The Modeling and Analysis for the Self-Excited Vibration of the Maglev Vehicle-Bridge Interaction System

Hindawi Publishing Corporation Mathematical Problems in Engineering Volume 2015, Article ID 709583, 10 pages http://dx.doi.org/10.1155/2015/709583 Research Article The Modeling and Analysis for the Self-Excited Vibration of the Maglev Vehicle-Bridge Interaction System Jinhui Li, Jie Li, Danfeng Zhou, and Lianchun Wang College of Mechatronics Engineering and Automation, National University of Defense Technology, Changsha 410073, China Correspondence should be addressed to Jie Li; Received 27 June 2014; Accepted 1 March 2015 Academic Editor: Miguel A. F. Sanjuan Copyright © 2015 Jinhui Li 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. This paper addresses the self-excited vibration problems of maglev vehicle-bridge interaction system which greatly degrades the stability of the levitation control, decreases the ride comfort, and restricts the cost of the whole system. Firstly, two levitation models with different complexity are developed, and the comparison of the energy curves associated with the two models is carried out. We conclude that the interaction model with a single levitation control unit is sufficient for the study of the self-excited vibration. Then, the principle underlying the self-excited vibration is explored from the standpoint of work acting on the bridge done by the levitation system. Furthermore, the influences of the parameters, including the modal frequency and modal damping of bridge, the gain of the controller, the sprung mass, and the unsprung mass, on the stability of the interaction system are carried out. The study provides a theoretical guidance for solving the self-excited vibration problems of the vehicle-bridge interaction systems. 1. Introduction Compared with the conventional rail-way systems, the electromagnetic maglev system (EMS) has advantages of lower noise, less exhaust fumes emission, less maintenance cost, and the ability to climb steeper slopes, which is a new kind of urban transport that has been widely concerned in recent years [1–3]. The rapid development and enormous advantages of maglev system sketch a bright future for its commercial applications. However, the self-excited vibration of bridge is a burning issue to be solved. It occurs when the vehicle is suspended upon the guideway, standing still or moving at very slow speed [4]. It degrades the safety of bridge and durability of bridge. Furthermore, the self-excited vibration deteriorates the stability of the levitation system. The American maglev technologies (AMT) system achieved successful levitation in Florida on a bridge mounted to the earth on a concrete foundation but later encountered difficulties in achieving stable levitation when the vehicle was moved to an elevated bridge installed on the Old Dominion University campus [5]. It was believed that the flexibility of the bridge, which employed 90-feet long, was the main reason that contributed to the difficulties of achieving a stable levitation. For the maglev CMS04 system of China, the self-excited vibration may occur if the parameters of PD controller are unsuitable. Figure 1 shows the recorded data from Tangshan maglev engineering base when the self-oscillation occurs. According to Figure 1, when the self-excited vibration occurs, the electromagnet will vibrate vertically. Furthermore, the vibrations of electromagnet will be transferred to the cabin of vehicle, which is harmful for the riding comfort for passengers. Besides, the current fluctuations of the electromagnet impact the power system violently, which may lead to its collapse. To explore the principle underlying the self-excited vibration and solve it, extensive researches have been reported. The derivation of bifurcation equations and numerical simulation using the center manifold method are carried out [6]. They believed that the bifurcations, such as the homoclinic, Hopf bifurcation, secondary Hopf bifurcation, and chaos are the causes of self-excited vibration. In [7] the authors believed that the self-excited vibration is due to the improper frequency relationship between various 2 Mathematical Problems in Engineering Gap (mm) 9.5 9 Acc. of EM (m/s2 ) Current (A) 8.5 0 0.5 1 Time (s) 1.5 2 0 0.5 1 Time (s) 1.5 2 0 0.5 1 Time (s) 1.5 2 30 25 20 2 0 −2 Figure 1: Recorded data from field when the self-oscillation occurred. components of the system. In [8] the authors believed that the self-excited vibration is more likely to occur if the difference between the modal frequency of bridge and the natural frequency of controller is sufficiently small. The influences of signal delay on the stability of nonlinear levitation system are studied by [9–11]. The analysis shows that when the time delay reaches a critical value, the system will undergo a subcritical bifurcation, and the periodic vibration will occur. The bridge in the maglev route is simply supported, and its span is large compared with other dimensions, and the vertical deflection of the bridges is small compared with its span when the self-excited vibration occurs. In almost all the published literatures [12, 13], the elevated bridge is modeled as a Bernoulli-Euler beam. A consensus in maglev field has been reached, which will be adopted in this paper. However, the development of the levitation model is relatively complicated. There have been many magnetic levitation models created for different purposes. Generally, these models are classified into two categories, one for the research of dynamics [14] and the other for the analysis and synthesis of the control system [15]. For the first kind, to obtain a precise and creditable dynamic response, the details of the system should be considered in all directions, so that this kind of model is relatively complicated. For the second kind, to simplify the process of analysis and synthesis of the control system, some inessential parts should be neglected and only the quintessential parts should be included. For the analysis and synthesis of the self-excited vibration, various models with different complexities are adopted in published literatures. In [2], it is believed that the secondary suspension system of the vehicle can be neglected in the analysis, and the model of a single levitation unit-bridge coupled system was adopted. However, in [16], the structure of a single levitation unit-bridge with secondary suspension system was included. In [17], a complicated interaction model with a levitation module and secondary suspension system are developed. Yet as of today, to the authors’ knowledge, no effective theoretical method has been reported for the selection of a suitable minimum interaction model. As we all known, the rather larger error may be introduced if an oversimplified levitation model is selected. On the contrary, the theoretical der (...truncated)