About ZAERO TM
Engineers’ Toolkit for Aeroelastic Solutions
ZONA Technology, Inc.
Unsteady Aerodynamics Flutter Nonlinear Flutter Trim Ejection Loads Maneuver Loads Gust Loads
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Page 2 ZONA Technology, Inc.
TABLE OF CONTENTS The ZAERO Software System/Architecture .......................... 3 UAIC: Unified Aerodynamic Influence Coefficients ............... 5 ZONA6: Subsonic Unsteady Aerodynamics ......................... 6 ZTRAN: Transonic Unsteady Aerodynamics ........................ 7 ZONA7: Supersonic Unsteady Aerodynamics ...................... 9 ZONA7U: Hypersonic Unsteady Aerodynamics .................. 10 ZSAP: Sonic Acceleration Potential Panel Method ............. 11 ZTAW: AIC Correction Method ............................................ 12 High Fidelity Geometry (HFG) Module ................................ 13 3D Spline Module ................................................................ 15 Bulk Data Input .................................................................... 16 Graphic Display ................................................................... 17 Flutter Module ...................................................................... 18 Parametric Flutter Analysis ................................................. 19 Static Aeroelastic/ Trim Module ........................................... 20 Aeroservoelasticity (ASE) Module ....................................... 21 Rational-Function Approximation of Unsteady Aerodynamics ...................................................................... 22 Aeroelastic State-Space Model ........................................... 23 Transient Maneuver Loads .................................................. 24 Transient Ejection Loads ..................................................... 25 Transient Discrete and Continuous Gust Loads ................. 26 Nonlinear Flutter Module ..................................................... 27
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The main features of the ZAERO system include:
• High Fidelity Geometry (HFG) module to model full aircraft with stores/nacelles (1)
• Flight regimes that cover all Mach numbers including transonic/hypersonic ranges (2)
• Unified Mach AIC (UAIC) matrices as archival data entities for repetitive structural design/analysis (3)
• Matched/non-matched point flutter solutions using K/g-methods with true damping (4)
• Built-in Flutter Mode Tracking procedure with structural parametric sensitivity analysis (5)
• State space Aeroservoelastic (ASE) analysis with continuous gust for SISO/MIMO control system (6)
• Trim analysis for static aeroelasticity/flight loads (7)
• Dynamic Loads Analysis including transient maneuver loads (MLOADS), ejection loads (ELOADS), and discrete gust loads (GLOADS) (8),(9),(10)
• 3D Spline module provides accurate FEM/Aero displacements and forces transferal (11)
• Modal Data Importer to process NASTRAN/I-DEAS/ELFINI/ANSYS/etc. modal output (12)
• Dynamic Memory & Database Management (ZDM) Systems establish subprogram modu-larity (13)
• Open architecture allows user direct access to data entities (14)
• Bulk Data Input minimizes user learning curve while relieving user input burden (15)
• Provides graphic display capability of aerodynamic models, CP’s, flutter modes and flutter curves for use with PATRAN/FEMAP/TECPLOT/ANSYS/EXCEL/etc. (16)
• Executive control allows massive flutter/ASE/Trim/Dynamic Loads inputs and solution out-puts (17)
• Nonlinear Flutter Analysis for open/closed loop system with structural nonlinearities using discrete time-domain state space approach (NLFLTR) (18)
• NASLINK module to export ZAERO aerodynamic data to MSC.NASTRAN (19)
Page 4
ZAERO Engineering Module Diagram
ZONA Technology, Inc.
UAIC Module Unsteady Aerodynamic
Data Generation (AIC) Matrices
FEM Module Modal Data Importer Executive Control Command
HFG Module Aerodynamic Model
Input
SPLINE Module Aerodynamic & FEM Model Interconnection
General Engineering Modules
FLUTTER/ FLTPRAM
Module ASE
TRIM Module MLOADS
Module ELOADS Module
GLOADSMFTGUST
Module
Discipline Engineering Modules NLFLTR Module
“ASSIGN FEM=“
Module
PLTAERO PLTAERO PLTC
P PLTC
PLTFLUT PLTFLUT PLTVG
PLTVG PLTMIST
PLTMIST PLTTRIM
PLTTRIM PLTTIME
PLTTIME MLDPRNT
MLDPRNT
PLTMODE PLTMODE
Graphical Post - Processing Output
Aeroelastic Analysis& Sensitivity
1
11
3
2 15
15
15
14
17
13
12
16
4
7
9
5
6
8
10
1819
Aeroelastic Analysis& Sensitivity
1
11
3
2 15
15
15
14
17
13
12
16
4
7
9
55
6
8
10
181819
Aerodynamic ModelDefinition
• CAERO7• BODY7
FEM/Aero Spline Input
• SPLINE1• SPLINE2
• SPLINE3• ATTACH
Flight ConditionDefinition
• MKAEROZ- Mach Numbers- List of reduced frequencies- Method flag for ZONA6, ZTRAN, ZONA7, ZONA7U
- Mean flow conditions in terms of α,β, p, q, r, and δ
HFGModule
Sensitivity
3D SplineModule
UAICModule
NASLINK
Flutter(g-method)
Aeroservoelasticity(ASE)
Flight Loads(TRIM)
Maneuver Loads(MLOADS)
Ejection Loads(ELOADS)
Gust Loads(GLOADS)
Nonlinear Flutter(NLFLTR)
User Direct Accessto Data Entities
Modal DataImporter
MSC.Nastran Structural Finite Element (FEM) Modal Output File(MSC, ASTROS, IDEAS, ELFINI, ANSYS, NE)
Graphic/Analysis Output(PATRAN, FEMAP, TECPLOT,
ANSYS, EXCEL, PEGASUS)
Executive Control
• FLUTTER • ASE • TRIM • NLFLTR• MLOADS • ELOADS • GLOADS
ZDM Database
• UAIC matrices of M, k pairs• Gust force vectors• Control surface aerodynamic
force vectors• 3-D spline matrix
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• ZONA6 generates steady/unsteady subsonic aerodynamics for wing-body/aircraft configurations with external stores/nacelles including body wake effects.
• ZTAIC generates unsteady transonic (modal) AIC’s using a transonic equivalent strip method.
• ZTRAN generates unsteady transonic wing-body AIC matrix using overset field-panel method.
• ZSAP generates steady/unsteady aerodynamics for wing-body configurations with external stores/nacelles at Mach number = 1.0.
• ZONA7 generates steady/unsteady supersonic aerodynamics for wing-body/aircraft configurations with external stores/nacelles (formerly ZONA51 for lifting surfaces).
• ZONA7U generates unified hypersonic and supersonic steady/unsteady aerodynamics for wing-body/aircraft configurations with external stores/nacelles.
