It is used to find the damping ratio of an underdamped system in the time domain. It is defined as the natural logarithm of the amplitude reduction factor. The amplitude reduction factor is the ratio of any two successive amplitudes(Peaks) on the same side of the mean position.
If x1 and x2 are successive values of the amplitude on the same side of the mean position, as shown in Fig.,
Then the amplitude reduction factor,
Where tp is the time period of forced oscillation (or) the time difference between two consecutive amplitudes. As per definition, logarithmic decrement,
In general, amplitude reduction factor,
Logarithmic decrement,
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