 mm. Subtracting twice the worm de­ dendum (11.65 mm), we get a cutter pitch diameter of 132.70 mm. The cutter thickness at the pitch line will be equal to the normal circular pitch minus the worm thread thickness. This comes out to 11.7777 mm. The first step in the calculation will be to determine the angle of rotation of the cutter ( 2 ) at which various points on the cutter cut the deepest into the worm profile (these are points of tangency with the worm profile). Table 17.9 shows this calculation. Values of h may be assumed to give as many points as desired. The five points shown are picked to be one addendum above and below pitch line, at pitch line, and two intermediate points. After 2 has been determined, a second calculation sheet is worked through to get the axial section of the worm. This calculation literally determines the position of a point on the cutter with respect to the axial section of the worm. If angles of 2 other than the critical value were used, it would be possible to plot the entire cutting curve of a point on the cutter. In fact, a solution can be obtained by assuming a series of 2 values for each line (30) value from Table 17.9 and then plotting individual curves to see

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Dudley’s Handbook of Practical Gear Design and Manufacture

TABLE 17.9 Calculation of Angle of Rotation for a Straight-Sided Cutter Cutting a Worm Given Data 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.

Number of worm threads Pitch diameter of worm Pitch diameter of cutter Axial pitch Shaft angle* Cutter pressure angle Cutter tooth thickness Distance a (assume) [(2) + (3)] × 0.50 Lead = (1) × (4) (10) ÷ 2π sin (5) cos (5) tan (6) –1 ÷ (14) (15) × (12) (15) × (13) (9) × (13) (9) × (16) (11) × (12) (11) × (17) (18) + (20) (22) × (22) (19) – (21) (24) × (24) (22) × (24) (7) ÷ 2 (3) ÷ 2 (8) × (14) (27) – (29) r2 = (28) + (8) (30) × (12) (31) × (16) (32) + (33) (34) × (34) (35) – (25) (23) + (35) (36) ÷ (37) (26) ÷ (37) (39) × (39) (38) + (40) (41) (39) + (42) Rotation angle = arcsin (43)

5 85.327 mm 132.7 mm 25.00 mm 25 25 11.78 mm Calculation Steps 7 3.5 109.0135 125.00 19.89437 0.422618 0.906308 0.466308 –2.14451 –0.906308 –1.94358 98.79978 –98.7998 8.407723 –38.6664 107.2075 11493.45 –60.1334 3616.029 –6446.75 5.89 66.35 3.264154 1.632077 2.625846 4.257923 73.35 69.85 1.109731 1.799476 –66.4777 –63.3056 –65.3679 –61.5061 4272.968 3783.003 656.9398 166.9747 15766.42 15276.45 0.041667 0.010930 –0.408892 –0.422006 0.167192 0.178089 0.208859 0.189019 0.457011 0.434763

0

–3.5

–7

0 5.89 66.35 2.489222 –60.1335 –57.6443 3322.865 –293.163 14816.31 –0.019787 –0.453112 0.189322 0.169536 0.411747

–1.63208 7.522077 62.85 3.178967 –56.9614 –53.7825 2892.555 –723.474 14386.00 –0.050290 –0.448127 0.200818 0.150528 0.387979

–3.26415 9.154154 59.35 3.868712 –53.7894 –49.9207 2492.072 –1123.96 13985.52 –0.080366 –0.460959 0.212483 0.132118 0.363480

0.048120 2.758124

–0.023365 –1.33881

–0.060148 –3.44831

–0.097479 –5.59404

0.012758 0.730968

* The shaft angle is usually set to be the same as the worm lead angle.

Design of Tools to Make Gear Teeth

985

which value of 2 does the deepest cutting on the worm. In some cases, it may not be possible to get a solution for 2 directly from the first calculation sheet. If this happens, a solution can still be obtained by plotting curves for each line (30) value from Table 17.9 and reading values at the point of deepest cutting. Table 17.10 shows the calculation of the worm axial section. Items (6), (7), and (8) for this calculation are taken from Table 17.9. Items (26) and (32) are the answers. They are rectangular coordinates of

TABLE 17.10 Calculation of Axial Section of Worm Cut with a Straight-Sided Cutter Given Data 1.

Number of worm threads

2.

Pitch diameter of worm

85.327 mm

5

3. 4.

Pitch diameter of cutter Axial pitch

132.7 mm 25.00 mm

5.

Shaft angle

25 Calculation Steps

6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26.

r 2 (line 31, Table 17.9) (line 30, Table 17.9) Rotation (line 44, Table 17.9) [(2) + (3)] × 0.50 Lead = (1) × (4) (10) ÷ 2π sin (5) cos (5) sin (8) cos (8) (6) × (14) (6) × (15) (12) × (16) (12) × (7) (13) × (16) (13) × (7) (9) – (17) (20) + (19) (21) – (18) [(22) × (22)] + [(23) × (23)] (25)

73.35 2.625846

69.85 4.257923

66.35 5.89

62.85 7.522077

59.35 9.154154

2.758124

0.730968

–1.33881

–3.44831

–5.59404

109.0135 125.00 19.89437 0.422618 0.906308 0.048120

0.012758

–0.023365

–0.060148

–0.097479

0.998842 3.529584

0.999919 0.891108

0.999727 –1.55024

0.998190 –3.78030

0.995238 –5.78540

73.26503

69.84432

66.33189

62.73621

59.06735

1.491667 1.109731

0.376599 1.799476

–0.655160 2.489222

–1.59763 3.178967

–2.44501 3.868712

3.198890

0.807618

–1.40499

–3.42612

–5.24335

2.379825 35.74847

3.858989 39.16918

5.338153 42.68161

6.817317 46.27729

8.296481 49.94615

4.308620

2.607095

1.084227

–0.247151

–1.37464

0.888158 1296.517

3.482390 1541.022

5.993313 1822.896

8.414942 2141.649

10.74149 2496.508

36.00719

39.25585

42.69538

46.27795

49.96506

27.

(23) ÷ (26)

0.119660

0.066413

0.025395

–0.005341

–0.027512

28. 29.

arcsin (27) (28) × π ÷ 180

6.872482 0.119947

3.807981 0.066462

1.455153 0.025397

–0.305994 –0.005341

–1.57652 –0.027515

30.

(29) × (11)

2.386278

1.322216

0.505261

–0.106248

–0.547402

31. 32.

(30) + (24) (31) – Z p *

3.274437 –3.22414

4.804606 –1.69397

6.498574 0

8.308694 1.810120

10.19409 3.695519

* Z p is the value from line 31 in the column in which r2 equals half of the cutter pitch diameter. In this case, it is the idle column, and Z p is 6.498574.

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Dudley’s Handbook of Practical Gear Design and Manufacture

points on the axial section profile. Item (26) is a radial dimension, while (32) is an axial dimension—with 0 at the approximate pitch diameter of the worm. The approximate pitch diameter is used because the point used in the beginning for the pitch point of the cutter does not reach in quite deep enough to give a point on the pitch line of the worm in this calculation. In general, it is best to check hobs and worms in the normal section. This permits the indicator to read normal to the surface being checked instead of at an angle. Table 17.11 shows the calculation of the normal section from the coordinates just obtained for the worm axial section. Items 19 and 23 are rectangular coordinates of the worm normal section.

TABLE 17.11 Calculation of Normal-Section Deviation from Known Axial Section Calculation Steps 1.

(line 32, Table 17.10)

–3.22414

2. 3.

Worm pitch diameter × π Lead (line 10, Table 17.10)

268.0627 125.00

–1.69397

0

1.810120

3.695519

4.

(3) ÷ (2)

0.466309

5. 6.

Lead angle = arctan (4) (line 26, Table 17.10)

25.00006 36.00719

39.25585

42.69538

46.27795

49.96506

7.

(6) ÷ (4)

77.21746

84.18423

91.56031

99.24313

107.1502

8. 9.

(3) ÷ 2 π (7) + (8)

19.89437 97.11183

104.0.786

111.4547

119.1375

127.0445

10.

(1) ÷ (9)

–0.033200

–0.016276

0

0.015194

0.029088

11. 12.

arcsin (10), degrees (11) × π ÷ 180

–1.90258 –0.033206

–0.932579 –0.016277

0 0

0.870559 0.015194

1.666876 0.029092

13.

(8) × (12)

–0.660620

–0.323812

0

0.302277

0.578776

14. 15.

(7) × (10) (13) + (14)

–2.56364 –3.22426

–1.37017 –1.69398

0 0

1.507854 1.810132

3.116824 3.695600

Note: If (150 does not equal (1), this approximation method is not good enough, and a trail-error procedure must be used. Assume a new value for (10) and repeat steps (11) to (15), until (15) equals (1). 16.

cos (11)

0.999449

0.999868

1.00

0.999885

17.

(6) × (16)

35.98734

39.25065

42.69538

46.27261

49.94392

18. 19.

(6) × (10) sin (5)

–1.19545 0.422619

–0.638923

0

0.703126

1.453403

20.

–(18) ÷ (19)

2.828664

1.511816

0

–1.66373

–3.43904

21.

24.40341

22.

Normal pressure angle [from Eq. (6.62), (6.63)] tan (21)

23.

(2) ÷ 2π

42.66350

24. 25.

(17) – (23) (24) × (22)

–6.67617 –3.02892

–3.41285 –1.54838

0.031882 0.014464

3.609110 1.637424

7.280421 3.303069

26.

(25) + (20)

–0.200258

–0.036565

0.014464

–0.026310

–0.135967

27. 28.

cos (21) Deviation = (26) × (27)

0.910659 –0.182367

–0.033298

0.013172

–0.023959

–0.123820

29.

Distance A = (24) ÷ (27)

–7.33114

–3.74767

0.035010

3.963185

7.994673

Note: For derivation curve, plot (28) against (29).

0.999577

0.453692

Design of Tools to Make Gear Teeth

987

Machines customarily used to check the profiles of worms are set at an inclination corresponding to the normal pressure angle. Since the curvature of the profile is not great, it is handy to measure the profile deviations from a straight line. This makes it desirable to continue the calculation from a straight line. Items 30 and 31 show the deviations from a straight line set at a normal pressure angle of 24.4072 . This angle was calculated from the cutter pressure angle using Eqs. (5.63) and (5.64). Figure 17.25 shows a plot of the normal section deviations. The solid curve is a plot of items 31 and 32. It will be noted that this curve does not read 0 at the point where A is 0. This is caused by the fact that the axial section was based on an approximate pitch diameter (see above). The dashed-line curve has been moved over by 0.013 mm to correct for this approximation. The dashed-line curve is the one that should be used for checking. The other curve could be used, but it would confuse the inspectors to have the checking line intersect the surface instead of passing tangent to the surface. Figure 17.25 shows that the normal section profile has a curvature of about 0.1 mm. Although this is not much distance, it is a significant amount when it comes to meshing of precision gear teeth. To get a worm that will operate smoothly under high speed and load, it will be necessary to hob its mating gear with a hob which has the same normal section curvature as the worm. The calculation just given is a method of getting both data to check the finished worm and data to check the hob used to finish the worm gear.

FIGURE 17.25

Curvature of worm normal section.

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Dudley’s Handbook of Practical Gear Design and Manufacture

17.5 GEAR-SHAVING CUTTERS When gears are made by the shaving process, the gear engineer may have frequent need to study the shaving tool. Essentially this tool represents a gear which has a conjugate tooth action with the part being shaved. The surfaces of the shaving tool are serrated with small rectangular grooves. The cutter is designed purposely to run on a shaft which lacks a few degrees of being parallel with that of the work. The out-of-parallel condition causes the cutter teeth to have a sliding motion across the gear teeth even at the pitch line. It is this sliding motion, together with the serrations and the fact that the tool and work are meshed together at a tight center distance, that causes the tool to “shave” off tiny slivers of material. (See Figure 17.26 for shaving-cutter examples and Figure 17.27 for how a shaving cutter works.) Since the shaving cutter is mounted on a shaft that is not parallel to the gear axis, the teeth of the shaving cutter and the teeth of the work run together like a pair of crossed-helical gears. The choice of shaft angle governs the cutting action of the cutter. In general, the higher the shaft angle, the faster the cutter cuts. The best control over helix angle, though, is gained with a low shaft angle. Since the choice of shaft angle is largely a matter of judgment in weighing different variables, there are no fixed rules. The general practice is outlined in Table 17.12. At the present time there are no general trade standards for shaving-cutter diameters, bore sizes, or face widths. Each manufacturer develops a practice which may agree in some respects and disagree in others with those of other cutter manufacturers. Light-duty shaving machines for fine-pitch gears up to 100 mm (4 in.) diameter use cutters from about 50 to 75 mm (2 to 3 in.) diameter. Medium-duty shaving machines for gears up to 450 mm (18 in.) use 150- to 280-mm (6- to 11-in.) cutters. Heavy-duty ma­ chines for gears up to 1.2 m (48 in.) or more use 175- to 300-mm (7- to 12-in.) cutters. Many shavingcutter users and manufacturers prefer a 200- to 225-mm (8-to 9-in.) cutter whenever the design permits. In general, it is desirable to have a “hunting” ratio between the number of cutter teeth and the number of gear teeth. This tends to make it desirable to design shaving cutters with prime numbers of teeth. A prime number will hunt with all numbers of teeth except itself. Thus, numbers of teeth like 37, 41, 43, 47, 53, 59, 61, 67, 73, etc., are popular for shaving cutters.

FIGURE 17.26 Shaving cutters of different pitches. (Courtesy of national Broach and Machine Co., a division of Lear Siegler, Inc., Detroit, MI, U.S.A.)

Design of Tools to Make Gear Teeth

FIGURE 17.27

989

Shaving-cutter action. (Courtesy of Ex-Cell-O Corp., Tool Products Div., Detroit, MI, U.S.A.)

TABLE 17.12 General Practice for Setting Value of the Shaft Angle Application

Shaft Angle, Degrees

Spur pinions, under 20 teeth

8–12

Spur pinions, 20 to 35 teeth Spur gears

10–15 10–15

Helical pinions, narrow face

8–12

Helical gears, narrow face Helical pinions, wide face

10–15 5–10

Helical gears, wide face

10–15

Internal gears

4–8

The normal circular pitch of a shaving cutter must be equal the normal circular pitch of the work. Likewise, the normal pressure angle of the cutter must equal that of the work. Ordinarily the helix angle of the cutter is opposite in hand to that of the work. When the hands of helix are opposite and the gear is external, the shaft angle is the difference between the helix angles. With spur gears or low-helix-angle

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gears, the cutter helix might be designed with either hand for a given hand on the workpiece. If the hands of the helices are the same and the gear is external, the shaft angle is the sum of the two helix angles. The conditions just mentioned usually make it impossible to have the pitch diameter of the cutter come out to some even figure like 200.0 mm or 8.000 in. The designer usually just chooses teeth and shaft angles so that the cutter diameter is very close to the desired diameter. Shaving cutters are sharpened on their involute surfaces. This makes the cutter change quite appre­ ciably in outside diameter during its life. Like shaper cutters, shaving cutters do not produce the same thickness on the work throughout their life. If the thickness of the work is held constant, the depth that the shaving cutter is fed into the work will vary. The tendency of the shaving cutter to cut a deeper depth as the teeth are made thinner in sharpening may be controlled by reducing the cutter outside diameter a proper amount each time the cutter is sharpened. To function properly, the shaving cutter should finish the involute at least as deep as the form diameter. Preferably, the cutter should finish the involute close to the diameter that corresponds to the deepest point of undercut left by the pre-shave tool. Perfect blending of root fillet with shaved profile can be obtained only when the shaving action stops in the middle of the undercut. As the shaving-cutter tip rolls out of mesh, it follows a path which takes it closer to the root diameter of the gear. This movement may cause trouble. The cutter tip may foul either the root diameter or the root fillet of the gear. It is necessary to size the cutter outside diameter so that it will clear the root fillet even when it is at its deepest point of penetration with the gear. This means that the working depth of the cutter must be always a certain amount less than the whole depth of the gear. The calculations necessary to design a shaving cutter are fairly complicated. They will not be given here.

17.6 PUNCHING TOOLS The tool used to punch gears usually has three working parts: the punch, the die, and the knockout piece. Sheet metal is placed between the die and the punch. The punch has the shape of the gear to be made. For punching an external gear, an external-toothed punch is used. The die has the shape of the stock that is left after the punching is removed from the sheet. An internally toothed die is used for making an external gear. In punching a gear, the first operation is to feed the sheet metal into the tool. The punch is driven into the die by the ram of the punching machine. This cuts out the gear and leaves it stuck in the die. The punching is removed from the die by a movement of the knockout tool. This tool has a shape very similar to that of the punch. The punching-tool assembly may be so built that spring action both retracts the punching tool and actuates the knockout piece. In this case, the punching machine has to furnish power only for the punching stroke (during which the springs are compressed). Only thin gears can be punched. The thickness of the sheet stock should not exceed the gear-tooth thickness at the pitch line. As a rule of thumb, the stock should not be thicker than 1.5 mt , (1.5/ Pd ). This means that a 1-module (25-pitch) gear could be made up to about 1.5 mm (0.06 in.) face width. The punching operation upsets the metal so that the tooth corners are rounded some on the die end. The face of the tooth is formed by a shearing operation. This shearing is not exactly at 90 to the axis of the gear. This means that a pair of new punched gears cannot be expected to have uniform contact across even their narrow face width. The accuracy of punched gears may be improved by secondary operations. A shaving type of punching operation may be used to take a small amount of metal off the gear-tooth faces. A coining operation may also be used to improve accuracy. In coining, the punching is rammed into the bottom of a die with so much pressure that the metal flows to conform completely to the shape of the cavity. It is also conceivable that conventional crossed-axis shaving might be used on punched parts. The fact that punched gears are essentially thin wafers pressed on a small shaft is the greatest drawback to shaving. The reactions of the shaving tool would tend to “cock” the gear on its shaft.

Design of Tools to Make Gear Teeth

FIGURE 17.28

991

Gear punching tool.

The power required to punch gears depends primarily on the cross-sectional area to be sheared. As a rule of thumb, the tonnage capacity of the punching machine should equal at least twice the force calculated as the product of the shear area times the ultimate shear strength. For instance, a 1-in. gear of 1/32-in. thickness would have a shear area at the teeth of about 2.5 × n × 1 in. × 1/32 in., or 0.245 in.2. If the material was steel with 60,000-psi shear strength, a press of about 15 tons capacity would be needed to cut the teeth. Additional power would be required if the gear had a large bore to be cut. All sizes of punching machines up to as large as 2000 tons are on the market. Figure 17.28 shows an example of a gear punching tool.

17.7 SINTERING TOOLS Before gears are sintered, the metal powder is compressed into a gear-shaped briquette. A complicated and expensive set of tools is required for the briquetting operation. After the briquette is made, it is sintered in an oven. The sintering consists of heating the briquette to a temperature almost up to the melting point. Carefully controlled atmosphere furnaces are used for this operation. Sintered gears are rather porous. This is an advantage from the lubrication standpoint, but a detriment to the strength of the part. Higher-strength gears can be obtained by filling most of the voids in the part by metal infiltration. This is done by placing a slug of lower-melting metal (such as copper) on top of the briquette when it is placed in the oven. Upon heating, the metal slug melts and soaks through the briquette. After heating and cooling, the briquette becomes an impregnated gear. In making sintered gears, the special tools are used in the briquetting part of the process rather than in the actual sintering. Figure 17.29 shows in schematic fashion the three tools used to make a briquette. The metal powder is poured into the cavity of a die barrel. The floor of the die barrel is the stripper. The power is pressed against the stripper by a ram. The die barrel has internal teeth, while the ram and the stripper have external teeth. After the powder is compressed, the briquette is ejected by withdrawing the ram, then pushing the briquette out with an upward stroke of the stripper. The die barrel is made by broaching internal teeth into a special die steel. After broaching, the die is hardened by furnace heat treatment. The die steel used must be one which will have very little di­ mensional change upon hardening. Air-hardening types of steel are often used for the die barrel. The ram

992

FIGURE 17.29

Dudley’s Handbook of Practical Gear Design and Manufacture

Sintering tools.

and the stripper have ground external teeth. Both these pieces are ground slightly larger than the expected size of the die barrel after heat treatment. The die barrel is lapped with a series of about three external toothed laps. This removes slight errors, polishes the die teeth to a mirror-like finish, and enlarges the die so that the ram and stripper will fit with an almost perfect size-to-size fit.

FIGURE 17.30

Surfaces of ram and die are ground true and fit with almost no clearance.

Design of Tools to Make Gear Teeth

993

The teeth of the ram and stripper are designed so that all surfaces can be ground and measured with great precision. The root fillets are often made as true arcs of circles. This facilitates measurement of root size. Also, it is easy to dress a true circular arc on the tip of a form-grinding wheel. The sides of the teeth are involute curves. No corner radii or chamfers are used at the tips of the teeth. Figure 17.30 shows the way the teeth of the ram and the die fit with each other. To keep the powder from leaking, it is necessary that the tooth surfaces fit so well that no clearance of even as little as 0.01 mm (0.0005 in.) exists anywhere between the contacting tooth surfaces. Great skill and care are required to build a set of sintering tools that will fit on all surfaces with this kind of precision.

