Dynamics of a 2D aerofoil

1. QUEEN MARY UNIVERSITY OF LONDON
Simulation of the
dynamics of a 2D
fluttering aerofoil with a
trailing edge flap
DEN410 Aeroelasticity
Reena Thapa
Student Id:-090242902
14th April 2014

2. Reena Thapa Page 1
Abstract
Understanding the deformation due to aerodynamics forces is vital when designing an aircraft,
especially on the aircraft wing. This report explores the wing divergence and bending-torsion
flutter based on computer aided simulations. As a simulation tool, MATLAB has been used
to simulate the two-dimensional aerofoil model with the trailing edge flap. Three set of
simulations, out of which two were carried out for the two degree of freedom model with and
without C(k) and the third simulation was carried out for three degree of freedom model that
included the trailing edge flap. It was found that the flutter was avoided for the simulation
without C(k) as U/b increases. However, the case was not similar for the simulation with C(k)
as some flutter were present. Finally for the three degree of model simulation, flutter was
present throughout the system.
Table of Contents
Abstract ......................................................................................................................................1
1.Introduction.............................................................................................................................2
1.1 Aims and Objectives ........................................................................................................2
2. Background Theory................................................................................................................2
3.Simulation Procedure..............................................................................................................9
4. Results..................................................................................................................................13
4.1 The Simulation results of two degree of freedom aerofoil model without C(k)...........13
4.2 The Simulation results of two degree aerofoil model with C(k)....................................15
4.3 The Simulation results of three degree of freedom aerofoil with flap ...........................17
5. Discussion............................................................................................................................22
6. Conclusion ...........................................................................................................................23
7. References............................................................................................................................23

3. Reena Thapa Page 2
1.Introduction
Aeroelasticity explores the interaction of inertial, structural and aerodynamic forces on
aircraft, building or any other surfaces. Aeroelastic problems would not exist if airplane
structures where perfectly rigid. Modern airplane structures are very flexible, and this
flexibility is fundamentally responsible for the various types of aeroelastic phenomena
(Bisplinghoff,1996). Flutter is a dangerous phenomena that is encountered in aircraft wings.
In an aircraft, as the speed of the wind increases, there may be a point at which the structural
damping is insufficient to damp out the motions which are increasing due to aerodynamic energy
being added to the structure. This vibration can cause structural failure and therefore considering
flutter characteristics is an essential part of designing an aircraft.
In aircraft wings, flutter may indeed become problematic due to the flap control. It is thus
important to predict the effects of the trailing edge control systems on the stability limits
(Bergami, 2008). Therefore, Stability is analysed through an aeroelastic investigation of a
representative 2D aerofoil section equipped with a trailing edge flap.
1.1 Aims and Objectives
The aim of the simulation is to investigate the factors governing the response of a three
degree of freedom aerofoil model with a trailing edge flap, in relation to basic analytical
techniques. In order to achieve this aim, the following objectives were implemented:
 Familiarisation with basic dynamics and control of a two-dimensional aerofoil
with a trailing edge flap.
 Implement the simulation tool called MATLAB to simulate the dynamics of a
two-degree of freedom and three-degree of freedom aerofoil model, both with
trailing edge flap.
 Assessment of the design of the controller and performance features.
 Understand the dynamics with reference to an aeroelastic system.
 Employ the simulation to predict the flow speed at which flap equipped system
may become unstable due to static and dynamic instabilities i.e. wing divergence
and bending-torsion flutter.
2. BackgroundTheory
In this investigation, the aerofoil section is modelled as a rigid flat plate which has a mass, 𝑚.
The aerofoil section do not deform but it moves in the plane. The three degree of freedom is
described by the torsion or pitch rotation of the aerofoil as the angle 𝛼 between the aerofoil
chord and the 𝑥 axis. . The wing stiffness is idealized and represented by two springs 𝐾𝐴 and
𝐾 𝐵. The chord of the wing is assumed to be 2𝑏. The origin of the co-ordinate system is
assumed to be located at the elastic centre of the cross-section which is assumed to be located
at EA. The co-ordinates on the system are given in dimensionless form, distances from the

