14. Analytical periodic solution for solving nonlinear vibration equations

M. Bayat1, I. Pakar2, M. Bayat3

1Young Researchers and Elites Club, Science and Research Branch, Islamic Azad University, Tehran, Iran

2, 3Department of Civil Engineering, Mashhad Branch, Islamic Azad University, Mashhad, Iran

1Corresponding author

E-mail: 1mbayat14@yahoo.com, 2iman.pakar@yahoo.com, 3mahdi.bayat86@gmail.com

(Received 10 April 2013; accepted 7 June 2013)

Abstract: In this study a new kind of analytical methods called He’s variational approach method is applied to solve strong nonlinear vibration equations. The He’s variational approach method is very easy and contrary to the other conventional methods, only one iteration leads to high accuracy of the solutions for the whole range of initial amplitudes and does not demand small perturbation. Some examples are given to illustrate the effectiveness and convenience of the methodology. The Runge-Kutta’s (RK) algorithm was also implemented to achieve the numerical solutions for the examples. The results reveal that the variational approach method is very effective and simple. It is predicted that the VAM can find wide application in engineering problems, as indicated in following examples.

Keywords: variational approach method, nonlinear vibration, analytical methods.


[1]        Ganji D. D., Sadighi A. Application of He’s Homotopy-pertubation method to nonlinear coupled systems of reaction-diffusion equations. Inter. J. of Nonlinear Sci. and Num. Sim., Vol. 7, Issue 4, 2006, p. 411‑418.

[2]        Mehdipour I., Ganji D. D., Mozaffari M. Application of the energy balance method to nonlinear vibrating equations. Current Applied Physics, Vol. 10, Issue 1, 2010, p. 104‑112.

[3]        Bayat M., Pakar I., Domaiirry G. Recent developments of some asymptotic methods and their applications for nonlinear vibration equations in engineering problems: a review. Latin American Journal of Solids and Structures, Vol. 9, Issue 2, 2012, p. 145‑234.

[4]        Fu Y., Zhang J., Wan L. Application of the energy balance method to a nonlinear oscillator arising in the microelectromechanical system (MEMS). Current Applied Physics, Vol. 11, Issue 3, 2011, p. 482 485.

[5]        Bayat M., Pakar I. Application of He’s energy balance method for nonlinear vibration of thin circular sector cylinder. International Journal of the Physical Sciences, Vol. 6, Issue 23, 2011, p. 5564‑5570.

[6]        He J. H. An improved amplitude-frequency formulation for nonlinear oscillators. International Journal of Nonlinear Sciences and Numerical Simulation, Vol. 9, Issue 2, 2008, p. 211‑212.

[7]        Marinca V., Herisanu N. A. Modified iteration perturbation method for some nonlinear oscillation problems. Acta Mechanica, Vol. 184, Issue 1, 2006, p. 231‑242.

[8]        Shen Y. Y., Mo L. F. The max–min approach to a relativistic equation. Computers & Mathematics with Applications, Vol. 58, Issues 11‑12, 2009, p. 2131‑2133.

[9]        Zeng Dq, Lee Yy Analysis of strongly nonlinear oscillator using the max–min approach. Int. J. Nonlinear Sci. Numer. Simul., Vol. 10, 2009, p. 1361‑1368.

[10]     Bayat M., Barari A., Shahidi M. Dynamic response of axially loaded Euler-Bernoulli beams. Mechanika, Vol. 17, Issue 2, 2011, p. 172‑177.

[11]     Bayat M, Pakar I. On the approximate analytical solution to non-linear oscillation systems. Shock and Vibration, Vol. 20, Issue 1, 2013, p. 43‑52.

[12]     Pakar I., Bayat M. Analytical study on the non-linear vibration of Euler-Bernoulli beams. Journal of Vibroengineering, Vol. 14, Issue 1, 2012, p. 216‑224.

[13]     Shou D. H. Variational approach for nonlinear oscillators with discontinuities. Computers & Mathematics with Applications, Vol. 58, Issue 11-12, 2009, p. 2416‑2419.

[14]     He J. H. Variational approach for nonlinear oscillators. Chaos, Solitons and Fractals, Vol. 34, Issue 5, 2007, p. 1430‑1439.

[15]     Jun-Fang L. He’s variational approach for nonlinear oscillators with high nonlinearity. Computers & Mathematics with Applications, Vol. 58, Issues 11‑12, 2009, p. 2423‑2426.

[16]     Shu-Qiang W. A variational approach to nonlinear two-point boundary value problems. Computers & Mathematics with Applications, Vol. 58, Issues 11‑12, 2009, p. 2452‑245.

[17]     Zhang W., Qian Y., Yao M., Lai S. Periodic solutions of multi-degree-of-freedom strongly nonlinear coupled Van Der Pol oscillators by homotopy analysis method. Acta Mechanica, Vol. 217, Issues 3‑4, 2011, p. 269‑285.

[18]     Chen S. S., Chen C. K. Application of the differential transformation method to the free vibrations of strongly non-linear oscillators. Nonlinear Anal.: Real World Appl., Vol. 10, Issue 2, 2009, p. 881‑888.

[19]     Pakar I., Bayat M. An analytical study of nonlinear vibrations of buckled Euler-Bernoulli beams. Acta Physica Polonica A, Vol. 123, Issue 1, 2013, p. 48‑52.

[20]     Nayfeh A. H. Problems in Perturbation. Second Edition, Wiley, New York, 1993.

[21]     Bayat M., Pakar I., Bayat M. On the large amplitude free vibrations of axially loaded Euler-Bernoulli beams. Steel and Composite Structures, Vol. 14, Issue 1, 2013, p. 73‑83.

[22]     Chen Y., Liu J. Homotopy analysis method for limit cycle flutter of airfoils. Applied Mathematics and Computation, Vol. 203, Issue 2, 2008, p. 854‑863.

[23]     Wang S. Q., He J. H. Nonlinear oscillator with discontinuity by parameter-expansion method. Chaos, Solitons & Fractals, Vol. 35, Issue 4, 2008, p. 688‑691.

[24]     Bayat M., Pakar I. Accurate analytical solution for nonlinear free vibration of beams. Structural Engineering and Mechanics, Vol. 43, Issue 3, 2012, p. 337‑347.

Cite this article

Bayat M., Pakar I., Bayat M. Analytical periodic solution for solving nonlinear vibration equations. Journal of Measurements in Engineering, Vol. 1, Issue 2, 2013, p. 86‑94.


Journal of Measurements in Engineering. June 2013, Volume 1, Issue 2
© Vibroengineering. ISSN Print 2335-2124, ISSN Online 2424-4635, Kaunas, Lithuania