Development and Design of a Form-Adaptive Trailing-Edge for … · 2015-02-13 · given to Prof. Dr.-sc.techn. Dirk Dahlhaus who introduced me to this interesting project. In addition,

Development and Design of aForm-Adaptive Trailing-Edge for

Wind Turbine Blades

By

Alireza Taheri

A Thesis submitted toFaculty of Engineering at

Cairo University and University of Kasselin Partial Fulfillment of the

Requirements for the Degree of

Master of Sciencein

RENEWABLE ENERGY AND ENERGY EFFICIENCY

University of Kassel - Kassel, GermanyFaculty of Engineering, Cairo University - Giza, Egypt

January 2015


Wind Turbine Blades

By

Alireza Taheri




Master of Sciencein


Under the Supervision of:

Prof. Dr. -Ing. Martin LawerenzUniversity of Kassel

Prof. Dr. -Ing. Siegfried HeierUniversity of Kassel

Dr. Basman El HadidiCairo University

January 2015


Wind Turbine Blades

By

Alireza Taheri




Master of Sciencein


Approved by the Examining Committee:

Prof. Dr. -sc.techn. Dirk DahlhausUniversity of Kassel

Prof. Dr. Adel KhalilFaculty of Engineering, Cairo University

Dr. Basman ElhadidyFaculty of Engineering, Cairo University

January 2015

I hereby confirm on my honour that I personally prepared the present master thesisand carried out myself all the investigations and calculations, directly involved withit, in the Institute of Thermal Energy Technology, Department of Turbomachinery,University of Kassel. I also confirm that I have used no resources other than thosedeclared. All formulations and concepts adopted literally or in their essential contentfrom internet sources have been cited according to the rules for academic works.The support provided during the work has been indicated in acknowledgements.The master thesis is submitted in printed and electronic form. I confirm that thecontent of the digital version is completely identical to the printed version.

Signature Kassel, 15.01.2015

Acknowledgements

I would like to express my deep gratitude to my supervisor, Prof. Dr.-Ing. MartinLawerenz with whom I worked on my thesis at Institute of Thermal Energy (ITE),Department of Turbomachinery in University of Kassel. However, special thanks aregiven to Prof. Dr.-sc.techn. Dirk Dahlhaus who introduced me to this interestingproject. In addition, I appreciate my colleagues in ITE who all provided a positiveatmosphere for me to work, particularly M.Sc. Irfan Ahmed for his comments dur-ing the project. I am also grateful to my other supervisors, Dr. Basman El-hadidiand Prof. Dr.-Ing. Siegfried Heier.Last but not least in importance, my family who always colour my life. I own all ofmy success to their kind and warm encouragements.

It is not the strongest of the species that survive, nor the mostintelligent, but the one most responsive to change.”

Charles Darwin

Contents

List of Figures vi

List of Tables ix

Abstract x

Nomenclature xi

1 Introduction 11.1 Current Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Thesis Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Methodology and Outline . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Literature Review 5

3 Theoretical Background 113.1 Wind Turbine Power . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.2 Geometric and Aerodynamic Characteristics of Airfoils . . . . . . . . 123.3 Control Mechanisms in Wind Turbines . . . . . . . . . . . . . . . . . 21

3.3.1 Pitch Control . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.3.2 Stall Control . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.3.3 Yaw Control . . . . . . . . . . . . . . . . . . . . . . . . . . . 243.3.4 Shape Adaptive Control . . . . . . . . . . . . . . . . . . . . . 25

3.4 The Beam Deflection Theory . . . . . . . . . . . . . . . . . . . . . . 26

4 Design of the Form-Adaptive Airfoil 294.1 Design of the Flexible Blade Skins . . . . . . . . . . . . . . . . . . . 294.2 Design of the Actuators . . . . . . . . . . . . . . . . . . . . . . . . . 314.3 Final Structural Design . . . . . . . . . . . . . . . . . . . . . . . . . 37

5 Modeling the Form-Adaptive Airfoil 40

6 Experimental Investigations 476.1 Preliminary Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 476.2 Final Investigation in Wind Tunnel . . . . . . . . . . . . . . . . . . . 51

iv

CONTENTS

7 Flow Simulation 557.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 557.2 Geometry Generation and Conversion in Xfoil . . . . . . . . . . . . . 567.3 Flow Analysis for Designed Airfoil . . . . . . . . . . . . . . . . . . . 577.4 Comparing the results with a Standard Airfoil Deflection . . . . . . 66

8 Conclusions and Recommendations 728.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 728.2 Recommendations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

Appendix A Matlab and Fortran Scripts 74A.1 Modeling the symmetric NACA Airfoils . . . . . . . . . . . . . . . . 74A.2 Thickness of a NACA Airfoil in a given position . . . . . . . . . . . . 75A.3 Pressure Distribution along the Airfoil . . . . . . . . . . . . . . . . . 75A.4 Comparing the Cantilever Beams Deflection by One Load . . . . . . 76A.5 Comparing the Cantilever Beams Deflection by Two Point Loads . . 77A.6 Modeling the Cantilever Beam Deflection with Distributed Loads . . 79A.7 Modeling the Deflection of Two Blade Skins with Two Actuators . . 84A.8 Modeling a Flexible NACA 0012 with Two Actuators . . . . . . . . 86A.9 Modeling Two Blade Skins Deflection with an Actuator Load . . . . 88A.10 Smoothing the Geometry of the Airfoil . . . . . . . . . . . . . . . . . 90A.11 Modeling a Form Variable Airfoil Deflected with One Actuator . . . 91

Bibliography 94

v

List of Figures

1.1 The idea to use form-adaptive airfoils for wind turbine blades . . . . 3

2.1 The adaptive-form wings were tested during the flight in 1989 [1] . . 52.2 Linkage mechanism of Mission Adaptive Wing in details [2] . . . . . 62.3 A gripper mechanism for deflecting the airfoils [3] . . . . . . . . . . . 62.4 Flexible TE model carried by FlexSys Inc. [3] . . . . . . . . . . . . . 72.5 Schematic of kinematic chain mechanism [4] . . . . . . . . . . . . . . 72.6 The flexible airfoil designed by 3 segments kinematic chains [5] . . . 82.7 Virtual prototype of a beam deflected by SMA actuators [5] . . . . . 82.8 Schematic of the volume stretching concept [4] . . . . . . . . . . . . 92.9 The airfoil is bent by inflated silicon actuators [6] . . . . . . . . . . . 92.10 Virtual prototype of an opposite lever arms [5] . . . . . . . . . . . . 10

3.1 Geometric parameters of an airfoil cross section[7] . . . . . . . . . . 133.2 Velocity vectors components described on an Enercon wind turbine [8] 133.3 The generated geometry of NACA 0012 . . . . . . . . . . . . . . . . 143.4 Pressure and shear stress over the airfoil surface [9] . . . . . . . . . . 153.5 Definition of lift and drag . . . . . . . . . . . . . . . . . . . . . . . . 153.6 Velocity gradient inside the boundary layer[9] . . . . . . . . . . . . . 173.7 Lift-coefficient variation with angle of attack [10] . . . . . . . . . . . 173.8 Separation point details[9] . . . . . . . . . . . . . . . . . . . . . . . . 183.9 Boundary layer properties[10] . . . . . . . . . . . . . . . . . . . . . . 183.10 Displacement thickness [10] . . . . . . . . . . . . . . . . . . . . . . . 193.11 Effective body [10] . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.12 Transition from laminar to turbulent flow [10] . . . . . . . . . . . . . 203.13 Pitch control[11] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.14 The principle of pitch control[12] . . . . . . . . . . . . . . . . . . . . 223.15 Flow separation [12] . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.16 Rotor blade twists [13] . . . . . . . . . . . . . . . . . . . . . . . . . . 233.17 The principle of active stall control[12] . . . . . . . . . . . . . . . . . 243.18 High lift devices on an airbus A-318 . . . . . . . . . . . . . . . . . . 253.19 Concentrated load along a cantilever beam . . . . . . . . . . . . . . 263.20 Two point loads on a cantilever beam . . . . . . . . . . . . . . . . . 27

4.1 Wind turbine blade upper and lower skins [14] . . . . . . . . . . . . 294.2 Schematic components of an airfoil [15] . . . . . . . . . . . . . . . . 304.3 A trial model of two straight cantilever beams as a form-adaptive TE 31

vi

LIST OF FIGURES

4.4 The schematic of shape-adapter actuators concept . . . . . . . . . . 324.5 The inflatable silicon actuator . . . . . . . . . . . . . . . . . . . . . . 334.6 The seat limits the inflation of the actuator from three sides . . . . . 334.7 The 15° deflection for lower surface by silicon actuator . . . . . . . . 344.8 No deflection for upper surface by silicon actuator . . . . . . . . . . 354.9 The inflatable actuator not requiring supporter or seat . . . . . . . . 354.10 Three L brackets are attached to the internal surface of the beams . 364.11 Testing the rubber actuator . . . . . . . . . . . . . . . . . . . . . . . 364.12 The assembling of form-adaptive airfoil . . . . . . . . . . . . . . . . . 384.13 The final designed form-adaptive airfoil . . . . . . . . . . . . . . . . 39

5.1 One point load on the position 60% of a cantilever beam . . . . . . . 415.2 The comparison for 0.5 millimeters change in beam’s thickness . . . 415.3 The comparison for the change in module of elasticity . . . . . . . . 425.4 The comparison for the change in position of load . . . . . . . . . . . 425.5 The comparison for the change in position of two loads . . . . . . . . 435.6 The predicted shape for a NACA 0012 deflection . . . . . . . . . . . 435.7 The modified deflection shape without gap in trailing edge . . . . . . 445.8 Scheme of designed airfoil . . . . . . . . . . . . . . . . . . . . . . . . 445.9 The coordination of designed airfoil before . . . . . . . . . . . . . . . 455.10 The modeled deflection shape of designed airfoil . . . . . . . . . . . . 46

6.1 The aluminium cantilever beams are deflected by designed actuator . 486.2 The predicted deflection for Aluminum (thickness 0.2 mm) . . . . . 486.3 The EPDM cantilever beams are deflected by designed actuator . . . 496.4 The predicted deflection for Soft rubber (thickness 2 mm) . . . . . . 496.5 The actuator’s drawbacks . . . . . . . . . . . . . . . . . . . . . . . . 516.6 Schematic of wind tunnel, the laboratory of Kassel University . . . . 526.7 The front view of test region . . . . . . . . . . . . . . . . . . . . . . 526.8 The cross view of test region (actuators’ position) . . . . . . . . . . . 536.9 The investigated deflection with the final model . . . . . . . . . . . . 54

7.1 Cp distribution for NACA 0012 at 4 degree AOA . . . . . . . . . . . 567.2 The coordination of airfoil with maximum deflection . . . . . . . . . 567.3 The deflected model is generated in xfoil . . . . . . . . . . . . . . . . 577.4 The pressure coefficient . . . . . . . . . . . . . . . . . . . . . . . . . 587.5 The pressure Distribution comparison . . . . . . . . . . . . . . . . . 597.6 The pressure distribution presented by vectors at 0 degree AOA . . . 597.7 Comparison of edge velocity for symmetric and deflected airfoil . . . 607.8 The Cl over α for symmetric and deflected airfoil . . . . . . . . . . . 607.9 The Cl over α for deflected airfoil with different Re numbers . . . . . 617.10 The points of inflection and lack of curvature in deflected airfoil . . . 617.11 The shape factor is changing between -1.7 and -1.6 AOAs . . . . . . 627.12 The Cd over α for symmetric and deflected airfoil . . . . . . . . . . . 627.13 The L/D over α for symmetric and deflected airfoil . . . . . . . . . . 637.14 The L/D over Cl for symmetric and deflected airfoil . . . . . . . . . . 637.15 The δ∗ for symmetric and deflected airfoil . . . . . . . . . . . . . . . 647.16 The θ for symmetric and deflected airfoil . . . . . . . . . . . . . . . . 64

vii

LIST OF FIGURES

7.17 The kinematic shape factor for symmetric and deflected airfoil . . . 657.18 Teh skin friction coefficient for symmetric and deflected airfoil . . . . 657.19 The deflected NACA 0010 is modeled by Matlab . . . . . . . . . . . 667.20 The deflected NACA is generated in Xfoil . . . . . . . . . . . . . . . 667.21 Pressure distribution is compared with deflected NACA . . . . . . . 677.22 Pressure coefficient for trial deflection is compared with deflected NACA 687.23 Cl over α is compared with deflected NACA . . . . . . . . . . . . . . 697.24 Cd over α is compared with deflected NACA . . . . . . . . . . . . . 697.25 l/d over α is compared with deflected NACA . . . . . . . . . . . . . . 707.26 L/D over Cl is compared with deflected NACA . . . . . . . . . . . . . 707.27 δ∗ is compared with deflected NACA . . . . . . . . . . . . . . . . . . 717.28 θ is compared with deflected NACA . . . . . . . . . . . . . . . . . . 717.29 Shape factor is compared with deflected NACA . . . . . . . . . . . . 71

viii

List of Tables

4.1 Specifications for the fabricated structure . . . . . . . . . . . . . . . 39

6.1 Lower beam deflection with a silicon sealed actuator . . . . . . . . . 476.2 Comparison the measured deflections with predicted Matlab amounts 506.3 Details of the measured deflection . . . . . . . . . . . . . . . . . . . . 506.4 Boundary condition for the wind tunnel . . . . . . . . . . . . . . . . 53

7.1 The geometric parameters are compared with deflected NACA . . . 67

ix

Abstract

Adaptive shape mechanisms have had important influences on improvement of theaerodynamic design and flow control system of the air planes and helicopters, sincelate twentieth century. Over the past decade, several studies have been carried outto apply the potential of form-adaptive blades also in wind turbines in order toovercome the drawbacks and limitations in current control mechanisms. But thecurrent shape adaptation concepts are still not compatible to the enormous size andweight of wind turbine blades. The objective of this project is, therefore, to designa form-adaptive airfoil which has a hollow and light structure which is more suitablefor wind turbine blades. The blade skin should be able to bend at the trailingedge, upward and downward, in order to control the geometry of the airfoil for loadreduction during high wind speeds, as well as more energy production when theoperational wind is slow. In order to design an airfoil with variable geometry, twopneumatic shape-adapter actuators are designed which can be inflated and deflatedwith air pressure. The actuators’ expansion force is transferred to internal upperand lower surface of the airfoil by some L-type brackets and the trailing edge isdeflected upward and downward around 10 degree. Deflection of the designed airfoilis modeled by MATLAB program and the aerodynamic consequences is studiedby XFOIL program. Consequently, the flow simulation has confirmed that a windturbine blade whose geometry is variable could improve the lift over drag whichit can function efficiently in a certain range of angles of attack as a load controlmechanism. This concept reduces the deleterious effects of stresses and fatiguesover the rotors and provides the possibility to design longer blade, which it meansmore wind energy capturing and less maintenance costs as well as greater life span.

x

Nomenclature

Roman Symbols

C Chord lineV∞ freestream velocityR Resultant forceM MomentL LiftD DragP Pressuref Skin frictionCP Coefficient of powerCp Coefficient of pressureCl Coefficient of liftCd Coefficient of dragCm Coefficient of momentumCf Total skin friction coefficientcf Local skin friction coefficientRe Reynolds numberMa Mach numbera speed of soundVw Wall(surface) velocityTw Wall(surface) temperaturePr Prandtl numberH Shape factorE Modules of ElasticityI The second moment of area

xi

Greek Symbols

α Angle of attackρ Densityδmax Maximum Displacement(Deflection)δ Velocity boundary layer thicknessδT Thermal boundary layer thicknessδ∗ Displacement thicknessθ Momentum thicknessν Kinematic viscosityµ Dynamic viscosityτ Surface shear stressβ The angle of beam’s deflection

Abbreviations

LE Leading edgeTE Trailing edgePS Pressure SideSS Suction SideAOA Angle Of AttackMAW Mission Adaptive WingHAWT Horizontal Axis Wind TurbineNACA National Advisory Committee for AeronauticsPRVS Pitch Regulation Variable SpeedEPDM Ethylene Propylene Diene MonomerSMA Shape Memory Alloys

Chapter 1

Introduction

At the beginning of the twentieth century windmills gradually transformed intowind turbines when the rotor was connected to an electric generator, and duringthe twenty first century, wind turbines are one of the fastest growing technologiesin energy production as they have gradually become an important industry withremarkable annual global growth. According to last statistics released by the WorldWind Energy Association (WWEA) at April 2014, the world wind energy capacityreached 318 GW by the end of 2013, after 282 GW in 2012, and wind power con-tributes close to 4% to the global electricity demand and a wind capacity of morethan 700 GW s predicted as possible for 2020[16]. The main controversial issue forthe current developed wind turbine industry is to capture the most possible energyfrom the wind power with the lowest operation costs to improve the wind’s powershare in global electricity supply. The main technical suggestion to capture morewind power is to build the longer blades to increase the swept area as well as thetaller towers to use higher wind velocities. That is why the size of recent multi-megawatt class wind turbines are huge. The biggest off-shore wind turbine at themoment is V164 with 8 MW rated power and 80 meter-long blades from MHI-Vestaswhich came online in January 2014 at the Danish national wind turbine test centerand the the biggest one which is designed for the onshore sector is Enercon E126with 7.5 MW power and 127-meter rotor diameter[17].

1.1 Current Problems

There are some barriers to increase the size of wind turbine blades as a possiblepathway to achieve the increment in energy production. The longer the blades,the heavier structure, the higher deleterious drag on blades, as well as the moremaintenance costs. The wind has naturally very strong power, as it can be seen inconsequences of a hurricane or a typhoon. The variable wind speeds cause highlyaerodynamic loads on the huge wind turbine’s structure. The increased length of theblades, however, produces much stresses and fatigues which can cause deleteriouseffects on wind turbine components over the time. Conventional control systems oncurrent wind turbines have been developed to reduce these stresses, nevertheless,it is unlikely that blade’s pitch or active stall control can fulfill the expectations toeliminate these fluctuating loads. The motivation for this research is then to improve

1

Introduction

the current wind turbine control systems which are unable to cope with the highlydynamic and non-uniform flow stream loads especially for the long wind turbinerotors. For the enormous size and weight of large blades as well as their naturalhigh elasticity, a uniform pitch angle variation is not sufficient to effectively adjustthe system to the variety of flow streams. Therefore, these reasons heavily limitsthe responsiveness of current wind turbine control systems to the blades stresses.

1.2 Thesis Objectives

The main idea behind the shape-adaptive wind turbine blades is inspired by innova-tions from the nature, where the birds change their wings camber during the flightto control the lift and drag in different situations during the hovering, take-off andlanding (figure 1.1). The concept of adaptive shape airfoils thus are under study as asolution for mentioned obstacles for the next generation of enormous wind turbines.Changing the camber of the blade’s section has significant potential to control aero-dynamic loads which appear mostly in gusty winds, thereby the blades’ shape canautomatically changes to adjust with the wind speed in a certain defined angles ofattack. In addition, the rotor size could be remarkably increased with this mecha-nism, since the aerodynamic stresses are directly a function of blade length.Changing the geometry of airfoils is applied in other turbomachinery like Helicopterand aircraft industries since last century by means of different shape-adapter mech-anism and high lift devices (chapter 2). However, these applied mechanisms arenot favourable for wind turbine blades. Since current conventional control systemsin wind turbines have major weaknesses to overcome the aerodynamic loads andcontrol the high drags over the rotating structure of blades, it is the main objectiveto also change the camber of wind turbine blades during the operation. The currentshape-adapter actuators should be adapt with hollow and light structure of windturbine blades and, however, the reasonable price, as it is always a controversialconcern to make a concept possible to be applied in market scale. The selectedsolution to apply the vary the shape of airfoil is bending the trailing edge withpneumatic shape adapter actuators in a hollow structure which let the maximumfree space inside the airfoil and results a light weight for the blades compared withother current actuation mechanisms (completely described in chapter 2). It is, byno means, the only solution in this direction, but it is nevertheless a fresh conceptas an alternative to achieve the form-adaptive blades and design the next generationof wind turbines. A large part of the development goals is expected to be met withthis mechanism for market scale production of wind turbines whose’ blade can adaptthe shape actively during the operation.

