## 3 Nonlinear Dynamic Analysis of the RBRS
### Influence of Friction Coefficient on Dynamic Characteristics of RBRS under Variable Speed  
Friction coefficient \(\mu\) has an important influence on the dynamic behavior of RBRS, and its size directly affects the dynamic characteristics of the coupling system. Figures 22-27 show the dynamic response (bifurcation diagram and three-dimensional spectrum diagram) of the system under different working conditions when the rotational speed is in the range of \(\omega\in[7.0,71.0]\) rad/s and the friction coefficient is \(\mu\) = 0.05, \(\mu\) = 0.3, and \(\mu\) = 0.78, respectively. By comparison it can be seen that when the friction coefficient is small, that is, \(\mu\) = 0.05, the RBRS in the interval of \(\omega\in[7.0,60.2]\) rad/s mainly exhibits the period-1 motion, and the response amplitude is relatively stable (when \(\omega\) = 51.2 rad/s, the coupling system behaves as chaotic motion; when \(\omega\) = 52.6 rad/s, the coupling system behaves as a period-3 motion), and there are many obvious discontinuous amplitude jumps. With the increase of the rotational speed, the system transits from periodic motion to chaotic motion through saddle-node bifurcation at \(\omega\) = 60.4 rad/s, and exhibits multiple motion states including periodic-1, periodic-2, periodic-n, Quasi-periodic, and chaotic motion. The coupling system appears as intermittent chaos, with the saddle-node bifurcation, the period-doubling bifurcation, and the inverse period-doubling bifurcation appearing alternately. In addition to the frequency multiplication component in the three-dimensional spectrum, there are also frequency demultiplication components (\(f_{\text{n}}\)/2, \(3\,f_{\text{n}}\)/25, \(f_{\text{n}}\)/7,...) and continuous spectrum components; when \(\mu\) = 0.3, the dynamic characteristics of the  
Figure 19: Periodic-n motion of RBRS at \(\omega\) = 19.2 rad/s: (**a**) Waveform, (**b**) Frequency, and (**c**) Axis orbit and Poincaré map. Figure 20: Chaotic motion of RBRS at \(\omega\) = 40.8 rad/s: (**a**) Waveform, (**b**) Frequency, and (**c**) Axis orbit and Poincaré map. coupling system become more complex and the chaotic movement is strengthened. At lower rotational speeds, the jump of the coupling system is weakened, and the motion state returns to the stable periodic motion at 21.0 \(<\omega<\) 27.8 rad/s again from the periodic motion in the range of \(\omega\in\) [7.0,20.0] rad/s through the chaotic motion in the narrow interval \(\omega\in\) [20.2,21.0] rad/s. When \(\omega>\) 27.8 rad/s, the system enters the chaotic region through the saddle-node bifurcation, and the periodic motion, the quasi-periodic motion, and the chaotic motion appear alternately until \(\omega=\) 61.4 rad/s. When 61.4 rad/s \(<\omega<\) 63.2 rad/s, RBRS is in stable period-4 motion and period-8 motion and is accompanied by the occurrence of inverse period-doubling bifurcation; the three-dimensional spectrum shows distinct frequency demultiplication components of \(f_{n}\)/4 and \(f_{n}\)/8, and RBRS re-enters into the chaotic region at \(\omega=\) 63.2 rad/s through the saddle-node bifurcation; when the friction coefficient is increased to \(\mu=\) 0.78, while \(\omega<\) 28.0 rad/s, the vibration system performs chaotic motion and periodic multiple motion alternatively only in the narrow interval \(\omega\in\) [21.8,24.8] rad/s, and in other cases perform stable periodic motion. With the increase of the rotational speed, the system enters the unsteady region through the saddle-node bifurcation at \(\omega=\) 28.2 rad/s. The system alternates frequently between the period-doubling motion, the counter-periodic motion, and the chaotic motion, and the jumping discontinuity of the response amplitude is intensified. Table 5 shows the intervals in which the frequency components change (where \(f_{n}\) is the frequency conversion). Figure 23: 3-D frequency spectrum of RBRS at \(\mu=\) 0.05. Figure 22: The bifurcation diagram of RBRS at \(\mu=\) 0.05. Figure 26: The bifurcation diagram of RBRS at \(\mu=0.78\).