Journal of Applied Nonlinear Dynamics
Approximate Analytical Solutions of A Nonlinear Oscillator Equation Modeling A Constrained Mechanical System
Journal of Applied Nonlinear Dynamics 6(1) (2017) 1726  DOI:10.5890/JAND.2017.03.002
Serge Bruno Yamgoué$^{1}$; Bonaventure Nana$^{1}$; François Beceau Pelap$^{2}$
$^{1}$ Department of Physics, Higher Teacher Training College Bambili, The University of Bamenda, P.O. Box 39 Bamenda, CAMEROON
$^{2}$ Laboratoire de Mécanique et de Mod´elisation des Systèmes Physiques (L2MSP), Département de Physique, Université de Dschang, BP 69 Dschang, CAMEROUN
Download Full Text PDF
Abstract
In this paper, we consider a class of nonlinear oscillators whose equations of motion are in the form of that of a cubic Duffing oscillator extended by a term which is a quadratic monomial in the velocity and whose coefficient is a rational function of the position. We apply a combination of harmonic balance and Newton method to seek analytical approximations to the periodic solutions to the equation. The analysis can be applied directly to the equation in its "natural" rational form or after reducing it to the same denominator and considering only the numerator. The advantages and drawback of these two usages of the method are also discussed.
Acknowledgments
We express our gratitude to the anonymous reviewers for drawing our attention to several relevant bibliographical papers used in this work.
References

[1]  Nayfeh, A.H. and Mook, D.T. (1979), Nonlinear Oscillation, John Wiley: New York. 

[2]  Amore, P. and Aranda, A. (2003), Presenting a new method for the solution of nonlinear problems, Physics Letters A, 316, 218225. 

[3]  Amore, P. and Aranda, A. (2005), Improved LindstedtPoincaré method for the solution of nonlinear problems, Journal of Sound and Vibration, 283, 11151136. 

[4]  Wu, B. and Li, P. (2001), A method for obtaining approximate analytical periods for a class of nonlinear oscillators, Meccanica, 36, 167176. 

[5]  Wu, B.S. and Li, P.S. (2001) A new approach to nonlinear oscillations, ASME Journal of Applied Mechanics 68, 951952. 

[6]  Wu, B.S. and Sun, W.P. (2011), Construction of approximate analytical solutions to strongly nonlinear damped oscillators, Archive of Applied Mechanics, 81, 10171030. 

[7]  Yamgoué, S.B. (2012), On the harmonic balance with linearization for asymmetric single degree of freedom nonlinear oscillators, Nonlinear Dynamics, 69, 10511062. 

[8]  Beléndez, A., Gimeno, E., Álvarez, M.L., Méndez, D.I., and Hernández, A. (2008), Application of a modified rational harmonic balance method for a class of strongly nonlinear oscillators, Physics Letters A, 372, 6047 6052. 

[9]  Gimeno, E. and Beléndez, A. (2009), Rationalharmonic balancing approach to nonlinear phenomena governed by pendulumlike differential equations, Zeitschrift für Naturforschung A, 64a, 819826. 

[10]  Yamgoué, S. B., Bogning, J. R., Kenfack Jiotsa, A., and Kofané, T.C. (2010), Rational harmonic balancebased approximate solutions to nonlinear singledegreeoffreedomoscillator equations, Physica Scripta, 81(3), 035003. 

[11]  Lai, S.K., Lim, C.W. and Wu, B.S. (2005), Accurate higherorder analytical approximate solutions to largeamplitude oscillating systems with a general nonrational restoring force, Nonlinear Dynamics, 42, 267281. 

[12]  Lim, C.W., Wu, B.S., and Sun, W.P. (2006), Higher accuracy analytical approximations to the duffingharmonic oscillator. Journal of Sound and Vibration, 296, 10391045. 

[13]  Beléndez, A., Méndez, D.I., Fernández, E., Marini, S., and Pascual, I. (2009), An explicit approximate solution to the duffingharmonic oscillator by a cubication method, Physics Letters A, 373, 28052809. 

[14]  Luo, A.C.J. (2012), Continuous Dynamical Systems, HEP/L&H Scientific: Beijing/Glen Carbon. 

[15]  Luo, A.C.J. and Huang, J.Z. (2012), Approximate solutions of periodic motions in nonlinear systems via a generalized harmonic balance, Journal of Vibration and Control, 18, 16611871. 

[16]  Luo, A.C.J. and Huang, J.Z. (2012), Analytical dynamics of periodm flows and chaos in nonlinear systems, International Journal of Bifurcation and Chaos, 22, Article No. 1250093 (29 pages). 

[17]  Luo, A.C.J. and Huang, J.Z. (2012), Analytical routines of period1 motions to chaos in a periodically forced Duffing oscillator with twinwell potential. Journal of Applied Nonlinear Dynamics, 1, 73108. 

[18]  Luo A.C.J. and Huang, J.Z. (2012), Unstable and stable periodm motions in a twinwell potential Duffing oscillator. Discontinuity, Nonlinearity and Complexity 1, 113145. 

[19]  Kovacic, I. and Rand, R. (2013), About a class of nonlinear oscillators with amplitudeindependent frequency, Nonlinear Dynamics, 74, 455465. 

[20]  Mickens, R.E. (2002), Fourier representations for periodic solutions of oddparity systems, Journal of Sound and Vibration, 258(2), 398401. 

[21]  Mickens, R.E. and Semwogerere, D. (1996), Fourier analysis of a rational harmonic balance approximation for periodic solutions, Journal of Sound and Vibration, 195(3), 528530. 

[22]  Spiegel, M.R. (1963), Theory and problems of advanced calculus SI(metric)edition, McGrawHill: New York. 

[23]  Gradshteyn I.S., and Ryzhik I.M. (2000), Table of Integrals, Series, and Products, 6th edition, Academic Press: San Diego. 

[24]  Kovacic, I. and Rand, R. (2013), Straightline Backbone Curve, Communications in Nonlinear Science and Numerical Simulation, 18, 22812288. 