Stability and coupling dynamic characteristics of a vibrating system with one internal degree of freedom and two vibrators

•Stability of a vibrating system with one internal degree of freedom.•Synchronization of a vibrating system driven by two vibrators.•Stable states in the different resonant regions.•Vibrating mills with high frequency. In present work the stability and coupling dynamic characteristics for a vibrating system with one internal degree of freedom, driven by two vibrators, are investigated, which is a further extension and innovation of a previous literature. In theory, the responses of the relative and absolute motion of the system are given; the synchronization criterion of the system is obtained based on the average method, and the stability criterion of the synchronous states is derived by Hamilton principle. According to the above theory results, the frequency-amplitude characteristics of the system are discussed qualitatively by numeric, as well as the stable states, phase relationships and corresponding motion types of the system in different resonant regions. The Runge-Kutta routine is employed to verify the feasibility of the used theory method. Finally, an experimental vibrating mill corresponding to the present dynamical model is designed and manufactured, and the experiments are carried out to further examine the validity of the present investigations. It is known that, in order to obtain the reasonable relative circular motion of two rigid frames, in engineering, not only the working point of the system, should be set in the sub-resonant region with respect to the natural frequency of the main vibrating system in x- and y- directions, but also the distance between the rotational center of each vibrator and the centroid of the system, must be adjusted to be larger to meet the above requirements. Only in this way, can the greater relative vibration amplitude of the system characterized by the circular motion be realized, the energy be saved, and the ideal isolation effect be satisfied.