Designing Wings with Fixed Twist for Minimum Induced Drag

Designing Wings with Fixed Twist for Minimum Induced Drag D. F. Hunsaker * and W. F. Phillips † Utah State University, Logan, Utah 84322-4130 J. J. Joo ‡ Air Force Research Laboratory, Wright-Patterson AFB, OH 45433 Minimum induced drag for a prescribed gross weight and wingspan is obtained from the elliptic lift distribution. Here it is shown that minimum induced drag for steady level flight is not obtained by imposing the constraints of prescribed gross weight and wingspan. Because wing structural weight is a function of wingspan and lift distribution, there exists an optimum lift distribution, weight, and wingspan for a given non-structural weight distribution that minimizes the induced drag in steady level flight. This optimum lift distribution can vary significantly from the elliptic lift distribution, depending on design constraints. Furthermore, it is shown that for any fixed lift and weight distributions, there is an optimum wingspan that minimizes the induced drag, which is based on the tradeoff between wingspan and wing structural weight. It is shown that a fully optimized design can reduce induced drag by up to 11.1 percent from that produced by a fixed elliptic lift distribution. Example developments of three such optimum lift distributions for rectangular wings are developed for varying sets of design constraints.

Nomenclature A An Bn b Cδ Cσ c Di E h hf hi I L ~ L ~ Mb na ng

= = = = = = = = = = = = = = = = = =

beam cross-sectional area Fourier coefficients in the lifting-line solution for the section-lift distribution, Eq. (1) Fourier coefficients in the lifting-line solution for the dimensionless section-lift distribution, Eq. (6) wingspan shape coefficient for the deflection-limited design, Eq. (74) shape coefficient for the stress-limited design, Eq. (41) local wing section chord length wing induced drag modulus of elasticity of the beam material height of the beam cross-section flange height (i.e., vertical thickness) of the I-beam cross-section inside height of the box beam cross-section beam section moment of inertia total wing lift local wing section lift local wing section bending moment load factor in g’s limiting load factor at the hard-landing design limit

*

Assistant Professor, Mechanical and Aerospace Engineering, 4130 Old Main Hill, AIAA Member. Emeritus Professor, Mechanical and Aerospace Engineering, 4130 Old Main Hill, AIAA Senior Member. ‡ Research Engineer, AFRL/RQVC, AIAA Senior Member. †

1 American Institute of Aeronautics and Astronautics Copyright © 2017 by Douglas F. Hunsaker and Warren F. Phillips. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

Hunsaker, Phillips, and Joo

nm S Sb

tmax V∞ W Wr Wn Ws ~ Wn ~ Ws w wi ww z

γ

δ δ max θ

κW ρ

σ max

= limiting load factor at the maneuvering-flight design limit = wing planform area = proportionality coefficient between the weight of the wing structure per unit span and the wing section bending moment, Eq. (15) = maximum airfoil section thickness = freestream airspeed = gross weight = weight of the non-structural components carried at the wing root = total weight of the non-structural components = total weight of the wing structure = weight of non-structural components per unit span carried within the wing = weight of the wing structure per unit span = width of the beam cross-section = inside width of the box beam cross-section = web width (i.e., horizontal thickness) of the I-beam cross-section = spanwise coordinate relative to the midspan = specific weight of the beam material = wing deflection = maximum wing deflection = change of variables for the spanwise coordinate, Eq. (1) = weight distribution coefficient, Eq. (35) = air density = maximum longitudinal stress

I. Introduction

F

ROM Prandtl’s classical lifting-line theory, an arbitrary spanwise section-lift distribution can be written as a Fourier sine series. Combining the Kutta–Joukowski law3,4 with the classical lifting-line solution for the spanwise section-circulation distribution yields 1,2

∞ ~ L (θ ) = 2bρV∞2 ∑ An sin( nθ ); n =1

θ ≡ cos −1 ( − 2 z b )

(1)

For a finite wing with no sweep or dihedral immersed in a uniform flow, Prandtl’s classical lifting-line equation relates this section-lift distribution to the spanwise chord-length and aerodynamic-angle-of-attack distributions. For any given wing planform, Prandtl’s lifting-line equation can be used to obtain the spanwise geometric- and/or aerodynamic-twist distribution required to produce any desired spanwise section-lift distribution.5–10 From the definitions of wing lift coefficient and aspect ratio, the classical lifting-line solution for the total wing lift coefficient10 can be written in terms of only the first Fourier coefficient in the sine series on the right-hand side of Eq. (1), i.e.,

π b 2A1 L = 2 1 S 2 ρ V∞ S

(2)

2 2 1 2 π b ρ V∞ A1

(3)

This can be rearranged to give

L =

From the definitions of wing drag coefficient and aspect ratio, the classical lifting-line solution for the induced drag coefficient10 can be written as 2 American Institute of Aeronautics and Astronautics

Hunsaker, Phillips, and Joo

πb2 Di = 2 1 S 2 ρ V∞ S

∞

∑ nAn2

(4)

n =1

With the application of Eq. (3), Eq. (4) can be rearranged to give

Di =

2( L b ) 2 π ρ V∞2

∞

A2

n =1

1

∑ n An2

(5)

From Eqs. (1) and (3), it is convenient to write an arbitrary dimensionless spanwise section-lift distribution as

~ ∞   bL (θ ) = 4 sin(θ ) + ∑ Bn sin( nθ ) ; π L n=2 

Bn ≡

An , A1

z ≡ −

b cos(θ ) 2

(6)

In steady level flight, the total wing lift L must equal the gross weight W. Thus, using the definition of Bn from Eq. (6), for steady level flight, Eq. (5) requires

Di =

∞  2(W b ) 2  1 + ∑ nBn2  2  π ρ V∞  n=2 

(7)

For a fixed weight and wingspan, this induced drag is minimized with the section-lift distribution having Bn = 0 for all n > 1. From Eq. (6), this yields the well known elliptic lift distribution introduced by Prandtl,2 i.e.,

~ bL (θ ) b cos(θ ) = 4 sin(θ ); z ≡ − π L 2

or

~ bL ( z ) = 4 1 − (2 z b) 2 L π

(8)

From Eq. (7), the induced drag in steady level flight for the elliptic lift distribution is

Di =

2(W b ) 2 π ρ V∞2

(9)

