Damped Oscillations

The oscillations of a body whose amplitude remains the same throughout the time are called undamped oscillations.

The oscillations of a body whose amplitude goes on decreasing with time due to the presence of dissipative forces are called damped oscillations.

ω = \[\sqrt {\frac {k}{m}\]

\[\omega^{\prime}=\sqrt{\frac{k}{m}-\left(\frac{b}{2m}\right)^2}\]

\[\frac{d^2x}{dt^2}+\omega^2x=0\]

\[m\frac{d^{2}x}{dt^{2}}+b\frac{dx}{dt}+kx\] = 0 & emsp; (b = damping constant)

x = A sin(ωt + α)

x = Ae^−bt/2m sin(ω′t + ϕ)

A = Ae^−bt/2m