Rigid body mechanics : mathematics, physics and applications

256 Pages·2006·10.895 MB·English

by  William B Heard

William B. Heard
Rigid Body Mechanics
William B. Heard

Rigid Body Mechanics

Math ematics, Physics and Applications

WILEY-VCH Verlag GmbH & Co. KGaA
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William B. Heard
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© 2006 WILEY-VCH Verlag GmbH & Co.
KGaA, Weinheim

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ISBN-13: 978-3-527-40620-3
ISBN-10: 3-527-40620-4
Thisbookisdedicatedto
Peggy
andthememoryofmy
MotherandFather
VII
Contents
Preface XI
1 Rotations 1
1.1 RotationsasLinearOperators 1
1.1.1 VectorAlgebra 1
1.1.2 RotationOperatorsonR3 7
1.1.3 RotationsSpecifiedbyAxisandAngle 8
1.1.4 TheCayleyTransform 12
1.1.5 Reflections 13
1.1.6 EulerAngles 15
1.2 Quaternions 17
1.2.1 QuaternionAlgebra 20
1.2.2 QuaternionsasScalar–VectorPairs 21
1.2.3 QuaternionsasMatrices 21
1.2.4 RotationsviaUnitQuaternions 23
1.2.5 CompositionofRotations 25
1.3 ComplexNumbers 29
1.3.1 Cayley–KleinParameters 30
1.3.2 RotationsandtheComplexPlane 31
1.4 Summary 35
1.5 Exercises 37
2 Kinematics,Energy,andMomentum 39
2.1 RigidBodyTransformation 40
2.1.1 ARigidBodyhas6DegreesofFreedom 40
2.1.2 AnyRigidBodyTransformationisComposedofTranslationand
Rotation 40
2.1.3 TheRotationisIndependentoftheReferencePoint 41
RigidBodyMechanics:Mathematics,PhysicsandApplications.WilliamB.Heard
Copyright©2006WILEY-VCHVerlagGmbH&Co.KGaA,Weinheim
ISBN:3-527-40620-4
VIII Contents
( )
2.1.4 RigidBodyTransformationsFormtheGroupSE 3 41
2.1.5 Chasles’Theorem 42
2.2 AngularVelocity 44
2.2.1 AngularVelocityinEulerAngles 47
2.2.2 AngularVelocityinQuaternions 49
2.2.3 AngularVelocityinCayley–KleinParameters 50
2.3 TheInertiaTensor 50
2.4 AngularMomentum 54
2.5 KineticEnergy 54
2.6 Exercises 57
3 Dynamics 59
3.1 VectorialMechanics 61
3.1.1 TranslationalandRotationalMotion 61
3.1.2 GeneralizedEulerEquations 62
3.2 LagrangianMechanics 68
3.2.1 VariationalMethods 68
3.2.2 NaturalSystemsandConnections 72
3.2.3 Poincaré’sEquations 75
3.3 HamiltonianMechanics 84
3.3.1 MomentaConjugatetoEulerAngles 86
3.3.2 AndoyerVariables 88
3.3.3 Brackets 92
3.4 Exercises 98
4 ConstrainedSystems 99
4.1 Constraints 99
4.2 LagrangeMultipliers 102
4.2.1 UsingProjectionstoEliminatetheMultipliers 103
4.2.2 UsingReductiontoEliminatetheMultipliers 104
4.2.3 UsingConnectionstoEliminatetheMultipliers 107
4.3 Applications 108
4.3.1 SphereRollingonaPlane 108
4.3.2 DiscRollingonaPlane 111
4.3.3 Two-WheeledRobot 114
4.3.4 FreeRotationinTermsofQuaternions 116
4.4 AlternativestoLagrangeMultipliers 117
4.4.1 D’Alembert’sPrinciple 118
4.4.2 EquationsofUdwadiaandKalaba 119
4.5 TheFiberBundleViewpoint 122
4.6 Exercises 127
Contents IX
5 IntegrableSystems 129
5.1 FreeRotation 129
5.1.1 IntegralsofMotion 129
5.1.2 ReductiontoQuadrature 130
5.1.3 FreeRotationinTermsofAndoyerVariables 136
5.1.4 PoinsotConstructionandGeometricPhase 140
5.2 LagrangeTop 145
5.2.1 IntegralsofMotionandReductiontoQuadrature 145
5.2.2 MotionoftheTop’sAxis 147
5.3 TheGyrostat 148
5.3.1 BifurcationofthePhasePortrait 149
5.3.2 ReductiontoQuadrature 150
5.4 KowalevskyTop 153
5.5 LiouvilleToriandLaxEquations 154
5.5.1 LiouvilleTori 155
5.5.2 LaxEquations 156
