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Numerical simulation of ship motion due to waves and manoeuvring

K R I S T I A N K O S K I N E N

Degree project in Naval Architecture

Second cycle Stockholm, Sweden 2012

KTH Engineering Sciences

ABSTRACT This is a master thesis conducted at KTH Centre for Naval Architecture in collaboration with Seaware AB and Wallenius Marine AB. Traditionally simulation of ship motion is divided into manoeuvring and seakeeping. In manoeuvring the plane motion in surge, sway and yaw degrees of freedom for a ship considered moving in calm water is simulated. For increased accuracy the roll degree of freedom can be included as it affects the plane motion. In seakeeping ship motion due to waves at a specific speed and course in 3 to 6 DOF (degrees of freedom), depending on the area of interest, is simulated. The motion that a ship undergoes at sea is however dependent on the interaction between the forces and moments due to waves as well as the forces and moments related to ship manoeuvring. Furthermore, analysis on e.g. methods for counteracting roll motion in waves with rudder movement requires the modelling of forces and moments due to waves and manoeuvring in several DOF. It is therefore desirable to develop a unified model that describes ship motion in several DOF with respect to waves and the effects of manoeuvring. To create a mathematical model, written in MATLAB script, for simulation of ship motion due to waves and the manoeuvring related forces in 5 DOF, a wave induced (Ovegård 2009) and manoeuvrability (Zachrissson 2011) ship motion model were integrated. The code was validated against a linear strip theory, a non-linear ship motion model as well as model experimental results. Results have shown that whilst the heave and pitch motions agree with the models and tank tests the roll motion is seen, in some cases, as creating larger response in comparison to other models and experimental tank tests.

3 Acknowledgements

ACKNOWLEDGEMENTS I would like to first and foremost thank my family, who by now are probably pretty tired of hearing how cool it is to be able to simulate ship motion with the help of mathematical models. Also, I would like to thank my supervisors Anders Rosén (KTH), Erik Ovegård (Seaware) and Carl-Johan Söder (Wallenius Marine AB) for their contribution during the master thesis project. Finally, I would like to acknowledge the members of the master thesis study group Manohara Ranganath Draksharam and Ragnvald Lökholm Alvestad for the interesting and frequently occurring discussions on topics reaching from religion to Einstein’s theory of relativity.

4

Contents Abstract ......................................................................................................................................................................... 2

Acknowledgements ...................................................................................................................................................... 3

1. Introduction .............................................................................................................................................................. 5

2. Introduction to mathematical modelling of ship motions ................................................................................ 7

3. Coordinate systems ................................................................................................................................................. 9

3.1 The global earth bound coordinate system................................................................................................... 9

3.2 The local translative coordinate system ......................................................................................................... 9

3.3 The ship fixed rotative coordinate system .................................................................................................. 10

3.4 The variable .................................................................................................................................................. 10

3.5 The transformation matrixes ......................................................................................................................... 10

3.6 Ship heading with respect to waves ............................................................................................................. 11

4. The unified ship motion model ........................................................................................................................... 11

4.1 The wave induced ship motion model ........................................................................................................ 11

4.1.1 Example in 1 DOF ................................................................................................................................. 12

4.1.2 The non-linear model ............................................................................................................................. 13

4.2 Ship manoeuvrability model .......................................................................................................................... 13

4.3 Integration of the models .............................................................................................................................. 15

4.3.1 Autopilot ................................................................................................................................................... 16

4.3.2 The surge degree of freedom ................................................................................................................ 17

4.3.3 The unified equation of motion ............................................................................................................ 18

5. Validation ................................................................................................................................................................ 19

5.1 Ro-ro vessel (SSPA 2733) .............................................................................................................................. 19

5.1.1 Results discussion .................................................................................................................................... 25

5.2 The box vessel ................................................................................................................................................. 27

5.2.1 Results discussion .................................................................................................................................... 31

6 Discussion and conclusion .................................................................................................................................... 32

References ................................................................................................................................................................... 33

Appendix A - Simulation results and input data ................................................................................................... 34

Appendix B - Verification of the unified model ................................................................................................... 35

Test ship .................................................................................................................................................................. 35

Verification tests .................................................................................................................................................... 35

Verification results ................................................................................................................................................ 38

5 1. Introduction

1. INTRODUCTION

From the beautiful ships that sailed up the river Nile bringing massive carved stone blocks to the building sites of the great pyramids around 2000 BC (see figure 1), to the fleet of Columbus that discovered the new world to the luxuries cruise liner RMS Titanic (see figure 1) that still fascinates our minds to the mighty war ships HMS Hood and Bismarck that ruled the seas in World War II had all been designed to serve a certain purpose.

