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Kolonay 1
CRD
Computational Aeroelasticity
The Cultural and Convention CenterMETU
Inonu bulvariAnkara, Turkey
Sponsored by:RTA-NATO
The Applied Vehicle Technology Panel
presented byR.M. Kolonay Ph.D.
General Electric Corporate Research & Development CenterAnkara, Turkey Oct.. 1-5, 2001
Kolonay 2
CRD
• Introduction- Fluid-Structure Interactions
•Aeroelasticity- Aeroelastic analysis/design in an MDA/MDO Environment
• Static Aeroelasticity
• Dynamic Aeroelasticity
• Commercial Programs with Aeroelastic Analysis/DesignCapabilities
Presentation Outline
Ko 3
D
nAn ered independently tored
om
ustors, Turbines
Transmission Lines
Introduction
Fluid Structure Interactioy system where the fluid and structure cannot be considict the response of the fluid, the structure, or both.
e Fields of Application
• Aerospace Vehicles- Aircraft, Spacecraft, Rotorcraft, Compressors, Comb
• Utilities- Hydroturbines, Steamturbines, Gasturbines, Piping,
• Civil Structures- Bridges, Buildings
• Transportations•Trains, Automobiles, Ships
lonay
CR
-p
S
K 4
RD
ie
Introduction
lds of Application (Continued)
• Medical- Blood flow in veins, arteries, and heart
• Marine- Submarines, Off-shore Platforms, Docks, Piers
• Computer Technology- High velocity flexible storage devices
olonay
C
F
Kolonay 5
CRD
Failure to recognize F-S InteractionTacoma Narrows Bridge #1 (Galloping Girtie)
- Chief Designer: Leon Moisseiff- Length: 5,939 ft.- 42 MPH winds induced vortical separated flow that lead to torsional flutter- Piers used in second bridge- 1992: National Historic Site (natural reef)- Photos taken by Leonard Coatsworth
Introduction
Kolonay 6
CRD
Aeroelasticity (sub-set of FS Int.)Aeroelasticity (British Engineers Cox and Pugsley credited with term) -Substantial inter-action among the aerodynamic, inertial, and structural forces that act upon and within theflight vehicle.
Introduction
Aerodynamic Forces
Inertial Forces
DynamicStability
Elasticity
DynamicAeroelasticity
Elastic Forces MechanicalVibration
Static Aero-
Kolonay 7
CRD
Early Aeroelastic Problems
• S. P. Langley’s Aerodome (monoplane)- 1/2 scale flew- October, 1903: Full scale failed, possibly due to wing torsional divergence- 1914 Curtis made some modification and flew successfully.
Introduction
Kolonay 8
CRD
After Langley’s failure the U.S. War Department reported -
“We are still far from the ultimate goal, andit would seem as ifyearsof constant work ...would still be necessary before we can hopeto produce an apparatus of practical utility
on these lines.”
9 Days Later ...
Introduction
Kolonay 9
CRD
December 17, 1903
Introduction
Kolonay 10
CRD
Early Aeroelastic Problems
• Hadley Page 0/400 bomber- Bi-plane tail flutter problems (fuselage torsion coupled with elevators)- DH-9 had similar problems- Solution was to add torsional stiffness between right and left elevators.
Introduction
Kolonay 11
CRD
Early Aeroelastic Problems
• Fokker D-8 (credited with last official kill of WW I)- D8 had great performance but suffered from wing failures in steep dives- Early monoplanes had insufficient torsional stiffness resulting in:
• wing flutter, wing-aileron flutter• loss of aileron effectiveness
- Solution: Increase torsional stiffness, mass balancing
Introduction
Kolonay 12
CRD
Computational Aeroelasticity
Early Theoretical Developments[1],[3].
• Wing divergence - Reissner (1926)
• Wing flutter - Frazer and Duncan (1929)
• Aileron reversal - Cox (1932)
• Unsteady aerodynamics and flutter - Glauert, Frazer, Duncan,Kussner, Theodorsen (1935)
• 3 DOF wing aileron flutter - Smlig and Wasserman (1942)
Introduction
By Early 1930’s Analytical methods existed to aid designers toconsider both static and dynamic aeroelastic phenomena
Kolonay 13
CRD
Computational Aeroelasticity
Designs from the 40’s-70’s “designed out” Aeroelastic Effects
• Accomplished by increasing structural stiffness or mass bal-ancing (always at weight cost)
70’s & 80’s brought technology developments in three key areas
• Structures, Controls, and Computational Methods- Advanced composite materials enabled aeroelastic tailoring- Fly By Wire and Digital Control Systems enabled statically unstable aircraft- FEM, CFD, Optimization, Computational Power enabled advanced designs.
