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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Complex solution (real part)Sinusoidal Drive
Try a solution
ir sin q
r cos q
The solution is known for a simple sinusoidal force.
Fourier series for total force
Apply superposition to steady state solutions.Fourier Series
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 stepsImpulsive Force
for t >t0
for t >t1
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