4. Reena Thapa Page 3
origin are in fact normalized with respect to half chord length, 𝑏. The co-ordinate system of a
rigid aerofoil section is illustrated in Figure 2.1.
Figure2.1: Aerofoil section in a co-ordinate system.
The line joining the elastic center of the cross section of the wing is assumed to be a straight
line and referred to as the elastic axis. The elastic axis (EA) is assumed to be located at a
distance ba aft from mid-chord and the centre of mass,𝐶𝑀 is assumed to be located at a
distance b𝑥 𝛼 aft from the elastic axis. The mass moment of inertia of the wing model about
the elastic axis is 𝐼 𝛼 .
Equations of motions can be derived using Lagrangian Method:
𝐿 = 𝑇 − 𝑉
Where 𝐿 is the Lagrangian, 𝑇is the total Kinetic Energy and 𝑉 is the total Potential Energy.
Deriving the total Potential energy of the system
The translational potential energy of the two springs is given by:-
𝑉1 =
1
2
𝑘ℎℎ2
The Potential Energy of the rotating aerofoil is given by:-
𝑉1 =
1
2
𝑘 𝛼 𝛼2
The Potential Energy of the trailing edge flap while rotation is given by:-
𝑉3 =
1
2
𝑘 𝛽 𝛽2
Therefore the total potential energy of the aerofoil section including the trailing edge flap is:

5. Reena Thapa Page 4
𝑉 = 𝑉1 + 𝑉1 + 𝑉3
𝑽 =
𝟏
𝟐
(𝒌 𝒉 𝒉 𝟐
+ 𝒃𝟐𝒌 𝜶 𝜶 𝟐
+ 𝒌 𝜷 𝜷 𝟐
)
Deriving the total Kinetic energy of the system
The translational vertical motion of the wing with the trailing edge flap included, gives the
Kinetic Energy at the centre of mass:
𝑇1 =
1
2
𝑚(ℎ̇ + 𝑏𝑥 𝛼 𝛼̇)2
A rotational component of the Kinetic Energy due to rotation of the aerofoil can be expressed
as:
𝑇2 =
1
2
𝐼𝐶𝑀 𝛼̇2
Where ICM = Mass moment of inertia
According to the parallel axis theorem the mass moment of inertia about the centre of mass is
related to the mass moment of inertia about the elastic axis. Their relation is defined as:
𝐼𝐶𝑀 = ( 𝐼 𝛼 − 𝑚( 𝑏𝑥 𝛼)2)
The total kinetic energy of the system is also affected by the trailing edge flap. Measured
from the centre of gravity of the flap, the vertical displacement of the trailing edge flap is
given as:
ℎ 𝛽 = ℎ + 𝛼( 𝑐 − 𝑎 + 𝑥 𝑏) 𝑏
The trailing edge flap is deflected by 𝑏𝑥 𝛽 𝛽.
The Total Kinetic Energy of the trailing edge flap due to the vertical displacement can be
expressed as:
𝑇3 =
1
2
𝑚 𝛽[(ℎ̇ 𝛽 + 𝑏𝑥 𝛽 𝛽̇)
2
− ℎ̇ 𝛽
2
]
Therefore, the total Kinetic Energy of the aerofoil and the flap together is given by:
𝑻 =
𝟏
𝟐
𝒎(𝒉̇ + 𝒃𝒙 𝜶 𝜶̇ ) 𝟐
+
𝟏
𝟐
𝑰 𝑪𝑴 𝜶̇ 𝟐
+
𝟏
𝟐
𝒎 𝜷[(𝒉̇ 𝜷 + 𝒃𝒙 𝜷 𝜷̇ )
𝟐
− 𝒉̇ 𝜷
𝟐
]
The equations of motion can be obtained by inserting the expressions for the total energy into
Langrange’s equation.
𝑑
𝑑𝑡
(
𝜕𝑇
𝜕𝑞̇
) −
𝜕𝐿
𝜕𝑞̇
= 0

