Table of Contents
International Journal of Computational Mathematics
Volume 2014 (2014), Article ID 939623, 7 pages
http://dx.doi.org/10.1155/2014/939623
Research Article

Approximate Periodic Solution for the Nonlinear Helmholtz-Duffing Oscillator via Analytical Approaches

1School of Mechanical Engineering, Iran University of Science and Technology, Narmak, Tehran 16846, Iran
2Department of Mechanical Engineering, Amirkabir University of Technology, Hafez Avenue, Tehran 15914, Iran

Received 1 June 2014; Accepted 17 September 2014; Published 29 September 2014

Academic Editor: Anh-Huy Phan

Copyright © 2014 A. Mirzabeigy et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. A. H. Nayfeh and D. T. Mook, Nonlinear Oscillations, Wiley-Interscience, New York, NY, USA, 1979. View at MathSciNet
  2. S. J. Liao, Beyond Perturbation: Introduction to Homotopy Analysis Method, Chapman & Hall/CRC Press, Boca Raton, Fla, USA, 2003. View at MathSciNet
  3. R. E. Mickens, Truly Nonlinear Oscillations: Harmonic Balance, Parametric Expansions, Iteration, and Averaging Methods, World Scientific, Singapore, 2010. View at Publisher · View at Google Scholar · View at MathSciNet
  4. I. Kovacic and M. J. Brennan, The Duffing Equation: Nonlinear Oscillators and Their Behaviour, John Wiley & Sons, New York, NY, USA, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  5. T. Pirbodaghi, S. H. Hoseini, M. T. Ahmadian, and G. H. Farrahi, “Duffing equations with cubic and quintic nonlinearities,” Computers and Mathematics with Applications, vol. 57, no. 3, pp. 500–506, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  6. Y. Khan, M. Akbarzade, and A. Kargar, “Coupling of homotopy and the variational approach for a conservative oscillator with strong odd-nonlinearity,” Scientia Iranica, vol. 19, no. 3, pp. 417–422, 2012. View at Publisher · View at Google Scholar · View at Scopus
  7. H. Askari, Z. S. Nia, A. Yildirim, M. K. Yazdi, and Y. Khan, “Application of higher order hamiltonian approach to nonlinear vibrating systems,” Journal of Theoretical and Applied Mechanics, vol. 51, no. 2, pp. 287–296, 2013. View at Google Scholar · View at Scopus
  8. S. Nourazar and A. Mirzabeigy, “Approximate solution for nonlinear Duffing oscillator with damping effect using the modified differential transform method,” Scientia Iranica, vol. 20, no. 2, pp. 364–368, 2013. View at Publisher · View at Google Scholar · View at Scopus
  9. M. Akbarzade and A. Kargar, “Accurate analytical solutions to nonlinear oscillators by means of the Hamiltonian approach,” Mathematical Methods in the Applied Sciences, vol. 34, no. 17, pp. 2089–2094, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  10. Y. Khan and A. Mirzabeigy, “Improved accuracy of He's energy balance method for analysis of conservative nonlinear oscillator,” Neural Computing and Applications, vol. 25, no. 3-4, pp. 889–895, 2014. View at Publisher · View at Google Scholar · View at Scopus
  11. A. Mirzabeigy, M. K. Yazdi, and A. Yildirim, “Nonlinear dynamics of a particle on a rotating parabola via the analytic and semi-analytic approaches,” Journal of the Association of Arab Universities for Basic and Applied Sciences, vol. 13, no. 1, pp. 38–43, 2013. View at Publisher · View at Google Scholar · View at Scopus
  12. V. Marinca and N. Herisanu, “Optimal homotopy asymptotic approach to nonlinear oscillators with discontinuities,” Scientific Research and Essays, vol. 8, no. 4, pp. 161–167, 2013. View at Google Scholar
  13. L. Cveticanin, M. K. Kalamiyazdi, H. Askari, and Z. Saadatnia, “Vibration of a two-mass system with non-integer order nonlinear connection,” Mechanics Research Communications, vol. 43, pp. 22–28, 2012. View at Publisher · View at Google Scholar · View at Scopus
