ABAQUS关于固有频率的提取方法

Abaqus固有频率提取6.3.5

Naturalfrequencyextraction

Products:

Abaqus/Standard

Abaqus/CAE

Abaqus/AMS

References“Procedures:overview,”

Section6.1.1“Generalandlinearperturbationprocedures,”

Section6.1.2“Dynamicanalysisprocedures:overview,”

Section6.3.1\t"v610keywin"*FREQUENCY\t"v610usiwin"“Configuringafrequencyprocedure”in“Configuringlinearperturbationanalysisprocedures,”

Section14.11.2oftheAbaqus/CAEUser'sManualOverview

Thefrequencyextractionprocedure:performseigenvalueextractiontocalculatethenaturalfrequenciesandthecorrespondingmodeshapesofasystem;willincludeinitialstressandloadstiffnesseffectsduetopreloadsandinitialconditionsifgeometricnonlinearityisaccountedforinthebasestate,sothatsmallvibrationsofapreloadedstructurecanbemodeled;willcomputeresidualmodesifrequested;isalinearperturbationprocedure;canbeperformedusingthetraditionalAbaqussoftwarearchitectureor,ifappropriate,thehigh-performanceSIMarchitecture(see

“UsingtheSIMarchitectureformodalsuperpositiondynamicanalyses”in“Dynamicanalysisprocedures:overview,”

Section6.3.1);andsolvestheeigenfrequencyproblemonlyforsymmetricmassandstiffnessmatrices;thecomplexeigenfrequencysolvermustbeusedifunsymmetriccontributions,suchastheloadstiffness,areneeded.

Eigenvalueextraction

Theeigenvalueproblemforthenaturalfrequenciesofanundampedfiniteelementmodelis

whereisthemassmatrix(whichissymmetricandpositivedefinite);

isthestiffnessmatrix(whichincludesinitialstiffnesseffectsifthebasestateincludedtheeffectsofnonlineargeometry);

istheeigenvector(themodeofvibration);and

M

and

Naredegreesoffreedom.

When

ispositivedefinite,alleigenvaluesarepositive.Rigidbodymodesandinstabilitiescause

tobeindefinite.Rigidbodymodesproducezeroeigenvalues.Instabilitiesproducenegativeeigenvaluesandoccurwhenyouincludeinitialstresseffects.Abaqus/Standardsolvestheeigenfrequencyproblemonlyforsymmetricmatrices.

Selectingtheeigenvalueextractionmethod

Abaqus/Standardprovidesthreeeigenvalueextractionmethods:LanczosAutomaticmulti-levelsubstructuring(AMS),anadd-onanalysiscapabilityforAbaqus/StandardSubspaceiteration

Inaddition,youmustconsiderthesoftwarearchitecturethatwillbeusedforthesubsequentmodalsuperpositionprocedures.Thechoiceofarchitecturehasminimalimpactonthefrequencyextractionprocedure,buttheSIMarchitecturecanoffersignificantperformanceimprovementsoverthetraditionalarchitectureforsubsequentmode-basedsteady-stateortransientdynamicprocedures(see

“UsingtheSIMarchitectureformodalsuperpositiondynamicanalyses”in“Dynamicanalysisprocedures:overview,”

Section6.3.1).Thearchitecturethatyouuseforthefrequencyextractionprocedureisusedforallsubsequentmode-basedlineardynamicprocedures;youcannotswitcharchitecturesduringananalysis.Thesoftwarearchitecturesusedbythedifferenteigensolversareoutlinedin

Table6.3.5–1.Table6.3.5–1

Softwarearchitecturesavailablewithdifferenteigensolvers.SoftwareArchitectureEigensolverLanczosAMSSubspaceIterationTraditional

SIM

TheLanczossolverwiththetraditionalarchitectureisthedefaulteigenvalueextractionmethodbecauseithasthemostgeneralcapabilities.However,theLanczosmethodisgenerallyslowerthantheAMSmethod.TheincreasedspeedoftheAMSeigensolverisparticularlyevidentwhenyourequirealargenumberofeigenmodesforasystemwithmanydegreesoffreedom.However,theAMSmethodhasthefollowinglimitations:AllrestrictionsimposedonSIM-basedlineardynamicproceduresalsoapplytomode-basedlineardynamicanalysesbasedonmodeshapescomputedbytheAMSeigensolver.See

“UsingtheSIMarchitectureformodalsuperpositiondynamicanalyses”in“Dynamicanalysisprocedures:overview,”

Section6.3.1,fordetails.TheAMSeigensolverdoesnotcomputecompositemodaldampingfactors,participationfactors,ormodaleffectivemasses.However,ifparticipationfactorsareneededforprimarybasemotions,theywillbecomputedbutarenotwrittentotheprinteddata(.dat)file.YoucannotusetheAMSeigensolverinananalysisthatcontainspiezoelectricelements.Youcannotrequestoutputtotheresults(.fil)fileinanAMSfrequencyextractionstep.Ifyourmodelhasmanydegreesoffreedomandtheselimitationsareacceptable,youshouldusetheAMSeigensolver.Otherwise,youshouldusetheLanczoseigensolver.TheLanczoseigensolverandthesubspaceiterationmethodaredescribedin\t"v610stmwin"“Eigenvalueextraction,”

Section2.5.1oftheAbaqusTheoryManual.

LanczoseigensolverFortheLanczosmethodyouneedtoprovidethemaximumfrequencyofinterestorthenumberofeigenvaluesrequired;Abaqus/Standardwilldetermineasuitableblocksize(althoughyoucanoverridethischoice,ifneeded).Ifyouspecifyboththemaximumfrequencyofinterestandthenumberofeigenvaluesrequiredandtheactualnumberofeigenvaluesisunderestimated,Abaqus/Standardwillissueacorrespondingwarningmessage;theremainingeigenmodescanbefoundbyrestartingthefrequencyextraction.Youcanalsospecifytheminimumfrequenciesofinterest;Abaqus/Standardwillextracteigenvaluesuntileithertherequestednumberofeigenvalueshasbeenextractedinthegivenrangeorallthefrequenciesinthegivenrangehavebeenextracted.See

“UsingtheSIMarchitectureformodalsuperpositiondynamicanalyses”in“Dynamicanalysisprocedures:overview,”

Section6.3.1,forinformationonusingtheSIMarchitecturewiththeLanczoseigensolver.Input

File

Usage:

\t"v610keywin"*FREQUENCY,EIGENSOLVER=LANCZOSAbaqus/CAE

Usage:

Stepmodule:

StepCreate:

Frequency:

Basic:

Eigensolver:

LanczosChoosingablocksizefortheLanczosmethodIngeneral,theblocksizefortheLanczosmethodshouldbeaslargeasthelargestexpectedmultiplicityofeigenvalues(thatis,thelargestnumberofmodeswiththesamefrequency).Ablocksizelargerthan10isnotrecommended.Ifthenumberofeigenvaluesrequestedis

n,thedefaultblocksizeistheminimumof(7,

n).Thechoiceof7forblocksizeprovestobeefficientforproblemswithrigidbodymodes.ThenumberofblockLanczosstepswithineachLanczosrunisusuallydeterminedbyAbaqus/Standardbutcanbechangedbyyou.Ingeneral,ifaparticulartypeofeigenproblemconvergesslowly,providingmoreblockLanczosstepswillreducetheanalysiscost.Ontheotherhand,ifyouknowthataparticulartypeofproblemconvergesquickly,providingfewerblockLanczosstepswillreducetheamountofin-corememoryused.ThedefaultvaluesareBlocksizeMaximumnumberofblockLanczossteps180250345≥435

Automaticmulti-levelsubstructuring(AMS)eigensolverFortheAMSmethodyouneedonlyspecifythemaximumfrequencyofinterest(theglobalfrequency),andAbaqus/Standardwillextractallthemodesuptothisfrequency.Youcanalsospecifytheminimumfrequenciesofinterestand/orthenumberofrequestedmodes.However,specifyingthesevalueswillnotaffectthenumberofmodesextractedbytheeigensolver;itwillaffectonlythenumberofmodesthatarestoredforoutputorforasubsequentmodalanalysis.TheexecutionoftheAMSeigensolvercanbecontrolledbyspecifyingthreeparameters:

,

,and

.Thesethreeparametersmultipliedbythemaximumfrequencyofinterestdefinethreecut-offfrequencies.

