# Vertical Dynamic Response Prediction of the Electromagnetic Levitation Systems
## 2 Vertical Dynamic Modeling of the Electromagnetic Levitation System
### Linear Model in the Frequency Domain
The track amplitude \(A_{r}\left(\lambda\right)\) is deduced from \(S_{v}\left(\lambda\right)\):  
\[A_{r}\left(\lambda\right)=k_{r}\sqrt{S_{v}\left(\lambda\right)} \tag{39}\]  
According to practical experience, we set \(k_{r}=0.0247\). **3. Vertical Dynamic Prediction under Track Irregularity**  
The characteristic equations of transfer function \(G_{s}\) is:  
\[ \begin{split}\Delta\left(s\right)&=\left(2L_{0}\;m _{e}\right)s^{4}+\left(\left(2R+k_{c}\right)m_{e}+5k_{c}k_{A}k_{i}\right)s^{3} \\ &\quad+\left(5k_{c}k_{D}k_{i}+3.5k_{i}^{2}\right)s^{2}+\left(5k_{ c}k_{p}k_{i}-3.5k_{z}\right)s\\ &\quad+\left(5k_{c}k_{l}k_{i}\right)\end{split}  \tag{40}\]  
We select parameters according to the actual parameters of the low-speed maglev train: \(m_{e}=1000\) kg, \(m_{b}=1300\) kg, \(N=360\), \(R=0.55\)\(\Omega\), \(A=0.0185\) m\({}^{2}\), \(l=0.66\) m, \(u_{0}=4\pi\times 10^{-7}\). The control parameters are: \(k_{p}=4000\), \(k_{D}=80\), \(k_{l}=20,000\), \(k_{a}=0.2\), and \(k_{c}=110\). Obviously, the parameters above satisfy the Routh criterion of Equation (40), which means the levitation system is stable. As the dynamic model is affected by the track wavelength \(\lambda\) and the vehicle speed \(v\) at the same time, to depict this influence, we have drawn a 3-D graph related to \(\lambda\) and \(v\). The 3-D amplitude spectrums of the levitation gaps are shown in Figure 2.  
According to Figure 2, the following conclusions can be drawn:  
1. When the speed is very low, the amplitudes of both levitation gaps are relatively small at different wavelengths, which means that the electromagnet module can strictly track the space trajectory of the track. With the increase of the speed, the amplitude spectrum values of the levitation gaps generally rise, which means that the tracking performance of the electromagnet module is reduced. Figure 2: Amplitude spectrums of levitation gaps. (**a**) Gap1 (the back), (**b**) Gap2 (the front). When the wavelength is close to 0, the amplitude spectrum values of both levitation gaps are relatively small at different speeds. In this case, the electromagnet module keeps its inertial position stable and does not track the space trajectory of the track. Therefore, the amplitude of the levitation gap is equal to the amplitude of the track. However, as the amplitude of the track at small wavelengths is small, the amplitude values of the levitation gaps are relatively small. Although this means that the tracking performance of the levitation control system is very weak, it is consistent with the phenomenon that the high-frequency signal is ignored, which is expected in the levitation control system. 3. The maximum value appears in the case of high speed and large wavelength, which is consistent with engineering experience. When the speed is very high, we need to pay special attention to the track irregularities at large wavelengths. Due to the larger amplitude of the long-wave track irregularities, the contributions of the long-wave track irregularities to the gap deviation is also large. 4. The amplitudes of the two levitation gaps are different. Moreover, in a broad range of wavelength and speed, the amplitudes of the back gap are larger than that of the front gap. As the input amplitudes of the two levitation gaps are the same, the difference in the transfer functions is the main reason for the dynamic response difference.