A Novel Rational Harmonic Balance Approach for Periodic Solutions of Conservative Nonlinear Oscillators

Gimeno, Encarnación; Álvarez, Mariela L.; Gallego, Sergi; Ortuño, Manuel; Méndez, David I.

doi:10.1515/ijnsns.2009.10.1.13

Cited by 16 publications

(14 citation statements)

References 58 publications

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“…This oscillator has been also solved using a novel rational harmonic balance method (RHBM) [17] and the relative error for the approximate frequency obtained was 0.24%, which is similar to that obtained using the second-order NHBM.…”

Section: Example 1 Anti-symmetric Constant Force Oscillatorsupporting

confidence: 57%

Solutions for Conservative Nonlinear Oscillators Using an Approximate Method Based on Chebyshev Series Expansion of the Restoring Force

et al. 2016

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Approximate solutions for small and large amplitude oscillations of conservative systems with odd nonlinearity are obtained using a "cubication" method. In this procedure, the Chebyshev polynomial expansion is used to replace the nonlinear function by a third-order polynomial equation. The original second-order differential equation, which governs the dynamics of the system, is replaced by the Duffing equation, whose exact frequency and solution are expressed in terms of the complete elliptic integral of the first kind and the Jacobi elliptic function cn, respectively. Then, the exact solution for the Duffing equation is the approximate solution for the original nonlinear differential equation. The coefficients for the linear and cubic terms of the approximate Duffing equation -obtained by "cubication" of the original second-order differential equation -depend on the initial oscillation amplitude. Six examples of different types of common conservative nonlinear oscillators are analysed to illustrate this scheme. The results obtained using the cubication method are compared with those obtained using other approximate methods such as the harmonic, linearized and rational balance methods as well as the homotopy perturbation method. Comparison of the approximate frequencies and solutions with the exact ones shows good agreement.

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Section: Example 1 Anti-symmetric Constant Force Oscillatorsupporting

confidence: 57%

Solutions for Conservative Nonlinear Oscillators Using an Approximate Method Based on Chebyshev Series Expansion of the Restoring Force

et al. 2016

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“…For system (2) this is so, because it has no critical points and it is well-known ( [9]) that periodic orbits of planar autonomous systems must surround a critical point. For equation (1) it is not difficult to see that any periodic solution x(t) must vanish for some t * ∈ R, that is x(t * ) = 0.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…The method used to obtain T (A) is based on standard procedures applied to 1-dimensional oscillatory physical systems, see [2,12,13,15,16,19], adapted to this singular setting. Item (ii) provides a new result.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Weak periodic solutions of xẍ + 1 = 0 and the Harmonic Balance Method

García-Saldaña

Gasull

2017

J. Phys.: Conf. Ser.

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Accurate approximate solution to nonlinear oscillators in which the restoring force is inversely proportional to the dependent variable A Beléndez, E Gimeno, E Fernández et al. Abstract. We prove that the differential equation xẍ + 1 = 0 has continuous weak periodic solutions and compute their periods. Then, we use the Harmonic Balance Method until order six to approximate these periods and to illustrate how the accuracy of the method increases with the order. Our computations rely on the Gröbner basis approach.

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“…Solving (42) it follows that where (43) has been taken into account and b 0 , b 1 , b 2 , and b 3 are given in (32) - (35). In Figure 1 we have plotted B 2 as a function of A.…”

Section: Solution Proceduresmentioning

confidence: 99%

“…Various generalizations of the harmonic balance method have been made and one of them is the rational representation proposed by Mickens [1, 32 -34]. In this method the approximate solution obtained approximates all of the harmonics in the exact solution [34,35], whereas the usual harmonic balance techniques provide an approximation to only the lowest harmonic components. For most cases, the application of the rational harmonic balance method leads to very complicated sets of algebraic equations with very complex nonlinearity that have to be solved even for the second-order approximation.…”

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

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A Novel Rational Harmonic Balance Approach for Periodic Solutions of Conservative Nonlinear Oscillators

Cited by 16 publications

References 58 publications

Solutions for Conservative Nonlinear Oscillators Using an Approximate Method Based on Chebyshev Series Expansion of the Restoring Force

Solutions for Conservative Nonlinear Oscillators Using an Approximate Method Based on Chebyshev Series Expansion of the Restoring Force

Weak periodic solutions of xẍ + 1 = 0 and the Harmonic Balance Method

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

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