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1、DYNAMICANALYSISUSINGMODESUPERPOSITIONTheModeShapesUsedToUncoupleTheDynamicEquilibriumEquationsNeedNotBeTheExactFree-VibrationModeShapes13.1EQUATIONSTOBESOLVEDThedynamicforceequilibriumEquation(12.4)canberewritteninthefollowingformasasetofNsecondorderdiff
2、erentialequations:JMu&&(t)(t)+Cu&+Ku(t)=F(t)=åfjg(t)j(13.1)j=1Allpossibletypesoftime-dependentloading,includingwind,waveandseismic,canberepresentedbyasumof“J”spacevectorsfj,whicharenotafunctionoftime,andJtimefunctionsg(t),whereJcannotbegreaterthanthenumb
3、erofjdisplacementsN.Thenumberofdynamicdegrees-of-freedomisequaltothenumberoflumpedmassesinthesystem.ManypublicationsadvocatetheeliminationofallmasslessdisplacementsbystaticcondensationpriortothesolutionofEquation(13.1).Thestaticcondensationmethodreducest
4、henumberofdynamicequilibriumequationstosolve;however,itcansignificantlyincreasethedensityandthebandwidthofthecondensedstiffnessmatrix.Inbuildingtypestructures,inwhicheachdiaphragm2STATICANDDYNAMICANALYSIShasonlythreelumpedmasses,thisapproachiseffectivean
5、disautomaticallyusedinbuildinganalysisprograms.Forthedynamicsolutionofarbitrarystructuralsystems,however,theeliminationofthemasslessdisplacementis,ingeneral,notnumericallyefficient.Therefore,themodernversionsoftheSAPprogramdonotusestaticcondensationinord
6、ertoretainthesparsenessofthestiffnessmatrix.13.2TRANSFORMATIONTOMODALEQUATIONSThefundamentalmathematicalmethodthatisusedtosolveEquation(13.1)istheseparationofvariables.Thisapproachassumesthesolutioncanbeexpressedinthefollowingform:uY(t)=F(t)(13.2a)WhereF
7、isan“NbyL”matrixcontainingLspatialvectorswhicharenotafunctionoftime,andY(t)isavectorcontainingLfunctionsoftime.FromEquation(13.2a)itfollowsthatuY&(t)=FF&(t)and(t)&&uY=&&(t)(13.2b)and(13.2c)Priortosolution,werequirethatthespacefunctionssatisfythefollowing
8、massandstiffnessorthogonalityconditions:TT2FM=IandFFK=FW(13.3)2whereIisadiagonalunitmatrixandWisadiagonalmatrixwhichmayormaynotcontainthefreevibrationfrequencies.Itshouldbenotedthatthefundamentalsofmathematicsplacenorestri
