Definition:Period of Underdamped Oscillation

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 as to make $S$ underdamped.


 * UnderdampedPeriodAmplitude.png

While the behaviour of $S$ is not strictly speaking periodic, its oscillations can be defined to have a "period" as follows:

Let $T$ be the smallest value of $t$ such that:
 * $x = 0$
 * $x' < 0$

Then $T$ is the period of the oscillation of $S$.

Also see

 * Period of Oscillation of Underdamped System is Regular