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Mathematical Problems in Engineering
Volume 2015, Article ID 582326, 10 pages
http://dx.doi.org/10.1155/2015/582326
Research Article

Element for Beam Dynamic Analysis Based on Analytical Deflection Trial Function

1College of Water Resources & Civil Engineering, China Agricultural University, Beijing 100083, China
2China Aerospace Construction Group Co., Ltd., Beijing 100071, China

Received 18 September 2014; Accepted 15 December 2014

Academic Editor: Chenfeng Li

Copyright © 2015 Qiongqiong Cao et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

For beam dynamic finite element analysis, according to differential equation of motion of beam with distributed mass, general analytical solution of displacement equation for the beam vibration is obtained. By applying displacement element construction principle, the general solution of displacement equation is conversed to the mode expressed by beam end displacements. And taking the mode as displacement trial function, element stiffness matrix and element mass matrix for beam flexural vibration and axial vibration are established, respectively, by applying principle of minimum potential energy. After accurate integral, explicit form of element matrix is obtained. The comparison results show that the series of relative error between the solution of analytical trial function element and theoretical solution is about and the accuracy and efficiency are superior to that of interpolation trial function element. The reason is that interpolation trial function cannot accurately simulate the displacement mode of vibrating beam. The accuracy of dynamic stiffness matrix method is almost identical with that of analytical trial function. But the application of dynamic stiffness matrix method in engineering is limited. The beam dynamic element obtained in this paper is analytical and accurate and can be applied in practice.