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.

中文翻译：

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部分填充流体的同轴圆柱壳的空间振动分析

本文致力于对水平定向的弹性同轴壳的固有振动进行数值研究，壳之间的环形间隙完全或部分填充有可压缩的粘性流体。该问题使用有限元方法在三维公式中解决。流体运动是根据速度势在声学近似中描述的。使用Bubnov-Galerkin 方法转换相关方程以及对应于润湿表面上完全接触的边界条件。流体动力是从粘性应力张量中找到的。薄壁结构动力学问题的数学公式基于虚拟位移的变分原理，其中包括流体施加在弹性体润湿部分的力的法向分量和切向分量。壳是通过假设它们的曲线表面被一组平面元素非常精确地近似来建模的，这些平面元素的应变是根据薄板的经典理论确定的。获得的结果已经通过将它们与已知的公布数据进行比较来验证，其中环形间隙的整个体积都充满了理想的流体。估计了液位和间隙尺寸对具有各种边界条件的同轴壳的固有频率和相应振动模式的影响。结果表明，部分填充导致固有振动频率的分裂，流体体积的减少促进了它们的最小值的增长。结果表明，在某些间隙尺寸下，混合振动模式不仅可以出现在子午方向，还可以出现在圆周方向。