PSI - Issue 33

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

517 9

j i

 

1

2

a



  

 





  n  

  



, ,   

sin

W

* p F





2

p



2

i 

G



1

n



j i

 

    

  

    



 



  n     2 sin 

  ,

  , dr g r   1 21

( ) dr e dp r     p

2 g r 21

(  

)

r



   

 

1

n



1



The case of steady-state oscillations is considered below. With this aim the substitution p i   , 2 2 p    was made ( p − Laplace transform parameter,  − circular frequency of steady -state oscillations).

2

a

a





  n      1 4 2 cos n G G    





  n  

  



, ;    

cos

;  

U

p

F



2 1

n



det

(20)

    

    



 

   n

  

   n

  

2

1

, ; r   

, ; r   

;

g

r dr g 

r dr



11

11

1

n



1



2

a

* 



  n     n       1 sin 2 sin n p G   g 





   2



, ;   

,  

W

F





n

det

    

    

 

   n

  

   n

  

2

1

, ; r   

, ; r   

;

r dr g 

r dr



21

21

1

n



1



                2 2 2 * 1 2 0 1 1 2 K K * 4 2



2

    0 2 1 1            2 1 2 1 1 K K 2 2 K K

det

   1     1 2 2 n F                   ;           (2) 2 2 1 2 0 1 1 2 * 0 2 1 1 ( ; ) 2 2 n F K K K K                1    2 2 2 * 1 * 2 1 1 ; 2 2  

2   2 1           * 2 * ;

2   1 2

;

1

  



  



  i i g r i   1 , ,

21 i n g

11 i n g

1, 2 i  are corresponding to functions

, ; ,

, ; r   ,

r  

Functions

, can be found

1, 2

2 2 p    .

in Appendix B with a replacement

6. Deriving and solving the integral equation. Since one boundary condition ( ,0, ) 0 U    remained unfulfilled, one requires its fulfilment, using expression (19). The integral equation is obtained in the form

   

( , ; ) ( )           ( ; ), 0 F r r dr f

(21)

n

1 0

n





[0,1) r  .

where unknown function ( ) r  

( , 0) p

U r

has been extended with zero in the interval

