Flutter Analysis- p-k Method

1. AAE556
Lectures 34,35
The p-k method, a modern
alternative to V-g
Purdue Aeroelasticity 1

2. Genealogy of the V-g or “k”
method
 Equations of motion for harmonic response (next
slide)
– Forcing frequency and airspeed are known parameters
– Reduced frequency k is determined from w and V
– Equations are correct at all values of w 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 invented V-g artificial damping to create an iterative
approach to finding the flutter point
Purdue Aeroelasticity 2

3. Equation #2, moment equilibrium
2
2 2 2 2 2
0
h
M M
h h
x r r
b b
     
w
w w  w  

   
     
   
   
 
2
1 1
2 2
h h
M M a L M a L
  
   
     
   
   
1 1
2 2
h h
M a L

 
  
 
 
3
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Divide by w2
2 2
2
2
1
0
h
h h
x r r M M
b b
    


  

w
w
   
     
   
   
Include structural damping
 
2 2
2
2
1
0
1 h
ig
h h
x r r M M
b b
    


w
 
w

   
     
   
  



4. The eigenvalue problem
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4
2
2
2
2
2
0 1
0
1
0
1
2
0
h
h h
h
h h
x
b b
x r
r
h
L L a L
b
M M


 


 
w
w
 
 
 
     
 
     
 

   
   
 
 
   
     
 
 
 
   
   
 
 
 
 
 
     
 
 
 
 

 
 
 
2
2
2
2
2 1
1 1
0
1
0
2
h
h
h h
h h
L L a L
x
b b
x r
M M
r


 





w
w
 
 
 
 

 
 
 
 
 
 
 
 
 


   
 
 
 
   
 
 
 
   
   
 
 
 
 
 
   
 
   
 
 
 



5. Return to the EOM’s before we
assumed harmonic motion
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Here is what we would like to have
The first step in solving the general stability problem
     
 
 
   
   
1
2
2 3
2
0
ij j ij j ij j
ij j ij j
p M K A
p A p A
  
 
 
   
 
     
   
  
   
     
   
   
   
1 2 3
0
ij j ij j ij j ij j ij j
M K A A A
    
     
   
    
         
    pt
j j e
 
 p j
 w
 
25-5

6. The p-k method casts the flutter
problem in the following form
       
2 2
1
0
2
ij ij ij ij
p M p B K V A
   
 
       
   
       
 
 
 
    pt pt
h
b
t e e
 

 
 
   
 
 
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…but first, some preliminaries
p j
 w
 
6

7. Setting up an alternative solution
scheme
h
x K
h h P
b b m b mb


  
2 2 2
a
x I K M
h
b b mb mb mb
  
 
  
2
2 2
1 0
0
h
a
K P
h
h
mb
m
b
b
I
K M
mb
m
x
b
x
mb






 
 
 
     
 
     
   
 
     
   
     
 
       
   
7
Purdue Aeroelasticity

8. The expanded equations
 
2
2 2
2
2
4 2
2
1 0
0
1 0
0
1
2
1
2
h
a
h
h h
K P
x h
h
mb
m
b
b
I
K
x M
mb
mb mb
K
x h
h
m
b
b
I
K
x
mb
mb
L L a L
b
mb













 w
 
 
 
     
 
     
   
 
     
   
     
 
       
   
 
 
   
 
   
    
   
   
   
 
     
 
 
 
 
 
 
 
 
 2
1 1 1 1
2 2 2 2
h h
a L a
h
b
M L L a
 

 
 
 
 
 
 
      
 
      
 
      
   
 
     
 
 
 


  





8
Purdue Aeroelasticity

9. Break into real and imaginary
parts
 
 
3 2
2
3 2
1
2
1 1 1 1 1
2 2 2 2 2
1
2
Real
1 1 1 1 1
2 2 2 2 2
h h
h h
h h
h h
L L a L
b
mb
a L M L a L a
L L a L
b
mb
a L M L a L a

 

 
 w
 w
 
 
 
 
 
 
 
 
 
 
 
 
 
      
 
      
 
      
   
 
      
   
 
 
 
 
 
 
 
 

 
    
      
    
 
    
 
 
