overdamped

Overdamped refers to a damped system in which the damping is strong enough to prevent oscillations as the system returns to equilibrium. In the standard second-order form, this occurs when the damping ratio ζ is greater than 1.

A common mechanical model is a mass-spring-damper system described by m x'' + c x' + k x = 0, with m > 0, c > 0, and k > 0. The characteristic equation m r^2 + c r + k = 0 has roots r1 and r2 given by (-c ± sqrt(c^2 - 4 m k)) / (2m). For overdamping, the discriminant is positive (c^2 > 4 m k), so the roots are real and negative. The general solution is x(t) = A e^{r1 t} + B e^{r2 t}, consisting of two decaying exponentials with no oscillation.

Damping ratio and natural frequency provide a compact description: ω_n = sqrt(k/m) is the undamped natural frequency, and ζ = c / (2 sqrt(m k)) = c / (2 m ω_n) is the damping ratio. Overdamping corresponds to ζ > 1. In this regime, the system returns to equilibrium monotonically, without overshoot, but typically more slowly than in the critically damped case.

Applications and examples include mechanical devices such as door closers and some vibration isolators, as well as electrical circuits like series RLC circuits, where overdamping occurs when the resistance is sufficiently high (for a series RLC, ζ > 1 translates to R > 2 sqrt(L/C)). Real systems may deviate from the ideal model due to nonlinearities, temperature effects, and frequency-dependent damping, but the characteristic feature of non-oscillatory, slower decay remains.