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Numerical analysis of wave-induced responses of fl oating bridge pontoons with bilge boxes

Espen KleppaJune 2017

DTU Mechanical Engineering

Section of Fluid Mechanics, Coastal and Maritime Engineering

Technical University of Denmark

Nils Koppels Allé, Bld. 403

DK-2800 Kgs. Lyngby

Denmark

Phone (+45) 4525 1360

Fax (+45) 4588 4325

www.mek.dtu.dk

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d

AbstractIn the offshore industry it has become more common to install damping ”devices” onlarge displacement structures like FPSO and SPAR platforms. The damping devicesare often circular discs on the bottom of the structure and it is installed to reducethe motion of the structure, mainly the heave motion. In this thesis a bottom discis installed on a bridge pontoon supposed to be located in a fjord in Norway. Theobjective is to study the motion behavior of the pontoon, study how it is dependenton the Cd-coefficient and develop a simple numerical model for estimating the KC-number by vertical oscillation velocity of the pontoon.

A surface model is created in the hydrodynamic analysis tool Ansys Aqwa and runfor several different disc ratios in a Sea State given by the Norwegian Public RoadAdministration (NPRA). The main purpose of the flange is to increase the addedmass of the structure and shift the Natural Period away from the wave spectrum.Several simulations with different disc ratios is performed and the natural period isshifted away from the wave spectrum when the disc ratio is increasing. However, adownside by adding the bottom disc is that a cancellation effect of the damping loadis occurring at certain periods.

By adopting KC-numbers and Cd-coefficients from earlier studies on similar struc-tures, a procedure is developed for estimating the KC-number for the pontoon. Byusing linear potential theory and Morison drag linearization, motion analysis is per-formed in Aqwa by adding Morison drag elements on to the disc to obtain the effectof drag loads on the pontoon.

A second and harsher sea state is tested for the same pontoon for comparison andthe same procedure is carried out for this sea state. It is found that the pontoonis quite independent of the Cd- coefficient for the initial moderate sea state, wherethe resulting vertical velocities are very small and are giving very small values ofKC-number. For the second sea state the peak period is shifted and interacting morewith the resulting motion RAOs of the pontoon, resulting in larger response andhence larger vertical velocities. The resulting KC-numbers for the second sea state iswithin range of the initially assumed KC-numbers.

ii

PrefaceThis Master thesis was prepared at the department of Engineering Design and AppliedMechanics at the Technical University of Denmark in fulfillment of the requirementsfor acquiring a Nordic Master degree in Maritime Engineering. The thesis is submittedat the Technical University of Denmark and Chalmers University of Technology forobtaining a double degree as part of the Nordic Master program.

Kongens Lyngby, June 23, 2017

Espen Kleppa (s151091)

iv

AcknowledgementsThis Master thesis is written as a final closure of my Nordic Master degree in MaritimeEngineering the spring of 2017. It is assumed that the reader has basic knowledge inhydrodynamic potential flow theory, viscous drag forces and motion response analysis.It is primarily intended to engineers working within the offshore and maritime indus-try, as well as researchers and students seeking to research the subject matter further.

In the process of writing this thesis, I have gotten support and guidance fromseveral people that I would like to mention.

I would sincerely like to thank my main supervisor Ass. Professor Yan-Lin Shaofor his patience, guidance and support while working on this project. We have hadmany interesting discussions during the last months.

Further I would like to thank my company supervisor Xu Xiang at Statens veg-vesen in Norway. He has taught me and guided me through the software used in thethesis and given good input and support on several matters.

I would also like to thank the people working at the fjord crossing department atStatens vegvesen for taking me in while I was working there for 6 weeks during theproject period.

A thank also goes to my co-supervisors, Ass.Professor Wengang Mao and ProfessorEmeritus Preben Terndrup Pedersen for their support during the project period.

Lastly I would like to thank my parents. They always have faith in me and havesupported me whenever I have needed it trough out my whole education.

vi

ContentsAbstract i

Preface iii

Acknowledgements v

Contents vii

1 Introduction 11.1 Scope of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Earlier studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Theory 72.1 Basic assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.2 Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.3 Viscous drag forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.4 The Equation of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . 132.5 Natural period . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3 Methodology 173.1 Ansys Aqwa software . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.2 Design phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.3 Aqwa Analysis Tool . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.4 Verification of method . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.5 Correlation Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.6 Environmental conditions . . . . . . . . . . . . . . . . . . . . . . . . . 243.7 KC-number estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.8 Cd-coefficient estimation . . . . . . . . . . . . . . . . . . . . . . . . . . 28

4 Results 314.1 Shifting of Natural period . . . . . . . . . . . . . . . . . . . . . . . . . 314.2 Motion analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364.3 Fictive sea state comparison . . . . . . . . . . . . . . . . . . . . . . . . 404.4 Sensitivity of Cd-coefficient . . . . . . . . . . . . . . . . . . . . . . . . 44

viii Contents

5 Discussion & Conclusion 47

Bibliography 51

A Matlab codes Motion Analysis 53A.1 Motion Response Analysis including KC-number estimation . . . . . . 53A.2 Natural period estimation . . . . . . . . . . . . . . . . . . . . . . . . . 103

B Verification calculation 109

C Correlation Analysis 117

D RAOs from extreme sea state simulation 121

E RAOs for Sea State 2 127

CHAPTER 1Introduction

As a part of the Norwegian Public road administrations (NPRA) project to improvethe road system between Kristiansand and Trondheim, there are a lot of fjords thathas to be crossed either by tunnels or bridges. One of the fjords to be crossed isBjørnafjorden between Stord and Os in Hordaland county. It is decided that a bridgeis to be built for this crossing. At the decided location of this crossing, the bridgehas to be more then 5 km wide and the depth of the fjord can reach over 500 meter,which implies that the best solution will be some kind of floating bridge instead ofpier foundations. Different bridge concepts has to be investigated to find the bestsolution for the crossing. One of the bridge designs that are the most promising, is along floating bridge supported by pontoons with span widths between 100 meter and200 meter.

((a)) Potential graphical appearance of thebridge over Bjørnafjorden Baezeni) (2017)

((b)) Existing bridge, Nordhordalandsbrua,over Salhusfjorden/Osterfjorden Salhusgjen-gen (2016)

Figure 1.1: Graphical appearance of the bridge concept investigated in this thesis.

In the south end, the girder is supported by a cable-stayed bridge to increase theheight from the sea surface to the girder, this to meet the requirement of a navigationchannel for large ships going up the fjord. This bridge type can either be side mooredto the seabed at different locations along the girder, or end anchored in both endsof the bridge Xiang et al.. In Figure 1.2 the bridge over Bjørnafjorden is showngraphically how it can potentially look compared to the existing Nordhordalandsbrua.

As Nordhordalandsbrua is showing, this type of bridge concept has been builtbefore so the concept is not new. However the dimensions are tripled compared to

2 1 Introduction

the length of the bridge and pontoon size and the sea environment is also harsherin Bjørnafjorden with wave periods between 3 - 7 seconds and wave heights up to 3meter.

1.1 Scope of the thesisIn the offshore industry it has been more and more common to install ”dampingdevices” on large displacement offshore structures. These ”damping devices” canbe damping plates on Spar or Tension Leg platforms, it can be bilge keels on shipshaped FPSO or it can be circular bottom flanges on a circular FPSO or a floatingwind turbine structure. The application of damping devices is wide and use full.

((a)) Eni Norges Sevan 1000 FPSO, Go-liat, installed with bottom flange. Thomas(2016)

((b)) Typical SPAR platform with dampingplates. News (2017)

Figure 1.2: Two different cases where damping devices are used in the offshore indus-try.News (2017)

The damping devices are installed on these structures to decrease the motionof the structure, most importantly the heave motion. The damping devices willtrigger flow separation, hence viscous damping on the pontoon motion similar toother floating vessels with bilge keels attached. The pontoons on Nordhordalandsbruahad no damping device installed when it was built in in the early 1990’s. It wasno need for it as the sea environment in the fjord was pretty modest compared tocommon north-sea offshore environment. For Bjørnafjorden the sea environment isestimated to be a bit harsher then the case for Nordhordalandsbrua, and NPRA isinvestigating the possibility of using a bottom flange on the pontoons for the bridgecrossing Bjørnafjorden.

In this thesis, the effects of the bottom flange on the pontoon will be investigatedmainly when it comes to heave motion response. An important parameter when it

1.2 Earlier studies 3

comes to heave motion analysis is the drag coefficient, i.e. Cd-coefficient. The Cd-coefficient is closely related to the Keulegian − Carpenter-number, or KC-number,which can be defined in Equation 1.1

KC = V T

D(1.1)

where V is the flow velocity oscillation amplitude, T is the period of the oscillationand D is the characteristic length dimension of the structure. The investigation willinclude motion response analysis and the effect of Cd-coefficient on the pontoon inaddition to developing a procedure for estimating KC-number of the pontoon basedon data from earlier similar structures. The investigation will also include the tuningof the natural period of the structure in regards to the width of the flange.

1.2 Earlier studiesThere have been several earlier studies where the effect of heave damping plates hasbeen investigated. To get an idea of the scope of the problem a literature study hasbeen done on several papers written by highly rated naval architects.

Figure 1.3: Behaviour of Damp-ing Coefficient Z against diame-ter ratio Dd/Dc from paper ofTao and Cai(2004)

Tao and Cai(2004) has done experiments ona circular cylinder with a bottom plate attachedwhich is similar to a Spar or TLP platform withbottom plate. In offshore environment with longwave periods, resonant heave oscillation might oc-cur and cause damage to risers and mooring sys-tems.

The goal for their study is to investigate theeffects of a systematic disc ratio variation, Dd/Dc,on the vortex shedding pattern, and study the cor-responding associated heave damping effect origi-nating from the vortex shedding of the cylinderand damping plate.

In the paper they are taking a close look on theKC- number and the frequency parameter β, andtheir effect on the vortex shedding pattern. It wasfound in the study that the diameter ratio of thedisc and cylinder was strongly influencing the re-sults of the vortex shedding and viscous damping,but also that the the diameter ratio was stronglydependent on the KC-number and that the form drag on the structure is highlydependent on the KC- number.

4 1 Introduction

In the case study presented in the study they did numerical simulation on a fullsize Spar platform with a draft of 198.1 meter, 39 meter diameter hull, and a range offive different disc diameters ranging form 42.1 meter to 78 meter. They run 6 differentconfigurations for the five ratios, and one bare hull configuration, for three differentamplitudes of oscillation and at a period of 28 sec. The corresponding KC-numberswhere 0.15, 0.44 and 0.74. The resulting behaviour of damping coefficient Z with thediameter ratio is shown in Figure 1.3. One can see from the curve that the Z againstDd/Dc show a weak non- linear trend at the different KC-numbers, and when the discdiameter is getting larger the curves seem to flatten out and converge to a specificZ-value.

The OMAE2016-55059 paper by Shao et al. (2010) considering motionanalysis of cylindrical floating structure with heave damping bilge boxes attached,more specifically the Sevan 1000 FPSO. The application used here is stochastic lin-earization of the drag term in Morison Equation, and an inhouse code is developedand verified by comparing with WADAM software results. The linearized drag forcesis added in to the standard panel method and generates RAO’s which is correspondingwell to the model test results of the FPSO shown in Figure 1.4.

((a)) Heave RAO Comparison, Hs=8.3m,Tp=12.6s, γ = 2.0. Cylindrical FPSO withbilge box, Ballast condition.

((b)) Heave RAO Comparison, Hs=17.2m,Tp=16.8s, γ = 3.0. Cylindrical FPSO withbilge box, Ballast condition.

Figure 1.4: RAO comparison from Shao et al. (2010) for The Sevan 1000 FPSO.

Figure 1.4 shows the model test RAO’s compared to the numerical analysis RAO’s.The red line has the estimated Cd included, and the green has both the Cd and apercentage of Cm added. The application presented in this paper is estimating theRAO’s quite well. Two sea states are considered in the analysis and they are shownin Table 1.1.

1.2 Earlier studies 5

Sea state Hs [m] Tp[m] Wave spectrum1 8.3 12.6 JONSWAP, γ = 2.02 17.2 16.8 JONSWAP, γ = 3.0

Table 1.1: Discription of sea state

The KC-numbers for thetwo sea states is calculated byusing the significant relativeheave motion at the centre ofthe FPSO, and they are quitelow. 0.21 for sea state 1 and0.42 for sea states 2, whichagrees well with other studies done for large displacement structures. The estimatedKC number and Cd-coefficients are used to compare to the study of TTao and Cai(2004) reported earlier in this thesis. The damping coefficients for KC = 0.74 showedgood agreement with the numerical results in Tao and Cai (2004), but for KC = 0.15and 0.44 they are underestimated.

In both of the studies of Tao and Cai(2004) and Shao et al.(2010), they concludethat flow separation viscous effect has an impact on the added mass and naturalperiod of the structure and that the KC-number clearly has e big influence on thedrag coefficient for large displacement structures. It would be interesting to studythese same aspects on the bridge pontoon with the bottom flange.

6

CHAPTER 2Theory

A floating structure has 6 degrees of freedom and for the pontoon investigated in thisthesis, the 6 degrees are oriented as shown in Figure 2.1.

((a)) Global body motion modes for thebridge et.al.

((b)) Local body motion modes for the pon-toon.

Figure 2.1: Definition of body motion modes for the global bridge and the localpontoon.

As can be seen from Figure 2.1 the orientation for the local coordinate systemon the pontoon is different then that for the global coordinate system on the bridge.The global coordinate system has defined its surge motion in the direction of thelongitudinal bridge girder, while the surge motion for the pontoon is defined perpen-dicular to the bridge girder in the longest length parameter of the pontoon. Thismake logic sense if the pontoon is considered as a ship in a hypothetical situation. Inthis thesis only the pontoon will be considered and the local coordinate system andmotion definition will be used.

8 2 Theory

2.1 Basic assumptionsThe following theory outlined in this chapter will have its reference mainly from theAqwa Theory Manual ANSYS (2013a) and the text book ”Sea loads on ships andoffshore structures” by O.M.Faltinsen (1990).

If a fluid is assumed incompressible, invicid and irrotational, potential theorycan be used for calculating a hydrodynamic problem. The pontoon in this thesis isassumed to be a large volume structure and thus the potential flow effects are thedominating effects. There will also be viscous effects present, but these are secondaryeffects and can be included empirical. It is convenient regarding mathematical analy-sis to use the velocity potential to describe the velocity vector in a irrotational fluid.In an incompressible fluid the velocity potential has to satisfy the Laplace equationin equation 2.1.

∂2ϕ

∂x2 + ∂2ϕ

∂y2 + ∂2ϕ

∂z2 = 0 (2.1)

The pressure in the fluid originates from Bernoulli′sEquation and can be writtenas in Equation 2.2 for a incompressible, invicid and irrptational flow, where C is aarbitrary function of time and V is the velocity vector V(x, y, x, t)=(u, v, w).

p + ρgz + ρ∂ϕ

∂t+ ρ

2V ∗ V = C (2.2)

Relevant boundary condition for solving problems in potential theory are theKinematic Free Surface Condition (KFSC) and the Dynamic Free Surface Condition(DFSC). The KFSC basically states that the fluid particles on the free surface remainon the free surface, while the DFSC states that the water pressure is equal to theconstant atmospheric pressure p0 on the free surface. In linear form based on thevelocity potential being proportional to the wave elevation, ξ, KFSC and DFSC canbe written:

∂ξ

∂t= ∂ϕ

∂zon z = 0 (kinematic condition) (2.3)

gξ = ∂ϕ

∂ton z = 0 (dynamic condition) (2.4)

The linear wave theory and its derivation can be seen in many books and papers.If a structure is considered floating in incident regular waves with amplitude of ξa

and the wave steepens is small. Basically linear theory means that the wave inducedmotion and load amplitudes are linear proportional to ξa. The derivation of the

2.2 Response 9

theory will not be done in this paper, but the reader is referred to take a look inO.M.Faltinsen (1990), page 16, Table 2.1 if it is of interest.

Moving on to the introduction of environmental conditions, statistical measureddata is important when it comes to designing a vessel for the correct conditions itwill operate in. It is assumed that the sea can be considered as a stationary randomprocess, and it can be referred to as a short term description of the sea. Calculatinga wave spectrum for a sea state is highly dependent on the significant wave heightand period. The method used describing the wave spectrum will be derived later inthe report.

2.2 ResponseThe response of the pontoon is considered operating in steady state waves, this impliesthat the linear dynamic loads and motions acting on the a floating structure areharmonically oscillating at the same frequency as the exciting wave loads on thestructure. The hydrodynamic wave problem can mainly be divided in to two kind offorces;

1. Wave excitation loads, also called Froud-Krylov force and Diffraction force.These loads occur when the structure is restrained from oscillating and there areincident regular waves acting on the structure.2. Added Mass, Damping and restoring forces are the forces and moments onthe structure when the structure is forced to oscillate at the same frequency as thewave excitation frequency. This counts for every single rigid-body motion mode, andthere are no incident waves.

2.2.1 Added Mass and DampingWhen a floating structure is forced to oscillate, the structure is generating radiationwaves that are outgoing from the structure. This forced motion results in an oscil-lating fluid pressure on the body surface, and by integrating this fluid pressure overthe body surface one can obtain the the resulting forces and moments on the body,also called added mass and damping loads. The added mass is the force due to thewater that has to be displaced as the structure oscillates, and the damping is theforce due to the energy carried away from the structure through radiated waves fromthe oscillating body.

In this thesis the heave motion of the pontoon is to be studied more thorough.To find the fluid motion when a structure is free to move in a fluid domain, it isconvenient to use the velocity potential. When the velocity potential is determined,the pressure can be found by using the linearized Bernoulli’s equation. By excludinghydrostatic pressure and integrating the remaining linearized pressure over the meanposition of the body, a vertical force on the body will be obtained. The linear forceobtained can be written as in Equation 2.5.

10 2 Theory

F3 = −A33d2η3

dt2 − B33dη3

dt(2.5)

where A33 is the added mass in heave direction and B33 is damping in heave direction.

2.2.2 Froude-Krylov and DiffractionThe unsteady fluid pressure form the incident waves when a structure is restrainedfrom moving can be divided into two effects. One is the unsteady pressure induced bythe undisturbed waves, while the other is the force due to the fact that the structureis changing this unsteady pressure field. The forces are called Froude-Krylov andDiffraction force respectively. The diffraction problem can be solved in a way similarto the added mass and damping problems where one has to solve the boundary valueproblem for the velocity potential. The boundary condition where the normal deriva-tive of the diffraction velocity potential for the submerged body has to be oppositeand of identical magnitude as for the normal velocity of the undisturbed wave system.This ensures that the normal component of the total velocity on the structure is zero.

2.3 Viscous drag forces

Figure 2.2: Illustration of the stripdz on the on the pontoon.

When it comes to calculating loads on circularand cylindrical structures or members whereviscous forces are active, using the Morison’sequation is a nice approach. O.M.Faltinsen(1990) is describing the Morison’s equation fora horizontal force acting on a strip of length dzof a circular cylinder to be:

dF = ρπD2

4dzCma1 + 1

2ρCdDdz|u|u (2.6)

where ρ is the mass density of water, D isthe cylindrical diameter, a1 and u is the accel-eration and velocity of the undisturbed horizontal fluid, Cm and Cd is the mass anddrag coefficients determined empirically by different parameters like Reynolds num-ber, KC-number etc. Applying this theory to the pontoon shown in Figure 2.2, anapproximation of dz will be the diameter/beam of the hull projected in the longitu-dinal x-direction. In a case where the structure is is moving, equation 2.6 can beextended to Equation 2.7.

2.3 Viscous drag forces 11

dF = ρπD2

4dz(Cm + 1)a1 − ρ

πD2

4dzCmη + 1

2ρCdDdz|u − η|(u − η) (2.7)

η is the displacement of the moving body at the strip dz, hence η and η are thevelocity and acceleration due to the body motion respectively, or in other words, therelative acceleration and velocity. In this thesis the numerical model will be run in apotential flow theory solving software, hence the vertical viscous effect on the flangehas to be modeled in. Therefore there will be added linearized Morison Elements inthe numerical model to act as the viscous effects on the flange of the pontoon. Thisvertical drag force due to the flange can be expressed with a Morison-like formula,using the area of the bottom disc:

dFdrag = 12

ρSCd|u|u (2.8)

where S = ((π*D2d)/4 + Dd*Ld) is the area of the whole bottom plate, where Dd

is the disc diameter and Ld is the length of the mid section of the bottom disc, Xianget al.(2017). The drag term in Morison’s equation is important in order to get correctviscous effect.

2.3.1 Added mass coefficient Cm

In Section 2.3 the Morison’s Equation was given as:

dF = ρπD2

4dz(Cm + 1)a1 − ρ

πD2

4dzCmη + 1

2ρCdDdz|u − η|(u − η) (2.9)

According to the paper of Shao et al.(2010) the first item in Equation 2.9 isconnected to the added mass of the structure while the middle item with the term(Cm + 1) in it is connected to excitation loads. Further, when the damping devicesare modeled into the numerical panel model it is important to not double book thepotential flow contribution of the added mass coefficient. The added mass coefficientcan be seen like this: Cm = Cpot

m + Cviscm . When the damping devices are modeled

into the numerical panel model, Cpotm = 0 should be used. This is indeed the case for

the pontoon investigated in this thesis. For larger displacement structures the addedmass can have great effect on the structures RAO’s and Natural period.

12 2 Theory

2.3.2 Morison Drag LinearizationThere are two linearization methods in the literature. One is Stochastic linearization,the other is regular wave linearization which is applied in this thesis. The quadraticrelative velocity term in the drag part of the Morison’s Equation can be written|ur|ur = |(uw − ub)|(uw − ub) where uw is the velocity of the water particle motion inthe fluid, and ub is the velocity of the floating body motion. In the ANSYS (2013a),the Morison Drag Linearization described for a submerged tube/cylinder, and it isdone by replacing the non-linear term |(uw − ub)| by a factor multiplied by the rootmean square of relative velocity in order to create an equivalent linear term. Thisfactor is according to Borgman , α = (8/π)1/2. The linearized drag force at a crosssection of a circular cylinder can then be expressed as

dFdrag = 12

ρDCDαurms(uw − ub)dl (2.10)

dFdrag = 12

ρDCDαurmsuwdl − 12

ρDCDαurmsubdl (2.11)

where urms is the root mean square of the transverse directional relative velocity,and practically the b = 1

2 ρDCDαurmsdl is the damping coefficient for the structure.The first item on the right hand side of Equation 2.11 is representing the excitationloads, while the last item on the right hand side is representing the damping loadson the structure in question. Then the resulting linearized drag force can be foundby integrating the linearized velocity terms over the whole submerged length of thestructure, where L1 is the draft of the tube and (L′ + L1) is the whole submergedtube:

dFdrag = 12

ρDCD

∫ L′+L1

L1

αurmsuwdl − 12

ρDCD

∫ L′+L1

L1

αurmsubdl (2.12)

2.4 The Equation of Motion 13

2.4 The Equation of MotionThe basic model for the wave-induced motions of a floating structure can be describedby the Linear Harmonic Oscillator. If a floating body can only respond in one singledegree of freedom, the vertical heave motion, the equation of motion can be writtenas:

Fex(t) = (m + a)x3(t) + bx3(t) + cx3(t) (2.13)

where Fex is the exciting force, m and a is the mass and added mass, b is damping,c is the hydrostatic stiffness and x3, x3 and x3 is the responding acceleration, velocityand displacement of the body respectively Bingham.

When a floating body is oscillating and it is assumed that both the force of thebody and the response of the body are time harmonic at frequency, the force andresponse can be written as:

Fex(t) = A[XRcos(ωt) − XIsin(ωt)] = ℜ

AXeiωt

(2.14)

x3(t) = ξR3 cos(ωt) − ξI

3sin(ωt) = ℜ

ξ3eiωt

(2.15)

By inserting Equation 2.14 and2.15 into Equation 2.13, the equation of motioncan be rewritten as:

AX = ξ3[−ω2(m + a) + iωb + c] (2.16)

and the resulting transfer function can be written as:

ξ2

A= X

−ω2(m + a) + iωb + c(2.17)

x2(t) from Equation 2.15 is the resulting displacement of the oscillation. Whendifferentiating it the velocity and acceleration of the oscillation can be found:

x3(t) = ℜ

iωξ3eiωt

(2.18)

x3(t) = ℜ

− ω2ξ3eiωt

(2.19)

14 2 Theory

2.5 Natural periodEvery structure has its own Natural Period and is an important parameter whenassessing a structure’s motion amplitudes. If a floating structure is oscillated by waveswith a period that lies in the vicinity of the structures natural period, the structuremight start to oscillate within the resonance period range, which can give dangerouslylarge motions to the structure. This is something that has to be avoided at all cost.Due to high damping or low excitation levels it might be hard to evaluate the responseat resonant periods. The natural period of a structure is highly dependent of theadded mass and the water plane area, and it is easier to get a high natural period fora semi-submersible then for a FPSO. According to O.M.Faltinsen (1990), the naturalperiod can be given for any structure in any motion mode as

Tni = 2π

√Aii + m

Cii(2.20)

where Aii is the added mass, m mass of the structure and Cii is the hydrostaticstiffness. For the pontoon in question, in addition to the heave hydrostatic stiffnessof the structure there has to be added a stiffness from the bridge girder resting onthe pontoon.

