Fang Pan, Hou Yongjun, Nan Yanghai
School of Mechanical Engineering, Southwest Petroleum University, Chengdu, China.
Department of Mechanical Engineering and Robotics, Universite Libre de Bruxelles, Brussel, Belgium.
PLoS One. 2015 May 19;10(5):e0126069. doi: 10.1371/journal.pone.0126069. eCollection 2015.
A new mechanism is proposed to implement synchronization of the two unbalanced rotors in a vibration system, which consists of a double vibro-body, two induction motors and spring foundations. The coupling relationship between the vibro-bodies is ascertained with the Laplace transformation method for the dynamics equation of the system obtained with the Lagrange's equation. An analytical approach, the average method of modified small parameters, is employed to study the synchronization characteristics between the two unbalanced rotors, which is converted into that of existence and the stability of zero solutions for the non-dimensional differential equations of the angular velocity disturbance parameters. By assuming the disturbance parameters that infinitely approach to zero, the synchronization condition for the two rotors is obtained. It indicated that the absolute value of the residual torque between the two motors should be equal to or less than the maximum of their coupling torques. Meanwhile, the stability criterion of synchronization is derived with the Routh-Hurwitz method, and the region of the stable phase difference is confirmed. At last, computer simulations are preformed to verify the correctness of the approximate solution of the theoretical computation for the stable phase difference between the two unbalanced rotors, and the results of theoretical computation is in accordance with that of computer simulations. To sum up, only the parameters of the vibration system satisfy the synchronization condition and the stability criterion of the synchronization, the two unbalanced rotors can implement the synchronization operation.
提出了一种新的机制来实现振动系统中两个不平衡转子的同步，该振动系统由双振体、两台感应电机和弹簧基础组成。利用拉普拉斯变换法，根据用拉格朗日方程得到的系统动力学方程，确定振体之间的耦合关系。采用一种解析方法，即修正小参数平均法，来研究两个不平衡转子之间的同步特性，将其转化为角速度扰动参数的无量纲微分方程零解的存在性和稳定性问题。通过假设扰动参数无限趋近于零，得到了两个转子的同步条件。结果表明，两台电机之间的剩余转矩绝对值应等于或小于其耦合转矩的最大值。同时，利用劳斯-赫尔维茨方法推导了同步稳定性判据，确定了稳定相位差区域。最后，通过计算机仿真验证了两个不平衡转子稳定相位差理论计算近似解的正确性，理论计算结果与计算机仿真结果一致。综上所述，只有当振动系统的参数满足同步条件和同步稳定性判据时，两个不平衡转子才能实现同步运行。
