Damped Harmonic Oscillator Fourier Expansion

In summary, the conversation discusses a damped harmonic oscillator subject to a periodic driving force and obtaining the Fourier expansion of the given function. It also asks to find the steady state response to the driving force for specific values of m, b, and k. The conversation also provides useful integrals and mentions possible errors in the solution.
  • #1
Niner49er52
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Homework Statement



A damped harmonic oscillator originally at rest and in its equilibrium position is subject to a periodic driving force over one period by F(t)=-[tex]\tau^2[/tex]+4t^2 for -[tex]\tau/2[/tex]<t<[tex]\tau/2[/tex] where [tex]\tau[/tex] =n[tex]\pi[/tex]/[tex]\omega[/tex]

a.) Obtain the Fourier expansion of the function in the integral given and show that it is F(t)=-8[tex]\pi^2[/tex]/3[tex]\omega^2[/tex]+[tex]\sum[/tex]16(-1)^n/(n[tex]\omega[/tex])^2 cos(n[tex]\omega[/tex]t)

b.)Find the steady state response x[tex]_{p}[/tex](t) to this driving force when the damped harmonic oscillator satisfies the equation: mx"+bx'+kx=F(t) when m=1kg., b=20kg/s, and k=400N/m. This should be an infinite sum.


Homework Equations


For a driven damped harmonic oscillator with a single diving force, F(t)=F[tex]_{0}[/tex]sin([tex]\omega[/tex]t) the steady state solution has the form:
x[tex]_{p}[/tex](t)=(F[tex]_{0}[/tex]/m)/[tex]\sqrt{4\beta^2\omega^2}[/tex]+([tex]\omega^2_{0}[/tex]-[tex]\omega^2[/tex])^2 sin([tex]\omega[/tex]t-[tex]\delta[/tex])
where [tex]\delta[/tex]=tan[tex]^{-1}[/tex](2[tex]\beta[/tex][tex]\omega[/tex]/[tex]\omega^2_{0}[/tex]-[tex]\omega^2[/tex])

Since the force is an infinite series, you should be able to write the response as an infinite series where each term would have a corresponding term to that above.

Useful Integrals:

[tex]\int[/tex]t^2sin(n[tex]\omega[/tex]t)dt=-t^2 cos(n[tex]\omega[/tex]t)/n[tex]\omega[/tex] +2t(sin(n[tex]\omega[/tex]t)/n[tex]\omega[/tex])^2 +2cos(n[tex]\omega[/tex]t)/(n[tex]\omega[/tex])^3

[tex]\int[/tex]t^2cos(n[tex]\omega[/tex]t)dt=t^2sin(n[tex]\omega[/tex]t)/n[tex]\omega[/tex]+2tcos(n[tex]\omega[/tex]t)/(n[tex]\omega[/tex])^2-2sin(n[tex]\omega[/tex]t)/(n[tex]\omega[/tex])^3


The Attempt at a Solution


it would take way too long and too much confusing stuff to put everything I've done here, but any helpful suggestions would be welcome. i have it for the most part but keep running into small errors that are throwing me off such as a cosine term ending up a sine term and missing some terms that are associated with [tex]\tau[/tex]. part b.) is where I'm having the most trouble
 
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  • #2
just realized i made some mistakes when posting, wherever there is an omega as an exponent it should not be there! it is just multiplied through
 

1. What is a damped harmonic oscillator?

A damped harmonic oscillator is a type of oscillating system in which the amplitude of the oscillations decreases over time due to the presence of damping forces. In other words, it is a system that oscillates back and forth around a central equilibrium position, but gradually loses energy and eventually comes to rest.

2. What is a Fourier expansion?

Fourier expansion, also known as Fourier series, is a mathematical technique for representing a periodic function as a sum of sinusoidal functions. It is based on the idea that any periodic function can be broken down into a series of sine and cosine functions with different frequencies and amplitudes.

3. How is Fourier expansion used in the study of damped harmonic oscillators?

In the study of damped harmonic oscillators, Fourier expansion is used to describe the behavior of the system over time. By expanding the displacement function of the oscillator into a series of sinusoidal functions, we can analyze the contributions of each frequency component to the overall motion of the oscillator. This allows us to understand how the damping forces affect the oscillatory behavior of the system.

4. What is the significance of the Fourier coefficients in a damped harmonic oscillator?

The Fourier coefficients in a damped harmonic oscillator represent the amplitudes and phases of each frequency component in the Fourier expansion. These coefficients can tell us how each sinusoidal component contributes to the overall motion of the oscillator and how the damping forces affect each component differently. The coefficients also help us determine the decay rate and natural frequency of the oscillator.

5. Can a damped harmonic oscillator have an infinite number of Fourier coefficients?

No, a damped harmonic oscillator can only have a finite number of Fourier coefficients. This is because the damping forces eventually cause the amplitude of each frequency component to decrease to zero, and thus, there will be no more contributions from higher frequency components. However, for practical purposes, a sufficient number of coefficients can be used to accurately describe the behavior of the oscillator.

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