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Damped Harmonic Oscillator Fourier Expansion

  1. Nov 6, 2008 #1
    1. The problem statement, all variables and given/known data

    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.


    2. Relevant 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


    3. The attempt at a solution
    it would take way too long and too much confusing stuff to put everything ive 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
     
  2. jcsd
  3. Nov 6, 2008 #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
     
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