1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Saw-tooth Wave and Fourier Series amplitude of oscillation

  1. Sep 29, 2013 #1
    1. The problem statement, all variables and given/known data

    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.

    2. Relevant 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.

    3. 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.

  2. jcsd
  3. Sep 29, 2013 #2


    User Avatar
    Staff Emeritus
    Science Advisor
    Homework Helper
    Education Advisor

    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).
  4. Sep 30, 2013 #3

    rude man

    User Avatar
    Homework Helper
    Gold Member

    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 dncos(2πnt/τ + θn) summed from n = 1 to ∞.

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

    The coefficients gn 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 = dn.

    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.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted