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Green's function for a critically damped oscillator

  1. Sep 10, 2011 #1
    1. The problem statement, all variables and given/known data
    Consider critically damped harmonic oscillator, driven by a force F(t)
    Find the green's function G(t,t') such that x(t) = ∫ dt' G(t,t')F(t') from 0 to T solves the equation of motion with x(0) =0 and x(T) =0


    2. Relevant equations
    x(t) = ∫ dt' G(t,t')F(t') from 0 to T


    3. The attempt at a solution

    Hi guys, im completely new to green's function.. need alot of help understanding the use, and how to use it >_<

    I've been doing some readings and this is what i understand so far

    x = [itex]x _{h}[/itex] + ∫G(t,t')f(t') dt'

    i.e. G(t,t') will be the particular solution to the ode with F(t) = δ(t-t')

    does this mean that I should let [itex]\ddot{x}[/itex] +2γ[itex]\dot{x}[/itex] + [itex]ω _{0}[/itex]² = δ(t-t') and solve this to get the green's function?
     
  2. jcsd
  3. Sep 10, 2011 #2

    Hootenanny

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    Yes. Except that there's a small typo in your expression. It should be

    [tex]\ddot{x} +2\gamma\dot{x} + \omega_{0}^2 x = \delta(t-t^\prime)[/tex]

    A nice way to solve the above equation is to write the RHS as the inverse of a Fourier transform, meaning the RHS becomes a harmonic function of time.
     
  4. Sep 10, 2011 #3
    Hello! thanks for you reply! :)

    yeah you are right, i missed out the x there

    not sure if i got it right but on the RHS, i got [itex]\frac{1}{2\pi}[/itex][itex]\frac{1}{i(t-t')}[/itex][itex]e^{i\omega(t-t')}[/itex]+C

    is this right?

    oh and (d²/dt² + 2γd/dt + ω0²) G(t,t') = δ(t-t')
    so i should be solving as usual 2nd order ode to get my answer for homogeneous and inhomo am i doing the steps right?
     
  5. Sep 10, 2011 #4
    Try taking the Fourier transform of both sides. To makes things simpler, recognize that the dependent of t and t' in G has this form: [itex]G(t, t^\prime) = G(t - t^\prime)[/itex]
     
  6. Sep 10, 2011 #5
    im really bad at fourier transform, so i didnt follow your advice..

    but i did try considering the force being written as a fourier transform
    f(t) = 1/2[itex]\pi[/itex]∫F(w)[itex]e^{iwt}[/itex] dw
    and the dirac delta
    δ(t-t')= 1/2[itex]\pi[/itex]∫ [itex]e^{iwt}[/itex][itex]e^{-iwt'}[/itex] dw

    so i went ahead to solve and i got that

    G(t,t') = [itex]x_{h}[/itex] + 1/2[itex]\pi[/itex] ∫ ([itex]e^{iwt}[/itex][itex]e^{-iwt'}[/itex]) / ([itex]-\omega^{2}[/itex] + 2i[itex]\gamma\omega[/itex] + [itex]\omega_{0}^{2}[/itex]) dw



    and im stuck at this integration lol
     
  7. Sep 11, 2011 #6

    Hootenanny

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    The integration is a little tricky. You're going to need to use complex analysis, which I'm assuming you've at least met. Since we are dealing with a critically damped oscillator, we can set [itex]\gamma=1[/itex]. What we're going to do now is extend the integral into the complex plane. Let [itex]C\subset\mathbb{C}[/itex] be a semi-circle in the upper half plane, that is, [itex]C=\{\omega:|\omega|< a, \omega\geq0\}[/itex]. So, we have

    [tex]I=\int_C \frac{e^{i\omega(t-t^\prime)}}{-\omega^2 + 2i\omega_0\omega + \omega_0^2}\;\text{d}\omega[/tex]

    Can you do the next step, maybe using Cauchy's Residue Theorem?
     
    Last edited: Sep 11, 2011
  8. Sep 11, 2011 #7
    Hmm nope I have never heard of the residue theorem. I tried wiki-ing it to see how it works, but it looks complicated. Is there any other way to do this?
     
  9. Sep 11, 2011 #8

    Hootenanny

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    I can't immediately see another straightforward method to evaluate the integral. Let me ask a few of the other homework helpers and get back to you.
     
  10. Sep 11, 2011 #9

    vela

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    The differential equation for the damped harmonic oscillator is usually
    [tex]\ddot{x} + 2\zeta\omega_0 \dot{x} + \omega_0^2x = 0[/tex]There's a factor of [itex]\omega_0[/itex] in the damping term, and with this convention, a critically damped oscillator would correspond to [itex]\zeta=1[/itex], so I think you actually want [itex]\zeta=\omega_0[/itex] here.
     
  11. Sep 11, 2011 #10

    Hootenanny

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    Good catch! I didn't notice that :redfaced:

    I will correct my earlier posts.
     
  12. Sep 16, 2011 #11

    vela

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    Another way to find the Green's function is to solve the differential equation in the two regions 0≤t<t' and t'<t≤T. Apply the boundary conditions, and then match the two solutions at t=t'. The Green's function needs to be continuous at the boundary, but its derivative is not. By integrating the differential equation from t'-ε to t'+ε and letting ε→0, you can find the discontinuity in the derivative.
     
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