Saw-tooth Wave and Fourier Series amplitude of oscillation

[BlackHole213]

Homework Statement

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

\F(t)=c(t-n\tau) for (n-0.5)\tau<t<(n+0.5)\tau

for each integer n. Find the amplitudes a_n of oscillation at the angular frequencies 2\pi n/\tau 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 \Sigma_{n}=a_{n}\cos(\frac{n\pi t}{L})+b_n\sin(\frac{n\phi t}{L}). and L=tau\2.

I also know a_{n}=\frac{1}{L}\int f(t)\cos(\frac{n\pi t}{L})dx and b_n 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)\tau and (n+0.5)\tau. 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 a_{n}=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.

 

[vela]

Don't mix up the $n$'s. You're using $n$ to mean two different things. One is in your definition of F(t); the other is as the label for the Fourier coefficients.

You're also using $a_n$ to mean two different things. One is the cosine Fourier coefficients for F(t), and the other is the amplitude of the response for the nth angular frequency. They're different things. Mathworld is correct about the Fourier coefficients $a_n$ vanishing because F(t) is an odd function. The book is referring to the system's response to F(t).

 

[rude man]

You need to start by formulating the ordinary differential equation (ODE) for the critically damped system. Then you need to find the Fourier series for the forcing function.

For the solution to the 2nd order ODE assume a solution of

x(t) = d₀ + 2Ʃ d_ncos(2πnt/τ + θ_n) summed from n = 1 to ∞.

For the Fourier series of the sawtooth write
F(t) = c{g₀ + 2Ʃ g_ncos(2πnt/τ + ψ_n)}.

The coefficients g_n for the sawtooth forcing function are found in the usual way. You should be able to look them up for a sawtooth wave with period τ.

The 2nd order ODE for a critically damped 2nd order system can be found in many texts & wikipedia.

Then, just solve the ODE knowing F(t) in Fourier series form. Amplitude of each harmonic = d_n.

I can't read what's after the free-period τ, is it τ'? In the general case the period of the forcing function does not have to be related in any particular way to the (natural) period of the undamped oscillator.