# Forced Vibrations and Resonance

## Objective

To study the phenomenon of resonance in great details.

## Introduction

Every system has certain natural frequencies of vibration. If started properly, it vibrate with one of these natural frequencies. A simple pendulum has only one natural frequency, \(\nu=2\pi\sqrt{g/l}\). But a string fixed at both ends has several natural frequencies given by \(\nu_n=\frac{n}{2l}\sqrt{T/\mu}\).

The natural frequency of a vibrating system depend on the geometry of the system and also on the material. In any physical vibrating system, there is inherent damping which gradually decreases the amplitude. A pendulum started once, continues to oscillate for long time, but if you watch it for hours you will see that its amplitude continuously decreases and finally it stops.

If a system is continuously forced by some external sinusoidal force, it vibrate with the frequency of this external force, forgetting about its own natural frequencies. Interestingly, the amplitude of vibration does not depend on what amplitude you give to start the motion. After very short time, it picks up an amplitude that depends on the peak value of the applied sinusoidal force, damping in the system, the frequency of the applied force and also the natural frequencies of the system besides mass etc of the system.

If the applied frequency is not close to any of the natural frequencies of the system, the vibration amplitude is quite small. If it is equal to one of the natural frequencies, the amplitude becomes very large. This phenomenon is called resonance.

## Apparatus

oscillator-vibrator, plastic rings.

## Procedure

Take a plastic strip of width about a centimeter or little less. Cut the strip in different sizes and make rings by using tape on two ends. Join all these plastic rings together at one place. Fix this assembly on some small object which can be fitted on the cap of the oscillator-vibrator.

Connect the oscillator to the vibrator and slowly increase the frequency from very low value. At some frequency the largest ring will start vibrating with large amplitude while the other rings just stay almost motionless. All the rings are mounted on the same vibrator. The vibrator is forcing the system to go with the frequency given by the oscillator. This frequency matches with the natural frequency of the largest ring and not with the natural frequencies of the other rings. Thus, the largest ring vibrates much more vigorously than the others showing the phenomenon of resonance.

Increase the oscillator frequency slowly. The amplitude of vibration of the largest ring is reduced. At some stage the second largest ring pick up the vibrations. Keep increasing the frequency and enjoy the resonance.

## Discussion

The natural frequencies of the strip may not have simple formula. Also the vibrations are not of well defined shape. But it is clear that only one ring vibrates vigorously while other keeps quite when the same force is applied to all. At some frequencies, you will find more than one ring vibrating. This is because rings also have overtones. The source frequency may be equal to natural frequency of one and overtone frequency of other.