Figure 1. a) Ancient Egyptian river boat. b) The RMS titanic.

A modern ship design procedure can be seen as an iterative process (Milchert 2000) where requirements, regulations and rules including, amongst others, analysis of seakeeping and intact stability criteria issued by the International Maritime Organization (IMO) (Rosén 2011) and ship design solutions are compared in order to achieve an optimum solution. The combination of the ever growing population together with high demand for goods and increasing oil prices have resulted in the design of merchant ships that are optimized for minimum resistance and maximum load capacity. The stability criteria issued by the IMO for merchant ships are based on statistical studies made in the middle of the twentieth century (IMO 2008) of righting lever curves for the ship in calm water as well as the areas under these. Modern merchant ships, seen today, differ however significantly from the ships that were used in the statistical studies upon which the IMO criteria are based, both in size and hullform (Kluwe 2009). Furthermore, the criteria do not include the dynamic intact stability which is of great importance for modern merchant ships. Accidents during the last years have shown that the existing intact stability criteria do not provide sufficient safety margins. An example of this is the M/S Finnbirch (see figure 2) that capsized and sank in November 2006 between the Swedish islands of Öland and Gotland. An unfavourable course and speed in combination with rough sea conditions with high and long waves resulting in partly loss of stability causing large rolling motions followed by shifting cargo have been seen as the cause of the accident (SHK 2008).

Figure 2. The M/S Finnbirch.

As a consequence of accidents as the above, steps taken towards the understanding and development of the field of ship dynamic stability have being made by different research groups such as the research group formed by the Centre for Naval Architecture at the Royal Institute of Technology (KTH) together

a) b)

6 1. Introduction

with Seaware and Wallenius Marine AB, in which this master thesis project is a part of. Moreover, the IMO is working on the development of a second generation intact stability criteria where several dynamic stability related aspects, such as pure loss of stability, parametric roll and surf-riding/broaching are taken into consideration. As phenomenon related to stability alternations in longitudinal and quartering waves, such as, parametric rolling and pure loss of stability cannot be foreseen by today’s stability standards, it is of great importance to develop tools that enable the prediction of ship motion with respect to different sea states. Certainly, prediction of ship motion can be done in many different ways. Testing several full scale ships would unquestionably give the best estimate, but would of course be too costly and practically impossible. Another way of analysing ship motion is by testing ship models in wave basins. Although a better option than testing full scale ships, it is often time consuming and costly. A third option is the prediction of ship motion by computer simulation. Computer simulations are done with respect to simplified models. These models represent physical reality to a degree that depends on the simplifications and assumptions made. Traditionally simulation of ship motion is divided into manoeuvring and seakeeping (Fossen 2005). In manoeuvring the plane motion in surge, sway and yaw degrees of freedom for a ship considered moving in calm water is simulated. For increased accuracy the roll degree of freedom can be included as it affects the plane motion. In seakeeping ship motion due to waves at a specific speed and course in 3 to 6 DOF (degrees of freedom), depending on the area of interest, is simulated. The motion that a ship undergoes at sea is however dependent on the interaction between the forces and moments due to waves as well as the forces and moments related to ship manoeuvring. Furthermore, analysis on e.g. methods for counteracting roll motion in waves with rudder movement requires the modelling of forces and moments due to waves and manoeuvring in several DOF. It is therefore desirable to develop a unified model that describes ship motion in several DOF with respect to waves and the effects of manoeuvring. During the last decades, several unified models describing ship motion due to manoeuvring in waves have been developed. A nonlinear unified state-space model for ship manoeuvring and control in seaway is presented by Fossen (2005), where the unified model is obtained by superimposing a manoeuvring and seakeeping model. The potential and viscous damping terms in the model established by Fossen (2005) are presented by a so called state-space approach where instead of using the convolution integral which is used to derive the damping forces in classic theory a linear reduced-order state-space model is used to approximate the damping forces. Thus, achieving the standard representation used in feedback systems. Regarding the wave excitation forces, they include the Froude-Krylov and diffraction forces (1st order wave loads) as well as the wave drift force (2nd order wave loads). Hua and Palmquist (1995) describe a time domain ship motion simulation program (SMS) where two mathematical models, a wave induced model and a manoeuvrability model, are incorporated into a unified model. The unification of the models is obtained by assuming that no interference between the turning motion introduced by ship manoeuvres and the velocity potential, diffraction or radiation waves is attained, thus making it possible to superimpose the given models. The damping forces, in contrast to the method described in Fossen (2005) are derived to the time domain through the convolution integral. As to the wave exciting forces, only the Froude-Krylov and diffraction (1st order wave loads) are included, where the Froude-Krylov force is treated nonlinearly. Furthermore, Min-Guk and Yonghwan (2011) introduced a unified model for ship manoeuvring in waves where the interaction between the manoeuvring and seakeeping model is done similar to the unified models presented above. That is, the seakeeping and manoeuvring problems are coupled and solved simultaneously. The emphasis of the program is on the 2nd order wave drift obtained by using a direct pressure calculation method which is seen as an important factor in the ship trajectory calculations. The aim of the present master thesis has been the expansion and development of a 3 DOF nonlinear seakeeping model (Ovegård 2009) by integrating a 4 DOF manoeuvrability model (Zachrissson 2011)

