医学物理学-chapter-4a-14.ppt

ChapterFourOscillation,wavemotionandsoundNewwordsandexpressions•simpleharmonicmotion(简谐运动);•spring(弹簧)，elastic(弹性的)，Phase(位相)•reciprocal(倒数)，unavoidable(不可避免的)•amplitude(振幅);damp(vt.阻尼,n.湿气);•deduce(vt.推论),deduction•vibration(振动)；oscillation(振动)•Resonance(共振）§4.1Simpleharmonicmotion(SHM)4.1.1EquationofSHM1.DefinitionofSHM•Simpleharmonicforce(简谐力):TheforceFonabodyisproportionaltoitsdisplacementxfromtheoriginandalwaysdirectedtowardstheorigin.F=-kx(4.1)kistheelasticconstant.ox•SHMIfabodymovesinastraightlineunderthesimpleharmonicforce,themotionofthebodyiscalledsimpleharmonicmotion.2.EquationofSHMIfabody’smassismandisexertedbyasimpleharmonicforce,itsequationofmotioncanbeobtainedbyusingNewton’ssecondlawofmotion.22dtxdmmaFOntheotherhand,consideringeq.(4.1),wehavekxdtxdm22Orxmkdtxd22Define2=k/mandwehave0222xdtxd(4.2)ThisisthedifferentialequationoftheSHM.Itssolutioncanbeexpressedas)cos(tAx(4.3)ThemotiondescribedbyacosineorsinefunctionoftimeisalsocalledSimpleHarmonicMotion.Differentiatingtheequation(4.3)withrespecttot,thevelocityandaccelerationoftheSHMcanbeobtained)sin()sin()(cos)(cos)]cos([tAAdtdddAdtdAtAdtddtdxv=t+d(cosx)/dx=-sinxd(sinx)/dx=cosx(4.4)xtAAdtdddAdtdAdtdvdtxda2222)cos()(cos)(sin)(sin(4.5)Soweget:0222xdtxdThisshows)cos(tAxisindeedthesolutionofdiff-equation4.1.2ThecharacteristicquantitiesofSHMIntheequationofSHM,A,andareconstantsandbuttheyaresoimportant.)cos(tAx1.AiscalledAmplitude(振幅).Itisthemaximumdisplacementofavibratingbodyfromequilibriumposition.2.Period(周期)andfrequency(频率)•Theperiod,denotedbyT,isthetimetakenforacompletevibrationwhichisindependentofthepositionchosenforthestartingpointofthecompletevibration.txFinditsamplitudeandperiod.•Thefrequency,denotedbyf,isthenumberofcompletevibrationspersecond,itisthereciprocal(倒数)oftheperiodTf1(4.6)TheangularfrequencyorangularvelocityisdefinedasTf22∴2f(4.7)3.Phaseandinitialphase（初位相）tiscalledthephaseofSHM,isthephaseatt=0,calledinitialphase.Att=0,equations(4,3)and(4.4)becomesrespectivelysincos00AvAx(4.8)Thisiscalledinitialcondition.Squaringbothsidesoftheaboveequations,theamplitudeoftheSHMcanbefound22220sinAv222222020)sin(cosAAvx2220cosAx∴kmvxvxA202022020Ontheotherhand,theinitialphasecanalsobeworkedoutfromequation(4.8),sowehave(4.9)tan00xv∴00arctanxv(4.10)Example1.Aparticlewithmassm=2.00×10-2kgisinSHMattheendofaspringwithspringconstantk=50.0N/m.Theinitialdisplacementandvelocityoftheparticleis3.00×10-2mand–1.32m/srespectively.Calculate:(1)theangularfrequency;(2)theinitialphase;(3)theamplitudeofthevibration;(4)theperiod;(5)thefrequency.Solution:theknownquantitiesare:m=2.00×10-2kgk=50.0N/mx0=3.00×10-2mv0=–1.32m/sNowusingtheformulaewehavelearned,theproblemcanbesolvedeasily.(1).theangularfrequency,wehave0.501000.20.502mk(rad/s)(2).Theinitialphaseofthevibrationcanbefoundusingtheinitialdisplacementandinitialvelocity.Att=0,weknowx0=Acos=3.00×10-2mv0=-Asin=-1.32m/sThecanbeobtainedbysolvingaboveequations.Ontheotherhand,itcanbecalculatedirectlyby3.410.501000.332.1arctanarctan200xv(3).TheamplitudecanbecalculatebytheformulamvxA2220201000.4(4).Theperiodcanbefoundthroughtherelationbetweentheangularfrequencyandtheperiod;sT126.00.5022(5)Tf14.1.3ThereferencecircleofSHMx=Acos(t+)TheequationofSHM.t+MM0APP0Fig.4.2thecircleofreferenceofSHM.OVectorrotationaroundafixedpointOMP4.1.4TheenergyofSHM)(sin21212122222tAmdtdxmmvEk)(cos21212222tAmkxEp∴2222121kAAmEEpk(4.11)Energy=kinetic+potential.xFig.4.3TotalenergyofthevibratingsystemxEkEpEEnergyisconservedinthesystem!Example4.2simplependulumfhsinmgcosmglmaFSelfstudysin22mgdtdmlmgdtdml22§4.2Damped(阻尼的)harmonicmotion,forcedvibration(受迫振动)andresonance4.2.1Thedampedharmonicmotion•Realvibratingsystemshavedampedforceorfriction.Howdotheyswingsohigh?!dampedvibrationordampedHarmonicmotion•Howtodeterminethedampedforce?•Ignorefrictions•consideringdampedforceonly•basedonexperiments:dampedforcespeedandoppositetovelocityvf:dampingcoefficient.Itsvaluedependsonthefactorsofitsshape,magnitude,surfaceconditionanditsmaterialproperty.dtdxVibrationalequation：F=ma.dtdxkxdtxdm22022022xdtdxdtxdSet:mkm20,2022dtdxkxdtxdm022dtdxmxmkdtxddampingelasticffF022022xdtdxdtxdThisisthedynamicequationfordampedvibrationanditisalinearhomogeneousdifferentialequation.iscalleddampedfactorordampedcoefficient!Whendampedforceissmall(0),ithasasolution:)cos(0teAxtWith220A0andaredeterminedbytheinitialconditions.DampingoscillationsCBatxo过阻尼临界阻尼Forexample:=1,0=3,A0=2and=/4,theequationofmotionbecomes)4/8cos(2text)4/8cos(2textWhenthedampedforceissmall,theperiodofthedampedvibrationisgivenby2202T(4.12)Fordampedvibrations,whichofthefollowingisnottrue?A.Themechanicalenergybecomessmaller.B.Theamplitudedecreases.C.Theperiodincreaseswithtime.D.The