Coupled longitudinal-transverse dynamics of an axially moving beam with an internal resonance

The forced nonlinear dynamics of an axially moving beam with coupled longitudinal and transverse displacements is numerically investigated in this paper with special consideration to the case with a three-to-one internal resonance. The two coupled nonlinear partial differential equations for the longitudinal and transverse motions are discretized via the Galerkin technique, and the resulting set of nonlinear ordinary differential equations is solved either by means of the pseudo-arclength continuation method or via direct time integration. Specifically, the frequency–response curves of the system are obtained using the pseudo-arclength continuation technique, and the bifurcation diagrams of Poincaré maps via direct time integration. The effect of system parameters on the above-mentioned diagrams is examined and the results are presented in the form of time histories, phase-plane portraits, Poincaré maps, and fast Fourier transforms (FFTs). It is shown that depending on the system parameters, the system displays a wide variety of rich dynamics.

► Coupled longitudinal-transverse dynamics of an axially moving beam is investigated.
► Frequency–response curves are plotted using the pseudo-arclength continuation technique.
► Bifurcation diagrams are constructed via direct time integration.