Resonance

External Force  Dynamical systems involve external forces. Arbitrary force F(q, t)Arbitrary force F(q, t) Small oscillations only F(t)Small oscillations only F(t)  The oscillator Lagrangian gains an extra term from the time-dependent force. Corresponds to workCorresponds to work Derivative is powerDerivative is power

Driven  There are solutions for the one-dimensional driven oscillator. Inhomogeneous equation of motionInhomogeneous equation of motion Separate transient and steady-state solutionSeparate transient and steady-state solution  The transient solution solves the undriven oscillator. Applies to damped as wellApplies to damped as well

Single Push  A step function driving force is like a push to a pendulum. Force F 0 = 1, for t > 0  Solution requires boundary conditions. q(t) is C 1 q =  m l F0F0 (t’ =  t)

Sinusoidal Drive  A sinusoidal drive can be represented by a complex force. Consider real partConsider real part Try a solutionTry a solution  The transient part damps out exponentially. A is complex determined by initial conditionsA is complex determined by initial conditions

Oscillator Energy  The energy is proportional to the steady state amplitude.  The energy peaks with frequency. This is resonance

Lorentzian  The resonant and driving frequencies are similar for large Q. Set both about equal to 1  This is a Lorentzian function. E  Linear oscillator 

Energy Width  One measure of the oscillator is the width of the Lorentzian peak. Full width at half maximum  Measuring the peak width gives Q. High Q is narrow resonance High Q is a slow decay of transients E  Linear oscillator  next