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This paper addresses the control strategy for the suppression of maglev vehicle-bridge interaction resonance, which worsens the ride comfort of vehicle and degrades the safety of the bridge. Firstly, a minimum model containing a flexible bridge and ten levitation units is presented. Based on the minimum model, we pointed out that magnetic flux feedback instead of the traditional current feedback is capable of simplifying the block diagram of the interaction system. Furthermore, considering the uncertainty of the bridge’s modal frequency, the stability of the interaction system is explored according to an improved root-locus technique. Motivated by the positive effects of the mechanical damping of bridges and the feedback channels’ difference between the levitation subsystem and the bridge subsystem, the increment of electrical damping by the additional feedback of vertical velocity of bridge is proposed and several related implementation issues are addressed. Finally, the numerical and experimental results illustrating the stability improvement are provided.

Compared with the conventional railway systems, the electromagnetic maglev system 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 [

However, when the maglev is suspended upon the bridge, standing still or moving at very slow speed, the bridge and vehicle may vibrate continuously with oversized amplitude, which is called maglev self-excited vibration. The self-excited vibration degrades the safety of the bridge and worsens the ride comfort of vehicle, which is a burning issue to be solved [

Up to the present, extensive investigations on the principle underlying the maglev self-excited vibration have been reported. Alberts et al. [

Recently, more works are focused on the engineering solutions of self-excited vibration. Generally, the solutions tend to be divided into two groups. The first group is to optimize the parameters of the bridge system, including enlarging the mass per meter [

To some extent, these optimizations are capable of avoiding the self-excited vibration. In engineering, the mass increment of bridge was widely adopted. However, it raises the initial cost significantly. The modal damping ratio is mainly determined by the bridge’s materials. Hence, the increment of modal damping ratio is a theoretical method and unavailable in engineering. Decreasing the modal frequencies of bridges is unsuitable for the completed maglev routes.

The other group is to improve the control strategy of levitation system, including optimization of the parameters and minimization of the time-delay of feedback channels [

To analyze the robustness of the control scheme to bridges with different modal frequencies, in this paper, the magnetic flux feedback instead of the traditional current feedback is proposed to simplify the block diagram of the vehicle-bridge interaction system, and an improved root-locus technique corresponding to the modal frequencies is explored.

Furthermore, to avoid the instability of the bridge with improper modal frequency and minor modal damping, a practical control scheme is explored from the perspective of theoretical source and engineering implementation. Finally, its validity is checked experimentally by the real and full-sized maglev system.

The research reported here is engineering-oriented. The purpose is to develop a practical control strategy that is capable of eliminating the self-excited vibration and is applicable to a real maglev system.

Considering the complexity of the self-excited vibration, an overall dynamic model of the interaction system with details, which is shown in Figure

The side views of CMS04 maglev system.

In this section, the maglev bridge is simplified as a Bernoulli-Euler beam due to the fact that the length of the bridge is much larger than the size of other dimensions. The nonlinearity behaviors of the bridge are neglected because the amplitude of the vibration is sufficiently small when compared with the span of bridge [

For the maglev vehicle, the dynamics of sprung mass and the coupling force between different electromagnets are neglected due to the isolation effect of air-spring and antiroll beams. Based on the above assumptions, the minimum interaction model is shown in Figure

The minimum model of maglev vehicle-bridge system.

Furthermore, the motion of bridge is described by the following differential equation [

For the simply supported concrete beam,

Here,

According to the observation of the low-speed maglev test base of china, the maglev self-excited vibration is mainly evoked by the first modal of bridge. Hence, the stability of the first modal of bridge should be emphasized and the higher modals may be neglected temporarily. In this case, (

Assuming that the number of levitation units suspended on the bridge is

In this section, the nonlinearity behavior of the bridge is neglected because the amplitude of the vibration is sufficiently small when compared to the span of bridge. According to [

Suppose the turns of a single electromagnet are

Here, variable

When examining the stability of the interaction system around the equilibrium point, the linearized model may be applied to simplify the analysis process without introducing noticeable errors. Considering the linear additive property of the linearized model, the expected current

Correspondingly, control voltage

In this case, the block diagram of the maglev vehicle-bridge interaction system is shown in Figure

The block diagram with current feedback.

