Logarithmic decrement

Logarithmic decrement, , 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:

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: