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

Gimeno, Encarnación; Álvarez, Mariela L.; Méndez, David I.; Hernández-García, Armando

doi:10.1016/j.physleta.2008.08.024

Cited by 17 publications

(7 citation statements)

References 21 publications

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“…Recently many effective methods were suggested to solve nonlinear oscillators, such as the variational iteration method [1], the homotopy perturbation method [2], the parameter-expansion method [3], energy balance method [4], harmonic balance method [5], a complete review on various analytical methods is available in Refs. [6,7].…”

Section: Introductionmentioning

confidence: 99%

A simple approach to nonlinear oscillators

Ren

2009

Physics Letters A

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Section: Introductionmentioning

confidence: 99%

A simple approach to nonlinear oscillators

Ren

2009

Physics Letters A

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“…It is noteworthy to observe that all the advantages [19] of the HB can be exploited within our proposed technique. For instance, the rational harmonic balance [5,28] or the combination of HB with Newton-like linearization (HBwL) [17,18,23,29] can be used for the construction of improved approximate solutions. The important point to lay emphasis on here is that each approximate solution or ansatz should satisfy the boundary conditions in Eq.…”

Section: Outline Of the Approachmentioning

confidence: 99%

Harmonic Balance for Non-Periodic Hyperbolic Solutions of Nonlinear Ordinary Differential Equations

Yamgoué

Lekeufack

Kofané

2017

Mathematical Modelling and Analysis

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In this paper, we propose a new approach for obtaining explicit analytical approximations to the homoclinic or heteroclinic solutions of a general class of strongly nonlinear ordinary differential equations describing conservative singledegree-of-freedom systems. Through a simple and explicit change of the independent variable that we introduce, these equations are transformed to others for which the original homoclinic or heteroclinic solutions are mapped into periodic solutions that satisfy some boundary conditions. Recent simplified methods of harmonic balance can then be exploited to construct highly accurate analytic approximations to these solutions. Here, we adopt the combination of Newton linearization with the harmonic balance to construct the approximates in incremental steps, thereby proposing both appropriate initial approximates and increments that together satisfy the required boundary conditions. Three examples including a septic Duffing oscillator, a controlled mechanical pendulum and a perturbed KdV equations are presented to illustrate the great accuracy and simplicity of the new approach.

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“…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%

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

show abstract

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

Cited by 17 publications

References 21 publications

A simple approach to nonlinear oscillators

A simple approach to nonlinear oscillators

Harmonic Balance for Non-Periodic Hyperbolic Solutions of Nonlinear Ordinary Differential Equations

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

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