Aero-Structural Wing Design Optimization for Small Civil Airplane

Hayato Yamaguchi*

Fuji Heavy Industries Ltd., Utsunomiya, 320-8564, Japan

Yoshifumi Ikeda†

Japan Aircraft Development Corporation, Tokyo, 105-0001, Japan

The methodology of aero-structural optimization is developed to reduce wing weight and to improve lift to drag ratio at cruse condition. Aerodynamic database and beam model are used for estimation of aerodynamic characteristics and loads of wing-body configuration with wing deflection. The aerodynamic database contains the sensitivities of design variables to aerodynamic coefficients and load distributions computed by Euler code. The beam model includes mass and stiffness for wing box. Cross-sectional areas of upper/lower panels, front/rear spars, and ribs are determined to resist critical loads. The stiffness for bending and twist is estimated by using the results of wing box sizing. Wing bending and twist are computed with mass and stiffness distributions. Statistical methods are applied to estimation of weight and drag for components except wing. This methodology can optimize lift distribution on elastic wing to minimize takeoff weight of airplane by modifying twist and camber distribution. The developed methodology is applied to aero-structural optimization of small civil airplane.

Nomenclature AMk = kth term coefficient of approximated function for bending moment BB

Dpk, BMk = kth term coefficient of approximated function for sensitivity CD = drag coefficient CDp,j = pressure drag coefficient at jth angle of attack CDp,0j = pressure drag coefficient of baseline at jth angle of attack CL = lift coefficient Cm = pitching moment coefficient ci = ith design variable for camber distribution Fl = allowable stress for lower panel of wing box h0.25, h0.8 = Hicks-Henne functions L/Dcr = lift to drag ratio at cruise Mj = bending moment at jth angle of attack n = load factor Pi = ith design variable q = dynamic pressure R = design range of airplane S = wing area ti = ith design variable for twist distribution W = weight of airplane z = camber distribution z0 = baseline camber distribution α = angle of attack φ = twist distribution

* Aerodynamics Engineer, Research & Laboratories Department. † Project Engineer, Commercial Airplane Group.

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35th AIAA Fluid Dynamics Conference and Exhibit6 - 9 June 2005, Toronto, Ontario Canada

AIAA 2005-4668

Copyright © 2005 by the authors. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

φ 0 = baseline twist distribution η = nondimensional span-wise location from side of body ΔCDp,ij = sensitivity of ith design variable to pressure drag coefficient at jth angle of attack Δz = difference of camber distribution from baseline Δφ = difference of twist distribution from baseline ΛEA = sweepback angle of elastic axis

I. Introduction ENERALLY, the mission of civil aircraft is to transport a given payload over a specific range with maximum efficiency. Less fuel for the mission and lighter structural weight will produce efficient flights. Lift distribution

on wing affects both of drag and aerodynamic loads. Trapezoidal lift distribution has small bending moment at the wing root though large drag. On the other hand, elliptic lift distribution has small drag though large bending moment. The lift distribution minimizing takeoff weight exists between trapezoidal and elliptic lift distributions. Thus, the aero-structural optimization by modification of the lift distribution is the key to the wing design for less fuel and lighter structural weight.

G

On the conceptual design phase, it is important to reduce the risk and the cost of the aircraft development. Improvement of the prediction accuracy for weight and aerodynamic performance will lead to the reduction in the development risk. The reduction in the design cycle time will relate to the development cost. Computational fluid dynamics (CFD) and finite element method for structural analysis (FEM) are powerful tools to improve the accuracy of prediction. However, it takes a lot of time and expense to obtain the optimum solution, because we usually need iteration to converge both of aerodynamic and structural design.

In this study, to reduce time and expense required for the aero-structural optimization, we developed the methodology with aerodynamic database based on the CFD results, and with beam model of wing box structure. This methodology can optimize lift distribution on elastic wing to minimize takeoff weight of airplane by modifying twist and camber distribution, and we can obtain the wing loft under 1g-flight condition. The developed methodology is applied to a configuration of small civil airplane shown in Fig. 1.

