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1、SolutionofeigenvalueproblemFreevibrationequationcanbeexpressedas(1)Whichrepresentasetofnsimultaneoushomogeneousdifferentialequationsofthetype(2)WeareinterestedinaspecialtypeofsolutionofEqs.,namely,thatinwhichallthecoordinatesexecutesynchronousmotion.Physically,thisimpliesamotioninwhichallthec
2、oordinateshavethesamedependence,andthegeneralconfigurationofthemotiondoesnotchange,exceptfortheamplitude,sothattheratiobetweenanytwocoordinateand,,remainsconstantduringthemotion.Mathematically,thistypeofmotioncanberepresentedby[1](3)Whereareconstantamplitudesandisafunctionoftime,thesamefuncti
3、onforallthecoordinate.Theinterestliesinthecaseinwhichthecoordinatesrepresentstableoscillation,whichimpliesthemustbebounded.InsertingEq.intoEq.andrecognizingthatthefunctiondoesnotdependontheindex,weobtain(4)Eq.canberewrittenintheform(5)Usingthestandardargument,weobtainthattheleftsideofEq.doesn
4、otdependentontheindex,whereastherightsidedoesnotdependontime,sothattworatiosmustbeequaltoaconstant,andinparticulartothesameconstant.Assumingthatisarealfunction,theconstantmustbearealnumber.Denotingtheconstantby,Eqs.yield(6)And(7)WeconsiderasolutionofEq.intheexponentialform(8)Introducingsoluti
5、oninEq.anddividingthroughby,weconcludethatsmustsatisfytheequation(9)Whichhastworoots(10)Ifisanegativenumber,thenandarerealnumber,equalinmagnitudebutoppositeinsign.Inthiscase,Eq.hastwosolutions,however,areinconsistentwithstableboundedmotion,sothatthepossibilitythatisnegativemustbediscardedandt
6、heonethatispositivemustberetained.Letting,whereisreal,Eq.yields(11)SothatthesolutionofEq.becomes(12)Becauseisarealfunction,mustbeconjugateof.Thensolutioncanbeexpressedintheform(13)WhereCisanarbitraryconstant,thefrequencyofharmonicmotionanditsphaseangle,allthreequantitiesbeingthesameforeveryco
7、ordinate.TocompletethesolutionofEqs.,wemustdeterminetheamplitude.ItsconvenienttowriteEqs.inthematrixfor(14)Eq.representstheeigenvalueproblemassociatedwithmatrixsandanditpossessesnontrivialsolutionsifandonlyofthedeterminantofthecoefficientsvan
