Damped Harmonic Oscillator

Under the influence of restoring force only, an object executes (linear) harmonic oscillations of constant amplitude about equilibrium position. The total mechanical energy of the system remains conserved, and in principle, the oscillations never die out. In practice, however, there are always external forces of friction present which dissipate away the energy of the oscillator and its motion eventually stops. Such a system is called a damped oscillator.

The force of fluid friction acts against the direction of motion and, for low velocity, its magnitude is proportional to speed:

F_ƒ = – a v

where a is frictional or damping coefficient. The action of frictional force is equivalent to a decrease in restoring force; the restoring force tries to pull back the object towards equilibrium position, while friction opposes the move. Hence, the time period of oscillations slowly increases, and finally approaches an infinite value, i.e. the object comes to a stop.

If friction is small and the object makes many oscillations before the motion dies out, it is called an under-damped oscillator. When the friction is very-large, there are no oscillations and the entire energy is spent in overcoming the frictional force as the object moves a part of the distance from displaced to equilibrium position. Such a system is called an over-damped oscillator. We shall now solve the equation of motion of a damped oscillator to determine the conditions for under or over damping.

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