Two-To-One Internal Resonance of Super-Critically Axially Moving Beams

Internal resonance is a common phenomenon in super-critically axially moving beams. It can lead to an energy exchange among associated modes and produce a large-amplitude response. However, the effect of the internal resonance on the dynamics of system is not clear. Therefore, in the present paper, we investigated the nonlinear dynamics of a super-critically axially moving beam with two-to-one internal resonance. By applying a proper transporting speed, the two-to-one internal resonance condition of the system is established. The method of multiple scales is employed to solve the governing equation so as to obtain the nonlinear response of the system. To analyze the effect of the quadratic and cubic nonlinearities, the perturbation solution is expanded up to three orders. The primary resonance of the first and second modes is investigated. Their steady-state solutions are solved, and the stability of these solutions is examined. Response of the system is demonstrated via the frequency response and force-response curves. Results show that jumping, saturation, and hysteresis phenomenon may occur. Moreover, the modulated motion can be found in the case of primary resonance of the first mode. The approximate analytical results are verified by the numerical results.