1、Road Vehicle Dynamics Problems and Solutions Rao V. Dukkipati Jian Pang Mohamad S. Qatu Gang Sheng Zuo Shuguang Road Vehicle Dynamics Problems and Solutions SIBK002_FM_pi-viii.indd i SIBK002_FM_pi-viii.indd i 3/9/10 7:34:23 AM 3/9/10 7:34:23 AMOther SAE titles of interest: Road V ehicle Dynamics By
2、Rao Dukkipati, Jian Pang, Mohamad Qatu, Gang Sheng, and Shuguang Zuo (Order No. R-366) For more information or to order a book, contact SAE at 400 Commonwealth Drive, Warrendale, PA 15096-0001; phone (724) 776-4970; fax (724) 776-0790; e-mail CustomerServicesae.org; website http:/store.sae.org. SIBK
3、002_FM_pi-viii.indd ii SIBK002_FM_pi-viii.indd ii 3/9/10 7:34:24 AM 3/9/10 7:34:24 AMRoad Vehicle Dynamics Problems and Solutions Rao Dukkipati Jian Pang Mohamad Qatu Gang Sheng Shuguang Zuo Warrendale, Pa. SIBK002_FM_pi-viii.indd iii SIBK002_FM_pi-viii.indd iii 3/9/10 7:34:24 AM 3/9/10 7:34:24 AM C
4、opyright 2010 SAE International eISBN: 978-0-7680-3361-8All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photo- copying, recording, or otherwise, without the prior written permission
5、 of SAE. For permission and licensing requests contact: SAE Permissions 400 Commonwealth Drive Warrendale, PA 15096-0001-USA Email: permissionssae.org Tel: 724-772-4028 Fax: 724-772-4891 Library of Congress Cataloging-in-Publication Data Road vehicle dynamics : problems and solutions / Rao Dukkipati
6、 . et al.p. cm. “SAE order no. R-386.”Includes bibliographical references and index.ISBN 978-0-7680-2051-91. AutomobilesDynamics. I. Dukkipati, Rao V .TL243.R63 2010629.231dc222009011865 SAE 400 Commonwealth Drive Warrendale, PA 15096-0001 USA Tel: 877-606-7323 (inside USA and Canada)724-776-4970 (o
7、utside USA) Fax: 724-776-1615 Email: CustomerServicesae.org Copyright 2010 SAE International ISBN 978-0-7680-2051-9 SAE Order No. R-386 Printed in USA SIBK002_FM_pi-viii.indd iv SIBK002_FM_pi-viii.indd iv 3/9/10 7:34:24 AM 3/9/10 7:34:24 AMContents Preface . . . . . . . . . . . . . . . . . . . . . .
8、 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii Chapter 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Chapter 2 Analysis of Dynamic Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 Chapter
9、 3 Tire Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 Chapter 4 Ride Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 Chapter 5 Vehicle Rollover Analysis . . . . . . . . . . . . . . . . . . . . . . . . . .
10、. . . . . . 131 Chapter 6 Handling Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 Chapter 7 Braking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 Chapter 8 Acceleration . . . . . . . . . . . . . . . . . . . . . .
11、 . . . . . . . . . . . . . . . . . . . . 189 Chapter 9 Total Vehicle Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219 Chapter 10 Accident Reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243 Appendix A Vector Algebra . . . . . . . . . . .
