Flexible Multibody Dynamics: Efficient Formulations and Applications

306 Pages·2016·8.9 MB·English

by  Arun K. Banerjee

FLEXIBLE MULTIBODY
DYNAMICS
EFFICIENT FORMULATIONS
AND APPLICATIONS
ArunK. Banerjee
FormerlyPrincipalResearchScientist
LockheedMartinAdvancedTechnologyCenter
PaloAlto,CA
USA
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LibraryofCongressCataloging-in-PublicationData
Banerjee,ArunK.,author.
Flexiblemultibodydynamics:efficientformulationsandapplications/ArunK.Banerjee.
pagescm
Includesbibliographicalreferencesandindex.
ISBN978-1-119-01564-2(hardback)
1.Machinery,Dynamicsof. 2.Multibodysystems–Mathematicalmodels. I.Title.
TJ173.B2352015
621.8′11–dc23
2015033623
AcataloguerecordforthisbookisavailablefromtheBritishLibrary.
Coverimages:Redrescuehelicoptermovinginbluesky:©aragami123345/iStock;Roboticprobeindeepspace,
computerillustration:VictorHabbickVisions/SciencePhotoLibrary/Getty
Setin10/12ptTimesbyAptaraInc.,NewDelhi,India
1 2016
Thisbookisdedicatedto:
Alpona,mywifewhosufferedoverthirty-five yearsasIworkedatmy
jobandonweekendsandnightsforpublications,andurgedmetowrite
thisbookafterIretired,butdidnotlivetoseeitpublished;
and
ProfessorThomasKane,mymentor, fromwhosemagnificentbooksI
learnedtododynamics, andwhostartedmeonthepathtopublished
researchthatthisbookembodies;
and
Ourdaughter, Onureena,andson, Abheek,whohavebeenmyblessings;
and
LockheedMartinSpaceSystemsCompany anditsmanagers,
particularlyDr.RonDotson,whogavemeafreehandtoworkonthe
algorithms reportedhere,whichledtoaflexiblemultibodydynamicscode
developedwithmajorhelpfrommycolleague,MarkLemak,andseveral
independentformulations.
Contents
Preface ix
1 DerivationofEquationsofMotion 1
1.1 AvailableAnalyticalMethodsandtheReasonforChoosingKane’sMethod 1
1.2 Kane’sMethodofDerivingEquationsofMotion 2
1.2.1 Kane’sEquations 4
1.2.2 SimpleExample:EquationsforaDoublePendulum 4
1.2.3 EquationsforaSpinningSpacecraftwithThreeRotors,FuelSlosh,
andNutationDamper 6
1.3 ComparisontoDerivationofEquationsofMotionbyLagrange’sMethod 11
1.3.1 Lagrange’sEquationsinQuasi-Coordinates 14
Reader’sExercise 15
1.4 Kane’sMethodofDirectDerivationofLinearizedDynamicalEquation 16
1.5 PrematurelyLinearizedEquationsandaPosterioriCorrectionbyadhoc
AdditionofGeometricStiffnessduetoInertiaLoads 19
1.6 Kane’sEquationswithUndeterminedMultipliersforConstrainedMotion 21
1.7 SummaryoftheEquationsofMotionwithUndeterminedMultipliers
forConstraints 22
1.8 ASimpleApplication 23
Appendix1.A GuidelinesforChoosingEfficientMotionVariablesin
Kane’sMethod 25
ProblemSet1 27
References 28
2 Deployment,Station-Keeping,andRetrievalofaFlexibleTether
ConnectingaSatellitetotheShuttle 29
2.1 EquationsofMotionofaTetheredSatelliteDeploymentfromthe
SpaceShuttle 30
2.1.1 KinematicalEquations 31
2.1.2 DynamicalEquations 32
2.1.3 SimulationResults 35
2.2 Thruster-AugmentedRetrievalofaTetheredSatellitetotheOrbitingShuttle 37
2.2.1 DynamicalEquations 37
Contents v
2.2.2 SimulationResults 47
2.2.3 Conclusion 47
2.3 DynamicsandControlofStation-KeepingoftheShuttle-TetheredSatellite 47
Appendix2.A SlidingImpactofaNoseCapwithaPackageofParachute
UsedforRecoveryofaBoosterLaunchingSatellites 49
Appendix2.B FormationFlyingwithMultipleTetheredSatellites 53
Appendix2.C OrbitBoostingofTetheredSatelliteSystemsby
ElectrodynamicForces 55
ProblemSet2 60
References 60
3 Kane’sMethodofLinearizationAppliedtotheDynamicsofaBeamin
LargeOverallMotion 63
3.1 NonlinearBeamKinematicswithNeutralAxisStretch,Shear,andTorsion 63
3.2 NonlinearPartialVelocitiesandPartialAngularVelocitiesfor
CorrectLinearization 69
3.3 UseofKane’sMethodforDirectDerivationofLinearized
DynamicalEquations 70
3.4 SimulationResultsforaSpace-BasedRoboticManipulator 76
3.5 ErroneousResultsObtainedUsingVibrationModesinConventionalAnalysis 78
ProblemSet3 79
References 82
4 DynamicsofaPlateinLargeOverallMotion 83
4.1 MotivatingResultsofaSimulation 83
4.2 ApplicationofKane’sMethodologyforProperLinearization 85
