## 3 Nonlinear Dynamic Analysis of the RBRS
### Effect of Radical Stiffness on Dynamic Characteristics of RBRS at Variable Speed
When the radial stiffness increases to \(k_{r}=2.6\times 10^{8}\) N/m, it can be found that the period doubling region widens, the period-doubling bifurcation and the inverse period-doubling bifurcation alternate frequently, and obvious frequency demultiplication components appear, and the transient quasi-periodic motion in high speed region. There is no chaos in the coupling system in the whole speed changing region. As the radial stiffness continues to increase to \(k_{r}=4\)\(\times\)\(10^{8}\) N/m, the coupling system exhibits complex nonlinear behavior and more unstable regions; different bifurcation paths, such as saddle-node bifurcation, period-doubling bifurcation, and inverse period-doubling bifurcation, and strong nonlinear phenomena, such as discontinuous amplitude jumps, appear. The chaotic region of the system is mainly concentrated in the three regions of \(\omega\in[18.8,52.2]\) rad/s, \(\omega\in[25.6,43.6]\) rad/s, and \(\omega\in[54.6,67.4]\) rad/s, and with the speed increasing, the chaotic motion region is widened, the intermittent chaotic phenomenon intensifies, and the continuum region increases  
Figure 17: The bifurcation diagram of RBRS at \(k_{r}=4\times 10^{8}\) N/m. Figure 18: 3-D frequency spectrum of RBRS at \(k_{r}=4\times 10^{8}\) N/m. in the three-dimensional spectrum. At the same time, the vibration system has obvious frequency demultiplication components; m/\({}_{n}\)/2, m/\({}_{n}\)/3, m/\({}_{n}\)/4, and m/\({}_{n}\)/6 (m: positive integer), and the frequency amplitude of \(f_{n}\)/2,\(f_{n}\)/3,\(f_{n}\)/4 is significantly larger than other frequency amplitudes. In the high-speed region, the system finally transits to a stable period 4 motion through the saddle-node bifurcation, and the jumping and discontinuity of amplitude is enhanced. Table 4 shows the intervals in which the frequency components change (where \(f_{n}\) is the frequency conversion). Comparing the bifurcation diagram and the three-dimensional spectrogram of RBRS, it can be seen that with the increase of radial stiffness, the motion complexity of the coupling system increases, the chaotic area increases, the response amplitude increases, and the vibration degree is enhanced. At the same time, in the high-speed region, the periodic motion region narrows and tends to disappear. In order to analyze more deeply, the related dynamic responses of the vibration system (\(k_{r}=4\times 10^{8}\) N/m) at \(\omega=19.2\) rad/s, \(\omega=40.8\) rad/s and \(\omega=69.8\) rad/s are displayed in Figures 19-21 through waveform, frequency, axis orbit, and Poincare map. When the rotational speed is small, i.e., \(\omega=19.2\) rad/s, the vibration amplitude of the system is small, and the corresponding main frequency domain components are frequency multiplication (\(f_{n}\), \(2f_{n}\), \(3f_{n}\),...) and m/\({}_{n}\)/2 (m: positive integer) with small amplitude. The coupling system behaves as periodic-2 motion, and its attractors are two points on the Poincare map. When \(\omega=59\) rad/s, the amplitude of the system vibration response increases and presents a non-periodic change; in the frequency domain, the continuous frequency spectrum components appear, corresponding to the chaotic axis orbit; the attractors on the corresponding Poincare map are irregular scattered points, and then the system motion is in chaotic state. When the rotational speed increases to \(\omega=69.8\) rad/s, the rotor returns to the periodic motion again, and the vibration response has different frequency characteristics;\(f_{n}\)/4 is the main frequency component and has obvious peak value. At the same time, four irregular closed curves are shown on the axis orbit map, and the attractors on the corresponding Poincare map are four isolated points.