Dynamic effect of internal resonance caused by gravity on the nonlinear vibration of vertical cantilever beams

Abstract

Although the transverse vibration of the cantilever structure is of great interest, the internal resonance of the vertical cantilever structure is always ignored. In this paper, nonlinear free vibration and 1/3 super-harmonic resonance of a hanging cantilever beam are firstly presented with 3:1 internal resonance caused by gravity. By employing the temporal multi-scale method, mode responses in these two vibrations are obtained. The harmonic balanced method is used for solving the excitation component and the whole deflection in super-harmonic resonance. For nonlinear free vibration, beat phenomenon is found and the effect of mode interaction is determined by gravity, damping and initial perturbation. For super-harmonic resonance, the amplitude of the first two modes can be quite large and exceed the excitation component if damping is not strong. Besides, the softening-type frequency curves are predominated by the inertia nonlinearity. Compared with the excitation component and the whole deflection, the mode responses are more easily affected by gravity. If the value of gravity parameter is a little bit lower than the one in the condition of strict internal resonance, the energy transmission, multiple solutions and bifurcations and three complex types of frequency curves can be found in mode responses. Hysteresis and saturation phenomena in mode responses can be discovered as well. Results from analytical methods are almost identical to those from numerical ways. In summary, the internal resonance of slender hanging cantilever structures should be aroused more attention, for hanging cantilevers are common in practical engineering and considerable mode responses can be induced by large initial perturbation in weakly damped free vibration or by low excitation frequency and large excitation amplitude in super-harmonic resonance. In addition, the complex dynamics can be worthwhile to judge the occurrence of internal resonance and its impact on structures. Since gravity has tensile effects on hanging cantilevers, these phenomena may also occur in beams with axial tension.