Spectral Galerkin Scheme for the Post-Critical Dynamics of a Geometrically Nonlinear, Inextensible, Cantilevered Beam Subjected to Non-Conservative Forces

A spectral Galerkin scheme is proposed for a nonlinear model describing the large deflection and planar motion of an inextensible cantilevered beam subjected to non-conservative forces, such as Beck’s beam and a cantilever immersed in axial airflow. It is established that the proposed scheme achieves spectral accuracy in space and second-order accuracy in time, relative to the sizes of the spatial and temporal grids, respectively. Furthermore, numerical simulations using the scheme are performed to investigate the post-critical limit cycle oscillations induced when the non-conservative forces exceed certain critical thresholds. The results provide a detailed analysis of how Kelvin-Voigt type internal damping destabilizes the system or increases the amplitude of the oscillation.