Semi-analytical Solution for Nonlinear Transverse Vibration Analysis of an Euler–Bernoulli Beam with Multiple Concentrated Masses Using Variational Iteration Method

Torabi, Keivan; Sharifi, D.; Ghassabi, M.; Mohebbi, Akbar

doi:10.1007/s40997-018-0168-7

Cited by 9 publications

(3 citation statements)

References 31 publications

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“…In addition, many techniques have been developed and improved in solving problems with nonlinear equations, and the applications can be found in many fields and subjects with rich histories [5][6][7]. On the other hand, the deformation and vibration of beams are of great interest to researchers because of the signatory roles and extensive results for comparisons and validation of solution methods and techniques [8][9][10][11][12][13][14][15][16]. The method and procedure can be used for the analysis of nonlinear wave propagation in homogeneous and functionally graded materials [5,17].…”

Section: Introductionmentioning

confidence: 99%

The Approximate Solutions of Large Deflection of a Cantilever Beam under a Point Load

Meng,

Lian,

Zhang

et al. 2023

Mathematical Problems in Engineering

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By integrating the physical and time domains with the Galerkin method and adopting approximate solutions satisfying the boundary conditions, a set of algebraic equations of nonlinear nature is obtained from the nonlinear differential equation of an elastic beam for the undetermined coefficients of solutions of the deflection. These coefficients are to be used with the approximate basis functions for the asymptotic and explicit solutions of the nonlinear differential equation of flexure of a beam widely known as an elastica. Taking advantage of powerful tools for the symbolic manipulation of algebraic equations, such a novel method and procedure offer a new and efficient approach and option with known linear solutions in dealing with increasingly complex nonlinear problems in practical applications of both static and dynamic nature.

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Section: Introductionmentioning

confidence: 99%

The Approximate Solutions of Large Deflection of a Cantilever Beam under a Point Load

Meng,

Lian,

Zhang

et al. 2023

Mathematical Problems in Engineering

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“…In recent years, research studies have shown much interest in applying He’s VIM to solve the free vibration problem of beams (Asoor et al, 2016; Chen et al, 2015; Chen et al, 2020; Heidari-Rarani et al, 2017; Lu and Ma, 2018; Martin, 2019; Mohammadian, 2017; Tazabekova et al, 2018; Torabi et al, 2018; Younesian et al, 2012; Yun-dong and Yi-ren, 2017). The first study that used the VIM to obtain the eigenvalues and mode shapes of a Bernoulli–Euler beam with several end conditions was introduced by Liu and Gurram (2009).…”

Section: Introductionmentioning

confidence: 99%

“…The natural frequencies and mode shapes of a rotating tapered Timoshenko beam were obtained by Chen et al (2015). Torabi et al (2018) used the VIM in solving the linear and nonlinear vibration of Bernoulli–Euler beams with multiple concentrated masses. El-Sayed and El-Mongy (2018) investigated the free vibration of a tapered single-span beam connected to a subsystem at both ends using the VIM.…”

Section: Introductionmentioning

confidence: 99%

A new numeric–symbolic procedure for variational iteration method with application to the free vibration of generalized multi-span Timoshenko beam

El-Sayed

El-Mongy

2021

Journal of Vibration and Control

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In this article, a novel approach is introduced for the free vibration analysis of beams based upon the variational iteration method. The new approach uses a numeric–symbolic procedure that tackles the problem of increased execution time involved in symbolic integrations. This drawback is usually encountered in solving complicated free vibration problems such as stepped beams connected to lumped parameter subsystems. The proposed procedure is applied for free vibration analysis of a generalized multi-span Timoshenko beam connected to multiple lumped subsystems. Each subsystem is represented by a two-degree-of-freedom spring–mass–damper system. Several verification examples are presented where the results of the proposed numeric–symbolic variational iteration method are compared with the conventional symbolic approach symbolic variational iteration method in terms of execution time. Special attention is given to the verification of the new results against finite element modeling results and exact solutions where possible. Based on the presented results, it is shown that the new numeric–symbolic variational iteration method procedure efficiently reduces the time required for solving the free vibration problem while maintaining the high accuracy and robustness of the variational iteration method. The new procedure presented here may facilitate solving some engineering problems in which the conventional symbolic approach usually fails to solve owing to extensive memory requirements. The study contributes toward further improvements of the variational iteration method and its application to sophisticated dynamic systems.

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A variational iteration method (VIM) for nonlinear dynamic response of a cracked plate interacting with a fluid media

Motaharifar

Ghassabi

Talebitooti

2020

Engineering with Computers

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Semi-analytical Solution for Nonlinear Transverse Vibration Analysis of an Euler–Bernoulli Beam with Multiple Concentrated Masses Using Variational Iteration Method

Cited by 9 publications

References 31 publications

The Approximate Solutions of Large Deflection of a Cantilever Beam under a Point Load

The Approximate Solutions of Large Deflection of a Cantilever Beam under a Point Load

A new numeric–symbolic procedure for variational iteration method with application to the free vibration of generalized multi-span Timoshenko beam

A variational iteration method (VIM) for nonlinear dynamic response of a cracked plate interacting with a fluid media

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