Wing Flutter Analysis With An Uncoupled Methodkoray/publis/KavukcuogluMSThesisPresentation.pdf · Wing Flutter Analysis With An Uncoupled Method ... h h 2 h h M s K rV Q h + - =

Wing Flutter Analysis With An Uncoupled Method

M. Sc. Thesis Presentation

Supervisor Prof. Dr. Mehmet A. AKGÜNCo-Supervisor Dr. Erdal OKTAY

Koray KAVUKCUOLU

August 18, 2003 Koray KAVUKCUOLU 2

• Introduction– Flutter

• Flutter Modeling– Simple Systems– Coupled and P-K Methods– Akgün’s Method

• Modal Analysis of AGARD Wing 445.6– Finite Element Model– Results

• Aerodynamic Analysis of AGARD Wing 445.6– USER3D– CFD Model– Solution Procedure– Grid Transformation– Force Calculation– Results

• Numerical Results– Discussion

Presentation Outline


Flutter

Dynamic instability of an elastic body in an airstream caused by the unsteady aerodynamic forces generated from elastic deformations of the structure.

Introduction


αK

α

2b

U

αK

hK

α

U

z

b b

h

x

c.g.ba αbx

yM

Flutter Modeling

1 – DOF System– Rigid Airfoil– Unit Span– Oscillation around the

Leading Edge

2 – DOF System– Rigid Airfoil– Unit Span– Pitching and Plunging Motion

yMKI =+ αα αα iwte0αα =

hh QhmwShm =++ 2αα

ααααα αα QwIIhS =++ 2


Flutter Modeling

[ ] [ ] ( , , , )M q K q f q q q t+ =

P-K Method– Forced Response Analysis– Eigenvalue Solution

[ ] [ ] [ ] 2 210

2h h h hM s K V Qρ η + − =

EOM for the system

– Harmonic displacement

– Transform to Modal Coordinates

Basic EV Flutter Eqn.

)()( ikpib

Vks +=+= γγ * Valid when γ = 0

*


Flutter Modeling

Coupled Methods– Solution of Aeroelastic Equations every time step– Grid Transformation– Coupling Algorithm

– Weakly Coupled (Separate CFD – CSD)– Strongly Coupled (Stability = Least Stable Code)– Fully Coupled (Iterative solution at each time step)

iiiiiii Qqqq =++ 22 ωωζ

– 2 SDOF Equations

– Decoupled by Diagonalization

– Solved at each time step


Flutter Modeling

Present Approach– Forced Response Analysis– Eigenvalue Solution– Aim is to decouple CSD and CFD

[ ] [ ] ( , , , )M q K q f q q q t+ = EOM for the system

[ ] i tq e ωη= Φ– Harmonic displacement

– Transform to Modal Coordinates

Unsteady AerodynamicForce Definition

[ ] ( )Im i tj jf A e ωφ=

[ ] j jr A φ=

• Substitute to EOM and get Eigenvalue flutter problem


Flutter Modeling

[ ] [ ] [ ] [ ] 02 =Φ−− ηω RMK TMM Basic EV Flutter Eqn.

– Similar to P-K method Formulation

– Generalized Aerodynamic Force (GAF) Matrix ( [R] ) is unsymmetric

– [R] = f(M,k) k : reduced frequency

• Transform into a Polynomial Eigenvalue Problem

[ ] [ ] [ ]R IR R i R= +

[ ]P

0

pR p

p

R c T=

=

[ ]P

0

pI p

p

R c Z=

=

– By time integration get real and

imaginary parts of aerodynamic force

– Fit Pth order polynomial to aerodynamic

force


Flutter Modeling

– Polynomial curve fitting in terms of reduced frequency (k)

P

0

0pp

p

k Q η=

=

Polynomial Eigenvalue Flutter Equation

R

I

ηη

η

=

Transform to a real augmented system[Q] is (8x8)

[ ] [ ][ ] [ ]0

00

M

M

KQ

K

=

[ ] [ ][ ] [ ]

2

2 2

00

M

M

MUQ Q

Mb∞

= −

p pQ Q=[ ] [ ][ ] [ ]

