Finding the ratio ω/ωo of an underdamped oscillator

HiggsBrozon

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

The amplitude of an underdamped oscillator decreases to 1/e of its initial value
after m complete oscillations. Find an approximate value for the ratio ω/ω0.


2. Relevant equations

x''+2βx'+ω₀²x = 0 where β=b/2m and ω₀=√(k/m)

x(t) = Ae^-βtcos(ω₁t-δ) where ω₁ has been defined as ω₀²-β²

3. The attempt at a solution

The initial amplitude is equal to A₀ = Ae^-βt
and the final amplitude after m oscillations is equal to A₀(1/e) = A_f = Ae^-(βt+1)

After this I honestly don't know where to go. I tried plugging in my A_f into the underdamped motion equation and solving for ω but that didn't seem to make any sense. I'm assuming that ω₀ will be equal to just √(k/m)?
Also, I thought that the frequency of an underdamped oscillator didn't change over time. So why would the angular frequency change?
If anyone could give me a push in the right direction that would be very helpful. I've been working on this for a quite while now.

bossman27

Normally, I think we'd have [itex] w^{2} = w_{0}^{2} - \beta ^{2} [/itex]. I'm actually more accustomed to writing [itex] w^{2} = w_{0}^{2} - \frac{\gamma ^{2}}{4} [/itex] , but regardless, you've goofed up your amplitude relationships.

After m periods, [itex] t = Tm [/itex] where [itex] T [/itex] is the period of oscillation in seconds. This also means the cosine term will have the same value as it does at [itex] t=0 [/itex].

Thus, your equation should be [itex] A(t=Tm) = A_{0} e^{-1} = A_{0} e^{-\beta T m} \Rightarrow \beta T m = 1 [/itex]

See if you can go from there.

bossman27

Also, what makes you think the angular frequency is changing? Even with your incorrect equation, I don't see how you would make that deduction.

rude man

First, without loss of generality, let δ = 0 since this is related to initial conditions & you weren't given any.

So the 1st (initial) amplitude is Aexp(-βt) with t=0.
And at the end of 1 cycle, which lasts t = ? seconds, what is the amplitude in terms of A, β and ω₁? And so on 'till at the end of 5 cycles?

And what did the problem say the amplitude after 5 cycles was as a % of the first amplitude?
So how about an equation in β and ω₁ plus the equation I corrected for you above (in red)?

HiggsBrozon

My apologies on the late reply guys. It was a late night and I ended up finishing the problem the following morning. Thanks for your help!

For my equation of motion I used x(t) = Ae^-βtcos(ω₁t) = Ae^-βmTcos(ω₁mT)
where ω₁ = √(ω₀²-β²) (and yes my original formula was missing a square root.)
and T = 2pi/ω₁

The cosine term is always equal to 1 so
x(t) = Ae^-βmT = A/e
Therefore, βmT = 1
→ β = 1/(mT) = ω₁/(2pim)

ω₁/ω₀ = √(ω₀²-β²)/ω₀ = √(1-β²/ω₀²) = 1 - β²/(2ω₀²) = 1 - 1/(8pi²m²)