Some New Analytical Techniques for Duffing Oscillator with Very Strong Nonlinearity

Document Type : Research Paper


Department of Mechanical Engineering Ferdowsi University of Mashhad


The current paper focuses on some analytical techniques to solve the non-linear Duffing oscillator with large nonlinearity. Four different methods have been applied for solution of the equation of motion; the variational iteration method, He’s parameter expanding method, parameterized perturbation method, and the homotopy perturbation method.
The results reveal that approximation obtained by these approaches are valid uniformly even for very large parameters and are more accurate than straightforward expansion solution.

The methods, which are proved to be mathematically powerful tools for solving the nonlinear oscillators, can be easily extended to any nonlinear equation, and the present paper can be used as paradigms for many other applications in searching for periodic solutions, limit cycles or other approximate solutions for real-life physics and engineering problems.


[1] Kerschen, G., Worden, K., Vakakis, A.F., and Golinval, J., “Past, Present and Future of Nonlinear System Identification in Structural Dynamics”, Journal of Mechanical Systems and Signal Processing, Vol. 20, pp. 505-592, (2006).
[2] Nayfeh, A.H., and Mook, D.T., “Nonlinear Oscillations”, Wiley-Interscience, New York, (1979).
[3] Strogatz, S.H., “Nonlinear Dynamics and Chaos: with Applications to Physics, Biology,   Chemistry, and Engineering”, Addison-wesley, Reading, MA, (1994).
[4] Verhulst, F., “Nonlinear Differential Equations and Dynamical Systems”, Second Edition, Springer, Berlin, (1999).
[5] Rand, R., “Lecture Notes on Nonlinear Vibrations”, Cornell, New York, USA, (2003).
[6] Caughey, T.K., “Equivalent Linearization Techniques”, Journal of the Acoustical Society of America, Vol. 35, pp. 1706–1711, (1963).
[7] Iwan, W.D., “A Generalization of the Concept of Equivalent Linearization”, International Journal of Nonlinear Mechanics, Vol. 8, pp. 279–287, (1973).
[8]  Rosenberg, R.M., “The Normal Modes of Nonlinear N-degree-of-freedom Systems”, Journal of Applied Mechanics, Vol. 29, pp. 7–14, (1962).
[9] Rosenberg, R.M., “On Nonlinear Vibrations of Systems with Many Degrees of Freedom”,      Advances in Applied Mechanics, Vol. 9 , pp. 155–242, (1966).
[10] Rand, R., “A Direct Method for Nonlinear Normal Modes”, International Journal of Non-       linear Mechanics, Vol. 9, pp. 363–368, (1974).
[11] Shaw, S.W., and Pierre C., “Normal Modes for Non-linear Vibratory Systems”, Journal of
       Sound and Vibration, Vol. 164, pp. 85–124, (1993).
[12] Vakakis, A.F., Manevitch, L.I., Mikhlin, Y.V., Pilipchuk, V.N. and Zevin, A.A., “Normal
       Modes and Localization in Nonlinear Systems”, Wiley, New York, (1996).
[13] Vakakis, A.F., “Non-linear Normal Modes and their Applications in Vibration Theory: an
        Overview”, Mechanical Systems and Signal Processing, Vol 11, pp. 3–22, (1997).
[14] Nayfeh, A.H., “Introduction to Perturbation Techniques”, Wiley-Interscience, New York, USA, (1981).
[15] O’Malley, R.E., “Singular Perturbation Methods for Ordinary Differential Equations”,
       Springer, New York, USA, (1991).
[16] Kevorkian, J., and Cole, J.D., “Multiple Scales and Singular Perturbation Methods”,
       Springer, New York, USA, (1996).
[17] He, J.H., “Variational Iteration Method - a kind of Non-linear Analytical Technique: Some
       Examples”, International Journal of Nonlinear Mechanics, Vol. 34, pp. 699–708, (1999).
[18] He, J.H., “Some Asymptotic Methods for Strongly Nonlinear Equations”, International Journal of Modern Physics B, Vol. 20, No. 10, pp. 1141-1199, (2006).       
[19] Inokuti, M., “General Use of the Lagrange Multiplier in Non-linear Mathematical
       Physics, in: S. Nemat-Nasser (Ed.) ”, Variational Method in the Mechanics of Solids,
       Pergamon Press, Oxford, pp. 156-162, (1978).
[20] He, J.H., “Variational Iteration Method for Non-linearity and its Applications”, Mechanics
       and Practice, Vol. 20, No. 1, pp. 30-32, (1998) (in Chinese).
[21] Finlayson, B.A., “The Method of Weighted Residuals and Variational Principles”,        Academic Press, New York, USA, (1972).
[22] He, J.H., “Bookkeeping Parameter in Perturbation Methods”, International Journal of
        Nonlinear Sciences and Numerical Simulation, Vol. 2, pp. 257–264, (2001).
[23] He, J.H., “New Interpretation of Homotopy Perturbation Method”, International Journal of
        Modern Physics B, Vol. 20, No. 18, pp. 2561–2568, (2006).
[24] He, J. H., “Modified Lindstedt-Poincare Methods for Some Strongly Non-linear Oscillations
        Part II: A New Transformation”, International Journal of Non-linear Mechanics, Vol. 37,   No. 2, pp. 315, (2002).