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Parametric Roll Maneuverability Analysis of a High-Aspect-Ratio-Wing Civil
Transport Aircraft
Conference Paper · January 2020
DOI: 10.2514/6.2020-1191
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This is a preprint of the following article:
Cristina Riso, Divya Sanghi, Carlos E. S. Cesnik, Fabio Vetrano, Patrick Teufel. Parametric Roll
Maneuverability Analysis of a High-Aspect-Ratio-Wing Civil Transport Aircraft, 2020 AIAA SciTech
Forum, Orlando, FL, January 6–10, 2020.
The published article may differ from this preprint and is available at:
https://arc.aiaa.org/doi/10.2514/6.2020-1191.
Parametric Roll Maneuverability Analysis of a
High-Aspect-Ratio-Wing Civil Transport Aircraft
Cristina Riso1 , Divya Sanghi1 , Carlos E. S. Cesnik1 , Fabio Vetrano2 , and Patrick Teufel2
1 Department of Aerospace Engineering, University of Michigan, Ann Arbor, MI, 48109
2 Airbus Operations S.A.S.
Abstract
Next-generation civil transport aircraft may take advantage of high-aspect-ratio wings for
lower induced drag. However, these high-aspect-ratio wings are very flexible and may
degrade aircraft roll maneuverability. This paper analyzes the roll maneuverability of a
high-aspect-ratio-wing civil transport aircraft derived from a contemporary configuration
with regular wing aspect ratio. Roll maneuverability is analyzed using the University of
Michigan’s Nonlinear Aeroelastic Simulation Toolbox for capturing geometrically nonlinear
effects in the aircraft aeroelastic response and their influence on the flight dynamic response.
Results for parametric variations of wing stiffness and mass distributions are presented to
discuss design guidelines for improving roll maneuverability in aircraft with high-aspect-ratio
wings.
Contents
1
2
3
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Theoretical Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
2
2.1
2.2
Fundamental Description . . . . . . . . . . . . . . . . . . . . . . . . . . .
Coupled Flight Dynamic and Nonlinear Aeroelastic Equations of Motion . .
3
4
Flexible and Very Flexible Aircraft Numerical Models . . . . . . . . . . . . . .
4
3.1
3.2
3.3
4
5
High-Fidelity Structural Models . . . . . . . . . . . . . . . . . . . . . . .
Low-Order Structural Models . . . . . . . . . . . . . . . . . . . . . . . . .
Low-Order Transonic Aerodynamic Models . . . . . . . . . . . . . . . . .
5
6
10
Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13
4.1
4.2
Static Roll Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Dynamic Roll Response . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13
16
Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20
.1
.2
24
25
Equivalent Beam Mass Model . . . . . . . . . . . . . . . . . . . . . . . .
Equivalent Beam Stiffness Model . . . . . . . . . . . . . . . . . . . . . . .
1
1
Introduction
The next-generation of civil transport aircraft may take advantage of high-aspect-ratio wings
(HARWs) that increase aerodynamic efficiency by developing lower induced drag. However,
these HARWs undergo larger structural deflections than conventional wings with regular
aspect ratios. Large structural deflections reduce the change in wing lift for given aileron
input, so degrading aircraft roll maneuverability [1, 2] to the point it may be inadequate
for certification standards. Therefore, it is desirable to investigate the impact of HARWs
on aircraft roll maneuverability and the benefits that can be achieved by tailoring the wing
stiffness and mass distributions.
Despite the recent interest in HARW aeroelasticity [3] and in passive aeroelastic tailoring [4], most of previous research focused on aircraft performance or stability [5, 6, 7, 8]
while only few studies investigated roll maneuverability. Gibson et al. [9] maximized the
aileron reversal speed of a wingbox by reinforcing its upper skin using topology optimization.
Kitson et al. [10] studied the static and dynamic roll response of a modified version of the
Common Research Model (CRM) [11] having a wing aspect ratio of 13.5. More specifically,
they observed an improved roll maneuverability when considering a tow-steered composite
HARW wingbox in place of a more flexible metallic wingbox with the same aspect ratio.
Motivated by the growing interest in HARW configurations for civil transport aviation,
this paper investigates the roll maneuverability of a HARW aircraft model representing a
hypothetical next-generation civil transport aircraft. To this aim, a very flexible aircraft (VFA)
test case with a HARW is developed from the Airbus XRF1 flexible aircraft (FA) test case. The
high-fidelity MSC Nastran global finite element model (GFEM) of each test case is converted
to a fully flexible equivalent beam-type representation for analyzing roll maneuverability
in the University of Michigan Nonlinear Aeroelastic Simulation Toolbox (UM/NAST) [12].
Firstly, the roll maneuverability of the FA and VFA test cases are compared to highlight the
impact of introducing a HARW in an otherwise identical aircraft design. Next, results for
parametric variations of wing stiffness and mass distributions are presented to identify design
guidelines for improving roll maneuverability in the next generation of HARW aircraft.
The paper is organized as follows. Section 2 summarizes the formulation implemented
into the UM/NAST framework [12]. Section 3 introduces the UM/NAST numerical models
of the FA and VFA test cases used in this study. Section 4 compares the FA and VFA roll
maneuverability and investigates the impact of parametric variations in the wing stiffness
and mass distributions. A section of concluding remarks summarizes the identified passive
aeroelastic tailoring guidelines for improving roll maneuverability in HARW aircraft.
2
Theoretical Background
This section summarizes the coupled flight dynamic and nonlinear aeroelastic formulation
implemented into UM/NAST [12], which is used as the computational framework for the roll
maneuverability studies of this paper.
2
wz
wx
p
Bx
Bz
By
wy
pB
Gz
Gx
Gy
Figure 1: Global, body, and local frames used for describing aircraft motion.
2.1
Fundamental Description
UM/NAST is a low-order computational framework to analyze aircraft in free flight by
considering fully coupled rigid-body dynamics, nonlinear structural dynamics, and unsteady
aerodynamics.
