A new analytical approach to the Duffing-harmonic oscillator. (English) Zbl 1055.70009

Summary: We present a new approach to solving the nonlinear Duffing-harmonic oscillator. It addresses the significant drawback in the classical harmonic balance method. By combining the linearization of the governing equation with the method of harmonic balance, we construct analytical approximations to the oscillation periods and periodic solutions for the oscillator. These analytical representations give approximations to the exact solutions in the whole range of oscillation amplitude. The new approach also avoids the necessity of numerically solving equations with complex nonlinearities as in the classical harmonic balance method.


70K42 Equilibria and periodic trajectories for nonlinear problems in mechanics
Full Text: DOI


[1] Stoker, J.J., Nonlinear vibrations, (1950), Interscience New York · Zbl 0035.39603
[2] Saaty, T.L.; Bram, J., Nonlinear mathematics, (1964), McGraw-Hill New York · Zbl 0198.00102
[3] Mickens, R.E., Oscillations in planar dynamic systems, (1996), World Scientific Singapore · Zbl 0840.34001
[4] Nayfeh, A.H., Perturbation methods, (1973), Wiley New York · Zbl 0375.35005
[5] Murdock, J.A., Perturbations: theory and methods, (1991), Wiley New York · Zbl 0810.34047
[6] Wu, B.S.; Zhong, H.X., Acta mech., 154, 121, (2002)
[7] Krylov, N.; Bogoliubov, N., Introduction to nonlinear mechanics, (1943), Princeton Univ. Press Princeton, NJ · Zbl 0063.03382
[8] Bogoliubov, N.N.; Mitropolsky, J.A., Asymptotic methods in the theory of nonlinear oscillations, (1963), State Press for Physics and Mathematical Literature, (in Russian), English translation: Hindustan Publishing Co
[9] West, J.C., Analytical techniques for nonlinear control systems, (1960), English Univ. Press London
[10] Mickens, R.E., J. sound vib., 94, 456, (1984)
[11] Mickens, R.E., J. sound vib., 111, 5l5, (1986)
[12] Delamotte, B., Phys. rev. lett., 70, 3361, (1993)
[13] Lim, C.W.; Wu, B.S.; He, L.H., Chaos, 11, 843, (2001)
[14] Mickens, R.E., Il nuovo cimento B, 101, 359, (1988)
[15] Mickens, R.E., J. sound vib., 244, 563, (2001)
[16] Wu, B.S.; Lim, C.W.; Ma, Y.F., Int. J. nonlinear mech., 38, 1037, (2002)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.