zaero brochure

About ZAERO TM

Engineers’ Toolkit for Aeroelastic Solutions

ZONA Technology, Inc.

Unsteady Aerodynamics Flutter Nonlinear Flutter Trim Ejection Loads Maneuver Loads Gust Loads

To Order Call: 480•945•9988


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


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)

ZAERO Engineering Module Diagram


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


• 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

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)



• 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)


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)


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)


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)


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

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)


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)


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)


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)




Functionality • Generates steady/unsteady supersonic aerodynamics for wing-body/aircraft configu-

rations with external stores/nacelles


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

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



Functionality • Generates steady/unsteady aerodynamics at sonic speed (M = 1.0) for wing-body/

aircraft configurations with external stores/nacelles.


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

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°



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.



ZAERO Morhing Aircraft model with 2062 boxes

ZAERO C-130 model with 3978 boxes

ZAERO Predator model with 1410 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


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


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


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


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.


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


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)


• 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

⎡ ⎤ ⎡ ⎤= +⎣ ⎦ ⎣ ⎦⎡ ⎤= ⎣ ⎦


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.


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:


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



0

0.4

0.8

1.2

1.6

0 4 8 12 16 20 24 28

tV/L

Kz







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.


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)


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)


U = 11.711 m/s

-5.0

-3.0

-1.0

1.0

3.0

5.0

7.0

β (d

eg)


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.


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.


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.


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

Discrete State-Space Approach,” AIAA Journal, Vol. 41, No. 9, September 2003.

ZTRAN 33. Chen, P.C., Gao, X.W., and Tang, L., “Overset Field-Panel Method for Unsteady Tran-

sonic Aerodynamic Influence Coefficient Matrix Generation,” AIAA Journal, Vol. 42, No. 9, September 2004.

34. Chen, P.C., “Flutter Studies of the Overset Field-Panel Method for Transonic Aeroelastic Applications,” International Forum on Aeroelasticty and Structural Dynamics, Munich Arebella Sheraton Grand Hotel, June 28—Jul 1, 2005

Phone: 480-945-9988 Fax: 480-945-6588 E-mail: [email protected] URL: www.zonatech.com

9489 E. Ironwood Square Drive Scottsdale, AZ 85258-4578

ZONA Technology, Inc. SM

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