IMPACT ASSESSMENT ON 2nd ORDER ROLL MOTIONmonografias.poli.ufrj.br/monografias/monopoli10031833.pdf · 3. Carregamento Estrutural do Turret. I. Paulo de Tarso Themistocles Esperança

IMPACT ASSESSMENT ON 2nd ORDER ROLL MOTION

Bernardo Ferreira Fortini Pimentel

Projeto de Graduação apresentado ao Curso de Engenharia Naval e Oceânica da Escola Politécnica, Universidade Federal do Rio de Janeiro, como parte dos requisitos necessários à obtenção do título de Engenheiro.

Advisor:

Claude Camps

Paulo de Tarso Themistocles Esperança

Rio de Janeiro

Junho de 2020

ii

IMPACT ASSESSMENT ON 2nd ORDER ROLL MOTION


PROJETO DE GRADUAÇÃO SUBMETIDO AO CORPO DOCENTE DO CURSO DE

ENGENHARIA NAVAL E OCEÂNICA DA ESCOLA POLITÉCNICA DA UNIVERSIDADE

FEDERAL DO RIO DE JANEIRO COMO PARTE DOS REQUISITOS NECESSÁRIOS

PARA A OBTENÇÃO DO GRAU DE ENGENHERIO NAVAL E OCEÂNICO.

Examinada por:

Prof. D.Sc. Paulo de Tarso Themistocles Esperança

Prof. D.Sc. Claudio Alexis Rodríguez Castillo

D.Sc. Thalles Carvalho Giangiarulo de Aguiar

RIO DE JANEIRO, RJ – BRAZIL

Junho 2020

ptarso

Imagem Posicionada

iii

Pimentel, Bernardo Ferreira Fortini

Impact Assessment on 2nd Order Roll Motion/

Bernardo Ferreira Fortini Pimentel. – Rio de Janeiro: UFRJ/

Escola Politécnica, 2020.

XI, 45 p.: il.; 29,7 cm.

Orientador: Paulo de Tarso Themistocles Esperança,

Claude Camps

Projeto de Graduação – UFRJ/ Escola Politécnica/

Curso de Engenharia Naval e Oceânica, 2020.

Referências Bibliográficas: p. 46.

1.Roll em baixa-frequência de FPSO/FLNG.

2.Comparação de Metodologias. 3. Carregamento

Estrutural do Turret. I. Paulo de Tarso Themistocles

Esperança. II. Universidade Federal do Rio de Janeiro,

Escola Politécnica, Curso de Engenharia Naval e

Oceânica. III. Impact Assessment on 2nd Order Roll Motion.

iv

Resumo do Projeto de Graduação apresentado à Escola Politécnica/ UFRJ como parte

dos requisitos necessários para a obtenção do grau de Engenheiro Naval e Oceânico.

ANÁLISE DO MOVIMENTO DE ROLL DE SEGUNDA ORDEM


Junho/2020

Orientador: Paulo de Tarso Themistocles Esperança

Claude Camps

Programa: Engenharia Naval e Oceânica

Para aplicações de engenharia offshore, embarcações como FPSOs e FLNGs podem

ter seu período de roll natural além de 20 segundos. Análises clássicas de movimento

envolvem apenas o movimento de roll de 1ª ordem o que pode não ser suficiente para

prever corretamente o movimento de roll, uma vez que a contribuição da 2ª ordem pode

ser significativa. Este trabalho desenvolve diferentes metodologias de avaliação do

movimento em roll de uma plataforma de produção de petróleo offshore a fim de

compreender os parâmetros mecânicos que levam a este movimento. O projeto finaliza

com uma metodologia a ser adotada em futuros projetos de por engenheiros de

ancoragem e o impacto do movimento de roll em baica frequência no carregamento

estrutural do turret.

Palavras-chave: Roll de segunda ordem, Movimento de Plataformas, Hidrodinâmica

Offshore.

v

Abstract of Undergraduate Project presented to POLI/UFRJ as partial fulfillment of the

requirements for the degree of Naval Architect and Marine Engineer.

IMPACT ASSESSMENT ON 2ND ORDER ROLL MOTION


June/2020

Advisor: Paulo de Tarso Themistocles Esperança

Claude Camps

Department: Naval Architecture and Marine and Ocean Engineering

For offshore engineering applications, vessels such as FPSOs and FLNGs can have

their natural roll period beyond 20 seconds. Classical motion analyzes involving only 1st

order roll motion may not be enough to adequately predict roll motion once the

contribution of 2nd order may be significant. This paper develops different methodologies

to assess roll motion of an offshore oil platform to understand the mechanical parameters

that lead to this motion. The project concludes with a procedure to be adopted in future

projects by mooring designers and the impact of the low-frequency roll motion on turret

structural loading.

Keywords: 2nd Order Roll Motion, Platform’s motion, Offshore Hydrodynamics

vi

Summary

Table of Figures ........................................................................................................................ viii

Tables ........................................................................................................................................... ix

Glossary ....................................................................................................................................... x

Acronyms ............................................................................................................................. x

Mathematical Symbols ....................................................................................................... x

1. Introduction .......................................................................................................................... 1

2. Motivation ............................................................................................................................. 1

3. Objectives ............................................................................................................................ 2

4. Theoretical Background ..................................................................................................... 3

4.1 Motion Equation .............................................................................................................. 4

4.2 2nd Order Wave Excitation Spectrum ........................................................................... 6

4.3 Roll Motion (1st and 2nd order)....................................................................................... 7

4.4 Quadratic Transfer Function ......................................................................................... 8

4.5 2nd Order Roll Moment Spectrum ............................................................................... 10

5. Approaches for Assessing Low-frequency Roll Motion .............................................. 11

5.1 Radiation/Diffraction Calculations: ......................................................................... 11

5.2 Roll Assessment ....................................................................................................... 12

5.2.1 Spectral Calculation ......................................................................................... 12

5.2.2 Time Domain Reconstruction ......................................................................... 16

5.2.3 Time Domain Analysis: .................................................................................... 16

5.2.3.1 OrcaFlex 10.2d.............................................................................................. 17

5.2.3.2 OrcaFlex 10.3d.............................................................................................. 18

5.2.4 Study Case ........................................................................................................ 18

5.2.5 Comparison ....................................................................................................... 24

5.2.6 Benchmark Comparison and Partial Conclusions ....................................... 27

6. Approach for Future Projects: Turret Load Design ..................................................... 29

6.1 Turret Loads - Formulation .......................................................................................... 30

6.2 Study Case .................................................................................................................... 32

6.2.1 Environmental Condition ................................................................................. 34

6.2.2 Orcaflex 10.3d – 6 DoF............................................................................................ 35

6.2.3 Hybrid Method ........................................................................................................... 38

7. Conclusion and Perspectives ......................................................................................... 42

vii

8. References ........................................................................................................................ 44

Appendix .................................................................................................................................... 45

Appendix I: Script Python ................................................................................................ 45

Appendix II: Force Coefficients ...................................................................................... 45

Appendix III: HydroStar Input Files ................................................................................ 47

Appendix IV: Meshing (hsmsh/hslec) ............................................................................ 48

Appendix V: Radiation/Diffraction Module (hsrdf) ....................................................... 48

Appendix VI: Mechanical Module (hsmcn) ................................................................... 49

Appendix VII: Quadratic Transfer Function Module (hsamg/hsqtf) ........................... 49

Appendix VIII: Wave Response Module (hspec) ......................................................... 50

Appendix IX: Roll Response Comparison .................................................................... 51

viii

Table of Figures

Figure 1: JONSWAP Wave Spectrum ..................................................................................... 6

Figure 2: 2nd Order Wave Spectrum ...................................................................................... 6

Figure 3: 2nd Order Wave Spectrum Comparison ................................................................. 7

Figure 4: Response motion of a moored structured Image extracted from [1] ................ 8

Figure 5: Benchmark Flowchart ............................................................................................. 11

Figure 6: Hydrostar convention for wave incidence direction Image extracted from [5] 12

Figure 7: Spectral Calculation Flowchart .............................................................................. 13

Figure 8: Verification of 2nd order roll spectrum moment .................................................. 15

Figure 9: Signal treatment using SigView ............................................................................. 17

Figure 10: OrcaFlex 10.2d Calculation Definition ................................................................ 18

Figure 11: HydroStar Hull Mesh ............................................................................................. 19

Figure 12: Roll RAO - GMt = 11.5 m ..................................................................................... 20

Figure 13: Roll RAO - GMt = 3 m ........................................................................................... 21

Figure 14: Roll FQTF - GMt = 11.5 m – Δω = 0 rad/s......................................................... 21

Figure 15: Roll FQTF - GMt = 11.5 m – Δω = 0.2 rad/s ..................................................... 22

Figure 16 : Roll FQTF - GMt = 3 m - 𝛥𝜔 = 0 rad/s .............................................................. 23

Figure 17: Roll FQTF - GMt = 3 m - Δω = 0.2 rad/s ............................................................ 23

Figure 18: Roll 2nd Order Moment Spectrum ...................................................................... 24

Figure 19: 2nd Order Roll Standard Deviation for 30 deg of Wave Incidence ................ 25



Figure 22: Roll Standard Deviation according to Wave Incidence ................................... 26

Figure 23: Gumbel Distribution of Roll Maxima ................................................................... 28

Figure 24: Turret Sketch ......................................................................................................... 30

Figure 25 : Hydrostar Hull Mesh ............................................................................................ 32

Figure 26: RAO and QTF of the FPSO ................................................................................. 33

Figure 27: OrcaFlex Modelling Scheme ................................................................................ 35

Figure 28: Definition of the analysis set on OrcaFlex 10.3d .............................................. 36

Figure 29: Wave excitation calibration .................................................................................. 36

Figure 30: Comparison between the wave incidence time series obtained with Ariane

and OrcaFlex ............................................................................................................................. 37

Figure 31: Comparison between the offset time series obtained with Ariane and OrcaFlex

..................................................................................................................................................... 37

Figure 32: Flowchart of the steps used to correct Ariane's roll motion ............................ 39

Figure 33: Roll correction - WF + LF ..................................................................................... 40

Figure 34: Horizontal loads acting on the chain table and the radial wheel .................... 40

Figure 35: Current Force Coefficients ................................................................................... 46

Figure 36: Wind Force Coefficients ....................................................................................... 47

Figure 37: Hydrostar Used Modules ...................................................................................... 47

ix

Tables

Table 1: Terms of the Motion Equation ................................................................................... 5

Table 2: FQTF Components ................................................................................................... 10

Table 3: Vessel Properties ...................................................................................................... 19

Table 4: Wave Properties ........................................................................................................ 20

Table 5 : Results Comparison between Methodologies ..................................................... 27

Table 6: Rayleigh Extrapolation of Maxima Roll Angle ....................................................... 27

Table 7: Vessel properties ...................................................................................................... 32

Table 8: Turret Properties ....................................................................................................... 33

Table 9: Sail and Current Projected Areas ........................................................................... 34

Table 10: Environmental condition ........................................................................................ 34

Table 11: Current speed profile .............................................................................................. 34

