Logarithmic decrement

The logarithmic decrement, $$\delta$$, is related to the damping ratio, $$\zeta$$ as follows:

\begin{align} \zeta & = \frac{1}{\sqrt{1 + \Big(\frac{2 \pi}{\delta}\Big)^{2}}} \\ \delta & = \frac{2 \pi \zeta}{\sqrt{1 - \zeta^{2}}} \end{align}