## 4 The Influence of Structural Parameters on Synchronization and Stability
### Parameters Influence on the Synchronization
For small \(r_{l}\), the maximum value of a dimensionless coupling torque and synchronization ability coefficient between an exciter and a CR remain largely unchanged with variations in \(r_{l}\); for large \(r_{l}\), one can notice a rapid increase of these two synchronization parameters when the value of \(r_{l}\) is increased. It must also be noted that the maximum value of a dimensionless coupling torque and synchronization ability coefficient between two CRs slightly decrease at first, but shortly increase, with the increasing \(r_{l}\). Recall, an increased value of \(T_{\rm cjjmax}\) results in an increase in the allowable residual torque difference between two coupled objects, Equations (29)-(31). That is to say, it is easier to satisfy the synchronization criterion and implement synchronization of an exciter and three CRs. Consequently, in this case, to make the system implement synchronization more easily and improve the ability of synchronization of the system, a smaller value of \(r_{m}\) and a greater value of \(r_{l}\) should be chosen. Figure 3: Maximum values of dimensionless coupling torques and synchronization ability coefficients versus \(r_{l}\) for different \(r_{m}\) when \(f_{R1}=0.08\), \(f_{R2}=0.08\), \(f_{R3}=0.08\): (**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. Figure 2: Maximum value of the necessary electromagnetic torque for the vibrating system: (**a**) the values of \(T_{\rm e01 max}\) for different values of \(f_{Ri}\) (\(i=1,\ 2,\ 3\)) versus \(r_{m}\); (**b**) the value of \(T_{\rm e01 max}\) with changes in \(f_{R2}\) and \(f_{R3}\); (**c**) the value of \(T_{\rm e01 max}\) with changes in \(\eta_{r}\) and \(\eta_{m}\). Consider now the cases of different friction coefficients. As depicted in Figures 4 and 5, the results show that the variation tendency of the synchronization parameters \(T_{cij\text{max}}\) and \(\zeta_{ij}\) with changes in \(r_{l}\) is about in line with that in the case of identical friction coefficients. From Figures 4 and 5, it is worth noting that, for large \(r_{l}\), the maximum values of dimensionless coupling torques and synchronization ability coefficients between two coupled objects are sensitive to replacement of the two coupled objects (for example, the switch to an exciter and CR 3 from an exciter and CR 1 or from CR 1 and CR 2 to CR 3 and CR 1) when \(r_{m}\) is fixed. The reason behind this is the substitution of one friction coefficient for another. In detail, it is worthwhile to mention that, from a comparison between Figures 4 and 5, one can realize that a larger value of the friction coefficient leads to a decrease in the maximum value of a dimensionless coupling torque and synchronization ability coefficient between an exciter and a corresponding CR; an increased difference value between two arbitrary friction coefficients results in a decrease in the maximum value of a dimensionless coupling torque and synchronization ability coefficient between two corresponding CRs. Next, assume that the three variable friction coefficients are identical, i.e., \(f_{R1}=f_{R2}=f_{R3}=f_{R}\). Figure 6 shows the relation of \(T_{cij\text{max}}\) and \(\zeta_{ij}\) to friction coefficient \(f_{R}\). As presented in Figure 6, it is noticeable that as \(f_{R}\) is increased, the synchronization coefficients \(T_{cij\text{max}}\) (\(ij=12,\ 13,\ 14\)) and \(\zeta_{ij}\) decrease gradually, and for \(T_{cij\text{max}}\) (\(ij=23,\ 42,\ 34\)), in contrast, there is a correlated rise. In a vibrating system, the residual torque difference between an exciter and a CR is always much higher as compared to that between two CRs.