REFERENCES [1] Radzevich, S. P., Gear cutting tools: Science and engineering (2nd ed.). CRC Press, 2017, 606p. [First edition: Radzevich, S. P., Gear cutting tools: Fundamentals of design and computation, 2010, 754p.] [2] Radzevich, S. P., Theory of gearing: Kinematics, geometry, and synthesis (2nd ed.). revised and expanded. CRC Press, 2018, 898p. [First edition: Radzevich, S. P., Theory of gearing: Kinematics, geometry, and synthesis, CRC Press, 2012, 743p.]

18

Dynamic Model of Technological System for Gear Finishing Michael Storchak

18.1 INTRODUCTION The processes of gear finishing with toothed rolling tools both with a rigid connection between the tool and the machined gear, and with free rolling are characterized by increased vibration activity. This significantly reduces tool life, productivity, and quality of machining, and, as a consequence, increases the cost of manufacturing gears. A significant increase in the efficiency of the gear finishing process with toothed rolling tools is provided, as shown in previous studies [13], due to the choice of the machine engagement geometry characteristics and geometric contact in a pair of tool-machined gear, as well as the dynamic state of the technological system. The developed models of the machine engagement geometry [11] and the teeth profile shaping [12] for the finishing processes of hardened gears provide for the interaction of idealized objects with absolutely rigid and precise surfaces and uniform relative motion. The interaction of the tool and the machined gear, and, con­ sequently, the result of machining, is significantly influenced in the real process by the dynamic state of the technological system. The dynamic state is determined by vibrations in the movement of all system links due to the uneven movement of the drives, periodic changes in the rigidity of the machine kinematic chain links coupling, the cutting process, errors in the manufacture and installation of links, and other factors. Vibrations of the tool and the machined gear lead to such undesirable phenomena as teeth impacts, contact separation, and change in relative motion, as a result of which the accuracy of the manufactured product, machining productivity, and tool life are significantly reduced. As a result, the required design and service properties of the machined gears are not provided. It is obvious that stabilization of the tool and the gear relative movement will significantly improve the technical performance of the finishing process. At the same time, knowledge of the interaction of real laws of a machine tool pair will allow synthesizing new machining methods and tools for their implementation. Naturally, an accurate description of the real interaction of the tool with the ma­ chined gear during its machining is extremely difficult due to the complexity of the physical phenomena occurring in contact, a significant number of technological system links, and many other factors that de­ termine the nature of the machine pair links movement. Therefore, it is necessary to postulate a number of assumptions on the basis of which the possibility of a certain idealization of the tool and the machined gear dynamic interaction will be provided. Moreover, such assumptions will not violate the qualitative nature of this interaction. Such idealization will make it possible to develop a model for the dynamic interaction of the technological system links. The using of this dynamic model will provide taking into account the influence of the real tool and machined gear movement on the choice of design parameters of the optimization system, which is part of the information system of the gear finishing process [11].

18.2 DEVELOPMENT OF A GENERALIZED DYNAMIC MODEL A dynamic model of a technological system describes the relationship of an equivalent elastic system with work processes [7]. As applied to gear finishing, it is most expedient to present the model as a 995

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Dudley’s Handbook of Practical Gear Design and Manufacture

closed single-loop dynamic system, including an equivalent elastic system and the cutting process, as one of the most important work processes. The main elements of the technological system for gear finishing are: drives of the tool and the machined gear, the kinematic chain for the main movement im­ plementation, the feeds movement and setting movements, the chain for the implementation of the tool and machined gear, a device for claiming the machined gear on the machine, as well as the tool and machined gear itself. In the general case, such a system is a system with distributed parameters and an infinite number of degrees of freedom [15]. The development of such a model is extremely difficult, and the results of its functioning are difficult to visualize. To implement the solution of such models, they are idealized by linearization based on the following postulated statements: • the system is holonomic, and the displacements of its links are virtual; • the system is a system with lumped parameters and a finite number of degrees of freedom. In this case, the masses of the links are concentrated in the places of their attachment. The shafts are massless, and their mass is reduced to the mass of the corresponding links. Elastic and dissipative elements of the system are weightless and have linear characteristics; • all links, except for the tool and machined gear, have only one degree of freedom in the torsion direction; • the tool and the machined gear have three degrees of freedom: in torsional, axial, and radial directions along the engagement line; • the engagement rigidity in the machine pair depends on the engagement phase and changes in proportion to the total contact length of the tool and the machined gear; • dissipative friction in contact between the tool and the machined gear is assumed to be viscous. The first two postulates are generally accepted and are widely used in the development of equivalent elastic system models [7], [15]. The validity of the third postulate is based on the fact that the rigidity of all system links in the radial direction is an order of magnitude greater than in the torsional one. The reliability of the fourth and fifth postulates has been confirmed experimentally [3], [6,8]. Based on the analysis of a significant number of technological systems for gear finishing (mainly processing systems with a rolling disk gear tool) and taking into account the postulated statements, a generalized scheme of the dynamic model of the technological system has been developed (Figure 18.1). This model is generalized in relation to the technological systems of gear finishing, since the inertia moment JR is the reduced inertia moment of all links of the tool movements kinematic chain, starting from the rotor of the engine to the tool mass, and the moment of inertia JF is the reduced inertia moment of all links of the kinematic chain of machined gear motion, including the communication chain of the tool and the machined gear relative movement. To obtain a dynamic model scheme of a specific tech­ nological system, it is necessary to deploy the given inertia moments JR and JF in accordance with the kinematic chain of a specific machine tool. To derive the equation of technological system links motion, the machine pair is connected to the moving Cartesian coordinates system. In Figure 18.2, the coordinate system X0O0Y0Z0 is associated with the tool, and the X0 axis is the axis of the tool rotation. The coordinate system X1O1Y1Z1 is associated with the machined gear. The counterclockwise rotation of the tool is taken as the positive direction of rotation of the machine engagement. Movements that match the posrtive direction of the coordinate axes are taken as the positive direction of linear movements. The scheme of the tool movements reduction is shown in Figure 18.3. The movement of the tool and the machined gear is determined by the constraint equation based on the continuity condition of the contact between the tool and the machined gear: n0

n1 = ,

(18.1)

Dynamic Model of Technological System

997

FIGURE 18.1 Generalized dynamic model of technological systems for gear finishing.

FIGURE 18.2 Coordinate systems associated with the tool and the machined gear.

n0 =

n1 =

R0y 0

cos

0y

cos

+ X0 y

R1y 1

cos

1y

cos

+ X1 y

1 sin

0y

cos

+ Z0 y

1 sin

1y

cos

+ Z1 y

1 sin

0y

(18.2)

cos

y

1 sin

1y

cos

(18.3) y

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Dudley’s Handbook of Practical Gear Design and Manufacture

FIGURE 18.3 Scheme of the tool movements reduction.

where δn0 and δn1 are the tool and machined gear displacement, Δ is the tool and machined gear meshing error of a harmonic nature, reduced to the base tangent, R0y and R1y are the radii-vectors of tool and machined gear displacement respectively, δφ0 and δφ1 are the torsional displacement of the tool and machined gear respectively, α0y and α1y are the pressure angle at the contact point of the tool and machined gear respectively, βy and βy are the angles of tooth line inclination of the tool and machined gear respectively. In accordance with the accepted postulates, the considered model has 16 degrees of freedom. However, taking into account the equation of the relationship between the displacements of the tool and the gear being machined leads to overdetermination of the system. Elimination of overdetermination is achieved by determining the displacements of the machined gear in the torsional direction through displacements in other directions:

1

=

R0y cos 0

1y

R1y cos cotg

y

0y

+ X0

cos

cos sin

1y

0y R1y

+ Z0

cotg

cos

y

cos

1y

0y R1y

X1

1y

cotg R1y

1y

Z1

cotg

y

R1y

(18.4)

R1y

In the absence of the sign of small displacements “δ” and the designation of the coefficients for the corresponding small displacements through “a0,” “b0,” “d0,” “a1,” “b1,” and “d1,” Eq. (18.4) is greatly simplified: 1

= a0

0

+ b0 x 0 + d 0 z 0

b1 x1

d1 z1

a1

(18.5)

Considering Eq. (18.5), the dynamic model has one degree of freedom less. To derive the equations of technological system links motion, the Lagrange equation of the second kind is used [4]: d dt

T [qi ]

T + [qi ]

R + [qi ]

P = [Qi ], [qi ]

(18.6)

Dynamic Model of Technological System

999

where T, R, and P are the kinetic, dispersion (Rayleigh function), and potential energy of the system respectively, [qi ], [qi ], [qi ] are the column-vectors of accelerations, velocities, and displacements, re­ spectively, [Qi ] is the column-vector of ith disturbances. The detailed equations of the corresponding energies for the consideration system are defined as follows: T=

R=

1 2

1 (JR 2

(h e (

0

2 R

+ J0

R)

2

0

2

2 1

+ J1

+ JF

F

2

+ m 0 x 0 2 +m 0 z 0 2 + m1 x12 +m1 z12 ),

+ h30 x 0 2 + h20 z 0 2 + h21 x12 + h21 z12 + he (b0 x 0 )2 + he (d0 z 0 )2+

+ he (b1 x1)2 + he (d1 z1)2 + he (a1 z1)2 + he (a1 P=

1 2

(cR (

0

R)

2

+ ce (a0

0)

2

+ ce (a1

1)

2

1)

2

+ h e (a 0

0)

2 ),

+ ce (b0 x 0 )2 + ce (d0 z 0 )2 + ce (b1 x1)2 +

ce (d1 z1)2 + c30 x 0 2 + c20 z 0 2 + c31 x12 + c21 z12 + cR (

1

F)

2)

(18.7)

(18.8)

(18.9)

where JR is the inertia moment of the reduced mass of kinematic chain links of the tool movements, J0 is the tool inertia moment, J1 is the inertia moment of the machined gear, JF is the inertia moment of the reduced mass of kinematic chain links of the machined gear movements, R is the torsional acceleration of the reduced mass of kinematic chain links of the tool movements, 0 torsional acceleration of the tool mass, 1 torsional acceleration of the machined gear mass, F torsional acceleration of the reduced mass of kinematic chain links of the machined gear movements, x 0 is the axial acceleration of the tool mass, x1 is the axial acceleration of the machined gear mass, z 0 is the acceleration of the tool mass in the tangential direction, z1 is the acceleration of the machined gear mass in the tangential direction, m0 is the tool mass, m1 is the machined gear mass, he is the damping coefficient in the engagement of the tool with the machined gear, h30 is the damping coefficient of the tool in the in the axial direction, h20 is the damping coefficient of the tool in the in the tangential direction, h31 is the damping coefficient of the machined gear in the axial direction, h21 is the damping coefficient of the machined gear in the tangential direction, R is the torsional speed of the reduced mass of kinematic chain links of the tool movements, 0 is the torsional speed of the tool mass, 1 is the torsional speed of the machined gear mass, x 0 is the axial speed of the tool mass, x1 is the axial speed of the machined gear mass, z 0 is the speed of the machined gear mass in the tangential direction, z1 is the speed of the machined gear mass in the tangential direction, φR is the torsional movement of the reduced mass of kinematic chain links of the tool movements, φ0 torsional movement of the tool mass, φ1 torsional movement of the machined gear mass, x0 is the axial movement of the tool mass,

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x1 is the axial movement of the machined gear mass, z0 is the movement of the tool mass in the tangential direction, z1 is the movement of the machined gear mass in the tangential direction, cR is the stiffness coefficient of the reduced mass of tool kinematic chain links in the engagement line direction, ce is the engagement stiffness coefficient in the contact of the tool with the machined gear, c30 is the stiffness coefficient of the tool in the axial direction, c20 is the stiffness coefficient of the tool in the tangential direction, c31 is the stiffness coefficient of the machined gear in the axial direction, c21 is the stiffness coefficient of the machined gear in the tangential direction, cF is the stiffness coefficient of the reduced mass of kinematic chain links of the machined gear in the engagement line direction. According to the first accepted postulate, the term T is equal to zero for all i. After substituting the qi

energy equation (18.7)–(18.9) into the Lagrange equation of the second kind (18.6), using the result of the first accepted postulate and performing the necessary transformations, a system of inhomogeneous differential equations of the second order is obtained: [A] [qi ] + [B] [qi ] + [D ] [qi ] = [Qi ],

(18.10)

where [A] inertial coefficient matrix, [B] damping coefficient matrix, [D] stiffness matrix. In this case, the error Δ, reduced to the common normal of the mating surfaces of the tool and the machined gear, is represented by the Fourier series [13]. It is assumed that the components of the Fourier series are only harmonics of the rotational and tooth engagement frequency. These harmonics are due to the eccentricity of the tool and the machined gear and the cyclic errors of their engagement: = D0 + D1 cos(p0 + D3 cos(p1

0

0

t ) + E1 sin(p0

t ) + E3 sin(p1

0

t) +

0

t ) + D2 cos(p0 z

D4 cos(p1z

0

0

t ) + E2 sin(p0 z

t ) + E4 sin(p1

z

0

t ),

0

t)

(18.11)

where D0, D1, D2, D3, D4, E1, E2, E3, E4, are the Fourier series coefficients, p0, p0z, p1, p1z are the harmonics of rotation and tooth components, = 2 n 0 , n0 is the tool rotation speed. The equations system (18.10) determines the law of technological system links motion, the magnitude of their displacement, and direction. This system is a mathematical interpretation of a dynamic model of a gear finishing technology system. The system of equations (18.10) with zero right-hand sides describes the free vibrations of the technological system links [4,15]: [A] [qi ] + [B] [qi ] + [D ] [qi ] = [I ],

(18.12)

where [I] is the null matrix. The equations system (18.12) is designed to determine the eigenfrequencies and eigenmodes of the modeled technological system vibration. To determine the eigenfrequencies, the equations system (18.12) is reduced to a system of first-order differential equations by substitution {qi} = { i} [5]. Solution in the form: {qi} = [X ] e{

i} t

,

(18.13)

Dynamic Model of Technological System

1001

where [X] is the column-vector of eigenvalues, λi is the eigenfrequency (in general a complex number), e is the base of natural logarithm, is substituted into Eq. (18.12). As a result, the equations system (18.12) is transformed into a linear equations system by reducing it by a common factor e{ i} t and performing elementary algebraic transformations: {T} [X ] = { i} [X ],

(18.14)

where [T] is the characteristic matrix. The equations system (18.14) has a solution if the determinant of the characteristic matrix [T] is equal to zero [5]. The calculation of the eigenfrequencies {λi} of the vibration system and the eigenmodes of vibrations [X] is based on this position.

18.3 DETERMINING THE MODEL PARAMETERS To implement the numerical solution of the developed dynamic model, it is necessary to determine the values of its parameters. The parameters of the dynamic model are the coefficients of the equations system (18.10): reduced inertial masses (masses of links and their moments of inertia), damping coefficients, and stiffness coefficients. The determination of the masses and inertia mo­ ments of the technological system links is carried out by calculation according to the detailed drawings of a specific machine tool in accordance with general recommendations [7,15]. The sought values of the masses and inertia moments were determined by conditionally dividing the parts into the simplest figures [15]. In this case, the masses and inertia moments of the shafts with the parts attached to them are taken into account together. The damping coefficients in the shaft bearings were determined from the damping time constant [14]. The damping coefficients of the connection be­ tween the tool and the machined gear with other technological system links were determined by the dissipation coefficients [9]. The calculated values were compared with the values determined ex­ perimentally. For this purpose, the impact of a modal hammer excited and recorded damped vi­ brations in the corresponding direction of the investigated system link. The scheme of the measuring setup for determining the dynamic characteristics of the mechanical elastic system of the machine tool is shown in Figure 18.4. Excitation of eigenvibrations of the machine tool elastic system is carried out by a modal hammer impacting. The input signal of the force sensor, located directly under the striker of the impact hammer, after being pre-amplified by the Nexus amplifier from Brüel & Kjær, is fed to the 16-channel analyzer LDS Dactron Focus. At the same time, this analyzer receives a response signal from the mechanical elastic system of the machine tool from a three-component piezoelectric ac­ celeration sensor type 8396A from Kistler. The analysis results of mechanical vibrations of the machine tool are visualized with an SSA 3032X oscilloscope and in parallel are fed to a PC for further processing and saving the results. The damping coefficient was determined by the damping decrement according to the relationship [10,15]: h=

where m is the link mass, T is the vibration period, Ai is the amplitude of the ith vibration, ln is the natural logarithm.

2m A ln( i ), T Ai+1

(18.15)

FIGURE 18.4 Set-up scheme for measuring the dynamic parameters of the technological system.

1002 Dudley’s Handbook of Practical Gear Design and Manufacture

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1003

Due to the complexity of the physical processes of cutting and friction occurring in the contact of the tool with the machined gear, the damping coefficient he in this contact was determined experi­ mentally in two stages. At the first stage, experimental studies of the forces during the gear finishing with a rolling gear tool were carried out. As an example, the process of diamond gear honing was investigated. At the second stage, the cutting-friction forces in the contact of the diamond gear hone with the machined gear were refined at a given contact point along the height of the teeth profile (along the length of the engagement line) on a stand that implements the roller analogy method. In turn, the experimental determination of forces during diamond gear honing was also carried out in two stages. At the first stage, the interaction forces of a gears reference pair were determined, simulating the engagement of a gear hone and a machined wheel. The second stage was devoted to the determination of the interaction forces in the real process of gear honing. The dependence of the cutting forces in the real process of diamond gear honing on the cutting modes with one- and two-profile finishing methods is shown in Figure 18.5. The cutting forces in the gear finishing were determined by excluding the interaction forces in the reference pair from the similar forces of interaction between the tool and the machined gear in the real finishing process. Subsequently, these forces were refined at the given points of contact between the profiles of the tool and the machined gear teeth (at a given point on the line of engagement) in the experimental studies using the roller analogy method. This method reproduces the contact conditions between the teeth of the tool and the machined gear at a certain point of their contact by contacting two cylindrical rollers. A scheme explaining the roller analogy method is shown in Figure 18.6. In this case, the radii of the rollers r0 and r1 correspond to the radii of tool ρ0 and the machined gear ρ1 teeth curvature at a given point along the height of the teeth profile (at a given point on the line of engagement). The change in the magnitude and sign of the cutting-friction force in the contact of the machine pair along the height of the machined gear teeth is not taken into account. To determine the damping coefficient he on the roller shaft simulating a given point of the tool profile teeth, the tangential force FZ and the radial force FY were measured by means of strain-gauge elastic beams. The strain gauges glued to these beams were connected to the measuring circuit according to a half-bridge circuit. The measured forces were used to determine the cutting-friction coefficient KC: KC = tan

=

FY , FZ

(18.16)

and by the coefficient of cutting-friction KC, the damping coefficient he in the contact of the tool with the machined gear was determined:

he =

KC F VS

(18.17)

where α is the friction angle, F is the normal force acting in contact between the tool and the machined gear (the force of pressing the rollers against each other), VS is the sliding speed at a given point between the teeth profiles in contact of the tool with the machined gear (sliding speed in the contact of the rollers). The stiffness coefficients of the shafts and the engagement of the kinematic chains gears of the machine (if they are present) were determined by the known dependencies:

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FIGURE 18.5 Dependence of the interaction forces between the tool and the machined gear during diamond gear honing on the cutting conditions.

• torsional stiffness of shafts [10,15]: C=

where d is the shaft diameter, G is the shear modulus, L is the shaft length,

d4 G , 32 L

(18.18)

Dynamic Model of Technological System

FIGURE 18.6

1005

Scheme of the roller analogy method.

• engagement rigidity of gears [1]: C=

E b , 11 … 13 cos b

(18.19)

where E is the elastic modulus, εα is the gear engagement factor, βb is the reduced angle of tooth line inclination, b is the reduced width of the gear. The connection rigidity between the tool and the machined gear with the elements of the technological system was determined experimentally, since the sought values depend on the contact rigidity of a significant number of parts. An example of the stiffness coefficient measurement scheme and some results of their determination are presented in Figure 18.7. To create a load on the tool spindle when measuring its rigidity, a rigid block is used (see Figure 18.7). The measurement of the load value is carried out by a single-axis piezoelectric sensor type 9321B from Kistler, and the movement of the mandrel fixed in the spindle is carried out by the con­ tactless CCD laser triangulate system from Keyence. The signals are recorded with an NI USB-6259 measurement card. An original measurement program was developed in the LabVIEW software en­ vironment from National Instruments to saving and processing the measured signals. The engagement stiffness coefficient of the machine pair was determined as the stiffness of the helical gearing in terms of specific stiffness, proportional to the total length of contact lines or the total length of contact between the cutting edges of the tool and the machined gear body: ce =

E lS = cS lS , 11 … 13

(18.20)

where cS is the specific gearing stiffness, lS is the total length of the contact lines or the total contact length of the tool cutting edges and the machined gear body.

