Mean power in driven, damped harmonic oscillators

[yjx]

Ok, there's a bit I don't understand in my lecture notes. The maths doesn't seem to quite work out. Any help would be appreciated.

Here's the section I'm confused about:

http://img228.imageshack.us/i/physy.jpg/

It's the transition from the second last line of working to the last line which I can't figure out.

It's probably just me being stupid but I can't see how the two are equivalent.

Thanks.

 

[AlephZero]

You mean getting from Real part of the long expression, to
m gamma |v_0|^2 ??

The reason is that the omegas are real quantities, so the term divided by (i omega) is imaginary and doesn't contribute to the real part.

What this means phyiscally is that the force only does work over a complete cycle against the displacement component that is 90 degrees out of phase with it (or the velocity component that is in phase with it). At resonance, the displacement is 90 degrees out of phase with the force, so the force can do work and maintain the amplitude of oscillation at a large value. A long way away from resonance, the displacement is either nearly in phase with the force or nearly 180 degrees out of phase. The force puts energy into the system for half of each cycle but the energy is given back during the other half, and the amount of work done over the complete cycle is small.

 

[yjx]

Ah of course, I'd read the equation wrongly. I thought that the gamma was within the division which it isn't. Now it makes perfect sense! I blame poor equation writing (obviously not my failure to count the number of brackets).

Thanks for the help!