DAMPED SIMPLE HARMONIC MOTION -

In the absence of frictional force, the total energy of the oscillation system remains unchanged during motion i.e system will oscillate with constant amplitude. Such oscillations are called undamped oscillations.

In a real situation, this is not true. There is always some friction force called “damping force” that acts on a system, this leads to the decrease in energy of the system and hence amplitude decreases continuously. Ultimately body comes to rest. Such oscillations are called damped oscillations.

i.e damped oscillations are periodic oscillations whose amplitude decreases gradually with time.

In damped oscillations, the damping force acts on a body which is proportional to the velocity of the body.

i.e $\fn_cm \large f\propto v$

$\fn_cm \large \left [ f=-bv \right ]$ (usually valid for small velocity)

where b is the damping constant which depends upon the nature of the medium.

i.e in damped oscillation, the net force on the body acting on the system is

$\fn_cm \large F=-kx-bv$

$\fn_cm \large \Rightarrow m\frac{d^2x}{dt^2}=-kx-b\frac{dx}{dt}$

$\fn_cm \large \left [ \frac{d^2x}{dt^2}+\left ( \frac{b}{m} \right )\frac{dx}{dt}+\left ( \frac{k}{m} \right )x=0 \right ]$

Which is the differential equation of a damped oscillation. The solution of above equation is given by

$\fn_cm \large x=Ae^{-\frac{bt}{2m}}cos(\omega_b t+\phi)$

where $\fn_cm \large \omega_b=\sqrt{\frac{k}{m}-\frac{b^2}{4m^2}}\;\;\;\;\;(constant)$

the time period of damped oscillation is

$\fn_cm \large T=\frac{2\pi}{\omega_b}$

$\fn_cm \large \left [ T=\frac{2\pi}{\sqrt{\frac{k}{m}-\frac{b^2}{4m^2}}} \right ]$

The total energy of damped oscillation is given by

$\fn_cm \large E=\frac{1}{2}K\left [ Ae^{\frac{-bt}{2m}} \right ]^2$

$\fn_cm \large \left [ E=\frac{1}{2}KA^2e^{-\frac{bt}{m}} \right ]$

NOTE

1. Damped simple harmonic motion is not strictly simple harmonic. It is approximately so only for time intervals much less than $\fn_cm \large \frac{2m}{b}$ where b is the damping constant.

Q.1> For the damped oscillator spring-mass system, the mass m of the block is 200 g, k = 90 N m–1 and the damping constant b is 40 g s–1. Calculate (a) the period of oscillation, (b) time taken for its amplitude of vibrations to drop to half of its initial value and (c) the time taken for its mechanical energy to drop to half its initial value.

Q.2> You are riding in an automobile of mass 3000 kg. Assuming that you are examining the oscillation characteristics of its suspension system. The suspension sags 15 cm when the entire automobile is placed on it. Also, the amplitude of oscillation decreases by 50% during one complete oscillation. Estimate the values of (a) the spring constant k and (b) the damping constant b for the spring and shock absorber system of one wheel, assuming that each wheel supports 750 kg.