Log-Normal Distribution

March 29, 2018

Many physical phenomena exhibit log-normal distribution. When a system’s response amplitude is the product of multiple independent parameters, the distribution of the response amplitude tends to be log-normal.

The log of the amplitude is the sum of the logs of the multiplicative terms. 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.

Rayleigh - Figure 24

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 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.

Rayleigh - Figure 25

Figure 3.24. Distribution of engine vibration levels.

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.Rayleigh - arms