This study investigates finite transient deformations in hyperelastic beam structures based on the Gent material model. To enable its application within beam formulations, the Gent model is expressed in a linearized form. A five-parameter beam element, incorporating two displacement variables, two difference parameters, and one thickness parameter, is adopted. The nonlinear dynamic response is solved using the implicit Newmark method, allowing efficient analysis of beams subjected to complex loading and boundary conditions. The results show that the proposed approach accurately captures the response of geometrically nonlinear beams and reproduces the behavior of neo-Hookean hyperelastic beams when the extensibility parameter of the Gent model approaches very large values. As the extensibility parameter decreases, the influence of chain extensibility becomes increasingly pronounced. This effect is particularly relevant for applications involving rubber and soft-robotic beams, bio-inspired flexible structures, and elastomeric beams subjected to large-strain vibrations, impact, and other transient loading conditions.