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 maximize the oscillation by minimizing the denominator term

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

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

the only term you can change is the drive frequency [math]\omega[/math] of your applied sinusoidal force.

if your set your sinusoidal force to a frequncy

[math]\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]A= \frac{f_0}{2\beta \omega_0}[/math]

Quality factor (Q)

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

If you look at the equation for the amplitude squared

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

for the case that

[math]\beta \lt \omega_0[/math]

You get a maximum amplitude when [math]\omega \approx \omega_0[/math]

[math]A^2 \approx \frac{f_0^2}{ 4 \beta^2 \omega_0^2}[/math]

The magnitude of [math]A^2[/math] is cut in half if the denominator becomes

[math]A^2 \approx \frac{f_0^2}{ 8 \beta^2 \omega_0^2}[/math]

returning back to the original form of the denominator

[math]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}[/math]

you can have the denominator be

[math]\Rightarrow(\omega_0^2 - \omega^2)^2 + 4 \beta^2 \omega^2 = 8 \beta^2 \omega_0^2[/math]

or

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

[math](\omega_0 - \omega)(\omega_0 + \omega) = \pm 2 \beta \omega_0[/math]

[math](\omega_0 - \omega)(2\omega_0 ) = \pm 2 \beta \omega_0[/math]

[math] \omega = \omega_0 \pm 2 \beta [/math]

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] Q \equiv \frac{\omega_0}{2 \beta}[/math]

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