Application of Homotopy Perturbation Method for Harmonically Forced Duffing Systems

Abstract:

In this paper, He’s homotopy perturbation method (HPM) is applied to solve harmonically forced Duffing systems. Non-resonance of an undamped Duffing system and the primary resonance of a damped Duffing system are studied. In the former case, the first-order analytical approximations to the system’s natural frequency and periodic solution are derived by HPM, which agree well with the numerical solutions. In the latter case, based on HPM, the first-order approximate solution and the frequency-amplitude curves of the system are acquired. The results reveal that HPM is an effective technique to the forced Duffing systems.

• Periodical: Applied Mechanics and Materials (Volumes 110-116)
• Pages: 2277-2283
• DOI: https://doi.org/10.4028/www.scientific.net/AMM.110-116.2277
• Online since: October 2011
• Authors: Xiang Meng Zhang, Ben Li Wang, Xian Ren Kong, A Yang Xiao
• Keywords: Duffing Systems, Harmonically Forced, Homotopy Perturbation Method, Non-Resonance, Primary Resonance

References

[1] Sun, W. and B. Wu, A new analytical approach to the Duffing-harmonic oscillator, Physics Letters A, vol. 311, no. 4-5, 2003, pp.365-373.

DOI: 10.1016/s0375-9601(03)00513-9

[2] He, J. H, Homotopy perturbation technique, Computer Methods in Applied Mechanics and Engineering, vol. 178, no. 3-4, 1999, pp.257-262.

DOI: 10.1016/s0045-7825(99)00018-3

[3] He, J. H, A coupling method of a homotopy technique and a perturbation technique for non-linear problems, International Journal of Non-Linear Mechanics, vol. 35, no. 1, 2000, pp.37-43.

DOI: 10.1016/s0020-7462(98)00085-7

[4] He, J. H, Some asymptotic methods for strongly nonlinear equations, International Journal of Modern Physics B, vol. 20, no. 10, 2006, pp.1141-1199.

DOI: 10.1142/s0217979206033796

[5] He, J. H, Homotopy perturbation method: a new nonlinear analytical technique, Applied Mathematics and Computation, vol. 135, no. 1, 2002, pp.73-79.

DOI: 10.1016/s0096-3003(01)00312-5

[6] He, J. H, Application of homotopy perturbation method to nonlinear wave equations, Chaos Solitons & Fractals, vol. 26, no. 3, 2005, pp.695-700.

DOI: 10.1016/j.chaos.2005.03.006

[7] Abbasbandy, S, Application of He's homotopy perturbation method to functional integral equations, Chaos Solitons & Fractals, vol. 31, no. 5, 2007, pp.1243-1247.

DOI: 10.1016/j.chaos.2005.10.069

[8] Chowdhury, M. S. H. and I. Hashim, Solutions of time-dependent Emden-Fowler type equations by homotopy-perturbation method, Physics Letters A, vol 368, no. 3-4, 2007, pp.305-313.

DOI: 10.1016/j.physleta.2007.04.020

[9] Ganji, D. D., G. A. Afrouzi and R. A. Talarposhti, Application of variational iteration method and homotopy-perturbation method for nonlinear heat diffusion and heat transfer equations, Physics Letters A, vol. 368, no. 6, 2007, pp.450-457.

DOI: 10.1016/j.physleta.2006.12.086

[10] Ganji, D. D., M. Rafei and J. Vaseghi, Application of homotopy-perturbation method for systems of nonlinear momentum and heat transfer equations, Heat Transfer Research, vol. 38, no. 4, 2009, pp.361-379.

DOI: 10.1615/heattransres.v38.i4.70

[11] Chowdhury, M. S. H. and I. Hashim, Analytical solutions to heat transfer equations by homotopy-perturbation method revisited, Physics Letters A, vol. 372, no. 8, 2008, pp.1240-1243.

DOI: 10.1016/j.physleta.2007.09.015

[12] Biazar, J. and H. Ghazvini, Exact solutions for non-linear Schrodinger equations by He's homotopy perturbation method, Physics Letters A, vol. 366, no. 1-2, 2007, pp.79-84.

DOI: 10.1016/j.physleta.2007.01.060