PSI - Issue 40

Kirill E. Kazakov et al. / Procedia Structural Integrity 40 (2022) 201–206 Kirill E. Kazakov / Structural Integrity Procedia 00 (2022) 000 – 000

204

4

( ) ( ) ,

in a r h z g z in  





t



* a z z 

*

* out

* in

*

* *

0 out 

,

,

,

,

,

( )

0 in

t

g z

















a





0

2 out

2 in

( ( E t E t in out

)

) ( , ) q z t

) a h z ) ( ) in 2 out 

 



 







* *

( , ) 2(1 * * * 

* *

out

( )

,

( ) (1 

,

,

c t

m z

q z t



)

( E t out

)





2(1



out

in

*

*

t

t

1   1

1 

* in

( ) *

* * * in

( , ) ( ) *   f

*

* out

( ) *

* * * out

( , ) ( ) *   f

*

V

V

(2)

,

,

f t

K t

d

f t

K t

d









in     r

,   



*

( ) *

( , ) ( ) * * * *   f d k z

*

( , ) 1 * * *  

F

,

f z

k z

k z





c



1



( ( E E t in in

) ( ) ( out out E

) )





 

out  

* * * in

( , ) * * * out      K t ) , in 0

in

( K t in

,

( K t out

,    out

) . 0

( , ) 

K t

 



out   



in

E t



in  



out

Then equation (1) will take the following dimensionless form:

( ) ( )( * * * * c t m z

* * * * in

( , ) * * * * * out q z t

( ), g z z * *

*

*

 I V

) I V F

) ( , ) ( q z t

[ 1,1],

1.

(3)

t

 

 

 



The resulting equation is written in operator form. It is also integral and contains operators of various types. The functions ( ) * * m z and ( ) * * g z in this equation, connect with the profiles of the contacting surfaces. 4. Solution of the problem Article by Kazakov and Manzhirov (2008) showed that the solution of such equations cannot be built using classical methods, since it includes functions associated with surface profiles that can be rapidly changing. This is due to the fact that researchers are forced to be limited to a relatively small number of members of the series, since the computational error accumulates in long series. But small number of terms are not sufficiently for correct series expansion of the functions describing surface profiles. To construct a solution of such equations, it is necessary to use a special approach, the basic theses of which will be described below. First of all, we note that the resulting equation (3) is similar to equation (1.2) of article by Manzhirov and Kazakov (2018). The differences are that now in the first term in left side of (3) there is a time operator, and in the right side of (3) there are no  ( t ) and  ( t ). Thus, to obtain a solution to the problem from this article, it is necessary to use a similar approach. We point out two basic steps that allow you to obtain a solution. First, it is necessary to look for a solution in the form

( ) ( , ) * * * * m z

. ( ) ( ) ( ) * * * * * * c t m z g z

Q z t

* * *

) * 1 in 

I V

( , )

(  

q z t



Secondly, it is necessary to use a special system of basis functions { p j 0 ( z )} constructed using orthonormalization of following system of linearly independent functions

j = 1,2,3,… in Hilbert space L 2 [ – 1,1]

   

.    

, ( ) m z z * 2 * *

* m z z

1

,

,



* *

* *

( )

( )

( )

m z

In orthonormalization, it is necessary to use a dot product calculated by the formula