Under Damping
If the restoring force is larger than friction, and the oscillation is classified as under-damped. In this case, ω is real and behaves exactly like the solution of an undamped oscillator. Hence, we have
q = [(A_{1} + A_{2}) cos ω t + i (A_{1} - A_{2}) sin ω t]= A_{0} sin (ω t + Ø) = A ( t ) sin (ω t + Ø)where we used, A_{1} + A_{2} = A sin Ø and i (A_{1} - A_{2}) = A cos ØThe above solution shows that particle performs oscillations with frequency ω, or time period T = 2 π/ω, but with a time dependent A ( t ) = A_{0} . In absence of friction, = 0, the amplitude remains constant, i.e. equal to A_{0}. In presence of friction, amplitude decays exponentially with time; the time t = τ = 1/ , when amplitude becomes 1/e of its initial value is called decay constant of the oscillator. That is,A( t ) = A_{0} e^{-t/τ}= A_{0}/e at t = τwhere A_{0} is the initial amplitude (or displacement) at t = 0.