Global Ice Ridge Ramming Loads Based on Full Scale Data and Specific

Global ice ridge ramming loads based on full scale data and speciﬁc energy approach

Job Kramers echnische Universiteit Delft T

GLOBALICERIDGERAMMINGLOADSBASED ON FULL SCALE DATA AND SPECIFIC ENERGY APPROACH

Job Kramers

in partial fulﬁllment of the requirements for the degree of

Master of Science in Offshore and Dredging Engineering

at the Delft University of Technology, to be defended publicly on November 25th, 2016 at 14:30.

Sponsor: Sustainable Arctic Marine and Coastal Technology (SAMCoT) Research Centre

Student number: 4011120 Date: ?, 2016 Supervisor: Prof. S. Løset NTNU Prof. dr. ir. A. V. Metrikine TU Delft M. A. van den Berg TU Delft Thesis committee: Prof. dr. ir. A. Metrikine TU Delf Prof. S. Løset NTNU

An electronic version of this thesis is available at http://repository.tudelft.nl/.

ABSTRACT

This thesis analyses the loads that occurred during an ice ridge ramming experiment with the icebreaker Oden. Sea ice ridges are formed due to breaking and deformation of the ice cover. Wind, current, thermal expansion and Coriolis forces induce compression and shear forces onto level ice which can break the ice into rubble. The blocks of ice rubble are pushed together, forming a wall of broken ice in hydrostatic equilibrium. This wall of broken ice forced up by pressure is deﬁned as an ice ridge. In general ice ridges are long, nonsymmetrical, curvilinear features with a wide variability of sizes and shapes.

In Arctic regions, sea ice ridges are often used to calculate the design load in the absence of icebergs. Ice ridges also play a major role in icebreaker efficiency, since an icebreaker might need several ‘rams’ to break through an ice ridge. Ice ridge actions on icebreakers are not completely understood. Complex ice behavior under rapidly applied stress, and the complex geometries of the bodies in contact makes it a challenging research topic. The dynamic behavior of the vessel during the ramming can be used to make an estimate of the ice loads that occurred. This thesis analyses the ice load that occurred during a ridge ramming experiment that was performed with icebreaker Oden during the ODEN AT research cruise project in 2013. To advance our understanding in the global ice ridge ramming loads, two models were developed: 1) a simulation model using the Specific Energy Absorption (SEA) of mechanical crushing of ice to calculate the global ice loads, 2) a load identification model using full-scale data to determine the global ice loads.

The simulation model was developed to enhance the understanding of relevant physical phenomena and parameters. During this process, specific energy principles of crushing of ice were identified as a promising although relatively unknown method for impact dynamics into ice. The Specific Energy Absorption (SEA) of mechanical crushing of ice is defined as the energy per unit mass of crushed ice, necessary to turn solid ice into crushed (pulverized) material. Besides the SEA value, the penetration velocity, density of ice, and volume of crushed ice, are required to calculate the ice load. A contact model was developed to determine the load location and direction on the hull.

The icebreaker Oden is represented by a nonlinear mass-damper-spring system. Maneuvering theory is applied, which means that the hydrodynamic variables are estimated at one frequency of oscillation. In the simulation model, a known thrust force is applied on the vessel, making it move forward in open water, and then penetrate the ice ridge. The simulation model calculates the ice loads and vessel’s motions (i.e. accelerations, velocity, and displacement). The load identification model combines the Kalman filter and a joint input-state estimate algorithm to estimate the state- and excitation vector from acceleration, velocity and displacement data in 3DOF (i.e. surge, heave, pitch). The joint input-state estimate algorithm combines measured data with an estimate of the state of the system in a way that minimizes the error. The full-scale data analyzed in this thesis includes a profile of a multi-year ice ridge, vessel characteristics, acceleration data from a motion reference unit (MRU), GPS data, and propulsion data.

From the results of the load identification model, we conclude that the current combination of model and data does not provide sufficient information to estimate the global ridge ramming loads with high reliability. The main reasons for this are the absence of additional MRU(s), the low sample frequency of the MRU, the data uncertainty, and the simplified hydromechanics. However, the suggested approach to calculate the global ice loads is reliable as long as the data is valid. This is verified by recalculating the ice loads from the data (i.e. motions), generated by the simulation model. Results indicate that the specific energy approach can be used to simulate an impact of a vessel into an ice ridge, under assumption that the ice fails purely due to crushing. This assumption is only valid during the beginning of the impact, as other failure modes often start to dominate as the penetration of the vessel into the ridge increases.

Cover picture: https://upload.wikimedia.org/wikipedia/commons/f/f8/Antarctica_--_Oden_the_ Icebreaker_-m.jpg

iii ACKNOWLEDGMENT

I would like to express my gratitude to professor A.V.Metrikine and my supervisors M.A. van den Berg at Delft University of Technology, and professor S. Løset of Norwegian University of Science and Technology. I would like to give a special thanks to Hongtao Li, Thorvald Grindstad, and Ekaterina Kim for their help and guidance during my research.

Then I would like to take the opportunity to thank my friends and colleagues at NTNU and Fjordgata 7, who made my stay in Trondheim so much fun. Finally, I could not have done this without the encouragement and support of my family and friends.

Job Kramers, Delft 20 October, 2016

iv ABBREVIATIONS

BFS Bottom Founded Structures CEP Circular error probability CM Contact Model CP Contact Point CPP Controllable Pitch Propeller CFD Computational Fluid Dynamics COB Centre of Buoyancy COG Centre of Gravity COR Centre of Rotation COS Centre of Shape DOF Degree of Freedom DWT Deadweight tonnage FPP Fixed Pitch Propeller FS Floating Structures GNSS Global navigation satellite system GLONASS Global navigation satellite system GPS Global positioning system HMI Human Machine Interface IMU Inertial Measurement Unit LOA Length Over All LWL waterline length MRU Motion Reference Unit RMS Root mean square IACS International Association of Classiﬁcation Societies SEA Speciﬁc Energy Absorption

v SYMBOLS

a acceleration [m/s2] aii added mass i=1,2,3,4,5,6 see below a11 added mass surge [kg] a22 added mass sway [kg] a33 added mass heave [kg] a added mass roll [kg m2] 44 · a added mass pitch [kg m2] 55 · a added mass yaw [kg m2] 66 · Ac system matrix 2 A j,i wake area normal to flow at location i [m ] 2 AN nominal contact area [m ] 2 Aw wetted surface [m ] 2 Awp contact area below water plane [m ] b hydrodynamic damping i=1,2,3 [N s/m] ii · b nonlinear surge damping [N s2/m2] 11,NL · b nonlinear heave damping [N s2/m2] 33,NL · B beam [m] Ba,f aft,front beam [m] Bc system matrix 2 B j,i hull area at location i [m ] C damping matrix C f friction coefficient [-] C f p flat plate friction coefficient [-] C f r residual friction coefficient [-] Cow open water friction coefficient [-] Ct thrust coefficient [-] Cw wake fraction [-] d displacement [m] 2 d Abow,btm change of indentation (bow, bottom) [m/s ] d(t) measurement data vector D diameter [m] Dr draft [m] f (t) excitation vector [N] F force [N] Fx1,ice horizontal global ice load [N] Fx3,ice vertical global ice load [N] Fb buoyancy force [N] FH horizontal load [N] FN normal load [N] FV vertical load [N] g gravitational acceleration [m/s2] Gc influence matrix GM metacentric height [m] GZ righting stability arm [m] h height [m] hk keel height [m] hs sail height [m] HR ridge height [m]

vi SYMBOLS vii

I second moment of inertia of water plane area [m4] I moment of inertia around y axis [kg m2] y · Jc direct transmission matrix k hydrostatic spring stiffness i 1,2,3,4,5,6 [N/m2],[N m/r ad] ii = · k nonlinear hydrostatic spring stiffness i 1,2,3,4,5,6 [N/m],[N m2/r ad 2] ii,NL = · k35 heave-pitch coupling hydrostatic spring stiffness k53 heave-pitch coupling hydrostatic spring stiffness K stiffness matrix KQ Torque coefficient [-] L length [m] LB j,i hull length i [m] Lpp characteristic traveled length of the fluid [m] Lm middle body length [m] m mass [kg] M mass matrix M pitching moment due to ice loads [N m] x5,ice · M pitching moment due to thrust force [N m] x5,T · M stability moment [N m] s · n numbers of revolutions [1/s] P pressure [Pa] p(t) force time histories pav pressure averaged over area [Pa] P error covariance matrix PD,E,in,out Power. delivered, effective, in, out [W] P/D pitch ratio [-] Pi tchα pitch angle [deg] P Px1,x3,x5(t) total excitation force in direction x1,x3,x5 Q torque [N m] · Q process noise covariance matrix 3 Q j flow rate [m /2] r radius [m] R measurement noise covariance matrix R resistance [N] RB ice breaking resistance [N] RF friction resistance [N] Ri ice resistance [N] Row open water resistance [N] RR ridge resistance [N] RS ice submerging resistance [N] Rn Reynolds number [-] S sensitivity [%] Sa,v,d,p selection matrix SEA specific energy absorption [J/kg] T thrust force [N] Tx1 horizontal component thrust force [N] Tx3 vertical component thrust force [N] v velocity [m/s] ve entrance velocity [m/s] v j mean wake velocity [m/s] vk measurement noise V volume [m3] viii SYMBOLS

w width [m wk keel width [m] wk stochastic system noise ws sail width [m] W work [J] x1 surge, forward direction positive [m] x2 sway, portside lateral direction positive [m] x3 heave, upward direction positive [m] x4 roll, positive if starboard goes up [rad] x5 pitch, positive if bow goes down [rad] x5,mean pitch angle with horizontal plane in hydrostatic equilibrium [rad] x6 yaw, positive when rotating counter clockwise in horizontal plane [rad] xˆk k 1 a priori state estimate | − xˆk k a posteriori state estimate | αk keel angle [deg] αs sail angle [deg] β (bow) angle [deg] βOden bow angle Oden [deg] β0.7R hydrodynamic pitch [deg] ∆ displacement [kg] or [m3] 1 ² loading rate [s− ] η porosity [-] η efﬁciency [-] θa,f slope aft,front [deg] µ friction coefﬁcient [-] ν kinematic viscosity [Pa s] · ρ density [kg/m3] φ angle [deg] displacement [m3] ∇ ω natural frequency [Hz] CONTENTS

1 Introduction 1 1.1 Background and motivation ...... 1 1.2 Problem outline...... 2 1.3 Objective ...... 3 1.4 Research design...... 4 1.5 Reading guide...... 4 2 Ice Actions and Ice Resistance 5 2.1 Ice features, ice actions and ice resistance ...... 6 2.1.1 Ice features...... 6 2.1.2 Ice properties ...... 8 2.1.3 Design scenario ...... 8 2.1.4 Interaction geometry ...... 9 2.1.5 Failure mode...... 9 2.1.6 Ice resistance ...... 9 2.2 Ice ridges ...... 11 2.2.1 Geometry and morphology ...... 11 2.2.2 Physical and Mechanical properties ...... 13 2.3 Icebreakers ...... 14 2.3.1 Ice breaker examples...... 18 2.3.2 Ice and Weather Conditions ...... 18 2.3.3 Ice operations ...... 21 2.3.4 Ice resistance ...... 24 2.3.5 Icebreaker design ...... 30 2.4 Ice ridge loads...... 35 2.4.1 Load patch ...... 37 2.4.2 Pressure ...... 38 2.4.3 Specific energy principles ...... 39 2.4.4 Previous research into vessel-ice ridge impact phenomena ...... 42 3 Theoretical Background on Load Identification and Hydrodynamics 44 3.1 Introduction ...... 44 3.2 Joint input-state estimation method for linear systems ...... 44 3.2.1 Mathematical formulation ...... 45 3.2.2 Joint input-state estimation algorithm ...... 46 3.3 Hydrodynamics ...... 48 3.3.1 Axis convention ...... 48 3.3.2 Hydrostatics ...... 48 3.3.3 Potential coefficients...... 49 3.3.4 Maneuvering theory ...... 50 3.3.5 Propulsion Theory ...... 51 4 Full-scale data 54 4.1 Introduction ...... 54 4.2 Experiment log and pictures ...... 55 4.3 Vessel characteristics ...... 56 4.3.1 Design loads ...... 57 4.3.2 Hydrodynamic surge resistance ...... 58 4.3.3 Oden’s Hydrostatic Nonlinearity for Heave motions ...... 60 4.3.4 Heave’s nonlinear frictional damping due to fluid-structure interaction ...... 62 4.3.5 Overview hydrodynamical vessel characteristics...... 63

ix x CONTENTS

4.4 Ice ridge characteristics ...... 64 4.5 Kongsberg Seapath 320+ System ...... 65 4.5.1 Motion Reference Unit ...... 66 4.5.2 Global Navigation Satellite System ...... 67 4.5.3 Full-scale Data ...... 67 4.6 Propulsion data ...... 68 4.6.1 Propeller characteristics ...... 68 4.6.2 Full scale propeller data ...... 69 5 Numerical models 70 5.1 Simulation Model: Global ice ridge ramming loads based on specific energy approach ...... 70 5.2 General introduction to the load identification model ...... 72 5.2.1 Data Synchronization ...... 72 5.2.2 Model Dimension and axis convention ...... 73 5.3 GPS data Conversion ...... 73 5.3.1 GPS raw Flat earth coordinates (mast) ...... 73 → 5.3.2 GPS Flat earth coordinates (mast) GPS Flat earth coordinates (COG) ...... 73 → 5.3.3 GPS Flat earth coordinates (COG) Rigid body reference frame ...... 74 → 5.4 Contact Model ...... 76 5.4.1 Triangle Interaction Algorithm ...... 77 5.4.2 State Identification and Simulation ...... 77 5.4.3 Load Case Conditions ...... 79 5.4.4 Shape identification of penetration ...... 79 5.4.5 Rotation Based Algorithm ...... 80 5.4.6 Indentation Volume and Contact Point Identification ...... 82 5.4.7 Benefits and Limitations ...... 82 5.5 Set up of the load identification model ...... 84 5.5.1 Equations of Motion ...... 84 5.5.2 Propeller Thrust ...... 85 5.5.3 State estimate and force estimate for 2DOF and 3DOF system ...... 86 5.5.4 Linearizing nonlinear terms ...... 88 5.5.5 Model setup overview ...... 90 5.5.6 Filter parameters and tuning...... 91 5.5.7 Coanda effect ...... 92 5.5.8 Calculation of Coanda effect in a numerical model ...... 93 5.6 Sensitivity and Uncertainty Analyses ...... 95 5.6.1 Parameter uncertainty ...... 96 5.6.2 Parameter uncertainty of thrust force ...... 96 5.6.3 Parameter uncertainty of global ice loads ...... 97 5.6.4 Sensitivity and uncertainty analyses of the thrust force ...... 99 5.6.5 Sensitivity analyses of global ice loads ...... 99 5.6.6 Uncertainty of the global ice loads ...... 99 6 Results and discussion 102 6.1 Contact model ...... 102 6.2 Results of simulation model...... 102 6.3 Results of load identification model...... 104 6.3.1 Hydromechanical parameters in state space model ...... 105 6.3.2 Filter parameters and tuning...... 105 6.3.3 Thrust force ...... 106 6.3.4 Contact model ...... 106 6.3.5 State estimate ...... 107 6.3.6 Estimate of the global ice loads ...... 107 6.3.7 Sensitivity and uncertainty analyses of the thrust force ...... 110 6.3.8 Sensitivity analyses of the global ice loads ...... 112 6.3.9 Uncertainty analyses of the global ice loads ...... 115 CONTENTS xi

6.3.10 Decomposition of hydrodynamic forces in the vertical direction during ice ridge ramming experiment ...... 117 6.3.11 State of heave and the predicted downward ice load at end of impact ...... 118 6.3.12 The Coanda effect ...... 120 7 Conclusions and recommendations 122 7.1 Conclusions...... 122 7.2 Recommendations ...... 125 A Icebreaker examples 127 A.1 MSV Fennica ...... 127 A.2 Oden ...... 127 A.3 50 Let Pobedy ...... 128 B Appendix A: Experiment log and pictures 131 B.1 Vessel log ...... 131 B.2 360 camera ...... 131 C Kongsberg Seapath 320+ System Data 142 D Vessel Characteristics 146 D.1 Propeller characteristics ...... 146 D.2 Thrust coefficient algorithm ...... 152 D.3 Structural requirements...... 153 D.4 Technical Drawings ...... 153 D.5 Nonlinear heave damping...... 157 D.6 Potential coefficients of heave and pitch motions ...... 157 E Ridge Characteristics 160 F Simulation model paper 163 G Influence of linearizing nonlinear terms in numerical model 183 G.1 Set up of the analyses ...... 183 G.2 Findings ...... 183 G.3 Conclusion ...... 184 H Decomposition of hydrodynamic forces during ice ridge ramming experiment 190 I Uncertainty Analyses of thrust force 193 I.1 Sensitivity analyses of thrust force ...... 193 I.1.1 Parameter sensitivity to thrust ...... 194 I.2 Thrust Uncertainty ...... 198 J Sensitivity Analyses of global ice loads 200 J.1 Sensitivity to vessel displacement ...... 200 J.1.1 Discussion ...... 200 J.2 Sensitivity to Draft ...... 201 J.3 Sensitivity to Waterline area...... 201 J.4 Sensitivity to Waterline Length ...... 201 J.5 Sensitivity to meta centric height GML ...... 201 J.6 Sensitivity to rotational inertia Iy ...... 201 J.7 Sensitivity to added mass for surge ...... 202 J.8 Sensitivity to added mass for heave ...... 202 J.9 Sensitivity to added mass for pitch ...... 202 J.10 Sensitivity to heave-pitch hydrostatic stiffness coupling terms ...... 202 J.11 Sensitivity to potential damping for heave motions ...... 202 J.12 Sensitivity to potential damping for pitch motions ...... 202 J.13 Sensitivity to nonlinear surge damping b11,NL ...... 203 J.14 Sensitivity to nonlinear heave damping b33,NL ...... 203 J.15 Sensitivity to Thrust force ...... 203 J.16 Sensitivity to the contact point of ice-vessel...... 203 CONTENTS 1

J.17 Sensitivity to the heave data...... 204 J.18 Sensitivity to the surge acceleration data ...... 204 J.19 Sensitivity to GPS position ...... 205 K Uncertainty Analyses of global ice loads 207 L Results of numerical model 213 L.0.1 GPS data conversion ...... 213 L.0.2 Thrust force ...... 213 L.0.3 Coanda effect ...... 215 L.0.4 State estimate ...... 216 L.0.5 Estimate of the global ice loads ...... 221 L.1 Pitching moment ...... 226 L.2 Predicted downward ice load at end of impact ...... 226 Bibliography 228 Bibliography 228 1 INTRODUCTION

1.1. BACKGROUNDANDMOTIVATION There is a great economical, and therefore political, interest in the Arctic. Land and territorial waters in the Arctic belong to one of the following countries: Russia, Canada, Norway, Denmark (Greenland), and the United States. Many of the countries around the Arctic region try to claim as much of the area as they can. Mostly because of the expected large reserve of undiscovered oil and gas resources in the Arctic area underneath the ice pack (Holm, 2012) [1].

There are two commercial ﬁelds where ice encounters are common. These areas are the merchant shipping industry and the offshore industry. In the offshore industry the world’s increasing energy demand and deplet- ing fossil fuels reserves increase the pressure to develop hydrocarbon ﬁelds in the Arctic areas. Apart from the drilling- and production- platforms many different types of ships are needed such as icebreakers, supply vessels, support vessels and oil tankers.

Both areas are affected by the melting ice in the Northern Arctic areas, where the ice condition deteriorate. The melting is expected to continue for decades. This impacts the Oil and Gas industry by increasing the fea- sibility of hydrocarbon recovery, hereby creating a demand for more structures and vessels in the Arctic areas. In the shipping industry merchandising ships can sail through a broader range of arctic regions. Melting of the Arctic ice caps will raise the commercial viability of the Northern Sea Route, which is located at the north coast of Siberia, connecting North-East Asia with North-Western Europe. This Northern alternative represents a sizeable reduction in shipping distances, and a decrease in the average transportation days compared to the currently used route. It will lower transportation costs, save fuel, decrease CO2 emissions, and save transportation time which may effectively force supply chains in industries between East Asia and Europe to change. The economic impact can be huge, hence an increased interest in Arctic Technology.

Besides the merchant shipping industry and the offshore industry other fields that benefit from studies about sea ice in the Arctic are: climate studies, fisheries (the ice edge areas are very productive), exploration of mineral deposits in the Arctic, the use of icebergs as a source of fresh water (Holm, 2012) [1].

1 2 1.I NTRODUCTION

1.2. PROBLEMOUTLINE Sea ice ridges are formed due to breaking and deformation of the ice cover. Wind, current, thermal expansion and Coriolis forces induce compression and shear forces onto level ice which can break the ice into rubble. The blocks of ice rubble are pushed together, forming a wall of broken ice in hydrostatic equilibrium. This wall of broken ice forced up by pressure is deﬁned as an ice ridge. The extent and thickness of ridges is related to wind and ocean dynamics. In areas where wind has a predominant direction, more dynamic effects are created, leading to more ridging. In general ice ridges are long, nonsymmetrical, curvilinear features with a wide variability of sizes and shapes.

In Arctic regions sea ice ridges are often used to calculate the design load in the absence of icebergs (Blanchet, 1998) [2]. It is essential to know the likelihood and characteristics of ice ridges that may be encountered in a certain Arctic area. Subsequently, the ice action of an ice ridge on a structure must be determined. Ice going vessels like icebreakers are one of the structures that encounter ice ridges. Ice ridges also play a major role in icebreaker efficiency, since an icebreaker might need several ‘rams’ to break through an ice ridge. Ice ridge actions on icebreakers are not completely understood. Complex ice behavior under rapidly applied stress, and the complex geometries of the bodies in contact makes it a challenging research topic. To acquire more insight in ice loads, full-scale and model test ice ridge ramming experiments are needed. The dynamic behavior of the vessel during the ramming can be used to make an estimate of the ice loads that occurred. To ensure a dependable ice load identification a profound understanding of all influencing components is required. Key components influencing the ice load are the ice ridge characteristics, vessel characteristics and hydromechanics.

Sea ice ridges can vary greatly in geometry, morphology and its physical and mechanical properties. Besides diversity between ice ridges, the ice ridges themselves can have significant variations in ice strength due to the inhomogeneous characteristics of ice. These variations in ice strength are found all across the ice and depend on crystallography, temperature, salinity and porosity. It is difficult to precisely determine all relevant ridge characteristics, thus parameter uncertainties may be significant. This complicates research into ice ridge loads. Several vessel characteristics are important during a sea ice ramming experiment. In dynamical actions like a ridge impact, mass is important. Major component here is the vessel mass. However, also ridge mass and hydrodynamic added mass due to vessel’s movements are expected to have significant influence. Vessel geometry, especially hull shape, will have an big impact on occurring ice actions (Jones, 2004) [3] (Riska, 2010) [4]. Bow location at which the impact occurred was found to be an important factor in the peak stresses in the vessel (Chen, 1990) [5]. The bow impact location can therefore influence the ice loads significantly. Propulsion systems will generate the thrust on the vessel. This thrust is the forward force, and generates the kinetic energy, which is required to break ice. However, thrust cannot be directly measured. It depends on several propulsion system components (hence vessel characteristics) and hydro mechanical phenomena (propulsion theory).

Studies show that it is difficult to conclude which ice failure mode occurs during an ice ridge ramming. Sev- eral ice failure modes are probable, and every failure mode will induce a different ice action. The ’real’ load magnitude gets characterized by the governing failure mode, or modes, occurring at each moment of time. Several failure modes might occur at the same time. The extend a failure mode contributes to ice failure, therefore to load magnitude, is difficult to validate or predict. Therefore developing theories to predict ice ridge loads is difficult.

Ice ridge action cannot be determined without understanding the hydromechanical behavior of the vessel. Relevant hydromechanical and hydrodynamical components must therefore be identified. Other phenomena might complicate the ice ridge load identification. Based on several studies the following phenomena are also identified as potential (meaningful) influences: size effect, ice edge spalling effect, internal damping, hull friction, snow cover, load patch identification, reliability and accuracy of measurement systems that might be used. (Loset et al. 2006) [6], (Riska, 2010) [4], (Chen, 1990) [5], (Palmer and Croasdale, 2013) [7], (Zou et al. 1996)[8].

Lastly, it must be emphasized that data analyses brings its own challenges. First corresponding key challenge is identification of parameter values. Not all parameters can directly be identified or, within the scope of this 1.3.O BJECTIVE 3 thesis, easily determined. An example are potential coefficients (e.g added mass, added damping), which requires potential theory software to get the desired parameters, which itself requires a detailed geometrical 3D model. As such it will require either an author’s investment in learning several types of software, or making justified assumptions which can be based on for example literature, crew experience, and full scale-data. Second key challenge is uncertainty analyses. This is challenging in the sense of identifying uncertainties of used parameters, which often are unknown, and also how each parameter contribute to the global load’s uncertainty. Identifying the result’s uncertainty is essential, as it determines to which extend that result can be trusted.

1.3. OBJECTIVE The aim of this thesis is to improve the understanding of global ice loads during the ramming of sea ice ridges. To grasp this phenomena, a model must be developed that enables ice load determination of an ice ridge during vessel impact based on full-scale ramming data, beforehand determined ice ridge properties, vessel properties and hydro mechanical effects. This model will be used to validate the multi-body ice-structure interaction simulation model developed at NTNU. As the name implies this body is based on multi-body dynamics. It consists of rigid ice bodies affecting one another and taking into account all inﬂuencing systems, like wind, hydromechanics, etc. This model will be used to simulate the ice structure interaction.

The general research question of this thesis is deﬁned as:

• Given full-scale data of an ice ridge ramming experiment, what were the global ice ramming loads?

Full-scale ramming data is effectuated by the ODEN AT research cruise Project 2013, made possible by Statoil and SAMCoT. Oden is a Swedish icebreaker built in 1988, which has participated in several other scientiﬁc expeditions. The data contains a proﬁled ice ridge, MRU data, GPS data, and propulsion data. However, before the general research question can be answered, it is critical to understand the governing physical phenomena that occur during ice ridge ramming impacts. Hence an improved understanding of arctic engineering, particularly into ice ridges and icebreakers, should be obtained. Furthermore, it will be a challenge to calculate the global loads using full-scale data. Hence, the following subquestions originated:

• Given the experiment’s Arctic environment, which ice features are expected to be encountered, and what are the main factors governing their ice actions?

• What are important icebreaker design aspects considering ice operations and environmental conditions?

• What is an ice ridge, and how can their corresponding failure load be calculated?

• Given a limited number of acceleration-, velocity-, and displacement-data, how can de state and excitation load of a structure be calculated?

Due to literature reviews and researcher’swork during the thesis, a more comprehensive understanding about relevant theories was acquired. During this process, speciﬁc energy principles of crushing of ice were identi- ﬁed as a very promising although relatively unknown method for impact dynamics into ice. It were Professor Sveinung Løset and Post-doc Ekaterina Kim, who guided the author towards this method as an optional approach to simulate ice ridge ramming loads. As such another subquestion was formulated:

• What are speciﬁc energy principles for crushing of ice and can they be used to calculate global ice loads during ramming of a sea ice ridge by a vessel? 4 1.I NTRODUCTION

1.4. RESEARCHDESIGN Before a model can be developed a detailed understanding of the interfacing systems is required. Therefore this thesis first presents and overview of ice features, ice properties, ice actions/resistance, icebreakers, ice ridges characteristics, vessel characteristics, theoretical background on load identification methods, and hydromechanical influences. These theories can be characterized as state of the art or literature research.

Under recommendation of my supervisors Professor Sveinung Løset and Marnix van den Berg, a simulation model was developed simultaneously with the literature review. Reasoning was, that simulating an ice ridge ramming impact would enhance understanding of relevant physical phenomena and parameters. In the simulation model, a known thrust force is applied on the vessel, making it move forward in open water, and then penetrate the ice ridge. In the simulation model, the calculated movements (i.e. accelerations, velocity, and displacement) of the vessel are saved as data. These simulated motions are later used by the load identiﬁca- tion model (with and without added noise), to recalculate the vessel’s state and excitation loads. Hereby, the simulation model validates the load identiﬁcation model. Matlab was chosen as programming software.

Several failure mode options are available to implement in the simulation model such as bending failure, crushing failure, plugging, etc. Investigating all options was out of the scope of the thesis, hence an assumption on failure mode was made. As noted in section 1.3 the speciﬁc energy principles of crushing of ice was chosen as a method to simulate the global ice ridge loads during ramming of an ice ridge. This approach implies a pure crushing failure assumption. To verify the methodology used the results will be compared to the full-scale data measurement.

Given specific energy principles, the simulation model requires calculation of indentation/penetration volume into the ridge (why, how, etc will be explained in the thesis). A consequence of this was the need for a contact model, that can calculate volumetric indentation, and ideally the contact point or the load patch, for the ice-structure interaction. To the author this contact model is one of the most challenging parts, as Matlab appears not to be the most ideal program for this seemingly easy task. A beneficial consequence of the contact model is that it is directly implementable in the load identification model. As such it was decided to finish the contact model.

The load identification model is developed after the simulation model. Given the full-scale data, the contact model, and the theory from literature review, this should be possible. The chosen approach to calculate the state and excitation load in the load identification model, is validated before the full-scale data is taken into account. This answers the following subquestion: "Given a limited number of acceleration-, velocity-, and displacement-data, how can de state and excitation load of a structure be calculated?". This is achieved, by implementing the simulated motions from the simulation model into the load identification model. A parameter sensitivity study is required to justify model assumptions, and to investigate parameter influences. Furthermore, an uncertainty analyses is executed to identify the uncertainty of the global ice loads and thrust force.

1.5. READINGGUIDE Chapter 2 presents a study into ice features and ice resistance. Chapter 3 presents a theoretical background on load identification and hydrodynamics. The full-scale data, including the calculated hydromechanical parameters, are presented in Chapter 4. Based on the theory and data presented in the chapters before, Chapter 5 presents the simulation model based on specific energy approach, and the load identification model based on full-scale data. Chapter 4 also includes a sensitivity and uncertainty analyses of the thrust force and global ice loads. In Chapter 6, the results are presented. Lastly, the conclusions and recommendations of this thesis are presented in Chapter 7. 2 ICE ACTIONSAND ICE RESISTANCE

The Arctic covers a large area where many diverse environments and ice scenarios are found. The types of sea ice features that are found in a certain area depend on climate, oceanography, interaction with seabed and interaction with land ice. Building offshore structures in the Arctic is in a lot of aspects the same as in other offshore areas. However, the low temperatures, the darkness in winter, and the remoteness, make the work here more challenging compared to most other locations. For humans the Arctic environment is harsh, making the outside operations more difﬁcult. The presence of ice will require offshore structures to be able to withstand ice actions. Another important engineering aspect is the environmental impact. The Arctic environment is much more sensitive to external inﬂuence such as oil spills and other hazards. It is of the utmost importance to protect this fragile environment. It is an engineers responsibility to make sure their design is safe and environmentally friendly. When an offshore structure is designed it is essential to know what type of ice can be found in the operational area, and what kind of ice actions they might induce.

Ice can impose large actions on rigs, platforms and vessels. It can also make access by vessels difﬁcult or impossible, like for support or supply vessels. The extend to which an offshore structure has to interact with ice depends on location and purpose of the structure. Some structures might simply avoid the ice, while others must be designed in such a way that they can operate without being damaged by the ice. In the case of ice going vessels, it is more accurate to talk about ice resistance than about ice actions. This because in the case of a vessel, it is the vessel imposing an action on the ice, rather than the ice inﬂicting an action on the vessel.

Chapter 2 concerns a study into ice actions and ice resistance. The information presented in this chapter gives readers a systematic understanding of the ice ridge ramming phenomenon by an icebreaker. Based on literature studies it was concluded that before this phenomenon can be understood, a profound understanding of the following key components must be achieved:

1. Ice features, ice actions and ice resistance. 2. Ice ridges 3. Icebreakers

In this chapter the mentioned topics are elaborated in separate sections. Section 2.1 presents background information on ice features, ice actions for general offshore structures and ice resistance for vessels. In this section an overview is given of the relevant components and parameters with respect to offshore structures in the Arctic. Therefore, it may be interesting to readers who only wish to know the basics of ice loads. Fur- thermore, this overview helps illustrate why components are further investigated in this thesis. For example, ice ridges and icebreakers should be looked into separately before ice ridge loads on an icebreaker can be understood, and are given in section 2.2 and 2.3 respectively. Section 2.4 shows a study of previous research into ice ridge loads on vessels.

5 6 2.I CE ACTIONSAND ICE RESISTANCE

2.1. ICE FEATURES, ICEACTIONSANDICERESISTANCE In this section the basics regarding ice features and ice actions is presented. The aim of this section is to give a description of ice features that might be encountered in the Arctic, followed by the main factors governing ice actions. This thesis does not go into details with respect to all ice actions, but rather presents an overview of the possible ice actions, followed by a more extensive description of the ice actions related to the thesis research.

It is necessary to ﬁnd the action required to fail an ice feature, given the structure form, the ice properties, and environmental conditions. This requires that all types of ice interaction and failure modes should be considered. The governing ice action, i.e. ice action with the lowest determined action, is assumed to lead to ice failure. According to ISO 19906 [9] the ice actions can include:

1. Static, quasi-static, cyclic and dynamic actions (Environmental Loading and Accidental Loading). 2. Cyclic and dynamic actions that can cause structural fatigue, liquefaction and personnel discomfort. 3. Spatial actions such as rubbling, pile-up, ride-up and similar ice behaviour that can hinder operations.

Two types of ice actions can be identiﬁed: global ice action and local ice action (Løset et al. 2006) [6]. A global ice action is the action exerted on the whole structure. Local ice action is exerted locally, and are important for the stability and the strength of a structure. Th effective pressure and the nominal contact area are often used to characterize the global action. The effective pressure is the pressure averaged over the nominal contact area at a time instance corresponding to the maximal level of action. The nominal contact area is the product of ice thickness and structure width. Using these deﬁnitions is often a convenient way of dealing with global ice loads as the approach to determine the ice action becomes easier. However, it is an idealization and thus does not correspond to reality: in general the pressure will differ on the location, and the real contact area will not be equal to the nominal contact area. The maximum local ice action will only occur at a limited part of the contact area and therefore, is characterized by local pressure and local contact area.

The structure’s shape and size, the ice conditions, and the environmental driving actions, can result in a number of different interaction scenarios, and failure modes. Together they determine the ice action. According to Løset et al. (2006) [6] the factors inﬂuencing the scenarios can be represented by Figure 2.1. However, Fig- ure 2.1 is developed for general offshore structures, and is not directly applicable to vessels. This is elaborated on below.

In the arctic three main types of offshore structures can be identified: bottom founded structures (BFS), floating structures (FS) and vessels. There is a fundamental difference between vessels and BFS/FS: vessels are the ones impacting ice (vessel action on ice) where BFS/FS are impacted by ice features (ice action on structure) A vessel possesses most of the kinetic energy while the ice is approximately motionless during ice-structure interaction, whereas BFS/FS are approximately motionless and ice possesses most of kinetic energy during ice-structure interaction. As a consequence, there is a big difference in the ice-structure interaction scenarios for vessels compared to BFS/FS. Therefore, it is common to talk about ice resistance instead of ice action, in the case of a vessel. This will be elaborated upon in section 2.1.6. In Figure 2.2 the floating drilling unit Prirazlomnaya and the vessel Mikhail Ulyanov is shown, it illustrates the different interaction scenarios in a typical ice field between a vessel and a FS.

2.1.1. ICE FEATURES The ﬁve ice features from Figure 2.1 are described as follows:

1. Level ice: A region of ice with relatively uniform thickness. Also called sheet ice. 2. Rafted ice: Ice feature formed from the superposition of two or more ice sheet layers. 3. Ice ridge: A linear feature formed of ice blocks created by the relative motion between ice sheets. 4. Ice rubble: Ice fragments or small pieces of ice that cover a larger expanse of area without any particular order to it. 5. Iceberg: Originates from land ice: glacial or shelf ice that has broken away from its source. It is characterized with a greater than 5 m freeboard. Icebergs can be freely ﬂoating or grounded and are sometimes deﬁned as tabular, dome, pinnacle, wedge or blocky shaped. 2.1.I CE FEATURES, ICEACTIONSANDICERESISTANCE 7

Figure 2.1: The major parameters affecting the ice action [6]

Figure 2.2: Mikhail Ulyanov oil vessel arrives at Prirazlomnaya to load oil from platform. The picture shows how ice is drifting against the floating structure Prirazlomnaya and how Mikhail Ulyanov is moving itself through the ice. The areas with large and small ice floes, rubble, ridges and open water, illustrate the large diversity of an ice field [10].

The ice features need to be further speciﬁed. ISO 19906 [9] suggests the use of Figure 2.3. The additional required information about an ice feature is:

• First/multi-year ice: Multi-year ice has distinct properties that distinguish it from first-year ice, based on processes that occur during the summer melt. Multi-year ice has survived at least one summer melt and is stronger than first-year ice. • Dimensions: Both in horizontal plane and the ice thickness. • Coverage: A measurement of the area of the ocean where there is at least some sea ice. One single piece of ice will interact differently with a structure than an ice field with several ice pieces, making coverage a factor that must be taken into account. • Pressure/confinement: Influences the maximum stress ice can endure before it breaks. • Velocity: Relevant with respect to kinetic energy of an ice feature. 8 2.I CE ACTIONSAND ICE RESISTANCE

Figure 2.3: Common physical parameters used to describe interaction scenarios [9].

2.1.2. ICEPROPERTIES The strength of the ice (compressive, flexure, tensile and shearing) depends on the ice properties: 1. Crystallography: Size of ice grains influences the strength of the ice. The smaller the grains, the stronger the ice. 2. Temperature: Ice gets stronger at colder temperature, but also more brittle. 3. Salinity: Influences mainly porosity. 4. Porosity: The lower porosity, the stronger the ice. 5. Surface tension: influences friction and adhesion, which can result into large ice actions on a structure.

2.1.3. DESIGNSCENARIO The ice action is a result of an ice feature impacting the structure and depends on the mass, initial velocity and properties, the environmental conditions, and the structure’s form and size. Different scenarios of interaction have to be considered in a design scenario. The general considered scenarios are: 1. Limit stress: The limit-stress mechanism is when ice failure processes adjacent to the structure (compressive, shear, tensile, flexure, buckling, splitting) govern the ice action. This means that the ice fails at the structure. 2. Limit momentum: Also called the limit-energy mechanism, occurs when the kinetic energy (or momentum) of the ice feature limits the ice action. The ice’s kinetic energy is consumed and ice feature stops. No or little failure of ice, no or little penetration of the structure into the ice. Examples include large isolated floe (e.g. multiyear ice in summer), ice island or iceberg impacts. Often occurs with structural damage. 3. Limit force: The limit-force mechanism occurs when actions from winds, currents and the surrounding pack ice on an ice feature in contact with the structure are insufficient to fail the ice against the structure. The ice field in front of the structure will therefore come to a rest. Instead of failing at the structure ice can fail far away from the structure, for example by ridging or rafting 4. Splitting: When ice stresses cause splitting of ice, hereby limiting the maximum ice action. An example is a light house with sidewards moving ice around it. 2.1.I CE FEATURES, ICEACTIONSANDICERESISTANCE 9

2.1.4. INTERACTIONGEOMETRY The structure’s geometry incorporates the type of the structure, its shape, its size. The size of the structure is one of the key factors governing ice actions. Several studies show that ice tends to fail at relatively lower loads for larger contact areas (Løset et al. 2006) [6], (Palmer and Croasdale, 2013) [7]. The aspect ratio is sometimes used in relation to interaction geometry, and can be deﬁned as the structure’s diameter divided by the ice thickness. Scenarios with a large aspect ratio tend to have different failure modes then opposed to scenarios with a low aspect ratio. For example, splitting seems to occur at high aspect ratios.

1. Single vs Muli-leg platforms: Offshore structures can have several legs. In multi-leg platforms the legs influence each other. The extend of influence depends on distance between the legs, also called spacing. Four legs do not necessarily experience the same ice load as four times a single leg would. If spacing is higher than six times the leg diameter a stand alone approach is used. 2. Out-of-plane shape: Structures with vertical walls near the waterline will experience higher loads than sloping structures, because sloping structures enhance ice failure due to bending, which typically occurs at lower loads. There are situations where a sloping structure is disadvantageous; i.e. construction is more complicated and it can create problems when mooring ships [6]. 3. Water depth: Ice might pile up when water depth is low. 4. Waterline shape: Can influence ice loads; round pile vs square pile. It must be noted that adfreeze should be taken into account as well. 50-60% additional adfreeze diameter due to high tides low tides effect has been experienced on bridge piles.

2.1.5. FAILUREMODE Different types of failure take place depending on the strengths level, stress distribution, ice velocity and structure shape:

1. Creep: Slow moving ice (1-3mm/s) ﬂows around the structure. Perfect contact between ice and structure. Ice yields continiously around the structure without forming of cracks in the ice. Creep is most relevant for narrow structures. 2. Crushing: Typical for high velocity interaction scenarios. Micro cracks (damage) followed by extru- sion of particles. Non-uniform pressure distribution on the contact area. Often together with splitting. Crushing tends to give the highest ice loads. 3. Bending: Typically for sloped structures. 4. Buckling: Typically for thin ice and wide structures. Often connected with radial or circumferential crack formation. 5. Spalling: A formation of out-of-plane horizontal cracks which grow away from the contact zone and divide the ice sheet into layers. The higher the velocity, the smaller the spalling pieces will be. Spalling is conﬁnement related. 6. Splitting: Requires a minimum stress level trying to split the ice and must be initiated (i.e. a crack). Splitting is associated with tensile failure. Radial cracks are a form of splitting: Radial cracks form at stress concentrated areas; i.e. corners. They develop especially at high aspect ratios. For rectangular structures they tend to radiate from the corners. Circumferential cracking occurs due to out of plane bending moment by eccentric loading.

2.1.6. ICE RESISTANCE Figure 2.1 is used for general offshore structures in the Arctic, which were identified as BFS, FS and ships. BFS and FS are often intended to stay at one location, e.g. as a drilling rig or production platform. FS can stay in place by using mooring systems or dynamic positioning systems. When an ice action occurs, the ice feature typically impacts the structure, i.e. the ice imposes an ice action. A vessel at the other hand, for example a icebreaker, are not intended to stay at a fixed location, but sail through the icy waters. Highest ice loads occur when the vessel is moving, not when it is kept at a certain position. The vessel imposes the action and ice loads are represented as a measure of ice resistance. As a consequence the overview presented in Figure 2.1 is found to be invalid for vessels. The question arises if the overview can be modified, to achieve an overview of the parameters contributing to ice resistance on vessels. Although extensive knowledge in vessel’s ice resistance is available, a general overview similar to Figure 2.1 for ice going vessels has not been developed. Therefore, the author constructed an overview with the major parameters affecting the ice resistance on an vessel, presented in Figure 2.4. 10 2.I CE ACTIONSAND ICE RESISTANCE

The major components affecting the ice resistance of vessels are concluded to be: ice features, ice properties, vessel properties (including interaction geometry) and failure modes. The same major ice features and ice properties are identiﬁed, with one exception: icebergs. Icebergs do not contribute to ice resistance. It is practically impossible to design a vessel to withstand icebergs, and a vessel will always try to avoid contact with an iceberg. However, several vessels have been sunk by an iceberg, like the famous Titanic. A designer must be be aware of the risk of icebergs.

For offshore structures (BFS and FS) the limiting mechanisms may be used to characterize two major inﬂu- ences on the ice action. First of all, to describe what type of mechanism actually limits the ice action. Sec- ondly, it can be used to characterize where the ice fails: at contact of structure and ice, away from structure, or not at all. These limiting mechanisms are not relevant for vessels, because there is no ice action which determines how the ice fails. Distinction of the location where ice fails is still very relevant. For example crushing occurs at the structure and bending failure does not occur at the structure. This distinction is already found in the failure modes. Therefore, it is concluded that limiting mechanism can safely be excluded from the overview.

Vessel properties will greatly influence the ice resistance. This includes vessel dimensions, mass, velocity, interaction geometry, and hydromechanical characteristics. The most important dimensions are the hull shape and size. With respect to interaction geometry, vessels do not have legs and therefore will not influence ice resistance. Neither will ice resistance depend on water depth. An important component, especially in highly dynamical scenarios, is added hydrodynamic mass. When an ice feature is impacted by a vessel, it has to deal with the complete inertia depending on vessel mass and hydrodynamic added mass. Also radiated waves from the vessel may have significant influence on ice resistance, especially for small ice features.

In the design of a ship, the failure modes creep and buckling can be ignored, they will not give the design load.

Figure 2.4: Major parameters affecting the ice resistance for a vessel.

This thesis concerns an icebreaker ramming an ice ridge. Using Figure 2.4 we can now identify the key parameters affecting the ice ridge load. The next step is to identify how all relevant parameters contribute to the ice resistance. This requires a profound understanding of ice ridges (the ice feature) and its ice properties. Fur- thermore, it is necessary to have an thorough understanding of the relevant structure: icebreakers. Especially how design parameters affect ice resistance. A comprehensive study into ice ridges is given in section 2.2. This includes ridge formation, morphology, physical and mechanical characteristics. Section 2.3 presents an extensive description of icebreakers. 2.2.I CERIDGES 11

2.2. ICERIDGES Sea ice ridges are formed due to breaking and deformation of the ice cover. Wind, current, thermal expansion and Coriolis forces induce compression and shear forces onto level ice which can break the ice into rubble. The blocks of ice rubble are pushed together, forming a wall of broken ice in hydrostatic equilibrium. This wall of broken ice forced up by pressure is deﬁned as an ice ridge. The extent and thickness of ridges is related to wind and ocean dynamics. In areas where wind has a predominant direction, more dynamic effects are created, leading to more ridging. In general ice ridges are long, non-symmetrical, curvilinear features with a wide variability of sizes and shapes.

In ice ridges three distinguishable parts are usually described: a sail, a consolidated layer and the rubble. A typical cross section of a first-year ridge is given in Figure 2.5. The sail is composed of blocks that accumu- late above the water level when ice floes fail against each other. These bocks pile up and freeze together by contact. Below the sail, and under water level, the consolidated layer is found. Blocks that initially pile up underwater form cavities, which fill up with water. As the season progresses, the water freezes in these cavities, contributing to the continuous consolidation of the ridge. This consolidated layer grows downward. The rubble below the consolidated layer consists of loose blocks partially refrozen together, with water trapped in between.

Figure 2.5: Cross section of ﬁrst-year ice ridge [11]

2.2.1. GEOMETRYANDMORPHOLOGY Sea ice ridges are a form of mechanically deformed ice. The distinction between ice ridges and level ice is often obvious since ridges are substantially thicker than level ice, with keel depths going up to 50m in the central Arctic. A 100 year keel depth of 37-41 meter was identified in Fram strate. This distinction between ridge and level ice becomes harder for old ridges. First year level ice is relatively homogeneous. Old level ice has substantially higher spatial ice thickness variations due to the inhomogeneous horizontal melt in the summer. A rubble field, consisting of accumulations of broken ice covering a large area, can be distinguished from a ridge by the ice ridge’s curvilinearity, i.e. an rubble field does not require a linear or sinuous linear appearance.

Distinction between first-year ice ridges and old ice ridges is commonly made since their differing characteristics. Second year and multi-year ice ridges both belong to the category old ice ridges. The definition of when a first-year ice ridges becomes a second-year ice ridges tends to differ in the literature. The World Me- teorological Organization states that a first-year ice survived only one winter, a second year ice ridge only one summer. The Canadian Ice Community (CIC) defines ice existing after 1 October as old ice. This paper defines a second-year ice ridge as a ridge that survived at least one summer; the transformation happens when the ridge starts to refreeze after summer melt. As a consequence multi-year ice ridges tend to have more or 12 2.I CE ACTIONSAND ICE RESISTANCE less a completely consolidated layer, whereas first-year ice ridges have a significant amount of rubble. This has significant implications on ice ridge morphology, geometry and ice ridge loads, hence their mutual distinction.

In geometrical sense the key characteristics of an ice ridge are keel depth, sail height, keel width, keel shape and block thickness. The key geometrical characteristics are visualized in Figure 2.6. Research showed that level ice thickness around the ice ridge can be used to approximate average ice ridge thickness (Høyland [12]). For ﬁrst-year ridges the consolidated layer is around 1.2-2.1 times as thick as the surrounding level ice.

Figure 2.6: Key dimensions of a ﬁrst-year ice ridge according to Strub-Klein (2012) [11].

The keel depth is the main ice ridge characteristics. It is the most distinguishable part of an ice ridge compared to level ice. Furthermore the keel generally contains a large part of the total ridge mass. The keel of multi-year ice ridges is broader and more ’rectangular’ in shape. Keel depths vary a lot and depends on location. According to Kvadsheim (2014) [13] the average keel depth in Fram Strait is 7.7 meter. Furthermore Kvadsheim stated that the probability of encountering ice ridges deeper than 20 m was 0.15%. It must be emphasized that Fram Strait has heavy ridging conditions due to ice inﬂow from the arctic by currents. Wind and currents have extensive inﬂuence on ice conditions. A visualization of the average multi-year ridge is given in Figure 2.7 by Timco and Burden (1997) [14].

Figure 2.7: Features of an average Multi-year ridge [11]

The sail height is not that important as a characteristic itself, but it does give valuable information about the keel depth: the keel depth can be approximated by sail height ratios. This sail height ratio for ﬁrst-year ice ridges is approximately 4-5 and for multi-year ice ridges around 3-4. Keel width is around 2-3 times the sail width. A higher air volume in the sail conﬁrms the ice is also formed from rain and refrozen snow. First-year ice ridges have an sail angle around 35-40 degrees and a keel angle around 32 degrees. Block sizes are measured in a wide range, with thicknesses around 0.2-0.6 meter and lengths of 0.6-4 meter (Sinha and 2.2.I CERIDGES 13

Shokr, 1994) [15], (Løset et al. 2006) [6], (Bruneau, 1996) [16]. Multi-year ice ridges are studied less intensively. According to Strub-Klein (2012) [11] ridge cross-sectional geometry can vary greatly along the length of a ridge.

The ridge structure keeps evolving during the seasons. Study showed that in the lifespan of first-year ice ridges three phases can be identified: 1) Initial phase, consisting of rubble with internal temperature differences with formation of freeze bonds, 2) Main phase, an consolidated layer can be identified which continues to freeze, 3) Decay phase, where the ridge melts due to warm environment (Høyland) [12], (Høyland, 2002) [17]. The most important processes responsible for this evolution are the consolidated layer growth and the erosion from surrounding currents and climate. Several studies in consolidation due to thermodynamics of ice ridges have been made, for example by Marchenko (2008) [18] and Høyland (2002) [17].

It must be noted that on top of a ridge snow can be found. Snow works like a thermal insulator and can greatly inﬂuence the ice growth/melt. Furthermore snow affects the friction between ice and structure. There are no rigorous ways of taking snow cover into account. Generally the snow load is represented by an equivalent ice thickness around 1/3 of the snow thickness.

2.2.2. PHYSICALAND MECHANICALPROPERTIES The physical properties of an ice ridge are permeability, temperature, salinity, density and ice texture in time and space [12]. The horizontal spatial variation is bigger in old ridges. Temperature changes in the cross section of a ridge. In the consolidated layer the temperature normally increases towards the bottom, with a bottom temperature equal to the sea water temperature. The keel has a constant temperature, equal to the surrounding seawater. This makes the consolidated layer identiﬁable with temperature measurements. In summer the top temperature is increased, causing melting of ice and increased brine drainage, enhancing the desalination process. This summer melt is responsible for the decreased permeability of old ridges. Dur- ing the ridge formation process ice blocks are rotated randomly, resulting in more random grain orientation in ice ridges compared to level ice. The ice texture evolves during the ridge life containing a mixture of granular and columnar ice. Ice density dependents on temperature. Cold temperatures make the ice harder and more brittle.

First-year ice ridges have a typical salinity of 4-8 ppt, and a keel porosity between 0.25 and 0.40. Multi-year ice ridges have a more or less completely consolidated layer with substantially lower salinity. In ﬁrst-year ice ridges the rubble is often characterized through a cohesion and the friction angle in a Coulomb-Mohr con- text. In situ measurements indicate an ice cohesion for the ice blocks of 30-60 kPa (Weiss, 2013) [19], which corresponds to the 50 kPa used by Shavrova (2008) [20]. Shavrova used a friction angle of 30 degrees. How- ever, since a ridge is far from pure ice, values for rubble are lower. Results indicate values in the range of 0-20 kPa for the cohesion and 8-70 for friction angle (Shavrova, 2007) [21].

The mechanical properties of the solid parts of the ridge are assumed to be close to that of level ice but with a more isotropic granular structure. Multi-year ice ridges are stronger than ﬁrst-year ice ridges. Measurments have shown that warm multi-year ice has a strength of approximately 20 MPa. Testing of mechanical properties can be done in ﬁeld, or in laboratory. Measurement and analyses methods are not discussed in this paper but more information can be found in Eicken and Salganek (2010) [22].

The most important mechanical parameters where found to be brine fraction, air fraction and and loading rate. In an ice ridge two levels of porosity can be identified: macro porosity and total porosity. The macro porosity is defined as the ratio of volume of any non-sea ice material to the total volume. The total porosity can be defined by including the porosity of the level ice from which the ridge is formed, i.e. it includes the brine pockets. Old ridges have a porosity similar to that of the surrounding level ice. The first-year ridges consist of a large rubble part which was found to have higher porosities [12]. Studies showed that the porosity has a profound effect on the ice strength.