7 2. Introduction to mathematical modelling of ship motions

resulting in a 5 DOF model in order to improve the simulation accuracy of the seakeeping model, in particular for relative wave directions differing from head and following seas. The report starts with a description of the general mathematical model for ship motion in six degrees of freedom followed by the description of the coordinate systems used to describe ship motion, ship geometry, forces, moments and waves. The report then continues with the theoretical description of the ship motion models and how they were integrated. Finally, the report ends with the validation of the unified model and discussion and conclusion chapter.

2. INTRODUCTION TO MATHEMATICAL MODELLING OF SHIP MOTIONS Physical reality is far too complex to mimic in full detail. However, by identifying what the major influences of a physical phenomenon are, a simplified model of reality can be made. A ship sailing at seas is influenced by numerous different variables that affect the motion of the ship. Possible shifting cargo, motion due to winds, water streams in the ocean and possible effects due to shallow water are a part of the factors behind the motion of a ship. Forces due to winds, water streams and shallow waters are however small compared to the forces due to waves and ship manoeuvring and are therefore not included in the model. Shifting cargo, which can have major effects on the stability of a ship, depending on the mass and distance that the cargo has shifted in comparison with the ship geometry and weight are not taken in consideration. Furthermore the deformation of the ship is not included in the current model. The ship hull is considered as a rigid body. The motion of the ship in the different degrees of freedom is described by Newton’s second law, also called the equation of motion,

(1)

where describes the ship motion in the surge, sway and heave degrees of freedom for j=1,2,3 ,see

figure 3, and roll, pitch and yaw degrees of freedom for j=4,5,6 .

Figure 3. Sign convention for the six degrees of freedom (Hua and Palmqvist 1995).

8

is the mass matrix including the mass in the translatory degrees of freedom as well as mass moment

of inertia in the rotational degrees of freedom at the centre of gravity of the ship

(2)

and is the force vector describing the forces for j=1,2,3 and moments for j=4,5,6 affecting the motion

of the ship. As the hydromechanic forces and moments affecting the motion of the ship and thus its seakeeping and manoeuvring behaviour are proportional to the pressure distribution of the fluid over the hull, propeller, rudder and probable fin Garme (2011) the left hand side of equation 1 can be written as

∑∯

∑∯

(3)

where p is the pressure distribution on the hull, propeller, probably fin and rudder, is the unit normal, the position vector from the centre of gravity to dS and S the wetted surface of the structure in question. Applying the principle of conservation of mass and Newton’s laws i.e. the conservation of momentum on a fluid element, resulting in the Navier-Stokes and continuity equations, the velocity and pressure fields of the fluid can be described. By assuming that the viscous effect in comparison to the gravitational is negligible as well as assuming that the fluid is irrotational the velocity field of the fluid can be described as a gradient of a scalar field i.e. the velocity potential (Lewandowski 2004). By further assuming that the fluid is incompressible the continuity equations can be expressed as the Laplace equation

(4)

for the velocity potential and the conservation of momentum as the Bernoulli equation

(5)

where is the density of the fluid, the fluid pressure, the velocity potential, the gravitational

constant and the water depth. Due to the linearity of the Laplace equation individual solutions can be superimposed, thus satisfying the Laplace equation. Before going into a detailed description of how these forces and moments are modelled a description of the different coordinate systems is needed.