ZAERO Unsteady Aerodynamic Methods
• The functionality of the UAIC module is to provide the AIC matrices needed for sub-sonic, transonic, supersonic, and hypersonic aeroelastic analysis. In addition, a ZONA Transonic AIC Weighting (ZTAW) module is available to correct the AIC matrix using the downwash weighting matrix method or the force correction matrix method.
NASTRAN
ZAERO/UAIC
Mach Number RangeSubsonic Transonic Supersonic Hypersonic
ZSA
Pat
M =
1.0
ZT
AIC
/ZT
RA
N
ZO
NA
7
ZO
NA
7U
ZO
NA
51
DL
M
Win
g/B
ody
with
Ext
erna
l Sto
res
Lifti
ng S
urfa
ce
Geo
met
ric F
idel
ity
ZO
NA
6
NASTRAN
ZAERO/UAIC
Mach Number RangeSubsonic Transonic Supersonic Hypersonic
ZSA
Pat
M =
1.0
ZT
AIC
/ZT
RA
N
ZO
NA
7
ZO
NA
7U
ZO
NA
51
DL
M
Win
g/B
ody
with
Ext
erna
l Sto
res
Lifti
ng S
urfa
ce
Geo
met
ric F
idel
ity
ZO
NA
6
Page 6
Functionality • Generates steady/unsteady subsonic aerodynamics for wing-body/aircraft configu-
rations with external stores/nacelles including the body-wake effect.
Main Features • Any combinations of planar/nonplanar lifting surfaces with arbitrary bodies includ-
ing fuselage+stores+tip missiles. • Higher-order panel formulation for lifting surfaces than the Doublet Lattice Method
(DLM). First case below shows the ZONA6 robustness over DLM. • High-order paneling allows high-fidelity modeling of complex aircraft with arbi-
trary stores/tip missile arrangement. Second case below shows a solution improve-ment.
70 Degree Delta Wing (M=0.8, k=0.5, ho=0.35cr) • Robust ZONA6 solutions are in
contrast to the breakdown of the DLM solutions
• High-order formulation of ZONA6 requires little care in paneling
SUBSONICUNSTEADYPRESSURES
40x10 panel cuts
Station 2
1.00.80.60.40.20.0-10.0
0.0
10.0
20.0
ZONA6DLM
x/c
Im(C
p)
10Station
-10.0
10.0
30.0
50.0
70.0
x/c
Im(C
p)
1.00.80.60.40.20.0
ZONA6DLM
40x10 panel cuts40x10 panel cuts
Station 2
1.00.80.60.40.20.0-10.0
0.0
10.0
20.0
ZONA6DLM
x/c
Im(C
p)
10Station
-10.0
10.0
30.0
50.0
70.0
x/c
Im(C
p)
1.00.80.60.40.20.0
ZONA6DLM
40x10 panel cuts
• No. of Wing Aero Boxes=90Tiptank Aero Boxes=264Store Aero Boxes=216
• ZONA6 shows improvement over NLR’s predicted results
UNSTEADYPRESSURES
ALONGSTORE
ZONA6
NLR Analysis
Test Data
NLR Wing-Tiptank-Pylon-Store (M=0.45, k=0.3055, q=157.5°, xo=0.15cr)
ZONA Technology, Inc.
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• Generates unsteady transonic AIC matrix that has the same form as AIC of ZONA6/ZONA7.
Functionality
• ZTRAN solves the time-linearized tran-sonic small disturbance equations using overset field-panel method.
• The surface box modeling is identical to that of ZONA6. Only a few additional input parameters are required to generate the volume cells.
• The variant coefficients in the time-linearized transonic small disturbance equation are interpolated from the Compu-tational Fluid Dynamics (CFD) steady solutions.
• The overset field-panel scheme allows the modeling of complex configurations with-out extensive field panel generation ef-forts.
Y
Z
X
Volume Block
Lift Surface
Volume CellY
Z
X
Volume Block
Lift Surface
Volume Cell
Lifting Surfaces
Y
Z
X
BODY7 Surface Boxes
Volume Block
Volume Cell
Y
Z
X
BODY7 Surface Boxes
Volume Block
Volume Cell
BodiesMain Features
Unsteady Pressure Validations
y/2b=47.5%
-10
0
10
20
30
40
50
60
0 0.2 0.4 0.6 0.8 1
X/C
Re
( ΔC
p)
ExperimentZONA6 (Linear)Present
y/2b=47.5%
-30
-25
-20
-15
-10
-5
0
5
10
0 0.2 0.4 0.6 0.8 1
X/C
Im ( Δ
Cp)
Experiment ZONA6 (Linear)Present
y/2b=81.7%
-20
-15
-10
-5
0
5
0 0.2 0.4 0.6 0.8 1
X/C
Im ( Δ
Cp)
Experiment ZONA6 (Linear)Present
y/2b=51.5%
-15
-10
-5
0
5
10
15
0 0.2 0.4 0.6 0.8 1
X/C
Im ( Δ
Cp)
Experiment ZONA6 (Linear)Present
F-5 wing at M = 0.9, K = 0.275
F-5 wing at M = 0.95, K = 0.264
Lessing wing at M = 0.9, K = 0.13
LANN wing at M = 0.822, K = 0.105
y/2b=51.5%
-5
0
5
10
15
20
25
0 0.2 0.4 0.6 0.8 1
X/C
Re
( ΔC
p)
ExperimentZONA6 (Linear)Present
y/2b=81.7%
-5
0
5
10
15
20
25
30
0 0.2 0.4 0.6 0.8 1
X/C
Re
( ΔC
p)
ExperimentZONA6 (Linear)Present
y/2b=50 %
0
2
4
6
8
10
12
14
0 0.2 0.4 0.6 0.8 1
x/c
Mag
nitu
de
Experiment (Test 1)Experiment (Test 2)ZONA6 (Linear)Present
y/2b=50%0
50
100
150
200
250
300
0 0.2 0.4 0.6 0.8 1
x/c
Phas
e A
ngle
(deg
)
Experiment (Test 1)Experiment (Test 2)ZONA6 (Linear)Present
Page 8
Flutter Validations
α = -2 (deg)
4
4.5
5
5.5
6
0.3 0.4 0.5 0.6 0.7 0.8 0.9
Mach Number
Flut
ter F
requ
ency
(Hz)
ExperimentZONA6 (Linear)Present
α = -2 (deg)
130
150
170
190
0.3 0.4 0.5 0.6 0.7 0.8 0.9
Mach Number
Dyn
. Pre
ssur
e (p
sf)
ExperimentZONA6 (Linear)Present
AGARD 445.6 weakened wing AGARD 445.6 solid wing PAPA wing at α= 1° PAPA wing at α= -2°
0.25
0.30
0.35
0.40
0.45
0.6 0.7 0.8 0.9 1.0 1.1 1.2
ZONA6 (Linear)
Experiment
Present
s
Ub αϖ μ
Mach Number
0.30
0.35
0.40
0.45
0.50
0.55
0.6 0.7 0.8 0.9 1.0 1.1 1.2
ZONA6 (Linear)
Experiment
Present
α
ϖϖ
Mach Number
0.45
0.50
0.55
0.60
0.65
0.6 0.8 1.0 1.2
ZONA6 (Linear)
Experiment
Present
Mach Number
s
Ub αϖ μ
0.45
0.50
0.55
0.60
0.65
0.6 0.8 1.0 1.2
ZONA6 (Linear)
Experiment
Present
Mach Number
α
ϖϖ
α = +1 (deg)
130
150
170
190
0.3 0.4 0.5 0.6 0.7 0.8 0.9
Mach Number
Dyn
. Pre
ssur
e (p
sf)
ExperimentZONA6 (Linear)Present
α = +1 (deg)
4
4.5
5
5.5
6
0.3 0.4 0.5 0.6 0.7 0.8 0.9
Mach Number
Flut
ter
Freq
uenc
y (H
z)
ExperimentZONA6 (Linear)Present
ZONA Technology, Inc.
