The Study of a Parametrically Excited Nonlinear Mechanical System with the Continuation Method

The dynamic behavior of a one-degree-of-freedom, parametrically excited nonlinear system is investigated. The Galerkin method is applied to the principal and fundamental parameteric resonance of the system. The continuation method is used to study the change of harmonic oscillation with respect to the variation of excitation frequency. The numerical stability analysis of the trivial solution is carried out and the stable and unstable regions of the trivial solution are given. They are found to agree with the results obtained by the analytical method of Galerkin. Periodic solutions are traced and the coexistence of multi-periodic solutions is observed With the change of excitation frequency the large amplitude periodic-2 oscillation is found to be in the same closed branch with the small amplitude periodic-2 solution. In addition, the bifurcation pattern of the trivial solution is found to change from subcritical Hopf bifurcation into supercritical Hopf bifurcation with the increase of excitation amplitude. Combined with the conventional numerical integration method, new complex dynamic behavior is detected.