  14. L. Xu, “A Hamiltonian approach for a plasma physics problem,” Computers and Mathematics with Applications, vol. 61, no. 8, pp. 1909–1911, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  15. Y. Khan, M. Madani, A. Yildirim, M. A. Abdou, and N. Faraz, “A new approach to van der Pol's oscillator problem,” Zeitschrift fur Naturforschung Section, vol. 66, no. 10-11, pp. 620–624, 2011. View at Publisher · View at Google Scholar · View at Scopus
  16. Y. Khan, H. Vázquez-Leal, and N. Faraz, “An efficient new iterative method for oscillator differential equation,” Scientia Iranica, vol. 19, no. 6, pp. 1473–1477, 2012. View at Publisher · View at Google Scholar · View at Scopus
  17. L.-L. Ke, J. Yang, and S. Kitipornchai, “An analytical study on the nonlinear vibration of functionally graded beams,” Meccanica, vol. 45, no. 6, pp. 743–752, 2010. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  18. F. Alijani, F. Bakhtiari-Nejad, and M. Amabili, “Nonlinear vibrations of FGM rectangular plates in thermal environments,” Nonlinear Dynamics, vol. 66, no. 3, pp. 251–270, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  19. F. Alijani, M. Amabili, K. Karagiozis, and F. Bakhtiari-Nejad, “Nonlinear vibrations of functionally graded doubly curved shallow shells,” Journal of Sound and Vibration, vol. 330, no. 7, pp. 1432–1454, 2011. View at Publisher · View at Google Scholar · View at Scopus
  20. A. Fallah and M. M. Aghdam, “Nonlinear free vibration and post-buckling analysis of functionally graded beams on nonlinear elastic foundation,” European Journal of Mechanics—A/Solids, vol. 30, no. 4, pp. 571–583, 2011. View at Publisher · View at Google Scholar · View at Scopus
  21. P. A. Sharabiani and M. R. H. Yazdi, “Nonlinear free vibrations of functionally graded nanobeams with surface effects,” Composites Part B: Engineering, vol. 45, no. 1, pp. 581–586, 2013. View at Publisher · View at Google Scholar · View at Scopus
  22. J. H. He, “Max-min approach to nonlinear oscillators,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 9, no. 2, pp. 207–210, 2008. View at Google Scholar · View at Scopus
  23. M. K. Yazdi, H. Ahmadian, A. Mirzabeigy, and A. Yildirim, “Dynamic analysis of vibrating systems with nonlinearities,” Communications in Theoretical Physics, vol. 57, no. 2, pp. 183–187, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  24. R. Azami, D. D. Ganji, H. Babazadeh, A. G. Dvavodi, and S. S. Ganji, “He's max-min method for the relativistic oscillator and high order duffing equation,” International Journal of Modern Physics B, vol. 23, no. 32, pp. 5915–5927, 2009. View at Publisher · View at Google Scholar · View at Scopus
  25. H. M. Sedighi, K. H. Shirazi, and A. Noghrehabadi, “Application of recent powerful analytical approaches on the non-linear vibration of cantilever beams,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 13, no. 7-8, pp. 487–494, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  26. H. M. Sedighi, K. H. Shirazi, and M. A. Attarzadeh, “A study on the quintic nonlinear beam vibrations using asymptotic approximate approaches,” Acta Astronautica, vol. 91, pp. 245–250, 2013. View at Publisher · View at Google Scholar · View at Scopus
  27. J.-H. He, “Hamiltonian approach to nonlinear oscillators,” Physics Letters A, vol. 374, no. 23, pp. 2312–2314, 2010. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  28. Y. Khan, Q. Wu, H. Askari, Z. Saadatnia, and M. Kalami-Yazdi, “Nonlinear vibration analysis of a rigid rod on a circular surface via hamiltonian approach,” Mathematical and Computational Applications, vol. 15, no. 5, pp. 974–977, 2010. View at Google Scholar · View at Scopus