Figure 2.3: Illustration of the bridge girder section for each pontoon.

2.5 Natural period 15

According to et.al., the girder can be considered as a simply supported beam andcalculated according to Equation 2.21. In Figure 2.3 is a simple illustration of howthe bridge girder is likely to look while resting on the pontoon.

kwb = 48EIy

L3 (2.21)

where E = 200 GPA is the elastic module for alloy steel, Iy = 18.7m2 is the areamoment about lateral axis and L = 200meters is the distance between two pontoons.All of these properties are fetched from the et.al.. For heave motion Equation 2.20can be rewritten in to Equation 2.22

Tn3 = 2π

√A33 + m

C33 + kwb(2.22)

The pontoon has a high added mass when the flange is added, and this addedmass can be used to shift the pontoons natural period away from the range of thepeak period (Tp) of the sea state. This way the risk of reaching resonance level willminimized.

16

CHAPTER 3Methodology

Calculating forces and motions in a dynamic system can be very challenging andthere are many factors that play a role on the final results. To estimate heave motionfor a relatively large full scale pontoon with a bottom flange with proper accuracy,obviously a CFD analysis is the best choice. For a master thesis it might be tocomputationally expensive to do a proper CFD analysis in 3-D at this design stage.Therefor it is more common to use an approach that involves the Morison Equationin early design stages to model the viscous contributions for a standard panel model.This method has shown itself to be very efficient with acceptable accuracy in thedesign stage of large volume offshore structures. With this method the analysis canoperate in frequency domain with linearized Morison drag. Some of the uncertaintiesof the adopted method will also be addressed in chapter 3 and 4.

3.1 Ansys Aqwa softwareIn order to get data to use in the evaluation of the pontoons characteristics, establish-ment of a proper panel model has to be done. The software used to generate motionanalysis results in this thesis is Ansys Aqwa. This is a Hydrodynamic Analysis tooland it is a modularised, fully integrated hydrodynamic analysis suite based around3-D diffraction/radiation methods and estimate the first order wave loads on a float-ing structure. Aqwa is assuming an ideal fluid with an existing velocity potentialand is using linear hydrodynamic theory described in Chapter 2. Ansys Workbenchimplementation provides a modern interface to develop and solve Aqwa panel models.The panel models are modeled in a sub-software called Spaceclaim and it gives outboth surface models and solid modeles. For the hydrodynamic diffraction analysis towork, surface panel models has to be used. Aqwa is using basic hydrodynamic poten-tial theory and definitions to calculate the motion results, and it also uses Morison’sEquation and Morison linearization to calculate the hydrodynamic forces, which isexplained in Section 2.2.

18 3 Methodology

3.2 Design phaseThrough this thesis period, 6 weeks has been spent in Norway at the offices of NPRA’s,getting input, inspiration and good discussions on many aspects of the thesis. As men-tioned earlier, the pontoon design and dimensions is not finally set when this thesis iswritten. At the modeling stage of the thesis it is decided on a set of design parametersthat will be used, with dimensions on both the the pontoon and flange. On studiesNPRA and their consulting firms has done, a flange width of between 5-6 meters hasbeen most commonly used. The design parameters used to model the pontoon inSpaceclaim is shown in Table 3.1.

Diameter hull 25 [m]Total length of hull 62 [m]Total width of hull 25 [m]

Draft 9.25 [m]Weight ca 14000 [ton]

Diameter flange 37 [m]Flange thickness 0.5 [m]Flange width 6 [m]

Table 3.1: Parameters pontoonFigure 3.1: Pontoon modeled inSpaceclaim with properties as in Table3.1

The model in Figure3.1 is the surface model given from Spaceclaim. Spaceclaimis pretty similar to many other modeling software’s, where a sketch is made and thesketch can be pulled/extruded to make a 3-D model.

Figure 3.2: Morison element circlebeams modeled in Spaceclaim.

It is possible to model as many surfacesas one requires, and it is possible to groupthe surfaces together. There where some im-portant functions in the modeling that hadto be done in order to make a functionalpanel model. The x-y-plane in the coordi-nate system seen in Figure 3.1 is correspond-ing to the fictive horizontal water plane inthe analyzing software Aqwa, so everythingmodeled in negative z-direction would tech-nical be submerged and a part of the draftof the structure. Another important model-ing function is to check that all the surfacenormal’s is pointing outwards.

The surface normal indicates what is theoutside and inside of the model, and the

3.3 Aqwa Analysis Tool 19

direction the normal arrow points is the outside. If some of the normal’s are pointinginwards, an error will occur when motion analysis is run on the model.

In order to account for the viscous effect in this potential theory based Aqwasoftware, one has to model some circle beam sketch elements in Spaceclaim. Theelements are sketched with 0.5 meter vertical clearance to the flange. The elementsare positioned in the middle of the flange on the longitudinal parts. The beams canonly be sketched in straight lines, hence this would cause some difficulties in thecircular ends of the pontoon. The way this was done was to split the half circle intothree equally long long beams and connect them as shown in Figure 3.3. Of coursethis elements could be split into several more smaller beams to fit the circular flangebetter, but due to the time consuming changing of parameters in a later stage of theanalysis it was found that splitting the beam into three beams was sufficient.

3.3 Aqwa Analysis ToolFurther when the surface model is ready, it can be used as input to the analysis toolAqwa. There are a lot of parameters that has to be assigned before running thediffraction analysis, which is using the source distribution method ANSYS (2013a) tocalculate the first order loads on a structure. By introducing the source distributionover the mean wetted surface, the fluid potential can be expressed:

ϕ(X) = 14π

∫S0

σ(ξ)G(X, ξ, ω)dS where X ∈ Ω ∪ S0 (3.1)

Here ϕ(X) s the velocity potential ξ is the position of a source, G, is the Green’sFunction and ω is the angular frequency. By applying the surface boundary conditionwhere:

∂ϕ

∂n=

−iωnj for radiation potential∂ϕ∂n for diffraction potential

in Equation 3.1, the source strength over the mean wetted hull surface can bedetermined. The Hess-Smith constant panel method is deployed into Aqwa to solvefor the source strength, and the mean wetted surface is divided into quadrilateral ortriangular panels.

It is assumed that the potential and the source strength within each panel areconstant and taken as the corresponding average values over that panel surface.

A point mass has to be assigned for the panel model to have any weight at all.one can assign a value or let Aqwa calculate it by the draft of the structure. Thenby solving for the hydrostatic solution, the mass of the structure is calculated. FromSpaceclaim the Morison elements has been modeled as simple circle line beams. InAqwa these circle line beams can be assigned height, width, weight, Inertia, AddedMass coefficient and Drag coefficient. It is important that the added mass and mass

20 3 Methodology

properties do not get double booked, so mass, inertia properties and Added Masscoefficient is set approximately to zero. The Drag Coefficient is set to whatever valueis desired. In Figure 3.3 the Morison Drag elements are visual as the gray rectangle’salong the flange. These plates will work as the viscous drag force on the structure inmotion analysis.

Figure 3.3: Rectangular MorisonDrag elements along the flange inAqwa.

It is important to set a correct and us-able frequency for which the analysis will runfor. The range of frequency and number offrequency that should be analyzed can be in-putted after preference, and can be programcontrolled. A frequency in the range of: ω ∈0.1< ω < 2.5 [rad/s] and 50 frequencies to es-timate for would be sufficient for this thesis.

In hydrodynamic diffraction analysis likethis where surface piercing hulls are considered,there is the occurrence of irregular frequencies.These irregularities might cause large errors inthe solution over a substantial frequency bandaround these frequencies. These errors causeabrupt variations in the calculated hydrody-namic coefficients, and this is the case in all forms of surface piercing hulls. InAqwa there is a function that can be selected to remove these irregular frequencies byusing the internal lid method for the source distribution approach. In this methodit is assumed that the fluid field is existing interior to the mean wet body surface,which is is satisfying the same free surface condition as the flouting structure. Thisinterior mean free surface is represented in Aqwa by a series of internal LID panels,removing the irregular frequencies, ANSYS (2013b).

3.4 Verification of method

As mentioned earlier, hydrodynamic diffraction analysis including the source distri-bution method is used to analyze the structure presented in this thesis. In order toidentify the validity of this method, a comparison of results from earlier studies has tobe done. In the Doctoral thesis of Dr. Yan-Lin Shao, there is a section on diffractionof a simple truncated vertical circular cylinder. Resulting amplitudes of the linearwave excitation forces in surge and heave directions is presented in plots, and com-pared to results from Kinoshita et al. (1997). In order to validate the hydrodynamicdiffraction analysis method used in this master thesis, the results from the referencetruncated vertical cylinder Shao has to be replicated in that method.

3.4 Verification of method 21

((a)) Excitation force Surge motion ((b)) Excitation force Heave motion

Figure 3.4: Results for Excitation force in surge and heave motion for a truncatedvertical cylinder in finite water depth Shao

Figure 3.5: Mesh of truncatedcylinder in finite water depthfrom Ansys Aqwa

The cylinder is analysed with the same pa-rameters as in the reference case: Radius = R,Draft = R, Water depth = 2R. The same methoodis used to present the results in the plots, non-dimensionalizing the resulting force on the Y-axiswith F/ρgR2A and the wave number multipliedwith radius on the X-axis. To get the curves tofit as good as they do, an approximation of wavedispersion relation from the paper of Zai-Jin Youwas used You 2008. The most commonly use wavedispersion relation can be written as in Equation3.3 and 3.2, and are derived under that assump-tion that there is no current, the water depth isconstant, the waves are linear and small and thatthe flow is irrotational. A more explicit solutionof the wave dispersion number was needed to fitthe curve accurate.

k0h = kh ∗ tan(h) ∗ kh (3.2)

ω2 = gk ∗ tan(h) ∗ kh (3.3)

where ω is the wave angular frequency, k is the wave number, k0 is the wavenumber in deep water, g is acceleration of gravity and h is the mean water depth.The following equation for the wave dispersion relation is one of the most accuratesolutions presented in the paper of Zai-Jin You, and the method is developed by J.N.Hunt You 2008.

22 3 Methodology

kh = (k0h)

√√√√1 +

[(k0h)

(1 +

6∑n=1

Dn(k0h)n

)]−1

(3.4)

where D1=0,6666666666, D2=0,3555555555, D3=0,1608465608, D4=0,0632098765,D5=0,0217540484, and D6=0,0065407983. kh is then multiplied with the radius ofthe cylinder and we obtain the the curves in Figure 3.6

((a)) Resulting excitation force Surge mo-tion

((b)) Resulting excitation force Heave mo-tion

Figure 3.6: Results for Excitation force in surge and heave motion for a truncatedvertical cylinder in finite water depth.

Seen from Figure 3.6 one can see that the results from the validation analysis isfitting the results from the paper of Shao(2010) quite good. The shallow water resultsis fitting almost perfectly. It is now safe to say that the hydrodynamic diffractionanalysis proceadure that is going to be used in this thesis will have good agreementwith real model test results and that the method used is valid.

3.5 Correlation AnalysisA mesh has to be defined and the computational time is heavily dependent on the sizeof the mesh. There are pointers given in O.M.Faltinsen (1990) of what are reasonablemesh sizes and number of elements. The number of elements should be between500-1500, and the mesh size should not be larger then 1/8 of the wave length. TheAqwa Theory Manual ANSYS (2013a) states the size relation ship should be 1/7 ofthe wave length, so its more conservative. For the sea state considered in this thesis,the wave length, λ, is calculated to be 61.6 meter at Tp = 6 sec. Then the elementlength can not be longer then 8/61.6 meter = 7.7 meter according to the pointer in

3.5 Correlation Analysis 23

O.M.Faltinsen (1990). A correlation test was done to see how different mesh sizeswas affecting the results.

The idea is to run a simulation in Aqwa for the same pontoon with mesh sizes withelement lengths of 1, 2, 3, 4, 5 [meter]. The simulation will be run for Froud-Krylovand Diffraction force, and Added Mass and the results will be presented in curves tobe analyzed. The results from the analysis is shown in Figure 3.7.

((a)) Resulting Added Mass for correlationanalysis

((b)) Resulting excitation Diffraction andFroud-Krylov force for correlation analysis

Figure 3.7: Results for Excitation force in surge and heave motion for a truncatedvertical cylinder in finite water depth.

Figure 3.8: Element size at 2 meterfor the mesh of the panel model inAqwa.

In Figure 3.7 a, there seem to be some dis-turbance on the results at approximately 0.2Hz. A sudden peak like this in the results mightindicate that the internal lid functions for re-moving irregular frequency was of when thisanalysis was done. A new simulation was runfor the 3 meter element length with the inter-nal lid on. The results from this simulation wasnice and even like it was expected to be, andthe results for this simulation can be Foundin Appendix ?? Obviously the simulation withthe smallest element length will give the mostaccurate results, but when ten’s simulations isgoing to be run, the computational time mat-ter. According to the resulting curves, the 3, 4and 5 meter length elements gives quite unre-liable results for the Added Mass, while the 2 meter length element is giving quitesimilar results as the 1 meter length element. For the Diffraction and Froud-Krylov

24 3 Methodology

force, the results are all over quite close to the 1 meter length element. It is ondecided on basis of the results for the Added Mass and computational time that themesh element length used in further simulations is 2 meter.

3.6 Environmental conditionsThis thesis is written in cooperation with NPRA, and the bridge is supposed to bedesigned for a specific fjord which has a set of more or less specific environmentalconditions. NPRA is still measuring the conditions in the fjord, but the preliminarydata given for this thesis is given in Table 3.2 and corresponds to waves with 100-yearreturn period for Bjørnafjorden.

Peak Period Tp 6 [sec]Significant wave height Hs 3 [m]

Non-dimensional peak shape parameter 1.8 - 2.3

Table 3.2: Sea state parameters in Bjørnafjorden (”Sea State 1”)

In NPRA’s and their consultants investigation on the bridge, they have given outseveral OMEA-papers regarding several aspects of the interest. In their analyses theJonswap Wave Spectrum is used to describe the sea state, a wave spectrum that willbe adopted in this thesis from the DNV-RP-C205 rules and regulations DNV-GL(2010). Definition of a wave spectrum is also required in Aqwa when running motionanalysis, and it is possible to choose which type to input. The Jonswap is based onthe Pierson-Moskowitz Spectrum which is defined as

SP M (ω) = 516

H2s ω2

p

ω5 exp(

− 54

( ω

ωp

)−4)(3.5)

where ω is the angular frequency and ωp=2π/Tp is the angular spectral peakfrequency. The Jonswap Spectrum can then be found according to Equation 3.6:

SJW (ω) = Aγ ∗ SP M (ω) ∗ γexp

(−0.5

(ω−ωp

σ ωp

))(3.6)

where Aγ is a normalizing factor, γ is the non-dimensional peak shape parameterand σ is the spectral width parameter. In general the Jonswap Spectrum has higherspectral density peak then the Pierson-Moskowitz Pectrum, and the peak varies inheight when γ is changed. From the wave spectrum it is possible to calculate the re-sponse spectrum of a floating body, by multiplying the wave spectrum by the squaredTransfer Function of the body.

3.7 KC-number estimation 25

3.7 KC-number estimationAs mentioned earlier the KC-number can be used to determine the drag coefficient of afloating structure, referring to Section 1.2. For a big structure with large displacement,as the pontoon, the KC- number is expected to be a low value, approximately

KC = 0.1 < KC < 0.5.Based on the literature studies in Section 1.2 and the theory outlined in Section

2.2, a procedure has been developed to estimate the KC-number for the pontoon. Theprocedure is visualized in Figure 3.9 and indicates that a KC-number first has to beassumed in order to perform the analysis. If a constant KC-number is to be used inirregular waves, there are no rational way to decide it because of the varying periodof the waves and motion of the structure. KC-numbers has to be assumed based onearlier literature and studies on similar structures.

Figure 3.9: Basics of the procedure used to identify the KC-number.

26 3 Methodology

The KC-number, or the Keulegian-Carpenter number, can be expressed:

KC = V T

D(3.7)

where V is the flow velocity oscillation amplitude, T is the period of the oscillationand D is the characteristic length dimension of the structure. The period and thelength dimension are given by the sea state parameters and design parameters. TheCharacteristic length dimension is taken to be the beam of the bottom disc, as a lotof the flow separation will happen perpendicular to the length of the pontoon.

The oscillating velocity is not known until it is solved for, and is a key componentto estimate the correct KC-number. There are several ways to determine a motionvelocity, and using linear theory to simulate irregular sea, it is possible to find sta-tistical estimates for the motion amplitude. In this thesis a 3 hour Most ProbableMaxima (MPM) and Significant Value (H 1

3) will be used for the motion amplitude.

Figure 3.9 is visualizing the procedure used to identify the KC-number:A sea state has been enlightened in Section 3.6, and the wave spectrum can be

used together with the heave motion RAO’s to estimate a frequency domain responsespectrum for heave motion shown in Equation 3.8:

SR(ω) = SJW (ω)|H3(ω)|2 (3.8)

Here S(ω) is the Jonswap wave spectrum and |H3(ω)2| is the heave motion RAO’sestimated in Aqwa. In all Equations in this section, ω represents the angular frequency.The spectral moments M0 and M2 can then be calculated from the response spectrum.

M0 =∫ ∞

0ω0(SR)dω (3.9)

M2 =∫ ∞

0ω2(SR)dω (3.10)

Spectral moments in 0th and 2nd order are useful to estimate statistical parame-ters. The Standard Deviation SD is calculated in Equation 3.11 and represents thevariation in the response data set.

SD =√

M0 (3.11)

The spectral moments can also be used to find the mean response period, T2 for theresponse data set.

T2 = 2π

√M0

M2(3.12)

3.7 KC-number estimation 27

With all the statistical parameters estimated, the MPM for a 3 hour duration canbe estimated assuming that the response is normal(or Gaussian) distributed and thewave spectrum is narrow-banded, using Equation 3.13 where t is the duration andlog is the natural logarithm. The significant value H 1

3of heave response can also be

estimated by using Equations 3.14.

MPM = SD

√2 ∗ log

( t

T2

)(3.13)

H 13

= 4√

M0 (3.14)

Now, Equation 3.7 gives a clear definition of the KC-number in a harmonicallyoscillatory flow, and the empirical Cd-coefficient used in this thesis will be dependenton the KC-number. Despite the fact that a sea state of 3 hour duration is consideredwhen calculating the MPM, the KC-number and hence the Cd-coefficient is heldconstant in the panel model analysis in Aqwa when obtaining RAOs, hence the Cd-coefficient is not dependent on time.

Resulting from the response in the Equation of Motion in Section 2.4, the verticalmotion amplitude can be differentiated from the displacement x2:

x3(t) = ℜ

ξ3eiωt

=⇒ x3(t) = ℜ

iωξ3eiωt

(3.15)

The motion amplitude x3 can then be implemented as the amplitude of the flow ve-locity oscillation of the structure. The KC-number can then be obtained by Equation3.16.

KCestimated = x3Tp

D(3.16)

28 3 Methodology

3.8 Cd-coefficient estimationIn order to investigate how the heave motion of the pontoon is dependent on theCd-coefficient, a range of Cd- coefficient had to be tested. In this thesis CFD analysiswill not be performed as the focus for the thesis is optimization of the pontoon andanalysis of response in irregular waves. For the pontoon configuration (hull+flange)investigated, there are no direct model test data available. It is worth mentioningthat NPRA is performing model tests at Marintek for the bridge pontoon designsthe summer of 2017. Given access to the test results would create many interestingMaster project in the years to come.

Regardless, the available data when this thesis is written, are data for a circularhull with circular bottom flange, typically a FPSO or a Spar platform. So in this thesisCd-coefficients for a cylindrical hull with circular bottom disc will be used. To havesome clue of which range to stay in regarding the KC-number and Cd-coefficient,earlier literature is used. Tao and Cai(2004) investigated how a circular cylinderwith bottom plate is dependent on the damping coefficient at different KC-number.Figure 3.10 (2016) represents Tao’s findings with the 3 curves for KC-number 0.15,0.44, 0.74. Also included in Figure 3.10 is the findings from Shao et al.(2016) wherethe Sevan 1000 FPSO geometry is tested for the same technique. The findings ofShao et al.(2016) are the three red points at ratio 1.24, and they are underestimatinga bit on two of the curves, but it seem to give a good indication for the range ofKC-number for the structure.

Figure 3.10: Behaviour of Damping Coefficient Z against diameter ratio Dd/Dc atdifferent KC-numbers from paper of Shao et.al. (2016)

3.8 Cd-coefficient estimation 29

Adopting this method and testing it for the pontoon would be very interesting.As can be seen from Figure 3.10 Tao and Cai has calculated the damping coefficientfor a range of bottom plate diameter and hull diameter ratios, Dd/Dc, and the curvesshow how the damping coefficient is changing from a non-linear state to a linear stateas the ratio is increasing. The pontoon should also be tested for this ratio.

In order to find the range of Cd-coefficients, the values from the curves in Figure3.10 is adopted by using the Spars parameters presented in Table 3.3, the equationsfor added mass, ma, and damping coefficient, Z, of the Spar presented in Equation3.17 and 3.20 respectively. The ratios that the damping coefficient will be estimatedfor is presented in Table 3.4, and the KC-numbers used are KC = [0.15, 0.44, 0.74].

Spar diameter (Dc) 39 [m]Total draft (Td) 198.1 [m]

Disc thickness (td) 0.475 [m]

Table 3.3: Dim. Spar

Disc diameter (Dd) [m] Ratio (Dd/Dc)39 142 1.07745.1 1.1648.4 1.2453 1.359

57.72 1.4865 1.6670 1.8

Table 3.4: Disc diameter and Ratio.

ma = ρD3d

3−[

πρD2c

8(Dd − D) + πρ(Dd − D)2(2Dd + D)

24

](3.17)

where

D =√

D2d − D2

c (3.18)

The mass can be found using the geometrical parameters where Rhull is the radiusof the cylindrical hull of the Spar andRflange is the radius of the bottom plate:

m = ρ(πR2hull(Td − tf ) + πR2

flangetf ) (3.19)

30 3 Methodology

Z = 13π2

D2dDc

D2cTd + D2

dtd

Cd

(m + ma)/m∗ KC (3.20)

Cd-coefficients can be found by rearranging Equation 3.20:

Cd = Z ∗ (3π2)(D2c Td + D2

dtd)((m + ma)/m)KC ∗ (D2

dDc)(3.21)

By using the three KC-numbers with corresponding damping-coefficients at agiven ratio, polynomial fitting can be used to find the Cd-coefficients for any givenKC-number between 0.15 and 0.74. It is chosen to evenly divide the KC-numberrange into 8 numbers with equal spacing ranging from 0.15 - 0.74.

Cd,i = p3 + p2KCi + p1KC2i (3.22)

where p1, p2 and p3 can be estimated by a polyfit function in Matlab. Now thatall of these Cd coefficients are obtained for each ratio, Aqwa can be run by inputtingthe Cd-coefficients on the Morison drag elements. The resulting RAO data will beused to calculate the MPM and H 1

3and a new set of KC-numbers can be calculated,

by inserting them into Equation 3.7.

CHAPTER 4Results

In Chapter 2 and 3 the theory and methods used in this thesis is explained thoroughly.The objective is to analyze the pontoon’s motion response when the flange is varyingin width.

4.1 Shifting of Natural periodAs mentioned earlier, adding the bottom will serve different purposes. Like raisingthe added mass and adding more damping in to the dynamic system. The pontoonhas been simulated for 4 different flange sizes: Flange width, Df = 0, 4, 6,8 [m].

Figure 4.1: Jonswap and Pierson-Moskowitz spectrum compared for the given metao-cean data.

32 4 Results

The Jonswap wave spectrum is shown in Figure 4.1 for the given properties inSection 3.6. The Jonswap spectrum has this extra peak factor γ to account forcontinuous growing waves. By using the Added Mass, mass and hydrostatic stiffnessestimated by Aqwa together with the estimated structural stiffness kwb, the NaturalPeriod for a structure can be calculated according to the theory stated in section 2.5for a given range.

Figure 4.2: Added mass comparison for different flange width.

In Figure 4.2 the Added Mass is increasing with the increasing flange width asexpected. The added mass of the structure is between 1.5 and 3 times the magnitudeof the pontoon itself. The circtangle shape of the pontoon and bottom disc mighthave an impact on these large magnitudes.

