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Shock and Vibration
Volume 2016 (2016), Article ID 4097123, 30 pages
Research Article

A Unified Spectro-Geometric-Ritz Method for Vibration Analysis of Open and Closed Shells with Arbitrary Boundary Conditions

1College of Mechanical and Electrical Engineering, Harbin Engineering University, Harbin 150001, China
2College of Shipbuilding Engineering, Harbin Engineering University, Harbin 150001, China

Received 18 September 2015; Accepted 16 December 2015

Academic Editor: Tai Thai

Copyright © 2016 Dongyan Shi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


This paper presents free vibration analysis of open and closed shells with arbitrary boundary conditions using a spectro-geometric-Ritz method. In this method, regardless of the boundary conditions, each of the displacement components of open and closed shells is represented simultaneously as a standard Fourier cosine series and several auxiliary functions. The auxiliary functions are introduced to accelerate the convergence of the series expansion and eliminate all the relevant discontinuities with the displacement and its derivatives at the boundaries. The boundary conditions are modeled using the spring stiffness technique. All the expansion coefficients are treated equally and independently as the generalized coordinates and determined using Rayleigh-Ritz method. By using this method, a unified vibration analysis model for the open and closed shells with arbitrary boundary conditions can be established without the need of changing either the equations of motion or the expression of the displacement components. The reliability and accuracy of the proposed method are validated with the FEM results and those from the literature.