II. Definition of Optimization Problem

A. General Optimization is performed using gradient-based method1. The optimization problem is to minimize the takeoff

weight (TOW) subject to side constraints, which limit the values of design variables. The problem can be written as

Figure 1. Configuration of small civil airplane

( )PTOWmin (1)

subject to

maxmin PPP ≤≤ (2)

B. Wing Design Variables In this study, the differences in the twist and camber distributions from the baseline loft are parameterized by 9

design variables. Modifying the twist distribution can change the wing lift distribution, which affects aerodynamic loads and induced drag. On the other hand, it also changes the characteristics of pitching moment and trimmed angle of attack. Using the twist and camber distributions as the design variables, we can optimize trimmed lift to drag ratio (L/D) at the same time.

The twist distribution of wing is defined by a set of three design variables (i.e., t1, t2, and t3). New twist distribution φ(η) (Fig. 2) is given as

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( ) ( ) ( )ηφηφηφ Δ+= 0 (3)

( ) 2

)

( ) ( ) ( )cxhccxhccxz /// +=

321 ηηηφ ttt ++=Δ (4)

where t1 represents the angle of wing incidence, and t2 and t3 are coefficients of a polynomial to express the wing twist distribution.

Generally, we need a large number of design variables to define arbitrary camber distribution. It takes a lot of time to optimize them. In this study, to reduce the time required in optimization, we can simplify the expression of camber distribution. At a cross section, the camber distribution in the chord direction is defined by a couple of design variables (i.e., c1 and c2). New camber distribution z(x/c) (Fig. 3) is given as

(5) ( ) ( ) ( cxzcxzcxz /// 0 Δ+=

Δ (6) 8.0225.01

The modification of camber distribution can be expressed by the sum of two Hicks-Henne functions (Fig. 4). The function h0.25 can change the front loading of pressure distribution for airfoil, and the function h0.8, the rear loading. Because we define the design variables (c1 and c2) at each section of wing root, kink, and tip, six design variables are required. At a cross section except the above sections, the modification of camber distribution is defined by linear interpolations.

root, kink or tip

z0(x/c) baseline

Δz(x/c) difference in camber distributionfrom baseline

z(x/c) new airfoil

root, kink or tip

z0(x/c) baseline

Δz(x/c) difference in camber distributionfrom baseline

z(x/c) new airfoil

C. Evaluation Function Takeoff weight of an airplane with satisfying design

range is adopted as the evaluation function of the optimization. The takeoff weight includes the structural weight and the fuel weight. The bending moment at each section affects the wing structural weight (i.e., structural design) The cruise L/D relates to the fuel weight (i.e., aerodynamic design). Therefore, we can perform the multidisciplinary optimization (MDO) by minimizing the takeoff weight. Moreover, the modification of lift distribution on the wing affects the drag and the

Figure 3. Modification of twist distribution.

Twis

t Dis

tribu

tion,

deg

ree

1Tip

φ(η) new twist distributionφ0(η) baseline twist distributionΔφ(η) difference in twist distribution from baseline

(specified by design variables)

φ0(η)

φ(η)Δφ(η)

Nondimensional Span Station η0Root

3

2

1

0

-1

-2

Twis

t Dis

tribu

tion,

deg

ree

1Tip

φ(η) new twist distributionφ0(η) baseline twist distributionΔφ(η) difference in twist distribution from baseline

(specified by design variables)

φ0(η)

φ(η)Δφ(η)

Nondimensional Span Station η0Root

3

2

1

0

-1

-20

0.2

0.4

0.6

0.8

1

1.2

0 0.2 0.4 0.6 0.8 1X/C

Hic

ks-H

enne

Fun

ctio

n h(

x/c)

( ) ( ) ⎟⎠⎞

⎜⎝⎛= 25.0log

5.0log

25.0 sin cxcxh π

( ) ( ) ⎟⎠⎞

⎜⎝⎛= 8.0log

5.0log

8.0 sin cxcxh π

Figure 4. Two Hicks-Henne functions formodification of camber distribution.

Figure 2. Modification of twist distribution.

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aerodynamic loads. Trapezoidal lift distribution has small bending moment at the root of the wing though large drag. Elliptic lift distribution has small drag though large bending moment. Usually, the lift distribution minimizing takeoff weight exists between trapezoidal and elliptic lift distribution, and it is obtained by the optimization of the design variables, which are defined in the previous section.