12、 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313 Appendix B Matrix Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321 Appendix C Laplace Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367 Appendix D Glossar
13、y of Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419 Appendix E Direct Numerical Integration Methods . . . . . . . . . . . . . . . . . . . . . . 427 Appendix F Units and Conversions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493 Appendix
14、 G Accident Reconstruction Formulae . . . . . . . . . . . . . . . . . . . . . . . . . 495 SIBK002_FM_pi-viii.indd v SIBK002_FM_pi-viii.indd v 3/9/10 7:34:24 AM 3/9/10 7:34:24 AMSIBK002_FM_pi-viii.indd vi SIBK002_FM_pi-viii.indd vi 3/9/10 7:34:25 AM 3/9/10 7:34:25 AMPreface Objective of This Book Roa
15、d V ehicle Dynamics: Problems and Solutions is a comprehensive supplement to our book, Road V ehicle Dynamics (R-366), which was published by SAE International in 2008. More than 300 example problems and solutions are presented in an organized and systematic manner to offer detailed explanations of
16、complex aspects of this exciting, challenging, and multidisciplinary eld. Road Vehicle Dynamics: Problems and Solutions presents vehicle dynamics methods based on mathematical fundamentals and stresses physical system modeling. The book is organized around the concepts of road vehicle dynamics as th
17、ey have been developed in the frequency and time domain for an introductory undergraduate or graduate course for engineering students of all disciplines. Generally speaking, road vehicle dynamics can seem to be a dif cult subject to under- stand and learn. Despite the availability of a few textbooks
18、 in this eld, students con- tinue to remain perplexed because of the outcomes of the numerous conditions that often must be kept in mind and correlated when solving a given problem. In addition, different possible interpretations of the terms used in road vehicle dynamics can contrib- ute to the dif
19、 culties experienced by students. The main objective of this book is to provide readers with the opportunity to improve their problem-solving skills by using systematic rules of analysis that may be followed in a step-by-step manner. In schools, students often experience dif culty in under- standing
20、 and learning road vehicle dynamics for the following reasons: No systematic rules of analysis that may be followed in a step-by-step manner have been formulated or developed to nd solutions to the problems at hand. Most available textbooks on the subject usually explain a given principle in an abst
21、ract manner, leaving students confused about the application of that principle. Too few examples are provided in textbooks, and they often do not provide suf- cient bases for students to solve the problems in their homework assignments or examinations. The examples presented in textbooks often are d
22、if cult to understand. Examples often do not include diagrams/graphs, where appropriate. Students often spend too many hours solving a single problem, sometimes by sim- ple guesswork or by trial and error. We believe that road vehicle dynamics is a subject that is best learned by allowing individual
23、s to review on their own the methods of analysis and solution techniques. This method of learning is similar to that practiced in various scienti c laboratories and in medical elds. The objective of this book is to introduce individuals from a variety of disciplines and backgrounds to the vast array
24、 of problems that are amenable to numerical solution in SIBK002_FM_pi-viii.indd vii SIBK002_FM_pi-viii.indd vii 3/9/10 7:34:25 AM 3/9/10 7:34:25 AMviii Road Vehicle Dynamics road vehicle dynamics. Emphasis is placed on application rather than on pure theory, which, although kept to a minimum, is pre
25、sented in mostly a heuristic and intuitive manner. This is deemed suf cient for individuals to fully understand the workings, ef ciencies, and shortcomings or failings of each technique. Because we intended this book as a rst course on road vehicle dynamics, the concepts have been applied in simple
26、terms, and the solution procedures have been explained in detail. Audience This book is intended for vehicle designers, developers, and evaluators, as well as senior undergraduate students and/or graduate students. No previous knowledge of road vehicle dynamics is assumed. This book is appropriate f
27、or several groups of audi- ences, including the following. Senior undergraduate and graduate students in mathematics, science, and engi- neering who are taking an introductory course on road vehicle dynamics will nd the book helpful in understanding the subject. The book can be adapted for a short p
28、rofessional course on road vehicle dynamics. Design and research engineers will be able to draw upon the book in selecting and developing road vehicle dynamics for analytical and design purposes. Practicing engineers and managers will be able to learn more about the basic principles and concepts inv
29、olved in road vehicle dynamics and how these can be applied to address their own workplace concerns. Content This book consists of ten chapters and an extended list of appendices. The chapters include an introduction, analysis of dynamic systems, vehicle forces and tire mechan- ics, ride dynamics, r
30、oll dynamics, handling and steering, braking, accelerating, total vehicle dynamics, and accident reconstruction. The appendices span vector and matrix algebra, Fourier series, Laplace transformation, vehicle dynamics terminology, direct numerical integration methods, conversion of units, and acciden
31、t reconstruction for- mulae. In this book, we pay particular attention to the issue of safety. In fact, safety considerations are included in most chapters of this book. In addition, two complete chapters are devoted to roll dynamics and accident reconstruction to address the legal issues that may r
32、esult from an automotive accident. Numerous worked examples of problems and solutions offer detailed explanations and guide readers through each set of problems to enable them to save a great deal of time and effort in arriving at an understanding of problems in road vehicle dynamics. This book is i
33、ntended to help students of road vehicle dynamics nd their way through complex material involving a diverse variety of concepts. It offers detailed illustrations of solution methods that may not be clearly apparent; representative and typical prob- lems given in homework, class work, and examination
34、s; step-by-step explanations; and the opportunity to save time and effort in arriving at an understanding of problems in road vehicle dynamics. These worked-out example problems and solutions should be of interest to a wide audience, including students, vehicle designers, developers, and evaluators.
35、 SIBK002_FM_pi-viii.indd viii SIBK002_FM_pi-viii.indd viii 3/9/10 7:34:25 AM 3/9/10 7:34:25 AMChapter 1 Introduction PROBLEM 1.1 For the vibrating system shown in Figure 1.1: a. Determine the number of degrees of freedom needed to specify the motion of the system. b. Identify a set of generalized co
36、ordinates. Figure 1.1 Vibrating system. m k 2 k 1 k 3 Rigid bar Solution to Problem 1.1 a. Here, the rotational and translational motions of the rigid bar are independent of each other. Also the motion of the hanging mass m is an independent motion. Hence, the system has three degrees of freedom. b.
