AAE 556
Aeroelasticity
The P-k flutter solution method
(also known as the “British” method)
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The eigenvalue problem from the
Lecture 33
h2
2 0 h b 1
2 1 ig
x
2
0
r
2
1 Lh
M h
2
h 2
2 b h
0
0
1
1 x
2
r
x h
b
2
r
1
h
L
a
Lh b 0
2
0
M
x 1 Lh
r 2
M h
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h
L 2 a Lh b
M
2
Genealogy of the V-g or “k”
method
Equations of motion for harmonic response (next
slide)
– Forcing frequency and airspeeds are is known parameters
– Reduced frequency k is determined from and V
– Equations are correct at all values of 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
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Go back to the original typical section equations of
motion, restricted to steady-state
harmonic response
2 AEOM
DEOM
h
BEOM F
b
EEOM M SC
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The coefficients for the
EOM’s
AEOM
2 2 L
1 h 2 h
BEOM
DEOM
1
1
x L Lh a
2
1
1
x M h Lh a
2
EEOM r2 1 2
Mh 1
M L
a
2
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Lh 1
a
a
2
2
2
5
The eigenvalue problem
AEOM
DEOM
2
h
BEOM 0
b
EEOM 0
AEOM
DEOM
h
BEOM 0
b
EEOM 0
AEOM
DEOM
h
BEOM 0
b
EEOM 0
2
2
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Another version of the eigenvalue
problem with different coefficents
AEOM
DEOM
h
h
BEOM
A B 0
b
b
EEOM D E 0
2 2 2
A 1 h h 1 ig Lh
1
B x L Lh a
2
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Definitions of terms for alternative set-up
of eigenvalue equations for “k-method”
h
A B 0
b
D E 0
1
D x M h Lh a
2
2
1
2
E r 1
1
ig
M
a
h
M
2
1
1
L a Lh a
2
2
2
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Return to the EOM’s before we
assumed harmonic motion
Here is what we would like to have
M ij j Kij j Aij1 j Aij 2 j Aij3 j 0
Here is the first step in solving the stability problem
pt
e
j j
p j
p 2 M ij j Kij j Aij1 j
2
3
p Aij j p 2 Aij j 0
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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 p Bij K ij V 2 Qij ,real 0
2
2
t e
pt
…but first, some preliminaries
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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
it
P L b Lh L a Lh e
2
b
3
2
1
h b it
P b Lh L a Lh e
2
3
2
h
P A11
P
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h
A12 b
11
This lift expression looks strange;
where is the dynamic pressure?
h
1
it
P b Lh L a Lh e
2
b
3
2
V 2
h
it
1
3 2
P 2 b Lh L a Lh e
2
V
b
V 2
b 2 2
P
b2 2
2
V
k
h
1
it
Lh L a Lh e
2
b
b
V
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Writing aero force in different notation
- more term definitions
P A11
h b
1
h b
2
A12 V Q11 Q12
2
Q11
Q11
2
A
Q12
2 11
V
A12
2 b3 2
1
Q12
Lh L a Lh
2
V
2
The Qij’s are complex numbers
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Aero force
in terms of the Qij’s
Q11
1
Q12 2b k Lh L Lh a
2
2
Q11 2 bk 2 Lh 2 b k 2 i 2kC k
1
Q12 2 bk L Lh a
2
2
k2
1
2
Q12 2 b ik 1 2C k 2C k k i 2kC k a
2
2
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Focus first on the term Q11
Q11 2 b k i 2kC k
2
Q11 2 b k 2 i 2k F iG
Q11 2 b k 2 2kG i 2kF
Q11 Q11,real iQ11,imaginary
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The second term
k2
1
2
Q12 2 b ik 1 2C k 2C k k i 2kC k a
2
2
k2
1
2
Q12 2 b ik 1 2 F iG 2 F iG k i 2k F iG a
2
2
3
1
2
Q12 2 b 2F k a 2kG a i k 2G 2kF a
2
2
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Let’s adopt notation from the controls
community to help with our conversion
2
Q11 2 b k 2kG j 2kF
Q11 Q11,real jQ11,imaginary
If we were to assume motion e pt and p j
p
then
1
j
Q11 Q11,real
p
jQ11,imaginary
j
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Continue working on the first term in the
aero force expression
p
Q11 Q11,real Q11,imaginary
Q11,imaginary
Q11 Q11,real p
P A11
h
A12 b
The expression for A11 reads
2
1
1
V
A11 V 2Q11,real p
Q11,imaginary
2
2
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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
V
A11 V 2Q11,real p
Q11,imaginary
2
2
1
Vb V
2
A11 V Q11,real p
Q11,imaginary
2
2 b
1
Vb Q11,imaginary
2
A11 V Q11, real p
2
2
k
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Finally, the exact expressions for each
term are as follows
1
Vb Q11,imaginary
2
A11 V Q11,real p
2
2
k
Q11 2 b k 2 2kG j 2kF
1
Vb
2
2
A11 V 2 b k 2kG p
2F
2
2
Both terms are real numbers, there is no j here.
