1. Driven Oscillator

2. 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

3. 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

4. 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

5. 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.

6. 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

7. 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

8. 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.

9. 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
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