Logarithmic decrement

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 "x"₀ 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.

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

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 be found using two peaks that are right next to each other with a slightly simplified variation on these equations. This method is not as accurate, but in general will give values within the specified range.

: $zeta\; =\; frac\{ln(x\_0/x\_1)\}\{sqrt\{4pi^2\; +\; (ln(x\_0/x\_1))^2$

where "x"_o is the left peak and "x"₁ is the first peak to the right of "x"_o.

See also

Damping

Damping ratio

Damping factor

Q factor