Synchronization of three rigid frames and two counter-rotating unbalanced rotors in a vibration system

This paper aims at investigating the synchronization of unbalanced rotors and rigid frames in a vibration system, consisting of three elastically connected rigid frames and two counter-rotating unbalanced rotors, which are mounted on different inner rigid frames. Using the Lagrange approach, the governing equations of motions are derived. Using the method of averaging, the existence condition of synchronization for two unbalanced rotors is determined. Applying Lyapunov stability theory, the condition of the system to get stable synchronous states is obtained. In order to analyze the selected motion characteristic of unbalanced rotors and rigid frames, numerical simulations are performed. Specifically, the phase difference between two unbalanced rotors and motion lag phases of rigid frames is studied by considering different synchronous frequencies and ratios of masses of unbalanced rotors. Furthermore, a series of experiments for different frequency regions of vibrations is carried out to validate the obtained theoretical results. The coupling motion characteristics of unbalanced rotors are studied by comparing the cases of excitation of a single and two unbalanced rotors. The present work and obtained results can serve as theoretical and experimental basis for the design of vibration jaw crushers in engineering.