# Rayleigh Distribution

March 29, 2018

Back to: Random Testing

There are several situations where the *Rayleigh* distribution occurs in random vibrations. The peak amplitudes of the response of resonance to a random excitation exhibit a Rayleigh distribution. The amplitudes of the spatial vibration patterns in complex systems have a Rayleigh distribution. This second case led John W. Strutt, 3rd Baron Rayleigh, to derive the formula for the probability distribution bearing his name. He considered the vibration amplitude to be a vector *r* with *a* and *b* components that are independent and Normally distributed with zero mean value and variance, *σ*_{o} ^{2}, as illustrated here:

The normalized magnitude of *r* is given by Equation 23:

(1)

Equation 23

The PDF is given in Equation 24 and displayed in Figure 3.19.

(2)

Equation 24

The Rayleigh PDF is not symmetrical and the mean value is not equal to 1. Instead, *r* ≅ 1.25 *σ** _{o}* and

*σ*

*≅ 0.655*

_{r}*σ*

*.*

_{o}As an example, Figure 3.20 displays the vibration response of the tip of a cantilever beam resonance to random excitation at the base. The histogram of the peak magnitudes, |Ai|, which are proportional to the peak stresses in the beam, is shown compared to the Rayleigh distribution in Figure 3.21. The results are then used in *fatigue analysis* (see Section 3.3.)

The Rayleigh distribution is closely associated with the *χ*_{2}^{2} distribution because the Rayleigh variables are the square root of the *χ*_{2}^{2} variables:

(3)

The confidence level “Not to be Exceeded” for the estimation of the peak level is displayed as the area *P* in the graph below. It is plotted as a function of the number of standard deviations from the mean in Figure 3.22.