simple harmonic motion

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1、Energy:SimpleHarmonicMotionCharlesWFayIVSimpleHarmonicMotionEnergyofaSimpleHarmonicMotionHarmonicOscillatorVelocityofMassSpringsystemUnit5-Lecture1VelocityofaHarmonicOscillatorDampedMotionForcedVibrationsandResonanceCharlesWFayIVOctober31,2012Energy:SimpleS

2、impleharmonicmotionHarmonicMotionCharlesWFayIVIHarmonicmotionismotionthatrepeatsataregularSimpleHarmonicinterval.MotionIResultsfromaforcethatrestoresthesystemtoanEnergyofaHarmonicOscillatorequilibriumposition.VelocityofMassSpringsystemIThisrepetitivemotioni

3、sknownasoscillation.VelocityofaHarmonicOscillatorDampedMotionITimeforonefullcycleisknownasthePeriod.ForcedVibrationsandResonanceINumberofcyclespersecond–Frequency.ISimpleHarmonicmotionismotionthatcanbedescribedbytheequation,x(t)=Asin(!t+)(1)Energy:SimpleSi

4、mpleHarmonicMotionHarmonicMotionCharlesWFayIVSimpleHarmonicMotionEnergyofaHarmonicOscillatorVelocityofMassSpringsystemVelocityofaHarmonicOscillatorDampedMotionForcedVibrationsandResonanceFigure:SimpleHarmonicOscillatorEnergy:SimpleEnergyofanOscillatorHarmon

5、icMotionCharlesWFayIVSimpleHarmonicMotionEnergyofaHarmonicOscillatorVelocityofMassSpringsystemVelocityofaHarmonicOscillatorDampedMotionForcedVibrationsandResonanceFigure:SimplePendulumEnergy:SimpleEnergyofanOscillatorHarmonicMotionCharlesWFayIVWritethekinet

6、icenergyasafunctionoftime,SimpleHarmonicMotion1212K=mv=m(!Asin(!t+))Energyofa22HarmonicOscillator11VelocityofMassSpring=m!2A2sin2(!t+)=kA2sin2(!t+)systemVelocityofaHarmonic22OscillatorDampedMotionWritethepotentialenergybecomes,ForcedVibrationsandResonanc

7、e1212122U=kx=k(Acos(!t+))=kAcos(!t+)222Addthepotentialandkineticenergiestofindthetotalenergy,121212E=mv+kx=kA222Energy:SimpleExample:VelocityofMassSpringsystemHarmonicMotionCharlesWFayIVWhatisthevelocityofamass-springsystemexhibitingSHMSimpleHarmonicforana

8、rbitrarydisplacement?MotionEnergyofaHarmonicOscillatorVelocityofMassSpringsystemVelocityofaHarmonicOscillatorDampedMotionForcedVibrationsandResonanceEnergy:SimpleExample:VelocityofMassSpringsystemHarmo