Two-to-one internal resonances in parametrically excited buckled beams

The response of a buckled beam possessing a two-to-one internal resonance to a principal parametric resonance of the higher mode is analyzed. The analysis assumes a unimodal static buckled deflection, considers quadratic nonlinearities only, and determines the amplitude and phase modulation equations via the method of multiple scales. Frequency- and force-response curves are generated. The force-response curves exhibit the saturation phenomenon. The equilibrium solutions of the modulation equations undergo saddle-node, pitchfork, and Hopf bifurcations. Dynamic solutions of the modulation equations are explored and a variety of bifurcation phenomena are documented, including period-doubling bifurcations, intermittency of type I, chaos, and crises.