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ISTANBUL TECHNICAL UNIVERSITY FACULTY OF AERONAUTICS AND ASTRONAUTICS

GRADUATION PROJECT

SEPTEMBER, 2020

AEROELASTIC ANALYSIS OF A FIXED WING

Thesis Advisor: Asst. Prof. Dr. Özge ÖZDEMİR

Gamze ÖZEN

Department of Astronautical Engineering

Anabilim Dalı : Herhangi Mühendislik, Bilim

Programı : Herhangi Program

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SEPTEMBER 2020

ISTANBUL TECHNICAL UNIVERSITY FACULTY OF AERONAUTICS AND ASTRONAUTICS

AEROELASTIC ANALYSIS OF A FIXED WING

GRADUATION PROJECT

Gamze ÖZEN

(110150302)

Department of Astronautıcal Engineering

Anabilim Dalı : Herhangi Mühendislik, Bilim

Programı : Herhangi Program

Thesis Advisor: Asst. Prof. Dr. Özge ÖZDEMİR

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Thesis Advisor : Asst. Prof. Dr. Özge ÖZDEMİR ..............................

İstanbul Technical University

Jury Members : Prof. Dr. Metin Orhan KAYA .............................

İstanbul Technical University

Prof. Dr. Zahit MECİTOĞLU ..............................

İstanbul Technical University

Gamze Özen, student of ITU Faculty of Aeronautics and Astronautics student ID

110150302, successfully defended the graduation entitled “AEROELASTIC

ANALYSIS OF A FIXED WING”, which she prepared after fulfilling the

requirements specified in the associated legislations, before the jury whose signatures

are below.

Date of Submission : 07 September 2020

Date of Defense : 14 September 2020

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To my family and my dearest friends,

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FOREWORD

I would like to express my gratitude to my advisor Asst. Prof. Dr. Özge Özdemir for

her kindness, understanding and support during the preparation process of this thesis

from the thesis selection to defence of it.

I also would like to thank my dear friend Kübra Tezcan for her endless support and

great knowledge about the Matlab. My special thanks to my lovely friends Elif

Yıldırım and Ruveyda Memiş who supported and cared me throughout the

preparation of this thesis.

I would like to express my sincere gratitude to my dear friend Merve Özdemir for her

caring when I felt alone and helpless.

Finally, I owe my gratitude to my mom and my sisters for their endless love.

September 2020

Gamze ÖZEN

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TABLE OF CONTENTS

Page

TABLE OF CONTENTS ..................................................................................... vii

SUMMARY ........................................................................................................... xi 1. INTRODUCTION ...............................................................................................1

1.1 Purpose of Thesis ........................................................................................... 1 1.2 Literature Review ........................................................................................... 1

2. THEORY OF DIVERGENCE ...........................................................................3 2.1 Purpose .......................................................................................................... 3

2.2 Uniform Lifting Surface ................................................................................. 3

3. THEORY OF FLUTTER ...................................................................................7 3.1 Purpose .......................................................................................................... 7 3.2 Two-Degree-of-Freedom Flutter ..................................................................... 7

3.3 k-Method ........................................................................................................ 8 3.4 Assumed Modes ............................................................................................10

4. AEROELASTIC TOOL ................................................................................... 12 4.1 Wing Geometry .............................................................................................12

4.2 Solution Method for Flutter ...........................................................................13 4.3 User Guide of Aeroelastic Tool .....................................................................15

5. VALIDATION OF THE AEROELASTIC TOOL .......................................... 17 5.1 Validation of the Divergence .........................................................................17

5.2 Validation of the Flutter.................................................................................19

6. CONCLUSIONS AND RECOMMENDATIONS ............................................ 21

REFERENCES ..................................................................................................... 22 APPENDICES ...................................................................................................23

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LIST OF TABLES

Page

Table 4.1 : Wing Geomerty of Ref [4] .................................................................... 12

Table 4.2 : Flight Condition of Ref [4].. ................................................................. 12

Table 5.1 : Comparison of Divergence Speed ......................................................... 18

