1. International Conference on Engineering and Technology of Machinery, VETOMAC IV
AERO-ELASTIC ANALYSIS OF STIFFENED COMPOSITE WING
STRUCTURE
B. Pattabhi Ramaiahc ∗ , B. Rammohan∗ , Vijay Kumar. S∗ , D. Satish Babu∗ ,
R. Raghunathan•
• Group Director, Sc’G’, Aeronautical Development Establishment, Bangalore.
∗ Scientists, Aeronautical Development Establishment (ADE), DRDO, Bangalore.
c
correspondence author: pattabhib@yahoo.com
ABSTRACT
The interaction of the Elastic, Inertia and Aerodynamic forces is a dynamic phenomena
resulting in flutter. Dynamic Aeroelasticity is critical for a high-speed subsonic class of
Aerial Vehicles. In the present work, dynamic Aero-elastic analysis of the Unmanned
Aerial Vehicles (UAVs) has been studied for a stiffened composite structure. The flutter
speed and the corresponding flutter frequencies are computed using the Velocity-
Damping (V-g) method. The V-g method is employed to estimate the flutter speed and
flutter frequencies, for a high-speed subsonic composite wing structure, designed and
developed at ADE. Also the improvement of the flutter frequencies over the existing
metallic wing structure is discussed. The work can be further extended to develop an
optimized composite structure with higher margins in flutter speeds.
Key words: Aeroelasticity, flutter, Velocity-damping method.
1. Introduction
The prime objective of the structural engineer is to design an airframe whose flight
envelope is limited by engine power rather than its structural limitations [1]. The
computation of the flutter speed for rectangular wing is discussed in Ref. [2]. A computer
program for the flutter analysis including the effects of rigid-roll and pitch of swept wing
subsonic aircraft is given in Ref. [3]. Interactive software for wing flutter analysis was
developed including the effects of change in Mach number, dynamic pressure, torsional
frequency, sweep and mass ratio in Ref. [4]. In the present work an attempt is made to
estimate the flutter speed using the well-known Velocity-damping (V-g) method for the
unmanned aerial vehicles inputting uncoupled bending and torsional frequencies
estimated from the numerical codes. A computer code is developed to extract the
complex Eigen value for the estimation of the flutter speed of the UAVs. Considerable
improvement in the flutter speed is seen compared to the existing wing structure.
2. List of Symbols
V = Critical flutter speed in fps
Kθ = Wing torsional stiffness
(antisymmetric) measured at 0.7S
S = Wing semi span in ft
D = Distance from the wing root to
equivalent tip (0.9S)
Cm = Wing mean chord in ft.
e = Position of inertia axis aft of LE
Xa = Position of flexural axis aft of LE
λ = Taper ratio
r = Stiffness ratio
k = Reduced frequency bω/u
C(k) = Theodorsen function
ωh = Bending frequency in rad/s
ωα = Torsional frequency in rad/s
ra = Radius of gyration about the
elastic axis
Λ = Sweepback in deg
1

2. International Conference on Engineering and Technology of Machinery, VETOMAC IV
f(M) = (1-M2
)/4 for 0 < M <0.8
= 0.775 for 0.8 <M<0.95
Where M = M0 cos and MΛ 0 is the
greatest forward Mach number at which
the UAV can achieve the maximum dive
speed
μ = Non-dimensional number m/πρb2
where m is the mass per unit length
b = Semi chord length in m
ρ = Sea level air density in kg/m3
a = Distance between the elastic axis and
the center of mass
3. Evaluation of the Flutter Speed by V-g Method
The Velocity-Damping method, abbreviated as V-g method, basically deals with strain
energy and kinetic energy of the structure and the aerodynamic damping. The structural
damping ratio g is plotted against the vehicle velocity for each vibration mode. The values
of the damping ratio are negative up to certain speeds and changes the sign as the structure
absorbs energy from the free stream, resulting into self-excited diverging oscillations. The
speed, which corresponds to zero value of the damping ratio g in the V-g curve, could be
taken as the critical flutter speed. The algorithm used to estimate the flutter speed is given
below,
The strain energy of the anisotropic plate
∫ ∫ ∇∇=
+
−
l c
c
wdydxV
0
2
2/
2/
2
2
1
(3.1)
and for the rectangular composite panel,
( )∫ ∫
+
− ⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
+++
++
=
l c
c XYXYYYXYXX
YYYYXXXX
dydx
WDWWDWWD
WDWWDWD
V
0
2/
2/
2
,66,,26,,16
2
,22,,12
2
,11
444
)(2)(
2
1
(3.2)
where
( ) ( )
∑=
−
⎥
⎦
⎤
⎢
⎣
⎡ −
=
n
k
kk
kijij
ZZ
QD
1
3
1
3
3
θ
(3.3)
The kinetic energy of the plate can be written as
∫ ∫
+
−
•
⎭
⎬
⎫
⎩
⎨
⎧
=
l c
c
dxdymT
0
2/
2/
2
2
1
ω (3.4)
where wwtm ρ= .
wρ is the specific gravity of the wing structure and is the thickness of the wing.wt
The Lagrange’s equations of motion, Ref. [8],
i
iii
Q
q
T
q
V
q
T
dt
d
=
∂
∂
−
∂
∂
+
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
∂
∂
•
(3.5)
Assuming sinusoidal motion for the vibration mode and neglecting warping stiffness of the
structure, the equations of motion can be written as
[ ] [ ]
⎭
⎬
⎫
⎩
⎨
⎧
=
⎭
⎬
⎫
⎩
⎨
⎧
+
⎭
⎬
⎫
⎩
⎨
⎧
− ti
ti
ijij
eQ
eQ
q
q
K
q
q
M ω
ω
ω
/
/
2
1
2
1
2
12
(3.6)
2

