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

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conical shell as a set of cylindrical shells with decreasing or increasing radius (because one of the unknowns is the radius) due to equations for a cylindrical shell (see fig. 2b). The corresponding expressions in Appendix A (A.1)-(A.8) should be supplemented by boundary conditions on the edge: 1 sin x w x M w k Q x         (15)

x x k M    

(16)

u x u k N  

(17)

M

x



v k N

  

(18)

v

x



xtg



, , , u v w k k k k 

Our solution can also take into account elastic boundary conditions. Therefore, these stiffness values

0 u v w k k k k      we obtain classical

are described as coefficients within a group of boundary conditions. If

boundary conditions. If a conical shell is represented a set of cylindrical shells, the boundary conditions will be the same as for the cylindrical shell.

Fig. 2. Cone as a set of cylinders (a) and cone (b), represented for solving using TM method.

We can use Wittrick- William’s algorithm to find the exact number of natural frequencies, especially it is important for spring stiffness boundary conditions. In this case, we have to go from transfer matrix shell representation to dynamic stiffness matrix. The same technique is showed in work Dubyk and Ishchenko (2021), but for calculations shell frequencies without initial stresses:     C t J J s     Κ (19) Here c J  is a number of natural frequencies for a clamped component and values of the main diagonal, after applying the Gauss elimination process for a stiffness matrix   t K  .     t s K  is a sign count of positive