Analytical study of self-synchronization in two unbalanced rotors based on energetic conditions

When self-synchronization occurs in a system in which several nonlinear self-excited oscillators with different natural frequencies are coupled, the frequencies of all of the oscillators are entrained due to their interaction. Although many researchers have examined the characteristics of this phenomenon, the mechanism has not yet been completely clarified. This paper treats a system comprising two co-rotating unbalanced rotors fixed to a rigid body and derives the two energetic conditions satisfied by energy dissipated in damping and energy transferred in the system from the energy integration of the equations of motion. Using these two conditions and the approximation of constant angular frequency, methods for obtaining an approximate solution and for deriving the existence limit of this solution are proposed. The two methods clarify the parameter effects on the phenomenon efficiently and are expected to contribute to the clarification of the mechanism and the application to vibration machines.