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Research Article
1. Introduction
In recent years, structural engineers have been gradually concerned with the analysis of cylindrical shells, which have non-circular
profiles and are found to be in many engineering applications, such as aerospace, mechanical, nuclear, petrochemical, modern passen-
ger airplanes, civil, and marine structures. The frequencies and mode shapes of the vibration of thin elastic shells essentially depend
on some determining functions such as the radius of the curvature of the neutral surface, the shell thickness, the shape of the shell
edges, and so forth. In simple cases when these functions are constant, the vibration deflection displacements occupy the entire shell
surface. If the determining functions vary from point to point of the neutral surface, then localization of the vibration modes lies near
the weakest lines on the shell surface, which has less stiffness. The kinds of these problems are found to be difficult, because the radius
of its curvature varies with the circumferential coordinate, closed-form or analytic solutions cannot be obtained, in general, for this
class of shells, numerical or approximate techniques are necessary for their analysis. Vibration problems in structural dynamics have
become more of problems in recent years because the use of high-strength material requires less material for load support structures,
and components have become generally more slender and are vibrate-prone. The vibration response of shells of revolution has been
studied by many researchers because the basic equations for this was established by Flügge [1], Love [2], and Rayleigh [3]. The best
collection of documents can be found in Leissa [4] in which more than 500 publications were analyzed and discussed in both linear
and non-linear vibration cases for circular cylindrical shells. Recently, other related references may be found in the well-known work of
Markus [5], Zhang et al. [6], Li [7], and Pellicano [8]. Some of researchers have considerable interest in the study of vibration behavior
of circular cylindrical shells with variable thickness such as [9–13] in which their investigations have been made into different forms
of thickness, that is, axial, circumferential, and step-wise thickness variation. A few researchers have devoted their studies for vibration
characteristics of non-uniform circular cylindrical shells with constant thickness such as [14–17] in which their investigations have been
made into different forms of radius of curvature, that is, oval, elliptical, and three-lobed and four-lobed cross sections. In contrast, the
vibration study of non-uniform cylindrical shells with variable thickness has received much less attention, but some of implementa-
tions are well documented by Suzuki and Leissa [18], Mitao et al. [19], and Khalifa [20] in which their treatments have been modeled
Department of Mathematics, Faculty of science, South Valley University, Qena, Egypt
*Correspondence to: Mousa Khalifa Ahmed, Department of Mathematics, Faculty of science, South Valley University, Qena, Egypt.
†E-mail: mousa@japan.com
Copyright © 2011 John Wiley & Sons, Ltd. Math. Meth. Appl. Sci. 2011,34 1789–1800
1789
Received 12 March 2011 Published online 8 August 2011 in Wiley Online Library
(wileyonlinelibrary.com) DOI: 10.1002/mma.1493
Simplified equations and solutions for the free
vibration of an orthotropic oval cylindrical
shell with variable thickness
Mousa Khalifa Ahmed*†
Communicated by W. Sprößig
Based on the framework of the Flügge’s shell theory, the transfer matrix approach and the Romberg integration method,
this paper presents the vibration behavior of an isotropic and orthotropic oval cylindrical shell with parabolically
varying thickness along its circumference. The governing equations of motion of the shell, which have variable coeffi-
cients are formulated and solved. The analysis is formulated to overcome the mathematical difficulties related to mode
coupling, which comes from variable curvature and thickness of shell. The vibration equations of the shell are reduced to
eight first-order differential equations in the circumferential coordinate and by using the transfer matrix of the shell, these
equations can be written in a matrix differential equation. The proposed model is adopted to get the vibration frequen-
cies and the corresponding mode shapes for the symmetrical and antisymmetrical modes of vibration. The sensitivity of
the frequency parameters and the bending deformations to the shell geometry, ovality parameter, thickness ratio, and
orthotropic parameters corresponding to different type modes of vibration is investigated. Copyright © 2011 John Wiley
&Sons,Ltd.
Keywords: vibration behavior; frequencies; transfer matrix approach; orthotropic oval shells; variable thickness; symmetric and
antisymmetric type-modes