6. Reena Thapa Page 5
𝑑
𝑑𝑡
𝜕 [[
1
2
𝑚(ℎ̇ + 𝑏𝑥 𝛼 𝛼̇)
2
+
1
2
𝐼𝐶𝑀 𝛼̇2
+
1
2
𝑚 𝛽 [(ℎ̇ 𝛽 + 𝑏𝑥 𝛽 𝛽̇)
2
− ℎ̇ 𝛽
2
]] + [
1
2
𝑘ℎℎ2
+
1
2
𝑘 𝛼 𝛼2
+
1
2
𝑘 𝛽 𝛽2
]]
𝜕ℎ̇
−
𝜕[[[
1
2
𝑚(ℎ̇ + 𝑏𝑥 𝛼 𝛼̇)
2
+
1
2
𝐼𝐶𝑀 𝛼̇2
+
1
2
𝑚 𝛽 [(ℎ̇ 𝛽 + 𝑏𝑥 𝛽 𝛽̇)
2
− ℎ̇ 𝛽
2
]]] + [
1
2
𝑘ℎℎ2
+
1
2
𝑏2
𝑘 𝛼 𝛼2
+
1
2
𝑏2
𝑘 𝛽 𝛽2
]]
𝜕ℎ
= 0
This should yield a set of two equations of the form:
𝑚ℎ̈ + 𝑚𝑏𝑥 𝛼 𝛼̈ + 𝑘ℎℎ = 0
𝑚𝑏𝑥 𝛼ℎ̈ + 𝐼 𝛼 𝛼̈ + 𝑘 𝛼 𝛼 = 0
Given that the aerodynamic restoring force and restoring moment about the elastic axis, are 𝐿
and 𝑀 respectively, the disturbance force and moment are 𝐿 𝐺 and 𝑀 𝐺, show that the
equations of motion in inertia coupled form are:-
𝑚ℎ̈ + 𝑚𝑏𝑥 𝛼 𝛼̈ + 𝑘ℎℎ + 𝐿 = 𝐿 𝐺
And
𝑚𝑏𝑥 𝛼ℎ̈ + 𝐼 𝛼 𝛼̈ + 𝑘 𝛼 𝛼 + 𝑀 = 𝑀 𝐺
Generally, the restoring lift and moment may be only expressed as convolution integrals.
However, by making certain constraining assumptions that the motion is purely simple
harmonic, it can be shown that:
[
𝐿
𝑀
] = 𝑀 𝑎 [ℎ̈
𝛼̈
] + 𝐶 𝑎 [ℎ̇
𝛼̇
] + 𝐾𝑎 [
ℎ
𝛼
]
Where
𝑴 𝒂 = 𝜋𝜌𝑏3
[
1
𝑏
−𝑎
−𝑎 𝑏 (𝑎2
+
1
8
)
]
𝑪 𝒂 = 𝜋𝜌𝑏3
𝑈 [
2𝐶( 𝑘)
𝑏
1 + 𝐶( 𝑘)(1 − 2𝑎)
−𝐶( 𝑘)(1 + 2𝑎) 𝑏 (
1
2
− 𝑎) (1 − 𝐶( 𝑘)(1+ 2𝑎))
]
𝑲 𝒂 = 𝜋𝜌𝑏𝑈2
𝐶( 𝑘) [
0 2
0 −𝑏(1 + 2𝑎)
]

