C-Class Catamaran Daggerboard: Analysis and Optimization Mechanical Engineering

C-Class Catamaran Daggerboard: Analysis and Optimization

Sara Filipa Felizardo Santos Silva

Thesis to obtain the Master of Science Degree in Mechanical Engineering

Supervisors: Prof. Virginia Isabel Monteiro Nabais Infante Prof. João Carlos de Campos Henriques

Examination Committee Chairperson: Prof. Luis Manuel Varejão de Oliveira Faria Supervisor: Prof. Virginia Isabel Monteiro Nabais Infante Member of the Committee: Prof. Luis Manuel de Carvalho Gato

November 2014 ii iii

The problem is not the problem; the problem is your attitude about the problem Capt. Jack Sparrow iv Acknowledgments

First and foremost, I would like to offer my special thanks to professor Joao˜ Henriques for the patience and for presenting me to the topic of hydrofoil improvement. I would like to express my appreciation to professor Virginia Infante for the guidance and support through the work and for introducing me to Optimal Structural Solutions company. Without them this dissertation would not exist.

Additionally, I would like to thank Engineers Antonio´ Reis and Andre´ Coelho from Optimal Structural Solutions for giving me the opportunity to work with their daggerboards and for always being available to answer my questions.

A special thanks to my family: my father, my mother, my sister, my grandmother and grandfather, for all the support and inspiration which allowed me to overcome this journey. Another special thank you to my friends and colleagues for their support along this journey.

Finally, I would like to thank my beloved Rui for all the support, patience and love that allowed me to proceed and succeed in this project.

v vi Resumo

Desde 2004 que a competiçao˜ international de catamarans de classe-C tem ganho adeptos e sus- citado o interesse dos engenheiros no desenvolvimento e optimizaçao˜ das embarcaçoes.˜ Um dos componentes que tem sido sujeito a mais estudos e´ o patilhao,˜ o leme central que atravessa o casco, e que permite o levantamento da embarcaçao˜ quando esta começa a ganhar velocidade.

Neste trabalho, o perfil alar do patilhao˜ foi optimizado de forma a que este crie a sustentaçao˜ sufi- ciente para levantar a embarcaçao˜ a uma velocidade reduzida. Para isto, foi utilizado o software Xfoil incorporado no programa principal onde foi desenvolvido o modelo Class-Shape-Transformation, de modo a gerar a geometria do perfil. Utilizou-se tambem´ o modelo de optimizaçao˜ Differential Evolution para os parametrosˆ das condiçoes˜ estabelecidas.

Procedeu-se a` analise´ estrutural do modelo tridimensional do patilhao˜ atraves´ da utilizaçao˜ do software Ansys. Foram comparados os resultados dos perfis NACA 2412 e NACA 5412 ja´ existentes, com o perfil optimizado gerado no ambitoˆ do programa de trabalhos da dissertaçao.˜ Inicialmente utilizou-se uma configuraçao˜ mais simples para o perfil, tendo sido verificado grande deformaçao˜ da estrutura, o que contribui para o aparecimento de fracturas devido a` fadiga. Por esta razao,˜ o perfil foi alterado de modo a evitar estas grandes deformaçoes.˜

Por fim, foi realizado um estudo do material atraves´ do software CES Edupack. Foram definidos os constrangimentos para a utilizaçao˜ do patilhao˜ em competiçao˜ e selecionado dois materiais de forma a reduzir o custo total da contruçao˜ da mesma.

Palavras chave: Patilhao,˜ Perﬁl Alar, Class-Shape-Transformation Method, Differential Evolution, Analise´ Estrutural, Analise´ de Materiais.

vii viii Abstract

Since 2004, the international C-class catamaran championship has gained fans and the interest of engineers for the development of these vessels to become faster and lighter. The daggerboard, a hydrofoil that creates lift to take the boat out of water when the speed increases, is one of the catamaran components that has been subject to greater development and studying.

In the present work, the hydrofoil profile was optimized in order to create enough lift to lift up the boat at a low velocity. It was used an interface between Xfoil software and the main program. The main program uses the Class-Shape-Transformation method to generate a profile geometry and an optimized program known as Differential Evolution. The interface between the three components allows the user to generate a profile for the desired conditions.

The structural analysis of the structure was made using the Ansys software. The three-dimensional structures are modeled with three different profiles: NACA 2412, NACA 5412 and the new profile created by the program developed under the dissertation’s scope. Initially, it was used a simple configuration which was modified along the study in order to create a final structure with mechanical properties that contributes to a non permanent deformation of the structure, thus avoiding fatigue related damage.

Finally, a study of materials was presented using the software CES Edupack. Constrains for the daggerboard’s material utilization were deﬁned to select two materials which can be applied to the structure in order to reduce the daggerboard’s construction cost.

Keywords: Daggerboard, Hydrofoil, Class-Shape-Transformation Method, Differential Evolution, Struc- tural Analysis, Material Analysis.

Contents

1 Introduction 1 1.1 Motivation and objectives...... 2 1.2 Structure of the thesis...... 2

2 State of Art 5 2.1 C-Class catamaran racing boat...... 5 2.2 Daggerboard proﬁles...... 7 2.3 Hydrofoil design...... 11 2.3.1 Eppler’s hydrofoil...... 11 2.3.2 Hydrofoil characteristics...... 12 2.3.3 Cavitation...... 16 2.4 Material evolution...... 17

3 Methodology 19 3.1 Project conditions and assumptions...... 21

4 Daggerboard design 23 4.1 Hydrofoil design...... 24 4.1.1 Class-Shape-Transformation Method...... 24 4.1.2 Differential Evolution...... 26 4.1.3 Hydrofoil Shape Generation...... 28 4.1.4 Case of Study...... 29 4.1.5 CST shape...... 29 4.1.6 Control coefﬁcients...... 32 4.1.7 Objective Function...... 33

5 Structural Analysis 37 5.1 Pre-study...... 37 5.1.1 Static analysis - blade...... 38 5.1.2 Modal analysis - blade...... 40 5.2 Daggerboard’s analysis...... 40 5.2.1 Static analysis...... 41

xi xii CONTENTS

5.2.2 Modal analysis...... 42 5.3 CST daggerboard improvement...... 45 5.3.1 Structural designing...... 46 5.3.2 Static analysis - improved CST daggerboard...... 47 5.3.3 Modal analysis - improved CST daggerboard...... 47

6 Material analysis 53

7 Conclusions 59

8 Future work 61

Bibliography 63

A Xfoil software 1

B Structural analysis - complementary5 B.1 Blade...... 5 B.2 Daggerboard...... 6 B.3 Original L daggerboard...... 9

C Material Complement 11 C.1 Daggerboard with different materials...... 11 C.2 Material data sheets...... 13 List of Figures

2.1 Team Cascais Catamaran [9]...... 6 2.2 Catamaran Components...... 7 2.3 Flyer - Team Hydros boat [1]...... 8 2.4 Daggerboard - degrees of freedom...... 8 2.5 Daggerboard Overview [4]...... 9 2.6 Alpha’s ﬁrst daggerboard...... 9 2.7 Forces Involved - V Proﬁle [10]...... 10 2.8 Airfoil’ forces [24]...... 12 2.9 Hydrofoil Nomenclature...... 13 2.10 Positive image...... 14 2.11 Negative image...... 14

2.12 NACA 0012 - cp distribution...... 15 ◦ 2.13 NACA 0012 - cp distribution - angle of attack = 3 ...... 15 2.14 Cavitation - Three Stages...... 16 2.15 Pressure Coefﬁcient vs Velocity...... 18

3.1 Daggerboard dimensions nomenclature...... 20 3.2 NACA 2412 - angle of attack = 0◦ ...... 21 3.3 NACA 5412 - angle of attack = 0◦ ...... 21

4.1 Bernstein polynomial decomposition...... 26 4.2 Two-dimensional cost function...... 27 4.3 D = 7 parameters, crossover example [33]...... 28 4.4 Hydrofoil shape - example...... 29 4.5 CL vs alpha - NACA and Eppler profiles...... 30 4.6 CD vs alpha - NACA and Eppler profiles...... 30 4.7 cp vs x/c - NACA and Eppler profiles...... 30 4.8 NACA 2412 - Re = 2.5 × 106 and Alpha = 3.5◦ ...... 31 4.9 NACA 5412 - Re = 2.5 × 106 and Alpha = 3.5◦ ...... 31 4.10 Hydrofoil Eppler 836 (black line) and initial shape (red line)...... 32 4.11 Eppler 836 and CST shape approximation...... 33

xiii xiv LIST OF FIGURES

4.12 CST shape with thickness in the trailing edge...... 33 4.13 L/D ratio evolution - E836...... 35

4.14 cp,min evolution - E836...... 35 4.15 CST ﬁnal shape...... 35 4.16 CST ﬁnal shape - pressure distribution...... 36 4.17 E836 - pressure distribution for the same conditions of CST...... 36

5.1 Mesh Reﬁnement - Stress...... 38 5.2 Mesh Reﬁnement - Displacement...... 38 5.3 Stress distribution - NACA 2412 blade...... 39 5.4 Paddle stress distribution - NACA 5412...... 39 5.5 Stress distribution - CST...... 39

5.6 Mesh detail - CSTinitial ...... 41

5.7 Total constrain - CSTinitial ...... 41

5.8 Pressure distribution - CSTinitial ...... 41

5.9 Deformation - CSTinitial ...... 42 5.10 Daggerboard stress distribution - NACA 2412...... 43 5.11 Daggerboard stress distribution - NACA 5412...... 43

5.12 Daggerboard stress distribution - CSTinitial ...... 44 5.13 NACA 2412 x-direction displacement vs time [s...... 45

5.14 CSTinitial x-direction displacement vs time [s...... 45 5.15 NACA 2412 y-direction displacement vs time [s...... 45

5.16 CSTinitial y-direction displacement vs time [s...... 45 5.17 NACA 2412 z-direction displacement vs time [s...... 45

5.18 CSTinitial z-direction displacement vs time [s...... 45

5.19 CSTimproved - trapezoidal blade...... 47 5.20 Stress distribution - CST improvement...... 48 5.21 Total displacement - CST improvement...... 48 5.22 graphic x direction vs frequency [Hz...... 49 5.23 graphic y-direction vs frequency [Hz...... 49 5.24 graphic z-direction vs frequency [Hz...... 49

5.25 Total displacement - CSTﬁnal ...... 50

5.26 Stress distribution - CSTﬁnal ...... 50

5.27 Graphic x-direction (CSTinitial and CSTﬁnal )...... 51

5.28 Graphic y-direction (CSTinitial and CSTﬁnal )...... 52

5.29 Graphic z-direction (CSTinitial and CSTﬁnal )...... 52

6.1 Ashby map - Young’s modulus vs density...... 56 6.2 Ashby map - Yield strength vs density...... 57 6.3 Ashby map - Elongation vs Price...... 58 LIST OF FIGURES xv

A.1 Load of the geometry coordinates...... 2 A.2 Input parameters...... 2 A.3 Minimum pressure coefﬁcient...... 3 A.4 Pressure distribution...... 4 A.5 Output ﬁle example...... 4

B.1 Displacement distribution - NACA 2412...... 6 B.2 Displacement distribution - NACA 5412...... 6 B.3 Daggerboard displacement distribution - NACA 2412...... 7 B.4 Daggerboard displacement distribution - NACA 5412...... 7 B.5 Daggerboard displacement distribution - CST...... 8 B.6 NACA 5412 x-direction displacement...... 8 B.7 NACA 5412 y-direction displacement...... 8 B.8 NACA 5412 z-direction displacement...... 8 B.9 Original L daggerboard dimensions...... 9

C.1 Daggerboard displacement- CFRP...... 11 C.2 Daggerboard stress distribution- CFRP...... 12 C.3 Daggerboard displacement- Titanium alloy...... 12 C.4 Daggerboard stress distribution- Titanium alloy...... 13 C.5 Composite T800 data sheet...... 14 C.6 Composite T800 data sheet...... 15 C.7 CFRP - data sheet...... 16 C.8 GFRP - data sheet...... 17 C.9 Titanium alloys - data sheet...... 18 C.10 Rigid Polymer Foam HD - data sheet...... 19 C.11 Rigid Polymer Foam MD - data sheet...... 20

List of Tables

2.1 ICCCC 2013 - Teams...... 6

4.1 Scenario of study...... 34 4.2 CST Final Geometry Characteristics...... 35

5.1 Material Properties - T800 material (data sheet in appendixC)...... 38 5.2 Stress and Maximum Displacement - Blades...... 39 5.3 Natural frequencies - blade...... 40 5.4 Maximum displacements and stress - rectangular blade...... 42 5.5 Natural frequencies - rectangular blade...... 44 5.6 CST blade improvement...... 46 5.7 Natural frequencies - CST proﬁle improvement...... 47 5.8 Final daggerboard results...... 51 5.9 Natural frequencies - NACA and CST proﬁle evolution...... 51

6.1 Selected materials and properties...... 54 6.2 T800, CFRP and Titanium Alloy results...... 54

B.1 Mesh Refinement - NACA 2412...... 5 B.2 Mesh Refinement - NACA 5412...... 6 B.3 Mesh Refinement - CST...... 6

xvii

Nomenclature

Symbol Units Description L N Lift force D N Drag force

CL − Lift coefﬁcient

CD − Drag coefﬁcient ρ kg/m3 Fluid density c m Chord length v m/s Velocity

cp − Pressure coefﬁcient

cp,min − Minimum pressure coefﬁcient

cp,neg − Negative pressure coefﬁcient

p∞ Pa Freestream static pressure

pv Pa Vapour pressure U − Sum of negative pressure α ◦ Angle of attack

σcavit − Cavitation number d m Depth length os structure A − Aspect ratio g m/s2 Gravitational acceleration A m2 Section area b m Span length m kg Mass x m x coordinates y m x coordinates β − Panel distribution angle n − Number of panels ψ − x non-dimensional coordinates ζ − y non-dimensional coordinates C − Class function S − Shape function

xix xx LIST OF TABLES

N − Shape parameters AU − Upper curvature coefficients AL − Lower curvature coefficients K − Binomial coefficient

Np − Number of parameters D − Vector dimensions G − Generation

xi,G − Target vector

vi,G+1 − Mutant vector

ui,G+1 − Third vector F − Ampliﬁcation control CR − Cut off parameter Re − Reynolds Number L/D − Lift-drag ratio

βOF − Weight value of objective function λ − Constant

σmax MPa Maximum stress

Sy MPa Yield Strength ν − Poisson coefﬁcient E GPa Young’s modulus

nSF − Safety factor

fn Hz Natural frequency

Cmaj m Major chord

cmin m Minor chord P N Weight

Sty MPa Tensile Strength √ KIC MPa m Fracture toughness ε % Strain

Cm EUR Cost Acronyms

Acronym Meaning 2D Two-dimensional 3D Three-dimensional CFD Computational Fluid Dynamics CFRP Carbon Fiber Reinforced Composite CST Class-Shape-Transformation method DE Differential Evolution method DOF Degrees of freedom GFRP Polyester Glass Reinforced Composite ICCCC International C-Class catamaran championship NACA National advisory commitee for aeronautics LSE Least Squares Error method RPF HD Rigid polymer foam HD RPF MD Rigid polymer foam MD

xxi xxii LIST OF TABLES Chapter 1

Introduction

The main purpose of this study is to improve the lift velocity performance of a Portuguese C-Class Catamaran by upgrading the design of the hydrofoil structure, known as daggerboard. The catamaran championship has been the leader in this ﬁeld allowing great developments of the catamaran’s main components: wing, hulls, rudders and daggerboards.