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Functionality • Generates steady/unsteady supersonic aerodynamics for wing-body/aircraft configu-
rations with external stores/nacelles
Main Features • Any combinations of planar/nonplanar lifting surfaces with arbitrary bodies includ-
ing fuselage+stores+tip missiles. • Panel formulation for lifting surface is identical to that of ZONA51 – now the indus-
trial standard method for supersonic flutter analysis in MSC.NASTRAN. • High-order paneling allows high-fidelity modeling of complex aircraft with arbitrary
stores/tip missile arrangement.
NACA Wing-Body (xo=0.35cr) ZONA7
WING + BODYWING ONLYBODY ONLY
TEST DATAR = 1.18 x 106
R = 1.89 x 106
MOMENTDERIVATIVES
IN-PITCH
NLR F-5 Wing with Underwing Missile (F=20Hz, k=0.1, xo=0.5cr)
ZONA7PP + LP + L + MB + AW
TEST DATAPYLON (P)P + LAUNCHER (L)P + L + MISSILE
BODY (MB) +AFT WINGS (AW)
P + L + MB + AW + CANARD FINS
UNSTEADYSIDE FORCE
AND YAWING MOMENT
Page 10
Functionality • Generates unified hypersonic and supersonic steady/unsteady aerodynamics for wing
-body/aircraft configurations with external stores/nacelles.
Main Features • Nonlinear thickness effects of ZONA7U yields good agreement with Euler solution
and test data. • Steady solutions approach linear and Newtonian limits. • Confirms hypersonic Mach independent principle. • Results/formulation are superior to Unsteady Linear Theory and Piston Theory. • ZONA7U usually results in more conservative flutter boundaries than other methods. • Unified with ZONA7 and is therefore applicable to all Mach numbers > 1.0. • Additional input to ZONA7 amounts to only wing root and tip sectional profile
thickness.
70 Degree Delta Wing • Thickness effect apparent a t
higher M• Thus, it yields more
conservative flutter boundaries
SUPERSONICFLUTTER
BOUNDARIES
Rectangular Wing with Wedge Profile(M=4.0, s=15°, xo=0.25c)
• ZONA7U solution compares well with Euler solution over a wide frequency range
• Piston Theory and Linear Theory (ZONA7) yield inferior results by comparison
HYPERSONIC/SUPERSONIC
GAF - CLα
ZONA Technology, Inc.
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Functionality • Generates steady/unsteady aerodynamics at sonic speed (M = 1.0) for wing-body/
aircraft configurations with external stores/nacelles.
Main Features • Any combinations of planar/nonplanar lifting surfaces with arbitrary bodies includ-
ing fuselage+stores+tip missiles. • Compute the steady/unsteady aerodynamics at exactly Mach one. • Paneling scheme is identical to that of ZONA6/ZONA7, i.e. ZSAP shares the same
aerodynamic model as ZONA6/ZONA7. • Computational time is on the same order.
Non-Planar Aerodynamics of a SAAB/Canard Wing
-2
-1
0
1
0 1 2 3 4 5 6
k
-0.5
0
0.5
1
1.5
0 1 2 3 4 5 6k
-2
-1
0
1
0 1 2 3 4 5 6
k
-0.5
0
0.5
1
1.5
0 1 2 3 4 5 6k
ReQ12
ImQ12
Box Number 10 X 10 for Canard Box Number 50 X 10 for Canard & 20 X 20 for Wing & 90 X 20 for Wing
ZONA7(M=1.01)Present (M=1.0)ZONA6 (M=0.99)
• Canard-Wing configuration in Canard Pitch
Motion about its Mid-Chord. • Lift on Wing is mainly induced by the
oscillatory wake from Canard. • Real and Imaginary parts of Lift (Re(Q12) &
Im(Q12)) at M=1.0 are contrasted with that of the Subsonic Lifting Surface Method (ZONA6) at M=0.99 and the Supersonic Lifting Surface Method (ZONA7) at M=1.01
• ZONA6 and ZONA7 require large number of Boxes for solution convergence whereas the Present Sonic Method does not.
AGARD standard 445.6 Weakened Wing (in Air) and Solid Wing (in Freon 12)
0.25
0.3
0.35
0.4
0.45
0.5
0.55
0.8 0.9 1 1.1 1.2Mach Number
TDT Test (Solid/Freon 12)
Present (Solid/Freon 12)
TDT Test (Weakened/Air)
Present (Weakened/Air)
0.3
0.35
0.4
0.45
0.5
0.55
0.8 0.9 1 1.1 1.2Mach Number
μωαsbU
αωω
•Comparison of Flutter Speed Index and Flutter Frequency Ratio with TDT wind tunnel measurements
Page 12
Functionality
Main Features
• Generates a corrected AIC matrix to match the given set of forces/moments or unsteady pressures.
• The AIC correction module computes the AIC weighting matrix using a ZONA Transonic AIC Weighting (ZTAW) method that adopts a successive kernel expan-sion procedure.
• The ZTAW method is an improved AIC correction method over the previous cor-rection methods such as the force/moment correction method by Giesing et al and the downwash weighting matrix (DWM) method by Pitt and Goodman. With in-phase pressures obtained by wind-tunnel measurement or CFD, ZTAW yields accurate out-of-phase and higher frequency pressures resulting in well-correlated aeroelastic solutions whereas the previous method yield erroneous out-of-phase pressure in terms of shock jump behavior.
• Four methods are incorporated in ZTAW: the steady downwash weighting matrix method, the unsteady downwash weighting matrix method, the steady force cor-rection matrix method, and the unsteady force correction matrix method.
