# Vertical Dynamic Response Prediction of the Electromagnetic Levitation Systems
## 4 Numerical Simulation  
We have carried out numerical simulations to verify the correctness of the theoretical analysis. By using the IFFT method, the track irregularity data obtained from the low interference German PSD function is shown in Figure 6. The spacial step of the data is 0.25 m, and the total length is 3.276 km. The maximum value is 3.02 mm, and the minimum value is \(-\)2.93 mm. The PSD diagram of simulation data and the low interference German PSD function are shown in Figure 7. As shown in Figure 7, the PSD value of the simulation data is basically the same as that of the German track within the set wavelength range of 0.05-  
The root mean square error (RMSE) function is adopted to measure the deviation degree of time-domain signal, which has the property of variance and the physical signal unit at the same time. The RMSE is defined as:  
\[\text{RMSE}=\sqrt{\frac{1}{n}\sum_{i=1}^{n}\left(\delta_{i}-\delta_{0}\right)^{ 2}} \tag{41}\]  
Here, \(\delta_{0}=9\) mm in Figure 8.  
According to Figure 8, the RMSE value of the back levitation gap is 74.2% larger than that of the front levitation gap. At the same time, the deviation range of the two levitation gaps are \([-2.547,2.256]\) mm (the back), and \([-1.541,1.364]\) mm (the front), respectively. These are consistent with the previous analysis result in Section 3.1 that the deviation degree of the back levitation gap is larger than that of the front levitation gap.  
According to Figure 9, the amplitude spectrums of the numerical simulation signals are highly approximate to the theoretical analysis results, which verifies the correctness of the theoretical analysis method. Due to the error of the IFFT method, there are local fluctuations in the amplitude spectrums in Figure 9.  
Figure 8: Levitation gap signals in numerical simulation at 80 km/h. (**a**) Gap1 (back), (**b**) Gap2 (front).  
Figure 9: Amplitude spectrums of levitation gap errors at 80 km/h. (**a**) Theoretical calculation, (**b**) numerical simulation.  
The RMSE values of levitation gap errors at different speeds in the numerical simulation are shown in Figure 10. The maximum values of levitation gap errors at different speeds are shown in Figure 11.  
According to Figure 10, it is found that the RMSE values of both levitation gap errors generally increase with the increase of vehicle speed and tend to be stable when the speed is greater than 200 km/h. The maximum value of the levitation gap error in Figure 11 has the same trend with the RMSE value. At the same time, it is found that these trends are consistent with that of the amplitude spectrum area in Figure 3, which means that the amplitude spectrum area in the theoretical calculation is the key indicator to present the deviation degree of the levitation gap signals.  
According to Figure 11, under the excitation of the german low interference track irregularity, the maximum value of the levitation gap error may exceed the safe range when the speed is larger than 250 km/h. The actual track irregularity condition may be worse than that of the low interference track. Due to the fact that the strength of the urban maglev track is weaker than that of the high-speed maglev track and the simple-supported beam is often used, the corresponding wavelength amplitude  
Figure 11: Maximum values of levitation gap errors vs. vehicle speed.  
Figure 10: RMSE values of levitation gap errors vs. vehicle speed.  
of the simple-supported beam is relatively large, which means the actual levitation system may exceed the safe range when the speed is smaller than 250 km/h.