Auto Math Handbook HP1554: Easy Calculations for Engine Builders, Auto Engineers, Racers, Students, and Per formance Enthusiasts [Revised] 1557885540, 9781557885548

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AUTO MATH HANDBOOK Easy Calculations for Engine Builders, Auto Engineers, Racers, Students, and Performance Enthusiasts Revised Edition

John Lawlor & Bill Hancock

HPBooks

HPBooks Published by the Penguin Group

Penguin Group (USA) Inc. 375 Hudson Street, New York, New York 10014, USA

Penguin Group (Canada), 90 Eglinton Avenue East, Suite 700, Toronto, Ontario M4P 2Y3, Canada (a division of Pearson Penguin Canada Inc.) Penguin Books Ltd., 80 Strand, London WC2R 0RL, England Penguin Group Ireland, 25 St. Stephen's Green, Dublin 2, Ireland (a division of Penguin Books Ltd.) Penguin Group (Australia), 250 Camberwell Road, Camberwell, Victoria 3124, Australia (a division of Pearson Australia Group Pty. Ltd.) Penguin Books India Pvt. Ltd., 11 Community Centre, Panchsheel Park, New Delhi—110 017, India Penguin Group (NZ), 67 Apollo Drive, Rosedale, Auckland 0632, New Zealand (a division of Pearson New Zealand Ltd.) Penguin Books (South Africa) (Pty.) Ltd., 24 Sturdee Avenue, Rosebank, Johannesburg 2196, South Africa

Penguin Books Ltd., Registered Offices: 80 Strand, London WC2R 0RL, England While the author has made every effort to provide accurate telephone numbers and Internet addresses at the time of publication, neither the publisher nor the author assumes any responsibility for errors, or for changes that occur after publication. Further, the publisher docs not have any control over and docs not assume any responsibility for author or third-party websites or their content. AUTO MATH HANDBOOK

Original edition copyright ©1991 by John Lawlor Revised edition copyright ©2011 by Michael Lutfy Cover design by Bird Studios All rights reserved. No part of this book may be reproduced, scanned, or distributed in any printed or electronic form without permission. Please do not participate in or encourage piracy of copyrighted materials in violation of the author's rights. Purchase only authorized editions. HPBooks is a trademark of Penguin Group (USA) Inc.

Original edition: April 1991 Revised edition: September 2011 ISBN: 978-1-55788-554-8 PRINTED IN THE UNITED STATES OF AMERICA

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NOTICE: The information in this book is true and complete to the best of our knowledge. All recommendations on parts and procedures arc made without any guarantees on the part of the author or the publisher. Tampering with, altering, modifying, or removing any cmissions-control device is a violation of federal law. Author and publisher disclaim all liability incurred in connection with the use of this information. We recognize that some words, engine names, model names, and designations mentioned in this book are the property of the trademark holder and arc used for identification purposes only. This is not an official publication.

CONTENTS Acknowledgments introduction Chapter Chapter Chapter Chapter Chapter Chapter Chapter Chapter Chapter Chapter Chapter Chapter Chapter Chapter Chapter Chapter Chapter Chapter Chapter Chapter Chapter

1: Displacement, Stroke and Bore 2: Compression Ratio 3: Piston Speed 4: Brake Horsepower and Torque 5: Indicated Horsepower and Torque 6: Air Capacity and Volumetric Efficiency 7: Weight Distribution 8: Center of Gravity 9: g Force and Weight Transfer 10: Moment of Inertia 11: Aerodynamics 12: Rolling Resistance 13: Shift Points 14: Quarter-Mile E.T. and MPH 15: Computer Programs 16: Instrument Error and Calibration 17: MPH, RPM, Gears and Tires 18: Tire Sizes and Their Effects 19: Average MPH and MPG 20: Fuel Economy and Cost of Ownership 21: Crankshaft Balancing

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1 8 16 20 27 31 35 41 45 50 54 59 61 65 69 73 77 80 84 88 94

APPENDICES

101

Conversion Factors Bibliography Index

102 115 118

ACKNOWLEDGMENTS When I sat down to process the words that became the Auro Math Handbook, 1 had the support and guidance of several valued friends and fellow editors, writers and photographers. Past editors Duane Elliot of Off-Road, Jim McGowan of Guide to Muscle Cars and Muscle Car-Classics, Spence Murray of Mini-Truck, Mike Parris of Off-Road, Ralph Poole of Trailer Boats, Lcroi “Tex” Smith of Hot Rod Mechanix and the late Tom Senter of Popular Hot Rodding encouraged me to write about automotive mathematics and accepted articles I did on the subject from 1971 through 1990. Those articles were really the start of this book. John Baker and the late Dick Cepek spurred my interest further by asking me to produce brief mathematical pieces for the catalogs issued by their respective firms, John Baker Performance Products of Webster, Wisconsin, and Dick Cepek Inc. ofCarson, California. Former Petersen Publishing Co. librarian Jane Barrett and former editors and writers Dean Batchelor, John Dinkcl, Al Hall, Jon Jay and Jim Ixjscc all suggested useful material for the book.

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Friends Richard Shedenhelm and James J. Scanlan, M.D., read portions of the manuscript and offered excellent criticisms. Gail Harrington, former managing editor of MotorHome Magazine, was an unfailing source of personal encouragement and enthusiasm during the many months I had to spend my evenings and weekends working on the book. Michael Lutfy of HPBooks had the unenviable task of editing the work of a fellow editor, and a highly opinionated one at that! Finally, there's Tom Madigan, a former editor of OffRoad and author of several books himself. He has given me his personal support, not only for this book but at several crucial points in my career. In gratitude, it is to Tom and his wife, Darlene, that Auto Math Handbook is dedicated. Thank you one and all. Without your help, the job of writing the book would have been much more difficult, and it would not have turned out as well as it did.

—John I.awlor

INTRODUCTION If you 're seriously interested in automobiles and how the)7 perform, sooner or later you'll have to deal with mathematics. Virtually all aspects of motorsports, from bore and stroke, through power and torque, to rime and speed, involve mathematical calculations. I recognized this as a young auto enthusiast in the 1950s, and was pleased when I discovered a booklet called Mechanics of Vehicles by Jaroslav J. Taborek. It was a collection of 14 articles about the mathematics of motor vehicle behavior, originally published by Machine Design magazine in 1957. Taborek was a professional engineer and he wrote for his colleagues, not for enthusiasts. Much of his work involved more complex mathematics and for many of us was simply too complex. Then in 1961, an article called "Madi and Formulas for Hot Rodders" by Don Francisco appeared in Hot Rod Magazine Number One. It was only five pages long, but it provided some genuinely usefid mathematics for the aspiring hot rodder, and none of it required more than a grade school background in math. To the best of my knowledge, it was rhe first such compilation especially for car enthusiasts. In the years leading up to the publication of the first edition of Auto Math Handbook, there had been numerous magazine articles and book chapters dealing with various aspects of auto math, but they all had been at one or the other of the extremes represented by Taborek's and Francisco's pioneering and long out-of-print efforts. They’ve been either ponderous professional tomes or frankly sketchy popular works. There had been no book­ length collection of practical, elementary math for auto enthusiasts of average education. That's a gap I sought to fill with this work by concentrating on math of genuine interest to the enthusiast, and avoiding anything too specialized. That dictated a particular emphasis on the engine and drivetrain, which arc the core of true hot

Although there's plenty of arithmetic, algebra and even a little geometry, there's no calculus, so you can relax. However, I did include the formulas for horsepower and torque, despite the fact that their measurement requires equipment not likely to be found in the average hot rodder's garage. The interrelationships of horsepower and torque arc among the most important principles of

automotive engineering, and can't be ignored in a book devoted to automotive mathematics. As a result, the work serves as a useful primer of auto engineering and performance fundamentals, as well as a handbook of auto math. Hopefully it will have particular appeal to younger enthusiasts who are just developing an interest in the technology’ of auto performance. Many of the formulas presented could be worked on a simple four-function arithmetic calculator. However, some of them will be much easier on an inexpensive scientific calculator, with pi and parentheses keys, and a few require a calculator which can find eidier square or cube roots. The problem examples in the text were worked our to eight digits, because that is the capacity of most inexpensive calculators. However, the solutions were generally’ rounded off to no more than three decimal places and sometimes to none at all, depending on the degree of precision that seemed appropriate in each case. In the text, single-digit whole numbers or integers arc followed by a decimal point and a zero, c.g., five is 5-0; numbers less than one—or 1.0—have a zero preceding the decimal point, e.g., five-tenths is 0.5. To enhance rhe value of the book as a reference and make it simpler to look up a specific formula, each chapter concludes with a table summarizing the formulas it has covered. Further, most of the formulas arc written in plain English or easily’ recognized abbreviations such as rpm and mph, rather than in algebraic symbols, to make them as clear as possible to the non-mathematician. A working engineer who happens to sec this volume may criticize the limited attention given to the metric system of measurement—or ro identify it more properly, the Systcmc International des Unites, or S.I. for short. The professional may work with such S.I. units as the merer for length, kilogram for weight and watt for power. However, the kind of enthusiast I had in mind when I wrote this book continues to measure in traditional feet, pounds and horsepower, not in their metric or S.I. equivalents.

—John Lawlor (1991)

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INTRODUCTION I was given the difficult task of trying to improve upon a bestselling classic by bringing it up to modern standards while holding true to rhe original intent of a basic beginner’s math handbook for auto enthusiasts. While die format has not changed, I chose to include formulas and equations from additional areas of interest. In this version, I have included discussions and formulas for vehicle aerodynamics and inertia. Some may want more advanced formulas but the intent was to keep the formulas simple and expose the beginner to the math necessary to build a basic foundation for vehicle performance. By using simple formulas, the reader will have a much better way to evaluate vehicle performance and understand or predict the results from potential modifications. When this book was originally written, dynamometer testing was just beginning to become available to the amateur enthusiast to rent and use. Today there are both engine and chassis dynamometers readily available throughout rhe country. Aerodynamic testing has now become available to the serious amateur and hill-scale wind tunnels are available for rent in selected areas of the country. For an enthusiast who wants to maximize the potential of their vehicle, these tools represent a huge step forward in their development plan.

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Perhaps the biggest recent step forward has occurred in the area of computers. Personal computers and the specialized software that is readily available for automotive applications have greatly expanded within the last five years. Heretofore, mathematical calculations involving complex formulas and iterative calculations have been outside the range of all but the most accomplished engineers and mathematicians. Now a serious enthusiast can access cutting-edge solutions very easily. The problem now becomes one of ensuring that the equation fits the particular solution rather than trying to manipulate a lengthy formula or struggle with the advanced mathematics involved. As technology advances, so do the tools that arc available. In many cases, today’s serious auto enthusiast has access to more computing power on their cell phone or PDA than did the engineers who put a man on the moon forty years ago. By applying die correct formulas and letting the computer do the mind-numbing repetitive calculations, we can solve complex problems in the blink of an eye, or at least in the time it takes to drink a cup of coffee. Have fun learning and using auto math. —Bill Hancock (2011)

Chapter 1

Displacement, Stroke and Bore

A clear understanding of the relationship between displacement, stroke and bore is essential for high performance engine building. Photo by Bill Hancock.

The displacement or cubic capacity of an automobile engine is the sum of the individual swept cylinder volumes. Think of the swept cylinder as a volume about the size of a coffee can, with the diameter being the size of your bore and the height of the can being the stroke of your engine. However, this swept volume is not the total volume of the cylinder. The total volume of a cylinder includes not only the swept volume but also the compressed volume above the piston when the piston is at top dead center. The compressed volume is comprised of the combustion chamber volume plus the volume between the top of the piston, the top or deck surface of the block, and plus the volume of the head gasket. Piston Displacement To find an engine’s overall swept volume, or piston displacement, or simply displacement as it's also called, you must know the engine's bore and stroke as well as the number of cylinders. The stroke is the distance the piston travels in the cylinder from top co bottom and the bore is the diameter of the cylinder. If you have the factor}' figures for the bore and stroke, you would probably have the displacement too. So why do you need a formula for finding it? One reason is that die displacement given in the factory-supplied spec sheets is usually rounded up or down to a whole number or integer by die factory. An integer is a whole number which can be either positive or negative but docs not have any additional fractions such as 1 1/2 or additional decimal components such as 1.562. When it comes to precise engine building, it can be instructive to double-check the stock specs and find the exact figure. If you happen to be building an engine to comply with

pi oRn What is pi? Well, for openers, it is nor a dessert served with ice cream. Pie is the pronunciation for the Greek letter, which in mathematics is called a constant. The rules of geometry tell us that the relationship between the diameter or distance across the widest parr of a circle and the circumference or the distance around the perimeter of the circle is constant. So, no matter how big or how small die circle may be, the value of pi is the same for all circles. The value of pi is 3.1415927. It is a dimensionless number, which means that there is no description such as feet or centimeters diat follows it. For our purposes, we can use 3.1416 and achieve sufficient accuracy for the calculations in this book.

the rules for a specific class, it is vitally important not to exceed the displacement limits for your particular class. More importantly, if you're building a racing engine to compete in a class with a specific displacement limit and you want to increase or decrease the displacement by modifying the stroke or bore, you need to know how to calculate die effects of particular modifications. If you change the stroke or bore, how much will the displacement change? To find the overall engine displacement, you must first find the swept volume of a single cylinder, and then multiply that figure by the number of cylinders. For that, you need to know the formula for die volume of a cylinder as a geometric shape, which is pi + 4 or 3.1415927 + 4, which equals 0.7853982. This number is then multiplied by the square of the diameter

Auto Math Handbook

SYMBOLS We all know what + plus and - minus mean. When we sec / it means divided by. When we see numbers followed by a / that also means divided by. The square of a number is when a number is multiplied times itself and is denoted by a small figure 2 above and to the right of the number or symbol, such as x2. So if D was rhe symbol for diameter, D2 would denote diameter squared or multiplied times itself. The square root of a number is a value, which if multiplied by itself would equal die original number, llic symbol for square root is V The symbol for the cube root of a number, or a value diat if multiplied by itself and then again by the value would equal the original number is 3\ . As we progress, we will define certain values by symbols. Learn what these arc. Not only will they be needed to solve the problems, but they will be useful in conversation. Supposed you heard two people discussing a new car and one said that it had a low "ccc dec." What they referred to was the Cd or drag coefficient of die vehicle which is another dimensionless number that puts a value on die overall aerodynamic shape efficiency, 'rhe old Dodge Daytona and Plymouth Superbird winged cars had a Cd of 0.27, while a cement truck probably has a Cd of 25 (just kidding—but the value is large enougn that nobody wants to measure it).

Cubic Inch Displacement

The swept volume in cubic inches of an individual cylinder is found by multiplying pi 14 by the bore in inches squared by the stroke in Inches. The overall swept volume or displacement of an engine is simply the volume of one cylinder multiplied by the total number of cylinders.

and by the height of the cylinder. In an engine, the diameter of a cylinder is tne bore, and the height is the stroke, so the formula for finding the cylinder volume in cubic inches is pi divided bv 4, multiplied by the bore squared in inches multiplied by the stroke in inches or: Cylinder Volume ■ pi + 4 x bore2 x stroke To find the overall piston displacement of the entire engine, you'd take the value of this equation and multiply that by the number of cylinders in the engine. Example—As an example, let's sec what the displacement would be for an 8-cylinder engine with a 4.0" bore and 3.48" stroke: Displacement = 0.7853982 x 42 x 3.48 x 8 The answer is 349.84776 cubic inches, which can be rounded up to an even 350, because the decimal figure is well over 0.5. Those happen to be the measurements of a common version of the Chevrolet small-block V-8. Decimal Conversion—If either die bore or stroke includes a common fraction, it must be converted to a decimal before being entered on an arithmetic calculator. To demonstrate, take an eight-cylinder engine with a bore of 3 7/8 inches and a stroke of 3 1/4 inches. In decimals, those figures would be 3.875" and 3.25" respectively, and would plug into the formula this way:

Displacement» 0.7853982 x 3.8752 x 3.25 x 8 We got those numbers by taking the fraction 7/8 and using our calculator dividing 7 by 8 to get a result of 0.875 or by taking 1/4 and dividing 1 by 4 to get 0.25. Here, the displacement is 306.62435 cubic inches or, rounded up, 307. This, too, is a variation of the Chevy small-block V-8. On a scientific calculator with parentheses keys, it wouldn't be necessary to prefigure the decimal equivalents of the fractions. Tnev could be entered as follows: Displacement ■ 0.7853982 x (3 + 7/8)2 x (3 + 1/4) x 8 The answer is again 306.62435. But note that plus [+] signs must be used between the integers and fractions. Rounding Up or Down—I mentioned earlier that it can be instructive to double-check factory displacement figures. As a case in point, from 1958 through 1966, Ford built a V-8 engine with a bore of 4.0" and a stroke of 3.5", and advertised its displacement as 352 cubic inches. Then, in 1969, the

Displacement, Stroke: and Bore

ROUNDING OFF When wc round off a number, we shorten the number and thereby decrease its accuracy. So why do we round off a number? We generally round off a number for simplicity and ease of calculation. Wc round oft in real life all the time. When your friend asks "how long does it take to get to the track,' die general answer is "2 to 2 1/2" hours not 2 hours, 12 minutes and 13.7 seconds. When we need accuracy it is there. We wouldn't refer to our car's time at the dragstrip as "something under a minute" unless of course we were racing our cement mixer. So how do we round oft? Let's take the number 3.1415927 (remember pi?). If we wanted to shorten it and still keep most of the accuracy, wc would round it off to four decimal places. To do this we would count five numbers to the right of the decimal point, which in this case would be the 9. If this number is 5 or greater, we would add one number to the value in die fourth place, which is the "5” in our example. Therefore our rounded off number would be 3.1416. Let's try rounding off 27.1237945 to two places. It would be 27.12, because rhe third number to the right of the decimal point is a 3, which is less than 5, so we just keep the second number to the right of the decimal the same.

company introduced a new, lighter-weight V-8 widi the same bore and stroke but, this time, claimed a displacement of 351 cubic inches. To find which figure is closer to the truth, try the formula with a 4.0" bore and 3.5" stroke:

Displacement = 0.7853982 x 42 x 3.5 x 8 l he actual displacement is 351.85838 cubic inches, which should be rounded up to 352 because the decimal is well over 0.5. However, Ford has chosen to round it down to 351 for the later, better­ performing engine, probably to avoid confusing it with the older, less efficient unit. Overboring—Staying for the moment with the earlier Ford 351—uh, 352—imagine that you have a well-worn specimen that needs a clean-up overbore of0.030". How would that aftcct the displacement? To find out, try the formula again with the bore increased to 4.03 Displacement = 0.7853982 x 4.032 x 3.5 x 8 The answer is 357.15604 cubic inches or, rounded

Stroke can be measured by carefully using a one-inch travel dial indicator and moving the crankshaft 1.000" at a time then resetting the indicator until the measurement is completed or use a special long travel indicator. Always double check your measurements. Photo by Bill Hancock.

When the crankshaft is out of the engine, this handy stroke checker uses a special 0-5" travel dial Indicator to make the measure­ ment By placing the two Vees on adjacent main journals and rotating the device, the total stroke length can be obtained. Photo by Bill Hancock.

Cylinder bore can be measured accurately with a dial-bore gauge. This is an essential tool for those who want to do their own boring or honing. Photo by Larry Shepard.

Airro Math Handbook

CYLINDER WALL THICKNESS

When boring or honing cylinders, a steel honing plate approximately 1.0" thick should be bolted to the top of the block using the intended engine fasteners to simulate cylinder head stress. Photo by Bill Hancock.

While you can increase the Bore by 0.140" it is highly recommended that you consider having the bore's thickness checked prior to ever considering a block, to ensure that the block has enough material for cylinder wall integrity after you arc finished. Cylinder wall integrity isn't just about having a cylinder that won’t leak water into the bore, but about a cylinder that has enough strength and rigidity to hold the additional cylinder pressure of" improved performance. It must also remain straight and round while doing so. If the bore is too thin in spots, it will nor be able to be honed properly and end up being crooked and oblong. This bore distortion will result in poor ring scaling, which creates blow by. Rknu by, as rhe name implies, is rhe volume of gases that blow by the piston rings and hence do no useful work on the top of rhe piston. These wasted gases end up in rhe crankcase and cither work their way out the breathers or are rerouted back through die PCV to the intake manifold to try another trip through the combustion precess. Today, both OEM and aftermarket manufacturers have a wonderful selection of highperformance or race cylinder blocks diat feature robust cylinder walls, thick decks, four Ixilt main caps and a host of other features to make them durable and user friendly to the performance enthusiast. While they may add weight and seem a bit expensive, in the end, they are far cheaper than a fully machined and carefully assembled engine that will blow out a thin wall on its first hard pass. A thick-bore race block will make more power because of its ability to maintain straight round bores and keep the deck sealed under die higher cylinder pressures that create higher output.

down, 357 cubic inches. Now, what if you wanted to know how much you could modify cither the stroke or the bore, yet stay within a specific cubic-inch limit? Stroke The formula for die stroke is the displacement, divided by one-fourth of pi, multiplied by the square of the bore, multiplied by the number of cylinders:

Stroke = displacement + (pi -e- 4 x bore2 x no. of cyl.) Prior to boring an engine, cylinder wall thickness should be measured on all cylinders to ensure adequate thickness for your intended overbore. This checker uses sonic waves to determine the thickness. Photo by Bill Hancock.

Example—Suppose you have a car with a latemodel Ford 352—uh, 351—V-8 and you want to

Displacement, Stroke and Bore

race in a class with a limit of 366 cubic inches. You want to keep the stock 4.0" bore but stretch the 3.5" stroke. How far can you go with it? The displacement you arc concerned with is 366, the bore 4.0" and the number of cylinders is, of course, 8:

Stroke = 366 * (0.7853982 x 42 x 8) The maximum allowable stroke within the 366 cubic-inch limit would be 3.6406693" or, rounded down, 3.64", 0.14” more than stock. Bore To find either the displacement or the stroke, you have been using the square of the bore. Conversely when you have the displacement, to find the bore, you will have to work with the square root of the other factors. The formula for the bore is the square root of the displacement, divided by one-fourth of pi, multiplied by the stroke, multiplied by the number of cylinders or: Bore » \ displacement + [(pi + 4) x (stroke x no. of cyl.)]

Example—If, when building the Ford V-8 for that 366-cubic-inch limit describedearlier, you decided to stick with the 3.5 stroke, how big a bore could you use? Bore = \ 366 + (0.7853982 x 3.5 x 8) The answer is 4.0795906”. That is the absolute maximum, so do not round it up to 4.08”, even though the decimal is over 0.5, or you will be over the limit dictated by the rulebookl To demonstrate that point, tty a bore of 4.08" and stroke of 3.5" in the formula for displacement: Displacement = 0.7853982 x 4.082 x 3.5 x 8 Those figures provide 366.07346 cubic inches— and an engine that is too big for a 366 cubic-inch class. As the National Hot Rod Association or NHRA states in its drag race rules, "Any part of a cubic inch is rounded off to the next highest inch." In other words, NHRA officials would consider your overbored engine to have 367 cubic inches, and would dierefore disqualify it. So take advantage of rounding up when you can. But, like Ford with its 352—uh, 351—know when to round down, too.

Metric Displacement In die metric system, the bore and stroke are usually given in millimeters and the displacement in cubic centimeters. If you use the displacement formula as is, and enter die bore and stroke in

The Sunnen CK-10 is a popular type of honing machine found in most serious engine-building shops. In the hands of a skilled operator, it's extremely accurate. Photo by Bill Hancock.

millimeters, the result will be in cubic millimeters, which must be divided by 1000 to be converted to cubic centimeters. You can accomplish diat by changing the formula to: Displacement (cc) « (pi + 4 x bore2) x (stroke x no. of cylinders) + 1000

Example—Let’s try this widi die 6-cylinder engine in Nissan's original Z car, the Datsun 240Z, offered from 1970 through 1973. It had a bore of 83 millimeters and a stroke of 73.7 millimeters, or, in the formula:

Displacement = 0.7853982 x 832 x 73.7 x 6 + 1000 1 hat provides a displacement of 2392.5708 or, rounded up, 2393 cubic centimeters. Converting—However, a simpler, more direct way to find the displacement in cubic centimeters would be to convert the bore and stroke to centimeters by dividing them bv 10 before entering them. In the 240Z, mat would change them from 83 to 8.3 and from 73.7 to 7.37: Displacement = 0.7853982 x 8.32 x 7.37 x 6 The result is again 2392.5708 or 2393 cubic centimeters. In using the formulas for bore and

Auto Math Handbook

stroke, you can work with the figures in centimeters and multiply the result by 10 to convert to millimeters. To continue with the Z car: Tn 1974, the 240Z became the 260Z, with the engine enlarged co 2565 cubic centimeters. The bore was still 83 millimeters or 8.3 centimeters, but the stroke had been increased. You can find the new stroke in centimeters by entering the displacement in cubic centimeters and the bore in centimeters:

Stroke - 2565 + (0.7853982 x 8.32 x 6) The result is 7.9011454 centimeters or, multiplied by 10 and rounded down, 79 millimeters. For 1975, the 260Z evolved into the 280Z, with the engine displacing 2753 cubic centimeters. The stroke was the same 79 millimeters or 7.9 centimeters as in the 260Z, but the bore had been enlarged. Here is die calculation to find the new cylinder dimension:

Bore - v2753 + (0.7853982 x 7.9 x 6) The bore was 8.5994167 centimeters or, multiplied by 10 and rounded up, an even 86 millimeters. Centiliters—If you wonder about the significance of those figures—240, 260 and 280—they represent the approximate engine displacement in centiliters, a centiliter being 1/100 of a liter. A liter, in turn, is 1000 cubic centimeters. Mercedes-Benz, is another make which uses centiliters of engine displacement to designate specific models. The Mercedes 300, for example, has a 3.0-liter engine and the 500 a 5.0-liter. Finally, there's the question of converting back and forth between the two systems of measurement, with inches and cubic inches on the one hand, and millimeters, cubic centimeters and liters on the other. Factors for these conversions and many others will be found in Appendix A beginning on page 102.

According to the NHRA, "Any part of a cubic inch is rounded off to the next highest inch," so be careful how you round off your figures. A few thousandths over could mean disqualification. Photo by Larry Shepard.

Disptacement, Stroke and Borf.

FORMULAS FOR DISPLACEMENT, STROKE AND BORE pi = 3.1415927 or 3.1416 pi 4- 4 = 0.7853982 or 0.7854

Cylinder Volume = pi 4 4 x bore2 x stroke

Displacement = pi 4 4 x bore2 x stroke x number of cylinders

Stroke = displacement 4- (pi 4 4 x bore2 x number of cylinders) Bore = \ displacement 4 (pi 4 4 x stroke x number of cylinders)

Chapter 2

Compression Ratio

The higher the compression ratio, the greater the combustion, which results in greater power. That's the credo most drag racers live by. Photo by Larry Shepard.

The compression ratio of an engine is rhe comparison between the swept volume of the cylinder plus the compressed volume when die piston is at bottom dead center versus the compressed volume when the piston is at top dead center. The compressed volume is the sum of the combustion chamber, the deck clearance volume and the head gasket volume. The higher the compression ratio is, the more the incoming air/fuel mixture will be compressed. And the more the mixture is compressed, the more powerful combustion will be. I lowcvcr, as the mixture is compressed, it gets hotter and there is a danger drat some of it may ignite prematurely. Thar phenomenon is called detonation or knock. Octane Rating The octane rating of a gasoline is a measure of its resistance to knock. The higher the octane is, the greater the knock resistance will be. The greater the resistance to knock, the higher the compression ratio can be. As with anything, there are limits, so if you arc contemplating raising your compression ratio, do some research before you start milling heads or adding high compression pistons. With the declining availability and the rising cost of higher octane fuels available at the pump, we arc limited to lower compression ratios for street driven vehicles.

Calculating Compression Ratio The compression ratio isn't difficult to calculate; it's equal to the stun of the cylinder volume plus the compressed volume divided by the compressed volume, or:

WHY CUBIC CENTIMETERS? The question is often asked "Why do we use cubic centimeters to measure volumes in our engines for compression ratio calculations?" The answer is really quite simple. The device for measuring volumes of this size is called a burette and they are commonly graduated in cubic centimeters. The volume of a cubic centimeter is handy to use and gives a fair amount of accuracy for die task at hand. Somewhere along the line, a burette graduated in cc’s was deemed size appropriate. It makes more sense than using droplets or gallons. Cubic inches could conceivably be used, but it is very difficult to find a burette graduated in cubic inches. So in the end, we use cubic centimeters and convert everything else in the engine to cubic centimeters. For metric engines, where the bore and stroke are already in millimeters, this makes compression ratio calculations really simple.

Compression Ratio = (cylinder vol. + compressed vol.) + compressed vol. The cylinder volume, as explained in the previous chapter, can be found with the formula: Cylinder Volume = (pi -e- 4) x bore2 x stroke There's one slight catch: Combustion chamber volume is usually measured in cubic centimeters or cc’s, so you'll also want the cylinder volume in cc’s. That means entering the bore and stroke into the formula in centimeters.

Compression Ratio

COMPRESSION IGNITION A diesel engine is classified as a Compression Ignition, or C.l. engine, where compression alone is used to ignite the mixture of air and fuel. Conversely, a gasoline engine which relics on a spark to provide the ignition is classified as a S.I. or Spark Ignition engine. Diesel engines routinely have compression ratios of over 20 to 1. By rapidly compressing a mixture of fuel and air, the heat rise within che cylinder is enough to cause the mixture to ignite. Sometimes during extremely cold starting conditions, diesel engines have to employ pre-heaters and an incoming air-fuel mixture to initiate the burn and get the engine started. Once the engine starts, the pre-heater is no longer needed. By virtue of the fact that a diesel engine uses compression to ignite the air-fuel mixture, the entire ignition system found on a gasoline engine is nor needed on a diesel. On rhe other hand, the diesel engine relies on a high pressure Riel deliver}' pump and a series of fuel injectors to deliver fuel directly into the combustion chamber at precisely die right ume to ensure proper engine operation at that particular load and throttle position. In the past this was accomplished with a sophisticated high pressure mechanical fuel pump diat umed die injector pulse and adjusted the volume for each point of the Riel map. Today, as in gasoline engines, this complex task is easily handled by electronic injectors and a digital engine controller. Because computers can handle a dazzling array of complex computations instantaneously, each revolution of an engine can have different parameters for the timing and fuel volume for the next injector pulse. The computer, or engine controller as it is sometimes called, looks at a variety of sensor inputs and then goes to a series of tables or maps in its memory in order to provide die precise amount of fuel and spark needed at exactly die correct time to match the needs of the engine.

If you already have the bore and stroke in inches, you simply multiply both by 2.54 to convert each to centimeters. If, on the other hand, you have die cylinder dimensions in millimeters, you only need to divide by 10 to find what they are in centimeters. Example—Let's use the 4.0" bore and 3.5" stroke of the Ford 351 V-8 as an example. Multiplied by the conversion factor of 2.54, those figures would become 10.16 and 8.89 centimeters respectively, and plug into the formula thusly:

10 Cubic Inch

TDC

T

90 Cubic Inch

Swept Volume 1 BDC

9.0:1 Compression Ratio The compression ratio is the relationship between the combined volume of the cylinder and compressed volume with the piston at bottom dead center and the compressed volume with the piston at top dead center.

A flat, 1/4" plate of clear plexiglass with a small hole in it, is placed over the combustion chamber, so the fluid used to measure chamber capacity can be seen clearly. Once the chamber is full, the burette is checked and the amount of fluid used is recorded. Photo by Bill Hancock.

Cylinder Volume = 0.7853982 x 10.162 x 8.89 That indicates a cylinder volume of 720.74072 cubic centimeters.

Measuring Chamber Volume Now conies the hard part: Finding the clearance or compressed volume. Because die combustion chamber is irregular in shape, its volume cannot be calculated with a simple formula. You will have to measure it physically and, to do that, you will need a burette marked in cc's and filled with a light oil, cleaning solvent or rubbing alcohol. Water is not

Alto Math Handbook

This illustration provided by Larry Shepard, formerly of Mopar Performance, shows rod, piston and cylinder dimen­ sions. Note the gasket thickness and deck height at the top. Slight as these may be, they affect mea­ surement of the combustion chamber volume.

recommended because it tends to rust the bo-res unless the bore is carefully cleaned and dried after each measurement. The measuring procedure is called, logically enough, cc-ing. CC-ing with Engine Assembled—With die engine fully assembled and mounted on a stand, it should be tilted so the spark plug hole in the cylinder to be measured is vertical. With the piston at top dead center, die valves closed and the spark plug removed,

pour liquid from the burette through the plug hole until it reaches die beginning of the plug threads. The amount poured from the burette will indicate the combustion chamber volume. When utilizing this method it is best to use a thicker fluid such as a light motor oil to prevent it from running past the ring gap in the cylinder. Example—Suppose you cc just one cylinder head in the Ford 351 engine I've been using as an example

Compression Ratio

and find chat the compressed volume is 92.5 cubic centimeters, and apply that in die formula for compression ratio:

Compression Ratio = (720.74072 + 92.5) + 92.5 The compression ratio would be 8.7917915 or, rounded up, 8.8:1. CC-ing with Engine Disassembled—If the engine is disassembled, measuring the combustion chamber volume is considerably more difficult. First, you have to cc the head. Then you have to calculate the added volume that will be provided by the gasket. If the engine had flat-topped pistons which, at top dead center, were perfectly even with the deck, or top of the block, that’s all you'd need. However, that isn't true of many engines. At top dead center, the piston may stop short of the deck height. In addition, if die piston top is dished, or concave, it will increase die compressed volume; if it’s domed, or convex, it will decrease the volume. 1/2" Downfill Method—In HPBooks' How to Hot Rod Snudl-Block Mopar Engines, Larry Shepard of Chrysler's Mopar Performance offers a technique for cc-ing the block he calls the "0.500” downfill" or "1/2' downfill" method. With this method, the head is removed and the engine is positioned so diat die cylinder to be measured is vertical, 'llicn rhe piston is lowered exactly 0.500 inch or 1.27 centimeters from top dead center. The distance is arbitrary. The point is simply to be sure that die entire piston, including the dome, is below die deck and fully witliin die block. Now, with the burette, you can find the amount ot fluid required to fill the volume above die piston and below the deck. By using the formula for cylinder volume, you can find what the volume would be if the piston were flat-topped and at zero deck clearance. The difference between that figure and the volume you find by cc-ing tells you how much a dished piston increases the overall combustion chamber volume or how much a domed piston decreases it. Adding Gasket Thickness—Okay, getting back to your Ford engine, suppose you cc die combustion chamber in one of the heads and find it has a volume of 75 cc's. Your gasket is 0.045" or 0.1143centimeters thick; to find how much that will add to the chamber volume, apply die formula for cylinder volume using the gasket thickness in place of the stroke: Volume = 0.7853982 x 10.162 x 0.1143 That works out to 9.2666664 cc. With flat-topped pistons even with the deck at top dead center, your total combustion chamber volume would be 75 plus

It's not unusual for normally aspirated race engines to run as high as an 12.5to-1 compression ratio. Photo by Bill Hancock.

This instrument, called a dial indicator bridge, is used to measure the deck height, i.e. the distance between the top of the piston and the top of the block. Photo by Bill Hancock.

9.2666664 or 84.266667 cc or, rounded up, 84.27 cc. What would that give you in the way or compression ratio? To find out, add the cylinder volume to the chamber volume, and divide by the chamber volume, or:

Compression Ratio = (720.74072 + 84.266667) + 84.266667 The answer is 9.5526606 or, rounded down, 9.55:1.

Auto Math Handbook

The dome of a piston can vary greatly in shape, complicating the task of determining compressed volume. One method of measuring the dome volume is with the 1/2" downfill technique described in the text Photo by Bill Hancock.

Calculating Chamber Volume If you can use compressed volume to find compression ratio, can you do the opposite and use compression ratio to find chamber volume? Indeed, you can, and it’s a useful thing to know how to do in order to find exactly what chamber volume you need for a specific compression ratio. To find combustion chamber volume, divide the cylinder volume by die compression ratio minus 1.0, or: Chamber Volume = (cylinder volume v (compression ratio - 1.0)

Example—If you wanted to increase the compression ratio of your Ford 351 to 10.5:1, what would the overall combustion chamber volume have to be?

Chamber Volume = 720.74072 + (10.5 - 1.0) or 720.74072 + 9.5 The answer would be 75.867444 or, rounded up, 75.87 cc. You can double-check those results by reverting to the formula for compression ratio:

Compression Ratio = (720.74072 + 75.867444) + 75.867444 Sure enough, that provides a compression ratio of 10.5:1, bringing you right back to where you started.

