Rayleigh Distribution

March 29, 2018

Rayleigh - photo

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

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, σo 2, as illustrated here:

Rayleigh - image1

The normalized magnitude of r is given by:

Rayleigh - givenby1

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

Rayleigh - equation23

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.

Rayleigh - Figure20

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)

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

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


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

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

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

Rayleigh - 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.

Rayleigh - image2

Rayleigh - Figure23

Figure 23. Confidence Intervals For “Not to Exceed Levels” with Rayleigh Distribution