3. International Conference on Engineering and Technology of Machinery, VETOMAC IV
Where
[ ]
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
=
12
0
0
5
4
mclI
mclI
Mij (3.7)
and
[ ]
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
=
cl
ID
l
ID
l
ID
l
cID
Kij
866
2
616
3
616
3
711
42
2
(3.8)
where I4, I5 , I6, I7 I8 are Non-dimensional integral expressions as given in Ref. [6].
The aerodynamic forces acting on the structure are, Ref. [8]
( ) ( ) tiE
E eiLL
b
w
iLLbL ω
απρω ⎥
⎦
⎤
⎢
⎣
⎡
+++= 4321
32
(3.9)
)(
2
121 kc
k
i
iLL −=+ (3.10)
( )[ )(211
)(2
243 kca ]
k
i
k
kc
aiLL −+++=+ (3.11)
( ) ( ) tiE
E eiMM
b
w
iMMbM ω
απρω ⎥
⎦
⎤
⎢
⎣
⎡
+++= 4321
42
(3.12)
)(
)21(
121 kc
k
ai
iMM
+
−=+ (3.13)
⎥
⎦
⎤
⎢
⎣
⎡
⎟
⎠
⎞
⎜
⎝
⎛
−−⎟
⎠
⎞
⎜
⎝
⎛
−+
+
+⎟
⎠
⎞
⎜
⎝
⎛
+=+ akca
k
i
k
kca
aiMM
2
1
)(2
2
1)()21(
8
1 2
2
2
43 (3.14)
( ) ( ) ti
e
c
qlI
iLL
b
qlI
iLLbQ ω
ωπρ ⎥
⎦
⎤
⎢
⎣
⎡
+++= 23
43
14
21
3
1 (3.15)
( ) ( ) ti
e
c
qlI
iMM
bc
qlI
iMMbQ ω
πρω ⎥
⎦
⎤
⎢
⎣
⎡
+++= 23
43
13
21
42
2 (3.16)
Now the Flutter problem can be formulated as
[ ]{ } 02
=− qAK ω (3.17)
where the stiffness matrix K and aerodynamic matrix A are defined below
3
711
11
l
ICD
K = (3.18)
2
616
12
2
l
ID
K = (3.19)
1221 KK = (3.20)
cl
ID
K 866
22
4
= (3.21)
( 414411 iLLbImclIA ++= )πρ (3.22)
3

4. International Conference on Engineering and Technology of Machinery, VETOMAC IV
( )41
3
3
12 iLL
c
Ilb
A +=
πρ
(3.23)
( 41
3
3
21 iMM
c
Ilb
A +=
πρ
) (3.24)
( 412
5
4
5
22
12
iMM
c
IlbmclI
A ++=
πρ
) (3.25)
The flutter frequencyω , the damping ratio g and the flutter speed u are extracted from
( )zRe
1
=ω (3.26)
( )
( )z
z
g
Re
Im
= (3.27)
k
b
u
ω
= (3.28)
4. Estimation of Flutter speed of an Airfoil
The Idealized Mathematical model of the wing to estimate the flutter speed is shown in
Figure.1.
Figure 1: Mathematical Idealization of Airfoil.
The wing airfoil is assumed to be thin and the motion is assumed as simple harmonic.
Hence the second order linear differential equation of the system, in generalized
coordinates can be written as [14]
(4.1)hh QhmShm =++ 2
....
ωαα
(4.2)ααααα αωα QIIhS =++ 2
....
and the flutter speed can be estimated from the determinant given below
4

