Design Variale Concepts 19 Mar 09 La 7 Lecture Notes Nomenclature W total weight (= W wing + W fuse + W pay ) reference area (wing area) wing aspect ratio c r root wing chord c t tip wing chord λ taper ratio (= c t /c r ) I o wing root ending inertia E Young s modulus wing root ending moment M o c T max C D c d CDA 0 κ o δ wing span average wing chord maximum thrust lift coefficient drag coefficient wing profile drag coefficient drag area of non-wing components wing root ending curvature tip deflection Design pace Design Variales are numers whose values can e freely varied y the designer to define a designed oject. As a very simple example, consider a rectangular wing with a pre-defined airfoil. It can e defined y deciding on the values of the following two design variales: {, c } (1) Placing these variales along orthogonal axes defines a design space, or set of all possile design options. Each point in the design space corresponds to a chosen design, as illustrated in Figure 1. c 3 1 4 8 1 Figure 1: Two-variale design space of a rectangular wing. 1
Variale et Choice Frequently, an alternative variale set can e defined in terms of the starting variale set. For example, we can define the same design space using the variale set {, } () with the following relations translating etween the two alternative variale sets: = /c = c = c = / (3) Figure shows the alternative {, } design space. Other variale sets are possile for c = 3 c = 4 = 1 16 8 = 8 c = 1 = 4 4 Figure : Alternative design variale set of a rectangular wing. 6 this case, such as {, }, {, }, etc. The est variale set is usually the one which gives the simplest or clearest means to evaluate the ojective function, or performance of the design, so that the est point in the design space can e selected. Example Ojective Function As an example, we wish to determine a wing design which will maximize the payload weight of an electric aircraft, whose motor and propeller can generate at most some given maximum thrust T max. We egin with the level flight force-equilirium relations, W = L T max = D = L C D (4) = W C D (5)
( CDA0 / T max = (W fuse + W wing + W pay ) + c d + C ) L π (6) Therefore, the ojective function to e maximized is otained y solving for the payload weight. W pay (, ) = CDA 0 / T max + c d + π W fuse W wing (7) ince and appear explicitly in this ojective function definition, these are proaly the est choices for the design variales. Ojective Function Contours In practice, the dependence of W pay on {, } is far more complex than what s explicitly visile in equation (7). For example, the wing weight W wing will clearly depend on {, }, as will c d via the chord Reynolds numer. Also, T max will likely depend on the flight speed, which is influenced y wing loading and hence y. Given quantitative models of all these effects, we can numerically determine the value of W pay for every {, } comination. The results might e as shown in Figure 3, which shows the ojective function as contours, or isolines. The point where the ojective function has a maximum represents the optimum design. 4 16 W W W pay pay Wpay pay = 1 = = 4 = 5 8 6 1 18 Figure 3: Ojective function contours (isolines) in design space of a rectangular wing. Black dot shows the optimum-design maximum payload weight point. Constraints In almost any real design optimization prolem, an ojective function such as given y equation (7) does not capture all considerations which might go into selection of a design. Frequently one has to account for constraints which rule out certain regions of the design space. One typical constraint which appears in wing design is the structural requirement of adequate strength or stiffness. 3
To incorporate a stiffness constraint, we first must express the stiffness requirement in terms of the chosen design variales, or {, } in this case. We will require that the tip-deflection/span δ/ not exceed some reasonale upper limit, say δ/ 0.05...0.10 (8) in 1G flight. The tip deflection can e estimated using simple eam theory. Assuming the eam curvature to e roughly constant across the span and equal to its value κ o at the root, the estimated tip deflection is given as follows. κ o = M o (9) EI o δ 1 ( ) κ o (10) To use this, we still need to estimate the root ending moment M o and root ending inertia I o. A conservative estimate is that the net local loading/span is proportional to the chord. For a simple-taper wing with taper ratio λ = c t /c r, the root ending moment is then otained y twice integrating this loading, giving M o = 1 1 + λ 1 1 + λ (W fuse + W pay ) (11) Assuming the wing is constructed out of a solid material the ending inertia of its root airfoil cross section is approximately I o 0.036 c r t r (t r + h r) = 0.036 c 4 r τ(τ + ε ) (1) where t r is the maximum root airfoil thickness, h r is the maximum root airfoil camer height, τ = t/c is the airfoil thickness/chord ratio, and ε = h/c is the airfoil camer/chord ratio. For a straight-taper wing of taper ratio λ, the root chord is related to the average chord y c r = c 1+λ (13) Comining equations (10), (11), (1), and (13) gives δ = 0.018 W fuse + W pay Eτ(τ + ε ) (1+λ)3 (1+λ) c 4 (14) Putting and c in terms of our chosen design variales {, } as given y (3), the deflection/span ratio finally ecomes δ = 0.018 W fuse + W pay Eτ(τ + ε ) (1+λ)3 (1+λ) 3 (15) In the design space, the isolines of δ/ are given y rearranging equation (15) into = [ 0.018 (δ/) ] W fuse + W pay Eτ(τ + ε ) (1+λ)3 (1+λ) 3 (16) 4
δ/ = 0.01 δ/ = 0.05 δ/ = 0. 4 16 8 6 1 18 Figure 4: Wing deflection/span contours (dashed) superimposed on ojective function contours (solid). The contour δ/ = 0.05 is the chosen constraint oundary. Black dot shows the constrained maximum-payload weight point. which is shown in Figure 4 for three values of δ/. All points aove the δ/ = 0.05 isoline satisfy a chosen deflection constraint (8), and hence constitute the feasile design space. The new constrained optimum design is the point of maximum ojective function which still lies in the feasile design space. Additional Design Variales Most practical design prolems have vastly more than the two design variales {, } assumed in the examples aove. A asic rule is that any adjustale quantity which is likely to have a strong effect on the constrained ojective function should e considered as a design variale. One such candidate is the wing taper ratio c t /c r =λ, which clearly has a powerful effect on the tip deflection in relation (15). If λ is chosen as a new design variale, the design space is now three dimensional as shown in Figure 5. {,, λ} (17) λ 1 0 Figure 5: Three-variale design space. As efore, each point represents a unique design. 5
In reality, there would also e other variales such as the modulus E of the construction material, and c d via airfoil shape, etc. The design space would then e Design pace licing {,, λ, E,, c d...} (18) Because the entire design space of many dimensions is impossile to visualize graphically, we typically attempt to get its character y slicing it with a plane defined y only two variales, y choosing unique values for all the others. For example, the D space in Figure 4 is the same as the 3D space in Figure 5 sliced along the λ = 1 plane. Two of the three possile slice orientations are also shown in Figure 6. λ = 4 λ λ = 0.4 Figure 6: Two D slices through a 3D design space. The variale(s) which are not along the axes of a slice are held fixed. Occasionally it is also useful to slice a design space using 1D lines rather than D planes. This allows plotting of a quantity of interest, such as W pay in the current example, along each slice line. This is an alternative means of locating the optimum design, in lieu of the contour technique shown in Figure 3. λ W pay λ = 0.8 = 16 = 1 = 8 Figure 7: Three line slices through a 3D design space. 6