7. Reena Thapa Page 6
Where C(k) is a complex function (the so called Theodorsen function) of the non-dimensional
parameter, 𝑘 = 𝜔𝑏/𝑈 , (known as the reduced velocity) and U is the velocity of the airflow
relative to the aerofoil. As the aerodynamic stiffness matrix alone is a function of the square
of the velocity𝑈2
, one may ignore the effects of aerodynamic inertia and damping in the first
instance.
Thus approximating C(k) as equal to unity, one may express the equations of motion as a set
of coupled second order matrix equations representing a vibrating system; i.e. in the form,
𝐌𝐱̈ + 𝐂𝐱̇ + 𝐊𝐱 = 𝐅(𝐭).
In the first instance one should identify the matrices, M, C, K, x and F(t).
Dividing each of the equations by 𝑚𝑏2
and it follows that in non-dimensional matrix form,
the equations of motion in the presence of an external disturbance force and an external
disturbance moment are:
[
1 𝑥 𝑎
𝑥 𝑎 𝐼 𝑎 𝑚𝑏2⁄
] [ℎ̈ 𝑏⁄
𝛼̈
] + [
𝑘ℎ 𝑚⁄ 0
0 𝑘 𝛼 𝑚𝑏2⁄
] [ℎ 𝑏⁄
𝛼
] + [ 𝐿̅ 𝑏
𝑀̅
] = [
𝐿̅ 𝐺 𝑏
𝑀̅ 𝐺
]
and the generalized aerodynamic restoring moments are,
[ 𝐿̅ 𝑏
𝑀̅
] =
𝜋𝜌𝑏2
𝑚
(
𝑈
𝑏
)
2
(𝑴̃ 𝒂 [
ℎ̈
𝑏
𝛼̈
] + 𝑪̃ 𝒂 [
ℎ̇
𝑏
𝛼̇
] + 𝑲̃ 𝒂 [
ℎ
𝑏
𝛼
])
Where
𝑴̃ 𝒂 = (
𝑏
𝑈
)
2
[
1 −𝑎
−𝑎 (𝑎2
+
1
8
)
]
𝑪̃ 𝒂 =
𝑏
𝑈
[
0 1
0 (
1
2
− 𝑎)
] +
𝑏
2𝑈
𝐶( 𝑘)[
4 2(1 − 2𝑎)
−2(1+ 2𝑎) −(1 − 2𝑎)(1 + 2𝑎)
]
𝑲̃ 𝒂 = 2𝐶( 𝑘)[
0 1
0 − (
1
2
+ 𝑎)
]
And, 𝐿̅ 𝐺 and 𝑀̅ 𝐺 are an external non-dimensional disturbance vertical force and an external
disturbance anti-clockwise moment due to a typical gust. The equations of motion of the
aerofoil including the trailing edge flap may be expressed as:
[
𝑚𝑏2 𝑚𝑏2 𝑥 𝛼 𝑚 𝛽 𝑏2 𝑥 𝛽
𝑚𝑏2 𝑥 𝛼 𝐼 𝛼 𝑚 𝛽 𝑏2 𝑥 𝛽( 𝑐 − 𝑎) + 𝐼𝛽
𝑚 𝛽 𝑏2 𝑥 𝛽 𝑚 𝛽 𝑏2 𝑥 𝛽( 𝑐 − 𝑎) + 𝐼𝛽 𝐼𝛽
]
[
ℎ̈
𝑏
𝛼̈
𝛽̈]
+ [
𝑘ℎ 𝑏2 0 0
0 𝑘 𝛼 0
0 0 𝑘 𝛽
][
ℎ
𝑏
𝛼
𝛽
][
𝐿𝑏
−𝑀
−𝑀𝛽
] = [
0
0
0
]

8. Reena Thapa Page 7
The last vector on the left hand side of above equation represents the generalised
aerodynamic restoring moments.
The generalised aerodynamic restoring moments can be expressed as:
[
𝐿𝑏
−𝑀
−𝑀𝛽
] = 𝜋𝜌𝑏3
𝑈2
(
𝑀̃ 𝑎
[
ℎ̈
𝑏
𝛼̈
𝛽̈]
+ 𝐶̃𝑎
[
ℎ
𝑏
̇
𝛼̇
𝛽̇]
+ 𝐾̃𝑎 [
ℎ
𝑏
𝛼
𝛽
]
)
Where
𝑴̃ 𝒂 = (
𝑏
𝑈
)
2
[
1 −𝑎
𝑇1
𝜋
−𝑎 (𝑎2
+
1
8
) 2
𝑇13
𝜋
𝑇1
𝜋
2
𝑇13
𝜋
−
𝑇3
𝜋2]
, 𝐶̃𝑎 = 𝐶̃𝑎𝑛𝑐 + 𝐶̃𝑎𝑐−𝑛𝑡𝑙 + 𝐶̃𝑎𝑐−𝑡𝑙 𝐶( 𝑘),
𝑪̃ 𝒂𝒏𝒄 =
𝑏
𝑈
[
0 1 −
𝑇4
𝜋
−1 0
𝑇15
𝜋
𝑇4
𝜋
−
𝑇15
𝜋
0 ]
, 𝐶̃𝑎𝑐−𝑛𝑡𝑙 =
𝑏
𝑈
[
0
1
−
𝑇4
𝜋
][1 (
1
2
− 𝑎)
𝑇11
2𝜋
]
𝑪̃ 𝒂𝒄−𝒕𝒍 =
𝑏
𝑈
[
2
−(1 + 2𝑎)
𝑇12
𝜋
][1 (
1
2
− 𝑎)
𝑇11
2𝜋
]
And
𝑲̃ 𝒂 = 𝑲̃ 𝒂𝒏𝒄 + 𝑲̃ 𝒂𝒄−𝒏𝒕𝒍 + 𝑲̃ 𝒂𝒄−𝒕𝒍 𝐶( 𝑘), 𝑲̃ 𝒂𝒏𝒄 =
[
0 0 0
0 −1
𝑇4
𝜋
0
𝑇4
𝜋
𝑇5
𝜋2]
𝑲̃ 𝒂𝒄−𝒏𝒕𝒍 = [
0
1
−
𝑇4
𝜋
][0 1
𝑇10
𝜋
] , 𝐾̃𝑎𝑐−𝑡𝑙 = 2
[
1
− (
1
2
+ 𝑎)
𝑇12
𝜋 ]
= [0 1
𝑇10
𝜋
]
With 𝑇𝑖 = 𝑇𝑖( 𝑐, 𝑎), i = 1,2,3… 15, defined inTable 1 and 𝐶(𝑘) is Theodorsen’s lift deficiency
function.

