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Improved accuracy of He’s energy balance method for analysis of conservative nonlinear oscillator

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Abstract

In this paper, the accuracy of He’s energy balance method for the analysis of conservative nonlinear oscillator is improved based on combining features of collocation method and Galerkin–Petrov method. In order to demonstrate the effectiveness of proposed method, Duffing oscillator with cubic nonlinearity, double-well Duffing oscillator, and nonlinear oscillation of pendulum attached to a rotating support are considered. Comparison of results with ones achieved utilizing other techniques shows improved energy balance method can very effectively reduce the error of simple energy balance method. Also, results show in large amplitude of oscillation, and improved energy balance method yields better accuracy rather than second-order energy balance method based on collocation and second-order energy balance method based on Galerkin method. Improved energy balance method can be successfully used for accurate analytical solution of other conservative nonlinear oscillator.

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References

  1. He JH (2002) Preliminary report on the energy balance for nonlinear oscillations. Mech Res Commun 29:107–111

    Article  MATH  Google Scholar 

  2. He JH (1999) Homotopy perturbation technique. Comput Met Appl Mechan Eng 178:257–262

    Article  MATH  Google Scholar 

  3. Mickens RE (1996) Oscillations in planar dynamics systems. World Sci, Singapore

    Google Scholar 

  4. He JH (2010) Hamiltonian approach to nonlinear oscillators. Phy Lett A 374:2312–2314

    Article  MATH  Google Scholar 

  5. Liao SJ, Cheung AT (1998) Application of homotopy analysis method in nonlinear oscillations. ASME J Appl Mechan 65:914–922

    Article  Google Scholar 

  6. He JH (2008) Max-min approach to nonlinear oscillators. Int J Nonlinear Sci Numer Simul 9:207–210

    Google Scholar 

  7. Herisanu N, Marinca V (2010) Explicit analytical approximation to large amplitude non-linear oscillations of a uniform cantilever beam carrying an intermediate lumped mass and rotary inertia. Meccanica 45:847–855

    Article  MATH  MathSciNet  Google Scholar 

  8. Khan Y, Wu Q (2011) Homotopy perturbation transform method for nonlinear equations using He’s polynomials. Comput Math Appl 61:1963–1967

    Article  MATH  MathSciNet  Google Scholar 

  9. Khan Y, Austin F (2010) Application of the Laplace decomposition method to nonlinear homogeneous and non-homogenous advection equations. Zeitschrift fur Naturforschung 65a:849–853

    Google Scholar 

  10. Rebelo PJ (2011) An approximate solution to an initial boundary value problem to the one-dimensional Kuramoto–Sivashinsky equation. Int J Numer Methods Biomed Eng 27:874–881

    Article  MATH  MathSciNet  Google Scholar 

  11. Akbarzede M, Langari J, Ganji DD (2011) A coupled homotopy-variational method and variational formulation applied to nonlinear oscillators with and without discontinuities. ASME J Vibration Acoust 133:044501

    Article  Google Scholar 

  12. Cveitcanin L (2006) Homotopy–perturbation for pure nonlinear differential equation. Chaos, Solitons Fractals 30:1221–1230

    Article  Google Scholar 

  13. Ozis T, Yildirim A (2007) Determination of the frequency-amplitude relation for a Duffing-harmonic oscillator by the energy balance method. Comput Mathe Appl 54:1184–1187

    Article  MathSciNet  Google Scholar 

  14. Marinca V, Herisanu N, Bota C (2008) Application of the variational iteration method to some nonlinear one-dimensional oscillations. Meccanica 43:75–79

    Article  MATH  Google Scholar 

  15. Herisanu N, Marinca V (2010) A modified variational iteration method for strongly nonlinear problems. Nonlinear Sci Lett A 1:183–192

    Google Scholar 

  16. Belendez A, Belendez T, Neipp C, Hernandez A, Alvarez ML (2009) Approximate solutions of a nonlinear oscillator typified as a mass attached to a stretched elastic wire by the homotopy perturbation method. Chaos, Solitons Fractals 39:746–764

    Article  MATH  Google Scholar 

  17. Herisanu N, Marinca V (2012) Optimal homotopy perturbation method for a non-conservative dynamical system of a rotating electrical machine. Z Naturforsch 67a:509–516

    Article  Google Scholar 

  18. Khan Y, Smarda Z (2012) Modified homotopy perturbation transform method for third order boundary layer equation arising in fluid mechanics. Sains Malays 41:1489–1493

    MATH  Google Scholar 

  19. Khan Y, Madani M, Yildirim A, Abdou MA, Faraz N (2011) A new approach to Van der Pol’s Oscillator Problem. Z Naturforsch 66a:620–624

