ZAEROde Zona Technology ZAERO es una arquitectura de software orientada a sistemas de vuelo para el calculo de condiciones aeroelasticas. El producto ha sido desarrollado por la empresa ZONA Technology, Inc.
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THE
ZAERO SOFTWARE SYSTEM / ARCHITECTURE
Main Features :
 High Fidelity Geometry module to model full aircraft with
stores/nacelles [1]
 Flight regimes that cover all Mach numbers including
transonic/hypersonic ranges [14]
 Unified Mach AIC matrices as archival data entities for
repetitive structural design/analysis [3]
 Matched/nonmatched point flutter solutions using K / PK / g
methods with true damping [5]
 Builtin Flutter Mode Tracking procedure with structural
parametric sensitivity analysis [13],[8]
 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]
 Nonlinear Flutter Analysis for open/closed loop system
using discrete timedomain state space approach (NLFLTR) [18].
 3D Spline module provides accurate FEM/Aero displacements
and forces transferal [2]
 Modal Data Importer to process all NASTRAN/FEM modal output
[4]
 Dynamic Memory and Database Management Systems establish
subprogram modularity [9]
 Open architecture allows user direct access to data
entities [10]
 Bulk Data Input minimizes user learning curve while
relieving user input burden [12]
 Provides graphic display capability of aerodynamic models,
C_{P}?s, flutter modes and flutter curves with
PATRAN/TECPLOT/EXCEL, etc. [16]
 Executive control allows massive flutter/ASE/TRIM/Dynamics
Loads inputs and solution outputs [17]
 NASLINK module to export ZAERO aerodynamic data to
MSC.NASTRAN [19]


The main body of the ZAERO system is comprised of eight
modules and the ZONA Database Management (ZDM) system. The modules are:

[1] HighFidelity Geometry (HFG) 
[4] Modal Data Importer 
[7] Gust 
[2] 3D Spline 
[5] Flutter 
[8] Sensitivity 
[3] Unified AIC (UAIC) 
[6] Aeroservoelasticity (ASE) 


For nearly one and a half
decades, ZONA Technology, Inc. (ZONA) has been directing its R&D
effort towards the development of viable unsteady aerodynamic methods
for aeroelastic applications. The first ZONA software product for
supersonic lifting surface unsteady aerodynamics is the ZONA51 code,
which is the outgrowth from the Harmonic Gradient Method (HGM) developed
by Chen and Liu in 1985 (Ref 1). In early 1991, ZONA reached an
agreement with the MacNealSchwendler Corporation (MSC) to provide a new
set of unsteady aerodynamic methods as the Aeroelastic Option II for
MSC/NASTRAN. Today, ZONA51 is the industrial standard method for
supersonic lifting surface unsteady aerodynamics in MSC/NASTRAN ? Aero
Option II with over 120 users worldwide.
 ZONA6 generates steady/unsteady subsonic aerodynamics for
wingbody/aircraft configurations with external stores/nacelles
including body wake effects.
 ZTAIC generates unsteady transonic (modal) AIC?s using
externallyprovided steady mean pressures.
 ZONA7 generates steady/unsteady supersonic aerodynamics for
wingbody/aircraft configurations with external stores/nacelles
(formerly ZONA51 for lifting surfaces).
 ZONA7U generates unified hypersonic and supersonic steady/unsteady
aerodynamics for wingbody/aircraft configurations with external
stores/nacelles.