Milled Heads One of the most popular ways to increase compression ratio is to remove material from die deck face of the cylinder head with a machine called a mill. Hence die term milled cylinder heads. The question then becomes, how much do you mill the heads for a specific increase in compression? We must first realize that only a few combustion

chambers have a round area at the deck surface where they meet die bore. Most cylinder heads have a fiat area called the squish area and a smaller irregularly shaped pocket where the valves arc located. In order to calculate the volume of this pocket, we cc'd it. Now we must calculate how much to remove from the deck face in order to reduce the volume of the pocket. The first step is to find the area of the pocket. Let's say that we did our compression ratio calculations and found that we nad to reduce the volume of rhe combustion chamber by 5-5 cubic centimeters. I he question then becomes: how much must we remove from the deck face of rhe head in order to reduce the volume by 5.5 cc's? When we measured the area of the combustion chamber, we came up widi an area of 6.95 sq. inches. How many thousandths should we remove? Let's start by converting cubic centimeters into cu lie inches. We look in the appendix of this book under cubic centimeters into cubic inches." We find that if we multiply rhe cubic centimeters times 0.0610237, wc will have cubic inches. 5.5 cc x 0.0610237 = 0.33563035 cubic inches = volume to be removed Now if wc take our cubic inches and divide by our total area, we will arrive ar the amount to remove. 0.33563035 cubic inches + 6.95 square inches = 0.048292” Or for simplicity's sake 0.048". If we measured another head but measured in square centimeters and found the area to be 49.05 square centimeters here's how we would do it: Let's say that our calculations revealed that we needed to remove 3.8cc from the combustion chamber volume. How much should we remove from the deck face of the cylinder head? Wc know that our area is 49.04 sq. cm. so we divide 3.8 cubic centimeters by 49.04 square centimeters and the result is 0.077 centimeters or 0.77 mm. Other Ways to Estimate the Area of the Combustion Chamber We can use a technique we call "counting the squares." Simply trace the combustion chamber onto a piece of square ruled graph paper and count the squares inside of die line. Begin by counting only the full squares. Put a dot in each square as you count it. Write down the number of full squares, then count and mark the squares that appear to lie half squares and jot down

Compression Ratio

that number. Next do quarter squares then estimate the rest. Add all of these up and see what you get. If you want to practice, take a compass and draw a small 3" diameter circle on a piece of ,4" ruled graph paper. Count the squares and compare your answer to the true area of 7.08 sq. in. to see how accurate you can be. You can increase your accuracy' by' using graph paper ruled to ten aduations per inch or 100 squares per square inch i ou wish. Displacement Ratio- t he displacement ratio is the cylinder volume divided by compressed volume, and it is always 1.0 less than the compression ratio. For example, the displacement ratio for a cylinder •with a 9.55:1 compression ratio is 8.55:1. Now let's see how we use die displacement ratio. Example—Suppose you want to raise the compression ratio in an engine from 9.55:1 to 11:1. How much do you mill the heads? The existing displacement ratio is, as you’ve seen, 8.55:1, while the new displacement ratio will be 11 minus 1.0 or 10. Begin by subtracting rhe old ratio from the new and dividing the result ov the product of the two ratios multiplied together. That figure is dicn multiplied by the stroke in order to find the amount the heads should be milled. Here's all that expressed as an equation: Amount to Mill = (new disp. ratio - old disp. ratio) + (new disp. ratio x old disp. ratio) x stroke

Milling in Inches—If the stroke on our engine was 3.5", the figures plugged into the formula would look like:

Amount to Mill ■ (10 - 8.55) + (10 x 8.55) x 3.5 = 1.45 + 85.5x3.5 That gives you 0.0169591 times 3.5 or 0.0593567" or, rounded up, 0.060" to mill the heads for our desired increase from 8.55:1 to 11:1. Milling in Millimeters—The formula will also work if you have the stroke in millimeters. Multiplied by a conversion factor of 25.4, a stroke of 3.5" would be 88.9 millimeters. To find how much the heads should be milled in millimeters for the same increase in compression ratio:

Amount to Mill = (10 - 8.55) + (10 x 8.55) x 88.9 = 1.45 + 85.5x88.9 That works out to 0.0169591 x 88.9 or 1.5076608 or, rounded up, 1.51 millimeters to mill the heads in order to increase the compression ratio from 8.55:1 to 11:1. Now, if you've been observant, you'll notice that if you calculated the amount to mill in inches first, you

Here is a polar compensating planimeter. It works very nicely to give surface areas for irregular shapes such as combustion chambers. Photo by Bill Hancock.

wouldn't necessarily need to re-compute the entire formula again to know what it would be in millimeters. All you have to do is multiply the answer in inches, 0.0593567, by the conversion factor, 25.4. The result is 1.5076608, exacdy the same as the answer in millimeters.

Using a Planimeter There is an ingenious device that has been a favorite tool of draftsmen and designers for centuries called the planimeter. I won't try to explain how it works, but suffice it to say that it has served us well until CAD, computer aided design, arrived a few years ago. In five minutes, you can use one of these devices to find the area of any irregularly shaped combustion chamber and armed with the area, you simply multiply by the depdi to get the volume of the surface outlined. Where this device comes in handy is when you try to figure our how much to remove from the head to reduce the chamber volume. loday, you can still buy a new planimeter, but given current sales and the state of the art, it is likely that the planimeter will soon go the way of the old-fashioned slide rule. ’Diev are available on eBay both new and used ranging from several dollars all the way to $1500 each. l*or what we are doing, a fairly simple one will suffice. It will not only let you measure tnc combustion chamber area but also other engine related areas such as die area of Intake or Exhaust ports. 'Hie first step is to determine what scale die device is reading. The instruction booklet will cover the adjustment settings. If you do not have a calibration arm, find a piece of graph paper and trace an area of 2" by 2" and sec what it yields. The answer should be 4 sq. inches or 25.81 sq. cm. If the value is different, adjust the arm and retry until your reading exactly

Alto Math Handbook

A tracing of the combustion chamber outline can be made by holding a piece of stiff paper over the combustion chamber while moving a soft pencil back and forth across the paper to shade the edge of the chamber. Once the shading is done, darken the resulting line representing the outline of the chamber, and make a start/finlsh mark. Photo by Bill Hancock.

Holding the device by the pointer end, guide the pointer around the outline of the chamber, making sure the measuring wheel glides evenly around the paper and doesn't get stuck. Photo by Bill Hancock.

matches your square or circle of a known area. We start with a tracing of the combustion chamber outline taken from the deck surface of the cylinder head. Initially, it is important to correctly position the planimeter, otherwise you will run out of travel and have to start all over again. Make sure that the tracer point can easily reach the farthest point on the chamber outline and still not be fully extended. At the farthest point it is good to still have no more than a 165 deg. angle between the tracer arm and the pole arm. Once you have determined an initial position, anchor the pole weight and hold the trace arm to guide it on a quick trip around the perimeter of the chamber shape. If all goes well and you have clearance, then adjust the dials so that they both read zero. Pick an arbitrary point on the perimeter of the chamber outline and make a small mark at right angles to the perimeter. This mark will serve as your starr/finish line as you travel around rhe perimeter. Carefully place the measuring arm down on your paper so tnat the tracer point is at die start/finish line and then check the dials to ensure that they are still reading zero. Carefully drag the tracer point around the perimeter, being careful to follow the line as closely as possible. Make sure that the paper template stays flat and does not move while you are making your measurements. When you get to the start/finish line, stop and take a reading, ana then without moving anything make a second lap. When you stop at the start/finisn line this time, die value shown on the dials should lie exactly twice what it was for one lap. It may take some practice initially for this to occur, but as you get familiar with the planimeter, you will grow to trust it. Always make at least two laps and average them to get a more accurate result.

Compression Ratio

FORMULAS FOR COMPRESSION RATIO Compression Ratio = (cylinder volume + compressed volume) + compressed volume Cylinder Volume = (pi + 4) x bore2 x stroke

Compressed Volume = (cylinder volume + compression ratio) - 1.0

Displacement Ratio = cylinder volume + compressed volume Amount to Mill = (new disp. ratio - old disp. ratio) + (new disp. ratio x old disp. ratio) x stroke

Chapter 3

Piston Speed

Drag racers need to be especially concerned with piston speed. If it gets too high, the piston will outrun flame front travel and lose power at best At worst, a rod or piston will break. Photo by Larry Shepard.

Piston speed is the rate at which the piston travels up and down in die cylinder, and is usually measured in feet per minute. The rate isn’t constant. At higher rpm, the piston may be going more chan 100 miles per hour near the middle of its Stroke. It slows as it approaches cither end of its cycle, it momentarily comes to a complete stop when it reaches top or bottom dead center, and then accelerates as it starts back in the other direction. In other words, the piston may be going from zero to over 100 mph and back to zero during each stroke—over a distance of only 2.0, 3.0 or 4.0 inches! Most people do not believe that the pistons are stopping twice per revolution in a running engine! The laws of physics tell us that in order to maintain the same path and reverse directions, an object must come to a full stop, even if only momentarily. If the piston speed gets too high, the primary hazard is that a piston or a rod—or in some cases both of diem—may break from the strain. Therefore, when designing an engine, piston speed will also help define a practical rpm limit. Designers all work within generally accepted guidelines for average piston speed in order to make sure that die engine is successful. If the piston speed calculations reveal that the piston speed is too high, the only alternative is to shorten the stroke and increase the bore to keep the displacement rhe same. If the bores get too big, the other alternative is to increase the number ol (Minders. If you study the Formula One engine designs over the years, you can sec these changes resulting from the demand for increasing rpm. A secondary problem is diat a piston may outrun flame front travel—running faster than the expanding air/fucl mixture is pushing it resulting in a drop in horsepower. That's not as serious as breakage, but it's not exactly desirable cither.

It's possible to determine the exact piston speed, as well as the rate of acceleration or deceleration, at any point during the cycle, but it takes differential calculus to do it. Fortunately, you don't have to worry about that. All you need to find is the average piston speed, and that can be done with a relatively simple formula.

Average Piston Speed A piston makes two Rill strokes, one up and one down, during each crankshaft revolution. Therefore, die average piston speed in inches per minute would be two times the stroke in inches, times the crankshaft revolutions per minute, or rpm. The result is divided by 12 to convert it to feet per minute or fpm, and the formula is: Piston Speed in fpm = (2 x stroke in inches x rpm) +12 If you divide both the numerator and denominator in the equation by 2, you can reduce that to: Piston Speed in fpm ■ (stroke in inches x rpm) + 6 In that form, it'll obviously be a little easier to work with. Example—In the early days of hot rodding, when die flarhcad Ford V-8 reigned supreme at the dry lakes and dirt tracks, 2500 feet per minute was considered the maximum practical piston speed—not just for Fords, but for all cars. How did the flathead Ford stack up against that norm of 2500 feet per minute? Introduced in 1932, the early Ford V-8 had a displacement of 221 cubic inches, with a bore of 3.06" and, more to our immediate point, a stroke of 3.75 • l et’s see what its piston speed was at 4000 rpm:

Piston Speed = (3.75 x 4000) + 6

Alto Math Handbook

ROD STRENGTH Most hot rodders know that as they increase the power by adding spark timing and increasing the compression ratio that the rod sees far greater force trying to compress the rod. On the power stroke the gases pushing down on top of the piston greatly increase the compressive load in the rod and hence die force is transferred to the crankshaft to make power. On the exhaust stroke the rod is being pushed upward in tnc cylinder by the crankshaft and it forces the piston up the bore causing the exhaust gas to leave the combustion chamber. As rpm increases however, the weight of the piston becomes detrimental, and as it races toward the top of the bore on die exhaust stroke both the piston and the rod see a tremendous load when they come up for a momentary stop at top dead center (TOO). For this reason, if you are going to increase the power and rpm in an engine, you should seriously consider upgrading or replacing your connecting rods with stronger ones. Some of the high-end engine simulation programs give a value for rod loading. II the load in tnc rod exceeds tne strength, the rod may either break in tension or typically bend in compression.

liston speeds have become possible, but there are still imits. British tech author A. Graham Bell in his xiok, Performance Timing in Theory and Practice, suggested what some of tne limits arc. With a stock cast-iron crankshaft and connecting rods, he recommends a maximum of 3500 feet per minute and, with a forged crank and heavy-duty rods and main bearing caps, between 3800 and 4000 feet per minute. At tnc outer limit, he believed that an all-out drag racing engine which is equipped with super­ duty components, run on racing iuel with fast flame front travel and revved to the max for only a few seconds ar a time, may get away with piston speeds as high as 5000 to 6000 feet per minute. As this is being written, we can watch NASCAR engines running in a 500-milc race at speeds of up to 9000 rpm for durations of up to four hours. With a 3.48" stroke, this translates into 5100 feet per minute.

Rev Limits What if you wanted to know how high you could rev an engine without exceeding a specified piston speed? The formula is:

Rpm = (piston speed in fpm x 6) + stroke in inches Let’s see what the engine speed of the Fairlane V-8 with its stroke of 2.87 inches would've been at the old piston speed limit of 2500 feet per minute: Rpm - (2500 x 6) + 2.87

It could've been run all the way up to 5226 rpm. And what of its potential against the modern piston speed standard for stock engines of 3500 feet per minute?

Rpm - (3500 x 6) + 2.87 That works out to 7317 rpm. Our comparison of the flathead and Fairlanc V-8s is instructive because it demonstrates that the same displacement is possible with radically different strokes. Let’s be honest, though, 221 cubic inches was small for a V-8, even by today’s slimmed-down standards, and the early Fairlane's 2.89" stroke was unusually short. It shouldn't have been surprising that the engine was capable of high rpm without excessive piston speed. Small-Block V-8 Piston Speeds But what about high-performance small-block V-8s typically found in Mustangs and Thunderbirds? Or, for that matter, in Camaros and Corvettes? Among the derivatives of that original Fairlane engine arc V-8s of 302 and 351—uh, 352—cubic inches. Chevrolet has had comparable engines of 302 and 350 cubic inches. 302$—The 302s were developed in the late 1960s to fit the 5.0-liter limit in the Sports Car Club of .America's Trans-Am series. Both Ford and Chevrolet used the same cylinder dimensions, a bore of 4.0" and stroke of 3.0". Therefore, at any given rpm, they had the same piston speed. For example, at 8000 rpm, the figures would be:

Piston Speed = (3 x 8000) ? 6 Or 4,000 feet per minute. 3501351—Bores of 4.0" arc also used in the Ford 351 and Chevy 350, but the strokes are 3.5’ and 3.48" respectively, so at any given rpm, the piston speeds will be similar but not the same. For example, at 8000 rpm, the figures for the Ford 351 would be: Piston Speed = (3-5 x 8000) * 6 Which would be 4667 feet per minute. For the Chevy 350:

Piston Speed = (3.48 x 8000) + 6 And that works out to 4640 feet per minute. In cither case, you'd be pressing your luck unless you used the highest quality internal parts possible. Drag Racing—For drag racing, you'll occasionally hear ol Chevy small-block V-8s that are run from 10to 12,000 rpm. However, these have been dcstroked,

Piston Speed

A piston makes two full strokes during each crankshaft revolution. Therefore, to And mean piston speed, multiply the stroke In inches by two, then multiply that figure by the rpm. Divide by 12 to find the average piston speed in feet per minute (fpm). Courtesy Larry Shepard.

That put it right at 2500 feet per minute! Flatheads may have been revved beyond 4000 rpm, bur not for very long, at least in stock form! In 1962, just 30 years after the debut of the flathead, Ford introduced another 221 cubic-inch V8. Called the Fairlane V-8 after the mid-size series car where it was first used, it was the forerunner of Ford's modern small-block V-8s and featured a

relatively big bore and short stroke, with a bore and stroke of 3.50 by 2.87 ' respectively. And what was its piston speed at 4000 rpm?

Piston Speed = (2.87 x 4000) + 6 That works out to only 1913 feet per minute, well under the traditional maximum of 2500. With modern advances in metallurgy, higher

Alto Math Handbook

ROD STRENGTH Most hot rodders know that as they increase the power by adding spark timing and increasing the compression ratio that the rod sees far greater force trying to compress the rod. On the power stroke the gases pushing down on top of the piston greatly increase the compressive load in the rod and hence die force is transferred to the crankshaft to make power. On the exhaust stroke the rod is being pushed upward in tnc cylinder by the crankshaft and it forces the piston up the bore causing the exhaust gas to leave the combustion chamber. As rpm increases however, the weight of the piston becomes detrimental, and as it races toward the top of the bore on die exhaust stroke both the piston and the rod see a tremendous load when they come up for a momentary stop at top dead center (TOO). For this reason, if you are going to increase the power and rpm in an engine, you should seriously consider upgrading or replacing your connecting rods with stronger ones. Some of the high-end engine simulation programs give a value for rod loading. II the load in tnc rod exceeds tne strength, the rod may either break in tension or typically bend in compression.

liston speeds have become possible, but there are still imits. British tech author A. Graham Bell in his xiok, Performance Timing in Theory and Practice, suggested what some of tne limits arc. With a stock cast-iron crankshaft and connecting rods, he recommends a maximum of 3500 feet per minute and, with a forged crank and heavy-duty rods and main bearing caps, between 3800 and 4000 feet per minute. At tnc outer limit, he believed that an all-out drag racing engine which is equipped with super­ duty components, run on racing iuel with fast flame front travel and revved to the max for only a few seconds ar a time, may get away with piston speeds as high as 5000 to 6000 feet per minute. As this is being written, we can watch NASCAR engines running in a 500-milc race at speeds of up to 9000 rpm for durations of up to four hours. With a 3.48" stroke, this translates into 5100 feet per minute.

Rev Limits What if you wanted to know how high you could rev an engine without exceeding a specified piston speed? The formula is:

Rpm = (piston speed in fpm x 6) + stroke in inches Let’s see what the engine speed of the Fairlane V-8 with its stroke of 2.87 inches would've been at the old piston speed limit of 2500 feet per minute: Rpm - (2500 x 6) + 2.87

It could've been run all the way up to 5226 rpm. And what of its potential against the modern piston speed standard for stock engines of 3500 feet per minute?

Rpm - (3500 x 6) + 2.87 That works out to 7317 rpm. Our comparison of the flathead and Fairlanc V-8s is instructive because it demonstrates that the same displacement is possible with radically different strokes. Let’s be honest, though, 221 cubic inches was small for a V-8, even by today’s slimmed-down standards, and the early Fairlane's 2.89" stroke was unusually short. It shouldn't have been surprising that the engine was capable of high rpm without excessive piston speed. Small-Block V-8 Piston Speeds But what about high-performance small-block V-8s typically found in Mustangs and Thunderbirds? Or, for that matter, in Camaros and Corvettes? Among the derivatives of that original Fairlane engine arc V-8s of 302 and 351—uh, 352—cubic inches. Chevrolet has had comparable engines of 302 and 350 cubic inches. 302$—The 302s were developed in the late 1960s to fit the 5.0-liter limit in the Sports Car Club of .America's Trans-Am series. Both Ford and Chevrolet used the same cylinder dimensions, a bore of 4.0" and stroke of 3.0". Therefore, at any given rpm, they had the same piston speed. For example, at 8000 rpm, the figures would be:

Piston Speed = (3 x 8000) ? 6 Or 4,000 feet per minute. 3501351—Bores of 4.0" arc also used in the Ford 351 and Chevy 350, but the strokes are 3.5’ and 3.48" respectively, so at any given rpm, the piston speeds will be similar but not the same. For example, at 8000 rpm, the figures for the Ford 351 would be: Piston Speed = (3-5 x 8000) * 6 Which would be 4667 feet per minute. For the Chevy 350:

Piston Speed = (3.48 x 8000) + 6 And that works out to 4640 feet per minute. In cither case, you'd be pressing your luck unless you used the highest quality internal parts possible. Drag Racing—For drag racing, you'll occasionally hear ol Chevy small-block V-8s that are run from 10to 12,000 rpm. However, these have been dcstroked,

Piston Speed

FORMULAS FOR PISTON SPEED Piston Speed (in fpm) = (stroke in inches x rpm) ■? 6

Rpm = (piston speed in fpm x 6) + stroke in inches

generally to 290 cubic inches, so to determine the stroke, apply the formula for stroke discussed in Chapter 1: Stroke = 290 + (pi * 4) x (42 x 8) You’ll find that a 290 cubic-inch V-8 with a 4.0' bore would have a stroke of 2.88". Therefore, at 10,000 rpm: Piston Speed ■ (2.88 x 10,000) + 6 The piston speed would be 4808 feet per minute. At 12,000 rpm, the piston speed would rise to 5760 feet per minute, and that’s about as far as anybody ought to go! This also assumes that the

valve train will function without false motion typically referred to as valve float. Getting a pushrod engine to reliably achieve speeds above 10,000 rpm requires sophisticated components and careful assembly. Between modern short-stroke engine designs and ongoing improvements in metallurgy, the recommended maximums in piston speed have become so high that some hot rodders don’t pay much attention to them any more. But it's still wise to be aware of them, because there is a point at which even the best bearings and rods can fail, particularly when a powerplant is run consistently at highcr-than-avcragc rpm.

Chapter 4

Brake Horsepower and Torque

Horsepower is defined as the measure of the ability to move a given weight a given distance in a given period of time. There couldn't be a more direct or dramatic example of what that really means than the performance of a Top Fuel dragster. This one will propel its 1200 lb weight from a standing start to the end of a quarter-mile in slightly less than 5 seconds. Photo by Larry Shepard.

Horsepower Horsepower is the measure of the ability to move a given weight a given distance—that is, to apply leverage—in a given period of time. Concept—The concept dates back to the 17th century and James Watt's development of the first practical steam engine. Watt first used his engine to pump water out of mines. Previously, such pumping had been done with draft horses, so it was logical to relate the work the steam engine could do to rhe number of horses it could replace. The account presented here of how Watt did that is adapted from Horben Arthur Kline's The Science ofMeasurement: A Historical Survey. Watt's Draft Horse—The horse plodded a circular path, pulling at a right angle on the end of a 12-foot lever projecting from a capsun at the center of the circle. The capstan, in turn, was geared to operate rhe pump. Watt estimated that the horse pulled with a force of 180 lb. 1'he circle it followed had a circumference of 2 times pi times a radius of 12 feet, or 75.398224 feet. The horse could make 144 trips around the circle in an hour or 2.4 trips a minute, for a speed of 180.95573 or about 181 feet per minute. To convert that demonstration of the horse's ability' into measurable leverage or what is commonly referred to as torque, Watt multiplied 180 pounds times 181 feet, obtaining 32,580 pounds-feec per minute. He rounded that figure up to 33,000 pounds-feet per minute, or 550 pounds-fcct per second, which became the norm for 1.0 horsepower. Today the measurement is included in the Systeme International des Unites (S.I.), which is a group of current measurements used internationally. Generating Force—Watt's draft horse generated force around the circumference of a circle and, as the animal pulled

on the lever, that force was applied to the capstan at the center of the circle. An automobile engine can be described as doing just the opposite. It delivers force ar the output end of the crankshaft. Envision a 1.0-foot lever attached at a right angle to the crankshaft at that point. As the crank rotates, the free end of the lever will follow a circle with a radius of 1.0 foot. Watt's definition of horsepower involved a force in pounds, applied over a distance in feet, for a time of 1.0 minute. Therefore, to convert the rotational force of the crankshaft into horsepower, you must know the distance the free end of the 1.0-foot lever will go in 1.0 minute. That, of course, would be the circumference of a circle with a 1.0-foot radius multiplied by the number of crankshaft revolutions per minute, or rpm. The circumference is pi multiplied by 2 multiplied by 1.0 foot or, more simply, 2 times pi, which is 6.2831853 feet. Therefore, the total distance me free end of the lever will go in 1.0 minute is 6.2831853 feet times rpm. The product of that calculation can be multiplied by the known torque of the engine to find the total pounds-feet of torque per minute. The result can then be divided by Watt's pounds-fcct (lb-ft) figure per minute for 1.0 horsepower (33,000) to find the engine's horsepower. That works out to the following formula:

Horsepower = (6.2831853 x rpm x torque) + 33,000 Dividing the right side of the equation by two times pi, you can eliminate 6.2831853 and reduce 33,000 to 5252.1131. By rounding down the latter figure, you can simplify the formula to: Horsepower = (rpm x torque) + 5252

Brake Horsepower and Torque

This is the formula used to determine horsepower when an engine is tested on a dynamometer. On modern computerized dynos, measurements and calculations arc done electronically. On older units, though, torque was found by measuring the resistance of a device known as a Prony Drake against the flywheel end of the crankshaft. Therefore, output figures at the flywheel are still called brake torque and brake horsepower, or bhp, after the old Prony brake. Gross vs. Net Ratings There are two forms of brake torque and brake horsepower, gross and net. The gross figures represent what the engine can do under ideal conditions, with laboratory’" intake and exhaust manifolding, and without the load of any auxiliary equipment except the fuel, oil and water pumps. They show the absolute maximum output at die flywheel. The net torque and power, on the other hand, represent what the engine can do as it's installed in a vehicle, with such auxiliary items as the air cleaner, alternator, fan and standard intake and exhaust systems in place. Musclecar Ratings—During the 1950s and 60s, automakers tried to outdo each other in gross horsepower claims. By die late '60s, that policy had begun to backfire as insurance companies began adding expensive surcharges to premiums on cars with high-powered engines. So the factories reversed themselves and actually underrated their hottest powerplants. They underrated the brake horsepower, that is, not the brake torque. A classic example was the street version of the Chrysler 426 cubic-inch Hcmi. Its advertised output was 425 bhp at 5000 rpm, and the dyno showed 446 lb-ft of torque at 5000 rpm. Applying die formula with those figures:

Here is the control screen for a DTS dyno. It gives all of the pertinent information an operator needs to safely test an engine. The screen can be easily reconfigured by the operator to highlight particular areas of interest Photo by Larry Shepard.

Horsepower = (5000 x 446) + 5252 The result is 424.60015, which of course can be rounded up to 425, Chrysler's bhp figure. But Chrysler never claimed that was the maximum output! On the dyno, the rpm continued to climb faster than the torque dropped and, at 6000 rpm, the reading was 412 lb-ft. Let’s try the formula again:

Horsepower = (6000 x 412) + 5252 That works our to 470.6778 or, rounded up, 471 bhp! And that is 46 bhp, or almost 11 percent more than Chrysler officially claimed! The practice of unrealistic engine output claims, whether exaggerated or underrated, began to fade in

This engine is being readied for testing on the dyno at Impastato Racing Engines. Photo by Bill Hancock.

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can determine the torque at that same rpm by transposing the formula to: Torque = (5252 x horsepower) + rpm Using the advertised output of the Hemi:

Torque = (5252 x 425) + 5000 The answer is with 446.42 lb-ft of torque at 5000 rpm, which is essentially the same as the reading produced on the dyno.

This DTS dyno is capable of handling over 1000 horsepower. Photo by Bill Hancock.

1971, when General Motors announced it was switching from gross to net horsepower and torque ratings in its advertised specifications. Within the next two years, all domestic automakers and most importers followed suit, thus ending claims based on gross output. For the 1971 model year, GM published both gross and net figures, tne only time the author is aware of any manufacturer doing so. "The differences were revealing. To take just one example, the base version of that year's 350 cubic-inch Chevrolet V-8 had a gross rating of 245 bhp, but a net rating of only 165 bhp—32 percent less! Present-day enthusiasts sometimes ask if there is a conversion factor to determine the net equivalent of the gross horsepower ratings used before die early 1970s. Unfortunately, there isn't, because rhe gross ratings varied so widely—some of them way above the true gross output and some way below—and no single conversion factor could take all die variations into account. Therefore, the only way to determine the original net output of a musclccar engine of the 1960s is to run that engine in its original form on a dyno. Calculating Torque If you know the horsepower at a given rpm, you

Brake Specific Fuel Consumption Another aspect of engine performance and efficiency measured on a dynamometer is fuel consumption. An engine on a dyno is stationary and its fuel consumption obviously can't be measured in miles per gallon. However, if the dyno is equipped with a fuel flow meter, it will show the rate of fuel flow in pounds per hour. The rate at any given rpm divided by the brake horsepower at the same rpm provides a figure called the brake specific fuel consumption, brake specific: for short or bsfc abbreviated. To restate that as a formula: BSFC = fuel pounds per hour + brake horsepower The lower the brake specific, the less fuel the engine is using to develop the horsepower in question; in other words, the more efficient the engine and the better the fuel economy. Example—Suppose you have an engine that shows a fuel-flow rate of 144 pounds per hour at die rpm where it develops 300 brake horsepower: BSFC - 144 + 300 The brake specific would be 0.48. Generally, the figure should be 0.50 or less. It won't be constant. It will lie at its lowest, i.c., most efficient, at or near rhe same rpm where peak torque is developed. At lower rpm, the airflow dirough die intake system is slower than it is at peak torque and, at higher rpm, there isn't time to maintain the same airflow as there is at peak torque. So at engine speeds below or above peak torque, airflow will not be as efficient and, as a result, the brake specific fuel consumption will lie greater.

Specific Gravity Specific gravity is defined as die weight per unit of volume for a substance compared to the weight of a similar volume of pure water. In other words, is the substance heavier or lighter than a similar volume of water? Put differently, anything with a specific gravity of less than 1.000 will float. For example,

Brake Horsepower and Torque

Here a calibrated float is used in a graduate to measure the specific gravity of fuel. Using the specific gravity of the fuel, and the flow rate in gallons, the user can determine fuel consumption in pounds per hour. Photo by Bill Hancock.

TESTING FOR WATER IN GASOLINE By using die difference in specific gravity between water and gasoline, we can find out very quickly if wc have any water in our gasoline. Drain a sample of the suspected fuel into a clear glass container and set aside for a few minutes until the fluids have settled out. If water is present, there will be a dividing line visible between rhe two liquids, with the gasoline being at the top and the water being at the bottom. This concept is the backbone of a process called fractional distillation. Refineries use fractional distillation to refine crude oil into gasoline and all of its byproducts.

gasoline has a specific gravin' of around 0.76, and therefore when gasoline and water arc mixed, the gasoline rises to the top. Density Density is die weight per unit of volume. Let's see how we could figure our weight per gallon of gasoline by using only the density. In our previous example, gasoline with a specific gravity of 0.763 means that it is roughly three quarters the weight of a similar volume of water. Wc know from the tables diat water weighs 8.334 lb/gal

0.763 x 8.334 lb/gal = 6.36 lb/gal So now we know that a gallon of the gas wc arc­ using weighs 6.36 lb. The various types and grades of gasoline each have different specific gravities, so be sure to check with your fiicl supplier to ger the exact specific gravity for your fuel. Fuel Check Now that we know how specific gravity is calculated, we can use this principle to check our fuel. Using a device known as a ITALhydromcter (see photo) which is simply a calibrated bobber, we can measure rhe specific gravity of a given fiicl sample and compare it to the standard specification for that brand of fuel and octane. If fuel has been diluted or has aged by being improperly stored, it can lose its potency and properties which arc so critical to the proper operation of a performance engine. While certainly not a definitive test of gasoline quality, a specific gravity test is better than no test at all. If a fuel won't pass a specific gravity test, stop right there. If a sample docs pass a specific gravity test, it still might not be correct. In the end, only a series of chemical tests and an actual motoring rest will

determine the true capability and rating of a given fuel. Unfortunately, these tests require sophisticated equipment and are typically only carried out at the refinery o-r research level due to the cost of the equipment.

Dyno Chart When an engine is tested on a dynamometer, a chan is created showing die brake torque in poundsfeet and the brake horsepower at specific rpm intervals. An example is shown on the next page in Fig. 4a. It shows the torque and horsepower for a modified 350 cubic-inch Chevy V-8 at intervals of 200 rpm from 3000 to 7000 rpm. Maximum torque, 350 lb-ft, occurs from 3800 to 4000 rpm and maximum horsepower, 343, at 6000 rpm. Those are all relatively high figures for a small V-8, indicating that the engine has been reworked for higher performance. Note how flat die torque output is between 3000 and 4200 rpm. Over diat 1200-rpm spread, it varies only 10 pounds-fcct. Beyond 4000 rpm, the torque gradually declines, though it hits another flat spot of 305 pounds-fcct from 5400 to 5800 rpm. However, as rpm continues to climb, so does horsepower, until the engine reaches its peak output at 6000 rpm. From that point on, torque starts to fall sharply, while horsepower drops more slowly. Most dyno data sheets will show much more than the chart on the next page. They'll include not only torque and horsepower, but fiicl flow, air flow and brake specific fiicl and air consumption, along with

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Fig. 4a. This chart shows the torque and horsepower for a modified 350 cubic-inch Chevy at intervals of 200 rpm from 3000 to 7000 rpm as measured on a dyno.

RPM 3000 3200 3400 3600 3800 4000 4200 4400 4600 4800 5000 5200 5400 5600 5800 6000 6200 6400 6600 6800 7000

LB-FT 340 340 345 345 350 350 340 335 330 325 315 310 305 305 305 300 280 255 240 190 160

BHP 194 207 223 236 253 .267 272 281 289 297 300 307 314 325 337 343 331 311 302 246 213

intake and exhaust specs, temperatures, and ignition settings. But the real foundation of dyno testing is in the relationship of torque and horsepower as a function of rpm.

Effects of Elevation A normally aspirated or NA engine is one without forced induction, such as turbocharging or supercharging. If you drove a car equipped with one of these engines up a tall mountain you would notice that the higher you go above sea level, die less power you have. As the air thins out, less of it will be drawn into the engine. There will be a decrease in volumetric efficiency (a topic covered further in Chapter 6) and, with it, a loss in engine output. The resulting drop in horsepower at any given rpm is approximately 3.0 percent for each 1000 feet of elevation. If you live in flatlands at or near sea level, none of this will lie of much significance to you. But, if you have ever driven over California's Sierras or Colorado's Rockies, you know what I'm talking about. Because of this, Pikes Peak in Colorado Springs, Colorado is the best example of the performance loss, since it is essentially one long uphill grade. As you wind your way up the hill, you can feel die effects of altitude on the engine performance. An engine that was fairly responsive at the bottom of the hill becomes pretty sluggish at the top.

These are dyno-testing headers, a 4-into-1 design. Note that they have been adjusted about 6" with a straight insert just before the transition—lower left of photo. Adjustability is a common feature of race headers. Photo by Larry Shepard.

Example—Let's consider a trip to Colorado in a car powered by a four-cylinder engine developing 100 net blip. That nice, even output figure will simplify the calculations. From the East or Midwest, your first major stop in Colorado is likely to lie the mile-high city of Denver. On the way there, you’ll begin to feel the effects of elevation as you leave the plains of the Midwest and stan climbing the Eastern slope of die Rockies. When you reach Denver, you'll be at over 5000 feet and, at a 3.0 percent loss per 1000 feet, that means you'll have lost 15 percent of your power. The 100bhp engine now has only 85 bhp. But that's just for openers. Heading west from Denver on Interstate 70, you'll cross the Rockies over passes at elevations of 10,000 and even 11,000 feet. And, at 10,000 feet, the engine will lose 30 percent of its output, which will decrease to 70 bhp. Or suppose you head South to Colorado Springs to try climbing Pikes Peak, one of the highest spots in the United States that can be reached by automobile. At the summit, you'll be more than 14,000 feet above sea level. That will cost you 42 percent of engine power, dropping the litde 100-bhp fourbanger's output to only 58 bhp! Of course, if the car's engine had been equipped

Brake Horsepower and Torque

FORMULAS FOR BRAKE HORSEPOWER AND TORQUE Horsepower = (rpm x torque) + 5252 Torque = (5252 x horsepower) e rpm Brake Specific Fuel Consumption = fuel pounds per hour + brake horsepower

Bhp loss= (elevation in feet + 1000) x (0.03 x blip at sea level)

with a turbo, it would be a different matter because a supercharged or turbocharged engine boosts rhe manifold pressure, by forcing more air into rhe engine than the surrounding atmosphere would.

Atmospheric Correction Factors An engine performs differently depending on the weather surrounding it. If in the example we just discussed, the engine is subjected to lower atmospheric pressure it will lose power compared to the sea level conditions. In addition to barometric pressure, ambient temperature, relative humidity, and vapor pressure will all affect the output. In an effort to be able to accurately compare performance under differing atmospheric conditions, it is necessary to measure the various values for pressure and temperature and apply a formula which lets us equate the performance back to a so-called standard day. The SAE has used a variety of standards over the years to accomplish this. Currently SAE standard J1349 appears to be the method of choice for most of die auto manufacturers. However, J607 a version used in the 1960's has been a favorite of rhe performance industry and hence is programmed into many of the computerized performance dynamometers. It also just nappens to yield the highest corrected numbers. When measuring or rating the output of an engine on a dynamometer, it is vitally important to be able to equate the performance of one set of data to another taken under different weather conditions. In most cases the correction factor, when applied, will raise the raw or uncorrectcd output data values of die engine. What this is really saving is that, if die engine in question was run on the theoretical perfect day,' the output would have been higher. In some cases however, die corrected numbers" arc lower and tliis indicates the engine was actually run on an exceptionally favorable day. A cool crisp dry day in

October with high barometric readings will often result in just such a case. Some people unfamiliar with corrected readings are prone to distrust diem and not use diem. As long as you carefully calculate your values and consistently apply rhe formulas, you will be much furdier ahead using corrected numbers in your engine development work. Without going into great detail about the applied theory for correction factors, let’s look at the formula for SAE J607. This standard allows us to correct our data back to a standard day with 60 deg. F dry air (zero percent relative humidity) at sea level and ar 29.92 inches of mercury barometric pressure.

CF- (29.92 in.Hg + Tp -Vp) x .CAT + 460 + 520 Where Tp is test pressure in inches of mercury, V is vapor pressure in inches of mercury (in.Hg), CAT is carburetor air temperature or inlet air temperature in deg. F

Tp or rest pressure is best measured by an accurate barometer compensated for temperature and geographic location. If that is unavailable, a call to the local airport with a request for station pressure will work just as well. The local weather broadcast is probably the least accurate, but still better than nothing. Vp or vapor pressure for air is a pressure where droplets of water are forming into liquid droplets as fast as they arc evaporating. This pressure varies with temperature, so we must use a chart to find this pressure. Harold Bettes, the author of HPBtxiks' Engine Airflow has provided the chart for us to use here. It is shown on page 26. Let's begin by defining Wet Bulb (WB) and Dry Bulb (DB) temperatures. Dry bulb refers to the tin of a mercury thermometer called the bulb. If the bulb is dry; the temperature will read the temperature of the

Auto Math Handbook

air. To understand wet bulb temperature, wet your fingertip and blow on it. Your fingertip will feel cooler as long as it is wet and you arc blowing air across it. We use a handheld device called a sling psychrometer, which employs a wet sock surrounding die bulb of the thermometer to determine the two distinct temperatures. Once we have the two temperatures, our next step is to get the vapor pressure. Let's suppose that we have a wet bull) (WB) temperature or 65° and a dry bulb (DB) temperature of 80°. Follow the line from 80° OB straight up until it intersects the WB line for 65°. Holding your finger on that point, go directly across to the right and read die vapor pressure of 0.45. Coincidentally, if you follow the curves for relative humidity, it also corresponds to 45% relative humidity.