9. Reena Thapa Page 8
Table 1: Table of Theodorsen’s T-functions,𝑇𝑖, 𝑖 = 1,2,3.3 … ,∅ 𝑐 = 𝑐𝑜𝑠−1
𝑐, 𝑐 =distance of
the control surface hinge line from mid-chord in semi-chords, 𝑎 =distance of the elastic axis
from mid-chord in semi-chords.
Also, The equations of motion can also be expressed as:
(𝐌 + 𝑺𝒒𝐌̃ 𝒂)𝐱̈ + (𝐂 + 𝑺𝒒𝐂̃ 𝒂)𝐱̇ + (𝐊 + 𝑺𝒒𝐊̃ 𝒂)𝐱 = −[ 𝐿̅ 𝐺 𝑀̅ 𝐺 𝑀̅ 𝛽𝐺 ]
𝑇
Where
𝑆 = 2𝜋𝑏2
, 𝑞 =
1
2
𝜌𝑈2
and 𝐱 = [ℎ/𝑏 𝛼 𝛽] 𝑇

10. Reena Thapa Page 9
3.SimulationProcedure
Step 1: In order to simulate the governing equations of motion of the two degrees of freedom
vibration model, they first were expressed as:
[ℎ̈ 𝑏⁄
𝛼̈
] = [
1 𝑥 𝑎
𝑥 𝑎 𝐼𝑎 𝑚𝑏2⁄
]
−1
+ {[
𝐿̅ 𝐺 𝑏
𝑀̅ 𝐺
] − [
𝑘ℎ 𝑚⁄ 0
0 𝑘 𝑎 𝑚𝑏2⁄
][
ℎ 𝑏⁄
𝛼
]}
This can also be written as:
[ℎ̈/𝑏
𝛼̈
] = [
1 𝑥 𝛼
𝑥 𝛼 𝑟𝛼
2]
−1
([
𝐿̅ 𝐺 𝑏
𝑀̅ 𝐺
] − [
𝜔ℎ0
2
0
0 𝑟𝛼
2
𝜔 𝛼0
2 ] [
ℎ/𝑏
𝛼
])
Or as:
[ℎ̈ 𝑏⁄
𝛼̈
] =
1
𝑟𝑎
2 − 𝑥 𝑎
2
[
𝑟𝑎
2
−𝑥 𝑎
−𝑥 𝑎 1
]
−1
{[
𝐿̅ 𝐺 𝑏
𝑀̅ 𝐺
] − [
𝜔ℎ0
2
0
0 𝑟𝑎
2
𝜔 𝑎0
2 ] [
ℎ 𝑏⁄
𝛼
]}
≡
1
𝑅 − 𝑆2
[
𝑅 −𝑆
−𝑆 1
]{[
𝐿̅ 𝐺 𝑏
𝑀̅ 𝐺
] − [
𝐾 0
0 𝑃
] [
ℎ 𝑏⁄
𝛼
]}
Step 2: Matlab SIMULINK was used to simulate the dynamics of a two-degree of freedom
vibration model with governing equations of motion given by:
[
1 𝑥 𝛼
𝑥 𝛼 𝐼 𝛼 /𝑚𝑏2] [ℎ̈/𝑏
𝛼̈
] + [
𝑘ℎ/𝑚 0
0 𝑘 𝛼/𝑚𝑏2] + [
ℎ/𝑏
𝛼
] = [
𝐿̅ 𝐺 𝑏
𝑀̅ 𝐺
]
As listed in Table 2, the parameters with their corresponding values were assumed and any