  29. S. Durmaz, S. A. Demirbag, and M. O. Kaya, “High order Hamiltonian approach to nonlinear oscillator,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 11, pp. 565–570, 2010. View at Google Scholar
  30. A. Yildirim, Z. Saadatnia, H. Askari, Y. Khan, and M. KalamiYazdi, “Higher order approximate periodic solutions for nonlinear oscillators with the Hamiltonian approach,” Applied Mathematics Letters, vol. 24, no. 12, pp. 2042–2051, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  31. L. Cveticanin, M. Kalami-Yazdi, Z. Saadatnia, and H. Askari, “Application of hamiltonian approach to the generalized nonlinear oscillator with fractional power,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 11, no. 12, pp. 997–1002, 2010. View at Google Scholar · View at Scopus
  32. Y. Farzaneh and A. Akbarzadeh Tootoonchi, “Global Error Minimization method for solving strongly nonlinear oscillator differential equations,” Computers & Mathematics with Applications, vol. 59, no. 8, pp. 2887–2895, 2010. View at Publisher · View at Google Scholar · View at Scopus
  33. A. Mirzabeigy, M. Kalami-Yazdi, and A. Yildirim, “Analytical approximations for a conservative nonlinear singular oscillator in plasma physics,” Journal of the Egyptian Mathematical Society, vol. 20, no. 3, pp. 163–166, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  34. A. Kargar and M. Akbarzade, “Analytical solution of nonlinear cubic-quintic duffing oscillator using global error minimization method,” Advanced Studies in Theoretical Physics, vol. 6, no. 9-12, pp. 467–471, 2012. View at Google Scholar · View at MathSciNet · View at Scopus
  35. M. K. Yazdi, A. Mirzabeigy, and H. Abdollahi, “Nonlinear oscillators with non-polynomial and discontinuous elastic restoring forces,” Nonlinear Science Letters A, vol. 3, pp. 48–53, 2012. View at Google Scholar
  36. Y. Khan and M. Akbarzade, “Dynamic analysis of nonlinear oscillator equation arising in double-sided driven clamped microbeam-based electromechanical resonator,” Zeitschrift für Naturforschung A, vol. 67, no. 8-9, pp. 435–440, 2012. View at Publisher · View at Google Scholar · View at Scopus
  37. M. Akbarzade and Y. Khan, “Nonlinear dynamic analysis of conservative coupled systems of mass-spring via the analytical approaches,” Arabian Journal for Science and Engineering, vol. 38, no. 1, pp. 155–162, 2013. View at Publisher · View at Google Scholar · View at Scopus
  38. J.-H. He, “He Chengtian's inequality and its applications,” Applied Mathematics and Computation, vol. 151, no. 3, pp. 887–891, 2004. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  39. A. Y. T. Leung and Z. Guo, “Homotopy perturbation for conservative Helmholtz-Duffing oscillators,” Journal of Sound and Vibration, vol. 325, no. 1-2, pp. 287–296, 2009. View at Publisher · View at Google Scholar · View at Scopus
  40. H. Askari, Z. Saadatnia, D. Younesian, A. Yildirim, and M. Kalami-Yazdi, “Approximate periodic solutions for the Helmholtz-Duffing equation,” Computers and Mathematics with Applications, vol. 62, no. 10, pp. 3894–3901, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  41. M. Akbarzade, Y. Khan, and A. Kargar, “Determination of periodic solution for the Helmholtz-Duffing oscillators by Hamiltonian approach and coupled homotopy-variational formulation,” International Journal of Physical Sciences, vol. 7, pp. 560–565, 2012. View at Google Scholar
  42. Z. Li, J. Tang, and P. Cai, “A generalized harmonic function perturbation method for determining limit cycles and homoclinic orbits of Helmholtz-Duffing oscillator,” Journal of Sound and Vibration, vol. 332, no. 21, pp. 5508–5522, 2013. View at Publisher · View at Google Scholar · View at Scopus