III. Methodology of Aero-Structural Optimization The methodology of aero-structural optimization consists of four major processes and three iteration loops

shown in Fig. 5. 1) Establishment of Aerodynamic Database 2) Structural Evaluation 3) Aerodynamic Evaluation 4) Gradient Evaluation Before the optimization is performed, aerodynamic database is established on the basis of the results of CFD

analysis (Euler code) for wing-body configurations. It consists of the sensitivities of the design variables to aerodynamic coefficients (i.e., lift, drag, pitching moment) and loads (i.e., shear load, bending moment, and torsion). The database is used to estimate the aerodynamic coefficients and loads, when the twist and camber distributions are modified and/or when the wing deflects under the flight conditions.

In the optimization process, first of all, the values of the design variables are given by a user to determine the initial distributions of twist and camber. In the structural evaluation on the first loop, cross-sectional areas of upper/lower panels, front/rear spars, and ribs are determined to resist the critical loads (e.g., 2.5g maneuver, discrete gust etc.). Using the results of sizing for wing box, we can estimate the bending and torsional stiffness. If the wing deflects under the loads, the aerodynamic load furthermore changes. Hence, the wing structural weight and the amounts of wing bending and twist must be converged in this process.

In the aerodynamic evaluation on the second loop, the cruise L/D and the takeoff weight of the airplane which is specified by the design variables, are estimated. The aerodynamic coefficients for the wing-body configurations are predicted on the basis of the aerodynamic database and the wing deflection under 1g-flight condition. Here, the mass and stiffness distributions of wing in the process of the structural evaluation are applied to the computing of 1g loft. The aerodynamic coefficients of tails and nacelles and the friction drag of entire airplane are estimated by using the basic dimensions (e.g., area, span, max diameter etc.) and statistical methods. To obtain the zero fuel weight, the weight except wing structural weight are also calculated by statistical methods. Breguet’s range equation gives the fuel weight to fly over the specified mission.

In the gradient evaluation on the final loop, the sequential quadratic programming (SQP) evaluates the gradients of takeoff weight to design variables, and determines a search direction to give new values of the design variables.

Aircraft Geometry

1. Aerodynamic Database

Establishment(CFD (Euler) )

Aerodynamic Database・Airload Sensitivity・Drag Sensitivity

2. StructuralEvaluation

(Wing Weight)

Wing WeightStiffness

Wing WeightConverged?

DesignVariables Pi

NO

YES

Cruise L/DTOW

TOWConverged?

3. AerodynamicEvaluation

(L/D & TOW)

4. GradientEvaluation(ΔTOW/ΔDi)

Optimum Solution

TOWMinimized?

Modifying values of twist and camber

YES

NO

YES

NO

Aircraft Geometry

1. Aerodynamic Database

Establishment(CFD (Euler) )

Aerodynamic Database・Airload Sensitivity・Drag Sensitivity

2. StructuralEvaluation

(Wing Weight)

Wing WeightStiffness

Wing WeightConverged?

DesignVariables Pi

NO

YES

Cruise L/DTOW

TOWConverged?

3. AerodynamicEvaluation

(L/D & TOW)

4. GradientEvaluation(ΔTOW/ΔDi)

Optimum Solution

TOWMinimized?

Modifying values of twist and camber

YES

NO

YES

NO

Figure 5. Methodology of aero-structural optimization.

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Moreover, SQP checks the convergence to the optimum. If satisfied, the iteration process terminates. Otherwise, the structural and aerodynamic evaluations are performed for the airplane, specified by the new values.

A. Aerodynamic Database Generally, aerodynamic coefficients and loads will vary under different angle of attack, wing deflection

(including bending and twist), and camber distribution. We need to know the aerodynamic coefficients and loads of airplane with arbitrary twist and camber distributions of wing in the optimization processes. However, if the performance of each airplane is estimated one by one using CFD, it takes a lot of time to converge to the optimum. If we can know the sensitivity of the flight condition and wing loft to the aerodynamic coefficients and loads, the aerodynamic performance (e.g., drag polar curve and bending moment distribution etc.) can be obtained for arbitrary flight conditions and lofts.

To avoid consuming time, the aerodynamic coefficients and loads of baseline loft and modified lofts for wing-body configuration are calculated by using Euler code. A modified loft has a wing shape specified by variation in a design variable. The aerodynamic sensitivities are established on the basis of the CFD results before the optimization are performed. In the optimization processes, the aerodynamic coefficients and loads for wing-body configuration are estimated using the database without CFD analysis.

