Damping

Damping is the loss of energy that makes an oscillation die out. For a mass on a spring with a damper, $m x'' + c x' + k x = 0$, written $x'' + 2\zeta\omega_0 x' + \omega_0^2 x = 0$ with the natural frequency $\omega_0 = \sqrt{k/m}$ and the damping ratio $\zeta = c/(2\sqrt{mk})$. Below 1 the motion is underdamped (it swings), at 1 critically damped, above 1 overdamped (it creeps). For a mass hanging vertically, gravity only moves the rest position down by mg/k; x is measured from there. The page releases the mass from rest at x = 1 and asks whether more damping stops it sooner.

The right of the picture is the root plane. The two roots multiply to $\omega_0^2$, so while the motion swings they lie on the circle of radius ω₀: the real part $-\zeta\omega_0$ sets the decay, the imaginary part $\omega_d = \omega_0\sqrt{1-\zeta^2}$ the frequency, so damping also slows each swing. At ζ = 1 the roots meet at −ω₀. Past it one root runs left and the other creeps back towards 0, about $-\omega_0/(2\zeta) = -k/c$ for large ζ. The slowest part of the motion decays like $e^{-\sigma t}$, and this decay rate σ, the distance of the rightmost root from the imaginary axis, is largest exactly at ζ = 1. On x(t) the dashed blue curve shows it: the envelope $\pm e^{-\zeta\omega_0 t}/\sqrt{1-\zeta^2}$ when underdamped (the curve touches it between its peaks), the slow root's term when overdamped.

“Stops” needs a rule. The page uses $T_{1\%}$, the earliest time after which |x| stays within 1 % of the release for good. Its strip under x(t) magnifies the band. The extrema sit at $t_n = n\pi/\omega_d$ with $|x(t_n)| = e^{-\zeta\omega_0 t_n}$, which decides exactly which peaks leave the band; the last return is then found by bisection on the closed-form solution, so the readout is numeric. A smaller ζ gives a slightly later settling time whenever one more peak leaves the band, so the curve of $T_{1\%}$ against ζ jumps. Its minimum is not at critical damping but at $\zeta^* = \ln 100/\sqrt{\pi^2 + \ln^2 100} \approx 0.826$, where the one overshoot touches −1 % exactly: 0.667 T₀ against 1.057 T₀. Critical damping is the fastest only among motions that never cross x = 0, and only from rest; another band moves ζ* (a 2 % band gives about 0.780).