Resonance

RESONANCE

It is a special case of forced vibration

The phenomenon of making a body vibrate with its natural frequency under the influence of another vibrating body with the same frequency is called resonance.

If the frequency of the external periodic force is equal to the natural frequency of oscillation of the system, we get oscillations of larger amplitude. This is known as resonance. Amplitude becomes larger if the two frequencies are exactly equal to each other. v = vʹ (Fig 3.8)

Theory of resonant vibrations

(a) Condition of amplitude resonance: In case of forced vibrations, we have 

The expression (1) shows that the amplitude varies with the frequency of the force p. For a particular value of p, the amplitude becomes maximum. This phenomenon is known as amplitude resonance.

The amplitude is maximum when

Thus the amplitude is maximum when the frequency of the impressed force becomes This is known as the resonant frequency.

This frequency of the system both in presence of damping,  and in the absence of damping i.e.

ρ = ω

Condition for Amplitude Resonance

Using equation (1), A is maximum only when, For negligible damping, b = 0 and

Sharpness of Resonance

The rate of change (fall) of amplitude with the change of forcing frequency on each side of resonant frequency is known as sharpness of resonance.

Figure 3.9 shows the variation of A with forcing frequency at different conditions of damping.

The curve (1) shows the amplitude when there is no damping, i.e., b = 0. In this case the amplitude becomes infinite at p = ω. This case is never attained in practice due to frictional resistance, as slight damping is always present.

The curve (2) and (3) shows the effect of damping on the amplitude. It is observed that the peak of the curve moves towards the left.

Examples

(i) Two tuning forks of same frequency are mounted on a suitable sound boards and arranged as shown in fig.3.10.

If one is made to vibrate by striking it with rubber hammer, it is found that the second fork is also set in vibrations.

If the vibrations of the first fork is stopped by touching with hand. The second fork continue to vibrate and the sound will be heard.

(ii) A column of army marching over a bridge can set forced vibrations of the bridge. If the frequency of foot steps matches with natural frequency of the bridge, due to resonance, the bridge may vibrate with a larger amplitude.

This may cause damage to the bridge. Hence soldiers are asked to break steps while crossing a bridge.

(iii) Radio receivers

Radio stations have their own broadcasting (carrier) frequencies. When we tune our radio set, the moment the radio tuner attains exactly the frequency of any broadcasting station, we start hearing the loud and clear sound of that particular radio station due to resonance.

(iv) Other Examples

The loud speaker diaphragm vibrates according to the amplifier circuit. Air column in resonance tube vibrates as per the vibrations of the tuning fork.

Simple experiment on Resonance

From an elastic chord a number of pendulums are suspended (Fig. 3.11). If the pendulum P is set into vibrations, other pendulums also vibrate because they are attached to the same chord. But their motions are not regular.

After sometime we will notice that Q which is the same length as P, vibrate with maximum amplitude and of same time period. They are said to be in resonance. Other pedulums of different lengths are not in resonance.