Table 12: Damping Coefficients ............................................................................................. 36

x

Glossary

Acronyms

BV Bureau Veritas CoG Center of Gravity DNV Det Norske Veritas DoF Degrees of Freedom FD Frequency Domain FLNG Floating Liquefied Natural Gas Unity FPSO Floating Production Storage and Offloading Unity FQTF Full Quadratic Transfer Function JIP Joint Industry Project JONSWAP Joint North Sea Wave Observation Project LF Low frequency MPM Most Probable Maximum MPM Most Probable Maxima QTF Newman Simplification of the Quadratic Transfer Function RAO Response Amplitude Operator TD Time Domain WF Wave-frequency

Mathematical Symbols

𝐹−⃗⃗⃗⃗ (2)

Low-frequency Second Order Load [N or N.m]

𝐹+⃗⃗ ⃗⃗

(2)

High-frequency Second Order Load [N or N.m]

𝑓−⃗⃗ ⃗(2)

Low-frequency terms from Difference Frequency FQTF

𝑓+⃗⃗⃗⃗ (2)

High-frequency terms from Sum Frequency FQTF

𝑆𝑀(2) 2nd order Roll Moment Spectrum

𝑓 (2)(𝜔𝑖, 𝜔𝑗) Full Quadratic Transfer Function

𝑓 (2)(𝜔) Newman Simplification of the FQTF

𝑓 𝑑 (𝜔) Normalized Drift Force

𝑛0⃗⃗⃗⃗ Versor perpendicular to the hull’s surface

𝐴𝑖,𝑗 i(j)-th Wave Amplitude [m]

𝐵𝑐 Critical Damping [Nms/rad]

𝐵𝑒𝑞 Linear Equivalent Damping [Nms/rad]

𝐵𝑙𝑖𝑛 Roll Linear Damping Coefficient [Nms/rad]

𝐵𝑞𝑢𝑎𝑑 Roll Quadratic Damping Coefficient [Nm(s/rad)2]

𝐹𝐻 Horizontal Force on the Chain Table [kN] 𝐹𝑀𝑒𝑎𝑛 Mean Drift Force [N] 𝐹𝑟𝑎𝑑 Radial Force on the Radial Wheels [kN]

𝐺𝑀𝑇̅̅ ̅̅ ̅̅ Transverse Metacentric Height [m]

𝐻𝑠 Significant Wave Height [m]

𝐼𝑥𝑥 Roll Inertia [t.m2]

𝐾ℎ Hull Hydrostatic Stiffness [N/m]

𝐾𝐺̅̅ ̅̅ Elevation of vessel’s CoG relative to the keel [m]

𝑃𝑖𝑗 In-phase Quadratic Transfer Function

𝑄𝑖𝑗 Out-phase Quadratic Transfer Function

𝑅𝑥𝑥 Radius of Gyration [m]

𝑅𝑦𝑦 Radius of Gyration [m]

xi

𝑆𝐽 JONSWAP Wave Energy Spectrum [m2 s]

𝑆𝑃𝑀 Pierson-Moskovitz Wave Energy Spectrum [m2 s]

𝑆𝑊 2nd order Wave Energy Spectrum [m4 s]

𝑇𝑃 Peak Wave Period [s]

𝑇𝑛 Roll Natural Period [s]

𝑇𝑧 Return Period [s]

𝑘𝑖 i-th Wave Number [m-1]

𝑩𝒍𝒊𝒏 Linear Damping Matrix [Nms/rad]

𝑩𝒒𝒖𝒂𝒅 Quadratic Damping Matrix [Nm(s/rad)2]

𝑭𝒆𝒏𝒗 Environmental Load Torsor [N or N.m]

𝑴𝒂𝒅𝒅 Added Mass Matrix [kg or kg.m2]

𝛼𝑖 i-th Phase Angle [deg]

�̇� Roll Speed [deg/s]

𝜎�̇� Standard Deviation of the Second Order Roll Velocity

[deg/s]

𝜎𝑟𝑜𝑙𝑙(2) Second Order Roll Standard Deviation [deg]

𝜔𝑑 Natural Frequency with Damping [rad/s]

𝜔𝑖,𝑗 i(j)th Wave Circular Frequency [rad/s]

𝜔𝑛 Roll Natural Circular Frequency [rad/s]

𝜔𝑝 Peak Frequency [rad/s]

𝜙𝐷𝑖 i-th Diffracted Velocity Potential

𝜙𝐼𝑖 i-th Incident Velocity Potential

𝜙𝑅𝑖 i-th Radiation Velocity Potential

^ Cross-Product

∆ Displacement [t]

∇ Volume [m3]

CoG Center of Gravity [m] t Time [s] 𝐵 Damping [Ns/m or Nms/rad]

𝐿𝐶𝐺 Longitudinal Center of Gravity [m]

𝑆 1st Order Energy Spectrum [m2 s]

𝑆 Hull Wetted Surface [m2]

𝑆(𝜔) Pierson-Moskowitz Spectrum [m2s]

𝑇𝑉𝐺 Transverse Center of Gravity [m]

𝑉𝐶𝐺 Vertical Center of Gravity [m]

𝑔 Gravity [m/s2]

𝑡 Time [s]

𝑩 Damping Matrix [N.s/m]

𝑲 Stiffness Matrix [N/m]

𝑴 Mass Matrix [kg or kg.m2]

𝚫 Laplace Operator

𝛾 Peak Enhancement Factor [-]

𝜂 Free Surface Elevation [m]

𝜃 Roll Angle [deg]

𝜌 Water Density [kg/m3]

𝜏 Damping Rate [ - ]

𝜙 Velocity Potential

1

1. Introduction

The content of this project is the outcome of a 6-month internship that took place

at SBM Offshore in Monaco and it is confidential.

This company is a contractor for offshore oil operators such as Petrobras or Shell.

Most of the SBM’s turnover is due to the leasing of FPSOs around the world, specially

alongside the West African’s and Brazilian’s coast. SBM’s strategy is to grow in size and

value within the offshore and gas industry, but it is also taking measures towards the

Offshore renewable energy sector. Currently, it is estimated that 1% of the world`s oil

production or 10% of the world’s offshore oil production comes from an SBM system.

FPSO is an acronym that stands for Floating Production Storage and Offloading

vessel. This vessel is, normally, a ship-shaped unity that allows not only the production

but also the storage of oil in its tanks. It has become an interesting investment since late

discoveries of oil reservoirs are found in deep waters. These discoveries have pushed

the borders of oil exploration further from the coastline where export logistics are more

complicated. Those factors are demanding a high capacity of oil storage in within the

unity.

SBM Offshore is one of the world leaders on Floating Production and mooring

systems employing over than 7.000 people. The company is mainly present in the

Netherlands, Brazil, West Africa, Malaysia, USA, and Monaco. It provides services from

design conception to installation and operation of oil platforms.

This project took place at Monaco headquarters. The office gathers different

expertise on project execution, such as: engineering, business development and

offshore sustainable energy with focus on offshore wind farms and wave energy.

In this sense, the technological development on the design of SBM floating units

is crucial to ensure the liability of its floating systems and comply with safety regulations.

The internship took place in the hydrodynamics department on the assessment

of 2nd order roll motion due to low-frequency forces. This load appears on moored ship-

shaped vessels. This phenomenon may lead to slow-drift which due to low damping may

yield to higher roll amplitude.

The assessment of 2nd order roll motion is now mandatory once the platform’s

natural period lies over 25s, several SBM’s projects now lies in this range. Therefore, it

provides the motivation for a full comprehension of the phenomenon and include its

impacts in the design phase of mooring projects.

2. Motivation

The activities developed during the internship at SBM Offshore sought to develop

an assessment on 2nd order roll motion focused on the vessel’s mechanical properties

and the environmental parameters that this phenomenon is sensitive to. Primarily, roll

slow-drift is related to the vessel’s natural period and the energy transported by 2nd order

2

wave loads. Considering a small damped uncoupled system, roll natural period may be

estimated by:

𝑇𝑛 = 2𝜋 √(𝐼𝑥𝑥 + 𝐼𝑥𝑥𝑎𝑑𝑑

)

∆ 𝑔 𝐺𝑀𝑇̅̅ ̅̅ ̅̅

(1)

1st order wave loads normally offer a good approach for computing roll motion of

units which natural period lie in the same period range as sea waves, commonly from 5s

to 20s. Despite that, FPSOs and FLNGs may present a natural period above 20s being

susceptible to motion from 2nd order loads. Classification’s Society now imposes that 2nd

roll motion should be assessed if the vessel’s roll natural period is above 25 seconds.

Several reasons can result to longer 𝑇𝑛 . FPSOs in harsh environmental

conditions (e.g. Norway, Canada, or North Sea) demand higher freeboard to avoid green

water damage on topside. This fact increases significantly 𝐾𝐺̅̅ ̅̅ ̅, decreasing 𝐺𝑀𝑇̅̅ ̅̅ ̅̅ and

finally increasing 𝑇𝑛. Similarly, FLNGs carrying low density hydrocarbons in its tanks and

having high topside facilities may also present roll motion excited by 2nd order low-

frequency loads.

The phenomenon of 2nd order roll motion has been faced by SBM engineers that

had to overcome it under the development of:

• Turret mooring system for vessels designed by third parties.

• A few vessels from their fleet.

In summary, on one hand high natural period will reduce 1st order roll response

but, on the other hand, it leads to increased roll motions due to 2nd order loads.

Over the past decades, research have been developed. A JIP research assessed

important features to understand and quantify 2nd order roll motion. SBM Offshore is

interested to set up a design method to assess potential impacts of 2nd order roll motion

on FPSOs.

Therefore, the objective of this project is to benchmark different calculation

methodologies, assess vessel’s mechanical properties where 2nd order low-frequency

motion is governing total roll response, and compute its impact on the vessel and the

design of mooring systems.

3. Objectives

The aim of this project is assessing 2nd order roll motion and the mechanical

properties that lead to large roll response. It will, therefore, help to grasp a further

understanding of the phenomenon and establish margins where slow drift is governing

total roll response.

This project was divided into two stages. Firstly, it aimed to benchmark different

calculation models to assess 2nd order roll motion: Spectral, TD reconstruction and TD

analysis. Secondly, it focused to develop a methodology to include low-frequency roll

motion on future mooring designs. It aimed primarily on turret equipped vessels.

3

4. Theoretical Background

This section is dedicated to crucial theoretical aspects of the work which will be

used in the following chapters. It also provides the basis for understanding the

phenomenon of 2nd order roll motion and how it has been treated.

According to experimental and theoretical observations, wave drift forces are

commonly treated as 2nd order loads.

The perturbation theory splits the contribution of each parameter’s order into the

final motion of a body piercing on waves. Perturbation theory states that fluid properties

can be further developed in a power series considering a small variation (𝜖 ≪ 1) from its

static value (𝐹(0)). The variation’s power number also represents the order of the

oscillation. The power series of an arbitrary property 𝐹 (i.e. wave height, velocity

potential, motion, pressures, etc.) can be expressed as:

𝐹 = 𝐹(0) + 𝜖𝐹(1) + 𝜖2𝐹(2) + 𝜖3𝐹(3) + ⋯ (2)

The index (1) means that this parameter is linearly related to wave amplitude (𝐴),

while 2nd order parameters are related to the square of the incident wave height indicated

with the index (2). One should bear in mind that the product between two first order

parameters is a second order component.

