PSI - Issue 40

Ksenia Syzrantseva et al. / Procedia Structural Integrity 40 (2022) 418–425 Ksenia Syzrantseva at al. / Structural Integrity Procedia 00 (2022) 000 – 000

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1. Introduction The rolling bearing calculations used worldwide were the first probability calculations for mechanical engineering objects. However, they only considered the dispersion of the dynamic load carrying capacity of bearings, assuming that the design load was deterministic according to works of Beizelman et al. (1967), Tomaszewski (2013) and Levanov et al. (2021). The method proposed by Reshetov et al. (1990) for estimating the probability of failure-free operation of a bearing is based on determining the mean load safety factor:

(1)

,

n C 1/  

p P L

on the premise that the random value of the safety factor follows the normal law (Halme and Andersson (2016), Zuo et al. (2015)). By substituting the mean value of the safety factor into the expression for normal distribution quantile, the probability of failure-free operation is determined by tables of Reshetov et al. (1990) and Kapur and Lamberson (1977). Despite the fact, that Reshetov et al. (1990) refer to a widely accepted (including standards GOST 18855-82 and ISO) assumption of bearing service life and dynamic load capacity distribution according to the Weibull law, they accept a significant simplification of calculations: the transition from Weibull distribution to normal distribution to describe the dynamic load capacity. Moreover, in their proposed method, the law of the equivalent load distribution is assumed to be normal, which corresponds to the average loading mode. The problem of the The approach to processing random values proposed in this paper is based on methods of nonparametric statistics, which initially assume that the type of distribution of a random value is either unknown or can only be determined approximately (Syzrantsev et al. (2008) and Syzrantseva et al. (2019)). At present, the methods of nonparametric statistics allow for solving practically the whole range of problems that used to be solved by the methods of parametric statistics but without imposing any restrictions on a random value distribution function and hence excluding the errors caused by the replacement of real distributions of a random value by close "convenient" known distributions that have an analytical description (Sonawane and Dhawale (2016), Liu et al. (2018) and Golofast (2020)). Professional mathematical processors developed by now offer the researcher a rich set of standard functions for solving equations, including the transcendental ones, methods for optimization of functions, a wide range of functions for implementation of user algorithms, and also convenient means of visualizing the diagrams of functions and experimental data (D'yakonov (200)). Implementation of methods for evaluation of the strength reliability of machine parts and units based on the mathematical apparatus of nonparametric statistics is related with the solution of two auxiliary problem:  Computer modeling of random values with laws known with the parameter accuracy and with laws determined by methods of nonparametric statistics  Restoring a distribution density function for a given sample of a random value Let us consider the solution of the first problem. Its urgency is determined by the necessity to determine the probability of failure-free operation of the part at different operation modes. According to GOST 21354-75 (1981), the external load is a random value in the general case. At the light, average normal, average equal probability, and heavy modes of gear operation the integral function for the external load is described by gamma, normal, equal probabilit y and β -distribution, respectively. Since these distribution laws are known with the accuracy up to parameters, the problem of generating random value samples according to them is implemented in the software package MathCad as random number gauges based on methods of nonparametric statistics; their implementation transition to light, heavy, or sample-defined loading modes remains unsolved in this case. 2. Description of the probability method for rolling bearing durability evaluation