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What is the Difference Between Free Vibration And Forced Vibration?
Practically all vibrations are damped vibrations. The vibrating body works against different resistive forces. So its energy diminishes and the amplitude gradually decreases. To maintain a steady vibration, energy from an external source is needed. If energy is supplied from an external source in such a way that the rate of supply of energy exactly balances the rate of loss of energy, then the amplitude of the body remains constant. The value of the amplitude is similar to that of free vibration of the body.
This type of vibration is called a forced vibration. For example, if we do not wind a pendulum clock, it will stop after a while due to damping. When we wind the clock, we compress a spring within the clock which stores potential energy and supplies that energy continuously. The pendulum oscillates continuously with constant amplitude and time period.
External means are required not only for maintaining the vibration, but also to vibrate a body that is initially at rest.
Examples of forced vibration:
i) If we strike a prong of a tuning fork, the intensity of the emitted sound ¡s not very high and so it cannot be heard from a distance. Now, if the handle of the vibrating turning fork is made to touch the surface of a table, the tuning fork sets the table surface into forced vibration. Sound is emitted from the table also. Hence, the sound is amplified.
It must be kept in mind that, according to the principle of conservation of energy, the total amount of energy cannot be increased. When the handle of the tuning fork is pressed against a table surface, a part of the energy from the tuning fork is transferred to the table. As the damping of the table is higher than that of the tuning fork, the energy transferred to the table by the tuning fork decays at a faster rate. Hence, the vibration of the table stops earlier and the intensity of the sound produced due to vibration of the table is comparatively higher.
In this example of forced vibration, the upper surface of the table is made to vibrate by the tuning fork. The vibration of the tuning fork is the driving vibration and the vibration of the upper surface of the table is the driven vibration.
ii) A thread is tied loosely between P and Q [Fig.]. Two pendulums C and D, having different lengths, are suspended from two points between P and Q. If the pendulum C is made to oscillate, it will continue its oscillation with its natural frequency. As the thread PQ is tied loosely, energy will be transferred from pendulum C to pendulum D through the thread. As a result, the pendulum D will begin to oscillate.
As the lengths of the pendulums are different, their natural frequencies are not the same. It is found that the pendulum D initially tries to oscillate at its natural frequency. But its vibration is damped quickly. Then pendulum D begins to oscillate at the natural frequency of C. In this example, pendulum C provides the driving vibration and the vibration of pendulum D is the driven vibration.
In these two examples, it is to be noted that the external driving forces are neither steady nor momentary. Rather, the force originating from the vibrating body is a periodic force. In fact, the spring of a clock exerts a periodic force on the pendulum.
Definition: If an external periodic force is applied to a freely vibrating body, the body tries to maintain its vibrations at its own frequency; but after some time, the body begins to vibrate with the frequency of the applied periodic force. Such a vibration of a body is called a forced vibration.
Equation of motion: A particle under forced harmonic vibration in one-dimension is subject to three types of forces:
A restoring force: The resultant of all the resistive forces acting on a particle is,
F2 = -k’v = -k’\(\frac{d x}{d t}\)
where v is the velocity of the particle and k’ is a constant.
Externally impressed periodic force: This is of the form, FF3 = F0sinωt
where ω is the circular frequency and F0 is the amplitude of the periodic force; both ω and F0 are constants.
The acceleration of the particle is, a = \(\frac{d v}{d t}\) = \(\frac{d^2 x}{d t^2}\). Then from the relation F = ma, we get
ma = F1 + F2 + F3
or, m\(\frac{d^2 x}{d t^2}\) = -kx – k’\(\frac{d x}{d t}\) + F0sinωt
or, m\(\frac{d^2 x}{d t^2}\) + k’\(\frac{d x}{d t}\) + kx = F0sinωt
or, \(\frac{d^2 x}{d t^2}\) + 2b\(\frac{d x}{d t}\) + \(\omega_0^2 x\) = a0sinωt …….. (1)
where, ω0 = ±\(\sqrt{\frac{k}{m}}\), b = ±\(\frac{k^{\prime}}{2 m}\) and a0 = ±\(\frac{F_0}{m}\)
The equation (1) is known as the equation of motion of a forced SHM. It is to be noted that, the natural frequency ω0 of the particle vibration is, in general, different from the frequency ω0 of the particle vibration is, in general, different from the frequency ω of the external periodic force. In the special case, when ω = ω0, the phenomenon of resonance comes into play.
Comparison between Free Vibration and Forced Vibration
Free Vibration | Forced Vibration |
1. A body has free vibration if no force other than the restoring force acts on it. | 1. If an external periodic force acts on a body, it executes a forced vibration. |
2. The frequency of free vibration of a body depends on density, shape and elasticity of its material. | 2. The frequency of forced vibration becomes equal to that of the applied periodic force. |
3. The total energy of the vibrating body remains conserved. | 3. The rate of loss of energy due to resistive forces is exactly compensated by the rate of supply of energy by the periodic force. |
4. Free vibration ideally never dies out. But in reality, due to presence of damping forces, it dies out gradually. | 4. Forced vibration continues as long as the applied periodic force acts on the body even in the presence of damping forces. |