JOURNAL OF AIRCRAFT
Vol. 50, No. 3, May–June 2013
Lift Distributions for Minimum Induced Drag with Generalized
Bending Moment Constraints
David J. Pate∗ and Brian J. German†
Georgia Institute of Technology, Atlanta, Georgia 30332
Downloaded by UNIVERSITY OF WITWATERSRAND on February 5, 2020 | http://arc.aiaa.org | DOI: 10.2514/1.C032074
DOI: 10.2514/1.C032074
The previous works of Prandtl, Jones, and Klein and Viswanathan addressed the problem of determining the lift
distribution that minimizes induced drag for a given lift and specified bending moment. In these formulations,
bending moment is considered to be a surrogate for wing weight. These classical methods require the bending
constraints to be imposed at the same lift coefficient at which drag is minimized. In practice, however, it is commonly
desired to minimize drag at a representative cruise lift coefficient while imposing the bending constraints at a limiting
structural load condition, such as a maneuver lift coefficient. This paper presents an approach to extend the classical
methods by allowing the bending constraints to be imposed at different lift coefficients than that at which induced drag
is minimized. An example for a wing planform similar to that of a Boeing 737 shows that the penalty for optimizing
induced drag at the maneuver lift coefficient as implied in the classical methods results in between a 1–10% increase in
drag at cruise compared to the results from this new approach. It is expected that the new approach will enable the
classical methods to be extended to practical applications in multidisciplinary wing design.
ρ∞
σ
Nomenclature
AR
b
CD;i
CIBM
Ck
CL
CL;cruise
CL;IBM
=
=
=
=
=
=
=
=
CL;RBM
CRBM
c
cl
L
l
q∞
S
v∞
y
y
y 0
y 0 0
Γ
γ
δ
εIBM
εRBM
η
η
Λ
λ
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
aspect ratio
wingspan, ft
induced drag coefficient
integrated bending moment coefficient
constant of integration
lift coefficient, L∕q∞ S
cruise lift coefficient
lift coefficient for the integrated bending moment
constraint
lift coefficient for the root bending moment constraint
root bending moment coefficient
wing chord, ft
section lift coefficient
lift, lb
local aspect ratio, b∕c
freestream dynamic pressure, lb∕ft2
wing planform area, ft2
freestream velocity, ft∕s
distance in span direction, ft
nondimensional distance in span direction
nondimensional location of the center of lift
nondimensional second moment of circulation
circulation, ft2 ∕s
nondimensional circulation, Γ∕v∞ b
induced drag parameter, πARCD;i ∕C2L − 1
integrated bending moment ratio
root bending moment ratio
distance in the span direction, ft
nondimensional distance in span direction
sweep angle of the wing quarter chord, deg
taper ratio
=
=
freestream density, slug∕ft3
maneuver ratio
Subscripts
a
a–d
=
=
d
des
given
IBM
RBM
=
=
=
=
=
γ additional y
associated with γ a , where γ a y
CL
associated with the mutual interaction between the γ a
and γ d distributions
def γ des y
associated with γ d , where γ d y
CL;des
related to the design circulation distribution
related to the given bending moment constraints
integrated bending moment
root bending moment
def
I.
I
Introduction
NDUCED drag is the streamwise force acting on a lifting body
caused by the downwash associated with lift. Prandtl [1] and
Munk [2] demonstrated that minimizing induced drag for a planar
wing at a given lift requires the downwash from the trailing vortices to
be constant. This condition corresponds to an elliptic spanwise lift
distribution. However, designing the lift distribution is not an issue of
induced drag alone; in general, it is a multidisciplinary problem in
which wing structural weight, stall characteristics, parasite drag,
wave drag, and aeroelastic response must also be considered.
Extensions to the classical problem of induced drag minimization
have incorporated considerations of structural weight by imposing
constraints on wing bending moment. Bending moment is a
convenient surrogate for wing weight because it can be calculated
directly from the lift distribution and is independent of details
of the structural design. Prandtl [3], solved the problem of
minimum induced drag subject to a constraint on integrated
bending moment. His result showed that the corresponding optimal
downwash distribution varies quadratically along the span. Jones [4],
specified root bending moment as a constraint and showed
that the corresponding downwash distribution is linear. Klein and
Viswanathan [5] combined these two bending moment constraints
into a unified formulation, such that one or both can be specified.
DeYoung [6], extended Jones’s formulation to allow the bending
moment to be specified at an arbitrary span location. Finally, Löbert
[7] developed a formulation to determine optimal lift distributions
constrained by the integral of section bending moment divided by
section thickness, and he obtained an analytical result for the special
case of linearly tapering thickness.
Received 19 August 2012; revision received 2 November 2012; accepted
for publication 13 November 2012; published online 16 April 2013.
Copyright © 2012 by David J. Pate and Brian J. German. Published by the
American Institute of Aeronautics and Astronautics, Inc., with permission.
Copies of this paper may be made for personal or internal use, on condition
that the copier pay the $10.00 per-copy fee to the Copyright Clearance Center,
Inc., 222 Rosewood Drive, Danvers, MA 01923; include the code 1542-3868/
13 and $10.00 in correspondence with the CCC.
*Graduate Research Assistant, Aerospace Engineering, 270 Ferst Drive.
Student Member AIAA.
†
Assistant Professor, Aerospace Engineering, 270 Ferst Drive. Senior
Member AIAA.