2

Introduction

Figure 1.1: The idea to use form-adaptive airfoils for wind turbine blades[18]

1.3 Methodology and Outline

The preferred method to change the shape of airfoil in this project is by means oftwo pneumatic shape adapter actuators, which can be inflated and deflated with airpressure. The leading edge remains fix without any deflection or change in geome-try. The trailing edge is assumed as two cantilever beams and two elastic adaptersare placed between the beams. Each adapters expansion force should be transferredto internal upper and lower surface of the airfoil by L-type brackets and bend theairfoil’s TE upward and downward depending on the direction of effective bendingmoment. Different adapters and beams are tested in pilot study to investigate theeffect of material, the free body diagram as well as the dimensions on deflection

3

Introduction

parameters in order to design the final model for the wind tunnel investigations.Numerical calculation of the desired deflected geometry is also modeled by Matlaband then compared with the real deflected shape of airfoil. To find the final resultsthus the air flow are simulated with the same boundary condition and Reynoldsnumber in Xfoil and compared after and before shape adaptation.

The thesis addresses the above mentioned objectives in the following chapters:

Chapter 2 reviews the history of adaptive form airfoils in related industries like air-plane, helicopter and turbomachinery and then compare the implemented conceptswith wind turbines. Furthermore, the recent efforts and news regarding flexible windturbine blades are reported properly.

Chapter 3 represents the basic theoretical knowledge for whatever is necessary forthe reader to knows before entering the technical chapters, including a short reviewon wind turbine components and control mechanisms, geometric and aerodynamiccharacteristics of wind turbine blades, as well as the required theory of beams de-flection.

Chapter 4 explains the practical procedure in order to design the final adaptiveshape adapters and flexible blade skins.

Chapter 5 provides the desired geometry of the designed flexible airfoil. The shapeof deflection is modeled by Matlab, calculated by the differential equation of can-tilever beams deflection, when the symmetric airfoil is deflated at the trailing edgecaused by the internal actuator forces.

Chapter 6 reports all the preliminary tests which have been applied in laboratorywith different materials and actuator’s alternatives. The experimental measure-ments have been carried out in two conditions, without and with wind speed. TheReynolds number, Mach number and angle of attack remain constant during theviscous investigation in wind tunnel.

Chapter 7 presents a two dimensional flow analysis according to the real coordina-tion of designed airfoil. The simulation has been done for a symmetric blade beforeand after deflection. The deflected designed airfoil is also compared with a NACAairfoil, in order to recognize the design weaknesses.

In chapter 8 finally the results are summarized and the recommendations are listedin case may the topic be followed in future.

4

Chapter 2

Literature Review

Early efforts to change the airfoil’s camber have been investigated in aircraft’s in-dustry, as a necessary aerodynamic characteristic during different flight conditionsby high lift devices. These efforts led to installing several shape adapter actuatorsmounted throughout the wing during the 1985-1989, Air Force Research Lab at theWright Patterson Air Force Base tested a deflected airfoil for a F-111 aircraft, ina project of NASA which called Mission Adaptive Wing (MAW). It was the firsttime which the wing of the F-111 modified so that the curvature of the leading andtrailing edges could be actively varied during the flight[1].

Figure 2.1: The adaptive-form wings were tested during the flight in 1989 [1]

Figure 2.2 shows the desired mechanical systems that were developed to change theshape of the leading and trailing edge surfaces of the F-111 wing. Inherent draw-backs in the design like increases in weight, complexity and lack of space halted theprogram from further development. After the MAW project, several other conceptshave been proposed by researchers in smart materials and smart structures; however,scalability and survivability of these approaches has always been an issue [2].

5

Literature Review

(a) leading edge section(b) Trailing edge section

Figure 2.2: Linkage mechanism of Mission Adaptive Wing in details [2]

The concept of active control on rotor blades by using adaptive shape structureshas been thoroughly studied in the field of helicopter technology. The interest forsmart rotor control in helicopter rises mainly because of the importance of vibrationand noise reduction at the rotor. Articles of Breitbach [19] or Stanewsky [20] aremain references focusing on active control of helicopter applications. During thefirst Phase of the Smart Wing Program funded by DARPA, in early 1995, a variablewing camber system was investigated the effects of changing the shape of a wingusing many small piezoelectric shape adapter actuators distributed throughout ahelicopter rotor [3].

In 1998, compliant mechanism technology was applied to the wing morphingproblem for first time when Sridhar Kota worked on project funded by Air Vehi-cle Directorate, a flexible 3-foot NACA 63418 profile was designed, fabricated, andtested. Wind tunnel test results showed a 51% increase in lift-to-drag ratio and a25 % increase in the lift coefficient for the 6-deg AOA [21]. Figure 2.3 illustratesthe distributed nature of compliance by a simple gripper mechanism(2.3a) next tothe actual designed patent(2.3b) working with piezoelectric actuators.

(a) A simple gripper mechanism(b) piezoelectric adaptiveform actuators

Figure 2.3: A gripper mechanism for deflecting the airfoils [3]

Research for developing successful variable geometry leading edge and trail-ing edge systems, must address many criteria including the required aerodynamicshapes, the required stiffness and dynamic response, and the weight and power re-quired to actuate the control surface. In parallel with the trailing edge design study,a prototype wind tunnel model was constructed to validate aerodynamic perfor-mance of the intended shape change. Figure 2.4 shows the wind tunnel test on amodel with the variable geometry compliant trailing edge. The FlexSys trailing edge

6

Literature Review

flap provides the variable geometry necessary to minimize drag throughout a longendurance reconnaissance mission. The flap deflects +10 to -10 degrees using inter-nal electromechanical actuators. This model was tested at the Ohio State Universitywind tunnel facility in the Fall of 2002. [3] Based on this mechanism, FlexSys Incor-poration has published recently a news about flexible wind turbine blades with thestatement introduction of Active FlexFoil flap on utility-scale wind turbine bladeswhich allows longer blades to be applied on wind turbines and capture more energyup to 13% [22]. Although many analysis show the feasibility and correct directionsof ongoing works about adaptive shape blades for wind turbines, the idea is still inpilot research and is still not fabricated in market scale.

(a) 10°upward deflection (b) 10°downward deflection

Figure 2.4: Flexible TE model carried by FlexSys Inc. [3]

Following, all the previous researches and investigations in Faculty of Turboma-chinery which are related to form-adaptive airfoils are represented in brief. Kine-matic Chain mechanism has been investigated for the purpose of the large deflectionsin turbomachinery applications. Based on this concept, the airfoil should be divideinto a few connected segments. The connector actuators provide a torsional momentwhen rotation of the first segment is transferred from segment to segment by thekinematics [4]. Each segment has to be actuated with one single joint. The rota-tion of driven element is then transferred gradually from segment to segment by thekinematic chains and this method results a large deflection in trailing edge for air-foil. These kinematics can be applied to unlimited number of segments, neverthelessminimum three segment is necessary to function [23]. Figure 2.5 shows a schematicof this concept with 5 segments.

MLE

TE

Segment 5

Segment 4

Segment 3

Segment 2

Segment 1

Figure 2.5: Schematic of kinematic chain mechanism [4]

7

Literature Review

The surface of the airfoil however can be covered with an additional layer toprovide the smooth deflection along the large deformations. Figure 2.6 shows thedesigned flexible airfoil before and after deflection by kinematic chain mechanismwhen the segments are covered by the skins layer. If this concept would be appliedon wind turbines, the main weakness of it is the heavy weight of bending jointsand segments. The hollow flexible structure of airfoil would be preferred for a windturbine blade, as far as the weight is one of the main considerations in wind turbinedesign.

(a) The symmetric airfoil

Flexible covering skin

M (Torsion)

Kinematic chain

Deflection

TE

LE

(b) The deflected airfoil

Figure 2.6: The flexible airfoil designed by 3 segments kinematic chains [5]

The another proposal for bending the profile shape is using the smart materialslike Shape Memory Alloys (SMA) to create the required bending moment. The figure2.7 presents how the airfoil can be bent where the series of SMA actuators are placedin the fin structure along the chord. The moments are applied on a skeleton structureactively when the actuators are expanding or shrinking by electrical heating [5].

Figure 2.7: Virtual prototype of a beam deflected by SMA actuators [5]

8

Literature Review

Furthermore, Volume Stretching is another interesting concept, which is calledfluid muscle in many literature . The airfoil should be designed with an elasticmaterial and integrated pressurized channels. The channels are considered alongthe flexible part of the chord in two rows, where the inflatable actuators (muscles)can be placed inside (figure 2.8). By expansion of actuators in one row, the trailingedge can be deflected up to 50°[5].

Elastic Material

Pressure channels

(a) The structure of pressure channels


Figure 2.8: Schematic of the volume stretching concept [4]

Figure 2.9: The airfoil is bent by inflated silicon actuators [6]

9

Literature Review

The volume stretching concept is investigated [5] at University of Kassel in 2003.Figure 2.9 illustrates how the elastic airfoil is bent as a consequence of inflatedupper row of silicon actuators pressurized with air line. The airfoil material is anelastic composite of Elastosil M 4601 and its cross-linker with a high stiffness andtemperature resistance. The airfoil then designed with 24 circular channels alongthe flexible part of airfoil. 12 upper channels are designed for downward deflectionand 12 lower channels for upward deflection [6].

As the most related literature, another alternative is also sketches as is repre-sented in figure 2.10, similar with previous volume stretching structure, where theinflatable actuator can expand as the muscles inside the cells and cause the requiredbending moment by transferring the expansion force to the opposite lever arms [5].

Internal pressure

(a) The structure of pressure channels

Flexible TE

Deflection

Actuators


Figure 2.10: Virtual prototype of an opposite lever arms [5]

Mentioned applications (airplane, helicopter or turbomachinery profiles) are some-how different in comparison with wind turbine blades in terms of operating condi-tions, size and weight. Between the above mentioned reviewed methods, the leverarms mechanism (which is developed recently in University of Kassel and still notinvestigated) is the closest concepts to be fulfill the requirements to be applied fora wind turbine blade, as it has a hollow structure. The hollow airfoil causes thelighter weight and then, can be controlled easier by pneumatic muscles. Therefore,it is considered as the main clue to design the actuators of this project (as far asit is not designed and not investigated before), so that some opposite brackets wereused to create the required bending moment in a hollow airfoil.

10

Chapter 3

Theoretical Background

3.1 Wind Turbine Power

The wind power captured by a turbine is commonly expressed as a function of theturbines swept area and a coefficient of power, the air density and the wind speed:

P = 1/2CPρAV3 (3.1)

Where:P is the mechanical power of the turbine in WattsCP is the dimensionless coefficient of powerρ is the air density in kg/m3

A is the swept area of the turbine in m2

V is the speed of the wind in m/s

All modern wind turbines consist of a number of rotating blades. If the bladesare connected to a vertical shaft, the turbine is called vertical-axis (VAWT) and ifthe shaft is horizontal, it is called a horizontal-axis wind turbine (HAWT). Most ofthe modern large wind turbines follow the three bladed horizontal axis design with orwithout gearbox and transmits the electricity to the grid network by a modern gen-erator with partial or full frequency conversion. The coefficient of power is relatedto the turbine design and defined as the ratio between the actual power obtainedand the maximum available power as given by the above equation. A theoreticalmaximum for this ratio is CPmax = 16/27 = 0.593, referred to Betz limit [24]. Inpractice, most wind turbines are rated for wind speeds from 8 to 12 m/s. The powerof a turbine is thereby directly proportional to the swept area, and it is proportionalto the blade length squared although the wind speed has the largest influence onturbine power. The ratio between the rotor diameter and the hub height is oftenapproximately one[24]. The tower height is important since wind speed increaseswith height above the ground and the rotor diameter is important as it gives thelarger swept area A.

Ideally a wind turbine rotor should always be perpendicular to the wind. Onmost wind turbines a wind vane is therefore mounted somewhere on the turbine tomeasure the direction of the wind. This signal is coupled with a yaw motor, whichcontinuously turns the nacelle into the wind. The nacelle of modern multi-megawatt

11


wind turbines weighs usually more than 300 metric tons [25] and is connected tothe tower via a yaw system. This system is responsible for the alignment of theturbine rotor with the incoming wind and it comprises of a roller or gliding bearingstructure which allows the rotation of the nacelle against the tower. The necessaryyawing moment is provided by a plurality of powerful electrical drives located insidethe nacelle. The rotor of the wind turbine comprises of a hub and the blades and isconnected to the wind turbine drive-train via the main shaft. The hubs of wind tur-bine rotors are very crucial as other components are mounted on hubs and thereforethey have to withstand all the aerodynamic, structural and operational loads thatare generated from the nacelle - blade interaction. The blades are mounted on thehubs in a way that they can be rotated along a pitch axis. Different materials havebeen tested for blades fabrication to find the best combination of requirements whichis sufficiently, strong and stiff, have a high fatigue endurance limit, and reasonableprice. Today, most common construction materials for blades are glass fiber rein-forced plastic, but other materials such as laminated wood are also used for small-and medium-size wind turbines.

3.2 Geometric and Aerodynamic Characteristics of Air-foils

In many flow analysis process it is common to focus on the two dimensional flowstream as the reasons are detailed in the chapter 7. Therefore, usually a crosssection of airfoil is considered to study of the aerodynamic characteristics. Thegeometry of cross section is illustrated in figure 3.1 for a cambered airfoil andthe main parameters is named. Chord line (C) is the reference dimension for anairfoil and defined as the straight line connecting the leading edge to the trailingedge. It divides the airfoil into an upper and a lower surfaces. Symmetric airfoilsare defined where the upper and lower surfaces has the same shape and the meancamber line is matched with the chord line. Otherwise the airfoil is called camberedor asymmetric. The camber line or mean line is an theoretical line midway betweenthe upper and lower surfaces. The camber is therefore, the maximum perpendiculardistance between the mean camber line and the chord line. Thickness is the distancebetween the upper and lower surfaces, and measured perpendicular to the chord line.Camber and thickness are most important parameters to describe the shape of theairfoil. The shape of the nose is usually circular, with a leading-edge radius ofapproximately 0.02 chord length [10]. The angle of attack is a key parameter in flowsimulations which is called α and defined as the angle between the relative velocityof the fluid (V∞) and the chord line.

12


Figure 3.1: Geometric parameters of an airfoil cross section[7]

The above mentioned relative velocity is defined as the freestream flow velocityeffected on the blade including the components of wind velocity and blade rotationalspeed. Figure 3.2 highlights these vectors for a HAWT.

Rotor axis

Real wind vel.

Peripheral vel. Effective wind vel.

componant

componant

Figure 3.2: Velocity vectors components described on an Enercon wind turbine [8]

The NACA (National Advisory Committee for Aeronautics) was founded onMarch 3, 1915, as an independent national agency and they research and develop-ments in early twentieth century lead to several series of NACA airfoils which arestill often used in airfoil manufacturing as the main standard of airfoil’s geometry[26]. The geometry of NACA airfoils are described using a series of digits followingthe word NACA?. The parameters in numerical code can be entered into equationsto precisely generate the cross-section of the airfoil.

NACA four-digit airfoils are often used as standard aerodynamic design. Thefour digits are in proportion as percentage of the chord:

13


• First digit describing maximum camber.

• Second digit describing the distance of maximum camber from the airfoil LE.

• Last two digits describing maximum thickness of the airfoil.

For example a NACA 0012 airfoil is symmetrical, the 00 indicating that it hasno camber. The 12 indicates that the airfoil has a 12% thickness to chord lengthratio: it is 12% as thick as it is long. The equation (3.2) gives the coordination ofevery symmetric NACA airfoil [27].

y = cT (1.4845

√x

c− 0.63

(xc

)− 1.7580

(xc

)2+ 1.4215

(xc

)3− 0.5075

(xc

)4) (3.2)

Where:

• c, is the chord length,

• x, is the position along the chord from 0 to c,

• y, is the half thickness at a given value of x (center line to surface), and

• T, is the maximum thickness as a fraction of the chord (the last two digits inthe NACA 4-digit denomination divided by 100).

For instance, figure 3.3 illustrates a plot of symmetric NACA airfoil. Havingthe chord length and NACA number, the coordination can be plotted by Matlabscripts A.1.

0 20 40 60 80 100 120 140 160 180 200

−60

−40

−20

0

20

40

60

x [mm]

y [m

m]

Figure 3.3: The generated geometry of NACA 0012

The velocity components always effects over the body surface and makes thepressure and shear stress over the blade skin (figure 3.4). Shear stress is defined asa component of stress which effects parallel to the surface of the airfoil [28].

14


Figure 3.4: Pressure and shear stress over the airfoil surface [9]

The net effect of the pressure and shear distributed over the complete airfoilssurface is a resultant aerodynamic force (F) and moment (M) on the body. Re-sultant of aerodynamic forces on the surface has two components which are shownin figure 3.5. Basically, the aerodynamic shape of the airfoils, makes force on thestreamlines to curve around the geometry. The pressure difference between upperand lower side cause a lifting force on the airfoil and if the both sides are affected bysame pressure, no lift would be resulted. Provided that the airfoil is almost alignedwith the flow, the boundary layer stays attached and the associated drag is mainlycaused by friction with the air.

F

D

L

M

c

Figure 3.5: Definition of lift and drag

When V∞ defined as a free-stream wind velocity; [10]

L ≡ lift ≡ component of R, perpendicular to V∞D ≡ Drag ≡ component of R, parallel to V∞

Then, lift and drag coefficients (Cl and Cd) are defined as:

Cl =L

1/2ρ∞V∞2c

(3.3)

Cd =D

1/2ρ∞V∞2c

(3.4)

15


Where ρ is the density and c is the chord length of the airfoil. The unit for thelift and drag in equations (3.3) and (3.4) is force per length (in N/m). The momentis positive when it tends to turn the airfoil clockwise like what is illustrated in thefigure 3.5 (nose up) and a moment coefficient is defined as: [24]

Cm =M

1/2ρ∞V∞2c2

(3.5)

It is widely common in the most aerodynamic fundamentals to study the pressuredistribution over the airfoil with coefficient instead of pressure itself. Indeed, thepressure coefficient (Cp ) is another similar parameter that can be added to previousdefined coefficients where p∞ is the free stream pressure:

Cp =p− p∞

1/2ρ∞V∞2 (3.6)

Skin friction coefficient is also frequently used quantity for boundary layer calcula-tions as a proportion of surface shear stress (τ) over freestream pressure:

Cf =τ

1/2ρ∞V∞2 (3.7)

The presented coefficients are functions of angle of attack, Reynolds and Machnumber. The angle of attack(α) defined as the angle between the chord line andthe effective free stream velocity (V∞). Free stream Reynolds number (Re) is adimensionless combination and one of the most powerful parameters in fluid dy-namics which defined by John Anderson as “a measure of the ratio of inertia forcesto viscous forces in a flow”[10].

Re =V∞c

ν∞=ρ∞V∞c

µ∞(3.8)

Where:ν is the kinematic viscosity of fluid.µ is dynamic viscosity of fluid.c is the travelled length of the fluid which for an airfoil it is equal to chord length.

The Mach number is also an important parameter in the study of gas dynamicsand defined as the ratio between flow velocity and the local speed of sound.

Ma =V∞a∞

(3.9)

Ludwig Prandtl described his theory in 1904 that an effect of friction cause theadjacent fluid to the aerodynamic surface to stick and check a thin region of layer [9].This layer is boundary layer where the flow is retarded by the influence of frictionbetween a solid surface and the flow. Although this layer is very thin in comparisonwith the size of airfoil, its effects on the drag and heat transfer is high to the extendthat in Prandtl’s own description is “it produces marked results” [9]. When outsidethe boundary layer the flow is inviscid (like figure 3.6), in a thin boundary layeradjacent to the surface, the effects of friction are immense. The blow-up of theboundary layer shows how the flow velocity changes, as a function of the distance

16


to the surface, from zero at the surface to the full inviscid-flow value at the outeredge.

