damped harmonic motion

Asked by saikapianharsh7079 | 21st May, 2022, 08:23: PM

Expert Answer:

A damped harmonic motion is an oscillatory motion in which the amplitude of oscillation is slowly decreasing

while there is negligible change in period of motion. Loss of amplitude is called damping .

While considering simple harmonic motion, it is assumed that there is no friction force against the motion.

If there is no friction force , an oscillating simple pendulum  or an oscillating mass-spring system will keep on

oscillate indefinitely . But in real situation  , we find amplitude of these oscillations is slowly decreasing .

There are many causes for damping like friction in the oscillating system, air resistance etc.,

For undamped motion , force acting on the oscillating mass is restoring force that is directly proportional to displacement

F = m a(t) = - k x(t)

Where m is mass of oscillating object , a(t) is acceleration that is function of time , k is force constant and

x(t) is displacement that is function of time .

For damped motion there is an additional damping force  that is proportional to velocity is acting on the system.

F = m a(t) =  - k x(t)  - b v(t)

By substituting , a(t) = d2x/dt2  and  v(t) = dx/dt , we write above equation as

 d2x/dt2 + ( b / m ) dx/dt  + ( k/m ) x(t) = 0

Solution of displacement to above equation is

begin mathsize 14px style x left parenthesis t right parenthesis space equals space x subscript o e to the power of negative bevelled t over tau end exponent cos left parenthesis omega apostrophe t space plus ϕ right parenthesis end style

where τ = 2m/b is damping time constant and ω' is frequency of damped motion

begin mathsize 14px style omega apostrophe space equals space square root of k over m space minus space open parentheses fraction numerator b over denominator 2 m end fraction close parentheses squared end root space end style

when damping constant b → 0 ,  displacement and frequecy becomes that of undamped simple harmonic motion.

 

 

Answered by Thiyagarajan K | 22nd May, 2022, 10:41: AM