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

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

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

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

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

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 $A$ is maximized.

since

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

then for a given forced amplitude $(f_0)$

you can maximize the oscillation by minimizing the denominator term

$(\omega_0^2 - \omega^2)^2 + 4 \beta^2 \omega^2$

since $\beta$ is the amount of friction being applied remove energy from the system and $\omega_0$ is the natural oscilaltion frequency (constants that characterize the system)

the only term you can change is the drive frequency $\omega$ of your applied sinusoidal force.

if your set your sinusoidal force to a frequncy

$\omega = \omega_0$

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

$A= \frac{f_0}{2\beta \omega_0}$

Resonance Width

While the natural frequency $(\omega_0)$ determine the frequency where the maximum oscillation can occur, the dampening force parameter $(\beta)$ determines the width of the resonance.

If you look at the equation for the amplitude squared

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

for the case that

$\beta \lt \omega_0$

You get a maximum amplitude when $\omega \approx \omega_0$

$A^2 \approx \frac{f_0^2}{ 4 \beta^2 \omega_0^2}$

The magnitude of $A^2$ is cut in half if the denominator becomes

$A^2 \approx \frac{f_0^2}{ 8 \beta^2 \omega_0^2}$

returning back to the original form of the denominator

$A^2=\frac{f_0^2} { (\omega_0^2 - \omega^2)^2 + 4 \beta^2 \omega^2 } =\frac{f_0^2}{ 8 \beta^2 \omega_0^2}$

you can have the denominator be

$\Rightarrow(\omega_0^2 - \omega^2)^2 + 4 \beta^2 \omega^2 = 8 \beta^2 \omega_0^2$

or

$(\omega_0^2 - \omega^2)^2 = 4 \beta^2 \omega_0^2$

$(\omega_0 - \omega)(\omega_0 + \omega) = \pm 2 \beta \omega_0$

$(\omega_0 - \omega)(2\omega_0 ) = \pm 2 \beta \omega_0$

$\omega = \omega_0 \pm 2 \beta$

The parameter $\beta$ determines the width of the resonance

Quality factor (Q)

The "sharpness" of the resonance is quantified in terms of a quality factor given by the ratio of the natural frequency to the dampening as

$Q \equiv \frac{\omega_0}{2 \beta}$

The above is proportions to the amount of energy stored in one cycle of the oscillation divided by the average energy dissipated in one cycle.

Forest_UCM_Osc#Resonance