Transcript AAE556 lecture 4 - Purdue University

AAE 556
Aeroelasticity
Lecture 12
Flexural axis
Control effectiveness
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1
Flexural axis (FA) concept
an attempt to explain aeroelastic effects
FA Definition - a line (locus of points) along which the
streamwise (or chordwise) angle of attack does not change
when a discrete load is applied there (and the air is off).
plus
 E     tan  0
upward load
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yo
xo
minus
2
What is the difference between flexural axis and elastic axis?
The air is off
b
x
upward
load
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   tan   0
y
 0
3
Locate the flexural axis by choosing an arbirary y
position and poking around the wing in the
x-direction
 K 0    M  
M    Pyo 

 0 K    M 




  

 
M
 Pxo




yo
xo
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Step #1
Compute angular displacements
  1  K
  
    0
0   Pyo 



K   Pxo 
 yo 
 K 
 
P  K yo 

 

  P x 
  K K  o  K xo 
 o 
 K 
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Use flexural axis definition to
solve for the x,y coordinates
 E     tan  0
xo
yo
 E  P( 

tan)  0
K K
 yo 
 K 
 
P  K yo 

 

  P x 
  K K  o  K xo 
 o 
 K 

 xo   K
    
tan    tan b
K

y
 o  

The flexural axis location is always forward of the elastic axis if the wing is swept
back. Increased sweepback rotates the flexural axis forward and increases the
distance between aero load and the flexural axis. The red dot is the position of
the resultant lift force. The dashed green line is the flexural axis. The dashed
green line is the reference for wing sweep.
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Example calculation
dependence of flexural axis angle
on stiffness ratio – notice that when the bending stiffness is
infinite then the flex axis and the elastic axis are identical
flexural
axis
degrees
inangle
angle
Flexural axis
degrees

 xo   K
    
tan    tan b
K

y
 o  

K
1
K
45
30
K
2
K
15
K

K
0
K
3
K
-15
-30
-45
-45
-30
Wing
-15
0
15
sweep angle
sweepdegrees
angle in
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30
45
degrees
7
How much wing sweep does it take to
exactly place the flexural axis on the
aero load position at y = b/2 and x = -e?

 xo   K
    
tan    tan b
K

y
 o  


 2e   K
tan  special   tan b
    

 b   K

tan  special
2e K

b K
tan  crit
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 e  c   K 
 2    

 c  b   K 
8
Switch gears - Control effectiveness
rigid wing/springs model
Lift  L  qn Sao    tan  qn Scl  o
 K11

 K 21


K12   


q
Sea

  n
o o
K 22   
 cl
 cl
 
cl b
cl 2e
 c cmac 
1 
 e cl









V
Vco
s
Q  qn Scl
a ile
r on

b tan 
K11  K  Q
2
K 21  Qe tan 
K12

b
 Q
2
K 22  K  Qe
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Invert the aeroelastic stiffness matrix
Then solve for the two deflections
Qe o

K 22 c1  K12 c2


Qe o

 K 21c1  K11c2




b tan 


  K K  Q K
 K e 
2


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cl b
c1 
cl 2e
cl  c cmac
1 
c2 
cl  e cl




Q  qn Scl
10
Swept wing lift due to full-span
aileron deflection
L  qn Sao    tan   qn Scl  o
cl 

 K Qec2  K Qec1 tan    
c
l


L  qn Scl
o


 b tan 

 K e  
 K K  Q  K

2




Q  qn Scl
 b tan 

  K K  Q  K
 K e 
2


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Lift expression algebra
 q cos2 Scl ccm
q cos2 Scl  o

1 
L
Q 
b tan 
K cl

1
 K e  
 K
K K 
2

Scl ccmac
q(cos2 )Scl  o 
2
L
1  q cos 

Q
K cl

1
QD
qSecl
Q

QD
K








 K b tan   2
1 
 cos 
 K 2e 
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12
Aileron effectiveness measured
as aileron lift effectiveness
Lrigid  (qScl cos ) o
2
L flex
Lrigid

 Secl
1  q 

 K


  c   cmac
   
  e   cl
q
1
qD
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 2 
 cos  



13
The reversal dynamic
pressure solution
 K
qR   
 Secl


  e   cl
 
  c   cmac



 e   cl
qR  qDo   
 c   cmac
 1 

  cos 2  

 1 

  cos 2  

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Example - effects of sweep on aileron effectiveness
unswept wing divergence at 250 psf. (qo)
Reversal at 150 psf.
1.00
unswept
aileron effectiveness
0.75
wing
0.50
45o sweep
0.25
15o sweep
0.00
-0.25
0
50
30o sweep
100 150 200 250 300 350 400 450 500
airspeed (ft/sec.)
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Summary-swept wing
control effectiveness
 wing
sweep changes the divergence dynamic
pressure and lift effectiveness
 sweepback stabilizing - slight sweep only
 Sweep forward is aeroelastically destabilizing
 sweepback reduces lift effectiveness and
control effectiveness
 Sweep forward increases lift effectiveness
and control effectiveness
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Homework for next week
2
K2
K1
1
W
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