Logarithmic decrement

The logarithmic decrement is defined as 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 overshoot (amplitude - final value) at time t and 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: : \zeta = \frac{\delta}{\sqrt{4\pi^2 + \delta^2}} Thus logarithmic decrement also permits evaluation of 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 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. ==Simplified variation==

The damping ratio can be found for any two adjacent peaks. This method is used when and is derived from the general method above: : \zeta = \frac{1}{\sqrt{1 + \left(\frac{2\pi}{\ln \left(\frac{x_0}{x_1}\right)}\right)^2}} where x0 and x1 are amplitudes of any two successive peaks. For system where \zeta \ll 1 (not too close to the critically damped regime, where \zeta \approx 1 ). : \zeta \approx \frac{\ln \left(\frac{x_0}{x_1}\right)}{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 is: : \mathrm{ 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 + \left( \frac{\pi}{\ln(\mathrm{OS})} \right)^2}} == See also ==