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Damped oscillator- graphical interpretation

  1. Oct 13, 2012 #1
    Hi!
    The damped oscillator equation is as follows:
    x(t)= A exp(γt/2) cos(ωt)

    where ω= √( (w0)^2 + (γ^2)/4 )

    I have attached a graph of a damped oscillator.
    The question is if I use graph to measure angular frequency, will it be w0 or ω?

    It should be w0 because if I put γ=0, I should be getting the normal undamped system. The enveloped curve would disappear since exp(γt/2) is 1. BUT then where is ω on the graph!!!! :grumpy:
     

    Attached Files:

  2. jcsd
  3. Oct 13, 2012 #2

    PhysicoRaj

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    How can u expect ω in an x-t graph? You will have to calculate.
     
  4. Oct 13, 2012 #3
    of course I know that. Let m rephrase. The time T between successive maxima is constant. So I consider the complete oscillations, k, for a given time, t. To get angular frequency, W= k* 2pi/t
    The question is, what is this this W? is it w0 or is it the angular frequency ω of the damped oscillator
     
  5. Oct 13, 2012 #4
    it is the angular frequency of damped oscillator.
     
  6. Oct 13, 2012 #5

    Philip Wood

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    The answer to your question is ω. Peaks in the damped sinusoid [itex]e^{-kx} cos(\omega t)[/itex]occur a little before the peaks in the pure sinusoid [itex]cos(\omega t)[/itex], but by the same amount each time, so the time between peaks is the same as that between the peaks in [itex]cos(\omega t)[/itex].
     
  7. Oct 13, 2012 #6

    AlephZero

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    An easier way to see the anwer is ##\omega## is to think about the times when x(t) = 0. They are the roots of ##\cos \omega t = 0##.
     
  8. Oct 13, 2012 #7

    Philip Wood

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    AlephZero. Agree, but thought rsaad (in post 3) was worried about maxima.
     
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