Figure 3.6: Velocity gradient inside the boundary layer[9]

Another “marked result” is the flow separation. Stall is defined as a conditionin aerodynamics when flow separates from the body(airfoil) surface due to passinga critical angle of attack. Usually lift and drag coefficient are depending from theangle of attack. Figure 3.7 shows the typical continuous changes of lift versus AOA.From lower value till moderate AOAs, Cl varies linearly with α as it is representedby lift slope in the figure. In this region, the flow moves smoothly and attached overthe airfoil as shown in the streamline at the left side of the figure. The maximumpossible lift reach normally in 15-20 ° AOA which is so-called stall angle. This crit-ical angle depends on Reynolds number and airfoil camber shape. Therefore, theCl,max is one of the key parameters of airfoil performance, because it determines thestalling speed of an airfoil. At the other extreme of the curve, the value of α whenlift equals zero is called the zero-lift angle of attack(αL=0). For a symmetric airfoil,the zero-lift always at zero AOA 1, whereas for all airfoils with positive camber,zero-lift is a negative value [10].

Figure 3.7: Lift-coefficient variation with angle of attack [10]

1αL=0 = 0° 17


Figure 3.8 shows how the boundary layer separates from top surface of theairfoil more in details. Inside this separated region, a part of flow is recirculating,and another part of the flow is moving in opposite direction of free stream so calledreserved flow. This separation is due to viscous effects and consequently, leads todramatic reduction in lift and a large increase in drag. The blow up shows the ve-locity profile above the separation point, as well.

Figure 3.8: Separation point details[9]

Since one of the main purposes of this project is to compare the properties ofboundary layer after and before shape adaptation of wind turbine blades (chap-ter 7), the thickness of this thin layer is exaggerated in figure 3.9 to describe thefundamental properties of viscous flow over a flat sketched plate.

Outer edge of velocity boundary layer

Outer edge of thermal boundary layer

uT

��

��

y

x

Velocity profile Temperature profile

Figure 3.9: Boundary layer properties[10]

Immediately at the surface, the velocity of flow (Vw) is zero and the temperature(Tw) is equal to the temperature of the surface. Above the surface, the flow veloc-ity increases in the y direction until the height δ, where it is equal to free streamvelocity (ue = V∞). In other words, the quantity δ is called the velocity boundarylayer thickness and defined as that distance above the airfoil where velocity at theouter edge of boundary layer reaches to the amount of freestream velocity. Similarly,

18


the flow temperature will increase above the airfoil’s surface, ranging between walltemperature(Tw) at y=0 to the edge temperature of the thermal boundary layer(Te) at y=δT which (Te = T∞) defined as thermal boundary layer thickness. Gen-erally, δT 6= δ and the relative thickness depend on the Prandtl number. In thephysical basis, the Prandtl number is an index which is proportional to the ratioof energy dissipated by friction to the energy transported by thermal conduction[10]:

Pr =µ∞cpk∞

∝ friction dissipation

thermal conduction(3.10)

If Pr=1, then δT = δ; if Pr > 1, then δT < δ; and if Pr < 1, δT > δ.As Prandtl number for air is 0.71, the thermal boundary layer in our case is thickerthan velocity boundary layer as is shown in figure 3.9.

The velocity gradient at the surface generates the shear stress(τw),

τw = µ

(∂u

∂y

)w

(3.11)

where (∂u/∂y)w is the velocity gradient at y=0.Similarly, the heat transfer at the wall (qw) is defined with the temperature gradient:

qw = k

(∂T

∂y

)w

(3.12)

where(∂T/∂y)w is the temperature gradient evaluated at y=0 (at the surface). Bothτw and qw are functions of distance from leading edge, shown on figure 3.9, andimportant in boundary layer calculation.

Displacement thickness δ∗ is also a frequent used property of boundary layer,defined as the distance by which an external flow streamline is displaced by bound-ary layer formation ( 3.10). The boundary-layer thickness is relatively larger thandisplacement thickness and typically δ∗ ≈ o.3δ [10].

��

��

Boundary layer

Hypothetical flow with no boundary layer

(invisid case)

Actual flow with a boundary layer

Streamline

External streamline

1

2

Velocity boundary layer

Figure 3.10: Displacement thickness [10]

The definition of displacement thickness, somehow is an interpretation for theconcept of an effective body. Such that an aerodynamic shape sketched in figure3.11, the flow-stream has a path like ac rather than the surface line ab. In the otherwords, The effective body is the actual body shape plus the displacement thicknessdistribution.

19


b

c

a

Effective body

Actual body surface

Figure 3.11: Effective body [10]

Another important property is momentum thickness(θ) . The simple explanationof θ is an index that is proportional to the decrement in momentum flow due to thepresence of boundary layer. In fact, the height of the hypothetical flow (Left sidein figure 3.10) is carrying the missing momentum flow before the formation ofboundary layer at free stream condition. The skin friction coefficient(Cf ), which isdefined earlier in equation (3.7), could also help somehow to describe this conceptbetter; whereas the momentum thickness is proportional to the integrated frictioncoefficient along the surface. For instance, θ evaluated at a given station x = x1, asa proportional to the integrated friction drag from the LE to x1:

θ(x1) =1

x1

∫ x1

0cf dx = Cf (3.13)

where cf is the local skin friction coefficient and Cf is the total skin friction coefficientfor the whole length of surface. At high Reynolds numbers, it is desirable to have alaminar boundary layer and due to the characteristic velocity profile of the laminarflow, it results a lower skin friction.

Kinematic shape factor is also used to determine the nature of the flow at whereis laminar or turbulent:

H =δ∗

θ(3.14)

The higher the value of H, the stronger the contrary pressure gradient. Convention-ally, a high contrary pressure gradient can reduce the Re at which transition fromlaminar into turbulent flow.

��

Turbulent

Transition region

Laminar

Figure 3.12: Transition from laminar to turbulent flow [10]

All the above mentioned boundary layer properties are general concepts for bothcompressible as well as incompressible flows and they can be applied to turbulentand laminar flows. As the flow eventually goes from laminar into turbulence duringboundary layer transition (figure 3.12, the increased momentum and energy exchange

20


that occur due to turbulence flow makes the boundary layer thicker than laminarboundary layer and less stable as well. The thicker boundary layer in turbulentflow, causes also the increment in thermal and velocity boundary layer thicknesses.Similarly, θ, H and cf are larger for turbulent flows while pressure drag reducesafter transition [10].

3.3 Control Mechanisms in Wind Turbines

Wind turbines are designed to produce electrical energy as cheaply as possible butthey are not designed to have maximum power output at stronger winds, becauseusually such strong winds are rare. To control or regulate a wind turbine, it usu-ally operates within a designed range, in order to keep the rotational speed and thepower output in a certain range, yaw the turbine, start or shutdown the turbine.Furthermore, the control system can ensure a smooth power output at lower windspeeds, also limit the power at high wind speeds. The stall regulation and pitch reg-ulation are in used as the the most common strategies for these purposes. However,the variety of shape adaptive control concepts are under research stages to controlthe aerodynamic loads as a supplementary control mechanism.

3.3.1 Pitch Control

3.3.1.1 Constant Rotational Speed

On a pitch controlled wind turbine, the turbine’s electronic controller checks thepower output of the turbine several times per second. When the power output be-comes too high, the rotor blades pitch actively to reduce simultaneously the anglesof attack along its entire length. Conversely, the blades are pitched back into thewind whenever the wind speed drops again. It is worth mentioning here that eachblade could be fitted with a small electrical motor to be pitched independently andit is not the only way to pitch all the blades together with a same amount [24].The rotor blades, however, should be able to pitch around their longitudinal axis asillustrated in the figure 3.13.

Figure 3.13: Pitch control[11]

21


A pitched blade can act as an aerodynamic brake as well as the power regulator.By pitching the leading edge of the blades up against the wind, the angle of attackdecreases and consequently, the power output will be reduced. In figure 3.14, thepitching mechanism is sketched simply how it can turn the blades to the lower angleof attack during high wind speeds to avoid entering the stall and reversely, the angleof attack is increased to a value close to the rated power of wind turbine by pitchcontrol whenever the wind speed is low.[12]

High wind Speeds

Low wind speeds

close to rated powerWind speeds

Figure 3.14: The principle of pitch control[12]

During normal operation the blades will pitch a certain fraction of a degree atunit time while the rotor is turning at the same time. Thereby, designing a pitchcontrolled wind turbine requires some complicated calculation to pitch the rotorblades exactly as much as required. The pitch mechanism is usually operated usinghydraulics or electric stepper motors [29].

3.3.1.2 Variable Speed

Many pitch regulated wind turbines are using asynchronous generators and oper-ating at a fixed rotational speed, therefore they have two generators, one which isefficient at lower wind speeds and another one which is fit at higher wind speeds.[24]If a variable speed generator had been used, it is able to run at different rotationalspeeds, and the turbine could be operated at the optimum rotational speed for eachwind speed. The pitch control concept is adjustable with variable rotor speed andis called pitch regulation variable speed (PRVS).Using a variable speed generator, the electrical output will be almost constant sincethe generator torque is constant, and the pitch system should control the rotationalspeed rather than the output power. As far as the change in rotational speed is slowdue to the high inertia of the rotor, there is plenty of time for the pitch mechanismto react. This will overcome the problem of the large fluctuations from a pitch regu-lated wind turbine operating at constant rotational speed. Therefore the PRVS canreduce fatigue loads and improve power quality and that for lower wind speeds canproduce slightly more power by running at the optimum power coefficient(CP ≈ 0.5).

22


On the other side, the extra cost of the control system and the necessary convertersalso has to be considered.

3.3.2 Stall Control

3.3.2.1 Passive Stall

Passive stall control is mechanically the most simple, since the blades are bolted tothe hub at a fixed angle and cannot be pitched and it is normally operated at analmost constant rotational speed. Thereby the angle of attack increases if the windspeed raises(figure 3.2). Consequently, by increasing wind speed the angle of attackraise to the extend which creates turbulence as is shown in the figure 3.15.

(a) Attached flow (b) Separated flow

Figure 3.15: Flow separation [12]

After stall, the lift coefficient decreases and the drag coefficient increase and thepower eventually decreases depends on the fixed pitch angle and distributed twistsalong the chord [24]. The blade of stall controlled wind turbines are twisted slightlyalong its longitudinal axis gradually from the root till tip in order to ensure thatstall occurs gradually rather than abruptly (figure 3.16). On a stall regulated windturbine an asynchronous generator is often used whereby the rotational speed isalmost constant and determined by the torque characteristic of the generator whichcan act both as a motor and as a generator (the motor mode is used to start theturbine).1

Figure 3.16: Rotor blade twists [13]

1In zero shaft torque, the rotational speed for a six-pole three-phase generator is 1000rpm or 1200 rpm respectively for 50 Hz and 60 Hz supply systems.

23


The stall control is normally used for wind turbines of up to 2 MW because inlarger cases, it exhibits undesirable vibrations and noises [12].

3.3.2.2 Active Stall

If the stall controlled blades are not bolted to the hub, they could reduce the poweroutput by actively increasing the angle of attack, rather than decreasing in highwind speeds. It forces the blades to go into deeper stall which is called activestall. Technically the active stall machines resemble pitch controlled machines, sincethey have pitchable blades. The main difference of active stall control with pitchcontrolled systems is that when the machine reaches to overloads wind speeds, theactive stall controlled blades will pitch in the opposite direction from what a pitchcontrolled does. It means the controller of an active stall control is not respondingto the output power like a classical pitch regulated, but it reacts directly to the windspeed [24].The basic concept of stall is illustrated in figure 3.17, which at low wind speeds theangle of attack is low. The turbine operates close to the rated power in a higherlocal angle of attacks when the wind speed is normal and then finally the higherwind speeds lead to stall which cause naturally less lift and more drag to control theoutput power.

Low wind speeds

close to rated powerWind speeds

High wind Speeds

Figure 3.17: The principle of active stall control[12]

The active stall regulation is that it can control the power output more accuratethan passive stall, to avoid overshooting the rated power of the machine. Anotherpositive point is that the turbine can be operated almost always at rated powerduring all high wind speeds; While normal passive stall controlled wind turbineusually has a drop in the electrical power output for higher wind speeds [29].

3.3.3 Yaw Control

it is common to have a yaw drive for almost all modern wind turbines, which isconstantly used to rotate the nacelle to minimize the yaw misalignment and get asmuch contact area of air through the rotor disc as possible. Furthermore, it is alsopossible to limit the output power by controlling the yaw of the turbine. In this case,

24


the rotor should be turned out of the wind in high wind speeds to limit the airflowthrough the rotor and obviously, limit the power extraction. This control system isa rare and old mechanism and the 1.5MW Italian wind turbine called GAMMA 60is an operating example for it [24].

3.3.4 Shape Adaptive Control

This method is widely used in air crafts, when using flaps change the geometry of thewings to provide extra lift at take-off and reversely, to create extra drag during thelanding. Assuming if an airfoil is designed for an aircraft, lift force should overcomethe gravity and higher the lift means the heavier mass that can be lifted off theground. To maintain a constant speed the drag must be balanced by a propulsionforce delivered from an engine, and the smaller the drag means the smaller requiredengine. It is the main concept of the adaptive shape control systems by changingthe geometry of airfoil using widely in aircraft wings so-called the high lift devices.High lift devices include slats for leading edge and flaps for trailing edge in order tochange the geometry of airfoil during the landing or take off. Figure 3.18 illustratesthese high lift devices for an airbus A318.

FlapSlat

Figure 3.18: High lift devices on an airbus A-318

Beside the above mentioned control system of wind turbines, the Form-AdaptiveAirfoil could be also applied to bring the required aerodynamic controls, in orderto capture more wind energy at the low wind speeds or react to brake during highoperational speeds. Moreover, the change in camber can control the stresses andfatigues where the blade faces the turbulence flow in large pitch angles. There aresome variety of ideas to alter the geometry of the blades based on different conceptswhich the chapter 2 reviews most of the patented ideas in related fields. However,this project is also following an idea to develop an airfoil with shape adaptive trailingedge and investigate its aerodynamic results.

25


3.4 The Beam Deflection Theory

As the deflection of the trailing edge of airfoil is the main destination of this project,the flexible parts of upper and lower surfaces of the airfoil are made-up as two can-tilever beams1 (figure 3.19). Therefore, it needs to be clarified the basic theory ofbeam deflection as well.

��

��

x

Fa

y

C

l

Adaptive−form trailing edge

Figure 3.19: Concentrated load along a cantilever beam

F is a point load on the beaml is the beam lengthc is the chord lengtha is the distance from the point load to the fixed endδmax is the maximum displacement(deflection) of the beam.

Equation (3.15) is the main differential equation of the bending moment causedby the shear force on a beam [30]. The differential equation solved by LeonhardEuler in the eighteenth century for some quite sophisticated end-loading conditionsso-called Euler-Bernoulli and describes the relationship between the bending momentand the displacement. If one integrates from this differential equation, the first stepgives the slope of the deflection and the second integration gives the amount of dis-placement, adjustable for all kind of beam deflection affected by whether point loador distributed loads.

d2y

dx2=−Mz

EIz(3.15)

Where:E is the module of elasticity :An elastic modulus, or modulus of elasticity, is a number that present the resistanceof material to elastically deformation. The unit is pa(or N/m2) and it is defined asthe slope of its stress-strain curve in the elastic deformation region. The more stiffmaterial, the higher elastic modulus [30].I is the second moment of area:In figure 3.19, since the deflection is in y axis, the beam cross section is bendingwith respect to z axis. Therefore the Iz should be calculated like following:

1A cantilever is a beam supported on only one end 26


area of the cross section

beam width (b)

(h)beam thickness

y

z

x

Iz =

∫Ay2dA =

∫ b/2

− b/2

∫ h/2

− h/2y2dydz =

∫ b/2

− b/2

1

3

h3

4dx =

bh3

12(3.16)

For a cantilever beam cause by a concentrated load at any point along the beam,the coordination of deflected beam and maximum deflection can be calculated re-spectively with equations (3.17) and (3.18) , which are solved from the mentioneddifferential equation [30].

y =

Fx2

6EIz(3a− x) for 0 < x < a

Fa2

6EIz(3x− a) for a < x < l

(3.17)

δmax =Fa2

6EIz(3l − a) (3.18)

The above mentioned equations then have been programmed with Matlab scripts(appendix A.4), which is able to provide all the deflection parameters like the shapeas well as all the deflection values like maximum displacement and the angle of de-flection, on a cantilever beam caused by a point load along the beam. Moreover, theFortran script in appendix A.9 is developed to calculate the deflection parametersfor a cantilever beam affecting by distributed loads.

Furthermore, based on equation (3.15), the differential integration is done tocalculate the cantilever beam’s deflection affected by more concentrated loads. Theadditional deflection then can be achieved by applying more number of point loads,if the deflection of the beam caused by previous loads is considered (figure 3.20).

��

��

x

a

y

l

b

F F21

Figure 3.20: Two point loads on a cantilever beam

The equation (3.19) provide the deflection parameters for two point loads at anyposition of the beam.

27


y =

F2x2

6EIz(3b− x) + F1x2

6EIz(3a− x) for 0 < x < a

F2x2

6EIz(3b− x) + F1a2

6EIz(3x− a) + |x− a| tanβ for a < x < b

F2x2

6EIz(3x− b) + F1a2

6EIz(3x− a) + |x− a| tanβ for b < x < l

(3.19)

Consequently, another Matlab script is developed which is represented in ap-pendix A.7, for modeling the geometry of two deflected beams with two point loads,as the beams are playing the role of upper and lower surface of wind turbine blade’sskin in the next steps.

28

Chapter 4

Design of the Form-AdaptiveAirfoil

4.1 Design of the Flexible Blade Skins

Since commercial wind turbines were presented to the market around 1980, theyhave grown in size remarkably. Considering the normal scales laws, weight of theblades increases with the blade length. Recent research carried out by the Depart-ment of Wind Energy at the Technical University of Denmark shows this relationbetween weight and blade length has been reduced considerably due to successfulimprovement in structural design, material optimization and process technology de-velopment during the history of wind turbines industry[12]. In addition to weight,aerodynamic loads must be considered in blade design. Therefore, the design of windturbine blade is a compromise between aerodynamic and structural aspects. Aero-dynamic design usually considers the outer two-third of the blade while structureconsiderations are more significant for the design of inner one-third (blades root).

Figure 4.1: Wind turbine blade upper and lower skins [14]

The wind turbine blade is normally a hollow that two shells are formed the outergeometry; one shell on the suction side and another on the pressure side as is shownin figure 4.1. The blade profile becomes excessively large at the rotor hub to carry

29

Design of the Form-Adaptive Airfoil

the high loads and reinforce it structurally at this load intensive region. Thereforethe root region of the blade is typically thick and round shape with low aerodynamicefficiency. But the rest two-third of the blade should be as slender as possible; asthe lift to drag ratio will be maximised[31]. As the step for design a blade skinin this section, the characteristics of a cross section is defined at the middle of theblade’s length where the skins should be designed with aerodynamic considerations.The blade cross-section typically called as airfoil in literature as a schematic airfoilis illustrated in figure 4.2. Although the rotor blade is an integral design in whichmost components work together, the load bearing components can be classified intwo main parts of the airfoil:

• Spar structure including shear webs and spar caps

• Aerodynamic shell including the coated skin layers

After Shear Web

Spar Caps

Structural adhesive

Forward Shear Web

LE

TE

Aerodynamic shape of skin

Figure 4.2: Schematic components of an airfoil [15]

Regarding to material requirements, deflection of the trailing edge with internalloads of shape adapter actuators has two clearly paradoxical requirements; On theone side, the airfoil has to be flexible enough and light to bend. On the other hand,the structure has to maintain a sufficient stiffness to resist the high aerodynamicload within high velocity of wind flow. Therefore, different parameters should beconsidered to design a flexible blade skin particularly the variable internal actuatorloads will make it more complicated. In early steps thus the simplification of airfoilstructure is necessary in order to study the aerodynamic and mechanical reactionsof the adaptive-form airfoil. As the upper and lower shells of wind turbine bladesare two slender structure, they can be studied as straight beams in the middle ofthe length. Hence, the trailing-edge-curvature of the airfoil is neglected as a pilotassumption in this project and consequently, the trailing edge of airfoil is assumedas two straight beams in all calculations. This assumption is justifiable accordingEulerBernoulli beam theory which is known also as simplified beam theory, since thechord length is itself long and slender enough compared with it’s curvature that canbe ignored. Therefore, the two assumed straight beams should be deflected withinternal actuator force as the main objective of the project. Accordingly, at least

30


one free end is required to bend the trailing edge of the blade. As the leading edge isfixed by structural spars, two cantilever beams play the role of the airfoil’s trailingedge and the free ends of beams can be deflected by a bending moment created bythe shape adaptive actuators in between (next section). The beams transfer theload to the fixed end (shear webs) where the total moment of the actuator’s forcesand shear stresses. The TE could be attached with an structural adhesive (like amagnet or a clips in pilot design steps) to close the TE, while the beams can slideover each other during the deflection (figure 4.3).

