Definition:Critically Damped

Definition
Consider a physical system $S$ whose behaviour can be described with the second order ODE in the form:
 * $(1): \quad \dfrac {\mathrm d^2 x} {\mathrm d t^2} + 2 b \dfrac {\mathrm d x} {\mathrm d t} + a^2 x = 0$

for $a, b \in \R_{>0}$.

Let $b = a$, so that the solution of $(1)$ is in the form:
 * $x = C_1 e^{-a t} + C_2 t e^{-a t}$

Then $S$ is described as being critically damped.


 * CriticallyDamped.png

Also see

 * Definition:Overdamped
 * Definition:Underdamped