## 2 The Dynamic Model of the RBRS
\right] \tag{11}\]  
The total damping matrix (Rayleigh damping form is applied to determine the viscous part) can be obtained by the following formula [30]:  
\[\mathbf{C}=\alpha\mathbf{M}+\beta\mathbf{K}+\mathbf{C}^{{}^{\prime}} \tag{12}\]\[\alpha=2\bigg{(}\frac{\xi_{2}}{\omega_{2}}-\frac{\xi_{1}}{\omega_{1}}\bigg{)} \bigg{(}\frac{1}{\omega_{2}^{2}}-\frac{1}{\omega_{1}^{2}}\bigg{)} \tag{14}\]  
\[\beta=2(\xi_{2}\omega_{2}-\xi_{1}\omega_{1})/(\omega_{2}^{2}-\omega_{1}^{2}) \tag{15}\]  
where \(\omega_{1}\) and \(\omega_{2}\) are the first and second natural frequencies of the system, respectively. \(\xi_{1}\) and \(\xi_{2}\) are the first and second modal damping ratios, respectively. \[\mathbf{C}^{\prime}=diag\Big{[}0,\ 0,\ c_{2\times x},\ 0,\ c_{2\times x},\ 0,\ 0,\ 0,\ 0,\ 0,\ c_{2\times y},\ 0,\ c_{2\times y},\ 0,\ 0,\ 0\Big{]} \tag{16}\]  
where \(c_{2\times x}\), \(c_{2\times x}\), \(c_{2\times y}\), and \(c_{2\times y}\) are the damping of the combination bearing and turbine guide bearing in \(x\) and \(y\) directions, respectively. \[ \begin{array}{l}\mathbf{F}_{\rho}=[m_{1}\rho_{1}\omega^{2}cos(\omega t),\ 0,\ 0,\ 0,\ 0,\ 0,\ 0,m_{4}\rho_{4}\omega^{2}cos(\omega t),\ 0,\\ m_{1}\rho_{1}\omega^{2}sin(\omega t),\ 0,\ 0,\ 0,\ 0,\ 0,m_{4}\rho_{4}\omega^{2} sin(\omega t),\ 0]^{T}\end{array}  \tag{17}\]  
where \(m_{1}\rho_{1}\) and \(m_{4}\rho_{4}\) denote eccentricity of unbalance mass of the generator rotor and the turbine runner. \[\mathbf{F}_{e}=\Big{[}F_{ex},\ 0,\ 0,\ 0,\ 0,\ 0,\ 0,F_{ey},\ 0,\ 0,\ 0,\ 0,\ 0,\ 0,\ 0,\ 0 \Big{]}^{T} \tag{18}\]  
where \(F_{ex}\) and \(F_{ey}\) are nonlinear electromagnetic force of the rotor in \(x\) and \(y\) directions. Nonlinear electromagnetic force can be expressed as:  
\[\left[ \begin{array}{c}F_{ex}\\ F_{ey}\end{array} \right]=\alpha k_{n}\delta_{0}(\varepsilon+\frac{5\varepsilon^ {3}}{4})\times\left[ \begin{array}{c}\cos\theta_{r}\\ sin\theta_{r}\end{array} \right] \tag{19}\]  
where \(k_{n}\), \(\varepsilon\) and \(\theta_{r}\) represent electromagnetic stiffness, relative eccentric, and rotation angle of the rotor. \(\alpha\) and \(\delta_{0}\) are variation coefficient of the electromagnetic stiffness and average air gap length. The functions \(k_{n}\), \(\varepsilon\), and \(\theta_{r}\) are respectively given in Equations (20)-(22):  
\[k_{n}=\frac{\pi\mu_{0}rL(K_{j}I_{j})^{2}}{2\delta_{0}^{3}} \tag{20}\]  
\[\varepsilon=\frac{\left(x^{2}+y^{2}\right)^{1/2}}{\delta_{0}} \tag{21}\]  
\[\cos\theta_{r}=\frac{x}{\left(x^{2}+y^{2}\right)^{1/2}},\sin\theta_{r}=\frac{y }{\left(x^{2}+y^{2}\right)^{1/2}} \tag{22}\]  
where \(\mu_{0}\), \(r\), \(L\), \(K_{j}\), and \(I_{j}\) are permeability of vacuum, rotor radius, rotor height, coefficient of air gap fundamental wave magnetomotive force, and excitation current of the generator, respectively. \[\mathbf{F}_{w}=\Big{[}0,\ 0,\ 0,\ 0,\ 0,\ 0,\ F_{wx},\ 0,\ 0,\ 0,\ 0,\ 0,\ 0,\ F_{wy},\ 0\Big{]}^{T} \tag{23}\]  
where \(F_{wx}\) and \(F_{wy}\) are the hydraulic force of the runner in \(x\) and \(y\) directions. Hydraulic force can be expressed as:  
\[\left[ \begin{array}{c}F_{wx}\\ F_{wy}\end{array} \right]=K_{w}\left[ \begin{array}{c}x\\ y\end{array} \right] \tag{24}\]  
where \(K_{w}\) is coefficient of the hydraulic force. \[\mathbf{F}_{r}=\Big{[}0,\ 0,\ F_{rx},0,\ 0,\ 0,\ 0,\ 0,\ 0,\ 0,\ F_{ry},0,\ 0,\ 0,\ 0,\ 0\Big{]}^{T} \tag{25}\]where \(F_{rx}\) and \(F_{ry}\) are nonlinear rubbing force of the combination bearing in \(x\) and \(y\) directions. Nonlinear rubbing force can be expressed as:  
\[\left[ \begin{array}{c}F_{x}\\ F_{y}\end{array} \right]=\left\{ \begin{array}{ll}-\frac{k_{r}(r-\delta)}{r} \left[\begin{array}{cc}1&-f\\ f&1\end{array}\right]\left[\begin{array}{c}x\\ y\end{array}\right],&(r\geq\delta)\\ \left[\begin{array}{c}0\\ 0\end{array}\right],&(r<\delta)\end{array} \right. \tag{26}\]  
where \(r\) represents vibration amplitude of the rotor. \(k_{r}\), \(\delta\), and \(f\) are radial stiffness of the stator, gap, and friction coefficient of rubbing.