Synchronous behavior of a rotor-pendulum 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 Poincar 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.