FIGURE 18.7 Scheme for measuring the stiffness coefficients of the tool and the machined gear connection.

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The value of lS is selected on the basis of numerical modeling of the shaping process [2,12]. This value changes with the change in the rotation angle of the machined gear, proportional to the processing time. Therefore, the stiffness of the engagement ce also depends on time. However, in order to simplify the cal­ culation of eigenfrequencies and eigenmodes vibration spectrum, the engagement rigidity is taken constant. In contact between the tool and the machined gear there are materials of different physical, me­ chanical, and chemical properties: hardened steel (the machined gear) and diamond-containing or abrasive-containing metal, polymer or rubber binder. The material elasticity modulus of the tool working layer (binder) is not smaller than two orders of magnitude less than the material elasticity modulus of the machine gear. In this regard, when determining the engagement rigidity ce of the tool with a machined gear, only the elastic modulus of the work layer tool material is used. In this case, the body of the machined gear is assumed to be rigid. The adequacy of the developed dynamic model of the technological system for the gear finishing was tested using an example of diamond gear honing of hardened gears on a machine model 5A913. The characteristic matrix for this specific dynamic model is presented in Figure 18.8. The size of the characteristic matrix [T] for the consideration case is 30×30 (see Figure 18.8). To analyze the eigenfrequencies and eigenmodes of vibration with such a size of the characteristic matrix, it is necessary to use numerical methods [4]. Therefore, to determine the spectrum of eigenfrequencies {λi} of eigenvibration modes [X], a software-implemented algorithm has been developed (Figure 18.9). The algorithm uses standard programs for reducing the matrix [T] to an almost triangular form—the Hessenberg form—and calculating the eigenvalues by the Francis double QR iteration method [5]. To check the correctness of the calculation, the eigenvalues of the matrix [T] were substituted into its main diagonal and the main determinant of the matrix [T] was calculated. If the eigenvalues are correctly

FIGURE 18.8 Characteristic matrix [T] for a dynamic model of diamond gear honing of hardened gears on a gear honing machine 5A913.

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FIGURE 18.9 The flowchart of the algorithm for calculating the spectrum of eigenfrequencies and eigenmodes.

calculated, this determinant should be equal to zero, which has been proven. The calculated and ex­ perimentally determined parameters of the dynamic model of diamond gear honing of hardened gears on a machine model 5A913 are shown in Table 18.1. The dependence of the engagement stiffness ce of the diamond gear hone with the machined gear on the engagement time t (the angle of tool rotation) is shown in Figure 18.10. Determination of the disturbing forces qualitative composition and the possible range of variation parameters and solutions of the system was carried out experimentally. For this, the results of studying the forces in the process of diamond gear honing were used [13]. A scheme of a measuring setup for studying the characteristics of the diamond gear honing process is shown in Figure 18.11. The action forces of the gear rolling tool (in the given example, the diamond gear hone) were measured by means of strain gauges located on the tool and the machined gear shafts. The signal from strain gauges after amplification at the type D 402 from HBM was fed to an NI USB-6259 measurement card, and then processed and visualized with the LabView software from National Instruments. The tool rotation angle was recorded by a tool shaft angular displacement transducer and, after the analyzer type CNT-90 and visualization on the oscilloscope type SSA 3032X, was fed to the measuring card and PC for processing. Signal from the sensor of the tool and the machined gear contact after the amplifier type 2626 from Brüel & Kjær was fed in a similar way for processing and visualization. An example of an oscillogram of the finishing gears characteristics obtained using the described measuring setup is shown in Figure 18.12. By processing such oscillograms, the column vector of the disturbing forces and the initial conditions for solving the system of equations (18.10) are determined. To determine the initial conditions (initial movement and initial speed) on the oscillogram of the finishing process characteristics, the force components FX, FY, and FZ, as well as the torque MR at any moment in time, were recorded (see Figure 18.12). Then, according to the

Dynamic Model of Technological System

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TABLE 18.1 Dynamic Model Parameters of the Technological System for Gear Honing Inertial Coefficients Sign

Unit

Numerical Value

Sign

Numerical Value Calculated

Experiment

Sign

Unit

Numerical Value

h30

kg m/sec

0.16

-

cR

N m/rad

2.01·104

h10

kg/sec

2.4

3.0

c23

N m/rad

9.998·106

J3 J4

−4

9.69·10 1.26·10−3

h20 hk

1.8 -

2.2 5.0

c3 c45

N m/rad N m/rad

0.94·105 0.77·107

J5

3.58·10−4

h11

3.0

-

c5

N m/rad

0.96·105

J6 J7

−3

4.83·10 4.83·10−3

h21 -

-

8.0 -

-

c67 c7

N m/rad N m/rad

0.4·107 0.42·105

J8

3.0·10−3

-

-

-

-

c89

N m/rad

0.22·107

J9 J0

−2

2.15·10 7.53·10−2

-

-

-

-

c9 c30

N m/rad N/m

0.36·105 0.58·107

J1

5.14·10−3

-

-

-

-

c20

N/m

0.76·107

JF m0

−4

kg

9.36·10 4.89

-

-

-

-

ce c11

N/m N/m

0.25·105 0.95·107

m1

kg

1.83

-

-

-

-

c21

N/m

0.25·108

-

-

-

-

-

-

-

cF

N/m

0.20·105

J2

3.0·10−4

Unit

Stiffness Coefficients

−4

JR

kg m2

Damping Coefficients

9.69·10

FIGURE 18.10 Dependence of the engagement stiffness of the diamond gear hone with the machined gear on the engagement time.

known force and rigidity of the corresponding link connection with the mechanical structure of the tech­ nological system (see Table 18.1), its displacement at the considered moment of time, taken as the initial one, was determined. The initial speed was determined from the measured force components and the link mass. The initial conditions for the coordinates of those system links, access to which is difficult, were determined by calculation using the known load on the first link of the kinematic chain. The initial conditions determined by the described method are given in Table 18.2.

FIGURE 18.11

Schematic diagram of a set-up for measuring forces and dynamic characteristics during diamond gear honing.

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Dynamic Model of Technological System

FIGURE 18.12

1011

Measuring signals the kinetic characteristics of the diamond gear honing process.

TABLE 18.2 Initial Conditions for Solving the Equations System (18.10) Initial Conditions

Initial movement Initial speed Initial movement Initial speed Initial movement Initial speed

Coordinates φR

φ2

φ3

φ4

φ5

5.0·10-4 rad

2.0·10-4 rad

8.0·10-4 rad

1.0·10-4 rad

2.3·10-4 rad

-3

1.0·10 rad/sec φ6

6.0·10 rad/sec φ7

1.0·10 rad/sec φ8

4.0·10 rad/sec φ9

2.0·10-3 rad φ0

2.4·10-4 rad

2.4·10-4 rad

2.4·10-4 rad

3.0·10-4 rad

1.0·10-4 rad

-3

-3

2.0·10 rad/sec z0

2.0·10 rad/sec x1

5.0·10 rad/sec z1

8.0·10-3 rad/sec φ8

0.8·10-4 m

0.5·10-4 m

0.4·10-4 m

0.2·10-4 m

2.0·10-4 rad

-4

-3

-3

-3

4.0·10-3 rad/sec

4.0·10 m/sec

-3

-3

2.0·10 rad/sec x0 5.0·10 m/sec

-3

-3

3.7·10 m/sec

-3

3.0·10 m/sec

The adequacy of the developed dynamic model to the real process of diamond gear honing of har­ dened gears was checked by comparing the experimental and calculated values of the eigenfrequencies of the system and the numerical values of the system amplitude-phase frequency response (Nyquist-plot). The experimental values of the system eigenfrequencies were determined by analyzing the free vibra­ tions of the system when excited by a impact of a modal hammer on one of the links of the system and subsequent harmonic analysis of the system recorded vibrations [13,14]. To carry out these studies, an

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TABLE 18.3 Comparison of the Calculated and Experimental Values of the First Six Eigenfrequencies of the Technological System for Gear Honing Number of the Eigenfrequency

f1

f2

f3

Calculated

25.8

88.6

163.8

Experiment Relative deviation

26.4 2.3

85.0 4.2

145.0 13.0

f4

f5

f6

172.9

193.3

287.6

150.0 15.3

170.0 13.7

235.0 22.4

experimental measuring setup was used, the scheme of which is shown in Figure 18.4. The need to conduct a harmonic analysis of the obtained oscillograms of system vibrations is due to the fact that the obtained curves are polyharmonic due to the large number of degrees of freedom of the technological system. Comparison of the calculated and experimental values of the system eigenfrequencies was carried out for the first six frequencies. The comparison results are shown in Table 18.3. The relatively good agreement between the experimentally determined and calculated first six ei­ genfrequencies of the technological system of diamond gear honing (the largest deviation is 22%) proves the adequacy of the developed dynamic model. The points of the amplitude-phase frequency response (Nyquist-plot) were determined when the system was loaded with a harmonic force. For this, instead of the tool, a disc with profiled grooves was fixed on the tool spindle, with which the roller was in contact. The dimensions of the profile grooves (their depth and pitch) on the disk provided a force corresponding to the forces acting in a pair of toolmachined gear. The frequency of the exciting force changed stepwise when replacing the disc with profiled grooves and when changing the number of tool spindle revolutions. The exciting force was recorded by a uniaxial piezoelectric sensor type 9321B from Kistler in contact with the roller. The output force was recorded by means of strain gauges located on the tool shaft (see Figure 18.11). The system displacement was calculated from the output force and from the previously known determined stiffness coefficients (see Table 18.01). The phase shift was determined from the displacement of the input (exciting force) and output force oscillograms on the tool shaft. According to the output force, the calculated displacement and phase shift, the amplitude-phase frequency response (Nyquist-plot) of a specific technological system was built (Figure 18.13). The largest difference between the calculated and experimental values of the amplitude-phase fre­ quency response (Nyquist-plot) was 48%. Taking into account the complexity and multichannel nature of the studied technological system, the conducted experimental studies on measuring the eigenfrequencies and the values of its amplitude-phase frequency characteristic prove the adequacy of the developed dynamic model to the real technological system of gear finishing. This makes it possible to use the developed generalized dynamic model of gear finishing in the information system [11] to select the optimal parameters of the tool and the machining process.

18.4 OBJECTIVE FUNCTIONS OF THE DYNAMIC MODEL The target functions (characteristics) of the dynamic model of the technological system are: the solution of the system of equations (18.10)—the equations of motion of the center of mass of the technological system links (mainly the tool and the machined gear) and the spectrum of the eigenfrequencies of the system—the solution of the system (18.14). Based on the analysis of the eigenfrequencies spectrum of the technological system, it is possible to significantly reduce the vibration activity of the finishing process and thereby significantly increase the quality and productivity of processing as well as tool life. Reducing the vibration activity of the interaction process between the tool and the machined gear is performed by detuning from resonance modes. For this, as a rule, a resonance diagram (beam diagram) is

Dynamic Model of Technological System

1013

FIGURE 18.13 Amplitude-phase frequency characteristic (harmonic locus) of radial oscillations of a diamond gear hone.

used [10]. The resonance diagram is a rectangular Cartesian coordinate plane, on the horizontal axis of which the values of the disturbing influences are plotted, and on the vertical plane—the values of the system eigenvibrations. The points of intersection of these two groups of beams: disturbing influences and eigenvibrations of the technological system determine the vibroactive zones of the machining process. For example, Figure 18.14 shows a resonance diagram (beam diagram) of the diamond honing process of hardened gears. By analyzing this diagram, areas of different intensity of vibration activity of the machining process are determined. In particular, the diagram highlights the possible areas of low vibroactivity, which should be aimed at when choosing the parameters and structure of technological systems. As a rule, the spectrum of eigenfrequencies is quite dense. Therefore, it is difficult to find a zone with low vibration activity. There are two ways to do this:

FIGURE 18.14

Resonance diagram (beam diagram) of the diamond honing process of hardened gears.

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• changing the rotation speed of the tool spindle and machined gear spindle; and • changing the parameters of the dynamic model of the technological system. Due to the dense spectrum of the system eigenfrequencies, the first way is ineffective, and the narrow range of the tool spindle speeds further aggravates the situation. In addition, the movement speed of the machined gear is related to the condition of rolling, limited by the required quality of the machined teeth surfaces, the design of the equipment, etc. Therefore, changing the machined gear speed will require significant costs, sometimes significantly larger in comparison with the achieved gain. The second way is more effective, since changing the dynamic model parameters, namely, the coefficients of inertia, damping, stiffness, and parameters of disturbing influences, can be easily controlled by changing the geometric and design parameters of the tool and other elements of the technological system. It is in­ comparably lighter and more effective in comparison with the first way by reducing the vibration activity of the machining process. For example, the mass and inertia moment of the tool, and the rigidity of its engagement with the machined gear, have a significant impact on the spectrum of eigenfrequencies of the technological system, which can be varied within wide limits. Figure 18.15 shows the dependences confirming the significant influence of the mass, inertia moment of the tool, its rigidity, and rigidity of engagement with the machined gear on the first five eigenfrequencies of the technological system during diamond gear honing. Thus, by changing these and other parameters of the technological system for gear finishing, it is possible to ensure low vibration activity of the machining process due to the displacement of this process into a resonance-free zone. A particular interest is the definition of the displacement law of the mass center of the technological system links not only as characteristics of the dynamic model, but also as separate functions. The im­ position of the forced movements of the tool and the machined gear on their ideal movements when shaping the teeth profile will allow to describe the real contact of the tool with the machined gear, as well as to determine the deterministic component of the engagement error and to establish ways to improve the accuracy and productivity of machining. However, solving such a problem causes significant diffi­ culties and is beyond the scope of this study. Determination of the forced movements of the tool and the machined gear by solving the equations system (18.10) is used to screen the influence of dynamic vibrations of the technological system links on the movement of the tool, to eliminate contact separation, mainly when finishing the gears by free rolling, as well as to reduce the time of action of transient processes. Screening of the dynamic vibrations influence of the technological system links is realized by determining the parameters of the technological system, at which the amplitude of tool and the machined gear vibrations, defined as the solution of the equations system (18.10), does not exceed the specified ones. These parameters include the stiffness of the connection between the tool and the tool spindle: c1R, c20, and c30 and damping factors: h1R, h20, and h30. These parameters of the dynamic model are de­ termined by the iterative approximation of the equations system (18.10) solutions to the given vibration amplitudes. Gear finishing without breaking the contact of the tool with the machined gear is determined from the relationship equation (18.1). The machining condition with permanent contact is defined by the fol­ lowing expression: u = n0

n1

0

(18.21)

The amplitude of the relative vibrations of the tool and the machined gear is used as a constraint for solving the system of equations (18.10). The parameters of the dynamic model of the technological system are determined by iterative approximations of this solution. Reducing the time of transient processes action is achieved by increasing the damping in the tech­ nological system [10,14]. For this purpose, it is necessary to change in the considered dynamic model the damping coefficients of the connection between the tool and the machined gear with the technological

Dynamic Model of Technological System

1015

FIGURE 18.15 Dependences of the dynamic model parameters influence on the spectrum of technological system eigenfrequencies.

system and the cutting-friction coefficient KC (18.16) in the contact of the machine pair. In order to increase damping in the system, various elastic-damping mounts are often used [9]. Using this method and the corresponding constrains on the relative vibration amplitude of the tool and the machined gear, the parameters of the tool working layer, for example, the diamond-containing layer for toothed hones, and the desired damping coefficients are determined.

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18.5 SYNTHESIS OF TOOLS AND PARAMETERS OF THE TECHNOLOGICAL SYSTEM The result of the developed dynamic model functioning is its objective functions (see Chapter 18.2). Determination of the objective function values that provide the specified requirements allow on, when optimizing the system, to find the values of the dynamic model coefficients that meet these re­ quirements. The use of the developed model of the dynamic interaction of the technological system links as an independent object that does not depend on other models of the information system [11] makes it possible to determine the ways of constructive support of the design parameter values de­ termined during optimization. This is done by analyzing the eigenfrequencies spectrum of the technological system and its forced vibrations. The coefficients of the dynamic model are used to determine the constructive and geometric parameters of the tool and the parameters of other tech­ nological system elements, for example, the processing modes that ensure low vibration activity of the machining process, the design and parameters of the connection between the tool and the ma­ chined gear with the elements of the technological system, etc. As an example, the developed dy­ namic model was used to synthesize the diamond gear honing process. To fulfill the recommendations for a significant reduction in the mass and inertia moment of the tool, as well as its rigidity, the tool gear rim is made separately from the tool hub and connected to it by elastic elements made of a material with significant vibration-absorbing properties, for example, rubber. The use of such a technique is known in toothed disc tools [13], so in gears [10]. The dimensions and grade of the elastic element material connecting the toothed ring of the hone with the tool hub are determined by the following relationships [13]: =

6G (1 + ( 2

ac bc )2 ) (ac + bc ) cc

cc

A

(18.22)

where с is the stiffness coefficient determined by the dynamic model operation, A is the contact area of elastic elements, ac, bc, and cc are the dimensions of rectangular elastic elements. It is possible to reduce the rigidity of the engagement of the gear hone with the machined gear in two ways: by reducing the rigidity of the entire teeth of the tool due to grooves in its cavity and by placing the working abrasive-containing working elements of the tool on an elastic base. An example of the im­ plementation of such a constructive solution is shown in Figure 18.16.

FIGURE 18.16 Diamond gear hone with reduced gearing stiffness (a) and elastic connection of the ring gear with the hub (b).

Dynamic Model of Technological System

FIGURE 18.17

1017

Fragments of diamond gear hone teeth with reduced gearing rigidity.

A fragment of a toothed rim of a diamond gear hone, realizing the second way, is shown in Figure 18.17. The elastic base of the tool working elements is, for example, an adhesive bond (see Figure 18.17) or, for example, the use of elastic rubber-containing bonds (see Figure 18.17). The use of synthesized tools can significantly reduce the vibration activity of the finishing process and eliminate contact separation during machining. This provides a significant increase in productivity and machining accuracy as well as tool life. Thus, the developed dynamic model provides a synthesis of tools and technological systems for gear finishing.

REFERENCES [1] Babichev, D., & Storchak, M. (2014). Synthesis of cylindrical gears with optimum rolling fatigue strength. Production Engineering, 9(1), 87–97. https://doi.org/10.1007/s11740-014-0583-6. [2] Babichev, D., & Storchak, M. (2018). Quality characteristics of gearing (Vol. 51, pp. 73–90). Mechanisms and machine science. https://doi.org/10.1007/978-3-319-60399-5_4. [3] Bergs, T. Bergs, T. Kolcke, F. Brecher, C. Schrank, M. G. Kampka M. Kiesewetter-Marko C. Lopenhaus C. Epple A. (2019). Prediction of process forces in gear honing. Gear Technology, 2019, 36(3), 34–39. [4] Cooke, D. J., & Bez, H. E. (1989). Computer mathematics (394p). Cambridge University Press. [5] Datta, B. N. (2010). Numerical linear algebra and applications (2nd Rev. ed., 554p). Society for Industrial & Applied Mathematics. [6] Heisel, U., Kushner, V., & Storchak, M. (2012). Effect of machining conditions on specific tangential forces. Production Engineering, 6(6), 621–629. https://doi.org/10.1007/s11740-012-0417-3. [7] Kudinov, V. A. (1967). Machine dynamics (359 с). Mashinostroyeniye. [8] Kushner, V., & Storchak, M. (2014). Determining mechanical characteristics of material resistance to deformation in machining. Production Engineering, 8(5), 679–688. https://doi.org/10.1007/s11740-014-05 73-8. [9] Pisarenko, G. S., Jakovlev, A. P., & Matveev, V. V. (1971). Vibration damping properties of construction materials (370p.). Naukova Dumka. [10] Singh, A. (2017). Fundamentals of machine design (842p.). Cambridge University Press. [11] Storchak, M. (2019). Optimization of geometrical engagement parameters for gear honing. In Stephen P. Radzevich (Ed.), Advances in gear design and manufacture (pp. 35–64). CRC Press, Taylor & Francis Group. [12] Storchak, M. (2020). Simulation of the teeth profile shaping during the finishing of gears (Vol. 81, pp. 365–384). Mechanisms and machine science. Springer. https://doi.org/10.1007/978-3-030-34945-5_16. [13] Storchak, M. (1994). Technological systems for finishing gears (448p.). Institute for Superhard Materials. [14] Sweeney, G. (1971). Vibration of machine tools (230p.). Machinery Publishing Company. [15] Uicker, J. J., Pennock, G. R., & Shigley, J. E. (2016). Theory of machines and mechanisms (5th ed., 976p). Oxford University Press.