Further development of
these ZONA codes in recent years led to the present developed product:
the ZAERO software system. A Unified Mach number AIC (UAIC) module is
created to include all ZONA unsteady aerodynamics codes listed above.
The functionality of the UAIC module is to provide the needed AIC matrix
according to the flight condition input for any given Mach numbers.
FUNCTIONALITY
 Generates steady/unsteady subsonic aerodynamics for
wingbody/aircraft configurations with external stores/nacelles
including the bodywake effect.
MAIN FEATURES
 Any combinations of planar/nonplanar lifting surfaces with arbitrary
bodies including fuselage+stores+tip missiles.
 Higherorder panel formulation for lifting surfaces than the Doublet
Lattice Method (DLM). First case below shows the ZONA6 robustness over
DLM.
 Highorder paneling allows highfidelity modeling of complex
aircraft with arbitrary stores/tip missile arrangement. Second case
below shows a solution improvement.
70 Degree Delta Wing (M=0.8, k=0.5, ho=0.35cr) Ref 10
 Robust ZONA6 solutions are in contrast to the breakdown of the DLM
solutions
 Highorder formulation of ZONA6 requires little care in paneling
NLR WingTiptankPylonStore, Ref 9
 No. of Wing Aero Boxes=90, Tiptank Aero Boxes=264, Store Aero
Boxes=216
 ZONA6 shows improvement over NLR?s predicted results
FUNCTIONALITY
 Generates unsteady transonic modal AIC using externallyprovided
steady mean pressure.
MAIN FEATURES
 While using steady pressure input (provided by measurement or
CFD):
 grid generation is not required
 the correct unsteady shock strength and position are ensured
 The modal AIC of ZTAIC as an archival data entity allows:
 repetitive aeroelastic analysis and structures design.
 the ease of application of the K / g methods for flutter
analysis.
 Readily integrated with ZONA6 as a unified subsonic/transonic AIC
method for complex aircraft configurations.
 Additional input to ZONA6 amounts to only the provided steady
pressure data.
AGARD Standard 445.6 Wing Refs 12,17
 Steady pressure input provided by CAPTSD
 Good agreement with test data
 At subsonic speed, ZTAIC results approach that of ZONA6, as
expected
 ZTAIC predicts the transonic flutter dip
Lessing Wing in FirstBending Oscillation (M=0.9, k=0.13, n=0.5 x
span) Ref 12
 Used Lessing?s test data as the steady pressure input
 Thus yields correct unsteady shock position and magnitude
FUNCTIONALITY
 Generates unsteady transonic AIC matrix that has the same form as
AIC of ZONA6/ZONA7.
MAIN FEATURES
 ZTRAN solves the timelinearized transonic small disturbance
equations using overset fieldpanel 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 timelinearized transonic small
disturbance equation are interpolated from the Computational Fluid
Dynamics (CFD) steady solutions.
 The overset fieldpanel scheme allows the modeling of complex
configurations without extensive field panel generation efforts.
Unsteady Pressure Validations
Flutter Validations
FUNCTIONALITY
 Generates steady/unsteady supersonic aerodynamics for
wingbody/aircraft configurations with external stores/nacelles
MAIN FEATURES
 Any combinations of planar/nonplanar lifting surfaces with arbitrary
bodies including fuselage+stores+tip missiles.
 Panel formulation for lifting surface is identical to that of ZONA51
? now the industrial standard method for supersonic flutter analysis
in MSC/NASTRAN.
 Highorder paneling allows highfidelity modeling of complex
aircraft with arbitrary stores/tip missile arrangement.
NACA WingBody (xo=0.35cr, Ref 4) Ref 15

ZONA7
WING + BODY
WING ONLY
BODY ONLY
TEST DATA
R = 1.18 x 106
R = 1.89 x 106

FUNCTIONALITY
 Generates unified hypersonic and supersonic steady/unsteady
aerodynamics for wingbody/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 Ref 13
 Thickness effect apparent at higher M
 Thus, it yields more conservative flutter boundaries
Rectangular Wing with Wedge Profile (M=4.0, sigma=15°, xo=0.25c) Ref
13
 ZONA7U solution compares well with Euler solution over a wide
frequency range
 Piston Theory and Linear Theory (ZONA7) yield inferior results by
comparison
FUNCTIONALITY
 Generates steady/unsteady aerodynamics at sonic speed (M = 1.0) for
wingbody/aircraft configurations with external stores/nacelles.
MAIN FEATURES
 Any combinations of planar/nonplanar lifting surfaces with arbitrary
bodies including 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.

 CanardWing configuration in Canard Pitch Motion about its
MidChord.
 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.


