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

2008 ◽  
Vol 56 (12) ◽  
pp. 3204-3220 ◽  
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

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

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.

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

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.

An Innovative Eigenvalue Problem Solver by Using Adomian Decomposition Approach

2012 ◽  
Vol 433-440 ◽  
pp. 6742-6750
Author(s):  
Jung Chang Hsu ◽  
Shao Shu Chu
Keyword(s):  
Eigenvalue Problem ◽  
Decomposition Method ◽  
Eigenvalue Problems ◽  
Adomian Decomposition Method ◽  
Problem Solver ◽  
Eigenvalues And Eigenfunctions ◽  
Decomposition Approach ◽  
Analytical Results ◽  
Adomian Decomposition ◽  
Mathematical Operations

The paper deals with eigenvalues and normalized eigenfunctions for a Strum-Liouville eigenvalue problem. The technique we have used is based on applying a Adomian decomposition method (ADM) to our eigenvalue problems. Doing some simple mathematical operations on the method, we can obtain ith eigenvalues and eigenfunctions one at a time. The computed results agree well with those analytical results given in the literature.

Vibration Analysis of a Stepped Beam by Using Adomian Decomposition Method

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.

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

2017 ◽  
Vol 23 (9) ◽  
pp. 1345-1363 ◽  
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.

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

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.

scholarly journals Modeling of vibration for functionally graded beams

2016 ◽  
Vol 14 (1) ◽  
pp. 661-672 ◽  
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.

Solving Free Vibration of Stepped Beam by Using the Adomian Decomposition Method

2010 ◽  
Author(s):  
Anooshiravan Farshidianfar ◽  
Rassoul Tabassian ◽  
Omid Kazemzadeh Khoee ◽  
Sayed Javadorreza Noei
Keyword(s):  
Free Vibration ◽  
Cross Sections ◽  
Decomposition Method ◽  
Natural Frequencies ◽  
Adomian Decomposition Method ◽  
High Accuracy ◽  
Mode Frequency ◽  
New Approach ◽  
Stepped Beam ◽  
Adomian Decomposition

This paper studies the free vibration of Euler-Bernoulli stepped beam with different cross-sections and also different materials for each section. In this work, a new approach called Adomian decomposition method (ADM) is used to deal with vibration problem. Natural frequencies of stepped beam are obtained with high accuracy using this method. Numerical results are validated by ANSYS.10 and proper convergence is observed between results. Effects of various parameters like step ratio on natural frequency are discussed. Applying this method on free vibration of stepped beam constructs a systematic procedure which is completely straightforward and could calculate both low and high mode frequency with appropriate accuracy.

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

2018 ◽  
Vol 25 (1) ◽  
pp. 41-51 ◽  
Author(s):  
Mohammad Hajhosseini ◽  
Saeed Ebrahimi
Keyword(s):  
Adomian Decomposition Method ◽  
Wide Band ◽  
Band Gaps ◽  
Geometrical Parameters ◽  
Vibration Band ◽  
Bernoulli Beam ◽  
Periodic Arrays ◽  
Beam Elements ◽  
Adomian Decomposition ◽  
Euler Bernoulli Beam

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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