## 3 Numerical Analysis
### Time Response Analysis  
The time histories of all working Conditions are obtained, and the tilting (orange line) and center point movement (blue line) of four discs at three Conditions are drawn in Figures 9-11. In each figure, the deformation of the spinning discs are increased from (a) to (d), and sub-figures (i) to (iii) show the responses at different rotation speeds. There are a few general observations from the 3 Conditions. First, the process of tilting and axial movement can be very short after the impulsion. Generally less than 1 s, take Figure 10d as an example, the motion of 15\({}^{\circ}\)-deformed disc under Condition 2 lasts the \begin{table}
\begin{tabular}{c c c} \hline \hline
**Group** & **Impulse Force** & **Boundary Distance** \\ \hline Condition 1 (reference group) & 5 N & 7 mm \\ Condition 2 & 7 N & 7 mm \\ Condition 3 & 5 N & 6 mm \\ \hline Deformation angle [\({}^{\circ}\)] & 0 (flat), 5, 10, 15 & \\ Angular velocity [rad/s] & 40, 120, 200 & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Simulation parameters. longest time, and the movements still ends within a second. Secondly, the tilting and axial movement of the disc are strongly correlated. In most cases they are hard to describe when two motions exist simultaneously such as Figure 10, which indicates that the discs constantly hit the boundary so the impact force keeps exerting on the discs. And once one movement stops, the other stops soon after. Additionally, the axial movement usually are very nonlinear, but when there is only tilting while the axial movement stops, as shown in Figure 10(iii),b(iii), the tilting amplitude shows linear decay until to zero, which suggests if there is no boundary as well as the external force, free tilting motion is a decay motion, and the disc always aligns back to the vertical position. Figures 9 and 10 show the time responses with different impulse forces. The increase in impact force makes the motions of all discs more intense and longer lasting. For example, in Figure 9a, the tilting and axial translation of the flat disc stops at an instant, meaning that no impact occurs between the flat disc and boundary at Condition 1. When the impulse force increases from 5 N to 7 N in Figure 10a, the flat disc has a chance to impact the boundary so the motion becomes chaotic and last longer than Condition 1. In addition, comparing Figures 9 and 11 show the influences of narrowing the boundary distance, somehow the effect is similar as to increase the impulsion, the motions of the four discs become more chaotic in Condition 3 (Figure 11) than in Condition 1 Figure 9. The reason is that the impact probability increases as the boundary distance decreases. Another influence is the boundary distance restrict the tilting angle. so in Figure 11, the amplitudes of the tilting angle of all discs are below \(15^{\circ}\), smaller than those in Condition 1 and 2. Notably, the final axial position of the discs is difficult to predict. It is not always at the middle of two boundaries; it varies with the rotation speed, deformation of the discs, impulse force and boundary distance. For instance, \(5^{\circ}\)-deformed disc stops very close to the middle of the boundary in Figure 9b, while in Figure 10b it stops at 2 mm, 3 mm and \(-1\) mm with different rotation speeds, and stops at 2 mm, \(-0.5\) mm and \(-0.5\) mm in Figure 11b. So the final steady displacement quite random which applies to all discs. As for the influence of the deformation, one observation from Figures 9-11 is that the discs with larger deformation angle has more intense and complex motions. Take \(5^{\circ}\)-deformed disc and \(10^{\circ}\)-deformed disc for comparison, in Condition 1 (Figure 9), the \(5^{\circ}\)-deformed disc return to the axial equilibrium position immediately after the impulsion, while the \(10^{\circ}\)-deformed disc tilts for 0.2 s at 40 rad/s and tilts even longer as angular velocity increases, and motions of \(15^{\circ}\)-deformed disc are more chaotic than \(10^{\circ}\)-deformed disc.