Rayleigh Distribution

March 29, 2018

Rayleigh - photo

Lord Rayleigh, 1842-1919 (John W. Strutt)

There are several situations where Rayleigh distribution occurs in random vibration. 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 also have a Rayleigh distribution. This second example 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, σo 2, as illustrated in the image below:

Rayleigh - image1

The normalized magnitude of r is given in Equation 23:

(1)   \begin{equation*} z=\frac{|r|}{\sigma_{0}}=\frac{\sqrt{a^2+b^2}}{\sigma_{0}} \end{equation*}

Equation 23

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

(2)   \begin{equation*} R(z)=ze^{-\frac{z^2}{2}} \end{equation*}

Equation 24

The Rayleigh PDF is not symmetrical and the mean value is not equal to 1. Instead, r ≅ 1.25 σo and σr ≅ 0.655 σo.

Raleigh distribution

Figure 3.19. The Rayleigh distribution.

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.

Figure 21. Random Vibration Response of the Fundamental Resonance of a Cantilever Beam

Figure 3.20. Random vibration response of the fundamental resonance of a cantilever beam.

 

Figure 22. PDF of Peak Vibration Response, |Ai|

The Rayleigh distribution is closely associated with the χ22 distribution because the Rayleigh variables are the square root of the χ22 variables:

(3)   \begin{equation*} r=\sqrt{a^2+b^2} \end{equation*}

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.

Rayleigh - image2

Rayleigh - Figure23

Figure 3.22. Confidence intervals for “not to exceed levels” with Rayleigh distribution.