Introduction
Kolonay 14
CRD
Aeroelastic Successes• DARPA sponsored X-29 (First flight 1984)
- Aeroelastic tailored (graphite epoxy) forward swept wing- Fly By Wire triple redundant digital and analog control system- Germany proposed FSW designs (He 162) in WWII
Introduction
Kolonay 15
CRD
Aeroelastic Successes• Active Aeroelastic Wing USAF/NASA (AAW)
- Use control surfaces (leading and trailing edge) as tabs to twist the wing formaneuvers
- Use TE surfaces beyond reversal- Produces lighter more maneuverable aircraft
Introduction
Kolonay 16
CRD Introduction
Distribution
Sal
esM
arke
ting
Aerodynamics
Cost
Heat T
rans
fer
Acoustics
StructuresD
ynamics
Ele
cto-
Mag
netic
s
Controls
Manufacture
Maintenance
Reliability
ProducibilityRobustness
MDA/MDO
Product Structural Design in an MDA/MDO Environment
Kolonay 17
CRD
Goal of Computational Aeroelasticity
To accuratelypredict static and dynamicresponse/stability so that it can be accountedfor (avoided or taken advantage of) early inthe design process.
Computational Aeroelasticity
Kolonay 18
CRD Computational Aeroelasticity
Aeroelastic Equations of Motion
Mu̇̇ Bu̇ Ku+ + F u u̇ u̇̇ t, , ,( )=
K Structural Stiffness–B Structural Damping–M Structural Mass–F u u̇ u̇̇ t,, ,( ) External Aerodynamic Loads–
Kolonay 19
CRD Computational Aeroelasticity
Discretization of EOM
• Structures - Typically, although not necessarily, rep-resented by Finite Elements in either physical or generalizedcoordinates. Derived in aLagrangian frame of reference.
• External Loads - Aerodynamic loads. Representa-tions range from Prandtl’s lifting line theory to full Navier-Stokes with turbulence modeling. Represented in physical andgeneralized coordinates in a (usually)Eulerianframe of refer-ence.
K B M, ,
F u u̇ t, ,( )
Kolonay 20
CRD Computational Aeroelasticity
Fluid-Structural Coupling Requirements
• Must ensure spatial compatibility - proper energy exchangeacross the fluid-structural boundary
• Time marching solutions require proper time synchronizationbetween fluid and structural systems
• For moving CFD meshes GCL[6] must be satisfied
If coupling requirements for time-accurate aeroelastic simula-tion are not met then dynamical equivalence cannot beachieved. That is, regardless of the fineness of the CFD/CSMmeshes and the reduction of time step to 0, the scheme may con-verge to the “wrong” equilibrium/instability point.[5]
Kolonay 21
CRD
General Modeling Comments
• Use appropriate theory to capture desired phenomena
- Fluids - Navier-Stokes vs. Prandtls’ lifting line theory- Structures - Nonlinear FEM vs. Euler beam theory
• Model the fluid and structure with a consistent fidelity
- For a wing don’t model the fluid with NS and the structure with beam theory
Computational Aeroelasticity
Kolonay 22
CRD
Aeroelastic Phenomena
Computational Aeroelasticity
Static Aeroelastic Phenomena
• Lift Effectiveness
• Divergence
• Control Surface Effective-ness/Reversal
• Aileron Effectiveness/Reversal
Dynamic Aeroelastic Phenomena
• Flutter
• Gust Response
• Buffet
• Limit Cycle Oscillations (LCO)
• Panel Flutter
• Transient Maneuvers
• Control Surface Buzz
Kolonay 23
CRD Static Aeroelasticity
Static Aeroelastic Phenomena
• Lift Effectiveness
• Divergence
• Control Surface Effectiveness/Reversal
• Aileron Effectiveness/Reversal
Kolonay 24
CRD Static Aeroelasticity
Static Aeroelastic Effects
• For trimmedflight aeroelastic effects change only load distri-bution.
- Lift- Drag- Pitching Moment- Rolling Moment
• Forconstrainedflight (wind tunnel models) aeroelastic effectschange both magnitude and distribution of loads.