4.1 Shifting of Natural period 33

As can be seen from Figure 4.4 and 4.3, the Natural periods are compared inrelation to the wave spectrum. Both natural period with and without bridge stiffnessare calculated and shown in Figure 4.3 and 4.4 respectively. The natural period canbe found using this graphic method. The wave spectrum is representing the energyin the waves at different periods, and the goal is to shift the natural period awayfrom the wave spectrum as far as possible in order to avoid a resonance effect. Theintersection point with the green intersection curve is representing the Natural periodof the different structures. As can be seen there are large differences when adding thebridge stiffness. Studying the plot with the stiffness included, the pontoon with noflange is inside the wave spectrum, which is not ideal at all and risk of resonance ispresent. By adding the flange and increasing it, the Natural period is shifting awayfrom the wave spectrum as predicted, and the risk of resonance is minimizing. Forthe plot where the bridge stiffness is excluded, all of the natural periods are safelyout of range.

Figure 4.3: Natural period without bridge stiffness.

34 4 Results

Figure 4.4: Natural period with bridge stiffness included.

The resulting Natural Periods are listed in Table 4.1. These periods are resultingnatural periods for head sea, hence sway, roll and yaw motion are not excited.

Flange width [m] Tn,3 incl. kwb [sec] Tn,3 excl. kwb [sec]0 5,96 9,44 6,9 10,96 7,5 128 8,2 13,4

Table 4.1: Resulting natural period for different flange sizes

When studying the damping coefficient on the structure, there seem to be a neg-ative side effect when adding a flange. As can be seen from Figure 4.5 the dampingload is getting cancelled out at a certain period when adding the flange and adjustthe width.

4.1 Shifting of Natural period 35

Figure 4.5: Damping load comparison for different flange width.

For the pontoon with no flange, the damping load starts at zero and gives nocancellation effect along the increasing period. As the flange size increases the cancel-lation effect start to occur and a pattern can be seen where the cancellation periodis increasing with the flange size. Without any damping loads the structure motionwill loose this decaying motion effect from the damping, and the structure might beexposed to resonance effects at these certain wave periods. It is also worth noticingthat for pontoons with 8 and 6 meter wide flange, the there is occurrence of negativedamping load. As regular damping has a decaying effect on the body motion, nega-tive damping will contribute to a resonance effect on the body. The occurrence forthis negative damping effect might be due to errors in the mesh at this period for thepontoon. They should be expected to be >=0.

36 4 Results

4.2 Motion analysis

Using the method explained in Section 4.4, a data base of Cd-coefficients can beestimated for each of the diameter ratios, Dd/Dc. As stated in Section 4.4, dragcoefficients will be calculated for 8 different ratios, and for 8 different KC numbers.This implies that 64 different Cd-coefficients will be calculated. However, data onlyfor Dd/Dc- ratio =[1.36, 1.48,1,66] with corresponding Cd = [3.8,5.31.7,87] will bepresented in this report. The RAO-data for each simulation is then collected andused to calculate the Response Spectrum for each Cd-coefficient.

KC–> 0.15 0.2343 0.318 0.4029 0.4871 0.5714 0.6557 0.7400Ratio:1 9.2414 8.6630 7.8304 7.0748 6.2342 5.5764 4.852 4.5019

Ratio:1,077 8.805 8.2544 7.4611 6.7411 5.9402 5.3134 4.6240 4.2896Ratio:1,16 8.4317 7.9040 7.1444 6.4549 5.6880 5.0878 4.4277 4.1075Ratio:1,24 8.1199 7.6117 6.8802 6.2162 5.4777 4.8997 4.2639 3.9556Ratio:1,36 7.8701 7.3775 6.6686 6.0250 5.3092 4.7489 4.1328 3.8339Ratio:1,48 7.6824 7.2015 6.5095 5.8813 5.1825 4.6357 4.0342 3.7424Ratio:1,66 7.5567 7.0837 6.4030 5.7850 5.097 4.5598 3.9682 3.6812Ratio:1,8 7.4930 7.0240 6.3490 5.7363 5.0548 4.5214 3.9347 3.6502

Table 4.2: Estimated Cd-coefficients for each ratio and KC-number.

The RAO data for Dd/Dc- ratio =[1.36, 1.48,1,66] for the pontoon is shown inFigure 4.6 and compared to the wave spectrum from sea state 1.

4.2 Motion analysis 37

Figure 4.6: Comparison of RAOs and wave spectrum for Dd/Dc- ratio =[1.36,1.48,1,66]

38 4 Results

In Figure fig:RES3 the response spectrum for the pontoon for Dd/Dc- ratio =[1.36,1.48,1,66] is displayed. In Figure 4.7 the interacting curves are zoomed in and onecan see there are not much energy for this sea state.

Figure 4.7: Highlighting the important peak points comparison of RAOs and wavespectrum for ratio 1.24.

Looking at Figure 4.7 and the the area under the curves, which represent theenergy in the spectra, it is obvious that the response spectra will consist of low ampli-tudes as well. By integrating the area under the curves along the period, the energyin the response can be found and thus the the heave velocity amplitude following theprocedure in Figure 3.9.

4.2 Motion analysis 39

Figure 4.8: Response spectrum for sea state 1

Now that the Response spectra is estimated, it is relatively safe to say that thedesign of the pontoon with varying flange size is fit to operate in Sea State 1. Followingthe method outlined in Section 3.7 the spectral moments M0 and M2 can be calculatedand eventually the response amplitude can be found. This amplitude is estimated bytwo different approaches, the most probable maxima (MPM) of the amplitude, andthe significant value of the amplitude (H 1

3). As expected the resulting MPM gives

very small values which will result in small vertical velocities as shown in Table 4.3.The resulting (H 1

3) and corresponding velocity amplitude is around the half of the

MPM results, and can be found in Appendix A.1.

Ratio 1.36 1.48 1.66MPM(KC = 0.15) 0,0955 0,0654 0,0796MPM(KC = 0.487) 0,0826 0,0660 0,0791MPM(KC = 0.74) 0,0791 0,0676 0,0784

Table 4.3: Resulting Velocity Amplitude assumed value KC=0.15 for KC-numberestimation, Sea state 1

In Table 4.3 the resulting vertical velocities is shown for a selected range of KC-number = [0.15, 0,487, 0,74]. KC-number = 0.15 gave the highest value for theresponse amplitudes, the largest velocities will appear here. Still the vertical velocitiesare very small, as expected. With this velocity amplitude, the Keulegian-Carpenter

40 4 Results

number for the structure can be estimated using Equation 3.7 The the estimatedKC-numbers will be presented in Section 4.4.

4.3 Fictive sea state comparison

In Section 4.2 the results of motion analysis for Sea State 1 is described, and concludedthat there are no immediate danger for large motions and resonance of the pontoon.If for instance NPRA would test this bridge concept for another fjord where theenvironment is harsher, it is believed that the wave spectrum will have more influenceon the motions of the structure. A fictive sea state is created and described in Table4.4.

Peak Period Tp 10 [sec]Significant wave height Hs 4.5 [m]

Non-dimensional peak shape parameter 2

Table 4.4: Sea state parameters for fictive sea state (”Sea State 2”).

Sea State 2 is still a pretty modest sea state compared to for instance NorthernAtlantic sea state. In Figure 4.9 the wave spectrum for Sea State 2 is shown comparedto Sea State 1 given by NPRA. From the curves one can obviously see that Sea State2, has more energy then Sea State 1. As the spectrum has a larger peak period theremight be more motion response in the vicinity of the peak period.

4.3 Fictive sea state comparison 41

Figure 4.9: Comparison of wave spectrum for Sea State 1 and 2.

Aqwa has been run for a number of chosen parameters described in Table 4.5. Theparameters are chosen on basis of obtaining as much information as possible fromfewer simulations, therefor one high, one small and one KC-number in between ischosen from the assumed range given in Table 4.2, with corresponding Cd-coefficients.The choice of flange ratio, Dd/Dc, is corresponding to a flange width of 4, 6 and 8meter, which has been used to study the Natural period of the Pontoon in Section2.5. Figure 4.10 shows the response spectrum compared to the wave spectrum fromSea State 2. It is obvious from the curves in 4.10 that the motion response for thepontoon will be larger then for Sea State 1 as the RAO curves has its peaks insidethe wave spectrum.

Flange Ratio 1.36 1.48 1.66KC-number 0.15 0.487 0.74

Cd - coefficient 7.87 5.31 3.81

Table 4.5: Parameters for testing fictive sea state.

42 4 Results

Figure 4.10: Comparison of RAOs and wave spectrum for Sea State 2.

Figure 4.11: Comparison of Response curves for Sea state 2.

4.3 Fictive sea state comparison 43

In Figure 4.11 the response spectrum is displayed for Sea State 2, and as predictedthe response motion is larger for Sea State 2 and the peak values are much higherthen for Sea State 1. In this response spectrum there is only one peak for each of theindividual curves, which makes sense since the RAO-curves is positioned at a periodlocated inside the wave spectrum. The MPM and significant amplitudes hare alsocalculated for the Sea State 2, and can be found in Appendix A.1. In Table 4.6 theVertical velocities are displayed, and the trend is still that The MPM is larger thenthe significant value. It is also worth noticing that the velocity is decreasing as theflange size is increasing, meaning that the damping plate is performing as intended.In addition, the vertical velocity and Cd-coefficient is behaving in the same pattern,controlling the drag force on the structure. With this new velocity amplitude, theKeulegian-Carpenter number for the structure can be estimated using Equation 4.1.The Resulting KC-numbers will be displayed in Section 4.4

Ratio 1.36 1.48 1.66xMP M (KC = 0.15) [m/s] 2,3075 2,0295 1,0903xMP M (KC = 0.487) [m/s] 2,5534 1,9409 1,0070xMP M (KC = 0.74)[m/s] 2,2974 1,7168 0,9251xH 1

3(KC = 0.15) [m/s] 1,2397 1,0952 0,5925

xH 13

(KC = 0.487) [m/s] 1,3683 1,0474 0,5469xH 1

3(KC = 0.74) [m/s] 1,2331 0,9265 0,5018

Table 4.6: Resulting Velocity Amplitude assumed value KC=0.15 for KC-numberestimation, Sea state 1

44 4 Results

4.4 Sensitivity of Cd-coefficient

In Section 4.2 and 4.3 motion analysis has been done for two different sea states. Inthis section the last step from the procedure given in Figure 3.9 will be performed, theestimation of KC-number based on the assumed KC-number starting point adoptedfrom similar earlier studies (Tao et.al. 2004). The KC-number is estimated by Equa-tion 4.1.

KCestimated = xTp

D(4.1)

Estimating the KC number for for both the MPM, and the significant value, H 13,

gives the resulting KC-numbers in Table 4.7 and 4.8 for Sea State 1 and Sea State 2respectively. All resulting KC-numbers can be found in Appendix A.1.

Ratio 1.36 1.48 1.66KCMP M (KC = 0.15) 0,0174 0,0106 0,0117KCMP M (KC = 0.487) 0,0150 0,0107 0,0116KCMP M (KC = 0.74) 0,0144 0,0110 0,0115

Table 4.7: Resulting Estimated KC-number(MPM) for KC=[0.15, 0.487, 0,74 ], Seastate 1

The estimated KC-numbers for Sea State 1 are very small as expected. KC-numbers equal to 0.011 and similar for all ratios does indicate that the pontoon isnot very dependent on Cd-coefficients. It does not seem that the design presented inthis thesis is very affected by the sea state it is supposed to be operating in. From adesigning point of view this is a good thing, it means that the bridge probably willwill be very comfortable to walk and drive on as the heave motion is very small.

However, for Sea State 2 the motions responses are larger and hence the resultingKC-numbers. Varying between 0.225 - 0.82, the estimated KC-numbers are withinrange of the initially assumed KC-numbers. There can be noticed a trend that theKC- number is decreasing as the flange size is increasing, indicating that the verticalvelocity is more dominant then the characteristic dimension of the structure whenestimating the KC-number for a given period.

4.4 Sensitivity of Cd-coefficient 45

Ratio Dd/Dc 1.36 1.48 1.66KCMP M (KC = 0.15) 0,6992 0,5485 0,2659KCMP M (KC = 0.487) 0,7737 0,5245 0,2456KCMP M (KC = 0.74) 0,6961 0,4640 0,2256KCH 1

3(KC = 0.15) 0,3757 0,2960 0,1445

KCH 13

(KC = 0.487) 0,4146 0,2831 0,1334KCH 1

3(KC = 0.74) 0,3737 0,2504 0,1224

Table 4.8: Resulting Estimated KC-number for KC=0.15, Sea state 1

In Figure 4.12 the estimate KC-numbers are plotted against the assumed KC-numbers they are based upon. The curves indicate how the KC-number is changingalongside the flange width. Using graphing method the final estimated KC-numberfor each flange size can be obtained when the KC-curves cross the interaction curve.The final estimated KC-numbers are displayed in Table 4.9. For the curves to crossthe interaction curve, extrapolation had to be used with the Matlab function ”polyfit”described in Section 4.4.

Figure 4.12: Comparison of Response curves for Sea state 2.

46 4 Results

KC-curve Final estimated KC-numberKCMP M (KC = 0.15) Sea State 2 0.71KCMP M (KC = 0.487) Sea State 2 0.519KCMP M (KC = 0.74) Sea State 2 0.26KCH 1

3(KC = 0.15) Sea State 2 0.415

KCH 13

(KC = 0.487) Sea State 2 0.295KCH 1

3(KC = 0.74) Sea State 2 0.145

Every estimation from Sea State 1 0.011

Table 4.9: Resulting Estimated KC-number from graphic readings.

First of all comparing the two sea states and the resulting KC-number it gives, itis rather surprising that the gap is that high. Both sea states are modest sea statescompared to the size of the structure, but clearly the change in the peak period ofthe sea state is of great importance to the response. The resulting KC-numbers forSea State 1 is very low for all ratios, indicating that the for this sea state the pontoonis clearly inertia dependent, and not affected by the drag forces all.

When adjusting the sea state up to Sea State 2, the results get more interesting.The KC-number is clearly depending on the flange size and the resulting KC-numbersare within of the range that was initially assumed. The KC-numbers are still smallwitch means that the structure still is more inertia dependent, but it indicates thatthe methodology used in this thesis show relatively good agreement with the paperof Tao and Cai (2004).

In addition to these results, simulations for a third sea state was run to test thepontoon for extreme conditions. The sea state has a Hs=14 meter and a Tp of 15second. It is expected for this sea state to give huge responses and large oscillatingvelocities, resulting in large KC-numbers. Unfortunately, due to time consumption,a proper analysis could not be done and presented with results here. However theresulting RAOs and Wave specter data is attached in Appendix D.

CHAPTER 5Discussion &

ConclusionIn this thesis the effect of a bottom flange on floating bridge pontoon, supposed tobe located in Bjørnefjorden Norway, has been investigated. The approach has beento develop a combined model in the hydrodynamic analysis software Ansys Aqwa,develop combined models for different flange sizes and analyze them to obtain load andmotion characteristics. The panel model has also been analyzed for its dependenceon the KC-number in different sea states. The theoretical background for describinghydrodynamic loads and responses has been addressed. After analyzing the results,some conclusions can be made.

The main goal for adding the bottom disc to the pontoon is to shift the Naturalperiod of the structure away from the peak period of the sea state, by increasingthe added mass of the structure. Four different flange sizes was analyzed and onecould clearly see the natural period of the pontoon shift away from the peak period.When calculating the natural period of the structure, the bridge girder stiffness istaken into account. However the bridge might be side-anchored and then a mooringstiffness should be added in the calculations. If the mooring stiffness is added theincreasing flange size is expected to have less impact on the shifting of the naturalperiod.

Studying the damping loads, a negative side effect surfaced when increasing theflange. The hydrodynamic damping started to cancel out at certain periods for thedifferent flange sizes, the cancelling period increasing alongside the increase of theflange. For the two largest flanges, there are also occurrence of relatively small loads ofnegative damping load. These negative damping loads might contribute as resonanceeffects at these specific periods for the structure. Another explanation can be thatnegative damping is due to a positive feedback load between the oscillation of thepontoon and the interaction of the incoming waves. But most likely it is due to errorsin the mesh in the combined model in Aqwa. Studying Figure 4.5 it is clear that thenegative damping is occurring at periods with good clearance of the 100-year returnpeak period for the sea state, hence it is very unlikely that the period for negativedamping will be exceeded.

48 5 Discussion & Conclusion

From the RAO data collected from Ansys Aqwa for Sea State 1, there is a cleartrend that the structure is responding at the various periods away from the wavespectrum and for all of the simulations. This implies that the Response spectra willhave very small energy and hence very small structural oscillating velocity amplitude.This is resulting in small KC-numbers for all flange sizes, referring to Table 4.7, whichis quite surprisingly different from the original assumed range of KC-numbers. It issafe to conclude that the structure is highly independent of the Cd-coefficient at thissea state, and the pontoon is more or less dominated by inertia forces. Only the heavemotion is considered in this thesis, but it gives a good indication that the pontoonhas solid characteristics against this sea state, and might have oversized dimensions.Changing the size and design of the pontoon could be profitable on a huge projectlike this.

A second sea state was introduced to investigate the response of the pontoon inharsher environment. The new wave spectrum immediately has a large impact onthe response and gives KC-numbers varying from 0.25 to 0.77 in value, referring toTable 4.8. This is still generally small values of the KC-number, but it is in thesame range as initially assumed range of 0.15-0.74. This is interesting regarding themethod used in this project. The initial Cd-coefficients was selected on the basis ofliterature studying large circular structures with bottom discs. With different geom-etry harmonically oscillating in irregular waves and simplified numerical calculations,relatively similar KC-numbers are obtained. Obviously there are factors which arenot taken into account in this thesis, and only two sea states was tested properly, butit gives an indication that the procedure works.

In this thesis the motion response analysis of the pontoon has been performedbased on simplified methods adopted from earlier studies on similar structure. Asthis is very relevant topic for NPRA and they will continue working on this designfor many years, it would be very interesting to investigate more on the design. Inthe summer of 2017, NPRA is performing model tests at Marintek in Trondheim onpontoon designs. It would be interesting to use the motion data from the modeltests to make a proper numerical model for tuning different force coefficients, likeCd-coefficient or Cm-coefficient.

Also the data can be used to study the added mass more thoroughly. In thepaper of Javier Moreno et.al. it is found after performing still water experiments ona cyllindrical hull with bottom disc, at lower KC values the added mass coefficientscould differ by 30%, which can affect the natural frequency estimates of a structure. Itwould be interesting to investigate the effect this 30 % loss could have on the naturalperiod on the pontoon.

In addition, in this thesis mainly the first order loads has been used to investigatethe motion of the pontoon. Further it could be interesting to look at secondaryloads, like the sum frequency load or the or the wave drift loads. These loads willhave smaller impact on the motion, but it would be interesting to investigate theircontribution.

5 Discussion & Conclusion 49

Lastly it would be interesting perform a study on heel angles of the pontoon. Fromthe offshore sector studying Spar platforms with bottom discs, strong currents in theMexico Golf has tended to heel the platforms radically. With a bottom disc attachedto the pontoon, incoming drift forces in the fjord and small occurring heel angle of thepontoon, it would be interesting to study the effect this would have on the pontoonand the bridge girder.

50 5 Discussion & Conclusion

F. S. . V. . Baezeni), June 2017. [Online]. Avail-able: https://www.uis.no/forskning-og-ph-d/nytt-fra-forskningen/skal-sikre-verdens-lengste-flytebru-mot-kraftig-vind-article115032-8389.html

Salhusgjengen, “Nordhordlandsbrua,” March 2016. [Online]. Avail-able: https://www.geocaching.com/geocache/GC6CDFD_nordhordlandsbrua?guid=c6d5c017-5ab0-4d08-b9a9-5a18d5ebbf5e

X. Xiang, Øyvind Nedrebø, M. E. Eidem, B. Sørby, E. Svangstu, B. Jakobsen, andP. N. Larsen, “Viscous damping modelling of floating bridge pontoons with heav-ing skirt and its impact on bridge girder bending moments, omae2017-61041,”OMAE2017-61041.

M. Thomas, “Giant challenge brings innovative solutions,” May 2016. [Online].Available: http://www.epmag.com/giant-challenge-brings-innovative-solutions-846676

H. News, “Helifuel delivers for extreme conditions,” May 2017. [Online]. Available:https://www.helifuel.no/helifuel-aasta-hansteen-delivery/

L. Tao and S. Cai, “Heave motion suppression ofa spar with a heave plate.”

Y.-L. Shao, J. You, and E. B. Glomnes, “Stochastic linearization and its application inmotion analysis of cylindrical floating structure with bilge boxes, omae2016-55059,”OMAE2016-55059.

A. G. F. et.al., “Omae2017-62698, hydrodynamical aspects of pontoon optimizationfor a side-anchored floating bridge,” OMAE2017-62698.

I. ANSYS, “Aqwa theory manual,” November 2013. [Online]. Available: http://148.204.81.206/Ansys/150/Aqwa%20Theory%20Manual.pdf

O.M.Faltinsen, Sea Loads on ships and offshore structures, 3rd ed. CambridgeUniversity Press, 1990.

L. Borgman, “Random hydrodynamic forces on objects’, the annals of mathematicalstatistics, [vol.38, no.1, pp37-51.].”

H. B. Bingham, “Supplemental lecture note for: “wave loads on ships and offshorestructures”.”

I. ANSYS, “Aqwa user’s manual,” November 2013. [Online]. Available: http://148.204.81.206/Ansys/150/Aqwa%20Users%20Manual.pdf

Y.-L. Shao, “Numerical potential-flow studies on weakly-nonlinear wave-body inter-actions with/without small forward speed,” Ph.D. dissertation, NTNU.

Z.-J. You, “A close approximation of wave dispersion relation for direct calculationof wavelength in any coastal water depth.”

5 Discussion & Conclusion 51

DNV-GL, “Recommended practice det norske veritas dnv-rp-c205 environmentalconditions and environmental loads,” October 2010. [Online]. Available:https://rules.dnvgl.com/docs/pdf/dnv/codes/docs/2010-10/rp-c205.pdf

52

APPENDIX AMatlab codes Motion

AnalysisA.1 Motion Response Analysis including KC-number

estimation• RAO from AQWA, DATA HANDLING• Wave spectrum to fit the final RAOs, Hs=3m, Tp=6sec• SW 2 to fit Final RAO data, sea state Hs=4,5m, Tp=10sec• Wave spectrum Hs=3m, Tp=6sec• WS 2, sea state, Hs=4,5m, Tp=10sec• RAO data from Aqwa data• Interpolating frequency data set to make a common frequency• Interpolating RAOs and SW to make common freqency data• Plotting RAOS• Plotting final RAOs:• response spectrum• Plotting response for all sea states• Velocity estimation for final RAOs test, sea state, Hs=3m, Tp=6sec• Velocity estimation for final RAOs, sea state, Hs=4,5m, Tp=10sec• Velocity estimation for sea state Hs=3m , Tp=6sec• Plotting final KC-comparison curve• Finding new CD-values using data from Tao et.al.• Plot of Cd-curve• Cd - coefficients for each ratio• Quadric fitting for every ratio

clc;clear all;close all;% Espen Kleppa% Master Thesis

54 A Matlab codes Motion Analysis

Tp = 6;Tp2 = 10;Hs = 3;Hs2 = 4.5;rho = 1.025;A = Hs/3;A2 =Hs2/2;D_flange = [39 42 45.1 48.4 53 57.72 65 70 78]; %meter

RAO from AQWA, DATA HANDLING[table1]=xlsread('5_75');RAO_heave = [table1(:,3) table1(:,2)];RAO_Omega = table1(2:end,3);RAO_amp = table1(:,2);RAO_period = table1(2:end,1);

% Data for different flange size 4, 6, 8, sea state 1 and 2% run for Cd: 3.18, 5.31 and 7.87:[table_S1_S2]= xlsread('RAO_final');

[table9]= xlsread('RAOs_2'); %data fo Sea state 1

Wave spectrum to fit the final RAOs, Hs=3m, Tp=6secwp= 2*pi/Tp;

%A_pm = trapz(f0,SW_pm_DNV);sigma = 0.09;sigma2 = 0.07;gamma = 2.05;

AG = 1-(0.287*log(gamma));

w2 = linspace(2.51954,0.1,100);w2=w2';%w1=RAO_Omega;SW_pm_DNV_f1 = 5/16 .* Hs.^2.*wp.^4 .*w2 .^(-5) .* exp(-(5/4)...

