Times-frequency synchronization of two exciters with the opposite rotating directions in a vibration system

Abstract Some vibration machines need composite harmonic motion syntheses, such as electromagnetic vibration motors and four-axis inertia shaking machines. In view of the insufficient theoretical and experimental research on the theory of times-frequency synchronization of these vibration systems, a kind of mechanical model of two exciters with the opposite rotating directions and different rotating speed is established. Firstly, the differential equations of the vibration system are obtained using Lagrange equations. Secondly, based on the average method of modified small parameters, a set of average angular velocity perturbation parameters e 0 and two sets of phase difference perturbation parameters e 1 and e 2 are introduced to derive the frequency capture condition of the system in the case of times-frequency synchronization. Thirdly, by using Lyapunov stability theory and partial order comparison in matrix theory and some other methods, we get the phase difference stable interval of the times-frequency vibration system under the parameters of angular velocity perturbation and phase difference perturbation is α 0 ∈ ( 0 , π 2 ) when n ≠ 1 and α 0 ∈ ( − π 2 , π 2 ) when n = 1 . Finally, the corresponding experiments for the times-frequency synchronization system are carried out to validate the theoretical analysis by taking the case of n  = 1 (same frequency experimental group) and n  = 0.5 (two times frequency experimental group). During the course of the experiment, we reveal the phenomenon of double stable values in a single stable interval and provide a reasonable explanation. In addition, based on the theoretical research and experiments analysis, we point out the application prospect of times-frequency synchronization theory in engineering for discussion.