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 Modern Mechanical Engineering (MME) , 2015, DOI: 10.4236/mme.2015.54011 Abstract: In the present study, finite element dynamic analysis or time history analysis of two-span beams subjected to asynchronous multi-support motions is carried out by using the moving support finite element. The elemental equation of the element is based on total displacements and is derived under the concept of the quasi-static displacement decomposition. The use of moving support element shows that the element is very simple and convenient to represent continuous beam moving, deforming and vibrating simultaneously due to support motions. The comparison between the numerical results and analytical solutions indicates that the FE result agrees with the analytical solution.
 Mathematical Problems in Engineering , 2011, DOI: 10.1155/2011/920651 Abstract: The electromagnetic launcher's rail can be modeled as a beam on elastic foundation with simply supported beam by moving load. In this paper, Euler beam theory is applied to build the mechanical model, and the analytical solution of the equation subjected to sinusoidal magnetic pressure is derived in detail, which has successfully avoided the errors which are caused by using the uniform pressure to approximately replace the variable force. Numerical analysis of the influences brought from the elastic coefficient, the damping coefficient, the mass of rail, and the load's velocity on the deformation of beam by the MATLAB software show that the elastic coefficient and the load's velocity have quite obvious effect on the deformation of the beam while the damping coefficient and the mass of rail have not obvious effect on the deformation of the beam.
 Modern Applied Science , 2010, DOI: 10.5539/mas.v4n8p127 Abstract: To solve the accurate calculation of force-deformation of the electromagnetic launcher’s rail , this is helpful to extend the rail life and improve the firing accuracy. Therefore, the electromagnetic launcher’s rail can be modeled as a beam on elastic foundation with simply supported beam by moving load. In this paper Euler beam theory is applied to build the Mechanical model and the analytical solution of the equation subjected to logarithmic magnetic pressure is derived in detail, which has successfully avoided the errors which are caused by using the uniform pressure to approximately replace the variable force. The numerical analysis brings from the elastic coefficient, the damping coefficient, the mass of rail and the load’s velocity have influence on the deformation of beam by the MATLAB software. The consequence shows that the elastic coefficient and the load’s velocity have quite obvious affect on the deformation of the beam while the damping coefficient and the mass of rail have not obvious affect on the deformation of the beam. It laid the foundation for solve the electromagnetic launcher’s rail subjected to magnetic pressure of arbitrary function and promote the practicality of the electromagnetic guns.
 Computer and Information Science , 2011, DOI: 10.5539/cis.v4n5p59 Abstract: In order to extend the rail life and improve the firing accuracy, the electromagnetic launcher’s rail can be modeled as a beam on elastic foundation with simply supported beam by moving load. Euler beam theory is applied to build the Mechanical model and the analytical solution of the equation subjected to cosine magnetic pressure is derived in detail, which has successfully avoided the errors which are caused by using the uniform pressure to approximately replace the variable force. Numerical analysis of the dynamic response on rail by the MATLAB software shows that as the exciting frequency increases, the peek of the maximal deflection and vibration velocity increase gradually. Taking the same speed of load into account, the dynamic response of rail is obviously smaller than constant force. Therefore the reliable theory basis is provided for the design and control the rail, which promotes the practical application of the electromagnetic Launcher.
 Latin American Journal of Solids and Structures , 2010, DOI: 10.1590/S1679-78252010000200005 Abstract: the resonance characteristics of a two-span continuous beam traversed by moving high speed trains at a constant velocity is investigated, in which the continuous beam has uniform span length. each span of the continuous beam is modeled as a bernoulli-euler beam and the moving trains are represented as a series of two degrees-of-freedom mass-springdamper systems at the axle locations. a method of modal analysis is proposed in this paper to investigate the vibration of two-span continuous beam. the effects of different influencing parameters, such as the velocities of moving trains, the damping ratios and the span lengths of the beam, on the dynamic response of the continuous beam are examined. the two-span continuous beam has two critical velocities causing two resonance responses, which is different from simple supported beam. the resonance condition of the two-span continuous beam is put forward which depends on the first and second natural frequency of the beam and the moving velocity.
