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Due to complex dynamic coupling relationships in the system, the research that couples pendulum and vibrators in the same system is rarely found. Especially in the vibrating system with spatial motion, the theoretical derivation may be too complex to be easy to implement. In this paper, we present a vibrator-pendulum coupling system with spatial motion characteristics; here, the pendulum is driven by the two vibrators mounted on it. The synchronization of the vibrators and the motion characteristics of the system are studied via theory, numeric, and experiment. In theory, the synchronization and stability criteria of the two vibrators are deduced. According to the theoretical results, some numerical qualitative analyses are given to further reveal the synchronous stable states and spatial motion characteristics of the system. The validity of the theoretical methods and the correctness of the numerical qualitative results are examined by simulation and experiment. Taking the pendulum as the working rigid frame, the proposed dynamic model and research results can be applied to design a new kind of vibrating jaw crusher with spatial motion characteristics. [ABSTRACT FROM AUTHOR]

In this paper, a dynamic model with two vibrator-driven pendulums is presented. Each pendulum is motivated by a pair of vibrators mounted on it. According to the deduced synchronization criteria of the pendulums and the vibrators, the synchronous states of pendulums are determined by the phase differences of vibrators. The two pendulums achieve anti-phase synchronization when the phase differences of arbitrary-two vibrators are all stabilized around zero. If the phase differences of the two vibrators mounted on the same pendulum are zero whereas those on different pendulums are π, the pendulums will arrive at the in-phase synchronous state. Besides, the system shows a relatively good isolative effect under the anti-phase synchronous motion of the two pendulums. Based on the theoretical results, some numerical qualitative analyses are presented and simulations are implemented for verification. The dynamic model in this paper can offer guidance to design a kind of vibrating jaw crusher. Here, the two pendulums serve as movable jaws and work by the anti-phase synchronous swing motion trajectory. [ABSTRACT FROM AUTHOR]

In present work, the double and triple-frequency synchronization behaviors of a vibrating mechanical system with two different driving frequencies, driven by three reversed rotating exciters, are investigated by theory, numeric, and experiment. Based on Lagrange's equations, the dynamic model corresponding to vibrating machine is proposed and motion differential equations are constructed. The Bogoliubov standard formal equations for three exciters are established, by introducing the asymptotic method, in which the synchronization problem is converted into that of the existence and stability of zero-valued solution of the average differential equations. The synchronization criterion of satisfying the synchronous operation is deduced. According to the Routh–Hurwitz criterion, the stability criterion of the synchronous states is achieved analytically. Based on the obtained theory results, the stability characteristics of the system, are numerically discussed in detail, including the stability ability coefficients and stable phase differences. Finally, simulations and experiments under the condition of two different driving frequencies, are performed to further examine the validity of the theoretical and numerical qualitative results. The present work can provide a theoretical reference for designing some new types of the vibrating machines with adjustable frequencies and motion trajectories. [ABSTRACT FROM AUTHOR]

This paper aims at investigating the times–frequency synchronization of the three exciters in a far super-resonant vibrating system. Based on the asymptotic method, the Bogoliubov standard formal equations of the three exciters are constructed, and the criterion of the double-frequency synchronization for the simplified form is deduced. Meanwhile, the stability criterion of the synchronous states is derived by complying with the Routh–Hurwitz principle. The stability characteristics of double-frequency synchronization are qualitatively discussed by numeric, including the stable phase differences and stability ability among the three exciters. Finally, the corresponding simulations and experiments for the double-frequency synchronization are carried out, respectively. According to the comparisons and analyses of results for theory, numeric, simulation and experiment, including the stable phase differences among the three exciters, responses of the system, synchronous rotational velocities, etc., it is shown that the feasibility of the used theory method and the validity of the obtained results are verified. This paper can also provide a theoretical reference for designing new types of vibrating equipment in engineering with two different driving frequencies. [ABSTRACT FROM AUTHOR]

A new dynamical model with five rigid frames (RFs), driven by two counter-rotating exciters, is proposed to explore the synchronization, stability, and motion characteristics of the system in this paper. The motion differential equations and the corresponding responses of the system are given firstly. Using the average method, the average torque balance equations for the two exciters are deduced. According to the relationship between the difference of the dimensionless effective output electromagnetic torques for the two motors and the coupling torques of the system, the theory condition of realizing synchronization is obtained. Based on the Hamilton's theory, the theory condition of stability of the system is deduced. The stability and motion characteristics of the system for different resonant regions are qualitatively discussed in numeric, including the stable phase difference of the two exciters, relative phase relationships among the five rigid frames, amplitude-frequency characteristics, stability coefficients, and the effective load torque between the two exciters. Simulations are carried out to further quantitatively validate the feasibility of the above theoretical and numerical qualitative results. It is shown that in engineering the reasonable working points of the system should be selected in Region II, only in this way, can the synchronous and stable relative linear motion of the system with the zero stable phase difference in vertical direction be realized, and in this case, the vibrations of the four inner rigid frames (IRFs) in the horizontal direction are compensated with each other, and the energy is also saved due to utilizing the resonant effect. Based on the present work, some new types of vibrating coolers/dryers or vibrating screening machines can be designed. [ABSTRACT FROM AUTHOR]

• Multi-cycle synchronization of the two pairs of reversed rotating exciters. • Stability of the multi-cycle synchronous states. • Vibrating machines with linear motion driven by two different driving frequencies. With the rapid development of science technology in vibration utilization engineering fields, more and more high demands are proposed to improve the functions and performances of machines, especially some new types of large scale vibrating equipments with linear motion driven by two different driving frequencies, designed by the multi-cycle synchronization theory of multiple exciters, have been the urgent desires for engineers and scholars. In the present work, taking a dynamical model driven by two pairs of reversed rotating exciters in a far super-resonant vibrating mechanical system for example, the multi-cycle synchronization and stability of the system are studied in detail by theory, numeric and experiment. Firstly the differential equations of motion of system are deduced by Lagrange equations. Introducing the asymptotic method, the theoretical conditions of implementing foundation frequency (1:1), double-frequency (1:2) and triple-frequency (1:3) synchronization among the two pairs of exciters, are obtained, as well as the stability conditions corresponding to the synchronous states based on the Routh-Hurwitz criterion. The synchronous stable regions of the system are numerically discussed. Simulations and experiments are carried out to further verify the feasibility of the used theory method and the obtained results. It is shown that, for the foundation frequency, double-frequency or triple-frequency synchronous states, under the precondition of satisfying the conditions of multi-cycle synchronization and stability, the phase differences among the two pairs of exciters, can be stabilized in the vicinity of zero of the Region II, in this case, the linear motion of the system with two different driving frequencies can be realized, which is of great significance in engineering. [ABSTRACT FROM AUTHOR]