Damping Effect

Damping

A force which opposes the motion of the oscillator is called damping force. Due to this force, the amplitude of oscillation decreases continuously and becomes zero at infinite time. This type of force may be frictional force or viscous force. For small oscillations, the damping force acting on the oscillator is directly proportional to its velocity ie.\begin{align*} m\frac{\mathrm{d^2x} }{\mathrm{d} t^2}\propto \frac{\mathrm{dx} }{\mathrm{d} t}\end{align*}\begin{align*} or, m\frac{\mathrm{d^2x} }{\mathrm{d} t^2}=-\gamma\frac{\mathrm{dx} }{\mathrm{d} t}\end{align*} where, \(\gamma\) is the proportionality constant called the damping constant and the negative sign indicates that there is decrease in velocity with time.

Damped Oscillator

An oscillator in which the motion is under the action of damping force is called a damped oscilator. In this oscillation, the amplitude of oscillation decreases exponentially with time. Force acting on the oscillator in case of damping is given by, \(F=-cx\). But in the case of damping, the total force acting on the oscillator is given by,\begin{align*} F=-cx-\gamma\frac{\mathrm{dx} }{\mathrm{d} t}\end{align*}\begin{align*} m\frac{\mathrm{d^2x} }{\mathrm{d} t^2}=-cx-\gamma\frac{\mathrm{dx} }{\mathrm{d} t}\end{align*}\begin{align*} \frac{\mathrm{d^2x} }{\mathrm{d} t^2}=-\frac{cx}{m}-\frac{\gamma}{m}\frac{\mathrm{dx} }{\mathrm{d} t}\end{align*}\begin{align*} \frac{\mathrm{d^2x} }{\mathrm{d} t^2}=-\omega _{0}^2x-\frac{1}{\tau}\frac{\mathrm{dx} }{\mathrm{d} t}\end{align*} where, \(\omega _{0}=\sqrt{\frac{c}m{}}\) is the natural frequency of oscillation and \(\tau=\frac{m}{l}\) is the relaxation time. It is the time at which the velocity of the oscillator becomes \(\frac{1}{e}\) times the initial velocity. So,\begin{align*} \frac{\mathrm{d^2x} }{\mathrm{d} t^2}+\omega _{0}^2x+\frac{1}{\tau}\frac{\mathrm{dx} }{\mathrm{d} t}=0 \end{align*} This is the second order differential equation of a damped oscilaltor. The general solution of it can be written as\begin{align*} x=\frac{x_{o}}{2}\exp \frac{-t}{2\tau}[(1+\frac{1}{2\tau\beta})\exp\beta t+(1-\frac{1}{2\tau\beta})\exp-\beta t]\end{align*} where, \(\beta=\sqrt{\frac{1}{4\tau^2}-\omega _{o}^2}\) Depending upon the values of \(\beta\), there are three cases.

Case I- Over damping

When \(\beta\) is real ie. \(\frac{1}{2\tau}>\omega_{o}\), the displacement decays exponentially without changing the direction and hence there is no oscillation> the motion is called aperiodic or dead beat.

Case II- Critically damped

When \(\beta\) is 0 ie. \(\frac{1}{2\tau}=\omega_{o}\), the system returns to its equilibrium position in a short time.

Case III Underdamped

When damping is so small that \(\frac{1}{2\tau}<\omega_{o}\), then \(\beta\) is imaginary. In this case, the frequency of the damped oscillator approaches its natural frequency.

Power Dissipation

The rate of loss of energy in the damped harmonic oscillator due to the damping force is called power dissipation. In case of damped harmonic oscillator, energy of oscillator will be decreased in each cycle ie.\begin{align*} power\space dissipation=-\frac{\mathrm{dE}}{\mathrm{d} t}=\frac{-E_{avg}}{\tau} \end{align*} where, \(E_{avg}\) is the average energy loss and \(\tau\) is the relaxation time.

A factor which measures the quality of the damped harmonic oscillator is called quality factor. The quality factor of an oscillator is defined as \(2\pi\) times the ratio of average energy stored per cycle to the average energy loss per cycle ie.\begin{align*} Q=2\pi=\frac{\frac{E_{avg}}{T}}{\frac{E_{avg}}{\tau}}=(\frac{2\pi}{T})\tau=\omega\end{align*} where, \(\omega\) is the frequency of oscillation and for small oscillation, \(\omega=\omega_{o}\), the natural frequency of the oscillator. Power factor is inversely proportional to the damping constant.

Power absorption

The average energy stored by the forced oscillator per unit time is called power absorption. The value of power absorption is numerically equal to the power dissipation in a damped harmonic oscillator.

Transient state and steady state

In case of damped harmonic oscillator, the amplitude of the oscillator goes on decreasing with time and finally becomes zero. But, when a certain energy is applied to the oscillator, it again oscillates with the certain amplitude. But the amplitude of the oscillator is not constant initially. This state of the oscillator is called as the transient state. Wen the external periodic force is continued, the oscillator oscillates with constant amplitude after the transient state. Ths state of the oscillator is called the steady state.

References

Adhikari, Pitri Bhakta. A Textbook of Physics Volume-I. Kathmandu: Sukunda Pustak Bhawan, 2015.

Feynman, Richard P. The Feynman Lectures on Physics Volume 1. Noida: Dorling Kindersley (India) Pvt. Ltd., 2014.

Mathur, D S. Mechanics. New Delhi: S. Chand & Company Pvt. Ltd., 2015.

Young, Hugh D, Roger A Freedman and A Lewis Ford. University Physics. Noida: Dorling Kindersley (India) Pvt. Ltd., 2014.