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Mathematical Problems in Engineering
Volume 2013 (2013), Article ID 247268, 9 pages
Research Article

A Large Span Crossbeam Vibration Frequencies Analysis Based on an Analogous Beam Method

College of Mechanical Engineering and Applied Electronics Technology, Beijing University of Technology, Beijing 100124, China

Received 22 November 2012; Revised 9 January 2013; Accepted 9 January 2013

Academic Editor: Igor Andrianov

Copyright © 2013 Zhifeng Liu 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.


The novel method of an analogous beam is studied, which the flexural rigidity and mass per unit length correspond was described as the reciprocal of the mass per unit and the reciprocal of the flexural rigidity of the beam. It is shown that both beams possess the same natural frequencies of flexural vibration. In order to approximate calculation of these frequencies, the continuously distributed mass of the original beam is substituted for a number of concentrated masses. The analogous beam then becomes a chain of rigid links connected by pins and equipped with springs restraining the relative rotation of adjacent links. The equations of motion for the analogous beam can be solved by a procedure which consists of assuming a value for the natural frequency and calculating the deflections successively from one end of the beam to the other. Under normal circumstances, there will be a certain error, and one boundary condition will not be satisfied. The procedure is repeated with different values of the frequency until the error is removed. The method is illustrated by an example of a Crossbeam for which the fundamental frequency is found.