Aircraft are described as systems of fully flexible, geometrically nonlinear beams following the strain-based formulation of Su and Cesnik [13]. The aircraft motion is described by
introducing a global inertial frame G, a global body frame B, and local frames w along the
flexible beam-type members representing the vehicle components. The origin and orientation
of these frames are illustrated in Fig. 1. The position of the B frame in the G frame is given
by the vector pB , which is resolved in the B frame. The B frame is defined such that Bx points
downstream, By points towards the right wingtip, and Bz is the cross product of Bx and By .
Each local frame w has origin at a point p along the reference beam axis of a vehicle member
and is oriented such that w x is along that axis, w y points to the leading edge, and wz is the
cross product of w x and w y .
The position and orientation of a point along the reference axis of vehicle component
with respect to the B frame are described by the following 12-component column vector:
h(s)T = p(s)T w x (s)T w y (s)T wz (s)T
(1)
where s is the local abscissa and p, w x , w y , and wz are resolved in the B frame. The formulation
of Su and Cesnik [13] solves for the structural dynamics using strains as the independent
degrees of freedom (DOFs). A strain-based beam-type finite element has three nodes and
four strain DOFs denoted by ε x , κ x , κ y , and κz . These are the local extensional strain and the
3
twist, out-of-plane, and in-plane bending curvatures of the reference beam axis, respectively,
which are assumed to be constant throughout the element. Once strains are known, nodal
positions and orientations are recovered by integrating kinematic relations [12, 13] from the
start to the end of each beam-type member.
2.2
Coupled Flight Dynamic and Nonlinear Aeroelastic Equations of
Motion
The coupled equations of motion for the rigid-body and elastic DOFs can be cast as [12]
MFF MF B εÜ
CFF CF B εÛ
KFF 0 ε
R
+
+
= F
(2)
MBF MBB βÛ
CBF CBB β
0 0 b
RB
1
ζÛ = − Ωζ ζ
2GB Û
PB = C 0 β
(3)
where the independent DOFs and their time derivatives are listed in the column vectors



ε



ε
q=
= pB
b

θB 

 



 εÛ 



εÛ
qÛ =
= vB
β

 ωB 

 





 εÜ 

εÜ
qÜ = Û = vÛ B
β


 ωÛ B 
 
(4)
In Eqs. (2) and (3), ε is the strain vector for the entire model, vB and ωB are the linear and
angular velocities of the B frame with respect to the G frame, the vector b consists of their
integrals, and pB and the vector of quaternions ζ give the position and orientation of the
B frame with respect to the G frame. The matrices MFF , MF B , MBF , and MBB in Eq. (2)
are generalized mass matrices, CFF , CF B , CBF , and CBB are generalized damping matrices,
and KFF = K is the constant stiffness matrix of the constitutive relation of the strain-based
formulation [13]. The generalized load vectors RF and RB consist of loads due to initial
strain, gravity, and point or distributed forces and moments. Aerodynamic loads are modeled
as distributed loads and can be computed using various formulations available in UM/NAST
such as the two-dimensional finite-state unsteady thin airfoil theory of Peters et al. [14], the
unsteady vortex lattice method [15], and the method of segments [16]. When aerodynamic
loads are computed using the Peters’ theory, Eqs. (2) and 3 are augmented with additional
first-order equations for describing the dynamics of inflow states that account for unsteady
wake effects. Finally, Ωζ is a matrix relating the quaternion vector with its derivative and
C GB is a rotation matrix from the B to the G frame. Details on the derivation of Eq. (2)
and (3) are reported in Refs. [12, 13].
3
Flexible and Very Flexible Aircraft Numerical Models
This section describes the FA and VFA test cases considered for the roll maneuverability
studies of this paper. Starting from the high-fidelity GFEMs of these test cases, low-order
beam-type representations are developed in UM/NAST to quantify roll maneuverability by
means of fully coupled flight dynamic and nonlinear aeroelastic analyses. The beam-type
4
(a) Wingbox model
(b) Full-vehicle model
Figure 2: GFEMs of the FA (black) and VFA (blue) test cases.
representations of the FA and VFA test cases are verified by comparing modal results and
aeroelastic trim results from UM/NAST with reference solutions from MSC Nastran modal
analyses and from MSC Nastran/CFD aeroelastic trim analyses.
3.1
High-Fidelity Structural Models
The FA test case used in this paper is the Airbus XRF1, an industrial standard multidisciplinary
research test case representing a typical contemporary configuration for a long-range widebody civil transport aircraft. A HARW derivative of the XRF1, named XRF1-HARW, is
developed from the available XRF1 GFEM to obtain a VFA test case representing a nextgeneration HARW vehicle of the same category. The FA and VFA GFEMs are shown in
Fig. 2 and differ only for the wingbox. The HARW wingbox in the VFA GFEM was designed
in a previous aerostructural optimization [17] conducted at the University of Michigan.
The inertial properties and natural frequencies of the VFA and FA test cases (in vacuum)
are compared in Tables 1 and 2 for a typical half-loaded mass case1. Trends are similar for
1OOP = out-of-plane bending, IP = in-plane bending.
5
Table 1: VFA to FA test case inertial property comparison (half-loaded case).
Property
Property variation
Mass
Longitudinal CG position
Lateral CG position
Vertical CG position
Roll moment of inertia
Pitch moment of inertia
Yaw inertia moment of inertia
+3.19%
0.60%
−3.10%
3.54%
44.37%
1.40%
10.13%
Table 2: VFA to FA test case free-free natural frequency comparison (half-loaded case).
Mode
Frequency variation
Wing OOP bending (sym)
Wing OOP bending (asym)
Wing/pylon OOP bending (sym)
Wing/pylon OOP bending (asym)
Wing IP bending (asym)
Wing/pylon/fuselage OOP bending (sym)
−41.42%
−33.40%
−20.43%
−18.43%
−18.20%
−14.47%
other mass cases. The frequency variations refer to the first six free-free elastic modes shown
in Fig. 3. The VFA shows +44% and +10% increases in the roll and yaw moments of inertia,
respectively, and significantly lower natural vibration frequencies.
