Theoretical and Experimental Investigation of the Nonlinear Vibration of a buckled beam to a subharmonic resonance

We investigate the nonlinear vibrations of a clamped-clamped buckled beam to a subharmonic resonance of order one-half of its first vibration mode. We use 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 solve the discretized equations using the method of multiple scales to obtain a second-order approximation, including the modulation equations governing its amplitude and phase. To investigate the local and global dynamics, we numerically integrate the discretized equations using a shooting method to compute periodic orbits and use Floquet theory to investigate their stability and bifurcations. We obtain interesting dynamics, such as a phase-locked motion, resulting from a Hopf bifurcation, snapthrough motions, and a sequence of period-doubling bifurcations leading to chaos. It is important to note that by using a single-mode Galerkin discretization, one cannot predict some of these nonlinear phenomena. We carry out an experiment and obtain results that are in good qualitative agreement with the theoretical results.