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Logarithmic Decrement

The intensity of damping, in case of under-damped motion is determined by how much the amplitude decreases as the particle completes one oscillation in time period T. That is, the damping is regarded as high or low depending upon the decrease in amplitude during one oscillation.

If A₁ and A₂ are the amplitudes at times t₁ and (t₁ + T) respectively, then we have

where a constant of motion, is called logarithmic decrement.

Another way in which logarithmic decrement can be interpreted is as following: let us consider the decrease in the amplitude during n complete oscillations, i.e. in time nT. We find

For n_θ = 1, A_n+1 = A₁/e; that is θ = 1/n is the inverse of periods in which the amplitude of oscillations decreases by a factor e (starting from any point of time). Hence, if θ = 0.01, the oscillation amplitude decreases to nearly 1/3rd (e = 2.72) in 1/θ = 100 oscillations; in 10 oscillations, A₁₁/A₁ = e-0.1 = 1/1.1, or A₁₁ is about 10 percent of A₁. Thus, we can take the above oscillations to be almost undamped if counting only a few oscillations.