PSI - Issue 33

O. Pozhylenkov et al. / Procedia Structural Integrity 33 (2021) 385–390 Author name / Structural Integrity Procedia 00 (2019) 000 – 000

388

4

{ (2 + 1) ′ ( ) + ( ) = − , = ∫ ( ) ∗ cos( ) 0 (0) = 0 ′ ( ) − ( ) = 0 ′ (0) − (0) = 0

, = (1 + ) ∗ (1 − 2 )

(5)

The initial problem in the transforms domain (5) is reformulated into the vector boundary problem: 2 [ ( )] = ′′ ( ) + ′ ( ) + ( ) 2 [ ( )] = 0 [ ( )] = , = 1,2

(6)

The vectors and matrices of the vector boundary problem are derived:

= ( 1 0 1 + 0 0 ) , = ( 0 − 0 0 0 ) , = ( − 2 − 2 0 + 1 12

2 0 0 − 2 + 1 22 2 ) ,

( ) = ( ( ) ( ) )

(7)

To solve this vector boundary problem, the fundamental solution matrix ( ) is constructed. To found the solution of the stated problem firstly the matrix (where the unit matrix) must be substituted into the equation (6). From the equality 2 ( ) = ( ) , one can derive the ( ) matrix: ( ) = ( 2 − 2 − 0 2 + 1 12 2 − 0 0 2 + 2 0 − 2 + 1 22 2 ) (9) The fundamental solution matrix is found with the help of formula ( ) = 2 1 ∮ −1 ( ) Gantmakher F. R. (1998). The calculation of the integral requires to know all poles of the under integral function. To do it the determinant of the matrix ( ) was found: ( ) = (1 + 0 )( − 1 )( + 1 )( − 2 )( + 2 ) (10) where 1 , − 1 , 2 , − 2 are roots of the ( ) . After contour integration procedure the four linear independent solutions of the matrix equation were derived: ( ) = 1 + 1 0 ∑ [ −1 ( )], ( − ) =0 ( ) = 1 + 1 0 ( 0 ( ) + 1 ( ) + 2 ( ) + 3 ( ))