# Vertical Dynamic Response Prediction of the Electromagnetic Levitation Systems
## 2 Vertical Dynamic Modeling of the Electromagnetic Levitation System
### Analysis of Typical Wavelengths  
There are typical periodical track irregularities in the maglev track, which are related to the sleeper pace, F-rail section, and rail beam span. Here, we select typical wavelengths: 1.2 m (sleeper pace), 6 m (F-rail section), 9 m (half-track beam span), and 18 m (typical simple-elevated track beam span). The amplitude spectrums of levitation gaps related to vehicle speed at different wavelengths are shown in Figure 4. The amplitudes of the periodical track irregularities \(A_{r}\) at different wavelengths are shown in Figure 4 too, which provide a reference to the amplitude spectrums of levitation gaps.  
According to Figure 4, for the short-wave wavelength of 1.2 m (sleeper pace), the maximum points of the two levitation gaps appear at 13 km/h (the back gap) and 12 km/h (the front gap), respectively. When the speed is lower than 13 km/h, the amplitudes of the front levitation gap are larger than that of the back levitation gap, which is different from the general trend. With the increase of speed, the amplitudes of the two levitation gaps tend to be equal to the track input amplitude at high speed.  
For the wavelength of 6 m (F-rail section), the maximum point of the back levitation gaps appears at 63 km/h. In general, the amplitudes of the back levitation gap increase first and then decrease, while the amplitudes of the front levitation gap experience twice the increase, with a local minimum point at around 100 km/h.  
For the wavelength of 9 m (half-track beam span), the maximum point of the back levitation gap appears at 124 km/h. The amplitudes of the front levitation gap also experience twice the increase, with a local minimum point at around 150 km/h.  
For the wavelength of 18 m (typical simple-elevated track beam span), the maximum points of the two levitation gaps appear at 263 km/h (the back gap) and 213 km/h (the front gap), respectively. In a certain speed range, the amplitude of the front levitation gap is larger than that of the back levitation gap. The tracking performance is strong when the amplitude of the levitation gap is less than the  
Figure 4: Amplitude spectrums of levitation gaps vs. vehicle speed. (**a**) at 1.2 m, (**b**) at 6 m, (**c**) at 9 m, (**d**) at 18 m.  
input amplitude of the track. Therefore, the electromagnet module has a good tracking ability for the wavelength of 18 m when the speed is less than 158 km/h.  
In general, the two levitation units have different amplitude responses at different wavelengths, which need to be analyzed specifically. The general trend is that at low speed, the amplitude response is smaller than the track input, and the tracking ability is strong. At high speed, the tracking ability is weak, and the amplitude response is even larger than the track input amplitude. Then with the continuous increase of speed, the amplitude response is equal to the track input amplitude. This phenomenon is consistent with the characteristic that the electromagnetic levitation system tracks the low-frequency signal and ignores the high-frequency signal. From the previous analysis, it can be concluded that the levitation system has good tracking performance for a track profile change below 2 Hz.