 AGARD standard 445.6 Weakened Wing (in Air) and Solid Wing
(in Freon 12)
 Comparison of Flutter Speed Index and Flutter Frequency
Ratio with TDT wind tunnel measurements

FUNCTIONALITY
 Generates a corrected AIC matrix to match the given set of
forces/moments or unsteady pressures.
MAIN FEATURES
 The AIC correction module computes the AIC weighting matrix using a
ZONA Transonic AIC Weighting (ZTAW) method that adopts a successive
kernel expansion procedure.
 The ZTAW method is an improved AIC correction method over the
previous correction methods such as the force/moment correction method
by Giesing et al and the downwash weighting matrix (DWM) method by
Pitt and Goodman. With inphase pressures obtained by windtunnel
measurement or CFD, ZTAW yields accurate outofphase and higher
frequency pressures resulting in wellcorrelated aeroelastic solutions
whereas the previous method yield erroneous outofphase 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 correction matrix method, and the unsteady force
correction matrix method.
Unsteady Pressure Validations
Flutter Validations
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 winglike and bodylike
definitions. Any modifications to the HFG module, such as input geometry
enhancements, will have minimal impact on other general modules.


Winglike components
include:
wings, tails, pylons, launchers store fins, etc. 

Bodylike components
include:
fuselage, underwing stores, missile bodies, etc. 
The 3D Spline module
establishes the displacement/force transferal between the structural
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 displacements and slopes in
three dimensions at all aerodynamic grids.
NACA L51F07 WingBody
Configuration with Three Structural Modes 
FEM
MODEL 
AERODYNAMIC
MODEL 


Rigid
Body Pitch Mode 


First
Wing Bending Mode 


First
Body Bending Mode 
NACA
L51F07 WingBody Configuration with Three Structural Modes. 
The ZONA Dynamic Memory
and Database Management System (ZDM) consists of the following five
parts:
 Matrix Entity Manager
The matrix entity manager is designed to store and retrieve very
large, often sparse, matrices. It minimizes disk storage
requirements while allowing algorithms to be developed that can
perform matrix operations of virtually unlimited size.
 Relational Entity Manager
Relational entities are essentially tables. Each relation has data
stored in rows (called entries) and columns (called attributes).
Each attribute is given a descriptive name, a data type, and
constraints on the values that the attributes may assume (i.e.
integer, real or character data). These definitions are referred to
as the schema of the relation.
 Unstructured Entity Manager
There are many times that a software module requires temporary, or
scratch, disk space while performing tasks. The data generated
within these tasks are generally "highlylocal" and, due
to the modular nature of the software, are not be passed through
arguments to other modules within the system. To effectively
accommodate the transfer of this type of data, ZDM supports an
unstructured database entity type composed of "records"
that may contain any arbitrary collection of data.
 Dynamic Memory Manager
The dynamic memory manager consists of a suite of utility routines
to allocate and release blocks of dynamic memory. The Dynamic Memory
Manager provides the capability of developing an engineering
software system which allows operations to be performed on data that
would normally exceed the size of available memory.
 Engineering Utility Modules
Engineering utility modules contain a pool of routines that perform
operations on matrix database entities. These operations include
matrix decomposition, eigenvalue solver, matrix multiplication,
matrix partitioning/merging, etc. These routines first check the
property of a given matrix and then select the appropriate numerical
technique to perform a particular matrix operation.
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

YRL

ZRL

RCH

LRCHD

ATTCHR



CONT

CONT

XTL

YTL

ZTL

TCH

LTCHD

ATTCHT




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




ZAERO allows for the
graphic interface with commercialized graphic packages. Graphical data
in output files containing the aerodynamic model, unsteady p ressures
(CP), interpolated structural modes and flutter modes can be displayed
via PATRAN or TECPLOT. Vg and Vf diagrams can be displayed via typical
XY plotting packages. An example of the F16 aerodynamic model with
external stores and the resulting Vg and Vf diagrams are shown below. 
Unsteady aerodynamic Model 
VG and VF Diagrams 
Animated Flutter Mode 
Verification of Spline 
Unsteady Pressure Display 
Transient Response 
The ZAERO flutter module
contains two flutter solution techniques: the Kmethod and the gmethod.
The gmethod is ZONA's newly developed flutter solution method (Ref 20)
that generalizes the Kmethod and the PK method for true damping
prediction. Ref 20 shows that the PK method is only valid at the
conditions of zero damping, zero frequency, or linear varying
generalized aerodynamic forces (Q) with respect to reduced
frequency. In fact, if Q is highly nonlinear, it is shown that
the PK method may produce unrealistic roots due to its inconsistent
formulation.
The flutter module has a builtin atmospheric table as an option to
perform matchedpoint flutter analysis. Sensitivity analysis with
respect to the structural parameters is also included in the gmethod.
Three Degrees of
Freedom Airfoil at M=0.0 (MSC/NASTRAN HA145 Test Case) 