3.2
Low-Order Structural Models
Roll maneuverability is analyzed using UM/NAST [12] for capturing geometrically nonlinear
effects in the aircraft aeroelastic response and their impact on the flight dynamic response.
This requires converting the GFEM of each aircraft test case to an equivalent beam-type
representation compatible with the UM/NAST strain-based formulation.
The Fem2Stick (F2S) code was developed at the University of Michigan [18] for extracting
spanwise stiffness and mass distributions from a GFEM for creating its beam-type counterpart
in UM/NAST [18]. F2S allows to effectively model wing-type flexible members as equivalent
beams [18, 10]. However, it assumes infinite axial stiffness and neglects couplings between
axial and bending/torsion flexibility. These assumptions make F2S not accurate when applied
to tube-type components like fuselage segments or pylons. For developing fully flexible
models of the FA and VFA test cases in UM/NAST, an enhanced GFEM-to-beam model
order reduction framework was developed [19] that also identifies the axial stiffness spanwise
distribution along with the distributions of the axial/bending and axial/torsion coupling terms.
Details on the GFEM-to-beam model order reduction process are provided in Appendix.
The GFEM-to-beam model order reduction procedure is applied to the isolated GFEM
of each FA member for obtaining the equivalent beam stiffness and mass properties. Next,
6
(a) Symmetric wing OOP bending (mode #1)
(b) Antisymmetric wing OOP bending (mode #2)
(c) Symmetric wing/pylon OOP bending (mode (d) Antisymmetric wing/pylon OOP bending
#3)
(mode #4)
(f) Symmetric wing/pylon/fuselage OOP bending
(e) Antisymmetric wing IP bending (mode #5) (mode #6)
Figure 3: VFA test case free-free mode shapes (half-loaded case).
Figure 4: Beam-type representation of the FA (black) and VFA (blue) test cases.
7
Table 3: MSC Nastran vs. UM/NAST normalized FA wing natural frequencies (half-loaded
case).
Mode #
Mode type
MSC Nastran
UM/NAST
Error
1
2
3
4
5
6
7
8
9
10
OOP bending
OOP bending
IP bending
OOP bending
Torsion
IP bending
OOP bending
Torsion
OOP bending
IP bending
1.00
2.76
3.21
6.14
6.89
8.12
9.81
11.94
13.71
16.01
0.99
2.73
3.22
6.11
7.06
8.54
10.17
11.98
15.13
17.33
−1.38%
−0.93%
0.37%
−0.48%
2.52%
5.17%
3.63%
0.32%
10.32%
8.24%
Table 4: MSC Nastran vs. UM/NAST normalized VFA wing natural frequencies (half-loaded
case).
Mode #
Mode type
MSC Nastran
UM/NAST
Error
1
2
3
4
5
6
7
8
9
10
OOP bending
OOP bending
IP bending
OOP bending
IP bending
Torsion
OOP bending
Torsion
IP bending
OOP bending
0.56
1.38
1.59
2.78
3.95
4.76
4.79
6.93
8.45
9.29
0.57
1.42
1.61
2.94
4.06
4.76
5.05
7.00
7.70
8.81
0.76%
2.82%
1.22%
5.42%
2.67%
0.05%
5.52%
1.09%
−8.78%
−5.15%
8
Table 5: MSC Nastran vs. UM/NAST normalized FA full-vehicle natural frequencies (halfloaded case).
Mode #
Mode type
MSC Nastran
UM/NAST
Error
1
2
3
4
5
6
7
8
9
10
Fuselage bending
Wing OOP bending (sym)
Wing OOP bending (asym)
Fuselage bending
Left wing/pylon OOP bending
Right wing/pylon OOP bending
Right pylon OOP bending
Left pylon OOP bending
Fuselage bending
Fuselage bending
1.00
1.08
1.09
1.10
1.34
1.34
1.65
1.65
1.72
1.74
1.01
1.07
1.07
1.08
1.34
1.34
1.88
1.88
1.54
1.83
0.98%
−0.34%
−1.09%
−1.97%
0.40%
0.33%
14.30%
14.16%
−10.28%
4.91%
Table 6: MSC Nastran vs. UM/NAST normalized VFA full-vehicle natural frequencies
(half-loaded case).
Mode #
Mode type
MSC Nastran
UM/NAST
Error
1
2
3
4
5
6
7
8
9
10
Wing OOP bending (asym)
Wing OOP bending (sym)
Fuselage bending
Right wing/pylon IP bending
Left wing/pylon IP bending
Fuselage bending
Right wing/pylon OOP bending
Left wing/pylon OOP bending
Wing OOP bending (asym)
Wing OOP bending (sym)
0.62
0.62
1.00
1.06
1.07
1.10
1.33
1.33
1.54
1.54
0.62
0.62
1.01
0.99
0.99
1.08
1.21
1.21
1.54
1.54
−0.51%
−0.51%
0.93%
−6.59%
−7.70%
−1.21%
−9.08%
−9.36%
0.46%
0.53%
9
property distributions for the individual members are combined for forming the UM/NAST
full-vehicle model. The VFA model is developed by replacing the FA wing beam model
with the beam-type representation of the HARW, while the other vehicle members remain
unvaried.
The beam-type representations of the VFA and FA test cases are shown in Fig. 4. The mass
models are verified by comparing the vehicle total mass, center of mass, and inertia tensors
against the corresponding quantities evaluated from the GFEM. Results match, because the
GFEM-to-beam mass model reduction is numerically exact (see Appendix). The stiffness
models are verified by comparing the natural frequencies to the ones evaluated for the GFEM.