The Portuguese catamaran’s components were made by Optimal Structural Solutions, a Portuguese company that develops projects on aeronautical and automotive ﬁelds. They are involved with the design work, composites manufacturing process selection, and testing. After the ﬁrst participation on the International C-Class Catamaran Championship (ICCCC), the team is now making new studies in order to optimize parts of the structure and to improve the catamaran’s performance.

Daggerboards are hydrofoils placed on the middle of the hulls that allow the catamaran to reduce drag and thus increase speed. Since there is less resistance in the air than underwater, the sooner the boat rises in the water, the earlier it increases its velocity, making the daggerboards the main components responsible for improving the catamaran’s performance on race.

The daggerboard has been developed along the years by changing its structural profile. The different configurations produce modifications on the catamaran’s behaviour by improving its stability and velocity. Since the creation of the C-Class championship, there have been significant advances on both the profiles and the materials. The lighter the boat, the bigger the increase in speed, leading to the appearance of new materials and profiles.

This work illustrates the developments already achieved until the present date, and contributes to this cause by making an investigation about the main daggerboard problems and by elaborating a methodology for the daggerboard’s improvement.

1 2 CHAPTER 1. INTRODUCTION

1.1 Motivation and objectives

The motivation was to ﬁnd efﬁcient solutions for the hydrofoil’s design, with the objective of increasing the catamaran’s performance on the race. Taking this into account, a new hydrofoil geometry was generated aiming to improve the lift and reduce the drag. A structural analysis was also performed in order to improve the daggerboard’s mechanical behaviour.

The two-dimensional hydrofoil improvement was done using a computational program written in Python. A three-dimensional daggerboard structure was then designed with the new hydrofoil section and a structural analysis was performed in order to create a daggerboard capable of supporting the load of the boat without permanent deformations. A study of material was also performed in this work. An alternative material was explored to be applied on the daggerboard’s structure.

1.2 Structure of the thesis

This work is divided into three core parts: two-dimensional, three-dimensional design, and study of material. There are 7 main chapters apart from the introduction:2 State-of-Art,3 Methodology,4 Daggerboard design,5 Structural analysis,6 Material analysis,7 Conclusions, and8 Future work.

Chapter 1: Introduction

This ﬁrst chapter carries out an overview of the work developed throughout this thesis as well as the main objectives that were set. It also shows how the thesis is organized.

Chapter 2: State of Art

In this chapter, literature research regarding hydrofoils, daggerboards, and material is performed. The daggerboards used in the last C-Class catamaran championship by the different teams were collected as well as the main advantages and disadvantages of these proﬁles found by each team. The history of hydrofoils and their materials are described as well as the main hydrofoils characteristics. Cav- itation is presented as one of the responsibles for the drag force increase and as the starting point for the new geometry generation.

Chapter 3: Methodology

The steps taken developing this work, and the project conditions for the new hydrofoil design, are described in this chapter. The methodology is applied for the hydrofoil’s geometry generation and for the 1.2. STRUCTURE OF THE THESIS 3 daggerboard’s structural improvement. The project conditions are described and summarized in order to collect the main goals for this work.

Chapter 4: Daggerboard design

The two-dimensional hydrofoil geometry is generated. The theory behind the design is described and the main characteristics resulting from this new shape are presented. A hydrodynamic characteristics comparison between the new and the used hydrofoil geometries is made in order to verify the performance improvement of the generated geometry.

Chapter 5: Structural analysis

A three-dimensional model is designed. Static and modal analysis are performed on the structure. First, it is made a pre-study only to the blade of the daggerboard. Then a structural analysis is performed to the daggerboard with the new geometry section and then it is compared to the daggerboard section used by the portuguese team in the last race. The daggerboard with the new section is then improved until the displacement values avoid great deformations on the structure.

Chapter 6: Material analysis

A material analysis is performed in order to ﬁnd different solutions for the structure, aiming for cost reduction. The mechanical parameters, as well as the environment conditions, are set in order to select the materials for the daggerboard’s structure.

Chapter 7: Conclusions

In this chapter, the main conclusions are summarized. The results obtained in each part of the work are analyzed and the ﬁnal structure’s characteristics are presented.

Chapter 8: Future work

Future work suggestions that are beyond the scope of this work are described in the ﬁnal chapter of this dissertation. A Computational Fluid Dynamics (CFD) analysis is the main suggestion for future work. Adding this analysis to the methodology presented in this work, it is possible to design a different daggerboard conﬁguration more alike the daggerboards that were used by other teams in the championship. 4 CHAPTER 1. INTRODUCTION Chapter 2

State of Art

In this chapter, it is described the C-Class Catamaran race and its rules as well as a literature research about the different existing types of hydrofoil most used in race. A brief description of the Eppler hydrofoils history and materials evolution is done in order to create a better understanding about this theme. The hydrofoil nomenclature and cavitation effect’s description are essential due to their importance for this work. Cavitation is one of the most problematic issues in the hydrodynamics field. It can cause serious damage in submarine machines and there is still no efficient solution to extinguish cavitation effects. This problem is used in this work to define a design limit to the hydrofoil geometry.

2.1 C-Class catamaran racing boat

The International C-Class Catamaran Championship (ICCCC) is a high speed boat race where com- petitors can show their creativity and engineering skills. Since 2004, this race contributes to the development of this catamaran class improving their performance year after year by constantly researching for new designs, new materials, and new solutions. The few existing rules challenge every participant to be creative with their designs and engineering.

As mentioned by the Canadian’s team designer, if you ask most designers what kind of project they enjoy, they will tell you it is the project with technical uncertainties that sketch the imagination, and the C- Class rules certainly leave ample room for imagination [16]. This challenge is composed by few teams of different parts of the world. This very last year, 2013, 8 teams participated on the championship, from Canada and USA to France, Switzerland, Great Britain, Spain, and Portugal (Tab. 2.1).

Team Cascais was born from a partnership between Tony Castro and Optimal Structural Solutions. The main purpose of this project was the participation on ICCCC with the ﬁrst Portuguese C-Class catamaran totally built in Portugal, illustrated in Fig. 2.1. The rules for this competition are described below [3].

5 6 CHAPTER 2. STATE OF ART

Table 2.1: ICCCC 2013 - Teams

Country Team

Canada Fred Eaton United States of America Cogito Project France Challenge France France Groupama Great Britain Team Invictus Portugal Team Cascais Spain Sentient Blue Switzerland Team Hydros

Figure 2.1: Team Cascais Catamaran [9].

1. A catamaran is deﬁned as a two-hulled sailing boat with essentially duplicate or mirror image hulls, ﬁxed in parallel positions. 2. Sail area shall not be more than 27.868 m2. 3. The overall length of the catamaran shall not be more than 7.62 m. The length shall be measured between perpendiculars to the extremities of the hulls with the catamaran in her normal trim. The measurement shall be taken parallel to the center line of the craft and shall exclude rudder hangings. 4. The extreme beam shall not be more than 4.267 m. 5. The crew shall be two people. 6. The C class emblem shall be carried on the mainsail and shall consist of the letter C over two parallel horizontal lines over national letters and sail numbers.

As can be seen, these rules only cover the measures of the sail, hulls and extreme beam. No rules for the design nor material types are available which means these are open ﬁelds to explore, aiming to improve the boat’s performance on race. 2.2. DAGGERBOARD PROFILES 7

2.2 Daggerboard proﬁles

In C-Class, the daggerboard has been a constant subject of investigation, consisting in ﬁnding and experimenting with different daggerboard proﬁles in order to get less drag at higher speeds. For better understanding, a brief demonstration of the several parts from a catamaran is provided in Fig.2.2.

Figure 2.2: Catamaran Components.

Each team that participated over the last years in the ICCCC made their decisions choosing different daggerboard proﬁles. All the teams had the same goal on daggerboard proﬁle design: generate higher lift so they could go faster.

L Foils - Team Hydros

In 2010, Switzerland’s team sailed the Patient Lady VI and it was the launchpad for Team Hydros challenge in 2013. This time, the team made improvements on foils in order to get a faster boat. They ended on the 2nd position last year with Flyer boat.

Thanks to interviews, it is possible to brieﬂy explain the daggerboard mechanism, which is all manual through a game of ropes. This team chose to use an L shape foil (Fig. 2.3). This decision is justiﬁed by the good results on stabilization of the boat at low speeds and the faster creation of lift when the speed increases [7].

This daggerboard has three degrees of freedom (DOF) through the hull illustrated in Fig. 2.4: vertical and horizontal along the hull and perpendicular to the ﬂow direction, represented by black and red row colors, respectively.

When the boat starts to move, a big angle of attack is obtained in order to guarantee enough lift to maintain the balance between the hydrodynamic and aerodynamic forces and to lift up the boat. The sailor moves the foil by changing both angles (x and y directions). By changing the foil position, the angle of attack also changes. By increasing speed, the produced lift also increases, making the boat fly 8 CHAPTER 2. STATE OF ART above the water. In order to keep this position, the angle of attack is then reduced, by the modification of both angles again. Beyond that, on this position the foil is able to take the side force more efficiently while the tip keeps doing the lift. This strategy allowed team Hydros to get the second place on the race.

Figure 2.3: Flyer - Team Hydros boat [1]

Figure 2.4: Daggerboard - degrees of freedom

S Foils - Team Groupama

Team Groupama decided to improve their foil profile to an S shape Fig. 2.5. This decision had as consequence the conditioning of the daggerboard’s DOF [6]. In contrast with team Hydros, this profile cannot change the horizontal position over the hull neither the horizontal position perpendicular to the flow direction. There is no mechanism to do this. In order to achieve this position, team Groupama has an S shape which only allows vertical movements. Beyond being the profile with better stability, it also has the advantage to get a simpler mechanism for the sailors. The main disadvantage is that this mechanism causes friction between the foil and the hull, which makes the vertical movement slower.

T foils - Team Canada

In 2007, Team Canada won ICCCC with the Alpha boat. The team made a previous study regarding the design applied to the two boats, Alpha and Rocker [28]. Despite the fact that Rocker was a stable boat, it could not match the 20 knot (10.3 m/s) plus speed of Alpha. 2.2. DAGGERBOARD PROFILES 9

Figure 2.5: Daggerboard Overview [4]

The team decided to construct a T foil for the Rocker boat. It seemed a good idea since it has good performance on motor boats where the main goal was only to reduce the drag force in order to reduce the fuel consumption. So they developed the T foil and achieved stable ﬂying. At the end, the Rocker was a stable and slow boat. They could not achieve competitive speeds around the course.

Figure 2.6: Alpha’s ﬁrst daggerboard [28]

The team’s conclusion also states that ”After this fairly careful experiment in foiling, we are com- fortable concluding that hydrofoils don’t perform in this configuration on a C-Class catamaran. There is no doubt that fine-tuning our parameters could increase Rocker’s speed, but it will take more than fine tuning to make the required leap in performance”, [28].

Alpha’s performance was different. The team developed a thinner foils section that outperformed the others - less drag with slightly greater maximum lift. The thin profile had the added benefit of less weight. Besides the straight foil profile, as shown in Fig. 2.6, this team was able to combine several factors to 10 CHAPTER 2. STATE OF ART produce success, which include lower overall weight, thinner foil section, and more time on the water to experiment enabling the creation of race tactics.

V Foils - Team Emirates

Looking to a different class, the A-Class, whose main differences for C-Class are the boat dimensions, there are some different solutions at the daggerboard’s ﬁeld with the same purpose as the C- Class. Using the Team Emirates example [10], they decided to use a curved daggerboard. When the board is lowered, the daggerboard gets a V shape instead of an L shape.

On an interview given by Team Emirates [10], the catamaran sits on the V so that when the boat speed increases, and the lift increases, the foil rises higher in the water with both legs of the V at an inclined angle instead of nearly horizontal ( Fig. 2.7). As team leader Morrelli says, ”The advantages/disadvantages are that this V is self-levelling. As it raises up to the surface at high-speed, it loses lift, because it stalls a little bit, and it settles back down where a true L will go completely out of the water, and you have to find another way of controlling the angle of attack and the amount of lift it creates”[10]. Because the foils extend to their tips at an upward angle instead of flat, as the foil reaches the surface, the portion of the foil with water flowing over it, generating lift, is reduced incrementally, and the boat comes down gradually to an equilibrium point. The idea is to balance the forces as speed changes, avoiding the all-or-nothing conditions that a more horizontal lifting foil encounters.

Figure 2.7: Forces Involved - V Proﬁle [10]

This is a foil shape example that could be used on C-Class but until now there are no records of this proﬁle being used on race. 2.3. HYDROFOIL DESIGN 11

Team Cascais

Team Cascais has less experience both in boat construction and racing performance. From last year, the team decided to make some changes in order to create a better equilibrium between the boat and team performance in race. For this, it was decided to change the way of controlling the angle of attack of the daggerboard. Instead of changing the angle by modifying the movement of the daggerboard, it is easier for the team, since there are only two persons sailing the boat, to control it only by the rudder. Besides, the rudders are in a better position of the boat, and they are lighter and easier to control. The daggerboard is maintained still while the rudders do the rest of the job.

Last year, Optimal company and Team Cascais chose a National Advisory Committee for Aeronautics (NACA) 2412 profile for the hydrofoil shape and an S profile for the structure of the daggerboard. Beyond this profile, Optimal has also developed an L daggerboard with NACA 5412 section.

In this work it was compared the NACA proﬁles used by Team Cascais to an Eppler proﬁle and then to a generated shape.

2.3 Hydrofoil design

Since the 1950’s, hydrofoils had great development both on design and performance. Engine boats were always a matter of concern mostly because of the needed power to move the boats. Increasing the velocity also increases the drag force and, therefore, the fuel consumption. In order to ﬁnd a good balance between speed and drag, new studies were performed and new hydrofoil proﬁles started to emerge.

It was concluded, as mentioned in [16], that it was possible to achieve higher speeds on the water by making the boats ﬂy over it. It would reduce the drag force and consequently the fuel consumption. This conclusion let engineers develop hydrofoils that created enough lift to get the boat up and maintain balance between hydrodynamic and aerodynamic forces while it is over the surface [14].

A hydrofoil is similar to an airfoil. The main difference lies on the fact that when choosing the profile it should be considered cavitation effects and the pressure distribution over the upper surface. There are several airfoil profiles designed for several specifications. For example, for aircraft wings it is common to use NACA profiles because of their good behavior in lift generation. For the vessels, it is prudent to choose a geometry that avoids cavitation, like an Eppler hydrofoil.

