# Forest UCM Osc Resonance

Oscillators driven by a source in resonance

## Complete Solution for the Sinusoidally Driven Damped oscillator

where

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 is maximized.

since

then for a given forced amplitude

you can maximize the oscillation by minimizing the denominator term

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

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

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

## Resonance Width

While the natural frequency determine the frequency where the maximum oscillation can occur, the dampening force parameter determines the width of the resonance.

If you look at the equation for the amplitude squared

for the case that

You get a maximum amplitude when

The magnitude of is cut in half if the denominator becomes

returning back to the original form of the denominator

you can have the denominator be

or

The parameter 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

Consider the case of the underdamped oscillator

for a lightly damped oscillator

since

then

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

where

The energy dissipated with time may be expressed as

The energy lost from one oscillation is then

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

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
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