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Effect of thickness and angle-of-attack on the aeroelastic stability of
supersonic fins
Article in Aeronautical Journal -New Series- · August 2012
DOI: 10.1017/S0001924000007272
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THE AERONAUTICAL JOURNAL
JULY 2012 VOLUME 116 NO 1181
XX
Effect of thickness and angle-ofattack on the aeroelastic stability
of supersonic fins
R. D. Firouz-Abadi
S. M. Alavi
firouzabadi@sharif.edu
smahdi.alavi@yahoo.com
Department of Aerospace Engineering
Sharif University of Technology
Tehran, Iran
ABSTRACT
This paper aims at developing an aeroelastic model for the instability analysis of supersonic thick
fins. To this aim the modal analysis technique is used for the structural dynamics modelling of
a fin with a general structure. An unsteady aerodynamic model is applied based on the
shock/expansion analysis over the flat surfaces of the fin along with local application of the
piston theory. Assuming a supersonic fin with an arbitrary polygonal cross-section, thickness
and initial angle-of-attack, the steady flow properties (e.g. Mach number, density and
temperature) are calculated over the flat surfaces of the fin. Then, assuming small amplitude
vibrations, the generalised aerodynamic forces are obtained in terms of the structural modal coordinates. Using the obtained model, the effect of thickness, initial angle-of-attack, taper ratio
and sweep angle on the aerodynamic derivatives and aeroelastic stability of the fin are studied
which show their remarkable effects on the instability Mach number and its type. Specially, the
presented results show that increasing the fin thickness dramatically diminishes the stability
margin mainly at low angles of attack. Also a sharp decrease of the divergence Mach number
is observed by increasing the fin’s incidence angle.
Paper No. 3702. Manuscript received 1 March 2011, revised version received 27 August 2011, accepted 19 October 2011.
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THE AERONAUTICAL JOURNAL
JULY 2012
NOMENCLATURE
α
β
γ
v
θ
θ
ρ
ω
ξ
Ca
e
e
fm
Ka
Ks
n
M
P
q
T
u
v
w
z
angle of the face.
oblique wave angle.
specific heat ratio.
Prandtl-Meyer function.
elastic rotation vector.
natural mode shapes of the elastic rotations.
density.
natural frequency.
generalised modal co-ordinates.
aerodynamic damping matrix.
elastic displacement vector.
natural mode shapes of the elastic displacements.
generalised aerodynamic forces.
aerodynamic stiffness matrix.
diagonal structural stiffness matrix.
outward unit normal of the undeformed surface.
Mach number.
unsteady pressure.
vector of generalised co-ordinates system.
temperature.
flow velocity.
unit direction vector of the air velocity.
displacements along the normal vector of each surface.
structural state vector.
1.0 INTRODUCTION
Aeroelasticity is a branch of aerospace science, concerning the mutual interactions between
aerodynamics and structural vibrations. This subject has been the concern of many researchers
and numerous studies have been performed on the topic. Nowadays, with the recent
improvements in aerodynamics and, therefore, the increased efficiency and the reduced drag
acting over flying vehicles, and also with the aid of various and powerful propulsive mechanisms,
which have made it possible to achieve supersonic and hypersonic speeds, aeroelastic analysis
of flying objects has become an essential requirement. There is a possibility that instability of
the control surfaces of the supersonic X-43 has been a major factor leading to its crash. Flight
with a high angle-of-attack and curved noses of such vehicles are the main factors causing
numerous complexities in aerodynamic loading, resulting in difficulties in aeroelastic analysis.
Therefore, one of the important issues concerned in today’s engineering problems at supersonic
and hypersonic regimes is calculation of the unsteady aerodynamic loads.
Various analytical methods and numerical algorithms have been developed in order to
overcome this problem. With the aid of the recently developed high speed processors, numerical
methods have found a vast application for the solution of the Euler and Navier-Stokes equations
for invscid and viscous flows. However, in spite of the high speed processors involved in the
studies, the numerical methods are still complicated and time consuming. Therefore, researchers
have mainly focused on developing analytical and semi-analytical engineering models, instead.
