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Energy in forced oscillation

During each cycle of its oscillation the driver delivers some energy to the driven oscillator. Some of this energy may be returned to the driver later in the cycle (for example, a spring which has been forced to stretch can pull back, helping the driver return in its oscillation) but, at resonance, the phase difference p /2 prevents this. Some energy is dissipated in overcoming damping. The rest is stored in the forced oscillator, increasing its amplitude.

The amounts of energy lost to damping in each cycle increases with the amplitude of the oscillation. The final amplitude of oscillation is that for which the energy delivered by the driver in each cycle is equal to the energy used to overcome damping. With heavy damping this happens at small amplitude, and even the resonant oscillations are not violent. With light damping large amplitudes of the driven oscillator are achieved. A lot of energy of the driver is stored in the driven oscillator, particularly at resonance.

The quality factor, Q
The quality factor, Q, of an oscillator can be formally defined like this:

However, there is a much more useful, though non-rigorous, description of Q: it is approximately equal to the number of free oscillations which occur before all the oscillator's energy is gone.

Q is related to the degree of damping of the oscillator, and to the sharpness of its resonance peak. Low values of Q are associated with heavily damped oscillations which do not resonate violently and which die away quickly if they are not forced. High values of Q are associated with light damping and sharp resonance.

Some typical values of Q are:
Car suspension                        1
Tethered trolley                      10
Simple pendulum                    1000
Guitar string                             1000
Quartz crystal of watch         105
Excited atom                           107
Excited nucleus                       1012

Consider the guitar string, for example. The energy is emitted as sound waves, with a fundamental frequency of, say, 512 Hz (the C above middle C). If Q = 1000, then roughly 1000 oscillations occur before all the energy is gone. Thus the plucked string will cease to oscillate after 1000/512  2 seconds: which agrees roughly with experience.