## 2 The Nonlinear Rotor System  
A brush seal rotor system is established from a flexible rotor with nonlinear brush seal forces. The discrete mapping method [19; 20] is used to transform the governing equations into a discretized nonlinear system. Consider the brush seal rotor system as  
\[\mathbf{M}\ddot{\mathbf{X}}+\mathbf{C}\ddot{\mathbf{X}}+\mathbf{K}\dot{ \mathbf{X}}=\widetilde{\mathbf{F}}_{s}+\mathbf{F}_{\omega}+\mathbf{G} \tag{1}\]  
where \(\mathbf{M}=\text{diag}(m,m)\) is the mass matrix of the nonlinear rotor system; \(\mathbf{C}=\text{diag}(c,c)\) is the damping coefficient matrix; \(\mathbf{K}=\text{diag}(k,k)\) is the stiffness matrix; \(\widetilde{\mathbf{F}}_{s}=\left[\overline{F}_{x},\overline{F}_{y}\right]^{T}\) is the nonlinear brush seal force; \(\mathbf{F}_{\omega}=me\omega^{2}[\cos\omega t,\sin\omega t]^{\text{T}}\) is the centrifugal force of the nonlinear rotor system; \(\mathbf{G}=\left[0,mg\right]^{\text{T}}\) is the weight vector. \(\mathbf{X}=\left[X,Y\right]^{\text{T}}\) represents the displacements in \(x\)- and \(y\)-directions. \(e\) is the mass eccentricity and \(\omega\) is the angular speed. On the rotor system, nonlinear forces are introduced by the brush seal, which is mounted on the frame around the rotor journal. Elasticity and beam theory are utilized to determine the force that a single brush bristle exerts on the rotor. The contact force between the bristle and the rotor can be derived in accordance with the following assumptions: (1) it is assumed that the contact between the bristle and the rotor is always in a point-to-point contact and the contact point is at the end point of the bristle; (2) the bristle is always in elastic deformation. The contact diagram that depicts the interaction of a single brush bristle and the rotor is illustrated in Figure 1, and the parameters are in Table 1.  
From Figure 1, consider the friction force between the bristle and the rotor as \(f_{N}=\mu N\). Then, the contact forces \(F_{\eta}\) and \(F_{\xi}\) in the \(\eta\)- and \(\xi\)-directions are obtained as:  
\[ \begin{split} F_{\eta}&=N\cos\psi+f_{N}\sin\psi=\sqrt {1+\mu^{2}}N\sin(\theta_{0}+\varphi+\arctan\mu),\\ F_{\xi}&=N\sin\psi-f_{N}\cos\psi=\sqrt{1+\mu^{2}}N \cos(\theta_{0}+\varphi+\arctan\mu)\end{split}  \tag{2}\]  
In Equation (2), \(F_{\xi}\) is much smaller than \(F_{\eta}\), so \(F_{\xi}\) is neglected. \begin{table}
\begin{tabular}{l l} \hline \hline
**Name** & **Symbols** \\ \hline Brush bristle deformation coordinate system & (\(\eta\), \(\xi\)) \\ Brush seal coordinate system & (\(x\), \(y\)) \\ Bristle length after deformation & \(s\) \\ Actual radius clearance & \(\delta\) \\ Bristle contact deflection angle & \(\varphi\) \\ The \(\eta\)-directional force of the bristle & \(F_{\eta}\) \\ The \(\xi\)-directional force of the bristle & \(F_{\xi}\) \\ Maximum turning angle of the bristle & \(\theta_{\max}\) \\ Sliding friction & \(f_{\mathrm{N}}\) \\ The projection of \(s\) on the radius direction & \(\alpha\) \\ Shaft radius & \(R\) \\ Bristle pre-rotation angle & \(\theta_{0}\) \\ Maximum deflection & \(\eta_{(\phi)}\) \\ Normal force on shaft journal & \(N\) \\ Normal force of a single brush bristle on the shaft journal & \(N_{1}\) \\ Friction coefficient of the rotor & \(\mu\) \\ Angle between normal force and sealing force & \(\psi\) \\ Eccentricity & \(e\) \\ Declination angle & \(\gamma\) \\ Minimum gap & \(\delta_{\max}\) \\ The angle between the circumferential direction and the negative direction of & \(\rho\) \\ the y-axis & Brush bristle diameter & \(R_{0}\) \\ Moment of inertia of the bristle cross section & \(I\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Brush seal parameter nomenclature. Figure 1: Contact diagram of the bristle and the rotor. Consider the bristle in the bending diagram in Figure 2. Based on elasticity and beam theory, we have:  
\[ \begin{array}{c}\theta(\xi)=-F_{\eta}\frac{\xi^{2}}{2 EI}+\frac{F_{\eta}s}{EI} \xi,\\ w(\xi)=-F_{\eta}\frac{\xi^{3}}{6 EI}+\frac{F_{\eta}s}{EI}\frac{\xi^{2}}{2}.\end{array}