936
937
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PATE AND GERMAN
When designing a wing using these approaches, the wing is twisted
such that the optimal lift distribution is achieved at a design lift
coefficient. However, the shape of the lift distribution for a twisted
wing changes with lift coefficient [8–10], and induced drag can
increase significantly at off-design lift coefficients [11]. Although it
is generally desirable to achieve minimum induced drag at or near a
representative cruise lift coefficient, the structural limits of the wing
must be designed to a high lift coefficient corresponding to a
maneuver or gust load. This leads to a discrepancy between the lift
coefficient at which minimum induced drag is desired and the
lift coefficient corresponding to the bending constraints. This
paper presents a solution that generalizes the Prandtl, Jones, and
Klein formulations to address this discrepancy by allowing the lift
coefficient for minimum induced drag to be specified separately from
the lift coefficients at which the bending constraints are imposed.
Modern methods for wing design have trended toward using
multidisciplinary numerical optimization instead of analytical
formulations with bending constraints. Kroo [12] developed a
computational method for the design and analysis of subsonic lifting
surfaces. The analysis included consideration of induced drag,
viscous drag, wing weight, trim, and stall. The circulation distribution and the corresponding twist distribution were optimized
for minimum total drag with constraints on lift, weight, and trim.
Structural weight was calculated by integrating section bending
moment divided by section thickness.
Craig and McLean [13] developed a computer program along
similar lines. The analysis includes the effects of aeroelastic twist and
viscous section drag. They calculated drag at the cruise condition,
but, differing from the approach in [12], they based the wing weight
on the local wing section bending moment at a critical condition, such
as a 2.5g maneuver.
McGeer [14] explored optimization of the lift, chord, and
thickness-to-chord ratio distributions to minimize induced drag
subject to constraints on structural weight, compressibility drag,
parasite drag, and section lift coefficient. The lift distribution is
represented as a Fourier series, and the equations are solved in an
iterative scheme. Weight is calculated based on the spanwise
integration of bending moment divided by thickness.
Wakayama and Kroo [15] conducted a multidisciplinary
optimization study for a subsonic planform by seeking minimum
drag subject to constraints on wing structural weight and maximum
lift. Structural load cases included multiple cruise flight conditions at
specific altitudes, Mach numbers, and aircraft weights, as well as a
maneuver condition and a taxi bump. The structural model used to
determine wing weight models the wing skin, spar, ribs, and stringers.
The spanload was not a design variable; instead, it was controlled by a
small number of twist design variables.
Iglesias and Mason [16] developed a numerical approach to find
the optimum spanload for a wing by minimizing induced drag subject
to constraints on lift coefficient, pitching moment coefficient, and
wing root bending moment. Root bending moment was used as the
driving parameter to define the shape of the lift distribution.
More recently, Ning and Kroo [17] conducted a conceptual wing
design study that includes analysis of induced drag, viscous drag,
wing weight, stall, and trim. They sought to minimize drag with fixed
values of lift, wing weight, and stall speed, and they found that the
results were sensitive to the ratio of maneuver lift coefficient to cruise
lift coefficient.
Donovan and Takahashi [18] considered the optimization of the lift
distribution and subsequent twist distribution to minimize mission
fuel burn for a fixed planform. Wing weight was held constant and the
changes in induced drag were computed as wing twist was varied.
The lift distribution was parameterized by the design lift coefficient
and root bending moment. In a later study, Donovan et al. [19]
explored the impact of root bending moment on the wing weight and
mission performance of a business jet and a larger passenger aircraft.
Changes in thickness were included in a detailed analysis of wing
structural weight.
The structure of a wing is typically designed to a limiting
load condition corresponding to a high lift coefficient maneuver.
Accordingly, Refs. [13–15,17] specify a maneuver condition
separately from the reference cruise condition. McGeer [14] points
out that minimizing drag at the maximum load condition can produce
“extraordinarily high drag at cruise CL ’s,” and this behavior is
demonstrated theoretically in [8,10]. More recently, Ning and
Kroo [17] state that significantly different results are obtained by
considering different lift coefficients for structural sizing.
In previous work discussed in the literature, only numerical
optimization methods have allowed bending moment constraints to
be imposed at independent lift coefficients. The purpose of the
present paper is to extend the classical analytical optimization
methods to enable a similar capability for specifying the bending
constraints at independent conditions. Analytical methods have the
advantage of providing insight and transparency that may be lost in
numerical optimization. Additionally, it is expected that the proposed
method can be used within a broader multidisciplinary design
optimization framework as a means to reduce the number of design
variables needed to control the twist distribution.
The paper begins by reviewing the work in [5] and then develops
compact new relations that transparently illustrate the geometric
nature of the bending moment problem. Next, the formulation is
generalized to allow specification of minimum drag at a different lift
coefficient than that of the bending moment constraint. Finally, this
method is applied to an example wing to illustrate its benefits.
II.
Overview of Previous Formulations
Prior to presenting the results of Prandtl, Jones, Klein, and
Viswanathan we introduce the notation that will be used to express
the resulting equations in a compact and intuitive form. For a
circulation distribution that is symmetric with respect to the x axis,
the coefficients of lift, root bending moment, integrated bending
moment, and induced drag are respectively expressed as
Z
CL AR
1
−1
dy 2AR
γy
Z
1
dy
γy
(1)
0
Z
1
CRBM AR
y
dy
yγ
(2)
dy
y 2 γy
(3)
0
1
CIBM AR
2
Z
1
0
and
CD;i 1
AR
2π
Z
1
Z
−1
1
−1
γy
γη
dη
y − η
dη dy
(4)
Here, γ Γ∕v∞ b is the nondimensional circulation, and y is a
nondimensional spanwise variable. Note that γ cl c∕2b from the
Kutta–Joukowski Theorem.
