## 2 The Dynamic Model of the RBRS  
In order to study the dynamic behavior of RBRS effectively, the finite element model of double rotor-bearing system of hydraulic turbine is modeled and simplified as four lumped mass points according to the structural characteristics of the system. The rotor and the runner are located at node 1 and node 4, respectively, and the combination bearing and turbine guide bearing are located at node 2 and node 3, respectively. The schematic diagram of RBRS is shown in Figure 1. The rotation center coordinates of the turbine rotor and the runner are \(O_{1}(x_{1},y_{1})\) and \(O_{4}(x_{4},y_{4})\), and the centroid coordinates of the rotor and the runner are \(G_{1}(x_{\rm g1},y_{\rm g1})\) and \(G_{4}(x_{\rm g4},y_{\rm g4})\). The dynamic equation of RBRS with 16 degrees of freedom can be deduced as follows:  
\[\mathbf{M\ddot{q}}+(\mathbf{C}+\mathbf{G})\mathbf{\dot{q}}+\mathbf{Kq}=\mathbf{F}_{\rho}+\mathbf{F}_{e}+\bm {F}_{w}+\mathbf{F}_{r} \tag{1}\]  
where \(\mathbf{M}\), \(\mathbf{C}\), \(\mathbf{G}\), \(\mathbf{K}\), \(\mathbf{q}\) are mass matrix, damping matrix, gyroscopic matrix, stiffness matrix, and displacement vector of RBRS, respectively. \[\mathbf{q}=\left[x_{1},\theta_{y1},x_{2},\theta_{y2},x_{3},\theta_{y3},x_{4}, \theta_{y4},y_{1},\theta_{x1},y_{2},\theta_{x2},y_{3},\theta_{x3},y_{4}, \theta_{x4}\right]^{T} \tag{2}\]  
where \(x_{i}\), \(y_{i}\) (\(i\) =1,2,3,4) and \(\theta_{xi}\), \(\theta_{yi}\) (\(i\) = 1,2,3,4) are the displacements of the rotor, combination bearing, turbine guide bearing, and runner in \(x\) and \(y\) directions and angles of orientation associated with the \(x\) and \(y\) axes, respectively. Suppose that RBRS is composed of an elastic shaft element and the generalized coordinates of the shaft element are the displacements of nodes at both ends. The rotor system in this study only considers bending deformation and torsional deformation in horizontal and vertical directions, ignoring axial deformation. \[\mathbf{M}_{ST}=\frac{\mu l}{420}\left[ \begin{array}{cccc}156&22l&54&-13l\\ 22l&4l^{2}&13l&-3l^{2}\\ 54&13l&156&-22l\\ -13l&-3l^{2}&-22l&4l^{2}\end{array} \right] \tag{3}\]  
Figure 1: (a) Schematic diagram of rotor-bearing-runner system; (b) Schematic of lumped mass point and its coordinate system. \[\mathbf{M}_{sR}=\frac{\mu r^{2}}{120l}\left[ \begin{array}{cccc}36&3l&-36&3l\\ 3l&4l^{2}&-3l&-l^{2}\\ -36&-3l&36&-3l\\ 3l&-l^{2}&-3l&4l^{2}\end{array} \right] \tag{4}\]  
\[\mathbf{I}_{s}=\frac{\mu r^{2}}{60l}\left[ \begin{array}{cccc}36&3l&-36&3l\\ 3l&4l^{2}&-3l&-l^{2}\\ -36&-3l&36&-3l\\ 3l&-l^{2}&-3l&4l^{2}\end{array} \right] \tag{5}\]  
\[\mathbf{K}_{s}=\frac{EI}{l^{3}}\left[ \begin{array}{cccc}12&6l&-12&6l\\ 6l&4l^{2}&-6l&2l^{2}\\ -12&-6l&12&-6l\\ 6l&2l^{2}&-6l&4l^{2}\end{array} \right] \tag{6}\]  
where \(\mathbf{M}_{sT}\), \(\mathbf{M}_{sR}\),\(\mathbf{G}_{s}\) (\(\mathbf{G}_{s}=\omega\mathbf{I}_{s}\)), and \(\mathbf{K}_{s}\) represent moving inertia matrix of the unit, rotational inertia matrix of the unit, rotation matrix of the unit, and stiffness matrix of the unit, respectively. \(\mu\), \(l\), and \(EI\) are the quality of the unit length, the length of the unit, and bending stiffness. \[\mathbf{M}=\left[ \begin{array}{cc}\mathbf{M}_{x}&0\\ 0&\mathbf{M}_{y}\end{array} \right] \tag{7}\]  
\[\mathbf{K}=\left[ \begin{array}{cc}\mathbf{K}_{x}&0\\ 0&\mathbf{K}_{y}\end{array} \right]+\mathbf{K}^{\prime} \tag{8}\]  
\[\mathbf{K}^{\prime}=diag\Big{[}0,\ 0,\ k_{22x},\ 0,\ k_{23x},\ 0,\ 0,\ 0,\ 0,\ 0,\ 0,\ k_{2y},\ 0,\ k_{23y},\ 0,\ 0,\ 0\Big{]} \tag{10}\]  
where \(k_{2x}\), \(k_{23x}\), \(k_{22y}\), and \(k_{23y}\) are the stiffness of the combination bearing and turbine guide bearing in \(x\) and \(y\) directions, respectively. \[\mathbf{G}=\left[ \begin{array}{cc}0&\mathbf{G}_{s}\\ -\mathbf{G}_{s}&0\end{array}