PSI - Issue 59

Andrii Senyk et al. / Procedia Structural Integrity 59 (2024) 502–507 Andrii Senyk at al. / Structural Integrity Procedia 00 (2023) 000 – 000

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2. Results and discussion As the result of the calculations, it is found that for all standard sizes of bushings, the condition λ от < λ о , is met, which means that the homogeneity hypothesis is accepted and indicates the technological process stability according to the criterion of the deviation from the roundness of the rolling bushings ICS. The results obtained from 30 circlegrams were tabulated and averaged for each of 48 positions and averaged circlegrams were obtained. For each of these circlegrams, the coordinates of the center       п п N i i N i i Y X Y X 1 0 1 0 , 2 2 and the radius of the base circle    N i п i R r N 1 0 , where i i X Y , і i r are the Cartesian coordinates and the radius vector of the і -th point of the averaged circlegram, respectively; 48  п N - the number of fixed positions, are determined by the method of R.S. Sprague. Then, on the averaged circlegrams shown in Fig. 1, we found the running deviations of the radius vector i R , which had the nature of random errors and were the periodic function with 2π period, the argument of which was assumed to be angle of rotation φ. In this case, the implementation of the deviation from the roundness of the reduced circlegram can be represented by trigonometric Fourier series:         1 0 ) sin ( cos 2 ( ) k k k kx b kx a a f x , where 0 a is a free member; k a and k b are trigonometric Fourier coefficients: f x kx dx a k    ( )cos 1 2 0   ; f x kx dx b k    ( )sin 1 2 0   . Carrying out numerical integration of the random function f(x) for the 48th values of the argument (every 7.5°), the coefficients of the trigonometric Fourier polynomial for the 10th order harmonics were determined using the following formulas: . In order to determine the coefficients of the trigonometric Fourier polynomial for 24 values of the argument in the area of folding seam ( 4 4       ); and in the area opposite to the folding seam ( 3 4 5 4      ), that is, when the period 2 l ≠ 2 π , where l is a half-period, and taking into account that 4   l we get: 24 ( ) cos  24 1 48 0 i    ik f x a i k  ; 24 ( ) sin  24 1 48 0 i    ik f x b i k 

1 24 0 i  

1 24 0 i  

 ik

 ik

( ) cos 

;

( ) sin 

.

a

f x

b

f x





k

i

k

i

12

12

12

12

The values of the total harmonic amplitudes k A of the Fourier series are equal to 2 2 k k k A a b   . Since the components of the amplitude spectrum are uncorrelated with each other, the dispersion of deviations from the roundness of bushings ICS is determined by formula     1 2 2 ( ) i k k D A A . The values ( ) k D A for different areas of the bushings ICS are given in Table 1.