FIROUZ-ABADI AND ALAVI EFFECT OF THICKNESS AND ANGLE-OF-ATTACK ON THE AEROELASTIC...
3
One of the simple and capable methods applied in this field is known as piston theory, developed
in the 1950s by Lighthill(1), Ashley and Zartarian(2). Piston theory describes the relation between
the unsteady pressure over a flat plate in terms of the local angle-of-attack. Due to simplicity
and good accuracy of this theory, it has been considered as a very common model for derivation
of the unsteady aerodynamic pressure over a flat plate for panel flutter analysis. The classic piston
theory is developed for flat plates and its application to cambered surfaces and nonzero angleof-attack leads to some errors in results. Therefore, researchers have tried to improve the
accuracy and the validity boundaries of this theory.
Liu(3) showed that the wing thickness has a considerable effect in supersonic governing
equations, leading to a forward shift in the location of the pressure center and thus a reduction
in flutter speed. Yates and Bennett(4) improved the accuracy of the dynamic instability prediction
of a winglet in supersonic and hypersonic regimes, using piston, modified Newtonian, and shock
wave theories. Combining the three methods, they covered the limitations of each individual
approach in consideration of the thickness, angle-of-attack, and Mach number. Hui(5,6,7,8) applied
the piston theory to calculate the stability derivatives in steady and unsteady flow fields.
Considering a small oscillatory motion for geometries like a flat plate and a wedge with an
arbitrary angle-of-attack, he modified the piston theory to include the effects of the thickness
and initial angle-of-attack. Ramsey(9) considered the effects of the curvature and thickness on
the aeroelastic stability of a fan subject to a supersonic flow. In this study, the piston theory and
Lane’s linear potential theory were applied simultaneously to modify the nonlinear effects of
the surface thickness and curvature.
Friedmann and et al(10) applied the piston theory, Euler method and Navier-Stokes approach
to model the aerodynamic loadings for the aeroelastic analysis of the double-wedge aerofoil.
Using the results of such an analysis, the calculations of the aerodynamic loads acting on a
reusable launch vehicle (RLV) were simplified. Further, they used a similar approach to
investigate the aerothermoelasticity of the RLV(11,12). Oppenheimer and Doman(13) used the piston
theory for the calculation of the unsteady pressure distribution over the body surfaces of a flying
vehicle in hypersonic regime subjected to rigid body oscillations. Thuruthimattam et al(14) studied
the hypersonic aeroelasticity of a generic hypersonic vehicle with a lifting-body fuselage and
canted fins. They used third order piston theory besides Euler aerodynamics and obtained the
aeroelastic damping and frequency using computational time response. They observed that the
3D flow effects captured using Euler aerodynamics lead to higher flutter Mach numbers in
comparison with the results of the nonlinear piston theory. Based on the solution of steady mean
flow by a CFD Euler method, Zhang and et al(15,16) used a local piston theory for the calculation
of stability derivatives and flutter boundaries of supersonic wings and reported good agreements
in comparison with the unsteady Euler method. They showed that using such approach increases
the accuracy of flutter analysis and considerably saves the computational costs of the unsteady
CFD calculations.
This paper aims at developing a simple fast algorithm for the aeroelastic modelling of thick
fins with nonzero angle-of-attack and general structure in supersonic flow. Assuming supersonic
fins with polygonal wing sections, a procedure is suggested for the solution of steady flow over
the fin using the analytical shock/expansion wave theory. Then, the piston theory is used
locally along with the modal analysis technique to obtain an aeroelastic model which avoids the
use of time consuming CFD solvers and the mesh generation problems. The achieved agreement
between the obtained results and those of the CFD Euler method is satisfactory and demonstrates
the efficiency of the present model as a fast aeroelastic analysis tool in the preliminary stages
of design and optimisation. Using the obtained fast solver, several case studies are performed
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THE AERONAUTICAL JOURNAL
JULY 2012
to investigate the effect of thickness and angle-of-attack on the aerodynamic derivatives and
aeroelastic stability margins of supersonic fins.
