Natural vibrations of truncated conical shells of variable thickness

The paper presents the results of studying the natural frequencies of circular truncated conical shells, the thickness of which varies along the length according to different laws. The behavior of elastic structure is described in the framework of the classical theory of shells based on the Kirchhoff-Love hypotheses. The corresponding geometric and physical relations together with the equations of motion are reduced to a system of ordinary differential equations for new unknowns. The solution to the formulated boundary value problem is found using Godunov's orthogonal sweep method involving the numerical integration of differential equations by the fourth order Runge-Kutta method. The natural frequencies of vibrations are evaluated using a combination of a step-wise procedure and subsequent refinement by the interval bisection method. The reliability of the obtained results is verified by making a comparison with the known numerical-analytical solutions. The dependences of the minimum vibration frequencies obtained at shell thicknesses subject to a power-law variation (linear and quadratic, having symmetric and asymmetric shapes) and harmonic variation (with positive and negative curvature) are investigated for shells with different combinations of boundary conditions (free support, rigid and cantilever fastening), cone angles and linear dimensions. The results of the study confirm the existence of configurations, which provide a significant increase in the frequency spectrum in comparison with shells of constant thickness under the same limitations on the weight of the structure.