2.3.1 Eppler’s hydrofoil

Airfoil design has been developed over the past century, beginning with copies of bird wings and cut-and-try shapes, some of which were tested in low-Reynolds number wind tunnels. NACA system- 12 CHAPTER 2. STATE OF ART atized this approach by perturbing successful airfoil geometries to generate series of related airfoils. This approach was not the most indicated if the main goal was to design an airfoil with speciﬁc characteristics. At that time, the airfoil design was developed by the application of the next doctrine: the desired boundary-layer characteristics result from the pressure distribution, which results from the airfoil shape.

The inversion of an airfoil analysis method provided the means of transforming the pressure distribution into an airfoil shape [22]. The transformation of the desired boundary-layer characteristics into a pressure distribution was left to Richard Eppler, who developed a computer code that provides a much more direct connection between the boundary-layer development and the pressure distribution.

With this code, instead of intuitively or empirically transforming the desired boundary-layer characteristics into a pressure distribution, the designer can determine directly the modiﬁcations to the pressure distribution that will produce the desired boundary-layer development at any given angle of attack.

The hydrofoils presented in this work belong to a set of several airfoils and hydrofoils developed by Eppler’s method. They have been tested over the years for different Reynold’s numbers and different environment conditions.

Summarizing, Richard Eppler developed an experimentally-veriﬁed, theoretical method that allows airfoils and hydrofoils to be designed for almost all subcritical applications. Since these hydrofoils and data are available for common use, an Eppler hydrofoil was chosen to be the base for the development of this work.

2.3.2 Hydrofoil characteristics

The hydrofoil creates a lift force, perpendicular to the flow direction, and a drag force, which has the same direction of the flow (Fig. 2.8). Since the purpose of this work was to make the boat fly over the water, the main goal was to increase the lift generated by the hydrofoil.

Figure 2.8: Airfoil’ forces [24]

The angle of attack is the angle between the ﬂow direction and the hydrofoil chord. If the angle of attack increases, the lift force also increases, while the drag force decreases. The main problem of 2.3. HYDROFOIL DESIGN 13 these daggerboards is the variation of the angle of attack, with the changes of the sea currents and the occurrence of waves. It is very dangerous for the sailors and for the vessel if the waves impact directly. At the beginning, with reduced velocities, it is necessary to have big angles of attack on the foils, in order to increase speed and, consequently, lift.

Lift (L) and drag (D) forces can be calculated by Eq. (2.1), respectively. Both equations have similar constants, like the ﬂow velocity, represented by v, the ﬂuid density, ρ, and the hydrofoil chord, c. Both lift

(CL) and drag (CD) coefﬁcients are dependent of the hydrofoil shape.

L C = (2.1a) L 1 2 2 ρcv D C = (2.1b) D 1 2 2 ρcv

The mean camber line deﬁnes the hydrofoil curvature, as shown in Fig. 2.9. Due to the changing of the curvature, the lift and drag forces values change as well.

Figure 2.9: Hydrofoil nomenclature [24]

The upper surface is where the velocity reaches high speeds and the static pressure reaches low values. The static pressure in the lower surface is higher than in the upper surface. The pressure gradient between both surfaces generates the lift force.

In this work it was not considered the free surface effects in the hydrofoil pressure distribution. The incorporation of this parameter could be done by considering two limiting cases: low Froude number and high Froude number.

In the ﬁrst case (Fig. 2.10), it can be simulated by calculating the ﬂow about the body and its mirror image in the z=0 plane, creating the positive image. Considering a low Froude number, the velocity is equal to zero, making the pressure in the free surface not constant.

In the second case (Fig. 2.11), having a high Froude number, the signs of the image singularities are reversed, creating a negative image. This simulation gives zero horizontal velocity on z=0, making the velocity there equal to free stream. In this case, the pressure drops near the surface making the lift coefﬁcient increase.

For this work, the second case is the most indicated to be considered since the Froude number obtained is a high number. However, this was not considered since this would require to change the 14 CHAPTER 2. STATE OF ART code of the software used to compute the hydrofoil pressure distribution which is beyond the scope of the current work.

Figure 2.10: Positive image [19]

Figure 2.11: Negative image [19]

The distribution of pressure over a hydrofoil is usually expressed by the pressure coefﬁcient,

p − p c = ∞ , (2.2) p 1 2 2 ρv where p is the local static pressure, p∞ is the freestream static pressure, ρ is the fluid density, and v is the flow velocity. In Fig. 2.12, a NACA 0012 is represented such as the cp evolution with the increasing of the angle of attack. This representation was obtained from the Xfoil software. This is a symmetric profile and it is commonly used for the design of wings. Note that the leading edge peak becomes more extreme as the angle increases. For equilibrium we must have a pressure gradient when the flow is curved. In the case shown here, the pressure must increase as we move further from the surface. This means that the surface pressure is lower than the pressures further away. This is why the cp is more negative in regions with curvature in this direction. The curvature of the streamlines determines the pressures and the net lift.

Fig. 2.13 represents the pressure coefﬁcient distribution in both upper and lower surfaces, for an angle of attack equal to 3 degrees. This airfoil is an example just to illustrate the dependence of the parameters from each other. NACA 0012 is an airfoil commonly used for ﬂying purposes and experiments due to the availability of many data sources for comparison. 2.3. HYDROFOIL DESIGN 15

Figure 2.12: NACA 0012 - cp distribution.

◦ Figure 2.13: NACA 0012 - cp distribution - angle of attack = 3 . 16 CHAPTER 2. STATE OF ART

2.3.3 Cavitation

Hydrofoils have an important issue to avoid: cavitation. Cavitation is the formation of vapor cavities within a ﬂowing liquid due to both the excessive decrease of local pressure and the moment when the critical speed is reached or exceeded, [14]. It is common to verify the cavitation phenomenon on hydraulic systems, like turbines or pumps. On the case of the hydrofoil, it also depends of the distance between the surface, the localization of the daggerboard underwater, and the minimum pressure value.

In this study, the effect on the ﬂow past hydrofoils was a matter of concern, since it was one of the identiﬁed problems in last race. This can be characterized into three stages:

• The inception stage. • A partial cavitation stage in which the vapor forms an attached cavity shorter than the chord. • A fully cavitation or supercavitation stage where the vapor cavity is distinctly longer than the chord.

Fig. 2.14 illustrates those three stages. Using the description of cavitation by [23], If the pressure above the liquid is reduced by any means, evaporation recommences until a new balance is reached. If the pressure is sufficiently lowered, the liquid boils when bubbles of vapour are formed in the fluid and rise to the surface, producing large volumes of vapour. In hydraulic engineering the vapour pressure of a liquid is of importance, for there may be places of low local pressure, particularly, when the liquid is flowing over a solid surface. If, in one of these places, the pressure is reduced until the liquid boils, then bubbles suddenly collapse. There, very rapid collapsing motions cause high impact pressures if they occur against portions of the solid surface, and may eventually cause a local mechanical failure by fatigue of the solid surface.

Figure 2.14: Cavitation - Three Stages [15]. 2.4. MATERIAL EVOLUTION 17

The cavitation number is commonly represented by σcavit and it is deﬁned by Eq. (2.3).

p − p σ = ∞ v , (2.3) cavit 1 2 2 ρv

The p and pv are the ambient and vapor pressure, ρ represents the ﬂuid density, v is the velocity of upstream ﬂow which corresponds to the boat speed.

The static pressure has a minimum value somewhere at the surface of the foil. The corresponding reduction of the static pressure is indicated by the pressure coefﬁcient, Eq. (2.4).

p − p c = min ∞ (2.4) p,min 1 2 2 ρv

For the critical condition of pmin = pv , it is possible to conclude that the critical cavitation number is represented by Eq. (2.5).

σcavit = −cpmin = |cpmin| (2.5)

This equation represents the beginning of cavitation. It can also be concluded by Eq. (2.14) that when velocity increases, the cavitation number decreases, which reduces the range of the pressure coefﬁcient allowed to avoid cavitation.

In cambered sections, there is an optimum lift coefficient, at which the streamlines meet the section nose smoothly [26]. If a thinner profile is chosen with a non-smooth nose it would have had a higher pressure coefficient (cp) contributing to higher cavitation and, consequently, increasing the drag. In

Fig. 2.15 it is possible to see an example of cp evolution with velocity for different depths from free surface.

The only way to avoid cavitation, although it will never be totally avoided, is to generate balance between the angle of attack, the velocity of the boat and the proﬁle pressure coefﬁcient. In [32], several studies were already made in order to create a ”formula” to avoid cavitation problems but there still isn’t a reliable solution.

2.4 Material evolution

Since 1906 [5], the hydrofoil boats have been developed in order to create more velocity. The ﬁrst hydrofoil was built by Enrico Forlanini [11], an engineer and aircraft pioneer who contributed to the heli- copters and hydrofoils development. Forlanini designed a classic ladder type construction with multiple struts. The material used for this ﬁrst hydrofoil was a metal, the same used in the boat construction. With 18 CHAPTER 2. STATE OF ART

Figure 2.15: Pressure coefﬁcient vs velocity. the world war, the hydrofoil proﬁles had advances in their development in order to improve the military ships’ velocity and decrease the fuel consumption, raising the chances to gain advantage over other boats, both tactically and economically, whilst maintaining the type of material.

Only when the boat racing started to emerge, the materials of these boats started to change. Thanks to racing rules, that provide limits for the components dimensions and boat total weight, the reduction of the weight of the boat started to be matter of concern between the teams. Engineers have joined the designing of the boats, improving it with new material solutions. Thanks to these modiﬁcations, the composites started to arise among the race participants boats and a new era for the materials started, in order to create better and faster boats. Chapter 3

Methodology

The work is divided in three parts: hydrofoil design, structural design and material analysis. In the ﬁrst part it was generated an improved two-dimensional (2D) geometry. In the second part, it was designed and analyzed a three-dimensional (3D) structure. Finally, a material analysis was performed.

Since one of the project requirements is to develop a new hydrofoil geometry, it was decided to choose an Eppler’s profile with similar characteristics that benefits the non-appearance of cavitation effect. Starting from this point, it was easier to provide a new profile with the necessary characteristics for the imposed conditions. It was used a set of programs, such as Xfoil software, that provides profile data such as lift, drag and pressure coefficients, and it was included an optimization method in the main program in order to improve the parameters.

The design approach implemented on this work [19] was performed by setting the mass and speed of the boat. Since the total mass is the sum of the boat mass and two regular persons, the equilibrium of the boat is very sensitive to the position of both sailors. This speed-weight combination must provide the maximum lift-drag ratio relation with minimum cavitation effects. Besides these ﬁxed variables, there were two global design variables deﬁned for the current project:

• Depth of the structure, d - since there were already two models developed, an L and S proﬁles, the depth of these structures was used as starting point of the new structure design. • Aspect ratio, A - deﬁned as a range of values and leading to the daggerboard’s dimensions.

The ﬁrst daggerboard structure was designed considering the original L daggerboard provided by Optimal company. The structural study started with a simpler geometry and then it was developed until the structural results were acceptable. The principal dimensions that deﬁne a daggerboard’s geometry are illustrated in Fig.3.1.

The design methodology can be described as follows:

1. By selecting a depth, the cavitation number can be calculated setting the minimum pressure coef-

19 20 CHAPTER 3. METHODOLOGY

ﬁcient of the structure before the occurrence of cavitation effect Eq. (3.1).

p + ρgd − p σ = atm v = −c (3.1) cavit 1 2 p,min 2 ρv

2. With the minimum pressure coefficient defined, it was possible to generate a hydrofoil profile and optimize the geometry until the maximum lift-drag ratio was achieved.

3. The correspondent CL, produced by the optimized geometry, can be used to calculate the section area of the hydrofoil Eq. (3.2). mg A = (3.2) 1 2 2 ρv CL 4. By deﬁning the aspect ratio, the span b of the hydrofoil was calculated Eq. (3.3).

b2 A = (3.3) A

5. Using the original dimensions and a simpler geometry for the blade, a daggerboard was designed and a static and modal analysis were performed. The analysis results were compared to the daggerboards with NACA 2412 and NACA 5412 proﬁle sections. 6. A redesign of the structure was made in order to improve the displacement results maintaining a safety factor above of the required by the Optimal company.

Since a ﬂuid dynamics analysis was not included in this study, it was not possible to provide the pressure distribution over the depth panel of the daggerboard. For this reason, only the L conﬁguration without any inclinations was studied in this work.

Figure 3.1: Daggerboard dimensions nomenclature. 3.1. PROJECT CONDITIONS AND ASSUMPTIONS 21

3.1 Project conditions and assumptions

As mentioned before, Team Cascais has two different hydrofoils, L and S profiles with two different sections. The L profile is composed by a NACA 5412 section, Fig. (3.3), and the S profile by a NACA 2412, Fig. (3.2).

Figure 3.2: NACA 2412 - angle of attack = 0◦ .

Figure 3.3: NACA 5412 - angle of attack = 0◦ .

Since these two proﬁles are used more often as airfoils, they are not the most indicated for underwater use because of the tendency for cavitation effects. In spite of this fact, the lift coefﬁcient produced is enough to lift the boat over the water.

For this conﬁguration, the boat gets out of the water at a velocity of 18.5 m/s. One of the goals for 22 CHAPTER 3. METHODOLOGY the present work was decreasing this velocity to 10 m/s. The ﬁrst vessel constructed by Optimal had a mechanism that allowed the movement of the daggerboard through the hull. Now this mechanism is different, the daggerboard must be still, and always in the same position. Since rudders are lighter than daggerboards, they are the ones who change the angle of attack, but they are also maintained on the same position most of the time. This corresponds to an angle of attack equal to 3.5◦ .

This structure had to have enough thickness to avoid fatigue and undesirable permanent deformations. The safety factor of, at least, 5 is a company requirement. All these conditions were taken into account in order to achieve a hydrofoil with better performance than the ﬁrst ones. To summarize,

• A new hydrofoil geometry was required. • The lift-velocity was 10 m/s. • The lift angle of attack is equal to 3.5◦ . • There are no constrains for the hydrofoil dimensions - depth, span and chord. • The cavitation number has to be controlled in order to avoid cavitation effects. • The safety factor of the structure must be above 5. Chapter 4

Daggerboard design

In this chapter, it is speciﬁed how a two-dimensional hydrofoil shape optimization was made and which tools were used to achieve the ﬁnal shape desired. Since a hydrofoil is similar to an airfoil, there are some methods in the literature that were already used to design a new shape and that can be applied to this work. However, cavitation must be considered and avoided in order to get less drag force and improve the boat’s lift velocity. It means that a special attention was given to the pressure drop on the upper and lower surfaces of the hydrofoil: it had to be under control in the optimization process in order to guarantee the best performance in race conditions.

It was used a hydrofoil shape already known to start the process. In order to get a smooth geometry, a Class-Shape-Transformation (CST) method was applied. It deﬁnes a basic shape, with the class function, and modiﬁes the geometry with the shape function.

Differential Evolution (DE) is a population based optimization algorithm developed to optimize real parameters and value functions. It is an easy method where it is possible to control the minimum and maximum value for each parameter in order to get the best set of parameters which minimize the objective function.