Table 5.2 : Comparison of Flutter Speed ................................................................ 20

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LIST OF FIGURES

Page

Figure 4.1 : 3D Wing with Zero Twist. .................................................................. 13

Figure 4.2 : Menu Section of the Aeroelastic Tool. ................................................ 16

Figure 5.1 : Matlab Solution of the Divergence Analysis. ...................................... 17

Figure 5.2 : Twist Angle with Divergence Speed. .................................................. 17 Figure 5.3 : Comparison of Twist Angle Along Non-dimensional Span. ................ 18

Figure 5.4 : Reduced Frequency Graph. ................................................................. 19 Figure 5.5 : Matlab Solution for the Flutter ............................................................ 19

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AEROELASTIC ANALYSIS OF A FIXED WING

SUMMARY

The aim of this study is to develop a code that can calculate the static and dynamic

instability elements on a fixed wing. In order to achieve this goal, aeroelastic

analyzes were made on a 3-dimensional, uniform and rectangular wing. First of all,

the wing to be studied was determined, the twist occurred on this wing was plotted

along the wing span, the wing divergence velocity and the resulting wing tip twist

angle-velocity graph were calculated. Thus, the analysis of the wing in terms of static

aeroelasticity has been completed.

Then, time-dependent physical factors were included in the problem and the flutter

speed and frequency were calculated. In this calculation, the torsion mode and

bending mode are used together with the approximate Theodorsen’s function and the

k-method. The necessary parameters for the approximate Theodorsen’s function

were calculated with approximate modes equations, and finally, the reduced

frequency value with zero damping coefficient was obtained. This value was used to

obtain the flutter speed and frequency by the k-method. The data were compared

with the previous study results, the accuracy of the improved aeroelastic tool was

tested. Thus, it has been concluded that the outputs of the tool are similar to other

methods.

Since this tool is low cost and easy to use, it can be applied to different wing

geometries for aeroelastic analysis in engineering. It can be a tool that students can

use to design wing in aircraft design projects, or if it is developed, it can contribute to

the academic literature.

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SABİT BİR KANATIN AEROELASTİK ANALİZİ

ÖZET

Bu çalışmanın amacı sabit bir kanat üzerinde oluşan statik ve dinamik kararsızlık

unsurlarını hesaplayabilen bir kod geliştirmektir. Bu amacı gerçekleştirebilmek içn 3

boyutlu, uniform ve dikdörtgen bir kanat üzerinde aeroelastik analizler yapılmıştır.

Öncelikle üzerinde çalışılacak kanat belirlenmiş, belirlenen bu kanat üzerinde

meydana gelen burulma kanat açıklığı boyunca grafiğe dökülmüş, kanat diverjans

hızı ve buna bağlı kanat ucu burulma açısı-hız grafiği hesaplanmıştır. Böylece statik

aeroelastisite açısından kanadın analizi tamamlanmıştır.

Daha sonra zamana bağlı fiziksel faktörler işleme dahil edilmiş, çırpınma hızı ve

frekansı hesaplanmıştır. Bu hesap yapılırken burulma modu ve eğilme modu,

yaklaşık Theodorsen fonksiyonu ve k-yöntemi ile birlikte kullanılmıştır. Yaklaşık

Theodorsen fonksiyonu için gerekli parametreler yaklaşık mod denklemleriyle

hesaplanmış en sonunda sönümleme katsayısının sıfır olduğu indirgenmiş frekans

değeri elde edilmiştir. Bu değer k-yöntemi ile çırpınma hızı ve frekansını elde

etmekte kullanılmıştır. Veriler önceki çalışma sonuçlarıyla karşılaştırılmış,

geliştirilmiş aeroelastik aracın doğruluğu test edilmiştir. Böylece aracın çıktılarının

diğer yöntemlerle benzerlik gösterdiği sonucuna varılmıştır.

Bu araç maliyeti düşük olduğu ve kullanımı kolay olduğu için mühendislik

eğitimlerinde aeroelastik analiz yapmak için farklı kanat geometrilerine

uygulanabilir. Uçak tasarımı projelerinde öğrencilerin kanat tasarımı yapmak için

kullanabileceği bir araç olabileceği gibi geliştirilirse akademik açıdan literatüre katkı

sağlayabilir.