2.0 STRUCTURAL DYNAMICS MODELLING
Using the modal analysis technique, the elastic displacement and rotation vectors at any point
of the wing can be written as;
N
e = ∑ en ξn (t )
. . . (1)
n =1
θ=
N
∑ θ ξ (t )
n n
. . . (2)
n =1
where ēn and θn denote the natural mode shapes of the elastic displacements and rotations, respectively and ξns are the generalised modal co-ordinates. Using the Lagrange equations along with
Equations (1) and (2) for determination of the kinetic and potential energies of the structure, and
applying the virtual work principle and the orthogonality relations of the natural modes, one
obtains the governing equations of the fin vibrations as;
ξ + Ksξ = f
. . . (3)
where and ξ are f vectors containing the generalised modal co-ordinates and forces, respectively
and Ks is the diagonal structural stiffness matrix that is defined as follows;
s
Kmm
= ωm2
. . . (4)
where ωm is the natural frequency of the mode m. Also the generalised forces due to the
unsteady aerodynamic pressure P over the fin surface S are calculated as follows;
fm = −∫ P (n·em )dS
. . . (5)
S
where n shows the outward unit normal of the undeformed surface.
3.0 AERODYNAMIC MODELLING OF THE FLEXIBLE FIN
According to first order piston theory, the unsteady pressure over a flat plate is expressed in terms
of its transverse deflection w as follows;
P=
2
⎛
⎞
⎜⎜w ′ + 1 M − 2 w ⎟⎟
2
u M − 1 ⎟⎠
M − 1 ⎝⎜
ρu 2
. . . (6)
2
where the over-dot shows the time derivation and the prime symbol indicates the slope of the
transverse deformation in the flow direction. Also M is the Mach number, ρ is the air density
and u is the airspeed over the surface.
For thin fins in zero angle-of-attack, weak compression and expansion waves occur over the
surface and thus the variations of flow properties (e.g. Mach number, temperature and density)
FIROUZ-ABADI AND ALAVI EFFECT OF THICKNESS AND ANGLE-OF-ATTACK ON THE AEROELASTIC...
5
O'i
surface i+1
surface i-1
t
t
M i-1
M i-1
__
M
3D effect
O'i-1
surfa
ce i
n
M i-1
Oi
Free Stream
Oi-1
Figure 1. Supersonic flow over the surfaces of a thick fin.
on the surface can be neglected. But, if the fin has considerable thickness or orients with a non
zero angle-of-attack, the flow properties change drastically behind the shock and expansion
waves. On this point of view, first, one can obtain the flow properties over the fin’s surfaces and
then apply the piston theory locally on each surface.
3.1 Steady supersonic flow over the fin
The flow properties behind shock and expansion waves over the fin surfaces can be numerically
calculated using steady CFD solvers. However, such computational tools are usually time
consuming due to their limitations for grid generation and iterative numerical schemes,
especially for 3D geometries. The shock/expansion waves theory is one of the simplest methods
which furnishes analytical relations for the solution of steady two or three dimensional
supersonic flows. Recalling that these waves are inherently two dimensional in nature, a
procedure is purposed to solve the steady flow over the fin.
Figure 1(a) shows a sketch of surfaces of a thick fin in supersonic flow. The surfaces upper
and lower the mid plane of the fin are numbered individually and the Mach number, density and
temperature in the surface number are denoted by Mi, ρi and Ti, respectively. The angle αi
between the two following surfaces i – 1 and i is obtained as;
αi = Cos–1(ni–1 ni)
. . . (7)
where ni–1 and ni are the outward normal vectors of the corresponding surfaces. When the angle αi
is negative an expansion wave occurs at the edge line Oi–1O′i–1, and when it is positive an oblique
shock wave occurs.
