integrated structural, flight dynamics and aeroelastic...
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American Institute of Aeronautics and Astronautics
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Integrated Structural, Flight Dynamics and Aeroelastic
Analysis of the ANCE X-3d as a Flexible Body
Luis A. Hernández1
Universidad Simón Bolívar, Caracas, Miranda, 1080-A, Venezuela
Pedro J. Boschetti2
Universidad Simón Bolívar, Naiguatá, Vargas, 1160, Venezuela
and
Pedro J. González3
Instituto Tecnológico de Aeronáutica, São José Dos Campos, SP, 12228-900, Brazil
The objective of this paper is to generate simplified structural configurations for the ANCE
X-3d by considering the influence of structural flexibility on the flight dynamic characteristics
and the aeroelastic phenomena. This aircraft consists of an unswept wing with double tail
boom structure and two vertical stabilizers. Two structures were designed by an analytical
approach and finite element models to create suitable structural arrangements for the wing,
tail booms, and stabilizers, and carbon-fiber composite materials were selected for this
purpose. Knowing the stiffness and mass properties of the main structural components,
reduced order aero-structural models were developed to quantify the influence of the
flexibility on the aircraft aerodynamics and stability characteristics. Flight dynamic
evaluation of the airplane considering the flexibility of the structure was performed at
different velocities and altitudes. The resultant flutter and divergence velocities fulfill the
design criteria.
Nomenclature
Ak = circulation Fourier mode coefficients
CD0 = minimum drag coefficient
CL0, CM0 = lift and pitching moment coefficients at zero angle of attack
CLq, CDq, CMq = variation of lift, drag, and pitching moment coefficients with pitch rate
CLα, CDα, CMα = lift, drag, and pitching moment slopes
CYβ, Cℓβ, Cnβ = variation of side force, rolling, yawing coefficients with sideslip angle
CYp, Cℓp, Cnp = variation of side force, rolling, yawing coefficients with roll rate
CYr, Cℓr, Cnr = variation of side force, rolling, yawing coefficients with yaw rate
D = control state vector
E = control error-integral vector
E = young modulus, Pa
EI = bending stiffness, Nm2
iF = beam force-stress resultant
JF = global beam force-stress resultant
G = shear modulus, Pa
GJ = torsional stiffness, Nm2
h0 = stick fixed neutral point
k = lift dependent drag factor or induced drag factor
1 Undergraduate student, Program in Mechanical Engineering, Sartenejas Valley. 2 Associate Professor, Department of Industrial Technology, Camurí Grande Valley, Senior Member AIAA. 3 PhD Candidate, Aerospace Engineering Division, São José dos Campos, Member AIAA.
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AIAA Scitech 2019 Forum
7-11 January 2019, San Diego, California
10.2514/6.2019-0824
Copyright © 2019 by Luis A. Hernández, Pedro J. Boschetti, Pedro J. González. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.
AIAA SciTech Forum
American Institute of Aeronautics and Astronautics
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h0 = stick fixed neutral point
iM = beam moment resultant
JM = global beam moment resultant
n = load factor
ir = beam -section location
ER = earth referenced global position
SS = shear strength, Pa
T = air temperature
iu = beam-section velocity
U , U = aircraft absolute position, acceleration
U, Uc = state vector, commanded state vector
Vs, Vcr, Vcar, VD= stall speed, cruise speed, Carson speed, dive speed
VF, VDIV = flutter speed, torsional divergence speed
Xc, Xt = compression strength, tensile strength
δ1, δ2, δ3 = aileron, elevator, rudder deflection
δF = control deflections
i = beam-section rotation
= aircraft orientation Euler angles vector
= system eigenvalue
ρ = air density
ζ = damping ratio
ω = frequency
ωF = flutter frequency
i = local beam-section rotation
, = aircraft absolute rotation, angular velocity
I. Introduction
IRCRAFT structural design is an interdisciplinary process. The airframe structural components must be capable
of withstanding the loads generated in critical flight conditions, to ensure the physical integrity of the aircraft in
the complete flight envelope and provide enough stiffness to reduce the influence of the structural flexibility on the
aerodynamic characteristics and dynamic response of the aircraft, as well as preventing the appearance of potentially
destructive aeroelastic phenomenona.1 In general, the dynamic and aeroelastic analysis of the complete airplane
represents a highly complex task and for this reason, it is treated as a separate problem in the design process with
direct influence on the structural sizing of the airframe. However, for the design team to accomplish a complete
rational analysis of the aircraft dynamic response, a significant effort must be done to develop accurate mathematical
models and obtain valuable experimental data to proceed with the validation of the numerical results and the
subsequent certification process of the aircraft, which demands a large amount of computational and experimental
engineering analyses.2 The aeroelastic phenomena represent a hazard to the safe operation of the aircraft, consequently
it is required mitigating the unstable behavior of the structure, which generally involves important modifications of
the structural design, increasing the financial risk of the project and affecting the initial time estimation for the design
phase.3 For this reason, it can be advantageous to generate a simplified model in an earlier phase of the structural
design process in order to gain valuable insight about the dynamic response of the aircraft that could serve as feedback
for the structure team and as input data to the aeroelastic and flight dynamic engineers. Another potential advantage
of this approach is that it allows exploring different structural configurations using low-cost computational models to
evaluate the sensitivity of the aircraft dynamic response to the structural characteristics of the main components and
the possible failure scenarios.2
The objective of this paper is to generate simplified structural configurations for the Unmanned Airplane for
Ecological Conservation (ANCE X-3d)4,5 by considering the influence of structural flexibility on the flight dynamic
characteristics and the initial analysis of aeroelastic phenomena as part of the structural design process. The ANCE
X-3d has been developed to patrol oil extraction areas in order to look for oil leaks to minimize the response time of
emergency squads and reduce the environmental damage that these events could produce to the ecosystem and to the
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conservation of the natural environment and wildlife.
