Volume 5, Issue 2, April 2017, Page: 70-77
Vibration Analysis of Non-uniform Beams Resting on Two Layer Elastic Foundations Under Axial and Transverse Load Using (GDQM)
Ramzy M. Abumandour, Basic Engineering Sciences Department, Faculty of Engineering, Menofia University, Menofia, Egypt
Islam M. Eldesoky, Basic Engineering Sciences Department, Faculty of Engineering, Menofia University, Menofia, Egypt
Mohamed A. Safan, Department of Civil Engineering, Faculty of Engineering, Menofia University, Menofia, Egypt
R. M. Rizk-Allah, Basic Engineering Sciences Department, Faculty of Engineering, Menofia University, Menofia, Egypt
Fathi A. Abdelmgeed, Department of Physics and Engineering Mathematics, Faculty of Engineering, Kafr El-Sheikh University, Kafr El-Sheikh, Egypt
Received: Jan. 1, 2017;       Accepted: Jan. 19, 2017;       Published: Mar. 4, 2017
DOI: 10.11648/j.ijmea.20170502.11      View  1804      Downloads  107
Abstract
The natural frequencies of non-uniform beams resting on two layer elastic foundations are numerically obtained using the Generalized Differential Quadrature (GDQ) method. The Differential Quadrature (DQ) method is a numerical approach effective for solving partial differential equations. A new combination of GDQM and Newton’s method is introduced to obtain the approximate solution of the governing differential equation. The GDQ procedure was used to convert the partial differential equations of non-uniform beam vibration problems into a discrete eigenvalues problem. We consider a homogeneous isotropic beam with various end conditions. The beam density, the beam inertia, the beam length, the linear (k1) and nonlinear (k2) Winkler (normal) parameters and the linear (k3) Pasternak (shear) foundation parameter are considered as parameters. The results for various types of boundary conditions were compared with the results obtained by exact solution in case of uniform beam supported on elastic support.
Keywords
Non-linear Elastic Foundation, Vibration Analysis, Non-uniform Beam, Mode Shapes and Natural Frequencies, GDQM and Newton’s Method
To cite this article
Ramzy M. Abumandour, Islam M. Eldesoky, Mohamed A. Safan, R. M. Rizk-Allah, Fathi A. Abdelmgeed, Vibration Analysis of Non-uniform Beams Resting on Two Layer Elastic Foundations Under Axial and Transverse Load Using (GDQM), International Journal of Mechanical Engineering and Applications. Vol. 5, No. 2, 2017, pp. 70-77. doi: 10.11648/j.ijmea.20170502.11
Copyright
Copyright © 2017 Authors retain the copyright of this article.
This article is an open access article distributed under the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/) which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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