×

zbMATH — the first resource for mathematics

A general solution of the Duffing equation. (English) Zbl 1121.70016
Summary: We describe an application of elliptic balance method to obtain a general solution of forced damped Duffing equation by assuming that the moduli of Jacobi elliptic functions are slowly varying as a function of time. From this solution, the maximum transient and steady-state amplitudes are determined for large nonlinearities and positive damping. The amplitude-time response curves obtained from our approximate solution are in good agreement with numerical solutions over the selected time interval.

MSC:
70K40 Forced motions for nonlinear problems in mechanics
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Elías-Zuñiga, A., ’On the elliptic balance method’, Mathematics and Mechanics of Solids 8, 2003, 263–279. · Zbl 1047.70001 · doi:10.1177/1081286503008003002
[2] Elías-Zuñiga, A., ’Application of Jacobian elliptic functions to the analysis of the steady-state solution of the damped Duffing equation with driving force of elliptic type’, Nonlinear Dynamics, 42, 2005, 175–184. · Zbl 1094.70014 · doi:10.1007/s11071-005-2554-0
[3] Stoker, J. J., Non-linear Vibrations in Mechanical and Electrical Systems, Intersciense, New York, 1950. · Zbl 0035.39603
[4] Nayfeh, A. H. and Mook, D. T., Non-linear Oscillations, Wiley, New York, 1973. · Zbl 0364.70018
[5] Amore, P. and Montes-Lamas, H., ’High order analysis of nonlinear periodic differential equations’, Phys. Lett. A 327, 2004, 158–166. · Zbl 1138.34318 · doi:10.1016/j.physleta.2004.05.016
[6] Struble, R. A., ’A discussion of the Duffing problem’, Journal of the Society for Industrial and Applied Mathematics 11(3), 659–666. · Zbl 0115.07704
[7] McCartin, B. J., ’An alternative analysis of Duffing’s equation’, SIAM Review 34(3), 1992, 482–491. · Zbl 0765.34022 · doi:10.1137/1034088
[8] Luo, A. C. J. and Han, R. P. S., ’A quantitative stability and bifurcation analysis of the generalized Duffing oscillator with strong nonlinearity’, Journal of the Franklin Institute 334B(3), 1997, 447–459. · Zbl 0868.34030 · doi:10.1016/S0016-0032(96)00089-0
[9] Rand, R. H., Lecture Notes on Nonlinear Vibrations, A free on-line book available at http://www.tam.cornell.edu/randdocs/nlvib36a.pdf , 2001.
[10] Elías-Zuñiga, A., ’Absorber Control of the Finite Amplitude Nonlinear Vibrations of a Simple Shear Suspension System’, University of Nebraska-Lincoln, Ph.D. Dissertation, 1994.
[11] Byrd, P. F. and Friedman, M. D., Handbook of Elliptic Integrals for Engineers and Physicists, Springer-Verlag, Berlin, 1953.
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.