Equation (2) is related to dimensional root bending moment as
Z b∕2
b
L 0 yy dy q∞ S CRBM
2
0
This indicates that root bending moment is equivalent to integrated
shear load as defined in [5]. Equation (3) is related to dimensional
integrated bending moment as
2
Z b∕2 Z b∕2
b
L 0 ηη − y dη dy q∞ S
CIBM
2
0
y
Next, consider the elliptic circulation distribution, which is defined as
q
γ 0 1 − y 2
γy
Equation (1) can be used to show
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PATE AND GERMAN
γ0 2CL
πAR
γy
and Eqs. (2) and (3) then indicate
2
CRBM γelliptic C
3π L
(5)
1
CIBM γelliptic CL
16
(6)
where C1 , C2 , and C3 are constants of integration. Rewriting the
logarithm in terms of the more compact inverse hyperbolic cosine
used by Jones gives
q
2C2
2
1 − y 2C1 γy
C3
π
2C
2
3
− C3 1 − y 2 2
2 y 2 cosh−1 1∕jyj
3
π
Finally, we define the following ratios:
def
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εRBM CRBM
CRBM γelliptic
q
2C
1 − y 2 2C1 2 C3
π
p C
1 − 1 − y 2
2
3
p − C3 1 − y 2 2
− 2 y 2 ln
3
π
1 1 − y 2
(7)
(11)
Substituting Eq. (11) into Eqs. (1–3) gives the following equations
relating C1 , C2 , and C3 to CL , CRBM , and CIBM :
def
εIBM CIBM
CIBM γelliptic
We call εRBM the root bending moment ratio and εIBM the
integrated
bending moment ratio. The expressions for CRBM γelliptic and
CIBM indicated in Eqs. (5) and (6) can be substituted into
γelliptic
Eqs. (7) and (8) for arbitrary lift distributions, allowing εRBM to be
expressed as
εRBM 3π CRBM
2 CL
(9)
CIBM
CL
(10)
2 3
π∕4 # C1
1∕5 4 C2 5
π∕48
C3
(12)
3
32
−4
24∕π
CL
15π∕2 −48 54 CRBM 5
CIBM
−24 160∕π
(13)
which can be inverted to give
3
2
6∕5π
C1
30
4 −2
4 C2 5 AR
C3
6∕π
2
Therefore, Eq. (11) can be expressed in terms of CL , CRBM , and CIBM .
Instead of substituting CL , CRBM , and CIBM for C1, C2 , and C3 , a
simpler and more natural expression can be obtained by using
Eqs. (9) and (10) to obtain
and εIBM to be expressed as
εIBM 16
3
" π
4∕3
CL
4 CRBM 5 AR 2∕3 1∕π
π∕16 1∕10
CIBM
2
(8)
Together, the root bending moment ratio and the integrated bending
moment ratio will be referred to as the bending ratios. A bending ratio
above unity indicates that the lift is shifted more toward the tips than
an elliptic distribution, and a bending ratio below unity indicates a
shift in loading toward the root. By relating the bending moments of a
distribution to that of an elliptic distribution the bending ratios
provide an intuitive reference. Additionally, the bending ratios
depend only on the shape of a distribution and not its amplitude,
which is defined by CL. The root bending moment ratio is similar to
the concept of root bending moment relief (RBMR) used by Donovan
and Takahashi in [18] and to the concept of root bending moment
reduction used by Iglesias and Mason in [16].
A. Klein Circulation Distribution
The work of Jones and Prandtl is effectively superseded by the
generalized formulation of Klein and Viswanathan [5]. We, therefore,
base our discussion on this general formulation, which specifies lift,
root bending moment, and integrated bending moment as constraints.
The results of Jones and Prandtl, respectively, correspond to the
special cases in which only lift and root bending moment or lift and
integrated bending moment are specified.
The constrained minimization problem is
min CD;i γ
γ
s:t: CL γ − CL;des 0
CRBM γ − CRBM;given 0
CIBM γ − CIBM;given 0
The details of the steps to find the solution using variational calculus
can be found in [5]. Klein and Viswanathan [5] showed that
the solution of the constrained optimization problem results in the
circulation distribution
γy
q
2CL
66 − 170εRBM 105εIBM 1 − y 2
πAR
302 − 5εRBM 3εIBM y 2 cosh−1 1∕jyj
3
− 203 − 8εRBM 5εIBM 1 − y 2 2
(14)
When εRBM 1 and εIBM 1 this expression becomes the elliptic
distribution. The form of Eq. (14) indicates that the scale of the
circulation distribution is determined by CL, and the shape is
determined by εRBM and εIBM . It is also noteworthy that Eq. (14),
is linear with εRBM and εIBM . Therefore,
although nonlinear with y,
any linear function of γ, such as section lift coefficient, will be linear
with εRBM and εIBM . Similarly, any quadratic function of γ will be
quadratic with εRBM and εIBM .
