COMPUTATIONAL DYNAMICS

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1 COMPUTATIONAL DYNAMICS THIRD EDITION AHMED A. SHABANA Richard and Loan Hill Professor of Engineering University of Illinois at Chicago A John Wiley and Sons, Ltd., Publication

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3 COMPUTATIONAL DYNAMICS

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5 COMPUTATIONAL DYNAMICS THIRD EDITION AHMED A. SHABANA Richard and Loan Hill Professor of Engineering University of Illinois at Chicago A John Wiley and Sons, Ltd., Publication

6 This edition first published , John Wiley & Sons Ltd Registered office John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex, PO19 8SQ, United Kingdom For details of our global editorial offices, for customer services and for information about how to apply for permission to reuse the copyright material in this book please see our website at The right of the author to be identified as the author of this work has been asserted in accordance with the Copyright, Designs and Patents Act All 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, photocopying, recording or otherwise, except as permitted by the UK Copyright, Designs and Patents Act 1988, without the prior permission of the publisher. Wiley also publishes its books in a variety of electronic formats. Some content that appears in print may not be available in electronic books. Designations used by companies to distinguish their products are often claimed as trademarks. All brand names and product names used in this book are trade names, service marks, trademarks or registered trademarks of their respective owners. The publisher is not associated with any product or vendor mentioned in this book. This publication is designed to provide accurate and authoritative information in regard to the subject matter covered. It is sold on the understanding that the publisher is not engaged in rendering professional services. If professional advice or other expert assistance is required, the services of a competent professional should be sought. Library of Congress Cataloging-in-Publication Data: Shabana, Ahmed A., Computational dynamics / Ahmed A. Shabana. 3rd ed. p. cm. Includes bibliographical references and index. ISBN (cloth) 1. Dynamics. I. Title. QA845.S dc A catalogue record for this book is available from the British Library. ISBN (Hbk) Typeset in 10/12pt Times by Laserwords Private Limited, Chennai, India. Printed in Singapore by Markono Print Media Pte Ltd.

7 To My Students

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9 CONTENTS PREFACE xi 1 INTRODUCTION Computational Dynamics / Motion and Constraints / Degrees of Freedom / Kinematic Analysis / Force Analysis / Dynamic Equations and Their Different Forms / Forward and Inverse Dynamics / Planar and Spatial Dynamics / Computer and Numerical Methods / Organization, Scope, and Notations of the Book / 18 2 LINEAR ALGEBRA Matrices / Matrix Operations / Vectors / Three-Dimensional Vectors / Solution of Algebraic Equations / Triangular Factorization / 55 vii

10 viii CONTENTS *2.7 QR Decomposition / 60 *2.8 Singular Value Decomposition / 74 Problems / 82 3 KINEMATICS Kinematics of Rigid Bodies / Velocity Equations / Acceleration Equations / Kinematics of a Point Moving on a Rigid Body / Constrained Kinematics / Classical Kinematic Approach / Computational Kinematic Approach / Formulation of the Driving Constraints / Formulation of the Joint Constraints / Computational Methods in Kinematics / Computer Implementation / Kinematic Modeling and Analysis / Concluding Remarks / 169 Problems / FORMS OF THE DYNAMIC EQUATIONS D Alembert s Principle / D Alembert s Principle and Newton Euler Equations / Constrained Dynamics / Augmented Formulation / Lagrange Multipliers / Elimination of the Dependent Accelerations / Embedding Technique / Amalgamated Formulation / Open-Chain Systems / Closed-Chain Systems / Concluding Remarks / 209 Problems / VIRTUAL WORK AND LAGRANGIAN DYNAMICS Virtual Displacements / Kinematic Constraints and Coordinate Partitioning / Virtual Work / Examples of Force Elements / 231

