## 2 Theoretical Analysis
### The Equation of Motion of Rod-Fastened Rotor (RFR)
It is known from the literature [36] that the differential equation of massless shaft segment with pressure is expressed as:  
\[\frac{\partial^{2}}{\partial z^{2}}\left(EI_{X}\frac{\partial^{2}y}{\partial z ^{2}}\right)+T_{0}\frac{\partial^{2}y}{\partial z^{2}}=0 \tag{24}\]  
It is known that \(M=EI_{X}\frac{\partial^{2}y}{\partial z^{2}}\), \(EI_{X}\) is the bending stiffness of the rotor shaft segment, so Equation (24) can be simplified as:  
\[\frac{\partial^{2}M}{\partial z^{2}}+\frac{T_{0}}{EI_{X}}M=0 \tag{25}\]  
By integrating Equation (25) and using the corresponding boundary conditions, the rotating angle and displacement in the \(x\)-\(z\) plane can be obtained as:  
\[\varphi_{j}=\varphi_{j-1}+\frac{l_{j}}{EI_{X}}(1-\frac{1}{6}\frac{T_{0}}{EI_ {X}}l_{j}^{2})(\mu M_{j-1}-M_{rj-1}+M_{rj})-\frac{l_{j}^{2}}{2 EI_{X}}(S_{j-1}- S_{rj-1}+S_{rj}) \tag{26}\]  
\[x_{j}=x_{j-1}+l_{j}\varphi_{j-1}+\frac{1}{2}\frac{l_{j}^{2}}{EI_{X}}(\mu M_{j -1}-M_{rj-1}+M_{rj})-\frac{1}{6}\frac{l_{j}^{3}}{EI_{X}}(S_{j-1}-S_{rj-1}+S_{rj}) \tag{27}\]  
From the above, we know that the flexural stiffness of the RFR shaft segment with the contact interface is different from that of the corresponding integral rotor. Therefore,  
Figure 5: The force analysis of the \(j\)th fastened rotor shaft segment. \((El_{x})_{eq}\) should be used in place of \(El_{x}\) to consider the influence of the contact interfaces. In the following, the flexural stiffness of the shaft segment with the contact interface is replaced by \((El_{x})_{eq}\). Therefore, the following matrix form can be written as:  
\[\left( \begin{array}{c}x\\ \varphi\\ M\\ S\end{array} \right)_{j}=\left( \begin{array}{ccc}1&l&\frac{\beta}{2(El_{x})_{eq} }\mu&\frac{-\beta^{3}}{6(El_{x})_{eq}}\\ 0&1&\frac{l(1-\frac{1}{2}\frac{T_{0}}{T_{0}})^{2}}{(El_{x})_{eq}}\mu&\frac{- \beta^{2}}{2(El_{x})_{eq}}\\ 0&0&1&-\frac{1}{2}\\ 0&0&\frac{l(1-\frac{T_{0}}{T_{0}})^{2})T_{0}}{(El_{x})_{eq}}\mu&(1-\frac{2T_{0 }}{T_{0}})_{j}\\ \end{array} \right)_{j}\left( \begin{array}{c}x\\ \varphi\\ M\\ S\end{array} \right)_{j-1}-\left( \begin{array}{ccc}0&0&\frac{\beta^{2}}{2(El_{x})_ {eq}}\mu&\frac{-\beta^{2}}{6(El_{x})_{eq}}\\ 0&0&\frac{l(1-\frac{1}{2}\frac{T_{0}}{T_{0}})^{2}}{(El_{x})_{eq}}\mu&\frac{- \beta^{2}}{2(El_{x})_{eq}}\\ 0&0&1&0\\ 0&0&0&1\end{array} \right)_{j}\left( \begin{array}{c}0\\ 0\\ M_{r}\\ S_{r}\end{array} \right)_{j-1} \tag{28}\]  
In the \(y\)-\(z\) plane, the same form can be obtained. The bending moment \(M_{bg}\), \(N_{bg}\) applied by the disc to the \(j\)th shaft segment, should be expressed as:  
\[\left( \begin{array}{c}M_{k}\\ N_{k}\end{array} \right)_{j}=\left( \begin{array}{cc}-\theta_{y}&0\\ 0&\theta_{x}\end{array} \right)_{j}\left( \begin{array}{c}\ddot{\varphi}\\ \dot{\psi}\end{array} \right)_{j}+\left( \begin{array}{cc}0&-\theta_{z}\omega \\ -\theta_{z}\omega&0\end{array} \right)_{j}\left( \begin{array}{c}\dot{\varphi} \\ \dot{\psi}\end{array} \right)_{j} \tag{29}\]  
Here, \(\theta_{x}\), \(\theta_{y}\), \(\theta_{z}\) are the moments of inertia of the disk. The force of the \(j\)th shaft segment can be expressed as:  
\[\left( \begin{array}{cc}\sum P_{x}\\ \sum P_{y}\end{array} \right)_{j}= \left( \begin{array}{cc}m&0\\ 0&m\end{array} \right)_{j}\left( \begin{array}{c}\ddot{x}\\ \ddot{y}\end{array} \right)_{j}+\left( \begin{array}{cc}d_{Fx}&d_{Fxy}\\ d_{Fyx}&d_{Fy}\end{array} \right)_{j}\left( \begin{array}{c}\dot{x}\\ \dot{y}\end{array} \right)_{j}+\left( \begin{array}{cc}d_{xx}&d_{xy}\\ d_{yx}&d_{yy}\end{array} \right)_{j}\left( \begin{array}{c}\dot{x}\\ \dot{y}\end{array} \right)_{j} \tag{30}\] \[+\left( \begin{array}{cc}k_{Fx}&k_{Fxy}\\ k_{Fyx}&k_{Fy}\end{array} \right)_{j}\left( \begin{array}{c}x\\ y\end{array} \right)_{j}+\left( \begin{array}{cc}k_{xx}&k_{xy}\\ k_{yx}&k_{yy}\end{array} \right)_{j}\left( \begin{array}{c}x\\ y\end{array} \right)_{j}+\left( \begin{array}{cc}P_{cx}\\ P_{cy}\end{array}