The aircraft is a twin-boom monoplane, with a maximum
takeoff mass of 182.055 kg, wingspan of 5.187 m,
geometric mean chord of 0.604 m, and wing area of
3.1329 m2. Figure 1 illustrates a sketch of the airplane.
Figures 2 and 3 show the flowchart corresponding to the
classic aircraft structural design process and that one
used in the present investigation, respectively. The main
difference in Fig. 3 with respect to the first diagram lies
in the fact that the structural sizing now is dependent on
the level of influence of the aircraft flexibility on the
overall flight characteristics.
In order to proceed with the dynamic analysis of the
ANCE X-3d as a flexible body, it was necessary to obtain
the stiffness and mass distributions of the airframe main
structural elements. For this purpose, two structures were
designed by an analytical approach and finite element
models to generate suitable structural arrangements for
the wing, tail booms, and stabilizers. Carbon-fiber
composite materials were selected because of their high
mechanical properties and weight-saving characteristics.
Reduced order aero-structural models were developed
with the objective of quantifying the influence of the
flexibility on the aircraft aerodynamics and stability
characteristics. Finally, an aeroelastic analysis was
carried out to predict the divergence and flutter velocities
to ensure that both phenomena occurred outside the flight
envelope of the ANCE X-3d, and that they achieved the
design criteria.
II. Structural Design
A. Load Estimation
The critical loads over the different components of
the vehicle were determined by evaluating the
aerodynamic loads over a range of flight conditions in
order to select the maximum load case for each of the
structural components. The maximum load factors were
computed using the program Coquivacoa, which
calculates in-flight states in time domain of a subsonic
airplane considering atmospheric and control
disturbances that may appear during flight, using a
classic four-order Runge-Kutta method.6 This software
calculates the critical load factors at asymmetric flight
conditions considering the maximum deflection of the control surfaces for each individual component, which is ±15
deg.7 Open-loop simulations were carried out to obtain the maximum load factor values using the corresponding flight
parameters of each flight condition. Flight envelope diagrams at sea level and for cruise altitude are drawn based on
the data attained by these simulations according to the FAR Part 23 (Ref. 8); these are shown in Fig. 4.
The vortex lattice models of this airplane previously used to simulate the flow field around the ANCE X-3d (Refs.
9,10) are employed to obtain the lift distribution over the aerodynamic surfaces (wing, horizontal stabilizer, and
vertical stabilizer) for the given flight conditions.
Figure 1. Axonometric view of the ANCE X-3d.
Figure 2. Flowchart for the classic aircraft structural
design process.
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B. Structural Configuration
The main structural components are designed using composite materials to take advantage of their elevated
mechanical properties and low weight to strength ratio.11 The composite material selected is ACP Standard Carbon
Fiber.12 Finite element models are developed using ANSYS software13 in order to compute the stress and deformation
distributions over the airframe structural elements for the critical load case determined in the previous section. The
computational model of the composite laminate is constructed using ANSYS ACP14 to generate an accurate
representation of the composite structure.
Figure 4. Velocity as function of load diagram at sea level (left) and for cruise altitude (right).
-2
-1
0
1
2
3
4
5
0 10 20 30 40 50 60
Lo
ad f
acto
r
Velocity, m/s
ManeuverGust 25 ft/sGust 50 ft/sEnvelope
-2
-1
0
1
2
3
4
5
0 10 20 30 40 50 60 70
Lo
ad f
acto
r
Velocity, m/s
Maneuver
Gust 25 ft/s
Gust 50 ft/s
Envelope
Figure 3. Flowchart for the modified aircraft structural design process.
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Two structural configurations were designed herein.
Table 1 presents a comparison of the mechanical
characteristics of both structural configurations. The
second design has a higher wing torsional stiffness in the
section located between the fuselage and the wing-tail
boom joint (sec. 2) and a lower bending stiffness in the
same section. The tail boom bending stiffness is also
increased for the first section compared to the first
structure. The second section’s (sec. 1) torsional and
bending stiffness are reduced for the wing and tail boom.
These variations in stiffness properties are mostly
achieved by changing the composite laminate
configuration using 0 deg, 90 deg and ±45 deg plies.
Table 2 shows the mass values of the main components
for both configurations, and it can be observed that the
mass of the second structure is 45.9% less than the mass of the first one. These values correspond only to the mass of
the structural elements and do not represent real mass distribution of the complete airframe. Figure 5 illustrates the
structure geometry and the location of the different sections.
