In principle, oscillations can go on for ever. In practice, however, the oscillations we observe around us do not. They die out, either rapidly or gradually. A child on a swing knows that the amplitude of her swinging will decline until eventually she will come to rest, unless she can put some more energy into the swinging to keep it going.
This happens because of friction. On a swing, there is friction where the swing is attached to the frame and there is friction with the air. The amplitude of the child’s oscillations decreases as the friction transfers energy away from her to the surroundings.
We describe these oscillations as damped. Their amplitude decreases according to a particular pattern.
This is shown in Figure 18.25.

The amplitude of damped oscillations does not decrease linearly. It decays exponentially with time. An exponential decay is a particular mathematical pattern that arises as follows. At first, the swing moves rapidly. There is a lot of air resistance to overcome, so the swing loses energy quickly and its amplitude decreases at a high rate. Later, it is moving more slowly. There is less air resistance and so energy is lost more slowly–the amplitude decreases at a lower rate. Hence, we get the characteristic curved shape, which is the ‘envelope’ of the graph in Figure 18.25.
Notice that the frequency of the oscillations does not change as the amplitude decreases. This is a characteristic of simple harmonic motion. The child may, for example, swing back and forth once every two seconds, and this stays the same whether the amplitude is large or small.

PRACTICAL ACTIVITY 18.2

 

Investigating damping

You can investigate the exponential decrease in the amplitude of oscillations using a simple laboratory arrangement (Figure 18.26). A hacksaw blade or other springy metal strip is clamped (vertically or horizontally) to the bench. A mass is attached to the free end. This will oscillate freely if you displace it to one side.
A card is attached to the mass so that there is significant air resistance as the mass oscillates. The amplitude of the oscillations decreases and can be measured every five oscillations by judging the position of the blade against a ruler fixed alongside.
A graph of amplitude against time will show the characteristic exponential decrease. You can find the ‘half-life’ of this exponential decay graph by determining the time it takes to decrease to half its initial amplitude (Figure 18.27). 
By changing the size of the card, it is possible to change the degree of damping, and hence alter the half-life of the motion.

Energy and damping

Damping can be very useful if we want to get rid of vibrations. For example, a car has springs (Figure 18.28) that make the ride much more comfortable for us when the car goes over a bump. However, we wouldn’t want to spend every car journey vibrating up and down as a reminder of the last bump we went over. So the springs are damped by the shock absorbers, and we return rapidly to a smooth ride after every bump.
Damping is achieved by introducing the force of friction into a mechanical system. In an undamped oscillation, the total energy of the oscillation remains constant. There is a regular interchange between potential and kinetic energy. By introducing friction, damping has the effect of removing energy from the oscillating system, and the amplitude and maximum speed of the oscillation decrease.