AAE556 Lecture_3435_pk_flutter
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AAE 556
Aeroelasticity
The P-k flutter solution method
(also known as the “British” method)
Purdue Aeroelasticity
1
The eigenvalue problem from the Lecture 33
ωh2
ω
2 ÷ 0 h b 1
−
+
÷( 1 + ig ) ωθ
ω
θ xθ
2
0
rθ
2
θ
2
1 Lh
+
µ
M θ h
ωθ2
0
h ω 2 ÷
1
Ω 2 b = h
x
1
θ 0
θ
rθ 2
xθ h
b
2
rθ θ
1
h
L
−
+
a
÷Lh b = 0
α 2
0
θ
M θθ
xθ 1 Lh
+
rθ 2 µ
M θ h
1
h
Lα − 2 + a ÷Lh b
θ
M θθ
2
Purdue Aeroelasticity
Genealogy of the V-g or “k” method
i
Equations of motion for harmonic response (next slide)
–
–
–
i
Forcing frequency and airspeeds are is known parameters
Reduced frequency k is determined from ω and V
Equations are correct at all values of ω and V.
Take away the harmonic applied forcing function
–
–
–
Equations are only true at the flutter point
We have an eigenvalue problem
Frequency and airspeed are unknowns, but we still need k to define the numbers to compute the
elements of the eigenvalue problem
–
We invent ed Theodorsen’s method or V-g artificial damping to create an iterative approach to finding
the flutter point
Purdue Aeroelasticity
3
Go back to the original typical section equations of motion, restricted to steady-state
harmonic response
2 AEOM
+ω
DEOM
h
BEOM F
b =
EEOM M SC
θ
4
Purdue Aeroelasticity
The coefficients for the EOM’s
AEOM
ω 2 ω 2 L
= − 1 − h ÷ θ2 ÷− h
ωθ ω ÷ µ
BEOM
DEOM
1
1
= − xθ − Lα − Lh + a ÷÷
µ
2
1
1
= − xθ − M h − Lh + a ÷÷
µ
2
(
EEOM = − rθ2 1 − ωθ2
)
Mh
+
µ
2
1
M α Lα 1
Lh 1
+
a
−
+
+
a
−
+
a
÷
÷
÷
2
µ
µ
2
µ
2
5
Purdue Aeroelasticity
The eigenvalue problem
2 AEOM
ω
DEOM
h
BEOM 0
b =
EEOM 0
θ
AEOM
DEOM
h
BEOM 0
b =
EEOM 0
θ
AEOM
( −µ )
DEOM
h
BEOM 0
b =
EEOM 0
θ
ω
ω2
2
6
Purdue Aeroelasticity
Another version of the eigenvalue problem with different
coefficents
AEOM
( −µ )
DEOM
h
h
BEOM
A B 0
b =
b =
EEOM D E 0
θ
θ
ω 2 ω 2 ω 2
A = µ 1 − h ÷ θ ÷ h ÷ ( 1 + ig ) ÷+ Lh
ωθ ω ωθ
÷
1
B = µ xθ + Lα − Lh + a ÷
2
7
Purdue Aeroelasticity
Definitions of terms for alternative set-up of eigenvalue equations for
“k-method”
h
A B 0
b =
D E θ 0
1
D = µ xθ + M h − Lh + a ÷
2
2
ω
θ
1
2
E = µ rθ 1 −
1
+
ig
÷−
M
+
a
+ Mα
(
)
h
÷
ω ÷
÷
2
2
1
1
− Lα + a ÷+ Lh + a ÷
2
2
8
Purdue Aeroelasticity
Return to the EOM’s before we assumed harmonic motion
Here is what we would like to have
&j } + K ij { η j } + Aij( 1) { η j } + Aij( 2) { η&j } + Aij( 3) { η&
&
M ij { η&
j } = { 0}
{η } = {η } e
j
j
Here is the first step in solving the stability problem
pt
p = σ + jω
p 2 M ij { η j } + Kij { η j } + Aij( 1) { η j }
2
3
+ p Aij( ) { η j } + p 2 Aij( ) { η j } = { 0}
9
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The p-k method will use the harmonic aero results to cast the stability
problem in the following form
1
p M ij { η } − p Bij { η } + K ij − ρV 2 Qij ,real { η } = { 0}
2
2
{η ( t )} = {η} e
…but first, some preliminaries
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10
pt
Revisit the original, harmonic EOM’s where the aero forces were still on the right hand side
of the EOM’s and we hadn’t yet nondimensionalized
h
1
iωt
P = − L = πρ b ω Lh ÷+ Lα − + a ÷Lh θ e
2
b
3
2
1
h b iωt
Lα − 2 + a ÷Lh ÷ e
θ
P = πρ b ω ( Lh )
3
2
h
V
airfoil chordline
b
ba
shear center
b
P
P = A11
h
A12 b
θ
11
Purdue Aeroelasticity
This lift expression looks strange; where is the dynamic
pressure?
