Logarithmic decrement: Difference between revisions

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{{Short description|Measure for the damping of an oscillator}}

{{multiple issues|

{{refimprovemore citations needed|date=February 2012}}

[[File:Dampingratio111.svg|thumb|400px|The logarithmic decrement can be obtained e.g. as ln(''x''₁/''x''₃).]]

{{Cleanup|date=February 2008}}

'''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]].

'''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:

==Method==

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

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:

where ''x''(''t'') is the overshoot (amplitude - final value) at time ''t'' and {{nowrap|''x''(''t'' + ''nT'')}} is the amplitudeovershoot of the peak ''n'' periods away, where ''n'' is any integer number of successive, positive peaks.

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

The damping ratio is then found from the logarithmic decrement by:

Thus logarithmic decrement also permits evaluation of the [[Q factor]] of the system:

: <math> Q\zeta = \frac{1\delta}{\sqrt{4\pi^2 + \zetadelta^2}}, </math>

: <math> Q = \frac{1}{2} \sqrt{1 + \left(\frac{n2\pi}{\ln \frac{x(t)}{x(t+nT)}}\right)^2}. </math>

Thus logarithmic decrement also permits evaluation of the [[Q factor]] of the system:

: <math> \omega_dQ = \frac{1}{2\pi}{Tzeta}, </math>

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

: <math> \zetaQ = \frac{1}{2} \sqrt{1 + \left(\frac{2n2\pi}{\deltaln \frac{x(t)}{x(t+nT)}}\right)^2}}. </math>

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

: <math> \omega_d = \frac{2\pi}{T}, </math>

: <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.

: <math> \omega_d = \frac{2\pi}{T} </math>

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.

: <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.

==Simplified Variationvariation==

The damping ratio can be found for any two adjacent peaks. This method is used when {{nowrap|1=''n'' = 1}} and is derived from the general method above:

: <math> \zeta = \frac{1}{\sqrt{1 + \left(\frac{2\pi}{\ln \left(\frac{x_0/}{x_1}\right)}}\right)^2},} </math>

where ''x''_''0'' and ''x''_''1'' are amplitudes of any two successive peaks.

And forFor system where <math> \zeta <<\ll 1 </math> (not too close to the critically damped regime, where <math> \zeta =\approx 1 </math>).

: <math> \zeta =\approx \frac{\ln \left(\frac{x_0/}{x_1}\right)}{2\pi}. </math>

==Method of fractional overshoot==

The method of fractional overshoot can be banana useful for damping ratios between about 0.5 and 0.8. The fractional overshoot ''{{math|OS''}} is:

: <math>\mathrm{ OS} = \frac{x_p - x_f}{x_f}, </math>

where ''x''_''p'' is the amplitude of the first peak of the step response and ''x''_''f'' is the settling amplitude. Then the damping ratio is

: <math> \zeta = \frac{1}{\sqrt{1 + \left(
\frac{\pi}{\ln (\mathrm{OS})}
\right)^2}.} </math>

== See also ==

* [[Damping]]

* [[Damping ratio]]

* [[Damping factor]]

* [[Q factor]]

==References==

{{reflist}}

* {{cite book|last=Inman|first=Daniel J.|title=Engineering Vibration|year=2008|publisher=Pearson Education, Inc.|___location=Upper Saddle, NJ|isbn=978-0-13-228173-27|pages=43-4843–48}}

[[Category:Kinematic properties]]

[[Category:Logarithms]]