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Forced Oscillations and Resonance

Since the free oscillations of a damped oscillator will eventually die out, it is necessary to supply energy to keep the system in oscillation. A problem of great practical importance is that of a damped harmonic oscillator driven by an externally applied harmonic force of the type F(t) = F0 sin σ t, where F0 is the magnitude of the force applied and σ its angular frequency. The oscillator is then said to execute forced (or maintained) oscillations. Very often a system is set into oscillation by linking (or coupling) it in some way with another oscillating system (which we will call the driver). For example, in a resonance tube, the air column vibrates because it is linked (by sound waves) to the vibrations of a tuning fork. The diaphragm of a loudspeaker vibrates because it is linked (by current oscillations) to the output circuit of an amplifier. The oscillator picks up energy from the driver and oscillates. Initially, the oscillator likes to oscillate at its own frequency but the driver would like it to oscillate at its own frequency and after some time it succeeds. At this time, the oscillator has forgotten its time-history and begins to oscillate at a constant amplitude at the frequency of the driver. The oscillations are then called steady-state forced oscillations. In the steady state the rate at which power is lost through friction equals the rate at which it is fed to the oscillator by the driver. When the frequency of the driver is varied, the amplitude of the forced oscillator also changes. In Figure 12.26 we have plotted the amplitude of the steady-state forced oscillation as the frequency of the driver is varied from zero to a large value (see Supplement at the end of the Chapter).

Figure: Displacement of a forced oscillator as frequency v' of the driver is varied. v is the frequency of free-damped oscillations
We notice that the displacement (and hence energy) of forced oscillations is very small if ν' << ν and ν' >> ν. But when ν' = ν the amplitude of the forced oscillations becomes very large. In other words, when the frequency of the driving force is equal to the frequency of free-damped oscillations, the oscillator responds most favourably to the driving force and extracts maximum energy from it. The case ν' = ν is called resonance and the oscillations are then called resonant oscillations.
Resonance can be best illustrated by a very simple experiment. From the same elastic string PQ (Fig. 12.27) are suspended four pendulums A, B, C and D.
The pendulum A (the driver) is set into oscillation. The pendulums B, C and D will also start oscillating because they are coupled or linked to A by the elastic string. Initially, the motion of B, C and D will be erratic. After some time (when the steady state is reached) the all pendulums B, C and D will all start oscillating with the frequency of the pendulum A. The oscillations of pendulums B, C and D are forced oscillations. We will notice that the pendulum B, whose frequency is much larger than that of A (since it is much shorter) and the pendulum D, whose frequency is much smaller than that of A (since it is much longer) have very small amplitude. The pendulum C which has the same length as the pendulum A (and hence the same frequency) oscillates with the largest amplitude. The oscillations of C are in resonance with those of A.

In physics we come across a variety of resonances. The air-column in a resonance tube resounds when its frequency agrees with that of the fork. The most familiar example of resonance is when we tune our radio to a particular broadcasting station. There are many stations sending radio waves of various frequencies causing forced oscillations in the circuit of the receiver. A particular setting of the tuner corresponds to a particular frequency of the circuit. When this frequency equals that of the waves from a particular broadcasting station, the power absorption is maximum and hence we hear only that station. A sodium chloride crystal, which consists of positively and negatively charged ions, can absorb energy if subjected to an oscillating electric field. When the frequency of the relative oscillations of ions matches the frequency of the electric field, the crystal absorbs maximum energy from the filed.

Some resonances can cause disaster. A column of army men marching over a bridge can set in a forced oscillation of the bridge. If the frequency of their footsteps happens to match one of the natural frequencies of the bridge (which is determined by the dimensions and elastic properties of the material of the structure) resonance will occur and the bridge will oscillate with a destructively large amplitude. This is the reason why soldiers break step when crossing a bridge.

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