Nonlinear dynamics of axially moving viscoelastic beams over the buckled state

This paper investigates the nonlinear forced dynamics of an axially moving viscoelastic beam in the supercritical speed regime, when the system is beyond the first instability. Both cases with and without internal resonances are examined. The beam configurations beyond the first bifurcation as well as the critical speeds are obtained analytically. The equation of motion about the buckled state is obtained by substituting the buckled configuration into the main equation of motion. A distributed external excitation load is then applied to the buckled system. The partial differential equation of motion about the buckled state is then discretized by means of the Galerkin method, yielding a set of second-order nonlinear ordinary differential equations with cubic and quadratic terms. The nonlinear resonant responses as well as bifurcation diagrams of Poincaré maps of the system are analyzed using the pseudo-arclength continuation technique along with direct time integration.