PSI - Issue 33

Anna Fesenko et al. / Procedia Structural Integrity 33 (2021) 509–527 Author name / Structural Integrity Procedia 00 (2019) 000 – 000

513

5

* p p W U      , (1) (1) 0

1 U W aG p        (1) (1)

 

(1)

U

3 1

1 1



 

,





*

p

p

p

p









1 0 0       

  

p d e d   

( , )cos     

p

P





(10)

p

n



For solving a one-dimensional boundary value problem (9), (10) a second-order matrix differential operator and the unknown vector of displacements’ transformations are set     1 2 2 2 1 2 * * 1 1 2 1 1 2 2 2 1 * * 1 1 p L p                                                                   , ( ) ( ) ( ) p p U W             y

The boundary functional corresponding to the boundary conditions (10) is written in the form

3 1        3 1 *

1 0 0 1

 

  

  U (1)

A

 

(1) 

(1)     y A y I y

 

I

,

,

 

0

 

*

In these notations the boundary value problem (9), (10) is derived   2 ( ) ( ), 1 , U (1) L         y f y γ

(11)

( )    

1 1 1 0 p aG p    

 

  

f

γ

( )

,

 

 

 

 

0

A general solution of the vector homogeneous equation in (11) relates to the solution of the matrix differential equation 2 ( ) 0, 1 L       Y (12) with the help of the auxiliary matrix

  

  

(1) 1

 ( )

H

0

   

H

( , )

(1) 0

 

H

0

( )

(1) ( ) m H z is the Hankel first order function,

0,1 m  , a relationship has been proven (Popov, 2013)

where





   

   

2

2

2

p

1 1

2

   



 

*

*

1



M

( )

 

2 ( , )         H H M , ( , ) ( ) L

(13)

2

2

2

p

2

1 1       *



*

1



( )  M has the form

The inverse matrix for