第10章 机械振动的电磁振荡

*MechanicalOscillationANDElectromagneticOscillationCHAPTER10ByDr.GYChen第十章机械振动和电磁振荡*Manyoscillationsexistinnature.OscillationofspringsPlayonaswing*§10-1SimpleHarmonicMotion(SHM)简谐振动Twoconditionsofmechanicaloscillation:惯性Freeoscillations:Ifthespringkismassless,SimpleHarmonicOscillatorSimpleHarmonicMotion(SHM)简谐振子弹性回复力Onlybytheelasticrestoringforce*§10-1SimpleHarmonicMotion(SHM)1characteristicsThesystem(blackspring)inFig.10-1iscalledasasimpleharmonicoscillator.Spring(k):massless;body(m);ProcessofSHM*ProcessofSHMFig.10-2*Equilibriumposition:pointO(F=0);物体：在平衡位置附近作周期性往复运动；Spring:deformation(形变)--x;Body:displacement(位移)--x;force:F=-kx(frictionless);坐标原点：通常取为原长位置O；AccordingtoNewton’sSecondLaw,Someconcepts*acceleration:whosesolutionis*作谐振动的物体的加速度，总是与其离开平衡位置的位移大小成正比，且两者方向相反。——运动学特征。作谐振动的物体所受合力大小，总是与其离开平衡位置的位移大小成正比，且两者方向相反。——动力学特征。*Notes:(1)Maximumvalues:(2)Thecurvesofx(t)、v(t)anda(t):*2Threecharacteristicquantities(特征量)ofSHM.Wecanseethat:(1)Thedisplacementisdeterminedbythreequantities:A,,;(2)x=Acos(ωt+φ)isaperiodicfunctionoftimet.*(1)Amplitude(A)A--themaximumdisplacementfromtheequilibriumposition.(2)Theperiod(T)andfrequency(ν)Period(周期)T:Thetimenecessaryforonecompleteoscillation(acompleterepetitionofthemotion).完成一次全振动所需的时间SIUnit:s*Frequency(频率)ν:Thenumberofcompleteoscillationfinishedbytheparticleorthesystemperunittime.单位时间内粒子或系统完成全振动的次数。Apparently:SIUnit:Hz1Hz=1s-1Angularfrequency(角频率)ω:Thenumberofcompleteoscillationfinishedbytheparticleorthesystemin2πseconds.HaveainverserelationshipwithTSIUnit:rad/s*(3)Phase(相位) IfA,ωandφareknown,themotionoftheoscillatorisdeterminedcompletelybythequantityΦ=ωt+φ,whichiscalledphase.当A、ω、φ已知时，谐振动的物体在任意时刻的运动状态由ωt+φ决定。ωt+φ称为相位。 Whent=0,Thephaseturnsintoφ,thisiscalledinitialphase(初相). 而φ是t=0时的相位，称为初相位，简称初相。反映了振动的初始状态。* Differenceofphases(相位差)are：△Φ=(ωt+φ2)-(ωt+φ1)=φ2-φ1equaltothedifferenceofinitialphase(4)ComparisonofPhases(相位的比较)Condition:thefrequencyshouldbethesame.两个频率相同的谐振动才能比较步调。Ifx1=A1cos(ωt+φ1)x2=A2cos(ωt+φ2)Note:*(5)Discussions： （1）如果ΔΦ>0,φ2>φ1，则称第二个谐振动超前于第一个谐振动的相位。 （2）如果ΔΦ=φ2-φ1=0或2π的整数倍，则两个谐振动同时到达正的最大位移、最小位移。任意时刻，振动方向相同，称两个谐振动同相或同步。 （3）如果ΔΦ=φ2-φ1=π的奇数倍，则一个物体到达正的最大位移，另一个物体正好到达负的最大位移。任意时刻，振动方向相反，称两个谐振动为反相。*3DeterminationofAandφ. Initialconditions:Conclusions：If*Examples（1）：一个质量为10g的物体作简谐振动，周期为4s,t=0时坐标为24cm速度为零。计算：（1）t=0.5s时物体的位置；(2)t=0.5s时物体受到的力的大小和方向；(3)从初始位置运动到x=-12cm所需的最少时间；(4)x=12cm时物体速度的大小。解：本 题 快递公司问题件快递公司问题件货款处理关于圆的周长面积重点题型关于解方程组的题及答案关于南海问题 已知物体作简谐振动。周期为T=4s,振幅为A=24cm,则：*由初始条件：（1）当t=0.5s时：*（2）当t=0.5s时，物体受力：与x轴正向相反。（3）从初始位置运动到x=-12cm的最少时间：*（4）x=-12cm物体的速度大小：注意：单位换算。*Example(2):Infigure,provethesimplependulum(单摆）isasimpleharmonicoscillatorwhenissmall.Prove:*Prove:(1)Displacement:θorx(2)Restoringtorque(恢复力矩):(3)Angular(角量)acceleration:Note:(linear线量)*4TherotatingvectorrepresentationofSHM(旋转矢量 表 关于同志近三年现实表现材料材料类招标技术评分表图表与交易pdf视力表打印pdf用图表说话 pdf 示法)AvectorwithalengthofAisrotatingaboutpointOatanangularvelocity(seeinFig).