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Help with oscillator problem before class please/thank you.

  1. Oct 16, 2011 #1
    I can't seem to figure out how to derive this relation, so a first step or any suggestions would be greatly appreciated. Thank you in advance.

    1. The problem statement, all variables and given/known data

    After four cycles the amplitude of a damped harmonic oscillator has dropped to 1/e of its initial value. Find the ratio of the frequency of this oscillator to that of its natural frequency (undamped value).

    2. Relevant equations

    I started off with two equations of motion:
    1. x=C*cos(ωt) for simple harmonic oscillator undamped
    2. x=C*e^(-Kt)*cos(ωdt) for under-damped since this seems to be the only case where four cycles would occur, although it is not specified.

    Also we have the equations for frequency for both cases:
    1. ω=2∏/T for simple harmonic oscillator
    2. ωd = (ω2-K2)^(1/2) for under-damped
    K=β/2m where β=damping coefficient

    3. The attempt at a solution
    I'm having trouble isolating the frequency from both equations, but I'm not sure there is any need to use the equations of motion for a simple ratio of the frequencies. I know that four cycles indicates 4T = 4*(2∏/ωd) for the under-damped.. but not sure where to go from there.
  2. jcsd
  3. Oct 17, 2011 #2
    Any ideas?
  4. Oct 17, 2011 #3


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    Science Advisor

    There is no reason to "isolate" the frequency, you are not asked for the frequency or the damping constant. You are asked for their ratio.

    YOu are told that "After four cycles the amplitude of a damped harmonic oscillator has dropped to 1/e of its initial value". Given frequency [itex]\omega[/itex], four cycles will require that [itex]\omega t= 8\pi[/itex] so that [itex]t= 8\pi/\omega[/itex].

    Find the Amplitude when [itex]t= 8\pi/\omega[/itex] and set it equal to C/e.
  5. Oct 17, 2011 #4
    So I plug in for t in the equations of motion and solve for the amplitude C, but I'm still stuck on how to show the ratio of frequency. I'm just not seeing the form with the frequency of one over the other =/
  6. Oct 17, 2011 #5
    I don't think you can find C from the equation of motion since we don't know what x is.. so I'm guessing there's another relation between frequency and amplitude, but I haven't been successful in finding it yet. Any help would be greatly appreciated.
    Last edited: Oct 17, 2011
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