Synchronization characteristics of a rotor-pendula system in multiple coupling resonant systems

Abstract

The problem is motivated by observations of a rotor-pendula system, which derived from a new shale shaker. To grasp the dynamic characteristics of the shale shaker, the key research is exploring the synchronous mechanism for the system, since synchronous state between rotors is closely related to the dynamic characteristics of the system. In this paper, the dynamic equation of the rotor-pendula system is firstly derived by applying Lagrange’s equations. Through Laplace’s transformation method, the approximate responses of the system in synchronous state are obtained, which is determined coupling coefficients and synchronous state of the system. Then, the synchronous balance equation and the stability criterion of the system are obtained with Poincaré method on which stable phase difference and synchronous behavior can be ascertained. To verify the correctness of the theoretical analysis, numerical simulations are implemented by Runge–Kutta method, and it is shown that the synchronous behavior is determined by the geometry parameters, coupling coefficients, and rotor rotation direction.

Theoretical Study of Synchronous Behavior in a Dual-Pendulum-Rotor System

“…Based on the approximate solutions above, the synchronization and stability of the system can be determined with Poincaré method [20,[25][26][27]. In addition, the phase angle of the rotors can be assumed by…”

“…(2) According to (20), coupling coefficients 12 , 21 , and 23 are fractions; thus parameters and with particular values lead to being zero. The second "critical point" is…”

“…Moreover, in light of (35) and (39), the rotation direction of the two rotors is opposite, and the stable phase difference between the two rotors is shown in Figure 2. According to (20) and (30), we have (− 11 + 23 ) sin = 0 when = −1, and the synchronous balance condition as in (17a), (17b), (17c) is simply expressed as sin = 0. Similarly, the synchronous stability criterion can be rewritten by (− 11 + 23 ) sin > 0.…”

“…The objects of the above-mentioned researches are mainly synchronization of the coupled pendula or the coupled rotors; however, the synchronization of the pendulumrotor system is less concerned. The extensive application of the pendulum-rotor system in aero-power plant, mining screening machines, parallel robots, and the like of the other rotation equipment is important for our industrial production and daily life [20][21][22]. The aim of this paper is to investigate a system composed of two unbalanced rotors coupled with two pendula in a horizontal plane.…”

“…But mass of the motor and the rotor, which can be relatively heavy or light, may affect the system motion. Therefore, the proposed problem is different from the dynamics of classical pendulum systems presented in papers [8][9][10][11] and classical rotor systems presented in papers [12][13][14][15][16][17][18][19][20][23][24][25][26][27]. We consider an ideal source of energy supply to asynchronous motors, which drag the unbalanced rotors.…”