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Physics Topics can be both theoretical and experimental, with scientists using a range of tools and techniques to understand the phenomena they investigate.
What are the Different Types of Damping Motion?
Definition: If resistive forces acts on a vibrating body in addition to the restoring force, its amplitude gradually diminishes. After some time, the body comes to rest at its position of equilibrium. This type of vibration is called damped vibration. The resistive effect is called damping.
In fact, free vibration does not exist in real life. All vibrating bodies ultimately come to rest after some time, i.e., all vibrations are damped. Different types of resistive forces act on different types of vibrating bodies. For a simple pendulum. the resistive force is the viscous force of air; in a moving coil galvanometer, the resistive force is electromagnetic damping. The vibrating body has to work against these resistive forces. So its energy decreases. When the energy of the body becomes zero, it comes to rest.
Graphical representation of free and damped vibration: The characteristics of free and damped vibrations can be understood easily with the help of displacement-time graphs. If the vibration of a body is free, it will vibrate for ever with its amplitude unchanged [Fig.(a)]. In real life, the amplitude gradually diminishes and ultimately the body comes to rest [Fig.(b)].
Beneficial examples of damped vibration: Damping plays a beneficial role in our modern day life. One such application of damped oscillation is the car suspension system. It makes use of damping to make our ride less bumpy and more comfortable by counteracting and hence reducing the vibrations of the car when it is on the road for optimal passenger comfort, the system is critically damped or slightly underdamped.
Different Types of damped motion:
i) If the damping is very weak, a body vibrates with almost its natural frequency. For example, if the bob of the simple pendulum is heavy enough, then the damping becomes insignificant. So the pendulum continues to oscillate for a long time. For example, if the bob of the simple pendulum is heavy enough, then the damping becomes insignificant. So the pendulum continues to oscillate for a long time. The time period and frequency of such a pendulum are almost equal to those of a free pendulum.
ii) If the damping is stronger, the vibration of a body does not continue for a long time. For example, if a light piece of wood is used as the bob of a simple pendulum and is allowed to oscillate, it comes to rest after a few oscillations. In this case the time period becomes very large, i.e., the frequency of vibration is much less than the natural frequency.
iii) In case of very strong damping, the vibrating body comes back to its position of equilibrium from its displaced position and stops there. So the body cannot move past its position of equilibrium. The body does not vibrate at all. This is called overdamped motion or aperiodic motion [Fig.]
iv) There is a particular state of damping, between small damping and overdamping, for which the body returns to its equilibrium position in the least time, but cannot travel past its position of equilibrium. This is called critical damping[Fig.]. In practical cases, if we want to stop the vibration of a body quickly, its damping is kept close to the state of critical damping.
v) If the damping is less than critical damping, the body oscillates with decreasing amplitude. This is known as underdamping [Fig.].
Decrement of amplitude in damped vibration: From the graph [Fig.] for damped vibration we get, A1 = initial amplitude of vibration, A2 = amplitude of vibration after one complete oscillation, A3 = amplitude of vibration after two complete oscillations.
A1 > A2 > A3.
Two important characteristics of such a damped vibration are:
i) The amplitude of vibration decreases in a constant ratio for each complete vibration.
That is, \(\frac{A_1}{A_2}\) = \(\frac{A_2}{A_3}\) = \(\frac{A_3}{A_4}\) = …….. = constant. This constant is called the decrement.
ii) From the very initiation of motion, damping comes into play. Therefore, even the first amplitude, A1 of damped vibration is less than the amplitude A0 of free vibration [Fig.].
Equation of motion: A particle under a damped harmonic vibration in one-dimension is subject to two types of forces:
A restoring force: F1 = -kx, where k is constant.
One or more resistive forces: Each resistive force is proportional to the velocity (v = \(\frac{d x}{d t}\)) of the particle, and acts in a direction opposite to that of the instantaneous velocity. So, the resultant of the resistive forces is,
F2 = -k’v = -k\(\frac{d x}{d t}\), where k’ is another constant.
The acceleration of the particle is, a = \(\frac{d v}{d t}\) = \(\frac{d^2 x}{d t^2}\). Thus from the relation F = ma, we get,
The equation (1) is known as the equation of motion of a damped SHM.
Numerical Examples
Example 1.
A seconds pendulum is shifted 4 cm away from its equilibrium position and then released. After 2 s the pendulum is 3 cm away from its position of equilibrium. What will be the position of the pendulum after another 2 s?
Solution:
Time period of a seconds pendulum is 2 s. In case of this damped vibration, let the change in time period be negligible. For the given pendulum the initial amplitude of vibration A1 = 4 cm. After one complete oscillation, the amplitude of vibration A2 = 3 cm.
So, if the amplitude of vibration after two complete oscillations is A3, then the decrement is:
\(\frac{A_1}{A_2}\) = \(\frac{A_2}{A_3}\) or, A3 = \(\frac{\left(A_2\right)^2}{A_1}\) = \(\frac{(3)^2}{4}\) = 2.25 cm
Example 2.
After 100 complete oscillations, a pendulum’s amplitude becomes \(\frac{1}{3}\)rd of its initial value. What will be its amplitude after 200 complete oscillations? [CBSE ’02]
Solution:
The pendulum has a damped vibration. So, the amplitude decreases at the same rate. Since, after 100 complete oscillations the amplitude of vibration becomes \(\frac{1}{3}\) the initial amplitude, after 200 complete oscillations, the amplitude will be \(\frac{1}{3}\) × \(\frac{1}{3}\) = \(\frac{1}{9}\)th of the initial amplitude.