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

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

Quality factor (Q)

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.

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

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

Forest_UCM_Osc#Resonance