## 3 Nonlinear Motions of the Brush Seal Rotor System
### Numerical Simulation  
For comparison of the nonlinear motions, numerical simulation is performed to compare the semi-analytical and the numerical results. The numerical results can be obtained by some numerical integration method such as the Runge-Kutta method or Euler method. This paper adopts an implicit mid-point method [20] for numerical results. The solid linesrepresent the numerical results, while the circles represent the semi-analytical motions. I.C. means the initial conditions. Figure 3 shows the semi-analytical motions and numerical results of the stable period-1 motion. It can be seen from Figure 3a,b that the displacements are simple sinusoidal curves. The phase diagrams are simple ellipses in Figure 3c,d. The semi-analytical motions of the period-1 motion are compared with complete agreement with the numerical results. The nonlinear rotor system needs two more harmonic terms to meet the engineering requirements if the accuracy required is \(\varepsilon<10^{-3}\) in Figure 3e,f. For the accuracy of \(\varepsilon<10^{-9}\), three harmonic terms are required to meet the theoretical design or control requirements. Figure 4 shows the comparison of the unstable period-1 motion of the brush seal rotor system. In Figure 4a,b, if the semi-analytical motions and the numerical results are set at the same initial conditions, the numerical motions are consistent with the analytic results in \(t=[0,\,1.5]\). But when t \(>1.5\), the numerical results gradually move away from the semi-analytical results and the dangerous motion happens. This confirms the unstable  
Figure 3: Numerical comparison of stable period-1 motion of brush seal rotor system: (**a**) displacement, \(x_{1}\); (**b**) displacement, \(y_{1}\); (**c**) phase diagram, \((x_{1},x_{2})\); (**d**) phase diagram, \((y_{1},y_{2})\); (**e**) harmonic spectrum, \(A_{i,j}\); (**f**) harmonic phases, \(\varphi_{i,j}\). (\(i=1\) and \(2\) for \(x\)- and \(y\)-directions, respectively; \(j=0,1,2,\cdots,5\)). motions. Figure 4c,d present the same unstable phenomenon. From Figure 4e,f, if the accuracy required is set for \(\varepsilon<10^{-3}\), the brush seal rotor system needs two harmonic terms to meet the engineering requirements. For accuracy \(\varepsilon<10^{-10}\), five harmonic terms are required to meet the theoretical design or control requirements. Figure 5 presents the stable period-2 motion of the brush seal rotor system at \(\Omega=10.2\). In Figure 5a,b, the numerical motions in solid lines match with the semi-analytical solutions in circles. This confirms stable period-2 motion solutions. Figure 5c,d present the displacement orbit and velocity orbit. The exact match of the numerical motions with the semi-analytical solutions validates the correctness of the obtained semi analytical motions. Figure 5e,f present the harmonic amplitudes and phases of the stable period-2 motion. For displacement \(x_{1}\), the most contributed harmonic amplitudes are \(A_{1,0}=0.1027\), \(A_{1,0,5}=0.1774\), \(A_{1,1}=0.4730\), and \(A_{1,1,5}=0.0136\). For displacement \(y_{1}\), the most contributed harmonic amplitudes are \(A_{2,0}=0.2007\), \(A_{2,0,5}=0.8378\), \(A_{2,1}=0.5680\), \(A_{2,1,5}=0.0367\), and  
Figure 4: Numerical comparison of unstable period-1 motion of brush seal rotor system: (**a**) displacement, \(x_{1}\); (**b**) displacement, \(y_{1}\); (**c**) phase diagram, \((x_{1},x_{2})\); (**d**) phase diagram, \((y_{1},y_{2})\); (**e**) harmonic spectrum, \(A_{i,j}\); (**f**) harmonic phases, \(\varphi_{ij}\). (\(i=1\) and \(2\) for \(x\)- and \(y\)-directions, respectively; \(j=0,1,2,\cdots,4\)). \(A_{2,2}=0.0129\). The brush seal rotor system needs twenty harmonic terms to meet the engineering requirements of the accuracy required, set for \(\varepsilon<10^{-12}\). Figure 6 presents the unstable period-2 motion of the brush seal rotor system at \(\Omega=7.38\). In Figure 6a,b, the numerical solutions in solid lines almost move along with the semi-analytical motions in circles. Even the period-2 motion is unstable.