Forest UCM Osc Resonance

Oscillators driven by a source in resonance

$\ddot x + 2 \beta \dot x + \omega^2_0x = f(t)$

Complete Solution for the Sinusoidally Driven Damped oscillator

$x(t) =x_h + x_p = C_1 e^{r_1 t} + C_2 e^{r_2 t} + A \cos(\omega t-\delta)$

where

$r_1 = - \beta + \sqrt{\beta^2 - \omega_0^2}$

$r_2 = - \beta + \sqrt{\beta^2 + \omega_0^2}$

$A=\frac{f_0} { \sqrt{(\omega_0^2 - \omega^2)^2 + 4 \beta^2 \omega^2 }}$

$\delta = \tan^{-1}\left ( \frac{2 \beta \omega}{(\omega_0^2- \omega^2)} \right )$

Quality factor (Q)

Forest_UCM_Osc#Resonance