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 maximize the oscillation by minimizing the denominator term
- [math]\displaystyle{ (\omega_0^2 - \omega^2)^2 + 4 \beta^2 \omega^2 }[/math]
since [math]\displaystyle{ \beta }[/math] is the amount of friction being applied remove energy from the system and [math]\displaystyle{ \omega_0 }[/math] is the natural oscilaltion frequency (constants that characterize the system)
the only term you can change is the drive frequency [math]\displaystyle{ \omega }[/math] of your applied sinusoidal force.
if your set your sinusoidal force to a frequncy
- [math]\displaystyle{ \omega = \omega_0 }[/math]
then the denominator is minimazed thereby maximizing the amplitude of the forced oscillation.
Resonance occurs when your applied sinusoidal force matches the natural frequency of the oscillaor.
This amplude is
- [math]\displaystyle{ A= \frac{f_0}{2\beta \omega_0} }[/math]
Quality factor (Q)
While the natural frequency [math]\displaystyle{ (\omega_0) }[/math] determine the frequency where the maximum oscillation can occur, the dampening force parameter [math]\displaystyle{ (\beta) }[/math] determines the width of the resonance.
Problem 5.41
The "sharpness" of the resonance is quantified in terms of a quality factor given by the ration of the natural frequency to the dampening as
- [math]\displaystyle{ Q \equiv \frac{\omega_0}{2 \beta} }[/math]