Dynamics of the Bogie of Maglev Train with Distributed Magnetic Forces

Hindawi Publishing Corporation Shock and Vibration Volume 2015, Article ID 896410, 12 pages http://dx.doi.org/10.1155/2015/896410 Research Article Dynamics of the Bogie of Maglev Train with Distributed Magnetic Forces Yaozong Liu,1 Wenxi Deng,2 and Pu Gong2 1 College of Mechatronics Engineering and Automation, National University of Defense Technology, Changsha, Hunan 410073, China Beijing Enterprises Holding Maglev Technology Development Co. Ltd., Changsha Branch, Changsha, Hunan 410073, China 2 Correspondence should be addressed to Yaozong Liu; Received 3 June 2015; Revised 3 August 2015; Accepted 6 August 2015 Academic Editor: Jeong-Hoi Koo Copyright © 2015 Yaozong 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. A dynamic model of the bogie of maglev train with distributed magnetic forces and four identical levitating controllers is formulated. The vertical, pitching, and rolling degree of freedom of the electromagnet modules and their coupling are considered. The frequency responses of the bogie to track irregularity are investigated with numerical simulation. The results tell us that there are resonances related to the first electromagnetic suspension whose frequencies are determined by the control parameters. A comparative analysis has been carried out between the models with distributed or concentrated magnetic forces. The comparison indicates that simplifying the distributed magnetic force to concentrated one degenerates the dynamic behavior of the maglev bogie, especially resulting in overestimated resonances of the first electromagnetic suspension of maglev trains. The results also indicate that those resonances only occur on specific wavelengths of irregularity that relate to the length of the electromagnets. 1. Introduction Due to its lower noise, less costly maintenance, and no danger of derailment, the maglev train is currently under rapid development around the world. A commercial line of high speed maglev trains has been operated in Shanghai, China, since 2003. Commercial lines of low speed maglev trains have been built in both Japan and Korea. Two new commercial lines of low speed maglev transportation are under construction in China. The maglev train is levitated by the magnetic forces between the electromagnets and the track which are adjusted by controllers in real time for stable levitation. Hence, the magnetic forces are the most fundamental elements while modeling the dynamics of maglev trains. The magnetic forces are obvious distributed along the electromagnets. But in the most of the literatures, they are simplified to concentrated forces for convenience of modeling. Liu et al. [1] proposed a proportional-differential (PD) controller with fractional orders to enhance the levitation stiffness around the operating point. The single degree of freedom (SDOF) maglev model with linearized magnetic force (LMF) was adopted. Zhou et al. [2] employed an improved least mean square algorithm with phase correction to suppress the self-excited vibration of the maglev train due to the flexibility of the track, who simplified the maglev model in the same way. Kong et al. [3] formulated sliding mode controllers (SMCs) for the whole vehicle model with three identical SDOF bogie models with LMFs to enhance the dynamic response of the maglev system for various speeds. Li et al. [4] derived feedback linearization controllers with acceleration feedback and disturbance observer for the maglev train with five bogies to improve robustness, who modeled the magnetic forces along one module of the bogie as two concentrated nonlinear forces. He et al. [5] designed a decoupling controller for the maglev module which was modeled as a rigid beam with two concentrated forces applied at fixed positions. Even in the more detailed virtual prototype simulation model of maglev train, the levitation forces are modeled as concentrated ones [6]. In earlier articles, the actively controlled electromagnetic forces are substituted with a sequence of equivalent springs and dashpots to represent the distribution [7, 8]. Owing to the simplicity, such dynamic models are significantly helpful for us to understand the dynamical behavior of maglev trains and to design and compare different control laws. But the 2 Shock and Vibration Antirolling and decoupling mechanisms Right module Antirolling and decoupling mechanisms Left module Figure 1: The three-dimensional structure of the maglev bogie. Fwr1 Ww Fwl1 Ww Fwr2 Fwl2 y z kb cb kb cb y 𝛽l Fbr1 𝛽r Wm Fzl1 Fzl2 𝛼r 1 Ground 2 3 zs1 Wgr Fzr1 Fzr2 ··· ··· (a) Forces of the maglev bogie from rear view n n + 1n + 2 zk 2n rs2 zs2 𝛿k ··· · · ·· · · rk 𝛿s1 Fzr1 Fzr2 Fbr2 Fbl2 rs1 Wgl z x Fbl1 z Fwr2 Lw Lg Fwr1 Fzrn ··· 𝛿s2 Fzrk · · · (b) Forces of the right module from side view Figure 2: Sketch of the maglev bogie on track. simplified dynamic model raises the robustness requirements of the controller and is not suitable for investigating the dynamic responses of the vehicle. In the full scale low speed maglev trains, the electromagnets are arranged in modules under the train along the track. There are four or five bogies under each vehicle. Each bogie has one electromagnet module in each side. The module is meters long and consists of several coils with common pole plates. The coils in one module are divided into two sets for the connivance of controlling, which results in the fact that each individually controlled electromagnetic force is distributed more than one meter along the pole plate. Since it is inversely proportional to the square of the levitation gap, the distributed magnetic force will produce very different moments to the module in contrast to the fabricated concentrated one, especially while the module pitched or yawed from the track. In this paper, a dynamic model of the bogie of maglev train with distributed magnetic forces is formulated. The frequency responses of the bogie to track irregularity in different traveling speed are presented, analyzed, and compared to those from the model with concentrated magnetic forces. 2. Modeling The maglev bogie is an independent unit to levitate and propel the cabin of maglev train. It consists of two rigid modules mounted with electromagnets and linear inductive motors and two antirolling and decoupling mechanisms (see Figure 1). The authors formulated the dynamic model of the maglev bogie with concentrated magnetic forces in [9]. The magnetic force of each controlled point is simplified as a fabricated concentrated one acting at the center of electromagnets. In this paper, we reformulated the dynamic model of the maglev bogie with distributed magnetic forces. The following assumptions are made while we formulate the dynamic equations: (1) The distribution (...truncated)