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Driven Oscillator

Driven Oscillator. One simple driving force is sinusoidal oscillation. Inhomogeneous equation Complex solution (real part). Sinusoidal Drive. Try a solution. Im. r. ir sin q. q. Re. r cos q. The solution given was a particular solution. Steady state solution

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Driven Oscillator

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  1. Driven Oscillator

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

  3. 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 General Solution underdamped

  4. 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 Superposition Given solutions q1 etc. Then there are solutions

  5. 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. Fourier Series

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

  7. 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 Impulsive Force for t >t0 for t >t1

  8. A small impulse permits a number of approximations. t1-t0small a(t1-t0) is constant A delta function Keep lowest order terms in the time interval. This returns to equilibrium for long time t. Narrow Spike

  9. 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. Green’s Method for tn < t < tn+1 next

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