 Shock and Vibration , 2014, DOI: 10.1155/2014/406093 Abstract: This study presents a technique that uses a model reduction method for the dynamic response analysis of a beam structure to a moving load, which can be modeled either as a moving point force or as a moving body. The nature of the dedicated condensation method tailored to address the moving load case is that the master degrees of freedom are reselected, and the coefficient matrices of the condensed model are recalculated as the load travels from one element to another. Although this process increases computational burden, the overall computational time is still greatly reduced because of the small scale of motion equations. To illustrate and validate the methodology, the technique is initially applied to a simply supported beam subjected to a single-point load moving along the beam. Subsequently, the technique is applied to a practical model for wheel-rail interaction dynamic analysis in railway engineering. Numerical examples show that the condensation model can solve the moving load problem faster than an analytical model or its full finite element model. The proposed model also exhibits high computational accuracy. 1. Introduction A structure subjected to moving loads is a common situation in mechanical and civil engineering. Some examples include a shaft workpiece to a lathe tool, a bridge to cars and vehicles, and a rail to cranes or rail vehicles. Various models have been proposed for this dynamic analysis. Although they may be applied in different engineering domains, these models share similar characteristics. Two approaches are generally employed to build a model, that is, an analytical method or a finite element (FE) method. Various analytical solutions to a uniform simply supported beam with a single span exist in the excellent monographs of Frba . Mamandi and Kargarnovin  modeled a continuous beam with Timoshenko beam theory which is applicable for beams with a large height to span ratio. For slender beams, Bernoulli-Euler beam theory is promising. Hayashikawa and Watanabe  used a continuum method of dynamic analysis for multispan continuous beams. Zheng et al.  modified beam vibration functions, which are used as assumed modes to analyze the vibration of multispan nonuniform beams. Lalthlamuana and Talukdar  investigated dynamic interaction between a single-span bridge and moving flexible vehicles using a semianalytical approach. Analytical methods can provide accurate solutions to simple moving load problems, but they are sometimes cumbersome for modeling complex practical structures. Alternatively, the FE method has been
 Journal of Engineering and Applied Sciences , 2012, Abstract: The response of non-initially stressed Euler-Bernoulli beam with an attached mass to uniform partially distributed moving loads was examined. The governing partial differential equations were analyzed for both moving force and moving mass in order to determine the dynamic behavior of the system. The response amplitude due to the moving force was greater than that of the moving mass. The response amplitude of the moving force problem with non-initial stressed increased as the mass of the load M was increased.
 Journal of Applied Sciences , 2004, Abstract: In this study, the dynamic behavior of laminated composites subjected to a single force traveling at a constant velocity has been investigated. Three-dimensional finite element model based on classical lamination theory was used. The dynamic analyses for simply supported composite beams under the action of moving loads are carried out by the finite element method. The dynamic magnification, which is an important parameter in the case of moving load, has been obtained for different load velocities and ply orientations. The results for laminated composite beams under the action of a moving load have been illustrated and compared to the results for an isotropic simple beam. The results have shown that the traveling velocities and ply orientations have significant influence on the dynamic responses of composite beams, specially for those with s layup.
 Folake Oyedigba Akinpelu Applied Mathematics (AM) , 2012, DOI: 10.4236/am.2012.33031 Abstract: The paper investigates the response of non-initially stressed Euler-Bernoulli beam to uniform partially distributed moving loads. The governing partial differential equations were analyzed for both moving force and moving mass problem in order to determine the behaviour of the system under consideration. The analytical method in terms of series solution and numerical method were used for the governing equation. The effect of various beam observed that the response amplitude due to the moving force is greater than that due to moving mass. It was also found that the response amplitude of the moving force problem with non-initial stress increase as mass of the mass of the load M increases.
 I.A. Adetunde Research Journal of Applied Sciences , 2012, Abstract: This study is concerned with dynamical behavior of Euler Bernoulli beam traversed by uniform partially distributed moving masses. The governing partial differential equation was systematically analyzed and the analytical numerical solution for classical boundary condition obtained. The deflection of the Euler- Bernoulli beam is calculated under various specified conditions and the results displayed graphically. It is found that moving force solution is not an upper bound for an accurate solution of the moving mass problem.
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