## 1 Introduction
[27; 28; 29; 30], the authors studied the synchronization regime of multiple identical or non-identical exciters in a single rigid base, and concluded that using multiple exciters instead of two exciters in a single rigid base cannot improve the effective power of a far-resonant vibrating system, since the exciting forces are mutually offset due to strong inner coupling. Now that the self-synchronization of multi-exciters cannot satisfy the requirements of more energy supply and highly efficient productive processing, the controlled and composite synchronization methods presented in Refs. [30; 31; 32; 33; 34] are feasible choices to meet these demands. Lately, the synchronization of two DC motors supported by a rectangular plate and the plate vibration control in synchronous states has been analyzed in Ref. [35], while [PERSON] et al. [36; 37] analyzed the Sommerfeld effect and self-synchronization of two or three non-ideal induction motors installed on a simply supported beam structure. In this paper, we consider the synchronization of an exciter and three CRs with non-identical friction coefficients in a far-resonant vibrating system driven by an exciter. The synchronization mechanism can be realized if an exciter and three CRs rotate synchronously around their rotating centers. Based on this synchronization mechanism, a new type of vibrating crusher or vibrating mill with multiple working chambers can be developed. The materials to break, such as ores, minerals, slags, and concretes, can be located in the clearance area between three CRs and their corresponding inner wall. During the synchronous operation process, three CRs rotate along the inner wall with the effective extrusion, and the goal of crushing the materials is achieved. In practical applications, the friction coefficient \(f_{R\dot{I}}\) between the surface of CR \(i\) and the surface of its cavity is inevitably different from the other two friction coefficients due to the variability in their manufacturing process. This results in an asymmetric coupling: the influence of the rigid platform on each CR will be different from the influence that the rigid platform exerts on the other two CRs. As a result of the lack of symmetry in the coupled system, the synchronization of rotating elements cannot be implemented in some coupled dynamical systems [11; 38; 39; 40; 41; 42]. Therefore, it is urgent to consider what may happen in the proposed vibrating system and whether an exciter and three CRs can synchronize or not as a result of the non-identical dry friction coefficients. We endeavor to provide an answer to these questions in our studies. In addition, the significance and necessity of undertaking such a study are stated as follows:  
(1) In a sense, the proposed coupled system in this paper can be treated as an improved model of that in Ref. [43]. In Ref. [43], the in-phase synchronization of two exciters is a fundamental precondition for the synchronous rotation of the CR. However, the \(r_{l}\) (\(r_{l}\) is the length ratio of the distance between the rotary center of an exciter and the mass center of the rigid frame to the equivalent rotating radius of the vibrating system about the centroid of the rigid frame) should be large enough to synchronize two exciters in in-phase. In this case, with a high value of \(r_{l}\), the vibrating system will most certainly come in an extra-large size, which is not actually desired in practice. The structural weakness confines the design and applications of vibrating machines. This paper introduces a new coupling structure, which generates an ever-present and stable synchronization regime under variation of the key parameter \(r_{l}\), resulting in a small-sized and compact structure very easy to design and implement in various engineering practices. (2) To reach the synchronous operation of the exciter and CRs, the induction motor would have to offer extra energy to the CRs indirectly since the direct driving sources of the CRs are absent.