Geometrically exact extreme vibrations of cantilevers

This paper examines the extremely large nonlinear vibrations of a cantilever subject to base excitation in primary and secondary resonance regions for the first time. To predict extremely large vibration amplitudes accurately, a geometrically exact continuous model of the cantilever is developed for the centreline rotation of the cantilever; the proposed model’s accuracy is verified for extremely large deformations through comparison to a nonlinear finite element model. The theory of Euler-Bernoulli, along with inextensibility assumption, and the Kelvin-Voigt material damping model are utilised to develop the geometrically exact model. The main feature of the geometrically exact model is that all nonlinear trigonometric terms in the model are kept intact before and after the discretisation process, which itself is performed utilising the Galerkin scheme. The numerical results show that the cantilever undergoes extremely large oscillations even at relatively small base excitation amplitudes. It is shown that for some cases the amplitude of the tip of the cantilever grows so large that it “bends backward”; a behaviour which can only be captured using the proposed geometrically exact model.