From the perspective of theoretical derivation, numerical simulation, and engineering application, the vibratory synchronization characteristics of a dual-mass vibrating system driven by two exciters, were studied. The differential motion equations of the total system were calculated using Lagrange’s equations, and the responses of the vibrating system in the steady state were derived by Laplace transform. The synchronization criterion between two exciters was deduced by using the averaging method. Based on the Hamilton principle, the stability criterion of the vibrating system in synchronous states is given. According to the theoretical results, the coupling characteristics between two exciters such as synchronization and stability were analyzed numerically. Some analyses of the numerical simulation of the system were carried out, which fully support the theoretical results. The rotational speed of two exciters, their phase difference, responses, and difference of responses of two rigid bodies were studied quantitatively in the subresonant state and super-resonant state of the system. This paper presents a practical example of vibratory synchronization of a dual-mass system driven by two exciters in engineering.

In this article, multifrequency controlled synchronization of four inductor motors by the fixed frequency ratio method in a vibration system is investigated. The electromechanical coupling dynamical model of the vibrating system is established. The synchronous condition of the vibrating system is obtained with the small parameter method. Through the theoretical derivation and numerical simulation, multifrequency self-synchronization of four induction motors in the vibration system can’t be realized. To achieve the purpose of multifrequency synchronization motion, the method of multifrequency controlled synchronization is proposed, and a fuzzy PID controlling method is introduced. The stability of the controlling system is certified by the Lyapunov criterion. An arbitrariness of the proposed controlling method which is applied to the vibration system is presented. To certify the accuracy of the theory and simulation, a vibrating test bench is constructed. Some experiments are operated to validate the effectiveness and the proposed controlled synchronization method.

Multifrequency controlled synchronization of two homodromy eccentric rotors is investigated. The vibrating screen with two rotors is a typical underactuated system. So, the model of the vibrating screen is converted to a synchronization motion problem. Firstly, the mechanical‐electromagnetic coupling dynamical model of the vibrating system is established. And then the fuzzy PID method is used for the two motors which are based on the master‐slave control strategy. The slave motor uses the method of the phase ratio to trace the master motor and achieve the synchronous motion. Through the simulation, the two rotors cannot reach the state of multifrequency self‐synchronization. And, then, it is realized by multifrequency controlled synchronization. The article presents the motion trails of the vibrating system between one and two times and realizes the zero phase difference after each period. Finally, some experiments are used to verify the accuracy and effectiveness of the conclusion in the simulation and analysis of the feature of the movement tracks.

Based on this paper, multifrequency composite synchronization of three inductor motors with the method of fixed speed ratio in a vibration system is investigated. The multifrequency composite synchronization consists of self-synchronization and multifrequency controlled synchronization. In the self-synchronization part, motor 1 and motor 2 can implement the self-synchronization motion in the vibration system. In the part of multifrequency controlled synchronization, motor 3 traces motor 1 with the fuzzy PID method based on a modified master–slave controlling strategy. To satisfy the need in engineering, the stable phase difference should be implemented in the vibration system. The feasibility of the theory is analyzed by the numerical simulation and some discussions about the arbitrariness of the multifrequency parameter are listed. And then, based on a vibrating test bench driven by three eccentric motors (ERs), some experiments are given to certify the effectiveness of the proposed method of multifrequency composite synchronization. Meanwhile, a priority in the multifrequency composite synchronization motion is proposed. The self-synchronization is first realized and then the controlled synchronization is. Finally, from the numerical and experiment results, it is inferred that the zero phase difference between motors 1 and motor 2 as well as the multifrequency phase difference between motors 1 and 3 can't be realized in the multifrequency self-synchronization. However, with the fixed frequency ratio method based on the fuzzy PID controlling method, the results above can be realized in the multifrequency composite synchronization.

In this article, an engineering problem of three co-rotating exciters with the circular distribution in a vibrating system is investigated. The dynamical model constructed by the motion differential equations is established. By introducing the small parameter averaged method in the dynamic equation, the synchronization and stability conditions of the electromechanical coupling dynamical model is derived. To illustrate the necessity of the controlling method, the self-synchronization of the vibrating system is firstly analyzed with the theory, numerical simulations and experiments. With the self-synchronization results, it is indicated that the ellipse trajectory which is needed in the industry can’t be realizefd by the self-synchronization motion of the vibrating system. And then, a fuzzy PID controlling method based on the master–slave controlling strategy is introduced in the vibrating system to realize the controlled synchronization. The Lyapunov stability criterion is given to certify the stability of the controlling system. Through some simulations and experiments, the effectiveness of controlled synchronization is illustrated in the discussion. Finally, the present work illuminates the feasibility and practicality for designing some new types of vibrating screens in the industry.

This article presents the dynamic model of three eccentric rotors driven by induction motors and respectively derives the response equations of the three directions with different frequency in a vibrating system. On account of the difficulty to realize multifrequency self-synchronization, the controlling method is used to realize the synchronization of the speed and phase. In this article, the fuzzy PID method is introduced to control the vibrating system which is based on a master–slave controlling strategy. And a tracking method based on the phase ratio is proposed. The effectiveness and stability are shown in the numerical simulation and the experiment validation is given to certify the accuracy of the theory. In the meanwhile, the arbitrary of the multifrequency is also demonstrated. Finally, some conclusions are summarized to present the significant in the engineering.