Damped oscillator consecutive amplitude ratio

Uniquebum

1. The problem statement, all variables and given/known data
Undamped oscillator's period [itex]T_0 = 12s[/itex]. Damped oscillator's angular frequency [itex]\omega_1 = \omega_0 * 97\%[/itex] where [itex]\omega_0[/itex] is the angular frequency of the undamped oscillator's. What is the ratio of consecutive maximum amplitudes?

2. Relevant equations
Equation of damped oscillator's motion:
[itex]x = e^{-\alpha t}A_0sin(\omega_1 t + \phi)[/itex]
where [itex]\alpha = \frac{b}{2m}[/itex] where [itex]b = [/itex]damping constant.


3. The attempt at a solution
Firstly, were' talking about maximums so we can disregard the sin() function.
Calculating [itex]\omega_1 = \omega_0 * 0.97 = \frac{2\pi}{T_0}0.97[/itex].
Thus for the damped oscillator [itex]T_1 = \frac{T_0}{0.97}[/itex]

Then we could write something as follows:
[itex]\frac{x_0}{x_1} = \frac{e^{-\alpha t_0}A_0}{e^{-\alpha t_1}A_0}[/itex]
but we have no clue of alpha nor about x_0 and x_1... Any help appreciated.

Stonebridge

This is useful
http://en.wikipedia.org/wiki/Logarithmic_decrement

Uniquebum

I ended up using the formula

[itex]\zeta = \sqrt{1-(\frac{\omega_1}{\omega_0})^2}[/itex]
And got approx 0.243 out of it. In my answer spreadsheet they claim the answer to be 0.21 however. Now i'm wondering whether i got it right or not... heh :) Thanks for the help either way.