III. Structural and Mass Model
A. Aswing Mathematical Modelling
The Aswing software is aimed at the overall evaluation of the aerodynamics, structural and control system
implementation on flexible aircraft of moderate to high aspect ratio.15,16,17 The program allows making quick design
modifications to get insight into structural failure, flight dynamic response, stability characteristics and aeroelastic
phenomena appearance in a wide range of flight conditions. The complete nonlinear system is solved by means of a
full Newton method. The structural model consists of nonlinear connected beams with arbitrary mass, inertia, and
stiffness distributions. The lifting-line based aerodynamic model allows considering general aerodynamic sections
with control-surface deflection. Compressibility effects are considered using the Prandtl–Glauert correction in wind
axes. The complete aircraft dynamics can be computed including the airplane response to atmospheric gust encounters
with a general state-feedback law governing control-surface deflections and thrust configuration.15
The general unsteady problem is represented using the two vectors shown in Eqs. (1) and (2), where E is a control-
error integral vector. The system is treated in nonlinear residual form as presented in Eq. (3), where Uc is the
commanded state vector.15
Table 1. Bending and torsional stiffness of the main structural components.
Design 1 Design 2
EIxx, Nm2 EIyy, Nm2 GJ, Nm2 EIxx, Nm2 EIyy, Nm2 GJ, Nm2
Wing beam sec. 1 56600 56600 32000 23672.4 23672.4 52893.3
Wing beam sec. 2 56600 56600 32000 13011.7 13011.7 3128.1
Tail boom sec. 1 19600 19600 14700 26778.3 26778.3 6351.2
Tail boom sec. 2 18100 18100 13600 9118.6 9118.6 3594.2
Horizontal tail* 2550 8210 234 2550 8210 234
Vertical tail* 14.4 230 17.4 14.4 230 17.4
Table 2. Total mass of the main structural components.
Design 1 Design 2
Mass, kg Total mass, kg Mass, kg Total mass, kg
Wing beam 2.023 4.045 1.754 3.509
Tailboom 7.732 15.465 3.225 6.449
Horizontal tail 0.804 0.804 0.804 0.804
Vertical tail 0.240 0.480 0.240 0.480
Total - 20.79 - 11.24
Figure 5. Simplified structural model in ANSYS.
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( )i i i i i i J J J J k Er M F u r M F A R U U = U E (1)
1 2( ...)F F =D (2)
( , , ; )c =R U U D U 0 (3)
The corresponding linearized system can be written as follows:15
+ + =
R R RU U D R
U DU (4)
This linearized system of equations in Eq. (4) is the base of the Aswing solution for steady state, time-domain,
frequency-response and eigenvalue analysis. Eigenvalue calculation is achieved by Eq. (5) to obtain the corresponding
nontrivial solutions X̂ for the unknown eigenvalues .15
ˆ ˆ =AX MX (5)
= = −
R RA M
U U (6)
An instability of eigenmode is indicated if ℝ ( ) 0 (Ref. 15). This could represent a flight instability as in the
common case of unstable spiral mode or a structural instability like flutter, which is an undesirable condition because
one goal during the design process is that the structural modes must be stable.
B. Aswing Structural and Mass Model
The structural and mass properties presented in Tables 1 and 2 were used to create the structural and mass models
used by Aswing. These data were also employed to build a simplified beam-like finite element model in ANSYS.
Natural frequencies and mode shapes of the structures are calculated by both methods and these are shown in Table
3, and Figs. 6 and 7 illustrate the corresponding mode shapes. For the first design, the natural frequencies achieved by
both methods present an agreement below 10% except for the fourth mode shape, which has a difference of 11.54%.
An analogous result is attained in the case of the second design, where the natural frequency of the second mode
presents a difference of 11.78% between both methods. The difference may be a direct result of the simplified beam-
like FEM model used for extracting the mechanical properties of the structural elements. It should be noted that for
the second design, it was not possible to obtain the corresponding sixth mode shape so the comparison is made using
the first five modes.
C. Aswing Aerodynamic Model
The geometry of the ANCE X-3d is modeled in Aswing using five surfaces and three bodies, and this is illustrated
in Figs. 6 and 7. The aerodynamic and stability coefficients estimated by Aswing are compared with those coefficients
presented in previous published works18-20 achieved by wind tunnel testing and using other numerical and empirical
methods. Table 4 shows the aerodynamic parameters and stability coefficients calculated for the airplane with Aswing
Table 3. Natural frequencies obtained for both proposed structures. Design 1 Design 2
Mode ANSYS, Hz Aswing, Hz Percental
difference ANSYS Hz Aswing, Hz
Percental
difference
1 2.13 2.11 1.33 6.88 7.5 9.07
2 6.48 6.43 0.7 8.69 9.71 11.78
3 6.64 6.66 0.25 12.53 12.67 1.13
4 19.08 21.28 11.54 21.3 20.82 2.32
5 20.01 18.03 9.88 22.21 21.1 5.23
6 20.78 19.71 5.13 - - -
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Figure 6a. Mode shapes (first to fourth) for the first structural configuration (D1) calculated with ANSYS (left)
and Aswing (right).
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Figure 6b. Mode shapes (fifth to sixth) for the first structural configuration (D1) calculated with ANSYS (left)
and Aswing (right).
Figure 7a. Mode shapes (first to second) for the second structural configuration (D2) calculated with ANSYS
(left) and Aswing (right).
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Figure 7b. Mode shapes (third to fifth) for the second structural configuration (D2) with ANSYS (left) and
Aswing (right).
Table 4. Aerodynamic parameters and stability coefficients obtained by different methods in rad-1.
