The study of a parametrically excited nonlinear mechanical system with the continuation method (Q1371140)

The authors study the one-degree-of-freedom nonlinear system subjected to harmonic parametric excitation: \(\ddot x(t)+ 2c\dot x(t)+ (\omega^2_0+ p\cos\omega t)x(t)+ ax^2(t)+ bx^3(t)= 0\). The Galerkin method is applied for obtaining the response of the system to principal and fundamental parametric resonance. The numerical analysis with the continuation method is carried out, and attention is focused on tracing the periodic solution with the variation of excitation frequency. The numerical stability analysis of the trivial solution is carried out, and the stable and unstable regions of trivial solution are computed. The numerical results with the continuation method are in good agreement with the results of the Galerkin method. The range of frequency for existence of trivial solution is also investigated. The tracing of the periodic solution verifies the existence of subcritical Hopf bifurcation. The bifurcation diagram is also obtained. The authors show that the period-2 small and large amplitude oscillations remain in the same closed branch, and report many results of calculations.