Forest UCM Osc Resonance
Oscillators driven by a source in resonance
- [math]\displaystyle{ \ddot x + 2 \beta \dot x + \omega^2_0x = f(t) }[/math]
Complete Solution for the Sinusoidally Driven Damped oscillator
- [math]\displaystyle{ 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]\displaystyle{ r_1 = - \beta + \sqrt{\beta^2 - \omega_0^2} }[/math]
- [math]\displaystyle{ r_2 = - \beta + \sqrt{\beta^2 + \omega_0^2} }[/math]
- [math]\displaystyle{ A=\frac{f_0} { \sqrt{(\omega_0^2 - \omega^2)^2 + 4 \beta^2 \omega^2 }} }[/math]
- [math]\displaystyle{ \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]\displaystyle{ A }[/math] is maximized.
since
- [math]\displaystyle{ A=\frac{f_0} { \sqrt{(\omega_0^2 - \omega^2)^2 + 4 \beta^2 \omega^2 }} }[/math]
then for a given forced amplitude [math]\displaystyle{ (f_0) }[/math]
you can maximuze the oscillation by minimizing the denominator term
- [math]\displaystyle{ (\omega_0^2 - \omega^2)^2 + 4 \beta^2 \omega^2 }[/math]