FOILS MOTH THESIS

Evaluation of the Performance of a Hydro-Foiled

Moth by Stability and Force Balance Criteriawith the Software Tool FutureShip Equilibrium

31. Symposium Yachtbau und YachtenwurfHamburg, November 2010

Christian Bögle (TU-Berlin)

Dr. Karsten Hochkirch (FurturShip)Heikki Hansen (FutureShip)

Gonzalo Tampier-Brockhaus (TU-Berlin)

Department of Ocean Engineering and Naval ArchitectureTU-Berlin

Berlin, Germany

Table of Content Zusammenfassung...........................................................................................3 Abstract............................................................................................................4 1 Introduction...................................................................................................5

1.1 Motivation.........................................................................................................5 1.2 Project Objectives............................................................................................7 1.3 Methodology.....................................................................................................7 1.4 Evaluation.........................................................................................................9

2 Velocity Prediction......................................................................................10 2.1 Force Balance................................................................................................10 2.2 Design Criteria and Design Parameters........................................................13 2.3 FS-Equilibrium................................................................................................16

2.3.1 Force Modules.................................................................................................16 2.3.2 Input Data ........................................................................................................19

2.4 Velocity Prediction and Result Interpretation.................................................21 2.4.1 Variation...........................................................................................................22 2.4.2 Results.............................................................................................................24

3 Stability.......................................................................................................28 3.1 Small Disturbance Theory..............................................................................29 3.2 In-stationary Simulation..................................................................................31 3.3 Stability Investigation and Results Interpretation...........................................33

4 Evaluation...................................................................................................38 5 Conclusions and Future Work ...................................................................43 Tables and References...................................................................................46

List of Figures........................................................................................................46 List of Tables.........................................................................................................48 Nomenclature........................................................................................................49 References............................................................................................................51

Appendices.....................................................................................................52 A Feedback control system (mathematical approach) ........................................52 B Preliminary foil system design tool...................................................................53 C FS-Equilibrium..................................................................................................58

C.1 Predefined Modules..................................................................................62 C.2 User Modules............................................................................................66 C.3 Input Data .................................................................................................74 C.4 Force Module Validation............................................................................98

D Foil Set-Up for VPP........................................................................................101 E VPP Results....................................................................................................104 F Stability Results...............................................................................................111

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Zusammenfassung

ZusammenfassungDie “International Moth” (Motte) ist eine der am weitesten entwickelten Bootsklassen der Welt. 1928 erfunden, war es die erste Bootsklasse, in der Tragflügel (Hydro-Foils) in den Klassenregeln erlaubt wurden. Diese Entwicklung hat 2001 mit V-Tragflügeln begonnen, heutiger stand der Technik sind Doppel-Tragflächen Systeme mit einer, durch einen Oberflächen Sensor gesteuerten, aktiv geregelten Haupt-Tragfläche am Schwert.

Die letzten Entwicklungen beschränkten sich hierbei hauptsächlich auf die Optimierung der Tragflächen Profile und Formen, um den Widerstand zu verringern und den Auftrieb zu vergrößern. Tiefergehende Untersuchungen der Flächenverhältnisse der Tragflächen, sowie Einflussgrößen des Sensor gesteuerten Kontroll-Systems wurden nicht durchgeführt, oder wurden nicht veröffentlicht.

Grundlage für den Vergleich verschiedener Tragflügel und Kontroll-Systeme ist die Bestimmung der Gleichgewichtszustände in Abhängigkeit von Kurs und Wind-geschwindigkeit. Ein Programm, mit dem die Gleichgewichtszustände bestimmt werden können, ist FS-Equilibrium. Im Vergleich zu anderen Software Lösungen, besteht bei FS-Equilibirium die Möglichkeit, eigene Module, die das Verhalten der Tragflächen wiedergeben, zu programmieren und implementieren.

Ziel dieser Arbeit ist der Vergleich verschiedener Tragflügelsysteme hinsichtlich Ihrer Stabilität und Geschwindigkeit. Hierfür wurden die Einflussgrößen des Kontrollsystems definiert und Grundprinzipien bezüglich des Verhaltens des Tragflügelsystems bei verschiedenen Bedingungen hergeleitet. Auf der Basis der Einflussgrößen wurden die mathematischen Beziehungen hergeleitet und in die offene modulare Umgebung von FS-Equilibrium eingebunden.

Nach der Verifikation der Module konnte ein Vergleich von verschiedenen Tragflügel-systemen, abhängig von Kurs und Windgeschwindigkeiten, durchgeführt werden. Für die Bestimmung der Einflußgrößen wurde im Vorfeld ein Tabellendokument erstellt, mit dem, unter Berücksichtigung stark vereinfachender Annahmen, das Verhalten der Motte für das Abheben simuliert werden konnte. Die Ergebnisse aus dieser Vereinfachung konnten mit den Ergebnissen aus FS-Equilibrium abgeglichen werden.

Die gefundenen Gleichgewichtszustände wurden auf Ihre Stabilität hin untersucht. Mit Hilfe der “Small Disturbance Theory” und einer in-stationären Simulation konnte gezeigt werden, dass viele gefundene Lösungen, speziell die am-Wind Kurse, hochgradig instabil sind.

Die Ergebnisse dieser Arbeit zeigen, dass, für die Auslegung des Tragflügelsystems, ein stärkerer Fokus auf die Stabilität gelegt werden sollte. Der Ansatz einer vereinfachten Tabellen Berechnung zur Bestimmung der Parameter für das Tragflächensystem hat sich bewährt, und kann als Ausgangspunkt für weitere Untersuchungen hinsichtlich Stabilität und für die Herleitung eines Stabilitätskriteriums genutzt werden. Zusätzlich hat sich gezeigt, dass sich FS-Equiblibrium gut für die Bestimmung der Gleichgewichtszustände und der Stabilität eignet. Für weitergehende Untersuchungen sollten noch zusätzliche Module definiert bzw. bestehende verfeinert werden, um weitere Kraftkomponenten zu berücksichtigen. Hierdurch können realere Ergebnisse erzielt werden und ein Abgleich mit realistischen Segelzuständen und den Erfahrungen von Seglern wäre besser möglich.

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Abstract

AbstractThe “International Moth” is one of the most advanced boat classes in the world. Invented in 1928, it was the first class with hydro-foils in their class rules. This development started in 2001 with v-shaped foils, today a two hydrofoil system with an active flap control for the main foil via a surface sensor is state of the art.

Recent investigations were mainly focused on the foil and the profile shape, for reducing the drag- and increasing the lift coefficient. In depth investigations concerning the plan form area of the rudder- and centreboard foil, as well as an investigation concerning the geometrical parameters of the control system have not been conducted, or at least not published yet.

To be able to perform a comparison of different parameters of the foil and the control system, the equilibrium states of different sailing conditions have to be found. One tool which is capable of performing this task is FutureShip Equilibrium. In comparison to other velocity prediction programs, additional modules representing foils etc. can be added and time dependent calculations can be performed.

The goal of this work is the comparison of different foil / control system configurations based on their velocity and stability. Therefore the basic geometric parameters, influencing the foil system, have been defined and the basic principles, how the control system should react, and the design principles of the foils have been found. On this basis the influence of these parameters was formulated by mathematical means and implemented into the open modular workbench FS-Equilibirium by different force modules.

After the validation of the force modules, the performance of different foil system configurations have been compared and evaluated due to their speed. For setting appropriate parameters defining the foil systems, a spread sheet has been set-up for the simulation of the take off behaviour of the Moth, on the basis of very simplified assumptions. The results of this spread sheet have been verified using the results of FS-Equilibrium.

Using the results of the Velocity Prediction Program, the obtained equilibrium conditions have been investigated due to their stability. Performing a dynamic investigation with the small disturbance theory and a fully in-stationary analysis showed, that especially the upwind courses are highly unstable.

Using the results from this work, a major focus must also be set on the stability and not only the velocity. As the spread sheet seems to be a good starting point, this can be the basis for a further investigation of the stability and the definition of a suitable stability criteria. In addition it is shown, that using the open modular workbench FutureShip Equilibrium is an appropriate way for the prediction of the velocity and the stability. For using this approach in any further investigations, additional force modules should be defined and existing ones up-dated to take additional force components into account. By this, more realistic results can be achieved and compared to real conditions and the experience of sailors.

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1 Introduction

1 IntroductionThe International Moth, invented in 1928 by Len Morris, is one of the most advanced boat classes in the World. At his time, Len built a cat rigged (single sail) flat bottom scow to sail on Anderson’s Inlet at Inverloch, 130 km from Melbourne. The boat was hard chined, 11 feet long and carried a 80 square feet single mainsail. After building another two boats the Inverloch Yacht club was formed out of these three boats. Restrictions for the class, known as the Inverloch Eleven Footer class, were then drawn up, with the distinguishing characteristic that of being not a one-design boat but rather that of a boat permitting development within the set of design parameters.

In 1929, Captain Joel Van Sant of Atlantic City, New Jersey set up another development class. The major difference between the Australian and American boats early on was that the American boat used only 72 square feet of sail area on a somewhat shorter mast. The US development class was formally organized in 1932 as the "National Moth Boat Association" and in 1935, due to increasing overseas interest, changed its name to the "International Moth Class Association" or IMCA.

In 1933 the US Moth class got recognized by the Australians by an article in an American magazine. The Australians noted the similarities between the two boats and intuitively realized that the name "Moth Boat" rolled more easily from the tongue than "Inverloch Eleven Footer Class", and changed the name of their class to Moth [1].

After some changes in the class rules over time, the rules can be summarized as followed:

• maximum length (without outrigger): 3355 mm

• maximum beam: 2250 mm

• maximum luff length: 5600 mm

• maximum sail area: 8 m2

• minimum displacement at DWL: 70 Kg

Those few rules makes the International Moth one of the fastest developing boat-classes in the world. Today boats have a hull beam of approx. 30 cm, fully built in carbon fibre, full batten pocket sails, and a total weight of less than 30 kg. The latest improvement was hydrofoils.

1.1 Motivation

Andy Paterson of Blodeaxe boats is wildly considered to have developed the first functional foiling moth. In 2001 Brett Burvill sailed a Moth skiff with surface piercing foil, at the world championship in Australia. Afterwards this hydrofoil configuration was rated as muultihull and therefore from the class. As a result, Gath and John Ilett developed a two hydrofoil system for the Moth with an active flap control for the main foil via a surface sensor [2].

5 Years after the first attempts of hydrofoil sailing, the brake through for foiled Moths came with the introduction of “Bladerider”, a hydro-foiled moth, developed by Andrew McDougall in Melbourne and built in China for volume production and worldwide sales. These boats were extensively tested and further developed by world class sailors. Subsequently, the Moth

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1 Introduction

became far more commercialised and development more professionalized and confidential. This makes it hard for the private Moth builder, to built a competitive boat or even improve the existing foil systems.

This leads to the question, “what makes a foils system a good foil system and how can a foil system be improved?”

There are two major requirements on a hydro-foiled Moth – the Moth should be fast and good to handle. This two aspects can be reduced to one common denominator - the stability. Stability in this case can be seen as static stability, having an equilibrium condition of all reacting forces at a maximum speed and knowing the tendency of the craft to return to its equilibrium position and the dynamic stability.

Using the system “hydro-foiled Moth”, two major influences on the stability can be identified, the mechanical components and the crew, where each can be subdivided in more influencing variables. This breakdown can be demonstrated by the tree diagram, shown in Figure 1-1:

Following the discussions in different forums, the main focus in the last years has been set on the evaluation of the best wing sections and foil shapes to minimize the drag of those components. Just little attempts had been made improving the overall performance, hence

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Figure 1-1: Aspects influencing the stability of a hydro-foiled Moth

1 Introduction

the static and dynamic stability. If the foil shape and the foil-section is regarded as a sub-component of the foils, the best way of improving the foil system is the reduction of the influence of the human component – the sailor – and improving the feedback control system, as a major component of the foil system.

Even though the development of hydro-foiled Moths is very evolved, today control systems are extensively tested by sailors, but the theoretical influence of the different aspects of a foil system on the performance, hence the stability, is not documented so far.

1.2 Project Objectives

Stability for conventional yachts and boats is mainly expressed by stability curves, which show the ability of the boat to create an upright moment. Hereby the centre of buoyancy of the hull is moved relative to the centre of gravity, due to the change of heel or pitch, creating an upright moment. For the stability of a hydro-foiled Moth additional force components have to be considered, while the stabilising forces components from the hull are missing.

The transverse stability is the sporty part for the sailor, while the longitudinal stability is dependant of the rudder- and the centreboard-foil and the acting forces. As the rudder foil is controlled by the sailor, whose influence on the stability should be minimized, main focus is set on the prediction and improvement of the centreboard foil system.

For the evaluation of the foil system, the geometrical parameters influencing the behaviour of the foil system have to be identified. The derivation of the parameters has been done by criteria, defining the different states and conditions of the hydro-foiled Moth. These design principles are also the basis for the evaluation of the foil systems static stability.

As the designer of a hydro-foiled moth is faced with many geometrical parameters, a set-up of the foil system, which should be close to most current set-ups, will be used as initial foil system. Afterwards single parameters or even a set of parameters will be varied, for derivation of the influence of this parameters on the velocity.

The equilibrium conditions are the basis for the evaluation of the static and dynamic stability. Hereby techniques known from aircraft design will be used for the prediction of the grade of stability.

1.3 Methodology

Basis for the stability - static and dynamic - is the prediction of the force components and the calculation of the equilibrium states. A common approach for calculating the force balance, are velocity prediction programs (VPP).

Most VPPs do not have the opportunity to define hydrofoils and a feedback control system. The VPP from FutureShip, FS-Equilibrium, is a open modular work bench, based on programmable force modules. Additional force modules can be programmed with common program languages or even a c++ based api program language, and added to the program. As the name “FS-Equilibrium” already implies, the used program calculates the equilibrium state of the forces and optimizes the trim variables and conditions to minimize a specific design value, normally the ship speed or the velocity made good (VMG).

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1 Introduction

As there is already a large number of predefined modules available within FS-Equilibrium, there is none for an actively controlled foil system. For the use of FS-Equi, these force modules have to be defined first.

The additional forces are mainly driven by the foils and thereby the control system, reacting on different conditions. State of art is the already mentioned foil system, invented by Gath and John Ilett from Fastacraft. This system consists of a sensor wand controlled main foil at the centreboard and a user controlled rudder foil. The concept is well illustrated in an article of David Schmidt at www.sailmagazin.com [3] with an illustration of Jason Lee in Figure 1-2.

With the knowledge of the function of the foil system, geometrical parameters, influencing the foil system, can be predicted and the mathematical relationship of these parameters, the conditions and resulting forces of the foils be derived.

As there are numerous geometrical parameters defining the influence of the feedback control system, the determination of the input data is not an easy task. Also most parameters are interacting with each other, making the prediction even more complicated. Therefore a preliminary foil design tool has been created on the basis of a spread sheet, giving a rough estimation of how a specific parameter influences the foil system. Together with the preliminary defined design criteria, the input data for the the specific modules can be created.

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Figure 1-2: Principle function of the foil system [3]

http://www.sailmagazin.com/

1 Introduction

In addition all geometrical, hydro- and aerodynamic data for the calculation of the different forces had to be determined. This has been done by using test data, theoretical approaches like 3D panel codes, or by estimation from theoretical thoughts.

With the definition of the input data, the calculation of the equilibrium conditions can be performed and the derived data can be used for the evaluation process.

1.4 Evaluation

Using the VPP with the defined model of a hydro-foiled Moth, the sail condition as well as optimised trim-variables can be calculated for a set of wind speeds (TWS) and courses (TWA). This data is used for the validation of the velocity dependant criteria used for the preliminary foil design, hence the preliminary foil design tool.

Thereby the validation of the design tool is important, as this is also used for the definition of an optimised foil set-up for specific requirements and conditions. In the following calculations the performance of different foils – plan-form areas – with and without optimised feedback control systems has been compared.

The equilibrium states, which has been found with the VPP are also the basis of evaluation of the dynamic stability. In a first step FS-Equilibrium gives the possibility to evaluate the small disturbance stability. This is a method known from aircraft design, linearising the equations of motion at a certain condition and apply a small disturbance in one degree of freedom. Afterwards the oscillation of the motion will be evaluated. For a dynamical stable set-up, the oscillation will dying out. The time needed for the amplitude to die out to half of its value is an indicator for the grade of stability.

Another approach for the evaluation of the stability is the usage of the in-stationary module of FS-Equilibrium. By taking a equilibrium condition as start point, the response of the boat on small disturbances or even the time dependent behaviour without any disturbances can be analysed. For both criteria the major interest is the self-stabilising ability of the system.

Concluding the status of the work and the method for the definition and the evaluation of a foil system are shortly discussed.

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2 Velocity Prediction

2 Velocity PredictionAs already mentioned, the prediction of the force balance is the basis for the evaluation of the static and the dynamic stability. Hereby the six condition variables are calculated by solving the set of linear equations defining the forces and moments. Commonly this is known as velocity prediction.

The forces and resulting condition are also influenced by additional trim variables, e.g. crew position, de-power of the sail. In a second loop these have to be optimized, minimizing a prior defined optimisation objective. In most cases this is the negative speed or velocity made good (VMG).

For conventional yachts, this is the major design criteria. For a hydro-foiled moth, maximum speed is not the only important aspect, but also sailing at different stages – non-foiling, take off, foiling at design speed and foiling at maximum speed. Therefore various design criteria have to be defined. Finding the best compromise of the geometrical influencing parameters is the main task of the designer.

Thus the knowledge of the forces and the influence of the geometrical parameters is an essential part.

2.1 Force Balance

Main focus by means of stability is set on the force components influencing the longitudinal behaviour of the Moth. Transversal stability is mainly driven by the centre of gravity, which is primarily influenced by the sailor, who makes up to 70% of the total weight, and the geometrical dimensions of the boat, which are fixed in the class rules – hence the maximum upright moment.

Using the transversal force balance in Figure 2-1, the body fixed lift components of the rudder and the centreboard-foil can be split up into a horizontal and vertical component . For a windward heel angle, this leads to a significant side force, counteracting the sail side force by little reduction of the vertical lift component. This is important as the side force from the struts will be reduced due to the change in plan-form area and the flight height, but also gives the possibility to sail without leeway angle and thereby reducing the induced drag of the struts. Also the vertical component of the sail side force will not be levelled out by the vertical component of the strut, giving an additional lift.

Besides the transversal force balance, Figure 2-1 also shows that the main lift components are the lift forces acting on the rudder- and the centreboard foil. These have to be in balance with the gravity force of the sailor and the boat. In addition the Pitch moments have to be balanced. These are influenced by the lift and gravity forces and additionally by the drag components of the foils and the struts and the driving force. Correspondingly the drag forces are influenced by the lift components, hence the the control mechanism of the boat.

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The lift at the rudder foil is controlled by the sailor. The first rudder foils had been built with a flap, however modern boats are turning the whole foil changing the angle of attack. The centreboard- or main foil is controlled by a feedback control system, which controls the flap angle of the main foil. The basic concept of the feedback control system is shown in Figure 1-2. For the prediction of the lift forces the relation between the flap angle and the sensor wand has to be evaluated.

The feedback control system contains a sensor wand, connected at the bow. This sensor is turned by the drag in the water, due to speed and the height of the centre of rotation (CoR). Additionally a rubber robe is attached to the the sensor wand, creating a counter moment as shown in Figure 2-2. The rotation angle of the sensor wand can be calculated by the force balance of the hydrodynamic drag and the robe tension force at the CoR.

