A new vibration approach of an elastic oval cylindrical shell with varying circumferential thickness

Abstract

A new vibration behavior is presented for an elastic oval cylindrical shell having circumferentially variable thickness with complex radius of curvature of an isotropic and orthotropic material. Based on the framework of the Flügge’s shell theory, the transfer matrix approach and the Romberg integration method, the governing equations of motion that have variable coefficients 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 frequencies and the corresponding mode shapes for symmetric and anti-symmetric 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 types of vibration modes of shells is investigated.

Full text

Article
A new vibration approach of an elastic
oval cylindrical shell with varying
circumferential thickness
Mousa Khalifa Ahmed
Abstract
A new vibration behavior is presented for an elastic oval cylindrical shell having circumferentially variable thickness with
complex radius of curvature of an isotropic and orthotropic material. Based on the framework of the Flu¨gge’s shell
theory, the transfer matrix approach and the Romberg integration method, the governing equations of motion that have
variable coefficients 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 frequencies and the corresponding mode shapes for symmetric and anti-symmetric 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 types of vibration modes of shells is investigated.
Keywords
Frequencies, orthotropic oval shells, symmetric and anti-symmetric type-modes, transfer matrix approach, variable
thickness, vibration behavior
Received: 1 January 2011; accepted: 9 March 2011
1. Introduction
In recent years, structural engineers have gradually
become concerned with the analysis of cylindrical
shells that have non-circular profiles and are found in
many engineering applications, such as aerospace,
mechanical engineering, nuclear, petrochemical,
modern passenger airplanes, civil and marine struc-
tures. The frequencies and mode shapes of vibration
of thin elastic shells essentially depend on some deter-
mining 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 dis-
placements occupy the entire shell surface. If the deter-
mining 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. These types of problems are
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; and numerical or approximate tech-
niques are necessary for their analysis. Vibration prob-
lems in structural dynamics have become more of a
problem in recent years because the use of high
strength material requires less material for load sup-
port-structures and components have become generally
more slender and vibration-prone. The vibration
response of shells of revolution has been studied by
many researchers since the basic equations for this
Faculty of Science at Qena, Department of Mathematics, South Valley
University, Egypt
Corresponding author:
Mousa Khalifa Ahmed, Faculty of Science at Qena, Department of
Mathematics, South Valley University, Egypt
Email: mousa@japan.com
Journal of Vibration and Control
18(1) 117–131
!The Author(s) 2011
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DOI: 10.1177/1077546311407234
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