Skip Navigation Links
Journal of Vibration Testing and System Dynamics

C. Steve Suh (editor), Pawel Olejnik (editor),

Xianguo Tuo (editor)

Pawel Olejnik (editor)

Lodz University of Technology, Poland


C. Steve Suh (editor)

Texas A&M University, USA


Xiangguo Tuo (editor)

Sichuan University of Science and Engineering, China


Modified Harmonic Balance Method for Solving Nonlinear Free Vibration Problem of Beam Resting on Nonlinear Elastic Foundation

Journal of Vcibration Testing and System Dynamics 3(2) (2019) 133--145 | DOI:10.5890/JVTSD.2019.06.003

M. Saifur Rahman$^{1}$, Ahmed Hossain$^{2}$

$^{1}$ Department of Mathematics, Rajshahi University of Engineering & Technology, Rajshahi-6204, Bangladesh

$^{2}$ Department of Civil Engineering, Rajshahi University of Engineering & Technology, Rajshahi-6204, Bangladesh

Download Full Text PDF



Free vibration of beam is an important issue in structural engineering because of its various physical applications in the real field. In this paper, the free vibration of a nonlinear beam rest on a nonlinear elastic foundation is studied. A mathematical modeling for the free vibration of beam rest on nonlinear elastic foundation is presented. A modified harmonic balance is used to investigate the nonlinear free vibration response of beam. The main advantage of the method is that the unknown coefficients are expressed in power series of a new parameter. However, in a classical harmonic balance method the values of the unknown coefficients are determined by solving a set of complicated algebraic equations truncating higher order nonlinear terms. The results obtained by the proposed method have been compared with corresponding numerical results to verify the accuracy of method. Besides, the effects of various elastic foundation coefficients have been examined.


