Symmetries and integrability of a fourth-order Euler–Bernoulli beam equation

2010 ◽  
Vol 51 (5) ◽  
pp. 053517 ◽  
Author(s):  
Ashfaque H. Bokhari ◽  
F. M. Mahomed ◽  
F. D. Zaman
Keyword(s):  
Fourth Order ◽  
Beam Equation ◽  
Bernoulli Beam ◽  
Euler Bernoulli Beam

scholarly journals Numerical Solution of Fourth-Order Boundary Value Problems for Euler-Bernoulli Beam Equation using FDM

2021 ◽  
Vol 2070 (1) ◽  
pp. 012052
Author(s):  
M. Adak ◽  
A. Mandal
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Boundary Value Problems ◽  
Fourth Order ◽  
Difference Method ◽  
Boundary Value ◽  
Stability And Convergence ◽  
Beam Equation ◽  
Bernoulli Beam ◽  
Euler Bernoulli Beam

Abstract Euler-Bernoulli beam equation is widely used in engineering, especially civil and mechanical engineering to determine the deflection or strength of bending beam. In physical science and engineering, to predict the deflection for beam problem, bending moment, soil settlement and modeling of viscoelastic flows, fourth-order ordinary differential equation (ODE) is widely used. The analytical solution of most of the higher order ordinary differential equations with complicated boundary condition that occur in any engineering problems is not easy way. Therefore, numerical technique based on finite difference method (FDM) is comparatively easy and important for solving the boundary value problems (BVP). In this study four boundary conditions (Neumann condition) are considered for solving BVP. Absolute error calculation, numerical stability and convergence are discussed. Two examples are considered to illustrate the finite difference method for solving fourth order BVP. The numerical results are rapidly converged with exact results. The results shows that the FDM is appropriate and reliable for such type of problems. Thus present study will enhance the mathematical understanding of engineering students along with an application in different field.

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.

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

2015 ◽  
Vol 137 (10) ◽  
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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