Logarithmic Decrement

a quantitative characteristic of the rate of damping of oscillations. The logarithmic decrement δ is equal to the natural logarithm of the ratio of two successive maximum deflections x of an oscillating quantity in the same direction: δ = log(x₁/₂). It is the inverse of the number of oscillations after which the amplitude attenuates by e times. For example, if δ = 0.01, then the amplitude will be reduced by e times after 100 oscillations. The logarithmic decrement characterizes the number of cycles in the course of which damping of oscillations takes place, not the time of such damping. The total damping time is determined by the ratio T /δ, where T is the oscillation period.

The usual magnitudes of the average values of the damping decrements of some systems are δ ≃ 0.1 for the acoustic oscillations of a system; δ ≃ 0.02-0.05 for an electrical circuit; δ ≃ 0.001 for a tuning fork; and δ ≃ 10i^-4- 10^5. for a quartz plate. Hence it can be seen that, for example, a tuning fork completes about 1,000 oscillations before the amplitude of the oscillations is reduced by a factor of 3, since e = 2.718 ~ 3.

In the theory of forced oscillations the concept of the quality Q of an oscillatory system, to which the logarithmic decrement is associated by the relationship , is usually used instead of the logarithmic decrement. For high quality factors, δ =π/Q