Analysis of vibration band gaps in an Euler–Bernoulli beam with periodic arrays of meander-shaped beams

In this study, an Euler–Bernoulli beam carrying periodic arrays of meander-shaped beams is introduced. Each meander-shaped beam consists of several connected beam elements. Two models with different number of beam elements are considered. In each model, the effects of geometrical parameters on the lower and upper edges of the first three band gaps are investigated using the Adomian decomposition method. Results show that the wide band gaps at low frequency ranges can be obtained by changing the geometrical parameters. Furthermore, the band gaps are very close to each other for specific values of the geometrical parameters. Another advantage of this periodic beam is that its length is shorter than other types of periodic beams. These features make this periodic beam very useful in different applications of the band gap phenomenon such as vibration absorption. The finite element simulation (ANSYS software) is used to validate the analytical results and good agreement is found.

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Numerical solution of large deflection beams by using the Laplace Adomian decomposition method

Engineering Computations ◽

10.1108/ec-01-2021-0044 ◽

2021 ◽

Vol ahead-of-print (ahead-of-print) ◽

Author(s):

Ming-Xian Lin ◽

Chia-Hsiang Tseng ◽

Chao Kuang Chen

Keyword(s):

Decomposition Method ◽

Large Deflection ◽

Adomian Decomposition Method ◽

Nonlinear Behavior ◽

Bernoulli Beam ◽

Rapid Convergence ◽

Content Type ◽

Adomian Decomposition ◽

Euler Bernoulli Beam ◽

Better Than

PurposeThis paper presents the problems using Laplace Adomian decomposition method (LADM) for investigating the deformation and nonlinear behavior of the large deflection problems on Euler-Bernoulli beam.Design/methodology/approachThe governing equations will be converted to characteristic equations based on the LADM. The validity of the LADM has been confirmed by comparing the numerical results to different methods.FindingsThe results of the LADM are found to be better than the results of Adomian decomposition method (ADM), due to this method's rapid convergence and accuracy to obtain the solutions by using fewer iterative terms. LADM are presented for two examples for large deflection problems. The results obtained from example 1 shows the effects of the loading, horizontal parameters and moment parameters. Example 2 demonstrates the point loading and point angle influence on the Euler-Bernoulli beam.Originality/valueThe results of the LADM are found to be better than the results of ADM, due to this method's rapid convergence and accuracy to obtain the solutions by using fewer iterative terms.

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Application of ADM Using Laplace Transform to Approximate Solutions of Nonlinear Deformation for Cantilever Beam

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/2016/5052194 ◽

2016 ◽

Vol 2016 ◽

pp. 1-5

Author(s):

Ratchata Theinchai ◽

Siriwan Chankan ◽

Weera Yukunthorn

Keyword(s):

Laplace Transform ◽

Cantilever Beam ◽

Adomian Decomposition Method ◽

Small Deformation ◽

Approximate Solutions ◽

Beam Equation ◽

Bernoulli Beam ◽

Initial Value ◽

Adomian Decomposition ◽

Euler Bernoulli Beam

We investigate semianalytical solutions of Euler-Bernoulli beam equation by using Laplace transform and Adomian decomposition method (LADM). The deformation of a uniform flexible cantilever beam is formulated to initial value problems. We separate the problems into 2 cases: integer order for small deformation and fractional order for large deformation. The numerical results show the approximated solutions of deflection curve, moment diagram, and shear diagram of the presented method.

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An innovative eigenvalue problem solver for free vibration of Euler–Bernoulli beam by using the Adomian decomposition method

Computers & Mathematics with Applications ◽

10.1016/j.camwa.2008.07.029 ◽

2008 ◽

Vol 56 (12) ◽

pp. 3204-3220 ◽

Cited By ~ 46

Author(s):

Hsin-Yi Lai ◽

Jung-Chang Hsu ◽

Cha’o-Kuang Chen

Keyword(s):

Eigenvalue Problem ◽

Free Vibration ◽

Decomposition Method ◽

Adomian Decomposition Method ◽

Problem Solver ◽

Bernoulli Beam ◽

Adomian Decomposition ◽

Euler Bernoulli Beam

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Comparison of Adomian Decomposition Method and Differential Transformation Method for Vibration Problems of Euler-Bernoulli Beam

Applied Mechanics and Materials ◽

10.4028/www.scientific.net/amm.157-158.476 ◽

2012 ◽

Vol 157-158 ◽

pp. 476-483

Author(s):

Zhi Feng Liu ◽

Chun Hua Guo ◽

Li Gang Cai ◽

Wen Tong Yang ◽

Zhi Min Zhang

Keyword(s):

Decomposition Method ◽

Adomian Decomposition Method ◽

Transformation Method ◽

Differential Transformation Method ◽

Mode Shapes ◽

Bernoulli Beam ◽

Differential Transformation ◽

Adomian Decomposition ◽

Vibration Problems ◽

Euler Bernoulli Beam

In this paper, we compare the Differential transformation method and Adomian decomposition method to solve Euler-Bernoulli Beam vibration problems. The natural frequencies and mode shapes of the clamped-free uniform Euler-Bernoulli equation are calculated using the two methods. The Adomian decomposition method avoids the difficulties and massive computational work inherent in Differential transformation method by determining the very rapidly convergent analytic solutions directly. We found the solution between the two methods to be quite close. According to calculation of eigenvalues, natural frequencies and mode shapes, we compare the convergence of Differential transformation method and Adomian decomposition method. The two methods can be alternative ways to solve linear and nonlinear higher-order initial value problems.

