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Closed-Form Expression for the Exact Period of a Nonlinear Oscillator Typified by a Mass Attached to a Stretched Wire

Published online by Cambridge University Press:  03 June 2015

Malik Mamode
Malik Mamode*
Affiliation:
Department of Physics, Laboratoire PIMENT, Equipe MASC, University of La Réunion, 97400 Saint-Denis, France
*
*Corresponding author. Email: [email protected]
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Abstract

The exact analytical expression of the period of a conservative nonlinear oscillator with a non-polynomial potential, is obtained. Such an oscillatory system corresponds to the transverse vibration of a particle attached to the center of a stretched elastic wire. The result is given in terms of elliptic functions and validates the approximate formulae derived from various approximation procedures as the harmonic balance method and the rational harmonic balance method usually implemented for solving such a nonlinear problem.

Keywords

34A3434C1534C25Nonlinear oscillatorexact periodelliptic integral
Type
Research Article
Information
Advances in Applied Mathematics and Mechanics , Volume 3 , Issue 6 , December 2011 , pp. 689 - 701
Copyright
Copyright © Global-Science Press 2011

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References

[1] Mickens, R. E., Oscillations in Planar Dynamic Systems, World Scientific Publishing, Singapore, 1996.Google Scholar
[2] Beléndez, A., Hernàndez, A., Beléndez, T., àlvarez, M. L., Gallego, S., OrtunñO, M. and Neipp, C., Application of the harmonic balance method to a nonlinear oscillator typified by a mass attached to a stretched wire, J. Sound Vib., 302 (2007), pp. 1018–1029.Google Scholar
[3] Sun, W. P., Wu, B. S. and Lim, C. W., Approximate analytical solutions for oscillation of a mass attached to a stretched elastic wire, J. Sound Vib., 300 (2007), pp. 1042–1047.Google Scholar
[4] Gimeno, E., àlvarez, M. L., Yebra, M. S., Rosa-Herranz, J. and Beléndez, A., Higher accuracy approximate solution for oscillations of a mass attached to a stretched elastic wire by rational harmonic balance method, Int. J. Nonlinear Sci. Numer., 10(4) (2009), pp. 493–504.Google Scholar
[5] Landau, L. D. and Lifshitz, E. M., Mechanics, 3rd ed, Pergamon, New York, 1976.Google Scholar
[6] Ganji, D. D., Malidarreh, N. Ranjbar and Akbarzade, M., Comparison of energy balance period with exact period for arising nonlinear oscillator equations (He’s energy balance period for nonlinear oscillators with and without discontinuities), Acta Appl. Math., 108 (2009), 353–362.Google Scholar
[7] Xu, L., Application of He’s parameter-expansion method to an oscillation of a mass attached to a stretched elastic wire, Phys. Lett. A, 368 (2007), pp. 259–262.Google Scholar
[8] Shou, D. H., Variational approach to the nonlinear oscillator of a mass attached to a stretched wire, Phys. Scripta, 77 (2008), 045006.CrossRefGoogle Scholar
[9] Gradshteyn, I. S. and Ryzhik, I. M., Table of Integrals, Series and Products, 6th ed, Academic Press, San Diego, 2000.Google Scholar
[10] Mamode, M., Some remarks on nonlinear oscillators: period, action, semiclassical quantization and Gibbs ensembles, J. Phys. A Math. Theor., 43 (2010), 505101.CrossRefGoogle Scholar