Parametric resonance of a translating beam with pulsating axial speed in the super-critical regime

In the super-critical regime, parametric resonance of an axially translating beam is investigated for the first time. Resonances are found to be generated while the pulsated frequency is equal to not only two times of the natural frequency, but also just the natural frequency. By using the method of multiple scales, parametric resonance responses are solved out. Validity of the analysis method is certified by simulations. The quadratic nonlinearity caused by nontrivial equilibrium configuration is taken into consideration with the original cubic nonlinearity. A high-order multi-scaled perturbed solution with multi-modal shapes is introduced to ensure the accuracy of the results. The effects of the principal parameters are studied to certify the conditional existence of the parametric resonance. The parametric resonance in the sub-critical regime is used to demonstrate the difference between these two conditions. Comparison indicates that the translating beam at super-critical speed is more sensitive and the vibration is harder to be restrained.