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 according to different laws. The behavior of the 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 variables. The solution to the formulated boundary value problem is found using Godunov orthogonal sweep method involving the numerical integration of differential equations by the Runge-Kutta fourth order 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 results is verified by 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, with symmetric and asymmetric shapes) and harmonic variation (with positive and negative curvature) are investigated for shells with different combinations of boundary conditions (simply supported, rigidly clamped, and cantilevered support), cone angles and linear sizes. The results of the study confirm the existence of configurations that provide a significant increase in the frequency spectrum compared to shells of constant thickness under the same limitations on the structure weight.