Analysis of the Spatial Vibrations of Coaxial Cylindrical Shells Partially Filled with a Fluid

This paper is devoted to a numerical study of the natural vibrations of horizontally oriented elastic coaxial shells the annular gap between which is completely or partially filled with a compressible viscous fluid. The problem is solved in a three-dimensional formulation using the finite element method. The fluid motion is described in the acoustic approximation in terms of the velocity potential. The relevant equations together with the boundary conditions corresponding to complete contact on the wetted surfaces are transformed using the Bubnov-Galerkin method. The hydrodynamic forces are found from the viscous stress tensor. The mathematical formulation of the problem of thin-walled structure dynamics is based on the variational principle of virtual displacements that includes the normal and tangential components of the forces exerted by the fluid on the wetted parts of elastic bodies. The shells are modeled by assuming that their curvilinear surfaces are approximated quite accurately by a set of plane elements whose strains are determined according to the classical theory of thin plates. The results obtained have been validated by comparing them with the known published data for the case where the entire volume of the annular gap is filled with an ideal fluid. The influence of the fluid level and gap size on the natural frequencies and the corresponding vibration modes of coaxial shells with a variety of boundary conditions is estimated. It is demonstrated that partial filling leads to a splitting of the natural vibration frequencies, with a decrease in the fluid volume promoting the growth of their minimum values. It is shown that at some gap size mixed vibration modes can appear not only in the meridional direction, but also in the circumferential one.