# Log-Normal Distribution

March 29, 2018

Many physical phenomena exhibit log-normal distribution. When the response amplitude of a system is the result of the multiplication of a number of independent parameters, the distribution of the response amplitude will tend to be log-normal. The log of the amplitude is the sum of the logs of the multiplicative terms, and the central limit theorem implies that the log of the amplitude will have a normal distribution.

For example, if x = A·B/C, then log(x) = log(A) + log(B) – log(C), with A, B, C positive values. Figure 3.23 shows the log-normal distribution plotted on both a linear and log x scale. Figure 3.23. Log-Normal distribution on a linear and log x scale.

In random vibration, this applies to the distribution of the vibration magnitude over an ensemble of random conditions. Figure 3.24 displays a graph of the histogram of the RMS acceleration levels (ARMS) measured in the vertical and transverse directions on the coupling housing of 40 nominally identical diesel engines running under nominally identical conditions. The measured data are plotted on a decibel scale [decibel level = 20 Log (ARMS/1 μG), dB], and compared to a normal distribution of the Log levels with a mean of 114.2 dB and σ = 3.5 dB. Figure 3.24. Distribution of engine vibration levels. 