# Rayleigh Distribution

March 29, 2018

Back to: Random Testing

The *Rayleigh* distribution occurs in random vibrations in several situations. The peak amplitudes of the response of a 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:

and the PDF is given in Eq. 23 and shown in Fig. 20.

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 illustration of this, Fig. 21 shows 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 Fig. 22. This result is used in *fatigue analysis* (see Section 3.3)

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

.

The *confidence level* for the estimation of the peak level “Not to be Exceeded” is given by the area P in the graph below and is plotted as a function of the number of standard deviations from the mean in Fig. 23.