So in this case we now have an equation that looks like this:

CF - (29.92 + Tp - 0.45) x \ (80 + 460) + 520 In this case, if we had a barometer reading of 29.75, we would solve the equation:

(29.92 + 29.75) - 0.45 x \ L038 29.3 x 1.019 29.92 -5- 29.857 CF = 1.0021 So if we had an engine that produced an indicated 627 horsepower we would multiply it by our CF and see die following:

627 x 1.0021 = 628.3 horsepower

Chapter 5

Indicated

Horsepower and Torque

Engines like this 426 Hemi A990 engine represent the pinnacle of the horsepower race during the muscle car era. The brake torque and horsepower of this engine can be measured on the dyno, but those numbers only represent output at the flywheel. Those figures do not account for losses within the engine due to inertia and friction. Therefore, the brake figures will always be less than the horsepower and torque actually developed within the cylinders. Once you know the flywheel and cylinder outputs, you can determine the engine's true mechanical efficiency. A measuring device called an indicator is used to measure cylinder pressure during each of the four strokes—intake, compression, combustion and exhaust—and from them the indicated mean effective pressure, or mep, can be determined. Once you know the mep, indicated horsepower and torque can be calculated. Photo by Bill Hancock.

As demonstrated in the previous chapter, brake torque can be measured on a dynamometer and, from it, brake horsepower can be calculated. However, those figures represent output at the flywheel and, because of losses within the engine primarily from friction and also from inertia, they will always be less than the horsepower and torque actually developed within the cylinders. And, once the output both at the flywheel and in rhe combustion cylinders is known, you can determine the engine's mechanical efficiency.

Indicated Mean Effective Pressure You can't measure the horsepower and torque developed within the cylinders directly. However, using a device called an indicator, you can measure cylinder pressure during each of die four strokes—intake, compression, combustion and exhaust—and, from them, you can find die indicated mean effective pressure, or mep, which is a form of output that occurs widiin the cylinders and is unaffected by friction and inertia. When the indicated mep is known, it s possible to calculate the indicated horsepower and torque within the cylinders. An indicator is not something die average hot rodder is likely to have readily available. Nonetheless, rhe serious performance enthusiast should be aware of the

interrelationships of mep, horsepower and torque. (It should be noted that the formulas involving these interrelationships are equally valid for either brake or indicated figures.) Indicated Horsepower There’s a formula for calculating horsepower from mep that's favored bv many engineering theorists because it involves a simple acronym, PLAN, that's easy to remember.

Horsepower = PxLxAlN r 33,000 P stands for mep in pounds per square inch or psi; L for rhe length of the stroke in feet; A for the top surface area of one piston in square inches; and N for the number of power strokes per minute. When these four factors are multiplied together, they show the total amount of torque the engine develops in one minute. That figure is then divided by 33,000—the number of pounds-fcct (lb-ft) per minute equal to one horsepower—to find die total horsepower. Part of the appeal of PLAN is that it focuses on the aspects of engine design that ultimately determine horsepower. When you modify an engine to improve performance, you ate really increasing P, L, A and/or N.

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Engines like this one used in NHRA pro stock typically have an extremely high specific output Photo by Larry Shepard.

Example—For example, when you raise the compression ratio, P is increased. When an engine is bored or stroked, A or L is increased. When you replace rhe camshaft and valve train to get higher rpm, you increase N. The formula is awkward in practical use, though, because it requires separate calculations to find L, A and N. Io find L, the stroke in inches is divided by 12 or:

L = stroke +12 l'o find A, divide pi by 4 then multiply it by the bore squared, or: A = (pi + 4) x bore2 And, to find N for a conventional four-stroke-cycle engine, divide the rpm by 2 and multiply the result by the number of cylinders, or: N = (rpm + 2) x no. of cylinders So our "simple” acronym eventually leads to a much more complicated formula:

Horsepower = (mep x stroke x bore2 x pi x rpm x no. of cylinders) + (12x4x2x 33,000) But look closely. Embedded in there arc the factors of the formula for displacement:

Displacement = (pi + 4) x bore2 x stroke x no. of cyl.

You can remove those and use the displacement instead, simplifying the raw formula to: Horsepower ■ (mep x displacement x rpm) + (12 x 2 x 33,000) And that, of course, can be reduced still further to:

Horsepower = (mep x displacement x rpm) + 792,000

Example—Let's take as an example a 302 cubicinch engine which has an indicated mep of 175 psi at 4200 rpm: Horsepower = (175 x 302 x 4200) * 792,000 Working the equation through gives an answer of 280.265 indicated horsepower. Indicated Torque The mep will be in direct proportion to the torque, and the peak mep will occur at the same rpm as the peak torque. To find indicated toraue, multiply the mep times the displacement, and divide the result by pi times 4 times 12 or: Torque = (mep x displacement) + (3.1415927 x 4 x 12) Multiplied, the constants become 150.79645, which can be rounded off to 150.8, reducing the formula to:

Indicated Horsepower and Torque

Torque = (mcp x displacement) -s- 150.8

Example—Assume that the 302 cubic-inch engine has a maximum indicated mcp of 177.5 psi at 3200 rpm—though the engine speed isn't immediately relevant: Torque - (177.5 x 302) + 150.8 And that would mean a maximum indicated torque of 355-5 lb-ft.

Brake Mean Effective Pressure Mean effective pressure occurs within an engine's cylinders and cannot be dirccdy measured at the flywheel, as brake torque is. However, you can find mep from either horsepower or torque. If you start with brake horsepower or brake torque, you will come up with a hypothetical brake mean effective pressure, or bmep. To find bmep from horsepower, the formula is: BMEP = (hp x 792,000) + (displacement x rpm) Example—Let's sec what the bmep would be at the 7000-rpm horsepower peak of that 426 cubicinch, 471-bhp Chrysler Hemi I described on page 21 in Chapter 4.

BMEP = (471 x 792,000) + (426 x 7000) The answer is 125 psi. The formula for finding bmep from torque is: BMEP = (torque x 150.8) -5- displacement Chrysler listed the 426 Hcmi's maximum brake torque as 490 lb-ft at 4000 rpm. Therefore, die input for bmep would be: BMEP = (490 x 150.8) + 426 That results in a figure of 173 psi. BMEP vs. BSFC—Note that the bmep is higher at peak torque than at peak horsepower. Like brake specific fuel consumption (bsfc), discussed in Chapter 4, brake mean effective pressure is at its best at peak toroue. Indeed, as measures of an engine's efficiency, the bsfc and bmep tend to reflect each other. Tnc bsfc is at its lowest or most efficient at peak torque, and the bmep is at its highest or most efficient at the same point. On eidier side of peak torque, the bsfc gets worse as it increases and the bmep gets worse as it decreases.

While bmep is a calculated, dieoretical form of output, it is useful for comparing the relative performances of different engines. It is entirely possible, for example, diat a small 4-cylinder sports car engine and a big stock car racing V-8 could have similar bmep figures, despite vast differences in their displacements and their horsepower and torque characteristics. Typical bmep figures are 130 to 145 psi for die engine in a standard passenger car, 165 to 185 psi in a high-performance or sports car, and 185 to 210 psi in a racing vehicle.

Mechanical Efficiency An important advantage of having both indicated and brake output figures is that they can be used to determine the engine's percentage of mechanical efficiency. The basic formula is the same, whether the figures used are horsepower, torque or even mep: Mechanical Efficiency = (brake output -5- indicated output) x 100

Mechanical Efficiencyfrom Horsepower—Let's take that 302 cubic-inch engine which was found to have 280.265 indicated horsepower at 4200 rpm and 355.5 lb-ft of torque at 3200 rpm, and assume that its brake output ratings arc 225 horsepower and 300 lb-ft at the same respective engine speeds. To find mechanical efficiency from the horsepower figures, you use:

Mechanical Efficiency = (225 + 280.265) x 100 And the result would be 80.28 percent mechanical efficiency. The difference between the two horsepower figures—indicated output minus brake output or, in this case, 55.265—is known as friction horsepower, because it is the amount lost between die cylinders and the flywheel from friction. Mechanical Efficiencyfrom Torque—To find mechanical efficiency from the brake and indicated torque ratings of our 302-cubic-inch engine, the figures would be: Mechanical Efficiency = (300 + 355.5) x 100 This time, you have 84.39 percent mechanical efficiency and 55.5 lb-ft of friction torque. And, once again, here's an example of greater efficiency at peak torque than at peak horsepower.

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FORMULAS FOR INDICATED HORSEPOWER AND TORQUE Horsepower = (mep x displacement x rpm) + 792,000

Torque = (mep x displacement) + 150.8 BMEP = (hp x 792,000) * (displacement x rpm) BMEP = (torque x 150.8) -r displacement Mechanical Efficiency = (brake output -5- indicated output) x 100

Chapter 6

Air Capacity and Volumetric Efficiency

To determine what size carburetor you'll need for your street engine, carburetion authorities recommend you assume a volumetric efficiency of 85% and use that in the formula for theoretical air capacity to find the proper carburetor flow in cfm. Photo by Bill Hancock.

Aii automobile engine is a form of air pump, and knowing its theoretical air capacity is necessary to detern line its volumetric efficiency, i.e., the relationship between the theoretical capacity and the actual airflow. In addition, on a carbureted engine, the air capacity may serve as a guide to choosing the proper carburetor size.

Air Capacity The air capacity is a product of rpm and displacement. In a conventional four-stroke engine, the volume displaced on intake strokes during each crankshaft revolution will be 1/2 of the overall cubic capacity. So, to find the air capacity in cubic inches per minute, multiply the rpm by the displacement in cubic inches and divide by 2, or:

Air Capacity » (rpm x displacement) -5- 2 In practice, calculating the air capacity in cubic inches per minute would result in unwieldy figures, so the measurement is converted to cubic feet per minute or cfm by dividing the displacement by 1728, the number of cubic inches in a cubic foot: cfm » [(rpm x displacement)+ 2] x 1728 That, in turn, can be simplified to:

cfm » (rpm x displacement) -e- 3456

Example—As an example, consider the theoretical air capacity of the 350 cubic-inch Chevy V-8 used for the dyno chart in Fig. 6a on the next page, the same chart used on page 24. You are interested in two particular engine speeds: die rpm at peak torque because, like other measurements of engine efficiency already discussed, volumetric efficiency is highest at that point; and the maximum rpm, because that's where die air capacity is its greatest. According to the chan, maximum torque is delivered at 4000 rpm, so to find the air capacity:

cfm = 4000 x 350 + 3456 Which works out to 405 cfm. Maximum rpm on die chart is listed as 7000 rpm, so to determine what this engine's greatest air capacity is:

cfm = 7000 x 350 + 3456 The answer is 709 cfm.

Volumetric Efficiency The actual airflow can be measured at each rpm and then divided by the theoretical capacity at the same rpm to find the engine's volumetric efficiency, or V.E. The resulting figure can be multiplied by 100 to convert it from a decimal to percent. Stated as an equation, that becomes:

Airro Math Handbook

Fig. 6a. This chart shows the torque and horsepower for a modified 350 cubic-inch Chevy at intervals of 200 rpm from 3000 to 7000 rpm as measured on a dyno.

RPM 3000 3200 3400 3600 3800 4000 4200 4400 4600 4800 5000 5200 5400 5600 5800 6000 6200 6400 6600 6800 7000

LB-FT 340 340 345 345 350 350 340 335 330 325 315 310 305 305 305 300 280 255 240 190 160

BHP 194 207 223 236 253 267 272 281 289 297 300 307 314 325 337 343 331 311 302 246 213

An engine has to have a great air capacity to run this pair of Holley 1050 cfm Dominator carburetors. Photo by Larry Shepard.

V.E. = (actual cfm + theoretical cfm) x 100 Let's suppose the actual airflow of our engine was 365 cfm at 4000 rpm and 565 cfm at 7000 rpm. The volumetric efficiency at 4000 would be:

V.E. = (365 + 405) x 100 Or just over 90 percent. And, at 7000 rpm with an actual airflow of 565 cfm: V.E. = 565 * 709 x 100 The volumetric efficiency at that rpm is just under 80 percent. According to Mike Urich, former engineering vice president of Holley Carburetors and author of HPBooks' Holley Carburetors, Manifolds and Fuel Injection'. An ordinary low-performance engine has a V.E. of about 75 percent at maximum speed; about 80 percent at maximum torque. A nigh-performance engine has a V.E. of about 80 percent at maximum speed; about 85 percent at maximum torque. An allout racing engine has a V.E. of about 90 percent at maximum speed; about 95 percent at maximum torque. A highly tuned intake and exhaust system with efficient cylinder-head porting and a camshaft ground to take Rill advantage of the engine's other equipment can provide such complete cylinder filling that a V.E. of 100 percent—or slightly higher—is obtained at the speed for which the system is tuned.’’ In practice, a highly tuned engine of the type Urich mentions may have a V.E. as much as 10 or 12 percent greater than the theoretical air capacity.

Carburetor Size In an era when all OEM high-performance engines are being equipped with fuel injection, carburetion may seem old-fashioned. But it's not dead yet, and it's likely to lie around for some time to come, although not on new cars. There are still a lot of hot rods being built with carburetors. There are older cars being restored, such as musclecars of the 1960s and 1970s, with carburetors. And in several forms of racing, there arc classes that continue to require carburetors rather than injectors. So, to a lot of performance enthusiasts, carburetion remains important. An engine’s air capacity can be a guide to choosing carburetor size, in terms of the car buretor's airflow in cfm, for a given combination of displacement and rpm. This time, though, we're concerned with practical rather than theoretical capacity, i.e., the actual airflow. Street Carb—But how do you estimate what size carburetor you need while you're building an engine, before you can measure the airflow? For a street

Air Capacity and Volumetric Efficiency

This Edelbrock Edelbrock 800 dm Performer carb is designed for performance, delivering crisp throttle response. Photo by Mike Mavrigian.

engine, Urich and most other authorities on carburetion recommend assuming a volumetric efficiency of 85 percent and plugging that figure into the formula for theoretical air capacity to find the proper carburetor flow in cfm:

According to the example in the text for calculating racing carburetor cfm, this Holley Dominator carburetor would be a good choice. Photo by Larry Shepard.

Street Carb cfm » [(rpm x displacement) 4- 3456] x 0.85 Example—Suppose you are building a street rod with a Chevy 350 V-8 that you don't expect to rev beyond 5000 rpm.

Street Carb cfm = [(5000 x 350) + 3456) x 0.85 Before taking die V.E. into account, that would work out to 506 cfm. Multiply by that 0.85 and the figure drops to only 430 cfm. Rating Carb—For a street engine, that formula is fairly reliable. But on a racing engine, as Urich points out, the volumetric efficiency can increase to 100 percent or more. Let's take the case of a highly tuned engine which can compress the intake air somewhat and increase the actual flow to 10 percent more than rhe theoretical air capacity. That's die same as saying 110 percent of, or 1.1 times, the theoretical figure: Racing carb cfm = [(rpm x displacement) + 3456] x 1.1 For this example, suppose you have a Chew 350 built for racing that you expect to be running up to 7000 rpm:

Racing carb cfm = [(7000 x 350) + 3456] x l.l This time, the initial figure is 709 cfm which, multiplied by 1.1, becomes 780 cfm. However, if you go very' far beyond the dieoretical air capacity in choosing carburetor size, you'll quickly reach a point of diminishing returns. Mike Urich cited an instructive illustration in a drag racing

The Dodge Challenger Drag Pak cars provide excitement wherever they go. Photo by Larry Shepard.

test of a Chevrolet Camaro with the 350 cubic-inch engine. Using a 650-cfin l lollcy' carburetor, die car ran die 1/4-mile in an elapsed time of 14.35 seconds with a terminal speed of 96.05 mph. Switching to a bigger 800 cfm Holley, the c.t. increased to 14.61 seconds and the speed dropped to 95.44 mph. Those may not seem like significant differences, but drag races arc often won or lost by even narrower margins.

Auto Math Handbook

Bigger Is Not Always Better Carburetors are not made in an infinite variety of sizes, so you're not likely to find one that corresponds exactly with your calculated carburetor cfm. In most cases, you'd probably be wiser co choose a carburetor the first available size down from your calculated figure rather than die first size up. As a case in point, suppose you determine chat you need a 720 to 730 cfm carburetor. However, the make and model of carb you want to use is available

in cither 700 or 750 cfm sizes, with nothing in between. Your temptation might be to go for the 750, but you'd probably be better off with the 700. For choosing a carburetor is one area where bigger is not always better.

FORMULAS FOR AIR CAPACITY AND VOLUMETRIC EFFICIENCY Theoretical cfm = (rpm x displacement) + 3456

Volumetric Efficiency' = (actual elm + theoretical cfm) x 100

Street Carb cfm = [(rpm x displacement)

3456] x 0.85

Racing Carb cfm = [(rpm x displacement) + 3456] x 1.1

Chapter 7

Weight

Distribution

These lightweight portable car scales are much more accurate for weighing a car than the larger heavy duty truck scales are. Photo by Bill Hancock.

Weight distribution is a statement of the percentages of a vehicle's overall weight divided lengthways between the front and rear wheels, or sideways between the left and right wheels. It is an important factor in the handling of all types of vehicles, from family sedans and race cars to motor homes and heavy’ trucks. The lengthways or front/rear weight distribution will vary greatly, depending on the type of vehicle. However, with the important exception of cars built for oval track racing, there's usually little if any significant difference in the sideways or left/right weight distribution.

Weighing the Vehicle To find how the weight of a car or truck is distributed, you must first find the weights at each wheel. The ideal method is to weigh the vehicle with each wheel on a separate flat scale. A professional race team will typically carry a set of four flat scales to check weights on the wheels when setting up a car for specific track conditions, but a set of such scales would be an extravagance for anyone not needing them regularly. Public Scale—A public scale, such as one of those used by professional truckers or moving companies, is a less costly alternative, though it may also be less accurate than individual flat scales. The public scale typically has a capacity of up to 100 tons in order to weigh heavy trucks and their cargos, and at the more modest weights of ordinary cars or

light trucks, the device’s readings may not be precise. For that reason, it isn’t really worth the trouble to use such a scale to find the weight at each wheel. Wheel Weights in Pairs—Fortunately, it isn’t really necessary. The formulas involving weights at the wheels that I'll be discussing in this chapter call for those weights in pairs—the fronts and rears and, for oval track cars, the lefts and rights—so those are the figures you should measure on the public scale. Measuring Weight—Start by finding the overall weight of the vehicle with all four wheels on the scale. Next, measure the weight with the wheels at one end of the vehicle on the scale and then the wheels at the other end. Finally, if you need to know die sideways weight distribution, weigh the wheels at one side of the vehicle and then the other side. If the ramps at the ends or sides of the scale platform aren't level, position the vehicle with the wheels being weighed as far on to the platform as possible, and the wheels not being weighed as close to the edge without touching it as possible. If the vehicle is tilted at the slightest angle, there will be enough weight transfer to the wheels off the scale to invalidate die readings. Isn't it redundant to weigh both ends? Couldn't you weigh the front wheels and then subtract that figure from the overall weight to find the weight at die rear wheels? Yes, you could. But by actually weighing both sets of wheels, you'll have an optimum check on the scale's accuracy because the

Alto Math Handbook

The best way to weigh a vehicle is with an individual flatbed scale at each wheel. The measurements should be taken on level ground, with all four wheels on scales at the same time. Photo by Bill Hancock.

This electronic individual scale system gives all four weights but also gives a variety of combinations such as front to rear and side to side weights.

and you don't want to be shuttling a race car back and forth on the scale platform with a line of impatient truckers waiting to check their loads before heading out on the highway. So it’s a good idea to check ahead of time with the scale operator to find when the facility isn’t likely to be busy.

Front/Rear Distribution To find the percentage of weight on a given set of wheels, divide the weight on those wheels by the overall vehicle weight, and multiply the result by 100, or:

Wheel Weight Percentage = (weight on wheels + overall weight) x 100 Example—Let's suppose you have a car that weighs 4000 lb overall with 2240 lb on the front wheels: A truck scale, while handy, is typically not as accurate as a scale designed for a lower range. Here we are weighing the vehicle with the front wheels on one scale and the rears on the other scale. This is handy for front to rear weight Photo by Bill Hancock.

overall weight and the sum of the front and rear weights should be the same. If you weigh the left and right wheels, their sum should be the same, too. It there arc any significant discrepancies, go to another scale and start over. Obviously, all this is going to be time-consuming

Wheel Weight Percentage = (2240 * 4000) x 100 The answer is 56 percent, which means that over one-half of the overall vehicle weight is located over the front wheels. If the scale were accurate, it would have shown 1760 lb on the rear wheels: Wheel Weight Percentage = (1760 + 4000) x 100 Or 44 percent. Of course, you could also have arrived at that figure by subtracting 56, the percentage on the front wheels, from 100. Typical Front/Rear Weights—That front/rear weignt ratio of 56/44 is typical for a full-size sedan

Weight Distribution

with a front engine and rear drive, like a Ford Crown Victoria. The engine is the heaviest single component in a motor vehicle and its location— front, rear or in between—strongly affects the weight distribution. On a front-engined, rear-drive sports or GT car with only two seats, like the Mazda Miata, Chevrolet Corvette or Mercedes-Benz SL, die smaller passenger compartment allows the engine to he set back somewhat for a better balanced front/rear weight ratio of 51/49 or 52/48. For a compact or mid-size car with a front engine and front drive, as is the case with most of today's popular cars, the front transaxle increases the weight on the front wheels, and the front/rear weight ratio is likelv to be between 60/40 and 65/35. Conversely, on a true rear-engined car, i.e., one with the engine behind the rear transaxle, like the Porsche 911 and its many variants, the front/rear weight ratio will be on rhe order of 40/60. On a mid-engined car, i.e., one with the engine behind the passenger compartment but ahead of the rear transaxle, such as the Toyota MR2 or Acura NSX, the proponion is slightly less severe, but only slightly, at about 42/58. Understeer dr Oversteer—It's widely believed that a vehicle's weight distribution affects its handling characteristics. A front-heavy car is supposed to understeer, i.e., its front wheels will drib toward the outside of a turn. A rear-heavy car, on the other hand, will reputedly oversteer, i.e., its rear wheels will drift to the outside of a turn. That's an oversimplification, though. A vehicle's tendency to understeer or oversteer is the product of the tire slip angles. Those, in turn, are affected not only by weight distribution, but also by such factors as suspension design and tire size. For example, rear-heavy cars like die 911, MR2 and NSX, often have bigger tires at the rear than at the front to help neutralize any tendency to oversteer and, in normal driving, they may even understeer somewhat. However, when any car is pushed beyond its limit of adhesion, centrifugal force is likely to take over and cause the tires at the heavier end to lose traction and skid or even spin toward the outside of a turn. Left/Right Distribution On an oval track race car with rear drive, there should be a weight bias toward the rear and toward the left side—toward the rear to increase weight transfer to the rear for better traction when

Here we are weighing the left side front and the left side rear, while the right side is off the scale. Photo by Bill Hancock.

Now we switched sides and we are weighing the right side while the left side is off the scale. When you are done, compare the left side weight plus the right side weight to the total weight The two total weights should be the same. Photo by Bill Hancock.

accelerating and to minimize weight transfer to the front when braking, and toward rhe left to decrease weight transfer to the right as the car follows a continuous series of turns to the left. During the 1970s, Chrysler Corporation produced a kit for building a Dodge Aspen- or Plymouth Volare-based stock car for oval track racing. According to rhe factor)’ parts catalog for this vehicle, 53 to 55 percent of the weight should be on the left side and 52 to 54 percent on rhe rear wheels.

Auto Math Handbook

If you are going to weigh your car very often, it will pay to get a set of portable scales that you can take to the track or use in your garage. Photo by Bill Hancock.

Adding Weight at Either End If you add weight at either end of a vehicle, it will obviously affect the weight distribution by increasing the percentage on the nearer pair of wheels and decreasing the percentage on the farther pair. Examples of this would be adding a winch at the front of a four-wheel-drive truck, or a trailer hitch at the rear of a tow vehicle, or ballast to a race car. With the added object in place, you can take the vehicle back to die scale. Or you can calculate the added weight on the nearer wheels. To do that, you must know the vehicle's wheelbase, the amount of added weight and the horizontal distance of the added weight from the centers of the nearer wheels. Specifically, you need to know the distance of the added weight's center of gravity from the centers of the wheels. CG ofAdd-Ons—The center of gravity or eg of an object is die point around which the weight of the object is evenly balanced in every direction. In the case of a symmetrical or regularly shaped object like a winch, you can assume the eg is at the center of the object. However, with a complex or irregularly-shaped object like a trailer hitch, estimating the eg is not so easy. One suggestion: Before the object is installed, try to balance it on a sawhorse. Once you have an estimate of the distance of the added object's eg from the centers of the wheels, you can find how much weight it's going to add to the nearer wheels with the formula:

For oval track racing, where the cars turn left only, there should be a weight bias toward the left as well as toward the rear. Photo by Tom Sturgeon.

Wheel Weight Increase ■ [(front wheels + wheelbase) x weight] + weight

To find the left/right weight ratio, use die same formula you did for the front/rear ratio, only this time start with the weight on the left wheels rather than on the front ones. Example—Let's consider a dirt track stock car weighing 3000 lb overall, with 1410 lb on the front wheels, 1590 lb on the rear, 1590 lb on the left, and 1410 lb on the right. In this instance, the weight distribution lengthways and sideways would be the same: Wheel Weight Percentage = (1590 + 3000) x 100 Which works out to 53 percent at the rear and left, within the parameters Chrysler recommended, and front/rear and left/right weight ratios bodi of 53/47.

Adding a Winch at Front—Suppose you have a sport-utility vehicle that has a 107'' wheelbase and, in standard form, weighs 4000 lb, with 55 percent or 2200 lb on die front wheels and 45 percent or 1800 lb on the rear wheels. If you install a winch at the front which weighs 110 pounds, and you estimate that the eg of the wincli is 45" ahead of the front wheel centers, the figures in the formula will be: Wheel Weight Increase ■ [(45 + 107) x 110] + 110 That works out to 156 lb added at the front wheels. The 110-lb weight of the winch has acted as a force on a lever, with the front wheel centers as the fulcrum, lifting a weight of 46 lb from the rear wheels and transferring it to the front ones. The

Weight Distribution

For trailering, proper understanding of the weights of both the tow vehicle and trailer is important The vehicle manufacturer will usually specify the maximum trailer weight the vehicle can handle. The tongue weight of the trailer, i.e. the weight at the hitch, should be between 10 and 15 percent of the trailer's overall weight. Photo by Bill Hancock.

overall vehicle weight is now 4410 lb, with 2356 lb on the front wheels and 1754 lb on die rear ones, and the weight distribution has become 57/43.

Trailer Tongue Weight One of the most useful applications of the formula for added weight at the front or rear of a vehicle is to find the effect of the tongue weight of a trailer on the weight distribution of a tow car or truck. I’m talking here about a vehicle with a weight­ carrying hitch, which suppons the entire tongue weight on the hitch ball, as opposed to a weight­ distributing hitch, which uses springs and levers to divide the tongue weight between the front and rear wheels of the tow vehicle; the weight will lower the vehicle's ride height slightly, but tne vehicle will remain level even with a heavy trailer attached. A weight-carrying hitch is satisfactory with smaller, lighter trailers, Dut a weight-distributing one is recommended for heavier towing. The tongue weight of a trailer should be between 10 and 15 percent of the overall weight. For example, a 1000-lb trailer should have a tongue weight between 100 and 150 lb. Using a Bathroom Scale—How can you find the trailer's tongue weight? If the trailer has a gross

weight rating of 2000 lb or less, the tongue weight should be no more than 300 lb and can De checked on an ordinary bathroom scale. However, if there's any reason to believe that the tongue weight is going to lie beyond the 300-lb capacity of the typical bathroom scale, you'll have to use a little trickery. l ake a brick or a block of wood the same height as the scale and a 2x4 more than three feet long. Place the block so that its center line is exactly one foot to one side of the center line of the jack extension on the trailer tongue. Place rhe scale so that its center line is two feet to the other side of the center line of die jack extension. Now place the

An ordinary bathroom scale with a capacity of 300 lb can be used to check trailer tongue weights as high as 900 lb by offsetting the scale from the point at which the weight is applied. The exact procedure is described In the text.

Auto Math Handbook

FORMULAS FOR WEIGHT DISTRIBUTION Percent of Weight on Wheels = (weight on wheels + overall weight) x 100 Increased Weight on Wheels = [(distance of eg from wheels + wheelbase) x weight] + weight

2x4 so that it stretches across both the Block and scale. Lower the jack until the trailer tongue drops far enough that its weight is supported by the 2x4. Read tne scale and multiply the figure it shows by the number of feet, in this case 3, between the centers of the block and scale. That will give you the tongue weight. The setup I’ve described will handle tongue weights up to 3 times the scale's capacity, or 900 lb with a 300-lb scale. For heavier tongue weights than that, you can use a longer 2x4, move rhe scale out another one or two feet, and change the multiplication factor accordingly. The factor must be the same as the distance in feet between the centers of the block and scale. Heavier Trailers—You can also take the trailer to a truck scale. I lowcvcr, because of such a scale's questionable accuracy with lower weights, you may not get satisfactory results by simply dropping the trailer tongue on the scale. In the words of Bill Estes, former editor and associate publisher of Trailer Life Magazine: 'You'll get a much better figure by subtracting the difference, say, between 4,000 and 4,500 lb than by weighing a 500-lb trailer tongue on a scale that may be calibrated for as much as 80,000 lb.' Estes describes the procedure for finding the tongue weight: "Position the car and trailer so the tongue jack and trailer wheels are on the scale but the car wheels are off. Get a weight figure on the trailer wheels with the tongue resting on the tow vehicle off die scale. Then, lower rhe tongue jack to the scale, raise the coupler off the ball, drive die vehicle away and leave the trailer on the scale. You'll have two readings. Subtract to determine hitch weight.” Subtract, that is, the weight of the trailer hitched from the weight unhitched. If the row vehicle has a weight-distributing hitch, the spring levers should be disconnected to

deactivate the weight-distributing feature and the back of the tow vehicle should be blocked or jacked so that the vehicle and trailer remain at their normal ride height. Okay, suppose you have a 3000-lb trailer and you find it has a tongue weight of 450 lb. For a tow vehicle, let's go back to that 107' wheelbase sport­ utility and assume that its 4,000-lb weight includes a weight-carrying hitch. The center line of the hitch ball is 52" aft of the center lines of the rear wheels. Plugging the appropriate figures into the formula: Wheel Weight Increase = [(52 + 107) x 450| + 450 You learn that 669 lb have been added to the rear wheels, of which 669 minus 450, or 219 lb, have been lifted from the front wheels. The front-wheel weight has been decreased from 2200 to 1981 lb and the rear-wheel weight has gone up from 1800 to 2469 lb. Overall weight is now 4450 lb. Most importantly, what's happened to the weight distribution? Let s check at die front wheels:

Wheel Weight Percentage = (1981 + 4450) x 100 That's 44.5 percent, which gives a front/rcar weight ratio of 44.5/55.5, almost the reverse of the original 55/45! That drastic a shift in weight distribution will cause the rear end of the tow vehicle to drop and the front end to rise, and the vehicle will ride at a noticeable angle instead of level. The added weight at the rear is likely to result in greater rear end sway around turns, making the vehicle more difficult to control. In addition, the headlights will point slightly upward, right into the eyes of drivers of oncoming vehicles. And that helps explain why, with a heavier trailer, a weight-distributing hitch is preferable to a simple weight-carrying one.

Chapter 8

Center of Gravity

In a race car, it’s important to know the position of the center of gravity—longitudinally, laterally and vertically—in order to determine such aspects of vehicle behavior on the track as weight transfer to the rear when accelerating or to the outside of a turn when cornering. Photo by Tom Sturgeon.

The center of gravity or eg of an object, as noted in Chapter 7, is the point around which die weight of the object is evenly balanced in every direction. In the previous chapter, the concern was with die eg of add-on devices like winches and hitches and die effects they have on weight distribution. In this chapter, I'm going to discuss the eg of the vehicle itself. This is especially important on a race car, where you need to know die exact location of the eg in order to predict several aspects of vehicle dynamics, such as how much weight transfer occurs to die rear while accelerating or to the outside of a turn while cornering.

Horizontal Position The horizontal position of the eg, both lengthways and sideways, is inversely proportional to the weight distribution. The horizontal position is measured in relation to the distances between the points at which the vehicle is weighed, i.c., the wheels. Lengthways Location—The lengthways location of the center of gravity is measured as a part of the wheelbase. Io find how far it is behind the front wheel centers, divide the weight on the rear wheels by the overall vehicle weight and then multiply rhe resulting decimal figure by the wheelbase, or:

eg location behind front wheels = (rear wheel weight + overall weight) x wheelbase As an example, go back to the oval track race car described in Chapter 7. That car weighs 3000 lb overall, with 1410 lb on the front and right wheels and 1590 lb on the rear and left wheels. Now suppose that the wheelbase is 108" and the track 63'.

The information needed to find the lengthways position of the eg is shown in Fig. 8a on page 42 and is applied in the formula thusly:

eg location behind front wheels = (1590

3000) x 108

eg location behind front wheels ■ 0.53 x 108 The eg is 57.24" behind the front wheel centers, as shown in Fig. 8b on page 42. Sideways Location—Just as the lengthways position of the eg can be measured as a part of the wheelbase, the sideways location can l>c measured as a part of the vehicle ITALtrack, which is the lateral distance between the centers of the treads of tires on cither side. However, the sideways location of the eg is usually described in terms of how far it is off-center toward the heavier side. To find that, divide the weight on the lighter side by the overall weight and multiply the resulting decimal by the track, then subtract that figure from 1/2 the track, or: eg location off-center on heavy = (track + 2) - [(weight on 1 ight side -r overall weight) x track] II there's a significant difference between the front and rear tracks, use die average of the two. The sideways or off-cener location on the heavy side in our oval track car would be:

eg location = (63 + 2) - [(1410 + 3000) x 63] eg location = 31.5 - (0.47 x 63) eg location heavy side = 1.89 inches In diis case, the eg is located 1.89 inches off-center to the left.

Alto Math Handbook

Fig. 8a. To find how far the eg is behind the front wheel centers, you need to know the vehicle's rear wheel weight, overall weight and wheelbase.

Scale 1:24 Fig. 8b. With weights of 1590 lb on the rear wheels and 3000 lb overall and a wheelbase of 108", the lengthways eg position would be 57.24" behind the front wheel centers.

57.24 inches

Scale 1:24 Vertical Position On an automobile with a front-mounted, pushrod V-8 engine and rear-wheel drive, which is die configuration of most traditional high-performance cars, the eg will usually be from 14 to 22 inches above the ground. On such a car, one rule of thumb is that the eg will be at about the same height as the camshaft. To find what that height is, use a yardstick to measure from the ground to the camshaft centerline at the front of the engine. However, to pinpoint the eg height more precisely, or to find it at all on a vehicle with other than a front V-8 engine/rcar-drivc layout, you'll have to do some jacking around—literally. Weights and Measures—Stan again by weighing the vehicle at all four wheels. You can do it back ar the truck scale, or you can use just two individual scales as described in Chapter 7. But, this time, it will be a lot easier with a set of four scales. Record the overall weight and die front- and rear­ wheel weights. Then raise one end—up to 24" if possible—with the wheels at die other end still on the scales, and note how much weight is transferred to that end. It doesn’t matter which end of the vehicle is raised and which is left on the scales. Use a heavy-duty hydraulic jack or an overhead chain hoist to lift the vehicle. Don't even think about using a bumper jack. It wouldn't lie likely to lift the vehicle

high enough and, as high as it did go, it wouldn't hold the vehicle safely while you take measurements underneath. At a truck scale, you would have to measure the overall weight, then drive the vehicle partway off' the scale platform, jack up die end that’s now clear and record the weight at the wheels still on the scale. That's a time-consuming activity not likely to endcar one to the scale operator—or to any truckers waiting in line. You may, however, be able to go use the scales of a moving company, although they may charge you a small fee. With only two individual scales, you would have to weigh each end of the vehicle separately. At the opposite end, you would need two blocks the same height as the scales to place under the wheels to assure that the vehicle is level while being weighed. Thar would mean additional jacking as you swap the scales and blocks, stretching out what is already a tedious procedure. So, if at all possible, a set of four scales should be used. Suspension and Tire Deflection—At the end of die vehicle that remains on the scales, die added weight when the other end is raised could deflect the suspension and tires enough to throw’ off the readings. In addition, the suspension and tires at the raised end could drop slightly and prevent an accurate measurement of just how far diat end has

Center of Gravity

Fig. 8c. To find the eg height, it's necessary to raise one end of the car at least two feet, or 24" as shown here, and to measure how much weight Is added to the scales at the other end. The ground­ level wheelbase must also be determined, either by measurement or geometric calculation. Applying the formula described in the text to the measurements in the drawing provides a eg height of approximately 17".

