Synchronous Stability of Four Homodromy Vibrators in a Vibrating System with Double Resonant Types

This paper aims at studying the synchronous stability of four homodromy vibrators in subresonant and superresonant states. The motion differential equations are established firstly. The simplified form of analytical expressions is yielded, and the stability criterion of synchronous states satisfies Routh–Hurwitz criterion. The coupling dynamic characteristics of the system are analyzed in detail numerically, such as the maximum of coupling torque, coefficients of ability of synchronization and stability, and phase differences. Based on the ratio of operating frequencies to natural frequencies of the system, the resonant regions are divided into two areas: subresonant and superresonant. It is shown that the phase differences among four vibrators in the subresonant state are stabilized about zero, and exciting forces of four vibrators are positively superposed, while in the superresonant one, the phenomenon of the diversity of the nonlinear system occurs, i.e., two groups of synchronous and stable solutions of the phase differences (pi and pi/2) are found, and in this case, the exciting forces of four vibrators are counteracted, the rigid frame embodies no vibration, and the minimum of dynamic load transferring to foundation is realized. The correctness of theoretical results is verified by numerical characteristic analysis and simulations. This paper can provide a theoretical reference for designing a type of new high-frequency vibrating mill