T Tp p

p T Tp p

T ZQ

Z Z

− Φ Φ = − Φ − Φ

for p = 1,3,4 … P


Modal Analysis of AGARD Wing 445.6

Finite Element Model

– Shell elements

– Distributed thickness

– Rotated element coordinate system

– In accordance with real wing

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.0 0.2 0.4 0.6 0.8 1.0

x/c

Thi

ckne

ss (

inch

) Kolonay UpKolonay DownPresent UpPresent DownNACA65004 UpNACA65004 Down


Modal Analysis

– Theoretical Aspects

– Comparison with Experimental Results

[ ] [ ] [ ] 0=++ qKqCqM

– Undamped System

– Harmonic Free Vibrations i tq u e ω= [ ] 0C =

[ ] [ ] 2 0M u K uω− + =


– Carried out with ANSYS


Modal Analysis of AGARD Wing 445.6Natural Frequency Comparison

%10.8

%12.2

%16.9

%13.0

-%8.69

-%0.39

-%2.57

% 0.31

-%6.24

% 0.59

-%0.65

%0.92

109.10

56.88

44.57

10.85

Li

89.94

50.50

37.12

9.63

Kolonay

92.358

50.998

37.854

9.688

Present Study

98.50

50.70

38.10

9.60

Experiment

4

3

2

1

Modes

Deflection Comparison

71.52-22.6267.182-22.824

31.38-27.1731.595-27.9233

25.09-45.4845.873-25.3752

27.92-0.08628.093-0.01251

KolonayPresent StudyModes



1

MN

MX

X

Y

Z

-.01225

3.7357.483

11.2314.977

18.72522.472

28.093

JUL 26 200317:42:58

NODAL SOLUTION

STEP=1SUB =1FREQ=9.688UZ (AVG)RSYS=0DMX =28.093SMN =-.01225SMX =28.093

1

MN

MX

X

Y

Z

-25.375

-15.876-6.376

3.12412.623

22.12331.623

45.873

JUL 26 200317:43:59

NODAL SOLUTION

STEP=1SUB =2FREQ=37.854UZ (AVG)RSYS=0DMX =45.873SMN =-25.375SMX =45.873

1

MN

MX

X

Y

Z

-27.923

-19.988-12.052

-4.1163.82

11.75519.691

31.595

JUL 26 200317:45:12

NODAL SOLUTION


Comparison of Mode Shapes

1

MN MX

X

Y

Z

-22.82

-10.821.181

13.18125.181

37.18149.182

67.182

JUL 26 200317:45:42

NODAL SOLUTION


– The results of the modal analysis are accepted for further use …

1st Mode 2nd Mode

3rd Mode 4th Mode


Aerodynamic Analysis of AGARD Wing 445.6

Aerodynamic Analysis

– USER3D

– Unstructured 3D Euler Solver

– Harmonic oscillation of the wing

– 120 time steps per period for each oscillation frequency

– Grid transformation

– Surface interpolation technique is adopted

– Pressure transformation

– Surface interpolation technique is adopted

– Force Calculation

– Using the pressure distribution on structural model



USER3D

– Parallel finite volume based unstructured 3D Euler solver

– Uses ALE (Arbitrary Lagrangian-Eulerian) formulation

– The flow variables are non-dimensionalised

0

0

0

0

0

0

20 0 0

'

'

'

'

'

'

' '

u u a

v v a

w w a

a a a

T T T

P P a P P P

ρ ρ ρ

ρ γ

= ⋅= ⋅= ⋅= ⋅= ⋅= ⋅

= ⋅ = ⋅

– Pressure is calculated from ideal gas relation

( )

++−−= )(21

1 222 wvuep ργ



USER3D

– Moving Mesh Algorithm

– Spring analogy

( ) ( ) ( )2 2 21

m

j i j i j i

kx x y y z z

=− + − + −

dSnWdVt s

Ω Ω∂

⋅=∂∂

– Geometric Conservation Law

– MPICH library (portable and high performance implementation of MPI)

is used in USER3D for parallelization



CFD Model

– Non-dimensionalised wrt root chord (21.96”)

– Unstructured grid (tetrahedral elements)

– I-DEAS is used to generate mesh

– Half-domain is meshed mirrored full-domain

– The wing profile is NACA 65A004

– 126380 elements & 26027 nodes totally

– 13254 elements & 6667 nodes on the wing

AGARD Wing 445.6



CALCULATEDEFORMED

MESH FOR NEXTTIME STEP

DOUNSTEADY

AERODYNAMICSOLUTION

STOREAERODYNAMIC

SOLUTION DATA

Solution Procedure– Mesh deformation according to mode shapes

– Sinusoidal oscillation for three periods at each frequency

– 10 different frequencies at first 4 mode shapes (40 solution cases)