5. International Conference on Engineering and Technology of Machinery, VETOMAC IV
⎪
⎪
⎭
⎪
⎪
⎬
⎫
⎪
⎪
⎩
⎪
⎪
⎨
⎧
⎟
⎠
⎞
⎜
⎝
⎛
++⎟
⎠
⎞
⎜
⎝
⎛
+⎟
⎠
⎞
⎜
⎝
⎛
+−
+⎥
⎦
⎤
⎢
⎣
⎡
−
⎭
⎬
⎫
⎩
⎨
⎧
⎟
⎠
⎞
⎜
⎝
⎛
+−+
⎭
⎬
⎫
⎩
⎨
⎧
⎟
⎠
⎞
⎜
⎝
⎛
+−+
⎪⎭
⎪
⎬
⎫
⎪⎩
⎪
⎨
⎧
+⎥
⎦
⎤
⎢
⎣
⎡
−
2
2
2
2
2
2
222
22
2
2
1
2
1
2
1
1
2
1
2
1
2
1
1
αα
ω
ω
πρ
α
πρ
α
πρωω
ωω
πρ
α
α
α
α
αα
α
α
h
a
h
hh
h
LL
M
b
m
r
L
b
m
x
LL
b
m
xL
b
m
(4.3)
Analytical code in MatLab has been developed to evaluate Eqn. (4.3). The flutter speed
can be obtained by letting the determinant to go to zero. Some of the variables used in the
code are listed below.
The Theoderson function
( ) ( )
( )
( )kiH
kH
kH
kC 2
02
1
2
1
+= (4.4)
where H is the Hankel functions [12]. The aerodynamic coefficients are given by [13]
( iGF
b
v
iLh +−= )
ω
21 (4.5)
( )[ ] ( iGF
b
v
iGF
b
v
iL +⎟
⎠
⎞
⎜
⎝
⎛
−++⎟
⎠
⎞
⎜
⎝
⎛
−=
2
221
2
1
ωω
α ) (4.6)
2
1
=hM (4.7)
⎟
⎠
⎞
⎜
⎝
⎛
−=
ω
υ
α
b
iM
8
3
(4.8)
5. Results and Discussion
Figure 2. Shows the finite element model of the composite wing developed in Hypermesh,
in order to estimate the normal modes.
Figure 2. Finite Element Model of the composite wing.
5

6. International Conference on Engineering and Technology of Machinery, VETOMAC IV
Free-Free boundary conditions are imposed on the wing and the Normal Modes analysis is
performed using MSc. Nastran. The first three modes are given in Figures 3, 4 and 5
respectively.
Figure 3. First Bending (50Hz) Figure 4.Second Bending (81.8Hz) Figure 5. Third Twisting (112.3Hz)
Knowing Natural frequencies of the FRP wing from the numerical model, the flutter speed
can be estimated using Eqn. (4.3). The various input parameters of reduced frequency for
composite and metallic wings are listed in Tables 1 and 2 respectively.
Sl.No. Parameter Values
1 Non-Dimensional Mass Ratio 105.04
2 Bending frequency – rad/s 314.78
3 Second Bending frequency – rad/s 513.96
4 Air density – kg/m3
1.2260
5 Radius of Gyration 0.5750
6 Distance between mid chord and Flexural axis in semi chord length 0.1500
7 Distance between mid chord and Center of mass in semi chord length -0.4000
Table 1. Input Data for High Speed Composite wing, used in MatLab Code
Sl.No. Parameter Values
1 Non-Dimensional Mass Ratio 64.430
2 Bending frequency – rad/s 150.79
3 Torsional frequency – rad/s 251.33
4 Air density – kg/m3
1.2260
5 Radius of Gyration 0.5750
6 Distance between mid chord and Flexural axis in semi chord length 0.1500
7 Distance between mid chord and Center of mass in semi chord length -0.4000
Table 2. Input Data for High Speed Metallic wing, used MatLab Code
From the data listed in Tables 1 and 2 the V-g diagrams have been plotted in MatLab and
the output is shown in Figures 6 and 7, for composite and metallic wing respectively. The
methodology is discussed in Ref. [15].
6

