Stability and nonlinear vibration of an axially moving isotropic beam

In this paper, the stability and nonlinear vibration of an axially moving isotropic beam with simply supported boundary conditions are investigated. The equation of motion of the axially moving beam is established using the assumed mode method and Hamilton's principle. For the linear system, the natural frequencies of the beam under different moving velocities are calculated by solving the generalized eigenvalue problem of the system to analyze the divergence and flutter velocities. For the nonlinear system, the bifurcation of the transverse displacement with respect to the moving velocity is analyzed and the vibration properties of the beam under harmonic excitation are presented. From the investigation, it can be seen that with the moving velocity increasing, the beam exhibits the divergence or the flutter types of instability. The beam may exhibit a transient stable state before flutter. The first order divergence velocity of the beam obtained from the linear analysis is accordance with the bifurcation velocity obtained from the nonlinear system. The nonlinear forced vibrations of the beam under different moving velocity are presented to show the effects of the moving velocity on the response amplitude. The existence of the transient stable state is also proved from the nonlinear forced vibration analysis.