From Eq. (4) and the definition of the induced drag parameter δ
def
δ
πARCD;i
−1
C2L
(15)
The induced drag parameter of this distribution is
δ 57 − 32εRBM 18εIBM 40ε2RBM
15ε2IBM − 48εRBM εIBM (16)
From the Kutta–Joukowski Theorem, the section lift coefficient can
be related to the local aspect ratio and the circulation distribution as
2lyγ
y
cl y
(17)
b∕cy
is the nondimensional
where the local aspect ratio ly
inverse of the chord. For a given y0 , Eqs. (14) and (17) can be
combined as
939
PATE AND GERMAN
q
cl y 0 4ly 0 66 − 170εRBM 105εIBM 1 − y20
CL
πAR
302 − 5εRBM 3εIBM y 20 cosh−1 1∕jy 0 j
3
2
2
− 203 − 8εRBM 5εIBM 1 − y 0 B. Jones and Prandtl Special Cases
For the special case in which only lift and root bending moment are
specified the solution can be found by omitting C3 from Eq. (11) and
conducting the same process for replacing the remaining constants,
C1 and C2 . This gives Jones’s solution:
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CL
AR
π
4∕3
C1
;
2∕3 1∕π C2
CRBM
CL
C1
3 3∕π −4
;
AR −2 3π
C2
CRBM
q
2CL
3 − 2εRBM 1 − y 2
γy
πAR
6εRBM − 1y 2 cosh−1 1∕jyj
a) Jones distributions for different values of
RBM
(18)
Using Eq. (3), the εIBM of this distribution can be expressed in terms
of εRBM as
εIBM 8εRBM − 3
5
(19)
Similarly, when only lift and integrated bending moment are
specified, C2 is omitted from Eq. (11) to give Prandtl’s solution:
CL
AR
π
π∕4
C1
;
π∕16 π∕48 C3
1 −12
CL
4
;
πAR −3 48
C3
CIBM
q
2CL
3εIBM − 2 1 − y 2
γy
πAR
3
41 − εIBM 1 − y 2 2
CIBM
C1
b) Prandtl distributions for different values of IBM
Fig. 1 Samples of the Klein distribution for the Jones and Prandtl
special cases.
def
y 0 (20)
Therefore, y 0 and εRBM may be considered interchangeable.
Similarly, the second moment of circulation y 0 0 is defined as
The corresponding εRBM is
εRBM 3εIBM 2
5
def
y 0 0 (21)
The induced drag parameter for the Jones distribution [Eq. (18)] is
δ 8εRBM − 12
(22)
and the induced drag parameter for the Prandtl distribution
[Eq. (20)] is
δ 3εIBM − 12
(23)
Sample circulation distributions are shown in Figs. 1a and 1b for
Eqs. (18) and (20), respectively. These figures illustrate the similarity
of the Prandtl and Jones distributions. The location of the center of lift
is marked for each distribution by the black curve in the inset
of Fig. 1a.
The center of lift, or moment arm of lift, of a general circulation
distribution is represented by the parameter y 0 and is defined using
the first moment as
R1
R
y
dy
y
dy
yγ
AR 01 yγ
2C
4
R0 1
R1
εRBM
1
RBM 3π
C
γ
y
d
y
AR
γ
y
d
y
L
0
−1
2
R1 2
R
dy AR 01 y2 γy
dy CIBM
y γy
1
R0 1
R1
1
εIBM
C
16
γ
y
d
y
AR
γ
y
d
y
L
0
−1
2
which implies that y 0 0 and εIBM are also interchangeable. More
specifically, y 0 and y 0 0 provide an alternative geometric interpretation
of εRBM and εIBM . Note that the apostrophes in y 0 and y 0 0 indicate the
degree of the moment and not differentiation.
C. Additional Discussion of Induced Drag
The solutions for the induced drag parameter given in Eqs. (16),
(22), and (23) are quadratic with εRBM and εIBM because induced drag
is a quadratic function of circulation, and circulation is a linear
function of εRBM and εIBM . Figure 2 shows contours of induced drag
parameter as a function of εRBM and εIBM from Eq. (16). The contours
form a paraboloid centered at εRBM ; εIBM 1; 1. The major
principal axis of the contours is shown as a solid gray line. When
εRBM and εIBM follow this line δ remains very low. However, when
εRBM and εIBM trend in a direction perpendicular to this line, parallel
to the minor principal axis, δ increases rapidly. The contours are
shown on a base 10 logarithmic scale due to the very large values of δ
that are possible.
940
PATE AND GERMAN
and
1.2
10
εIBM 1.0
1.1
IBM
0.10
0.01
1
0.9
0.8
0.8
0.9
1
1.1
1.2
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RBM
Fig. 2 Contours of Eq. (16) and the induced drag parameter of the Klein
distribution.
Figures 3a and 3b demonstrate the distribution shapes that
correspond to traversing the major or minor principal axes of
Eq. (16). Each curve is labeled by its εRBM ; the corresponding εIBM is
not shown but can be found for the major and minor principal axes,
respectively, using
εIBM 25 p
2929
εRBM − 1 1
48
(24)
25 −
p
2929
εRBM − 1 1
48
Figure 3a shows distributions that are very similar to the Jones and
Prandtl distributions in Figs. 1a and 1b, whereas Fig. 3b shows
distributions that are radically different. The values of εRBM and εIBM
can be specified independently in the Klein distribution, but if they do
not correlate closely to align with the major principal axis defined in
Eq. (24), a very inefficient distribution will result.
The slope of the major principal axis is approximately 1.648. The
corresponding slopes of Eq. (19) and Eq. (21) are 1.6 and 1.667,
respectively. The fact that these three slopes and Figs. 3a, 1a, and 1b
are so similar indicates that there is little practical difference in
reasonable instantiations of the Klein distribution as compared to the
Prandtl and Jones distributions. Additionally, the Prandtl and Jones
distributions themselves are quite similar to one another.
III.
Off-Design Relations
The circulation distributions described in Sec. II are defined by
specifying bending moment constraints at the same lift coefficient at
which the induced drag is minimized. To generalize these methods to
include bending moment constraints at off-design CL s, the off-design
characteristics of a twisted wing must be considered. We provide a
summary of off-design behavior in this section.