Fig. 8d. Now it's possible to graph the lengthways or longitudinal position, 57.24" behind the front wheels, and the vertical position, 17". The sideways or lateral position can also be calculated easily, using the weight on the left or right wheels, the weight overall and the vehicle track.

been lifted, a figure you'll need for calculating eg height. To deactivate the suspension at cither end, the shock absorbers can be replaced with solid metal rods of the same length, while the tires on the scales can simply be overinflated to minimize deflection. Full or Empty Fuel Tank—There arc differences of opinion as to how much fuel should be aboard tine vehicle while it is being weighed to find eg height. Some say that the tank should be full and others that it should be empty. Still others compromise by saying it should be half full—or half empty, depending on your point of view. One of the best suggestions I've heard is to go through the whole procedure with the tank empty, then repeat it with the tank full. That will provide the two extremes in eg height that can occur in normal vehicle operation. Generally, on most conventional vehicles, the eg height is slightly lower with a hill tank than it is with an empty one. Necessary Dimensions—Once you know die weight of the vehicle and the amount of weight transferred when one end is raised, there are three dimensions in inches you need. The first is the wheelbase with the vehicle level, which you would

most likely already know; the second is die wheelbase at ground level with one end raised; the third, as indicated earlier, is the distance that one end has been raised. The figures needed for the eg height on our oval track car arc all shown in Fig. 8c. To find eg height, multiply the wheelbase with the vehicle level by the wheelbase with one end of die vehicle raised at least 24" by the added weight shown on the scales with rhe one end raised. Then divide die product of that calculation by the distance the one end has been raised multiplied by the overall vehicle weight. Or, stated as an equation:

eg Height = (level wheelbase x raised wheelbase x added weight on scales) t (distance raised x overall weight) As noted earlier, you would probably already know the level wheelbase of the vehicle and, of course, you could use a yardstick to check how high the one end has been raised. Measuring Ground Level Wheelbase—To find the wheelbase at ground level with one end of the vehicle raised, you can either measure or calculate it. To measure it, drop a plumb bob from the bottom of

Auto Math Handbook

FORMULAS FOR CENTER OF GRAVITY CG Location Behind Front Wheels = (rear wheel weight + overall weight) x wheelbase

CG Location Off-Center to Heavy Side = [(track + 2) - (weight on light side + overall weight)] x track CG Height = (level wheelbase x raised wheelbase x added weight on scales) + (distance raised x overall weight)

one of the raised tires. iVlakc a chalk mark where the plumb bob strikes die ground and, with a tape measure, find how far it is from the center of the ground level wheel on the same side of the vehicle. Calculating Ground Level Wheelbase—To calculate the wheelbase at ground level, note in Fig. 8c that the two wheelbase measurements and the distance that one end of die vehicle has been raised form a right triangle. You would already know the measurements of two sides of that triangle—the 108 " wheelbase with the vehicle­ level and die 24" distance that one end has been raised. You can find the third side by applying rhe Theorem of Pythagoras, which states that, in a right triangle, die square of the side opposite the right angle equals the sum of the squares of the odicr two sides. The 108 " wheelbase is the side opposite the right angle, and 108 squared is 11664. The odier known side is the 24” lift, and 24 squared is 576. The square of die diird side is 11664 minus 576, or 11088. The measurement of the third side would be the square root of 11088, or 105.3.

Added Weight—Finally, to find how much weight has been added on the scales, simply find the difference between the reading when the vehicle was level and the reading after the oilier end was raised. In our example, the first reading was 1410 lb and die second 1518 lb, a difference of 108 lb.(It's simply coincidence in this case that die level wheelbase in inches and the added weight in lb both happen to be 108.) CG Height Let's plug our figures into the formula and find die eg height of our oval track car:

eg Height = (108 x 105.3 x 108) + (24 x 3000) = 1,228,219.2 + 72,000 The answer is 17.0586", which, of course, can Ik rounded down to an even 17", a fairly typical figure for the type of car in question. In Fig. 8d, the eg position both lengthways and vertically has been plotted. As you can sec, the position of the eg isn't difficult to calculate. But getting the data needed to find the position, especially vertically, can be difficult and timeconsuming.

Chapter 9

g Force and Weight Transfer

When a high performance car leaves the starting line under hard throttle in first gear, the amount of force applied at the drive wheels can be phenomenal. Drag racers need to set up the car carefully for proper launching, otherwise they could smoke the tires and lose the race. Photo by Larry Shepard.

weight transfer during acceleration or cornering. Weight transfer, in turn, is critical because it can influence how a vehicle's chassis should be set up. g Force In order to find weight transfer during a particular maneuver, there arc factors needed other than the position of the eg. One of the most important is the g (for gravity) force acting on the vehicle during the maneuver. Here on earth, a free-falling object will gain speed every second by 32.174 feet per second (fps) or, as a physicist would say, it accelerates at 32.174 feet per second per second. That can also be written 32.174 feet per second squared. Thar figure is 1.0 g and is the scientific norm for measuring the acceleration of any moving object, not just one in free fall. When you're in a rapidly accelerating vehicle and you feel as if you're being thrust back into your seat, you're experiencing g force. To find the g force acting on a car while it's accelerating, you need to know the thrust in pounds being applied by the drive wheels to the road surface. And to find the thrust at the drive wheels, you need to know the torque at the wheels and the rolling radius of the wheels and tires. Here, things begin to get a little complicated. Drive Wheel Torque—In discussing indicated versus brake engine output in Chapter 5, we pointed our that friction and inertia within the engine cause losses in horsepower and

torque between the combustion chambers and the flywheel. Well, friction and inertia make some further claims between the flywheel and the drive wheels, with the transmission and final-drive assembly taking a particular toll on horsepower and torque. If you have engine dynamometer numbers, you can find the results of those losses by testing the vehicle on a chassis dynamometer, which measures output at the drive wheels. Or you can simply estimate die losses at about 15 percent, which is what they re likely to be on most modern cars. In other words, the drivetrain should be about 85 percent efficient. A car which has 100 brake horsepower at the flywheel should have about 85 horsepower at tile drive wheels. To find the maximum torque in lb-ft at the drive wheels, you have to multiply the torque at the flywheel by both the first-gear ratio and die final-drive ratio, and by our 0.85 efficiency factor, or:

Drive Wheel Torque = (flywheel torque x first gear x final drive) x 0.85 Let’s demonstrate that by using a late-model Chevrolet Corvette with a 350 cubic-inch engine which has a maximum torque of 330 lb-ft. The vehicle also has a five-speed manual transmission with a 2.88:1 first gear and a 3.07:1 final-drive ratio. Applying those figures in die formula, you have: Drive Wheel Torque = (330 x 2.88 x 3.07) x 0.85 That provides a figure of 2480.0688 or, rounded down, 2480 lb-ft at the wheels. That's right—2480! No wonder it's

Auto Math Handbook

g Force ■ wheel thrust + weight Given a weight of 3292 lb for the Corvette, you would divide that into die thrust figure of 2362 lb :

A Mustang leaves the starting line. The parachute and rear spoiler give a subtle clue to the real performance of this package. Photo by Bill Hancock.

easy to smoke a 'Vette's rear tires in first gear! Wheel Thrust—As noted in Chapter 4, torque can be described as a force in pounds applying leverage over a distance in feet; hence our definition of it in lb-ft. Ar the drive wheels, that distance is determined by the tire size or, to be more precise, by the tire's rolling radius. That's the vertical measurement from the center of die wheel to the tire’s point of contact on the ground. Because the weight of the vehicle flattens the tires slightly, the rolling radius is usually slightly less than die horizontal radius, lhc resulting radius is referred to as the static loaded radius or SLR for short To find the thrust in pounds the drive wheels apply to the pavement, divide die torque at the wheels in Ib-ft by the rolling radius in feet: Wheel Thrust = drive wheel torque + rolling radius Using a yardstick or tape measure, it's easier to get an accurate reading of the rolling radius in inches and convert it to feet, rather dian trying to measure it directly in feet. On the 'Vette, suppose the rolling radius is 12.6 inches. Io convert that to feet, divide by 12, giving you a figure of 1.05 to divide into the drive wheel torque: Wheel Thrust = 2480 + 1.05 The thrust at die drive wheels is 2361.9048 or, rounded up, 2362 lb. Calculatingg Force—To find the g force during acceleration, you simply divide die dirust in pounds by the vehicle weight, or:

g Force - 2362 + 3292 The Vette's potential rate of acceleration would be 0.717497 or, rounded down, 0.72 g. We already know that 1.0 g equals 32.174 feet per second per second. Multiplying that by 0.72, the 'Vette's potential acceleration could also be expressed as 23.16528 or rounded off, 23 feet per second per second. Those arc strictly theoretical figures, though, that don't rake into account such variables as rolling resistance or aerodynamic drag. Nonetheless, if you know rhe maximum g forces for a variety of vehicles, they do have comparative value. Weight Transfer Weight transfer is especially important in drag racing. As a car leaves the starting line on a quarter­ mile run, the weight that shifts momentarily to the rear will apply force to the drive wheels that should improve traction; die greater the weight transferred, rhe better the bite. To find a vehicle's maximum weight transfer during acceleration, multiply the overall weight by the height of the eg, divide that by the wheelbase, and then multiply the result by the g force, or:

Weight Transfer = (weight x eg height) + (wheelbase x g) You already have a weight of 3292 lb and a g force of 0.72 for the Corvette. Its eg height would be approximately 18 inches and its wheelbase 96.2 inches. So, to apply the formula: Weight Transfer « (3292 x 18) + (96.2 x 0.72) The maximum potential weight transferred to die rear wheels during acceleration would be 443.49605 or, rounded down, 443 lb. To show the effect of eg height on weight transfer, suppose you wanted to rebuild the Corvette for drag racing. You were able to iack it up enough that the eg was raised 6.0" to a heiglit of 24 while, for the sake of simplicity, the other critical specs were kept the same: Weight Transfer ■ 3292 x 24 + 96.2 x 0.72 That would increase weight transfer to 590.10124 or, rounded down, 590 lb, a gain of over 33 percent or one-third!

G Force and Weight Transfer

In actual practice, if the car were being prepared for drag racing, rhe engine would have been modified for higher output and a numerically higher final-drive ratio installed. With greater torque and stronger gearing, the potential g force would have been raised and added still funner to the amount of potential weight transfer. That explains why some drag cars, in classes which permit it, arc built as high off the ground as they are. Lateral Acceleration In most forms of motorsports other than drag racing, weight transfer is kept as low as possible for steady, consistent handling. In both road and oval track racing, for example, a low center of gravity and, with it, minimum weight transfer are desirable for cornering stability. During straightaway acceleration on a drag strip, as you've seen, the key to the force being applied to the vehicle is the thrust in lb at the drive wheels. During cornering on an oval track or road course, rhe key is the g force acting on the vehicle and attempting to push it sideways as it goes around the turn. This sideways g force is called lateral acceleration, a factor you'll need in order to calculate lateral weight transfer. Calculating—To calculate lateral acceleration, you'll need two factors that can only be determined by testing the vehicle on a skid pad. As Fred Puhn explains in HPBooks' How to Make Your Car Handle-. A skid pad is a flat piece of pavement with a circle painted on it. The car is driven around the circle, keeping the center of the car right on the line. By measuring the time it takes to make one lap of the circle, the lateral acceleration can be computed. To do this, you need to know the radius of tne circle and the rime for one lap at maximum speed." The radius of the circle should be in feet and the time for one lap in seconds. The raw formula for the lateral acceleration in feet per second per second is the square of 2.0 times pi, multiplied t>y the radius divided by the square or the time, or:

Unlike drag racing, weight transfer is kept as low as possible in most other forms of motorsports for steady, consistent handling. A low center of gravity, and, with It, minimum weight transfer are desirable for maximum cornering ability. Photo by Tom Sturgeon.

(2.0 x pi)2 + 32.174 - 6.28318532 + 32.174 39.478418 «■ 32.174 which works out to 1.2270286 or, rounded down, 1.227. Now the formula for lateral acceleration in g's can be simplified io:

Lateral Acceleration = (1.227 x radius) -e- time2

Example—Let's suppose you test the Corvette on a skid pad with a radius of 150 feet and it turns a lap in 14.5 seconds:

Lateral Acceleration = [(2.0 x pip x radius] + time2 To find lateral acceleration directly in g force, the value of 1.0 g in feet per second per second—which, of course, is 32.174—-can be plugged into the formula: Lateral Acceleration = [(2.0 x pi)2 -r 32.174) x radius] + time2 The figures involving pi and g can be reduced to a single constant:

During fast cornering, lateral acceleration tends to force a vehicle sideways, out of the turn. To find lateral acceleration in g force, the car must be timed in seconds at the limits of its adhesion around a skid pad of known radius. Photo by Bill Hancock.

Alto Math Handbook

In skid pad testing, a car should be kept centered on the circumference of the skid pad circle. This diagram is adapted from HPBooks' How to Make Your Car Handled Fred Puhn, an excellent reference on chassis engineering for highperformance cars.

Lateral Acceleration = (1.227 x 150) -s- 14.52 Lateral Acceleration = (1.227 x 150) + 210.25 The result would be a lateral acceleration figure of 0.8753864 or, rounded down, 0.875 g.

Lateral Weight Transfer To find the weight transfer during cornering, the formula is essentially the same as the one for weight transfer during acceleration, except that the vehicle's wheel track is used instead of its wheelbase: lateral Weight Transfer = (weight x eg height) * (wheel track x g) You already know the weight of the Corvette in our ongoing example is 3292 lb and its eg height is 18". The Corvette has a front track of 59.6" and a rear track of 60.4 ' for an average of an even 60”. So to find its sideways weight transfer:

Lateral Weight Transfer = (3292 x 18) + (60 x 0.875) The answer is 864.15 lb. You can reduce the amount of weight transferred in a turn by decreasing the weight and/or the eg height and/or by increasing die track—none or which is particularly easy to do.

Centrifugal Force Civen the vehicle's weight and die g force determined in a skid pad rest, you can also find the centrifugal force in pounds acting on the vehicle while cornering with a simple formula: Centrifugal Force = weight x g For the Corvette, multiply 3292 x 0.875, which works out to a centrifugal force of 2880.5 lb. That, however, is mainly a point of academic interest, and not of the same practical significance as the g force or weight transfer values.

G

Force and Weight Transfer

FORMULAS FOR G FORCE AND WEIGHT TRANSFER Drive Wheel Torque = (flywheel torque x first gear x final drive) x 0.85 Wheel Thrust = drive wheel torque •? rolling radius g = wheel thrust 4- weight

Weight Transfer = (weight x eg height) + (wheelbase x g) Lateral Acceleration = (1.227 x radius) 4- time2

Lateral Weight Transfer = [(weight x eg height) + wheel track] x g Centrifugal Force = weight x g

Chapter 10

Moment of Inertia

With extremely light weight, these dragsters have an extremely low vehicle inertia. Photo by Larry Shepard.

Inertia is defined as the resistance to change in an object’s velocity. In other words, an object with high inertia would resist the force trying to change its velocity with a great deal of force. If rhe object were at rest, the velocity would l>c zero but die resistance would still be diere. Mass obviously plays a big part in inertia calculations as we will see, but also the shape of the mass or more correctly where the mass is centered relative to the axis of rotation, when looking at rotating objects. A vehicle itself has an equivalent moment of inertia, which in general terms, depends largely on die following factors. • vehicle weight • driveshaft rpm • mph To get a better understanding of vehicular inertia consider two perfeedy identical cars with different final drive ratios being the only difference. Obviously the one with the numerically higher final drive ratio will theoretically accelerate easier. Be careful here. Notice I said easier, not necessarily better. To choose the best ratio selection one will have to consider the traction limits and engine speed and torque curve characteristics in order to optimize the final drive ratio. Let's look at two vehicles in the chart above: Which car will accelerate easier? Well let's apply die formula for equivalent vehicle inertia and sec what the results yield. The formula for equivalent vehicle inertia (1) is:

I = (6.098 x vehicle weight) * (driveshaft rpm + mph)I2 We begin by finding out die driveshaft rpm for car A. To do that, wc start w'ith the tire diameter. Using the formula below; wc will calculate the number of revoluuons it takes the tire to

Table A Car A CarB

Weight

Gear Ratio

Tire Dia.

3800 lb 3200 lb

4.11

32"

3.23

32"

MPH 70 70

cover one mile. We all know that there arc 5280 feet per mile. The circumference is the distance around the tire. We find it with the following formula: pi x diameter ■ circumference 3.1416x32 = 100.53” Next wc convert inches into feet by dividing die previous answer by 12.

100.53 + 12 « 8.3775 feet Io ger the number of revoluuons per mile, simply divide the number of feet in a mile (5280) by the tire diameter measured in feet. In the case: 5280 + 8.3775 = 630.26 revoluuons Now to get the driveshaft revolutions, or the number of times the driveshaft has to rotate per mile, wc multiply the tire revoluuons umes die rear axle ratio. 630.26 x 4.11 - 2590.37 rev/mile Since wc arc going 70 mph, die driveshaft revolves 70 times our value for 1 mile, or:

Moment of Inertia

70 x 2590.33 - 181,323.1 rev/hour Our driveshaft is rotating 181,323 times in an hour at 70 miles per hour. If we want to convert this into revolutions per minute, or rpm, we simply divide the hourly rate by the number of minutes in an hour (60) to get rpm. Remember, our formula for equivalent inertia was:

181,323.1 + 60 = 3022 rpm Now wc have all of the values we need to find our inertia: I ■ (6.098 x weight) + (driveshaft rpm + mph)2 I - (6.098 x 3800) + (3022 + 70)2 I = 23,172.4 + (43.17)2 I = 23,172.4 + 1864 I = 12.43 slug-fit for Car A Now let's quickly do Car B the same way we did car A. Wc will leave out all of the steps, since the steps are identical, with only the rear axle value

CarC CarD

Weight

Table B Axle Ratio

Tire Dia.

Speed

3800 lb 3200 lb

4.11 3.23

32" 29"

70 65

Finding Tire Diameter Using the Roll-Out Method If you need to find your tire diameter while the tire is on the vehicle, try this method. Find a section of flat level pavement and make a chalk mark on die tire and the pavement at die 6 o'clock position. Proceed to carefully roll the car forward until the mark on the tire lines up in the 6 o'clock position again. Make another mark on the pavement where the original mark on the tire meets the pavement. Measure between the two marks on the pavement and you will have the circumference of the tire. Once you have the circumference in inches, divide that number by pi (3.1416) to get the tire diameter in inches.

1 = 20.12 slug-ft for Car B If we recalculated the values for car B using yet a different rear axle ratio ( 3.91), let's sec what difference it would make: 1= 13.72 slug-ft What a dramatic improvement! Ar this point, we should be able to understand that a car with a higher numerical ratio can accelerate easier, but suppose we now had to compare two similar cars but with different axle ratios, different tires and different weights. How can we use this? Which car would have the higher moment of inertia? Let's apply our formula and see using the specs in Table B above. Using our formula, wc find that Car C has an equivalent inertia of 16.93 slug-ft and that Car D has an equivalent inertia of 16.32 slug-ft. So in this case wc sec that two seemingly different cars have a verv close inertia. In this case Car C will be at a slight disadvantage. Using a few formulas is certainly easier titan changing a rear axle ratio! Flywheel Comparison Not only vehicles, but all masses, have moment of inertia. Let's discuss just what this means in automotive terms and how wc can calculate these values. Let's look at a flywheel. For classification terms, a typical flywheel is best categorized as a flat disc, however, as we will show, two flywheels having the same overall weight and the same diameter can have two distinctly different moments of inertia. The cross sections of two flywheels arc shown in Fig. 10a

and Fig. 1 Ob on the next page. The flywheel in Fig. 10a has its weight spread evenly across the entire flywheel, while the second one in Fig. 10b might concentrate its mass in a thick band or ring of metal near die outer edge just inboard of the ring gear. While both flywheels weigh die same, die second flywheel with the concentration of mass near the outer diameter will have greater inertia. In the vehicle, if the power is the same, it will take more time to accelerate than die first flywheel with its mass located evenly across the diameter. In any form of racing, acceleration plays a big part of the performance equation, since the ability to get up to speed after a comer or in the case of drag racing, after leaving the starting line and after each shift, is critical. Sometimes there arc cases where wc may want

Auto Math Handbook

Cross Sections of Steel Flywheels

R = 10" diameter ■? 2 R = 5 inches So the area of our flywheel is:

Figure A

3.1416 x 25 sq. inches = 78.54 sq. inches To get the volume wc multiply the area by the thickness T of the disc:

7.00"

Figure B

more inertia. In drag racing, a heavy car with a small displacement engine without much torque will bog or slow down dramatically when the clutch is engaged at die starting line. When this occurs, the rpm falls well below the torque curve and once diat happens, it takes seemingly forever to regain the original rpm. Here a heavy flywheel will store the kinetic energy of the engine revving at the starting line and when the car leaves die line, the flywheel releases its stored energy and prevents the rpm from falling too far. On the same token however, the engine accelerates die heavy flywheel through each gear more slowly than a similarly equipped car with a fight flywheel due to the increased flywheel inertia. In tiring to find the best answer for your car. you can quickly get lost if you use flywheel weight instead of using the flywheel's inertia value. Let's see how we can calculate these values and how we could measure them. Put more simply, inertia defines not only weight, but where the weight is concentrated. The formula for die moment of inertia in a plain flat disc is: J0=l/2MR2 where Jq = Moment of Inertia M= Mass R = Radius Let's calculate the moment of inertia for a flat steel disc 10" in diameter and 0.75 thick. First, we go to the appendix and look up die weight for steel. Steel weighs 0.28383 lbs. per cubic inch. So knowing that, let's calculate the volume of steel involved. Using the formula for the area of a circle: a- riR2 where A = area; I~1 (pi) = 3.1416; R = radius Wc plug in our values for pi if fl = 3.1416

T = 0.75 78.54 x 0.75 = 58.905 cubic inches Now that wc have the volume, we can get its weight by multiplying by 0.28383 lb/cubic inch, which is the weight per cubic inch of steel. 58.905x0.28383- 16.72 lb Now we can calculate the mass:

M = W+g where M = Mass; W = weight; g = acceleration due to gravity = 32.2 ft/scc.2; M = 16.72 + 32 = M = 0.519. Using this number, we apply the formula for inertia Jo and we get:

Jo- 1/2 (0.519) x52 Jq = 6.49 slug-ft Now what if wc have a 10" diameter flywheel that has a thick outer rin^ but has a thin center and also weighs 16.72 lb? Let s look at rhe difference in the moment of inertia for this flywheel and compare it to the flat disc flywheel above. As wc see from the drawing, our new 10" flywheel has an outer ring that has an inner diameter of 7" and is 1.00" thick. The inner flange area of the flywheel is 7.00" in diameter and 0.490" thick. To find the total flywheel inertia, we divide the flywheel into sections to get the individual inertias, then add the individual inertias to obtain the total inertia. First, wc calculate the inertia of the flat disc that makes up the inner flange area. We begin by finding the area:

riR2-A 3.1416 x (3.5)2 - 38.4845 sq. inches Then wc multiply by the thickness to get the volume: 38.4845 x 0.490" = 18.8574 cubic inches We multiply the volume times the unit weight of steel to get rhe weight of the center flange:

18.8574 cubic inches x 0.28383 lb/cubic inch = 5.35 lb Next we calculate the inertia of die flange area, using rhe formula from above.

Moment oe Inertia

Jo = 1/2 MR2 Wc begin by finding the mass of die flange area:

5.35 lb -5-32.2 ft/sec2 = 0.1661 lb mass Jo = 1/2 (0.1661) (3.5)2 Jo = 1.0174 slug-ft Next wc calculate the inertia of the outer ring of our flywheel. When do this by taking the total inertia of a 10" diameter flat disc that is 1.00" thick and subtract the inertia of a 7" disc of the same thickness. We already know our area from above: R2 = area = 78.5398 sq. inches Our thickness is 1.00", which makes our volume: Volume = area x height 78.5398 x 1.00 ■ 78.5398 cubic inches Now wc multiply by the unit weight of steel to get our part weight: 78.5398 x 0.28383 = 22.29 lbs We get the mass by dividing by 32.2 ft/ sec2:

22.29 + 32.2 = 0.692 lb mass Again, we use the formula for inertia: Jo =1/2 MR2 Jo - 1/2 (0.692) (5)2 1/2x0.692x25-8.65 slug-ft Now wc do the same inertia calculation for a 7.00" diameter disc that represents the middle. Then we find the inertia of the outer ring by subtracting the inertia of the center disc from the total flywheel inertia. So the area of the center is: 3.1416 x 3.52 = 38.48 sq. inches 78.54 - 38.48 = 40.06 square inches Now we multiply that by the thickness of 1.00" to get the cubic inches of the ring: 38.48x1.00 = 38.48 cubic inches Then multiply by 0.28383 to get the weight:

38.48x0.28383 = 10.92 lb Wc convert the weight to mass: 10.92 4 32.2 = 0.3391 lb mass Next wc calculate the inertia of the center disc, using the formula from above:

Jo= 1/2 (0.3391 )(3.5)2 Jq = 2.077 slug-ft Now if we subtract the inertia of the inner disc from the inertia of the total thick flywheel, we should

MOMENT OF INERTIA FORMULAS Theseformulas will help with calculationsfor Moment ofInertia MPH mph = (D x rpm) + 336 x Gr where: D = diameter of tire in inches; rpm = revolutions per minute; Gr = drive axle ratio

l ire Diameter D = (336 Gr x mph) + rpm where: D = diameter of tire in inches; rpm = revolutions per minute; Gr = drive axle ratio

The rotational inertia of this traditional steel flywheel can be calculated and compared to other alternatives by using some simple formulas. Whenever considering the inertia of a part, never rely on the weight of the part alone to determine the inertia because sometimes parts which weigh less can have higher inertia and vise versa. Photo by Bill Hancock.

have the inertia of just the thick outer ring. 8.65 - 2.077 - 6.573 slug-ft To this, we add the value of the thin center flange that wc found earlier:

6.573 + 1.0174 = 7.5904 slug-ft Now we compare this to our original flywheel that was 0.750" thicK and weighed die same.

Flywheel A: Flywheel B:

Inertia

Weight

6.49 slug-ft 7.59 slug-ft

16.72 lb 16.72 lb

What a difference! 1.1 slug-feet may not seem like a lot, but try a back-to-back comparison in die car by switching flywheels and y'ou will feel the difference, but more imporranrly, so will the car.

Chapter 11

Aerodynamics

The Outlaw dirt track cars use massslve wings about as large as the entire chassis to create huge amounts of downforce. The downforce is required to allow the tires to gain traction or grip. Photo by Bob Bolles.

Vehicle aerodynamics has become increasingly important in studying the effects of air resistance and lift on vehicles. As a vehicle moves in a given direction, it is essentially creating a hole in the air by moving the air aside. When the vehicle passes by any given point, the hole that was initially created by the vehicle closes back up. This passage creates drag or resistance on die vehicle. Put very simply, there arc two major components of this resistance. Shape, referred to as Cd (a dimensionless number called the coefficient ofdrag) and Size Af {frontalarea measured in square feet) when multiplied together create CdA. CdA is a composite number which gives a value for overall resistance. Land speed cars at the Bonneville Salt Flats come in all shapes and sizes. The small frontal area of a streamliner, coupled with its smooth shape, create a vehicle that requires far less horsepower than it's full-bodied cousins to go very fast. Obviously a Class A motor Coach represents the opposite end of the scale, a large frontal area with the aerodynamic attributes or Cd of a cinder block. No matter how swoopy you make the body, the fact drat you are trying to punch a huge hole in die air requires a tremendous amount of horsepower. As the speed increases, the resistance increases as rhe square of die velocity. R« Af 0.0025 V2 where: R = Resistance in Ibf Af= Frontal area V = Velocity in mph

Horsepower Required Obviously, as a vehicle increases in speed, more and more horsepower is required. Until the late '60s, the method for making a car go fast was to create more horsepower in the engine. Then aerodynamic cars like the Dodge Daytona and Plymouth Superbird were created as ways to achieve the faster speeds without having to create more engine horsepower. The racers soon realized that the way to reduce the required horsepower or aerodynamic horsepower is to reduce cither the coefficient of drag or reduce the frontal area or both.

HP = (V3 x Cd x Af) + 146,600 where: HP = horsepower V = velocity in mph Cd = coefficient of drag Af - frontal area To illustrate, consider a race car with a frontal area of 21 sq. fit. running at Daytona while testing two engines. The two engines are separated by 1.5 mph lap speeds. The first engine averages 187.6 mph while the second averages 189.1 mph. So for the first engine: HP = (6,602,349.4 x 21) + 150,000 HP = 924 hp And for the second engine: HP = (6,761,991.0 x 21) + 150,000 HP = 946.68 hp A difference of 22.36 hp. So to go from 187.6 mph to 189.1 mph requires 22.36 hp for that particular car. Now let's do it

Aerodynamics

The Dodge and Plymouth winged cars of the early 70s were perhaps the most effective use of aerodynamics for stock cars. They were soon legislated out of existence. Here we see that the frontal area while already small is also aided by the fact that the car is low to the ground. The size of the hole that this car makes decreases as the car gets lower. It becomes apparent why cars are checked for height after the race as well as before. Photo by Tom Sturgeon.

again with the same cars but at 100 mph and 101.5 mph: Carl HP - (1,000,000 x 21) + 150,000 HP = 140.0 hp

Car 2 HP = (1,045678.4 x 21) + 150,000 HP-146.4 hp A difference of 6.4 hp. This illustrates how much horsepower is required just to run 187 mph, and how much less is required to run 100 mpn. It also should illustrate just how much more horsepower it rakes to incrementally raise the speed from 187.6 mph to 189.1 mph. Frontal Area To get the true frontal area of a vehicle, the blueprints must be carefully measured and the incremental areas all addeef up. For a person just trying to do some rough calculations multiply the vehicle height by the width times 0.8 shown in the

following formula: WxHxO.8 = Af W = width in feet H = height in feet Af- frontal area in sq. ft. How Does Frontal Area Affect Horsepower Required?—Let s take die previous example where weliad two engines and use the more powerfill engine at 946.68 hp and die car widi a frontal area of 21 sq. ft. We know that it ran 189.1 mph. Let’s suppose that the crew' chief lowered die car 1.5" for qualifying. How' fast w’ould it go? The car originally was 74.11 inches wide and 51 inches high: 74.11 x 51 = 3779.6 square inches. To convert back to square feet, divide by 144 or 26.24 square feet. Then we multiply by 0.8 to get our adjusted frontal area of 21 square feet. Now let's do this again but this time we will use 49.5 inches for die height and see what happens:

49.5x74.11 = 3915.95 sq. in.

Auro Math Handbook

Yaw is difficult to accurately measure since it is a dynamic phenomenon. Today most teams use a yaw meter in their onboard data instrumentation package to capture the yaw angle as the car goes around the track. Photo by Tom Sturgeon.

We divide this by 144 to get 25.47 sq.ft. Now wc multiply by 0.8 to get our adjusted frontal area of 20.38 sq. ft. Now let's rerun the original horsepower calculation to see the difference in speed.

hp = (mph3 x Af) -r 150,000 If we rewrite the formula and solve for mph, it becomes:

mph3 = (150,000 x hp) * Ap mph3 = (150,000 x 946.68) ♦ 30.36 mph3 = 6,974,558 If we use our calculator and take the cube root of 6,974,558, we end up with 191.0. So we just learned that by dropping the car 1.5 ’ wc were able to pick up the speed by almost 2 mph. Knowing this, it should come as no surprise that NASCAR inspectors insist that all cars meet all the templates and vehicle heights exacdy. It also explains why vehicles are measured for height AFTER die race to make sure they didn't accidently drop an inch or two during the race. Roll, Pitch and Yaw In addition to creating resistance on a car going down the road, aerodynamics play a big pan in tnc handling as well. Wc have all seen spoilers and wings on cars over the years, now let's examine what they

really do and how they affect the vehicle handling. Before we can begin our discussion, let’s define these three basics ot vehicular motion. Originally associated with aircraft and boats, the automotive engineers have adopted them to better describe and quantify the motions as applied to road vehicles. Think of cars as airplanes that taxi really fast but never (hopefully) get off the ground. Roll—Roll is defined as rotational movement around the longitudinal axis which enters the grille and exits through the trunk while passing through the center of gravity and staying parallel to the ground. Typically we see roll when a car enters a corner and rolls to one side. Pitch—Imagine an axis that enters the driver's side and exits from the passenger side of the car while passing through the center of gravity. This axis is perpendicular to the roll axis. The resultant positive and negative rotation could be compared to die motion a rocking horse would produce. Yaw—The axis enters the roof and exits from die bottom of the vehicle while passing through the center of gravity and while staying perpendicular to both die roll and pitch axis. A vehicle diat spins out could best illustrate extreme yaw. Interestingly, most oval track cars are in a constant state of yaw as they proceed around the track Vehicular aerodynamics usually do not play a big

Aerodynamics

HANDLING BASICS Forget aerodynamics for the moment. As a vehicle increases speed, the cornering forces increase due to centrifugal force. Tires provide resistance to prevent the car from spinning out. The force that the tire can generate to counteract these centrifugal forces depends on not only the size and compound of the tire, hut more importantly on the amount of normal or downward force applied to the tire. This can best be demonstrated by taking a partially deflated basketball and dragging it across a glass table top. If you simply drag the ball across the table, it has some resistance, but not much. If you then have a friend push down on the ball while you try to drag it across die table, it becomes clear that normal force or downforce is critical. As with anything, there can be too much and when rhat happens, rhe tires become overloaded and actually lose tractive effort as the weight increases. We typically see overloading when a car suddenly spins out in a corner. Usually after careful analysis, you will find that the car suspension has bottomed out causing the load on a particular tire to increase dramatically. Ar this point, die tire loses its ability

part in roll, but they have an incredible effect on pitch and yaw. Cornering forces typically induce roll. Pitch is the most obvious place to begin. Viewed from the side, when air moves over and under a car, it creates forces which act on the vehicle. These forces translate to the suspension and either remove or add force or weight to the tires. At speed, some cars tend to lift the front end while others tend to lift the rear end. Lift can have a positive or negative value. Negative lift is often referred to as downforce. By using spoilers on die front end, called chin spoilers and spoilers or wings on the deck lid, the engineers and vehicle aerodynamicists do what is called balance die car. The goal is to create a vehicle which has a steady and predict able ratio of front and rear down force, while maintaining a low drag profile. The ratio of front to rear down force is called the pitch couple. Saying it like that makes it sound simple. Making a car that is well balanced aerodynamically while still maintaining low drag resistance can take months of wind tunnel and track testing. As the vehicle increases in speed the front and rear lift forces tend to increase. Sometimes, the change front to rear is disproportionate and the handling becomes compromised at a given speed range. When weight is

to maintain a high tractive load and the reduction in cornering resistance causes the car to lose traction and either oversteer or understeer. Our goal with aerodynamics is to provide additional downforce in proportional amounts as the speed increases, hopefully without adversely affecting the drag. At this point, you should understand that aerodynamics can become a game of compromises. When you finally get the pitch couple in. a workable range, the drag becomes high and the car slows down. When you get the drag reduced so the car can go fast, it may not want to handle. In the end, when you get the car so it is fast and so it handles well, then the driver complains that it is fine until he gets around other cars in traffic or until he goes past the grandstands and catches a side wind. Aerodynamics is a science and as such is best left to the truly qualified aerodynamicists. Casually adding a spoiler or wing can have huge unintended and deadly consequences. Be especially careful, or as die fine print says, 'Don't try this yourself.'

removed from the front tires for example, the tractive effort of those tires is decreased, so in this case, the car would understeer or "push.' In the same way, added downforce on the rear might also cause the car to understeer because the car would lift the front end due to the added weight on the rear. The goal of aerodynamic tuning is to maintain the same pitch couple or the correct balance of front to rear forces. Finding the comfortable balance of front to rear forces comes from wind tunnel data followed by ontrack testing. As the cars become more competitive the qualifying aero setup differs from the race setup. During a race, the car must be stable when other cars arc surrounding it and affecting its aero balance. Yaw plays a part of the aerodynamic equation when tne vehicle tries to move from side to side or during cornering. An unstable car will spin out very quickly, whereas a car like the old Plymouth Superbirds and Dodge Daytonas with their twin upright tail fins will go through a corner like it is on rails. Yaw also creates an additional problem of increasing the frontal area. To understand this take a model car on a table top and twist it slightly sideways then move it down the table in its sideways position. The true exposed frontal area increases as the yaw

Auto Math Handbook

This Dodge Charger has a number of aerodynamic features to aid In handling and drag reduction. Photo by Tom Sturgeon.

angle increases. Years ago, some of the sharper NASCAR teams realized this and mounted the bodies slightly twisted on the car with what could best be termed reverse yaw. As a result, when the car chassis proceeded around the track mostly in vaw, the body was perfectly aligned aerodynamically with the direction of travel anc hence used less power. Going back to pitch: what the engineers strive for is the correct pitch couple. Pitch couple is the sum of die forces applied to the front and rear of the car which while acting together try to rotate the car around the pitch axis. The goal becomes to measure these forces and then adjust the spoilers and other aerodynamic aids to ensure that both the front and rear tires have the proper loading so rhe car exhibits neutral handling where it is neither oversteering nor understeering. Roll, pitch and yaw do not really have a great effect on vehicle handling ar speeds below 60 mph. However, drag has an increasing effect on anything going over 20 mph. Widi fuel economy being so critical, all of rhe major vehicle manufacturers have spent millions working on improving their drag coefficients. As we have seen, frontal area is the key. Once you get the car as low and as narrow as possible, then you must focus on the drag coefficient. While shape does matter some things do not. One of the most frustrating things about aerodynamics is that we cannot sec the air. Often in pictures you see the cars being tested in the wind tunnel and smoke being used for flow visualization.

