# Study on Beat Vibration of a High Temperature Superconducting EDS Maglev Vehicle at Low Speed
## 3 Theory Vertical Equilibrium Point  
Because the vehicle is running at a low speed of approximately 14 mps while the _MMF_ is approximately 310 kAt, we sought to discover the location of the theory vertical equilibrium point. Its location is a balance point where the levitation force is equal to gravity, and is useful for designing the initial conditions in multi-body dynamics. The levitation force is decomposed into the oscillation part and average part, which is used to determine the theory vertical equilibrium point, as in Ref. [9]. The levitation force \(F_{z}^{i}\) is of a periodic quality in Figure 5 and can be divided into the average part \(F_{z\_\text{mean}}^{i}(z)\) and oscillation part \(F_{z\_\text{osc}}^{i}(x,z)\), as shown in Figure 6a,b. Based on the Fourier technique, the levitation force \(F_{z}^{i}(x,z)\) on the \(i\)-th SCM can be decomposed as in Equation (15):  
\[F_{z}^{i}(x,z)\approx F_{z\_\text{mean}}^{i}(z)+F_{z\_\text{osc}}^{i}(z,x)\ (i=1,2,3,4) \tag{15}\]  
where \(F_{z\_\text{mean}}^{i}(z)\) is the average part of the levitation force, and the oscillation part of the levitation force is:  
\[F_{z\_\text{osc}}^{i}(z,x)=\sum_{n=1}^{N}A_{n}^{i}(z)\sin(\frac{2\pi}{Pitch} nx+\varphi_{n}^{i}(z)) \tag{16}\]  
The average part of the levitation force \(F_{z\_\text{mean}}^{i}(z)\) in Equation (15) is a non-linear function of \(z\) and serves as a non-linear spring force, as shown in Figure 6b. The oscillation part functions as the exciting force. To decipher the theory vertical equilibrium point, assuming \(\gamma(t)=0\), which is according to the calculation of the levitation force, the force diagram of the vehicle in Figure 3c can be simplified as in Figure 6c or Figure 6d. Considering Equation (13), Equation (8) vanishes and Equation (7) should be solved. Substituting Equation (15) into Equation (7), the vertical dynamics equation becomes:  
\[m\ddot{z}(t)=mg-\sum_{i=1}^{4}2F_{z\_\text{mean}}^{i}(z)-\sum_{i=1}^{4}2F_{z\_ \text{osc}}^{i}(z,x) \tag{17}\]  
or  
\[m\ddot{z}(t)+F_{z\_\text{all}}(z)=mg-F_{z\_\text{osc}}(x,z) \tag{18}\]  
Figure 6: Decomposition of levitation force: (**a**) \(i\)-th levitation force \(F_{z}^{i}(x,z)\); (**b**) decomposition of \(i\)-th levitation force; (**c**) force diagram of vehicle with \(\gamma(t)=0\); (**d**) decomposition of force diagram of vehicle with \(\gamma(t)=0\). where:  
\[F_{x\_\_\_\text{all}}(z)=\sum_{i=1}^{4}2F_{x\_\_\_\text{mean}}^{i}(z) \tag{19}\]  
\[F_{x\_\_\text{osc}}(x,z)=F_{x\_\_\text{osc}}(vt,z)=\sum_{i=1}^{4}2\sum_{n=1}^{N} A_{n}^{i}(z)\sin(\frac{2\pi}{Pitch}nvt+\varphi_{n}^{i}(z))) \tag{20}\]  
The average part of all the levitation forces \(F_{x\_\_\text{all}}(z)\) depends on z and plays the role of the non-linear spring force. \(F_{x\_\text{osc}}(x,z)\) depends on both z and \(x\) functions as the exciting force. When treating the levitation force in the simulation, experts may make \(N=1\) to reduce the modelling difficulties in the simulation, as in Ref. [9]. Based on the curve fitting method, \(F_{x\_\_\text{all}}(z)\) can be fitted by three-order polynomial expression:  
\[F_{x\_\_\text{all}}(z)=a_{1}z^{3}+a_{2}z^{2}+a_{3}z \tag{21}\]  
where \(a_{1},\ a_{2}\), and \(\ a_{3}\) are the fitting parameters. There is no constant term on the right side of Equation (21) as \(F_{x\_\_\text{all}}(0)=0\). From Equation (18), the theory vertical equilibrium point \(z_{0}\) (\(0\) mm \(<z_{0}<80\) mm) satisfies Equation (22):  
\[F_{x\_\_\text{all}}(z_{0})\approx a_{1}z_{0}^{3}+a_{2}z_{0}^{2}+a_{3}z_{0}=mg \tag{22}\]  
\(F_{x\_\_\text{all}}(z)\) nonlinearly depends on z (\(z\in[0,2,4,\cdots,100]\) mm) and the increment of z is as small as 2 mm, so we may roughly identify the equilibrium point from Figure 7a at different velocities. In fact, we carry out curve fitting through four points, which are the three local points near the equilibrium point \(z_{0}\) and the original point \((0,0)\) to improve the accuracy.