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

Hindawi Publishing Corporation Mathematical Problems in Engineering Volume 2013, Article ID 247268, 9 pages http://dx.doi.org/10.1155/2013/247268 Research Article A Large Span Crossbeam Vibration Frequencies Analysis Based on an Analogous Beam Method Zhifeng Liu, Bing Luo, Yongsheng Zhao, Wentong Yang, and Ligang Cai College of Mechanical Engineering and Applied Electronics Technology, Beijing University of Technology, Beijing 100124, China Correspondence should be addressed to Yongsheng Zhao; lb 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. 1. Introduction The heavy-type numerical control milling planer mainly consists of some critical functional and structural components such as crossbeam, column, slip board, slippery pillow, milling head, and worktable. It is an economic machine tool which has the characteristics of large span and high efficiency in modern large-sized workpiece machining equipments. It can realize profile milling surface processing and obtain a high machining accuracy. The crossbeam, which is a significant support component, is divided into fixed girder and dynamic beam. The crossbeam we studied has characteristics of large span and heavy load. In addition, it connects with columns and other components, bearing complex loads in working conditions. Therefore, one urgent problem that arises is how to evaluate the static and dynamic performances of crossbeam that have a great influence on machine performance and machining quality. B. P. Zhang and N. S. Zhang adopted a self-evolutionary compensation approach to reduce the deformation induced by the gravity of an 8.8 m long crossbeam [1]. Xu et al. conducted a simulation research about a 6.3 m long crossbeam and analyzed the influence of the junction plane parameter changes on static and dynamic performance of the crossbeam sliding box system [2]. Xie et al. discussed the influence of internal stiffened plate layout on the dynamic performance of crossbeam, took vibration modal relative displacement as the reference basis for design improvement, and put forward several suggestions on crossbeam improvement [3]. Zatarain et al. utilized finite element method for the machine model by the modal analysis. Finally, he selected the reasonable structure through the comparison of several schemes [4]. Guo et al. [5] analyzed a large span and heavy load crossbeam by simulation analysis and experiment research. The analogy of Christian Otto Mohr (1835–1918) allowed the computation of displacements and sloped in a linear elastic Euler-Bernoulli beam as bending moments and shear forces in a beam loaded by auxiliary forces and with modified support conditions. Since displacements and slopes can be obtained from static considerations, the analogy had found widespread attention in the engineering community. Williams’ book [6] showed that the column analogy method provided the most useful means for the determination of fixed-end moments, stiffness, and carryover factors. Ellakany et al. [7] provided the analysis of composite beams which is carried out using a combination of the transfer matrix and 2 the analog beam methods. An extension of Mohr’s analogy to bending of shear-deformable beams with eigenstrain-type actuation by Irschik and Naderb [8]. Using Mohr’s analogy, it was shown that the auxiliary loading of the adjoint beam must form a self-equilibrated system of loading in order to achieve the latter goal. Gamer [9] studied the application of the Mohr method to bending of beams with elastic joints. Irschik [10] presented a review on static and dynamic shapes control of structures by piezoelectric actuation. A simplified grillage beam analogy was performed to investigate the behaviour of railway turnout sleeper system with a low value of elastic modulus on different support moduli by Manalo et al. [11]. This study aimed at determining an optimum modulus of elasticity for an emerging technology in railway turnout application-fibre composites sleeper. Refined theories into Mohr’s analogy had been motivated by the work of Aldraihem and Khdeir [12], who pointed out that situations may occur for which the beam behavior in the region of the patches must be accurately described by higher-order beam theories. Sato et al. [13] presented the mathematical hypothesis that a beam on equidistant elastic support (BOES) can be considered as a beam on an elastic foundation (BOEF) in static and free vibration problems. A unifying numerical method was provided by Rubin in [14]. For a general discussion of the influence of shear on the deflection of beams including thermal loading, see the book by Mang and Hofstetter [15]. This class under consideration was identified by Irschik [16], who showed that various shear deformable beam theories and one-dimensional versions of plate theories can be put into a common mathematical form. Al-Sarraf and Ali [17] researched vibration analysis of plates using beam-column analogy the percentage of error depends on mesh size. ElMously [18] established a Timoshenko beam on Pasternakfoundation model that was developed for the analysis of thin elastic cylindrical shells. Szyszkowski and Grewal [19] solved optimal control problems for linear dynamic systems with quadratic performance index using the beam analogy. Dong and Meng [20] presented the thermal analogy method to predict the dynamic behavior of complex structures with piezoelectric actuators. A practical efficiency factor of circular and spiral shear reinforcements for solid and hollow core circular shear (...truncated)