SHM can only continue indefinitely if energy is not lost from the system, that is to say if energy is not used up by work being done against dissipative forces such as friction.
Damping is therefore another term for resistance. As energy is lost so will the amplitude of the oscillations die away. This is called Damped Harmonic Motion and, since all systems are damped to a greater or lesser extent, this is the kind of harmonic motion that we are most used to.
For example the note from a tuning fork dies away after a few seconds because of the resistance of the air and internal losses in the metal.
Similarly for the note made by a stretched string. On a piano persisting notes can be a nuisance so once the pianist lifts his finger off the note a pad, called a damper, comes into contact with the string. If the pianist actually requires the note to be sustained he presses the loud pedal which lifts all the dampers off the strings.
What then is Forced Harmonic Motion?
Forced harmonic motion is simply that motion which results when applied forces are also involved (i.e. forces other than the damping or the inherent restoring forces that we have discussed). It is easy to see that the resultant motion depends very much on the interplay between the restoring force and the applied force.
How then does this link up with resonance?
Let us consider a child’s swing. Damping due to friction in the ‘hinge’ and the resistance of the air cause any oscillations to diminish in amplitude if the swing is unattended. To overcome such losses requires energy to be put back into the swing; hence the parent. However, for this to work effectively the parent has to make sure that the ‘push’ is applied in the same direction as the instantaneous velocity. This is most conveniently and safely done at the point of maximum displacement. Hence the frequency of the push is the same as that of the natural frequency of the swing and the correctness of the timing means that the push has to be in phase with the oscillation. This leads to a state of dynamic equilibrium whereby the maximum amplitude of the swing stays at some constant value. This is an example of not only forced harmonic motion but also of resonance because the applied force is at the same frequency and in the correct phase with the natural motion.
A more enthusiastic parent might decide to put more energy in with the result that a new equilibrium is established with a larger amplitude of swing resulting. In any equilibrium situation the energy put in per unit time by the parent matches the energy lost per unit time from the system.
We should remember that a larger amplitude of swing occurring at the same natural frequency must imply greater accelerations.
If we had forced harmonic motion in a system in which there were no losses there is no way to lose energy so in a resonance situation the amplitude of oscillation would grow and grow with time, as would the accelerations involved. Ultimately the forces would become infinite and the system would tear itself apart.
This does not happen because we do not have frictionless systems and under these circumstances the maximum displacement (and hence the maximum forces) are determined by the magnitude of the applied force and the magnitude of the dissipative force.
What is resonant amplification?
Let us assume in our example of the swing that, starting from rest, the parent imparts a force giving 1 unit of acceleration at every push, so that after say 5 pushes the swing has accrued 5 units of acceleration. This will not be entirely true because the swing will have lost some acceleration due to the frictional forces; let us say the accumulated acceleration is 3.5 units at this stage. The swing continues to accumulate acceleration until the frictional losses increase to match the applied gains, that is to say until the frictional losses are themselves equivalent to 1 unit of acceleration, and dynamic equilibrium is reached. For arguments sake let the accumulated acceleration at this stage be 8.2 units.
We say that the resonant amplification, or Q factor, is 8.2. It is simply a number. Had the input been 2 units per swing then the final accumulated acceleration would have been 16.4 units, to a first approximation.