19

Powder Metal Gears Anders Flodin

19.1 INTRODUCTION Powder metal (PM) gears have been manufactured since the 1930s. One of the first applications was in oil pumps as pump gears. Since then, materials, production technology, and designs have evolved and today powder metal gears can be made in more advanced and more optimized designs than what is commonly available using the conventional machining route. The Achilles heel of powder metal gears has traditionally been its lower strength. Today that is remedied by new processes such as roll densification, forging, or hot isostatic pressing techniques. However the inherent porosity has side effects that in certain applications are advantageous such as lubricity, less weight, and improved acoustic properties. Today powder metal gears are used in automotive transmissions, not only to drive an oil pump but to transfer power to the wheels and propel the car forward. Another aspect of PM gears is that the production process of gears and the material is less resource demanding making it more attractive from a sustainability perspective. In PM gearing the: 1. Materials, 2. Manufacturing and 3. Design are very interconnected and that will be reviewed in this chapter as well as explanations of the processes and design aspects.

19.2 PM MATERIALS FOR GEARS There are thousands of PM materials and blends tailored for different applications and part producers, and for many applications there are different variants depending on the processes used to make a certain part and then there are variants within a certain alloying group depending on allowable cost/benefit ratio. But to keep things manageable when discussing PM gears it is the density and heat treatment that has to match a specific designed strength and durability. This in turn determines the material and the manu­ facturing steps involved to make the gears. There are several ways to manufacture the material: Gas atomizing for high alloyed steels such as tool steels or stainless steels. Water atomizing for low to medium alloyed steels, typically PM gear steels are within this group. Sponge steel for very low oxygen levels and lower density applications with high green strength.

There are a couple of more exotic methods but for PM gear steels the atomized powders are the most commonly used materials and that is the focus of this chapter. Materials for Additive Manufacturing (AM) of gears is made with a different process, see Chapter 20 that discusses AM materials for gears. Generally the chemical composition of the PM gear steel follows the wrought gear steel equivalent. The biggest difference is the Manganese that is often avoided in PM materials due to the affinity of oxygen to Mn which has a negative effect. For wrought steels Mn is not an issue since the Mn atoms are inside the atomic lattice but for PM as a powder the Mn atoms becomes more exposed to oxygen and oxygen is an unwanted component in gear steels since it forms inclusions and poor bonding between PM steel particles in the sintering process. Generally PM gears are made: 1. As sintered (S) 2. Sinter hardened (SH) 1019

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3. Quenched and tempered (QT) 4. Induction hardened (IH) 5. Case carburized and tempered (CQT) For every one of these five variants a different material composition will be chosen. Metal Powders Industry Federation (MPIF) have developed generic names and nomenclature that is used together with common brand names for the different materials such as Astaloy Mo or Distaloy AQ.

19.2.1 AS SINTERED (S) As sintered (S) gears can be made with a number of materials. Since no hardening is required, the stresses have to be low or life expectancy is non critical. A typical material would be and Iron-Carbon (Fe-C) or Iron-Copper-Carbon (Fe-Cu-C) material such as F-0000 and FC-0200 or ASC 100.29 + 0.8% C with Yield strengths of around 450 MPa at 7.0 g/cc density.

19.2.2 SINTER HARDENED (SH) Sinter hardened (SH) gears normally use a material with higher carbon content as the method to reach higher hardness through martensitic transformation during the cooling step in the sinter furnace. There is no additional process step needed unless tempering is specified. The hardening is done inside the sin­ tering furnace that is equipped with a chamber at the end of the furnace with a fan that blows the gas in the furnace onto the components. The gas is cooled in a water to gas heat exchanger. Typical sintering gases are a mixture of Nitrogen and Hydrogen according to: • Reducing-decarburizing type: Hydrogen (H2), cracked ammonia (75% H2, 25% N2) • Carburizing type: Endogas (32% H2, 23% CO, 0%–0.2% CO2, 0%–0.5% CH4, bal. N2) • Neutral type: Cryogenic nitrogen (N2), if desirable with minor additions of H2 to take care of residual oxides or of methane or propane to restore carbon losses Typically materials need 2% Cu or 1% Mo with 0.4%–0.8% C to form a Martensitic structure in a sinter harden process. The cooling speed is varied by controlling the speed of the fan. Typical cooling rates vary between 1°C/s and 10°C/s. For more information on the sintering process see [1].

19.2.3 QUENCH

AND

TEMPER (QT)

The quench and temper process for powder metal gears is similar to the process used for wrought steel gears. After sintering, the gears are heated up again to around 850°C and quenched in a cooling medium, such as water, oil, or gas. The tempering process following the quenching is typically done at 180°C–200°C for one hour. This will add ductility without sacrificing too much hardness. Typical materials for this process needs to be alloyed for hardenability and since there is no carbon containing atmosphere the material needs a higher admixed graphite content compared to materials that are being case carburized. Most Fe-C-Cu and Mo-containing materials will respond to quench and temper as long as graphite levels of 0.4%–0.8% are admixed. In some cases Sinter hardening can replace the quench and temper process but normally higher hardness and deeper martensitic layers are accomplished using quench and temper since the cooling speed is faster using oil than with gas. For smaller components with less mass, a substitution may be possible and sinter hardening will be sufficient. One thing to keep in mind is that when quenching in oil, the oil will enter the pores if the density is not high enough, typically 7.3–7.4 is needed for closed porosity. It is difficult to get all the oil out if that is required for instance if the part is welded later in the process chain, and then sinter hardening or gas quenching can be the better alternative.

Powder Metal Gears

1021

19.2.4 INDUCTION HARDENING (IH) Induction hardening of powder metal gears is commonly used in the gear industry. It is a very fast process just like with wrought steel gears. Powder metal, due to its lower inductance, needs dual fre­ quency coils and higher energy input. The materials used is often Molybdenum and/or Nickel alloyed with 0.4%–0.8% admixed graphite such as Astaloy Mo. Since the IH process is very fast the oxidation within the pores (if open porosity) will not affect the properties of the steel. Typically sprockets are made in powder metal with subsequent induction hardening as well as gears where low distortion is required.

19.2.5 CASE CARBURIZING

AND

TEMPERING (CQT)

There are several ways to case carburize PM gears. Low pressure furnaces with gas quenching or gas carburizing with oil quenching are the most common types of furnaces used. Common for them both is that the process time is reduced due to the much faster penetration of carbon into the PM material due to the pore system. The higher the density of the PM gear the longer the processing time and at full density the processing time will be the same as for wrought steel. Typically for a 7.2 g/cc gear the processing time will be 50%–60% of the wrought steel processing time. When measuring hardness, irrespectively of heat treatment type, micro Vickers measurement tech­ nique should be used. Hundred gram weight is a rule of thumb even though 200 g can be found in literature. The reason for this is that the porosity will influence the result when using a regular Vickers cone or Brinell ball and the numbers will come out very low when compared to wrought steel gears. The macro hardness is referred to as apparent hardness and when macro hardness is defined on a drawing for a wrought gear, it will be much higher than what can be obtained on a porous PM gear. See Figure 19.1 for good hardness curves on a PM gear after case carburizing measured with the micro Vickers method. The recommended case depth is the same as for a wrought steel gear tooth with the same modulus. Equation (19.1) can be used as a rule of thumb [2] to set the proper case depth. Eht = 0.4

0.15m n

where

FIGURE 19.1

Hardness curves using micro Vickers with 100 g weight (HV0.1).

(19.1)

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Eht is the case depth in millimeters. Mn is the normal modulus of the gear tooth in millimeters. Typical materials for case carburizing contains 1.5%–2% of alloying materials such as Molybdenum, Chromium, and Nickel. As an example Astaloy CrA in Figure 19.1 and Astaloy Mo as well as Astaloy 85Mo for smaller modulus gears (mn < 1.2 mm) are common materials for case carburizing. The ad­ mixed graphite content is kept at 0.2%–0.25% in order to get the volume expansion in the surface from the adsorbed carbon atoms in the processing gas. Increasing the graphite content will increase case depth but it will also lead to less compressive residual stress due to less difference in volume expansion in case and core. A greater unbalance between case depth in the root and in the flank will also follow if the base carbon content is too high in the material. Typically values above 0.35% C could result in a through hardened flank but with a root that still exhibits a core less than 550HV0.1. Recent developments in batch furnaces as well as part to part furnaces have improved the dimensional stability of the parts. These furnaces, sometimes referred to as 2D furnaces since the batches are laid out on trays with no stacking of parts on top of each other, works very well with powder metal. The Koyo high speed low distortion process using part-wise induction heating to 1200°C with a Nitrogen and Methane mix has proven extra advantageous due to the porosity in the PM material. First the carbon diffusion is extremely fast at 1200°C–1300°C, then the grain growth is nonexistent due to the short process time and presence of pores, eliminating the normalizing step that is needed for wrought steel gears using the same process. The distortion, due to the short time at high temperature is also sig­ nificantly lower than for conventional carburizing processes. For a representative wrought steel gears the process time is 27.5 minutes, which is very fast and will render a case depth of 0.5 mm, for the same PM gear the process time is nine minutes which is mainly the result of not having to normalize the steel after quenching. Most heat treatment processes for wrought steels can be used with powder metal steel. The process parameters will be different, often the process time is shorter with PM due to the porosity. This is because gas penetration and reactivity is higher and grain growth is reduced by the pores which saves time in the normalizing step. See [3–5] for more information. The point is that there is no need for special equipment to heat treat PM, just an adjustment of process parameters.

19.3 MANUFACTURING The idea of sintered parts and gears is to utilize as few manufacturing steps as possible to achieve sufficient strength and tolerances of the part. This means that there are several ways to make PM gears depending on the requirements that is set in the design phase. The strength can be tailored for the PM gears by adapting the manufacturing process. This means that a gear that sees less stress cycles can be made at lower total cost than the gear that needs more durability which is obtained by more processing for higher density or more advanced materials. Examples are given in [4]. The basic manufacturing chain of PM gears consists of: 1. Compaction During compaction powder is gravity fed into the tool cavity making up the gear shape. It is compacted, sometimes at elevated temperature (40°C–150°C) to change the viscosity of the lu­ bricant that is present in the powder. The lubricant is there to protect the tooling and part during ejection of the part and to reduce inter-particle friction which increases part density and thereby strength. Adding excessive compaction force reduces life of the tools and can create cracks in the parts during ejection. Normally the compaction pressure varies from 500 to 1000 MPa with re­ sulting densities from 6.8 to 7.5 g/cc. Compaction pressures as a high as 1000 MPa demands special considerations in tool design and powder material. 2. Sintering

Powder Metal Gears

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During sintering the particles in the gear are bonded together. It is not a melting process but a diffusion process that requires time, temperature, and atmosphere. Typical process data is 1120°C–1280°C for twenty minutes. Before that there is a de-lubrication step where the solid lubricant in the powder and part is burnt off. This has to be done under controlled time and temperature to allow the lubricant to turn into gas and leave the part. The whole sintering process takes around three to four hours and is commonly done in a continuous mesh belt furnace or a batch furnace when the temperature is closer to 1280°C. The physics involved is well described in [1]. 3. Post sinter processing Depending on strength requirement different heat treatments can be applied, grinding or honing for tolerances, densification for more strength, peening, etc. But after compaction and sintering the result is a gear with strength properties sufficient for power tools, starter motor gears, and most lawn and garden machinery gears.

19.3.1 COMPACTION The compaction press in combination with the tooling and material set the boundaries for what is possible to manufacture. It is possible to estimate if a part is compactable or not, the steps below outline the process. 1. The surface area of the gear in combination with the density required sets the compaction force necessary. Assume that for 7.2 density gear, 800 MPa compaction pressure is necessary. If the surface area of the gear is 100 cm2 the required force needed is pressure times area and the press force has to be matched by the press. Common press sizes are 500–800 tons but there are a few 1200–1600 ton presses that has the necessary capability for advanced gear production. The in­ dustry uses tons as a measure for press force or press size. 2. The height of the gear is limited by the density variation that is acceptable in the compaction direction together with the fill height available in the tool. As a rule of thumb the non-compacted powder requires a tool cavity height (aka fill height) that is 2.5 times the finished component height. It is calculated as the ratio between finished density divided by fill density. Typical values could be 2.6 g/cc fill density and 7.0 g/cc finished part density, see Figure 19.2 which illustrates how the compaction pressure in the powder column is reduced with the height of the column. The admixed lubricant that is in the powder will reduce the axial variation of (x ) in Figure 19.2. 3. Shallow helix angles can be compacted without an active drive system of the upper punch, but tools with helix angles higher than approximately 5° benefits from an active drive system. The height of the gear also plays a role, the higher the component (wider face width) the more urgent will an active drive system be to avoid wear of the tooling, ideally there should be no contact between punches and tool die. The active drive system synchronizes the punch into the die but also controls the rotating motion of the punch. The entry of the punch into the die has to be syn­ chronized also for compaction without a drive system, clearance is a few micrometers between tool parts and die, any interference while the punch is entering the die will destroy the tooling. See Figure 19.3. 4. There is also a limitation to how high a component can be to be successfully ejected. That height is around 55 mm but higher gears exists and with die wall lubrication (DWL) heights of 65 mm can be achieved. One way around this limitation is to compact two or even three parts and join them during sintering by brazing or by welding in the sintered state. DWL will also increase density since lubricant level in material mix can be reduced. 5. Maximum helix angle is determined through experience and geometry. If the part is very high then 36° helix angle becomes difficult to eject out of the die, but for a gear height of 30 mm then helix angles of 32° are common. Tool makers experienced in helical gearing have this knowledge.

1024

FIGURE 19.2 punch.

Dudley’s Handbook of Practical Gear Design and Manufacture

Axial stress in a powder column as a function of distance x from the face of the upper compaction

6. Cross holes, meaning holes perpendicular to the press direction are normally machined, often before sintering which is very economical since tool wear on drill bits and cutting inserts is very low. There exists solutions to create holes and features perpendicular to press direction but they are quite rare in production and have limitations. 7. Ring gear with internal teeth for epicyclical gear trains are commonly made with PM technology. For bigger ring gears in for instance automotive transmissions where demands and tolerances are high a drive system for the lower punch is necessary (as opposed to outside gear teeth were the upper punch is driven). This requires a sophisticated press and tooling and without it the density distribution in the gear will affect the distortions of the gear after sintering as well as durability. Since internal gear grinding adds cost to correct this and to ensure structural integrity of the part it is strongly advised to ensure that the tools and press are fit for the job. Figure 19.4 shows a cross section of a ring gear with typical uneven density originating from poor compaction lacking the proper drive system of the tool parts. Powder forging is a very common method for making fully dens gears and parts. The process requires special material grades that have tighter specifications on impurities and nonmetallic inclusions. The process steps involved are: 1. Compaction of powder to a preform, sometimes with gear teeth sometimes without. 2. Sintering but without allowing the part cool down transfer to-. 3. Hot forging in a separate forging press with tool lubricated with graphite slurry. Gear teeth and other features are sometimes formed in the tool so not only compaction for porosity removal. 4. Machining of any features not formed by the tool. 5. Hardening. 6. Hard finishing.

Powder Metal Gears

FIGURE 19.3

1025

Helical gear tool part movement, note that the die is moving as well.

Helical, spur, and spiral bevel gears are made with this process for automotive transmissions and final drives. Advantage is higher strength than with conventional machining and less scrap and chips are produced. Double compaction and double sintering (2P2S) is a method to increase density in a part where the part is first compacted and sintered at a low temperature (850°C thirty minutes) and then compacted again and sintered at normal temperature and time. To reach 7.25 g/cc in a part a compaction pressure of 800 MPa is needed. This puts a lot of stress on the tools and the tool has to be built without any thin sections that could potentially buckle or break. Tool cross sections less than 4 mm in width should be avoided at these pressures. To be able to make a thin rimmed gear with cross sections that require punches that are 2.5 mm thick at 7.25 g/cc, compaction at 490 MPa, sintering at 850°C and then re­ pressing at 490 MPa will achieve the same final density (7.25 g/cc). Another scenario could be that the press size is not available that can reach 800 MPa, then 2P2S can be a solution. The drawback is of course increased manufacturing cost. In order to increase density warm die compaction or warm compaction, where the powder is heated as well, can be used, see Figure 19.5. Of the two methods, the warm die compaction is the easiest to work with and after the temperature stabilizes in the die, normally twenty to forty parts, the friction energy

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FIGURE 19.4 Example of uneven density in ring gear as result of poor compaction.

from the compaction cycle is sufficient for heating the die and for certain parts the die will even require water cooling. Compaction at higher temperature (60°C–100°C) will give 0.1–0.2 g/cc higher density with the corresponding increase in strength, see Figure 19.6.

19.3.2 SINTERING The sintering process has been touched upon previously in this chapter. Here, a more in depth review is given. Sintering is a thermal process and is governed by the following parameters: • • • • •

Temperature and time Geometrical structure of the powder particles Composition/cleanliness of the powder mix Density of the powder compact Composition of the protective atmosphere in the sintering furnace

Powder Metal Gears

FIGURE 19.5

1027

Different temperature assisted compaction processes and their relative density increase.

FIGURE 19.6 Green density and green strength of Premix powders for room temperature (RT) compaction and Densmix powder for warm compaction containing Distaloy AQ + 0.6% graphite + 0.6/0.8% lubricant.

Of these five parameters, three relate to the material; structure, composition/cleanliness, and density. Density is a function of lubricant, structure, and cleanliness/oxygen level when the compaction pressure is held constant so the result from sintering is very much dependent on the quality of the material used as well as the ability of the furnace to keep stable temperature and atmosphere. Temperature and time regulates power consumption, maintenance, and cost. Higher temperature shortens the time but cost increases as well as geometrical distortions, however the particles sinter together better at higher temperatures and pores become rounder which improves mechanical properties. The geometrical structure of the powder particle affects in that smaller particles sinter better but are harder to compact. The composition affects the speed of the diffusion of alloying elements. The cleanliness in the sense of presence of oxides, affects the diffusion process negatively and will affect the strength of the diffusion bonds. The density affects the sintering with the size and number of connection points (sintering necks). The higher the density the more surface area is in direct contact between particles and facilitates diffusion of atoms. There is no melting of the steel, when Copper is added the Copper particles melt but the steel is not melting, it is a diffusion process.

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The composition of the protective atmosphere has to be balanced with the material composition. Reduction of oxides is desired but without decarburizing the surface of the part. The most common examples were given under the sinter hardening discussion. The sintering process is very important for gear durability, toughness, and geometric quality. Most of the distortion takes place in the sintering process and less happens in subsequent heat treatments, whatever they may be, see [6].

19.3.3 POST PROCESSING Post processing of PM gears involve many of the common steel gear processes such as hardening, peening, hard finishing, the list could be made long just as in conventional gear technology. The most common processes used will be dealt with here as well as some processes that should be used with care when working with PM gears. The more generic PM processes such as steam treatment and tumbling that is sometimes used with PM gears will not be covered here, they are well described in both MPIF-35 and AGMA 6008. The focus will be on more gear specific processes. 19.3.3.1 Roll Densification The most common method of removing surface porosity to improve durability of the gears is the roll densification. Millions of gears have been made for oil pumps in heavy duty vehicles, timing gears in cars and the roll densification has also proven itself in car transmissions. The process has been researched thoroughly and there are plenty of references [7–9]. In this process the PM gear is rolled between two master gears that will form the involute and lead by plastic deformation, into the desired pre heat treatment shape. The introduced geometric modifications are on the sub millimeter scale. Compensation for heat treatment distortion is introduced as well as removal of pores in the surface. The bottom land between the gear teeth is normally not touched but root will be densified which improves bending fatigue. Also contact fatigue is improved and a densified gear will have very similar fatigue properties to a wrought steel gear. The process is fully automated with cycle times of around eight to ten seconds depending on the diameter of the gear. Helical gears, spur gears, and internal gears can be processed with this technology. The setup can be seen in Figure 19.7. Another aspect of this process is the very smooth surface finish that is acquired, close to mirror like. This is beneficial for low noise generation in a rolling sliding contact. See Figure 19.8 for surface comparison of a broached car transmission ring gear and the ground (for getting right pre-form during development, not ground in production) and surface rolled planet gear. Chromium alloyed PM materials densify particularly well, followed by Astaloy Mo mate­ rials. Carbon content should be low since carbon reduces the ductility of the material, which is needed for the repeated plastic deformation that occurs in the rolling process. Typically around 0.2% C works well but 0.4% C will have a negative impact on the densified layer thickness. This means that heat treatment processes that require higher carbon content does not work so well with the rolling process. 19.3.3.2 Hard Finishing Grinding and power honing are common processes for both powder steel and wrought steel gears. The process parameters are the same for conventional steel and PM and protuberance, and stock material to be removed on the flank, can be kept the same as for conventional steel (often 0.1 mm stock). Normally the root is not touched unless the process is validated and retains the compressive stress state and hardness that is normally present after case carburizing. The tolerance class achieved is the same as for the equivalent wrought steel gear. Usually ISO 5-7 depending on which parameter that is studied. 19.3.3.3 Peening and Shot Blasting Peening and shot blasting can be used with PM gears in order to clean and/or improve fatigue strength. The process will induce compressive stresses in the material and the dial in process to set the intensity and coverage for the media used is the same as for a solid steel gear. Peening in the non-hardened state is

Powder Metal Gears

1029

FIGURE 19.7 Roll densification set up. PM gear in center. Process time is typically eight to ten seconds depending on size of gear.

possible but renders a surface with poor surface roughness that is not suitable as a contacting surface. Peening will seal up the surface porosity and create a dens layer of around 0.05–0.1 mm. 19.3.3.4 Hardening Some hardening processes were mentioned earlier and of particular interest in gearing is Case carbur­ izing (CQT), Nitriding, Nitrocarburizing, Sinter hardening, and Induction hardening. The CQT process was discussed earlier, in addition to the regular the CQT process there is the low pressure carburizing process (LPC) which is commonly in use in automotive industry for gear hardening. This process works very well with PM materials and is especially suitable for Chromium alloyed PM steels that would be prone to oxidation in the CQT process. There is also the option of a combined furnace that does sintering and LPC as one unit which is a very short process chain to make high end gears. 1. Compaction. 2. Sinter and LPC as one process. 3. Hard finishing. Such a process produces gears with the same ISO quality class as ground wrought steel gears with 75% pitting and bending fatigue resistance at high production speeds (6–7 gears/minute) and low cost. If equal strength to solid steel gears strength is required a roll densification process would have to be incorporated between the sinter and the LPC process. Nitriding is commonly used with PM. Plasma nitriding is preferred since the diffusion of nitrogen can be controlled much better with the plasma. Ring gears in automotive applications typically require this due to tolerance demands that can’t be met with regular CQT. See [4] for more information. Nitrocarburizing is a variant of the nitriding process where the process gas contains both nitrogen and carbon that will allow for hardening on the surface. The plasma process is preferred due to the higher control of deposition of atoms. Salt bath techniques tend to create a network of nitrides in the part and not a defined surface layer.

