ResonanceExternal ForceDrivenSingle PushSinusoidal DriveOscillator EnergyLorentzianEnergy WidthResonanceResonanceExternal ForceExternal ForceDynamical systems involve Dynamical systems involve external forces.external forces.•Arbitrary force Arbitrary force FF((qq, , tt))•Small oscillations only Small oscillations only FF((tt))The oscillator Lagrangian The oscillator Lagrangian gains an extra term from the gains an extra term from the time-dependent force.time-dependent force.•Corresponds to workCorresponds to work•Derivative is powerDerivative is power2221221qqEmmjmmjjmmjqtFqqVqqGL )(2121 qqqqqqqdtdE22)(tFqVqGmjmjjmj)(tFqdtdEDrivenDrivenThere are solutions for the There are solutions for the one-dimensional driven one-dimensional driven oscillator.oscillator.•Inhomogeneous equation of Inhomogeneous equation of motionmotion•Separate transient and Separate transient and steady-state solutionsteady-state solutionThe transient solution solves The transient solution solves the undriven oscillator.the undriven oscillator.•Applies to damped as wellApplies to damped as well)(2tFqq tsqqq )(2tFqqss02ttqqSingle PushSingle PushA step function driving force A step function driving force is like a push to a pendulum.is like a push to a pendulum.•Force Force FF0 0 = 1= 1, for , for tt > 0 > 0Solution requires boundary Solution requires boundary conditions.conditions.•qq((tt)) is C is C11q = ml001)0()0(1)0(21abqaqqF011 qqQq(t’ = t)01 qqQqteqQtcos21teqQtsin2202)0()0()0(21bQaqbqaqtQtetqQtsin21cos1)(2Sinusoidal DriveSinusoidal DriveA sinusoidal drive can be A sinusoidal drive can be represented by a complex represented by a complex force.force.•Consider real partConsider real part•Try a solutionTry a solutionThe transient part damps out The transient part damps out exponentially.exponentially.•AA is complex determined by is complex determined by initial conditionsinitial conditionstieqtq0)( Qieqtis21tieqqQq1tiQtteAeq2tsqqq Oscillator EnergyOscillator EnergyThe energy is proportional to The energy is proportional to the steady state amplitude.the steady state amplitude.The energy peaks with The energy peaks with frequency.frequency.•This is resonanceThis is resonance0ddE2221 QieqEtis 222211QE 0214112222222QQ 22214 Q2211QrLorentzianLorentzianThe resonant and driving The resonant and driving frequencies are similar for frequencies are similar for large large QQ..•Set both about equal to 1Set both about equal to 1This is a Lorentzian function.This is a Lorentzian function.)1(2)1)(1(122221QQELinear oscillator 224111QEEnergy WidthEnergy WidthOne measure of the One measure of the oscillator is the width of the oscillator is the width of the Lorentzian peak.Lorentzian peak.•Full width at half maximumFull width at half maximumMeasuring the peak width Measuring the peak width gives Q.gives Q.•High Q is narrow resonanceHigh Q is narrow resonance•High Q is a slow decay of High Q is a slow decay of transientstransientsELinear
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