Numerical simulation of natural vibration of a partially fluid-filled truncated conical shell in a gravity field

In this article, the behavior of circular truncated conical shells partially filled with an ideal fluid is studied taking into account the gravitational effects on its free surface. The mathematical formulation of the problem of the elastic body dynamics is developed using the variational principle of virtual displacements and the relations of the classical theory of shells, which are based on the Kirchhoff-Love hypotheses. The behavior of the compressible fluid is described by the equations of a potential theory, which together with the boundary conditions are converted to a weak form using the Bubnov-Galerkin method. The hydrodynamic pressure exerted by the fluid on the inner surface of the shell is calculated using the linearized Bernoulli equation. The problem is solved in the framework of axisymmetric formulation based on a semi-analytical version of the finite element method. The natural frequencies are calculated using the QR transform. The identification of coupled vibration modes in the full frequency spectrum is accomplished through an iterative dynamic condensation procedure based on the Muller method. The reliability of the results obtained within the framework of the developed numerical model is confirmed by comparison with known data using examples of elastic and rigid shells. The influence of the cone angle on the lower frequencies of sloshing and coupled vibration modes is analyzed for shells with different boundary conditions (rigidly clamped, simply supported and cantilevered shells), different linear dimensions and levels of filling of the shell with fluid. It has been demonstrated that the character of the change in lower frequencies is determined by the magnitude of the angle at the apex and is determined by its geometric parameters for coupled modes and is independent of them in the case of sloshing modes.