Natural frequencies of nonlinear transverse vibration of axially moving beams in the supercritical regime

Natural frequencies are investigated for transverse vibration of axially moving beams in the supercritical ranges. In the supercritical transport speed regime, the straight equilibrium configuration becomes unstable and bifurcate in multiple equilibrium positions. The transverse motion of axially moving beams can be governed by a nonlinear partial-differential equation or a nonlinear integro-partial-differential equation. For motion about each bifurcated solution, those nonlinear equations are cast in the standard form of continuous gyroscopic systems by introducing a coordinate transform. The first four natural frequencies are investigated for the beams via the 8-term Galerkin method and the finite difference method to truncate the corresponding governing equations without nonlinear parts into an infinite set of ordinarydifferential equations under the simply supported boundary conditions. Numerical results indicate that the nonlinear coefficient has little effects on the natural frequency from the nonlinear partial-differential equation, and the two nonlinear models predict qualitatively the same tendencies of the natural frequencies with the changing parameters. Quantitative comparisons demonstrate that results of the 4-term Galerkin method for the natural frequency for axially moving beams in the supercritical range are with rather high precision.