Digital
DATCOM20 AVL18 PANAIR18 CMARC20
Wind
tunnel19
Aswing
Rigid First structure Second structure
k 0.0364 0.0506 0.0493 0.0488 0.0464 0.0541 0.0579 0.0558
h0 0.792 0.826 0.893 0.958 - 0.9422 0.7935 0.9195
CD0 - - - - 0.0317 0.0295 0.0264 0.0253 CL0 0.311 0.3111 0.3089 0.3273 0.3424 0.2775 0.3 0.2592
CM0 -0.081 -0.1 -0.009 -0.096 - -0.0557 -0.0901 -0.0746
CDα 0.128 0.152 0.13 0.173 - 0.1332 0.0968 0.1182
CLα 5.6379 4.8128 4.2743 5.4202 4.34302 4.4347 2.7864 4.0868
CLq 8.474 10.157 - 6.221 - 11.28533 -5.27969 7.2287
CMα -3.054 -2.773 -2.75 -3.839 - -3.07 -1.515 -2.736
CMq -26.075 -28.13 - -18.663 - -35.4193 -15.7436 -25.46
CYβ -0.761 -0.527 - -0.327 - -0.33695 -0.32642 -0.3287 CYp -0.083 -0.055 - -0.005 - -0.05698 -0.02627 -0.092
CYr - 0.455 - 0.421 - 0.2529 0.24221 0.2538
Cℓβ -0.059 -0.057 - -0.138 - -0.01309 -0.0195 -0.0331
Cℓp -0.53 -0.546 - -1.127 - -0.52969 -0.52611 -0.5319
Cℓr 0.069 0.091 - 0.211 - 0.11902 0.10184 0.1081
Cnβ 0.175 0.132 - 0.189 - 0.11632 0.11289 0.1116
Cnp -0.031 -0.001 - -0.032 - -0.03512 -0.03067 -0.0298
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as a rigid body and for both proposed structures at Carson speed (45 m/s) at sea level. Figure 8 shows the lift coefficient
as a function of angle of attack and the drag polar curve of the airplane. A very good correlation can be observed
between the values of induced drag factor, lift slope and stability coefficients achieved by experimental and numerical
methods and those calculated by Aswing for the rigid body model. The minimum drag coefficient estimated by Aswing
is 6.9% smaller than the one achieved by wind tunnel testing because the Aswing aerodynamic model does not have
landing gear, engine and camera. The stability coefficients obtained for the first structure present significant
differences respect to those ones of the second structure and the airplane as a rigid body. The pitch slope attained for
the first structure is 65.51% and 44.62% smaller than the one estimated for a rigid body and for the second structure,
respectively, representing a reduction of static stability. The lift and drag slopes obtained for the first structure is
37.2% and 27.3% less than those estimated for the airplane as a rigid body, respectively. Hantrais-Gervois and
Destarac21 describe that CLα rotates around the zero-lift point because of the wing elastic twist distribution and if the
wing twist variation is moderate, a drag polar invariance with flexibility is expected at the cruise point for airplanes
of moderate aspect ratio (RA≈9). The lift and drag slopes for the second structure show a decrease of 8.51% and 12.63%
when compared with the rigid body results, respectively; however, it is noticed that there is no significant rotation of
the lift curve for the flexible body, which could be explained by the increase of torsional stiffness in the first section
of the wing structure. As expected, the drag polar curve shows no significant sensitivity to the flexibility of the airplane
structure, according to the observation in Ref. 21, but the induced drag factor calculated for the second structure is
smaller compared to the one for the first structural configuration. The resultant maximum lift-drag ratio value of the
airplane as a rigid body estimated by Aswing is 12.52. For the first and second structures the resultant maximum lift-
drag ratio are 12.79 and 13.31, respectively. As flexibility increases, the maximum lift-drag ratio also rises.
IV. Flight Dynamics
Table 5 presents the eigenvalues of the dynamic modes of the airplane as both a rigid and a flexible body computed
by Aswing in cruise altitude (2438 m) and velocity equal to 51 m/s (Carson speed). The eigenvalues obtained for the
airplane as a rigid body using Aswing for phugoid, short period, and Dutch roll are complex and have negative real
parts, indicating that after a disturbance the response would decay sinusoidally in time. The root for the roll mode is
real and negative, representing a stable and heavily damped rolling motion. The root for spiral motion is positive,
indicating a slightly divergent spiral motion. The eigenvalues obtained in previous studies present similar behavior,
expected for spiral mode. Because of the mass model employed by Aswing herein only has the mass of the main
structure, there are no correlations between the values estimated by this model and those obtained in previous studies,
which used inertial properties estimated for the complete aircraft. Table 5 shows a moderate variation of the damping
and frequency of the flight dynamic modes for the second structure, and for the first structure it is observed a smaller
sensitivity of the eigenvalues concerning aircraft flexibility.
Figure 8. Lift coefficient as a function of angle of attack (left) and drag polar curve (right).
