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1、EFFECTIVEMODALMASS&MODALPARTICIPATIONFACTORSRevisionEByTomIrvineEmail:tomirvine@aol.comOctober6,2010_______________________________________________________________________IntroductionTheeffectivemodalmassprovidesamethodforjudgingthe“significance”ofavibrationmode.Modeswithr
2、elativelyhigheffectivemassescanbereadilyexcitedbybaseexcitation.Ontheotherhand,modeswithloweffectivemassescannotbereadilyexcitedinthismanner.Consideramodaltransientorfrequencyresponsefunctionanalysis?viathefiniteelementmethod.??Alsoconsiderthatthesystemisamulti-degree-of-f
3、reedomsystem.?Forbrevity,onlyalimitednumberofmodesshouldbeincludedintheanalysis.?Howmanymodesshouldbeincludedintheanalysis??Perhapsthenumbershouldbeenoughsothat?thetotaleffectivemodalmassofthemodel?isatleast90%oftheactualmass.?DefinitionsTheequationdefinitionsinthissection
4、aretakenfromReference1.Consideradiscretedynamicsystemgovernedbythefollowingequation(1)whereMisthemassmatrixKisthestiffnessmatrixistheaccelerationvectoristhedisplacementvectoristheforcingfunctionorbaseexcitationfunction10Asolutiontothehomogeneousformofequation(1)canbefoundi
5、ntermsofeigenvaluesandeigenvectors.Theeigenvectorsrepresentvibrationmodes.Letbetheeigenvectormatrix.Thesystem’sgeneralizedmassmatrixisgivenby(2)Letbetheinfluencevectorwhichrepresentsthedisplacementsofthemassesresultingfromstaticapplicationofaunitgrounddisplacement.Defineac
6、oefficientvectoras(3)Themodalparticipationfactormatrixformodeiis(4)Theeffectivemodalmassformodeiis(5)ExampleConsiderthetwo-degree-of-freedomsystemshowninFigure1,withtheparametersshowninTable1.10m1k1m2k2k3x1x2yFigure1.Table1.ParametersVariableValue2.0kg1.0kg1000N/m2000N/m30
7、00N/mThehomogeneousequationofmotionis(6)Themassmatrixis(7)10Thestiffnessmatrixis(8)TheeigenvaluesandeigenvectorscanbefoundusingthemethodinReference2.Theeigenvaluesaretherootsofthefollowingequation.(9)Theeigenvaluesare(10)(11)(12)(13)(14)(15)Theeigenvectormatrixis(16)Theeig
8、envectorswerepreviouslynormalizedsothatthegeneralizedmassistheidentitymatrix.(17)10(18)(1