In order to perform the hydrofoil shape study in a viscous environment, Xfoil software was included on the main program. Xfoil is a program for the design of airfoil shapes. The geometry’s coordinates and the Reynolds number must be speciﬁed in order to get pressure distribution in upper and lower surfaces, as have to be the lift and drag coefﬁcients for a given angle of attack, or a set of them.

The ﬁnal hydrofoil geometry is presented, as well as the main characteristics. The improvement process and results are compared to NACA 2412, the daggerboard section used by team Cascais in last race.

23 24 CHAPTER 4. DAGGERBOARD DESIGN

4.1 Hydrofoil design

The main goal of hydrofoil design proceeds from a knowledge of the boundary layer properties and the relation between geometry and pressure distribution. On this work, the main goal was to create a hydrofoil with a good lift-drag ratio, to get a maximum amount of lift while producing low drag, and maintain a constant pressure distribution over the hydrofoil surface.

The design approach for this work consisted on choosing an existing hydrofoil, that was already studied and analysed for similar projects, whose goals coincide with the present work goals. The main advantage of this approach is that there is test data available making the prediction of the hydrofoil behaviour easier in similar conditions. The approximation of the known geometry to a new one was done by using the Least Squares Differences (LSD) method between them, generating the new hydrofoil coordinates by the CST method. Then, an optimization method, differential evolution, was applied.

4.1.1 Class-Shape-Transformation Method

Before using the CST method, it was necessary to deﬁne the x coordinates of the hydrofoil. In most of the cases involving airfoils, a denser panelling is used near the leading and trailing edges, where the radius of curvature is smaller. A frequently used method for dividing the chord into panels with larger density near edges is the Full Cosine method. With this method the x coordinate was obtained from Eq. (4.1).

c x = (1 − cos β) (4.1) 2

The chord is represented by c and, for n chordwise panels needed, β is given by Eq. (4.2), where i is from 1 to n+1.

π β = (i − 1) (4.2) n

The CST method was developed for aerodynamic design optimization by [20], and it can be used to generate two and three-dimensional shapes. For this work, it was only used for the two-dimensional generation. Any geometry can be represented by this method. The class function defines which type of geometry it will produce. Since it was defined to generate an airfoil or hydrofoil, the only thing that differentiates one shape from another is a set of control coefficients that is built into the defining shape equations. These control coefficients allow the local modification of the shape of the curvature until the desired shape is achieved.

This method is based on Bezier curves with an added Class function. The non-dimensional coordinates are deﬁned in Eq. (4.3). 4.1. HYDROFOIL DESIGN 25

x ψ = (4.3a) c y ζ = (4.3b) c

The upper and lower surface deﬁning equations are represented as follows,

ζU (ψ) = CN1 (ψ) SU (ψ) + ψ ∆ζ (4.4a) N2 U

ζL(ψ) = CN1 (ψ) SL(ψ) + ψ. 4 ζ (4.4b) N2 L

Eq. (4.5) represents the class function where, for a round-nose hydrofoil, the parameters N1 and N2 must be equal to 0.5 and 1, respectively.

CN1 (ψ) = ψN1 (1 − ψ)N2 (4.5) N2

As mentioned before, CST method allows to represent a hydrofoil only by deﬁning the class function. In order to achieve the desired shape, it is necessary to deﬁne the shape function,

NU U X U S (ψ) = Ai Si (ψ) (4.6a) i=0

NL L X L S (ψ) = Ai Si (ψ) (4.6b) i=0

where NU and NL are the order of Bernstein polynomial for upper and lower surface, respectively. In this U L work NU = NL = N and they are equal to one less than the number of curvature coefﬁcients (A and A ) used. S is the component shape function and it is represented by

N i N−1 Si (ψ) = Ki ψ (1 − ψ) (4.7)

N where Ki is the binomial coefﬁcient, that is related to the order of the Bernstein polynomials used. It is deﬁned as follows N! K N = (4.8) i i!(N − i)!

In Fig. 4.1, it is represented a series of Bernstein polynomials in the form of Pascal’s triangle.

The complete equations, for upper and lower surfaces by CST method, are presented in Eq. (4.9a) 26 CHAPTER 4. DAGGERBOARD DESIGN

Figure 4.1: Bernstein polynomial decomposition [20]. and Eq. (4.9b), respectively. The last term, ψ. 4 ζ, represents the tail thickness.

0.5 1.0 N i NU −1 ζU (ψ) = ψ (1 − ψ) Ki ψ (1 − ψ) ] + ψ. 4 ζU (4.9a)

0.5 1.0 N i NL−1 ζL(ψ) = ψ (1 − ψ) Ki ψ (1 − ψ) ] + ψ. 4 ζL (4.9b)

Given that the control coefﬁcients AU and AL were the only unknown terms, it was used an approximation method to a known geometry to obtain the respective control coefﬁcients, creating a smooth shape to begin the study.

4.1.2 Differential Evolution

Since the maximization of the lift-drag ratio was one of the goals to achieve, and it was granted by finding the control coefficients of the shape, it was used a method that optimizes these control coefficients independently and in parallel, minimizing the time of these calculations.

Differential Evolution (DE) is an optimization method to minimize the function value, by the definition of a range of values for every single variable of the function. DE uses a number of parameters in vectors of dimension D to optimize a population of each generation, G, i.e. for each iteration of the minimization process, [33]. The number of optimization parameters, Np, does not change during the minimization process. The initial population is randomly chosen and it should cover the entire domain of research. This space has inferior and superior limits, which should be defined, and it corresponds to the project parameters. In the present work, it represented each control coefficient of the hydrofoil shape.

For each generation, a new population is born using three stages: mutation, crossing and selection. DE generates new vectors with parameters by adding a weighted difference between the two previous vectors to a third vector of the same population - mutation operation. The mutated vector’s parameters are then mixed with the parameters of the target vector, to yield the third vector. This mixing stage is called crossover. If the result of the objective function is reduced with this new vector, the vector remains and it is used in the next generation (iteration). If the result of the objective function is superior than the target vector, the vector is not replaced - selection operation. 4.1. HYDROFOIL DESIGN 27

Mutation

For each target vector xi,G, a mutant vector is generated according to,

vi,G+1 = xr1,G + F(xr2,G − xr3,G) with i = 1, ... , Np (4.10)

with random indexes r1, r2, ... ∈ 1, 2, ..., Np, which are chosen to be different from the running index i, so that Np must be greater or equal to four to allow for this condition. F controls the ampliﬁcation of

(xr2,G − xr3,G) and F > 0. An example is illustrated in Fig. 4.2.

Figure 4.2: Two-dimensional cost function showing its contour lines and the process for generating [33].

Crossover

The third vector is presented as,

ui,G+1 = (u1i,G+1, u2i,G+1, ..., uDi,G+1) (4.11)

The crossover operation crosses two vectors, xi,G and vi,G+1 and generates the third vector, ui,G+1. For each vector component, it generates a random number in range U[0, 1], randj . Cut off, CR, parameter is introduced and it is between zero and one. If randj < CR,

ui,G+1 = vi,G+1 (4.12)

Otherwise,

ui,G+1 = xi,G (4.13)

In order to guarantee the existence of at least one crossover, a ui,G+1 is randomly chosen to be part of vector vi,G+1. This operation is illustrated in Fig. 4.3. 28 CHAPTER 4. DAGGERBOARD DESIGN

Figure 4.3: D = 7 parameters, crossover example [33].

Selection

In order to decide whether or not it should become a member of generation G+1, the third vector ui,G+1 is compared to the target vector xi,G using the greedy criterion. If vector ui,G+1 yields a smaller cost function value than xi,G then xi,G+1 is set to ui,G+1. Otherwise, the old value is retained, xi,G.

The study performed by [33], where different optimizing methods are compared to DE, concludes that DE outperformed all the other minimization methods in terms of required number of function evaluations necessary to locate a global minimum of the test functions. DE can be used in this type of problem as it requires few robust control variables.

4.1.3 Hydrofoil Shape Generation

The CST method provides a smooth geometry by the definition of the control coefficients. Just to illustrate the shape generation, it was introduced the following coefficients.

AU = [1, 1, 1, 1] (4.14a)

AL = [1, 1, 1, 1] (4.14b)

These coefﬁcients generated a hydrofoil geometry (Fig. 4.4).

This was not the most indicated shape to begin the study, so a known hydrofoil was selected and all the process started from this point. 4.1. HYDROFOIL DESIGN 29

Figure 4.4: Hydrofoil shape - example.

4.1.4 Case of Study

A hydrofoil with desirable characteristics, such as low pressure coefﬁcient (in order to avoid cavitation) in a viscous environment and a good lift-drag ratio, was chosen. Hydrofoil Eppler 818 (E818) was a good hydrofoil to start our approximation, since it had a constant pressure value in both surfaces with a small area between them. However, Eppler 836 (E836) was also a good starting geometry as it can be seen by comparison between the already used proﬁles, NACA 2412 and NACA 5412, and E817 and E836, in Fig. 4.7, Fig. 4.5 and Fig. 4.6.

In Fig. 4.7, it is possible to visualize the pressure distribution only on the upper surface where the load of the vessel is distributed. The more constant the pressure distribution, the more constant the load distribution. It means that the daggerboard will suffer less oscillation in race, while the vessel gets out of water. In spite of this, the E818 generates more lift, and produces much more drag than E836 or even than NACA proﬁles. Even though both generate constant pressure distributions, the E836 has an inferior value for the minimum cp value, which provides less cavitation effect appearance. For this reason, the E836 proﬁle was chosen to begin the shape generation and the optimization process.

4.1.5 CST shape

With the known geometry and the CST method, it was possible to generate a CST shape and approximate it to the E836 shape. The LSE method was used in order to minimize the error between both curves. By finding the E836 control coefficients it was defined the first set of control coefficients to begin 30 CHAPTER 4. DAGGERBOARD DESIGN

Figure 4.5: CL vs alpha - NACA and Eppler proﬁles with Re = 2.5×106.

Figure 4.6: CD vs alpha - NACA and Eppler proﬁles with Re = 2.5×106.

Figure 4.7: cp vs x/c - NACA and Eppler proﬁles with Re = 2.5×106 and α = 3.5◦ .

the hydrofoil shape optimization. The used Differential Evolution code was taken from [25]. 4.1. HYDROFOIL DESIGN 31

Xfoil software performs analysis to the airfoils in viscous conditions by introducing the Reynolds number (Re). The Re was calculated with the formula represented in Eq. (A.1). The original chord has 0.23 m of length. For initial analysis it was used a chord of 0.25 m. The lift velocity was equal to 10 m/s and a density of 1025 kg/m3 was considered.

The corresponding lift-drag ratio and cp,min values are 114 and -1.115 for NACA 2412, 169 and -

1.2467 for NACA 5412, respectively. The correspondent cp distribution for both proﬁles in Fig. 4.8 and Fig. 4.9.

Figure 4.8: NACA 2412 - Re = 2.5 × 106 and Alpha = 3.5◦ .

Figure 4.9: NACA 5412 - Re = 2.5 × 106 and Alpha = 3.5◦ .

The new geometry, which from now on is called CST shape, had to achieve the lift-drag ratio and had cp,min value as its highest reference, in order to avoid cavitation. 32 CHAPTER 4. DAGGERBOARD DESIGN

4.1.6 Control coefﬁcients

First of all, it was used the LSD method to approximate both shapes. In Fig. 4.10, the E836 shape (black line) and the initial geometry (red line) are presented. The main goal of this approximation was to reduce the error between them until the initial geometry becomes equal to E836 shape and thereby achieved the correspondent control coefﬁcients.

Figure 4.10: Hydrofoil Eppler 836 (black line) and initial shape (red line).

The approximation of the CST shape and E836 was accomplished. The coefﬁcients obtained for E836 geometry are represented as follows.

AU = [0.10825, 0.15195, 0.15925, 0.14782, 0.31366] (4.15a)

AL = [0.14485, 0.14163, 0.14873, 0.15288, 0.31482] (4.15b)

In Fig. 4.11 both shapes are presented. As can be seen, there is a slight difference between both shapes, 1.36% minimum error. Since the CST shape control coefﬁcients were found, it was constructed an objective function based on the main goals to be achieved. It was added some thickness, 0.5 mm, in the trailing edge, as can be seen in Fig. (4.12) inside the red circle, in order to prepare the geometry to perform structural analysis. 4.1. HYDROFOIL DESIGN 33

Figure 4.11: Eppler 836 and CST shape approximation.

Figure 4.12: CST shape with thickness in the trailing edge.

4.1.7 Objective Function

Before deﬁning the objective function it was important to recall the main goals to be achieved.

• (Cp,min)CST ≥ (cp,min)NACA 2412;

• (L/D)CST ≥ (L/D)NACA 2412; • The pressure distribution should be the most ﬂat possible.

Since the daggerboard used in the last competition was the S proﬁle with NACA 2412, the ﬁrst set of analysis were performed using only these geometry values as reference.

The objective function is deﬁned in Eq. (4.18). It was used an initial βOF = 0.50 and then this value was increased to βOF = 0.75 and βOF = 0.90.

The first attempt to define an objective function focused on the pressure distribution of the hydrofoil. Since there was a minimum pressure value defined by the cavitation number, it was decided to design the pretended pressure distribution using only the negative pressure values. For this, it was defined a vector U which included the sum of negative pressure values, sum(cp,neg) and the pressure limit to be achieved, -0.5. The result equation is presented in Eq. (4.16).

2 U = (cp,neg − 0.5) (4.16)

Eq. (4.16) is then added to lift-drag ratio, deﬁning the objective function, Eq. (4.17).

L ObjFunc = − β U − (β − 1) λ (4.17) pre OF OF D 34 CHAPTER 4. DAGGERBOARD DESIGN

However, this combination was decreasing the minimum pressure but not the lift-drag ratio. For this reason, it was decided to use the minimum pressure of the proﬁle and maximize it, Eq. (4.18). This last decision worked, and lead to an increase in the lift-drag ratio and the minimum pressure.

L ObjFunc = − β c − (β − 1) λ (4.18) OF p,min OF D

The optimization program minimizes the objective function. In this case, Eq. (4.18) is negative in order to be maximized. The lift-drag ratio has to be multiplied by a constant λ to have the same order magnitude of cp,min. It was used λ = 0.01.

As the boat starts to increase the velocity, it tends to lift up of the water, making the depth of the hydrofoil decrease, until the weight of the vessel equalizes the lifting force provided by the daggerboards. Considering that the cavitation effect is one of the responsible parameters for the drag increase, it was established to give a biggest importance to the cp,min control by giving it a higher weight in the objective function formula along the iteration process.

Using the Eq. (2.3) described in the previous chapter, and defining a depth of 1.8 m, and a velocity equal to 10 m/s, the real cavitation number is 2.285 (Tab.4.1). This value provides a limit for the pressure coefficient, Eq. (4.19), and it allows a better control on the pressure coefficient obtained from each iteration.

Table 4.1: Scenario of study

velocity [m/s] depth [m] σcavit

10 1.8 2.285

The pressure value allowed must respect Eq. (4.19).

