Nonlinear vibrations of axially moving Timoshenko beams under weak and strong external excitations

In this paper, nonlinear vibrations under weak and strong external excitations of axially moving beams are analyzed based on the Timoshenko model. The governing nonlinear partial-differential equation of motion is derived from Newtonʹs second law, accounting for the geometric nonlinearity caused by finite stretching of the beams. The complex mode approach is applied to obtain the transverse vibration modes and the natural frequencies of the linear equation. The method of multiple scales is employed to investigate primary resonances, nonsyntonic excitations, superharmonic resonances, and subharmonic resonances. Some numerical examples are presented to demonstrate the effects of a varying parameter, such as axial speed, external excitation amplitudes, and nonlinearity, on the response amplitudes for the first and second modes, when other parameters are fixed. The stability of the response amplitudes is investigated and the boundary of instability is located.