Stability and control of maglev vehicle–girder coupled system considering torsional vibration of the girder

Highlights

• This paper investigates the principle of the maglev vehicle–girder coupled torsional resonance.

• A maglev girder (with track) modeling method with a searching algorithm is proposed.

• We find that the velocity feedback may lead to girder torsional resonance.

• A distributed virtual tuned mass damper scheme is proposed to suppress the girder torsional resonance.

• The proposed method can suppress the vehicle–girder coupled torsional resonance.

Abstract

The EMS (Electro-Magnetic Suspension) maglev vehicle uses controlled electromagnetic forces to achieve stable levitation; however, the interactions between the vehicle and the flexible girder may induce strong coupled resonances. Recently, a new type of girder resonance has been encountered, and field measurements indicate that this resonance is caused by torsional vibration of the girder. However, up to now, the principle of the girder torsional resonance has not been studied, and related control strategies have not been developed either. In this paper, a typical maglev girder with a steel track is investigated through finite element analysis, and a searching algorithm is proposed to obtain a best fit of the girder transfer function. Then, a maglev vehicle with twenty levitation control units is considered, and together with the girder model, the vehicle–girder coupled system is established. Factors that affect the coupled resonances are analyzed, and it is found that the velocity feedback of the electromagnet, which is effective to suppress the bending resonance of the girder, may lead to torsional resonance of the girder. This conclusion is validated by experiments in a real maglev system. To solve this problem, a distributed virtual tuned mass damper scheme is discussed, and suggestions to avoid the torsional resonances are also presented. This work not only interprets the principle underlying the vehicle–girder coupled torsional resonance, but also presents a methodology for dealing with the maglev vehicle–girder coupled resonance problems.

Introduction

The maglev transport gained firm progress in the past ten years. In May, 2015, the first urban maglev system in China – the Changsha Maglev Express – began its commercial operation in Changsha, Hunan province. Later in December, 2017, another urban maglev system – the Beijing S1 maglev line – was opened to the public. After years of commercial demonstration, the maglev system is attracting more and more application interests in crowded cities due to its low noise and prominent climbing capability. To further extend the application of the maglev system, the total cost of the maglev project needs to be reduced. A feasible way is to reduce the cost of the guideway system, which is estimated to take up 60%$\sim$80% of the total initial capital [1]. One suggestion is to cut the weight and volume of the maglev girder, which is also helpful to make the maglev guideway more artistic; another suggestion is to apply steel girders instead of concrete girders. However, both of these schemes decrease the stiffness of the girder, which, as a result, may worsen the maglev vehicle–girder coupled resonance. In history, this problem has caused many serious problems to the maglev systems around the world. It was reported that in early development period, the HSST maglev system and the TR04 maglev system experienced obvious vehicle–guideway coupled vibration. Due to the importance of the guideway to the stability of the maglev system, the first commercial high speed maglev line in Shanghai, China, employed rather heavy elevated girders to support the train, and it was reported that the mass of the concrete girder reached 7 t/m. Meanwhile, the coupled resonance problem also occurred in some other medium-low speed maglev systems, which utilize rather lighter girders [2], [3]. From 2008 to 2012, both the CMS-03A and the CMS-04 maglev vehicle experienced serious vehicle–girder coupled resonances on a short elevated bridge on the Tangshan maglev test line, and the large amplitude vibration of the girder even caused the electromagnets to clash with the track. Based on these experiences, the maglev track design is still an important aspect to be cautiously considered when a new maglev line is built.

