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 canis then be used to find the natural frequency ''ω''_''n'' of vibration of the systemfound from the damped naturallogarithmic frequencydecrement ''ω''_''d''''by:''

: <math> \omega_dzeta = \frac{2\pidelta}{T\sqrt{4\pi^2 + \delta^2}}, </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.

The method ofThus logarithmic decrement becomesalso lesspermits andevaluation less precise asof the damping[[Q ratiofactor]] increases past about 0.5; it does not apply at all for a damping ratio greater than 1.0 becauseof the system is overdamped.:

: <math> Q = \frac{1}{2\zeta} </math>

==Simplified Variation==

: <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 be found for any two adjacent peaks. This method is identical to the above, but simplified for the case of n=1:

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.

==Simplified Variationvariation==

The damping ratio can be found for any two adjacent peaks. This method is identicalused towhen the{{nowrap|1=''n'' above,= but1}} simplifiedand foris derived from the casegeneral ofmethod n=1above:

: <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.

For 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>

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

==Method of fractional overshoot==

The method of fractional overshoot can be 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 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]]