Transverse equilibrium and vibration of axially moving beams with fixed boundaries in the supercritical regime

The equilibria, the bifurcation, and the first two supercritical natural frequencies of nonlinear transverse free vibrations of axially moving beams with fixed boundaries are investigated. In the supercritical transport speed ranges, the pattern of equilibria consists of the straight configuration and non-trivial solutions that bifurcate with transport speed. The equilibrium solutions are performed analytically for an axially moving beam with the fixed ends. The analysis results illustrate the tendencies of the critical speeds with the changing physical parameters. For motion about each non-trivial equilibrium configuration, the nonlinear integro-partial-differential equation is cast in the standard form of continuous gyroscopic systems by introducing a coordinate transform in the supercritical regime. Numerical schemes are presented for the governing equation via the finite difference method and the discrete Fourier transform for the natural frequencies of transverse vibrations. Numerical results indicate that the first natural frequencies of axially moving beams increase with the growth of axial speed. And the calculation results are confirmed qualitatively via the Galerkin method to truncate the corresponding governing equations without nonlinear parts.