For a wing with any prescribed lift and wingspan, the elliptic lift distribution given in Eq. (8) will always result in minimum induced drag. However, when designing a wing to minimize the induced drag in steady level flight, imposing the constraints of prescribed gross weight and prescribed wingspan does not yield an absolute minimum in the induced drag. From Eq. (7), it is well known that the induced drag in steady level flight can be decreased by increasing the wingspan and/or decreasing the gross weight. Obviously, to minimize the induced drag for any given lift distribution, we want to design an airplane with the wingspan as large as possible and the gross weight as small as possible, within the existing design constraints. However, we cannot increase the wingspan arbitrarily, because for any fixed lift and weight distributions, increasing the wingspan increases the weight of the wing structure required to support the wing section bending moments. Hence, for any given lift distribution and wing structural design; there is an optimum wingspan for minimizing the induced drag, which is based on the tradeoff between wingspan and gross weight. Furthermore, any section-lift distribution that produces lower wing section bending moments than those produced by the elliptic lift distribution will allow the implementation of a larger wingspan for a given gross weight. Because minimum possible gross weight is a function of both the wingspan and the lift distribution, designing a wing to minimize the induced drag in steady level flight requires solving a variational problem in which the lift distribution, wingspan, and gross weight are all allowed to vary. The variational problem associated with designing a wing, which yields an absolute minimum in induced drag, was first considered by Ludwig Prandtl in 1933.11 In this paper, Prandtl emphasized that the elliptic lift distribution yields minimum induced drag only under the somewhat arbitrary constraints of prescribed lift and wingspan. In the same paper, Prandtl also conceded that it would be very difficult to formulate and solve the variational problem associated with attaining an absolute minimum in induced drag with no constraints placed on the lift distribution, wingspan, or gross weight. Instead, Prandtl set forth the simpler task of obtaining an analytical solution for the lift 3 American Institute of Aeronautics and Astronautics

Hunsaker, Phillips, and Joo

distribution, which minimizes the induced drag under the constraints of prescribed gross lift and prescribed moment of inertia of gross lift, but with no constraint placed on the wingspan. Prandtl’s solution11 for minimizing induced drag under these particular constraints yields the dimensionless section-lift distribution given by

~ bL ( z ) = 16 [1 − ( 2 z b ) 2 ]3 2 L 3π

(10)

Using the same trigonometric change of variables presented in Eq. (1), the section-lift distribution given in Eq. (10) can be written equivalently as

~ bL (θ ) = 16 sin 3(θ ); L 3π

z ≡ −

b cos(θ ) 2

(11)

After applying the trigonometric identity 4 sin3θ = 3 sinθ − sin3θ , Prandtl’s 1933 lift distribution can be written in exactly the same form as Eq. (6), i.e.,

~ bL (θ ) = 4 [sin(θ ) − 13 sin(3θ )]; L π

z ≡ −

b cos(θ ) 2

(12)

Comparing Eq. (12) with Eq. (6), yields B2 = 0, B3 = −1 3, and Bn = 0 for all n > 3. Using these Fourier coefficients in Eq. (7) yields the induced drag in steady level flight for Prandtl’s 1933 lift distribution

Di =

8(W b ) 2 3π ρ V∞2

(13)

It should be emphasized that Prandtl made no claim that the lift distribution given in Eq. (10) yields an absolute minimum in induced drag for any specific case of a physical wing.11 This lift distribution minimizes induced drag only under the particular constraints of prescribed gross lift and prescribed moment of inertia of gross lift. It would be very difficult to argue that these constraints are any less arbitrary than the constraints of prescribed gross lift and prescribed wingspan, for which the elliptic lift distribution yields minimum induced drag. Prandtl11 developed the non-elliptic lift distribution presented in Eq. (10) as that which minimizes induced drag under the assumptions that 1. the weight of the wing structure per unit span at each section of the wing is proportional to the wing section bending moment at that wing section; 2. the proportionality coefficient between the weight of the wing structure per unit span and the wing section bending moment is independent of the spanwise coordinate; 3. the wingspan is allowed to vary, while holding constant the gross weight and the proportionality coefficient between the weight of the wing structure per unit span and the wing section bending moment; and 4. the wing section bending moment at each section of the wing is a function of only the lift as given by

~ M b ( z) =

b 2

~

∫ L ( z ′)( z ′ − z ) dz ′,

for z ≥ 0

(14)

z ′= z

The gross weight of an airplane can be divided into two parts; the weight of the wing structure, denoted here as Ws, and the weight of the non-structural components, denoted here as Wn. Let the weight of the wing structure per ~ unit span at the spanwise coordinate z be denoted as Ws ( z ). From Prandtl’s assumption11 that the weight of the wing structure per unit span at each section of the wing is proportional to the wing section bending moment at that wing section, we can write

~ ~ M b ( z) Ws ( z ) = Sb 4 American Institute of Aeronautics and Astronautics

(15)

Hunsaker, Phillips, and Joo

where Sb is a proportionality coefficient with the units of length squared. From Prandtl’s assumption11 that the wing section bending moment at each section of the wing is given by Eq. (14), Eq. (15) can be written as

~ Ws ( z ) = 1 Sb

b 2

~

∫ L ( z ′)( z ′ − z ) dz ′,

for z ≥ 0

(16)

z ′= z

The total weight of the wing structure is obtained from the integral

Ws =

b 2

~

∫ Ws ( z ) dz

(17)

z = −b 2

Both the elliptic lift distribution given in Eq. (8) and Prandtl’s 1933 lift distribution given in Eq. (10) are spanwise symmetric. Hence, with the application of Prandtl’s assumption11 that the proportionality coefficient between the weight of the wing structure per unit span and the wing section bending moment is independent of the spanwise coordinate, Eqs. (16) and (17) yield

Ws = 2 Sb

b 2 b 2

~

∫ ∫ L ( z ′)( z ′ − z ) dz ′dz

(18)

z = 0 z ′= z

Because the total lift L, in steady level flight, must equal the gross weight W, Eq. (18) can be written equivalently as b 2 b 2 ~ bL ( z ′) ( z ′ − z ) dz ′dz Ws = 2W ∫ ∫ S b b z = 0 z ′= z L

(19)

Using the lift distribution from Eq. (8) in Eq. (19) and carrying out the integration, the total weight of the wing structure required to support the elliptic lift distribution under the constraints of Prandtl’s assumptions11 can be expressed as 2 Ws = Wb 32 S b

(20)

Similarly, using Eq. (10) in Eq. (19), the total weight of the wing structure required to support Prandtl’s 1933 lift distribution under the constraints of his assumptions11 is 2 Ws = Wb 48 S b

(21)

Comparing Eqs. (20) and (21), we see that, for a given weight, the ratio of the wingspan allowed by Prandtl’s 1933 lift distribution to that allowed by the elliptic lift distribution is (3 2)1 2. In other words, within the constraints of his assumptions, Prandtl’s 1933 lift distribution allows a 22.5 percent increase in the wingspan over that allowed by the elliptic lift distribution for the same gross weight. Combining Eqs. (20) and (21) with Eqs. (9) and (13), we find that, for a given weight, the ratio of the induced drag produced by Prandtl’s 1933 lift distribution to that produced by the elliptic lift distribution is 8 9. That is, within the constraints of Prandtl’s assumptions, his 1933 lift distribution produces 11.1 percent less induced drag than that produced by the elliptic lift distribution for the same gross weight. Nevertheless, we should not lose sight of the fact that a direct comparison between Eqs. (9) and (13) reveals that Prandtl’s 1933 lift distribution produces 33.3 percent more induced drag than that produced by the elliptic lift distribution, for the same gross weight and wingspan. It is also important to remember that, when designing a wing, the weight of the wing structure is not determined by the bending moments encountered in steady level flight. The constraining loads are typically applied to the wing either during a hard landing or during high-load-factor maneuvers. Furthermore, the designer is not constrained to 5 American Institute of Aeronautics and Astronautics