In the present work, 1st and 2nd order loads are assessed. 1st order loads are a

pre-condition for 2nd order motion analysis. According to [1], 1st order wave loads are

obtained considering a body floating in its static position (zero-order) - meaning that the

waves are approaching a restrained body. On the other hand, the derivation of the 2nd

order loads is based on the following assumptions:

• The body is floating in small amplitude waves (𝐴

𝜆≪ 1).

• 2nd order loads are exciting a body in waves carrying a 1st order harmonic

motion forced by wave frequency loads. Therefore, expressions obtained for

the 2nd order wave loads may contain wave exciting loads.

Assuming a fluid: inviscid, irrotational, homogeneous and incompressible; the

fluid motion can be described as a function of its velocity potential (𝜙), defined in the

earth-bounded system. The fluid satisfies the Laplace equation (𝚫𝜙 = 0), a condition

also valid for its components.

𝚫𝜙(1) = 0 and 𝚫𝜙(2) = 0 (3)

This section will firstly discuss the vessel’s motion equation followed by the

definition of the low-frequency wave energy spectrum. This section will later discuss on

low-frequency motion of moored structures and the Quadratic Transfer Functions (𝑓 (2)).

Finally, it finishes with the description of the motion response spectrum due to low-

frequency wave excitation.

4

4.1 Motion Equation

According to DNVGL-ST-0111, the motion equation of a vessel can be described as

follows:

(𝑴 + 𝑴𝒂𝒅𝒅)�̈� + 𝑩𝒍𝒊𝒏�̇� + 𝑩𝒒𝒖𝒂𝒅 𝑓(�̇�) + 𝑲𝑥 = 𝑭𝒆𝒏𝒗(𝑡, 𝑥, �̇�) (4)

The equation can be simplified according to the vessel’s mode of operation.

Generally, for moored and station keeping/moored vessels, it is assumed a low-

frequency mathematical model.

A hypothesis due to the complexity of the problem was made considering

quadratic damping (𝐵𝑞𝑢𝑎𝑑). Quadratic damping is a function of the vessel’s velocity and

it has been linearized in spectral calculations. Quadratic damping is proportional to the

square of the velocity vector (𝑓(�̇�)) expressed below:

𝑓(�̇�) = �̇� /�̇�/ (5)

According to [2], the resultant equivalent moment damping from linear and quadratic contributions are expressed by:

𝑀𝐵 = 𝐵𝑙𝑖𝑛�̇� + 𝐵𝑞𝑢𝑎𝑑�̇�|�̇�| (6)

A stochastic linearization is recommended for irregular seas condition considering an equivalent linear system to the non-linear.

The difference between linear and non-linear system is:

𝛿 = 𝐵𝑙𝑖𝑛�̇�(𝑡) + 𝐵𝑞𝑢𝑎𝑑�̇�(𝑡)|�̇�| − 𝐵𝑒𝑞�̇�(𝑡) (7)

And the variance expressed as:

𝐸[𝛿2] = (𝐵𝑙𝑖𝑛 − 𝐵𝑒𝑞)

2𝐸[�̇�2] + 2(𝐵𝑙𝑖𝑛 − 𝐵𝑒𝑞𝐵𝑞𝑢𝑎𝑑)𝐸[�̇�(𝑡)2|�̇�(𝑡)|]

+ 𝐵𝑞𝑢𝑎𝑑𝐸[�̇�(𝑡)2|�̇�(𝑡)|2]

(8)

Assuming a Gaussian distribution it is possible to define an equivalent linear damping combining both linear and quadratic contributions:

𝐵𝑒𝑞 = 𝐵𝑙𝑖𝑛 + √8

𝜋𝜎�̇�𝐵𝑞𝑢𝑎𝑑

(9)

Table 1 presents a description of the different components and how they have

been considered and determined throughout this project.

5

Table 1: Terms of the Motion Equation

Component Determination Comments

Inertia Terms

Inertia Matrix 𝑴 Weight distribution 6 x 6 Matrix

Added Mass 𝑴𝒂(𝝎) Via HydroStar

Radiation module 6 x 6 function of motion frequency

Damping Terms

Linearized Equivalent Damping

𝑩𝒆𝒒

Linear Damping Effects 𝐵𝑙𝑖𝑛

CFD, Tests

FD

𝐵𝑒𝑞 = 𝐵𝑁 + √8

𝜋𝜎�̇�𝐵𝑞𝑢𝑎𝑑

TD 𝐵𝐸𝑄

= −𝐵𝑙𝑖𝑛𝜔𝑥

− 𝐵𝑞𝑢𝑎𝑑𝜔𝑥|𝜔𝑥|

Quadratic damping effects 𝐵𝑞𝑢𝑎𝑑

Forced Oscillations tests

Stiffness Matrix 𝑲

Linear Hydrostatic Stiffness 𝑲𝒉

via HydroStar As defined by [3]. It is related to the hull geometry

Anchoring Stiffness 𝑲𝒂

Mooring system software

-

Additional 𝑲𝒕𝒂𝒏𝒌𝒔 Additional tank

stiffness Due to free-surface effects

Excitation Forces

Wave Depending on

mode of calculation (FD or TD)

Software Benchmark Section 5.2

Current* TD From drag coefficient from wind tunnel tests

Wind* TD From drag coefficient from wind tunnel tests

*whenever used on TD calculation

For a damped mass spring system, the natural circular frequency (𝜔𝑑) is

determined as:

𝜔𝑑 =√4 𝐾 𝑀 − 𝐵2

2𝑀

(10)

For a damping rate 𝜏 =𝐵

𝐵𝑐 << 1 the natural circular frequency may be determined

as 𝜔𝑛 = √𝐾

𝑀

Critical damping (𝐵𝑐) is defined as:

𝐵𝑐 = 2 √𝐾 𝑀 (11)

Normally, environmental load force is given by the combination of wave, wind and

current.

6

4.2 2nd Order Wave Excitation Spectrum

According to wave theory, the sea can be modelled as the sum of different

harmonic waves following a stochastic process.

𝜂(𝑡) = ∑𝐴𝑖 cos(𝜔𝑖𝑡 + 𝛼𝑖)

𝑖

(12)

In this sense, to describe the ocean energy, various wave energy power spectra

have been proposed (Pierson-Moskowitz, JONSWAP, Torsethaugen…) for the offshore

industry. They provide the possibility to work on frequency domain and to extrapolate to

time-domain wave signals. JONSWAP spectrum (𝑆𝐽) is illustrated in Figure 1. This

spectrum is a function of the significant wave height (𝐻𝑆), wave frequency (𝜔) and

angular spectral peak frequency (𝜔𝑝).

JONSWAP Spectrum

𝑆𝐽(𝜔) = 𝐴𝛾 𝑆(𝜔) 𝛾exp (−0.5(

𝜔−𝜔𝑝

𝜎 𝜔𝑝 )

2

)

Where,

𝑆(𝜔) =5

16 𝐻𝑠 𝜔𝑝

4 𝜔−5exp (−5

4∗ (

𝜔

𝜔𝑝

))

Figure 1: JONSWAP Wave Spectrum

On the other hand, according to [1], based on the assumption that the wave

elevation follows a Gaussian process it can be shown that the 2nd order wave energy

spectrum can be obtained using:

𝑆𝑊(∆𝜔) = 8∫ 𝑆𝐽(𝜔) 𝑆𝐽(𝜔 + ∆𝜔) 𝑑𝜔∞

0

Figure 2: 2nd Order Wave Spectrum

7

These results have been obtained from a JOWNSAP spectrum with 𝐻𝑠 = 16.2 𝑚

and 𝑇𝑝 = 18 𝑠.

It can be observed from Figure 2 that the spectrum range follows a monotonically

decreasing curve with energy peak in small circular frequency values. Therefore, it

explains why moored or anchored vessels with low metacentric transverse height and

consequently high natural period values are susceptible to 2nd order roll motion.

Moreover, it can also be concluded that the 2nd order spectrum energy is sensitive

to the parameters which express wave spectrum 𝐻𝑠 and 𝑇𝑝. It can be observed that for

higher peak period more the 2nd order wave energy will be accumulated in the spectrum`s

low frequency. In Figure 3, a comparison of 2nd order has been held between two wave

spectrums described by a JONSWAP Spectrum with:

• 𝐻𝑠 = 15𝑚;

• 𝑇𝑝 = 13 𝑠 and 𝑇𝑝 = 18 𝑠

Figure 3: 2nd Order Wave Spectrum Comparison

4.3 Roll Motion (1st and 2nd order)

2nd order roll motion is a nonlinear behavior related to 2nd order wave drift

moment. 2nd order forces and moments are more apparent on horizontally restrained

structures such as moored vessels. The response analyses for an anchored or moored

structure on irregular seas follow three important components as it can be observed in

Figure 4:

• An oscillatory displacement excited by the wave-frequency bound region (~ 5s -

20s) due to linear motion with harmonic characteristic. In this situation, the

vessel’s position varies around its hydrostatic equilibrium position.

• Mean drift: caused by nonlinear effects. This force is generally obtained by

Newman’s simplification of the quadratic function.

0

1000

2000

3000

4000

5000

6000

7000

0 0.1 0.2 0.3 0.4 0.5 0.6

S_LF

[m

^4.s

]

Difference Frequency [rad/s]

2nd Order Wave Spectrum

Tp = 13s Tp = 18s

8

𝐹𝑀𝑒𝑎𝑛 = 2 ∗ ∫ 𝑆(𝜔) ∗ �⃗⃗� (2)

(𝜔) 𝑑𝜔+∞

𝜔=0

(13)

• Slow-drift: an oscillatory displacement motion caused by low-frequency drift

forces. Due to its small damping, this motion may present large amplitudes

All three motion modes are illustrated at Figure 4.

Figure 4: Response motion of a moored structured Image extracted from [1]

4.4 Quadratic Transfer Function

As seen on 4.3, roll motion can be decomposed into 3 components:

• Two components due to 2nd order loads.

• One due to 1st order loads.

While 1st order motion corresponds to the linear motion commonly described by

the RAO operator, 2nd order loads are described by their Quadratic Transfer Function

and are proportional to the square of the wave amplitude as observed in Section 4.

Quadratic loads are commonly used to assess wave drift forces (Surge, Sway

and Yaw) crucial for the design of anchoring systems.

