# Free, damped and forced oscillations

There are three main types of simple harmonic motion:
(a) free oscillations – simple harmonic motion with a constant amplitude and period and no external influences
(b) damped oscillations – simple harmonic motion but with a decreasing amplitude and varying period due to external or internal damping forces
(c) forced oscillations – simple harmonic motion but driven externally

## (a) Free oscillations

The amplitude remains constant as time passes, there is no damping. This type of oscillation will only occur in theory since in practice there will always be some damping. The displacement will follow the formula x = r sinwt where r is the amplitude. It is these types of oscillation that we have looked at already.

Addition of free simple harmonic motions
It is interesting to look at the superposition of two simple harmonic motions, such as a pendulum that is set swinging and then pushed sideways at an angle to its original motion. However to see the motion clearly it is best done on the oscilloscope or using a computer. You can also show this motion quite easily by using a can with a hole in the bottom and full of sand as the pendulum "bob". As the can swings they sand dribbles out tracing a pattern of the motion on the floor below.

(i) If two simple harmonic motions act along the same direction with the same frequency, then their resultant is a simple harmonic motion with the same frequency along that line. The amplitude will be constant but will depend on the phase difference between the two simple harmonic motions. For example if the driving forces were p out of step there would be no motion at all. A phase difference of p means that one driving force would be trying to move the object in one direction while the other would be trying to move it in exactly the opposite direction – they would cancel and so the net result would be no motion.

(ii) If their frequencies are different but they still act along the same line then beats will be produced, the variation in amplitude depending on the difference in frequency. You will be able to find out about beats in more detail in the section on wave motion. Simply, they give rise to the warbling sound that you get if two instruments that are slightly out of tune compared to each other are played together. The closer the two frequencies come the smaller is the beat frequency and when they are exactly in tune the beat frequency is zero and the effect disappears.

(iii) If they act in perpendicular directions there are two sets of possibilities:

1. The frequencies are the same and of equal amplitude:
a phase difference 0 gives a straight line, a phase difference of p/2 gives a circle, and a phase difference of p gives zero oscillation.

2. The frequencies are different but of equal amplitude:
this gives Lissajous figures, three examples of which are shown in the accompanying diagram. for a phase difference of p/2. The numbers of loops in the x and y directions can be counted, and this will give the frequency ratio of the two s.h.m.s:

Frequency ratio (fx/fy) = number of loops in x-direction/ number of loops in y-direction

Student investigation
The damping of the oscillations of a system can be very important. Investigate the damping in the two following examples.

(a) Air damping The effect of air damping on the oscillations of a helical spring may be carried out using a large disc of light but rigid cardboard fixed to the spring. You should displace the spring by a given amount and then record the amplitude of the subsequent oscil¬lations. It may be possible to investigate the dependence of the damping on the size of the cardboard. Plot suitable linear graphs to present your results. Would a card with turned-up or turned-down edges be as good or better than the flat card?

(b) Liquid damping Once again a spring may be used, but this time a metal cylinder should be fixed to the end. This cylinder should be allowed to oscillate in a cylindrical container of liquid. As before, attempt to record the variation in amplitude of the oscillations. Investigate the dependence of the damping on (i) the liquid in the cylindrical container, (ii) the diameter of the cylindrical container.

## (b) Damped oscillations

These are oscillations where energy is taken from the system and so the amplitude decays. They may be of two types:

(i) Natural damping, examples of which are:
internal forces in a spring,
fluids exerting a viscous drag.

(ii) Artificial damping, examples of which are:
electromagnetic damping in galvanometers, the coating of panels in cars to reduce vibrations, shock absorbers in cars, interference damping - gun mountings on ships.
Artificial damping can be light, in which case the system oscillates about the midpoint (Figure 3(a)), heavy, in which the system takes a long time to reach equilibrium (Figure 3(b)) or critical, where the system reaches equilibrium in a short time compared with T with no overshoot where T is the natural period of vibration of the system (Figure 3(c)).

A good example of damping can be seen in the moving coil galvanometer. Electromagnetic damping is used here: the coil moves in a magnetic field and the current flowing in it can be shorted with a resistor, thus varying the damping. The system is either
(i) dead beat — that is, critically damped, or
(ii) ballistic — the damping is as small as possible.
With reasonably light damping the period is unchanged but as the damping is increased the time period is increased and the oscillations die away more rapidly.

Damping is also important in a weighing machine (balance) such as in a shop or a checkout at a supermarket where a true reading of the mass of an object placed on the scale pan is needed quickly. If the damping was light and the pan oscillated you could clearly get a bargain by choosing to pay when the reading was low. If the damping was heavy you would obviously have to wait a long time before the true reading was reached. Some possible variations of reading (displacement from the final correct reading) are shown in Figure 4)

(c) Forced oscillations
These are vibrations that are driven by an external force. A simple example of forced vibrations is a child's swing: as you push it the amplitude increases and if the driving frequency is the same as the natural frequency of the swing resonanace occurs. A loudspeaker is also an example of forced oscillations; it is made to vibrate by the force on the magnet on the current in the coil fixed the speaker cone.