Unsteady Pressure Validations
y/2b=64.1%
-15
-10
-5
0
5
10
15
20
0 0.2 0.4 0.6 0.8 1
X/C
Re
( ΞC
p )
Experiment ZTAWDWM
y/2b=64.1%
-10
-8
-6
-4
-2
0
2
4
6
0 0.5 1
X/C
Im ( Ξ
Cp )
ExperimentZTAWDWM
F-5 Wing at M = 0.95 and k = 0.264
y/2b=65%
-10
0
10
20
30
40
50
0 0.2 0.4 0.6 0.8 1
X/C
Re
( ΞC
p )
Experiment ZTAWDWM
y/2b=65%
-25
-20
-15
-10
-5
0
5
10
0 0.5 1
X/C
Im ( Ξ
Cp )
ExperimentZTAWDWM
LANN Wing at M = 0.822 and k = 0.105
Flutter Validations
12.0
14.0
16.0
18.0
20.0
22.0
24.0
0.6 0.7 0.8 0.9 1Mach
Freq
uenc
y (H
z)
DWMZONA 6ExperimentZTAW
60.0
70.0
80.0
90.0
100.0
110.0
120.0
130.0
140.0
0.6 0.7 0.8 0.9 1Mach
Dyn
. Pre
ss.
DWMZONA 6ExperimentZTAW
α = -2 (deg)
130
150
170
190
0.3 0.4 0.5 0.6 0.7 0.8 0.9
Mach
Dyn
. Pre
ssur
e (p
sf) Experiment
Zona 6ZTAW
α = -2 (deg)
4
4.5
5
5.5
6
0.3 0.4 0.5 0.6 0.7 0.8 0.9
Mach
Flut
ter F
requ
ency
(Hz)
ExperimentZona 6ZTAW
AGARD 445.6 Weakened Wing PAPA Wing at α=2°
ZONA Technology, Inc.
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The HFG module is capable of modeling any full aircraft configuration with stores and/or nacelles. A complex aircraft configuration can be represented by the HFG module by means of wing-like and body-like definitions. Wing thickness effects and in-flow of the inlet and out-flow of the nozzle effects can be included in the boundary condition.
ZAERO F-15 model with 4258 boxes
Body-Like Components
Wing Macroelements
CAERO7Thickness Distributionfor Hypersonic Flow
PAFOIL7
Steady Cp Inputfor Transonic Flow
CAERO7
Wing-Like Components
Body Macroelements
BODY7Body SurfaceGrid Definition
SEGMESH
Body-Wake ModelEngine Inlet Model
PBODY7
Wing-Like Components Include:-Wings, Tails, Pylons,-Launchers, -Store Fins, etc.
Body-Like Components Include:-Fuselage,-Underwing Stores, -Missile Bodies, etc.
Body-Like Components
Wing Macroelements
CAERO7Thickness Distributionfor Hypersonic Flow
PAFOIL7
Steady Cp Inputfor Transonic Flow
CAERO7
Wing-Like Components
Body Macroelements
BODY7Body SurfaceGrid Definition
SEGMESH
Body-Wake ModelEngine Inlet Model
PBODY7
Wing-Like Components Include:-Wings, Tails, Pylons,-Launchers, -Store Fins, etc.
Body-Like Components Include:-Fuselage,-Underwing Stores, -Missile Bodies, etc.
Page 14 ZONA Technology, Inc.
ZAERO F-18 model with 2530 boxes
ZAERO Morhing Aircraft model with 2062 boxes
ZAERO C-130 model with 3978 boxes
ZAERO Predator model with 1410 boxes
ZAERO F-16 model with 4002 boxes
The 3D Spline module establishes the displacement/force transferal between the struc-tural Finite Element Method (FEM) model and the ZAERO aerodynamic model. It consists of four spline methods that jointly assemble a spline matrix. These four spline methods include: (a) Thin Plate Spline; (b) Infinite Plate Spline; (c) Beam Spline and (d) Rigid Body Attachment methods. The spline matrix provides the x, y and z dis-placements and slopes in three dimensions at all aerodynamic grids.
Page 15 To Order Call: 480•945•9988
FEM Model Aerodynamic Model
Rigid Body Pitch Mode
First Wing Bending Mode
First Wing Torsion Mode
Page 16 ZONA Technology, Inc.
ZAERO utilizes the bulk data input format, similar to that of NASTRAN and ASTROS. This type of input format has the advantage of: (a) minimizing the user learning curve; (b) relieving user input burden and (c) automated input error detection. An example of this type of input format is shown below. Flow charts are also shown demonstrating some of the ZAERO bulk data interdependencies.
• Example of ZAERO Bulk Data Input Format 1 2 3 4 5 6 7 8 9 10
CAERO7 WID LABEL ACOORD NSPAN NCHORD LSPAN ZTAIC PAFOIL7 CONT
CONT XRL YLR ZRL RCH LRCHD ATTCHR ACORDR CONT
CONT XTL YTL ZTL TCH LTCHD ATTCHT ACORDT
CAERO7 101 WING 8 5 4 20 0 0 ABC
+BC 0.0 0.0 0.0 1.0 10 4 DEF
+EF 0.0 1.0 0.0 1.0 11 0
• Bulk Data Interrelationship for Aerodynamic Geometry Input
CAERO7
BODY7
AEROZ Aerodynamic Reference Parameters
ACOORD
ZTAIC Transonic strip
method
MACHCP
CHORDCP
PAFOIL7
AEFACT
PBODY7
SEGMESH
PAFOIL7 SID IPBODY7 IDMESH
SID
PLTAERO Plot the Aerodynamic
model
Surface Box Generation
CELLWNG CELLBDY CELLBOX
ZTRAN Overset Field - Panel Method
Wing components
Body components
AEROZ Aerodynamic Reference Parameters
AEFACT
Mach-Steady Cp relation
Steady Cp Input
Transonic Aerodynamics with Steady Pressure Input
Define airfoil shape
Aerodynamics with airfoil thickness/camber distribution input
- X-coordinate - Airfoil camber - Airfoil half-thickness
Aero-coordinate system
Wake/Inlet Panels
Body segment definition
AEFACT
ZTAIC Spanwise/chordwise divisions of wing
Coordinate location of circumferential points for arbitrary body
Page 17 To Order Call: 480•945•9988
ZAERO allows for the graphic interface with commercialized graphic packages. Graphical data in output files containing the aerodynamic model, unsteady pressures (CP), interpolated structural modes, and flutter modes can be displayed via PATRAN, FEMAP, IDEAS, PEGASUS or TECPLOT. V-g and V-f diagrams can be displayed via typical X-Y plotting packages (e.g., Excel). An example of the F-16 aerodynamic model with external stores and the resulting V-g and V-f diagrams are shown below.
Unsteady aerodynamic Model V-G and V-F Diagrams
FEM Model
Aerodynamic Model
Animated Flutter Mode Verification of Spline
-4
-3
-2
-1
0
1
2
3
4
0 0.25 0.5 0.75 1
β (d
eg)
ExperimentalConner et al.ZAERO
U = 11.711 m/s
Unsteady Pressure Display Transient Response
Page 18 ZONA Technology, Inc.