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FIGURE 19.8 Surface comparison of broached (upper measurement) and rolled (lower measurement) gear flank. Same scale in both measurements. The rolled surface has a mirror like finish with roughness values in the lower range of a super-finished gear.

Powder Metal Gears

1031

Sinter hardening was discussed earlier. One aspect of sinter hardening is that it is done at relatively high temperature and for relatively long time which affects distortion of the parts. This can be controlled to a certain extent using the material composition as well as using jigs to support the parts. More info on the thermal stability of parts during sinter hardening is given in [10]. 19.3.3.5 Wire EDM Wire EDM should be done with care. The surface tends to decarburize by the discharge between wire and material and that leads to lower than expected performance. Leave 0.5 mm or more material that has to be machined off if the wire cut surface is under stress. 19.3.3.6 Welding Welding of PM is done on industrial scale using laser, ion beam, or capacity discharge. There are a few pitfalls to consider. • If machining of the part has been done before welding and cutting fluids have been used, removal of cutting fluids from the pore system is important. • High carbon levels should be avoided just as when welding wrought steels. This happens when synchronizers are welded to the case hardened gear for instance. Carbon levels in the case har­ dened surface are much too high for easy welding. Solution is to allow for extra material that is machined away before welding. • The joint between parts is sometimes designed in the mold with some extra material to fill up for the removal of the pores in the weld zone. • Welding steel to PM is straight forward. • Welding in vacuum is more forgiving and renders less pores in the weld. Electron Beam (EB) welding will therefore require very little process tuning as compared to laser welding that is not done in protective atmosphere. • Adding material during the welding process is possible. • The higher the density of the PM gear the better will the weld be. • It is possible to just put parts in contact in the sintering furnace and they will sinter together and form a strong bond, brazing and welding is stronger still, see Figure 19.9.

19.4 TOLERANCES The tolerance class achieved after each production step varies. Sometimes the process improves the tolerances like repressing or rolling, sometimes it will deteriorate the tolerances, like with thermal processes. In [8] an investigation of the tolerances of PM gears are made and gives information on

FIGURE 19.9

Examples of two parts designed to be sintered together.

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materials and processes that has lower or higher distortions. Generally the more punches that is in the tool the wider will the tolerances have to be since the runout is negatively affected. This is because all tool parts needs clearance and when those clearances all line up, the core rod forming the bore of the gear is pushed off-center affecting the results for other parameters such as spacing error and form when measured. See Figure 19.10. Typical clearance between tool parts is 15 microns, so this means that theoretical maximum run out will be 65 microns for the gear in Figure 19.11 below. The gear in Figure 19.11 needs three upper punches and three lower punches plus a core rod forming the spline. The runout will be somewhat improved when grinding the flanks and tolerances could be met for this particular gear in the application. In the PM toolbox there is a sizing process which is a variant of the 2P2S process where certain parts of the gear needs tighter tolerances and are therefore put in a tool for the second compaction that targets the high tolerance areas of the gear (as opposed to the whole gear) and plastically deforms the part into tighter tolerances and increased density. Coining is a sizing operation that also increases density, there is some confusion around the differ­ ences between coining, sizing, and 2P2S. AGMA 6008 mentioned below has more information on this.

19.5 PRODUCTIVITY Powder metal gears are attractive because of relatively lower cost. The lower cost comes from: • Low scrapping levels, the material cost for a complete PM gear and a wrought steel gear is normally about the same even though the price per kilogram for PM steel is roughly twice that for traditional gear steels. • High manufacturing speed, the gear in Figure 19.11 can be compacted at 6.5 gears/minute only requiring gear tooth grinding after heat treatment. • Lower number of manufacturing steps compared to conventional gear manufacturing. • This leads to second order effects such as smaller foot print of shop, less conveyors, and robots to transport semi-finished parts. Less scrap that needs handling, less area to clean, heat/cool, and maintain. The PM compaction and sintering process is a very robust process, with good process control the scrap rates come out very low, the process can run automated using a minimal crew to monitor process data and provide material for the compaction press. Tool wear is relative to a hob or broaching tool orders of magnitude lower and a tungsten die insert lasts for several hundred thousand parts, sometimes more than a million parts can be made from the same die. The punches can be reground three to five times and will last for several hundred thousand parts with some re-grinding over time. They are usually made of high speed steel but rarely coated like with cutting tools. This makes refurbishment quicker and normally the manufacturer or tool maker has one set of punches in store to keep production going during tool maintenance. Running a press two or three shift with 6.5 strokes/minute will produce 200,000 gears per month which is what is needed for a year’s production of many car transmissions. The press can be re-tooled and reprogrammed in a few hours to run the mating the gear or any other part. This is why large scale PM parts manufacturers wants high volume orders, to be competitive the press occupancy level should be 80% or higher. The investment cost is relatively high for a top notch CNC compaction press, on the same order as the combined cost for a multi axis lathe and a gear hobbing machine. The compaction tool for a helical gear is also significantly higher than for a hob and cutting inserts but actually on the same order of magnitude as a 250 mm diameter ring gear broach. To be able to pay for the investments, high press occupancy is needed.

Powder Metal Gears

1033

FIGURE 19.10 Exaggerated clearances between tool parts and how they can line up and create run out error. The more tool parts the larger the run out can potentially be since the clearances add up. Upper sketch shows all tool parts aligned concentrically while the lower sketch shows what could happen under the compaction as a worst case scenario.

1034

FIGURE 19.11

Dudley’s Handbook of Practical Gear Design and Manufacture

Net shaped PM optimized transmission gear for a six speed manual.

19.6 POWDER METAL GEAR MACRO DESIGN There are design opportunities with PM gears that comes with the tooling and compaction process. Generally the gear body can be divided into three sections 1. Hub 2. Web 3. Rim Each of these section can be manufactured with its own punch provided the press has the capability. This means that the hub can have its own height and thickness and can be given keyways, slots, spline, or any other type of geometry that will attach it to a shaft. The web can be positioned anywhere axially, it can also be tapered, spoked, or with holes for lower weight or better cooling flow when stacked in a furnace for hardening or other processing. Its thickness can be controlled and ensure that material is put where it is needed. This also means that less material is used which directly affects cost of part. The rim can be designed for optimum stiffness, a radius can be introduced between rim and web. Also part numbers can be imprinted in the parts with the punches. What has to be kept in mind is that cross sections cannot be too thin, 3–4 mm is needed for robustness of the punches. See Figure 19.12 for some examples.

19.7 POWDER METAL MICRO DESIGN Due to the lower Young’s modulus (E) and Poisson’s number (ν) in PM materials, deflection of a tooth under load will be different for a PM gear compared to a wrought steel gear. The relation between E, ν and density is given by Eqs. (19.2) and (19.3) 3.4

E = E0

(19.2) 0

Powder Metal Gears

FIGURE 19.12

1035

Powder metal gear samples, courtesy of Alvier AG.

0.16

=

(1 +

0)

1

(19.3)

0

where: E0 is the value for Young’s modulus for steel typically 206 GPa ρ0 is 7.86 g/cc for steel. ρ is the density of the PM part ν0 is Poisson’s number for steel, typically 0.3. The variation in Poisson’s number for PM has little to no effect on the deflection of the gear teeth under load but for clarity it is mentioned as function of density and can be incorporated into a material model. The Young’s modulus should be accounted for in the micro design of the gear flanks to ensure good contact alignment as well as when investigating the sensitivity to perturbations in manufacturing. The most robust design for a solid steel gear might not coincide with the best solution for a PM gear. Normally for an external helical gear the micro design modifications are shifted around ±5 microns for the lead crowning and tip relief as an example when working with 7.2 g/cc PM gears. The robust design of PM gears is explored more in depth in [11] and [12]. PM Ring gears with internal teeth can also be optimized in the same way as external cylindrical gears. For thin rimmed gears which are common in ring gear designs, care has to be taken when root optimizing since that tends to create a sharper radius at the bottom land. This increases stresses in the rim, since the material underneath is of insufficient volume, and becomes higher stressed than the actual gear root, see Figure 19.13. There are still optimizations that can be made in the design of the root, bottom land, and rim thickness that will come for free since built in to compaction tooling. But care has to be taken since the increased bottom land stress does not show up unless FEM is used when improving the root form with an elliptic shape.

FIGURE 19.13

Stress increase in thin rimmed internal ring gear at the bottom land after root optimization.

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19.7.1 ROOT OPTIMIZATION Optimizing the root requires computationally intense FEA, the process is described by Kapelevich [13] but there are simpler methods that will reduce bending stress that are built in to some gear software, such as elliptic root shapes. This is not a true optimization but will reduce bending stress in cylindrical gears with 5%–20%. A true optimization would push the stress reduction a few more percent but would also require longer run times in the computer as well as more complex modeling activities. Kapelevich [14] also demonstrated that with the right modifications the contact ratio can be increased due to the increased deflection in PM gears under load. The increased contact ratio will lower stresses once sufficient deflection of the gear tooth is obtained. This design method will also improve the transmission error when a certain deflection is reached since contact ratio increases, see Figure 19.14 where the effective contact ratio as a function of load is depicted.

19.8 STRESS CALCULATIONS OF PM GEARS Calculating bending stress and contact stress for PM gears is similar to working with solid steel gears. Generally speaking, any method for gear stress analysis may be used as long as the PM material is accounted for by adjusting the Young’s modulus and Poisson’s number. The material is treated as a continuum despite the porosity. There is however differences to the governing laws of plasticity of PM compared to wrought steel. ISO 6336 can be used and worth noting is that in ISO 6336 there are several factors rating the gears and the load, most of them will not be influenced by the use of PM but some will be tweaked to better fit the PM material behavior in the gear application. The most important parameter to change is ZE that factors in Young’s modulus and Poisson’s number for the materials that are being used. To calculate allowable bending and contact stress the Method B can be used for case hardened PM steels by using the S–N graphs in Figure 19.15. For other processes and densities the material supplier should be consulted to obtain S–N data or to recalculate the S–N data for a slightly different density. There is also software available that has S–N data for PM gears in their material database. Finite element methods can be used as well as long as the calculations are within the linear elastic domain. Again, use a Young’s modulus and Poisson’s number corresponding the intended density of the

FIGURE 19.14 Effective contact ratio as a function of load for different designs with 2:1 gear ratio using Direct Gear Design method.

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FIGURE 19.15 S–N curves for case hardened PM steel gears. Generated on FZG back to back gear testing machine and pulsator rig. One percent failure probability and 7.2 g/cc density is plotted based on more than forty data points for each curve.

part. There is no special materials model needed for linear elastic calculations, the material is treated as a continuum and no special care is taken to model the pores. For calculations in the plastic regime the right material model has to be employed such as Ponte-Castaneda. Gurson model may be used and is built into certain FEM software but will be less accurate than the Ponte Castaneda. Material models for plasticity of PM is discussed in [15]. AGMA 2001 may also be used but AGMA has a specific PM gear rating standard that follows AGMA 2001, see below.

19.9 PM SPECIFIC STANDARDS There exists a few standards on PM materials and PM gearing. The most useful ones are the MPIF standard 35, AGMA 6008-A88, AGMA 944-A19, AGMA 942-A12, AGMA 930-A05.

19.9.1 MPIF 35 The MPIF standard 35 is a material and process reference manual. It is not aimed at gearing but powder metal in general. The Standard lists materials but in a generic format and does not take into account any difference in material behavior depending on manufacturer, it only relates a code to a steel chemistry. The material nomenclature in MPIF 35 is often used on drawings to specify the material but the spe­ cification is quite wide and in reality a much more tightly specification is needed from the gear and

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material supplier since in practice five different materials can be designed from the same MPIF desig­ nated material. Here the quality control function should be involved to make sure what the specifications should be and how to guarantee that they are fulfilled. Example: FC0208-30 is an iron-carbon mix with minimum yield strength 30,000 PSI. The data sheet in MPIF35 will give general information on the mechanical properties that should be expected from this material at certain density. The MPIF 35 will also contain information around post sinter treatments such as steam treatment and machining as well as corrosion resistance and many other properties. The information is generic and material suppliers often have their own data that will be more specific to their product as well as dependencies on density, lubricant and processes used.

19.9.2 AGMA 6008 AGMA 6008 is an overview of PM gearing and contains a lot of information on processes, design aspects, tolerances, production methods, inspection of PM gears. MPIF 35 is focused on materials and AGMA 6008 is focusing on the design and manufacturing.

19.9.3 AGMA 944 AGMA 944 is a standard that relates different failure modes that can occur with PM gears and what the root cause can typically be. Most of the failures are the same as for wrought steel gears but in some cases the root cause maybe different due to porosity or the production methods.

19.9.4 AGMA 942 AGMA 942 deals with nomenclature around PM gearing and different methods to characterize PM gears such as density and hardness. A grading system of PM gears is also included in the standard.

19.9.5 AGMA 930 AGMA 930 is a standard for calculating the load capacity of external PM gears in bending. The pro­ cedure described closely follows ANSI/AGMA 2001--C95, with changes made for the special properties of PM materials, gear proportions, and types of applications.

REFERENCES [1] Höganäs handbook for sintered components 2. Production of sintered components. https:// www.hoganas.com/handbooks [2] ISO 6336-5. Calculation of load capacity of spur and helical gears. Part 5: Strength and quality of mate­ rials, p. 21. [3] Danninger, H., & Dlapka, M. (2018). Heat treatment of sintered steels – What is different? Journal of Heat Treatment and Materials, 73(3), 117–130. https://10.3139/105.11035. [4] Radzevich, S. (2019). Advances in gear design and manufacture.(pp. 329–362). CRC Press. ISBN-13 9781138484733. [5] Toda, K. (2018). High efficiency and high accuracy heat treatment by ultra rapid carburizing process. Mechanical surface tech (44 ed., pp. 34) [in Japanese]. [6] Flodin. A. (2018). Powder metal through the process steps. Gear Technology, Sep/Oct, pp. 50–57. [7] Peng, J., Zhao, Y., Chen, D., Li, K., Lu, W., & Yan, B. (2016). Effect of surface densification on the

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[8] [9] [10]

[11] [12]

[13] [14]

[15]

Dudley’s Handbook of Practical Gear Design and Manufacture microstructure and mechanical properties of powder metallurgical gears by using a surface rolling process . Materials, 9(10), 846. Flodin. A. (2016). Powder metal gear technology: A review of the state of the art. Power Transmission Engineering, March, 38–43. Klocke, F., Gorgels C., Kauffmann P., & Gräser E. (2013). Influencing densification of PM gears. InG. Schuh, R. Neugebauer, E. Uhlmann (Eds.), Future trends in production engineering. Springer. Warzel, R., & Neilan, A. (2013). Effect of composition and processing on the precision of sinter hardening of powder metal (PM) steels. Advances in powder metallurgy and particulate materials (compiled by D. Christopherson & R. M. Gaisor, Part 5, pp 62–73). Metal Powder Industries Federation. Henser, J. (2010). Micro geometry optimization of PM gears [MSc Thesis]. WZL-RWTH. Mackaldener, M., Flodin, A., & Andersson, S. (2001). Robust noise characteristics of gears due to their applications, manufacturing errors and wear. In: JSME International conference on motion and power transmission, MPT 2001, Fukuoka, Japan, pp. 21–26. Kapelevich, A. (2019). Asymmetric gearing. (pp. 119–137). ISBN 978-1138554443. CRC Press. Kapelevich, A., & Flodin, A., (2017). February 28 –March 3. Contact ratio optimization of powder metal gears: Stress and transmission error reduction. Proceedings of MPT2017-Kyoto The JSME International Conference on Motion and Power Transmissions, Kyoto, Japan. Angelopoulos, V. (2017). Improved PM gear rolling simulations using advanced material modelling. Journal of Advanced Mechanical Design, Systems, and Manufacturing. 11(6).

20

3D Printed Gears Anders Flodin

20.1 INTRODUCTION 3D printing or additive manufacturing (AM) is a relatively new technology for manufacturing not only gears but parts in general. The AM idea has been around since the 1970s as a science fiction concept but already in the 1980s functional lab machines were developed. In this chapter the focus will be on plastic and metal 3D printed gears but the AM playing field is very diverse with concrete house printers, food printers, multi material printers and there are many more variants, technologies, and materials available, too many to be in any way near possible to cover in this chapter. The advantage of AM is that theoretically no tooling is required. In current reality this is not entirely true, and this will be elaborated on further in this chapter. Also, one machine can theoretically print both the gears and the housing, and the shafts, it is however not that simple. In this chapter the layer by layer process, which is the most common process, will be explained. It is used in both plastic and steel type prints and has certain characteristics that are useful to be aware of.

20.2 PLASTIC GEARS The most common material for 3D printed gears is plastic. There are a multitude of materials and technologies to choose from. Some of the most common materials are PLA, ABS, and PET-G, the reason for that is that they are easy to print, and most printers can work with them. There are also more advanced materials that are glass or carbon fiber reinforced through chopped of continuous fiber, or more flexible materials such as TPU.

20.2.1 FUSED FILAMENT DEPOSITION (FFD) Most common method for printing plastic gears and parts is the filament method where a plastic filament on a spool is fed into an extruder head that is heated and the molten plastic is laid down in layers on top of one another to build the part from bottom to top. Typical layer thickness is 0.1 mm but it is adjustable in what is called the slicing software that creates the NC code for the 3D printer. These types of printers are commonly available, and many AM enthusiasts have them in their homes. Common Materials are ABS and PLA but there are hundreds to choose from, from different suppliers. The layers are clearly visible and the surface roughness perpendicular to the layers is higher than along the layers. For a gear this means that the surface roughness will be better in the rolling/sliding direction on the flanks of a cylindrical gear than perpendicular, unless the gear was printed standing up which does not make sense. The tolerances are not very good, but it will be functional and there are plenty of examples in the maker community of functioning 3D printed gear trains made this way. The weaknesses of these prints are the tolerances (around 0.05–0.1 mm) and the intra layer adhesion, meaning the prints don’t exhibit the same mechanical properties in all directions. They tend to break between layers rather than perpendicular to them.

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20.2.2 STEREO LITHOGRAPHY (SL) This is one of the first methods to be commercialized and is used for more professional prints. The equipment is significantly more expensive than the filament method and it is also a bit messier with a bath of photopolymer liquid. UV light is directed into the bath and draws up, layer by layer, the part which cures by being exposed to the UV light. It offers higher resolution printing than many other 3D printing technologies, allowing users to print parts with fine details and surface finishes. Smaller low-cost machines are now available which makes this a more available technology, but it requires some hands-on work to handle the liquid and the fumes.

