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### Computational Dynamics

**Engineering Applications and Design**

**Engineering Mathematics **

Computational Dynamics

Ahmed A. Shabana

Richard and Loan Hill Professor of Engineering

545 Pages

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**Computational Dynamics**

Preface

Computational dynamics has been the subject of extensive research over the last three decades. This subject has grown rapidly as a result of the advent of high-speed digital computers and also as a response to the need for simulation and analysis computational capabilities for physics and engineering systems that consist of interconnected bodies. These systems are highly nonlinear in nature and their analysis requires the use of matrix, numerical, and computer methods. It is the objective of this book to present an introduction to the subject of computational dynamics at a level suitable for senior undergraduate and first-year graduate students. The book introduces students to concepts, definitions, and techniques used in the field of multibody system dynamics. To achieve this goal, classical approaches are first discussed in order to help students review some of the fundamental concepts and techniques in the general field of mechanics. The book then builds on these concepts to demonstrate the use of the classical methods as a foundation for the study of computational dynamics. Various computational methodologies that are used in the computer-aided analysis of multibody systems are presented. In the analysis presented in this book, only rigid body dynamics is considered.

TOC

1 INTRODUCTION 1

1.1 Computational Dynamics / 2

1.2 Motion and Constraints / 3

1.3 Degrees of Freedom / 6

1.4 Kinematic Analysis / 9

1.5 Force Analysis / 11

1.6 Dynamic Equations and Their Different Forms / 11

1.7 Forward and Inverse Dynamics / 13

1.8 Planar and Spatial Dynamics / 15

1.9 Computer and Numerical Methods / 16

1.10 Organization, Scope, and Notations of the Book / 18

2 LINEAR ALGEBRA 21

2.1 Matrices / 22

2.2 Matrix Operations / 24

2.3 Vectors / 33

2.4 Three-Dimensional Vectors / 42

2.5 Solution of Algebraic Equations / 48

2.6 Triangular Factorization / 55

*2.7 QR Decomposition / 60

*2.8 Singular Value Decomposition / 74

Problems / 82

3 KINEMATICS 87

3.1 Kinematics of Rigid Bodies / 88

3.2 Velocity Equations / 92

3.3 Acceleration Equations / 94

3.4 Kinematics of a Point Moving on a Rigid Body / 95

3.5 Constrained Kinematics / 97

3.6 Classical Kinematic Approach / 104

3.7 Computational Kinematic Approach / 124

3.8 Formulation of the Driving Constraints / 126

3.9 Formulation of the Joint Constraints / 128

3.10 Computational Methods in Kinematics / 141

3.11 Computer Implementation / 150

3.12 Kinematic Modeling and Analysis / 161

3.13 Concluding Remarks / 169

Problems / 170

4 FORMS OF THE DYNAMIC EQUATIONS 177

4.1 D’Alembert’s Principle / 178

4.2 D’Alembert’s Principle and Newton–Euler Equations / 182

4.3 Constrained Dynamics / 186

4.4 Augmented Formulation / 190

4.5 Lagrange Multipliers / 191

4.6 Elimination of the Dependent Accelerations / 193

4.7 Embedding Technique / 195

4.8 Amalgamated Formulation / 197

4.9 Open-Chain Systems / 197

4.10 Closed-Chain Systems / 203

4.11 Concluding Remarks / 209

Problems / 209

VIRTUAL WORK AND LAGRANGIAN DYNAMICS 211

5.1 Virtual Displacements / 212

5.2 Kinematic Constraints and Coordinate Partitioning / 215

5.3 Virtual Work / 225

5.4 Examples of Force Elements / 231

5.5 Workless Constraints / 246

5.6 Principle of Virtual Work in Statics / 247

5.7 Principle of Virtual Work in Dynamics / 257

5.8 Lagrange’s Equation / 262

5.9 Gibbs–Appel Equation / 267

*5.10 Hamiltonian Formulation / 267

5.11 Relationship between Virtual Work and Gaussian Elimination / 274

Problems / 276

6 CONSTRAINED DYNAMICS 283

6.1 Generalized Inertia / 284

6.2 Mass Matrix and Centrifugal Forces / 289

6.3 Equations of Motion / 294

6.4 System of Rigid Bodies / 296

6.5 Elimination of the Constraint Forces / 300

6.6 Lagrange Multipliers / 309

6.7 Constrained Dynamic Equations / 317

6.8 Joint Reaction Forces / 323

6.9 Elimination of Lagrange Multipliers / 326

6.10 State Space Representation / 329

6.11 Numerical Integration / 332

6.12 Algorithm and Sparse Matrix Implementation / 340

6.13 Differential and Algebraic Equations / 342

*6.14 Inverse Dynamics / 349

*6.15 Static Analysis / 351

Problems / 352

7 SPATIAL DYNAMICS 359

7.1 General Displacement / 360

7.2 Finite Rotations / 361

7.3 Euler Angles / 369

7.4 Velocity and Acceleration / 371

7.5 Generalized Coordinates / 376

7.6 Generalized Inertia Forces / 380

7.7 Generalized Applied Forces / 392

7.8 Dynamic Equations of Motion / 401

7.9 Constrained Dynamics / 405

7.10 Formulation of the Joint Constraints / 408

7.11 Newton–Euler Equations / 417

7.12 D’Alembert’s Principle / 418

7.13 Linear and Angular Momentum / 419

7.14 Recursive Methods / 422

Problems / 438

8 SPECIAL TOPICS IN DYNAMICS 445

8.1 Gyroscopes and Euler Angles / 445

8.2 Rodriguez Formula / 450

8.3 Euler Parameters / 454

8.4 Rodriguez Parameters / 456

8.5 Quaternions / 459

8.6 Rigid Body Contact / 462

8.7 Stability and Eigenvalue Analysis / 468

Problems / 472

9 MULTIBODY SYSTEM COMPUTER CODES 475

9.1 Introduction to SAMS/2000 / 476

9.2 Code Structure / 478

9.3 System Identification and Data Structure / 479

9.4 Installing the Code and Theoretical Background / 481

9.5 SAMS/2000 Setup / 483

9.6 Use of the Code / 484

9.7 Body Data / 486

9.8 Constraint Data / 492

9.9 Performing Simulations / 496

9.10 Batch Jobs / 498

9.11 Graphics Control / 500

9.12 Animation Capabilities / 503

9.13 General Use of the Input Data Panels / 503

9.14 Spatial Analysis / 506

9.15 Special Modules and Features of the Code / 509

REFERENCES 515

INDEX 521