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.
| Published in | International Journal of Mechanical Engineering and Applications (Volume 5, Issue 2) |
| DOI | 10.11648/j.ijmea.20170502.11 |
| Page(s) | 70-77 |
| Creative Commons |
This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited. |
| Copyright |
Copyright © The Author(s), 2024. Published by Science Publishing Group |
Non-linear Elastic Foundation, Vibration Analysis, Non-uniform Beam, Mode Shapes and Natural Frequencies, GDQM and Newton’s Method
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APA Style
Ramzy M. Abumandour, Islam M. Eldesoky, Mohamed A. Safan, R. M. Rizk-Allah, Fathi A. Abdelmgeed. (2017). 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, 5(2), 70-77. https://doi.org/10.11648/j.ijmea.20170502.11
ACS Style
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). Int. J. Mech. Eng. Appl. 2017, 5(2), 70-77. doi: 10.11648/j.ijmea.20170502.11
AMA Style
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). Int J Mech Eng Appl. 2017;5(2):70-77. doi: 10.11648/j.ijmea.20170502.11
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Download@article{10.11648/j.ijmea.20170502.11,
author = {Ramzy M. Abumandour and Islam M. Eldesoky and Mohamed A. Safan and R. M. Rizk-Allah and Fathi A. Abdelmgeed},
title = {Vibration Analysis of Non-uniform Beams Resting on Two Layer Elastic Foundations Under Axial and Transverse Load Using (GDQM)},
journal = {International Journal of Mechanical Engineering and Applications},
volume = {5},
number = {2},
pages = {70-77},
doi = {10.11648/j.ijmea.20170502.11},
url = {https://doi.org/10.11648/j.ijmea.20170502.11},
eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijmea.20170502.11},
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.},
year = {2017}
}
TY - JOUR T1 - Vibration Analysis of Non-uniform Beams Resting on Two Layer Elastic Foundations Under Axial and Transverse Load Using (GDQM) AU - Ramzy M. Abumandour AU - Islam M. Eldesoky AU - Mohamed A. Safan AU - R. M. Rizk-Allah AU - Fathi A. Abdelmgeed Y1 - 2017/03/04 PY - 2017 N1 - https://doi.org/10.11648/j.ijmea.20170502.11 DO - 10.11648/j.ijmea.20170502.11 T2 - International Journal of Mechanical Engineering and Applications JF - International Journal of Mechanical Engineering and Applications JO - International Journal of Mechanical Engineering and Applications SP - 70 EP - 77 PB - Science Publishing Group SN - 2330-0248 UR - https://doi.org/10.11648/j.ijmea.20170502.11 AB - 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. VL - 5 IS - 2 ER -
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