The MSC Nastran and UM/NAST natural frequencies for the first ten wing modes,
normalized by the first FA wing frequency, are compared in Tables 3 (FA) and 4 (VFA). Results
are for a typical half-loaded mass case. The MSC Nastran and UM/NAST natural frequencies
for the first ten full-vehicle modes, normalized by the first FA full-vehicle frequency, are
compared in Tables 5 (FA) and 6 (VFA). Results also are for a typical half-loaded mass case.
The MSC Nastran full-vehicle natural frequencies are computed by clamping the fuselage
cross-section at the wing reference axis location, because that point is kept fixed in the
UM/NAST modal analysis. The frequency error in the wing-dominated modes is about 15%, which increases to 10-15% for fuselage- and pylon-dominated modes. The larger errors
for these modes are due to the less accurate equivalent beam stiffness properties evaluated
for tube-type members due to their more complex geometries, features such as cut-outs,
and lower stiffness resulting in cross-sectional deformations that are not captured by a beam
representation. However, the full-vehicle natural frequency comparisons are acceptable for
the analyses of this paper that focuses on roll maneuverability.
3.3
Low-Order Transonic Aerodynamic Models
The FA and VFA aerodynamics is modeled using a low-order surrogate-based transonic aerodynamic model based on the method of segments [16]. This approach allows one to capture
three-dimensional transonic-flow effects with sufficient accuracy for roll maneuverability
analyses and low computational cost that facilitates parametric studies.
The method of segments describes a flexible lifting member as a collection of twodimensional rigid cross-sections along the span which undergo arbitrary three-dimensional
translations and rotations. The aerodynamic loads on each cross-section are provided by a
kriging surrogate model that returns the lift, drag, and moment coefficients for given local
instantaneous Mach number and effective angle of attack. The kriging surrogate model for
each cross-section is created by fitting a database of aerodynamic coefficients computed
from full-vehicle steady rigid CFD simulations for various flight Mach number M and body
angle of attack α conditions. Although the kriging surrogates are based on steady CFD data,
they are used with the local flow conditions during simulations. Therefore, this description
captures the effect of rigid-body motion and structural deflections on the aerodynamic loads,
although it lacks the modeling of unsteady lags due to compressibility and shed wake.
The kriging surrogates for the FA and VFA test cases are based on coefficient values from
steady rigid CFD solutions for M ∈ [0.6, 0.89] and α ∈ [−4, +5] deg and are created in the
Matlab DACE toolbox [20] using a zero-th order regression model and a linear correlation
model. The kriging surrogates for the lift, drag, and pitching moment coefficients at the
10
-0.4
-0.2
0
0.2
0.4
0.6
0.8
0.01
0.03
0.04
0.05
0.06
0.07
0.08
0.1
Coefficient value
1
Coefficient value
0.02
0.5
0
-0.5
5
0.05
0
5
0.9
0
Local AoA (deg)
0.9
0
0.8
-5
0.7
-10
0.6
0.8
-5
0.7
-10
Local AoA (deg)
Mach number
(a) Lift coefficient
0.6
Mach number
(b) Drag coefficient
-0.16
-0.14
-0.12
-0.1
-0.08
-0.06
Coefficient value
-0.05
-0.1
-0.15
-0.2
5
0.9
0
0.8
-5
Local AoA (deg)
0.7
-10
0.6
Mach number
(c) Moment coefficient
Figure 5: Kriging surrogates for the aerodynamic coefficients at the FA wing midspan node.
11
0.15
Normalized z coordinate
Normalized z coordinate
0.15
0.12
MSC Nastran/CFD
0.09
UM/NAST
0.06
Undeformed
0.03
0
0
0.2
0.4
0.6
0.8
0.12
0.09
MSC Nastran/CFD
0.06
Undeformed
0.03
0
1
UM/NAST
0
0.2
0.4
0.6
0.8
Normalized y coordinate
Normalized y coordinate
(a) Empty case
(b) Fully loaded case
1
Figure 6: MSC Nastran/CFD vs. UM/NAST FA aeroelastic trim deflection.
Table 7: MSC Nastran/CFD vs. UM/NAST FA aeroelastic trim parameters (empty case).
Empty weight
Parameter
MSC Nastran/CFD
UM/NAST
Error
Angle of attack (deg)
Elevator (deg)
Tip z-displ. (% semispan)
0.55
−2.11
2.36
0.51
−1.66
2.38
−7.22%
−21.18%
0.56%
FA wing midspan cross-section are shown in Fig. 5 as an example. The surrogates for each
spanwise location are included into the FA and VFA UM/NAST models and the complete
aeroelastic models are verified by comparing UM/NAST trim results with reference MSC
Nastran/CFD results. Comparisons for the FA are shown in Fig. 6 and Table 7 for a typical
cruise trim condition and two mass cases corresponding to the minimum and maximum
weight. The aircraft wing deformation is well captured by the UM/NAST model. The
difference in the trim parameters is motivated by the absence of fuselage aerodynamics in
UM/NAST, but it is acceptable for the scope of this study.
Table 8: MSC Nastran/CFD vs. UM/NAST FA aeroelastic trim parameters (fully loaded
case).
Fully loaded
Parameter
MSC Nastran/CFD
UM/NAST
Error
Angle of attack (deg)
Elevator (deg)
Tip z-displ. (% semispan)
2.37
−3.79
5.58
2.78
−4.25
5.65
17.31%
12.33%
1.35%
12
-0.1
WR2
0
Normalized x coordinate
Normalized x coordinate
-0.1
WR3-1
0.1
WR3-2
WR1
0.2
WR3-3
WR3-4
WR3-5
0.3
0.4
0.5
0.6
WR3-6
0
0.2
0.4
0.6
0.8
WR3-1
0.1
WR3-2
WR1
0.2
WR3-3
WR3-4
0.3
WR3-5
0.4
0.5
0.6
1
WR2
0
0
0.2
0.4
0.6
WR3-6
0.8
1
Normalized y coordinate
Normalized y coordinate
(a) FA wing
(b) VFA wing
Figure 7: FA and VFA wing planform subdivision for parametric studies.
