# Dynamic Characteristics Analysis of a Rod Fastening Rotor System Considering Contact Roughness
## 4 Results and Discussion
### Dynamic Characteristics of Rod Fastening Rotor System  
In this section, the dynamic characteristics of the rod fastening rotor system are revealed. According to reference [36], the main structure parameters of the system are given in Table 2. When the two discs are connected by the bolt preload, the dynamic characteristics of the rod fastening rotor system are mainly determined by the disc mass, shaft bending stiffness and the equivalent bending stiffness of the joint's interface.  
Based on the above analysis of the contact mechanism of the joint's interface and reference [34], the bending stiffness of the interface can be expressed as  
\[K_{G}=kl_{a}\eta_{ra} \tag{35}\]  
where \(l_{a}\) denotes the actual second moment of area, \(\eta_{ra}\) denotes the ratio of real contact area to the nominal contact area and \(k\) denotes the elastic constraint of the equivalent distributed spring shown in Figure 3, which obeys the following form:  
\[k=\frac{K_{r}}{A_{a}} \tag{36}\]  
According to reference [36], for No. 45 steel, the elastic modulus of the joint's interface is 115.4 GPa, the surface hardness is 2.058 GPa, the fractal dimension \(D\) is 1.34 and the characteristic length scale \(G\) is \(7.075\times 10^{-11}\) m. Thus, the relation between the bending stiffness of the interface and the preload of the bolt is shown in Figure 8. It is clear that due to the increase in the preload of the bolt, the bending stiffness of the interface gradually increases.  
\begin{table}
\begin{tabular}{c c} \hline Disc mass \(m\) (kg) & 29.18 \\ \hline Disc radius \(R\) (mm) & 100 \\ \hline Shaft length \(L\) (mm) & 448.8 \\ \hline Shaft radius \(r\) (mm) & 30 \\ \hline Shaft elastic modulus \(E_{rs}\) (GPa) & 210 \\ \hline \end{tabular}
\end{table}
Table 2: Main parameters of rotor system [36].  
Figure 7: Relationship between dimensionless normal contact stiffness \(K_{r}^{*}\) and characteristic length scale \(G\).  
Figure 9 suggests that the preload of the bolt has an assignable influence on the natural frequency of the rod fastening rotor system. The natural frequency of the system increases with the change in preload. This trend of natural frequency is similar to that of the bending stiffness shown in Figure 8. Meanwhile, the inner relationship between the natural frequency and the grade of asperity can be observed in Figure 9 as well.  
At last, the key parameters (i.e., fractal dimension \(D\) and characteristic length scale \(G\)) are taken into consideration. Since the disc contact stiffness appears sensitive to normal contact stiffness and the joint's interface is sensitive to the fractal dimension and the characteristic length scale, the change in the natural frequency is further discussed in detail. From Figure 10, it is clear that the vibration frequency of the rotor system becomes large with the increase in the fractal dimension. However, it will decrease with an increase in the characteristic length scale. The main reason for this is that the total stiffness of the system is closely related to the bending stiffness of the shaft and the equivalent bending stiffness of the interface. Overall, the trend of the two curves shown in Figure 10 is approximately linear. The physical characteristics of the rod fastening rotor system are mainly determined by the material of the shafts, discs and fastener bolts and the surface finish of the discs.  
Figure 8: Relation between bending stiffness of interface \(K_{G}\) and preload of bolt \(F_{p}\).  
Figure 9: Effects of the preload and frequency index on the natural frequency of the rod fastening rotor system.