What is "logarithmic decrement"

Logarithmic decrement, δ, is used to find the damping ratio of an underdamped system in the time domain. The logarithmic decrement is the natural log of the ratio of the amplitudes of any two successive peaks:

$\delta = \frac{1}{n} \ln \frac{x(t)}{x(t+nT)},$

where x(t) is the amplitude at time t and x(t+nT) is the amplitude of the peak n periods away, where n is any integer number of successive, positive peaks. The damping ratio is then found from the logarithmic decrement:

$\zeta = \frac{1}{\sqrt{1 + (\frac{2\pi}{\delta})^2}}.$

Thus logarithmic decrement also permits to evaluate the Q factor of the system:

$Q = \frac{1}{2\zeta},$ $Q = \frac{1}{2} \sqrt{1 + (\frac{n2\pi}{\ln \frac{x(t)}{x(t+nT)}})^2}.$

The damping ratio can then be used to find the natural frequency ω of vibration of the system from the damped natural frequency ω:

$\omega_d = \frac{2\pi}{T},$ $\omega_n = \frac{\omega_d}{\sqrt{1 - \zeta^2}},$

where T, the period of the waveform, is the time between two successive amplitude peaks of the underdamped system.

The method of logarithmic decrement becomes less and less precise as the damping ratio increases past about 0.5; it does not apply at all for a damping ratio greater than 1.0 because the system is overdamped.