Cantilever beamsSpar Stracture

Actuator’s brackets

Slidable Magneted TE

Figure 4.3: A trial model of two straight cantilever beams as a form-adaptive TE

4.2 Design of the Actuators

Variety of mechanisms could be used to bend an airfoil Trailing edge as some of themrepresented in literature review. The main challenge of this project is to design theshape adapter actuators which can provide the required deflection of the airfoil trail-ing edge. The concept is the pneumatic forces which are generated by air pressure insome elastic materials so-called actuator and then transfer the force to the internalsurface of airfoil by means of some L-type brackets. Since the brackets are connectedto the internal surface of the airfoil (see figure 4.3), a bending moment which isresulted by transferred forces thus has one single direction for both upper and lowersurfaces to bend them a together (as the airfoil’s TE). Figure 4.4 simply illustratesthe schematic of described concept, how the expansion of an elastic actuator canaffect a single cantilever beam to bend downward.

31


��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

M

The Load−Transformer Bracket

Bending Moment

F

Deflated actuator

Inflated actuator

c

a

r(Transfered Force)

P (The air pressure applied by expansion)

(M) = Pr = Fa

Figure 4.4: The schematic of shape-adapter actuators concept

Furthermore, to make the calculations easier, the beams are placed perpendicu-lar to the earth, to avoid affecting the loads by the structure’s weight (see figure 4.7).On the other hand, since the wind flow is not applied on the structure in pilot tests,the external atmospheric air pressure is the same on both sides of the beam andthere is no need to consider external air pressure in calculations. Therefore, theonly effective pressure have been calculated for beam’s deflection are internal actu-ator point loads. The beam’s material is selected rubber for early design step as itis flexible and soft enough to be deflected with small amount of loads. Two piece ofwoods are used to keep rubber plates in between and support the fixed end of thecantilever beams. Therefore, the required free body diagram of described conceptin figure 4.4, is modeled in early stages of the project as it is described in figure 4.7.

The first selected alternative to apply the above mentioned concept for shapeadaptation purpose was an elastic material which can be inflated and expand onlyfrom one side. The required properties was sent to a related manufacturer (ITA-profile company [32]), in order to build the actuator with our dimension (11 mmthickness and 198 mm length). The actuator is fabricated as shown in figure 4.5.The material is silicon and the elastic characteristics is provided by manufacturer todeclare which amount is the maximum expansion limit by how much air pressure.Accordingly, with applying 0.4 bar air pressure inside the actuator, the transferredforce would be 3.5 kg/m and with 0.6 bar internal air pressure it transfers 7 kg/mforce to the designed brackets.

32


Figure 4.5: The inflatable silicon actuator

Based on the designed structure, the expansion of this actuator should be lim-ited from three sides of the outer surface and then the desired expansion would beresulted along one direction. (figure 4.6)

��

��

��

��

��

��

��

��

��

��

��

��

(Transfered Force)F

c

P

Actuator’s Seat

Figure 4.6: The seat limits the inflation of the actuator from three sides

Then the required groove is prepared on a cube piece of cushioning to supportthe silicon actuator from the three mentioned sides. Then, silicon actuator is fittedinto the groove and is connected to the air pressure line of the laboratory for infla-tion. The figure 4.7 shows the mentioned structure before and after the expansionof the actuator.

The first alternative then was fixed to the workshop’s table by it’s seat andis placed close to the L-brackets which are connected to the beams. The L-type-

33


brackets thereby transfer the actuator expansion to the beam. This concept hasbeen tested first so the rubber plate is installed as lower surface of the blade skin;such that the beam was deflected 15 ° as a lower surface. Figure 4.7 displays thementioned deflection and more investigated data are reported in chapter 6 aboutthis test.

Straight rubber beam (Before Deflection)

as a cantilever beam

The first 40 % of the beam is supported

120 [mm]80 mm

Silicon actuator bofore inflation

(a) Rubber beam before inflation

L−type brackets

Inflated Silicon Actuator

Air pressure hose

Rubber plate (Deflected cantilever beam)

The beam’s supprter

Maximum

Deflection

F (The transfered force to the internal surface)

P (The air pressure is applied by expansion)

(b) Rubber beam after deflection

Figure 4.7: The 15° deflection for lower surface by silicon actuator

This actuator is also tested when the brackets are fixed other way around to benda rubber plate as a upper surface as well. Although enough force has been appliedon the internal surface, this actuator expansion cause no deflection for upper side,since the supporter of actuator (seat) was an obstacle for beam deflection. (figure4.8).

34


M

Obstacle for deflection

Figure 4.8: No deflection for upper surface by silicon actuator

Since the seat (which is obstacle for upper surface’s deflection) is necessary forthe concept of one side expansion, the application of this concept would be muchmore difficult inside the slender structure of real airfoils. Therefore using this con-cept was rejected as an actuator for shape adaptation in this project.

The airfoil thickness is a small value in the thin wind turbine blades accordingto aerodynamic reasons. This space limitation should be always considered duringthe design steps of shape-adapter actuators. The lack of space between two upperand lower sides of airfoil, as well as the limitation of free body diagram in orderto upper beam’s deflection, led to develop the second concept of shape adapter ac-tuators. The actuator would be placed between upper and lower surfaces and canbe expanded without any required supporter or seat for limitation. This concept ismodeled with the same required dimension mentioned for previously ordered actu-ator and the result is illustrated at figure 4.9.

Figure 4.9: The inflatable actuator not requiring supporter or seat

The expansion of this actuator is also tested; This time the actuator should be

35


placed between two beams without any seat and transfers the internal air pressureby two rows of brackets rather than one one row of brackets (figure 4.10). Threebrackets are then attached to the internal surface of the beams with a liquid glueand the different parts of the beams are marked for assembling with the actuatorand supporters. The contact area between the brackets and the beams was not thewhole width of the beam in this steps, since it was just preliminary tests to choosethe good function of actuators in early steps.

120 mm

60 %

Flexible Part

40 %

Fixed Part

30 mm

Upper Surface

Contact area with actuator

Contact area with actuator

Lower Surface

Figure 4.10: Three L brackets are attached to the internal surface of the beams

But the resulted expansion was not enough to deflect the beams at least 10°.Therefore, these tests lead to the fact that, the more expansion of the actuator,the more pressure transfer by the inflation. Therefore, based on this experiences,a bicycle tube was tested in next step as the inflatable actuator which was inflatedby air pressure through its valve and gives much more expansion as it has higherelasticity. (figure 4.11)

Figure 4.11: Testing the rubber actuator

36


An important concern was the diameter of bicycle tube, due to the small thick-ness of the real wind turbine blades. The second concern was the possibility ofmovement for the actuator as the beams will move according to their deflection.The actuator then should move with them. Therefore, a normal bicycle tube is cho-sen considering these concerns, with 18 millimeters width, which was the narrowestavailable one in the market for the race bicycles. The tube is then cut around itsvalve and sealed from both ends in order to move easier with beams displacements.Figures 6.2 and 6.3 shows a top view from the modified designed Free-Expansionactuator, as the result of the study to choose an actuator. Thereby this concept isselected with an acceptable performance for the trial model of flexible airfoil. Theresulting deflections by this concept was successfully logged and the investigationsare reported properly as the preliminary tests of the project in chapter 6.

4.3 Final Structural Design

According to the mentioned effective parameters on flexibility of a wind turbineblade, the following variables are the most important considerations in order to de-sign an adaptive form airfoil:

• Air pressure inside the actuator actuators

• Number and position of the shape-adaptive actuators

• Material of the beams (Elasticity module and Inertia moment of area)

• Dimension of the beams, actuators and brackets

A structure is thus designed based on above described concept to apply the ideaof shape adaptation. Considering the required stiffness and flexibility a stainlesssteel of type 1.4301 is selected with 1 mm thickness as the airfoil’s upper and lowersurface which has the elasticity module E = 193Gpa(193e9N/m2) [33]. Two steelplates are cropped as an upper and lower blade’s shells. The hollow spar structureis screwed to the the upper and lower shells. The same material is rolled and screwedto the spars for leading edge, however it has not a standard aerodynamic design asdesired for nose. The same steel plate is also rolled at TE to connect the upper andlower skins of the airfoil together. The connector clips at the TE is fixed to the lowersurface by glue (figure 4.12c). The upper beam is then connected so that is just fixedin one end by spar structure while at the free end it can easily slide over the lowerbeam. Therefore both plates can easily bend with one free end nevertheless they areconnected by that clips. In basic design tests, it has been observed that if the beamsare bent by inflation of actuators, the backward motion needs another set of actua-tors, since in real case it is necessary to change the shape again to the straight shapeor even bend it in other direction to decrease the lift and brake. Therefore, anothercouple of brackets are considered in order to create reverse bending moment andstraight the beams’ TE backward, while the first actuator are deflating simultane-ously by its separate valve. In other words, the structure has two actuators betweentop and bottom surfaces: one actuator can bend the airfoil downward to increase

37


the lift and another one is able to bend it back upward. The tube width is 18 mmrather than 30 mm the previously tested in last section, that it is the most narrowbicycle tube which is found in the market belong to the racer bicycles. Furthermore,the brackets are connected along the whole plate’s width which is provided morecontact area with actuators and transfer more loads. Figure 4.12 depicts how theseparate upper and lower skins are assembled to the spar structure with a close viewof actuators and brackets.

(a) Upper skin

(b) Lower skin

(c) Connecting the LE to the flexible skins of TE

Figure 4.12: The assembling of form-adaptive airfoil

The assembled airfoil is illustrated in figure 4.13, where all the components anddimensions are indicated. All the specifications are also listed in table 4.1.

38


Table 4.1: Specifications for the fabricated structure

Description Value

Chord length 440 mm

Nose radius 6 mm

TE radius 3 mm

Maximum thickness 44 mm (10 % of chord length)

Width of airfoil 198 mm

Beams thickness 1 mm

Elasticity module E= 193 Gpa (Stainless Steel)

Number of actuatorsTop side 1Down side 1

Actuators’ diameter 18 mm

Contact area between actuators and brackets 198 mm * 30 mm

Position of brackets on upper side 60 % of chord

Position of brackets on lower side 53% and 93 % of chord

Fixed part 108 mm (25 % of chord)

Flexible part 332 mm (75 % of chord)

Lower Brackets

Chord 440 [mm]

Fixed Part, Flexible Part, 332 [mm]108 [mm]

Lower Surface Shell

Spar StructureUpper Brackets

air pressure valve

Nose radius

Slidable TE

3 [mm]TE radius

air pressure hoseInflatable Actuators

Max. thickness

44 mm

Upper Surface Shell

Internal Sliding Reinforcement

6 [mm]

Figure 4.13: The final designed form-adaptive airfoil

39

Chapter 5

Modeling the Form-AdaptiveAirfoil

The following assumptions are used in this chapter to develop a program for thedeformation of the symmetric airfoils’ geometry:

• The only transferred pressure on brackets is the bending moment caused bythe inflation of actuators.

• The thickness of the actuators does not change in due to inflation by airpressure.

• The contact area between actuator tubes and brackets remain constantly 2*120square millimeters during tube inflation.

• The effect of beam’s curvature is neglected and the deflection calculation isdone for two cantilever beams as the flexible trailing edge of airfoil.

• The elasticity of materials does not change by atmosphere temperature .

• The selected materials are not affected by any plastic change in the shape bydeflection.

According to simplified beam theory, two straight beams are assumed as theupper and lower surface of the symmetric airfoil in chapter 4. The basic beamdeflection equations thus are enough to model the geometry of deformation and thencalculate the amount of required force analytically or even predict the thicknessand stiffness of required materials subsequently if the desired amount of bendingis given. Deflection calculation of cantilever beams could cause by one or moreactuator loads. Based on the mentioned equations for beam deflection in chapter 3,the Matlab scripts A.4 is written to study the important parameters which areaffecting on the geometry of beams’ deflection. For example, figure 5.1 simulatesthe deflection shape of a rubber beam affected by a 3 N concentrated load at positionof 60 % length which results the maximum deflection of 61.9 millimeters and 26.1°atthe TE. The module of elasticity is E = 109N/m2 [34] for soft rubber and beam’sdimensions are assumed 198*1.5*200 respectively for width, thickness and length.

40

Modeling the Form-Adaptive Airfoil

0 20 40 60 80 100 120 140 160 180

−100

−80

−60

−40

−20

0

20

40

x [mm]

y [m

m]

Figure 5.1: One point load on the position 60% of a cantilever beam

The equations (3.17) shows how the amount and position of force as well as thestiffness of material effect on the beam deflection. For this reason Matlab scripts A.4is used to compare the effect of these variables on deflection shape. Figure 5.2illustrate a comparison when the second beam (dash-red line) has all the samementioned properties of mentioned rubber beam (the blue line), except the thicknesswhich is 2 millimeters rather 1.5 millimeters; while the maximum deflection and theangle of deflection at TE (27.3 mm and 12.3°) almost halved compared with thementioned values for 1.5 thickness (61.9 mm and 26.1°) .

0 20 40 60 80 100 120 140 160 180

−100

−80

−60

−40

−20

0

20

40

x [mm]

y [

mm

]

1.5 mm thickness

2 mm thickness

Figure 5.2: The comparison for 0.5 millimeters change in beam’s thickness

The effect of elasticity is studied as well, when the material of the beam ischanged to PVC (E = 2.9 ∗ 109N/m2 [34]) rather than soft rubber with the sameother properties. In result, the maximum bending is roughly one third to 21.3 mmand 10°at the TE and the deflection shape is changed as figure 5.3.

41


0 20 40 60 80 100 120 140 160 180

−100

−80

−60

−40

−20

0

20

40

x [mm]

y [

mm

]

Soft rubber

PVC

Figure 5.3: The comparison for the change in module of elasticity

The position of point load can also change the amount of deflection. The figure5.4 depicts when the same load (3 N) is implied on 80 % instead of 60 % of the chordlength, and deflection amount increase to 100.9 mm and 40°at the TE (dash-red line).

0 20 40 60 80 100 120 140 160 180

−120

−100

−80

−60

−40

−20

0

20

x [mm]

y [

mm

]

Load on 60 %

Load on 80 %

Figure 5.4: The comparison for the change in position of load

Furthermore, for more than one point load, the deflection caused by previousloads should be also superimposed to final bending amounts and shape [35]. Thenthe Matlab script A.5 is developed to calculate the effect of two point loads andcompare the deflection parameters on two different cantilever beams. Due to projectdestination, the first 40% of beams are considered fixed (as leading edge of the windturbine blades) without the possibility of deflection. Figure 5.5, shows comparisonin the position of point loads where two 7 newtons point loads is applied on 50 and60 % on the chord length (dash-red line) compared with the deflection shape wherethe same loads are applied on 70 and 90 % of chord length (blue line); while both

42


are rubber beams with the same stiffness and the same length. First beam deflectsmaximum 21.04 mm and 9.94°but the second one has maximum 38.42 mm deflectionand 17.7°at TE.

0 20 40 60 80 100 120 140 160 180 200

−80

−60

−40

−20

0

20

40

x [mm]

y [m

m]

Two point loads are placed on 50 and 60 % of beam length

Two point loads are placed on 70 and 90 % of beam length

Figure 5.5: The comparison for the change in position of two loads

In the next step, in order to model the airfoils deflection, it is enough to super-impose the above mentioned calculation on the coordination of airfoil rather thana straight beam. The mentioned scripts then can be modified and developed topredict the shape of the deflected airfoils. As an example, the Matlab scripts A.8 iswritten to calculate the deflection of a NACA 0012 when two actuators are placedon 60 and 70 % of the chord length and 40 % of the leading edge is fixed. Figure5.6 represents the resulting modeled shape for this airfoil deformation. The internalactuators’ air pressure is selected as 5 bars and the transferred forces on internalsurfaces are calculated consequently, with bending momentum equations which isdescribed in theoretical background of the thesis.

0 20 40 60 80 100 120 140 160 180 200

−80

−60

−40

−20

0

20

40

60

x [mm]

y [

mm

]

NACA 0012

Deflected NACA 0012

Figure 5.6: The predicted shape for a NACA 0012 deflection

43


As is shown in figure 5.6, there is a gap at trailing edge which is due to differentposition of effecting actuator loads on internal surfaces. This conditions at the TEis described in details during the design steps how it should be solved when theupper and lower skin are capable of sliding and deflecting together when a clips wasconnected to lower part and upper part slide inside the clips. After connecting theend point of the beams, the deflected shape can be modified as figure 5.7 withoutany gap in coordination of the trailing edge.

0 20 40 60 80 100 120 140 160 180 200

−80

−60

−40

−20

0

20

40

60

x [mm]

y [

mm

]

NACA 0012

Modified Deflected NACA 0012

Figure 5.7: The modified deflection shape without gap in trailing edge

In order to develop a program and predict the geometry of the deflection, at firstthe geometry of non-deflected airfoil is required. Since the fabricated airfoil in thisproject had not an standard pre-defined coordination, the outer shape is measuredmanually and then is drawn by Xfig 3.2 as it is illustrated in figure 5.8.

��

��

��

��

332 [mm]

Flexible PartFixed Part

Lower Brackets

Inflatable Adapters

Upper Brackets

Spar Structure

108 [mm]

Chord 440 [mm]

Thickness

Upper Surface Shell

Lower Surface Shell

Internal Sliding Reinforcement

Nose radius

Slidable TE

TE Radius3 [mm]

44 [mm]

6 [mm]

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

Figure 5.8: Scheme of designed airfoil

44


Since the trailing edge shells are connected as it is drawn in schematic of theouter shape, the two dimensional coordination can be measured manually. Thenthanks to the bezier function in Matlab program (appendix A.10), the coordinationof drawn airfoil interpolated and smoothed like figure 5.9.

−0.1

−0.05

0

0.05

0.1

0 0.2 0.4 0.6 0.8 1

y/c

[−

]

x/c [−]

Actual geometry

−0.1

−0.05

0

0.05

0.1

0 0.2 0.4 0.6 0.8 1

y/c

[−

]

x/c [−]

Actual geometrySmoothed Coordination

Figure 5.9: The coordination of designed airfoil before

Therefore, provided that the coordination of the symmetric airfoil 5.9, and themaximum deflection at the TE are known (also all the other required variables arelisted in table 4.1 page 39), the deflection differential equations is solved in orderto model the deflected shape of the designed airfoil for both upward and downwarddeflections with the Matlab script A.11. The written script script code is thus canbe applied for future projects in order to predict the deflection shape of flexibleairfoils and model the deflection shape of any given airfoil, when it is affected byone or more bending moments. Figure 5.10 illustrates the modeled bent shapewhich the geometry is similar with the shape of deflection during the experimentalinvestigations (figure 6.9 page 54). The required internal forces to create this amountof displacements is calculated so 49.1 N force on upper surface and 77.2 N force onlower surface is required for downward deflection; and for upward deflection, thesame calculation gives 21. 3 N force on upper surface and 62.5 N force on lowersurface1.

