AAE556-Lecture 34,35 p-k flutter-1

AAE 556
Aeroelasticity
The P-k flutter solution method
(also known as the “British” method)
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1
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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
1
    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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3
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  r2 1  2

Mh  1
 M  L


  a 
 2

 
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1
 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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8
Return to the EOM’s before we
assumed harmonic motion
Here is what we would like to have
 M ij   j    Kij   j    Aij1   j    Aij 2   j    Aij3   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    Aij1   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
   it
P   L   b   Lh     L    a  Lh    e
2
  
  b  
3
2


1
   h b  it
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
   it
P   b   Lh     L    a  Lh    e
2
  
  b  
3
2
V 2 
h  
  it
1


3 2
P   2 b   Lh     L    a  Lh  e
2
  
V 
 b 
V 2
b 2 2
P
b2  2
2
V
k
 h  
1
   it
 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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
14
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


Purdue Aeroelasticity
25
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
Purdue Aeroelasticity



26
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
Purdue Aeroelasticity
27
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
Purdue Aeroelasticity
28
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
Purdue Aeroelasticity
29
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
Purdue Aeroelasticity
30
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 }=
Purdue Aeroelasticity
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
Purdue Aeroelasticity
32
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
Purdue Aeroelasticity
Repeat
process for
each 
33
What should we expect?
 0
 Aij   
1

M
K


M 1B 
I

Right half-plane

Root locus plot
Purdue Aeroelasticity
34
Back-up slides for Problem 9.2
Purdue Aeroelasticity
35
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 

Purdue Aeroelasticity
36
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
Purdue Aeroelasticity

1
   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 

Purdue Aeroelasticity
38
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 

Purdue Aeroelasticity
39
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.
Purdue Aeroelasticity
40
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.
Purdue Aeroelasticity
41