According to [3], in order to assess the vessel`s FQTF from an irregular sea state,

wave components are combined into a bichromatic wave surface elevation:

𝜂(1)(𝑥, 𝑡) = 𝐴1 cos(𝑘1𝑥 − 𝜔1𝑡 ) + 𝐴2 cos (k2𝑥 − 𝜔2𝑡) (14)

The first order velocity potential from the radiation-diffraction problem is defined

by [3] as:

9

𝜙(1)(𝑥, 𝑦, 𝑧, 𝑡) = ℝ{(𝜙𝐼1(1)

+ 𝜙𝐷1(1)

− ∑𝑖𝜔1 𝑥1𝑗(1)

𝜙𝑅1𝑗

6

𝑗=1

)𝑒−𝑖𝜔1𝑡

+ (𝜙𝐼2(1)

+ 𝜙𝐷2(1)

− ∑𝑖𝜔2 𝑥2𝑗(1)

𝜙𝑅2𝑗

6

𝑗=1

)𝑒−𝑖𝜔2𝑡}

(15)

According to [3], to compute the second order loading, the final load is the result

of the pressure integration around the hull and it is written as:

𝐹(2)⃗⃗ ⃗⃗ ⃗⃗ ⃗ = ∫1

2𝜌𝑔 (𝜂(1) − 𝜁(1))

2�⃗� 0𝑑Γ

Γ0

+ 𝐴(1) ^∬ −𝜌 𝜙𝑡(1)

𝑛0⃗⃗⃗⃗ 𝑑𝑆𝑆𝐶0

+ ∬ − 𝜌 [𝜙𝑡(2)

+1

2(∇𝜙(1)2 + 𝑃0𝑃⃗⃗⃗⃗⃗⃗ ⃗(1) ∇𝜙𝑡

(1))] 𝑛0⃗⃗⃗⃗ 𝑆𝐶0

𝑑𝑆

(16)

Furthermore, [3] deduces from Equation 16 a mathematical development of the

second order efforts using the quadratic transfer function:

𝐹 (2)(𝑡) = 𝐴12𝑓 𝑑(𝜔1) + 𝐴2

2𝑓 𝑑(𝜔2)

+ ℝ {𝐴12 𝑓+⃗⃗⃗⃗

(2)(𝜔1, 𝜔1) 𝑒

−2𝑖𝜔1𝑡 + 𝐴22𝑓+⃗⃗⃗⃗

(2)(𝜔2, 𝜔2) 𝑒

−2𝑖𝜔2𝑡

+ 2 𝐴1 𝐴2 𝑓−⃗⃗ ⃗(2)

(𝜔1, 𝜔2) 𝑒−𝑖(𝜔1−𝜔2)𝑡

+ 2 𝐴1 𝐴2 𝑓+⃗⃗⃗⃗ (2)

(𝜔1, 𝜔2) 𝑒−𝑖(𝜔1+𝜔2)𝑡}

(17)

To determine the FQTF implies on calculating two different types of modes: High-

frequency and Low-frequency.

• High-frequency (𝑓+⃗⃗⃗⃗ (2)

): high-frequency FQTF are limited to stiff offshore systems,

e.g. vertical resonance of TLP’s (springing)

𝐹+⃗⃗ ⃗⃗

(2)(𝑡) = ℝ{∑∑𝐴𝑖𝐴𝑗 𝑓+⃗⃗⃗⃗

(2)(𝜔𝑖, 𝜔𝑗, 𝛽) e𝑖[−(𝜔𝑗+𝜔𝑖)𝑡+𝛼𝑖+𝛼𝑗]

𝑗𝑖

} (18)

• Low-frequency (𝑓−⃗⃗ ⃗(2)

): Low-frequency wave drift load comes from the difference

mode. It has a much wider use range and includes much of the moored offshore

structure’s behavior, e.g. Surge, Sway, Yaw, and Roll of floating moored vessels.

𝐹−⃗⃗⃗⃗ (2)

(𝑡) = ℝ{∑∑𝐴𝑖𝐴𝑗 𝑓−⃗⃗ ⃗(2)

(𝜔𝑖, 𝜔𝑗, 𝛽) e𝑖[−(𝜔𝑖−𝜔𝑗)𝑡+𝛼𝑖+𝛼𝑗]

𝑗𝑖

} (19)

10

This study will focus on the low-frequency mode of the quadratic transfer function

once High-frequency mode does not excite ship-shaped unities.

The components of the full quadratic transfer function matrix are defined in Table

2 :

Table 2: FQTF Components

(𝑓 (2) (𝜔1, 𝜔1) ⋯ 𝑓 (2) (𝜔1, 𝜔𝑛)

⋮ ⋱ ⋮

𝑓 (2) (𝜔𝑛, 𝜔1) ⋯ 𝑓 (2) (𝜔𝑛, 𝜔𝑛)

)

Dynamic Wave Drift load

Diagonal Values Mean drift

(force/moment)

Mean drift load from Newman’s

simplification (Equation 13)

Near off-diagonal terms*

Low-frequency drift loads Harmonic

Response Far off-diagonal terms

High-frequency drift loads

Newman’s approximation is largely used on low-drift force and are commonly

used on mooring line sizing since it provides good approximation on Sway and Surge

drift forces. Throughout, the current project, Newman’s approximation corresponds the

main diagonal of the FQTF matrix as Equation 20.

𝑓 (2)(𝜔𝑖, 𝜔𝑗) ≅ 𝑓 (2)(𝜔𝑖, 𝜔𝑖) (20)

There are two formulations capable of providing the quadratic part of the roll QTF

(𝐹𝑄) : near-field and middle-field. Middle-field formulation may present instability due to

boundary condition (free surface condition) and it is not recommended while treating

vertical force or moments, as defined on [2]. Therefore, only near-field is described

below.

Near-field formulation has been described on [4] and it is based on the integration of

the pressures around the hull. It is recommended for assessments on roll quadratic

functions and therefore is the methodology chosen.

4.5 2nd Order Roll Moment Spectrum

It has been shown that 2nd order wave excitation spectrum has its energy

restrained to low circular difference-frequency wave. Nonetheless, according to [5], it is

possible to combine spectrally wave excitation (Figure 2) and the vessel’s non-linear

response to obtain the 2nd order load spectrum response:

𝑆𝑀(2)(𝜔) = 8 ∫ 𝑆(𝜔) 𝑆(𝜔 + ∆𝜔) | 𝑓−⃗⃗ ⃗

(2)(𝜔 + ∆𝜔)| 2 𝑑𝜔

∞

0

(21)

11

5. Approaches for Assessing Low-frequency Roll

Motion

This section summarizes the methodology used during the internship to assess

motion from 2nd order roll.

The details of each calculation methodology are presented in the following sub-

sections. Figure 5 shows a schematic view of the process. Due to late changes in the

consideration of damping of low-frequency motion, two versions of OrcaFlex software

were assessed: 10.2d and 10.3d

Figure 5: Benchmark Flowchart

5.1 Radiation/Diffraction Calculations:

The outputs from radiation/diffraction calculation are:

• RAO & FQTF for 6 DoF motion at CoG.

• Added mass.

HydroStar have been selected for performing the calculations on

radiation/diffraction. As reported in [4], among the radiation-diffraction software capable

of providing the same results for RAO and FTQF are: AQWA (v15 or +), Hydrostar (v7

or +), Wamit and Hobem. Due to the team’s expertise and software availability,

HydroStar was chosen.

The following calculation criteria were input to determine vessel hydrodynamical

data:

• Near Field method as recommended by [2].

• Infinite water depth - unless otherwise mentioned.

12

• Varying the vessels heading from 0 to 180 degrees assuming a symmetrical

geometry by step of 10 deg.

• First Order Calculations: frequency range from 0.025 rad/s to 2 rad/s with a

discretization of 0.025 rad/s.

• FQTF Calculation: difference-frequency range from 0.025 rad/s to 0.5 rad/s

with a discretization of 0.025 rad/s.

Calculations were launched considering the vessel’s hull geometry (meshed

according to panel method), its inertia properties and an equivalent linear roll damping

equal to 3% of critical damping – unless otherwise mentioned.

Hydrostar wave convention system is defined with respect to the vessel’s

longitudinal axis as shown in Figure 6.

Figure 6: Hydrostar convention for wave incidence direction Image extracted from [5]

An example of the import commands for each module used on Hydrostar is

described on Appendix III: HydroStar Input Files. All the results were taken with respect

to the CoG.

5.2 Roll Assessment

5.2.1 Spectral Calculation

This methodology was implemented on a Python script. It aims to allow an agile

approach when dealing with 2nd order roll motion in conceptual phase design. Figure 7

offers an overview of the methodology that has been used on past project to assess

spectrally low-frequency roll motion.

Spectral calculation has the following limitations intrinsic to FD approach:

• Fixed wave relative heading (with respect to the vessel).

• No wind induced roll.

• No mooring induced roll.

• Assumption on the extrapolation of 1st and 2nd order roll maxima.

• Assumption of short-term extrema (Rayleigh) but the distribution is not

appropriate due to system’s non-linearity.

13

Figure 7: Spectral Calculation Flowchart

Roll RAOs and FQTFs are dependent on the (linear) damping values used to

compute them. Damping is, therefore, varied as follows: 1.5% 𝐵𝑐, 3% 𝐵𝑐, 6% 𝐵𝑐, 9% 𝐵𝑐

and 12% 𝐵𝑐. A calculation loop between Equation 25 and Equation 26 makes sure that

the damping used for the radiation/diffraction outputs converge with the final roll

response.

FD approach is based on mean vessel’s heading. The following approximations

were made:

• The 2nd order roll moment response spectrum is narrow-banded, with a pronounced energy peak at the roll natural frequency. The 2nd order moment spectrum can thus be computed only for one difference-frequency, equal to the roll natural frequency, reducing computational effort.

• 2nd order roll motion is assumed narrow-banded and probability density function of the response ranges follows a Rayleigh distribution.