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Figure 2-1: Force balance of a hydro-foiled Moth


Also connected to the sensor wand is the control wire, which controls the pushrod in the daggerboard and thereby the flap angle. With the assumption that the forces of the flap onto the pushrod are small, the relation can be expressed purely geometrically and is illustrated in Figure 2-3.

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Figure 2-2:Sensor force balance

Figure 2-3: Geometrical relation between sensor ans flap angle


The geometrical parameters, shown in Figure 2-2 and Figure 2-3 have a big influence on the lift force at the main foil, hence the stability of the Moth. A major task of the designer is the right prediction and evaluation of these parameters.

In appendix A all formulas, used for the prediction of the lift-force, are extensively explained.

2.2 Design Criteria and Design Parameters

For the design of the feedback control system, four different stages have to be considered. First stage is the non foiling state. Here the foil system should generate no lift, reducing the induced drag at the foils. Second stage is the take off speed. At this speed the foils are creating enough lift force, to heave the boat out of the water. Next stage is the travel speed for a certain condition. At this stage the drag is ideally minimized and a maximum speed can be achieved for a range of wind speeds. The last stage is maximum speed. Here the lift of the foil is minimized and the foil cannot pierce the water surface.

Due to this four stages, the response function of the flap angle due to speed and sinkage/ altitude have to be calculated.

• Non-foiling: at non foiling stage the lift at the foils should be minimized. Influencing parameter therefore is the pretension of the robe, but also has an influence on the sensitivity of the foil systems.

• Take off Speed: The first professional build moth by means of profit – the bladerider – has a take off wind speed of approx. 3 Bf. Assuming that the take-off speed is close to wind speed and the lift of the main foil is about 60 to 70 % of the overall lift, the main foil must create a lift of 70 to 80 kg at a boat speed of 3-4 m/s and a altitude equal to the draft of the canoe body of the hull. But it should also create a significant lift while sailing at DWL. This criteria should be updated and validated through the further calculations and is just a starting point at this stage.

• Design speed: This should be the speed range, at which the control system together with the main foil operates at an optimum. Lift off effects can be neglected, as boat is already in flying state. As the driving force and the drag forces at the foil system produce a significant pitch moment, it can be assumed that the lift is only created by the main foil, and the rudder foil produces no lift or even a down force for keeping the boat horizontal. The optimum design speed may vary due to condition and region.

• Maximum speed: At this speed the moth should still be able to be handled properly, but any further speed will lead to suction or ventilation at the main foil and therefore uncontrollable conditions. At this speed, the minimum flap angle will be reached. The rudder foil also produces no lift.

Looking at the given parameters / possibilities influencing the feedback control system, it is hard to find the ideal set-up, as many possible combinations have to be considered. Using a simplified spread sheet, the lift characteristic at the main foil can be calculated for the different stages, and a start-up of the Moth can be very rudimentary simulated. Basis for the prediction of the forces are formulas given by Speers [4] and Abott [5].

Starting with the prediction of the flap angle, the lift force of the centreboard is calculated due to speed, sink and pitch for all 4 stages. Hereby the speed, sink and pitch can be set

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manually to get the desired value for the lift at the main foil. Now the drag at the centreboard foil is calculated at the take-off condition. Using a predicted value for the lift at the rudder foil of approximately 35%, the drag at the rudder foil can be back-calculated by the section drag, the induced drag and the geometry of the foil. Ideally the leeway angle is zero, hence no induced drag at the struts. Now the total drag at take off condition is calculated and used as a constant driving force. With this force, and very simplified assumption concerning the drag and lift of the hull, an in-stationary lift-off simulation can be performed. Thereby it is assumed that heel and pitch are constant, hence just the weight of the sailor, but not the position will be considered.

Together with the drag components of the hull, and the centreboard foil geometry, it can now be simulated, if the specific parameters of the feedback control system may lead to take off of the Moth. Result is a plot, as in Figure 2-4, of the speed and the z-offset versus the time, whereas the time may not be realistic as not all force components and conditions are considered.

Input parameters of the control system, used in Figure 2-2 and Figure 2-3 are renamed for the use within FS-Equilibrium. The feedback control system can now be defined by the parameters from Table 2-1. The corresponding lift and drag coefficients are calculated by the foil section and the shape of the foil, as well as the plan-form area.Table 2-1: Parameter names of feedback control system within FS-Equilibrium

CoR Sensor.CoR Centre of rotation (DWL)

lS Sensor.Length Length of sensor wand

DS Sensor.Diam Diameter of sensor

cD Sensor.DragCoeff Drag coefficient of sensor

lR Robe.DistCoR Distance robe attachment to CoR of the sensor

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Figure 2-4: Boat speed and z-offset vs. time plot


FR Robe.Tension Pretension of robe

lCW Flap.SensorDistCoR Distance from sensor CoR to the control wire cable attachment

δS0 Flap.DeltaSensor_0 Angle offset for the sensor cable attachment

f Flap.fDeltaFlap Reduction factor between sensor robe and vertical pushrod in the daggerboard (R2/R1)

lFl Flap.FlapDistCoR Distance pushrod attachment to flap CoR

δFL0 Flap.deltaFlap_0 Flap angle at 0° sensor angle

The influence of these parameters on the operation of the feedback control system are briefly discussed:

• sensor length: The length of the sensor is responsible at which speed the control system starts to work – at which speed a significant lift will be created. As long as the robe moment MR > Mhydro, the minimum flap angle will be used, producing small, or even negative lift at the main foil. A small lift coefficient is reducing the induced drag. The maximum lift angle as well as the maximum z-offset (altitude) are influenced by the sensor length as well. For a long sensor the height and the maximum flap angle will be increased.

• sensor diameter and DragCoeff: Together with the length of the sensor, the diameter and the drag coefficient influencing the starting point of the control system. But also the sensitivity of the system can be changed – a long sensor with a small diameter will be more sensitive than a short sensor with a big diameter.

• Robe distCoR and Tension: The distance of the tension robe and the pretension of the robe are responsible for the speed at which the feedback control system starts to react.

• Flap sensorDistCoR, fdeltaFlap, FlapDistCoR: Those parameters are defining the reduction between the sensor distance and the flap control distance. The distances to the CoR at the sensor and the flap should be set by real means. The reduction factor fdeltaFlap should be set to a value, that the flap reaches the maximum Lift coefficient needed for take off.

• DeltaSensor_0: This parameter defines the angle offset of the cable attachment and the sensor. Having no angle offset defined, a Moth in flying mode will react very sensitively on a change of the sensor angle, as the sensor angle is low and the steering cable changes the flap by the sinus of the sensor angle (∆S1). By

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Figure 2-5: Angle offset DeltaSensor_0


definition of an angle offset, this movement can be reduced (∆S2) hence the boat reacts less sensitive at a higher flight height.

• deltaFlap_0: Minimum flap angle – low drag in low-rider mode, minimum necessary lift coefficient at maximum speed

2.3 FS-Equilibrium

The program used for the prediction of the static stability, and hereby the prediction of the velocity, is the open modular workbench FS-Equilibrium from FutureShip. The advantage of FS-Equi is the possibility of integration of user programmed force modules, in different program languages. For this work the C++ based internal api-language is used, giving the possibility of using graphical user interfaces as well as standard information output for debugging or additional informations during a run. Also many predefined force modules are available. A brief introduction is given in Apendix C on the basis of a paper from Richards [6].

The prediction of the speed is done by solving the set of linear equations defining the motion. This is done using a Newton-Raphson solver which represents the inner loop of the solution. In an outer loop, the trim-variables are optimised by means of maximizing the speed. Trim-variables are dependant on the chosen predefined force modules or the self programmed force modules. For the hydro-foiled Moth four trim possibilities have been used – de-powering of the sail, called FLAT, crew position, defined by MassMove1 and MassMove2, and a trim-variable for controlling the flap angle of the rudder foil which is called deltaFlap.

With the modular structure of FS-Equilibrium, an arbitrary number of force modules can be defined, influencing the grade of details, for the investigation of special effects. To give an overview, the used force modules are listed and shortly discussed in the following section. An detailed representation is listed in appendix C.1 and C.2 for predefined and user-defined modules.

2.3.1 Force Modules

The predefined force modules can be categorized in three different groups – gravity-, aerodynamic- and hydrodynamic forces.

Gravity force modules: The gravity force modules used for the Moth, are the fixed mass of the boat and the flexible positioned mass of the crew. Hereby two different force modules can be used – MoveableMass2D and ControlledMass, where in the first one, the position is defined as a trim variable, in the second as control variable, influencing the solution technique.

Aerodynamic force modules: the major aerodynamic force acting on a sail boat are the lift and drag forces from the sail. Hereby the possibility of de-powering the sail is implemented as the trim-variables FLAT. Additional REEF and TWIST are included, deactivated in this case. Another force component is the windage – the aerodynamic drag due to the hull and other appendices. As shown in the paper of Beaver and Zseleczky [7] for a hydro-foiled Moth the aerodynamic drag of the Moth is about 70% of the hydro-foil

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drag for upwind courses. This makes windage a major drag component of a hydro-foiled vessel.

Hydrodynamic force module: predefined module in this category is the buoyancy force module. Here the buoyancy force and moments are calculated due to heel, pitch and sinkage of the hull. In addition the viscous drag is calculated by the wetted area and the form-factor, due to the ITTC57 drag coefficient calculation.

For the velocity prediction of a hydro-foiled vessel, an additional hydrodynamic force module has been defined. Hereby the main focus was set on the flight-height dependent struts – daggerboard and rudder – and the foils together with the control mechanism. An additional penalty drag module has been written for adding drag due to heel, pitch and leeway angle. The main focus here was stabilising the trim- and the balance algorithm, but realistic values can be set due to the use of response functions.

UserRudder / centreboard (struts): For the calculation of the side force and drag of the rudder and the centreboard, almost the same routines had been implemented. The difference therefore is the consideration of the rudder angle for the calculation of the lift and the drag. For the prediction of the lift and drag forces, 2-dimensional lift and drag coefficient-curves have to be defined and are recalculated to 3-dimensional lift and drag force by the aspect ratio. In addition spray and parasitic drag coefficients can be declared. Input values are the same for both types of struts and are summarised in Table 2-2.

Table 2-2: Input data centreboard and rudder

TopLE Top leading edge point of the centreboard (CB) / rudder

Length Strut lengthThíckness Thickness in % of the chord lengthChord Chord length of CB / ruddervCoE Vertical centre of effort as distance from geometric

CoA normalized by immersion vs. immersion normalized by lenght

alpha Forward angle of CB / rudderfAReff Factor for calculating effective aspect ration (oswald

factor e included) vs. immersion normalized by length+ Foil CoP2D Centre of pressure of foil (% chord)

cL Lift coefficient vs. abs(angle of attack) cD Drag coefficient vs. abs(angle of attack)cDSpray Spray drag coefficient cDPara Parasitic drag coefficient

Options Debug Additional output in command windowUserTrim Use trim variable instead of rudder angle (rudderFoil

only)

Rudder- / centreboard foil: Other than the struts, the lift and drag curves are implemented for the whole foil / wing, giving the possibility to calculate the data with 3-D

17


panel codes, ranse codes or test data. Thus no surface piercing can be regarded – this should be prevented by setting the right boundaries, as this is not an aspired condition. The curves have to be defined as 3-D response surfaces, with the lift-/drag-coefficient versus the pitch and the control angle – the flap angle or the rotation angle of the whole foil. Difference between the the rudder-foil and the centreboard-foil force module is the implementation of the feedback control system within the centreboard foil module, and a trim-/control variable for the rudder foil. Therefore the modules are named as ControlSysFoil and TrimVarFoil. In addition wave drag, intersection drag and parasitic drag can be considered. The summary of the input data for the ControlSysFoil module is shown in Table 2-3. For the TrimVarFoil module, all data concerning the feedback control system (Sensor, Robe, Flap) can be neglected. Instead a trim-variable for the foil-control is defined.

Table 2-3: Input data ControlSysFoil / TrimVarFoil

+ Foil CoP Centre of pressure – equiv. acting point of pressure forces

Planform Plan-form area of foilAReff Effective aspect ratioThickness t Thickness in % of chord lengthChord c Chord lenghtcL 3-dim. response function (cL vs. angle of attack and flap

angle)cD 3-dim. response function (cD vs. aoa and deltaFlap)cDwave Wave drag coefficientcDInter Interference drag coefficientcDPara Parasitic drag coefficientcritAoA Critical angle of attack (warning message option only)

+ Sensor CoR Centre of rotation – connection point of sensorLength Length of sensorDiam Sensor diameter (constant)DragCoeff Drag coefficient of sensor geometry (rod → approx. 0.5)

+ Robe DistCoR Distance robe attachment to CoR (lever arm)Tension Tension force

+ Flap sensorDistCoR Distance to flap attachment at sensorDeltasensor_0 Angle offset between flap attachment and sensorfDeltaFlap Reduction factor between sensor and flap (DSD/DSFL)FlapDistCoR Distance flap attachment to CoR at flapdeltaFlap_0 Flap angle at 0° sensor angle (minimum angle)

Options Debug Detailed output for debug purposeWarning messages Warning if flap angle exceed specified max. angleUse_FlapRespFctn Option for using flap response function – flap angle vs.

speed and z-offset – instead of predefined sensor

18


Penalty drag module: As already mentioned, the penalty drag module is mainly done for stabilising the balance algorithm. The calculation is done using a separate response function drag coefficient versus the specific angle. The reference area is a response function versus the sinkage. The minimum drag condition can be established using heel and pitch parameters defined as reference angles. As a result of using this extra parameters the entire drag curve will shift. Due to the use of response function, measured data for the dependency of the heel, pitch and yaw on the drag can be defined. Here also a specific wave drag, dependant from heel, pitch, yaw and sinkage can be defined by its components, in case measured data is available.

The explanation of the predefined modules and the mathematical implementation of the user defined modules are listed in the Appendices C.1 and C.2 .

2.3.2 Input Data

A major task within this work was the determination and derivation of the input data. In the opinion of the author, the validation and prediction of more accurate data and therefore better and more realistic results is one of the main tasks that should be performed in the future. On this basis more realistic force modules could be implemented.

The data for this work is derived from various resources. For a brief overview the different approaches are shortly discussed. A detailed description is given in Appendix C.3 . The description is separated into the three different force groups.

Gravity forces:

• Moth mass: Data for the MassModule was derived by using an FE-model of a hydro-foiled Moth. Hereby the properties where defined in accordance to mass data researched from the internet or predicted by the total mass. Masses of the different parts, the centre of gravity and the mass moment of inertia has been calculated by the pre-/post-processor ALTAIR Hyperworks.

• Crew (MassMove2D): The mass of the sailor is set to a weight of 80 Kg. Position is set to a fixed height in the body fixed coordinate system, predicted from the geometry of a Moth, to 0.4 meters. The position in x and y direction is flexible. For the calculation of the mass moment of inertia, the body was split in easy to calculate geometrical bodies – tube, box, bowl – for which the mass moment of inertia can be predicted and recalculated to the centre of gravity of the sailor [8].

Aerodynamic forces:• FWindage: The prediction of the windage data is done using the raw data from

Beaver and Zseleczky [7]. Hereby the lift force from the hull is subtracted from the forces components to predict the drag coefficients. These are constant for all courses. The lift-force of the hull is neglected at this stage. For further investigations this force components should be predicted and modelled, as this is a significant side force, especially for upwind courses. Principal data can be found in [9]. The reference area is predicted by the basic geometrical dimensions of the sailor and the boat.

• FRig: For the calculation of the aerodynamic sail data, more information is necessary. On one hand, this are the lift and drag curves for the sail, but also the centre of effort

19


and the basic geometrical data from the rig and the boat needs to be given. For the prediction of the lift and drag curves, two sets of data have been used. First set is the data given for a yacht, DYNA from the TU-Berlin, documented by Hansen [9], which has been transformed the the upwind course data, provided by Chris Williams, calculated with the north sails panel code FLOW. This seems to be a good approach as no whole sail-polar is measured so far. The centre of effort is back calculated from the FLOW raw data, and fit to the curvature of the DYNA data from [9] . All other data is measured from actual boat designs.

Hydrodynamic forces:• Buoyancy force: Hull offset data is imported from a CAD file. The design is made in

many discussions by Juryk Henrichs and the author, for the basic shape and finalized by Juryk in 2008. Design data is given in Appendix C.3.6. As no reliable wave drag data is available, a “high” form factor of 1.3 as been used for the frictional drag. Wave drag is not implemented at this stage. Therefore the influence of the heel, pitch and yaw, as well as sink would have to be considered and would exceed the scope of the report.

• Struts (centreboard / rudder): The struts – rudder and centreboard – are defined by the geometry data and the lift and drag characteristic. This is calculated in 2D with XFLR5, a C based and GUI supported version of XFOIL. This data is internally recalculated due to the change in geometry by the flight height into 3 dimensional lift and drag coefficients and lift and drag forces. Section geometry is set fixed to NACA 63-012 for the centreboard and NACA 0012 for the rudder. Coefficients for the additional drag components are predicted by Beaver and Zseleczky [7], on the basis of Hoerner in [10] and [11], or scaled with the overall forces also provided in [7].

• Foils: Other than the struts, the lift and drag data of the foils is already calculated as 3D data with the wing module of XFLR5, in which the 3 dimensional characteristic can be predicted with an implemented 3D panel code. The foil section is set to a NACA 64-412 for the centreboard foil. Most rudder-foils also have a unsymmetrical foil section, but in the opinion if the author, also a down-force is needed for high speed foiling. Therefore a symmetric NACA 63-012 section is used. The shape of the foil was set to semi-elliptical shape for all foil geometries. The shape and profile is an aspect for foil optimization, but not considered in this work.

• Feedback control system: The characteristic of the feedback control system is calculated with the already explained preliminary foil design tool. Here also the plan-form area is defined. For the main foil an initial area of 0.11 m2 is used which will be varied. The plan-form area for the rudder is set to 0.08 m2.

Before using FS-Equilibrium for the velocity prediction, the implemented routines were verified, using the input data. The signs of the forces had been checked for a first overview. Afterwards the absolute values were evaluated by plotting the selected force components versus the true wind speed (TWS) and the true wind angle (TWA). All validation is printed in appendix C.4.

20


2.4 Velocity Prediction and Result Interpretation

A typical form of the velocity prediction is the polar plot. Hereby the data, commonly the ship speed is plotted versus the TWA. Plotting different wind speed in one polar plot, gives a very good overview on how the boat reacts at different conditions. A typical polar plot for the speed of a Moth is shown in Figure 2-6.

In the case of the hydro-foiled Moth, the change in speed is a clear indicator for the craft reaching foiling state. Different foil / -control configurations can now be analysed and compared to each other. Not given by the velocity prediction is the take off behaviour of the Moth. Assuming that the drag of the Moth is maximized shortly before take off and will drop to a certain level after take off, the VPP does not give the answer if the Moth can actually reach the foiling state at the given course and wind speed. This must be done with a in-stationary

21

Figure 2-6: Moth polar plot


simulation, considering the sailor position and trim changes during take off. A simple approach has been done with the preliminary Moth deign tool, which will also be taken for defining different foil set-up's.

2.4.1 Variation

It is unlikely to find the best set-up with the criteria for the foil-system in a single approach. Rather getting a feeling of the influence of the parameters, that can be changed and optimised during the design process. Therefore a initial foil-system has been defined.

As already discussed in the previous chapter, some parameters, which, by themselves, would be worth an optimisation, are taken as fixed values. Main focus is set onto the control system and the plan-form area of the main foil. Also the plan-form area of the rudder should be optimised, but is also set fixed at the beginning, as this characteristic can easily be updated at a later stage.

Initial plan-form is set to 0.11 m2 which can be taken as a very common value as shown in [7]. The shape was set to a straight tailing edge and a elliptical leading edge. When changing the plan-form area, the aspect ratio is kept constant.

