Damping is a condition in which external force operates in such a manner that it impedes the motion of oscillatory body. As a matter of fact, all real time harmonic motions that we consider to be simple are actually damped SHM. We consider them SHM only as an approximation. The motion of a block hanging from a spring, for example, is not SHM as air works to oppose the motion – at every instant. As a consequence, the span (amplitude) of the motion keeps decreasing every cycle. Diminishing amplitude is the characterizing feature of damped oscillation. A typical displacement – time plot looks as shown here.
| Damped oscillation |
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The real time situation may be a bit complex to describe damped oscillation mathematically. Here, we consider one simplified situation in which air resistance can be considered to be proportional to the velocity of the oscillatory body. In such case, net force on the oscillating body is the resultant of restoring and damping force (with a negative sign) :
In terms of displacement derivatives :
For small damping constant “b”, the solution of this differential equation yields :
The amplitude of the oscillation is a decreasing function in time, which tends to become zero :
A damped oscillation thus dies down as the case with most of the oscillatory systems, which are not provided with external energy to compensate energy dissipated due to damping. On the other hand, the angular frequency of damped oscillation depends on additional factor of proportionality constant “b” :
Clearly, angular frequency of damped oscillation is lesser than corresponding angular frequency if damping is absent i.e. b = 0,







