Computational Flexible Multibody Dynamics: A Differential-Algebraic Approach

254 Pages·2013·3.217 MB·English

by  Bernd Simeon (auth.)

DAE-F
Diff erential-Algebraic Equations Forum
Bernd Simeon
Computational
Flexible Multibody
Dynamics
A Diff erential-Algebraic Approach
Differential-Algebraic Equations Forum
Editors-in-Chief
AchimIlchmann(TUIlmenau,Ilmenau,Germany)
TimoReis(UniversityofHamburg,Hamburg,Germany)
EditorialBoard
LarryBiegler(CarnegieMellonUniversity,Pittsburgh,USA)
SteveCampbell(NorthCarolinaStateUniversity,Raleigh,USA)
ClausFührer(LundsUniversitet,Lund,Sweden)
RoswithaMärz(HumboldtUniversitätzuBerlin,Berlin,Germany)
StephanTrenn(TUKaiserslautern,Kaiserslautern,Germany)
PeterKunkel(UniversitätLeipzig,Leipzig,Germany)
RicardoRiaza(UniversidadPolitécnicadeMadrid,Madrid,Spain)
VuHoangLinh(VietnamNationalUniversity,Hanoi,Vietnam)
MatthiasGerdts(UniversitätderBundeswehrMünchen,Munich,Germany)
SebastianSager(Otto-von-Guericke-UniversitätMagdeburg,Magdeburg,Germany)
SebastianSchöps(TUDarmstadt,Darmstadt,Germany)
BerndSimeon(TUKaiserslautern,Kaiserslautern,Germany)
EvaZerz(RWTHAachen,Aachen,Germany)
Differential-Algebraic Equations Forum
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Forfurthervolumes:
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Bernd Simeon
Computational
Flexible Multibody
Dynamics
A Differential-Algebraic Approach
BerndSimeon
Felix-Klein-ZentrumfürMathematik
TechnischeUniversitätKaiserslautern
Kaiserslautern,Germany
ISBN978-3-642-35157-0 ISBN978-3-642-35158-7(eBook)
DOI10.1007/978-3-642-35158-7
SpringerHeidelbergNewYorkDordrechtLondon
LibraryofCongressControlNumber:2013942394
MathematicsSubjectClassification(2010): 74S05,65M12,65L80
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knowledge makeththyworth.
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Preface
Flexible multibody dynamics is a rapidly growing field with various applications
invehicleanalysis,aerospaceengineering,robotics,andbiomechanics.Overall,the
mathematical models in this field seem reasonably settled and, in computational
terms,greatstrideshavebeenmadeoverthelasttwodecadesassophisticatedsoft-
warepackagesarenowadayscapableofsimulatinghighlycomplexstructureswith
rigidanddeformablecomponents.
Written from the perspective of a numerical analyst, this monograph provides
comprehensiveinformationonboththemathematicalframeworkaswellasthenu-
mericalmethodsforflexiblemultibodydynamics.Suchapresentationofthesubject
appears to be needed and should benefit not only graduate students and scientists
working in the field but also those interested in time-dependent partial differen-
tial equations and heterogeneous problems with multiple time scales. At the same
time, a number of open issues at the frontiers of research are addressed by taking
a differential-algebraic approach and extending it to the notion of transient saddle
pointproblems.
Theresultsinthisbookaretheproductofresearchworkundertakenduringthe
pastfifteenyears.Partsofthematerialwerealsocoveredingraduatecoursesoffered
attheUniversitätKarlsruhe,theTechnischeUniversitätMünchen,andtheTechnis-
che Universität Kaiserslautern. Though it is self-contained in many respects, the
textnonethelesscallsforsomebasicknowledgeofdiscretizationschemesforordi-
nary differential equations and of the finite element method. Simulation examples
illustratethemathematicalmodelsandthecomputationaltechniques.
The interplay of modeling and numerics is a key feature in flexible multibody
dynamicsandmainlydeterminesthemethodologychoseninthismonograph.This
isreflectedbytheorganizationintotwomainpartswithfourchapterseach.While
thefirstpartdevelopsadetailedmathematicalframeworkforsystemsofrigidand
deformablebodies,thesecondintroducesdiscretizationmethodsinspaceandtime
andconcentratesinparticularontheproblemofdifferenttimescales.
Though this outline seems to indicate a purely mathematical treatment of the
subject,thisbookisdefinitelynotintendedfortheexclusiveuseofspecialistsinnu-
mericalanalysis.Onthecontrary,itshouldpromotethecommunicationbetweenthe
vii
viii Preface
engineerandthemathematician,thusleadingtoatrulyinterdisciplinarycooperation
withmutualbenefits.
Theoriginofthisbookgoesbackalongway.AfterIhadcompletedthePh.D.,
itwasmyadvisorPeterRentropwhofirstdrewmyattentiontothefieldofflexible
multibodydynamics.Itisapleasuretothankhimforhiscontinuinginterestinthe
subject and for his support over the years. The next colleague and friend I would
like to thank is Claus Führer, with whom I had various inspiring discussions and
whocarefullyreadanearlierversionofthemanuscript.Manyothercolleagueshave
alsocontributedbyprovidingsuggestionsandvaluableremarksonspecificissues:
Martin Arnold, Peter Betsch, Severiano Gonzalez Pinto, Ernst Hairer, Domingo
HernándezAbreu,LaurentJay,ChristianLubich,PanosPapadopoulos,LindaPet-
zold,WernerC.Rheinboldt,JohnStrain,andBarbaraWohlmuth.Moreover,Iwish
to sincerely thank all my master and Ph.D. students who worked in this or related
fields for their collaboration, their effort, and their patience. Special thanks go to
