Shock and Vibration

Volume 2016 (2016), Article ID 9748135, 14 pages

http://dx.doi.org/10.1155/2016/9748135

## Comparison Study on the Exact Dynamic Stiffness Method for Free Vibration of Thin and Moderately Thick Circular Cylindrical Shells

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

Received 3 September 2016; Revised 21 November 2016; Accepted 24 November 2016

Academic Editor: Marcello Vanali

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.

#### Abstract

Comparison study on free vibration of circular cylindrical shells between thin and moderately thick shell theories when using the exact dynamic stiffness method (DSM) formulation is presented. Firstly, both the thin and moderately thick dynamic stiffness formulations are examined. Based on the strain and kinetic energy, the vibration governing equations are expressed in the Hamilton form for both thin and moderately thick circular cylindrical shells. The dynamic stiffness is assembled in a similar way as that in classic skeletal theory. With the employment of the Wittrick-Williams algorithm, natural frequencies of circular cylindrical shells can be obtained. A FORTRAN code is written and used to compute the modal characteristics. Numerical examples are presented, verifying the proposed computational framework. Since the DSM is an exact approach, the advantages of high accuracy, no-missing frequencies, and good adaptability to various geometries and boundary conditions are demonstrated. Comprehensive parametric studies on the thickness to radius ratio () and the length to radius ratio () are performed. Applicable ranges of are found for both thin and moderately thick DSM formulations, and influences of on frequencies are also investigated. The following conclusions are reached: frequencies of moderately thick shells can be considered as alternatives to those of thin shells with high accuracy where is small and is large, without any observation of shear locking.

#### 1. Introduction

Free vibration analysis of shell structures is important for further dynamic analysis in engineering. Among all shell types, circular cylindrical shells are the most widely used in a good variety of applications, for example, tubes, containment vessels, and aircraft fuselages. Therefore, exact modal characteristic analysis of circular cylindrical shells is of considerable scientific and engineering significance.

It is well known that shell theories can be categorised into thin, moderately thick, and thick forms. Since the Kirchhoff-Love hypothesis for thin elastic shell theory was established [1], much effort was spent on its vibration development. Arnold and Warburton [2] analysed the flexural vibration of thin cylindrical shells with general boundary conditions using Rayleigh’s principle. Leissa [3] provided a systematic review and summary on the vibration of different shell theories and shell types. Bardell et al. [4] investigated the free vibration of isotropic open cylindrical shell panels by applying an version finite element method. Tan [5] introduced a substructuring method for predicting the natural frequencies of circular cylindrical shells with arbitrary boundary conditions.

On the other hand, moderately thick shell theory which derives from the Reissner-Mindlin hypothesis is less frequently used in analysing the free vibration behaviour of shells due to its complexity. Naghdi and Cooper [6] conducted vibration analysis on circular cylindrical shells with the consideration of shear deformation and rotary inertia by employing elastic wave method. Mirsky and Herrmann [7] studied the nonaxisymmetric vibration of moderately thick circular cylindrical shells. Tornabene and Viola [8] proposed an approach to analyse free vibration of parabolic shells using Generalised Differential Quadrature method (GDQ) based on the first-order shear deformation theory. Mantari et al. [9] applied a new accurate higher-order shear deformation theory to perform free vibration analysis of multilayered shells.

Although the free vibration of shells has been extensively examined, there are continuing research efforts made with new methods and shell types, for example, -version mixed finite element by Kim et al. [10]; GDQ method for doubly curved shells of revolution by Tornabene et al. [11, 12]; higher-order theory for thin-walled beams by Pagani et al. [13]; and three-dimensional analysis method by Ye et al. [14, 15].

The dynamic stiffness method (DSM) is an exact method since the dynamic stiffness is computed directly from the exact vibration governing equations. It is suitable for computational analysis of free vibration of continuous systems with infinite degrees of freedom. In conjunction with the Wittrick-Williams (W-W) algorithm [16, 17] and the recursive Newton’s method [18, 19], the DSM has been employed in free vibration of skeletal structures, for example, nonuniform Timoshenko beams [20], three-layered sandwich beams [21], and functionally graded beams [22]. Pagani et al. [23] employed the exact dynamic stiffness elements to investigate the free vibration of thin-walled structures. A recent paper [24] applied the DSM to the free vibration analysis of thin circular cylindrical shells. The present authors used to employ the DSM to investigate the free vibration of shells of revolution [25], and the complete theory was given in [26] based on the thin shell assumption. Further extension of the DSM to moderately thick circular cylindrical shells [27] was made with the consideration of transverse shear deformation and rotary inertia.

As an exact method, the dynamic stiffness method has the following advantages:(1)The method is exact and theoretically solutions can be obtained with any desired accuracy.(2)The original eigenvalue problems of two-dimensional partial differential equations (PDEs) are degraded to the eigenvalue problems of a set of one-dimensional ordinary differential equations (ODEs) by introducing the circumferential vibration modes. Since the method is exact, frequencies and vibration modes of any order can be calculated with a small number of elements.(3)The Wittrick-William algorithm is mathematically strict and ensures that no eigenvalue will be omitted.(4)The DSM formulation is general and versatile, making it capable of solving the free vibration of circular cylindrical shells with a variety of geometries and arbitrary boundary conditions.Though the free vibration formulation using the DSM has been established for both thin and moderately thick circular cylindrical shells, there is no comprehensive comparison study available currently to address the applicability of DSM formulations between the two different shell theories. This paper attempts to show the compatibility on the free vibration of circular cylindrical shells between thin and moderately thick DSM formulations. The DSM formulations of both thin and moderately thick circular cylindrical shells are well compared in Sections 2 and 3. Afterwards, the Wittrick-Williams algorithm is presented, providing a solution to computing the number of clamped-end natural frequencies . Four numerical examples are given in Section 5, demonstrating the capability and reliability of the proposed method. Extensive parametric studies are performed in Section 6 on the ratios of thickness to radius () and length to radius () to find out the applicable ranges of each theory. Differences between frequencies from thin and moderately thick DSM formulations are analysed. The following conclusions are reached in the final: the compatibility of DSM between thin and moderately thick formulations is proven without shear locking.

#### 2. Vibration Equations

This section theoretically presents a comparison study on DSM formulations for free vibration of both thin and moderately thick circular cylindrical shells. Figure 1 shows the coordinate system of a homogeneous, isotropic, and circumferentially closed circular cylindrical shell. The axial and circumferential directions of the shell are termed as and , respectively. The length of the shell is , radius is , and the thickness is . Young’s modulus is , Poisson’s ratio is , and the density is .