σcavit ≥ |cp,min|CST (4.19)

Several iterations were made by changing the weight of the objective function. In the ﬁrst iterations it was used a βOF = 0.5 and then it was increased until βOF = 0.9. The iteration’s evolution is illustrated in Fig. 4.13 and Fig. 4.14.

Eq. (4.20) clarifies the final control coefficients for the CST final shape.

AU = [0.109363, 0.271069, 0.028628, 0.543274, 0.015074] (4.20a)

AL = [0.060382, 0.018628, 0.007965, 0.000362, 0.012472] (4.20b) 4.1. HYDROFOIL DESIGN 35

Figure 4.13: L/D ratio evolution - E836. Figure 4.14: cp,min evolution - E836.

The lift-drag ratio values converged to 158 and the cp,min to -0.80157. These were acceptable values since the modulus of the pressure coefficient stayed below the cavitation number defined as 2.2846. The lift-drag ratio had an increase of 38 % and the minimum pressure coefficient had an increase of 39 % when compared to NACA 2412 lift-drag ratio and pressure coefficient values, respectively, which fulfills one of the goals of this work.

The CST ﬁnal shape and its pressure distribution are presented in Fig. 4.15 and Fig. 4.16, cor- respondingly. The CST hydrofoil has a ﬂat pressure distribution when compared to E836 (Fig. 4.17) pressure distribution, considering the same conditions.

Figure 4.15: CST ﬁnal shape.

For the scenario presented in Tab. (4.1) the ﬁnal shape has the following hydrofoil characteristics, Tab. (4.2).

Table 4.2: CST Final Geometry Characteristics

◦ Angle [ ] Velocity [m/s] CL CD L/D ratio Cpmin

3.5 10 0.642 0.004 158 -0.802 36 CHAPTER 4. DAGGERBOARD DESIGN

Figure 4.16: CST ﬁnal shape - pressure distribution.

Figure 4.17: E836 - pressure distribution for the same conditions of CST. Chapter 5

Structural Analysis

In chapter4 an improved two-dimensional hydrofoil was generated. In this chapter, this hydrofoil was used to design a new three-dimensional model which was then improved using Ansys software until the displacement distribution results were satisfactory.

First, it was done a pre-study to the blade of the daggerboard. With this study the main characteristics of the different proﬁles (NACA 2412, NACA 5412 and CST) in three-dimensional structure are presented. The natural frequency results from CST blade were compared to the results from an experimental study by Marco and MacGillivray [29].

The ﬁrst daggerboard structure has an initial rectangle blade geometry with similar dimensions to the original. Static and modal analysis were performed to CSTinitial and NACA 2412 daggerboards and the displacement results were compared.

The CSTinitial blade daggerboard was modiﬁed to a trapezoidal geometry, like the original one has been. This daggerboard is named CSTimproved. The dimensions of this improved blade were calculated following the methodology described in chapter3. After the blade’s improvement, the depth panel also suffered a thickness modiﬁcation in order to decrease the maximum displacement of the daggerboard.

This last modiﬁcation changes the daggerboard’s name to CSTﬁnal.

5.1 Pre-study

Without a ﬂuid analysis to the structure it is not possible to get the right pressure distribution over all the structure. Instead of making assumptions for the pressure distribution it was decided to perform analysis with real values and get acceptable results. For this reason, a pre-study of the blade was performed in order to start the daggerboard structural analysis.

The element chosen for the analysis was SOLID185 which is used for the modelling of solid structures

37 38 CHAPTER 5. STRUCTURAL ANALYSIS and is deﬁned by eight nodes having three DOF at each node. The material simulated was T800, a composite used by Optimal company. The main mechanical characteristics of the T800 material are presented in Tab. 5.1.

Table 5.1: Material Properties - T800 material (data sheet in appendixC)

Young Modulus Yield Strength Poisson coefﬁcient Density 3 E [GPa] Sy [MPa] ν ρ [kg/m ]

170 2650 0.3 1810

The mesh refinement was elaborated for the three profiles (NACA 2412, NACA 5412 and CST) in order to find a good compromise between the mesh length and computational time waste. The convergence for stress and displacement is visible in Fig. 5.1 and Fig. 5.2, respectively.

Figure 5.1: Mesh Reﬁnement - Stress. Figure 5.2: Mesh Reﬁnement - Displacement.

The correspondent values of each graphic are presented in appendixB.

5.1.1 Static analysis - blade

The three profiles have the same dimensions: chord of 0.25 m and length equal to 0.5 m. The pressure distribution was obtained by the Xfoil software and distributed over the blade surface. Since the pre-study was only to understand the main differences between the three different profiles, they are totally constrained on one side. A static analysis was performed and stress distribution results are illustrated in Fig. 5.3, Fig. 5.4 and Fig. 5.5. Since the CST profile is thinner than the NACA profiles, it was expected a higher displacement of the CST profile blade. The results from these analysis are summarized in Tab. 5.2. 5.1. PRE-STUDY 39

Table 5.2: Stress and Maximum Displacement - Blades

Proﬁle Reﬁnement [m] σmax [MPa] Max. displacement [m]

NACA 2412 0.006 57 0.001341 NACA 5412 0.006 87 0.001841 CST 0.004 109 0.003125

The CST profile presents an increase in displacement of 133% and 69% when compared with NACA 2412 and 5412, respectively. It also developed a higher stress distribution over the blade with an increase of 91% and 25% when compared with NACA 2412 and 5412, respectively. The maximum stress verified in the CST profile provides a safety factor of 24, Eq. (5.1), which is higher than the project requirement.

Sy 2650 nSF,CSTblade = ⇔ nSF, CST blade = = 24.3 (5.1) σmax 109

Figure 5.3: Stress distribution - NACA 2412 blade. Figure 5.4: Stress distribution - NACA 5412 blade.

Figure 5.5: Stress distribution - CST. 40 CHAPTER 5. STRUCTURAL ANALYSIS

5.1.2 Modal analysis - blade

In the modal analysis it was used the Block Lanczos [31] method to perform the analysis. Using an experimental work elaborated by Marco and MacGillivray in [29], a blade vibrational study was done in order to get a range of natural frequencies underwater and in vacuum. This experimental study aims to compare the natural frequencies obtained for both situations. Since the experimental results are significant for the present work, they are used and compared to CST blade results to conclude if the behaviour is similar between computational and experimental environments. Taking into account the results, a refinement was done and only the five modes were considered. The natural frequencies were obtained and presented in Tab. 5.3. The experimental results are also presented in Tab. 5.3. The CST frequencies are much higher than the experimental results. Only the first mode frequency is similar or at least at the same magnitude of the experimental. This fact is due to the different material used in the experimental, which is a metallic material, and CST blade. Therefore, since the CST blade has higher natural frequencies for the same modes when compared to the metallic blade, it means that CST has a better resonance resistance which prevents large structural oscillations and consequently, fatigue fractures.

Table 5.3: Natural frequencies - blade

Proﬁle f1 [Hz] f2 [Hz] f3 [Hz] f4 [Hz] f5 [Hz]

CST blade 6.3 29.9 36.8 79.2 80.8 Experimental [29] 6.4 14.1 15.1 33.0 54.9

Since the material used by the article is not speciﬁed, a comparison study using the same material was not performed.

5.2 Daggerboard’s analysis

The main goal of this study was to understand the behaviour of the L daggerboard with the CSTinitial profile and compared it with NACA 2412 and NACA 5412. Since the CST thickness is lower than the NACA’s profiles, the deformation should be higher. This fact had already been verified in the pre-study, but since the applied constrains are different, it was necessary to perform a full structure static analysis.

The profiles dimensions are based on the original ones. The depth considered in chapter4 is equal to 1.8 m, the initial span is around 0.5 m and the chord equal to 0.25 m. The material and element type were maintained, such as the mesh refinement. At this time, the constrains were applied on the top of the daggerboard, simulating the hull fitting. The pressure distribution was applied only over the blade.

The static analysis was performed for the three proﬁles. In order to have more data about the CSTinitial 5.2. DAGGERBOARD’S ANALYSIS 41 daggerboard behaviour, a modal analysis was also performed.

The ﬁnal mesh, the constrain, and the pressure applied are illustrated in Fig. 5.6,Fig. 5.7 and Fig. 5.8, respectively.

Figure 5.6: Mesh detail- CSTinitial.

Figure 5.7: Total constrain - CSTinitial. Figure 5.8: Pressure distribution - CSTinitial.

5.2.1 Static analysis

After the pressure distribution and constrains applied in the structures, the CSTinitial daggerboard deformation is illustrated in Fig. 5.9 and the stress distribution over the daggerboards are presented in

Fig. 5.10, Fig. 5.11 and Fig. 5.12. It is visible that the CSTinitial daggerboard has higher stress distribution over all the surface. The NACA 5412 daggerboard has lower stress value distribution over the depth panel due to its higher thickness which allows lower deformation in this zone. The critical stress point is common in all structures. This is a curved zone where stress tends to concentrate and it can lead to critical situations like fatigue. The daggerboard has a higher deformation in the y-direction. The CSTinitial proﬁle has a higher displacement due to its thickness. To prevent this fact, one of the possible solutions is to increase the thickness in the depth panel since it is where the structure has the higher stresses.

The CSTinitial structure has a security factor of almost 17, Eq. (5.2). To summarize the results of this analysis, the displacement, and stress values for each proﬁle are presented in Tab. 5.4 42 CHAPTER 5. STRUCTURAL ANALYSIS

Table 5.4: Maximum displacements and stress - rectangular blade

Proﬁle Displacement [m] σmax [MPa]

NACA 2412 0.055 85.6 NACA 5412 0.072 137 CSTinitial 0.118 158

Sy 2650 nSF, CSTinitial = ⇔ nSF, CSTinitial = = 16.77 (5.2) σmax 158

Figure 5.9: Deformation - CSTinitial.

5.2.2 Modal analysis

The static analysis was just a first step to perform a daggerboard behaviour study. A modal analysis allowed to obtain the natural frequencies of the structure and respective modes of vibration. Initially were chosen ten modes for the analysis but since the structure begins to deform in a non-sense way, the number of modes was reduced to five. This range of frequencies deforms the daggerboard in different directions. For a better understanding of the structure’s deformation, a node was picked in the critical displacement zone which was coincident on the three profiles. It was presented the correspondent graphics of this node displacement for the different profiles in order to verify if the deformation direction is similar on each. The different natural frequencies, fn, for each mode and profile are presented in Tab.5.5.

It is necessary to pay special attention to the graphic time lecture, since the range of the natural frequencies in NACA 2412 proﬁle was higher than in CSTinitial proﬁle, the time considered was also higher. For this reason, all the graphics were compared for a time range between 6 and 80 Hz.

Since the NACA proﬁles have a similar pressure distribution between them and the original L daggerboard has a NACA 5412 proﬁle, it is only presented the NACA 2412 graphic results. The NACA 5412 5.2. DAGGERBOARD’S ANALYSIS 43

Figure 5.10: Daggerboard stress distribution - NACA 2412.

Figure 5.11: Daggerboard stress distribution - NACA 5412. 44 CHAPTER 5. STRUCTURAL ANALYSIS

Figure 5.12: Daggerboard stress distribution - CSTinitial.

Table 5.5: Natural frequencies - rectangular blade

Proﬁle f1 [Hz] f2 [Hz] f3 [Hz] f4 [Hz] f5 [Hz]

NACA 2412 7.82 38.31 45.63 88.86 101.02 NACA 5412 8.40 39.06 49.08 88.53 107.93 CSTinitial 6.32 29.99 36.82 79.24 80.83

results can be consulted in appendixB.

Both proﬁles have different displacement movements in the same direction. While NACA 2412

(Fig.5.13) moves in the positive x-direction, CSTinitial profile moves into negative direction (Fig.5.14). It is due to the profile geometry and how it distributes the pressure along its curve. The maximum displacement over x-direction is around 0.9 m and 1.05 m for CSTinitial and NACA 2412 profiles, respectively.

Despite of both having similar displacement paths, in the vertical direction, (Fig. 5.15 and Fig. 5.16)

CSTinitial profile tends to have a smaller amplitude of movement. It contributes to prevent fatigue in the structure. Finally, in z-direction the displacement of CSTinitial profile was flatter than NACA 2412 profile. This profile is able to distribute the load uniformly over the surface (Fig. 5.18) in order to have a more linear deformation of the structure when compared to NACA 2412 (Fig. 5.17). 5.3. CST DAGGERBOARD IMPROVEMENT 45

Figure 5.13: ] Figure 5.14: ] NACA 2412 x-direction displacement vs time [s]. CSTinitial x-direction displacement vs time [s].

Figure 5.15: ] Figure 5.16: CSTinitial y-direction displacement vs NACA 2412 y-direction displacement vs time [s]. time [s].

Figure 5.17: ] Figure 5.18: ] NACA 2412 z-direction displacement vs time [s]. CSTinitial z-direction displacement vs time [s].

5.3 CST daggerboard improvement

The static and modal analysis results demonstrated the CSTinitial behaviour when the pressure and the natural frequencies were applied on the blade. In this section, the geometric dimensions were modiﬁed in order to get a daggerboard structure similar to the one constructed by Optimal company, in which the blade has a trapezoidal geometry instead of a rectangular one. 46 CHAPTER 5. STRUCTURAL ANALYSIS

First of all, structural modiﬁcations were done in order to decrease the displacement of the blade to avoid vibrations on it when the boat increases the velocity. So, in order to obtain an optimized daggerboard it was designed a new blade structure and compared it to the CSTinitial daggerboard. The methodology described in chapter3 was applied on this section.

In chapter4, it is found the final profile geometry by the definition of a cp,min limit regarding the cavitation number from the correspondent velocity. Since the lift coefficient is defined as 0.6423, and recalling the Eq. (5.3), it is possible to define a section area for the daggerboard. As mentioned in chapter 2, the mass considered for the boat is around 300 kg, where 150 kg corresponds to the vessel and two sailors with 75 kg each. The boat is supported by two daggerboards and two rudders. It was considered that the rudders supports around 400 N each, according to Optimal company calculations. From this point, it was also considered a conservative approach. When the catamaran changes direction, it tends to lift up one side of the boat, the weight is placed totally on one daggerboard. For this reason, the weight considered was equal to the total weight of the boat, around 2543 N without counting on rudders.

5.3.1 Structural designing

With the weight and the section area deﬁned, Eq. (5.3) the same aspect ratio (A) of original daggerboards is used, which is equal to 3. The Eq. (5.4) calculates the CSTimproved span.

mg P A = ⇔ A = with g = 9.81 m/s2 (5.3) 1 2 1 2 2 ρv CL 2 ρv CL

b2 A = with A = 3 (5.4) A

Since the sectional area of the original structures is a trapezium, it was deﬁned that the major chord was twice the minor chord, i.e cmin = 0.5Cmaj , which leads to Eq. (5.5).

C + c C + 0.5C 2A A = maj min b ⇔ A = maj maj b ⇔ C = (5.5) 2 2 maj 1.5b

Using the equations above, the dimensions for the ﬁnal blade using both conservative methods are presented in Tab. 5.6.

