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In this paper, the nonlinear governing equations of motion for an axially moving cylindrical shell with free ends which modeled as a free-free beam are established by the generalized Hamilton’s principle for the first time. The Galerkin method and multiple-scale method are adopted to solve the governing equations. Stability of the motion determined by the velocity and axial stiffness is investigated according to the linear equation firstly. Next, multiple-scale approach method is used to investigate the frequency response by adjusting the detuning parameters of the first two natural frequencies considering the internal resonance. Stability of periodic motion is studied through the trajectories of eigenvalues and phase of trajectories with adjusting the detuning parameter. Influences of its velocity and flexible force of the moving beam on the periodic motion and its stability are addressed through bifurcation diagrams and Lyapunov exponents. Correctness of the presented method for investigating dynamic behavior of the moving beam has been validated through Runge–Kutta numerical calculation. The goal of the research lies in the application of the method in the free-free moving beam and revealing the nonlinear dynamic behavior of the beam.
Yanhong et al. (Mon,) studied this question.