Deriving energy loss rate for damped oscillator

[Bill12]

Hi,

I do not know how to drive an experession for energy loss of damped oscillator.I know that:

X(t)=A exp(-Beta*t)cos(wt-delta)
and:
v=dx/dt...
I found E=K+U
but it seems to be so messy. It is like:

E=(1/2)*m*(A^2)*exp(-2*beta*t)[ beta^2 (cos(wt-delta))^2)+beta*

sin 2(wt-delta)+w^2 ]

I do not know if it is right or not but also I do not know how to get the energy loss from it.

I will thank for help.

 

[krab]

Do you want loss, or loss rate? Loss rate is easy. dE/dt is proportional to v^2. Integrating results indeed in a complicated expression. But if you are in an underdamped regime (many oscillations before the movement decays away), the expression represents an exponential decay with a small modulation on top of it. If you are not interested in the small modulation, the expression is very simple. You can find it for example by finding 1/2 mv^2 at those times when x goes through zero.

 

[Bill12]

Finally solved it.
Thanks for help.