To begin, we consider the additional circulation distribution
γ additional , which is the circulation distribution produced by the
untwisted wing. It is assumed that γ additional and CL both scale linearly
with angle of attack. Furthermore, because γ additional is determined by
the untwisted wing, it vanishes as CL vanishes. Therefore, we can
define the distribution γ a as
def
γ a y
a) Major principal axis
(25)
γ additional y
CL
which implies that γ a is fixed for a given planform and can be found
by calculating the circulation distribution of the untwisted wing at
a nominal angle of attack and then dividing by the resulting lift
coefficient. The subscript a will be used to denote parameters
associated with γ a .
When a circulation distribution is specified [e.g., Eq. (14)] it is
called the design distribution γ des , and the corresponding lift
coefficient is called CL;des . The wing is twisted to achieve γ des at
CL;des , and the scale of this twist is determined by CL;des. When
Eq. (14) is used as a design distribution, εRBM and εIBM are called
εRBM;des and εIBM;des , respectively.
We define γ d as
def
γ d y
γ des y
CL;des
Dividing γ des by CL;des removes the dependency on CL;des , which
serves to scale γ d to a lift coefficient of 1. The subscript d will be used
to denote parameters associated with γ d .
Finally, from the derivation presented in [11], with the
presumption of potential flow and a linear lift curve, a general
circulation distribution can be represented as
CL;des γ d y
CL − CL;des γ a y
γy
b) Minor principal axis
Fig. 3 Klein distributions following the major and minor principal axes
of Eq. (16).
(26)
In on-design conditions (CL CL;des ), Eq. (26) simplifies to
γ des y,
and in off-design conditions (CL ≠ CL;des ), Eq. (26)
γy
includes a contribution from γ a .
The implementation of Eq. (26) is shown in Figs. 4a and 4b for a
Jones distribution with εRBM 0.9 and a trapezoidal wing with
AR 7, λ 1, and Λ 30 deg. The curves in Fig. 4a were
generated by varying CL;des while holding CL constant, and the
curves in Fig. 4b were generated by varying CL while holding CL;des
constant. The moment arm of each curve is indicated by the thick
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PATE AND GERMAN
and
cl y 0 CL;des cl;d y 0 CL − CL;des cl;a y 0 (29)
B. Nonlinear Relations
Except for the special case in which the design bending ratios
(εRBM;des and εIBM;des ) are equal to the bending ratios of the untwisted
wing (εRBM;a and εIBM;a ) the off-design bending ratios will be
nonlinear functions of CL .
Using Eq. (9) to replace CRBM in Eq. (27) with εRBM leads to
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εRBM CL;des
C
εRBM;d 1 − L;des εRBM;a
CL
CL
(30)
Note that εRBM;des does not depend on CL;des , which implies
εRBM;des εRBM;d . If εRBM;des εRBM;a then this equation simplifies
to imply that εRBM is always equal to εRBM;a . Otherwise, εRBM is
a nonlinear function of CL . The behavior of this relation is
demonstrated in Fig. 5 for a rectangular wing with an aspect ratio of
12. The label on each line indicates how the design root bending ratio
is related to the untwisted root bending ratio of this wing. The
integrated bending moment ratio can be expressed in the same way as
a) Design distributions for various design lift coefficients
with CL = 1
εIBM
CL;des
CL;des
ε
1−
εIBM;a
CL IBM;d
CL
(31)
C. Induced Drag
As shown in [11], the expression for a general circulation
distribution [Eq. (26)] can be used to write a general form of the
induced drag polar as
CD;i C2L
1 δ
πAR
(32)
where
b) Off-design distributions for various lift coefficients with
CL,des = 1
Fig. 4 The implementation of Eq. (26) for the Jones distributions.
black line labeled y 0 . In Fig. 4b, the moment arms of the curves with a
small lift coefficient are negative and are not shown.
A. Linear Relations
will be linear
Equation (26) implies that linear functions of γy
with CL and CL;des . For example, the root bending moment of a
general circulation distribution can be decomposed into contributions
from γ a and γ d as
CL;des 2
C
C
δd 2 L;des 1 − L;des δa–d
CL
CL
CL
2
C
1 − L;des δa
CL
δ
(33)
The terms δd and δa are the induced drag parameters of γ d and γ a ,
respectively, and δa–d is the induced drag parameter of the mutual
interaction between γ a and γ d (in the context of the mutual induced
drag theorem).
Z1
y
dy
yγ
CRBM AR
0
Z1
L;des γ d y
CL − CL;des γ a y
dy
yC
AR
0
Z1
Z1
d y
a y
dy CL − CL;des AR
dy
yγ
yγ
CL;des AR
0
0
Considering Eq. (2) this becomes
CRBM CL;des CRBM;d CL − CL;des CRBM;a
(27)
where CRBM;d and CRBM;a are the root bending moment coefficients
of γ d and γ a , respectively. Integrated bending moment and section lift
coefficient can be decomposed similarly as
CIBM CL;des CIBM;d CL − CL;des CIBM;a
(28)
Fig. 5 Off-design εRBM of a rectangular wing with AR 12 and
εRBM;a 1.073. Each line corresponds to a different εRBM;des with
CL;des 0.5.
942
PATE AND GERMAN
coefficients for the root bending constraint and the integrated bending
constraint as CL;RBM and CL;IBM , respectively, and the cruise lift
coefficient at which induced drag should be minimized as CL;cruise .
We define the maneuver ratio for the root bending moment
constraint as
σ RBM CL;RBM
CL;cruise
and we define the maneuver ratio for the integrated bending moment
constraint as
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σ IBM a) The
RBM
CL;IBM
CL;cruise
If the bending constraints are imposed at the same maneuver lift
coefficient, or when only one is specified, the subscript is omitted,
and σ is referred to as simply the maneuver ratio.