Table 5.6: CST blade improvement

Weight Section Area Span Major chord Minor chord 2 P [N] A [m ] b [m] Cmaj [m] cmin [m]

2543 0.077 0.48 0.21 0.11 5.3. CST DAGGERBOARD IMPROVEMENT 47

The ﬁnal CSTimproved blade dimensions described in Tab. 5.6 are illustrated in Fig. 5.19. With this new conﬁguration a static and a modal analysis were performed.

Figure 5.19: CSTimproved - trapezoidal blade.

5.3.2 Static analysis - improved CST daggerboard

The stress distribution and total displacement of the structure are illustrated in Fig. 5.20 and Fig. 5.21, respectively. The maximum displacement decreased to 0.115 m from 0.118m˙ which was not a signiﬁcant modiﬁcation. However, the curved zone of the daggerboard was not a critical zone anymore, which prevents fracture. The safety factor of the improved daggerboard remained at 17.

5.3.3 Modal analysis - improved CST daggerboard

By performing a modal analysis of this structure, the range of natural frequencies increases when compared to the CSTinitial daggerboard which means that this conﬁguration has a higher capability to avoid oscillations and, consequently, to prevent fractures due to fatigue. The modal analysis results of the structures are shown in Tab. 5.7.

Table 5.7: Natural frequencies - CST proﬁle improvement

Proﬁle f1 [Hz] f2 [Hz] f3 [Hz] f4 [Hz] f5 [Hz]

CSTinitial 6.32 29.99 36.82 79.24 80.83 CSTimproved 6.07 37.12 38.39 71.33 84.62

In Fig.5.22, Fig.5.23 and Fig.5.24 the movements of the same point on both structures are compared,

CSTinitial, and CSTimproved daggerboard. It is notorious that there are almost no modiﬁcations in the amplitude of movements. In fact, in the three directions the amplitude of movements higher. 48 CHAPTER 5. STRUCTURAL ANALYSIS

Figure 5.20: Stress distribution - CSTimproved.

Figure 5.21: Total displacement - CSTimproved. 5.3. CST DAGGERBOARD IMPROVEMENT 49

Figure 5.22: - CSTinitial and CSTimproved]graphic x direction vs time [s] - CSTinitial and CSTimproved.

Figure 5.23: - CSTinitial and CSTimproved]graphic y-direction vs time [s] - CSTinitial and CSTimproved.

Figure 5.24: - CSTinitial and CSTimproved]graphic z-direction vs time [s] - CSTinitial and CSTimproved.

Since the displacement values were not satisfied, it was decided to improve the depth panel by increasing the thickness, this new configuration is named as CSTfinal. The maximum displacement decreased drastically from 11 cm to 3 cm, Fig.5.25. The maximum stress verified was around 116 MPa, Fig.5.26, which leads to the increase of the safety factor to 23.

The natural frequencies range also increased, meaning a structure more resistant to vibrations. The 50 CHAPTER 5. STRUCTURAL ANALYSIS

Figure 5.25: Total displacement - CSTﬁnal.

Figure 5.26: Stress distribution - CSTﬁnal. 5.3. CST DAGGERBOARD IMPROVEMENT 51

movements of the node were once again compared to CSTinitial. The amplitude of movements, considering the same time of analysis, decreased, which was the main goal of this structural improvement.

The ﬁnal results of the CSTﬁnal daggerboard can be consulted in Tab.5.8. The modal results for the

NACA’s proﬁles, CSTinitial and CSTﬁnal are summarized and presented in Tab. 5.9.

Table 5.8: Final daggerboard results

Proﬁle Displacement [m] Stress [MPa] Safety factor

CSTinitial 0.118 158 17 CSTﬁnal 0.030 116 23

Table 5.9: Natural frequencies - NACA and CST proﬁle evolution

Proﬁle f1 [Hz] f2 [Hz] f3 [Hz] f4 [Hz] f5 [Hz]

NACA 2412 7.82 38.31 45.63 88.86 101.02 NACA 5412 8.40 39.06 49.08 88.53 107.93 CSTinitial 6.32 29.99 36.82 79.24 80.83 CSTﬁnal 15.58 57.37 73.38 104.8 168.1

Figure 5.27: Graphic x-direction vs time [s] - CSTinitial and CSTﬁnal. 52 CHAPTER 5. STRUCTURAL ANALYSIS

Figure 5.28: Graphic y-direction vs time [s] - CSTinitial and CSTﬁnal.

Figure 5.29: Graphic z-direction vs time [s] - CSTinitial and CSTﬁnal. Chapter 6

Material analysis

Engineering designs must have a part where a material study is performed in order to choose the material which better ﬁts the loading conditions. A material selection is a necessary process that must be done carefully, in order to enable the structure to be able to support the project’s requirements.

In this work, the material requirements were deﬁned by the environment and loading conditions applied on the structure. The daggerboard is underwater most of the time, appearing in the surface only when the boat is changing direction. It has to support the sum of weight of the vessel and sailors, and maintain the boat in a stable position, contradicting the moment force generated by the movements of the boat. It must avoid deformations and fractures due to fatigue.

The mechanical properties required for the design were:

• High Young’s modulus, E.

• High yield and tensile strength, Sy and SUTS, respectively.

• High fracture toughness, KIC. • High strain, ε. • Low density, ρ.

• Low relative cost, Cm. • High resistance to salt and fresh water.

CES Edupack is a software that creates maps of Ashby using the mechanical or material properties deﬁned. It provides a material and processes database which can be achieved by the input of the desired material properties[17].

• E1/3/ρ, for the stiffness-limited design at minimum mass. 1/2 • Sy /ρ, maximizing the strength limit and minimizing mass.

• ε/Cm, with a limit for elongation of 10% and a price between 10 and 100 EUR/kg.

53 54 CHAPTER 6. MATERIAL ANALYSIS

To illustrate the input constrains, the maps of Ashby are presented in Fig. 6.1, Fig. 6.2 and Fig. 6.3. The ceramic and electrical components categories were removed from the study.

The actual material considered for this study is a composite usually used in sport fields. Composites are made by embedding fibers in a continuous matrix of a polymer, a metal, or a ceramic. The development of high-performance composites is one of the great material developments of the last decades [18]. The daggerboard is constructed layer by layer of T800 material fiber.

By crossing the information of each property requirement, the selected materials are presented in Tab. 6.1. The data sheets of the materials can be consulted in appendixC.

The Carbon Fiber Reinforced composite (CFRP) offers more stiffness and strength and is also lighter than Polyester-Glass Reinforced composite (GFRP), but the cost is higher. Titanium alloy has mechanical properties similar to CFRP. Although it is a heavy material, it can be introduced in a composite matrix of CFRP creating a ﬁnal material ideal for the daggerboard. The Rigid Polymer Foam HD (RPF HD) and Rigid Polymer Foam MD (RPF MD) are materials with a yield strength very low which develops a safety factor below the imposed limit of 5. For these reasons, only the CFRP and titanium alloy were considered.

Table 6.1: Selected materials and properties

√ 3 Material E [GPa] Sy [MPa] SUTS [MPa] ε [%] KIC [MPa m] ρ [kg/m ] Cost [EUR/kg]

T800 170 2650 2650 1.6 - 1810 66.3 CFRP 150 1050 1050 0.35 20 1600 33.1 GFRP 28 192 241 0.95 23 1900 27.4 Titanium alloys 120 1200 1450 10 70 4800 21.9 RPF HD 0.5 12 12.4 10 0.09 470 19.8 RPF MD 0.2 3.5 5.1 5 0.05 165 19.8

In order to check the daggerboard’s behaviour with these materials, a static analysis was performed for both, the CFRP and titanium alloy, Tab.6.2.

Table 6.2: T800, CFRP and Titanium Alloy results

Material Displacement [m] Stress [MPa] Safety Factor Total Mass [kg] Cost [EUR]

T800 0.0300 116 23 21.19 1405 CFRP 0.0346 116 9 17.55 523 Titanium alloys 0.0431 115 11 51.50 1025

The composite T800 had a better mechanical behaviour, allowing a maximum displacement of 30 mm. However, the composite CFRP and titanium alloy had a displacement of 34.6 mm and 43.1 mm, respectively, which were not a signiﬁcant increase. The main difference between these materials were the total daggerboard weight and the material cost. For a better performance on race, a lightweight 55 material like T800 material should be chosen. If the cost was the project priority, CFRP should be the selected material instead. The titanium alloy remains as a suggestion for the composite matrix [30].

56 CHAPTER 6. MATERIAL ANALYSIS

10000

Titanium alloys

1000

CFRP, epoxyCFRP, (isotropic) matrix

Density (kg/m^3)

GFRP, GFRP, epoxy (isotropic) matrix

Rigid Polymer Foam (HD) Rigid

100

Rigid Polymer Foam (MD) Rigid

Rigid Polymer Foam (LD) Rigid

0.1

100

0.01

1000 0.001 Young's modulus (GPa) modulus Young's

Figure 6.1: Ashby map - Young’s modulus vs density.

10000

Titanium alloys

1000

CFRP, epoxyCFRP, (isotropic) matrix

Density (kg/m^3)

GFRP, GFRP, epoxy (isotropic) matrix

Rigid Polymer Foam (HD) Rigid

100

Rigid Polymer Foam (MD) Rigid

Rigid Polymer Foam (LD) Rigid

0.1

100

0.01 1000 Yield strength (elastic limit) (MPa) limit) (elastic strength Yield

Figure 6.2: Ashby map - Yield strength vs density.

58 CHAPTER 6. MATERIAL ANALYSIS

10000

1000

Rigid Polymer Foam (HD) Rigid

Titanium alloys

GFRP, GFRP, epoxy (isotropic) matrix

100

CFRP, epoxyCFRP, (isotropic) matrix

Price (EUR/kg)

Rigid Polymer Foam (MD) Rigid

0.1

100

1000 Elongation (% strain) (% Elongation

Figure 6.3: Ashby map - Elongation vs Price. Chapter 7

Conclusions

The main goal for this project consisted in creating a new hydrofoil shape that could increase the lift velocity of a C-Class Catamaran. The new CST hydrofoil shape is able to lift the catamaran at a velocity of 10 m/s without cavitation effects. The lift-drag ratio increased in 39% and the minimum pressure decreased in 72%, when compared to NACA 2412 values for the same conditions which fulfills the project conditions defined. For a depth defined as 1.8 m the cavitation appears at a pressure coefficient equal to 2.285. Since the CST hydrofoil has a minimum pressure coefficient of 0.8016 it means that the boat is able to increase the velocity up to 16 m/s without crossing the cavitation limit. The final CST hydrofoil geometry characteristics is presented in Tab. 4.2. Despite of the fact that the main goals were achieved, the pressure distribution at the trailing edge generates a negative lift which must be improved.

A pre-study of the blade was performed for the three sections: NACA 2412, NACA 5412 and CST. Structural analysis demonstrated a higher displacement for the CST blade when compared to the other two proﬁles(Tab. 5.2). In order to verify the modulation results given by Ansys, the CST blade natural frequencies range was compared to an experimental work developed by Marco and MacGillivray [29]. The results were of the same order of magnitude but the CST proﬁle demonstrated a better resonance resistance (Tab. 5.3).

The three-dimensional daggerboard was designed with the CST section. The improving of the daggerboard structure was developed until the maximum displacement verified became lower than the daggerboard’s configurations with NACA’s sections. The final CST daggerboard has an L configuration, a trapezoidal blade and a depth panel with variation of thickness. The safety factor of the final daggerboard configuration increased from 17 to 23, which fulfills the safety factor requirement (Tab. 5.8). Beyond the displacement improvement, the natural frequency range is also higher than the NACA’s daggerboard, meaning a better fatigue damage tolerance due to a better resonance resistance.

After the three dimensional proﬁle was developed, the two dimensional hydrofoil was again analyzed in order to conclude if the changes on the chord due to the geometry modiﬁcation changed the hydrofoil’s characteristics. It was concluded that the lift-drag ratio had a decrease of 4.5% and the minimum

59 60 CHAPTER 7. CONCLUSIONS pressure coefﬁcient remains at -0.8016 that keeps this proﬁle able for the purpose.

The study of material was elaborated by defining the main mechanical properties to be maximized, minimizing the density and cost of the material. With these constrains, two materials were selected: CFRP and titanium alloy. By performing a structural analysis to the final daggerboard with these new materials, it was concluded that the maximum displacement verified had no significant change. The main modification over the structure involved the total weight and the total cost of daggerboard for each material (Tab. 6.2). The main conclusion is that the T800 material should be selected if the main goal was to increase the total weight. On the other hand, if the cost was the main concern, then the composite matrix of CFRP and titanium alloy should be considered for the daggerboard construction. Chapter 8

Future work

Despite of all the work presented in this study, there is still room for improvements and research in the hydrofoil study ﬁeld. In this chapter, some future work is suggested aiming for the continuous optimization of daggerboards.

A sensibility study should be done to the proﬁle in order to verify the new hydrofoil geometry’s behaviour when the angle of attack is changed. By adding an angle of attack parameter to the objective function, it is guaranteed that the hydrofoil is prepared for the changes in the angle without losing its main characteristics.

Future studies and researches can be made considering the free surface effects. While doing this, the constant pressure line, simulating the atmosphere pressure between the underwater daggerboard and the hull, can be considered. This case can be simulated using computational fluid dynamics (CFD) study in order to understand the real forces generated by the flow over the structure. For this, a refined structural mesh must be carefully done as well as the choice of the turbulence modal. From the research done for the present work, the K- turbulence model is the most common use for hydrofoil’s study [27]. This model uses two equations, which relate the energy of turbulence (K) with the turbulent dissipation ( ). The method used for the hydrofoil generation creates a smooth geometry which allows us to modify the hydrofoil shape locally. In order to achieve a better design in future research the thickness parameter improvement should be included in the objective function.

The skin friction is also a matter of concern in the hydrofoil study and it was not included in this work because it is out of its scope. The drag force has a skin friction component. Future work will include skin friction in the drag study of the structure.

The geometry for the daggerboard structure adopted has the simplest configuration, an L structure without any inclination. With the CFD study performed over the structure, the main forces can be calculated for any daggerboard configuration. The next step for this study is to design an S profile, perform the CFD study and follow the methodology described in this work for the static and modal studies.

61 62 CHAPTER 8. FUTURE WORK

In addition to this daggerboard work, a hydrodynamic and aerodynamic study of the catamaran could be performed, including the daggerboard’s material modification to understand if this change influences, for the best or for the worst, the boat’s performance. In this work, it was concluded that the main advantage by using the two new materials, composite CFRP and Titanium alloy, is delimited by the final price of the daggerboard. T800 material allows the construction of a lighter daggerboard with less deformation. However, if the new mechanical system is applied to the new catamaran, where the daggerboard position remains still, the sailors won’t have to move it during the race, making no difference if the daggerboard is heavy. Due to this modification, the total weight of the boat increases. Since there is not a weight limit in the rules for the boat, a study of all of the boat will be done in order to quantify how does the weight difference influences the catamaran’s performance.

