The theory on synchronization of two exciters is more widely used in engineering, while that of more than two exciters is less considered. So it is of great significant to investigate synchronization of three exciters. Firstly by introducing the average method of modified small parameters, the dimensionless coupling equations(DCE) of three exciters are derived, which convert the problem of synchronization into that of existence and stability of zero solutions for the DCE and lead to the construction on criterions of synchronization and stability in the simplified form for three exciters. Then the synchronization criterion is discussed numerically, as well as the abilities of synchronization and stability, some results thereof indicate that the synchronization ability increases with the increase of the coupling moment among three exciters, but decreases with that of their phase differences. Finally, an experiment on synchronization with three exciters is carried out. Through the comparison and analysis of experimental data on phase differences among three exciters, responses of system, and phases of three exciters recorded by high-speed camera, the parameters of system satisfying the above two criterions can ensure the synchronous and stable operation of three exciters. As a result, the average method of modified small parameters can be used as a theoretical apparatus studying reasonably the synchronization mechanism of three exciters, it is also proved to be useful and feasible by numeric and experiment. The present research lays the foundation and guidance for the establishment of synchronization theory system with multi-exciter and engineering design.
In this paper, the synchronization problem of three homodromy coupled exciters in a non-resonant vibrating system of plane motion is studied. By introducing the average method of modified small parameters, we deduced dimensionless coupling equation of three exciters, which converted the problem of synchronization into that of the existence and stability of zero solutions for the average differential equations of the small parameters. Based on the dimensionless coupling torques and characteristics of the cor- responding limited functions, the synchronization criterion for three exciters was derived as the absolute value of dimensionless residual torque difference between arbitrary two motors being less than the maximum of their dimensionless coupling torques. The stability criterion of its synchronous state lies in the double-condition that the inertia coupling matrix is positive definite and all its elements are positive as well. The synchronization determinants are the coefficients of synchronization ability, also called as the general dynamical symmetry coefficients. The double-equilibrium state of the vibrating system is manifested by numeric method, and the numeric and simulation results derived thereof indicate the indispensable and crucial role the structural parameters of the vibrating system play in the stability criterion of synchronous operation. Besides, by adjusting its structural parameters, the elliptical motion of the vibrating system successfully met the requirements in engineering applications.
