Driven Harmonic Motion
Forces and Motion

Energy in forced oscillation

Teaching Guidance for 14-16 PRACTICAL PHYISCS

During each cycle of its oscillation the driver transfers 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 transferred 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 transferred from the driver is then 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:

Q = 2π × energy storedenergy dissipated per cycle

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 dissipated.

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 suspension1
Tethered trolley10
Simple pendulum1,000
Guitar string1,000
Quartz crystal of watch105
Excited atom107
Excited nucleus1,012

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

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