## 3 Nonlinear Dynamic Analysis of the RBRS
### Influence of Excitation Current on Dynamic Characteristics of RBRS at Variable Speed  
Nonlinear electromagnetic force has an important influence on the inherent characteristics and dynamic response of RBRS, which is one of the important reasons for the nonlinear characteristics of RBRS. In order to study the influence of different excitation currents of the generator \(I_{j}\) on the dynamic characteristics of RBRS, Figures 4 and 5, respectively, give the bifurcation diagram and 3-D frequency spectrum of RBRS with rotational speed \(\omega\) (\(\omega\) = 7.0-71 rad/s) as control parameter at \(I_{j}\) = 1000 A. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Rotating Speed (rad/s)**} & \multicolumn{2}{c}{**Reference [31]**} & \multicolumn{2}{c}{**This Paper**} \\ \cline{2-5}  & **Main Frequency Feature** & **Rubbing** & **Main Frequency Feature** & **Rubbing** \\ \hline \(7\leq\omega<23\) & \(f_{n1}\) & 0 & \(f_{n1}\) & 0 \\ \hline \multirow{3}{*}{\(23\leq\omega<59\)} & \(f_{n1}\), \(2f_{n1}\), \(3f_{n1}\), \(4f_{n1}\), \(5f_{n1}\), & & \(f_{n1}\), \(2f_{n1}\), \(3f_{n1}\), \(4f_{n1}\), \(5f_{n1}\),\(f_{n1}\)/2, & \\  & unobvious frequency & 1 & \(3f_{n1}\)/2, \(5f_{n1}\)/2, \(7f_{n1}\)/2, continuous & 1 \\  & demultiplication components & & frequency components & \\ \hline \multirow{3}{*}{\(59\leq\omega\leq 71\)} & \multirow{3}{*}{\(f_{n1}\), \(2f_{n1}\)} & \multirow{3}{*}{0} & \(f_{n1}\), \(2f_{n1}\), \(3f_{n1}\), \(4f_{n1}\), \(5f_{n1}\), \(f_{n1}\)/2, & \\  & & & \(3f_{n1}\)/2, \(5f_{n1}\)/2, \(7f_{n1}\)/4, \(3f_{n1}\)4, & \\  & & & frequency components & \\ \hline \hline \end{tabular} Note: “\(0\)” represents no rubbing “\(1\)” represents rubbing.  
\end{table} 
Table 2: Comparison of the frequency characteristics. Figure 3: (**a**) Adapted from reference [31]; (**b**) Model in this paper. When the rotational speed is small, the motion state of RBRS is periodic motion, and the response amplitude is relatively stable. At the same time, the frequency component is not very obvious, mainly based on one times the frequency conversion \(f_{\rm n}\). Figure 6 shows the characteristics of periodic motion at \(\omega\) = 19.4 rad/s, through waveform, frequency, axis orbit, and Poincare map. With the increase of the rotational speed, the system passes through saddle-node bifurcation at \(\omega\) = 20 rad/s and enters chaotic motion. In the range of \(\omega\) \(\in\) [20.0,59.8] rad/s, the system mainly exhibits chaotic motion, except for a few points where the vibration system presents periodic-n motion at \(\omega\) = 24.4 rad/s, \(\omega\) = 24.6 rad/s, \(\omega\) = 42.6 rad/s, and \(\omega\) = 42.8 rad/s, and has different bifurcations such as saddle-node bifurcation, period-doubling bifurcation, inverse period-doubling bifurcation; in the three-dimensional spectrum, the continuous and discrete spectral components alternate and the amplitude changes nonlinearly. The vibration responses of the RBRS at \(\omega\) = 33.6 rad/s are given in Figure 7. It can be seen from the figure that the amplitude of the system vibration response increases and changes aperiodically; in the frequency domain, besides the frequency conversion and frequency multiplication components, the continuous frequency components also appear, and the axis orbit of the system is disorderedly. On the corresponding Poincare map, the attractors are irregular discrete points, and the system exhibits chaotic motion.