Shape, Structure, and Kinematic Parameterization of a Power-Optimal Hovering Wing

Optimal thickness distributions of aeroelastic flapping shells

“…(23) at a given time step, the relevant terms can be inserted into Eq. (24), and everything needed to evaluate dG i /dx in Eq. (9) is then available.…”

“…Furthermore, the thickness of each finite element (x) must lie between side constraints. The second optimization study, perhaps more relevant to flapping wing design [24], is:…”

“…Previous work in this area is relatively rare: the optimal thickness distribution of a flapping beam is considered by Isogai and Harino [12], Stanford et al [24], and Thomson [27]. Ho et al [11] discuss the optimal stiffness distribution (by controlling the elastic modulus at different patches along the wing) of a flapping shell.…”

“…The former directly computes the derivative of the system response with respect to the design variables via a linear system of equations with multiple right-hand sides (one per design variable). This linear set of equations can be integrated forward in time, along with the nonlinear set of aeroelastic equations [24]. Conversely, the adjoint method requires the solution of a linear set of equations with multiple right-hand sides (one per constraint, and thus independent of the number of design variables [8].…”

“…Beam models have been utilized for some three-dimensional flapping studies (see Isogai and Harino [12], Beran et al [1], Stanford et al [24], among others), but an inability to handle chordwise deformations and pressure gradients (an important source of thrust generation via adaptive feathering [11]) may limit their usefulness for design. The inclusion of a threedimensional elastic shell model can afford an important understanding into the complex interaction between elastic, inertial, and aerodynamic forces/power [26], and how these relationships are shifted by tailoring the stiffness/mass throughout the shell.…”