According to actual Moth design, the take-off speed is set to 3.5 m/s. Modern Moth start foiling at a wind speed of 2-3 Bft., which is approximately 3 to 4 m/s. Assuming that the Moth can be sailed as fast as wind speed for non-foiling, this leads to an take off speed of 3.5 m/s.

The first calculations have been done using one set-up for the feedback control system and changing the plan-form area of the main foil. This set-up is “optimised” for the initial foil plan-form area of 0.11 m2 and the predicted take off speed of 3.5 m/s. To get a good feeling how changes will in the plan-form area will effect the behaviour of the Moth, the area is set to 0.08 m2 and to 0.14 m2 in further calculations.

While estimating the values for the control system, one effect becomes clear. For reaching the foiling state a higher driving force is needed as the actual drag force when just reaching flying state. Therefore a drag-bump must be resolved, shorty before take off. As the drag force versus the time is also plotted in the preliminary foil design tool, this effect is clearly visible and illustrated in Figure 2-7.

22

Figure 2-7: Drag vs. time - default control system 0.11 m2 main foil

0 10 20 30 40 50 60 700

10203040506070

Total drag vs. time

Time [s]

Drag

forc

e [N

]


An experienced sailor would change course to get a higher driving force, get airborne and head back when foiling.

For the next calculations a separate set-up for each plan-form area is created. Hereby it will be assumed that, while the take off speed for the default area of 0.11 m2 is 3.5 m/s, the take off speed will be affected as well. Therefore the take off speed was set to 4 m/s for the 0.08 m2 foil, and 3 m/s for the 0.14 m2 foil. It is obvious that this will lead to different drag forces, as the plan-form area is included linearly in the equation for calculating the drag force, and the speed by the square. For getting an overview over all set and calculated values, these are summarized in Table 2-4. The full set of parameters is listed in Appendix C.3 . Here also the velocity-/ offset plots are documented.

Table 2-4: Foil system set-up

Parameter / Condition Foil set-up 01 Foil set-up 02 Foil set-up 02

Est. take off seed 3.5 m/s 4.0 m/s 3.0 m/s

Est. total drag at take off 64.0 N 69.6 N 60.47 N

+ Foil Plan-form 0.11 m 0.08 m 0.14 m

+ Sensor Length 0.9 m 0.9 m 0.9 m

+ Robe Tension 10 N 10 N 10 N

+ Flap DeltaSensor_0 35° 35° 35°

fDeltaFlap 0.125 0.140 0.118

deltaFlap_0 -5° -5° -4°

Looking at the calculated values for the total drag at take off speed shows the clear influence of the speed onto the drag. Now it seems to be an advantage of having a big centreboard foil, as the drag is much lower for the big foil. But the size of the foil will have an effect onto the maximum speed that can be reached. Using the same spread sheet, but setting a constant driving force, shows the difference for the maximum speed. The driving force was set well above the force needed for take off to 80 N.

Table 2-5: Max. speed and flight height at constant driving force

Plan-form

Set-up

0.11 m2

01

0.08 m2

02

0.14 m2

03

Driving force 80 N 80 N 80 N

Max. Speed 5,38 m/s 7,18 m/s 4,98 m/s

Flight height 0.58 m 0.62 m 0.55 m

Time to take off 9 s 13 s 8 s

23


The corresponding results indicate, that a foil system is always a compromise between take off speed and maximum speed. It must be considered that the values in Table 2-3 can not be regarded as absolute values, as not all drag components and pitch influence are included in the calculations. Also the limits like stall flap angle and crew position are not regarded by this calculation. Restrictions in the calculation are discussed in more detail in Appendix B .

Taking the three different control system configurations and the three plan-form areas, leads to seven variations. Table 2-6: Variation of plan-form area vs. foil set-up

Plan-form area Foil set-up 01 Foil set-up 02 Foil set-up 03

0.11 m2 X X X

0.08 m2 X X

0.14 m2 X X

2.4.2 Results

As the feedback control system is mainly set up by the take off condition, first glance is at a TWS of 3.5 and 4 m/s where all set-ups have their first flight condition. As has been mentioned the foil with a plan-form area of 0.14 m2 will first tend to take off, Figure 2-8 shows that this is different. For the reference foil with a plan-form area of 0.11 m2 a first flight condition is found at a TWS of 3.5 m/s and a TWA of 105°. The 0.14 m2 foil follows at 85°. All other set-ups have their first take off condition at a TWS of 4 m/s and a TWA of 110 ± 5°.

At this stage it should be mentioned that finding the equilibrium states is done with a variation of solver settings. Due to the significant change in drag in foiling state, values change over a big range. This may lead to problems for the solver while finding the optimum condition. Therefore one solver or at least the settings for the solver might be good for foiling state, but do not converge in low rider mode. Thereby the take off stage is always critical as values varies by the biggest range and results may vary by ±5°.

Another point that surprises is the behaviour of the feedback control system. Leaving the plan-form area and changing the control system leads to a slightly different behaviour at take off and “landing”, while top speed and low rider speed is as good as not affected. This behaviour is visible at all true wind speeds. One reason therefore are the minor changes at the different control systems, mainly focusing on a similar take off condition. Hereby the factor defining the ratio between the sensor wand control wire and the main-foil pushrod was mainly adjusted together with the minimum flap angle.

Although the plan-form area shows a clear visible effect on the speed at low-rider and foiling condition. Having a bigger foil reduces the speed in low-rider and in foiling state, but also decreases the take off wind speed. Here the first theoretical foiling conditions are possible at a TWS of 3.5 m/s for the plan-form area of 0.11 m2 and 0.14 m2 while the foil system with a main foil of 0.08 m2 does starts foiling at a 4 m/s. It's also very clear that by using similar geometric dimensions of the sensor wand, the maximum reachable speed seems to depend by the plan-form area only. Also visible is the circumstance, that bigger foils tend to stay

24


longer in foiling condition while heading windward at low wind speeds as shown in Figure 2-8.

Considering the progression of the heel angle versus the TWA shows, that while heading windward the heel to weather increases, as could be seen in Figure 2-9 (positiv heel windward).

25

Figure 2-8: Moth Polars for take off condition, a) TWS = 3.5 m/s, b) 4 m/s)

Figure 2-9: Heel angle vs. TWA for 0.11 m2 / 0.08 m2 / 0.14 m2 main foil


This effect occurs while the remaining strut can not, or at least not with minimum drag, create enough side force due to the reduction of the wetted surface area with increasing flight-height of the Moth. While heeling windward, a significant horiziontal side force component can be created by the lift force of the foil, in fact by the sinus of the heel, but also the vertical lift component will be reduced by the cosine (Figure 2-1). Together with delimited range of crew position and the angle of the rudder foil control surface, not enough lift will be available while heading windward.

As this effect can also result from a badly designed feedback control system, the heel angle should be considered in the preliminary foil design.

For high wind speeds the foiling range is, with some exception, the same. According to the polars in figure Figure 2-10, the foil with the plan-form area of 0.08 m2

seems to have a clear advantage in comparison to the bigger foils.

Using the VPP FS-Equilibrium has shown some difficulties with the mathematical formulation of the force modules as well as the solver settings. Problem hereby is optimisation of four trim-variables, which, on one hand, have a relatively wide range, and on the other hand are interfering with each other. Thereby MassMove2 and Flat are counteracting as well as MassMove1 and deltaFlap. MassMove1 tends to have a more backward position, which seems to be better by means of maximising the speed.

26

Figure 2-10: Moth polars - travel- and max. speed (TWS = 6 m/s / 10 m/s)


By considering three different plan-form areas and three different set-up for the feedback control system, even the changes at the feedback control system are minor, some clear trends can be found:

• Take off condition: Foils size seems to have an optimum between a) 0.08 m2 and b) 0.14 m2 – where take off speed will be reached at a later stage due to a) too small plan-form area and b) to much drag.

• Foiling range at low wind speed: Small foils do not have enough vertical lift at upwind courses due to increase of heel angle.

• Cruising speed, high wind speeds: small foils have a better performance by means of speed at all courses.

• Feed back control system: small changes (deltaFlap_0, initial flap angle, and ratio) effects take off speed, travel and maximum speed are hardly effected. For changing the performance at high wind speeds, the sensor should also be changed.

27

3 Stability

3 StabilityKnowing the equilibrium conditions for a set of TWS and TWA shows the best velocity for the Moth, but it does not show anything about how stable the moth performs at this specific conditions.

Using an approach from aircraft design, the stability of an aircraft is measured by two types of stability, the static stability and the dynamic stability. The static stability shows the tendency of the craft to return to its equilibrium position. Three conditions of the static stability can be distinguished – stable, indifferent and unstable. This three conditions are sketched in Figure 3-1.

In addition to the static stability, the craft must also be dynamically stable. A craft can be considered dynamically stable if, after being disturbed from its equilibrium condition, the ensuing motion diminishes with time [12].

For the prediction of the static stability of an aircraft, the motions can be divided into longitudinal and lateral motion, where only the desired force of each plane can be considered. Looking at the force balance of a hydro-foiled Moth in Figure 2-1 shows, that, while having a big side force component at the sail, the influence of the yaw, drift and heel on the longitudinal motion is much bigger than at an aircraft. Therefore the prediction of the stability has to be done three dimensionally, which would exceed this work and the brain of the author. Instead the numerical possibilities of FS-Equilibrium will be used.

With the use of FS-Equilibrium there are two ways for investigating the stability at the found equilibrium states – the prediction of the dynamic stability with the linearised equations of motion and the small disturbance theory and by solving the equations of motion in a time dependant scheme with the in-stationary mode.

The equations of motion can be obtained from Newtons second law, which states that the summation of all external forces acting on a body is equal to time rate of change of the momentum of the body, and the summation of the external moments acting on the body is equal to the time rate of change of the moment of momentum (angular momentum) [12]. This can be expressed in the vector equations:

∑ F=d

dtm⋅v

∑M=ddt

r X v ⋅m(1)

28

Figure 3-1: Static stability conditions

3 Stability

3.1 Small Disturbance Theory

Basis for the use of the small disturbance theory is the assumption, that the motion of the craft consist of small deviations from a reference condition of steady motion. With this assumption the equation of motion can be linearised and be brought into the following form [13]:

x t =A⋅x t b⋅u t (2)

Where x(t) is the vector of the state variables or state vector, u(t) is the control vector and A and B are the linearised system matrices at the specific condition.

The solution of this first order differential equation is of the form

x t =X 0⋅e⋅t (3)

Where x0 is an eigenvector and λ is an eigenvalue of the system. The solution contains of n eigenvalues and eigenvectors whereby n is the number of the degree of freedom of the system. Since any of the eigenvalues can provide a soultion, and the equation is linear, the most general solution is the sum of all x(t):

x t =∑iX 0i⋅e

i⋅t (4)

The eigenvalues λ contains from a real and a complex part. The conjugate pair of the eigenvalues describes an oscillation mode that, depending on the sign of real part of the eigenvalue, grows or decays. Thereby it can bee seen that the question of stability can be answered just by the sign of the real part of the eigenvalues. If all real parts are negative, the system is stable. For the prediction of the handling quality of an aircraft not just the qualitative characteristics are of importance, but also the quantitative. For an aircraft these are the period T, the time to double or half the amplitude and the number of cycles to double or half the amplitude.

Using FS-Equilibrium for the prediction of the stability, the equations of motions can be linearised at every equilibrium condition. The result of the linearisation is shown in the output window in the form of the system-matrix (A), the trim matrix (B), the control vector b and a noise matrix describing the change in aerodynamic values, as well as the eigenvalues and the eigenvectors of the system.

Together with the eigenvalues, the decay time, the damping and the period is directly given in the output window. The output of an arbitrary condition is shown in Figure 3-2. Another benefit of FS-Equilibrium is the ability of directly visualising the reaction of the boat due to an applied disturbance within the motion viewer. Therefore some predefined motions will be applied to the system and the response will directly be calculated at the linearised condition.

29

Figure 3-2: Eigenvalue output in FS-Equilibrium

3 Stability

Looking at the eigenvalues it is obvious that the system is unstable as eigenvalue 9 has a positive sign. Using the motion viewer of FS-Equilibrium to analyse the response due to different perturbations it becomes clear, that the heel is the unstable degree of freedom (DoF). This is obvious as a foiled Moth does not have any self-stabilising moments. Making a new linearisation with a system decoupled from the heel (disable of heel as degree of freedom), the system turned out to be stable as all real parts of the eigenvalues are negative or zero.

The response of the conditions can also be plotted over the time. Figure 3-3 shows the response ot the pitch due to an disturbance of rising the pitch angle from 0.23° to 1.23°, Figure 3-4 shows the same for the z-coordinate and an offset from -0.627 m to -0.527 m. Both curves show that the amplitude diminishes – hence the system reacts stable to a disturbance in pitch and z when the heel is fixed.

30

Figure 3-3: Pitch response ∆pitch = 1°

Figure 3-4: Z-response for ∆Z = - 0.1 m

3 Stability

While these graphs are showing very clearly the damping and oscillation of the response, the creation is very time consuming. The same results can be achieved by comparing the eigenvalues of the different systems. FS-Equilibrium directly calculates the decay time, the damping coefficient and the period if an imaginary part exists. Looking at the eigenvalues of the above configuration in Figure 3-5, most eigenvalues have a very short decay time and even no imaginary part , hence no oscillation at all.

For a Moth only those modes are of interest, which are actually effecting the sailor. Therefore the short period and highly damped motions are not used for comparison of the set-ups.

The investigation has been done with a set of preselected conditions. The first set affects the take off condition. This might be the most unstable condition as the boat is just out of the water and not a high reserve by the driving force is given. Next condition will be set to cruising speed, as this is the most common sailed condition. For cruising condition a higher wind-speed is considered, a speed of 6 m/s seems to be a very common condition. Thereby two courses has been taken into account, a fore wind course at a TWA of 120° and a upwind course at 60° TWA. Also considered for those two courses is a high wind speed of 10 m/s.

3.2 In-stationary Simulation

In addition to the small disturbance investigation, the in-stationary simulation has been performed, in which the motions are described via the equations of motions including all external forces, calculated by their respective force modules. The equation of motion will not be linearised, as by using the small disturbance theory, but solved due to the change of the condition. Hereby FS-Equilibrium gives a range of possibilities for controlling the behaviour of the craft. Manoeuvring simulations for instance operate with changing rudder angles. Thereby a fixed course can be set, or may also be controlled by the use of predefined manoeuvres. A PID-controller is implemented to keep the desired coarse.

As during the stationary simulation the trim-variables will not be updated automatically, some restriction have to be done. As already mentioned at a earlier stage, a Moth in low rider mode have limited, in foiling mode no possibilities of stabilising itself in transversal direction. The Moth must be levelled out by the crew. As transverse stability is not of interest for the evaluation of the foil- and the feedback control system, the heel will be set to a fixed value, which is given through the initial condition, knowing that the side-force is influenced by the heel.

Using the auto-pilot to keep to keep the moth on the initial course, the plot of the speed, the pitch and the sinkage as in Figure 3-6 is given as output.

31

Figure 3-5: Eigenvalue output with fixed heel

3 Stability

Comparing the result of the dynamic response of the small disturbance theory of the shown initial condition and the in-stationary simulation shown in Figure 3-6, the condition seem not very stable at a first glance. Speed is dropping instantly by approx. 37.5 % of the initial speed, as well as the pitch changes from 0.23° to 1.8° and down to -2.4°. Sinkage also increases (height drops) by approx. 36%. But the craft tends to get stabilised as, after first dropping, the speed increases and pitch as well as sinkage converges. This condition seems to be stable by means of static stability, as the system shows the ability to return to its initial condition.

If the chosen conditions converge, the trim variables can be updated by a predefined value, in case of the investigation of the longitudinal motion and stability, the trim-variable MassMove1, the x-position of the sailor, and deltaFlap, the rudder-foil control angle. The change of this two variables will be done, creating either an positive or an negative pitch angle.

As the the size of the change is dependant on the boundaries of the trim-variable, the disturbance is set to about 5% of the range. This is approximately a ∆X of ± 0.1 m for the crew and / or ± 1° for the flap angle. Using the in-stationary simulation from Figure 3-6, the disturbance will be applied at approximately 30 s of simulation time with an macro, so both changes are applied together. The result of the disturbance look like Figure 3-7.

32

Figure 3-6: Output of in-stationary simulation - speed / pitch / sink vs. time

Figure 3-7: In-stationary simulation with disturbance

3 Stability

Starting with the initial condition, the speed drops first to a certain level, but the motion is stabilizing and the speed increasing. Hereby the pitch and the sinkage converges as the amplitude drops over the time and tends toward a fixed value. Giving a disturbance at 30 seconds, here negative movement of the crew plus decreasing of the lift at the rudder-foil, the pitch drops instantly and the speed is reduced due to the increase in drag of the foils. Its clearly visible that the disturbance destabilises the Moth or at least, cannot be stabilised within the given boundaries as the sinkage tends toward the draft of the canoe body and speed drops below take-off speed.

For the consideration of other inertia effects than the pure mass moments of inertia, an additional mass- and damping matrix can be defined with an extra module within FS-Equilibrium. If the behaviour shows such an instant drop from the initial condition, other damping effects should be considered, and the option of defining this extra damping matrix should be used.

Basis for the conditions of the in-stationary simulation are the conditions used for the small disturbance theory. As the simulation and evaluation of all conditions used for the small disturbance investigations would be very time consuming, the critical conditions are used for verifying the results from the small disturbance theory. These are the diverging conditions in the first place. For a cross-check, the condition with the minimum damping ratio, hence the longest decay times, should also be investigated.

3.3 Stability Investigation and Results Interpretation

Starting with the already mentioned conditions for the investigation of the stability by means of dynamic stability, the resulting eigenvalues show, that especially the upwind conditions are showing an unstable condition. Thereby the destabilizing degree of freedom is the rotation in z direction. Although the rudder, as well as the heel is influencing the side forces, the values will be assumed to be fixed while the influence of the feedback control system is of interest.

Performing the stability investigation with constrained heel and yaw, the number of unstable conditions dropped from 13 to 3. So the majority of the investigated conditions seem to be stable.

As the main criteria for defining the parameters of the feedback control system was the take-off condition, first closer look is set on this results. Hereby just the 0.14 m2 foil with the set-up 01 (optimized for 0.11 m2) shows an instability with quite a long decay time compared to the other amplitudes. The results are summarized in Table 3-1.

Table 3-1: Results dynamic stability – take off condition

Identifier Plan-form/

Set-Up

Condition (TWS [m/s] /

TWA [°])

Eigenvalue Decay Time Damping Period

Real Imaginary

P0.11 / F.01 3.5 / 105 -0.267 0.00 2.592 1.00 0.00

P0.11 / F.02 4.0 / 115 -0.326 0.00 2.127 1.00 0.00

P0.11 / F.03 4.0 / 115 -0-267 0.00 2.013 1.00 0.00

P0.08 / F.01 4.0 / 115 -0.337 0.00 2.049 1.00 0.00

33

3 Stability

P0.08 / F.02 4.0 / 115 -0.344 0.00 2.021 1.00 0.00

P0.14 / F.01 4.0 / 110 0.012 0.00 -59.223 -1.00 0.00

P0.14 / F.03 3.5 / 85 -0.230 0.00 3.013 1.00 0.00

Including the other investigated conditions, two more unstable conditions at TWS of 6 m/s and a TWA of 60° can be identified. Hereby the foils with a plan-form area of 0.08 m2 and 0.11 m2 have a positive real part in its eigenvalue. As this conditions are both using foil set-up F02, one conclusion could be, that this set-up is either not suitable for the plan-form areas or not well predicted at all. But it is remarkable, that these foil set-up are stable at a TWS of 10 m/s, leading to the conclusion, that the trim configuration, maximising the speed, by itself is unstable.