unlisted parameters were appropriately assumed.
[ 𝐿̅ 𝑏
𝑀̅
] =
𝜋𝜌𝑏2
𝑚
(
𝑈
𝑏
)
2
(𝑀̃ 𝑎 [
ℎ̈
𝑏
𝛼̈
] + 𝐶̃𝑎 [
ℎ̇
𝑏
𝛼̇
] + 𝐾̃ 𝑎 [
ℎ
𝑏
𝛼
])
Parameter Value
𝑥 𝛼 0.2
𝑟𝛼
2
= 𝐼 𝛼 /𝑚𝑏2
0.25
𝜔ℎ0
2
= 𝑘ℎ/𝑚 3300
𝜔 𝛼0
2
= 𝑘 𝛼/𝐼 𝛼 10000
𝜇 = 𝑚/𝜋𝜌𝑏2
10
𝑎 -0.4
𝑚 𝛽/𝑚 0.2
𝑥 𝛽 0.15
𝑘 𝛽/𝐼𝛽 40000
Table 2: Parameters and their corresponding values
Step 3: The aerodynamic forces and moments were written as generalised aerodynamic
restoring moments which are defined by the relations:

11. Reena Thapa Page 10
[ 𝐿̅ 𝑏
𝑀̅
] =
𝜋𝜌𝑏2
𝑚
(𝐌̃ 𝑎 [
ℎ̈
𝑏
𝛼̈
] + 𝐂̃ 𝑎−𝑛𝑐 [
ℎ̇
𝑏
𝛼̇
] +
𝜋𝜌𝑏2
𝑚
[
𝐿̃ 𝑐 𝑏
𝑀̃ 𝑐
]),
𝐌̃ 𝑎 = [
1 −𝑎
−𝑎 (𝑎2
+
1
8
)
], and 𝐂̃ 𝑎−𝑛𝑐 =
𝑈
𝑏
[
0 1
0 −(
1
2
− 𝑎)
]
And
[
𝐿̃ 𝑐 𝑏
𝑀̃ 𝑐
] = 2 [
1
− (
1
2
+ 𝑎)
] 𝐶( 𝑘)(
𝑈
𝑏
[1 (
1
2
− 𝑎)]
𝑑
𝑑𝑡
[
ℎ
𝑏
𝛼
] + (
𝑈
𝑏
2
) [0 1] [
ℎ
𝑏
𝛼
])
Figure 3.1: The two degree of freedom model without C(k)
Step 4: The Theodoresen function C(k) was assumed as equal to zero and the generalised
aerodynamic restoring moments were included into the SIMULINK model of the governing
dynamics.
𝐶( 𝑘) ≈ 1 −
0.165( 𝑖𝑘)
𝑖𝑘 + 0.0455
−
0.335( 𝑖𝑘)
𝑖𝑘 + 0.3
The fact that ik is equivalent to bs/U was employed where s is the Laplace transform
variable and using only pure integrators, a SIMULINK model for a system was set up with a
input/output transfer function that is given by:
𝑇( 𝑠) = 1 −
0.165𝑠
𝑠 +
0.0455𝑈
𝑏
−
0.335𝑠
𝑠 +
0.3𝑈
𝑏

