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This page intentionally left blank P1:VSA/BSA P2:VSA/BSA QC:VSA February17,2005 16:21 CB851/Shabana CB851FM Dynamics of Multibody Systems ThirdEdition DynamicsofMultibodySystems,ThirdEdition,introducesmultibodydynamics,with an emphasis on flexible body dynamics. Many common mechanisms such as auto- mobiles,spacestructures,robots,andmicromachineshavemechanicalandstructural systemsthatconsistofinterconnectedrigidanddeformablecomponents.Thedynam- icsoftheselarge-scale,multibodysystemsarehighlynonlinear,presentingcomplex problemsthatinmostcasescanonlybesolvedwithcomputer-basedtechniques.The bookbeginswithareviewofthebasicideasofkinematicsandthedynamicsofrigid anddeformablebodiesbeforemovingontomoreadvancedtopicsandcomputerim- plementation. This revised third edition now includes important new developments relatingtotheproblemoflargedeformationsandnumericalalgorithmsasappliedto flexiblemultibodysystems.Thebook’swealthofexamplesandpracticalapplications willbeusefultograduatestudents,researchers,andpracticingengineersworkingon awidevarietyofflexiblemultibodysystems. Ahmed A. Shabana is a Professor in the Department of Mechnical and Industrial Engineering at the University of Illinois, Chicago. Dr. Shabana received his Ph.D. inmechanicalengineeringfromtheUniversityofIowa.Hisactiveareasofresearch interestareindynamics,vibration,andcontrolofmechanicalsystemsconsistingof rigidanddeformableinterconnectedbodies.Heisalsotheauthorofthreeotherworks, TheoryofVibration:AnIntroduction,VibrationofDiscreteandContinuousSystems, andComputationalDynamics. i P1:VSA/BSA P2:VSA/BSA QC:VSA February17,2005 16:21 CB851/Shabana CB851FM ii P1:VSA/BSA P2:VSA/BSA QC:VSA February17,2005 16:21 CB851/Shabana CB851FM DYNAMICS OF MULTIBODY SYSTEMS Third Edition Ahmed A. Shabana UniversityofIllinoisatChicago iii P1:VSA/BSA P2:VSA/BSA QC:VSA February17,2005 16:21 CB851/Shabana CB851FM Tomyfather andto thememoryofmymother    Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo Cambridge University Press The Edinburgh Building, Cambridge  ,UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridg e.org /9780521850117 © Cambridge University Press 2005 This publication is in copyright. Subject to statutory exception and to the provision of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published in print format 2005 - ---- eBook (NetLibrary) - --- eBook (NetLibrary) - ---- hardback - --- hardback Cambridge University Press has no responsibility for the persistence or accuracy of s for external or third-party internet websites referred to in this publication, and does not guaranteethatanycontentonsuchwebsitesis,orwillremain,accurateorappropriate. iv P1:VSA/BSA P2:VSA/BSA QC:VSA February17,2005 16:21 CB851/Shabana CB851FM Contents Preface ix 1 INTRODUCTION 1 1.1 MultibodySystems 1 1.2 ReferenceFrames 3 1.3 ParticleMechanics 6 1.4 RigidBodyMechanics 11 1.5 DeformableBodies 15 1.6 ConstrainedMotion 18 1.7 ComputerFormulationandCoordinateSelection 22 1.8 ObjectivesandScopeofThisBook 25 2 REFERENCEKINEMATICS 28 2.1 RotationMatrix 29 2.2 PropertiesoftheRotationMatrix 35 2.3 SuccessiveRotations 39 2.4 VelocityEquations 47 2.5 AccelerationsandImportantIdentities 55 2.6 RodriguezParameters 59 2.7 EulerAngles 63 2.8 DirectionCosines 68 2.9 The4×4TransformationMatrix 72 2.10 RelationshipbetweenDifferentOrientationCoordinates 80 Problems 82 v P1:VSA/BSA P2:VSA/BSA QC:VSA February17,2005 16:21 CB851/Shabana CB851FM vi CONTENTS 3 ANALYTICALTECHNIQUES 85 3.1 GeneralizedCoordinatesandKinematicConstraints 86 3.2 DegreesofFreedomandGeneralizedCoordinatePartitioning 94 3.3 VirtualWorkandGeneralizedForces 102 3.4 LagrangianDynamics 115 3.5 ApplicationtoRigidBodyDynamics 123 3.6 CalculusofVariations 129 3.7 Euler’sEquationintheCaseofSeveralVariables 135 3.8 EquationsofMotionofRigidBodySystems 142 3.9 Newton–EulerEquations 150 3.10 ConcludingRemarks 154 Problems 156 4 MECHANICSOFDEFORMABLEBODIES 159 4.1 KinematicsofDeformableBodies 160 4.2 StrainComponents 164 4.3 PhysicalInterpretationofStrains 168 4.4 RigidBodyMotion 169 4.5 StressComponents 172 4.6 EquationsofEquilibrium 175 4.7 ConstitutiveEquations 178 4.8 VirtualWorkoftheElasticForces 183 Problems 186 5 FLOATINGFRAMEOFREFERENCEFORMULATION 188 5.1 KinematicDescription 189 5.2 InertiaofDeformableBodies 200 5.3 GeneralizedForces 213 5.4 KinematicConstraints 219 5.5 EquationsofMotion 223 5.6 CouplingbetweenReferenceandElasticDisplacements 228 5.7 ApplicationtoaMultibodySystem 231 5.8 UseofIndependentCoordinates 241 5.9 DynamicEquationswithMultipliers 244 5.10 GeneralizedCoordinatePartitioning 248 5.11 OrganizationofMultibodyComputerPrograms 251 5.12 NumericalAlgorithms 254 Problems 263 6 FINITE-ELEMENTFORMULATION 267 6.1 ElementShapeFunctions 268 6.2 ReferenceConditions 276 P1:VSA/BSA P2:VSA/BSA QC:VSA February17,2005 16:21 CB851/Shabana CB851FM CONTENTS vii 6.3 KineticEnergy 278 6.4 GeneralizedElasticForces 287 6.5 CharacterizationofPlanarElasticSystems 288 6.6 CharacterizationofSpatialElasticSystems 294 6.7 CoordinateReduction 300 6.8 TheFloatingFrameofReferenceandLarge DeformationProblem 304 Problems 307 7 THELARGEDEFORMATIONPROBLEM 309 7.1 Background 310 7.2 AbsoluteNodalCoordinateFormulation 314 7.3 FormulationoftheStiffnessMatrix 318 7.4 EquationsofMotion 322 7.5 RelationshiptotheFloatingFrameofReferenceFormulation 323 7.6 CoordinateTransformation 325 7.7 ConsistentMassFormulation 328 7.8 TheVelocityTransformationMatrix 331 7.9 LumpedMassFormulation 332 7.10 ExtensionoftheMethod 335 7.11 ComparisonwithLargeRotationVectorFormulation 339 Problems 342 APPENDIX:LINEARALGEBRA 345 A.1 MatrixAlgebra 345 A.2 EigenvalueAnalysis 349 A.3 VectorSpaces 350 A.4 ChainRuleofDifferentiation 353 A.5 PrincipleofMathematicalInduction 354 Problems 355 REFERENCES 357 INDEX 369 P1:VSA/BSA P2:VSA/BSA QC:VSA February17,2005 16:21 CB851/Shabana CB851FM viii