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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 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]].
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The damping ratio is then found from the logarithmic decrement by:
: <math> \zeta = \frac{1}{\sqrt{1 + (\frac{2\pi}{\delta})^2}} </math>
Thus logarithmic decrement also permits evaluation of the [[Q factor]] of the system:
: <math> Q = \frac{1}{2\zeta} </math>
: <math> Q = \frac{1}{2} \sqrt{1 + \left(\frac{n2\pi}{\ln \frac{x(t)}{x(t+nT)}}\right)^2} </math>
The damping ratio can then be used to find the natural frequency ''ω''<sub>''n''</sub> of vibration of the system from the damped natural frequency ''ω''<sub>''d''</sub>'':''
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==References==
{{reflist}}
* {{cite book|last=Inman|first=Daniel J.|title=Engineering Vibration|year=2008|publisher=Pearson Education, Inc.|___location=Upper Saddle, NJ|isbn=0-13-228173-2|pages=
[[Category:Logarithms]]
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