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This paper investigates stability of the synchronous states for three homodromy exciters in the vibrating system, some theoretical analyses and simulation results on which are given. Based on Lagrange equations, the differential equations of motion of the system are obtained. Using the average method yields the dimensionless coupling torque balanced equations of three exciters and the simplified analytical expressions for synchronization criterion of the system are deduced, the stability criterion of the synchronous states complies with Routh–Hurwitz principle. The dynamic characteristics of the system for different frequency ratios are discussed numerically. In order to verify the validity of theoretical methods, the simulation results by a Runge–Kutta routine are carried out; it indicates that the motion state of the vibrating system can be classified into two types: sub-resonant state and super-resonant state. The motion type of the rigid frame is strong positive superposition vibration in a sub-resonant state; while in a super-resonant state, the exciting forces of three exciters are mutually cancelled and the rigid frame is motionless. During the design process of the vibrating machines, the ideal working points are only selected in a sub-resonant state, which makes the vibrating machines work efficiently.

Investigations on stability of the synchronous states for three homodromy exciters in the vibrating system with double resonant types