12. Reena Thapa Page 11
Figure 3.2: The two degree of freedom model with C(k)
Figure 3.3:The block diagram for C(k)
Step 5: A SIMULINK model was developed for four inputs-two output system with inputs
ℎ
𝑏
, 𝛼,
ℎ̇
𝑏
𝑎𝑛𝑑 𝛼̇, and outputs 𝐿̃ 𝑐 𝑏 and 𝑀̃ 𝑐.
Step 6: The sub system developed in Step 4 was employed and the simulation of the two-
degrees of freedom flutter model including the generalised aerodynamic restoring moments
was carried out.
The equations of motion were written as:
1
Out1
1
s
Integrator1
1
s
Integrator
J
Gain3
L
Gain2
0.335
Gain1
0.165
Gain
1
In1

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[
1 +
1
𝜇
𝑥 𝛼 −
𝑎
𝜇
𝑥 𝛼 −
𝑎
𝜇
𝑟𝛼
2 +
𝑎2
𝜇
+
1
8𝜇]
[ℎ̈/𝑏
𝛼̈
] +
𝑈
𝜇𝑏
[
0 1
0 (
1
2
− 𝑎)] [ℎ̇/𝑏
𝛼̇
] + [
𝜔ℎ
2
0
0 𝑟𝛼
2 𝜔 𝛼
2
][
ℎ/𝑏
𝛼
] + [
𝐿̅ 𝑐 𝑏
𝑀̅ 𝑐
] = [
𝐿̅ 𝐺 𝑏
𝑀̅ 𝐺
]
After including aerodynamic inertia and aerodynamic damping,
[
ℎ̈
𝑏
𝛼̈
] =
1
𝑅𝐼 − 𝑆2
[
𝑟𝛼
2
+
𝑎2
𝜇
+
1
8𝜇
−𝑥 𝛼 +
𝑎
𝜇
−𝑥 𝛼 +
𝑎
𝜇
1 +
1
𝜇 ]
x
{[
𝐿̅ 𝐺 𝑏
𝑀̅ 𝐺
] − [
𝐿̅ 𝑐 𝑏
𝑀̅ 𝑐
] −
𝑈
𝜇𝑏
[
0 1
0
1
2
− 𝑎
] ℎ̇ 𝑏⁄
𝛼̇
− [
𝐾 0
0 𝑃
][
ℎ 𝑏⁄
𝛼
]}
Where 𝑅 = 𝑟𝛼
2
+
𝑎2
𝜇
+
1
8𝜇
, 𝑆 = 𝑥 𝑎 −
𝑎
𝜇
, 𝐼 = 1 +
1
𝜇
, 𝐾 = 𝜔ℎ
2
and 𝑃 = 𝑟𝛼
2
𝜔 𝛼
2
.
Step 7: Assuming the parametric values Listed in Table 2 and assuming the non-listed
parameter values appropriately, the system was simulated for 𝑈̅ =
𝑈
𝑏
= 0, 10, 20, 50, 100, 150,
200.
Step 8: Finally, the Step 1-Step 7 was employed and the effect of the trailing edge flap that is
hinged at the three quarter chord point (𝑐 = 0.5) was included in the SIMULINK model of
the two-degree of freedom flutter model.
Figure 3.4: The three degree of freedom model with the trailing edge flap.

14. Reena Thapa Page 13
4. Results
4.1 The Simulation results of two degree of freedom aerofoil model without
C(k)
For U/b=0
Figure 4.1(a) h/b against Time (b)Alpha against Time
For U/b=10
Figure 4.2(a) h/b against Time (b)Alpha against Time
For U/b=20
Figure 4.3(a) h/b against Time (b)Alpha against Time
For U/b=50

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Figure 4.4(a) h/b against Time (b)Alpha against Time
For U/b=100
Figure 4.5(a) h/b against Time (b)Alpha against Time
For U/b=150
Figure 4.6(a) h/b against Time (b)Alpha against Time

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For U/b=200
Figure 4.7(a) h/b against Time (b)Alpha against Time
4.2 The Simulation results of two degree aerofoil model with C(k)
For U/b=0
Figure 4.8(a) h/b against Time (b)Alpha against Time
For U/b=10
Figure 4.9(a) h/b against Time (b)Alpha against Time
For U/b=50

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Figure 4.10(a) h/b against Time (b)Alpha against Time
For U/b=100
Figure 4.11(a) h/b against Time (b)Alpha against Time
For U/b=150
Figure 4.12(a) h/b against Time (b)Alpha against Time