11 CONTENTS ix 5.5 Workless Constraints / Principle of Virtual Work in Statics / Principle of Virtual Work in Dynamics / Lagrange s Equation / Gibbs Appel Equation / 267 *5.10 Hamiltonian Formulation / Relationship between Virtual Work and Gaussian Elimination / 274 Problems / CONSTRAINED DYNAMICS Generalized Inertia / Mass Matrix and Centrifugal Forces / Equations of Motion / System of Rigid Bodies / Elimination of the Constraint Forces / Lagrange Multipliers / Constrained Dynamic Equations / Joint Reaction Forces / Elimination of Lagrange Multipliers / State Space Representation / Numerical Integration / Algorithm and Sparse Matrix Implementation / Differential and Algebraic Equations / 342 *6.14 Inverse Dynamics / 349 *6.15 Static Analysis / 351 Problems / SPATIAL DYNAMICS General Displacement / Finite Rotations / Euler Angles / Velocity and Acceleration / Generalized Coordinates / Generalized Inertia Forces / Generalized Applied Forces / Dynamic Equations of Motion / Constrained Dynamics / Formulation of the Joint Constraints / Newton Euler Equations / 417

12 x CONTENTS 7.12 D Alembert s Principle / Linear and Angular Momentum / Recursive Methods / 422 Problems / SPECIAL TOPICS IN DYNAMICS Gyroscopes and Euler Angles / Rodriguez Formula / Euler Parameters / Rodriguez Parameters / Quaternions / Rigid Body Contact / Stability and Eigenvalue Analysis / 468 Problems / MULTIBODY SYSTEM COMPUTER CODES Introduction to SAMS/2000 / Code Structure / System Identification and Data Structure / Installing the Code and Theoretical Background / SAMS/2000 Setup / Use of the Code / Body Data / Constraint Data / Performing Simulations / Batch Jobs / Graphics Control / Animation Capabilities / General Use of the Input Data Panels / Spatial Analysis / Special Modules and Features of the Code / 509 REFERENCES 515 INDEX 521

13 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. CONTENTS The third edition of the book is organized into nine chapters that cover the basic concepts and computational methods in kinematics and dynamics of multibody systems. Simple examples are used in most chapters to demonstrate the basic ideas and procedures presented. The problem sets presented at the end of each chapter are intentionally selected to be simple in order to focus on the main concepts and computational methodologies discussed in the book. In developing the materials of this book, modest demands are made on the expertise of the reader in mathematics and dynamics. xi

14 xii PREFACE In Chapter 1, some basic definitions that are used repeatedly in the book are introduced. The materials presented in this chapter also serve as a brief introduction to the materials covered in subsequent chapters. The organization of the book and the notation used are discussed at the end of this chapter. The reader is encouraged to read this introductory chapter before reading subsequent chapters. Chapter 2 is devoted to a review of some concepts and operations in matrix and vector algebra. Matrix and vector properties and identities as well as methods for solving systems of algebraic equations are among the topics discussed in Chapter 2. QR decomposition and singular value decomposition, which are used in the dynamic analysis of constrained multibody systems to obtain a minimum set of independent differential equations, are discussed in two sections of this chapter. The reader with a background in linear algebra will find that most of the material presented in Chapter 2 is familiar. In Chapter 3, the kinematics of constrained multibody systems is discussed. The position, velocity, and acceleration equations are developed, and the use of these equations in the kinematic analysis of multibody systems that consist of interconnected bodies is demonstrated. The number of degrees of freedom of a multibody system depends on the number and types of joints that connect the system components. Several mechanical joints and the formulation of their kinematic constraint equations are presented. It is shown that the mobility of the system depends on the number of linearly independent constraint equations of its joints. The conditions of the joint connectivity between interconnected bodies are formulated using a set of nonlinear algebraic constraint equations that depend on the system coordinates. The configuration of the system is determined by solving these nonlinear algebraic equations using numerical and computer methods. There are several computer techniques that are currently used for the dynamic analysis of multibody systems. Some of these techniques lead to a relatively large system of loosely coupled differential and algebraic equations; others lead to smaller systems of strongly coupled equations. In Chapter 4, different forms of the dynamic equations are presented. In this chapter, the basic multibody system equations are developed using Newtonian mechanics without the need for using analytical Lagrangian techniques such as the principle of virtual work. The use of D Alembert s principle to derive Euler equation is demonstrated, and the concept of Lagrange multipliers, which can be used to define the generalized constraint forces, is introduced using Newtonian mechanics and simple examples. Systematic analytical procedures for developing some of the forms of the dynamic equations presented in Chapter 4 are described in detail in the following two chapters. In Chapter 5, the principle of virtual work, which represents the cornerstone for developing many of the existing dynamic formulations, is presented. The concepts of virtual displacement and generalized forces that are necessary for using the principle of virtual work are introduced. A systematic procedure based on the principle of virtual work for eliminating the constraint forces from the static and dynamic equations is outlined. Among the topics that are discussed in Chapter 5 are Lagrange s equation, thegibbs-appel equation, and the canonical form of the equations of motion. Chapter 5 is concluded by discussing the relationship between virtual work and Gaussian elimination. Chapter 6 covers computational methods in dynamics. Several computer methodologies for formulating the equations of motion are discussed in this chapter. In one approach,

