Periodic response of an axially high-speed moving beam under 3:1 internal resonance

In the supercritical speed range, nonlinear forced vibration of an axially moving viscoelastic beam in the presence of 3:1 internal resonance is investigated. The straight beam becomes buckled due to the supercritical moving speed. The governing equation is cast for motion around buckled configuration by using a coordinate transform. Moreover, the first two modes of the buckled beam are set to 3:1 by adjusting the axial moving speed. Then the corresponding equation is approximately analyzed by utilizing the multi-scale method. For the beam subjected to the primary resonances and super-harmonic resonance with internal resonance, frequency-amplitude relationship of steady-state responses is constructed. Numerical examples discovered the influence of internal resonance on the nonlinear dynamic characteristics of the axially moving beam. Specifically, the energy transfer between the first two order modes is found for an axially supercritical moving beam. Moreover, several typical nonlinear phenomenon, such as double jumping phenomenon, hysteretic phenomenon and saturation-like phenomenon, are discovered in the nonlinear vibration of the axially moving beam. By comparing with numerically simulative results via the finite difference method and the Galerkin method, it is confirmed that the method of multi-scale in the present work is quite credible.