# Study on Beat Vibration of a High Temperature Superconducting EDS Maglev Vehicle at Low Speed
## 4 Multi-Body Dynamic Model of the EDS Vehicle  
From Figure 5, the average parts of the levitation forces on the third and fourth SCM are larger than that on the first and second SCM, so the vehicle would pitch. This section shows a new numerical multi-body dynamic model developed to study the vertical displacement and pitch angle in SIMPACK script. The levitation forces are handled by MATLAB/SIMULINK and the tool MATSIM is used to exchange the data.  
(1) Decoupling of Levitation forces  
From Section 2.2, when calculating the levitation force, \(F_{z}^{i}(x,z)\) on the \(i\)-th SCM depends on \(x\) and \(z\) with the constant \(v\) and \(MMF\). The levitation forces on the SCMs are coupled with each other through \(z\), which is the origin of the body coordinate system \(O-XYZ\). To evaluate the pitch angle, this paper decouples the levitation forces from each other. In other words, \(F_{z}^{i}(x,z)\) is changed to \(F_{z}^{i}(x,z_{i})\), where \(z_{i}\) is the coordinate of \(i\)-th SCM.  
The levitation force \(F_{z}^{i}(x,z_{i})\) is treated as the exciting force, not decomposed as the right hand of Equation (15). Practically, the levitation force \(F_{z}^{i}(x,z_{i})\) can be interpolated from a 2D matrix \(\mathbf{F}_{z}^{i}(\mathbf{x},\mathbf{z}_{i})\), where \(\mathbf{x}=[0,\cdots,200]\) m and \(\mathbf{z}_{i}=[0,2,\cdots,100]\) mm. For \(v\in[13,14,15]\) mps and \(MMF\in[300,310,320]\) kAt, a certain levitation force matrix \(\mathbf{F}_{z}^{i}(\mathbf{x},\mathbf{z}_{i})\) is formed by Equation (23):  
\[\mathbf{F}_{z}^{i}(\mathbf{x},\mathbf{z}_{i})=\begin{bmatrix}\mathbf{F}_{z}^ {i}(x)\ for\ z_{i}=0\\ \mathbf{F}_{z}^{i}(x)\ for\ z_{i}=2\\ \vdots\\ \mathbf{F}_{z}^{i}(x)\ for\ z_{i}=100\end{bmatrix}(i=1,\ 2,\ 3,\ 4) \tag{23}\]  
(2) Model elements in multi-body dynamic model  
Figure 8 shows a multi-body dynamic model of the vehicle with six DOF in SIMPACK script. Eight sensor elements obtain the \(x\) and \(z_{i}\) values of each SCM in order to decipher the levitation force. Eight control elements (MATSIM) exchange the data between SIMPACK and MATLAB/SIMULINK, which stores the levitation force matrix \(\mathbf{F}_{z}^{i}(\mathbf{x},\mathbf{z}_{i})\). Eight force elements apply the levitation force value from the control elements to the vehicle. Four contact force elements are used to model the guiding wheels.  
Figure 8: Multi-body co-simulation dynamic model with eight levitation force elements.  
(3) Simulation conditions  
Because it is not easy to exactly control the experimental velocity, initial vertical displacement, and MMF, it is better to study the effect of these factors on the dynamic performance of the vehicle to obtain an overall impression. Of course, when all other problems, such as the control system, are solved, an experimental investigation can be carried out. The goal of this paper is to uncover the phenomenon of beat vibration by multi-body dynamics. We assume that the vehicle is running at a steady speed with different initial conditions. Three factors are considered: the initial vertical displacement \(z(0)\), velocity \(v\), and _MMF_. The levels of these factors are shown in Table 3. Using the traversing method, considering the influence of these three parameters, the multi-body dynamics model runs \(21\times 3\times 3=189\) times.  
(4) Solver setting  
The simulation time is 10 s; the sampling rate is 2000 Hz; and the integration method is SODASKT 2.