Journal of Mechanical Science and Technology 32 (9) (2018) 4017~4024 www.springerlink.com/content/1738-494x(Print)/1976-3824(Online) DOI 10.1007/s12206-018-0801-9 A reduced time-varying model for a long beam on elastic foundation under moving loads † Guiming Mei 1 , Caijin Yang 1,* , Shulin Liang 1 , Jiangwen Wang 1 , Dong Zou 1 , Weihua Zhang 1 , Yunshi Zhao 2 , Zhong Huang 1 , Shuqi Song 1 , Mengying Tan 1 , Yao Cheng 1 and Bingrong Miao 1 1 State Key Laboratory of Traction Power, Southwest Jiaotong University, Chengdu 610031, China 2 Institute of Railway Research, University of Huddersfield, Huddersfield, HD1 3DH, UK (Manuscript Received May 21, 2017; Revised May 8, 2018; Accepted May 11, 2018) ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- Abstract Dynamics of a long beam on the elastic foundation subjected to moving loads is studied in the present paper. The sliding window tech- nique is used to dynamically truncate the long beam and a reduced time-varying beam system is obtained. The Hamilton’s principle is employed to establish the equations of motion of the reduced system. The variable separation method is adopted to solve dynamical re- sponses of the reduced system. Examples of a long simply supported Timoshenko beam on the nonlinear foundation subjected to a single moving load and multiple loads are included. Numerical results of the reduced model compared with the ones obtained from the moving element model adapted in literature are carried out to show the validity and the good efficiency of the method proposed in the present paper. Keywords: Long beam; Moving load; Reduction; Time-varying; Dynamic response; Moving element method ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- 1. Introduction The dynamic analysis of beams on elastic foundations under moving loads has been a topic of interest in many engineering fields, especially in railway engineering. A vast amount of literature exists on this topic. Among these previous efforts, three main methods are widely used to tackle the moving load problems, namely the Fourier transform method, the finite element method and the mode superposition method. The Fourier transform method is a frequency domain ap- proach, which can produce the analytical solutions. The Fou- rier transform method was sought by many researchers. Mathews [1, 2] investigated the dynamic response of an infi- nitely long beam resting on an elastic foundation under an arbitrary moving load using this method. This method was also used by Trochanis et al. [3], Ono and Yamada [4] and Sun [5-7] to solve different problems. However, the Fourier transform method has two major limitations, as pointed out by Yu and Yuan [8]. In Ref. [8], they improved it and derived the analytical solutions of an infinite Euler–Bernoulli beam on a visco-elastic foundation subjected to arbitrary dynamic loads. Unlike the Fourier transform method, the finite element method and mode superposition method are numerical ap- proaches. In the finite element method, the beam is discretized into a finite number of elements with given lengths. Based on nodal coordinates and assumed shape function of the element, mass and stiffness matrices of all elements are computed and then assembled to form the resulting equations of motion of the beam. Motion equations of the beam are numerically solved to yield dynamical responses of the system by means of several time-stepping methods such as the Newmark’s method, the Wilson’s method and the RKF45 method, etc. Filho [9] presented a review on the application of the finite element method to dynamic problems of a uniform beam trav- ersed by a moving load. Andersen [10] obtained the finite element solution of the problem with the loads moving uni- formly along an infinite Euler beam supported by a linear elastic Kelvin foundation with linear viscous damping. Qiu [11] numerically analyzed the dynamic response of a flexible beam floating in an unbounded water domain under the effect of moving loads using the finite element method. Wu et al. [12] discussed the dynamic response of different structures to moving loads based on the finite element techniques. Chang [13] presented dynamic finite element analysis of a nonlinear beam subjected to a moving load. Of the three methods, the mode superposition method may be most widely used in practice. It attempts to approximate the solutions by using the first few natural modes of the beam, instead of all modes, thus it is often faster and less expensive. * Corresponding author. Tel.: +86 13980459159 E-mail address: ycj78_2012@163.com † Recommended by Associate Editor Sungsoo Na © KSME & Springer 2018