4.3 SimulationAlgorithm 90
4.4 Conclusion 92
Appendix4.A SpecializedModalIntegrals 93
ProblemSet4 94
References 96
5 DynamicsofanArbitraryFlexibleBodyinLargeOverallMotion 97
5.1 DynamicalEquationswiththeUseofVibrationModes 98
5.2 CompensatingforPrematureLinearizationbyGeometricStiffnessdueto
InertiaLoads 100
5.2.1 RigidBodyKinematicalEquations 104
5.3 SummaryoftheAlgorithm 105
5.4 CrucialTestandValidationoftheTheoryinApplication 106
Appendix5.A ModalIntegralsforanArbitraryFlexibleBody 112
ProblemSet5 114
References 114
6 FlexibleMultibodyDynamics:DenseMatrixFormulation 115
6.1 FlexibleBodySysteminaTreeTopology 115
6.2 KinematicsofaJointinaFlexibleMultibodyBodySystem 115
vi Contents
6.3 KinematicsandGeneralizedInertiaForcesforaFlexibleMultibodySystem 116
6.4 KinematicalRecurrenceRelationsPertainingtoaBodyandIts
InboardBody 120
6.5 GeneralizedActiveForcesduetoNominalandMotion-InducedStiffness 121
6.6 TreatmentofPrescribedMotionandInternalForces 126
6.7 “RuthlessLinearization”forVerySlowlyMovingArticulating
FlexibleStructures 126
6.8 SimulationResults 127
ProblemSet6 129
References 131
7 ComponentModeSelectionandModelReduction:AReview 133
7.1 Craig-BamptonComponentModesforConstrainedFlexibleBodies 133
7.2 ComponentModesbyGuyanReduction 136
7.3 ModalEffectiveMass 137
7.4 ComponentModelReductionbyFrequencyFiltering 138
7.5 CompensationforErrorsduetoModelReductionbyModal
TruncationVectors 138
7.6 RoleofModalTruncationVectorsinResponseAnalysis 141
7.7 ComponentModeSynthesistoFormSystemModes 143
7.8 FlexibleBodyModelReductionbySingularValueDecompositionof
ProjectedSystemModes 145
7.9 DerivingDampingCoefficientofComponentsfromDesired
SystemDamping 147
ProblemSet7 148
Appendix7.A MatlabCodesforStructuralDynamics 149
7.10 Conclusion 159
References 159
8 Block-DiagonalFormulationforaFlexibleMultibodySystem 161
8.1 Example:RoleofGeometricStiffnessduetoInterbodyLoadonaComponent 161
8.2 MultibodySystemwithRigidandFlexibleComponents 164
8.3 RecurrenceRelationsforKinematics 165
8.4 ConstructionoftheDynamicalEquationsinaBlock-DiagonalForm 168
8.5 SummaryoftheBlock-DiagonalAlgorithmforaTreeConfiguration 174
8.5.1 FirstForwardPass 174
8.5.2 BackwardPass 174
8.5.3 SecondForwardPass 175
8.6 NumericalResultsDemonstratingComputationalEfficiency 175
8.7 ModificationoftheBlock-DiagonalFormulationtoHandle
MotionConstraints 176
8.8 ValidationofFormulationwithGroundTestResults 182
8.9 Conclusion 186
Appendix8.A AnAlternativeDerivationofGeometricStiffnessdueto
InertiaLoads 187
Contents vii
ProblemSet8 188
References 189
9 EfficientVariables,RecursiveFormulation,andMulti-PointConstraints
inFlexibleMultibodyDynamics 191
9.1 SingleFlexibleBodyEquationsinEfficientVariables 191
9.2 MultibodyHingeKinematicsforEfficientGeneralizedSpeeds 196
9.3 RecursiveAlgorithmforFlexibleMultibodyDynamicswithMultiple
StructuralLoops 201
9.3.1 BackwardPass 201
9.3.2 ForwardPass 207
9.4 ExplicitSolutionofDynamicalEquationsUsingMotionConstraints 209
9.5 ComputationalResultsandSimulationEfficiencyforMoving
Multi-LoopStructures 210
9.5.1 SimulationResults 210
Acknowledgment 215
Appendix9.A Pseudo-CodeforConstrainednb-Bodym-LoopRecursive
AlgorithminEfficientVariables 216
ProblemSet9 220
References 220
10 EfficientModelingofBeamswithLargeDeflectionandLarge
BaseMotion 223
10.1 DiscreteModelingforLargeDeflectionofBeams 223
10.2 MotionandLoadsAnalysisbytheOrder-nFormulation 226
10.3 NumericalIntegrationbytheNewmarkMethod 230
10.4 NonlinearElastodynamicsviatheFiniteElementMethod 231
10.5 ComparisonoftheOrder-nFormulationwiththeFiniteElementMethod 233
10.6 Conclusion 237
Acknowledgment 238
ProblemSet10 238
References 238
11 Variable-nOrder-nFormulationforDeploymentandRetractionof
BeamsandCableswithLargeDeflection 239
11.1 BeamDiscretization 239
11.2 Deployment/RetractionfromaRotatingBase 240
11.2.1 InitializationStep 240
11.2.2 ForwardPass 240
11.2.3 BackwardPass 243
11.2.4 ForwardPass 244
11.2.5 Deployment/RetractionStep 244
