14.10 Resonance

14.10 Resonance

In damped oscillation, the oscillator cannot maintain its natural frequency for long duration due to the resistive forces and the amplitude of the oscillation decreases gradually with time. But we can maintain constant amplitude by applying a periodic external force which is called a driving force. Thus, when the oscillating body is subjected to a periodic driving force, then such oscillation is called forced oscillation and its frequency is called driving frequency. The vibration of a vehicle caused by the running of engine is an example of forced vibration. In forced oscillation, the amplitude of the oscillation depends upon the relation between the driving frequency and the natural frequency of the body.

If the frequency of the driving force is same as the natural frequency of the oscillating body, the amplitude of vibration is very much increased. This phenomenon is known as resonance. It also occurs when the applied force has frequency an integral multiple of the natural frequency of the body. If $f$ is the natural frequency of a body, then resonance will take place at:

$f_{2} = 2f_{1}$, $f_{3} = 3f_{1}$, $f_{4} = 4f_{1}$, $\dots$ $f_{n} = nf_{1}$

To demonstrate the resonance phenomenon, we perform a simple experiment. The experimental setup consists of two pair of pendulums A & B and C and D such that the length of A and B is $\ell_{1}$, and length of C and D is $\ell_{2}$. All the pendulums are suspended by a horizontal rod (Fig. 14.17).

Now we introduce another pendulum P whose length can be varied i.e. either $\ell_{1}$ or $\ell_{2}$. Consider the case when the length of pendulum P is equal to $\ell_{1}$.

If the pendulum 'P' is set into vibration, this vibration reaches the other pendulums through the rod. Then the pendulums A and B receive a driving force through the rod and they also start vibration and their amplitude increases due to the resonance phenomenon because their lengths, natural frequency and natural periods are same. At the same time, the pendulums C and D whose natural frequencies are different from natural frequency of P do not oscillate i.e. they continue to remain at rest. If the length of the pendulum P is made equal to $\ell_{2}$, and allowed to vibrate, then the pendulums C and D start vibration due to resonance while pendulum 'A and B' remain at rest.

(Fig. 14.17: A set of five different pendulums of different lengths)

Interesting Information

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The resonance is the oscillation up and down back and forth motion caused by seismic waves. During an earth quake, buildings oscillate. An earthquake occurred in Pakistan magnitude $7.6$ on reactor scale on $8$ October, $2005$ in Azad Jammu and Kashmir, a territory under Pakistan. It was centered near the city of Muzaffarabad. Over $86000$ people died during this earthquake.