Finally, the daggerboard and rudder interaction must also be analyzed to understand their influence on each other. This study is important to comprehend how much the rudder configuration influences the daggerboard’s performance and vice-versa. Bibliography

[1] Bob odge. http://www.catsailingnews.com. Accessed 28 February 2014.

[2] Cavitation and bubbles ﬂows. http://cav.saﬂ.umn.edu/. Accessed 4 March 2014.

[3] Championship rules. http://www.restronguetsc.org. Accessed 24 February 2014.

[4] Groupama c-class launch. http://www.catsailingnews.com/2013/08/groupama-c-class- launched.html. Accessed 28 February 2014.

[5] Hydrofoil history. http://www.hydrofoil.org/history.html. Accessed 24 February 2014.

[6] International c class catamaran championship -groupama walkthrough with designer martin ﬁscher. http://vimeo.com/76473008. Accessed 28 February 2014.

[7] International c class catamaran championship -hydros walkthrough with mischa and bastiaan. http://vimeo.com/76441159. Accessed 28 February 2014.

[8] Material properties. http://www.tpub.com/doematerialsci/materialscience22.htm. Accessed 15 September 2014.

[9] A new challenger coming from portugal. http://theﬂyingboats.com. Accessed 24 February 2014.

[10] Shaping the foils that re-shaped the america’s cup: Part 2 - etnz designer gino morrelli (and pete melvin). http://www.cupinfo.com/en/americas-cup-gino-morrelli-foils-multihulls-13144.php. Accessed 28 February 2014.

[11] Storia di milano - enrico forlanini. http://www.storiadimilano.it/Personaggi/Milanesi Accessed 24 February 2014.

[12] Swiss c-class catamarans prepare for little america’s cup. http://www.sailweb.co.uk/multihull/2237/swiss-c-class-catamarans-prepare-for-little-americas-cup. Accessed 18 March 2014.

[13] (1998). New advances in sailing hydrofoils. Saint-Cloud Cedex.

[14] (2010). L’hydroptere: How multidisciplinary scientiﬁc research may help break the sailing speed record. Royal Institution of Naval Architects.

63 64 BIBLIOGRAPHY

[15] (2013). Cavitation and its effects - a case study. MIT .

[16] Acosta, A. (1973). Hydrofoils and hydrofoil craft. California Institute of Technology.

[17] Ashby, M. (2010). Materials selection in mechanical design.

[18] Ashby, M., H. Shercliff, and D. Cebon (2007). Materials, engineering, science, processing and design. Elsevier Ltd.

[19] Besnard, J., A. Schimtz, G. Tzong, K. Laups, H. Hefazi, J. Hess, H. Chen, and T. Cebeci (1998). Hydrofoil design and optimization for fast ships. CSULB Foundation.

[20] Brenda, M. and Kulfan (2008). Universal parametric geometry representation method. Journal of Aircraft, Vol.45, No.1.

[21] Drela, M. (2001). MIT Aero and Astro Harold Youngren, Aerocraft, Inc. User Primer.

[22] Eppler, R. (1990). Airfoil design and data.

[23] Francis, J. (1971). A textbook of ﬂuid mechanics. Edward Arnold.

[24] Henriques, J. (2013). Fluid Mechanics slides. Instituto Supeior Tecnico.

[25] Henriques, J. C. C., M. F. P. Lopes, R. P. F. Gomes, L. M. C. Gato, and A. F. O. F. ao (2012). On the annual wave energy absorption by two-body heaving wecs with latching control. Renewable Energy.

[26] Hoerner, S. F. (1965). Fluid-Dynamic Drag.

[27] Ingvarsdottir, H., C. Ollivier-Gooch, and S. Green. Cfd modeling of the ﬂow around a ducted tip hydrofoil.

[28] Killingi, S. (2009). Alpha and rocker - two design approaches that led to the successful challenge for 2007 international c-class catamaran championship. Institute of Physics Publishing, Measurement Science and Technology.

[29] Marco, J. and I. MacGillivray (2007). Modal analysis of large structures sudmerged in a ﬂuid. International Congress on sound and vibration.

[30] Mileiki, S., A. Rudnev, and M. Gelachov (1995). Carbon-ﬁbre/titanium silicide interphase/titanium- matrix composites: fabrication, structure and mechanical properties. Composites Science and Tech- nology.

[31] Ren-Cang, L. and Z. Lei-Hong (2013). Convergence of block lanczos method for eigenvalue clus- ters. Mathematics preprint series.

[32] Sedlar, M., P. Zima, and M. Muller (2009). Cfd analysis of cavitation erosion potential in hydraulic machinery.

[33] Storn, R. and K. Price (1997). Differencial evolution - a simple and efﬁcient heuristic for global optimization over continuous spaces. Journal of Global Optimization, vol.11, 341 - 359. Appendix A

Xfoil software

Xfoil is used as ﬂow solver in the present work. It was developed in MIT by Mark Drela in 1980s. It is a free software commonly used for aircraft wing section design. This software is based on high order panel methods with the fully-coupled viscous/inviscid interaction method used in the code developed by Drela. In this work, a viscous analysis is performed with a Reynolds number equal to 2.5 × 106 and a ﬁxed angle of attack of 3.5◦ . The boundary layers (BL) and wake are described with a two-equation lagged dissipation integral BL formulation and an envelope en transition criterion.

According to [21], the entire viscous solution (boundary layers and wake) is strongly interacted with the incompressible potential ﬂow via the surface transpiration model (the alternative displacement body model is used in ISES code). This permits calculation of limited separation regions.

The drag is determined from the wake momentum thickness far downstream. The total velocity at each point on the airfoil surface and wake, with contributions from the freestream, the airfoil surface vorticity, and the equivalent viscous source distribution, is obtained from the panel solution with the Karman-Tsien correction added.

If lift is specified, then the wake trajectory for a viscous calculation is taken from an inviscid solution at the specified lift. In the other hand, if alpha is specified, then the wake trajectory is taken from an inviscid solution at that alpha.

First of all, the ﬁle with the geometry coordinates is load at the software Fig. (A.1). For a speciﬁc

Reynolds number(Eq. (A.1)) and alpha, Fig. (A.2), Xfoil provides the pressure distribution,cp, the lift and drag coefficients. It is able to give a minimum pressure coefficient, too Fig. (A.3). Summarizing, Xfoil finds the flow around the hydrofoil for the given angle of attack and a window pops up showing the pressure distribution, the section lift coefficient, the section moment coefficient and the angle of attack. The drag coefficient and the lift-drag ratio are also presented. Both viscous and inviscid flow distribution are shown on the pop up window. The dashed lines represent the inviscid flow distribution, Fig. (A.4). This provides an easy way to compare viscous and inviscid flow. In addition, numerous boundary layer

1 2 APPENDIX A. XFOIL SOFTWARE parameters are calculated. Transition was modeled by the en method with n=9.

ρcv Re = (A.1) µ

Figure A.1: Load of the geometry coordinates.

Figure A.2: Input parameters.

All the results of each iteration are saved in an output ﬁle. The polar, with the angle of attack, lift and drag coefﬁcients and additional information is presented in Fig. (A.5).

In case of no convergence, no value is written in the output ﬁle. In most of the cases, convergence can not be achieved due to boundary layer separation and stall regions. Moreover, speciﬁc points can lead to numerical error (no convergence). In this case, preliminary computations have to be performed to force the convergence (if it is possible). Indeed when performing viscous analysis calculations, it is 3

Figure A.3: Minimum pressure coefﬁcient. always a good idea to sequence runs so that alpha (the angle of attack) does not change too drastically from one case to another.

For this work, it is speciﬁed the hydrofoil coordinates, introduced in Xfoil software that performed a viscous analysis with a speciﬁc alpha. By the interaction between Xfoil and the optimized program, several iterations are performed until the lift-drag ratio, given by the xfoil, reaches the maximum value allowed by the geometry. It is possible to get information over the pressure distribution by graphic interface and by coordinates. This information is implemented in Ansys software to create a pressure distribution over the daggerboard in order to study the load effects over the structure. As can be concluded, this software is very useful to perform this type of studies, by allowing the calculation of the main hydrofoil characteristics in a fast and trustfully way. 4 APPENDIX A. XFOIL SOFTWARE

Figure A.4: Pressure distribution.

Figure A.5: Output ﬁle example. Appendix B

Structural analysis - complementary

In this chapter, the information that is not presented in the work is illustrated and better explained in this appendix. First, the mesh reﬁnement information that is presented by graphics in section 5.1, is presented in tables in this section. The displacement distribution and the displacement in the three directions over the NACA 5412 daggerboard is also illustrated. The relations between the chords and the aspect ratio used in the methodology adopted in subsection 5.3.1 are also explained.

B.1 Blade

In section 5.1 of chapter5, a mesh refinement is made in order to have the best mesh and perform the analysis with minimum computer time waste. In Tab.B.1, Tab.B.2 and Tab.B.3 are presented the mesh refinement and respective stress and displacement values in each iteration for the three profiles.

The displacement distributions over the blades are illustrated in Fig.B.1 and Fig.B.2. The maximum displacement values are collected and placed in tables in section 5.1.

Table B.1: Mesh Reﬁnement - NACA 2412

Mesh Reﬁnement [m] σ [MPa] Displacement [m]

1 0.05 31 0.0005 2 0.025 42 0.0010 3 0.0125 53 0.0012 4 0.006 57 0.0013 5 0.005 57 0.0013 6 0.004 57 0.0013

5 6 APPENDIX B. STRUCTURAL ANALYSIS - COMPLEMENTARY

Table B.2: Mesh Reﬁnement - NACA 5412

Mesh Reﬁnement [m] σ [MPa] Displacement [m]

1 0.05 46.5 0.0008 2 0.025 63 0.0013 3 0.0125 79.1 0.0017 4 0.006 85.1 0.0018 5 0.005 85.6 0.0018 6 0.004 85.8 0.0018

Table B.3: Mesh Reﬁnement - CST

Mesh Reﬁnement [m] σ [MPa] Displacement [m]

1 0.05 50 0.0010 2 0.025 74 0.0020 3 0.0125 94 0.0028 4 0.006 109 0.0030 5 0.005 109 0.0031 6 0.004 109 0.0031

Figure B.1: Displacement distribution - NACA 2412.Figure B.2: Displacement distribution - NACA 5412.

B.2 Daggerboard

For the first daggerboard configuration, it is performed a static and a modal analysis. The displacement distributions over the three profiles are illustrated in Fig.B.3, Fig.B.4 and Fig.B.5. Since the original daggerboard uses a NACA 2412 profile, only the modal results from this daggerboard are presented and compared to CST daggerboard results. However, since the NACA 5412 has been analyzed in this work, its results are presented in Fig.B.6, Fig.B.7 and Fig.B.8. B.2. DAGGERBOARD 7

Figure B.3: Daggerboard displacement distribution - NACA 2412.

Figure B.4: Daggerboard displacement distribution - NACA 5412. 8 APPENDIX B. STRUCTURAL ANALYSIS - COMPLEMENTARY

Figure B.5: Daggerboard displacement distribution - CSTinic.

Figure B.6: NACA 5412 x-direction displacement vsFigure B.7: NACA 5412 y-direction displacement vs frequency [Hz]. frequency [Hz].

Figure B.8: NACA 5412 z-direction displacement vs frequency [Hz]. B.3. ORIGINAL L DAGGERBOARD 9

B.3 Original L daggerboard

The aspect ratio and major and minor chord’s relation presented in subsection 5.3.1, are obtained from the original structure illustrated in Fig.B.9. The major chord from the original daggerboard is equal to 0.225 m and the minor chord to 0.112 m. The relation between both is presented in Eq. (B.1) and it is used to calculate the new chord length for the new CST daggerboard.

c 0.112 r = original ⇔ r = ⇔ r = 0.5 (B.1) Coriginal 0.225

The section area is obtained using the original chord lengths and the original span. The aspect ratio is then calculated in Eq. (B.2). This value is also used for the CST blade geometry calculation.

b2 0.52 [h!]A = original ⇔ A = ⇔ A = 3 (B.2) original A original 0.225+0.112 original trapezium 2 × 0.5

Figure B.9: Original L daggerboard dimensions. 10 APPENDIX B. STRUCTURAL ANALYSIS - COMPLEMENTARY Appendix C

Material Complement

In this appendix, the displacement and stress distributions of the materials selected in chapter6 are illustrated. The information about materials selected are also presented as data sheets provided by software CES Edupack.

C.1 Daggerboard with different materials

In this section, the displacement and stress distributions of the daggerboard with the two selected materials, CFRP and titanium alloy, are illustrated in Fig. C.1, Fig. C.2, Fig. C.3, Fig. C.4.

Figure C.1: Daggerboard displacement- CFRP

11 12 APPENDIX C. MATERIAL COMPLEMENT

Figure C.2: Daggerboard stress distribution- CFRP

Figure C.3: Daggerboard displacement- Titanium alloy C.2. MATERIAL DATA SHEETS 13

Figure C.4: Daggerboard stress distribution- Titanium alloy

C.2 Material data sheets 14 APPENDIX C. MATERIAL COMPLEMENT

TECHNICAL ¨ DATA SHEET No. CFA-007 T800H DATA SHEET Intermediate modulus, high tensile strength fiber, with excellent balanced composite properties. Designed and developed to meet the weight saving demand of aircraft. Has been used in primary structure of aircraft, including vertical fin and horizontal stabilizer.

FIBER PROPERTIES

English Metric Test Method Tensile Strength 796 ksi 5,490 MPa TY-030B-01 Tensile Modulus 42.7 Msi 294 GPa TY-030B-01 Strain 1.9 % 1.9 % TY-030B-01 Density 0.065 lbs/in3 1.81 g/cm3 TY-030B-02 Filament Diameter 2.0E-04 in. 5 µm

Yield 6K 6,679 ft/lbs 223 g/1000m TY-030B-03 12K 3,347 ft/lbs 445 g/1000m TY-030B-03

Sizing Type 40A, 40B 1.0 % TY-030B-05 & Amount 50B 1.0 % TY-030B-05

Twist Twisted, Untwisted

FUNCTIONAL PROPERTIES

CTE -0.56 α⋅10-6/˚C Specific Heat 0.18 Cal/g⋅˚C Thermal Conductivity 0.0839 Cal/cm⋅s⋅˚C Electric Resistivity 1.4 x 10-3 Ω⋅cm Chemical Composition: Carbon 96 % Na + K <50 ppm

COMPOSITE PROPERTIES*

Tensile Strength 380 ksi 2,650 MPa ASTM D-3039 Tensile Modulus 25.0 Msi 170 GPa ASTM D-3039 Tensile Strain 1.5 % 1.5 % ASTM D-3039

Compressive Strength 230 ksi 1,570 MPa ASTM D-695 Flexural Strength 235 ksi 1,620 MPa ASTM D-790 Flexural Modulus 22.0 Msi 150 GPa ASTM D-790

ILSS 14.0 ksi 10 kgf/mm2 ASTM D-2344 90˚ Tensile Strength 9.0 ksi 63 MPa ASTM D-3039

* Toray 250˚F Epoxy Resin. Normalized to 60% fiber volume.