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using a single lift distribution during all flight phases. Variable geometric and/or aerodynamic twist can be used to implement different lift distributions during different flight phases.6,7,9,12–14 For example, Prandtl’s 1933 lift distribution could be implemented during landing and high-load-factor maneuvers; and the elliptic lift distribution could be implemented during steady level flight. Within the constraints of Prandtl’s assumptions,11 this would allow a 22.5 percent increase in the wingspan over that allowed by a fixed elliptic lift distribution, without increasing the gross weight or imposing any induced-drag penalty during steady level flight. Within the accuracy of the associated assumptions, implementing this variable lift distribution would result in a 25 percent reduction in induced drag over that produced by a fixed-twist design using Prandtl’s 1933 lift distribution; and it would result in a 33.3 percent reduction in induced drag over that produced by a design using a fixed elliptic lift distribution. Unfortunately, the assumptions made by Prandtl11 in the development of Eq. (10) are extremely constraining. In general, the total weight of the wing structure required to support the wing bending-moment distribution is a complex function of wingspan. The proportionality coefficient between the weight of the wing structure per unit span and the wing section bending moment depends on the airfoil section thickness. Even for a rectangular wing, the airfoil section thickness depends on the wingspan and planform area. If the airfoil chord length and thickness vary with the spanwise coordinate, then the proportionality coefficient between the weight of the wing structure per unit span and the wing section bending moment will also vary with the spanwise coordinate. The most constraining of Prandtl’s assumptions is that the wing section bending moment at each section of the wing is a function of only the lift as given by Eq. (14).11 In general, the wing bending-moment distribution depends on both the lift distribution and the weight distribution. Even if all non-structural components within the wing were weightless, the weight of the wing structure is distributed over the wingspan; and, in steady level flight, the wing section bending moment at each section of the wing would be given by

~ M b ( z) =

b 2

~

~

∫ [ L ( z ′) − Ws ( z ′)]( z ′ − z ) dz ′,

for z ≥ 0

(22)

z ′= z

which differs significantly from Eq. (14). More typically, a significant portion of the weight of the non-structural components is also distributed over the wingspan. For example, engines are commonly mounted on the wing and fuel is usually distributed in wing tanks outboard of the wing root. Let the weight per unit span of non-structural components carried within the wing at the ~ spanwise coordinate z be denoted as Wn ( z ). The total weight of non-structural components is then given by

Wn = Wr +

b 2

~

∫ Wn ( z ) dz

(23)

z = −b 2

where Wr is the weight of the non-structural components carried at the wing root. At every section of a wing, the wing structure must be sufficient to support the wing bending moment at that particular wing section. For any given section lift distribution and normal acceleration, the section bending-moment distribution, including the effects of wing mass, is given by

~ M b ( z) =

b 2

~

~

~

∫ [ L ( z ′) − naWn ( z ′) − naWs ( z ′)]( z ′ − z ) dz ′,

for z ≥ 0

(24)

z ′= z

where na is the load factor in g’s. For typical weight distributions, the bending moment predicted from Eq. (24) differs substantially from that predicted from Eq. (14). Because the relation used by Prandtl for computing the wing section bending-moment distribution11 does not accurately account for wing mass, the resulting minimum-drag analysis does not apply to a wing with an arbitrary weight distribution. However, as will be demonstrated in the following section, there is one particular weight distribution for which Eq. (24) reduces to a result proportional to that given by Eq. (14). Approaches similar to that of Prandtl have been taken by others to find analytical solutions to this complex, variational, optimization problem. For example, Jones15 looked at minimizing the induced drag for a given lift and bending moment. Jones’s results show that an optimum lift distribution for a given root bending moment results in an increase in span of 15% and a decrease in induced drag of 15% from that of an elliptic lift distribution having the 6 American Institute of Aeronautics and Astronautics

Hunsaker, Phillips, and Joo

same total lift and root bending moment. Klein and Viswanathan16,17 have also considered the problem of a given lift and root bending moment,16 and have extended the theory to include a given structural weight.17 More recently, Bowers18 has revisited Prandtl’s11 work and evaluated the application of his 1933 lift distribution for avian flight. It is interesting to note that Prandtl’s 1933 lift distribution has been found to have the favorable property of producing proverse yaw.18

II. Minimizing Induced Drag for Stress-Limited Designs The wing structure at each section of a wing must be sufficient to support the wing bending-moment distribution at the design limits for both maneuvering flight and a hard landing. For the maneuvering-flight design limit, the limiting load factor will be denoted here as nm and the total lift is given by

L ≡

b 2

~

∫ L ( z ) dz

b 2

= nmWr +

z = −b 2

~

~

∫ nm [Wn ( z ) + Ws ( z )]dz

= nmW

(25)

z = −b 2

Then from Eq. (24), the wing bending-moment distribution for the maneuvering-flight design limit is

~ M b ( z ) = nm

  L~ ( z ′) ~ ~ ∫ W L − Wn ( z ′) − Ws ( z ′)  ( z ′ − z ) dz ′, for z ≥ 0  z ′= z  b 2

(26)

Similarly, for the hard-landing design limit, the limiting load factor will be denoted here as n g and the total lift is b 2

~

L ≡

∫ L ( z ) dz

z = −b 2

= Wr +

b 2

~

~

∫ [Wn ( z ) + Ws ( z )]dz

= W

(27)

z = −b 2

Thus, the wing bending-moment distribution from Eq. (24) for the hard-landing design limit is

~ M b ( z ) = − ng

~ ~ ~ W L ( z ′)  ( z ′ − z ) dz ′, for z ≥ 0 ′ ′ ( ) ( ) + − W z W z s ∫  n n g L  z ′= z  b 2

(28)

The wing bending moment is typically positive for the maneuvering-flight design limit and negative for the hardlanding design limit. Obviously, negative bending moments do not correspond to negative structure weight. It is the absolute value of the bending moment that determines the wing structure. Either Eq. (26) or Eq. (28) could provide the structural design limit, depending on the weight distribution. Clearly, the constraining wing bending-moment distribution depends heavily on the weight distribution as well as the lift distribution. For example, if only the maneuvering-flight design limit were considered, then the section bending-moment distribution could be minimized with Wr = 0 and the weight of all non-structural components distributed over the wingspan such that

