Definition:Underdamped

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:
 * $x = e^{-b t} \left({C_1 \cos \alpha t + C_2 \sin \alpha t}\right)$

where $\alpha = \sqrt {a^2 - b^2}$.

Then $S$ is described as being underdamped.


 * Underdamped.png

Also see

 * Definition:Overdamped
 * Definition:Critically Damped