## 2 Theoretical Analysis
### The Displacement and Load Transfer Equations of the Tie Rods  
The rotor shaft segments and tie rods are in a parallel relationship in the structure, and the displacement and load transfer of the tie rods will directly influence the dynamic performance of the rotor. Therefore, the displacement and load transfer equations of the tie rods are given below in detail. For the rod-fastened rotor, the tie rods and the rotor are both discretized at the corresponding cross-sections, which is favorable for analyzing the interaction relationship between the tie rods and the rotor. The preload of the rotor is the compressing force \(T_{0}\), and the preload of the tie rods is the pulling force \({T_{0}}^{\prime}=-T_{0}\), and the minus sign means that their directions are opposite. The preload is always tangent to the shaft segment. The force analysis of the \(j\)th shaft segment of the tie rod in the \(x\)-\(z\) plane and the \(y\)-\(z\) plane is shown in Figure 4. \(x_{rj}\), \(y_{rj}\) are the displacement of rod shaft segment in the \(x\) and \(y\) direction, \(z\) is the axial coordinate, \(l_{j}\) is the length of rod shaft segment, \(\varphi_{rj}\),\(\psi_{rj}\) are the rotating angle of tie rods in the \(x\)-\(z\) plane and \(y\)-\(z\) plane, \(M_{rj}\),\(N_{rj}\) are the bending moment of tie rod shaft segment, \(S_{rj}\),\(Q_{rj}\) are the shear force of tie rod shaft segment, and \(\omega\) is the rotational angular velocity. It can be seen that the balanced equation of the force and bending moment of the tie rod shaft segment in the \(x\)-\(z\) plane are written as:  
\[S_{rj}-{T_{0}}^{\prime}\varphi_{rj-1}=S_{rj-1}-{T_{0}}^{\prime}\varphi_{rj} \tag{10}\]  
\[M_{rj}=M_{rj-1}-l_{j}S_{rj-1} \tag{11}\]  
It is known from the literature [36] that the differential equation of massless shaft segment with pre-tension is given by:  
\[\frac{\partial^{2}}{\partial{x^{2}}}\bigg{(}EI_{x}\frac{\partial^{2}y}{ \partial{x^{2}}}\bigg{)}-T_{0}^{\prime}\frac{\partial^{2}y}{\partial{x^{2}}}=0 \tag{12}\]  
It is known that the bending moment of tie rods \(M_{r}=EI_{rx}\frac{\partial^{2}y}{\partial{x^{2}}}\). \(EI_{rx}\) is the bending stiffness of the tie rod shaft segment. Thus Equation (12) can be simplified as:  
\[\frac{\partial^{2}M_{r}}{\partial{x^{2}}}-\frac{T_{0}}{EI_{rx}}M_{r}=0 \tag{13}\]  
By integrating Equation (13) and using the corresponding boundary conditions, the rotating angle and displacement in the \(x\)-\(z\) plane can be obtained as:  
\[\varphi_{rj}=\varphi_{rj-1}+\frac{l_{j}}{EI_{rx}}(1+\frac{1}{6}\frac{{T_{0}} ^{\prime}}{EI_{rx}}l_{j}^{2})M_{rj-1}-\frac{{l_{j}}^{2}}{2 EI_{rx}}S_{rj-1} \tag{14}\]  
\[x_{rj}=x_{rj-1}+l_{j}\varphi_{rj-1}+\frac{1}{2}\frac{{l_{j}}^{2}}{EI_{rx}}M_{ rj-1}-\frac{1}{6}\frac{{l_{j}}^{3}}{EI_{rx}}S_{rj-1} \tag{15}\]  
Figure 4: The force analysis of the \(j\)th tie rod shaft segment. Therefore, the following matrix form is written as:  
\[\left( \begin{array}{c}x_{r}\\ \varphi_{r}\\ M_{r}\\ S_{r}\end{array} \right)_{j}=\left( \begin{array}{cccc}1&l&\frac{l^{2}}{2 EL_{r _{r}}}&\frac{-l^{3}}{6 EL_{r_{r}}}\\ 0&1&\frac{l(1+\frac{1}{6}\frac{T_{0}}{2 EL_{r}}l^{2})}{EL_{r_{r}}}&\frac{-l^{2}} {2 EL_{r_{r}}}\\ 0&0&1&-l\\ 0&0&\frac{l(1-\frac{1}{6}\frac{T_{0}^{\prime}}{2 EL_{r}}l^{2})T_{0}^{\prime}}{EL _{r_{r}}}&(1+\frac{l^{2}T_{0}^{\prime}}{2 EL_{r_{r}}})\end{array} \right)_{j}\left( \begin{array}{c}x_{r}\\ \varphi_{r}\\ M_{r}\\ S_{r}\end{array} \right)_{j-1} \tag{16}\]  
In the \(y\)-\(z\) plane, the same form can be obtained. For the \(j\)th shaft segment of the tie rod, from Equation (16), the following can be obtained:  
\[\left( \begin{array}{c}x_{r}\\ \varphi_{r}\end{array} \right)_{j}=\left( \begin{array}{cc}1&l\\ 0&1\end{array} \right)_{j}\left( \begin{array}{c}x_{r}\\ \varphi_{r}\end{array} \right)_{j-1}+\left( \begin{array}{c}\frac{l^{2}}{2 EL_{ r_{r}}}&\frac{-l^{2}}{6 EL_{r_{r}}}\\ \frac{l(1+\frac{1}{6}\frac{T_{0}}{2 EL_{r}}l^{2})}{EL_{r_{r}}}&\frac{-l^{2}}{2 EL_ {r_{r}}}\end{array} \right)_{j}\left( \begin{array}{c}M_{r}\\ S_{r}\end{array} \right)_{j-1} \tag{17}\]  
\[\left(