This unstable conditions have been taken for a closer investigation. By using the in-stationary simulation within FS-Equilibrium, the results from the dynamic stability investigation can be validated.

For each of the unstable set-up, a stable configuration will be used for a direct comparison. This might make it possible to get a direct link between a specific variable and the unstable behaviour. For the unstable set-up at a TWS of 6 m/s the corresponding condition at 10 m/s will be used. The unstable configuration at take off, which is P014F01 (plan-form 0.14 m2, foil set-up 01), has been compared with P014F03, which shows a quite similar condition.

The chosen set-up's and conditions are confronted in Table 3-2. The unstable conditions are marked with light grey, below, in white, is the compared, dynamically stable condition.Table 3-2: Used set-up's / conditions for in-stationary simulations

ID TWS TWA Vs Heel Pitch Lee Sink Rudder Mass-Move1

Mass-Move2

delta-Flap

Flat

[m/s] [°] [m/s] [°] [°} [°] [m] [°] [m] [m] [°] [-]

P014F01 4 110 5.34 2.11 0.23 1.21 -0.6 -1.59 -1.46 -0.72 6.18 0.89

P014F03 4 115 5,92 2.05 1.13 1.4 -0.63 -2.06 -1.66 -0.9 6.28 1.

P011F02 6 60 4.73 5.4 -1.84 2.47 -0.52 -2.54 -1.21 -1.4 3.59 0.65

P011F02 10 60 6.83 8.5 0.61 1.61 -0.61 -3.33 -1.26 -1.4 2.43 0.28

P008F02 6 60 4.69 7.99 -1.82 1.95 -0.43 -2.52 -1.21 -1.3 3.59 0.65

P008F02 10 60 6.97 7.41 0.75 1.69 -0.6 -3.73 -1.63 -1.4 3.37 0.27

Comparing the conditions from TWS = 6 m/s and 10 m/s shows, that the pitch angle has a difference of over 2 degrees, for take off condition still approx. 1 degree. Having a small or negative pitch angle seems to de-stabilise the Moth. Assuming that the condition by itself is stable and shows a static stability, the disturbance should be applied in a “nose up” way, which seems to be more conservative.

34

3 Stability

Performing an in-stationary analysis shows, that all conditions and set-ups considered for the simulation are unstable. For the prediction of the stability the sinkage is plotted versus the time as shown in Figure 3-8.

This can also be done with the pitch, showing the same results, while the ship speed might be misleading, as heel and yaw are not regarded in the stability investigation, and therefore tend to be higher as the predicted results from the velocity prediction. The plots shown in Figure 3-8 are indicating the unstable behaviour, as flight height drops immediately from the initial height to more or less zero meters, hence low rider condition.

These results shows that a dynamically stable condition does not automatically denote that the condition is also static stable. This has already been mentioned by Cook [14] for the prediction of the stability of an aircraft. For a hydro-foiled Moth, the dynamical behaviour according to the small disturbance theory, seems to play a minor role in comparison to the static stability, therefore an in-stationary simulation has to be performed.

Leaving the question if the set-up of the foil system is badly chosen or the condition which might lead to the maximum speed is unstable by itself. Therefore the initial foil system is investigated in the preliminary defined five conditions – TWS 6 / 12 m/s at upwind and downwind and at take-off condition.

The upwind and take off condition of the stability reacts like in the already discussed conditions – highly unstable. The height and the speed immediately drops to low-rider condition while the pitch rises very fast. In contrast the down-wind conditions behave very stable. Also applying a disturbance at approximately 10 s simulation time will keep the system stable. Hereby a disturbance of 1° rudder foil angle is applied, in either plus or minus. The sailor position was kept constant, as already positioning at the rear end and no further

35

Figure 3-8: Ship speed vS vs. time

3 Stability

distortion can be applied to the range limitation. For the reference foil system this is demonstrated in Figure 3-9 for the sinkage versus the time. Same behaviour can be seen at other chosen foil systems configurations.

The decrease of sinkage can be drawn back on the increase in speed due to constrained heel and yaw. Looking at the speed shows an instant increase from initial to a maximum speed before falling back to a higher level in accordance to the sinkage.

The results from the stability investigations regarding the static and dynamic stability can be outlined by the following points:

• Dynamic stability with constrained heel and pitch achieved in most cases

• Proof of static stability given by in-stationary simulation in a mathematical sense.

• Dynamic stability might be given for static unstable conditions.

• Up wind force equilibrium conditions predicted by maximising the boat speed seemed highly unstable.

• For down-wind courses a high grade of stability can be achieved.

36

Figure 3-9: Dynamic response – sink vs. time

4 Evaluation

4 EvaluationUsing the program FS-Equilibrium for the prediction of the velocity and the stability shows clearly, that the prediction of the parameters for the feedback control system is a central point for the design of a foil system. The main focus hereby is set on an optimised set of parameters for a given plan-form area of the main foil, meeting the requirements for reaching take off at a predefined take off speed. Also considered is the cruising speed and the maximum speed, which has to suppress the tendency for the main foil to break through the water surface instantly destroying lift.

Hereby the preliminary foil design tool, described in Appendix B, was not just helpful, it was absolutely necessary, as the prediction of the parameters could not have been handled properly in another way. Using the tool during the process of velocity prediction also showed, how effective the tool is working in a design process, but also showed the limits of the tool. At the stage of the preliminary foil design, some values had to be estimated, concerning the conditions defined as design criteria. The take off speed plays a major role for the definition of the parameters. Other parameters influencing take off, is the pitch angle at take off and the lift ratio, defining the load on the main foil. The load on the main foil is defined by the ratio of lift at the main foil in relation to the over all lift, which has to level out the weight of the crew and the boat. Cruising and maximum speed have been regarded as, more or less, a result from the foil system a pitch angle of 0° and 100% load on the main foil.

By means of these preliminary defined values, the results of FS-Equilibrium can now be discussed in correlation to the estimated results from the preliminary design tool.

Take off speed: The initial take off speed for the reference foil system with a plan-form area of 0.11 m2 is set to 3.5 m/s. This value results from the known take off wind speed of actual designs and the assumption, that a moth can be sailed near to wind speed. Also assumed is the circumstance that a foil with a smaller plan-form area needs a higher wind speed for take off, while a bigger foil needs a slower one. Therefore the take off speed for the 0.08 m2 foil is set to 4 m/s and for the 0.14 m2 foil to 3 m/s.

Plotting the sinkage, or flight-height for the use of positive values, as an indicator for take off versus the boat speed shows the actual take off condition. Thereby the actual take off speed of the three foil configurations in Figure 4-1, predicted with FS-Equilibirum, show a good accordance with the assumed values from the preliminary foil design tool. The predicted values are slightly higher, which may results from additional drag components, not taken into account by the preliminary design tool.

As outcome it can be accepted, that the take off behaviour is mainly controlled by the set-up of the feedback control system in accordance with the plan-form area of the main foil.

37

4 Evaluation

Lift ratio: Together with the take-off speed, the lift ratio of the main-foil relative to the total lift has been set. As a starting point this value was set to a constant value of 65%. Looking at the lift ratio of the foiling conditions in Figure 4-2 suggests, that the assumption of a fixed value for all foils is not best chosen as the results vary between 80% and 50%.

The bigger foils seem to have more load on the main foil with an average value of 65% to 70%, the small foil has approx. 60% with a rising tendency for the higher speeds. As the rudder foil for the small foil creates enough lift to carry up to 50% of the load, it should be

38

Figure 4-1: Take off speed vs. speed for 0.11 m2 / 0.08 m2 / 0.14 m2 main foil

Figure 4-2: Lift ratio vs. speed for 0.11 m2 / 0.08 m2 / 0.14 m2 main foil

4 Evaluation

reduced in size for the bigger foils. This is also mentioned by Beaver and Zseleczky [7], as the main foil is able to produce enough lift. Therefore the assumption, that the main foil carries up to 100% of the load can not be confirmed. The lift ratio does hardly reach more than 80%.

Pitch angle: The last assumption made for the preliminary foil design is the pitch angle for creating enough lift during take off and at high speeds. This value is not easily verified, especially for take off, as the dynamic take off behaviour of the craft is not simulated at this stage. It's known that sailors “pump” the boat out of the water by dynamically decreasing the pitch (nose up) to shortly create more lift, while, due to inertia, speed can be seen constant. Still the pitch angle of the three foil set-ups, plotted in Figure 4-3 versus the boat speed, shows, that the pitch tends to be in a nose up position for take off, while being in a positive pitch range in stationary conditions.

Cross-checking the pitch, the lift ratio and the crew position of the three set-ups may lead to the assumption, that the range of the main foil flap angle, is not well chosen. Looking at the crew-position in Figure 4-4 shows, that the sailor tends to be at the stern for an optimised boat speed. This increases the load on the rudder foil, while the load on the main foil, and hereby the induced drag, is decreased by a positive pitch, showing that the minimum flap angle is to high.

Also using Figure 4-4 it is visible, that the crew positions is already at the stern for relatively low boat speeds, while for the bigger foil, the crew positions varies in a wider range. This can be interpreted, that more combinations of conditions and trim-variables are possible for reaching a near to maximum velocity. Using this effect, it should be able to reduce the range of the sailor movement and therefore the influence of the sailor.

Another point affecting the pitch is the robe tension, which controls the break away moment of the sensor wand. Increasing the robe tension and therefore keeping the flap in minimum

39

Figure 4-3: Pitch vs. speed for 0.11 m2 / 0.08 m2 / 0.14 m2 main foil

4 Evaluation

position may decrease the positive (nose down) pitch as the foil will have a lower induced drag. This may lead to a “harmonized” pitch distribution over the speed range, also reducing the movement of the sailor and / or control surface at the rudder foil.

Outlining the results from the velocity prediction in comparison to the preliminary foil design tool, shows the advantage of using such a tool in a early design stage but also the restrictions. One major point of using the design tool is to put a stronger focus on the maximum speed. This should also be done using the time dependant simulation by setting a initial condition (speed, sinkage) and a fixed driving force, which can be estimated by the maximum speed. Hereby the flap-range, hence Centreboard.Foil.deltaFlap_0 and the sensor wand control wire – pushrod ratio for a good take off behaviour can be verified.

According to the velocity prediction, the foil design tool should be extended by the following points for a better understanding of the foil system in the preliminary design stage:

• The heel should be considered in the preliminary design stage. According to Figure 2-9 (page 25) the heel changes with the course. For up-wind courses the side-force of the sail increases. By heeling windward, the foils create a significant horizontal side-force, which can compensate the side-force of the sail, but also the vertical lift component will be decreased. By considering the heel, the decrease of lift will be taken into account and foiling state can be extended at up-wind courses.

• For the evaluation of the maximum speed, the driving force as well as the initial condition for the time dependant calculation should be user definable. Hereby the range for the flap angle and the reduction factor at the pushrod, set for take off condition, can be verified.

• Pitch moment should be considered. For a given pitch angle either the crew position for a given lift ratio or the lift ratio for a given crew position can be back calculated.

• By calculating the pitch moment, control surface angles for the rudder-foil can be predicted and plan-form area optimised.

40

Figure 4-4: Long. crew position vs. speed for 0.11 m2 / 0.08 m2 / 0.14 m2 main foil

4 Evaluation

Concerning the stability by means of static and dynamic stability, it is clearly visible that dynamic stability is reached more easily than static stability. With constrained heel and yaw most conditions are dynamically stable. But the static stability seems to be more of importance, as the results shows a completely different behaviour.

Thereby the in-stationary simulation shows that especially the up-wind conditions are highly unstable. This shows that not only the performance by means of speed is important, but also a stability investigation is necessary. For the proper design of a foil system the stability, especially the static stability, should be considered at an early stage.

Basis for this kind of investigation can be a stability consideration as made for an aircraft. Starting at an equilibrium point, the components should create an upright moment when changing the pitch to either side – nose up or nose down. This behaviour is sketched in Figure 4-5.

Using this criteria in aircraft design shows, that the gravity force should act in front of the lift force for a wing only design. Considering a similar behaviour for the Moth, the reason for the in-stability of the moth could be found in an equilibrium condition, where the crew position is at the stern of the Moth. As the moments of the driving force and the drag forces must also be considered within the moment equation, the best crew position might be not in front of the main foil, but tends to be in a more forward position than predicted by the VPP for best velocity.

Altogether the velocity prediction with an additional stability investigation is an appropriate method for the evaluation of the performance of a hydro-foiled moth. But also conditions near the optimum, concerning the velocity, should be considered in any further investigation. The variation of the results shows, that many different combination of conditions / trim-variables are possible to achieve a good solution. Thereby many solutions, which may be more stable by means of static stability are not considered in the stability investigation. This result may also be achieved by setting appropriate boundaries. These boundaries can be predicted by theoretical thoughts concerning stability, the already calculated results and by the experience of current Moth sailors.

41

Figure 4-5: Static stability criteria for a craft

5 Conclusions and Future Work

5 Conclusions and Future Work Using a VPP for the prediction of the performance of a sail-boat is a common approach within yacht-development and improvement of a given design. Therefore additional force modules have been programmed for the use within the open modular workbench FS-Equilibrium.

For using a VPP, the parameters influencing the the foil system and especially the feedback control system have been evaluated and the influence on the forces mathematically formulated for the implementation in the different force modules. While defining the force modules, it became clear, that additional forces should be taken into account for further investigations. This is especially necessary when the experience of Moth sailors should be considered for the definition of more realistic ranges for the trim variables and conditions.

Especially the wave drag from the hull is a missing force component. This should have a major influence on the take-off condition. As this force is dependant of at least 4 degrees of freedom, this will be a substantial task. Another force component that should be included is the aerodynamic lift of the hull/frame. The drag component is considered within the windage force module, but the prediction of the drag coefficients showed, that there is also high side force due to aerodynamic lift at upwind courses. Other components are the drag at the sensor wand, the influence of the pressure field of the struts onto the pressure field of the foils and more. Also the flap hinge moment at the main foil can influence the working way of the feedback control system and is not considered at this stage.

The formulas for the calculation of the lift at the main foil were the basis for the definition of a design criteria which consist of the four stages for a hydro-foiled Moth – low rider, take off-, cruising- and maximum speed. These four stages and the mathematical formulation have been used for creating a preliminary foil system design tool, for the simulation of the take off behaviour and prediction of the cruising and maximum speed for a set of given parameters. Reducing the problem to two degrees of freedom – speed and sinkage – good results can be achieved for the take-off condition.

Also some tendencies have been found, which define the best conditions for maximising the speed. Hereby it is clearly visible that the heel angle has a major influence, especially at upwind courses with a high side force component. Therefore the heel should be taken into account within the preliminary foil design. For the prediction of cruising condition and maximum condition, the tool should be extended by setting a user defined start condition, which contains the speed, sink and a fixed pitch.

Hereby the initial foil system with a plan-form area of 0.11 m2 shows the best over all perofamce – low take off speed, good range in foiling mode at lower wind speeds and only a little lower maximum speed. Using the consideration made by the results of the velocity prediction in chapter 2.4, it should be possible to reduce the influence of the sailor by updating the parameters of the foil system. Therefore a final analysis has been made with a slightly smaller foil. The following values has been changed:

• MassMove1 (crew x-position) is set to a range of -0.25 to -0.75 m

• Plan-form area decreased to 0.1 m2

• deltfaFlap_0 at the main foil is -10°

42


• reduction factor adjusted according to the range of the flap angle at the main foil to 0.16

• Robe tension is increased to 25 N

• Sensor wand increased to 1.0 m while reducing the diameter to 5 mm.

The polars in Figure 5-1 show that by updating the foil system a quite similar behaviour of the moth can be achieved, even as the crew position will only change in a range of 0.5 m. This gives the designer the chance to built a foil system, which is far better to handle. Just the take off at a TWS of 3.5 m/s will not be reached with the modified foil. But “low rider” boat speed has been improved by the lower minimum flap angle and higher robe tension.

Looking at the stability of the equilibrium conditions shows the necessity to consider these at an early stage within the design process. Hereby the small disturbance theory shows good results with constrained heel and yaw. Static stability, assessed by an in-stationary simulation, showed that this criteria is hardly reached. Just down wind courses can be seen as statically stable. Therefore a criteria, which can be applied directly while designing the foil system, should be evaluated.

43

Figure 5-1: Moth polars - reference vs. updated foil system


As the use of a VPP is standard for yacht design, it can only be seen as a basis for the design of a hydro-foiled Moth. For an optimised design, more criteria than best speed on different courses are necessary. According to the stability investigations, made in chapter 3, no foil system, on the theoretical bases made in this work, could be found which reaches static and dynamic stability on an up-wind course. On the other hand, modifying a foil system that reaches similar performance by means of speed is much easier to achieve. Therefore a major focus should be set on the development of a stable foil system and the required stability criteria.

It can be assumed that an experienced sailor may achieve a higher speed with a more unstable Moth, but still - a slow foiling Moth will be faster than a unstable capsized one.

