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 minimized 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 ratio of the energy stored in the oscillator to the energy dissipated is defined as the "Quality" (Q) factor of the oscillation

$Q = \frac{\mbox{Energy stored in oscillator}}{\mbox{Energy dissipated}}$

Consider the case of the underdamped oscillator

$x = Ae^{- \beta t} \cos(\omega_1 t -\delta)$

$\dot x = -\omega_1 Ae^{- \beta t} \left [\sin(\omega_1 t -\delta) + \frac{\beta}{\omega_1}\cos(\omega_1 t -\delta) \right ]$

for a lightly damped oscillator

$\beta \lt \omega_0$

since

$\omega_1 \equiv \sqrt{\omega^2_0- \beta^2 } \approx \omega$

then

$\dot x = -\omega Ae^{- \beta t} \left [\sin(\omega t -\delta) \right ]$

The energy stored int the oscillator may be written in term of the maximum kinetic energy as

$E = K.E. (max) = \frac{1}{2} m \dot{x}^2 = \frac{1}{2} m \omega^2 Ae^{- 2\beta t} = E_0 e^{- 2\beta t}$

where

$E_0 =\frac{1}{2} m \omega^2 A$

The energy dissipated with time may be expressed as

$\Delta E = \left | \frac{dE}{dt} \right | \delta t = \left | -2 \beta E_0 e^{- 2\beta t} \right | \delta t = 2 \beta E \delta t$

The energy lost from one oscillation is then

$\Delta t = \frac{1}{\omega}$

$Q = \frac{\mbox{Energy stored in oscillator}}{\mbox{Energy dissipated}} = \frac{E}{\Delta E}$

$= \frac{E}{2 \beta E \frac{1}{\omega}}$

$= \frac{\omega }{2 \beta }$

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

$Q \equiv \frac{\omega_0}{2 \beta}=\frac{\frac{1}{\beta}}{2 \frac{1}{\omega_0}}=\frac{\tau_d}{2 \frac{1}{\omega_0}}$

$=\frac{\pi \tau_d}{2\pi \frac{1}{\omega_0}}$

$=\frac{\pi \tau_d}{\tau}=\pi \frac{\mbox{decay time}}{\mbox{period}}$

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.

Some Q values for several oscilaltors

Accelerator Settings

Q	System
5-10	rubberbands and loud speakers
$10^3$	tuning forks and violin strings
$10^4$	microwave cavity
$10^7$	excited atoms
$10^{10}$	CEBAF's accelerator RF cryomodule
$10^{12}$	excited nuclei
$10^{14}$	gas lasers

Forest_UCM_Osc#Resonance