The upstream air velocity in surface i – 1 can be decomposed into tangential and normal
components with respect to the edge Oi–1O′i–1, which related Mach number are shown by Mti–1
and Mti–1, respectively. Assuming that the tangential component of the velocity remains constant
across the wave, the downstream Mach number M behind an expansion wave can be obtained
by the Prandtl-Meyer theory as following;
αi = ν(M) − ν(Mni−1 )
. . . (8)
where v is the Prandtl-Meyer function
ν(M) =
γ +1
γ −1 2
Tan−1
(M − 1) − Tan−1 M2 − 1
γ −1
γ +1
. . . (9)
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THE AERONAUTICAL JOURNAL
JULY 2012
where γ is the specific heat ratio. Furthermore, the downstream Mach number behind an
oblique shock is obtained by the following relation;
2 + (γ − 1) (Mni−1 ) Sin2 βi
1
2
Sin(βi − αi ) 2γ (Mni−1 ) Sin2 βi − γ + 1
2
M=
. . . (10)
where βi is the oblique wave angle that is calculate by solving the following equation;
(Mni−1 ) Sin2βi − 1
2
(Mni−1 ) (γ + Cos2βi ) + 2
2
Tanαi = 2Cotβi
. . . (11)
By determining the Mach number over the surface i, the density, temperature and air velocity
direction over the surface can be evaluated. Behind an oblique shock the air density and
temperature are obtained as;
⎛ (γ + 1)Mi2−1Sin2 βi ⎞⎟
⎟
ρi = ρi−1 ⎜⎜
⎜⎝ 2 + (γ − 1)M2 Sin2 β ⎟⎟⎠
i −1
i
⎛ 2γ Mi2−1Sin2 βi + 1 − γ ⎞⎟ ⎛⎜ 2 + (γ − 1)Mi2−1Sin2 βi ⎞⎟
⎟
Ti = Ti−1 ⎜⎜
⎟⎟ ⎜⎜
⎜⎝
γ +1
⎠ ⎝ (γ + 1)Mi2−1Sin2 βi ⎟⎟⎠
. . . (12)
. . . (13)
and for an expansion wave they are;
⎛⎜ γ
⎞
−1⎟⎟⎟
⎠⎟
⎛ 2 + (γ − 1)Mi2−1 ⎞⎟⎜⎜⎝ γ −1
⎟
ρi = ρi−1 ⎜⎜
⎜⎝ 2 + (γ − 1)M2 ⎟⎟⎠
. . . (14)
⎛ 2 + (γ − 1)Mi2−1 ⎞⎟
⎟
Ti = Ti−1 ⎜⎜
⎜⎝ 2 + (γ − 1)M2 ⎟⎟⎠
i
. . . (15)
i
Also, the unit direction vector of the air velocity vi on the surface number is determined as;
vi =
Mti−1ti−1 + M(ni × ti−1 )
(Mti−1 )
2
+ (M)
2
. . . (16)
where ti–1 is the unit direction vector of the edge Oi–1O′i–1. The procedure is continued for the
next surface similarly. For this purpose the tangential and normal component of Mach vector
with respect to the edge OiO′i are obtained as;
Mni = MCos(Λi ) + Mti−1Sin(Λi )
. . . (17)
Mti = MSin(Λi ) + Mti−1Cos(Λi )
. . . (18)
where Ai = Cos–1 (ti–1·ti) is the angle between the edges Oi–1O′i–1. and OiO′i. Starting from the
leading edge and the free-stream surface, in which the flow properties are known, this procedure
FIROUZ-ABADI AND ALAVI EFFECT OF THICKNESS AND ANGLE-OF-ATTACK ON THE AEROELASTIC...
7
is individually applied on the surfaces upper and lowers the mid plane of the fin and is
terminated at the trailing edge.
3.2 Local piston theory
Assuming very small elastic deflection of the fin, the velocity and slope of the transverse
deformations on each surface of the fin is obtained as;
ω′ = (n × θ)·v
.
. . . (19)
.
