Accurate approximate solution to nonlinear oscillators in which the restoring force is inversely proportional to the dependent variable

Gimeno, Encarnación; Fernández, Elena; Méndez, David I.; Álvarez, Mariela L.

doi:10.1088/0031-8949/77/06/065004

Cited by 21 publications

(24 citation statements)

References 41 publications

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“…The Fourier representation of (44) is given in (45). The following result was obtained by Beléndez et al [36] for the Fourier coefficients a 2n+1 :…”

Section: Resultsmentioning

confidence: 60%

“…The main purpose of this paper is to construct an analytical approximation to the solution of (1) using a modified RHBM introduced by Beléndez et al [36,37] and which has been applied for truly conservative nonlinear oscillators with good results. To solve (1) by the modified RHBM, a new independent variable τ = ω t is introduced.…”

Section: Solution Proceduresmentioning

confidence: 99%

“…In this paper a modified rational harmonic balance method is investigated in terms of the truncation terms of the series expansion of the nonlinear differential equations. This new procedure has been proved to be effective for various conservative nonlinear oscillations with odd nonlinearities [36,37] and now it is proposed for constructing approximate analytical periodic solutions for nonlinear phenomena governed by pendulum-like differential equations.…”

Section: Introductionmentioning

confidence: 99%

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Rational-Harmonic Balancing Approach to Nonlinear Phenomena Governed by Pendulum-Like Differential Equations

Gimeno

2009

Zeitschrift Für Naturforschung A

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This paper presents a new approach for solving accurate approximate analytical solutions for nonlinear phenomena governed by pendulum-like differential equations. The new approach couples Taylor series expansion with rational harmonic balancing. An approximate rational solution depending on a small parameter is considered. After substituting the approximate solution into the governing differential equation, this equation is expanded in Taylor series of the parameter prior to harmonic balancing. The approach gives a cubic equation, which must be solved in order to obtain the value of the small parameter. A method for transforming this cubic equation into a linear equation is presented and discussed. Using this approach, accurate approximate analytical expressions for period and periodic solutions are obtained. We also compared the Fourier series expansions of the analytical approximate solution and the exact one. This allowed us to compare the coefficients for the different harmonic terms in these solutions. These analytical approximations may be of interest for those researchers working in nonlinear physical phenomena governed by pendulum-like differential equations in fields such as classical

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“…The Fourier representation of (44) is given in (45). The following result was obtained by Beléndez et al [36] for the Fourier coefficients a 2n+1 :…”

Section: Resultsmentioning

confidence: 60%

Section: Solution Proceduresmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Rational-Harmonic Balancing Approach to Nonlinear Phenomena Governed by Pendulum-Like Differential Equations

Gimeno

2009

Zeitschrift Für Naturforschung A

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“…However, for large x, the equation of motion is that of a truly nonlinear oscillator in which the restoring force is inversely proportional to the dependent variable and   2 / 2 A  1.25331/ A [20][21][22], which tends to zero when A decreases. Consequently the angular frequency  increases from 1 to 1.25331/A as the initial value of x(0) = A increases.…”

Section: Formulation and Solution Methodsmentioning

confidence: 99%

Application of a modified rational harmonic balance method for a class of strongly nonlinear oscillators

et al. 2008

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An analytical approximate technique for conservative nonlinear oscillators is proposed.This method is a modification of the rational harmonic balance method in which analytical approximate solutions have rational form. This approach gives us the frequency of the motion as a function of the amplitude of oscillation. We find that this method works very well for the whole range of parameters, and excellent agreement of the approximate frequencies with the exact one has been demonstrated and discussed.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 with complex nonlinearities.

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“…The most commons and most widely studied methods of all approximation methods for nonlinear differential equations are perturbation methods [1]. Some of other techniques include variational and variational iteration methods [4][5][6][7][8][9][10][11][12], exp-function [13,14], homotopy perturbation [15][16][17][18][19][20][21][22], equivalent linearization [23,24], standard and modified Lindstedt-Poincaré [25][26][27][28][29], artificial parameter [30,31], parameter expanding [32][33][34], harmonic balance methods [1,[35][36][37][38][39][40], etc. Surveys of the literature with numerous references and useful bibliography and a review of these methods can be found in detail in [28] and [41].…”

Section: Introductionmentioning

confidence: 99%

Linearized Harmonic Balancing Approach for Accurate Solutions to the Dynamically Shifted Oscillator

Rodes¹,

Rosillo²,

García³

et al. 2009

International Journal of Nonlinear Sciences and Numerical Simulation

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The analytical approximate technique developed by Wu et al for conservative oscillators with odd nonlinearity is used to construct approximate frequency-amplitude relations and periodic solutions to the dynamically shifted oscillator. This nonlinear oscillator is described by an equation of motion which includes a linear restoring force and an anti-symmetric, constant force which is a nonlinear force depending only upon the sign of the displacement. By combining Newton's method with the method of harmonic balance, analytical approximations to the oscillation frequency and periodic solutions are constructed for this oscillator and the approximate periods obtained are valid for the complete range of oscillation amplitudes. Excellent agreement of the approximate frequencies and periodic solutions with the exact ones are demonstrated and discussed and the results reveal that this technique is very effective and convenient for solving this class of conservative nonlinear oscillatory systems.

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Accurate approximate solution to nonlinear oscillators in which the restoring force is inversely proportional to the dependent variable

Cited by 21 publications

References 41 publications

Rational-Harmonic Balancing Approach to Nonlinear Phenomena Governed by Pendulum-Like Differential Equations

Rational-Harmonic Balancing Approach to Nonlinear Phenomena Governed by Pendulum-Like Differential Equations

Application of a modified rational harmonic balance method for a class of strongly nonlinear oscillators

Linearized Harmonic Balancing Approach for Accurate Solutions to the Dynamically Shifted Oscillator

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