44

Tables and References


List of Figures

Figure 1-1: Aspects influencing the stability of a hydro-foiled Moth.........................................6Figure 1-2: Principle function of the foil system [3]..................................................................8Figure 2-1: Force balance of a hydro-foiled Moth..................................................................11Figure 2-2:Sensor force balance...........................................................................................12Figure 2-3: Geometrical relation between sensor ans flap angle...........................................12Figure 2-4: Boat speed and z-offset vs. time plot..................................................................14Figure 2-5: Angle offset DeltaSensor_0 ................................................................................15Figure 2-6: Moth polar plot ...................................................................................................21Figure 2-7: Drag vs. time - default control system 0.11 m2 main foil ....................................22Figure 2-8: Moth Polars for take off condition, a) TWS = 3.5 m/s, b) 4 m/s)..........................25Figure 2-9: Heel angle vs. TWA for 0.11 m2 / 0.08 m2 / 0.14 m2 main foil............................25Figure 2-10: Moth polars - travel- and max. speed (TWS = 6 m/s / 10 m/s)..........................26Figure 3-1: Static stability conditions.....................................................................................28Figure 3-2: Eigenvalue output in FS-Equilibrium...................................................................30Figure 3-3: Pitch response ∆pitch = 1°..................................................................................30Figure 3-4: Z-response for ∆Z = - 0.1 m................................................................................31Figure 3-5: Eigenvalue output with fixed heel........................................................................31Figure 3-6: Output of in-stationary simulation - speed / pitch / sink vs. time..........................32Figure 3-7: In-stationary simulation with disturbance.............................................................33Figure 3-8: Ship speed vS vs. time........................................................................................35Figure 3-9: Dynamic response – sink vs. time.......................................................................36Figure 4-1: Take off speed vs. speed for 0.11 m2 / 0.08 m2 / 0.14 m2 main foil....................39Figure 4-2: Lift ratio vs. speed for 0.11 m2 / 0.08 m2 / 0.14 m2 main foil..............................39Figure 4-3: Pitch vs. speed for 0.11 m2 / 0.08 m2 / 0.14 m2 main foil...................................40Figure 4-4: Long. crew position vs. speed for 0.11 m2 / 0.08 m2 / 0.14 m2 main foil.............41Figure 4-5: Static stability criteria for a craft..........................................................................42Figure 5-1: Moth polars - reference vs. updated foil system..................................................44Figure B-1: Influence of flap angle on lift curves....................................................................54Figure B-2: cL vs. angle of attack - analytical approach........................................................55Figure B-3:Preliminary foil design – FS-Equi input data........................................................56Figure B-4: Preliminary foil design -common input data........................................................56Figure B-5: Condition spread sheet input data .....................................................................57Figure B-6: Boat speed and z-offset vs. time plot..................................................................58Figure B-7: Drag vs. time plot................................................................................................58Figure C-1: FS-Equilibrium GUI ............................................................................................59Figure C-2: Solution sequence FS-Equilibrium [6].................................................................61Figure C-3: Instationary manoeuvring simulation with FS-Equi [6]........................................62Figure C-4: Offset data definition in FS-Equi buoyancy module............................................64Figure C-5: Bladerider dimensions........................................................................................75Figure C-6: FE-model property region plot ..........................................................................77Figure C-7: Geometrical bodies for estimation of mass moment of inertia............................79

45


Figure C-8: Aerodynamic force gage axes [BeZc09].............................................................81Figure C-9: Wind velocity ratios.............................................................................................82Figure C-10: Centre of effort rudder / centreboard................................................................84Figure C-11: Vertical centre of effort distribution....................................................................85Figure C-12: Polars - NACA 0012.........................................................................................86Figure C-13: Polars - NACA 63-012......................................................................................86Figure C-14: NACA 0012 cL vs. AoA – cD vs. AoA................................................................87Figure C-15: NACA 63-012 cL vs. AoA – cD vs. AoA.............................................................87Figure C-16: NACA 62-012 foil section with flap....................................................................88Figure C-17: Daggerbord foil (wing) definition in XFLR5.......................................................88Figure C-18: Graphical output of XFLR5...............................................................................89Figure C-19: Daggerboard foil cL vs. flap angle and AoA......................................................89Figure C-20: Daggerboard foil cD vs. flap angle and AoA.....................................................89Figure C-21: Rudder foil definition in XFLR5.........................................................................90Figure C-22: Rudder foil cL vs. flap angle and AoA...............................................................90Figure C-23: Rudder foil cD vs. flap angle and AoA..............................................................90Figure C-24: Daggerboard T-Foil drag components..............................................................92Figure C-25: Rudder T-foil drag components.........................................................................92Figure C-26: Centre of area moth sail - North Sails, Chris Williams......................................93Figure C-27: Sail lift curve cL vs. AWAeff..............................................................................94Figure C-28: Sail aspect ratio................................................................................................95Figure C-29: Sail drag curve cD vs. AWAeff..........................................................................96Figure C-30: Sail centre of effort vs. AWAeff.........................................................................96Figure C-31: Offset definition of moth hull by Juryk Henrichs................................................97Figure C-32: Curve of sectional area of hull..........................................................................97Figure C-33: Lift and drag at main foil vs. sinkage.................................................................99Figure C-34: Lift and side force vs. heel and speed..............................................................99Figure C-35: Lift at foils vs. pitch.........................................................................................100Figure C-36: Rudder lift and drag vs. deltaFlap ..................................................................100Figure D-1: Speed / sinkage vs. time set-up F01.................................................................101Figure D-2: Speed / sinkage vs. time set-up F02.................................................................102Figure D-3: Speed / sinkage vs. time set-up F03.................................................................102Figure D-4: Speed / sinkage vs. time set-up F04.................................................................103Figure E-1: Moth polar plan-form area 0.11 m2 – set-up F01..............................................104Figure E-2: Moth polar plan-form area 0.11 m2 – set-up F02..............................................105Figure E-3: Moth polar plan-form area 0.11 m2 - set-up F03...............................................106Figure E-4: Moth polar plan-form area 0.08 m2 - set-up F01...............................................107Figure E-5: Moth polar plan-form area 0.08 m2 - set-up F02...............................................108Figure E-6: Moth polar plan-form area 0.14 m2 - set-up F01...............................................109Figure E-7: Moth polar plan-form area 0.14 m2 - set-up F03...............................................110

46


List of Tables

Table 2-1: Parameter names of feedback control system within FS-Equilibrium....................14Table 2-2: Input data centreboard and rudder......................................................................17Table 2-3: Input data ControlSysFoil / TrimVarFoil.................................................................18Table 2-4: Foil system set-up.................................................................................................23Table 2-5: Max. speed and flight height at constant driving force..........................................23Table 2-6: Variation of plan-form area vs. foil set-up..............................................................24Table 3-1: Results dynamic stability – take off condition .......................................................34Table 3-2: Used set-up's / conditions for in-stationary simulations.........................................35Table C-1: Standard link of degree of freedom to free variables [6].......................................60Table C-2: Weight estimation of a International Moth............................................................77Table C-3: Density of materials..............................................................................................77Table C-4: FE-weight due to estimated properties ...............................................................78Table C-5: Mass moment of inertias of basic bodies.............................................................80Table C-6: Proportion of weight of a human body [8].............................................................81Table C-7: Centreboard and rudder data...............................................................................84Table C-8: Reynolds numbers for centreboard an rudder.....................................................85Table C-9: Additional drag coefficients...................................................................................92Table C-10: Moth sail data FLOW – Chris Williams, North Sails............................................95Table C-11: Main parameters of hull......................................................................................99Table C-12: Mathematical signs and tendencies of force modules........................................99Table D-1: Geometrical parameter for feedback control systems........................................102Table F-1: Linearisation results for one fixed DoF (heel)......................................................112Table F-2: Linearisation results for one fixed DoF (heel, yaw).............................................113

47


Nomenclature

aoa Angle of attack

AR, AReff Aspect ratio, effective aspect ratio

AWA, AWAeff Apparent wind speed, effective apparent wind speed

β, βeff Apparent wind angle, effective apparent wind angle

Bft Beaufort

cD Drag coefficient

cL Lift coefficient

CoE Centre of Effort

CoR Centre of rotation

Cs Separation constant

DCB, DCBF Drag centreboard, drag centreboard foil

∆δFl Actual flap angle from minimum flap angle

∆δS0 Angle offset sensor wand attachment

deltaFlap Trim-variable flap angle at rudder foil

δFl Actual flap angle from neutral position

δFl0 Minimum flap angle from neutral position

DoF Degree of Freedom

DR, DRF Drag rudder, drag rudder foil

δS Sensor angle to vertical

DSail Driving force sail

∆sCW Actual travel of control wire from zero position

∆sFl Actual travel of flap / pushrod from zero position

DWL Design water line

e Oswald factor

εA Aerodynamic efficiency

εH Hydrodynamic efficiency

f Reduction factor from control wire travel to pushrod travel

FHydro Hydrodynamic force (sensor)

FR, FRobe Robe tension force

48


ϕ Heel angle

k Form factor

λ Leeway angle

LCB, LCBF Lift centreboard, lift centreboard foil

lCW Distance control wire attachment to sensor CoR

lFl Distance flap attachment to flap CoR

LH,CBF/RF Horizontal component of lift force at centreboard-/rudder-foil

lR Distance robe attachment to centre of rotation at sensor

LR, LRF Lift rudder, lift rudder foil

lS Sensor wand length

LV,CBF/RF Vertical component of lift force at centreboard-/rudder-foil

LV,dra Dry vertical length (sensor)

Lvim Vertical immersed length

m Mass

MCoR Moment at centre of rotation (sensor)

MR Moment due to robe tension force

MS Sensor wand moment due to hydrodynamic drag

ν Kinematic viscosity

Re Reynolds number

SSail Side force sail

TWA True wind angle

TWS True wind speed

VMG Velocity made good

VPP Velocity prediction program

Vs Ship speed

WSA Wetted surface area

49


References

[1] Wikipedia, “Moth (dinghy)”, http://en.wikipedia.org/wiki/Moth_%28dinghy%29, 2010.

[2] Wikipedia, “Sailing Haydrofoiled”, http://en.wikipedia.org/wiki/Sailing_hydrofoil, 2010.

[3] Schmidt, D., “Learning to Fly”, www.sailmagazine.com, 2009.

(http://www.sailmagazine.com/racing/learning_to_fly)

[4] Speer, T. E., “Aerodynamic of Teardrop Wingmasts”, Des Moines, Washington, USA.

[5] Abbott, I. H., VonDoernhoeff, A. E., “Theory of Wing Sections”, Dover Publications Inc., 1959.

[6] Richards, T., Harries, S., Hochkirch, K., “Maneuvering Simulations for Ships and Sailing Yachts using FS-Equilibrium as an Open Modular Workbench”, Papar, FS-Systems, Potsdam.

[7] Beaver, B., Zseleczky, J., “Full Measurement on a Hydrofoil International Moth”, 19th Chesapeake Sailing Symposium, March 2009.(www.moth-sailing.org/download/CSYSPaperFeb09.pdf)

[8] Saß, Dr. M., “Moment of Inertia”, Eperiment tutorial part 1, Physik Department, TU-Munich, Munich, 2004.

[9] Hansen, H., “Enhanced Wind Tunnel Techniques and Aerodynamic Force Models for Yacht Sails”, Phd-Thesis, University of Auckland, Auckland, New Zealand, 2006.

[10] Hoerner, S. F., “Fluid-Dynamic Lift”, Published by Author, 1975.

[11] Hoerner, S. F., “Fluid-Dynamic Drag”, Published by Author, 1965.

[12] Nelson, R. C., “Flight Stability and Automatic Control”, McGraw-Hill, Inc., USA, 1989.

[13] Edkin, B., Reid, L.D., “Dynamics of Flight, Stability and Control”, John Wiley and Sons, Inc., New York, Toronto, 1996.

[14] Cook, M. V., “Flight Dynamic Principles”, Second Edition, Elsevier, Amsterdam, Boston, Heidelberg, 2007.

[15] Marchaj, C., A., “Aero- Hydrodynamics of Sailing”, 3rd Edition, Tiller Publishing, St. Michaels, 2000.

50

http://www.moth-sailing.org/download/CSYSPaperFeb09.pdf

http://www.sailmagazine.com/racing/learning_to_fly

http://en.wikipedia.org/wiki/Sailing_hydrofoil

http://en.wikipedia.org/wiki/Moth_(dinghy)

Appendices

Appendices

A Feedback control system (mathematical approach) The feedback control mechanism is driven by a sensor wand attached at the bow, by which the flap angle at the foil is set due to speed and altitude of the flying moth. The mechanical concept is shown in Figure 1-2.

The sensor angle δD can be computed with the sum of moment at the centre of rotation (CoR). The drag-moment must be the same as the moment from the robe attached at the top of the sensor, hence:

M CoR=M RM S=0 (5)

MCoR is the moment at the centre of rotation, MR the moment due to robe tensions and MS the sensor moment.

The length of the robe is assumed to be big in comparison to the distance to the centre of rotation. Therefore the moment can be calculated with a constant force FR:

M R=−cos S⋅l R⋅F R (6)

The sensor moment is related to the drag force of the sensor and its lever arm. This can be expressed by the vertical dry length of the sensor lVdry, the sensor rotation angle δS and the hydrodynamic relevant data - the diameter DS, the drag coefficient cD and the speed v.

M S=12⋅l S

2⋅cos2S −lVdry⋅cD⋅2⋅v2⋅DS (7)

Hereby the vertical dry length of the sensor can be written as the quotient of the vertical position of the centre of rotation and the cosine of the heel angle.

With the robe moment MR the result is the following quadratic equation:

cos2S −2⋅l R⋅F R

l S⋅cD⋅2⋅v2⋅DS⋅lS

⋅cos S −lVdry

2

l S2 =0 (8)

The sensor angle is limited by the vertical position of the sensor. This condition is given if the initial brake away moment is not reached or the sensor does not touch the water surface due to sink, heel and pitch. This means mathematically, that the robe moment is bigger than the sensor moment and can be expressed by

14⋅l S

2−lVdry2 ⋅cD⋅

2⋅v2⋅DS lR⋅F R (9)

To control the flap angle, the flap is connected at the distance lFl to the vertical pushrod, which is also connected to the sensor wand control wire at the foil control bell crank on top of the daggerboard. The sensor wand control wire is attached at the distance lS from the centre of rotation.

51

Appendices

For a more sensitive reaction of the sensor at “fly height”, it is state of the art methodology to attach the sensor cable with an angle offset to the sensor ∆δS0. The influence of the angle offset shown in Figure 2-5 on page 15. Hereby the distance ∆SS, given by the sensor, can be expressed by

sS=l s⋅sin S−S0l s⋅sin S0 (10)

To compensate the difference in the connection length of the control cable at the sensor lS

and the Flap lFL, the distance ∆SD is reduced by a factor f which is the quotient of the turning radius R2 and R1. At a moth this will be the connection point of the sensor cable and the vertical pushrod in the daggerboard as shown in Figure 2-3 on page 12.

The flap angle can now be computed by

Fl=Fl0Fl with

tan Fl =S Fl

lFl=

f⋅ S S

lFl (11)

Now the lift and drag coefficient can be determined with the angle of attack and the flap angle δFL as a function of speed and altitude.

The lift and drag coefficient is implemented as a 3-dimensional response function, with the δFL and the angle of attack (α) as dependant variables.

B Preliminary foil system design toolThe characteristic of the feedback control system is influenced by many parameters, which are interacting each other. This makes it hard to achieve a proper set-up for the feedback control system. Therefore a spread-sheet had been programmed, in which the interaction of the basic parameters can be simulated.

Basis of this simulation is the assumption that a constant driving force is given, independent from the ship speed. This driving force is estimated by the take off condition. For calculation of the sink and the ship speed, only the x and z forces are considered. The drag components are reduced to the major components, hull drag, foil drag and strut drag. The induced drag of the struts is neglected, as in ideal condition a moth can be sailed with minimum leeway angle.

Starting point is the calculation of the driving force. For a target take off speed and the lift at take off speed, which is the draft of the hull, the flap angle of the main foil could be calculated with the formulas given in Appendix A. Now the lift coefficient of the main foil had to be estimated. The lift coefficient of a symmetric foil can be calculated due to Speers [4] by:

a =dc L

=

a0

1180

⋅a0

⋅AReff⋅e

with AReff⋅e = AR⋅fAReff (12)

Here a0 is the lift curve slope for an 2 dimensional analysis. As most main foils have a non-symmetric shape, the lift coefficient at zero angle off attack had to be added to the lift coefficient.

52

Appendices

The influence of the flap on the lift coefficient can be calculated by comparing the lift curves of different flap angles. The lift curve slope a can be determined by freeware panel codes, here XFLR5 from Andre Deperrois, a c++ programmed and gui supported version of the famous XFOIL from Marc Drela.

As could be seen from the curves in Figure B-1, the influence of the flap angle could be regarded as linear. This made it possible to calculate the lift coefficient with a flap angle, scaled by the factor fδ to be directly converted to an additional angle of attack.

The over all lift coefficient can now be determined by:

c L = cL0a0

1180

⋅a0

⋅AReff⋅e

⋅ f ⋅Flaoa (13)

Where fδ is the factor for scaling the flap angle which had been calculated by

f =aoa Fl , c L=0−aoa Fl=0, cL=0

Fl (14)

Figure B-2 shows the corresponding curves for a NACA 63-412 foil with the necessary values.

53

Figure B-1: Influence of flap angle on lift curves

Appendices

Next the parameters of the feedback control system could be approximated, with a resulting lift at the main foil is approx. 60% to 70% of the over-all weight. The additional lift must be achieved by the rudder foil.

The drag force of the foil is calculated by

cD = cD0 cL

2

⋅AReff⋅e (15)

With the back calculation of the lift coefficient at the rudder foil, the total Drag of the Moth can be calculated at take off condition, by adding the rudder strut/foil drag and the centreboard strut/foil drag.

For the calculation of the hull displacement and wetted surface area, two polynomial fitted curves for WSA vs. sink and displacement vs. sink had been implemented in the spread sheet. This is necessary to calculate the lift force and the resistance of the hull as a function of ship speed and sinkage. As the wave drag of a hull is hard to estimate, a hight form factor K~1.3 had been used, to take wave drag into account.

Now the take off behaviour can be calculated by a time dependant scheme. For a given condition at t = 0 s the over-all lift and drag forces are calculated and subtracted from the weight and the driving force at take off speed. As the force is equal to the product of mass and acceleration, the acceleration, and by integration over the time, the speed and the offset in z direction could be calculate, leading to a new condition for the next time step. Result is the speed and the altitude versus the time.

54

Figure B-2: cL vs. angle of attack - analytical approach

3,11 1,41

Appendices

For the use within the spread sheet a input mask was created. The first part ,shown in FigureB-3, is for defining the parameters concerning the feedback control system. This data could also be exported into a macro by an additional sheet .

55

Figure B-3:Preliminary foil design – FS-Equi input data

Figure B-4: Preliminary foil design -common input data

Appendices

The additional data, concerning the calculation of the drag components, will be defined in the common data block in Figure B-4. This includes the lift characteristic of the main foil, as this will be implemented as a 3 dimensional response function within FS-Equilibrium.

After the definition of the control system and the common parameters, the different conditions have to be defined. Therefore 4 lines are reserved within the spread sheet where the different speeds, trim angles and offsets have to be set. As an Result the flap angle deltaFlap and the lift in N is given.

“State 0” is the non foiling condition. This line is for checking the flap angles at different speed and altitudes, that no separation or stall might occur. The next line is important, defining the take off condition. At this condition the driving force is calculated. Here the take off speed and the draft should be set. Depending on the foils, a pitch angle of about 1 to 3 degrees might be necessary. The next 2 lines are for checking the flight conditions at travel and defined maximum speed at which hight the boat will be levelled out.

Next part is the definition of a time step, where the values range is about 0.1 to 0.25 seconds, and a damping coefficient to prevent oscillation when choosing a higher time step.

For the calculation of the total lift and drag the fraction of the lift at the main foil has to be set. This value is set for a starting point to 65%.

The result is a graph showing the speed of the boat and the z-offset (altitude) vs. the time. As not all drag components are considered, the absolute values might differ from reality. But its assumed to be a good approach to understand the influence of the different parameters and to get a set good starting values. As the driving force is also calculated without those missing drag components, the error can be assumed to be small.

56

Figure B-5: Condition spread sheet input data

Appendices

As an additional information the total drag is plotted vs. the time. Its shows clearly how the drag rises until take off, the dropping to certain level before raising again due to increase of speed.

C FS-EquilibriumThe description of FS-Equilibrium had been already made for different papers and thesis, hence the following description is taken from Maneuvering Simulations for Ships and Sailing Yachts using FRIENDSHIP-Equilibrium as an Open Modular Workbench by Tanja Richardt, Stefan Harries and Karsten Hochkirch, with some minor modifications [6].

57

Figure B-6: Boat speed and z-offset vs. time plot

Figure B-7: Drag vs. time plot

Appendices

FutureShip-Equilibrium (formerly FRIENDSHIP-Equilibrium) is an advanced workbench for the analysis of stationary and instationary modes of motion of both ships and sailing yachts. The external forces acting on a vessel for a given state are calculated via various force modules. Each force type like buoyant forces, gravitational forces, rudder forces, keel forces, hull resistance, aerodynamic forces, added resistance in waves, windage etc. is calculated in specific modules. All forces are added up by the program to determine the resulting forces on the vessel.

These modules can be added to and taken out from the simulations individually depending on the design task at hand. A wide selection of force modules is already available. In addition any kind of force acting on the vessel can be defined in a high level language such as FORTRAN, C or C++ and introduced at run time via individual modules. In the new versions also a C++ based API programming can be used for adding GUI supported force modules. The modular structure thus allows for the integration of user provided modules. A prominent application for such a user specific adaptation is the Real-Time VPP at the Twisted Flow Wind Tunnel (TFWT) in Auckland were a force module links the aerodynamic forces measured by the wind tunnel’s balance to the program, see Hansen et al. (2003).

Figure C-1 shows an example window of FRIENDSHIP-Equilibrium when used for a velocity prediction of a sailing yacht. The added modules are declared in the ’Input Modules’ window.

