- #1

- 34

- 0

## Homework Statement

An oscillator with free period [itex]\tau[/itex] is critically damped and subjected to a force with the saw-tooth form

[itex]\F(t)=c(t-n\tau)[/itex] for (n-0.5)[itex]\tau[/itex]<t<(n+0.5)[itex]\tau[/itex]

for each integer n. Find the amplitudes a_n of oscillation at the angular frequencies [itex]2\pi n/\tau[/itex] if c is a constant.

## Homework Equations

The textbook, Classical Mechanics 5th edition by Kibble, gave an example of such a problem with a square tooth wave and used a Fourier series.

I know that a Fourier series is in the form of [itex]\Sigma_{n}=a_{n}\cos(\frac{n\pi t}{L})+b_n\sin(\frac{n\phi t}{L})[/itex]. and L=tau\2.

I also know [itex]a_{n}=\frac{1}{L}\int f(t)\cos(\frac{n\pi t}{L})dx[/itex] and [itex]b_n[/itex] is the same, but with a sine instead of a cos.

## The Attempt at a Solution

I'm thinking that my limits of my two integrals would be (n-1/2)[itex]\tau[/itex] and (n+0.5)[itex]\tau[/itex]. However, the question is throwing me off. It asks for a_n, yet when I evaluate the integral with WolframAlpha, I get a value of 0. The book says that the answer is c/m(ω^2)n(1+n^2). Furthermore, I looked at MathWorld and it said that the Fourier series of a saw-tooth wave had [itex]a_{n}[/itex]=0 Since I haven't used Fourier series before, I have no idea if I'm even on the right track. I'm also not sure if I'm interpreting the question correctly.

Thanks.