Analytical analysis of traveling wave vibration for rotating bi-directional dual-functionally graded carbon nanotube reinforced composite stepped thin cylindrical shells

In this paper, considering both carbon nanotubes and the matrix are distributed axially and radially, the traveling wave vibration for the bi-directional dual-functionally graded carbon nanotube reinforced composite stepped thin cylindrical shells is researched. To begin with, the effective material properties of stepped shells are obtained by the improved law of mixtures. In the meantime, the energy expressions of the shell are derived by considering the centrifugal and the Coriolis effect generated from rotation on the basis of the first-order shear deformation theory. Through varying the stiffness value of the artificial spring, the boundary constraints on the shell and the connections between each segment are simulated in the actual working conditions. Furthermore, the Gegenbauer-Ritz method, a novel analytical approach derived from the conventional Ritz method, is chosen to analyze this model and its validity is verified through comparisons with the existing literature and the finite element method. Ultimately, the influences of the carbon nanotubes distribution types, rotating speed, and other parameters on traveling wave vibration of the rotating stepped shells under classical and elastic boundary conditions are discussed in axial and radial directions, respectively. It is shown that gradually increasing the content of carbon nanotubes towards the weak areas of the components can effectively enhance their stiffness to maximize service life. Meanwhile, when the length-to-radius ratio is large, the axial type is superior to the radial type for the reinforcement of the overall stiffness of the shell.