.*(w2./wp).^(-4)) ;%DNV-RP-C205

SW_jw_DNV_f1 = AG.*SW_pm_DNV_f1.*gamma.^(exp(-0.5.*...((w2-wp)./(sigma*wp)).^2));

A.1 Motion Response Analysis including KC-number estimation 55

SW 2 to fit Final RAO data, sea state Hs=4,5m, Tp=10secwp2= 2*pi/Tp2;T_w2 = (2*pi)./w2;

SW_pm_DNV_f12 = 5/16 .* Hs2.^2.*wp2.^4 .*w2 .^(-5) .* exp(-(5/4)....*(w2./wp2).^(-4)) ;%DNV-RP-C205

SW_jw_DNV_f12 = AG.*SW_pm_DNV_f12.*gamma.^(exp(-0.5.*...((w2-wp2)./(sigma*wp2)).^2));

Wave spectrum Hs=3m, Tp=6secwp= 2*pi/Tp;

%A_pm = trapz(f0,SW_pm_DNV);sigma = 0.09;sigma2 = 0.07;gamma = 2.05;

AG = 1-(0.287*log(gamma));

w1 = linspace(2.4783,0.1,100);w1=w1';%w1=RAO_Omega;SW_pm_DNV1 = 5/16 .* Hs.^2.*wp.^4 .*w1 .^(-5) .* exp(-(5/4)...

.*(w1./wp).^(-4)) ;%DNV-RP-C205

figure()plot(w1, SW_pm_DNV1)%axis ([0 3 0 3])title('PM specter')legend('PM')

SW_jw_DNV1 = AG.*SW_pm_DNV1.*gamma.^(exp(-0.5.*((w1-wp)./(sigma*wp)).^2));

figure()plot(w1,SW_jw_DNV1,w1,SW_pm_DNV1,'LineWidth',1.5)title('Jonswap vs Pierson-Moskowitz specter')legend('Jonswap','Pierson-Moskowitz' )xlabel('\omega [rad/s]')ylabel('Spectral density [m^2/(rad/s)]')

56 A Matlab codes Motion Analysis

WS 2, sea state, Hs=4,5m, Tp=10secwp2= 2*pi/Tp2;T_w1 = (2*pi)./w1;

SW_pm_DNV12 = 5/16 .* Hs2.^2.*wp2.^4 .*w1 .^(-5) .* exp(-(5/4)....*(w1./wp2).^(-4)) ;%DNV-RP-C205

figure()plot(w1, SW_pm_DNV12)%axis ([0 3 0 3])title('PM specter')legend('PM')

SW_jw_DNV12 = AG.*SW_pm_DNV12.*gamma.^(exp(-0.5.*((w1-wp2)./...(sigma*wp2)).^2));

figure()plot(w1, SW_jw_DNV1,w1,SW_jw_DNV12,'LineWidth',1.5 )title('Comparison of Jonswap-Spectrum')legend('Sea state 1: Hs=3m, Tp=6sec. ','Sea state 12: Hs=4.5m, Tp=10sec')xlabel('\omega [rad/s]')ylabel('Spectral density [m^2/(rad/s)]')

RAO data from Aqwa dataRAO_final_P = table_S1_S2(:,1);% 4m flange, Sea state 1RAO_final_41 = table_S1_S2(:,2:4);

% 4m flange, Sea state 2RAO_final_42 = table_S1_S2(:,5:7);

% 6m flange, Sea state 1RAO_final_61 = table_S1_S2(:,8:10);

% 6m flange, Sea state 2

A.1 Motion Response Analysis including KC-number estimation 57

RAO_final_62 = table_S1_S2(:,11:13);

% 8m flange, Sea state 1RAO_final_81 = table_S1_S2(:,14:16);

% 8m flange, Sea state 2RAO_final_82 = table_S1_S2(:,17:19);

RAOs_015 = table9(:,1:8);RAOs_02343 = table9(:,9:16);RAOs_0318 = table9(:,17:24);RAOs_04029 = table9(:,25:32);RAOs_04871 = table9(:,33:40);RAOs_05714 = table9(:,41:48);RAOs_06557 = table9(:,49:56);RAOs_074 = table9(:,57:64);

Interpolating frequency data set to make a common frequencyv= RAOs_015(:,8);x=RAO_period;vq1 = interp1(x,v,T_w1);RAOs_015_1(:,8)=vq1;

Interpolating RAOs and SW to make common freqency datax1=RAO_final_P;%Final RAOsv_final_1=RAO_final_41(:,1:3);v_final_2=RAO_final_42(:,1:3);v_final_3=RAO_final_61(:,1:3);v_final_101112=RAO_final_62(:,1:3);v_final_4=RAO_final_81(:,1:3);v_final_5=RAO_final_82(:,1:3);

vq_final_1 = interp1(x1,v_final_1,T_w2);vq_final_2 = interp1(x1,v_final_2,T_w2);vq_final_3 = interp1(x,v_final_3,T_w1);vq_final_101112_1 = interp1(x,v_final_101112(:,1:2),T_w1);vq_final_101112_2 = interp1(x1,v_final_101112(:,3),T_w2);vq_final_4 = interp1(x1,v_final_4,T_w2);vq_final_5 = interp1(x1,v_final_5,T_w2);

58 A Matlab codes Motion Analysis

RAOs_final_41=vq_final_1(2:99,:); %SW1RAOs_final_42=vq_final_2(2:99,:); %SW2RAOs_final_61=vq_final_3(2:99,:); %SW1RAOs_final_101112=[vq_final_101112_1(2:99,:)...

,vq_final_101112_2(2:99,:)]; %SW2RAOs_final_81=vq_final_4(2:99,:); %SW1RAOs_final_82=vq_final_5(2:99,:); %SW2

%sea state Hs=3,Tp=6v1=RAOs_015(:,1:8);v2=RAOs_02343(:,1:8);v3=RAOs_0318(:,1:8);v4=RAOs_04029(:,1:8);v5=RAOs_04871(:,1:8);v6=RAOs_05714(:,1:8);v7=RAOs_06557(:,1:8);v8=RAOs_074(:,1:8);

x=RAO_period;

vq1 = interp1(x,v1,T_w1);vq2 = interp1(x,v2,T_w1);vq3 = interp1(x,v3,T_w1);vq4 = interp1(x,v4,T_w1);vq5 = interp1(x,v5,T_w1);vq6 = interp1(x,v6,T_w1);vq7 = interp1(x,v7,T_w1);vq8 = interp1(x,v8,T_w1);

RAOs_015=vq1(2:99,:);RAOs_02343=vq2(2:99,:);RAOs_0318=vq3(2:99,:);RAOs_04029=vq4(2:99,:);RAOs_04871=vq5(2:99,:);RAOs_05714=vq6(2:99,:);RAOs_06557=vq7(2:99,:);RAOs_074=vq8(2:99,:);

x=RAO_period;

T_w1 = T_w1(2:99,:);

A.1 Motion Response Analysis including KC-number estimation 59

T_w2 = T_w2(2:99,:);RAO_period = T_w1;RAO_final_p = T_w2;w1 = w1(2:99,:);w2 = w2(2:99,:);RAO_Omega = w1;RAO_final_Omega = w2;SW_jw_DNV1 = SW_jw_DNV1(2:99,:);SW_jw_DNV12 = SW_jw_DNV12(2:99,:);

Plotting RAOS%figure rao comparison with WS hs=3, Tp=6 , sea state 1

figure()plot ( RAO_period,RAOs_015(:,8),RAO_period,RAOs_02343(:,8),...

RAO_period,RAOs_0318(:,8),RAO_period,RAOs_04029(:,8),...RAO_period,RAOs_04871(:,8),RAO_period,RAOs_05714(:,8),...RAO_period,RAOs_06557(:,8), RAO_period,RAOs_074(:,8),...RAO_period,SW_jw_DNV1,'LineWidth',1.5)

axis([0 25 0 2.5])title('Wave spectrum and RAO- comparison for ratio 1.78')legend('RAO=0.15','RAO=0.2343','RAO=0.318','RAO=0.4029','RAO=0.4871',...

'RAO=0.5714','RAO=0.6557','RAO=0.74','Wave spectrum')xlabel('Period [sec]')ylabel('Spectral density [m^2/(rad/s)]')

figure()plot ( RAO_period,RAOs_015(:,7), RAO_period,RAOs_02343(:,7),...

RAO_period,RAOs_0318(:,7), RAO_period,RAOs_04029(:,7),...RAO_period,RAOs_04871(:,7), RAO_period,RAOs_05714(:,7),...RAO_period,RAOs_06557(:,7), RAO_period,RAOs_074(:,7),...RAO_period,SW_jw_DNV1,'LineWidth',1.5)

axis([0 25 0 2.5])title('Wave spectrum and RAO- comparison for ratio 1.66')legend('RAO=0.15','RAO=0.2343','RAO=0.318','RAO=0.4029','RAO=0.4871',...

'RAO=0.5714','RAO=0.6557','RAO=0.74','Wave spectrum')xlabel('Period [sec]')ylabel('Spectral density [m^2/(rad/s)]')

figure()plot ( RAO_period,RAOs_015(:,6), RAO_period,RAOs_02343(:,6),...

60 A Matlab codes Motion Analysis

RAO_period,RAOs_0318(:,6), RAO_period,RAOs_04029(:,6),...RAO_period,RAOs_04871(:,6), RAO_period,RAOs_05714(:,6),...RAO_period,RAOs_06557(:,6), RAO_period,RAOs_074(:,6),...RAO_period,SW_jw_DNV1,'LineWidth',1.5)

axis([0 25 0 2.5])title('Wave spectrum and RAO- comparison for ratio 1.48')legend('RAO=0.15','RAO=0.2343','RAO=0.318','RAO=0.4029','RAO=0.4871',...

'RAO=0.5714','RAO=0.6557','RAO=0.74','Wave spectrum')xlabel('Period [sec]')ylabel('Spectral density [m^2/(rad/s)]')

figure()plot ( RAO_period,RAOs_015(:,5), RAO_period,RAOs_02343(:,5),...

RAO_period,RAOs_0318(:,5), RAO_period,RAOs_04029(:,5),...RAO_period,RAOs_04871(:,5), RAO_period,RAOs_05714(:,5),...RAO_period,RAOs_06557(:,5), RAO_period,RAOs_074(:,5),...RAO_period,SW_jw_DNV1,'LineWidth',1.5)

axis([0 25 0 2.5])title('Wave spectrum and RAO- comparison for ratio 1.36')legend('RAO=0.15','RAO=0.2343','RAO=0.318','RAO=0.4029','RAO=0.4871',...

'RAO=0.5714','RAO=0.6557','RAO=0.74','Wave spectrum')xlabel('Period [sec]')ylabel('Spectral density [m^2/(rad/s)]')

figure()plot ( RAO_period,RAOs_015(:,4),RAO_period,RAOs_02343(:,4),...

RAO_period,RAOs_0318(:,4),RAO_period,RAOs_04029(:,4),...RAO_period,RAOs_04871(:,4),RAO_period,RAOs_05714(:,4),...RAO_period,RAOs_06557(:,4),RAO_period,RAOs_074(:,4),...RAO_period,SW_jw_DNV1,'LineWidth',1.5)

axis([0 25 0 2.5])title('Wave spectrum and RAO- comparison for ratio 1.24')legend('RAO=0.15','RAO=0.2343','RAO=0.318','RAO=0.4029','RAO=0.4871',...

'RAO=0.5714','RAO=0.6557','RAO=0.74','Wave spectrum')xlabel('Period [sec]')ylabel('Spectral density [m^2/(rad/s)]')

figure()plot ( RAO_period,RAOs_015(:,3), RAO_period,RAOs_02343(:,3),...

RAO_period,RAOs_0318(:,3), RAO_period,RAOs_04029(:,3),...RAO_period,RAOs_04871(:,3), RAO_period,RAOs_05714(:,3),...RAO_period,RAOs_06557(:,3), RAO_period,RAOs_074(:,3),...

A.1 Motion Response Analysis including KC-number estimation 61

RAO_period,SW_jw_DNV1,'LineWidth',1.5)axis([0 25 0 2.5])title('Wave spectrum and RAO- comparison for ratio 1.16')legend('RAO=0.15','RAO=0.2343','RAO=0.318','RAO=0.4029','RAO=0.4871',...

'RAO=0.5714','RAO=0.6557','RAO=0.74','Wave spectrum')xlabel('Period [sec]')ylabel('Spectral density [m^2/(rad/s)]')

figure()plot ( RAO_period,RAOs_015(:,2), RAO_period,RAOs_02343(:,2),...

RAO_period,RAOs_0318(:,2), RAO_period,RAOs_04029(:,2),...RAO_period,RAOs_04871(:,2), RAO_period,RAOs_05714(:,2),...RAO_period,RAOs_06557(:,2), RAO_period,RAOs_074(:,2),...RAO_period,SW_jw_DNV1,'LineWidth',1.5)

axis([0 25 0 2.5])title('Wave spectrum and RAO- comparison for ratio 1.077')legend('RAO=0.15','RAO=0.2343','RAO=0.318','RAO=0.4029','RAO=0.4871',...

'RAO=0.5714','RAO=0.6557','RAO=0.74','Wave spectrum')xlabel('Period [sec]')ylabel('Spectral density [m^2/(rad/s)]')

figure()plot ( RAO_period,RAOs_015(:,1), RAO_period,RAOs_02343(:,1),...

RAO_period,RAOs_0318(:,1), RAO_period,RAOs_04029(:,1),...RAO_period,RAOs_04871(:,1),RAO_period,RAOs_05714(:,1),...RAO_period,RAOs_06557(:,1),RAO_period,RAOs_074(:,1),...RAO_period,SW_jw_DNV1,'LineWidth',1.5)

axis([0 25 0 2.5])title('Wave spectrum and RAO- comparison for ratio 1.0')legend('RAO=0.15','RAO=0.2343','RAO=0.318','RAO=0.4029','RAO=0.4871',...

'RAO=0.5714','RAO=0.6557','RAO=0.74','Wave spectrum')xlabel('Period [sec]')ylabel('Spectral density [m^2/(rad/s)]')

62 A Matlab codes Motion Analysis

A.1 Motion Response Analysis including KC-number estimation 63

64 A Matlab codes Motion Analysis

A.1 Motion Response Analysis including KC-number estimation 65

Plotting final RAOs:%Sea state 2figure()plot ( RAO_final_p,RAOs_final_42(:,1),RAO_final_p,RAOs_final_42(:,2),...

RAO_final_p,RAOs_final_42(:,3),RAO_period,RAOs_final_101112(:,1),...

66 A Matlab codes Motion Analysis

RAO_period, RAOs_final_101112(:,2),RAO_final_p,RAOs_final_101112(:,3),...RAO_final_p,RAOs_final_82(:,1),RAO_final_p,RAOs_final_82(:,2),...

RAO_final_p,RAOs_final_82(:,3),RAO_final_p,SW_jw_DNV_f12(2:99,:),...'LineWidth',1.5)

axis([0 20 0 5])title('Wave spectrum and RAO comparison for Sea state 2')legend('F-size 4m, Cd=3.83','F-size 4m, Cd=5.31','F-size 4m, Cd=7.87',...

'F-size 6m, Cd=3.83','F-size 6m, Cd=5.31','F-size 6m, Cd=7.87',...'F-size 8m, Cd=3.83','F-size 8m, Cd=5.31','F-size 8m, Cd=7.87',...'Wave spectrum')

xlabel('Period [sec]')ylabel('Spectral density [m^2/(rad/s)]')

RAOs_final_S1 =[RAOs_final_41,RAOs_final_81];RAOs_final_S12 =RAOs_final_61;RAOs_final_S2 =RAOs_final_101112;RAOs_final_S22 =[RAOs_final_42,RAOs_final_82];

response spectrum%sea state 1N=8;for i = 1:NR1(:,i) = SW_jw_DNV1.*RAOs_015(:,i).^2;R2(:,i) = SW_jw_DNV1.*RAOs_02343(:,i).^2;R3(:,i) = SW_jw_DNV1.*RAOs_0318(:,i).^2;R4(:,i) = SW_jw_DNV1.*RAOs_04029(:,i).^2;R5(:,i) = SW_jw_DNV1.*RAOs_04871(:,i).^2;R6(:,i) = SW_jw_DNV1.*RAOs_05714(:,i).^2;R7(:,i) = SW_jw_DNV1.*RAOs_06557(:,i).^2;R8(:,i) = SW_jw_DNV1.*RAOs_074(:,i).^2;end

%sea state 2for i=1:6Res_final_S1(:,i) = SW_jw_DNV_f1(2:99,:).*RAOs_final_S1(:,i).^2;end

for i=1:3Res_final_S12(:,i) = SW_jw_DNV1.*RAOs_final_S12(:,i).^2;end

A.1 Motion Response Analysis including KC-number estimation 67

for i=1:3Res_final_S2(:,i) = SW_jw_DNV_f12(2:99,:).*RAOs_final_S2(:,i).^2;end

for i=1:6Res_final_S22(:,i) = SW_jw_DNV12.*RAOs_final_S22(:,i).^2;end

Plotting response for all sea statesfigure()plot(T_w1,R1(:,:),'LineWidth',1.5) %T_w1,R4(:,4),T_w1,R8(:,4),T_w1,R1(:,8),...

% T_w1,R4(:,8),T_w1,R8(:,8),axis([2 14 0 0.0015])title('Response spectrum KC = 0.15')legend('Response ratio=1',...

'Response, ratio=1.077',...'Response, ratio=1,16',...'Response, ratio=1,24',...'Response, ratio=1,36',...'Response, ratio=1,48',...'Response, ratio=1,66',...'Response ratio=1,78')

xlabel('Period [sec]')ylabel('Spectral density [m^2/(rad/s)]')

figure()plot(T_w1,R5(:,:))%,T_w1,R4(:,4),T_w1,R8(:,4),T_w1,R1(:,8),...

%T_w1,R4(:,8),T_w1,R8(:,8),'LineWidth',1.5)axis([2 14 0 0.0015])title('Response spectrum KC=0.4871')legend('Response ratio=1',...

'Response, ratio=1.077',...'Response, ratio=1,16',...'Response, ratio=1,24',...'Response, ratio=1,36',...'Response, ratio=1,48',...'Response, ratio=1,66',...'Response ratio=1,78')

xlabel('Period [sec]')ylabel('Spectral density [m^2/(rad/s)]')

figure()

68 A Matlab codes Motion Analysis

plot(T_w1,R8(:,:))%,T_w1,R4(:,4),T_w1,R8(:,4),T_w1,R1(:,8),...%T_w1,R4(:,8),T_w1,R8(:,8),'LineWidth',1.5)

axis([2 14 0 0.0015])title('Response spectrum KC=0.74')legend('Response ratio=1',...

'Response, ratio=1.077',...'Response, ratio=1,16',...'Response, ratio=1,24',...'Response, ratio=1,36',...'Response, ratio=1,48',...'Response, ratio=1,66',...'Response ratio=1,78')

xlabel('Period [sec]')ylabel('Spectral density [m^2/(rad/s)]')

figure()plot(RAO_final_p,Res_final_S2,T_w1,Res_final_S22,'LineWidth',1.5)axis([7 17 0 11])title('Response spectrum for sea state Hs=4.5, Tp=10')xlabel('Period [sec]')ylabel('Spectral density [m^2/(rad/s)]')legend('F-size 6m, Cd=3.83','F-size 6m, Cd=5.31','F-size 6m, Cd=7.87',...

'F-size 4m, Cd=3.83','F-size 4m, Cd=5.31','F-size 4m, Cd=7.87',...'F-size 8m, Cd=3.83','F-size 8m, Cd=5.31','F-size 8m, Cd=7.87')

Velocity estimation for final RAOs test, sea state, Hs=3m, Tp=6sec%not the final resultsfor i=1:6

M0_final_S1(:,i)= abs(trapz(RAO_final_Omega.^0.*Res_final_S1(:,i),...RAO_final_Omega));

M2_final_S1(:,i)= abs(trapz(RAO_final_Omega.^2.*Res_final_S1(:,i),...RAO_final_Omega));

end

for i=1:3M0_final_S12(:,i)= abs(trapz(RAO_Omega.^0.*Res_final_S12(:,i),...

RAO_Omega));M2_final_S12(:,i)= abs(trapz(RAO_Omega.^2.*Res_final_S12(:,i),...

RAO_Omega));end

for i=1:6

A.1 Motion Response Analysis including KC-number estimation 69

SD_final_S1(i) = sqrt(M0_final_S1(i));T1_final_S1(i) = 2*pi*(M0_final_S1(i)/M2_final_S1(i));Tz_final_S1(i) = (2*pi) * sqrt(M0_final_S1(i)./(M2_final_S1(i)));

t = 3*60*60;

MPM_final_S1(i) =SD_final_S1(i) .* sqrt(2*log(t/Tz_final_S1(i)));%MPM from equ 2.35 faltinsen

H_13_final_S1(i) = 4*sqrt(M0_final_S1(i));%significant wave height equ. 2.26

PA_final_S1(i) = (A/M0_final_S1(i))*exp((A^2)/(2*M0_final_S1(i)));%probability density function to

end

for i=1:3SD_final_S12(i) = sqrt(M0_final_S12(i));T1_final_S12(i) = 2*pi*(M0_final_S12(i)/M2_final_S12(i));Tz_final_S12(i) = (2*pi) * sqrt(M0_final_S12(i)./(M2_final_S12(i)));

t = 3*60*60;

MPM_final_S12(i) =SD_final_S12(i) .* sqrt(2*log(t/Tz_final_S12(i)));%MPM from equ 2.35 faltinsen

H_13_final_S12(i) = 4*sqrt(M0_final_S12(i));%significant wave height equ. 2.26

PA_final_S12(i) = (A/M0_final_S12(i))*exp((A^2)/(2*M0_final_S12(i)));%probability density function to

end

X_vel_1final_S1 = RAO_final_Omega*MPM_final_S1(1,1);X_vel_2final_S1 = RAO_final_Omega*MPM_final_S1(1,2);X_vel_3final_S1 = RAO_final_Omega*MPM_final_S1(1,3);X_vel_4final_S1 = RAO_final_Omega*MPM_final_S1(1,4);X_vel_5final_S1 = RAO_final_Omega*MPM_final_S1(1,5);X_vel_6final_S1 = RAO_final_Omega*MPM_final_S1(1,6);

70 A Matlab codes Motion Analysis

X_vel_1final_S12 = RAO_Omega*MPM_final_S12(1,1);X_vel_2final_S12 = RAO_Omega*MPM_final_S12(1,2);X_vel_3final_S12 = RAO_Omega*MPM_final_S12(1,3);

X_vel_MPM_final_S1 = [X_vel_1final_S1,X_vel_2final_S1,X_vel_5final_S1,...X_vel_4final_S1,X_vel_5final_S1,X_vel_6final_S1];

X_vel_MPM_final_S12 = [X_vel_1final_S12,X_vel_2final_S12,X_vel_3final_S12]

Amp_MPM_test_final_S1 = X_vel_MPM_final_S1(62,:);Amp_MPM_test_final_S12 = X_vel_MPM_final_S12(61,:);

Dim= 37;Dim2 = 41;Dim3 = 33;KC_new_MPM_final_WS1_4m = (Amp_MPM_test_final_S1(:,1:3)*Tp)/Dim3;KC_new_MPM_final_WS12_6m = (Amp_MPM_test_final_S12(:,1:3)*Tp)/Dim;KC_new_MPM_final_WS1_8m = (Amp_MPM_test_final_S1(:,4:6)*Tp)/Dim2;

X_vel_MPM_final_S12 =

0.1585 0.1599 0.16370.1570 0.1583 0.16210.1554 0.1568 0.16050.1539 0.1552 0.15890.1523 0.1536 0.15730.1508 0.1521 0.15570.1492 0.1505 0.15410.1476 0.1489 0.15250.1461 0.1474 0.15090.1445 0.1458 0.14930.1430 0.1443 0.14770.1414 0.1427 0.14610.1399 0.1411 0.14450.1383 0.1396 0.14290.1368 0.1380 0.14130.1352 0.1364 0.13970.1337 0.1349 0.13810.1321 0.1333 0.13650.1306 0.1317 0.13490.1290 0.1302 0.13320.1275 0.1286 0.13160.1259 0.1270 0.13000.1244 0.1255 0.1284

A.1 Motion Response Analysis including KC-number estimation 71

0.1228 0.1239 0.12680.1213 0.1223 0.12520.1197 0.1208 0.12360.1182 0.1192 0.12200.1166 0.1176 0.12040.1151 0.1161 0.11880.1135 0.1145 0.11720.1120 0.1129 0.11560.1104 0.1114 0.11400.1089 0.1098 0.11240.1073 0.1083 0.11080.1058 0.1067 0.10920.1042 0.1051 0.10760.1027 0.1036 0.10600.1011 0.1020 0.10440.0996 0.1004 0.10280.0980 0.0989 0.10120.0964 0.0973 0.09960.0949 0.0957 0.09800.0933 0.0942 0.09640.0918 0.0926 0.09480.0902 0.0910 0.09320.0887 0.0895 0.09160.0871 0.0879 0.09000.0856 0.0863 0.08840.0840 0.0848 0.08680.0825 0.0832 0.08520.0809 0.0816 0.08360.0794 0.0801 0.08200.0778 0.0785 0.08040.0763 0.0769 0.07880.0747 0.0754 0.07720.0732 0.0738 0.07560.0716 0.0723 0.07400.0701 0.0707 0.07240.0685 0.0691 0.07080.0670 0.0676 0.06920.0654 0.0660 0.06760.0639 0.0644 0.06600.0623 0.0629 0.06440.0608 0.0613 0.06270.0592 0.0597 0.06110.0577 0.0582 0.05950.0561 0.0566 0.0579

72 A Matlab codes Motion Analysis

0.0546 0.0550 0.05630.0530 0.0535 0.05470.0515 0.0519 0.05310.0499 0.0503 0.05150.0484 0.0488 0.04990.0468 0.0472 0.04830.0452 0.0456 0.04670.0437 0.0441 0.04510.0421 0.0425 0.04350.0406 0.0409 0.04190.0390 0.0394 0.04030.0375 0.0378 0.03870.0359 0.0363 0.03710.0344 0.0347 0.03550.0328 0.0331 0.03390.0313 0.0316 0.03230.0297 0.0300 0.03070.0282 0.0284 0.02910.0266 0.0269 0.02750.0251 0.0253 0.02590.0235 0.0237 0.02430.0220 0.0222 0.02270.0204 0.0206 0.02110.0189 0.0190 0.01950.0173 0.0175 0.01790.0158 0.0159 0.01630.0142 0.0143 0.01470.0127 0.0128 0.01310.0111 0.0112 0.01150.0096 0.0096 0.00990.0080 0.0081 0.0083

Velocity estimation for final RAOs, sea state, Hs=4,5m, Tp=10sec%final resultsfor i=1:3M0_final_S2(:,i)= abs(trapz(RAO_Omega.^0.*Res_final_S2(:,i),RAO_Omega));M2_final_S2(:,i)= abs(trapz(RAO_Omega.^2.*Res_final_S2(:,i),RAO_Omega));

end

for i=1:6M0_final_S22(:,i)= abs(trapz(RAO_final_Omega.^0.*Res_final_S22(:,i),...

