S. A. Ahmadsani, and H. Nahvi, Vibration analysis of a nonlinear 2-DOF system in the simultaneous resonant case, Iranian Journal of Mechanical Engineering Transactions of the ISME, in Persian, Vol. 16, No. 1, pp. 21-42 (2014).
To accurately predict vibration behavior of structures, a complete mathematical modeling is necessary. In nonlinear structural vibration analysis, using the linear model may lead to wrong results. In this paper, the nonlinear vibrations of a two degrees of freedom system consists of the main system and an absorber is studied at simultaneous secondary and internal resonances. The frequency response equations are obtained by solving equations of motion using the method of multiple time scales (MTS) technique. The effects of system parameters on the amplitude of the main system is investigated. Stability analysis is performed by the method of Andronov and Vitt and the saddle-node bifurcation points are detected. The results show that in the simultaneous super-harmonic and internal resonance case, system shows an almost linear behavior. Also, the nonlinear parameters of the absorber have insignificant effect on the amplitude of the main system. Unlike the linear case, when the nonlinear stiffness of the main system increases, the system amplitude increases as well. Finally, the domain of the detuning parameter with three responses, resulted in the jump phenomenon in the response, is determined. At the end, the possibility of existence of bifurcation phenomenon in the response is studied. By choosing a control parameter, bifurcation diagrams are drawn in order to detect periodic and chaotic responses. Poincaré map diagram, phase- plane and the time response for different values of control parameters are drawn to distinguish periodic, quasi-periodic and chaotic motions.