Synchronization behaviors of a vibrating mechanical system with adjustable frequencies and motion trajectories

In present work, the double and triple-frequency synchronization behaviors of a vibrating mechanical system with two different driving frequencies, driven by three reversed rotating exciters, are investigated by theory, numeric, and experiment. Based on Lagrange's equations, the dynamic model corresponding to vibrating machine is proposed and motion differential equations are constructed. The Bogoliubov standard formal equations for three exciters are established, by introducing the asymptotic method, in which the synchronization problem is converted into that of the existence and stability of zero-valued solution of the average differential equations. The synchronization criterion of satisfying the synchronous operation is deduced. According to the Routh-Hurwitz criterion, the stability criterion of the synchronous states is achieved analytically. Based on the obtained theory results, the stability characteristics of the system, are numerically discussed in detail, including the stability ability coefficients and stable phase differences. Finally, simulations and experiments under the condition of two different driving frequencies, are performed to further examine the validity of the theoretical and numerical qualitative results. The present work can provide a theoretical reference for designing some new types of the vibrating machines with adjustable frequencies and motion trajectories.