ω = n·e
. . . (20)
Introducing Equations (19) and (20) into Equation (6), the local piston theory can be formulated
over the fin surfaces as follows;
P=
⎡
1 M2 − 2 ⎤
⎢(n × θ)·v +
n·e ⎥
u M2 − 1 ⎥⎦
M2 − 1 ⎢⎣
ρu 2
. . . (21)
4.0 AEROELASTIC MODELLING
The unsteady aerodynamic pressure on the fin surfaces is obtained in terms of the generalised
modal co-ordinates from substituting the modal expansion series from Equations (1) and (2) into
Equation (21) as follows;
N
N
n =1
n =1
P = ρu 2 ∑ C θn ξn (t ) + ρu ∑ C en ξn (t )
. . . (22)
where the coefficients Cen and Cθn are;
C en =
M2 − 2
en ·n
(M2 − 1)1.5
C θn =
(n × θn )·v
M2 − 1
. . . (23)
. . . (24)
Using Equation (22) in Equation (5), yields the following expression for the generalised
aerodynamic forces;
N
N
n =1
n =1
fm = −∑ Kamn ξn − ∑ Camn ξn
. . . (25)
where and Kamn are Camn respectively the elements of the aerodynamic stiffness and damping
matrices which are obtained by calculating the following integrals over the whole surfaces of
the fin, out of the Mach wave cone region.
a
Kmn
=
∫C
S
θn
(n·em )dS
. . . (26)
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THE AERONAUTICAL JOURNAL
a
Cmn
=
∫C
S
en
(n·em )dS
JULY 2012
. . . (27)
In order to obtain the aerodynamic stiffness and damping matrices, the fin surface is discretised
into small elements as shown in Fig. 2 and the integrals in Equations (26) and (27) are evaluated
numerically. For this aim, the integrands of Equations (26) and (27) are approximated using their
the nodal values and the element shape functions within each surface element and the Gauss
quadrature is used for the numerical integration. Combination of Equations (3) and (25) gives
the aeroelastic model of the fin as follows;
q + Ca q + (Ks + Ka )q = 0
. . . (28)
which is expressed in the state-space form as;
⎡Ca
z = ⎢⎢
⎢⎣ I
.
−Ks − Ka ⎤
⎥z
⎥
0
⎥⎦
. . . (29)
where zT = [qT qT]T .
In 3D wings, the pressure difference between the upper and lower surfaces must vanish at the
wing tip, which in turn, causes the tip vortex and loss in lift. Based on this issue, the aerodynamic
model can be improved to capture the 3D effects on the unsteady pressure distribution over the
wing surfaces. To this aim, the above introduced procedure can be modified by eliminating the
integrals of Equations (26) and (27) over the aerodynamic elements which are inside the Mach
1
wave cone as illustrated in Fig. 1(b) with cone angle .μ = Sin−1
M
Y
Z
X
Figure 2. Discretisation of the fin surfaces into small elements.
FIROUZ-ABADI AND ALAVI EFFECT OF THICKNESS AND ANGLE-OF-ATTACK ON THE AEROELASTIC...
9
5.0 VALIDATION AND NUMERICAL EXAMPLES
5.1 Effect of thickness and angle-of-attack on the aerodynamic
coefficients
In order to examine the validity of the used aerodynamic model, a general fin with aspect ratio
of 1.5 and double-wedge section is considered as shown in Fig. 2. The geometric parameters
of the fin are given in Table 1.
Table 1
Geometric and mass characteristics of the 2Dof rigid fin
Parameter
Description
Value
Kh
plunging spring stiffness
25kN/m
Kα
torsional spring stiffness
800N.m
Iyy
mass moment of inertia
1·5kg.cm2
m
a
fin mass
the elastic axis location,
positive rearward
0·15kg
b
semi-chord at the root
7·3cm
xα = e – a
static unbalance parameter
–0·03
0·13
The lift coefficient slope CLα and the pitching moment coefficient slope CMα about the elastic
axis of the fin are obtained using the present local piston theory and compared with the 2D- and
3D-Euler CFD results. Figure 3 shows the obtained results versus the fin thickness at Mach
number 4 and zero angle-of-attack. Also the variation of the aerodynamic derivatives with respect
to the initial angle-of-attack for a fin with t/c = 10% at Mach number 4 is depicted in Fig. 4.
The good agreement between the present results and those of CFD Euler confirms that the
combination of the local piston and shock/expansion wave theories yields reliable results in
prediction of aerodynamic coefficients which are used in the aeroelastic analysis.
