Logarithmic decrement
Encyclopedia
Logarithmic decrement, δ, 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:
where x0 is the greater of the two amplitudes and xn is the amplitude of a peak n periods away.
The damping ratio is then found from the logarithmic decrement:
The damping ratio can then be used to find the undamped natural frequency ωn of vibration of the system from the damped natural frequency ωd:
where T, the period of the waveform, is the time between two successive amplitude peaks.
The damping ratio can also be found using a slightly simplified variation on these equations for two adjacent peaks. This method is identical to the above, but simplified for the case of n equal to 1:
where x0 is the left peak and x1 is the first peak to its right.
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.
The method of fractional overshoot can be useful for damping ratios between about 0.5 and 0.8. The fractional overshoot OS is
where xp is the amplitude of the first peak of the step response and xf is the settling amplitude. Then the damping ratio is
Damping ratio
[[Image:Damped spring.gif|right|frame|Underdamped [[spring–mass system]] with ζ 1 , and is referred to as overdamped.*Underdamped:If s is a complex number, then the solution is a decaying exponential combined with an oscillatory portion that looks like \exp...
of an underdamped system in the time domain. The logarithmic decrement is the natural log
Natural logarithm
The natural logarithm is the logarithm to the base e, where e is an irrational and transcendental constant approximately equal to 2.718281828...
of the ratio of the amplitudes of any two successive peaks:
where x0 is the greater of the two amplitudes and xn is the amplitude of a peak n periods away.
The damping ratio is then found from the logarithmic decrement:
The damping ratio can then be used to find the undamped natural frequency ωn of vibration of the system from the damped natural frequency ωd:
where T, the period of the waveform, is the time between two successive amplitude peaks.
The damping ratio can also be found using a slightly simplified variation on these equations for two adjacent peaks. This method is identical to the above, but simplified for the case of n equal to 1:
where x0 is the left peak and x1 is the first peak to its right.
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.
The method of fractional overshoot can be useful for damping ratios between about 0.5 and 0.8. The fractional overshoot OS is
where xp is the amplitude of the first peak of the step response and xf is the settling amplitude. Then the damping ratio is