58

Figure C-1: FS-Equilibrium GUI

Appendices

Active modules are marked. Forces which should not be summed up can be declared non-active. The states to be calculated are defined in the ’Cycle Range’ window. For the given cycle range the equilibrium conditions are output in the ’Velocity Prediction’ table. In the force window, the forces of all modules are displayed for the selected state.

For all calculations the desired degree of freedom can be chosen by the user. Up to all six degrees of freedom may be considered in the simulations. The standard link between the degrees of freedom and the free variables is summarized in Table C-1.

Table C-1: Standard link of degree of freedom to free variables [6]

Free Variable Condition Comment

Symbol Name Equilibrium

1 Vs Ship speed ΣFX= 0 basic forces along the centerline, e.g. resistance andaerodynamics, propeller thrust

2 ϕ Heel angle ΣMY = 0 heeling and righting moments

3 λ Leeway angle ΣFY = 0 aerodynamic side force to be compensated by hydrodynamiccomponents

4 δr Rudder angle ΣMZ = 0 yawing moment to be equalized

5 θ Pitch angle ΣMY = 0 trimming moments to be compensated

6 dz sinkage ΣFZ = 0 vertical forces – especially by adding loads as additionalcargo, crew, gear or water ballast – as well asdynamic effects to be compensated by additional sinkage

7 δt Trim tab angle ΣFY = 0 trim angle to be set to a value giving optimum speed

Figure C-2 illustrates the structure of FutureShip-Equilibrium. The used modules and the required parameters are defined in one or several input files. The acting forces are determined for specified environmental conditions. Depending on the force modules these conditions are part of the required input. For instance the force calculation on sailing yachts requires the wind speed and the wind direction as input.

59

Appendices

Three simulation modes are offered by the program for different applications:

• stationary mode

• hydrostatic mode

• instationary mode

In the stationary mode the steady state of the ship will be determined for specified environmental conditions. The program will resolve the equilibrium in which the sum of all forces add up to zero in the defined degrees of freedom by a nonlinear equation solver. The balance will be computed by means of a Newton-Raphson algorithm. Velocity predictions in steady conditions may also be calculated within this mode. Depending on the considered degrees of freedom, the output contains the state variables displayed in Figure C-1.

For the analysis of motions the program offers an instationary mode. For instationary analysis the excitation forces have to become part of the input. Manoeuvring simulations for instance operate with changing rudder angles. Fixed rudder angles can be set. The rudder angles may also be controlled by the use of predefined manoeuvres A PID-controller (Proportional Integral Derivative) is implemented as autopilot to keep a desired course. An additional manual maneuvering module offers the possibility to steer the boat interactively with a joystick. Either way, steering changes are accounted for while running the process.

The motions are described via the equations of motions including all external forces, calculated by their respective force modules. Additional added mass and damping forces are determined via a further force module and are then considered as part of the excitation force.

60

Figure C-2: Solution sequence FS-Equilibrium [6]

Appendices

A module for linear coefficients of added mass and damping is readily available. Alternatively, an additional module which describes these forces can be defined by the user. Nonlinear coefficients as used by Masayuma et al. (1995) for instance, may be implemented in more advanced modules. The accelerations are calculated by solving the equation of motions via a time-step method with a chosen integration routine. FutureShip-Equilibrium currently makes available a fourth-order Runge-Kutta scheme, a fifth-order Runge-Kutta-Feldberg scheme as well as a fifth-order Cash-Karp Runge-Kutta variable time step scheme. In the variable time step integration the time step is adjusted so that the difference of the result using a fourth order Runge-Kutta scheme and the result using a fifth order scheme is less than a selected tolerance for each of the state variables. A time scale can be set such that the maneuver is executed in real time or a specified fraction of that. The process flow of the time-stepping procedure is sketched in Figure C-3. All state variables are computed as functions of time. The trajectory is displayed in global coordinates. Moreover, any desired state variable can be plotted during the operation.

C.1 Predefined ModulesWithin FS-Equilibrium a lot of predefined as well as experimental force modules are available. In the following chapters the modules and the input data will shortly be described. Explanation to the implemented modules are taken from the the module description within FS-Equilibrium.

C.1.1 BuoyancyForce moduleThe BuoyantForce Module describe the buoyant forces of submerged components of the vessel. The total displacement of the vessel can be expressed as one or more BuoyantForce modules. The geometry of a component is described in an offset file

As an option the hydrodynamic mass and damping terms can be computed by means of a strip theory approach using conformal mapping with lewis transformation of the heeled hull.

61

Figure C-3: Instationary manoeuvring simulation with FS-Equi [6]

Appendices

The module provides the private shared keys Volume, Lwl, Bwl, WPA, WSA, XCB, YCB, ZCB, XWL, YWL, CB, CP, AM, ALat, Draft and the global key ':TotalVolume', the latter being the sum of all previous Volumes.

Input Data:

+ Edit Offset Spec

OffsetGroup Offset data

Coomands Command sequence to achieve the desired offset group

Transform Transformation of the offset group

Import Import geometric objects (IGES, SHF)

FormFactor Form factor if viscous forces shall be approximated

Fluid Fluid properties

Impermeability Impermeability of the compartment. 1 for all closed and 0 for totally flooded compartments when computing damage cases via the lost buoyancy method.

Plate Thickness

If defined an additional displacement is added depending on the wetted surface

Added Mass Compute added mass matrix for the body

WaveHeight Wave hight as a function of x position (stability effects due to stationary wave system)

WaveScale Factor to scale wave height

SactionalDrag Sectional drag coefficients (computation of viscous damping based on a strip theory approach)

+ Options DEBUG

Bohlmann

Munk

ViscDamp

NoWLIntersection

The definition of them hull is done by section-points which can be imported directly or be created by an imported surface within the module. The imported hull can be visualized as shown in Figure C-4.

62

Appendices

C.1.2 FRig HansenThe FRig module calculates the aerodynamic force components acting at the sail centre of effort. The FRig module uses lift, parasitic drag, longitudinal and vertical centre of effort in a standardized form so that the size of a rig with the same or similar proportions can be scaled without modifying the input curves. The sails are de-powered with the trim parameter TWIST and EASE (FLAT) as introduced by Hansen et al.. The trim parameter REEF is also included in the model to describe the 'physical' reefing. It should only be used manually for reducing the sail area but it should not be an active trim parameter.

Input Data:DWL Design waterline length of yachtMastHeight Height of mast above waterBoomHeight Height of boom above WL (for REEF param. only)SailArea Reference sail areaCentreOfArea Centre of area (rel. to DWL)Efficiency Increase of induced resistance compared to ellip. loadingSeparationCon Separation constantTwistWeight Influence of twist parameterTwistWeightLift Influence of the twist parameter on lift

63

Figure C-4: Offset data definition in FS-Equi buoyancy module

Appendices

Sopt Optimum position of CoE as fraction of mast heightFluid Fluid property to useCL Lift coefficient vs. effective wind angle (EWA)CDp Parasitic drag coefficient vs. effective wind angle (EWA)XCE Longitudinal centre of effort relative to CentreOfArea,

normalized by DWL vs. EWAZCE Centre of effort height relative to CentreOfArea, normalized

by MastHeight vs. EWAREEF Trim Parameter REEFFLAT Trim Parameter FLATTWIST Trim Parameter TWIST+ Options Debug

Euler

C.1.3 FWindageThe windage module calculates the parasitic aerodynamic drag from the drag coefficients in the three planes and the projected areas in the three planes. The centre of area is also required so that the moments and the wind velocity at the height of the feature can be calculated. The type parameter indicates if the windage should be included in the sum of the aerodynamic or hydrodynamic forces. It does not have any influence on the performance of the yacht.

Input Data:Area Projected Area in each planeCd Drag coefficient in each planeCentre Centre of areaType Force type (aero- or hydrodynamic)Flui Fluid property to use+ Options Debug

Euler

C.1.4 MassThe mass module allows to calculate the forces due to a point mass located at a point in body fixed coordinates. If dynamic simulations are considered the gyradii referred to the centre point need to be specified as well. Any number of mass items can be included in a model.

The centre may be scaled and translated on input using the keywords 'scale' and 'origin', however only the transformed centre will be saved.

Input Data:Mass The mass in KgCentre Centre of gravity in body fixed coordinates

64

Appendices

Gyradius The gyradii with repect to the centre of gravityCrossProductGyradius The radii of the products of inertia with respect to the

centre of gravity+ Options Debug

Euler

C.1.5 ControlledMass / MoveableMass2DThe mass module allows to calculate the forces due to a point mass located at a point in body fixed coordinates. If dynamic simulations are considered the gyradii referred to the centre point need to be specified as well. Any number of mass items can be included in a model.

The centre may be scaled and translated on input using the keywords 'scale' and 'origin', however only the transformed centre will be saved.

This specific version allows to modify the centre of gravity as expressions of the current condition

Input Data:Mass The mass in KgCentre Centre of gravity in body fixed coordinatesGyradius The gyradii with repect to the centre of gravityCrossProductGyradius The radii of the products of inertia with respect to the

centre of gravity+ Options Debug

EulerdX_Expression Expression for displacement of dX componentdY_Expression Expression for displacement of dY componentdZ_Expression Expression for displacement of dZ component

C.2 User ModulesWithin FS-Equilibrium is the possibility to implement user defined force modules. This could be c, c++ or fortran based programs, as well as a c++ based api language which creates GUI supported force modules. This programming language had been used to add the necessary force modules, needed for the simulation of a hydro-foiled moth.

C.2.1 Centreboard / RudderThe centreboard and rudder module calculate the drag and side forces of a vertical profile with a additional foil at the bottom. The plan-form area and the centre of effort will be recomputed due to the z-offset. Also the effective aspect ratio can be varied over the sinkage. The lift and drag data is calculated out of 2-dimensional data.

65

Appendices

Input data:TopLE Top leding edge point of the centreboard (CB) / rudderLength Strut lengthThickness Thickness in % of the chord lengthChord Chord length of CB / ruddervCoE Vertical centre of effort as distance from geometric

CoA normalized by immersion vs. immersion normalized by lenght

alpha Forward angle of CB / rudderfAReff Factor for calculating effective aspect ration (oswald

factor e included) vs. immersion normalized by lenght+ Foil CoP2D Centre of pressure of foil (% chord)

cL Lift coefficient vs. abs(angle of attack) cD Drag coefficient vs. abs(angle of attack)cDSpray Spray drag coefficient cDPara Parasitic drag coefficient

Options Debug Additional output in command windowUserTrim Use trim variable instead of rudder angle (rudderFoil

only)Calculation:

// CACLULATION OF GEOMETRICAL PARAMETERS

// Top of leading edge (LE) relative to waterline (WL)TopLE.tranformBATopLE [2]+= zOffset

// sensor vertical length and vertical immersionvLenght=lenght⋅cos alpha

vImmers=vLenght−TopLE [2]cos heel ∗sin pitch

// Parameters dependant on altitude – planform area, vertical Centre of effort, spray

drag switch (1 – foiling, 0 – non-foiling) if vImmersvLenght then // lowriderPlan=lenght⋅chordvCoE=0.5−RespFctn vCoE ⋅vLenghtfSpray=0

else // foilingPlan=vImmers⋅chordvCoE=vLenght−vImmers0.5−RespFctn vCoE⋅vImmersfSpray=1

end// CoE, aspect ratio AR

66

Appendices

CoE [2]=TopLE [2 ]−vCoECoE [1]=0CoE [0 ]=TopLE [0]−Chord⋅CoP2DCoE [2 ]⋅sin alpha

AReff =PlanformChord 2 ⋅fAReff=

PlanformChord 2 ⋅ fctnvImmerse

// CALUCATION OF FORCES AND MOMENTS

// Angle of attack (aoa)

aoa= tanlocalFlow CoE [1]localFlow CoE [2]

// For calculation of rudder forces:aoa=aoaRudderAngle

// Lift and drag coefficients (section / induced / spray)c L= fctn aoacD0= fctnaoa

cDInducd=cL

2

⋅AReff// Lift and drag force components

Force [0]=cL⋅roh2

⋅vS2⋅Planform

Force [2]=cD0cDInducedcDParasitic⋅roh2

⋅vS2⋅PlanformcDSpray⋅

roh2

⋅vS2⋅Thickness2

Force [3]=0// calculate moments and transform to global (A) coordinate system

Force.setCentreOfEffort CoE Force.tranform TransformationMatrixBA

Limitations:

• Influence of T-foil pressure field neglected

C.2.2 ControlSystemFoilThe control system foil force module calculates the the lift and drag forces of a sensor controlled T-foil. The default usage is the sensor, described in Chapter 1.1 and the mathematical formulation in Appendix A. As an option, a 3 dimensional control-field, flap angle vs. ship speed and height, can be specified.

Input data:+ Foil CoP Centre of pressure – equiv. acting point of pressure

forcesPlanform Plan-form area of foilAReff Effective aspect ratio

67

Appendices

Thickness t Thickness in % of chord lengthChord c Chord lengthcL 3-dim. response function (cL vs. angle of attack and

flap angle)cD 3-dim. response function (cD vs. aoa and deltaFlap)cDwave Wave drag coefficientcDInter Interference drag coefficientcDPara Parasitic drag coefficientcritAoA Critical angle of attack (warning message option only)

+ sensor CoR Centre of rotation – connection point of sensorLength Length of sensorDiam sensor diameter (const.)DragCoeff Drag coefficient of sensor geometry (rod → approx.

0.5)+ Robe DistCoR Distance robe attachment to CoR (lever arm)

Tension Tension force + Flap SensorDistCoR Distance to flap attachment at sensor

Deltasensor_0 Angle offset between flap attachment and sensorfDeltaFlap Reduction factor between sensor and flap (DSD/DSFL)FlapDistCoR Distance flap attachment to CoR at flapdeltaFlap_0 Flap angle at 0° sensor angle (minimum angle)

Options Debug Detailed output for debug purposeWarning messages Warning if flap angle exceed specified max. angleUse_FlapRespFctn Option for using flap response function – flap angle vs.

speed and z-offset – instead of predefined sensor

Calculation:

// COMMON PARAMETERS

// Lokal flow at centre of pressure in fixed coordinate system (CSYS) and angle of attack

lokalFlow=lokalVelocityB CoP va=lokalFlow [0]2lokalFlow [2]2

aoa=−atanlokalFlow[2 ]lokalFlow[0] // Centre of rotation at current position in absolute CSYS relative to waterline

CoRAWL=CoR.transform TransformationMatrixBACoRAWL [2]=−SINK

// FLAP ANGLE due to speed and zCoR (CoRAWL[2])

// vertical position of CoR in body-fixed system

68

Appendices

vCoRB=CoRAWL[2 ]∣cos heel∣

// Robe- and hydrodynamic initial breakaway torqueM Robe=Robe.distCoR⋅Robe.Tension

M Hydro* =Sensor.cD⋅2⋅va

2⋅12⋅Ditcher.Lenght 2−vCoRB

2

// Check boundariesif vCoRBSensor.lenght thenD=0.

else if M Hydro* M Robe thenD=0.

elseD= fctn va , vCoRB

end if// Calculate δFlap from δsensor

S D=Flap.SensorDistCoR⋅sin D−D0sin D0

Flap=Flap.deltaFlap0atanFlap.fdeltaFlap⋅Delta S D

Flap.FlapDistCoR // FORCES AND MOMENTS

// Lift and drag coefficientsc L=RespFctn aoacD0=RespFctn aoa

cDInduced=c L

2

⋅AReff⋅e=c L

2

⋅fAR⋅AR// Lift and drag forces

lift=cL⋅2⋅va

2⋅Foil.Planform

drag=cD0cDInducedcDWave⋅2⋅va

2⋅Foil.PlanformcDInter⋅2⋅va

2⋅Foil.Thickness

// Force components in absolute CSYSforce [1]=drag

force [2]=liftsin Heel

force [3]=liftcosHeel

// Transform CoP in absolute CSYSFoil.CoP.transform TransformationMatrixBA

// Calculate moments

69

Appendices

forces.setCentreOfEffort Foil.CoP

Limitation:

• Rope tension force is assumed to be constant, as the total length of robe >> change of robe length

• force action from the flap on the sensor is neglected due to lack of data. This can be important for damping purpose.

• Influence of the centreboard pressure field on foil pressure field is neglected.

C.2.3 TrimVarFoil (ControlVarFoil)Input Data:+ Foil CoP Centre of pressure – equiv. acting point of pressure

forcesPlanform Plan-form area of foilAReff Effective aspect ratio (not used yet)Thickness t Thickness in % of chord lenghtChord c Chord lengthcL 3-dim. response function (cL vs. angle of attack and

flap angle)cD 3-dim. response function (cD vs. aoa and deltaFlap)cDwave Wave drag coefficientcDInter Interference drag coefficientcDPara Parasitic drag coefficientcritAoA Critical angle of attack (warning message option only)

Options Debug Detailed output for debug purpose

Calculation:

// COMMON PARAMETERS

// Local flow at centre of pressure in fixed coordinate system (CSYS) and angle of attacklokalFlow=lokalVelocityB CoP va=lokalFlow [0]2lokalFlow [2]2

aoa=−atanlokalFlow[2 ]lokalFlow[0] // Centre of rotation at current position in absolute CSYS relative to waterline

CoRAWL=CoR.transform TransformationMatrixBACoRAWL [2]=−SINK

// FLAP ANGLE

70

Appendices

// Get flap angle as trim variable (or control variable)// Get Flap angleFlap=trimvar deltaFlap

// FORCES AND MOMENTS

// Lift and drag coefficientsc L=RespFctn aoacD0=RespFctn aoa

cDInduced=c L

2

⋅AReff⋅e=c L

2

⋅fAR⋅AR// Lift and drag forces

lift=cL⋅2⋅va

2⋅Foil.Planform

drag=cD0cDInducedcDWave⋅2⋅va

2⋅Foil.PlanformcDInter⋅2⋅va

2⋅Foil.Thickness

// Force components in absolute CSYSforce [1]=drag

force [2]=liftsin Heel

force [3]=liftcosHeel

// Transform CoP in absolute CSYSFoil.CoP.transform TransformationMatrixBA

// Calculate momentsforces.setCentreOfEffort Foil.CoP

Limitation:

• Influence of the centreboard pressure field on foil pressure field is neglected.

C.2.4 PenaltyDragThe penalty drag modules calculates a penalty drag due to heel, pitch, yaw and sinkage. Thereby the influence of heel pitch and leeway is defined by a drag coefficient which is implemented 2D response function cD vs. heel / pitch / leeway. The reference area needed for the calculation of the dreg is dependent from the sinkage.

This module is used for stabilising the solver but also for creating “smooth” limits for e.g. frame drag due to high heel in low rider mode.

Input Data:RefArea Reference area vs. SINKrefHeel Reference heel (min. drag)CDHeel Drag coeff. vs. heel angle (at refHeel)refPitch Reference pitch-angle

71

Appendices

CDPitch Drag coeff pitch vs. pitch angle (at refPitch)CDLeeway Drag coeff. leeway vs. leeway angle Options Debug Detailed output for debug purpose

Calculation:

// COMMON PARAMETERSAref = fctn SINK cD,HEEL = fctn HEEL−refHeel cD,PITCH = fctn PITCH−refPitchcD,LEE = fctn LEEWAY

// FORCE CALULATION

drag = cD ,HEELcD , PITCHcD , LEE ⋅roh2

⋅ v2 ⋅ Aref

force [1]=dragforce [2 ... 6]=0.

Limitation:

• Main purpose for stabilising the solver.

