## 3 Dynamics Modeling of Shafting of Hydropower Units  
The main shaft-system structure of the hydropower-generating units is shown in Figure 8a. \(O_{1}\) and \(O_{2}\), respectively, represent the geometric centers of the generator rotor and the turbine runner, and \(B_{1}\), \(B_{2}\), and \(B_{3}\), respectively, represent the geometric centers of the upper guide bearing, the lower guide bearing, and the water guide bearing. The centroid of the rotating part is shown in Figure 8b. \((x_{i},y_{i})\) is used to represent the centroid coordinates of the generator rotor and the turbine rotor, respectively, and \((x_{i}+e\sin\omega t,y_{i}+e\cos\omega t)\) represents the centroid coordinates. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Wear** & \(A_{1x}\) **(N)** & \(A_{2x}\) **(N)** & \(A_{3x}\) **(N)** & \(B_{x}\) **(N)** \\ \hline
0 & 321.53 & 469.24 & 472.44 & 2619.09 \\
3.209 & 656.85 & 416.28 & 807.47 & 1750.09 \\
9.626 & 719.05 & 501.78 & 786.84 & \(-\)13.86 \\
16.043 & 767.76 & 407.77 & 952.44 & \(-\)284.20 \\
22.461 & 981.82 & 529.61 & 1263.59 & \(-\)631.86 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: x direction excitation force parameter. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Wear** & \(A_{1y}\) **(N)** & \(A_{2y}\) **(N)** & \(A_{3y}\) **(N)** & \(B_{y}\) **(N)** \\ \hline
0 & 378.29 & 572.44 & 558.14 & 2160.96 \\
3.209 & 606.03 & 428.50 & 866.00 & 1824.45 \\
9.626 & 631.62 & 453.98 & 513.98 & \(-\)686.21 \\
16.043 & 709.94 & 355.04 & 792.64 & \(-In order to facilitate the analysis, this paper adopts appropriate simplified assumptions: (1) In the modeling stage, the torsional motion and axial vibration of shafting are ignored, and only the radial displacement and inclination of the spindle system are analyzed. (2) Assume that all components of the spindle system behave isotropic during operation. In the rotation process of the disk, the angle of its rotation is defined as, where \(\omega\) is the angular speed of the disk, and t is the time variable. Based on the coordinate function of the center of mass, the velocity coordinate function can be obtained by obtaining the first derivative of time, and the general expression of the translational kinetic energy of the center of mass of a disk can be derived:  
\[T_{m}=\frac{1}{2}m_{i}\Big{[}\big{(}\dot{x}_{i}-e_{i}\dot{\varphi}\sin\varphi \big{)}^{2}+\big{(}\dot{y}_{i}+e_{i}\dot{\varphi}\cos\varphi\big{)}^{2}\Big{]}, \tag{7}\]  
When considering the tilt of the shafting of hydropower units, the simplified expression of the rotational kinetic energy can be obtained according to the rotor dynamics and the principle of conservation of the momentum moment [22]:  
\[T_{r}=\frac{1}{2}\Big{[}J_{d_{i}}\Big{(}\dot{\theta}_{x_{i}}^{2}+\dot{\theta} _{y_{i}}^{2}\Big{)}+\Big{(}J_{p_{i}}+m_{i}e_{i}^{2}\Big{)}\Big{(}\dot{\varphi} ^{2}-2\dot{\varphi}\dot{\theta}_{y_{i}}\theta_{x_{i}}\Big{)}\Big{]}, \tag{8}\]  
By combining Equations (7) and (8), the total kinetic energy equation of the shafting of hydropower units can be obtained, as follows:  
\[ \begin{array}{rl}T=&\frac{1}{2}m_{1}\Big{(}\dot{x}_{1}^{2}+\dot{y}_{1}^{2}+ e_{1}^{2}\dot{\varphi}^{2}+2e_{1}\dot{\varphi}\dot{y}_{1}\cos\varphi-2e_{1} \dot{\varphi}\dot{x}_{1}\sin\varphi\Big{)}+\\ &\frac{1}{2}m_{2}\Big{(}\dot{x}_{2}^{2}+\dot{y}_{2}^{2}+e_{2}^{2}\dot{\varphi} ^{2}+2e_{2}\dot{\varphi}\dot{y}_{2}\cos\varphi-2e_{2}\dot{\varphi}\dot{x}_{2} \sin\varphi\Big{)}+\\ &\frac{1}{2}\big{(}J_{p_{1}}+m_{1}e_{1}^{2}\big{)}\Big{(}\dot{\varphi}^{2}-2 \dot{\varphi}\dot{\theta}_{y_{1}}\theta_{x_{1}}\Big{)}+\frac{1}{2}\big{(}J_{ p_{2}}+m_{2}e_{2}^{2}\big{)}\Big{(}\dot{\varphi}^{2}-2\dot{\varphi}\dot{\theta}_{y_{ 2}}\theta_{x_{2}}\Big{)}+\\ &\frac{1}{2}J_{d_{1}}\Big{(}\dot{\theta}_{x_{1}}^{2}+\dot{\theta}_{y_{1}}^{2} \Big{)}+\frac{1}{2}J_{d_{2}}\Big{(}\dot{\theta}_{x_{2}}^{2}+\dot{\theta}_{y_{ 2}}^{2}\Big{)}\end{array} , \tag{9}\]  
where \(m_{1}\) and \(m_{2}\) are the masses of the generator rotor and the turbine runner, respectively. \(J_{p}\) is the polar moment of inertia.