7. International Conference on Engineering and Technology of Machinery, VETOMAC IV
0 100 200 300 400 500 600 700
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
Velocity (m/s)
Damping
0 2 4 6 8 10 12
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
Frequency ratio
Damping
0 0.5 1 1.5 2 2.5
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
Reduced frequency
Damping
100 200 300 400 500 600 700
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
Flutter frequency (rad/s)
Damping
Figure 6. Output for High speed Composite UAV
0 100 200 300 400 500
-1.2
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
Velocity (m/s)
Damping
0.5 1 1.5 2 2.5 3
-1.2
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
Frequency Ratio
Damping
0 5 10 15 20
-1.2
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
Strouhl Number
Damping
140 160 180 200 220 240 260 280
-1.2
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
Flutter Frequency (rad/s)
Damping
Figure 7. Output for High speed UAV
7

8. International Conference on Engineering and Technology of Machinery, VETOMAC IV
Method Flutter speed (m/s) Flutter-Dive Speed ratio
Ug-Method (Composite wing) 283.40 1.31
Ug- Method (Existing Wing) 264.56 1.225
Table 3. Comparison of flutter speeds from for Metallic and Composite wings.
6.0 Conclusions
An attempt has been made here to replace the existing wing with that of the composite
wing (GFRP) and the flutter speed of the wing was found to be 283.40 m/s as listed in
Table 3, and the corresponding flutter frequency of the wing was found to be 54.5 Hz. This
is a considerable improvement in the flutter speed of the wing, which was found to be
264.56 m/s and the corresponding flutter frequency of 33 Hz, estimated using u-g method,
AVP-970 standards. The composite wing can be tailored aero elastically to study the wing
divergence (static instability) and the control reversal effects.
7.0 References
1. Weishaar, T. A., and Foist, B.L., 1983, “Vibration and Flutter of Advanced Lifting
surfaces”, Proceedings of 24th
AIAA/ASME/ASCE/AHS structures, structural
dynamics and Materials conference, Lake Tahoe, Nev., part 2, pp.498-508.
2. Bennet, G., 1995, “Modeling of wing weight for High Altitude long Endurance
Aircraft”, Part 1, Unmanned systems, 13(1).
3. Houser. J.M., and Manual Stein., 1974, “Flutter Analysis of Swept wing subsonic
Aircraft with parameter studies of composite wings”, NASA TN D-7539.
4. Vivek Mukhopadhyay., 1996, “An interactive software for conceptual wing flutter
analysis and parametric study”, NASA TM-110276.
5. 1963, “Aviation practices standard–970”, Aero-Elasticity, vol. (1), leaflet 500/3.
6. Howell.S.J, 1981, “Aeroelastic Flutter and Divergence of Graphite/Epoxy Cantilevered
plates with bending torsion coupling”, M.S. Thesis, Department of Aeronautics and
Astronautics, M.I.T.
7. Aston, J.E., and Whitney, J.M., 1970, “Theory of Laminated Plates”, Technomic
publishing co., Stanford, conn.
8. Dugundji., Brain J. Ladsberger., 1985, “Experimental Aeroelastic behaviour of
unswept and forward swept cantilever graphite/epoxy wings”., J. aircraft ., pp 679-686
9. NISA., 2004, “Aeroelasticity manual display IV”, EMRC, vol.1,
10. V.Prabhakaran, et.al., 1999 , “Composite wing Design for Falcon Airframe”, ADE/IR.
11. Upadhyaya, A.R., et. Al., 1990, “Modal Analysis of a cropped Delta wing of an
Unmanned Aircraft”, NASAS-90, pp. 1-21.
12. Fung, Y.C, 1955, “The theory of Aeroelasticity”. Galcit Aeronautical series, John
Wiley & Sons,
13. Scanlan., and Rosenbaum., 1951, “Introduction to the study of vibration Aircraft
Vibration and Flutter”, The Macmillan Company, Newyork.
14. Bishplingoff, R.L., Ashley, H., and Halfman, R. L., 1955, “Aeroelasticity”, Addison-
Wesley publishing Co., Reading, Mass.
15. Richardson. J.R, Aug. 30–Sept. 1, 1965, “A more realistic method for routine flutter
calculations”, AIAA symposium on structural dynamics and aeroelasticity, Boston/
Massachussets, pp. 10-17.
8