Approximate solutions for the nonlinear pendulum equation using a rational harmonic representation

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Title: Approximate solutions for the nonlinear pendulum equation using a rational harmonic representation
Authors: Beléndez, Augusto | Arribas Garde, Enrique | Ortuño, Manuel | Gallego, Sergi | Márquez, Andrés | Pascual, Inmaculada
Research Group/s: Holografía y Procesado Óptico | GITE - Física, Óptica y Telecomunicaciones
Center, Department or Service: Universidad de Alicante. Departamento de Física, Ingeniería de Sistemas y Teoría de la Señal | Universidad de Alicante. Departamento de Óptica, Farmacología y Anatomía | Universidad de Alicante. Instituto Universitario de Física Aplicada a las Ciencias y las Tecnologías | Universidad de Castilla-La Mancha. Departamento de Física Aplicada
Keywords: Nonlinear pendulum equation | Approximate solutions | Tension | Padé approximants | Rational harmonic representation
Knowledge Area: Física Aplicada | Matemática Aplicada
Date Created: 23-Sep-2011
Issue Date: 1-Sep-2012
Publisher: Elsevier
Citation: BELÉNDEZ VÁZQUEZ, Augusto, et al. "Approximate solutions for the nonlinear pendulum equation using a rational harmonic representation". Computers and Mathematics with Applications. Vol. 64, No. 6 (2012). ISSN 0898-1221, pp. 1602–1611
Abstract: The exact expression for the maximum tension of a pendulum string is used to obtain a closed-form approximate expression for the solution of a simple pendulum in terms of elementary functions. This approximate solution has a rational harmonic expression and depends on an unknown function, which must be determined. This unknown function is expanded using the Padé approximant and two new parameters are introduced which are determined by means of a term-by-term comparison of the power series expansion for the approximate maximum tension with the corresponding series for the exact one. Using this approach, accurate approximate analytical expressions for the periodic solution are obtained. We also compared the Fourier series expansions of the approximate solutions and the exact ones. This allowed us to compare the coefficients for the different harmonics in these solutions. We also compared the approximate and exact solutions as a function of time for several oscillation amplitudes. Finally, in this procedure we used some of the approximate expressions for the simple pendulum frequency which can be found in the bibliography; however, the procedure can be applied using other approximate frequencies.
Sponsor: This work was supported by the Generalitat Valenciana of Spain under the project PROMETEO/2011/021 and by the ‘Vicerrectorado de Tecnología e Innovación Educativa’ of the University of Alicante, Spain, under the project GITE-09006-UA.
ISSN: 0898-1221 (Print) | 1873-7668 (Online)
DOI: 10.1016/j.camwa.2012.01.007
Language: eng
Type: info:eu-repo/semantics/article
Peer Review: si
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