AAE 556
Aeroelasticity
Lecture 10
Swept wing aeroelasticity
Purdue Aeroelasticity
10-1
Swept wing static aeroelasticity
Read Sections 3.1 through 3.4
 Goals
– Show that bending deformation changes
aeroelastic loads on swept wings
– Develop simple aerodynamic and stiffness models
V
C

1
2
1
D
A
 B
1
sin 
Purdue Aeroelasticity
section
1-1
V
10-2
The letter N shows how bending
deformation changes air loads
V
C

1
2
1
D
A
 B
1
sin 
section
1-1
Purdue Aeroelasticity
V
10-3
Computing the lift on a flexible swept
wing “flying door” model
Use the airspeed component
normal to the wing to define
dynamic pressure
K1
d
o
f
K2

C
V
V cos 
C
B
A
b
c
B
A
Purdue Aeroelasticity
10-4
The wing structure idealization has two degrees of freedom,
both involve rotation that is resisted by two springs
representing wing torsional and bending stiffness
K  K1 f 2
K  K2d 2
K1
d
o
f
K2

C
V
V cos 
C
Purdue Aeroelasticity
B
A
A
b
c
B
springs resist upward
and downward wing
movement
10-5
We have two possible references for lift
Aerodynamic forces
are estimated by matching “downwash”
Streamwise
coordinates
are the
favorite of
aero people,
but I’m a
structures guy
What is “downwash?”
V

Vcos
y


ref
ere
nce
a
x
B
B
xis
A
x streamwise
direction
y
A
A-A chordwise view
c
z(x,y)
view A-A
Vcos
vn
B-B
freestream view
s
v
z(x,y)
view B-B
Purdue Aeroelasticity
v
vn
V
10-6
“Preserving” downwash velocity – angle of
attack equivalence
What angle of attack, when multiplied by YOUR
definition of dynamic pressure gives the correct
airloads?
v
 freestream    o   cos    sin 
V
o
v
 cos   sin 
 c   chordwise 



V cos  cos  cos 
cos 
 structural     tan 
( chordwise )
Purdue Aeroelasticity
10-7
The lift on the deformable wing model is
found by integrating aerodynamic forces
along the wing axis
qn  q cos 
2
Use the airspeed and
dynamic pressure
components normal to
the wing
 o

Lift  L  qn cbao 
    tan  
 cos 

Wing semi-span is b.
Purdue Aeroelasticity
10-8
Equations for bending and torsional static
equilibrium due to an initial angle of attack
“Bending” equilibrium about the x-axis ()
K1
d
o
f
K2

C
V
V cos 
C
B
A
K 2 d   K    ly  dy
2
b
b
c
B
A
o
b2   o

K  qn ccl
    tan  

 2
 cos 

Purdue Aeroelasticity
10-9
Twisting moment equilibrium about
y-axis ()
K1
d
o
f
K2

C
V
V cos 
C
B
A
b
K1 f 2  K     le  dy
c
B
A
 o

K   qn ccl eb 
    tan  
 cos 

torsional static equilibrium
Purdue Aeroelasticity
10-10
Define swept wing problem parameters and
write equations in matrix form
Q  qn cbcl  qn cbao t  tan 
static equilibrium equations with initial angle of attack
b
tb b 


 K 0    Q o  
 


 0 K     cos   2   Q  2 2   
 
  



e

t
e
e
 


or
tb b  
b


  Q o  
  K 0   Q  2


2  
2



  0 K 

    cos   

 te e  
e 
Purdue Aeroelasticity
10-11
Combine structural and aerodynamic
stiffness matrices - final result

Qtb

K


 
2 

Qte 

or
b
  Qb     Q 


o  
2

  
2
K  Qe    cos   e 
  Q o 
b 2 

K ij   
 

cos


 
e

 
Purdue Aeroelasticity
10-12
Compute the determinant (D) of the aeroelastic
stiffness matrix
Qbt 

2 bet
D  K   K 
K  Qe   Q
2 
2

bt


D  K K   Q  K
 K e 
2


Purdue Aeroelasticity
10-13
Divergence condition
D0
QD 
so that
K K
bt
eK  K
2
K
qD 
Q  qn cbao
Seao
  b  K
cos 1    
  e  K


2
 tan  


 2 


Only one divergence q for a 2 DOF system?
Purdue Aeroelasticity
10-14
Eliminate divergence?
Just sweep the wing
Set denominator equal to zero
 b  K
1  
 e  K
tan  crit
 tan  critical

0

2

 e  c  K
 2  
 c  b  K
  e  c  K
No divergence if   tan  2  
  c  b  K
 

1
Purdue Aeroelasticity




 


10-15
example
e  0.1
c
Critical sweep angle
vs.
stiffness ratio
sweep angle for divergence suppression
(degrees)
10
b  4,5,6
c
aspect ratio
b/c=4
8
aspect ratio
b/c=5
6
Torsional
stiffening is still
good
4
aspect ratio
b/c=6
2
0
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
Kphi/Ktheta
Purdue Aeroelasticity
10-16
example
qD 
K
K
3
qD
K

Seao
1
cos 2   10 sin  cos 
nondimensional divergence dynamic
pressure vs. wing sweep angle
b 6
c
nondimensional divergence
dynamic pressure
e  0.1
c
2.0
1.5
sweep back
sweep forward
1.0
5.72 degrees
0.5
0.0
-0.5
-1.0
-1.5
-2.0
-90 -75 -60 -45 -30 -15
0
15
30
45
60
75
90
sweep angle
(degrees)
Purdue Aeroelasticity
10-17
Summary
 Bending
deflection creates lift reduction
 Divergence does not occur if the sweep
angle is more than about 5 degrees
 Lift effectiveness is reduced
Purdue Aeroelasticity
10-18