The unreasonable dynamic characteristics results in decrease of screening efficiency of the vibrating screen. However, the synchronous behavior of motors is key factor to determine dynamic characteristics of the screens. In this paper, two unbalanced rotors actuated with motors in a three-dimensional space are proposed. To understand the synchronous mechanism, the dynamic equation of the system is firstly confirmed based on Lagrangian formulation; meanwhile, synchronization condition of the system is calculated with average and small parameter method; then, synchronization stability of the system is explored by Lyapunov method; finally, some numerical simulations are given to validate the theoretical computations. It is found that, to implement the stable synchronous rotation between the rotors, the values of the parameter in this system must be satisfied by synchronous condition and synchronous stability; the synchronous state is determined by the rotation direction, the damping ratio, the frequency ratio, and the motor position; the system is a planar motion when the identical mass rotors oppositely actuated, but the system is a spatial motion in the other cases.

中文翻译：

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三维空间和远共振系统中不平衡转子的同步状态

不合理的动态特性导致振动筛筛分效率降低。然而，电机的同步行为是决定屏幕动态特性的关键因素。在本文中，提出了在三维空间中由电机驱动的两个不平衡转子。为理解同步机理，首先基于拉格朗日公式确定系统的动力学方程；同时，采用平均小参数法计算系统的同步条件；然后，利用Lyapunov方法探索系统的同步稳定性；最后，给出了一些数值模拟来验证理论计算。发现，为了实现转子之间的稳定同步旋转，该系统参数取值必须满足同步条件和同步稳定性；同步状态由旋转方向、阻尼比、频率比和电机位置决定；当质量相同的转子作相反的驱动时，系统是平面运动，但在其他情况下系统是空间运动。