The paper aim to understand the nonlinear vibration response characteristics of axially moving beams with intermediate support constraints. The dynamic equation of the system is established based on Hamilton principle, and the motion equation is discretized into a nonlinear ordinary differential equation using the Galerkin method. Combining the improved arclength method extension method with the incremental harmonic balance method, the dynamic equations of the system were solved, and the dynamic response characteristics of the system under different parameters were analyzed. Research has shown that the overall system response exhibits a hard spring characteristic. The support spring has a significant impact on the dynamics of the axially moving beam. In addition, when the axial movement speed is within a certain range, the system exhibits internal resonance, and the system's response has more complex response.