Different from traditional wheel–rail system, the guideway resonance phenomena occur more frequently and they arouse worldwide research interests on the maglev vehicle–guideway interaction problems. For the high speed maglev system, early works mainly focused on the vertical vibration problems of the girders induced by a moving maglev vehicle [4], [5], [6], and fundamental works on the modeling of maglev girders were developed at this stage. Later, it was found that the guideway may suffer from large amplitude vibration when the maglev train was suspending in a standstill, which was known as the self-excited vibration problem and received increasing attentions [7], [8], [9], [10]. In these researches, the flexible guideway model plays an important role in the study of the maglev vehicle–guideway interaction problem. For the elevated bridge, researchers incline to simplify the bridge as a Bernoulli–Euler beam, and simply supported beam [1], [2], [7], [11] or multi-span beam [12] are usually employed. This model treats the flexible beam as a superposition of mass–spring–damper​ resonators; therefore, researchers design equivalent test equipment to study the vehicle–guideway dynamic problems. In [13], a vibration test bench was designed to investigate the maglev vehicle–guideway coupled dynamics with flexible track support and track irregularity, and concluded that when the match of the track support stiffness and the track beam mass is inappropriate, the vehicle-track resonance may occur. On the other hand, the steel track, which is paved on top of maglev girders, is another factor that may cause resonance problems. In [14], the influence of the steel track on the maglev train–bridge system was investigated, and it was concluded that the effect of the track should not be neglected. In [15], a multi-span Euler beam model with flexible supports was established to obtain a mathematical expression of the track model. Based on this model, the amplitude of the track resonance was investigated in [16], which agreed well with practical measurement. Recently, an analytical model of the full scale track, which consists of the sleepers, F-rails, and flexible track supports, was established [17]. This model is more accurate in investigating the coupling between different levitation units. For complex structures, the finite element analysis (FEA) method is used. In [18], the high speed maglev guideway model is firstly established in the FEA software to give an initial model of the guideway, which is further updated by an incremental iteration procedure to improve its precision. In [19], the FEA method is applied in the study of the maglev vehicle–guideway–tunnel–soil interaction problem when a high speed EMS (Electro-Magnetic Suspension) maglev vehicle travels through a tunnel.

As for the principle of the maglev vehicle–guideway coupled resonances, some researchers believe that the mode frequency of the guideway being close to the frequency of the controller would result in self-excited vibration problem [13]. Meanwhile, Zhou et al. [7] concluded that for low damping girders, the mode frequencies of the girder being higher than the critical frequency of the levitation control system would cause instability problem. Time delay is also a factor that affects the stability of the coupled system [11], [20]. Xu et al. [11] studied the time delay in the levitation gap and velocity feedback paths, and the stable domain constrained by the delays of the two variables was obtained. To solve the maglev vehicle–guideway​ coupled resonances, several procedures have been proposed, including control parameter tuning scheme [21], vibration energy absorber scheme [7], [22], adaptive vibration cancellation method [15], etc.

So far, the above mentioned researches mainly concern the vertical bending vibration of the girder/track. However, recently, a different type of girder resonance has been encountered, and the waveforms of the levitation gap and the current are shown in Fig. 1. The data was recorded when the fully loaded CMS-04 train was slowly passing a steel switch girder on the Tangshan maglev test line. It can be seen that the levitation gap and the levitation current began to oscillate dramatically, and at last the electromagnet clashed with the track, resulting in a levitation failure. The frequency of the vibration was 28.5 Hz, much higher than the ordinary vertical bending mode frequency of the girder (around 13.5 Hz). Further tests suggested that this resonance was caused by the torsional vibration mode of the girder. However, the maglev vehicle–girder coupled torsional vibration problem has not been studied before. Yet this problem must be solved before the next generation maglev system, which is expected to adopt lighter girders, is put into commercial application. Therefore, the motivation of this work is to go deep into the reasons why the torsional resonance occurs, and to develop a suitable control method to eliminate the maglev vehicle–girder coupled torsional resonance.

The major work of this paper includes: (1) modeling of the full scale maglev girder. A typical steel box girder is taken as an example, its frequency responses are to be investigated using the FEA method; then its transfer functions, including the torsional modes, are obtained by an optimal searching algorithm; (2) the multi-point levitation control system of the urban maglev vehicle is established, and together with the girder model, the maglev vehicle–girder coupled model is established; (3) the principle of the self-excited vibration problem is studied based on the model established in the previous step, and the reason that causes girder torsional instability problem is discussed; (4) To solve the girder torsional resonance problem, a distributed virtual tuned mass damper scheme is discussed, and some suggestions to avoid the torsional vibration of the girder is put forward.