No abstract

An approach is proposed to investigate an axisymmetric system consisting of an infinite thin elastic cylindrical shell that contains a potential flow of perfect compressible fluid and a periodically vibrating spherical inclusion. The approach emerged as part of a project devoted to developing methods to bring plugged oil wells back into production by the Vibration Theory Department of the S. P. Timoshenko Institute of Mechanics. This mathematical approach allows transforming the general solutions to equations of mathematical physics from one coordinate system to another to obtain an exact analytical solution (in the form of Fourier series) to interaction problems for systems of rigid and elastic bodies Keywords: Kirchhoff-Love hypotheses, wave equation, thin-walled elastic cylindrical shell, perfect compressible fluid, spherical inclusion, potential flowIntroduction. The behavior of structural members interacting with a compressible liquid or gas has recently become of great interest. Among the variety of structure-fluid interaction problems, there are problems that are not restricted to the interaction of various structures (bodies) with the fluid, but are also concerned with the interaction between these structures.The interaction of bodies of similar geometry in one medium or another has been studied most adequately because the fact that components of the general solution are described in the same coordinate system makes it much easier to find this solution. This allows using the summation theorems for special functions to express the general solution in the coordinate systems fixed to the bodies. Techniques of solving such problems are outlined, for example, in [1,10,14,19,23].A completely different situation arises with the general solution for interacting bodies of different geometries in one medium or another that is at rest or moving with a certain velocity. Since the components of the general solution are described in different coordinate systems, the summation theorems for special functions are not sufficient to express the general solution in one coordinate system. For this purpose, use is made of expressions that relate cylindrical and spherical wave functions. The validity of these relations was proved in [11,12]. Issues associated with the interaction of differently shaped bodies in a fluid were addressed, for example, in [13,16,18,20,22].The present paper studies an axisymmetric system consisting of an infinitely long, thin-walled, elastic, cylindrical shell that contains a potential flow of perfect compressible fluid and a vibrating spherical inclusion. Our analysis of the phenomena occurring in this system will be mainly based on the monographs and publications [1, 3, 5, 7, 8, 13, 20, etc.].To analyze the dynamic behavior of the hydroelastic system, we will simultaneously solve equations describing the motion of the fluid and the shell. These equations are taken from the linear theory of potential flows and the theory of thin elastic shells based on the Kirchhoff-Love hypotheses. The solution can...

No abstract

The paper studies the interaction of a harmonically oscillating spherical body and a thin elastic cylindrical shell filled with a perfect compressible fluid and immersed in an infinite elastic medium. The geometrical center of the sphere is located on the cylinder axis. The acoustic approximation, the theory of thin elastic shells based on the Kirchhoff-Love hypotheses, and the Lamé equations are used to model the motion of the fluid, shell, and medium, respectively. The solution method is based on the possibility of representing partial solutions of the Helmholtz equations written in cylindrical coordinates in terms of partial solutions written in spherical coordinates, and vice versa. Satisfying the boundary conditions at the shell-medium and shell-fluid interfaces and at the spherical surface produces an infinite system of algebraic equations with coefficients in the form of improper integrals of cylindrical functions. This system is solved by the reduction method. The behavior of the hydroelastic system is analyzed against the frequency of forced oscillations.Many applied problems necessitate analyzing the dynamic interaction of solids, both rigid and elastic, filled (partially or completely) with a fluid. Multiply connected problems of hydroelasticity were addressed by several authors. Shenderov [13] studied the diffraction of stationary acoustic waves. The authors of [2, 3] present the results on diffraction of waves in an elastic medium. These and other publications solved diffraction problems for similar surfaces: parallel cylinders or spherical (spheroidal) bodies.The papers [6, 7] derive a potential that describes the pulsation of a sphere in a compressible (acoustic approximation) or incompressible fluid filling a circular cylindrical cavity. It is assumed that the spherical source is on the cavity axis (axial symmetry). Olsson [17] solved a similar problem for a compressible fluid in an elastic cylindrical shell by the zero-field method. The papers [8][9][10][11]16] addressed the internal problem for a spherical body interacting with a cylinder (rigid and a thin elastic cylindrical shell) filled with an incompressible fluid. The authors of [15] solved the axisymmetric problem for a sphere oscillating inside a cylindrical shell filled with and immersed in a fluid. The authors of [14] constructed a potential describing the motion of a spherical segment in an incompressible fluid contained in a semiinfinite elastic shell and presented the solutions of specific problems.In the present paper, we will derive a potential describing the harmonic oscillations of a spherical body in a perfect compressible fluid. The fluid is contained in a thin elastic cylindrical shell of thickness h. The shell is immersed in an infinite elastic medium. The spherical source is located on the shell axis (axial symmetry).The analysis of the dynamic behavior of the hydroelastic system reduces to the simultaneous solution of the equations that model the motion of the fluid, shell, and medium, namely the acoustic approximation, the...