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.(adsbygoogle = window.adsbygoogle || []).push({});

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(ωfor under-damped since this seems to be the only case where four cycles would occur, although it is not specified._{d}t)

Also we have the equations for frequency for both cases:

1. ω=2∏/T for simple harmonic oscillator

2. ω_{d}= (ω^{2}-K^{2})^(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∏/ω_{d}) for the under-damped.. but not sure where to go from there.

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# Homework Help: Help with oscillator problem before class please/thank you.

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