Therefore, the standard deviation of the roll response is given by:

𝜎𝑟𝑜𝑙𝑙(2)2 = ∫

𝑆𝑀(2)(𝜔)

(𝐾 − (𝑀 + 𝑀𝑎𝑑𝑑)𝜔2)2 + (𝐵𝜔)2𝑑𝜔

∞

0

(22)

14

Assuming a Newman simplification and the hypothesis expressed before:

𝜎𝑟𝑜𝑙𝑙(2)2 = 𝑆𝑀

(2)(𝜔𝑛)∫1

(𝐾 − (𝑀 + 𝑀𝑎𝑑𝑑)𝜔2)2 + (𝐵𝜔)2𝑑𝜔 ≅

𝜋 𝑆𝑀(2)(𝜔𝑛)

2 𝐵 𝐾

∞

0

(23)

The equivalent damping from 1st order calculation:

𝐵𝑒𝑞(1) = 𝐵𝑙𝑖𝑛 + √8

𝜋 𝐵𝑞𝑢𝑎𝑑 𝜎𝑟𝑜𝑙𝑙(1)

2 𝜋

𝑇𝑧(1)

(24)

The linearized equivalent damping derived from 2nd order calculation

:

𝐵𝑒𝑞(2) = 𝐵𝑙𝑖𝑛 + √8

𝜋 𝐵𝑞𝑢𝑎𝑑 𝜎𝑟𝑜𝑙𝑙(2)

2 𝜋

𝑇𝑧(2) (25)

Finally, combining Equation 23 and Equation 25, the standard deviation of the 2nd

order roll motion can be determined from real root of the following 3rd order polynomial

equation:

ℝ( [2 𝐾 √8

𝜋𝐵𝑞𝑢𝑎𝑑

2𝜋

𝑇𝑧] 𝜎𝑟𝑜𝑙𝑙(2)

3 + [2 𝐾 𝐵𝑙𝑖𝑛]𝜎𝑟𝑜𝑙𝑙(2)2 − 𝜋𝑆𝑀

(2)(𝜔𝑛)) = 0 (26)

Considering that 1st order roll response is excited by a Gaussian wave moment;

it can be showed that 1st order roll response follows a Gaussian distribution and its

maxima may be accurately defined using a Rayleigh’s distribution by. Assuming a short-

term statistic with 3 hours duration, the maximum 1st order roll angle is expressed by:

𝜃𝑊𝐹 (1) = 2 𝜎𝑟𝑜𝑙𝑙(1)√2 ln (

∆𝑡

𝑇𝑛) (27)

According to [5], 2nd order roll response does not necessarily follow a Rayleigh

distribution due to non-linearity. Despite that, the Rayleigh distribution can provide an

estimation of maximum roll, which can be valuable during conception phase. Therefore,

assuming a short-term statistic with 3 hours duration, the maximum 2nd order roll angle

is expressed as below:

𝜃𝐿𝐹 (2) = 2 𝜎𝑟𝑜𝑙𝑙(2)√2 ln (

∆𝑡

𝑇𝑛) (28)

15

Mean 2nd order roll angle may be estimated dividing Equation 13 to the vessel’s

roll hydrostatic stiffness:

𝜃𝑀𝑒𝑎𝑛 =2 ∗ ∫ 𝑆(𝜔) ∗ 𝑓 (2)(𝜔) 𝑑𝜔

+∞

𝜔=0

∆ 𝑔 𝐺𝑀𝑇̅̅ ̅̅ ̅̅

(29)

Finally, to estimate final maximum roll angle, the three angles from each mode

can be summed:

𝜃𝑚𝑎𝑥 = 𝜃𝑊𝐹 (1) + 𝜃𝑚𝑒𝑎𝑛

(2) + 𝜃𝐿𝐹 (2) (30)

Starspec Spectral Response

Spectral parameters have been calculated using the module Starspec from

Hydrostar V8 to serve as input data to the Python script as elucidated in Figure 7.

Therefore, Starspec should provide statistical parameters such as:

• 2nd order Roll moment spectrum.

• 1st order moment distribution spectrum and statistical parameters.

A benchmark calculation was held to validate 2nd order roll moment (𝑆𝑀) and

ensure that any discrepancy that may arise from roll final response were mitigated.

Figure 8 presents a graph comparing 𝑆𝑀 from Starspec and a Python script from a barge

once the moment spectrum response is expressed as:

𝑆𝑀(∆𝜔) = 8 ∫ 𝑆(𝜔) 𝑆(𝜔 + ∆𝜔) | 𝐹𝑄𝑇𝐹 (𝜔 + Δ𝜔) |2 𝑑𝜔∞

0

(31)

Figure 8: Verification of 2nd order roll spectrum moment

0

2E+16

4E+16

6E+16

8E+16

1E+17

1.2E+17

1.4E+17

1.6E+17

0 0.1 0.2 0.3 0.4 0.5 0.6

Ro

ll M

om

ent

Spec

tru

m

Frequency [rad/s]

Starspec Python Script verification

16

5.2.2 Time Domain Reconstruction

As stated on [2], 2nd order roll phenomenon is not described by a Rayleigh distribution and it explains the interest on performing a time series reconstruction. Hydrostar V8 by its Staspec module is able of reconstructing the roll time signal on time domain. Intrinsic limitations of this analysis lay on:

• No wind/mooring induced roll.

• Recombination on 1st and 2nd order roll maxima is done but assessed separately.

Newton`s second law is used to analyze the 2nd order roll motions of the vessel.

The following motion equation is derived from [5].

{−(Δ𝜔)2(𝑴 + 𝑴𝑨(𝚫𝝎)) − 𝑖Δ𝜔 𝑩(𝚫𝝎) + 𝑲} 𝑿 = 𝑭𝟐(𝚫𝝎) (32)

This method is based on BV`s time series generation available on Hydrostar/Staspec

V8.0. According to [5], the maxima of the 2nd order roll motion does not follow a Rayleigh

distribution and that justify the generation of a time series signal from spectrum response.

• 𝑴 : inertia matrix of the body.

• 𝑴𝑨(𝚫𝝎) : additional mass matrix coming from radiation problem solution.

• 𝑩(𝚫𝝎) : damping matrix coming from the radiation problem solution and

additional damping defined by the user.

• 𝑲 : stiffness matrix coming from the hydrostatic properties of the

body or additional stiffness due to mooring system or liquid in tanks.

• 𝑿 : motion vector of the body.

• 𝑭𝟐 : low-frequency load (either moment or force)

Expressing the Quadratic Transfer Function as complex exponential function as 𝐴

being the wave amplitude and 𝜑 the phase. 2nd order time series of the roll moment may

be reconstructed using Equation 33 from [5]:

𝜃(2)(𝑡) = ∑∑𝐴𝑖 𝐴𝑗 𝑟−(2)(𝜔𝑖, 𝜔𝑗, 𝛽) cos [(𝜔𝑖 − 𝜔𝑗)𝑡 − 𝜑𝑖 + 𝜑𝑗 + 𝛼−

(2)(𝜔𝑖, 𝜔𝑗, 𝛽)]

𝑗𝑖

(33)

5.2.3 Time Domain Analysis:

TD analysis have been performed using OrcaFlex 10.2d and its newest version

OrcaFlex 10.3d. This latest version presents updates and new calculations method for

damping low-frequency motion. In this phase of the project, a spread moored FPSO was

modeled using soft mooring links to fix vessel’s heading and minimize its impact on roll

motion, the arrangement of the lines chosen were:

• Symmetrically disposed over the bow/aft and portside/starboard.

• Attached to the center line.

• Same height as the CoG.

To avoid coupling with others DoFs, only Roll QTF was imported to OrcaFlex and

Roll, Sway and Yaw RAOs were neglected.

The roll angle signal was treated with SigView, an in-house software that allows

the treatment of TD domain results and extract statistical properties such as: MPM, Mean

value, and standard deviation. It also allows to filter the signal with low and high

17

bandpass filters. Figure 9 shows an illustrative case of signal treatment of the roll signal

calculated with OrcaFlex 10.3d for 𝐺𝑀𝑡 = 4.5 𝑚 and 30 𝑑𝑒𝑔 of wave incidence. It is

possible to observe from the second picture that the main response of the system lies

around the natural frequency (0.15 rad/s).

Figure 9: Signal treatment using SigView

5.2.3.1 OrcaFlex 10.2d

OrcaFlex 10.2d has as limitation the fact that damping is only included to the

motion equation once the Primary Motion of the vessel was treated as wave frequency

response. Therefore, to include roll damping, the motion had to be treated as “Wave

Frequency” even though the vessel’s main longitudinal rotation was due to 2nd order

motion. First order loads from RAO were not considered and set to zero to assess

exclusively low-frequency motion.

18

Figure 10: OrcaFlex 10.2d Calculation Definition

OrcaFlex 10.2d offers on “Other Damping” the possibility to include damping on

vessel’s wave-frequency in relation to its reference point. According to [6], the resultant

damping force/moment of “Other damping” is the sum of linear and quadratic modes.

The roll damping (𝑚𝑥) for this version is expressed as:

𝐵 = −𝐵𝑙𝑖𝑛𝜔𝑥 − 𝐵𝑞𝑢𝑎𝑑𝜔𝑥|𝜔𝑥| (34)

Where 𝜔𝑥 corresponds to the wave-frequency part of the vessel primary motion

velocity relative to the earth fixed axis at the reference origin.

5.2.3.2 OrcaFlex 10.3d

OrcaFlex 10.3d allows a different treatment regarding the damping of low-

frequency motions. In this version, OrcaFlex is able damp 1st and 2nd order motion

treating Primary Motion as Both low and wave frequency. The Dividing period parameter

allows OrcaFlex to establish a frontier between what will be considered as wave or low-

frequency.

OrcaFlex uses the same equation but it is able to set the damping force/moment

to be applied to the “Total Motion”. Therefore, 𝜔𝑥 is the primary motion velocity defined

as being the wave and low-frequency mode.

5.2.4 Study Case

The present subsection aims to discuss the results that have been obtained by

the 4 calculation methods explained before: Spectral calculation, Time Generation,

OrcaFlex 10.2d and OrcaFlex 10.3d of a study case.

The study case is based on a FLNG modelled by an equivalent barge (Figure 11)

with the same mechanical properties as the real vessel. It is fixed in place by a soft

mooring system which prevents vessel from Yawing. The main vessel`s properties are

shown in Table 3.

19

Table 3: Vessel Properties

Parameter Unit Value

Geometry

L [m] 486.0

B [m] 75.0

T [m] 19.0

Inertia Properties

Displacement [t] 709864

LCG from AP [m] 274.17

TCG [m] 0

VCG from keel [m] 32

Rxx [m] 33.0

Ryy [m] 134.3

Rzz [m] 134.3

Ixx [t.m2] 759.9 E+6

Iyy [t.m2] 12.8 E+9

Izz [t.m2] 12.8 E+9

In order to assess the hydrodynamics properties of the vessel – added mass,

hydrostatic stiffness, RAO, FQTF - a HydroStar V8 model was created. As recommended

by [4], near-field method is used in the diffraction module - as explained in section 4.4.

Nb. of cells 6408

Average panel Size

2.17

Figure 11: HydroStar Hull Mesh

The vessel’s natural period has been varied from 25.7s to 63.8s by modifying its

transverse metacentric height (𝐺𝑀𝑡). Roll inertia (𝐼𝑥𝑥) and displacement (Δ) were kept

constant. Damping coefficients, 𝐵𝐿𝑖𝑛 and 𝐵𝑄𝑈𝐴𝐷 have been obtained via HydroStar for

each model. Wave incidence has been varied according to 30 deg, 70 deg and 90 deg.

RAO were assessed using different damping factors as explained on section 5.1.

As it can be observed from Figure 12 and Figure 13, the damping factor plays an

important role on the linear response of the vessel (RAO) especially when excitation

frequency (𝜔) lies close to the vessel’s natural period reducing the peak response. These

calculations used a JONSWAP wave spectrum:

20

Table 4: Wave Properties


𝐻𝑠 [m] 16.2

𝑇𝑝 [s] 18.6

𝛾 3

• 𝐺𝑀𝑡 = 11.5 𝑚 and 𝑇𝑛 = 25.7 𝑠:

Figure 12: Roll RAO - GMt = 11.5 m

• 𝐺𝑀𝑡 = 3.0 𝑚 and 𝑇𝑛 = 63.78 𝑠:

21

Figure 13: Roll RAO - GMt = 3 m

On the other hand, the roll quadratic transfer function (Figure 14 and Figure 15)

presents some particularities. Once again, damping is important for difference-frequency

excitations (Δ𝜔) nearby the natural frequency (𝜔𝑛). Despite that, near-off terms from the

main diagonal - see section 4.4 – express more sensitivity to damping factor than the

QTF, when Δ𝜔 = 0. The impact of the damping over the FQTF was expected once

damping modifies the 1st order velocity potential (Φ(1)).