Kolonay 25
CRD Static Aeroelasticity
Shear Center/Center of TwistL
MAC
Aerodynamic Centere
Shear Center/Center of Twist - Applied Shear force results in no moment or twist- Applied moment produces no shear force or bending
Aerodynamic Center - Pitching moment independent of angle of attack
Center of Pressure - Total Aerodynamic Moment equal zero (AC=SC for symm. airfoil)
- 0.25c for subsonic, 0.5c for supersonic
e - Eccentricity
Useful 2-D Section Definitions
Kolonay 26
CRD Static Aeroelasticity
Effect of Swept Wing Bending on StreamwiseAerodynamic Incidence
AA
U
Rigid Wing
Flexible Wing
A-A
A A
A-A
Flexible Wing
Rigid Wing
“wash out” “wash in”
ASW FSW
Kolonay 27
CRD
EOM
(1)
- rigid body accelerations only, used for inertial relief and trim - Steady aerodynamic forces can be represented as
or
Now (1) can be written as
(2)
K[ ] u{ } M[ ] u̇̇{ }+ F u( ){ }=u̇̇{ }
F u( )
F u( ) q G[ ]T AIC[ ] GS[ ] u{ } q G[ ]T AIRFRC[ ] δ{ }+=
F u( ) q AICS[ ] u{ } q Pa[ ] δ{+=
K qAICS–[ ] u{ } M[ ] u̇̇{ }+ q Pa[ ] δ{ }=
Linear Static Aeroelasticity
For Linear Aerodynamics [AIC] & [AIRFRC] depend only on Mach Number (M)
Kolonay 28
CRD
Steady Aerodynamic Loads
•
• - Spline matrix which transforms forces from Aerodynamic DOF (ADOF) to
Structural DOF (SDOF).
• - Spline matrix which transforms SDOF (displacements) to ADOF (panel slopes)
•
• - Aerodynamic Influence Coefficient Matrix. Relates forces on ADOF (panels)due to unit perturbations of the ADOF (slopes)
• - Unit Rigid body aerodynamic load vectors. One vector for each
• - Vector of aerodynamic configuration parameters (angle of attack, elevator angle,aileron deflection, roll rate, pitch rate etc.)
F u( ) q G[ ]T AIC[ ] Gs[ ] u{ } q G[ ]T AIRFRC[ ] δ{ }+=
q Free stream dynamic pressure=
G[ ]T
Fs{ } G[ ]TFa{ }=
Gs[ ]
αa{ } Gs[ ] u{ }=
AIC[ ]
AIRFRC[ ] δi
δ{ }
Linear Static Aeroelasticity
Kolonay 29
CRD
-2 0 2 4 6 8
-0.02
0
0.02
0.04
0.06
0.08
0.1
0.12
Aeroelastic Effects on Swept Wing Forces andMoments
Linear Static Aeroelasticity
2 0 2 4 6 8
-0.02
0
0.02
0.04
0.06
0.08
0.1
0.12
Rigid ASWFlex ASW
Rigid FSW
Flex FSW
Induced DragCDαAngle of Attackα Pitching MomentCMα
Coe
ffici
ent o
f Lift
CL
α
-0.00100.0010.002
-0.03
-0.01
0.01
0.03
0.05
0.07
0.09
0.11
Kolonay 30
CRD
Divergence of a Constrained Vehicle• When the aerodynamic stiffness becomes greater than
the structural stiffness , the structure fails or diverges.
• The divergence dynamic pressure for a restrained vehicle canbe found by solving the eigenvalue problem (static stability)
(3)
• Lowest eigenvalue represents the divergence dynamic
pressure
• The eigenvector represents the divergent shape
• Divergence is independent of initial angle of attack
qAICSK
K qAICS–[ ] u{ } 0{ }=
qD
uD{ }
Linear Static Aeroelasticity
Kolonay 31
CRD Linear Static Aeroelasticity
0 5 100
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0.2 0.4 0.6 0.80
2
4
6
8
10
12
14
16
18
20
Dynamic Pressure (psi) Dynamic Pressure (psi)
CL
α
qD
ASW FSW
Affect of Sweep on Lift Effectiveness(M=0.7)
Eq.