*Theprojection(投影)Pofthisrotatingvectorinx-axisisgivenbywhichisassameastheequationofSHM.ArotatingvectorSHM*Discussions: Determinedtheinitialphasebytherotatingvectorrepresentation.(1)过平衡位置沿x轴正向运动;(2)过平衡位置沿x轴负向运动;*§10-2TheEnergyofSHMThepotentialenergyofthesystemis:anditskineticenergyisequalto:1TotalenergyofSHM*ThetotalenergyofSHMisconstant.（1）弹簧振子作简振动过程中机械能守恒。（2）对于一定的弹簧振子，谐振动的机械能与振幅平方成正比。振幅越大，振子的总机械能越大。Consideringk=mω2,thetotalenergyConclusions:*Averagevalueofenergyduringaperiod Conclusion:谐振动在一个周期内的平均势能和平均动能相等,均为kA2/4。2AverageenergyofSHM*3TransformationoftheenergyofSHM振动过程中，动能和势能大小时刻改变并且相互转化，但总能量保持不变。Conclusions:*Example(1):Amassof100gvibrateshorizontallyinSHMwithafrequencyof20Hzandanamplitudeof15cm.Calculate:(1)Thetotalenergyofthemotion.(2)Thevelocityofthemasswhenitis10cmfromthecenterpoint.*Solution:*Example(2): P15.例10.6、10.7*§10-3DampedVibration&ForcedVibrationResonance(阻尼振动受迫振动共振)1.DampedVibrationThebodyissubjecttothedampingforce,anditsamplitudeAandenergyEwilldecreasetozerogradually.振动的物体在实际运动过程中，会受到阻碍其运动的力的作用，能量逐渐受到损失，振幅逐渐衰减的振动称为阻尼振动或减幅振动。Fig.10-7*2.ForcedVibration(受迫振动) Thesystemissubjecttoaperiodicexternalforce(系统受周期性强迫力作用下的振动)3.Resonance(共振)Phenomenathatanmaximumofamplitudeofadampingvibrationappears.*§10-4ElectromagneticOscillation1.LCcircuitchargeFig.10-8*2.OscillationofLCcircuitFig.10-9*§10-5SuperpositionoftwoSHMs1.Thecompositionoftwoharmonicvibrationswiththesamedirectionandfrequency.Thedisplacementsresultingfromtwoharmonicvibrationsinthesamestraightline(x-axis)are:*Itcanbeprovedthatthedisplacementofcompositionvibrationisgivenby:withthesameangularfrequency.Itiseasytoprovebyusingtherotatingvectors:*Specialexamples: (1)whenφ2-φ1=2kπ,A=A1+A2,Areachesmaximum(Enhancing) (2)whenφ2-φ1=2kπ±π,A=|A1-A2|,Areachesminimum(Weakening) (3)Generally,|A1-A2|<A<A1+A2*ThetwoharmonicvibrationsareFindtheircompositionvibration.Example:Plottherotatingvectorsofx1,x2,x=x1+x2.Solution:→φ1=π/2,φ2=0.*Then:Hence:*2.Thecompositionoftwoharmonicvibrationswiththesamedirectionanddifferentfrequencies.§10-5SuperpositionoftwoSHMsThetotalamplitudewillvariesperiodicallywiththetime，whoseChinesenameis拍.3.Thecompositionoftwoharmonicvibrationswiththesamefrequencyandverticaldirections.Theorbitoftheendofthecompositevibrationvectorwillbealine,aellipseoracircle,whichdependsonthephasedifferenceofthetwovibrations.Applications：polarizedlight偏振光*§10-5SuperpositionoftwoSHMs4.Thecompositionoftwoharmonicvibrationswithverticaldirectionsanddifferentfrequencies.Theorbitoftheendofthecompositevibrationvectorwillbecomplexshapes.Animportantandtypicalcaseofthemiscalledlissajousfigures.(李萨如图形)ItisprovedthatanycomplexvibrationcanbedecompositedintomanySHMs.SHMisthesimplestandbasicvibration.*习题课Example(1):轻弹簧原长l0，平衡时伸长了x0，现手拉下L后无初速释放，请写出物体的运动方程。*Solution:*Thesimplependulum(单摆） Prove: ThemotionofthesimplependulumisSHM.Example(2):