37、 The choice of generalized coordinates is not unique, and several sets of possible choices exist, as shown in Figures 1.1(a) through (c). SIBK002_Ch01_p1-42.indd 1 SIBK002_Ch01_p1-42.indd 1 3/8/10 9:13:17 AM 3/8/10 9:13:17 AM2 Road Vehicle Dynamics PROBLEM 1.2 For the vibrating system shown in Figur
38、e 1.2: a. Determine the number of degrees of freedom needed to specify the motion of the system. b. Identify a set of generalized coordinates. m k 2 k 1 k 3 x 1 x 2 x 3 m k 2 k 1 k 3 x x 3 m k 2 k 1 k 3 x 1 x 3 Figure 1.1 (c) A third free body diagram. Figure 1.1 (b) Another free body diagram. Figur
39、e 1.1 (a) One possible free body diagram. SIBK002_Ch01_p1-42.indd 2 SIBK002_Ch01_p1-42.indd 2 3/8/10 9:13:18 AM 3/8/10 9:13:18 AM Introduction 3 Solution to Problem 1.2 a. The system shown in Figure 1.2 has two degrees of freedom. m k k AB 2m Rigid bar 2l Figure 1.2 Vibrating system. Figure 1.2 (a)
40、Free body diagram. m 2m G 2l l b. One choice of a set of generalized coordinates is , which is the clockwise angular displacement of the rigid bar from the equilibrium position of the system, and x, which is the downward displacement of point G from the equilibrium position of the system, as shown i
41、n Figure 1.2. PROBLEM 1.3 Develop a mathematical model of a washing machine as a lumped-parameter or dis- crete parameter system. Consider a washing machine standing on elastomeric (rubber) mounts, with the drum rotating in the vertical plane with constant angular acceleration. Also assume that all
42、components or elements of the washing machine undergo no elas- tic deformations. Solution to Problem 1.3 Figure 1.3(a) shows a washing machine that is mounted on rubber pads. The drum is assumed to be rotating in a vertical plane relative to the body of the machine with a constant angular velocity.
43、The following assumptions can be made: a. The body of the machine and the drum do not undergo elastic deformations. b. The clothes are distributed uniformly around the drum. c. Inertia properties remain unchanged with time. SIBK002_Ch01_p1-42.indd 3 SIBK002_Ch01_p1-42.indd 3 3/8/10 9:13:18 AM 3/8/10
44、 9:13:18 AM4 Road Vehicle Dynamics Figure 1.3(b) shows the mathematical model of the washing machine, where the mass m represents the combined mass of the body of the machine, the drum, and the clothes. Also, x(t) represents the vertical displacement of mass m. The rubber pads can act in parallel as
45、 springs and dashpots. Assuming symmetry of the machine, the spring con- stants and the viscous damping coef cients on the left and right supports are repre- sented by k2and c2 , respectively. Figure 1.3 (a) A washing machine, and (b) a mathematical model of the washing machine. (b) (a) m k/2 c/2 k/
46、2 c/2 x(t) PROBLEM 1.4 Develop a mathematical model of a motorcycle with a rider for use in investigating vibration in the vertical direction. The elasticity of the tires, elasticity and viscous damping of the struts, wheel masses, mass, elasticity, and damping of the rider are to be considered in t
47、he model. Solution to Problem 1.4 Let m eq= equivalent mass of the system (wheels, motorcycle, and rider)k eq= equivalent stiffness of the system (tires, struts, and rider)c eq= equivalent damping of the system (struts and rider)k s= stiffness of the strutc s= damping of the strutm c= mass of the mo
48、torcyclem R= mass of the riderk t= stiffness of the tirewm w= mass of the wheel If we consider the equivalent values for mass, stiffness, and damping, then we see that a single degree of freedom model can be modeled as shown in Figure1.4(b). The masses of the wheels, elasticity of the tires, and ela
49、sticity and damping of the struts separately result in a model that can be represented as shown in Figure 1.4(c). Similarly, when the SIBK002_Ch01_p1-42.indd 4 SIBK002_Ch01_p1-42.indd 4 3/8/10 9:13:18 AM 3/8/10 9:13:18 AM Introduction 5 elasticity and the damping of the rider are considered, the resulting model is as shown in Figure 1.4(d). If one combines the spring constants of the front and rear tires and the masses of the front and rear wheels, and if the spring and damping constants of