A.1 Motion Response Analysis including KC-number estimation 73

RAO_final_Omega));M2_final_S22(:,i)= abs(trapz(RAO_final_Omega.^2.*Res_final_S22(:,i),...

RAO_final_Omega));end

for i=1:6SD_final_S22(i) = sqrt(M0_final_S22(i));T1_final_S22(i) = 2*pi*(M0_final_S22(i)/M2_final_S22(i));Tz_final_S22(i) = (2*pi) * sqrt(M0_final_S22(i)./(M2_final_S22(i)));

t = 3*60*60;

MPM_final_S22(i) =SD_final_S22(i) .* sqrt(2*log(t/Tz_final_S22(i)));%MPM from equ 2.35 faltinsen

H_13_final_S22(i) = 4*sqrt(M0_final_S22(i));%significant wave height equ. 2.26

PA_final_S22(i) = (A/M0_final_S22(i))*exp((A^2)/(2*M0_final_S22(i)));%probability density function to

end

for i=1:3SD_final_S2(i) = sqrt(M0_final_S2(i));T1_final_S2(i) = 2*pi*(M0_final_S2(i)/M2_final_S2(i));Tz_final_S2(i) = (2*pi) * sqrt(M0_final_S2(i)./(M2_final_S2(i)));

t = 3*60*60;

MPM_final_S2(i) =SD_final_S2(i) .* sqrt(2*log(t/Tz_final_S2(i)));%MPM from equ 2.35 faltinsen

H_13_final_S2(i) = 4*sqrt(M0_final_S2(i));%significant wave height equ. 2.26

PA_final_S2(i) = (A/M0_final_S2(i))*exp((A^2)/(2*M0_final_S2(i)));%probability density function to

end

X_vel_1final_S22 = RAO_Omega*MPM_final_S22(1,1);

74 A Matlab codes Motion Analysis

X_vel_2final_S22 = RAO_Omega*MPM_final_S22(1,2);X_vel_3final_S22 = RAO_Omega*MPM_final_S22(1,3);X_vel_4final_S22 = RAO_Omega*MPM_final_S22(1,4);X_vel_5final_S22 = RAO_Omega*MPM_final_S22(1,5);X_vel_6final_S22 = RAO_Omega*MPM_final_S22(1,6);

X_vel_1final_S2 = RAO_final_Omega*MPM_final_S2(1,1);X_vel_2final_S2 = RAO_final_Omega*MPM_final_S2(1,2);X_vel_3final_S2 = RAO_final_Omega*MPM_final_S2(1,3);

X_vel_H13_1final_S22 = RAO_Omega*H_13_final_S22(1,1);X_vel_H13_2final_S22 = RAO_Omega*H_13_final_S22(1,2);X_vel_H13_3final_S22 = RAO_Omega*H_13_final_S22(1,3);X_vel_H13_4final_S22 = RAO_Omega*H_13_final_S22(1,4);X_vel_H13_5final_S22 = RAO_Omega*H_13_final_S22(1,5);X_vel_H13_6final_S22 = RAO_Omega*H_13_final_S22(1,6);

X_vel_H13_1final_S2 = RAO_final_Omega*H_13_final_S2(1,1);X_vel_H13_2final_S2 = RAO_final_Omega*H_13_final_S2(1,2);X_vel_H13_3final_S2 = RAO_final_Omega*H_13_final_S2(1,3);

%Resulting vertical velocitiesX_vel_MPM_final_S22 = [X_vel_1final_S22,X_vel_2final_S22,X_vel_3final_S22,...

X_vel_4final_S22,X_vel_5final_S22,X_vel_6final_S22]X_vel_MPM_final_S2 = [X_vel_1final_S2,X_vel_2final_S2,X_vel_3final_S2]X_vel_H13_final_S22 = [X_vel_H13_1final_S22,X_vel_H13_2final_S22,...

X_vel_H13_3final_S22,X_vel_H13_4final_S22,...X_vel_H13_5final_S22,X_vel_H13_6final_S22]

X_vel_H13_final_S2 = [X_vel_H13_1final_S2,X_vel_H13_2final_S2,...X_vel_H13_3final_S2]

Amp_MPM_test_final_S2 = X_vel_MPM_final_S2(77,:);Amp_MPM_test_final_S22 = X_vel_MPM_final_S22(77,:);Amp_H13_test_final_S2 = X_vel_H13_final_S2(77,:);Amp_H13_test_final_S22 = X_vel_H13_final_S22(77,:);

Dim= 37;Dim2 = 41;Dim3 = 33;KC_new_MPM_final_WS22_4m = (Amp_MPM_test_final_S22(:,1:3)*Tp2)/Dim3;KC_new_MPM_final_WS2_6m = (Amp_MPM_test_final_S2(:,1:3)*Tp2)/Dim;

A.1 Motion Response Analysis including KC-number estimation 75

KC_new_MPM_final_WS22_8m = (Amp_MPM_test_final_S22(:,4:6)*Tp2)/Dim2;

KC_new_H13_final_WS22_4m = (Amp_H13_test_final_S22(:,1:3)*Tp2)/Dim3;KC_new_H13_final_WS2_6m = (Amp_H13_test_final_S2(:,1:3)*Tp2)/Dim;KC_new_H13_final_WS22_8m = (Amp_H13_test_final_S22(:,4:6)*Tp2)/Dim2;

KCf=[0.15,0.4871,0.74]pf1 = polyfit(KCf, KC_new_MPM_final_WS22_4m, 2);pf2 = polyfit(KCf, KC_new_MPM_final_WS2_6m, 2);pf3 = polyfit(KCf, KC_new_MPM_final_WS22_8m, 2);pf4 = polyfit(KCf, KC_new_H13_final_WS22_4m, 2);pf5 = polyfit(KCf, KC_new_H13_final_WS2_6m, 2);pf6 = polyfit(KCf, KC_new_H13_final_WS22_8m, 2);

pf = [pf1;pf2;pf3;pf4;pf5;pf6];pf = pf';

KCfinal=linspace(0.15,1,10)for i=1:length(KCfinal)KC_poly1(i) = pf1(3) + pf1(2).*KCfinal(i) + pf1(1).*KCfinal(i).^2;KC_poly2(i) = pf2(3) + pf2(2).*KCfinal(i) + pf2(1).*KCfinal(i).^2;KC_poly3(i) = pf3(3) + pf3(2).*KCfinal(i) + pf3(1).*KCfinal(i).^2;KC_poly4(i) = pf4(3) + pf4(2).*KCfinal(i) + pf4(1).*KCfinal(i).^2;KC_poly5(i) = pf5(3) + pf5(2).*KCfinal(i) + pf5(1).*KCfinal(i).^2;KC_poly6(i) = pf6(3) + pf6(2).*KCfinal(i) + pf6(1).*KCfinal(i).^2;end

%resulting KC-numbers for Sea State 2:

KC_ESTIMATED_RESULTS_SS2=[KC_poly1,KC_poly2,KC_poly3,KC_poly4,...KC_poly5,KC_poly6]

X_vel_MPM_final_S22 =

9.0106 9.9711 8.9711 4.2577 3.9323 3.61268.9224 9.8735 8.8833 4.2160 3.8938 3.57728.8342 9.7759 8.7955 4.1744 3.8553 3.54198.7460 9.6783 8.7077 4.1327 3.8168 3.50658.6578 9.5807 8.6199 4.0910 3.7783 3.47118.5696 9.4831 8.5321 4.0493 3.7398 3.43588.4814 9.3855 8.4443 4.0077 3.7013 3.40048.3932 9.2879 8.3565 3.9660 3.6628 3.36508.3050 9.1903 8.2686 3.9243 3.6244 3.3297

76 A Matlab codes Motion Analysis

8.2168 9.0927 8.1808 3.8826 3.5859 3.29438.1286 8.9951 8.0930 3.8410 3.5474 3.25908.0404 8.8975 8.0052 3.7993 3.5089 3.22367.9522 8.7999 7.9174 3.7576 3.4704 3.18827.8640 8.7023 7.8296 3.7159 3.4319 3.15297.7758 8.6047 7.7418 3.6742 3.3934 3.11757.6876 8.5071 7.6540 3.6326 3.3549 3.08227.5994 8.4095 7.5661 3.5909 3.3164 3.04687.5112 8.3119 7.4783 3.5492 3.2779 3.01147.4230 8.2143 7.3905 3.5075 3.2394 2.97617.3348 8.1167 7.3027 3.4659 3.2010 2.94077.2466 8.0191 7.2149 3.4242 3.1625 2.90547.1584 7.9215 7.1271 3.3825 3.1240 2.87007.0702 7.8239 7.0393 3.3408 3.0855 2.83466.9820 7.7263 6.9515 3.2992 3.0470 2.79936.8938 7.6287 6.8636 3.2575 3.0085 2.76396.8056 7.5311 6.7758 3.2158 2.9700 2.72856.7174 7.4335 6.6880 3.1741 2.9315 2.69326.6292 7.3359 6.6002 3.1325 2.8930 2.65786.5410 7.2383 6.5124 3.0908 2.8545 2.62256.4528 7.1407 6.4246 3.0491 2.8161 2.58716.3646 7.0431 6.3368 3.0074 2.7776 2.55176.2764 6.9455 6.2490 2.9658 2.7391 2.51646.1882 6.8479 6.1611 2.9241 2.7006 2.48106.1000 6.7503 6.0733 2.8824 2.6621 2.44576.0118 6.6527 5.9855 2.8407 2.6236 2.41035.9236 6.5551 5.8977 2.7991 2.5851 2.37495.8354 6.4575 5.8099 2.7574 2.5466 2.33965.7472 6.3599 5.7221 2.7157 2.5081 2.30425.6590 6.2623 5.6343 2.6740 2.4696 2.26895.5708 6.1647 5.5465 2.6324 2.4312 2.23355.4826 6.0671 5.4586 2.5907 2.3927 2.19815.3944 5.9695 5.3708 2.5490 2.3542 2.16285.3062 5.8719 5.2830 2.5073 2.3157 2.12745.2180 5.7743 5.1952 2.4657 2.2772 2.09215.1298 5.6767 5.1074 2.4240 2.2387 2.05675.0417 5.5791 5.0196 2.3823 2.2002 2.02134.9535 5.4815 4.9318 2.3406 2.1617 1.98604.8653 5.3839 4.8440 2.2989 2.1232 1.95064.7771 5.2863 4.7561 2.2573 2.0847 1.91524.6889 5.1887 4.6683 2.2156 2.0462 1.87994.6007 5.0911 4.5805 2.1739 2.0078 1.84454.5125 4.9935 4.4927 2.1322 1.9693 1.80924.4243 4.8959 4.4049 2.0906 1.9308 1.7738

A.1 Motion Response Analysis including KC-number estimation 77

4.3361 4.7983 4.3171 2.0489 1.8923 1.73844.2479 4.7007 4.2293 2.0072 1.8538 1.70314.1597 4.6031 4.1415 1.9655 1.8153 1.66774.0715 4.5055 4.0536 1.9239 1.7768 1.63243.9833 4.4079 3.9658 1.8822 1.7383 1.59703.8951 4.3103 3.8780 1.8405 1.6998 1.56163.8069 4.2127 3.7902 1.7988 1.6613 1.52633.7187 4.1151 3.7024 1.7572 1.6229 1.49093.6305 4.0175 3.6146 1.7155 1.5844 1.45563.5423 3.9199 3.5268 1.6738 1.5459 1.42023.4541 3.8223 3.4390 1.6321 1.5074 1.38483.3659 3.7247 3.3512 1.5905 1.4689 1.34953.2777 3.6271 3.2633 1.5488 1.4304 1.31413.1895 3.5295 3.1755 1.5071 1.3919 1.27873.1013 3.4319 3.0877 1.4654 1.3534 1.24343.0131 3.3343 2.9999 1.4238 1.3149 1.20802.9249 3.2367 2.9121 1.3821 1.2764 1.17272.8367 3.1391 2.8243 1.3404 1.2380 1.13732.7485 3.0415 2.7365 1.2987 1.1995 1.10192.6603 2.9439 2.6487 1.2571 1.1610 1.06662.5721 2.8463 2.5608 1.2154 1.1225 1.03122.4839 2.7487 2.4730 1.1737 1.0840 0.99592.3957 2.6511 2.3852 1.1320 1.0455 0.96052.3075 2.5535 2.2974 1.0903 1.0070 0.92512.2193 2.4559 2.2096 1.0487 0.9685 0.88982.1311 2.3583 2.1218 1.0070 0.9300 0.85442.0429 2.2607 2.0340 0.9653 0.8915 0.81911.9547 2.1631 1.9462 0.9236 0.8530 0.78371.8665 2.0655 1.8583 0.8820 0.8146 0.74831.7783 1.9679 1.7705 0.8403 0.7761 0.71301.6901 1.8703 1.6827 0.7986 0.7376 0.67761.6019 1.7727 1.5949 0.7569 0.6991 0.64221.5137 1.6751 1.5071 0.7153 0.6606 0.60691.4255 1.5775 1.4193 0.6736 0.6221 0.57151.3373 1.4799 1.3315 0.6319 0.5836 0.53621.2491 1.3823 1.2437 0.5902 0.5451 0.50081.1609 1.2847 1.1558 0.5486 0.5066 0.46541.0727 1.1871 1.0680 0.5069 0.4681 0.43010.9845 1.0895 0.9802 0.4652 0.4297 0.39470.8963 0.9919 0.8924 0.4235 0.3912 0.35940.8081 0.8943 0.8046 0.3819 0.3527 0.32400.7199 0.7967 0.7168 0.3402 0.3142 0.28860.6317 0.6991 0.6290 0.2985 0.2757 0.25330.5435 0.6015 0.5412 0.2568 0.2372 0.2179

78 A Matlab codes Motion Analysis

0.4553 0.5039 0.4533 0.2152 0.1987 0.1826

X_vel_MPM_final_S2 =

7.9411 7.5945 6.71787.8633 7.5201 6.65207.7855 7.4457 6.58627.7077 7.3713 6.52047.6300 7.2969 6.45467.5522 7.2225 6.38887.4744 7.1481 6.32307.3966 7.0737 6.25727.3188 6.9994 6.19137.2410 6.9250 6.12557.1633 6.8506 6.05977.0855 6.7762 5.99397.0077 6.7018 5.92816.9299 6.6274 5.86236.8521 6.5530 5.79656.7743 6.4786 5.73076.6966 6.4042 5.66496.6188 6.3299 5.59916.5410 6.2555 5.53336.4632 6.1811 5.46756.3854 6.1067 5.40176.3076 6.0323 5.33596.2299 5.9579 5.27016.1521 5.8835 5.20436.0743 5.8091 5.13855.9965 5.7347 5.07275.9187 5.6604 5.00695.8409 5.5860 4.94115.7631 5.5116 4.87535.6854 5.4372 4.80955.6076 5.3628 4.74375.5298 5.2884 4.67795.4520 5.2140 4.61215.3742 5.1396 4.54635.2964 5.0652 4.48055.2187 4.9909 4.41475.1409 4.9165 4.34895.0631 4.8421 4.28314.9853 4.7677 4.2173

A.1 Motion Response Analysis including KC-number estimation 79

4.9075 4.6933 4.15154.8297 4.6189 4.08574.7520 4.5445 4.01994.6742 4.4701 3.95414.5964 4.3958 3.88834.5186 4.3214 3.82254.4408 4.2470 3.75674.3630 4.1726 3.69094.2853 4.0982 3.62514.2075 4.0238 3.55934.1297 3.9494 3.49354.0519 3.8750 3.42773.9741 3.8006 3.36193.8963 3.7263 3.29613.8185 3.6519 3.23033.7408 3.5775 3.16453.6630 3.5031 3.09873.5852 3.4287 3.03293.5074 3.3543 2.96713.4296 3.2799 2.90133.3518 3.2055 2.83553.2741 3.1311 2.76973.1963 3.0568 2.70393.1185 2.9824 2.63813.0407 2.9080 2.57232.9629 2.8336 2.50652.8851 2.7592 2.44072.8074 2.6848 2.37492.7296 2.6104 2.30912.6518 2.5360 2.24332.5740 2.4616 2.17752.4962 2.3873 2.11172.4184 2.3129 2.04592.3407 2.2385 1.98012.2629 2.1641 1.91432.1851 2.0897 1.84852.1073 2.0153 1.78272.0295 1.9409 1.71691.9517 1.8665 1.65111.8739 1.7921 1.58531.7962 1.7178 1.51951.7184 1.6434 1.45371.6406 1.5690 1.38791.5628 1.4946 1.3221

80 A Matlab codes Motion Analysis

1.4850 1.4202 1.25631.4072 1.3458 1.19051.3295 1.2714 1.12471.2517 1.1970 1.05891.1739 1.1227 0.99311.0961 1.0483 0.92731.0183 0.9739 0.86140.9405 0.8995 0.79560.8628 0.8251 0.72980.7850 0.7507 0.66400.7072 0.6763 0.59820.6294 0.6019 0.53240.5516 0.5275 0.46660.4738 0.4532 0.40080.3961 0.3788 0.3350

X_vel_H13_final_S22 =

9.6815 10.6858 9.6307 4.6275 4.2708 3.91939.5868 10.5812 9.5364 4.5822 4.2290 3.88099.4920 10.4766 9.4421 4.5369 4.1872 3.84259.3972 10.3720 9.3479 4.4916 4.1454 3.80429.3025 10.2674 9.2536 4.4464 4.1036 3.76589.2077 10.1628 9.1593 4.4011 4.0618 3.72749.1130 10.0582 9.0651 4.3558 4.0200 3.68919.0182 9.9536 8.9708 4.3105 3.9782 3.65078.9234 9.8490 8.8765 4.2652 3.9364 3.61248.8287 9.7444 8.7823 4.2199 3.8946 3.57408.7339 9.6399 8.6880 4.1746 3.8528 3.53568.6391 9.5353 8.5937 4.1293 3.8110 3.49738.5444 9.4307 8.4995 4.0840 3.7692 3.45898.4496 9.3261 8.4052 4.0387 3.7274 3.42058.3548 9.2215 8.3109 3.9934 3.6856 3.38228.2601 9.1169 8.2166 3.9481 3.6438 3.34388.1653 9.0123 8.1224 3.9028 3.6020 3.30558.0705 8.9077 8.0281 3.8575 3.5602 3.26717.9758 8.8031 7.9338 3.8122 3.5184 3.22877.8810 8.6985 7.8396 3.7669 3.4766 3.19047.7862 8.5939 7.7453 3.7216 3.4348 3.15207.6915 8.4893 7.6510 3.6763 3.3929 3.11367.5967 8.3847 7.5568 3.6310 3.3511 3.07537.5019 8.2801 7.4625 3.5857 3.3093 3.03697.4072 8.1755 7.3682 3.5404 3.2675 2.9986

A.1 Motion Response Analysis including KC-number estimation 81

7.3124 8.0709 7.2740 3.4951 3.2257 2.96027.2176 7.9663 7.1797 3.4498 3.1839 2.92187.1229 7.8617 7.0854 3.4045 3.1421 2.88357.0281 7.7571 6.9912 3.3593 3.1003 2.84516.9333 7.6525 6.8969 3.3140 3.0585 2.80676.8386 7.5479 6.8026 3.2687 3.0167 2.76846.7438 7.4433 6.7084 3.2234 2.9749 2.73006.6490 7.3387 6.6141 3.1781 2.9331 2.69166.5543 7.2341 6.5198 3.1328 2.8913 2.65336.4595 7.1295 6.4256 3.0875 2.8495 2.61496.3647 7.0249 6.3313 3.0422 2.8077 2.57666.2700 6.9204 6.2370 2.9969 2.7659 2.53826.1752 6.8158 6.1428 2.9516 2.7241 2.49986.0804 6.7112 6.0485 2.9063 2.6823 2.46155.9857 6.6066 5.9542 2.8610 2.6405 2.42315.8909 6.5020 5.8599 2.8157 2.5987 2.38475.7961 6.3974 5.7657 2.7704 2.5569 2.34645.7014 6.2928 5.6714 2.7251 2.5151 2.30805.6066 6.1882 5.5771 2.6798 2.4733 2.26975.5118 6.0836 5.4829 2.6345 2.4314 2.23135.4171 5.9790 5.3886 2.5892 2.3896 2.19295.3223 5.8744 5.2943 2.5439 2.3478 2.15465.2275 5.7698 5.2001 2.4986 2.3060 2.11625.1328 5.6652 5.1058 2.4533 2.2642 2.07785.0380 5.5606 5.0115 2.4080 2.2224 2.03954.9432 5.4560 4.9173 2.3627 2.1806 2.00114.8485 5.3514 4.8230 2.3175 2.1388 1.96284.7537 5.2468 4.7287 2.2722 2.0970 1.92444.6589 5.1422 4.6345 2.2269 2.0552 1.88604.5642 5.0376 4.5402 2.1816 2.0134 1.84774.4694 4.9330 4.4459 2.1363 1.9716 1.80934.3746 4.8284 4.3517 2.0910 1.9298 1.77094.2799 4.7238 4.2574 2.0457 1.8880 1.73264.1851 4.6192 4.1631 2.0004 1.8462 1.69424.0904 4.5146 4.0689 1.9551 1.8044 1.65583.9956 4.4100 3.9746 1.9098 1.7626 1.61753.9008 4.3055 3.8803 1.8645 1.7208 1.57913.8061 4.2009 3.7861 1.8192 1.6790 1.54083.7113 4.0963 3.6918 1.7739 1.6372 1.50243.6165 3.9917 3.5975 1.7286 1.5954 1.46403.5218 3.8871 3.5032 1.6833 1.5536 1.42573.4270 3.7825 3.4090 1.6380 1.5118 1.38733.3322 3.6779 3.3147 1.5927 1.4699 1.34893.2375 3.5733 3.2204 1.5474 1.4281 1.3106

