第10章 机械振动的电磁振荡

1MechanicalOscillationANDElectromagneticOscillationCHAPTER10ByDr.GYChen第十章机械振动和电磁振荡3Manyoscillationsexistinnature.OscillationofspringsPlayonaswing4§10-1SimpleHarmonicMotion(SHM)简谐振动Twoconditionsofmechanicaloscillation:elasticrestoringforceinertia惯性Freeoscillations:Ifthespringkismassless,SimpleHarmonicOscillatorSimpleHarmonicMotion(SHM)简谐振子Fig.10-1弹性回复力Onlybytheelasticrestoringforce5§10-1SimpleHarmonicMotion(SHM)

1characteristics

Thesystem(blackspring)inFig.10-1iscalledasasimpleharmonicoscillator.Spring(k):massless;

body(m);Fig.10-1ProcessofSHM6(1)(2)(3)(4)(5)(6)ProcessofSHMFig.10-27Equilibriumposition: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);xFig.10.1km坐标原点：通常取为原长位置O；AccordingtoNewton’sSecondLaw,Someconcepts8acceleration:whosesolutionis9

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。

10Notes:(1)Maximumvalues:(2)Thecurvesof

x(t)、v(t)anda(t):OTtx、v

、ax2A

v>0

<0<0>0a<0

<0

>0>0减速加速减速加速AA-A-A-2Avaφ=0Fig.10-3112Threecharacteristicquantities(特征量)ofSHM.Wecanseethat:(1)Thedisplacementisdeterminedbythreequantities:A,,;(2)x=Acos(ωt+φ)isaperiodicfunctionoftimet.12(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:s13Frequency(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/s14(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。

15Differenceofphases(相位差)

are：

△Φ=(ωt+φ2)-(ωt+φ1)=φ2-φ1

equaltothedifferenceofinitialphase(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:16(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。

173DeterminationofAandφ.Initialconditions:Conclusions：If18Examples（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：本题已知物体作简谐振动。周期为T=4s,振幅为A=24cm,则：19由初始条件：（1）当t=0.5s时：注意单位换算！20（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：注意单位换算！24cm-12cmFig.10-421（4）x=-12cm物体的速度大小：注意：单位换算。22Example(2):Infigure,provethesimplependulum(单摆）isasimpleharmonicoscillatorwhenissmall.Prove:Fig.10-523Prove:(1)Displacement:θorx

(2)Restoringtorque(恢复力矩):(3)Angular(角量)

acceleration:Note:(linear线量)244TherotatingvectorrepresentationofSHM(旋转矢量表示法)

AvectorwithalengthofAisrotatingaboutpointOatanangularvelocity(seeinFig).25Theprojection(投影)Pofthisrotatingvectorinx-axisisgivenbywhichisassameastheequationofSHM.Arotatingvectoronetoone一一对应SHM26Discussions: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;27§10-2TheEnergyofSHMThepotentialenergyofthesystemis:anditskineticenergyisequalto:1TotalenergyofSHMFig.10-128ThetotalenergyofSHMisconstant.（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:29AveragevalueofenergyduringaperiodConclusion: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。

2AverageenergyofSHM303TransformationoftheenergyofSHM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:31Example(1):Amassof100gvibrateshorizontallyinSHMwithafrequencyof20Hzandanamplitudeof15cm.Calculate:(1)Thetotalenergyofthemotion.(2)Thevelocityofthemasswhenitis10cmfromthecenterpoint.32Solution:33Example(2):P15.例10.6、10.734§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.10-7352.

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.36§10-4ElectromagneticOscillation1.LCcircuitLCLCKchargeFig.10-837LCKεt=0LCKεiit=T/4LCKεt=T/2LCKεt=3T/2ii2.OscillationofLCcircuitFig.10-938§10-5

SuperpositionoftwoSHMs

1.Thecompositionoftwoharmonicvibrationswiththesamedirectionandfrequency.Thedisplacementsresultingfromtwoharmonicvibrationsinthesamestraightline(x-axis)are:39Itcanbeprovedthatthedisplacementofcompositionvibrationisgivenby:withthesameangularfrequency.Itiseasytoprovebyusingtherotatingvectors:40Specialexamples:

(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+A241ThetwoharmonicvibrationsareFindtheircompositionvibration.Example:Plottherotatingvectorsofx1,x2,x=x1+x2.Solution:→φ1=π/2,φ2=0.42Then:Hence:432.Thecompositionoftwoharmonicvibrationswiththesamedirectionanddifferentfrequencies.§10-5

SuperpositionoftwoSHMs

Thetotalamplitudewillvariesperiodicallywiththetime，whoseChinesenameis拍.3.Thecompositionoftwoharmonicvibrationswiththesamefrequencyandverticaldirections.Theorbitoftheendofthecomposite