Table Of ContentAUTO MATH
HANDBOOK
Easy Calculations for Engine Builders, Auto Engineers,
Racers, Students, and Performance Enthusiasts
Revised Edition
John Lawlor & Bill Hancock
HPBooks
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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
10
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 Iv
introduction V
Chapter 1: Displacement, Stroke and Bore 1
Chapter 2: Compression Ratio 8
Chapter 3: Piston Speed 16
Chapter 4: Brake Horsepower and Torque 20
Chapter 5: Indicated Horsepower and Torque 27
Chapter 6: Air Capacity and Volumetric Efficiency 31
Chapter 7: Weight Distribution 35
Chapter 8: Center of Gravity 41
Chapter 9: g Force and Weight Transfer 45
Chapter 10: Moment of Inertia 50
Chapter 11: Aerodynamics 54
Chapter 12: Rolling Resistance 59
Chapter 13: Shift Points 61
Chapter 14: Quarter-Mile E.T. and MPH 65
Chapter 15: Computer Programs 69
Chapter 16: Instrument Error and Calibration 73
Chapter 17: MPH, RPM, Gears and Tires 77
Chapter 18: Tire Sizes and Their Effects 80
Chapter 19: Average MPH and MPG 84
Chapter 20: Fuel Economy and Cost of Ownership 88
Chapter 21: Crankshaft Balancing 94
APPENDICES 101
Conversion Factors 102
Bibliography 115
Index 118
ACKNOWLEDGMENTS
When I sat down to process the words that became the Friends Richard Shedenhelm and James J. Scanlan,
Auro Math Handbook, 1 had the support and guidance of M.D., read portions of the manuscript and offered
several valued friends and fellow editors, writers and excellent criticisms.
photographers. Gail Harrington, former managing editor of
Past editors Duane Elliot of Off-Road, Jim McGowan of MotorHome Magazine, was an unfailing source of personal
Guide to Muscle Cars and Muscle Car-Classics, Spence encouragement and enthusiasm during the many months
Murray of Mini-Truck, Mike Parris of Off-Road, Ralph I had to spend my evenings and weekends working on the
Poole of Trailer Boats, Lcroi “Tex” Smith of Hot Rod book.
Mechanix and the late Tom Senter of Popular Hot Rodding Michael Lutfy of HPBooks had the unenviable task of
encouraged me to write about automotive mathematics editing the work of a fellow editor, and a highly
and accepted articles I did on the subject from 1971 opinionated one at that!
through 1990. Those articles were really the start of this Finally, there's Tom Madigan, a former editor of Off-
book. Road and author of several books himself. He has given
John Baker and the late Dick Cepek spurred my interest me his personal support, not only for this book but at
further by asking me to produce brief mathematical pieces several crucial points in my career. In gratitude, it is to
for the catalogs issued by their respective firms, John Tom and his wife, Darlene, that Auto Math Handbook is
Baker Performance Products of Webster, Wisconsin, and dedicated.
Dick Cepek Inc. ofCarson, California. Thank you one and all. Without your help, the job of
Former Petersen Publishing Co. librarian Jane Barrett writing the book would have been much more difficult,
and former editors and writers Dean Batchelor, John and it would not have turned out as well as it did.
Dinkcl, Al Hall, Jon Jay and Jim Ixjscc all suggested
useful material for the book. —John I .aw lor
iv
INTRODUCTION
If you 're seriously interested in automobiles and how automotive engineering, and can't be ignored in a book
the)7 perform, sooner or later you'll have to deal with devoted to automotive mathematics.
mathematics. Virtually all aspects of motorsports, from As a result, the work serves as a useful primer of auto
bore and stroke, through power and torque, to rime and engineering and performance fundamentals, as well as a
speed, involve mathematical calculations. handbook of auto math. Hopefully it will have particular
I recognized this as a young auto enthusiast in the appeal to younger enthusiasts who are just developing an
1950s, and was pleased when I discovered a booklet called interest in the technology’ of auto performance.