Klaus Dressler and his coworkers at the Fraunhofer ITWM in Kaiserslautern for
theirhintsonpracticalaspectsandforsupplyingtherealisticexampleinSect.8.5.
Lastbutnotleast,IexpressmygratitudetoDorisHemmer-KolbandKirstenHöffler
forthefinalproofreadingandtoMarioAignerfromSpringerVerlagfortheefficient
handlingofthemanuscript.
Kaiserslautern BerndSimeon
April2013
Contents
PartI MathematicalModels
1 APointofDeparture . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.1 ElasticPendulum . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.1.1 SmallElasticDisplacement . . . . . . . . . . . . . . . . . 5
1.1.2 FloatingReferenceFrame. . . . . . . . . . . . . . . . . . 6
1.1.3 GalerkinProjection . . . . . . . . . . . . . . . . . . . . . 7
1.1.4 TimeIntegration. . . . . . . . . . . . . . . . . . . . . . . 8
1.2 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.2.1 TreatmentofConstraints . . . . . . . . . . . . . . . . . . 9
1.2.2 DiscretizationinSpaceandTime . . . . . . . . . . . . . . 10
1.3 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2 RigidMultibodyDynamics . . . . . . . . . . . . . . . . . . . . . . . 13
2.1 EquationsofMotion . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.1.1 Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.1.2 Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.1.3 RelationtoHamiltonianSystems . . . . . . . . . . . . . . 20
2.1.4 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.2.1 SliderCrank . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.2.2 Newton–EulerEquations . . . . . . . . . . . . . . . . . . 23
2.2.3 WheelSuspension . . . . . . . . . . . . . . . . . . . . . . 26
2.3 Differential-AlgebraicEquations . . . . . . . . . . . . . . . . . . 28
2.3.1 BasicTypesofDAEs . . . . . . . . . . . . . . . . . . . . 28
2.3.2 TheIndex . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.3.3 ConstraintManifoldandLocalStateSpaceForm. . . . . . 35
2.3.4 SolutionInvariantsversusConstraints. . . . . . . . . . . . 36
2.3.5 Differential-AlgebraicEquationsinSaddlePointForm. . . 38
2.3.6 RemarksandFurtherReferences . . . . . . . . . . . . . . 39
ix
x Contents
2.4 AnalysisoftheEquationsofConstrainedMechanicalMotion . . . 39
2.4.1 IndexandExistenceofSolutions . . . . . . . . . . . . . . 40
2.4.2 MinimaxCharacterizationofConstraints . . . . . . . . . . 42
2.4.3 InfluenceofPerturbations . . . . . . . . . . . . . . . . . . 43
2.4.4 AlternativeFormulations . . . . . . . . . . . . . . . . . . 44
2.4.5 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3 ElasticMotion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.1 BasicEquationsofanElasticBody . . . . . . . . . . . . . . . . . 53
3.1.1 VariationalFormulation . . . . . . . . . . . . . . . . . . . 53
3.1.2 EquationsofMotion . . . . . . . . . . . . . . . . . . . . . 58
3.1.3 SmoothnessandAppropriateFunctionSpaces . . . . . . . 59
3.1.4 TheTraceSpace . . . . . . . . . . . . . . . . . . . . . . . 64
3.2 ConstraintsinLinearElasticity . . . . . . . . . . . . . . . . . . . 67
3.2.1 VariationalFormulation . . . . . . . . . . . . . . . . . . . 68
3.2.2 AppropriateFunctionSpaces . . . . . . . . . . . . . . . . 69
3.3 MathematicalAnalysis. . . . . . . . . . . . . . . . . . . . . . . . 70
3.3.1 ExistenceofSolutions . . . . . . . . . . . . . . . . . . . . 71
3.3.2 InfluenceofPerturbations . . . . . . . . . . . . . . . . . . 76
3.3.3 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
3.4 RelatedMathematicalModels . . . . . . . . . . . . . . . . . . . . 80
3.4.1 DomainDecomposition . . . . . . . . . . . . . . . . . . . 80
3.4.2 DynamicContact . . . . . . . . . . . . . . . . . . . . . . 82
3.4.3 IncompressibleElasticBody . . . . . . . . . . . . . . . . 84
4 FlexibleMultibodyDynamics . . . . . . . . . . . . . . . . . . . . . . 87
4.1 FloatingReferenceFrame . . . . . . . . . . . . . . . . . . . . . . 87
4.1.1 VariationalFormulation . . . . . . . . . . . . . . . . . . . 88
4.1.2 EquationsofUnconstrainedMotion . . . . . . . . . . . . . 90
4.1.3 ExtensionsandSpecialCases . . . . . . . . . . . . . . . . 93
4.2 Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
4.2.1 WeakFormofEquationsofConstrainedMotion . . . . . . 97
4.2.2 ModelingJoints . . . . . . . . . . . . . . . . . . . . . . . 99
4.2.3 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
4.3 FlexibleMultibodySystem . . . . . . . . . . . . . . . . . . . . . 104
4.3.1 VariationalFormulation . . . . . . . . . . . . . . . . . . . 104
4.3.2 EquationsofMotion . . . . . . . . . . . . . . . . . . . . . 106
4.4 SpecialBodies . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
4.4.1 PlaneStrainandPlaneStress . . . . . . . . . . . . . . . . 109
4.4.2 BeamModel . . . . . . . . . . . . . . . . . . . . . . . . . 110
4.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
4.5.1 SliderCrank . . . . . . . . . . . . . . . . . . . . . . . . . 112
4.5.2 TruckModel . . . . . . . . . . . . . . . . . . . . . . . . . 115
4.5.3 PantographandCatenary . . . . . . . . . . . . . . . . . . 117
4.6 SummaryofKeyFormulasinPartI . . . . . . . . . . . . . . . . . 120