Damped Oscillator and Operating Regimes

Adding viscous friction

In the presence of fluid friction proportional to speed (force = -h.v), the equation of motion becomes:

m.x''(t) + h.x'(t) + k.x(t) = 0

We introduce the quality factor Q = sqrt(m.k)/h (or equivalently the damping coefficient lambda = h/(2.m)), which measures how many times the system oscillates before the amplitude dies out.

The three regimes

Depending on the sign of the discriminant of the characteristic equation, we distinguish:

Regime	Condition	Behavior
Underdamped (pseudo-periodic)	lambda < omega0	Oscillations of decreasing amplitude
Critical	lambda = omega0	Fastest return to equilibrium without oscillation
Overdamped	lambda > omega0	Slow return, without oscillation

The underdamped regime in detail

In this case, the solution is written x(t) = A.exp(-lambda.t).cos(omega.t + phi), with omega = sqrt(omega0^2 - lambda^2) < omega0: damping slightly slows the pseudo-angular frequency compared to omega0.

Practical application

The critical regime is sought in car shock absorbers or automatic doors: a fast return to equilibrium is wanted, without bouncing or spurious oscillation.

Common pitfall

Do not confuse omega0 (natural angular frequency without damping) with omega (pseudo-angular frequency with damping): omega is always strictly less than omega0 as soon as there is friction, never equal.