1.8 Forced oscillation

Simple harmonic motion is the simplest form of oscillation. The periodic attributes like frequency and time period are independent of disturbance that sets the periodic motion. The system oscillates at its natural frequency typical of the set up involving a restoring mechanism. Any system having moving part has inherent natural frequency or frequencies depending on the degree of freedom – numbers of possible oscillatory motions.

An oscillatory system, however, can be subjected to external force which may alter the nature of oscillation altogether. For example, a system capable of oscillation can be set to oscillate at an altogether different frequency other than natural frequency. Since we are investigating oscillatory motion, we shall study the impact of external force which is itself periodic.

Consider, for example, the oscillation of spring – block system attached to a rigid support as shown in the figure. What if the rigid support itself oscillates (up and down) at a certain frequency? What would be the motion of the block? It would be mix up of two oscillations – (i) external oscillation of the rigid support and (ii) natural oscillation of the spring – block system. In the beginning, the block will oscillate in a varying manner, but soon it settles to oscillate with frequency at which the rigid support is made to oscillate. We can describe such forced oscillation by harmonic function :

$$x=A\sin \left({\omega }_{e}t+\phi \right)$$

where “ ${\omega }_e$ ” is the angular frequency of external force being applied on the system.

Damping

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.

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) :

$$F_{\text{net}}=-kx-bv$$

$$ma+kx+bv=0$$

In terms of displacement derivatives :

$$\Rightarrow m\frac{d^{2}x}{d{t}^2}+b\frac{dx}{dt}+kx=0$$

For small damping constant “b”, the solution of this differential equation yields :

$$x=A_0{e}^{-bt/2m}\sin \left({\omega }_{b}t+\phi \right)$$

The amplitude of the oscillation is a decreasing function in time, which tends to become zero :

$$A=A_0{e}^{-\frac{bt}{2m}}$$

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” :