Vseawater η (2.1) = V V V seawater + pureice + brinepockets Vseawater Vbrinepockets ηtot + (2.2) = V V V seawater + pureice + brinepockets 14 2.I CE ACTIONSAND ICE RESISTANCE

Ice strength during compression directly relates to the loading rate, with a maximum strength around a load- 3 1 ing rate ² 10− s− . Furthermore the loading rate helps identify how the ice fails: brittle or ductile. For low = temperatures and higher indentation rates ice fails brittle. The strength dependency on loading rate is given in Figure 2.8.

Complemented to the listed properties that influence ice strength, research showed that strength is also influences by grain size and boundary conditions. Smaller grains give stronger ice. The boundary conditions influence the failure modes. When looking at ice ridges it was discovered that the ice strength variations in a sea ice ridge can have significant variations due to its inhomogeneous characteristics. The variations in ice strength are found all across and throughout the ice and depend on crystallography, temperature, salinity and porosity. For first-year ice ridges a high strength heterogeneity of 40-55% has been identified (Shafrova, 2008) [20].

Figure 2.8: Strength dependens on loading rate (Løset, 2006) [6].

2.3. ICEBREAKERS With this section the author strives to give the reader a basic understanding of icebreakers: why we use icebreakers, where we use icebreakers, how we use icebreakers, and what the important icebreaker design aspects are. This section includes a historical background, a description of the weather and ice conditions where icebreakers operate, ice operations, several type of ice actions icebreakers encounter, and an introduction into icebreaker design.

An icebreaker is a special-purpose vessel designed to move and navigate through ice-covered waters and, as the name suggests, break a passage through the ice. Icebreakers are intended to assist other ships in ice. They must be able to operate at all ice conditions occurring in their operational area. In general icebreakers are expensive to build, expensive in fuel and heavy for their size. Different from most vessels, icebreakers are required to have a strengthened hull, an ice-clearing shape and the power to push through sea ice. Ice strengthened ships have strengthening and some ice performance but need icebreaker escort in heavy ice conditions. This strengthening can be applied to all manner of ships like tankers, containers ships, warships, supply ships, etc. These ships commonly have an ull designed for open water performance. Ice-strengthened ships can usually cope with continuous one year old ice about 50cm - 100cm thick. Ice going ships are de- ﬁned as merchant ships that do not need icebreaker escort and therefore sail independently.

Most icebreakers are needed to keep trade routes open where there are either seasonal or permanent ice conditions. In these regions merchant vessels are often not capable of delivering enough power to manage the ice themselves. Merchant vessels often sail in convoys to use icebreakers most efﬁciently. Besides assistance to other ships, icebreakers are often used for ice management and used to transport goods in heavy ice conditions. According to Riska (2010) [4] the Baltic is the most active sea area for ice navigation.

Traditionally icebreakers use two alternative methods to break the ice. The ﬁrst method is continuously pushing through the ice sheet, hence relying on their thrust. The second method is the ramming of ice features, 2.3.I CEBREAKERS 15 hereby relying on the vessel’s inertia. The latter method is required in the case of very thick ice, for example an ice ridge. In such cases icebreakers often drive their bow onto the ice, using the mass of the ship to break the ice. Sometimes icebreakers need to ram a speciﬁc multiple amount of times before an ice feature breaks. Occurring loads differ greatly between these ice breaking methods.

BASICSREGARDINGICEBREAKERTERMINOLOGY This section outlines some basic terminology regarding icebreakers. Readers who are familiar with these terms can proceed to the next section.

• Astern: Behind or towards the rear of a vessel. • Beam: A measure of the width of the vessel. • Bow: the bow is located at the front end of the ship. • Bollard pull: The zero speed pulling capability of the vessel. • Coverage: Portion of sea surface covered by ice. • Dead Weight Tonnage: A measure of how much mass a ship is carrying or can safely carry. • Framing: The frames support the hull shell plates and resist the loads on the shell by bending and shear deformations. • Freeboard: The distance from the waterline to the upper deck level. • Icebreaker: Any vessel whose operational profile may include escort or ice management functions, whose powering and dimensions allow it to undertake aggressive operations in ice-covered waters. • Ice strengthened ships: Ships that have a strengthened hull but need icebreaker escorts in heavier ice conditions. • Ice going ships: Merchant ships that do not require icebreaker escort and therefore sail independently. • Ice management: Activities undertaken with the objective of reducing or avoiding interaction with any kind of ice features [23]. This can include but is not limited to: a) Detection, tracking and forecasting of ice. b) Evaluation of the threat posed. c) Physical ice management, such as ice breaking and iceberg towing. d) Procedures for disconnection of offshore installations. • Metacentric height: The vertical distance between the centre of gravity and the metacentre (the point at which the force of buoyancy will cut the centre line at a given angle of heel). • Parallel mid body: Describes the middle side area of the vessel’s exterior hull which is flat and usually vertical. • Portside: As one looks forward, portside side is at one’s left. • Reamers: a hull that is wider in the bow than in the stern. These so-called "reamers" increase the width of the ice channel and improve the icebreaker’s maneuverability in ice. • Shoulder crushing: This phenomenon is created if an icebreaker bow breaks a narrower channel than the icebreaker beam. Consequently the vessel must force itself into this channel. The icebreaker needs to crush the rest of the channel width up to its maximum beam (sometimes called ’forward shoulders’). Shoulder crushing significantly increases ice resistance. • Starboard: As one looks forward, starboard side is at one’s right. • Stem: The most forward part of a vessel’s bow. It is an extension of the keel itself. • Trim: The difference between the draft forward and the draft aft. • Typical hull shapes: Three typical different hull shapes for icebreakers are identified: wedge shape, spoon shape, and square shape. These hull shapes are shown in Figure 2.11. 16 2.I CE ACTIONSAND ICE RESISTANCE

HISTORICALBACKGROUND For early wooden wind powered ships the best way to deal with ice was to not be in the water, they would just stop using their vessels in ice. It was possible to ice strengthen the ship. These ice strengthened ships have been around for hundreds of years. They were originally wooden and based on existing designs, but with a reinforcing hull. Still, these ships were no match for the frozen water way. That came to change with the perfection of the steam engine by James Watt in the late seventeen hundreds and their subsequent steam powered ships. The steam engine did not immediately result in icebreaking ships. Most of vessels with steam engine still had sails, and even with that combination the vessels were to underpowered to push trough ice. Captains who blundered getting stuck into heavy ice had few options. Crew would try to free their ships with sledgeham- mers (or anything else they could ﬁnd). Or winching themselves forward by putting an anchor on the ice and use capsizing to break the ice. Depending on the atmospheric condiments the ship could be stuck until fall or mercilessly crushed by the ice. A famous example is Ernest Shackleton’s loss of the Endurance. Ernest Henry Shackleton was a polar explorer who led three British expeditions to the Antarctic. On 19 January 1915, icebreaker Endurance became frozen fast in an ice ﬂoe. Realizing that Endurance would be trapped until the following spring, Shackleton ordered to abandon the ship. While stuck, Endurance drifted slowly northward until spring, when extreme pressures on the vessel crushed the hull. Figure 2.9 shows Endurance’s remains in spring.

Figure 2.9: The vessel Endurance got crushed after it became stuck in the ice

The first ice breaking ships appeared mid 1840’s in Hudson River in the US and in the Elbe River in Germany. In 1870s and 1880s whaling and sealing became an overly good business. Better strengthened ships were needed resulting in an improved ice strengthened design. These improved first icebreakers appeared in 1860’s and 1870’s in the St. Petersburg and Hamburg harbours. The first detailed description of ice breaking ships was written by Robert Runeberg in 1888/1889 (Jones, 2004)[3]. Runeberg discussed both the continuous and the ramming ice breaking method. He derived expressions for the vertical pressure at the bow, thickness of broken ice pieces, and the total elevation at the bow during continuous ice breaking scenarios. Runeberg recognized the importance of hull angle and the friction between ice and hull on resistance. He suggested to make the vertical component of the ice force as high as possible by introducing a very sloping bow. Up to this day icebreakers are constructed with small bow angles. Late 1920s, shipbuilders were making icebreakers combining strength, power and design features. But they were expensive. Increased interest in icebreaker design occurred with the interest in fuel. Houses in the USA previously got heated by coals, which were brought by rail. The new heating of houses by fuel required barges on the rivers to transport this fuel. During WWII icebreakers became vital in the frozen north of the Atlantic. These waters were used to supply Russia. Furthermore there was the strategic importance of Greenland: valuable mines and German weather stations. To remain for longer periods of time above the polar ice circle the ships needed to be both strong and large enough to carry the necessary supplies. These icebreakers can be seen as military icebreakers. During the cold war Russia started to make nuclear icebreakers. To the Soviet Union the polar waters were 2.3.I CEBREAKERS 17 important: it is relatively nearby their borders, and it contained the northern sea route. With these nuclear icebreakers Russia would be able to supply oil and other materials throughout the year. Nuclear icebreakers would become the fastest and strongest icebreakers of all time. After the cold war, icebreakers were mainly used as scientific vessels. Kashteljan et al. (1968) are credited with the first detailed attempt to analyze level ice resistance by breaking it down into components: resistance of breaking level ice, resistance due to submersing ice, resistance due to passage through broken ice, water friction and wave making resistance (Ashton, 1986) [24], (Jones, 2004) [3]. White (1969) [25] worked on the prediction of dynamically developed force at the bow of an icebreaker. His major contribution was to identify those qualities of a bow that would be desirable for (a) improved continuous icebreaking, (b) improved ramming and (c) improved extraction ability. White concluded that there were only three qualities that would improve all three capabilities simultaneously: (1) decrease of spread angle complement, (2) decrease of friction coefficient, (3) increase of thrust. White proposed a bow shape for a polar icebreaker, shown in Figure 2.10. The Manhattan voyage in 1969, rise in oil prices in 1973 and in 1979, led

Figure 2.10: White’s recommended icebreaking bow (1969) [3]. to a promise of extensive Arctic development, and therefore interest in icebreaker design. It let to icebreakers with unconventional bow forms. Instead of the classical wedge-shaped bow the spoon-shaped and Thyssen- Waas bow (squared-bow) were developed (see Figure 2.11). Oden and Canmar Kigoriak are an example of a spoon-shaped bow, the converted Mudyuq has this Thyssen-Waas bow. These designs all have small stem angles around 20 degrees. Icebreaker’s design has been developed signiﬁcantly in the period 1985-2015. This may be contributed to a more scientiﬁc approach to modeling of ships in ice with extensive model testing, and recently, numerical methods [3].

Figure 2.11: Three different hull types for icebreakers [26].

Icebreaker machinery has experienced many changes since the early icebreakers which contained steam engines and fixed pitch propellers. Classic icebreaker propulsion systems used a single propeller. Increased engine power resulted in more propeller solutions. Swedish IB Ymer was the first diesel-electric icebreaker, completed in 1933. The diesel-electric machinery is more expensive than a direct diesel drive but the torque performance of a fixed pitch propeller with a direct drive is not good. Solution for this is the use of controllable pitch propellers.

Bow propellers were introduced in icebreakers in the end of 19th century. These bow propellers reduce the 18 2.I CE ACTIONSAND ICE RESISTANCE required force to break the ice and also reduce the friction by lubrication. Development of propulsion systems led finally to Urho class icebreakers with two bow and two stern propellers, two rudders and diesel electric propulsion. These icebreakers where completed mid 1970’s and can be considered the last conventional icebreakers. Only lately the bow propellers have been made superfluous by the introduction of so called Z- drives (azimuthing propulsion units). Finnish multi-purpose icebreaker Fennica (completed in 1993) is the first icebreaker with azimuthing propulsion. Russia completed the first nuclear icebreaker in 1957, and was called NS Lenin.

2.3.1. ICEBREAKEREXAMPLES Three icebreaker examples are given in Appendix A:

1. MSV Fennica, a Finnish multipurpose icebreaker and platform supply vessel 2. Oden, a Swedish icebreaker. It was the ﬁrst non-nuclear surface vessel to reach the North Pole, together with the German research icebreaker RV Polarstern 3. 50 Let Pobedy, a Russian, former Soviet, class of nuclear-powered icebreakers. Up to this day they are the largest and most powerful icebreakers ever constructed.

The examples include a brief description of the icebreaker, the general characteristics, and a picture.

2.3.2. ICEAND WEATHER CONDITIONS THE ARCTICENVIRONMENT There are several arctic areas with different ice conditions. Ice covered seas are located mostly in the high northern or southern latitudes, the Arctic and Antarctic. Temperatures are low here, and the daylight hours in winter are short. Different ice conditions are mainly caused by currents, wind, sea and air temperature, interaction with the seabed and interaction with the coast.

Fast ice is directly connected to the land. Outside the fast ice region the ice cover is broken and moving. The region where the effect of the coastline is felt is called the transition zone. An example of such a sea is the Beaufort Sea. If the ice cover is converging, the ice coverage tends to be high with heavy ridging. If the ice cover is diverging, the coverage and ridging is less intense. Outside the transition zone the pack ice zone is found. A visualization is presented in Figure 2.12. Observing the difference can be difﬁcult, but this is not important for icebreaker design. According to Riska (2010) [4] the main weather parameters that form the design basis for icebreakers are:

• Description of ice cover • Sea and air temperatures • Winter or ice season lengths

Figure 2.12: The ice zones in a sea ice cover according to Riska (2010) [4]. 2.3.I CEBREAKERS 19

Icebreakers may encounter several different ice scenarios. This includes different ice features, but also ice thickness, ice concentration and openings in the ice (for example a crack or fracture). Vessels navigating in the middle of the sea rarely encounter level ice, because driving forces from wind or currents break the ice cover, causing rafting and ridging when the ice ﬁeld is converging. A typical sea ice ﬁeld is therefore composed of zones of level ice, open water and ridges. Ice ridges form the largest obstacle for shipping as even smaller ice ridges stop a merchant vessel. To describe ice conditions standard codes like for example the WMO Sea Ice Nomenclature [23] can be used.

With respect to the description of an ice cover, the most important data for a ship transit through an ice cover are:

• Ice coverage • Level ice thickness • Density of ice ridges • Average maximum thickness of ice ridges

The ridge resistance is often higher than the icebreaker thrust, and therefore must be penetrated using icebreaker’s inertia. The statistics of ridges have been studied intensively and most often it is concluded that the ridge size and density follow an exponential probability distribution. Sometimes level ice performance and ridge penetration are coupled by using the so called equivalent level ice thickness. This is basically the average thickness of all ice in the area. This method to describe icebreaker performance is used in general ice conditions where the ridges are smaller than those that would stop the vessel. For more information the reader is referred to Riska (2010) [4].

Ice of land origin It must be noted that not all ice features originate from the sea. In some areas land based ice like icebergs, bergy bits, growlers and even ice islands can be found. Icebergs are large masses of floating ice originating from glaciers. Ice islands are vast tabular icebergs originating from floating ice shelves. Big piaces of ice can be dangerous as they can cause considerable damage to a vessel that collides with them. A vessel’s navigations is adjusted due to these features. An additional problem occurs at the smaller pieces like bergy bits and growlers. They are often difficult to detect, both by sight and radar. Especially in storm conditions or when surrounded by pack ice, detection becomes hard. Approximately 60-90% of an iceberg’s mass is located below the waterline (Johnson, 2007) [27].

ICING The operational temperature must be taken into account in an icebreaker design. Low temperature inﬂu- ences the construction materials and also implies requirements for deck machinery, accommodation and main machinery. A typical requirement is that a ship must be able to operate in a temperature of -35 degrees Celsius [4]. A combination of water droplets and cold temperatures can cause ice accretion, also called icing. Icing is the freezing of water droplets on the deck and superstructures. An example of this is shown in Figure 2.13.

Icing is a complicated process which depends upon meteorological conditions, condition of loading, vessel behavior, and the size and location of superstructures and rigging. Wave slamming causes sea spray which then can be blown on board of the vessel. This sea spray can freeze on decks and superstructures and has the highest contribution to icing. Sea spray generation is enhanced in open water conditions. Icing may also occur in conditions of snowfall, sea fog, drastic decrease in ambient temperature, and from the freezing of raindrops on contact with the icebreaker. It may be stated that the severity of vessel icing due to sea spray is a function of environmental factors (water and air temperature, wind speed, sea state) and vessel characteristics (course relative to the wind, course relative to wave propagation, speed, bow design, freeboard, exposed superstructures). The most intensive ice formation takes place when wind and sea come from ahead.

Icing is a serious hazard for marine operations. One important hazard is potential vessel instability due to the ice accretion. However, many hazards are related to the operational aspects of icebreakers in arctic conditions. Icing has a profound effect on the operations of an icebreaker, this includes operational problems like malfunctioning machinery, rigging damage and blocked vents. It is for example important to maintain the anchor windlass free of ice. This so that an anchor can be dropped in case of emergencies. However, another major operational hazard is the physical danger for the crew: they need to work in more severe conditions 20 2.I CE ACTIONSAND ICE RESISTANCE

Figure 2.13: Ice accretion on a vessel superstructure [23] outside, and the crew often needs to take extra activities to rectify the icing problems. At present, there are few methods available to prevent icing and to clear it quickly should it occur [23].

SEAICEACCUMULATIONONTHEHULL Sticking Sometimes sea ice accumulation on the hull of the vessel occurs. This phenomenon sometimes is characterized as sticking, typically happens when the ambient air temperatures fall just below zero. It can be observed in colder ambient air temperatures as well, but it is more rare. Ice features commonly linked to sticking are young ice and ﬁrst-year thin ice. Sticking intensity on the hull depends on air temperature and icebreaker hull’s roughness. Sea ice accumulation increases hull friction, reducing the icebreaker speed and maneuvering capabilities.

Shoulder crushing Shoulder crushing is a form of ice accumulation. It can be identiﬁed by the crushed ice extruded on top of the ice, an example is given in Figure 2.14. This shoulder crushing increases ice resistance signiﬁcantly. Shoulder crushing should be prevented in icebreaker design. Model tests are a suggested method to test a new icebreaker design on this phenomena.

Figure 2.14: Shoulder crushing at the side of a vessel [4]

Snow Another factor influencing the friction forces is the snow cover on top of the ice. Generally snow does not have a noticeable effect on ice loads, but does have an effect on the performance. This frictional effect of snow is amplified for flat bows, which are common for icebreakers [4]. 2.3.I CEBREAKERS 21

2.3.3. ICE OPERATIONS It must be known why and where icebreakers are needed. This depends mainly on where and why ice capable tonnage is needed. The two most commercial areas are those of merchant shipping industry and the offshore industry. Operational areas where merchant ships regularly encounter ice are the Baltic Sea, Great Lakes, St. Lawrence Seaway, Russian western Arctic and Okhotsk Sea [4]. Figure 2.15 shows the Baltic Sea, which contains many ports located in Sweden, Finland, Estonia, Russia, Latvia, Lithuania, Poland, Denmark and Germany.

Icebreakers escort other vessels in the areas with regular winter trade. When merchant vessels choose to travel with an icebreakers escort, the ice strengthening of these merchant vessels is allowed to be less than that of the vessels navigating independently. The merchant industry therefore has to evaluate which ice navigation alternative is best. Also the requirements placed by the maritime authorities play a role, as higher ice strengthened merchant vessels require less icebreakers. A visualization of this balance between merchant ﬂeet, icebreakers and maritime regulations is given in Figure 2.16. In general no icebreaker service is offered in areas where only occasional visits are made.

Figure 2.15: The Baltic Sea

ICEBREAKING ESCORT OPERATIONS Escort is one of the main tasks of icebreakers. Most important components for an icebreaker escort are discussed below. Figure 2.17 shows an icebreaker escorting a vessel through ice.

• Convoys: Convoys of ships may be formed by the Commanding Officer of the icebreaker, after con- sultation with the appropriate shore authority. Feasible size of a convoy depends on ice conditions, icebreaker capability and number of icebreakers available. • Track width: Progress through ice depends to a great extent on the width of the track made by the icebreaker. After the track is made by the icebreaker the track may fill itself up with ice. Track width therefore is influenced by ice conditions, escort speed and distance between vessels. • Icebreaker beam: The icebreaker beam influences how big of a track width the icebreaker can make. • Minimum escort distance: The minimum distance will be determined by the Commanding Officer of the icebreaker. This will be achieved on the basis of distance required by the escorted ship(s) to come to a complete stop. It is the responsibility of vessels under escort to see that this minimum escort distance is maintained. If the escorted vessel is unable to maintain the minimum escort distance, the icebreaker should be informed at once to avoid the collision hazards. 22 2.I CE ACTIONSAND ICE RESISTANCE

Figure 2.16: The interacting components in a winter navigation system [4].

• Maximum escort distance: Maximum distance is determined on the basis of ice conditions and the distance at which the track will remain open or nearly so. If the escorted vessel is unable to maintain the maximum escort distance, the icebreaker should be informed at once. • Maintaining the escort distance: This distance is dictated by ice conditions and the risk of collision by the escorted vessel. Progress made by an escort depends greatly on the correct escort distance being maintained. • Ice concentration: If coverage is very high the track will generally close quicker. • Ice pressure: Ice pressure influences how fast a track width can be closed. The effect of ice pressure depends greatly on ice concentration. • Effect of escort distance on width of track: When an icebreaker makes a track, it causes outward movement of the floes. Track width depends on the extent of this outward movement together with the amount of open water available for floe movement. A longer escort distance allows for a longer period of movement that may result in a bigger track width. • Icebreaker speed: Icebreaker speed influences how ice breaks. It might be possible to push ice floes sideways. At the other hand an ice floe might be broken into smaller peaces of which a part will get into the icebreaker track. At low speed ice floes will slide along the hull and remain relatively intact. At high speeds the floes will be shattered into many pieces. Most tracks made by icebreakers will contain ice rubble, which may also contain floes, which could damage an escorted vessel at excessive speed. • Escorted ship beset: When a ship under escort has stopped for any reason, the icebreaker should be notified immediately. • Freeing a beset vessel: It is the icebreakers job to free a beset vessel. This is usually carried out by backing down the track, cutting away the ice on either bow of the beset ship, and passing astern along the vessel’s side before moving ahead. • Towing in ice: This procedure would only be undertaken in emergencies because of the risk of damage to both vessels. The Commanding Officer of an icebreaker who receives a request for a tow will judge whether or not the situation calls for such extreme measures. • Anchoring in ice: In general anchoring only happens in an emergency. • Icebreaker stop; red warning lights and air horn: Warning lights and air horns may be used to indicate that an icebreaker has stopped. During close escort work, on the escorted vessel a lookout shall always be kept for the flashing red light and warning horn. 2.3.I CEBREAKERS 23

Figure 2.17: Russian icebreaker "Krasin" escorts a supply ship into the McMurdo base, Antartica.

ICEBREAKERICECLASSES Designing a new vessel generally happens by following rules that classification societies, and also some maritime authorities (like Finnish and Swedish Maritime Administrations and Transport Canada) have developed. The classification societies are responsible for approving the design and supervising the construction of individual vessels to ensure conformity with the standards set by international conventions and by the classification of that vessel (Sodi, 1995) [28]. Several classification societies have developed standards for Arctic vessels, including icebreakers. These Classification Societies include:

1. Finnish-Swedish ice class 2. IACS Polar Classes 3. American Bureau of Shipping Arctic Ice Classes 4. Det Norske Veritas Arctic Ice Classes 5. Lloyd’s Register Arctic Ice Classes 6. Russian Maritime Register of Shipping Arctic Ice Classes 7. Transport Canada CAC Deﬁnitions

Ice classification depends on its capability to resist damage while navigating in ice under normal handling conditions. At present there are three main sets of ice class rules: the Finnish-Swedish Ice Class Rules, the Russian Maritime Register of Shipping ice rules and the unified Polar Class rules of the International Associ- ation of Classification Societies (IASC).

The Finnish-Swedish ice class is an ice class assigned to a vessel operating in ﬁrst-year ice in the Baltic Sea and calling Finnish or Swedish ports. Ships are divided into six ice classes based on requirements for hull structure design, engine output and performance in ice according to the regulations issued by the Finnish Transport Safety Agency. Design point in the Finnish-Swedish ice class rules is the elastic limit.

Russian Maritime Register of Shipping ice rules consist of nine ice classes, and additionally four icebreaker ice classes. The ice rules contain three parts: hull, machinery and powering. Design point in Russian Mar- itime Register of Shipping ice rules is the full plastic response for plating and frames.

The International Association of Classification Societies published a set of Unified Requirements for Polar Class Ships. A overview is given in table 2.1. Due to the wide range of requirements, comparison between ice notations is difficult. Figure 2.18 shows a comparison made by the Russians. This figure should be used with caution, it is recommended that the relevant Maritime Administration are consulted about the acceptance and equivalence of ice notations. The IACS Polar Ship Rules have been developed to provide harmonization between classification society requirements [23]. 24 2.I CE ACTIONSAND ICE RESISTANCE

Table 2.1: IACS Polar Ice Classes

Polar Ice class Ice Description (based on WMO’s Sea Ice Nomenclature) PC 1 Year-round operation in all Polar waters PC 2 Year-round operation in moderate multi-year ice conditions PC 3 Year-round operation in second-year ice which may include multi-year ice inclusions. PC 4 Year-round operation in thick first-year ice which may include old ice inclusions. PC 5 Year-round operation in medium first-year ice which may include old ice inclusions. PC 6 Summer/autumn operation in medium first-year ice which may include old ice inclusions. PC 7 Summer/autumn operation in thin first-year ice which may include old ice inclusions.

Figure 2.18: Approximate comparisons between ice classes (By Russian Federation) [23].

2.3.4. ICE RESISTANCE Understanding how ice and cold temperature act on a ship is the basis for all designs. Encounterable ice features and the way the icebreaker is operated determine the vessel-ice interaction. Hull design and propulsion machinery strength is based on evaluating the ship-ice interaction scenarios.

When looking at ice loads it is important to distinguish local and global loads, as discussed in Chapter 2. Local forces are most important for the structural design. Global ice loading is important in determining the icebreaker performance in ice. Global ice forces contribute to the ice resistance which is the time average of the global ice load. Maxima included in the time history of the ice load are not that important for performance as icebreaker inertia smoothens their effect. According to Riska (2010) [4] a division of the resistance forces into components of similar origin can be made as follows:

• Forces from breaking the ice • Forces from submerging the broken ice • Forces from friction along the ship hull (both ice pieces breaking and sliding along the hull) • Hydrodynamic forces

As discussed in section 2.1.5 several ice failure modes exist (i.e. creep, crushing, bending, buckling, splitting, spalling, shear), and each failure mode should be investigated individually. It must be stressed that not all ice forces work on the bow only. When maneuvering in ice, sidewards loads will be implemented on the icebreaker. Another example is the case an icebreaker stops in a converging ice ﬁeld, which can induce large forces on the side hull. An icebreaker must be able to withstand these compressive ice forces.

The total resistance in ice is assumed to be the sum of the pure ice resistance Ri and open water resistance Row as shown in equation 2.3. R R R (2.3) iTOT = i + ow

Both the total resistance component RiTOT and open water resistance Row can be determined by experiments. The pure ice resistance therefore is determined by subtracting Row from RiTOT . Then the ice resis- 2.3.I CEBREAKERS 25 tance may be divided further into the earlier deﬁned ice resistance forces:

R R R R (2.4) i = B + s + F where the resistance components are the breaking, submerging and friction resistance components, respectively. Most methods used to calculate ice resistance are based on regression on full-scale and model-scale data (Riska, 2010) [4]. Regression assumes that ice resistance is linear with ship speed and consists of these three components. In practice this is not exactly true, and the methods should be used cautiously, especially outside the range of validity. For a complete icebreaker design lots of different ice scenarios must be investigated. This thesis presents the key scenarios: forces from level ice, forces from ice ﬂoes and forces from ice ridges.

This sections looks into the ice resistance of level ice, ice ﬂoes and ice ridges. Presented equations on ice loads in this section are based on the Finnish-Swedish ice class rules.

RESISTANCE IN LEVEL ICE The most efficient way of breaking level ice is by bending the ice downwards. At first contact the ice edge gets locally crushed. Crushing continues up to the point the vertical force component on the hull becomes large enough to make the ice fail by bending. This means that the applied bending force exceeds the bending strength of the level ice. Due to the bending crack an ice floe is created, which must be extruded from the icebreaker bow by moving downward and sidewards. However, ice floes float, and therefore ice floes sliding downward along the hull interact significantly with the hull.

On the ice several forces worked during the icebreaker-ice interaction. Main forces are the hydrodynamic support force from water, the crushing/contact force and the frictional force opposing the relative motion of the broken up ice ﬂoes. These ice forces can be divided into three components, distinction based on the origin of the loads:

1. Breaking forces (crushing, bending and turning) 2. Submerging forces 3. Sliding forces

See Figure 2.19 for a visualization of the three components and their force contributions. The top view of the ice breaking process is shown in Figure 2.20.

Figure 2.19: The forces in level ice breaking [4].

In the Finnish-Swedish ice class it is assumed that for a constant level ice thickness the contact area remains constant. This area is narrow in vertical direction and long in horizontal direction. The load path may be more irregular in shape when it impacts a multi-year ice of rounded shape. Then it is assumed that the pressure is equal on the load patch. What this average pressure should be is not straight forward. The ice properties vary highly in an ice feature, therefore also the ice pressure varies a lot. In reality the ice pressure is not working homogeneously on the load patch but rather contains different local pressures on different locations. For design it therefore is common to use empirical determined formulas to deﬁne this average design pressure. One of these options is to use Sanderson’s (1988) pressure area relationship, shown in equation 2.5. Here pg 26 2.I CE ACTIONSAND ICE RESISTANCE

Figure 2.20: Top view of the ice breaking process [4].

2 represents the pressure in [MPa], AN the nominal contact area in [m ]. The nominal contact area is the load patch normal to the vessel hull. 0.57 p 8.1 A− (2.5) av = · N With average pressure and load patch determined, it is possible to determine the level ice load.

F p A (2.6) n = av · N

Where Fn represents global ice load normal to the hull in [N]. This force represents the load required to fail this level ice. It is convenient to split the normal force into vertical and horizontal components. Subsequently ice friction is added to these horizontal force and vertical force. If the bow angle with vertical equals β [rad] and the friction coefﬁcient µ [-], the total horizontal force FH [N] and vertical force Fv [N] can be calculated as in equation 2.7. Equation 2.7 assumes that the friction force can be calculated with F µ F . f r iction = · n

FH FN cos(β) µ FN sin(β) = + · (2.7) F F sin(β) µ F cos(β) V = N − · N Equation 2.7 shows that frictional forces reduce the vertical component, which is responsible for causing bending in the ice cover. This is especially true when the surface is rough (µ large i.e. more than around 0.1) [4]. A rough surface therefore does not only signiﬁcantly increase resistance as a friction force, but also limits the bending failure capabilities of the vessel. For a good icebreaker design surface roughness must therefore be as low as practically possible.

Assuming a constant level ice thickness and icebreaker velocity, then approximately equal size ice ﬂoes are broken off continuously. There is a repetitive breaking pattern as can be seen in Figure 2.21. When turning in level ice, the ice breaking sequence at the bow is similar to that when going straight ahead. However, the vessel gets also pushed sidewards into the ice (created by rudder or azimuthing thrusters). An analogy for this phenomena is a big truck making a turn, the truck edges will swing out further then when going straight ahead. The magnitude of this sidewards load depends on the rotation radius. See in Figure 2.22 how the vessel aft will have to break additional ice. 2.3.I CEBREAKERS 27

Figure 2.21: Typical icebreaker resistance in level ice [4].

Figure 2.22: Turning in level ice induces sidewards ice loads [4].

RESISTANCE IN ICE FLOES When looking at ice floes a very important factor are ice floe size and coverage. In a very open ice field, thus low coverage, the floes may be pushed in front or sideways of the icebreaker. When the coverage is very high the ice field starts to behave more similar to a level ice scenario, thus bending becomes more important. In general a combination of pushing forward and bending downwards occurs. Small floes are not broken in bending, but displaced along and around the hull. It must be stressed that this does not mean that ice floes impose relatively lower loads. Inertial forces related to rigid body motion become higher compared to that of level ice. This difference is enhanced at higher velocities. The lower the coverage, the more open water, the higher icebreaker velocities can be expected. Figure 2.23 shows an example of an ice field filled with ice floes.

Determining the ice loads occurring in an ice field with several ice floes is less straight forward than that of level ice. Determining the contact force of a single ice floe is hard, even if the bending of the ice floe is ignored 28 2.I CE ACTIONSAND ICE RESISTANCE

Figure 2.23: A typical ice field with several ice floes and the ice floe is assumed to move as a rigid body. Also icebreaker movements affect ice floe movements, especially if confinement is low. Another difficulty occurs because the interaction becomes a multi-body dynamical problem. Each ice floe will have different characteristics (geometrical, physical and mechanical) and ice floes will influence each other by hydrodynamical effects and mutual impacts. Each ice floe movement can move in 6 DOF.In the case of very large ice floes, similar to a multi-year ice ridge, and straight on impact (meaning that the icebreaker rams the floe perpendicular to the ice edge), only three ship motion components (heave, pitch and surge) are to be accounted for [4]. For more information about hydrodynamic effects of marine operations in broken ice fields, the reader is referred to (Tsarau, 2015) [29].

One suggested way to approximate ice floe loads is using equation 2.8. This equation presented in Riska (2010) [4], defined by Daley (2001), assumes a constant pressure and is based on energy approach. The ship collides with a smaller ice floe and the collision area is on one side of the bow. Equation 2.8 essentially is one dimensional, and therefore must be seen as a simplified approach to determine the ice floe load.

F C p0.36 v1.28 0.64 (2.8) n = · 0 · shi p · 4 where Fn is the ice floe load normal to the hull in [N], C is a factor containing the geometric information at the contact, p the constant from the pressure-area relationship [N], v the vessel velocity [m/s2] and 0 shi p 4 ship displacement [m3]. Alternatively, equation 2.9 can be used, which was defined by Riska (2010) [4]. This equation has been used to determine the normal ice load on a multi-year ice floe. This case includes ice edge crushing, followed by beaching of the vessel onto the ice. Here it is assumed that the ice mass is large compared to vessel mass. In equation 2.9 C is a constant containing the dependency on ice strength, Awp the contact area below water level. q F C sin(ϕ)0.2 A v (2.9) n = · · 4 · wp · shi p For oblique impacts the ice floe’s contact force typically shows two peaks. This occurs mostly due to the ship heeling motion: at first impact the icebreaker heels away from the floe but when the contact force decreases the ship heels back causing the second peak.

RESISTANCE IN ICE RIDGES For icebreakers, ice ridges are the largest common obstacle to encounter in polar waters. The ridge needs to be broken using the icebreaker’s inertia. Beaching on an ice ridge is common, and to be able to continue, the icebreaker must extricate herself from the ridge. If ice ridges are deep, around the draft of the icebreaker or larger, it will impose conﬁnement on the vessel. Ice cannot be effectively cleared from the hull, which increases the ice loads. Subsequently this enhances the probability of propeller-ice interaction.

The ice ridge resistance depends on ridge thickness, which is different at each location along the hull. There- fore, for a ship at location x, the resistance from a ship segment with length x equals: 4 R R (H (x)) x (2.10) 4 R = r R · 4 where Rr (HR (x)) is the ridge resistance in ridge thickness Hr per unit ship length. Equation 2.10 takes ridge resistance as independent of vessel speed (v). This implies that the speed dependency is allocated to the 2.3.I CEBREAKERS 29 open water resistance. Thus it is possible to write resistance as: Z RR,TOT RR,B (HR (xbow )) RR (HR (x)) dx Row (v) (2.11) = + Lpar · + with RR,B as the ridge breaking resistance acting at ship bow, Lpar as the parallel midbody length of the ship. Besides the ridge resistance magnitude, the knowledge where on the ship hull the loads acts is also very important. Since ridge thickness changes, the ridge resistance will change in time. It therefore may be more suitable to speak of the energy required to penetrate certain size of a ridge. A name to characterize this method is the speciﬁc energy approach. A typical energy consumption during an ice ridge impact is visualized in Figure 2.24. Subsequent chapter will go into more details into this subject, and the reader is referred to this chapter for more information.

Figure 2.24: Energy consumed in penetrating ice ridges, ER , based on ice model tests (Riska, 2010) [4].

Ridge behavior First-year ice ridge loads can be divided into two parts: consolidated layer load and keel load. The action of sails can normally be assumed to be negligible (Dalane et al. 2015) [30]. The consolidated layer load is commonly approximated by treating it as level ice. For the keel often material behavior is used to calculate its global ice load, for example Coulomb-Mohr failure. This can be applied because the keel can be seen as an accumulation of discrete ice pieces and that can be interpreted as a granular material. Dalane et al. (2015) [30] did model tests with a moored ﬂoater to look into its ridge behavior. Figure shows a cross-sectional sketch of a typical ﬁrst-year ridge. Consolidated layer and keel are indicated, as well as keel depth and sail height. Results showed that with respect to the consolidated layer:

1. Consolidated layer always fails in bending in part 1 of the ridge (with part 1 Defined in Figure 2.25) 2. Due to the morphology of the ice ridges used in the model test there was no failure of the consolidated layer in part 2. 3. Two different behaviors of the consolidated layer were observed for part 3: Either the consolidated layer breaks in bending by any direct interaction between ridge and floater. Or the floater pushed rubble from the keel forwards, forcing the consolidated layer upwards and therefore making the consolidated layer fail by upward bending failure.

Another interesting ﬁnding by Dalane et al. (2015) [30] is that the consolidated layer and the surrounding level ice always acted as conﬁnement on rubble in the keel.

It must be emphasized that methods used for estimating ridge loads on offshore structures give too high values and therefore require modiﬁcation for ship-ice ridge interactions. Section 2.4 goes into more details on ice ridge loads. 30 2.I CE ACTIONSAND ICE RESISTANCE

Figure 2.25: Cross-sectional sketch of a typical ﬁrst-year ridge. Marked areas deﬁne sail, consolidated layer, keel and level ice. (Dalane et al. (2015) [30].

2.3.5. ICEBREAKERDESIGN ICEBREAKERPERFORMANCE Ice performance in ice consists of ability to break ice and to maneuver in ice. Capability of breaking ice is measured in uniform ice conditions. It therefore depends on velocity and ice thickness. With respect to ridges or thick ice floes the performance can be measured by the ability to penetrate the ice feature. These capabilities should be defined in the functional specification of an icebreaker.

Icebreaker velocity in an ice ﬁeld is a function of ice resistance determined by ice properties, hull shape, vessel dimensions and the thrust provided by the propulsion system. Maneuvering performance is similarly determined by the transverse forces provided by the propulsion system and the resisting forces (mainly due to ice). Turning performance can be measured by identifying the turning circle, and time it takes. According to Finnish-Swedish ice class rules the turning circle diameter should be less than 5 times the length of the icebreaker.

When designing an icebreaker the resisting forces must be minimized and propulsion efﬁciency maximized. Most important parameters for this are hull shape and propulsion design [4]. Their design will be discussed in subsequent sections. As demonstrated in section 2.3.4, global ice loads relate to ice resistance, therefore performance, and are divided in the following categories:

1. Breaking forces 2. Submerging forces 3. Sliding forces

At low speed in level ice around 50 % of the resistance is due to the breaking of ice category. However, this percentage varies for different ice features and scenarios. A representation of ice resistance experienced by an icebreaker is given in Figure 2.21. As the ﬁgure shows, maximum load depends on the bending failure. The ice ridge resistance can be measured and will depend on ridge thickness. Besides a bow force also a friction force at the parallel mid-body occurs. It is common to speak of energy required to penetrate a certain ice ridge. During impact the icebreaker’s kinetic energy is consumed. Ice ridges may require several ramming attempts. According to Riska (2010) [4] a large tanker has an energy of about 1 k J/m2, this is energy per cross sectional area of the ice ridge.

The resistance does not depend on ice loads only, but also on open water resistance. In general icebreakers have worse open water performance than conventional vessels. Ice load calculations are complex, and ice model tests are necessary at the end of the conceptual design phase to verify the design. 2.3.I CEBREAKERS 31

PERFORMANCE ENHANCING SYSTEMS Performance enhancing systems are designed to reduce the power necessary for propulsion and will increase icebreaker maneuverability through ice. Prevention of sea ice accumulation at the hull, and the effect of snow and sea ice accumulation on the friction forces can be mitigated by the following systems:

• Low friction coatings to the shell plating, reducing hull roughness. • Compressed air bubble systems. This system uses air compressors and nozzles to blow air bubbles below the waterline. The air bubbles rise to the surface together with entrained water, lubricating the interface between ice and icebreaker hull. This lubrication happens both above and below the waterline. This prevents ice formation and lowers the friction. • (Heated) Water-wash system. This system uses pumps and nozzles to lubricate the hull from above the water line. This water may be heated. The objective is to ﬂood the ice with water, thereby lubricating the interface between ship and ice, and to wash away any snow cover from the ice to be broken. • Heeling systems Purposely generating heeling movements (mainly roll) may break the ice around the hull, lowering ice accumulation. This system is especially helpful if the icebreaker gets stuck in pres- sured ice, or when beached on an ice feature.

HULLDESIGN The primary consideration in choice of the hull form is the lowest power required to make progress in ice. Open water performance are of the less importance for an icebreaker. The hull shape has a profound effect on the icebreaker performance and must be designed in the most optimal way. The icebreaker hull shape design strives at:

• Minimizing the ice resistance • Good operational (manoeuvring) characteristics • Enabling the icebreaker to go astern • Optimal propeller performance by minimizing propeller-ice interaction

The hull shape will inﬂuence the ice resistance and maneuvering characteristics. The bow must be shaped in a way it can break level ice effectively, and at continuous speed. Several bow types can be used, i.e. wedge- shaped, spoon-shaped, square shaped. Furthermore, an icebreaker bow must be developed to ride up on the ice, when encountering pressure ridges or similar ice features that will not break on the ﬁrst ramming. The most important parameters on the hull design are beam and stem angle. A classical bow shape of a modern icebreaker is shown in Figure 2.26.

Figure 2.26: Classical bow shape of a modern icebreaker [4].

An assessment must be made on the beam magnitude. Commonly the channel width made by an icebreaker is about two ice thicknesses wider than the beam of the icebreaker. Larger beam enables wider channels, but will also increase ice resistance. The typical largest icebreaker beams at present are about 26 meter. With respect to hull shape’s inﬂuence on maneuvering performance, the shoulder area is crucial for good manoeuvring characteristics. Icebreakers will encounter additional ice forces at the sides when making turns in ice. This effect can be mitigated by using reamers. For example the vessel Oden has reamers. 32 2.I CE ACTIONSAND ICE RESISTANCE

A stem angle around 20 degrees is common for icebreakers. Because the angle is small it improves the ice bending failure mode. Nowadays the stem is rounded which decreases crushing at the stem. Good backing performance is important when beached, this is enhanced by avoiding blunt lines at the stern. The stern angle normally is of less importance.

All icebreakers must be able to move astern in ice. Some only have to do this in the channel with broken ice. Other icebreakers are purposely designed in a way that they can break ice when moving astern. This latter type requires additional breaking and deﬂecting capabilities at the stern. While moving astern one of the main concerns is getting ice blocks into the propellers.

The icebreaker propeller may encounter ice floes. The hull shape influences the amount of ice that gets below the icebreaker bottom and therefore influences propeller-ice interaction. One way to decrease this interaction is by using a bow plough, indicated in Figure 2.26. Floes moving along the hull will hit the bow plough which pushes them aside. The bow plough’s effectiveness decreases with increasing ice thickness. A disadvantage is lowered open water performance. For icebreakers the advantages outrank the disadvantages.

Masts, rigging, superstructures, deckhouses and other items on deck must be designed and arranged so that excessive icing is avoided. Rigging should be kept at a minimum, as should the surfaces of erections on deck.

MACHINERY LAYOUT The icebreaker machinery is responsible for producing the required thrust. Key components of the propulsion machinery are main engine, power transmission and the propeller.

Machinery alternatives . The machinery layout can be realized in several ways. A diesel engine or engines with a direct shaftline transmission (can be with or without gears) can be used. This can be realised with a fixed pitch propeller (FPP) or controllable pitch propeller (CPP). As the name indicates CCP allowes the propeller pitch to be changed, which gives you an advantage: it allows the ship to change direction without first stopping, and then reversing the engines. One alternative engine type is the diesel-electric engine: power transmission is electric and separate electric propulsion motors supply the propellers with torque. Here the propeller shaft is spun by the powerful electric motors, who utilize magnetic fields for propulsion as opposed to being directly connected to the diesel powerplant which would require additional gears and mechanical linkages. This is an advantage in the case an ice block hits the propeller. When a piece of ice hits the propeller there is a shock going through the shaft. If a propeller would get jammed in the ice, a diesel motor with direct connection would get it harder: more likely to be damaged. Russia possesses several icebreakers which use nuclear energy for power. Sometimes gas turbines were build on the vessel. These gas turbines were so powerful they could only be used for short bursts.

The performance depends on ice conditions. Propeller–ice interaction threatens the integrity of the propulsion and decreases the propulsion efficiency. When much ice is acting on the propeller, ice torque can exceed the engine produced torque, as a consequence the propeller slows down. Diesel engines have a relatively small RPM range where they can deliver full power. Direct drive diesel solutions may stall in situations with heavy ice loads. This often leads to an engine stop. It is more convenient to use CCP at icebreakers. CCP can adjust to the decreased torque by decreasing the propeller pitch and in this way maintain the RPM. The electric motors can maintain the torque in a large RPM range, thus the diesel-electric machinery is a more efficient engine for icebreakers. In general the required torque is increased, while produced thrust is decreased if ice-propeller interaction occurs. The diesel electric propulsion requires a control system that allows an over-torque. Over-torque occurs when the torque exceeds the maximum torque absorbed by the propeller. An immediate RPM adjusting will reduce icebreaker performance. Allowing over-torque for short periods of time enhances thrust efficiency. Riska (2010) [4] noted that several knots in ship speed can be gained by allowing 40 % of over-torque. This was proven by full-scale trials with icebreaker Fennica.

Propulsion design The propulsion systems have chanced signiﬁcantly in the past century. Single propeller solutions were improved with more propeller solutions and then to additional bow propeller solutions. Bow 2.3.I CEBREAKERS 33 propellers decrease ice resistance by decreasing breaking resistance and lubrication effects.

In icebreaker design bow propellers have been made superﬂuous by the introduction of azimuthing propulsion units, also called Z-drives. Since the construction of Fennica and Nordica, azimuthing thrusters have become the most common propulsion system in icebreakers. An azimuth thruster is a propeller placed in pods that can be rotated to any horizontal angle, making a rudder unnecessary. Azimuthing thrusters offer a superb manoeuvring capability and they also replace bow propellers because the propeller wash of the azimuthing thrusters can be used to lubricate along the icebreaker hull. Azimuth thruster increased icebreakers have better maneuverability compared to a ﬁxed propeller and rudder system. Typical manoeuvres that can be accomplished with ships having azimuthing thrusters are a turn on spot and ridge breaking moving astern. A modern alternative for azimuthing thrusters is the azipod system. Here the electric propulsion motor is in the hub of the propeller. Consequentially smaller space is required for the engine room. First manufacturer of podded drives was ABB Azipod.

Adding nozzles to the thruster adds thrust at low velocities (approximately 35%). However, these nozzles may get full of ice, which is undesirable. If a nozzle gets stuck azimuthing thrusters need to be turned around. Icebreakers Fennica and Nordica were one of the ﬁrst icebreakers using nozzles. Figure 2.27 and Figure 2.28 show examples of the azimuth and azipod thruster respectively.

During ice trials with icebreaker Fennica it was discovered that moving the thrusters from side to side can enhance performance in thick ridges. The twisting helps ice breaking loose. This phenomenon has gives even better results when moving astern due to additional lubrication. This observation resulted in dual mode ships that go forward in open water or light ice conditions but go astern at heavy ice conditions. In icebreaker design both bow hull and stern hull can be optimized to speciﬁc conditions. Diesel electric machinery with azimuth or azipod thrusters gives the most advantages. This solution however is also most expensive.

Figure 2.27: Azimuth thruster with Nozzle

HULLANDMACHINERYSTRENGTH A detailed description of icebreaker design is not presented in this thesis. However, some key components found in DNV Ships for navigation in ice (2009) [31] and Riska (2010) [4], directly related to ice loads on icebreaker hull and propeller are discussed in this section.

Hull strength Different modes of operation and ice regimes will generate different magnitudes of ice loads. Design of hull structures requires knowledge of the ice loads acting on different regions of the ship hull. An icebreaker’s performance in ice can be limited by the hull structure’s capability to withstand ice impacts. Using ice classes a designer can determine the ice loads and required hull structure strength. With respect to force magnitude, ramming operations generate the largest forces on the icebreaker. These impacts can be infrequent like a MY ice ridge or repetitive like ice floes in an ice field, which can result in direct damage or cumulative damage respectively. For the Finnish-Swedish ice class the following key steps with respect to hull strength design are identified: 34 2.I CE ACTIONSAND ICE RESISTANCE

Figure 2.28: Azipod thruster without Nozzle. The electric motor is inside the submerged pod

• Determine the Load patch. Idealize load patch to be used for structural design. • Determine ice pressure. Both average ice pressure as well as local ice pressure is of importance. Pres- sure depends on ice failure mode (crushing giving highest ice pressure), dimensions (Sanderson 1988 Pressure-Area relationship), conﬁnement and coverage. • Determine design point. Design point includes the allowed structural response and how frequently it is reached. Allowed structural response can be based on maximum stress or maximum yield. Statistics of the loading must be presented.

Most important design loads according to DNV Ships for navigation in ice (2009) [31] are:

• Ice impact forces. Mainly on the bow, but may also be important at stern and sides. For different ice features • Beaching forces. At larger ice features like ice ridges. • Ice compression loads amidships. Side hull must be able to withstand compressive ice forces, for example from converging ice ﬁelds or maneuvering. • Local ice pressure. Highest pressures are identiﬁed at local hot spots. Important for structural strength. • Allowed accelerations. Substructures, equipment and supporting structures must withstand accelerations arising as a result of impacts with ice features.

When designing special attention should be paid to the hull material. Fracture toughness of steel depends on operating temperature and on the loading rate.

Machinery loading All ship machinery must have sufficient strength, which is determined by hydromechanical forces, hydrodynamical forces and ice forces. Ice-structure interaction might occur at the propulsion system when a piece of ice hits it. For example the ice floes which are submerged below the icebreaker bottom. Another ice load case may occur at beaching situations. In large ice ridges with keel depth in the range of icebreaker draft or larger the hull is confined. Ice cannot be effectively cleared. The icebreaker hull becomes entirely surrounded by ice. In this case propeller-ice interaction can not be avoided. Ice floes under the ship bottom can hit propellers and rudders.

Design point of the propeller blades is an impact with an individual ice ﬂoe. Ice load magnitude depends on: force on the propeller, mass of the ice ﬂoe, velocity icebreaker, propeller rotation rate and propeller diameter. Ice loads on the propeller can be prevented by designing an adequate ice clearance and stern frame clearance, see Figure 2.29. This means that the propeller must be submerged deep enough for a good ice clearance. Propeller clearance, or stern frame clearance, must be large to lower the ice loading. A narrow clearance between the propeller blade tip and the stern frame or the bottom of the level ice sheet should be avoided as a small clearance will cause high loads on the propeller blade tip. According to Finnish-Swedish ice class rules, stern frame clearance should be at least 0.5 meter and ice clearance should be positive when the level ice thickness is taken as stated in the rules [32].

Design forces of the propeller blades are determined by the size of the impacting ice ﬂoe. Research on these propeller blades has been done by for example Marquis & al. (2008) [33]. A formulation for the rules can be found in ice rules. 2.4.I CERIDGELOADS 35

Figure 2.29: Deﬁnitions of ’Stern frame clearance’ and ’Ice clearance’

2.4. ICERIDGELOADS Vessel performance in ice depends on the ability to break ice and to manoeuvre in ice. Ice ridges are distinct ice features in arctic areas, which tend to give high ice loads when they interact with a vessel. Vessel performance in ice therefore depends on the ability to break ice ridges. This ice ridge breaking ability is measured by the ability to penetrate the ridge. The total resistance in ice is assumed to be the sum of open water resistance and pure ice resistance. Distinguish can be made between breaking resistance at the bow, and the parallel mid-body resistance. Parallel mid-body describes the middle side area of the vessel’s exterior hull which is ﬂat and usually vertical. Parallel mid-body resistance depends on the friction coefﬁcient for the interaction between hull and ice.

The ramming of an ice ridge is a process consisting of one or several ice crushing impacts followed by beach- ings. The ramming is completed when the ship breaks through or is stopped in the ridge. When a vessel cannot completely break through, the vessel will slide backwards. The ridge resistance depends on the ridge thickness, which is different at each location along the hull. Therefore the ridge resistance will change in time during a ramming.

When looking at the hull loads it is important to differentiate between global and local forces. Local forces can refers to two doings, ﬁrstly it can be the total force of any single hull structural element, secondly it can refer to loading that is part of the total load. The global load refers to the total contact force from any single interaction scenario. In this case the total force is the sum of all the ice loads acting simultaneously at the ship.

When calculating ridge loads there often is a lack of knowledge in the internal structure, deformation modes and full scale data. Some full scale data is available from offshore structures like the Molikpaq platform, Kemi 1 lighthouse (Finland), Nordstrømsgrund lighthouse (Sweden) and Confederation bridge (Canada). Some full scale data from ice ramming experiments is collected by Chen et al.(1990) [5] by measuring the hull gauge strains. Model tests was executed by T. Leiviskä (1999) [34] to ﬁnd the ridge resistance.

