Abstract
The mathematical models of multi-degree-of-freedom (MDOF) strongly nonlinear dynamical systems are described by coupled second-order differential equations. In general, the exact solutions of MDOF strongly nonlinear dynamical systems are frequently unavailable. Therefore, efforts have been mainly concentrated on the approximate analytical solutions. The homotopy analysis method (HAM) is a useful analytic tool for solving strongly nonlinear dynamical systems, and it provides a simple way to ensure the convergence of solution series by means of a convergence-control parameter \({\hbar}\). Unlike the classical perturbation techniques, this method is independent of the presence of small parameters in the governing equations of motion. In this paper, the HAM is applied to formulate the analytical approximate periodic solutions of MDOF strongly nonlinear coupled van der Pol oscillators. Within this research framework, the frequency and the displacements of two-degree-of-freedom (2-DOF) strongly nonlinear systems can be explicitly obtained. For authentication, comparisons are carried out between the results obtained by the homotopy analysis and numerical integration methods. It is shown that the fourth-order or eighth-order solutions of the present method provide excellent accuracy. Illustrative examples of three-degree-of-freedom (3-DOF) strongly nonlinear coupled van der Pol oscillators are also presented and discussed. Finally, the optimal HAM approach is used to accelerate the convergence of the solutions.
Similar content being viewed by others
References
Nayfeh A.H., Mook D.T.: Nonlinear Oscillations. Wiley, New York (1979)
Odani K.: The limit cycle of the van der Pol equation is not algebraic. J. Differ. Equ. 115, 146–152 (1995)
Liao S.J.: An analytic approximate approach for free oscillations of self-excited systems. Int. J. Non-Linear Mech. 39, 271–280 (2004)
Chen Y.M., Liu J.K.: A study of homotopy analysis method for limit cycle of van der Pol equation. Commun. Nonlinear Sci. Numer. Simul. 14, 1816–1821 (2009)
Liao, S.J.: The proposed homotopy analysis techniques for the solution of nonlinear problems. PhD dissertation, Shanghai Jiao Tong University, Shanghai (1992)
Allan F.M., Syam M.I.: On the analytic solution of the nonhomogeneous Blasius problem. J. Comput. Appl. Math. 182, 362–371 (2005)
Abbasbandy S.: The application of homotopy analysis method to nonlinear equations arising in heat transfer. Phys. Lett. A 360, 109–113 (2006)
Hayat T., Sajid M.: On analytic solution for thin flow of a fourth grade fluid down a vertical cylinder. Phys. Lett. A 361, 316–322 (2007)
Liao S.J., Chwang A.T.: Application of homotopy analysis method in nonlinear oscillation. ASME J. Appl. Mech. 65, 914–922 (1998)
Liao S.J.: On the homotopy analysis method for nonlinear problems. Appl. Math. Comput. 147, 499–513 (2004)
Allan F.M.: Derivation of the Adomian decomposition method using the homotopy analysis method. Appl. Math. Comput. 190, 6–14 (2007)
Abbasbandy S.: The application of homotopy analysis method to solve a generalized Hirot-Satsuma coupled KdV equation. Phys. Lett. A 361, 478–483 (2007)
Abbasbandy S.: Solitary wave solutions to the Kuramoto-Sivashinsky equaton by means of the homotopy analysis method. Nonlinear Dyn. 52, 35–40 (2008)
Hayat T. et al.: On the MHD flow of a second grade fluid in a porous channel. Comput. Math. Appl. 54, 407–414 (2007)
Hayat T., Shahzad F., Ayub M.: Analytical solution for the steady flow of the third grade fluid in a porous half space. Appl. Math. Modell. 31, 2424–2432 (2007)
Fakhari A., Domairry G., Domairry G.: Approximate explicit solutions of nonlinear BBMB equations by homotopy analysis method and comparison with the exact solution. Phys. Lett. A 368, 64–68 (2007)
Yabushita K., Yabushita M., Tsuboi K.: An analytic solution of projectile motion with the quadratic resistance law using the homotopy analysis method. J. Phys. A Math. Theor. 40, 8403–8416 (2007)
