Under damping, Mechanical Engineering

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Under Damping

If  2251_download.png  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 =  469_download (1).png [(A1 + A2) cos ω t + i (A1 - A2) sin ω t]

= A0  469_download (1).png  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  469_download (1).png . In absence of friction,  1255_download (2).png  = 0, the amplitude remains constant, i.e. equal to A0. In presence of friction, amplitude decays exponentially with time; the time t = τ = 1/ 1255_download (2).png , 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.


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