(defaultvalueof5)controlsthecutofffrequencyforsubstructureeigenproblemsinthereductionphase,while

and

(defaultvaluesof1.7and1.1,respectively)controlthecutofffrequenciesusedtodefineastartingsubspaceinthereducedeigensolutionphase.Generally,increasingthevalueof

and

improvestheaccuracyoftheresultsbutmayaffecttheperformanceoftheanalysis.RequestingeigenvectorsatallnodesBydefault,theAMSeigensolvercomputeseigenvectorsateverynodeofthemodel.Input

File

Usage:

\t"v610keywin"*FREQUENCY,EIGENSOLVER=AMSAbaqus/CAE

Usage:

Stepmodule:

StepCreate:

Frequency:

Basic:

Eigensolver:

AMSRequestingeigenvectorsonlyatspecifiednodesAlternatively,youcanspecifyanodeset,andeigenvectorswillbecomputedandstoredonlyatthenodesthatbelongtothatnodeset.Thenodesetthatyouspecifymustincludeallnodesatwhichloadsareappliedoroutputisrequestedinanysubsequentmodalanalysis(thisincludesanyrestartedanalysis).Ifelementoutputisrequestedorelement-basedloadingisapplied,thenodesattachedtotheassociatedelementsmustalsobeincludedinthisnodeset.Computingeigenvectorsatonlyselectednodesimprovesperformanceandreducestheamountofstoreddata.Therefore,itisrecommendedthatyouusethisoptionforlargeproblems.Input

File

Usage:

\t"v610keywin"*FREQUENCY,EIGENSOLVER=AMS,NSET=nameAbaqus/CAE

Usage:

Stepmodule:

StepCreate:

Frequency:

Basic:

Eigensolver:

AMS:

LimitregionofsavedeigenvectorsControllingtheAMSeigensolverTheAMSmethodconsistsofthefollowingthreephases:Reductionphase:

InthisphaseAbaqus/Standardusesamulti-levelsubstructuringtechniquetoreducethefullsysteminawaythatallowsaveryefficienteigensolutionofthereducedsystem.Theapproachcombinesasparsefactorizationbasedonamulti-levelsupernodeeliminationtreeandalocaleigensolutionateachsupernode.Startingfromthelowestlevelsupernodes,weuseaCraig-Bamptonsubstructurereductiontechniquetosuccessivelyreducethesizeofthesystemasweprogressupwardintheeliminationtree.Ateachsupernodealocaleigensolutionisobtainedbasedonfixingthedegreesoffreedomconnectedtothenexthigherlevelsupernode(thesearethelocalretainedor“fixed-interface”degreesoffreedom).Attheendofthereductionphasethefullsystemhasbeenreducedsuchthatthereducedstiffnessmatrixisdiagonalandthereducedmassmatrixhasunitdiagonalvaluesbutcontainsoff-diagonalblocksofnonzerovaluesrepresentingthecouplingbetweenthesupernodes.Thecostofthereductionphasedependsonthesystemsizeandthenumberofeigenvaluesextracted(thenumberofeigenvaluesextractediscontrolledindirectlybyspecifyingthehighesteigenfrequencydesired).Youcanmaketrade-offsbetweencostandaccuracyduringthereductionphasethroughthe

parameter.Thisparametermultipliedbythehighesteigenfrequencyspecifiedforthefullmodelyieldsthehighesteigenfrequencythatisextractedinthelocalsupernodeeigensolutions.Increasingthevalueof

increasestheaccuracyofthereductionsincemorelocaleigenmodesareretained.However,increasingthenumberofretainedmodesalsoincreasesthecostofthereducedeigensolutionphase,whichisdiscussednext.Reducedeigensolutionphase:

InthisphaseAbaqus/Standardcomputestheeigensolutionofthereducedsystemthatcomesfromthepreviousphase.Althoughthereducedsystemtypicallyistwoordersofmagnitudesmallerinsizethantheoriginalsystem,generallyitstillistoolargetosolvedirectly.Thus,thesystemisfurtherreducedmainlybytruncatingtheretainedeigenmodesandthensolvedusingasinglesubspaceiterationstep.ThetwoAMSparameters,

and

,defineastartingsubspaceofthesubspaceiterationstep.Thedefaultvaluesoftheseparametersarecarefullychosenandprovideaccurateresultsinmostcases.Whenamoreaccuratesolutionisneeded,therecommendedprocedureistoincreasebothparametersproportionallyfromtheirrespectivedefaultvalues.Recoveryphase:

Inthisphasetheeigenvectorsoftheoriginalsystemarerecoveredusingeigenvectorsofthereducedproblemandlocalsubstructuremodes.Ifyourequestrecoveryatspecifiednodes,theeigenvectorsarecomputedonlyatthosenodes.

