Nonlinear modeling and experimental verification of Gannet-inspired beam systems during diving

A nonlinear model is proposed to answer at which diving speeds and beak angles will cause injury to Gannet-inspired beam systems during plunge-diving. In doing so, the critical velocities at which buckling occurs with various types of boundary conditions are first obtained for vertical dives and the resulting forces at the point of impact are determined. The Gannet-inspired system is modeled as an Euler-Bernoulli beam to represent the neck and body of the Gannet, while the head of the Gannet is modeled as a cone with varying half-angles. The experimental investigations of Gannet-like diving systems are first introduced to present the varying parameters and assumptions of the simplified model. Next, the resulting forces during impact are investigated and a study is conducted to compare various approximations of the drag coefficient for the cone-shaped head. Considering the mid-plane stretching nonlinearity, the equations of motion for the structural system under various types of boundary conditions are derived using the Hamilton's principle. The characteristic equations, buckled configurations, and critical velocities are determined for each set of boundary conditions. The results show that the system with the smallest half-beak angle and thus the lowest drag force and beam length delays the critical velocity and is most representative of a Gannet during diving. The obtained results demonstrate great agreement with the conducted experiments. For clamped-clamped boundary conditions, the critical velocity is found to be the greatest because of the increased stability at both ends of the beam. It is also noted that a nonlinear approximation for the coefficient of drag offers the best fit with the provided experimental values when compared to a hyperbolic tangent approximation, which predicts the coefficient of drag to be less than that obtained in experiments, and thus predicts that the systems will buckle at higher velocities.