FUNDAMENTALS OF ACOUSTICS(3)

FUNDAMENTALSOFACOUSTICS(3)ParticlesVibratingSystemsxmDREVIEW:Simpleharmonicmotion)3.1(222oxdtxdo)4.1(sincos0201tAtAx00cos()(1.5)xAmk2022(1.2)dxmDxdt000sindxvAtdt220002cosdxaAtdtxavxav0t1-2DAMPEDOSCILLATIONSWheneverarealbodyissetintooscillation,dissipative(frictional)forcesarise.Theseforcesareofmanytypes,dependingontheparticularoscillatingsystem.buttheywillalwaysresultinadampingoftheoscillations-adecreaseintheamplitudeofthefreeoscillationswithtime.LetusfirstconsidertheeffectofaviscousfrictionalforceSuchaforceisassumedtobeproportionaltothespeedofthemassandtobedirectedsoastoopposethemotion.ItcanbeexpressedasdtdxRfmrrfOnasimpleoscillatorWhereRmisapositiveconstantcalledthemechanicalresistanceofthesystem.•Itisevidentthatmechanicalresistancehastheunitsofnweton-secondpermeter(N.s/m)•Iftheeffectofresistanceisincluded,theequationofmotionofanoscillatorbecomes022DxdtdxRdtxdmm•Dividingthroughbymandrecallingthat20DmWehave022022xdtdxdtxdmRm2002fmD•Thisequationmaybesolvedbythecomplexexponentialmethod.AssumeasolutionoftheformtteCeCx2121μ1、μ2Aretwosolutionsoffollowequation02202Wecanobtain:20221,1.When≥220,Rm2≥4mDtteCeCx2121IfthemechanicalresistanceRmislargeenough,thesystemisnolongeroscillatory•Inmostcasesofimportanceinacoustics,themechanicalresistanceRmissmallenoughsothat＜220,μiscomplex.Now,μisgivenby22021,jDefininganewconstantΩbyNowμisgivenby2002201j21,AndΩisseentobethenaturalangularfrequencyofthedampedoscillator.•NoticethatΩisalwayslessthanthenaturalangularfrequencyω0ofthesameoscillatorwithoutdamping.Thecompletesolutionisthesumofthetwosolutionsobtainedabove,Oneconvenientformofthisgeneralsolutionistetatatxsincos)(21000()cos()costxtAetAttWhereAandφarerealconstantsdeterminedbytheinitialconditions.Theamplitudeofthedampedoscillator,nowdefinedasteAtA0)(A(t)isonlongerconstant,butdecreasesexponentiallywithtime.Aswiththeundampedoscillator,thefrequencyisindependentoftheamplitudeofoscillation.•Onemeasureoftherapiditywithwhichtheoscillationsaredampedbyfrictionisthetimerequiredfortheamplitudetodecreaseto1/eofitsinitialvalue.ThistimeiscalleddecaymodulusandisgivenbymRm21Thequantityδiscalledtheresistancecoefficient.2T0()tAtAe2T()()cos()xtAtttX(t)Fig.displaysthetimehistoryofthedisplacementofadampedharmonicoscillatorENERGYOFVIBRATIONtmvtDxE222121000coscostxtAetAtt00()sin()()cos()dxtvtAttAttdt)cos()sin()()(cos)(21)(sin)(21cos21222222222tttAmttAmttAmttDAE)(21)(211)(220tmvtDAEdtTtET1.3FORCEDOSCILLATIONS1.Asimpleoscillator,whendrivenbyanexternallyappliedforceF,thedifferentialequationforthemotionbecomes:22dxmFDxdtIfF=Fmcosωt,anditwillbeadvantageoustoreplacetherealdrivingforceFmcosωtbyitsequivalentcomplexdrivingforcef=Fexp(jωt)wecanwriteintheform:tjmemFxdtxd202200220()cos()cos()mFxtAttmω≠ω00()0tt00tdvdt220200cos0()sin0mFAmA)(220mFAmForA≠0，φ0=0022000220()coscos()2sinsin()22mmFxtttmFttmInitialconditionsThisspecialpatternofmotionisknownasthebeatingphenomenon.Soundwavesofslightlydifferentfrequencieswillalsogiverisetobeats.0002020sin()()sin()2sin2mmtFtttmtFttm2.Inadampingoftheoscillations,thedifferentialequationforthemotionbecomes22jtmmdxdxmRDxFedtdt22022jtmFdxdxxedtdtmmRm2mD012()()()xtxtxt(1-3)10()cos()txtAet0Ω≈ω0100()cos()txtAe2()jtmxtAemmmFADjRm)(2WhenAssumeasolutionoftheformAndsubstituteintoequationtoobtainjmmmmmmmmeZjXRDmjRZZjFA)()(2DmjRjFDjRmFAmmmmmWeobtainZmiscalledthecomplexmechanicalimpedance,Rmiscalledthemechanicalresistance;XmiscalledthemechanicalreactancemDmmmmRmtgRXtgDmRZ1122)(ThemechanicalimpedancejmmZZeHasmagnitudeAndphaseangle•Thedimensionsofmechanicalimpedancearethesameasthoseofmechanicalresistanceandareexpressedinthesameunits,N.s/m,oftendefinedasmechanicalohms.•Itistobeemphasizedthat,althoughthemechanicalohmisanalogoustotheelectricalohm,thesetwoquantitiesdonothavethesameunits.•Theelectricalohmhasthedimensionsofvoltagedividedbycurrent;•Themechanicalohmhasthedimensionsofforcedividedbyspeed.)(2jmmjmmmeZFeZjFA2()cos()sin()2mmmmFFxtttZZ00()cos()sin()tmmFxtAettZThesolutionofequationisThesumoftwoparts:atransienttermasteady-statetermForthecaseofasinusoidaldrivingforcef(t)=Fmcos(ωt)appliedtotheoscillatoratsomeinitialtime,thesolutionof(1-3)isthesumoftwoparts–atransienttermcontainingtwoarbitraryconstantsandasteady-statetermwhichdependsofFandωbutdoesnotcontainanyarbitraryconstants.ThetransienttermisobtainedbysettingFequaltozero.Thearbitraryconstantsaredeterminedbyapplyingtheinitialconditionstothetotalsolution.Afterasufficienttimeinterval,Thedampingtermmakesthisportionofthesolutionnegligible.Leavingonlythesteady–statetermwhoseangularfrequencyωisthatofthedrivingforceENERGYOFVIBRATIONdAdxFFvdtdtcos()cosmmmFdv