## 3 Nonlinear Motions of the Brush Seal Rotor System
### Bifurcation Diagrams  
For a better understanding of stable and unstable motion switching, the nonlinear motions and bifurcations are projected in 2D planes in Figure 7. The solid lines mean stable motions and the dashed lines mean unstable motions. Between stable and unstable motions, bifurcations happen. The stable and unstable displacement \(x_{1}\) is depicted by Figure 7a. When the mass eccentricity is set to be e = 0.0005, the stable displacement experiences a jumping at \(\Omega=1.0955\) because of a saddle node bifurcation where a nonlinear resonance happens. The resonance is more obvious at the \(y\)-direction in Figure 7c. The dimensionless rotor system is characterized with a linear resonance frequency of \(\Omega^{\prime}=1\). For the original  
Figure 6: Numerical comparison of unstable period-2 motion of brush seal rotor system: (**a**) displacement, \(x_{1}\); (**b**) displacement, \(y_{1}\); (**c**) displacement plane, \((x_{1},y_{1})\); (**d**) velocity plane, \((x_{2},y_{2})\); (**e**) harmonic spectrum, \(A_{i,j}\); (**f**) harmonic phases, \(\varphi_{i,j}\). (\(i=1\) and \(2\) for \(x\)- and \(y\)-directions, respectively; \(j=0,1,2,\cdots,10\)). system, the linear resonance frequency is around 1027 r/min. Because of the nonlinearity of the brush seal forces, the resonance frequency will shift. The nonlinear resonance of the rotor system increases with the increase in the mass eccentricity. So do the resonance frequencies. The motion jumps at speed \(\Omega\) = 1.1787 when the eccentricity grows to \(e\) = 0.0015 where a saddle node bifurcation occurs. Because of saddle node bifurcations, the nonlinear rotor system may experience chatter and non-smooth vibration during running. When the rotor runs to \(\Omega\) = 2.0500, the stable period-1 motion changes to stable period-2 motion because of a period-doubling bifurcation. When the rotational speed increases to \(\Omega\) = 2.0600, a period-doubling bifurcation occurs and the stable period-2 motion disappears. The bifurcations on the velocity in the \(x\)-direction of the nonlinear rotor system are illustrated in Figure 7b. On the stable motion, there is an increase in the magnitude of the velocity along the \(x\)-direction. It is important to note that the velocity and the displacement in the \(x\)-direction have the same stable and unstable speed ranges, as well as the bifurcations. When the speed is operated to much lower than the critical speed, the nonlinear motion of the rotor remains steady regardless of the mass eccentricities. The vibration velocities approach almost zeros. The period-doubling bifurcations are \(\Omega\) = {1.98, 2.05, 2.06} for \(e\) = 0.0005, \(\Omega\) = {1.96,197, 2.04, 2.1} for \(e\) = 0.0015; \(\Omega\) = {1.93,1.96, 2.04, 2.14} for \(e\) = 0.0025; \(\Omega\) = {1.87, 1.93, 2.05, 2.18} for \(e\) = 0.0035; \(\Omega\) = {1.66, 1.8, 2.12, 2.29} for \(e\) = 0.0055; \(\Omega\) = {1.56, 1.64, 2.17, 2.35} for \(e\) = 0.0065. The displacement \(y_{1,k}\) of the nonlinear motions in the brush seal rotor system is presented in Figure 7c. At the location where the saddle node bifurcation is located, the displacement chatters in the \(y\)-direction. Figure 7d illustrates the velocity \(y_{2,k}\) varying with the rotational speed in the \(y\)-direction. The nonlinear phenomenon of \(y_{2,k}\) is similar to that in the \(x\)-direction. For obtaining the period-2 motions, the eigenvalue analysis of the brush seal rotor system is performed in Figure 8. The solid lines mean the stable motions and the dashed lines mean unstable motions. The nonlinear motion interacts with stable and unstable  
Figure 7: Semi-analytical Poincare map of the nonlinear period-1 motions in the rotor system with various eccentricities: (**a**) Poincare displacement, \(x_{1,k}\); (**b**) Poincare velocity, \(x_{2,k}\); (**c**) Poincaré displacement, \(y_{1,k}\); (**d**) Poincare velocity, \(y_{2,k}\). (\(k=0,1,2,\cdots,+\infty\)). branches. For clear observation, the eigenvalue analysis and the bifurcations are illustrated in two separate speed ranges.