In the present study, the nonlinear forced dynamics of an axially moving beam is investigated numerically taking into account the in-plane and out-of-plane motions. The nonlinear partial differential equations governing the motion of the system are derived via Hamilton's principle. The Galerkin scheme is then introduced to these partial differential equations yielding a set of second-order nonlinear ordinary differential equations with coupled terms. This set is transformed into a new set of first-order nonlinear ordinary differential equations by means of a change of variables. A direct time integration technique is conducted upon the new set of equations resulting in the bifurcation diagrams of Poincaré maps of the system. The dynamical characteristics of the system are investigated for different system parameters and presented through use of time histories, phase-plane portraits, Poincaré sections, and fast Fourier transforms.