The ZAERO flutter module contains two flutter solution techniques: the K-method and the g-method. The g-method is ZONA’s newly developed flutter solution method (Ref 20) that generalizes the K-method and the P-K method for true damping predic-tion. Ref 20 shows that the P-K method is only valid at the conditions of zero damp-ing, zero frequency, or linear varying generalized aerodynamic forces (Q) with re-spect to reduced frequency. In fact, if Q is highly nonlinear, it is shown that the P-K method may produce unrealistic roots due to its inconsistent formulation. The flutter module has a built-in atmospheric table as an option to perform matched-point flutter analysis. Sensitivity analysis with respect to the structural parameters is also included in the g-method.
• Three Degrees of Freedom Airfoil at M=0.0 (MSC/NASTRAN HA145 Test Case) • A non-zero frequency “dynamic diver-
gence speed” is well predicted by the g-method, the P-K method and the tran-sient method (a time-domain method).
• Both the g-method and the transient method capture two aerodynamic lag roots which are absent in the P-K method solution.
• The frequency vs. velocity (V-f) dia-grams of the g-method and the transient method are in good agreement. The frequency of the free-free plunge mode computed by the P-K method remains zero. This results in poor correlation in the V-f diagram with the g-method and transient method.
-1.0
-0.8
-0.5
-0.3
0.0
0.3
0.5
0
Dam
ping
50 100 150 200 250 300Velocity (ft/s)
0.0
1.0
2.0
3.0
4.0
5.0
0 50
Freq
uenc
y (H
z)
100 150 200 250 300
Transient Method
Velocity (ft/s)
g-mode 1
g-mode 2g-mode 3
g-aero lag 1
g-aero lag 2pk-mode 1
pk-mode 2
pk-mode 3
Direct Method Mass Increment Method
Modal Analysis
General-ized
Matrices
Flutter Equations
assumes assumes
Page 19 To Order Call: 480•945•9988
Functionality
Main Features
• Performs parametric flutter analysis by executing a massive number of flutter/ASE analyses for various mass and stiffness distributions.
• Massive flutter analyses of open/closed loop systems with various mass and stiffness in the structures using the mass increment method.
• For n aircraft/store configurations, the flutter equation in physical coordinates {x} reads:
where MB and KB are the mass and stiffness matrices of a baseline structure and ΔM and ΔK are the incremental changes of mass and stiffness from the baseline structure to the ith structure of interest.
[ ]{ } [ ]{ } [ ]{ } 0, 1,B i B iM M x K K x q AIC x i n∞+ Δ + + Δ − = =
[ ] [ ] { }2 0,
1,B i B i iM M K K
i n
ω φ⎡ ⎤− +Δ + +Δ =⎣ ⎦=
[ ] [ ] { }2 0B B BM Kω φ⎡ ⎤− + =⎣ ⎦
[ ][ ][ ]
`
`
`
`
, 1,
Ti i B i i
Ti i B i i
Ti i i
M M M
K K K
Q AIC i n
φ φ
φ φ
φ φ
⎡ ⎤ = + Δ⎣ ⎦⎡ ⎤ = + Δ⎣ ⎦⎡ ⎤ = =⎣ ⎦ [ ] , compute only once
TB B B B
TB B B B
TB B B
M M
K K
Q AIC
φ φ
φ φ
φ φ
⎡ ⎤ =⎣ ⎦⎡ ⎤ =⎣ ⎦⎡ ⎤ =⎣ ⎦
{ } [ ]{ }{ }2 0
i d
i i i d
x
S M K q Q
φ ξ
ξ∞
=
⎡ ⎤+ − =⎣ ⎦
{ } [ ]{ }{ }2
0
B B
T TB B i B B B i B B B
x
S M M K K q Q
φ ξ
φ φ φ φ ξ∞
=
⎡ ⎤⎡ ⎤ ⎡ ⎤+ Δ + + Δ − =⎣ ⎦ ⎣ ⎦⎣ ⎦
500 1000 1500 2000 2500 3000 3500 4000
250
500
750
1000
1250
1500
1750
2000
550
750
700
750
700
750
850
900
800
9501150
8501000
115012008501000
950900
900900
850950
950
950
1150
900
10501100
1200
1000
950
1000
1050
1150
1050
9501000
11001250 1050
1300
1300
1100
10001100
1050 10501350
11001150
1200
10001100
1100
1250
1100
1100
1150
1150
1050
1100
11001200
1100
1200
1100
1300
1150
1100
1100
1100
1100
1200
1250
13001250
1200
1250
1200
1250
1200
1250
1250
1250
1300
1300
1300
1350
1350
V(KEAS)140013501300125012001150110010501000950900850800750700650600550500450
Weight (lbs)
Pitc
hIn
ertia
(slu
g-ft2 )
• Data mining the massive flutter results by automatically searching for the ve-locity-damping curve crossing at user-specified damping levels.
• Ease for post-processing using off-the-shelf graphic tool such as TECPLOT. Shown in the figure is the flutter speed vs. various pitch inertia and weight dia-gram of the store.
• Flags to indicate the severity of the flut-ter instability
Page 20 ZONA Technology, Inc.
Performs the static aeroelastic/trim analysis for solving the trim system and computing the flight loads.
StaticAeroelasticDeformation
StressDistribution
W ind Tunnel M odelASTR OS - LIFT TR IMAOA = 1 D eg., M =0.9V=12053 in/sec
-25638.4
-25638.4
-20630.1
-25638.4
-20630.1
4411.4
-20630.1
-20630.1
-560
5 .2
-5605.2
-20630.1
ASTROS R ESULTM = 1.2, q = 350 psfAOA = 5 Deg.VSS/ON
F-18 at 4-G Pull-Up Maneuver at Mach 1.2 and Altitude = 10,000 ft.
Main Features • It employs the modal approach for solving the trim system of the flexible aircraft.
The modal approach formulates a reduced-order trim system that can be solved with much less computer time than the so-called “direct method”.
• It is capable of dealing with the determined trim system as well as the over-
determined trim system (more unknowns than the trim equations). The solutions of the over-determined trim system are obtained by using an optimization technique which minimizes a user-defined objective function while satisfying a set of constraint functions.
• For a symmetric configuration (symmetric about the x-z plane), it requires only the
modeling of one half of the configuration even for the asymmetric flight conditions. • It generates the flight loads on both sides of the configuration in terms of forces and
moments at the structural finite element grid points in terms of NASTRAN FORCE and MOMENT bulk data cards for subsequent detailed stress analysis.
