An accurate closed-form approximate solution for the quintic Duffing oscillator equation. (English) Zbl 1201.34019

Summary: An accurate closed-form solution for the quintic Duffing equation is obtained using a cubication method. In this method the restoring force is expanded in Chebyshev polynomials and the original nonlinear differential equation is approximated by a cubic Duffing equation in which the coefficients for the linear and cubic terms depend on the initial amplitude. The replacement of the original nonlinear equation by an approximate cubic Duffing equation allows us to obtain explicit approximate formulas for the frequency and the solution as a function of the complete elliptic integral of the first kind and the Jacobi elliptic function cn, respectively. Excellent agreement of the approximate frequencies and periodic solutions with the exact ones is demonstrated and discussed and the relative error for the approximate frequency is lower than 0.37%.


34A45 Theoretical approximation of solutions to ordinary differential equations
34C25 Periodic solutions to ordinary differential equations
65L99 Numerical methods for ordinary differential equations
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[1] Nayfeh, A.H., Problems in perturbations, (1985), Wiley New York
[2] Mickens, R.E., Oscillations in planar dynamics systems, (1996), World Scientific Singapore · Zbl 1232.34045
[3] Lai, S.K.; Lim, C.W.; Wu, B.S.; Wang, C.; Zeng, Q.C.; He, X.F., Newton-harmonic balancing approach for accurate solutions to nonlinear cubic – quintic Duffing oscillators, Appl. math. modelling, 33, 852-866, (2009) · Zbl 1168.34321
[4] Ramos, J.I., On lindstedt – poincaré techniques for the quintic Duffing equation, Appl. math. comput., 193, 303-310, (2007) · Zbl 1193.65142
[5] Lim, C.W.; Xu, R.; Lai, S.K.; Yu, Y.M.; Yang, Q., Nonlinear free vibration of an elastically restrained beam with a point mass via the Newton-harmonic balancing approach, Int. J. nonlinear sci. numer. simul., 10, 661-674, (2009)
[6] Denman, J.H., An approximate equivalent linearization technique for nonlinear oscillations, J. appl. mech., 36, 358-360, (1969) · Zbl 0179.40902
[7] Jonckheere, R.E., Determination of the period of nonlinear oscillations by means of Chebyshev polynomials, ZAMM Z. angew. math. mech., 55, 389-393, (1971) · Zbl 0223.34036
[8] Bravo Yuste, S., Cubication of non-linear oscillators using the principle of harmonic balance, Int. J. non-linear mech., 27, 347-356, (1992) · Zbl 0766.70016
[9] Beléndez, A.; Álvarez, M.L.; Fernández, E.; Pascual, I., Cubication of conservative nonlinear oscillators, Eur. J. phys., 30, 973-981, (2009) · Zbl 1257.65048
[10] Beléndez, A.; Méndez, D.I.; Fernández, E.; Marini, S.; Pascual, I., An explicit approximate solution to the Duffing-harmonic oscillator by a cubication method, Phys. lett. A, 373, 2805-2809, (2009) · Zbl 1233.70008
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