The Equivalent Linearization Method with a Weighted Averaging for Solving Undamped Nonlinear Oscillators

Hieu, Dang Van; Hai, Ninh Quang; Hung, Duong The

doi:10.1155/2018/7487851

Cited by 19 publications

(21 citation statements)

References 30 publications

(53 reference statements)

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“…[20,21], Anh et al had developed this method by using the weighted averaging instead of the convenial (classical) averaging [22]. The equivalent linearization method with a weighted averaging has been used to analyze responses of strong nonlinear oscillation problems and this method gives the solutions with much better accuracy than the classical method [21,[23][24][25]. In this section, we will find the approximate analytical solution of Eq.…”

Section: Solution Proceduresmentioning

confidence: 99%

Analysis of Nonlinear Vibration of Euler-Bernoulli Beams Subjected to Compressive Axial Forcevia the Equivalent Linearization Method with a Weighted Averaging

2019

International Journal of Scientific and Innovative Mathematical

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In this paper, the equivalent linearization method with a weighted averaging is applied to analyze nonlinear vibration of Euler-Bernoulli beam resting on the elastic foundationand subjected to the compressive axial force. Frequency-Amplitude relationships of beams with simply supported and clampedclamped end conditions are given in closed-forms. The present results are comparedwithonesachieved byusing He's Variational Approach method and the 4th-order Runge-Kutta method, comparison shows accuracy of the obtained solution. Effects of the compressive axial force and the coefficient ofelastic foundation on the nonlinear vibration behavior of beam are investigated in this work.

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Section: Solution Proceduresmentioning

confidence: 99%

Analysis of Nonlinear Vibration of Euler-Bernoulli Beams Subjected to Compressive Axial Forcevia the Equivalent Linearization Method with a Weighted Averaging

2019

International Journal of Scientific and Innovative Mathematical

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“…With this new proposal, accuracy of the ELM has been greatly improved. The new proposed method has been applied very effectively in analysis of deterministic strong nonlinear oscillations [22][23][24][25]. In this paper, the proposed method is further extended in analysis of a Duffing-harmonic oscillator.…”

Section: Original Research Articlementioning

confidence: 99%

“…The averaging value of function f(ωt) does not depend on the time t but depends on the parameter s (in expression of the weighted coefficient function h(t)). The parameter s is called the adjustment parameter, accuracy of the obtained solution depends on the choice of value of the parameter s. The choice of value of the parameter s has been examined by authors in many cases [22][23][24][25], the obtained results will be very accurate when s = 2.…”

Section: The Nonlinear Duffing-harmonic Oscillatormentioning

confidence: 99%

An Approximate Solution for a Nonlinear Duffing – Harmonic Oscillator

Dang

2019

ARJOM

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The equivalent linearization method introduced by Caughey is a powerful tool for analyzing random oscillations. The method is also easy to apply for deterministic oscillations. However, with strong nonlinear systems, the error of this method is usually quite large and even not acceptable. In conjunction with a weighted averaging, the equivalent linearization method has shown more accuracy than the classical one in which the conventional averaging value is used. Combining advantages of the classical equivalent linearization method and accuracy of the weighted averaging, the proposed method has shown that it is a useful tool for analyzing nonlinear oscillations including strong nonlinear systems. In this paper, the proposed method is applied to analyze a nonlinear Duffing – harmonic oscillator. The present results are compared with the results obtained by using other analytical methods, exact results and numerical results.

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“…In [44,45], Anh et al had developed this method by using the weighted averaging instead of the convenial (classical) averaging [43,46]. The equivalent linearization method with a weighted averaging was then used effectively to analyze strong nonlinear oscillations [45,47,48].…”

Section: Analysing Of Postbuckling and Free Nonlinear Vibrationmentioning

confidence: 99%

“…Substituting the averaging values in (48) and (49) into 41, we get the approximate frequency of oscillation:…”

Section: Analysing Of Postbuckling and Free Nonlinear Vibrationmentioning

confidence: 99%

Postbuckling and Free Nonlinear Vibration of Microbeams Based on Nonlinear Elastic Foundation

Hieu

2018

Mathematical Problems in Engineering

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In this paper, post-buckling and free nonlinear vibration of microbeams resting on nonlinear elastic foundation subjected to axial force are investigated. The equations of motion of microbeams are derived by using the modified couple stress theory. Using Galerkin’s method, the equation of motion of microbeams is reduced to the nonlinear ordinary differential equation. By using the equivalent linearization in which the averaging value is calculated in a new way called the weighted averaging value, approximate analytical expressions for the nonlinear frequency of microbeams with pinned–pinned and clamped–clamped end conditions are obtained in closed-forms. Comparisons with previous solutions are showed accuracy of the present solutions. Effects of the material length scale parameter and the axial compressive force on the frequency ratios of microbeams; and effect of the material length scale parameter on the buckling load ratios of microbeams are investigated in this paper.

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The Equivalent Linearization Method with a Weighted Averaging for Solving Undamped Nonlinear Oscillators

Cited by 19 publications

References 30 publications

Analysis of Nonlinear Vibration of Euler-Bernoulli Beams Subjected to Compressive Axial Forcevia the Equivalent Linearization Method with a Weighted Averaging

Analysis of Nonlinear Vibration of Euler-Bernoulli Beams Subjected to Compressive Axial Forcevia the Equivalent Linearization Method with a Weighted Averaging

An Approximate Solution for a Nonlinear Duffing – Harmonic Oscillator

Postbuckling and Free Nonlinear Vibration of Microbeams Based on Nonlinear Elastic Foundation

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