Logarithmic decrement

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

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

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:

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

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

${\displaystyle Q={\frac {1}{2\zeta }}}$

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

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:

${\displaystyle \omega _{d}={\frac {2\pi }{T}}}$

${\displaystyle \omega _{n}={\frac {\omega _{d}}{\sqrt {1-\zeta ^{2}}}}}$

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

The 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:

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

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

For system where ${\displaystyle \zeta \ll 1}$ (not too close to the critically damped regime, where ${\displaystyle \zeta \approx 1}$).

${\displaystyle \zeta \approx {\frac {\ln({\frac {x_{0}}{x_{1}}})}{2\pi }}}$

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

${\displaystyle OS={\frac {x_{p}-x_{f}}{x_{f}}}}$

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

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

• Damping
• Damping factor

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