## 4 Bifurcation Analysis, Stability Charts, and Control Performance
### Influence of the Derivative Gain (\(d\)) on the Rotor System Dynamics
Accordingly, to investigate the stability boundaries of the derivative control gain as a function of either the rotor eccentricity \(f\) or the detuning parameter \(d\), the stability chart of the twelve-pole system is established in both \(d-\sigma\) plane and \(d-f\) plane, as illustrated in Figure 15. It is clear from Figure 15a that the system may lose its stability at a small derivative gain (i.e., \(d<0.0045\)) if the rotor angular speed \(\Omega\) is closed to or equal the natural frequency \(\omega\) (i.e., when \(\sigma\to 0\)). However, increasing the derivative control gain in a way so that \(d>0.0045\), guarantees the stability of the twelve-pole system oscillation regardless of the rotor angular speed. Figure 15b shows the stability margin of the derivative gain as a function of the eccentricity \(f\). The figure illustrates that the increase in derivative gain makes the twelve-pole system withstand the large eccentricities without losing its stability. Figure 15: Stability chart of the twelve-pole system: (**a**) \(d-\sigma\) plane when \(f=0.01\) and \(p=1.22\), and (**b**) \(d-f\) plane when \(p=1.22\) and \(\sigma=0\). Figure 16: Numerical simulation for the twelve-pole system according to the marked point \(p_{1}\) in Figure 14b (i.e., when \(d=0.004,f=0.01\)): **(a,b)** temporal oscillation, (c) orbital motion, (**d,e)** Poincaré return map, and (**f,g**) frequency spectrum. Figure 17: Numerical simulation for the twelve-pole system according to the marked point \(p_{2}\) in Figure 14b (i.e., when \(d=0.006,f=0.01\)): **(a.b)** temporal oscillation, (c) orbital motion, (**d.e)** Poincaré return map, and (**f.g**) frequency spectrum. Figure 18: Numerical simulation for the twelve-pole system according to the marked point \(p_{3}\) in Figure 14b (_i.e._, when \(d=0.006,f=0.02\)): **(a.b)** temporal oscillation, (c) orbital motion, (**d.e**) Poincaré return map, and (**f.g**) frequency spectrum. To validate the accuracy of the obtained stability charts shown in Figure 15, a numerical simulation for the rotor system according to the three marked points (i.e., the points \(p_{1},\ p_{2}\) and \(p_{3}\)) in Figure 15b is performed, as shown in Figures 16-18. Figure 16 simulates the temporal oscillations of the rotor system according to the point \(p_{1}\) (i.e., \(d=0.004,\ f=0.01,\ p=1.22\), and \(\sigma=0.0\)) that is marked in Figure 15b within the unstable solutions region. Figure 16a,b shows the instantaneous vibrations of the twelve-pole system in \(X\) and \(Y\) directions, while Figure 16c shows the corresponding steady-state orbital motion. In addition, the Poincare-map and frequency spectrum are shown in Figure 16d-g. Figure 16a-c confirm that the system performs unstable periodic motion when simulated according to the point \(p_{1}\) that agrees with the stability chart given in Figure 15b. Moreover, Figure 16d-g demonstrate that the rotor system performs a quasiperiodic motion. The numerical simulations of the rotor temporal oscillations according to the points \(p_{2}\) and \(p_{3}\) shown in Figure 15b have been illustrated in Figures 17 and 18, respectively. Now, by comparing Figures 16-18 with the three marked points (\(p_{1},\ p_{2}\) and \(p_{3}\)) shown in Figure 15b, we can confirm the accurate correspondence among the obtained stability chart and the numerical simulations. Finally, the bifurcation behaviors of the twelve-pole system when utilizing the eccentricity \(f\) as the bifurcation control parameter is discussed through Figures 19-21, where Figure 19a,b shows the system amplitudes (\(a\) & \(b\)) versus the rotor eccentricity \(f\) when \(\sigma=0.0\) at various values of the proportional gain \(p\) (i.e., when \(p=1.22\) and \(p=1.5\)). It is clear from Figure 19a,b that the twelve-pole system exhibits complex bifurcations behavior at \(p=1.22\). In addition, the figure confirms that the system loses its stability at critical values of the eccentricity.