4. ANKARA INTERNATIONAL AEROSPACE CONFERENCE
27-29 September, 2007 - METU, Ankara
AIAC-2007-109
AEROELASTIC ANALYSIS OF A TAPERED AIRCRAFT WING
S. Durmaz1, O. Ozdemir Ozgumus 2, M. O. Kaya3
Istanbul Technical University, Faculty of Aeronautics and Astronautics, 34469, Maslak, Istanbul, Turkey
ABSTRACT
In this study, aeroelastic flutter analysis of a tapered aircraft wing that is modeled as an Euler-Bernoulli
beam is carried out. Applying the Hamilton’s principle to the kinetic and the potential energy expressions, the
governing differential equations of motion and the boundary conditions are obtained. The desired results are
obtained by applying both one-term and two-term Galerkin Method. However, only the two-term Galerkin
Method is introduced here since it already covers the one-term method. Additionally, an example that studies
the Goland wing is taken from open literature in order to validate the calculated results and a very good
agreement between the results is observed.
WING MODEL
In this study, a tapered aircraft wing that is given in Fig.1 is considered. Here, the x -axis coincides with
the elastic axis, which is assumed to be straight, and the distance between the elastic axis and inertia axis at
any point x is denoted by y x  . The speed of the air flow relative to the wing denoted by U is assumed to
be constant.
U
Elastic Axis
x
y
Inertia Axis
x
y
Figure 1. Elastic axis and inertia axis for a tapered cantilever aircraft wing in steady air flow
The wing deformation can be measured by a bending deflection w in the z -direction and a rotation 
about the elastic axis. The angle  is referred to the local angle of attack. The bending deflection, w , is
positive downward while the rotation,  , is positive if the leading edge is up. The chordwise distortion will be
neglected. The combined bending and rotational vibration of the cantilever wing in steady air flow is shown in
Figs.2 (a) and 2 (b). Here the chord length is given by c x  2b x .

1

Graduated Student in Aeronautics Department, E-mail: durmazseher@yahoo.com
Research Assistant in Aeronautics Department, E-mail: ozdemirozg@itu.edu.tr
3
Associate Professor in Aeronautics Department, E-mail: kayam@itu.edu.tr
2
AIAC-2007-000
Durmaz, Ozdemir Ozgumus, Kaya
 U
x
w
y
x
c
z
w
y
z
(a)
(b)
Figure 2. (a) Bending deflection of the elastic axis (b) Rotation of the wing cross section
FORMULATION
Equations of Motion and the Boundary Conditions
The kinetic and the potential energy expressions of an Euler-Bernoulli Beam that experience bendingtorsion coupling are taken from Ref.[4]. Applying the Hamilton’s principle to these energy expressions, the
governing differential equations of motion and the boundary conditions are obtained as follows
Equations of Motion:
2
x 2


2w 
2w
 2
 EI


m

my
 L  0

x 2 
t 2
t 2

(1a)
 
 
2w
 2

I
M  0
 GJ
  my

x 
x 
t 2
t 2
(1b)
where the lift force, L  and the aerodynamic moment, M  are defined as follows

1
 
  U  ba
L'  2UbCk  w  U  b  a    b 2 w
2






(2a)


1

1
 
  U  ba 
M '  b  a 2UbC k  w  U  b  a    b 2 w
2

2
 




(2b)
1
1 a 
  U  b  
  b  w
8 2 
2
3
GJ is the torsional stiffness, m mass per unit length, I  is the mass
moment of inertia per unit length,  is the air density, C L is the local lift coefficient and t is the time.
Here, EI is the bending stiffness,
Boundary Conditions:
w
  0
x
At
x0
w
At
xL
w  w   2 w   2 w 

0
 EI
  2  EI 2   GJ
x  x  x  x 
x
(3a)
2
Ankara International Aerospace Conference
(3b)
AIAC-2007-000
Durmaz, Ozdemir Ozgumus, Kaya
Exponential Solution Function:
A sinusoidal variation of w( x, t ) and  ( x, t ) with a circular natural frequency
functions are approximated as exponential solutions.

