Help with oscillator problem before class please/thank you.

[Fjolvar]

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.

Homework Statement

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).

Homework 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(ω_dt) 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. ω_d = (ω²-K²)^(1/2) for under-damped
K=β/2m where β=damping coefficient

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.

 

[Fjolvar]

Any ideas?

 

[HallsofIvy]

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 \omega, four cycles will require that \omega t= 8\pi so that t= 8\pi/\omega.

Find the Amplitude when t= 8\pi/\omega and set it equal to C/e.

 

[Fjolvar]

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 \omega, four cycles will require that \omega t= 8\pi so that t= 8\pi/\omega.

Find the Amplitude when t= 8\pi/\omega and set it equal to C/e.

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 =/

 

[Fjolvar]

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.