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Mathematical Problems in Engineering
Volume 2016, Article ID 4513520, 12 pages
Research Article

Free Vibration Analysis for Shells of Revolution Using an Exact Dynamic Stiffness Method

1School of Naval Architecture and Civil Engineering, Jiangsu University of Science and Technology, Zhangjiagang 215600, China
2Department of Civil Engineering, Tsinghua University, Beijing 100084, China

Received 2 May 2016; Accepted 3 July 2016

Academic Editor: Francesco Tornabene

Copyright © 2016 Xudong Chen and Kangsheng Ye. 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.


An exact generalised formulation for the free vibration of shells of revolution with general shaped meridians and arbitrary boundary conditions is introduced. Starting from the basic shell theories, the vibration governing equations are obtained in the Hamilton form, from which dynamic stiffness is computed using the ordinary differential equations solver COLSYS. Natural frequencies and modes are determined by employing the Wittrick-Williams (W-W) algorithm in conjunction with the recursive Newton’s method, thus expanding the applications of the abovementioned techniques from one-dimensional skeletal structures to two-dimensional shells of revolution. A solution for solving the number of clamped-end frequencies in the W-W algorithm is presented for both uniform and nonuniform shell segment members. Based on these theories, a FORTRAN program is written. Numerical examples on circular cylindrical shells, hyperboloidal cooling tower shells, and spherical shells are given, and error analysis is performed. The convergence of the proposed method on is verified, and comparisons with frequencies from existing literature show that the dynamic stiffness method is robust, reliable, and accurate.