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AcomponentmodesynthesisalgorithmformultibodydynamicsofwindturbinesK.Holm-Jørgensen,S.R.K.Nielsen1DepartmentofCivilEngineering,AalborgUniversity,Sohngaardsholmsvej57,DK-9000Aalborg,DenmarkarticleinfoArticlehistory:Received29September2008Receivedinrevisedform4May2009Accepted6May2009HandlingEditor:M.P.CartmellAvailableonline5June2009abstractAsystemreductionschemerelatedtoamultibodyformulationofwindturbinedynamicsisdevised.Eachsubstructureisdescribedinitsownframeofreference,whichismovingfreelyinthevicinityofthemovingsubstructure,inprinciplewithoutanyconstraintstotherigidbodypartofthemotionofthesubstructure.Thesystemreductionisbasedonacomponentmodesynthesismethod,wheretheresponseoftheinternaldegreesoffreedomofthesubstructureisdescribedasthequasi-staticresponseinducedbytheboundarydegreesoffreedomviatheconstraintmodessuperimposedincombinationtoadynamiccomponentinducedbyinertialeffectsandinternalloads.Thelattercomponentismodelledbyatruncatedmodalexpansioninfixedinterfaceundampedeigenmodes.Theselectedmodalvectorbasefortheinternaldynamicsensuresthattheboundarydegreesoffreedomaccountfortherigid-bodydynamicsofthesubstructure,andexplicitlyrepresentthecouplingdegreesoffreedomattheinterfacetotheadjacentsubstructures.Themethodhasbeendemonstratedforabladestructure,whichhasbeenmodelledastwosubstructures.Twomodellingmethodshavebeenexaminedwherethefirstisbyuseoffixed–fixedeigenmodesfortheinnermostsubstructureandfixed–freeeigenmodesfortheoutermostsubstructure.Theotherapproachisbyuseoffixed–freeeigenmodesforbothsubstructures.Thefixed–fixedmethodshowsgoodcorrespondencewiththefullFEmodelwhichisnotthecaseforthefixed–freemethodduetoincompatibledisplacementsandrotationsattheinterfacebetweenthetwosubstructures.Moreover,theresultsfromthereducedmodelbyuseofconstantconstraintmodesandconstantfixedinterfacemodesoveralargeoperatingareaforthewindturbinebladearealmostidenticaltothefullFEmodel.&2009ElsevierLtd.Allrightsreserved.1.IntroductionFlexiblemultibodybasedsimulationsofthedynamicbehaviourofawindturbinerequiresadiscretizationinspaceforeachsubstructureofthesystem.Typically,thisisdonebyanFEmethod,ofteninvolvingmanydegreesoffreedomforeachsubstructure.Inordertoreducethecomputationaleffort,reducedordermodelsofthesubstructuresneedtobeimplemented.Especially,thisisnecessaryinstochasticanalysesbasedonMonteCarlosimulations,orduringthedesignphaseofawindturbine,wheremultipleloadcasesneedtobeanalysed.Areducedordermodelisalsonecessaryinsomeactivevibrationcontrolalgorithms,wherethestructuralmodelmustbeprocessedinrealtime.DuetothegeometricContentslistsavailableatScienceDirectjournalhomepage::10.1016/j.jsv.2009.05.007Correspondingauthor.E-mailaddresses:khj@civil.aau.dk(K.Holm-Jørgensen),soren.nielsen@civil.aau.dk(S.R.K.Nielsen).1Tel.:+4599408451;fax:+4598142555.JournalofSoundandVibration326(2009)753–767complexityoftheblades,whichotherwiserequiresmanyelementstomodel,thesystemreductioninthispaperisfocusedontheblades.Thebasicideaofflexiblemultibodydynamicsistointroduceamovingframeofreferencetoeachsubstructure.Relativetothemovingframeelasticdisplacementsarerelativelysmall,renderinglinearanalysispossible.Hence,nonlinearitiesareconfinedtothedescriptionofthemovingframe.Thisisdefinedbyapositionvectorandaparametervector,alsoknownasapseudovector,definingtheoriginandrotationofthemovingframerelativetoafixedframeofreference.Thestandardformulationofmultibodymethodsrequiresthatthereisnorigid-bodymotionbetweenthesubstructureanditsmovingframe.InAgrawalandShabana[1]anautomatedmethodisderivedtoeliminatetherigid-bodymotionofthebodyrelativetothemovingframe.Thisisdonebyimposingreferenceconditionsbyuseofabooleanmatrixontheshapefunctionswherebythedeformationmodesbecomeconsistentwiththeboundaryconditions.InShabana[2]itisdemonstratedthattwosetsofdeformationmodesassociatedwithtwodifferentsetsofboundaryconditionse.g.simplysupportedandfree–freecanbeusedtoobtainthesamesolutionprovidedthatthemovingframeisproperlyselected.ThepositionandorientationofthemovingframeisdefinedbyasetofLagrangiancoordinatesthatdescribetherigid-bodytranslationandrotation.Hereby,thesecoordinatesbecomeapartofthedegreesoffreedomofthemultibodysystem,seee.g.Nikravesh[3],Garcı´aandBayo[4],Ge´radinandCardona[5]andShabana[6].Theuseofsuchamixedsetofreferentialandelasticcoordinatesleadstohighlynonlinearsystemequations.Further,asaresultoftheinertialcouplingbetweenthesaiddegreesoffreedomthemassmatrixdependsonthereferentialcoordinates,evenwhenformulatedinthemovingframe.TocircumventthesedifficultiesKawamotoetal.[7–10]suggestedtoletthemovingframeofreferencefloatinacontrolledwayrelativetothemovingsubstructure,sothesearealwayssufficientlyclosetoeachother,inorderforthesmalldisplacementassumptiontobefulfilled.Hereby,thesystemmatricesdonotdependonthegeneralizedcoordinatesbyexplicitlypredictingtherigid-bodymotion.Toreduceoreliminatethegapbetweenthepredictedandactualmotion,itisnecessarytoregularlyupdatethemotionofthemov