He J.H.: Asymptotology by homotopy perturbation method. Appl. Math. Comput. 156, 591–596 (2004)
He J.H.: Homotopy perturbation method for bifurcation of nonlinear problems. Int. J.Nonlinear Sci. Numer. Simul. 6, 207–208 (2005)
He J.H.: Some asymptotic methods for strongly nonlinear equations. Int. J. Mod. Phys. B 20, 1141–1199 (2006)
He J.H.: An elementary introduction to recently developed asymptotic methods and nanomechanics in textile engineering. Int. J. Mod. Phys. B 22, 3487–3578 (2008)
Chowdhury M., Hashim I.: Solutions of time-dependent Emden-Fowler type equations by homotopy perturbation method. Phys. Lett. A 368, 305–313 (2007)
Odibat Z., Momani S.: Modified homotopy perturbation method: Application to quadratic Riccati differential equation of fractional order. Chaos Solitons Fractals 36, 167–174 (2008)
Sajid M., Hayat T., Asghar S.: Comparison between the HAM and HPM solutions of thin film flows of non-Newtonian fluids on a moving belt. Nonlinear Dyn. 50, 27–35 (2007)
Hashim I., Chowdhury M.: Adaptation of homotopy-perturbation method for numeric-analytic solution of system of ODEs. Phys. Lett. A 372, 470–481 (2008)
Abbasbandy S., Tan Y., Liao S.J.: Newton-homotopy analysis method for nonlinear equations. Appl. Math. Comput. 188, 1794–1800 (2007)
Wen J.M., Cao Z.C.: Sub-harmonic resonances of nonlinear oscillations with parametric excitation by means of the homotopy analysis method. Phys. Lett. A 371, 427–431 (2007)
Wen J.M., Cao Z.C.: Nonlinear oscillations with parametric excitation solved by homotopy analysis method. Acta Mechanica Sinica 24, 325–329 (2008)
Pirbodaghi T. et al.: On the homotopy analysis method for nonlinear vibration of beams. Mech. Res. Commun. 36, 143–148 (2009)
Chen Y.M., Liu J.K.: Homotopy analysis method for limit cycle flitter of airfoils. Appl. Math. Comput. 203, 854–863 (2008)
Wang Z., Zou L., Zhang H.Q.: Applying homotopy analysis method for solving differential-difference equation. Phys. Lett. A 369, 77–84 (2007)
Akyildiz F.T., Vajravelu K., Liao S.J.: A new method for homoclinic solutions of ordinary differential equations.. Chaos Solitons Fractals 39, 1073–1082 (2009)
Allan F.M.: Construction of analytic solution to chaotic dynamical systems using the Homotopy analysis method. Chaos Solitons Fractals 39, 1744–1752 (2009)
Bataineh A.S., Noorani M.S.M., Hashim I.: Solving systems of ODEs by homotopy analysis method. Commun. Nonlinear Sci. Numer. Simul. 13, 2060–2070 (2008)
Bataineh A.S., Noorani M.S.M., Hashim I.: On a new reliable modification of homotopy analysis method. Commun. Nonlinear Sci. Numer. Simul. 14, 409–423 (2009)
Ganjiani M., Ganjiani H.: Solution of coupled system of nonlinear differential equations using homotopy analysis method. Nonlinear Dyn. 56, 159–167 (2009)
Alomari A.K., Noorani M.S.M., Nazar R.: Solution of delay differential equation by means of homotopy analysis method. Acta Applicandae Mathematicae 108, 395–412 (2009)
Liao S.J.: Notes on the homotopy analysis method: Some definitions and theorems. Commun. Nonlinear Sci. Numer. Simul. 14, 983–997 (2009)
Nayfeh A.H.: Introduction to Perturbation Techniques. Wiley, New York (1981)
Bogoliuboff N., Mitropolsky Y.A.: Asymptotic Methods in the Theory of Nonlinear Oscillations. Gordon & Breach, New York (1962)
Liao S.J.: An optimal homotopy-analysis approach for strongly nonlinear differential equations. Commun. Nonlinear Sci. Numer. Simul. 15, 2003–2016 (2010)
Liang S.X., Jeffrey D.J.: Comparison of homotopy analysis method and homotopy perturbation method through an evolution equation. Commun. Nonlinear Sci. Numer. Simul. 14, 4057–4064 (2009)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Zhang, W., Qian, Y.H., Yao, M.H. et al. Periodic solutions of multi-degree-of-freedom strongly nonlinear coupled van der Pol oscillators by homotopy analysis method. Acta Mech 217, 269–285 (2011). https://doi.org/10.1007/s00707-010-0405-7
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00707-010-0405-7