Page 21 To Order Call: 480•945•9988
Constructs state-space equations for the open-loop or closed-loop aeroelastic system and performs stability analysis
ZAEROUAIC Module
Baseline FEModel
StructuralVariations
GeneralizedMatrices
Rational AerodynamicApproximations
ControlMode
GustModel
ControlMargins
GustResponse
State SpaceASE Model
Open/Closed-loop
Flutter
AnalysisResults
SensitivityAnalysis
ASE Module
Main Features
• Rational-function approximation of the unsteady aerodynamic coefficient matrices • State-space MIMO formulation • Modular linear control modeling of most-general architecture • Open- and closed-loop flutter analysis • Open-closed gain and phase margins • Input and output singular values • Augmentation of continuous-gust dynamics • Structural gust response in statistical terms • Fixed-modes parametric studies • Sensitivity of flutter and control margins with respect to structural and control vari-
ables • Frequency-domain stability analysis without rational function approximation
Page 22 ZONA Technology, Inc.
The unsteady frequency domain aerodynamic force coefficient matrices are approxi-mated by a rational matrix function in the Laplace domain. The approximation for-mula is either the classic Roger’s formula where p is the non-dimensional Laplace variable p = sb/V, or the more general mini-mum-state formula that results with significantly less subsequent aerodynamic states per desired accuracy.
The approximation roots are selected by the user or determined by the code based on the frequency range of the input matrices. A direct least-square solution is used for Roger’s approximation, and a non-linear least-square is used for the minimum-state approximation.
22
0 1 223
( )ln
lll
pQ p A A p A p Ap γ
+
−=
⎡ ⎤ = + + +⎡ ⎤ ⎡ ⎤⎡ ⎤ ⎡ ⎤⎣ ⎦ ⎣ ⎦⎣ ⎦ ⎣ ⎦⎣ ⎦ +∑
( ) 120 1 2( )Q p A A p A p D I p R E p
−⎡ ⎤ = + + + −⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦⎣ ⎦⎣ ⎦
Computed Approximated
-300 -250 -2000
20
40
60
80
Real
Imag
inar
y
Q (3,2)
-100 -50 0 50-250
-200
-150
-100
-50
0
Real
Imag
inar
y
Q (3,4)
-50 0 50 100-100
-80
-60
-40
-20
0
Real
Imag
inar
y
Q (5,4)
-150 -100 -50 00
50
100
150
200
250
Real
Imag
inar
y
Q (2,2)
Page 23 To Order Call: 480•945•9988
• Open-loop Aeroelastic State-Space Equation − The generalized structural matrices and the aerodynamic approximation coefficient
matrices are used to construct the time-domain state-space equation of motion of the open-loop aeroelastic system excited by control-surface motion
− Augmentation of control actuators of at least third order yields the plant equations
• Control System Model − Single-Input-Single-Output (SISO)
elements defined by s-domain trans-fer functions
− Multi-Input-Multi-Output (MIMO) elements defined by individual state-space matrices [Ac], [Bc], [Cc], [Dc] that may be imported from external control synthesis codes.
• Closed-loop ASE Model − The plant and control models
are interconnected by the fol-lowing scheme:
− Stability analyses of open- and closed-loop systems are based on system eigenvalues. Sensi-tivity computations are based on analytical expressions.
− Performed with respect to these gains.
− Junction elements (JNC) which are actually zero-order elements connecting some inputs with some outputs by {yj} = [Dj]{uj}.
− Variable control gains which form the control gain matrix when the system is closed. Control margins, singular values and sensitivity analyses are performed with respect to these gains.
{ } { } { }{ } { }
p p p p p
p p p
x A x B u
y C x
⎡ ⎤ ⎡ ⎤= +⎣ ⎦ ⎣ ⎦⎡ ⎤= ⎣ ⎦
Page 24 ZONA Technology, Inc.
The transient maneuver loads module performs the transient maneuver loads analysis due to the pilot input command.
Idealized Forward Swept Wing Test Case (ZAERO and P-Transform correlate well while the quasistatic method does not due to low-frequency approximation)
Main Features:
• It is formulated in the state space form for either the open loop or closed loop sys-tem. The rigid body degrees of freedom are transformed into the airframe states so that the sub-matrices associated with the airframe states in the state space matrices are in the same definition with those of the flight dynamics.
• It allows the users to replace the program-computed sub-matrices associated with the airframe states by those supplied by the flight dynamic engineers. This can ensure that the time response of the airframe states is in close agreement with those of the flight dynamic analysis.
• It computes the time histories of the maneuver loads of flexible airframe in the presence of control system. These maneuver loads include the time histories of component loads, grid point loads, etc. Based on these time histories of loads, the user can identify the critical maneuver load conditions.
• It outputs the transient maneuver loads at each time step in terms of NASTRAN FORCE and MOMENT bulk data cards either by the mode displacement method or the mode acceleration method for subsequent detailed stress analysis.
Page 25 To Order Call: 480•945•9988
The transient ejection loads module performs the transient ejection loads analysis due to store separation.
Aircraft Response due to Ejection Force (x 5)0
0.2
0.4
0.6
0.8
1
1.2
0 0.1 0.2 0.3 0.4 0.5
Time (sec)
Sto
re E
ject
or F
orce
/Fm
ax
-5-4-3-2-1012345
0 0.1 0.2 0.3 0.4 0.5Time (sec)
Win
g Ti
p Fw
d G
/ Fm
ax (*
1000
)
F light TestZAERO
-5-4-3-2-1012345
0 0.1 0.2 0.3 0.4 0.5Time (sec)
Win
g Ti
p A
ft G
/ Fm
ax (*
1000
)
F light TestZAERO
Advanced Fighter Test Case (ZAERO versus Flight Test)
• It allows multiple store ejections (in sequential scheduling) while the aircraft is maneuvering due to pilot input commands.
• It accounts for the effects of the sudden reduction in aircraft weight due to the separation of the stores from the aircraft.
• It is formulated in the state-space form for either an open-loop or closed-loop sys-tem.
• It outputs the transient loads at each time step in terms of NASTRAN FORCE and MOMENT bulk data cards either by the mode displacement method or the mode acceleration method for subsequent detailed stress analysis.
Main Features:
Page 26 ZONA Technology, Inc.
The gust loads module performs transient discrete and continuous gust analysis for either open-loop or closed-loop system.