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Aerodynamic moment expression
M aero
M
1 1
h
a
Lh
b
2 2
4 2
b
2
M L 1 1 a Lh 1 a
2
2
2
3 1
i
8 k
M aero A21
h
A22 b
1 1
A21 b 4 2 a Lh
2 2
2
1
1
1
4 2
A22 b M L a Lh a
2 2
2
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The Qij’s
2
A
Q22
2 21
V
Q21
Q21, real
2
Real
A
2 21
V
Q22, real
2
Real
A
2 22
V
A22
Q21,imaginary
2
Imag
A
2 21
V
Q22,imaginary
2
Imag
A
2 22
V
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The p-k process
Step
1
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
Choose
an air density (altitude) and
airspeed (V)
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Perform this computation
2
Qij
A
2 ij
V
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Compute the aerodynamic damping matrix,
defined as
1 Vb
Bij
Qij ,imaginary
2 k
Qij ,imaginary
1
Bij Vb
2
k
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Take the results and insert them into an
eigenvalue problem that reads as follows
1
p M ij p Bij K ij V 2 Qij ,real 0
2
2
2
Qij ,real
real ( Aij )
2
V
pt
t
e
Q
1
Bij Vb
2
ij ,imaginary
k
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Summary
1
p M ij p Bij K ij V 2 Qreal ,ij 0
2
2
Choose k=b/V arbitrarily
Choose altitude (, and airspeed (V)
Mach number is now known
Compute AIC’s from Theodorsen formulas or others
Compute aero matrices-B and Q matrices are real
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Solving for the eigenvalues
Convert the “p-k” equation to first-order state vector form
1
2
p M ij p Bij K ij V Qij ,real 0
2
2
displacement vector x j
velocity vector v j x j
м
ь
x
п
п
j
п
п
z
=
э
State vector = { j } н
п
п
v
j
п
о п
ю
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State vector elements are
related
{x j }= {v j } {x j }= {v j }
The equation of motion becomes
йM ij щ{v j }кл ъ
ы
йBij щ{v j }+
кл ъ
ы
йK ij щ{x j }= {0}
кл ъ
ы
Solve for
- 1
{v j }= - йклM ij щ
ъ
ы
йй
{v j }= кклл
йKij щ{x j }+
кл ъ
ы
{v j }
- 1
йM ij щ
кл ы
ъ
йBij щ{v j }
к ъ
л
ы
м
ь
xjп
п
- 1
- 1
щ
щ
й
щ
п
п
йM ij щ йKij щ
йM ij щ йBij щ
э
ъ
к
ъън
кл ъ
к
ъ
ы л ыылкл ъ
ы кл ъ
ыы
п
п
v
ып
j
о п
ю
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State vector eigenvalue
equation – the “plant” matrix
м
x j ьп
п
п
{z j }= нпv пэп =
п jю
п
о
й [0]
I ] щм
[
x j ьп
к
ъп
п
п
н
кй- M - 1K щ M - 1Bъпv эп =
клкл
ъ
п jю
п
ъ
ы
ыо
{
йAij щ{z j }
кл ъ
ы
}
pt
z
t
=
z
e
(
)
{ j}
Assume a solution
j
Result
{z j }= p {z j }=
йAij щ{z j }
кл ы
ъ
Solve for eigenvalues (p) of the [Aij] matrix (the plant)
Plot results as a function of airspeed
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1st order problem
Mass matrix is
diagonal if we
use modal
approach – so
too is structural
stiffness matrix
Compute p roots
– Roots are
either real
(positive or
negative)
– Complex
conjugate
pairs
0
Aij
1
M K
1
M B
I
1
2
Kij Kij V Qreal ,ij
2
{z j }= p {z j }=
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йAij щ{z j }
кл ы
ъ
31
Eigenvalue roots
pi pi ,real jpi ,imaginary
pi i g i j
k
b
V
kV
b
g
is the estimated system damping
There are “m” computed values of at the
airspeed V
You chose a value of k=b/V, was it correct?
