Logarithmic decrement

Logarithmic decrement, $\delta$ , 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}}.

Thus logarithmic decrement also permits to evaluate the Q factor of the system:

Q = \frac{1}{2\zeta},

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

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:

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

\zeta = \frac{1}{\sqrt{1 + (\frac{2\pi}{\ln (x_0/x_1)}})^2},

where x₀ and x₁ are any two successive peaks.

And for system $\zeta << 1$ (not too close to the critically damped regime, where $\zeta =1$ ).

\zeta = \frac{\ln (x_0/x_1)}{2\pi}.

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 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

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

References

Inman, Daniel J. (2008). Engineering Vibration. Upper Saddle, NJ: Pearson Education, Inc. pp. 43–48. ISBN 0-13-228173-2.

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Logarithmic decrement

Simplified Variation

Method of fractional overshoot

See also

References