The ridge load depends on the ice action. For ice ridges the most important ice actions are: 1. Crushing 2. Bending 3. Spalling 4. Splitting 5. Shearing If the driving force is sufﬁciently large crushing can occur. Crushing is a general description of ice failure into small particles. The highest values of ice pressure are associated with crushing. As ice breaks the particles must be removed from the contact area, and therefore requires a ﬂow moving away from the contact center. Studies show crushing occurs at high indentation rates and without perfect contact between structure and ice. High pressure zones are formed on the contact zones resulting in local crushing on these zones. High pressure point locations change in time and are non-simultaneous. However research showed that the high pressure points develop at the center. These areas can be seen as critical zones, and will carry most of the ice load. For ship ramming this critical zone sometimes is simulated as a line-like load (Riska, 2010) [4]. 36 2.I CE ACTIONSAND ICE RESISTANCE

Spalling occurs due to out-of-plane horizontal cracks growing away from the contact zone. When a vessel is ramming through ice, ice fractures at the free edge bordering the actual contact area between ice and structure. Because of these fractures ice pieces are spalling off, reducing the contact area between vessel and ice. According to Zou et al. (1996) [8] spalling governs the variation of size, number and location of high pressure zones during the interaction process. It was found that the crack would propagate at loads lower than those found using damage analysis only. The crack propagating resulting in spalling will induce a reduction of the ice force. They showed that crack propagation increases in zones of low confinement, which are located near the free surface. Figure 2.30 shows these spalls and the critical zone in which the high pressure points occur. How well Figure 2.30 is comparable to reality greatly depends on the interaction between ice and structure. The size and geometry of both ice and structure influence the confinement, which determines the likelihood of spalling. Level ice represents a totally different situation than an ice ridge, as a ridge will imply higher contact areas, confinement, etc. In Figure 2.30 an angle between wall and ’undamaged’ ice can be imagined. The spallings must be moved away from the region composed by this angle and wall, i.e. the ice pieces must ’escape’ between hull and ice. For smaller angles escaping becomes more problematic, therefore pieces of ice broken off will be crushed against the hull before they can escape. This reasoning is also applicable on velocity. High velocity will increase the entrapment of broken pieces of ice, increasing the amount of crushing. Spalling therefore plays a key role in the formation of critical zones.

Bending failure is enhanced by using sloping structures. In the case of a vessel hitting an ice feature it will induce a bending moment into the ice that can break the ice. This is known as bending failure. After breaking the broken pieces are removed which contributes to the ice load. For a ice going vessel this means that ice pieces are pushed sideways or underneath the bow. From a structural design point of view bending failure has the beneﬁt that bending action loads are typically lower than the other failure modes.

Splitting is fracturing of ice. Crack formation can be in-plane, or out-of-plane. Out-of-plane failure contains both radial and circumferential cracking. This failure mode is highly applicable on ice floes. Lu (2014) [35] identified two requirements before cracking is initiated: 1) crack forming must be initiated, i.e. there must be an initial crack, 2) a sufficiently high lateral force must be present. Lu showed that splitting failure depends on the confinement and size of the ice feature. Ice brittleness will influence the splitting mode as well. As ridges have a relatively high confinement compared to ice floes the likelihood of splitting failure is lower. To enhance splitting failure it is possible to mitigate confinement conditions by clearing the ice behind a ridge. The lateral force is largely dependet on the icebreaker hull design.

Shearing, or plug failure, on ice ridges has been observed at vertical structures, for example at the lighthouse Norströmsgrund (Kärnä et al. 2003) [36], (ISO19906, 2010) [9]. As the name suggests an ice plug is formed around the width of the structure due to shear failure, which then can be pushed out of the ridge alignment. With respect to vessel-ice interactions plugging is a realistic failure mode, especially for small ridges. How ice breaks during a ramming experiment depends on the normative ice action. Visual observations can

Figure 2.30: Visualization of a critical zone with spalls (Zou, 1996) [8]. 2.4.I CERIDGELOADS 37 help in the identification of ice actions. However, distinguishing to what extend a specific load action contributes to the ice load, while changing in time, is difficult. Physical uncertainties in ice tend to be significant, and it is hard to identify what was exactly happening in the ice and structure during structure-ice interaction.

In Section 2.3.4 it was found that ridge resistance can be described by: Z RR,TOT RR,B (HR (xbow )) RR (HR (x)) dx Row (v) (2.12) = + Lpar · + with RR,TOT as total pure ice resistance, Rr (HR (x)) is the ridge resistance in ridge thickness Hr per unit ship length, RR,B is the ridge breaking resistance acting at ship bow, Lpar is the parallel midbody length of the ship, v is vessel speed, and Row is the open water resistance.

It was found that bending is a very common failure type in ice ridges. An ice breaking ship cannot break the ice ridge depending on its thrust, as the resistance of even small ridges causes resistance higher than the thrust. The ship penetrates ridges by using its inertia. The kinetic energy of the vessel is consumed depending on penetration rate and ridge dimensions. According to Riska (2010) [4], and Tsuprik (2013) [37], energy principles a more convenient way to deal with ice ridge resistance, and will therefore require to be speciﬁed further. More information on speciﬁc energy principles is found in Section 2.4.3.

Lastly, it was found that the keel can be modeled as a granular material, implying that response can be modeled using the material behavior theory. Since this thesis focuses on multi-year ice ridges, keel resistance becomes signiﬁcantly lower compared to ﬁrst-year ice ridges. Therefore this thesis will not present further details on keel loads. If additional information is required more information can be found in: Dalane et al. (2015) [30], Løset et al. (2006) [6], and Palmer (2013) [7].

2.4.1. LOAD PATCH When calculating ice loads a certain load patch is commonly assumed, visualized in Figure 2.31. A load patch can be defined as an area of non-zero ice pressure normal to the vessel bow. In the case of a ridge impact this load patch is assumed to be narrow in vertical direction and long in horizontal direction. As Figure 2.31 shows the load patch depends on the ice ridge geometry. A load patch implies that the ice ridge is idealized. The impact of this load patch assumption on the ice load is hard to define. However, data from ship ramming experiments into multi-year ice indicate that the global ice action is random (ISO19906, 2010) [9]). Even seemingly identical looking rams will give different loads. Further research is required to determine these observed variations. Maybe the inhomogeneous character of ice can explain this observation. Ice contains many defects such as cracks, inclusions, pores, grain boundaries and other weaknesses. The size and location of these defects are random. According to Shafrova (2008) [20] the high strength heterogeneity in first-year ice ridges is 40-55%. To what extent ice strength variations influence the ice load should be further investigated.

The load patch assumption’s error on the load will differ from vessel to vessel and ice feature to ice feature. For practical purposes a load patch assumption is more obvious, as calculations with a simple rectangular area are easier compared to those with complex shapes. Furthermore, the geometry of the ice ridge is never completely known, even with extensive measurements. Hence, a basic load patch idealization is always required.

Assuming the load patch suggests that there are three quantities describing the local ice load: pressure pc , load height hc and load length L. It must be noted that pressure is never measured directly. The ice force F is measured on a certain gauge area Ag so the pressure is presented as F /Ag . With the force F normal on the area. F p h L (2.13) = c · c · 38 2.I CE ACTIONSAND ICE RESISTANCE

Figure 2.31: Actual load patch and its idealization (Riska, 2010) [4].

2.4.2. PRESSURE There is quite large controversy as how to treat ice pressure. Often ice pressure is described as the average pressure on the considered area. However, local ice pressure magnitudes vary considerably inside the nominal area (the load patch). One additional problem for ice pressure is the observation that ice temperature and indentation rate inﬂuence the failure mode. Ice starts to fail in a brittle fashion for lower temperatures and higher indentation rates.

According to ISO19906, the global pressure of sea ice can be described by equation 2.14. Where pg represents the pressure in MPa, w the width of the structure in metres, h the ice thickness in metres, h1 is a reference thickness of 1m, m and n are empirical coefficients and Cr is the ice strength coefficient in MPa. However, for ship ramming tests ISO19906 suggests the use of equation 2.15, where pg represents the pressure in MPa, 2 AN the nominal contact area in m , CP and DP are random coefficients.

µ h ¶n ³ w ´m pg Cr (2.14) = h1 h

p C ADP (2.15) g = P N Equation 2.15 includes a certain relation between pressure and area. The relation between pressure and area has been intensively studied and helped formulate design rules like equation 2.16. Equation 2.16 has been suggested by Sanderson (1988) and gives the upper limit for this pressure and area relationship.

0.57 p 8.1 A− (2.16) av = · N The pressure-area relationship deﬁned in equation 2.16 is the one used most. However, researchers should be careful using this ’pressure-area relationship’. Strictly speaking the pressure-area relationship can refer to three different relationships. This distinction is not made consistently. To prevent misconceptions the differences are presented as:

1. Pressure area process During indentation in ice the area and pressure will change in time (P(t), A(t)). As the indenter moves through the ice, identical areas will not necessarily display the same pressure. The indentation will cause ice damages which can be seen as ’softening’ of the ice structure, making it easier to further damage the ice, hence effecting the pressure. It is a continuous process which can be plotted as P(t) vs A(t), i.e. a pressure-area relationship.

2. Statistically defined pressure-area relationship As for example equation 2.16 suggests a lower pressure can be assumed for a larger area during design. The relationship is defined based on a statistical fit curve to all the available data points. It is empirical with little physical basis. That a relationship can be defined from the curve fit does not prove the existence of a underlaying mechanism causing the relationship. The data is achieved from mutually very different structure-ice interactions and failure mechanisms.

3. Spatial pressure-area relationship During an impact local pressure variations can be identiﬁed. This spatial pressure-area distribution changes in time, i.e. each moment in time has it’s own pressure distribution on the total interaction area. 2.4.I CERIDGELOADS 39

In this thesis, the pressure-area relationship is referring to the statistically defined pressure-area relationship. There are several reasonable explanations for the relation. Size effect is one of the reasons why the average global pressure generally decreases as a function of the nominal contact area. The size effect depends on several components such as: flaws in the ice and flaw hierarchy, ice inhomogeneity, non-simultaneous failure, fracture mechanism and boundary conditions. Due to this size effect relatively lower average pressures are expected for larger nominal areas. Another possible explanation is based on the observation that within the nominal contact area there is a line-like feature along which the ice pressure is transmitted. The line is produced by a flaking process leaving a line on which a high pressure is acting. The flakes seem to be created so that the line of high pressure is directed towards the corners of the nominal contact area (Riska, 2010) [4], Kujala, 1994) [38]. See Figure 2.32 for a visualization of this crushing and flaking failure process.

Figure 2.32: Division of the failure process into crushing and ﬂaking [38].

2.4.3. SPECIFIC ENERGY PRINCIPLES In 1968 Y.M. Popov suggested to use ‘specific energy of mechanical crushing ice’ to calculate the ice strength. This method assumes a solid body crushing into ice, and hereby spending all kinetic energy into ice crushing. Measurements were established by drop-weight tests and indentation tests. The specific energy is defined as energy per unit mass of crushed ice necessary to turn solid ice into crushed (pulverized) material. It must be emphasized that the study into specific energy principles is limited to the specific energy absorption during the local ice crushing process, e.g. fragmentation of the ice.

A team of researchers led by D.E. Kheisin (in Sint Petersburg 1967-1969) did the drop-weight experiment on lake ice to obtain numerical values for specific energy. By implementing these values in a model, known as Hydrodynamic model (GMD) of Kurdumov and Kheisin, they calculated the load on icebreakers. Several icebreakers and ice navigation-type vessels are build according to icebreaker design rules defined by this method (mainly Russia). In 1973-1978 in Vladivostok a group of scientist lead by N.G. Khrapaty continued research in this method with new experiments and phenomenological models (Tsuprik, 2013) [37]). The technique of using specific energy of mechanical crushing ice as a parameter of ice strength is still used today [5][4][37] [39][40][41].

Tsuprik (2013) [37] identified some shortcomings in their experiments on specific energy. These shortcomings include: Badly measured ice characteristics before and after impact, ignored vessel vibrations, ignored flexural vibrations of the ice cover, ignored heat release during impact, ignored kinetic energy of flying pieces of ice. But most importantly the specific energy’s dependency on temperature was not established. Not know- ing this relation makes the practical use of the results difficult. This observation corresponds with Kim (2014) [41], where is stated that the specific energy of the ice crushing process (of solid ice) depends on:

• Temperature of ice • Geometry • Size

She also stated that ice type (i.e. mechanical state) is important when making practical use of this theory, i.e. sea ice instead of solid fresh water ice. 40 2.I CE ACTIONSAND ICE RESISTANCE

Specific Energy Absorption. In Kim (2014) [41] is presented how the theory can be used, using a term called Specific Energy Absorption (SEA). The Specific Energy Absorption is defined as the energy absorbed per unit mass of crushed material. The applied work W , i.e. loss of kinetic energy, used to define Specific Energy Absorption (SEA) in J/kg reads:

Z u(t) 1 2 W mv F du SEA ρVcr ushed (2.17) = 2 = 0 = · where v is the indenter speed at impact in m/s, m is the mass of indenter in kg; F is the applied load in N, u is the penetration in meter, g is the gravitational acceleration in m/s2, ρ is the average ice density in kg/m3 3 and Vcr ushed is the volume of crushed ice in m . The deﬁnition assumes a constant indentation speed and homogeneous ice properties. The pressure calculated p F /A is constant per crushed mass. =

Dependent on temperature. Based on experiments Tsuprik defined a relationship between the specific energy on temperature which is shown in Figure 2.33. For more information about the relationship of the specific energy on temperature is referred to Tsuprik (2013) [37] and Ekaterina (2014) [41].

Figure 2.33: Speciﬁc energy of mechanical fracture of sea ice, depending on the temperature [37].

Independent of interaction scenario. Results in Kim (2014) [41] indicate that the speciﬁc energy of the ice crushing process is independent of the interaction scenario: drop-weight tests and indentation tests give similar energies. This knowledge of scale transition is important because tests are often conducted on miniature ice specimens.

Independent of intender velocity and mass. It was concluded that the mass of intender and velocity are not inﬂuencing the speciﬁc energy for crushing (Tsuprik, 2013) [37].

Dependent on mechanical state of ice. Sea ice can be very different than one solid piece of fresh water ice. In [41] speciﬁc energy absorption during crushing of (laboratory grown) freshwater ice was compared with lake ice and iceberg ice. According to this research ice type is an important factor. 2.4.I CERIDGELOADS 41

Size- and scale-independent. Kim (2014) [41] investigated the size and scale in-dependency of the specific energy during crushing. In [41] the laboratory freshwater ice was investigated at small scale, the lake ice and iceberg ice at medium scale. It was concluded that specific energy for crushing has a size- and scale- independent characteristic under certain geometrical structure and ice feature conditions. These conditions are related to the grain size; if the ice grains are sufficiently small compared to the penetration depth and indenter size, and the sample size (width and thickness) is sufficiently large compared to the indenter size and the penetration depth, the specific energy is a size- and scale-independent characteristic.

Confinement due to size and geometry of the ice It must be emphasized that the experiments by Tsuprik (2013) [37] are accomplished with a spherical intender body. This body is geometrically very different from the hull of some vessels (Like for example icebreaker Oden). When looking at impact, studies show that this geometry is one of the components that is important as it effects the confinement. Different boundary conditions resulting from ice-structure deformations, local shapes of the structure and ice, global shape and size of the ice mass will promote or suppress splitting of ice (Kim, 2014) [40]. Brittle-like failure of ice via tensile spalling occurs at relatively low degrees of confinement (this implies an easier ’escape’ of spallings as discussed before). With an increasing level of confinement the ice exhibits Coulombic and plastic faulting. Looking at impact problems it therefore is essential to realize the most critical loading conditions. The local ice shape is a concern and it can be assumed that the degree of confinement will affect the energy consumption during crushing, hence affecting the specific energy for crushing value (Kim, 2014) [40]). This is shown in Figure 2.34.

Figure 2.34: Speciﬁc energy of mechanical fracture of sea ice, depending on the conﬁnement and angle of attack [40].

SEA as a physical property? A state variable is one variables of the set of variables that are used to describe the mathematical "state" of a dynamical system. These are important in experiments: stress, stress rate, strain, strain rate and temperature. Material properties characterize the material but are independent of sample size and boundary conditions, it describes the state of the material. Then there are different types of material properties: physical properties and mechanical properties. Physical properties can be observed or measured without changing the composition of matter, for example: density, thermal conductivity, latent heat, porosity, salinity, grain size, grain orientation, etc. A mechanical property is a property which is the behavior of the material when it is linked to the application of force such as strength. A mechanical property therefore tells how a material behaves under force, think about: elastic limit, viscosity, hardness, fracture toughness, etc. Given that specific energy of ice crushing is independent of; intender velocity, intender mass, interaction scenario, and (generally) also independent of scale and size, it raises the question if specific energy principles may be defined as a physical characteristic of ice. To the authors knowledge this is not confirmed, but if correct it presents a promising theory for ice impacts under the assumption of pure crushing/fragmentation. 42 2.I CE ACTIONSAND ICE RESISTANCE

Speciﬁc energy principles during ice ridge ramming eperiments Summarized, for a vessel to ridge impact the speciﬁc energy depends on:

• Temperature of ice • Conﬁnement due to size and geometry of the ice • Mechanical state of the ridge

2.4.4. PREVIOUSRESEARCHINTOVESSEL-ICERIDGEIMPACTPHENOMENA The analysis of ice ridge impact phenomena has been restricted due to the complex ice behavior under conditions of rapidly applied stress and the complex geometries of the bodies in contact. This field has not yet been developed to the point at which design standards based on analytical considerations can be firmly established [4]. However, ice ridge load calculation and identification has been carried out before which gives some feeling for problem complexity, possible approaches and load expectations.

This Section discusses the analytical model by Chen et al. (1990) [5], model test executed by T. Leiviskä (1999) [34] and a FEM model by Ørjan Fredriksen (2012) [42]. These include research into ice ridge loads on vessels.

THEANALYTICALMODELDEVELOPEDBY CHENETAL. (1990) Chen et al. (1990) [5] tried to develop an analytical model capable of describing the ship-ice impact interaction. The analytical model used was the Hydrodynamic Model impact model developed by Kheisin and Kurdyumov (and therefore using the speciﬁc energy principles). In their model Chen et al. distinguish between the two phases discussed earlier: the initial phase of impact penetration into ice accompanied by ice crushing, and the subsequent beaching phase. Besides the analytical model full scale measurements were done. The recorded data of the ramming forces, their durations and time histories, ramming velocity records, and the locations of the ramming forces, were analyzed and compared with those predicted by the analytical model. Bending strain gauge measurements on the hull girder were used to estimate the longitudinal bending moment distribution of their ice breaking vessel during impacts with ice ridges.

Chen et al. concluded that the analytical model was capable of predicting the impact loads, as the model gave reasonable estimates of the force magnitudes. Looking at the force records several peaks where found for each ramming. Chen et al. suggests that the total ramming exists out of several impacts against separate, or loosely connected, ice features. According to their study almost al submitted records contain a peak force exceeding 14 MN. They also concluded that the impact location on the bow was an important factor in the peak stress values. Higher peak stresses were measured when the force was applied more towards the forward end. For the construction of ice breaking vessels this force location therefore, can be very important. In their paper Chen at al. noted that the analytical model’s accuracy should be improved, but that this was not possible due to lack of ice ridge data and velocity data.

In the analytical model the ice load was calculated by equation 2.18. Where Fn is the global impact force normal to the stem in [MN], KT a coefficient, D displacement measured in [kg], V the velocity in [m/s], A a parameter characterizing the dynamic crushing strength of ice in [MPa(s/m3)1/4], exponents (d, v, a) are coefficients depending on ice ridge shape, S is an shape factor depending on particular slopes of the ship bow and the ice feature, F (t) the force time history factor [ ]. The used parameters seem empirically determined − as they are presented as values for square, round and spoon shaped ridges. But there is a physical basis behind the parameters. Hence for the listed geometries the parameters are analytically precise. For complex shaped ridges the parameter determination becomes difficult and approximations are required.

F K Dd V v Aa S F (t) (2.18) n = F · · · · ·

SHIP’SRIDGERESISTANCEINMODELICE T. Leiviskä (1999) [34] studied the ship ridge resistance at model scale. Of particular interest was the ratio of the components for bow and parallel mid-body. The model test was performed in the ice tank of the Ship Labratory at the Helsinki Univercity of Technology. The model ice used was granular ﬁne-grained ice, with a model ice temperature of -15 degrees Celcius. The model ridges were manufactured by breaking a level ice sheet into pieces and then pusing them into a long pile. The consolidated layer effect was studied by freezing the ridges between tests. The used model scale was 1:20. 2.4.I CERIDGELOADS 43

T. Leiviskä [34] suggests the use of Malmberg’s (1983) equation for ice ridge loads, given in equation 2.19. This equation takes into account both bow and parallel mid-body.

µ 1 ¶ RR ρ gT (1 n) {Kp B H tan(ψ) cos(α) = M − · 2 + · · (µ cos(φ) sin(ψ) sin(α)) (2.19) · H · + · µ H 1 ¶ µH Lpar K0H ( )B } + · + T − 2 where Rr is the ridge resistance in Newton, n the porosity, ρM the difference in density between water and 3 2 ice in kg/m , g is the gravitational acceleration in m/s , T the draught in meter, B the beam in meter, Kp the coefficient of passive stress, H the ridge thickness, µH coefficient of friction, K0 the coefficient of lateral stress, Lpar the length of parallel mid-body, α the waterline entrance angle, φ the stem angle and ψ defined in equation 2.20. µ tan(φ) ¶ ψ (2.20) = sin(α) According to Leiviskä’s study the proportion of the resistance component for the parallel mid-body was 16 to 32 percent of the total resistance. Furthermore the thicker and harder the consolidated layer, the lower the resistance for the parallel mid-body. This implies that the ridge resistance for MY-ice ridges should depend mainly on the bow-ridge interaction. The results also indicate that parallel mid-body resistance is barely influenced by the consolidated layer. Results confirm that ice ridge resistance increases with velocity.

The model test shows that Malmberg’s equation gives quite close results for the bow resistance. For the parallel mid-body resistance however it works poorly, the resistance gets underestimated. A second disadvantage of equation 2.19 is that it does not take velocity into account. According Leiviskä’s study this velocity inde- pendence is wrong.

FEM MODELBY ØRJAN FREDRIKSEN. Ørjan Fredriksen (2012) [42] made Two FEM models to determine ice-induced loading on a ship hull during ramming. The models consist of a beam element model and a detailed shell element. Quasi-static and dynamic response analyses for ice ridge impact loading are carried out, where the duration of the load pulse is varied systematically from 0.25 s to 2.0 s. When comparing the models, differences of up to 40% were identiﬁed for the peak response, where the beam model predicts a larger peak response than that of the shell model. When compared to the full scale data large discrepancies are identiﬁed, especially for the shell model. In the study this is mitigated by modifying the models, therefore losing the reliability of the model, at least in physical sense.

GLOBALRESPONSEOFSHIPHULLDURINGRAMMINGOFHEAVYICEFEATURES. Bromas (2013) [43] aims at analysing ice- induced global forces that affect a ship’s hull when colliding with heavy ice features. A model was developed that used recorded motions in order to calculate the global forces that affect the ship hull. The motion data was collected by DNV during the Coldtech research project on board the Norwegian Coast Guard vessel KV Svalbard. In Bromas (2013) [43] was concluded that the vessel’s motions can be used to calculate the global forces colliding with ice features. Main problems regarding this research is that only one MRU was used, which means that measurements could not be veriﬁed. Furthermore, it is unclear what type of ice feature is rammed, and what their size was. Bromas concluded that the forces on a vessel’s hull due to hitting of large ice features, are mostly governed by the acceleration of the vessel body. 3 THEORETICAL BACKGROUNDON LOAD IDENTIFICATION AND HYDRODYNAMICS

3.1. INTRODUCTION This chapter presents the basics regarding load identification and hydrodynamics of marine craft. Both are fundamental components in the aim of finding the global ice loads. Section 3.2 presents background on the load identification method: joint input-state estimation method for linear systems. This section also briefly discusses the Kalman filter. Section 3.3 presents the background on hydromechanics, which includes axis convention, hydrostatics, potential coefficients, maneuvering theory and propulsion theory.

3.2. JOINTINPUT-STATE ESTIMATION METHOD FOR LINEAR SYSTEMS This section presents an algorithm for jointly estimating the input and state of a structure from a limited number of acceleration measurements. The algorithm extends an existing joint input-state estimation filter, derived using linear minimum-variance unbiased estimation (Lourens et al. 2012) [44]. The Kalman filter, briefly discussed in section 3.2.2, looks similar to this filter, except that the true value of the input is replaced by an optimal estimate. The joint input-state estimation algorithm assumes no prior information on the dynamic evolution of the input forces, and no regularization is required. It must be emphasized that state estimation algorithms exist for linear and nonlinear systems. This algorithm is applicable to linear systems. For nonlinear systems the extended Kalman filter can be used (Corigliano, 2004) [45].

A collection of variables that completely characterizes the motion of a system is called the state of a dynamical system [46]. The purpose of the state is to predict future motions, e.g. displacements, velocities. State variables are gathered in a vector which is called the state vector. The set of all possible states is called the state space. State estimation can be deﬁned as the process of identifying, based on a system model, quantities that allow a complete description of the system state from vibration response data.

Many theories assume that the input forces are either known or broadband, this way they can be modeled as a zero mean stationary white process. However, often no measurements of the input force are available or the broadband assumption is violated (Lourens et al. 2012) [44], (Laurens et al. 2012) [47]. A subsequent problem occurring is that the algorithms assume that the location of the forces is given, which often is not the case. In this thesis that problem is solved using a combination of the contact model presented in section 5.4, and an algorithm that first uses a predictor location followed by an iteration to a more certain location. Elaboration about contact point identification is given in chapter 5. The logarithm presented in Lourens et al. (2012) [44] enables identification of a set of equivalent forces, meaning that the acting points of the forces are arbitrarily chosen and equivalent forces, that would produce the same measured response, are identified at all chosen locations. The methodology presented in this chapter assumes that locations of forces are known, and therefore enables to jointly estimate the states and input forces at the correct location.

44 3.2.J OINTINPUT-STATE ESTIMATION METHOD FOR LINEAR SYSTEMS 45

3.2.1. MATHEMATICAL FORMULATION EQUATION OF MOTION The continuous-time governing equations of motion for a linear system discretized in space is given by:

Mu¨(t) Cu˙(t) K u(t) f (t) S (t)p(t) (3.1) + + = = p nDOF nDOF nDOF where u(t) R is the vector of displacement, M, C, and K R × are the mass, damping and stiff- ∈ ∈ ness matrix respectively. f (t) is the excitation vector, which is factorized into an input force inﬂuence matrix nDOF np np S R × , and the vector p(t) R representing n force time histories. Each column of the matrix p ∈ ∈ p Sp gives the spatial distribution of the load time history in the corresponding element of the vector p(t). In case of a point load, the column of Sp has only a limited number of non-zero entries corresponding to the distribution of the load over the structure mesh.

CONTINUOUS-TIME STATE-SPACEMODEL ns ns Consider the state vector x(t) R × with n 2n ∈ s = DOF µu(t)¶ x(t) (3.2) = u˙(t)

The second-order equation of motion can then be written as a ﬁrst-order continuous-time state equation by:

x˙(t) A x(t) B p(t) (3.3) = c + c ns ns ns np where the system matrices A R × and B R × are deﬁned as: c ∈ c ∈ · 0 I ¸ Ac 1 1 (3.4) = M − K M − C − − · 0 ¸ Bc 1 (3.5) = M − Sp Subsequently, consider the measurement data vector d(t) Rnd . This data vector contains n observed quan- ∈ d tities, which are expressed as a linear combination of the displacement, velocity and acceleration vectors:

d(t) S u¨(t) S u˙(t) S u(t) (3.6) = a + v + d n nDOF where S , S , S R d × are selection matrices (for acceleration, velocity and displacement respectively). a v d ∈ These data selection matrices can be used to specify the measurements and/or difference relations. In state space form equation 3.6 can be written as:

d(t) G x(t) J p(t) (3.7) = c + c n ns n np where G R d × is the inﬂuence matrix, and J R d × the direct transmission matrix, which can be de- c ∈ c ∈ ﬁned as:

1 1 Gc [Sd Sa M − KSv Sa M − C] = − − (3.8) 1 J [S M − S ] c = a p The continuous-time state space model for full-order system described by equation 3.1, can now be represented by equation 3.3 and equation 3.6.

DISCRETE-TIME STATE-SPACEMODEL Using a sampling rate of 1/ t the state-space model can be discretized to yield it’s discrete-time equivalent 4

xk 1 Axk Bpk (3.9) + = + d Gx Jp (3.10) k = k + k where x x(k t), p p(k t), d d(k t), k 1,....N, and k = 4 k = 4 k = 4 = Ac t 1 Ac e 4 , B [A I]A− Bc , = = − c (3.11) G G , J J = c = c 46 3.T HEORETICAL BACKGROUNDON LOAD IDENTIFICATION AND HYDRODYNAMICS

3.2.2. JOINTINPUT-STATE ESTIMATION ALGORITHM The idea of the joint input-state estimation algorithm is to jointly estimate the input and state of a structure, in a way that minimizes the mean of the squared error. The linear system under consideration is the discrete- time state-space system of equation 3.9 and equation 3.10, supplemented with stochastic system noise (w k ∈ Rns ) and stochastic measurement noise (v Rnd ), giving: k ∈

xk 1 Axk Bpk wk (3.12) + = + + d Gx Jp v (3.13) k = k + k + k It is assumed that the stochastic system noise and measurement noise are to be mutually uncorrelated, zero- mean, white noise signals with known covariance matrices Q E[w wT ] 0 and R E[v vT ] 0. For infor- = k k ≥ = k k > mation concerning a case that stochastic system noise and measurement noise are correlated, the reader is referred to Lourens (2012) [44]. The process noise covariance Q and measurement noise covariance matrices might change with each time step or measurement (Welch and Bishop, 2006) [48]. It is assumed that the rank of the direct transmission matrix J equals the number of applied forces. This condition is required to enable the existence of an unbiased input estimator, implying n n . Thus the number d ≥ p of forces that can be identiﬁed is limited to the number of acceleration data available.

Consider xˆk k 1 to be the a priori state estimate of xk at step k given knowledge of the process prior to step k, | − and xˆk k to be the a posteriori state estimate at step k given measurement dk k . Similarly, the error covariance | | matrices: a priori error covariance matrix Pk k 1 [(xk xˆk k 1)(xk xˆk k 1)] and a posteriori error covari- | − = − | − − | − ance matrix Pk k [(xk xˆk k 1)(xk xˆk k )]. It is assumed that the initial state and it’s variance are known. | = − | − − | The Joint input-state estimation algorithm computes the force and state estimates in three steps: input estimation, measurement update, time update. The algorithm is presented below:

Input estimation:

T Rˆk GPk k 1G R (3.14) = | − + T 1 1 T 1 M (J Rˆ− J)− J Rˆ− (3.15) k = k k pˆk k Mk (dk Gxˆk k 1) (3.16) | = − | − T 1 1 Pp[k k] (J Rk− J)− (3.17) | = Measurement update:

T ˆ 1 Lk Pk k 1G Rk− (3.18) = | − xˆk k xˆk k 1 Lk (dk Gxˆk k 1 Jpˆk k ) (3.19) | = | − + − | − − | ˆ T T Pk k Pk k 1 Lk (Rk JPp[k k] J )Lk (3.20) | = | − − − | Pxp[k k] Ppx[k k] Lk JPp[k k] (3.21) | = | = − | Time update:

xˆk 1 k Axˆk k Bpˆk k (3.22) + | = | + | · ¸· T ¸ £ ¤ Pk k Pxp[k k] A Pk 1 k AB | | T Q (3.23) + | = Ppx[k k] Pp[k k] B + | |

The above system matrices are not indexed (for ease), but the algorithm can be applied to time-variant systems by simply adding the appropriate subscripts. Equation 3.16 identiﬁes the load at step k given measurement data up to step k. It uses equation 3.15, which can be seen as a weight factor of the trustworthiness of the actual measurement and the prediction of the measurement, with the purpose to identify the load pk k with maximum accuracy. The same reasoning ap- | plies to the a posteriori state estimate, presented in equation 3.19. Equation 3.19 uses Lk which can be seen as a gain or blending factor that minimizes the a posteriori error covariance. If error covariance R approaches 3.2.J OINTINPUT-STATE ESTIMATION METHOD FOR LINEAR SYSTEMS 47

zero, the actual measurement dk is trusted more and more, while the predicted measurement, related to error covariance matrix Pk k 1, is trusted less. Likewise, if the a priori estimate error covariance approaches zero, | − the actual measurement is trusted less and less, while the prediction for the next step is trusted more and more. Mk has therefore a similar purpose as Lk , but in relation to the load identiﬁcation (pˆk k ). | Equation 3.20 calculates the update of the error covariance at step k, i.e. a posteriori error covariance. Equa- tion 3.23 predicts the error covariance for the next step, i.e. equal to the a priori covariance for the next step (k 1). In equation 3.22 the a priori for the next time step is determined. +

FILTERPARAMETERSANDTUNING The choice of the diagonal elements of the covariance matrices Q and R should be based on the order of magnitude of the state vector and the accuracy of the sensors, respectively. Measurement noise covariance R is usually measured prior to operation of the filter, and this in general practically possible. The determination of the process noise covariance Q is generally more complicated, since it is difficult to directly observe the process that is being estimated. Generally superior filter performance can be obtained by tuning the filter parameters Q and R.

KALMAN FILTER The Kalman filter is very similar to the described algorithm above and can be defined as a recursive linear state estimator designed to be optimal in a minimum-variance unbiased sense. When the joint input-state estimation algorithm is applied with B=J=0, the Kalman filter is obtained. An ongoing Kalman filter cycle uses two different steps: time update and measurement update. The time update projects the current state estimate ahead in time. The measurement update adjusts the projected estimate by an actual measurement at that time. After each time and measurement update pair, the process is repeated with the previous a posteriori estimates used to project or predict the new a priori estimates. The Kalman filter is not discussed in much detail, but a visualization is presented in Figure 3.1. Figure 3.1 uses some different definitions: instead of Lk the figure shows Kk (called the Kalman gain), H instead of G, and zk instead of dk . For more information the reader is referred to G.Welch and G. Bishop (2006) [48].

Figure 3.1: A picture of the operation of the Kalman ﬁlter as presented in (Welch and Bishop, 2006) [48]. 48 3.T HEORETICAL BACKGROUNDON LOAD IDENTIFICATION AND HYDRODYNAMICS

3.3. HYDRODYNAMICS This section presents the theory on hydrodynamics. The pressure carrying a vessel can be divided into hydrostatic and hydrodynamic pressure. Buoyancy force corresponds to the hydrostatic pressure and is proportional tot the displacement of the vessel. The hydrodynamic force is approximately proportional to the square of the relative speed to the water (Fossen, 2011) [49].

The dynamical system in this thesis is not linear, e.g. hydrodynamic resistance behaves non linear, and Oden’s sloping waterline makes the behavior of the vessel nonlinear. However, it is assumed that these terms can be linearized, allowing the use of theories that assume that the system possesses linear characteristics (i.e. in a linear system the ship’s motions behave linearly). A consequence of a linear system is that at each frequency that vessel motions occur, the ratios between the motion amplitudes and phase shifts between motions are constant, i.e. doubling input force will result in a doubled output.

Since the system is implemented linearly in the numerical model, resulting motions in waves can be seen as a superposition of the motion of the body in still water and the forces on the restrained body in waves [50]. The hydromechanical forces and moments are induced by the harmonic oscillations of the rigid body, moving through the undisturbed surface of the sea. Wave exciting forces and moments are produced by waves coming in on the restrained body.

In this thesis the vessel is represented as a mass-spring-damper system. Motions of and about COG are presented in section 3.3.1. Motion superposition is applied in case motions in any other point on the structure are required/given. This requires transformation matrices for all three axes (x1, x2, x3).

Section 2.3 outlines basic terminology regarding icebreakers, which includes basic vessel terminology. Read- ers who are familiar with these can proceed.

3.3.1. AXISCONVENTION In this thesis a distinction is made between an Earth-bound coordinate system, and a rigid body-bound coordinate system. In the ﬁrst case the origin of the axis system lies in the still water surface. In the latter case the system is connected to the vessel with its orign located at its COG. The positive axes and rotational deﬁnitions are:

• Surge along axis, x1, in the longitudinal forward direction positive. • Sway along axis, x2, in the portside lateral direction positive. • Heave along axis, x3, is perpendicular to the horizontal still water surface plane, upwards positive. • Roll, x4, is the rotation around the x1 axis. Right hand rule is used for deﬁnition, giving positive roll if starboard goes down. • Pitch, x5, is the rotation around the x2 axis, giving positive pitch if bow goes down. • Yaw, x6, is the rotation around the x3 axis, giving positive yaw when the vessel moves counter clockwise in the horizontal plane when looking down along negative x3 axis.

Given deﬁnitions above are illustrated in Figure 3.2.

3.3.2. HYDROSTATICS This section only discusses the hydrostatic properties of a structure. Hydrostatic properties are properties with the condition that dynamic effects can be ignored. To comply with this condition it is assumed that any disturbance to the equilibrium state will be brought slowly enough so that all dynamic effect may be ignored.

Hydrostatic pressure is calculated using Archimedes’s law. Using this law buoyancy forces are then calculated as follows: F ρg (3.24) b = ∇ where F represents the buoyancy force in Newton, and the volume of submerged part of considered ob- b ∇ ject [m3]. The center of buoyancy (COB) is located in the center of displaced water. Static stability of a vessel describes the up-righting properties of the vessel when it is brought out of equilibrium. Vertical equilibrium uses equation 3.24. In static equilibrium COG and COB are located in one vertical line, with the buoyancy moment and all acting forces equal to zero. This is characterized by rotational equilibrium. 3.3.H YDRODYNAMICS 49

Figure 3.2: Ship motions axis system [51].

If one adds a heeling moment onto the vessel, it will heel towards equilibrium under a certain heel angle φ. As a result of this heeling, the shape of the displacement will change, moving the COB. An equilibrium will ∇ be achieved when a stability moment equals the heeling moment. The stability moment Ms is given by:

M ρg GZ (3.25) s = ∇ · where GZ is the righting stability lever arm. This arm depends on the metacenter, located at a certain arm above COG: the metacentric height (GN): GZ GN sin(φ) (3.26) = Metacentric height [m] is geometrically deﬁned and depends on the center of buoyancy KB (height above the keel), to center of Gravity KG (height above the keel), and BM as follows:

GM KB BM KG (3.27) = + − where BM depends on displacement [m3], and second moment of inertia I of the water plane area [m4]: ∇ BM I . = ∇ More information can be found in [50] and [49]. It must be emphasized that vessels are in general not perfectly symmetric, hence depending on geometry coupling terms may exist. For example heave and pitch tend to have signiﬁcant coupling terms due to fore- and aft-body asymmetry.

3.3.3. POTENTIAL COEFFICIENTS In order to determine added mass and added damping, potential theory should be applied [49]. In potential theory three assumptions are made: inviscid fluid (no viscosity), irrotational flow, and incompressible fluid. Given these assumptions, the frequency dependent coefficients consists of two components:

• Hydrodynamic inertia force, e.g. added mass • Potential damping forces, e.g. added damping

There are several methods to determine the coefficients, the most popular methods are 2D strip theory and panel methods. However, these methods require computer software to enable coefficient determination [49]. The author likes to emphasize that applying this theory using this software is the conventional, and recommended, method. As this was out of the scope of the thesis, other approaches are required to determine these values. In this thesis the potential flow coefficients are approximated at one frequency of oscillation. This simplified approach is a consequence of applying the maneuvering theory, which is discussed in section 3.3.4. Following on this, a full-scale sea trial and reference vessel is used to determine the potential flow coefficients. In section 4.3.2, the full-scale sea trial is used to determine the added mass for surge motions, and the hydrodynamic resistance in surge direction. The hydrodynamic resistance for surge motions in maneuvering theory, has no potential damping. The hydrodynamic surge resistance at a constant velocity is a 50 3.T HEORETICAL BACKGROUNDON LOAD IDENTIFICATION AND HYDRODYNAMICS result of viscous effects, which are neglected by the potential theory. This viscous damping is implemented as a nonlinear friction force. Also nonlinear viscous damping for heave motions is taken into account, this is presented in 4.3.4. The potential damping and added mass for heave and pitch motions are determined in section 4.3.5, using a reference vessel that is analyzed using strip theory.

3.3.4. MANEUVERINGTHEORY Maneuvering theory is a more idealized hydrodynamics approach compared to the Seakeeping theory [49]. Motions of ships at zero or constant speed in waves can be analyzed using the Seakeeping theory. In this theory, hydrodynamic coefficients and wave forces are computed as a function of wave excitation frequency, hull geometry, and mass distribution. In maneuvering theory, frequency depended added mass and potential damping coefficients are approximated by constant values. As a result the hydrodynamic forces and moments can be approximated at one frequency of oscillation such that the fluid-memory effects can be neglected. (Fossen, 2011 [49]). Maneuvering theory presents a nonlinear mass-damper-spring system with constant coefficients. It is convenient to represent hydrodynamic systems with frequency independent quantities, since these reduce model complexity.

POTENTIAL COEFFICIENTS In maneuvering models potential coefficients for the horizontal motions (surge, sway, yaw) at forward speed can be described by a zero-frequency model. The matrix for added mass and damping are defined in equation 3.28 and 3.29 respectively.   a11(0) 0 0 [1,2,6] MA A (0)  0 a22(0) a26(0) (3.28) = = 0 a62(0) a66(0) D B [11,22,66](0) 0 (3.29) p = = One limitation of the 3DOF system presented in equation 3.28 and 3.29 is that heave, roll and pitch are not included, which is often required. For heave, roll, and pitch the constant frequency models need to be formulated at their natural frequencies. This is because they are the dominating ones. Hence damping forces will dominate the potential damping terms at low frequency. Fluid memory effects can be neglected at higher velocities [49]. The natural frequencies for the decoupled motions in heave, roll and pitch are given by the implicit equations : s k33 ωheave ω33 (3.30) = = m a (ω ) + 33 33 s k44 ωr oll ω44 (3.31) = = I a (ω ) x + 44 44 s k55 ωpi tch ω55 (3.32) = = I a (ω ) y + 55 55 General equation of motion described by the potential coefficients is given by: [M A(ω)]ξ¨ [B(ω)]ξ˙ K ξ 0 (3.33) RB + + + = Assuming no coupling motions between the surge, heave–roll–pitch and the sway–yaw subsystems, and using equations 3.28 and 3.29, this results in the following added mass and potential damping matrix [49]:   a11(0) 0 0 0 0 0  0 a (0) 0 0 0 a (0)  22 26     0 0 a33(ω33) 0 0 0  MA   (3.34) ≈  0 0 0 a44(ω44) 0 0     0 0 0 0 a55(ω55) 0  0 a62(0) 0 0 0 a66(0) 0 0 0 0 0 0 0 0 0 0 0 0     0 0 b33(ω33) 0 0 0 Dp   (3.35) ≈ 0 0 0 b44(ω44) 0 0   0 0 0 0 b55(ω55) 0 0 0 0 0 0 0 3.3.H YDRODYNAMICS 51

VISCOUSDAMPING Hydrodynamic damping has a linear and nonlinear characteristic ([49]). It is mainly caused by:

• Potential damping • Skin friction • Wave drift damping • Damping due to vortex shedding • Damping due to lifting forces

Potential damping takes into account added mass and potential damping. However, this is under assumption of a inviscid fluid. D’Alembert proved that there is no resultant drag force, when a constant uniform potential flow is present [50]. This is the reason why in maneuvering theory (i.e. inviscid flow, and time independent parameter), the potential damping for surge motion is zero. The added mass for surge is not zero, as this acts when the vessel accelerates. The rest of the force originates from the vicious effects, and is described by the skin friction force. Skin friction has a linear skin friction component, and a high-frequency contribution due to a turbulent boundary layer. This latter part is ofter referred to as nonlinear skin friction. Wave drift damping can be understood by the added resistance when a vessel advances in waves.

Taking into account viscous damping Bv (ω) (a non-potential theory coefﬁcient) equation 3.33 turns into:

[M A(ω)]ξ¨ [B (ω) B(ω)]ξ˙ K ξ 0 (3.36) RB + + v + + = In this thesis, only a nonlinear skin friction term for surge and heave is implemented in the model, these are elaborated in section 4.3.2 and 4.3.4.

3.3.5. PROPULSION THEORY The basic purpose of propellers is to deliver thrust. Energy from an engine is transformed into torque and rotation, which is then transformed into thrust and translation. The propeller blades are attached to the hub, which is itself attached to the propeller shaft. Each propeller has a pressure side and a suction side. Basic propeller geometry is illustrated in Figure 3.3.

Figure 3.3: Sketch of a propeller [50].

To get an basic understanding into propeller theory the following terminology is presented:

• Kinetic energy loss occurs due to the energy transform from propeller to the water, where the propeller generates velocities in the wake. This is described by the efficiency, see equation 3.37. Efficiency can vary greatly between types of propellers. • Pitch (P) in propeller technology refers to the helical progress along a cylindrical surface. The dimension of pitch is length. • We define the nose-tail pitch line as the straight line between the most inner and outer part of the propeller blade section. • The nose-tail pitch line makes an angle with the propeller shaft, hence an angle can be identified. This angle is defined as pitch angle. The pitch angle definition is given in equation 3.38. • The angle of attack is the angle between the nose-tail line and the undisturbed flow (relative to the blade). 52 3.T HEORETICAL BACKGROUNDON LOAD IDENTIFICATION AND HYDRODYNAMICS

• The entrance velocity Ve , should not be confused with rotational velocity (2πnr ). The latter one is the tangential velocity of the water particles at a radius r. • The pitch ratio (P/D) is the ratio between pitch and it’s propeller diameter. • Significant radius is a conventionally used representative for propellers. In general taken as 0.7R. Half of the propeller disk is located within this 0.7R radius, and half outside of it, hence the definition. • The propeller cannot transform all energy to advance energy. A certain difference occurs, which is defined as slip. Slip is visualized in Figure 3.4. • The hydrodynamic pitch β0.7R is the angle at which the incident flow encounters the blade section and is a hydrodynamic inflow rather than a geometric property of the propeller. This point is not taken at the outer edge of the propeller, but at the significant radius (0.7R). Hydrodynamic pitch is defined in equation 3.39.

Pout T V e η · (3.37) = P = Q 2πn in · µ Pi tch ¶ Pi tchα ar ctan (3.38) = 2πr µ V e ¶ β0.7R ar ctan (3.39) = 0.7π nD · where η represents efﬁciency, T thrust [N], Q torque [N m], n number of revolutions [1/s], V e is the entrance · velocity [m/s], D the diameter [m], and r the radius [m].

Figure 3.4: Deﬁnition of pitch. Note that slip is marked in the right ﬁgure. [50]

PROPELLERMECHANICS This section assumes an uniform flow for the open water characteristics, while the vessel is moving forward with a constant velocity. Given this assumption and definitions from section 3.3.5, the generated thrust is given by: 2 1 πD ¡ 2 2¢ T Ct (β0.7R ,P/D) ρ V (0.7πnD) (3.40) = · 2 4 e + with thrust coefficient C t(β,P/D) [-], water density ρ [kg/m3], diameter D [m], number of revolutions n [1/s], and entrance velocity V e [m/s]. Equation 3.40 can be understood as how ’hard’ the blades are pushing onto the water, i.e. how hard the blades can accelerate the water particles compared to the entrance velocity. The thrust coefficient connects the hydrodynamic pitch and pitch ratio to a thrust factor. Therefore Ct also depends on Ve, n, and Pi tchα: The higher the entrance velocity, the lower the relative velocity between blades and water, hence the lower the thrust. The higher n, the higher the relative velocity between blades and water, hence the higher the thrust. Pitch angle determines how much rotational energy is transformed into forward energy, i.e. how much water is accelerated aftwards compared to the rotational velocity of the blades.

A torque coefﬁcient K [-] may be deﬁned to characterize the effectiveness from torque Q [N m] and revolu- Q · tions n [1/s] to thrust T [N] by: Q KQ (3.41) = ρD5n2

Delivered power PD [W] and effective power PE [W] are given by equation 3.42 and equation 3.43 respectively:

P Q 2πn (3.42) D = · 3.3.H YDRODYNAMICS 53

P T V (3.43) E = · e A propeller’s capability of generating thrust depends also on the vessel characteristics. When given a certain amount of revolutions per second, the thrust a vessel can generate will generally decrease with higher entrance velocity; the faster the vessel sails, the less efficiently it can ’push’ itself forward. The term ’generally’ must be elaborated since propeller geometry, pitch angle, and direction of vessel propagation are also influencing factors. This also explains why the highest thrust coefficients are not required to be identified at β 0. 0.7R = The entrance velocity of the water to the propeller is disturbed by the presence of the vessel. This includes the hull, and also complex structures on it (appendages). As a result, the mean water velocity at the propeller is usually less than the boat velocity. This must be implemented in the thrust calculations. This is expressed mathematically by the wake fraction Cwake :

V V C (3.44) e = s · wake where Ve is the entrance velocity, Vs the ship velocity, and Cwake equals: µ ¶ Vs Ve Cwake 1 − (3.45) = − Vs

The non dimensional thrust coefficients and torque coefficients need to be determined empirically, via panel methods, or via RANS (CFD) methods [52]. Systematic series of propeller models have been investigated in the Wageningen Propeller Series [53]. Curves of open water propellers where created, on which noted coefficients are determined in a range of hydrodynamic pitch and pitch ratios. β0.7R value varies over 360 degrees. These results are presented as for quadrant measurements, as presented in Figure 4.10 in chapter 4. The results were corrected to a standard Reynolds number of Rn 2 106. From 0 until approximately 40 = · degrees, the curves are the regular open water diagrams. 4 FULL-SCALEDATA

4.1. INTRODUCTION The ice ridge ramming experiment was executed around 20:30 in the evening on the 26th of August 2013, 78.9 degrees North, and 8.89 degrees West. The location is shown in Figure 4.1. The extent of encountered ice depends strongly on the currents in this area. The North Atlantic East Current keeps the west coast of Svalbard relatively open of ice, as it contains relatively warm water. The west coast of Greenland however, has an inﬂux of ice from the arctic. In this region multi year ice and heavy ridging can be found. The experiment was realized in the latter region, where the SAMCoT team found a multi year ice ridge up to 12 meters thick. The weather during the experiment was calm with wind speed relative to the ship around 3 m/s, and an air temperature of about 1 degree Celsius. The area has a high ice coverage with level ice around 1.7 meters thick.

Figure 4.1: Ridge location given in latitude and longitude.

The full-scale data used and presented in this chapter includes:

• Description of the experiment procedure, including visual observations. • Vessel characteristics. • Ice ridge characteristics (of the proﬁled MY ice ride). • Seapath 320 system, which is a positioning, altitude and heading sensor delivering MRU and GPS data. • Propulsion data.

54 4.2.E XPERIMENTLOGANDPICTURES 55

4.2. EXPERIMENTLOGANDPICTURES The profiled ridge was rammed four times. After the profiled first ridge seven other ridges were rammed. There is always a conflict between the captain’s intention in protecting the ship and researchers intention to test the limit of the ridge (this was stated in the experiment log). The captain reversed several times in the ridge ramming so as to obtain sufficient inertia. Nevertheless, for researcher’s purpose it is preferred to run through a ridge with constant speed to easily estimate the ridge resistance. This because a vessel has to increase its thrust force, to continue with the same velocity as it did in open water conditions. This additional required thrust force, is equal to the ice resistance. In general a ram consumes all the kinetic energy (i.e. with the exceptions the vessel breaks through the ridge). Increasing the kinetic energy at impact (i.e. increase impact velocity), introduces the risk of too large loads on the vessel hull. Designing an icebreaker to be able to push through ridges of this size is not practical, which is best illustrated by calculating the design loads of the profiled ridge. In section 4.3.1 a method to determine the design load for Oden is presented.

The full-scale data corresponds to the log description. This is visualized in Figure 4.2, which shows acceleration measurements in surge direction during the experiment (more about acceleration data in Section 4.5). In Figure 4.2 the impacts of the ridges are indicated by stars, impact on (big) ice ﬂoes by squares. The brown circle presented is pointed out because, although ice loads are large, the force is not applied by the ridge, as the vessel already has completely penetrated the proﬁled ridge. The vessel log is presented in Appendix B.

360 degrees camera During the experiment two cameras were used to make pictures from the mast of the vessel. This data gives valuable information about surrounding ice features and can potentially help identify ice failure modes like bending, radial cracking, etc. Before the ridge is rammed Oden clears the ice in front of the ridge resulting in a open water channel. Behind the ridge level ice is identiﬁed, conﬁning the ridge. The icebreaker then backs up and starts with the ramming experiment.

With respect to load identiﬁcation, it is advantageous to hit a ridge from normal direction, as it creates a symmetric load case on the bow. However, pictures indicate that the ice ridge was not hit completely from a normal direction. Furthermore the pictures show that impact 1, impact 2 and impact 3 give a penetration around 30, 40 and 60 meters respectively. This means that impact 3 was approximately enough to penetrate the length of the ridge, but a fourth impact was required to get the vessel through the ridge. This might be due to the thick consolidated layer of the proﬁled ice ridge. After impact 4, the vessel backs up and continues its journey through the icy waters where it encountered several other ridges as well. The relevant pictures made by the 360 camera are presented in Appendix B.