-1.2
-1.0
-0.8
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
1.2
-20 -15 -10 -5 0 5 10 15 20
Lif
t co
effi
cien
t
Ange of attack, deg
Wind Tunnel
ASWING rigid
ASWING flex. D1
ASWING flex. D2
-1.0
-0.8
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
1.2
0.02 0.04 0.06 0.08 0.10
Lif
t co
effi
cien
t
Drag coefficient
Wind Tunnel
ASWING rigid
ASWING flex. D1
ASWING flex. D2
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Table 6 compares the dynamic characteristics of the airplane when it is considered a rigid body and using the first
and second structures, and it is observed that the modification in the stiffness and mass distributions has an important
influence on the short period and Dutch roll frequencies and on the roll mode. A moderate variation of the short period
and Dutch roll modes for the second structure (D2) is appreciated in Figs. 9 and 10 at sea level and cruise altitude,
respectively. For the first structure (D1), the short period and Dutch roll modes do not present significant changes.
The phugoid and spiral modes remain practically invariant for both structures at sea level and cruise altitude.
Figures 11 to 13 illustrate the variation of the flight mode eigenvalues with velocity for both structural
configurations. The results indicate that the effect of flexibility increases with velocity, however, all the flight modes
are stable in the velocity range of the ANCE X-3d. For the phugoid mode, the damping diminishes with velocity until
56 m/s and 66 m/s for the first and second structures, respectively, at sea level, and then this value remains relatively
constant. The damping value for phugoid mode reduces until 50 m/s and 70 m/s for the first and second structures,
respectively, for the cruise altitude. Figures 14 to 17 present a similar behavior in the root-locus plot. Figures 18 to 20
compare the damping as a function of velocity at sea level and cruise altitude for the first and second structural
configurations. The phugoid mode shows a slight variation in damping between both structural configurations, and
the damping for Dutch roll and short period modes are quite different.
V. Aeroelastic Analysis
Figures 21 to 26 illustrate the root-locus and damping as a function of velocity at sea level, for cruise altitude, and
service ceiling, respectively, for the first structural configuration (D1), and Fig. 27 to 32 for the second structure (D2).
For the first structural configuration, the results show that flutter occurs outside the flight envelope of the vehicle
fulfilling the design requirements. The flutter velocity is equal to 86 m/s at sea level, 80 m/s for cruise altitude and
78 m/s for service ceiling. Additionally, the flutter mode corresponds to the asymmetric torsion of the wing structure.
Figures 22, 24, and 26 present damping modes as a function of velocity of the critical aeroelastic modes, and a fast
reduction of damping for high velocities values can be observed. For cruise altitude, flutter velocity is 4.17% higher
than the maximum permissible value of 1.2VD established in Ref. 22. Figures 27, 29 and 31 show the symmetric
in-plane bending (SYM. B IP), symmetric out-of-plane bending (SYM. B OP), asymmetric in-plane bending (ASYM.
B IP), asymmetric out-of-plane bending (ASYM. B OP) and torsional structural modes at sea level, for cruise altitude
and at service ceiling. It is observed that the torsional mode of the horizontal tail structure becomes unstable at 79 m/s,
77 m/s, and 93 m/s at sea level, for cruise altitude, and at service ceiling, respectively, which represents a drastic
reduction of flutter velocity compared to the results of the first structure. This result also suggests a lack of torsional
Table 5. Dynamic modes eigenvalues at cruise flight condition.
Digital
DATCOM20 AVL20 CMARC20
Aswing
Rigid First structure Second
structure
Phugoid Real -0.0093 -0.0093 -0.0097 -0.04109 -0.03387 -0.02922
Im (±) 0.2308 0.2306 0.244 0.303 0.30315 0.25093
Short Period Real -2.0737 -1.9172 -1.912 -3.6207 -3.04246 -4.4274
Im (±) 4.1976 4.0325 4.7212 6.09775 5.71142 9.61222
Dutch Roll Real -0.4701 -0.3545 -0.454 -1.055 -1.01497 -1.66534
Im (±) 3.127 2.6987 2.4785 3.57246 3.59889 5.0911
Roll Real -4.2369 -4.3371 -6.7322 -16.9461 -20.309 -58.223
Spiral Real -0.0037 0.0042 -0.0289 0.01353 0.01213 0.01228
Table 6. Comparison of the dynamic characteristics of the airplane
Rigid First
structure
Second
structure
Difference
rigid-1st
str.
Difference
rigid-2nd
str.
Difference
1st str-2nd
Phugoid ζ -0.1344 -0.11103 -0.11565 21.05% 16.2% 4.17%
ω, rad/s 0.303 0.30315 0.25093 0.05% 20.75% 17.23%
Short period ζ -0.8598 -0.88259 -0.90828 2.58% 5.33% 2.91%
ω, rad/s 6.09775 5.71142 9.61222 6.76% 36.56% 68.3%
Dutch roll ζ -0.2832 -0.27143 -0.3109 4.34% 8.9% 14.54%
ω, rad/s 3.57246 3.59889 5.0911 0.73% 29.83% 41.46%
Roll Real -16.946 -20.309 -58.223 16.56% 70.89% 186.68%
Spiral Real 0.01353 0.01213 0.01228 11.51% 10.18% 1.21%
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Figure 9. Eigenvalues obtained at Carson speed (45 m/s) at sea level.
