Dynamic interaction of an oscillating sphere and an elastic cylindrical shell filled with a fluid and immersed in an elastic medium

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...

Dynamic interaction of a spherical radiator in a fluid-filled cylindrical borehole within a poroelastic formation

“…Similarly, the reflected cylindrical wave in the fluid-filled cavity,ũ c , may simply be expressed (in the cylindrical coordinates) as (Kubenko and Dzyuba, 2004) u c ðR; z; xÞ ¼…”

“…In a series of papers, Kubenko and coworkers employed the axi-symmetric cylindrical to spherical wave transformations to study the acoustic field in a rigid cylindrical vessel excited by an internal finite spherical oscillator (Kubenko and Dzyuba, 2000), the dynamic interaction between an oscillating sphere and an encapsulating thin elastic cylindrical shell filled with (immersed in) a compressible liquid (Kubenko and Dzyuba, 2001;Dzyuba and Kubenko, 2002), the dynamic interaction of a semi-infinite elastic cylindrical shell with a liquid containing a vibrating spherical segment (Kubenko and Savin, 2002), and the dynamic interaction of an oscillating sphere and an encapsulating fluid-filled elastic cylindrical shell embedded in an elastic medium (Kubenko and Dzyuba, 2004). In a closely related problem, Linton (1995) used multipole expansions to solve the problem of the acoustic scattering of an arbitrary mode in a circular cylindrical waveguide by a sphere situated on its axis.…”

Nonaxisymmetric interaction of a spherical radiator in a fluid-filled permeable borehole

“…He considered both hard and soft boundary conditions for both the guide and the sphere. In a series of papers, Kubenko and coworkers employed the axisymmetric cylindrical to spherical wave transformations to study the acoustic field in a rigid cylindrical vessel excited by an internal finite spherical oscillator (Kubenko and Dzyuba, 2000a), the dynamic interaction between an oscillating sphere and an encapsulating thin elastic cylindrical shell filled with (immersed in) a compressible liquid (Kubenko and Dzyuba, 2001;Dzyuba and Kubenko, 2002), the dynamic interaction of a semi-infinite elastic cylindrical shell with a liquid containing a vibrating spherical segment (Kubenko and Savin, 2002), and the dynamic interaction of an oscillating sphere and an encapsulating fluid-filled elastic cylindrical shell embedded in an elastic medium (Kubenko and Dzyuba, 2004). In a closely related problem, Lee (2003) used Fourier transforms to solve the axisymmetric wave equation associated with scattering of torsional elastic waves by a spherical cavity located symmetrically in an infinitely long circular cylinder.…”

Dynamic interaction of an eccentric multipole cylindrical radiator suspended in a fluid-filled borehole within a poroelastic formation

“…In contrast to the numerous research works which treat a point source in an acoustic borehole, there are very few theoretical analyses relating a finite size (spherical or cylindrical) source in a cylindrical cavity (waveguide). In a series of papers, Kubenko and coworkers employed the axisymmetric cylindrical to spherical wave transformations to study the acoustic field in a rigid cylindrical vessel excited by an internal finite spherical oscillator [6], the dynamic interaction between an oscillating sphere and an encapsulating thin elastic cylindrical shell filled with (immersed in) a compressible liquid [7,8], the dynamic interaction of a semi-infinite elastic cylindrical shell with liquid containing a vibrating spherical segment [9], and the dynamic interaction of an oscillating sphere and an encapsulating fluid-filled elastic cylindrical shell embedded in an elastic medium [10]. Considering a uniform circular cylinder of unlimited length suspended in a fluid-filled cylindrical cavity as an idealized model of an acoustic logging tool [11], Poterasu [12] investigated dynamic coupling effects for a pulsating source in a fluid-filled cavity embedded within an (ideal) elastic infinite media by the boundary element method.…”