Regular and Chaotic Vibrations of a Parametrically and Self-Excited System Under Internal Resonance Condition

Vibrations of a parametrically and self-excited system with two degrees of freedom have been analysed in this paper. The system is constituted by two parametrically coupled oscillators characterised by self-excitation and nonlinear Duffing’s type nonlinearities. Synchronisation phenomenon has been determined near the principal resonances in the neighbourhood of the first p₁ and the second p₂ natural frequencies, and near the combination resonance (p₁+p₂)/2. Vibrations have been investigated for parameters which satisfy the internal resonance condition p₂/p₁=3. The existence and break down of the synchronisation phenomenon have been revealed analytically by the multiple time scale method, whilst transition of the system to chaotic motion has been carried out numerically.