PSI - Issue 47

Yaroslav Dubyk et al. / Procedia Structural Integrity 47 (2023) 863–872 Yaroslav Dubyk et al./ Structural Integrity Procedia 00 (2023) 000 – 000

870 8

Fig. 5. S-S conical shell: α=20 0 , x

2 /x 1 =1,78, h/R=0,0041, R=123,7 mm, E=207 GPa, ρ=8200 kg/m

3 , µ=0,3; (a) with F

compress =86 942 N (b) with

P external =0.2 MPa & F compress =86 942 N. ( ○ ) m=1; ( □ ) m=2; (Δ) m = 3; (◊) m = 4 : Ansys results,

our results.

Analyzing fig. 4-5 results, our solution showed good convergence at small numbers n for pressure loading. However, good convergence is also seen at high frequencies under the compressive force tests. 4. Conclusions Formulas for describing the dynamic behavior of a conical shell according to Donnell-Mushtari theory were obtained. Such formulas did not have simplifications at the stage of problem setting, their solution was found in ordinary polynomials. Both classical and elastic boundary conditions are given for which it is possible to apply the obtained solution. When using such boundary conditions, it was necessary to modernize the Williams-Wittrick counter for its use in shells. Experimental values of natural frequencies for steel cones vibrations with different boundary conditions were used for testing. Also, numerical results obtained by FE code calculation are compared with the presented methodology. The results show the possibility of using proposed formulas in free vibrational analysis of conical and cylindrical shells with initial prestress. Appendix A. Differential equations system Conical shell can describe by a system of eight ordinary differential equations, which solved by polynomials:                2 2 1 sin sin x x x n x n dn x n x v x n w x Eh u x u x h dx x x tg x                    (A.1)

   

   

   

   

 

 

 





 

2 h n

2

x dn x 

x n x 

1

12

x n x

m x

2





h

x

1  

n























sin

dx

x

x



2 2 x tg h  

2

2 2 x tg h  

2

2 2 x tg h  

2

2 12

12

cos 12 

   3



   

   

2 sin 2 

2 n n 

2

 

v x Eh

 

2 2 2 x htg

12

v x

2





( ) u x Ehn

h

(A.2)

1





















2

2 2 x tg h  

2

sin

12

2 2 x tg h  

2

2

2

2 2 x tg h  

2

x



2 12

2 1 sin 12 x   



 

  w x Eh n 3





2 sin 2

2 2

2

n tg

 



 

3 x Eh n

( )

( ) sin



w x Ehn

x





  1 12

  1 12





  

2

x

tg  

2 3

2 2 x tg h  

2

2 2 x tg h  

2

2 sin x

cos

tg   

x



