On the Nonlinear Dynamics of a Buckled Beam Subjected to a Primary-Resonance Excitation

We investigate the nonlinear response of a clamped-clamped buckled beam to a primary-resonance excitation of its first vibration mode. The beam is subjected to an axial force beyond the critical load of the first buckling mode and a transverse harmonic excitation. We solve the nonlinear buckling problem to determine the buckled configurations as a function of the applied axial load. A Galerkin approximation is used to discretize the nonlinear partial-differential equation governing the motion of the beam about its buckled configuration and obtain a set of nonlinearly coupled ordinary-differential equations governing the time evolution of the response. Single- and multi-mode Galerkin approximations are used. We found out that using a single-mode approximation leads to quantitative and qualitative errors in the static and dynamic behaviors. To investigate the global dynamics, we use a shooting method to integrate the discretized equations and obtain periodic orbits. The stability and bifurcations of the periodic orbits are investigated using Floquet theory. The obtained theoretical results are in good qualitative agreement with the experimental results obtained by Kreider and Nayfeh (Nonlinear Dynamics 15, 1998, 155-177).