underdæmpede

Underdæmpede, or underdamped in English, describes a physical or engineered system whose damping is insufficient to prevent oscillations after a disturbance. In a linear second-order system this corresponds to a damping ratio between zero and one (0 < ζ < 1).

The standard model is a mass–spring–damper described by m x'' + c x' + k x = 0, where m is mass, c is damping coefficient, and k is stiffness. The natural frequency is ω_n = sqrt(k/m), and the damping ratio is ζ = c/(2 sqrt(m k)). The damped oscillation frequency is ω_d = ω_n sqrt(1 − ζ^2).

Solutions take the form x(t) = e^{−ζ ω_n t} [A cos(ω_d t) + B sin(ω_d t)]. Thus the motion oscillates with decaying amplitude. For a unit step input, the transient response exhibits overshoot and a gradual approach to steady state; the percentage overshoot depends on ζ via OS = exp(−π ζ / sqrt(1 − ζ^2)).

Applications and contexts include mechanical vibrations (vehicle suspensions, machinery, buildings subject to wind or earthquakes) and electrical circuits (series RLC circuits). In all cases underdamping produces noticeable oscillations before settling, unlike critically damped (ζ = 1) or overdamped (ζ > 1) systems which settle without oscillation or more slowly without overshoot.

Relationship to Danish usage: in Danish engineering literature, underdæmpede describes the same concept, with the adjective used for systems that visibly oscillate before stabilizing.