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 =   [(A₁ + A₂) cos ω t + i (A₁ - A₂) sin ω t]

= A₀    sin (ω t + Ø) = A ( t ) sin (ω t + Ø)

where we used, A₁ + A₂ = A sin Ø    and     i (A₁ - A₂) = A cos Ø

The above solution shows that particle performs oscillations with frequency ω, or time period T = 2 π/ω, but with a time dependent A ( t ) = A₀   . In absence of friction,    = 0, the amplitude remains constant, i.e. equal to A₀. 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 ) = A₀ e^-t/τ

= A₀/e    at t = τ

where A₀ is the initial amplitude (or displacement) at t = 0.