Past exam of the mathematical methods I course of the University of Oxford / 2019 / 10 / c / Solution

In the steady periodic response, the cycle average of the mechanical energy derivative vanishes. The applied average power therefore equals the power dissipated by the viscous force:Using solution givesThe same result follows from averaging the instantaneous input power $F(t)\dot{z}(t)$, using the response phase relative to the drive.

For positive $m,k,\gamma$ and $\omega >0$, rewrite this asThe last factor is at most one, with equality precisely when $k-m{\omega }^2=0$. Thus the power peak occurs at ${\omega }_0=\sqrt{k/m}$ andThis power-resonance angular frequency is exactly ${\omega }_0$; it need not equal the angular frequency maximizing displacement amplitude. Positive damping is essential: at $\gamma =0$ and exact resonance there is no bounded steady periodic response to nonzero forcing, so the divergent expression is not a finite-power solution.