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.

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.

scholarly journals On the Solutions of Nonlinear Higher-Order Boundary Value Problems by Using Differential Transformation Method and Adomian Decomposition Method

2011 ◽  
Vol 2011 ◽  
pp. 1-19 ◽  
Author(s):  
Che Haziqah Che Hussin ◽  
Adem Kiliçman
Keyword(s):  
Boundary Value Problems ◽  
Decomposition Method ◽  
Nonlinear Differential Equations ◽  
Boundary Value ◽  
Adomian Decomposition Method ◽  
Transformation Method ◽  
Higher Order ◽  
Differential Transformation Method ◽  
Differential Transformation ◽  
Adomian Decomposition

We study higher-order boundary value problems (HOBVP) for higher-order nonlinear differential equation. We make comparison among differential transformation method (DTM), Adomian decomposition method (ADM), and exact solutions. We provide several examples in order to compare our results. We extend and prove a theorem for nonlinear differential equations by using the DTM. The numerical examples show that the DTM is a good method compared to the ADM since it is effective, uses less time in computation, easy to implement and achieve high accuracy. In addition, DTM has many advantages compared to ADM since the calculation of Adomian polynomial is tedious. From the numerical results, DTM is suitable to apply for nonlinear problems.

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.

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.

The Combination of Multi-Step Differential Transformation Method and Finite Element Method in Vibration Analysis of Non-Prismatic Beam

2017 ◽  
Vol 09 (01) ◽  
pp. 1750010 ◽  
Author(s):  
Ryszard Hołubowski ◽  
Kamila Jarczewska
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Transformation Method ◽  
Differential Transformation Method ◽  
Algebraic Equations ◽  
Differential Transformation ◽  
Prismatic Beam ◽  
Time Histories ◽  
Euler Bernoulli Beam ◽  
Element Method

The paper presents a new algorithm being the combination of multi-step differential transformation method (MsDTM) and finite element method (FEM) as a powerful tool for solving variety dynamic problems. The proposed algorithm, named as differential transformation finite element method (DTFEM), transforms partial differential equation into a set of recursive algebraic equations. The final form of a solution is a piecewise function which in general case may be a symbolic function. High effectiveness and accuracy of DTFEM is demonstrated on the example of forced vibrations of non-prismatic Euler–Bernoulli beam. Computed time histories of displacements, velocities and accelerations are highly consistent with results obtained by Newmark method.

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