20.2.3 SELECTIVE LASER SINTERING (SLS) Selective Laser Sintering (SLS) is a 3D printing technology that produces highly accurate and durable parts that are capable of being used directly by the user. It is similar to the LPBF process for steel discussed below but instead of fusing steel particles it fuses polymer powder particles. The parts will require cleaning after printing since it is all in a powder bed. Intricate or hollow shapes have to be designed with an opening so that plastic powder trapped inside a hollow sphere for instance, can be removed by compressed air or brushing. The cleaning is often a manual labor even though there is ongoing work to automate it for series production. Also, the material that is not printed sometimes needs to be discarded since it can be considered contaminated after it has been cleaned away from the part(s). There are a few more variants of these three examples but not all can be mentioned in this context. There is however one technology that stands out and that is the continuous fiber printing technology where a continuous carbon-, glass-, or aramid fiber is laid down with the plastic filament meaning two print heads. The fiber can be cut and applied at high stress areas such as the root of a tooth or around the circumference of the gear as a continuous loop. This will create a gear that will outperform aluminum in stiffness, strength, and weight and per unit weight it will be stronger than steel.

20.3 STEEL GEARS There are several ways to make 3D printed gears in steel. Each has their advantages and disadvantages. Three principal methods can be discerned, the filament method (FFD), the Laser/LED/Electron beam (LPBF) method, and a glue method (BJ).

20.3.1 FUSED FILAMENT DEPOSITION (FFD) In principle the printing is similar to printing plastic gears with a filament. The printer can be a regular off the shelf filament printer. The filament materials that are available at the time of printing are not true steel gear materials but more biased towards tool steels and stainless steels, but there is nothing preventing lower alloyed gear relevant filaments to be produced, the process will work for almost any material that allows itself to be sintered. The filament is similar in composition to Metal injection molding (MIM) material where the binder is a polymer-wax with a high percentage of powder metal that is alloyed to work with the consecutive sintering and hardening processes. Figure 20.1 demon­ strates the principle of FFD. The advantage of this process is a low investment cost but the dis­ advantages are similar to MIM fabrication disadvantages such as slow de-binding and large shrinkage of the gears. On the upside it can create hollow structures without the need for an exit hole for cleaning and requires no machining to remove gears from the build plate. Figure 20.2 shows the inside of a hollow filament printed gear, the print quality is not very good and shows poor adhesion within the layer but the honeycomb structure is easily accomplished resulting in a very light weight gear. Figure 20.2 Illustrates one of the pitfalls in 3D printing, adhesion between layers, it is not as simple as pushing the print button, yet.

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FIGURE 20.1 Principal sketch of a general Filament. 3. Print head with feeder wheels 3a and 3b. 4. Heater element. 5. Print nozzle. 6. Printed part. 7. Heated bed.

FIGURE 20.2 Inside filament printed gear. The honeycomb structure is created by the slicing software and can be adjusted. The print is far from good with poor adhesion between the layers in the cut plane.

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The geometric quality and surface roughness are slightly better with the FFD method than for the laser printed gears discussed below but the print and sinter parameters need to be improved for better adhesion in the example in Figure 20.2.

20.3.2 LASER

AND

ELECTRON BEAM METHODS (LPBF, EB-PBF)

These two beam methods work with a steel powder bed where the particles are sinter-bonded by the beam through local temperature elevation/melting. Several beams can work simultaneously to speed up the process. It is a layered process meaning the parts are built up layer by layer. So, for each printed layer a new layer will be put on top of it using a scraper or roller. Layer thickness is given by the layer of powder that is distributed by the scraper. Figure 20.3 shows the principle build-up and print parameters. As well as the principle of printing and how each layer of powder is distributed with a roller or scraper and then printed on. So also with this method, layer adhesion has to be monitored to avoid poor bonding and reducing anisotropy. There is a balance when printing between the printing parameters in Figure 20.3. Focal diameter, speed, hatching distance all contribute to surface roughness, temperature build up in bed, distortions, phase transformations in the steel, particle bonding, productivity etc. Figure 20.4 shows a build plate after Laser Powder Bed Fusion (LPBF) printing. All the powder covering the parts during printing has been cleaned off. After printing, the whole build plate should be annealed to release the stress build up that has been introduced by the heat input from the laser. The parts have to be cut off, often with a band saw and the parts will be turned since the sawed surface will be quite rough. The gears now have an ISO quality of 10–12 so to be able to use them, further machining is required of the flanks and bore. Figure 20.5 shows two gears, one that is filament printed in steel and one that is LPBF printed. The different surface texture generated with the two methods is visible. Since every layer is scraped/rolled over the last printed layer, any designed cavities in the gear (for weight reduction or cooling channels, etc.) will be filled with powder. This means that a hollow structure like in Figure 20.2 will be filled with powder. The gear body has to have an open honey comb structure which could be potentially undesirable due to the splash losses when in an oil bath, or designed in such a

FIGURE 20.3 Typical print parameters: layer thickness 20 µm, scan speed 1100 mm/s, hatching distance 0.09 mm and bulk exposure was 234 W. Every layer was rotated 67°.

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FIGURE 20.4

Printed build plate, twenty-four hours print time.

FIGURE 20.5 printed.

Close up of as printed surface on gears. Left gear is LBPF printed and the right gear is filament

way that powder can be cleaned out, and oil from running the gears in an oil bath has to be kept out as well. But it opens up for ideas on filling the voids with a polymer for vibration reduction for instance. Also, the powder that is not printed act as support for overhangs and can’t easily be reused without proper closed loop and climate-controlled systems in place for material handling.

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Most materials available are not traditional case hardening materials for gears such as stainless steel, Titanium and Tool steels. This is changing rapidly and at the time of printing there is a 16MnCr5 material available, see [1]. Klee et al. [2] has written an extensive report on the processing of 16MnCr5 FZG gears including powder material, metallography, surface roughness, and strength analysis.

20.3.3 BINDER JET (BJ) The binder jet process has some interesting features that sets it apart from the Laser and Electron beam methods. It is still a powder bed type printing process but instead of a laser, fusing particles together, a binder or glue is sprayed on the particles where they are supposed to bond, similar to an ink jet printer head sprinkling ink on a paper. This allows for very fine details to be printed, see Figure 20.6, much finer than what is doable with laser or filament. The build volume is also smaller than what is normal for laser and filament, typically the parts are not bigger than 50 mm and the build volume is around 200 × 180 ×

FIGURE 20.6 Functioning whistle with small ball inside, on a USB drive for reference.

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FIGURE 20.7 Topographic comparison of BJ printed gear flank (left) and compacted and sintered gear flank (right). Printed with permission [ 3].

70 mm. Surface roughness is around Ra 6 µm after printing and depending on materials the ultimate tensile strength is from 550 MPa for 316L to 1200 MPa for DM427 which is a highly alloyed material. Klee et al. [3] investigated BJ printed gear strength, metallography, cost, and topography. The por­ osity in BJ prints are generally higher than in LPBF. The shrinkage during sintering does not compensate for the relatively low density of the bed material during printing. Figure 20.7 shows a 3D topography image of a compacted and sintered gear flank and the corresponding BJ printed flank. Relatively speaking the sintered gear flank is much smoother but the BJ printed flank is quite good and could be used as is for lower demanding gears, tumbling is sufficient to polish the gears without detrimental alteration of the profile.

20.4 STEELS FOR POWDER BED PRINTING There are a range of materials available for LPBF that is designed for case hardening, generally they have the same chemical composition as the equivalent wrought materials. The powder morphology is quite different for print powder compared to press powders discussed in Chapter 19. When making powder steel through gas or water atomizing the particle size follows a distribution curve, typically like in Figure 20.8. The material can be sieved into fractions or, as in Figure 20.8, measured by laser diffraction. For press powders the size variation that is acceptable is quite wide which means that only the very fine and very coarse particles are removed by sieving, and the rest can be used to make parts. For bed printing the material must be significantly finer and the size variation significantly narrower meaning that the atomizing process for producing these particles has to be designed to give a much narrower distribution curve than what would be necessary in the production of press powders. The usable spread in the distribution curve is normally around 50–150 µm for an AM material, but it depends on the process. For fine detail printing as in the BJ process a finer powder is desired (20–60 µm), while LBPF processes often, but not always as exemplified in Table 20.1, work with somewhat coarser powders as mentioned above.

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FIGURE 20.8

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Typical size distribution for press powders.

TABLE 20.1 Sample Powder Properties for LPBF Material [4] Particles > 63 µm Max 1% Particles < 20 µm Max 8% Particles 20–63 µm Balance Hall Flow Rate 14 seconds/50 g Apparent Density 4.5 g/cm3 Inner Porosity < 0.2%

Figure 20.9 shows typical material particles for powder bed printing with a reference picture to a press powder particle. The AM particles are more spherical and they are, relative to press powders, more homogenous in size and shape and significantly smaller. The level of detail that can be printed is partly determined by the particle size and size of the melt pool from the laser or electron beam. Table 20.1 shows AM material properties and besides what is listed in Table 20.1 there are other factors important to the powder as well, such as the absence of agglomerates and cross contamination. Table 20.2 shows typical mechanical properties from printed test bars. For LPBF gears, a low alloyed case hardening composition works well. 16MnCr5 derivatives are available. Carbon content needs to be low since the LPBF process is essentially a welding process and high carbon content (>0.22% C) is detrimental since it promotes cracks during cooling, see [5]. This would also mean that materials for induction hardening that requires high carbon content (0.4%–0.6% C) would require attention to the process parameters to print well with LPBF.

20.5 POST PROCESSING OF AM STEEL GEARS Due to the relatively poor tolerances after printing steel gears as well as the anisotropic properties, see Table 20.2, post processing is necessary to achieve functional gears. To improve tolerances, machining is necessary after heat treatment. Any hard finishing process valid for wrought steel gears will also work with AM gears, soft machining processes will also work both for steel but also for other metals such as

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FIGURE 20.9 Left: SEM photo of material particles for 3D printing. Right: Typical press powder particle. N.B. The particles are not exactly scaled to one another, but the press powder particles are significantly bigger than the AM particles.

TABLE 20.2 Sample Properties for a LPBF Printed Test Bar, as Printed, No Heat Treatment [4] X-Y Plane Normal

Z Direction

Young’s Modulus

195 GPa

165 GPa

Yield Strength

770 MPa

755 MPa

Tensile Strength Break Elongation

1000 MPa 17%

930 MPa 14%

bronze or aluminum. For steels a soft annealing process may be required to reduce tool wear depending on what phases that are present in the steel, [5] reports hardness levels after printing of 330 HV10. To improve the anisotropic material properties, Hot Isostatic Pressing (HIP), is often used as well as cold isostatic pressing (CIP), see [8]. The HIP process will also change the microstructure of the steel which is often desired after using melt type printing (Laser, Laser diode, Electron beam) since the structure of the steel is difficult to control. This is because the phase transformations that occurs during cooling of the printed and localized melted metal is often unknown and can vary in the part that is printed. HIP also removes any residual porosity and dissolves/removes oxides.

20.6 FINAL REMARKS Making metal gears using additive manufacturing technologies is, by the time of writing this chapter, under development and it is likely going to be a technology for niche gears in the near future. Spare part

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gears on demand, specialty gears, very small fine pitch gears are discussed as the first to market. As the technology develops, the market will follow. It is still not as easy as pushing a print button but the engineers are working towards simplifying the process and making it more accessible, robust, and easy to use.

REFERENCES [1] Gear Technology. (2020, June). Feature article: Additive manufacturing–An update, pp. 24–29. [2] Klee, et al. (2020). Powder characterization and parameter optimization for additive gear manufacturing by LPBF. 61st Conference “Gear and Transmission Research” of the WZL 2020. [3] Klee, et al. (2020), March. Binder jetting – PM gears in small series production. DDMC2020 Fraunhofer Direct Digital Manufacturing Conference, Berlin. ISBN 978-3-8396-1521-8 [4] Material datasheet repository at Hoganas AB Sweden. https://www.hoganas.com/en/powder-technologies/ additive-manufacturing-metal-powders/3d-printing-metal-powders/steel-am-metal-powders/ [5] Reinhart, G., Schmitt, M., Kamps, T., Siglmüller, F., Winkler, J., Schlick, G., Seidel, C., Tobie, T., Stahl, K., & Reinhart, G. (2020). Laser-based powder bed fusion of 16MnCr5 and resulting material properties. Additive Manufacturing, 35, 101372, Elsevier. [6] Cuesta, I. I., Martínez-Pañeda, E., Díaz, A., & Alegre, J. M. et al. (2019). Cold isostatic pressing to improve the mechanical performance of additively manufactured metallic components. Materials, 12(15), 2495. [7] Seede, R., Mostafa, A., Brailovski, V., Jahazi, M., & Medraj, M. (2018). Microstructural and microhardness evolution from homogenization and hot isostatic pressing on selective laser melted inconel 718: Structure, texture, and phases. Journal of Manufacturing and Materials Processing, 2(2), 30.

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Gear Noise and Vibration (NVH) Joshua Götz, Sebastian Sepp, Michael Otto, and K. Stahl

21.1 FUNDAMENTALS OF GEAR NOISE Gear noise may be perceived as tonal sound that is related to one or a few clearly audible frequencies. The noise typically increases in frequency when the speed of the gears is increased and it increases in intensity when transmitted power of the gearbox is increased. Along a certain range of operating conditions, noise may change in intensity. In general, gear noise is dependent on the transmitted load and the running speed of the gears. In most applications, gear noise is unwanted, and has to be prevented. Reasons range from comfort requirements like in automotive vehicles or motor yachts over regulations like for wind turbines to safety like in marine applications. Dramatically increasing dynamic forces may not only yield unwanted noise but also lead to the premature end of lifetime by surpassing the maximum load capacity of the gear mesh. Major efforts in engineering are concentrated on low noise design on one hand and on noise analysis in operation on the other hand. In gear design low noise generation is valued as a major quality criterion, the gear engineer bears a special responsibility for the transmission performance. Some design rules concerning gear main geometry or simple shapes of recommended micro geometry are available to the engineer. Nevertheless, a significant portion of developing silent gears is an iterative trial and error activity. That makes sound understanding of the underlying basics mandatory for today’s gear engineer.

21.1.1 TRANSFER PATH

OF

VIBRATION

The noise and vibration characteristics of gearboxes are mainly determined by the tooth engagement since the gear mesh is considered as the main cause for the vibration excitation of transmissions. Those vibrations are transmitted by the wheel bodies, shafts, and bearings to the gearbox housing, which is subsequently excited to vibrate and excite the ambient air to vibrations that are perceived by the human ear as sound. The noise impression is dominated by the sensitivity of the human ear and by the com­ position of the sound. The latter is evaluated in the field of psychoacoustics. The typical vibration transfer path is exemplarily shown for a single-stage gearbox in Figure 21.1a. The fundamental equation for machine acoustics in level notation is shown in Figure 21.1b [1]. It describes the noise transfer from the standpoint of physics and shows the potential for improving NVHbehavior. To reduce the noise emission of a gearbox it is possible to optimize the radiation level of the housing, for example by increasing housing stiffness with ribs and reducing flat and thin walled areas. Also the structure-borne noise level may be adapted by measures, like increasing housing mass, which usually requires higher wall thickness or material with higher densitiy. These measures are selected not often due to the competing goal of light weight design. An effective way to limit gear noise is to reduce the level of gear excitation. Due to the narrow-banded frequency components of the excitation spectrum of gears, damping, insulation, and design measures for optimizing the radiation behavior of the housing offer a contribution to the general influence of the noise emission behavior, but it is not possible to eliminate the tonality of gearboxes, which is often the most dominant noise portion in perception. Therefore, a reduction of the gear excitation is generally an effective measure for NVH optimization.

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FIGURE 21.1 (A) Transfer path scheme for air-borne sound radiation due to structure-borne sound excitation in the gear mesh [ 2] and (b) corresponding fundamental equation for machine acoustics in logarithmic level notation [ 1].

21.1.2 GEAR NOISE EXCITATION The main source of gear noise is the gear mesh with its engaging teeth. Under continuous loaded operation the source of noise can be attributed to several properties of the gears. These may provide leverage to the gear designer to improve the noise behavior. The following influencing factors on the gear excitation are central: • • • • • • •

Variation of mesh stiffness along path of contact Deviations of the tooth flank as well as pitch and runout errors Extended tooth contact under load Deformation of the shaft-bearing-system Variation of the friction force and its reversal at the pitch point Roughness and surface structure of the tooth flank Periodically changing bending moments of the gears due to the moving contact lines in the meshing area • Air pocketing and churning in fast running gearboxes One of the main influence factors on gear mesh excitation is the characteristic of the gear mesh stiffness while the teeth run through the mesh. The basic influence relates to different number of teeth that share the load in successive meshing positions. Figure 21.2 illustrates the principle for a spur gear

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FIGURE 21.2 The dark teeth are in contact and transmit the load. The resulting mesh stiffness is related to the number of teeth that share the load in one position. The light grey line below illustrates the resulting teeth de­ formation, the variation of deformation is the Transmission Error (TE) [ 3,4].

mesh. Regarding gears with an integer profile contact ratio, nominally always the same number of teeth are in contact. This relates well to the fact that gears with high contact ratios like εα = 2.0 show a low noise excitation. For helical gears the total contact ratio, which indicates the mean number of teeth in contact, is usually larger than two. Also, for helical gears a variation of mesh stiffness has to be expected (see Figure 21.3). Due to the inclined line of contact in a helical gear mesh, the tooth pair stiffness shows a wide variation. Since usually several pairs of teeth share the load in helical gear meshes, the stiffness variation of each tooth pair is of lower influence on the total mesh stiffness than in spur gear meshes. Especially for helical gears with an integer value for the overlap ratio εβ the sum of the length of all lines of contact is the same in each meshing position which explains their low noise behavior. Considering the importance of the load sharing between several tooth pairs in contact, it is clear that also flank deviations have an impact on noise excitation. This also implies a high impact of manufacturing precision which is generally confirmed in many analyses. Standard calculations relate the gear quality to the amount of dynamic load increase during operation (AGMA 2101). The tooth flank shape may be modified to deviate from the involute by design. Flank modifications are applied to balance load deformations of teeth and to prevent edge loading, in fact the modification also is a main optimization criterion to reach low noise excitation. Figure 21.4 illustrates by example how different flank modifications influence the noise exciting tooth force variation. Figure 21.4 shows that unmodified flanks typically lead to an increasing noise ex­ citation level with increasing load. Modification without regard to noise but, e.g., load capacity may show a similar behavior as an unmodified mesh, however the general noise level may increase. Specially designed modifications for low noise excitation may reach the same noise level as an unmodified gear mesh at high loads, but will show a noise level minimum at a specific load, the design load of the modification. For low loads this modification can show a less advantageous behavior. Tooth deformation may affect profile contact ratio, if the amount of deformation is larger than the amount of tip relief under high loads. In this case, the tooth deformation leads to contact between the

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FIGURE 21.3 Principle of changing mesh stiffness. cze is tooth pair stiffness, cz is the resulting mesh stiffness, czγ is the mean mesh stiffness [ 5].

FIGURE 21.4 Example for effect of different tooth flank modifications on gear noise excitation [ 6]. (For definition of tooth force level see Section 21.2.5.)

tooth flank of the following tooth and the tooth tip edge of a tooth of the mating gear (see Figure 21.5). A similar effect is observed at the end of contact. That extended tooth contact increases the profile contact ratio and thereby influences meshing stiffness and noise excitation. Another important influence factor is the deformation of the shaft-bearing-system and its effect on the gear contact. Shaft bending and torsion directly influence the load distribution and should be considered to maintain a full contact pattern. Low noise gear designs with integer number of overlap ratio are known to operate well as long as the tooth contact extends over the full flank. Shaft deformation is taken into account by static simulation calculation. Further impact factors on gear noise are the variation of the friction force which changes direction at the pitch point, roughness, and surface structure of the tooth flank, periodically changing bending mo­ ments of the gears due to the moving contact lines in the meshing area as well as air pocketing and churning in fast running gearboxes and further more. These factors are usually not focused in gear design, but may be scrutinized when noise issues occur during operation.

Gear Noise and Vibration (NVH)

FIGURE 21.5

1055

Extended tooth contact due to deformation of loaded teeth [ 7].

21.1.3 EIGENFREQUENCY

AND

RESONANCE

To determine the dynamic behavior of a gearbox, the gear excitation as well as the dynamic behavior of the structure have to be considered. Every structure shows a specific dynamic behavior with char­ acteristic eigenfrequencies. In areas where the eigenfrequencies coincide with the gear excitation, the resulting system vibrations are increased. In Figure 21.6 a typical resonance curve of a single stage gearbox is shown. If the excitation frequency equals the eigenfrequency fE, higher vibration amplitudes A occur in this area of operation. The higher harmonics of the gear excitation with harmonic orders n = 2,

FIGURE 21.6

Exemplary resonance curve of a single-stage gearbox.

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FIGURE 21.7 Approximation of the torsional gearing eigenfrequency (in Hz) as a function of the center distance. N (Valid for cylindrical steel gears and average gear stiffness c = 20 mm m ) [ 9,10].

