Underdamped

Summary

A damped harmonic oscillator $x'' + 2 \zeta \omega_0 x' + \omega_0^2 x = 0$ with $\omega_0 > 0$ is underdamped when its damping ratio satisfies $0 < \zeta < 1$. Then the characteristic roots are $-\zeta \omega_0 \pm i \omega_1$ with $\omega_1 = \omega_0 \sqrt{1 - \zeta^2}$. Each nonzero solution has the form $x(t) = A e^{-\zeta \omega_0 t} \cos(\omega_1 t + \varphi)$ with $A > 0$: an oscillation whose amplitude decays exponentially.

Remarks

The other cases are critical damping, $\zeta = 1$, and overdamping, $\zeta > 1$. In these cases the solution changes sign at most one time. The damped frequency $\omega_1$ is smaller than the undamped frequency $\omega_0$. The zeros of an underdamped solution are spaced $\pi / \omega_1$ apart.

Explanations

L1 human

An underdamped system overshoots its rest position again and again. Each swing is smaller than the previous one, by the same factor. A guitar string or a car on soft springs behaves like this.