2-D Thin Airfoil Subjectedto a Sharp-Edged Gust
Comparison Between Sear’s Function and the Gust Forces Computed by ZONA6
Comparison Between Wagner’s Function and ZAERO State-Space Equations
Comparisons Between ZAERO Results and Analytical Solution for a 2-D Airfoil Encountering
Sharp-Edged Gust
z
x
-∞
b
1 =V
WG
b
b = 1 ftM = 0.0V = 100 ftρ = 0.0002 slug/ft3
q∞ = 1 psf
z
x
-∞
b
1 =V
WG
b
b = 1 ftM = 0.0V = 100 ftρ = 0.0002 slug/ft3
q∞ = 1 psf
k = 0
0.04
0.100.20
0.40
0.60
0.80
1.00
1.20
1.60
2.0
Values of k
2.5
3.0
3.5
4.0
5.05.0
6.0
7.0
8.0
9.0
10.0
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
0.25
0.3
-0.4 -0.2 0 0.2 0.4 0.6 0.8 1 1.2
Real
Imag
inar
y
Sears Function
ZAERO/ZONA6Aerodynamics
0.4
0.5
0.6
0.7
0.8
0.9
1
0 4 8 12 16 20
tV/L
Wagner's Fuction
ZAERO/ASE
Wagner's Fuction
ZAERO/ASE
⎟⎠⎞
⎜⎝⎛
LtVφ
0.4
0.5
0.6
0.7
0.8
0.9
1
0 4 8 12 16 20
tV/L
Wagner's Fuction
ZAERO/ASE
Wagner's Fuction
ZAERO/ASE
⎟⎠⎞
⎜⎝⎛
LtVφWagner's Fuction
ZAERO/ASE
⎟⎠⎞
⎜⎝⎛
LtVφ
0
0.4
0.8
1.2
1.6
0 4 8 12 16 20 24 28
tV/L
Kz
μ = 5Analytical SolutionZAERO
μ = 15Analytical SolutionZAERO
μ = 100Analytical SolutionZAERO
0
0.4
0.8
1.2
1.6
0 4 8 12 16 20 24 28
tV/L
Kz
μ = 5Analytical SolutionZAERO
μ = 15Analytical SolutionZAERO
μ = 100Analytical SolutionZAERO
μ = 5Analytical SolutionZAERO
μ = 15Analytical SolutionZAERO
μ = 100Analytical SolutionZAERO
Validation of the Discrete Gust Module with 2-D Classical Theory (Excellent agreement is seen while NASTRAN fails to provide satisfactory results)
Main Features: • It includes various options for defining the discrete gust profile such as one-minus-
cosine, sine, sharp-edged gust, and arbitrary gust profiles for discrete gust and Dryden’s or Von Karman’s gust spectrum for continuous gust.
• For the discrete gust analysis, it includes three options to model the gust profile; the frequency-domain approach, the state-space approach, and the hybrid approach where the discrete gust loads are obtained by inverse Fouier transform and the sys-tem matrix by state-space formulation.
• Its state space equations provide accurate displacement time history thereby cir-cumventing the unreasonably large displacement response problem of the Fourier transform method in NASTRAN.
• It outputs the transient loads at each time step in terms of NASTRAN FORCE and MOMENT bulk data cards either by the mode displacement method or the mode acceleration method for subsequent detailed stress analysis.
Page 27 To Order Call: 480•945•9988
The nonlinear flutter module is a simulation tool for the transient response of open/closed-loop aeroelastic systems that include (1) nonlinear structures (2) nonlinear control system (3) large-amplitude unsteady aerodynamics (externally imported from other CFD code).
-4
-3
-2
-1
0
1
2
3
4
5
0 0.25 0.5 0.75 1Time (s)
β (d
eg)
ExperimentalConner et al.ZAERO
U = 6.453 m/s
-4
-3
-2
-1
0
1
2
3
4
0 0.25 0.5 0.75 1
β (d
eg)
ExperimentalConner et al.ZAERO
U = 11.711 m/s
-5.0
-3.0
-1.0
1.0
3.0
5.0
7.0
β (d
eg)
ExperimentalConner et al.ZAERO
U = 17.447 m/s
3 d.o.f. Airfoil with Free-Play
Excellent Agreement with Analytical and Experimental Results
Main Features:
• Nonlinearities can be specified as a function of multiple user defined nonlinear parameters such as displacements, velocities, accelerations, element forces, modal values and control system outputs.
• Discrete time-domain state space equations at each distinct value of the nonlinear parameters are pre-computed. During the time-integration computation, updated state-space equations are obtained by interpolation.
• It outputs the NASTRAN FORCE and MOMENT bulk data cards at a given time step for subsequent stress analysis.
Page 28 ZONA Technology, Inc.
ZONA51 1. Chen, P.C. and Liu, D.D., "A Harmonic Gradient Method for Unsteady Supersonic Flow
Calculations," Proceedings of the 24th AIAA/ASME/ASCE/AHS Structures, Structural Dynamics and Materials Conference, Lake Tahoe, Nevada, May 2-4,1983, AIAA Paper No.. 83-0887-CP. Also Journal of Aircraft, Vol. 22, No. 15, May 1985, pp. 371-379.
2. Liu, D.D., James, D.K., Chen, P.C. and Pototzky, A.S., "Further Studies of Harmonic Gra-dient Method for Supersonic Aeroelastic Applications, " DGLR/AAAF/RAeS European Forum on Aeroelasticity and Structural Dynamics, Aachen, FRG, April 17-19, 1989, Paper No. 89-068. Also Journal of Aircraft, Vol. 28, No. 9, September, 1991, pp. 598-605.
3. Johnson, E.H., Rodden, W.P., Chen, P.C. and Liu, D.D., Comment on "Canard-Wing Inter-action in Unsteady Supersonic Flow," Journal of Aircraft, Vol. 29, No. 4 July-August, 1992, p. 744.
ZONA7 4. Chen, P.C. and Liu, D.D., "Unsteady Supersonic Computations of Arbitrary Wing-Body
Configurations Including External Stores," AIAA/ASME/ASCE/AMS/ASC 29th Struc-tures, SDM Conference, Williamsburg, Virginia, April 18-20, AIAA Paper No. 88-2309CP. Also Journal of Aircraft, Vol. 27, No. 2, February 1990, pp. 108-116.
5. Garcia-Fogeda, P. and Liu, D.D., "Analysis of Unsteady Aerodynamics for Elastic Bodies in Supersonic Flow," AIAA 24th Aerospace Sciences Meeting, January 6-9, 1986, Reno, Nevada, AIAA Paper No. 86-0007. Also Journal of Aircraft, Vol. 24, No. 12, December 1987, pp. 833-840.
6. Garcia-Fogeda, P. and Liu, D.D., "Supersonic Aeroelastic Applications of Harmonic Poten-tial Panel Method to Oscillating Flexible Bodies," Journal of Spacecraft and Rockets, Vol. 25, No. 4, July-August 1988, pp. 271-277.
7. Liu, D.D., Garcia-Fogeda, P. and Chen, P.C., "Oscillating Wings and Bodies with Flexure in Supersonic Flow--Applications of Harmonic Potential Panel Method," International Council of Aeronautical Sciences, London, U.K., September 7-12, 1986, I.C.A.S. Paper No. 86-2.9.4. Also Journal of Aircraft, Vol. 25, No. 6, June 1988, pp. 507-514.
8. Garcia-Fogeda, P., Chen, P.C. and Liu, D.D., "Unsteady Supersonic Flow Calculations for Wing-Body Combinations Using Harmonic Gradient Method," AIAA 26th Aerospace Sci-ences Meeting, Reno, Nevada, January 11-14, 1988, AIAA Paper No. 88-0568. Also AIAA Journal, Vol. 28, No. 4, April 1990, pp. 635-641.