SubspaceiterationmethodForthesubspaceiterationprocedureyouneedonlyspecifythenumberofeigenvaluesrequired;Abaqus/Standardchoosesasuitablenumberofvectorsfortheiteration.Ifthesubspaceiterationtechniqueisrequested,youcanalsospecifythemaximumfrequencyofinterest;Abaqus/Standardextractseigenvaluesuntileithertherequestednumberofeigenvalueshasbeenextractedorthelastfrequencyextractedexceedsthemaximumfrequencyofinterest.Input

File

Usage:

\t"v610keywin"*FREQUENCY,EIGENSOLVER=SUBSPACEAbaqus/CAE

Usage:

Stepmodule:

StepCreate:

Frequency:

Basic:

Eigensolver:

SubspaceStructural-acousticcoupling

Structural-acousticcouplingaffectsthenaturalfrequencyresponseofsystems.InAbaqusonlytheLanczoseigensolverfullyincludesthiseffect.InAbaqus/AMSandthesubspaceeigensolvertheeffectofcouplingisneglectedforthepurposeofcomputingthemodesandfrequencies;thesearecomputedusingnaturalboundaryconditionsatthestructural-acousticcouplingsurface.Anintermediatedegreeofconsiderationofthestructural-acousticcouplingoperatoristhedefaultinAbaqus/AMSandtheLanczoseigensolver,whichisbasedontheSIMarchitecture:thecouplingisprojectedontothemodalspaceandstoredforlateruse.Structural-acousticcouplingusingtheLanczoseigensolverwithouttheSIMarchitectureIfstructural-acousticcouplingispresentinthemodelandtheLanczosmethodnotbasedontheSIMarchitectureisused,Abaqus/Standardextractsthecoupledmodesbydefault.Becausethesemodesfullyaccountforcoupling,theyrepresentthemathematicallyoptimalbasisforsubsequentmodalprocedures.Theeffectismostnoticeableinstronglycoupledsystemssuchassteelshellsandwater.However,coupledstructural-acousticmodescannotbeusedinsubsequentrandomresponseorresponsespectrumanalyses.Youcandefinethecouplingusingeitheracoustic-structuralinteractionelements(see

“Acousticinterfaceelements,”

Section29.14.1)orthesurface-basedtieconstraint(see

“Acoustic,shock,andcoupledacoustic-structuralanalysis,”

Section6.10.1).Itispossibletoignorecouplingwhenextractingacousticandstructuralmodes;inthiscasethecouplingboundaryistreatedastraction-freeonthestructuralsideandrigidontheacousticside.Input

File

Usage:

Usethefollowingoptiontoaccountforstructural-acousticcouplingduringthefrequencyextraction:\t"v610keywin"*FREQUENCY,EIGENSOLVER=LANCZOS,ACOUSTICCOUPLING=ON(defaultiftheSIMarchitectureisnotused)Usethefollowingoptiontoignorestructural-acousticcouplingduringthefrequencyextraction:\t"v610keywin"*FREQUENCY,EIGENSOLVER=LANCZOS,ACOUSTICCOUPLING=OFFAbaqus/CAE

Usage:

Stepmodule:

StepCreate:

Frequency:

Basic:

Eigensolver:Lanczos,toggle

Includeacoustic-structuralcouplingwhereapplicableStructural-acousticcouplingusingtheAMSandLanczoseigensolverbasedontheSIMarchitectureForfrequencyextractionsthatusetheAMSeigensolverortheLanczoseigensolverbasedontheSIMarchitecture,themodesarecomputedusingtraction-freeboundaryconditionsonthestructuralsideofthecouplingboundaryandrigidboundaryconditionsontheacousticside.Structural-acousticcouplingoperators(see

“Acoustic,shock,andcoupledacoustic-structuralanalysis,”

Section6.10.1)areprojectedbydefaultontothesubspaceofeigenvectors.Contributionstotheseglobaloperators,whichcomefromsurface-basedtieconstraintsdefinedbetweenstructuralandacousticsurfaces,areassembledintoglobalmatricesthatareprojectedontothemodeshapesandusedinsubsequentSIM-basedmodaldynamicprocedures.User-definedacoustic-structuralinteractionelements(see

“Acousticinterfaceelements,”

Section29.14.1)cannotbeusedinanAMSeigenvalueextractionanalysis.Input

File

Usage:

Useeitherofthefollowingoptionstoprojectstructural-acousticcouplingoperatorsontothesubspaceofeigenvectors:\t"v610keywin"*FREQUENCY,EIGENSOLVER=AMS,ACOUSTICCOUPLING=PROJECTION(defaultfortheAMSeigensolver)or\t"v610keywin"*FREQUENCY,EIGENSOLVER=LANCZOS,SIM,ACOUSTICCOUPLING=PROJECTION(defaultinSIM-basedanalysis)Usethefollowingoptiontodisabletheprojectionofstructural-acousticcouplingoperators:\t"v610keywin"*FREQUENCY,ACOUSTICCOUPLING=OFFAbaqus/CAE

Usage:

Usethefollowingoptiontoprojectstructural-acousticcouplingoperatorsontothesubspaceofeigenvectors:Stepmodule:

StepCreate:

Frequency:

Basic:

Eigensolver:AMS,toggleon

Projectacoustic-structuralcouplingwhereapplicableUsethefollowingoptiontodisabletheprojectionofstructural-acousticcouplingoperators:Stepmodule:

StepCreate:

Frequency:

Basic:

Eigensolver:AMS,toggleoff

Projectacoustic-structuralcouplingwhereapplicableProjectionofstructural-acousticcouplingoperatorsusingtheLanczoseigensolverbasedontheSIMarchitectureisnotsupportedinAbaqus/CAE.SpecifyingafrequencyrangefortheacousticmodesBecausestructural-acousticcouplingisignoredduringtheAMSandSIM-basedLanczoseigenanalysis,thecomputedresonanceswill,inprinciple,behigherthanthoseofthefullycoupledsystem.Thismaybeunderstoodasaconsequenceofneglectingthemassofthefluidinthestructuralphaseandviceversa.Forthecommonmetalandaircase,thestructuralresonancesmayberelativelyunaffected;however,someacousticmodesthataresignificantinthecoupledresponsemaybeomittedduetotheair'supwardfrequencyshiftduringeigenanalysis.Therefore,Abaqusallowsyoutospecifyamultiplier,sothatthemaximumacousticfrequencyintheanalysisistakentobehigherthanthestructuralmaximum.Input

File

Usage:

Useeitherofthefollowingoptions:\t"v610keywin"*FREQUENCY,EIGENSOLVER=AMS,,,,,,acousticrangefactoror\t"v610keywin"*FREQUENCY,EIGENSOLVER=LANCZOS,SIM,,,,,,acousticrangefactorAbaqus/CAE

Usage:

Stepmodule:

StepCreate:

Frequency:

Basic:

Eigensolver:AMS,

Acousticrangefactor:

acousticrangefactor

SpecifyingafrequencyrangefortheacousticmodeswhenusingtheSIM-basedLanczoseigenanalysisisnotsupportedinAbaqus/CAE.EffectsoffluidmotiononnaturalfrequencyanalysisofacousticsystemsToextractnaturalfrequenciesfromanacoustic-onlyorcoupledstructural-acousticsysteminwhichfluidmotionisprescribedusinganacousticflowvelocity,eithertheLanczosmethodorthecomplexeigenvalueextractionprocedurecanbeused.IntheformercaseAbaqusextractsreal-onlyeigenvaluesandconsidersthefluidmotion'seffectsonlyontheacousticstiffnessmatrix.Thus,theseresultsareofprimaryinterestasabasisforsubsequentlinearperturbationprocedures.Whenthecomplexeigenvalueextractionprocedureisused,thefluidmotioneffectsareincludedintheirentirety;thatis,theacousticstiffnessanddampingmatricesareincludedintheanalysis.Frequencyshift

FortheLanczosandsubspaceiterationeigensolversyoucanspecifyapositiveornegativeshiftedsquaredfrequency,

S.Thisfeatureisusefulwhenaparticularfrequencyisofconcernorwhenthenaturalfrequenciesofanunrestrainedstructureorastructurethatusessecondarybasemotions(largemassapproach)areneeded.Inthelattercaseashiftfromzero(thefrequencyoftherigidbodymodes)willavoidsingularityproblemsorround-offerrorsforthelargemassapproach;anegativefrequencyshiftisnormallyused.Thedefaultisnoshift.IftheLanczoseigensolverisinuseandtheuser-specifiedshiftisoutsidetherequestedfrequencyrange,theshiftwillbeadjustedautomaticallytoavalueclosetotherequestedrange.Normalization