In order to clearly realise the effect of the fin thickness and angle-of-attack on the lift coefficients CLα ,CLα. ,CLh., and moment coefficients CMα ,CMα. ,CMh. several case studies were carried
out. The fin is considered in the following free-steam flow states
M∞ = 4, P∞ = 1Atm, ρ∞ = 1 ⋅ 225kg/m 3 ,T∞ = 298K
Figures 5 and 6 illustrates the variation of the aerodynamic coefficients with the thickness ratio
for zero angle-of-attack as well as with the angle-of-attack for t/c = 10%. The obtained results
illustrate the remarkable effect of thickness and initial angle-of-attack on the aerodynamic
derivatives. Also the results show that the sensitivity of the aerodynamic derivatives to the Mach
number decreases at higher Mach numbers.
5.2 Effect of thickness and angle-of-attack on the aeroelastic stability
For the verification of the present aeroelastic model, a clamped delta wing was considered
as shown in Fig. 7 and the obtained results of flutter analysis are compared with those from
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THE AERONAUTICAL JOURNAL
JULY 2012
0.4
Local piston theory
Euler (2D)
Euler (3D)
Local piston theory
Euler (2D)
Euler (3D)
1.8
0.3
CMD
CL D
1.6
1.4
0.1
1.2
1
0.2
0
5
10
D0(deg)
15
0
20
0
5
10
D0(deg)
(a)
15
20
(b)
Figure 3. Comparison of the local piston theory and Euler method for calculation of
CLα and CMα versus the thickness ratio at M = 4 and zero angle-of-attack.
0.4
Local piston theory
Euler (2D)
Euler (3D)
1.8
Local piston theory
Euler (2D)
Euler (3D)
0.3
CMD
CL D
1.6
1.4
0.2
0.1
1.2
1
0
5
10
D0(deg)
(a)
15
20
0
0
5
10
D0(deg)
15
20
(b)
Figure 4. Comparison of the local piston theory and Euler method for calculation
of CLα and CMα versus the initial angle-of-attack at M = 4 and t/c = 10%.
the wind-tunnel tests carried out at the NASA Langley wind tunnel by Tuovila and
McCarty(17).
The experiment was carried out at Mach 3 with the air density of 0·37kg/m3. The wing is
made from magnesium plate with 0·86mm thickness, elastic modulus of 42·8GPa, density
of 17,66kg/m3 and Poisson ratio of 0·35. Using a finite element code, the first three natural
modes and frequencies of the wing are obtained and then the flutter speed and frequency of
the wing are determined using the present model. The variation of damping and frequency
of the aeroelastic modes is shown in Fig. 8. Table 2 represents the good agreement between
the obtained flutter results in comparison with the experimental data which confirms the
validity of the present model.
FIROUZ-ABADI AND ALAVI EFFECT OF THICKNESS AND ANGLE-OF-ATTACK ON THE AEROELASTIC...
10-5
CL
10-3
0
M=12
M = 12
M=3
1.5
11
-1
.
CL h
CL
1
-0.5
-2
.
-3
M=3
-1
M = 12
-4
0.5
M=3
0
4
5
10
t/c
15
20
25
0
5
10-1
10
15
t/c
20
-1.5
0
25
-5
10
0
M =12
5
10
t/c
15
20
25
10-4
M = 12
-0.5
3
.
-1
2
M=12
h
.
CM
CM
CM -1
M=3
-2
-1.5
M=3
1
5
10
t/c
15
20
25
-3
M=3
-2
0
0
5
10
15
0(deg)
0
5
10
t/c
15
20
25
Figure 5. Variation of the aerodynamic derivatives versus thickness ratio at zero angle-of-attack.
-3
10
10-5
M=12
M = 12
M=3
-1
CL
1.5
-0.5
.
1
0.5
CL h
CL
.
-2
-1
M = 12
-3
0
5
10
0(deg)
15
10-1
M=3
-1.5
M=3
0
5
10
0
15
0(deg)
-5
10
5
0(deg)
10
10-4
M =12
2.5
15
M = 12
-0.5
-0.5
M=3
-1
.
CM h
CM
CM
.