1The distributed force along the bracket is assumed as a point load at the middle ofthe brackets

45


0 50 100 150 200 250 300 350 400

−150

−100

−50

0

50

100

150

x [mm]

y [

mm

]

Deflected

Symmetric

(a) Downward deflection

0 50 100 150 200 250 300 350 400

−100

−50

0

50

100

150

x [mm]

y [

mm

]

Deflected

Symmetric

(b) Upward deflection

Figure 5.10: The modeled deflection shape of designed airfoil

46

Experimental Investigations

Chapter 6


6.1 Preliminary Tests

Based on described simplification of beam theory for the airfoil’s T.E, some basictests are applied on straight beams with different materials before final investiga-tions in order to check the designed the actuators, pressure required and internalspace limitations between two beam’s surfaces. During the design of the actuatorin chapter 4, the first tests were described as the first alternative of shape-adapteractuators shown in figure 4.7. The air pressure makes perpendicular load on thecantilever beam which is fixed at one end. The bending results for lower surfacecaused by one inflatable silicone actuator are logged as table 6.1 while the beam’smaterial is a common soft rubber so-called EPDM. The comparison shows the struc-ture has a limit value for maximum deflection, which is around 11°(25 millimeters)and increasing the air pressure does not lead to the more deflection.

Table 6.1: Lower beam deflection with a silicon sealed actuator

air pressure (mbar) maximum deflection(mm)

400 air line1 24

110.8 air pump 25

The next preliminary tests are done to deflect two beams when the actuatoris placed in between as the structure of airfoil’s skin and the concept of actuatorsare described in chapter 4, section 4.2. The figure6.1 presents the shape andamounts of deformation for upper and lower surfaces of blade skins, when the mate-rial is Aluminum (EAluminum = 69∗109N/m2) with thickness 0.2 millimeters and theshape-adapter actuators took place at 66.6 % of the beam length which is 200 mil-limeters. Dimension of upper and lower beams are 2*198*200 millimeters, and theyare placed with 40 millimeters distance in between. Three L brackets with dimension32*32*40 millimeters and 2 mm thickness are implementing the load transferring tothe beams’ internal surfaces based on the concept of described mechanism in designof the shape-adapter actuators. The beams bending are also calculated for upper

1The line pressure is supplied by the central compressor of the laboratory. 47


and lower beams and the predicted shapes and amounts of deflection are shown infigure 6.2, modeled by Matlab (scripts A.9).

(a) before inflation

(b) after inflation

Figure 6.1: The aluminium cantilever beams are deflected by designed actuator

0 20 40 60 80 100 120 140 160 180 200

−100

−80

−60

−40

−20

0

20

40

x [mm]

y [

mm

]

Lower beam

Upper beam

Figure 6.2: The predicted deflection for Aluminum (thickness 0.2 mm)

48


Furthermore, another material is also investigated as the upper and lower shellsof wind turbine blade. The second tested material is the usual soft rubber (EPDM)with a lower module of Elasticity compared with Aluminum (EEPDM = 108N/m2)while the thickness is 10-fold more (2 millimeters). The actuator is placed at thesame position and the prediction is also done with the same Matlab scripts (figures6.3 and 6.4).

(a) before inflation

(b) after inflation

Figure 6.3: The EPDM cantilever beams are deflected by designed actuator

0 20 40 60 80 100 120 140 160 180 200

−100

−50

0

50

x [mm]

y [

mm

]

Lower beam

Upper beam

Figure 6.4: The predicted deflection for Soft rubber (thickness 2 mm)

49


The maximum measured deflections for rubber and aluminum are compared intable 6.2 with Matlab predicted amounts.

Table 6.2: Comparison the measured deflections with predicted Matlab amounts

Material predicted max. deflection(mm) measured max. deflection(mm)

upper beam lower beam upper beam lower beam

EPDM 27.5 49.9 33 44

Aluminum 39.9 72.3 19 63

The different value of the maximum deflection for upper beams and lower beamshows this structure of actuators has still some limitations and cause to harvest theresults of the tests in order to find out the reasons. The investigations illustratefor both studied materials that the upper beams are bent less than lower beams,since it was also predicted by beams’ deflection calculations in plots 6.2 and 6.4.One reason is the position of transferred force, when the adapter-shape actuator isat 66 % of chord length, the center point of effective loads would be at 57.5% onupper beams and 75.7 % on lower beams. Therefore, the lower beam is supposed tobend more as the reason is described in theory of beam deflection (section 3.4) byfigure 5.4 page 42. The more details of displacements are provided then in table 6.3with exact inflated air pressures inside the actuator. The same value of deflectionscompared with the two different pressures in this table for the EPDM material (500mbar and 160 mbar), as by increasing the pressure and inflation the maximum de-flections for upper and lower beams remain the same value.

Table 6.3: Details of the measured deflection

Material pressure(mbar) max. deflection(mm)

upper beam lower beam

EPDM 500 (air line) 33 44

EPDM 160 (air pump) 33 44

As investigated further, another reason of this different deflection is related tolack of the contact area between brackets and inflated actuators, when the contactarea between L brackets and inflated actuator changes during the deflection. Thefigure 6.5 represent this drawback which helps us to improve the further designs.One idea is, it might be improve if the designed actuator could move during thedeflection with the upper and lower surfaces.

50


Figure 6.5: The actuator’s drawbacks

6.2 Final Investigation in Wind Tunnel

Final investigation is done in wind tunnel in order to provide the deflected geometryof the airfoil and test whether the designed actuators function on the final designedmodel or not, when a high wind flow is acting on surfaces. In other words, to provethe ability of bending the flexible TE by inflation of first actuator and then bend itbackward for brake system with deflection of first actuator and inflation of secondactuator, as the concept is described completely in design step (chapter 4.

The wind tunnel test took place at the main campus of Kassel University. Itis a linear cascade wind tunnel belongs to the department of Turbomachinery. Ithas three parallel loops, namely, a calibration tunnel , a linear cascade wind tun-nel, and an annular cascade wind tunnel. A 610-Watt radial-compressor suppliesthe required air with maximum pressure ratio of 1.8 and a maximum flow rate of8 kg/s. The compressor is connected to a heat-exchanger and bypass valve. Thebypass valve can distribute the air through the venturi tube and then, into any ofthree mentioned wind tunnels. The investigation of our project is done in the linearcascade as it is highlighted in the schematic view of wind tunnel (figure 6.6). Thelinear cascade consists of a duct and a throttle valve, a settling camber, a connectingduct and finally a rectangular test rig.

51


5 m

Test rigSettling chamber

Ventury tube

Suction−nozzle

Heat exchanger

Experimental setup:

Linear cascade wind tunnel

Duct

Linear cascade wind tunnel

Drive motor

Compressor

Transformer

Annular cascade wind tunnel

Calibration tunnel

Figure 6.6: Schematic of wind tunnel, the laboratory of Kassel University

Figure 6.7 illustrate how the designed airfoil is attached at the outer edge ofthe test region and a clipboard is set behind the airfoil. The coordination of zeroposition, as well as the maximum downward and upward deflection of airfoil beforestarting the wind flow is drawn on the clipboard.

Figure 6.7: The front view of test region

52


Figure 6.8, is a side view of test region, showing the actuators function withthe air pump manually during the test. Each tube can be inflated and deflatedseparately using two-way valves inside the transparent air hoses.

Figure 6.8: The cross view of test region (actuators’ position)

The boundary condition for investigation are presented in table 6.4.

Table 6.4: Boundary condition for the wind tunnel

Description Parameter Value

Angle of attack α 0 °

Atmosphere Pressure P 1008.9 mbar

Atmosphere Temperature T 21.8 °C

Mach number Ma 0.05

Reynolds number Re 5e5

Figure 6.9 shows the maximum downward and upward deflection after startingthe wind flow, which is investigated at the tip for downward direction 55 millimetersand 9.4°as well as 70 millimeters and 11.9°for upward deflection. The maximumdeflection is exactly the same amount compared with marked amount measuredwithout wind flow. The only different is the required pressure for inflation of the ac-tuators during the wind flow which is more than required pressure in atmosphere airflowstream. (The amount of difference between required pressures is not logged dur-ing this investigation as the only available air pump which can pump the actuatorsmanually was not calibrated)

53


(a) Maximum downward deflection

(b) Maximum upward deflection

Figure 6.9: The investigated deflection with the final model

54

Flow Simulation

Chapter 7

Flow Simulation

7.1 Introduction

Wind turbine blades are long and slender structures as the spanwise velocity com-ponent is much lower than the streamwise component, and it is therefore assumed inmany aerodynamic models that the flow at a given radial position is two dimensional[24]. Therefore the two-dimensional flow analysis is enough for theis project objec-tives and if the airfoil’s coordinate is generated in X and Y direction, the velocitycomponent in the z-direction is thus assumed zero in simulations .

XFOIL is a open-source software originally written by Mark Drela at MIT (Mas-sachusetts Institute of Technology) in 1986 [36]. XFOIL is chosen for this projectas it is free access and gives the results much more quickly than more advancedCFD programs, however the results are still accurate enough. Nevertheless, XFOILhas some limitations for instance it is only effective at low Reynolds numbers andincompressible flows1. The different commands are available listed in the main in-terface and can be executed by typing their corresponding code. XFOIL uses theairfoil coordinate file over unit chord (xc ,

yc ) to generate it’s two-dimensional geom-

etry. The airfoil coordination can be imported from a text file or be selected as astandard NACA number. The user can then define the inviscid/viscous flow andproperties of viscous flow such as Mach number and Reynolds number. XFOIL willthen use the input data to simulate the flowstream at certain defined ranges of AOAand provide the lift coefficient (Cl), drag coefficient (Cd), moment coefficient (Cm)and many other parameters of boundary layer as output results of flow analysis inthe form of a plot file ot text file.

For instance, a NACA 0012 has been loaded by Xfoil at 4 degree AOA and theviscous flow is applied with Re=2e6 and Ma=0.08. Figure 7.1 shows the pressure co-efficient distribution for this example. The Cp has the higher value for pressure sideof the airfoil (red line), while the pressure coefficient is negative values distributedalong the suction side (blue line). Moreover, The area between pressre distributionof suction side and pressure side presents the resulting lift due to 4 degree AOA.

1The flow in which the density ρ is constant is called incompressible [10]. 55

Flow Simulation

Figure 7.1: Cp distribution for NACA 0012 at 4 degree AOA

7.2 Geometry Generation and Conversion in Xfoil

As mentioned before, generating an airfoils in Xfoil is done either by selecting aNACA number or importing its coordination. Since the airfoil which modeled inthis project was not a NACA airfoil, the smoothed coordination for non-deflectedshape (figure 5.9) is loaded for flow simulation before the deflection. For deflectedshape, however, the actual deflected geometry is also required. Therefore, with thesame procedure for non-deflected shape mentioned in chapter 5, the coordinationof deflected shape which is illustrated in figure 6.9a 54 is also measured manuallyand smoothed like figure 7.2. (To avoid repeating the work, the flow analysis is doneonly for downward deflected shape.)

−0.15

−0.1

−0.05

0

0.05

0 0.2 0.4 0.6 0.8 1

y/c

[−

]

x/c [−]

Actual geometry

−0.15

−0.1

−0.05

0

0.05

0 0.2 0.4 0.6 0.8 1

y/c

[−

]

x/c [−]

Actual geometrySmoothed Coordination

Figure 7.2: The coordination of airfoil with maximum deflection

56

Flow Simulation

The smoothed coordination of designed airfoil is then loaded in Xfoil with com-mand “load”. Although the outer shape was smoothed (by Matlab bezier), therewere still error which the flow “is not converged” in viscous mode, since the airfoilhad not an aerodynamic and standard design, particularly over the suction side atthe position between 10 till 20 percent of chord length. Therefore, the coordinationagain is smoothed in Xfoil with the command “PANE” to fill in the coarse pointsspacing and result a better aerodynamic shape for flow analysis. Moreover, it issmoothed again with commands “FILT” and “ECEC” under MDES (modificationof surface speed distribution). Then numerical convergence has been achieved byincreasing the number of iteration (command ”ITER“). The numerical calculationis converged when the amount of error is minimum based on the defined criteria inXfoil. This criteria is defined by the amounts of Cl, Cd and Cm as they are nume-ically calculated by a certain number of iteration. When the different between valuesof mentioned coefficients are minimum, that whould be acceptable as a “converged”results. The generated model as well as deflected shape is illustrated in figure 7.3with it’s properties like the camber line, maximum thickness, internal area, noseradius and trailing edge angle of outflow.

Figure 7.3: The deflected model is generated in xfoil

7.3 Flow Analysis for Designed Airfoil

In order to simulate the viscous boundary layer we need to calculate Reynolds num-ber, having a Mach number 0.05 comes from our boundary conditions in wind tunnelwhen the atmosphere temperature is +15°:

Mach = 0.05 = Va

a = 340.27m/s(speed of sound) ⇒ V = 17.01m/s

Re = ρV cµ = V c

ν

ν = 1.48e− 5m2/sc = 0.44m (chord length) ⇒ Re = 5.08e5

57

Flow Simulation

Figure 7.4 illustrate the Cp versus unit chord length, at zero AOA before and after thedeflection. Before deflection (figure 7.4a) there is no lift as is expected for a symmetricairfoils and the area between pressure distributions of PS and SS is almost zero. However,the pressure coefficient distribution for deflected shape (figure 7.4b) has an area betweenPS and SS and lift force appears due to deflection (Cl = 1.18). It is worth to notice, theCp = 1 for both shapes at the leading edge, where the velocity is zero and the pressureis maximum. The plot of pressure coefficient provides also the boundary layer behaviouraround the airfoil geometry for both viscous and inviscid flow (dashed line). Moreover. themoment coefficient (Cm) is a negative value, which means the airfoil tends to turn counter-clockwise after deflection.

(a) Pressure coefficient before deflection

(b) Pressure coefficient for maximum deflection

Figure 7.4: The pressure coefficient

According the pressure coefficient distribution at zero AOA, the aerodynamic pressuredistribution is also calculated and plotted in Matlab with script A.3 based on the equation(3.6). The figure 7.5 illustrates how pressure distribution is almost the same for both suctionand pressure sides in symmetric shape but not in deflected shape.(Green line is for pressure

58

Flow Simulation

side and blue line for suction side). The integration of area between SS and PS is the amountof lift force due the deflected shape of airfoil.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

1.0132

1.0132

1.0132

1.0133

1.0133

1.0133

1.0133

1.0133

1.0133

1.0133x 10

5

x/c [m]

Pre

ssure

[pa]

Suction side

Pressure side

(a) Pressure Distribution on symmetric airfoil

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 11.0132

1.0132

1.0132

1.0132

1.0132

1.0133

1.0133x 10

5

x/c [m]

Pre

ssure

[pa]

Suction side

Pressure side

(b) Pressure Distribution on maximum deflected airfoil

Figure 7.5: The pressure Distribution comparison

The figure 7.6 illustrates how the different pressure distributions of lower and uppersurfaces makes the resulting lift after deflection by pressure vectors.

Figure 7.6: The pressure distribution presented by vectors at 0 degree AOA

59

Flow Simulation

According to equation (3.6) in page 16, the inverse proportional of pressure distributionand velocity is expected. As it is illustrated by figure 7.7, the velocity rise at where pres-sure distribution shows dropped down in figure 7.5. In this figure the velocity distribution isprovided as a unit-less proportion of Ue

V∞where V∞ = 17.01m/s and Ue is the edge velocity of

the boundary layer. Generally, for the symmetric airfoil the velocity is the same at suctionand pressure sides, but the effect of deflection is clear, as the velocity increased at suctionside and decreased for pressure side after deflection.

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2

0 0.2 0.4 0.6 0.8 1

Ue /

V∞

[−

]

x/c [−]

Symmetric suction sideSymmetric pressure side

Deflected suction sideDeflected pressure side

Figure 7.7: Comparison of edge velocity for symmetric and deflected airfoil

The installed airfoil in Wind Tunnel didn’t give us the possibility to change the AOA.Thanks to Xfoil, the polar is available for a range of AOA calculated for both straightand deflected airfoil. The maximum lift coefficient increased from 1.21 to 1.39 after thedeflection, while the stall also shifts to the earlier point from 14.2 degree to 8.3 degree AOA.Comparing the two lift coefficient for both shapes (figure 7.8), it could be a positive resultprovided that the lift coefficient increases caused by deflected shape.

−1.2

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

−20 −18 −16 −14 −12 −10 −8 −6 −4 −2 0 2 4 6 8 10 12 14 16 18 20

Cl [−

]

α [° ]

Symmetric AirfoilDeflected Airfoil

Figure 7.8: The Cl over α for symmetric and deflected airfoil

In order to find the cause of the jumps in lift coefficient over α for deflected geometry,one of the sharpest one is targeted which is between -1.7 and -1.6 degree AOA, Reynolds

60

Flow Simulation

number set 2e6 rather than 5e5 Mach number remains the same (0.05). The results (fig-ure 7.9) shows the jump is still there and just shifted from -1.7°to 2°, lift coefficient showsincrement from 1.4 to the 1.6 with this change in Re number while the flow separationis happening at almost same α. As a result for this comparison, the lower Re cause theseparation of boundary layer appear earlier and the higher Re can produce larger lift andsmoother separation (area between 10°and 16°in figure 7.9).

−1.2

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

−20 −18 −16 −14 −12 −10 −8 −6 −4 −2 0 2 4 6 8 10 12 14 16 18 20

Cl [−

]

α [° ]

Deflected Re=2e6Deflected Re=5e5

Figure 7.9: The Cl over α for deflected airfoil with different Re numbers

The shifted jump in the curve therefore may be the effect of flow stream transition due tolack of the camber in trial designed airfoil, which is defined as region number 1 in figure 7.10.This free space can cause the early laminar separation due to design drawbacks.

−0.15

−0.1

−0.05

0

0.05

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1

y/c

[−

]

x/c [−]

Deflected designed airfoil

2

3

1

4

Figure 7.10: The points of inflection and lack of curvature in deflected airfoil

Moreover, the comparison is done between -1.7 and -1.6 AOA for the kinematic shapefactor of the boundary layer over deflected design (figure 7.11). The kinematic shape factormight be the best factor which can illustrate the behavior of flow stream over the deflectedbody. The most clear change between this two angle is happened at position 0.9 of chordlength in pressure side, which the flow has a point of inflection over the body (region 4 infigure 7.10). Also, the kinematic shape factor shows how the flow reacts to the observeddrawbacks in regions 1 and 3, with illuminated picks in figure 7.11 at the LE. These jumpthus might be the response to these mentioned weaknesses of the geometry.

61

Flow Simulation

0

2

4

6

8

10

12

14

16

18

0 0.2 0.4 0.6 0.8 1

H [

−]

x/c [−]

Deflected suction side −1.7 AOADeflected pressure side −1.7 AOADeflected suction side −1.6 AOA

Deflected pressure side −1.6 AOA

Figure 7.11: The shape factor is changing between -1.7 and -1.6 AOAs

The Cd also shows response to our prediction while the stall points for both shape canbe easily seen in the above mentioned range of AOAs.

0

0.03

0.06

0.09

0.12

0.15

0.18

0.21

0.24

0.27

0.3

−20 −18 −16 −14 −12 −10 −8 −6 −4 −2 0 2 4 6 8 10 12 14 16 18 20

Cd [

−]

α [° ]


Figure 7.12: The Cd over α for symmetric and deflected airfoil

The L/D ratio of symmetric and deflected airfoils are compared in figures 7.13 and 7.14over α and Cl in a wide range. As a positive result, the graphs clearly shows the amountof difference in lift over drag in our target range like -5 to 5 degree AOA that could be asan efficient and ideal working range for an adaptive shape control system. A bump which isillustrated in these graphs (ca. -4 to -1°AOAs or 0.7 to 0.9 lift coefficients) could be improvedby design a standard airfoil since the comparison are provided with a deflected NACA 0010in figures 7.25 and 7.26.

62

Flow Simulation

−70

−60

−50

−40

−30

−20

−10

0

10

20

30

40

50

60

70

80

−20 −15 −10 −5 0 5 10 15 20

L/D

[−

]

α [° ]


Figure 7.13: The L/D over α for symmetric and deflected airfoil

−70

−60

−50

−40

−30

−20

−10

0

10

20

30

40

50

60

70

80

−1.4 −1.2 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4

L/D

[−

]

Cl [−]


Figure 7.14: The L/D over Cl for symmetric and deflected airfoil

Since we have an form-adaptive surface, the flow simulation calculation for this projectis an iterative solution to calculate the boundary layer properties, accurately. This is donedue to define the strategy of designing the airfoil shape and the required actuators loadingin future steps of the project. The comparison starts with δ∗ along the chord length. Figure7.15 shows the displacement thickness of boundary layer is increasing along the airfoil asexpected for both symmetric and deflected shapes although it shows the comparably higherrise in value for deflected shape for both pressure and suction sides in last 20 percent of chordlength. The reason lay in the fact that explained in theoretical backgrounds, appearanceof turbulent flow and increment of boundary layer thickness after transition area. However,along the whole chord length, the value of δ∗ for pressure side become less than symmetricshape, and for suction side is more which was also expected as the boundary layer thicknessafter deflection is thinner in pressure side.

