Extreme parametric resonance oscillations of a cantilever: An exact theory and experimental validation

Abstract

Flexible cantilevered beams are capable of undergoing motions of extremely large amplitudes which are relevant to both conventional and emerging applications. However, modelling and capturing such behaviour is not a straightforward task and requires the application of a geometrically exact model. In this study, the extreme parametric resonance responses of a cantilever with a tip mass are captured through carefully conducted experimental measurements and compared to the predictions of a centreline-rotation-based geometrically exact cantilever beam model. For the experimental part, a state-of-the-art in-vacuo base excitation set-up consisting of a vacuum chamber and a long-stroke shaker is utilised to excite the cantilever in the axial direction near parametric resonance and drive the cantilever to very large oscillation amplitudes. This set-up is accompanied by a high-speed camera to capture footage of the deformed configurations of the cantilever in a period of steady-state oscillations at each excitation frequency. The cantilever motion amplitudes are then extracted from the videos through use of a robust image processing code. For the theoretical modelling and numerical analysis parts of the study, the cantilever is modelled via the Euler–Bernoulli beam theory assuming an inextensible centreline, while considering the centreline rotation as the main motion variable. All nonlinear terms in the model are kept intact throughout the derivation, discretisation, and simulation procedures to ensure accurate model predictions. Comparisons between experimental and theoretical results are conducted for three axial base-acceleration levels. It is shown that the geometrically exact model’s predictions are in excellent agreement with experimental results at various oscillation amplitudes.