9 3. Coordinate systems

3. COORDINATE SYSTEMS

In order to describe the translative and rotative motions of a ship, the forces and moments involved and the position of the ship in the wave system three coordinate systems, transformation matrixes and a position vector is needed. The different coordinate systems used in the model are:

the global, earth fixed, coordinate system

the local, translative coordinate system

the local, ship fixed, rotative coordinate system

3.1 THE GLOBAL EARTH BOUND COORDINATE SYSTEM

The global earth bound coordinate system , seen in figure 4, with its origin at the starting point of a

simulation describes the platform on which the ship motion is simulated. The -plane describes the still

water plane where the -axis points at the main direction of the wave propagation and the -axis points upwards i.e. in the opposite direction of gravity.

Figure 4. The coordinate systems and the position vector s (Hua and Palmqvist 1995).

3.2 THE LOCAL TRANSLATIVE COORDINATE SYSTEM

The local, translative, coordinate system , with its origin in the centre of gravity, CG, of the ship, see

figure 3 and 4, only translates with the ship. The -axis points in the main direction of the ship, the -axis

at the port side and the -axis is parallel to the -axis. The equation of motion is described in this coordinate system.

10 3. Coordinate systems

3.3 THE SHIP FIXED ROTATIVE COORDINATE SYSTEM

The local, ship fixed, coordinate system , seen in figure 4, with its origin in the centre of gravity,

CG, of the ship only rotates with the ship. The - axis is parallel to the keel line of the ship, the - axis

points to the port side of the ship and the -axis points upwards. The geometry of the ship is expressed in this coordinate system.

3.4 THE VARIABLE

The vector , used as an extra variable to ease the transformations between the different coordinate systems , including the position related to the constant and instant motions, is written as,

[

]

(6)

where is a vector including the instant motions. U is a constant speed that represents the speed in the

surge direction of the ship in the and coordinate systems. This simplification is based on

that the yaw angle is kept small. VCG the vertical centre of gravity, T the ship draught and the angle

between the and coordinate systems.

3.5 THE TRANSFORMATION MATRIXES

As the equations of motion are expressed in the forces and moments expressed in differing

coordinate systems are to be transformed to . Vector transformations between and

are done with the transformation matrix

[

]

(7)

where the rotative Euler angles

[

]

define the angular difference between the two coordinate systems, where is the yaw angle, the pitch

angle and the roll angle. The Euler angle notation is equal to the vector notation used in equation 1 to express the ship rotation.

11

With the help of the transformation matrix in equation 7 a vector can be transformed from to

by

and due to the orthogonality of the transformation matrix the vector can be transformed back by

.

3.6 SHIP HEADING WITH RESPECT TO WAVES

The relative angle between the waves and the ship, , in the mathematical model is defined as 00 following seas and 1800 for head seas (illustration see figure 5).

Figure 5. Definition of the relative wave angles (Perez 2005).

4. THE UNIFIED SHIP MOTION MODEL

Based on the assumption that no major interference between the turning motion of the ship and the incident velocity potential, radiation and diffraction waves arise the mathematical model for describing ship motion in waves in the sway, heave, roll, pitch and yaw degrees of freedom is derived and implemented in MATLAB by integrating the wave induced ship motion model (Ovegård 2009) and the ship manoeuvring model (Zachrisson 2011).

4.1 THE WAVE INDUCED SHIP MOTION MODEL

Based on the linear potential flow theory the motion of a ship in waves is modelled by separately analysing the hydromechanical radiation and restoring forces as a rigid body oscillates in still water and the Froude-Krylov and diffraction exciting forces produced by waves on a rigid body. A more extensive

12 4. The unified ship motion model

and thorough explanation concerning the methods of calculation from a numerical point of view is found in Ovegård (2009). As the model is able to simulate non-linear behaviour, such as parametric rolling, a part of the forces are modelled non-linearly. Before specifically explaining what parts of the forces are modelled linearly and non-linearly the physics behind the different forces are explained by looking at wave induced ship motion in 1 DOF.