Figure 4.2: Surge acceleration data of the ramming experiment. In the figure: red star presents the impacts into the profiled ridge, green star presents impacts into other ridges, purple square presents impacts into (large) ice floes, round circle present the ice force directly after the (fully penetrated) profiled ridge. 56 4.F ULL-SCALE DATA

4.3. VESSELCHARACTERISTICS An introduction to icebreaker Oden is presented in section 2.3.1. Based on the theory presented in Chapter 2 and Chapter 3, this section presents the vessel characteristics which are required in the numerical model. The vessel characteristics include the geometrical and physical aspects (e.g. weight, gross tonnage, rotational inertia, draft, etc). Subsequently, this section presents the hydrodynamical parameters. The determination of the hydrodynamic resistance for surge motions is presented in section 4.3.2. Furthermore, it was found that Oden has some nonlinear characteristics for heave, which are discussed in section 4.3.3 and 4.3.4.

The key vessel characteristics are tabulated as in Table A.2. The propeller characteristics can be seen as part of the vessel characteristics, but are discussed separately in section 4.6. They are required for thrust determination, given the full-scale propeller data.

Table 4.1: Key characteristics of icebreaker Oden

Ice class DNV POLAR-20 Mass 11.5 106 [kg] · LOA (Length overall) 107.75 [m] Waterline area 2624 [m2] Wetted surface 5000 [m2] Draft 8 [m] Beam (max) 31.2 [m] Beam (parallel-mid body) 25 [m] Bow angle (αbow ) 22 [degrees] Max speed (open water) 16.0 [knots] Max total thrust 2.4 106 [N] · Longitudal metacentric height (GML) 168 [m] Transverse metacentric height (GMT ) 10.4 [m] Rotational inertia (pitch) 1.202 1010 [kg m2] · · Wetted surface (A ) 2.19 105 [m2] w · Hull resistance coefﬁcient (Cr,ow ) 0.0017 [-] Number of thrusters 2 Wake fraction (propellers) 0.8

An idealization of Oden’s geometrical characteristics is presented in Figure 4.3, which is based on Figure D.9 and Figure D.10 (Appendix D). These ﬁgures also show the location of Oden’s thrusters, more speciﬁc; propeller’s COR (Center Of Rotation). Given the coordinate system in Figure 4.3, the following coordinates are presented:

 66.0 3.05  £ ¤ − COG 0 0 T 1  26.0 9.00  Oden = = − − 19.5 9.00 − (4.1)  66.0 3.05  £ ¤ − CORPr op. 48.5 5.70 T 2  19.5 9.00  = − − = − 49.3 3.05 4.3.V ESSELCHARACTERISTICS 57

Figure 4.3: The idealized Geometry of Oden, as it is simulated in the numerical model by the contact model. Oden’s Center of Gravity (COG) taken as axis origin. The vessel hull is represented by two triangles (T1,T2), propeller location given in COR, height of propeller is on scale (D 4.5 m). =

4.3.1. DESIGNLOADS The vessel would never be able to penetrate the ridge by thrust. If the vessel would be able to do this, it has to be developed to withstand huge design loads, and it would require extreme thrust forces. This can be illustrated, by calculating the design loads. For example the DNV design codes can be used. Remember that in Chapter 2 was found that the consolidated layer of ice ridges are commonly approximate by treating it as level ice. Thereby, the ridge is basically simulated as thick level ice. To determine the ice load a certain average pressure and load patch is assumed. For the keel often material behavior is used to calculate its global ice load, for example Coulomb-Mohr failure. This can be applied because the keel can be seen as an accumulation of discrete ice pieces and that can be interpreted as a granular material. The keel is neglected in this level ice approach calculation, due to the thick consolidated layer from this MY ice ridge.

Given DNV standards for Polar Class ships [54] the design force, design line load and design pressure of Oden were determined, with the requirement that the vessel must be able to break ice as thick as the ridge. The ridge itself is presented in section 4.4. The design ice load is characterized by an average pressure uniformly distributed over a rectangular load patch of height and width. In this case, the load patch has a depth equal to the maximum depth of the ridge, and width equal to the width of Oden. A summary of how to calculate these structural requirements for Oden is found in Appendix D.3. For a comprehensive explanation it is referred to [54]. In this section the key equations and results are presented:

• Design load F 36.1[MN] i = • Design line load Q 35.1[MN/m] i = • Design Pressure P 2.49[MPa] i = These loads are to large too overcome with thrust alone. Therefore, the vessel’s inertia is used to penetrate the ridge. Furthermore, several ramming attempts are required to penetrate the ridge. There is quite large controversy how to treat ice pressure. When using for example Sanderson’s relation (equation 2.16) and a load patch assumption using ridge depth (12m max), it results in an approximation of the ridge load equal to 103 MN. This shows, that using Sanderson’s relation results in a huge overestimation of the load, and really presents an upper limit.

Lastly, the author would like to emphasize two things:

1. For vessels, the methods for estimating the ridge loading on offshore structures give too high values. 2. Bending failure is a common failure mode for ice ridges, at least on ﬂoaters. This is concluded for example in Dalane et al. (2015) [30]. Bending failure is not taken into account in the calculations above. 58 4.F ULL-SCALE DATA

4.3.2. HYDRODYNAMICSURGERESISTANCE Given open water trials, it was possible to determine the surge hydrodynamic resistance depending on the vessel velocity. Initially it was assumed that the following components might inﬂuence this hydrodynamic resistance: added mass, potential damping, linear hull resistance and nonlinear hull resistance. The latter two terms can be understood as friction coefﬁcients, taking into account hull roughness and viscous effects. Hence wetted surface (Aw ) is required to calculate the total hydrodynamic/friction resistance. Potential damping in surge direction for a vessel within maneuvering theory is equal to zero, given equation 3.29.

To determine the hydrodynamic resistance, it is convenient to have data from open water trials (thus without ice in the water) at different velocities. However, given the delivered data it is concluded that only one open water trial (hence one velocity) is available. This data is presented in Figure 4.4 and discussed in more detail later. Based on this data only, it is not possible to simultaneously determine linear and nonlinear resistance terms. It was found that the hydrodynamic resistance can be simulated using added mass and the nonlinear hull friction only. This does raise the question what the inﬂuence of this assumption on the results might be, especially at high velocities. Given that the impact velocities are never very large, it is assumed that this effect is negligible.

If hull resistance is determined, then an acceleration trial can be used for identiﬁcation of added mass. It should be noted that currents and wind can impose signiﬁcant extra loads. However, given the calm conditions during experiment, they were allowed to be neglected.

An open water resistance model is developed to calculate the hydrodynamic damping terms. In this model it is assumed that the hydrodynamic resistance can be calculated by equation 4.2:

1 2 R C ρAw v (4.2) = 2 f

3 2 where R is the total resistance in [N], ρ the water density in [kg/m ], Aw the wetted surface of the vessel [m ], C f is the resistance coefficient [-], and velocity v [m/s]. According to [49] this friction coefficient C f can be split into two parts: flat plate friction C [-], and residual friction C [-] (i.e. hull roughness pressure ∝ f p ∝ f r ∝ resistance, wave making resistance, wave breaking resistance etc). As a result the velocity dependent friction coefficient C f (v) can be written as follows:

C (v) C (v) C (4.3) f = f p + f r

As equation 4.3 shows, the ﬂat plate friction depends on Reynolds number (Rn), hence velocity, and can be calculated according to equation 4.4 [49]. C f r is a constant.

0.075 C f p (v) (4.4) = ¡log(R ) 2¢2 n −

vLpp Rn (4.5) = υ where Rn is Reynolds number [-], v the relative velocity [m/s], Lpp the characteristic traveled length of the fluid [m], and υ the kinematic viscosity [Pa s]. This kinematic viscosity depends on temperature: at 20 and 0 6 · 6 degrees Celsius this gives a υ of 1.002 10− [Pa s] and 1.787 10− [Pa s] respectively. Besides temperature, · · · · the salinity in seawater influences the kinematic viscosity as well. Taking into account a salinity of 25 [ppt] 6 the kinematic viscosity at 0.1 degrees Celsius equals 1.8629 10− [Pa s]. · · Proposed equation 4.4 blows up at low velocities, i.e. log(0) . It is therefore needed to insert a minimum = −∞ allowable Reynolds number in the numerical model, which is taken at R 106. n,min = Figure 4.4 shows the results of an open water trial, where the vessel starts of at a velocity equal to zero. Both the starboard and port side propeller data are given. The yellow line gives the values determined by equation 4.2. At a certain moment in time the resistance and propeller forces become constant, with an acceptable error. At this point the vessel is sailing with a constant speed of 6.8 m/s. Equation 4.2 is now applied on the open water trials, with aim to identify the C f r . Nonlinear friction term for flat plate friction is equal to 4.3.V ESSELCHARACTERISTICS 59

C 0.0017, thus: f p =≈ 1 2 R C (v)ρAw v = 2 f 1 2 R (C (v) C )ρAw v (4.6) = 2 f p + f r 1 9.9 105 2.24 105 C 1025 5000 6.82 · = · + 2 · f r · · ·

Hence C f r must be approximately equal to 0.0066. The nonlinear hydrodynamic resistance based on this trial is implemented in the model and prevents the vessel from breaching its maximum velocity. Instead of making this distinction it is also possible to assume a constant nonlinear hydrodynamic resistance coefﬁcient term: b 2.141 104. It was concluded that both methods give practically the same value for the given 11,NL ≈ · full-scale data. This is visualized in Figure 4.5.

Figure 4.4: The propeller forces [N] in time with the calculated resistance based on equation 4.2. Propeller forces (green, green), resistance force (yellow), difference/error (black).

Using the acceleration experiment, the added mass was changed to ﬁt the acceleration. The equation of motions for surge can be written as:

(m a )x¨ (t) b x˙ (t) x˙ (t) T (t) (4.7) + 11 1 + 11,NL| 1 | 1 = where m is the vessel mass, a11 is the added mass for surge (to be determined), b11,NL the just determined nonlinear surge damping coefﬁcient, and thrust T . Based on the sea trial (Figure 4.4) this added damping was concluded to be 20%. To validate this value, the model is run with different values: a m i with ≈ 11 = · i=0.01,0.02,...0.40. Subsequently the following equation determines the added mass for surge:

mean ¡(m a )x ¨(t) b x˙ (t) x˙ (t) T (t) R(t) F (t)¢ 0 (4.8) data + 11 1 + 11,NL| 1 | 1 − + + ice,x ≈ If the added mass was indeed constant, and the model was perfect, then the proposed mean in equation 4.8 should be equal to zero. From the given full-scale data, it was found that the best ﬁt occurs for a m 0.21. 11 = · 60 4.F ULL-SCALE DATA

Figure 4.5: Hydrodynamic resistance calculated in two different ways ( C vs b ) ∝ f ∝ 11,NL

4.3.3. ODEN’S HYDROSTATIC NONLINEARITYFOR HEAVE MOTIONS This thesis assumes a system that possesses linear characteristics, i.e. ship motions behave linearly. However, Oden’s sloping waterline makes the behavior of the vessel nonlinear. This is mainly the case for hydrodynamic stiffness for heave (k33). Hence this section discusses the nonlinear hydrodynamics for heave motions. Hy- drodynamic damping for surge (b11,NL) is nonlinear as well, and discussed in section 4.3.5.

It must be investigated what the influence of the assumption of linearity on hydrostatic loads is, e.g. the error it causes when ignored. If this error is significant, the assumption of linear hydrostatics does not hold, and a solution needs to be implemented in the model. For simplicity the idealized shape of Oden is taken in the analyses, illustrated in figure 4.3. The bottom hull can be generalized into three parts: left (aft to bottom hull, sloping), middle (bottom hull, flat), and right (bottom hull to bow, sloping). The aft part has a slope of approximately θ 16.7 degrees (0.2915 radians), the bow part has a slope of approximately θ 22 degrees a = b = (0.3805 radians). For the middle-body width B 25 meters, and for the middle-body length L 45.5. The m = m = aft width B 25, and a bow width B 31.2 meters. It is assumed that the vessel is at hydrostatic equilibrium a = a = at a draft Dr equal to 8 meters. The buoyancy force with draft Dr in hydrostatic equilibrium will be equal to: µ Dr 2 Dr 2 ¶ F b(Dr ) ρg 0.5 Ba Lm Dr Bm 0.5 Bb (4.9) = · · tan(θa) · + · · + · tan(θb) · where the buoyancy force is given in [N], ρ the water density in [kg/m3], Dr in [m]. For a draft of 7, 8, and 9 meters, this gives buoyancy forces equal to 120 MN, 143 MN, and 169 MN respectively. Therefore, an increase of 1 meter draft results in 1.6 MN more buoyancy compared to the loss of buoyancy resulting from a decrease of 1 meter draft. In relation to a draft of 8m: • F b(7m) 0.835 F b(8m) = · • F b(8m) 1.000 F b(8m) = · • F b(9m) 1.176 F b(8m) = · For the analyses the buoyancy force is split into a linear part ( A ) and a nonlinear part ( Fb(Dr)). Linear ∝ wl ∝ buoyancy force due to heave motions is taken into account by k33: k ρg A (4.10) 33 = wl Hydrostatic force due to heave (x3) is calculated by: F b k x (4.11) L = 33 · 3 Based on idealized geometry and a draft of 8 m A 2454 m2, thus k 24.68 MN/m. Nonlinear buoyancy wl = 33 ≈ force is taken into account by: dF b (Dr ) F b(Dr ) F b(Dr ) (4.12) NL = − eq Given hydrostatic equilibrium at 8 meters draft (Dreq ), the input for above forces is defined as: Dr Dr x 8 x (4.13) = eq − 3 = − 3 4.3.V ESSELCHARACTERISTICS 61

Conclusion on Assumption of nonlinearity: Now it can be validated if equation 4.11 is accurate enough compared to equation 4.14; Given a heave motion of 1 meter, equation 4.11 gives an extra buoyancy force of − F b 24.7 MN, a heave motion of +1 meter results in 24.7 MN buoyancy loss. By contrast, results of equa- L ≈ ≈ tion 4.14 give: F b (9m) 27.0 MN, F b (7m) 25.4 MN. Subsequently 27.0 100% 109%. Given these NL ≈ NL ≈ − 24.7 · ≈ results it is concluded that using a linear buoyancy force brings about a signiﬁcant error that should not be ignored. At each time step in the numerical model the nonlinear hydrostatic force will be linearized so that it can be implemented in linear models (discussed theories in previous chapters). Furthermore, inﬂuence of pitch movements (x5) should ideally be taken into account as well.

It must be stressed that these calculations are an approximation, since displacement of Oden equals 11500 tonnes, where this calculation based on idealized shape results in an overestimation of the displacement 2 (14600 tonnes). Furthermore the waterline area (Awl,real ) of Oden equals 2624 m while idealized geometry 2 results in an Awl,ideali zed of 2454 m (Awl,real is based on the technical drawings presented in Appendix D). Hence a conversion factor is required. The question arises if this factor should be based on displacement or waterline area. Since hydrodynamic loads in the numerical model depend on hydrodynamic stiffness, which itself depends on waterline area, the latter relation is chosen. Assuming that this relation is linear, a conversion factor equal to 2624 1.0693 can be used: 2454 =

Awl,real dF bNL,real dF bNL,ideali zed (4.14) = · Awl,ideali zed

HYDROSTATIC NON-LINEARITYFORHEAVEINTHEMODEL The vessel’s middle area (i.e. part with length 45.5 meter and width 25 meter) is completely linear, and by using equation 4.10 this component contributes as follows:

ρg Awl,middle Awl,real k (4.15) 33,middle = cos(x x ) · A 5,mean + 5 wl,ideali zed The front and aft of the vessel have a linear and nonlinear contribution. Pitch motions are neglected in the start. For convenience C1 is deﬁned as: µ ¶ 0.5 Ba 0.5 Bb C1 ρg · · (4.16) = tan(θa) + tan(θb) with C 8.1106 105. Subsequently equation 4.12 can be rewritten, and the hydrostatic buoyancy force for 1 = ∗ aft and bow dF ba,b, depending on heave (x3) becomes:

dF b (x ) F b(Dr x ) F b(Dr ) a,b 3 = eq − 3 − eq dF b (x ) C (Dr x )2 C (Dr )2 (4.17) a,b 3 = 1 · eq − 3 − 1 · eq dF b (x ) C ( 2 Dr x x2) a,b 3 = 1 · − · eq · 3 + 3

Following on this, the linear part k33,ab,L and nonlinear part k33,ab,NL of the bow and aft can be deﬁned in as follows: dF b (x ) k x k x2 a,b 3 = − 33,ab,L · 3 − 33,ab,NL · 3 k 2 C Dr (4.18) 33,ab,L = · 1 · eq k C 33,ab,NL = − 1 The reaction force from heave spring stiffness (dF b) can be described by a linear and nonlinear part:

dF b (x ) ¡ k k ¢ x k x2 (4.19) a,b 3 = − 33,middle − 33,ab,L · 3 − 33,ab,NL · 3

By also taking into account pitch motions and the conversion factor, the linear (k33,L) and nonlinear contributions (k33,NL) are equal to: µ ¶ ¡ 2 Dreq ρg 0.5 Ba 0.5 Bb ρg Awl,real ¢ Awl,real k33,L · · · · · = cos(x5,mean x5) tan(θa) + tan(θb) + cos(x5,mean x5) · Awl,ideali zed + µ ¶ + (4.20) ¡ ρg 0.5 Ba 0.5 Bb ¢ Awl,real k33,NL − · · = cos(x x ) tan(θ ) + tan(θ ) · A 5,mean + 5 a b wl,ideali zed 62 4.F ULL-SCALE DATA

At hydrostatic equilibrium this gives a total spring stiffness k 2.6385 107 [N/m]. 33 = ·

When Oden is in hydrostatic equilibrium, there is a horizontal arm between areal center point of Awl and COG. Therefore, a negative heave motion downwards will result in extra buoyancy, with its COB not in a vertical line with the COG. As a result there will be a coupled motion where the vessel will rotate (pitch) until COB and COG are in one vertical line. Vice versa, a pitch motion will induce a heave motion. These are taken into account by the heave-pitch coupling terms k35 and k53, and these are described by:

k k ρ g A r (4.21) 35 = 53 = · · wl · where r presents the horizontal arm in meters, between COG and the areal center point of Awl. For Oden the arm was determined using the technical drawings. The arm between COG and areal center point of Awl was found to be equal to 4.575 m in direction of the aft. Given the presented axis system in section 3.3.1, a positive heave results in a negative pitch moment, and a positive pitch results in negative heave motion. If k would be linear, k k k r . However, due to k there is also a nonlinear part for k and k : 33 35 = 53 = 33 · 33,NL 35 53 k x ¡k k r x ¢ r x 35 · 5 = − 33,L + 33,NL · · 5 · · 5 2 2 k35 x5 k33,L r x5 k33,NL r x · = − · · − · · 5 (4.22) k x ¡k k x ¢ r x 53 · 3 = − 33,L + 33,NL · 3 · · 3 k x k r x k r x2 53 · 3 = − 33,L · · 3 − 33,NL · · 3 At hydrostatic equilibrium k k 1.2071 108 [N/r ad]. Also a nonlinear damping term is included in 35 = 53 = − · the numerical model. This is presented in section 4.3.4.

4.3.4. HEAVE’S NONLINEAR FRICTIONAL DAMPING DUE TO FLUID-STRUCTUREINTERACTION For a complete description of the heave motion, a nonlinear damping term is required in the numerical model. This term represents the frictional damping due to fluid-structure interaction. It are the viscous forces along the hull that cause the friction damping for heave motions. At a low frequency of oscillation, these viscous forces contribute very little to the damping of the vessel. At high frequencies however, the viscous component becomes large. Therefore, for a highly dynamical system, this should be implemented. In Ankudinov (1991) [55] it is confirmed that nonlinear damping occurs, and that implementing this alongside the linear damping can make predictions closer to experiments. The flat plate friction is used to approach the nonlinear damping for heave motions b33,NL. It must be emphasized that given analyses are based on the flat plate of the same area as the ship’s wetted surface. In general this is an accepted assumption in ship resistance calculations [55]. In this thesis b33,NL takes into account the nonlinear frictional damping for heave, where the linear heave damping b33 takes into account the potential damping.

The hydrodynamic damping in the numerical model is calculated as shown in equation 4.2. The resistance coefﬁcient can be calculated by equation 4.3, 4.4, and 4.5. The friction term is assumed to depend only on 6 ﬂat plate friction, hence C 0. In equation 4.5, v x˙ (t) [m/s], υ 1.8629 10− [Pa s], and characteristic f r = = 3 = · · travel length Lpp is calculated by: L Dr x (4.23) pp = eq − 3 where Dreq represents the draft in [m] while in hydrostatic equilibrium. Furthermore, since the wetted sur- 2 face Awl in [m ] depends on the draft of the vessel (hence heave), Awl is calculated by

A A x (2 LW L B B ) (4.24) wl = wl,eq − 3 · · + a + f where Awl,eq is the wetted surface in hydrostatic equilibrium, LWL the waterline length in [m], Ba the vessel width at aft in [m], and Bb the vessel width at bow in [m]. Following on this, b33,NL is deﬁned as the hydrody- 2 namic heave resistance divided by x˙3 : 1 b33,NL C ρAw (4.25) = 2 f b33,NL is a dynamic coefﬁcient that changes over time, for a visualization the reader is referred to Figure D.11. The value is found to be approximately b 1.2 104 [N s2/m2]. 33,NL ≈ · · 4.3.V ESSELCHARACTERISTICS 63

4.3.5. OVERVIEWHYDRODYNAMICALVESSELCHARACTERISTICS HYDROSTATIC CHARACTERISTICS In Table 4.2 an overview of the hydrostatic characteristics (at hydrostatic equilibrium) is presented. Theory of Chapter 3 is applied to identify the hydrostatic characteristics for Oden. k11, k22, and k66 equal zero. k33, k35 and k53 are derived in section 4.3.3, with nonlinear terms included in k33 and k53 due to Oden’s sloping bow. Rotation displaces the center of buoyancy, giving a moment for roll and pitch which can be respectively calculated by k ρg GM , and k ρg GM (using equation 3.25 and 3.26). Roll is neglected in this 44 = ∇ T 55 = ∇ L thesis, with exception of the GPS data conversion that is presented in section 5.3.

Table 4.2: Spring stiffness parameters of Oden

k11 0 [N/m] k22 0 [N/m] k 2.6385 107 [N/m]* 33 · k 5.97 108 [N m/r ad] 44 · · k 1.89 1010 [N m/r ad] 55 · · k 0 [N m/r ad] 66 · k 1.2071 108 [Nm/r ad] 35 − · k 1.2071 108 [Nm/m]* 53 − · note*: k33, and k53 are nonlinear, and only presented for static equilibrium. For more information the reader is referred to section 4.3.3. Here the equations to calculate k33, k35, and k53 are presented as well.

HYDRODYNAMICCHARACTERISTICS Water acts dynamically on the vessel, which is taken into account as added mass and damping. The main source of energy loss by potential damping is surface wave formation which is almost independent of viscous properties of the ﬂuid [55]. These potential theory coefﬁcients are frequency depended. However, as discussed in Chapter 3, they are taken as constant frequency independent parameters in the numerical model. These values are presented in Table 4.3.

In Chapter 3 is shown that in maneuvering theory, the damping coefﬁcients in the horizontal plane b11, b22, and b66 are equal to zero. Potential damping for roll motions b44 is neglected. The potential coefﬁcients for heave and pitch are determined using Ankudinov (1991) [55] (added mass for heave a33, added mass for pitch a55, added damping for heave b33, and added damping for pitch b55). The vibration analyses are based on strip theory. It must be emphasized that the hull form of an icebreaker may be quite different from other ships, and therefore will have different hydrodynamic characteristics. However, based on the literature study, these give the best available estimates, and are presented in Figure D.12 and D.13 in Appendix D. In this thesis, a22, a44, and a66 were based on experience from crew members.

Beside the potential coefﬁcients there are also friction coefﬁcients taken into account. These are the nonlinear surge and heave damping, b11,NL and b33,NL respectively. Since b33,NL depends on heave motion, its magnitude will changes over time. b11,NL and b33,NL were determined in section 4.3.3 and 4.3.4 respectively.

Given mass, added mass and hydrostatic stiffness, the following natural frequencies can now be determined using equations 3.30, and 3.32: ω 0.99 [rad/s], and ω 0.96 [rad/s]. heave ≈ pi tch ≈ 64 4.F ULL-SCALE DATA

Table 4.3: Potential ﬂow and hydrodynamic resistance parameters of Oden (for numerical model).

a 2.3 106 [kg] 11 · a 9.2 106 [kg] 22 · a 15.5 106 [kg] 33 · a 2.4 109 [kg m2] 44 · · a 8.625 109 [kg m2] 55 · · a 13 109 [kg m2] 66 · · b11 0 [N s/m] b22 0 [N s/m] b 6.60 106 [N s/m] 33 · b44 0 [N s/m] b 7.273 109 [N s/r ad] 55 · b66 0 [N s/m] b 2.141 104 [N s2/m2] 11,NL · b 1.2 104 [N s2/m2] 33,NL ·

4.4. ICERIDGECHARACTERISTICS The ridge has been proﬁled extensively before the ramming. 16 proﬁles were drilled across the ridge. The deepest keel was –10.30 meters, and a surrounding level ice was 1.7 meters thick. The drilling scheme is presented in Figure E.1. Two cross sections where drilled, visualized in Figure E.2 and Figure E.3. Due to the thick consolidated layer it is concluded that this ridge is a multi year ice ridge. For more information is referred to Appendix E.

Sea ice hardness was recorded along the drilling depth. The following scale was used:

• 0 - water or air gap • 1 – slush • 2 – soft ice • 3 – hard ice

Ice hardness is determined according to the feeling of the one who drilled the ice. In other words, based on the experience of the user of the driller. Given above deﬁnitions, water means there is no ice at all. Slush, means that some ice can be observed at very beginning of freezing. However, the lattice or grains are not connected, or scatter distributed ice particles. It does not carry loads at all, but it affects properties of ﬂuid. Soft ice, is somehow real ice. It has lattice and other ice characteristics. From this stage, ice is sort of perfect. In this condition the brine is average distributed. It starts carrying loads, but it cannot carry much load. Hard ice comes after soft ice, after a certain thermal cycle, the distribution of brine is not averaged any more. This ice is way harder and can carry substantial loads.

During profiling no hard ice was encountered. This corresponds to theory of specific energy principles (the ice is quite ’warm’ at moment of profiling). Proof of this can be found using SEA values depending on temperature: Figure 2.33. The profiling data tables of the two cross sections are presented in Figure E.1. In this thesis the ice ridge is idealized as a symmetrical ice ridge of semi-infinite length, i.e. the two cross sections need to be idealized towards one average cross section. Geometrical data results of this idealization are shown in Table 4.4. Figure 4.6 visualizes the idealized ridge, as it is included in the contact model (more about the contact model in section 5.4). 4.5.K ONGSBERG SEAPATH 320+ SYSTEM 65

Table 4.4: Geometry data of the idealized ice ridge

Figure 4.6: Idealized and discretized (for contact model) ice ridge. In the ice ridge white triangles represent the consolidated layer, magenta represents the keel layer. At the right level ice with a thickness of 1.7 m.

4.5. KONGSBERG SEAPATH 320+ SYSTEM The product description of the Kongsberg Seapath 320+ System is based on the Kongsberg Seapath 320+ User Manual [56]. Basic information is presented in this section, but for more detailed information the reader is referred to the manual.

The Kongsberg Seapath 320+ is a positioning, altitude and heading sensor. The product combines inertial technology together with GPS and GLONASS satellite signals. Core components in the product are the MRU 5+ inertial sensor, the two combined GPS/GLONASS receivers, the Processing Unit and the HMI Unit. These components are elaborated on this section.

PURPOSE AND APPLICATIONS. The product is developed speciﬁcally for hydrographic and other high precision applications where heading, position, roll, pitch, heave and timing are critical measurements. This Seapath product offers combination of global navigation satellite system (GNSS) signals and inertial measurements for demanding operations in challenging environments. It enables the use of GLONASS (global navigation satellite system) in addition to the GPS satellites, which signiﬁcantly increases satellite availability, provides robust integrity monitoring and results in more precise solutions, particularly in highly obstructed environments.

This Seapath product is a two-module solution with a processing and an HMI unit connected via ethernet. The processing unit runs all critical computations independent of user interface on the HMI Unit to ensure continuous and reliable operation. Several HMI units can be connected to the same processing unit in a networked architecture. 66 4.F ULL-SCALE DATA

SYSTEM COMPONENTS Seapath 320 system architecture is shown in Figure 4.8. Most important components with their location on Oden are illustrated in Figure 4.7. This Seapath system comprises the following main components, which are physically separated [56]: • Processing Unit for I/O and calculations • Human Machine Interface (HMI) Unit with monitor, keyboard and PC mouse • Motion Reference Unit (MRU), model 5+ inertial sensor: This is the Inertial Measurement Unit (IMU) within the system measuring dynamic linear motion and altitude. An MRU consists of gyros and accelerometers. An IMU measures six degrees of freedom for the motions of the ship. The sensor is controlled by an embedded computer, which was connected to the ship’s network and contained a real time clock for synchronization of measurements from different systems (e.g. propulsion systems, camera system, etc). See section 4.5.1 for a more comprehensive description. • MRU 5th generation, ﬂoor mounting bracket • MRU junction box with three metres of cable for interfacing to the MRU • Antenna Bracket with two Global navigation satellite system (GNSS) antennas: On Oden the antenna is installed in the mast at approximately 35 meter height. • RTK corrections on RTCM or Trimble CMR format

Figure 4.7: Sideview Oden with measurement system locations.

Figure 4.8: Seapath 320 system architecture.

4.5.1. MOTION REFERENCE UNIT Motion Reference Unit model 5+ (MRU 5+) inertial sensor is speciﬁcally designed for motion measurements in marine applications. A MRU consists of gyros and accelerometers. The unit incorporates 3-axis sensors 4.5.K ONGSBERG SEAPATH 320+ SYSTEM 67 for linear acceleration and angular rate, along with complete signal processing electronics and power supply. The MRU 5+ outputs absolute roll and pitch. Dynamic acceleration in the MRU axes direction as well as velocity and relative position, are also provided. The MRU achieves high reliability by using sensors with no rotational or mechanical wear-out parts. When the MRU is used within this product, only raw angular rate and linear acceleration data are output from the unit. All processing of these signals to roll, pitch, heave and velocity measurements is performed in the Kalman ﬁlter inside the Processing Unit. See Figure 4.9 for the functional modules of the MRU 5+.

Figure 4.9: Functional modules of Motion Reference Unit model 5.

4.5.2. GLOBAL NAVIGATION SATELLITE SYSTEM GNSS is a generic term for satellite navigation systems providing autonomous geo-spatial positioning with global coverage. The Global Positioning System (GPS) is a satellite-based navigation system made up of a network of satellites placed into orbit by the U.S. Department of Defence. GPS was originally intended for military applications but in the 1980’s the government made the system available for civilian use. The Global Navigation Satellite System (GLONASS) is a Russian satellite based navigation/positioning system. GPS is the only GNSS with full constellation. However, GLONASS is operable but does not have full constellation.

4.5.3. FULL-SCALE DATA Most data are determined either by input of MRU or the GNSS antennas, as shown below. However, true heading is provided by integrating the best signal characteristics of the MRU yaw rate and differential carrier phase measurements between two GNSS antennas, thus uses both MRU and GNSS input. Heave is determined by the vertical acceleration (MRU): vertical acceleration is high pass ﬁltered and integrated twice over time to heave position. Heave velocity is computed with one integration over time of ﬁltered vertical acceleration. Full-scale data sampling rate is 10 Hz. The following acquired data are used in this thesis:

• (MRU) Acceleration data of surge, sway and heave in [cm/s2]. • (GNSS) North velocity, East velocity and Heave velocity [cm/s] • (MRU) Roll rate, pitch rate and yaw rate [degrees/s] • (GNSS) Latitude [degrees], longitude [degrees] and height [cm] • (MRU) Heave [cm] • (MRU) Roll and pitch [degrees] • (GNSS,MRU) Heading/yaw [degrees]

The accuracy of the data, provided by the Seapath 320+ Installation Manual [57], is given to be:

• Roll and pitch accuracy for 5 degrees amplitude: 0.02 degrees RMS • Scale factor error in roll, pitch and heading: 0.08% RMS • Heading accuracy: 0.065 degrees RMS • Heave accuracy: 5 cm or 5%, whichever is highest • Position accuracy: 0.5 m RMS or 1 m 95 % CEP • Velocity accuracy: 0.03 m/s RMS or 0.07 m/s 95 % CEP

The following full-scale data are presented in Appendix C: surge acceleration, sway acceleration, heave acceleration, heave displacement, heave velocity (down velocity), pitch, pitch rate, speed over ground (depending on north and east velocity), and heading. 68 4.F ULL-SCALE DATA

4.6. PROPULSION DATA Before the propulsion data can be presented and analyzed, the propeller characteristics need to be identiﬁed. These propeller characteristics are discussed in this section 4.6.1. Thereafter, available full-scale propulsion data is discussed and presented. It must be emphasized that Oden has two thrusters. Maximum power is 9000 KW on each shaft. One thruster can be coupled to one or two engines.

4.6.1. PROPELLERCHARACTERISTICS The open-water characteristics of Oden’s thrusters are based on the Wageningen Nozzle 37 with the Ka4-70 series, which were fared by means of harmonic analyses (Roddy, 2006 [53]). These characteristics enable the identification of the thrust coefficient depending on hydrodynamic pitch and P/D value. The resulting harmonic analysis coefficients for CT∗, CT∗ n, and CQ∗ are presented in the form of:

30 1 X CT∗ [A(k)cos(kβ) B(k)sin(kβ)] = 100 k 0 + = 30 1 X CT∗ n [A(k)cos(kβ) B(k)sin(kβ)] (4.26) = 100 k 0 + = 30 1 X CQ∗ − [A(k)cos(kβ) B(k)sin(kβ)] = 1000 k 0 + = where CT∗ is the thrust coefficient, or total thrust coefficient of ducted propeller system , CT∗ n is the thrust coefficient due to the duct, and CQ∗ is the torque coefficient. The resulting harmonic analysis coefficients are presented in frequency domain in Appendix D (Figure D.1, Figure D.2, Figure D.3, Figure D.4, and Figure D.5). Matlab was chosen to execute the Inverse Fast Fourier Transform to go to the time domain of hydrodynamic pitch (β). Following results of the thrust coefficient are presented in Figure D.6 and Figure D.7 (Appendix D). The coefficients are visualized in time domain in Figure 4.10. For more detailed information the reader is referred to Appendix D and (Roddy, 2006) [53].

Figure 4.10: Four Quadrant Estimate for Nozzle 37 with Ka4-70 Propeller Series. Symbols = Predictions, Solid Lines = Measured Data. (Roddy, 2006 [53]) 4.6.P ROPULSION DATA 69

4.6.2. FULL SCALE PROPELLER DATA Using the theory presented in section 3.3, the propeller characteristics given above, and the full-scale propulsion data, the thrust can be calculated. The following full-scale propeller data for starboard and portside, to be used in the numerical model, are available:

• Revolution rate (RPMr aw ) [1/minutes] • Pitch angle of propeller blades (Pi tchα) [degrees] • Rudder angle [degrees] • Torque [N m] · • Power [W]

Given the measurement systems and data analyses, it was concluded that some data were measured in arbitrary values, rather than in conventional quantities. The data needed to be transformed to obtain the required data. Transformation required data are found to be:

• Revolution rate: the vessel RPM indicator was found to be above maximum capability of the vessel, i.e. an offset. Hence a conversion factor was suggested. The engines should run constantly on RPM of 139, but measurements showed the value of 150 RPM. Resulting in a conversion factor of 139 0.93. 150 = • Pitch angle of propeller blades: Raw pitch reading have a variation for -50 to 64, while the actual pitch can vary between -15 to 27 degrees. Readings logged on the system need to be related to actual measurement as observed on the Pitch OD box. A conversion factor of 27/64 is introduced. • Torque: Measured with a very old (1989) system that is considered to be unprecise. Based on experience from the vessel crew, data can only be used as relative comparison. • Power: Like torque stated to be unprecise. When applying equation 3.42 to calculate power ( RPM, ∝ torque), unrealistic values appeared given that maximum power on each shaft is 9000 kW.

For the revolution rate and pitch angle of propeller blades conversion calculations are applied in equation 4.27 and 4.28 respectively. In equation 4.27, n is the number of revolutions per second, and the factor 0.93 is the conversion factor. In equation 4.28, Pi tchα is the pitch angle in radians, pi tchr aw is the raw (arbitrary) 27 value, and 64 is the conversion factor. With respect to the numerical model, it must be validated that the maximum total thrust of 2.4 MN is never exceeded.

1 n RPMr aw 0.93 (4.27) = · · 60 27 π Pi tchα pi tchr aw (4.28) = · 64 · 180 5 NUMERICALMODELS

This chapter discusses two numerical models:

1. A simulation model using the Speciﬁc Energy Absorption (SEA) of mechanical crushing of ice to calculate the global ice loads 2. A load identiﬁcation model using full-scale data to determine the global ice loads.

The simulation model was developed before the load identification model. The purpose of the simulation model is to enhance the understanding into the governing physical phenomena that occur during ice ridge ramming impacts, and to answer one of the research subquestions (What are specific energy principles for crushing of ice and can they be used to calculate global ice loads during ramming of a sea ice ridge by a vessel?). The simulation model is discussed in section 5.1. Starting at section 5.2, the load identification model is presented.

Key components of the load identiﬁcation model that are discussed in Chapter 5 are: data synchronization (5.2.1), model dimension and axis convention (5.2.2), GPS data conversion (section 5.3), Contact model (section 5.4), the load identiﬁcation model (section 5.5), thrust force calculation algorithm (section 5.5.2), Coanda effect (section 5.5.7), and uncertainty analyses (section 5.6). At the end of Chapter 5, a sensitivity and uncertainty analyzes are presented.

5.1. SIMULATION MODEL:GLOBALICERIDGERAMMINGLOADSBASEDON SPECIFIC ENERGY APPROACH A simulation model was developed simultaneously with the literature review, thus before the load identification model was developed. Reasoning was, that simulating an ice ridge ramming impact would enhance the understanding of relevant physical phenomena and parameters. This section briefly discusses this simulation model. It must be noted that at the stage of writing this thesis, the author’s knowledge into the ice ramming phenomenon increased compared to the moment of writing of the simulation model. That is why the theory from the paper concerning the simulation model is presented first, followed by some adjustments, based on lessons learned by the author at a later stage. The general research question of the paper is the same as that of the research’s subquestion: What are specific energy principles for crushing of ice and can they be used to calculate global ice loads during ramming of a sea ice ridge by a vessel? For a comprehensive explanation of the simulation model is referred to the paper in Appendix F.

The paper investigates the use of speciﬁc energy principles of the crushing of ice as a method to calculate the global ice ridge loads during ramming of an ice ridge. Besides the speciﬁc energy principles and the simulation model, the paper goes into ice ridge characteristics, and ice ridge loads on vessels. These topics are discussed in Chapter 2, and not discussed any further in this section.

The simulation model is a two dimensional time-domain model with three degrees of freedom: surge, heave and pitch. The model simulates rigid body motions including hydrodynamics, propulsion systems and ice loads. The model strives to reconstruct the time responses of the real system, in this case the ramming of an

70 5.1.S IMULATION MODEL:GLOBAL ICE RIDGE RAMMING LOADS BASED ON SPECIFIC ENERGY APPROACH 71 ice ridge. Matlab is used as a simulation program.

During the simulation the vessel will start in open water at a certain distance from the ice ridge. Due to thrust the vessel will move towards the ridge, in which the open water movements are simulated. At impact a contact model is used to deﬁne the ice load magnitude, location and direction on the vessel’s hull. Simulation stops when the vessel impact stops, i.e. when the vessel stops in/on the ridge or starts gliding backwards into the water.

The simulation model has been based on several assumptions. The most important assumptions are the geometrical idealizations of vessel and ridge, the ice failure mode and the simpliﬁed hydrodynamics. The ice loads are determined by a contact model under assumption of pure crushing failure. This allows the use of the speciﬁc energy principles.

The Specific Energy Absorption is defined as the energy absorbed per unit mass of crushed material. In section 2.4.3 is presented how the Specific Energy Absorption (SEA) can be used. Due to its importance to the simulation model we repeat Equation 2.17 below. The applied work W , i.e. loss of kinetic energy, used to define Specific Energy Absorption (SEA) in J/kg reads:

Z u(t) 1 2 W mv F du SEA ρVcr ushed (5.1) = 2 = 0 = · where v is the indenter speed at impact in m/s, m is the mass of indenter in kg; F is the applied load in N, u is the penetration in meter, g is the gravitational acceleration in m/s2, ρ is the average ice density in kg/m3 3 and Vcr ushed is the volume of crushed ice in m . The definition assumes a constant indentation speed and homogeneous ice properties. The pressure calculated p F /A is constant per crushed mass. = As shown in section 2.4.3, for a vessel to ridge impact the specific energy depends on temperature of ice, confinement due to size and geometry of the ice, and mechanical state of the ridge. Furthermore it was concluded that SEA is independent of interaction scenario, independent of intender velocity, mass, and that it is size- and scale-independent. The SEA value can be determined using Tsuprik’s (2013) [37] relation between temperature and specific energy, shown in Figure 2.33. With a relative warm ice temperature of approximately - 1 degrees Celsius average, this gives a specific energy value of 1.9833 J/cm3. This gives a SEA value of approximately 2175 J/kg (assuming an ice density of 917 kg/m3). This SEA value can be implemented in equation 2.17 to calculate the work, which is used in the simulation model.

The ice load determination requires the model to clarify the loss of volume due to crushing in time (i.e. increase of indentation volume in time). This requires the simulation model to know the location and orientation of the vessel, velocity, penetration speed normal to the hull, contact area, penetration volume, and the specific energy for crushing. The orientation, location and velocity of the vessel are simulated in 3DOF taking into account propeller forces and hydromechanics. Thereby, also hull velocity in normal direction is established. The last required components, the penetration volume, the contact area, and contact point, are determined by the contact model. This contact model is discussed in section 5.4. After the vessel and ridge idealization the contact model calculates the exact volumetric ridge penetration. The model will split vessel and ridge geometry in triangles, to enable the model to calculate penetration volumes. To get the ice load in time, the volume loss in time is computed. Besides time steps, the accuracy depends on the geometry idealization. The idealization is directly related to the triangle size and the number of triangle elements. In the simulation model the geometry idealization of the vessel and the ridge characteristics are more simple compared to that in the load identification model. However, to answer the research question these simplification should not matter.

The contact model calculates the center of mass of the volume of crushed ice in each time step, which is implemented as the contact point for structure-ice interaction. The ice load direction is defined to be normal to the hull. An ice friction force along the hull is also implemented, which is calculated as: F f r iction = µ F (see equation 2.7). Where µ is the friction coefficient, taken as 0.15 [4]. The model used several · nor mal,ice simplifications and assumptions, but at that time it had two key limitations: 1) the crushed mass was not removed, which implies that the model can not differentiate between crushed an non-crushed ice. 2) the rammed ridge is very thick, deeper than Oden, which implies that the assumption of ice loads working purely 72 5.N UMERICALMODELS on the hull is not correct as soon as the keel bottom starts penetrating the ridge. These two key limitation, part of the contact model, were mitigated during the development of the load identification model. Hydro- dynamic resistance is taken into account as discussed in section 4.2. For used parameters is referred to the paper. It must be stressed that some of these parameters are different than those used in the load identification model. A more comprehensive summary of the assumptions made in the simulation model is shown below: 1. An ice ridge impact can be simulated by a rigid body simulation in 3DOF. 2. Geometry of vessel and ridge are idealized. Oden is idealized as a rigid body with a flat bottom and, constant bow angle of 22 degrees. Influence of complex geometrical shapes and components on the hull, for example bow plough and thrusters, are completely ignored. 3. Since the hull is broader at the bow than at the mid section it is assumed that only the hull has this higher width. This is to prevent underestimation of ice loads. 4. The ice ridge is assumed to have a homogeneous ice strength, density (917 kg/m3) and constant SEA of 2175 J/kg. 5. Crushing is the only failure mode of the ice. 6. The ice ridge cannot move or rotate during impact. 7. The ramming is a head on collision, with completely symmetrical cross sections at the hull. 8. Several given or approximated vessel characteristics are assumed to be constant, also during impact. These include: location COG, metacentric height, vessel inertia, hydrodynamic mass, hydrodynamic open water resistance, hydrostatic forces. 9. Environmental forces can be ignored. 10. There is no thrust during impact (this is not the case in the load identification model). 11. The ice load acts only on the vessel hull in the direction normal to the hull area plus the friction between ice and hull dependent on the normal ice load. Results of the simulation model will be discussed later. However, the author likes to emphasize that the SEA values presented in Figure 2.33 can be adjusted. Reasoning for this lowering is discussed in section 6.2. To what extend changing is allowed, could not be justified using literature reviews. Initially the SEA value will therefore not be changed.

5.2. GENERAL INTRODUCTION TO THE LOAD IDENTIFICATION MODEL In the case of very large ice floes, like a multi-year ice ridge, and straight on impact (meaning that the icebreaker rams the floe perpendicular to the ice edge), only three ship motion components (heave, pitch and surge) are to be accounted for [4]. As such, the load identification model in this thesis is a two dimensional time-domain model with three degrees of freedom: surge, heave and pitch. The icebreaker Oden is idealized as a rigid body and represented by a nonlinear mass-damper-spring system. Maneuvering theory is applied, which means that the hydrodynamic variables are estimated at one frequency of oscillation. Forces acting on the vessel that are taken into account are hydrodynamical forces, thrust forces, ice forces, and the Coanda effect. The model strives to reconstruct the time responses (i.e. rigid body motions and excitation loads) of the ridge ramming experiment from full -scale data from the Oden AT research cruise project 2013. The full-scale data analyzed in this thesis includes a profile of a multi-year ice ridge, vessel characteristics, acceleration data from a motion reference unit (MRU), GPS data, and propulsion data.

The load identification model combines the Kalman filter and a joint input-state estimate algorithm to estimate the state- and excitation vector from acceleration, velocity and displacement data in 3DOF (i.e. surge, heave, pitch). The joint input-state estimate algorithm combines measured data with an estimate of the state of the system in a way that minimizes the error. Propulsion theory is applied to calculate the thrust force in section 5.5.2. A contact model is developed to identify the contact point between vessel and ice, and is also capable of calculating the penetration volume into the profiled rige. Before the load identification model can be used, several conversions have to be applied to the data, e.g data synchronization over time, and GPS data conversion.

5.2.1. DATA SYNCHRONIZATION Before the load identiﬁcation model is presented, all full-scale data are synchronized in time. Propulsion data were given in 2 Hz sample frequency, IMU data in 10 Hz. Interpolation is applied for propulsion data to match that of the IMU data. 5.3.GPS DATA CONVERSION 73

5.2.2. MODEL DIMENSIONANDAXISCONVENTION Several data conversions are applied to transform the raw scale data to conventional physical quantities. As an example several measurements were given in binary form, or in non conventional dimensions. Further- more data must comply to chosen axis/coordinate system, thus towards the dimensions used in the load identification model. The defined coordinate system of the load identification model is presented in figure 3.2. After the transformation, data are implemented in the load identification model as follows: • Displacements are given in [m]. (With surge direction positive towards the bow, sway positive to portside, and heave positive upwards). • Velocities are given in [m/s]. • Accelerations are given in [m/s2]. • Angular displacements given in [rad]. (Roll positive if starboard goes down, pitch positive if bow goes down, and yaw positive if bow rotates anti clockwise) • Angular velocities given in [rad/s]. • Heading [rad] is defined as the angle between the direction of the vessels’s nose, and the true east. In load identification model anti-clockwise rotations are taken positive.

5.3. GPS DATA CONVERSION Compared to MRU/IMU data, GPS data require some additional conversions before it can be implemented in the load identiﬁcation model. GPS data that requires conversion: • Latitude* (degrees) • Longitude* (degrees) • height* (cm) • North velocity (m/s) • East velocity (m/s) Following steps are presented in section 5.3.1, 5.3.2, and 5.3.3.

*Note: Beforehand raw displacement data have been transformed from binary format to latitude (degrees), longitude (degrees) and height (m) format.

5.3.1. GPS RAW FLAT EARTH COORDINATES (MAST) Data present surge and→ sway displacement in latitude and longitude coordinates. In the load identification the vessel, simulated as a rigid body, moves in a flat earth coordinate system. Hence a conversion is required to get to the new reference frame. It must be emphasized that this requires a choice of origin position. It is important to define the origin location near the positions of interest (ridge), as the accuracy of this transformation decreases with increasing distance from the chosen origin.

5.3.2. GPSFLAT EARTH COORDINATES (MAST) GPSFLAT EARTH COORDINATES (COG) After synchronization and the dimensional data conversion,→ some additional conversions are applied. One of the main reasons is that GPS data is gathered in the mast of the vessel. This means that the data must be transformed to the displacements experienced in the COG instead of that what is measured by the GPS system 35 meters up in the mast, see ﬁgure 4.7. To transform the data from ’mast data’ to ’COG’ data, transformation matrices were used. These are presented in equation 5.2. In equation 5.2 Ax , Ay , and Az correspond to the rotation matrices due to roll (x4), pitch (x5) and yaw (x6) respectively.

1 0 0   cos(x ) 0 sin(x ) cos(x ) sin(x ) 0 5 5 6 − 6 Ax 0 cos(x4) sin(x4), Ay  0 1 0 , Az sin(x6) cos(x6) 0 (5.2) = − = = 0 sin(x ) cos(x )) sin(x ) 0 cos(x ) 0 0 1 4 4 − 5 5 Rotation around the heave axes does not need to be taken into account, as it does not change position in the ﬂat earth coordinate system. Using equation 5.2, we deﬁne the rotational transformation matrix: Rr ot :   cos(x5) 0 sin(x4) Rr ot  0 cos(x4) sin(x4)  (5.3) = − sin(x ) sin(x4) cos(x )cos(x ) − 5 4 5 74 5.N UMERICALMODELS

Followed by the transformation to determine the difference between mast reference frame and COG reference frame by:       dx1mast COG x1mast x1mast − Rotation e f f ect dx2mast COG  Rr ot x2mast  x2mast  (5.4) − = − = · − dx3mast COG x3mast x3mast − where dx1mast COG , dx2mast COG , and dx3mast COG represent the displacement difference between mast − − − and COG. To get the actual COG coordinates:       x1COG x1mast dx1mast COG − x2COG  x2mast  dx2mast COG  (5.5) = − − x3COG x3mast dx3mast COG −

Figure 5.1: Route of experiment, ﬂat earth coordinate system. Left an overview, at the right zoomed in with the 4 impacts marked.

5.3.3. GPSFLAT EARTH COORDINATES (COG) RIGIDBODYREFERENCEFRAME During the ramming experiment the vessel moves in 6DOF.For→ surge and sway these are now in a course over ground reference system, located at COG. Course over ground is the actual direction of progress of a vessel, between two points, with respect to the surface of the earth. However, the load identification model uses rigid body movements. It requires input of total experienced displacement in surge direction, total displacement in sway direction, velocity in surge direction, and velocity in sway direction. These displacements and velocities depend on the heading. Reason for this requirement is that the load identification model is in 3DOF,and needs input on total displacement in the surge direction. After this conversion, the load identification model is able to identify the motion state when a vessel moves laterally, forward or backwards.

A lateral velocity is not allowed to be confused with a forward velocity, as that situation is fundamentally different. For example, it is possible for a vessel to move only Northwards with a heading of 0.5π radians, meaning 100% laterally, i.e. pure sway movement with no forwards movements at all.

Given COG coordinates at two time steps (ti and ti 1), and their mutual differences: −

dx1COG,i x1COG,i x1COG,i 1 = − − (5.6) dx2COG,i x2COG,i x2COG,i 1 = − − then the displacement with respect to the rigid body reference frame in dt is given by:

dx1i dx1COG,i cos(headingi ) dx2COG,i sin(headingi ) = · + · (5.7) dx2 dx1 sin(heading ) dx2 cos(heading ) i = COG,i · − i + COG,i · i 5.3.GPS DATA CONVERSION 75 and the total displacement with respect to the rigid body reference frame (x1,x2):

x1i x1i 1 dx1i = − + (5.8) x2i x2i 1 dx2i = − +

Besides displacement conversions, North and East velocities (e.g. vNor th, vEast ) are transformed to surge and sway velocities (e.g. x˙1i and x˙2i ), given the heading(headingi ): µ µ ¶¶ vNor th,i x˙1i vh cos arctan headingi = · vEast,i − µ µ ¶¶ (5.9) vNor th,i x˙2i vh sin arctan headingi = · vEast,i − where vh represents the horizontal velocity over ground. At this stage all data is ready to be used in the load identiﬁcation model to determine the global ice loads. 76 5.N UMERICALMODELS

5.4. CONTACT MODEL Identifying the Contact Point (CP) of ice structure interaction is the Contact Model’s primary purpose in this thesis. Instead of assuming a load patch it enables determination of a physically more correct location, because CP is directly linked to the volumetric indentation into the ice. The numerical model will be able to deﬁne acting ice moments using this CP.However, the Contact Model (CM) also delivers volumetric indentation, contact area, and hull penetration velocities in normal direction, which makes it highly applicable for simulation purposes. The volumetric indentation for example can be used combined with the speciﬁc energy principles to determine ice loads. Another example is using the CM for ice friction simulations. It’s the author’s aim to validate the CM using the full scale data presented in Chapter 4.

Figure 5.2: Contact Model discretizes vessel, ice ridge, and level ice into triangles. At the left idealized Oden, in the middle the ice ridge, at the right level ice with a thickness of 1.7 m. In the ice ridge white triangles represent the consolidated layer, magenta represents the keel layer.