• 𝐺𝑀𝑡 = 11.5 𝑚 and 𝑇𝑛 = 25.7 𝑠:

Figure 14: Roll FQTF - GMt = 11.5 m – Δω = 0 rad/s

22

Figure 15: Roll FQTF - GMt = 11.5 m – Δω = 0.2 rad/s

Furthermore, the roll FQTF for loading cases with longer natural period are less

affected by the damping (Figure 16 and Figure 17).

• 𝐺𝑀𝑡 = 3.0 𝑚 and 𝑇𝑛 = 63.78 𝑠:

23

Figure 16 : Roll FQTF - GMt = 3 m - 𝛥𝜔 = 0 rad/s

Figure 17: Roll FQTF - GMt = 3 m - Δω = 0.2 rad/s

In this model, loads from wind and current have not been considered. Only wave

loads were included to the model, so no other roll influence were considered - e.g. wind

induced roll angle. A JONSWAP spectrum was used according to Table 4. Such

condition was chosen for SBM past experiences and since theoretically it could yield to

a higher roll response as seen on Section 4.2.

As attended roll moment spectrum response (𝑆𝑀) - obtained as expressed on

Section 5.2.1 - it is impacted by damping since its respective FQTF is impacted:

24

𝐺𝑀𝑡 = 11.5 𝑚 and 𝑇𝑛 = 25.7 𝑠 𝐺𝑀𝑡 = 3.0 𝑚 and 𝑇𝑛 = 63.78 𝑠

Figure 18: Roll 2nd Order Moment Spectrum

Moreover, observing the right-hand image from Figure 18, it can be argued the

narrow-banded hypothesis assumed during spectral calculation (Equation 23).

Comparing the results from Figure 21 to Figure 18, it is possible to conclude that the

spectral calculation for this case do not converge with the results obtained with the other

methods.

The influence relative wave incidence was assessed according varying vessel’s

heading: 30 deg, 70 deg and 90 deg.

5.2.5 Comparison

The present results were obtained comparing four hydrodynamic models:

Spectral Calculation, Fully Coupled analysis (OrcaFlex 10.2d & OrcaFlex 10.3d) and

Time Reconstruction via HydroStar. The results are based on the 2nd order roll standard

deviation.

30 deg 2nd order Roll Amplitude Standard Deviation

GMt [m] Tn [s] Spectral

Calculation OrcaFlex 10.2d OrcaFlex 10.3d Hydrostar

3 63.78 1.3 2.1 1.9 2.1

4.5 44.3 0.9 1.2 0.9 0.7

5.3 39.2 0.8 0.9 0.7 0.6

6.0 35.2 0.9 0.9 0.6 1.3

11.5 25.73 0.3 0.3 0.1 0.3

25

Figure 19: 2nd Order Roll Standard Deviation for 30 deg of Wave Incidence

70 deg 2nd order Roll Standard Deviation



3 63.78 1.9 2.9 2.1 2.5

4.50 44.3 1.8 2.2 1.6 2.0

6.0 35.2 2.0 1.9 1.3 2.7

11.5 25.73 0.9 0.8 0.6 1.1


0.0

0.5

1.0

1.5

2.0

2.5

0 10 20 30 40 50 60 70

Stan

dar

d D

evia

tio

n [

deg

]

Tn [s]

Standard Deviation - 30 deg

Spectral Calculation OrcaFlex 10.2d Hydrostar OrcaFlex 10.3d

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

0 10 20 30 40 50 60 70

Stan

dan

d d

evia

tio

n [

deg

]

Tn [s]



26

90 deg 2nd order Roll Standard Deviation



3 63.78 1.6 2.5 1.6 2.3

4.50 44.3 1.4 2.2 1.3 1.7

6 35.2 1.6 2.0 1.2 2.0

11.5 25.73 1.3 1.3 0.7 1.8


One important result that can be derived from the calculations is the fact that

contrary to 1st order roll motion, the low-frequency roll motion may not be maximized on

beam seas but rather on quartering seas (around 60/70 deg) of wave incidence as

represented in Figure 22.

Figure 22: Roll Standard Deviation according to Wave Incidence

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0 10 20 30 40 50 60 70

Stan

dan

d d

evia

tio

n [

deg

]

Tn [s]



27

5.2.6 Benchmark Comparison and Partial Conclusions

Important conclusions can be withdrawn from the results obtained during

benchmark from the behavior of 2nd order physics and the methodologies response and

they will be divided onto two types: Radiation/Diffraction results and final roll response.

Radiation/Diffraction Results and spectral response:

1) The impact of the damping rate (𝜏) to RAO is limited to the peak response around

the vessel’s natural roll frequency, as shown in Figure 12 and Figure 13.

FQTFs are also sensitive to 𝜏 around the vessel’s natural period. Vessels with

small 𝑇𝑛 are more affected than ones with long natural period as illustrated in

Figure 14 to Figure 17 . This is evidenced through the 2nd order roll moment

spectrum.

2) Differently from 1st order roll motion, maximum low-frequency roll response is not

found at beam seas but rather quartering seas (around 60 degrees), see Figure

22. Spectral calculation is not suitable for turret moored vessels with large yaw

motion amplitude “fish tail” since it assumes a fixed heading.

3) The QTF cannot be used to describe 2nd order roll motion; therefore, it makes

necessary to compute full quadratic transfer function. Table 5 presents the results

obtained for the barge with environmental condition described in Table 4 (wave

incidence of 30 deg) with the following mechanical properties:

• 𝐺𝑀𝑡 = 6 𝑚 and 𝑇𝑛 = 35.5 𝑠

Table 5 : Results Comparison between Methodologies

4) Loading cases with longer natural periods will present higher roll motion due to

low-frequency loads as shown from Figure 19 to Figure 21;

5) As stated on [5] and [3], Rayleigh extrapolation is not indicated for obtaining 2nd

order roll motion due to the systems non-linearity. On the other hand, no other

type of extrapolation has been proposed. Nevertheless, for pre-design

calculations such as those using the spectral methodology it offers an estimation.

Table 6: Rayleigh Extrapolation of Maxima Roll Angle

MethodologyMean Wave

[deg]

Standard Deviation

[deg]

Python Spectral Calculation 0.09 0.9

OrcaFlex - 10.2d 0.09 0.9

OrcaFlex - 10.3d 0.09 0.6

OrcaFlex - 10.3d - Newman 0.09 0.3

HydroStar - Time Generation 0.09 1.3

Wave Incidence - 30 deg

28

A common method to extrapolate roll maxima angle is by the usage of

Gumbel Cumulative Distribution. The MPM value is obtained after launching 20

similar sea-states varying the wave seed parameter. The Most Probable Maxima

corresponds to the value associated with 37% of the cumulative probability.

Comparing Table 6 and Figure 23 Rayleigh extrapolation gives a superior value

of MPM when compared with Gumbel Distribution from TD analysis on OrcaFlex

10.3d. These values were assessed for the case where the barge has a 𝑇𝑛 =

44.1 𝑠. The MPM value from TD analysis is 3.2 deg while the Rayleigh assumption

extrapolates maximum roll response for 5.7 deg after statistical post-processing.

MPM Value = 3.2 deg

Figure 23: Gumbel Distribution of Roll Maxima

6) Spectral calculation, as explained on section 5.2.1, it has a strong hypothesis

considering 2nd order roll moment response spectrum (𝑆𝑀). As it can be observed

in Figure 18, 𝑆𝑀 does not present a pronounced peak at roll natural frequency

and therefore the narrow-banded hypothesis can be argued since it does not

seem suitable.

OrcaFlex:

1) OrcaFlex 10.2d does not apply damping to low-frequency motion. The LF is

determined by a dividing period to be specified on ships motion calculation that

serves to dissociate by filtering LF motions and WF motions. The dividing period

had to be increased so that it would include low-frequency motion and therefore

damp the total motion.

• Dividing period = 60 s.

• Primary Motion = LF + WF.

As it can be seen in Figure 19 to Figure 21 it has provided the highest values of

2nd order roll standard deviation and, consequently, the highest roll amplitudes;

Rayleigh


[deg]

Standard Deviation 2

[deg]MPM [deg]

Python Spectral Calculation 0.16 0.9 5.7

OrcaFlex - 10.2d 0.16 1.2 7.8

OrcaFlex - 10.3d 0.16 0.9 5.7

OrcaFlex - 10.3d - Newman 0.16 0.8 5.0

HydroStar - Time Generation 0.16 0.7 4.8


29

2) OrcaFlex 10.3d is suitable to assess 1st and 2nd roll without once damping settings

for 10.3d can be applied for total primary motion, defined as LF + WF. See section

5.2.3.2.

Comparison:

1) All four methods (Spectral Calculation, Time Reconstruction OrcaFlex 10.2d and

OrcaFlex 10.3d) provide the same mean roll angle as observed in Table 5.

2) Spectral calculation tends to give in the overall conservative results on standard

deviation when compared to results from OrcaFlex 10.3d and OrcaFlex 10.2d;

especially for vessel’s natural period up to 40 seconds. This range encompasses

most of the today’s and near-future offshore FPSO/FLNG.

3) Spectral calculation can assess quickly statistical properties of 2nd order roll

motion. Although this method can assess roll response, it is not recommended to

be used during final stages of a mooring design. It may be mainly used during

preliminary project.

4) The validity of the Spectral Method is questionable since the 2nd Order Roll

Moment Spectrum was proved not to be narrow-banded in all cases.

5) A satisfactory conclusion could not be reached for the discrepancy observed in

Figure 19 at 63s of vessel’s natural period. It is believed to be related with the

narrow-banded hypothesis of the 𝑆𝑀.

6) OrcaFlex 10.3d is capable of damping 2nd order motion from cutting period.

7) After extensive analysis and lack of experimental results, it has been considered

that OrcaFlex 10.3d has the more coherent roll response.

6. Approach for Future Projects: Turret Load Design

The turret mooring system consists on a fixed structure integrated to FPSO/FLNG

responsible for fixing the vessel to the seabed by means of its mooring lines. The turret

is equipped with axial bogies and radial wheels that allows the platform to weathervane.

This structure minimizes the mooring system’s load once the vessel is passively aligned

to the minimum drift environmental load (wave, current and wind).

These loads for will then be used by the structural engineer while designing the

mechanical connections:

• Axial Boogies.

• Radial Wheels.

• Chain table.

It is not the objective of this section to comment on how the environmental loads

were computed but rather on how the platform’s motion affects the loads acting on the

turret and how they were computed. This section aims to present the methodology

30

adopted to assess vessel motion and turret loads. A study case a FPSO equipped with

an internal turret will be analyzed.

As seen in section 5 current software limitations have been mapped. The main

goal of this section, therefore, it is to develop a methodology to be used in future projects

once previous data has not yet been benchmarked.

