## 4 The Influence of Structural Parameters on Synchronization and Stability
### Parameters Influence on the Synchronization  
The critical structural parameters are changed to investigate the influence on the synchronization characteristics. The physical meaning of \(T_{\rm e01}\) can be seen in Equation (33), i.e., \(T_{\rm e01}\) is the necessary electromagnetic torque of the vibrating system. \(T_{\rm e01}\) is a limited function of the phase differences and must not be more than its rated value (\(T_{\rm eN1}\)) at any time. Here, \(T_{\rm e01 max}\) is defined as the maximum value of \(T_{\rm e01}\) when the values of the phase differences vary. At the stage of unstable state during the entire operational process of the system, the dynamic coupling torques are assumed to manage to perform the stable synchronization by regulating the phase differences. This can easily result in the wide fluctuations of the rotational velocities and the phase differences; consequently, \(T_{\rm e01}\) fluctuates as well and even reaches its all time maximum value (\(T_{\rm e01 max}\)). To sum up, the inequation \(T_{\rm e01 max}\leq T_{\rm eN1}\) is the necessary condition of achieving synchronization with sufficient energy. Figure 2a presents maximum values of the necessary electromagnetic torque for the given sets of friction coefficients under different values of \(r_{m}\) when \(r_{R}=0.005\) m, \(m_{R}=4\) kg. The value of \(r_{m}\) can be varied by changing \(m_{1}\). It can be seen that the increasing \(r_{m}\) brings the increasing required value of \(T_{\rm e01 max}\). It also should be noted that the larger the sum of a set of friction coefficients, the nearer a curve of \(T_{\rm e01 max}\) comes to the value of \(T_{\rm eN1}\). When the sum of a set of friction coefficients is equal to another, their corresponding curves overlap with each other. By fixing an arbitrary friction coefficient (here, \(f_{R1}=0.08\)), Figure 2b exhibits a linear growth of \(T_{\rm e01 max}\) with the increasing \(f_{R2}\) and \(f_{R3}\). It is worth noting that each of the three friction coefficients has the same effect on the variation of \(T_{\rm e01 max}\). Accordingly, based on the necessary condition of achieving synchronization with sufficient energy, the chosen values of \(r_{m}\) and \(f_{Ri}\) should be relatively smaller to ensure that the inequation \(T_{\rm e01 max}\leq T_{\rm eN1}\) holds. Figure 2c shows a plot of \(T_{\rm e01 max}\) with changes in \(\eta_{r}\) and \(\eta_{m}\). The values of \(\eta_{r}\) and \(\eta_{m}\) can be varied by changing \(r_{R}\) and \(m_{R}\), respectively. In Figure 2c, the parameters of the vibrating system are set to \(m_{1}=6\) kg and \(f_{R1}=f_{R2}=f_{R3}=0.08\). The area I indicates the regions where \(T_{\rm e01 max}\leq T_{\rm eN1}\), i.e., the regions where an exciter and three CRs can reach synchronization, whereas some other area denotes the regions of unsynchronized behavior. Next, with the following set of physical parameters \(m_{R}=4\) kg, \(r_{R}=0.005\) m, the maximum values of dimensionless coupling torques (\(T_{\rm cjjmax}\)) and synchronization ability coefficients (\(\zeta_{ij}\)) are calculated and plotted as the functions of \(r_{l}\) for the different \(f_{Ri}\) (\(i=1\), \(2\), \(3\)) and \(r_{m}\) in Figures 3-5. At first, consider the simplified case of the identical friction coefficients. Assume that \(f_{R1}=f_{R2}=f_{R3}=0.08\). In Figure 3, it can be found that even with variations in \(r_{m}\) and \(r_{l}\), equations \(T_{\rm c12 max}=T_{\rm c13 max}=T_{\rm c14 max}\), \(T_{\rm c23 max}=T_{\rm c42 max}=T_{\rm c34 max}\), \(\zeta_{12}=\zeta_{13}=\zeta_{14}\), \(\zeta_{23}=\zeta_{42}=\zeta_{34}\) will constantly hold. The reason is that the system parameters are completely symmetrical in this situation. It is noticed that, increasing \(r_{m}\) leads to the decreasing of all synchronization coefficients \(T_{\rm cjjmax}\) and \(\zeta_{ij}\) (see Figures 3-5).