Energetic Consideration on the Occurrence Condition of Self-Synchronization in Two Unbalanced Rotors

When 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. In order to provide new insights into the mechanism of this phenomenon, this paper considers a system comprising two counter-rotating unbalanced rotors and analyzes the energy condition that must be satisfied in a synchronized state. From the derived conditions, it is possible to show analytically the range of synchronous frequencies and the range of applied voltages at which synchronization can occur. Furthermore, the synchronous frequency and the phase difference can be obtained numerically for given system parameters. The results are in fairly good agreement with accurate solutions obtained from the shooting method, showing the effectiveness of the proposed energy-based analysis.