Table of Contents
International Journal of Computational Mathematics
Volume 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.

Abstract

The conservative Helmholtz-Duffing oscillator is analyzed by means of three analytical techniques. The max-min, second-order of the Hamiltonian, and the global error minimization approaches are applied to achieve natural frequencies. The obtained results are compared with the homotopy perturbation method and numerical solutions. The results show that second-order of the global error minimization method is very accurate, so it can be widely applicable in engineering problems.

1. Introduction

Mathematical modeling and frequency analysis of the nonlinear vibrational systems are an important and interesting field of mechanics. A lot of researchers have worked in this field and have proposed a lot of methods for demonstrating the dynamics responses of these systems [14]. They have developed this field of science and have analyzed the responses of the nonlinear vibration problems such as Duffing oscillators [510], nonlinear dynamics of a particle on a rotating parabola [11], nonlinear oscillators with discontinuity [12], oscillators with noninteger order nonlinear connection [13], the plasma physics equation [14], and van der Pol oscillator [15, 16]. The Helmholtz-Duffing equation is a nonlinear problem with the quadratic and cubic nonlinear terms. Surveying the literature shows that this equation has wide applications in the engineering problems. For example, due to different vibration behavior of functionally graded materials (FGMs) at positive and negative amplitudes, the governing equations of FGM beams, plates, and shells are conduced to a second-order nonlinear ordinary equation with quadratic and cubic nonlinear terms [1720]. Moreover, Sharabiani and Yazdi [21] obtained a Helmholtz-Duffing type equation within studying of nonlinear free vibrations of functionally graded nanobeams with surface effects. On the other hand, they revealed application of this equation in FG nanostructures.

In this paper, the frequency-amplitude relationship of the conservative Helmholtz-Duffing oscillator is obtained by means of the max-min [2226], Hamiltonian [2731], and global error minimization methods [3235]. The Hamiltonian approach is a kind of energy method and is proposed by He [27]. It is a simple method and can be used for the conservative nonlinear equations. Recently, it is applied for dynamic analysis of an electromechanical resonator [36]. Moreover, Akbarzade and Khan [37] employed the second-order Hamiltonian approach for nonlinear dynamic analysis of conservative coupled systems of mass-spring. The max-min approach is made on the base of Chengtian’s inequality [38]. It is a valuable method for obtaining the frequency responses of the nonlinear problems, and many researchers are attracted to use this method for studying the nonlinear systems. The global error minimization method is a modified type of variational approach and converts the nonlinear equation to an equivalent minimization problem.

The Helmholtz-Duffing equation is analyzed by many researchers. For instance, Leung and Guo [39] have applied the homotopy perturbation method (HPM) to this equation and have obtained accurate responses. Askari et al. [40] studied approximate periodic solutions of this equation using He’s energy balance method (HEBM) and He’s frequency-amplitude formulation (HFAF). Akbarzade et al. [41] used the first-order of the Hamiltonian approach and coupled homotopy-variational formulation to study the periodic solutions of the Helmholtz-Duffing oscillator; they also discussed the stability of the system for selected constant parameters. Recently, Li et al. [42] determined limit cycles and homoclinic orbits of this oscillator via a generalized harmonic function perturbation method. In the next sections, the max-min, the Hamiltonian, and the global error minimization methods are applied for evaluating the dynamics responses of the Helmholtz-Duffing equation. The results of these methods are compared with the exact ones and HPM solutions.

2. Solution Procedure

Let us consider the Helmholtz-Duffing equation The response oscillates between an asymmetric limit zone , where both and are positive. One can determine in the form of [39] where

2.1. Max-Min Approach (MMA)

We can rewrite (1) in the following form:

By choosing as a trial function that satisfied the initial conditions, the maximum and minimum values of can be calculated with Maple software as and , respectively, so we can write

By using Chengtian interpolation [38], we have where , are weighting factor, . Therefore the frequency can be approximated as Then the solution of (1) can be written:

By using the approximate solution, (4) can be written in the following form:

If by chance (8) is the exact solution of (1), then the right-hand side of (9) vanishes. According to [22], we set where . By substituting (8) into (10), we obtain the following expression for :

Substituting (11) into (7) and after some simplification, we have

2.2. Hamiltonian Approach
2.2.1. The First-Order Hamiltonian Approach (FHA)

The Hamiltonian of (1) is constructed as

Assume the first approximate solution of (1) as Then integrating (13) with respect to time from 0 to , we have Substituting (14) in (15) yields

Set

Finally, The result of first-order Hamiltonian and max-min approaches is alike.

2.2.2. The Second-Order Hamiltonian Approach (SHA)

For the second-order Hamiltonian approach, we consider the following equation as the response of the system: Whereas (19) must satisfy the initial condition we have Substituting (19) into (15) yields

Set Since finding a relation between different parameters from (22) is not easy, the value of other parameters is determined in Table 1 for several values of .

tab1
Table 1: Frequency-amplitude relationships for several values of .
2.3. Global Error Minimization Method
2.3.1. The First-Order Global Error Minimization Method (FGEM)

Based on standard procedure of modified variational approach, the minimization problem is For first-order approximation, consider a trial function as follows: Substituting (24) into (23) yields The solution of (25) could be found through By some simplifications, the following equation is obtained: One may find the first-order approximation by solving (27) as

2.3.2. The Second-Order Global Error Minimization Method (SGEM)

For the second-order approximation, consider a trial function as Whereas (29) must satisfy the initial condition, we have Substituting (29) into (23) yields

The solution of (31) could be found through which yield the following:

Finally, by simultaneously solving (30) and (33) unknown parameters are determined for different values of and as the second-order global error minimization method (SGEM) solution.

3. Results

The relative errors of the max-min, first- and second-orders of the Hamiltonian, and the global error minimization methods are shown in Tables 2, 3, and 4 for , , and , respectively. It is seen that the second-order global error minimization method results are closer to the exact ones than the other mentioned methods for large initial amplitudes.

tab2
Table 2: Comparison between MMA, FHA, SHA, FGEM, and SGEM results with exact ones and HPM responses ().
tab3
Table 3: Comparison between MMA, FHA, SHA, FGEM, and SGEM results with exact ones and HPM responses ().
tab4
Table 4: Comparison between MMA, FHA, SHA, FGEM, and SGEM results with exact ones ().

The differences of obtained responses and velocities with exact ones are plotted in Figures 1 and 2, respectively. It is found that the second-order global error minimization method illustrates a better accuracy than the aforementioned techniques.

939623.fig.001
Figure 1: Difference between obtained responses and the numerical solution (,  ).
939623.fig.002
Figure 2: Difference between obtained velocities and the numerical solution (,  ).

4. Conclusion

The Helmholtz-Duffing equation is investigated via the analytical approaches; the accuracy and validity of the obtained results have been examined by comparing to the exact ones and HPM solutions. The second-order of the global error minimization method achieved better approximate solutions for this equation. In present study, it is demonstrated that higher order of the modified variational approach is accurate and simple for solving asymmetric nonlinear conservative oscillatory systems.

Conflict of Interests

The authors declare that they have no competing interests.

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