The governing equation of motion for free vibration of a uniform Euler-Bernoulli beam is given as (original) (raw)
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A numerical method for solving free vibration of Euler-Bernoulli beam
A method of using He's variational iteration method to solve free vibration problems of Euler-Bernoulli beam under various supporting conditions is presented in this paper. By employing this technique, the beam's natural frequencies and mode shapes can be solved and a rapid convergent sequence is obtained during the solution. The obtained results are the same as the results obtained by the Adomian decomposition method. It is verified that the present method is accurate and it provides a simple and efficient approach in solving the vibration problems of uniform Euler-Bernoulli beams. A robust and efficient algorithm is also programmed using Matlab based on the present method, which can be easily used to solve Euler Bernoulli beam problem. Keywords—Euler-Bernoulli beam, He's variational iteration method, free vibration, natural frequency, mode shape. I. INTRODUCTION The vibration problems of uniform Euler-Bernoulli beams have been solved by different approaches. Smith et a...
This paper presents a way of using He's variational iteration method to solve free vibration problems for an Euler-Bernoulli beam under various supporting conditions. By employing this technique, the beam's natural frequencies and mode shapes can be obtained and a rapidly convergent sequence is obtained during the solution. The results obtained are the same as the results obtained by the Adomian decomposition method. It is verified that the present method is accurate and it provides a simple and efficient approach for solving vibration problems for uniform Euler-Bernoulli beams. A robust and efficient algorithm is also programmed using Matlab based on the present method, which can be easily used to solve Euler Bernoulli beam problems.
Effect of Free Vibration Analysis on Euler-Bernoulli Beam with Different Boundary Conditions
2020
This paper presents an analysis of the effect of free vibrations of a free-free beam, fixed-fixed beam and simply supported beam using the series solution. It was found that the mode shape for each of the modes has effects on the displacement or deflection of such beam so that the deflection increases as the increase of the mode. Also, a Simply-Supported beam has a lower displacement compared to the free-free beam and fixed-fixed beam which almost have the same displacement. At mode one, it is seen that a Simply Supported beam has a higher amplitude, followed by a free-free beam and then a fixed-fixed beam.
Transverse Vibration Analysis of Euler-Bernoulli Beams Using Analytical Approximate Techniques
Advances in Vibration Analysis Research, 2011
The vibration problems of uniform and nonuniform Euler-Bernoulli beams have been solved analytically or approximately [1-5] for various end conditions. In order to calculate fundamental natural frequencies and related mode shapes, well known variational techniques such as Rayleigh_Ritz and Galerkin methods have been applied in the past. Besides these techniques, some discretized numerical methods were also applied to beam vibration analysis successfully. Recently, by the emergence of new and innovative semi analytical approximation methods, research on this subject has gained momentum. Among these studies, Liu and Gurram [6] used He's Variational Iteration Method to analyze the free vibration of an Euler-Bernoulli beam under various supporting conditions. Similarly, Lai et al [7] used Adomian Decomposition Method (ADM) as an innovative eigenvalue solver for free vibration of Euler-Bernoulli beam again under various supporting conditions. By doing some mathematical elaborations on the method, the authors obtained i th natural frequencies and modes shapes one at a time. Hsu et al. [8] again used Modified Adomian Decomposition Method to solve free vibration of non-uniform Euler-Bernoulli beams with general elastically end conditions. Ozgumus and Kaya [9] used a new analytical approximation method namely Differential Transforms Method to analyze flapwise bending vibration analysis of double tapered rotating Euler-Bernoulli beam. Hsu et al. [10] also used Modified Adomian Decomposition Method, a new analytical approximation method, to solve eigenvalue problem for free vibration of uniform Timoshenko beams. Ho and Chen [11] studied the problem of free transverse vibration of an axially loaded non-uniform spinning twisted Timoshenko beam using Differential Transform Method. Another researcher, Register [12] found a general expression for the modal frequencies of a beam with symmetric spring boundary conditions. In addition, Wang [13] studied the dynamic analysis of generally supported beam. Yieh [14] determined the natural frequencies and natural www.intechopen.com
Non-linear vibration of Euler-Bernoulli beams
2011
In this paper, variational iteration (VIM) and parametrized perturbation (PPM) methods have been used to investigate non-linear vibration of Euler-Bernoulli beams subjected to the axial loads. The proposed methods do not require small parameter in the equation which is difficult to be found for nonlinear problems. Comparison of VIM and PPM with Runge-Kutta 4th leads to highly accurate solutions.
Scientia Iranica, 2017
In this research, the combination of Fourier sine and cosine series is employed to develop an analytical method to conduct the free vibration analysis of an Euler-Bernoulli beam with varying cross-sections, fully or partially supported by a variable elastic foundation. The foundation sti ness and cross-section of the beam are considered as arbitrary functions in the beam's length direction. The idea behind the proposed method is to superpose Fourier sine and cosine series to satisfy the general elastically end constraints; therefore, no auxiliary functions are required to supplement the Fourier series. This method provides a simple, accurate and exible solution to various beam problems and is also able to be extended to other cases whose governing di erential equations are nonlinear. Moreover, this method is applicable to plate problems with di erent boundary conditions if twodimensional Fourier sine and cosine series are taken as a displacement function. Numerical examples are carried out, illustrating the accuracy and e ciency of the presented approach.
Nonlinear Free Transverse Vibration Analysis of Beams Using Variational Iteration Method
2018
In this study, Variational Iteration Method is employed so as to investigate the linear and non-linear transverse vibration of Euler-Bernoulli beams. This method is a very powerful approach with a high convergence speed providing an analytical and semi-analytical solution to the linear equations and is able to be extended to present semi-analytical solution to the non-linear ones. In this method, firstly, Lagrange`s multiplier and Initial Function should be chosen. The suitable choice of these two elements would effectively affect the convergence speed. In this attempt, in addition to presenting a discussion on how to choose these two functions appropriately, the calculated frequencies in the non-linear state are compared with the available results in the literature, and the accuracy and convergence speed are studied, as well.
Modal Analysis of Vibration of Euler-Bernoulli Beam Subjected to Concentrated Moving Load
Iraqi journal of science, 2020
This paper investigates the modal analysis of vibration of Euler-Bernoulli beam subjected to concentrated load. The governing partial differential equation was analysed to determine the behaviour of the system under consideration. The series solution and numerical methods were used to solve the governing partial differential equation. The results revealed that the amplitude increases as the length of the beam increases. It was also found that the response amplitude increases as the foundation increases at fixed length of the beam.
MATEC Web of Conferences
This paper deals with the geometrically nonlinear free and forced vibration analysis of a multi-span Euler Bernoulli beam resting on arbitrary number N of flexible supports, denoted as BNIFS, with general end conditions. The generality of the approach is based on use of translational and rotational springs at both ends, allowing examination of all possible combinations of classical beam end conditions, as well as elastic restraints. First, the linear case is examined to obtain the mode shapes used as trial functions in the nonlinear analysis. The beam bending vibration equation is first written in each span. Then, the continuity requirements at each elastic support are stated, in addition to the beam end conditions. This leads to a homogeneous linear system whose determinant must vanish in order to allow nontrivial solutions to be obtained. Numerical results are given to illustrate the effects of the support stiffness and locations on the natural frequencies and mode shapes of the B...