# Vertical Dynamic Response Prediction of the Electromagnetic Levitation Systems
## 2 Vertical Dynamic Modeling of the Electromagnetic Levitation System  
### Modeling of the Electromagnetic Levitation System  
As the module levitation systems in the maglev train are decoupled with each other, we take a single module levitation system as the typical analysis object, which can fully reflect the vertical dynamic difference of the levitation system. Therefore, the vertical dynamic model, including a rigid track and a rigid electromagnet module, is established, which is shown in Figure 1. The electromagnet module moves at a constant speed \(v\) to the right, which is the positive direction of the X-axis. The origin \(O\) of the coordinate system of the electromagnet corresponds to the right terminal of the electromagnet. There are two degrees of freedom of the electromagnetic module: vertical movement in the Y-axis and rotation around the center of the mass. Variable \(\theta\) is the rotation angle and its positive direction is presented in Figure 1.  
In Figure 1, variables \(y_{r}\) and \(y_{e}\) are the vertical displacement in the Y-axis of the track and the electromagnet, respectively. Variables \(\delta_{1}\) and \(\delta_{2}\) are the levitation gaps measured by the two gap-sensors, which are configured at the two terminals of the electromagnet module. There are four coils in an electromagnet module, which are connected by series and divided into two units controlled by two voltage-chopping controllers, separately. Each controller uses the measured levitation gap \(\delta\) to control the coil current \(i\), thus achieving stable levitation gap control. Variables \(F_{e1}\) and \(F_{e2}\) are the electromagnetic attraction forces produced by the electrified electromagnet and the magnetic track, which are evenly distributed along the pole face. Variable \(m_{e}\) is the equivalent mass of the electromagnet module, and \(m_{b}\) is the equivalent mass of the vehicle body transmitted through air-springs. According to Figure 1, the vertical displacements of \(y_{r0}\) and \(y_{r1}\) are deduced by \(y_{r2}\) :  
\[y_{r0}\left(t\right)=y_{r2}\left(t-2l/v\right) \tag{1}\]  
\[y_{r1}\left(t\right)=y_{r2}\left(t-4l/v\right) \tag{2}\]  
where \(l\) is the length of a single coil. The measured levitation gaps \(\delta_{1}\left(t\right)\) and \(\delta_{2}\left(t\right)\) are:  
\[\delta_{1}\left(t\right)=y_{e1}\left(t\right)-y_{r2}\left(t-4l/v\right) \tag{3}\]  
\[\delta_{2}\left(t\right)=y_{e2}\left(t\right)-y_{r2}\left(t\right) \tag{4}\]  
The average gaps \(\delta_{1}\) and \(\delta_{2}\) are the mean value of the gap integration along the pole surface of the electromagnet:  
\[\delta_{1}\left(t\right)=\frac{3}{4}y_{e1}\left(t\right)+\frac{1}{4}y_{e2} \left(t\right)-\frac{v}{2l}\int_{-4l/v}^{-2l/v}f_{r}\left(t+\tau\right)d\tau \tag{5}\]  
\[\delta_{2}\left(t\right)=\frac{1}{4}y_{e1}\left(t\right)+\frac{3}{4}y_{e2} \left(t\right)-\frac{v}{2l}\int_{-2l/v}^{0}f_{r}\left(t+\tau\right)d\tau \tag{6}\]  
The relationship between controlled voltage \(u\left(t\right)\) and current \(i\left(t\right)\) for each single coil of two units are as follows:  
\[\frac{u_{1}\left(t\right)}{2}=Ri_{1}\left(t\right)+\frac{u_{0}N^{2}A}{2\delta _{1}\left(t\right)}i_{1}\left(t\right)-\frac{u_{0}N^{2}Ai_{1}\left(t\right)}{ 2\delta_{1}^{2}\left(t\right)}\delta_{1}\left(t\right) \tag{7}\]  
Figure 1: Vertical dynamic model of the electromagnet module and the track. Figure 1.0  
where \(R\) is the resistance of a single-coil, \(N\) is the turn number of a single-coil, \(A\) is the pole area corresponding to the length of a single-coil, and \(u_{0}\) is the magnetic permeability of a vacuum. If the magnetic leakage and the magnetic resistance of the track and electromagnet are ignored, the average electromagnetic forces provided by each two coils can be expressed as follows:  
Figure 1.1  
Figure 1.2  
The vertical movement equation can be expressed as:  
Figure 1.3  
where \(g\) is the acceleration of gravity. The rotation equation is:  
Figure 1.4  
where \(I_{e}=4m_{e}l^{2}/3\) is the inertia moment of the electromagnet module.