In this paper, we consider the problem of stability of a thin-walled shell structure with two hemispherical bottoms of the same thickness, partially filled with liquid, which is immersed in an external liquid medium and is under hydrostatic pressure. The dynamic characteristics of such a structure containing a limited volume of liquid under internal pressure and hydrostatic pressure are obtained. The developed program for calculating the dynamic characteristics of axisymmetric shell structures containing liquid is based on the finite element method. The finite elements have an annular shape when rotated around the axis of symmetry. The program is implemented in Excel spreadsheet using Visual Basic for Applications (VBA). It allows to calculate the natural frequencies of thin-walled elastic structures interacting with an arbitrary number of liquids, considering the influence of the static stress-strain state caused by hydrostatic and internal pressure and other external forces that do not violate the axial symmetry of the problem. At a fixed value of the internal pressure, the calculation of the lowest natural frequencies of vibrations with different numbers of waves along the circumference is performed. By successive refinement, the critical thickness of the shell is determined, at which at least one of the natural frequencies reaches zero. The internal pressure p varies from 0 to 1 atm. in increments of 0,1 atm. and the calculations are repeated to obtain each critical value. At each pressure value, curves are plotted on the graph "number of waves — natural frequency". On the coordinate plane "pressure — shell thickness" the boundary of the instability region is constructed.
Пак Сонги, Григорьев В.Г. Моделирование динамической устойчивости тонкостенных конструкций, частично заполненных жидкостью, при гидростатическом воздействии. Математическое моделирование и численные методы, 2022, № 3, с. 3–17.