(20
)
Kolo 32
CRD
onsitin
(4)
ing can be cast in thea-t as
(5)
Static Aeroelastic Trim Equatig equation (2) in thef-set (Reference Appendix A) yields
the procedure in Appendix A for Guyan reduction equation (4)
K ff qAICS–[ ]u f M ff u̇̇ f+ Pfaδ=
or
K ffa
u f M ff u̇̇ f+ Pfaδ=
Kaaa
ua Maau̇̇a+ Paaδ=
with
Kaaa
Kaaa
Kaa
Goa
–=
Paa
Paa
Kaoa
Kooa 1–
Poa
–=
Maa Maa MaoGo Goa T
MoaT
Goa T
MooGoa
+ + +=
Linear Static Aeroelasticity
nay
Wr
Usse
Ko 33
D
qu
(6)
s w rigid body transfor-
ati nt of ther-set (i.e. sup-ort
(7)
sin an be cast in the fol-
win
la
la
δ
y
ation (5) can now be partitioned into ther-set and thel-setto
ith the inertial relief formulation where is the
on matrix. To produce stability derivatives that are independepoint) an orthogonality condition is imposed in the form
g the orthogonality condition and equation (6) c
g form
K lla
K lra
Krla
Krra
ul
ur
M ll M lr
M rl M rr
u̇̇l
u̇̇r
+P
P
=
u̇̇l Du̇̇r= D
DT
IM ll M lr
M rl M rr
ul
ur
0=
u̇̇l Du̇̇r=
Linear Static Aeroelasticit
lonay
CR
E
A
mp
U
lo
Ko 34
D
(8)
qu adding it to the secondw. to yield the followingst
(9)
Pla
Pra
0
δ
DT
DT
Pla
0
Pla
Pra
+
δ
ty
ation (8) can be solved by multiplying the first row by and The new second row is interchanged with the third equationem of equations.
K lla
K lra
M ll D M lr+
Krla
Krra
M rl D M rr+
DT
M ll M rl+ DT
M lr M rr+ 0
ul
ur
u̇̇r
=
DT
K lla
K lra
M ll D M lr+
M ll M rl+ DT
M lr M rr+ 0
K lla
Krla
+ DT
K lra
Krra
+ mr
ul
ur
u̇̇r
DT
=
Linear Static Aeroelastici
lonay
CR
Erosy
Ko 35
D
he rigid body mass
atr
(10)
olv second and third rows
e o
(11)
ith
Linear Static Aeroelasticity
re is defined as the
ix. Using a simplifying notation equation (9) becomes
ing the first row of equation (10) for and substituting in the
btain the trim equations in the form
mr DT
Mll D DT
Mlr Mrr+ +=
R11 R12 R13
R21 R22 R23
R31 R32 R33
ul
ur
u̇̇r
Pla
0
DT
Pla
Pra
+
=
ul
K11 K12
K21 K22
u1
u2
P1
P2
δ{ }=
lonay
CR
W
m
S
w
w
Ko 36
D
(12)
olv ts and accelerations
sp
Linear Static Aeroelasticity
ing equation (11) for and the rigid body displacemen
ectively yields
K11 R22 R21R111– R12–=
K12 R23 R21R111– R13–=
K21 R32 R31R111– R12–=
K22 R33 R31R111– R13–=
P1 R21R111– Pl
a–=
P2 DTPla Pr
a R31R111– Pl
a–+=
u1 ur=
u2 u̇̇r=
u1 u2
lonay
CR
S
re
Ko 37
D
(13)
(14)
qu lysis. There is one equa-
n the vector of structural
cce rameters. Partitioningqua r set (subscriptsk,s)lu
]δ
Linear Static Aeroelasticity
timization problem.