In CM vessel geometry and ice ridge geometry are discretized into a 2D triangle mesh. Accuracy depends on the time step and the complexity of the interacting geometries. Simple structures can be modeled with just a few triangles, where complex structures tend to require more, and smaller, triangles to produce a more correct idealization. The total mesh size is not limited, except that it will inﬂuence computation time. The CM geometry idealization is shown in Figure 5.2. Figure 5.2 shows both the consolidated layer and keel layer in the ice ridge. Note that also level ice is modeled. The thickness of the consolidated layer, keel, and level ice were determined by proﬁling data. In Appendix E the ridge idealization is presented in more detail. This Appendix also includes the idealized ice ridge coordinates.

As soon as structure and ice are discretized, CM can be used. The basic idea is that the model identiﬁes how a certain triangle (lets say Ta) interacts with another triangle (Tb), i.e. to what extend does Ta penetrate Tb? Hereby CM needs to determine areal overlap of the triangles (which is related to volumetric penetration under assumption of symmetry in width), and intersection coordinates between structure and ridge mesh. When the vessel moves inwards the ice, indentation volume, contact area, and intersecting coordinates will change. This change can be used to identify the increase of indentation, and therefore identify the loss of ice mass. Theoretically this loss of mass should directly relate to ice loads, but in reality that is not always the case. Furthermore, assuming that the load acts at the mass center point of the lossed indentation volume in a certain time step, then CM will identify this Contact Point of ice-structure interaction.

Summarized, the most important assumptions of the CM are:

1. Structure and ice may be simulated as rigid bodies. 2. Ice-structure interaction always occurs head on; completely symmetrical cross sections at the hull. dV olume d Area Beam, where d Area is calculated in CM and Beam is the width of the structure. = × 3. Any 2D structure and/or piece of ice can be simulated by a mesh of triangles. 4. Inﬂuence of complex geometrical shapes and components on the hull, for example bow plough and thrusters, are allowed to be neglected. 5. Loss of mass relates directly to ice loads. 6. The Contact Point (CP) between structure and ice is assumed to be located in the mass center point of the penetrated volume (per time step).

Speciﬁc assumptions related to CM’s application within this thesis are:

1. Contact model takes into account 3DOF: surge, heave, and pitch. 5.4.C ONTACT MODEL 77

2. Oden’s hull width differs in length, with a hull that is broader at the bow than at the mid section, it is assumed that only the bow has this higher width. This to prevent underestimation of ice loads. 3. The ice ridge cannot rotate during impact. (Although possible as a (future) implementation in CM). 4. (When used for simulation purposes) Ice load direction is deﬁned to be normal to the hull, plus the friction force along the hull (normal to ice load). 5. In case no CP can be determined, a proper approximation must be made.

Contact Model includes several key components to achieve the required results, which are discussed in the coming sections. These key components include the following:

1. Triangle Interaction Algorithm 2. State Identiﬁcation and Simulation 3. Load Case Categorization 4. Shape identiﬁcation of penetration 5. Rotation Based Algorithm 6. Penetration Volume and Contact Point Determination

It must be stressed that this CM is developed in Matlab. Matlab script is available for delivering intersection points between 2 polygons. However, CM must identify which coordinates build up the overlap area (penetration), and which don’t. Furthermore it must differentiate which part is crushed and which part is not crushed ice. Initially these functions seemed straightforward, but required a new developed algorithm to get to the desired results. In this thesis this algorithm is named the ’Triangle Interaction Algorithm’. More about this in section 5.4.1.

5.4.1. TRIANGLE INTERACTION ALGORITHM At the base of CM is the Triangle Interaction Algorithm. This algorithm determines intersection points between two triangles (Ta, Tb). Basics steps are given in this section. The basic steps of the Triangle Interaction Algorithm are:

1. Input: deﬁne coordinates of the two triangles in accordance with coordinates counterclockwise. (=input) 2. Simulate geometry of Ta and Tb. 3. Identify coordinates of Tb inside Ta. This is based on a linear combination of 2 vectors; is a point within or outside the vector space? It determines which points are within the space enveloped by the vectors simulated between two triangle coordinates. Every coordinate is checked for each vector, resulting in a 3x3 matrix that contains the information about which coordinates of Tb are within/outside Ta. 4. Given step 3, calculate all coordinates of Tb inside Ta. 5. Calculate their mutual intersection coordinates. Differing algorithms for the four possible solution of previous step (0,1,2, or 3 coordinates of Tb within Ta). 6. Identify and sort coordinates that make the overlap areashape. Sort these CP-coordinates in a counterclockwise order (=output). 7. Calculate overlap area between the triangles (=output).

The Triangle Interaction Algorithm is applied until all vessel triangles are compared with all ridge triangles. If nvesselT represents the number of triangles used in the mesh to simulate the vessel, and nr idgeT the number of triangles used in the mesh to simulate the ice ridge, then the Triangle Interaction Algorithm is applied n n times. Mesh size is therefore recommended to be tuned, to achieve an optimal equilibrium vesselT × r idgeT between computation speed ( n n ) and accuracy ( idealization made). ∈ vesselT × r idgeT ∈

5.4.2. STATE IDENTIFICATION AND SIMULATION As the vessel moves the state of the system changes in time. To calculate the change of indentation volume, information of two time steps is required (t1 and t2). The state vector at t1 and t2 are the input variables of the CM. The state vectors in CM must include: surge displacement (x1), heave displacement (x3), and pitch angle (x5). For convenience it is assumed that the vessel always moves rightward into the ridge, as visualized in the ﬁgures in this section.

During the State Identification and Simulation procedure, the vessel is simulated in the state at t1 and t2: defined as state1 and state2 respectively. This means that for both states the vessel and ridge are simulated 78 5.N UMERICALMODELS with their defined triangles. To explain a simplified example is shown in Figure 5.3, where the two red triangles represent a vessel with bow pitched upwards (T1 and T2), and the three blue triangles the ice ridge (Tr1, Tr2, Tr3). A different state vector (x1, x3, x5) will result in a different location and orientation of the triangles.

Figure 5.3: Simpliﬁed example of a vessel (left,red) and ice ridge (right, blue) in the CM. The CM calculates the total penetration (marked yellow).

Figure 5.3 illustrates a ridge impact, with the vessel already penetrated into the ridge. Assuming the vessel moves, state 1 and state 2 will generally give a different total penetration and Contact Coordinates. The contact coordinates at the hull are deﬁned as: P1, P2, P3, P4, PB1, and PB2. Here state1 delivers P3, P4, PB2. Thus state2 delivers P1, P2, and PB1. These coordinates are marked in Figure 5.4. In the most general case of an impact, the ridge only interacts with the bow since the keel bottom is quite deep. Only very thick ridges might interact with the bottom hull of the vessel. The general case is visualized in Figure 5.5. Further deﬁned in CM: total penetration volume due to T1 at t1 = Abtm1, with ’btm’ referring to ’bottom hull’. Total ’bow’ penetration, i.e. penetration volume due to T2 at t1, = Abow1. Similar for state2: Abtm2 and Abow2.

Given Abtm1, Abow1,Abtm2, and Abow2 the change of indentation for bow and bottom hull are given by equation 5.10: d Abow d Abow2 d Abow1 = − (5.10) d A d A d A btm = btm2 − btm1 and d Abow (P1, P2, P3, P4) ∈ (5.11) d A (PB2, PB1, P1, P4) btm ∈

Figure 5.4: Simpliﬁed example of state1 and state2 of a vessel in the CM. State 2 is moved to the right, downwards and rotated counter clockwise. If the vessel would be totally enveloped in ice, then the yellow area would be the newly penetrated area.

Figure 5.5: Simpliﬁed example of the most general case of an ridge impact as in CM. A vessel (left) penetrates an ice ridge (grey). The penetration in the time step is marked yellow. 5.4.C ONTACT MODEL 79

5.4.3. LOAD CASE CONDITIONS CM makes a distinction between bow and bottom loads. To enable this, CM needs to comply to the following two conditions:

• Bow-Condition: Bow loads can physically only occur if the bow is in the ridge. In CM this is identiﬁed when the keel stem of the vessel at time t2 is not the most right* coordinate when added to the set of coordinates representing d Abow (P1, P2, P3, P4). If otherwise, the bow moved out of the ridge, meaning no loads can occur. • Bottom-Condition: Bottom hull loads can only occur if the bottom hull is inside the ridge.

Physically an ice load can only act if there is a loss and/or displacement of ice. Question arises where the load acts if the CM determines no penetration, but the numerical model still identiﬁes a load. This might very well occur since this thesis analyzes full scale data, which includes uncertainties and errors. This decreases CM’s accuracy. Thus, in the case CM identiﬁes zero or negative penetration, then the CP value must be approximated:

• CP-Condition: In the case no volumetric indentation is calculated, even though physically incorrect, an approximation to CP must be determined to ensure a smooth working numerical model.

To guarantee a CP,the ﬁrst approach applied by CM in case of no-CP is a ’level ice approach’. This means that CM simulates level ice at the bow. No situations have been observed during CM validations that this point was not identiﬁed after this level-ice-approximation.

*note: in the numerical model, the vessel moves to the right during ramming. ’Most right’ coordinate may be translated to ’deepest penetrated’ coordinate.

5.4.4. SHAPE IDENTIFICATION OF PENETRATION The Contact Coordinates (P1, P2, P3, P4, PB1, PB2) form a certain shape for both bow and bottom. The 2D areal size of these shapes equals d Abow and d Abtm. These shapes are always one of the following in the CM: 1. Quadrilateral shape 2. Triangular shape 3. Line shape 4. Point shape 5. No shape (meaning that bow and ridge do not interact with each other)

Quadrilateral shape is for example the case in Figure 5.5. CM uses Contact Coordinates to identify which shape is simulated. For the bow the following shape options are possible:

1. Quadrilateral shape: set of identified CP coordinates includes 4 nonidentical coordinates (P1 P2 6= 6= P3 P4) 6= 2. Triangular shape: set of identified CP coordinates includes 3 nonidentical coordinates (P1 P4 or = P2 P3) = 3. Line shape : set of identified CP coordinates includes 2 nonidentical coordinates (P1 P4 and P2 P3) = = 4. Point shape: set of identified CP coordinates includes 1 nonidentical coordinate (P1 P4 = P2 P3) = = 5. No shape

For the bottom hull the same theory applies, but then with other coordinates. Line shape, point shape, and no shape correspond (physically) to a zero ice load. CP-condition is then applied. For line shapes it is assumed that CP is exactly in the middle between these Contact Coordinates. In case of point shape, this point equals CP.If there is no shape identiﬁed, then CM simulates the surrounding level ice and recalculates the CP again for that situation. This approach was characterized in section 5.4.3 as the ’CP-condition’ with ’Level ice approximation’.

Triangular shape occurs with a set of identiﬁed CP coordinates that include 3 nonidentical coordinates. In general this is the case when P1 P4 or P2 P3. However, for a correct working CM, all possible deﬁnitions = = are checked. These include:

1. P1 = P4 2. P2 = P3 80 5.N UMERICALMODELS

3. P1 or P4 is not identiﬁed. (CP coordinates are a set of 3) 4. P2 or P3 is not identiﬁed. (CP coordinates are a set of 3) 5. P1 = P2 6. P3 = P4

In case of triangular shape or quadrilateral shape an ice load may be simulated. The question is: does the vessel penetrate into the ridge, retreat out of it, or do both phenomena occur simultaneously? To answer this question a Rotation Based Algorithm is developed, presented in section 5.4.5. The quadrilateral and triangular shapes can differ tremendously in time.

5.4.5. ROTATION BASED ALGORITHM To differentiate between penetration and retreat it must be identiﬁed if the Contact Coordinates from state 2 moved forwards compared to state 1. CM can not directly compare which coordinate is located most ’right’,as pitch movement’s rotation can induce situations where this comparison is false. Therefore another approach was developed. In this Rotation Based Algorithm the Center Of Shape (COS) is determined. Subsequently it is checked if the coordinate translation were penetration or retreat. The methodology is required for quadrilateral shape and triangular shape. Summarized the general steps that the Rotation Based Algorithm executes:

1. Insert CP coordinates of the identiﬁed shape (during Shape identiﬁcation in section 5.4.4) 2. Calculate Center Of Shape (COS) 3. Generate location based axis system with COS as origin 4. Simulate vectors from COS to all Contact Coordinates and calculate the tangent. 5. Given the tangent and Contact Coordinates, determine if translation was penetration/retreat.

Rotation Based Algorithm uses the centroid formula. This equation requires coordinates of a triangle (C1, C2, C3), and determines the centroid or geometric center:

C1x C2x C3x COSx + + = 3 (5.12) C1y C2y C3y COSy + + = 3

Where COSx is the COS x-coordinate, and COSy is the COS y-coordinate. Figure 5.6 illustrates the ﬁrst four steps of the Rotation Based Algorithm. The coordinates C1 and C2 are variables. Note that they might be in any different orientation, including mutually changed. The Rotation Based Algorithm calculates what the rotational translations direction from C1 to C2 is. This rotational direction from C1 to C2 (R(C1 C2)) can → either be clockwise, or anti-clockwise: R(C1 C2) α α (5.13) → = 2 − 1 For two coordinates the rotation R(C1 C2) is anticlockwise if α α is positive, and clockwise if α α is → 2 − 1 2 − 1 negative.

Figure 5.6: Step 1,2, 3 and 4 for a triangle that was simulated by three coordinates. Note COS, simulates vectors (to C1 and C2), and the tangent of these two vecors (α1 and α2). 5.4.C ONTACT MODEL 81

TRIANGULARSHAPE Triangular shape occurs with a set of identiﬁed Contact Coordinates that include 3 nonidentical coordinates. In general this is identiﬁed when P1 P4 or P2 P3, this is illustrated in Figure 5.7. Figure 5.7 shows the = = difference between penetration (marked yellow) and retreat (marked white).

Figure 5.7: Most general identiﬁed triangular shapes with their COS. A blue arrow corresponds to positive translation, red arrow to retreat translation. The positive translation correspond to penetration, hence this is marked in yellow

After the ﬁrst four steps of the Rotation Based Algorithm, it must be identiﬁed if the translation was penetration or retreat. It must be emphasized that this rotation is around COS. Using equation 5.13 the following is stated:

P4 P1 is indentation if R(P4 P1 ) posi ti ve bow → bow bow → bow = P3bow P2bow is indentation if R(P3bow P2bow ) neg ati ve → → = (5.14) PB2 PB1 is indentation if R(PB2 PB1 ) posi ti ve btm → btm btm → btm = P4 P1 is indentation if R(P4 P1 ) neg ati ve btm → btm btm → btm = Given this information, CM is aware of which areas are penetrated and which are retreated.

QUADRILATERAL SHAPE Depending on the translation of the intersection coordinates (P4 to P1, and P3 to P2) four different Quadri- lateral shape categories can be identiﬁed. This is illustrated in Figure 5.8. The procedure is quite similar to that of triangular shapes. Summarized, the general steps that the Rotation Based Algorithm executes are:

1. Insert CP coordinates of identiﬁed shape (during Shape identiﬁcation in section 5.4.4). 2. Split Quadrilateral shape into two triangles. 3. Calculate the COS of the two triangles (COST1 and COST2). 4. Determine the area of the two triangles. 5. Calculate Center Of Shape (COS) by calculating the weighted average of the COST1 and COST2. 6. Generate the location based axis system with COS as origin. 7. Simulate vectors from COS to all CP coordinates and calculate the tangent. 8. Given the tangent and CP coordinates, determine if translation was penetration/retreat.

It is recommended to take a look at Figure 5.9. The quadrilateral shapes differ in time. The two triangles obtained by separation made in step 2 of Figure 5.9 are in general not equal in size. Applying the Rotational Based Algorithm as in equation 5.13, and using the same deﬁnition as in equation 5.14. At this stage CM is aware of which part is penetrating into, or retreating out of. 82 5.N UMERICALMODELS

Figure 5.8: Quadrilateral alternatives showing penetration areas (yellow), retreat (red) and unchanged areas (white), deﬁned around COS.

5.4.6. INDENTATION VOLUMEAND CONTACT POINT IDENTIFICATION The last required step is to identify how much ice was crushed, and what the Center Of Mass (COM) of the crushed ice is. In this thesis, the discussed COM is the Contact Point (CP) during the ice-structure interaction.

Figure 5.9 presents an overview of the steps applied for a quadrilateral shape. As the most right phase shows, there is a signiﬁcant difference between COS of the quadrilateral shape and the actual COM. CM’s capability to differentiate between crushed and uncrushed ice, is found to be beneﬁcial during simulations. For convenience CM determines and saves: retreated areas, COM of retreated areas, total penetration in ice (for bow hull and keel hull).

Figure 5.9: Quadrilateral procedure of determining the CP of penetration. Showing penetrated areas (yellow), retreated (red) and unchanged areas (white), deﬁned around COS. From the left; 1) Split into two triangles with their COS; 2) Determine COS of complete quadrilateral shape; 3): Generate vectors from COS to Contact Coordinates, calculate their tangent, and apply to determine if translation was penetration/retreat; 4) Calculate the COM of the crushed ice (= CP,result CP delivered from total CM)

For both bow and bottom hull a Contact Point is calculated. During simulations it can be used to apply loads on different locations. However, with respect to this thesis just one CP is used. It can be seen as an weighted average, or virtual CP.CP depends on the bow CP and bottom CP.The indentation volume per bow and hull is taken into account in relation to the total indentation volume, thereby deﬁning to what extent either CP must be taken into account.

5.4.7. BENEFITS AND LIMITATIONS BENEFITS • Accuracy of the Contact Model is precise with high possibilities for simulations. • Contact Point no longer needs to be assumed. It will mitigate the amount of idealizations and assumptions made on structure-ice interaction (such as load patch), which will enhance accuracy of calculations. • Contact Point is physically quantiﬁed. Different from many methods, CM relates Contact Point directly to the lost/moved mass of ice, which enables a physically more correct quantiﬁcation. 5.4.C ONTACT MODEL 83

• Rotation Pitch motion is taken into account too. • Penetration vs retreat differentiation: CM can differentiate between crushed and non crushed ice, mitigating the chance of overestimations of penetration volume. • Triangle mesh properties: The triangles can be given different characteristics. For example at an ice ridge keel, and consolidated layer can be taken into account differently. • Multi-applicable: Although designed for a vessel impacting an ice ridge, CM can be used for basically any areal two dimensional penetration. i.e. it is not limited to one purpose. • Load Identification and CP. With this CM a justified value of CP can be used. Thus not only loads will be identified in the model, but also the location of the global load. It enables you to see how the load evolved through the ice. • Can take into account bottom hull loads too.

LIMITATIONS • 3DOF maximum: It works two dimensional with 3DOF maximum (with the assumption of symmetrical loading). This is often not compatible to reality. • Assumptions: CM is based on assumptions, and executed idealizations. As with all engineering problems, the more assumptions, the higher the uncertainty. • Differentiation between crushed and non-crushed ice. CM is able to differentiate between crushed and crushed ice under condition that the penetration velocity is higher than 0.05 m/s. This CM property is usefull, as it mitigates errors compared to contact models that cannot differentiate. For lower penetration velocities (v 0.05 m/s) CM cannot guarantee this property, and therefore presents a lim- < itation; when penetration velocity is slow combined with a fast rotation around CP an error might occur, i.e. only at low penetration velocity a situation occurs in which a rotational displacement might be higher then translational displacement. That is the vessel must penetrate sufficiently fast into the ice. The value 0.05 m/s can be seen as an empirically determined value, and might decrease when sample frequency is higher. This limitation should not cause a problem. Under low velocities ice loads will be low as well, and the vessel will not have enough momentum to impose a significant error. This justifies the limitation quan- titatively: small velocity, small load, small error. Another way of explaining is that these velocities are simply out of the interest area of engineering purposes. A second situation that will introduce the same limitation occurs when the CP does not change (i.e. the vessel does not penetrate), but the vessel does still pitch around this CP.To the author’s knowledge this situation is extremely rare, and it was not observed when testing CM on full scale data. It can be stated that it therefore is statistically justified: given that the probability of limiting mechanism occurrence is low, then the expected error will be low. Furthermore, crushing failure is observed at high impact velocities. • Discretion algorithm is missing. At the moment of writing this thesis, CM can not yet discretize geometries itself. This means that it has to be put in manually by the user. For simple structures, this is not a problem, but for very complex and/or large mesh required structures it will take a long time to get the CM starting. A discretization algorithm that can do this automatically is a solution, which might be an idea for future work. • Computation time. Under the condition that geometrical idealization and mesh discretization were applied correctly, CM may be characterized as precise. However, direct, or ’live’, application is not suitable (yet). An example where live application would be handy is a vessel that uses radar to identify ice thickness, and then uses IMU’s to determine live loads. Reason why CM is not suitable is a) the discretization algorithm is missing (as discussed above), b) computation time. The discretization algorithm problem is expected to be solvable, that computation time can only be decreased by more advanced computers. It’s the author’s expectation that CM will not be able to handle high sample frequency measurements with conventional computers. 84 5.N UMERICALMODELS

5.5. SET UP OF THE LOAD IDENTIFICATION MODEL This chapter discusses the numerical model that was build in Matlab for load identiﬁcation. The theoretical background that is applied is discussed in previous chapters. Full scale data were presented in Chapter 4. The aim is to combine theory with full scale data, with the purpose of identifying what actually happened during the ramming experiments (e.g. state of the system and load identiﬁcation).

The numerical model requires the Contact Model, data synchronization, and GPS data conversion as input. This chapter will present the overview, using theory and algorithms that were presented separately.

5.5.1. EQUATIONS OF MOTION Given the axis convention presented in section 3.3.1, this section presents the location and directional deﬁ- nition of acting forces (e.g. thrust T (t), horizontal ice load Fx1,ice (t), vertical ice load Fx3,ice (t) and following total pitching moment PM (t) M (t), M (t)). x5 ∝ x5,T x5,ice Thrust is deﬁned as positive when applying a forward moving force on the vessel. Subsequently, ice loads will counteract this motion and push the bow upwards. For an illustration the reader is referred to Figure 5.10.

The 3DOF equations of motion for the system are presented for surge (x1), heave (x3) and pitch (x5):

X (m a )x¨ (t) b x˙ (t) x˙ (t) F (t) (5.15) + 11 1 + 11,NL| 1 | 1 = x1 X (m a )x¨ (t) b x˙ (t) b x˙ (t) x˙ (t) k x (t) k ¡x (t) x ¢ F (t) (5.16) + 33 3 + 33 3 + 33,NL| 3 | 3 + 33 3 − 35 5 − 5,mean = x3 X (I a )x¨ (t) b x˙ (t) k ¡x (t) x ¢ k x (t) M (t) (5.17) y + 55 5 + 55 5 + 55 5 − 5,mean − 53 3 = x5 With total excitation forces and moment defined as: X F (t) T (t) F (t) (5.18) x1 = x1 − x1,ice X F (t) T (t) F (t) (5.19) x3 = x3 + x3,ice X M (t) M (t) M (t) (5.20) x5 = x5,T + x5,ice Equation 5.15, 5.16 and 5.17 take into account the potential coefficients (a11, a33, a55, b11, b33, b55), nonlinear surge resistance coefficient (b11,NL), nonlinear heave resistance coefficient (b33,NL) and hydrostatic coefficients (k33, k55, coupling terms: k35, k53). Values were presented in Chapter 4. x5,mean is implemented because of the fact that Oden has a certain pitch angle in static equilibrium. This value is determined using the full scale MRU data: x 0.0243[rad]. 5,mean = −

Horizontal ice load Fx1,ice (t) is taken positive in aft-wards direction, vertical ice load Fx3,ice (t) is taken positive upwards. Tx1(t) and Tx3(t) represent the continuous-time thrust in surge and heave direction respectively. These depend on pitch, as shown in equation 5.21. A forward and upward thrust are taken positive. Mx5,T (t) is the continuous-time pitch moment due to thrusters, and Mx5,ice (t) the moment due to ice. The pitching moment due to thrust is positive when rotational direction is taken with the bow moving downwards. These loads are illustrated in Figure 5.10.

Mx5,ice (t) depends on the Contact Point (CPx ,CPz ), as illustrated in equation 5.22. The moment due to horizontal ice load Mx5,ice,x1 depends on the vertical distance between CP and COG, and Mx5,ice,x3 on the horizontal distance between CP and COG. Mx5,T (t) is calculated similarly, except that the propeller location is ﬁxed on the vessel. The propeller location is based on Oden’s geometrical characteristics presented in Figure D.9 and Figure D.10 (Appendix D), and coordinates were presented in equation 4.1.

Tx1(t) T cos(x5(t) x5,mean) = · + − (5.21) T (t) T sin(x (t) x ) x3 = · − 5 − 5,mean

Mx5,ice (t) Mx5,ice,x1 Mx5,ice,x3 = + (5.22) M (t) F (CP COG ) F (CP COG ) x5,ice = − x1,ice · z − z − x3,ice · x − x 5.5.S ET UP OF THE LOAD IDENTIFICATION MODEL 85

Figure 5.10: Simpliﬁed example of a forward moving vessel under ice loading and deﬁned pitching moment. Forces: red arrow T = thrust, blue arrow Fx1,ice = horizontal ice load, blue arrow Fx3,ice = vertical ice load. Locations: COG, Contact Point ice, CPT = propeller location.

5.5.2. PROPELLER THRUST The load identiﬁcation model takes thrust into account as presented in section 3.3.5, including the wake factor of 0.8. For convenience, equation 3.40 for thrust is repeated:

2 1 πD ¡ 2 2¢ T Ct (β0.7R ,P/D) ρ V (0.7πnD) (5.23) = · 2 4 e +

Given the propulsion data in 4.6, it is concluded that Oden did not only work in the (β0.7R ,P/D) range presented in (Roddy, 2006) [53], described in section 4.6. An algorithm that is applicable to all (β0.7R ,P/D) ranges is presented in this section. This algorithm works for both starboard and portside thrusters.

The thrust coefficient depends on β0.7R and pitch ratio (P/D) and is only presented in the range of P/D between 0.6 to 1.4 in the Wageningen Four Quadrant Estimate (Figure 4.10). Furthermore, after Inverse Fast Fourier Transform of the thrust coefficient data, a matrix of thrust coefficients is generated, which was presented in Appendix D. Noted matrix can directly be applied in equation 5.23, and contains 500 values, each at a specific hydrodynamic pitch and P/D value. All values in between are determined using interpolation, assuming linear relations. Other approximations were required to identify C t(β,P/D) for values of β0.7R and P/D outside the range. General range is defined as:

• 0 β 2π radians, 0.6 P/D 1.4 < 0.7R < < < The following ranges outside the general range are identiﬁed:

• P/D < 0.6 and 0 β 12.37 < 0.7R < • P/D < 0.6 and 12.37 β 180 < 0.7R > • P/D < 0.6 and 180 β 0 − < 0.7R < A comprehensive description of the algorithm is presented in Appendix D.2. As an example, the procedure to determine range (0 β 12.37 and P/D<0.6) is given below: < 0.7R < Statement: if 0 β 12.37 and P/D=P/D<0.6: < 0.7R < 1. Identify the β , with P/D 0.6, for which C t becomes equal to 0: C t(β ?,P/D 0.6) 0 0.7R = 0.7R = = = 2. Given above, assume a linear relationship, and ﬁnd the β0.7R for which C t would be zero given the P/D value: C t(β ?,P/D P/D) 0. 0.7R = = = 3. Identify C t(β 0,P/D 0.6) 0) 0.7R = = = 4. Given the results of point 2 and 3, assume a linear relationship between these, and interpolate to the C t value by: C t(β β ,P/D P/D) C t(β β ,P/D 0.6) P/D 0.7R = 0.7R = = 0.7R = 0.7R = ∗ 0.6 86 5.N UMERICALMODELS

5.5.3. STATE ESTIMATE AND FORCE ESTIMATE FOR 2DOF AND 3DOF SYSTEM Applied forces can be split into hydrodynamic forces, the thrust force, and the global ice forces. Location and direction of these forces change due to surge, heave and pitch motions of the vessel. The location of the thrusters is constant, direction depends on pitch angle. Coupling terms k35 and k53 depend on heave and pitch. Location and direction of the global ice forces (Fx1,ice (t), Fx3,ice (t), Mx5,ice (t)) are determined by the Contact Model.

Provided theory presented in section 3.2, it is concluded that the joint input-state estimate algorithm limits the number of forces that can be identified in the load identification model. Given two acceleration data P P (surge, heave), two excitation loads can be identified with the algorithm: Px1(t), and Px3(t). However, P the excitation moment for pitch motions ( Px5(t)) is a function of the loads defined in the equations of motion for surge and heave. This raises the question if the joint input-state estimate algorithm can be directly applied to the 3DOF system. It is concluded that this is not possible, hence the joint input-state estimate algorithm can only be used for the 2DOF system (i.e. surge and heave motions). As a consequence, another approach is required to solve the complete 3DOF system which also contains pitch motions. The reason why the joint input-state estimate cannot be used for the 3DOF system is elaborated on below. To find the state of the 3DOF system a combination of the joint input-state estimate algorithm, Kalman filter, Contact Model, and iterations taking into account an a priori pitch estimate, and full scale data, is used.

The aim of this thesis is to ﬁnd the global ice loads which are located at the CP between ice and vessel. This CP depends on the state of surge, heave, and pitch motions at two adjacent time steps. To calculate the pitching moment due to the ice, the CP is required, which itself depends on the state of the system. This state depends on the excitation loads of which only two out of three can be calculated without the input of the CP. This implies a loop-like problem: CP depends on a load that itself depends on the CP,which cannot be calculated simultaneously. In this model this is mitigated by iterations using an a priori estimate for pitch, CM, and the Kalman ﬁlter. This iteration is applied until the change of CP between iterations is considered to be small enough to have a negligible effect.

Before explaining how the state and input can be estimated for the 3DOF system, it should be understood why the state and input for the 2DOF should be calculated before pitch motions can be taken into account. As stated in Chapter 4, the MRU device is located at the COG. This means that the joint input-state estimate algorithm can calculate excitation forces that are located at the COG. If the excitation moment for pitch can not be calculated directly, by the joint input-state estimate algorithm, as is possible for the surge and heave excitation force, than this pithing moment must be a function of the heave and surge excitation force, and the arm from force to COG. This arm between COG and locations where the excitation forces act, is equal to zero, i.e. no matter what the excitation forces in the COG for surge and heave motions are, the moment at COG as a function of these excitation forces is per deﬁnition equal to zero, as long as there are only data measured at the COG. In case the IMU was not located at the COG, the joint input-state estimate algorithm requires input of the arm between COG and location of MRU, and an excitation moment can be calculated. Reliability of the joint input-state estimate algorithm will enhance when data is available at more than one location (i.e. several MRU measurements will be required). For completeness, a mathematical proof is given below. For the mathematical proof we start with equation 3.1: Mu¨(t) Cu˙(t) K u(t) f (t) Sp (t)p(t), where f(t) + + = = P P is the excitation vector. It is assumed that we want to determine the three loads instantly ( Px1(t), Px3(t), P Px5 (t)), but only have two acceleration data (x¨1, x¨3). In state-space form, this means that the inﬂuence matrix (Sp (t)) and force time history vector (p(t)) become:

  P    1 0 0 Px1(t) x1, x1˙(t), x1¨(t) P ∝ Sp 0 1 0, p(t)  Px3(t)  x3, x3˙(t), x3¨(t) (5.24) = = = ∝ 0 0 1 PP (t) x , x ˙(t), x ¨(t) x5 ∝ 5 5 5 5.5.S ET UP OF THE LOAD IDENTIFICATION MODEL 87

8 1 Now given data vector d(t) R × , the following selection matrices (Sa, Sv. Sd) are deﬁned:         x1(t) 0 0 0 0 0 1 0 0 x ˙(t) 0 0 1 0 0 0 0 0  1        x ¨(t) 1 0 0 0 0 0 0 0  1                x3(t) 0 0 0 0 0 0 1 0 d(t)  , Sa  , Sv  , Sd   (5.25) = x˙3(t) = 0 0 = 0 1 0 = 0 0 0         x¨3(t) 0 1 0 0 0 0 0 0         x5(t) 0 0 0 0 0 0 0 1 x˙5(t) 0 0 0 0 1 0 0 0 As equation 5.25 shows Sa has a different size than Sv, and Sd (the acceleration data for pitch is missing). The P direct consequence is that Px5(t) cannot be determined by the joint input-state estimate algorithm. Math- 1 ematically this can be understood via equation 3.8; J [Sa M − Sp ]. Given that we want to determine three nDOF np 3=3 3 3 force time histories (np=3), then Sp R × R × . Furthermore M R × . Then J can only be com- n 3 ∈ n 2 =∈ ∈ n np puted if Sa R d × , which it is not: Sa R d × due to the missing acceleration data. Jc R d × , therefore Jc ∈8 3 ∈ ∈ should be R × to make the algorithm work.

In case Sa is implemented with the right size (i.e. last column only zeros), then the algorithm will not work T 1 1 T 1 either. This can mathematically be explained by equation 3.15: M (J Rˆ− J)− J Rˆ− . When applying this k = k k T ˆ 1 equation (J Rk− J) is a singular matrix, i.e. not invertible. This due to the fact that the matrix J’s last column has only zeros, so the determinant will be zero.

JOINTINPUT-STATE ESTIMATE ALGORITHM TO SOLVE 2DOF SYSTEM First the state and input is calculated for the 2DOF system at the COG. This 2DOF system includes the states of surge and heave motions (x1 and x3), and excitation forces (Px1 and Px3). The equations of motions to be solved: X (m a )x¨ (t) b x˙ (t) x˙ (t) P (t) (5.26) + 11 1 + 11,NL| 1 | 1 = x1 X (m a )x¨ (t) k x (t) b x˙ (t) b x˙ (t) x˙ (t) P (t) (5.27) + 33 3 + 33 3 + 33 3 + 33,NL| 3 | 3 = x3 Then the nonlinear terms are linearized, so that it can be implemented in the joint input-state estimate algorithm. How linearizing is achieved, is discussed in section 5.5.4. This linearizion changes the continuous-time state space model, hence the discrete-time space model. Therefore a reformulation of the state space model is applied at each time step. Following on this, the joint input-state estimate algorithm is applied and we ﬁnd: state surge, state heave, excitation force in surge direction (Px1), and excitation force in heave direction (Px3). The a priori estimate in the algorithm, always uses the state space model that belongs to that time step. The error due to linearizing is investigated in section 5.5.4.

SOLVING THE 3DOF SYSTEM The results of the joint input-state estimate algorithm are implemented in the 3DOF system. Given Px1 and Px3, these total excitation forces can be split into the following parts: X X P (t) F (t) T (t) F (t) (5.28) x1 = x1 = x1 − x1,ice X X ¡ ¢ Px3(t) Fx3(t) k35 x5(t) x5,mean X = + − (5.29) P (t) T (t) F (t) k ¡x (t) x ¢ x3 = x3 + x3,ice + 35 5 − 5,mean The thrust force and a priori estimate of pitch is known. This enables calculation of the global ice loads Fx1,ice and Fx3,ice . It must be emphasized, that the deviation between the a priori and a posteriori state estimate of pitch was found to be low enough, to directly use the a priori estimate.

The state of pitch depends on the pitching moment, which depends on the vertical global ice load Fx3,ice , which again depends on the state of pitch. Therefore, iterations are used to mitigate the error in pitch state. This is elaborated on below. The contact model calculates the contact point for the ice loads, using identified surge state, identified heave state, a priori estimate for pitch, and pitch data at given time step. With the CP, Mx5,ice is determined, and thereby also Mx5. This enables the Kalman filter to identify the state of pitch with the equations of motions as in equation 5.15, 5.16, 5.17, 5.18, 5.19, and 5.20. 88 5.N UMERICALMODELS

Contact Point with Kalman Filter Iteration : (At a certain time step i) pitch state is required to determine CP,CP is required to determine the force Mx5, and Mx5 is required to determine the pitch state. This presents a loop-like problem. This is mitigated by iteration:

1. iteration 1: CP depends on the identiﬁed state x1i 1, x1i , x3i 1, x3i , x5i 1 + a priori estimate of x5i . − − − Here is taken into account the distance traveled in 0.5 dt, which is x˙ ,x˙ ,x˙ . · ∝ 1 3 5 • The results are used with the Kalman ﬁlter to identify an ’improved a priori estimate’ of pitch at time (x5i ).

2. iteration 2: CP depends on identiﬁed state x1i 1, x1i , x3i 1, x3i , x5i 1 + improved a priori estimate of − − − x5i .

• The results are used with the Kalman ﬁlter to identify the state of pitch at time i (x5i ).

3. iteration 3 (optional): CP depends on identiﬁed state x1i 1, x1i , x3i 1, x3i , x5i 1 x5i . − − − • This iteration is only used to verify if the difference between the CP locations between iteration 1 and iteration 2, is small enough.

5.5.4. LINEARIZINGNONLINEARTERMS It was found that some terms should be implemented nonlinearly in the numerical model. However, the used Kalman ﬁlter and joint input-state estimate algorithm are intended for linear systems. Therefore nonlinear terms cannot directly be used. In the numerical model this problem is solved by linearizing each nonlinear term at each time step. These nonlinear terms include the hydrostatic spring stiffness k , k k , 33 35 ∝ 33 k k , and hydrodynamic damping terms b and b . There must be a continuity between the 53 = 35 11,NL 33,NL nonlinear terms and linearized terms. How terms are linearized, and what the imposed error is due to this linearizion, is presented in this section.

A reaction force to a nonlinear spring stiffness kNL is given by:

F k x2 (5.30) = NL · Linearizing at displacement x gives: δF kNL 2x δx = · · (5.31) δF k δx = L · where kL is the linearized spring stiffness. The nonlinear term is linearized at each moment of time. When the model is discretized, the linearized relationship between two subsequent forces Fi and Fi 1 is given by: +

Fi 1 Fi kL(xi 1 ) ∆x + = + + 2 · F k x2 (5.32) i = NL · i ∆x xi 1 xi = + − where i is the time step, xi the displacement of the spring at time step ti , xi 1 the displacement of the spring + at time step ti 1, kL(xi 1 ) the linearized spring stiffness at a displacement xi 1 . It is assumed that at ti the + + 2 + 2 state xi and input Fi are known, and xi 1 and Fi 1 are unknown (i.e. to be determined by load identiﬁcation + + model). To determine the state and input of the next time step the linearized spring kL(xi 1 ) is required. This + 2 cannot be calculated, since it depends on the unknown displacement xi 1 (no evolution of xi 1 is assumed, + + xi 1 can be smaller, equal to, or larger than xi ). The best estimate of the linearized spring stiffness at ti 1 is + + 2 at ti : kL(xi 1 ) kL(xi ) + 2 ≈ (5.33) k (x ) k 2 x L i = NL · · i As a result equation 5.32 can be rewritten:

Fi 1 Fi kL(xi ) ∆x + ≈ + · 2 ¡ ¢ Fi 1 kNL xi kNL 2 xi xi 1 xi (5.34) + ≈ · + · · · + − 2 Fi 1 2 kNL xi xi 1 kNL xi + ≈ · · · + − · 5.5.S ET UP OF THE LOAD IDENTIFICATION MODEL 89

The continuous time equation of motion for surge and heave are found in equation 5.15 and 5.17 respectively. In the discretized equation of motion for surge and heave, the nonlinear damping b11,NL and b33,NL are linearized for ti 1 as follows: +

b11,NL x˙1(ti 1) x˙1(ti 1) 2 b11,NL x˙1(ti ) x˙1(ti 1) b11,NL x˙1(ti ) x˙1(ti ) · | + | + ≈ · · | | + − · | | (5.35) b33,NL x˙3(ti 1) x˙3(ti 1) 2 b33,NL x˙3(ti ) x˙3(ti 1) b33,NL x˙3(ti ) x˙3(ti ) · | + | + ≈ · · | | + − · | |

The nonlinear part k33,NL of the heave’s spring stiffness k33 is linearized as follows:

2 2 k33,NL x3(ti 1) 2 k33,NL x3(ti ) x3(ti 1) k33,NL x3(ti ) (5.36) · + ≈ · · · + − ·

In equation 4.22 it was found that k53 in the equation of motion for pitch has a nonlinear part as well. There- fore, linearizing k x is achieved by: 53 · 3 2 k53 x3(ti 1) k33,L r x3(ti 1) k33,NL r x3(ti 1) · + = − · · + − · · + (5.37) 2 k53 x3(ti 1) k33,L r x3(ti 1) 2 k33,NL r x3(ti 1) k33,NL r x3(ti ) · + ≈ − · · + − · · · + + · ·

Similarly, for k35:

2 2 k35 x5(ti 1) k33,L r x5(ti 1) k33,NL r x5(ti 1) · + = − · · + − · · + (5.38) 2 2 2 k35 x5(ti 1) k33,L r x5(ti 1) 2 k33,NL r x5(ti 1) k33,NL r x5(ti ) · + ≈ − · · + − · · · + + · ·

INFLUENCE OF LINEARIZING NONLINEAR TERMS IN NUMERICAL MODEL The question arises if this imposes (signiﬁcant) errors. This section strives to quantify the imposed error that occurs due to the linearized terms. Furthermore, the linearized nonlinear terms are compared to a linear approximation of the nonlinear terms, as the linear approximation might impose a smaller error than the linearized nonlinear terms.

First a simulation is executed. This simulation takes into account the equations of motion, as they are presented in equation 5.15, 5.16, and 5.17. The simulation model works with nonlinear terms (no need to lin- earize). The loads are exactly known, and are chosen to be in the same order of magnitude as the identified loads in the numerical model. The basic idea is to generate data with the simulation model, and use that data to recalculate the loads with the numerical model. If the algorithm is perfect, the identified loads will be equal to the simulated loads. If there is an error, it means that the linearized terms impose an error in the model. It must be emphasized that no measurement noise was simulated, to enable pure quantification of the error due to linearized terms.

First results indicate a surprisingly large error in the vertical global ice load, both for linear and linearized terms. It was found that the highest contribution to the error in the vertical global ice load was due to the error in identiﬁed pitch state, and not due to the error from nonlinearity. The pitch-heave coupling is important, and here arises a problem for the numerical model: there is no pitch acceleration data, and as a consequence the calculated pitch excitation moment will have a large error, and therefore the state estimate of pitch will have a large error as well. This is elaborated on below. Furthermore, it was found that state iden- tiﬁcation of surge and heave have high accuracy, also for nonlinear spring stiffness. This was a condition of the test (no noise generated).

To identify a correct vertical global ice load, the correct state of pitch must be identified. For the numerical model, a good state estimate of pitch requires: a) the joint input-state estimate algorithm and an input of pitch acceleration data, or b) the Kalman filter when the applied pitching moment is correct. The numerical model must use the Kalman filter. The error in the pitch estimate depends on the error in the pitch excitation moment. The main reasons for the error in the pitch excitation moment are:

1. The correct excitation moment depends on the forces taken into account and the forces not taken into account (for example wind, currents, etc). There is an error in the excitation moment because not all forces are now taken into account. 2. The loads that are taken into account have an error them self, causing an error in the pitch excitation moment. 90 5.N UMERICALMODELS

3. The arm to the loads have an error. This causes an error in the pitch excitation moment. To optimize the numerical model, it is necessary to take the pitch data as very accurate compared to the pitch estimate by the model. In other words, for a good estimate of pitch motions the process noise must be large enough compared to the measurement noise. With respect to the numerical model, this means that the noise covariance matrices must be implemented in such a way that the state estimate of pitch is always very close to that of the pitch measurements. This condition is implemented in the numerical model.

Based on the ﬁndings above a new simulation and identiﬁcation model were developed, that had no pitch- heave coupling terms. This approach makes the global vertical ice load independent of pitch motions, and thereby it is possible to identify the error purely from linearizing the nonlinear terms. Based on these models, the error found due to linearizing nonlinear terms were 1% and 1% for the horizontal and vertical global ice load respectively. The linear terms result in a larger error than the linearized nonlinear terms, as illustrated in Figure 5.11.

It therefore is concluded that the nonlinear terms, proposed in the numerical model result in an error, but that this error is smaller than when using only linear approximations for these nonlinear terms. As a result nonlinear terms are linearized at each time step in the numerical model. For a more comprehensive description of the analyzes the reader is referred to Appendix G.

Figure 5.11: The vertical ice load as from the simulation model (correct, black line). Furthermore, two lines: the vertical ice load as from the identiﬁcation model using linearized nonlinear terms (red line), the vertical ice load as from the identiﬁcation model using linear terms (blue line)

5.5.5. MODELSETUPOVERVIEW The model setup is summarized as: 1. Data synchronization in time 2. Data synchronization in dimensions 3. GPS data conversion (GPS raw data to rigid body reference frame) 4. Determine continuous-time state-space model using Equations of Motions 5. Split the system to: (a) 2DOF system with joint input-state estimate algorithm (b) 3DOF system with Kalman ﬁlter and Contact Model iteration 5.5.S ET UP OF THE LOAD IDENTIFICATION MODEL 91

6. Determine discrete-time state space model for both 2DOF and 3DOF system as shown in sections 3.2 and 3.2.2. 7. Parameter implementation (e.g. ship parameters, potential coefficients, etc) and determine data vectors required for: 2DOF join input-state estimate algorithm, and 3DOF Kalman filter algorithm 8. Specify the input for joint input-state estimate algorithm and the Kalman filter state estimate algorithm according to the theory in section 3.2 and 3.2.2 (e.g. covariance matrix, location matrix, initial conditions, etc) 9. Start the state and input estimate algorithms (thus for each time step): (a) Apply join input-state estimate algorithm to identify the state of surge, state of heave, global force P P in surge direction ( Px1,dt ), global force heave direction ( Px3,dt ). This includes linearizion, and reformulation of state matrices as presented in section 3.2 and 3.2.2. (b) Given the propeller data, calculate the total thrust followed by decoupling the thrust in surge and heave direction (Tx1(t), Tx3(t)). (Applying theory of 3.3.5) (c) Identify applied forces for surge and heave, by working out 5.28 and 5.29 (d) Define the 3DOF system; with EM as in equation 5.15, 5.16, 5.17, 5.18, 5.19, and 5.20. (e) Apply Contact Point with Kalman Filter Iterations to determine the pitching moment and the state of pitch 10. (Save results)

5.5.6. FILTERPARAMETERSANDTUNING According to the theory of section 3.2 and 3.2.2, the choice of the diagonal elements of the covariance matrices Q and R should be based on the order of magnitude of the state vector and the accuracy of the sensors, respectively. Measurement noise covariance R can be determined prior to operation of the filter, since it depends on the measurement systems. Generally superior filter performance can be obtained by tuning the filter parameters Q and R. For the 3DOF system the data vector is represented by:   d(x1(t)))  d(x˙ (t))   1   d(x¨ (t))   1     d(x3(t))  d(t)   (5.39) =  d(x˙3(t))     d(x¨3(t))     d(x5(t))  x˙5(t)) where d(x1),d(x3), and d(x5) represent data of displacement in surge, heave and pitch respectively. The uncertainty of these parameters is given by the measurement noise vk : T £ ¤ vk3do f vx1 vx˙1 vx¨1 vx3 vx˙3 vx¨3 vx5 vx˙5 = (5.40) vkT £ 1.0 0.07 0.005 0.05 0.07 0.005 0.02 pi()/180 0.0027¤ 3do f = ∗ The parameters depend on the measurement system’s accuracy, which are presented in Chapter 4. Accelera- tion data has an accuracy of 1cm. Therefore, an uncertainty of 0.5 cm is implemented. Using equation 5.40, the measurement covariant matrix equals R E[v vT ]. = k k Even though there is a theoretical equation of motion that precisely predicts the state of position and velocity, using the data, that may not be true in reality. The predictions may be wrong if some extra parameters come into play that are not taken into account (for example wind, extra force, etc). Therefore we use the process noise covariance Q, which uses a system noise of w : Q E[w wT ]. There is no easy way to define these k = k k values. In this thesis the noise values were empirically determined, i.e. the system noise values were changed until state identification seemed most reasonable. A way to look at it, is to reach a smooth curve fit on the data, while taking into account the accuracy given in equation 5.40. As a result the following process noise vector is presented:

T £ ¤ wk3do f wx1 wx3 wx5 wx˙1 wx˙3 wx˙5 = (5.41) T £ 2 5 6 5 6 3¤ wk 5 10− 1 10− 5.5 10− 1 10− 1 10− 1.5 10− 3do f = · · · · · · For initial conditions the covariance matrix P is assumed to be equal to Q. 92 5.N UMERICALMODELS

5.5.7. COANDA EFFECT The Coanda effect obtains its name from aeronautical engineer Henri Coanda. The Coanda effect is the phenomenon in which a jet ﬂow attaches itself to a nearby (convex) surface and remains attached even when the surface curves away from the initial jet direction, as visualized in ﬁgure 5.12 (Thrustmaster’s Houston, 2015) [58].

A thruster jets out a flow of water. In free surroundings, this jet of water entrains and mixes with its surroundings as it flows away. When a nearby surface is close enough, it restricts the entrainment in that region. What happens is that a flow accelerates to try to balance the momentum transfer. A difference in this momentum causes a pressure difference across the jet, which deflects the jet closer to the surface, possibly attaching the jet to the surface. With respect to thrust this often is undesirable because: a) the jet stream flowing along the bottom plating of the structure enhances friction due to it’s high velocity, b) the pressure difference can cause significant forces on the structure or it’s surroundings.

Figure 5.12: The Coanda effect makes the jet ﬂow attach itself to a nearby surface [58].

Identifying the influence of the Coanda phenomena on a structure is generally accomplished using CFD analysis (Computational Fluid Dynamics) and/or model tests [58]. This is out of the scope of this thesis. Fur- thermore, it is expected that the Coanda effect has little influence on the vessel’s performance because the thrusters are located at the aft of the ship, which entails that the nearby surface is small. The phenomena might have a significant influence when moving backwards.

It is the author’s suggestion to approximate the Coanda effect in bollard pull conditions. If the effect is negligible it does not need to be implemented in the numerical model. Approximating the effect can be achieved using propeller theory (to approximate jet velocities) and Bernoulli’s principles on incompressible ﬂow:

v2 P g z constant (5.42) 2 + · + ρ =

With velocity (v) in m/s, gravitational acceleration (g), pressure P (N), density of the ﬂuid at all points in the ﬂuid (ρ), and z as the elevation of the point above a reference plane, with the positive z-direction pointing upward.

The flow velocity in a jet flow depends on the thrust applied, the distance from the thrusters, and the location in the jet area perpendicular to the flow (wake area A j ). The velocity at the propellers depends on the thrust applied by the thrusters and the area of water this thrust is applied on, e.g the area of the propellers. According to Cozijn et al. (2010) [59], the flow behind the thruster can be described in two separate areas, being the initial developing zone ( x 7 8) and the fully developed zone ( x 7 8). The wake flow in the initial developing D < − D > − zone has a core with lower axial velocities (near the propeller hub), surrounded by a higher velocity region (near the propeller blades). Close to the propeller, the peak in the axial velocity is found at approximately r 0.8R. The wake flow in the fully developed zone has a single peak, at the propeller centre line (r 0). = = The flow pattern in the fully developed zone resembles a swirling turbulent jet. Mean velocity v j [m/s] in the propeller wake can be described by equation 5.43 [59]. s 2T v j (5.43) = ρA j 5.5.S ET UP OF THE LOAD IDENTIFICATION MODEL 93

5.5.8. CALCULATION OF COANDA EFFECT IN A NUMERICAL MODEL In the load identification model, mass balance is applied in the increasing wake area, with a wake divergence angle of 13 degrees [58]. It is assumed that the wake area increases equally into all lateral directions up to the point the flow reaches the vessel’s hull. Starting from first hull contact, it is assumed that the flow follows the hull with a wake area that can only increase in the horizontal plane. This assumption is in agreement with Thrustmaster’s Houston, (2015) [58], which states that this height remains constant or decreases. Since Oden has two thrusters, the flow is constrained in the middle of the ship. It is assumed that flows are symmetrical, and that the particle velocity inside a wake area is uniform (no local variations). Using vessel geometry the wake velocity over the hull can be determined, and by using equation 5.42 the pressure differences can be determined.

The calculation of the Coanda effect starts with a geometrical idealization of Oden’s hull. It is assumed that port and starboard side are identical, and that the Coanda effect is symmetrical. Along the hull, the normal wake area A j is calculated at seven different positions: A j,i , with i=0...6. The length LB j (i=0..5) is between two adjacent wake areas from A j,i to A j,i 1. Given these lengths, Oden’s hull can be split into six hull areas: + B j,i , with i=0...5. These six areas are shown in Figure 5.13. The bow itself, which has on upwards slope, is assumed to have no Coanda effect. This is justiﬁed because: a) the effect will be small due to the relative slow water particle velocities in the jet at the bow, b) the upwards slope will cause such divergence that Coanda effect is expected to be insigniﬁcant.

Figure 5.13: Geometry used in Coanda effect calculations in the load identification model. In the top of the figure, the side view is presented. Here, the black arrows in the blue jet flow are a measure of water particle velocity. The larger the water velocity, the larger the under pressure. In the bottom the top view of the hull is presented, where the bow areas (B j,i ) with corresponding lengths (L ji ) are shown. Here it can be seen that at A j,3 the jet flows arrive at the middle of the vessel, where as a consequence the jet diversion stops. At A j5, the side hull is reached. A j,6b is the wake area used in the mass balance equation (to determine pressure), while A j,6 is the actual wake area underneath the vessel hull.