~ ~ L ( z) ~ Wn ( z ) = W − Ws ( z ) L

(29)

Using Eq. (29) in Eq. (26), for this weight distribution, we see that the section bending moment vanishes completely at every section of the wing, independent of the gross weight and load factor. However, this weight distribution is far from optimal when the hard-landing design limit is also considered. Because the wing bending-moment distribution depends on the weight distribution, the variational problem associated with minimizing induced drag for an arbitrarily specified weight distribution, with no constraint placed on the wingspan, will most likely need to be solved numerically. Although this can be done, the existence of Prandtl’s analytical solution,11 which is based on the bending-moment distribution given in Eq. (14), raises an important question. Is there any particular weight distribution for which the bending-moment distributions given in Eqs. (26) and (28) both yield a result proportional to that given in Eq. (14)? To answer this question, consider a weight distribution where the non-structural components carried within the wing are distributed so that 7 American Institute of Aeronautics and Astronautics

Hunsaker, Phillips, and Joo

~ ~ L ( z) ~ − Ws ( z ) Wn ( z ) = (W − Wr ) L

(30)

~ Equation (30) alone does not completely specify the weight distribution Wn ( z ). It simply provides one relation ~ ~ ~ between the five design parameters, Wn ( z ), W, Wr, Ws ( z ), and L ( z ) L. Equation (30) could be applied in the early ~ stages of preliminary design, if no other constraint is placed on the weight distribution. However, Wn ( z ) can not be evaluated from Eq. (30) until the other four parameters in Eq. (30) have been determined from other means. Using Eq. (30) in Eq. (26), the section bending-moment distribution at the maneuvering-flight design limit is

~ M b ( z ) = nmWr

~ L ( z ′) ∫ L ( z ′ − z ) dz ′, for z ≥ 0 z ′= z b 2

(31)

Similarly, using Eq. (30) in Eq. (28), the section bending-moment distribution at the hard-landing design limit is

~ M b ( z ) = − [( n g − 1)W − n gWr ]

~ L ( z ′) ∫ L ( z ′ − z ) dz ′, for z ≥ 0 z ′= z b 2

(32)

Equating the absolute value of the section bending-moment obtained from Eq. (31) to that obtained from Eq. (32) yields the optimum value of Wr for the weight distribution specified by Eq. (30), i.e.,

Wr =

ng − 1 W nm + n g

(33)

This gives the optimum weight distribution, which minimizes the bending moment required for the constraining design limit. Using Eq. (33) in Eq. (32) yields a bending-moment distribution for the hard-landing design limit, which is exactly the negative of that required for the maneuvering-flight design limit. If Wr is larger than the value given by Eq. (33), then the limiting bending-moment distributions obtained from Eqs. (31) and (32) become more positive and Eq. (31) provides the structural design limit. On the other hand, if Wr is smaller than the value given by Eq. (33), the limiting bending-moment distributions from Eqs. (31) and (32) become more negative and Eq. (32) provides the structural design limit. In either case, if the lift is positive over the entire semispan, the structural design limit for the wing bending moment can be written as

~ M b (z) =

where

κW

~ L ( z ′) ( z ′ − z ) dz ′, for z ≥ 0 L z ′= z b 2

κW Wr ∫

ng − 1  W Wr ≥ nm , nm + n g ≡  ( n g − 1) W − n g , Wr < n g − 1 W nm + n g Wr 

(34)

(35)

For any given structural design, the required weight of the wing structure per unit span is a known function of the absolute value of the wing section bending moment at the structural design limit. For example, if the section bending moment is supported by any vertically symmetric beam and the airfoil chord length and thickness vary with the spanwise coordinate z, the maximum longitudinal stress in any cross-section of the beam can be written as

σ max =

h( z ) ~ M b ( z) 2I ( z)

(36)

where h is the height of the beam cross-section and I is the section moment of inertia. Minimum structural weight for any given maximum allowable stress is attained when, at the design limit, the maximum stress at each section of the wing is equal to the maximum allowable stress. For a vertically symmetric beam, this requires that the section moment of inertia varies with the spanwise coordinate z such that 8 American Institute of Aeronautics and Astronautics

Hunsaker, Phillips, and Joo

I ( z) =

h( z ) ~ M b ( z) 2σ max

(37)

The weight of any beam per unit span is the beam cross-sectional area A multiplied by the specific weight of the beam material γ , i.e.,

~ W s ( z ) = γ A( z )

(38)

Combing Eqs. (37) and (38), the weight of the beam per unit span could be written as

~ γ A( z ) h ( z ) ~ Ws ( z ) = M b ( z) 2σ max I ( z )

(39)

The cross-sectional area for a given cross-section increases with the product of the width w and the height h; whereas the moment of inertia increases with the product of the width and the height cubed, wh3. Hence, it is convenient to scale the structural cross-section such that A h2 I remains constant along the span, and Eq. (39) can be written as 2 ~ ~ γ Ws ( z ) = Ah M b ( z) 2 I h ( z )σ max

(40)

Because the airfoil thickness is typically a fixed fraction of the chord length and the beam height must fit within the thickness of the airfoil at each section of the wing, it is convenient to write Eq. (40) as

~ ~ | M b ( z) | 2 I ( h t max ) C (t c ) c ( z )σ max , Cσ ≡ ; S b ( z ) ≡ σ max Ws ( z ) = Sb ( z ) γ Ah 2

(41)

where c(z) is the airfoil section chord length distribution and t max c is the maximum-thickness-to-chord-length ratio for the airfoil sections. The shape coefficients Cσ for three common beam cross-sections are rectangular beam: Cσ = I-beam: Cσ =

(1 − wi hi3 wh 3 )( h t max ) ( h t max ) , , box beam: Cσ = 6(1 − wi hi wh ) 6

[ 2( h f h ) 3 + 6( h f h )(1 − h f h ) 2 + ( ww w)(1 − 2 h f h ) 3 ]( h t max ) 6[ 2 h f h + ( ww w)(1 − 2 h f h )]

For the box beam, wi is the inside width and hi is the inside height; and for the I-beam, ww is the web width and hf is the flange height (i.e., its vertical thickness). Using Eq. (41) in Eq. (17), for a maximum-stress constraint with spanwise symmetric wing loading, the total weight of the wing structure required to support the bending moment distribution at the design limit can be expressed as

Ws = 2

b 2

∫

z =0

~

| M b ( z) | Sb ( z )

dz ; S b ( z ) ≡

Cσ (t max c ) c ( z )σ max

γ

, Cσ ≡

2 I ( h t max ) Ah 2

(42)

