underdamping

Under damping, or underdamping, refers to a regime of a dissipative system described by a second-order linear differential equation in which the damping ratio ζ lies between 0 and 1. In mechanical contexts this yields oscillatory motion with gradually diminishing amplitude. The standard model is m x'' + c x' + k x = 0, with natural frequency ω_n = sqrt(k/m) and damping ratio ζ = c / (2 sqrt(mk)). For ζ < 1 the unforced response is x(t) = e^{-ζ ω_n t} [A cos(ω_d t) + B sin(ω_d t)], where ω_d = ω_n sqrt(1 - ζ^2) is the damped natural frequency.

The time-domain behavior is a decaying oscillation about the equilibrium position. Amplitude reduces roughly as e^{-ζ ω_n t}, and the system may overshoot the final steady state before settling. The smaller the damping ratio (closer to zero), the more oscillations occur before stabilization; as ζ approaches 1 from below, oscillations diminish and the response approaches critical damping.

In design terms, underdamping is a balance between speed and overshoot. It yields the fastest possible response without fully eliminating oscillations. In frequency terms the system exhibits a resonance near ω_d, with a quality factor Q increasing as damping decreases. Applications span mechanical structures, automotive suspensions, electronic RLC circuits, and control systems, where ζ is chosen to meet performance criteria for rise time, settling time, and overshoot.

Examples include a car responding to a road irregularity with decaying body motion; a mass on a spring with modest viscous damping; and damped oscillations in certain musical instrument contexts.