11/01/11 - Harmonically Driver Oscillators and Resonance

			Driven oscillator with no damping and ω ≠ ω0

The general harmonically forced oscillator equation with no damping is:

where ω0 is the natural frequency of the homogeneous system (no driving function), and ω is the angular frequency of the driving function.

The solution of this equation assuming ω ≠ ω0 is:

where the constants B and C depend on the initial conditions x(0) and x'(0).

A graph of this response is shown below for the parameter values indicated:

Another graph of this response is shown for a driving frequency closer to the natural frequency:

You can see from these graphs that we get "beats". When the driving frequency and the natural frequency are in phase, the amplitude of the response grows large, and when the two frequencies are out of phase, the amplitude of the response decreases significantly. It turns out that the beat frequency is the difference between the driving frequency and the natural frequency. As the frequencies get closer and closer together, the beats grow in amplitude and spread out in time.

			Movie of driven oscillator with amplitude beats at frequency near the natural frequency

			Driven oscillator with no damping and ω = ω0

The general harmonically forced oscillator equation with no damping is:

where the system is now driven at its natural frequency ω0.

The solution of this equation is:

where the constants B and C depend on the initial conditions x(0) and x'(0).

A graph of this response is shown below for the parameter values indicated:

Note that the driving function and the system are now always in phase, the amplitude of the response grows linearly without limit -- until something breaks.

For a real-life demonstration of this type or resonant response, watch the Tacoma Narrows Bridge videos at:

http://www.youtube.com/watch?v=HxTZ446tbzE&fmt=18

and

http://www.youtube.com/watch?v=3mclp9QmCGs&fmt=18

			Driven oscillator with damping

The general forced harmonic oscillator equation with damping is:

The general solution for the underdamped case (c < ω0) is:

The values of the constants B and C depend on the initial conditions, x(0) and x'(0). The first two terms represent the transient (temporary) part of the solution which decays with a time constant TC = 1/c. Note that the frequency of the transient part of the solution has been shifted slightly from the natural frequency due to the presence of damping. The last term represents the steady-state response. The addition of damping to the system keeps the response from increasing without limit for any frequency, including the resonant frequency, because the denominator of the steady-state part of the response has no real roots.

Note that the response can also be written in the form:

where G (called the transfer function of the system or the ratio of output/input amplitudes, also called the gain) is:

and the phase shift of the system is:

The resonant frequency ωres (at which maximum gain occurs) is found by taking the derivative of G with respect to ω, setting the result to zero and solving for ωres. The result is:

The maximum gain is then:

and the phase shift at resonance is:

The curve below shows the gain G of the system as a function of driving-function frequency (normalized to 1 for ω = ω0) for different amounts of damping:

Note the leftward shift in the resonant frequency (indicated by the black dot •) as the damping increases.