## 4 The Influence of Structural Parameters on Synchronization and Stability
### Parameters Influence on the Synchronization
For purpose of making it easier to satisfy the synchronization criterion and implement synchronization, a larger value of \(T_{cij\text{max}}\) (\(ij=12,\ 13,\ 14\)) has more significant effect on the implementation of synchronization. Furthermore, the larger the value of \(\zeta_{ij}\), the stronger the synchronization ability. Consequently, a small \(f_{R}\) is needed in this case. In Figure 7, the  
Figure 4: Maximum values of dimensionless coupling torques and synchronization ability coefficients versus \(r_{l}\) for different \(r_{m}\) when \(f_{R1}=0.08\), \(f_{R2}=0.04\), \(f_{R3}=0.08\): (**a**) maximum value of a dimensionless coupling torque between an exciter and a CR; (**b**) maximum value of a dimensionless coupling torque between two CRs; (**c**) synchronization ability coefficient between an exciter and a CR; (**d**) synchronization ability coefficient between two CRs. contour plots of the maximum values of dimensionless coupling torques and the synchronization ability coefficients versus \(\eta_{r}\) and \(\eta_{m}\) are pictured. The system parameters are set to \(m_{1}=6\)kg and \(f_{R1}=f_{R2}=f_{R3}=0.08\). Looking at Figure 7, which gives insight into variation of the synchronization effect, one sees that there is a marked rise in the synchronization coefficients \(T_{cj\text{max}}\) and \(\zeta_{ij}\), with the increasing \(\eta_{r}\) and \(\eta_{m}\). Therefore, the dimensionless parameters \(\eta_{r}\) and \(\eta_{m}\) are often preferably high to enhance synchronization ability and make it easy to implement synchronization, but should be reduced when the coordinate \(\{\eta_{r},\eta_{m}\}\) is not in the area I (can be seen in Figure 2) to satisfy the synchronization criterion. In general, to have the best synchronization effect, an appropriate selection of each structural parameter must be implemented, and further understanding of the structural parameters influence on the stability is also needed. Figure 5: Maximum values of dimensionless coupling torques and synchronization ability coefficients versus \(r_{l}\) for different \(r_{m}\) when \(f_{R1}=0.08\), \(f_{R2}=0.06\), \(f_{R3}=0.04\): **(a)** maximum value of a dimensionless coupling torque between an exciter and a CR; **(b)** maximum value of a dimensionless coupling torque between two CRs; **(c)** synchronization ability coefficient between an exciter and a CR; **(d)** synchronization ability coefficient between two CRs. Figure 6: Maximum values of dimensionless coupling torques and synchronization ability coefficients versus \(f_{R}\) (\(f_{R1}=f_{R2}=f_{R3}=f_{R}\)): (**a**) \(T_{\text{cj/max}}\) (\(ij=12,\,13,\,14\)) value versus \(f_{R}\); (**b**) \(T_{\text{cj/max}}\) (\(ij=23,\,42,\,34\)) value versus \(f_{R}\); (**c**) \(\zeta_{ij}\) values versus \(f_{R}\). In Figure 8, it is shown that the consistent emerging values of stable phase differences are 'immune' to changes in \(r_{l}\). As depicted in Figure 8a, the constant phase differences are: \(2\alpha_{1}=204^{\circ}\) (or \(-156^{\circ}\)), \(2\alpha_{2}=0^{\circ},2\alpha_{3}=0^{\circ}\); in Figure 8b, the constant phase differences are: \(2\alpha_{1}=206^{\circ}\) (or \(-154^{\circ}\)), \(2\alpha_{2}=344^{\circ}\) (or \(-16^{\circ}\)), \(2\alpha_{3}=16^{\circ}\); in Figure 8c, the constant phase differences are: \(2\alpha_{1}=206^{\circ}\) (or \(-154^{\circ}\)), \(2\alpha_{2}=352^{\circ}\) (or \(-8^{\circ}\)), \(2\alpha_{3}=352^{\circ}\) (or \(-8^{\circ}\)); in Figure 8d, the constant phase differences are: \(2\alpha_{1}=192^{\circ}\) (or \(-168^{\circ}\)), \(2\alpha_{2}=0^{\circ}\), \(2\alpha_{3}=0^{\circ}\). From this figure, it is evident that the three CRs synchronize in-phase for three identical friction coefficients and synchronize with a non-zero phase difference for three non-identical friction coefficients.