Dynamics of a super-critically axially moving beam with parametric and forced resonance

In the super-critical regime, steady-state responses of an axially moving beam are analyzed subjected to parametric combined with forced excitations. By employing the method of multiple scales, the primary resonance is investigated. Steady-state resonances exist unless the parametric frequency and the external frequency are commensurable. Natural modes are triggered when the parametric frequency is close to two times of or just the natural frequency. For the case of the first one, the combined excitation deduces a response curve with twin resonance peaks. Distance of them is determined by the parametric excitation, and the widths are depended on the external force. Double jumping is found in the response curve. For the case of the second one, the combined excitation produces a simple resonance in the form of a typical forced vibration. The response curve is superimposed by each of the excitations. With the numerical method, the incommensurability of excitation frequencies is found to produce beats and quasi-periodicity vibrations.