Application of the harmonic balance method to a nonlinear oscillator typified by a mass attached to a stretched wire

Beléndez, Augusto; Hernández-García, Armando; Beléndez, Tarsicio; Álvarez, Mariela L.; Gallego, Sergi; Ortuño, Manuel; Neipp, Cristian

doi:10.1016/j.jsv.2006.12.011

Cited by 86 publications

(75 citation statements)

References 13 publications

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“…Considerable attention has been directed towards the study of strongly nonlinear oscillators and several methods have been used to find approximate solutions to nonlinear oscillators [1][2][3][4][5][6][7][8][9][10]. In general, given the nature of a nonlinear phenomenon, the approximate methods can only be applied within certain ranges of the physical parameters and to certain classes of problems.…”

Section: Introductionmentioning

confidence: 99%

An explicit approximate solution to the Duffing-harmonic oscillator by a cubication method

et al. 2009

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The nonlinear oscillations of a Duffing-harmonic oscillator are investigated by an approximated method based on the 'cubication' of the initial nonlinear differential equation. These explicit formulas are valid for all values of the initial amplitude and we conclude this cubication method works very well for the whole range of initial amplitudes. Excellent agreement of the approximate frequencies and periodic solutions with the exact ones is demonstrated and discussed and the relative error for the approximate frequency is as low as 0.071%. Unlike other approximate methods applied to this oscillator, which are not capable to reproduce exactly the behaviour of the approximate frequency when A tends to zero, the cubication method used in this paper predicts exactly the behaviour of the approximate frequency not only when A tends to infinity, but also when A tends to zero. Finally, a closedform expression for the approximate frequency is obtained in terms of elementary functions.To do this, the relationship between the complete elliptic integral of the first kind and the arithmetic-geometric mean as well as Legendre's formula to approximately obtain this mean are used.

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Section: Introductionmentioning

confidence: 99%

An explicit approximate solution to the Duffing-harmonic oscillator by a cubication method

et al. 2009

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“…It is very difficult to solve nonlinear problems and, in general, it is often more difficult to get an analytic approximation than a numerical one for a given nonlinear problem. There are several methods used to find approximate solutions to nonlinear problems, such as perturbation techniques [1][2][3][4][5][6], harmonic balance based methods [6][7][8][9] or other techniques [10][11][12][13][14][15][16][17][18]. An excellent review on some asymptotic methods for strongly nonlinear equations can be found in detail in references [19] and [20].…”

Section: Introductionmentioning

confidence: 99%

Solution for an anti-symmetric quadratic nonlinear oscillator by a modified He’s homotopy perturbation method

Beléndez

Pascual

Beléndez

et al. 2009

Nonlinear Analysis: Real World Applications

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In this paper He's homotopy perturbation method has been adapted to calculate higherorder approximate periodic solutions of a nonlinear oscillator with discontinuity for which the elastic force term is anti-symmetric and quadratic term. We find He's homotopy perturbation method works very well for the whole range of initial amplitudes, and the excellent agreement of the approximate frequencies and periodic solutions with the exact ones has been demonstrated and discussed. Only one iteration leads to high accuracy of the solutions with a maximal relative error for the approximate period of less than 0.73%for all values of oscillation amplitude, while this relative error is as low as 0.040% when the second iteration is considered. Comparison of the result obtained using this method with those obtained by harmonic balance method reveals that the former is very effective and convenient.

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“…In particular the study of nonlinear oscillators is of great interest to many researchers [2,3]. There are several methods used to find approximate solutions to nonlinear oscillators, such as perturbation techniques [4][5][6][7][8][9][10][11][12][13][14] or harmonic balance based methods [15][16][17][18][19][20][21][22]. Surveys of the literature with numerous references and useful bibliographies have given in [3,23].…”

Section: Introductionmentioning

confidence: 99%

An Improved 'Heuristic' Approximation for the Period of a Nonlinear Pendulum: Linear Analysis of a Classical Nonlinear Problem

Hernández-García¹,

Beléndez²,

Márquez³

et al. 2007

International Journal of Nonlinear Sciences and Numerical Simulation

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A new analytical approximate expression for the period of the simple pendulum is obtained by using a heuristic but pedagogical derivation. This formula depends on two parameters obtained by comparing term-by-term the power-series expansions of the approximate and exact expressions for the period. This formula is compared with others in the literature and the numerical results obtained show that the published approximations are not nearly as good as the new expression proposed in this paper.

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Application of the harmonic balance method to a nonlinear oscillator typified by a mass attached to a stretched wire

Cited by 86 publications

References 13 publications

An explicit approximate solution to the Duffing-harmonic oscillator by a cubication method

An explicit approximate solution to the Duffing-harmonic oscillator by a cubication method

Solution for an anti-symmetric quadratic nonlinear oscillator by a modified He’s homotopy perturbation method

An Improved 'Heuristic' Approximation for the Period of a Nonlinear Pendulum: Linear Analysis of a Classical Nonlinear Problem

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