3, 4, … result in increased amplitudes at the excitation frequencies f = 1/2*fE, 1/3*fE, 1/4*fE, …. The coincidence of the basic gear excitation (order number n = 1) with the eigenfrequency fE is called the resonance point. Typically, the resonance point shows the highest vibration amplitudes and therefore represents a critical operation point regarding the excitation and noise behavior of the gearbox. To avoid an operation in critical running conditions such as the resonance point, it is important to know the main eigenfrequencies. The most important eigenfrequency of a gear pairing is the torsional gearing eigenfrequency, where both gears vibrate in torsional direction against each other [7]. The main influence on the torsional gearing eigenfrequency of a gear pair are the center distance and the gear transmission ratio. Knowing these two parameters, an approximation for the torsional gearing eigen­ frequency is possible (see Figure 21.7). For example, the eigenfrequency for a gear pairing with center distance 250 mm and transmission ratio 3 is approximately 3250 Hz. The approximation is only valid for cylindrical steel gears for which the gear’s body width is identical to the toothing width and with an N average gear stiffness c = 20 mm m according to ISO 6336 [8].

21.2 CALCULATION METHODS TO EVALUATE GEAR NOISE EXCITATION Calculation methods to evaluate the gear noise excitation are essential since they offer the possibility to characterize the NVH behavior of the gearbox already during the design process in a time and cost-efficient way. Different methods suitable for different kind of development stages exist. Those methods and their associated boundary conditions are described in this chapter.

Gear Noise and Vibration (NVH)

21.2.1 DIFFERENTIAL EQUATION

1057 FOR

VIBRATION PHENOMENA

There are two basic possibilities for determining the excitation and vibration behavior of gearboxes by calculation. The quasi-static and the dynamic consideration. In both cases the solution of the general vibration differential equation Eq. (21.1) [1] is in the focus, but under different boundary conditions M _ x + _C x = x _ x¨ + D

(21.1)

where M

is the mass matrix

x¨

is the acceleration

D _

is the damping matrix

x

is the velocity

_C

is the stiffness matrix

x

is the displacement

F

is the force

Within the dynamic calculation, stiffness C, damping D, and mass inertia M influences are taken into account and thus, the complete dynamic operating behavior is recorded. Since the mesh stiffness is timevariable, Eq. (21.1) describes a parameter-excited oscillation in terms of gearboxes. The solution of the general vibration differential equation is performed without simplified boundary conditions with the help of a time step integration method (explicit or implicit). This method requires much computing time and therefore is not suitable for a large number of calculations. For the quasi-static calculation, the boundary conditions are suitably selected, so that the mass inertia and damping influences are omitted and the general vibration differential equation is simplified to Eq. (21.2) [9]. C x =F

(21.2)

where: C x

is the stiffness matrix is the displacement

F

is the force

21.2.2 TRANSMISSION ERROR

BY

QUASI-STATIC APPROACH

The transmission error under load corresponds to the variation of the deformation of the gear mesh in contact direction and results from a very slow rotation of the gears. In that case, the mass inertia of the components involved and the damping have a negligible influence. The deformation x in a specific meshing position results as a function of the time t from the mesh force F, taking into account the mesh stiffness czi and the deviations xfi from each contact point according to Eq. (21.3) [9]. x (t ) =

F

i czi (t )

xfi (t )

i czi (t )

where x is the transmission error F is the force in the tooth mesh

czi is the mesh stiffness at the specific engagement point i x fi is the deviation of the tooth flank at a specific engagement point i

(21.3)

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The transmission error is mainly used for the comparison of gears with the same main geometry but different flank modifications or deviations, since the amount of transmission error is related to the size of the gears. The design of micro geometries for single gear stages based on the calculated transmission error is a common and widely used approach, which is also comprehensively validated experimentally [5,11,12].

21.2.3 TOOTH FORCE EXCITATION

BY

QUASI-STATIC APPROACH

The force excitation is obtained by assuming a constant deformation of the teeth and no transmission error. The basic approach is opposite to that of the transmission error, under the premise of no mass inertia and direct correlation of elasticity and deformation. The assumption for the tooth force excitation is no deformation and direct correlation of elasticity and tooth force. Under these conditions, the de­ formation of the mesh in contact direction x is constant. Any change in the mesh stiffness leads to a change in the tooth force, thus the force excitation is calculated according to Eq. (21.4) [9]. F (t ) =

(cz í (t ) (x + x fi (t )))

(21.4)

i

where F is the force excitation x is the static transmission error

czi is the mesh stiffness at the specific engagement point i x fi is the deviation of the tooth flank at a specific engagement point i

The force excitation can be used to compare gears with different main geometry and also forms the basis for further level value formation such as the tooth force level or the application force level (see Section 21.2.5) [13].

21.2.4 DYNAMIC

APPROACH

For the computationally efficient evaluation of the excitation and acoustic emission behavior, the transmission error (Section 21.2.2) and the force excitation (Section 21.2.3) are suitable characteristic values. Under real operation conditions, however, natural frequencies as well as dynamic vibration amplitudes and forces also influence the vibration behavior of gearboxes. A critical operating condition occurs if the tooth mesh frequency or another excitation frequency is close to one of these natural frequencies. In this case, a resonance condition is present and the magnitude of the vibration amplitudes depends on the distance between the excitation frequency and the natural frequency, the intensity of the excitation, and on the effective damping. The position of the natural frequencies is mainly determined by the distribution and the size of the masses and stiffnesses of the individual gear elements. The number of considered natural frequencies depends on the frequencies of the excitations. The higher and the broader the excitation frequency spectrum, the more natural frequencies have to be recorded in fine resolution. The most basic substitute model of a helical single-stage gearbox only considers the torsional degrees of freedom ψ1 and ψ2 of pinion and wheel, see Figure 21.8. These degrees of freedom of the two-wheel bodies are coupled by the gear mesh, symbolized in Figure 21.8 by the stiffness c and the parallelconnected damper element k. The rotational motion of the two-wheel bodies is described by the two following differential equations: [14] 1

1

+kx

2

2

+kx

db1 2 db2 2

+ cm x + cm x

db1 2 db2 2

= T1 = T2

(21.5)

Gear Noise and Vibration (NVH)

1059

with: x=

db1 + 2

1

2

db 2 2

where: is the reduced mass is the angle of rotation T is the torque

is the damping k cm is the mean mesh stiffness db2 is the base diameter

The introduced coordinate x describes the deflection of the tooth spring in the direction of the tooth normal force and thus corresponds to the transmission error of the pinion and gear reduced to the basic circles. By transforming both equations, the two degrees of freedom of rotation can be reduced to the translation degree of freedom x on the base circle. The resulting differential Eq. (21.6) [14] corresponds to the differential equation of a single-mass oscillator in Figure 21.8. m x + k x + cm x =

2 T1 db1

with: 1

m= 2

+

1

2

(db2 / db1)2

÷

db1 2

2

(21.6)

The single-mass oscillator according to Eq. (21.6) is equivalent to the substitute model of a single-stage cylindrical gear mesh according to Figure 21.8. The natural frequency of this model is calculated according to Eq. (21.7) [9].

FIGURE 21.8 Single-stage gearbox with torsional DOF of pinion and wheel and associated single-mass oscillator [ 14].

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fE =

1 2

cm m

(21.7)

where: fE is the eigenfrequency m is the reduced mass

cm is the mean mesh stiffness

The scope of dynamic calculations depends strongly on the scope and number of degrees of freedom of the model. As shown in Eqs. (21.5)–( (21.7), a single spur gear stage can be easily described using two torsional degrees of freedom. As soon as several stages, shafts, and translational degrees of freedom are considered, the complexity increases significantly. In this case, it is useful to switch to multi-body simulation programs.

21.2.5 EVALUATION

BY

CHARACTERISTIC VALUES

The calculation methods described in Section 21.2 can be used to calculate the mesh excitation and the vibration response of a gearbox. In order to allow the engineer a fast and reliable evaluation of the vibration excitation, the tooth force level LFZ is defined, which only considers the gear mesh and not the vibration response of the complete drive unit. Due to the direct correlation between vibration ex­ citation and vibration response, it is still possible to make evaluative statements, since a reduction of the excitation leads directly to a qualitatively and quantitatively similar reduction of the vibration response. The tooth force level is based on the force excitation (Section 21.2.3) and its Fourier components of the time course and is calculated according to Eq. (21.8) [9]. LFZ = 10 log[

n

1 (f0 F0 )2

i =1

(i fz Fi )2] dB

(21.8)

where: LFZ F0 fz n

f0 i Fi

is the tooth force level is the reference force (1 N) is mesh frequency

is the reference frequency (1 Hz) is the mesh order is the Fourier amplitude of order i

is the total number of Fourier components

The i-th Fourier component is weighted with the order i, since in subcritical gear systems higher harmonics inevitably come into the range of the natural frequency of the gear mesh during operation. By this procedure, pre-resonances are taken into account in the calculation of the characteristic value. For multi-stage gear units, the bearing force level is a suitable characteristic value for measuring the cross influence of adjacent gear stages on an intermediate shaft. Bearing force means an evaluation of the fluctuating load supported by the bearings. For this purpose, the force excitation of the respective adjacent stages must be vectorially added to the bearing force excitation and then the bearing force level LFL is determined with Eq. (21.9) [9]. LFL = 10 log[

1 (f0 F0 )2

n i =1

(i fz,1 FLi )2] dB

where: LFL F0

is the bearing force level is the reference force (1 N)

f0 i

is the reference frequency (1 Hz) is the mesh order

(21.9)

Gear Noise and Vibration (NVH) fz,1 n

1061 FLi

is the mesh frequency of first gear stage

is the Fourier amplitude of order i

is the total number of Fourier components

Relating the operational range of the drive unit with the characteristic value is the purpose of the application force level. The basis for this characteristic value is a Campbell diagram, which can be created from different databases (simulation or measurement). Figure 21.9 shows an example of a measured Campbell diagram for the torsional acceleration level of a cylindrical gear stage (a) unmodified (b) flank-modified. In red, the operating range to be evaluated is defined by speed, frequency, and order limits and is included in the calculation according to Eq. (21.10) [13]. with: LA, F = 10. log

F¯ (Ord ) =

1 ordo F (Ord )2 db F02 i0rd = ord , u

no (Ord ) i = n u (Ord ) Fi (Ord )

no (Ord ) nu (Ord ) no

no (Ord ) (1)

nu (Ord ) nu

(21.10)

(2)

where: LA, F is the F¯ (Ord ) is the Fi (Ord ) is the F0 is the iOrd is the

application force level mean and operating area rated amplitude mesh force amplitude of order Ord and speed i reference force (1 N) mesh order

ordo, u no (Ord ) nu (Ord ) no nu

is is is is is

the the the the the

upper/lower mesh order maximum speed of order Ord minimum speed of order Ord maximum speed of all orders minimum speed of all orders

Within the framework of the example in Figure 21.9, the application force level LA,F can be reduced by 8 dB by means of an excitation-optimized flank modification [13]. This confirms the suitability of the generated characteristic value for a simple comparison of different gear geometries and flank modification variants.

21.3 MEASUREMENT OF VIBRATION The following section summarizes possible ways to measure and evaluate gearbox vibrations. The measurement of vibration always involves a sensing device that detects the vibratory motion, re­ cording equipment that records the signals from the sensing device, and data processing equipment that converts the raw data to some kind of analytical data in the form of vibration amount at different frequencies.

21.3.1 SENSORS

AND

MEASUREMENT RESULTS

There are different types of commonly used sensing devices to measure the vibrations of a system, e.g., the proximity measurement, the velocity measurement, the acceleration measurement, and the angle measurement. Especially in gear systems the acceleration and angle measurement are widely used methods. q. Accelerometers are often piezoelectric sensors with a crystal, which will produce an electrical charge proportional to the stress induced in the crystal. The output of the sensor is a time-varying voltage signal. Knowing the sensitivity of the accelerometer (e.g., 100 mV/g), the electrical signal can be

FIGURE 21.9 Exemplary Campbell Diagram with predefined area for application force level calculation (a) unmodified (b) flank-modified.

1062 Dudley’s Handbook of Practical Gear Design and Manufacture

Gear Noise and Vibration (NVH)

1063

converted in a common unit of the acceleration like g or m/s2. Figure 21.10 shows a sampled acceleration signal in time domain. For a detailed investigation of the excitation and noise behavior it is often useful to convert the time-varying signal into the frequency domain by a spectrum analysis (see Figure 21.11). One possible way to calculate the spectrum of a sampled signal is the FFT-method. The level of the acceleration in dB is calculated according to Eq. (21.11). La = 10 log

a2 a dB = 20 log dB a0 a02

(21.11)

where: La a a0

is the level of the acceleration is the acceleration is the reference acceleration (1 10

6

m /s2)

The acceleration measurements are often performed during continuous speed run-ups. A typical way to picture the results of run-up measurements are Campbell-Diagrams (see Figure 21.12). CampbellDiagrams may be plotted according to the frequency (Frequency-Diagram) or according to the order (Order-Diagram). The order specifies the measured frequency referenced to a reference frequency. In most cases the reference frequency is either equal to the shaft revolution or to the shaft revolution multiplied by the number of teeth. In Figure 21.12 the reference frequency is the pinion shaft revolution. In the Campbell-Diagram the x-axis shows the measured frequency or the order, the y-axis shows the rotational speed. Based on the continuous run-up measurement, a FFT is performed for every rotational speed. The individual spectra are plotted row by row against the rotational speed. Thus, every row in the Campbell-Diagram represents one spectrum for an specific rotational speed. The amplitudes of the spectra are color-coded and plotted as level in decibel. In Frequency-Diagrams the excitations are visible

FIGURE 21.10

Acceleration signal in the time domain.

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FIGURE 21.11

Acceleration signal in the frequency domain.

as lines through the origin, whereas the system eigenfrequencies are vertical lines. Thereby the Frequency-Diagram is suitable for investigations on the system behavior. In Order-Diagrams the resolution-dependent excitations are vertical lines and the system eigenfrequencies are hyperbolic lines. The Order-Diagram has its benefits in the examination of the excitations. Angle Measurement Another widely used method for evaluating the noise excitation behavior of a gear mesh is the angle measurement. The angle measurement is often done with the help of incremental angular encoders, although other techniques and sensors exist. By measuring the angular position at pinion and wheel shaft, the non-uniform ratio of the rotary motion can be calculated. With knowledge of the involute base diameters of the meshing gears and the time-dependent angular position signals, the loaded transmission error (LTE) can be derived according to Eq. (21.12). x (t ) =

where: x 1

db1 2

db2

is is is is is

the the the the the

loaded transmission error angular position of the pinion involute base diameter of gear 1 angular position of the wheel involute base diameter of gear 2

1 (t )

db1 2

2 (t )

db 2 2

(21.12)

FIGURE 21.12

Campbell-Diagram as Frequency- and Order-Diagram.

Gear Noise and Vibration (NVH) 1065

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Figure 21.13 shows an extract of a measured loaded transmission error covering five mesh periods. The signal has been high-pass-filtered to remove low-frequency signal components, resulting from, e.g., shaft imbalance. Typically, the loaded transmission error is measured at low revolution speed and relates to the quasi-static approach with very slow rotation of the gears in the simulation of the transmission error [see Eq. (21.3)]. In this example the measurement is performed at a revolution speed of ap­ proximately 60 min-1. Similarly, to the acceleration measurement the time-varying loaded transmission error can be converted to the frequency domain by a spectrum analysis. In Figure 21.14 the spectrum of the loaded transmission error is plotted against the mesh order as well as the mesh frequency. The highest amplitudes occur at the mesh order and the higher harmonics.

21.3.2 POSITIONING

OF

SENSORS

AND

OPERATING RANGE

The quality of the acceleration and angle measurements depend on the positioning of the sensors. In Figure 21.15 a test rig with applied accelerometer and angle measuring systems at the pinion and wheel shaft is shown. In operation the mesh force respectively the value of the force excitation of a gear stage

FIGURE 21.13

Loaded transmission error in the time domain.

FIGURE 21.14

Loaded transmission error in the frequency domain.

Gear Noise and Vibration (NVH)

FIGURE 21.15

1067

Test rig with applied accelerometer and angle measuring systems at the pinion and wheel shaft.

[see Eq. (21.4)] oscillates in the direction of its line of action. The vibrations of the gear mesh are then transmitted by the shafts and bearings to the housing surface. To measure the excitation behavior of the gear mesh with an acceleration measuring system, it is reasonable to attach the accelerometer as close as possible to one bearing outer ring and in direction of the line of action for each gear stage. The ac­ celerometer should not be positioned on large free oscillating areas, since the accelerometer will then measure the surface vibration instead of the gear excitation. To reduce the effect of damping, the angle measuring systems should be mounted as close as possible to the gear mesh. With the measurement of loaded transmission error (LTE), the quasi-static excitation for very low rotational speeds of gear meshes can be identified. Due to the low speeds, the LTE is mostly independent of dynamic effects and the dynamic properties of the surrounding system. For this reason, the LTE is primarily suitable for evaluating the noise excitation behavior at low to medium rotational speeds. The measurement of the acceleration retrieve the dynamic characteristics of the gear meshes over the speed. Therefore, the acceleration measurement is rather practical for medium to high rotational speeds.

21.3.3 SOURCES

OF

POSSIBLE ERRORS

The quality of acceleration measurements depends on several parameters of the signal recording and data processing system. A very critical point for acceleration measurements is the digital sampling of the analog sensor signal. In the following, some typical sources of errors will be mentioned briefly. Nyquist–Shannon Sampling Theorem For digital sampling of an analog sensor signal, the Nyquist–Shannon sampling theorem must be fulfilled. Meaning the highest frequency included in the recorded vibration signal must not exceed half of the digital sampling frequency. If the theorem is not satisfied due to high frequencies in the recorded signal or a too low sampling frequency, the frequency spectrum will contain additional false pseudo frequencies. This effect is called Aliasing. Since numerous higher harmonic gear mesh excitations exist in typical gearboxes, the application of an analog low-pass-filter to suppress too high frequencies could be necessary. Filtering the already sampled signal by a software-implemented filter will not work to prevent Aliasing. Aliasing Like already mentioned, Aliasing appears if the Nyquist–Shannon sampling theorem is not fulfilled. Then false pseudo frequencies appear in the frequency spectrum. Figure 21.16 shows the Frequency-

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FIGURE 21.16

Exemplary Frequency-Diagram with Aliasing.

Diagram of an exemplary acceleration measurement with Aliasing. Especially in the higher frequencyarea, there are false frequencies which appear as reflected excitations on the right boarder of the diagram. Electromagnetic Compatibility Both the acceleration and angle measuring systems are susceptible to electromagnetic perturbation. Therefore the measuring systems need an appropriate separation of the power electronic and other electrical sources of disturbance. If the electromagnetic compatibility is of low quality, an increased noise-level will compromise the results of measurements in a negative way. Further Environment Influences The measurement results of the noise excitation behavior of a gearbox are always affected by further en­ vironment influences such as, e.g., the peripheral devices. With the knowledge of these effects, the results could be evaluated on an appropriate way. In the following a few influences of the peripheral devices are listed: • • • • •

Non-uniform rotary motion of the drive unit Vibrations of additional transmission gearings Vibrations of the machine foundation Influences of the frequency converter Vibrations of ventilators, oil units, etc.

21.4 CONDITION MONITORING Automatic damage detection in gearboxes in time before the damages lead to a complete system failure is one major goal of the condition-based maintenance. In Figure 21.17 the service life of a machine part with different strategies of maintenance concepts is shown [15,16]. The preventive maintenance covers concepts where parts are generally replaced or repaired before they are critically damaged. The cor­ rective maintenance covers concepts where parts are replaced or repaired after a failure. In contrast to both other maintenance concepts, the condition-based maintenance allows longer maintenance intervals and less downtime by trying to identify the current state and thus the remaining service life of machine parts as accurate as possible.

Gear Noise and Vibration (NVH)

FIGURE 21.17

1069

Different maintenance approaches [ 3].

For gear condition monitoring approaches the previously introduced acceleration measuring as well as the angle measuring can be used. Due to the simple application of the accelerometer the most industrial condition monitoring systems rely on acceleration data. Using the acceleration to monitor the gear condition is a frequently used method to identify specific gear damage mechanisms like pitting, mi­ cropitting, scuffing, or wear.

EVALUATION

OF

VIBRATION MEASUREMENT DATA

For evaluating the vibration measurement data with the target of detecting different gear damage me­ chanisms, numerous methods are documented and have been practically applied. In principle the pro­ cessing of the obtained signal (acceleration or LTE) is possible in the time or frequency domain. However the time domain based methods are very basic and in the course of increasing computing capacities replaced by more complex methods in the frequency domain or by methods which use both the time and frequency domain to provide time variant spectral representations. The monitoring approach for most of the methods in the frequency domain is based on a comparison of the spectrum of a running machine to one or more spectra of a machine state known as not faulty. The most basic frequency domain condition monitoring approaches just compares the amplitudes of two or more spectra. In acceleration signals an amplitude increase of more than 6 dB is considered as sig­ nificant, an increase of more than 20 dB as serious [16,17].