is assumed and the
wx, t   wx e it
(4a)
 x, t    x e it
(4b)
where
wx   A1 f1 x eit  A2 f 2 x eit
(5a)
 x   B11 x eit  B2 2 x eit
(5b)
In this study, both one-term and two term Galerkin Methods are applied. However, only the two-term
Glerkin Method is introduced here since it already covers the one-term method.
Substituting Eqs. (4a)-(5b) into Eqs. (1a) and (1b) gives
 d 2  d 2 f1 

b2

A1  Z 2  EI

mf

2

C k if 1   b 2 f1  
1

2 
k
dx 
 dx 

 d 2  d 2 f2 

b2

A2  Z 2  EI

mf

2

C k if 2   b 2 f 2  
2

2 
k
dx 
 dx 



b3
b3
b3
1

B1  my  1  2 2 C k  1  2 C k   a i 1   i 1   b 3 a 1  
k
k
k
2





b3
b3
b3
1

B2  my  2  2 2 C k  2  2 C k   a i 2   i 2   b 3 a 2   0
k
k
k
2






1

1

A1   2 my f 1  2Ub 2 C k   a if 1   2 b 3   a  f 1   2   b 3 f 1  
2
2

2






1

1

A2   2 my f 2  2Ub 2 C k   a if 2   2 b 3   a  f 2   2   b 3 f 2  
2
2

2



 d 
d 1 
1

1
 1

B1   GJ
   2 I   1  2U 2 b 2 C k   a  1  2Ub 3 C k   a   a i 1 
dx 
2

2
 2

 dx 
1

1

1 a 
 b 3U   a i 1   2 b 4 a  a  1   b 3Ui 1   2 b 4    1  
2

2

8 2 
 d 
d 2 
1

1
 1

B 2   GJ
   2 I   2  2U 2 b 2 C k   a  2  2Ub 3 C k   a   a i 2 
dx 
2

2
 2

 dx 
1

1

1 a 
 b 3U   a i 2   2 b 4 a  a  2   b 3Ui 2   2 b 4    2   0
2

2

8 2 
where k 
b
U
and Z 
1
2
1  ig  .
3
Ankara International Aerospace Conference
(6a)
(6b)
AIAC-2007-000
Durmaz, Ozdemir Ozgumus, Kaya
Tapered Beam Formulas:
In this study, a beam model that tapers in one plane, xy plane, is considered. Therefore, the general
bx  , the cross-sectional area, Ax , the distance between the
elastic axis and inertia axis, y x  , the elastic axis location (positive rearward), ax  , the moment of inertia,
I y  x  , the polar moment J x and the mass moment of inertia per unit length I  of a beam, are given by
equations for the half of the veter length,
[2, 3].
x
x
x
x




bx   b0 1  cb  , a x   a 0 1  cb  , y  x   y 0 1  cb  , A x   A0 1  cb 
L
L
L
L




x
x
x



I y  x   I y 0  1  c b  , J  x   J 0  1  c b  , I   x   I  0 1  c b 
L
L
L



Here, the subscript
 0
denotes the values at the root section of the tapered beam and
(7)
cb represents the
breadth taper ratio which can be expressed as follows
cb  1 
b
b0
(8)
These tapered beam formulas are used in the related parts of the expressions given by Eqs. (6a) and
(6b).
APPLICATION OF THE GALERKIN METHOD
In this section, application procedure of the two-term Galerkin Method is described. Following the
procedure, Eq. (6a) is multiplied by f1 and the resultant expression is integrated from 0 to L which gives
the following equation
L
L
L
 L d 2  d 2 f1 

b2
2
2
2 2



A1   Z 2  EI
f
dx

mf
dx

2

C
k
if
dx


b
f
dx

1
1

1

1
0
0
0
dx 2 
k
 0 dx 

L
L
L
L
2
2
2


d  d f2 
b
2



 A2   Z 2  EI
f
dx

mf
f
dx

2

C
k
if
f
dx


b
f
f
dx

1
1
2

1
2

1
2
0
0
0
dx 2 
k
 0 dx 

L
L
 L
b3
b3
1

 B1   my 1 f1dx   2 2 C k 1 f1dx   2 C k   a i1 f1dx
k
k
2

0
0
 0
L
L
L

 L
b3
b3
3
   i1 f1dx    b a1 f1dx   B2   my  2 f1dx   2 2 C k  2 f1dx
k
k
0
0
0

 0
L
L
L

b3
b3
1

  2 C k   a i 2 f1dx    i 2 f1dx    b 3 a 2 f1dx   0
k
k
2

0
0
0

(9)
Writing the terms of Eq.(9) as matrix coefficients gives
i
i
1





A1  Za111  a112  C k  a113  a114   A2  Za121  a122  C k  a123  a124   B1  b111  C k  2 b112 
k
k
k





