## 3 Dynamic Model of a Rotor Bearing System
### Equations of Motion  
In this paper, the rotor bearing system is established in Figure 5, and the gyroscopic effect, nonlinear ball bearing forces and UMP are taken into consideration. The mass of the shaft is ignored and the parameters involved are expressed in Table 2. Based on [PERSON]'s second law, the equations of motion of the rotor bearing system can be deduced as:  
\[M\ddot{\mathbf{u}}^{\text{T}}+(\Omega\mathbf{J}+\mathbf{C})\dot{\mathbf{u}}^{\text{T}}+\mathbf{K} \mathbf{u}^{\text{T}}=\mathbf{F} \tag{19}\]  
where \(\mathbf{u}=\left(x,\theta_{y},x_{A},x_{B},y,\theta_{x},y_{A},y_{B}\right)\) is the generalized coordinate vector, \(\mathbf{M}=\left(\begin{array}{cc}M_{1}&\mathbf{0}\\ \mathbf{0}&\mathbf{M}_{1}\end{array}\right)\) is the mass matrix, \(\mathbf{J}=\left(\begin{array}{cc}\mathbf{0}&\mathbf{J}_{1}\\ -\mathbf{J}_{1}&\mathbf{0}\end{array}\right)\) is the gyroscopic matrix, \(\mathbf{K}=\left(\begin{array}{cc}\mathbf{K}_{1}+\mathbf{K}_{b}&\mathbf{0}\\ \mathbf{0}&\mathbf{K}_{1}+\mathbf{K}_{b}\end{array}\right)\) is the stiffness matrix and \(\mathbf{C}=\left(\begin{array}{cc}\mathbf{C}_{1}&\mathbf{0}\\ \mathbf{0}&\mathbf{C}_{1}\end{array}\right)\) is the damping matrix. \(M_{1},\mathbf{J}_{1},\mathbf{K}_{1},\mathbf{K}_{b}\) and \(\mathbf{C}_{1}\) can be expressed as follows:  
\[\mathbf{M}_{1}=\left(\begin{array}{ccc}m_{d}&&&\\ &J_{d}&&\\ &&m_{A}&\\ &&&m_{B}\end{array}\right)\quad\quad\mathbf{J}_{1}=\left(\begin{array}{ccc}0&&&\\ &J_{p}&&\\ &&0&\\ &&&0\end{array}\right)\]  
\[\mathbf{K}_{1}=\left(\begin{array}{ccc}\mathbf{K}_{c}&-\mathbf{K}_{c}\mathbf{\Phi}\\ -\mathbf{\Phi}^{\text{T}}\mathbf{K}_{c}&\mathbf{\Phi}^{\text{T}}\mathbf{K}_{c}\mathbf{\Phi}\end{array} \right)\quad\mathbf{K}_{b}=\left(\begin{array}{ccc}0&&&\\ &0&&\\ &k_{A}&\\ &&k_{B}\end{array}\right) \tag{20}\]  
\[\mathbf{C}_{1}=\left(\begin{array}{ccc}c&0&-\frac{c(-a+l)}{l}&-\frac{ac}{l}\\ 0&c&\frac{c}{l}&-\frac{c}{l}\\ -\frac{c(-a+l)}{l}&\frac{c}{l}&\frac{c}{l^{2}}+\frac{c(-a+l)^{2}}{l^{2}}+c_{b}&- \frac{c}{l^{2}}+\frac{ac(-a+l)}{l^{2}}\\ -\frac{ac}{l}&-\frac{c}{l}&-\frac{c}{l}&-\frac{c}{l^{2}}+\frac{ac(-a+l)}{l^{2}} &\frac{c}{l^{2}}+\frac{ac^{2}c}{l^{2}}+c_{b}\end{array}\right)\]where \(J_{p}=J_{d}/2\) and \(J_{d}=m_{d}R^{2}/2\). \(\mathbf{\Phi}\) is the offset matrix which can be obtained by the geometrical relations of the rotor position:  
\[\mathbf{\Phi}=\frac{1}{l}\left(\begin{array}{cc}l-a&a\\ -1&1\end{array}\right) \tag{21}\]  
\(\mathbf{K}_{c}\) is the stiffness matrix when the supports are rigid, and can be obtained by the flexibility matrix method [32]:  
\[\mathbf{K}_{c}=\left\{\frac{1}{3 EIl}\left(\begin{array}{cc}a^{2}(l-a)^{2}&a(l-a) (l-2a)\\ a(l-a)(l-2a)&l^{2}-3 la+3a^{2}\end{array}\right)\right\}^{-1} \tag{22}\]  
The external exciting force vector \(\mathbf{F}\) of Equation (19) includes the unbalanced mass force, the ball-bearing force and the UMP. Therefore, the vector \(\mathbf{F}\) can be written as  
\[\mathbf{F}=\mathbf{F}_{1}+\mathbf{F}_{2}+\mathbf{F}_{3} \tag{23}\] \[\mathbf{F}_{1}=\left(\begin{array}{cccccccc}m_{d}e_{0}\Omega^{2} \cos\Omega t&0&0&0&m_{d}e_{0}\Omega^{2}\sin\Omega t&0&0&0\end{array}\right)^{ \mathrm{T}}\] \[\mathbf{F}_{2}=\left(\begin{array}{cccccccc}0&0&F_{Ax}&F_{Bx}&0&0&F_ {Ay}&F_{By}\end{array}\right)^{\mathrm{T}}\] \[\mathbf{F}_{3}=\left(\begin{array}{cccccccc}F_{x\text{UMP}}&0&0&0&F_ {y\text{UMP}}&0&0&0\end{array}\right)^{\mathrm{T}}\]