| Peer-Reviewed

Vibration Analysis of Non-uniform Beams Resting on Two Layer Elastic Foundations Under Axial and Transverse Load Using (GDQM)

Received: 1 January 2017     Accepted: 19 January 2017     Published: 4 March 2017
Views:       Downloads:
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.

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), 2017. Published by Science Publishing Group

Keywords

Non-linear Elastic Foundation, Vibration Analysis, Non-uniform Beam, Mode Shapes and Natural Frequencies, GDQM and Newton’s Method

References
[1] Lai Y. C., Ting B. Y., Lee W. S., Becker WR. Dynamic response of beams on elastic foundation. J. Struct. Eng. ASCE 1992; 118: 853-58.
[2] Thambiratnam D., Zhuge Y., Free vibration analysis of beams on elastic foundations. Comput Struct 1996; 60 (6): 971-80.
[3] Ayvaz Y., Ozgan K., Application of modified Vlasov model to free vibration analysis of beams resting on elastic foundations. J Sound Vib 2002; 255 (1): 111-27.
[4] Irie T., Yamada G., Takahashi I., Vibration and stability of a nonuniform Timoshenko beam subjected to a flower force. Sound Vib 1980; 70: 503-12.
[5] Gutierrez R. H., Laura P. A. A., Rossi R. E., Fundamental frequency of vibration of a Timoshenko beam of nonuniform thickness. J Sound Vib 1991; 145: 241-5.
[6] Chen C. N., Vibration of prismatic beam on an elastic foundation by the differential quadrature element method. Comput Struct 2000; 77: 1-9.
[7] Chen C. N., DQEM Vibration analysis of non-prismatic shear deformable beams resting on elastic foundations. J Sound Vib 2002; 255 (5): 989-99.
[8] Malekzadeh P., Karami G., Farid M., DQEM for free vibration analysis of Timoshenko beams on elastic foundations. Comput Mech 2003; 31: 219-28.
[9] Karami G., Malekzadeh P., Shahpari S. A. A., DQEM for free vibration of shear deformable nonuniform beams with general boundary conditions. Eng Struct 2003; 25: 1169-78.
[10] Chen WQ, Lu C. F., Bain Z. G., A mixed method for bending and free vibration of beams resting on Pasternak elastic foundations. Appl Math Model 2004; 28: 877-90.
[11] Pellicano F., Mastroddi F., Nonlinear dynamics of a beam on elastic foundation. Nonlinear Dyn 1988; 14: 335-55.
[12] Qaisi M. I., Nonlinear normal modes of a continuous system. J Sound Vib 1998; 209 (4): 561-69.
[13] Nayfeh A. H., Nayfeh S. A., On nonlinear modes of continuous systems. Vib Acous ASME 1994; 116: 129-36.
[14] Maccari A., The asymptotic perturbation method for nonlinear continuous systems. Nonlinear Dyn 1999; 19: 1-18.
[15] Coskun I., Engin H., Nonlinear vibrations of a beam on an elastic foundation. J Sound Vib 1999; 223 (3): 335-54.
[16] Balkaya M., Kaya M. O., and Sağlamer A., Analysis of the vibration of an elastic beam supported on elastic soil using the differential transform method, Archive of Applied Mechanics, 2009; 79 (2):135-146.
[17] B. Ozturk, S. B. Coskun, The Homotopy Perturbation Method for free vibration analysis of beam on elastic foundation, Structural Engineering and Mechanics, 2011; 37 (4): 415-425.
[18] Avramidis I. E., Morfidis K., Bending of beams on three-parameter elastic foundation, International Journal of Solids and Structures, 2006; 43: 357–375.
[19] Abrate, S., Vibrations of Non-Uniform Rods and Beams, Journal of Sound and Vibration, 1995; 185: 703-716.
[20] Gottlieb, H. P. W., Comments on Vibrations of Non- Uniform Beams and Rods, Journal of Sound and Vibration, 1996; 195: 139-141.
[21] Naguleswaran, S., Comments on Vibration of Non- Uniform Beams and Rods, Journal of Sound and Vibration, 1996; 195: 331-337.
[22] Hodges, D. H., Chung, Y. Y. and Shang, X. Y., Discrete Transfer Matrix Method for Non-Uniform Rotating Beams, Journal of Sound and Vibration, 1994; 169: 276-283.
[23] Sharma, S. P. and DasGupta, S., Bending Problem of Axially Constrained Beams on Nonlinear Elastic Foundations, International Journal of Solid and Structures, 1975; 11: 853-859.
[24] Beaufait, F. W. and Hoadley, P. W., Analysis of Elastic Beams on Nonlinear Foundation,” Computers & Structures, 1980; 12: 669-676.
[25] Kuo, Y. H. and Lee, S. Y., “Deflection of Nonuniform Beams Resting on a Nonlinear Elastic Foundation,” Computers & Structures, 1994; 51: 513-519.
[26] Chen, C. N., Solution of Beam on Elastic Foundation by DEQM, Journal of Engineering Mechanics, 1998; 124: 1381-1384.
[27] Bagheri S., Nikkar A., Ghaffarzadeh H., Study of nonlinear vibration of Euler-Bernoulli beams by using analytical approximate techniques. Latin American Journal of Solids and Structures, 2014; 11: 157-168.
[28] Nikkar A., Bagheri S., Saravi M., Study of nonlinear vibration of Euler-Bernoulli beams by using analytical approximate techniques. Latin American Journal of Solids and Structures, 2014; 11: 320-329.
[29] Li, L., Zhang, D., Dynamic analysis of rotating axially FG tapered beams based on a new rigid–flexible coupled dynamic model using the B-spline method. Composite Structures, 2015; 124: 357-367.
[30] Ramzy M. Abumandour, Islam M. Eldesoky, Mohamed A. Safan, Rizk-Allah R. M. and Fathi A. Abdelmgeed, Deflection of Non-Uniform Beams Resting on a Non-linear Elastic Foundation using (GDQM), International Conference on Materials and Structural Integrity (ICMSI2016), Paris, France, June 2-4, 2016.
[31] Ramzy M. Abumandour, Kamel M. H. and Bichir S., Application of the GDQ Method to Structural Analysis, International Journal of Mathematics and Computational Science, 2016; 2 (1): 8–19.
[32] Ramzy M. Abumandour, Kamel M. H. and Nassar M. M., Application of the GDQ method to vibration analysis, International Journal of Mathematics and Computational Science, 2015; 1 (5): 242-249.
[33] Chang. S., Differential Quadrature and its application in engineering,” Springer, (2000).
[34] Qiang Guo and Zhong, H., Non-linear Vibration Analysis of Beams by a Spline-based differential quadrature. Journal of Sound and Vibration, 2004; 269: 405–432.
[35] Blevins, R. D., Formulas for Natural Frequency and Mode Shapes. Malabur, Florida. Robert E. Krieger, 1984.
Cite This Article
  • 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

    Copy | Download

    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

    Copy | Download

    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

    Copy | 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}
    }
    

    Copy | Download

  • 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  - 

    Copy | Download

Author Information
  • Basic Engineering Sciences Department, Faculty of Engineering, Menofia University, Menofia, Egypt

  • Basic Engineering Sciences Department, Faculty of Engineering, Menofia University, Menofia, Egypt

  • Department of Civil Engineering, Faculty of Engineering, Menofia University, Menofia, Egypt

  • Basic Engineering Sciences Department, Faculty of Engineering, Menofia University, Menofia, Egypt

  • Department of Physics and Engineering Mathematics, Faculty of Engineering, Kafr El-Sheikh University, Kafr El-Sheikh, Egypt

  • Sections