AAE556 Lecture 33 Vg Method revisited

AAE556
Lecture 33
V-g Method revisited

Purdue Aeroelasticity

1


Final EOM’s for forced response

h 
Fo
A   + B θ =
mb
b

ω is known because we pre-select it

ω Lh
2
2
A=−
− ω + ωh
µ
2

ω
B = −ω xθ −
µ
2

2


1
 
 Lα −  2 + a  Lh 

 

2

Purdue Aeroelasticity


Moment equilibrium equation

h 
D  ÷+ Eθ = 0
b
2
2
ω
ω
1

2
D = −ω xθ −
Mh +
Lh  + a 

µ
µ 2

2
2
ω
1
ω


E = −ω 2 rθ2 + ωθ2 rθ2 +
Mh + a −
Mα
µ
2
 µ
2
ω2  1
 ω
1

+
Lα  + a  −
Lh  + a 
µ
2
 µ
2


2

3
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Theodorsen’s method
The system is self-equilibrating

h 
Fo
A  ÷+ Bθ =
=0
mb
b

ω Lh ω ω
=− 2 − 2 +
ω µ ω ω
2

ATM

BTM

2

ω2
ω2
= − 2 xθ −

2
ω
µω

2
h
2


1
 
 Lα −  2 + a ÷Lh 

 

4

Purdue Aeroelasticity


Moment equilibrium equation

h 
D  ÷+ Eθ = 0
b
ω
ω
ω
1


= − 2 xθ −
Mh +
L
+ a÷
2
2 h
ω
µω
µω
2

2

DTM
ETM

2

2

2
ω 2 2 ωθ2 2 ω 2
1
 ω
= − 2 rθ + 2 rθ + 2 M h  + a ÷− 2 M α
ω
ω
ω µ
2
 ω µ

2

ω
1
 ω
1

+ 2 Lα  + a ÷− 2 Lh  + a ÷
ω µ 2
 ω µ 2

2

2

5
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Eigenvalue Equation of Motion #1
2
 h 
h
h
ω
1
 
2
2
2

−ω
− ω xθθ + ωh −
 Lh +  Lα −  + a ÷Lh  θ
b
b µ  b 
2
 
Divide by ω


÷= 0


2

ωh2 h 1  h 
h
1
 
− − xθ θ + 2 −  Lh +  Lα −  + a ÷Lh  θ
b
ω b µ b 
2
 


÷= 0


Include structural damping

 ωh2 
h
h 1 h 
1
 
− − xθ θ +  2 ÷( 1 + ig ) −  Lh +  Lα −  + a ÷Lh  θ
b
b µ b 
2
 
ω 
6
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
÷= 0



Equation #2, moment equilibrium
2



h
ω
h
2
2 2

2 2
−ω xθ  ÷− ω rθ θ + ωθ rθ θ −
 M θθ θ + M θ h ÷ = 0
µ 
b
b
2

M θθ

1

1

= M α −  + a ( Lα + M h ) +  + a  Lh
2

2

Divide by ω

Mθ h

1 1

= −  + a ÷Lh
2 2


2

h  2
ωθ2 2
1
h
− xθ  ÷− rθ θ + 2 rθ θ −  M θθ θ + M θ h ÷ = 0
ω
µ
b
b
Include structural damping

h  2
ωθ2
1
h
2
− xθ  ÷− rθ θ + 2 ( 1 + ig ) rθ θ −  M θθ θ + M θ h ÷ = 0
ω
µ
b
b
7
Purdue Aeroelasticity


Matrix equations
 ωh2  h 1  h 
 ωθ2 
h

1
 
− − xθ θ +  2 ÷( 1 + ig )  2 ÷ −  Lh +  Lα −  + a ÷Lh  θ
b
2
 
ω 
 ωθ  b µ  b 


÷= 0


h  2
ωθ2
1
h
2
− xθ  ÷− rθ θ + 2 ( 1 + ig ) rθ θ −  M θθ θ + M θ h ÷ = 0
ω
µ
b
b
2




ω
h

2
0
 ωθ 
 2 ÷
 h b   1
−  2 ÷( 1 + ig )  ωθ 
   + x
ω


2
 0
 θ   θ
r
θ 



1  Lh
+ 
µ
 M θ h

xθ  h 
b
2
rθ   θ 
 



1
   h 
L
−
+
a
÷Lh    b  0 
 α 2

    =  

0 
θ





M θθ

8

Purdue Aeroelasticity


The eigenvalue problem
 ωh2 

ω 
 2 ÷ 0  h b   1

−
 + 
÷( 1 + ig )  ωθ 

ω 
θ   xθ
2
 0

rθ 

2
θ
2


1  Lh
+ 
µ
 M θ h
 ωθ2 

0

h   ω 2 ÷
  1
Ω 2  b  =  h 
 x



1
 θ   0
 θ

rθ 2  


xθ   h 
b
2
rθ   θ 
 


1
   h 
L
−
+
a
÷Lh    b  = 0 
 α 2

     

0
  θ   
M θθ



xθ  1  Lh
+
rθ 2  µ 
 M θ h


1
    h 
 Lα −  2 + a ÷Lh     b 

   

   θ 
M θθ

9
Purdue Aeroelasticity


Another look at it
This should be easy for a 6

 ωθ2 

0

h   ω 2 ÷
  1
Ω 2  b  =  h 
 x



1
 θ   0
 θ

rθ 2  



xθ  1  Lh
+
rθ 2  µ 
 M θ h

th

grader with MATLAB


1
    h 
 Lα −  2 + a ÷Lh     b 

   

   θ 
M θθ

  ωθ2   Lh   ωθ2  

Lα Lh  1
  
−  + a ÷÷ 
1 + ÷  2 ÷ xθ +
h    ωh 2 ÷
µ   ωh  
µ µ 2
     h 


2

b
Ω  b=
 
1 

Mθ h 
1  2 M θθ 
 θ   
 θ 


x
+
r
+
÷
2  θ
  rθ 2  θ

µ ÷
r
µ
 


θ 

10
Purdue Aeroelasticity


The flutter problem – a complex eigenvalue with flutter frequency and airspeed
unknown

a = elastic axis location (shear center)

ωh

= bending-torsion frequency ratio
ωθ
S
xθ = θ
= dimensionless static unbalance
mb
rθ = dimensionless radius of gyration about SC
µ =density ratio
ω = frequency
k=reduced frequency

11
Purdue Aeroelasticity