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

We investigate the nonlinear response of a clamped-clamped buckled beamto a primary-resonance excitation of its first vibration mode. The beamis subjected to an axial force beyond the critical load of the firstbuckling mode and a transverse harmonic excitation. We solve thenonlinear buckling problem to determine the buckled configurations as afunction of the applied axial load. A Galerkin approximation is used todiscretize the nonlinear partial-differential equation governing themotion of the beam about its buckled configuration and obtain a set ofnonlinearly coupled ordinary-differential equations governing the timeevolution of the response. Single- and multi-mode Galerkinapproximations are used. We found out that using a single-modeapproximation leads to quantitative and qualitative errors in the staticand dynamic behaviors. To investigate the global dynamics, we use ashooting method to integrate the discretized equations and obtainperiodic orbits. The stability and bifurcations of the periodic orbitsare investigated using Floquet theory. The obtained theoretical resultsare in good qualitative agreement with the experimental results obtainedby Kreider and Nayfeh (Nonlinear Dynamics 15, 1998, 155–177.