We see from Eq. (42) that, for any spanwise symmetric wing loading, the weight of the wing structure required to support a maximum-stress constraint is proportional to the integral of the bending-moment distribution divided by the chord-length distribution. Because the relation used by Prandtl for computing the weight of the wing structure is based on the assumption that Sb is independent of z,11 the resulting minimum-drag analysis does not apply to the stress-limited design of any wing with a chord length and thickness that vary with the spanwise coordinate. However, Prandtl’s 1933 minimum-drag analysis11 could be applied to the stress-limited design of a rectangular wing with the weight distribution specified by Eq. (30). 9 American Institute of Aeronautics and Astronautics

Hunsaker, Phillips, and Joo

Using Eq. (34) in Eq. (42), for the stress-limited design of a rectangular wing with spanwise symmetric wing loading and the weight distribution specified by Eq. (30), if the lift is positive over the entire semispan, the weight of the wing structure is found to be proportional to that obtained from the relation used by Prandtl11 and presented here in Eq. (18), i.e.,

Ws =

2κ W Wr Sb

~ L ( z ′) Cσ t maxσ max ∫ ∫ L ( z ′ − z ) dz ′dz; Sb ≡ γ z = 0 z ′= z b 2 b 2

(43)

Using Eq. (6) in Eq. (43) yields

Ws =

κW Wr b 2 π Sb

π

π

∞   sin(θ ′) + ∑ Bn sin( nθ ′) [cos(θ ) − cos(θ ′)] sin(θ ) sin(θ ′) dθ ′dθ n=2  2 θ ′=θ 

∫ ∫

θ =π

(44)

Using the trigonometric identity sin(2θ ) = 2 cos(θ ) sin(θ ), Eq. (44) can be written

Ws =

κW Wr b 2 2π S b

π

π

∞   sin(θ ′) + ∑ Bn sin( nθ ′) [sin( 2θ ) sin(θ ′) − sin( 2θ ′) sin(θ )]dθ ′dθ n=2  2 θ ′=θ 

∫ ∫

θ =π

(45)

For any spanwise symmetric lift distribution, the Fourier coefficients Bn are exactly zero for all even n. Integration of the odd terms in Eq. (45) is easily shown to give exactly zero for all terms with n > 3. Finally, if the lift is positive over the entire semispan, integration of the remaining terms gives

Ws =

κW Wr b 2 (1 + B 32 S b

3)

(46)

Notice from Eq. (7) that all of the Fourier coefficients Bn make a positive contribution to the induced drag. However, we see from Eq. (46) that only B3 contributes to the required weight of a rectangular wing with spanwise symmetric wing loading and the weight distribution specified by Eq. (30). For a fixed weight and wingspan, Eq. (7) shows that minimum induced drag is attained with an elliptic lift distribution, i.e., Bn = 0 for all n > 1. Minimizing induced drag for a rectangular wing with spanwise symmetric wing loading and the weight distribution specified by Eq. (30) requires a lift distribution having Bn = 0 for all n > 3, because these terms make positive contributions to the drag but do not contribute to the required weight. Equation (46) shows that a lift distribution having B3 < 0 could produce less induced drag than the elliptic lift distribution, because it allows a larger wingspan for a given structural weight. Rearranging Eq. (46), for the stress-limited design of a rectangular wing with an all-positive spanwise symmetric lift distribution and the weight distribution specified by Eq. (30), we obtain

b2 =

32 S bWs

κW Wr (1 + B3 )

(47)

Using Eq. (47) in Eq. (7) gives

Di =

κW Wr

16π ρ V∞2

∞ W 2 (1 + B )1 + nB 2  3  n  ∑ S bWs   n=2

(48)

Clearly, the induced drag predicted from Eq. (48) is minimized if Bn = 0 for n = 2 and all n > 3. Enforcing this condition on Eq. (48) yields

Di =

κW Wr

16π ρ V∞2

W 2 (1 + B + 3 B 2 + 3 B 3 ) 3 3 3 S bWs

10 American Institute of Aeronautics and Astronautics

(49)

Hunsaker, Phillips, and Joo

If the weight and the coefficient Sb are held constant while the wingspan is allowed to vary, then the value of B3 that minimizes the induced drag predicted from Eq. (49) is obtained from the simple quadratic relation

1 + 6 B3 + 9 B32 = (1 + 3 B3 ) 2 = 0

(50)

Hence, if the weight and the coefficient Sb are held constant, the induced drag predicted from Eq. (49) is minimized using a lift distribution having

B2 = 0; B3 = −1 3; Bn = 0, for n > 3

(51)

Using Eq. (51) in Eqs. (6), (47), and (48) yields the all-positive spanwise symmetric lift distribution

~ bL (θ ) = 4 [sin(θ ) − 13 sin(3θ )]; L π b =

Di =

z ≡ −

b cos(θ ) 2

48 S bWs κW Wr

κW Wr

18π ρ V∞2

(52) (53)

W2 S bWs

(54)

The lift distribution given in Eq. (52) is exactly Prandtl’s 1933 lift distribution11 as arranged in Eq. (12). Because this lift distribution was obtained by minimizing Di with respect to B3 for constant Sb, we see from Eq. (43) that minimizing induced drag with this lift distribution requires holding the airfoil thickness constant as the wingspan is varied. Because airfoil thickness is typically a fixed fraction of the chord length and c = S b for a rectangular wing, this requires holding the chord length constant and increasing the planform area as the wingspan is increased. Prandtl developed his 1933 lift distribution11 as that which minimizes the induced drag under the constraints of prescribed gross lift and prescribed moment of inertia of gross lift, but with no constraint placed on the wingspan. However, Prandtl did not relate his constraints and assumptions to the weight distribution and geometry of a physical wing. From the analysis presented here, it has been shown that Prandtl’s 1933 lift distribution11 does not minimize induced drag for an arbitrary wing planform, or even for a rectangular wing with an arbitrary weight distribution. It does, however, minimize induced drag for a rectangular wing with spanwise symmetric lift and the weight distribution specified by Eq. (30), under the constraints of prescribed gross weight, prescribed maximum stress, and prescribed chord length. The wing loading for an airplane, W S , is typically fixed by airspeed requirements. Hence, for most stresslimited designs, it is more appropriate to minimize induced drag under the constraint of prescribed wing loading than under the constraint of prescribed chord length. This leads to an optimum lift distribution that does not match either the elliptic lift distribution or Prandtl’s 1933 lift distribution.11 For the stress-limited design of a rectangular wing with the weight distribution specified by Eq. (30) and any all-positive spanwise symmetric lift distribution, the total weight of the wing structure required to support the bending-moment distribution at the design limit is given by Eq. (46). Using the definition of Sb from Eq. (43) in Eq. (46) yields

Ws =

κW Wr b 2γ

32Cσ t maxσ max

(1 + B3 )

(55)

For the variational problem associated with minimizing induced drag under the constraint of prescribed wing loading, it is convenient to use the relation c = S b and rearrange Eq. (55) as

Ws =

γ (W S ) κW Wr b 3 (1 + B ) 3 32Cσ (t max c )σ max W

11 American Institute of Aeronautics and Astronautics

(56)

