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Hybrid Electric Vehicles: Principles and Applications with Practical Perspectives, 2nd Edition
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Hybrid Electric Vehicle System Modeling and Control, 2nd Edition
Wei Liu
Thermal Management of Electric Vehicle Battery Systems
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Automotive Aerodynamics
Joseph Katz
The Global Automotive Industry
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Vehicle Dynamics
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Modelling, Simulation and Control of Two‐Wheeled Vehicles
Mara Tanelli, Matteo Corno, Sergio Saveresi
Vehicle Gearbox Noise and Vibration: Measurement, Signal Analysis, Signal Processing and Noise Reduction Measures
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Modeling and Control of Engines and Drivelines
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Advanced Composite Materials for Automotive Applications: Structural Integrity and Crashworthiness
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This edition first published 2019
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Library of Congress Cataloging‐in‐Publication Data
Names: Minaker, Bruce P., 1969‐ author.
Title: Fundamentals of vehicle dynamics and modelling : a textbook for
engineers with illustrations and examples / Bruce P. Minaker, University
of Windsor, ON, CA.
Description: Hoboken, NJ : Wiley, 2020. | Series: Automotive series |
Includes bibliographical references and index. |
Identifiers: LCCN 2019020054 (print) | LCCN 2019021720 (ebook) | ISBN
9781118980071 (Adobe PDF) | ISBN 9781118980088 (ePub) | ISBN 9781118980095
(hardcover)
Subjects: LCSH: Motor vehicles-Dynamics-Textbooks. | Motor
vehicles-Mathematical models-Textbooks.
Classification: LCC TL243 (ebook) | LCC TL243 .M555 2020 (print) | DDC
629.2/31-dc23
LC record available at https://lccn.loc.gov/2019020054
Cover Design: Wiley
Cover Images: Background: © solarseven/Shutterstock, Left: © Denis Vorob'yev/iStockphoto, Middle: © Henrik5000/Getty Images, Right: © zorazhuang/Getty Images
To my wife Beth Anne, for everything
is defined as the ratio of the slip speed at the c...
, ...
is defined as the angle between the direction the ...
, but a...
is a function of speed. Note that ...
is a function of speed. Not...
and steer angle 
of the rigid rider bicycle in re...
This is my first attempt at writing a book. After many years of effort, I realize now that I was unprepared for such a task when I began, but I did it in the hopes that some of you may find it useful. What prompted this foray into the textbook market? Over many years of university teaching, I have found many texts on this topic and others that, while providing good relevant coverage, add much more content than necessary. I believe that this level of breadth is not in the reader's best interest, with many left overwhelmed by the prospect of mastering so much material, and eventually resigning themselves to never finishing. It is intimidating for a student to contemplate absorbing over a thousand pages in a twelve‐week semester.
My objective was to produce a book that is suitable for a single‐semester senior undergraduate or early graduate level mechanical engineering course, perhaps in a program with some focus on automotive engineering. It assumes that the student has some foundation in mathematics, particularly linear algebra and differential equations, and basic rigid body dynamics. The book does not aim to be a complete reference, but rather to give a solid foundation while generating enthusiasm in the student reader. I have found vehicle dynamics to be a most intellectually rewarding topic, and if I can share some of that with my readers, then I have accomplished my goal.
The book opens with some brief general discussion of the topic, and follows with an introduction to tire modelling. It includes material on longitudinal dynamics, lateral dynamics, and vertical dynamics. I have included many of the classic models that first drew me in, problems small enough to solve by hand on paper, without the use of a computer. This leads into some larger problems, tied to my work on automatic generation of the equations of motion of multibody systems, and its application to vehicle dynamics. The text concludes with a chapter expanding the necessary mathematical background for those readers who may require it.
I would like the reader to come away with answers to the three following questions: First, what are the concepts and tools used to generate a mathematical model of vehicle motion? Next, what are the fundamental vehicle ride and handling behaviours that we can predict from well‐established models? Finally, how has the arrival of multibody dynamics and computer aided engineering changed what we can do now? Hopefully, this will remove some of the mystery behind many of the software tools used in industry today, and how they generate at their results.
Astute readers may notice what seems like an odd combination of British and American English styles in the text. This stems from my education in and use of Canadian English. You can expect ‘‐our’ rather than ‘‐or’, but ‘‐ize’ instead of ‘‐ise’. In any instance where there was flexibililty in spelling, punctuation or grammar, I have tried to emulate those pre‐war vintage textbooks that I enjoy collecting.
Bruce P. Minaker
Windsor, Ontario, Canada
A mathematical style consistent with formatting standards is used wherever possible throughout the text. This has proven to be surprisingly difficult, as the symbols that have typically been associated with certain quantities tend to be inconsistent between fields, or overlapping. As a result, a few concessions and stylistic choices have been made, e.g., the character 
has been used to indicate a damping matrix, in place of the more commonly used 
, as the latter is also used in the standard state space form. The lowercase 
is still used to indicate a damping coefficient in the scalar case.
