结构动力学课件-2dyanmicsofstructures-ch.pptx

CHAPTER3.RESPONSETOHARMONICLOADING3-1UNDAMPEDSYSTEMBeforeconsideringthisviscouslydampedcase,itisinstructivetoexaminethebehaviorofanundampedsystemascontrolledbywhichhasacomplementarysolutionofthefree-vibrationParticularSolutionThegeneralsolutionmustalsoincludetheparticularsolutionwhichdependsupontheformofdynamicloading.Inthiscaseofharmonicloading,itisreasonabletoassumethatthecorrespondingmotionisharmonicandinphasewiththeloading;thus,theparticularsolutionisThenwehave,

CHAPTER3.RESPONSETOHARMONICLOADINGGeneralSolution

CHAPTER3.RESPONSETOHARMONICLOADING3-2SYSTEMWITHVISCOUSDAMPINGThecomplementarysolutionofthisequationisthedampedfree-vibrationresponseTheparticularsolutioninwhichthecosinetermisrequiredaswellasthesinetermbecause,ingeneral,theresponseofadampedsystemisnotinphasewiththeloading.Thenwehave,Inordertosatisfythisequationforallvaluesoft,itisnecessarythateachofthetwosquarebracketquantitiesequalzero;thus,oneobtains

CHAPTER3.RESPONSETOHARMONICLOADINGThenwehavethegeneralsolution,

CHAPTER3.RESPONSETOHARMONICLOADINGOfgreatinterest,however,isthesteady-stateharmonicresponsegivenbythesecondtermTheratiooftheresultantharmonicresponseamplitudetothestaticdisplacementwhichwouldbeproducedbytheforcewillbecalledthedynamicmagnificationfactorD;thusItisseenthatboththedynamicmagnicationfactorDandthephaseanglevarywiththefrequencyratioandthedampingratio.PlotsofDvs.andvs.AreshowninFigs.3-3and3-4,respectively,fordiscretevaluesofdampingratio,.

CHAPTER3.RESPONSETOHARMONICLOADINGFIGURE3-2Variationofdynamicmagnificationfactorwithdampingandfrequency.FIGURE3-4Variationofphaseanglewithdampingandfrequency.

CHAPTER3.RESPONSETOHARMONICLOADINGExampleE3-1.Aportableharmonic-loadingmachine