Solution of a Duffing-harmonic oscillator by the method of harmonic balance

Hu, H.; Tang, Jiacheng

doi:10.1016/j.jsv.2005.12.025

Cited by 78 publications

(37 citation statements)

References 3 publications

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“…It is important to point out that the exact behaviour of the approximate frequency when A tends to zero is not obtained when other approximate methods as used including the harmonic balance method [17][18][19], the homotopy perturbation method [15,16], the energy balance method [20], the variational iteration method [21], a modified iteration procedure [22] or the Ritz procedure [23].…”

Section: Comparison With the Exact And Other Approximate Solutionmentioning

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: Comparison With the Exact And Other Approximate Solutionmentioning

confidence: 99%

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

et al. 2009

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“…There are a large variety of approximate methods commonly used for solving nonlinear oscillatory systems including perturbations [1][2][3][4], parameterexpansion [5,6], variational [7], variational iteration [8], homotopy perturbation [9][10][11][12], harmonic balance [1,[13][14][15][16][17] methods, etc. Surveys of the literature with numerous references and useful bibliography and a review of these methods can be found in detail in [2] and [18].…”

Section: Introductionmentioning

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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“…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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Solution of a Duffing-harmonic oscillator by the method of harmonic balance

Cited by 78 publications

References 3 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

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

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

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