5.6 Exercises 159
6 NumericalMethods 161
6.1 ClassicalODEIntegrators 161
6.2 SymplecticODEIntegrators 168
6.3 LieGroupMethods 170
6.4 Differential–AlgebraicSystems 179
6.5 WobblestoneCaseStudy 182
6.6 Exercises 187
7 Applications 189
7.1 PrecessionandNutation 189
7.1.1 GravitationalAttraction 190
7.1.2 PrecessionandNutationviaLagrangianMechanics 192
7.1.3 AHamiltonianFormulation 195
7.2 GravityGradientStabilizationofSatellites 197
7.2.1 KinematicsinTermsofYaw-Pitch-Roll 198
7.2.2 RotationEquationsforanAsymmetricSatellite 198
7.2.3 LinearStabilityAnalysis 200
7.3 MotionofaMultibody: ARobotArm 203
7.4 MolecularDynamics 210
X Contents
Appendix
A SphericalTrigonometry 213
B EllipticFunctions 218
B.1 EllipticFunctionsViatheSimplePendulum 218
B.2 AlgebraicRelationsAmongEllipticFunctions 222
B.3 DifferentialEquationsSatisfiedbyEllipticFunctions 224
B.3.1 TheAdditionFormulas 224
C LieGroupsandLieAlgebras 225
C.1 InfinitesimalGeneratorsofRotations 225
C.2 LieGroups 227
C.2.1 Examples 230
C.3 LieAlgebras 231
C.3.1 Examples 232
C.3.2 AdjointOperators 234
C.4 LieGroup–LieAlgebraRelations 237
C.4.1 TheExpMap 237
( )
C.4.2 TheDerivativeofexp A t 238
D Notation 241
References 243
Index 247
XI
Preface
Thisisatextbookonrigidbodymechanicswrittenforgraduateandadvanced
undergraduatestudents of science and engineering. The primary reasonfor
writing the book was to give an account of the subject which was firmly
grounded in both the classical and geometrical foundations of the subject.
The book is intended to be accessible to a student who is well prepared in
linear algebra and advanced calculus, who has had an introductory course
in mechanics and who has a certain degree of mathematical maturity. Any
mathematicsneededbeyondthisisincludedinthetext.
Chapter1dealswiththerotations,thebasicoperationinrigidbodytheory.
Rotationsarepresentedinseveralparameterizationsincludingaxisangle,Eu-
lerangle,quaternion,andCayley–Kleinparameters. TherotationsformaLie
groupwhichunderliesallofrigidbodymechanics.
Chapter 2 studies rigid body motions, angular velocity, and the physical
conceptsofangularmomentumandkineticenergy. Thefundamentalideaof
angular velocity is straight from the Lie algebra theory. These concepts are
illustratedwithseveralexamplesfromphysicsandengineering.
Chapter 3 studies rigid body dynamics in vector, Lagrangian, and Hamil-
tonian formulations. This chapter introduces many geometric concepts as
dynamics occurs on differentialmanifolds and for rigid body mechanics the
manifoldisoftenaLiegroup. Theideaoftheadjointactionisseentobebasic
totherigidbodyequationsofmotion. Thischaptercontainsmanyexamples
fromphysicsandengineering.
Chapter 4 considers the dynamics of constrained systems. Here Lagrange
multipliers are introduced and several ways of determining or eliminating
them are considered. This topic is rich in geometrical interactions and there
areseveralexamples,somestandardandsomenot.
Chapter5considerstheintegrableproblemsoffreerotation,Lagrange’stop,
andthegyrostat. TheKowalevskytopandLaxequationsarealsoconsidered.
GeometricaltopicsincludethePoinsotconstruction,thegeometricphase,and
RigidBodyMechanics:Mathematics,PhysicsandApplications.WilliamB.Heard
Copyright©2006WILEY-VCHVerlagGmbH&Co.KGaA,Weinheim
ISBN:3-527-40620-4