Saw-tooth Wave and Fourier Series amplitude of oscillation

In summary, the problem involves a critically damped oscillator with a saw-tooth force of the form F(t)=c(t-nτ) for (n-0.5)τ<t<(n+0.5)τ. The amplitudes of oscillation at the angular frequencies 2πn/τ can be found by solving the ordinary differential equation for the system and using the Fourier series for the forcing function. The coefficients for the Fourier series of the saw-tooth function can be looked up for a period τ. The solution for the system's response can be written as x(t)=d0+2Ʃdncos(2πnt/τ+θn) where dn represents the amplitude of each harmonic.
  • #1
BlackHole213
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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.
 
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  • #2
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).
 
  • #3
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.
 

Related to Saw-tooth Wave and Fourier Series amplitude of oscillation

1. What is a saw-tooth wave?

A saw-tooth wave is a specific type of waveform that is characterized by a sharp rise and a gradual decline in amplitude. It resembles the shape of a saw blade, hence the name. It is a periodic function that repeats itself over time.

2. What is the amplitude of oscillation in a saw-tooth wave?

The amplitude of oscillation in a saw-tooth wave is the maximum displacement of the wave from its mean or equilibrium position. In other words, it is the distance between the highest and lowest points on the wave.

3. What is Fourier series?

Fourier series is a mathematical tool used to represent periodic functions, such as the saw-tooth wave, as a combination of simple sine and cosine functions. It allows us to break down a complex waveform into simpler components and analyze its frequency and amplitude.

4. How is the amplitude of oscillation related to the Fourier series of a saw-tooth wave?

The amplitude of oscillation in a saw-tooth wave is directly related to the coefficients of the Fourier series. The Fourier coefficients determine the amplitude of each harmonic component in the wave, and the sum of these components determines the overall amplitude of oscillation.

5. Can the amplitude of oscillation in a saw-tooth wave be changed?

Yes, the amplitude of oscillation in a saw-tooth wave can be changed by altering the coefficients of the Fourier series. By adjusting the coefficients, we can manipulate the amplitude of each harmonic component and thus change the overall amplitude of the wave.

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