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

420

3

algorithms are described in detail by Syzrantsev et al. (2008). Fig.1 shows the result of sampling generation for the random value (radial force) of length k=1000 under the heavy duty mode of the bearing load. The second auxiliary problem can be solved by the empirical risk method, or based on Parzen-Rosenblatt estimation (Parzen (1962), Syzrantseva and Chernaya (2019)).

Fig. 1. Density function for generating the random value of the radial force in the interval [0;1] under the heavy duty mode of the bearing load.

3. Computer simulation of durability evaluation for bearing 2207 Let us illustrate the development of the method proposed by Reshetov et al. (1990) for rolling bearing reliability evaluation by means of the nonparametric statistics apparatus and computer simulation by the example of the rolling bearing 2207 loaded with a random radial force with an average value of 6900 N and the required life of 63 million revolutions. In the first step, we will simulate the dynamic load capacity of the bearing in accordance with the Weibull distribution law. The probability function is described by the expression:

m

0 t C e P C      1 ( ) 1



 ln0.9 m

/ 90

C C

e

(2)

,

where С 90 is the nominal 90% dynamic load capacity of the chosen bearing, С 90 =25600 N; m is the parameter of the distribution shape, according to recommendations of GOST 18855-82 m =1.5; t 0 is the scale parameter calculated by transformations of the equation (2) by the formula:

m

C

.

1 39009523.8

t





ln 0.9 90

0



Then the density function of the dynamic load capacity of the bearing 2207 will be as follows:

m

0 t C



(3)

1 C e   m

( ) t P C m 

0

The sampling of a random value – the dynamic load capacity – is generated by a non-parametric gauge; the result of the developed program is shown in Fig. 2. To verify the result of a random value simulation, it is reasonable to calculate 90% quantile by means of MathCad integration: the probability that the value of the dynamic load capacity will exceed the value of С90 is: