Computational Aeroelasticity - METUyyaman/avt086/Kolonay/Ray_Kolonay_1.pdf · Computational Aeroelasticity The Cultural and Convention Center METU Inonu bulvari Ankara, Turkey Sponsored

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


• 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


• 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


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


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.


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


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


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


Kolonay 22

CRD

Aeroelastic Phenomena


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


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


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


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{ }


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


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{ }


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

+ + +=


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


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


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


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
lonay
CR

E

tio

aeva

Ko 38

D

(15)

ote

(16)

e

k

ces

rfaces

faces


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)


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]


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)


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


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


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

--------------------------=


Kolonay 45


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


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


% 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


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-------

-------------=


Kolonay 51


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


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


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

2S

3A

4A

5AIn

61

71

8A

Ko 56

D

. G for Flutter and Strengthna ry and Application,”,FF

0. H Flutter Equation byete er 1971, pp. 885-889.

1. N MacNeal-Schwendlerorp 1999.

lonay

CR

9AA

1D

1C

rumman Aerospace Corporation, “An Automated Procedure lysis and Optimization of Aerospace Vehicles Volume I. TheoDL-TR-75-137.

assig, H.J., “An Approximate True Damping Solution of the rminant Iteration,” Journal of Aircraft, Vol. 8, No. 11, Novemb

eill, D.J., “MSC/Flight Loads and Dynamics Training,”, The oration, 815 Colorado Boulevard, Los Angeles, CA, August

References

Documents

Computational Aeroelasticity - METUyyaman/avt086/Kolonay/Ray_Kolonay_1.pdf · Computational Aeroelasticity The Cultural and Convention Center METU Inonu bulvari Ankara, Turkey Sponsored