Wing Flutter Analysis With An Uncoupled Method

27 Pages·2003·5.44 MB·English

by  Koray Kavukcuoglu

Wing Flutter Analysis With
An Uncoupled Method
(cid:1)
Koray KAVUKCUO LU
M. Sc. Thesis Presentation
Supervisor Prof. Dr. Mehmet A. AKGÜN
Co-Supervisor Dr. Erdal OKTAY
Presentation Outline
• 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
August 18, 2003 Koray KAVUKCUO(cid:1)LU 2
Introduction
Flutter
Dynamic instability of an elastic body
in an airstream caused by the
unsteady aerodynamic forces
generated from elastic deformations
of the structure.
August 18, 2003 Koray KAVUKCUO(cid:1)LU 3
Flutter Modeling
1 – DOF System
K
a
– Rigid Airfoil
U
– Unit Span
a
– Oscillation around the
Leading Edge
2b
(cid:1)(cid:1) iwt
I a+ K a = M a =a e
a a y 0
2 – DOF System
z
b b
– Rigid Airfoil
U
x
a – Unit Span
h
Ka – Pitching and Plunging Motion
My
(cid:1)(cid:1)
c.g. (cid:1)(cid:1)
ba bxa mh + S a+ mw2h = Q
a h h
Kh
(cid:1)(cid:1)
(cid:1)(cid:1)
2
S h + I a+ I w a= Q
a a a a a
August 18, 2003 Koray KAVUKCUO(cid:1)LU 4
Flutter Modeling
P-K Method
– Forced Response Analysis
– Eigenvalue Solution
[ ]{(cid:1)(cid:1)} [ ]{ } { (cid:1) (cid:1)(cid:1) }
M q + K q = f (q,q,q,t) EOM for the system
– Harmonic displacement
– Transform to Modal Coordinates
(cid:1) 1 (cid:2)
[M ]s2 +[K ]- rV 2 [Q ] {h } = 0 Basic EV Flutter Eqn.
* (cid:3) (cid:4)
h h h h
(cid:5) 2 (cid:6)
Vk
g
s = (g + i) p = k(g + i) * Valid when = 0
b
August 18, 2003 Koray KAVUKCUO(cid:1)LU 5
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)
(cid:1)(cid:1) (cid:1)
2
q + 2zwq +w q = Q
i i i i i i i
– 2 SDOF Equations
– Decoupled by Diagonalization
– Solved at each time step
August 18, 2003 Koray KAVUKCUO(cid:1)LU 6
Flutter Modeling
Present Approach
– Forced Response Analysis
– Eigenvalue Solution
– Aim is to decouple CSD and CFD
[ ]{(cid:1)(cid:1)} [ ]{ } { (cid:1) (cid:1)(cid:1) }
M q + K q = f (q,q,q,t) EOM for the system
– Harmonic displacement
{ } [ ]{ } iwt
q = F h e
– Transform to Modal Coordinates
{ } ([ ]{ } iwt )
f = Im A f e
j j
Unsteady Aerodynamic
Force Definition
{ } { }
[ ]
r = A f
j j
• Substitute to EOM and get Eigenvalue flutter problem
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Flutter Modeling
{ }
[K ]-w2[M ]- [F]T [R]{h}= 0 Basic EV Flutter Eqn.
M M
– Similar to P-K method Formulation
– Generalized Aerodynamic Force (GAF) Matrix ( [R] ) is unsymmetric
– [R] = f(M,k) k : reduced frequency
[ ] [ ] [ ]
R = R + i R
R I
– By time integration get real and
P
(cid:7)
[ ] p (cid:1) (cid:2)
R = c T imaginary parts of aerodynamic force
(cid:5) (cid:6)
R p
p=0
– Fit Pth order polynomial to aerodynamic
P
[ ] (cid:7) p (cid:1) (cid:2) force
R = c Z
(cid:5) (cid:6)
I p
p=0
• Transform into a Polynomial Eigenvalue Problem
August 18, 2003 Koray KAVUKCUO(cid:1)LU 8
Flutter Modeling
– Polynomial curve fitting in terms of reduced frequency (k)
(cid:8) (cid:9)
P
(cid:7)
(cid:10) k pQ (cid:11){h} = {0} Polynomial Eigenvalue Flutter Equation
p
(cid:12) (cid:13)
p=0
(cid:1){ }(cid:2)
h
Transform to a real augmented system
{ } R
h = (cid:3) (cid:4)
{h } [Q] is (8x8)
(cid:5) (cid:6)
I
(cid:1)[K ] [0] (cid:2) (cid:8)U (cid:9)2 (cid:1)[M ] [0] (cid:2)
Q = (cid:3) M (cid:4) Q = Q -(cid:10) ¥ (cid:11) (cid:3) M (cid:4)
0 [0] [K ] 2 2 (cid:12) b (cid:13) [0] [M ]
(cid:5) (cid:6) (cid:5) (cid:6)
M M
(cid:1)-[F]T (cid:1)T (cid:2) [F]T (cid:1)Z (cid:2) (cid:2)
(cid:5) (cid:6) (cid:5) (cid:6)
(cid:3) p p (cid:4)
Q = Q Q =
for p = 1,3,4 … P
p p p (cid:3) (cid:4)
-[F]T (cid:1)Z (cid:2) -[F]T (cid:1)Z (cid:2)
(cid:5) (cid:5) (cid:6) (cid:5) (cid:6)(cid:6)
p p
August 18, 2003 Koray KAVUKCUO(cid:1)LU 9
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
h) Kolonay Up
c
n 0.2 Kolonay Down
s (i Present Up
s 0.0
e Present Down
kn 0.0 0.2 0.4 0.6 0.8 1.0 NACA65004 Up
c-0.2
hi NACA65004 Down
T
-0.4
-0.6
x/c
August 18, 2003 Koray KAVUKCUO(cid:1)LU 10