Hunsaker, Phillips, and Joo

For a given structural design, the first terms on the right-hand side of Eq. (56) are typically independent of both the weight and the wingspan. To express the induced drag in terms of the weight and the coefficient B3, it is convenient to rearrange Eq. (56) in terms of the ratio W b, i.e.,

(W b ) 3 =

(1 + B3 )γ (W S ) 32Cσ (t max c )σ max

κW WrW 2

(57)

Ws

Here again, because all of the Fourier coefficients Bn make a positive contribution to the induced drag but only B3 contributes to the structure weight, the induced drag predicted from Eq. (7) is minimized if Bn = 0 for n = 2 and all n > 3. Enforcing this condition while using Eq. (57) to eliminate the ratio W b from Eq. (7) yields

2(1 + 3 B32 )  (1 + B3 )γ (W S ) Di =  π ρ V∞2  32Cσ (t max c )σ max

κW WrW 2 

23

(58)

 

Ws

The variation of this drag with B3 is proportional to (1 + 3 B32 )(1 + B3 ) 2 3. For fixed weight and wing loading, the value of B3 that minimizes the induced drag predicted from Eq. (58) is obtained from the quadratic relation

1 + 9 B3 + 12 B32 = 0

(59)

Equation (59) has two real roots. Choosing the root that minimizes the function (1 + 3 B32 )(1 + B3 ) 2 3, we find that the induced drag predicted from Eq. (58) is minimized using a lift distribution having

B2 = 0; B3 = − 3 8 + 9 64 − 1 12 ; Bn = 0, for n > 3

(60)

Using Eq. (60) in Eqs. (6), (56), and (58); for a rectangular wing with all-positive spanwise symmetric lift and the weight distribution specified by Eq. (30), under the constraints of prescribed gross weight, prescribed maximum stress, and prescribed wing loading; we obtain the optimum results

~ bL (θ ) = 4 [sin(θ ) − 0.13564322 sin(3θ )]; L π b =

3

32Cσ (t max c )σ max

z ≡ −

b cos(θ ) 2

WsW

(61) (62)

0.86435678γ (W S ) κ W Wr

 0.86435678 γ (W S ) κ W WrW 2  Di = 2.11039450   Ws π ρ V∞2  32Cσ (t max c )σ max 

23

(63)

For this same wing geometry, weight distribution, and design constraints, the elliptic lift distribution results in the wingspan and induced drag given by

b =

Di =

2 π ρ V∞2

3

32Cσ (t max c )σ max WsW γ (W S ) κW Wr

 γ (W S )  32C (t σ max c )σ max 

(64)

κW WrW 2  Ws

23

 

and Prandtl’s 1933 lift distribution11 results in

12 American Institute of Aeronautics and Astronautics

(65)

Hunsaker, Phillips, and Joo

b =

Di =

3

8 3π ρ V∞2

48Cσ (t max c )σ max WsW γ (W S ) κW Wr

 γ (W S )  48C (t σ max c )σ max 

(66)

κW WrW 2  Ws

23

 

(67)

Comparing Eqs. (62) and (63) with Eqs. (64) and (65), we see that, for this wing geometry, weight distribution, and design constraints, the lift distribution given by Eq. (61) results in a 4.98 percent increase in the wingspan and a 4.25 percent decrease in the induced drag over that obtained for the elliptic lift distribution. On the other hand, comparing Eqs. (66) and (67) with Eqs. (64) and (65), Prandtl’s 1933 lift distribution results in a 14.5 percent increase in the wingspan and a 1.75 percent increase in the induced drag over that obtained for the elliptic lift distribution, for the same gross weight. Prandtl’s 1933 lift distribution11 given in Eq. (52) and the lift distribution given in Eq. (61) were developed to minimize induced drag under different design constraints. However, both of these lift distributions were developed for the weight distribution specified by Eq. (30). Although this weight distribution is that which minimizes the wing section bending moments, it is not always practical to maintain this weight distribution due to required engine placement, fuel tank requirements, and/or other design constraints. In general, numerical methods must be employed to evaluate optimum lift distributions that minimize induced drag under a prescribed maximum-stress constraint for more complex weight distributions.

III. Minimizing Induced Drag for Deflection-Limited Designs Although some maximum-stress constraint must always be satisfied for any wing design, the structural design of a wing is not always stress limited. Sometimes wing deflection imposes the critical constraint on wing design. This leads to an optimum lift distribution different from the elliptic lift distribution, Prandtl’s 1933 lift distribution,11 and the lift distribution given by Eq. (61). As is the case for constraining stresses, constraining deflections typically occur either during high-load-factor maneuvers or during a hard landing. For example, consider the deflection-limited design of a rectangular wing with spanwise symmetric lift and the weight distribution specified by Eq. (30). For this weight distribution, the structural design limit for the wing bending moment distribution is given by equation (34), where κ W is defined in Eq. (35). The deflection of a beam can be obtained from the relation

~ d 2δ ( z ) M b ( z) = EI ( z ) dz 2

(68)

where E is the modulus of elasticity. For spanwise symmetric lift and weight distributions, Eq. (68) is subject to the boundary conditions

δ ( 0) = 0 and dδ

dz

z =0

= 0

(69)

For this particular example, we will use the same wing structure that was used for the stress-limited design in Sec. II. Hence, the section moment-of-inertia distribution is related to the section bending-moment distribution through Eq. (37). If the sign of the bending moment does not change over the wing semispan, for the special case of a rectangular wing supported by a beam of height that does not vary with z, using Eq. (37) in Eq.(68) yields

d 2δ ( z ) 2σ max = 2 Eh dz

(70)

Because, for this particular structural design, the moment of inertia varies with z to maintain constant maximum longitudinal stress across the wingspan, Eq. (70) can be integrated twice subject to the boundary conditions given in Eq. (69) to give the deflection distribution 13 American Institute of Aeronautics and Astronautics

Hunsaker, Phillips, and Joo

δ ( z) =

σ max z 2

(71)

Eh

From this relation, the maximum deflection, which occurs at the wingtip, is

δ max = σ max b 2

(72)

4 Eh

Equation (72) provides a useful relation between the maximum stress and the maximum deflection in any wing supported by a beam with a height and maximum stress that do not vary with the spanwise coordinate z. Rearranging Eq. (72), the maximum stress can be written as a function of the maximum deflection, i.e.,

σ max = 4 Eh δ max 2

(73)

b

Using Eq. (73) and the relation c = S b in Eq. (42), for this particular deflection-limited design we obtain

Ws = 2

b 2

∫

~

| M b ( z) | Sb

z =0

dz ; S b ≡

8 I ( h t max ) 2 Cδ E (t max c ) 2 δ max W 2 , C ≡ δ Ah 2 b4 γ (W S ) 2

(74)

