FUNDAMENTALS OF ACOUSTICS(5)

BASICACOUSTICS(5)ANALOGIESBETWEENMECHANICALANDELECTRICALSYSTEMS•Wecandevelopadeeperunderstandingofwhatwehavesaidabouttheharmonicoscillatorbydrawingparallelswithsimpleelectriccircuit.•Theseanalogiescanhelpourstudyofmorecomplexvibratingsystems,andwewilleventuallyextendthemtoapplyinacousticproblemsaswell.Manyvibratingsystemsaremathematicallyequivalenttocorrespondingelectricalsystems.•Wehavealreadystudiedresonanceinaseriescircuit,muchoftheprecedingdiscussionshouldseemquitefamiliar.•Fromtheanalogiesyoucouldhavepredictedthatthenaturalfrequency0011/mbecomesDmLCmCD=1/Cm,CmiscalledmechanicalcomplianceFDxdtdxRdtxdmm22FvdtCvRdtdvmmm12.Inadampingoftheoscillations,thedifferentialequationforthemotionbecomesFmmRmCRLCVConsiderasimpleserieselectricalcircuitcontaininginductanceL,resistanceR,andcapacitanceC,drivenbysinusoidalvoltageVconwt,assuggestedinFig.(a)(a)EquivalentseriessystemsVCQdtdQRdtQdL22VIdtCRIdtdIL1ThedifferentialequationforthechargeqisThedifferentialequationforthecurrentI=dq/dtisWeseethattheelectricalcircuitofFig.aisthemathematicalanalogofthedampedharmonicoscillatorofFig.bFmmRmC(b)ThecurrentIintheelectricalsystemisequivalenttothespeedvinthemechanicalsystemRLCV•ThechargeQisequivalenttothedisplacementx•andthetwosystemhavesimilarforms,withthemechanicalresistanceRmanalogoustotheelectricalresistanceR,themassmanalogoustotheelectricalinductanceL,andthespringconstant(orcomplianceCm)analogoustotheelectricalcapacitanceC.MechanicalsystemElectricalsystemDisplacementxVelocityvAppliedforceFMechanicalresistanceRmMassmComplianceCm=1/DChargeQCurrentIVoltageVResistanceRInductanceLCapacitanceCjv0DfDxmDffRmfRvjmFvZej1vajvvxj0Themechanicalsystemji0CeLCeeReRijeEiZeLej1idijidtiQj0Theelectricalsystem•TheelementsintheelectricalsystemFig.aaresaidtobeinseriesbecausetheyexperiencethesamecurrent.•Similarly,theelementsinthemechanicalsystemFig.bcanberepresentedbytheseriescircuitofFig.b:theyexperiencethesamedisplacementand,therefore,thesamespeed.FmmRmC1x2xIfasimplemechanicaloscillatorisdrivenbyasinusoidalforceappliedtothenormallyfixedendofthespringassuggestedbyFig.c,thenthemassandthespringexperiencethesameforceandthiscombinationrepresentedbyaparallelcircuit,asshowninFig.d.c12()DfDxx12()DFfDxx2222mdxdxFRmdtdt2222mdxdxmRFdtdtFmmRmC1x2xForthespringForthem1222()mFDvvdtdvmRvFdt12112221121()()()mmFvvZvvjCjmvRvZvvSubstitutingvforx,timeFFForthecaseofasinusoidaldrivingforce112()FZvv22112()ZvZvv121;mmZZRjmjC1111msmFvFZjmRjCWheretheZ1andZ2areObtainthespeedv11212sZZZZZ1111IVjLRjCWhichhasthesameformasaserieselectricalsystem.RLCV2I1IRLCV2I1IFmmRmC2v1vThespeedofthedrivenendofthespringisequivalenttothecurrententeringtheparallelcircuit,andthespeedv2ofthemassisequivalenttothecurrentflowingthroughtheinductorEquivalentseries-parallelsystems•Theelementsinthemechanicalsystemaresaidtobeinserieswhentheyexperiencethesamedisplacement,•Theelementsinthemechanicalsystemaresaidtobeinparallelwhentheyexperiencethesameforce.(mandspring)F3m3R2C2m2R1C1m1RF1m1R1C2m3m2R3R2CProblemsF2mR2C1m1CF1m1C2mR2CHomework•P2741-16Transversemotion-thevibratingstringVibrationsofextendedsystems•Inthepreviouschapteritwasassumedthatthemassmovesasarigidbodysothatitcouldbeconsideredconcentratedatasinglepoint.•However,mostvibratingbodiesarenotsosimple.Aloudspeakerhasitsmassdistributedoveritssurfacesothattheconedoesmoveasaunit.Apianosting.