ZONA6 9. Chen, P.C., Lee, H.W., and Liu, D.D., "Unsteady Subsonic Aerodynamics for Bodies and
Wings with External Stores including Wake Effect", presented at the Aerospace Flutter and Dynamic Council Meeting, November 14-15,1990, San Antonio, Texas, and paper pre-sented at the international Forum on Aeroelasticity and Structural Dynamics, Aachen, June 3-6, 1991. Also Journal of Aircraft, Vol. 30, No. 5, Sept-Oct. 1993, pp. 618-628.
10. Liu, D.D., Chen, P.C., Yao, Z.X. and Sarhaddi, D., "Recent Advances in Lifting Surface Methods," Paper No. 4, Proceeding of International Forum on Aeroelasticity and Structural Dynamics, Manchester, U.K., June 1995. Also in The Royal Aeronautical Journal, Vol. 100, No. 998, Oct. 1996, pp. 327-339.
Page 29 To Order Call: 480•945•9988
ZTAIC 11. Liu, D.D., Kao, Y.F. and Fung, K.Y., "An Efficient Method for Computing Unsteady
Transonic Aerodynamics of Swept Wings with Control Surfaces," Journal of Aircraft, Vol. 25, No. 1, January 1988, pp. 25-31.
12. Chen, P.C., Sarhaddi, D. and Liu, D.D., "Transonic AIC Approach for Aeroelastic and MDO Applications," presented at the Euromech Colloquium 349 at DLR, Göttingen, Germany, Sept. 16-18, 1996. Also, Journal of Aircraft, Vol. 37, No. 1, Jan.-Feb. 2000,
ZONA7U 13. Liu, D.D., Yao, Z.X., Sarhaddi, D., and Chavez, F., “Piston Theory Revisited and Further
Applications,” ICAS Paper 94-2.8.4, presented at the 19th Congress of the International Council of the Aeronautical Sciences, Sept. 1994, also Journal of Aircraft, Vol. 34, No. 3, May-June 1997, pp. 304-312.
14. Chen, P.C., and Liu, D.D., “Unified Hypersonic/Supersonic Panel Method for Aeroelastic Applications to Arbitrary Bodies,” Journal of Aircraft, Vol. 39, No. 3, May-June 2002.
ZAERO/UAIC for MDO 15. Chen, P.C., Liu, D.D., Sarhaddi, D., Striz, A.G., Neill, D.J. and Karpel, "Enhancement of
the Aeroservoelastic Capability in ASTROS," STTR Phase I Final Report WL-TR-96-3119, Sept. 1996.
16. Chen, P.C., Sarhaddi, D. and Liu, D.D., “A Unified Unsteady Aerodynamic Module for Aeroelastic and MDO Application,” AGARD Structures and Material Panel (SMP)-Workshop 2 “Numerical Unsteady Aerodynamics and Aeroelastic Simulation,” Alborg, Denmark, Oct. 13-17, 1997.
17. Chen, P.C., Sarhaddi, D., Liu, D.D. and Karpel, M., “Unified Aerodynamic-Influence-Coefficient Approach for Aeroservoelastic and Multidisciplinary Optimization Applica-tions,” AIAA Paper No. 97-1181-CP. Also, Journal of Aircraft, Vol. 37, No. 2, Mar.-Apr. 2000, pp. 260-265.
18. Chen, P.C., Sarhaddi, D., Liu, D.D., Karpel, M., Striz, A.G. and Jung, S.Y., “A Unified Unsteady Aerodynamic Module for Aeroelastic, Aeroservoelastic and MDO Applica-tions,” CEAS, Vol. 2, Rome, Italy, Jun. 17-20, 1997.
19. Chen, P.C., Sarhaddi, D., Liu, D.D., Ratwani, M. and Minahen, T., “Aeroelastic/Aeroservoelastic Tailoring for Hinge Moment Minimization of Missile Fins,” SBIR Phase I Final Report (N68936-97-C-0151), Dec. 1998.
g-METHOD for FLUTTER 20. Chen, P.C., “A Damping Perturbation Method for Flutter Solution: The g-Method,” paper
presented at the “International Forum on Aeroelasticity and Structural Dynamics,” Wil-liamsburg, VA, Jun. 22-25, 1999. Also, AIAA Journal, Vol. 38, No. 9, Sept. 2000.
Page 30 ZONA Technology, Inc.
AEROSERVOELASTICITY (ASE) 21. Karpel, M., “Design for Active Flutter Suppression and Gust Alleviation Using State-
Space Aeroelastic Modeling,” Journal of Aircraft, Vol. 19, No. 3, 1982, pp. 221-227. 22. Karpel, M., “Time-Domain Aeroservoelastic Modeling Using Weighted Unsteady Aero-
dynamic Forces,” Journal of Guidance, Control, and Dynamics, Vol. 13, No. 1, pp. 30-37, 1990.
23. Karpel, M., “Extension to the Minimum-State Aeroelastic Modeling Method,” AIAA Journal, Vol. 29, No. 11, 1991, pp. 2007-2009.
24. Karpel, M. and Hoadley, S.T., “Physically Weighted Approximations of Unsteady Aero-dynamic Forces Using the Minimum-State Method,” NASA TP-3025, 1991.
25. Karpel, M. and Strul, E., “Minimum-State Unsteady Aerodynamic Approximations with Flexible Constraints,” Journal of Aircraft, Vol. 33, No. 6, pp. 1190-1196.
26. Karpel, M., “Sensitivity Derivatives of Flutter Characteristics and Stability Margins for Aeroelastic Design,” Journal of Aircraft, Vol. 27, No. 4, 1990, pp. 368-375.
27. Karpel, M. and Wieseman, C.D., “Modal Coordinates for Aeroelastic Analysis with Large Local Structural Variations,” Journal of Aircraft, Vol. 31, No. 2, 1994, pp. 396-403.
ZSAP 29. Chen, P.C., and Liu, D.D., “Unsteady Sonic Aerodynamics Using Acceleration Potential
Approach,” 44th AIAA/ASME/ASCE/AHS Structures, Structural Dynamics, and Materi-als Conference, Norfolk, VA, 7-10 April 2003, AIAA paper number 2003-1404.
30. Chen, P.C., and Liu, D.D., “Unsteady Wing-Body Aerodynamics for Aeroelastic Applica-tions at Mach One,” AIAA Journal, Vol. 44, No. 8, pp. 1709, August 2006.
GUST LOADS 31. Karpel, M., Moulin, B., and Chen, P.C., “Dynamic Response of Aeroservoelastic Systems
to Gust Excitations,” International Forum on Aeroelasticity and Structural Dynamics 2003, Amsterdam, June 4-6, 2003
NONLINEAR FLUTTER 32. Chen, P.C., and Sulaeman, E., “Nonlinear Response of Aeroservoelastic Systems using
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