For example, the sensitivity of a design variable P to pressure drag ΔCDp is approximated by a quadratic polynomial at a fixed angle of attack.

(7) iijDpiijDpijDp PBPBC ,12

,2, +=Δ

Subscript i means a kind of the design variables (i.e., i=1-9, wing incidence, camber of root section etc.). CFD analyses are performed at three angles of attack (i.e., j=1-3). Fig. 6 shows the example of ΔCDp for variation in the

design variable P1 (=t1, wing-incidence). BDp in Eq. (7) are coefficients of a polynomial and they are computed on the basis of the CFD results at each angle of attack. The coefficients must be remained as the aerodynamic database. In the optimization processes, the drag coefficient for wing-body configuration with arbitrary twist and camber distributions can be estimated as

0

0.02

0.04

0.06

0.08

0.1

-2 -1 0 1 2Design Variable P1, degree

CD

p

α=3.0deg. (j=3)

α=1.5deg. (j=2)

α=0.0deg. (j=1) ΔCDp13

∑=

Δ+=n

iijDpjDpjDp CCC

1,0,,

( ) ∑=

=3

0,

k

kijMkij AM ηη

(8)

Using the database about aerodynamic coefficients, we can know the coefficients of lift, drag, and pitching moment at three angles of attack, and the longitudinal characteristics of airplane are obtained. (i.e., CL-α, CL-CD, and Cm-CL curves)

Figure 6. Variation in pressure drag for wingincidence.

024

68

1012

1416

0 0.2 0.4 0.6 0.8 1Span-wise Location η

Bend

ing

Mom

ent,

106 lb

f-in P1 =+1.5deg.P1 = 0.0deg.P1 = -1.5deg.

M12 = -6.21x106η3 + 2.79x107η2

- 3.63x107η + 1.47x107

Aerodynamic loads also are estimated in the similar way to aerodynamic coefficients. However, aerodynamic loads have their distributions along the elastic axis of the wing. For the present analysis, a span-wise distribution (e.g., bending moment M) is expressed by a cubic polynomial (k=0-3) as shown in Fig. 7.

(9)

AMk are coefficients of a polynomial. In the same way as ΔCDp, these sensitivities of a design variable P to AMk (i.e., ΔAMk) are given as Figure 7. Variation in bending moment for wing

incidence (α=1.5degree).

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(10) iijMkiijMkijMk PBPBA ,12

,2, +=Δ

The coefficients BMk are stored as the aerodynamic database of loads. In the optimization processes, the distribution of bending moment along elastic axis on wing can be written as

(11) ( ) ∑ ∑= =

⎟⎠

⎞⎜⎝

⎛Δ+=

3

0 1,0,

k

kn

iijMkjMkj AAM ηη

This methodology is applied to the estimation of shear loads and torsion distributions on wing.

B. Wing Structural Weight The weight of wing box and secondary structure is separately estimated. We employ the analytical method to

obtain the weight of wing box, which consists of front/rear spars, upper/lower panels, and ribs.2,3 The structural model of wing box used in this study is a simple beam type model. Fig. 8 shows a cross section perpendicular to the elastic axis of the wing. The shape of the wing box is assumed to be rectangular. The upper and lower panels are located at mean height of each panel between front spar and rear spar.

The upper panel represents skin and stringer. The former carries compression/tension loads due to bending, and shear loads due to torsion. The latter carries compression/tension loads only. To simplify the structural consideration, we assume the ratio of stringer area to skin area is set to 0.5 at each cross section. The required cross-sectional area of the upper panel is determined to resist the design loads (e.g., 2.5g maneuver, discrete gust etc.) without failure and buckling.

The lower panel carries compression/tension loads and shear loads as well as the upper panel. However, it is the most important to consider the fatigue through the ground-air-ground (GAG) cycle. Figure 9 shows the relationship between the design range of airplane and the allowable stress of aluminum alloy.4 The figure indicates the allowable stress of the lower panel for short-range airplane is lower level than that for long-range airplane, because short-range airplane has a lot of takeoff and landing frequencies (i.e., GAG cycle). The required cross-sectional area of lower panel is obtained from the design loads and the allowable stress Fl. Here Fl in ksi is given as

(2024 Al Alloy) (12) 0.133117.56R=lF

(7075 Al Alloy) (13) 0.102220.59R=lF

where R is the design range of airplane in nautical miles. The front/rear spar webs carry shear loads. The required cross-sectional areas of the webs are calculated in a

similar way in the case of the upper panel. For ribs, it is assumed that the cross-sectional area of rib is equal to that of spar web at each section.

