Nonlinear responses of buckled beams to subharmonic-resonance excitations

We investigated theoretically and experimentally the nonlinear response of a clamped-clamped buckled beam to a subharmonic resonance of order one-half of its first vibration mode. We used a multi-mode Galerkin discretization to reduce the governing nonlinear partial-differential equation in space and time into a set of nonlinearly coupled ordinary-differential equations in time only. We solved the discretized equations using the method of multiple scales to obtain a second-order approximate solution, including the modulation equations governing its amplitude and phase, the effective nonlinearity, and the effective forcing. To investigate the large-amplitude dynamics, we numerically integrated the discretized equations using a shooting method to compute periodic orbits and used Floquet theory to investigate their stability and bifurcations. We obtained interesting dynamics, such as phase-locked and quasiperiodic motions, resulting from a Hopf bifurcation, snapthrough motions, and a sequence of period-doubling bifurcations leading to chaos. Some of these nonlinear phenomena, such as Hopf bifurcation, cannot be predicted using a single-mode Galerkin discretization. We carried out an experiment and obtained results in good qualitative agreement with the theoretical results.