The trained acrodvnamicists can make subtle changes that the uninitiated would never think of by merely looking at their wind tunnel results. After a few trips to the wind tunnel, the novice aerodynamicists soon learn that "if it looks good, it probably isn't." The trained aerodynamicists make seemingly simple changes and subtle shape modifications which make all the difference in the world. The gifted vehicle aerodynamicists can simply visualize air and manage it. When working with an aerodynamicist, make sure they have wheeled vehicle experience. While all of the basic principles hold true, wheeled vehicles have a different set of aerodynamic parameters and goals, since hopefully they never fly but merely taxi quickly and efficiently. Ram Air Pressure With many vehicles using hood scoops, the question becomes how much pressure is generated in the hood scoop when the vehicle is at speed. Obviously as tnc inlet air pressure increases, the engine tuning will change, since more air is being forced into the engine by the higher pressure. A handy formula can be found in the HPBooks' Engine Airflow by Harold Bettes.

Rjur - mph2 + 56,725 where Rajr = ram air presssure in psi mph = vehicle velocity in mph

Chapter 12

Rolling Resistance

Drag cars like this Hemi Challenger driven by JC Beatty have reduced their rolling resistance by making sure the alignment is perfect, the brakes don’t drag and the drivetrain friction is minimal. Photo by John DiBartolomeo.

A vehicle has resistance to rolling that comes from a variety of places. For our needs, these can be best quantified by looking at the tires, gearing, brake drag, wheel alignment and bearing loading. These forces typically rise with increasing vehicular weight. For comparison, lets look at a child's roller skate. It has hard wheels, oversize ball bearings, no gear train and non-stccrable axles. In addition it has only the weight of a small child supported by four hard wheels. It stands to reason that the rolling resistance of a roller skate is very low. An 18wheclcr loaded to 120,000 lb with a few undcrinflated tires could have huge rolling resistance. In order to put a measured number on this resistance, we could tow the vehicle with very sophisticated instruments and measure what is called die force to maintain velocity. In its basic form, this involves towing the subject vehicle and measuring the forces required to maintain various speeds. Once you look deeper, you realize that in order to eliminate die aerodynamic factors, a really long tow rope is required. This becomes very impractical as well as downright dangerous. Do not tty this! A much simpler method is to measure coast down.

traffic count, and at least two miles long. Decide on an upper and lower speed value. Enter the stretch at a safe speed, take your foot off the gas, carefully shift into neutral and start a stopwatch when the car passes down through your upper limit speed value. Stop die watch when you pass through the lower limit speed value. Turn around and repeat going in the opposite direction on the same stretch of highway. By running both ways, you will cancel the wind and grade effects if any. Average the two numbers and you have a beginning data point. You should practice this several times until your results are very close. Fry making your change and then repeat the test. Factors such as wheel alignment, tire pressure, vehicle attitude or rake angle, vehicle height, brake drag and other losses can be easily identified. ’Fhe only ironclad rule is to make only one change at a time. The beauty of a coast down test is that it combines all of the factors into one very simple but repeatable test. In order to make the test more reliable, be sure to repeat the test exactly for a minimum of three times, or until the data is repeatable within two percent. If possible, use a GPS device for speed indication, since they typically arc very accurate and will measure in increments of one mph.

Coast Down The principle here is to measure the combined effect of the various forces creating resistance on die vehicle. Aerodynamic resistance and rolling resistance will be the major contributors. This is by far the safest and most practical method. Almost everybody employs this method in one form or another to evaluate the overall drag due to rolling resistance as well as aerodynamic drag. Find a deserted stretch of straight level road with a low

Tire Pressure Obviously tire pressure affects rolling resistance as well as vehicle height. Try varying the tire pressure while staying within the vehicle manufacturer’s recommended pressures. The recommended pressures are derived from extensive testing to ensure the best and safest handling within all vehicle applications.

Au ro Math Handbook

RAKE ANGLE

This magnetic inclinometer is attached to a flat spot on the rocker panel to measure the rake. If your rocker doesn’t have a flat spot, use a straight 2x4 to average out the bumps and measure from it Photo by Bill Hancock.

Rake angle is die angle dtat typically the underside of the rocker panel makes with rhe ground plane. This angle is measured in degrees and is usually anywhere from 2 to 4 degrees negative, meaning that die front of the vehicle is pointed slighdy downward when viewed from the side. There arc two measurements for rake. Static rake is the angle measure when die vehicle is at rest on a flat level surface. Dynamic rake is the attitude that the vehicle assumes when the vehicle is at speed. Static and dynamic rake angles are rarely if ever the same. Dynamic rake is influenced by the positive and negative lift at both ends of die vehicle as well as the drivetrain torque effects. When testing a vehicle in the wind tunnel, you must carefully rake rhe attitude into consideration in order to tune for the true attitude as opposed to the static attitude. Vehicle handling and tractive effort depend greatly

Wheel Alignment Proper wheel alignment is obviously a great contributor to rolling resistance. If the tires arc not properly aligned, the scrubbing can result in added rolling resistance. Be sure to check bodi the front and rear alignment, especially for toe-in or toe-out. Cars with independent rear suspensions are more vulnerable to misalignment than those having traditional one-piece, rear-axle housings. If your vehicle has ever Been in a collision, be sure to have the alignment checked by a competent shop. Sometimes just pulling a car out of a ditch after it has become stuck will affect the wheel alignment. As suspension pares wear, alignment also changes.

on unit tire loading at speed. If for example a car has too much front end lift it may result in an understeering condition due to the fact that the tires don’t have sufficient weight or downforce on them to produce a meaningful lateral tractive force. Standing still on a set of scales in the pits, a car may show adequate wheel weight, but out on the track, once aerodynamic lift occurs, it may not want to turn at all. Always keep a close eye on rake angle and carcfiilly record the value for each step of your testing. Measuring Rake Angle—Rake angle is measured by finding an absolutely flat and level area larger than your vehicle and placing a straight edge on the bottom side of rhe rocker panel. Either place a protractor on the bottom side of the straight edge, and read the degrees of rake directly or simply find a flat spot on the floor that is relatively level and make two marks on the straight edge that arc 57.3 inches apart. Have your helper hold die straight edge on the bottom of die rocker panel while you make two measurements to the ground, one from each of the previous marks. Subtract die front measurement from the rear measurement and the number of inches will equal the degrees of rake. This is a simple use of the Sine function. If you diink of the rocker panel as being the hypotenuse and the ground being the base of a right triangle, then the height of the triangle divided by the hypotenuse is the Sine of die angle, while it is not perfect, it will suffice for what you are doing in the pits.

Weight When doing coast down testing, always try to have the same fuel load and overall weight, since weight affects rolling resistance as well as vehicle rake and height. Brakes Check the brakes to make sure that they7 are properly adjusted and do not have any undue drag, since this will greatly affect rolling resistance. Disc brakes often will not retract all the way after every stop causing parasitic friction.

Chapter 13

Shift

Points

Even the most humble doorslammer can be driven to quicker ETs and faster terminal speeds if the driver knows when to shift for optimum performance. But that takes homework! Photo by Larry Shepard.

To ger rhe best acceleration out of a high-performance vehicle during shifts, hot rodders say you should stay "on the cam." What they mean is that you should keep the engine within an rpm range where it is delivering optimum torque before and after each shift. As the vehicle speed increases, shifting to a higher gear will allow the engine to operate in this high torque part of the performance curve. The first step toward finding what that range might be is to use a dyno chart. So, once more, let's turn to the dyno chart in Fig. 13a, which is the same one used in Chapter 4. In order to calculate shift points, though, you'll be primarily concerned with rpm and torque, not horsepower, because it's torque that accelerates an automobile. Next, you need to know the loss or gain in rpm when you shift from one gear to another. Example—Suppose you have a Chevrolet powered by the modified 350 cubic-inch engine with output specifications shown in Fig. 13a, with a Warner 1-10 four-speed gearbox which has ratios of 2.20 in 1st, 1.66 in 2nd, 1.31 in 3rd and direct 1.00 drive in 4th. When you drag race, you shift at 6000 rpm. How much rpm do you lose during the shift from, say, 1st to 2nd? You can find out by dividing the ratio in 1st gear into the ratio in 2nd. The result will be a percentage which, when multiplied by the rpm in 1st, will provide the equivalent rpm in 2nd. Or, expressed as an equation: RPM After Shift = (ratio shift into before shift

ratio shift from) x rpm

With the Warner T-10, divide the 2.20 lst-gcar ratio into the 1.66 2nd-gear ratio and multiply by 6000: RPM After Shift = (1.66 + 2.20) x 6000 = 0.7545455 x 6000 The engine speed in 2nd will be 4527 rpm. Subtracting that figure from 6000, you'll find you have a drop of 1473 rpm or about 25 percent. Obviously, you can also apply the formula to shifts from 2nd to 3rd and from 3rd to 4th. From 2nd to 3rd, the engine speed will drop from 6000 to 4735 rpm, a loss of 1265 rpm or 21 percent; from 3rd to 4th, it will fall from 6000 to 4580 rpm, losing 1420 rpm or 27 percent.

Driveshaft Torque But, given the torque characteristics shown on the dyno chart, is 6000 rpm the best point at which to upshift? To answer that question, you need to know the driveshaft torque being delivered to the rear wheels before and after each shift. That's simply a matter of multiplying the brake torque at the flywheel by the transmission ratio, or: Driveshaft Torque = flywheel torque x transmission ratio

According to Fig. 13a, you have 300 lb-ft of torque at the flywheel at 6000 rpm. With a 1st gear ratio of 2.20, that becomes 660 lb-ft being delivered from the transmission via the driveshaft to rhe drive wheels. What about the friction mentioned in Chapter 12? Won't the output from the transmission to the driveshaft be slightly

Auto Math Handbook

Reducing the time that the engine is not delivering power to the driveline is the key to quicker ETs. Photo by Bill Hancock.

less than rhe inpur ro rhe transmission from the flywheel? Yes, it will be. But it doesn't really affect the comparative validity of our driveshaft torque figures so, to simplify your calculations, you needn't take it into account here. All right, you have 660 lb-ft of driveshaft torque at 6000 rpm in 1st gear. When you shift into 2nd, the rpm drops to 4527. According ro Fig. 13a, the engine has 330 lb-ft of torque at 4600 rpm. Multiply that by a 2nd gear ratio of 1.66 and the result would be 548 lb-ft of driveshaft torque. During the shift from 1st to 2nd, you've lost 112 lb-ft of driveshaft torque or about 17 percent. A 17pcrccnt drop in torque during an upshift doesn't sound like a good way to win a drag race. Ideal Shift Points To find the ideal shift points for this particular combination of Chevy' engine and Warner gearbox, the two formulas discussed in this chapter were used to construct the charts shown in Fig. 13b on page 63. There's a separate chart for each shift, from 1st to 2nd, 2nd to 3rd, and 3rd to 4th. The shift points arc shown at 200-rpm intervals from 6000 to 7000 rpm. The rpm immediately before the shift is in the 1st column. In the 2nd is the brake flywheel torque at that rpm, as shown on the engine dyno chart, and in the 3rd is the driveshaft torque ar that rpm. The 4th column displays the rpm immediately after the shift. The 5tn shows the flywheel torque at the rpm on the dyno chart closest to the rpm after the shift, and die 6th the driveshaft torque after the shift. (Note that in 4th gear, which is direct drive,

RPM 3000 3200 3400 3600 3800 4000 4200 4400 4600 4800 5000 5200 5400 5600 5800 6000 6200 6400 6600 6800 7000

LB-FT 340 340 345 345 350 350 340 335 330 325 315 310 305 305 305 300 280 255 240 190 160

BHP 194 207 223 236 253 267 272 281 289 297 300 307 314 325 337 343 331 311 302 246 213

Fig. 13a. This chart shows the torque and horsepower for a modified 350 cubic-inch Chevy at Intervals of 200 rpm from 3000 to 7000 rpm as measured on a dyno.

there's no difference between the flywheel and driveshaft torque figures.) In the 7th column is the change, minus or plus, in driveshaft torque after the shirt. After each of the three shifts, the least change in the 7th column occurs at 6600 rpm. During the shift from 1st to 2nd, there is a loss of only 5.0 lb-ft at that engine speed; from 2nd to 3rd, a gain of 8.0 lb-ft; and from 3rd to 4th, a gain of 1.0 lb-ft. Below 6600 rpm, there are much greater drops in torque after the shift. Above 6600, there are gains in torque after the shift but at die price of losing torque and, with it, momentum before the shift. So 6600 rpm seems to be the best shift point across the board. There’s a drag racer's rule of thumb that the best shift point is at an engine speed 10 percent beyond the horsepower peak. The dyno chan shows the horsepower peak at 6000 rpm and, of course, 6600 rpm is exactly 10 percent beyond that. However, that should be regarded as an interesting coincidence and not conclusive proof of the rule of thumb. Downshifts—The formulas can also be used to calculate changes in rpm and torque during downshifts. Let's say you're running a T-I0 equipped sports car in a road race. You're heading

Shift Points

toward a turn at 4000 rpm in 3rd gear. How high should the engine be revved for a downshift to 2nd? Divide 1.31 into 1.66 and multiply by 4000. The engine should be turning a theoretical 5069 rpm as you go from 4th into 3rd; "theoretical" because not even a pro Indycar or NASCAR driver can coordinate a tach reading and throttle pressure closely enough to achieve that precise an engine speed! You would gain a theoretical 1069 rpm, an increase of approximately 27 percent. During that

shift—using the torque figures from Fig. 13a at 4000 and 5000 rpm—thcre'd be a gain of 64 lb-ft in driveshaft torque. Here, too, it would be instructive to develop charts showing the changes at different shift points. These formulas aren't complicated but, obviously, you could spend a lot of time with them, calculating different combinations of torque, rpm and gearing. And those calculations might just win a race or two.

RPM 6000 6200 6400 6600 6800 7000

brake lb-ft 300 280 255 240 190 160

Shift from 2.20 1st to 1.66 2nd shaft lb-ft RPM brake lb-ft 660 330 4527 616 4678 330 561 4829 325 528 4980 315 418 310 5131 352 5282 310

RPM 6000 6200 6400 6600 6800 7000

brake lb-ft 300 280 255 240 190 160

Shift from 1.66 2nd to 1.31 3rd shaft lb-ft RPM brake lb-ft 498 4735 325 465 4893 325 423 5051 315 398 5200 310 315 5366 305 266 5524 305

shaft lb-ft 426 426 413 406 400 400

loss/gain -72 -39 -10 +8 +85 + 134

RPM 6000 6200 6400 6600 6800 7000

brake lb-ft 300 280 255 240 190 160

Shift from 1.31 3rd to 1.00 4th shaft lb-ft RPM brake lb-ft 4580 393 330 367 4753 325 334 4885 325 314 5038 315 249 5191 310 210 5344 305

shaft lb-ft 330 325 325 315 310 305

loss/gain -63 -42 -9 +1 +61 +95

shaft lb-ft 548 548 540 523 515 515

loss/gain -112 -68 -21 -5 +97 + 165

Fig. 13b. By using engine dyno figures and gear ratios to plot driveshaft torque before and after upshifts at different rpm, it's possible to determine the shift point at which the loss or gain in torque is lowest

Auto Math Handbook

FORMULAS FOR SHIFT POINTS Rpm After Shift ■ (ratio shift into

ratio shift from) x rpm before shift

Driveshaft Torque = flywheel torque x transmission ratio

Chapter 14

QuarterMile E.T. and MPH

The best way to find a car's quarter-mile elapsed time and terminal speed is to run it at the drag strip. However, Patrick Hale's Quarter and Quarter jr. computer programs can predict quarter-mile performance with surprising accuracy. Even his simplified formulas for e.t and mph can provide useful comparative data for different combinations of power and weight Photo by Bob Johnson.

Computer Programming In 1985, Patrick Hale, a drag racer, engineer and computer programmer, combined his interests and developed computer software daat predicts how a car should perform in the quarter mile. Called Quarter the program predicts elapsed rimes and speeds accurately, because it takes into account practically every mathematical variable that can affect acceleration, from gearing and shift points to drag coefficient and polar moment of inertia. It also shows what modifications will help a car's performance and what will hurt it—before the owner spends cold, hard cash for actual parts. All you need is an IBM compatible PC. The current version is now called Quarter Pro. For some racers, the program was too good. It required data that those running on limited budgets simply didn't have and couldn't get. So, a year later, Hale introduced a simpler, lower-priced program called QuarterJr.; it focuses on readily available specs and provides its own built-in estimates of rhe more advanced details. Both editions arc available at quarterjr.com. At either programming level, the two variables that arc the most important are horsepower and weight. The higher the power and/or the lower the weight, the faster and quicker rhe car will go. In the course of developing his computer programs, Hale came up with formulas for computing quarter-mile elapsed time or e.t. and terminal speed in mph from just power and weight. Because they don't consider any other variables, these formulas can’t and don't predict performance as

precisely as Quarter or Quarter Jr. But they do provide useful—and comparative—estimatcs.

Elapsed Time The formula for elapsed time (e.t.) involves the cube root of the weight-to-power ratio multiplied by a constant of 5.825, while the formula for terminal mph calls for the cube root of the powcr-to weight ratio multiplied by a constant of 234. Note an important distinction here: For e.t., you want weight-to-power, i.c., lb per horsepower. For mph, you want f>ower-to-weight, he., horsepower per pound. Here's the e.t. ormula in proper mathematical form: c.t.= ^'(weight + hp) x 5.825 Suppose you have a Corvette that weighs 3440 lb, complete with fuel and the driver aboard, with a 245-hn engine. According to the formula, its quarter-mile e.t. should be 14.05 seconds. Road & Track once tested just such a combination and posted an elapsed time of 14.6 seconds. Without considering any other variables, Hale's formula has come within 4.0 percent of an actual test run. Power or Weightfrom E. T.—From the formula for e.t., formulas can be derived to find either power or weight, when the other is known. To find how much hp would be needed to propel a car of a given weight to a given e.t., divide the weight by the cube of the e.t. divided by 5.825: HP = weight -r (e.t. ■? 5.825P For the 3440-lb 'Vctte to post an e.t. of 14.05 seconds, it

Auto Math Handbook

Power or Weightfrom MPH—Again, formulas can be derived to find either power or weight. To find the hp, the cube of the mph divided by 234 should be multiplied by the weight: HP » (mph + 234)3 x weight To propel the 3440-lb 'Vette to 97 mph in the quarter-milc, you’d need—yes—245 hp. To find the weight, the cube of 234 divided by the mph should be multiplied by the hp: The advantage of using computer performance predictions is that tracks are not always open or available for testing. The computer program will allow you to eliminate most of the poor combinations. Photo by Bill Hancock.

This Mustang Is hooked up and has virtually all of the weight on the rear wheels as it leaves the starting line. Consistent traction conditions are important when testing. Photo by Bill Hancock.

would need—surprise!—245 lip. If the unknown were the weight, you could find it with the cube of the e.t. divided by 5.825, multiplied by the hp:

Weight = (c.t. + 5.825)3 x hp And how much should the 245-hp, 14.05-second Vette weigh? The answer is 3438 lb. (The calculator lost a couple of pounds in rounding errors.)

Miles Per Hour Patrick Hale's formula for mph at the end of a quarter-mile acceleration run is: mph = 3\ (hp + weight) x 234 For the Corvette, the speed would be 97 mph. Road & Track's test figure was 95.5 mph. So, this time, the formula's error is less than 1.6 percent!

Weight ■ (234/mphp x hp Here, die weight works out to 3439.5 lb, only 1/2-lb off.

Realistic Input = Realistic Output As I said earlier, these formulas aren't as precise as Hale's computer programs, and they won’t show the effects of any modifications in areas other than power and weight. But, considering their simplicity and how easy they arc to work, they come remarkably close to real-life figures. How close depends on rhe accuracy of the input, and that can be a problem. Drino Miller, a race car builder and driver once remarked that he’d tell anyone whatever they wanted to know about any of his cars except two things, their power and their weight, because, in his words, "Everybody lies about those. He had a point. Exaggerated claims about power and weight are among the most common tactics racers use in their constant efforts to psych out one another. It was in the interest of accuracy that I used Road & Track test figures to demonstrate Hale’s quartermilc formulas. Road & Track actually weighs its test vehicles, while many other magazines simply print curb weights provided by the auto makers, and those arc often highly optimistic. But, as I've tried to show, with realistic input, the formulas can provide surprisingly realistic output. Gearing for Quarter-Mile Speed One of the tacit assumptions of Patrick Hale's formulas is that the vehicle is properly geared and that, of course, may or may not be true. It's possible that the gearing might be too low—or too high—to enable the car to reach the quarter-mile speed indicated by die formula. However, once the potential speed has been calculated, there’s a formula from another source for determining the optimum overall gearing. It's from Larry Shepard, formerly of Mopar Performance

QUARlER-MlLE E.T. AND MPH

and, for a car with a manual gearbox, it is:

Overall Gear Ratio = (tire dia. -5- 340) x (rpm + mph) The Corvette discussed earlier has 275/40ZR17 tires, which would have a diameter of 25.66 inches. (How we found the diameter is explained in Chapter 18.) The engine could be revved easily to 5500 rpm, and Hale s formula predicted a speed at the end of the quarter mile of 97 mph. Plugging the necessary figures into Shepard's formula: Overall Gear Ratio = (25.66 + 340) x (5500 + 97) Overall Gear Ratio » 0.0754706 x 56.7 The recommended overall gear ratio would be 4.279183, or about 4.28. How does diac compare with the Corvette’s actual gearing? Well, it has a 3.07 final drive but, as it cleared the end of the quarter, it was still in rhe 3rd of its 4 gears, with a ratio of 1.34. Multiplying 3.07 by 1.34, the overall ratio is 4.11. That's just slightly more than 4.0 percent off the formula's recommendation! With an automatic transmission, the constant 335 replaces 340:

A consistent launch will provide good repeatable data for comparison. Try to launch the car the same each time when doing your testing. Photo by Bill Hancock.

Overall Gear Ratio = tire dia. 335 x rpm + mph Had the Vette been equipped with a Turbo Hydra-Matic, the recommended overall gear ratio to reach a quarter-mile terminal speed of 97 mph would've been 4.3431297 or, rounded off, 4.34.

Constant Source Where do formulas like these come from? In particular, where did Patrick Hale get his constants 5.825 and 234 and Larry Shepard his 340 and 335? The answer is that the constants were derived empirically. That's a fancy way of saying by trial and error—a lor of trial and error! In Larry Shepard's case, vehicles were run using only engines where accurate dyno data was available and a true weight was measured before each run. Additionally the runs were corrected for atmospheric and wind conditions. Additionally segmented times were monitored to ensure consistent traction.

Performance for cars like this altered which have a high power to weight ratio can be easily predicted using one of the many PC based programs. Photo by Bill Hancock.

Alto Math Handbook

FORMULAS FOR QUARTER-MILE E.T. AND MPH E.T.=

(weight * hp) x 5-825

IIP = weight -r (e.t. v 5.825)3

Weight = (e.t. + 5.825P x hp MPH = 3\ (hp + weight) x 234

HP = (mph -r 234)3 x weight Weight = (234 + mph)3 x hp Overall Gear Ratio (manual transmission) = (tire diameter + 340 mph) x (rpm v mph)

Overall Gear Ratio (automatic transmission) = (tire diameter -r 335 mph) x (rpm + mph)

Chapter 15

Computer

Programs

With a personal computer, you can design and run a virtual engine using software like Dynomation, that is readily available and reasonably priced. Photo by Bill Hancock.

Today there are many computer programs that nor only predict vehicle performance but also simulate engine operation while capable of running on a laptop computer with amazing results. Other programs aid in engineering calculations and chassis design. Prior to the personal computer, there were some sophisticated vehicle performance programs which took massive and expensive computers such as the CRAY in order to run. Then in rhe mid-’70s, programmers like Curtis Leaverton with Dynomation, and Alan Lockheed with Engine Expert (latest Windows version available through audictcch.com), hit the market with sophisticated but easily run programs that did an amazing job of predicting performance engine output. There are many inexpensive programs diat will also run on a laptop and solve complex problems with relative case. There are a few like the two previously mentioned that arc highly detailed and will produce incredibly accurate results. The ProRacing Sim division of Comp Cams ([email protected]) offers many different programs ranging from the introductory engine performance software to sophisticated software which is extremely accurate and able to manage a wide array of inputs. To run most of the more sophisticated programs, you will need to supply input data such as compression ratio, bore and stroke, cylinder head flow figures, component weights, and engine specific information such as valve type lobe profile and camshaft centerline. However, the entry-level programs arc wonderful in that they do not require a lot of user input. The user is able to

choose from a menu of camshafts and exhaust systems as well as compression ratios. The prices start at under $100 and go up from there. In each case, the cost of the gaskets alone to build one engine is more than the price of the program to sec if the engine combination you have designed will work. It takes literally minutes to sec how a particular combination works. Once you find a combination that suits your needs, you can cont inue to refine and rerun the system with very minor changes until you maximize the output. Find one of these programs that fits your needs ana learn how to use it. The results will amaze you, and in the end save an incredible amount of time and money If you arc just starting out and don't know where to go, many of the manufacturers offer help on their web sites and some still offer a telephone help line. All of die OEM's and now many of the professional engine builders use this type of program to give them a quick way to design and qualify engine concepts before committing any time or money to the project.

Mitchell Software Wm. C. Mitchell (MitchcllSoftware.com) offers a whole range of programs to design and develop vehicle suspension and handling. Teams ranging from entry-level autocross racers up to and including Formula 1 and NASCAR Sprint Cup teams have used his offerings with great success. Until you have actually tried to design a suspension system, you will not appreciate the work that is saved, much less the speed and accuracy of the final results. Gone are the days of

Auto Math Handbook

Laptop computers have become very powerful and extremely versatile. With the influx of new software, the users can explore solutions never before attainable. You do not necessarily have to know the actual engineering behind the software; userdriven menus will help you find the solution. Photo by Bill Hancock.

trying to design steering geometries using paper and string. This software puts an incredible amount of design and development expertise in your hands. Until recently, the only way to have this amount of expertise was to hire a chassis designer. There are user groups available online where you can rradc ideas and concepts if you so choose. While the formulas for much of the afore­ mentioned are available, they involve a fair degree of higher math, but more importantly, unless you use tne formulas everyday, the chances of making a mistake either in application or execution arc great. Look around at the pros; they all use computer simulation in one form or another. The exercise today is in finding a suitable version for your needs.

Today cell phones and PDAs have numerous inexpensive applications or "apps" that can be downloaded and provide instant and convenient answers to many common problems the automotive enthusiast encounters. Photo by Bill Hancock.

Handheld Computers Computech Systems, Inc. has a whole system with specific applications for racing and tuning that fits on a personal digital assistant (PDA) and utilizes tne Palm Operating System. You can walk around with an incredible amount of computing power held literally in your hand. Many of the applications discussed in this book are available already preloaded into the system. Many racers and tuners use the popular weather correction system to adjust their jetting or fuel map at the track as the weather changes. The system docs require inputs such as barometric pressure, and wet and dry bulb temperatures. In order to provide accurate readings, you have to be careful where you take the data. Try to take all of the data such as ambient temperature and barometric pressure at die track and within 30 minutes of your test. Do not rely on television weather reports for accurate barometric readings, since the data may be from a different area and up to several hours old. All of the good programs include a detailed user manual which covers the application of the programs. Take some time and learn how to use the program before you go to the track. Practice using the program at home and in the garage so when you get to the track you are familiar with the operation and don't become confused and make a costly mistake. If you test on a dyno, use your tuning programs and since you have an instantaneous result you will be able to confirm your predictions. This exercise will provide the practice you need to feel comfortable with the program.

Port Area Many engine simulation programs require data on intake and exhaust port areas. The area they arc looking for is the minimum cross sectional area in the port itself. This area may also be referred to as the choke area of the port. The smallest crosssectional area of the port typically occurs somewhere near the valve guide boss inside die port. The valve guide and its support bosses require a fair bit of area, and hence reduce the overall area of the port in that location. How do you measure this, much less even find it? Begin by removing a cylinder head from the engine, or if you arc just in the design phase of your project, take a flow model and remove the valves. Clean up a pair of ports and valves using ScotchBrite or some comparable product to remove the carbon from the interior of the port. Next,

Computer Programs

One of the inputs that you will need for most serious engine performance programs is the choke diameter of the intake port. Here is a rubber mold being cut at the smallest area and perpendicular to the centerline of the port Photo by Bill Hancock.

install die valve and either spray or rub some release agent over everything in the port and backside of the valve head and stem surface. You will need to acquire some rubber molding product used in metrology. A product called Blu-Sil CMC (cold molding compound), made by the Perma-Flex Mold Company, Inc. has served well in this application for years. These products arc often referred to as metrology casting materials. They arc usually designated as a two part silicone based molding compound. Carefully weigh the components using a postal scale and using a tongue depressor stick, thoroughly mix the material together in an old plastic butter container according ro the directions. Prior to mixing the material, have the head with the valve installed and the entire interior surface coated with release agent, supported on a bench with the appropriate port flange facing up. Pour the material into the port slowly to eliminate air bubbles. Do not fill the port fully. Leave at least four inches to be filled. 1 lave a relatively large diameter dowel about eight inches long, coated with release agent ready. Hang the dowel so that it is suspended deep into the center of the port, then continue to fill the port all the way up. Leave the filled port to set for at least 30 hours at room temperature. Once the mold has hardened, remove the dowel and then carefully remove the mold. The reason for the dowel becomes apparent. Once it is removed, the resultant hole allows the mold to collapse and greatly eases removal from the port. It will still require some careful work to

Here is the area we are interested in. Once you have this area, simply take the rubber mold and trace around the perimeter of the section, then use a planimeter or the squares method to get the area. Photo by Bill Hancock.

remove the mold without breaking it, but if you were careful to coat all of the surfaces with release agent in the beginning it should be able to be done. Once tlie port mold is removed and cleaned, look at it from the side and visualize a line running down the center of the port. Get a long razor knife or an electric kitchen knife and carefully cut sections of the port which arc perpendicular to the centerline of the port. Next, use a piece of paper and a pencil to trace around the sections of port and then measure them with a planimeter until you find the one with the smallest area. The smallest one determines the choke area or smallest area of the port. For some programs, you will need to know the volume of the port/s. This can be done much like the previous example, except once the port is cleaned, apply a small amount of light grease on the valve scat and assemble the valve with a retainer and light spring to keep it firmly closed. Support the head again so the flange is facing up and somewhat level. Using a burette and a plate to seal the top of the port, fill the port through a hole in the plate with fluid until the port is full. The plate is not required, it just makes the process easier and more accurate, since the flange docs not have to be perfectly level. You will have to repeat this for the exhaust port as well.

Other Programs Available Computer programs arc available for sophisticated camshaft analysis as well as vehicle performance predictions. The important thing to remember when using any computer program is to

Auto Math Handbook

SEMINARS Today there arc many opportunities to learn from respected experts in the field of racing and performance through seminars and user groups. Annual seminars like the AETC—Advanced Engine Technology Conference, offer a three-day format where 300-500 participants typically have ten to twelve formal technical presentations conducted^by leaders in the field as well as the opportunity to mingle and converse with the speakers during meals and in between presentations. These seminars are open to the general public and do not require any al filiation or requirement beyond the attendance fee. The AETC offers a rare opportunity to ask questions of the experts who at any other point in time arc typically too engaged in their own projects to spend time on general questions.

make sure you understand the inputs so yo u are able to enter or load the correct data. If you make an error here, the answer will be wrong and you may never know ir. The advantage of the programs is their wealth of complex computing power, using some fairly long and involved equations. Some very intelligent programmers spent a great deal or time refining the programs so they would produce worthwhile results very quickly. You are able to profit from their efforts without having to go through all of the math or engineering involved in finding and applying the proper formula to fit the application. There are new applications being added every day. Begin by going online and looking at a particular area of interest. For example, there arc programs available which let you predict your initial carburetor jetting very precisely. Once you have the engine running, you can refine your jetting selections either on the dyno or at the track. There are programs which let you keep track of the

optimum fuel air ratio as well as the resulting vehicle performance by tracking the weather correction factor. With the rising popularity of smart phones, numerous free or relatively inexpensive applications or programs have become available which let the user merely download the application to their phone. These applications are growing by leaps and bounds. The popularity and wide availability of onboard data acquisition coupled with the rise in performance analysis programs has forced all of the serious competitors to invest in the technolog}’ if they expect to become or remain competitive. With the sheer number of engine and chassis component choices available in the market, there is no way to carefully evaluate them all. Computerized analysis lers us sort through a number of camshafts for example without ever having to leave our computer. Seminars and on-line user groups serve to educate the interested participants about rhe finer points of the various programs and lead to an even greater knowledge base. The users who take advantage of this vast knowledge bank can ultimately become quite successful. The Internet has broadened the knowledge base as well. Many of racing's technical secrets are now openly discussed and debated at length. With this transparency comes the caveat that just because somebody has posted something as factual docs not mean that it is. With digital photography and image manipulation, you can see almost any phenomenon on YouTube today. The potential user should ensure that the proposed solutions or the results are valid before trying it themselves. The laws of physics and thermodynamics still govern many segments of life. Do not try to circumvent them!

Chapter 16

Instrument Error and Calibration

The tachometer can be tested for accuracy on an electronic diagnostic machine at the neighborhood garage, while the speedometer and odometer can be checked against measured miles on the highway or against a GPS unit Photo by Bill Hancock.

When a car's performance is measured at a drag strip, oval track or road course, it's normally done with the racing facility's timing equipment, using instruments that arc finely calibrated for accuracy. After all, races are often won or lost by hundredths and sometimes even thousandths of a second. Bur when a car's performance is measured on the street or highway, the driver usually has to rely on the vehicle's own instruments—specifically its tachometer, speedometer and odometer—ana they're not likely to be all that finely calibrated! Testing Instruments The tachometer is the easiest to check for accuracy. At your neighborhood garage, there's likely to be an electronic diagnostic machine that can be hooked up to show engine rpm, and you can simply ask the technician to compare your tachometer's reading to that of the machine. 1 he speedometer and odometer arc more problematic and, to check them, you may have to take the car to a shop specializing in automotive instruments. Or, if you belong to a car club, they may have a set of in-car instruments to gather vehicle performance data that die members can borrow. Today, by far the most readily available and accurate speed check is a hand-held GPS (Global Positioning System). By utilizing one of these units while running your tests you can be assured of extremely accurate readings. Not only will they give speed but also some of them are equipped to read elevation changes.

Speedometer It's also possible to check rhe speedometer with a stopwatch. For the test, you'll need a stopwatch, a level stretch of highway with a marked measured mile, and cither a cruise control or a well-disciplined right foot. In some states, there arc posted speedometer checks along major highways, with 5 or 10 marked miles. But even where there aren't such posted checks, there arc often less conspicuous mile markers along the roadside. With the stopwatch, time the vehicle over the marked mile at a steady indicated speed. To find your actual speed in mph, divide the number of seconds it takes you to drive the mile into 3600, the number of seconds in an hour: Actual mph = 3600 -e- seconds per mile If the indicated speed is more than the actual speed, the speedometer reads fast. If the indicated speed is less than the actual speed, the speedometer reads slow. Example—If you drive a measured mile at an indicated 50 mph and it takes 72 seconds: Actual mph » 3600 -e- 72 The actual speed would be the same as the indicated, 50 mph. At that speed, at least, the speedometer would be correct. In other words, there is no error. Now, let's suppose the measured mile at an indicated 50 mph takes 75 seconds: Actual mph = 3600 + 75

Auto Math Handbook

In the example where you went an actual 51.06 mph at an indicated 50 mph, the difference was 1.06:

Percent Error = (1.06 + 51.06) x 100 At an indicated 50 mph, the speedometer was 2.08 percent slow.

On highways where there are no posted speedometer checks, there are often less conspicuous mile markers along the roadside, such as surveyors1 marks, but It may take some cruising to find them. Try to use newer highways, since the mile markers are still fresh and have not been replaced after something mowed them down and their original location was lost Photo by Bill Hancock.

The actual speed would be 48 mph. That 's 2 mph less than the indicated speed, so the speedometer reads Fast. Okay, suppose the time was only 70.5 seconds:

Actual mph = 3600 * 70.5 According to the calculator, that would be 51.06383 mph, which should be rounded down to 51.06. That’s 1.06 mph more than the indicated 50, so the speedometer reads slow. Speedometer Error Percentage—If the speedometer error is substantial, you should have tne speedometer gear replaced with one that will provide more accurate readings. To choose the new gear, the speedometer shop will need to know the percentage of error with the present gear. To determine the percentage, start with the difference between tnc indicated and actual mph by subtracting the smaller figure from the larger. Then divide that figure by the actual mph, and multiply the resulting decimal by 100 to convert it to percent: Percent Error = (actual speed - indicated speed) + (actual speed x 100) When you went an actual 48 mph at an indicated 50 mph, the difference was 2:

Percent Error = (2 + 48) x 100 At an indicated 50 mph, the speedometer was 4.17 percent fast.

Odometer Error To find the odometer error, it’s desirable to use more than 1 measured mile—5 miles at least, 10 if possible. The greater the distance, the more accurate the check will be. This time, though, you don't have to drive the vehicle at a steady speed or to time it with the stopwatch. Note the odometer reading at the beginning of the measured distance and again at the end. To find the indicated distance, subtract the first figure from the second, or: Indicated Distance - Reading at finish - Reading at start On an analog (as opposed to digital) odometer, pay particular attention to the reel at die right end, indicating tenths of a mile. If the number isn’t centered right in line with the odometer’s other numbers, interpolate the reading out to hundredths. For example, if at the start or finish of a run, the reel’s display is midway between 2 and 3, it’s showing 0.25 mile. If it’s 2/3 of the way between those two numbers, it s showing approximately 0.27 mile. If the indicated distance shown on the odometer is more than the measured distance, the odometer is fast. If the indicated distance is less, the odometer is slow. Odometer Error Percentage—To find the percentage of odometer error, use the same formula you did tor speedometer error, but with actual and indicated distances rather than speeds, or:

Percent Error = (difference between actual and indicated distances) + (actual distance x 100) Suppose that over a measured five-milc stretch, the odometer actually records 5.15 miles. To find the percentage of error, divide the 0.15 difference between the actual and indicated figures by the actual figure and, of course, multiply by 100: Percent Error ■ (0.15 + 5) x 100 The odometer is 3.0 percent fast.