Upright characters indicate a mathematical constant (
, 
, 
), italics are used to indicate a quantity is a mathematical variable, (
, 
), and boldface is used to indicate a vector quantity, (
, 
). Lowercase characters are chosen wherever possible, with the exception of matrices, which are always set in bold upright uppercase (
, 
). When the appropriate lowercase character is already in use, an uppercase character may be substituted. Historically, certain quantities (e.g., scalar components of force and moment vectors) have been set in uppercase; these choices are maintained when appropriate. No distinction is made between Latin and Greek characters when formatting; the traditional Greek characters are maintained for several quantities. Multicharacter names are avoided, to eliminate confusion with products of variables. Note that these rules are also applied to subscripts, i.e., if a quantity is marked with an italic subscript, that subscript is itself a variable. An upright subscript indicates a specific named instance of the variable. Any character modified with a dot above indicates the time derivative of that quantity (
, 
). A vector modified with a circumflex indicates a unit vector (
). The standard basis vectors are 
, 
, and 
. A vector modified with a tilde indicates the skew symmetric matrix of the vector (
). A pair of vertical bars around a character is used to indicate the absolute value of a scalar, the magnitude of a vector or a complex number, or the amplitude of a time varying sinusoid (
, 
). The transpose of a matrix or a vector is indicated using a prime (
). Column vectors may be presented as a row by using transpose notation for the space savings offered.
In 1976, the Society of Automotive Engineers (SAE) published standard J670e, establishing a convention for terminology and notation for vehicle dynamics. In 1991, the International Organization for Standardization (ISO) published a vehicle dynamics vocabulary, ISO 8855. The SAE J670e and ISO 8855 standards are incompatible in several aspects, the most notable being the axis systems defined in the two documents. The SAE standard utilizes an axis system based on aeronautical practice, with the 
axis positive forward, the 
axis positive to the driver's right, and the 
axis positive down. The ISO standard utilizes an axis system with the 
axis positive forward, the 
axis positive to the driver's left, and the 
axis positive up. In this text, the SAE standard and the corresponding historical aeronautical notation will be used in the discussion of the vehicle models in Chapters 3, 4, and 5, as they were developed well before 1991. In Chapter 6, an ISO‐style axis system will be used when discussing a modern multibody dynamics approach.
| Symbol | Description |
 |
longitudinal distance from mass centre to front axle |
 |
longitudinal distance from mass centre to rear axle |
 |
damping coefficient, or tire cornering coefficient |
 |
drag coefficient |
 |
arbitrary constant coefficient |
 |
arbitrary constant coefficient |
 |
longitudinal distance from mass centre to trailer hitch |
 |
longitudinal distance from trailer mass centre to trailer hitch |
 |
force, or arbitrary function |
 |
gravitational acceleration |
 |
longitudinal distance from trailer mass centre to trailer axle, or time step |
 |
centre of mass height |
 |
counter increment |
 |
spring stiffness |
| Symbol | Description |
 |
pitch radius of gyration |
 |
yaw radius of gyration |
 |
length |
 |
mass |
 |
counter end, or dimension |
 |
angular velocity,  axis direction |
 |
angular velocity,  axis direction |
 |
angular velocity,  axis direction |
 |
eigenvalue, or exponent coefficient |
 |
time, or track width |
 |
linear velocity,  axis direction |
 |
linear velocity,  axis direction |
 |
linear velocity,  axis direction |
 |
location,  axis direction |
 |
location,  axis direction |
 |
location,  axis direction |
 |
centre of mass |
 |
roll moment of inertia |
 |
pitch moment of inertia |
 |
yaw moment of inertia |
 |
frontal area |
 |
moment,  axis direction |
 |
moment,  axis direction |
 |
moment,  axis direction |
 |
power |
 |
cornering radius |
 |
time step size, discrete time |
 |
force,  axis direction |
 |
force,  axis direction |
 |
force,  axis direction |
| Symbol | Description |
 |
acceleration vector |
 |
force vector |
 |
moment vector |
 |
location and orientation vector |
 |
radius vector |
 |
spin vector |
 |
input signal |
 |
unit vector |
 |
linear and angular velocity vector |
 |
state vector, location vector, eigenvector |
 |
output vector |
 |
vertical location vector |
 |
system matrix |
 |
input matrix |
 |
output matrix |
 |
feedthrough matrix |
 |
descriptor matrix |
 |
input force matrix |
 |
input rate force matrix, or transfer function matrix |
 |
deflection Jacobian matrix |
 |
identity matrix |
 |
inertia matrix |
 |
constraint Jacobian matrix |
 |
stiffness matrix |
 |
damping matrix |
 |
mass matrix |
 |
transformation matrix |
 |
rotation matrix |
 |
angular velocity transformation matrix |
 |
orthogonal complement matrix |
 |
orthogonal complement matrix |
 |
velocity matrix |
| Symbol | Description |
 |
tire slip angle |
 |
body slip angle |
 |
trailer sway angle |
 |
steer angle |
 |
camber angle |
 |
damping ratio |
 |
efficiency |
 |
pitch angle |
 |
friction gradient |
 |
wavelength |
 |
coefficient of friction |
 |
jacking force angle |
 |
density |
 |
real component of eigenvalue, or tire slip ratio |
 |
time constant |
 |
roll angle, or phase angle |
 |
yaw angle |
 |
imaginary component of eigenvalue, or angular frequency |
 |
angular acceleration vector |
 |
elastic deflection vector |
 |
angular position vector |
 |
Lagrange multiplier |
 |
linear velocity vector |
 |
constraint equation vector |
 |
angular velocity vector |
This book is accompanied by a companion website:
www.wiley.com/go/minaker/vehicle‐dynamics
The website includes:
Scan this QR code to visit the companion website.