4
Results
This section discusses static and dynamic roll maneuverability results for the FA and VFA
(half-loaded cases) with baseline wing properties and for parametric variations of the wing
stiffness and mass distributions.
For the parametric studies, the FA and VFA wings are subdivided into three regions denoted by WR1, WR2, and WR3, respectively. These regions are shown in Fig. 7 (coordinates
are normalized by the semispan of each wing). The WR1 and WR2 regions correspond to the
wing segments from the vehicle centerline to the wing-fuselage interface (WR1) and from the
wing-fuselage interface to the wing-pylon interface (WR2). The pylon is placed at the same
dimensional spanwise location in the FA and VFA, and thus at a more inboard normalized
location in the VFA. The WR3 region spans from the pylon to the wingtip and it is subdivided
into six smaller subregions named WR3-1 to WR3-6. The inboard and outboard ailerons are
located in WR3-4 and WR3-5, respectively, which occupy the same normalized spanwise
locations in the FA and VFA. Wing stiffness and mass properties are varied in the entire WR3
and in its subregions individually while property variations in the WR1 and WR2 regions do
not impact roll maneuverability.
4.1
Static Roll Response
The static roll response is analyzed by quantifying the roll aerodynamic moment increment
on the FA and VFA isolated wings with root-clamped boundary conditions for given antisymmetric inboard/outboard aileron deflection. Results are for Mach number M = 0.83,
altitude h = 8484 m, and root angle of attack α = 2 deg.
Results for the FA and VFA wings with baseline properties are shown in Fig. 8a. The
VFA wing experiences a smaller roll aerodynamic moment increment for given aileron
deflection even in the presence of a longer moment arm of the ailerons compared to the
FA. This is because the VFA wing is more flexible and thus each half-wing undergoes
larger positive/negative out-of-plane bending displacements due to downward/upward aileron
13
5
x10
VFA
outboard
aileron
Normalized spanwise center of lift
2
Roll moment (N-m)
0
FA
outboard
aileron
-2
FA
inboard
aileron
-4
VFA
inboard
aileron
-6
-8
-10
0
1
2
3
4
5
0.44
0.42
FA right half-wing
FA left half-wing
0.4
0.38
VFA right half-wing
0.36
VFA left half-wing
0.34
0.32
0
1
Aileron deflection (deg)
2
3
4
5
Aileron deflection (deg)
(a) Roll moment (inboard/outboard aileron deflec- (b) Center of lift position (inboard aileron deflection)
tion)
50
Aileron effectivness variation (%)
Aileron effectivness variation (%)
Figure 8: Static roll response of the baseline FA and VFA wings.
40
30
20
10
Reversed out-of-plane
bending/torsion
coupling
Zero out-of-plane
bending/torsion coupling
Torsion
0
-10
-20
Out-of-plane
bending/torsion
coupling
Out-of-plane
bending
-30
-40
-50
0.85
0.9
0.95
1
1.05
Scaling factor
1.1
1.15
(a) FA
50
40
Out-of-plane
bending
30
20
10
0
Out-of-plane
bending/torsion
coupling
Torsion
Zero out-of-plane
bending/torsion coupling
-10
-20
Reversed out-of-plane
bending/torsion
coupling
-30
-40
-50
0.85
0.9
0.95
1
1.05
Scaling factor
1.1
1.15
(b) VFA
Figure 9: Inboard aileron effectiveness variation for uniform stiffness scaling in the FA and
VFA WR3 regions.
14
deflections. For an aft-wing, the twist of aerodynamic cross-sections due to out-of-plane
bending displacements contrasts the lift variation commanded by the ailerons. Therefore,
the VFA wing develops a smaller roll aerodynamic moment (in magnitude) for given aileron
deflection compared to the FA. This trend is also shown by the variation in the center of lift
position along the span reported in Fig. 8b. For both configurations, inboard ailerons are
more effective than outboard ailerons due to their larger area and lower out-of-plane bending
displacements. The VFA wing also shows a moderately nonlinear static roll response when
deflecting either pair of ailerons which is not observed for the FA wing (see Fig. 8a).
Parametric static analyses focus on variations in the FA and VFA wing stiffness properties.
This is because mass properties only influence the static response through the self-weight,
which does not significantly impact the results. The varied stiffness properties are the torsion
and out-of-plane bending stiffness terms and the out-of-plane bending/torsion coupling term,
which have the highest influence on the static roll response.
The effect of stiffness property variations is quantified in terms of inboard aileron effectiveness, that is, the slope in the curves in Fig 8a. For the VFA, the aileron effectiveness
is computed by averaging slopes at different values of the inboard aileron deflection due to
the nonlinearity in the aileron response (see Fig. 8). The variation in aileron effectiveness
for uniform stiffness scaling in the entire WR3 is shown in Fig. 9. Increasing the torsion or
the out-of-plane bending stiffness improves aileron effectiveness for both the FA and VFA
wings. However, increasing the out-of-plane bending stiffness results in a more significant
improvement compared to increasing torsion, as expected for an aft-swept wing. Increasing
the out-of-plane bending stiffness reduces structural displacements in the presence of aileron
deflections, so mitigating the loss in aileron effectiveness due to the twist of aerodynamic
cross-sections induced by bending. These trends are observed for both the FA and VFA
wings. The VFA aileron effectiveness is more sensitive to variations in the out-of-plane
bending and torsion stiffness, as expected due to higher wing flexibility.
Varying the magnitude of the out-of-plane bending/torsion coupling does not have a significant impact, while reversing its sign improves the FA aileron effectiveness as shown in
Fig. 9. This is because the out-of-plane bending/torsion coupling term is mainly negative
in the FA WR3 region, resulting in a wash-out effect which is detrimental for aileron effectiveness. On the other hand, a positive coupling term causes a positive elastic twist for
upward out-of-plane bending, which is beneficial for aileron effectiveness. The out-of-plane
bending/torsion coupling term is already positive in most of the VFA WR3 region by design,
although improving aileron effectiveness was not a design goal [17]. Therefore, reversing the
coupling term sign reduces aileron effectiveness as shown in Fig. 9. As a further verification
of these trends, results for zero out-of-plane bending/torsion coupling are also shown in Fig. 9.
