Free vibration of two coupled nonlinear oscillators

An analytical investigation is carried out on the free vibration of a two degree of freedom weakly nonlinear oscillator. Namely, the method of multiple time scales is first applied in deriving modulation equations for a van der Pol oscillator coupled with a Duffing oscillator. For the case of non-resonant oscillations, these equations are in standard normal form of a codimension two (Hopf-Hopf) bifurcation, which permits a complete analysis to be performed. Three different types of asymptotic states-corresponding to trivial, periodic and quasiperiodic motions of the original system-are obtained and their stability is analyzed. Transitions between these different solutions are also identified and analyzed in terms of two appropriate parameters. Then, effects of a coupling, a detuning, a nonlinear stiffness and a damping parameter are investigated numerically in a systematic manner. The results are interpreted in terms of classical engineering terminology and are related to some relatively new findings in the area of nonlinear dynamical systems.