Logarithmic decrement

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Logarithmic decrement, , is used to find the damping ratio of an underdamped system in the time domain.

The logarithmic decrement can be obtained e.g. as ln(x1/x3).

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

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

 

where x(t) is the overshoot (amplitude - final value) at time t and x(t + nT) is the overshoot 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:

 

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

 
 

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

 
 

where T, the period of the waveform, is the time between two successive amplitude peaks of the underdamped system.

Simplified variation

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

 

where x0 and x1 are amplitudes of any two successive peaks.

For system where   (not too close to the critically damped regime, where  ).

 

Method of fractional overshoot

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The method of fractional overshoot can be useful for damping ratios between about 0.5 and 0.8. The fractional overshoot OS is:

 

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

 

See also

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References

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  • Inman, Daniel J. (2008). Engineering Vibration. Upper Saddle, NJ: Pearson Education, Inc. pp. 43–48. ISBN 978-0-13-228173-7.