4.1.1 Example in 1 DOF

The linear potential flow theory derived by assuming that the wave amplitudes are small compared to the

wave length resulting in the exclusion of the

term in the Bernoulli equation together with the

assumption the wave and ship motions are much smaller in comparison to the ship dimensions, thus satisfying the given boundary condition that no fluid is transported through the hull the motion of the ship can be divided into:

Hydromechanical forces and moments induced by the harmonic oscillation of the rigid body in

still water. (see figure 6)

Wave exciting forces due to waves on a restrained body. (see figure 6)

Figure 6. Division of ship motion due to waves in 1 DOF (Journèe and Pinkster 2002).

The hydromechanical forces and moments due to the harmonic oscillation of the body are traditionally divided into radiation and restoring forces. The radiation forces, referring to the waves created by the oscillating body moving radially from it, include forces due to the movement of the added mass of water particles that are set into motion as the hull oscillates. Furthermore, the radiation forces include damping forces proportional to the velocity of the body motion. The restoring force is proportional to the change of position and behaves similarly to a spring force. The exciting forces include the Froude-Krylov forces and the diffraction forces, where the Froude-Krylov forces are the forces on the restrained body due to the undisturbed wave potential. As a wave comes in contact with a body it gets disturbed or diffracted and is modelled by the diffraction forces. The basic idea and physical meaning described above for the example describing motion in waves for 1 DOF is valid for the non-linear model. The difference is that the motion is in 5 DOF and that the Froude-Krylov, restoring and roll-damping forces are modelled non-linearly.


4.1.2 The non-linear model

Now, as can be understood a ship in physical reality is not restrained as in the example given above, but rather following the motion of the encountered waves in any direction to an extent dependent on the hull geometry, speed and position of the centre of gravity. This is taken into consideration, in particular in the Froude-Krylov and restoring calculations, which are obtained through integration of the static and dynamic fluid pressure in the normal direction over the momentary wetted surface of the ship hull. The damping forces in the radiation forces are divided into memory and viscous roll-damping forces. The division is done due to the difference of which the damping consists of. Viscous effects are the dominant factors behind the roll-damping in comparison to the memory forces which consider damping due to loss of energy needed to create the radiating waves. The viscous roll-damping forces are obtained by a semi-empirical method according to (Ikeda, Himeno, Tanaka (1978)) where the damping coefficients are obtained through a roll decay test. The frequency dependent memory forces are obtained by transforming the damping coefficients from the frequency domain to the time domain according to Lewandowski (2004) with the help of the inverse Fourier cosine transform function and thus obtaining the memory function which is then integrated over a predetermined timespan. The damping and added mass coefficients, with the exception of the roll-damping coefficients, are

derived according to de Jong (1973) with Lewis forms, which are integrated along the -axis according to (Salveson, Tuck, Faltinsen, 1970). The exciting diffraction forces are modelled, as in the case of the memory forces, as a function of the hull geometry dependent added mass and damping coefficients and a moving entity. The difference between these models is that the moving entities in the diffraction model are the wave water particles as supposed to the oscillating hull. In view of equation 1 the equation of motion is then written as,

(8)

4.2 SHIP MANOEUVRABILITY MODEL

In comparison to the wave-induced ship motion model, where the ship motion is modelled in waves, the manoeuvrability model by Zachrisson (2011) describes ship motion in calm water with respect to the forces and moments obtained from the hull and rudder. The degrees of freedom in the model include the surge, sway, roll and yaw. The velocity in surge is kept as constant in the unified model and therefore the force in surge that is modelled in the manoeuvrability model is left out. The coordinate system used in the manoeuvring model is plane i.e. two dimensional. Surge points at the absolute heading of the ship, sway at the star board side and yaw downwards in the direction of gravity. By leaving out the surge the equations of motion is expressed as,


[ ]

[ ]

(9)

(

)

where is the mass of the ship and the mass moment of inertia in the yaw degree of freedom.

and are the added mass and added moment of inertia in the respective degrees of

freedom. The added mass and mass moment of inertia are obtained according to Hooft and Pieffer (1998) and are based on empirical and semi-empirical formulas that are proportional to the hull geometry, block coefficient and draught.

is the sum of the forces in sway and can be written as

(10)

where and

are the hull and rudder forces. The hull forces are derived through semi-empirical formulas based on Annon (2002) with modifications made seen in Zachrisson (2011). The hydrodynamic coefficients in the semi-empirical formulas are based on (Lee T. et al 2003).

