## 4 Verification of Spatial Geometric Characteristic
### Track and Rotor Non-Coplanar (Skew-Mounted or Deflection Installation)
#### 4.2.2 Numerical Verification  
For numerical verification, the dimensionless parameters used are defined as follows:  
\[\begin{array}{ccccc}x=\frac{\tau_{x}}{R},&y=\frac{\
u_{x}}{R},&M=m+2m_{b},&\mu_ {b}=\frac{m_{b}}{M},&\eta=\frac{R\,m_{b}}{\varepsilon\,m},&\omega_{n}=\sqrt{ \frac{k}{M}},&\Omega=\frac{\omega_{n}}{\omega_{n}},\\ &\kappa=\frac{k_{T}}{R^{2}\,k^{2}},&\zeta=\frac{c}{2\,M\,\omega_{n}},&\zeta_{ T}=\frac{c_{T}}{2\,M\,R^{2}\,\omega_{n}},&\zeta_{b}=\frac{c_{T}}{2\,m_{b}\,R^{2}\, \omega_{n}}\end{array} \tag{38}\]  
Under the conditions of system parameters \(\theta=5^{\circ}\), \(\phi=30^{\circ}\), \(\Omega=3\), \(\kappa=0.2\), \(\mu_{b}=0.1\), \(\zeta=0.2\) and \(\zeta_{b}=\zeta_{T}=0.1\), the relationship between \(\eta\) and the ball balance positioning numerically, as shown in Figure 8. The solid line is the solution obtained by satisfying the spatial-geometric relationship of the axis that is perpendicular to the balls and the track center axis. The part with \(\eta<0.5\) has two real solutions (and a pair of conjugate complex solutions). Starting from \(\eta\geq 0.5\), four real solutions appear, and as the value of \(\eta\) gradually increases (about 1.93), two real solutions remain still. The hollow marked points in the figure are the ball balance positions obtained through numerical integration using the governing equation listed in [9]. It is worth noting that among the two equilibrium solutions in the \(\eta<0.5\) interval, only one is a stable solution, so the two balls gather together. When \(\eta=0.5\), although there are four equilibrium solutions, only one of them is stable. Two of the four equilibrium solutions in the interval \(0.5<\eta<1.93\) are stable. When \(\eta>1.93\), the only two equilibrium solutions are stable.  
Figure 8: Ball balance positioning under different \(\eta\) for the system with skew-mounted ABB.  
Figure 7: Ball balance positions: When the spin axis (**a**) passes through the track center; and does not pass through the track center (**b**,**c**).  
In addition, take \(\eta\) = 0.2 and \(\eta\) = 1 in Figure 8 for numerical examination. Plot the transient responses of the system and label their steady-state equilibrium values, as illustrated in Figures 9 and 10, respectively.  
Taking \(\eta\) = 0.2 as an example, the numerical results when the system is in a steady state are described in the spatial coordinates mentioned in Section 4, it can be seen that the radius of curvature \(R_{i}\), the positional coordinates of the curvature center \((x_{c_{i}},y_{c_{i}},z_{c_{i}})\), the pitch angle \(\phi_{1}\) and the yaw angle \(\phi_{2}\) of the ball track are \(R_{i}\) = 1, \(x_{c_{i}}=-0.275\), \(y_{c_{i}}=-0.192\), \(z_{c_{i}}=0\), \(\phi_{1}\) = 1.15\({}^{\circ}\) and \(\phi_{2}\) = 3.672\({}^{\circ}\), respectively. By using the spatial geometric characteristic \(\Delta_{i}\cdot(\mathbf{L}_{1i}\times\mathbf{L}_{2i})\) = 0 of the ball positioning, the possible equilibrium positions of the balls can be found at 34.57\({}^{\circ}\) and \(-144.69^{\circ}\). Similarly, when \(\eta\) = 1, the possible equilibrium positions of the ball are found to be 30.10\({}^{\circ}\), 120.70\({}^{\circ}\), 148.76\({}^{\circ}\), and \(-90.25^{\circ}\). Through comparison, it can be seen that the results derived from the spatial geometry can indeed denote the balls' possible balance positions.