Natural frequencies and mode shapes of variable thickness elastic cylindrical shells resting on a Pasternak foundation

Abstract

According to the framework of the Flügge’s shell theory, the Winkler and Pasternak foundations model, the transfer matrix approach and the Romberg integration method, the vibration behavior of an isotropic and orthotropic cylindrical shell with variable thickness is investigated. The governing equations of the shell based on the Pasternak foundation model are formulated and solved. The analysis is formulated to overcome the mathematical difficulties related to mode coupling which comes from the variable curvature and thickness of the shell. The vibration equations of the shell are reduced to eight first order differential equations in the circumferential coordinate. 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 the symmetrical and antisymmetrical modes of vibration. The sensitivity of the frequency parameters and the bending deformations to the Winkler and Pasternak foundations moduli, the thickness ratio, and the orthotropic parameters are demonstrated.

Full text

Article
Natural frequencies and mode shapes of
variable thickness elastic cylindrical shells
resting on a Pasternak foundation
Mousa Khalifa Ahmed
Abstract
According to the framework of the Flu¨gge’s shell theory, the Winkler and Pasternak foundations model, the transfer
matrix approach and the Romberg integration method, the vibration behavior of an isotropic and orthotropic cylindrical
shell with variable thickness is investigated. The governing equations of the shell based on the Pasternak foundation
model are formulated and solved. The analysis is formulated to overcome the mathematical difficulties related to mode
coupling which comes from the variable curvature and thickness of the shell. The vibration equations of the shell are
reduced to eight first order differential equations in the circumferential coordinate. 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 the symmetrical and antisymmetrical modes of vibration. The
sensitivity of the frequency parameters and the bending deformations to the Winkler and Pasternak foundations
moduli, the thickness ratio, and the orthotropic parameters are demonstrated.
Keywords
Orthotropic cylindrical shells, symmetric and antisymmetric modes, transfer matrix method, variable thickness, vibration
behavior, Winkler-Pasternak foundation
1. Introduction
Cylindrical shells which have variable thickness in con-
tact with elastic foundations are found in many engin-
eering applications, such as aerospace, mechanical, civil
and marine structures. The frequencies and mode
shapes of vibration 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, the elastic media, and so on. In simple cases
when these functions are constant, the vibration deflec-
tion 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 are
less stiff. In general, for this class of shells, numerical or
approximate techniques are necessary for their analysis.
In some practical applications, these shells are laid on a
soil medium as the foundation, and more attention paid
to the analysis of the shell’s behavior embedded in soil
simulated with two elastic parameters through the
Winkler-Pasternak model. Hereby, there are a few
researchers interested in the study of vibration behavior
of homogeneous, isotropic and orthotropic circular
cylindrical shells of uniform thickness under elastic
foundations (Paliwal and Bhalla, 1993a,b; Paliwal and
Srivastava, 1994; Paliwal et al., 1996; Paliwal and
Pandey, 1998; Gunawan et al., 2004, 2006; Golovko
et al., 2007) and their studies are conducted for the
vibration behavior of isotropic cylindrical shells on
the Winkler and Pastemak-type foundations. Whereas
Department of Mathematics, Faculty of Science at Qena, South Valley
University, Egypt
Corresponding author:
Mousa Khalifa Ahmed, Department of Mathematics, Faculty of Science at
Qena, South Valley University, Egypt.
Email: mossa@dr.com
Received: 9 December 2013; accepted: 31 January 2014
Journal of Vibration and Control
2016, Vol. 22(1) 37–50
!The Author(s) 2014
Reprints and permissions:
sagepub.co.uk/journalsPermissions.nav
DOI: 10.1177/1077546314528229
jvc.sagepub.com
by guest on December 31, 2015jvc.sagepub.comDownloaded from