Geometrical nonlinearities in a curved cantilever beam: a condensation model and inertia-induced nonlinear features

Subjected to high level forcing, flexible and curved beams exhibit pronounced geometrical nonlinearities. In particular, intrinsic nonlinearities of cantilevers are different from their counterparts with end-constrained boundaries and the combination of the enhanced nonlinear-inertia effects with initial curvature creates harsh demand on the modeling, numerical simulation and understanding of associated physical phenomena. This paper investigates the salient nonlinear features in a curved cantilever beam, with particular attention paid to the inertia-induced effects through both linear and nonlinear analyses. An inextensible condensation model, with the consideration of the initial curvature, is proposed based on a geometrically exact model for an Euler–Bernoulli cantilever beam. The free boundary of the cantilever gives rise to more significant longitudinal motion, which increases the inertia effects in the beam vibration which is in turn enhanced by the initial curvature. Specific techniques are proposed to numerically implement the developed model with increased accuracy and robustness. Numerical simulations are then conducted to validate the proposed model through comparisons with the finite element method, examine the assumptions underpinning the model and explore the salient physical features, in particular the inertia-induced effects in both linear and nonlinear cases. Results show a decrease in the natural frequencies due to the initial curvature effect, a transition of the first mode from hardening to softening caused by enhanced curvature-induced inertia effect, and a pronounced asymmetry of the higher order modes with respect to frequencies.