63

Flow Simulation

0

0.005

0.01

0.015

0.02

0.025

0.03

0 0.2 0.4 0.6 0.8 1

δ*

/c [

−]

x/c [−]



Figure 7.15: The δ∗ for symmetric and deflected airfoil

The changes is illuminated for θ along the airfoil in figure 7.16 as well. The momentumthickness behaves more or less similar to the explained trend of δ∗. Furthermore, at thesame positions which the points of inflection and lack of the camber have been shown infigure 7.10, the fluctuations can be seen in δ∗ and θ along the deflected airfoil at the regions1, 2, 3 and 4 due to mentioned drawbacks of the design.

0

0.003

0.006

0.009

0 0.2 0.4 0.6 0.8 1

θ/c

[−

]

x/c [−]



Figure 7.16: The θ for symmetric and deflected airfoil

64

Flow Simulation

0

2

4

6

8

10

12

14

16

18

0 0.2 0.4 0.6 0.8 1

H [

−]

x/c [−]



Figure 7.17: The kinematic shape factor for symmetric and deflected airfoil

The value of kinematic shape factor is marked out for both symmetric and deflectedshapes in figure 7.17. It shows almost same value for both side of symmetric airfoil. Thegraph presents a dramatic growth after deflection only on pressure side of the airfoil. Thereason is directly related to the similar rise in the last 10% of chord for δ∗, which was seenin figure 7.15 due to turbulence in TE, as the δ∗ is the numerator of shape factor and it’sdenominator is θ. The illustrated peaks at the LE are also considerable since at that positionthe turbulent flow is not expected. As it is described in Theoretical part of thesis, kinematicshape factor shows the most clear trend of streamline along the geometry of airfoil’s body.In both symmetric and deflected shapes, the mentioned picks whom are observed at theLE, appear at the same position of δ∗ again with the same shapes. The reason then mostprobably is laminar separation bubble which might have occurred for our designed airfoil,as it has not enough camber in this area. Another reason for this laminar separation ofstreamline might be the points of inflection in this positions. (figure 7.10)

Figure 7.18 also shows the increment of frictions due to the probable laminar separationbubble which might be due to the lack of camber at the LE which is marked as regionnumber 1 in figure 7.10.

−0.01

0

0.01

0.02

0.03

0.04

0.05

0.06

0 0.2 0.4 0.6 0.8 1

Cf

[−]

x/c [−]



Figure 7.18: Teh skin friction coefficient for symmetric and deflected airfoil

65

Flow Simulation

7.4 Comparing the results with a Standard Airfoil De-flection

As the designed airfoil model has not a standard geometry and the objective of project was todeflect a standard symmetric airfoil, the comparison is done between deflected NACA 0010and the deflected shape of our trial airfoil in order to study the drawbacks of the designedairfoil and analysis the effects of shape adaptation in terms of aerodynamic output variables.Figure 7.19 illustrates the he coordination of symmetric and deflected NACA 0010 whichis modeled by the developed program according to the explained details in chapter 5. Theamount of deflection, the thickness and the material is kept the same values with the testedairfoil of this project. The flexible part of TE also has the same part of chord length(75%)in both cases.

0 50 100 150 200 250 300 350 400

−150

−100

−50

0

50

100

150

x [mm]

y [

mm

]

Deflected NACA 0010

NACA 0010

Figure 7.19: The deflected NACA 0010 is modeled by Matlab

Accordingly, the coordination of deflected NACA 0010 produced by Matlab is generated inXfoil as figure 7.20.

Figure 7.20: The deflected NACA is generated in Xfoil

66

Flow Simulation

It could be then a logical comparison since they have almost similar thickness and lengthin symmetric shape, however both of symmetric airfoils are deflected 55 millimeters withthe same modeling program. The main properties of deflected geometry is also provided byXfoil and compared with the properties of the trial deflected airfoil in table 7.1 .

Table 7.1: The geometric parameters are compared with deflected NACA

Parameter Deflected NACA Deflected trial airfoil

Camber / c 0.03761 0.03711

Internal area (xc ,yc ) 0.06708 0.07561

Maximum thickness / c 0.10002 0.10118

Nose radius ( rLEc ) 0.04224 0.01415

Trailing Edge angle (∆θTEc ) 12.43° 42.20°

The pressure distribution is compared with the deflected NACA airfoil in figure 7.21,provided by the pressre vectors affected on body:

(a) Pressure vectors over deflected NACA 0010

(b) Pressure vectors over deflected trial airfoil

Figure 7.21: Pressure distribution is compared with deflected NACA

The pressure coefficient is plotted in figure 7.22 and compared with the deflected de-

67

Flow Simulation

signed model at zero AOA for the same Re and Mach number. The suction side shows lessfluctuation in LE and the boundary layers behavior could be also correlated in this figure.It is worth to mention that the L/D ratio for NACA 0010 is almost 4 units less than thedesigned airfoil.

(a) Pressure coefficient for deflected NACA 0010

(b) Pressure coefficient for deflected trial airfoil

Figure 7.22: Pressure coefficient for trial deflection is compared with deflectedNACA

The following polar curves show simply the situation of lift and drag for deflected NACA0010, compared with our trial deflected airfoil. In figure 7.23, the deflected NACA 0010shows a linear trend in lift coefficient within our desired working range (-5 to 5 AOA),where the deflected trial airfoil has undesirable response due to design’s weaknesses. By theway, the maximum lift coefficient is almost the same in both cases.

68

Flow Simulation

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

−20 −18 −16 −14 −12 −10 −8 −6 −4 −2 0 2 4 6 8 10 12 14 16 18 20

Cl [−

]

α [° ]

Deflected NACA Deflected trial airfoil

Figure 7.23: Cl over α is compared with deflected NACA

figure 7.24, provides the comparison for drag over the wide range of AOAs. The graphshows earlier stall for deflected NACA, but at the mentioned range of AOAs which is, atthe end, the operational boundary conditions of the designed airfoil (-5 to 5 AOA), the dragloads related to NACA airfoil have the same value with our designed model.

0

0.03

0.06

0.09

0.12

0.15

0.18

0.21

0.24

0.27

0.3

−20 −18 −16 −14 −12 −10 −8 −6 −4 −2 0 2 4 6 8 10 12 14 16 18 20

Cd [

−]

α [° ]


Figure 7.24: Cd over α is compared with deflected NACA

Figures 7.25 and 7.26 also compared the changes of L/D with the deflected NACA, overα and Cl, respectively. The remarkable point is, exactly at every angles of attack and liftcoefficient values, that the value of L/D shows drop for designed airfoil, the NACA airfoilresponses reversely, with higher lift over drag. The reason might be the shape factor andskin coefficients at that illustrated regions.

69

Flow Simulation

−40

−30

−20

−10

0

10

20

30

40

50

60

70

80

90

−20 −15 −10 −5 0 5 10 15 20

L/D

[−

]

α [° ]


Figure 7.25: l/d over α is compared with deflected NACA

−40

−30

−20

−10

0

10

20

30

40

50

60

70

80

90

−0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4

L/D

[−

]

Cl [−]


Figure 7.26: L/D over Cl is compared with deflected NACA

The comparison is done for boundary layer properties as well. The δ∗ and θ thicknessesalong the deflected NACA airfoil clearly show how the mentioned fluctuations and abnor-malities of the flow through our trial deflected airfoil could be improved with the deflectedNACA in figures 7.27 and 7.28. This improvement is highly remarkable at some regions,for instance, at the pressure side of δ∗ and shape factor. As was expected, since the pointof inflection at pressure side (regions number 3 in figure 7.10) and the lack of camber atsuction side (regions number 1 in figure 7.10) are not over the geometry of deflected NACA,the picks of kinematic shape factor are disappeared at 15% of chord in suction side and 25%of chord at pressure side, if the airfoil would be a standard NACA rather than our trialdesigned airfoil.

70

Flow Simulation

0

0.005

0.01

0.015

0.02

0.025

0.03

0 0.2 0.4 0.6 0.8 1

δ*/c

[−

]

x/c [−]

Deflected NACA suction sideDeflected NACA pressure side

Deflected trial airfoil suction sideDeflected trial airfoil pressure side

Figure 7.27: δ∗ is compared with deflected NACA

0

0.003

0.006

0.009

0 0.2 0.4 0.6 0.8 1

θ/c

[−

]

x/c [−]



Figure 7.28: θ is compared with deflected NACA

0

2

4

6

8

10

0 0.2 0.4 0.6 0.8 1

H [

−]

x/c [−]



Figure 7.29: Shape factor is compared with deflected NACA

71

Chapter 8

Conclusions andRecommendations

8.1 Conclusions

Current trend in wind industry is to achieve the maximum cost reduction for the energyproduction of wind turbines. As the market pushes to reduce the cost of energy, manufac-turers have increasingly sought to increase the size of turbines and design longer rotor bladesto capture more wind power. Economically, and environmentally, it makes good sense, butincrease in turbine size leads to high loads with negative and deleterious effects on rotorblades. Hence, load reduction on wind turbine blades has the vital role to increase thereliability and efficiency which is also the focus of this thesis work.

Main goals of this thesis work were to create a detailed model for the flexible trailing-edgefor wind turbine blades. Therefore, all investigations have been carried out to have controlon the geometry of the airfoil for load reduction during gusty winds, as well as more en-ergy production when the operational wind has low speed. As the current available shapeadaptation mechanisms are reviewed, they do not have a favourable structure fit to thewind turbine blades, in terms of internal structure and weight limitations, since they weredeveloped for other turbomachinery such as helicopter and airplane industries. In order toimplement a multi-variable control technique for load reduction on wind turbine structure,following issues were addressed to successfully achieve the prime target:

• Development a concept wherewithal the airfoil can be bent at the trailing edge, seam-less and without any flap

• Development and design of a set of pneumatic actuators, which can bend the airfoilat trailing edge.

• Development of a detailed analytical program which can calculate the deflection pa-rameters and model the shape of the deflected airfoil at trailing edge.

• Fabrication of a symmetric flexible airfoil and implementing the active shape adapta-tion in wind tunnel.

• Analysis of the aerodynamic results and comparisons of the characteristics for bothdesigned deflected and symmetric airfoils as well as a standard NACA airfoil.

The mission of the project fulfilled by two inflatable rubber actuators. The designed actu-ators in this project provides a sufficient camber in a symmetric airfoil, with 9.4°downwarddeflection and 11.9°upward deflection. In addition, the structure of the suggested airfoil can

72

Conclusions and Recommendations

meet the design expectations for a wind turbine blade, as it is light enough with a hollowbody. About the blade skin shells, a suitable connecting mechanism at the trailing edgeis offered by this project, wherewith, wind turbine blade skins can slide over each otherduring the upward and downward deflections. Therefore, the main objectives of the projectis met in flow simulations, which is the increment of lift over drag in a certain desired rangeof angles of attack so that directly leads to more power output and less deleterious drag.Consequently, the reduction in aerodynamic loads and stresses, leads to reduce the main-tenance costs, since the stresses on the rotating structures like bearings and gearbox woulddecrease, and additionally, the less loads over the blades would provide the possibility ofmanufacturing longer blades. The mentioned results would be indeed attractive for bothinvestors and manufacturers, as a wind turbine with form-adaptive airfoils can capture morepower by longer blades and has greater lifetime resulted by load reductions.

8.2 Recommendations

This designed model for load control approach gives efficiently the desired results for thewind turbine blades, however, the shape accuracy issue remains as the main weakness of ourinvestigations, mostly due to the fact that the designed airfoil was a pilot model for primetests. The main drawbacks of the designed airfoil have been mentioned in flow simulations,in which regions were observed the lack of camber and the points of inflection along theupper and lower surfaces. Therefore, in order to meet an improved geometry of the de-flection, a standard design is highly recommended for further investigations as a standardNACA airfoil is studied and the advantages are simulated by Xfoil program, compared withthe designed airfoil. In future, the material of the skin’s shells should be studied more,since elastic behaviour and lighter material with cheaper price are always desirable for windturbine blades. Also about inflatable actuator’s material, the next designs can be followedto investigate other materials which have probably more load-transmission efficiency. In ad-dition, providing the possibility of actuators movement is recommended for next steps. Asthe contact area between brackets and actuators was unstable, it reduces the reliability ofactuators. An internal glide track, for instance, could provide the invariant contact area inbetween with the possibility of required motions for actuators, wherewith, the actuators canmove during the inflation and deflation with the airfoil’s deflections, simultaneously. Lastbut not least in importance, the inflation of actuators in this step was done manually withan air pump, while the study of the automatic pneumatic control is indeed required for finaldesign, in order to deflect the airfoil automatically by inflation and deflation of actuators asa reaction to the affected loads on wind turbine blade skin.

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Appendix A

Matlab and Fortran Scripts

A.1 Modeling the symmetric NACA Airfoils

Following script is written to extract the coordination for every symmetric NACA airfoilsplus its plot. The chord length as well as the maximum thickness of the airfoil should bedefined. The code is based on the equation (3.2) page 14:

func t i on y = naca ( c , n )c l o s e a l lc l c%n= the l a s t two d i g i t s o f NACA a i r f o i l%c= chord l engthf o r i =1:1001

x=(i −1)∗( c /1000) ;

T= n /100 ;b0 = 1 . 4 8 4 5 ;b1 = 0 . 6 3 ;b2 = 1 . 7 5 8 0 ;b3 = 1 . 4 2 1 5 ;b4 = 0 . 5 0 7 5 ;y l = c∗T∗( b0∗ s q r t ( x/c ) − b1∗ ( x/c ) − b2 ∗( x/c ) . ˆ 2

+ b3 ∗( x/c ) . ˆ 3 − b4 ∗( x/c ) . ˆ 4 ) ;

y c ( i ,1)= −y l ;x c ( i ,1)=x ;

end

f o r j =1:1001z=(j −1)∗( c /1000) ;

T= n /100 ;b0 = 1 . 4 8 4 5 ;b1 = 0 . 6 3 ;b2 = 1 . 7 5 8 0 ;b3 = 1 . 4 2 1 5 ;b4 = 0 . 5 0 7 5 ;yu = c∗T∗( b0∗ s q r t ( z/c ) − b1∗ ( z/c ) − b2 ∗( z/c ) . ˆ 2

+ b3 ∗( z/c ) . ˆ 3 − b4 ∗( z/c ) . ˆ 4 ) ;

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y d ( j ,1)= yu ;x d ( j ,1)= z ;

end

f i g u r e ;p l o t (x , yu , ’−b ’ , x , yl , ’−b ’ , ’ LineWidth ’ , 3 ) ;a x i s equalx l a b e l ( ’ x [mm] ’ , ’ FontSize ’ , 14)y l a b e l ( ’ y [mm] ’ , ’ FontSize ’ , 14)saveas ( gcf , ’ a i r f o i l . eps ’ , ’ psc2 ’ )

A.2 Thickness of a NACA Airfoil in a given position

Following code is able to calculate the thickness of NACA airfoils. The required input is theNACA number and chorl length.

func t i on t=x2t 2 (Pc , n , c )T=n /100 ;%n= the l a s t two d i g i t s o f NACA a i r f o i l%Pc= the percentage o f chord l i n e ,

%( p o s i t i o n which i s d e s i r e d f o r t h i c k n e s s c a l c u l a t i o n )%c= chord l engthx=Pc∗c /100 ;b0 = 1 . 4 8 4 5 ;b1 = 0 . 6 3 ;b2 = 1 . 7 5 8 0 ;b3 = 1 . 4 2 1 5 ;b4 = 0 . 5 0 7 5 ;yt = c∗T∗( b0∗ s q r t ( x/c ) − b1∗ ( x/c ) − b2 ∗( x/c ) . ˆ 2

+ b3 ∗( x/c ) . ˆ 3 − b4 ∗( x/c ) . ˆ 4 ) ;yc = 0 ; %Bcs NACA0012 i s symmetrict e t a = atan (0 ) ; %symmetric dyc/dx=0yu = yc + ( yt ∗ cos ( t e ta ) ) ;y l = yc − ( yt ∗ cos ( t e ta ) ) ;t = yu − y l ;

A.3 Pressure Distribution along the Airfoil

Since XFOIL program has not the output for the pressure distribution, following code cal-culates it just having the atmosphere pressure, flow stream velocity, flow density and thepressure coefficient (Cp) distributed in a unit length of airfoil based on the equation (3.6)page 16:

c l e a r a l lc l cc l o s e a l li d u=importdata ( ’ upperx . txt ’ ) ;data u=s t r u c t 2 c e l l ( id u ) ;matrix1=ce l l 2mat ( data u ( 1 , 1 ) ) ;A=matrix1 ( : , 1 ) ;B=matrix1 ( : , 2 ) ;xu=f l i p u d (A) ;

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yu=f l i p u d (B) ;

i d l=importdata ( ’ lowerx . txt ’ ) ;d a t a l=s t r u c t 2 c e l l ( i d l ) ;matrix2=ce l l 2mat ( d a t a l ( 1 , 1 ) ) ;x l=matrix2 ( : , 1 ) ;y l=matrix2 ( : , 2 ) ;

c =200;r0= 1.225 ;p0= 101325;v0= 2 7 . 8 4 ;

Xu i=xuX l i=x lPu i =(0.5∗yu∗ r0 ∗v0 ˆ.2)+ p0P l i =(0.5∗ y l ∗ r0 ∗v0 ˆ.2)+ p0p lo t ( Xu i , Pu i , ’−bx ’ , ’ LineWidth ’ , 2 , ’ DisplayName ’ , ’ Suct ion s ide ’ )hold onp lo t ( Xl i , P l i , ’− ro ’ , ’ LineWidth ’ , 2 , ’ DisplayName ’ , ’ Pres sure s ide ’ )h l= legend ( ’ Location ’ , ’ northeast ’ ) ;l egend ( ’ boxof f ’ )s e t ( h l , ’ FontSize ’ , 1 4 ) ;x l a b e l ( ’ x/c [m] ’ , ’ FontSize ’ , 15)y l a b e l ( ’ Pressure [ pa ] ’ , ’ FontSize ’ , 15)saveas ( gcf , ’ p r e s su r e4 . eps ’ , ’ psc2 ’ )

A.4 Comparing the Cantilever Beams Deflection by OneLoad

With this Matlab code, the deflection shape of two cantilever beams can be compared bychanging the bending parameters like beams dimension, material, or the position of the forceaffecting on the beam based on the equation (3.17) page 27. It gives also the maximum de-flection and the angle of the deflected beam at the free end.

func t i on beamcompare1 ( pc1 , pc2 , c )c l o s e a l lc l c%pc1= The p o s i t i o n o f po int load on f i r s beam%pc2= The p o s i t i o n o f po int load on second beam% F i r s t beam parametersE1=10ˆ3;%N/ m mBt1=1.5;%mmBw1=198;%mmI1= ( Bt1ˆ3∗Bw1) / 1 2 ;a l 1= pc1∗c /100 ;p l1= 3;%N

% Second beam parametersE2=10ˆ3;%N/ m m

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Bt2=2;%mmBw2=198;%mmI2= ( Bt2ˆ3∗Bw2) / 1 2 ;a l 2= pc2∗c /100 ;p l2= 3 ;

f o r i =1:1000x=(i −1)∗ .200+0;i f x<a l1y c ( i ,1)= pl1 ∗x . ˆ 2 . ∗ ( 3 ∗ al1−x )/(6∗E1∗ I1 ) ;e l s ey c ( i ,1)= pl1 ∗ a l1 . ˆ 2 . ∗ ( 3 ∗ x−a l1 )/(6∗E1∗ I1 ) ;end

x c ( i ,1)=x ;end

f o r j =1:1000

z=(j −1)∗ .200+0;i f z<a l2y d ( j ,1)= pl2 ∗z . ˆ 2 . ∗ ( 3 ∗ al2−z )/(6∗E2∗ I2 ) ;e l s ey d ( j ,1)= pl2 ∗ a l2 . ˆ 2 . ∗ ( 3 ∗ z−a l2 )/(6∗E2∗ I2 ) ;end

x d ( j ,1)= z ;end

p1= p lo t ( x c ,−y c , ’ b ’ , ’ LineWidth ’ , 3 , ’ DisplayName ’ , ’ 1 . 5 mm th icknes s ’ )hold onp2 = p lo t ( x d ,−y d ,’−−r ’ , ’ LineWidth ’ , 3 , ’ DisplayName ’ , ’ 2 mm th icknes s ’ )hold o f fh l= legend ( ’ Location ’ , ’ southwest ’ ) ;l egend ( ’ boxof f ’ )s e t ( h l , ’ FontSize ’ , 1 4 ) ;

Tl = ( atan (max( y c ) / ( 0 . 6∗ c ) ) )∗180/ p i % degreeTu = ( atan (max( y d ) / ( 0 . 6∗ c ) ) )∗180/ p i % degree

Dlmax=max( y c )Dumax=max( y d )

a x i s equalx l a b e l ( ’ x [mm] ’ , ’ FontSize ’ , 15)y l a b e l ( ’ y [mm] ’ , ’ FontSize ’ , 15)%t i t l e ( ’ bending o f two beam with two actuator ’ )saveas ( gcf , ’ compare1 . eps ’ , ’ psc2 ’ )end

A.5 Comparing the Cantilever Beams Deflection by TwoPoint Loads

Following Matlab Script is developed by equation (3.19) page 28, in order to compare thedeflection shape with different position of forces on two different beams.

func t i on y = twobeam2forces ( Pc1 , Pc2 , Pc3 , Pc4 , c )

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c l o s e a l lc l c%pc : p o s i t i o n o f po int load over the beam

% F i r s t beam parametersEl=10ˆ3;%N/ m mBtl=2;%mmBwl=198;%mmI l= ( Btl ˆ3∗Bwl ) / 12 ;a l 1= Pc1 ;au1= Pc3 ;p l1= 20;%Npl2= 20 ;

% Second beam parametersEu=10ˆ3;%N/ m mBtu=2;%mmBwu=198;%mmIu= ( Btuˆ3∗Bwu) / 1 2 ;a l 2= Pc2 ;au2= Pc4 ;pu1= 20;%Npu2= 20 ;

t e ta1= ( pl1 ∗ a l1 ˆ2/(2∗ El∗ I l ) ) ;t e ta2= ( pu1∗au1 ˆ2/(2∗Eu∗ Iu ) ) ;

g1= tan ( te ta1 ) ;g2= tan ( te ta2 ) ;

f o r i =1:1001x=(i −1)∗( c /1000) ;

i f (x<(40∗ c /100))y c ( i ,1)= 0 ;

e l s e i f ( (40∗ c/100)<=x ) && (x<=(a l1 +80))y c ( i ,1)= pl2 ∗(x−80) .ˆ2 .∗ (3∗ al2−(x−80))/(6∗ El∗ I l )

+pl1 ∗(x−80) .ˆ2 .∗ (3∗ al1−(x−80))/(6∗ El∗ I l ) ;

e l s e i f ( ( a l 1+80)<x ) && (x<=(a l2 +80))y c ( i ,1)= pl2 ∗(x−80) .ˆ2 .∗ (3∗ al2−(x−80))/(6∗ El∗ I l )

+pl1 ∗ a l1 . ˆ 2 . ∗ ( 3 ∗ ( x−80)−a l1 )/(6∗ El∗ I l )+g1∗abs ( ( x−80)−a l1 ) ;

e l s e i f ( ( a l 2+80)<x)&&(x<=c )y c ( i ,1)= pl2 ∗ a l2 . ˆ 2 . ∗ ( 3 ∗ ( x−80)−a l2 )/(6∗ El∗ I l )

+pl1 ∗ a l1 . ˆ 2 . ∗ ( 3 ∗ ( x−80)−a l1 )/(6∗ El∗ I l )+g1∗abs ( ( x−80)−a l1 ) ;

endx c ( i ,1)=x ;

end

f o r j =1:1001z=(j −1)∗( c /1000) ;

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i f ( z<(40∗ c /100))y d ( j ,1)= 0 ;

e l s e i f ( (40∗ c/100)<=z ) && ( z<=(au1+80))y d ( j ,1)=pu2 ∗( z−80) .ˆ2 .∗ (3∗ au2−(z−80))/(6∗Eu∗ Iu )

+pu1 ∗( z−80) .ˆ2 .∗ (3∗ au1−(z−80))/(6∗Eu∗ Iu ) ;

e l s e i f ( ( au1+80)<z ) && ( z<=(au2+80))y d ( j ,1)=pu2 ∗( z−80) .ˆ2 .∗ (3∗ au2−(z−80))/(6∗Eu∗ Iu )

+pu1∗au1 . ˆ 2 . ∗ ( 3 ∗ ( z−80)−au1 )/(6∗Eu∗ Iu)+g2∗abs ( ( z−80)−au1 ) ;

e l s e i f ( ( au2+80)<z)&&(z<=c )y d ( j ,1)=pu2∗au2 . ˆ 2 . ∗ ( 3 ∗ ( z−80)−au2 )/(6∗Eu∗ Iu )

+pu1∗au1 . ˆ 2 . ∗ ( 3 ∗ ( z−80)−au1 )/(6∗Eu∗ Iu)+g2∗abs ( ( z−80)−au1 ) ;

endx d ( j ,1)= z ;

end

Tl = ( atan (max( y c ) / ( 0 . 6∗ c ) ) )∗180/ p i % degreeTu = ( atan (max( y d ) / ( 0 . 6∗ c ) ) )∗180/ p i % degree

Dlmax=max( y c )Dumax=max( y d )

f i g u r e ;p l o t ( x c ,−y c , ’ g ’ , x d ,−y d , ’ b ’ )a x i s equalx l a b e l ( ’ beams length mm’ )y l a b e l ( ’ d e f l e c t i o n mm’ )

saveas ( gcf , ’ twode f l ect ioncompare . eps ’ , ’ psc2 ’ )save ( ’ d e f l e c t i o n . mat ’ )

A.6 Modeling the Cantilever Beam Deflection with Dis-tributed Loads

This Fortran program is able to calculate the deflection of a cantilever beam affected bydistributed loads in every position along the beam, by solving the differential integrationbased on the main bending moment equation (3.15) page 26. It is solved by numericallyinterpolation in 4 steps beginning with then shear force, then bending moment, slope ofthe displacement, and finally the value of displacement, if the the pressure distribution overthe beam is given. (The attached subroutine of numerical interpolation is written by Prof.Dr.-Ing. M. Lawerenz in 2014.)

i n t e g e r n dimparameter ( n dim=1001)

r e a l x ( n dim ) ! x−coord inate along the beam a x i sr e a l y ( n dim ) ! d i sp lacement in the y d i r e c t i o nr e a l q ( n dim ) ! s l ope o f the d isp lacement

79


r e a l prs ( n dim ) ! p r e s su r er e a l s f r c ( n dim ) ! shear f o r c er e a l bmnt( n dim ) ! bending momentr e a l emod( n dim ) ! e l a s t i c i t y moduler e a l imoa ( n dim ) ! i n e r t i a moment o f arear e a l bwth ( n dim ) ! width o f the beam in the z d i r e c t i o nr e a l thns ( n dim ) ! t h i c k n e s s o f the beam in the y d i r e c t

r e a l d e l t a xr e a l d e l t a y

Inputr e a l x i ( n dim ) ! x−coord inate along the beam a x i sr e a l p r s i ( n dim ) ! p r e s su r er e a l emod i ( n dim ! e l a s t i c i t y moduler e a l bwth i ( n dim ! width o f the beam in the z d i r e c t i or e a l t h n s i ( n dim ! t h i c k n e s s o f the beam in the y d i r e

i n t e g e r n inp ! number o f input datai n t e g e r i i n p ! loop index inputi n t e g e r n c l c ! number o f mesh po in t si n t e g e r i c l c ! loop index c a l c u l a t i o n

i n t e g e r i e r ! e r r o r f l a g

read (11 ,∗ ) n inpread (11 ,∗ ) n c l c

do i i n p= 1 , n inpread ( 11 ,∗ ) x i ( i i n p ) , p r s i ( i i n p ) , emod i ( i i n p ) ,

bwth i ( i i n p ) , t h n s i ( i i n p )end do

mesh gene ra t i ond e l t a x =( x i ( n inp)− x i ( 1 ) ) / f l o a t ( n c l c −1)do i c l c =1, n c l c

x ( i c l c )= x i (1)+ ( x i ( n inp)− x i ( 1 ) )∗f l o a t ( i c l c −1)/ f l o a t ( n c l c −1)

c a l l i n t c s ( n inp , x i , p r s i , x ( i c l c ) , prs ( i c l c ) , i e r )i f ( i e r . ne . 0 ) then

wr i t e (∗ ,∗ ) ’ i n t c s e r r o r i e r : ’ , i e rstop

end i fc a l l i n t c s ( n inp , x i , emod i , x ( i c l c ) , emod( i c l c ) , i e r )c a l l i n t c s ( n inp , x i , bwth i , x ( i c l c ) , bwth ( i c l c ) , i e r )c a l l i n t c s ( n inp , x i , thns i , x ( i c l c ) , thns ( i c l c ) , i e r )

imoa ( i c l c )=bwth ( i c l c )∗ thns ( i c l c )∗∗3/12 .0end do

p r e d i c t i o n o f the shear f o r c es f r c (1)=0.0

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do i c l c =2, n c l cs f r c ( i c l c )= s f r c ( i c l c −1)+

0 . 5∗ ( prs ( i c l c )+prs ( i c l c −1))∗0 . 5∗ ( bwth ( i c l c )+bwth ( i c l c −1))∗ d e l t a x

end dodo i c l c =1, n c l c

s f r c ( i c l c )= s f r c ( i c l c )− s f r c ( n c l c )end do

p r e d i c t i o n o f the bending momentbmnt(1)=0.0do i c l c =2, n c l c

bmnt( i c l c )= bmnt( i c l c −1)− s f r c ( i c l c )∗ d e l t a xend dodo i c l c =1, n c l c

bmnt( i c l c )=bmnt( i c l c )−bmnt( n c l c )end do

p r e d i c t i o n o f the s l opeq (1)=0.0do i c l c =2, n c l c

q ( i c l c )=q ( i c l c −1) −0 . 5∗ ( bmnt( i c l c −1)/(emod( i c l c −1)∗ imoa ( i c l c −1)) +

bmnt( i c l c )/ ( emod( i c l c )∗ imoa ( i c l c ) ) )∗ d e l t a xend dop r e d i c t i o n o f the d isp lacementy (1)=0.0do i c l c =2, n c l c

y ( i c l c )=y ( i c l c −1) + 0 . 5∗ ( q ( i c l c −1) + q ( i c l c ) )∗ d e l t a xend do

do i c l c =1, n c l cwr i t e (21 ,∗ ) i c l c , x ( i c l c ) , q ( i c l c ) , y ( i c l c ) , bmnt( i c l c ) ,

s f r c ( i c l c )end doend================================================================subrout ine i n t c s (n , x , y , xq , yq , i e r )−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−( c ) M. Lawerenz 2014Cubic I n t e r p o l a t i o n

i m p l i c i t r e a l ( a−h , o−z )

dimension x (n ) , y (n)i e r =0

i f (n . eq . 1 ) thenyq=y (1)re turn

e l s e i f (n . eq . 2 ) thenyq=y(1)+(y(2)−y ( 1 ) ) / ( x(2)−x ( 1 ) )∗ ( xq−x ( 1 ) )

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re turne l s e i f (n . eq . 3 ) then

( parabe l n−2,n−1,n)j=n−1c2=xq−x ( j−1)c3=xq−x ( j )c4=xq−x ( j +1)

b23=x ( j−1)−x ( j )b24=x ( j−1)−x ( j +1)b34=x ( j )−x ( j +1)

p22= c3∗ c4/b23/b24p23=−c2∗ c4/b23/b34p24= c2∗ c3/b24/b34yq=y ( j −1)∗p22+y ( j )∗ p23+y ( j +1)∗p24return

e l s emehr a l s 3 s t u e t z s t e l l e n

i f ( xq . l t . x ( 2 ) ) thenrand l i n k s ( parabe l 1 , 2 , 3)j=1c1=xq−x ( j )c2=xq−x ( j +1)c3=xq−x ( j +2)

b12=x ( j )−x ( j +1)b13=x ( j )−x ( j +2)b23=x ( j +1)−x ( j +2)

p11= c2∗ c3/b12/b13p12=−c1∗ c3/b12/b23p13= c1∗ c2/b13/b23

yq=y ( j )∗ p11+y ( j +1)∗p12+y ( j +2)∗p13return

e l s e i f ( xq . gt . x (n−1)) thenrand r e c h t s ( parabe l n−2,n−1,n)j=n−1c2=xq−x ( j−1)c3=xq−x ( j )c4=xq−x ( j +1)

b23=x ( j−1)−x ( j )b24=x ( j−1)−x ( j +1)b34=x ( j )−x ( j +1)


yq=y ( j −1)∗p22+y ( j )∗ p23+y ( j +1)∗p24return

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e l s ez e n t r a l e r be r e i ch

i n t e r v a l l−bestimmungdo k=2,n−2

i f ( x ( k ) . l e . xq . and . x ( k+1). ge . xq ) thenj=kgoto 1

end i fend doi n t e r v a l l n i cht gefundeni e r =2returncont inue

i n t e r p o l a t i o n im z e n t r a l e n be r e i ch

c1=xq−x ( j−1)c2=xq−x ( j )c3=xq−x ( j +1)c4=xq−x ( j +2)

b12=x ( j−1)−x ( j )b13=x ( j−1)−x ( j +1)b23=x ( j )−x ( j +1)b24=x ( j )−x ( j +2)b34=x ( j +1)−x ( j +2)



y1=y ( j −1)∗p11+y ( j )∗ p12+y ( j +1)∗p13y2=y ( j )∗ p22+y ( j +1)∗p23+y ( j +2)∗p24

sq=(xq−x ( j ) ) / ( x ( j +1)−x ( j ) )fq =((6.∗ sq−15.)∗ sq +10.)∗ sq∗ sq∗ sq

fq s t e l l t d i e kopplung der beiden parabeln ( j −1, j , j +1) und( j , j +1, j +1) im i n t e r v a l l ( j , j +1) her :

yq=y1∗(1.0− fq )+y2∗ fq

f u e r fq=f ( s ) g i l t : s= 0 : fq= 0 ; fq ’= 0 ;s= 1 : fq= 1 ; fq ’= 0 ,

i f ( fq . l t . 0 . ) fq =0.i f ( fq . gt . 1 . ) fq =1.

yq=y1∗(1.− fq )+y2∗ fq

83


re turnend i fend i f

end

A.7 Modeling the Deflection of Two Blade Skins withTwo Actuators

The following code is written to calculate the deflection of two cantilever beams, when onepneumatic actuator is inflating between them and the point loads are transferred on theupper and lower beams by L-type brackets based on equation (3.19) page 28.

func t i on y = bendingm22 14 ( Pc1 , Pc2 , c )c l o s e a l lc l ct1= x2t ( Pc1 ) ;t2= x2t ( Pc2 ) ;

%E= 10ˆ9Gpa (N/ m ) ;E=10ˆ3;%N/ m mBt=2;%mmBw=198;%mmI= ( Btˆ3∗Bw) / 1 2 ;Fl1= 100 ; %N/ m m =10 barFl2= 100 ; %N/ m m =10 barFu1= 100 ; %N/ m m =10 barFu2= 100 ; %N/ m m =10 barm=4;%mmo=4;%mmdpipe=4;%mmLt=2;%mma1=(Pc1−40)∗c /100 ;a2=(Pc2−40)∗c /100 ;a l 1= a1+(dpipe /2)+m/2 ;a l2= a2+(dpipe /2)+m/2 ;au1= a1−(dpipe /2)−m/2 ;au2= a2−(dpipe /2)−m/2 ;r1= t1/2−Bt−Lt /2 ;r2= t2/2−Bt−Lt /2 ;p l1= Fl1∗ r1 / a l1 ;p l2= Fl2∗ r2 / a l2 ;pu1= Fu1∗ r1 /au1 ;pu2= Fu2∗ r2 /au2 ;t e ta1= ( pl1 ∗ a l1 ˆ2/(2∗E∗ I ) ) ;t e ta2= ( pu1∗au1 ˆ2/(2∗E∗ I ) ) ;g1= tan ( te ta1 ) ;g2= tan ( te ta2 ) ;f o r i =1:1001

x=(i −1)∗( c /1000) ;i f (x<(40∗ c /100))y c ( i ,1)= 0 ;

84


e l s e i f ( (40∗ c/100)<=x ) && (x<=(a l1 +80))y c ( i ,1)= pl2 ∗(x−80) .ˆ2 .∗ (3∗ al2−(x−80))/(6∗E∗ I )

+pl1 ∗(x−80) .ˆ2 .∗ (3∗ al1−(x−80))/(6∗E∗ I ) ;

e l s e i f ( ( a l 1+80)<x ) && (x<=(a l2 +80))y c ( i ,1)= pl2 ∗(x−80) .ˆ2 .∗ (3∗ al2−(x−80))/(6∗E∗ I )

+pl1 ∗ a l1 . ˆ 2 . ∗ ( 3 ∗ ( x−80)−a l1 )/(6∗E∗ I )+g1∗abs ( ( x−80)−a l1 ) ;

e l s e i f ( ( a l 2+80)<x)&&(x<=c )y c ( i ,1)= pl2 ∗ a l2 . ˆ 2 . ∗ ( 3 ∗ ( x−80)−a l2 )/(6∗E∗ I )

+pl1 ∗ a l1 . ˆ 2 . ∗ ( 3 ∗ ( x−80)−a l1 )/(6∗E∗ I )+g1∗abs ( ( x−80)−a l1 ) ;

endx c ( i ,1)=x ;

end

f o r j =1:1001z=(j −1)∗( c /1000) ;i f ( z<(40∗ c /100))y d ( j ,1)= 0 ;

e l s e i f ( (40∗ c/100)<=z ) && ( z<=(au1+80))y d ( j ,1)=pu2 ∗( z−80) .ˆ2 .∗ (3∗ au2−(z−80))/(6∗E∗ I )

+pu1 ∗( z−80) .ˆ2 .∗ (3∗ au1−(z−80))/(6∗E∗ I ) ;

e l s e i f ( ( au1+80)<z ) && ( z<=(au2+80))y d ( j ,1)=pu2 ∗( z−80) .ˆ2 .∗ (3∗ au2−(z−80))/(6∗E∗ I )

+pu1∗au1 . ˆ 2 . ∗ ( 3 ∗ ( z−80)−au1 )/(6∗E∗ I )+g2∗abs ( ( z−80)−au1 ) ;

e l s e i f ( ( au2+80)<z)&&(z<=c )y d ( j ,1)=pu2∗au2 . ˆ 2 . ∗ ( 3 ∗ ( z−80)−au2 )/(6∗E∗ I )

+pu1∗au1 . ˆ 2 . ∗ ( 3 ∗ ( z−80)−au1 )/(6∗E∗ I )+g2∗abs ( ( z−80)−au1 ) ;

endx d ( j ,1)= z ;

endTl = ( atan (max( y c ) / ( 0 . 6∗ c ) ) )∗180/ p i % degreeTu = ( atan (max( y d ) / ( 0 . 6∗ c ) ) )∗180/ p i % degree

f i g u r e ;p l o t ( x c ,−y c , ’ g ’ , x d ,−y d , ’ b ’ )a x i s equalx l a b e l ( ’ a i r f o i l l ength mm’ )y l a b e l ( ’ d e f l e c t i o n mm’ )t i t l e ( ’ bending o f two beam with two actuator ’ )saveas ( gcf , ’ beambended . eps ’ , ’ psc2 ’ )save ( ’ d e f l e c t i o n . mat ’ )

i f Pc1< 40 ;e r r o r ( ’ f i r s t Actuators must be a f t e r 40% of a i r f o i l ’ )

e l s e i f Pc2< 40 ;e r r o r ( ’ second Actuators must be a f t e r 40% of a i r f o i l ’ )

e l s e i f t1< (2∗Bt+2∗m) ;

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e r r o r ( ’ f i r s t Actuator i s not in p o s s i b l e po s i t i on ’ )e l s e i f t2< (2∗Bt+2∗m) ;

e r r o r ( ’ second Actuator i s not in p o s s i b l e po s i t i on ’ )e l s e i f r1 < 0 ;

e r r o r ( ’ not enough space f o r f i r s t Actuator ’ )e l s e i f r2 < 0 ;

e r r o r ( ’ not enough space f o r second Actuator ’ )

e l s ed i sp ( ’ v a l i d input ’ )

end

A.8 Modeling a Flexible NACA 0012 with Two Actua-tors

The code is able to model the deflection of every symmetric NACA airfoil, when two pneu-matic actuators are inflating between them inside the body of airfoil and the point loads aretransferred by L-type brackets to the internal surface of the airfoil based on equation (3.19)page 28.

func t i on y = Geometry de f l ected ( Pc1 , Pc2 , c )c l o s e a l lc l ct1= x2t ( Pc1 ) ;t2= x2t ( Pc2 ) ;% t1= 15 ;% t2= 15 ;%E= 10ˆ8Gpa (N/ m ) ;E=10ˆ3;%N/ m mBt=2;%mmBw=198;%mmI= ( Btˆ3∗Bw) / 1 2 ;Fl1= 100 ; %N/ m m =5 barFl2= 100 ; %N/ m m =5 barFu1= 100 ; %N/ m m =5 barFu2= 100 ; %N/ m m =5 barm=4;%mmo=4;%mmdpipe=4;%mmLt=2;%mma1=(Pc1−40)∗c /100 ;a2=(Pc2−40)∗c /100 ;a l 1= a1+(dpipe /2)+m/2 ;a l2= a2+(dpipe /2)+m/2 ;au1= a1−(dpipe /2)−m/2 ;au2= a2−(dpipe /2)−m/2 ;r1= t1/2−Bt−Lt /2 ;r2= t2/2−Bt−Lt /2 ;p l1= Fl1∗ r1 / a l1 ;p l2= Fl2∗ r2 / a l2 ;pu1= Fu1∗ r1 /au1 ;pu2= Fu2∗ r2 /au2 ;t e ta1= ( pl1 ∗ a l1 ˆ2/(2∗E∗ I ) ) ;

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t e ta2= ( pu1∗au1 ˆ2/(2∗E∗ I ) ) ;g1= tan ( te ta1 ) ;g2= tan ( te ta2 ) ;

f o r i =1:1001x=(i −1)∗( c /1000) ;

T= 0 . 1 2 ;b0 = 1 . 4 8 4 5 ;b1 = 0 . 6 3 ;b2 = 1 . 7 5 8 0 ;b3 = 1 . 4 2 1 5 ;b4 = 0 . 5 0 7 5 ;y l = c∗T∗( b0∗ s q r t ( x/c ) − b1∗ ( x/c ) − b2 ∗( x/c ) . ˆ 2

+ b3 ∗( x/c ) . ˆ 3 − b4 ∗( x/c ) . ˆ 4 ) ;

i f (x<(40∗ c /100))y c ( i ,1)= −y l ;

e l s e i f ( (40∗ c/100)<=x ) && (x<=(a l1 +80))y c ( i ,1)=−yl−(p l2 ∗(x−80) .ˆ2 .∗ (3∗ al2−(x−80))/(6∗E∗ I )

+pl1 ∗(x−80) .ˆ2 .∗ (3∗ al1−(x−80))/(6∗E∗ I ) ) ;

e l s e i f ( ( a l 1+80)<x ) && (x<=(a l2 +80))y c ( i ,1)=−yl−(p l2 ∗(x−80) .ˆ2 .∗ (3∗ al2−(x−80))/(6∗E∗ I )

+pl1 ∗ a l1 . ˆ 2 . ∗ ( 3 ∗ ( x−80)−a l1 )/(6∗E∗ I )+g1∗abs ( ( x−80)−a l1 ) ) ;

e l s e i f ( ( a l 2+80)<x)&&(x<=c )y c ( i ,1)=−yl−(p l2 ∗ a l2 . ˆ 2 . ∗ ( 3 ∗ ( x−80)−a l2 )/(6∗E∗ I )

+pl1 ∗ a l1 . ˆ 2 . ∗ ( 3 ∗ ( x−80)−a l1 )/(6∗E∗ I )+g1∗abs ( ( x−80)−a l1 ) ) ;

endx c ( i ,1)=x ;

end

f o r j =1:1001z=(j −1)∗( c /1000) ;

T= 0 . 1 2 ;b0 = 1 . 4 8 4 5 ;b1 = 0 . 6 3 ;b2 = 1 . 7 5 8 0 ;b3 = 1 . 4 2 1 5 ;b4 = 0 . 5 0 7 5 ;yu = c∗T∗( b0∗ s q r t ( z/c ) − b1∗ ( z/c ) − b2 ∗( z/c ) . ˆ 2 + b3 ∗( z/c ) . ˆ 3

− b4 ∗( z/c ) . ˆ 4 ) ;

i f ( z<(40∗ c /100))y d ( j ,1)= yu ;

e l s e i f ( (40∗ c/100)<=z ) && ( z<=(au1+80))y d ( j ,1)=yu−(pu2 ∗( z−80) .ˆ2 .∗ (3∗ au2−(z−80))/(6∗E∗ I )

+ pu1 ∗( z−80) .ˆ2 .∗ (3∗ au1−(z−80))/(6∗E∗ I ) ) ;

87


e l s e i f ( ( au1+80)<z ) && ( z<=(au2+80))y d ( j ,1)=yu−(pu2 ∗( z−80) .ˆ2 .∗ (3∗ au2−(z−80))/(6∗E∗ I )

+ pu1∗au1 . ˆ 2 . ∗ ( 3 ∗ ( z−80)−au1 )/(6∗E∗ I )+g2∗abs ( ( z−80)−au1 ) ) ;

e l s e i f ( ( au2+80)<z)&&(z<=c )y d ( j ,1)=yu−(pu2∗au2 . ˆ 2 . ∗ ( 3 ∗ ( z−80)−au2 )/(6∗E∗ I )

+pu1∗au1 . ˆ 2 . ∗ ( 3 ∗ ( z−80)−au1 )/(6∗E∗ I )+g2∗abs ( ( z−80)−au1 ) ) ;

endx d ( j ,1)= z ;

endTl = ( atan (max( abs ( y c ) ) / ( 0 . 6 ∗ c ) ) )∗180/ p i % degreeTu = ( atan (max( abs ( y d ) ) / ( 0 . 6 ∗ c ) ) )∗180/ p i % degree

f i g u r e ;p l o t ( x c , y c , ’−b ’ , x d , y d , ’−b ’ , ’ LineWidth ’ , 3 )a x i s equalx l a b e l ( ’ x [mm] ’ , ’ FontSize ’ , 14)y l a b e l ( ’ y [mm] ’ , ’ FontSize ’ , 14)% t i t l e ( ’ bending o f a i r f o i l NACA0012 with two actuator ’ )saveas ( gcf , ’ bended 60 70 . eps ’ , ’ psc2 ’ )save ( ’ bended 60 70 . mat ’ )save ( ’ c oo rd ina t i on . dat ’ )


e l s e i f Pc2< 40 ;e r r o r ( ’ second Actuators must be a f t e r 40% of a i r f o i l ’ )

e l s e i f t1< (2∗Bt+2∗m) ;e r r o r ( ’ f i r s t Actuator i s not in p o s s i b l e po s i t i on ’ )

e l s e i f t2< (2∗Bt+2∗m) ;e r r o r ( ’ second Actuator i s not in p o s s i b l e po s i t i on ’ )

e l s e i f r1 < 0 ;e r r o r ( ’ not enough space f o r f i r s t Actuator ’ )

e l s e i f r2 < 0 ;e r r o r ( ’ not enough space f o r second Actuator ’ )


end

A.9 Modeling Two Blade Skins Deflection with an Ac-tuator Load

Following Matlab Script is developed by equation (3.17) page 27, in order to compare thedeflection shape of Aluminum and EPDM when they are effected by one single point loadsof the actuator in between two straight cantilever beams in experimental investigation. Theamount of defection then can be predicted with this program for also other materials anddimensions.

f unc t i on y = beamtest11 ( Pc1 , c )c l o s e a l l

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c l c

E=69000;%N/ m mBt=0.2;%mmBw=198;%mmI= ( Btˆ3∗Bw) / 1 2 ;l =2; %mmP=0.05; %N/ m m =0.5∗10ˆ5∗10ˆ−6A=120∗ l ;

Fl1= P∗A;

Fu1= P∗A;

m=32;%mmo=32;%mmdpipe=2;%mmLt=2;%mma1=(Pc1−40)∗ c /100 ;

a l 1= (75.7−40)∗ c /100 ;

au1= (57.5−40)∗ c /100 ;r1= m/2 ;p l1= Fl1∗ r1 / a l1 ;pu1= Fu1∗ r1 /au1 ;

f o r i =1:1001x=(i −1)∗( c /1000) ;

i f (x<(40∗ c /100))y c ( i ,1)= 0 ;

e l s e i f ( (40∗ c/100)<=x ) && (x<=(a l1 +80))y c ( i ,1)=−( p l1 ∗(x−80) .ˆ2 .∗ (3∗ al1−(x−80))/(6∗E∗ I ) ) ;

e l s e i f ( ( a l 1+80)<x)&&(x<=c )y c ( i ,1)=−( p l1 ∗ a l1 . ˆ 2 . ∗ ( 3 ∗ ( x−80)−a l1 )/(6∗E∗ I ) ) ;

endx c ( i ,1)=x ;

end

f o r j =1:1001z=(j −1)∗( c /1000) ;i f ( z<(40∗ c /100))y d ( j ,1)= 0 ;

e l s e i f ( (40∗ c/100)<=z ) && ( z<=(au1+80))y d ( j ,1)=−(pu1 ∗( z−80) .ˆ2 .∗ (3∗ au1−(z−80))/(6∗E∗ I ) ) ;

89


e l s e i f ( ( au1+80)<z)&&(z<=c )y d ( j ,1)=−(pu1∗au1 . ˆ 2 . ∗ ( 3 ∗ ( z−80)−au1 )/(6∗E∗ I ) ) ;

endx d ( j ,1)= z ;

endTl = ( atan (−min ( y c ) / ( 0 . 6∗ c ) ) )∗180/ p i % degreeTu = ( atan (−min ( y d ) / ( 0 . 6∗ c ) ) )∗180/ p i % degree

f i g u r e ;p l o t ( x c , y c , ’ g ’ , x d , y d , ’ b ’ )% a x i s equalx l a b e l ( ’ a i r f o i l l ength mm’ )y l a b e l ( ’ d e f l e c t i o n mm’ )t i t l e ( ’ bending o f two rubber beams with one actuator ’ )saveas ( gcf , ’ bended . eps ’ , ’ psc2 ’ )save ( ’ d e f l e c t i o n . mat ’ )



end

A.10 Smoothing the Geometry of the Airfoil

This Script gives the smoothed geometry using ”bezier“ function for interpolation. Thecoordination distributed in three parts and in output, the smoothed coordination will beextracted.

c l e a r a l lc l cc l o s e a l lid u1=importdata ( ’ o r i g i n a l 1 . txt ’ ) ;data u1=s t r u c t 2 c e l l ( id u1 ) ;matrix1=ce l l 2mat ( data u1 ( 1 , 1 ) ) ;A1=matrix1 ( : , 1 ) ;B1=matrix1 ( : , 2 ) ;y c1=B1 ;x c1=A1 ;input xy c1 =[ x c1 , y c1 ] ;[ bc1 , intcyy1 ] = b e z i e r ( input xy c1 , 60 , [ ] , 1 ) ;

id u2=importdata ( ’ o r i g i n a l 2 . txt ’ ) ;data u2=s t r u c t 2 c e l l ( id u2 ) ;matrix2=ce l l 2mat ( data u2 ( 1 , 1 ) ) ;A2=matrix2 ( : , 1 ) ;B2=matrix2 ( : , 2 ) ;y c2=B2 ;x c2=A2 ;input xy c2 =[ x c2 , y c2 ] ;[ bc2 , intcyy2 ] = b e z i e r ( input xy c2 , 60 , [ ] , 2 ) ;

90


id u3=importdata ( ’ o r i g i n a l 3 . txt ’ ) ;data u3=s t r u c t 2 c e l l ( id u3 ) ;matrix3=ce l l 2mat ( data u3 ( 1 , 1 ) ) ;A3=matrix3 ( : , 1 ) ;B3=matrix3 ( : , 2 ) ;y c3=B3 ;x c3=A3 ;input xy c3 =[ x c3 , y c3 ] ;[ bc3 , intcyy3 ] = b e z i e r ( input xy c3 , 60 , [ ] , 3 ) ;

save ( ’ s1 matlab2 . txt ’ , ’ bc1 ’ , ’− a s c i i ’ )save ( ’ s2 matlab2 . txt ’ , ’ bc2 ’ , ’− a s c i i ’ )save ( ’ s3 matlab2 . txt ’ , ’ bc3 ’ , ’− a s c i i ’ )

A.11 Modeling a Form Variable Airfoil Deflected withOne Actuator

The following code is written by Matlab based on the equation (3.17) page 27, in order tomodel the deflection shape of every non-standard airfoil without the standard NACA design,with a internal actuator force.

Downward deflection:

c l e a r a l lc l cc l o s e a l l

E=193000;%N/ m mBt=1;%mmBw=198;%mmI= ( Btˆ3∗Bw) / 1 2 ;a l 1 =160;au1=125;d l =55;Fl1= (6∗ dl ∗E∗ I )/ ( a l1 ˆ2∗(3∗332− a l1 ) )Fu1= (6∗ dl ∗E∗ I )/ ( au1ˆ2∗(3∗332−au1 ) )id u1=importdata ( ’ s l ow e r . txt ’ ) ;data u1=s t r u c t 2 c e l l ( id u1 ) ;matrix1=ce l l 2mat ( data u1 ( 1 , 1 ) ) ;A1=matrix1 ( : , 1 ) ;B1=matrix1 ( : , 2 ) ;y1=B1 ;x1=A1 ;

f o r i =1:90x=( i −1)∗(440/89) ;

i f (x<108)y c ( i ,1)= y1 ( i , 1 ) ;e l s e i f (108<=x ) && (x<=268)y c ( i ,1)= y1 ( i ,1)−( Fl1 ∗(x−108) .ˆ2 .∗ (3∗ al1−(x−108))/(6∗E∗ I ) ) ;e l s e i f (268<x ) && (x<=440)y c ( i ,1)= y1 ( i ,1)−( Fl1∗ a l1 . ˆ 2 . ∗ ( 3 ∗ ( x−108)− a l1 )/(6∗E∗ I ) ) ;

91


endx c ( i ,1)= x1 ( i , 1 ) ;

end

id u1=importdata ( ’ s upper . txt ’ ) ;data u1=s t r u c t 2 c e l l ( id u1 ) ;matrix1=ce l l 2mat ( data u1 ( 1 , 1 ) ) ;A2=matrix1 ( : , 1 ) ;B2=matrix1 ( : , 2 ) ;y2=f l i p u d (B2 ) ;x2=f l i p u d (A2 ) ;

f o r j =1:90z=(j −1)∗(440/89) ;i f ( z<108)y d ( j ,1)= y2 ( j , 1 ) ;e l s e i f (108 <=z ) && ( z<=233)y d ( j ,1)= y2 ( j ,1)−(Fu1∗( z−108) .ˆ2 .∗ (3∗ au1−(z−108))/(6∗E∗ I ) ) ;e l s e i f (233<z)&&(z<=440)y d ( j ,1)= y2 ( j ,1)−(Fu1∗au1 . ˆ 2 . ∗ ( 3 ∗ ( z−108)−au1 )/(6∗E∗ I ) ) ;end

x d ( j ,1)= x2 ( j , 1 ) ;endTl = ( atan (55/332))∗180/ p i % degreef i g u r e ;p l o t ( x c , y c , ’ b ’ , x d , y d , ’ b ’ , A1 , B1 , ’ k ’ , A2 , B2 , ’ k ’ )a x i s equalx l a b e l ( ’X [mm] ’ , ’ FontSize ’ , 14)y l a b e l ( ’Y [mm] ’ , ’ FontSize ’ , 14)saveas ( gcf , ’ de f l ect ion downward . eps ’ , ’ psc2 ’ )

Upward deflection:

c l e a r a l lc l cc l o s e a l l

E=193000;%N/ m mBt=1;%mmBw=198;%mmI= ( Btˆ3∗Bw) / 1 2 ;a l 1 =160;au1=300;d l =70;Fl1= (6∗ dl ∗E∗ I )/ ( a l1 ˆ2∗(3∗332− a l1 ) )Fu1= (6∗ dl ∗E∗ I )/ ( au1ˆ2∗(3∗332−au1 ) )id u1=importdata ( ’ s l ow e r . txt ’ ) ;data u1=s t r u c t 2 c e l l ( id u1 ) ;matrix1=ce l l 2mat ( data u1 ( 1 , 1 ) ) ;A1=matrix1 ( : , 1 ) ;B1=matrix1 ( : , 2 ) ;y1=B1 ;x1=A1 ;

f o r i =1:90x=( i −1)∗(440/89) ;

92


i f (x<108)y c ( i ,1)= y1 ( i , 1 ) ;e l s e i f (108<=x ) && (x<=268)y c ( i ,1)= y1 ( i ,1)+( Fl1 ∗(x−108) .ˆ2 .∗ (3∗ al1−(x−108))/(6∗E∗ I ) ) ;e l s e i f (268<x ) && (x<=440)y c ( i ,1)= y1 ( i ,1)+( Fl1∗ a l1 . ˆ 2 . ∗ ( 3 ∗ ( x−108)− a l1 )/(6∗E∗ I ) ) ;endx c ( i ,1)= x1 ( i , 1 ) ;

endid u1=importdata ( ’ s upper . txt ’ ) ;data u1=s t r u c t 2 c e l l ( id u1 ) ;matrix1=ce l l 2mat ( data u1 ( 1 , 1 ) ) ;A2=matrix1 ( : , 1 ) ;B2=matrix1 ( : , 2 ) ;y2=f l i p u d (B2 ) ;x2=f l i p u d (A2 ) ;

f o r j =1:90z=(j −1)∗(440/89) ;i f ( z<108)y d ( j ,1)= y2 ( j , 1 ) ;e l s e i f (108 <=z ) && ( z<=408)y d ( j ,1)= y2 ( j ,1)+(Fu1∗( z−108) .ˆ2 .∗ (3∗ au1−(z−108))/(6∗E∗ I ) ) ;e l s e i f (408<z)&&(z<=440)y d ( j ,1)= y2 ( j ,1)+(Fu1∗au1 . ˆ 2 . ∗ ( 3 ∗ ( z−108)−au1 )/(6∗E∗ I ) ) ;endx d ( j ,1)= x2 ( j , 1 ) ;

endTu = ( atan (70/332))∗180/ p i % degreef i g u r e ;p l o t ( x c , y c , ’ b ’ , x d , y d , ’ b ’ , A1 , B1 , ’ k ’ , A2 , B2 , ’ k ’ )a x i s equalx l a b e l ( ’X [mm] ’ , ’ FontSize ’ , 14)y l a b e l ( ’Y [mm] ’ , ’ FontSize ’ , 14)saveas ( gcf , ’ de f l ect ion downward . eps ’ , ’ psc2 ’ )

93

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Documents

Development and Design of a Form-Adaptive Trailing-Edge for … · 2015-02-13 · given to Prof. Dr.-sc.techn. Dirk Dahlhaus who introduced me to this interesting project. In addition,