Use of the Riesz method to calculate axisymmetric vibrations of composite shells of revolution supported by rings and filled with a liquid

• Published in: Mechanics of solids 2016-05, Vol.51 (3), p.357-369
• Main Authors: Grishanina, T. V., Shklyarchuk, F. N.
• Format: Article
• Language: English
• Subjects: Classical Mechanics; Physics; Physics and Astronomy
• ISSN: 0025-6544; 1934-7936
• DOI: 10.3103/S0025654416030134

We consider the axisymmetric vibrations of a composite structure shaped as a system of thin shells of revolution connected by rings and filled with an ideal incompressible liquid. The structure is divided into independent shell blocks and frame rings. According to the Riesz method, the displacements of each free block treated as a momentless shell are represented as a series in prescribed functions supplemented with local functions of the shell boundary bending. According to the method of variations in displacements, the axisymmetric vibrations of a liquid in an elastic shell of revolution are described by plane displacement and deplanation of the liquid cross-sections. The plane displacement of the liquid is integrally expressed in terms of the shell normal displacements, and the deplanation is represented as a series in prescribed functions of the axial coordinate. The potential and kinetic energies of the system are first written in terms of generalized coordinates of independent free shell and frame blocks filled with the liquid and with free surfaces at the ends. Then the kinematic conditions of conjugation of the shell edges with the frame and the liquid surfaces are used to eliminate a part of generalized coordinates. Moreover, the generalized coordinates representing the deplanation of the liquid cross-sections in the cavities are also eliminated as cyclic coordinates. As a result, the potential and kinetic energies of the systemare written in terms of the basic generalized coordinates of the composite structure as a whole. As an example, the natural axisymmetric vibrations are calculated for a tank filled with a liquid, which consists of a cylindrical shell, spherical bottom shell, and the frame connecting these shells. The Riesz method convergence is estimated by the number of prescribed functions, as well as the influence of the deplanation of the liquid cross-sections and the shape of the transverse cross-section of the frame.