Two methodologies will be described:

• 6 DoF fully coupled analysis using OrcaFlex.

• Hybrid method correcting Ariane platform’s motion.

The usage of these two methodologies are justified from both software limitation

and differences in computation hypothesis. The methodology commonly used in the

industry for a single point moored vessel is to extract from an Ariane’s model the platform

motion and impose it to an OrcaFlex 6 DoF fully coupled calculation. OrcaFlex is then

used to extract tension in lines or connection forces. The problem of the direct usage of

a 6 DoF fully coupled analysis from OrcaFlex, lies over the fact that Ariane 7 motion has

already been benchmarked while OrcaFlex DoF has not. Ariane’s hypothesis differs from

OrcaFlex mainly in:

• Molin’s Moment on Yaw.

• Ariane 7 line’s dynamics are not considered for vessel motion while in

OrcaFlex it does.

Moreover, Ariane 7 most recent version is not yet capable of calculating 2nd order

low-frequency roll motion. It leads to an impasse once the assessment of the usage of

this software while designing the anchoring system of vessels subject LF roll motion is

required. Those limitations are to be assessed, overcame, and interpreted in this chapter.

Ariane 7 motion results were provided by SBM Offshore and shall be used to

compare the final results.

6.1 Turret Loads - Formulation

Figure 24: Turret Sketch

Where,

a: distance between chain table keel and radial wheel

b: distance between chain table keel and boogie*

31

D: Bearing diameter

*it can be considered 𝑎 = 𝑏 for hydrodynamic purposes

The chain table is responsible for connecting the mooring lines to the turret.

Therefore, this structure must withstand the loads transmitted by the mooring lines. It

corresponds to the lowest part of the turret in the keel line. The following equations define

the force and moment acting on the chain table:

𝐹ℎ𝐶𝐻 = √𝐹𝑥𝐶𝐻2 + 𝐹𝑦𝐶𝐻

2 𝑀𝐶𝐻 = √𝑀𝑥𝐶𝐻2 + 𝑀𝑦𝐶𝐻

2 (35)

Radial load acting on the turret is the result of inertia, anchoring and riser loads

being transferred to the turret via radial bearings. It corresponds to the total horizontal

force acting onto the turret.

𝐹𝑟 = √𝐹𝑥2 + 𝐹𝑦

2 (36)

According to this formulation, 2nd order roll is expected to impact radial load (𝐹𝑟𝑎𝑑)

for a set of environmental loads and high environmental return periods.

Axial bogies provide the vertical support of the turret - at its elevation 𝑏:

• vertical forces (mooring system, entrapped water*, turret mass and risers).

𝐹𝑧 = 𝐹𝑧𝑚+ 𝐹𝑧𝑟

+ 𝐹𝑧𝑡+ 𝐹𝑧𝑒

(37)

• horizontal moments.

𝑀𝑥𝐵= 𝑀𝑥𝑚

+ 𝑀𝑥𝑟+ 𝑀𝑥𝑡

+ 𝑀𝑥𝑒+ 𝑎𝐹𝑦

(38)

𝑀𝑦𝐵= 𝑀𝑦𝑚

+ 𝑀𝑦𝑟+ 𝑀𝑦𝑡

+ 𝑀𝑦𝑒+ 𝑎𝐹𝑥

𝑀𝑟 = √𝑀𝑥𝑏2 + 𝑀𝑦𝑏

2

* Entrapped water consists on an inertia force due to the water present into the cavity from the keel line to the water surface in between the vessel and the turret.

Two indicators are constructed according to OrcaFlex’s load signal

convention, 𝐹𝑏_𝑚𝑎𝑥 , 𝐹𝑏_𝑚𝑖𝑛. They correspond respectively to:

• Maximum equivalent vertical load at bogies; it is an indicator of the reaction

load.

• Minimum equivalent vertical load at bogies. It is an indicator of the

compression force. It provides an idea of the upper lift force once 𝐹𝑏_𝑚𝑖𝑛 >

0 and risk of the turret falls from its rotational trail.

They are defined by:

32

𝐹𝑏_𝑚𝑎𝑥 = 4𝑀𝑟

𝐷− 𝐹𝑧 𝐹𝑏_𝑚𝑖𝑛 = −(4

𝑀𝑟

𝐷+ 𝐹𝑧) (39)

In the present project the focus will be on the horizontal forces 𝐹𝑟𝑎𝑑 and 𝐹ℎ.

6.2 Study Case

In order to assess the impact of 2nd roll motion into turret load design, a study

case of a FPSO has been analyzed. Table 7 shows the vessel’s main geometry and

inertial dimensions. Risers were not included in the model to reduce calculation time.

Table 7: Vessel properties


Geometry

L [m] 273.2

B [m] 24.8

T [m] 19.96

Inertia Properties

Displacement [t] 260262

LCG from AP [m] 136.6

TCG [m] 0

VCG [m] 21.51

Rxx [m] 19.1

Ryy [m] 66.5

Rzz [m] 65.1

Ixx [t.m2] 9.00E+07

Iyy [t.m2] 1.09E+09

Izz [t.m2] 1.05E+09

Since this FPSO has a natural period of 31.1 seconds, it is compulsory to assess

2nd order roll motion according to DNVGL.

To assess the hydrodynamics properties of the vessel - added mass, hydrostatic

stiffness, RAO, FQTF – a panel model was set using Hydrostar V8. Fluid domain does

not have to be defined since near-field method is used in the diffraction module (see

section 4.4). Equivalent linear viscous damping of 3% was used to comply with client’s

specification.

Nb of cells 10295

Average panel Size

2.28

Figure 25 : Hydrostar Hull Mesh

33

RAO and FQTF (Figure 26) were extract and exported.

Figure 26: RAO and QTF of the FPSO

The vessel is equipped with an internal turret. Dimensions and inertia properties are described in Table 8.

Table 8: Turret Properties


Geometry

Diameter (D) m 20.0

Turret Cylinder Diameter m 17.8

Elevation of radial wheels

m 23.5

Inertia Properties

Mass t 6145

CoG – x coordinate m 0

CoG – y coordinate m 0

CoG – z coordinate m 32.5

Rxx m 19.5

Ryy m 19.5

Rzz m 8

Ixx t.m2 2.33 E+06

Iyy t.m2 2.33 E+06

Izz t.m2 3.89 E+05

34

6.2.1 Environmental Condition

In this model, environmental loads include wind, wave and current. Table 9

provides information on the longitudinal and lateral projected areas. Yaw area is in m3

once it includes the lever arm.

Table 9: Sail and Current Projected Areas

Item Unit Wind Current

Projected Front Area [m2] 3139 1089

Project Side Area [m2] 11474 5158

Yaw Area [m3] 3134697 1489779

Current and wind coefficients have been provided from wind tunnel tests for

Surge, Sway, and Yaw. The convention system of the environmental loads is the

OrcaFlex’s convention system. Force coefficients on Surge, Sway, and Yaw are

presented in Appendix II: Force Coefficients.

A standard DNV combination of the environmental loads in site was used to

determine the direction of the loads. Table 10 provides the environmental description of

the environment.

Table 10: Environmental condition

JONSWAP Spectrum Constant Constant

Wave Dir Hs Tp Wind Dir. Wind

Speed Current Dir. Wave Speed

[deg] [m] [s] [deg] [m/s] [deg] [m/s]

255 13 13.4 225 32 210 1.23

The current speed profile is shown in Table 11.

Table 11: Current speed profile

Wave Dir. [deg]

210

Depth Speed

[m] [m/s]

0 1.23

13 1.20

23 1.17

49 0.92

83 0.84

150 0.75

193 0.70

254 0.68

310 0.65

372 0.52

35

6.2.2 Orcaflex 10.3d – 6 DoF

An OrcaFlex model was created to compare the motion from Ariane and

OrcaFlex. The aim was to conclude whether they were compatible or not. For that, a

weathervane FPSO had to be defined.

Two modelling “Vessel Object” were defined:

• FPSO

• Turret. Turret inertia loads and mooring lines loads are assessed by two

different “Buoy Object” connected to the Vessel Turret (type “Lumped

Buoy”).

This modelling allows to uncouple turret inertia forces to mooring loads. Figure

27 provides a flowchart to illustrate the mode used. A bearing connection between the

FPSO and Turret allows the vessel to weathervane.

Figure 27: OrcaFlex Modelling Scheme

As explained in section 6, fully coupled 6 DoF analysis from OrcaFlex is not the

normal manner used in the offshore industry to calculate the mooring systems’ loads. As

explained, OrcaFlex assume different calculation hypothesis from Ariane 7 which may

lead to differences in the results. Since for this project it was used the newest OrcaFlex,

version 10.3d, it worth the time to analyze whether it could provide compatible motion

results when compared to Ariane 7. Figure 28 shows the calculation methods set. No roll

quadratic functions were included.

36

Figure 28: Definition of the analysis set on OrcaFlex 10.3d

The linearized damping coefficients from linear and quadratic contributions are described in Table 12.

Table 12: Damping Coefficients

Damping Coefficient

Unit Value

𝐵𝐿𝐼𝑁 [(kN m) / (rad/s)] 1.23E+06

𝐵𝑄𝑈𝐴𝐷 [(kN m) / (rad/s) ^2] 210E+06

To ensure the same level of excitation, the free surface elevation (𝜂(𝑡)) and wave

direction from Ariane 7 and OrcaFlex had calibrated. Normally, they do not present the

same convention system. Free surface elevation and wave direction are shown with

respect to OrcaFlex axis in Figure 29.

Free Surface Elevation [m] - 𝜂(𝑡) Wave Direction [deg]

Figure 29: Wave excitation calibration

37

Results:

1. As expected, OrcaFlex and Ariane 7 consider differently the Molin’s Moment

leading to different motions. Figure 30 compares the vessel Yaw time series

of OrcaFlex 10.3d and Ariane.

Figure 30: Comparison between the wave incidence time series obtained with Ariane and OrcaFlex

2. Figure 31 shows the vessel’s offset to the initial reference system. It can be

clearly inferred from these images that the motion from both software are not

equal.

Figure 31: Comparison between the offset time series obtained with Ariane and OrcaFlex

Ariane

OrcaFlex

Ariane

OrcaFlex

38

Achieving the equivalence of the results from each software is not the aim of this

project and the usage of OrcaFlex 6 DoF fully coupled analysis for anchoring systems

was not satisfactory. Therefore, since this method did not present good results for

describing the vessel’s motion it was not fully analyzed. A hybrid methodology has been

developed to overcome this limitation.

6.2.3 Hybrid Method

The Hybrid methodology overcomes Ariane 7’s limitation by correcting roll

motion. It includes LF roll contribution externally keeping constant the others DoF. In

other words, LF roll motion is uncoupled to the rest of the movement being calculated

separately. The model is similar to the one illustrated in Figure 27 using two “Lumped

Buoys” to assess separately inertia to mooring lines’ loads; a bearing connection as well

as two “Vessel Object” were set.

After determining the vessel data (Figure 25) and the environmental conditions

(Table 10 and Table 11), the following stages are executed:

1. Run an Ariane 7 simulation to extract motion without LF roll contribution.

2. Calculate with a python script for each time step the associated 2nd order low-

frequency roll moment using Equation 40.

