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Damped Oscillations

The energy of a damped oscillator is spent in overcoming the frictional forces and hence continuously decreases with time. Consequently the amplitude of oscillation decreases. The kinetic energy of the oscillator, at any time t, is given by

where,

The instantaneous potential energy is given by

Now, the time average value of kinetic energy per cycle is given by,

where, we assumed a low damping so that the time dependent amplitude A (t) = is regarded as constant over one cycle; hence, it is taken out from integral. Further, we used the fact that the time average of sin² (ωt + Ø) and cos² (ωt + Ø) is ½, and that of sin² (ωt + Ø) is zero.

Similarly, the average value of potential energy is given by,

Thus, we get the relations similar for undamped oscillator except for the fact that amplitude A₀ is replaced by

Average total energy is therefore

We see that the above power loss is exactly equal to average rate of work done per cycle against frictional force (F_ƒ = – av); the average work done during one oscillation is,

Hence, average rate of work done is