 A nonzero frequency "dynamic divergence speed" is well
predicted by the gmethod, the PK method and the transient method
(a timedomain method).
 Both the gmethod and the transient method capture two aerodynamic
lag roots which are absent in the PK method solution.
 The frequency vs. velocity (Vf) diagrams of the gmethod and the
transient method are in good agreement. The frequency of the
freefree plunge mode computed by the PK method remains zero. This
results in poor correlation in the Vf diagram with the gmethod and
transient method.


ZAERO provides analytical
sensitivities of flutter parameters (frequencies and damping) with
respect to structural design variables. Using the same set of
generalized coordinates, such as the normal modes of a baseline
structure, massive sensitivities can be computed without returning to
the finiteelement model.
Symbolically, the analytical sensitivity can be expressed as : 

which
is the sensitivity of damping of the jth mode ( y_{j }) with
respect to the ith design variable ( v_{i }). 

which
is the sensitivity of flutter reduced frequency of the jth mode ( k_{j
}) with respect to the ith design variable ( v_{i }). 
where A and B can be
derived using the orthogonality of the right and left eigenvectors of
the flutter equation. Computing A and B requires additional structural
information from the finite element code; the derivatives of the
stiffness matrices
and the mass matrices
with respect to the design variables. These matrices are imported from
the finite element code through a bulk data input card DMIG. 
Performs the static
aeroelastic/trim analysis for solving the trim system and computing the
flight loads.
Unsteady aerodynamic Model 
VG and VF Diagrams 
F18 at 4G PullUp 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 reducedorder trim
system that can be solved with much less computer time than the
socalled ?direct method?.
 It is capable of dealing with the determined trim system as well as
the overdetermined trim system (more unknowns than the trim
equations). The solutions of the overdetermined trim system are
obtained by using an optimization technique which minimizes a
userdefined objective function while satisfying a set of constraint
functions.
 For a symmetric configuration (symmetric about the xz 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.
The ASE
software developed by Prof. Moti Karpel of Technion ? Israel
Institute of Technology is integrated in the ZAERO software
system as shown below. 

Main Features
of the ASE Module :
 Rationalfunction approximation of the unsteady
aerodynamic coefficient matrices
 Statespace MIMO formulation
 Modular linear control modeling of mostgeneral
architecture
 Open and closedloop flutter analysis
 Openclosed gain and phase margins
 Input and output singular values
 Augmentation of continuousgust dynamics
 Structural gust response in statistical terms
 Fixedmodes parametric studies
 Sensitivity of flutter and control margins with respect to
structural and control variables
The unsteady
aerodynamic force coefficient matrices are approximated by a
rational matrix function in the Laplace domain. The
approximation formula is either the classic Roger?s formula
where p is the nondimensional Laplace variable p = sb
/ V, or the more general minimumstate 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 leastsquare solution is used for Roger?s
approximation, and a nonlinear leastsquare is used for the
minimumstate approximation.
A physicalweighting algorithm may be used to weight the data
terms according to aeroelastic measures of importance.



