Computational Dynamics

522 Pages·2001·3.927 MB·English

by  Ahmed Shabana

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Computational Dynamics
Second Edition
Ahmed A. Shabana
Department of Mechanical Engineering
University of Illinois at Chicago
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Copyright © 2001 by John Wiley & Sons, Inc. All rights reserved.
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Contents
Preface xi
1 1
Introduction
1.1 Computational Dynamics 2
1.2 Motion and Constraints 4
1.3 Degrees of Freedom 6
1.4 Kinematic Analysis 9
1.5 Force Analysis 12
1.6 Dynamic Equations and Their Different Forms 12
1.7 Forward and Inverse Dynamics 14
1.8 Planar and Spatial Dynamics 16
1.9 Computer and Numerical Methods 18
1.10 Organization, Scope, and Notations of the Book 20
2 22
Linear Algebra
2.1 Matrices 23
2.2 Matrix Operations 25
2.3 Vectors 35
2.4 Three-Dimensional Vectors 45
2.5 Solution of Algebraic Equations 52
2.6 Triangular Factorization 60
*2.7 QR Decomposition 65

viii CONTENTS
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*2.8 Singular Value Decomposition 81
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Problems 90
3 KINEMATICS 95
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3.1 Mechanical Joints 96
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3.2 Coordinate Transformation 104
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3.3 Position, Velocity, and Acceleration Equations 105
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3.4 Kinematics of a Point Moving on a Rigid Body 124
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3.5 Constrained Kinematics 132
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3.6 Formulation of the Joint Constraints 136
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3.7 Computational Methods in Kinematics 150
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3.8 Computer Implementation 159
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3.9 Kinematic Modeling and Analysis 171
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3.10 Concluding Remarks 180
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Problems 181
4 FORMS OF THE DYNAMIC EQUATIONS 188
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4.1 D’Alembert’s Principle 189
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4.2 Constrained Dynamics 194
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4.3 Augmented Formulation 196
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4.4 Elimination of the Dependent Accelerations 197
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4.5 Embedding Technique 199
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4.6 Amalgamated Formulation 202
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4.7 Open and Closed Chains 203
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4.8 Concluding Remarks 215
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Problems 216
5 VIRTUAL WORK AND LAGRANGIAN DYNAMICS 217
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5.1 Virtual Displacements 218
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5.2 Kinematic Constraints and Coordinate Partitioning 221
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5.3 Virtual Work 233
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5.4 Examples of Force Elements 240
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5.5 Workless Constraints 256
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5.6 Principle of Virtual Work in Statics 257
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5.7 Principle of Virtual Work in Dynamics 268
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5.8 Lagrange’s Equation 274
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5.9 Gibbs–Appel Equation 279
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*5.10 Hamiltonian Formulation 280
CONTENTS ix
5.11 Relationship between Virtual Work and Gaussian
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Elimination 286
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Problems 288
6 CONSTRAINED DYNAMICS 295
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6.1 Generalized Inertia 295
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6.2 Mass Matrix and Centrifugal Forces 301
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6.3 Equations of Motion 307
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6.4 System of Rigid Bodies 309
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6.5 Elimination of the Constraint Forces 314
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6.6 Lagrange Multipliers 323
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6.7 Constrained Dynamic Equations 332
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6.8 Joint Reaction Forces 339
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6.9 Elimination of Lagrange Multipliers 342
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6.10 State Space Representation 345
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6.11 Numerical Integration 349
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6.12 Differential and Algebraic Equations 358
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*6.13 Inverse Dynamics 368
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*6.14 Static Analysis 371
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Problems 372
7 SPATIAL DYNAMICS 378
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7.1 General Displacement 379
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7.2 Finite Rotations 380
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7.3 Euler Angles 388
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7.4 Velocity and Acceleration 391
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7.5 Generalized Coordinates 397
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7.6 Generalized Inertia Forces 401
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7.7 Generalized Applied Forces 414
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7.8 Dynamic Equations of Motion 423
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7.9 Constrained Dynamics 427
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7.10 Formulation of the Joint Constraints 430
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7.11 Newton–Euler Equations 439
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7.12 Linear and Angular Momentum 441
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7.13 Recursive Methods 443
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Problems 460
x CONTENTS
8 OTHER TOPICS IN SPATIAL DYNAMICS 467
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8.1 Gyroscopes and Euler Angles 467
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8.2 Rodriguez Formula 472
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8.3 Euler Parameters 476
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8.4 Rodriguez Parameters 479
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8.5 Quaternions 481
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8.6 Rigid Body Contact 485
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Problems 491
REFERENCES 493
INDEX 497
xiv PREFACE
ACKNOWLEDGMENTS
I would like to acknowledge the help I received from many of my graduate stu-
dents and research associates, who made significant contributions to the devel-
opment of this text. I mention, in particular, J. H. Choi, M. Gofron, K. S.
Hwang, Z. Kusculuoglu, H. C. Lee, M. Omar, T. Ozaki, M. K. Sarwar, M.
Shokohifard, and D. Valtorta. I thank Toshikazu Nakanishi of Komatsu, Ltd.
for providing the DAMS simulation results of the tracked vehicle model pre-
sented in Chapter 6. I would like also to gratefully acknowledge the support
received from the U.S. Army Research Office for our research in the area of
computational dynamics.
Thanks are due to Ms. Denise Burt for the excellent job in typing some
chapters of the manuscript. Mr. Frank Cerra and Mr. Bob Argentieri, the Senior
Engineering Editors; Ms. Kimi Sugeno, the Assistant Managing Editor; Mr. Bob
Hilbert, the Associate Managing Editor; and the production staff at John Wiley
deserve special thanks for their cooperation and thoroughly professional work in
producing the first and second editions of this book. The author is also grateful
to his family for their patience during the years of preparation of this book.
AHMED A. SHABANA
COMPUTATIONAL
DYNAMICS
CHAPTER 1
INTRODUCTION
Modern mechanical and aerospace systems are often very complex and con-
sist of many components interconnected by joints and force elements such as
springs, dampers, and actuators. These systems are referred to, in modern lit-
erature, as multibody systems. Examples of multibody systems are machines,
mechanisms, robotics, vehicles, space structures, and biomechanical systems.
The dynamics of such systems are often governed by complex relationships
resulting from the relative motion and joint forces between the components of
the system. Figure 1 shows a hydraulic excavator, which can be considered as
an example of a multibody system that consists of many components. In the
design of such a tracked vehicle, the engineer must deal with many interrelated
questions with regard to the motion and forces of different components of the
vehicle. Examples of these interrelated questions are the following: What is the
relationship between the forward velocity of the vehicle and the motion of the
track chains? What is the effect of the contact forces between the links of the
track chains and the vehicle components on the motion of the system? What
is the effect of the friction forces between the track chains and the ground on
the motion and performance of the vehicle? What is the effect of the soil–track
interaction on the vehicle dynamics, and how can the soil properties be charac-
terized? How does the geometry of the track chains influence the forces and the
maximum vehicle speed? These questions and many other important questions
must be addressed before the design of the vehicle is completed. To provide
a proper answer to many of these interrelated questions, the development of a
detailed dynamic model of such a complex system becomes necessary. In this
book we discuss in detail the development of the dynamic equations of complex
multibody systems such as the tracked hydraulic excavator shown in Fig.1. The
1