h
1
iωt
P = πρ b ω Lh ÷+ Lα − + a ÷Lh θ e
2
b
3
2
V 2
h
i ωt
1
3 2
P = 2 πρb ω Lh + Lα − + a Lh θ e
2
V
b
ρV 2
b 2ω 2
P=
b( 2π ) 2
2
V
ωb
k=
V
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12
h
1
i ωt
Lh + Lα − + a Lh θ e
2
b
Writing aero force in different notation
- more term definitions
P = A11
h 1
h
2
A12 b = ρV Q11 Q12 b
θ 2
θ
Q11
Q11
2
A
Q12 =
2 11
ρV
2πρ b3ω 2
Q12 =
( Lh )
2
ρV
A12
1
Lα − + a ÷Lh ÷
2
The Qij’s are complex numbers
13
Purdue Aeroelasticity
Aero force
in terms of the Qij’s
Q11
Q12 = 2bπ k ( Lh )
1
Lα − Lh + a ÷÷
2
2
(
Q11 = 2π bk 2 Lh = 2π b k 2 − i 2kC ( k )
)
1
Q12 = 2π bk Lα − Lh + a ÷÷
2
2
k2
1
2
Q12 = 2π b − ik ( 1 + 2C ( k ) ) − 2C ( k ) ÷− k − i 2kC ( k ) + a ÷÷
2
2
÷
(
)
14
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Q11
Focus first on the term
(
Q11 = 2π b k − i 2kC ( k )
2
(
Q11 = 2π b k 2 − i 2k ( F + iG )
(
)
Q11 = 2π b k 2 + 2kG − i [ 2kF ]
)
Q11 = Q11,real + iQ11,imaginary
15
Purdue Aeroelasticity
)
The second term
k2
1
2
Q12 = 2π b − ik ( 1 + 2C ( k ) ) − 2C ( k ) ÷− k − i 2kC ( k ) + a ÷÷
2
2
÷
(
)
k2
1
2
Q12 = 2π b − ik ( 1 + 2 ( F + iG ) ) − 2 ( F + iG ) ÷− k − i 2k ( F + iG ) + a ÷÷
2
2
÷
(
)
3
1
2
Q12 = 2π b −2 F − k a + 2kG + a ÷+ i −k − 2G + 2kF − + a ÷÷÷÷
2
2
÷
16
Purdue Aeroelasticity
Let’s adopt notation from the controls community to help with our
conversion
(
Q11 = 2π b k 2 + 2kG − j [ 2kF ]
)
Q11 = Q11,real + jQ11,imaginary
If we were to assume motion e pt and p ≅ jω
p
then
≅1
jω
Q11 = Q11,real
p
+
÷ jQ11,imaginary
jω
17
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Continue working on the first term in the aero force expression
p
Q11 = Q11,real + ÷Q11,imaginary
ω
Q11,imaginary
Q11 = Q11,real + p
ω
P = A11
h
A12 b
θ
The expression for A11 reads
2
1
1
ρ
V
A11 = ρV 2Q11,real + p
Q11,imaginary
2
2 ω
18
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The term with the p in it looks like a damping term so let’s work on it
2
1
1
ρ
V
A11 = ρV 2Q11,real + p
Q11,imaginary
2
2 ω
1
ρVb V
2
A11 = ρV Q11, real + p
Q11,imaginary
2
2 bω
1
ρVb Q11,imaginary
2
A11 = ρV Q11,real + p
2
2
k
19
Purdue Aeroelasticity
Finally, the exact expressions for each term are as follows
1
ρVb Q11,imaginary
2
A11 = ρV Q11,real + p
2
2
k
(
Q11 = 2π b k 2 + 2kG − j [ 2kF ]
)
1
ρVb
2
2
A11 = ρV 2π b k + 2kG − p
[ 2F ]
2
2
Both terms are real numbers, there is no j here.
20
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Aerodynamic moment expression
M aero
Mα =
1 1
h
− + a ÷Lh ÷
b
2 2
4 2
= πρ b ω
2
+ M α − Lα + 1 ÷ 1 + a ÷+ Lh 1 + a ÷ ÷θ
÷
2
2
2
3 1
−i
8 k
M aero = A21
h
A22 b
θ
1 1
A21 = πρ b 4ω 2 − + a ÷Lh ÷
2 2
2
1
1
1
4 2
A22 = πρ b ω M α − Lα + ÷ + a ÷+ Lh + a ÷ ÷
2 2
2
÷
21
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÷
÷
÷
÷
÷
The Qij’s
2
Q21 Q22 =
A
2 21
ρV
Q21,real
2
= Real
A
2 21 ÷
ρV
Q22,real
2
= Real
A
2 22 ÷
ρV
A22
Q21,imaginary
2
= Imag
A
2 21 ÷
ρV
Q22,imaginary
2
= Imag
A
2 22 ÷
ρV
22
Purdue Aeroelasticity
The p-k process
i
Step 1
i
Choose a value of k and compute all four complex aerodynamic
coefficients
– These are the complex Aij’s with the Theodorsen Circulation function in
them
– These will be a set of complex numbers, not algebraic expressions
i
Choose an air density (altitude) and airspeed (V)
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23
Perform this computation
2
Qij =
A
2 ij
ρV
24
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Compute the aerodynamic damping matrix, defined as
1 Vb
Bij = ρ
Qij ,imaginary
2 k
Qij ,imaginary
1
Bij = ρVb
2
k
25
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