For the FA, zero coupling increases aileron effectiveness, although not as much as in the case
of reversed coupling (meaning positive coupling for this configuration). The opposite occurs
for the VFA where the coupling term is positive by design, and thus setting it to zero reduces
aileron effectiveness.
Results for fixed stiffness scaling factors applied to WR3 subregions are shown in Tables 9
and 10. Varying the torsion stiffness in one subregion at a time has a limited impact on
aileron effectiveness, particularly for the FA wing. This is expected because varying the
torsion stiffness in the entire WR3 does not have a large impact either. The largest aileron
effectiveness improvement is achieved by increasing the out-of-plane bending stiffness in the
15
Table 9: Inboard aileron effectiveness variation for stiffness scaling in FA WR3 subregions.
Aileron effectiveness variation (%)
Torsion stiffness
(10% increase)
Out-of-plane
bending stiffness
(10% increase)
Out-of-plane
bending/torsion
coupling
(reversed sign)
Uniform
1.93
6.76
13.70
WR3-1
WR3-2
WR3-3
WR3-4
WR3-5
WR3-6
0.55
0.87
−0.01
0.15
0.03
0.00
2.13
3.46
0.66
−0.13
−0.08
0.00
2.15
7.72
−0.02
−0.14
−0.16
0.00
WR3-1 and WR3-2 regions (around the midspan) for both the FA and VFA wings. Varying
the properties in other WR3 subregions has a smaller or negligible effect. Considering that
the FA wing has a negative out-of-plane bending/torsion coupling, reversing its sign in WR31 and WR3-2 also improves aileron effectiveness. In contrast, reversing the coupling term
sign in these regions degrades aileron effectiveness for the VFA wing.
4.2
Dynamic Roll Response
The dynamic roll response is analyzed by comparing the time histories of the roll rate and
roll angle2 developed for an antisymmetric deflection of the inboard ailerons applied to the
FA and VFA full-vehicle models trimmed for level flight at M = 0.83 and h = 8484 m.
Results for FA and VFA with baseline wing properties are shown in Fig. 10. Roll rates
and angles follow the flight dynamic convention (roll axis towards the vehicle nose). The FA
develops a larger roll rate and roll angle (in magnitude) at a given simulation time compared to
the VFA due to higher wing stiffness and smaller vehicle roll moment of inertia (see Table 1).
Furthermore, the roll rate of the VFA slightly increases at t = 1 s when the aileron starts to
deflect before becoming negative following the applied command, and it shows oscillations
that are not observed for the FA. These oscillations are also present in the wingtip vertical
displacement and can be explained by observing the in-plane tip displacement of the FA and
VFA wings. The in-plane tip displacement of the VFA wing has a higher amplitude and
smaller damping associated to it which causes the oscillations observed in the roll rate and
vertical tip displacement.
Dynamic parametric analyses for uniform scaling in the wing mass distribution are shown
in Fig. 11. Results for uniform scaling in other inertial parameters such as mass offset
components and moments of inertia are not reported because they have no significant impact
on the results. Scaling the mass distribution uniformly also has only a slight impact on the
2The quantitative measures of roll rate and roll angle are based off an assumed value of Clδ and as such the
results are for a qualitative comparison only.
16
Aileron deflection (deg)
6
5
4
3
2
1
0
0
1
2
3
4
5
Time (s)
(a) Right inboard aileron input
2
0.5
0
0
Roll rate (deg/s)
Roll angle (deg)
VFA
-0.5
-1
FA
-1.5
-2
VFA
FA
-2
-4
-6
-2.5
-3
0
1
2
3
4
-8
5
2
3
(b) Roll rate
(c) Roll angle
4
5
0.2
Tip displacement (% semi-span)
Tip displacement (% semi-span)
1
Time (s)
8
6
VFA
4
2
FA
0
-2
0
Time (s)
0.15
0.1
FA
0.05
VFA
0
-0.05
0
1
2
3
4
-0.1
5
Time (s)
0
1
2
3
4
5
Time (s)
(d) Out-of-plane wingtip displacement (with re- (e) In-plane wingtip displacement (with respect to
spect to trim)
trim)
Figure 10: Dynamic roll response of the baseline FA and VFA in free flight.
17
Table 10: Inboard aileron effectiveness variation for stiffness scaling in VFA WR3 subregions.
Aileron effectiveness variation (%)
Torsion stiffness
(10% increase)
Out-of-plane
bending stiffness
(10% increase)
Out-of-plane
bending/torsion
coupling
(reversed sign)
Uniform
10.56
20.88
−38.20
WR3-1
WR3-2
WR3-3
WR3-4
WR3-5
WR3-6
2.50
3.95
0.57
−1.70
0.21
−0.00
6.32
11.77
2.87
0.32
−0.57
0.00
−24.70
−10.74
−3.62
0.19
0.08
0.00
roll responses for both the FA and VFA. Therefore, it is concluded that the roll response is
primarily influenced by the wing stiffness, and region studies with respect to mass property
variations are not performed.
Next, the dynamic roll response is analyzed for variations in the wing stiffness properties
to confirm the static analysis trends. Results for uniform stiffness scaling in WR3 are shown
in Figs. 12 to 14. As in the static studies, practical variations in the torsion stiffness have
limited impact on the roll response. However, reducing the torsion stiffness causes the FA to
become unstable at the assumed flight condition, as shown by the growing oscillations in the
roll rate time history in Fig. 12a. Increasing the out-of-plane bending stiffness improves the
roll dynamic response for both the FA and VFA, resulting in a larger roll rate and roll angle
(in magnitude) developed at a given simulation time. Figure 14 compares the FA and VFA
results with reversed sign of the out-of-plane bending/torsion coupling term. As observed
in the static analyses, the FA roll response improves when reversing the coupling term sign,
which is negative in the outer wing. In contrast, the VFA response degrades when reversing
the coupling term sign since this is already positive in the baseline design.