ation (14) is the basic equation for static aeroelastic trim ana
for each rigid body degree of freedom (6 DOF trim). is
lerations at the support point and is a vector of trim pation (14) into free or unknown (subscriptsf,u) values and known o
es and gathering all unknown values to the left yields
u1 K111– P1δ K12u2–[ ]=
K22 K21K111– K12–[ ]u2 P2 K21K11
1– P1–[=
or
LHSA[ ] u2{ } RHSA[ ] δ{ }=
or
L[ ] u2{ } R[ ] δ{ }=
u2{ }
δ{ }
Note: System can be over-specified producing trim op
lonayCR
E
tio
aeva
Ko 38
D
(15)
ote
(16)
e
k
ces
rfaces
faces
Linear Static Aeroelasticity
ntial values for are given in equation (16)
L ff Rfu–
Lkf Rku–
u2 f
δu
L– fk Rfs–
L– kk Rks–
u2k
δs
=
u2k andδ
u2
NX - longitudinal acceleration
NY - lateral acceleration
NZ - vertical acceleration
PACCEL - roll acceleration
QACCEL - pitch acceleration
RACCEL - yaw acceleration
∈ δ
BASE - reference stat
ALPHA - angle of attac
BETA - yaw angle
PRATE - roll rate
QRATE - pitch rate
RRATE - yaw rate
δsym{ }- symmetric surfa
δanti{ }- antisymmetric su
δasym{ }- asymmetric sur
∈
lonay
CR
P
Ko 39
D
elerations and
(17)
n
(18)
Linear Static Aeroelasticity
Rigid Trim Equationsrom equation (9) considering only rigid body accads yields
d the rigid trim equations as
LHSArigid R33 mr= =
RHSArigid P2 DT
Pla
Pra
+= =
LHSArigid[ ] u̇̇r{ } RHSArigid[ ] δ{ }=
lonay
CR
Flo
a
Ko 40
D
si for andm ue to unit param-
te
(17)
(18)
δ}
P1]
Linear Static Aeroelasticity
Stability Derivativesng equation (14) and using an identity vector ploying the rigid body mass matrix forces d
r values can be determined as
{mr
F mr K22 K21K111–K12–[ ]
1–P2 K21K11
1––[
=
F
Fx
Fy
Fz
Mx
My
Mz
∈
Thrust/Drag
Side Force
Lift
Roll Moment
Pitch Moment
Yaw Moment
=
lonay
CR
Ue
e
Ko 41
D
as derivatives are
(19)
ives (free-free)
Linear Static Aeroelasticity
Stability Derivativesed on equation (18) non-dimensional stability
• Note: These are “unrestrained” stability derivat
Surface Parameters
CD
Fx
qS------=
CS
Fy
qS------=
CL
Fz
qS------=
Cl
Mx
qSb---------=
Cm
My
qSc---------=
Cy
Mz
qSb---------=
Rate Parameters
CD
Fx
qSc---------=
CS
Fy
qSb---------=
CL
Fz
qSc---------=
Cl
Mx
qSb2
------------=
Cm
My
qSc2
------------=
Cy
Mz
qSb2
------------=
lonay
CR
B
Ko 42
D
s forro
(20)
ie
α
0
0
0
Linear Static Aeroelasticity
Example Stability Derivativem equations (14) and (17)
lding etc.
Fx
Fy
Fz
Mx
My
Mz
α
mr[ ] LHSA[ ] 1–RHSA[ ]
δ0 0=
δα 1.0=
δβ 0=
δPRATE 0=
δQRATE =
δRRATE =
δ{ }surface =
=
CDαCSα
CLαClα
CMα, , , ,
lonay
CR
F
Y
Ko 43
D
satives
)
ed or Fixed
Linear Static Aeroelasticity
M rivatives a given
Stability Derivative Type• There are four varieties offlexible stability deriv
- Unrestrained (orthogonality and inertia relief included
- Restrained (orthogonality, no inertial relief)
- Supported (no orthogonality, but inertial relief)
- Fixed (no orthogonality, no inertial relief)
• For wind tunnel comparison use eitherRestrain
ake sure you know which type of stability de program produces
lonay
CR
Kolonay 44
CRD
Lift Trim Analysis• For straight and level flight i.e. equation (14)
produces a single equation with one free parameter (say )
or in terms of stability derivatives
u2{ } NZ=
α
LHSA NZ× RHSA α×=
α LHSA NZ×( )RHSA
----------------------------------=
mr NZ× αqSCLα=
αmr NZ×( )
qSCLα
--------------------------=
Linear Static Aeroelasticity
Kolonay 45
CRD Linear Static Aeroelasticity
Aeroelastic Trim ( ) Eq. (14)α 2.61°= Rigid Trim ( ) Eq. (18)α 1.29°=
Aeroelastic and Rigid Trimmed Pressures
( )M 0.7 q, 5.04 psi, nz = 1g= =
Kolonay 46
CRD Linear Static Aeroelasticity
0 0.25 0.5 0.75 10
0.5
1
1.5
2
2.5Rigid Trim 0% chordRigid Trim 50% chordAeroelastic Trim 0% chord 0% spanAeroelastic Trim 50% chord
Non-Dimensional Semi-Span
Pre
ssur
e (p
si)
Rigid and Aeroelastic Trim Pressures vs. Span( )M 0.7 q, 5.04 psi, nz = 1g= =
Kolonay 47
CRD
0 0.25 0.5 0.75 1
-4
-3
-2
-1
0
1
2
Linear Static Aeroelasticity
% Semi-Span
Rel
ativ
e Tw
ist A
ngle
(de
g.)