For secondary structure, it is difficult to estimate its weight analytically, because the aspect of maintenance

40

45

50

55

60

0 1 2 3 4 5 6 7Range, 103 nm

Allo

wab

le S

tress

, ksi

2024 Al Alloy

7075 Al Alloy

-0.08

-0.06-0.04

-0.020

0.02

0.040.06

0.08

0 0.5x/c

z/c

1

Front Spar

Rear Spar

Lower Panel

Upper Panel

Figure 8. Structural model of wing box. Figure 9. Allowable stress for lower panel.

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affects the weight beside the strength and the stiffness of material. Therefore, to estimate the weight of secondary structure, we employ the statistically method established from the data of existing airplanes. The second structure includes fixed leading/trailing edges, ailerons, spoilers, slats, flaps, and so on.

C. Wing Deflection The amount of wing deflection is determined using structural model and loads on wing. The structural model

consists of the beam elements on the elastic axis of wing and concentrated masses representing wing structural weight and fuel weight shown in Fig 10. It is assumed that the elastic axis is in the middle of wing box. The bending stiffness EI and torsional stiffness GJ of a beam element are estimated on the basis of the result of wing box sizing in the estimation of the wing structural weight. We assume that the upper/lower panels (including skins and stringers) contribute to the EI value, and upper/lower skins and front/rear spar webs contribute to the GJ value.

The bending deflection along the elastic axis changes the angle of attack for the strip in the flow direction, because the wing for the present analysis has a sweepback angle. The variation in the angle of attack for the strip Δα is derived from the bending angle Δθ along the elastic axis and twist angle Δφ around the elastic axis.

EAEA ΛΔ−ΛΔ=Δ sincos θφα (14)

In the optimization processes, the aerodynamic coefficients and loads with considering Δα due to fight loads are computed using the aerodynamic database.

Engine Mass

Concentrated Mass

Beam Element

nt Spar

Rear Spar

D. Cruise L/D To calculate trimmed L/D of airplane on the cruise condition, we assumed the model of forces and moments

acting on airplane.5 The model consists of body, wing, tails, and nacelles shown in Fig. 11 and Table 1. Here, the symbol “X” in Table 1 means components to be considered in calculating lift, drag, and pitching moment of airplane. Equations (15-17) express the balance of forces and moment in Fig. 11.

WW

HTHTLWBLAirplaneL qS

nWSS

CCC =+= ,,, (15)

W

HTHTDiWBDpWBDfNACVTHTDAirplaneD S

CCCCC ,,,)(,0, +++= ++

S (16)

0,,,,, =+−+=mac

EAirplaneD

mac

HT

W

HTHTL

mac

CGAirplaneLWBmAirplanem c

CcS

Cc

CCC zxSx (17)

On a flight condition, lift coefficients of wing-body and horizontal tail can be determined so that the pitching moment around the aerodynamic center becomes zero. Each coefficient has to be obtained through the iteration,

Front Spar

Rear Spar

Concentrated Mass

Beam Element

Fro

Engine Mass

Figure 10. Structural model of wing box.

nW

LHT

LWB

MWB

TE

zE

xCG

xHT a.c.

Table 1. Contribution of components to forces and moment.

Component C C CL D m

Body X X X Wing X X X

Horizontal Tails X X X Vertical Tail - X -

Nacelles - X X

Figure 11. Force model of airplane.

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because the pitching moment of airplane contains the contributions of engine thrust and horizontal tail lift. The lift, pressure drag, and pitching moment of wing-body configuration with deflected wing are derived from the aerodynamic database and the beam model. Statistical methods apply to the estimation of the friction drag of wing and the profile drag of tails and nacelles.6-8

E. Takeoff Weight Approximately, the takeoff weight of an airplane WTO is determined as

FPAYOETO WWWW ++= (18)

The operational empty weight WOE consists of the weight of structure, propulsion, fixed equipment, operational items, and crew. In this study, the wing structural weight is computed by the above-mentioned method. For the rest of structure and the propulsion, the weight of each component (i.e., body, tails etc.) is estimated by statistical methods (see Ref. 6), which are derived from statistical data of existing airplanes. The airplanes have conventional layout with a semi-monocoque aluminum alloy structural framework. The weight of fixed equipment is assumed to be 14% of the takeoff weight.

