Logarithmic decrement

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Logarithmic decrement, δ, is used to find the damping ratio of an underdamped system in the time domain. The logarithmic decrement is the natural log of the ratio of the amplitudes of any two successive peaks:

\delta ={\frac {1}{n}}\ln {\frac {x(t)}{x(t+nT)}},

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:

\zeta ={\frac {1}{{\sqrt {1+({\frac {2\pi }{\delta }})^{2}}}}}.

The damping ratio can then be used to find the natural frequency ωn of vibration of the system from the damped natural frequency ωd:

\omega _{d}={\frac {2\pi }{T}},
\omega _{n}={\frac {\omega _{d}}{{\sqrt {1-\zeta ^{2}}}}},

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

The damping ratio can also be found using a slightly simplified variation on these equations for two adjacent peaks. This method is identical to the above, but simplified for the case of n equal to 1:

\zeta ={\frac {1}{{\sqrt {1+({\frac {2\pi }{\ln(x_{0}/x_{1})}})^{2}}}}},

where x0 is the left peak and x1 is the first peak to its right.

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 OS is

OS={\frac {x_{p}-x_{f}}{x_{f}}},

where xp is the amplitude of the first peak of the step response and xf is the settling amplitude. Then the damping ratio is

\zeta ={\frac {1}{{\sqrt {1+({\frac {\pi }{lnOS}})^{2}}}}}.

See also

References

    Inman, Daniel J. (2008). Engineering Vibrations. Upper Saddle, NJ: Pearson Education, Inc. p. 53. ISBN 0-13-228173-2. 

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