82 A Matlab codes Motion Analysis

3.1427 3.4687 3.1262 1.5021 1.3863 1.27223.0479 3.3641 3.0319 1.4568 1.3445 1.23392.9532 3.2595 2.9376 1.4115 1.3027 1.19552.8584 3.1549 2.8434 1.3662 1.2609 1.15712.7636 3.0503 2.7491 1.3209 1.2191 1.11882.6689 2.9457 2.6548 1.2756 1.1773 1.08042.5741 2.8411 2.5606 1.2304 1.1355 1.04202.4793 2.7365 2.4663 1.1851 1.0937 1.00372.3846 2.6319 2.3720 1.1398 1.0519 0.96532.2898 2.5273 2.2778 1.0945 1.0101 0.92702.1950 2.4227 2.1835 1.0492 0.9683 0.88862.1003 2.3181 2.0892 1.0039 0.9265 0.85022.0055 2.2135 1.9950 0.9586 0.8847 0.81191.9107 2.1089 1.9007 0.9133 0.8429 0.77351.8160 2.0043 1.8064 0.8680 0.8011 0.73511.7212 1.8997 1.7122 0.8227 0.7593 0.69681.6264 1.7951 1.6179 0.7774 0.7175 0.65841.5317 1.6905 1.5236 0.7321 0.6757 0.62001.4369 1.5860 1.4294 0.6868 0.6339 0.58171.3421 1.4814 1.3351 0.6415 0.5921 0.54331.2474 1.3768 1.2408 0.5962 0.5503 0.50501.1526 1.2722 1.1465 0.5509 0.5085 0.46661.0578 1.1676 1.0523 0.5056 0.4666 0.42820.9631 1.0630 0.9580 0.4603 0.4248 0.38990.8683 0.9584 0.8637 0.4150 0.3830 0.35150.7735 0.8538 0.7695 0.3697 0.3412 0.31310.6788 0.7492 0.6752 0.3244 0.2994 0.27480.5840 0.6446 0.5809 0.2791 0.2576 0.23640.4892 0.5400 0.4867 0.2338 0.2158 0.1981

X_vel_H13_final_S2 =

8.5704 8.1962 7.25088.4865 8.1159 7.17988.4025 8.0356 7.10888.3186 7.9554 7.03778.2346 7.8751 6.96678.1507 7.7948 6.89578.0667 7.7145 6.82477.9828 7.6342 6.75367.8988 7.5539 6.68267.8149 7.4737 6.61167.7309 7.3934 6.5406

A.1 Motion Response Analysis including KC-number estimation 83

7.6470 7.3131 6.46967.5631 7.2328 6.39857.4791 7.1525 6.32757.3952 7.0722 6.25657.3112 6.9920 6.18557.2273 6.9117 6.11447.1433 6.8314 6.04347.0594 6.7511 5.97246.9754 6.6708 5.90146.8915 6.5906 5.83046.8075 6.5103 5.75936.7236 6.4300 5.68836.6396 6.3497 5.61736.5557 6.2694 5.54636.4717 6.1891 5.47526.3878 6.1089 5.40426.3038 6.0286 5.33326.2199 5.9483 5.26226.1359 5.8680 5.19126.0520 5.7877 5.12015.9680 5.7074 5.04915.8841 5.6272 4.97815.8001 5.5469 4.90715.7162 5.4666 4.83605.6322 5.3863 4.76505.5483 5.3060 4.69405.4643 5.2257 4.62305.3804 5.1455 4.55205.2964 5.0652 4.48095.2125 4.9849 4.40995.1285 4.9046 4.33895.0446 4.8243 4.26794.9607 4.7440 4.19684.8767 4.6638 4.12584.7928 4.5835 4.05484.7088 4.5032 3.98384.6249 4.4229 3.91274.5409 4.3426 3.84174.4570 4.2623 3.77074.3730 4.1821 3.69974.2891 4.1018 3.62874.2051 4.0215 3.55764.1212 3.9412 3.48664.0372 3.8609 3.4156

84 A Matlab codes Motion Analysis

3.9533 3.7807 3.34463.8693 3.7004 3.27353.7854 3.6201 3.20253.7014 3.5398 3.13153.6175 3.4595 3.06053.5335 3.3792 2.98953.4496 3.2990 2.91843.3656 3.2187 2.84743.2817 3.1384 2.77643.1977 3.0581 2.70543.1138 2.9778 2.63433.0298 2.8975 2.56332.9459 2.8173 2.49232.8619 2.7370 2.42132.7780 2.6567 2.35032.6940 2.5764 2.27922.6101 2.4961 2.20822.5261 2.4158 2.13722.4422 2.3356 2.06622.3583 2.2553 1.99512.2743 2.1750 1.92412.1904 2.0947 1.85312.1064 2.0144 1.78212.0225 1.9341 1.71111.9385 1.8539 1.64001.8546 1.7736 1.56901.7706 1.6933 1.49801.6867 1.6130 1.42701.6027 1.5327 1.35591.5188 1.4525 1.28491.4348 1.3722 1.21391.3509 1.2919 1.14291.2669 1.2116 1.07181.1830 1.1313 1.00081.0990 1.0510 0.92981.0151 0.9708 0.85880.9311 0.8905 0.78780.8472 0.8102 0.71670.7632 0.7299 0.64570.6793 0.6496 0.57470.5953 0.5693 0.50370.5114 0.4891 0.43260.4274 0.4088 0.3616

A.1 Motion Response Analysis including KC-number estimation 85

KCf =

0.1500 0.4871 0.7400

KCfinal =

Columns 1 through 7

0.1500 0.2444 0.3389 0.4333 0.5278 0.6222 0.7167

Columns 8 through 10

0.8111 0.9056 1.0000

KC_ESTIMATED_RESULTS_SS2 =

Columns 1 through 7

0.6992 0.7406 0.7661 0.7755 0.7690 0.7466 0.7081

Columns 8 through 14

0.6537 0.5834 0.4971 0.5485 0.5484 0.5431 0.5327

Columns 15 through 21

0.5173 0.4968 0.4711 0.4404 0.4046 0.3637 0.2659

Columns 22 through 28

0.2610 0.2554 0.2493 0.2427 0.2354 0.2277 0.2193

Columns 29 through 35

0.2104 0.2009 0.7513 0.7947 0.8213 0.8311 0.8242

Columns 36 through 42

0.8005 0.7600 0.7027 0.6286 0.5377 0.5920 0.5918

Columns 43 through 49

86 A Matlab codes Motion Analysis

0.5861 0.5749 0.5583 0.5361 0.5085 0.4754 0.4368

Columns 50 through 56

0.3927 0.2890 0.2836 0.2775 0.2708 0.2635 0.2556

Columns 57 through 60

0.2470 0.2378 0.2280 0.2175

Velocity estimation for sea state Hs=3m , Tp=6sec\colorblack%final resultsfor i=1:NM0_R1(:,i)= abs(trapz(RAO_Omega.^0.*R1(:,i),RAO_Omega));M0_R2(:,i)= abs(trapz(RAO_Omega.^0.*R2(:,i),RAO_Omega));M0_R3(:,i)= abs(trapz(RAO_Omega.^0.*R3(:,i),RAO_Omega));M0_R4(:,i)= abs(trapz(RAO_Omega.^0.*R4(:,i),RAO_Omega));M0_R5(:,i)= abs(trapz(RAO_Omega.^0.*R5(:,i),RAO_Omega));M0_R6(:,i)= abs(trapz(RAO_Omega.^0.*R6(:,i),RAO_Omega));M0_R7(:,i)= abs(trapz(RAO_Omega.^0.*R7(:,i),RAO_Omega));M0_R8(:,i)= abs(trapz(RAO_Omega.^0.*R8(:,i),RAO_Omega));

M2_R1(:,i)= abs(trapz(RAO_Omega.^2.*R1(:,i),RAO_Omega));M2_R2(:,i)= abs(trapz(RAO_Omega.^2.*R2(:,i),RAO_Omega));M2_R3(:,i)= abs(trapz(RAO_Omega.^2.*R3(:,i),RAO_Omega));M2_R4(:,i)= abs(trapz(RAO_Omega.^2.*R4(:,i),RAO_Omega));M2_R5(:,i)= abs(trapz(RAO_Omega.^2.*R5(:,i),RAO_Omega));M2_R6(:,i)= abs(trapz(RAO_Omega.^2.*R6(:,i),RAO_Omega));M2_R7(:,i)= abs(trapz(RAO_Omega.^2.*R7(:,i),RAO_Omega));M2_R8(:,i)= abs(trapz(RAO_Omega.^2.*R8(:,i),RAO_Omega));end

M0_R = [M0_R1,M0_R2,M0_R3,M0_R4,M0_R5,M0_R6,M0_R7,M0_R8];M2_R = [M2_R1,M2_R2,M2_R3,M2_R4,M2_R5,M2_R6,M2_R7,M2_R8];

N =length(M0_R)for i=1:NSD(i) = sqrt(M0_R(i));

A.1 Motion Response Analysis including KC-number estimation 87

T1(i) = 2*pi*(M0_R(i)/M2_R(i));Tz(i) = (2*pi) * sqrt(M0_R(i)./(M2_R(i)));end

t = 3*60*60;

for i=1:NMPM_1(i) =SD(i) .* sqrt(2*log(t/Tz(i))); %MPM from equ 2.35 faltinsen

H_13(i) = 4*sqrt(M0_R(i)); %significant wave height equ. 2.26

PA(i) = (A/M0_R(i))*exp((A^2)/(2*M0_R(i))); %probability density functionend % to the maxima A with rayleigh dist.

X_vel_1 = RAO_Omega*MPM_1(1,1);X_vel_2 = RAO_Omega*MPM_1(1,2);X_vel_3 = RAO_Omega*MPM_1(1,3);X_vel_4 = RAO_Omega*MPM_1(1,4);X_vel_5 = RAO_Omega*MPM_1(1,5);X_vel_6 = RAO_Omega*MPM_1(1,6);X_vel_7 = RAO_Omega*MPM_1(1,7);X_vel_8 = RAO_Omega*MPM_1(1,8);

X_vel_sig1 = RAO_Omega*H_13(1,1);X_vel_sig2 = RAO_Omega*H_13(1,2);X_vel_sig3 = RAO_Omega*H_13(1,3);X_vel_sig4 = RAO_Omega*H_13(1,4);X_vel_sig5 = RAO_Omega*H_13(1,5);X_vel_sig6 = RAO_Omega*H_13(1,6);X_vel_sig7 = RAO_Omega*H_13(1,7);X_vel_sig8 = RAO_Omega*H_13(1,8);

%results of vertical velocitie for Sea state 1, only results for one Ratio%is output because every result is equal wit a wrong margin of 0.01 m/sX_vel_MPM = [X_vel_1,X_vel_2,X_vel_3,X_vel_4,X_vel_5,X_vel_6,...

X_vel_7,X_vel_8]X_vel_sig = [X_vel_sig1,X_vel_sig2,X_vel_sig3,X_vel_sig4,...

X_vel_sig5,X_vel_sig6,X_vel_sig7,X_vel_sig8]

Amp_MPM = X_vel_MPM(61,:);Amp_sig = X_vel_sig(61,:);

Dim= 37;

88 A Matlab codes Motion Analysis

KC_new_MPM = (Amp_MPM*Tp)/DimKC_new_sig = (Amp_sig*Tp)/Dim

N =

64

X_vel_MPM =

Columns 1 through 7

0.1663 0.1654 0.1647 0.1641 0.1637 0.1634 0.16320.1647 0.1638 0.1631 0.1625 0.1621 0.1618 0.16160.1631 0.1622 0.1615 0.1609 0.1605 0.1602 0.16000.1614 0.1606 0.1599 0.1593 0.1589 0.1586 0.15840.1598 0.1590 0.1583 0.1577 0.1573 0.1570 0.15680.1582 0.1573 0.1567 0.1561 0.1557 0.1554 0.15520.1565 0.1557 0.1550 0.1545 0.1541 0.1538 0.15360.1549 0.1541 0.1534 0.1529 0.1525 0.1522 0.15200.1533 0.1525 0.1518 0.1513 0.1509 0.1506 0.15040.1517 0.1509 0.1502 0.1497 0.1493 0.1490 0.14880.1500 0.1492 0.1486 0.1481 0.1477 0.1474 0.14720.1484 0.1476 0.1470 0.1465 0.1461 0.1458 0.14560.1468 0.1460 0.1454 0.1449 0.1445 0.1442 0.14400.1451 0.1444 0.1438 0.1433 0.1429 0.1426 0.14240.1435 0.1428 0.1421 0.1416 0.1413 0.1410 0.14080.1419 0.1411 0.1405 0.1400 0.1397 0.1394 0.13920.1403 0.1395 0.1389 0.1384 0.1381 0.1378 0.13760.1386 0.1379 0.1373 0.1368 0.1365 0.1362 0.13600.1370 0.1363 0.1357 0.1352 0.1349 0.1346 0.13440.1354 0.1347 0.1341 0.1336 0.1332 0.1330 0.13280.1337 0.1330 0.1325 0.1320 0.1316 0.1314 0.13120.1321 0.1314 0.1309 0.1304 0.1300 0.1298 0.12960.1305 0.1298 0.1292 0.1288 0.1284 0.1282 0.12800.1289 0.1282 0.1276 0.1272 0.1268 0.1266 0.12640.1272 0.1266 0.1260 0.1256 0.1252 0.1250 0.12480.1256 0.1249 0.1244 0.1240 0.1236 0.1234 0.12320.1240 0.1233 0.1228 0.1224 0.1220 0.1218 0.12160.1224 0.1217 0.1212 0.1208 0.1204 0.1202 0.12000.1207 0.1201 0.1196 0.1192 0.1188 0.1186 0.11840.1191 0.1185 0.1180 0.1175 0.1172 0.1170 0.1168

A.1 Motion Response Analysis including KC-number estimation 89

0.1175 0.1169 0.1163 0.1159 0.1156 0.1154 0.11520.1158 0.1152 0.1147 0.1143 0.1140 0.1138 0.11360.1142 0.1136 0.1131 0.1127 0.1124 0.1122 0.11200.1126 0.1120 0.1115 0.1111 0.1108 0.1106 0.11050.1110 0.1104 0.1099 0.1095 0.1092 0.1090 0.10890.1093 0.1088 0.1083 0.1079 0.1076 0.1074 0.10730.1077 0.1071 0.1067 0.1063 0.1060 0.1058 0.10570.1061 0.1055 0.1051 0.1047 0.1044 0.1042 0.10410.1044 0.1039 0.1034 0.1031 0.1028 0.1026 0.10250.1028 0.1023 0.1018 0.1015 0.1012 0.1010 0.10090.1012 0.1007 0.1002 0.0999 0.0996 0.0994 0.09930.0996 0.0990 0.0986 0.0983 0.0980 0.0978 0.09770.0979 0.0974 0.0970 0.0967 0.0964 0.0962 0.09610.0963 0.0958 0.0954 0.0951 0.0948 0.0946 0.09450.0947 0.0942 0.0938 0.0934 0.0932 0.0930 0.09290.0931 0.0926 0.0922 0.0918 0.0916 0.0914 0.09130.0914 0.0909 0.0905 0.0902 0.0900 0.0898 0.08970.0898 0.0893 0.0889 0.0886 0.0884 0.0882 0.08810.0882 0.0877 0.0873 0.0870 0.0868 0.0866 0.08650.0865 0.0861 0.0857 0.0854 0.0852 0.0850 0.08490.0849 0.0845 0.0841 0.0838 0.0836 0.0834 0.08330.0833 0.0828 0.0825 0.0822 0.0820 0.0818 0.08170.0817 0.0812 0.0809 0.0806 0.0804 0.0802 0.08010.0800 0.0796 0.0793 0.0790 0.0788 0.0786 0.07850.0784 0.0780 0.0777 0.0774 0.0772 0.0770 0.07690.0768 0.0764 0.0760 0.0758 0.0756 0.0754 0.07530.0751 0.0748 0.0744 0.0742 0.0740 0.0738 0.07370.0735 0.0731 0.0728 0.0726 0.0724 0.0722 0.07210.0719 0.0715 0.0712 0.0710 0.0708 0.0706 0.07050.0703 0.0699 0.0696 0.0693 0.0692 0.0690 0.06890.0686 0.0683 0.0680 0.0677 0.0676 0.0674 0.06730.0670 0.0667 0.0664 0.0661 0.0660 0.0658 0.06570.0654 0.0650 0.0648 0.0645 0.0644 0.0642 0.06410.0638 0.0634 0.0631 0.0629 0.0627 0.0626 0.06250.0621 0.0618 0.0615 0.0613 0.0611 0.0610 0.06090.0605 0.0602 0.0599 0.0597 0.0595 0.0594 0.05930.0589 0.0586 0.0583 0.0581 0.0579 0.0578 0.05780.0572 0.0569 0.0567 0.0565 0.0563 0.0562 0.05620.0556 0.0553 0.0551 0.0549 0.0547 0.0546 0.05460.0540 0.0537 0.0535 0.0533 0.0531 0.0530 0.05300.0524 0.0521 0.0519 0.0517 0.0515 0.0514 0.05140.0507 0.0505 0.0502 0.0501 0.0499 0.0498 0.04980.0491 0.0488 0.0486 0.0485 0.0483 0.0482 0.04820.0475 0.0472 0.0470 0.0469 0.0467 0.0466 0.0466

90 A Matlab codes Motion Analysis

0.0458 0.0456 0.0454 0.0452 0.0451 0.0450 0.04500.0442 0.0440 0.0438 0.0436 0.0435 0.0434 0.04340.0426 0.0424 0.0422 0.0420 0.0419 0.0418 0.04180.0410 0.0407 0.0406 0.0404 0.0403 0.0402 0.04020.0393 0.0391 0.0390 0.0388 0.0387 0.0386 0.03860.0377 0.0375 0.0373 0.0372 0.0371 0.0370 0.03700.0361 0.0359 0.0357 0.0356 0.0355 0.0354 0.03540.0344 0.0343 0.0341 0.0340 0.0339 0.0338 0.03380.0328 0.0326 0.0325 0.0324 0.0323 0.0322 0.03220.0312 0.0310 0.0309 0.0308 0.0307 0.0306 0.03060.0296 0.0294 0.0293 0.0292 0.0291 0.0290 0.02900.0279 0.0278 0.0277 0.0276 0.0275 0.0274 0.02740.0263 0.0262 0.0261 0.0260 0.0259 0.0258 0.02580.0247 0.0246 0.0244 0.0244 0.0243 0.0242 0.02420.0231 0.0229 0.0228 0.0228 0.0227 0.0226 0.02260.0214 0.0213 0.0212 0.0211 0.0211 0.0210 0.02100.0198 0.0197 0.0196 0.0195 0.0195 0.0194 0.01940.0182 0.0181 0.0180 0.0179 0.0179 0.0178 0.01780.0165 0.0165 0.0164 0.0163 0.0163 0.0163 0.01620.0149 0.0148 0.0148 0.0147 0.0147 0.0147 0.01460.0133 0.0132 0.0132 0.0131 0.0131 0.0131 0.01300.0117 0.0116 0.0115 0.0115 0.0115 0.0115 0.01140.0100 0.0100 0.0099 0.0099 0.0099 0.0099 0.00980.0084 0.0084 0.0083 0.0083 0.0083 0.0083 0.0082

Column 8

0.16300.16140.15990.15830.15670.15510.15350.15190.15030.14870.14710.14550.14390.14230.14070.13910.1375

A.1 Motion Response Analysis including KC-number estimation 91

0.13590.13430.13270.13110.12950.12790.12630.12470.12310.12150.12000.11840.11680.11520.11360.11200.11040.10880.10720.10560.10400.10240.10080.09920.09760.09600.09440.09280.09120.08960.08800.08640.08480.08320.08170.08010.07850.07690.07530.07370.07210.07050.06890.0673

92 A Matlab codes Motion Analysis

0.06570.06410.06250.06090.05930.05770.05610.05450.05290.05130.04970.04810.04650.04490.04330.04180.04020.03860.03700.03540.03380.03220.03060.02900.02740.02580.02420.02260.02100.01940.01780.01620.01460.01300.01140.00980.0082

X_vel_sig =

Columns 1 through 7

0.1736 0.1727 0.1719 0.1713 0.1708 0.1705 0.1703

A.1 Motion Response Analysis including KC-number estimation 93

0.1719 0.1710 0.1702 0.1696 0.1692 0.1688 0.16860.1702 0.1693 0.1685 0.1679 0.1675 0.1672 0.16690.1685 0.1676 0.1669 0.1663 0.1658 0.1655 0.16530.1668 0.1659 0.1652 0.1646 0.1641 0.1638 0.16360.1651 0.1642 0.1635 0.1629 0.1625 0.1621 0.16190.1634 0.1625 0.1618 0.1612 0.1608 0.1605 0.16030.1617 0.1608 0.1601 0.1596 0.1591 0.1588 0.15860.1600 0.1591 0.1584 0.1579 0.1575 0.1571 0.15690.1583 0.1574 0.1568 0.1562 0.1558 0.1555 0.15530.1566 0.1558 0.1551 0.1545 0.1541 0.1538 0.15360.1549 0.1541 0.1534 0.1529 0.1524 0.1521 0.15190.1532 0.1524 0.1517 0.1512 0.1508 0.1505 0.15030.1515 0.1507 0.1500 0.1495 0.1491 0.1488 0.14860.1498 0.1490 0.1483 0.1478 0.1474 0.1471 0.14690.1481 0.1473 0.1467 0.1462 0.1458 0.1455 0.14530.1464 0.1456 0.1450 0.1445 0.1441 0.1438 0.14360.1447 0.1439 0.1433 0.1428 0.1424 0.1421 0.14190.1430 0.1422 0.1416 0.1411 0.1407 0.1405 0.14030.1413 0.1405 0.1399 0.1394 0.1391 0.1388 0.13860.1396 0.1389 0.1383 0.1378 0.1374 0.1371 0.13690.1379 0.1372 0.1366 0.1361 0.1357 0.1354 0.13530.1362 0.1355 0.1349 0.1344 0.1340 0.1338 0.13360.1345 0.1338 0.1332 0.1327 0.1324 0.1321 0.13190.1328 0.1321 0.1315 0.1311 0.1307 0.1304 0.13030.1311 0.1304 0.1298 0.1294 0.1290 0.1288 0.12860.1294 0.1287 0.1282 0.1277 0.1274 0.1271 0.12690.1277 0.1270 0.1265 0.1260 0.1257 0.1254 0.12530.1260 0.1253 0.1248 0.1244 0.1240 0.1238 0.12360.1243 0.1236 0.1231 0.1227 0.1223 0.1221 0.12190.1226 0.1220 0.1214 0.1210 0.1207 0.1204 0.12030.1209 0.1203 0.1197 0.1193 0.1190 0.1188 0.11860.1192 0.1186 0.1181 0.1176 0.1173 0.1171 0.11690.1175 0.1169 0.1164 0.1160 0.1157 0.1154 0.11530.1158 0.1152 0.1147 0.1143 0.1140 0.1138 0.11360.1141 0.1135 0.1130 0.1126 0.1123 0.1121 0.11190.1124 0.1118 0.1113 0.1109 0.1106 0.1104 0.11030.1107 0.1101 0.1096 0.1093 0.1090 0.1087 0.10860.1090 0.1084 0.1080 0.1076 0.1073 0.1071 0.10690.1073 0.1067 0.1063 0.1059 0.1056 0.1054 0.10530.1056 0.1051 0.1046 0.1042 0.1039 0.1037 0.10360.1039 0.1034 0.1029 0.1026 0.1023 0.1021 0.10190.1022 0.1017 0.1012 0.1009 0.1006 0.1004 0.10030.1005 0.1000 0.0996 0.0992 0.0989 0.0987 0.09860.0988 0.0983 0.0979 0.0975 0.0973 0.0971 0.0969

94 A Matlab codes Motion Analysis

0.0971 0.0966 0.0962 0.0958 0.0956 0.0954 0.09530.0954 0.0949 0.0945 0.0942 0.0939 0.0937 0.09360.0937 0.0932 0.0928 0.0925 0.0922 0.0921 0.09190.0920 0.0915 0.0911 0.0908 0.0906 0.0904 0.09030.0903 0.0898 0.0895 0.0891 0.0889 0.0887 0.08860.0886 0.0882 0.0878 0.0875 0.0872 0.0871 0.08690.0869 0.0865 0.0861 0.0858 0.0856 0.0854 0.08530.0852 0.0848 0.0844 0.0841 0.0839 0.0837 0.08360.0835 0.0831 0.0827 0.0824 0.0822 0.0820 0.08190.0818 0.0814 0.0810 0.0808 0.0805 0.0804 0.08030.0801 0.0797 0.0794 0.0791 0.0789 0.0787 0.07860.0784 0.0780 0.0777 0.0774 0.0772 0.0770 0.07690.0767 0.0763 0.0760 0.0757 0.0755 0.0754 0.07530.0750 0.0746 0.0743 0.0741 0.0738 0.0737 0.07360.0733 0.0729 0.0726 0.0724 0.0722 0.0720 0.07190.0716 0.0713 0.0709 0.0707 0.0705 0.0704 0.07030.0699 0.0696 0.0693 0.0690 0.0688 0.0687 0.06860.0682 0.0679 0.0676 0.0673 0.0672 0.0670 0.06690.0665 0.0662 0.0659 0.0657 0.0655 0.0654 0.06530.0648 0.0645 0.0642 0.0640 0.0638 0.0637 0.06360.0631 0.0628 0.0625 0.0623 0.0621 0.0620 0.06190.0614 0.0611 0.0608 0.0606 0.0605 0.0603 0.06030.0597 0.0594 0.0592 0.0590 0.0588 0.0587 0.05860.0580 0.0577 0.0575 0.0573 0.0571 0.0570 0.05690.0563 0.0560 0.0558 0.0556 0.0555 0.0553 0.05530.0546 0.0544 0.0541 0.0539 0.0538 0.0537 0.05360.0529 0.0527 0.0524 0.0523 0.0521 0.0520 0.05190.0512 0.0510 0.0508 0.0506 0.0504 0.0503 0.05030.0495 0.0493 0.0491 0.0489 0.0488 0.0487 0.04860.0478 0.0476 0.0474 0.0472 0.0471 0.0470 0.04690.0461 0.0459 0.0457 0.0455 0.0454 0.0453 0.04530.0445 0.0442 0.0440 0.0439 0.0437 0.0437 0.04360.0428 0.0425 0.0423 0.0422 0.0421 0.0420 0.04190.0411 0.0408 0.0407 0.0405 0.0404 0.0403 0.04030.0394 0.0391 0.0390 0.0388 0.0387 0.0387 0.03860.0377 0.0375 0.0373 0.0372 0.0371 0.0370 0.03690.0360 0.0358 0.0356 0.0355 0.0354 0.0353 0.03530.0343 0.0341 0.0339 0.0338 0.0337 0.0336 0.03360.0326 0.0324 0.0322 0.0321 0.0320 0.0320 0.03190.0309 0.0307 0.0306 0.0305 0.0304 0.0303 0.03030.0292 0.0290 0.0289 0.0288 0.0287 0.0286 0.02860.0275 0.0273 0.0272 0.0271 0.0270 0.0270 0.02690.0258 0.0256 0.0255 0.0254 0.0254 0.0253 0.02530.0241 0.0239 0.0238 0.0237 0.0237 0.0236 0.0236