TORAY CARBON FIBERS AMERICA, INC.

Figure C.5: Composite T800 data sheet C.2. MATERIAL DATA SHEETS 15

Composite T800/3900-2 Unidirectional Prepreg (Fv=60%) segunda-feira, 1 de Dezembro Toray Composites (America), Inc. - Carbon/Epoxy de 2014

Please Note: This symbol denotes data that is available when you purchase this datasheet , or subscribe to Prospector:Composites.

General Information Product Description T800/3900-2 is a carbon/epoxy unidirectional prepreg with high-strain fibers and a high-toughness matrix. General Generic Name • Carbon/Epoxy Fiber • T800 Matrix • 3900-2 Fiber Supplier • Toray Composites (America), Inc. Matrix Supplier • Toray Composites (America), Inc. Fiber Volume Fraction • 60 % Form(s) • Unidirectional Prepreg Material Status • Commercial: Active • Asia Pacific • Latin America Availability • Europe • North America • 0,00586 in ( Cured Thickness • 0,149 mm ) Data Source • Journal Article 1 Data Rating • **

Technical Properties 2 Mechanical Nominal Value (English) Nominal Value (SI) Compressive Modulus E11 - Longitudinal 22,6 msi 156 GPa E22 - Transverse 1,29 msi 8,89 GPa Shear Modulus (G12 - In-Plane) 0,745 msi 5,14 GPa Poisson's Ratio (ν12 - In-Plane) 0,300 0,300 Additional Information Mechanical data was normalized to a fiber volume of 60%. This number was found on the product data sheet titled "Torayca T800H," Toray Composites (America) Inc., Santa Ana, CA.

Notes

1 2 Properties are not to be construed as design specifications.

Copyright © 2014 IDES Inc. (www.ides.com), and Firehole Technologies Inc. (www.fireholetech.com) Information for this material was last updated: 12-12-2013 The information presented on this datasheet was acquired by Firehole Technologies Inc., www.fireholetech.com, from various sources. IDES and Firehole make substantial efforts to assure the accuracy of this data. However, IDES and Firehole assume no responsibility for the data values and strongly encourages that upon final material selection, data points are validated. Page: 1 of 1

Figure C.6: Composite T800 data sheet 16 APPENDIX C. MATERIAL COMPLEMENT

CFRP, epoxy matrix (isotropic) Description The material Carbon fiber reinforced composites (CFRPs) offer greater stiffness and strength than any other type, but they are considerably more expensive than GFRP (see record). Continuous fibers in a polyester or epoxy matrix give the highest performance. The fibers carry the mechanical loads, while the matrix material transmits loads to the fibers and provides ductility and toughness as well as protecting the fibers from damage caused by handling or the environment. It is the matrix material that limits the service temperature and processing conditions. Composition (summary) Epoxy + continuous HS carbon fiber reinforcement (0, +-45, 90), quasi-isotropic layup. General properties Density 1.5e3 - 1.6e3 kg/m^3 Price * 29.8 - 33.1 EUR/kg Date first used 1963 Mechanical properties Young's modulus 69 - 150 GPa Shear modulus 28 - 60 GPa Bulk modulus 43 - 80 GPa Poisson's ratio * 0.305 - 0.307 Yield strength (elastic limit) 550 - 1.05e3 MPa Tensile strength 550 - 1.05e3 MPa Compressive strength 440 - 840 MPa Elongation * 0.32 - 0.35 % strain Hardness - Vickers * 10.8 - 21.5 HV Fatigue strength at 10^7 cycles * 150 - 300 MPa Fracture toughness * 6.12 - 20 MPa.m^0.5 Mechanical loss coefficient (tan delta) * 0.0014 - 0.0033 Thermal properties Glass temperature 99.9 - 180 °C Maximum service temperature * 140 - 220 °C Minimum service temperature * -123 - -73.2 °C Thermal conductor or insulator? Poor insulator Thermal conductivity * 1.28 - 2.6 W/m.°C Specific heat capacity * 902 - 1.04e3 J/kg.°C Thermal expansion coefficient * 1 - 4 µstrain/°C Electrical properties Electrical conductor or insulator? Poor conductor Electrical resistivity * 1.65e5 - 9.46e5 µohm.cm Optical properties Transparency Opaque Processability Moldability 4 - 5 Machinability 1 - 3 Eco properties Embodied energy, primary production * 453 - 500 MJ/kg CO2 footprint, primary production * 32.9 - 36.4 kg/kg Recycle False

Figure C.7: CFRP - data sheet C.2. MATERIAL DATA SHEETS 17

GFRP, epoxy matrix (isotropic) Composites are one of the great material developments of the 20th century. Those with the highest stiffness and strength are made of continuous fibers (glass, carbon or Kevlar, an aramid) embedded in a thermosetting resin (polyester or epoxy). The fibers carry the mechanical loads, while the matrix material transmits loads to the fibers and provides ductility and toughness as well as protecting the fibers from damage caused by handling or the environment. It is the matrix material that limits the service temperature and processing conditions. Polyester-glass composites (GFRPs) are the cheapest and by far the most widely used. A recent innovation is the use of thermoplastics at the matrix material, either in the form of a co-weave of cheap polypropylene and glass fibers that is thermoformed, melting the PP, or as expensive high-temperature thermoplastic resins such as PEEK that allow composites with higher temperature and impact resistance. High performance GFRP uses continuous fibers. Those with chopped glass fibers are cheaper and are used in far larger quantities. GFRP products range from tiny electronic circuit boards to large boat hulls, body and interior panels of cars, household appliances, furniture and fittings. General properties Density 1.75e3 - 1.97e3 kg/m^3 Price * 19.4 - 27.4 EUR/kg Date first used 1935 Mechanical properties Young's modulus * 15 - 28 GPa Shear modulus * 6 - 11 GPa Bulk modulus 18 - 20 GPa Poisson's ratio * 0.314 - 0.315 Yield strength (elastic limit) * 110 - 192 MPa Tensile strength * 138 - 241 MPa Compressive strength * 138 - 207 MPa Elongation * 0.85 - 0.95 % strain Hardness - Vickers * 10.8 - 21.5 HV Fatigue strength at 10^7 cycles * 55 - 96 MPa Fracture toughness * 7 - 23 MPa.m^0.5 Mechanical loss coefficient (tan delta) * 0.0028 - 0.005 Thermal properties Glass temperature 147 - 197 °C Maximum service temperature * 140 - 220 °C Minimum service temperature * -123 - -73.2 °C Thermal conductor or insulator? Poor insulator Thermal conductivity * 0.4 - 0.55 W/m.°C Specific heat capacity * 1e3 - 1.2e3 J/kg.°C Thermal expansion coefficient * 8.64 - 33 µstrain/°C Electrical properties Electrical conductor or insulator? Good insulator Electrical resistivity * 2.4e21 - 1.91e22 µohm.cm Dielectric constant (relative permittivity) * 4.86 - 5.17 Dissipation factor (dielectric loss tangent) 0.004 - 0.009 Dielectric strength (dielectric breakdown) * 11.8 - 19.7 1000000 V/m Optical properties Transparency Translucent Processability Moldability 4 - 5 Machinability 2 - 3 Eco properties Embodied energy, primary production * 150 - 170 MJ/kg CO2 footprint, primary production * 9.5 - 10.5 kg/kg Recycle False

Figure C.8: GFRP - data sheet 18 APPENDIX C. MATERIAL COMPLEMENT

alloys.pdf

Titanium alloys Description The material Titan was a Greek god, remarkable for his size and strength. His name has been appropriated many times, not always aptly (think of the Titanic). But the alloys of titanium merit the association: the strongest of them have the highest strength-to-weight ratio of any structural metal, about 25% greater than the best alloys of aluminum or steel. Titanium alloys can be used at temperatures up to 500 C - compressor blades of aircraft turbines are made of them. They have unusually poor thermal and electrical conductivity, and low expansion coefficients. The alloy Ti 6%Al 4% V is used in quantities that exceed those of all other titanium alloys combined. The data in this record describes it and similar alloys. Composition (summary) Ti + alloying elements, e.g. Al, Zr, Cr, Mo, Si, Sn, Ni, Fe, V General properties Density 4.4e3 - 4.8e3 kg/m^3 Price * 19.9 - 21.9 EUR/kg Date first used 1952 Mechanical properties Young's modulus 110 - 120 GPa Shear modulus 40 - 45 GPa Bulk modulus 96 - 102 GPa Poisson's ratio 0.35 - 0.37 Yield strength (elastic limit) 750 - 1.2e3 MPa Tensile strength 800 - 1.45e3 MPa Compressive strength 750 - 1.2e3 MPa Elongation 5 - 10 % strain Hardness - Vickers 267 - 380 HV Fatigue strength at 10^7 cycles * 589 - 617 MPa Fracture toughness 55 - 70 MPa.m^0.5 Mechanical loss coefficient (tan delta) 5e-4 - 0.002 Thermal properties Melting point 1.48e3 - 1.68e3 °C Maximum service temperature 450 - 500 °C Minimum service temperature -273 °C Thermal conductor or insulator? Poor conductor Thermal conductivity 7 - 14 W/m.°C Specific heat capacity 645 - 655 J/kg.°C Thermal expansion coefficient 8.9 - 9.6 µstrain/°C Electrical properties Electrical conductor or insulator? Good conductor Electrical resistivity 100 - 170 µohm.cm Optical properties Transparency Opaque Processability Castability 3 Formability 2 - 4 Machinability 1 - 3 Weldability 4 - 5 Solder/brazability 1 - 2 Eco properties Embodied energy, primary production * 651 - 720 MJ/kg CO2 footprint, primary production * 44.1 - 48.7 kg/kg Recycle True

Figure C.9: Titanium alloys - data sheet C.2. MATERIAL DATA SHEETS 19

Polymer Foam.pdf

Rigid Polymer Foam (HD) Description The material Polymer foams are made by the controlled expansion and solidification of a liquid or melt through a blowing agent; physical, chemical or mechanical blowing agents are possible. The resulting cellular material has a lower density, stiffness and strength than the parent material, by an amount that depends on its relative density - the volume-fraction of solid in the foam. Rigid foams are made from polystyrene, phenolic, polyethylene, polypropylene or derivatives of polymethylmethacrylate. They are light and stiff, and have mechanical properties the make them attractive for energy management and packaging, and for lightweight structural use. Open-cell foams can be used as filters, closed cell foams as flotation. Self-skinning foams, called 'structural' or 'syntactic', have a dense surface skin made by foaming in a cold mold. Rigid polymer foams are widely used as cores of sandwich panels. General properties Density 170 - 470 kg/m^3 Price * 9.9 - 19.8 EUR/kg Date first used 1931 Mechanical properties Young's modulus 0.2 - 0.48 GPa Shear modulus 0.055 - 0.195 GPa Bulk modulus 0.2 - 0.48 GPa Poisson's ratio 0.27 - 0.33 Yield strength (elastic limit) 0.8 - 12 MPa Tensile strength 1.2 - 12.4 MPa Compressive strength 2.8 - 12 MPa Elongation 2 - 10 % strain Hardness - Vickers 0.28 - 1.2 HV Fatigue strength at 10^7 cycles * 0.84 - 9.6 MPa Fracture toughness 0.0236 - 0.0905 MPa.m^0.5 Mechanical loss coefficient (tan delta) * 0.005 - 0.15 Thermal properties Glass temperature 66.9 - 171 °C Maximum service temperature 66.9 - 167 °C Minimum service temperature -113 - -73.2 °C Thermal conductor or insulator? Good insulator Thermal conductivity 0.034 - 0.063 W/m.°C Specific heat capacity 1e3 - 1.91e3 J/kg.°C Thermal expansion coefficient 22 - 70 µstrain/°C Electrical properties Electrical conductor or insulator? Good insulator Electrical resistivity 1e16 - 1e20 µohm.cm Dielectric constant (relative permittivity) 1.21 - 1.45 Dissipation factor (dielectric loss tangent) 8e-4 - 0.008 Dielectric strength (dielectric breakdown) 6.02 - 11 1000000 V/m Optical properties Transparency Opaque Processability Castability 1 - 3 Moldability 3 - 4 Machinability 3 - 4 Weldability 1 - 2 Eco properties Embodied energy, primary production * 96.6 - 107 MJ/kg CO2 footprint, primary production * 3.68 - 4.07 kg/kg

Figure C.10: Rigid Polymer Foam HD - data sheet 20 APPENDIX C. MATERIAL COMPLEMENT

Polymer FoamMD.pdf

Rigid Polymer Foam (MD) Description The material Polymer foams are made by the controlled expansion and solidification of a liquid or melt through a blowing agent; physical, chemical or mechanical blowing agents are possible. The resulting cellular material has a lower density, stiffness and strength than the parent material, by an amount that depends on its relative density - the volume-fraction of solid in the foam. Rigid foams are made from polystyrene, phenolic, polyethylene, polypropylene or derivatives of polymethylmethacrylate. They are light and stiff, and have mechanical properties the make them attractive for energy management and packaging, and for lightweight structural use. Open-cell foams can be used as filters, closed cell foams as flotation. Self-skinning foams, called 'structural' or 'syntactic', have a dense surface skin made by foaming in a cold mold. Rigid polymer foams are widely used as cores of sandwich panels. General properties Density 78 - 165 kg/m^3 Price * 9.9 - 19.8 EUR/kg Date first used 1931 Mechanical properties Young's modulus 0.08 - 0.2 GPa Shear modulus 0.0236 - 0.069 GPa Bulk modulus 0.08 - 0.2 GPa Poisson's ratio 0.27 - 0.33 Yield strength (elastic limit) 0.4 - 3.5 MPa Tensile strength 0.65 - 5.1 MPa Compressive strength 0.95 - 3.5 MPa Elongation 2 - 5 % strain Hardness - Vickers 0.095 - 0.35 HV Fatigue strength at 10^7 cycles * 0.455 - 2.8 MPa Fracture toughness 0.0066 - 0.0486 MPa.m^0.5 Mechanical loss coefficient (tan delta) * 0.005 - 0.15 Thermal properties Glass temperature 66.9 - 157 °C Maximum service temperature 66.9 - 157 °C Minimum service temperature -113 - -93.2 °C Thermal conductor or insulator? Good insulator Thermal conductivity 0.027 - 0.038 W/m.°C Specific heat capacity 1.12e3 - 1.91e3 J/kg.°C Thermal expansion coefficient 20 - 70 µstrain/°C Electrical properties Electrical conductor or insulator? Good insulator Electrical resistivity 1e17 - 1e21 µohm.cm Dielectric constant (relative permittivity) 1.1 - 1.19 Dissipation factor (dielectric loss tangent) 8e-4 - 0.008 Dielectric strength (dielectric breakdown) 5.61 - 6.76 1000000 V/m Optical properties Transparency Opaque Processability Castability 1 - 3 Moldability 3 - 4 Machinability 3 - 4 Weldability 1 - 2 Eco properties Embodied energy, primary production * 96.6 - 107 MJ/kg CO2 footprint, primary production * 3.68 - 4.07 kg/kg

Figure C.11: Rigid Polymer Foam MD - data sheet