Mechanics of Vehicles by Jaroslav J. Taborek. It was a Many of the formulas presented could be worked on a
collection of 14 articles about the mathematics of motor simple four-function arithmetic calculator. However,
vehicle behavior, originally published by Machine Design some of them will be much easier on an inexpensive
magazine in 1957. Taborek was a professional engineer scientific calculator, with pi and parentheses keys, and a
and he wrote for his colleagues, not for enthusiasts. Much few require a calculator which can find eidier square or
of his work involved more complex mathematics and for cube roots.
many of us was simply too complex. The problem examples in the text were worked our to
Then in 1961, an article called "Madi and Formulas for eight digits, because that is the capacity of most
Hot Rodders" by Don Francisco appeared in Hot Rod inexpensive calculators. However, the solutions were
Magazine Number One. It was only five pages generally’ rounded off to no more than three decimal
long, but it provided some genuinely usefid mathematics places and sometimes to none at all, depending on the
for the aspiring hot rodder, and none of it required more degree of precision that seemed appropriate in each case.
than a grade school background in math. To the best of In the text, single-digit whole numbers or integers arc
my knowledge, it was rhe first such compilation especially followed by a decimal point and a zero, c.g., five is 5-0;
for car enthusiasts. numbers less than one—or 1.0—have a zero preceding
In the years leading up to the publication of the first the decimal point, e.g., five-tenths is 0.5.
edition of Auto Math Handbook, there had been To enhance rhe value of the book as a reference and
numerous magazine articles and book chapters dealing make it simpler to look up a specific formula, each
with various aspects of auto math, but they all had been at chapter concludes with a table summarizing the formulas
one or the other of the extremes represented by Taborek's it has covered. Further, most of the formulas arc written
and Francisco's pioneering and long out-of-print efforts. in plain English or easily’ recognized abbreviations such as
They’ve been either ponderous professional tomes or rpm and mph, rather than in algebraic symbols, to make
frankly sketchy popular works. There had been no book them as clear as possible to the non-mathematician.
length collection of practical, elementary math for auto A working engineer who happens to sec this volume
enthusiasts of average education. That's a gap I sought to may criticize the limited attention given to the metric
fill with this work by concentrating on math of genuine system of measurement—or ro identify it more properly,
interest to the enthusiast, and avoiding anything too the Systcmc International des Unites, or S.I. for short.
specialized. That dictated a particular emphasis on the The professional may work with such S.I. units as the
engine and drivetrain, which arc the core of true hot merer for length, kilogram for weight and watt for power.
However, the kind of enthusiast I had in mind when I
Although there's plenty of arithmetic, algebra and even wrote this book continues to measure in traditional feet,
a little geometry, there's no calculus, so you can relax. pounds and horsepower, not in their metric or S.I.
However, I did include the formulas for horsepower and equivalents.
torque, despite the fact that their measurement requires
equipment not likely to be found in the average hot —John Lawlor (1991)
rodder's garage. The interrelationships of horsepower and
torque arc among the most important principles of
v
INTRODUCTION
I was given the difficult task of trying to improve upon Perhaps the biggest recent step forward has occurred in
a bestselling classic by bringing it up to modern standards the area of computers. Personal computers and the
while holding true to rhe original intent of a basic specialized software that is readily available for
beginner’s math handbook for auto enthusiasts. While die automotive applications have greatly expanded within the
format has not changed, I chose to include formulas and last five years. Heretofore, mathematical calculations
equations from additional areas of interest. In this involving complex formulas and iterative calculations
version, I have included discussions and formulas for have been outside the range of all but the most
vehicle aerodynamics and inertia. accomplished engineers and mathematicians. Now a
Some may want more advanced formulas but the intent serious enthusiast can access cutting-edge solutions very
was to keep the formulas simple and expose the beginner easily. The problem now becomes one of ensuring that
to the math necessary to build a basic foundation for the equation fits the particular solution rather than trying
vehicle performance. By using simple formulas, the reader to manipulate a lengthy formula or struggle with the
will have a much better way to evaluate vehicle advanced mathematics involved.
