# Study on Beat Vibration of a High Temperature Superconducting EDS Maglev Vehicle at Low Speed
## 5 Results and Discussion  
### Simulation Results  
Figure 9 shows the response of \(z(t)\) and \(\gamma(t)\) when \(v=14\) mps, \(MMF=310\) kAt, and \(z(0)=40\) mm. The \(z(t)\) and \(\gamma(t)\) show the phenomenon of beat vibration, which means there are two components with two closing frequencies. Figure 10 shows the \(\ddot{z}\) and \(\ddot{\gamma}\) response, where the phenomenon of beat vibration exists. The results in Figures 9 and 10 illustrate that the work in this paper successfully predicted the phenomenon of beat vibration. Furthermore, the experimental records in Figure 2 also verify the validity of the dynamic circuit model and mechanical model in this paper.  
The harmonic exciting frequencies in Equation (16) are \(\frac{nv}{pitch}=\frac{nv}{0.36}\) Hz \((n=1,2,\cdots,N)\), i.e., \(38.9n\) Hz when \(v=14\) mps. These exciting frequencies are too high to be confirmed only with one's sense of sight. There are two natural vibration components besides the forced vibration components from the vibration theory. The two natural frequencies of the vehicle are \(\lambda_{1}\) and \(\lambda_{2}\) because the vertical displacement \(z\) and pitch angle \(\gamma(t)\) are coupled with each other.  
Figure 11 is the FFT transformation of the vertical displacement acceleration \(\ddot{z}\) and pitch angle acceleration \(\ddot{\gamma}\). The \(\lambda_{1}=2\) Hz and \(\lambda_{2}=2.4\) Hz are close to each other and are the two natural frequencies, where \(z\) and \(\gamma\) are coupled with each other. This is the reason for the phenomenon of beat vibration. The frequencies \(38.9\) Hz and \(77.8\) Hz correspond to the first order harmonic exciting frequency \(\frac{v}{pitch}=\frac{14}{0.36}\) Hz and the second order harmonic exciting frequency \(\frac{2 nv}{pitch}=\frac{2\times 14}{0.36}\) Hz due to the arrangement of figure-eight- shaped coils, respectively. At approximately \(38.9\) Hz and \(77.8\) Hz, there is a small frequency band where there are local peak frequencies of \([\frac{nv}{pitch}-\lambda_{2},\frac{nv}{pitch}-\lambda_{1},\frac{nv}{pitch},\frac{nv}{pitch}+\lambda_{1},\frac{nv}{pitch}+\lambda_{2}]\)\((n=1,2,\cdots,N)\). This is due to the amplitude and phase of the \(n\)-th harmonic levitation force components \(A^{i}_{n}(z)\) and \(\phi^{i}_{n}(z)\)\((n=1,2,\cdots,N)\) depending on \(z\) in Equation (16), while \(z\) has the harmonic components of the eigenvalues \(\lambda_{1}\) and \(\lambda_{2}\). If \(A^{i}_{n}(z)\) and \(\phi^{i}_{n}(z)\) are invariant with \(z\) near the equilibrium point, there must be no such small frequency bands near the \(n\)-th harmonic exciting frequency \(\frac{nv}{pitch}\). The outstanding frequency order of the levitation exciting force is two, as shown in Figure 11. Therefore, it is better to set \(N=2\)  
\begin{table}
\begin{tabular}{l l l} \hline \hline
**Factor** & **Value** & **Description** \\ \hline \(z(0)\) & \([30,31,\cdots,50]\) mm & Initial vertical displacement, level number is 21. \\ \hline \(v\) & \([13,14,15]\) mps & Velocity of vehicle, level number is 3. \\ \hline _MMF_ & \([300,310,320]\) kAt & MMF of SCM onboard, level number is 3. \\ \hline \hline \end{tabular}
\end{table}
Table 3: Simulation factors and levels.