1MechanicalOscillationANDElectromagneticOscillationCHAPTER10ByDr.GYChen第十章機械振動和電磁振蕩3Manyoscillationsexistinnature.OscillationofspringsPlayonaswing4§10-1SimpleHarmonicMotion(SHM)簡諧振動Twoconditionsofmechanicaloscillation:elasticrestoringforceinertia慣性Freeoscillations:Ifthespringkismassless,SimpleHarmonicOscillatorSimpleHarmonicMotion(SHM)簡諧振子Fig.10-1彈性回復力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);物體：在平衡位置附近作周期性往復運動；Spring:deformation(形變)--x;Body:displacement(位移)--x;force:F=-kx(frictionless);xFig.10.1km坐標原點：通常取為原長位置O；AccordingtoNewton’sSecondLaw,Someconcepts8acceleration:whosesolutionis9

作諧振動的物體的加速度，總是與其離開平衡位置的位移大小成正比，且兩者方向相反。——運動學特征。

作諧振動的物體所受合力大小，總是與其離開平衡位置的位移大小成正比，且兩者方向相反。——動力學特征。

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

x(t)、v(t)anda(t):OTtx、v

、ax2A

v>0

<0<0>0a<0

<0

>0>0減速加速減速加速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).完成一次全振動所需的時間SIUnit:s13Frequency(頻率)ν:Thenumberofcompleteoscillationfinishedbytheparticleorthesystemperunittime.單位時間內粒子或系統完成全振動的次數。Apparently:

SIUnit:Hz1Hz=1s-1Angularfrequency(角頻率)ω:Thenumberofcompleteoscillationfinishedbytheparticleorthesystemin2πseconds.

HaveainverserelationshipwithTSIUnit:rad/s14(3)Phase(相位)

IfA,ωandφareknown,themotionoftheoscillatorisdeterminedcompletelybythequantityΦ=ωt+φ,whichiscalledphase.當A、ω、φ已知時，諧振動的物體在任意時刻的運動狀態由ωt+φ決定。ωt+φ稱為相位。Whent=0,Thephaseturnsintoφ,thisiscalledinitialphase(初相).而φ是t=0時的相位，稱為初相位，簡稱初相。反映了振動的初始狀態。

15Differenceofphases(相位差)

are：

△Φ=(ωt+φ2)-(ωt+φ1)=φ2-φ1

equaltothedifferenceofinitialphase(4)ComparisonofPhases(相位的比較)

Condition:thefrequencyshouldbethesame.

兩個頻率相同的諧振動才能比較步調。Ifx1=A1cos(ωt+φ1)x2=A2cos(ωt+φ2)Note:16(5)Discussions：（1）如果ΔΦ>0,φ2>φ1，則稱第二個諧振動超前于第一個諧振動的相位。（2）如果ΔΦ=φ2-φ1=0或2π的整數倍，則兩個諧振動同時到達正的最大位移、最小位移。任意時刻，振動方向相同，稱兩個諧振動同相或同步。（3）如果ΔΦ=φ2-φ1=π的奇數倍，則一個物體到達正的最大位移，另一個物體正好到達負的最大位移。任意時刻，振動方向相反，稱兩個諧振動為反相。

173DeterminationofAandφ.Initialconditions:Conclusions：If18Examples（1）：一個質量為10g的物體作簡諧振動，周期為4s,t=0時坐標為24cm速度為零。計算：（1）t=0.5s時物體的位置；(2)t=0.5s時物體受到的力的大小和方向；(3)從初始位置運動到x=-12cm所需的最少時間；(4)x=12cm時物體速度的大小。解：本題已知物體作簡諧振動。周期為T=4s,振幅為A=24cm,則：19由初始條件：（1）當t=0.5s時：注意單位換算！20（2）當t=0.5s時，物體受力：與x軸正向相反。（3）從初始位置運動到x=-12cm的最少時間：注意單位換算！24cm-12cmFig.10-421（4）x=-12cm物體的速度大小：注意：單位換算。22Example(2):Infigure,provethesimplependulum(單擺）isasimpleharmonicoscillatorwhenissmall.Prove:Fig.10-523Prove:(1)Displacement:θorx

(2)Restoringtorque(恢復力矩):(3)Angular(角量)

acceleration:Note:(linear線量)244TherotatingvectorrepresentationofSHM(旋轉矢量表示法)

AvectorwithalengthofAisrotatingaboutpointOatanangularvelocity(seeinFig).25Theprojection(投影)Pofthisrotatingvectorinx-axisisgivenbywhichisassameastheequationofSHM.Arotatingvectoronetoone一一對應SHM26Discussions:Determinedtheinitialphasebytherotatingvectorrepresentation.(1)過平衡位置沿x軸正向運動;(2)過平衡位置沿x軸負向運動;27§10-2TheEnergyofSHMThepotentialenergyofthesystemis:anditskineticenergyisequalto:1TotalenergyofSHMFig.10-128ThetotalenergyofSHMisconstant.（1）彈簧振子作簡振動過程中機械能守恒。（2）對于一定的彈簧振子，諧振動的機械能與振幅平方成正比。振幅越大，振子的總機械能越大。

Consideringk=mω2,thetotalenergyConclusions:29AveragevalueofenergyduringaperiodConclusion:諧振動在一個周期內的平均勢能和平均動能相等,均為kA2/4。

2AverageenergyofSHM303TransformationoftheenergyofSHM振動過程中，動能和勢能大小時刻改變并且相互轉化，但總能量保持不變。Conclusions:31Example(1):Amassof100gvibrateshorizontallyinSHMwithafrequencyof20Hzandanamplitudeof15cm.Calculate:(1)Thetotalenergyofthemotion.(2)Thevelocityofthemasswhenitis10cmfromthecenterpoint.32Solution:33Example(2):P15.例10.6、10.734§10-3DampedVibration&ForcedVibrationResonance

(阻尼振動受迫振動共振)1.DampedVibrationThebodyissubjecttothedampingforce,anditsamplitudeAandenergyEwilldecreasetozerogradually.振動的物體在實際運動過程中，會受到阻礙其運動的力的作用，能量逐漸受到損失，振幅逐漸衰減的振動稱為阻尼振動或減幅振動。Fig.10-7352.

ForcedVibration(受迫振動)Thesystemissubjecttoaperiodicexternalforce(系統受周期性強迫力作用下的振動)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