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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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Presentation Transcript
sinusoidal drive
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

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

General Solution

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

Superposition

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.

Fourier Series
sawtooth
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

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

Impulsive Force

for t >t0

for t >t1

narrow spike
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
green s method
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

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