- #1

Redwaves

- 134

- 7

- Homework Statement
- Calculate quality factor of a damped oscillation from a graph

- Relevant Equations
- ##x(t) = A_0 e^{\frac{-xt}{2}}cos(\omega_d t - \alpha)##

I'm trying to find the quality factor of a damped system.

I know 3 points from the graph, ##(t,x): (\frac{\pi}{120},0.5), (\frac{\pi}{80},0), (\frac{\pi}{16},0)##

From this I found that ##T = \frac{\pi}{20}##

##\omega_d = \frac{2\pi}{T} = 40 rad##

Then, from the solution ##x(t) = A_0 e^{\frac{-xt}{2}}cos(\omega_d t - \alpha)##

##cos(\omega_d t - \alpha)## must be 0 when x = 0

##\omega_d t - \alpha = \frac{\pi}{2}##

##\alpha = 0##

I also know that ##Q = \frac{\omega_0}{\gamma}##

and

##\omega_0^2 = \omega_d^2 + \frac{\gamma^2}{4}##

I need some help to find ##\gamma##

I know 3 points from the graph, ##(t,x): (\frac{\pi}{120},0.5), (\frac{\pi}{80},0), (\frac{\pi}{16},0)##

From this I found that ##T = \frac{\pi}{20}##

##\omega_d = \frac{2\pi}{T} = 40 rad##

Then, from the solution ##x(t) = A_0 e^{\frac{-xt}{2}}cos(\omega_d t - \alpha)##

##cos(\omega_d t - \alpha)## must be 0 when x = 0

##\omega_d t - \alpha = \frac{\pi}{2}##

##\alpha = 0##

I also know that ##Q = \frac{\omega_0}{\gamma}##

and

##\omega_0^2 = \omega_d^2 + \frac{\gamma^2}{4}##

I need some help to find ##\gamma##