( )0Q Q sin tω= ×

Mode # Maximum Tip Deflection(inch)

1 0.015

2 0.03

3 0.004

4 0.0015



Grid Transformation– 2D Structural Model 3D CFD Model

– Mode shape transformation

– Surface interpolation by Akima (IMSL)

X (inch)

Y(in

ch)

0 10 20 30 400

5

10

15

20

25

30

35

40 5.6E-03

5.2E-03

4.9E-03

4.5E-03

4.1E-03

3.7E-03

3.4E-03

3.0E-03

2.6E-03

2.2E-03

1.9E-03

1.5E-03

1.1E-03

7.5E-04

3.7E-04

Mode 2 Pe rce nt Error Dis tribution

Frame 002 12 Aug 2003 ERROR DATA

X (inch)

Y(in

ch)

0 10 20 30 400

5

10

15

20

25

30

35

40 4.4E-02

4.1E-02

3.8E-02

3.5E-02

3.2E-02

2.9E-02

2.6E-02

2.4E-02

2.1E-02

1.8E-02

1.5E-02

1.2E-02

8.8E-03

5.9E-03

2.9E-03

Mode 3 Pe rcent Error Dis tribution


X (inch)

Y(in

ch)

0 10 20 30 400

5

10

15

20

25

30

35

40 1.7E-02

1.6E-02

1.5E-02

1.4E-02

1.2E-02

1.1E-02

1.0E-02

9.0E-03

7.9E-03

6.8E-03

5.6E-03

4.5E-03

3.4E-03

2.3E-03

1.1E-03

Mode 4 Pe rce nt Error Dis tribution


X (inch)

Y(in

ch)

0 10 20 30 400

5

10

15

20

25

30

35

40 2.9E-02

2.7E-02

2.5E-02

2.3E-02

2.1E-02

1.9E-02

1.7E-02

1.5E-02

1.3E-02

1.1E-02

9.5E-03

7.6E-03

5.7E-03

3.8E-03

1.9E-03

Mode 1 Pe rcent Error Dis tribution

Frame 001 12 Aug 2003 ERROR DATAFrame 001 12 Aug 2003 ERROR DATA

Max % error

X (in)

Y(in

)

0 10 20 30 400

5

10

15

20

25

30

35

40

Mo de 1 Comparis onMode 1 12 Aug 2003 MODE SHAP E DATA

X (in)

Y(in

)

0 10 20 30 400

5

10

15

20

25

30

35

40

Mode 2 Comparis onMode 2 12 Aug 2003 MODE SHAPE DATA

X (in)

Y(in

)

0 10 20 30 400

5

10

15

20

25

30

35

40

Mo de 3 Comparis onMode 3 12 Aug 2003 MODE SHAP E DATA

X (in)

Y(in

)

0 10 20 30 400

5

10

15

20

25

30

35

40

Mode 4 Comparis onMo de 4 12 Aug 20 03 MODE SHAPE DATAMo de 4 12 Aug 20 03 MODE SHAPE DATA


Aerodynamic Analysis of AGARD Wing 445.6Force Calculation

– Calculate dimensional pressure distribution

– Use surface interpolation to transfer pressures

CFD Structure mesh at mid points of elements

– Transform pressures at mid-elements to forces at nodes

Pa

Pd

Pb

PcA B

C D

F = Pa(A) + Pb(B) + Pc(C) + Pd(d)

X

X (in)

Y(in

)

0 5 10 15 20 25 30 35 40 450

5

10

15

20

25

30

35

404946433936333026232016131073

Pres s ure Difference Comparis on for the 1s t Mode

DP(Pa)

Frame 001 12 Aug 2 003

X (in)

Y(in

)

0 5 10 15 20 25 30 35 40 450

5

10

15

20

25

30

35

40375350325300275250225200175150125100755025

Pres s ure Difference Comparis on for the 2nd Mode

DP(Pa)

Frame 00 1 12 Aug 200 3

X (in)

Y(in

)