Block “

According to (

As we all know, except for (

Correspondingly, control voltage

The block diagram with magnetic flux feedback.

The stability of the levitation system itself is a necessary condition for the suppression of the self-excited vibration of the maglev vehicle-bridge interaction system.

In light of Figure

When considering the flexibility of the bridge, the transfer function of the maglev vehicle-bridge interaction system from electromagnetic force

Generally, the characteristic roots may be denoted as

However, according to (

To explore the relationship between the varying modal frequency and the three real parts, the root-locus method is adopted manually by the commercial software MATLAB2012a. When the parameters are set as

The three real parts corresponding to varying modal frequency

When modal damping ratio

Pang et al. and Hong and Li pointed out that enlarging the modal damping of bridge is beneficial for avoidance of the self-excited vibration [

The three real parts corresponding to varying modal frequency

The mechanical damping of bridges is determined by its material. To a certain extent, the bridge’s stability problem could not be solved by enlarging its mechanical damping. Even so, it provides us some inspiration to avoid the self-excited vibration by improving the damping characteristics of the bridge subsystem.

Besides, according to Figure

Motivated by the positive effects of the mechanical damping and the difference of the feedback channels, we guess that the feedback of the bridge’s vertical velocity is available to improve the stability of the maglev vehicle-bridge interaction system. In this case, the expected magnetic flux is updated as

Variable

The block diagram with the bridge’s velocity feedback.

To explore the validity of control scheme (

The real parts of characteristic roots when

Hence, we conclude that the addition of the bridge’s velocity feedback to control scheme (

In light of (

In a real maglev system, two real-time signals, including levitation gap

Theoretically, the velocity signal of the electromagnet may be obtained by the integration of its acceleration signal, and the derivation of the levitation gap may be acquired by its differentiation:

However, in maglev engineering, owing to the leakage flux of electromagnet and the pulse of the chopper, levitation gap signal

Besides, as for acceleration signal

Theoretically, the addition of the bridge’s velocity feedback to control scheme (

To obtain creditable conclusions, the engineering conditions should be simulated at great length. Firstly, the overall nonlinear maglev vehicle-bridge model with detail, including the vehicle body, the secondary, the bogies and the levitation modules, the saturation of control voltages, the misalignment distribution between the actuators and sensors, is adopted. Besides, considering the eddy gap sensor is polluted by the high-frequency magnetic field, the reasonable amount of noise is applied to the gap sensors. Besides, the direct component of the acceleration transducer is set as 0.2 m/s^{2}.

In this subsection, the parameters of controller are set as

The numerical verification for vibration suppression method, which is activated at

When the time sequence is less than 1 s, expected levitation gap

Expectedly, the self-excited vibration occurs. The amplitude of electromagnet’s acceleration is up to 2 m/s^{2}, which transfers to the vehicle and degrades the ride comfort. The fluctuation of levitation gap is about 0.5 mm, which impacts the stability of the levitation system.

According to Figure

The experiments were conducted on the maintenance platform of the maglev test line, as shown in Figure

Field experiments on a full-scale maglev train at Tangshan maglev engineering base.

All the experimental data was acquired through the Ethernet based levitation monitoring network and a laptop based monitoring terminal. The data sampling rate was 200 samples per second.

Figure

The experimental verification for vibration suppression method, which is activated at

To check the validity, the improved control scheme was activated at

Considering that the maglev train consists of five bogies, we believe that the self-excited vibration will die away at last if the proposed suppression method of vibration is applied to five bogies and all activated at

Firstly, a minimum model containing a flexible bridge and ten levitation units is presented. Based on the minimum model, we conclude that the magnetic flux feedback is capable of simplifying the block diagram of the vehicle-bridge interaction system. Secondly, considering the uncertainty of the bridge’s modal frequency, the stability of the levitation system itself and the interaction system are explored according to the real parts of the three characteristic roots. Furthermore, motivated by the positive effects of the mechanical damping of bridges and the feedback channels’ difference between the levitation subsystem and the bridge subsystem, the increment of electrical damping by the additional feedback of vertical velocity of bridge is proposed and several related implementation issues are addressed. Finally, the numerical and experimental validities illustrating the stability improvements are carried out.

The authors declare that they have no competing interests.

This work was financially supported in part by the National Natural Science Foundation of China under Grant 112002230 and Grant 11302252.