Definition:Logarithmic Decrement

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

Let $T$ be the period of oscillation of $S$.

Let $x_1$ and $x_2$ be successive local maxima of $x$.

From Ratio of Successive Local Maxima for Underdamped Free Vibration:
 * $\dfrac {x_1} {x_2} = e^{b T}$

The logarithmic decrement of $S$ is defined as:
 * $\ln \left({\dfrac {x_1} {x_2} }\right) = b T$