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.

The accurate measurement of position and orientation for shearers is a key technology in realizing an automated, fully-mechanized, coal mining face. Since Global Positioning System (GPS) signal cannot arrive at the coal mine underground, wireless sensor network positioning system cannot operate stably in the coal mine; thus a strap-down inertial navigation system (SINS) is used to measure the position and orientation of the shearer. Aiming at the problem of the SINS accumulative error, this paper proposes a positioning error correction method based on the motion constraint-aided SINS zero velocity updated (ZUPT) model. First of all, a stationary state detection model of the shearer is built with median filter based on the acceleration and angular rate measured by the SINS. Secondly, the motion of the shearer is analyzed using coal mining technology, then the motion constraint model of the shearer is established. In addition, the alternate action between the motion constraint model and the ZUPT model is analyzed at the process of movement and cessation of the shearer, respectively; hence, the motion constraint-aided SINS ZUPT model is built. Finally, by means of the experimental platform of the SINS for the shearer, the experimental results show that the maximum position error with the positioning model proposed in this paper is 1.6 m in 180 s, and increases by 92.0% and 88.1% compared with the single motion constraint model and single ZUPT model, respectively. It can then restrain the accumulative error of the SINS effectively.

This work is a continuation and verification of the original literature by using experimental strategy. Based on the published paper, in order to avoid anti-phase synchronization with two co-rotating rotors system, a vibrating system with two co-rotating rotors installed with nonlinear springs have been proposed, and then, the synchronous condition and the synchronous criterion of the system are theoretically derived. From the analysis mentioned, it is shown that the synchronous state is mainly determined by the structural parameters of the coupling unit, coupling coefficients and positional parameters of the two exciters, etc. The main objective of the present work is to investigate the synchronous mechanism by experiments and simulations in this paper. Some simulation computations are firstly implemented to explain the synchronous mechanism of the system. Additionally, an experimental strategy with synchronous tests and dynamic characteristic tests of the vibrating system are carried out to validate the correctness of the simulation analysis. The simulations and experiments demonstrate that the nonlinear springs can overcome the difference of residual torques of the two motors to realize the synchronization of near zero phase difference under the condition of in-phase difference between two exciters. Finally, the error analysis results among the dynamic testing, synchronous testing results and simulations are discussed. This research can provide theoretical reference for designing large-sized and heavy-duty Vibrating Screens.

In this paper, a novel design for vibrating screens, called the triple-rotor-pendula system, is presented in a dual-super-far resonance system, which makes it possible to produce wanted vibrations with self-adjustable of the synchronization state between unbalanced rotors. To grasp the synchronization characteristics of the system, the dynamics equation of the triple-rotor-pendula system is primarily derived by adopting Lagrange’s equations. Next, the displacement responses of the system in steady state are obtained by Laplace’s transform method. Meanwhile, considering the average method with revised small parameter, the synchronization implement of the system is ascertained on condition that the absolute values of residual electromagnetism torques between two arbitrary motors are less than or equal to their load torque differences. Then, according to Routh–Hurwitz theorem and generalized Lyapunov equation, the criterion of synchronization stability of the system is obtained. Finally, the simulation computations confirm the results of analytical investigations, which show that the theoretical analysis for the synchronization behavior of the triple-rotor-pendula system is feasible. It can be known that the synchronization state of the system is influenced by mass ratio coefficients, structure parameters, rotating directions, and frequency ratios.

In-phase self-synchronization of two eccentric rotors with common rotational axis is hardly implemented in far-resonance system. In this article, a dual motor coaxially coupling with a torsion spring is proposed to obtain in-phase synchronization between the eccentric rotors. To explore the dynamic and synchronous characteristics of the proposed system, the mechanical model is first established with Lagrangian formulation. Second, the steady response of the system is calculated based on differential motion equations. Subsequently, the synchronous mechanism between the eccentric rotors is discussed by averaged small parameter method. Finally, some numerical computations are further implemented to verify correctness of theoretical analysis. The result shows that the synchronous state is determined by stiffness of torsion spring, masses of eccentric rotors, and distance between the motors. When axial distance between the motor is smaller, “critical stiffness of in-phase synchronization” is gradually enlarged as the masses of the eccentric rotors are increased and approached to equality, but in-phase synchronization is permanently maintained when the axial distance of the motor is far; in this situation, the synchronous state is hardly affected by variation of stiffness of torsion spring and masses of eccentric rotors. When the stiffness of the torsion spring is smaller, “critical distance $r_{\psi }(r_{\delta })$ of in-phase synchronization” is also enlarged as the masses of the eccentric rotors are increased and approached to equality; otherwise, the synchronous state is always locked in in-phase synchronization. When the stiffness of the torsion spring is smaller, “critical distance $r_{\theta }$ of anti-phase synchronization” is decreased as the masses of eccentric rotors are increased and approached to equality; otherwise, the synchronous state is always locked in in-phase synchronization.