(10)
i
1
i



C k  b113  b114  b115   B2  b121  C k  2 b122  C k  b123  b124  b125   0
k
k
k



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Ankara International Aerospace Conference
AIAC-2007-000
Durmaz, Ozdemir Ozgumus, Kaya
where
d 2  d 2 f1 
 EI 2  f1dx
dx 2 
dx 
0
L
L
a111  
0
0
d 2  d 2 f2 
 f dx
a121   2  EI
2  1
dx
dx


0
L
L
a114   b f dx
2
L
a113   2b 2 f12 dx
a112   mf12 dx
2
1
0
L
L
a123   2b 2 f1 f 2 dx
a124   b 2 f1 f 2 dx
0
0
1

b113   2b   a 1 f1dx
2

0
L
L
b112   2b 1 f1dx
3
3
0
L
L
b115   b3a1 f1dx
b121   my  2 f1dx
0
0
1

b123   2b   a  2 f1dx
2

0
L
3
L
b124   b  2 f1dx
3
0
As described above, Eq. (6a) is multiplied by
which gives the following equation
L
a122   mf1 f 2 dx
0
L
b111   my 1 f1dx
0
L
b114   b31 f1dx
0
L
b122   2b3 2 f1dx
0
L
b125   b3a 2 f1dx
0
f 2 and the resultant expression is integrated from 0 to L
i
i
1





A1  Za 211  a 212  C k  a 213  a114   A2  Za 221  a 222  C k  a 223  a 224   B1  b211  C k  2 b212 
k
k
k





(11)
i
1
i



C k  b213  b214  b215   B2  b221  C k  2 b222  C k  b223  b224  b225   0
k
k
k



Eq. (6b) is multiplied by
following equation
1 and the resultant expression is integrated from 0 to L which gives the
i
i




A1  a 311  C k  a 312  a 313  a 314   A1  a 322  C k  a 322  a 323  a 324   B1  b311  b312 
k
k




1
i
i
i
1


C k  2 b313  C k  b314  b315  b316  b317  b318   B 2  b321  b322  C k  2 b323 
k
k
k
k
k


(12)
i
i
i

C k  b324  b325  b326  b327  b328   0
k
k
k

Eq. (6b) is multiplied by
following equation
 2 and the resultant expression is integrated from 0 to L which gives the
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Ankara International Aerospace Conference
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Durmaz, Ozdemir Ozgumus, Kaya
i
i




A1  a 411  C k  a 412  a 413  a 414   A2  a 422  C k  a 422  a 423  a 424   B1  b411  b412 
k
k




1
i
i
i
1


C k  2 b413  C k  b414  b415  b416  b417  b418   B 2  b421  b422  C k  2 b423 
k
k
k
k
k


(13)
i
i
i

C k  b424  b425  b426  b427  b428   0
k
k
k

Eqs. (10)-(13) can be written in a 4  4 matrix form as follows :
i

 Za111  a112  C k  k a113  a114

 Za  a  C k  i a  a
213
114
 211 212
k

 a311  C k  i a312  a313  a314

k

i
 a411  C k  a412  a413  a414
k

i
Za121  a122  C k  a123  a124
k
i
Za221  a222  C k  a223  a224
k
i
 a322  C k  a322  a323  a324
k
i
 a422  C k  a422  a423  a424
k
 b111  C k 
1
i
b  C k  b113  b114  b115
2 112
k
k
1
i
 b211  C k  2 b212  C k  b213  b214  b215
k
k
1
i
i
i
 b311  b312  C k  2 b313  C k  b314  b315  b316  b317  b318
k
k
k
k
1
i
i
i
 b411  b412  C k  2 b413  C k  b414  b415  b416  b417  b418
k
k
k
k
 b121  C k 
1
i
b  C k  b123  b124  b125
2 122
k
k
1
i
 b221  C k  2 b222  C k  b223  b224  b225
k
k
1
i
i
i
 b321  b322  C k  2 b323  C k  b324  b325  b326  b327  b328
k
k
k
k
1
i
i
i
 b421  b422  C k  2 b423  C k  b424  b425  b426  b427  b428
k
k
k
k