The hull area B j,i depends on its hull length LB j , and the width of their adjacent wake areas A j,i and A j,i 1. + Initially the wake areas start in a circle, with a radius equal to the thruster’s diameter. This area is assumed to increase linearly in all lateral direction, given the divergence angle of 13 degrees. Just out of the thruster (A j,0), there is a gap between the hull and wake, thus no Coanda effect. At a certain point the wake will hit the vessel’s hull (A ), at a distance LB 3.75 meters. From this location forward, the Coanda effect is working j,1 0 = on the vessel. A j,1 is still a circle, and it defines the constant height of the jet flow along the hull, which is equal to 5.35 meters. A j,2 is defined as the average area from the circular and quadrilateral shape, as illustrated in Figure 5.15. The other areas that are located more forward have a quadrilateral shape with a height of 5.35 meters (i.e. A j,3, A j,4, A j,5, and A j,6). A j,3 is the location where the diverging flow hits the middle of ship, and where the divergence in inwards direction stops. Outwards, the jet is still diverging. A j,4 is where the aft slope ends, and the keel starts. A j,5 is the location where the diverging wake arrives at the sides of the hull, meaning that the wake area has a width equal to the width of the vessel. As a result, the hull area B j,5 has 94 5.N UMERICALMODELS a width smaller than the jet flow underneath it.

See Figure 5.14 for the side view of Oden’s idealized aft when the thrusters are reversed. Figure 5.14 shows how the constant height of 5.35 meters is determined. Figure 5.13 shows the idealized geometry that is used. For the dimensions of Oden’s hull that are used for the Coanda effect calculations, the reader is referred to 5.1.

Figure 5.14: Side view of Oden’s idealized aft when the thrusters are reversed. The figure illustrates where A j,0, A j,1, and A j,4 are located. Furthermore, it shows where and how the jet flow diverges in height until the jet flow makes contact with the hull. From thereon the flow is assumed to have a constant height of 5.35 m.

Figure 5.15: The wake area A j,2 is the average area of the circular and quadrilateral shape.

Given the idealized geometry presented in Figure 5.13, the force due to the Coanda effect can be calculated. Using equation 5.43, the mean velocity at the wake at A j,0 is calculated. Following on this, the ﬂow rate Q j in 3 [m /2] at A j,0 is calculated by: s 2T Q j A j0 (5.44) = ρA j,0 ·

Using the mass balance, it was assumed that this ﬂow rate Q j remains constant in all wake areas. Therefore, the uniform velocity v j,i (i=0..6) at each wake area equals:

Q j v j,i (5.45) = A j,i

The average velocities v j on the hull areas B j,i are then calculated by:

v j,i v j,i 1 v j (B j,i ) + + (5.46) = 2 5.6.S ENSITIVITYAND UNCERTAINTY ANALYSES 95

Table 5.1: Dimensions of Oden’s hull that are used for the Coanda effect calculations. Shown are hull area B j,i, which has a hull length LB j,i and is enclosed by two adjacent wake areas A j,i and A j,i 1. Note that A j,6b is required for the mass balance. + 2 2 2 i B j,i [m ] LB j,i [m] A j,i [m ] A j,i 1 [m ] + 0 - 3.75 15.9 22.48 1 12.78 2.25 22.48 26.94 2 21.29 4.1 26.94 36.36 3 143.88 14.9 36.36 54.57 4 114.64 10.1 54.57 66.88 5 568.75 45.5 66.88 66.88 5b A 122.51 j,6b =

It must be emphasized that in reality the velocity profile is decreasing quadratically. For simplicity, velocity is assumed to decrease linearly between two adjacent wake areas. The next step is calculating the pressure Pi at each hull area using equation 5.42. The force is then calculated by F P B = i · j,i However, the presented calculations are only valid under a fully developed and constant jet flow. The jet flow depends on the thrust force, which changes over time. In the load identification model, it is assumed that the jet velocity profile depends only on the thrust force at a given time step, i.e. the jet flow changes instantly over the hull.

The development over time is implemented in the model. Given are the geometry, the velocity proﬁle as determined by the thrust force, and the moment of time tT 1 when the thrust force was reversed. The present moment of time is called t , meaning that the Coanda effect is working t t seconds. By taking equation T 2 T 2 − T 1 5.46, and the hull length LB j , it is calculated how far the most forward particles have traveled. The time dti for a water particle to travel over the length LB j is calculated by:

LB j,i dti (5.47) = v j (B j,i ) with i 0...5. Starting at the thrusters the jet ﬂow is developing, moving forward towards the bow. The total = jet ﬂow length that is in contact with the hull is called LCoanda, meaning that the Coanda effect acts from LB1 to LCoanda along the hull. Here, LCoanda in [m] is calculated by:

i 1 X− dti L v (B ) dt v (B ),i 0...5 (5.48) Coanda = j j,i · i + Pi 1 · j j,i = n 0 (tT 2 tT 1) n− 0 dti = − − = Pi 1 where (tT 2 tT 1) n− 0 dti has a minimum value of zero. − − = Based on assumptions, idealization and calculation elaborated above, it is now possible to calculate the force and pitching moment due to the Coanda effect. Since the wake velocity is much larger at the thrusters, the aft will undergo a larger under pressure compared to the bow. As a result it is expected that the Coanda effect will pitch the bow upwards.

5.6. SENSITIVITYAND UNCERTAINTY ANALYSES An uncertainty analyses is effectuated to investigate to what extend the results can be trusted. Furthermore, it is desirable to know which assumptions and/or parameters cause the biggest uncertainties. This is worked out in the sensitivity analyses. Several parameters have an unknown uncertainty, which makes the uncertainty analyses more difficult. It was concluded that the uncertainties of the parameters that influence the thrust force could be determined with sufficient accuracy. This accuracy enabled a more detailed study into the thrust uncertainty, which is effectuated by a Monte Carlo analyses. These thrust influencing parameters are the wake fractor, diameter of propellers, revolution rate, propeller pitch angle, and the vessel velocity. The thrust uncertainty analyses is discussed in section 5.6.4.

For the global ice loads the most important influencing parameters are presented first. Subsequently, by a sensitivity analyses is investigated which of these parameters’ uncertainties have the largest influence on the 96 5.N UMERICALMODELS global ice load uncertainty. Lastly, by taking into account the most influencing parameters, the upper and lower boundary of the global ice loads is approximated. This uncertainty analyses of the global ice loads gives mainly insight in the sensitivities rather than an accurate value of the uncertainty, since not all parameter uncertainties are included, probability functions of parameters are not well defined, model uncertainty is not known accurately, and uncertainty in the standard deviations are stated to be too high. The analyses still gives insight in the global ice load uncertainty, and is found is found in section 5.6.6.

Before the uncertainty analyses for the thrust force and global ice loads is presented, the most important parameter uncertainties are discussed in section 5.6.1. A more extensive overview on the uncertainty analyses for the thrust force and global ice loads can be found in Appendix I and Appendix J respectively.

5.6.1. PARAMETER UNCERTAINTY Uncertainty of the IMU data was discussed in chapter 4.5. This data was measured by the Kongsberg Seapath 320+ System and reads: acceleration, velocity, and displacement in time. The joint input-state estimate algorithm takes into account both process and measurement noise, and calculates the optimal estimate given these uncertainties. However, this algorithm uses the equations of motion, in which several other parameters are used (e.g. mass, added mass, spring stiffnesses in 3DOF, etc). These parameters have their own uncertainty as well, and may have signiﬁcant effect on the results. To what extent is presented in subsequent sections. First the uncertainties of the most important parameters need to be determined. When an uncertainty of a parameter could not be determined an approximation is made. The calculated or approximated uncertainties are presented in this section, including the reasoning behind the values. The parameters inﬂuencing the thrust force and global ice loads are discussed in section 5.6.2, and section section 5.6.3 respectively.

5.6.2. PARAMETERUNCERTAINTYOFTHRUSTFORCE It is assumed that the uncertainties of the thrust inﬂuencing parameters can be determined accurately enough to enable a Monte Carlo simulation, where uncertainty can be given in the 68% and 95% accuracy boundary layers. This under the assumption of normal distributed parameters, as such 68% of values drawn from a normal distribution are within one standard deviation away from the mean, and proximately 95% of the values lie within two standard deviations. The thrust uncertainty analyses requires input of all parameters by a mean value µ and a standard deviation σ.

1. Propeller Diameter - The diameter is very precisely presented in the technical drawings, and therefore a small uncertainty equal t0 1% of its value is taken into account. As a result the standard deviation for the propeller diameter is equal to 0.045 meter. 2. RPM -The revolution speed of the propellers should stay constant (139), but the data shifts between 133-140. Mean value for RPM was found to be 137.8, and the data has a standard deviation of σ 1.02. = To be at the save side a standard deviation of 2 RPM is taken into account. 3. Propeller’s wake fraction - The flow around a propeller is affected by the flow of water around the vessel’s hull. As a result the average speed of the water particles through the propeller plane is generally different than that of the vessel’s velocity. These wake fraction can generally be determined quite accurate. However, the presence of ice disrupts the flow of water around the propeller of a ship as well. In Dick et al. (1995) [60] is stated that the ice has a significant overall influence on the water velocity around the propeller, but the magnitude of this influence was found to be unclear. To be at the safe side, the author implemented an uncertainty of 10% on the wake fraction. As is concluded in the sensitivity analyses this effect appears not to be large, and therefore it is assumed that the 10% accuracy is a justified approximation of the uncertainty of the wake fraction. 4. Propeller pitch - The uncertainty of the propeller pitch is most difficult to quantify, since a conversion factor of 27/64 was required to make the data fit with expected values. However, under full thrust this conversion factor should be very accurate, as the propeller pitch under full thrust is accurately known. The author’s suggestion is to take into account an uncertainty on the conversion factor. To be at the safe side, σ 0.093 (10%). = 5. Vessel velocity - Uncertainty of vessel velocity is presented in section 4.5, and is equal to 0.03 m/s. The data shows an average velocity of around 1.5 m/s, and a maximum velocity lower than 4.5 m/s. Given this data, it is stated that the uncertainty of vessel velocity can be implemented as an error of approximately 2%. To be at the safe side, a standard deviation of 4% is taken into account. 5.6.S ENSITIVITYAND UNCERTAINTY ANALYSES 97

5.6.3. PARAMETERUNCERTAINTYOFGLOBALICELOADS There is not a complete description of all model and parameter uncertainties. Therefore, the most important inﬂuencing parameters are presented in this section with their model value, minimum value, and maximum value. These parameters will be taken into account in the uncertainty analyses.

1. Vessel displacement - Oden’s mass is equal to the displacement, which in the model equals 11500 tonnes. According to Barras (2014) [61], displacement is the weight of volume of water that the ship displaces, which includes the lightweight plus deadweight. Lightweight is the weight of the ship itself when empty, deadweight is the weight that the vessel carries (like fresh water, water in ballast tanks, food and other provisions, fuel, etc). Oden’s displacement can vary between 11000 and 13000 tonnes. This range was presented by the crew and Swedish maritime administration, and takes into account the approximately 4900 tonnes of deadweight. 2. Draft - The Swedish Maritime Administration states that the draft of Oden can differ between 7 and 8.5m. In the numerical models the draft equals 8 meters. Therefore, the minimum value equals 7 m, and the maximum value equals 8.5 m. 3. Oden’s bow angle - Oden’s bow angle is presented in the technical drawings, which is very precise. It is stated that the bow angle uncertainty may be ignored. 4. Waterline Length - Waterline Length is presented accurately in the technical drawing, and therefore has a small uncertainty. However, the waterline Length depends on draft and therefore it can vary significantly. The main components that the waterline length will influence are the metacentric height, and the waterline area Awl . Given the draft the waterline length can be calculated. For minimum draft, the minimum value for the waterline length is equal to 102.45 m. For maximum draft, the maximum value for the waterline length is equal to 110.42 m. 5. Beam - In Appendix D the geometry of Oden can be found. Here is found that the hull of Oden at the sides is very steep, and constant. An increase of draft will have very little influence on a change in width. What is left are roll motions, which will change the width at waterline level. When assuming an angle of 5 [deg], the width will be 0.9962 times it’s original value. This is used as an approximation for the uncertainty for beam length: 0.25 meter and 0.31 meter for hull at the parallel-mid body and the bow section respectively. However, beam size has mainly effect on waterline area (e.g. hydrostatic spring stiffness), metacentric height, and rotational inertia of the vessel. Since these parameters are already taken into account separately, with a bigger deviation than a change in beam introduces, it is assumed that beam width can further be neglected in the uncertainty analyses. 6. Waterline area - the waterline area mainly influences the metacentric height and hydrostatic spring stiffness k33, k55, k35 and k35. The waterline area depends mainly on the beam and length of the vessel, and therefore their minimum and maximum values were used to calculate the minimum and maximum value for Awl . The increase of waterline area is implemented in the model as an increase of the parallel mid-body of the vessel, with a maximum beam (25 meter plus uncertainty) This results in a minimum value of: 2624 (107.75 102.45) (25 0.25) 2490 [m2] , and in a maximum waterline area − − · + = value of 2624 (110.42 107.75) (25 0.25) 2691 [m2]. An additional 5% uncertainty is taken into + − · + = account resulting in a minimum value of 2366 [m2], and maximum value of 2797 [m2]. 7. Metacentric height - Meta centric height depends on the geometry of the submerged hull. Its metacentric height should be within the values calculated when using a bigger and smaller submerged hull. This is visualized in 5.16. Based on this simplified geometry, the minimum and maximum values can 1 bh3 be calculated by: GM KB BM KG (see equation 3.27), with BM 12 , where b and h represent = + − = the width and length of the waterline area when looking along the heave∇ axis downwards respectively. The minimum value is approximately 16.3 m, the maximum value 228.3 m. 8. Rotational inertia for pitch - The uncertainty of rotational inertia of pitch is approximated and equal to 10% of its original value. To be at the safe side an uncertainty of 20% is taken into account. The minimum value is equal to 9.61e+09 [kg m2], and maximum value equal to 1.44e+10 [kg m2]. · · 9. Coupling terms heave-pitch - The coupling terms k35 and k53 depend on the added buoyancy (i.e. k33), and the arm between COG and areal center point of Awl in the horizontal plane. The uncertainty of k33 is taken into account by the uncertainty of Awl . The uncertainty of the arm must be taken into account as well. A large uncertainty of the arm of 3 m is taken into account, resulting in a minimum value of 1.575 meters, and maximum value of 7.575 meters. 10. Location MRU and COG - The location MRU and COG mainly influences the coupling terms between heave and pitch, occurring moments (the arm to forces changes), metacentric height, and rotational 98 5.N UMERICALMODELS

inertia. Apart from the occurring moments these components are already investigated in the sensitivity analyses. The moment due to the thrusters is found to be small. The arm to the CP is large (approximately 45 m), hence a large change would be required to have a significant effect. For these reasons it is assumed that uncertainty of location of IMU and COG is already included in the uncertainty of the other components. 11. Non linear surge damping - An uncertainty of 30% may be used, to make sure that the error will be 4 2/m2 within the required confidence. The minimum value b11,NL 1.665 10 in [N s ], mean value 2 = · · 2 b 2.378 104 in [N s2/m ], and the maximum value b 3.093 104 in [N s2/m ]. 11,NL = · · 11,NL = · · 12. Non linear heave damping - It must be emphasized that the nonlinear heave damping b33,NL was based on a purely flat plate resistance calculation. According to the theory, there should also be a residual resistance which cannot be determined without full scale testing. This nonlinear heave damping term can therefore be seen as a lower boundary of b33,NL. It seems justified to use the nonlinear surge damping to approximate an upper boundary value for b33,NL. For the estimate it is assumed that the relationship is purely depended on the normal area. This normal area for heave motions equals the waterline area Awl . An additional 10% is taken into account for the upper boundary. Nonlinear surge damping 2 b 2.38 104 in [N s2/m ], with a normal area of B D 31.2 8, where B represents the beam 11,NL ≈ · · f × f = × b at the bow and Dr the draft. This results in an upper boundary value for the nonlinear heave damping 2 b equal to 6.34 107 in [N s2/m ]. The calculation is shown in equation 5.49. 33,NL · · 13. Contact point - the contact model determines the contact point for ice-structure interaction. It is expected that the horizontal and vertical global ice load in the load identification model will not be sensitive to this CP. CP will mainly influence the pitching moment, as it defines the arm between CP and COG. An error of 20% is taken as uncertainty. 14. Added mass - It is assumed that the added mass for pitch and heave can be estimated at 20%. Added mass in surge direction is smaller, and directly relates to the hydrodynamic surge damping. For this reason a larger uncertainty is taken into account of 30%. It must be emphasized that the added mass for surge is depended on vessel mass via its interdependency to the vessel’s draft (see section 4.3.5). 15. Potential damping for heave and pitch motions - The potential damping coefficients for heave and pitch motions (i.e. b33 and b55 respectively) are approximated by an analyses on a non-icebreaker vessel. The hull form of an ice breaker can be quite different from other ships. Therefore, it is expected that the uncertainty of the potential damping coefficients is large. An uncertainty of 25% is taken into account to be at the safe side. 16. Thrust uncertainty - Thrust uncertainty is based on the uncertainty analyses presented in section 5.6.4. It was found that a minimum value will equal 76%, upper value 100% thrust. 17. Coanda effect - There is a downwards directed ice load predicted by the model. This behavior is repetitive in other ramming attempts. The Coanda effect was investigated with the purpose to identify to what extend it could make the vessel move downwards. The calculations give an estimate of the Coanda effect with a large uncertainty. The maximum boundary setting does not take into account the Coanda effect. For the minimum boundary load case, 30% additional effect is taken into account.

Below equation shows the upper boundary value for the nonlinear heave damping. Here Bb represents the beam at the bow and Dr the Draft.

Awl b33,NL,upper b11,NL 1.1 (5.49) = · B Dr · b ×

Figure 5.16: Geometries used to calculate the metacentric height of Oden. At the left the idealized shape of Oden, in the middle the geometry that will deliver the lower boundary of metacentric height, at the right the geometry that will deliver the upper boundary of metacentric height 5.6.S ENSITIVITYAND UNCERTAINTY ANALYSES 99

5.6.4. SENSITIVITYANDUNCERTAINTYANALYSESOFTHETHRUSTFORCE The aim of the sensitivity analyses is to investigated how much inﬂuence a parameter uncertainty (i.e. input) has on the results (i.e. output). This is applied on all inﬂuential parameters. The aim of the uncertainty analyses is to quantify the accuracy of the results, when taking into account the uncertainty of all parameters. The complete sensitivity and uncertainty analyses is presented in Appendix I.

The sensitivity is a measure of how much the result will change when one parameter is changed with a certain value. In this case, the result refers to thrust force. A parameters is changed with one standard deviation in this sensitivity analyses. The sensitivity is calculated as a deviation to the mean, presented in percentage. This enables the comparison between sensitivity between parameters, as the sensitivity is normalized to the mean result. A certain moment of time t i is chosen for the comparison between sensitivities of different parameters. = This time step may not be arbitrarily chosen, it should be at a moment where thrust force is around the maximum (i.e. as is the situation when the vessel is ramming the ridge). This can be illustrated as follows: Ã ¯ ¯ ! ¯Resul tµ σ(t i) Resul tµ(t i)¯ S σ + = − = 1 100% + = Resul t (t i) − · µ = Ã ¯ ¯ ! ¯Resul tµ σ(t i) Resul tµ(t i)¯ (5.50) S σ − = − = 1 100% − = Resul t (t i) − · µ = wi th i 1,2,3,....n = where n is the number of time steps, S σ is the sensitivity [%] of a parameter when using µ σ instead of + + µ, and S σ is the sensitivity [%] of a parameter when using µ σ instead of µ. t 1370 [s] in the sensitivity − − = analyses. The approach for the sensitivity analyses is elaborated on in more detail in section I.1.

Uncertainty analyses can be characterized by a Monte Carlo simulation [62]. It is a method to determine the accuracy by varying parameters within statistical constraints. During the analyses a normal distribution is assumed for the parameters when taking into account their uncertainty. Hence, every parameter is presented with a mean value (µ) and standard deviation (σ). About 68% of the values drawn from a normal distribution are within one standard deviation away from the mean. Furthermore approximately 95% of the values lie within two standard deviations. The uncertainty analyses is presented in section I.2. An additional (conservative) model uncertainty was implemented as an uncertainty in the thrust coefﬁcient with a standard deviation equal to 0.02. A 1000 simulations were used to determine this uncertainty.

Given the results, the sensitivity of the thrust force can be approximated, so that it can be used in the sensitivity and uncertainty analyses of the global ice loads. The minimum value and maximum thrust value are taken from the 95% accuracy boundary layers at t=1305 [s].

5.6.5. SENSITIVITYANALYSESOFGLOBALICELOADS The sensitivity of the most important parameters and their uncertainties to the global ice loads, which were discussed in section 5.6.3, are investigated. The sensitivity analyses is executed by ﬁrst running the model with normal/conventional settings, and following on this running the model with one parameter changed. As a result the ratio between both calculated global ice loads is calculated and presented in percentages. Therefore, the sensitivity is a measure of how much percentage a parameter change has on the global ice loads. To enable a valid comparison between parameter sensitivities, each parameter is increased by 10% of its own magnitude. Results and a more comprehensive explanation of the sensitivity analyses is presented in Appendix J. Most important ﬁndings are presented in section 6.3.8, for a more comprehensive description is referred to Appendix J.

5.6.6. UNCERTAINTYOFTHEGLOBALICELOADS Given the results of the sensitivity analyses, two sets of parameters can be deﬁned which will result in an upper and lower boundary of the global ice loads, depending on the parameter uncertainties. A distinction can be made into data uncertainties and model uncertainties. The data uncertainties taken into account in the uncertainty analyses are: vessel mass, beam, draft, and waterline length, heave acceleration, heave velocity, and heave position. Model uncertainties that are taken into account in the uncertainty analyses are: metacentric height, rotational inertia, waterline area (function of draft), added mass, heave-pitch coupling 100 5.N UMERICALMODELS

terms k35 and k53, potential damping heave, potential damping pitch, nonlinear surge damping, nonlinear heave damping, thrust force, and the Coanda effect. An additional 1% model uncertainty should be expected due to inﬂuence of the nonlinear terms that are linearized at each time step. This is elaborated in section 5.5.4.

The parameters that are considered most important for the uncertainty analyses of the global ice loads were discussed in section 5.6.3. Section 5.6.3 also presents the mean and maximum values of the parameters, given their uncertainties. Based on the sensitivity study of section 5.6.5, two case studies are developed: the lower boundary case-study and the upper boundary case-study. The lower boundary case-study represent the set of parameters that result in the lowest global ice loads. This is enabled by implementing the parameter uncertainties in such a way that all the implemented uncertainties lower the load, compared to the global ice loads that result from the mean value of all parameters. Respectively, the upper boundary case-study represent the set of parameters that result in the highest global ice loads. The parameters for the mean value case-study (i.e. normal setting), lower boundary case-study, and upper boundary case-study, are presented in Table 5.2. The results are presented in section 6.3.9. 5.6.S ENSITIVITYAND UNCERTAINTY ANALYSES 101 ] ] ] 2 2 ] ] 2 ] ] 2 m · / m m / m 2 [kg] [kg] · / m / m ] 2 s / r ad 2 s · N [ kg · [ N s [ [ kg N s N o N [ [ es 33 [N] 6 [ n y 10 5 [m] .3 [m] 9 4 k 000 [t] .25 [m] .31 [m] 7 T 97 [ m · 0.42 [m] 11, mean 33, mean 10 8. · 10 16 10 10 a a 13 25 31 10 27 · · · · · 11 55, mean · 575 25 3 2 a 44 · 09 34 8. 8. 1. 1. per boundary value 093 2 1. 9. 6. 3. 1. up 4.25 ) ] ] 2 2 ] ] ] ] 2 m · / m m ] (eq. 2 [kg] [kg] / m / m ] · 2 s / r ad 2 · N [ kg [N] [ N s [ / m N s N [ kg o T [ [ 2 es 33 6 · 9 n s y 0 [m] 9 4 k 000 [t] .75 [m] .69 [m] 8.3 [m] · 66 [ m · 2.45 [m] 11, mean 33, mean 10 7. 76 10 · 10 10 a a 11 · N 24 30 22 23 · · · · 10 [ 0. 55, mean 575 95 7 8 a 4 61 · 46 0. 4. 0. 0. 665 9. 8 10 wer boundary value 5. · 1. 0. lo 2 1. ≈ 4.25 ) ] ] ] ] 2 ] ] 2 2 ) m m ] (eq. 2/ m · / m / m ] / r ad · 2 s 2 [kg] · N 5.23 [kg] [ N s 6 N s [ [ kg / m 6 N [ [ kg [m] [m] 2 [ es 33 6 9 10 no s y 10 0 [m] 10 4 k 500 [t] · 10 · 24 [ m · an value · 7.75 [m] 7.86 [m] 10 25 31 8. 10 · 10 3 .5 10 11 · N 10 26 · · 10 16 [ · [N] (eq. 2. me 11 575 60 4 2 T 20 38 273 4. 6. 1. 10 1. 2. 7. · 2 1. ≈ 53 k = 35 k NL NL y 33, 11, I 33 b b 55 b b 33, mean 11, mean 55, mean a a a meter h-heave coupling spring stiffness rust eta centric height aterline length aterline area raft onlinear surge damping onlinear heave damping oanda effect [N] eam at parallel mid-body eam at bow otational inertia for pitch dded mass surge dded mass heave dded mass pitch otential damping heave otential damping pitch essel displacement ncrease uncertainty heave data [N] para D V B B W W M R Pitc N N A A A P P I Th C able 5.2: The parameters for the mean value case-study (i.e. normal setting), lower boundary case-study, and upper boundary case-study. These are used in the uncertainty analyses T 6 RESULTS AND DISCUSSION

This chapter contains the results from the simulation model and the load identification model. The simulation model is discussed in section 6.2, and is based on specific energy principles of crushing of ice to simulate the ice load. The load identification model is used to jointly estimate the state and loads from full-scale data, and is presented in section 6.3. For completeness the structural requirements of Oden, according to DNV standards for Polar Class ships, were calculated and are presented in Appendix D.3.

The contact model is used in the simulation model and in the load identiﬁcation model. The general results of the contact model are discussed in section 6.1. In section 6.3.4 the contact points during the ﬁrst ramming experiment are presented. Also, a decomposition of the hydrodynamic forces was executed, to investigate how the vertical global ice load relates to the hydrodynamic forces during the ramming experiment (section 6.3.10). Lastly, the model predicts a downwards directed vertical global ice load at the end of a ramming. This repetitive behavior is found to be strange, and discussed in section 6.3.11.

6.1. CONTACT MODEL Identifying the Contact Point of ice structure interaction is the Contact Model’s primary purpose in this thesis. However, the Contact Model delivers also volumetric indentation, contact area, and hull penetration velocities in normal direction, which makes it highly applicable for simulation purposes. Instead of assuming a load patch it enables determination of a physically more correct location, because CP is directly linked to the volumetric indentation into the ice. This only holds for a purely crushing assumption, which is correct at the ﬁrst stages of the ridge impact.

It is concluded that the contact model works well, in the simulation model and in the load identiﬁcation model. The model makes a successful distinction between ice-hull interaction at the bow and bottom (i.e. keel) hull, resulting in their corresponding indentation volume and contact point. The contact model is found to work for all deﬁned penetration shapes: no shape, point shape, line shape, triangular shape, and quadrilateral shape (see section 5.4.4).

Vessel and ridge were idealized and implemented in the contact model. For the idealized ice ridge is referred to Figure 4.6. More information regarding the contact model is found in section 5.4.

6.2. RESULTS OF SIMULATION MODEL A simulation model was developed to simulate the global ice loads when using the specific energy approach. It was found that specific energy principles can be used to simulate an impact of a vessel into an ice ridge, under assumption that ice fails purely by crushing. In reality this is not the case, as other failure modes will come into play. Several studies and visual observations confirm that ice ridges fail by different failure modes than the crushing. However, at the beginning of impact the methodology is reliable.

The simulation model was build in an earlier state compared to the load identiﬁcation model. The reader is referred to the paper in Appendix F, for a comprehensive description of the simulation model. The most

102 6.2.R ESULTS OF SIMULATION MODEL 103 important results are listed below:

• The model simulates open water conditions correctly. • Simulations proof that the contact model works correctly and that it is an ideal tool to be used with simulations using specific energy approach. The key limitations of the contact model, as discussed in Appendix F, were mitigated after writing of the paper. • The simulated vessel movements during ridge impact seem reasonable, although conservative. An impact velocity of 3 m/s will result in a penetration of approximately 3.5 meters. Full scale data indicate that this penetrations should be around 10 meters. • The simulation model clearly shows two ice load peaks, shown in Figure 6.1. This force works normal to the hull and depends purely on crushing failure of ice. In horizontal and vertical direction this corresponds to an ice load of 10 and 19 MN respectively. The load identification model predicts loads around 5 and 11 MN for the horizontal and vertical global ice load respectively. The specific energy approach therefore overestimates the ice load. • Finding two load peaks is reasonable. Firstly, because full-scale data calculations clearly show two peaks as well. Secondly, at impact the bow will experience a large upward directed acceleration, due to the high ice load, resulting in high pitch motion. This motion decreases the ridge penetration speed resulting in a lower ice load. Subsequently, the vessel bow gets lifted upwards, increasing potential energy until the pitch rate changes direction. From this point positive pitch velocities are observed, increasing penetration velocity, resulting in the second peak. • Identification of equal magnitude in the two ice load peaks, is not reasonable. The full-scale data shows that the second peak should be significantly lower. This complies to the theory, as at a later stage of the ridge impact, the likelihood of other failure modes starts to increase. A substantial lower amount of ice will be crushed and bending failure is expected. This lowers the load at which ice fails. Hence a lower global ice load on the vessel, giving a lower second peak. • The simulation model gives valuable information about the energy dissipation in the first stages of impact until another failure mode than crushing takes over. At moment of impact most ice will be crushed, which implies the model’s methodology holds. The model’s credibility will fail as soon crushing does not predominantly happen any more. This moment might happen before the peak force is reached. Peak force found with this model therefore can be seen as an upper boundary for the global ice force. • During simulations it was discovered that the first peak load is highly influenced by pitch movements, and the second peak relatively little. This pitch dependency is reasonable. Positive pitch velocity induces a faster crushed volume increase compared to the straight-on condition. Hence, a higher global ice loads occurs. Negative pitch velocity implies slower crushed volume increase, resulting in relatively lower global ice loads. • It is concluded that the primarily used specific energy absorption of crushing of ice (the SEA value), is assumed to be too large. The SEA value (2175 [J/kg]), presented in Figure 2.33 can be adjusted. To what extend lowering is allowed, could not be justified using literature reviews. Reasoning for this lowering of SEA value was justified for the following reasons: 1. The morphology of an ice ridge is different from the solid ice used in the DBT (Drop Ball Test). It is reasonable to state that the ridge’s ice fails more easy, then the solid ice. 2. Spherical intender body is geometrically very different from a vessel hull, affecting the confinement. 3. Compared to an ice ridge, the ice used in the DBT has a high confinement. This latter point is illustrated in Figure 2.34. It shows that SEA will quickly decrease with decreasing confinement, increasing spalling and splitting failure. Lower confinement compared to the DBT justifies lowering of SEA. Less energy will be spend on ice crushing, lowering the global ice loads. • Since there are justifiable reasons to lower the specific energy of crushing of ice in an ice ridge, it can be concluded that careful adjustment will improve the simulation. Lowering the SEA value results into more realistic penetration depth, vessel motions, and global ice loads. Lowering with a factor 2 will give a total horizontal ridge penetration of approximately 6 meter. The global ice load becomes 16 MN for the first peak, which corresponds to a vertical global ice load around 15 MN. These values seem more realistic, since the load identification model predicts loads in that range. It must be emphasized that a physical basis to adjust the SEA value to enhance representation of full-scale observations does not exist. Adjusting the value will give results that seem better, but it still assumes that the loss of energy per volume is constant. This is not the case, as the energy dissipation per indentation volume in the 104 6.R ESULTS AND DISCUSSION

start of impact should be higher because of the occurrence of other failure modes. • The pressure area relation might affect ice loads and ridge penetration as well. Size effect, like flaws in the ice and flaw hierarchy, ice inhomogeneity, non-simultaneous failure, fracture mechanism and boundary conditions, will make average global pressure decreases with increase of nominal contact area. Hence, energy consumption will be relatively higher at the start of ridge impact. Specific energy for crushing of ice is independent of size and scale, but it is reasonable to assume that ice loads will decrease with increasing nominal contact area. Due to this effect, ice loads in the simulation model might be allowed to be lowered. This is not implemented in the model. • It might be possible to adjust the SEA value to an extend that it starts approximately simulating penetration in ice ridges. It might be physically incorrect, but maybe an empirical relationship can be identified. Furthermore, it might be possible to make the energy loss per volume depended on velocity, confinement and contact area, which are the driving factors influencing the reliability of crushing. The change of energy loss per volume might change according to a certain trend when these parameters are adjusted. It must be emphasized that this suggestion is purely hypothetical. But if a relation can be found, it becomes more feasible to get a realistic simulation of vessel impacts into ice ridges.

Figure 6.1: The global ice ridge load working normal on the bow.

6.3. RESULTS OF LOAD IDENTIFICATION MODEL In this section the results of the load identification model are presented. The most important results for the load identification model include the calculated thrust force (section 6.3.3), state estimate (section 6.3.5), global ice load identification (section 6.3.6), and uncertainty analyses (Section 6.3.9). The results regarding the modeling of the Coanda effect modeling is presented in section 6.3.12. But first some general results are discussed below.

Before the load identification model starts with the state and load estimate, some steps on the data were executed. These steps include the data synchronization in time, data synchronization in dimensions, and GPS data conversion. The GPS data conversion started with GPS raw data which were conversed to flat earth coordinates in the mast, then to flat earth coordinates in the COG, and lastly to the rigid body reference frame. All steps are executed successfully. For more information the reader is referred to section 5 and 5.3.

In the load identification model, the complete 3DOF dynamical system was split into a 2DOF system for the joint input-state estimate algorithm, and 3DOF system for the Kalman filter. This was required to enable the estimate of all states, inputs, and for an optimal estimate for the global ice loads. It is found that this approach 6.3.R ESULTS OF LOAD IDENTIFICATION MODEL 105 is successful, and can be used for other state and input estimates with a limited amount of acceleration measurements. However, accuracy of the state and input estimate, will be significantly increased by using more measurement systems. With already one additional MRU, validation of the data will be enabled. Other benefits are for example that it enables estimate of structural damping, or estimate on the vibrations coming from different origins than the ridge impact. Furthermore, when using several MRUs, and by placing them on strategic places, more useful information can be gathered. The recommended additional locations are at the portside, starboard, bow and aft of the vessel. The largest benefit is the rotational acceleration data that can be calculated. This enables a direct use of the joint input-state estimate algorithm on roll, pitch and yaw motions. By improving these rotational states, the reliability of identified loads will significantly increase. Now the Kalman filter is used combined with an estimated pitching moment. This pitching moment is based on forces found in the horizontal and vertical. This estimate of pitching moment has a significant error which would be mitigated with more measurement systems. The reason for this large error in pitching moment estimate are: a) not all forces working on the system are taken into account (think about wind, currents, etc), b) the loads that are used to compose the pitching moment have an error them self which is propagated in the moment error, c) the arm to the loads that are taken into account have an error which also propagates into the pitching moment error.

Optimizing the load identiﬁcation model for global ice load estimate, is possible by tuning the process noise covariance matrix. This is elaborated in section 6.3.2. For more information about splitting the model, the reader is referred to section 5.5.3 and 5.5.4.

6.3.1. HYDROMECHANICAL PARAMETERS IN STATE SPACE MODEL In section 3.3 was found that for an ice ridge ramming experiment maneuvering theory can be used. In maneuvering theory the frequency depended added mass and potential damping coefficients are approximated by constant values. As a result the hydrodynamic forces and moments can be approximated at one frequency of oscillation such that the fluid-memory effects can be neglected. It is convenient to represent hydrodynamic systems with frequency independent quantities, since these reduce model complexity. Maneuvering theory presents a nonlinear mass-damper-spring system with constant coefficients.

The hydrostatic parameters correspond to the buoyancy force and is proportional to the displacement of the vessel. The hydrostatic spring stiffness for heave (k33), and heave-pitch coupling (k35 and k53) must be implemented nonlinear due to Oden’s sloping waterline. How this can be implemented in a model is presented in section 4.3.3.

The hydrodynamic parameters in the load identification model include the potential coefficients and frictional damping. Based on maneuvering theory, the damping coefficients in the horizontal plane b11, b22, and b66 are equal to zero. The hydrodynamic surge resistance includes added mass a11 and nonlinear surge damping b11,NL. In section 4.3.2 was found that a11 and b11,NL can be calculated with a full-scale acceleration sea trial. The potential coefficients for heave and pitch (a33, a55 b33, b55) are determined using Ankudinov (1991) [55], using results from a strip theory analyses. As discussed in section 4.3.4, nonlinear heave friction should be taken into account as well. This is achieved by a flat plate friction approximation, which is a general accepted assumption in ship resistance calculations.

For more information the reader is referred to section 4.3.

6.3.2. FILTERPARAMETERSANDTUNING The Kalman filter and joint input-state estimate algorithm use process noise covariance matrices. The choice of the diagonal elements of the covariance matrices Q and R should be based on the order of magnitude of the state vector and the accuracy of the sensors, respectively. Measurement noise covariance matrix R was based on the MRU and GPS measurement system. The process covariance matrix Q was determined by tuning, until data seemed most reasonable. This is achieved by finding a smooth fit through the data points, taking into account the uncertainty of the data points. This tuning therefore depends on the user. Working with the load identification model, it was found that careful choice in these matrices enables a superior filter performance. The choice of noise may differ per data set. In section 5.5.6, the noise vectors used are presented.

In the load identiﬁcation model, the vertical global ice load is found to be sensitive to the pitch state. To op- 106 6.R ESULTS AND DISCUSSION timize the load identiﬁcation model, it is necessary to take the pitch data as very accurate compared to the pitch estimate by the model. Therefore, the process noise must be large enough compared to the measurement noise. It must be emphasized, that this optimization is not required when a high accuracy in pitching moment is established. This can for example be achieved using more MRU systems. For more information the reader is referred to AppendixG.

6.3.3. THRUSTFORCE The propulsion theory can be used to calculate the thrust force with given data. The thrust force is calculated using data of the pitch ratio P/D, forward velocity (i.e. surge velocity), revolution rate, thrust coefﬁcient, and hydrodynamic pitch β0.7R . The revolution rate is approximately constant at 139 rpm. Several conversion factors were required before the full-scale data could be used.

The open-water characteristics of Oden’s thrusters are based on the Wageningen Nozzle 37 with the Ka4-70 series, which were faired by means of harmonic analyses (Roddy, 2006 [53]). These characteristics enable the identiﬁcation of the thrust coefﬁcient depending on hydrodynamic pitch and P/D value. The thrust force algorithm is found to work correctly over all β0.7R and P/D ranges, also the ranges outside of the presented data by Roddy (2006) [53]. This enables direct thrust force calculations in all thruster settings of Oden.

For the thruster at portside, the relationship between the thrust coefﬁcient, the pitch ratio, and the hydrodynamic pitch, is illustrated in Figure L.2. Hereby Figure L.2 proves that the thrust force algorithm works in all possible propeller settings. The relationship between thrust vs thrust coefﬁcient and forward velocity is shown in Figure 6.2. In Figure 6.2 can be seen that the maximum thrust of Oden (2.4 MN) is never exceeded. This was one of the boundary conditions of the thrust algorithm. For more information concerning the results of the thrust force, the reader is referred to Appendix L. The sensitivity and uncertainty analyses are discussed in section 6.3.7.

Figure 6.2: The relationship between thrust vs thrust coefﬁcient and forward velocity. Note the maximum thrust of Oden (i.e. 2.4 MN).

6.3.4. CONTACT MODEL It is concluded that the contact model works well, under the pure crushing assumption. Given this assumption and the full-scale data delivered, the vessel’s progress through the ridge can be identified. Figure 6.3 shows how the contact point between vessel-ridge was during the first impact of the profiled MY ice ridge. Figure 6.4 shows the volumetric penetration of the vessel into the ridge, and the weight of the crushed ice (when assuming a pure crushing failure of ice). 6.3.R ESULTS OF LOAD IDENTIFICATION MODEL 107

Figure 6.3: The contact point between vessel and ridge during the ﬁrst ridge impact.

Figure 6.4: The top graph shows the volumetric penetration of the vessel during the ﬁrst ridge ramming. The bottom graph shows the weight of this crushed volume of ice during the ﬁrst ridge ramming.

6.3.5. STATE ESTIMATE The load identification model is capable of identifying the state of the system accurately, given the uncertainties presented by the measurement system. The load identification model is especially accurate in identification of surge and heave state. The higher uncertainty in pitch state is due to the dependency on pitching moment, which was found to have a significant error. As discussed in section 5.5.4, the pitch state should not have a too large error. Figure L.11 shows that the state estimate of pitch is sufficient.

Results of state estimate are presented in section L.0.4 in Appendix L. It must be emphasized that the sample rate of 10Hz is low. A higher sample rate will result in a smoother state, and in higher accuracy. In the analyses of the results there are some concerns to the heave data. This is elaborated on in section 6.3.11.

6.3.6. ESTIMATEOFTHEGLOBALICELOADS The load identiﬁcation model is capable of calculating global ice loads. The global loads are illustrated in Fig- ure 6.5. For convenience Figure 6.5 shows the surge velocity as well. There were four impacts into the proﬁled MY ice ridge impact 1 (t=1280s), impact 2 (t=1375s), impact 3 (t=1520s) and impact 4 (t=1640s).

Readers who are only interested in the ramming experiment of the proﬁled MY ice ridge are referred to Figure L.15 and Figure L.16. The horizontal loads that are calculated for the whole data range are presented in Figure L.17. Figure L.17 shows the thrust force, hydrodynamic resistance, and horizontal global ice load. The main vertical loads that were calculated in the load identiﬁcation model for the whole data range, are presented in Figure L.18. The main vertical forces that are presented include the vertical global ice load, the force due to the heave-pitch motion coupling, and the force due to the Coanda effect. Several components from the vertical loads are not shown in Figure L.18 due to their low magnitude. This is elaborated in section 6.3.10.

The identified maximum global ice loads by the load identification model for the profiled MY ice ridge, are 108 6.R ESULTS AND DISCUSSION shown in Table 6.1. After the penetration of the profiled MY ice ridge, seven other ridges were rammed. Of these seven other ice ridges, two of the ridges were large. The global ice loads of these two additional ice ridges are also shown in Table 6.1. For an overview of all the ridge impacts, and their global ice loads resulting from the load identification model, the reader is referred to Table 6.2 or Table L.1. Note that the peaks of the global loads do not necessarily occur at the same moment of time.

Based on the results it is found that the vertical ice load has two peaks when ramming an ice ridge. This observation is repetitive in the data. This occurs mostly due to the ship’s heeling motion: at first impact the icebreaker pitches away from the ice ridge, but when the contact force decreases the ship pitches back causing the second peak. In general this second peak is lower than the first peak. This because the velocity is larger at the first impact peak, than at the moment of the second peak.

The surge motions are not coupled with heave or pitch motions. As a result, a relatively simple model can be developed to identify the horizontal ice load, or horizontal ice resistance. A disadvantage, is that the largest ice loads contribute to the vertical.

Table 6.1: The maximum global ice loads during a speciﬁc ridge impact, based on the suggested load identiﬁcation model. Shown are time step of ridge impact, horizontal global ice load Fx1,ice , vertical global ice load Fx3,ice , pitching moment Mx5,ice , and the impact velocity.

Ridge t [s] Fx1,ice Fx3,ice Mx5,ice v [m/s] [MN] [MN] [MN m] · Profiled MY ridge, 1295 3.8 9.1 -271 2.6 impact 1 Profiled MY ridge, 1377 4.4 11.5 -305 2.1 impact 2 Profiled MY ridge, 1520 4.8 12.9 -355 3.5 impact 3 Profiled MY ridge, 1635 4.2 6.3 -188 3.1 impact 4 ridge 7 (large) 2199 3.8 9.3 -327 4.1 ridge 8 (large) 2328 4.9 10.2 -310 3.1

Figure 6.5: The global ice loads (top) and surge velocity (bottom). 6.3.R ESULTS OF LOAD IDENTIFICATION MODEL 109

LOAD RATIO AND THE FRICTION COEFFICIENT Now the global ice loads are identified, it might be possible to determine the friction coefficient µ [-]. But first, we validate the relationship between the horizontal and vertical global ice load. This is achieved by looking to the load ratio. The bow angle of Oden is equal to 22 degrees. Therefore, the vertical ice load should be approximately 2.47 times as large as the horizontal ice load. Vice versa, the angle between the identified horizontal and vertical ice load should always be around the bow angle of Oden βOden. A large deviation from βOden indicates that the model’s loads might be wrong. The ratio and angle between the global ice loads are presented in Table 6.2.

Table 6.2 shows that the ratio in most cases is correct. The largest errors are found at the small ridges (i.e. ridge 2,3,4,5,6). Their larger deviation from βOden might be explained by the vessel’s impact direction or the occurring failure mode of the ice. For example, Figure B.11, B.15, and B.17 in Appendix B show that the rams into ridge 2, ridge 3, and ridge 4, are not from a normal direction into the ridge at all. For ridge 2, splitting failure is observed, with cracks parallel to the length of vessel and two cracks perpendicular behind ridge. Also ridge 4, shows clearly that splitting failure occurred. It is reasonable to ﬁnd different loads with different failure modes of ice. For comparison, in Figure B.18 can be seen that the large ridge 7, is rammed from a normal direction. Table 6.2 shows that the ratio of ridge 7 is close to βOden. More relevant pictures made by the 360 camera, are presented in Appendix B.

Table 6.2: The maximum global ice loads during a speciﬁc ridge impact, based on the suggested load identiﬁcation model. Shown are time step of ridge impact, horizontal global ice load Fx1,ice , vertical global ice load Fx3,ice , ratio between the global loads Fx3,ice /Fx1,ice , and the angle between the global ice loads atan(Fx3/Fx1).

Ridge t [s] Fx1,ice Fx3,ice Fx3,ice /Fx1,ice [- atan(Fx3/Fx1) [MN] [MN] ] [deg] Profiled MY ridge, 1295 3.8 9.1 2.39 22.7 impact 1 Profiled MY ridge, 1377 4.4 11.5 2.61 20.9 impact 2 Profiled MY ridge, 1520 4.8 12.9 2.69 20.4 impact 3 Profiled MY ridge, 1635 4.2 6.3 1.50 33.7 impact 4 ridge 2 1914 2.6 4.1 1.58 32.4 ridge 3 1990 3.0 5.8 1.93 27.3 ridge 4 2082 4.8 5.9 1.23 39.1 ridge 5 2158 2.2 5.1 2.32 23.3 ridge 6 2173 3.0 4.9 1.63 31.5 ridge 7 (large) 2199 3.8 9.3 2.45 22.2 ridge 8 (large) 2328 4.9 10.2 -2.08 25.7

Equation 2.7 is used to determine the friction coefﬁcient µ [-]. For convenience this equation is repeated below:

FH FN cos(β) µ FN sin(β) = + · (6.1) F F sin(β) µ F cos(β) V = N − · N where the friction force F µ F . The bow angle with the vertical equals β. The pitch angle x is taken f r iction = · n 5 into account by: β β x . The two unknowns to be solved in equation 6.1, are the friction coefﬁcient = Oden − 5 µ [-] and the normal load on the bow FN in [N]. The results for the rams into the proﬁled my-ice ridge are shown in Figure 6.6.

In Riska (2010) [4] it is found that the value of the friction coefﬁcient should be larger than 0.1 for level ice, with 0.15 as a typical value. Therefore, the results indicate unrealistic values for the friction coefﬁcient, i.e. negative and too large values. However, there are several reasons why equation 6.1 is not accurate in the case of a ridge ramming. 110 6.R ESULTS AND DISCUSSION

Figure 6.6: When using equation 6.1 the normal load FN in [N], and friction friction coefﬁcient µ [-] can be determined. These are illustrated for the rams into the my-ice ridge, with FN and µ in the top and bottom ﬁgure respectively.

First of all, equation 6.1 assumes that the normal ice load can be calculated by F p A, with p assumed N = av · av to be constant. This assumption is wrong. The average pressure and contact area will rapidly change during the ridge impact, with local pressure variations over the contact area. The error due to this assumption increases with increase of the ice thickness. It is therefore reasonable to state, that equation 6.1 is less valid for ice ridges compared to level ice. Secondly, an ice ridge consist of multiple ice bodies, rather than one ice body. The bodies will not only interact with the vessel and the water, but also with each other. It will therefore give a different reaction force, making equation 6.1 less applicable on ice ridges. Thirdly, a ridge ramming is a more dynamic phenomena compared to sailing through level ice or an ice ﬂoe. To get through the ridge, the vessel uses its inertia rather than the thrust force. If is fair to state, that equation 6.1 becomes less accurate when inertia forces increase. Lastly, it should be remembered that not only the bow experiences friction. The keel and sides of the vessel also experience friction forces, which are now not taken into account. As discussed in section 2.4.4, T. Leiviskä (1999) [34] discovered from model tests, that the parallel mid-body contributed around 16 to 32 percent to the total resistance.

There is one exception in the measured data, where it is reasonable to calculate the friction coefficient with equation 6.1. This is at the last stage of the first ridge ramming, where the thrust force, velocity, and horizontal ice load, remain approximately constant over time. This period is from t=1302 to t=1312 [s], and can be seen in Figure 6.7. During this period, the vessel is more pushing than penetrating the ridge. The inertia effects do not contribute since the velocity is constant. The contact area remains constant, and the average pressure will be fairly constant as well. Therefore the friction coefficient during this period is calculated. The results are illustrated in Figure 6.8. In Figure 6.8 two lines are shown. The second line is based under the assumption that 32% of the identified horizontal global ice load, originated from the parallel mid-body friction. As such, a friction force during the first ridge impact, is calculated to be in de range of 0.20-0.67 [-].

6.3.7. SENSITIVITYANDUNCERTAINTYANALYSESOFTHETHRUSTFORCE Given the sensitivities in section 5.6.1 it is concluded that uncertainties of the parameters propeller wake fraction, RPM, and vessel velocity, have a small inﬂuence on the thrust force. Their sensitivities to the thrust force were all within in the range of 1% to 4%. The uncertainty of propeller diameter and propeller pitch were found to have a large inﬂuence on the thrust force, with sensitivities up to 15.1% and 17% respectively.

A Monte Carlo analyses was executed to ﬁnd the thrust uncertainty, with a 1000 simulations. The result is illustrated in Figure 6.9. About 68% of values drawn from a normal distribution are within one standard de- 6.3.R ESULTS OF LOAD IDENTIFICATION MODEL 111

Figure 6.7: This ﬁgure shows the normal load FN in [N] during the ﬁrst ram into the my-ice ridge, when using equation 2.7. Here can be seen that between t=1302 to t=1312 [s], the normal load, thrust force and horizontal ice load remain approximately constant.

Figure 6.8: This figure shows the friction friction coefficient µ [-] at the end of the first ram into the profiled my-ice ridge, when using equation 2.7. Here the friction coefficient is presented for the period between t=1302 to t=1312 [s], which is a period in which the normal load, thrust force and horizontal ice load remain approximately constant over time. Two lines are shown, the red line takes into account an 32% friction contribution due to the parallel mid-body. Therefore, it assumes that F 0.68 F . H ≈ · x1,ice viation away from the mean. Furthermore proximately 95% of the values lie within two standard deviations. It must be stressed that the accuracy presented in Figure 6.9 is based on two important assumptions: a) the parameter uncertainties equal the values presented in section 5.6.2, b) the parameters follow a normal distribution. These two assumptions might be wrong. However, all uncertainties were implemented conser- vatively, hence the confidence ratio of the 68% and 95% accuracy boundary layers can only increase.

The results from the uncertainty analyses of the thrust force, are used in the uncertainty analyses of the global ice loads.The minimum value and maximum thrust value are taken from the 95% accuracy boundary layers at t=1305 [s]. Corresponding sensitivities can be calculated, and equal -24% and 24% for the minimum and maximum value respectively. Therefore, the minimum thrust value can be taken into account by 76% thrust force. For more information the reader is referred to Appendix I. 112 6.R ESULTS AND DISCUSSION

Figure 6.9: Thrust uncertainty based on 1000 different simulations, taking into account parameter uncertainties. Results are visualized by 5 lines: Mean value (expected value), the two adjacent lines present the boundary for 68% accuracy (1 standard deviation), the two outer lines present the boundary of 95% accuracy (2 standard deviations).

6.3.8. SENSITIVITYANALYSESOFTHEGLOBALICELOADS From the results of the sensitivity analyses, it is concluded that the global ice loads are practically not influenced by the rotational inertia Iy , potential heave damping, potential pitch damping, nonlinear surge damping, nonlinear heave damping, added mass for pitch a55, and the contact point. Their sensitivities are around 0 %. It must be emphasized that the nonlinear heave damping b33,NL is a lower boundary value, and simply to low to impose significant influence on the global loads. The maximum value, as presented in section 5.6.3, has a sensitivity around -8% on the vertical global ice load. Therefore, the nonlinear heave damping may not be neglected.

High influence on the horizontal global ice loads are the vessel mass and thrust force, with sensitivities around 6% and 5% respectively. High influence on the vertical global ice loads are the waterline area (8%), waterline length (5%), meta centric height (3%), heave-pitch stiffness coupling terms (2.7%). It must be emphasized that the influence of added mass for surge and heave will be significant, as their uncertainties are significantly larger then the 10% change used in the sensitivity analyses. The added mass for surge will influence the horizontal global ice load around 3% at the peak, and added mass for heave will influence the vertical global ice load around 3-5% at the peak.