11.3 NumericalSimulationofDeploymentandRetraction 246
11.4 DeploymentofaCablefromaShiptoaManeuveringUnderwater
SearchVehicle 247
viii Contents
11.4.1 CableDiscretizationandVariable-nOrder-nAlgorithmfor
ConstrainedSystemswithControlledEndBody 248
11.4.2 HydrodynamicForcesontheUnderwaterCable 254
11.4.3 NonlinearHolonomicConstraint,Control-ConstraintCoupling,
ConstraintStabilization,andCableTension 255
11.5 SimulationResults 257
ProblemSet11 261
References 267
12 Order-nEquationsofFlexibleRocketDynamics 269
12.1 Introduction 269
12.2 Kane’sEquationforaVariableMassFlexibleBody 269
12.3 MatrixFormoftheEquationsforVariableMassFlexibleBodyDynamics 274
12.4 Order-nAlgorithmforaFlexibleRocketwithCommandedGimbaled
NozzleMotion 275
12.5 NumericalSimulationofPlanarMotionofaFlexibleRocket 278
12.6 Conclusion 285
Acknowledgment 285
Appendix12.A SummaryAlgorithmforFindingTwoGimbalAngle
TorquesfortheNozzle 285
ProblemSet12 286
References 286
AppendixA EfficientGeneralizedSpeedsforaSingleFree-FlyingFlexibleBody 287
AppendixB AFORTRANCodeoftheOrder-nAlgorithm:Applicationto
anExample 291
Index 301
Preface
Thisbookisbasedonmypublishedresearchonderivingcomputationallyefficientequations
of motion of multibody systems with rotating, flexible components. It reflects my work of
over thirty-five years done mostly at Lockheed Missiles & Space Company, and also at
MartinMariettaandNorthropCorporations.Thecoverofthebookdepictstwoexamplesof
flexiblemultibodysystems:theGalileospacecraft,whichwassenttoJupiter,withitsrotating
antennadishonaninertially-fixedbasewithadeployedtruss,andahelicopterinflight.Other
examplesofmultibodysystems,apartfromthehumanbodyitself,areroboticmanipulators,
a space shuttle deploying a tethered subsatellite, and a ship reeling out a cable to a vehicle
doing sea floor mine searches. Formulation of equations of motion is the first step in their
simulation-baseddesign.
Inthisbook,IchoosetouseKane’smethodofderivingequationsofmotion,fortworeasons:
efficiencyinreducinglaborofderivingtheequations,andsimplicityofthefinalequationsdue
to a choice of variables that the method allows. However, the contribution of the book goes
beyondadirectformulationofKane’sequationstomorecomputationallyefficientalgorithms
like block-diagonal and order-n formulations. Another major contribution of this book is in
compensatingforerrorsofprematurelinearization,inherentwiththeuseofvibrationmodes
inlargeoverallmotionproblems,byusinggeometricstiffnessduetoinertialoads.
Ahighlightofthisbookistheapplicationofthetheorytocomplexproblems.InChapter1,
I explain Kane’s method, first with a simple example and then by applying it to a realistic
problem of the dynamics of a three-axis controlled spacecraft with fuel slosh. Presented
separately are Kane’s method of direct linearization of equation of motion and a method
of a posteriori compensation for premature linearization by adding geometric stiffness due
to inertia loads; in the Appendix, a guideline for choosing variables that simplify equations
of motion is provided. In Chapter 2, Kane’s method is used to derive nonlinear dynamical
equations for tethered satellite deployment, station-keeping and retrieval, and a problem of
impact dynamics of a nose cap during ejection of a parachute for recovery of a booster
launchingasatellite.Thenexttwochapterscoverlargeoverallmotionofbeamsandplatesthat
illustratetheapplicationofKane’smethodofdirectlinearization.Chapter5givesaderivation
ofequationsoflargeoverallmotionofanarbitraryflexiblebody,withamethodofredeeming
prematurely linearized equations by adding motion-induced geometric stiffness. Chapter 6
incorporatesthemotion-inducedgeometricstiffnessintothedynamicsofasystemofflexible
bodies in large overall motions. Chapter 7 is a review material from structural dynamics,
based mainly on the book by Craig, with some additional work on mode selection done at
Lockheed. Chapter 8 produces an algorithm for dynamical equations, with block-diagonal