A.1 Motion Response Analysis including KC-number estimation 95

0.0224 0.0222 0.0221 0.0221 0.0220 0.0220 0.02190.0207 0.0206 0.0205 0.0204 0.0203 0.0203 0.02030.0190 0.0189 0.0188 0.0187 0.0187 0.0186 0.01860.0173 0.0172 0.0171 0.0170 0.0170 0.0170 0.01690.0156 0.0155 0.0154 0.0154 0.0153 0.0153 0.01530.0139 0.0138 0.0137 0.0137 0.0136 0.0136 0.01360.0122 0.0121 0.0121 0.0120 0.0120 0.0120 0.01190.0105 0.0104 0.0104 0.0103 0.0103 0.0103 0.01030.0088 0.0087 0.0087 0.0087 0.0086 0.0086 0.0086

Column 8

0.17020.16850.16680.16520.16350.16180.16020.15850.15680.15520.15350.15180.15020.14850.14680.14520.14350.14180.14020.13850.13680.13520.13350.13180.13020.12850.12690.12520.12350.12190.12020.1185

96 A Matlab codes Motion Analysis

0.11690.11520.11350.11190.11020.10850.10690.10520.10350.10190.10020.09850.09690.09520.09350.09190.09020.08850.08690.08520.08350.08190.08020.07860.07690.07520.07360.07190.07020.06860.06690.06520.06360.06190.06020.05860.05690.05520.05360.05190.05020.04860.04690.0452

A.1 Motion Response Analysis including KC-number estimation 97

0.04360.04190.04020.03860.03690.03520.03360.03190.03030.02860.02690.02530.02360.02190.02030.01860.01690.01530.01360.01190.01030.0086

KC_new_MPM =

Columns 1 through 7

0.0111 0.0111 0.0110 0.0110 0.0110 0.0109 0.0109

Column 8

0.0109

KC_new_sig =

Columns 1 through 7

0.0116 0.0116 0.0115 0.0115 0.0114 0.0114 0.0114

Column 8

0.0114

98 A Matlab codes Motion Analysis

Plotting final KC-comparison curve

KC = linspace(0.15,0.74,8);KC_final = [0.15, 0.4871, 0.74]Amp_initial = (KC*Dim)/Tp;

KC_xx = linspace(0,1.4,10)KC_yy = linspace(0,1.4,10)xcor = linspace(0,0.15,5)ycor = KC_new_MPM(1,1:5)

figure()plot(KC,KC_new_MPM,...

KCfinal,KC_poly1,KCfinal,KC_poly2,...KCfinal,KC_poly3,KCfinal,KC_poly4,...KCfinal,KC_poly5,KCfinal,KC_poly6,...KC_xx,KC_yy,xcor,ycor,'o-','LineWidth',1.5)

axis([0 1.05 0 1.05])title('Comparison of Estimated KC-number for varius flanges and sea states')xlabel('Assumed KC-number')ylabel('Estimated KC-number')legend('All KC and Cd, SS. 1',...

'F-size 4m, MPM, SS.2',...'F-size 6m, MPM, SS.2',...'F-size 8m, MPM, SS.2',...'F-size 4m, H13, SS.2',...'F-size 6m, H13, SS.2',...'F-size 8m, H13, SS.2','Interaction line')

KC_final =

0.1500 0.4871 0.7400

KC_xx =

Columns 1 through 7

A.1 Motion Response Analysis including KC-number estimation 99

0 0.1556 0.3111 0.4667 0.6222 0.7778 0.9333

Columns 8 through 10

1.0889 1.2444 1.4000

KC_yy =

Columns 1 through 7

0 0.1556 0.3111 0.4667 0.6222 0.7778 0.9333

Columns 8 through 10

1.0889 1.2444 1.4000

xcor =

0 0.0375 0.0750 0.1125 0.1500

ycor =

0.0111 0.0111 0.0110 0.0110 0.0110

Finding new CD-values using data from Tao et.al.

L_hull = 1; %meterB_hull = 1; %meterT_hull = 198.1; %meterD_hull = 39; %meterR_hull = D_hull/2; %meterW_flange = 0 ; %meterT_flange = 0.475; %meterD_flange = [39 42 45.1 48.4 53 57.72 65 70 78]; %meter%D_flange = 57.72;

100 A Matlab codes Motion Analysis

R_flange = D_flange/2; %meter

Dd_Dc = D_flange/D_hull;Dd_Dc_1 = linspace(1,2,70);td_Dd = T_flange./D_flange;D = sqrt((D_flange.^2) - (D_hull^2));

DRatio =D_flange/D_hull;

m = rho*((R_hull^2) * pi * (T_hull-T_flange) + (R_flange.^2) *...pi * T_flange);

ma = ((rho*(D_flange.^3))/3) - ((((pi*rho)/8)*((D_hull^2)*...(D_flange-D))) + (((pi*rho)/24).*(((D_flange-D).^2) .*...(2.*D_flange + D))));

% Damping ratio Z for KC = 0.15 when using diamter ratio

KC = [0.15, 0.44, 0.74];Z = [0.01; 0.0256; 0.04 ];KC1= 0.15;KC2= 0.44;KC3= 0.74;Z1 = 0.01;Z2 = 0.0254;Z3 = 0.04;

for i=1:length(D_flange)

Cd1(i) = (Z1./KC1).*((3*pi^2) * ((D_hull^2)*T_hull + ...(D_flange(i).^2)*T_flange)*((m(i)+ma(i))/m(i)))..../((D_flange(i).^2)*D_hull);

Cd2(i) = (Z2./KC2).*((3*pi^2) * ((D_hull^2)*T_hull + ...(D_flange(i).^2)*T_flange)*((m(i)+ma(i))/m(i)))..../((D_flange(i).^2)*D_hull);

Cd3(i) = (Z3./KC3).*((3*pi^2) * ((D_hull^2)*T_hull + ...(D_flange(i).^2)*T_flange)*((m(i)+ma(i))/m(i)))..../((D_flange(i).^2)*D_hull);

end

A.1 Motion Response Analysis including KC-number estimation 101

Plot of Cd-curveD_flange = [39 42 45.1 48.4 53 57.72 65 70 78]; %meterR_flange = D_flange/2; %meterD = sqrt((D_flange.^2) - (D_hull^2));Dd_Dc = D_flange/D_hull;CD = [Cd1;Cd2;Cd3];

Cd - coefficients for each ratioCD_1 = [Cd1(1,1),Cd2(1,1),Cd3(1,1)]; % rato 1CD_2 = [Cd1(1,2),Cd2(1,2),Cd3(1,2)];CD_3 = [Cd1(1,3),Cd2(1,3),Cd3(1,3)];CD_4 = [Cd1(1,4),Cd2(1,4),Cd3(1,4)];CD_5 = [Cd1(1,5),Cd2(1,5),Cd3(1,5)];CD_6 = [Cd1(1,6),Cd2(1,6),Cd3(1,6)]; % Cd at ratio Dd/Dc = 1.48CD_7 = [Cd1(1,7),Cd2(1,7),Cd3(1,7)];CD_8 = [Cd1(1,8),Cd2(1,8),Cd3(1,8)];CD_9 = [Cd1(1,9),Cd2(1,9),Cd3(1,9)]; %ratio = 2

% Cd1 is KC = 0.15, Cd2 is KC = 0.44, Cd3 is KC = 0.74

CD = [CD_1;CD_2;CD_3;CD_4;CD_5;CD_6;CD_7;CD_8;CD_9];

Quadric fitting for every ratioKCx = linspace(0.15,0.74,8);

p1 = polyfit(KC, CD_1, 2); %for ratio = 1p2 = polyfit(KC, CD_2, 2);p3 = polyfit(KC, CD_3, 2);p4 = polyfit(KC, CD_4, 2);p5 = polyfit(KC, CD_5, 2);p6 = polyfit(KC, CD_6, 2);p7 = polyfit(KC, CD_7, 2);p8 = polyfit(KC, CD_8, 2); %for ratio = 1.8

p = [p1;p2;p3;p4;p5;p6;p7;p8];p = p';

%final Cd-coefficientsfor i=1:length(KCx)

for k=1:length(p)

102 A Matlab codes Motion Analysis

Cd_poly(i,k) = p(3,k) + p(2,k).*KCx(i) + p(1,k).*KCx(i).^2;end

end

%testfor i=1:length(KCx)Cd_poly_1(i) = p1(1,3) + p1(1,2).*KCx(i) + p1(1,1).*KCx(i).^2;Cd_poly_5(i) = p5(1,3) + p5(1,2).*KCx(i) + p5(1,1).*KCx(i).^2;end

for i=1:length(KC_new_MPM)for k=1:length(p)

Cd_poly_MPM(i,k) = p(3,k) + p(2,k).*KC_new_MPM(i) +...p(1,k).*KC_new_MPM(i).^2;

Cd_poly_sig(i,k) = p(3,k) + p(2,k).*KC_new_sig(i) +...p(1,k).*KC_new_sig(i).^2;end

end

Natural Period

A.2 Natural period estimation 103

A.2 Natural period estimation• INPUT• Data from Aqwa• Bridge/girder stiffness, kwb• Wave spectrum• NATURAL PERIOD• Input pontoon 0 meter flange• Input pontoon 4 meter flange• Input pontoon 6 meter flange• Input pontoon 8 meter flange• Plotting the natural period againdst the wave spectrum• Reading of the natural periods manually gives:

clc;clear all;close all;% Espen Kleppa - s151091% Master Thesis%Calculating Natural Period of structure

INPUTf0 = linspace(2.5,0.1,50);P_f0 = (2*pi)./f0;w = f0*2*pi;g = 9.81;

Hs = 3;Tp = 6;

a = 0.0081;b = 0.74;wp= 2*pi/Tp;

liney = [-500000,40000000];linex = [6,6];

liney2 = [0,0];linex2 = [0,14];

Data from Aqwa[table1]= xlsread('damp_0'); %damping for pontoon, no flange

104 A Matlab codes Motion Analysis

[table2]= xlsread('damp_4'); %damping for pontoon, 4 m flange[table3]= xlsread('damp_6'); %damping for pontoon, 6 m flange[table4]= xlsread('damp_8'); %damping for pontoon, 8 m flange[table5]= xlsread('AM_0'); %Added Mass for pontoon, 0 m flange[table6]= xlsread('AM_4'); %Added Mass for pontoon, 4 m flange[table7]= xlsread('AM_6'); %Added Mass for pontoon, 6 m flange[table8]= xlsread('AM_8'); %Added Mass for pontoon, 8 m flange

%damping comparison with varying flange, periodfigure()plot(table1(:,1),table1(:,5),table2(:,1),table2(:,5),...

table3(:,1),table3(:,5),table4(:,1),table4(:,5),linex,liney,...linex2,liney2)

axis([2.5,14,-500000,4000000])title('Damping comparison for different Flange Width')xlabel('Period [s]')ylabel('Radiation Damping, B_33 [N/(m/s)]')legend('No Flange','4 m Flange','6 m Flange','8 m Flange','Tp = 6 sec.')

%damping comparison with varying flange, omegafigure()plot(table1(:,2),table1(:,5),table2(:,2),table2(:,5),...

table3(:,2),table3(:,5),table4(:,2),table4(:,5))

%Added mass comparison with varying flange, periodfigure()plot(table5(:,1),table5(:,5),table6(:,1),table6(:,5),...

table7(:,1),table7(:,5),table8(:,1),table8(:,5))axis([2.5,14,10000000,50000000])title('Added Mass comparison for different Flange Width')xlabel('Period [s]')ylabel('Added MAss, A_33 [kg)]')legend('No Flange','4 m Flange','6 m Flange','8 m Flange')

%Added mass comparison with varying flange, omegafigure()plot(table5(:,2),table5(:,5),table6(:,2),table6(:,5),...

table7(:,2),table7(:,5),table8(:,2),table8(:,5))

A.2 Natural period estimation 105

Bridge/girder stiffness, kwbE_st = 190e9;%PaL = 200;%mIy = 18.1;%m^4

kwb_st = (48*E_st*Iy)/L^3;

Wave spectrumTt = linspace(0,20,50);

SW_pm_DNV = 5/16 .* Hs.^2.*wp.^4 .* f0.^(-5) .* exp(-(5/4).*(f0./wp).^-4);

A_pm = trapz(f0,SW_pm_DNV);sigma = 0.09;gamma = 2.05;

AG = 1-(0.287*log(gamma));SW_jw_DNV = AG.*SW_pm_DNV.*gamma.^(exp(-0.5.*((f0-wp)./(sigma*wp)).^2));

% Wave spectrum in reportfigure()plot(f0,SW_pm_DNV,f0,SW_jw_DNV);%axis ([0 0.6 0 2.5])title('Wave spectrum')legend('Pierson-Moskowits', 'Jonswap')xlabel('\omega [rad/s]')ylabel('Spectral density [m^2/(rad/s)]')

figure()plot(P_f0,SW_pm_DNV,P_f0,SW_jw_DNV);axis ([0 20 0 2.5])title('Wave spectrum')legend('Pierson-Moskowits', 'Jonswap')xlabel('\omega [rad/s]')ylabel('Spectral density [m^2/(rad/s)]')

% Wave spectrum to fit the natural periodSW_jw_DNV2 = SW_jw_DNV.*((2*pi)./f0);

figure()plot(Tt,SW_jw_DNV2)%axis ([0 0.6 0 2.5])

106 A Matlab codes Motion Analysis

legend('Jonswap')

NATURAL PERIOD

Input pontoon 0 meter flangem_p_0 = 13417630; % [kg] from Aqwam_girder = 4082000; % from SVVm0 = m_p_0 + m_girder;C33_0 = 14225083; %from AQWAA33_0 = table5(:,5); %from AQWAomega_0 = table5(:,2); %from AQWA

Tn_0 = 2*pi*sqrt((A33_0+m0)/(C33_0+kwb_st)); %equ. 3.49 from faltinsenTn_0_f = 2*pi*sqrt((A33_0+m0)/(C33_0)); %equ. 3.49 from faltinsen

Input pontoon 4 meter flangem_p_4 = 13780190; % [kg] from AQWAm4 = m_p_4 + m_girder;C33_4 = 14252747; %[n/m]from AQWAA33_4 = table6(:,5); %from AQWAomega_4 = table6(:,2); %from AQWA

Tn_4 = 2*pi*sqrt((A33_4+m4)/(C33_4+kwb_st)); %equ. 3.49 from faltinsenTn_4_f = 2*pi*sqrt((A33_4+m4)/(C33_4)); %equ. 3.49 from faltinsen

Input pontoon 6 meter flangem_p_6 = 13944000; %[kg] from AQWAm6 = m_p_6 + m_girder;C33_6 = 14224653; %from AQWAA33_6 = table7(:,5); %from AQWAomega_6 = table7(:,2); %from AQWA

Tn_6 = 2*pi*sqrt((A33_6+m6)/(C33_6+kwb_st)); %equ. 3.49 from faltinsenTn_6_f = 2*pi*sqrt((A33_6+m6)/(C33_6)); %equ. 3.49 from faltinsen

Input pontoon 8 meter flangem_p_8 = 14146753; %[kg] from AQWAm8 = m_p_8+m_girder;C33_8 = 14225828; %from AQWA

A.2 Natural period estimation 107

A33_8 = table8(:,5); %from AQWAomega_8 = table8(:,2); %from AQWA

Tn_8 = 2*pi* sqrt((A33_8+m8)/(C33_8+kwb_st)); %equ. 3.49 from faltinsenTn_8_f = 2*pi* sqrt((A33_8+m8)/(C33_8)); %equ. 3.49 from faltinsen

Plotting the natural period againdst the wave spectrumT = (linspace(0,20,50))';Tz2 = (linspace(0,20,50))';

figure()plot(Tz2,Tn_0,Tz2,Tn_4,Tz2,Tn_6,Tz2,Tn_8,T,T,P_f0(:,1:45),...

SW_jw_DNV2(:,1:45))axis([0 20 0 14])title('Natural Period with bridge stiffness')xlabel('Natural Period [s]')ylabel('Period [s]')legend('No Flange','4m flange', '6m flange','8m flange',...

'Period intersection','JW- Wave Spectrum')

figure()plot(Tz2,Tn_0_f,Tz2,Tn_4_f,Tz2,Tn_6_f,Tz2,Tn_8_f,T,T,...

P_f0(:,1:45), SW_jw_DNV2(:,1:45))title('Natural Period without bridge stiffnes')xlabel('Natural Period [s]')ylabel('Period [s]')legend('No Flange','4m flange', '6m flange','8m flange',...

'Period intersection','JW- Wave Spectrum')

Reading of the natural periods manually gives:NP_0m = 5.96 ;NP_4m = 6.9 ;NP_6m = 7.5 ;NP_8m = 8.2 ;

NP = [NP_0m, NP_4m, NP_6m, NP_8m,];

108

APPENDIX BVerificationcalculation

Diffraction and Froude-Krylov (Force/Moment vs Frequency)Line A: Diffraction and Froude-Krylov - Global X, -180° (AQW) [N/m]Line B: Diffraction and Froude-Krylov - Global Z, -180° (AQW) [N/m]input data*Frequency (Hz),Line A (N/m),Line B (N/m)

0 1592 206 2,69501E+11 79110 19107 2490 5625 71110 36622 4751 2265625 54550 54137 6594 70166015625 34570 71653 6633 1123046875 17260 89168 4851 74951171875 6961 6683 3248 48559570313 2471 24198 2199 62036132813 811 41714 1517 419921875 241 59229 1087 85571289063 51 76744 799 765625 11 94259 600 9423828125 02 11775 464 9521484375 02 2929 366 1,73615E+11 02 46805 291 1,36963E+11 02 64321 243 6,32599E+11 12 81836 182 86897277832 02 99351 161 9,05853E+11 183 16866 145 8,62289E+11 03 34382 87 1,07323E+12 0

700683593759033203125

3110351562552246093754599609375

6,79016E+117,14249E+114,16496E+122,17226E+126,32732E+139,60689E+131,61968E+149,42882E+131,48528E+138,20291E+132,08898E+135,58351E+141,65049E+125,28388E+141,86183E+14

0,1592 206,269500732422 7911,700683593750,19107 2490,5625 7111,90332031250,36622 4751,2265625 5455,311035156250,54137 6594,70166015625 3457,52246093750,71653 6633,1123046875 1726,45996093750,89168 4851,74951171875 696,6790161132811,6683 3248,48559570313 247,7142486572271,24198 2199,62036132813 81,41649627685551,41714 1517,419921875 24,21722602844241,59229 1087,85571289063 5,632732391357421,76744 799,765625 1,960688948631291,94259 600,9423828125 0,1619676202535632,11775 464,9521484375 0,942881703376772,2929 366,173614501953 0,148528113961222,46805 291,136962890625 0,820291399955752,64321 243,632598876953 1,208897948265082,81836 182,86897277832 0,5583510398864752,99351 161,905853271484 18,16504859924323,16866 145,862289428711 0,5283880233764653,34382 87,1073226928711 0,186183005571365

Hz Force X-dir Force Z-dir0,01592 206,2695007 7911,7006840,19107 2490,5625 7111,903320,36622 4751,226563 5455,3110350,54137 6594,70166 3457,5224610,71653 6633,112305 1726,4599610,89168 4851,749512 696,67901611,06683 3248,485596 247,71424871,24198 2199,620361 81,416496281,41714 1517,419922 24,217226031,59229 1087,855713 5,6327323911,76744 799,765625 1,9606889491,94259 600,9423828 0,161967622,11775 464,9521484 0,9428817032,2929 366,1736145 0,148528114

2,46805 291,1369629 0,82029142,64321 243,6325989 1,2088979482,81836 182,8689728 0,558351042,99351 161,9058533 18,16504863,16866 145,8622894 0,5283880233,34382 87,10732269 0,186183006

From AQWA

rho 1025g 9,81R 0,5A 1H 1

kR DWRad/s K-deep water K-shallow water kR-deep water

0,10002831 0,031936582 0,031936582 0,0159682911,200528217 0,146918247 0,383299161 0,0734591232,301028123 0,539727872 0,73466174 0,2698639363,40152803 1,179448821 1,086024319 0,589724411

4,502090768 2,066138765 1,437406959 1,0330693835,602590675 3,19969646 1,788769538 1,599848236,703090581 4,580165478 2,140132117 2,2900827397,803590488 6,207545821 2,491494696 3,103772918,904153226 8,081951547 2,842877336 4,04097577310,00465313 10,20316863 3,194239915 5,10158431711,10515304 12,57129705 3,545602494 6,28564852312,20565295 15,18633678 3,896965073 7,59316839113,30621568 18,04845829 4,248347713 9,02422914614,40671559 21,15733477 4,599710292 10,5786673915,5072155 24,51312258 4,951072871 12,25656129

16,60777824 28,11603445 5,302455511 14,0580172217,70827814 31,965659 5,65381809 15,982829518,80877805 36,06219487 6,005180669 18,0310974419,90927796 40,40564207 6,356543249 20,2028210421,00984069 44,99626972 6,707925888 22,49813486

kR F-X-dir F-Z-dir0,6 2,45 1,550,8 2,75 1,1

1 2,65 0,761,2 2,45 0,561,4 2,2 0,361,6 1,9 0,26

kR-shallow water Conv. Xdir Conv. Zdir0,015968291 0,082054449 3,1472914880,19164958 0,9907511 2,8291303830,36733087 1,890048109 2,170134421

0,543012159 2,623386454 1,3754098450,718703479 2,638666291 0,6867894730,894384769 1,930036354 0,2771404061,070066058 1,292254532 0,0985412591,245747348 0,875013694 0,0323876571,421438668 0,603632897 0,0096336641,597119958 0,432751334 0,0022407131,772801247 0,31814848 0,0007799661,948482537 0,239056168 6,44311E-052,124173857 0,184958961 0,000375082,299855146 0,145664649 5,90848E-052,475536436 0,115814908 0,0003263142,651227756 0,09691757 0,0004809022,826909045 0,072745669 0,0002221133,002590335 0,064406495 0,0072260953,178271624 0,058024331 0,0002101943,353962944 0,03465148 7,4064E-05

Non dimentionalize force

From num

kR SWk0h kh kR

0,031936582 0,1796649 0,089832450,146918247 0,392941576 0,1964707880,539727872 0,807637652 0,4038188261,179448821 1,349976672 0,6749883362,066138765 2,129325823 1,0646629123,19969646 3,214820002 1,607410001

4,580165478 4,583308762 2,2916543816,207545821 6,208234562 3,1041172818,081951547 8,082120632 4,04106031610,20316863 10,20321542 5,10160771212,57129705 12,57131149 6,28565574615,18633678 15,18634169 7,59317084618,04845829 18,04846011 9,02423005321,15733477 21,15733549 10,5786677524,51312258 24,51312288 12,2565614428,11603445 28,11603458 14,05801729

31,965659 31,96565906 15,9828295336,06219487 36,0621949 18,0310974540,40564207 40,40564209 20,2028210444,99626972 44,99626973 22,49813487

0

0,5

1

1,5

2

2,5

3

0 1 2 3

Fz/ρ *g*R^

2*A

kR

Heave mo3on

AQWA Results Deep Water

0

0,5

1

1,5

2

2,5

3

0 1 2 3

Fz/ρ *g*R^

2*A

kR

Surge mo3on

AQWA Results Deep Water

116

APPENDIX CCorrelation Analysis

ADDED MASS Z5m mesh 4m mesh 3m mesh 2m mesh 1m meshLine C (kg) Line C (kg) Line C (kg.m²/°)Line C (kg) Line C (kg)