-12
-10
-8
-6
-4
-2
0
2
4
6
8
10
12
-6 -5 -4 -3 -2 -1 0
Imag
inar
y p
art
Real part
Dutch Roll Rigid
Dutch Roll D1
Dutch Roll D2
Short Period Rigid
Short Period D1
Short Period D2
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
-0.03 -0.02 -0.01 0 0.01 0.02
Imag
inar
y p
art
Real part
Phugoid Rigid
Phugoid D1
Phugoid D2
Spiral rigid
Spiral D1
Spiral D2
Figure 10. Eigenvalues obtained at Carson speed (51 m/s) in cruise altitude (2438 m).
-13
-11
-9
-7
-5
-3
-1
1
3
5
7
9
11
13
-6 -5 -4 -3 -2 -1 0
Imag
inar
y p
art
Real part
Dutch Roll Rigid
Dutch Roll D1
Dutch Roll D2
Short Period Rigid
Short Period D1
Short Period D2
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
-0.03 -0.02 -0.01 0 0.01 0.02
Imag
inar
y p
art
Real part
Phugoid Rigid
Phugoid D1
Phugoid D2
Spiral rigid
Spiral D1
Spiral D2Dow
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Figure 11. Variation of Dutch roll eigenvalues with velocity at sea level (left) and cruise altitude (right).
0
2
4
6
8
10
12
14
16
-4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0
Imag
inar
y p
art
Real part
B.H Rigid
B.H Flexible D1
B.H Flexible D2
0
2
4
6
8
10
12
14
16
-4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0
Imag
inar
y p
art
Real part
B.H Rigid
B.H Flexible D1
B.H Flexible D2
Figure 12. Variation of short period eigenvalues with velocity for sea level (left) and cruise altitude (right).
0
5
10
15
20
25
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0
Imag
inar
y p
art
Real part
S.P Rigid
S.P Flexible D1
S.P Flexible D2
0
2
4
6
8
10
12
14
16
18
20
-9 -8 -7 -6 -5 -4 -3 -2 -1 0
Imag
inar
y p
art
Real part
S.P Rigid
S.P Flexible D1
S.P Flexible D2
Figure 13. Variation of phugoid eigenvalues with velocity at sea level (left) and cruise altitude (right).
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
-0.05 -0.04 -0.03 -0.02 -0.01 0.00
Imag
inar
y p
art
Real part
Phugoid Rigid
Phugoid D1
Phugoid Flexible D20.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
-0.06 -0.05 -0.04 -0.03 -0.02 -0.01
Imag
inar
y p
art
Real part
Phugoid Rigid
Phugoid D1
Phugoid Flexible D2
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Figure 14. Root-Locus of Dutch roll, short period and phugoid with velocity at sea level for the first structure
(D1).
-12
-10
-8
-6
-4
-2
0
2
4
6
8
10
12
-6 -5 -4 -3 -2 -1 0
Imag
inar
y p
art
Real Part
26 m/s
36 m/s
46 m/s
56 m/s
66 m/s
76 m/s
86 m/s
96 m/s
Dutch
Short period
-0.40
-0.35
-0.30
-0.25
-0.20
-0.15
-0.10
-0.05
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
-0.10 -0.08 -0.06 -0.04 -0.02 0.00
Imag
inar
y p
art
Real part
26 m/s
36 m/s
46 m/s
56 m/s
66 m/s
76 m/s
86 m/s
96 m/s
Phugoid
Figure 15. Root-Locus of Dutch roll, short period and phugoid with velocity at sea level for the second structure
(D2).
-25
-20
-15
-10
-5
0
5
10
15
20
25
-6 -5 -4 -3 -2 -1 0
Imag
inar
y p
art
Real part
26 m/s36 m/s46 m/s56 m/s66 m/s76 m/s86 m/s
Dutch roll
Short period
-0.40
-0.30
-0.20
-0.10
0.00
0.10
0.20
0.30
0.40
-0.08 -0.06 -0.04 -0.02 0.00
Imag
inar
y p
art
Real part
26 m/s
36 m/s
46 m/s
56 m/s
66 m/s
76 m/s
86 m/s
96 m/s
Phugoid
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Figure 16. Root-Locus of Dutch roll, short period and phugoid with velocity for cruise altitude for the first
structure (D1).
-12
-7
-2
3
8
-5 -4 -3 -2 -1 0
Imag
inar
y p
art
Real part
30 m/s
40 m/s
50 m/s
60 m/s
70 m/s
80 m/s
90 m/s
Short period
Dutch roll
-0.40
-0.30
-0.20
-0.10
0.00
0.10
0.20
0.30
0.40
-0.10 -0.08 -0.06 -0.04 -0.02 0.00Im
agin
ary p
art
Real part
30 m/s
40 m/s
50 m/s
60 m/s
70 m/s
80 m/s
90 m/s
Phugoid
Figure 17. Root-Locus of Dutch roll, short period and phugoid with velocity for cruise altitude for the second
structure (D2).
-25
-20
-15
-10
-5
0
5
10
15
20
25
-6 -5 -4 -3 -2 -1 0
Imag
inar
y p
art
Real part
30 m/s 40 m/s
50 m/s 60 m/s
70 m/s 80 m/s
90 m/s 100 m/s
Dutch roll
Short period
-0.40
-0.30
-0.20
-0.10
0.00
0.10
0.20
0.30
0.40
-0.08 -0.06 -0.04 -0.02 0.00
Imag
inar
y p
art
Real part
30 m/s
40 m/s
50 m/s
60 m/s
70 m/s
80 m/s
90 m/s
100 m/s
Phugoid
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Figure 18. Dutch roll damping as a function of velocity at sea level (left) and cruise altitude (right).
