## 3 Nonlinear Dynamic Analysis of the RBRS
### Influence of Excitation Current on Dynamic Characteristics of RBRS at Variable Speed
With the speed increasing further, when \(\omega\) = 59.6 rad/s, the system moves from a short quasi-periodic motion to a stable periodic motion via the Hopf bifurcation; except for the frequency multiplication components (\(f_{\rm n}\), 2\(f_{\rm n}\), 3\(f_{\rm n}\),...) in the 3-D frequency spectrum, there are also frequency demultiplication components (\(f_{\rm n}\)/3, 2\(f_{\rm n}\)/3, 4\(f_{\rm n}\)/3, 5\(f_{\rm n}\)/3,...), in which \(f_{\rm n}\)/3 is the main frequency  
Figure 4: The bifurcation diagram of RBRS at \(I_{j}\) = 1000 A.  
Figure 5: 3-D frequency spectrum of RBRS at \(I_{j}\) = 1000 A.  
component, and the amplitude gradually increases and is significantly larger than other frequency components. The vibration responses of the RBRS at \(\omega\) = 59.8 rad/s are displayed in Figure 8. As can be seen from the figure, the coupling system is quasi-periodic motion. The main frequency components are \(f_{n}\), \(2f_{n}\), and \(3f_{n}\). Meanwhile, the attractors form a closed circle in the Poincare map. In order to further comprehensively study the effects of different excitation current of the generator \(I_{j}\) on RBRS, Figures 9-12, respectively, give the bifurcation diagram and 3-D frequency spectrum of RBRS at \(I_{j}\) = 700 A and \(I_{j}\) = 1200 A. By comparing the bifurcation diagrams and the 3-D frequency spectrum, it can be seen that the system shows different vibration forms and different bifurcation paths under various excitation currents. When the excitation current of the generator \(I_{j}\) is small, the dynamic behavior of the system is relatively simple. When \(\omega\) < 20.2 rad/s, the vibration system behaves as a periodic motion, and the frequency components is not only the frequency conversion \(f_{n}\), but also the unobvious frequency multiplication component. As the rotational speed increases, the system transits from short-term periodic motion, chaotic motion, and quasi-periodic motion to periodic motion, successively undergoing saddle-node bifurcation, period-doubling bifurcation, inverse period-doubling bifurcation, and Hopf bifurcation. Frequency multiplication is the main frequency component in the 3-D frequency spectrum, and the peak value appears at \(7f_{n}\) when \(\omega\) = 21.0 rad/s. Figure 8: Quasi-periodic motion of RBRS at \(\omega\) = 59.8 rad/s: (**a**) Waveform, (**b**) Frequency, and (**c**) Axis orbit and Poincaré map. Figure 6: Periodic-2 motion of RBRS at \(\omega\) = 19.4 rad/s: (**a**) Waveform, (**b**) Frequency, and (**c**) Axis orbit and Poincaré map. Figure 7: Chaotic motion of RBRS at \(\omega\) = 33.6 rad/s: (**a**) Waveform, (**b**) Frequency, and (**c**) Axis orbit and Poincaré map. When the rotational speed increases to \(\omega\) = 54.8 rad/s, the system enters the chaotic motion and transits to stable periodic motion through the addle-node bifurcation, and the amplitude of response at \(\omega\) = 68.0 rad/s appears obvious jumping discontinuity. Comparing Figures 4 and 5, Figures 9 and 10, and Figures 11 and 12 shows that the dynamic behavior of the system is more complicated, and the discontinuous jump is obviously increased. As the rotational speed increases, the chaotic region becomes wider. In the range of \(\omega\) \(\in\) [20.0,22.4] rad/s, except for the stable period-doubling motion in a small region, the system is of continuous chaotic motion or intermittency chaos motion. When \(\omega\) = 29.0 rad/s, the system enters the chaotic motion through the saddle-node bifurcation, and in the range of \(\omega\) > 28.8 rad/s, the vibration system exhibits strong nonlinear characteristics of periodic motion and chaotic motion frequently alternating. With the increase of the rotational speed, different bifurcation paths, such as saddle-node bifurcation, period-doubling bifurcation, and inverse period-doubling bifurcation appear in the system. The corresponding vibration amplitude increases first, then decreases and then increases nonlinearly.