Finally, Fig. 15 compares the dynamic roll responses of the FA and VFA obtained
considering the best-case stiffness properties from the previous studies. For the FA, the
out-of-plane bending stiffness is increased by 15% uniformly in WR3 and the sign of the outof-plane bending/torsion coupling term is reversed. For the VFA, the out-of-plane bending
stiffness is increased by 15% and with no sign change in the out-of-plane bending/torsion
coupling term. The applied stiffness variations improve the roll dynamic response for both
the FA and VFA, in line with the previous results. However, the VFA roll response remains
significantly slower than the one of the FA. This suggests that passive aeroelastic tailoring
alone is not sufficient to ensure adequate roll maneuverability in HARW aircraft for practically
achievable variations in the stiffness properties, but it has to be complemented with other
design modifications. For instance, this study assumed inboard and outboard ailerons to be
at the same normalized spanwise locations in the FA and VFA wings. Since the static results
showed that inboard ailerons are more effective for both configurations, a more inboard
18
0.5
0.5
Baseline
Scaling factor = 0.85
Scaling factor = 1.15
0
0
-0.5
Roll rate (deg/s)
Roll rate (deg/s)
-0.5
-1
-1.5
-1.5
-2
-2.5
-2
-2.5
0
1
2
3
4
-3
5
1
2
3
Time (s)
(a) FA roll rate
(b) VFA roll rate
2
2
0
0
-2
-4
-6
-8
0
Time (s)
Roll angle (deg)
Roll angle (deg)
-3
-1
4
5
4
5
-2
-4
-6
0
1
2
3
4
-8
5
0
1
2
3
Time (s)
Time (s)
(c) FA roll angle
(d) VFA roll angle
Figure 11: Dynamic roll response in free flight for uniform mass scaling in the FA and VFA
WR3 regions.
19
0.5
0
0
-0.5
Roll rate (deg/s)
-0.5
Roll rate (deg/s)
0.5
Baseline
Scaling factor = 0.85
Scaling factor = 0.90
Scaling factor = 1.10
Scaling factor = 1.15
-1
-1.5
-1.5
-2
-2.5
-2
-2.5
0
1
2
3
4
-3
5
1
2
3
Time (s)
(a) FA roll rate
(b) VFA roll rate
2
2
0
0
-2
-4
-6
-8
0
Time (s)
Roll angle (deg)
Roll angle (deg)
-3
-1
4
5
4
5
-2
-4
-6
0
1
2
3
4
-8
5
0
1
2
3
Time (s)
Time (s)
(c) FA roll angle
(d) VFA roll angle
Figure 12: Roll dynamic response in free flight for uniform torsion stiffness scaling in the FA
and VFA WR3 regions.
aileron placement in the VFA wing can help mitigating the roll maneuverability degradation
due to a longer wing span.
5
Concluding Remarks
This paper analyzed the roll maneuverability of a VFA civil transport aircraft model with
a HARW derived from a contemporary FA configuration. Analyses were conducted in the
UM/NAST framework to capture geometrically nonlinear effects in the aircraft aeroelastic
response and their impact on the flight dynamic response.
The aileron effectiveness and free-flight dynamic roll response of the FA and VFA test
cases were first compared to study the impact of introducing a HARW in an otherwise
identical aircraft design. Next, parametric variations of wing stiffness and mass distributions
were applied to the FA and VFA wings to study their impact on roll maneuverability.
The static analyses showed that the VFA wing has reduced aileron effectiveness compared
20
Baseline
Scaling factor = 0.85
Scaling factor = 0.90
Scaling factor = 1.10
Scaling factor = 1.15
0.5
0
0.5
0
-0.5
Roll rate (deg/s)
Roll rate (deg/s)
-0.5
-1
-1.5
-1.5
-2
-2.5
-2
-2.5
0
1
2
3
4
-3
5
1
2
3
Time (s)
(a) FA roll rate
(b) VFA roll rate
2
2
0
0
-2
-4
-6
-8
0
Time (s)
Roll angle (deg)
Roll angle (deg)
-3
-1
4
5
4
5
-2
-4
-6
0
1
2
3
4
-8
5
0
1
2
3
Time (s)
Time (s)
(c) FA roll angle
(d) VFA roll angle
Figure 13: Roll dynamic response in free flight for uniform out-of-plane bending stiffness
scaling in the FA and VFA WR3 regions.
21
0.5
0.5
0
0
-0.5
-0.5
Roll rate (deg/s)
Roll rate (deg/s)
Reversed coupling
-1
-1.5
Reversed coupling
-2.5
-2
-2.5
0
1
2
3
4
-3
5
1
2
3
Time (s)
(a) FA roll rate
(b) VFA roll rate
2
2
0
0
-2
Baseline
-4
Reversed coupling
-6
-8
0
Time (s)
Roll angle (deg)
Roll angle (deg)
-3
-1
-1.5
Baseline
-2
Baseline
4
5
Reversed coupling
Baseline
-2
-4
-6
0
1
2
3
4
-8
5
0
1
2
3
4
Time (s)
Time (s)
(c) FA roll angle
(d) VFA roll angle
5
Figure 14: Roll dynamic response in free flight for uniform out-of-plane bending/torsion
coupling term reversal in the FA and VFA WR3 regions.
22
0
0
-0.5
-0.5
Roll rate (deg/s)
0.5
Roll rate (deg/s)
0.5
-1
-1.5
-2
-3
Best stiffness
properties
0
1
2
3
4
-2
-2.5
-3
5
0
1
2
3
4
Time (s)
Time (s)
(a) FA roll rate
(b) VFA roll rate
2
5
0.5
0
0
Baseline
-0.5
Roll rate (deg/s)
Roll angle (deg)
Best stiffness
properties
-1
-1.5
Baseline
-2.5
Baseline
-2
Baseline
-4
-1.5
Best stiffness
properties
-6
Best stiffness
properties
-1
-2
-2.5
-8
0
1
2
3
4
-3
5
0
1
2
3
4
Time (s)
Time (s)
(c) FA roll angle
(d) VFA roll angle
5
Figure 15: Roll dynamic response in free flight for uniform out-of-plane stiffness scaling by
15% in the FA and VFA WR3 regions and reversed coupling term in the FA WR3 region
only.
