Rayleigh Distribution

March 29, 2018

There are several situations where Rayleigh distribution occurs in random vibration. The peak amplitudes of a resonance‘s response to a random excitation exhibit a Rayleigh distribution. The amplitudes of the spatial vibration patterns in complex systems also have a Rayleigh distribution.

The second example led John W. Strutt to derive the formula for the Rayleigh probability distribution. He considered the vibration amplitude to be a vector r with a and b components that are independent and normally distributed with a zero mean value and variance, σ_o ².

Rayleigh - image1

Equation 23 gives the normalized magnitude of r.

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

Equation 23

Equation 24 gives the PDF, which is 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 σ_o and σ_r ≅ 0.655 σ_o.

Figure 3.19. The Rayleigh distribution.

Example

Figure 3.20 displays the vibration response of a cantilever beam’s resonance to random excitation at the base. Figure 3.21 compares the histogram of the peak magnitudes, |Ai|, to the Rayleigh distribution. The peak magnitudes are proportional to the peak stresses in the beam. Application-wise, engineers can use the results in fatigue analysis.

Figure 20. 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 21. PDF of Peak Vibration Response, |Ai|

Confidence Intervals

The Rayleigh distribution has a close association 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 area P in the graph below displays the confidence level “not to be exceeded” for the peak level estimation. Figure 3.22 is a plot as a function of the number of standard deviations from the mean.

Rayleigh - image2

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