• Reference area only a function of the sinkage

C.2.5 DynMovMassThe module “Dynamic Movable Mass” defines a counter torque for the in-stationary simulation of the moth (or any narrow hull). As the static equilibrium state of a flying moth, or a boat with a slender hull, not stabilizing itself, is unstable, the boat would always capsize in a in-stationary simulation. This counter torque which is normally done by small movements of the crew or the sailor, is calculated by the heel angle and a virtual torque radius. As the movements are small in comparison to the crew position, the mass moment of inertia can be seen constant and are remaining unchanged.

Input Data:Mas Crew mass as defined in MassMove2Drmax Maximum moving radius - maximum torque that can

be appliedvmax Maximum moving speed – calculation of the virtual

radiusrefheel Reference Heel angle – heel offset for calculation of

counter torque

Options Debug Detailed output for debug purpose

- turning speed dheel

- virtual torque radius

72

Appendices

Calculation:// virtual radius

r virtual =vmax ,crewdheel

dheel : actual turning speed

// calculation of counter torque

M dyn ,rotX = ∣heel−heel reference∣⋅180

⋅r virtual⋅Mass⋅g

// checking if dynamic torque > max. dynamic torqueif M dyn ,rotX rmax⋅Mass⋅gthen M dyn , rotX = rmax⋅Mass⋅g

// checking signsif AWA0 thenif heelheel reference then M dyn ,rotX=−M dyn ,rotX

elseif heelheel reference then M dyn ,rotX=−M dyn ,rotX

Limitation:• Module for in-stationary use only

C.3 Input Data

C.3.1 Geometrical DimensionsAs already mentioned, one type of boat set a new state of the art in moth building – the Bladerider. This boat not only was the first really industrial build boat, but also went through a real development phase. Therefore it can be seen as a very good reference concerning geometric dimensions and positions.

As CAD data could not be obtained, a photo from the internet will be used to measure the basic dimensions of the Bladerider.

73

Appendices

Today other dimensions and positions than those shown in Figure B-5 might be in favour, but they represent a good starting point to perform the velocity prediction and the stability investigations

C.3.2 Masses and gyradii

C.3.2.1 Boat

Modern Moth Skiffs approximately weight about 30 Kg. As no exact data can be provided at this stage, an approximation of the weight has been done as shown in the following table. The values for take over parts are from manufactures or moth pages in the internet. Some other data are estimated by the thickness of the laminate and checked with the FE-model of a user designed moth.

With this FE model also the mass moment of inertia will be obtained, as these values are much more accurate than geometric approximation as it will be done for the sailor.

74

Figure C-5: Bladerider dimensions

Appendices

Table C-2: Weight estimation of a International Moth

Part Weight Reference

Hull 9.15 Kg www.fastacraft.com

Frame/ trampoline/ fittings

6.00 kg estimated

Mast / spreader ,

boom, fittings etc.

3.0 Kg / 3.0 Kg www.imoth.de -

Sail 3.00 Kg estimated

Centreboard / -foil 1.50 Kg / 1.50 Kg www.fastacraft.com (~1 Kg/m +

mechanical parts)

Rudder/ -foil 0.85 Kg / 1.00 Kg www.fastacraft.com (see above)

Outrigger 0.65 Kg estimated

Sum 29.65 Kg

The shape of the hull was designed by Juryik Henrichs. This task was planned already for 2008 but was delayed. Thus just the construction of the mould was finished recently. The FE-model is set up with the FE-tool Hyperworks, with special permission by Advanced Composite Engineering (ACE), the current employer of the author.

For the calculation of the weight matrix of the FE-model, a basic laminate lay-up is defined. Other thickness has been estimated or scaled due to known values. For the calculation of the mass, the densities in Table C-3 has been used. The density of Glass-Fibre-Reinforced-Plastic (GFRP) and Carbon-Fibre-Reinforced-Plastic (CFRP) are calculated for a fibre volume fraction of 50%. The value for polyester is used for the sail, robes and the trampoline. For sail and trampoline a thickness between 0.25 mm and 0.5 mm is assumed. Steel robes have a diameter of 2 mm, polyester of 5 mm.

Table C-3: Density of materials

Steel Adhesive Polyester CFRP GFRP Core (C70)

[kg / m3] [kg / m3] [kg / m3] [kg / m3] [kg / m3] [kg / m3]

7900 1000 1100 1500 1880 70

A colour plot of the different property regions is shown in Figure C-6:

75

http://Www.fastacraft.com/

http://www.fastacraft.com/

http://www.imoth.de/

http://www.fastacraft.com/

Appendices

Using the estimated property sets, the following weight distributions have been achieved:Table C-4: FE-weight due to estimated properties

Component / Part FE weight

Hull 9.15 Kg

Wings

Frame 4.14 Kg

Trampoline + fittings 1.88 Kg

Rig

Mast 2.31 Kg

Boom 1.84 Kg

Fittings, fore-stay, shrouds 0.33 Kg

Sail 4.56 Kg

Centreboard / centreboard-foil 2.27 Kg

Rudder / rudder-foil 2.16 Kg

76

Figure C-6: FE-model property region plot

Appendices

Outrigger 0.58 Kg

Total mass 29.23 Kg

As a result, the centre of gravity and the mass moment of inertia tensor, with respect to the centre of gravity, is printed in Hypermesh (Hyperworks preprocessor).

xCoGyCoGzCoG = −0.26

0.000.85 and I XX I XY I XZ

I YY I YZI ZZ = 90.22 0.00 −2.19

114.45 0.0031.47 (16)

With the mass moment of inertia the gyradii and the cross product can be calculated with the formula:

iij = sign I ij ⋅ ∣I ij∣m (17)

Where iij is the resulting gyradius / cross product, Iij the term in the mass moment of inertia matrix and m the over all weight. For the calculation of the diagonal terms, the signum and absolute values are not necessary, as those terms can not become negative.

The result is the following matrix with the gyradii and the cross products:

i XX i XY iXZiYY iYZ

iZZ = 1.76 0.0 −0.2741.98 0.00

1.04

C.3.2.2 Crew

For the calculation of the mass moment of inertia and the cross product, the body will be divided in simple geometric bodies (Figure C-7), from which the mass moment can be calculated by basic functions.

The input of the gyradii and the cross products is done with respect to the centre of gravity of the sailor and parallel to the body fixed coordinate system of the skiff (marked with x,y,z).

77

Appendices

As for a modern skiff the position of the crew is not fixed and highly dynamic, especially on a 30 cm wide moth, a “normal” sailing position is assumed: the sailor sitting on the side frame.

First, the mass moments of the different body parts will be calculated with respect to their centre of gravity (xi´´, yi´´, zi´´) along the main axis of the body. The formulas are given in table Table C-5.

Table C-5: Mass moment of inertias of basic bodies

Tube

(in rotation axis)J z = m⋅R2

2(18)

Tube

(perpenticular to rotation axis)

J Z = m⋅R2

4h2

12 (19)

Box

(along height with a x b base area)

J Q = m12

⋅a2b2 (20)

Bowl J K = 23⋅m⋅R2 (21)

78

Figure C-7: Geometrical bodies for estimation of mass moment of inertia

Appendices

With this formulas principal moment of inertia tensor in the main axis of the body can be calculated:

I ´ ´= I XX 0 00 I YY 00 0 I ZZ (22)

The rotation of the inertia tensor by two angle as shown in figure is done with a transformation matrix and the transposed rotation matrix.

I '=RT⋅I⋅R (23)

with

R=cos⋅cos −sin cos⋅sinsin⋅cos cos sin⋅sinsin 0 cos (24)

and

RT = cos⋅cos sin⋅cos sin−sin cos 0cos⋅sin sin⋅sin cos (25)

Now the inertia tensor has to be moved into the centre of gravity of the sailor. This is done with the Huygens-Steiner theorem. The theorem describes the translation of a tensor along a vector (a1, a2, a3)T:

I=I ´m⋅a22a3

2 −a1⋅a2 −a1⋅a3

−a1⋅a2 a12a3

2 −a2⋅a3

−a1⋅a3 −a2⋅a3 a12a2

2 (26)

The calculated moment of inertia tensor can now be transformed in the necessary gyradii and the cross product (10).

The proportion of weight will be estimated with the values given in Table C-6:Table C-6: Proportion of weight of a human body [8]

Body part Weight percantage Absolut weight

Head 7,30% 5.84 Kg

Torso 48,90%

Upper arm ( x 2 ) 2 x 2.70%

Forearm + hands ( x 2 ) 2 x 2.50%

Thigh ( x 2 ) 2 x 9.70%

Calves + feet ( x 2 ) 2 x 7.00%

Sum 100,00% 80 Kg

79

Appendices

Now for every body part, the moment tensor with respect of centre of gravity and rotation axis is calculated and summarized to the over all moment tensor. Recalculated the gyradii and the cross product are:

i XX i XY iXZiYY iYZ

iZZ = 0.36 0.0 0.00.3 0.24

0.35

C.3.3 Windage dataAs already mentioned, the aerodynamic drag of a fully foiled moth is an important component within the overall resistance. For the prediction of the forces of the a moth, Bill Beaver and John Zseleczky did a full scale measurement of a moth and its components. In this test series, also a full scale moth was tested on its aerodynamic resistance. Therefore a platform holding the moth with a dummy sailor was mounted to the towing carriage at the U.S. Naval Academy Hydrodynamics Lab in Annapolis, Maryland. The full paper can be downloaded at www.moth-sailing.org/download/CSYSPaperFeb09.pdf.

The test configuration was set up for an apparent wind angle of 24°, with a leeway angle of 4°, a true wind speed of 12 kn and a boat speed of 12 kn as well. This leads to an apparent wind speed of 11.88 kn or 6.11 m/s. Three different heel angles where tested (0°, 15°, 30°).

As the force axis are aligned with the “course made good” coordinates shown in Figure C-8,forces and velocity has been recalculated due to apparent wind speed AWA, the leeway angle λ and the heel angle ϕ.

80

Figure C-8: Aerodynamic force gage axes [BeZc09]

http://www.moth-sailing.org/download/CSYSPaperFeb09.pdf

Appendices

As the forces in FS-Equi are aligned to the body fixed coordinate system, the forces and velocity ratios have to be recalculated with the leeway angle λ, the apparent wind angle (AWA) and the heel angle ϕ.

The velocity components are calculated as shown in Figure C-9 by the AWA and the leeway angle, whereas the force first have to be split up into the aerodynamic drag and -lift. Hereby the drag coefficient is a constant value for all wind angles, whereas the aerodynamic lift will change by AWA. The drag coefficient can be predicted by using the aerodynamic efficiency εA, defined in Marchaj [15] by:

tan−1A =LA

DA (27)

Where LA is the aerodynamic lift and DA the desired aerodynamic drag. The measured lift and drag can be coupled by the hydrodynamic efficiency εB:

tan−1B = side forcedrag (28)

Those two terms are related by the apparent wind speed AWA or β, as used in literature by:

= A B (29)

Now the aerodynamic drag can be expressed by the square-root of the lift and drag force, and the aerodynamic efficiency.

For the calculation of the drag coefficients the projected areas in the three body-fixed axis had been estimated by basic dimensions of a moth and a sailor:

Ax [m2] Ay [m2] Az [m2]

0,89 2,44 4,49

81

Figure C-9: Wind velocity ratios

Appendices

Using the first set of values from [7] with a heel angle of 0°, the drag coefficients cDx and cDy

can directly be calculated, giving the result of cDx=1.343 and cDy=1.099. For the calculation of the drag component in Z-direction is not enough data available, as here also the lift components in all three dimensions have to be considered. Therefore the value is estimated by 1.0. Due to the lack of data at this stage, the aerodynamic lift will not be considered at this stage.

C.3.4 Foil and strut geometry and dataThe geometrical data of the rudder and the centreboard have been obtained from Figure C-5. Both, the rudder and the centreboard chord length were set to 125 mm. Input Data can be summarized by the data in Table C-7:

Table C-7: Centreboard and rudder data

Centreboard Rudder

Profile NACA 63-012 NACA 0012

Length 1.09 m 0.96 m

Width 0.125 m 0.125 m

Angle 6.34° fwd 2°

Thickness t 12% of chord 12% of chord

Chord length c 0.125 m 0.125 m

To reduce the potential number of variation, the profile geometry was set to standard NACA profiles, as used in the first generation of hydro-foil. Centreboard and rudder are NACA 63-012 and NACA 0012 foils, the centre board foil is a NACA 63-412. The rudder foil is a symmetrical NACA 0012 profile, which will be rotated for control purpose.

The necessary data was calculated with XFLR5 from Andre Deperrois, a gui supported and in C++ rewritten Version of the fortran written panel code XFOIL from Marc Drela. This program is published under the open source licence and can be downloaded at http://xflr5.sourceforge.net/xflr5.htm.

Additionally XFLR5 can calculate wings and since version 4.0 also planes with a 3D panel method. Wings can be calculated with the lifting line theory, the vortice lattice method or the 3D panel method.

The centreboard and rudder data will be calculated using the 2D foil module. The recalculation for a 3D wing will be within the FS-Equi force module. For the centreboard – and rudder foil the 3D wing module had been used.

For determination of the lift and drag curves of the profile, the Reynolds number has to be calculated:

Re =V s⋅l

(30)

82

Appendices

The formula shows the influence of the boat speed. Therefore lift and drag coefficients should be given for all desired speed. A 3-dimensional response surface could be theoretically implemented in the module, but at this stage averaged values has been used.

With the length l = 0.125 m, the kinematic viscosity of water at 20°C ν = 1.0038E-6 and a speed range from 2 m/s to 10 m/s the Reynolds numbers can be calculated to:Table C-8: Reynolds numbers for centreboard an rudder

Speed [m/s] 2 3 4 5 6 7 8 9 10

Re [E-06] 0,25 0,37 0,5 0,62 0,75 0,87 1,00 1,12 1,25

C.3.4.1 Centreboard and rudder

The Foil data used for rudder and centreboard are pure 2 dimensional data, calculated with the direct foil analysis module in XFLR5. Hence the lift and drag forces and moments has to be recalculated using the formulas in Appendix B.

For calculation of the moments, the centre of effort is given as a the distance from the geometric centre of area normalized by the immersion of the foil vs. the immersion normalized by the length. Hence the CoE can be recalculated for the different states of the moth – low rider and foiled. The data implemented into the FS modules results from pure theoretical thoughts and should be verified in the future.

The approach for the prediction of the CoE is the lift distribution of a daggerboard. Using a standard daggerboard configuration, the lift distribution will drop to zero at the tip of the foil, whereas on the root a significant lift remains. Using a non-foiled T-foil daggerbaord, will also result in a significant lift at the tip, whereas in foiled state, the lift will drop to zero at the root.

This gives us the theoretical change of centre of effort as implemented in FS-Equilibrium.

83

Figure C-10: Centre of effort rudder / centreboard

Appendices

The same thoughts are the basis for calculation of the effective aspect ratio AReff. For a non-foiled standard profile, AReff can be assumed to be twice the geometric aspect ratio, using the DWL of the hull as mirroring plane. The T-foil also works like a mirroring plane, which would theoretically lead to an infinite aspect ratio. Using a factor of 1.5 for the T-foil and a factor of 2 for the hull, leads to a factor of approx. 2.5 to 3 for the non-foiling T-foil daggerboard. In foiling state, the factor of 1.5 remains as the surface has a much lower effect on AR, as the lift drops to zero at the surface.

The polar data for the centreboard and rudder are calculated for Reynolds numbers from 0.20E-06 to 1.00E-06, which is a speed range of approx. 2 to 10 m/s. The corresponding curves are given in Figure C-13 for a NACA 0012 profile used for the rudder and a NACA 63-012 used for the daggerboard in Figure C-14.

84

Figure C-11: Vertical centre of effort distribution

Appendices

Within the range of 4 to 6 m/s the difference of the cL. vs. cD abd the cL vs. angle of attack is very small for the NACA 0012, especially within a range of +/- 5°.

The NACA 63-012 shows the same characteristic. The cL/cD vs. angle of attack curves are spread up wider than at the NACA 0012 profile, but in the range of 4 m/s to 6 m/s the difference is small.

For both profiles the Re = 0.50E06 data was used, as here the cL/cD ratio is small, hence more conservative. For the use in FS-Equilibrium, the curve were converted into cL, cD vs. angle of attack (AoA).

85

Figure C-13: Polars - NACA 63-012

Figure C-12: Polars - NACA 0012

Appendices

The following cL and cD curves has been used as standard curves for all analyses. The red line represents the response function. In this case Cspline data interpolation was used.

C.3.4.2 Centreboard - / Rudder Foil

The lift / drag characteristic of the main foil is calculated using the wing module of XFLR5. In a first step, lift and drag data for the different sections are calculated with the “Foil Direct Design” module, also used for the daggerboard and the rudder.

At this point, it has to be decided if the rudder foil is flap controlled, or the whole foil will be rotated, hence no foil analysis with different flap angles has to be performed.

The centreboard foil is a flap controlled foil, where lift and drag are functions of pitch- and flap angle. The pitch-angle (angle of attack) varies from -3° to +5° , the flap angle was set to 15°, 5°, 2.5°, 0°, -2.5° and -5°. The hinge line of the flap is at the 30% line from the tailing edge, the centre of rotation set to 80% of the thickness from the bottom surface.

86

Figure C-15: NACA 63-012 cL vs. AoA – cD vs. AoA

Figure C-14: NACA 0012 cL vs. AoA – cD vs. AoA

Appendices

As the chord length varies over the span of the foil, the section lift and drag coefficients have to be calculated for different Reynolds numbers. This is done using the batch analysis ability of XFLR5. The data for not calculated Reynolds numbers will be interpolated. Thus polars for the maximum and the minimum Reynolds-number must be provided for the calculation of the wing. Otherwise a failure message occurs.

The definition of the wing was done section wise, with a straight tailing edge. Figure C-17 shows the input mask with the data from the 0.11 m2 hydro-foil. The speed for performing the wing analysis has been set to 5 m/s, as the calculated lift and drag data for the rudder and

87

Figure C-16: NACA 62-012 foil section with flap

Figure C-17: Daggerbord foil (wing) definition in XFLR5

Appendices

centreboard. Pitch angle range has been set to -3° to 5° with a step size of 0.5° and the analysis code set to 3D panel method. The result is exported to a text file and sorted for data import within FS-Equi. As the optimization of the plan-form area and the foil shape is not part of this work, it should be mentioned, that also a graphical post-processing is possible within XFLR5. Lift, drag distribution can be shown graphically, as well as down-wash, cP, surface-velocity and streamlines.

Subsequently the polar data was exported to a text file and organized by flap angle and angle of attack. The daggerboard foil shows the following response surface for lift and drag. Data-points are shown in red, the blue mesh represents the response surface.

The rudder foil is also a NACA 63-412 without a flap. The span has been set to 0.8 m with a root chord of 0.125m. The result is a foil with a plan-form area with 0.08 m2 and an aspect ratio of ~8.

88

Figure C-18: Graphical output of XFLR5

Figure C-20: Daggerboard foil cD vs. flap angle and AoA

Figure C-19: Daggerboard foil cL vs. flap angle and AoA

Appendices

The resulting lift and drag data is also implemented as 3D data. The corresponding response surfaces are shown in Figure C-22 and Figure C-23.

C.3.4.3 Additional drag components

Even thought section and induced drag are the major drag components, additional components should be considered as well, even if those are small. Drag components mentioned by [7] are strut spray/wave drag, foil wave drag and interference drag. Also mentioned is the surface finish and the gap at the flap for the main foil.

89

Figure C-21: Rudder foil definition in XFLR5

Figure C-22: Rudder foil cL vs. flap angle and AoA

Figure C-23: Rudder foil cD vs. flap angle and AoA

Appendices

Strut spray drag and intersection drag can be calculated by

F = cD⋅roh2⋅v2⋅t Strut

2 (31)

Hoerner gives a cDSpray value of 0.3, but the measurement of [7] shows a better correspondence at a value of 0.35. The intersection drag coefficient is given by Hoerner as 0.1.

