PSI - Issue 50

Kirill E. Kazakov et al. / Procedia Structural Integrity 50 (2023) 125–130 Kirill E. Kazakov / Structural Integrity Procedia 00 (2022) 000 – 000

129

5

Further, it is necessary to construct the eigenfunctions of the operator F ~ using the obtained basis. This will allow us immediately to reduce the resulting system of equations for the functional expansion coefficients to diagonal form. Since, as was mentioned above, the problem with a similar equation has already been solved by Kazakov and Parshin (2019), we present only the final formulas for the functions q i *( z *, t *):

( ) 1 *

  

[1, ),          i t z *





( ) *     K z t n ij ml i km k m k   0 0    

* * * i

( ) ( ) * * *  z c t i 

( ) , 1 1, * *

i km

 p z i m

1, , n

( , )

q z t



m z i

( ) ( ) ( ) * m j j 

* in

) ( ) *  c t

j

 I V

(

p

  



1

( ) ( *

I W

, 

)

d

z t k

 

l

k

( ) *

c t





1



, 1 , 0   i j m l

k

(6)

*

1

t





( ) *

( , ) ( ) * * * * f

, d J *

k

/ ( ) , m d d i  

W

1,

f t

R t k

   







, k i

1,

k

i



1

1



J

J

J

J

0,    i

, k i 0,      J i

, k i

, k i

1

( ) *

 p z i k

,

,

0,1,2, , 

k

d









, k i

d d

( ) * z

k

1

J

, k i k i 1, 



2 ,

k i

i km  are determined from the solution of the system

where  k and

1 0     n j l

ij ml

j kl

i k km

( ,    i j

1, , ,

0,1,2, ) 

,

K

n k m







* 1/2 * * * ( , )[ ( ) ( )]    j i ij k z m z m over the

ij ml K are the expansion coefficients of the kernel

in which the coefficients

* * *  R t k

* { ( )} i m z p

( , )

, and

are the resolvents of

basis

  1,2, , ; 0,1,2,   i nm

c t K t ( ) ( , ) * * * out *

* * * in

 ( , )

K t  



 ( , ) * *

.

k K t

k



( ) *

c t





k

The inverse change of variables leads to dimensional formulas for the contact pressures. 4. Discussion and conclusions

Let us pay attention to formulas (6). In these formulas, the functions m i ( z *),  i * ( z *) and, accordingly, the function h ( z ) and the combination ( r out + h ( z ) – g i ( z )) are represented by separate factors. Precisely these functions are associated with the description of the profiles of the contacting bodies and can change rapidly. All other functions ( ( ) * z t k , ( ) * c t  and polynomials ( ) * p z i m  ) are sufficiently smooth. This allows us, in real calculations, to use partial sums of series with a small number of terms (approximately 20 – 30 terms of the series). This is unattainable for other methods. Classical approaches such as, for example, series expansions in trigonometric or polynomial vector functions lead to problems in calculations. Thus, in this work, it is shown by what a method we can construct an analytical solution for one more class of contact problems for coated bodies, so that it allows accurate calculations even in the cases where the shapes of the objects are presented by rather complicated functions. Acknowledgements The present work was supported by the Ministry of Science and Higher Education within the framework of the Russian State Assignment under contract No. AAAA-A20-120011690132-4.