## 3 Nonlinear Dynamic Analysis of the RBRS
### Effect of Radical Stiffness on Dynamic Characteristics of RBRS at Variable Speed  
The nonlinear radial stiffness (\(k_{r}\)) has an important influence on the stable operation and dynamic characteristics of RBRS. In order to study the dynamic response characteristics of the coupling system with nonlinear rubbing forces at different speeds, Figures 13-18 give the bifurcation diagram and 3-D frequency spectrum of RBRS with different radial stiffness (\(k_{r}=1\times 10^{8}\) N/m, \(k_{r}=2.6\times 10^{8}\) N/m, and \(k_{r}\) = \(4\times 10^{8}\) N/m). All other parameters are fixed. Figure 13: The bifurcation diagram of RBRS at \(k_{r}=1\times 10^{8}\) N/m. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Excitation** & **Rotating Speed** & \\
**Current \(I_{j}\) (A)** & **\(\omega\) (rad/s)** & **Main Frequency Feature (m: Positive Integer)** \\ \hline \multirow{8}{*}{\(I_{j}=1000\)} & \(7.0\leq\omega\leq 16.2\) & \(f_{\rm n}\) \\  & \(16.2<\omega\leq 18.6\) & \(f_{\rm n},2f_{\rm n},3f_{\rm n},4f_{\rm n},\ldots\ldots\) \\  & \(18.6<\omega\leq 19.8\) & \(f_{\rm n},2f_{\rm n},3f_{\rm n},4f_{\rm n},\ldots,\) \\  & \(19.8<\omega\leq 23.6\) & \(f_{\rm n},2f_{\rm n},3f_{\rm n},4f_{\rm n},\ldots,\) continuous frequency components \\  & \(23.6<\omega\leq 24.6\) & \(f_{\rm n},2f_{\rm n},3f_{\rm n},4f_{\rm n},\ldots,\) \\  & \(24.6<\omega\leq 42.4\) & \(f_{\rm n},2f_{\rm n},3f_{\rm n},4f_{\rm n},\ldots,\) continuous frequency components \\  & \(42.4<\omega\leq 42.8\) & \(f_{\rm n},2f_{\rm n},3f_{\rm n},4f_{\rm n},\ldots,\) \\  & \(42.8<\omega\leq 54.6\) & \(f_{\rm n}/2,f_{\rm n},3f_{\rm n}/2,2f_{\rm n},5f_{\rm n}/2,3f_{\rm n},7f_{\rm n }/2,4f_{\rm n},\) continuous frequency components \\  & \(54.6<\omega\leq 59.6\) & \(f_{\rm n},2f_{\rm n},3f_{\rm n},\) continuous frequency components \\  & \(59.6<\omega\leq 71\) & \(f_{\rm n},2f_{\rm n},3f_{\rm n},\) unf\({}_{\rm n}/3\) \\ \hline \multirow{8}{*}{\(I_{j}=1200\)} & \(7.0\leq\omega\leq 15.4\) & \(f_{\rm n}\) \\  & \(15.4<\omega\leq 18.8\) & \(f_{\rm n},2f_{\rm n},3f_{\rm n},4f_{\rm n},\ldots,\) \\  & \(18.8<\omega\leq 20\) & \(f_{\rm n},2f_{\rm n},3f_{\rm n},4f_{\rm n},\ldots,\) \\  & \(20<\omega\leq 22.4\) & \(f_{\rm n},2f_{\rm n},3f_{\rm n},4f_{\rm n},\ldots,\) \\  & \(22.4<\omega\leq 27.2\) & \(f_{\rm n},2f_{\rm n},3f_{\rm n},4f_{\rm n},\ldots,\) \\  & \(27.2<\omega\leq 28.8\) & \(f_{\rm n},2f_{\rm n},3f_{\rm n},4f_{\rm n},\ldots,\) \\  & \(28.8<\omega\leq 65.2\) & \(f_{\rm n},2f_{\rm n},3f_{\rm n},4f_{\rm n},\ldots,\) \\  & \(65.2<\omega\leq 71\) & \(f_{\rm n},2f_{\rm n},3f_{\rm n},\) continuous frequency components \\ \hline \hline \end{tabular}
\end{table} 
Table 3: _Cont._Figure 16: 3-D frequency spectrum of RBRS at \(k_{r}=2.6\times 10^{8}\) N/m. Figure 14: 3-D frequency spectrum of RBRS at \(k_{r}=1\times 10^{8}\) N/m. Figure 15: The bifurcation diagram of RBRS at \(k_{r}=2.6\times 10^{8}\) N/m. It can be seen from Figures 13 and 14 that when the radial stiffness is small, the coupling system behaves as period 1 and a double-cycle motion, and only the bifurcation path of the period-doubling bifurcation and the inverse period-doubling bifurcation appears, and discontinuous jumps occur in the range of \(\omega\in[39.6,52.2]\) rad/s. In the three-dimensional spectrum, when the rotational speed is small, the frequency components are relatively simple, mainly the frequency multiplication components (\(f_{n}\), \(2f_{n}\), \(3f_{n,m}\),...). With the gradual increase of the rotational speed, besides the frequency multiplication component, the coupling system has unobvious frequency demultiplication components (\(f_{n}\)/\(2\),\(f_{n}\)/\(3\), \(f_{n}\)/\(4\),...) and the frequency demultiplication components alternately appear. The amplitude of the main frequency components increases first and then decreases nonlinearly (when \(\omega=58.0\) rad/s, the c amplitude peak appears at \(f_{n}\)).