Logarithmic decrement: Difference between revisions

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'''Logarithmic decrement''', <math> \delta </math>, is used to find the [[damping ratio]] of an [[underdamped]] system in the time ___domain. The logarithmic decrement is the [[natural logarithm|natural log]] of the ratio of the amplitudes of any two successive peaks:

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]].

==Method==

The logarithmic decrement is defined as the [[natural logarithm|natural log]] of the ratio of the amplitudes of any two successive peaks:

: <math> \delta = \frac{1}{n} \ln \frac{x(t)}{x(t+nT)}, </math>

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 by:

: <math> \zeta = \frac{1}{\sqrt{1 + (\frac{2\pi}{\delta})^2}}. </math>

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: <math> \omega_n = \frac{\omega_d}{\sqrt{1 - \zeta^2}}, </math>

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.

==Simplified Variation==

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== See also ==

* [[Damping]]

* [[Damping ratio]]

* [[Damping factor]]

* [[Q factor]]

==References==