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1. INTRODUCTION

Aeroelasticity is a field that investigates aerodynamic, elastic and inertial forces on a

lifting surface or a flight vehicle. An aircraft can encounter static and dynamic

instability that results deformation and failure of the structure. Thus, these

instabilities can cause one of the most dangerous situations that an aircraft can

encounter during flight which are divergence and flutter. Therefore, investigation of

divergence and flutter characteristics of structures is crucial in early stages of the

design.

1.1 Purpose of Thesis

The basic divergence and flutter analysis procedure of a uniform fixed wing had been

discussed through the report. Firstly, the divergence speed was calculated and twist

distribution along non-dimensional span were obtained. Secondly, flutter speed and

frequency were calculated for the same wing. Gathered results were compared with

the previous studies since no experimental data is available for this wing in order to

validate with results. Therefore, whether the aeroelastic tool is suitable for aeroelastic

analysis in early stage designs can be tested.

1.2 Literature Review

Lorem In 1934, Theodore Theodorsen laid out the theory of aeroelastic flutter for a

typical section with three degrees of freedom and provided a method for its practical

solution. At the time, this solution method offered the only means of solving the

flutter problem in an exact closed-form way.

Theodorsen and Garrick published a report that includes mathematical explanation of

the flutter in 1940. In this report, experimental datas and calculations are shown

about flutter speed. In 1942, Theodersen and Garrick stated that mass balancing is

not important for the aeroelastic analysis. They also showned profound effect of the

internal structure of damping.

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In 1945, Goland analyzed the flutter speed of a uniform cantilevered wing by

integrating of the differential equations for the wing motion with using approximate

method. In the study, compressiblity of air and aerodynamic wing loading are

omitted due the complexity of the dynamic problem. Goland stated that the influence

of the compressiblity of the air and wing loading are minor extent.

In 1948, analysis of the divergence of swept untapered and tapered wings with

stiffness had been performed experimentally by Diederich and Budiansky. In the

study, the location of the elastic axis is found tp affect of the divergence speed. Also,

wing geometry parameters effected the divergence speed.

Fung developed the approximate Theodorsen’s function in 1955. With this new

approximate function, more accurate flutter results can be obtained. In 1958, Fung

also used the Galerkin Method to give fine details about the nature of flutter.

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2. THEORY OF DIVERGENCE

2.1 Purpose

The concept of aeroelasticity is divided into two: Static aeroelasticity and dynamic

aeroelasticity. The concept of divergence is the static instability that occurs on the

lifting-surface under static aeroelasticity field. Divergence is the term given to the

twist angle of the lifting-surface goes to infinity. In accordance with the study, the

theory of divergence on a 3D fixed wing will be examined.

2.2 Uniform Lifting Surface

For an unswept uniform elastic lifting surface that is modelled as a beam with the

spanwise coordinate along the elastic axis is denoted by y, is presumed fixed at the

root and free at the tip. The distributed lift force and pitching moment per unit span

exerted by aerodynamic forces along a slender beam-like wing must be calculated.

For isotropic beams, the incidence angle may be a function of the spanwise

coordinate because of the possibility of elastic twist. The total applied, distributed,

twisting moment per unit span about elastic axis is denoted as M'(y), which is

positive leading-edge-up and given by

(2.1)

Where L' and M'ac are the distributed spanwise lift and pitching moment, mg is the

spanwise weight distribution, and N is the normal load factor for the case in which

wing is the level. Thus, N can be written as

(2.2)

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Where Az is the z component of the wing’s inertial acceleration, W is the total weight

of the aircraft, and L is the total lift.

The distributed aerodynamic loads can be written in coefficient form as

(2.3)

(2.4)

where the freestream dynamic pressure, q, is

(2.5)

The sectional lift and pitching-moment coefficients can be related to the angle of

attack α by an appropriate aerodynamic theory as some functions and ,

where the functional relationship generally involves integration over the planform.

However, for smalle values of α simpler form can be used which the lift-curve slope

ssumed to be a constant along the span.

(2.6)

where a denotes the constant sectional lift–curve slope, and the sectional-moment

coefficient is assumed to be a constant along the span.

The angle of attack is represented by two components. The first is a rigid

contribution, αr, from a rigid rotation of the surface. The second component is the

elastic angle of twist θ(y). Hence

(2.7)

To analyze the static behaviour of the wing it is appropriate to simplify the

fundamental constitutive relationship of torsional deformation,

(2.8)

where GJ is the effective torsional stiffness and T is the twisting moment about the

elastic axis.

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A static equation of moment equilibrium about the elastic axis can be obtained by

equating the rate of change of twisting moment to the negative of the applied torque

distribution.

(2.9)

With substituting equations, finally inhomogeneous, second-order, ordinary

differential equation with constant coefficients are obtained.

(2.10)

With the boundary conditions;

(2.11)

(2.12)

To simplify the notation, let

(2.13)

(2.14)

The static-aeroelastic equilibrium equation now can be written as

(2.15)

The general solution to this linear ordinary differential equation is

(2.16)

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Applying the boundary conditions, with the condition

(2.17)

(2.18)

The elastic-twist distribution becomes

(2.19)

While becomes infinite as approaches π/2. This phenomenon is called “torsional

divergence” and dynamic pressure equals to the divergence dynamic pressure at this

point.

The divergence dynamic pressure and velocity is;

(2.20)

(2.21)

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3. THEORY OF FLUTTER

3.1 Purpose

The interaction of inertial, structural and aerodynamic forces in dynamic

aeroelasticity can be investigated with time-dependent elements, while in static

aeroelasticity, time-dependent physical elements are not studied. The most dangerous

of the interactions described above is flutter, which is one of the self-induced forms

of vibration and ultimately causes structural distortion. In other words, flutter is a

state of aeroelastic instability caused by the coincidence of the bending and torsion

modes of the control surfaces of aircraft and missiles. There are four methods to

determine the flutter velocity and flutter frequency which will be studied.

3.2 Two-Degree-of-Freedom Flutter

The equations of motion for multi-degree-of-freedom flutter,

(3.1)

(3.2)

the motion is simple harmonic as represented by

(3.3)

(3.4)

The corresponding lift and moment can be written as

(3.5)

(3.6)

Substituting these time-dependent functions into the equations of motion, apair of

algebraic equations for the amplitudes of h and θ in the form are obtained.

(3.7)

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(3.8)

Finally a pair of homogeneous, linear, algebraic equations for and are obtained.

(3.9)

(3.10)

The inertia terms are symbolically simplified by defining the dimensionless

parameters,

(3.11)

(3.12)

The flutter determinant is

(3.13)

Where , the root of the quadratic equation of the flutter determinant.

3.3 k-Method

Subsequent to Theodorsen’s analysis of the flutter problem, numerous schemes were

devised to extract the roots of the “flutter determinant” and thereby identify the

stability boundary. To incorporate this form of structural damping into the analysis

(3.14)

(3.15)

where the dissipative structural damping terms are

(3.16)

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(3.17)

(3.18)

(3.19)

The damping coefficients and have representative values from 0.01 to 0.05

depending on the structural configuration.

The flutter determinant is,

Where

The two unknowns of this quadratic equation are complex, denoted by

The primary difference is that the numerical results consist of two pairs of real

numbers, and , which can be plotted versus airspeed U or a suitably

normalized value such as or “reduced velocity” 1/k.

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Plots of the damping coefficients g_1 and g_2 versus airspeed can indicate the

margin of stability at conditions near the flutter boundary, where g_1 or g_2 is equal

to zero.

3.4 Assumed Modes

In industry it is typical to use the finite element method as a means to realistically

represent aircraft structural dynamics. Although it is certainly possible to conduct

full finite-element flutter analyses, flutter analysis based on a truncated set of the

modes of the stucture is examined in the study.

For a beam with bending rigidity EI and torsional rigidity GJ, the strain energy is,

(3.20)

The assumed modes is the set of uncoupled cantilevered beam, free-vibration modes

for bending and torsion, such that

(3.21)

(3.22)

Where and are the numbers of modes used to represent bending and torsion,

respectively; and are the generalized coordinates associated with bending and

torsion, respectively; and and are the bending and torsion mode shapes,

respectively.

Let with the Theodersen’s theory, the equations of motion obtained

after several intermediate processes are as follows.

(3.23)

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(3.24)

, , , and are defined the quantities on the Theodersen’s theory.

And the fundamental bending and torsion frequencies are

(3.25)

(3.26)

In this section, bending and torsion mode shapes are discussed. With this method,

flutter analysis will give a good prediction. However, addition of the higher modes

would make the analysis more accurate.

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4. AEROELASTIC TOOL

4.1 Wing Geometry

The model is very high-aspect ratio wing.

Table 4.1: Wing Geomerty of Ref [4].

Parameter Value

Half Span 16 m

Chord 1 m

Mass/Unit Length 0.75 kg/m

Moment of Inertia 0.1 kg.m

Spanwise Elastic Axis 50% Chord

Center of Gravity 50% Chord

Bending Rigidity

Torsional Rigidity

Table 4.2: Flight Condition of Ref [4].

Property Value

Air Density 0.0889

Air Speed 25 m/s

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In 2001 Patil et. al [4] produced the geometric properties and flight condition

parameter for the HALE wing are given in Table 4.1 and Table 4.2. For the 3D

configuration, usage of Naca0012 airfoil is determined for its symmetricity.

Figure 4.1: 3D Wing with Zero Twist.

The geometric structure of the wing with zero twist angle are shown in figure 4.1.

4.2 Solution Method for Flutter

For flutter analysis bending and torsion modes are used with the approximate

Theodorsen’s function and k-method.

The equation for bending and torsion modes are as follows.

(4.1)

(4.2)

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are obtained for clamped-free beam. [2]

The approximate Theodorsen’s function is as follows.

(4.3)

The assumed modes for the Theodorsen’s theory,

(4.4)

(4.5)

(4.6)

(4.7)

The flutter determinant is,

(4.8)

With the help of the k-method, the reduced frequency value are obtained for the zero damping

coefficient. Thus, the real of the determinant for this reduced frequency value are used to obtain

the flutter frequency.

(4.9)

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(4.10)

(4.11)

4.3 User Guide of Aeroelastic Tool

First of all, the user must determine what aeroelastic analysis type to run. In the figure 4.2 the

menu section are shown for this determination: Static or Dynamic.

To calculate divergence speed and twist distribution along the wing span, flutter speed and

frequency, wing geometry parameters and flight conditions must be entered before run the

code. The angle of attcak value must be in degree.

Figure 4.2: Menu Section of the Aeroelastic Tool

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5. VALIDATION OF THE AEROELASTIC TOOL

5.1 Validation of the Divergence

After choosing the static aeroelasticity, the results are as follows.

Figure 5.1: Matlab Solution of the Divergence Analysis

Figure 5.2: Twist Angle with Divergence Speed

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Figure 5.3: Comparison of Twist Angle along Non-dimensional Span

Table 5.1: Comparison of Divergence Speed

Parameter Aeroelastic

Tool

Analysis of Ref

[4]

Analysis of Ref

[3]

Divergence

Speed, m/s 38.0205 37.29 37.15

Compared with reference analyzes, it is obtained 1.9590 and 2.3432 percentage of

error respectively.

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5.2 Validation of the Flutter

After choosing the static aeroelasticity, the results are as follows.

The damping coefficient is 0.0003991 for the reduced frequency 0.315.

Figure 5.4: Reduced Frequency Graph

Figure 5.5: Matlab Solution for the Flutter

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Table 5.2: Comparison of Flutter Speed

Parameter Aeroelastic Tool Analsis of Ref [4] Analysis of Ref [3]

Flutter Speed, m/s 33.6028 32.21 32.51

Flutter Frequency,

rad/s 21.1698 22.61 22.37

For the flutter speed compared with previous studies, it is obtained 4.3241 and 3.3614

percentage of error respectively.

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6. CONCLUSIONS AND RECOMMENDATIONS

This thesis is dedicated to develop an aeroelastic tool for the anlaysis of a fixed wing both

statically and dynamically. Gathered results are compared with the previous studies, it is seen

that the errors are not exceed 5% which means aeroelastic tool gives noticeably reasonable

results. However, the errors should be reduced to obtain more realistic results. Since the

compared studies are high-fidelity analysis, the results are less realistic. Using of assumed

modes method with higher modes for the flutter prediction would be given more accurate and

realistic results. Also, the approximate Theodorsen’s function can be the reason of these errors.

Thus, new approximate functions that are developed by Fung and Peters would give more

realistic results.

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REFERENCES

[1] Hodges, D.H and Pierce, G.A. (2011). Introduction to Structural Dynamics and

Aeroelasticity. Cambridge University Press.

[2] Hallissy, B. and Cesnik, C. (2011). High-fidelity Aeroelastic Analysis of Very Flexible

Aircraft. Collection of Technical Papers - AIAA/ASME/ASCE/AHS/ASC

Structures, Structural Dynamics and Materials Conference. 10.2514/6.2011-

1914.

[3] Patil, M. J. (1997). “Aeroelastic Tailoring of Composite Box Beams,” Proceedings of the

35th Aerospace Sciences Meeting and Exhibit, AIAA Paper 97-0015, Reston,

VA.

[4] Patil, M., Cesnik, C. & Hodges, D. (2001). Nonlinear Aeroelasticity and Flight Dynamics

of High-Altitude Long-Endurance Aircraft. Journal of Aircraft - J AIRCRAFT.

38. 88-94. 10.2514/2.2738.

[5] Bisplinghoff, R. L., Ashley, H. & Halfman, R. L. (1955). Aeroelasticity.

[6] Barmby, J.G., Cunningham, H.J. & Garrick, I.E. (1951). Study of Effects of Sweep on

the Flutter of Cantilever Wings.

[7] Theodorsen, T. (1935). General Theory of Aerodynamic Instability and the Mechanism of

Flutter. NACA TR 496

[8] Goland, M. and Buffalo, N. Y. (1946). The Flutter of a Uniform Cantilever Wing.

23

APPENDICES

APPENDIX A: Aeroelastic Tool

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APPENDIX A

%CHOOSING WHAT PARAMETER TO CALCULATE Parameter = menu( 'Choose the Aeroelastic Analysis Type', 'Static',... 'Dynamic'); if Parameter == 1

%WING PARAMETERS c = 1; GJ = 10^4; EI = 2*10^4; alfar = 2*(pi()/180); U = 25; l = 16; ec = 0.5; CLbeta = 0.8; CMbeta = -0.5; rho = 0.0889; CLalfa = 6; e = 0.25;

%DIVERGENCE PRESSURE AND SPEED qd = (GJ/(e*c*CLalfa))*(pi()/(2*l))^2 Ud = sqrt((2*qd)/rho)

%Variables for MATLAB tetatip = zeros(1,41); U_variable = linspace(0,Ud,41); tetal = zeros(1,41); ly = linspace(0,l,41);

%Equations for teta q = (1/2)*rho*(U^2); lambda = (q*c*CLalfa*e/GJ)^(1/2);

%TWIST CALCULATION for i=1:41 tetal(i) = ((alfar)*(tan(lambda*l)*sin(lambda*... ly(i))+cos(lambda*ly(i))-1))*180/pi; end %TIP TWIST for i=1:41 q = (1/2)*rho*(U_variable(i)^2); lambda = (q*c*CLalfa*e/GJ)^(1/2); tetatip(i) = ((alfar)*(tan(lambda*l)*sin(lambda*... l)+cos(lambda*l)-1))*180/pi; end

subplot(2,1,1) plot(U_variable,tetatip,'p') hold on

hold off ylim([0 50]) xlabel('Velocity [m/s]') ylabel('Theta [deg]') title('Divergence')

subplot(2,1,2)

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plot(ly/l,tetal, 'p') xlabel('{\itNondimensional Span }') ylabel('{\itTwist}, \theta [\circ]') title('Twist Distribution Along The Span')

elseif Parameter == 2 %%FLUTTER CALCULATION FOR THE 3D UNIFORM WING %WING GEOMERTY AND MASS PARAMETERS b = 0.5;%Semi-chord a = 0;%Spanwise elastic axis parameter e = 0;%Center of gravity parameter l = 16;%Span EI = 2*10^4;%Bendig rigidity GJ = 10^4;%Torsional rigidity m = 0.75;%Mass/span rho = 0.0889;%Air density Ic = 0.1;%Moment of inertia about center of mass

%FLUTTER DETERMINANT PARAMETERS mu = m/(rho*pi()*b^2); A11 = 0.958641; xteta = e-a; alfa1_L = 1.87510;%Obtained from Table 3.1 Ip = Ic+m*b^2*xteta^2;%Moment of inertia about reference point r = (Ip/(m*(b^2)))^0.5;%Radius of gyration Ww = (alfa1_L)^2*(EI/(m*l^4))^0.5;%Bending coefficient Wteta = (pi()/2)*(GJ/(m*b^2*r^2*l^2))^0.5;%Torsion coefficient

k = 0.0001:0.0001:1; i = sqrt(-1);

for n = 1:1:10000 %Approximate Theodorsen Function C = (0.01365+0.2808*i*k(n)-0.5*k(n)^2)/(0.01365+0.3455*i*k(n)-k(n)^2);

lw = 1-(2*i*C)/k(n); lteta = a+i*(1+(1-2*a)*C)/k(n)+2*C/(k(n)^2); mw = a-(i*(1+2*a)*C)/k(n); mteta = a^2+1/8-(i*(0.5-a)*(1-(1+2*a)*C))/k(n)+(1+2*a)*C/(k(n)^2);

A = mu^2*r^2*Wteta^2*Ww^2; B = -(mu^2*r^2*Wteta^2 + mu^2*r^2*Ww^2 + mu*Ww^2*mteta + mu*r^2*... Wteta^2*lw); D = mu^2*r^2 + mu*mteta + mu*r^2*lw + mteta*lw - mu^2*xteta^2*... A11^2 + mu*mw*xteta*A11^2 + mu*xteta*lteta*A11^2 - mw*lteta*A11^2;

Z = [A B D] ; Roots_of_det = roots(Z) ; %DAMPING COEFFICIENTS g1(n) = imag(Roots_of_det(1))/real(Roots_of_det(1)); g2(n) = imag(Roots_of_det(2))/real(Roots_of_det(2)); end

find_g1 = find(g1>0,1); find_g2 = find(g2>0,1);

if length(find_g1) ~= 0 k_value = k(find_g1-1)

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else k_value = k(find_g2-1) end kf = 0.315; C = (0.01365+0.2808*i*kf-0.5*kf^2)/(0.01365+0.3455*i*kf-kf^2);

lw = 1-(2*i*C)/kf; lteta = a+i*(1+(1-2*a)*C)/kf+2*C/(kf^2); mw = a-(i*(1+2*a)*C)/kf; mteta = a^2+1/8-(i*(0.5-a)*(1-(1+2*a)*C))/kf+(1+2*a)*C/(kf^2);

A = mu^2*r^2*Wteta^2*Ww^2; B = -(mu^2*r^2*Wteta^2 + mu^2*r^2*Ww^2 + mu*Ww^2*mteta + mu*... r^2*Wteta^2*lw); D = mu^2*r^2 + mu*mteta + mu*r^2*lw + mteta*lw - mu^2*xteta^2*... A11^2 + mu*mw*xteta*A11^2 + mu*xteta*lteta*A11^2 - mw*lteta*A11^2;

Z = [A B D] ; Roots_of_det = roots(Z);

w_flutter = (1/real(Roots_of_det(2)))^0.5 U_flutter = b*w_flutter/kf

end