###### Abstract  
To solve the limitation of vibration synchronization, this paper investigates the dynamics of a vibration system driven by three homodromy eccentric rotors (ERs) using control synchronization. According to the synchronous condition and the stability condition, the changes of the phase differences of the three-ERs system and the two-third ERs system are obtained. Based on the electromechanical mathematicatic model of the vibration system and the master-slave control strategy, the synchronous target that three ERs achieve the same phase motion is converted into velocity and phase tracking of ERs. Considering the coupling characteristic of the self-adjusting of the system in the state of vibration synchronization, the velocity and phase controllers are designed by employing discrete-time sliding mode control. An experimental system for control synchronization is designed, including hardware composition and software programming. For the feedback signal used to match control targets, a method of calculating velocity and the phase difference is proposed from engineering. Recording the rotational velocities, phase differences, and system response, three group experiments with different control schemes are achieved. According to the experimental data, the coupling characteristic of the vibration system adopting control synchronization is analyzed, which can provide the basis for designing vibration machines using control synchronization. synchronization; vibration system; eccentric rotor; control; phase difference Article  
[PERSON], [PERSON], [PERSON], L.  
Dynamics of the Vibration System Driven by Three Homodromy  
2021  
phase is their resultant force the largest. However, it is difficult to achieve the synchronous motion of ERs with the same phase by applying the technology of vibration synchronization, because the synchronization state is limited by the stability condition [11; 12]. Meanwhile, forced synchronization is no longer the best choice to achieve the synchronous motion of ERs with the same phase. To achieve the purpose of power maximization, the idea of control synchronization applying in the vibration system is proposed by introducing the control method into synchronous research [9; 14]. At present, the control strategy of multi-motors is mainly focused on several methods, such as master-slave control [14; 15; 16], cross-coupling control [17; 18], virtual shaft control [19], ring coupling control [20], and so on. From the view of calculative complexity and the hardware system, the master-slave control is the most direct and effective method compared with other methods. Because the vibration system has coupling dynamic characteristics such as the Sommerfeld effect [21], its control synchronization is different from those of other mechanical systems [15; 22; 23]. For this type of non-linear system, the most commonly used control algorithm is PID control [5; 24], the sliding mode control (SMC) [14; 15; 16], adaptive feed forward control [15; 20; 25], and so on. To satisfy the requirements of the maximum resultant force of the vibration machine, several scholars studied control synchronization of the vibration system. [PERSON] [14] proposed a method of control synchronization employing SMC on ERs, which numerically proves the feasibility of the control method. Adopting PI control, [PERSON] [24] investigated control synchronization of two ERs in one degree of freedom vibration system. [PERSON] [26] proposed a fuzzy PID method to investigate the multiple-frequency synchronization of ERs in the vibration system. Applying two PI algorithms separately, [PERSON] [27] studied the multiple control synchronization for a three-rotor vibration unit with varying payload. [PERSON] [5] studied control synchronization of dual rotors in the vibration system by applying PI control, which is used to achieve independent adjustment between frequency and amplitude. To sum up, the above scholars' research results promote the development of control synchronization of the vibration system.