In the late eighteenth century, humans discovered the first pterosaur fossil remains and have been fascinated by their existence ever since. Pterosaurs exploited their membrane wings in a sophisticated manner for flight control and propulsion, and were likely the most efficient and effective flyers ever to inhabit our planet. The flapping gait is a complex combination of motions that sustains and propels an animal in the air. Because pterosaurs were so large with wingspans up to eleven meters, if they could have sustained flapping flight, they would have had to achieve high propulsive efficiencies. Identifying the wing motions that contribute the most to propulsive efficiency is key to understanding pterosaur flight, and therefore to shedding light on flapping flight in general and the design of efficient ornithopters.
This study is based on published results for a very well-preserved specimen of Coloborhynchus robustus, for which the joints are well-known and thoroughly described in the literature. Simplifying assumptions are made to estimate the characteristics that can not be inferred directly from the fossil remains. For a given animal, maximizing efficiency is equivalent to minimizing power at a given thrust and speed. We therefore aim at finding the flapping gait, that is the joint motions, that minimize the required flapping power.
The power is computed from the aerodynamic forces created during a given wing motion. We develop an unsteady three-dimensional code based on the vortex-lattice method, which correlates well with published results for unsteady motions of rectangular wings. In the aerodynamic model, the rigid pterosaur wing is defined by the position of the bones. In the aeroelastic model, we add the flexibility of the bones and of the wing membrane. The nonlinear structural behavior of the membrane is reduced to a linear modal decomposition, assuming small deflections about the reference wing geometry. The reference wing geometry is computed for the membrane subject to glide loads and pretension from the wing joint positions.
The flapping gait is optimized in a two-stage procedure. First the design space is explored using a binary genetic algorithm. The best design points are then used as starting points in a sequential quadratic programming optimization algorithm. This algorithm is used to refine the solutions by precisely satisfying the constraints. The refined solutions are found in generally less than twenty major iterations and constraints are violated generally by less than 0.1%.
We find that the optimal motions are in agreement with previous results for simple wing motions. By adding joint motions, the required flapping power is reduced by 7% to 17%. Because of the large uncertainties for some estimates, we investigate the sensitivity of the optimized flapping gait. We find that the optimal motions are sensitive mainly to flight speed, body accelerations, and to the material properties of the wing membrane. The optimal flight speed found correlates well with other studies of pterosaur flapping flight, and is 31% to 37% faster than previous estimates based on glide performance. Accounting for the body accelerations yields an increase of 10% to 16% in required flapping power. When including the aeroelastic effects, the optimal flapping gait is only slightly modified to accommodate for the deflections of stiff membranes. For a flexible membrane, the motion is significantly modified and the power increased by up to 57%.
Finally, the flapping gait and required power compare well with published results for similar wing motions. Some published estimates of required power assumed a propulsive efficiency of 100%, whereas the propulsive efficiency computed for Coloborhynchus robustus ranges between 54% and 87%.