2
1.5
M=3
-2
0
5
0(deg)
10
-1.5
-1.5
M=12
15
0
-1
M=3
-2
5
10
0(deg)
15
0
5
10
0(deg)
Figure 6. Variation of the aerodynamic derivatives versus initial angle-of-attack at t/c = 10%.
15
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THE AERONAUTICAL JOURNAL
JULY 2012
10-4
-5
10
10-1
M =12
2.5
M = 12
-0.5
-0.5
M=3
.
-1
CM h
CM
CM
.
-1
2
1.5
M=3
-2
0
5
0(deg)
-1.5
-1.5
M=12
10
15
0
M=3
-2
5
10
0(deg)
15
0
5
(a)
Figure 7. Geometry of the delta
wing in supersonic flow.
15
(b)
Figure 8. Variation of the aeroelastic damping and
frequency versus the airspeed for the delta wing.
4
Mach Number
10
0(deg)
D= 0
Flutter
Divergence
q
3.5
3
q
5
2.5
q
10
2
0.05
0.1
0.15
0.2
t/c
(a)
(b)
Figure 9. Wing section and geometry of the 2DOF rigid fin.
Figure 10. Variation of the flutter
and divergence Mach number for
2DOF rigid fin with thickness at
various angle-of-attack.
Table 2
The obtained flutter results for the delta wing in comparison with experiment
Ref. 17
present
error
ω1(Hz)
ω2(Hz)
ω3(Hz)
ωflutter(Hz)
ωflutter(Hz)
49
49
–
183
185
1·1%
257
256·5
0·2%
159
145
8·8%
2,030
1,962
3·3%
As is seen in Figs. 5 and 6, the fin’s thickness and initial angle-of-attack have considerable effects
on the aerodynamic derivatives and consequently can change the aeroelastic stability margins.
This section is focused on the investigation of this topic using the fin of Fig. 2 that is constrained
by a pitching and a plunging spring. The structural parameters of the fin are given in Table 1.
Figure 10 shows the flutter and divergence speed of the fin versus the thickness ratio. The
results show that there is a remarkable reduction in the aeroelastic stability margin when the
thickness increases, particularly in the low angles of attack. The incidence angle is the other
effective factor in the aeroelastic stability of the supersonic fin. Increasing the initial angle-ofattack, causes a forward shift in the location of the pressure centre and thus leads to a sharp
decrease in the divergent Mach number. Figure 11 shows the variation of the flutter and
divergence Mach numbers with the initial angle-of-attack at different thicknesses of the fin. The
results show that increasing in the initial angle-of-attack causes the flutter Mach number of the
fin to grow at high thicknesses. On the other hand, according to the forward shift of the
pressure centre, the divergent Mach number reduces with increasing angle-of-attack. From the
FIROUZ-ABADI AND ALAVI EFFECT OF THICKNESS AND ANGLE-OF-ATTACK ON THE AEROELASTIC...
10
Flutter
Divergence
t/c=10%
8
3
2.7
2.4
15%
10%
20%
7
15%
20%
3
4
6
D(deg)
8
10
2
Figure 11. Variation of flutter and divergence Mach
number for 2DOF rigid fin with angle-of-attack at
various thickness.
20%
5
4
2
t/c=10%
6
15%
2.1
0
Flutter
Divergence
9
Mach Number
Mach Number
3.3
-5
0
5
10
15
Figure 12. Variation of flutter and divergence Mach
number for 2DOF rigid fin with sweep angle.
7.50mm
Flutter
Divergence
t/c=10%
5
Mach Number
13
15%
4
d base
20%
10%
10
5
15%
3
20%
45mm
85
2
0
0.2
0.4
0.6
0.8
ratiodivergence Mach
Figure 13. Variation of Taper
flutter and
number for 2Dof rigid fin with taper ratio.
1
15mm
125mm
150mm
Figure 14. Geometry and dimensions
of the elastic supersonic fin.
results, it can be seen that, based on the fin thickness and its angle-of-attack, the instability type
may change from flutter to divergence at high angles of attack.