For any structural cross-sections, Cδ can be obtained from Cδ ≡ 4Cσ ( h t max ). We see from Eq. (74) that, for this particular structural design with t max c and W S held constant, S b is not constant for a maximum-deflection constraint, but varies directly with gross weight squared and inversely with the fourth power of the wingspan. Using the definition of Sb from Eq. (74) in Eq. (46), the total weight of the wing structure required to support the bending moment at the design limit for this deflection-limited design of a rectangular wing with all-positive spanwise symmetric lift and the weight distribution specified by Eq. (30) is

Ws =

γ (W S ) 2

32Cδ E (t max c ) 2 δ max

κW Wr b 6 (1 + B

(75)

κW WrW 4

(76)

3)

W2

Rearranging Eq. (75) yields

(W b ) 6 =

(1 + B3 )γ (W S ) 2 32Cδ E (t max c ) 2 δ max

Ws

Using Eq, (76) in Eq. (7) gives

Di =

2 2  (1 + B3 )γ (W S ) 2 2  π ρ V∞  32Cδ E (t max c ) δ max

κW WrW 4  Ws

 

13

∞  2  1 + ∑ nBn    n=2

(77)

If the weight and wing loading are held constant while the wingspan is allowed to vary, then the Fourier coefficients Bn that minimize the induced drag predicted from Eq. (77) are

B2 = 0; B3 = − 3 7 + 9 49 − 1 21; Bn = 0, for n > 3

(78)

Using Eq. (78) in Eqs. (6), (75), and (77); for a rectangular wing with all-positive spanwise symmetric lift and the weight distribution specified by Eq. (30), under the constraints of prescribed gross weight, prescribed maximum deflection, and prescribed wing loading; we obtain the optimum results

~ bL (θ ) = 4 [sin(θ ) − 0.05971587 sin(3θ )]; L π

z ≡ −

b cos(θ ) 2

14 American Institute of Aeronautics and Astronautics

(79)

Hunsaker, Phillips, and Joo

b =

6

32Cδ E (t max c ) 2 δ max WsW 2 0.94028413γ (W S ) 2 κ W Wr

 0.94028413γ (W S ) 2 Di = 2.02139591  2 π ρ V∞  32Cδ E (t max c ) 2 δ max

(80)

κW WrW 4  Ws

13

 

(81)

It should be emphasized that the results presented in Eqs. (79) – (81) do not apply to all deflection-limited wing designs. The development of these relations was presented here simply as an example, to demonstrate how the application of a prescribed maximum-deflection constraint differs from the application of a prescribed maximumstress constraint. As was the case for the stress-limited examples presented in Sec. II, the lift distribution given in Eq. (79) was developed to minimize induced drag for the weight distribution specified by Eq. (30). Again, it is not always practical to maintain this optimum weight distribution. In general, numerical methods must be used to evaluate optimum lift distributions, which minimize induced drag under prescribed maximum-deflection constraints for more practical and/or complex weight distributions.

IV. Minimizing Induced Drag with Wingspan and Wing Weight Minimizing induced drag by varying the wingspan and lift distribution while holding the gross weight constant is not the only variational problem suggested by Eq. (7). Because gross weight increases with increasing wingspan for any fixed lift and weight distributions, Eq. (7) also suggests that the induced drag could be minimized by varying the wingspan and gross weight while holding the lift distribution constant. Because the weight of the wing structure depends on both the wingspan and the lift distribution, in general, Ws depends on b and all of the Fourier coefficients Bn. Because gross weight is simply the sum of the weight of the non-structural components and the weight of the wing structure, for an arbitrary wing design, Eq. (7) can be written

Di =

2  Wn + Ws (b, Bn )    b π ρ V∞2  b 

2

∞   1 + ∑ nBn2    n=2

(82)

For any prescribed non-structural weight, the term Wn b always decreases with increasing wingspan; and for typical design constraints, the term Ws (b, Bn ) b increases with increasing wingspan. For example, the design constraints that led to Prandtl’s 1933 lift distribution yield Ws proportional to b2 as given in Eq. (46); the design constraints that led to the lift distribution given in Eq. (61) yield Ws proportional to b3 as given in Eq. (56); and the design constraints that led to the lift distribution given in Eq. (79) yield Ws proportional to b6 as given in Eq. (75). For any fixed lift and weight distributions, there is an optimum wingspan for minimizing the induced drag, which is based on the tradeoff between wingspan and wing weight. For example, for the stress-limited design of a rectangular wing with the weight distribution specified by Eq. (30) and any all-positive spanwise symmetric lift distribution, the total weight of the wing structure required to support the bending-moment distribution at the design limit is given by Eq. (56). The gross weight is the sum W = Wn + Ws. Hence, using Eq. (56) in Eq. (7), the induced drag can be written as

2  Wn + (1 + B3 )γ (W S ) Di =  π ρ V∞2  b 32Cσ (t max c )σ max

κW Wr b 2  W

2

∞  2  1 + ∑ nBn    n=2 

(83)

For any given value of the ratio κ W Wr W, the function in the square brackets of Eq. (83) can be minimized with respect to b, based on the tradeoff between wingspan and wing weight. To minimize the ratio κ W Wr W for any given wingspan, the weight distribution given by Eq. (33) can be used. The gross weight is W = Wn + Ws. Hence, using Eq. (33) in Eqs. (35) and (56) yields κ W ≡ nm and

W = Wn +

(1 + B3 )γ (W S ) nm ( n g − 1) 3 b 32Cσ (t max c )σ max nm + n g

15 American Institute of Aeronautics and Astronautics

(84)

Hunsaker, Phillips, and Joo

From Eq. (83) the induced drag is

Di =

2  Wn + (1 + B3 )γ (W S ) nm ( n g − 1) b 2    π ρ V∞2  b 32Cσ (t max c )σ max nm + n g 

2

∞   1 + ∑ nBn2     n=2 

(85)

The wingspan that minimizes this induced drag is

b =

3

16Cσ (t max c )σ maxWn nm + n g (1 + B3 )γ (W S ) nm ( n g − 1)

(86)

Using Eq. (86) in Eq. (84), the gross weight that minimizes this induced drag is

W = 3 Wn 2

(87)

Using Eq. (86) in Eq. (85), the associated minimum induced drag is

Di =

9 2π ρ V∞2

 (1 + B3 )γ (W S )Wn2 nm ( n g − 1)     16Cσ (t max c )σ max nm + n g 

23

∞   1 + ∑ nBn2    n=2

(88)