𝑀𝑟𝑜𝑙𝑙(2)

= ∑∑𝐴𝑖(1)

𝐴𝑗(1)

𝑃𝑖𝑗 𝑐𝑜𝑠[(𝜔𝑖 − 𝜔𝑗) 𝑡 + (𝛼𝑖 − 𝛼𝑗 )] − 𝐴𝑖(1)

𝐴𝑗(1)

𝑄𝑖𝑗 𝑠𝑖𝑛[(𝜔𝑖 − 𝜔𝑗) 𝑡 + (𝛼𝑖 − 𝛼𝑗 )]

𝑁

𝑗=1

𝑁

𝑖=1

(40)

3. Import and impose calculated low-frequency roll moment to a soft moored

OrcaFlex model to assess roll motion with according damping.

4. Extract low-frequency roll angle in each time step (𝜃𝐿𝐹(𝑡)) and sum it to the

respective wave frequency roll angle from Ariane (𝜃𝑊𝐹(𝑡)) and Ariane motion .txt

file to correct final roll angle (𝜃(𝑡)).

𝜃(𝑡) = 𝜃𝑊𝐹(𝑡) + 𝜃𝐿𝐹(𝑡) Corrected Roll Angle WF roll angle from Ariane 7 LF roll angle from OrcaFlex soft-

moored vessel

5. Once Ariane motion .txt file is corrected, it is imported to the final OrcaFlex turret

moored vessel imposing its motion the simulation is run and post-treated

finalizing the calculation.

Figure 32 shows the flowchart of the methodology implemented in a python script

command all 5 previous steps.

39

Figure 32: Flowchart of the steps used to correct Ariane's roll motion

Using the soft-moored vessel from Figure 32 it is possible to impose 2nd order roll

moment (step 2) to assess the final roll motion (step 3 and 4). A low-frequency filter can

be used to extract LF roll motion and correct Ariane’s motion Time Series. The

comparison between the original roll angle signal and the corrected one is shown in

Figure 33. Discrepancies observed are mainly because coefficients 𝑄𝑖𝑗 and 𝑃𝑖𝑗 from

equation 40 have not been interpolated for both heading and inner difference frequency.

40

Figure 33: Roll correction - WF + LF

Once step 4 is completed the motion can be imposed to the vessel launching step

5. The results can be post-treated to assess, for example, chain table horizontal force

and radial force acting on the radial wheel. Final values are compared to Ariane original

motion in Figure 34.

Fh [kN] Chain table

Frad [kN] Radial Wheel

Figure 34: Horizontal loads acting on the chain table and the radial wheel

Ariane Motion

Ariane Corrected Motion

41

The following results have been obtained comparing both peak responses on

both loads:

The impact of the 2nd order roll motion can be then assessed in terms of anchoring

loads. Focusing on the horizontal forces showed previously, it is logical to withdrawn

conclusions over the loads. Firstly, radial force on acting on the radial wheels may

increase while the horizontal force on the chain table decreased. Secondly, the

decrement of FH is because a part of the horizontal force is now converted to vertical

force once roll angle are greater and FH is calculated in relation to the boat’s axis.

42

7. Conclusion and Perspectives

This project is the outcome of a research project on 2nd order roll motion of offshore platforms.

As it has been observed in Section 5, each methodology has its own hypothesis and limitations. Despite that, some limitations were overcame leading to the results presented in that same section. Partial conclusions were made in Section 5.2.6 but it worth restating the following conclusions:

• The energy of the 2nd order wave spectrum accumulated in the low-frequency range which excites vessels with larger roll natural period.

• Spectral Calculation assumes a narrow-banded Roll Spectrum Response and its usage is questionable. On the other hand, it may be used during preliminary calculation.

• A new method for extrapolating maximum roll angle from the standard deviation may be envisaged.

• OrcaFlex 10.2d is not capable of damping low-frequency motion and a special attention should be pay on the dividing period.

• OrcaFlex 10.3d seems to present good results but should be confirm on model test on tanks. For this project good experimental observations were not available.

In the second part of the research, two methodologies were analyzed to include low-frequency roll motion on a turret mooring design:

• OrcaFlex 6 DoF fully coupled analysis.

• Hybrid method gathering Ariane 7 and OrcaFlex by a Python script.

The usage of OrcaFlex 6 DoF fully coupled analysis to extract ship motion turret loads is questionable. First, because normally even for WF roll driven vessels this method is not used. Ariane 7 is commonly used to assess vessel’s motion and it has already been benchmarked. The main difference between those two methodologies are related to Molin’s damping moment and the dynamic of the lines.

Therefore, a more practical and conservative approach has been suggested, the Hybrid Method. It consists on a method that corrects Ariane’s motion including the low-frequency roll mode on final motion. This corrected motion is then imposed to the vessel on OrcaFlex. The results obtained by this second method converged with expectations, thus recommended for future endeavors.

I envisage for future continuation of his project the benchmark of this result or similar models to their equivalent model scale tests. This will provide a real system to benchmark the results and provide definitive conclusions. Moreover, a set of different sea-states should be used to conclude the impact of this motion on turret loads.

It is also envisaged the interpolation on the Python script of coefficients 𝑄𝑖𝑗 and

𝑃𝑖𝑗 from Equation 40. This should mitigate discrepancies and decrease the level of

uncertainty presented in the Hybrid Method.

Despite that, some overall tendencies can be withdrawn and should be bear in mind for future works on assessing 2nd order roll motion:

• High natural periods beyond 25 s.

• Low values of 𝐺𝑀𝑡.

• Large values of 𝐻𝑆 and 𝑇𝑝.

• Large separation between wind and waves (for weathervane units).

• Large values of wind speed combined with large vessel transverse area

(weathervane units).

43

• 2nd order roll has a minor impact on horizontal forces, but environmental

conditions should be varied and the other loads described on section 6.1

should be assessed.

44

8. References

[1] J. J. a. W. Massie, OFFSHORE HYDROMECHANICS, Delft: Delft University of Technology,

2001.

[2] JIP, "Guidance Note - JIP non linear roll," 2016.

[3] B. Molin, Hydrodynamique des Structures Offshore, Paris: TECHNIP, 2002.

[4] A. C. d. O. X.-b. C. F. M. Flavia C. Rezende, "A Comparison of Different Approximations for

Computation of Second Order Roll Motions for a FLNG," in OMAE, Nantes, France, 2013.

[5] "How to compute second order roll with HydroStar," Bureau Veritas, Neully sur Seine,

March 13th 2014.

[6] Orcina, "Documentation for OrcaFlex (version 10.2d)," 2018.

[7] Y. LIU, "On Second-Order Roll Motions Of Ships," in OMAE2003-37022, Cancun, Mexico,

2003.

[8] A. N. S. Fabio Tadao Matsumoto, "Predicting the Second-order Resonant Roll Motions of an

FPSO," in OMAE, San Francisco, USA, 2014.

45

Appendix

Appendix I: Script Python

Appendix II: Force Coefficients

-1

-0.5

0

0.5

1

1.5

0 50 100 150 200 250 300 350 400

Forc

e C

oe

ff.

Angle of Incidence [°]

Current Coefficients

Cfx

Cfy

46

Figure 35: Current Force Coefficients

-8.00E-02

-6.00E-02

-4.00E-02

-2.00E-02

0.00E+00

2.00E-02

4.00E-02

6.00E-02

0 100 200 300 400

Mo

men

t co

eff.

Ang. of Incidence [°]

Current Moment Coefficient

Cmz

-2.0000

-1.5000

-1.0000

-0.5000

0.0000

0.5000

1.0000

1.5000

2.0000

2.5000

0 100 200 300 400Forc

e c

oe

ff.

Ang. of Incidence (°)

Wind Coefficients

Cfx

Cfy

47

Figure 36: Wind Force Coefficients

Appendix III: HydroStar Input Files

Figure 37: Hydrostar Used Modules

-0.1500

-0.1000

-0.0500

0.0000

0.0500

0.1000

0.1500

0 100 200 300 400

Mo

men

t co

eff.

Ang. of Incidence [°]

Wind Moment Coefficient

Cmz

48

Appendix IV: Meshing (hsmsh/hslec)

Appendix V: Radiation/Diffraction Module (hsrdf)

49

Appendix VI: Mechanical Module (hsmcn)

Appendix VII: Quadratic Transfer Function Module (hsamg/hsqtf)

50

Appendix VIII: Wave Response Module (hspec)

51

Appendix IX: Roll Response Comparison • 30 deg

GMt [m] = 3 B crit.

Tn [s] = 63.8 2E+11

wn [rad/s] = 0.10


[deg]


[deg]


OrcaFlex - 10.2d 0.4 2.1

OrcaFlex - 10.3d 0.4 1.9



GMt [m] = 4.5 B crit.

Tn [s] = 44.313 3E+11

wn [rad/s] = 0.14


[deg]


[deg]


OrcaFlex - 10.2d 0.16 1.2

OrcaFlex - 10.3d 0.16 0.9




52

• 70 deg

• 90 deg


Tn [s] = 25.7 5E+11

wn [rad/s] = 0.24


[deg]


[deg]


OrcaFlex - 10.2d 0.01 0.3

OrcaFlex - 10.3d 0.01 0.1




GMt [m] = 3 B crit.

Tn [s] = 63.78 2E+11

wn [rad/s] = 0.10


[deg]


[deg]


OrcaFlex - 10.2d 0.16 2.9

OrcaFlex - 10.3d 0.18 2.1




Tn [s] = 44.3 3E+11

wn [rad/s] = 0.14


[deg]


[deg]


OrcaFlex - 10.2d 0.03 2.2

OrcaFlex - 10.3d 0.03 1.6





Tn [s] = 25.732 5E+11

wn [rad/s] = 0.24


[deg]


[deg]


OrcaFlex - 10.2d -0.06 0.8

OrcaFlex - 10.3d -0.06 0.6

OrcaFlex - 10.3d - Newman -0.06 -0.1

HydroStar - Time Generation -0.06 1.1


53

GMt [m] = 3 B crit.

Tn [s] = 63.78 2E+11

wn [rad/s] = 0.10


[deg]


[deg]

Pyhton Spectral 0.3 1.6

OrcaFlex - 10.2d -0.3 2.5

OrcaFlex - 10.3d -0.3 1.6




Tn [s] = 44.3 3E+11

wn [rad/s] = 0.14


[deg]


[deg]


OrcaFlex - 10.2d -0.2 2.2

OrcaFlex - 10.3d -0.2 1.3




Tn [s] = 25.7 5E+11

wn [rad/s] = 0.24


[deg]


[deg]


OrcaFlex - 10.2d -0.1 1.3

OrcaFlex - 10.3d -0.1 0.7



Documents

IMPACT ASSESSMENT ON 2nd ORDER ROLL MOTIONmonografias.poli.ufrj.br/monografias/monopoli10031833.pdf · 3. Carregamento Estrutural do Turret. I. Paulo de Tarso Themistocles Esperança