Three-dimensional nonlinear extreme vibrations of cantilevers based on a geometrically exact model

In this paper, a three-dimensional geometrically exact model is developed for cantilevers, for the first time, which allows for accurate analysis of oscillations of extremely large amplitude. The proposed exact model is on the basis of three Euler angles describing the centreline motion. Euler–Bernoulli beam theory is utilised together with Kelvin–Voigt material damping, while utilising the centreline-inextensibility assumption. Hamilton's principle is utilised to derive the equations of motion for the three rotational motions, while keeping all terms geometrically exact. Galerkin discretisation is performed on the three partial differential equations of motion and the resultant set of discretised equations is solved using a continuation technique. The unique feature of the proposed exact model is its capability to capture very large-amplitude vibrations accurately, for both two-dimensional and three-dimensional motions. The dynamical response of the cantilever is examined in detail in the primary resonance region, highlighting the effect of the one-to-one internal resonance between the in-plane and out-of-plane transverse motions. A comparison between the geometrically exact model and the third-order nonlinear model is conducted to better showcase the significance of the proposed exact model, and the limitations of a truncated third-order model.