## 3 Results and Discussion
### The Influence of Various Factors on the Dynamic Performance of RFR
#### 3.2.1 The Effect of Preload  
The RFR with circumferential tie rods depends on the tie rods to fasten the disks together. The existence of preload not only affects the contact state of the contact interface but also affects the transmission of the rotor displacement and load. These factors will affect the flexural stiffness of the rotor, which in turn affects the natural frequency of the rotor. Suppose the RFR is simplified as the corresponding integral rotor, the natural frequencies of the rotor system are shown in Table 2. When only considering the influence of preload on the contact state of the contact interface, it corresponds to situation A. The rate of change of natural frequencies of the RFR is shown in Figure 12. The rate of change of each order natural frequency of the RFR is written as:  
\[rF_{i}=\frac{F_{Ai}-F_{0}}{F_{0}}\quad i=1,2,3. \tag{38}\] \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & & **The Influence** & & & **The Influence** \\
**Situations** & **Contact** & **Coefficient** & **Factor of** & **The Tie** & **Coefficient of** \\
**Interfaces** & **of the** & **Coordinate** & **Rods** & **the Preload of** \\  & & **Preload of** & **Deformation** & **El\({}_{r}\)** & **the Tie Rods** \\  & & **the Rotor \(\beta\)** & **of Tie Rods \(\mu\)** & & \(\beta^{\prime}\) \\ \hline A & Considering & \(\beta=0\) & \(\mu=1\) & \(El_{r}=0\) & \(\beta^{\prime}=0\) \\ B & Considering & \(\beta\
eq 0\) & \(\mu=1\) & \(El_{r}=0\) & \(\beta^{\prime}=0\) \\ C & Considering & \(\beta=0\) & \(\mu\
eq 1\) & \(El_{r}=0\) & \(\beta^{\prime}=0\) \\ D1 & Considering & \(\beta=0\) & \(\mu=1\) & \(El_{r}=0\) & \(\beta^{\prime}=0\) \\ D2 & Considering & \(\beta=0\) & \(\mu=1\) & \(El_{r}=0\) & \(\beta^{\prime}\
eq 0\) \\ E & Considering & \(\beta\
eq 0\) & \(\mu\
eq 1\) & \(El_{r}=0\) & \(\beta^{\prime}\
eq 0\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The different situations. Figure 11: The rotor model. Here, \(F_{0}\) is the corresponding integral rotor, and \(F_{Ai}\) is the \(i\)-order natural frequency of the RFR corresponding to situation A. As can be seen from Figure 12, the preload affects the contact stiffness of the contact interface between the disks, which in turn affects the natural frequency of each order of the rotor. The greater the preload, the closer the natural frequency of the RFR to the natural frequency of the corresponding integral rotor. It also can be seen that the preload has a larger influence on the first-order natural frequency of the rotor and a smaller influence on the second-order natural frequency, which is determined by the structure of the rotor and the mode shape of the rotor. When considering the influence of preload on the contact state of the contact interface and the displacement and load transmission of the shaft segments at the same time, this corresponds to situation B. The rate of change of natural frequencies of the RFR is shown in Figure 13. In order to study the influence of the transmission of the rotor displacement and load caused by preload on the natural frequencies of the RFR, it is necessary to compare situation A and situation B. When comparing two different situations, such as situation A and situation B, the rate of change of each order natural frequency of the RFR is written as  
\[rF_{i}=\frac{F_{Bi}-F_{Ai}}{F_{Ai}}\quad i=1,2,3. \tag{39}\]  
Here, \(F_{Ai}\), \(F_{Bi}\) are the \(i\)-order natural frequency of \(RFR\) corresponding to situation A and situation B, respectively. The definition of the rate of change of the natural frequencies is similar in other comparisons. By comparing situation B with situation A in Figure 12, the amplitude of the rate of change of each order natural frequency increases with the increase of the preload. This means that the preload affects the displacement and load transmission of the shaft segment of the rotor, which slightly reduces the flexural stiffness of the rotor and results in each order natural frequency having a certain decrease. The effect is gradually increasing with the increase of the preload.