The sensitivity study shows, that the vertical global ice load is signiﬁcantly less sensitive to the vessel mass than the horizontal global ice load. This is due to the reaction forces from the hydrodynamics. In horizontal direction, the largest reaction force that is calculated depends on the mass and acceleration data. In surge direction there is no spring stiffness. As a result, the horizontal global load is independent on surge position data. The nonlinear hydrodynamic resistance for surge motions acts as the nonlinear surge damping, thus depends on surge velocity. A small part of the horizontal global load is found to depend on this surge damping. In the vertical direction, the largest reaction force that is calculated depends on heave displacement and hydrostatic stiffness k33. The acceleration data and vessel mass give a relatively smaller contribution to the vertical global load, which explains the lower sensitivity.

With respect to the vertical global ice load the waterline and the pitch-heave coupling terms k35 and k53 will cause a large uncertainty. The combination of minimum waterline area value 2366 [m2] and minimum arm of 0.575 in [m], will cause a sensitivity to vertical global ice load around -20%. The combination of maximum waterline area value 2797 [m2] and maximum arm of 8.575 in [m] will impose a sensitivity to vertical global 6.3.R ESULTS OF LOAD IDENTIFICATION MODEL 113 ice load of around 30%.

The most important results from the sensitivity study are elaborated on below, for more information is referred to Appendix J.

Sensitivity to vessel displacement

• the sensitivity for the horizontal global ice load S 10%(t 1286) 6.13 [%] + = = • the sensitivity for the vertical global ice load S 10%(t 1286) 0.48 [%] + = = The reason that the sensitivity equals 6% and not 10% for the horizontal global ice loads can be explained by the contributions from the thrust force and surge damping. The EM for surge reads:

(m a )x¨ (t) T (t) b x˙ (t) x˙ (t) F (t) (6.2) + 11 1 = x1 − 11,NL| 1 | 1 − x1,ice where T (t) b x˙ (t) x˙ (t) 1.25e 06. The acceleration from measurements at t-1286 gives x¨ 0.15 x1 − 11,NL| 1 | 1 = + 1 = − [m/s2], velocity data gives 2.09 [m/s]. Furthermore, F (m a )x¨ (t) 2.048e 06, and F global,x = + 11 1 = − + x1,ice = 3.4050e 06. Increasing the mass with 10% increases F to 2.33e 06. Note hereby that the increase + global,x − + of mass also inﬂuences the added mass a11. Therefore:

2.33e 06 T (t) b x˙ (t) x˙ (t) F (t 1286) − + = x1 − 11,NL| 1 | 1 − x1,ice = Which gives F 3.59e 06, and thereby F /F 3.59e 06/3.4050e 06 1.054. x1,ice = + x1,ice,new x1,ice,old = + + ≈ Why 5.4% and not 6%? This must have to do with the difference between data and state estimate of surge. For example, if the acceleration data would have been -0.1514 instead of -0.1500, then the change of horizontal global ice load would be 6%. This difference in acceleration data is well within the uncertainty of acceleration data.

Conclusion on vessel displacement: based on the raw measurement data an increase of vessel mass by 10% causes the horizontal global ice load to increase with 5.4%. Furthermore, the state estimate depends on both the state space model, and the uncertainty of the data. A different mass in the model, can inﬂuence the state estimate, which can explain the difference in increase of 6% vs 5.4%. For more information regarding the decomposition of hydrodynamic forces during ice ridge ramming experiment, the reader is referred to 6.3.10.

Sensitivity to waterline area . Awl is implemented as 100% and 110% Awl . The sensitivity of Awl is illustrated in Figure J.2. The following sensitivities were identiﬁed:

• the sensitivity for the horizontal global ice load S 10%(t 1286) 0 [%] + = = • the sensitivity for the vertical global ice load S 10%(t 1286) 7.55 [%] + = =

Sensitivity to waterline length

• the sensitivity for the horizontal global ice load S 10%(t 1286) 0 [%] + = = • the sensitivity for the vertical global ice load S 10%(t 1286) 5.78 [%] + = =

Sensitivity to meta centric height GML

• the sensitivity for the horizontal global ice load S 10%(t 1286) 0 [%] + = = • the sensitivity for the vertical global ice load S 10%(t 1286) 3.58 [%] + = = −

Sensitivity to added mass for surge

• the sensitivity for the horizontal global ice load S 10%(t 1286) 1.06 [%] + = = • the sensitivity for the vertical global ice load S 10%(t 1286) 0 [%] + = =

Sensitivity to added mass for heave

• the sensitivity for the horizontal global ice load S 10%(t 1286) 0 [%] + = = • the sensitivity for the vertical global ice load S 10%(t 1286) 0.76 [%] + = = 114 6.R ESULTS AND DISCUSSION

Sensitivity to heave-pitch hydrostatic stiffness coupling terms . The sensitivity to heave-pitch coupling is investigated by changing the arm between areal center point of Awl and COG. This arm equals 4.575 meters in the numerical model. The model is run with 10% extra arm to identify the sensitivity of k35 and k53. Results for the sensitivity analyses are as follows:

• the sensitivity for the horizontal global ice load S 10%(t 1286) 0 [%] + = = • the sensitivity for the vertical global ice load S 10%(t 1286) 2.73 [%] + = =

Sensitivity to nonlinear heave damping b33,NL

• the sensitivity for the horizontal global ice load S 10%(t 1286) 0 [%] + = = • the sensitivity for the vertical global ice load S 10%(t 1286) 0 [%] + = = It must be emphasized that the nonlinear heave damping b33,NL was based on a purely ﬂat plate resistance calculation. According to theory, there should also be a residual resistance which cannot be determined without full scale testing. This nonlinear heave damping term can therefore be seen as a lower boundary of b33,NL. It seems justiﬁed to use the nonlinear surge damping to approximate an upper boundary value for b33,NL. For the estimate it is assumed that the relationship is purely depended on the normal area. This normal area for heave motions equals the waterline area Awl . An additional 10% is taken into account,for the 2 upper boundary. Nonlinear surge damping b 2.38 104 in [N s2/m ], with a normal area of B D 11,NL ≈ · · f × f = 31.2 8, where B represents the beam at the bow and Dr the Draft. As a result: × b Awl b33,NL,upper b11,NL 1.1 (6.3) = · B Dr · b × This results in an upper boundary value for the nonlinear heave damping b equal to 6.34e 107 in [N 33,NL · · s2/m2].

Sensitivity to the thrust force

• the sensitivity for the horizontal global ice load S 10%(t 1286) 4.06 [%] + = = • the sensitivity for the vertical global ice load S 10%(t 1286) 0.12 [%] + = = −

Sensitivity to the heave data During the sensitivity analyses it was concluded that the data of heave acceleration seems to be of poor quality, illustrated in Figure 6.13. The question rises if it can be trusted, and therefore if heave position and heave velocity may be trusted. It is investigated what inﬂuence an increased uncertainty in heave acceleration, velocity, and displacement imposes on the global ice loads. Findings are elaborated on below.

Increasing uncertainty in heave acceleration by a factor 2 results in the following sensitivities

• the sensitivity for the horizontal global ice load S 200%(t 1286) 0 [%] + = = • the sensitivity for the vertical global ice load S 200%(t 1286) 1.05 [%] + = = Increasing uncertainty in heave velocity by a factor 2 results in the following sensitivities

• the sensitivity for the horizontal global ice load S 200%(t 1286) 0 [%] + = = • the sensitivity for the vertical global ice load S 200%(t 1286) 0.36 [%] + = = Increasing uncertainty in heave displacement by a factor 2 results in the following sensitivities

• the sensitivity for the horizontal global ice load S 200%(t 1286) 0 [%] + = = • the sensitivity for the vertical global ice load S 200%(t 1286) 1.77 [%] + = = − It was found that an additional uncertainty can have significant influence, for the results is referred to Ap- pendix J. There are some reasons why it is justified to change the uncertainty of heave data. As the ice- structure impact is quite far from the COG, some vibrations might be damped due to vessel’s structure. Also the vessel has its own vibrations, from for example the engines. This can be taken into account as an extra uncertainty. It is stated that the uncertainty of heave acceleration data can be a factor 3 times as high (uncertainty acceleration data is equal to 0.005 [m/s2]). Given the integrations over time, the uncertainty of heave velocity increases by a factor 31, and of heave displacement with 32. This results in a sensitivity of: 6.3.R ESULTS OF LOAD IDENTIFICATION MODEL 115

• the sensitivity for the horizontal global ice load S(t 1286) 0.02 [%] = = − • the sensitivity for the horizontal global ice load S(F ) 0.01 [%] ice,max = − • the sensitivity for the vertical global ice load S(t 1286) 7.38 [%] = = • the sensitivity for the vertical global ice load S(F ) 14.16 [%] ice,max =

Sensitivity to the surge acceleration data . Like the heave data, there might be an additional error in the acceleration data of surge. This was tested, and the following sensitivities were identiﬁed:

• the sensitivity for the horizontal global ice load S 200%(t 1286) 0.34 [%] + = = − • the sensitivity for the vertical global ice load S 200%(t 1286) 0.07 [%] + = = −

Sensitivity to GPS position . As the vessel is at the Arctic there will be less satellite coverage. It is considered that the uncertainty might be higher than given uncertainties. It was found that increasing uncertainty in surge displacement and velocity by a factor 2 results in the following sensitivities

• the sensitivity for the horizontal global ice load S 200%(t 1286) 0.34 [%] + = = • the sensitivity for the vertical global ice load S 200%(t 1286) 0.06 [%] + = = Increasing uncertainty in heave displacement by a factor 2 results in the following sensitivities

• the sensitivity for the horizontal global ice load S 200%(t 1286) 0.0 [%] + = = • the sensitivity for the vertical global ice load S 200%(t 1286) 0.34 [%] + = = The sensitivity to GPS position (i.e. surge position and velocity) and to surge acceleration data is investigated as well. Based on their sensitivity studies it is concluded that an additional uncertainty here, is not required as sensitivity was found to be low.

6.3.9. UNCERTAINTY ANALYSES OF THE GLOBAL ICE LOADS The uncertainty analyses of the global ice loads is discussed in detail in section 5.6. The results of the uncertainty analyses are given in Appendix K. As an example the uncertainty of the horizontal and vertical global ice load are given during the ﬁrst and second impact of the multi-year ridge. The uncertainty in the horizontal and vertical ice load are presented in Figure 6.10 and Figure 6.11 respectively. An additional 1% model uncertainty should be expected due to inﬂuence of the nonlinear terms that are linearized at each time step. This is elaborated on in section 5.5.4. 116 6.R ESULTS AND DISCUSSION

Figure 6.10: Uncertainty analyses horizontal global ice load during ridge impact 1 (t=1280s) and impact 2 (t=1375s)

Figure 6.11: Uncertainty analyses vertical global ice load during ridge impact 1 (t=1280s) and impact 2 (t=1375s) 6.3.R ESULTS OF LOAD IDENTIFICATION MODEL 117

6.3.10. DECOMPOSITIONOFHYDRODYNAMICFORCESINTHEVERTICALDIRECTIONDURING ICERIDGERAMMINGEXPERIMENT It is investigated how the vertical global ice load relates to the hydrodynamic forces during the ramming experiment. In the numerical model, the following equation of motion is taken into account for heave motion x3(t): X (m a )x¨ (t) b x˙ (t) b x˙ (t) x˙ (t) k x (t) k ¡x (t) x ¢ F (t) T (t) F (t) + 33 3 + 33 3 + 33,NL| 3 | 3 + 33 3 − 35 5 − 5,mean = x3 = x3 + x3,ice (6.4) P Fx3(t) represents the excitation force that includes the global vertical ice load Fx3,ice (t) that we want to calculate with the algorithm. Tx3(t) represents the vertical component of the thrusters, which was later found to be insigniﬁcantly small compared to the other loads. A problem arises, as the joint input-state estimate algorithm cannot directly take into account pitch motions (x5(t)) (see section 5.5.3). Therefore, the joint input-state estimate algorithm therefore takes into account the following equation of motion: X (m a )x¨ (t) k x (t) b x˙ (t) b x˙ (t) x˙ (t) P (t) (6.5) + 33 3 + 33 3 + 33 3 + 33,NL| 3 | 3 = x3 where the excitation vector Px3 is described as follows X X ¡ ¢ Px3(t) Fx3(t) k35 x5(t) x5,mean X = + − (6.6) P (t) T (t) F (t) k ¡x (t) x ¢ x3 = x3 + x3,ice + 35 5 − 5,mean The numerical model starts with equation 6.5 and the joint input-state estimate algorithm, it identiﬁes state x3 and input Px3. Following this, we import the estimate of pitch x5. We can then calculate the vertical global ice load by: X F (t) P (t) T (t) k ¡x (t) x ¢ (6.7) x3,ice = x3 − x3 − 35 5 − 5,mean

DECOMPOSITIONOFHYDRODYNAMICFORCES Two ﬁgures are presented in Appendix H: Figure H.1 and H.2.

In Figure H.2 is shown how all components contribute to the predicted ice load by the numerical model. These components that are taken into account are: the data (d, v, a) combined with hydrodynamics, vertical component thrust force Tx3, force due to Coanda effect Fcoanda, and force due to pitch-heave coupling (k35 and pitch state). Summarized:

• an increase of heave displacement increases the Fx3,ice . • an increase of heave velocity increases the Fx3,ice . • an increase of heave acceleration increases the Fx3,ice . • an increase of pitch displacement lowers the Fx3,ice . • an increase of Tx3(t) lowers the Fx3,ice . • an increase of Fcoanda lowers the Fx3,ice .

Or as in the following equation: X ¡ ¢ Fx3,ice (t) Px3(t) Tx3(t) k35 x5(t) x5,mean Fcoanda = − − − − ¡ ¢ Fx3,ice (t) (m a33)x¨3(t) k33x3(t) b33x˙3(t) b33,NL x˙3(t) x˙3(t) Tx3(t) k35 x5(t) x5,mean Fcoanda = + + + + | | − − − − (6.8) Figure 6.12 is based on Figure H.2. As can be seen in Figure 6.12, the vertical global ice load depends mostly on the reaction force from displacement data and the pitch-heave coupling (thus pitch data). It is also found that damping, nonlinear damping, and added mass have very little inﬂuence on the vertical global ice load. 118 6.R ESULTS AND DISCUSSION

Figure 6.12: Decomposition of hydrodynamic and external forces during the first ramming attempt into the profiled MY ice ridge. This to identify to what extend the hydrodynamical components influence the vertical global ice load. Upwards forces are positive, negative forces are negative.

6.3.11. STATE OF HEAVE AND THE PREDICTED DOWNWARD ICE LOAD AT END OF IMPACT The heave velocity and heave position are calculated from the heave accelerations. The acceleration data is shown in Figure 6.13. The identified heave position by the load identification model is illustrated in Figure 6.14. It is concluded that the heave data and state of heave that are identified with the model have a small error. This proves that the state identification works well. However, as discussed in section 5.6.5, the heave data seems to be of poor quality. This is a concern, since it was later found that the main contributor to the global ice load depends on the reaction force from heave spring stiffness k33 and the heave data. An subsequent issue is that the load identification model identifies a negative heave directly after the impact of the profiled MY-ice ridge. This is a repetitive finding for all four impacts. As a result the load identification model calculates a negative ice load. This raises the question, how can ice induce negative buoyancy on a downward sloping structure? This strange observation, caused by the negative heave position that is calculated by the load identification model, should be explained. The four possible explanations are as follows:

1. The data is wrong 2. The load identiﬁcation model is wrong 3. There was a downwards working ice load working on the vessel 4. Another phenomena, not taken into account in the load identiﬁcation model, is responsible for the negative heave measurements

It seems reasonable to state that the data is wrong. The heave acceleration is high pass ﬁltered and integrated twice over time to heave position, and heave velocity is computed with one integration over time of heave acceleration. The heave ﬁlter should remove static and slowly varying errors according to the manual [56]. The acceleration data for heave motions is of poor quality, as can be seen in Figure 6.13. If the acceleration data would be wrong, then the heave data would be wrong as well. On the other hand, there is a repetitive- ness observed in the other ramming attempts. This suggests that the heave data is measured wrongly in a consequent manner, or that there is a repetitive phenomenon occurring.

The explanation might be found in the heave ﬁlter. One of the purposes of this ﬁlter, is to make sure that the vessel in hydrostatic equilibrium has a zero heave displacement. This will be at different drafts during the expedition. For example fuel consumption will make the vessel lighter, and therefore the draft will decrease towards the end of the expedition. When the vessel rams the ridge, it will be lift upwards, and beach on the ridge. The vessel can be beached for quite some time. At the end of the ramming attempt the vessel is barely 6.3.R ESULTS OF LOAD IDENTIFICATION MODEL 119

Figure 6.13: Heave’s acceleration data in [m/s2] during the four ridge impacts penetrating the ice ridge, although the vessel’s thrusters are still pushing forward. At this stage the vessel is slowly pushing the ridge forward. When the captain decides to reverse the thrusters, it will take some time before the backwards thrust will be strong enough to pull the vessel backwards, into the water and off the ice ridge. During this ’beaching time’ the vessel is lift up, hence a positive heave displacement that remains practically constant. The heave ﬁlter might start ﬁltering out this static ’error’, which actually was the real displacement.

As noted, another phenomena not taken into account in the load identiﬁcation model, might have been responsible for the negative heave. Some possible explanation that the author looked into:

1. Heave vibration after gliding back from the beached position on the ridge 2. Coupling motion between heave and pitch 3. Coanda effect 4. Ventilation and backﬁll effect during ice-sloping structure interactions 5. Ice cannot hold the vessel’s weight and the ice starts to fail underneath the hull, resulting in a sudden downwards motion

These possible explanations are discussed in section L.2 in Appendix L. Unfortunately, no certain explanation was found that could explain the data.

The Coanda effect was one of the key focus points in the analyses of this measured negative heave. The model predicts an heave decrease of 6.5cm due to pure Coanda effect. The phenomenon is found to be significant, but is not sufficiently large to explain the data. According to the author, the most promising explanations regarding other phenomena, are the loss of buoyancy due to volume loss of the ridge, and that of the backfill phenomenon. This is briefly elaborated in section 6.3.11 below. However, the most reasonable explanation seems to be that the data is wrong, due to the heave filter.

LOSS OF BUOYANCY AND BACKFILL PHENOMENON One of the explanations looked into is the loss of buoyancy due to volume loss of the ridge, and that of the backﬁll phenomenon. The two approaches elaborated on below, might work simultaneously.

When the vessel is beached (or almost beached) the ridge has lost volume, hence a decrease of buoyancy of the ridge. This volume is replaced by the vessel, and practically lays on top of the ridge. Together they have a lower total buoyancy compared to just before the ridge impact, while their total mass is still the same. To increase the system’s buoyancy and to get to hydrostatic equilibrium, the complete vessel-ridge system will 120 6.R ESULTS AND DISCUSSION

Figure 6.14: Calculated heave state with load identiﬁcation model move downwards which is measured as a negative ice loading. In reality this might actually be the downwards moving motion due to the loss of buoyancy.

Another way to approach this problem, is to use the backfill phenomena. The vessel penetrates the ridge at relatively high velocity. Because of this, no backfill of the water occurs. At the end of the ridge impact, the velocity is low. According to theory, backfill occurs at low velocities, meaning that the water might backfill to the void around the destroyed ice. This by the vessel created void, originally contained ice, and is now filling with water. As water is around 10% heavier than ice, this will induce an additional weight on the ridge (or loss of buoyancy), that will move the ridge and vessel downwards at the end of the ridge impact.

6.3.12. THE COANDA EFFECT One of the reasons the Coanda effect was investigated was to what extend this phenomenon might be responsible for the negative heave motions that are observed at the end of a ridge impact. Based on the results (F 1. MN), and taking into account k 26 MN/m, it is concluded that the Coanda effect’s vertical force coan ≈ 33 ≈ is responsible for approximately 0.065 m. The moment due to the Coanda effect Mcoanda has an influence too. The observations and calculations show that the reverse thrusters will cause the aft of the ship to move downwards. However, it must be stressed that calculation are still very simplified. The author does not want to claim that the Coanda effect must be responsible for the negative heave motion’s observation. The results do imply that the effect might have influence and, therefore might be interesting for future research purposes. The vertical loading and the pitching moment due to the Coanda effect, just after the first impact, is shown in Figure L.4. 6.3.R ESULTS OF LOAD IDENTIFICATION MODEL 121

Figure 6.15: The vertical force and pitching moment that occurs due to the Coanda effect. 7 CONCLUSIONSANDRECOMMENDATIONS

7.1. CONCLUSIONS The objective of this thesis is to determine the global ice ridge ramming loads based on specific energy approach and full-scale data. The icebreaker Oden is represented by a nonlinear mass-damper-spring system. Maneuvering theory is applied, which means that the hydrodynamic variables are estimated at one frequency of oscillation. Two models are developed, the first being a simulation model using the specific energy approach, and the second being the load identification model using the full-scale data. The full-scale data analyzed in this thesis includes a profile of a multi-year ice ridge, vessel characteristics, acceleration data from a motion reference unit (MRU), GPS data, and propulsion data.

To reach the objective, an improved understanding in the governing physical phenomena that occur during ice ridge ramming impacts had to be achieved. As such, a study into ice features and ice resistance was executed. Here, it was found which ice features are expected to be encountered in the Arctic environment, and what their main factors governing their ice actions are. In addition, the important icebreaker design aspects considering ice operations and environmental conditions were established. Following this, ice ridges were studied in more detail, and it was identiﬁed how their corresponding failure load could be calculated.

Studies show that the impact location on the bow is an important factor in the peak stress values in the hull. A contact model was developed to calculate the volumetric penetration, contact area, and load location on the hull, under assumption of pure crushing of ice. This assumption is reliable at the ﬁrst stages of a ridge impact. The contact model’s credibility fails as soon as ice failure does not predominantly happen by crushing, although it is concluded to be an improvement compared to the commonly assumed load patch.

The Specific Energy Absorption (SEA) of mechanical crushing of ice is defined as energy per unit mass of crushed ice, necessary to turn solid ice into crushed (pulverized) material. For an ice ridge ramming phenomenon, this SEA value was found to depend on the temperature of the ice, the mechanical state of the ridge, and confinement due to the size and confinement of the ridge. A simulation model was developed to calculate the global ice loads by using the specific energy approach. A known thrust force is applied, causing the vessel’s open water motions that are calculated by the simulation model. As soon as the vessel interacts with the ridge, the contact model and specific energy principles determine the ice load. Results indicate that the specific energy approach can be used to simulate an impact of a vessel into an ice ridge, under assumption that the ice fails purely due to crushing. The principles are only valid during the beginning of the impact, as other failure modes often start to dominate as the penetration of the vessel into the ridge increases. It was established that the literature based specific energy values for crushing are determined during very different indentation ice-interaction scenarios. It is concluded that careful adjustment of this value is reasonable, resulting in more realistic ice loads. The simulated ice loads and vessel’s motions are used to validate the load identification model.

122 7.1.C ONCLUSIONS 123 heave, pitch). The joint input-state estimate algorithm combines measured data with an estimate of the state of the system in a way that minimizes the error. The algorithm assumes no prior information on the dynamic evolution of the input forces, and assumes that the stochastic system noise and measurement noise are to be mutually uncorrelated, zero mean, white noise signals with known covariance matrices. The number of forces that can be identified by joint input-state estimate algorithm is limited to the number of acceleration data available. The icebreaker Oden contained one MRU, located at the COG. No pitch acceleration data is measured. As a consequence, the numerical model was split into a 2DOF system (i.e. surge, heave) and 3DOF system, to be solved with the joint input-state estimate algorithm and Kalman filter respectively. Propulsion theory is applied to calculate the thrust force from the propulsion data. The excitation pitching moment is composed of forces identified in the horizontal and vertical plane. Superior filter performance is achieved by tuning the measurement noise covariance matrices.

The suggested approach to calculate the global ice loads, is reliable when full-scale data is valid. The vessel’s motions that are calculated by the simulation model during the ridge impact, were used by the load identification model to recalculate the ice load. Accuracy of identified loads decrease with increase of measurement noise. However, an exact match is found when the mass-damper-spring system is linear and no noise is added. An error in load identification appears when using nonlinear terms, most influential in this thesis being the nonlinear surge damping and heave’s hydrostatic spring stiffness. In this thesis this error is found to be around 1% at the peak load. The developed thrust force algorithm is found to work well for the whole range of possible thruster settings (i.e. the pitch angle of propeller blades, revolution rate, and entrance velocity). Furthermore, it was found that the vertical global ice load is sensitive to pitch motions, via the pitch-heave coupling hydrostatics. As such, minimizing the error in pitch state is fundamental for an accurate estimate of the vertical global ice load. To achieve this, the model is optimized by tuning the process noise covariance matrix in such a way that pitch data is taken very accurately compared to the process estimate.

The vertical ice ridge ramming loads are found to consist of two peaks. This occurs due to vessel’s heeling motion: at first impact the vessel pitches away from the ice ridge, but when the contact force decreases the vessel pitches back, causing the second peak. In general the first peak is larger than the second peak. It took Oden four attempts to break through the profiled MY ice ridge, which had a maximum thickness of 9.5 m, and sail width around 40 m. The impact velocities were between 2.1 and 3.5 m/s. From the results of the contact model, we conclude that the maximum volumetric penetration during the first ram is approximately 7000 m3. This volume relates to approximately 6400 tonnes of ice. The first three identified ramming load are found to be in the range of 3.8-4.8 MN and 9.1-12.9 MN, for the horizontal and vertical global ice load respectively. Two other large ice ridges give loads in the same range. These values result from the conventional model settings. A larger range is found when taking into account the results of the uncertainty analyses, this is elaborated on below. Furthermore, it is reasonable to state that the vessel had practically broken through the MY ridge after the third attempt. This is validated by penetration distance, and by comparing the identified loads. The ratio between the identified global loads is investigated too, because the vertical ice load should be approximately 2.47 times as large as the horizontal ice load, due to Oden’s bow angle of 22 degrees. From the results it is concluded, that this ratio is correct for the first three rams into the profiled ridge, with ratios between 2.39-2.69. A bigger deviation from the load ratio is found for some of the other ridges, which seems to be related to the loading direction and the ice’s failure mode. In the measurements one period of time is found, where it is reasonable to calculate the friction coefficient for the ice-structure interaction during the ramming. The chosen approach to calculate the friction coefficient for the other rams, is concluded to be not reliable. Main reasons for the low reliability are the rapidly changing local pressure variations over the contact area during the impact, the morphology of the ridge, the hydrodynamic behavior of the ice at large impact velocities (including the backfill and ventilation phenomena), and the unknown friction forces at the keel and parallel mid-body of the vessel. The period of time that is reliable, is found at the last phase of the first ridge ram. Here the thrust force, velocity, and horizontal ice load, remain approximately constant over time. The results show a friction coefficient in the range of 0.20-0.67 [-] in this period of time.

A sensitivity and uncertainty analyses was effectuated to the thrust force and global ice loads, based on the presented data- and model- uncertainty. A low reliability on the calculated thrust force was found due to the high uncertainty in the propulsion data. Thrust uncertainty was found to be around 24% at full thrust. For the global ice loads, a lower- and upper- boundary case-study was executed, where most parameter values were defined by their interdependency to vessel displacement. As a result, we conclude that the first three 124 7.C ONCLUSIONSANDRECOMMENDATIONS identified ramming loads are found to be in the range of 2.6-5.4 MN and 5.8-18.0 MN, for the horizontal and vertical global ice load respectively. The author would like to emphasize that the three subsequent rams have different loads within this range. The upper/lower boundary case-studies result in deviations from the conventional model setting around 10-16% for the horizontal, and 20-56% for the vertical global ice load. From the results of the uncertainty analyses, it is concluded that the reliability of the horizontal load is high, and for the vertical ice load low. Based on the uncertainties of the parameters, and their sensitivities to the global ice loads, this conclusion is justifiable. However, before this is discussed in more detail, the author likes to emphasize that we cannot just claim a high reliability on the identified ice loads. This is elaborated upon below. In the uncertainty analyses is found, that the horizontal global ice load depends mainly on the thrust force, and acceleration data (thus vessel mass). In the sensitivity analyses it is found that an additional 10% vessel mass and an additional 10% thrust force, results in a 6% and 4% increase in the horizontal ice load, respectively. In the lower boundary-case study the vessel mass is decreased by 4%, and thrust force is taken as 24% lower. In the upper boundary-case study the vessel mass is increased by 13%, and thrust force is not increased (due to the model’s boundary condition). Together, the thrust force and vessel mass contribute to approximately 8-10% of the calculated uncertainty, and the remainder of the parameters contribute all together just 2-6%. As such, a high accuracy is found for the horizontal ice load. The identified vertical global ice load depends mainly on the reaction forces coming from the hydrostatics, and barely on the inertia (i.e. little acceleration measured at the COG). The vertical load’s sensitivity to the hydrostatic parameters is high, and the uncertainty of these parameters is large. The waterline area and heave-pitch coupling terms alone, already introduce an uncertainty of 20% and 30%, for the lower- and upper-boundary case-study respectively. Several other parameters have a significant influence as well, e.g. the metacentric height, rotational inertia, the added mass for heave, and potential damping for heave.

From the results of the load identification model, we conclude that the current combination of model and data does not provide sufficient information to estimate the global ridge ramming loads with high reliability. The main problems are a too low sample frequency of the data, and the absence of an additional MRU. The biggest concern with the data is that a negative heave motion is measured at the end of the ridge impacts, which could not be physically explained. Without an additional MRU, the data cannot be verified, and thereby it is impossible to estimate the reliability of this data. Furthermore, it is impossible to determine the pitch acceleration without at least one additional MRU. As a consequence the calculated pitch excitation moment will have a large uncertainty, resulting in a large uncertainty in pitch state estimate, which lowers the reliability of the calculated vertical global ice load. Subsequently, there is a lack of reliable estimates of extra- and damped- vibrations in the vessel. The ice-structure impact is quite far from the COG, some vibrations might be damped due to vessel’s structure. Also some vibrations might be measured from different origins than the ridge impact (e.g. from the engines). Additional problems are the uncertainty of the vessel characteristics, the uncertainty of propulsion data, and the simplified hydrodynamics.

Lastly, this thesis addresses the Coanda effect. This is a phenomenon in which a jet ﬂow attaches itself to a nearby surface. This phenomenon causes a pressure difference on the hull when the vessel’s thrusters are reversed, causing a heeling motion that pitches the bow upwards, and a downwards movement of the vessel. This phenomenon could not sufﬁciently explain the negative heave motion that is measured at the end of a ridge impact. 7.2.R ECOMMENDATIONS 125

7.2. RECOMMENDATIONS For both models it is recommended to determine the vessel characteristics with greater certainty, and to improve the hydrodynamics. Several potential coefﬁcients were now determined using a reference vessel, while the hull form of an icebreaker is quite different from conventional ships. Subsequently, more open water sea trials should be executed at different velocities, to get a better estimate of the nonlinear surge damping. Fur- thermore, it is unknown what the assumption of maneuvering theory has on the hydrodynamic loads.

From an engineering point of view, the method of specific energy approach is useful. Studies show that the peak force tends to happen at the start of ridge impacts. That is the moment that the probability of crushing failure is high, and is known for giving the highest ice loads on structures. Therefore, a model based on specific energy principles will give valuable information about the upper boundary of global ice loads during impact. It is recommended to implement more failure modes to the model, starting with bending failure. Hydrodynamic effects in the model are simplified, and are recommended to be improved.

The specific energy principles assume that the loss of energy per volume is constant. This is experimentally proven in Drop Ball Tests (DBT) and indentation tests. However, it was found that lowering the SEA value is justified because the morphology of an ice ridge is different from the solid ice used in the DBT (an ice ridge is not homogeneous), intender geometry in the DBT is very different from a vessel hull, and confinement for an ice ridge is lower. Neither is the pressure-area relationship taken into account, which shows that relatively lower loads are needed to fail ice for larger areas. To what extend the SEA value can be lowered is unknown. This is recommended for future research. Energy dissipation per indentation volume at the start of impact could be higher because of the occurrence of other failure modes. It might be possible to make the energy loss per volume of ice dependent on parameters as for example velocity, contact area and confinement. If empirically determined SEA values can be found, it may allow prediction of ice ridges loads. A contact model, as developed in this thesis, is a suggested tool for this.

One of the main concerns regarding the load identification, is that there is just one MRU, with a low sample frequency of 10Hz. Especially the heave acceleration data seems to be of poor quality, and there are concerns with the heave displacement data, i.e. a negative heave displacement is measured at the end of a ridge impact. The consequence of this measured negative heave displacement is elaborated below. With already one additional MRU, validation of the data will be enabled. When using several MRUs, and by placing them on strategic places, more useful information can be gathered. The recommended additional locations are at the portside, starboard, bow and aft of the vessel. The largest benefit is the rotational acceleration data that can be calculated. This enables a direct use of the joint input-state estimate algorithm on roll, pitch and yaw motions. By improving these rotational states, the reliability of identified loads will significantly increase. Other benefits are for example that it enables an estimate of structural damping, or an estimate on the vibrations coming from different origins than the ridge impact. The MRU at the COG is recommended to always have a high accuracy and sample frequency. Additional MRU’s may be of less accuracy and sample frequency, since they are mainly for verification of data, and for gathering the rotational data (which don’t change as fast as the translational data during an ridge impact).

The negative heave displacement is a repetitive observation in the full-scale data. The load identification model predicts a downwards vertical global ice load, due to the reaction force from this negative heave and hydrostatic stiffness k33. This raises the question, how can ice induce negative buoyancy on a downward sloping structure? This is unreasonable and should be investigated further. Identified are three other possible explanations: 1) the data is wrong, 2) the numerical model is wrong, 3) another phenomenon, not taken into account in the numerical mode, is responsible for the negative heave measurements. The author looked into some possible phenomena, amongst others the Coanda effect, and ventilation and backfill effect. No certain explanation was identified. However, it seems reasonable to state that the data might be wrong, due to the heave filter in the MRU. The heave filter should remove static and slowly varying errors. As such, it makes sure that the vessel in hydrostatic equilibrium has a zero heave displacement, which will be at different drafts during an expedition (e.g. due to fuel consumption). During the vessels ’beaching time’ on the ridge, the vessel is lift up. Hence a positive heave displacement that remains practically constant over some period of time. The heave filter might start filtering out this static ’error’, which actually was the real displacement. It is highly recommended to investigate this negative heave displacement in more detail, because it is one of the leading issues regarding this thesis. 126 7.C ONCLUSIONSANDRECOMMENDATIONS

Global ice loads relate to the ice resistance, which can be divided in the ice breaking forces, the ice submerging forces, the friction forces, and the hydrodynamic forces. The author would like to emphasize that for a ridge impact, the ventilation process is likely to increase the reaction forces from the ﬂuid, and thereby the global ice resistance. To the authors knowledge, this effect has not been studied in the case of a vessel-ridge impact.

Looking at studies where vessels ram into ice ridges, it was found that a certain load patch is commonly assumed, which implies an idealization, hence an error in the ice loads. Effect of this assumption on the ice load is not exactly known. The average ice pressure on a contact area, depends on the size of the contact area. Sanderson’s pressure-area relationship is the one mostly used to determine this average pressure. However, it is concluded that the pressure-area relationship can refer to three different relationships, which stresses the need to deﬁne which is used. Furthermore, Sanderson’s relationship is a statistical curve ﬁt. It did not take into account temperature, ice velocity, surface friction and experimental set-up (e.g. rig stiffness). For practical purposes the relationship and load patch assumption are convenient, but carefulness is advised.

Lastly, in ISO 19906 [9], an overview is found of the factors inﬂuencing the interaction scenarios between various ice features and general offshore structure (Figure 2.1). It is a useful tool for ﬂoating structures and bottom founded structures, but found to be invalid for vessels. The author would like to recommend the constructed overview as is presented in Figure 2.4, which shows the major parameters affecting ice resistance of a vessel. A ICEBREAKEREXAMPLES

A.1. MSVFENNICA MSV Fennica is a Finnish multipurpose icebreaker and platform supply vessel. Built in 1993 by Finnyards in Rauma. MSV Fennica is an pioneer in the sense she is the first icebreaker with azimuthing propulsion system. Furthermore she was the first Finnish icebreaker designed to be used as an escort icebreaker in the Baltic Sea during the winter months and in offshore construction projects during the open water season. Using a combination of a spoon-shaped bow and a narrow stern allows the icebreaker to maintain reasonable sea keeping properties. Fennica and het sister ship Nordica represent the advancement of ships that are not only efficient icebreakers, but also well suited for offshore open water missions, making it a useful multipurpose vessel.

MSV Fennica has bow reamers, stern reamers and a v-shaped prow (bow plough). See Table A.3 for the general characteristics. When MSV Fennica was in the first ice trials, it was noted that the ship can maintain a continuous speed in thick ridges by moving the thrusters from side to side. This way the ice floes forming the ridge break loose and the ridge can be dispersed. The effect is enhanced when moving astern as icebreaker wake flushes the ice floes very efficiently. It led to the concept of dual mode ships and therefore MSV Fennica is important in icebreaker history.

Table A.1: General characteristics of icebreaker MSV Fennica

Build year 1993 Ice class DNV POLAR-10 Icebreaker Shape hull spoon-shape DWT 1,650 to 4,800 Beam 26 m Length 116 m Draft 7–8.4 m Propulsion system Diesel-electric (AC/AC), Two Aquamaster US ARC 1 azimuth thrusters (2 × 7,500 kW), Three Brunvoll bow thrusters (3 × 1,150 kW) Installed power 2 × Wärtsilä 16V32D (2 × 6,000 kW) 2 × Wärtsilä 12V32D (2 × 4,500 kW) Max speed 16.5 knots Crew 77

A.2. ODEN The Swedish icebreaker Oden was built in 1988 for the Swedish Maritime Administration. It is named after the Norse god Odin. Initially, it was build to clear a passage through the ice of the Gulf of Bothnia for cargo ships. Nowadays it works as a research vessel. In 1991 it was the ﬁrst non-nuclear surface vessel to reach the North Pole, together with the German research icebreaker RV Polarstern. It has participated in several scien-

127 128 A.I CEBREAKEREXAMPLES

Figure A.1: Icebreaker MSV Fennica tiﬁc expeditions in the Arctic and Antarctica. In full scale operation Oden has moved at a continuous speed of about 3 knots in 2 meter thick ice (Johansson, 2004) [63]. Figure A.2 shows the icebreaker. See table A.2 for the general characteristics.

Oden has a relatively ﬂat spoon-shape bow. The circular shape on such a bow will maximize the shearing force used to break ice and reduce high resistance resulting from crushing. A spoon-shape bow will typically reduce resistance in ice by twenty to forty percent (Jones, 2004) [3]. To enhance vessel performance, Oden has a water-wash system that lubricates the ice from above the water level. This reduces resistance and makes it more easily for Oden to ride up on top of the ice. The icebreaker has wide bow reamers to enhance maneuvering capabilities. These reamers also help push ice ﬂoes down. Below the hull a bow plough is build which makes sure the ship propellers are ice free. Stern reamers are in place to break ice beyond the beam of mid-body so that the discontinuous bow does not wedge and crush ice.

Oden has a very fast heeling system to help improve progress in heavy ice (National Research Council, 2007) [64], (Johansson, 2004) [63]. According to Jones (2008) [65] this heeling system can pump 800 tonnes of water from one side to the other in 25 seconds.

Table A.2: General characteristics of icebreaker Oden

Build year 1988 Ice class DNV POLAR-20 Shape hull spoon-shape DWT 4,906 Beam 31 m (max), 25 m (parallel-mid body) Length 107.8 m Draft 7-8.5m Propulsion system 2 × LIPS CPP Installed power Main engines: 4 × Sulzer 8ZAL4OS (4 × 4,500 kW) Auxiliary generators: 4 × Sulzer 6AT25H (4 × 1,270 kW) Max speed 16.0 knots (open water) Crew 15 65 +

A.3. 50 LET POBEDY Figure A.3 shows the 50 Let Pobedy (translated as 50 Years of Victory or Fiftieth Anniversary of Victory) Arktika-class icebreaker. A.3.50L ET POBEDY 129

Figure A.2: Icebreaker Oden, seen from helicopter

The Arktika class is a Russian, former Soviet, class of nuclear-powered icebreakers. Up to this day they are the largest and most powerful icebreakers ever constructed. Ships of the Arktika class are owned by the federal government. Of the ten civilian nuclear-powered vessels built by Russia six have been of this type. They are used for escorting merchant ships in the Arctic Ocean north of Siberia and for scientiﬁc and recreational expeditions to the Arctic. Arktika has three ﬁxed-pitch propellers which can deliver a combined bollard pull of 480 tons with 18-43 MW.

Table A.3: General characteristics of icebreaker MSV Fennica

Build year 1993 Ice class Arktika-class icebreaker Shape hull wedge-shape DWT 3,505 Beam 30 m Length 150 m Draft 11 m Propulsion system Nuclear-turbo-electric. Three shafts, 52 MW (combined) Installed power Two OK-900 nuclear reactors (2 × 171 MW) Max speed 20.6 knots Crew 140 128 passengers + 130 A.I CEBREAKEREXAMPLES

Figure A.3: Nuclear icebreaker 50 Let Pobedy, 50 Years of Victory B APPENDIX A:EXPERIMENTLOGAND PICTURES

B.1. VESSELLOG The vessel report gives the following observations:

• 20:27 took course on the ridge (approx. 200 m from edge of the sail) • 20:28 reached the edge of ice floe • 20:29 stopped approx. 40 m before the edge of the sail • 20:30 moved back (half of the ship length) and forward (maximum speed reached 2.8 knots), then stopped approx. 7m before sail edge. Circular crack formed in front of the sail. • 20:31 moved back and forward, got on the ridge (max. speed 5.7 nm). One long crack appeared along the ridge length and two long cracks behind the ridge in the direction close to Oden’s heading • 20:34 rammed through the ridge at speed approx. 5.2 knots. • 20:35 stopped approx. 30 meters before “100 meters” mark on the ramming path • 20:36 moved back and forward, hit remaining part of the ridge. • 20:38 got on “100 meters” mark • 20:39 hit 2nd ridge(one crack parallel to the length and two cracks perpendicular behind the ridge) • 20:41 hit 3rd ridge with an opening on the left side (two cracks behind the ridge and a wedge went into the opening) • 20:43 4th small ridge (cracked in two parts along ship’s heading) • 20:44 5th ridge, felt tough hit • 20:46 6th ridge with open channel on the right side (one crack 11 o’clock from heading) • 20:46 7th (quite big, surrounding ice approx. 4m thick). At first circumferential cracks formed, afterwards 3 long cracks behind the ridge appeared. Cracks went towards melting points either open channel far on the right • 20:49 8th ridge with two branches (V - shape) and open channel on the right side. One crack formed along the ridge length and three cracks behind the last branch. • 20:54 reached end of the floe.

B.2. 360 CAMERA The relevant pictures made by the 360 camera includes the following:

• Figure B.1: 20:24:00 - Vessel with nose at ridge (before ramming experiment) • Figure B.2: 20:27:50 - Vessel maximally pulled back and ready for impact 1. • Figure B.3: 20:29:10 - Penetration due to impact 1 into profiled ice ride (approx 30 m). • Figure B.4: 20:31:00 - Penetration due to impact 2 into profiled ice ride (approx 4 m). • Figure B.5: 20:33:00 - Impact 3 into profiled ice ride (in progress), no level ice failure behind ridge. • Figure B.6: 20:33:20 - Impact 3 into profiled ice ride (in progress), splitting failure of level ice behind ridge is observed

131 132 B.A PPENDIX A:EXPERIMENTLOGANDPICTURES

• Figure B.7: 20:33:40 - Penetration into profiled ice ridge due to impact 3 (approximately 60 m, full, penetration), splitting failure of ice ridge more profound. • Figure B.8: 20:35:50 - Penetration due to impact 4 into profiled ice ride, vessel is completely through ridge. • Figure B.9: 20:37:50 - Vessel pulls back after impact 4 into profiled ice ride to continue journey. • Figure B.10: 20:39:20 - Crack due to ice floe failure, corresponding load peak identified at 20:39:00. Ridge 2 is observed. • Figure B.11: 20:39:50 - Ridge 2 failure initiated. Observed are cracks parallel to the length of vessel and two cracks perpendicular behind the ridge. • Figure B.12: 20:40:00 - Maximum penetration in ridge 2, with propagation of cracks. • Figure B.13: 20:40:50 - Cracks appeared after hitting the second ice floe. • Figure B.14: 20:41:10 - Vessel just before hitting ridge 3 with an opening on the left side. • Figure B.15: 20:41:40 - Vessel just after full penetration ridge 3 (two cracks behind the ridge and a wedge went into the opening). • Figure B.16: 20:43:10 - Impact on the small ridge 4 is felt. • Figure B.17: 20:43:20 - Cracks are observed after failure of ridge 4: ice is cracked in two parts along ship’s heading. • Figure B.18: 20:47:10 - Full penetration of ridge 7. Ridge 7 is quite big, with surrounding ice approx. 4m thick. At first circumferential cracks formed, afterwards 3 long cracks behind the ridge appeared.

Note It was concluded that pictures related to ice ridge 5, ice ride 6, and ice ridge 8, do not need to be included.

Figure B.1: 20:24:00 - Vessel with nose at ridge. B.2.360 CAMERA 133

Figure B.2: 20:27:50 - Vessel pulled back maximally and ready for impact 1. Note the relative open channel in front of the ridge.

Figure B.3: 20:29:10 - Penetration due to impact 1 into proﬁled ice ride (approx 30m). 134 B.A PPENDIX A:EXPERIMENTLOGANDPICTURES

Figure B.4: 20:31:00 - Penetration due to impact 2 into proﬁled ice ride (approx 40m).

Figure B.5: 20:33:00 - Impact 3 into proﬁled ice ride (in progress), no level ice failure behind ridge. B.2.360 CAMERA 135

Figure B.6: 20:33:20 - Impact 3 into proﬁled ice ride (in progress), splitting failure of level ice behind ridge is observed

Figure B.7: 20:33:40 - Penetration into proﬁled ice ridge due to impact 3 (approximately 60m, full, penetration), splitting failure of ice ridge more profound. 136 B.A PPENDIX A:EXPERIMENTLOGANDPICTURES

Figure B.8: 20:35:50 - Penetration due to impact 4 into proﬁled ice ride, vessel is though ridge.

Figure B.9: 20:37:50 - Vessel pulls back after impact 4 into proﬁled ice ride to continue journey. B.2.360 CAMERA 137

Figure B.10: 20:39:20 - Crack due to ice ﬂoe failure (red arrow), corresponding load peak identiﬁed at 20:39:00. Ridge 2 is observed (blue arrows).

Figure B.11: 20:39:50 - Ridge 2 failure initiated. Note the cracks parallel to the length of vessel and two cracks perpendicular behind the ridge. 138 B.A PPENDIX A:EXPERIMENTLOGANDPICTURES

Figure B.12: 20:40:00 - Maximum penetration in ridge 2, note propagation of cracks.

Figure B.13: 20:40:50 - Cracks appeared after hitting an ice ﬂoe. B.2.360 CAMERA 139

Figure B.14: 20:41:10 - Vessel just before hitting ridge 3 with an opening on the left side.

Figure B.15: 20:41:40 - Vessel just after full penetration ridge 3 (two cracks behind the ridge and a wedge went into the opening). 140 B.A PPENDIX A:EXPERIMENTLOGANDPICTURES

Figure B.16: 20:43:10 - Impact on the small ridge 4 is felt.

Figure B.17: 20:43:20 - Cracks are observed after failure of ridge 4: ice is cracked in two parts along ship’s heading. B.2.360 CAMERA 141

Figure B.18: 20:47:10 - Full penetration of ridge 7. Ridge 7 is quite big, with surrounding ice approx. 4m thick. At ﬁrst circumferential cracks formed, afterwards 3 long cracks behind the ridge appeared. C KONGSBERG SEAPATH 320+ SYSTEM DATA

Figure C.1: Top figure shows the absolute velocities with: blue as North velocity, green as East velocity, and red as SOG. Bottom figure shows the heading, with absolute North defined as 0 rad, and clockwise as positive

142 143 igure C.2: Acceleration data of surge, sway and heave respectively F 144 C.K ONGSBERG SEAPATH 320+ SYSTEM DATA igure C.3: Heave displacement and heave velocity F 145 igure C.4: Pitch and pitch rate F D VESSEL CHARACTERISTICS

D.1. PROPELLERCHARACTERISTICS are key parameters, as they determine the thrust. The open-water characteristics of Oden’s thrusters are based on the Wageningen Nozzle 37 with the Ka4-70 series, which were faired by means of harmonic analyses (Roddy, 2006 [53]). These characteristics enable the identification of the thrust coefficient depending on hydrodynamic pitch and P/D value. The resulting harmonic analysis coefficients for CT∗, CT∗ n, and CQ∗ are presented in the form of: 30 1 X CT∗ [A(k)cos(kβ) B(k)sin(kβ)] = 100 k 0 + = 30 1 X CT∗ n [A(k)cos(kβ) B(k)sin(kβ)] (D.1) = 100 k 0 + = 30 1 X CQ∗ − [A(k)cos(kβ) B(k)sin(kβ)] = 1000 k 0 + = where CT∗ is the Thrust Coefficient, or Total Thrust Coefficient of Ducted Propeller System , CT∗ n is the Thrust Coefficient due to the Duct, and CQ∗ is the Torque Coefficient. The resulting harmonic analysis coefficients are presented in frequency domain in figure D.1, figure D.2, figure D.3, figure D.4, and figure D.5 [53]. Matlab was chosen to do the Inverse Fast Fourier Transform to go to the time domain of hydrodynamic pitch (β). Following results are presented in figure D.6 and figure D.7. The coefficients are visualized in time domain in figure D.8. For more detailed information the reader is referred to (Roddy, 2006 [53]).

Geometry. In ﬁgure D.9 and ﬁgure D.10 the geometry of Oden is presented, including some basic parameters (e.g dimensions, weights, gross tonnage, painted area).

146 D.1.P ROPELLERCHARACTERISTICS 147

Figure D.1: Coefﬁcients for Nozzle 37 with Ka4-70, P/D=0.6. [53]

Figure D.2: Coefﬁcients for Nozzle 37 with Ka4-70, P/D=0.8. [53] 148 D.V ESSEL CHARACTERISTICS

Figure D.3: Coefﬁcients for Nozzle 37 with Ka4-70, P/D=1.0. [53]

Figure D.4: Coefﬁcients for Nozzle 37 with Ka4-70, P/D=1.2. [53] D.1.P ROPELLERCHARACTERISTICS 149

Figure D.5: Coefﬁcients for Nozzle 37 with Ka4-70, P/D=1.4. [53] 150 D.V ESSEL CHARACTERISTICS

Figure D.6: Coefﬁcients for Nozzle 37 with Ka4-70. Results presented in time domain after IFFT in Matlab. D.1.P ROPELLERCHARACTERISTICS 151

Figure D.7: Coefﬁcients for Nozzle 37 with Ka4-70. Results presented in time domain after IFFT in Matlab. 152 D.V ESSEL CHARACTERISTICS

D.2. THRUST COEFFICIENT ALGORITHM The general range of β0.7R and P/D uses the Wageningen Four Quadrant Estimate data. This range is deﬁned as:

•0 β 2π degrees, 0.6 P/D 1.4 < 0.7R < < < The following ranges outside the general range are identiﬁed:

• P/D < 0.6 and 0 β 12.37 < 0.7R < • P/D < 0.6 and 12.37 β 180 < 0.7R > • P/D < 0.6 and 180 β 0 − < 0.7R < The last itemized range (P/D < 0.6 and 180β 0) is not that relevant for a ramming experiment, therefore − 0.7R < not summarized. A more comprehensive algorithm is presented in the matlab code.

If 0 β 12.37 and P/D=P/D<0.6: < 0.7R < 1. Identify the β , with P/D 0.6, for which Ct becomes equal to 0 ( C t(β ?,P/D 0.6) 0) 0.7R = 0.7R = = = 2. Given above, assume linear relationship, and ﬁnd the β0.7R for which Ct would be zero given the P/D value: C t(β ?,P/D P/D) 0. 0.7R = = = 3. Identify C t(β 0,P/D 0.6) 0) 0.7R = = = 4. Given results of point 2 and 3, assume a linear relationship between these, and interpolate to the Ct value by: C t(β β ,P/D P/D) C t(β β ,P/D 0.6) P/D 0.7R = 0.7R = = 0.7R = 0.7R = ∗ 0.6 If 0 β 12.37 and P/D=P/D<0.6: < 0.7R > 1. Identify the β , with P/D 0.6, for which Ct becomes equal to 0 ( C t(β ?,P/D 0.6) 0) 0.7R = 0.7R = = = 2. Given above, assume linear relationship, and ﬁnd the β0.7R for which Ct would be zero given the P/D value: C t(β ?,P/D P/D) 0. 0.7R = = = 3. Identify C t(β 0,P/D 0.6) 0) 0.7R = = = 4. Given results of point 2 and 3, assume a linear relationship between these, and interpolate to the Ct value by: C t(β0.7R β0.7R ,P/D P/D) C t(β0.7R 0,P/D P/D) C t(β0.7R 0,P/D P/D) β0.7R = = = = = − = = ∗ C t(β0.7R ?,P/D P/D) 0) = = = D.3.S TRUCTURALREQUIREMENTS 153

D.3. STRUCTURALREQUIREMENTS Given DNV standards for Polar Class ships [54] the design Force, design Line load and design pressure were determined. The design ice load is characterized by an average pressure uniformly distributed over a rectangular load patch of height and width. In this case the load patch has a depth equal to the maximum depth of the ridge, and width equal to the width of Oden. For a comprehensive explanation it is referred to [54]. In this section the key equations and results are presented:

• F 36.1[MN] i = • ARi 0.13[ ] = − • Q 35.1[MN/m] i = • P 2.49[MPa] i = Parameters:

• 11.5 [kt] ∇ = • x 3 [m] ≈ • L 107.75 [m] = • α 22 [deg] i = • β 20 [deg] i ≈ • CF 17.69 [-] [54] [-] C = • CF 2.01 [-] [54] [-] D = • CF 68.6 [-] [54] [-] F = Design load in [MN] is given by: F f ai CF 0.64 (D.2) i = · C · ∇ where CF equals 17.69 [-] [54], equals 11.5 [kt], fai is the shape coefﬁcient, which is the minimum of (fai,1 C ∇ fai,2 fai,3): αi f ai,1 (0.097 0.68 (x/L 0.15)2) = − · − · 2 βi (D.3) f ai,2 1.2 CF /(sin(β ) CF ( )0.64) = · F i · C · ∇ f ai,3 0.6 = where x represents distance from forward perpendicular [m], L ship length [m], factor CFF [-] [54], αi waterline angle [deg], and normal frame angle βi [deg].