performance and understand or predict the results from As technology advances, so do the tools that arc
potential modifications. available. In many cases, today’s serious auto enthusiast
When this book was originally written, dynamometer has access to more computing power on their cell phone
testing was just beginning to become available to the or PDA than did the engineers who put a man on the
amateur enthusiast to rent and use. Today there are both moon forty years ago. By applying die correct formulas
engine and chassis dynamometers readily available and letting the computer do the mind-numbing repetitive
throughout rhe country. Aerodynamic testing has now calculations, we can solve complex problems in the blink
become available to the serious amateur and hill-scale of an eye, or at least in the time it takes to drink a cup of
wind tunnels are available for rent in selected areas of the coffee.
country. For an enthusiast who wants to maximize the Have fun learning and using auto math.
potential of their vehicle, these tools represent a huge step
forward in their development plan. —Bill Hancock (2011)
vi
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
oRn
is the sum of the individual swept cylinder volumes. Think of pi
the swept cylinder as a volume about the size of a coffee can, What is pi? Well, for openers, it is nor a dessert served
with the diameter being the size of your bore and the height of with ice cream. Pie is the pronunciation for the Greek
the can being the stroke of your engine. However, this swept letter, which in mathematics is called a constant. The rules
volume is not the total volume of the cylinder. The total of geometry tell us that the relationship between the
volume of a cylinder includes not only the swept volume but diameter or distance across the widest parr of a circle and
also the compressed volume above the piston when the piston is the circumference or the distance around the perimeter of
at top dead center. The compressed volume is comprised of the circle is constant. So, no matter how big or how small
the combustion chamber volume plus the volume between the die circle may be, the value of pi is the same for all circles.
top of the piston, the top or deck surface of the block, and The value of pi is 3.1415927. It is a dimensionless number,
plus the volume of the head gasket. which means that there is no description such as feet or
centimeters diat follows it. For our purposes, we can use
Piston Displacement 3.1416 and achieve sufficient accuracy for the calculations
To find an engine’s overall swept volume, or piston in this book.
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 the rules for a specific class, it is vitally important not to
cylinder from top co bottom and the bore is the diameter of exceed the displacement limits for your particular class.
the cylinder. More importantly, if you're building a racing engine to
If you have the factor}' figures for the bore and stroke, you compete in a class with a specific displacement limit and you
would probably have the displacement too. So why do you want to increase or decrease the displacement by modifying
need a formula for finding it? One reason is that die the stroke or bore, you need to know how to calculate die
displacement given in the factory-supplied spec sheets is effects of particular modifications. If you change the stroke or
usually rounded up or down to a whole number or integer by bore, how much will the displacement change?
die factory. An integer is a whole number which can be either To find the overall engine displacement, you must first find
positive or negative but docs not have any additional fractions the swept volume of a single cylinder, and then multiply that
such as 1 1/2 or additional decimal components such as figure by the number of cylinders. For that, you need to know
1.562. When it comes to precise engine building, it can be the formula for die volume of a cylinder as a geometric shape,
instructive to double-check the stock specs and find the exact which is pi + 4 or 3.1415927 + 4, which equals 0.7853982.
