Mathematical modeling of free vibrations of a three-layer circular shell supported by longitudinal rigidity ribs.

The paper considers the construction of a mathematical model and the development of an algorithm for studying the free vibrations of a three-layer circular shell with a light transverse isotropic aggregate, supported by longitudinal rigidity ribs under load in the middle surface of the outer layers and ribs. The ribs are symmetrical about the middle surface of the three-layer circular shell. The rigidity of the ribs and the distance between them were considered the same. Kirchhoff-Lev hypotheses were accepted for the outer bearing layers, and a linear law of variation of tangential displacements in thickness was used for the aggregate and ribs, and the bending of the ribs in the vertical plane was taken into account. The system of differential equations of free vibrations of the reinforced three-layer shell, which is obtained on the basis of the Ostrogradsky-Hamilton variational principle, is simplified by introducing the functions F and ψ. The boundary conditions of the area of the shell closed between the ribs are set. With the help of the boundary transition, the conditions along the lines of the ribs are obtained, taking into account and without taking into account the shear deformations in the ribs, but without taking into account the torsional rigidity of the ribs. The equation of motion of a reinforced three-layer shell is obtained. The frequencies of free vibrations were studied and the values of the parameter of the first frequency of free vibrations for the circular shell supported by one and three longitudinal rigidity ribs depending on physical and mechanical properties of materials and geometric dimensions of the shell, curvature parameter and aggregate rigidity parameter were calculated. The tables for determining the coefficient of the first frequency of free vibrations ω for the shell supported by one and three rigidity ribs at δ_ω = 0.2 and different values of the shear parameter k_о and the rigidity parameter γ are given. [ABSTRACT FROM AUTHOR]