The foil wave drag is a function of the lift and the immersion. The drag coefficient is given by:

cDWave = cL2 ⋅ c

h⋅CDH

CLH2 (32)

The value of CDH/CL2H was determined as mentioned in [7] by plotting cD over different

immersion and interpolating the curve to zero immersion, hence no wave drag. Now the wave drag component could be back calculated for the given immersions. The values are in a range of 0.22 and 0.26 and where set to 0.25 for the velocity prediction.

Another component which is mentioned by [7] is the surface roughness. Also the gap for the flap will have influence on the drag. All those effects are summarized as parasitic drag within the FS-Equi modules. As is mentioned in various news groups, the drag components calculated by XFLR5 are quite optimistic. Therefore the drag components have been scaled with the parasitic drag using the data from from [7].

All together the additional drag coefficients can be summarized to:Table C-9: Additional drag coefficients

Centrebaord Rudder

Strut parasitic 0,001 0,001

Strut wave/spray 0,35 0,35

Strut / foil interference 0,1 0,1

Foil parasitic 0,0025 0,0025

Foil wave (CDH/CL2

H) 0,25 0,25

C.3.4.4 Validation of Drag Components

By summation of the different foil and strut drags the drag components of the analytical solution can be compared to the measured data in [7] as shown in Figure C-24 and Figure C-25.

90

Appendices

The daggerboard drag, sacled by the parasitic drag, shows a good correlation to the experimental data. The drag forces of the rudder T-foils showing a wider tolerance, but the analytical solution is well situated in the middle.

C.3.5 Sail dataThe Sail Data used is partly from the work of Heikki Hansen, especially for the downwind course, and from Chris Williams from North Sails, who was so kind providing the data from the internal CFD tool FLOW, for the upwind courses with an apparent wind angle of 15°, 18°, 21°, 24° and 27°, a true wind speed of 6,173 m/s and the actual sail geometry with the basic dimensions. Afterwards the downwind data from Hansen is transformed to have a smooth curve for lift and parasitic drag.

91

Figure C-24: Daggerboard T-Foil drag components

Hungry Beaver Vendor 1 Vendor 2 Analytical0.0000

5.0000

10.0000

15.0000

20.0000

25.0000

30.0000

35.0000

40.0000

45.0000

50.0000

Daggerboard T-foil drag components

Strut spray dragStrut section dragFoil wave dragFoil induced dragFoil section drag etc.

Figure C-25: Rudder T-foil drag components

Hungry Beaver Vendor 1 Vendor 2 Analytical0.0000

5.0000

10.0000

15.0000

20.0000

25.0000

30.0000

35.0000

Rudder T-foil drag components

Strut spray dragStrut section dragFoil wave dragFoil induced dragFoil section drag etc.

Appendices

As the centre of effort is calculated relative to the centre of area, the sail centre of area has to be determined. Therefore the plan-form sail shape is divided into triangles (Figure C-26) from which the local area and centre of area are calculated. Afterwards the over all CoE can be obtained by

CoAX=∑ x i⋅Ai

∑ Ai; CoAy=

∑ y i⋅Ai

∑ Ai (33)

For the calculation of the forces within the Frig force module, the effective coordinate system is used. Therefore the data predicted with FLOW had been recalculated in the effective coordinate system. The effective wind angle (AWAeff or βeff) is related to the heel Φ, the pitch θ and the apparent wind angle (AWA or βA) by:

eff = tan−1 tan⋅cos − sin⋅sincos (34)

The effective wind speed Veff can be obtained by:

V eff =V A ⋅ sinA⋅cos−cos A⋅sin⋅sin 2cos2A⋅cos2 (35)

The details about the derivation of these formulas are well documented in the PhD thesis of Heikki Hansen [9].

92

Figure C-26: Centre of area moth sail - North Sails, Chris Williams

Appendices

The calculated data from Chris Williams is given basically in 2 different formats. The first one is the raw data log file. The second one a summary file, which results can directly be imported in a North Sails rig module within FS-Equilibrium. This module is developed with North Sails and during an optimisation, the best sail-trim condition will be used from the input data. As only one heel angel is given, this module can not be used at this stage.

The raw data will be used, as the results are given in different coordinate systems. Within FLOW, the Y-coordinate system is the heeled coordinate system in leeway direction of the yacht. As leeway angle is assumed to be 0°, the data already represents the effective data. In this case the flow direction and apparent wind speed have to be recalculated with the formulas given above. The FLOW data for a heel angle of 30 degrees can be summarised as follows:Table C-10: Moth sail data FLOW – Chris Williams, North Sails

AWAeff veff Lift [N] Drag [N] cL cD

13.06 10.46 421.56 45.47 0.769 0.08315.72 10.30 462.96 53.17 0.870 0.10018.39 10.12 499.41 63.17 0.973 0.12321.09 9.91 529.97 75.28 1.076 0.15323.81 9.69 554.37 89.15 1.179 0.190

For the determination of the lift curve, the data of Chris Williams was used up to an effective wind angle of 24.8°. From 50° to 180° the data of the wind-tunnel test of the DYNA yacht has been used and scaled by a factor of 0.85. Below 13.1° two more data points has been added and one at 50° to achieve a smooth curve progression.

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Figure C-27: Sail lift curve cL vs. AWAeff

0 20 40 60 80 100 120 140 160 180 2000

0.4

0.8

1.2

1.6

cL HansencL scaledcL FLOW

Effective wind angle EWA

Lift

coef

ficie

nt

Appendices

The result is the orange curve. In addition the curve determined by Hansen and the FLOW data points are also included.

The implemented drag curve consist of the parasitic drag, where the section drag is included. The additional drag components, induced drag and separation drag are calculated within the module. For the derivation of the moth sail drag curve, the induced drag will be calculated by:

cDI =cL

2

⋅AReff⋅e (36)

The effective aspect ratio AReff is obtained using the wing-module of XFLR5. Here the aspect ratio is determined as a symmetrical “wing” with its symmetry plane at the bottom, for a sail at its tack. Regarding a real sail the aspect ratio is half the value of XFLR5 without having a mirroring plane. Using the waterline level mirroring plane, the effective aspect ratio is increase by a factor of 1.2. Same result is achieved by setting the efficiency e to 0.6 and using the AR given from XFLR5.

The separation drag was calculated by cL and the separation constant CS which is set to 0.016 by default. The separation drag-coefficient is then calculated by:

cDS = cL2 ⋅ CS (37)

Now the drag components are added to the total drag. To get a good data fit to the FLOW drag data, the parasitic drag curve from Hansen has been translated by value of -0.07 leading to a parasitic drag coefficient cP of 0.05 at 0° AWAeff, which seems to be a reasonable section drag for a full batten pocket sail.

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Figure C-28: Sail aspect ratio

Appendices

The graph shows the parasitic drag from Hansen (blue), as well as the the drag components resulting in the total drag. The data points from flow are shown as yellow dots, and are showing a good correlation to the calculated total drag (orange). The continuous green curve represents the input data for the FRig module.

In addition the change centre of effort as a function of the apparent wind angle is defined, for calculation of the moments. The values are defined relative to the centre of area, normalized by DWL / mast height.

The values from Flow show a large discrepancy to the wind tunnel data of DYNA. Nevertheless these values have been used as an initial starting point and have been fitted to a similar distribution correlating to the measured data.

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Figure C-29: Sail drag curve cD vs. AWAeff

0 20 40 60 80 100 120 140 160 180 2000

0.4

0.8

1.2

1.6

cD HansencDI (induced) cP (parasitic)cDs (separation)cD (total)cD FLOW

Effective wind angle EWA

Dra

g co

effic

ient

Figure C-30: Sail centre of effort vs. AWAeff

Appendices

C.3.6 BuoancyForce module (hull)The hull has been implemented within FS-Equilibrium with a step file. The design is actually made by Juryk Henrichs with little influence of the author.

The curve of the sectional area is given in . Thereby the origin is at the stern, pointing forward, as for all data given in Table C-10.

The main parameters of the hull can be summarised by Table C-10:

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Figure C-31: Offset definition of moth hull by Juryk Henrichs

Figure C-32: Curve of sectional area of hull

Appendices

Table C-11: Main parameters of hull

Draft Amidships m 0.166Displacement t 0.105WL Length m 3.353Beam max extents on WL m 0.31Wetted Area m^2 1.565Waterpl. Area m^2 0.776Prismatic coeff. (Cp) 0.69Block coeff. (Cb) 0.607Max Sect. area coeff. (Cm) 0.909Waterpl. area coeff. (Cwp) 0.746LCB from zero pt. (+ve fwd) m 1.5LCF from zero pt. (+ve fwd) m 1.428

C.4 Force Module ValidationAs programming is always a task where small errors might have a big effect but are hard to find and to quantify, the self programmed routines must be checked and validated. Therefore the data from Appendix C.3 is implemented into the different force modules and some quality checks are performed.

The first check is the verification of the mathematical signs and the tendency of the different forces. Using the coordinate system from Figure 2-1 the forces TWA from 40° to 180° and a positive leeway angle must have the following mathematical signs:Table C-12: Mathematical signs and tendencies of force modules

Module Change of variable Force

controlledMass CrewX < 0

CrewY < 0

My < 0

Mx > 0

Centreboard / Ruddder

LEE > 0

Decrease SINK

Rudder Y > 0

Fy < 0, Mx < 0

Decraese Fy

Fy < 0, Mx < 0, Mz > 0

Centreboard-foil / Rudder-foil

Increase Vs

Decrease SINK

DeltaFlap > 0

Increase Fz

Decrease Fz

Fz > 0, My > 0

For a more detailed verification of the modules, the resulting forces are plotted versus the conditions influencing the specific module / force component. Main focus hereby are the force modules for the struts and the foils – centreboard and rudder.

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Appendices

For the validation of the foils, the method, used in Appendix C.3 (Foil Data) can be used, but also visualised by plotting the the lift and drag force versus the sinkage for a TWS of 4, 5, 6 an 8 m/s in .

While the effect of the heel angle is very important at upwind courses, the change of lift and side force is plotted for a TWS of 4, 5 and 6 m/s versus the heel angle in .

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Figure C-33: Lift and drag at main foil vs. sinkage

Figure C-34: Lift and side force vs. heel and speed

Appendices

Hereby the effect of creating a significant side force while minor lift reduction at sailing with windward heel becomes clear when comparing the data of the side force to the lift.

The forces of the struts vs. speed, heel, sinkage and rudder are shown in for a TWS of 4, 5, 6 and 8 m/s, all showing a very harmonic run of the curves. The absolute values are also reasonable and correspond to the given lift and drag coefficients.

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Figure C-35: Lift at foils vs. pitch

Figure C-36: Rudder lift and drag vs. deltaFlap

Appendices

D Foil Set-Up for VPPThe data used for the different foil set-up's is generated with the preliminary foil design tool. Basis for the set-up has been the take off behaviour. This can be visualised by the velocity and the flight height within the tool. For an overview the complete set of input-data is compared in Table D-1 and the corresponding curves are shown in the additional figures of the sinkage and the fight height versus the time.

Table D-1: Geometrical parameter for feedback control systems

Foil 01 Foil 02 Foil 03 Foil 04*Foil CoP (Z-comp.) [m] -1.04 -1.04 -1.04 -1.04

Planform [m2] 0.11 0.08 0.14 0.1Thickness t [%] 0.12 0.12 0.12 0.12Chord l [mm] 110 110 110 110AReff [-] 10.8 10.8 10.8 10.8

Sensor CoR [m] 0.25 0.25 0.25 0.25Length [m] 0.9 0.9 0.9 1Diam [m] 0.0075 0.0075 0.0075 0.005DragCoeff [-] 0.75 0.75 0.75 0.5

Robe DistCoR [m] 0.025 0.025 0.025 0.025Tension [N] 10 10 10 25

Flap SensorDistCoR [m] 0.05 0.05 0.05 0.05DeltaSensor_0 [°] 35 35 35 25fDeltaFlap [-] 0.125 0.14 0.118 0.16FlapDistCoR [m] 0.02 0.02 0.02 0.02deltaFlap_0 [°] -5 -5 -4 -10

*used in final analysis in Chapter 5

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Figure D-1: Speed / sinkage vs. time set-up F01

0 10 20 30 40 50 60 700

1

2

3

4

5

6

-0,100,10,20,30,40,50,60,7

Speed Vs [m/s]Zoffset [m]

Time [s]

Spe

ed [m

/s]

z-of

fset

[m]

Appendices

101


0 10 20 30 40 50 60 700

1

2

3

4

5

6

-0,1

0

0,1

0,2

0,3

0,4

0,5

0,6


Time [s]

Spe

ed [m

/s]

z-of

fset

[m]


0 10 20 30 40 50 60 700

0,51

1,52

2,53

3,54

-0,1-0,0500,050,10,150,20,250,3


Time [s]

Spe

ed [m

/s]

z-of

fset

[m]

Appendices

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0 10 20 30 40 50 60 700

1

2

3

4

5

6

7

-0,100,10,20,30,40,50,60,7


Time [s]

Spe

ed [m

/s]

z-of

fset

[m]

Appendices

E VPP Results

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Figure E-1: Moth polar plan-form area 0.11 m2 – set-up F01

Appendices

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Figure E-2: Moth polar plan-form area 0.11 m2 – set-up F02

Appendices

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Figure E-3: Moth polar plan-form area 0.11 m2 - set-up F03

Appendices

106


Appendices

107


Appendices

108


Appendices

109


Appendices

F Stability ResultsThe investigations of the dynamic stability with the small disturbance theory has been done with constrained degrees of freedom. The first run with constrained heel shows the results in Table F-1. By also constrained leeway angle, the results in Table F-2 can be achieved.

Table F-1: Linearisation results for one fixed DoF (heel)

NAME TWS TWA Real Imaginary Decay Time Damping PeriodP008F01 10.0 120.0 -0.089 0.261 7.791 0.322 24.057P008F02 10.0 120.0 -0.089 0.263 7.828 0.319 23.857P011F01 10.0 120.0 -0.078 0.284 8.836 0.266 22.119P011F02 10.0 120.0 -0.072 0.314 9.659 0.223 20.006P011F03 10.0 120.0 -0.068 0.319 10.211 0.208 19.686P014F01 10.0 120.0 -0.057 0.331 12.074 0.171 18.976P014F02 10.0 120.0 -0.042 0.366 16.513 0.114 17.180P008F01 10.0 60.0 0.008 0.000 -91.773 -1.000 0.000P008F02 10.0 60.0 0.052 0.000 -13.253 -1.000 0.000P011F01 10.0 60.0 -0.025 0.000 27.850 1.000 0.000P011F02 10.0 60.0 0.006 0.000 -111.553 -1.000 0.000P011F03 10.0 60.0 0.001 0.000 -506.545 -1.000 0.000P014F01 10.0 60.0 -0.047 0.000 14.806 1.000 0.000P014F02 10.0 60.0 0.013 0.000 -53.584 -1.000 0.000P008F01 6.0 120.0 -0.092 0.251 7.495 0.346 25.045P008F02 6.0 120.0 -0.089 0.252 7.806 0.333 24.980P011F01 6.0 120.0 -0.088 0.272 7.914 0.307 23.120P011F02 6.0 120.0 -0.080 0.297 8.715 0.259 21.159P011F03 6.0 120.0 -0.079 0.212 8.805 0.349 29.685P014F01 6.0 120.0 -0.081 0.243 8.568 0.316 25.833P014F02 6.0 120.0 -0.069 0.260 10.040 0.257 24.185P008F01 6.0 60.0 0.077 0.000 -8.978 -1.000 0.000P008F02 6.0 60.0 0.259 0.000 -2.681 -1.000 0.000P011F01 6.0 60.0 -0.010 0.000 71.297 1.000 0.000P011F02 6.0 60.0 0.272 0.000 -2.550 -1.000 0.000P011F03 6.0 60.0 0.061 0.000 -11.343 -1.000 0.000P014F01 6.0 60.0 0.021 0.000 -33.202 -1.000 0.000P014F02 6.0 60.0 0.067 0.000 -10.347 -1.000 0.000P011F02 4.0 115.0 -0.093 0.253 7.422 0.346 24.818P008F02 4.0 115.0 -0.095 0.205 7.259 0.422 30.618P011F03 4.0 115.0 -0.097 0.269 7.119 0.340 23.319P014F01 4.0 110.0 0.046 0.000 -15.184 -1.000 0.000P008F01 4.0 15.0 -0.096 0.179 7.234 0.473 35.162P011F01 3.5 105.0 -0.030 0.129 23.039 0.227 48.613P014F02 3.5 85.0 0.111 0.000 -6.233 -1.000 0.000

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Appendices

Table F-2: Linearisation results for one fixed DoF (heel, yaw)

NAME TWS TWA Real Imaginary Decay Time Damping PeriodP008F01 10.0 120.0 -0.297 0.000 2.049 1.000 0.000P008F02 10.0 120.0 -0.300 0.000 2.314 1.000 0.000P011F01 10.0 120.0 -0.305 0.000 2.271 1.000 0.000P011F02 10.0 120.0 -0.303 0.000 2.286 1.000 0.000P011F03 10.0 120.0 -0.307 0.000 2.259 1.000 0.000P014F01 10.0 120.0 -0.303 0.000 2.289 1.000 0.000P014F02 10.0 120.0 -0.302 0.000 2.292 1.000 0.000P008F01 10.0 60.0 -0.482 0.000 1.438 1.000 0.000P008F02 10.0 60.0 -0.470 0.000 1.474 1.000 0.000P011F01 10.0 60.0 -2.249 0.000 0.308 1.000 0.000P011F02 10.0 60.0 -0.487 0.000 1.423 1.000 0.000P011F03 10.0 60.0 -0.493 0.000 1.406 1.000 0.000P014F01 10.0 60.0 -0.473 0.000 1.464 1.000 0.000P014F02 10.0 60.0 -0.567 0.000 1.223 1.000 0.000P008F01 6.0 120.0 -0.366 0.000 1.895 1.000 0.000P008F02 6.0 120.0 -0.370 0.000 1.872 1.000 0.000P011F01 6.0 120.0 -0.376 0.000 1.842 1.000 0.000P011F02 6.0 120.0 -0.395 0.000 1.756 1.000 0.000P011F03 6.0 120.0 -0.391 0.000 1.774 1.000 0.000P014F01 6.0 120.0 -0.409 0.000 1.693 1.000 0.000P014F02 6.0 120.0 -0.419 0.000 1.654 1.000 0.000P008F01 6.0 60.0 -0.543 0.000 1.275 1.000 0.000P008F02 6.0 60.0 0.213 0.000 -3.255 -1.000 0.000P011F01 6.0 60.0 -0.910 0.000 0.762 1.000 0.000P011F02 6.0 60.0 0.386 0.000 -1.794 -1.000 0.000P011F03 6.0 60.0 -0.692 0.000 1.002 1.000 0.000P014F01 6.0 60.0 -0.705 0.000 0.983 1.000 0.000P014F02 6.0 60.0 -0.839 0.000 0.826 1.000 0.000P011F02 4.0 115.0 -0.326 0.000 2.127 1.000 0.000P008F01 4.0 115.0 -0.338 0.000 2.049 1.000 0.000P008F02 4.0 115.0 -0.343 0.000 2.021 1.000 0.000P011F03 4.0 115.0 -0.344 0.000 2.013 1.000 0.000P014F01 4.0 110.0 0.012 0.000 -59.233 -1.000 0.000P011F01 3.5 105.0 -0.267 0.000 2.592 1.000 0.000P014F02 3.5 85.0 -0.230 0.000 3.013 1.000 0.000

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