-0.99
-0.97
-0.95
-0.93
-0.91
-0.89
-0.87
-0.85
20 40 60 80 100
Rea
l p
art
Velocity, m/s
Dutch Roll D1
Dutch Roll D2
-1.00
-0.98
-0.96
-0.94
-0.92
-0.90
-0.88
20 40 60 80 100
Rea
l p
art
Velocity, m/s
Dutch Roll D1
Dutch Roll D2
Figure 19. Short period damping with velocity at sea level (left) and cruise altitude (right).
-0.98
-0.96
-0.94
-0.92
-0.90
-0.88
-0.86
-0.84
-0.82
20 40 60 80 100
Rea
l p
art
Velocity, m/s
Short Period D1
Short Period D2
-1.00
-0.98
-0.96
-0.94
-0.92
-0.90
-0.88
-0.86
-0.84
-0.82
20 40 60 80 100
Rea
l p
art
Velocity, m/s
Short Period D1
Short Period D2
Figure 20. Phugoid damping with velocity at sea level (left) and cruise altitude (right).
-1.000
-0.998
-0.996
-0.994
-0.992
-0.990
-0.988
25 35 45 55 65 75 85
Rea
l p
art
Velocity, m/s
Phugoid D1
Phugoid D2
-1.000
-0.998
-0.996
-0.994
-0.992
-0.990
35 45 55 65 75 85
Rea
l p
art
Velocity, m/s
Phugoid D1
Phugoid D2Dow
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Figure 21. Root-locus as a function of velocity at sea level for the first structure (D1).
S.P, D.R, phugoid
0
25
50
75
100
125
150
175
200
-50 -40 -30 -20 -10 0
Imag
inar
y p
art
Real part
26 m/s
36 m/s
46 m/s
56 m/s
66 m/s
76 m/s
86 m/s
96 m/s
Wing asymmetric
torsion
Tailboom symmetric
bending
Unstable
40
42
44
46
48
50
52
54
56
-12 -8 -4 0 4 8 12
Imag
inar
y p
art
Real part
26 m/s
36 m/s
46 m/s
56 m/s
66 m/s
76 m/s
86 m/s
96 m/s
Figure 22. Damping as a function of velocity at sea level for the first structure (D1).
-10
-8
-6
-4
-2
0
2
4
0 10 20 30 40 50 60 70 80 90 100
Dam
pin
g,
s-1
Velocity, m/s
Tailboom symmetric bending
Wing asymmetric torsion
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Figure 23. Root-locus as a function of velocity for cruise altitude for the first structure (D1).
S.P, D.R, phugoid
0
25
50
75
100
125
150
175
200
-50 -40 -30 -20 -10 0 10
Imag
inar
y p
art
Real part
30 m/s
40 m/s
50 m/s
60 m/s
70 m/s
80 m/s
90 m/s
Wing asymmetric
torsion
Tailboom symmetric
bending
Unstable
40
42
44
46
48
50
52
54
56
-12 -8 -4 0 4 8 12
Imag
inar
y p
art
Real part
30 m/s
40 m/s
50 m/s
60 m/s
70 m/s
80 m/s
90 m/s
Figure 24. Damping as a function of velocity for cruise level for the first structure (D1).
-15
-10
-5
0
5
10
0 10 20 30 40 50 60 70 80 90 100
Dam
pin
g,
s-1
Velocity, m/s
Tailboom symmetric bending
Wing asymmetric torsion
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Figure 25. Root-locus as a function of velocity at service ceiling for the first structure (D1).
0
25
50
75
100
125
150
175
200
-125 -100 -75 -50 -25 0 25 50
Imag
inar
y p
art
Real part
47 m/s
57 m/s
67 m/s
77 m/s
87 m/s
97 m/s
107 m/s
Tailboom symmetric
bending
Wing asymmetric
torsion
Unstable
15
20
25
30
35
40
45
50
-10 -8 -6 -4 -2 0 2 4
Imag
inar
y p
art
Real part
47 m/s
57 m/s
67 m/s
77 m/s
78 m/s
Figure 26. Damping as a function of velocity at service ceiling for the first structure (D1).
-6
-5
-4
-3
-2
-1
0
1
2
40 45 50 55 60 65 70 75 80
Dam
pin
g,
s-1
Velocity, m/s
Tailboom symmetric bending
Wing symmetric torsion
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Figure 27. Root-locus as a function of velocity at sea level for the second structure (D2).
Unstable H.T
Torsion
40
60
80
100
120
140
160
180
-40 -30 -20 -10 0 10 20 30
Imag
inar
y p
art
Real part
26 m/s
36 m/s
46 m/s
56 m/s
66 m/s
76 m/s
79 m/s
96 m/s
Tailboom SYM.
B (IP)
Tailboom SYM. B (OP)
Tailboom
ASYM. B (OP)
Unstable
250
270
290
310
330
350
370
390
-10 -8 -6 -4 -2 0 2
Imag
inar
y p
art
Real part
26 m/s
36 m/s
46 m/s
56 m/s
66 m/s
76 m/s
86 m/s
96 m/s
Wing ASYM. B (IP)
Figure 28. Damping as a function of velocity at sea level for the second structure (D2).
