A unified strong spectral Tchebychev solution for predicting the free vibration characteristics of cylindrical shells with stepped-thickness and internal–external stiffeners

Abstract

In this paper, a strong spectral Tchebychev (ST) solution is developed to investigate the free vibration behaviors of the cylindrical shell with stepped-thickness (CS-S) and cylindrical shell with internal–external stiffeners (CS-IES) subject to arbitrary boundary constraints. The spring parameter technique is used to impose combined connections and arbitrary boundaries. The potential and kinetic energy expressions of CS-S and CS-IES are derived based on the framework of the first-order shear deformation theory (FSDT). Finally, the unified governing differential equations of CS-S and CS-IES are acquired by the Hamilton’s variational principle. According to the Gauss–Lobatto point discretization process, the Tchebyshev polynomials are used to perform spectral expansion on the axial variables of admissible displacement functions. Circumferential variables are represented by sine and cosine series. In this way, the unified free vibration mathematical model of CS-S and CS-IES is obtained. On the basis of confirming that the model results have reached the state of convergence, the numerical results are compared with those calculated by the reference and the finite element method, which fully validates that the strong spectral Tchebychev model obtained in this paper is feasible for analyzing the free vibration characteristics of CS-S and CS-IES subject to arbitrary boundary constraints. Last but not least, the paper carries out the parametric analysis of how the change of the geometric parameters of the stiffeners will affect the natural frequencies of CS-IES.