# Study on Beat Vibration of a High Temperature Superconducting EDS Maglev Vehicle at Low Speed
## 5 Results and Discussion
### Vertical Displacement and Pitch Angle Performance under Different Conditions  
Based on the traversing method, the vertical displacement \(z(t)\) and pitch angle \(\gamma(t)\) are calculated by multi-body dynamics 189 times. Four parameters are defined to provide an overview of the vertical movement and pitch angle performance:  
1. The mean value of the vertical displacement \(z_{\text{mean}}\), which indicates that the vertical balance point is defined as: \[z_{\text{mean}}=\text{mean}(z(t))\] (24)
2. The amplitude of the vertical displacement \(z_{\text{amp}}\) is defined as: \[z_{\text{amp}}=\frac{\text{max}(z(t))-\text{min}(z(t))}{2}\] (25)
3. The mean value of the pitch angle of the vehicle \(\gamma_{\text{mean}}\), which indicates that the mean pitch angle is defined as: \[\gamma_{\text{mean}}=\text{mean}(\gamma(t))\] (26)
4. The amplitude of the pitch angle of the vehicle is defined as \(\gamma_{\text{amp}}\): \[\gamma_{\text{amp}}=\frac{\text{max}(\gamma(t))-\text{min}(\gamma(t))}{2}\] (27)  
\(z_{\text{mean}}\), \(z_{\text{amp}}\), \(\gamma_{\text{mean}}\), and \(\gamma_{\text{amp}}\) for the various initial vertical displacement \(z(0)\), velocity \(v\), and the \(MMF\) of the SCM are plotted in Figures 12-14.  
Figure 14: Simulation results when \(MMF=320\) kAt: (**a**) \(z_{\text{mean}}\); (**b**) \(z_{\text{amp}}\); (**c**) \(\gamma_{\text{mean}}\); (**d**) \(\gamma_{\text{amp}}\).  
Because the sum of the levitation forces on the third and fourth SCMs are larger than the sum of that on the first and second SCMs and C.G is on top of the origin of the body system, the negative pitch angle of the vehicle (Figures 12c, 13c and 14c) and the mean value of each \(z\) displacement \(z_{\text{mean}}\) (Figures 12a, 13a and 14a) are slightly larger than the theory equilibrium point \(z_{0}\) calculated in Section 3. This indicates a good accuracy of the theory equilibrium point.  
Influence of initial vertical displacement \(z(0)\)  
The mean value of the vertical displacement \(z_{\text{mean}}\) shows a trend of first decreasing and then increasing as the initial vertical displacement \(z(0)\) increases, as shown in Figures 12a, 13a and 14a. The amplitude of the vertical displacement \(z_{\text{amp}}\) shows a trend of first decreasing and then increasing as the initial vertical displacement \(z(0)\) increases, as shown in Figures 12b, 13b and 14b. Additionally, the vertical displacement becomes larger when \(z(0)\) is away from the theory vertical equilibrium point.  
The absolute mean value of the pitch angle \(|\gamma_{\text{mean}}|\) shows a trend of first increasing and then decreasing as the initial \(z(0)\) increases, as shown in Figures 12c, 13c and 14c. The amplitude of the pitch angle \(\gamma_{\text{amp}}\) shows a trend of first decreasing and then increasing as the initial \(z(0)\) increases, as shown in Figures 12d, 13d and 14d.  
Influence of velocity of vehicle \(v\)  
As the velocity \(v\) increases, the mean value of the vertical displacement \(z_{\text{mean}}\), the amplitude value of the vertical displacement \(z_{\text{amp}}\), and the absolute mean value of the vertical displacement of vehicle \(|\gamma_{\text{mean}}|\) show a trend of decreasing, as shown in Figure 12a-c to Figure 14a-c.  
The amplitude of the pitch angle \(\gamma_{\text{amp}}\) shows a trend of first decreasing and then increasing as the initial \(z(0)\) increases in Figures 12d, 13d and 14d.  
Influence of MMF of SCM  
As the MMF of the SCM increases, a decreasing trend is depicted in Figures 12-14, which mention the variables of the mean value of the vertical displacement \(z_{\text{mean}}\), the amplitude value of the vertical displacement \(z_{\text{amp}}\), the absolute mean value of the vertical displacement \(|\gamma_{\text{mean}}|\), and the amplitude of the pitch angle \(\gamma_{\text{amp}}\). This means that an increasing MMF of the SCM is good for vibration control.