Post-buckling bifurcations and stability of high-speed axially moving beams

The nonlinear forced dynamics of an axially moving beam in the supercritical speed regime is numerically investigated in this paper, with special consideration to the case possessing an internal resonance between the first two modes. At critical speed, the initial equilibrium configuration of the beam becomes unstable and a new stable non-trivial equilibrium solution together with its symmetric counterpart arises. A closed-form solution for the post-buckling configuration is introduced and the system is assumed to be subject to a transverse harmonic excitation load at its buckled state. The equation of motion is cast into new coordinates which gives the equation governing the motion of the beam about the buckled state. This equation is discretized via the Galerkin method which yields a set of nonlinear ordinary differential equations (NODEs) with quadratic and cubic nonlinear terms. The set of NODEs are solved either via the pseudo-arclength continuation technique or by means of direct time integration so as to obtain frequency–response curves as well as bifurcation diagrams of Poincaré sections about the buckled state. Results are shown through time histories, phase-plane diagrams, and Poincaré sections.