FUNDAMENTALS OF ACOUSTICS(4)

BASICACOUSTICS(4)MechanicalResonance00()cos()sin()tmmFxtAettZThesolutionofequationisThesumoftwoparts:atransienttermasteady-statetermForaforceoscillations,thegeneralsolutionForthecaseofasinusoidaldrivingforcef(t)=Fmcos(ωt)appliedtotheoscillatoratsomeinitialtime,thesolutionof(1-3)isthesumoftwoparts–atransienttermcontainingtwoarbitraryconstantsandasteady-statetermwhichdependsofFandωbutdoesnotcontainanyarbitraryconstants.ThetransienttermisobtainedbysettingFequaltozero.Thearbitraryconstantsaredeterminedbyapplyingtheinitialconditionstothetotalsolution.Afterasufficienttimeinterval,Thedampingtermmakesthisportionofthesolutionnegligible.Leavingonlythesteady–statetermwhoseangularfrequencyωisthatofthedrivingforceZmiscalledthecomplexmechanicalimpedance,Rmiscalledthemechanicalresistance;XmiscalledthemechanicalreactancemDmmmmRmtgRXtgDmRZ1122)(ThemechanicalimpedancejmmZZeHasmagnitudeAndphaseangleENERGYRELATIONTheinstantaneouspower,suppliedtothesysteminthesteadystateisequaltheproductoftheinstantaneousdrivingforceandtheresultinginstantaneousspeed.ENERGYRELATIONdAdxFFvdtdtcos()cosmmmFdxvtdtZFFt2cos()cosmmFdAttdtZSubstitutingtheappropriaterealexpressionsforforceandspeed20221cos()cos11cos22TmMmmmmmFWttdtTZFvRZmmZRcosmmmFvZmmvx2212MmmWxRInmostsituationstheaveragepowerismoresignificancethantheinstantaneouspower.MechanicalResonance•Inthesteadystate,thedisplacementisequalto:)sin(2cos)(tZFtZFtxmmmm)cos()(tZFtvmmSpeedgivesmv1mR2mR0)cos()(tZFtvmmmZmR0Ifthespeedasgivenbyaboveisplottedasafunctionofthefrequencyofadrivingforce,acurveisobtainedasfollowThe(angular)frequencyofmechanicalresonanceisdefinedasthatatwhichthemechanicalreactanceXmvanishes,thisisthefrequencyatwhichadrivingforcewillsupplymaximumpowertotheoscillator.Itwasalsofoundtobethefrequencyoffreeoscillationofasimilarundampedoscillator.•AtthisfrequencythemechanicalimpedancehasitsminimumvalueofZm=Rm•Itisalsothefrequencyofmaximumspeedamplitude.•NotethatThisfrequencydoesnotgivethemaximumdisplacementamplitude.Iftheaveragepowersuppliedtothesystemasgivenby212mmmWvRmmmZFv2000222221111221()()1mmmmmmDDmmmRmRFFWRRTherefore•When0,maxMMWW2max12mMmFWR0022max11MffMmffWWQmaxmmww1.00.5f1mQ5mQ10mQ0ffbasedonequation,acurveisobtainedThecurvehasamaximumvalueofF2/2Rmattheresonancefrequencyandfallsatlowerandhigherfrequencies.mmRmQ0HereQmisthequalityfactorofthesystemThesharpnessofthepeakofthepowercurveisprimarilydeterminedbyQm.Ifitislarge,thecurvefallsoffveryrapidly-asharpresonance.If,ontheotherhand,itissmall,thecurvefallsoffmoreslowlyandthesystemhasabroadresonance.max12MMWWmmQffffQffff112002011021001mfffffQforWeobtainWheref1andf2arethetwofrequencies,respectivelyaboveandbelowresonancefrequencyforwhichtheaveragepowerhasdroppedtoone-halfitsresonancevalue.22210220021)(1)(ffffQDmmmmxmRFx011)(2202020ffffQddmff200211mQffThedisplacementamplitudeThefrequencythatmakesdisplacementamaximum:00.511.5200.511.522.533.50mxx0ffmQ3mQ1.5mQ2mQDisplacementfrequencycurveisshownasfollowTRANSIENTRESPONSEOFANOSCILLATOR•Wewillconsidertheeffectofsuperimposingthetransientresponseonthesteady-statecondition.•ThecompletegeneralsolutionistRFteAtxmmtm0010sin)cos()(0)0(,0)0(vxAsaspecialcaseletusassumethatSubstituteintoaboveequationtoobtainteRFtxtmm00sin)1()(Theeffectofthetransientisapparentintheleft-handportionofthecurve,butneartheright-handendthetransienthasbeensodampedthatthefinalsteadystateisnearlyreached.HomeworkP2731-5and1-12•1-5Thediaphragmofaloud–speakerweighs1g,andthedisplacementofitsdrivingrod1mmfromequilibriumrequiresaforceof106dynes.Thefrictionalforceopposingmotionisproportionaltothediaphragm’svelocityandis300dyneswhenthevelocityis1cmpersec.Ifitisassumedthatthediaphragmmoveslikeasimpleoscillator,whatwillbeitsnaturalfrequency,andwhatitsmodulusofdecay?•Thedrivingrodisdrivenbyaforceof100,000coswtdynes.Plotacurveoftherealandimaginarypartsofthemechanicalimpedanceofthediaphragmasfunctionoffrequency,fromf=0tof=1000Hz.Equivalentelectricalcircuitforasimpleoscillator