 A   0 
 1   
  A2   0 
 B    0 
 1   
  B2   0 



(14)
SOLUTION PROCEDURE
Determinant of Eq.(14) is taken and made equal to zero. Solving this characteristic equation for a
certain k value, four roots which are Z 1 , Z 2 , Z 3 , Z 4 , are found. If the imaginary part of any root is
positive, the wing experience flutter.
Assuming that the imaginary part of the second root,
Z 2 , is positive and knowing that Z 
1
2
1  ig  ,
it can be written that
Z 2    i 
1
2
1  ig 
(15)
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Ankara International Aerospace Conference
AIAC-2007-000
Durmaz, Ozdemir Ozgumus, Kaya
Considering Eq.(15), the frequency value at which flutter occurs is calculated as follows
1

2
    flutter 
1

rad. / sec.
(16)
Using the calculated flutter frequency and knowing that k 
k
b
U
 U flutter 
 flutterb
k
b
U
, the flutter speed is found as follows
ft / sec.
(17)
RESULTS
The computer package Mathematica is used to write a computer program for the expressions obtained in
the previous sections. Calculation of the flutter speed is made both for a uniform and for a tapered wing
model. The effects of several parameters are examined and in order to validate the calculated results, datas
of the Goland’s wing whose parametric values are given below are used.
GJ 0  2.39  10 6 lb ft2
EI 0  23.6  10 6 lb ft2
I  0  1.943 slug ft
a0  1 / 3
b0  3 ft
y 0  0.19973 ft
L  20 ft
m  0.746 slug/ft
Firstly, the flutter frequency and the flutter speed of a uniform Goland wing are found as
k  0.47 and
U flutter  446.462 ft/sec. which are very close to the ones found by Ref.[5]. In order to validate the
calculated results, a comparison is made in Table 1.
Table 1. Flutter Speed, Flutter Frequency and Reduced Frequency of a Uniform Goland Wing
Flutter Speed
Flutter Frequency
 f rad / sec 
Reduced Frequency
(k )
447
69
0.47
One-Term
Galerkin
443.610
69.883
0.4727
Two-Term
Galerkin
446.462
70.035
0.4706
U f  ft / sec 
Analytical Study
(Ref. [4])
Present
Study
In Fig. 3, variation of both the bending and the torsion frequencies of a uniform wing with respect to the
free stream velocity, U is introduced. Here it is noticed that as the free stream velocity increases, the
frequencies get closer to each other which increases the possibility of flutter.
In Fig. 4, variation of the flutter speed, U f with respect to the location of the elastic axis,
a , is
introduced. Here it is noticed that as the elastic axis gets closer to the leading edge (forward), it gets more
possible for the wing to get into flutter because the flutter speed gets lower.
In Fig. 5, variation of the flutter speed, U f with respect to the taper ratio,
cb is introduced. Here it is
noticed that as the taper ratio increases, the flutter speed increases which makes it less possible for the wing
to experience fluttr.
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Ankara International Aerospace Conference
AIAC-2007-000
Durmaz, Ozdemir Ozgumus, Kaya
100
80
Bending Frequency
60
Torsional Frequency
40
20
U
200
400
Figure 3.
Uf
600
800
U   plot
800
700
600
500
400
300
-0.4
-0.2
0
Figure 4. Elastic Axis Location, a , and Flutter Speed, U f , Relation
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Ankara International Aerospace Conference
0.2
a
AIAC-2007-000
1200
Durmaz, Ozdemir Ozgumus, Kaya
Uf
1000
800
600
400
cb
0.0
0.2
0.4
0.6
0.8
REFERENCES
[1] Ozdemir Ozgumus, O. and Kaya, M.O., Energy expressions and free vibration analysis of a
rotating double tapered Timoshenko beam featuring bending–torsion coupling, International Journal of
Engineering Science, 45, 562-586, 2007.
[2] Özdemir Ö. ve Kaya M.O., Flapwise bending vibration analysis of a rotating tapered cantilevered
Bernoulli-Euler beam by differential transform method, Journal of Sound and Vibration, 289, 413–420, 2006.
[3] Hodges, D.H. ve Pierce, G.A., Introduction to Structural Dynamics and Aeroelasticity, Cambridge
Aerospace Press, 2002.
[4] Lin, J. And Lliff, K.W., Aerodynamic Lift and Moment Calculations Using A closed Form Solution of
The Possio Equation, NASA, 2000
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Ankara International Aerospace Conference