figure. If you happen to be building an engine to comply with This number is then multiplied by the square of the diameter
Auto Math Handbook
SYMBOLS and by the height of the cylinder. In an engine, the
We all know what + plus and - minus mean. When we sec / it diameter of a cylinder is tne bore, and the height is
means divided by. When we see numbers followed by a / that also the stroke, so the formula for finding the cylinder
means divided by. The square of a number is when a number is volume in cubic inches is pi divided bv 4, multiplied
multiplied times itself and is denoted by a small figure 2 above and by the bore squared in inches multiplied by the
to the right of the number or symbol, such as x2. So if D was rhe stroke in inches or:
symbol for diameter, D2 would denote diameter squared or
multiplied times itself. The square root of a number is a value, which Cylinder Volume ■ pi + 4 x bore2 x stroke
if multiplied by itself would equal die original number, llic symbol To find the overall piston displacement of the
for square root is V The symbol for the cube root of a number, or a entire engine, you'd take the value of this equation
value diat if multiplied by itself and then again by the value would and multiply that by the number of cylinders in the
equal the original number is 3\ . As we progress, we will define engine.
certain values by symbols. Learn what these arc. Not only will they Example—As an example, let's sec what the
be needed to solve the problems, but they will be useful in displacement would be for an 8-cylinder engine with
conversation. Supposed you heard two people discussing a new car a 4.0" bore and 3.48" stroke:
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 Displacement = 0.7853982 x 42 x 3.48 x 8
dimensionless number that puts a value on die overall aerodynamic The answer is 349.84776 cubic inches, which can
shape efficiency, 'rhe old Dodge Daytona and Plymouth Superbird be rounded up to an even 350, because the decimal
winged cars had a Cd of 0.27, while a cement truck probably has a figure is well over 0.5. Those happen to be the
Cd of 25 (just kidding—but the value is large enougn that nobody measurements of a common version of the Chevrolet
wants to measure it). 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
cubic inches or, rounded up, 307. This, too, is a
Inch variation of the Chevy small-block V-8.
Displacement 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
The swept volume in cubic inches of an individual cylinder is found by through 1966, Ford built a V-8 engine with a bore of
multiplying pi 14 by the bore in inches squared by the stroke in Inches. The
4.0" and a stroke of 3.5", and advertised its
overall swept volume or displacement of an engine is simply the volume of one
cylinder multiplied by the total number of cylinders. displacement as 352 cubic inches. Then, in 1969, the
Displacement, Stroke: and Bore
Stroke can be measured
ROUNDING OFF
by carefully using a
When wc round off a number, we shorten the one-inch travel dial
indicator and moving
number and thereby decrease its accuracy. So why
the crankshaft 1.000" at
do we round off a number? We generally round
a time then resetting
off a number for simplicity and ease of the indicator until the
calculation. Wc round oft in real life all the time. measurement is
When your friend asks "how long does it take to completed or use a
special long travel
get to the track,' die general answer is "2 to 2
indicator. Always double
1/2" hours not 2 hours, 12 minutes and 13.7
check your
seconds. When we need accuracy it is there. We measurements. Photo
wouldn't refer to our car's time at the dragstrip as by Bill Hancock.
"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
When the
would round it off to four decimal places. To do crankshaft is out
this we would count five numbers to the right of of the engine,
the decimal point, which in this case would be this handy stroke
checker uses a
the 9. If this number is 5 or greater, we would add
special 0-5"
one number to the value in die fourth place,
travel dial
which is the "5” in our example. Therefore our Indicator to make
rounded off number would be 3.1416. Let's try the measure
rounding off 27.1237945 to two places. It would ment By placing
the two Vees on
be 27.12, because rhe third number to the right
adjacent main
of the decimal point is a 3, which is less than 5, so
journals and
we just keep the second number to the right of rotating the
the decimal the same. device, the total
stroke length can
be obtained.
Photo by Bill
Hancock.
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
Cylinder bore can
figure is closer to the truth, try the formula with a
be measured
4.0" bore and 3.5" stroke: accurately with a
dial-bore gauge.
Displacement = 0.7853982 x 42 x 3.5 x 8 This is an
essential tool for
l he actual displacement is 351.85838 cubic
those who want
inches, which should be rounded up to 352 because
to do their own
the decimal is well over 0.5. However, Ford has boring or honing.
chosen to round it down to 351 for the later, better Photo by Larry
performing engine, probably to avoid confusing it Shepard.
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
