S.Ya. Grodzenskiy1

1 MIREA – Russian University of Technology (Moscow, Russia)

In 1927, M. Frechet derived the probability distribution, which is used to analyze extreme phenomena related to the environment. P. Rozin and E. Rambler in 1933, this distribution is used to obtain a formula for the distribution of coal particles by size. In 1939, V. Weibull, when studying the issues of fatigue of materials and analyzing failures caused by the wear of ball bearings, used a distribution with a reliability function of the form, where T, s, t0 – are the scale, shape, and shift parameters.

In 1942, B.V. Gnedenko established that the distribution described by Weibull is one of the types of distribution of minima. Since the mid-1950s, interest in this distribution function has been growing rapidly, and the name "Weibull distribution" has been assigned to it. The dependence of the failure rate of technical objects on time is generally U-shaped. There are three periods: I – running–in, II – "normal operation" and III - wear. The Weibull distribution makes it possible to describe any of them: during the period of "childhood illnesses" s <1, during the period of "normal operation" s = 1, during the period of wear s > 1. By the value of s, it is possible to make an assumption about the nature of the malfunction and put forward proposals for its elimination. At the same time, the Weibull distribution is not flexible enough to describe the entire U-shaped dependence.

The article analyzes some generalized Weibull distributions and shows that in this case the dependence l(t) is not U-shaped, but rather V-shaped – there is no region of constancy of the failure rate. Of interest is the five-parametric logarithmically normal modified Weibull distribution, the concept of which is based on the division of failures into sudden, obeying the logarithmically normal distribution, and gradual − modified Weibull distribution. A model of resource consumption is proposed, according to which, in period I, the reliability of an object is due to its initial quality, determined by the intensity decreasing over time, and in period III − the rate of accumulation of irreversible changes with increasing intensity. Based on this interpretation of the "lambda characteristic", it is proposed to use a mixture of exponential and Weibull distributions

Grodzenskiy S.Ya. Weibull disytibution: history and modernity. Nonlinear World. 2023. V. 21. № 3. P. 54-65. DOI: https://doi.org/10.18127/j20700970-202303-06 (In Russian)

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