Equilibrium bifurcation of high-speed axially moving Timoshenko beams

In this paper, equilibrium bifurcations of an axially moving Timoshenko beam are studied in the supercritical region. For the first time, Timoshenko beam theory is applied to investigate nonlinear dynamics of high-speed axially moving structures. The static equilibrium equation is deduced from the governing equation of transverse vibration of the axially moving Timoshenko beam. Two kinds of boundary conditions are considered. The non-trivial equilibrium solutions are analytically determined. Moreover, the equilibrium equations are discretized by using the finite difference method. Therefore, equilibrium configurations are numerically verified by proposing an iterative scheme. This investigation shows that non-trivial equilibrium solutions of Timoshenko beams bifurcate with axially moving speed. By comparing with Euler-Bernoulli beam theory, this study finds that the critical speed, determined by the Timoshenko beam, is remarkably smaller. Nevertheless, the equilibrium deformation of the moving Timoshenko beam is obviously larger. Furthermore, the present work derives the critical speed of the axially moving Timoshenko beam. At last, the effects of the system parameters on the equilibrium bifurcation and the critical speed are presented.