The sweep angle is one of the fundamental parameters in the design of supersonic wings.
Increasing the sweep angle results in weaker shock/expansion waves on the wing and thus
increases the aeroelastic stability. Figure 12 illustrates the effect of sweep angle on the flutter
and divergence Mach numbers of the considered fin for different thickness ratios. The obtained
results identify the increase of the aeroelastic stability with the growth of the sweep angle. Further
it is observed that at high sweep angles the instability type changes from flutter to divergence.
Using wing taper is also a common design feature for the supersonic fins. The taper ratio
reduces the undesired effects of the wing tip and improves the aerodynamic efficiency of the
supersonic wings. The variation of the flutter and divergence speeds of the considered fin versus
the taper ratio is depicted in Fig. 13 for several thickness ratios. The taper ratio is applied on
both the chord and thickness of the fin. The obtained results implies that the aeroelastic
instability margins of the fin reduces as the taper ratio increases.
14
THE AERONAUTICAL JOURNAL
JULY 2012
(a) ω1 = 5,24Hz
(b) ω1 = 1,053Hz
(c) ω1 = 2,050Hz
(e) ω1 = 2,492Hz
(f) ω1 = 3,253Hz
(f) ω1 = 4,176Hz
Figure 15. The first six natural mode shapes of the elastic fin and corresponding natural frequencies.
10 4
1
1.6
0
1.4
Frequency
Damping
10 2
-1
-2
-3
1.2
1
0.8
0.6
-4
3
4
5
6
7
8
Mach Number
3
4
5
6
7
8
Mach Number
Figure 16. Plots of aeroelastic damping and frequency for the elastic fin at zero angle-of-attack.
5.3 Aeroelastic analysis of an elastic fin
In order to demonstrate the application of the proposed model for the aeroelastic analysis of
general supersonic thick fins and to analyse the effect of initial angle-of-attack on the stability,
a flexible fin with t/c = 10% and the geometry as shown in Fig. 14 is considered. The fin is
clamped to a rigid base and made of a composite shell with uniform thickness of 0·5mm, which
constituted its aerodynamic surfaces and a mold less foam core with low density, great
absorption energy and good thermal stability and damping capabilities. The fin surface was a
four layer Carbon/epoxy composite shell with symmetric cross-ply orientation. Table 3 shows
the materials properties of the fin. The aeroelastic model is obtained based on the first six natural
vibration modes of the fin which are extracted using a finite element model and are shown in
Fig. 15. Figure 16 shows the variation of damping and frequency of the aeroelastic modes with
Mach number for flight at sea level. The obtained results show that the flutter Mach number of
FIROUZ-ABADI AND ALAVI EFFECT OF THICKNESS AND ANGLE-OF-ATTACK ON THE AEROELASTIC...
15
6
Table 3
Material properties of
the elastic thick fin
Mach Number
5.9
5.8
5.7
5.6
5.5
5.4
0
5
10
15
(deg)
20
25
Carbon-epoxy
140
10
0·3
5·2
5·2
3·4
1,550
Foam
2·0
–
0·3
–
–
–
100
Figure 17. Variation of the flutter Mach number of the
elastic fin versus the initial angle-of-attack.
the fin is 5·97 while ignoring the fin thickness results that the fin is always stable. Also the
variation of the flutter Mach number versus the initial angle-of-attack is shown in Fig. 17 which
illustrates that the stability of the fin decreases at higher angles of attack.
6.0 CONCLUSIONS
A simple fast algorithm was presented for aeroelastic modelling of supersonic thick fins with
initial angle-of-attack. Shock/expansion theory was utilised along with local piston theory to
derive an unsteady aerodynamic model for a flexible fin and based on the modal analysis
technique, an aeroelastic model was obtained. Based on several case studies using the present
model, the effect of thickness and incidence angle of fin on the aerodynamic derivatives and
aeroelastic stability are highlighted. The results show that increasing the fin thickness or angleof-attack, diminishes the stable Mach envelope and may change the type of instability from
flutter to divergence. Also the results show that increasing the taper ratio or decreasing the sweep
angle of a thick fin results in the same trend.
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