The payload weight WPAY is determined from the number of passengers flying over the design range.

pasPAY NW 200= (19)

where the weight of each passenger including baggage is 200 lb. Breguet’s range equation gives the fuel weight WF to be carried.

( ) ⎥⎦

⎤⎢⎣

⎡−⎟

⎠⎞

⎜⎝⎛

××

+= 1/

expVDL

SFCESARWWW PAYOEF (20)

L/D used in Eq. (20) is the value for the airplane with deflected wing under the 1g-flight condition. ESAR is the equivalent still air range. It is the total range to be allowed for in the fuel weight estimation, and it includes influences of non-cruise flight segments (i.e., takeoff, climb, descent, and landing), the fuel reserves (i.e., diversion and hold), and other contingencies (e.g., wind). The expression in Ref. 6 is employed for the present analysis.

RESAR 063.1568 += (21)

IV. Optimization Results The developed methodology of aero-structural optimization is applied to small civil airplane configuration. Fig.

12. shows the airplane with about 100 seats. Nine design variables (i.e., wing incidence, twist and camber distributions) were optimized to minimize the takeoff weight. The aerodynamic characteristics of the wing-body

Table 2. Flight conditions for structural and aerodynamic evaluations.

Discrete Gust Cruise Weight MZFW Mid Cruise Point Speed 350ktEAS Mach 0.8

Altitude 20,000 ft 39,000 ft Load Factor 2.61 g 1.0 g

Figure 12. 100 seats class airplane.

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configuration are computed on the basis of the aerodynamic database and the beam model. During the optimization, the structural evaluation is performed on the condition of the discrete gust shown in Table 2. In the process of aerodynamic evaluation, cruise L/D is estimated on the conditions of Mach 0.8 and 39,000 ft.

Table 3 shows the values of the design variables for the optimum. The wing incidence of the optimum increases by 0.9 degree compared with that of the baseline, and the amount of twist-down from root to tip also increase. Fig. 13 shows the comparison of airfoils between the baseline and the optimum at wing root, trailing edge kink, and wing tip. The increment of wing- incidence and camber brings the higher lift to the optimum compared with the baseline at the same angle of attack. Therefore, the angle of attack at cruise is 1.9 degree for the baseline, and 0.7 degree for the optimum shown in Table 4. Fig. 14 is the comparison between results of Euler analysis and estimation by present method for the optimum solution. The result of estimation using aerodynamic database shows good agreement with that of Euler analysis. The strength of the shock wave on the upper surface of the wing is weakened by the optimization, especially between kink of trailing edge and wing tip (Fig. 15). It contributes to the reduction in pressure drag of wing-body configuration.

The wing-body of the optimum shows a tendency to increase pitching down due to increment of rear loading on airfoil. Though the lift coefficient of horizontal tail and the trim drag in cruising are increasing, the reduction in pressure drag of wing-body by the optimization is larger than the increment of the trim drag. Consequently, cruise L/D is improved by 3.5% shown in Table 4. The improvement reduces the fuel weight required for the design mission by 1,043 lb, and the takeoff weight reduces by 1,306 lb. This result indicates that the developed methodology can improve cruise performance without increasing the wing structural weight.

V. Conclusion An aero-structural optimization methodology for wing design has been developed and presented. The

methodology is based on aerodynamic database and beam model of wing box. The database consists of the sensitivities of the design variables to aerodynamic coefficients and loads, and is established on the basis of the results of Euler analysis for wing-body configurations. The weight and stiffness of wing box are estimated using rectangular beam model and analytical method. The required cross-sectional area of upper panel is determined to resist the design loads without failure and buckling. The fatigue through the ground-air-ground cycle is important for

Table 3. Design variable values for the optimum solution. Design Variable Optimum

