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.

If you look at the equation for the amplitude squared

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

for the case that

[math]\displaystyle{ \beta < \omega_0 }[/math]

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

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

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

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

returning back to the original form of the denominator

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

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]

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