Instrument Error and Calibration

Inconsistencies In the examples of how to find speedometer error, die figures used were for an indicated speed of 50 mph. I'he percentage of error won't necessarily be the same at other speeds and a professional speedometer check will cover a broad range of speeds. Similarly, the percentage of odometer error won't necessarily correspond with the speedometer error. If one reads fast, the other probaoly reads fast, too, but not always to the same degree. A real world example of such discrepancies is shown in a chart in Fig. 16a. When the final drive gears in my own car were replaced by gears with a numerically higher ratio, the speedometer and odometer were thrown way off. The figures in the first two columns of the chart are the actual and indicated readings recorded at an auto club facility. The figures in die other column show the differences between the actual and indicated readings and the percentages of error ar the various speeds checked. At the bottom of the chart are the figures for the odometer. Note that its percentage of error is considerably less than any of the percentages for the speedometer. Finally, to answer an obvious question: yes, the speedometer drive gear has since been replaced with one that's brought die readings back down closer to reality! If you have a cable driven speedometer or odometer and there are no gears available for the correction you need, you may want to consider a compact transmission which fits in line with die transmission and speedo cable. This transmission allows you to change die ratio by using the supplied gears.

Speedometer Indicated Speed 25 30 35 40 45 50 55 60 65 70 75 80

Actual Speed 20 25 28 33 38 42 46 50 54 58 62 66

Error DifFcrcncc/Pcrcent 5/25 5/20 7/25 7/21.2 7/18.4 8/19 9/19.6 10/20 11/20.3 12/20.6 13/20.9 14/21.2

Odometer Indicated Distance 100

Actual Distance 88

Error Difference/Percent 12/13.6

Fig. 16a

Here is a speedometer transmission. This fits inline in the speedometer cable on older cars. It can be helpful when there is not an appropriate speedometer gear available to make the correction. These transmissions are available at speedometer shops or online. Photo by Bill Hancock.

Al io Math Handbook

FORMULAS FOR INSTRUMENT ERROR Actual MPH = 3600 + seconds per mile Speedometer Error Percent = (difference between actual and indicated speeds) + (actual speed x 100) Indicated Distance = (odometer reading at finish) + odometer reading at start

Odometer Error Percent = (difference between actual and indicated distances) + (actual distance x 100)

Chapter 17

MPH, RPM,

Gears and Tires

The formulas for engine speed In rpm, vehicle speed in mph, overall gear ratio and tire diameter can be useful for analyzing the behavior on the road, track or strip of high-performance cars like this Barracuda T/A. Photo by Bill Hancock.

There are four significant, interrelated specifications— speed in miles per hour or mph, engine revolutions per minute or rpm, overall gear ratio and tire diameter in inches. Given any three of these, it's possible to determine what the fourth is—or should be. Determining the values for these four areas can be useful for analyzing the behavior of high-performance cars on the road, track or strip. Miles per Hour The raw formula for finding vehicle mph involves multiplying engine rpm by 60 (the number of minutes in an hour) by pi (the constant 3.1415927) times the tire diameter (which provides the circumference of the tire in inches) and then dividing by die overall gear ratio times 63,360 (the number of inches in a mile), or:

mph = (rpm x 60) x (pi x tire diameter) -e- (gear ratio x 63,360) The constants in the numerator, 60 and pi or 3.1415927, can be multiplied together to become 188.49556. That figure, in turn, can be divided into both die numerator and denominator, eliminating it from the numerator and reducing the constant in the denominator to 336.13524. That can be rounded down to 336, and the formula becomes a much more manageable: mph « (rpm x tire diameter) 4 (gear ratio x 336)

Example—Io demonstrate that formula, let's suppose you have a Ford Mustang with a 5.0-liter V-8 and a five-speed transmission with ratios of 3.35 in 1st, 1.93 in 2nd, 1.29 in 3rd, direct 1.00 in 4th and overdrive 0.68 in 5th. The final­ drive ratio is 3.08 and the tires have a diameter of 26 inches. In the quarter-mile, you run through 1st, 2nd and 3rd, shifting at 5500 and you want to know what speed you reach in each gear. For 1st gear, multiply 5500 by 26 and then divide by the 1st gear ratio of 3.35 times the final drive ratio of 3.08 times the constant 336:

mph - 1(5500 x 26) * (3.35 x 3.08 x 336)] mph = 143,000 + 3466.848 By completing the division, you'll find that at 5500 rpm in 1 st gear, the Mustang will be going 41.247842 mph or, rounded up, 41.25 mph. Similarly, the formula will show that at 5500 rpm in 2nd, the little Ford will be doing 71.60 mph, and in 3rd, 107.12 mph. Revolutions per Minute (rpm) Now let's suppose you want to know the rpm at a given mph. For that, the formula would be:

rpm = [(mph x gear ratio) x 336] + tire diameter What would the Mustang's rpm be out on a rural highway, running in 5th gear at the legal max of 65 mph? rpm = [(65 x 0.68) x (3.08 x 336)1 + 26 rpm = 45741.696 * 26

Auto Math Handbook

How do changes in gearing or tire size affect the engine or vehicle speed of a vehicle like this street rod? You can find out by applying the formulas in the text. Photo by Bill Hancock.

The answer is 1759 rpm. The engine is loafing when cruising in overdrive, and that means good fuel economy.

Overall Gear Ratio But suppose you'd rather gear rhe car for maximum response at highway speed. At 65 mph, you want to he able to downshift from 5th to 4th and be at the Mustang V-8's torque peak, which happens to be 3000 rpm, in order to have optimum passing ability. The overall gear ratio, which in direct drive 4th gear would simply be the final drive ratio, is now the unknown and the formula is: Gear Ratio = (ipm x tire diameter) * (mph x 336) The figures for the Mustang would be: Gear Ratio = (3000 x 26) + (65 x 336) Gear Ratio - 78,000 + 21,840 You should have a 3.57 final drive ratio in order to reach 3000 rpm at 65 mph in 4th gear. Final drive gears don t come in an infinite variety of ratios, though, and the closest you're likely to come to the ideal figure would be 3.54. To sec what the rpm would be with that ratio, let's go back to the formula for rpm:

rpm » (65 x 3.54) x (336 + 26) rpm = 77,313.6 -e- 26 That gives you 2973.6 rpm, which is certainly close enough for all practical purposes.

Tire Diameter Finally, there's the formula for tire diameter: Tire Diameter ■ [(mph x gear ratio) x 336] + rpm Going back to modifying the Mustang to turn 3000 rpm at 65 mph in 4tn gear, you could achieve a similar effect by changing the tire size instead of the final drive ratio: Tire Diameter a [(65 x 3.08) x 336] + 3000 Tire Diameter = 67.267.2 + 3000

The Lowdown—According to the formula, you’d need tires with a diameter of 22.4 inches to do the job. That would convert the Mustang into something of a low rider, which might have ground clearance problems on steep driveways. But, as a quick and dirty way of achieving the wanted relationship between mph and rpm, it would work. There is another problem, though. With the change in tire size, die speedometer will no longer read accurately. You can find what the error is with the techniques described in Chapter 16, or you can figure out what it should read with a formula presented in Chapter 18. Meanwhile, you'll find the formulas that have been discussed here—especially those for mph, rpm and gear ratio—among the most useful you can know, and I bet you'll be turning to them often. When you get used to working with them regularly, you'll probably wonder how you ever got along without them.

MPH, RPM, Gfars and Tires

FORMULAS FOR MPH, RPM, GEARS AND TIRES MPH = (rpm x tire diameter) -r (gear ratio x 336) RPM = [(mph x gear ratio) x 336] + tire diameter

Gear Ratio = (rpm x tire diameter) * (mph x 336) Tire Diameter = [(mph x gear ratio) x 336] * rpm

Chapter 18

Tire Sizes and

Their Effects

This lifted SUV has a raised eg and tons of ground clearance. By using much taller tires, the speedometer will have to be carefully calibrated to ensure accuracy. Photo by Bill Hancock.

Suppose you're planning to replace the tires on your car or truck with larger diameter ones. Say, for example, you have a high-performance car and need bigger rubber at the rear for better traction at the drag strip. Or you have a four-wheeldrive vehicle and want to increase its ground clearance for oil­ road driving. Whatever your reason for wanting bigger tires, there arc a couple of important questions to consider. First, what effect will the new tires have on the overall gearing? The vehicle may not respond the way you're used to, and you may find it necessary to downshift more frequently. You may even have to change the final-drive ratio in order to retain—or regain—the original level of performance. Second, the speedometer will read too slow. On the highway, you may be deceived into thinking you're cruising at the legal maximum, when actually you're well over it and ripe for a speeding ticket. Tire Diameter Fortunately, if you know the diameters of both the new and old tires and the vehicle's existing final-drive ratio, you can calculate the effects the bigger tires will have ahead of time. Your local tire dealer or the tire manufacturer's web site should have charts showing the diameters of the various tires they carry. Or you can simply apply a tape measure to one of the tires currently on your car or truck, and to one of those you're considering as replacements. Make sure that the tires arc properly inflated when you make these measurements. Section Height and Width—You may also be able to figure out the diameters of the tires from their respective sizes. In the old days when 6.00 x 16 was the standard size on many popular cars, it was easy. The tire's section height—the

distance between the edge of die rim and the face of die tread—and the section width—the distance between the sidewalls on cither side—were about the same. In the case of a 6.00 x 16, the section height and width were both alxjut 6.0' and the tire was mounted on a 16 wheel. To find rhe diameter, you simply multiplied the section height, 6.0", by 2, giving you 12", and added that to the wheel rim diameter, 16", for an overall figure of 28". For modern heavy-duty truck tires, the sizing system is even more straightforward. For example, a 31x11.50x15 tire will have a nominal diameter of 31" and width of 11.5 , and will fit a 15' wheel. Frankly, though, we are hedging widi that word "nominal,'’ because tire industry standards allow up to a 7.0 percent variation from specified dimensions. Aspect Ratio—Modern passenger car tires and light-duty truck tires aren’t sized quite so simply. In most cases, their section height and width arc no longer alike. The height is usually much less than the width, and the relationship between the two—the aspect ratio—is an important part of their specs. The aspect ratio is the percentage the section height is of the section width. Generally speaking, passenger car and light truck tires arc now produced in metric sizes that indicate the section width in millimeters, the aspect ratio in percent and the wheel rim diameter in inches. Example—As a case in point, let’s take an LT235/75R15 tire. The LT means it's a light truck tire; if it were a passenger car unit, it would have the initial P instead. Similarly, the R means it's a radial, while a B would indicate bias-belted construction. The 235 is the section width in millimeters and the 75 is the aspect ratio, indicating that the section height is

Tire Sizes and Their Effects

75 percent of the section width. Finally, the 15 is the rim diameter in inches. Metric Tire Diameter To find the diameter in inches of a metric size tire, you must first find the section height in inches. To do that, you convert the section width, 235 millimeters in our example, to inches by dividing it by 25.4, the number of millimeters in an inch. Then you convert the aspect ratio, 75, to a decimal figure by dividing it by 100. Multiply the quotients of these two calculations together to find the section height in inches. Double that figure and add the wheel rim diameter, which is already given in inches, and rhe result will lie the diameter of rhe tire in inches. Expressed as a formula, that would be: Tire Diameter ■ [2 x (section width + 25.4) x (aspect ratio + 100) + rim dia.] That can be simplified somewhat to: Tire Diameter = 2 x [(section width x aspect ratio) + 2540] + rim dia. Plug in the appropriate specs for a LT235/75R15 tire: Tire Diameter = 2 x 1(235 x 75) + 2540] + 15 Tire Diameter ■ (2 x 6.9) + 15 That would work out to a section height of 6.9 and an overall diameter of 28.88”, which, of course, could be rounded up to 28.9”. Checking the specs for three different BFGoodrich radial light truck tires available in the LT235/75R15 size—the Sport Truck T/A, All Terrain T/A and Mud-’lerrain T/A—you would find dieir diameters are listed as 28.9, 28.98 and 29.09", respectively, all of them within less than 1.0 percent of our calculated figure for that size. I have assumed the wheel rim meets a tire industry standard requiring a rim width that is 70 percent of the tire section width. For every 0.5' increase or decrease in rim width, there would be a corresponding 0.2 increase or decrease in the section width of the mounted tire.

Effective Drive Ratio 16 find what the effective overall drive ratio would be with a given increase in tire diameter, the formula is: Effective Ratio = (old tire diameter + new tire diameter) x original ratio

The Chevy Suburban has larger diameter tires that provide obvious benefits in ground clearance for rough, off-highway terrain. But the taller tires also require different axle gearing and recallbration of the speedometer and odometer. Photo by Bill Hancock.

Example—Suppose you have a set of 28.9” LT235/75R15s on a four-wheel-drive truck with a 3.08 final-drive ratio and, to increase the ground clearance, you want to replace them with 33' 33x12.50x15s. To find the effective drive ratio with the bigger tires, the figures would be: Effective Ratio ■ 28.9 + 33 x 3.08 Effective Ratio = 0.8757576 x 3.08 With die bigger tires, the effective ratio is only 2.70! That’s enough of a change to cause a noticeable loss in responsiveness. In order to keep the same overall ratio, you would need to change the rear axle ratio.

Equivialent Drive Ratio To find the final-drive ratio needed with the new tires to provide die equivalent of the vehicle's performance with the original tires, the formula is: Equivalent Ratio = (new tire diameter + old tire diameter) x original ratio Note that the positions of the tire diameters in this formula are reversed from their positions in rhe formula for effective ratio. In the case of die switch from 28.9 to 33" tires on a vehicle with a 3.08 final drive, the figures would be:

Equivalent Ratio = (33 + 28.9) x 3.08 Equivalent Ratio » 1.1418685 x 3.08 That works out to 3.51, so a set of final-drive gears in the 3.50-plus range would be needed to restore the lost responsiveness.

Auro Math Handbook

Tires come in a variety of sizes and more importantly tread patterns and diameters. Even changing tire diameter to a smaller size can have a dramatic effect on final drive ratio and your speedometer accuracy. Photo by Bill Hancock.

meantime, though, it would be helpful to know what the indicated speed would be at an actual 65 mph, using die formula: Indicated mph » (old tire diameter + new tire diameter) x actual mph Or, using the figures in the ongoing example: Indicated mph » (29.8 + 33) x 65 Indicated mph » 0.8757576 x 65 At an actual 65 mph, the indicated speed would be 56.9 mph. If you keep die speedometer reading under that figure, you won't get stopped for breaking the 65 mph limit.

Downsize Tires Although most auto enthusiasts would be more likely to want an increase in tire size than a decrease, the formulas for effective and equivalent drive ratios and for actual and indicated mph would be equally valid for a change to smaller tires. As an example, consider that improbable Mustang low rider described in Chapter 17. To recall its pertinent specs, its old tires were 26" in diameter and its new ones 22.4", while its final drive ratio was 3.08. 'Io find its effective drive ratio with the smaller tires, die equation would be:

This handy speedometer transmission shown in Chapter 16 attaches In­ line with the speedometer cable and allows you to compensate for different gear ratios and tire sizes. Photo by Bill Hancock.

Effective Ratio = (26 + 22.4) x 3.08 Effective Rado ■ 1.1607143 x 3.08 And the effective ratio would be 3.575. To find the equivalent ratio:

Speedometer Correction Assuming the speedometer was accurate with the original tires, the formula for determining what the actual speed is at any indicated speed with die bigger tires is:

Equivalent Ratio ■ (26 + 22.4) x 3.08 Equivalent Ratio = 0.8615385 x 3.08 The theoretical ratio would be 2.6535385 or, rounded down 2.65, and the closest gearset to that in the Mustang parts catalog is a 2.73. With an uncorrected speedometer, what would the actual speed be at an indicated 65 mph?

Actual mph = (new tire diameter + old tire diameter) x indicated mph Following the swap from 28.9" to 33" tires, what would the actual speed be at an indicated 65 mph?

Actual mph= (26 + 22.4) x 26 Actual mph = 0.8615385 x 65 The answer is an even 56 mph. Finally, what would the indicated speed be at an actual 65 mph?

Actual mph = (33 + 28.9) x 65 Actual mph ■ 1.1418685 x 65 The actual speed, rounded down, would be 74.2 mph. In many parts of the country, that's more than enough to attract the attention of the state highway patrol. Of course, you can get a new speedometer drive gear, as suggested in Chapter 18. In the

Indicated mph « (26 * 22.4) x 65 Indicated mph = 1.1067143 x 65 And the answer this time is 75.446429 or, rounded up, 75-45 mph. Note that earlier, with taller tires, the speedometer was slow. Now, with smaller or shorter ones, the instrument is fast, meaning that the speedometer indicates a faster

Tire Sizes and Their Effects

speed than the vehicle is actually traveling. WJry the Diameter?—Mathematically, the relationship Between final-drive ratio or speedometer reading with different tires is in direct proportion to the differences in the sizes of the tires. The formulas would work equally well with the respective rolling radii or circumferences. You could also use the

respective revolutions per mile of the tires; in that case, though, the results would be in inverse rather than direct proportion. But, generally, the tires’ diameters are among the easiest specs to find, even with the mumbo jumbo needed to calculate the diameter with a metric size, and that’s why they've been used here.

FORMULAS FOR TIRE SIZES AND THEIR EFFECTS Tire Diameter « 2 x [(section width x aspect ratio) Effective Ratio = (old tire diameter

f

f

2540] + rim diameter

new tire diameter) x original ratio

Equivalent Ratio = (new tire diameter f old tire diameter) x original ratio

Actual mph = (new tire diameter + old tire diameter) x indicated mph Indicated mph = (old tire diameter

f

new tire diameter) x actual mph

Chapter 19

Average MPH and MPG

Most drivers are already familiar with the formulas for miles per gallon and miles per hour. Knowing how to manipulate them properly can add to both the efficiency and enjoyment of highway travel. Photo by Bill Hancock.

The formulas for average fuel mileage and average highway speed arc familiar to most drivers. In fact, the expressions used to describe these two quantities—miles per gallon and miles per hour—are statements of their respective equations. Miles per gallon means the distance driven in miles divided by the amount of fuel used in gallons, or: Miles per Gallon = miles traveled + gallons consumed Miles per hour means the distance driven in miles divided by the time of rhe trip in hours, or:

Miles per Hour - miles traveled 4 hours Now lets take a closer look at average fuel mileage. Miles per Gallon (mpg) Suppose you drive from Boston to Washington, D.C. Your first stop is in New York City, which is 208 miles from Boston, and your car uses 10.4 gallons of Riel. What would the average Riel mileage be?

Miles per Gallon = 208 + 10.4 You got an even 20 miles per gallon. Fuel Range—Suppose your car has an 18-gallon tank. If you hadn't refilled in the Big Apple, how far could you have gone before you had to stop for Riel? In other words, what would your fuel range be in miles?

Miles = miles per gallon x number of gallons Which, in this case, would be: Miles = 20 x 18 Or 360 miles. You’ve already driven 208 miles, so you could’ve gone up to 152 miles farther before running out of fuel. You would've probably had to start looking for a filling station that would accept one of your credit cards when you passed through Philadelphia, which is 106 miles from New York. You wouldn’t have made it as far as Baltimore, which is another 96 miles, or 202 miles Rom New York, 'khat’s 50 miles beyond your range. Predicting Fuel Consumption—Washington is 239 miles from New York. At 20 miles per gallon, how much fuel will the car use for the trip? To find out, divide die miles by die miles per gallon: Gallons = miles + miles per gallon Or, in this case: Gallons = 239 + 20 Which would be 11.95 gallons. You’ve already used 10.4 gallons getting from Boston, so your overall Riel consumption would be 22.35 gallons for the total distance of 447 miles from Boston to Washington.

Miles per Hour (mph) On the stretch from Boston to New York, you took 4-1/4 hours to cover rhe 208 miles. To find your average speed, you convert that 4-1/4 to 4.25 and plug it into the formula for average miles per hour:

Average MPH and MPG

Miles per Hour = 208 + 4.25 On an 8-digit calculator, that would be 48.941176, which can be rounded up to 49 mph. What if your time was 4 hours and 35 minutes, or 4:35? Can you run that as:

And Y works out ro lie 21.6 minutes. That 0.6 of a minute is, of course, 6/10 of a minute, not 6.0 seconds. But by now, you know how to convert that 6/10 of a minute to seconds. Don’t you? In case you forgot, here is what you need to do:

Miles per Hour = 208 + 4.35 No you can’t, lliat 4:35 is 4 and 35/60, not 4 and 35/100. However, with a calculator, you can figure it this way:

0.6 x 60 seconds = 36 seconds

Miles per Hour ■ 208 + [4 + (35 + 60)] That would be the equivalent of: Miles per Hour ■ 208 + 4.5833333 Which would work out to 45.381818 or, rounded down, 45.38 mph. Distance and Time—Suppose you want to know how far you've gone if you drive at a given speed for a given number of hours. Multiply die miles per hour by the hours: Miles = miles per hour x hours If you drive at an average of 45 mph for 2-1/2 hours, how far would you go?

Miles = 45 x 2.5 You would cover 112.5 miles. If you drive at a given speed for a given number of miles, how many hours will it take? To find out, divide the miles by the miles per hour:

Hours = miles * miles per hour As an example, suppose you continue the 106 miles from New York to Philadelphia at an average of 45 miles per hour: Miles = 106 + 45 That works out to 2.3555556, which can be rounded up to 2.36 hours. Hundredths to Minutes—That 2.36 hours is 2 and 36/100 hours, not 2 hours and 36 minutes. To convert hundredths of an hour to minutes, or sixtieths of an hour, i.e., to convert from a centestimal to a sexagesimal fraction, you can set up a simple algebraic equation, using the letter Y as die unknown, to find how many sixtieths the hundrcddis would equal: (36 + 100) = (Y + 60) That would be:

(100 x Y) = (36 x 60) or (100 x Y) = 2160 or: Y = 2160 + 100

Moving Time vs. fravel Time Many people fail to take stops into account when computing travel time. They assume that if they average 60 mph for a trip oi 480 miles that it should take exactly 8 hours. They fail to factor in the time for rest stops, meal stops and fuel stops. In the end, the 480-mile trip may take nine hours and fifteen minutes resulting in an average speed of:

480 miles + 9.25 hours = 51.9 mph Most people look at two numbers: Moving time and travel time. Ba- looking at both numbers you can get a good idea of where you are spending your time. By looking at the average moving speed you might see that you arc averaging 61.2 mph while you are moving, but the travel time average is 49.8 mph. What this means in our trip of480 miles is that: Your moving time was 480 miles + 61.2 mph = 7.84 hours Our travel time is 480 miles + 49.8 = 9.64 hours Subtracting moving time from travel time: 9.64 - 7.84 - 1.8 hours not moving. Obviously you must stop occasionally, so in order to help your moving average try to make one stop for multiple reasons. Stopping for fuel and using die rest room, then picking up a sandwich for lunch could take three separate stops or be combined into one stop. Obviously tne number of stops you make is a purely personal matter, but each stop means more idling time and stop and go driving, so die fuel economy numbers will suffer accordingly. Over the road trucks have large fuel tanks for a number of reasons. First, the truck docs not have to waste time stopping for fuel as often. Second, fuel costs can be optimized by stopping at points where fuel is cheaper. Stops can be planned to match driver rest periods and terminals where adequate parking is available. With rising fuel costs, savvy truck drivers even watch the weather and adjust their route to take advantage of tail winds and avoid head winds.

Raceway Lap Times and Average Speeds A similar set of formulas can be used to calculate performance on an oval track or road course. Average

Auto Math Handbook

You can calculate lap speed in mph around a track by multiplying the lap distance—at Indy that is 2.5 miles—by 3600 and dividing by the lap time in seconds. Photo by Michael Lutfy.

lap speed in miles per hour can be found by multiplying the lap distance in miles by 3600—the number ofseconds in an hour—and dividing the result by the lap time in seconds:

lap time would you have to clock in order to average 230 mph? To find that, again multiply the lap distance by 3600, but this time, divide by the speed you want to reach:

Miles per Hour = (miles x 3600) -s- seconds Example—Suppose you’re running Indianapolis and averaging 40 seconds a lap around the 2.5 mile track. What's your average speed?

Seconds = (miles x 3600) mph To find the lap time needed for 230 mph at Indy:

Mph = (2.5 x 3600) + 40 Mph = 9000 40 You're haulin' at an even 225 mph. What kind of

Seconds = (2.5 x 3600) + 230 Seconds - 9000 + 230 You must get around the track in 39.130435 seconds.

Average MPH and MPG

FORMULAS FOR AVERAGE MPH AND MPG Miles per gallon = miles + gallons

Miles = miles per gallon x gallons Gallons = miles +• miles per gallon

Miles per hour = miles -e- hours

Miles = miles per hour x hours

Hours = miles +■ miles per hour Raceway miles per hour = (miles x 3600) -r seconds Seconds = (miles x 3600) + mile per hour

Chapter 20

Fuel Economy and Cost of Ownership 1

■■■■ ■■mil

While today's hybrids offer some interesting technological solutions, the true cost of owning and operating a car should include not only the acquisition costs but also the operating costs.

In Chapter 19 we discussed the measurement of fuel economy. Today, with rising fuel costs, fuel economy plays a big part in many car buying decisions. In the 70s, when most of the maximum speed limits were mandated at 55 mph, manufacturers tuned their vehicles ro deliver optimum fuel economy at 55 mph. Today, with higher maximum highway speeds and rising fuel prices all of the manufacturers strive to deliver their best fuel economy on the federally mandated EPA drive cycles, which arc comprised of both city and highway driving conditions. The published EPA fuel economy ratings for City and Highway driving form the popular basis for comparisons spanning different makes and models. Because these ratings arc supposed to represent a composite of the average person's daily driving experience, the ratings give the consumer a chance to compare one vehicle to another based on fuel economy alone. In the fine print, both the manufacturers and the government warn that your mileage may vary.'' The composite driving cycle will probably not match your exact driving route, but since all of the cars are tested on the same cycle the EPA number forms an equal basis for comparison. If you are serious about measuring your own fuel economy, you might want to run a scries of controlled experiments to more closely replicate your personal driving cycle and conditions. Try to start with a round trip consisting of at least two hours going to and from a destination along the same highway. By choosing a round trip of at least two hours duration hopefully you will be able to cancel out any traffic effects such as signals or slow downs. Choose a trip, like one to a relative’s house, that you make frequently so each time you make the trip you can make a change and evaluate it. By

taking the same route in both directions, hopefully you will cancel out anv grade or wind effects. Before leaving on your drive, carefully rill the tank at a convenient station and take note of which pump vou use. Let the automatic fill feature stop the pump. Carefully make note of your mileage and reset your trip odometer to zero if your vehicle is so equipped. Make your trip and return along the same route at the same speed, to the original station and use the same pump to refill the tank, again using the automatic fill feature. Record the mileage and divide the total miles run by the gallons used since the previous fill up. Establish a Baseline You should stan your test by establishing a baseline. I recommend that you start with the vehicle as you arc currently driving it with no changes. Once you have made the trip several times and established a good baseline, you may want to make some improvements to your vehicle. For your first experiment, you might want to consider running the air pressure in the tires at the high end of the vehicle manufacturer’s recommended range and have your front end alignment checked and corrected if necessary to ensure lower rolling resistance. If there is an opportunity to run at different speeds, such as on an Interstate highway, try making the trip at several different average speeds to determine at which speed your vehicle operates most efficiently. Once you have found a fuel efficient speed range, practice staying within it and not making any sudden speed changes. Using the vehicle speed control can be helpful as long as you arc traveling on flat level roads. When you encounter grades however, the speed control will force the

Fuel Economy and Cost of Ownership

This Toyota Prius hybrid offers high fuel mileage by using an electric motor with an onboard battery and a gasoline engine.

engine to wide open throttle in order to maintain the precise preset speed. This sudden application of throttle often coupled with a downshift results in increased fuel consumption compared to a driver who accepts a slight decrease in speed when climbing a grade and is willing to lose some speed momentarily to save fuel. Some drivers utilize a vacuum gauge to measure intake manifold vacuum and try to maintain the highest vacuum at all times. Whenever the throttle opening increases under load, the vacuum drops until there is no vacuum under full load at WOT (Wide open throttle). To get a clearer picture of fuel consumption ar different speeds, calculate the fuel consumption in miles per gallon. Since you are traveling the same distance, make the trip several times at different average speeds and compare the fuel consumption rates at tnc various average speeds. We all know that if you were to run the two-hour trip to grandmother's house at 35 mph, it would take far less fuel than it would if you were to average 90 mph for the same trip. As it turns out, neither of these speeds are practical, unless you want to ger rear-ended by someone doing the legal speed limit or spend some time cooling off in the local jail for speeding. Suppose that you could save one gallon of gas over the course of your trip by going three mph faster? Now before you get all excited, this may not happen in all cases, but there are points in an engine's fuel efficiency curve where the engine produces the same power but uses less fuel. In

engine testing, this measurement of pounds of fuel per horsepower hour is called brake specificJuel consumption or BSFC and is a measure of engine efficiency. By running some careful experiments, you can determine the best target speed in order to realize maximum fuel efficiency within a speed range. Once you find this point, then you can work to maximize its value. By keeping your car well tuned and making sure you have a clean air filter, you can ensure that the engine is delivering its maximum economy. Using the proper oil also reduces engine friction while still protecting the engine from wear. Perhaps the most important thing to remember with newer cars is to make sure that the oxygen sensor is working properly. This sensor (or in many cases, more than one) is located in the exhaust system. It samples the oxygen content of the exhaust and constantly compares it to the required content by using the engine computer. If die mixture is too rich or too lean, the engine management system makes the subtle changes in the spark or fuel delivery to correct the oxygen content. If the oxygen sensor is not functioning, the engine management system will typically force the system to operate in a rich condition in order to protect rhe engine. The "check engine” light will be turned on, which should be your clue to get your vehicle serviced. If you continue to operate with a rich mixture, you will suffer from decreased performance as well as decreased fuel economy and excess emissions. With the higher price of fuel and

Auto Math Handbook

Honda offers the hybrid Civic which boasts 47 mpg.

the potential for damaging other components in your system, it pays to repair and maintain your vehicle's engine management system as soon as it needs attention. The on-board diagnostic systems do an amazing job of monitoring the day to day engine performance. Proper tire inflation and correct front end alignment are perhaps the largest contributors to parasitic losses. However, if your vehicle has independent rear suspension, is older or may have ever suffered a collision, it pays to have die rear axle alignment checked also. Sometimes just having your vehicle towed out of a ditch from the rear can cause rear wheel misalignment leading to increased rolling resistance and tire wear. Here, is where a coast down test comes in handy. By performing one of these simple checks every 6 months, it will allow you to quickly monitor your car's rolling resistance. If you arc in doubt, find a friend with a similar vehicle with the same drive train and help them run a coast down test, then compare results. Factors Affecting Fuel Economy Many people think of fuel economy as strictly an engine function. In fact, die following factors all play a significant part in determining the overall fuel economy: Weight—1 he vehicle weight is important because it plays a significant part in the energy required to get the vehicle up to speed after every stop, as well as changing from one speed to another. Also, the more a vehicle weighs, the higher the rolling resistance.

When doing fuel economy testing, be sure to maintain a constant weight to ensure accuracy. Inertia—The vehicle equivalent inertia includes the vehicle weight but also factors in the axle ratio and tire diameter. If you spent most of your time driving in the city doing a lot of stop and go driving with very little constant high speed driving, you might consider a numerically higher final drive rado. This would help the vehicle accelerate from each stop. On the other hand, if you drove long distances at highway speeds with very little stop and go driving, a numerically lower final drive ratio would significantly reduce the revolutions per mile, and hence the fuel consumption. Remember though, if the final drive ratio is numerically too low, the engine will lug and be forced to operate at an inefficient point on the torque curve. Most vehicles today are delivered with final drive ratios designed to optimize the combined city/highway fuel economy rating. If the majority oi your driving is predominantly either highway or city, chances arc that you could improve your mileage with some careful option selections or modifications. By changing the final drive ratio it may be possible to improve the fuel economy. To avoid a less economical solution, you must weigh the cost of the change against the expected fuel economy gains over the expected lifetime of the vehicle. It would make little sense to spend $850 to change final drive ratios that would result in a $200 fuel savings over the life of die vehicle. Rolling Resistance—The rolling resistance or

Fuel Economy and Cost of Ownership

FIGURE 1 Vehicle A Purchase price: $40,000 Average mpg: 48 mpg Annual mileage: 12,000 miles per year x 5 years = 60,000 miles Fuel usage: 60,000 miles + 48 mpg = 1250 gal 1,250 gal x $3 gal: = $3,750 Total cost of ownership = $40,000 + $3,750 = $43,750

coast down performance is a simple test that you can perform which combines the effects of vehicle inertia, rolling resistance and drive train friction, as well as aerodynamic drag. Aerodynamics—Aerodynamic factors affecting fuel economy in a passenger car include the drag coefficient (Cd) and the frontal area (Af) neither of which change with vehicle speed, however, the aerodynamic horsepower requirement increases as the cube of vehicle speed. It is also important to remember that a fuel efficient engine does not necessarily guarantee good fuel economy. The proof of that is to see how the same engine is rated for fuel economy when installed in different vehicles, each having differing options affecting gross vehicle weight and final drive ratio.

Cost of Ownership In most cases, fuel economy alone is not the determining factor in a vehicle purchase. People choose their vehicles based on a variety of criteria. Some want a utilitarian form of transportation, while others want some performance and precise handling, or just want to make a social or environmental statement. Most people however, just want something economical, safe and reliable that will get them back and forth to work comfortably. Some drivers look at their vehicle with the same enthusiasm as they view their toaster. As long as it starts and runs they are happy. On the other end of the scale, some drivers define their lives by which car they own or drive. Be honest with yourself and really look at the overall vehicle and try to match your needs and desires with the available vehicles before purchasing. If it is important for your vehicle choice to make a social and environmental statement, at least choose a vehicle whose price and performance can be supported by numbers. Don't fall for the best fuel

Vehicle B Purchase price: $25,000 Average mpg: 32 mpg AnnuS mileage 12,000 for 5 years: = 60,000 miles 60,000 miles + 32 mpg: = 1,875 gal 1,875 gal x $3 per gallon = $5,625 Total cost of ownership: $25,000 + $5,625 = $30,625

economy number and base your decision on that alone. Set some criteria for what you want the vehicle to do. If you need to carry two adults and two children, then a 4-door sedan will probably work best for you. If, on the other end, you need to be able to haul a few sheets of plywood or tow a trailer, a light pickup truck or SUV may work boner. Once we have narrowed down the choice to a single type of vehicle, we often find ourselves trying to accidc between a vehicle (A) that costs $40,000 and gets 48 mpg or a vehicle (B) that costs $25,000 and gets 32 mpg. With the advent of hybrids and some of their impressive mileage claims, let's compare the cost of vehicle ownership. Using the two cars A and B above, let's look at rhe cost of ownership for five years with 12,000 miles driven per year. We will do this for several fuel prices and see how the numbers vary. Differing fuel prices greatly affect what we call the break-even point. To simplify this comparison we will assume that both vehicles arc under frill warranty and all operating costs except fuel are equal. In order to simplify the numbers, we will leave out other costs such as maintenance, tires and oil. In order to be totally accurate, a true comparison would have to include all of the following: finance costs (if the car were being financed), repair and maintenance costs (oil, tires, filters, brakes, battery, exhaust, etc.). If the vehicle was a hybrid, the cost of recharging the car would also have to be included, as well as batten' replacement and the disposal fee for the old battery, if that was required during the service life of the car. We have also left out the residual value (what you would get for the vehicle if you sold it at the end of the five-year period) and the cost of insurance for the period. We see that in Figure 1 the car with the pcxircr fuel mileage is actually cheaper to operate over a five-year period.

Auto Math Handbook

FIGURE 2 Vehicle A Purchase price: $40,000 Average mpg: 48 mpg Annual mileage: 60,000 miles Fuel usage: 1250 gal Fuel cost: 1,250 gal x $6 gal: = $7,500 Total cost of ownership = $40,000 + $7500 = $47,500

Vehicle B Purchase price: $25,000 Average mpg: 32 mpg Annual mileage: 60,000 miles Fuel usage: 1,875 gal Fuel cost: 1,875 gal x $6 per gallon = $11,250 Total cost of ownership: $25,000 + $11,250 = $36,250

While this Audi may not get the top fuel economy, it might be more fun to drive than a hybrid. Looking at more features than just the fuel mileage number Is something everybody should do when considering a vehicle purchase.

Now in Figure 2, we've compared these two cars again except with the cost of gasoline at $6.00 per gallon. You can sec that even at $6 per gallon, the cost of ownership based on fuel mileage and acquisition cost for the less expensive car with poorer fuel economy is still a more economical purchase over a five-year period. Now let's use some auto math and look at this question another way. Let's write an equation and see just how high the price of gas would have to go before the costs were equal. Our equation looks like this:

$40,000 + 60,000 miles 4- 48mpg (X price per gal) = $25,000 + 60,000 miles + 32 mpg (X price per gal) I^t's simplify the equation: $40,000 + 1250x = 25,000 + 1875x $40,000 - $25,000 = 1875x - 1250x $15,000 = 625x X-S24

So what this says is that in this case, until gasoline gets to $24/gallon the cheaper car with poorer fuel economy costs less to own and operate for the fiveyear period, with all other factors being equal. Now using some auto math let's loot at the same data a different way. Let's see how far we would have to drive the cars before they were equal in purchase and fuel costs, keeping fuel cost constant at S3 per gallon.