Equation (32) indicates that twisting a wing for minimum drag at a
high CL;RBM will likely cause unnecessarily high-induced drag at a
lower CL;cruise due to the quadratic term. However, it is possible to
translate the lift coefficient for the bending constraint by leveraging
the linearity of CRBM and CIBM with CL , which is demonstrated by
Eqs. (27) and (28). In particular, the bending constraint CRBM;given at
CL;RBM can be translated to CRBM;des at CL;des as
derivative from Eq. (34)
CRBM;des CRBM;given ∂CRBM
CL;des − CL;RBM ∂CL
(35)
and, similarly, the bending constraint CIBM;given at CL;IBM can be
translated to CIBM;des at CL;des as
CIBM;des CIBM;given b) The
Fig. 6
IBM
derivative from Eq. (34)
According to [11], the parameter δa–d varies linearly with γ d and,
therefore, it depends linearly on εRBM and εIBM . Additionally, δa–d is
zero when both εRBM 1 and εIBM 1. The implication is that, for a
given wing, the relationship between δa–d , εRBM , and εIBM for the
Klein distribution [Eq. (14)] can be expressed in the form
δa–d
(36)
Translating the bending constraints represents determining the
bending moment at CL;des that corresponds to the specified bending
moment at the desired maneuver CL. Equations (27) and (28) imply
that the partial derivatives ∂CRBM ∕∂CL and ∂CRBM ∕∂CL in Eqs. (35)
and (36) can be expressed as
The derivatives from Eq. (34) for several trapezoidal planforms.
∂δ
∂δ
εRBM − 1 a–d εIBM − 1 a–d
∂εRBM
∂εIBM
∂CIBM
CL;des − CL;IBM ∂CL
(34)
The two derivatives in Eq. (34) depend only on the additional
distribution, which means that they depend only on the wing
planform. Figures 6a and 6b show the value of these derivatives for
several trapezoidal planforms. The data for these figures were
calculated using a discrete Weissinger model [10,20]. Calculating
these derivatives required only one execution of the model using a
finite difference scheme based on the fact that δa–d 0 when εRBM 1 and εIBM 1.
To summarize, Eq. (32) gives the induced drag of a given wing for
any Jones, Prandtl, or Klein distribution at any CL and CL;des . The
∂δa–d
a–d
terms δa , ∂ε∂δRBM
, and ∂ε
can be calculated from three seperate
IBM
aerodynamic analysis executions.
∂CRBM
2
ε
CRBM;a 3π RBM;a
∂CL
(37)
∂CIBM
1
CIBM;a εIBM;a
16
∂CL
(38)
where εRBM;a and εIBM;a are the bending ratios of γ a . Therefore, these
two partial derivatives can be calculated for a given planform using
one calculation from an aerodynamics model to compute γ a . The
parameters εRBM;a and εIBM;a depend only on the planform, and some
example values are demonstrated in Figs. 7a and 7b.
When the constraints are defined using the bending ratios and the
maneuver ratios, Eqs. (35) and (36) become
εRBM;des σ RBM εRBM;given 1 − σ RBM εRBM;a
(39)
εIBM;des σ IBM εIBM;given 1 − σ IBM εIBM;a
(40)
Equations (39) and (40) can be directly substituted into Eq. (14) along
with CL;des to give the solution for minimum induced drag at CL;des
subject to bending constraints at independent lift coefficients.
A. Method of Generalization
IV.
Generalization for Constraints at Arbitrary
Lift Coefficients
Typically, the limiting root bending moment occurs at a maneuver
condition corresponding to a high lift coefficient, but the induced
drag should be minimized at a lower lift coefficient, such as a
representative cruise condition [14,17]. We will refer to the lift
The method for generalizing the problem of induced drag
minimization to allow bending constraints at arbitrary lift coefficients
is summarized as follows:
1) Given a typical cruise condition (CL;cruise ) and a limiting
maneuver condition (εRBM;given at CL;RBM and εIBM;given at CL;IBM ).
2) Use Eqs. (2), (3), (9), and (10) to calculate εRBM;a and εIBM;a .
This requires one calculation from an aerodynamics model, such as a
943
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PATE AND GERMAN
a)
RBM,a
b)
IBM,a
Fig. 7
for use in Eq. (39)
for use in Eq. (40)
εRMB;a and εIBM;a for a set of trapezoidal planforms.
vortex lattice method, for the untwisted planform at a nonzero angle
of attack to obtain γ a .
3) Use Eqs. (39) and (40) to translate the bending constraints to
εRBM;des and εIBM;des at CL;des , which is likely chosen to be CL;cruise .
4) Use Eq. (14) with CL;des , εRBM;des , and εIBM;des to find the design
circulation distribution that minimizes induced drag at CL;des subject
to the bending moment constraints.
5) Finally, twist the wing to obtain the design distribution, where
the spanwise twist angle is the sum of the geometric and aerodynamic
twist as measured by the zero lift angle of attack relative to the root.
When using a lifting line method, the geometric and aerodynamic
twist can be represented explicitly in the governing equation (e.g.,
[8,21]). When employing a method that is not based on lifting line
theory, a technique, such as that presented in [22], can be used to
determine the required twist.
The outcome of this method is, therefore, a design lift distribution
and a corresponding wing twist distribution. Because only the total
twist is defined, there is still freedom to control the balance between
aerodynamic and geometric twist as needed for other wing design
considerations.
B. Mach Number
In some cases, the Mach numbers for the cruise and maneuver
flight conditions may coincide, but in general, the Mach numbers
may differ. Equations (39) and (40) use the linear relationship
between the bending moments and CL , but the bending moments are
nonlinear with Mach number. According to the Prandtl–Glauert
compressibility correction [23], the
effect of compressibility is to
p
stretch the wing by a factor of 1∕ 1 − M2 in the stream direction.