    Article  Google Scholar 

  20. Rebelo PJ (2012) An approximate solution to an initial boundary value problem: Rakib–Sivashinsky equation. Int J Comput Math 89:881–889

    MATH  MathSciNet  Google Scholar 

  21. Saha Ray S, Patra A (2013) Haar wavelet operational methods for the numerical solutions of fractional order nonlinear oscillatory Van der Pol system. Appl Math Comput 220:659–667

    MathSciNet  Google Scholar 

  22. Sardar T, Saha Ray S, Bera RK, Biswas BB (2009) The analytical approximate solution of the multiterm fractionally damped Van der Pol equation. Physica Scr 80:025003

    Google Scholar 

  23. Daeichin M, Ahmadpoor MA, Askari H, Yildirim A (2013) Rational Energy Balance Method to Nonlinear Oscillators with Cubic term. Asian-Eur J Math 06:1350019

    MathSciNet  Google Scholar 

  24. Ma X, Wei L, Guo Z (2008) He’s homotopy perturbation method to periodic solutions of nonlinear Jerk equations. J Sound Vib 314:217–227

    Google Scholar 

  25. Belendez A, Mendez DI, Belendez T, Hernandez A, Alvarez ML (2008) Harmonic balance approaches to the nonlinear oscillators in which the restoring force is inversely proportional to the dependent variable. J Sound Vib 314:775–782

    Google Scholar 

  26. Pirbodaghi T, Hoseini SH, Ahmadian MT, Farrahi GH (2009) Duffing equations with cubic and quintic nonlinearities. Comput Math Appl 57:500–506

    MATH  MathSciNet  Google Scholar 

  27. Durmaz S, Demirbag SA, Kaya MO (2010) High order He’s energy balance method based on collocation method. Int J Nonlinear Sci Numer Simul 11:1–5

    MathSciNet  Google Scholar 

  28. Sfahani MG, Barari A, Omidvar M, Ganji SS, Domairry G (2011) Dynamic response of inextensible beams by improved energy balance method. Proc Inst Mech Eng Part K: J Multi-body Dyn 225:66–73

    Google Scholar 

  29. Durmaz S, Kaya MO (2012) High-order energy balance method to nonlinear oscillators. J Appl Math 2012:518684

    MathSciNet  Google Scholar 

  30. Yazdi MK, Khan Y, Madani M, Askari H, Saadatnia Z, Yildirim A (2010) Analytical solutions for autonomous conservative nonlinear oscillator. Int J Nonlinear Sci Numer Simul 11:979–984

    Google Scholar 

  31. Micknes RE (1986) A generalization of the method of harmonic balance. J Sound Vib 111:515–518

    Google Scholar 

  32. Wu BS, Sun WP, Lim CW (2007) Analytical approximations to the double-well Duffing oscillator in large amplitude oscillations. J Sound Vib 307:953–960

    Google Scholar 

  33. Momeni M, Jamshidi N, Barari A, Ganji DD (2011) Application of He’s energy balance method to Duffing-harmonic oscillators. Int J Comput Math 88:135–144

    MATH  MathSciNet  Google Scholar 

  34. Ghafoori S, Motevalli M, Nejad MG, Shakeri F, Ganji DD, Jalaal M (2011) Efficiency of differential transformation method for nonlinear oscillation: comparison with HPM and VIM. Curr Appl Phys 11:965–971

    Google Scholar 

  35. Yazdi MK, Ahmadian A, Mirzabeigy A, Yildirim A (2012) Dynamic analysis of vibrating systems with nonlinearities. Commun Theor Phys 57:183–187

    MATH  Google Scholar 

  36. Younesian D, Askari H, Saadatnia Z, Yazdi MK (2011) Periodic solutions for nonlinear oscillation of a centrifugal governor system using the He’s frequency-amplitude formulation and He’s energy balance method. Nonlinear Sci Lett A 2:143–148

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Zhejiang University, Hangzhou, 310027, China

    Y. Khan

  2. Department of Mechanical Engineering, Amirkabir University of Technology, Hafez Ave, 15914, Tehran, Iran

    A. Mirzabeigy

Authors
  1. Y. Khan
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  2. A. Mirzabeigy
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Correspondence to Y. Khan.

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Khan, Y., Mirzabeigy, A. Improved accuracy of He’s energy balance method for analysis of conservative nonlinear oscillator. Neural Comput & Applic 25, 889–895 (2014). https://doi.org/10.1007/s00521-014-1576-2

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  • DOI: https://doi.org/10.1007/s00521-014-1576-2

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