Equation (88) gives the minimum possible induced drag for the stress-limited design of a rectangular wing with the weight distribution specified by Eq. (30) and any given all-positive spanwise symmetric lift distribution. However, even though Eq. (33) was used to minimize the ratio κ W Wr W in Eq. (83), Eq. (88) does not provide an absolute minimum in induced drag for the specified design constraints and weight distribution, unless the optimum lift distribution is also used. From Eq. (88), we see that the variation of this drag with the Fourier coefficients Bn is proportional to (1 + ∑ nBn2 )(1 + B3 ) 2 3 . Minimizing this function yields the Fourier coefficients given in Eq. (60) and the optimum lift distribution given in Eq. (61). Similarly, for the deflection-limited design of this same rectangular wing with any all-positive spanwise symmetric lift distribution and the weight distribution specified by Eqs. (30) and (33), the total weight of the wing structure required to support the bending-moment distribution at the design limit is given by Eq. (75). Hence, using Eqs. (33), (35), and (75) with the relation W = Wn + Ws yields

W = Wn +

(1 + B3 )γ (W S ) 2 nm ( n g − 1) b 6 32Cδ E (t max c ) 2 δ max nm + n g W

(89)

Equation (89) is easily solved for the gross weight to give

W =

(1 + B3 )γ (W S ) 2 nm ( n g − 1) 6 Wn Wn2 + + b 2 4 32Cδ E (t max c ) 2 δ max nm + n g

(90)

Using this gross weight in Eq. (7) gives 2  2 nm ( n g − 1) 4 2  Wn + Wn + (1 + B3 )γ (W S ) Di = b 2  2b 2 2 4b 32Cδ E (t max c ) δ max nm + n g π ρ V∞ 

   

2

∞   1 + ∑ nBn2    n=2

The wingspan that minimizes this induced drag is

16 American Institute of Aeronautics and Astronautics

(91)

Hunsaker, Phillips, and Joo

b =

6

10Cδ E (t max c ) 2 δ maxWn2 nm + n g nm ( n g − 1) (1 + B3 )γ (W S ) 2

(92)

Using Eq. (92) in Eq. (90), the gross weight that minimizes this induced drag is

W = 5 Wn 4

(93)

Using Eq. (92) in Eq. (91), the associated minimum induced drag is 4 2 25  (1 + B3 )γ (W S ) Wn nm ( n g − 1)  Di =   8π ρ V∞2  10Cδ E (t max c ) 2 δ max nm + n g 

13

∞   1 + ∑ nBn2    n=2

(94)

Here again, even though Eq. (33) was used to minimize the structural weight for any given wingspan, Eq. (94) does not provide an absolute minimum in induced drag for the specified design constraints and weight distribution, unless the optimum lift distribution is also used. From Eq. (94), we see that the variation of this drag with the Fourier coefficients Bn is proportional to (1 + ∑ nBn2 )(1 + B3 )1 3. Minimizing this function yields the Fourier coefficients given in Eq. (78) and the optimum lift distribution given in Eq. (79). The results shown in Eqs. (86)–(88) and (92)–(94) are based on implementing the weight distribution given by Eqs. (30) and (33), which minimizes the bending moment required for any given wingspan at the constraining design limit. However, the reader is reminded that this weight distribution is not always practical due to various design constraints. Numerical methods can be used to evaluate the optimum wingspan and wing weight to minimize induced drag for other weight distributions.

V. Conclusions It has been shown that minimum induced drag for steady level flight is not obtained by imposing the constraints of prescribed gross weight and prescribed wingspan. Rather, because the wing structural weight is a function of wingspan and lift distribution, there exists an optimum lift distribution, weight, and wingspan for a given nonstructural weight distribution that minimizes the induced drag given in Eq. (7). This optimum lift distribution can vary significantly from the elliptic lift distribution, depending on design constraints. For example, Prandtl developed his 1933 lift distribution11 as that which minimizes the induced drag under the constraints of prescribed gross lift and prescribed moment of inertia of gross lift, but with no constraint placed on the wingspan. Within the constraints of his assumptions, Prandtl’s 1933 lift distribution given in Eq. (52) allows a 22.5 percent increase in the wingspan and produces 11.1 percent less induced drag than that of the elliptic lift distribution for the same gross weight. An optimum lift distribution subject to the constraints of prescribed gross weight, maximum stress, and wing loading is presented in Eq. (61). This lift distribution results in a 4.98 percent increase in wingspan and a 4.25 percent decrease in induced drag over that obtained for an elliptic lift distribution. Finally, an optimum lift distribution subject to the constraints of prescribed gross weight, maximum deflection, and wing loading is presented in Eq. (79). Each of the optimum lift distributions given in Eqs. (52), (61), and (79) was obtained by assuming a nonstructural weight distribution given in Eq. (30). Note that these solutions differ only in the magnitude of the third Fourier coefficient, B3, in the infinite series used in Eq. (6). As shown in Eq. (46), B3 is the only Fourier coefficient in the infinite series that contributes to the required weight of a rectangular wing with an all-positive spanwise symmetric lift distribution and the weight distribution specified by Eq. (30). It has been shown that there exists an optimum value for the non-structural weight at the wing root that minimizes the bending moment required for both maneuvering flight and landing. This optimum non-structural weight at the root is a function of the load factor to be sustained during maneuvering flight and the load factor encountered during a hard landing. For the weight distribution given in Eq. (30), the optimum non-structural weight at the wing root is given by Eq. (33). Additionally, it has been shown that for any fixed lift and weight distributions, there is an optimum wingspan that minimizes the induced drag. This optimum is based on the tradeoff between wingspan and wing weight. For example, Eq. (86) gives the optimum wingspan for the stress-limited design of a rectangular wing with the weight 17 American Institute of Aeronautics and Astronautics

Hunsaker, Phillips, and Joo

distribution specified by Eq. (30) and any given all-positive spanwise symmetric lift distribution. Equation (92) gives the optimum wingspan for the deflection-limited design of a rectangular wing with the weight distribution specified by Eq. (30) and any given all-positive spanwise symmetric lift distribution. For the analysis presented in this paper, we have considered only rectangular wings. Given a maximum-stress constraint with all-positive spanwise symmetric lift and the weight distribution specified by Eq. (30), this provided the great simplification of allowing us to carry out the integration in Eq. (42) for the arbitrary lift distribution given in Eq. (6) to produce the very simple result for the weight of the wing structure given in Eq. (46). When the airfoil chord length and thickness vary with the spanwise coordinate, we can no longer use Eq. (46) to compute the weight of the wing structure for this maximum-stress constraint. Instead, we must return to the more general relation given in Eq. (42). For arbitrary wing planforms and complex non-structural weight distributions, Eq. (42) must be integrated numerically. Hence, for most practical applications, numerical methods must be used to obtain optimum lift distributions and wingspans that minimize induced drag. However, the solutions presented in this work provide significant insight into the aerodynamic and structural coupling of wing design.

Acknowledgement This work was partially performed during a summer faculty fellowship sponsored by the Air Force Research Laboratory. The authors wish to express their appreciation for the support of AFRL. This paper has been cleared for public release, Case Number: 88ABW-2016-6087.

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18 American Institute of Aeronautics and Astronautics