This change in effective geometry causes a change in the bending
a)
b)
Fig. 8 The variation of the bending moments with Mach number for
various trapezoidal wings with CL 1.
moments obtained at a given lift coefficient. However, this change is
relatively small and, therefore, is not included in the presented
method. The variation in εRBM and εIBM with Mach number is shown
for several trapezoidal wings in Figs. 8a and 8b. The bending
moments are calculated at CL 1. Note that the labels on the vertical
axis do not go to zero, as the variation with Mach number is small.
V.
Example Application
Recall that the original problem statement in [5] was to minimize
induced drag at a specific lift coefficient at which bending moment
constraints were also specified. The generalization in Sec. IV allows
for induced drag to be minimized at a different lift coefficient that is
independent of the lift coefficients of the bending constraints. The
purpose of this section is to illustrate the drag penalty for minimizing
induced drag at the maneuver lift coefficient, which is the condition at
which the structural constraints are likely to be specified.
Section III developed relations for the bending ratios and induced
drag for an off-design lift coefficient. Whereas Eqs. (27) and (28)
showed that CRBM and CIBM are linear with CL , Eqs. (30) and (31)
showed that εRBM and εIBM are not. Figure 5 illustrated this nonlinear
behavior for εRBM for off-design values of CL . Using these values of
εRBM in Eq. (22) gives the on-design induced drag behavior for the
Jones distribution and is shown in Fig. 9. These curves do not
represent the drag that would be realized by a wing with fixed twist.
Rather, they each describe the locus of minima for a wing with a given
root bending moment, meaning the induced drag parameter that
would be realized if CL;des had been chosen at a given CL . This
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944
PATE AND GERMAN
Fig. 9 On-design induced drag parameter of a rectangular wing with
AR 12 and εRBM;a 1.073. Each curve corresponds to a different
εRBM;des with CL;des 0.5.
a) Induced drag parameter for several values of CL ,des
on-design induced drag serves as a basis for comparing the off-design
induced drag.
Figure 2 demonstrated that εRBM and εIBM must correlate closely to
avoid large drag penalties, and because Sec. II.B showed that the
Jones and Prandtl distributions are similar, the following example
illustrates only the Jones distribution and a corresponding root
bending moment constraint. Without loss of generality CL;RBM is,
henceforth, selected to be unity.
For the example, we consider the planform shown in Fig. 10 that is
similar to that of a Boeing 737, with an approximate aspect ratio of
9.2, taper ratio of 0.26, and quarter-chord sweep angle of 25 deg. The
aerodynamic parameters for this planform were calculated with a
discrete Weissinger model [10,20].
A. Matching the Root Bending Moment of the Untwisted Planform
The first case considered is to twist the wing using a Jones
distribution, such that the root bending moment is the same as that of
the untwisted planform. Figures 11a and 11b show the results for
twisting the wing to four different design lift coefficients: 0.25, 0.5,
0.75, and 1, which correspond to maneuver ratios of 4, 2, 4∕3, and 1.
The curve corresponding to CL;des 1 represents twisting the wing
to minimize drag at the maneuver lift coefficient, which does not
involve using the generalization presented in Sec. IV. The dotted line
describes the induced drag parameter of the on-design distribution at
the corresponding CL and serves as the greatest lower bound for
the twisted wings. For comparison, the dashed line shows the induced
drag parameter of the untwisted wing. Figure 11a shows the induced
drag parameter and Fig. 11b shows the percent increase in induced
drag as compared to the on-design induced drag (the dotted line in
Fig. 11a). The curve corresponding to CL;des 1 shows the penalty
for not using the proposed generalization and, instead, optimizing
drag at the same CL as the bending constraint. The penalty for
minimizing induced drag at CL;des 1 when cruising at CL 0.5
would be approximately a 1.2% increase.
b) Percent difference between CD,i of the twisted wings and the
CD,i of the design distribution
Fig. 11 Comparison of four design lift coefficients for the B-737 with
εRBM equal to that of the untwisted wing.
the figure. The on-design drag is not shown because it is
indistinguishable from the other curves. Figure 12b indicates that the
penalties for induced drag are similar to those from Fig. 11b.
C. Examining a Range of Root Bending Moment Reductions
Instead of observing off-design behavior by varying CL and CL;des ,
we now examine the penalty for minimizing drag at the maneuver lift
coefficient instead of the cruise lift coefficient. In particular, we
compare the drag for CL;des CL;cruise to the drag for CL;des CL;RBM 1 for a range of specified root bending moment ratios, all
calculated at CL CL;cruise . Figure 13 displays these penalties for
CL;cruise 0.25, 0.5, and 0.75, which correspond to σ 4, 2, and
4∕3 (with CL;des CL;cruise ). The penalty is calculated as
B. Reducing the Root Bending Moment of the Untwisted Planform
The second case considers a 10% reduction in root bending
moment from that of the untwisted wing. As suggested previously by
Fig. 9 the on-design induced drag parameter is no longer independent
of CL , because εRBM varies nonlinearly with CL . This explains the
different behavior seen in Fig. 12a as compared to the previous case.
Although the curves for each CL;des appear to be very close together,
the vertical scale is large. Differences can be seen in the inset in
Fig. 10 B-737 planform.