15 PREFACE xiii the differential equations of motion are expressed in terms of the independent variables using the embedding techniques or the recursive methods. This approach leads to a set of ordinary differential equations in which the constraint forces are eliminated automatically. Use of QR decomposition and singular value decomposition to obtain a minimum set of independent differential equations is also discussed. In another computer approach, the equations of motion of the multibody systems are formulated in terms of both dependent and independent coordinates. This approach leads to a large system of loosely coupled equations in which the constraint forces appear explicitly. These constraint forces can be expressed in terms of Lagrange multipliers, leading to a mixed system of differential and algebraic equations that can be solved using matrix and computer methods. The numerical methods and sparse matrix formulations used for solving mixed systems of differential and algebraic equations are also discussed in this chapter. The spatial kinematics and dynamics of rigid body systems are discussed in Chapter 7. The general displacement of unconstrained rigid bodies in space is defined in terms of the independent translation and rotation coordinates. Methods for defining the orientation of rigid bodies in space are described. Formulation of the spatial kinematic velocity and acceleration equations is discussed and examples of constraint equations that describe spatial mechanical joints are presented. Formulation of the augmented and recursive dynamic equations that govern the constrained spatial motion of rigid body systems is also discussed in Chapter 7. In Chapter 8, several topics in dynamics are discussed. These topics include the gyroscopic motion, the Rodriguez formula, Euler parameters, Rodriguez parameters, quaternions, and rigid body contact. The study of the multibody system stability using the eigenvalue analysis is also discussed in Chapter 8. Chapter 9 is devoted to the description of general purpose multibody system codes and their capabilities. As an example, the multibody system computer code SAMS/2000 (Systematic Analysis of Multibody Systems) is used. The structure of the code and the procedure of its use are described in the cases of planar and spatial kinematic and dynamic problems. SAMS/2000 allows the user to develop systematically virtual models of multibody systems that consist of interconnected bodies. The readers of the book are encouraged to solve many of the examples and exercise problems using multibody system codes. The use of these codes along with the study of the formulations presented in the book is necessary in order to have a good understanding and appreciation of the methods and algorithms used in developing multibody system software that are widely used in industry. The educational version of SAMS/2000 is limited to only four rigid bodies and does not include flexible body and rail simulation capabilities. It does not also include some other simulation options that are discussed in Chapter 9. Several sections are marked with an asterisk, both in the table of contents and in the text. Some of these sections, such as Sections 2.7 and 2.8, can be omitted during a first reading of the book. Others, such as Section 5.10, do not contribute to the development of the main ideas presented in the book, and therefore, can be omitted entirely, since the remaining chapters do not make use of the development presented in these sections. The goal of computational and multibody dynamics is to develop general algorithms that can be applied to a large number of applications. The exercise problems are designed with this goal in mind.

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