# Study on Beat Vibration of a High Temperature Superconducting EDS Maglev Vehicle at Low Speed
## 2 Governing Equations
### Levitation Force Based on Dynamic Circuitry Theory  
In this study, the figure-eight-shaped coils are not connected with each other, so the levitation force on the \(i\)-th left SCM is equal to the levitation force on the \(i\)-th right SCM. We only need to determine the levitation force on each SCM on the left side of the vehicle. As shown in Figure 4, assuming the vehicle moves at a constant velocity of \(v\) and \(z\) along the tracks with \(\gamma=0\), based on the dynamic circuitry theory, the current in the \(j\)-th figure-eight-shaped coil \(i_{j}\) satisfies Equation (9):  
\[\begin{bmatrix}R_{1}&&\\ &R_{2}&&\\ &&\cdots&\\ &&R_{n}\end{bmatrix}\begin{bmatrix}i_{1}\\ i_{2}\\ i_{n}\end{bmatrix}+\begin{bmatrix}L_{11}&L_{12}&\cdots&L_{1n}\\ L_{21}&L_{22}&&L_{2n}\\ \cdots&\cdots&\cdots&\cdots\\ L_{n1}&\cdots&\cdots&L_{nn}\end{bmatrix}\begin{bmatrix}i_{1}\\ i_{2}\\ i_{n}\end{bmatrix}=-v_{x}\begin{bmatrix}G_{11}&G_{12}&\cdots&G_{1m}\\ G_{21}&G_{22}&&G_{2m}\\ \cdots&\cdots&\cdots&\cdots\\ G_{n1}&\cdots&\cdots&G_{nm}\end{bmatrix}\begin{bmatrix}I_{1}\\ I_{2}\\ I_{m}\end{bmatrix} \tag{9}\]  
where \(I_{j}=I_{\text{SCM}}\) (\(j=1,\cdots,m\); \(m=4\)) is the current in the SCM aboard the vehicle; \(L_{ij}(i,j=1,\cdots,n)\) is the mutual inductance between the \(i\)-th and \(j\)-th loop coils on the guideway; and \(R_{i}\) (\(i=1,\cdots,n\)) is the resistance of the \(i\)-th loop coil. Both \(L\) and \(R_{i}\) are constant if the loop coils are all identical. The term \(G_{ij}(i=1,\cdots,n\) and \(j=1,\cdots,m)\) is the derivative with respect to \(x\) of the mutual inductance \(M_{ij}\) between the \(i\)-th loop coil on the guideway and the \(j\)-th superconducting coil on the vehicle, and \(M_{ij}\) is a function of the space location. Based on the virtual displacement principle, the magnetic forces on the \(j\)-th SCM in the \(x\)-direction, \(y\)-direction, and \(z\)-direction are:  
\[F_{x}^{j}(t)=\sum_{p=1}^{n}i_{p}I_{j}\frac{\partial M_{pj}}{\partial x}\ (j=1,2,3,4) \tag{10}\]  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Scheme** & **X/m** & **Y/m** & **Z/m** \\ \hline
1 st left SCM & \(1.35\) & \(-1.033\) & \(0\) \\
2 nd left SCM & \(0.81\) & \(-1.033\) & \(0\) \\
3 rd left SCM & \(-0.81\). & \(-1.033\) & \(0\) \\
4 th left SCM & \(1.35\) & \(-1.033\) & \(0\) \\
1 st right SCM & \(1.35\) & \(1.033\) & \(0\) \\
2 nd right SCM & \(0.81\) & \(1.033\) & \(0\) \\
3 rd right SCM & \(-0.81\) & \(1.033\) & \(0\) \\
4 th right SCM & \(-1.35\) & \(1.033\) & \(0\) \\ \hline \hline \end{tabular}
\end{table}
Table 2: Location of SCMs onboard.  
\[F_{y}^{j}(t) =\sum_{p=1}^{n}i_{p}p_{j}\frac{\partial M_{pj}}{\partial y}\ (j=1,2,3,4) \tag{11}\] \[F_{2}^{j}(t) =\sum_{p=1}^{n}i_{p}I_{j}\frac{\partial M_{pj}}{\partial z}\ (j=1,2,3,4) \tag{12}\]  
The higher the MMF of the SCM, the larger the current in the SCM and the larger the levitation force. The time-dependent or space-dependent levitation forces on the \(i\)-th left SCM and \(i\)-th right SCM are the same and satisfy Equations (13) and (14):  
\[F_{x\_{\text{left}}}^{i}(t)=F_{x\_{\text{right}}}^{i}(t)=F_{\text{z}}^{i}(t) \ (i=1,2,3,4) \tag{13}\]  
\[F_{\text{z}}^{i}(x)=F_{\text{z}}^{i}(vt)\ (i=1,2,3,4) \tag{14}\]  
In this paper, the space-dependent levitation force \(F_{\text{z}}^{i}(x)\ x=[0,\cdots,200]\ \text{m}\) on the \(i\)-th SCM \((i=1,2,3,4)\) is calculated under the condition of \(z\in[0,2,\cdots,100]\ \text{mm}\), \(v\in[14,15,16]\ \text{mps}\), and \(MMF\in[300,310,320]\ \text{kAt}\) when pitch \(\gamma(t)=0\). When the vehicle is moving at a constant velocity, the magnetic forces are at a period of the pole pitch. Figure 5 shows in detail the \(x\)-dependent levitation force \(F_{\text{z}}^{i}(x)\) on the \(i\)-th SCM for a different \(z\) when \(v=13\ \text{mps}\) and \(MMF=300\ \text{kAt}\).  
Figure 4: Levitation force on each SCM (\(F_{x\_{\text{left}}}^{i}(t)=F_{x\_{\text{right}}}^{i}(t)=F_{x}^{i}(t)\)).