2
3 2
2
1
2
Imag
1 1 1 1 1
2 2 2 2 2
h h
h h
L L a L
b
j
mb
a L M L a L a

 
 w
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
   
 
  
   
 
 
      
   
 
      
 
      
 
 
 
 
      
   
 
 
9
Purdue Aeroelasticity

10. Recognize the mass ratio
 
 
2
2
2
1
2
Real
1 1 1 1 1
2 2 2 2 2
1
2
Imag
1 1 1 1
2 2 2
h h
h h
h h
h
L L a L
a L M L a L a
L L a L
j
a L M L

 

 
w

w

 
 
 
 
 
 
 
 
 
 
 
 
 
  
 
 
 
      
 
 
      
 
      
 
 
 
 
      
   
 
 
 
 
 
 
 
 
 
 
 
 
   
 
   
   
 
   
 
2
1
2 2
h
a L a
 
 
 
 
 
 
 
 
 
   
 
 
  
 
   
 
 
 
   
 
 
 
10
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11. Multiply and divide real part by dynamic
pressure
Multiply imaginary part by p/jw
 
 
2
2
2
2
2
1
2
1 2
Real
2 1 1 1 1 1
2 2 2 2 2
1
2
Imag
1 1
2 2
h h
h h
h h
L L a L
k
V
b
a L M L a L a
L L a L
p
j
j

 



w
 w
 
 
 
 
 
 
 
 
 
 
 
   
   
  
 
   
 
   
      
   
 
      
 
      
 
 
 
 
      
   
 
 
 
 
 
 
 
 
 
  
   
 
 
 
2
1 1 1
2 2 2
h h
a L M L a L a
 
 
 
 
 
 
 
 
 
 
 
      
 
 
    
 
      
 
 
 
 
      
   
 
 
11
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12. Multiply and divide imaginary
part by Vb/Vb
 
 
2
2
2
2
1
2
1 2
Real
2 1 1 1 1 1
2 2 2 2 2
1
2
Imag
1 1
2 2
h h
h h
h h
L L a L
k
V
b
a L M L a L a
L L a L
V k
p
b
a

 




 
 
 
 
 
 
 
 
 
 
 
   
   
  
 
   
 
   
      
   
 
      
 
      
 
 
 
 
      
   
 
 
 
 
 
 
 
 
 
 
 
   
   
 
2
1 1 1
2 2 2
h h
L M L a L a
 
 
 
 
 
 
 
 
 
 
 
     
 
 
    
 
      
 
 
 
 
      
   
 
 

13. Define Aij and Bij matrices
 
 
2 2
2
2
1
2
Real
1 1 1 1 1
2 2 2 2 2
1
2
Imag
h h
ij
h h
h h
ij
L L a L
V k
A
b
a L M L a L a
L L a L
V k
B
b

 



 
 
 
 
 
 
 
 
 
 
 
    
 
    
  
   
 
 
      
    
 
      
 
      
 
 
 
 
      
   
 
 
 
 
 
 
 
 
 
 
 
     
 
  
2
1 1 1 1 1
2 2 2 2 2
h h
a L M L a L a
 
 
 
 
 
 
 
 
 
 
 
      
 
 
      
 
      
 
 
 
 
      
   
 
 

14. Place aero parts into EOM’s
Note the minus signs
 
 
2 2
2
1
2
Real
1 1 1 1 1
2 2 2 2 2
1
2
Imag
h h
ij
h h
h
ij
L L a L
V k
A
b
a L M L a L a
L L a L
V k
B
b

 



 
 
 
 
 
 
 
 
 
 
   
   
   
      
 
 
   
   
   
      
   
   
      
 
      
 
 
 
 
      
   
 
 
 
 
 
 
 
 
     
 
  
2
1 1 1 1 1
2 2 2 2 2
h
h h
a L M L a L a
 
 
 
 
 
 
 
 
 
 
 
 
 
 
      
 
 
      
 
      
 
 
 
 
      
   
 
 
2 0
0
ij ij ij ij
h
b
p M p B K A

   
 
 
     
   
   
     
 
 
 
 

