Two-to-One Internal Resonances in Buckled Beams

The vibrations of buckled beams with two-to-one internal resonances (ω₂ ≈ 2ω₁) about a static buckled position are analyzed. General boundary conditions and harmonic excitations (frequency Ω) in both the transverse and axial directions are considered. 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. The following specific cases are treated: Ω ≈ 2ω₂, Ω ≈ ω₁ + ω₂, and Ω ≈ ω₁. From the modulation equations for a primary resonance of the second mode (i.e., Ω ≈ ω₂), one-mode and two-mode stable equilibrium solutions are found in addition to dynamic solutions caused by Hopf bifurcations. In the region of dynamic solutions, a variety of phenomena are documented, including period-doubling bifurcations, intermittency, chaos, and crises.