Damping

Damping removes energy through velocity

A damper resists relative motion, so its force reverses when velocity reverses. Damping changes amplitude and phase and can move a system from oscillatory to non-oscillatory response.

Classify the roots

For m x¨ + c x˙ + kx = 0, try x = e^{rt}. The characteristic equation is mr² + cr + k = 0, with roots r = [−c ± √(c² − 4mk)]/(2m). If c² < 4mk, roots are complex and motion oscillates with decaying amplitude; equality is critical damping; larger c gives overdamped return.

For m = 1 kg and k = 9 N/m, critical damping is c_c = 2√(mk) = 6 N·s/m. A damper c = 2 is underdamped; c = 8 is overdamped. The classification predicts shape before solving constants.

Damping is model-dependent

Viscous damping assumes force proportional to speed. Coulomb friction gives nearly constant opposing force and produces different decay, while fluid drag can be quadratic.

Work it: Compute the damping ratio ζ = c/c_c for m = 0.25 kg, k = 100 N/m, c = 3 N·s/m. Predict whether oscillations occur and justify from the discriminant.