Analytical approximate solutions for conservative nonlinear oscillators by modified rational harmonic balance method

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Title: Analytical approximate solutions for conservative nonlinear oscillators by modified rational harmonic balance method
Authors: Beléndez, Augusto | Gimeno, Encarnación | Alvarez, Mariela L. | Yebra Calleja, María Soledad | Méndez Alcaraz, David Israel
Research Group/s: Holografía y Procesado Óptico
Center, Department or Service: Universidad de Alicante. Departamento de Física, Ingeniería de Sistemas y Teoría de la Señal
Keywords: Nonlinear oscillators | Analytical approximate solutions | Trully nonlinear oscillators | Conservative oscillators | Rational harmonic balance method
Knowledge Area: Física Aplicada | Matemática Aplicada
Date Created: Feb-2008
Issue Date: 22-Jun-2009
Publisher: Taylor & Francis
Citation: BELÉNDEZ VÁZQUEZ, Augusto, et al. "Analytical approximate solutions for conservative nonlinear oscillators by modified rational harmonic balance method". International Journal of Computer Mathematics. ISSN 1029-0265, First published on 22 June 2009
Abstract: An analytical approximate technique for conservative nonlinear oscillators is proposed. This method is a modification of the generalized harmonic balance method in which analytical approximate solutions have rational form. This approach gives us not only a truly periodic solution but also the frequency of the motion as a function of the amplitude of oscillation. Three truly nonlinear oscillators including cubic Duffing oscillator, fractional-power restoring force and anti-symmetric quadratic nonlinear oscillators are presented to illustrate the usefulness and effectiveness of the proposed technique. We find that this method works very well for the cubic oscillator, and excellent agreement of the approximate frequencies with the exact one has been demonstrated and discussed. For the second-order approximation we have shown that the relative error in the analytical approximate frequency is as low as 0.0046%. We also compared the Fourier series expansions of the analytical approximate solution and the exact one. This has allowed us to compare the coefficients for the different harmonic terms in these solutions. For the other two nonlinear oscillators considered the relative errors in the analytical approximate frequencies are 0.098% and 0.066%, respectively. The most significant features of this method are its simplicity and its excellent accuracy for the whole range of oscillation amplitude values and the results reveal that this technique is very effective and convenient for solving conservative truly nonlinear oscillatory systems.
Sponsor: This work was supported by the "Ministerio de Educación y Ciencia", Spain, under project FIS2005-05881-C02-02. The authors express their gratitude to the reviewers for their useful suggestions and for their comments, which significantly improved the original manuscript.
ISSN: 0020-7160 (Print) | 1029-0265 (Online)
DOI: 10.1080/00207160802380942
Language: eng
Type: info:eu-repo/semantics/article
Rights: This is an electronic version of an article published in the International Journal of Computer Mathematics ©2008 Copyright Taylor & Francis; International Journal of Computer Mathematics is available online at
Peer Review: si
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