Driven Oscillator

Sinusoidal Drive One simple driving force is sinusoidal oscillation. Inhomogeneous equation Complex solution (real part) Try a solution Im r ir sin q q Re r cos q

General Solution The solution given was a particular solution. Steady state solution The general solution combines that with the solution to the homogeneous equation. Adds a transient effect underdamped

Superposition Linear operators have well defined properties. Scalar multiplication Distributive addition Solutions to a linear operator equation can be combined. Principle of superposition Extend to a set of solutions Given solutions q1 etc. Then there are solutions

Fourier Series The harmonic oscillator equation can be expressed with a linear operator. The solution is known for a simple sinusoidal force. Fourier series for total force Apply superposition to steady state solutions.

Sawtooth Example Find the Fourier coefficients for a sawtooth driving force. For real-only coefficients there are both sine and cosine terms. F T A/2 t -A/2

Impulsive Force Assume the force is applied in a short time compared to the oscillator period. F(t)/m = 0, t < t0 F(t)/m = a, t0 < t < t1 F(t)/m = 0, t1 < t Transient effects dominate since t1 - t0 << T. Particular solution constant Initial conditions applied Superpose two steps for t > t0 for t > t1

Narrow Spike A small impulse permits a number of approximations. t1 - t0 small a(t1 - t0) is constant A delta function Keep lowest order terms in the time interval. This returns to equilibrium for long time t.

Green’s Method next A series of spike impulses can be superposed. Superpose the solutions As the impulses narrow the sum becomes an integral. The function G is Green’s function for the linear oscillator. for tn < t < tn+1 next