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The synchronization and stability of two unbalanced rotors (URs) separately driven by induction motors rotating in opposite directions, in a new mechanism system with four rigid frames (RFs), are investigated in present work. Applying Lagrange's equation, the differential equations of motion of the system are deduced. The criteria of synchronization and stability of the synchronous states are derived analytically by using the average method and Hamilton's principle, respectively. Based on the theoretical results, the coupling dynamic characteristics of the system are given by numeric, it is shown that the abilities of synchronization and stability, are the best under the condition that the parameters of the system are completely symmetrical. The stable states and the corresponding motion types of the system in different resonant regions, are clarified in detail. A Runge-Kutta simulation routine is employed to verify the validity of the theoretical results, as well as the feasibility of the used theory method. The selecting principle of the stable region with zero phase difference for the two URs, can provide a novel reference for designing a new type of vibrating feeder with the anti-blocking function in engineering.

In this paper, a dynamic model is adopted to investigate the stability and response characteristics of a vibrating system driven by four vibrators placed on two different rigid frames (RFs). Using the equations of motion of the system derived, the conditions for synchronization and stable operation of the system are studied by the average method and Hamilton’s rules, respectively. Based on the theoretical results obtained, some factors are further studied concerning the stable phase differences (SPDs), the coefficients for ensuring stability, and the vibration amplitudes of the two RFs in different resonant regions. These serve to reveal the stability and response characteristics of the system that determine the ultimate function of the vibrating machine. Finally, numerical simulations are carried out to examine the validity of the theoretical methods and numerical qualitative results. Based on the results from the theory and simulation analyses, it is suggested that the working region of the system should be selected in the sub-resonant region corresponding to the natural frequency (NF) of the main vibrating system in the [Formula: see text]- and [Formula: see text]-directions. In this case, the ideal relative circular motion for two RFs with a well isolation effect can be achieved, and the energy is saved.

The present work investigates the coupling synchronization principle and stability in a vibrating system with two pairs counter-rotating unbalanced rotors (also called exciters). Based on Lagrange equations, the dimensionless coupling differential equations of motion of the system are deduced. The synchronization criterion of two pairs exciters stems from the averaging method, it satisfies the fact that the absolute value of dimensionless residual torque difference between arbitrary two driving motors is less than or equal to the maximum of their dimensionless coupling torques. The stability criterion of the synchronous states complies with Routh-Hurwitz principle. The coupling dynamic characteristics of the system are numerically analyzed in detail, including synchronization and stability ability, maximum of the coupling torque and phase relationship, etc. Some simulation results applying the Runge-Kutta algorithm are performed, it is shown that the motion states of the system can be classified into two types: sub-resonant state and super-resonant state. Generally in engineering, the ideal working points should be selected in sub-resonance region, in this case the expended energy can be saved relatively by 1/5–1/3, which is less than that in super-resonance region under the precondition of the same vibration amplitude value.

As a continuous work of the previous literatures, a special dynamical model with one cylindrical roller driven by a single exciter and one outer ring, is taken for example to explore the vibratory synchronization transmission (VST) of the system considering sliding dry friction in this paper. The motion differential equations of the system, are given firstly. Using the average method, the theory condition of implementing VST is obtained. The VST characteristics are qualitatively discussed in numerical, which are further quantitatively verified by simulations. It is shown that, the vibration amplitudes of the roller in the horizontal and vertical directions are basically identical, and less affected by the friction coefficient, but the stable phase difference between the exciter and the outer ring is affected too much by it. Based on the present work, some new types of vibrating equipments, such as vibrating crushers/mills, can be designed. INDEX TERMSSliding dry friction, vibratory synchronization transmission, cylindrical roller, exciter.

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