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For U/b=200
Figure 4.13(a) h/b against Time (b)Alpha against Time
4.3 The Simulation results of three degree of freedom aerofoil with flap
For U/b=0
Figure 4.14: h/b against Time
Figure 4.15:Alpha against Time

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Figure4.16: Beta against Time
For U/b=20
Figure 4.17: h/b against Time
Figure 4.18:Alpha against Time

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Figure4.19: Beta against Time
For U/b=50
Figure 4.20: h/b against Time
Figure 4.21:Alpha against Time

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Figure4.22: Beta against Time
For U/b=100
Figure 4.23: h/b against Time
Figure 4.24:Alpha against Time

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Figure4.25: Beta against Time
For U/b=150
Figure 4.26: h/b against Time
Figure 4.27:Alpha against Time

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Figure4.28: Beta against Time
5. Discussion
The graphs from Figure 4.1- Figure 4.28 are the results obtained from the 3 sets of
simulations:(1)two degree of freedom aerofoil with C(k) (2)two degree of freedom aerofoil
without C(K) and (3) three degree of freedom aerofoil with flap. Each set of results are
obtained for a varying values of 𝑈̅ =
𝑈
𝑏
= 0, 10, 20, 50, 100, 150, 200. The y-axis consist of
non- dimensional amplitudes caused by translational disturbance, h/b, non-dimensional
amplitudes caused by pitch angle disturbance, 𝛼 and non-dimensional amplitudes caused by
the flap angle disturbance, β.
The results in section 4.1 are obtained from first set of simulations when the complex
function, C(k) are not considered. Figure 4.1(a) shows the zero amplitudes displacement until
1 s and the amplitude displacement can be observed upto 0.0006 throughout till 10 s when
U/b=0. This shows high instability. Figure 4.1(b) shows the amplitude displacement caused
by the angle of attack when U/b=0. There is zero displacement until 1 s, after which serious
fluctuations in its amplitude displacement can be observed upto a magnitude of 0.0002
throughout till 10 s. In both cases, the instability remains throughout the total time. However,
when U/b= 10, the amplitude displacement can be observed up to a 0.00055 at Figure 4.2(a)
and upto a 0.0002 at Figure 4.2(b) which dies eventually with time. This is caused by
damping and making the system stable eventually with time. As U/b increases, it can be
observed that the damping occurs earlier in the system making the system stable.
For the second set of simulations, even though the complex function is considered, the trends
of graphs in section 4.2 are similar to that obtained in section 4.1. However the trend changes
when U/b=100. The instability begins to occur at 6s which the system fails to dampen. At
U/b=200, as seen in Figure 4.13, the instability begins to occur at a much later time i.e. 9.8 s.

24. Reena Thapa Page 23
For the third set of simulations, that was run for the three degree of freedom of aerofoil with
the trailing edge flap, the instability or flutter can be observed throughout the system as seen
from Figure 4.14-4.28. Also the amplitudes are found to be displaced with higher magnitude
as U/b increases, indicating the dangerous and unstable vibration regime of the system.
6. Conclusion
The fluttering analysis of the aerofoil was carried out employing three particular cases of
simulation for two degrees and three degrees of freedom. In the first set of simulation,
Complex function, C(k) was assumed to be zero for the flutter analysis of aerofoil for two
degrees of freedom, which appeared to be almost stable. The second set of simulation was
carried out which was similar to the first simulation but complex function was considered. As
a result, Some instabilities were added to the system due to extra forces and moments.
Whereas the set of simulation was operated for the three degree of freedom aerofoil which
included the trailing edge flap. This made the system very unstable causing the flutter
throughout the system, which can be assumed to be caused by the deflection of the flap by
angle 𝛽.
7. References
Bergami, L.,2008. Aeroservoelastic Stability of a 2D Airfoil Section equipped with a Trailing
Edge Flap. Roskilde, Denmark: Technical University of Denmark.
Bisplinghoff, R.L., Carter, A.H. & Halfman, R.L.,1996. Aeroelasticity. Addison-Wesley.
Vepa, R,2014. Computer Aided Simulation Tutorial Exercise. Lab handout. London: Queen
Mary University of London.

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