FortheLanczosandsubspaceiterationeigensolversbothdisplacementandmasseigenvectornormalizationareavailable.Displacementnormalizationisthedefault.MassnormalizationistheonlyoptionavailableforSIM-basednaturalfrequencyextraction.Thechoiceofeigenvectornormalizationtypehasnoinfluenceontheresultsofsubsequentmodaldynamicsteps(see

\t"v610bmkwin"“Linearanalysisofarodunderdynamicloading,”

Section1.4.9oftheAbaqusBenchmarksManual).Thenormalizationtypedeterminesonlythemannerinwhichtheeigenvectorsarerepresented.Inadditiontoextractingthenaturalfrequenciesandmodeshapes,theLanczosandsubspaceiterationeigensolversautomaticallycalculatethegeneralizedmass,theparticipationfactor,theeffectivemass,andthecompositemodaldampingforeachmode;therefore,thesevariablesareavailableforuseinsubsequentlineardynamicanalyses.TheAMSeigensolvercomputesonlythegeneralizedmass.DisplacementnormalizationIfdisplacementnormalizationisselected,theeigenvectorsarenormalizedsothatthelargestdisplacemententryineachvectorisunity.Ifthedisplacementsarenegligible,asinatorsionalmode,theeigenvectorsarenormalizedsothatthelargestrotationentryineachvectorisunity.Inacoupledacoustic-structuralextraction,ifthedisplacementsandrotationsinaparticulareigenvectoraresmallwhencomparedtotheacousticpressures,theeigenvectorisnormalizedsothatthelargestacousticpressureintheeigenvectorisunity.Thenormalizationisdonebeforetherecoveryofdependentdegreesoffreedomthathavebeenpreviouslyeliminatedwithmulti-pointconstraintsorequationconstraints.Therefore,itispossiblethatsuchdegreesoffreedommayhavevaluesgreaterthanunity.Input

File

Usage:

\t"v610keywin"*FREQUENCY,NORMALIZATION=DISPLACEMENTAbaqus/CAE

Usage:

Stepmodule:

StepCreate:

Frequency:

Other:

Normalizeeigenvectorsby:

DisplacementMassnormalizationAlternatively,theeigenvectorscanbenormalizedsothatthegeneralizedmassforeachvectorisunity.The“generalizedmass”associatedwithmode

is

where

isthestructure'smassmatrixand

istheeigenvectorformode

.Thesuperscripts

N

and

M

refertodegreesoffreedomofthefiniteelementmodel.

Iftheeigenvectorsarenormalizedwithrespecttomass,alltheeigenvectorsarescaledsothat

=1.Forcoupledacoustic-structuralanalyses,anacousticcontributionfractiontothegeneralizedmassiscomputedaswell.Input

File

Usage:

\t"v610keywin"*FREQUENCY,NORMALIZATION=MASSAbaqus/CAE

Usage:

Stepmodule:

StepCreate:

Frequency:

Other:

Normalizeeigenvectorsby:

MassModalparticipationfactorsTheparticipationfactorformode

indirection

i,

,isavariablethatindicateshowstronglymotionintheglobal

x-,

y-,or

z-directionorrigidbodyrotationaboutoneoftheseaxesisrepresentedintheeigenvectorofthatmode.Thesixpossiblerigidbodymotionsareindicatedby

,

2,

,

6.Theparticipationfactorisdefinedaswhere

definesthemagnitudeoftherigidbodyresponseofdegreeoffreedom

N

inthemodeltoimposedrigidbodymotion(displacementorinfinitesimalrotation)oftype

i.Forexample,atanodewiththreedisplacementandthreerotationcomponents,

iswhere

isunityandallother

arezero;

x,

y,and

z

arethecoordinatesofthenode;and

,

,and

representthecoordinatesofthecenterofrotation.Theparticipationfactorsare,thus,definedforthetranslationaldegreesoffreedomandforrotationaroundthecenterofrotation.Forcoupledacoustic-structuraleigenfrequencyanalysis,anadditionalacousticparticipationfactoriscomputedasoutlinedin

\t"v610stmwin"“Coupledacoustic-structuralmediumanalysis,”

Section2.9.1oftheAbaqusTheoryManual.

ModaleffectivemassTheeffectivemassformode

associatedwithkinematicdirection

i

(,

2,

,

6)isdefinedasIftheeffectivemassesofallmodesareaddedinanyglobaltranslationaldirection,thesumshouldgivethetotalmassofthemodel(exceptformassatkinematicallyrestraineddegreesoffreedom).Thus,iftheeffectivemassesofth