Initial cost + (miles 4- 48 mpg) x $3/gal = 25,000 + (miles + 32 mpg) x $3/gal We write our equation using X as the unknown to denote the miles at the break-even point: $40,000 + (X + 48) x 3 = 25,000 + (X + 32) x 3 40,000 + (3X + 48) = 25,000 + (3X + 32) 15,000 = (3X + 32) - (3X + 48) 15,000 = 0.09375X - 0.0625X 15,000 = 0.03125X

Fuel Economy and Cost of Ownership

FIGURE 3

Fuel economy: Fuel tank volume: Fuel tank range:

Vehicle A 20 mpg 25 gal 500 miles

X= 480,000 miles By looking at the answers above, it should become very obvious that better fuel economy numbers alone arc only a small part of the true ownership cost. Fuel Range vs. Fuel Economy In order to sell cars with poorer fuel economy, the manufacturers often advertise and promote fuel range instead of fuel economy. The less sophisticated consumer often gets range and fuel mileage confused. Range is defined as the distance a vehicle can travel on a single tank of fuel, or in the case of a hybrid, on a single battery charge and a tank of fuel. How do we make this comparison? Again let's look at two vehicles with different fuel economy and different size fuel tanks. We only want to determine which vehicle is more fuel efficient. For this example in Figure 3, our choice has been narrowed down to a pair of vehicles:

Vehicle B 48 mpg 10 gal 480 miles

Vehicle A is advertised as being able to go 500 miles on a tank of gas. Vehicle B is advertised as getting 48 miles to a gallon. As the old saying goes, this is like comparing apples to oranges. We must investigate and get more data before we can make a true comparison. In this case, we will look up the published combined fuel economy rating on vehicle A and the find the tank capacity tor vehicle B in the specifications section of the owners manual. Now we compare the numbers. Clearly, if both. cars will fill your needs for passenger space and performance, vehicle B is a letter choice. It will cost far less to operate over the ife of the vehicle. If vehicle B is a hybrid however, the cost of charging each evening and the cost of bancry replacement and disposal, might tilt the economic scales in favor of the seemingly less efficient vehicle.

FORMULAS FOR AVERAGE MPG Miles per Gallon = miles -r gallons

Range in Miles = miles per gallon x gallons Gallons = miles ? miles per gallon Gallons per Hour = gallons used + hours

Range = miles -r tank full

Chapter 21

Crankshaft Balancing

This is what you need to weigh before starting a balance job. The rod, piston, pin, rings, and bearings will all have to be weighed and corrected if necessary before calculating the bobweight Photo by Bill Hancock.

Math plays a big part in correctly balancing an engine. If yon get the calculations wrong, the whole job will be a disaster and unfortunately, you will never realize the mistake until the engine is assembled and vibrates when it is run. At that point it is too late, the engine is assembled and the damage may have been done. It pays to understand the math that goes into a balance job and to know how to double check it. It is impossible to look at a crankshaft and tell if it has been balanced correctly. The only way to know for sure is to either assemble the engine and run it or have someone check the balance on a balancing machine. Crankshaft balancing is probably one of the least understood operations in engine building. All but the most professional engine builders rely on the local balancing shop to balance their rotating assembly, simply because it doesn’t make sense to own the expensive balancing equipment that you will nor use very often. As an engine enthusiast however, you should at least understand the principles and necessity for balancing. Friction is the enemy of performance and durability in an engine. If a crankshaft is straight and true, it will spin freely once it is correctly assembled in an engine with proper main bore alignment and the correct bearing clearances. In order to maintain that straightness throughout the entire rpm range, the crankshaft must be properly balanced. If the crankshaft is poorly balanced or tne components have been changed and the crank has not been rebalanced, a standing wave in the crankshaft while running could result. This bending would only occur while the crankshaft was in motion at higher rpm. Once the engine returns to idle, the crankshaft straightens out because the offsetting loads caused

by centrifugal force have gone. The bent crankshaft creates friction, robs power, creates undue stress and ultimately wears out the bearings. We will outline the procedure to balance a 90-degree V-8 engine. A Typical V-8 Balance Job Let’s begin by looking ar the components that need to be included in a typical balance job for a V-8 engine. They arc: Rods (bolts, bearings, bushings) Pistons (pin, rings, clips) Damper Flywheel or flcxplate Crankshaft Lets take these components and separate them into two groups based on their motion within the engine, rotating and reciprocating, and then carefully weigh them.

Parts in Motion The piston assembly (piston, pin, clips, and rings) goes up and down in the bore, therefore this would be classified as reciprocating motion. The connecting rod has both rotating and reciprocating motion. The small end of the rod is connected to the piston, which is reciprocating, while the large end of the rod is bolted to the crankshaft pin, which rotates. To address this, we weigh both ends of the rod separately and get a rotating and reciprocating component of the total rod weight. As a check, the rotating weight plus the reciprocating weight of the rod should equal the total weight of the rod. The crankshaft, flywheel and damper or harmonic balancer, have purely rotating morion.

Crankshaft Balancing

Each connecting rod is weighed for small end weight (shown), big end weight and total weight. A special hanging fixture is used to weigh small and big ends. When weighing either end of the rod, the centerline of each end (small and big) should be kept level. Pistons, pins, rings are also weighed separately. Photo by Mike Mavrigian.

CRANKSHAFT TERMINOLOGY Let’s review basic crankshaft terminology before we go too far and lose somebody. The rodjournal, is offset from the main bearing journal, or main as it is called. The main bearing journal is located at the axial center of the crankshaft. The rod journal is also referred to by some as the crank pin or crank throw. This journal is located at an offset distance from the crankshaft axial centerline. This offset distance is known as the radius of the crankshaft, which is half of the crankshaft stroke. Obviously if you rotate the crank around the central axis the rod pin rotates in a circle with a radius of R. If wc hooked a rod to the crank pin and constrained the rod’s free end in one plane, it would go back and forth or reciprocate. 1'his is how a crankshaft converts reciprocal motion into rotary motion. If we double the radius, we get whar is referred to as the stroke of the crankshaft, or the distance that the piston travels up and down in the cylinder as the crankshaft rotates. Adjacent to the crank pins is a disc or pair of discs that in addition to structurally connecting the pins to the mains, serve as a counterweight to balance the crank pin and all of the components (rod piston, pin, rings etc.) weights. The forward end of the crank sometimes called the nose or snout is typically a smaller diameter than either the main or rod journal and serves to drive the camshaft and accessories through a gear, hub, pulley or a harmonic damper if rhe engine has one. The rear end of the crankshaft has a disc called z flange with a series of holes spaced radially from the center. The flywheel mounts to the flange using a series of fasteners.

Weighing the Parts—Io begin a competition balance job, all components are laid out in an orderly fashion and marked with a felt tip marker to indicate their final assembled position and starting weight. Typically each rod and piston is reduced in weight to match the weight of the lightest part in the series. For example, if the pistons range in weight from 394 gm to 398 gm, the seven heaviest pistons will have to be carefully machined to remove weight necessary to make them all equal to 394 gm. This process is repeated for both ends of each rod. The piston rings, piston pins, rod bolts and bearings arc typically very close in value and need no correction.

Calculations—Once the weighing is done, we perform rhe calculations, which is one of the most important parts of the balancing procedure. If you make a math error here, you will never know it until you run down rhe track and feel a horrible vibration trying to tear your engine apart. For that reason, it pays to double-check your figures, or have a friend do the math independently, then compare answers. Be sure to use a balance worksheet similar to Fig. 21a or like in the photo on page 96. Fill in the blanks, as you go, with your finished numbers. Once you have filled in the individual weights, add up the reciprocating numbers for each pair of cylinders then divide the result by two. This gives you half or 50% of the reciprocating weight for each pair of cylinders. Next, we calculate a weight for all of die rotating components for that pair of cylinders (large end of born rods with fasteners, four rod bearing shells, and about eight grams for oil that resides in the crank and between the bearings and rod journal of the crankshaft). If you add half of the reciprocating weight and all of the rotating weight for each pair of cylinders together, you get a total number for the pair of cylinders,

Auto Math Handbook

Fig. 21a

V-8 Balance Sheet

#1

#2

#3

#4

#5

#6

#7

#8

Piston

Retainers

Ring Set Rod small end

Reciprocating weight

Rod big end

Bearing shells (2)

Rotating weight

Bobweight 1/2 reciprocating + rotating

Once all components (rod bearings, rods, pistons, pins, rings and pin locks) are weighed, the bobweight card Is filled out, allowing the balance operator to determine the required bobweights needed to check crankshaft balance. Photo by Mike Mavrigian.

-W._ °JLa'-*OWaNce PtsK>N

PIN

*«NOS — r, *AJLS/AKDB ^rroNs_21_S' z ’'ODPECjpk, RQBU FIGHTTTL

called the bobweight. A typical V-8 crankshaft has four crank pins spaced 90 degrees apart when viewed from the end of the crankshaft. These crankpins each have a pair of rods attached, which allow a pair of cylinders to deliver power to the crankshaft. When we balance a crankshaft, we cannot spin the crank with the rods and pistons attached, so we use a small weight that clamps to each rod pin or journal. This weight, also called a bobtveight, simulates the weight of the rotating and reciprocating components. The key to a good balance job starts with calculating the bobweight. Next, you assemble this weight that attaches to the corresponding throw of the crankshaft and simulates the load for balancing. Complete and attach all four bobweights to the crankshaft.

Crankshaft Baiancing

Here is a rod journal, crank pin or crank throw depending on what you want to call it Photo by Bill Hancock.

Crank Balancer—The damper and flywheel arc bolted to the crank and the whole assembly is placed in a crank balancing machine. A balancing machine or crank balancer is a device that measures the amount of imbalance at each end of the crankshaft and then locates each imbalance angularly, so corrections can be made to the counterweights to bring the crankshaft within specifications. Typically the imbalance is measured in inch-ounces. A suitable imbalance limit is 0.25 inch-ounces for a high-speed crankshaft. To understand balance and the terminology; let’s think about two weights placed on a kid’s playground see-saw. We will begin when we verify that die sec-saw is perfectly balanced with no weight on it. Now if we add 10 lb to each end of the see-saw at exactly five feet from the pivot or fulcrum point, the see-saw will remain balanced. Suppose we wanted to remain balanced but we only had a 20-lb weight? If we moved the 20-lb weight toward the fulcrum point or center of the sec-saw until it was 2.5 feet away from the fulcrum, the see­ saw would again be balanced. So rhe lesson here is that the length of the lever arm, multiplied by the weight at the end of each lever arm should be equal if the see-saw is to remain balanced. So in this case, each side has 50 ft-lb. Lever Arm The see-saw is the classic example of a force applied through a lever arm. In the first example, we multiplied the length of the lever aim times the weight, wc would get 5 ft x 10 lb or 50 ft-lb. In a

The crankshaft is checked for main and rod journal cleanliness, and mounted on the crankshaft balancing machine's clean (and oiled) nylon V-blocks. The bobweights are then attached to the rod journals. The bobweights will simulate the weight of the components that were previously weighed. Photo by Mike IMavrigian.

We utilize a set of four adjustable weights that clamp onto each throw or rod journal of the crankshaft. These weights are called the bobweight and simulate the weight of the rods and pistons. Photo by Bill Hancock.

crankshaft, wc have weights located ar various distances away from the center of rotation. When we use lighter components like connecting rods, we take away weight at the crank pin, therefore we must remove weight at the counterweight which is exactly 1B0 degrees away from the crank pin. The farther away from the center wc remove the weight,

Auto Math Handbook

Here we see the fully bobwelghted crank in the balancer waiting to be spun. Note that the last counterweight on the right has tape holding in a slug of heavy metal to make sure that it Is enough before welding it in place. Photo by Bill Hancock.

the less we have to remove. In most performance balance jobs, the new parts are lighter than the production pans they arc replacing, so weight must be subtracted from the crankshaft counterweights to maintain the factory balance. If the crankshaft is new and has never been balanced, it may require a significant amount of weight to be removed . Wc begin by spinning the crank on slow speed to see where the imbalance is located and to measure how much imbalance is present. If a crankshaft shows up as being heavy in the counterweight area, you will have to remove weight from that area. For example, if the balancer showed that the imbalance was 6.8 inch-ounces out of balance, we would begin by measuring the radius of the counterweight. Let’s say that the radius measured 3.4 inches from tire centerline of the crankshaft. First, we know that we cannot remove all of the weight from the outer edge of the counterweight, so we use a 3.00-inch radius for a starting point. If we have 6.8 inch-ounces of imbalance wc use the following formula: 6.8 inch-ounces = 3.00 inches x ounces 6.8 + 3.00 = 2.27 ounces If a crankshaft has been stroked or if the new parts weigh more than the old parts, weight may need to be added to the counterweight to offset the additional weight of the new parts, or their increased distance from the center. This is where the math gets a bit tricky. Since we can’t just add weight indiscriminately wherever wc want and we can’t remove any weight from the pin area or heavy side for fear of weakening the crankshaft or

components, there may be only one avenue left. First we should try to fill any holes that have been previously drilled to balance the crankshaft. This is done by drilling new holes parallel to die axis of the crankshaft. People often make the mistake of trying to fill the existing holes with steel plugs which arc welded in radially or at 90 degrees to the crankshaft centerline. As long as the weld holds, everydiing is fine but if the weld ever breaks and the metal flics out, the slug of metal can rip a huge hole in the engine block or oil pan and cause serious damage and devastating results. Do not take this chance! Take the extra care and drill and ream the more difficult holes and then press the weight in parallel to the crankshaft centerline, then weld it in place.

Heavy Metal High density or heavy metal has a weight roughly 90 percent greater than a piece of steel measuring rhe same dimensions. Heavy metal has become tne solution for balance jobs that require additional weight. Typically, heavy metal is an alloy of tungsten and copper that is easily machined and can De welded without problems. To use heavy metal, holes must be drilled parallel to the crankshaft centerline, and then reamed to size. The heavy' metal must then be cut to length and pressed into the reamed hole, then welded to ensure that it does not move. Now lets look at the math involved in this job. Calculating How Much Weight to Add—Let’s begin with a balance job where the crankshaft is light and out of balance by 38 grams positioned 3.25 inches from the centerline of the crankshaft on the outside diameter of the counterweight. This would result in an imbalance of: 38 gm x 3.25 inches or 123.5 inch-grams We will start by making a sketch and doing somerough calculations to ensure that one slug of heavy metal will be sufficient. Wc need to end up with the counterweight being slightly heavier than it needs to be, so once we add the slug of heavy metal, wc can still remove a small amount of weight and fine tunc the location of the imbalance. In order to do this we will begin by drilling out a 1.25-inch hole through the counterweight located 2.5 inches from the center. We chose 2.5 inches, because this dimension will let us drill and ream a 1.25 hole through the counterweight without breaking through to die outer edge of the counterweight. Since wc arc going to be making our correction centered at 2.5 inches from the center, we will need

Crankshaft Balancing

co add more weight, since the weight is closer to the center and hence not as effective. If we divide our total imbalance of 123.5 inch-grams by 2.5 inches, we come up with 49.4 grams that we will have to add. We begin by finding the area of a circle 1.25 inches in diameter:

riR2 = a R ■ 1.25 + 2 ■ 0.625 inches Pi (3.1416) x (0.625)2 = 1.227 sq. inches Next, multiply the area times the height to calculate the volume of the cylindrical slug of heavy metal. In this case, the height is the thickness of the counterweight or 1.25 inches. A x H = Volume 1.227 x 1.25 ■ 1.534 cubic inches We look up the weight of steel in the appendix and find that it is 0.28383 Ib/cubic inch (in3). Therefore, if we multiply this value times our volume, we get the following: 0.28383 lb/in3 x 1.534 in3 = 0.435395 lb Next we look up the conversion for pounds to grams in the appendix and find that 1 lb = 453.6 gm. So wc multiply that value times our cylinder weight: 453.6 gm/lb x 0.435395 lb » 197.495 grams Now we know that we are going to remove 197.5 grams of steel from the counterweight and we are going to replace it with heavy metal that weighs approximately 190%, or almost twice the weight of steel.

197.495x 1.9 = 375.2 gm Therefore our piece of heavy metal will weight approximately 375 grams. If we subtract the original weight of steel, we will have the additional weight added. 375.2 - 197.495 - 177.705 grams This is more than the weight our calculations said

Sometimes the operator gets lucky and nails it on the first try, while in some cases, weight correction must be "chased" several times. If adding heavy metal Is needed, the options are to drill a hole and install the tungsten slug on either the outer edge of the counterweight or 90 degrees (through a counterweight). Whenever feasible, the drill-through method is preferred, since this eliminates the remote possibility of the weight ever being slung out during engine operation. Either way, if a tungsten slug is added, balancing shops generally secure the slug with a small tig weld. Photo by Mike Mavrigian.

we needed, but it will allow us some margin to drill out the counterweight and do the final correction and still be safe. It is far easier to add slightly more weight initially while the crankshaft is out of the balancer, than having to go back and add another piece later on. At this point, the balance job becomes fairly simple in that all you have to do is drill holes to remove the weight of metal where the balancer indicates. You complete this procedure for each end of the crankshaft. When you complete the second end of the crankshaft, you must go back and check the first end again to ensure that it has remained unchanged as a result of the corrections you made to the second end. Once both ends arc within spec, die job is done.

FORMULAS FOR CRANKSHAFT BALANCING Imbalance = lever arm x weight

Bobweight = 50% reciprocating + 100% rotating

Auto Math Handbook

BALANCING TIPS Adapted from Building the Chevy LS2 by Mike Mavrigian

• Perform all machining to the crankshaft, flywheel, pistons and rods before balancing. Alterations to these parts after balancing will negate your balancing work, and you'll have to start over. For instance, if connecting rod beams arc to be smoothed and polished, do this before the balancing work begins. • If you find a gross difference in crankshaft spin-up weight from front to rear of the shaft, chances arc good that the crank is bent. To avoid wasting time, always check the crankshaft for runout before taking the time to spin balance the crank. With the crank laying on rhe balancer’s twin Vblocks, set up a dial indicator and check runout on the main journals. • By the same token, when a crank is ground, it s imperative that rhe stroke doesn't change from rod dirow to rod throw. The centerline of the rod journals should be identical (centerline of main to centerline of each rod journal). Also, and this is especially critical if the ignition system is crank-fired, the index of each rod journal must be correct. Although the stroke may be OK, if the index is out of whack, the engine's timing will be grossly out of sition. Just remember that crank grinding can affect lance, il the stroke or index is altered. • If the crankshaft was stored improperly, or if the crank was chucked off center in the crank grinder, or if it was chucked under tension, rhe axis of the crank mains could be untrue. Also, the effects of heat can create small deflections in the crank, so make sure the crank reaches room temperature before performing a spin-balance reading. • Replacement oversize pistons arc not necessarily lightened to match the weight of the OE piston, so never assume that the balance won't be affected even though you're changing a complete set of pistons from old to new. This is why the balancer was invented...to verify what you're dealing with and allow you to correct any mass problems! Replacement pistons are usually boxed by the maker as a matched set (weighing within X-grams of each other). However, you should never assume that all pistons of the same part number will weigh within diat acceptable match range (although today's performance aftermarket forged ana CNC-finishcd pistons are typically extremely well weight-matched). Forged and hypcreutectic pistons will likely be more closely matched in weight from the start. • Weigh the pistons and pins separately. You can then match pistons to pins to "even out” the piston/pin set weights, thereby reducing the time needed to machine weight from pistons (match the lightest piston with the heaviest pin, etc.).

S

• Try to maintain piston/pin weights to within one gram, from cylinder location to cylinder location. There’s really no need to make yourself crazy by trying to create a tighter tolerance range. A 1 -gram tolerance is perfectly acceptable (we say this not to make it easier, but because it's impractical to try to achieve tight-number tolerances. When you consider the changing forces that act upon the engine during operation, with oil slinging around, clinging to areas in a sometimes random pattern, you'll simply drive yourself nuts for no additional benefit if you try to create ultra-tight gram matching). • Some crankshafts—like those from the Chevy LS family of engines—arc internally balanced. That means that weight corrections are made on the crankshaft itself, without regard to damper or flywheel. The damper and flywheel for an internally balanced crankshaft should be zero balanced independently. If rhe engine is externally balanced (where the flywheel is an integral part of the crank's balance), consider future flywheel replacements. This is especially important with race motors that will routinely be rebuilt or repaired. Once the assembly is balanced, remove the flywheel and spin this up separately, and document the state of balance of that flywheel. In this way, future replacement flywheels can be balanced to those exact specs (duplicating the first flywheel), without die need to rebalance rhe crank! • When removing weight from a piston, don’t blindly remove stock from the underside of the dome. Measure the dome thickness first. As far as balancing is concerned, here's what you need to know: if rhe engine is an internally balanced design (where all of the crankshaft balancing occurs at the counterweights and the damper and flywheel arc zero-balanced on their own), the viscous damper itself is already balanced, so there’s no need to perform any balancing work on the damper at any time. If the engine is an externally balanced design (where die front damper and the flywheel are integral components of crank balance), the viscous damper will consists of two parts...an outer damper ring and a center hub. Disassemble the damper to separate the ring from the hub. Mount only the hub to the crank snout (along with the flywheel at the rear crank flange) for crankshaft balancing. Do not attach the viscous damper ring to the hub for balancing! In short, never install a viscous-type damper or damper ring to a crank for spin-balancing, since the centrifugal internal action of the damper's fluid will serve to mask some of the crank's harmonic disturbances, and will result in a false spin-balance reading.

APPENDICES

APPENDIX A: CONVERSION FACTORS

The following is a list of conversion factors for U.S. and S.I. units of measure that could be of use to the automobile enthusiast. Also included are factors for two widely used British measures: British thermal units, or Btus, and imperial gallons. As is true elsewhere in the book, abbreviations have been kept to a minimum and figures are carried out to a maximum of eight digits. Factors for converting between two specific units of measure are reciprocals, i.c., if multiplied together, the}' have a product of 1.0. For example, the factor for converting from gallons to quarts is 4.0, while the factor for converting from quarts to gallons is 0.25. Multiplied together, 4.0 and 0.25 equal 1.0. So, if you know die factor for converting in one direction but not for the other, you can find the latter by dividing the known factor into 1.0. As a case in point, the list includes a factor of 42 for multiplying a number of barrels of oil to find the equivalent in gallons. However, no factor is given for multiplying gallons to find barrels, because that isn’t a conversion many people would need to make. The 42-gallon barrel is an arbitrary unit of measure of oil used in international commerce and not the real size of a container. But suppose you did want to know how many barrels a given number of gallons of oil would equal. You can find the reciprocal of 42 by dividing it into 1.0, or 1/42, which would be 0.0238095. On a scientific calculator, you could enter 42 and then press the reciprocal key, marked 1 /x. Another way would be to divide 42 into gallons to convert to barrels. Mathematically, switching from multiplication in one direction to division in the other is the same thing as switching from multiplying by one conversion factor to multiplying by its reciprocal. When one factor is simpler than its reciprocal, you can save time and effort by knowing when to divide instead of to multiply. In our example, it would obviously be quicker and easier to divide the gallons by 42 than to multiply the barrels by 0.0238095.

In the case of very large numbers, the reciprocal may not have enough significant digits within an eignt-digit limit to be of much value. As an example, take the factor for converting from kilowatt-hours to joules, which is 3600000.0. On an eight-digit calculator, the reciprocal for converting from joules to kilowatt-hours would be 0.0000003. But, if you enter 0.0000003 and press the 1 /x key, you’ll get a reciprocal of 3333333.3, an error of more than 7.4 percent. The reciprocal has been rounded to only one significant digit, and that’s not enough for an accurate conversion back to the original factor, nor is it enough for accurate calculations. Consequently, reciprocals with only one or two significant digits arc not included in the list. If you had to convert joules to kilowatt-hours, you’d get more accurate results by first converting the joules to kilojoules (multiplying the joules by 0.001 or dividing by 1000) and then convening rhe kilojoules ro kilowatt-hours (multiplying the kilojoules by 0.0002778 or dividing by 3600). Another way to deal with very large or very small numbers is to use scientific notation, a form of mathematical shorthand that eliminates the need for a large number of digits. A scientific calculator will have a key marked either EXP (for exponent) or EE (for exponent entry) for using scientific notation, and instructions for working with it will be found in the calculator’s manual. In the list, where the terms gallons, miles and ounces are used without qualification, they mean U.S. gallons, statute miles, and avoirdupois ounces, respectively. Similarly, horsepower and torque mean SAF. horsepower and torque, while water means fresh water. Factors involving water are as measured at 4.0 degrees Celsius or 39.2 degrees Fahrenheit. That’s the temperature of water at its maximum density, which serves as the international standard for measuring the relative density or specific gravity of other liquids. Finally, factors which are exact figures are indicated by an asterisk (*).

Appendix A: Conversion Factors

TO CONVERT FROM:

MULTIPLY BY:

A atmospheres to bars atmospheres to inches of mercury atmospheres to inches of water atmospheres to kilograms per square centimeter atmospheres to kilopascals atmospheres to millibars atmospheres to pounds per square inch

1.01325* 29.921256 406.80172 1.0332275 101.325* 1013.25* 14.695949

B barrels, non-oil liquid, to cubic feet barrels, non-oil liquid, to cubic meters barrels, non-oil liquid, to gallons barrels, non-oil liquid, to liters barrels, oil, to cubic feet barrels, oil, to cubic meters barrels, oil, to gallons barrels, oil, to liters bars to atmospheres bars to inches of mercury bars to inches of water bars to kilograms per square centimeter bars to kilopascals bars to millibars bars to pounds per square foot bars to pounds per square inch Btus to calories Btus to horscpowcr-hours Btus to joules Btus to kilogram-meters Btus to kilowatt-hours Btus to kilojoules Btus to pounds-fcct Btus to watt hours Btus per gallon to megajoules per cubic meter Btus per gallon to megajoules per liter Btus per minute to horsepower Btus per minute to kilowatts Btus per pound to joules per kilogram Btus per pound to kilojoules per kilogram Btus per pound to megajoules per kilogram

4.2109376 0.1192405 31.5* 119.24047 5.6145833 0.1589873 42.0* 158.98729 0.9869233 29.529983 401.48716 1.0197162 100.0* 1000.0* 2088.5434 14.503774 251.99576 0.000393 1055.0559 107.58576 0.0002931 1.0550559 778.16927 0.2930711 0.279 0.000279 0.0235809 0.0175843 2326.* 2.32* 0.002326*

c calorics to Btus calories to joules calorics to kilogram-meters calories to pounds-feet calories to watt hours centiliters to deciliters centiliters to liters centimeters to feet

0.0039683 4.1868* 0.4269348 3.0880252 0.001163 0.1* 0.01* 0.0328084

Appendix A: Conversion Factors

0 CONVERT FROM: centimeters to hands centimeters to inches centimeters to meters centimeters to microns centimeters to millimeters centimeters to mils centimeters to yards centimeters per second to feet per second centimeters per second to kilometers per hour centimeters per second to miles per hour centimeters per second per second to feet per second per second centimeters per second per second to g centimeters per second per second to meters per second per second circles to circumferences circumferences to circles circumferences to degrees circumferences to grades circumferences to minutes circumferences to quadrants circumferences to radians circumferences to seconds cubic centimeters to cubic inches cubic centimeters to cubic meters cubic centimeters to gallons cubic centimeters to liters cubic centimeters to milliliters cubic centimeters to ounces, fluid cubic centimeters to pints cubic centimeters to quarts cubic feet to cubic centimeters cubic feet to cubic inches cubic feet to cubic meters cubic feet to cubic yards cubic feet to gallons cubic feet to liters cubic feet to ounces, fluid cubic feet to pints cubic feet to quarts cubic feet, water, to pounds cubic inches to cubic centimeters cubic inches to cubic feet cubic inches to gallons cubic inches to liters cubic inches to ounces, fluid cubic inches to pints cubic inches to quarts cubic inches, water, to pounds cubic meters to cubic centimeters cubic meters to cubic feet cubic meters to cubic yards

MULTIPLY BY: 0.0984252 0.3937008 0.01* 10000.0* 10.0* 393.70079 0.0109361 0.0328084 0.036* 0.0223694 0.0438084 0.0010197 0.01 1.0* 1.0’ 360.0* 400.0* 21600.0’ 4.0’ 6.2831853 129600.0’ 0.0610237 0.000001* 0.0002642 0.001* 1.0* 0.03381 0.0021134 0.0010567 28316.847 1728.0’ 0.0283168 0.037037 7.4805195 28.316866 957.50649 59.844156 29.922078 62.424215 16.387064 0.0005787 0.004329 0.0163871 0.5541126 0.034632 0.017316 0.0361251 1000000.0* 35.314667 1.3079506

Appendix A: Conversion Factors

0 CONVERT FROM: cubic meters to gallons cubic meters to liters cubic meters, water, to kilograms cubic meters, water, to pounds cubic yards to cubic feet cubic yards to cubic meters cubic yards to gallons cubic yards to liters

MULTIPLY BY: 264.17205 1000.0* 999.94004 2204.4903 27.0* 0.7645549 201.97403 764.55486

D deciliters to centiliters deciliters to liters degrees to circumferences degrees to grades degrees to minutes degrees to quadrants degrees to radians degrees to seconds

10.0* 0.1* 0.0027778 1.1111111 60.0* 0.0111111 0.0174533 3600.0*

F fathoms to feet fathoms to meters fathoms to yards feet to centimeters feet to fathoms feet to furlongs feet to inches feet to hands feet to kilometers feet to meters feet to mils feet to yards feet per second to centimeters per second feet per second to feet per minute feet per second to kilometers per hour feet per second to knots feet per second to meters per second feet per second to miles per hour feet per second per second to centimeters per second per second feet per second per second to g feet per second per second to meters per second per second furlongs to feet furlongs to meters furlongs to miles furlongs to yards

6.0* 1.8288* 2.0* 30.48* 0.1666667 0.0015152 12.0* 3.0* 0.0003048 0.3048* 12000.0* 0.3333333 30.48* 60.0* 1.09728* 0.5924838 0.3048* 0.6818182 30.48* 0.0310809 0.3048’ 660.0’ 201.168* 0.125* 220.0*

G g to centimeters per second per second g to feet per second per second g to meters per second per second gallons to cubic centimeters gallons to cubic feet

980.665* 32.174049 9.80665’ 3785.4118 0.1336806

Appendix A: Conversion Factors

TO CONVERT FROM: gallons to cubic inches gallons to cubic meters gallons to cubic yards gallons to liters gallons to ounces, fluid gallons to pints gallons to quarts gallons, acetone, to pounds gallons, castor oil, to pounds gallons, ethanol (ethyl alcohol), to pounds gallons, ether, to pounds gallons, gasoline, to pounds gallons, Kerosene, to pounds gallons, liquid propane, to pounds gallons, methanol (methyl alcohol), to pounds gallons, naptha, to pounds gallons, nitromethane, to pounds gallons, oil (crude), to pounds gallons, oil (refined), to pounds gallons, turpentine, to pounds gallons, water, to pounds gallons, water (sea), to pounds gallons, imperial, to liters gallons, imperial, to gallons, U.S gallons, U.S., to gallons, imperial gallons per horsepower-hour to liters per kilowatt-hour gons to grades grades to circumferences grades to degrees grades to gons grades to minutes grades to radians grades to quadrants grades to seconds grains to grams grains to milligrams grains to ounces grams to grains grams to kilograms grams to milligrams grams to ounces grams to pounds grams per centimeter to kilograms per meter grams per cubic centimeter co pounds per cubic foot grams per cubic centimeter to pounds per cubic inch grams per cubic centimeter to pounds per gallon grams per cubic centimeter to kilograms per cubic meter grams per kilowatt-hour to pounds per horsepower-hour H hands to centimeters hands to feet

MULTIPLY BY: 231.0* 0.0037854 0.0049511 3.7854118 128.0* 8.0* 4.0* 6.6 8.1 7.6 6.2 6.0 6.6 4.25 6.7 5.6 9.4 7.5 7.0 7.3 8.3449037 8.6 4.54609* 1.2009499 0.8326742 0.0793181 1.0* 0.0025* 0.9* 1.0* 54.0* 0.015708 0.01 3240.0* 0.0647989 64.798911 0.0022857 15.432358 0.001* 1000.0* 0.035274 0.0022046 10.0* 62.42796 0.0361273 8.3454044 1000.0* 0.001644

10.16* 0.3333333

Appendix A: Conversion Factors

TO CONVERT FROM: hands to inches horsepower to pounds-feet per minute horsepower to pounds-feet per second horsepower, metric, to horsepower, SAE horsepower, metric, to kilogram-meters per second horsepower, metric, to kilowatts horsepower, SAE, to horsepower, metric horsepower, SAE, to kilowatts horsepower-hours to Btus horsepower-hours to calorics horsepower-hours to kilojoules horsepower-hours to kilowatt-hours horsepower-hours to mega joules

MULTIPLY BY: 4.0* 33000.0* 550.0* 0.9863201 75.0* 0.7354988 1.0138697 0.7456999 2544.4336 641186.48 2684.52* 0.7456999 2.68452*

I inches to centimeters inches to feet inches to hands inches to meters inches to microns inches to millimeters inches to mils inches to yards inches of mercury to atmospheres inches of mercury to Bars inches of mercury to inches of water inches of mercury to kilograms per square centimeter inches of mercury to kilopascals inches of mercury to millibars inches of mercury to pounds per square foot inches of mercury to pounds per square inch inches of water to atmospheres inches of water to bars inches of water to inches of mercury inches of water to kilograms per square centimeter inches of water to kilograms per square meter inches of water to kilopascals inches of water to millibars inches of water to pounds per square foot inches of water to pounds per square inch

J

joules to Btus joules to calories joules to kilogram-meters joules to kilojoules joules to megajoules joules to newton-meters joules to ounces-inches joules to pounds-feet joules to pounds-inchcs joules to watt-hours

2.54* 0.0833333 0.25* 0.0254* 25400.0* 25.4* 1000.0* 0.0277778 0.0334211 0.0338639 13.595915 0.0345316 3.3863886 33.863886 70.726197 0.4911541 0.0024582 0.0024907 0.0735515 0.0025398 25.398476 0.249074 2.4907397 5.2020179 0.0361 251

0.0009478 0.2388459 0.1019716 0.001* 0.000001* 1.0* 141.61193 0.7375622 8.8507457 0.0002778

Appendix A: Conversion Factors

TO CONVERT FROM: joules to watt-seconds joules per gram to kilojoules per kilogram joules per kilogram to Btus per pound

MULTIPLY BY: 1.0 1.0* 0.0004299

K kilograms to grams kilograms to newtons kilograms to pounds kilograms to tons, long kilograms to tons, short kilograms, water, to liters kilograms per cubic meter to grams per cubic centimeter kilograms per cubic meter to pounds per cubic foot kilograms per cubic meter to pounds per gallon kilograms per kilowatt-hour to pounds per horsepower-hour kilograms per liter to pounds per gallon kilograms per meter ro grams per centimeter kilograms per meter to pounds per foot kilograms per meter to pounds per inch kilograms per square centimeter to atmospheres kilograms per square centimeter to bars kilograms per square centimeter to inches of mercury kilograms per square centimeter to inches of water kilograms per square centimeter to kilograms per square meter kilograms per square centimeter to kilopascals kilograms per square centimeter to millibars kilograms per square centimeter to pounds per square foot kilograms per square centimeter to pounds per square inch kilograms per square meter to kilograms per square centimeter kilograms per square meter to pounds per square foot kilogram-meters to Btus kilogram-meters to calories kilogram-meters to joules kilogram-meters to pounds-feet kilogram-meters to watt-hours kilojoules to Btus kilojoules to horsepower-hours kilojoules to joules kilojoules to kilowatt-hours kilojoules to megajoules kilojoules per ki ogram to Btus per pound kilojoules per ki ogram to joules per gram kilometers to meters kilometers to miles, nautical kilometers to miles, statute kilometers per hour to centimeters per second kilometers per hour to knots kilometers per hour to meters per second kilometers per hour to miles per hour kilometers per liter to miles per gallon, imperial kilometers per liter to miles per gallon, U.S. kilopascals ro atmospheres

1000.0* 9.80665* 2.2046224 0.0009842 0.0011023 1.00006 0.001* 0.062428 0.0083454 1.6439879 8.3454064 0.1 0.671969 0.0559974 0.9678411 0.980665* 28.959021 393.7244 10000.0* 98.0665* 980.665* 2048.1614 14.223343 0.0001* 0.2048161 0.0092949 2.3422781 9.80665* 7.2330139 0.0027241 0.9478171 0.0003725 1000.0' 0.0002778 0.001 0.4299226 1.0 1000.0 0.5399568 0.6213712 27.777778 0.5399568 0.2777778 0.6213712 2.8248094 2.3521459 0.0098692

Appendix A: Conversion Factors

TO CONVERT FROM: kilopascals to bars kilopascals to inches of mercury kilopascals to inches of water kilopascals to kilograms per square centimeter kilopascals to millibars kilopascals to pascals kilopascals to pounds per square foot kilopascals to pounds per square inch kilowatts to Btus per minute kilowatts to horsepower, metric kilowatts to horsepower, SAE kilowatts to newton-meters per second kilowatt-hours to Btus kilowatt-hours to horscpowcr-hours kilowatt-hours to joules kilowatt-hours to kilojoules kilowatt-hours to mega joules kilowatt-hours to watt-hours knots to feet per minute knots to feet per second knots to kilometers per hour knots to miles per hour

MULTIPLY BY: 0.01* 0.2952998 4.0148716 0.0101972 10.0* 0.001* 20.885434 0.1450377 56.869027 1.3596216 1.3410221 1000.0* 3412.1416 1.3410221 3600000.0* 3600.0* 3.6* 1000.0* 101.26859 1.6878099 1.852* 1.1507794