  1. [1]  Mustafa, O.Y., Murat, A., and S¨uleyman, A. (2014), An efficient analytical method for vibration analysis of a beam on elastic foundation with elastically restrained ends, Journal of Shock and Vibration, Article ID 159213, 7 pages.
  2. [2]  Wei, J., Cao, D., Yang, Y., and Huang, W. (2017), Nonlinear dynamical modeling and vibration responses of an L-shaped beam-mass structure, Journal of Applied Nonlinear Dynamics, 6(1), 91-104.
  3. [3]  Nayfeh, A.H. (1993), Introduction to Perturbation Techniques, John Wylie.
  4. [4]  Liao, S.J. (1992), The proposed homotopy analysis technique for the solution of nonlinear problems, Ph.D. Thesis, Shanghai Jiao Tong University.
  5. [5]  Sedighi, H M., Shirazi, K.H., and Zare, J. (2012), An analytic solution of transversal oscillation of quintic non-linear beam with homotopy analysis method, International Journal of Non-Linear Mechanics, 47, 777- 784.
  6. [6]  Moeenfard, H., Mojahedi, M., and Ahmadian, M.T. (2011), A homotopy perturbation analysis of nonlinear free vibration of Timoshenko micro beams, Journal of Mechanical Science and Technology, 25, 557-565.
  7. [7]  Younesian, D., Saadatnia, Z., and Askari, H. (2012), Analytical solutions for free oscillations of beams on nonlinear elastic foundations using the variational iteration method, Journal of Theoretical and Applied Mechanics, 50, 639-652.
  8. [8]  Plata, A.R.G. and de Oliveira, E.C. (2018), Variational iteration method in the fractional burgers equation, Journal of Applied Nonlinear Dynamics, 7(1), 189-196.
  9. [9]  Mickens, R.E. (1980), A generalization of the method of harmonic balance, Journal of Sound and Vibration, 111, 515-518.
  10. [10]  Sze, K.Y., Chen, S.H., and Huang, J.L. (2005), The incremental harmonic balance method for nonlinear vibration of axially moving beams, Journal of Sound and Vibration, 281, 611-626.
  11. [11]  Rahman, M.S., Haque, M.E. and Shanta, S.S. (2010), Harmonic balance solution of nonlinear differential equation (non-conservative), Journal of Advances in Vibration Engineering, 9, 345-356.
  12. [12]  Foda, M.A. (1995), Analysis of large amplitude free vibrations of beams using the KBM method, Journal of Engineering and Applied Science, Faculty of Eng., Cairo University, 42, 125-138.
  13. [13]  Coskun, I. and Engin, H. (1999), Non-linear vibrations of a beam on an elastic foundation, Journal of Sound and Vibration, 223(3), 335-354.
  14. [14]  Wickert, J.A. (1992), Non-linear vibration of a traveling tensioned beam, International Journal of Non-Linear Mechanics, 27, 503-517.
  15. [15]  Peng, J.S., Liu, Y., and Yang, J. (2010), A semi-analytical method for nonlinear vibration of Euler-Bernoulli beams with general boundary conditions, Mathematical Problems in Engineering, 2010: 1-17. Article ID 591786.
  16. [16]  Pirbodaghi, T., Ahmadianand, M.T., and Fesanghary, M. (2009), On the homotopy analysis method for non-linear vibration of beams, Mechanics Research Communications, 36, 143-148.
  17. [17]  Motallebi, A.A., Poorjamshidian, M., and sheikh, J. (2014), Vibration analysis of a nonlinear beam under axial force by homotopy analysis method, Journal of Solid Mechanics, 6, 289-298.
  18. [18]  Sedighi, H.M. and Daneshmand, F. (2014), Nonlinear transversely vibrating beams by the homotopy perturbation method with an auxiliary term, Journal of Applied and Computational Mechanics, 1, 1-9.
  19. [19]  Baghani, M., Jafari-Talookolaeiand, R.A., and Salarieh, H. (2011), Large amplitudes free vibrations and postbuckling analysis of un-symmetrically laminated composite beams on nonlinear elastic foundation, Applied Mathematical Modelling, 35, 130-138.
  20. [20]  Fallahand, A. and Aghdam, M.M. (2011), Nonlinear free vibration and post-buckling analysis of functionally graded beams on nonlinear elastic foundation, European Journal of Mechanics A/Solids, 30, 571-583.
  21. [21]  Yaghoobi, H. and Torabi, M. (2013), An Analytical Approach to Large Amplitude Vibration and Post- Buckling of Functionally Graded Beams Rest on Non-Linear Elastic Foundation, Journal of Theoretical and Applied Mechanics, 51(1), 39-52.
  22. [22]  Yaghoobi, H. and Torabi, M. (2013), Post-buckling and nonlinear free vibration analysis of geometrically imperfect functionally graded beams resting on nonlinear elastic foundation, Applied Mathematical Modelling, 37, 8324-8340.
  23. [23]  Kanani, A.S., Niknam, H., Ohadiand, A.R., and Aghdam, M.M. (2014), Effect of nonlinear elastic foundation on large amplitude free and forced vibration of functionally graded beam, Composite Structures, 115, 60-68.
  24. [24]  Azrar, L., Benamarand, R., and White, R.G. (2002), A semi-analytical approach to the nonlinear dynamic response problem of beams at large vibration amplitudes, Part II: multimode approach to the steady state forced periodic response, Journal of Sound and Vibration, 255(1), 1-41.
  25. [25]  Hasan, A.S.M.Z., Lee, Y.Y. and Leung, A.Y.T. (2014), The multi-level residue harmonic balance solutions of multi-mode nonlinearly vibrating beams on an elastic foundation, Journal of Vibration and Control, DOI: 10.1177/1077546314562225.
  26. [26]  Lee, Y.Y. (2015), Analytic Solution for Nonlinear Multimode Beam Vibration Using a Modified Harmonic Balance Approach and Vieta’s Substitution, Journal of Shock and Vibration, Article ID 3462643, 6 pages.
  27. [27]  Hasan, A.S.M.Z., Rahman, M.S., Lee, Y.Y., and Leung, A.Y.T. (2016), Multi-level residue harmonic balance solution for the nonlinear natural frequency of axially loaded beams with an internal hinge, Mechanics of Advanced Materials and Structures, DOI: 10.1080/15376494.2016.1227492.
  28. [28]  Rahman, M.S. and Lee, Y Y. (2017), New modified multi-level residue harmonic balance method for solving nonlinearly vibrating double-beam problem, Journal of Sound and Vibration, 406, 295-327.
  29. [29]  Mohamed, N., Eltaher, M.A., Mohamed, S.A. and Seddek, L.F. (2018), Numerical analysis of nonlinear free and forced vibrations of buckled curved beams resting on nonlinear elastic foundations, International Journal of Non-Linear Mechanics, 101, 157-173.
  30. [30]  Hosen, M.A., Rahman, M.S., Alam, M.S., and Amin, M.R. (2012), An analytical technique for solving a class of strongly nonlinear conservative systems, Applied Mathematics and Computation, 218, 5474-5486.