15. What are the features of the new
EOM’s?
We still need k defined before we can
evaluate the matrices
Airspeed, V, appears.
The EOM is no longer complex
We can calculate the eigenvalue, p, to
determine stability
2 0
0
ij ij ij ij
h
b
p M p B K A

   
 
 
     
   
   
     
 
 
 
 

16. The p-k problem solution
 Choose k=wb/V arbitrarily
 Choose altitude (, and airspeed (V)
 Mach number is now known (when appropriate)
 Compute AIC’s from Theodorsen formulas or others
 Compute aero matrices-Bij and Aij matrices are real
 Convert “p-k” equation to first-order state vector form
2
2
0
0
0
0
ij
ij ij ij ij
ij ij K
h
b
p M p B K A
h
b
p M p B


   
 
 
     
   
   
     
 
 
 
 
   
 
 
   
  
   
   
  
 
 
 
 

17. A state vector contains
displacement and velocity “states”
   
j j
velocity vector v x
 
{ } j
j
j
x
z
v
ì ü
ï ï
ï ï
= í ý
ï ï
ï ï
î þ
State vector =
 
j
displacement vector x

Purdue Aeroelasticity

18. Relationship between state
vector elements
{ } { }
j j
x v
=
{ } { } { } { }
0
ij j ij j ij j
M v B v K x
é ù é ù é ù
- + =
ê ú ê ú ê ú
ë û ë û ë û
{ } { } { }
{ }
1 1
1 1
j ij ij j ij ij j
j
j ij i ij i
j j
j
v M K x M B v
x
v M K M B
v
- -
-
-
é ù é ù é ù é ù
= - +
ê ú ê ú ê ú ê ú
ë û ë û ë û ë û
ì ü
ï ï
é ù
é é ù
é ù é ù
ê ú
ê ú ê ú
ë û ë û
ë û
ù ï ï
é ù é ù
= - í ý
ê ú
ê ú
ê ú ê ú
ë û ë û ï ï
ë û
ë û
ï ï
î þ
An equation of motion with
damping becomes
Purdue Aeroelasticity

19. Use an identify relationship
for the other equations
Purdue Aeroelasticity
19
{ } [ ][ ]{ }
0 0 1 0
0
0 0 0 1
j j
j i
j j
x x
z I z
v v
ì ü ì ü
é ù
ï ï ï ï
ï ï ï ï é ù
ê ú
= = =
í ý í ý ë û
ê ú
ï ï ï ï
ë û
ï ï ï ï
î þ î þ

20. State vector eigenvalue
equation
{ }
[ ] [ ]
{ }
1 1
0
j j
j ij j
j j
I
x x
z Q z
v v
M K M B
- -
é ù
ì ü ì ü
ï ï ï ï
ê ú
ï ï ï ï é ù
= = =
í ý í ý ê ú
ê ú ë û
é ù
ï ï ï ï
-
ê ú
ï ï ï ï
ê ú
î þ î þ
ë û
ë û
z(t)
   z
 est
Assume a solution
Result
Solve for eigenvalues (p) of the [Q] matrix (the plant)
Plot results as a function of airspeed
{ } { } { }
j j ij j
z p z Q z
é ù
= = ê ú
ë û
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21. 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
1 1
0
ij
ij ij ij
I
Q
M K M B
K K A
 
 
    
  
 
 
     
 
     
 
{ } { } { }
j j ij j
z p z Q z
é ù
= = ê ú
ë û
Purdue Aeroelasticity

22. Eigenvalue roots
 wg is the estimated system damping
 There are “m” computed values of w at the
airspeed V
 You chose a value of k=wb/V, was it correct?
– “line up” the frequencies to make sure k, w and V
are consistent
 
real imaginary
p p jp
p j
w g
 
 
Purdue Aeroelasticity

23. Procedure
Input k and V
Compute
eigenvalues
 
i i i
p j
w g
 
i
i
b
k
V
w

?
i input
k k 
 
yes real i i
imaginary i
p
p
w g
w


Repeat
process for
each w
No, change k
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24. P-k advantages
Lining up frequencies eliminates need
for matching flutter speed to Mach
number and altitude
p-k approach generates an
approximation to the actual system
aerodynamic damping near flutter
p-k approach finds flutter speeds of
configurations with rigid body modes
Purdue Aeroelasticity