Three-dimensional nonlinear planar dynamics of an axially moving Timoshenko beam

The three-dimensional nonlinear planar dynamics of an axially moving Timoshenko beam is investigated in this paper by means of two numerical techniques. The equations of motion for the longitudinal, transverse, and rotational motions are derived using constitutive relations and via Hamilton’s principle. The Galerkin method is employed to discretize the three partial differential equations of motion, yielding a set of nonlinear ordinary differential equations with coupled terms. This set is solved using the pseudo-arclength continuation technique so as to plot frequency-response curves of the system for different cases. Bifurcation diagrams of Poincaré maps for the system near the first instability are obtained via direct time integration of the discretized equations. Time histories, phase-plane portraits, and fast Fourier transforms are presented for some system parameters.