Structurally Consistent Class/Shape Transformation Equations for Morphing Airfoils

Parametric optimization for morphing structures design: application to morphing wings adapting to changing flight conditions

“…where ζ A and ζ P are the non-dimensional distances perpendicular to the chord to the active and passive surfaces, ψ is the non-dimensional distance along the chord (ψ ∈ [0, 1]), A i and P i are the shape function coefficients, and n is the order of the Bernstein polynomials. Leal and Hartl (2019) modified the CST equations so that children airfoils (i.e., morphed configurations) can be generated from any parent airfoil (i.e., original configuration) while considering kinematic constraints related to the internal structure; thus, all of the explored designs herein are known to be physically feasible for low strains (≤ 4%), even without recourse to full structural analysis. In the model it is assumed that the overall morphing structure has or maintains the following: rigid body spars, constant leading edge radius, constant passive surface length, and constant angles between spars and passive surface.…”

“…If all coordinates (ψ c A,j , ζ c A (ψ c A,j )) can be calculated, F is a square tensor and invertible, and the values of the active children shape coefficients A c i are determined by solving the linear system of equations (5). Therefore, the only free As detailed in Leal and Hartl (2019), a set of kinematic constraints is utilized to guarantee structural feasibility of the morphed configurations and to establish coordinates (ψ c A,j , ζ c A (ψ c A,j )) as a function of A p , P p , P c , and the locations and heights of spars. In the context of real aircraft, the parent airfoil and much of its internal structure are fixed and known.…”

“…All other aspects of the problem, such as the initial NACA 0012 shape and spar locations are held constant to allow clarity of presentation. CST equations of order n = 4 are utilized based on a convergence study performed by Leal and Hartl (2019). Depending on the requirements of the design exploration, n can be increased to allow more subtle topological manipulations and structural constraints (e.g., spars).…”

“…Some have addressed (i) using thickness (Liem et al 2017;Liu et al 2015) and volume (Morris et al 2010;Hicks and Henne 1978) constraints on the OML that reproduce the inner structure restrictions. Recently, Leal and Hartl have addressed both (i) and (ii) using a structurally consistent approach based on Class/Shape Transformation (CST) equations (Leal and Hartl 2019). Additionally, parametric optimization addresses (iii) by treating flight conditions as parameters.…”