0 5 10 15 20 25 30 35 40 450

5

10

15

20

25

30

35

409875431

-0-1-3-4-5-7-8-9

Pres s ure Diffe rence Comparis on for the 4th Mode

DP(Pa)

Frame 0 01 12 Aug 20 03

X (in)

Y(in

)

0 5 10 15 20 25 30 35 40 450

5

10

15

20

25

30

35

402927252321191715131197420

Pres s ure Difference Comparis on for the 3rd Mode

DP(Pa)

Frame 001 12 Aug 2 003 Frame 001 12 Aug 2 003



CFD Results– Total Lift Coefficient Convergence

– Pressure Distribution

– Total Lift Coefficient Comparison

– Phase DifferenceIte ration

CL

400 450 500 550 600 650 700 750

-0.004

-0.003

-0.002

-0.001

0

0.001

0.002

0.003

0.004

1s t Mode2nd Mode3rd Mode4th Mode

Fra me 0 01 12 Aug 2 00 3 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | |Fra me 0 01 12 Aug 2 00 3 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | |

Tip De flection

CL

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

-0.0006

-0.0005

-0.0004

-0.0003

-0.0002

-0.0001

0

0.0001

0.0002

0.0003

0.0004

0.0005

0.0006

1200 s teps /period120 s teps /period

Frame 0 01 12 Au g 2003 Frame 0 01 12 Au g 2003

9.688 Hz

Angle of Attack vs Spanwise Location

-2.0E-02

-1.5E-02

-1.0E-02

-5.0E-03

0.0E+00

5.0E-03

0 10 20 30 40

Y (in)

AOA (

deg)

Mode 1Mode 3Mode 4

Angle of Attack vs Span

-1.8E-01

-1.6E-01

-1.4E-01

-1.2E-01

-1.0E-01

-8.0E-02

-6.0E-02

-4.0E-02

-2.0E-02

0.0E+000 10 20 30 40

Y (in)

AOA (

deg)

Mode 2

Tip Defle ction

CL

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

-0.0018

-0.0015

-0.0012

-0.0009

-0.0006

-0.0003

0

0.0003

0.0006

0.0009

0.0012

0.0015 1200 steps /perio d120 steps /period

Frame 001 12 Aug 2003 Frame 001 12 Aug 2003

92.358 Hz



X

Y

0 0.25 0.5 0.75 1 1.25 1.5 1.75 20

0.25

0.5

0.75

1

1.25

1.5 296382923528832284302802727624272212681926416260132561125208248052440324000

PRES S URE DIS TRIBUTION ON THE WING

P (Pa)

F rame 001 12 Aug 2003 F rame 001 12 Aug 2003

– Pressure distribution

X

PR

ES

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.425000

26000

27000

28000

29000

30000

31000

32000

33000

PRES S URE DIS TRIBUTION ON THE WING AT Y = 0.51 s t Mode at 1 s t Natural Fre que ncy

uppe rlowe r

Fram e 001 18 Aug 2003 Fram e 001 18 Aug 2003

–Total Lift Coefficient Comparison

Ite ration

CL

400 450 500 550 600 650 700 750

-0.0006

-0.0004

-0.0002

0

0.0002

0.0004

0.0006

Force CalculationUS ER3D

Frame 001 12 Aug 2003 Frame 001 12 Aug 2003

NODE

PH

AS

E(D

EG

)

50 100 150 200

50

100

150

200

250

300

350

runM1 12runM1 2runM1 22runM2 12runM2 2runM3 12runM3 2runM3 22runM3 32runM4 2

2nd Mode Phas e Angle VariationFra me 0 01 2 9 Jul 20 0 3 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | |

NODE

PH

ASE

(DE

G)

50 100 150 200

50

100

150

200

250

300

350

run M11 4run M12 4run M14run M21 4run M24run M31 4run M32 4run M33 4run M34run M44

4th Mode Phas e Angle Variation

Fra me 0 01 2 9 Jul 20 0 3 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | |

NODE

PH

AS

E(D

EG

)

50 100 150 200

50

100

150

200

250

300

350

ru nM11 3ru nM12 3ru nM13ru nM21 3ru nM23ru nM31 3ru nM32 3ru nM33ru nM33 3ru nM43

3rd Mode Phas e Angle Variation

Frame 0 01 2 9 Jul 20 0 3 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | |

NODE

PH

AS

E(D

EG

)

50 100 150 200

100

150

200

250 runM11runM111runM121runM21runM211runM31runM311runM321runM331runM41

1s t Mode Phas e Angle Variation

Os cillation Fre que ncy Incres e s

Frame 0 01 2 9 Jul 20 0 3 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | |Frame 0 01 2 9 Jul 20 0 3 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | |

– Phase Difference



CFD Results– Total Lift Coefficient Convergence

– Verified through the Total Lift Coefficient Histograms

– Pressure Distribution

– Order of magnitude comparison to results by simpler methods

– Total Lift Coefficient Comparison

– Around 25 % error is introduced from force calculation

– Phase Difference

– Phase difference exists between each node as expected


Numerical Results

Eigenvalue Solution– Carried out with MATLAB®

– Element Mass and Stiffness Matrices extracted from ANSYS®

– Substructuring analysis

– User Programmable Features of ANSYS®

– Modal Analysis repeated with MATLAB ®

– Mode shapes and natural frequencies are compared

X (in)

Y(in

)

0 10 20 30 400

5

10

15

20

25

30

35

40Z

27.9226 .0524 .1922 .3220 .4518 .5816 .7214 .8512 .9811 .12

9 .257 .385 .523 .651 .78

-0 .09

Mode # 1

Frame 00 1 1 2 Aug 2 00 3

X (in)

Y(in

)

0 10 20 30 400

5

10

15

20

25

30

35

40Z

25.420.615.911.1

6.41.6

-3.1-7.9

-12.6-17.4-22.1-26.9-31.6-36.4-41.1-45.9

Mode # 2

Fra me 0 01 12 Aug 20 03

X (in)

Y(in

)

0 10 20 30 400

5

10

15

20

25

30

35

40Z

27.9223.9519.9916.0212.05

8.084.120.15

-3.82-7.79

-11.75-15.72-19.69-23.66-27.62-31.59

Mode # 3

Frame 0 01 12 Aug 20 03

X (in)

Y(in

)

0 10 20 30 400

5

10

15

20

25

30

35

40Z

22.8216 .8210 .82

4 .82-1 .18-7 .18

-13 .18-19 .18-25 .18-31 .18-37 .18-43 .18-49 .18-55 .18-61 .18-67 .18

Mode # 4

Frame 001 1 2 Aug 2 003 Frame 001 1 2 Aug 2 003

Mode # ANSYS MATLAB

1 9.688 9.688

2 37.854 37.854

3 50.998 50.998

4 92.358 92.358


Numerical Results

Eigenvalue Solution

– Polynomial curve fitting to real and imaginary parts of aerodynamic

forces

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8-0.04

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0.04

Reduced Frequency (k)

Aerodynamic Force (lb)

Comparis on of Polynomial Fit for Mode #2 Node #11

Real DataReal FitImaginary DataImaginary Fit

1,1 1,

231,1 231,

l ln

l

l ln

t t

T

t t

=

1,1 1,

231,1 231,

l ln

l

l ln

z z

Z

z z

=

[ ] 3 3 2 2 1 0T T k T k T k T = + + +

[ ] 3 3 2 2 1 0Z Z k Z k Z k Z = + + +


Numerical Results

Eigenvalue Solution

– Calculate generalized aerodynamic forces

– Calculate flutter frequency

High

Normal

Deflection

%26.8113.0(rad/s)

82.6897(rad/s)

0.678

%26.9113.0(rad/s)

82.58(rad/s)

0.678

ErrorExperimentPresent StudyMach #

For high deflection case, tip deflection

values are taken from Reference 8


Conclusions & Recommendations– A new approach was applied to AGARD Wing 445.6

– Modal analysis of the wing is performed

– CFD analysis of the wing is performed

– Surface interpolation technique is implemented

– Grid Transformation

– Pressure Transformation

– Polynomial curve fitting to aerodynamic forces is implemented

– Significant decrease in computational time for flutter prediction

– Significant error compared to experiment

– Force calculation technique should be revised

– All experimental cases should be studied

– GAF’s should be verified with literature

Documents

Wing Flutter Analysis With An Uncoupled Methodkoray/publis/KavukcuogluMSThesisPresentation.pdf · Wing Flutter Analysis With An Uncoupled Method ... h h 2 h h M s K rV Q h + - =