REFERENCES [1] Kollmann, F.G. (2000). Maschinenakustik. Springer. DOI: 10.1007/978-3-642-57229-6_7 [2] Höhn, B.-R., Heider, M., Stahl, K., Otto, M., & Bihr, J. (2011). Assessment of the vibration excitation and optimization of cylindrical gears. Proceedings of the ASME 2011 International Design Engineering Technical Conferences & Computers and Information in Engineering Conference. [3] Fromberger, M., Sendlbeck, S., Rothemund, M., Götz, J., Otto, M., & Stahl, K. (2019). Comparing data sources for condition monitoring suitability. Forschung im Ingenieurswesen, 83, 521–527. DOI: 10.1007/s10010-01900331-y [4] Götz, J., Weinberger, U., Siglmüller, F., Otto, M., & Stahl, K. (2020, July 12–16). Experimental and theoretical assessment of mesh excitation in mechanical drive trains. 27th International Congress on Sound and Vibration, Prague. [5] Kohn, B., Otto, M., & Stahl, K. (2014, June 2–6). Meeting NVH requirements by low noise mesh design for a wide load range. FISITA 2014-NVH-040, Maastricht, NL. [6] Otto, M., Zimmer, M., & Stahl, K. (2013, September 15–17). Striving for high load capacity and low noise excitation in gear design. AGMA FTM, Indianapolis, USA. [7] Niemann, G., & Winter, H. (2003). Maschinenelemente - machine elements - Volume 2: General transmis­ sions, gearboxes - Fundamentals, spur gears. Band 2: Getriebe allgemein, Zahnradgetriebe - Grundlagen, Stirnradgetriebe. Springer.

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[8] ISO 6336-2019 (2019). Calculation of load capacity of spur and helical gears - Part 1: Basic principles, introduction and general influence factors. Beuth Verlag GmbH. Standard ISO 6336-1. [9] Heider, M. (2012). Schwingungsverhalten von Zahnradgetrieben. Beurteilung und Optimierung des Schwingungsverhaltens von Stirnrad und Planetengetrieben [Dissertation]. FZG, TU München. [10] Houser, D., & Blankenship, G. (1989). Methods for measuring gear transmission error under load and at operating speeds. SAE Technical Paper Series - SAE International 400 Commonwealth Drive, Warrendale. [11] Davoli, P., Gorla, C., Rosa, F., Rossi, F., & Boni, G. (2007, September 4–7). Transmission error and noise emission of spur gears. A theoretical and experimental approach. ASME (Hg.): 10th ASME International Power Transmission and Gearing Conference 2007. International Power Transmission and Gearing Conference 2007, Las Vegas, S. 1–7. [12] Wagaj, P., & Kahraman, A. (2002). Impact of tooth profile modifications on the transmission error ex­ citation of helical gear pairs. Proceedings of ESDA. [13] Utakapan, T., Kohn, B., Fromberger, M., Otto, M., & Stahl, K. (2017). Evaluation of gear noise behavior with application force level. Forschung im Ingenieurswesen, 81, 59–64. DOI 10.1007/s10010-017-0228-y. [14] Gerber, H. (1986). Geräusche als Schwinger in Antriebssystemen. Schwingungen, Geräusche und Laufverhalten von Zahnradgetrieben TAE. [15] Kolerus, J., & Wassermann, J. (2017). Zustandsüberwachung von Maschinen. expert-Verlag. [16] Randall, R. B. (2011). Vibration-based condition monitoring: Industrial, aerospace and automotive ap­ plications. Wiley. [17] Schweigert, D., Gwinner, P., Otto, M., & Stahl, K. (2018). Noise and efficiency characteristics of high-rev transmissions in electric vehicles. E-MOTIVE Expert Forum Electric Vehicle Drives, Conference Paper.

22

Planetary Gear Trains Kiril Arnaudov and Dimitar P. Karaivanov

22.1 INTRODUCTION Planetary gear trains (PGTs) are most often coaxial gear trains (extremely rarely non-coaxial), which are designed to have at least one gear, most often more, called planets, mounted in an element called a carrier (Figure 22.1). The planets perform double rotation—around their geometric axis, and together with this axis around the main (central) axis of the gear train. Thus, the complex motion of the planets is a combination of two rotations: 1. Revolution with angular velocity H performed by the carrier H around the main axis of the gear train. 2. Relative rotation (spin) with angular velocity 2 rel , performed by each planet 2 with respect to the carrier H. This typical planet motion, resembling the motion of the planets around the Sun, is the reason why these gear trains are called planetary. It is their typical distinctive kinematic characteristic. Because PGTs with external meshing first appeared historically, in which the planets executed epi­ cyclic movement, the term epicyclic gear trains (drives) have gained popularity (see [1]). Considering the fact that there is a hypocyclic movement in the case of internal meshing, and in some cases peri­ cyclic, it should be borne in mind that the term planetary is more general and more correct for gear trains with a movable geometric axis of one of the gears [2–4].

22.2 TYPES OF SIMPLE PLANETARY GEAR TRAINS As simple (elementary) single-carrier PGTs [5] are meant the gear trains, possessing the following mandatory characteristics: 1. The existence of only one carrier H (Figure 22.2), which generally rotates and must have an outlet (external shaft), i.e., an external torque acts upon it. 2. The existence of one or more gear wheels with movable axes of rotation, called planets 2, that rest in the carrier H and perform the complicated rotation motions in question. 3. The existence of one or two (not more!) central gear wheels 1 and 3 (with external toothing—sun gear, or with internal toothing—ring gear). 4. The existence of three shafts, coming out of the gear train and loaded with torques, which means the existence of a three-shaft gear train. It does not matter if some of the three shafts are fixed (PGT works with F = 1 degree of freedom), or the three of them are movable (PGT works with F = 2 degrees of freedom). To designate the different types of PGTs, the type of their meshing is chosen here [2,6]: 1. A—for external meshing. 2. I—for internal meshing. 3. In case of one-rim planet—a line above the symbol is placed (for example II , AI —see Figures 22.3 and 22.4). 1071

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О2 2 rel

2

H H

ОH

FIGURE 22.1 Revolution and relative rotation (spin) of the planets in a PGT. (From Arnaudov, K. and Karaivanov, D. CRC Press, 2019. With permission.)

Н 2

2

1

3

1

Н 3

2

2

3

3

Н 2

Н

2

1

i0

0

2 1

i0

0

FIGURE 22.2 Examples of some simple (elementary) PGTs: 1—small central gear wheel; 2 and 2′—planets (with one or two rims); 3—big central gear wheel; H—carrier.

There is a wide variety of PGTs (even simple ones). The most commonly used are cylindrical PGTs, less commonly used are the other types—bevel and worm ones. In Figures 22.3 and 22.4 are shown only the most frequently used cylindrical PGTs from the big number of simple (single-carrier) PGTs. They are two groups depending on the sign of their basic speed ratio i0 . It is the ratio of the angular velocities of the two central gears in their relative rotation to the carrier and has a significant effect on the kinematics of the gear train. The basic speed ratio can be positive (when the directions of rotation of the two central wheels coincide) or negative (when they rotate in opposite directions). Unlike non-planetary gear trains, PGTs can operate in four different ways depending on the degrees of freedom and the direction of power: At F = 1 degree of freedom: 1. As a speed reducer. 2. As a speed multiplier. At F = 2 degrees of freedom: 3. As a summation differential (e.g., at a tween-motor drive). 4. As a division differential. This is illustrated in Figure 22.5 with the most commonly used AI -PGT.

Planetary Gear Trains

2

2

1

3

1073

Н

2 1

1 2

3

1

3

2

Н

3

1

3

2

1

3

2 2

Н

2 1

1

Н

3

3

2

Н

Н

2

2

Н

3 2 1

2

3

4 Н

Н

4

3 Н 2 1

FIGURE 22.3 Positive-ratio simple cylindrical PGTs (i0 > +1). (From Arnaudov, K. and Karaivanov, D. CRC Press, 2019. With permission.)

22.3 SPECIFIC CONDITIONS OF PLANETARY GEAR TRAINS Here are the three specific conditions that must be met in the most commonly used AI -PGT: 1. Mounting (assembly) condition 2. Coaxiality condition 3. Adjacent condition Similar conditions apply to other simple PGTs, but the formulae are different.

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Н 2

2

3 1

Н 2 1

3 Н 2

2 1

The most often used type simple PGT

2

3

3

2

4

Н

2

3

1

4

Н Н

1

3 4 1 3 2

2

1

4

2

3

Н

Н

Н

4

1

Н 2 3

FIGURE 22.4 Negative-ratio simple cylindrical PGTs (i0 < – 1). (From Arnaudov, K. and Karaivanov, D. CRC Press, 2019. With permission.)

Planetary Gear Trains

1075 3=

(a)

0

H=

1=

0

0

РА

РВ

РА

РВ

РА

РВ

РВ

РА

РВ

РА

РВ

РА

(b)

(c) РВ

РА1

РВ

РAH

РA3 РА1

РВ

РAH

РА3

(d) РВ3 РВ1

РА

РВH РА

РВ1

РА

РВ3

РВH

FIGURE 22.5 Working modes of a simple PGT: With F = 1 degree of freedom: (a) as a reducer (reduce the speed); (b) as a multiplier (multiply the speed). With F = 2 degrees of freedom (that is, as differential): (c) as a summation PGT; (d) as a division PGT. (From Arnaudov, K. and Karaivanov, D. CRC Press, 2019. With permission.)

MOUNTING (ASSEMBLY) CONDITION If the PGT is with multiple planets (in principle, all PGTs are at least with k = 2, usually with k = 3 planets), the number of teeth of central gear wheels (sun gear 1 and ring gear 3) cannot be selected at random. They must be selected so that during installation the teeth of all the planets fall against the tooth space of both central wheels. For equally spaced planets (the central angle between the planet centers is the same = 2 / k = const ), the mounting (assembly) condition can be defined as:

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Dudley’s Handbook of Practical Gear Design and Manufacture 2 k

2 k FIGURE 22.6 Uneven positioning of even number of planets (k = 4). (From Arnaudov, K. and Karaivanov, D. CRC Press, 2019. With permission.)

z1 + z 3 = an integer. k

(22.1)

The formula derivation can be found in the specialized literature [2,7]. It is possible to avoid this condition with asymmetric positioning of planets (non-uniform angle ), as shown in Figure 22.6.

COAXIALITY CONDITION This condition requires equality of center distance a w12 and a w23 of both pairs of mesh gears—the sun gear 1 and the planet 2 as well as the planet 2 and the ring gear 3. In the case of gears without profile shift the condition is as follows: z2 =

z3

z1 2

.

(22.2)

However, thanks to the profile shift, this condition can be avoided by choosing z2 less than the one defined above: z2 =

z3

z1 2

z2,

(22.3)

z2 = 0.5, 1, or 1.5 [2].

wherein, as the case

ADJACENT CONDITION This condition ensures that the planets do not interfere when they lie in one plane. It can be seen from Figure 22.7 that the tip circles of the planets (with diameter da2) will not touch at:

da2

p k

aw

Dl

FIGURE 22.7 Planet spacing according adjacent condition. (From Arnaudov, K. and Karaivanov, D. CRC Press, 2019. With permission.)

Planetary Gear Trains

i0

1077

z3 z1 11.7

4.8 3.1 2.5

FIGURE 22.8 Maximum basic speed ratio i0 depending on the number of planets k in case of one-plane planet spacing. (From k Arnaudov, K. and Karaivanov, D. CRC Press, 2019. With permission.)

2 1

2

3

4

5

6

da2 < 2a w sin

k

.

(22.4)

Figure 22.8 shows the maximum basic speed ratio i0 max for different number of planets k .

22.4 MESHING GEOMETRY OF PLANETARY GEAR TRAINS In an AI -PGT, planets 2 are involved in two meshings—an external one with sun gear 1 and an internal one with ring gear 3. In this case, the gear parameters—module m, number of teeth z1, z2, and z3, and profile shift coefficients x1, x2, and x3, cannot be selected independently of each other. Tooth geometry is very specific and has been discussed in detail in the literature [2,7,8]. Figure 22.9 shows three cases of profile shift that occur in AI -PGT. The use of profile shift in general in PGTs gives wide possibilities to the designer.

22.5 TORQUE METHOD FOR KINEMATIC AND POWER ANALYSIS OF PLANETARY GEAR TRAINS In contrast to the methods of Willis [9] and Kutzbach [10] which use the angular velocities ω (or the rotation speed n) and peripheral velocities ν, for the analysis of the PGTs, with the method revealed here, the torques of the three central elements (sun gear 1, ring gear 3, and carrier H) are used. The torque method [2,11–18] is suitable for the analysis of simple PGTs, but reveals its advantages especially in the analysis of complex, compound PGTs [19–22]. It is characterized by the following (Figure 22.10): 1. The well-known symbol of Wolf [23] is used, but modified [2,12,13,16] based on the lever analogy and using a new concept torque ratio t. At AI -PGT, this is the ratio of the torques of sun gear T1 and ring gear T3: t=

T3 z =+ 3 = T1 z1

i0 > +1.

(22.5)

The ideal external torques can be expressed by t as follows: T3 = t T1 and TH =

(1 + t ) T1.

(22.6)

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1. Case Shifted external meshing (x1 + x2 > 0); Null internal meshing: x3 – x2 = 0. aw = a23 > a12

2. Case Shifted external meshing (x1 + x2 > 0); Equal-shifted internal meshing (x3 – x2 = 0). aw = a23 > a12

3. Case Shifted external meshing (x1 + x2 > 0); Shifted internal meshing (x3 - x2 < 0). a12 < aw < a23

FIGURE 22.9 Possible cases of meshing depend on profile shift corrections (modifications). (From Arnaudov, K. and Karaivanov, D. CRC Press, 2019. With permission.)

2. The circumstance is used that the three ideal external torques are in equilibrium, i.e., Ti = T1 + T3 + TH = T1 + t T1

(1 + t ) TH = 0.

(22.7)

In case of work with F = 1 degree of freedom, this equilibrium refers to the torques of the input (TA), output (TB ), and fixed (TC ) shafts, i.e., Ti = TA + TB + TC = 0.

(22.8)

Relationships 22.7 and 22.8 are very convenient for easy verification of calculations, especially for complex compound PGTs. 3. It also uses the fact that the three ideal external torques are in constant ratio, regardless of the mode of operation of the gear train (see Section 22.2), i.e., T1: T3: TH = +1: + t:

(1 + t ).

(22.9)

Planetary Gear Trains

1079 3

Н

2

Т3

ТН

1

FН ТН

t 0

Т1

Т3

F3

1

F1

t

Т1

Precondition:

Torques:

Forces:

FIGURE 22.10 AI-PGT—kinematic scheme, ideal external torques, modified symbol of Wolf, and lever analogy. (From Arnaudov, K. and Karaivanov, D. CRC Press, 2019. With permission.)

4. The ideal external torques—input , output , and reactive (of the fixed element)—are used to determine the speed ratio i in the work of the PGT with F = 1 degree of freedom (not with F = 2!) as follows: A

i=

B

=

TB . TA

(22.10)

5. The real external torques 1, 3, and , respectively and , with the consideration of the losses in the gear train, are used to determine the efficiency of the PGT working with F = 1 degree of freedom: =

PB = PA

T T

B

B

A

A

T B /T A T /T T /T = B A = B TB / TA i A/ B

=

A

.

(22.11)

If the input torque is known in the analysis, which means equality of ideal and real torques T A = TA, the efficiency will be determined by the two input torques—the ideal TB and the real T B , as of course T B < TB : =

T B < 1. TB

If the output torque is known in the analysis, which means equality of ideal and real torques T efficiency will be determined by the two input torques T A > TA: =

TA < 1. T A

(22.12) B

= TB , the

(22.13)

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Dudley’s Handbook of Practical Gear Design and Manufacture

In Figure 22.11, the six possible cases of AI -PGT working with F = 1 degree of freedom are considered (see Figure 22.5) and the corresponding speed ratios i0 are determined. In case of work with F = 1 degrees of freedom, it is not the gear ratio i that is sought, but the resulting output angular velocity B in the twin-motor drive. For example, if the sun gear 1 and the ring gear 3 are driving, i.e., 1 > 0 and 3 > 0, the angular velocity of the output carrier B H will be determined as follows [2]: B

H

=

T1

1

+ T3 TH

3

,

(22.14)

derived from the power’s equilibrium P1 + P3 + PH = 0. The above considerations are for -PGT, but the method is universal. It is enough to determine the magnitudes of the torques of studied PGT: the smaller difference torque TD min (the smallest torque, above it is T1), the larger difference torque TD max (above it is T3) and the summation torque T (maximum torque, above this is TH ). It should be taken into account that in some types of PGTs at different number of teeth of the gears, different central elements have the smaller torque, i.e., the shafts change their role in the gear train. In simple PGTs, this occurs, for example, in AA - and II -PGT [2,24], and in compound PGTs—in many cases [2].

22.6 TYPE OF POWERS, LOSSES, AND BASIC EFFICIENCY OF PLANETARY GEAR TRAINS Unlike non-planetary gear trains, PGTs have the following types of power [2,5,6,25,26]: Input (total, absolute, transmitting) power PA is divided into two types of power (Figure 22.12) from which the output power PB is obtained at the output of the gear train: 1. Coupling power Pcoup , which transmits by coupling movement when PGT rotates as a coupling (as a whole) without relative rotation of gears to the carrier H. Consequently, this power is assumed to be transmitted without internal losses (the losses in the central element bearings are neglected). In the case of AI -PGT with input sun gear 1 and output carrier H, this power can be represented as follows: Pcoup =

PA = TA PB = TA

A H

= T1 = T1

1

at

H

1

=

3

=

H

> 0.

(22.15)

2. Relative (rolling) power Prel , or the power in the mesh that is transmitted by relative movement of the gears with respect to the carrier H: Prel = T1 (

1

H)

= T1

1 rel .

(22.16)

It is the relative (rolling) power that causes the losses in the meshings, bearings, seals, etc. These losses are considered by the basic loss factor 0 , respectively by the basic efficiency 0 . In [27] the following approximate formula for the basic efficiency of AI -PGT is given, which considers mainly meshing losses:

0

=1

0

1

0.15

1 1 1 + + 0.2 z1 z2 z2

1 + z3

,

(22.17)

Planetary Gear Trains

1081 Working as a reducer

Working as a multiplier

i>1

PB PA

i0 3 = T1. t t 0

(22.19)

T1 =

0

T3 T 1 < 3 = T1, resp. T3 = t T1 > t T1 = T3. t t 0

(22.20)

3,

resp. T1 =

Driven sun gear:

In the above formulae T1 and T3 are the ideal torques (determined without taking into account the internal losses) and 1 and 3 are the real torques taking into account the internal losses. Real torques are used to determine the efficiency of the gear train (see Section 22.5, formulae (22.11), (22.12), and (22.13)) and are in equilibrium as ideal torques (see formulae (22.7) and (22.8)) Ti = T1 + T3 + TH = 0,

resp.

(22.21)

Planetary Gear Trains

1083 .Prel 1

Prel

3

0 .Prel

PA

vН

2

v1 vtr

1 Ptr

vrel

1

FIGURE 22.12 Types of power and peripheral velocities in AI-PGT. (From Arnaudov, K. and Karaivanov, D. CRC Press, 2019. With permission.)

Ti = TA + TB + TC = 0

(22.22)

22.7 EFFICIENCY OF PLANETARY GEAR TRAINS Basic efficiency 0 of PGT (with fixed carrier) gives possibility to calculate efficiency for six possible modes of work of PGT with F = 1 degree of freedom (as reducer or multiplier) as well as with F = 2 degrees of freedom (as summing or devising differential), see Figure 22.5. This is relatively easy with the use of torques according to formula (22.11). Real torques are determined by taking into account the direction of relative (rolling) power (formulae (22.19) and (22.20)).

Prel PA 1.0

Prel

0.8

PA

= f (i1H(3))

0.6

0.4

Prel

0.2

PA

4

7

= f (i3H(1))

10

z t= i0 = z3 1

FIGURE 22.13 Changing the ratio Prel /PA of an -PGT with F = 1 degree of freedom depending on the torque ratio t , resp. on the basic speed ratio i0 . (From Arnaudov, K. and Karaivanov, D. CRC Press, 2019. With permission.)

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FIGURE 22.14 Efficiency of an -PGT working with F = 1 degree of freedom as a function of torque ratio t , resp. of basic speed ratio i0 , in an established mode (TB = const), taking into account the scattering of the basic efficiency 0 , resp. the coefficient of friction in meshing z . (From Arnaudov, K. and Karaivanov, D. CRC Press, 2019. With permission.)

Prel

FIGURE 22.15 Possible directions of relative (rolling) power Prel . (From Arnaudov, K. and Karaivanov, D. CRC Press, 2019. With permission.)

When working with F = 1 degree of freedom, depending on the fixed element, efficiency of defined as follows: H

Work as a reducer i > 1

i13(H ) = 13(H )

3

A B

=

0

=

TB TA

=

t=

=0

Work as a multiplier i < 1 z3 z1