Accurate periodic solution for non-linear vibration of dynamical equations

  • Pakar, Iman (Young Researchers and Elites Club, Mashhad Branch, Islamic Azad University) ;
  • Bayat, Mahmoud (Department of Civil Engineering, College of Engineering, Mashhad Branch, Islamic Azad University) ;
  • Bayat, Mahdi (Department of Civil Engineering, College of Engineering, Mashhad Branch, Islamic Azad University)
  • Received : 2014.03.30
  • Accepted : 2014.06.22
  • Published : 2014.07.31


In this paper we consider three different cases and we apply Variational Approach (VA) to solve the non-natural vibrations and oscillations. The method variational approach does not demand small perturbation and with only one iteration can lead to high accurate solution of the problem. Some patterns are presented for these three different cease to show the accuracy and effectiveness of the method. The results are compared with numerical solution using Runge-kutta's algorithm and another approximate method using energy balance method. It has been established that the variational approach can be an effective mathematical tool for solving conservative nonlinear dynamical equations.


  1. Bayat, M. and Pakar, I. (2011b), "Application of He's energy balance method for nonlinear vibration of thin circular sector cylinder", Int. J.Phy. Sci., 6(23), 5564-5570.
  2. Andrianov, I.V., Awrejcewicz, J. and Manevitch, L.I. (2004), Asymptotical Mechanics of thin -walled Structures, Springer - Verlag Berlin Heidelberg, Germany.
  3. Alicia, C., Jose L., H., Eulalia, M. and Juan, R.T. (2010), "Iterative methods for use with nonlinear discrete algebraic models", Math.Comput.Model., 52(7-8),1251-1257.
  4. Bayat, M. and Pakar, I. (2011a), "Nonlinear free vibration analysis of tapered beams by hamiltonianapproach", J.Vib. Eng., 13(4), 654-661.
  5. Bayat, M., Pakar, I. and Shahidi, M. (2011c), "Analysis of nonlinear vibration of coupled systems with cubic nonlinearity", Mechanika, 17(6), 620-629.
  6. Bayat, M. and Pakar, I. (2012 a), "Accurate analytical solution for nonlinear free vibration of beams", Struct. Eng. Mech., 43(3), 337-347.
  7. Bayat, M., Pakar, I. and Domaiirry, G. (2012b), "Recent developments of some asymptotic methods and their applications for nonlinear vibration equations in engineering problems: A review", Latin Am. J. Solid. Struct., 9(2), 145-34 .
  8. Bayat, M., Pakar, I. and Bayat, M. (2013a), "Analytical solution for nonlinear vibration of an eccentrically reinforced cylindrical shell", Steel Compos. Struct., 14(5), 511-521.
  9. Bayat, M. and Pakar, I. (2013b), "Nonlinear dynamics of two degree of freedom systems with linear and nonlinear stiffnesses", Earthq. Eng.Vib., 12(3), 411-420 .
  10. Bayat, M. and Pakar, I. (2013c), "On the approximate analytical solution to non-linear oscillation systems", Shock Vib., 20(1), 43-52.
  11. Bor-Lih,K. and Cheng-Ying,L. (2009), "Application of the differential transformation method to the solution of a damped system with high nonlinearity", Nonlinear Anal., 70(4),1732-1737.
  12. Bayat, M., Pakar, I. and Cveticanin, L. (2014a), "Nonlinear free vibration of systems with inertia and static type cubic nonlinearities : an analytical approach", Mech. Machine Theory ,77, 50-58
  13. Bayat, M, Pakar, I. and Cveticanin, L.(2014b), "Nonlinear vibration of stringer shell by means of extended Hamiltonian approach", Arch. Appl. Mech., 84(1),43-50.
  14. Bayat, M., Bayat, M. and Pakar, I. (2014c), "Nonlinear vibration of an electrostatically actuated microbeam", Latin Am. J. Solid. Struct., 11(3), 534-544.
  15. Dehghan, M. and Tatari, M. (2008), "Identifying an unknown function in a parabolic equation with over specified data via He's variational iteration method", Chaos. Soliton. Fract., 36(1),157-166.
  16. He, J.H. (2007), "Variational approach for nonlinear oscillators", Chaos. Soliton. Fract., 34(5), 1430-1439.
  17. He, J.H. (2008), "An improved amplitude-frequency formulation for nonlinear oscillators", Int. J. Nonlinear Sci., 9(2), 211-212.
  18. He, J.H. (2002), "Preliminary report on the energy balance for nonlinear oscillators", Mech. Res.Commun., 29(2), 107-111.
  19. Mehdipour, I., Ganji, D.D. and Mozaffari, M. (2010), "Application of the energy balance method to nonlinear vibrating equations", Current Appl. Phys., 10(1), 104-112.
  20. Nayfeh, A.H. (1981), "Introduction to perturbation techniques", John Wiley & Sons, New York.
  21. Nayfeh, A.H. and Mook, D.T. (1995), Nonlinear Oscillations, Wiley, New York.
  22. Pakar, I. and Bayat, M. (2012b), "Analytical study on the non-linear vibration of Euler-Bernoulli beams", J.Vib. Eng., 14(1), 216-224.
  23. Odibat, Z., Momani, S. and Suat Erturk, V. (2008), "Generalized differential transform method: application to differential equations of fractional order", Appl. Math. Comput., 197(2), 467-477.
  24. Pakar, I. and Bayat, M. (2011a), "Analytical solution for strongly nonlinear oscillation systems using energy balance method", Int. J.Phy. Sci., 6(22), 5166-5170.
  25. Pakar, I., Bayat, M. and Bayat, M. (2012a), "On the approximate analytical solution for parametrically excited nonlinear oscillators", J.Vib. Eng., 14(1), 423-429.
  26. Pakar, I. and Bayat, M. (2013a), "An analytical study of nonlinear vibrations of buckled Euler_Bernoulli beams", Acta. Phys. Polonica A., 123(1), 48-52.
  27. Pakar, I. and Bayat, M. (2013b), "Vibration analysis of high nonlinear oscillators using accurate approximate methods", Struct. Eng. Mech., 46(1), 137-151.
  28. Shaban, M., Ganji, D.D. and Alipour, A.A. (2010), "Nonlinear fluctuation, frequency and stability analyses in free vibration of circular sector oscillation systems", Current Appl. Phys., 10(5), 1267-1285.
  29. Shen, Y.Y. and Mo, L.F. (2009), "The max-min approach to a relativistic equation", Comput. Math. Appl., 58(11), 2131-2133.
  30. Wu, G. (2011), "Adomian decomposition method for non-smooth initial value problems", Math. Comput. Model., 54(9-10), 2104-2108.
  31. Xu, N. and Zhang, A. (2009), "Variationalapproachnext term to analyzing catalytic reactions in short monoliths", Comput. Math. Appl., 58(11-12),2460-2463.
  32. Xu, L. (2008), "Variational approach to solution of nonlinear dispersive K (m, n) equation", Chaos, Soliton. Fract., 37(1), 137-143.
  33. Zeng, D.Q. and Lee, Y.Y. (2009), "Analysis of strongly nonlinear oscillator using the max-min approach", Int. J. Nonlinear Sci., 10(10), 1361-1368.