Forest UCM Osc Resonance

Oscillators driven by a source in resonance

[math] \ddot x + 2 \beta \dot x + \omega^2_0x = f(t)[/math]

Complete Solution for the Sinusoidally Driven Damped oscillator

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

where

[math]r_1 = - \beta + \sqrt{\beta^2 - \omega_0^2}[/math]

[math]r_2 = - \beta + \sqrt{\beta^2 + \omega_0^2}[/math]

[math]A=\frac{f_0} { \sqrt{(\omega_0^2 - \omega^2)^2 + 4 \beta^2 \omega^2 }}[/math]

[math]\delta = \tan^{-1}\left ( \frac{2 \beta \omega}{(\omega_0^2- \omega^2)} \right )[/math]

The first two terms in the solution are exponentially decaying and tend to damp the oscillation.

The last term in the solution is the response of the system to a sinusoidal driving force.

Thus you can apply a force to prevent the oscillations from dying out.

Resonance

Resonance is the condition that your applied sinusoidal force is set to a frequency that will maximize the damped oscillations.

This means that the amplitude [math]A[/math] is maximized.

since

[math]A=\frac{f_0} { \sqrt{(\omega_0^2 - \omega^2)^2 + 4 \beta^2 \omega^2 }}[/math]

then for a given forced amplitude [math](f_0)[/math]

you can maximuze the oscillation by minimizing the denominator term

[math](\omega_0^2 - \omega^2)^2 + 4 \beta^2 \omega^2 [/math]

Quality factor (Q)

Forest_UCM_Osc#Resonance