Super-harmonic resonance and multi-frequency responses of a super-critical translating beam

With 3:1 internal resonance, super-harmonic resonances of a super-critical axially moving beam is studied for the first time. The response produced by 1/3 super-harmonic resonance is numerically determined and it is found to be very intense. Therefore, the direct multi-scale method is applied to fully investigate steady-state response of the super-harmonic resonance. In the super-critical speed range, the governing equation contains quadratic and cubic nonlinearity at the same time. Accordingly, the perturbation solution by applying the multiple scale method contains three time scales. The analytical solutions conclude that not only the primary resonance is contained, but also several harmonic responses are involved. Moreover, the perturbation analysis shows more harmonic responses should be involved. Furthermore, the 3:1 internal resonance investigated here indicates that energy transmits from the first mode to the second one. The amplitude-frequency response shows that nonlinearity of the super-critically translating beam is soft. Besides, the analytical process with internal resonance proves that the direct method of multiple scales can be applied to investigate the multi-mode responses.