– “line up” the frequencies to make sure k, and V
are consistent
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p-k computation procedure
Input k and V
Compute
eigenvalues
No, change k
ki kinput ?
yes
ki
pi i g i j
i b
V
preal ig i i
pimaginary i
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Repeat
process for
each
33
What should we expect?
0
Aij
1
M
K
M 1B
I
Right half-plane
Root locus plot
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Back-up slides for Problem 9.2
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A comparison between V-g and p-k
h2
0
2
h b 1
2 1 ig
x
2
0
r
2
1 Lh
M h
1
x
2
x h
b
r 2
1
h
L
a
Lh b 0
2
0
M
x h h2
0 h
b
b
2
2 2
r 0 r
V 2 1 Lh
2
2 V
M h
2
2
1
h
L
a
Lh b 0
2
0
M
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A comparison between V-g and p-k
1
2
x
x h h2
0 h
b
b
2
2 2
r 0 r
V 2 1 Lh
2
2
V
M h
2
1
2
x
2
1
h
L 2 a Lh b 0
0
M
x h h2
0 h
b
b
2
2 2
r 0 r
V 2k b Lh
2
2
b
m
M h
2
2
2
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h
L 2 a Lh b 0
0
M
37
A comparison between V-g and p-k
1
2
x
x h h2
0 h
b
b
2
2 2
r 0 r
V 2k b Lh
2
2
b
m
M h
2
1
x
2
2
1
h
L 2 a Lh b 0
0
M
2
x h h2
0 h
b
b
2
2 2
r 0 r
V 2 k Lh
2 m
M h
2
2
1
h
L
a
Lh b 0
2
0
M
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A comparison between V-g and p-k
1
p2
x
h
x
1
2
k
Lh
b
p
Vb
Imag
r 2
2
m
M h
2
2
2
0
V 2 k Lh
h
Re
2 2
2
m
0
r
M h
1
h
L 2 a Lh b
M
1
h
L
a
Lh b 0
2
0
M
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Flutter in action
Accident occurred APR-27-95 at STEVENSON, AL Aircraft: WITTMAN O&O, registration: N41SW
Injuries: 2 Fatal.
REPORTS FROM GROUND WITNESSES, NONE OF WHOM ACTUALLY SAW THE AIRPLANE,
VARIED FROM HEARING A HIGH REVVING ENGINE TO AN EXPLOSION. EXAMINATION OF
THE WRECKAGE REVEALED THAT THE AIRPLANE EXPERIENCED AN IN-FLIGHT
BREAKUP. DAMAGE AND STRUCTURAL DEFORMATION WAS INDICATIVE OF AILERONWING FLUTTER. WING FABRIC DOPE WAS DISTRESSED OR MISSING ON THE AFT
INBOARD PORTION OF THE LEFT WING UPPER SURFACE AND ALONG THE ENTIRE
LENGTH OF THE TOP OF THE MAIN SPAR. LARGE AREAS OF DOPE WERE ALSO MISSING
FROM THE LEFT WING UNDERSURFACE. THE ENTIRE FABRIC COVERING ON THE UPPER
AND LOWER SURFACES OF THE RIGHT WING HAD DELAMINATED FROM THE WING
PLYWOOD SKIN. THE DOPED FINISH WAS SEVERELY DISTRESSED AND MOTTLED. THE
FABRIC COVERING HAD NOT BEEN INSTALLED IN ACCORDANCE WITH THE POLY-FIBER
COVERING AND PAINT MANUAL; THE PLYWOOD WAS NOT TREATED WITH THE POLYBRUSH COMPOUND.
Probable Cause
AILERON-WING FLUTTER INDUCED BY SEPARATION AT THE TRAILING EDGE OF AN
UNBONDED PORTION OF WING FABRIC AT AN AILERON WING STATION. THE DEBONDING
OF THE WING FABRIC WAS A RESULT OF IMPROPER INSTALLATION.
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Things you should know
Royal Aircraft Establishment
The RAE started as HM Balloon Factory. From 1911-18 it was called the
Royal Aircraft Factory, but was changed its name to Royal Aircraft
Establishment to avoid confusion with the newly established Royal Air
Force.
Farnborough was known as a center of excellence for aircraft research.
Major flutter research was conducted there. Famous R&M’s such as the
“flutter bible” came from this facility. The RAE played a major role in both
World Wars. So confident was Hitler that he could occupy England with
relative ease that he spared the RAE from bombing in the hope of
benefiting from its research.
Recently the RAE (now known as the Royal Aerospace Establishment)
was absorbed into the DRA (Defence Research Agency), itself renamed
as DERA (Defence Evaluation and Research Agency). The world famous
initials are no more.
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