-30
-25
-20
-15
-10
-5
0
5
0 10 20 30 40 50 60 70 80 90 100
Dam
pin
g,
s-1
Velocity, m/s
Tailboom SYM. B (in plane)Tailboom - ASYM. B (out of plane)H.T Torsion
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Figure 29. Root-locus as a function of velocity for cruise altitude for the second structure (D2).
Unstable H.T
Torsion
40
60
80
100
120
140
160
180
-40 -30 -20 -10 0 10 20
Imag
inar
y p
art
Real part
30 m/s40 m/s50 m/s60 m/s70 m/s80 m/s77 m/s100 m/s
Tailboom SYM. B (OP)
Unstable
250
270
290
310
330
350
370
390
-10 -8 -6 -4 -2 0 2
Imag
inar
y p
art
Real part
30 m/s
40 m/s
50 m/s
60 m/s
70 m/s
80 m/s
90 m/s
100 m/sWing ASYM. B (IP)
Figure 30. Damping as a function of velocity for cruise altitude for the second structure (D2).
-25
-20
-15
-10
-5
0
5
0 10 20 30 40 50 60 70 80 90 100 110
Dam
pin
g,
s-1
Velocity, m/s
Tailboom SYM. B (in plane)Tailboom ASYM. B (out of plane)H.T Torsion
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Figure 31. Root-locus as a function of velocity at service ceiling level for the second structure (D2).
Unstable H.T
Torsion
40
60
80
100
120
140
160
180
-40 -30 -20 -10 0 10 20
Imag
inar
y p
art
Real part
47 m/s
57 m/s
67 m/s
77 m/s
87 m/s
97 m/s
107 m/s
Tailboom SYM. B (OP)
Unstable
250
270
290
310
330
350
370
390
-20 -15 -10 -5 0 5
Imag
inar
y p
art
Real part
47 m/s
57 m/s
67 m/s
77 m/s
87 m/s
97 m/s
107 m/s
Wing ASYM. B (IP)
Figure 32. Damping as a function of velocity at service ceiling level for the second structure (D2).
-18-16-14-12-10
-8-6-4-2024
40 50 60 70 80 90 100 110
Dam
pin
g,
s-1
Velocity, m/s
Tailboom SYM. B (in plane)Tailboom ASYM. B (out of plane)H.T Torsion
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stiffness for the horizontal
stabilizer. The flutter velocity
remains outside the flight
envelope, and the flutter velocity
is 0.26% higher to 1.2VD for
cruise altitude.
The wing torsional divergence
velocity is calculated by an
analytical approach1 at different
altitudes. Table 7 shows that the
resultant torsional divergence
velocities are higher than the
maximum aircraft velocity at
each specific flight condition.
However, the divergence velocity
exhibits a significant reduction for the second structure, which could be explained due to the torsional stiffness
distribution presented in Table 1. Table 7 summarizes the torsional divergence and flutter velocities for both structures.
VI. Conclusions
A general methodology is presented to account for flight dynamic response and aeroelastic phenomena
characteristics on the initial structural design of the ANCE X-3d main components using low-detail reduced aero-
structural models. This procedure proved to be useful in obtaining an overall evaluation of the aircraft flight dynamic
sensitivity to the structural flexibility affected by earlier modifications in the design process. The aeroelastic analysis
allowed obtaining valuable insight into the nature of the unstable aeroelastic modes and the stiffness properties linked
with the appearance of structural instabilities.
The simplified structural design of the ANCE X-3d proved to be a useful approximation to obtain an initial
structural definition of the airframe, significantly reducing computational modelling time by neglecting secondary
components that do not perform a structural function. The low computational cost of the reduced order beam-like
model also represents an important advantage due to the possibility of evaluating the physical response of the structure
for different geometric configurations, without the need to take special care of the airframe details that have a
negligible impact on the final structural properties.
The results have shown that structural flexibility does not have a significant influence on the aerodynamic and
stability characteristics of the ANCE X-3d. The aerodynamic performance has a moderate sensitivity to the wing
torsional stiffness in the section located between the fuselage and the wing-tail boom joint, but do not present an
important variation regarding the rigid body model. The ANCE X-3d flight dynamic modes are stable in the flight
envelope and they do not show a drastic modification as a consequence of aircraft flexibility, which satisfies the design
criteria.
The initial aeroelastic analyses suggest that the ANCE X-3d can operate in any condition located inside the flight
envelope with enough margin against aeroelastic phenomena. The results of the structural sensitivity analysis show
that flutter appearance is closely related to the wing first section torsional stiffness.
References 1Bisplinghoff, R. L., Ashley, H., and Halfman, R. L. Aeroelasticity. Dover Books on Aeronautical Engineering, New York,
1996, Chaps. 1, 8. 2Taylor, M.R., Weisshaar, A.T. and Surukhanov, V., “Structural Design Process Improvement Using Evolutionary Finite
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Table 7. Torsional divergence and flutter velocities for both structural
configurations.
First structure Second structure
Sea level
ωF, rad/s 43.21 141.49
VF, m/s 86 79
VDIV, m/s 467 188.6
Cruise
altitude
ωF, rad/s 43.40 140.65
VF, m/s 80 77
VDIV, m/s 525 212.4
Service
ceiling
ωF, rad/s 43.10 140.70
VF, m/s 78 93
VDIV, m/s 663 268
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