-0.1

-0.05

0

0.05

0.1

0 0.2 0.4 0.6 0.8 1x/c

z/c

OptimumBaseline

Wing-body Incidence t1, deg +0.86 Twist t2, deg -1.25 t3, deg +0.62 Camber c , %c +0.92 1,root

c , %c +0.32 2,root

c1,kink, %c +0.15 c2,kink, %c +0.27

a) Wing root c1,tip, %c +0.27 c2,tip, %c +0.65

-0.1

-0.05

0

0.05

0.1

0 0.2 0.4 0.6 0.8 1x/c

z/c

OptimumBaseline

a) Kink of trailing edge

-0.1

-0.05

0

0.05

0.1

0 0.2 0.4 0.6 0.8 1x/c

z/c

OptimumBaseline

c) Wing tip

Figure 13. Airfoils for the baseline and the optimum solution.

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Table 4. Weight and cruise L/D for the baseline and the optimum solution.

sizing of lower panel. The required cross-sectional area of lower panel is obtained on the basis of the relationship between the design range of airplane and the allowable stress of aluminum alloy.

Wing lofts are parameterized using quadratic polynomials for twist distribution and two Hicks-Henne functions for camber distributions. The optimum lift distribution for an elastic wing is obtained by controlling the design variables though optimization.

Results for takeoff weight minimization of small civil airplane shows improvement of cruise L/D by 3.5%. The strength of the shock wave on the upper surface of the wing is weakened by the optimization, especially between trailing-edge kink and wing tip. It contributes to the reduction in pressure drag of wing-body configuration. The improvement reduces the required fuel weight for the design mission by 1,043 lb, and the takeoff weight reduces by 1,306 lb. This result indicates that the developed methodology can improve cruise performance without increasing the wing structural weight.

Acknowledgments Support for this work from Japan Aircraft Development Corporation is gratefully acknowledged.

Baseline Optimum

Wwing, lb 11,021 11,027 +0.1% Wfuel, lb 25,983 24,940 -4.0% WTO, lb 115,893 114,587 -1.1% L/Dcr 14.48 14.98 +3.5%

αcr, deg 1.9 0.7 -

-1.5

-1

-0.5

0

0.5

10 0.2 0.4 0.6 0.8 1

x/c

Cp

OptimumBaseline

0

0.2

0.4

0.6

0.8

1

1.2

0 0.02 0.04 0.06 0.08CDp,WB

CL,

WB

Euler AnalysisEstimation by Present Method

a) η=0.15

-1.5

-1

-0.5

0

0.5

10 0.2 0.4 0.6 0.8 1

x/c

Cp

OptimumBaseline

Figure 14. Comparison between the results of Euler analysis and estimation by present method for the optimum solution. b) η=0.5

-1.5

-1

-0.5

0

0.5

10 0.2 0.4 0.6 0.8 1

x/c

Cp

OptimumBaseline

c) η=0.8

Figure 15. Pressure distributions for the baseline and the optimum solution using Euler analysis.

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References 1DOT, Design Optimization Tools Users Manual, Ver. 5.0, Vanderplaats Research & Development, Inc., Colorado Springs,

CO, 1999. 2Ardema, M. D. et al., “Analytical Fuselage and Wing Weight Estimation of Transport Aircraft,” NASA TM-110392, 1996. 3Niu, M. C. Y., Airframe Structural Design, 2nd ed., Hong Long Conmilit Press Ltd., Hong Kong, 1999, Chap. 3, 8. 4Niu, M. C. Y., Airframe Stress Analysis and Sizing, 2nd ed., Hong Long Conmilit Press Ltd., Hong Kong, 1999, Chap. 15. 5Lomax, T. L., Structural Loads Analysis for Commercial Transport Aircraft: Theory and Practice, AIAA Education Series,

AIAA, Virginia, 1996, Chaps. 2, 10. 6Jenkinson, L. R., Simpkin, P., and Rhodes, D., Civil Jet Aircraft, AIAA Education Series, AIAA, Virginia, 1999, Chaps. 7,8. 7Raymer, D. P., Aircraft Design: A Conceptual Approach, AIAA Education Series, AIAA, Virginia, 1999, pp. 170, 173. 8Torenbeek, E., Synthesis of Subsonic Airplane Design, Delft University Press, Delft, The Netherlands, 1982, Chap. 8.

American Institute of Aeronautics and Astronautics

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