## 2 Theoretical Analysis
### The Equation of Motion of Rod-Fastened Rotor (RFR)
\right)_{j}\]  
In Equation (30), \(d_{xx}\), \(d_{yy}\), \(d_{yy}\) are the damping coefficient of bearing, \(k_{xx}\), \(k_{xy}\), \(k_{yx}\), \(k_{yy}\) are the stiffness coefficient of bearing, \(d_{Fx}\), \(d_{Fxy}\), \(d_{Fyx}\), \(d_{Fy}\) are the externally applied damping coefficient, \(k_{Fx}\), \(k_{Fxy}\), \(k_{Fyx}\), \(k_{Fy}\) are the externally applied stiffness coefficient, \(P_{cxy}\), \(P_{cxy}\) are the external force exerted on the shaft. From Equation (28), the following can be obtained:  
\[\left( \begin{array}{c}x\\ \varphi\end{array} \right)_{j}=\left( \begin{array}{cc}1&l\\ 0&1\end{array} \right)_{j}\left( \begin{array}{c}x\\ \varphi\end{array} \right)_{j-1}+\left( \begin{array}{cc}\frac{\beta}{2(El_{x})_ {eq}}\mu&\frac{-\beta^{3}}{6(El_{x})_{eq}}\\ \frac{l(1-\frac{1}{6}\frac{T_{0}}{T_{0}})^{2}}{(El_{x})_{eq}}\mu&\frac{- \beta^{2}}{2(El_{x})_{eq}}\end{array} \right)_{j}\left( \begin{array}{c}M \\ S\end{array} \right)_{j-1} \tag{31}\] \[-\left( \begin{array}{cc}\frac{l^{2}}{2(El_{x})_{eq}}\mu&\frac{ -l^{3}}{6(El_{x})_{eq}}\\ \frac{l(1-\frac{1}{6}\frac{T_{0}}{T_{0}})^{2}}{(El_{x})_{eq}}\mu&\frac{-l^{2}}{2 (El_{x})_{eq}}\end{array} \right)_{j}\left( \begin{array}{c}M_{r}\\ S_{r}\end{array} \right)_{j-1}+\left( \begin{array}{cc}\frac{l}{2(El_{x})_{eq} }\mu&\frac{-l^{3}}{6(El_{x})_{eq}}\\ \frac{l(1-\frac{1}{6}\frac{T_{0}}{T_{0}})^{2}}{(El_{x})_{eq}}\mu&\frac{-l^{2}}{2 (El_{x})_{eq}}\end{array} \right)_{j}\left( \begin{array}{c}M_{r}\\ S_{r}\end{array} \right)_{j}\]  
Therefore, from Equations (28) and (31), the following can be obtained:  
\[\left( \begin{array}{c}M\\ S\end{array} \right)_{j}=\left(\begin{array}{cc}\boldsymbol{A}^{*-1}_{j+1}+ \boldsymbol{B}^{*-1}_{j+1}+\boldsymbol{B}^{*-1}_{j+2}\left( \begin{array}{cc}1&l \\ 0&1\end{array} \right)_{j+2}\right)\left( \begin{array}{c}x\\ \boldsymbol{\varphi}\end{array} \right)_{j+1}-\left(\begin{array}{cc}\boldsymbol{A}^ {*-1}_{j+1}+\boldsymbol{B}^{*-1}_{j+1}\right)\left( \begin{array}{cc}1&l\\ 0&1\end{array} \right)_{j+1}\left( \begin{array}{c}x\\ \varphi\end{array} \right)_{j}-\boldsymbol{B}^{*-1}_{j+2}\left( \begin{array}{c}x \\ \varphi\end{array} \right)_{j+2} \tag{32}\]  
\[\text{Here, }\boldsymbol{A}^{*-1}_{j}=\left( \begin{array}{cc}\frac{6(El_{x})_{eq} }{2(1+2\beta)\mu}&\frac{-2(El_{x})_{eq}}{l(1+2\beta)\mu}\\ \frac{12(El_{x})_{eq}(1-\beta)}{l^{3}(1+2\beta)}&\frac{-6(El_{x})_{eq}}{l^{2}(1+ 2\beta)}\\ \end{array} \right)_{j}\quad,\beta=\frac{1}{6}\frac{T_{0}}{(El_{x})_{eq}}l^{2}.\]  
In fact, \(\beta\) is the influence coefficient of the compressing force \(\text{T}_{0}\) on the displacement and load transmission of the fastening rotor shaft segments. Similarly, for the \((j+1)\)-th shaft segment, this can be written as:  
\[\left( \begin{array}{c}M\\ S\end{array} \right)_{j}=(A^{*-1}_{j+1}+B^{*-1}_{j+1}+B^{*-1}_{j+2}\left( \begin{array}{cc}1&l\\ 0&1\end{array} \right)_{j+2})\left( \begin{array}{cc}x\\ \varphi\end{array} \right)_{j+1}-(A^{*-1}_{j+1}+B^{*-1}_{j+1})\left( \begin{array} []{cc}1&l\\ 0&1\end{array} \right)_{j+1}\left( \begin{array}{cc}x\\ \varphi\end{array} \right)_{j}-B^{*-1}_{j+2}\left( \begin{array}{cc}x\\ \varphi\end{array} \right)_{j+2} \tag{33}\]  
From Equation (28), the balanced equation of the force and bending moment of the corresponding mass points is:  
\[\left( \begin{array}{c}-M_{k}\\ \Sigma\,P_{x}\end{array} \right)_{j}\ -\left( \begin{array}{c}M\\ S\end{array} \right)_{j}\ +\left( \begin{array}{cc}1&-\frac{l}{\mu}\\ \frac{l(1-\frac{1}{\mu}\frac{T_{0}}{\mu(Ek)_{\mu q}}\mu}{(Ek)_{\mu q}}\mu&1- \frac{T_{0}l^{2}}{2(Ek)_{\mu q}}\end{array} \right)\quad\left( \begin{array}{ c}M\\ S\end{array} \right)_{j-1}\ -\left( \begin{array}{c}M_{r}\\ S_{r}\end{array} \right)_{j-1}+\left( \begin{array}{c}M_{r}\\ S_{r}\end{array} \right)_{j}=0 \tag{34}\]  
\[\left( \begin{array}{c}-N_{k}\\ \Sigma\,P_{y}\end{array} \right)_{j}\ -\left( \begin{array}{c}N\\ Q\end{array} \right)_{j}\ +\left(