## 5 Results and Discussion  
### Influences of Centrifugal Force  
Under the condition of high speed, the multi-layer interference cylinders must ensure that contact pressure is always greater than the minimum contact pressure to transmit required torque [21]. According to Equation (11), the contact pressure between the contact surfaces can be obtained under four rotation gradients of 0, 200, 400, and 600 rad/s. The result is shown in Figure 7.  
As shown in Figure 7, the contact pressure of multi-layer interference cylinder decreases with the increase of rotational speed. For the same layer, the contact pressure slip rate is slower before 400 rad/s. When the rotational speed reaches 600 rad/s, the contact pressure will decrease significantly. Analyzing the four models, with the increase of layer number of interference, the contact pressure between the same contact surfaces rises significantly, and the contact pressure gradually decreases from inside to outside [22], that is, \(p_{1}>p_{2}>p_{3}>p_{4}\). It indicates that the contact pressure at the outermost surface is the minimum. Under the influence of centrifugal force, the outermost contact pressure is more easily reduced to below the critical value, which results in the loosening of the outer interference components and the failure of torque transmission. In order to further explore the effect of centrifugal force on the contact pressure of the multi-layer interference cylinder, the contact pressure on contact surfaces of model \(M_{4}\) is analyzed separately. In Figure 8, comparing with contact pressure \(p_{1}\), \(p_{2}\), \(p_{3}\), \(p_{4}\) of \(M_{4}\), the drop of contact pressure decreases from inside to outside with the increase of speed. \(p_{1}\) is from 176 MPa to 134 MPa; \(p_{2}\) is from 153 MPa to 116 MPa; \(p_{3}\) is from 92 MPa to 65 MPa; \(p_{4}\) is from 44 MPa to 29 MPa. Combining with Table 2, the slope of contact pressure of the same surface increases gradually with the increase of speed. For example, the slope of \(p_{1}\) is \(-0.024\) when the rotational speed is 0 rad/s, when the rotational speed reaches 600 rad/s, the slope is increased to \(-0.119\). Figure 7: Contact pressure of each model under the centrifugal force. Under the centrifugal force, the contact surface of multi-layer interference cylinder will have different deformations, which is the fundamental reason for the change of contact pressure. With the increase of speed, the interference state between contact surfaces gradually decreases. In Figure 4, the cylinder \(C_{1}\) of model \(M_{4}\) is squeezed by the other four cylinders, so the interference state is the best. The cylinder \(C_{5}\) is located at the outermost layer, with the smallest contact pressure, leading to the worst interference state. This shows that the outermost contact surface is most likely to be detached when the speed is large enough. In order to verify the accuracy of matrix model for multi-layer interference cylinders under centrifugal force, taking model \(M_{4}\) as an example, comparing the analytical solution with the numerical solution under 600 rad/s, the results are shown in Figure 9. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Comparison Parameters** & \multicolumn{4}{c}{**Numerical Value**} \\ \hline Rotational Speed (rad/s) & 0 & 200 & 400 & 600 \\ Contact pressure (MPa) & 176 & 172 & 158 & 134 \\ Slope & \(-\)0.024 & \(-\)0.048 & \(-\)0.095 & \(-\)0.119 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The slope of \(p_{1}\) with the increase of speed. Figure 8: Contact pressure of \(M_{4}\) at different rotational speed. Figure 9: Comparison between numerical solution and analytical solution. In Figure 9, the numerical solutions of contact pressures \(p_{1}\), \(p_{2}\), \(p_{3}\), and \(p_{4}\) under centrifugal force are gradually reduced, which is in agreement with the analytical results.