This work provides comparative modeling approaches to determine the velocities and natural frequencies of plunge-diving bird and bioinspired drone systems when entering water. These systems are chosen to further explain the survival of diving birds as they impact water and to provide insight into the design geometry and material choice of bioinspired diving drones. A nonlinear reducedorder model is developed and utilized to analyze the dive at impact considering both Timoshenko and Euler-Bernoulli beam theories. Using Hamilton's principle, the equations of motion are first derived. Then, static and dynamic buckling analyses are conducted. For this study, a geometrically simplified cone-beam system is considered, where the cone represents the head and the beam represents both the neck and body of the plunge-diving systems. The first study is to analyze the effects different diving drone materials and cone dimensions play on the sensitivity of the system. The second study applies geometric parameters to the cone-beam system representative of a plungediving bird (Northern gannet) and a surface-diving bird (Double-crested cormorant). The results show that choosing a material with a higher Young's modulus and a cone with a smaller half angle increase the velocity at which buckling occurs. The buckling velocities of the predicted Northern gannet model appear to be much greater than the average recorded diving speeds, suggesting that the bird is capable of plunge-diving at more extreme conditions. The natural frequencies are found for the aforementioned plunge-diving systems to predict failure if any external frequencies are known to act on the system while on a mission, such as conditions dependent on the climate or environment. It is shown in all buckling studies that the Euler-Bernoulli beam theory consistently overestimates the responses when compared with the Timoshenko beam theory. In the dynamic responses, Euler-Bernoulli beam theory overestimates for the pre-buckling region, then underestimates at the start of the post-buckling region until a point where the two theories cross paths. The amount of error with Euler-Bernoulli beam theory depends heavily on the slenderness ratio of the beam due to the theory being a simplification of the Timoshenko beam theory. It is noted that as the development of a more realistic bird model improves, the thickness will become significant and the use of Euler-Bernoulli beam theory at the point of impact will no longer be a valid assumption.
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 midplane 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 clampedclamped 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.
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