Iterative homotopy harmonic balancing approach for conservative oscillator with strong odd-nonlinearity

Guo, Zhao; Leung, A.Y.T.; Yang, H.X.

doi:10.1016/j.apm.2010.10.004

Cited by 33 publications

(24 citation statements)

References 35 publications

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“…Moreover, the present method gives better result than other second-order approximation obtained by [27].…”

Section: Introductionmentioning

confidence: 87%

“…But the first, second even third-order approximations are not desired result as compared with the exact solution. Furthermore, Guo et al [27] obtained the analytical periodic solutions of the oscillator up to third-order approximation.…”

Section: Introductionmentioning

confidence: 99%

“…On the other hand, in this case 1  A , the system Eq. (1) will represent a large oscillations (see [20,[26][27]). …”

Section: Introductionmentioning

confidence: 99%

“…A cubic-quintic Duffing oscillator of a conservative autonomous system can be described by the following differential equation, with cubic-quintic nonlinearities [20,[26][27]:…”

Section: Introductionmentioning

confidence: 99%

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Coupling of energy and harmonic balance method for solving a conservative oscillator with strong odd-nonlinearity

Razzak

2017

Scientia Iranica

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In this paper, a new analytical technique, combining the energy balance method (EBM) with harmonic balance method (HBM), is presented to obtain higher-order approximations of a conservative oscillator with strong odd-nonlinearity. To show the accuracy of the present method, one nonlinear oscillator named as cubic-quintic Duffing oscillator is investigated.The results obtained in this paper are compared with those results determined by other methods and exact solutions. The results give high accuracy and also provide better result than other existing results for both small and large amplitudes of oscillation. The main advantage of the present paper is its simplicity, which contains a few harmonic terms with lower order terms and these terms make the solution quickly converges. The present technique can be used to other nonlinear oscillators.Keywords: Nonlinear oscillation; Odd-nonlinearity; Cubic-quintic Duffing oscillator; Energy balance method; Harmonic balance method. IntroductionThe study of nonlinear oscillations is a necessary issue in engineering, physical science,

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“…Moreover, the present method gives better result than other second-order approximation obtained by [27].…”

Section: Introductionmentioning

confidence: 87%

Section: Introductionmentioning

confidence: 99%

“…On the other hand, in this case 1  A , the system Eq. (1) will represent a large oscillations (see [20,[26][27]). …”

Section: Introductionmentioning

confidence: 99%

“…A cubic-quintic Duffing oscillator of a conservative autonomous system can be described by the following differential equation, with cubic-quintic nonlinearities [20,[26][27]:…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Coupling of energy and harmonic balance method for solving a conservative oscillator with strong odd-nonlinearity

Razzak

2017

Scientia Iranica

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“…Duffing oscillators are one of the most important nonlinear oscillation problems and described by nonlinear differential equations. The Duffing oscillators have various applied areas, such as modelling of free vibrations of a restrained uniform beam with intermediate lumped mass 1 2 [1,2], magneto-elastic mechanical systems [3], fluid flow induced vibration [4] and nonlinear dynamics of slender elasticity [5] and so on.…”

Section: Introductionmentioning

confidence: 99%

A local differential transform approach for the cubic nonlinear Duffing oscillator with damping term

Tunc

Sarı

2018

Scientia Iranica

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Nonlinear behaviour of various problems is described by the Duffing model interpreted as a forced oscillator with a spring which has restoring force. In this paper, a new numerical approximation technique based on differential transform method has been introduced for the nonlinear cubic Duffing equation with and without damping effect. Since exact solutions of the corresponding equation for all initial guesses do not exist in the literature, we have first produced an exact solution for specific parameters by using the Kudryashov method to measure the accuracy of the currently suggested method. The innovative approach has been compared with the semi-analytic differential transform method and the fourth order RungeKutta method. Although the semi-analytic differential transform method is valid only for small time intervals, it has been proved that the innovative approach has ability to capture nonlinear behaviour of the process even in the long-time interval. In a comparison way, it has been shown that the present technique produces more accurate and computationally more economic results than the rival methods.

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An Iterative Approach to Obtaining the Nonlinear Frequency of a Conservative Oscillator with Strong Nonlinearities

Mohammadian

Pourmehran

2018

Int Appl Mech

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An iterative procedure is suggested for obtaining the higher-order approximate solutions of a conservative system comprising an oscillator with cubic and quintic restoring force function. The proposed method is similar to the traditional harmonic balance methods but unlike them the obtained from the previous step errors are considered in the present step to increase the accuracy of the solution. A comparison of results with those obtained by exact solution and other approximate analytical techniques confirms an accuracy of the method. It is shown that the achieved approximate solutions are valid for both small and large amplitudes of oscillation and can meet the exact solutions with a high level of accuracy in the lower-order of approximations. Furthermore, using the obtained analytical solutions, the effect of cubic and quintic terms on the frequency is discussed.

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Iterative homotopy harmonic balancing approach for conservative oscillator with strong odd-nonlinearity

Cited by 33 publications

References 35 publications

Coupling of energy and harmonic balance method for solving a conservative oscillator with strong odd-nonlinearity

Coupling of energy and harmonic balance method for solving a conservative oscillator with strong odd-nonlinearity

A local differential transform approach for the cubic nonlinear Duffing oscillator with damping term

An Iterative Approach to Obtaining the Nonlinear Frequency of a Conservative Oscillator with Strong Nonlinearities

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