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Modeling of vibration for functionally graded beams

Open Mathematics ◽

10.1515/math-2016-0057 ◽

2016 ◽

Vol 14 (1) ◽

pp. 661-672 ◽

Cited By ~ 2

Author(s):

Gülsemay Yiğit ◽

Ali Şahin ◽

Mustafa Bayram

Keyword(s):

Initial Conditions ◽

Adomian Decomposition Method ◽

Expansion Method ◽

Variable Coefficients ◽

Functionally Graded ◽

Fourier Series Expansion ◽

Bernoulli Beam ◽

Adomian Decomposition ◽

Graded Material ◽

Euler Bernoulli Beam

AbstractIn this study, a vibration problem of Euler-Bernoulli beam manufactured with Functionally Graded Material (FGM), which is modelled by fourth-order partial differential equations with variable coefficients, is examined by using the Adomian Decomposition Method (ADM).The method is one of the useful and powerful methods which can be easily applied to linear and nonlinear initial and boundary value problems. As to functionally graded materials, they are composites mixed by two or more materials at a certain rate. This mixture at a certain rate is expressed with an exponential function in order to try to minimize singularities from transition between different surfaces of materials as much as possible. According to the structure of the ADM in terms of initial conditions of the problem, a Fourier series expansion method is used along with the ADM for the solution of simply supported functionally graded Euler-Bernoulli beams. Finally, by choosing an appropriate mixture rate for the material, the results are shown in figures and compared with those of a standard (homogeneous) Euler-Bernoulli beam.

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Vibration Analysis of a Stepped Beam by Using Adomian Decomposition Method

Applied Mechanics and Materials ◽

10.4028/www.scientific.net/amm.160.292 ◽

2012 ◽

Vol 160 ◽

pp. 292-296

Author(s):

Qi Bo Mao ◽

Yan Ping Nie ◽

Wei Zhang

Keyword(s):

Decomposition Method ◽

Natural Frequencies ◽

Adomian Decomposition Method ◽

Free Vibrations ◽

Continuity Condition ◽

Mode Shapes ◽

Bernoulli Beam ◽

Stepped Beam ◽

Adomian Decomposition ◽

Euler Bernoulli Beam

The free vibrations of a stepped Euler-Bernoulli beam are investigated by using the Adomian decomposition method (ADM). The stepped beam consists two uniform sections and each section is considered a substructure which can be modeled using ADM. By using boundary condition and continuity condition equations, the dimensionless natural frequencies and corresponding mode shapes can be easily obtained simultaneously. The computed results for different boundary conditions are presented. Comparing the results using ADM to those given in the literature, excellent agreement is achieved.

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Vibration analysis of a uniform pre-twisted rotating Euler–Bernoulli beam using the modified Adomian decomposition method

Mathematics and Mechanics of Solids ◽

10.1177/1081286517720843 ◽

2017 ◽

Vol 23 (9) ◽

pp. 1345-1363 ◽

Cited By ~ 13

Author(s):

Desmond Adair ◽

Martin Jaeger

Keyword(s):

Differential Equations ◽

Boundary Conditions ◽

Vibration Analysis ◽

Decomposition Method ◽

Adomian Decomposition Method ◽

Free Vibration Analysis ◽

Published Data ◽

Bernoulli Beam ◽

Adomian Decomposition ◽

Euler Bernoulli Beam

The governing equations for a pre-twisted rotating cantilever beam are derived and used for free vibration analysis of a pre-twisted rotating beam whose flexural displacements are coupled in two planes. First differential equations of motion of a rotating twisted beam, including terms due to centrifugal stiffening, are derived for an Euler–Bernoulli beam undergoing free natural vibrations. The general solutions of these equations are obtained on applying the Adomian modified decomposition method (AMDM). The AMDM allows the governing differential equations to become recursive algebraic equations and the boundary conditions to become simple algebraic frequency equations suitable for symbolic computation. With additional simple mathematical operations on the model, the natural frequencies and corresponding closed-form series solution of the mode shape can be obtained simultaneously. Two main advantages of the application of the AMDM are, for the cases considered here, its fast convergence rate to the solution with the high degree of accuracy. As the AMDM technique is systematic, it is found straight-forward to modify boundary conditions from one case to the next. Comparison of results with published data showed the present calculations to be in reasonable agreement.

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Comparison of Methods for Modeling a Hydraulic Loader Crane With Flexible Translational Links

Journal of Dynamic Systems Measurement and Control ◽

10.1115/1.4030801 ◽

2015 ◽

Vol 137 (10) ◽

Cited By ~ 12

Author(s):

Henrik C. Pedersen ◽

Torben O. Andersen ◽

Brian K. Nielsen

Keyword(s):

Order Approximation ◽

Mode Shapes ◽

Beam Equation ◽

Flexible Structure ◽

Bernoulli Beam ◽

Finite Segment ◽

Single Beam ◽

Beam Elements ◽

Euler Bernoulli Beam ◽

Mode Order

When modeling flexible robots and structures for control purposes, most often the assumed modes (AMs) method is used to describe the deformation in combination with a floating reference frame formulation. This typically has the benefit of obtaining a low-order, but accurate model of the flexible structure, if the number of modes and AMs are properly chosen. The basis for using this method is, however, that the vibrations (deflections) are time and position independent, i.e., the expression is separable in space and time. This holds for the classic Euler–Bernoulli beam equation, but essentially does not hold for translational links. Hence, special care has to be taken when including flexible translational links. In the current paper, different methods for modeling a hydraulic loader crane with a telescopic arm are investigated and compared using both the finite segment (FS) and AMs method. The translational links are approximated by a single beam, respectively, multiple beam elements, with both one and two modes and using different mode shapes. The models are all validated against experimental data and the comparison is made for different operating scenarios. Based on the results, it is found that in most cases a single beam, low mode order approximation is sufficient to accurately model the mechanical structure and this yields similar results as the FS method.

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