Comments on the stability of the synchronous states of three vibrators in a vibrating system with two rigid frames

This paper focuses on the stability of three homodromy vibrators in a vibrating system with two rigid frames (RFs). The motion differential equations of the system are established. Using the average method yields the average coupling torque balanced equations of three vibrators, and the simplified analytical expressions for synchronization and stability criterions of the system were derived. The coupling dynamic characteristics were numerically analyzed, including frequency-amplitude response, stable phase differences, synchronization and stability ability, and phase relationships. The simulations to verify the validity of theoretical results were carried out. It is shown that the stable states of the system are classified into three types, thereof the stability of phase differences among vibrators is similar in sub-resonant and super-resonant states, which results in no vibration of the system. While in near sub-resonant state the system reflects strong useful positive superposition vibration with energy saving, which is exactly the desire in engineering design.