Analysis of natural vibrations of truncated conical shells of variable thickness filled with fluid

The article presents the results of numerical studies of natural vibrations of truncated straight conical shells of revolution completely filled with an ideal compressible fluid. The shell thickness is not constant along the generatrix and changes according to various laws. The behavior of the elastic structure and liquid medium is described in the framework of the classical shell theory, which is based on the Kirchhoff–Love hypotheses and the Euler equations. The equations of shell motion together with the corresponding geometric and physical relations are reduced to a system of ordinary differential equations with respect to new unknowns. The acoustic wave equation written with respect to the hydrodynamic pressure is transformed to a system of differential equations using the method of generalized differential quadrature. The solution of the formulated boundary value problem is developed by the Godunov orthogonal sweep method and is reduced to the calculation of natural vibrational frequencies. To this end, a step-by step computational procedure is applied in combination with the subsequent refinement of the found values in the obtained range by the Muller method. The validity of the results obtained is verified by comparison with the known numerical solutions. For shells with different cone angles and combinations of boundary conditions (free support, rigid clamping and cantilevered support), the dependence of the lowest vibration frequencies obtained with a power (linear and quadratic, having symmetric and asymmetric forms) and harmonic (with positive and negative curvature) thickness change were investigated. The influence of boundary conditions on the possibility of the existence of configurations (cone angle, law of thickness variation, ratio of maximum or minimum cross-section thickness) that ensured an increase in the fundamental frequency compared to shells of constant thickness with restrictions on the weight of the structure was estimated.