High Order He's Energy Balance Method based on Collocation Method

Durmaz, Seher; Demirbağ, Sezgin Altay; Kaya, Metin O.

doi:10.1515/ijnsns.2010.11.s1.1

Cited by 15 publications

(12 citation statements)

References 10 publications

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“…Usually, a set of algebraic equations with complex nonlinearities appears when EBM is formulated for determining higher-order approximations. On the contrary, some authors [23][24] extended the energy 2 balance method to obtain higher-order approximations, but the algebraic equations are not solved analytically.…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…Recently, several authors [23][24] have extended the energy balance method to determine higher-order approximations. The limitation of the articles [23][24] is that, they were not analytically solved algebraic equations; they only solved these algebraic equations numerically. On the other hand, Khan et al [20] used coupled method of He's homotopy perturbation method [7] and variational formulation [18] to obtain higher-order approximations for nonlinear cubic-quintic Duffing equations.…”

Section: Introductionmentioning

confidence: 99%

“…In this paper, a new analytical technique, combining the energy balance method [23] with harmonic balance method [17], has been presented to obtain higher-order approximations for nonlinear cubic-quintic Duffing equations. Generally, the second-order approximate frequency and the corresponding periodic solution have been determined which contains a few harmonic terms with lower order terms.…”

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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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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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“…There have been several classical approaches employed to solve the governing nonlinear differential equations to study the nonlinear vibrations including perturbation methods [18], He's Max-Min Approach (MMA) [2], He's Energy Balance Method [19], Combined Homotopy Variational Approach [20], Iteration perturbation method [21], Homotopy perturbation method (HPM) [22,23], Multistage Adomian Decomposition Method [24], Variational iteration method [3], Multiple scales method [25], Monotone iteration schemes [26], ADM-Padé technique [27], Navier and Levy-type solution [28], Hamiltonian approach [29], Parameter Perturbation Method [30], Differential Transform method [31], Laplace Transform method [32] . The application of new equivalent function for deadzone and preload nonlinearities on the dynamical behavior of beam vibration using PEM has been investigated by [6][7][8].…”

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confidence: 99%

High precise analysis of lateral vibration of quintic nonlinear beam

Sedighi

Reza

2013

Lat. Am. j. solids struct.

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This article intends to achieve a new formulation of beam vibration with quintic nonlinearity, including exact expressions for the beam curvature. To attain a proper design of the beam structures, it is essential to realize how the beam vibrates in its transverse mode which in turn yields the natural frequency of the system. In this direction, new powerful analytical method called Parameter Expansion Method (PEM) is employed to obtain the exact solution of frequency-amplitude relationship. Afterwards, it is clearly shown that the first term in series expansions is sufficient to produce a highly accurate approximation of mentioned system. Finally, preciseness of the present analytic procedures is evaluated in contrast with numerical calculations methods, giving excellent results. Nonlinear vibration of beams is of substantial interest for engineers and has been studied, considerably. From the engineering outlook, structures like helicopter rotor blades, space craft antennae, flexible satellites, airplane wings, robot arms, high-rise buildings, long-span bridges, drill strings and vibratory drilling can be modelled as a beam-like member. The problem of the transversely vibrating beam was recently formulated in terms of the partial differential equation of motion by many researchers [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17] with different boundary conditions. Sedighi et al. [1] presented the advantages of recent modern analytical approaches applied on the governing equation of transversely vibrating cantilever beams. A comprehensive study on the analytical investigation of cubic nonlinear vibrating tapered beams has been conducted by Bayat et al. [2]. The analytical expression for geometrically non-linear vibration of clamped-clamped Euler-Bernoulli beams including cubic non-linear strain-displacement relationship has been obtained by Barari et al. [3]. The application of new analytical approaches on the dynamical behavior of beams vibration with different boundary conditions has been investigated by Sedighi et al. [5][6][7][8]. Closed form solution to free vibration of beams with mixed boundary conditions has been proposed by Motaghian et al. [9]. Nikkhah Bahrami et al.[10] used modified wave approach for calculation of natural frequencies and mode shapes of arbitrary non-uniform beams. Non-linear modal analysis of a rotating beams studied by [11,12]. When the vibration amplitudes are moderate or large, the geometric nonlinearity must be included.It is very important to provide an accurate analysis towards the understanding of the non-linear vibration characteristics of these structures. Most models dealing with nonlinear dynamics of flexible

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Improved accuracy of He’s energy balance method for analysis of conservative nonlinear oscillator

Khan

Mirzabeigy

2014

Neural Comput & Applic

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High Order He's Energy Balance Method based on Collocation Method

Cited by 15 publications

References 10 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

High precise analysis of lateral vibration of quintic nonlinear beam

Improved accuracy of He’s energy balance method for analysis of conservative nonlinear oscillator

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