# Log-Normal Distribution

March 29, 2018

Many physical phenomena exhibit a Log-Normal distribution. When the response amplitude of a system results from the multiplication of a number of independent parameters, the distribution of the response amplitude will tend to be Log-Normal. This is because 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.  Fig. 24 shows the Log-Normal distribution plotted on both a linear and log x scale. Figure 24. Log-Normal Distribution

In random vibrations this applies to the distribution of the vibration magnitude over an ensemble of random conditions. This is illustrated in Fig. 25 with 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, plotted on a decibel scale [Decibel Level = 20 Log (ARMS/1 μG), dB], is compared to a Normal distribution of the Log levels with a mean of 114.2 dB and σ = 3.5 dB. Figure 25. Distribution of Engine Vibration Levels 