39780544 40506484 39973852 39124348 3946201239898052 40606796 39975504 38760964 3783365238773732 39271564 37824748 35454088 3308484036499996 36797808 34821572 32331836 3200049234152096 34362396 32466304 31239538 3370271232319626 32599484 31444490 32039864 3435602031317550 31809120 31737374 33234894 3376061631181046 31957100 32739898 33744940 3377316431698804 32724128 33657872 33639772 3348864832512130 33633548 34110740 35293044 3359850833256976 34293116 34181460 33350736 3392772033693992 34603744 33917028 33092626 3386930833821924 34662784 34969052 33072916 3393836833793632 34457112 33722592 33168942 3398384033484250 38933224 33567744 33336930 3408945636406736 34366980 33509092 33368082 3414230833812404 34054556 33642892 33433450 3417331233739604 33956568 33669796 33475406 3419089633369888 33923464 33824364 33498644 3420460833338796 34003100 33793200 33551886 34214160

0

5000000

10000000

15000000

20000000

25000000

30000000

35000000

40000000

45000000

0 0,1 0,2 0,3 0,4 0,5 0,6

kg

Frequency (Hz)

Added mass Z

mesh_5m

mesh_4m

mesh_3m

mesh_2m

mesh_1m

DIFFRACTION AND FROUD-KRYLOV, Z5m mesh 4m mesh 3m mesh 2m mesh 1m meshLine C (kg) Line C (kg) Line C (kg.m²/°)Line C (kg.m²/°)Line C (kg.m²/°)

13668511 13672781 13684132 13699339 1370107312514157 12432754 12124534 11566018 1025198810828579 10622377 9840505 8460756 5582584

8891134 8553425 7308802,5 5255192 1669125,56882700,5 6441513 4875688 2497663,75 245528,54968966,5 4463011 2754857,5 571113,313 802613,188

3250888 2727556,75 1106923,25 331023,594 1096173,881796249,88 1324459,88 291748,469 737481,25 635778,188

680030 423086,156 516882,719 1049990,88 501209,188188755,469 424621,531 846570,5 1022440,81 274597,313379034,938 689224,813 1099061 644732,125 56693,8984

552347,5 960833,625 1159874,38 569360,313 21608,1445749915,438 1166624,25 841022,25 392082,625 11253,9248896089,688 1220220,25 709888,313 253643,922 9091,13281957988,938 1010316 643084,5 126926,492 14956,4277679803,813 822124,5 502834,406 84093,8359 5640,12891774033,313 684463,625 243561,766 2293,71997 4880,46924769064,125 656508,813 264652,219 15145,2051 4954,77295734528,563 521935,344 143582,203 64523,668 13952,2783531758,688 349478 130615,945 11201,7344 15105,6719

-­‐2000000

0

2000000

4000000

6000000

8000000

10000000

12000000

14000000

16000000

0 0,1 0,2 0,3 0,4 0,5 0,6

Kg.m

Frequency (Hz)

DiffracGon and F-­‐K -­‐ Z

mesh_5m

mesh_4m

mesh_3m

mesh_2m

mesh_1m

120

APPENDIX DRAOs from extremesea state simulation

RAOs Including Morison Drag (Distance/Rotation vs Frequency)

Line A: RAOs Including Morison Drag - Global Z, -180° (AQW-1) [m/m]

*Wave Period (s),Line A (m/m)

62,83379,1,00010347366333 4,55748,7,36704701557755E-03

42,11767,1,00040721893311 4,40051,6,66394084692001E-03

31,67564,1,00126135349274 4,25398,5,17080537974834E-03

25,38264,1,00369024276733 4,11691,3,52736748754978E-03

21,17567,1,00869393348694 3,98839,2,56933737546206E-03

18,16497,1,01849353313446 3,86765,2,62443837709725E-03

15,90381,1,03649914264679 3,75401,2,79908976517618E-03

14,14327,1,06762540340424 3,64685,2,54078023135662E-03

12,73366,1,11701571941376 3,54564,1,88778853043914E-03

11,57957,1,1795117855072 3,4499,1,14847486838698E-03

10,61729,1,19206082820892 3,3592,7,52927095163614E-04

9,80263,0,989611804485321 3,27314,8,1240723375231E-04

9,10412,0,619489133358002 3,19138,8,29675700515509E-04

8,49852,0,329903841018677 3,1136,6,50055473670363E-04

7,96847,0,159623831510544 3,03953,3,65315994713455E-04

7,50065,6,75022602081299E-02 2,96889,1,68851853231899E-04

7,08473,2,11565028876066E-02 2,90147,2,21039052121341E-04

6,7125,7,74182844907045E-03 2,83704,2,41722169448622E-04

6,37743,1,47724049165845E-02 2,77541,1,68672471772879E-04

6,07423,1,71460881829262E-02 2,7164,6,00453968218062E-05

5,79854,1,55121302232146E-02 2,65985,4,63982687506359E-05

5,5468,1,16862282156944E-02 2,6056,7,79410765971988E-05

5,316,7,43530504405499E-03 2,55353,6,20594146312214E-05

5,10365,4,81728371232748E-03 2,50349,2,23984498006757E-05

4,9076,5,46582322567701E-03

4,72606,6,91423052921891E-03

RAOs Including Morison Drag (Distance/Rotation vs Frequency)

Line A: RAOs Including Morison Drag - Global Z, -180° (AQW) [m/m]

*Wave Period (s),Line A (m/m)

62,83379,1,00019764900208 4,78392,1,24631114304066E-02

42,30118,1,00075387954712 4,61343,1,19480341672897E-02

31,88267,1,00233256816864 4,45468,0,010419218800962

25,58199,1,00655233860016 4,30649,8,21195915341377E-03

21,36113,1,01515913009644 4,16784,5,96030987799168E-03

18,3355,1,03172326087952 4,03784,4,52487450093031E-03

16,06064,1,06142747402191 3,9157,4,1987937875092E-03

14,28796,1,10972380638123 3,80074,4,10738494247198E-03

12,86786,1,16905856132507 3,69233,3,58243379741907E-03

11,70439,1,15701341629028 3,58994,2,61909095570445E-03

10,73387,0,897165477275848 3,49307,1,59387593157589E-03

9,91203,0,523415505886078 3,4013,1,02175946813077E-03

9,20707,0,26936349272728 3,31422,1,04758585803211E-03

8,59572,0,129359871149063 3,23149,1,05075316969305E-03

8,06051,5,53166717290878E-02 3,15279,8,15768435131758E-04

7,58804,1,59732699394226E-02 3,07783,4,50023158919066E-04

7,16789,7,39068584516644E-03 3,00635,1,67727281223051E-04

6,79183,0,021122207865119 2,93812,2,32617312576622E-04

6,45326,2,80005894601345E-02 2,87291,2,73299374384806E-04

6,14684,2,90512666106224E-02 2,81054,1,98980007553473E-04

5,8682,0,026324575766921 2,75082,7,5298325100448E-05

5,61373,2,15852819383144E-02 2,69358,2,86675385723356E-05

5,38041,1,69537235051394E-02 2,63868,7,52580453990959E-05

5,16571,1,37018235400319E-02 2,58597,6,77036514389329E-05

4,96749,1,25185940414667E-02 2,53532,3,38801910402253E-05

RAOs Including Morison Drag (Distance/Rotation vs Frequency)

Line A: RAOs Including Morison Drag - Global Z, -180° (AQW-2) [m/m]

*Wave Period (s),Line A (m/m)

62,83379,1,00032830238342 4,70841,1,47427385672927E-02

42,06275,1,00129127502441 4,54041,1,28448093309999E-02

31,61255,1,00413691997528 4,38398,0,010177287273109

25,32158,1,01155304908752 4,23797,7,39157479256392E-03

21,11888,1,02701020240784 4,10137,5,28744328767061E-03

18,11266,1,05663883686066 3,9733,4,47187386453152E-03

15,8559,1,10572779178619 3,85299,4,21379320323467E-03

14,099,1,15171599388123 3,73975,3,59726580791175E-03

12,69261,1,04143738746643 3,63298,2,51418352127075E-03

11,54135,0,668295860290527 3,53213,1,36203854344785E-03

10,58156,0,350839465856552 3,43674,7,29364866856486E-04

9,76921,0,186175361275673 3,34635,8,21495137643069E-04

9,07266,0,110573329031467 3,26061,8,21796944364905E-04

8,46883,7,25153386592865E-02 3,17914,5,65444526728243E-04

7,94036,4,28283177316189E-02 3,10165,2,136087132385E-04

7,47397,1,87342818826437E-02 3,02785,7,17414877726696E-05

7,05933,2,44137272238731E-02 2,95747,2,0207317720633E-04

6,68828,3,66115570068359E-02 2,8903,1,85979981324635E-04

6,35429,4,11012955009937E-02 2,8261,9,38325756578706E-05

6,05206,3,91138009727001E-02 2,7647,3,56541277142242E-05

5,77728,3,39304395020008E-02 2,70591,6,88936415826902E-05

5,52637,2,77639534324408E-02 2,64956,6,55705662211403E-05

5,29635,2,19778344035149E-02 2,59552,3,55484444298781E-05

5,08471,1,79144144058228E-02 2,54363,1,71222618519096E-05

4,88933,1,59965343773365E-02 2,49378,2,71998760581482E-05

* Irregular Wave 1: JONSWAP (Hs) (Hs = 14 m; Gamma = 2,8; Peak Period = 15 s)

; Direction = 0.0°; Wave Spreading: None (Long-Crested Waves); with Cross Swell: None

*Period (s),Spectral Ordinate (m²/s)

Direction (°) >,0.0

Sub-Weight >,1 6,40808190204011,0,214919911654951

3,59710478190005,0,03946737837353 6,61485010221992,0,235073345470881

3,66134837592614,4,16073765642427E-02 6,83540758617687,0,257666764970598

3,72792871907066,4,39041936443468E-02 7,07117944132534,0,28303493672661

3,79697511866915,4,63723259744828E-02 7,32379718613975,0,311552951596633

3,86862745356768,4,90278911229274E-02 7,59513316739038,0,34363447339354

3,94303638532576,5,18889218251268E-02 7,88734905838913,0,379723114736129

4,0203634976372,5,49754649832059E-02 8,20294888271274,0,420273561317997

4,10078421160161,5,83099407909247E-02 8,5448578850173,0,465716341856857

4,18448796371639,6,19175292998784E-02 8,91650970304524,0,51638998291345

4,2716803258037,6,58264050976155E-02 9,32196100507282,0,572426581191698

4,36258331509264,7,00682604703056E-02 9,76604132368149,0,633556397751627

4,45743932689225,7,46788113764521E-02 10,2545483103972,0,698793894416265

4,55651233892732,7,96984049386288E-02 10,794502011823,0,765997940264129

4,66008913202932,8,51725927760353E-02 11,3944763412605,0,832570105502662

4,76848438789993,9,11530406305281E-02 12,0650707537855,0,910191956225715

4,88204235102412,9,76983692519516E-02 12,8195350239492,1,09432879433174

5,00114133210449,0,104875360251 13,6746501866651,1,63439190706597

5,12619603378755,0,1127600485325 14,6519980944681,2,32008526589877

5,25766517376852,0,12143925918423 15,7798057471953,1,61486796239337

5,39605526664033,0,131012309512991 17,0957106259986,0,64471424324661

5,54192818042998,0,141592996286355 18,6510570677487,0,284208385688876

5,69590644740548,0,153311881050235 20,5177330266042,9,25621408087719E-02

5,85868557819064,0,166318770319491 22,7996139444942,1,27747921309393E-02

6,0310423109007,0,180785779542426 25,6525655743411,3,26581009757282E-04

6,21384829019708,0,196910649245794

126

APPENDIX ERAOs for Sea State 2

period 4,s1,cd:3,84,s1,cd:5,314,s1,cd:7,874,s2,cd:3,84,s2,cd:5,31 4,s2,cd:7,87 6,s1,cd:3,8

2,50349 1,34E-05 1,31E-05 1,27E-05 1,33E-05 1,46354E-05 1,78019E-05 1,02E-05

2,55353 3,48E-05 4,22E-05 3,93E-05 4,31E-05 4,73149E-05 5,43331E-05 3,37E-03

2,6056 5,12E-05 5,99E-05 5,69E-05 6,8E-05 6,49642E-05 7,1468E-05 1,45E-02

2,65985 4,98E-05 4,87E-05 4,94E-05 4,86E-05 4,80818E-05 4,72039E-05 2,29E-02

2,7164 5,82E-05 5,2E-05 5,37E-05 5,18E-05 5,14735E-05 5,37715E-05 8,64E-03

2,77541 0,000199 0,000124 0,000118 0,000126 0,000134982 0,000151049 7,98E-02

2,83704 0,000163 0,000188 0,000179 0,00019 0,000227116 0,000222392 0,517403185

2,90147 0,000189 0,0002 0,000197 0,000274 0,000258205 0,000213711 2,21242547

2,96889 0,000298 0,000191 0,000199 0,00019 0,000182999 0,000173903 1,641716838

3,3953 0,000296 0,000551 0,000299 0,000512 0,000315043 0,000597323 1,28105092

3,1136 0,000448 0,000679 0,000484 0,000686 0,000543808 0,000778769 1,149979234

3,19138 0,000663 0,00076 0,000655 0,000763 0,000722031 0,000797071 1,089719892

3,27314 0,000736 0,000864 0,000756 0,000835 0,000775962 0,000769346 1,05779767

3,3592 0,000851 0,000328 0,000831 0,000347 0,000788561 0,000337306 1,039258361

3,4499 0,001454 0,001399 0,001477 0,001426 0,001583726 0,001973989 1,02776134

3,54564 0,001385 0,00152 0,001982 0,001536 0,002196825 0,001753597 1,020270705

3,64685 0,00183 0,00258 0,001475 0,002818 0,001616926 0,002379043 1,015192866

3,75401 0,002274 0,00245 0,002661 0,002468 0,002643276 0,002692469 1,011638165

3,86765 0,002657 0,002642 0,002918 0,002643 0,002747367 0,002647477 1,009081721

3,98839 0,003179 0,002829 0,002394 0,002815 0,002555578 0,002657026 1,007200122

4,11691 0,003444 0,003347 0,003399 0,003347 0,00335705 0,003418248 1,005784512

4,25398 0,003966 0,004235 0,004139 0,004272 0,004461339 0,004799582 1,0046978

4,40051 0,004555 0,005205 0,004948 0,005273 0,005604919 0,00613991 1,003843904

4,55748 0,005172 0,005931 0,00559 0,006152 0,006344002 0,006881233 1,003154159

4,72606 0,005752 0,0062 0,00591 0,006237 0,006414438 0,006704642 1,002577543

4,9076 0,006226 0,006349 0,00593 0,006931 0,005885034 0,005726855 1,002079964

5,10365 0,006611 0,005897 0,005996 0,00583 0,005520707 0,005150352 1,001631975

5,316 0,006888 0,006412 0,006499 0,006403 0,006426841 0,006741924 1,00168097

5,5468 0,006956 0,007645 0,00742 0,007777 0,00848935 0,009825514 1,001454592

5,79854 0,006802 0,008996 0,008341 0,009265 0,015753364 0,012683569 1,001267314

6,7423 0,006173 0,007637 0,008367 0,00792 0,011270546 0,011157554 1,001111388

6,37743 0,004609 0,006147 0,006519 0,00616 0,009216131 0,006516641 1,000979424

6,7125 0,005462 0,009371 0,005875 0,009705 0,006311365 0,01365947 1,000870347

7,8473 0,019704 0,062242 0,021575 0,062692 0,022417014 0,068844199 1,000776052

7,50065 0,056779 0,021521 0,061681 0,02164 0,065150015 0,023542877 1,000696659

7,96847 0,041126 0,049773 0,048737 0,150441 0,15417619 0,159920752 1,000628948

8,49852 0,132598 0,132741 0,169623 0,132771 0,166706192 0,133228526 1,00056839

9,10412 0,172344 0,169026 1,343168 0,168625 1,175731778 0,16389398 1,000517368

9,80263 1,591541 1,30648 1,549113 1,281004 1,387278318 1,523825884 1,000473142

10,61729 1,76611 1,514982 1,329979 1,490721 1,270970941 1,259085894 1,000435591

11,57957 1,391772 1,318722 1,327388 1,310278 1,329639167 1,213704586 1,000399828

12,73366 1,192083 1,169651 1,173211 1,166836 1,153136134 1,131232381 1,000369072

14,14327 1,09 1,09 1,09 1,09 1,09 1,07 1,000343323

15,90381 1,04 1,01 1,01 1,01 1,04 1,04 1,00E+00

18,16497 1,02 1,01 1,01 1,01 1,02 1,02 1,00E+00

21,17567 1,01 1,001 1,001 1,001 1,01 1,01 1,00E+00

25,38264 1,001 1,001 1,001 1,001 1,001 1,001 1,00E+00

31,67564 1 1 1 1 1 1 1,00E+00

42,11767 1 1 1 1 1 1 1,00E+00

62,83379 1 1 1 1 1 1 1,00E+00

6,s1,cd:5,316,s1,cd:7,876,s2,cd:3,86,s2,cd:5,316,s2,cd:7,878,s1,cd:3,8 8,s1,cd:5,31 8,s1,cd:7,87

1,24E-05 1,63E-05 1,63E-05 1,81E-05 2,28E-05 2,73929E-05 2,97753E-05 3,83446E-05

3,45E-03 3,59E-03 3,62E-03 3,69E-03 4,69E-05 2,54861E-05 2,3189E-05 2,52507E-05

1,44E-02 1,41E-02 1,43E-02 1,42E-02 5,64E-05 1,78949E-05 1,43623E-05 2,88675E-06

2,34E-02 2,45E-02 2,46E-02 2,50E-02 2,55E-05 3,88354E-05 4,61745E-05 4,3324E-05

7,17E-03 4,87E-03 3,04E-03 2,28E-03 5,31E-05 6,16946E-05 6,83741E-05 6,65557E-05

8,60E-02 9,76E-02 0,10061 0,107184 0,000153 5,76852E-05 5,566E-05 5,70226E-05

0,521855 0,526954 0,523564 0,525137 0,00022 2,18727E-05 4,89556E-06 2,51827E-05

1,950361 1,63461 1,530674 1,426779 0,000242 3,90536E-05 5,99941E-05 7,39828E-05

1,58663 1,494249 1,456612 1,411904 0,000169 7,14131E-05 8,93388E-05 0,000156108

1,269081 1,246404 1,236113 1,22303 0,000382 3,72267E-05 3,64453E-05 4,37006E-05

1,145801 1,13753 1,133577 1,128483 0,000692 8,4027E-05 0,000113978 0,000135554

1,087866 1,084102 1,082235 1,079831 0,000928 0,000271776 0,000323325 0,000370719

1,05685 1,054887 1,053883 1,052597 0,001424 0,000476433 0,00052448 0,000579123

1,038725 1,037602 1,037012 1,03626 0,001569 0,000642626 0,000655498 0,000671224

1,027439 1,026751 1,02638 1,025911 0,001503 0,000823356 0,000817504 0,000779598

1,020066 1,01962 1,019375 1,019068 0,00235 0,001223881 0,001283151 0,001280134

1,015056 1,014756 1,014587 1,014376 0,003249 0,001944692 0,002875337 0,00216818

1,011544 1,011334 1,011214 1,011065 0,003896 0,002880062 0,003424909 0,003154835

1,009015 1,008863 1,008775 1,008667 0,004269 0,003858746 0,003968626 0,004242442

1,007151 1,007039 1,006973 1,006892 0,004761 0,004871276 0,004890835 0,004770984

1,005748 1,005663 1,005612 1,005551 0,00591 0,006502649 0,006377293 0,005838422

1,00467 1,004605 1,004566 1,004518 0,007647 0,007476781 0,007541367 0,007465813

1,003822 1,003771 1,00374 1,003702 0,009496 0,009859598 0,009309845 0,009475647

1,003137 1,003097 1,003072 1,003041 0,011076 0,011011598 0,011371709 0,011691349

1,002564 1,002532 1,002511 1,002487 0,012173 0,013422899 0,013711541 0,013933097

1,002069 1,002043 1,002026 1,002006 0,013157 0,016503977 0,01664241 0,016418476

1,001623 1,001602 1,001588 1,001571 0,014882 0,024821229 0,024027705 0,019713471

1,001674 1,001655 1,001643 1,001629 0,017609 0,025209574 0,024989262 0,024110459

1,001448 1,001433 1,001423 1,001411 0,028334 0,029741004 0,029511323 0,028848974

1,001262 1,001249 1,00124 1,00123 0,023902 0,033298612 0,033262279 0,033146571

1,001107 1,001096 1,001088 1,001079 0,025292 0,035715718 0,035994146 0,036495917

1,000976 1,000966 1,000959 1,000952 0,023672 0,036605597 0,036698572 0,037567604

1,000867 1,000858 1,000853 1,000846 0,017308 0,03390928 0,033559229 0,034233961

1,000773 1,000766 1,000761 1,000755 0,005754 0,027951077 0,026813803 0,026282083

1,000694 1,000688 1,000683 1,000678 0,014425 0,028679606 0,027599378 0,024913343

1,000627 1,000621 1,000617 1,000612 0,047957 0,045401432 0,044813883 0,041933026

1,000567 1,000561 1,000558 1,000553 0,1153 0,065859496 0,059552729 0,057772707

1,000515 1,000511 1,000507 1,000504 0,251996 0,057548326 0,057356779 0,06195724

1,000471 1,000467 1,000464 1,000461 0,526697 0,036807816 0,053718984 0,082075112

1,000434 1,00043 1,000427 1,000424 0,993856 0,158681616 0,189454436 0,225631043

1,000399 1,000395 1,000392 1,00039 1,299124 0,682149768 0,716880202 0,706254244

1,000368 1,000365 1,000362 1,00036 1,239235 2,437095404 2,648915768 1,663106799

1,000342 1,000339 1,000337 1,000334 1,136078 1,556979656 1,496574163 1,422807455

1,00E+00 1,000315 1,000313 1,000311 1,071038 1,205275536 1,193527937 1,17989254

1,00E+00 1,00E+00 1,000293 1,000291 1,0352 1,08 1,08 1,08

1,00E+00 1,00E+00 1,000273 1,00E+00 1,016383 1,01 1,01 1,01

1,00E+00 1,00E+00 1,000256 1,00E+00 1,696123 1,001 1,001 1,001

1,00E+00 1,00E+00 1,00024 1,00E+00 1,002457 1 1 1

1,00E+00 1,00E+00 1,000226 1,00E+00 1 1 1 1

1,00E+00 1,00E+00 1,000214 1,00E+00 1 1 1 1

8,s2,cd:3,88,s2,cd:5,318,s2,cd:7,87

2,82E-05 2,78E-05 2,47E-05

2,16E-05 2,11E-05 2,59E-05

9,95E-06 4,95E-06 4,99E-06

3,93E-05 3,84E-05 3,58E-05

5,95E-05 5,93E-05 5,81E-05

4,83E-05 4,67E-05 4,59E-05

3,58E-06 1,49E-05 2,92E-05

5,77E-05 7,23E-05 9,27E-05

8,41E-05 9,86E-05 0,000124

2,84E-05 3,4E-05 5,78E-05

0,000117 0,00013 0,000127

0,000323 0,000355 0,000385

0,000527 0,000564 0,000613

0,000668 0,00068 0,00071

0,000837 0,000816 0,000776

0,001279 0,001284 0,001242

0,002581 0,002117 0,002138

0,003654 0,003825 0,003139

0,00394 0,003966 0,003968

0,004877 0,004809 0,004669

0,00618 0,0059 0,005658

0,007483 0,007433 0,007276

0,009192 0,009284 0,009332

0,011213 0,011402 0,01158

0,0136 0,013722 0,013811

0,01658 0,016476 0,016193

0,020442 0,028541 0,019371

0,025088 0,024616 0,023732

0,029582 0,029192 0,028508

0,033192 0,033034 0,032749

0,035659 0,035736 0,035801

0,035987 0,036079 0,036457

0,032233 0,031898 0,031874

0,024567 0,023064 0,021327

0,025435 0,023122 0,01937

0,044014 0,04273 0,041264

0,061037 0,064288 0,063331

0,061773 0,068287 0,082908

0,071438 0,094488 0,130146

0,206155 0,238581 0,281076

0,705615 0,702812 0,680965

1,80485 1,547938 1,286314

1,457308 1,392598 1,300368

1,187185 1,174268 1,153393

1,08 1,08 1,08

1,01 1,01 1,01

1,001 1,001 1,001

1 1 1

1 1 1

1 1 1