## 3 Results and Discussion  
In this section, first, a simple rotor shaft segment with an annular contact surface is used as an example to investigate the effect of the parameters of the contact surface on EFS of the contact interface and the effect of the preload on EFS of the shaft segment with the contact interface. Then, for a RFR, a dynamic model considering all factors brought by the tie rods is built, which includes the contact interface, preload, the coordinate deformation of the tie rods, the parallel relationship between the tie rods and rotor, etc. The influence of various factors on the dynamic performance of the rotor is investigated in detail. In addition, the effect of the tie rod parameters on the rotor system is discussed.  
As shown in Figure 3, the contact interface in the RFR is a pair of annular contact surfaces with an inner radius of 29 mm and an outer radius of 44 mm. The parameters of the contact surface measured in [35] are used here. On the nominal contact surface, RMS of the micro-convex body height distribution is \(\sigma\) = 2.01 \(\upmu\)m, the average radius of curvature of the top of the micro-convex body is \(\beta_{0}\) = 95 \(\upmu\)m, and the micro-convex body distribution density is \(\eta\) = 0.5625 \(\times\) 10\({}^{8}\) L/m\({}^{2}\).  
Based on the above parameters, the equivalent bending stiffness of the contact interface between the shaft segments at different preloads and rotation angle can be calculated by numerical calculation methods. The result is shown in Figures 6 and 7. When the rotation angle \(\theta\) is 1 \(\times\) 10\({}^{-7}\) rad, and the preload increases from 0 to 5 \(\times\) 10\({}^{7}\) N, the equivalent flexural stiffness of the contact interface is shown in Figure 6. It can be seen that when the preload is small, EFS of the contact interface increases rapidly. As the further increase of the preload, EFS of the contact interface will slowly increase to a constant value. When the preload is \(8\times 10^{3}\) N, \(5\times 10^{4}\) N, \(5\times 10^{5}\) N, respectively, changing the rotation angle from \(1\times 10^{-8}\) rad to \(5\times 10^{-4}\) rad, change of EFS of the contact interface is shown in Figure 7. It can be observed that when the rotation angle is small, EFS of the contact interface is almost unchangeable. When the rotation angle is larger than \(0.5\times 10^{-4}\) rad, EFS of the contact interface varies exponentially with the rotation angle. In the dynamic analysis of the rotor, the rotation angle is very small, so it is correct to ignore the influence of the rotation angle on the equivalent bending stiffness of the contact interface. Therefore, the influence of the rotation angle on EFS of the contact interface will be ignored in the subsequent calculations.  
Figures 8 and 9 show the changes of EFS of the contact interface with RMS of the micro-convex body distribution \(\sigma\) and the average micro-convex body radius of curvature of the top \(\beta_{0}\).  
Figure 6: EFS of the contact surface changes with preload.  
Figure 7: Equivalent flexural stiffness (EFS) of the contact surface changes with the rotation angle.  
segment with the contact interface increases rapidly with the preload. As the preload increases further, the increasing speed of EFS of the shaft segment with the contact interface gradually becomes slower. When the preload reaches \(10^{7}\) N, EFS of the shaft segment with the contact interface tends to be consistent with the bending stiffness of the corresponding integral shaft segment. A further increase of the preload will no longer change EFS of the shaft segment.