23
to the FA wing due to its higher flexibility, which results in larger out-of-plane bending
displacements for a given aileron input. The aileron effectiveness of both the FA and VFA
wings improved by increasing the out-of-plane bending stiffness primarily around the wing
midspan. A smaller improvement was also achieved by increasing the torsion stiffness in the
same region. A positive out-of-plane bending/torsion stiffness coupling term is beneficial for
aileron effectiveness. For the FA where the out-of-plane bending/torsion stiffness coupling
term is mainly negative along the span, aileron effectiveness improved by reversing the
coupling term sign. In contrast, the VFA wing already has a mainly positive out-of-plane
bending/torsion stiffness coupling term along the span by design with a beneficial effect on
aileron effectiveness.
The dynamic analyses showed that the VFA full vehicle develops smaller roll rate and
angle (in magnitude) for a given simulation time when subject to the same aileron input
as the FA. Uniformly scaling the wing mass properties did not significantly impact the
roll response, while uniformly scaling the wing stiffness properties confirmed the trends
observed in the static analyses. However, the VFA did not recover the roll maneuverability
of the FA for practical stiffness property variations. Future work will investigate the benefits
of combined wing passive aeroelastic tailoring and varying aileron placement for improving
roll maneuverability in HARW aircraft.
Acknowledgments
The material of this paper is based upon work supported by Airbus in the frame of the AirbusMichigan Center for Aero-Servo-Elasticity of Very Flexible Aircraft. The authors would like
to thank Airbus for providing the XRF1 test case as a mechanism for demonstrating the
approaches presented in this paper.
Appendix: GFEM-to-Beam Model Order Reduction
This appendix details the procedure for converting high-fidelity GFEMs into equivalent strainbased beam-type representations compatible with the UM/NAST formulation. The procedure
is implemented into a GFEM-to-beam model order reduction framework developed at the
University of Michigan [19].
.1
Equivalent Beam Mass Model
A GFEM can be represented as a collection of lumped masses mi with positions xi + δi in
the global reference frame, where xi are the GFEM nodes positions and δi the mass offsets,
and lumped inertia tensors Ji with respect to xi + δi also given in the global reference frame
(i = 1, . . . , N, where N is the number of nodes).
The equivalent mass mref , mass offset δref , and inertia tensor Jref at a generic beam
reference axis node are obtained by summing the contributions from the Nnb GFEM nodes
in the nearest neighborhood:
mref =
Nnb
Õ
i=1
mi
δref
Nnb
1 Õ
mi (xi + δi − xref )
=
mref i=1
24
Jref =
Nnb
Õ
i=1
(Ji + Ji,tr )
(5)
where xref is the position of the reference beam axis node (in the global reference frame)
and Ji,tr is the transport moment of mi from xi + δi to xref . The information in Eq. (5) can
be directly included in UM/NAST by defining rigid-bodies attached to the beam element
nodes [12].
.2
Equivalent Beam Stiffness Model
The stiffness module generates the strain-based cross-sectional stiffness matrix k that relates
the internal load vector f at a reference axis node to the local strain vector :
f = k
(6)
In Eq. (6), f and are 6-component vectors listing the internal force and moment resultants
and the strain components, respectively, all resolved in the local reference beam at the current
beam cross-section. Following the UM/NAST formulation [12], this has the x axis along the
beam, the y axis towards the leading edge, and the z oriented as the cross-product of x and y.
The 6×6 matrix k describes the axial, shear, torsion, and bending behaviors of the beam. For
the strain-based formulation used in UM/NAST [12, 13], the k matrix is reduced to a 4×4
matrix that retains only the terms associated with extension, torsion, and bending, along with
the related couplings. This improves the previous F2S implementation [18], which assumed
infinite axial stiffness and neglected the related coupling terms.
The equivalent beam stiffness matrix k for an aircraft component is computed as follows.
The component GFEM is subject to six static independent tip load cases, each considering
only one force or moment component in the global reference frame. A linear static analysis
with root-clamped boundary conditions is solved for each load case and the 3D displacement
field of the GFEM is output has a 1D field using interpolation elements. The 1D displacement
fields are next transformed from the GFEM global reference frame into the local frames
defined by each pair of consecutive reference beam axis nodes A and B. These frames have
x axis aligned from A to B, y axis towards the leading edge, and z axis upward, consistently
with the local frames of the UM/NAST formulation [12, 13]. Considering two consecutive
points A and B, one can write
f B = K∆u
(7)
In Eq. (7), f B is the load vector at node B due to the applied tip load and ∆u := u B −u A, where
u A and u B are the displacements vectors of the nodes A and B (listing nodal translations and
rotations). Both f B and ∆u are resolved in the local frame associated to the segment AB.
Equation (8) gives 6 scalar equations per load case for a total of 36 equations that allow to
solve for K. Once K is known, k is obtained by solving
K −1 Q−1 = k −1 HQ−1 + E k −1
(8)
for k −1 using the method of Ref. [21]. The matrices Q, E, and H in Eq. (8) are constant for
each segment AB and function of the segment length. Once k −1 is evaluated, it is reduced
to 4×4 and inverted again to obtain the cross-sectional stiffness matrix compatible with the
UM/NAST strain-based formulation. The process is repeated for each pair of nodes along
the reference beam axis for obtaining the equivalent beam stiffness distributions of the entire
component. Next, the process is repeated for each component and the obtained stiffness (and
25
mass) distributions are assembled together for forming the UM/NAST model of the entire
vehicle.
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