100
CD;i C
L;des CL;RBM
CD;i C
− CD;i C
L;des CL;cruise
L;des CL;cruise
CL CL;cruise
Figure 13 demonstrates the fact that very large penalties in induced
drag occur when the cruise lift coefficient is much lower than the
maneuver lift coefficient. On the other hand, if the two lift coefficients
are close, the penalty is small. Because the penalty depends on the
maneuver ratio σ CL;RBM ∕CL;cruise , the curve marked by σ 4
would also indicate the penalty when CL;cruise 0.375 and
CL;RBM 1.5, for example. The dotted line in Fig. 13 marks the root
bending moment ratio of the untwisted wing (εRBM;a ). This figure
indicates that potentially large induced drag penalties can be expected
945
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PATE AND GERMAN
a) Induced drag parameter
Fig. 14 Variation of section lift coefficient with εRBM;given for the B-737
planform.
presented in Sec. IVon the section lift coefficient, consider Eq. (29),
which is restated for convenience here:
cl y 0 CL;des cl;d y 0 CL − CL;des cl;a y 0 The section lift coefficient cl;d y0 is calculated based on the Jones
distribution as
q
4ly 0 cl;d y 0 3 − 2εRBM;des 1 − y 20
πAR
6εRBM;des − 1y 20 cosh−1 1∕jy 0 j
(41)
b) Percent difference between CD,i of the twisted wings and the
CD,i of the design distribution
Fig. 12 Comparison of four design lift coefficients for the example wing
with εRBM reduced 10% from that of the untwisted wing.
if the suggested generalization is not used, and the wing is instead
twisted to minimize drag at the maneuver lift coefficient.
D. Section Lift Coefficient
Along with wing weight, it is also important to consider the section
lift coefficients near the wingtip to indicate whether the design should
be altered to avoid tip stall. To examine the effect of the generalization
Fig. 13 The percent increase in induced drag at CL CL;cruise in terms
of σ (maneuver ratio) and εRBM;given as compared to a design for σ 1.
where εRBM;des is found from Eq. (39) with CL;RBM , CL;des , and
εRBM;given . The section lift coefficient cl;a y 0 results from analysis of
the untwisted wing. Figure 14 illustrates the results for the sample
wing with y 0 0.85, CL;RBM 1, and CL;des 0.5. If a constraint
on section lift coefficient is specified for a certain condition, such as
the maneuver, Fig. 14 and the previous equations can be used to
determine the limit for εRBM;given. For example, if cl 0.85 < 0.9 at
CL 1, then εRBM;given < 0.9.
VI.
Conclusions
Prandtl, Jones, and Klein and Viswanathan each solved a variation
of the problem of determining the design circulation distribution to
minimize induced drag subject to bending moment constraints,
which are assumed to be surrogates for constraints on wing weight. In
this paper, these solutions were presented in terms of the bending
ratios εRBM and εIBM , which, respectively, relate the root and
integrated bending moments to those of an elliptic distribution with
the same lift. A significant disadvantage of these classical formulations is the fact that the bending moment constraints are applied at
the same lift coefficient for which induced drag is minimized. In
practical design contexts, the bending constraints should be applied
at a maneuver lift coefficient, and induced drag should be minimized
at a representative cruise or loiter lift coefficient, which is much
lower. A wing with fixed twist can potentially pay a large penalty in
induced drag if the classical methods are used in a way that does not
manage the discrepancy between these two lift coefficients.
This paper has presented a generalization of these classical
methods to allow the bending moment constraints to be specified at
independent lift coefficients, thereby increasing the applicability of
these techniques for aircraft design. First, εRBM and εIBM were
introduced as natural parameters to define the distributions given by
Prandtl, Jones, and Klein and Viswanathan. Next, it was shown that
the induced drag of a given wing twisted to a Klein distribution for
any CL, CL;des , εRBM , and εIBM can be expressed using only three
executions of an aerodynamics model, such as a vortex lattice
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946
PATE AND GERMAN
method. The generalization was then derived by using the linearity of
CRBM and CIBM with CL to translate the bending constraints from
CL;RBM and CL;IBM to CL;des . Finally, the penalty for applying the
classical techniques without this generalization was demonstrated for
a wing similar to that of a Boeing 737. The resulting induced drag
penalty for minimizing drag at a maneuver lift coefficient of 1.0
instead of a cruise lift coefficient of 0.5 was approximately 1.2%.
This new generalized method can be used in the early phase of
wing design when the planform size and shape is still being
determined. Given a mission and maneuver condition, wing twist can
now be controlled directly by just one or two design variables (εRBM ,
εIBM , or both), instead of representing wing twist as a piecewiselinear composition or curve defined by several control points. In
addition to reducing the number of design variables, the formulation
also serves to eliminate the need for numerical optimization for
minimizing induced drag because the optimal results are presented
analytically in closed-form algebraic expressions. Although the
present method considers changing only wing twist, it could be used in
the context of a more general wing planform optimization approach.
The method involves several presumptions that do not
fundamentally limit its generality. It is presumed that the wing is
rigid, operates in a potential flow, and has a linear lift curve. It is also
presumed that the bending moment ratios of the untwisted wing do
not vary significantly with Mach number. The most significant
presumption in the application of these methods to problems in
aircraft multidisciplinary design is that root bending moment or
integrated bending moment suffices as a surrogate for wing weight.
However, this presumption does not influence the validity of the
mathematical approach for the generalization presented in the
paper. The concept of translating bending constraints is valid for
other linear functions of the circulation distribution. Additionally, the
generalized method allows multiple load conditions to be imposed at
independent lift coefficients and then translated to a common lift
coefficient to find the single constraint that is active.
Acknowledgment
The authors would like to recognize Rob McDonald of California
Polytechnic State University, San Luis Obispo, for his helpful
suggestions throughout this work.
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