The plant and
control models are interconnected by the following scheme:


Stability
analyses of open and closedloop systems are based on system
eigenvalues. Sensitivity computations are based on analytical
expressions.
 Gust filters are designed to represent Dryden?s and Von
Karman?s power spectral density functions of atmospheric
gust velocities.
 Augmentation of a gust filter yields dynamic equations
with random whitenoise excitation process,
 Meansquare response parameters are obtained by solving
the Lyapunov equation
 and then calculating the output covariance matrix
 Similar expressions are used for the sensitivity of the
response values to structural and control variables.
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 PTransform
correlate well while the quasistatic method does not due to
lowfrequency approximation)
MAIN FEATURES
 It is formulated in the state space form for either the open
loop or closed loop system. The rigid body degrees of freedom
are transformed into the airframe states so that the
submatrices 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 programcomputed
submatrices 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) 



Advanced Fighter Test Case (ZAERO versus Flight Test)
MAIN FEATURES
 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 statespace form for either an
openloop or closedloop system.
 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 transient discrete gust loads module performs a transient
response analysis of an aircraft structure when the aircraft
encounters a discrete gust.
2D Thin Airfoil Subjected to a SharpEdged Gust 
Comparison Between Sear?s Function and the
Gust Forces Computed by ZONA6 
Comparison Between Wagner?s Function and ZAERO
StateSpace Equations 
Comparisons Between ZAERO Results and Analytical
Solution for a 2D Airfoil Encountering SharpEdged Gust 
Validation of the Discrete Gust Module with 2D 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 oneminuscosine, sine, sharpedged 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 frequencydomain approach, the
statespace approach, and the hybrid approach where the
discrete gust loads are obtained by inverse Fouier transform
and the system matrix by statespace formulation.
 Its state space equations provide accurate displacement time
history thereby circumventing 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/closedloop aeroelastic systems that
include (1) nonlinear structures (2) nonlinear control system
(3) largeamplitude unsteady aerodynamics (externally imported
from other CFD code).
3 d.o.f. Airfoil with
FreePlay




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 timedomain state space equations at each distinct
value of the nonlinear parameters are precomputed. During the
timeintegration computation, updated statespace equations
are obtained by interpolation.
 It outputs the NASTRAN FORCE and MOMENT bulk
data cards at a given time step for subsequent stress
analysis.
Exports the ZAERO Aerodynamic data to MSC.Nastran.
Unsteady aerodynamic Model 
Excellent Agreement with Analytical and Experimental Results
MAIN FEATURES
 It automatically creates a directory to store the AIC
matrices, spline matrix, control surface modes, aerodynamics
geometry, and load modes for component loads.
 The user can user MSC.Nastran solution 144 for static
aeroelastic analysis, solution 145 for aerodynamic flutter,
solution 146 for aeroelastic response or solution 200 for
design optimization but using ZAERO unsteady aerodynamics.
F16 MA41 Limit Cycle Oscillation (LCO)
 F16 MA41 configuration experiences LCO from Mach 0.6 to
1.0 while the MA43 configuration is free from LCO.
 ZAERO (ZONA6 linear method) predicts a hump mode damping
curve that falls within 12% damping (M=0.6 to 1.0) for the
MA41 configuration while the MA43 case falls below 1%.
 The onset of LCO as predicted by ZAERO correlates very
well with flight data.
Case MA41 LCO Occurred
at M = 0.6 ? 1.0 in Flight
LAU129 Launcher 
Case MA43
No LCO Occurred in Flight
16S210 Launcher 
F16 Case MA41 and Case
MA43 Configurations


ZAERO Flutter Solutions of F16 Case MA41 at M =
0.6, 0.8, 0.9 and 1.2 
ZAERO Flutter Solutions of F16 Case MA43 at M =
0.6, 0.8, 0.9 and 1.2 
F18 Limit Cycle Oscillation (LCO)
 ZAERO transonic unsteady aerodynamic method (ZTAIC) is
used to perform a flutter analysis.
 Steady pressure input to ZTAIC is obtained by the CFL3D
NavierStokes solver.
 Unstable damping of Case 1 approximately ranges from M =
0.85 to 1.0 with a frequency of 5.6 Hz which correlates very
well with the flight test data.
 Similarly, good correlation is found for Case 2 (LCO
region at M>0.9 with frequency 8.8 Hz)
Two F18/Store LCO Configurations 
ZAERO Flutter Solution of F/A18 LCO Case 1 (at
Altitude = 0 kft) 
ZAERO Flutter Solution of F/A18 LCO Case 2 (at
Altitude = 0 kft) 