Load patch aspect ratio (ARi): ARi 7.46 sin(β ) 1.3 (D.4) = · i > Line Load Qi []MN/m]: Qi F 0.61 CF /ARi 0.35 (D.5) = i · D with factor CFD ans aspect ratio ARi.

Design Pressure Pi [MPa]: Pi F 0.22 CF 2 ARi 0.3 (D.6) = i · D ·

D.4. TECHNICAL DRAWINGS 154 D.V ESSEL CHARACTERISTICS

Figure D.8: Four Quadrant Estimate for Nozzle 37 with Ka4-70 Propeller Series. Symbols = Predictions, Solid Lines = Measured Data. (Roddy, 2006 [53]) D.4.T ECHNICAL DRAWINGS 155 igure D.9: Technical drawing of Oden, number 1 F 156 D.V ESSEL CHARACTERISTICS igure D.10: Technical drawing of Oden, number 2 F D.5.N ONLINEAR HEAVE DAMPING 157

D.5. NONLINEAR HEAVE DAMPING

Figure D.11: The dynamic value b33,NL, depending on heave velocity and vessel’s draft.

D.6. POTENTIAL COEFFICIENTS OF HEAVE AND PITCH MOTIONS The potential coefﬁcients for heave and pitch are determined using Ankudinov (1991) [55]. The analyses is based on strip theory. It must be emphasized that the hull form of an ice breaker may be quite different from other ships, and therefore will have different hydrodynamic characteristics. However, based on the literature study, these give the best available estimate, and are presented in Figure D.12 and D.13. 158 D.V ESSEL CHARACTERISTICS igure D.12: Potential damping and added mass for heave motions for different dimensions of length L and beam B. F D.6.P OTENTIAL COEFFICIENTS OF HEAVE AND PITCH MOTIONS 159 igure D.13: Potential damping and added mass for pitch motions for different dimensions of length L and beam B. F E RIDGE CHARACTERISTICS

The ridge has been proﬁled extensively before the ramming. 16 proﬁles were drilled across the ridge. The deepest keel was –10.30 meters, and a surrounding level ice was 1.7 meters thick. The drilling scheme is presented in Figure E.1. Two cross sections where drilled, visualized in Figure E.2 and Figure E.3.

Figure E.1: Drilling scheme of the proﬁled MY ice ridge.

Figure E.2: Cross section 1 of the proﬁled MY ice ridge.

The thickness of the consolidated layer, keel, and level ice where determined by profiling data. Figure E.4 shows the idealized ice ridge, as put into the Contact Model. Given coordinate system as presented at figure E.4 (COG of ridge is at x=0, z=0 at waterline level), the ice ridge triangle coordinates are given below. Here the Matlab code is defined as: Tr i angle [x z ;x z ;x z ;]. = coordinate1 coordinate1 coordinate2 ccoordinate2 coordinate3 coordinate3 It must be noted that triangle coordinates must be implemented counter clockwise in Matlab. Consolidated layer Triangles (Matlab Code): Tr01= [-30 0.425; -20 0.290; -30 -1.750]

160 161

Figure E.3: Cross section 2 of the proﬁled MY ice ridge.

Table E.1: Proﬁling data of cross section 1 and 2.

Tr02= [-30 -1.750; -20 0.290; -20 -2.935] Tr03= [-20 0.290; -10 -3.475; -20 -2.935] Tr04= [-20 0.290; -10 1.125; -10 -3.475] Tr05= [-10 1.125; 0 8/3; -10 -3.475] Tr06= [-10 -3.475; 0 8/3; 0 -7.833] Tr07= [0 8/3; 10 -2.65; 0 -7.833] Tr08= [0 8/3; 10 1.0; 10 -2.650] Tr09= [10 1.0; 20 -0.01; 10 -2.650] Tr10= [10 -2.65; 20 -0.01; 20 -3.650] Tr11= [20 -0.01; 30 -1.69; 20 -3.650] Tr12= [20 -0.01; 30 0.41; 30 -1.690]

Keel Triangles (Matlab Code): Tk1= [-20 -2.935; -10 -3.475; -10 -7.875] Tk2= [-10 -3.475; 0 -7.833; -10 -7.875] Tk3= [ 0 -7.833; 10 -2.650; 10 -6.350] Tk4= [ 10 -2.650; 20 -3.650; 10 -6.350] 162 E.R IDGE CHARACTERISTICS

Figure E.4: Idealized and discretized (for CM) ice ridge. In the ice ridge white triangles represent the consolidated layer, magenta represents the keel layer. At the right level ice with a thickness of 1.7 m.

Tk5= [ 10 -6.350; 20 -3.650; 20 -6.110] Tk6= [ 20 -3.650; 30 -3.440; 20 -6.110] Tk7= [ 20 -3.650; 30 -1.690; 30 -3.440] F SIMULATION MODEL PAPER

163 164 F.S IMULATION MODEL PAPER 165 166 F.S IMULATION MODEL PAPER 167 168 F.S IMULATION MODEL PAPER 169 170 F.S IMULATION MODEL PAPER 171 172 F.S IMULATION MODEL PAPER 173 174 F.S IMULATION MODEL PAPER 175 176 F.S IMULATION MODEL PAPER 177 178 F.S IMULATION MODEL PAPER 179 180 F.S IMULATION MODEL PAPER 181 182 F.S IMULATION MODEL PAPER G INFLUENCE OF LINEARIZING NONLINEAR TERMSINNUMERICALMODEL

It was found that some nonlinear terms should be included in this thesis. This is implemented in the numerical model by linearizing these nonlinear terms each time step. The question arises if this imposes (signiﬁcant) errors. To quantify this error a simulation model was developed, on which the load was known. Generated data was the input for the numerical model, with the purpose to identify the load. If the error would be zero, an exact match of the simulated load should be identiﬁed. Linearized terms in the numerical model are: b11,NL, b33,NL, k33, k35, and k53.

G.1. SET UP OF THE ANALYSES The idea of the analyses is to simulate a loading on the vessel and save the data (i.e. displacement, velocity, acceleration). This data is then used in the load identification model to recalculate the state (displacement, velocity, acceleration) and input (excitation load, which is known and has the same order of magnitude as the global ice loads.). The load identification model uses the joint input-state estimate algorithm and Kalman filter. These require input of a measurement and process noise covariance matrix. Without added noise the algorithms should be able to identify state and load accurately. For simplicity no noise was added, and the same measurement noise is taken into account for surge, heave, and pitch, with high accuracy on the data.

In the model, the following equations of motion are taken into account:

(m a )x¨ (t) b x˙ (t) x˙ (t) T (t) F (t) (G.1) + 11 1 + 11,NL| 1 | 1 = x1 − x1,ice (m a )x¨ (t) k x (t) b x˙ (t) b x˙ (t) x˙ (t) k ¡x (t) x ¢ T (t) F (t) (G.2) + 33 3 + 33 3 + 33 3 + 33,NL| 3 | 3 − 35 5 − 5,mean = x3 + x3,ice X (I a )x¨ (t) b x˙ (t) k ¡x (t) x ¢ k x (t) M (t) (G.3) y + 55 5 + 55 5 + 55 5 − 5,mean − 53 3 = x5 where Mx5(t) is a function of the global ice loads and an arbitrary chosen (known) arm to the contact point. Parameters are as presented in the thesis. The implemented load is arbitrary chosen load, and completely known.

First, it should be noted that the found error was very large. It suggested that there had to be another error in the model. The chosen approach, is rebuilding the simulation model and the load identification model. This is time consuming, but it enables validation of the model. If the results are completely the same, it means that the conclusions were correct. If the results differ, it implies that at least one of the two models must have an error. The old models are defined as: simulation model 1 and identification model 1. The new models are defined as: simulation model 2 and identification model 2.

G.2. FINDINGS Summarized, the following ﬁndings were found during the analyses:

183 184 G.I NFLUENCE OF LINEARIZING NONLINEAR TERMS IN NUMERICAL MODEL

1. Identification model 1 and identification model 2 give the same results when using the same data set. 2. Simulation model 1 and simulation model 2 do not result in the same data set, although the difference is small. The mistake found: simulation model 1 took a squared velocity wrongly into account. wrong way: F damping (x˙)2. correct way: F damping x˙ x˙. = · = · | | · 3. Both identification models calculate a large error in the vertical ice load, and the pitching ice moment. A study is executed to identify if the cause can be found in linearizing the nonlinear spring stiffness for heave:

(a) What is the error when using the linearized heave spring stiffness term k33 instead of the heave nonlinear spring stiffness term k33NL ? - The error is very small; see Figure G.3 which shows the heave motion’s hydrostatic contribution (Force=stiffness x z) when using a nonlinear spring stiffness and linearized spring stiffness. (b) The nonlinear heave spring stiffness consists of a linear and nonlinear part. How large is the contribution of the nonlinear part in the linearized spring stiffness, compared to the linear part in the linearized spring stiffness? - See Figure G.4, which shows that the nonlinear part has a small contribution to the vertical load. This suggests that the error due to linearizing nonlinear terms should be small, not as large as the found error. 4. If the error is not due to linearizing nonlinear terms, then another cause should be identified: (a) Is the state of heave identified incorrectly? - No, state identification is practically perfect. It was found that state identification of surge and heave have high accuracy, also for nonlinear spring stiffness. This was an condition of the test (no noise generated). The state of pitch has a relatively larger error. States are illustrated in Figure G.7, G.8, and G.5. (b) The vertical ice load also depends on pitch-heave coupling (see equation 6.8), does the model make an error here? - Yes, it was found that the state estimate of pitch has a significant error, this is illustrated in Figure G.5. As a consequence, the calculated vertical ice load has a significant error too. 5. Based on the findings, the load identification model was altered and rerun. Instead of taking into account the identified state of pitch, it only uses the exact pitch data. It was found that this was the reason for found error. Figure G.1 shows the difference between simulated and identified loads when pitch state is identified ’correctly’. This is further investigated: (a) Would the error disappear if pitch acceleration data would be available for the Kalman filter? - No, error is found to be similar (b) Would the error disappear if the measurement noise or process noise is altered? - The uncertainty of pitch data and the model is increased and decreased, and results are compared. It was possible to improve load identification (see Figure G.6). A finding, is that the measurement noise of the data cannot just simply be taken with the same magnitude. This makes sense, as surge and heave displacement go in meters, while pitch is in radians (smaller value). So, first of all the measurement noise of pitch should be smaller than that of surge and heave. Secondly, by tuning the measurement noise in such a way that the pitch data is taken as very accurately, the identified state of pitch becomes more accurate (logically..). As a results, the error in the global is mitigated. (c) It must be validated how well the measurement noise setting of the numerical model in the thesis would work with given simulated data. - When using the measurement noise covariance matrix as used in the numerical model, the results improve significantly. It suggest that the settings as in the numerical model are correct, and it emphasizes carefulness when implementing noise covariance matrices in a model.

G.3. CONCLUSION The aim was to identify the error due to linearizing nonlinear terms in the load identiﬁcation model, in each time step. Based on the analyses in this chapter, and as illustrated in Figure G.1, this error is small compared to using only linear terms. Therefore, the nonlinear terms should be implemented as linearized nonlinear terms.

In the analyses is found that the state of pitch identiﬁcation was initially not very good, and that this causes a high error in identiﬁed vertical global ice load. The pitch-heave coupling is important, and here lays a problem for the numerical model: there is no pitch acceleration data, and as a consequence the calculated pitch G.3.C ONCLUSION 185 excitation moment will have a larger error, and therefore the state estimate of pitch will have a larger error as well. This is elaborated below.

To identify a correct vertical global ice load, the correct state of pitch must be identified. For the numerical model, a good state estimate of pitch requires: a) the joint input-state estimate algorithm and a input of pitch acceleration data, or b) the Kalman filter when the applied pitching moment is correct. The numerical model must use the Kalman filter. The error in pitch estimate depends on the error in the pitch excitation moment. The main reasons for the error in pitch excitation moment are:

To optimize the numerical model, it is necessary to take the pitch data as very accurate compared to the pitch estimate by the model. In other words, for a good estimate of pitch motions the process noise must be large enough compared to the measurement noise. With respect to the numerical model in the thesis, this means that the noise covariance matrices must be implemented in such a way that the state estimate of pitch is always very nearby that of the pitch measurements.

Figure G.1: The vertical ice load as from simulation model (correct, black line). Furthermore two lines: the vertical ice load as from identiﬁcation model using linearized nonlinear terms (red line), the vertical ice load as from identiﬁcation model using linear terms (blue line) 186 G.I NFLUENCE OF LINEARIZING NONLINEAR TERMS IN NUMERICAL MODEL

Figure G.2: Load identiﬁcation of vertical global ice load. An improved load identiﬁcation is found in measurement noise setting as in numerical model.

2 Figure G.3: Two loads: F k33 z and F k z = · = 33NL · G.3.C ONCLUSION 187

Figure G.4: The contribution of the linear and nonlinear part of the linearized spring stiffness to the vertical load F k33 z: Load = · contribution due to linearized k33,nLoad contribution due to linear part of linearized k33,Load contribution due to nonlinear part of linearized k33.) gr

Figure G.5: Recalculating state of pitch has an signiﬁcant error 188 G.I NFLUENCE OF LINEARIZING NONLINEAR TERMS IN NUMERICAL MODEL

Figure G.6: Load identiﬁcation of vertical global ice load. An improved load identiﬁcation is found when taking measurement noise on pitch as very small.

Figure G.7: State identiﬁcation of simulated load (surge) G.3.C ONCLUSION 189

Figure G.8: State identiﬁcation of simulated load (heave) H DECOMPOSITIONOFHYDRODYNAMIC FORCESDURINGICERIDGERAMMING EXPERIMENT

It is investigated how the vertical global ice load relates to the hydrodynamic forces during the ramming experiment. Two ﬁgures are presented: Figure H.1 and H.2.

Figure H.1, shows how the calculated excitation force (Px3) by joint input-state estimate algorithm relates to the displacement, velocity, and acceleration data of heave (via k, c, m). The best way to interpret the ﬁgure, is by taking equation 6.5. with the upwards forces presented positive, and downwards forces presented negative. In Figure H.2 is shown how all components contribute to the predicted ice load by the numerical model. These components that are taken into account are: the data (d, v, a) combined with hydrodynamics, vertical component thrust force Tx3, force due to Coanda effect Fcoanda, and force due to pitch-heave coupling (k35 and pitch state). Summarized:

190 191 igure H.1: Decomposition of hydrodynamic forces, to identify to what extend hydrodynamics and heave data contribute to the excitation force. Upwards forces are positive, negative forces are negative. F 192 H.D ECOMPOSITIONOFHYDRODYNAMICFORCESDURINGICERIDGERAMMINGEXPERIMENT igure H.2: Decomposition of hydrodynamic and external forces, to identify to what extend they inﬂuence the vertical global ice load. Upwards forces are positive, negative forces are negative. F I UNCERTAINTY ANALYSES OF THRUST FORCE

This appendix consists of two parts: the sensitivity analyses and the uncertainty analyses of the thrust force. The aim of the sensitivity analyses is to investigated how much inﬂuence a parameter uncertainty (i.e. input) has on the results (i.e. output). This is applied on all inﬂuential parameters. The aim of the uncertainty analyses is to quantify the accuracy of the results, when taking into account the uncertainty of all parameters. In literature uncertainty analyses are often characterized by a Monte Carlo simulation [62]. It is a method to determine the accuracy by varying parameters within statistical constraints. During the analyses a normal distribution is assumed for the parameters when taking into account their uncertainty. Hence, every parameter is presented with a mean value (µ) and standard deviation (σ). About 68% of values drawn from a normal distribution are within one standard deviation away from the mean. Furthermore proximately 95% of the values lie within two standard deviations. The sensitivity Analyses is found in section I.1, the uncertainty analyses is presented in section I.2.

I.1. SENSITIVITYANALYSESOFTHRUSTFORCE This appendix shows the sensitivity analyses of used parameters to the thrust force. The sensitivity is a measure of how much the result will change when one parameter is changed with a certain value. In this case, the result refers to thrust force. A parameters is changed with one standard deviation in this sensitivity analyses.

First step of the sensitivity analyses is taking a parameter, and deﬁning its mean value µ and standard deviation σ. How the standard deviation, i.e. accuracy, was determined or approximated is discusses in section 5.6.1. It is assumed that these values are determined before the sensitivity analyses is started. The second step is to run the model with given parameter three times, one time with parameter equal to its mean (µ), one time with parameter equal to mean minus standard deviation (µ σ), and one time with parameter equal − to mean plus standard deviation (µ σ). The output from the model will consist of three different results: + Resul t(µ), Resul t(µ σ), and Resul t(µ σ). − + These results can be plotted, resulting in three lines (mean, lower boundary, and upper boundary line), which illustrate the sensitivity. However, making conclusions on these ﬁgures alone is hard. In the last step therefore, a sensitivity is calculated as a deviation to the mean, presented in percentage. This enables the comparison between sensitivity between parameters, as the sensitivity is normalized to the mean result. This is elaborated below.

The deviation between upper boundary vs mean, and lower boundary vs mean, changes over time. The question arises how to quantify this deviation to a sensitivity, so that it can easily be compared to other parameters. The approach that is chosen, is to calculate the deviation thus sensitivity at a speciﬁc moment of time t i. This time step may not be arbitrary chosen, it should be at a moment where thrust force is around =

193 194 I.U NCERTAINTY ANALYSES OF THRUST FORCE the maximum (i.e. as is the situation when the vessel is ramming the ridge). This can be illustrated as follows: Ã ¯ ¯ ! ¯Resul tµ σ(t i) Resul tµ(t i)¯ S σ + = − = 1 100% + = Resul t (t i) − · µ = Ã ¯ ¯ ! ¯Resul tµ σ(t i) Resul tµ(t i)¯ (I.1) S σ − = − = 1 100% − = Resul t (t i) − · µ = wi th i 1,2,3,....n = where n is the number of time steps, S σ is sensitivity [%] of parameter when using µ σ instead of µ, and + + S σ is sensitivity [%] of parameter when using µ σ instead of µ. − −

I.1.1. PARAMETERSENSITIVITYTOTHRUST The parameter sensitivity to the thrust force is presented in this section. The sensitivity in [%] is calculated at t 1370 [s]. It must be emphasized that in the analyses it is assumed that the parameter uncertainty occurs = simultaneously at both thrusters. In reality this will probably not occur. However, this has no inﬂuence on the sensitivity, it only inﬂuences the results Resul t(µ), Resul t(µ σ), and Resul t(µ σ). − +

Propeller wake fraction Uncertainty was stated to be 10% its value: 0.04. Hence the model is run with wake factor equal to 0.72, 0.80, 0.88. The results are shown in Figure I.1. • µ 0.80 [-] = • σ 0.08 [-] = • S σ 2.14 [%] − = • S σ 1.18 [%] + = • S 2.25 [%] max =

Diameter of propellers. The propeller diameter of 4.5 [m] has an uncertainty equal to 0.045 [m]. See Figure I.2, and results below: • µ 4.5 [m] = • σ 0.045 [m] = • S σ 13.8 [%] − = − • S σ 15.1 [%] + =

RPM. It was concluded that the RPM uncertainty is equal to approximately 2 [RPM]. See Figure I.3, and results below: • µ var i able = • σ 2 [RPM] = • S σ 3.31 [%] − = − • S σ 3.36 [%] + =

Propeller pitch This uncertainty is approximated to be 0.093 [-] on the conversion factor (10%). See Figure I.4, and results below: • µ var i able = • σ 0.0932 [-] = • S σ 15.1 [%] − = − • S σ 17.0 [%] + =

Water velocity into propeller (vessel velocity). This uncertainty may be different for the whole range of vessel velocities. Therefore, it assumed that uncertainty can be approximated by a percentage set of from its value: 4% of the velocity. See Figure I.5, and results below: • µ var i able = • σ 5 [%] = • S σ 0.81 [%] − = • S σ 0.83 [%] + = − I.1.S ENSITIVITYANALYSESOFTHRUSTFORCE 195

Figure I.1: Uncertainty wake factor

Figure I.2: Uncertainty propeller diameter. 196 I.U NCERTAINTY ANALYSES OF THRUST FORCE

Figure I.3: Sensitivity RPM, σ 2 [RPM] =

Figure I.4: Sensitivity propeller pitch conversion factor, σ 0.093 [-] = I.1.S ENSITIVITYANALYSESOFTHRUSTFORCE 197

Figure I.5: Sensitivity water velocity, σ 5 [%] = 198 I.U NCERTAINTY ANALYSES OF THRUST FORCE

I.2. THRUST UNCERTAINTY A Monte Carlo simulation is applied to calculate the uncertainty of the results. A normal distribution is assumed for the parameters given their mean value (µ) and standard deviation (σ). Uncertainty is assumed to be equal to the standard deviation (σ). Using Matlab, parameter values are calculated using normal distribution randomizer. The thrust force equation is based on physical laws, however, the thrust coefficient is empirically determined. These values have an error itself, but the most significant error will occur due to the calculation of thrust coefficients at pitch ratios and hydrodynamic pitch values that were not measured experimentally. Linear interpolation between data points in the numerical model was assumed, resulting in a error. Looking at Figure 4.10 this error is small, as the data was presented with sufficient small step size for hydrodynamic pitch. An error in thrust coefficient larger then 0.02 seems unrealistic. To be at the safe side, this model uncertainty is therefore implemented as a uncertainty in the thrust coefficient with a standard deviation equal to 0.02.

In Figure I.6 the thrust force over time is presented from 100 simulations. It becomes a bit hard to make sense out of ﬁgures similar to Figure I.6, because it is somewhat unreadable. The ﬁgures become more chaotic with an increase of simulations. Chosen solution for this, is to calculate the standard deviation for the results at each moment in time. Given the standard deviation the 68% and 95% accuracy boundary layers can be visualized (i.e. 1 standard and 2 standard deviations respectively). The result is illustrated in Figure I.7. Given the results, the sensitivity of the thrust force can be approximated, so that it can be used in the sensitivity and uncertainty analyses of the global ice loads. The minimum value and maximum thrust value are taken from the 95% accuracy boundary layers at t=1305 [s]. The minimum value at t=1305 is equal to 1.8 MN, and the maximum value is equal to 2.9 MN. Corresponding sensitivities can be calculated, and equal -24% and 24% for the minimum and maximum value respectively. However, the maximum thrust Oden can deliver is equal to 2.4 MN. Hence it seems that the error is overestimated at least by a factor 1.2 for the maximum thrust value. Carefulness is advised because the thrust force is calculated by an algorithm that uses conversion factors for the full scale data. The maximum thrust was one of the boundary conditions of this algorithm, hence a model thrust higher than 2.4 MN could not occur in this dataset. It therefore is assumed that the maximum thrust is maximum 2.4 MN at any moment of time. Therefore, the minimum thrust value can be taken into account by 76% thrust force.

Figure I.6: Thrust calculated in 100 different simulations, taking into account parameter uncertainties. I.2.T HRUST UNCERTAINTY 199

Figure I.7: Thrust uncertainty based on 1000 different simulations, taking into account parameter uncertainties. Results are visualized by 5 lines: Mean value (expected value), the two adjacent lines present the boundary for 68% accuracy (1 standard deviation), the two outer lines present the boundary of 95% accuracy (2 standard deviations). J SENSITIVITY ANALYSES OF GLOBAL ICE LOADS

In the sensitivity analyses of global ice loads the sensitivity of a parameter to the global ice loads is investigated. To enable quantiﬁcation of this sensitivity, the average value of a parameter µ that is increased with 10% is calculated at a time step t=i:

µ Resul tµ 110%(t) Resul tµ 100%(t) ¶ S 10%(t) × − × 100% (J.1) + = Resul tµ 100%(t) · × where S 10% is the sensitivity under a parameter change of +10%. t=1286 [s] is chosen in the analyses, this is + during the ﬁrst ramming of the ridge. The maximum value of a global ice load is most important. Therefore, the sensitivity of a parameter to the maximum global ice load is calculated as follows: Ã ¡ ¢ ! max Resul tµ 110% S 10%(Fice,max ) ¡ × ¢ 1 100% (J.2) + = max Resul tµ 100% − · ×

J.1. SENSITIVITYTOVESSELDISPLACEMENT The displacement in the numerical model equals 11500 tonnes. For the sensitivity analyses the model is run with 100% vessel displacement, and 110% vessel displacement. The most important parameters that change due to the change of the vessel’s displacement are the rotational inertia, added masses and buoyancy force. Based on this data:

• the sensitivity for the horizontal global ice load S 10%(t 1286) 6.13 [%] + = = • the sensitivity for the horizontal global ice load S 10%(Fice,max ) 6.09 [%] + = • the sensitivity for the vertical global ice load S 10%(t 1286) 0.48 [%] + = = • the sensitivity for the vertical global ice load S 10%(Fice,max ) 0.41 [%] + = −

J.1.1. DISCUSSION An increase of vessel mass by 10% causes the horizontal global ice load to increase with 6%, at t=1286s. Why not 10%? The EM for surge reads:

(m a )x¨ (t) T (t) b x˙ (t) x˙ (t) F (t) (J.3) + 11 1 = x1 − 11,NL| 1 | 1 − x1,ice where T (t) b x˙ (t) x˙ (t) 1.25e 06. The acceleration from measurements at t-1286 gives x¨ 0.15 x1 − 11,NL| 1 | 1 = + 1 = − [m/s2], velocity data gives 2.09 [m/s]. Furthermore, F (m a )x¨ (t) 2.048e 06, and F global,x = + 11 1 = − + x1,ice = 3.4050e 06. + Increasing the mass with 10% increases F to 2.33e 06. Note hereby that the increase of mass also global,x − + inﬂuences the added mass a11. Therefore:

2.33e 06 T (t) b x˙ (t) x˙ (t) F (t 1286) − + = x1 − 11,NL| 1 | 1 − x1,ice =

200 J.2.S ENSITIVITYTO DRAFT 201

Which gives F 3.59e 06, and thereby F /F 3.59e 06/3.4050e 06 1.054. x1,ice = + x1,ice,new x1,ice,old = + + ≈ Why 5.4% and not 6%? This must have to do with the difference between data and state estimate of surge. For example, if the acceleration data would have been -0.1514 instead of -0.1500, then the change of horizontal global ice load would be 6%. This difference in acceleration data is well within the uncertainty of acceleration data.

Conclusion: based on the raw measurement data an increase of vessel mass by 10% causes the horizontal global ice load to increase with 5.4%. Furthermore, the state estimate depends on both the state space model, and the uncertainty of the data. A different mass in the model, can inﬂuence the state estimate, which can explain the difference in increase of 6% vs 5.4%.

J.2. SENSITIVITYTO DRAFT When increasing the draft in the numerical model it will mainly inﬂuence the CP, LWL, and waterline area Awl . Heave damping will be inﬂuenced a little. Added mass for surge and hydrodynamic resistance of surge are calculated using full scale data, and not dependent on draft in the numerical model. Since noted parameters (CP,LWL, and Awl ) are tested separately, the sensitivity to draft is not investigated. However, in the uncertainty analyses the draft will be taken into account, as it delivers the upper and lower boundary values for CP,LWL, and Awl .

J.3. SENSITIVITYTO WATERLINE AREA The waterline area Awl is non linear in the model. For the sensitivity analyses Awl is implemented as 100% and 110% Awl . The sensitivity to the global ice loads is illustrated in Figure J.2. Based on this data:

J.4. SENSITIVITYTO WATERLINE LENGTH Waterline Length is presented accurately in the technical drawing, and therefore has a small uncertainty. However, waterline Length depends on draft and therefore it can vary signiﬁcantly. The main components the waterline length will inﬂuence are rotational inertia, meta centric height, and waterline area Awl . When assuming a 10% increase of waterline length, the rotation inertia increases from 1.20e+10 to 1.45e+10 [kg/m2], and meta centric height from 168 m to 232 m. Awl is non linear, and is implemented as if 10% extra length is added at the middle part of the vessel. The effect of this extra length on Awl is smaller than a 110% Awl . Based on this data:

J.5. SENSITIVITYTOMETACENTRICHEIGHT GML The meta centric height GML is increased with 10%. Results for the sensitivity analyses are as follows:

J.6. SENSITIVITY TO ROTATIONAL INERTIA I y The rotational inertia Iy is increased with 10%. Results for the sensitivity analyses are as follows:

• the sensitivity for the horizontal global ice load S 10%(t 1286) 0 [%] + = = 202 J.S ENSITIVITY ANALYSESOFGLOBALICELOADS

• the sensitivity for the horizontal global ice load S 10%(Fice,max ) 0 [%] + = • the sensitivity for the vertical global ice load S 10%(t 1286) 0.60 [%] + = = • the sensitivity for the vertical global ice load S 10%(Fice,max ) 0.57 [%] + = −

J.7. SENSITIVITYTOADDEDMASSFORSURGE The vessel mass in the numerical model equals 11500 tonnes, the added mass in surge direction is equal to 0.21 times the vessel mass. For the sensitivity analyses the model is run with 100% a11, and 110% a11. Results for the sensitivity analyses are as follows:

• the sensitivity for the horizontal global ice load S 10%(t 1286) 1.06 [%] + = = • the sensitivity for the horizontal global ice load S 10%(Fice,max ) 1.06 [%] + = • the sensitivity for the vertical global ice load S 10%(t 1286) 0 [%] + = = • the sensitivity for the vertical global ice load S 10%(Fice,max ) 0 [%] + =

J.8. SENSITIVITYTOADDEDMASSFORHEAVE The vessel mass in the numerical model equals 11500 tonnes, the added mass in surge direction is equal to 1.0 times the vessel mass. For the sensitivity analyses the model is run with 100% a33, and 110% a33. Results for the sensitivity analyses are as follows:

J.9. SENSITIVITYTOADDEDMASSFORPITCH For the sensitivity analyses the model is run with 100% a55, and 110% a55. Results for the sensitivity analyses are as follows:

J.10. SENSITIVITY TO HEAVE-PITCH HYDROSTATIC STIFFNESS COUPLING TERMS The sensitivity to heave-pitch coupling is investigated by changing the arm between areal center point of Awl and COG. This arm equals 4.575 meters in the numerical model. The model is run with 10% extra arm to identify the sensitivity of k35 and k53. Results for the sensitivity analyses are as follows:

J.11. SENSITIVITYTOPOTENTIALDAMPINGFORHEAVEMOTIONS Results for the sensitivity analyses to potential damping coefﬁcient b33 are as follows:

• the sensitivity for the horizontal global ice load S 10%(t 1286) 0.02 [%] + = = − • the sensitivity for the horizontal global ice load S 10%(Fice,max ) 0.02 [%] + = − • the sensitivity for the vertical global ice load S 10%(t 1286) 0 [%] + = = • the sensitivity for the vertical global ice load S 10%(Fice,max ) 0.44 [%] + = −

J.12. SENSITIVITYTOPOTENTIALDAMPINGFORPITCHMOTIONS Results for the sensitivity analyses to potential damping coefﬁcient b55 are as follows:

• the sensitivity for the horizontal global ice load S 10%(t 1286) 0.02 [%] + = = − J.13.S ENSITIVITYTONONLINEARSURGEDAMPING b11,NL 203

• the sensitivity for the horizontal global ice load S 10%(Fice,max ) 0.02 [%] + = − • the sensitivity for the vertical global ice load S 10%(t 1286) 0.25 [%] + = = − • the sensitivity for the vertical global ice load S 10%(Fice,max ) 0.53 [%] + = −

J.13. SENSITIVITYTONONLINEARSURGEDAMPING b11,NL Results for the sensitivity analyses are as follows:

• the sensitivity for the horizontal global ice load S 10%(t 1286) 0.18 [%] + = = − • the sensitivity for the horizontal global ice load S 10%(Fice,max ) 0.15 [%] + = − • the sensitivity for the vertical global ice load S 10%(t 1286) 0 [%] + = = • the sensitivity for the vertical global ice load S 10%(Fice,max ) 0 [%] + =

J.14. SENSITIVITYTONONLINEARHEAVEDAMPING b33,NL Results for the sensitivity analyses are as follows:

• the sensitivity for the horizontal global ice load S 10%(t 1286) 0 [%] + = = • the sensitivity for the horizontal global ice load S 10%(Fice,max ) 0 [%] + = • the sensitivity for the vertical global ice load S 10%(t 1286) 0 [%] + = = • the sensitivity for the vertical global ice load S 10%(Fice,max ) 0 [%] + = It must be emphasized that the nonlinear heave damping b33,NL was based on a purely ﬂat plate resistance calculation. According to theory, there should also be a residual resistance which cannot be determined without full scale testing. This nonlinear heave damping term can therefore be seen as a lower boundary of b33,NL. It seems justiﬁed to use the nonlinear surge damping to approximate an upper boundary value for b33,NL. For the estimate it is assumed that the relationship is purely depended on the normal area. This normal area for heave motions equals the waterline area Awl . An additional 10% is taken into account,for the 2 upper boundary. Nonlinear surge damping b 2.38 104 in [N s2/m ], with a normal area of B D 11,NL ≈ · · f × f = 31.2 8, where B represents the beam at the bow and Dr the Draft. As a result: × b Awl b33,NL,upper b11,NL 1.1 (J.4) = · B Dr · b × 7 This results in an upper boundary value for the nonlinear heave damping b33,NL equal to 6.34e 10 in [N 2 · · s2/m ].

J.15. SENSITIVITYTO THRUSTFORCE Results for the sensitivity analyses are as follows:

• the sensitivity for the horizontal global ice load S 10%(t 1286) 4.06 [%] + = = • the sensitivity for the horizontal global ice load S 10%(Fice,max ) 5.32 [%] + = • the sensitivity for the vertical global ice load S 10%(t 1286) 0.12 [%] + = = − • the sensitivity for the vertical global ice load S 10%(Fice,max ) 0.17 [%] + = −

J.16. SENSITIVITYTOTHECONTACTPOINTOFICE-VESSEL To investigate the sensitivity to the contact point, the distance between contact point and COG is increased with 10% for both horizontal and vertical location. This can be seen described by:

CP 10%(x) (CP(x) COG(x)) 1.1 COG(x) + = − · + (J.5) CP 10%(z) (CP(z) COG(z)) 1.1 COG(z) + = − · + Results for the sensitivity analyses are as follows:

J.17. SENSITIVITY TO THE HEAVE DATA The data of heave acceleration seems to be of poor quality, illustrated in Figure J.1. The question rises if it can be trusted, and therefore if heave position and heave velocity may be trusted. (The heave acceleration is high pass ﬁltered and integrated twice over time to heave position, and heave velocity is computed with one integration over time of heave acceleration. The heave ﬁlter should remove static and slowly varying errors according to the manual [56].) It is therefore suggested to investigate what an increased uncertainty in heave acceleration, velocity, and displacement imposes. Findings are elaborated below.

Figure J.1: Heave’s acceleration data in [m/s2] during the four ridge impacts

Increasing uncertainty in heave acceleration by a factor 2 results in the following sensitivities

• the sensitivity for the horizontal global ice load S 200%(t 1286) 0 [%] + = = • the sensitivity for the horizontal global ice load S 200%(Fice,max ) 0 [%] + = • the sensitivity for the vertical global ice load S 200%(t 1286) 1.77 [%] + = = − • the sensitivity for the vertical global ice load S 200%(Fice,max ) 2.00 [%] + = − There are some reasons why it is justiﬁed to change the uncertainty of heave data. As the ice-structure impact is quite far from the COG, some vibrations might be damped due to vessel’s structure. Also the vessel has its own vibrations from for example the engines. This should be taken into account as an extra uncertainty. It is stated that the uncertainty of heave acceleration data can be a factor 3 times as high (uncertainty equal to 0.015 [m/s2]). Given the integrations over time, the uncertainty of heave velocity increases by a factor 31, and of heave displacement with 32. This results in a sensitivity of: • the sensitivity for the horizontal global ice load S(t 1286) 0.02 [%] = = − • the sensitivity for the horizontal global ice load S(F ) 0.01 [%] ice,max = − • the sensitivity for the vertical global ice load S(t 1286) 7.38 [%] = = • the sensitivity for the vertical global ice load S(F ) 14.16 [%] ice,max =

J.18. SENSITIVITY TO THE SURGE ACCELERATION DATA Like the heave data, there might be an additional error in the acceleration data of surge. This was tested, and the following sensitivities were identiﬁed: J.19.S ENSITIVITYTO GPS POSITION 205

• the sensitivity for the horizontal global ice load S 200%(t 1286) 0.34 [%] + = = − • the sensitivity for the horizontal global ice load S 200%(Fice,max ) 0.32 [%] + = − • the sensitivity for the vertical global ice load S 200%(t 1286) 0.07 [%] + = = − • the sensitivity for the vertical global ice load S 200%(Fice,max ) 0.10 [%] + = −

J.19. SENSITIVITYTO GPS POSITION As the vessel is at the Arctic there will be less satellite coverage. It is considered that the uncertainty might be higher than given uncertainties. It was found that increasing uncertainty in surge displacement and velocity by a factor 2 results in the following sensitivities

• the sensitivity for the horizontal global ice load S 200%(t 1286) 0.34 [%] + = = • the sensitivity for the horizontal global ice load S 200%(Fice,max ) 0.33 [%] + = • the sensitivity for the vertical global ice load S 200%(t 1286) 0.06 [%] + = = • the sensitivity for the vertical global ice load S 200%(Fice,max ) 0.10 [%] + = Increasing uncertainty in heave displacement by a factor 2 results in the following sensitivities

• the sensitivity for the horizontal global ice load S 200%(t 1286) 0.0 [%] + = = • the sensitivity for the horizontal global ice load S 200%(Fice,max ) 0.0 [%] + = • the sensitivity for the vertical global ice load S 200%(t 1286) 0.34 [%] + = = • the sensitivity for the vertical global ice load S 200%(Fice,max ) 0.33 [%] + =

Figure J.2: Horizontal global ice load under 100% and 110% Awl . 206 J.S ENSITIVITY ANALYSESOFGLOBALICELOADS

Figure J.3: Sensitivity of contact point to the pitching moment. K UNCERTAINTY ANALYSES OF GLOBAL ICE LOADS

207 208 K.U NCERTAINTY ANALYSESOFGLOBALICELOADS ] ] ] 2 2 ] ] 2 ] ] 2 m · / m m / m 2 [kg] [kg] · / m / m ] 2 s / r ad 2 s · N [ kg · [ N s [ [ kg N s N o N [ [ es 33 [N] 6 [ n y 10 5 [m] .3 [m] 9 4 k 000 [t] .25 [m] .31 [m] 7 T 97 [ m · 0.42 [m] 11, mean 33, mean 10 8. · 10 16 10 10 a a 13 25 31 10 27 · · · · · 11 55, mean · 575 25 3 2 a 44 · 09 34 8. 8. 1. 1. per boundary value 093 2 1. 9. 6. 3. 1. up 4.25 ) ] ] 2 2 ] ] ] ] 2 m · / m m ] (eq. 2 [kg] [kg] / m / m ] · 2 s / r ad 2 · N [ kg [N] [ N s [ / m N s N [ kg o T [ [ 2 es 33 6 · 9 n s y 0 [m] 9 4 k 000 [t] .75 [m] .69 [m] 8.3 [m] · 66 [ m · 2.45 [m] 11, mean 33, mean 10 7. 76 10 · 10 10 a a 11 · N 24 30 22 23 · · · · 10 [ 0. 55, mean 575 95 7 8 a 4 61 · 46 0. 4. 0. 0. 665 9. 8 10 wer boundary value 5. · 1. 0. lo 2 1. ≈ 4.25 ) ] ] ] ] 2 ] ] 2 2 ) m m ] (eq. 2/ m · / m / m ] / r ad · 2 s 2 [kg] · N 5.23 [kg] [ N s 6 N s [ [ kg / m 6 N [ [ kg o [m] [m] 2 [ es 33 6 9 10 n s y 10 0 [m] 10 4 k 500 [t] · 10 · 24 [ m · an value · 7.75 [m] 7.86 [m] 10 25 31 8. 10 · 10 3 .5 10 11 · N 10 26 · · 10 16 [ · [N] (eq. 2. me 11 575 60 4 2 T 20 38 273 4. 6. 1. 10 1. 2. 7. · 2 1. ≈ 53 k = 35 k NL NL y 33, 11, I 33 b b 55 b b 33, mean 11, mean 55, mean a a a meter h-heave coupling spring stiffness rust eta centric height aterline length aterline area raft onlinear surge damping onlinear heave damping oanda effect [N] eam at parallel mid-body eam at bow otational inertia for pitch dded mass surge dded mass heave dded mass pitch otential damping heave otential damping pitch essel displacement ncrease uncertainty heave data [N] para D V B B W W M R Pitc N N A A A P P I Th C able K.1: The parameters for the mean value case-study (i.e. normal setting), lower boundary case-study, and upper boundary case-study. These are used in the uncertainty analyses T 209

Figure K.1: Uncertainty analyses horizontal global ice load during ridge impact 1 (t=1280s) and impact 2 (t=1375s) 210 K.U NCERTAINTY ANALYSESOFGLOBALICELOADS

Figure K.2: Uncertainty analyses horizontal global ice load during ridge impact 3 (t=1520s) and impact 4 (t=1640s)

Figure K.3: Uncertainty analyses vertical global ice load during ridge impact 1 (t=1280s) and impact 2 (t=1375s) 211

Figure K.4: Uncertainty analyses vertical global ice load during ridge impact 3 (t=1520s) and impact 4 (t=1640s)

Figure K.5: Uncertainty analyses global ice moment during ridge impact 1 (t=1280s) and impact 2 (t=1375s) 212 K.U NCERTAINTY ANALYSESOFGLOBALICELOADS

Figure K.6: Uncertainty analyses global ice moment during ridge impact 3 (t=1520s) and impact 4 (t=1640s) L RESULTSOFNUMERICALMODEL

L.0.1. GPS DATA CONVERSION Data conversion of GPS data was used to go from a ﬂat earth coordinate system towards a rigid body reference frame. Theory was presented in section 5.3, results are illustrated in ﬁgure L.1.

Figure L.1: GPS data conversion. East and North velocity (ﬂat earth coordinate system), vs horizontal velocities in rigid body reference frame.

L.0.2. THRUSTFORCE The thrust force is calculated using data of the pitch ratio P/D, forward velocity (i.e. surge velocity), revolution rate, thrust coefficient, and hydrodynamic pitch. The revolution rate is approximately constant at 139 rpm. The relationship for the thruster at portside between thrust coefficient, and pitch ratio and hydrodynamic pitch, is illustrated in Figure L.2. The relationship between thrust vs thrust coefficient and forward velocity is shown in Figure L.3. In Figure L.3 can be seen that the maximum thrust of Oden (2.4 MN) is never exceeded. This was one of the boundary conditions of the thrust algorithm.

213 214 L.R ESULTSOFNUMERICALMODEL

Figure L.2: The relationship for the thruster at portside between thrust coefﬁcient vs pitch ratio and hydrodynamic pitch

Figure L.3: The relationship between thrust vs thrust coefﬁcient and forward velocity. Note the maximum thrust of Oden (i.e. 2.4 MN). 215

L.0.3. COANDA EFFECT When the thrusters are reversed they start and pulling the vessel backwards. If this reversed thruster setting occurs long enough, the Coanda effect is observed. Reason for this effect is the under pressure that was caused due to the water particle velocities in the thruster’s vortex. The Coanda effect pulls the vessel downwards and enhances the pitch motions (i.e. the bow upwards motion is enhanced). The vertical loading and the pitching moment due to the Coanda effect, just after the ﬁrst impact, is shown in Figure L.4. The relationship between thrust force, surge velocity, and the vertical loading due to the Coanda effect for the four ridge impacts into the MY ice ridge are visualized in Figure L.5.

One of the reasons the Coanda effect was investigated was to what extend this phenomenon might be responsible for the negative heave motions that are observed at the end of a ridge impact. Based on the results (F 1.7 MN), and taking into account k 26 MN/m, it is concluded that the Coanda effect’s vertical coan ≈ 33 ≈ force is responsible for approximately 0.065 m. The moment due to the Coanda effect Mcoanda has an influence too. The observations and calculations show that the reverse thrusters will cause the aft of the ship to move downwards. However, it must be stressed that calculation are still very simplified. The author does not want to claim that the Coanda effect must be responsible for the negative heave motion’s observation. The results do imply that the effect might have influence and, therefore might be interesting for future research purposes.

Figure L.4: The vertical force and pitching moment that occurs due to the Coanda effect. 216 L.R ESULTSOFNUMERICALMODEL

Figure L.5: The relationship between thrust force, surge velocity, and the vertical loading due to the Coanda effect for the four ridge impacts into the MY ice ridge.

L.0.4. STATE ESTIMATE This section presents:

• State estimate of velocities (surge, heave, and pitch) - Figure L.6 • State estimate of surge velocity with surge velocity data - Figure L.7 • State estimate of surge velocity with surge velocity data, when zoomed in on impacts • State estimate of heave with heave data, when zoomed in on impacts - Figure L.9 • State estimate of heave velocity with heave velocity data, when zoomed in on impacts - Figure L.10 • State estimate of pitch with pitch data during impact 1 - Figure L.11 • State estimate of pitch velocity with pitch velocity data - Figure L.12 217

Figure L.6: State estimate of velocities (surge, heave, and pitch).

Figure L.7: State estimate of surge velocity vs surge velocity data at arbitrary moment of time. 218 L.R ESULTSOFNUMERICALMODEL

Figure L.8: State estimate of surge velocity during ridge impacts.

Figure L.9: State estimate of heave with heave data, when zoomed in on impact 1 and impact 2 219

Figure L.10: State estimate of heave velocity vs heave velocity data at ridge impacts.

Figure L.11: State estimate of pitch vs pitch data 220 L.R ESULTSOFNUMERICALMODEL

Figure L.12: State estimate of pitch velocity vs pitch velocity data, during impact 1 into the ice ridge. 221

L.0.5. ESTIMATEOFTHEGLOBALICELOADS Readers who are only interested in the ramming experiment of the proﬁled MY ice ridge are referred to Figure L.15 and Figure L.16. An overview of the global ice loads alone is presented in Figure L.14. For convenience the loads are illustrated in Figure L.13, this ﬁgure is taken from section 5.5.

Figure L.13: Simpliﬁed example of a forward moving vessel under ice loading and deﬁned pitching moment. Forces: red arrow T = thrust, blue arrow Fx1,ice = horizontal ice load, blue arrow Fx3,ice = vertical ice load. Locations: COG, Contact Point ice, CPT = propeller location.

The horizontal loads that are calculated in the numerical model are presented in Figure L.17. Figure L.17 shows the thrust force, hydrodynamic resistance, horizontal global ice load, and the total global load (i.e. P Px1, see section 5.5) in surge direction. The vertical loads that were calculated in the numerical model are presented in Figure L.18. The vertical forces that are presented include the vertical global ice load, the force due to the heave-pitch motion coupling, the force due to the Coanda effect, and the total global load (i.e. P Px3).

Figure L.14: The global ice loads Fx1,ice and Fx3,ice . Note that direction of forces are as presented in Figure 5.10 in section 5.5, Fx1,ice is positive in aft wards direction, Fx3,ice is positive when upwards.

The identified maximum global ice loads by the numerical model for the profiled MY ice ridge, are shown in Table 6.1. After the penetration of the profiled MY ice ridge, seven other ridges were rammed. Of these eight, two of the ridges were large, with global ice loads around 4.5 and 10 MN for the horizontal and vertical global 222 L.R ESULTSOFNUMERICALMODEL

Figure L.15: The horizontal loads during the four rammings, with forces taken positive when looking along the surge axis forward. Shown are: thrust force, hydrodynamic resistance, horizontal global ice load. ice loads respectively. A overview of the ridge impacts, and their global ice loads resulting from the numerical model, are presented in Table L.1.

Table L.1: The maximum global ice loads during a speciﬁc ridge impact, based on the suggested numerical model. Shown are time step of ridge impact, horizontal global ice load Fx1,ice , vertical global ice load Fx3,ice , pitching moment Mx5,ice , and the impact velocity.

Ridge t [s] Fx1,ice Fx3,ice Mx5,ice v [m/s] [MN] [MN] [MN m] · Profiled MY ridge, 1295 3.8 9.1 -271 2.6 impact 1 Profiled MY ridge, 1377 4.4 11.5 -305 2.1 impact 2 Profiled MY ridge, 1520 4.8 12.9 -355 3.5 impact 3 Profiled MY ridge, 1635 4.2 6.3 -188 3.1 impact 4 ridge 2 1914 2.6 4.1 x x ridge 3 1990 3.0 5.8 x x ridge 4 2082 4.8 5.9 x x ridge 5 2158 2.2 5.1 x x ridge 6 2173 3.0 4.9 x x ridge 7 (large) 2199 3.8 9.3 -327 4.1 ridge 8 (large) 2328 4.9 10.2 -310 3.1 223

Figure L.16: The horizontal loads during the four rammings, with forces taken positive when looking along the heave axis upwards. Shown are: the vertical global ice load, the force due to the heave-pitch motion coupling, the force due to the Coanda effect, and the P total global load (i.e. Px3). 224 L.R ESULTSOFNUMERICALMODEL horizontal global ice load. igure L.17: The horizontal loads that were calculated in the numerical model, with forces taken positive when looking along the surge axis forward. Shown are: thrust force, hydrodynamic resistance, and F 225 due to the heave-pitch motion coupling, and the force due to the Coanda effect. igure L.18: The largest vertical loads that were calculated in the numerical model, with forces taken positive when looking along the heave axis upwards. Shown are: the vertical global ice load, the force F 226 L.R ESULTSOFNUMERICALMODEL

L.1. PITCHINGMOMENT The moments that are taken into account are those due to the ice loading, thrust force, and Coanda effect. Results are illustrated in Figure

Figure L.19: Pitching moments that are calculated in the numerical model.

L.2. PREDICTEDDOWNWARDICELOADATENDOFIMPACT The numerical model identifies a negative heave directly after the impact of the profiled MY ice ridge. This is a repetitive finding for all four impacts. As a results the numerical model calculates a negative ice load. This raises the question what is, how can ice induce negative buoyancy on a downward sloping structure? This anomaly, caused by the negative heave position that is calculated by the numerical model, must be explained. The four possible explanations are as follows:

1. the data is wrong 2. the numerical model is wrong 3. there was a downwards working ice load working on the vessel 4. an other phenomena, not taken into account in the numerical model, is responsible for the negative heave measurements.

Some possible explanation that the author looked into:

1. Heave vibration after gliding back from the beached position on the ridge - First thought was that the motion could be caused by the vessel gliding back into the water after beaching. The heave motion going from -0.18 to -0.3 m during the first impact might be due to the gliding back motion. However, the decrease is observed before it glides back, i.e. something else must be happening (too). Furthermore, it is unrealistic that the a vessel behaved as a negatively damped vessel, i.e. that the heave displacement can increase without adding energy to the system. 2. Coupling motion between heave and pitch motions - This coupling effect was not sufficient to explain the data. 3. Coanda effect - The author likes to emphasize that the Coanda effect has it’s influence as well, although not by far enough to be responsible for the observation of negative heave. However, it can be concluded that the Coanda effect causes a downwards motion of the vessel, together with a negative pitching moment. The Coanda effect will cause the vessel to move downwards around 6 cm based on the simplified model presented in section 5.5.7. L.2.P REDICTEDDOWNWARDICELOADATENDOFIMPACT 227

4. Ventilation and backfill effect during ice-sloping structure interactions - For an sloping structure, the interaction process between structure and ice can be categorized in three main the ice breaking phase, ice rotating phase and ice accumulation phase. Ventilation is a phenomenon that usually occurs at high interaction speeds, i.e. dynamic processes. During ventilation, the freshly broken ice piece is rotated by the ship at relatively high speed, where as a consequence the water has not enough time to flush over the ice piece (Wenjun Lu, 2012) [66]. This leads to a void area above the ice piece which causes large reaction forces from the fluid. At relatively low interacting speed between structure and ice, water tends to backfill the void space above the rotating ice. This process is called the backfill effect. Accordingly to Wenjun Lu (2012) [66], during the ice rotating phase, there exist these two competing phenomena. The ventilation process is stated to increase the global resistance. The backfill process reduces the global resistance. It was concluded that this phenomena could not directly explain the data. 5. Ice cannot hold the vessel’s weight and the ice starts to fail underneath the hull, resulting in a sudden downwards motion - When the vessel is beached, and the ice underneath the hull suddenly fails, will cause the vessel to move downwards. This downwards motion can be measured as negative heave. However, this downwards motions can not be larger than the initial excitation. Making this an unrealistic explanation. 6. The impacts destroys ice ridge’s buoyancy - The vessel hits the ice ridge and the ice that gets destroyed gets pushed sideways and forward. Due to thickness of the MY ridge it can be expected that a large ratio of this destroyed ice cannot escape into the water, hence gets accumulated around the hull or on top of the ridge. It is expected that the bow hull gets enveloped in ice. The ice that is removed decreases the buoyancy of the ridge, making the vessel move downwards.

Figure L.20: The contact point of vessel and ridge, presented for vertical direction (heave) changing in time. BIBLIOGRAPHY

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