L liters to centiliters liters to cubic centimeters liters to cubic feet liters to cubic inches liters to cubic yards liters to deciliters liters to gallons, imperial liters to gallons, U.S liters to milliliters liters to ounces, fluid liters to pints liters to quarts liters to quarts liters, water, to kilograms liters, water, to pounds liters per kilowatt-hour to gallons per horsepower-hour liters per kilowatt-hour to quarts per horsepower-hour liters per kilowatt-hour to pints per horsepower-hour

100.0* 1000.0* 0.0353147 61.023744 0.0013079 10.0* 0.2199692 0.2641721 1000.0* 33.814023 2.1133764 0.879877 1.0566882 0.99994 2.2044903 12.607459 3.1518648 1.5759324

M megajoules to joules mega joules to kilojoules megajoules to kilowatt-hours megajoules to horscpowcr-hours megajoules per kilogram to Btus per pound meters to fathoms meters to feet meters to furlongs

1000000.0* 1000.0* 0.2777778 0.3725061 429.92261 0.5468067 3.2808399 0.004971

Appendix A: Conversion Factors

TO CONVERT FROM: MULTIPLY BY: meters to inches 39.370079 meters to kilometers 0.001 meters to mils 39370.079 meters to yards 1.0936133 meters per second to kilometers per hour 3.6* meters per second to feet per second 3.2808399 meters per second per second to centimeters per second per second 100.0 meters per second per second to feet p-er second per second 3.2808399 meters per second per second to g 0.1019716 microinches to micrometers 0.0254* micrometers to microinches 39.370079 micrometers to microns 1.0* microns to centimeters 0.0001* microns to inches 0.0000394 microns to microinches 39.370079 microns to micrometers 1.0* microns to millimeters 0.001* microns to mils 0.0393701 miles, nautical, to feet 6076.1155 miles, nautical, to kilometers 1.852* miles, nautical, to miles, statute 1.1597794 miles, nautical, to yards 2025.3718 miles, statute, to feet 5280.0* miles, statute, to furlongs 8.0* miles, statute, to kilometers 1.609344* miles, statute, to miles, nautical 0.8689762 miles, statute, to yards 1760.0* miles per gallon, imperial, to kilometers per liter 0.3540062 miles per gallon, imperial, to miles per gallon, U.S 0.8326742 miles per gallon, U.S., to kilometers per liter 0.4251437 miles per gallon, U.S., to miles per gallon, imperial 1.2009499 miles per hour to centimeters per second 44.704* miles per lour to feet per minute 88.0* miles per hour to feet per second 1.4666667 miles per hour to kilometers per hour 1.609344* miles per hour to knots 0.8689762 millibars to atmospheres 0.0009869 millibars to bars 0.001* millibars to inches of mercury 0.02953 millibars to inches of water 0.4014872 millibars to kilograms per square centimeter 0.0010197 millibars to kilopascals 0.1* millibars to millimeters of mercury 0.7500617 millibars to pascals 100.0* millibars to pounds per square foot 2.0885434 millibars to pounds per square inch 0.0145038 milligrams to grains 0.0154324 milligrams to grams 0.001* milli grams to ounces 0.0000353 milli iters to cubic centimeters 1.0* milli iters to liters 0.001* milli iters to ounces, fluid 0.033814

Appendix A: Conversion Factors

0 CONVERT FROM: millimeters to centimeters millimeters to inches millimeters to meters millimeters to microns millimeters to mils mils to centimeters mils to feet mils to inches mils to meters mils to microns mils to millimeters mils to yards minutes to degrees minutes to grades minutes to radians minutes to quadrants minutes to seconds

MULTIPLY BY: 0.1* 0.0393701 0.001* 1000.0* 39.370079 0.00254* 0.0000833 0.001* 0.0000254* 25.4* 0.0254’ 0.0002778 0.0166667 0.0185185 0.0002909 0.0001852 60.0*

N newtons to kilograms newtons to ounces newtons to pounds newtons per meter to newtons per millimeter newtons per meter to pounds per foot newtons per millimeter to newtons per meter newtons per millimeter to pounds per foot newtons per millimeter to pounds per inch newtons per square meter to pascals newton-meters to joules newton-meters to pounds-fcct

0.1019716 3.5969431 0.2248089 0.001* 0.0685218 1000.0* 68.52178 5.7101471 1.0* 1.0* 0.7375622

0 ounces to grains ounces to grams ounces to kilograms ounces to milligrams ounces to newtons ounces to pounds ounces, fluid, to cubic feet ounces, fluid, to cubic inches ounces, fluid, to gallons ounces, fluid, to milliliters ounces, fluid, to pints ounces, fluid, to quarts ounces-inches to joules or newton-meters ounces-inches to pounds-feet ounces-inches to pounds-inchcs

437.5* 28.35* 0.0283495 28349.523 0.2780139 0.0625* 0.0010444 1.8046875 0.0078125* 29.573529 0.0625* 0.03125* 0.0070616 0.0052083 0.0625*

P pascals to bars pascals to inches of mercury pascals to inches of water

0.00001* 0.0002953 0.0040149

Appendix A: Conversion Factors

TO CONVERT FROM: pascals to kilograms per square centimeter pascals to kilopascals pascals to millibars pascals to newtons per square meter pascals to pounds per square foot pints to cubic centimeters pints to cubic feet pints to cubic inches pints to gallons pints to liters pints to ounces, fluid pints to quarts pints per horsepower-hour to liters per kilowatt-hour pounds to grains pounds to grams pounds to kilograms pounds to newtons pounds to ounces pounds to tons, metric pounds, water, to cubic feet pounds, water, to cubic inches pounds, water, to cubic meters pounds, water, to gallons pounds, water, to liters pounds per cubic foot to grams per cubic centimeter pounds per cubic foot toot to pounds per cubic inch pounds per cubic foot to Kilograms per cubic meter pounds per cubic foot to pounds per gallon pounds per cubic inch to grams per cubic centimeter pounds per cubic inch to pounds per cubic foot pounds per cubic inch to kilograms per cubic meter pounds per cubic inch to pounds per gallon pounds per foot to kilograms per meter pounds per foot to newtons per meter pounds per foot to newtons per millimeter pounds per foot to pounds per inch pounds per gallon to grams per cubic centimeter pounds per gallon to pounds per cubic foot pounds per gallon to kilograms per cubic meter pounds per gallon to kilograms per liter pounds per gallon to pounds per cubic inch pounds per norsepower-hour to grams per kilowatt-hour pounds per horsepower-hour to grams per megajoule pounds per horsepower-hour to kilograms per kilowatt-hour pounds per inch to kilograms per meter pounds per inch to newtons per millimeter pounds per inch to ounces per inch pounds per inch to pounds per foot pounds per square foot to atmospheres pounds per square foot to bars pounds per square feet to inches of mercury pounds per square foot to inches of water

MULTIPLY BY: 0.0000102 1000.0* 0.01* 1.0* 208.85434 473.17647 0.0167101 28.875 0.125* 0.4731765 16.0* 0.5* 0.634545* 7000.0* 453.6 0.4535924 4.4482217 16.0* 0.0004536 0.0160194 27.681566 0.0004536 0.1198336 0.4536196 0.0160185 0.0005787 16.018464 0.1336806 27.679905 1728.0* 27679.905 231.0* 1.488164 14.5939 0.0145939 12.0 0.1198264 7.4805195 119.82643 0.1198264 0.004329 608.2774 168.9659 0.6082774 17.857968 0.1751268 16.0 0.0833333 0.0004725 0.0004788 0.014139 0.1922331

Appendix A: Conversion Factors

TO CONVERT FROM: pounds per square foot to kilograms per square centimeter pounds per square foot to kilograms per square meter pounds per square foot to kilopascals pounds per square foot to millibars pounds per square feet to pascals pounds per square foot to pounds per square inch pounds per square inch to atmospheres pounds per square inch to bars pounds per square inch to inches of mercury pounds per square inch to inches of water pounds per square inch to pounds per square foot pounds per square inch to kilograms per square centimeter pounds per square inch to kilopascals pounds per square inch to millibars pounds-feet to Btus pounds-feet to calorics pounds-feet to joules or newton-meters pounds-feet to kilogram-meters pounds-feet to ounccs-inchcs pounds-feet to pounds-inches pounds-feet to watt-hours pounds-feet per minute to horsepower pounds-inchcs to joules or newton-meters pounds-inches to ounces-inchcs pounds-inchcs to pounds-feet

MULTIPLY BY: 0.0004882 4.8824277 0.0478803 0.4788026 47.880259 0.006944 0.068046 0.0689476 2.0360207 27.681566 144.0* 0.070307 6.8947574 68.947574 0.0012851 0.3238316 1.3558179 0.138255 192.0* 12.0* 0.0003766 0.0000303 0.1129848 16.0* 0.0833333

Q

quadrants to circumferences quadrants to degrees quadrants to grades quadrants to minutes quadrants to radians quadratics to seconds quarts to cubic centimeters quarts to cubic feet quarts to cubic inches quarts to gallons quarts to liters quarts to ounces, fluid quarts to pints quarts per horsepower-hour to liters per kilowatt-hour

4.0* 90.0* 100.0* 5400.0* 1.5707963 3240000.0* 946.35295 0.0334201 57.75 0.25* 0.9463529 32.0* 2.0* 0.3172725

R radians radians radians radians radians

0.0159154 57.295779 63.661978 3437.7468 206264.81

to circumferences to degrees to grades to minutes to seconds

s seconds to degrees seconds to grades

0.0002778 0.0003086

Appendix A: Conversion Factors

TO CONVERT FROM: seconds to minutes seconds to radians square centimeters to square feet square centimeters to square inches square centimeters to square yards square feet to square centimeters square feet to square inches square feet to square meters square feet to square yards square inches to square centimeters square inches to square feet square inches to square meters square inches to square millimeters square inches to square yards square meters to square feet square meters to square inches square meters to square yards square yards to square feet square yards to square inches square yards to square meters

MULTIPLY BY: 0.0166667 0.0000049 0.0010764 0.1550003 0.0001196 929.0304 144.0* 0.092903 0.1111111 6.4516* 0.0069444 0.0006452 645.16* 0.0007716 10.76391 1550.0031 1.19599 9.0* 1296.0* 0.8361274

T tons, tons, tons, tons, tons, tons, tons, tons, tons, tons,

long, to kilograms long, to pounds long, to tons, short metric, to kilograms metric, to pounds metric, to tons, short short, to kilograms short, to pounds short, to tons, long short, to tons, metric

1016.0469 2240.0* 1.12* 1000.0* 2204.6226 1.1023113 907.18475 2000.0* 0.8928571 0.9071847

w watt-hours to Btus watt-hours to calorics watt-hours to joules watt-hours to kilogram-meters watt-hours to kilowatt-hours watt-hours to pounds-fcct

3.4121416 859.84523 3600.0* 367.09784 0.001* 2655.2237

Y yards yards yards yards yards yards yards

to centimeters to fathoms to feet to furlongs to inches to meters to mils

*Exact figure

91.44* 0.5* 3.0* 0.0045455 36.0* 0.9144* 36000.0*

APPENDIX B: BIBLIOGRAPHY Adler, U., editor-in-chief. Automotive Handbook, 2nd English edition. Stuttgart, Germany: Robert Bosch CimbH, 1986.

Carmichael, Robert D., and Edwin R. Smith. Mathematical Tables and Formulas. New York: Dover Publications, 1962.

Alston, Chris. Drag Race Chassis Tuning Manual. Sacramento, California: Alston Industries, 1985.

Chevrolet Power, 4th edition. Warren, Michigan: Chevrolet Motor Division, General Motors Corporation, 1980.

Anand, Dev. "Dyno Secrets," Car Craft Annual 1988. Los Angeles: Petersen Publishing Company, 1988, pp. 167-169.

Christy, John, editor. Supertuning. New York: New American Library, 1966.

---------- ."Street Machine Reference Guide," Car Craft, Vol. 38, No. 2, February 1990, pp. 67-68, 73-74.

Anderson, Edwin P. Gas Engine Manual, 2nd edition revised by Ted Pipe. Indianapolis: Theodore Audel and Company, 1977. Auth, Joanne Buhl. Deskbook ofMath Formulas and Tables. New York: Van Nostrand Reinhold Company, 1985.

Bell, A. Graham. Performance Tuning in Theory and Practice: Four Strokes. Somerset, England: Haynes Publishing Group, 1981.

Bird, J.O. Newnes. Engineering Science Pocket Rook. London: William Heinemann Limited, 1987. Bishop, Owen. Yardsticks of the Universe. New York: Peter Bedrick Books, 1982.

Chrysler Kit Car. Catalog SP11. Detroit: Chrysler Corporation, 1977. Corn, Juliana, and Tony Behr. Technical Mathematics through Applications. Philadelphia: Saunders College Publishing, 1982.

Csere, Csaba. "Torque of the'lown,'' Car and Driver, Vol. 36, No. 1, July 1990, pp. 24-25.

Emiliani, Cesare. The Scientific Companion: Exploring the Physical World with Facts, Figures, and Formulas. New York: John Wiley and Sons, 1988. Estes, Bill. The RVHandbook. Agoura, California: Trailer Life Books, 1991. ---------- ."Weigh Your RV," RVDo It Yourself. Calabasas, California: Trailer Life Books, 1975, pp. 26-29.

"Figure It Out!" Drag Racing USA, Vol. 6, No. 9, June 1970, pp. 45-47, 60-61, 70.

Blocksma, Mary. Reading the Numbers: A Survival Guide to the Measurements, Numbers and Sizes Encountered in Everyday Life. New York: Penguin Books, 1989.

Fitch, James W. Motor Truck Engineering Handbook, 3rd edition. Anacortes, Washington: Published by the author, 1984.

Brianza, David. Beginning Technical Mathematics Made Easy. Blue Ridge Summit, Pennsylvania: TAB Books, 1990.

Flammang, James. Understanding Automotive Specifications and Data. Blue Ridge Summit, Pennsylvania: TAB Books, 1986.

Campbell, Colin. The Sports Car: Its Design and Performance, 4th edition. Cambridge, Massachusscts: Robert Bentley, 1978.

Francisco, Don. "Math and Formulas for Hot Rodders," Hot Rod Magazine Yearbook Number One. Los Angeles: Petersen Publishing Company, 1961, pp. 35-39.

---------- . The Sports Car Engine: Its Tuning and Modification. Cambridge. Massachusetts: Robert Bentley, 1964.

Greenberg, Leon A. Alco-Calculator: An Educational Instrument. Piscataway, New Jersey: Rutgers University Center of Alcohol Studies, 1983.

Appendix B: Bibliography

Hale, Patrick. Drag Strip Dyno," The First 60 Feet: A Newsletterfor the Quarter Users Group, July 1987.

McFarland, Jim. The Great Manifold Bolt-On. El Segundo. California: Edclbrock Corporation, 1982.

---------- Quarter. Jr. Computer software. Phoenix, Arizona: Racing Systems Analysis, 1987. Apple, Commodore and IBM disks.

---------- ."Hot Rod Shop Series: Basic Mathematics for the Car Enthusiast," Hot Rod Magazine, Vol. 33, No. 1, January 1980, pp. 54-59.

High Performance Engines. Dearborn, Michigan: Ford Motor Company, 1969.

Hills, Herbert Impco Carburetion. Cerritos, California: Impco Carburetion, n.d. Hudson, Ralph G., S.B. The Engineers' Manual, 2nd edition. New York: John Wiley and Sons, 1979. Huntington, Roger. American Supercar. HPBooks, 1983. ---------- ."True Power," Car Life, Vol. 17, No. 4, May 1970, pp. 10-13.

Jarman, Trant. "Lets Torque," Automobile, Vol. 2, No. 3, June 1987, pp. 40-41. Jennings, Gordon. Two-Stroke Tuner's Handbook. HPBooks, 1973.

Jute, .Andre. Designing and Building Special Cars. London: B.T. Batsford, 1985. Klein, Herbert Arthur. The Science of Measurement: A Historical Survey. New York: Dover Publications, 1988.

Landis, Bob. "Racer Arithmetic," The Pacers Complete Reference Guide. Santa Ana, California: Steve Smith Autosports, 1976, pp. 196-199. Losee, Jim. "Guide to Formulas and Conversions," Car Craft, Vol. 37, No. 12, December 1989, pp. 52-55. Ludlam, F.W. The Elementary Theory ofthe Internal Combustion Engine, 3rd edition. London and Glasgow: Blackie and Sons, 1947.

Martin, Mike. Mopar Suspensions. Brea, California: S-A Design Books, 1984.

---------- ."Power Theory," Engines, Hot Rod High Performance Series, Vol. 4, No. 1, 1987, pp. 24-26. ---------- ."Street Machine Math," Car Craft Yearbook, Los Angeles: Petersen Publishing Company, 1987, pp. 87-89. Metric Conversion Tables. Woodbury, New York: Barrens Educational Series, 1976. Moore, Claude S., Bernie L. Griffin, and Edwar d C. Polhamus Jr. Applied Math for Technicians, 2nd edition. Englewood Cliffs, New Jersey: Prentice-Hall, 1982.

Newton, K., W. Steeds and T.K. Garrett. The Motor Vehicle, 11th edition. London: Butterworths, 1989. NHRA Drag Rules. Glendora, California: National Hot Rod Association, published annually.

Oddo, Frank. "Thumbs Up on Engine Life," Popular Cars, Vol. 8, No. 6, June 1986, p. 66. Patterson, G.A. Engine Thermodynamics with a Pocket Calculator, 2nd edition. Palos Verdes Estates, California: Basic Science Press, 1983.

Pitt, Jerry. "Time for Torque," Car Craft, Vol. 36, No. 11, November 1988, pp. 34-37. Puhn, Fred. How to Make Your Car Handle. Los Angeles, California: Price Stern Sloan/HPBooks, 1976. Richmond, Doug. Metrics for Mechanics. Berkeley, California: Dos Reales Publishing, 1974.

Appendix B: Bibliography

Roc, Doug. Rochester Carburetors, revised edition. Los Angeles, California: Price Stern Sloan/HPBooks, 1986.

Rogowski, Stephen J. Computers for Sea and Sky. Morristown, New Jersey: Creative Computing Press, 1982. SAF. Handbook, 4 vols. Warrendale, Pennsylvania: Society of Automotive Engineers, published annually. Schimizzi, Ned V. Mastering the Metric System. New York: New American Library, 1975.

Schofield, Miles. Basic Engine Math," Basic Engine Hot Rodding. Los Angeles: Petersen Publishing Company, 1972, pp. 14-17. Shepard, Larry S. How to Hot Rod Small-Block Mopar Engines. New York: HPBooks, 2003.

---------- .Mopar Chassis Speed Secrets. Farmington Hills, Michigan: Chrysler Corporation Direct Connection, 1984. ---------- .Mopar Oval Track Modifications. Farmington Hills, Michigan: Chrysler Corporation Direct Connection, 1983.

Smith, Philip H. The Design and Tuning of Competition Engines. 6th edition revised bv David N. Wenner. Cambridge, Massachusetts: Robert Bentley, 1977.

Smith, Steve. Advanced Race Car Suspension Development, revised edition. Santa Ana. California: Steve Smith Autosports, 1975. Storer, Jay. "What Is Horsepower?" Hot Rod Yearbook Number 13.1-os Angeles: Petersen Publishing Company, 1973, pp. 114-117.

Taborek, Jaroslav J. Mechanics of Vehicles. Cleveland, Ohio: Pen ton Publishing Company, 1957.

Taxel, I. Conversion Factors with Metric Calculator. Woodmere, New York: Published by the author, 1964. Zmr Inflation and Substitution/Performance Formulas. Akron, Ohio: BFGoodrich, 1986. Titus, Rick. "The Pursuit of Control," Hot Rod 1986 Annual. Los Angeles: Petersen Publishing Company, 1985, pp. 146-151. Urich, Mike, and Bill Fisher. Holley Carburetors, Manifolds & Fuel Injection. New York/New York: Penguin Group (USA)/HPBooks, 1994.

---------- . Tech Tips, ” Mopar Performance News, "Finding the Head CCs," Vol. 8, No. 3, April 1987, p. 8; "CC’ing the Block," Vol. 8, No. 4, May 1987, pp. 8-9; ".500" Down Fill Volumes,' Vol. 8, No. 5, June 1987, p. 10; "Compression Ratio," Vol. 8, No. 6, July 1987, pp. 14-15; "Gearing," Vol. 10, No. 9, October 1989, p. 11.

Van Valkcnburgh, Paul. Race Car Engineering and Mechanics, 2nd edition. Seal Beach, California: Published by rhe author, 1986.

Sherman, Don. "Pfederstarke and Other Horsepower Secrets Revealed, " Car and Driver, Vol. 30, No. 12, June 1985, pp. 26-27.

Von Hclmolt, Ken. "Differential Equations," 4x4 Answer Book. Canoga Park, California: Four Wheeler Publishing, 1987, pp. 69-71.

Simanaitis, Dennis. 'Technical Tidbits, " Road & Track, Vol. 38, No. 2, October 1986, pp. 148-149.

Wallace, Dave. "Chassis Tune-Up," Hot Rod 1986Annual. Los Angeles: Petersen Publishing Company, 1985, pp. 88-98.

---------- . "Technical Correspondence," Road & Track, "Lets Ask the Professor," Vol. 37, No. 10, June 1986, pp. 206-207; "Engine Output and Altitude," Vol. 39, No. 9, May 1988, pp. 166-168; "Calculating RPM and Speed," Vol. 39, No. 10, June 1988, pp. 158-160, 162; "Predicting Horsepower," Vol. 42, No. 2, October 1990, pp. 170-171. Smith, Carroll. Tune to Win. Osceola, Wisconsin: Classic Motorbooks, 1978.

Wilson, Waddell, and Steve Smith. Racing Engine Preparation. Santa Ana, California: Steve Smith Autosports, 1975. Winchell, Frank. "A Lesson in Basic Vehicular Physics," Automobile Magazine, Vol. 2, No. 9, December 1987, pp. 45-50.

INDEX Aerodynamics frontal area and, 55-56 handling basics and, 57 horsepower requirements for, 54-55 pitch couple in, 58 ram air pressure and, 58 roll, pitch, yaw in, 56-58 Air capacity carburetor size and, 32-34 formulas for, 31 volumetric efficiency and, 31-34 Amount to mill formula, 15 Atmospheric correction factors, 25-26 Average miles per gallon, 84-87, 93 B Bore, 3-6 formula for, 7 Brake horsepower and torque, 21-22 formulas for, 25 Brake mean effective pressure, 29 Brakes, rolling resistance and, 60 Brake specific fuel consumption, 22, 25, 29 c Carburetor, 32-34 cc-ing, 10-11 Center of gravity, 38 added weight and, 44 calculating ground level wheelbase for, 44 formulas for, 44 fuel tank foil or empty for, 43 height of, 43, 44 horizontal position and, 41 lengthways location of, 41 measuring ground level wheelbase for, 43-44 necessary dimensions for, 43 sideways location of, 41 suspension and tire deflection and, 42-43 vertical position and, 42-44 weights and measures and, 42 Centrifugal force, 48, 57 cfm. See Cubic feet per minute Chamber capacity measurement, 9 Combustion chamber area estimation of, 12-13 milled heads and, 12 planimeters for, 13-14 volume calculation of, 12 volume measurement of, 9-11 Compressed volume formula, 15 Compression ignition, 9

Compression ratio, 8-9, 15 Computer programs, 69, 72 handheld computers and, 70 port areas and, 70-71 quarter-milc c.t. and mph and, 65, 66 Conferences, 72 Crankshaft balancing, 100 heavy metal in, 98-99 lever arm in, 97-98 >arts in motion in, 94-97, 95, 96 Cu ■)ic centimeters, 8 Cubic feet per minute (cfm), 31, 34 Cy inder, 3-4, 7, 15

D Density, 23 Displacement, 1, 2, 3-4, 7 metric, 5-6 ratio, 13, 15 Downshifts, 62-63 Driveshaft torque, 61-62, 63 Drive wheel torque, 45-46, 49 Dynamometer, 21, 22 Dyno chart, 23-24

E Elapsed time (c.t.), 65-68 Elevation, 24-25 c.t. See Elapsed time F Flywheel comparison, moment of inertia and, '51-53 Formula(s), 67 for air capacity, 31 for amount to mill, 15 for average miles per gallon, 87, 93 for average mph, 87 for bore, 7 for brake horsepower and torque, 25 for brake horsepower loss, 25 for brake specific fuel consumption, 25 for center of gravity, 44 for centrifugal force, 48 for compressed volume, 15 for compression ratio, 15 for crankshaft balancing, 99 for cylinder volume, 7, 15 for displacement, 7 for displacement ratio, 15 for gear ratio, 79 for g force and weight transfer, 49 for indicated horsepower and torque, 30

Index

for inertia, 50-53 for instrument error, 76 for lateral acceleration, 49 for lateral weight transfer, 49 for mean effective pressure, 30 for mechanical efficiency, 30 for moment of inertia, 53 for mph, 68, 79, 83, 87 for overall gear ratio, 68 for piston speed, 19 for quarter-mile e.t. and mph, 68 for racing carb cfm, 34 for revolutions per minute, 19, 79 for shift points, 64 for street carb cfm, 34 for stroke, 7 for tire diameter, 79 for tire sizes and their effects, 83 for torque, 25 for volumetric efficiency, 34 for weight distribution, 38 Frontal area, aerodynamics and, 55-56 Front/rear distribution, 36-37 Fuel, 22, 25, 29, 43 check, 23 economy, 88-93 G Gear ratio, 79 overall, 66-68, 78 g force, 45^6, 49 g force and weight transfer, 45-46 centrifugal force in, 48 formulas for, 49 lateral acceleration in, 47-49 lateral weight transfer in, 48, 49 Ground level wheelbase, 43-44 H Headers, dyno-testing, 24 Honing, 4-5 Horsepower, 20-21, 25, 27-30, 54-55 Hybrids, 88, 89, 90 I Indicated horsepower and torque, 27-30 Indicated mean effective pressure, 27 Indicated torque, 28-29 Indicator, 27 Inertia, 52, 53, 90 formulas for, 50-53 Instrument error formulas for, 76 inconsistencies in, 75

speedometer error, 73-74, 75, 76, 82 testing for, 73-76 L Lateral acceleration, 47, 49 calculating, 47-48 Left/right distribution, 37-38

M Mean effective pressure, 27-30 Mechanical efficiency, 29-30 Micrometer, 4 Miles per gallon, 84 average, formulas for, 87, 93 Miles per hour (mph), 65-67 average, 84-87 formulas for, 68, 79, 83, 87 indicated and actual, 82-83 moment of inertia and, 50, 53 Milled heads, 12 Moment of inertia flywheel comparison in, 51-53 mph and, 50, 53 speed and, 51 tire diameter and, 50-51, 53 Moving time vs. travel time, 85 mph. See Miles per hour N Numbers, 1, 2 rounding off, 3, 6

0 Odometer error, 73-74, 75, 76 Overall gear ratio, 66-68, 78 Oversteer, 37 P pi, 1,7 Piston dome measurement, 12 Pistons, 100 Piston speed average, 16-18 problems from, 16 rev limits and, 18 small-block V-8, 18-19 Pitch, 56-57 couple, 58 PLAN, horsepower and, 27-28 Planimeters, 13-14 Public scale, 35

Index

Quarter-mile e.t. and mph, 68 computer programs and, 65, 66 gearing for speed in, 66-67 R Raceway lap times, average speeds and, 85-86, 87 Racing carb, 33 cfm, formula for, 34 Rake angle, 60 Ram air pressure, 58 Revolutions per minute, 19, 77-79 Rod strength, 18 Roll, 56-58 Rolling resistance, 90-91 brakes and, 60 coast down and, 59 rake angle and, 60 tire pressure and, 59 weight and, 60 wheel alignment and, 60 Roll-out method, for finding tire diameter, 51

S Shift points driveshaft torque and, 61-62 formulas for, 64 ideal, 62-63 Snap gauge, 4 Spark ignition, 9 Specific gravity, 22-23 Speed, 51. See also Piston speed average, raceway lap times and, 85-86, 87 centrifugal force and, 57 Speedometer error, 73-74, 75, 76, 82 Street carb cfm, formula for, 34 size of, 32-33, 34 Stroke, 1,2 checker, 3 formula for, 7 Suspension and tire deflection, 42-43 Symbols, 2

T Temperature, 25-26 Theoretical cfm, 31, 34 Tire diameter, 50, 53, 78-79 downsize, 82-83 effective drive ratio and, 81, 83 equivalent drive ratio and, 81, 83 metric, 81 roll-out method for finding, 51 section height and width in, 80-81 Torque, 21—22, 25 driveshaft, 61-62, 63 drive wheel, 45-46, 49 indicated, 28-29 indicated horsepower and, 27-30 mechanical efficiency from, 29 frailer tongue weight, 39—40 U Understeer, weight distribution and, 37

V Volumetric efficiency, 31-34 W Weight, 42, 44, 50, 60, 68, 90. See also g force and weight transfer -cartying hitch, 39 -distributing hitch, 39-40 from e.t., 65-66 Weight distribution, 41 adding weight at either end in, 38-39 center of gravity of add-ons and, 38 front/rcar distribution, 36-37 left/right distribution, 37-38 trailer tongue weight and, 39-40 understeer and oversteer and, 37 Weight of vehicle portable car scales for, 35, 38 problems measuring, 35-36 public scale for, 35 truck scales for, 35, 36 wheel weights in pairs and, 35

ABOUT THE AUTHORS John Lawlor John I-awlor was an automobile enthusiast since boyhood. He started writing about cars while still in college, with a weekly motoring column in the campus newspaper at Loyola University in Los Angeles, California. He became a professional journalist in the late 1950s, when he went to work for Petersen Publishing Co., where he became a senior editor of Motor Trend and later, an editor in the firms book division. During the 1960s, he moved to Bond Publishing Company, where he was an assistant editor of Car Life and a contributor to the parent magazine, Road & Track. He has also served as the managing editor of Popular Hot Rodding and Speed Age magazines. In 1967, he served as public relations director for the

Bill Hancock Bill Flancock was chosen to write the revision of John’s original work. As a mechanical engineer, Bill worked for Chrysler Corporation in their 70s NASCAR program for ten years and then left to start and run his own company, Arrow Racing Engines, Inc, which specializes in high performance engine development in the Detroit area. After selling the business in 2008, Bill currently writes and does consulting work for the performance industry. This revision incorporates several new chapters on emerging areas of interest in the performance calculations arena.

Inaugural Mexican 1000, the first off-road race down Mexico’s Baja California peninsula. His efforts resulted in more event coverage by motor enthusiasts’ magazines than any previous motorsport event of any kind. As a result, in 1979 Lawlor became the first journalist or publicist elected to the Off-Road Hall of Fame, sponsored by the Specialty Equipment Market Association (SEMA). In 1989, he was similarly honored as one of the first ten inductees into the Dune Buggies and Hot VWs Magazine Hall of Fame. He was the author of three additional books: Hou> to Talk Car, a dictionary of automotive slang; Inside Full-Time FourWheel Drive, a guide to the New Process 203 system published by Chrysler in the mid-70s; and HPBooks Auto Dictionary, with John Edwards.

HPBooks GENERAL MOTORS Big-Block Chevy Engine Buildups: 978-1-55788-484-8/HP1484 Big-Block Chevy Performance: 978-1-55788-216-5/HP1216 Building the Chevy LS Engine: 978-1-55788-559-3/HP1559 Camaro Performance Handbook: 978-1-55788-057-4/HP1057 Camaro Restoration Handbook ('61-'81): 978-0-89586-375-1/HP758 Chevy LS Engine Buildups: 978-1-55788-567-8/HP1567 Chevy LS Engine Conversion Handbook: 978-1-55788-566-1/HP1566 Chevy LS1/LS6 Performance: 978-1-55788-407-7/HP1407 Classic Camaro Restoration, Repair & Upgrades: 978-1-55788-564-7/HP1564 The Classic Chevy Truck Handbook: 978-1-55788-534-0/HP1534 How to Rebuild Big-Block Chevy Engines: 978-0-89586-175-7/HP755 How to Rebuild Big-Block Chevy Engines, 1991-2000: 978-1-55788-550-0/HP1550 How to Rebuild Small-Block Chevy LT-1/LT-4 Engines: 978-1-55788-393-3/HP1393 How to Rebuild Your Small-Block Chevy: 978-1-55788-029-1/HP1029 Powerglide Transmission Handbook: 978-1-55788-355-1/HP1355 Small-Block Chevy Engine Buildups: 978-1-55788-400-8/HP1400 Turbo Hydra-Matic 350 Handbook: 978-0-89586-051-4/HP511

FORD Classic Mustang Restoration, Repair & Upgrades: 978-1-55788-537-1/HP1537 Ford Engine Buildups: 978-1-55788-531-9/HP1531 Ford Windsor Small-Block Performance: 978-1-55788-558-6/HP1558 How to Build Small-Block Ford Racing Engines: 978-1-55788-536-2/HP1536 How to Rebuild Big-Block Ford Engines: 978-0-89586-070-5/HP708 How to Rebuild Ford V-8 Engines: 978-0-89586-036-1/HP36 How to Rebuild Small-Block Ford Engines: 978-0-912656-89-2/HP89 Mustang Restoration Handbook: 978-0-89586-402-4/HP029

MOPAR Big-Block Mopar Performance: 978-1-55788-302-5/HP1302 How to Hot Rod Small-Block Mopar Engine, Revised: 978-1-55788-405-3/HP1405 How to Modify Your Jeep Chassis and Suspension For Off-Road: 978-1-55788-424-4/HP1424 How to Modify Your Mopar Magnum V8: 978-1-55788-473-2/HP1473 How to Rebuild and Modify Chrysler 426 Hemi Engines: 978-1-55788-525-8/HP1525 How to Rebuild Big-Block Mopar Engines: 978-1-55788-190-8/HP1190 How to Rebuild Small-Block Mopar Engines: 978-0-89586-128-5/HP83 How to Rebuild Your Mopar Magnum V8: 978-1-55788-431-5/HP1431 The Mopar Six-Pack Engine Handbook: 978-1-55788-528-9/HP1528 Torqueflite A-727 Transmission Handbook: 978-1-55788-399-5/HP1399

IMPORTS Baja Bugs & Buggies: 978-0-89586-186-3/HP60 Honda/Acura Engine Performance: 978-1-55788-384-1/HP1384 How to Build Performance Nissan Sport Compacts, 1991-2006: 978-1 -55788-541-8/HP1541

How to Hot Rod VW Engines: 978-0-91265-603-8/HP034 How to Rebuild Your VW Air-Cooled Engine: 978-0-89586-225-9/HP1225 Porsche 911 Performance: 978-1-55788-489-3/HP1489 Street Rotary: 978-1-55788-549-4/HP1549 Toyota MR2 Performance: 978-155788-553-1/HP1553 Xtreme Honda B-Series Engines: 978-1-55788-552-4/HP1552

HANDBOOKS Auto Electrical Handbook: 978-0-89586-238-9/HP387 Auto Math Handbook: 978-1-55788-554-8/HP1554 Auto Upholstery & Interiors: 978-1-55788-265-3/HP1265 Custom Auto Wiring & Electrical: 978-1-55788-545-6/HP1545 Electric Vehicle Conversion Handbook: 978-1-55788-568-5/HP1568 Engine Builder's Handbook: 978-1-55788-245-5/HP1245 Fiberglass & Other Composite Materials: 978-1 -55788-4985/HP1498 High Performance Fasteners & Plumbing: 978-1-55788-5234/HP1523 Metal Fabricator's Handbook: 978-0-89586-870-1/HP709 Paint & Body Handbook: 978-1-55788-082-6/HP1082 Plasma Cutting Handbook: 978-1-55788-569-2/HP1569 Practical Auto & Truck Restoration: 978-155788-547-0/HP1547 Pro Paint & Body: 978-1-55788-563-0/HP1563 Sheet Metal Handbook: 978-0-89586-757-5/HP575 Welder's Handbook, Revised: 978-1-55788-513-5

INDUCTION Engine Airflow: 978-155788-537-1/HP1537 Holley 4150 & 4160 Carburetor Handbook: 978-0-89586-0477/HP473 Holley Carbs, Manifolds & F.I.: 978-1-55788-052-9/HP1052 Rebuild & Powertune Carter/Edelbrock Carburetors: 978-155788-555-5/HP1555 Rochester Carburetors: 978-0-89586-301-0/HP014 Performance Fuel Injection Systems: 978-1-55788-557-9/HP1557 Turbochargers: 978-0-89586-135-1 /HP49 Street Turbocharging: 978-1-55788-488-6/HP1488 Weber Carburetors: 978-0-89589-377-5/HP774

RACING & CHASSIS Advanced Race Car Chassis Technology: 978-1-55788-562-3/HP562 Chassis Engineering: 978-1-55788-055-0/HP1055 How to Make Your Car Handle: 978-1-91265-646-5/HP46 How to Build a Winning Drag Race Chassis & Suspension: 978-155788-462-6/HP1462 The Race Car Chassis: 978-1-55788-540-1/HP1540 The Racing Engine Builder's Handbook: 978-1-55788-492-3/HP1492

STREET RODS Street Rodder magazine's Chassis & Suspension Handbook: 9781-55788-346-9/HP1346 Street Rodder's Handbook, Revised: 978-1-55788-409-1/HP1409

ORDER YOUR COPY TODAY! All books are available from online bookstores (www.amazon.com and www.barnesandnoble.com) and auto parts stores (www.summitracing.com or www.jegs.com). Or order direct from HPBooks at www.penguin.com/hpauto. Many titles are available in downloadable eBook formats.