Question

What is the steady state power dissipation for a driven damped harmonic oscillator with sinusoidal input? Where does the 2 pi factor come from in the equation \[ Q = 2\pi \frac{Energy stored}{Energy dissipated per cycle}\]

Answer 1

\[Q = \frac{m\omega_0}{b}\] Where b is the damping constant, such that \[F_{damping} = -bv\]
The problem I am having is
\[Power = Fv = -bv^2\]
Where
\[v = -\omega A \sin(\omega t + \theta)\]

\[E = \int\limits_{0}^{2\pi/\omega} Power(t)dt= -b\omega^2A^2\int\limits_{0}^{2\pi/\omega}\sin^2(\omega t + \theta)dt = -b\omega^2A^2/2\]
There's no 2 pi there, and my book indicates there should be a factor of 2pi, and the power of omega should be one, not two. However, there is no period in the book's expression, so substituting 2pi/T for one omega won't do it.

Answer 2

\[\frac{2\pi}{cycle}=1\]

Answer 3

I confused the time integral of sin^2 with its average value. The time integral over a period should be\[\pi/\omega\]