Free Oscillations, Forced Oscillations and Resonance

Free oscillations

If an object is allowed to oscillate or vibrate on its own, it does so with its natural frequency (or with one of its natural frequencies). For example, if the bob of a simple pendulum of length $l$ is displaced and released, it oscillates only with the frequency $$n=\dfrac{1}{2\pi}\sqrt{\dfrac{g}{l}}$$ which is called its natural frequency, and the oscillations are free oscillations.

Forced oscillations

By applying a periodic force, the same pendulum can be made to oscillate with a different frequency. The oscillations are then forced oscillations, and the frequency is the driver frequency or forced frequency.

The four-pendulum experiment

Diagram (Fig. 5.15): four pendula A, B, C and D are tied to a common horizontal string. A and C are of the same length, B is shorter and D is longer. Pendulum A has a solid rubber ball as its bob and acts as the driver or source pendulum; B, C and D have hollow rubber balls as their bobs and act as the driven pendula.

• As A and C are of the same length, their natural frequencies are the same.
• B has a higher natural frequency, its length being shorter than A's.
• D has a natural frequency less than that of A and C.

Pendulum A is set into oscillation in a plane perpendicular to the plane of the paper. In the course of time the other three pendula also start oscillating in the same plane, because the vibrational energy is transferred through the string. The oscillations of A are free oscillations; those of B, C and D are forced oscillations of the same frequency as that of A.

Among B, C and D, pendulum C oscillates with the maximum amplitude and the other two with smaller amplitudes. As the energy depends on the amplitude, C has absorbed the maximum energy from the source pendulum A while the other two absorbed less.

Resonance

An object having the same natural frequency as that of the source absorbs maximum energy from the source. In such a case it is said to be in resonance with the source. For unequal natural frequencies on either side (higher or lower), the energy absorbed — and hence the amplitude — is less.

Graph (Fig. 5.16): if the activity is repeated for a set of pendula of different lengths, and the squares of their amplitudes are plotted against their natural frequencies, the curve rises to a sharp peak and falls away on either side. The peak occurs when the forced frequency matches the natural frequency, i.e. at the resonant frequency.

Resonance appears again later in the course: most traditional musical instruments use the principle of resonance (superposition of waves), and resonance in L–C circuits is discussed in AC circuits.

Standard results

• Free oscillations occur at the natural frequency: $n=\dfrac{1}{2\pi}\sqrt{\dfrac{g}{l}}$ for a pendulum, $n=\dfrac{1}{2\pi}\sqrt{\dfrac{k}{m}}$ for a spring–mass system
• Forced oscillations occur at the driver frequency, whatever the natural frequency of the driven body
• Resonance: driver frequency $=$ natural frequency, giving maximum energy transfer and maximum amplitude
• The plot of (amplitude)$^2$ against natural frequency peaks sharply at the resonant frequency