Spanwise Twist Due to Swept Wing Deformations
Flex Trim
Rigid Trim
Rigid
( )M 0.7 q, 5.04 psi, nz = 1g= =
Kolonay 48
CRD Linear Static Aeroelasticity
Aeroelastic Trimmed Displacements Rigid Trimmed Displacements
Swept Wing Aeroelastic Effects on Trimmed Displacements
max z-disp. = 5.4 in. max z-disp. = 11.4 in.
Kolonay 49
CRD
Control Surface Effects
Static Aeroelasticity
β0
β0
Incremental MomentIncremental Lift
Kolonay 50
CRD
Roll Trim Analysis (wing with aileron)Steady state roll (PACCEL = 0) for given (aileron deflection)
or in stability derivative form
β
LHSA44 PACCEL× RHSA43 β× RHSA44 PRATE×+=
PRATERHSA43 β×
RHSA44-------------------------------=
qSb Clββ Cl pb
2V-------
PRATE+ I rollPACCEL=
for steady roll and a givenβ
PRATE
Clββ
Cl pb2V-------
-------------=
Linear Static Aeroelasticity
Kolonay 51
CRD Linear Static Aeroelasticity
0 0.5 1 1.5
-50
-30
-10
10
30
50
70
Rigid TECS ASWFlex TECS ASW
Rigid TECS FSWFlex TECS FSW
Rigid LECS ASW
Flex LECS ASWRigid LECS FSW
Flex LECS FSW
Dynamic Pressure (psi)
Rol
l Rat
e (d
eg/s
ec)
Roll Rate vs. Dynamic Pressure forβ 1.0°=
qR FSW_TE
qR ASW_TE
Kolonay 52
CRD
Aileron Effectiveness
Static Aeroelasticity
0 1000 2000 3000 4000 5000
0 0.5 1 1.5
-0.1
-0.05
0
0.05
0.1
0.15
Dynamic Pressure (psi)
Velocity (in/sec)
Clβ( )
f
Cl pb2V-------
f-------------------------–
vs.Vvs.q
ReversalV
Reversal q
Kolonay 53
CRD
Aeroelastic Effects on Roll Rate Pressures
0.0520.0460.0390.0330.0260.0190.0130.0060.000
-0.007-0.014-0.020-0.027-0.033-0.040
0.0320.0280.0240.0200.0150.0110.0070.003
-0.001-0.005-0.010-0.014-0.018-0.022-0.026
0.0120.0100.0090.0070.0060.0040.0020.001
-0.001-0.002-0.004-0.006-0.007-0.009-0.010
Linear Static Aeroelasticity
pp p
qrigid = 27 (deg/sec) qrigid = 46 (deg/sec) qrigid = 59 (deg/sec)
qrigid = 0 (deg/sec) qrigid = -28 (deg/sec)qrigid = 16 (deg/sec)
M=0.7
q 0.78 (psi)= q 1.5 (psi)=q 0.28 (psi)=
Kolonay 54
CRD
Rolling Wing Deformations
Linear Static Aeroelasticity
M 0.7 q, 1.5 psi= =
Ko 55
D
. B ations, Addison-Wes-y P
. W ty”, Purdue Universitycho .
. S ional Flutter Theory toircr
. N cements, Vol III-ST
. B ts for Time-Accurateero , Aeroelasticity, Flow-du
. Fa ARD-R807, October995
. M P-221(01), April,971
. I.E aft Flutter,” Journal ofircr
References
isplinghoff, Ashley and Halfman “Aeroelasticity”, Dover Publicublishing Company, Inc., 1995.
eisshaar, “Fundamentals of Static and Dynamic Aeroelasticiol of Aeronautics and Astronautics, West Lafayette, IN 1992
milg, B. and Wasserman, L. S., “Application of Three Dimensaft Structures”, USAAF TR 4798, 1942.
eill, D.J., Herendeen, D.L., Venkayya, V.B., “ASTROS EnhanROS Theoretical Manual”, WL-TR-95-3006.
endiksen, Oddvar O., “Fluid-Structure Coupling Requiremenelastic Simulations”, AD-Vol.53-3, Fluid-Structure Interactionced Vibration and Noise, Volume III ASME, 1997.
rhat, C., “Special course on Parallel Computing in CFD”, AG.
acNeal, R. H., “The NASTRAN Theoretical Manual,” NASA-S.
. Garrick and W.H. Reed, III “Historical Development of Aircraft, Vol. 18, No. 11, November 1981.
lonay
CR
1le
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