1. ## Damped Oscillations

Hey,

I'm a bit stuck on these questions about damped oscillations. It's actually from a physics course but the question is very maths based.

I'm basically stuck (well can't do) sections a) b) and c). Normally I can do differentiation but this one is a real beast. I've tried doing the basic stuff in part a) (produce rule and all that) but i'm getting no where and end up with a huge equation which i don't know if its right or not. I must be doing something wrong.

Any help would be greatly appreciated!

2. Originally Posted by Cobra150
Hey,

I'm a bit stuck on these questions about damped oscillations. It's actually from a physics course but the question is very maths based.

I'm basically stuck (well can't do) sections a) b) and c). Normally I can do differentiation but this one is a real beast. I've tried doing the basic stuff in part a) (produce rule and all that) but i'm getting no where and end up with a huge equation which i don't know if its right or not. I must be doing something wrong.

Any help would be greatly appreciated!
(a) find $\displaystyle \dot{x}$

$\displaystyle x(t)=C e^{-\gamma t/2} \sin(w_f t+\phi)$

Using the product rule:

$\displaystyle \dot{x}(t)=C (-\gamma/2) e^{-\gamma t/2} \sin(w_f t+\phi)+C~ w_f~e^{-\gamma t/2} \cos(w_f t+\phi)$$\displaystyle = -\gamma ~x(t)/2 +C~ w_f~e^{-\gamma t/2} \cos(w_f t+\phi)$

Hence (again using the product rule):

$\displaystyle \ddot{x}(t) = -\gamma ~\dot{x}(t)/2 +C~(-\gamma/2) w_f~e^{-\gamma t/2} \cos(w_f t+\phi) - C~ (w_f)^2~e^{-\gamma t/2} \sin(w_f t+\phi)$

............. $\displaystyle =-\gamma ~\dot{x} (t)/2 -(w_f)^2 \dot{x}(t)+C~(-\gamma/2) w_f~e^{-\gamma t/2} \cos(w_f t+\phi)$

............. $\displaystyle =-\gamma ~\dot{x} (t)/2 -(w_f)^2 \dot{x}(t)+(-\gamma/2)[\dot{x}(t)+\gamma ~x(t)/2]$

Now check this (there is probably a mistake somewhere) and simplify.

RonL