Recent developments of some asymptotic methods and their applications for nonlinear vibration equations in engineering problems: a review

Bayat, Mahmoud; Pakar, Iman; Domairry, G.

doi:10.1590/s1679-78252012000200003

Cited by 65 publications

(52 citation statements)

References 143 publications

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“…(52) and with note that parameter s is chosen equal to 2, we get the approximate frequency of this oscillator: Comparison of the approximate frequencies ω in Eq. (54) and the approximate frequencies obtained by Parameter-Expansion Method (PEM) ωPEM (Bayat et al, 2012) in Eq. (56) with exact frequencies ωex in Eq.…”

Section: Examplementioning

confidence: 99%

“…The approximate frequency obtained by PEM as follows (Bayat et al, 2012): The exact frequency of this oscillator is : …”

Section: Examplementioning

confidence: 99%

“…In recent years, many powerful methods are used to find approximate solution as well as the amplitude- frequency relationship to the nonlinear differential equations. Some of these methods are Homotopy Perturbation Method (HPM) (He, 1999;He, 2004a;He, 2004b;He, 2004c;Turgut et al, 2007;Bayat et al, 2012), Max-Min Approach (MMA) (He, 2008;Chen et al, 2011;Dumaz et al, 2011;Yazdi et al, 2012;Bayat et al, 2012), Variational Iteration Method (VIM) (Bayat et al, 2012), Energy Balance Method (EBM) (Ganji et al, 2009;Khah et al, 2010;Younesian et al, 2010;Bayat et al, 2012), Amplitude-Frequency Formulation (AFF) (Chen et al, 2011;Jouyburi et al, 2014;Bayat et al, 2012), Parameter Expansion Method (PEM) (Kayaa et al, 2009;Dumaz et al, 2011;Darvishia et al, 2008;Zhao, 2009;Bayat et al, 2012 ), Homotopy Analysis Method (HAM) (He, 2004c;Bayat et al, 2012, Shahram Shahlaei-Far et al, 2016, Modified Homotopy Perturbation Method (MHPM) (Jouybari et al, 2014), Equivalent linearization Method (ELM) (Krylov et al, 1943;Caughey, 1959;Iyengar, 1988;Anh et al, 1995;Anh et al, 1997;Elishakoff et al, 2009;Anh, 2015) and combining Newton's Method with the Harmonic Balance Method (Lim et al, 2006).…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

The Equivalent Linearization Method with a Weighted Averaging for Analyzing of Nonlinear Vibrating Systems

Anh¹,

Hai²,

Hieu

2017

Lat. Am. j. solids struct.

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In this paper, the Equivalent Linearization Method (ELM) with a weighted averaging, which is proposed by Anh (Anh, 2015), is applied to analyze some vibrating systems with nonlinearities. The strongly nonlinear Duffing oscillator with third, fifth, and seventh powers of the amplitude, the other strongly nonlinear oscillators and the cubic Duffing with discontinuity are considered. The results obtained via this method are compared with the ones achieved by the Min-Max Approach (MMA), the Modified Lindstedt -Poincare Method (MLPM), the Parameter -Expansion Method (PEM), the Homotopy Perturbation Method (HPM) and 4 th order RungeKutta method. The obtained results demonstrate that this method is very convenient for solving nonlinear equations and also can be successfully exerted to a lot of practical engineering and physical problems.

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Section: Examplementioning

confidence: 99%

“…The approximate frequency obtained by PEM as follows (Bayat et al, 2012): The exact frequency of this oscillator is : …”

Section: Examplementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

The Equivalent Linearization Method with a Weighted Averaging for Analyzing of Nonlinear Vibrating Systems

Anh¹,

Hai²,

Hieu

2017

Lat. Am. j. solids struct.

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show abstract

“…Recent applications of some asymptotic methods such as the variational iteration, the homotopy perturbation, the energy balance, the parameter-expansion, the variational approach, the improved amplitude frequency formulation, the max-min approach, the Hamiltonian approach, and the homotopy analysis, to name a few, have been used to obtain approximate solutions of highly nonlinear problems in which the traditional perturbation methods have some limitations [1,2].…”

Section: Introductionmentioning

confidence: 99%

Enhanced Multistage Homotopy Perturbation Method: Approximate Solutions of Nonlinear Dynamic Systems

Olvera

Elı́as-Zúñiga

2014

Abstract and Applied Analysis

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We introduce a new approach called the enhanced multistage homotopy perturbation method (EMHPM) that is based on the homotopy perturbation method (HPM) and the usage of time subintervals to find the approximate solution of differential equations with strong nonlinearities. We also study the convergence of our proposed EMHPM approach based on the value of the control parameter h by following the homotopy analysis method (HAM). At the end of the paper, we compare the derived EMHPM approximate solutions of some nonlinear physical systems with their corresponding numerical integration solutions obtained by using the classical fourth order Runge-Kutta method via the amplitude-time response curves.

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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].…”

mentioning

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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Recent developments of some asymptotic methods and their applications for nonlinear vibration equations in engineering problems: a review

Cited by 65 publications

References 143 publications

The Equivalent Linearization Method with a Weighted Averaging for Analyzing of Nonlinear Vibrating Systems

The Equivalent Linearization Method with a Weighted Averaging for Analyzing of Nonlinear Vibrating Systems

Enhanced Multistage Homotopy Perturbation Method: Approximate Solutions of Nonlinear Dynamic Systems

High precise analysis of lateral vibration of quintic nonlinear beam

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