Effect of Free Vibration Analysis on Euler-Bernoulli Beam with Different Boundary Conditions (original) (raw)
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The governing equation of motion for free vibration of a uniform Euler-Bernoulli beam is given as
2017
EVERAL techniques have been used to carry out the vibration analysis of beams with a view to determining their vibration characteristics. Lai, et al. [1] analysed the free vibration of uniform Euler-Bernoulli beam with different elastically supported conditions using Adomian decomposition method (ADM). Li [2] had earlier studied the vibration characteristics of a beam having general boundary conditions. The displacement of the beam was sought in form of a linear combination of a Fourier series and an auxiliary polynomial function. Kim and Kim [3] also applied Fourier series to determine the natural frequencies of beams having generally restrained boundary conditions. Later, Liu and Gurram [4] adopted the He’s variational iteration method to estimate the vibration frequencies of a uniform Euler-Bernoulli beam for various supporting end conditions. Natural frequencies for the first six modes of vibration were presented in their work. Malik and Dang [5] employed the differential transf...
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...
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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.
Free Vibration of Beam on Continuous Elastic Support by Analytical and Experimental Method
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ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik, 2018
Free vibration of an Euler‐Bernoulli beam with rigid bodies connected to the beam ends by both revolute joints and torsional springs is considered. The mass centers of rigid bodies have both the transverse and the axial eccentricity relative to the neutral axis of the undeformed beam. The coupling of the partial differential equations of axial and bending vibrations of the beam due to boundary conditions is considered. The frequency equation and the mode shape orthogonality condition of the system are derived. In order to illustrate the effect of the transverse and the axial eccentricity on the vibration behavior of the beam, a numerical example is provided.
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
Chinese Journal of Engineering, 2013
The study of the dynamic properties of beam structures is extremely important for proper structural design. This present paper deals with the free in-plane vibrations of a system of two orthogonal beam members with an internal elastic hinge. The system is clamped at one end and is elastically connected at the other. Vibrations are analyzed for different boundary conditions at the elastically connected end, including classical conditions such as clamped, simply supported, and free. The beam system is assumed to behave according to the Bernoulli-Euler theory. The governing equations of motion of the structural system in free bending vibration are derived using Hamilton's principle. The exact expression for natural frequencies is obtained using the calculus of variations technique and the method of separation of variables. In the frequency analysis, special attention is paid to the influence of the flexibility and location of the elastic hinge. Results are very similar with those o...
Exact Solution for Free Vibration Analysis of FGM Beams
Revue des composites et des matériaux avancés, 2020
This study relates the exact solution for free-vibration analysis of beams in material gradient (FGMs) subjected to the different conditions of support using the Euler Bernoulli theory (CBT). It is assumed that the material properties continuously change across the thickness of the beam according to the exponential function (E-FGM). The equations of motion are obtained by applying the principle of virtual works on beams and fundamental frequencies are found by solving the equations governing the eigenvalue problems. Numerical results are presented to describe the influence of the material on the fundamental frequencies of the beam for different state boundaries.