转子耦合摆系统的同步行为理论研究

转子耦合摆系统广泛应用于航空动力装置、矿业筛分机械和并联机器人等高速旋转设备.但是对转子耦合摆系统同步行为(稳定相位差值)研究甚少,系统同步行为通常影响系统的工作执行精度.基于这一特殊背景,提出了转子与摆耦合系统的简化物理模型.利用庞加莱法研究转子耦合摆系统的同步问题,进一步揭示了该类系统同步现象的本质机理.首先运用拉格朗日方程建立了转子同向和反向旋转的系统动力方程,随后将系统动力方程转换为无量纲方程.然后利用拉普拉斯法对无量纲方程解耦,计算出了系统各个自由度的近似稳态响应解.继而采用庞加莱法导出了系统实现同步的平衡方程和稳定准则.只有系统的物理参数同时满足系统同步平衡方程和稳定准则时,系统才能实现稳定的同步行为.通过理论研究发现,系统的同步行为主要受弹簧刚度、摆杆安装倾角和转子旋转方向的影响.同时系统同步"临界点"会造成相位差角无解,导致系统动态特性表现为混沌.最后,使用计算机模拟验证了理论计算的正确性,两者的结果相符.

Synchronous b ehavior of a rotor-p endulum system

Rotor-pendulum systems are widely applied to aero-power plants, mining screening machineries, parallel robots, and other high-speed rotating equipment. However, the investigation for synchronous behavior (the computation for stable phase difference between the rotors) of a rotor-pendulum system has been reported very little. The synchronous behavior usually affects the performance precision and quality of a mechanical system. Based on the special background, a simplified physical model for a rotor-pendulum system is introduced. The system consists of a rigid vibrating body, a rigid pendulum rod, a horizontal spring, a torsion spring, and two unbalanced rotors. The vibrating body is elastically supported via the horizontal spring. One of unbalanced rotors in the system is directly mounted in the vibrating body, and the other is fixed at the end of the pendulum rod connected with the vibrating body by the torsion spring. In addition, the rotors are actuated with the identical induction motors. In this paper, we investigate the synchronous state of the system based on Poincaré method, which further reveals the essential mechanism of synchronization phenomenon of this system. To determine the synchronous state of the system, the following computation technologies are implemented. Firstly, the dynamic equation of the system is derived based on the Lagrange equation with considering the homonymous and reversed rotation of the two rotors, then the equation is converted into a dimensionless equation. Further, the dimensionless equation is decoupled by the Laplace method, and the approximated steady solution and coupling coefficient of each degree of freedom are deduced. Afterwards, the balanced equation and the stability criterion of the system are acquired. Only should the values of physical parameters of the system satisfy the balanced equation and the stability criterion, the rotor-pendulum system can implement the synchronous operation. According to the theoretical computation, we can find that the spring stiffness, the installation title angle of the pendulum rod, and the rotation direction of the rotors have large influences on the existence and stability of the synchronous state in the coupling system. Meanwhile, the “critical point” of synchronization of the system can lead to no solution of the phase difference between the two rotors, which results in the dynamic characteristics of the system being chaotic. Finally, computer simulations are preformed to verify the correctness of the theoretical computations, and the results of theoretical computation are in accordance with the computer simulations.