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 damping ratio is related to the natural log of any two amplitudes.

$\delta = \frac{1}{n} \ln \frac{x_0}{x_n},$

where x₀ is the greater of the two amplitudes and x_n is the amplitude of a peak n periods away. The damping ratio is then found as follows.

$\zeta = \frac{\delta}{\sqrt{4\pi^2 + \delta^2}}.$

The damping ratio can then be used to find the damped natural frequency of vibration of the system.

$\omega_d = \omega_n \sqrt{1 - \zeta^2}.$

The damping ratio, ζ, can also be found using two peaks that are right next to each other. This method is not as accurate, but in general will give values within the specified range.

$\zeta = \frac{\ln(x_0/x_1)}{\sqrt{4\pi^2 + (\ln(x_0/x_1))^2}}$