Damping

Remarks

With $\omega_0 = \sqrt{k/m}$ the equation reads $\ddot x + 2 \zeta \omega_0 \dot x + \omega_0^2 x = 0$, whose characteristic roots are $\omega_0 \big(-\zeta \pm \sqrt{\zeta^2 - 1}\big)$; the four cases correspond to imaginary, complex, double real and distinct real roots; see the damped harmonic oscillator. The energy $E = \frac{1}{2} m \dot x^2 + \frac{1}{2} k x^2$ satisfies $\dot E = -c \dot x^2 \le 0$.

Example: $m = 1$, $c = 2$, $k = 5$ gives $\zeta = 2 / (2 \sqrt 5) = 1/\sqrt 5 < 1$, so the oscillator is underdamped; its solutions are $e^{-t}(c_1 \cos 2t + c_2 \sin 2t)$.

Non-example: a negative coefficient, $c = -2$ with $m = 1$, $k = 5$, is not damping: the solutions $e^{t}(c_1 \cos 2t + c_2 \sin 2t)$ grow, and the energy increases.

Other kinds of damping, such as dry (Coulomb) friction or quadratic drag, are not linear in $\dot x$ and are not covered by this definition.