Undamped and Damped simple harmonic oscillations

Undamped oscillations:

When a Simple harmonic system oscillates with a constant amplitude which does not change with time , its oscillations are called Undamped simple harmonic oscillations .
e.g. Bob of simple pendulum oscillates in vacuum .

Damped oscillations:

The oscillations of a body whose amplitude goes on decreasing with time are defined as damped oscillations.

If ‘v’ be the velocity of the oscillator then damping force Fd = -b v ; where ‘b’ is damping constant.

The resulting force acting on damped harmonic oscillator is ,

F = Frestoring + Fdamping

F = -k x – b v

$\large m\frac{d^2 x}{d t^2} + b \frac{dx}{dt} + k x = 0 $

Solution to above differential equation is ,

$\large x = x_o e^{-bt/2m} cos(\omega’ t + \phi)$

Where , $\large x_o e^{-bt/2m}$ is the amplitude of oscillator which decreases with time & ω’ is the angular frequency of damped oscillator .

$\large \omega’ = \sqrt{\frac{k}{m} – (\frac{b}{2m})^2}$

$\large \omega’ = \sqrt{\omega^2 -\gamma^2}$

In these oscillations the amplitude of oscillations decreases exponentially due to damping force like frictional force, viscous force, hysteresis etc.

In these oscillations the frequency of the oscillations decreases .

Time period of the oscillator , $\large T = \frac{2 \pi}{\sqrt{\omega^2 -\gamma^2}}$ ;this is greater than the time period of the harmonic oscillator $\large T_0 = \frac{2\pi}{\omega}$

The body undergoing damped oscillation is known as damped harmonic oscillator.

Due to decrease in amplitude, the energy of the oscillator also goes on decreasing exponentially,

$\large E  = \frac{1}{2}k x_o^2 e^{-bt/m}= E_o e^{-bt/m} $ ; Where Eo = Maximum energy of oscillator .

Illustration: The amplitude of a damped oscillator decreases to 0.9 times its original value in 5s. In another 10s it will decreases to α times its original magnitude, where α is

Sol: $\large A = A_0 e^{-bt/2m} $

after 5 sec⁡ , $\large 0.9 A_0 = A_0 e^{-5b/2m} $

After 10 more sec , (i.e., t = 15 sec) its amplitude becomes α A0.

Hence, $\large \alpha A_0 = A_0 e^{-15b/2m} $

$\large \alpha = (e^{-5b/2m})^3 = (0.9)^3$

α = 0.729

Also Read :

Stable , Unstable & Neutral Equilibrium
S.H.M :Linear SHM & Angular SHM
Analytical Treatment to SHM
Kinetic Energy & Potential Energy & Total Energy in SHM
Average Value of P.E. & K.E. of Harmonic Oscillator
SHM as a projection of circular motion
Simple Pendulum in Inertial & Non Inertial Frame
Time period of a Long Pendulum
SHM of Spring Mass System
Physical Pendulum & Torsional Pendulum

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