## 6 Conclusions  
In the present work, we have focused on the study of synchronization in an exciter and three CRs with non-identical friction coefficients. It has been demonstrated that the three coupled CRs are seen to synchronize in-phase for friction coefficient matched conditions, and show partial synchronization when there is a small difference in their friction coefficients. There are two essential components of synchronization criterion (refer to Equations (29)-(31) and (33)). The two components play an analogous role in calculating the numerical values of phase differences, stability coefficients, and clinging ability coefficients at the steady-state. Since the CRs are not equipped with direct driving sources, they obtain energy from the induction motor through the vibrating rigid platform (with the help of dynamic coupling torques). Hence, the induction motor offers energy not only to the exciter but also to the CRs, for the purpose of reaching synchronous rotation of the exciterand CRs. The second component of the synchronization criterion (Equation (33)) takes into consideration the total energy balance and introduces the concept of the necessary electromagnetic torque; as a result, it can be used to determine whether the induction motor can offer adequate energy to achieve synchronization. Interestingly, in the simulation cases, for an appropriate value of \(T_{\rm e01 max}\) (\(T_{\rm e01 max}\leq T_{\rm eN1}\)), the synchronous motion is robust against mismatches in friction coefficients. However, for a large value of \(T_{\rm e01 max}\) (\(T_{\rm e01 max}>T_{\rm eN1}\)), the synchronous swing of three CRs seems to be sensitive to differences in friction coefficients. In this interval of \(T_{\rm e01 max}\) three CRs can swing synchronously only when the friction coefficients for each CR are identical. Results presented in this paper, although numerical, indicate how important the proper choice of system parameters and coupling structure's properties is in order to obtain desired, regular behavior in real, practical implementations of engineering problems. The friction coefficient \(f_{Ri}\) and dimensionless variables \(r_{m}\), \(r_{l}\), \(\eta_{r}\), and \(\eta_{m}\) can all affect the synchronization and stability and thus can be used to tune the synchronization performance to the ideal case conveniently at the design and optimization stage of vibrating machines. The combination of all these key parameters will eventually lead to a specific type of synchronous motion. The proposed vibrating system can produce an ideal synchronization regime that is ever-present and stable under variation of the structural parameter \(r_{l}\), leading to a small-sized and compact structure very easy to design and implement in various engineering practices. The dynamic coupling torques act as a self-adjusting synchronization controller, which can automatically maintain the original rotational velocities and phase differences and, thus, the obtained simulation results show robust motion synchronization in the exciter-CR vibrating system against the disturbance. The results of this study demonstrate an innovative solution for designing a type of desired vibrating crusher in various engineering practices by employing this synchronization mechanism. Crushing devices can hold materials between a CR's surface and an inner wall's surface. During the process of synchronization, a CR rotates along an inner wall and applies sufficient force to bring the surfaces together to generate enough energy within the materials being crushed, so that its molecules separate from each other. Without a doubt, the materials needing breaking will influence the synchronization between CRs and an exciter. This study can be further improved by addressing the issue of materials' effects on the synchronous and stable operation of the proposed vibrating system, and this issue will be investigated in future work. The manuscript was written through contributions of all authors (D.G., J.Z., Q.W., P.S., B.P., B.W.). Conceptualization, D.G. and B.W.