The term

is the force in sway due to the velocity in surge and yaw rate. The rudder model is

based on a flap rudder from Becker Marine Systems. The method for calculating the rudder forces is done according to Kijima, et al. (1993).

is the sum of the moments in yaw and is written as

(11)

where and

are the moments in yaw with respect to the ship hull

and rudder. Similar to the forces the

moments are derived from semi-empirical formulas described in Zachrisson (2011). The equations of motion in the manoeuvrability model are located at the centre of gravity, but the forces

are described at the mid ship position which results in the term - , where is the lever arm

from the mid-ship position to the centre of gravity. The lever arm , seen in figure 7, is measured from

the mid-ship position of the ship and positive ahead.


Figure 7. Rudder coordinate system definition.

is the sum of the moments in roll, obtained by the product of the sum of the hull and rudder

forces and their respective lever arms and , seen in figure 8, where is the vertical distance between

the centre of gravity and the centre of lateral rudder force and is the vertical distance between the centre of gravity and the centre of lateral hull force. d, in figure 8, is the mean draft and KG the distance from the keel to the centre of gravity.

Figure 8. The rudder and hull forces and the respective lever arms.

4.3 INTEGRATION OF THE MODELS

In this part of the report the integration process of the two mathematical models i.e. the wave-induced non-linear model and the manoeuvring model is explained. From a programing structural point of view, the two simulation models can roughly be described as consisting of:

A main part where initial values, such as, ship geometry, load case and sea state are initiated

Functions describing the different forces with respect to given input values

An equation of motion

A numerical solver.


The integration of the two models required the establishment of an EOM that included the forces and moments from the wave-induced and manoeuvring model, a unified numerical solver, a unified coordinate system, coordinate transformation between the two models and an autopilot as well as a unified main script. As the EOM and the numerical solver in the wave-induced model were already prepared for simulating motion in 6 DOF, namely the matrix size of the EOM, they were chosen as the basis for the unified solver and EOM. Furthermore, the coordinate systems in the wave-induced model, was chosen as the base for the unified coordinate system as it consisted of a well-established and complex method of keeping track of the ship position with respect to given waves. With the EOM, numerical solver and coordinate systems chosen the two models could be integrated. The integration of the two models was done by connecting the manoeuvring model via its EOM function, after required modifications, to the EOM function of the wave-induced model by simply calling the manoeuvre part as a regular function. The modifications included the change of output values from velocities and accelerations which are normally the input values to a solver to forces, coordinate system transformation and the establishing of an autopilot. The integration of the two models is also described in the flow chart seen in figure 10. Although the coordinate systems where the EOM are defined in the two models differ in that the z and y-axis point in opposite directions (see figure 9 ), no transformation is needed, with the exception of the roll degree of freedom. The coordinate systems in the two models are both defined according to the right hand rule and therefore the relations between the degrees of freedom in the plane are equal. Hence, no translation is made for surge, sway or yaw.

Figure 9. a) Definition of the coordinate system in the EOM. a) wave-induced model, b) manoeuvrability model.

Regarding the roll degree of freedom, data transformation is needed. The roll-moment is modelled as the

product of hull, rudder and fin forces and a lever arm with the constant distance of

. As the

forces point in the opposite direction, with respect to the x-axis, the roll-moment is multiplied with -1 in order to obtain the correct sign.

4.3.1 Autopilot

As the aim is to simulate ship motion in waves with respect to a certain wave direction an autopilot is needed. The autopilot steers the rudder of the ship and is written as

(12)

where is the angle of the rudder, is the yaw angle and

the yaw rate. and are proportional

to the rudder area. The negative signs in front of the terms in equations 12 are due to the definition of the rudder angle with respect to the ship coordinate system. The relation between the yaw and rudder angle, as defined in Zachrisson (2011), is that a positive rudder angle will turn the vessel to the starboard side


which is equivalent to a positive yaw angle in the manoeuvrability model (see figure 7). However, in order for the autopilot to work the yaw angle is to be opposed which is accomplished by the negative signs.

4.3.2 The surge degree of freedom

Due to the complexity of simulating the surge force or surge-induced forces, which would have required the development of 3-D hydrodynamic models, it was left out of this project. Instead the velocity in surge

is kept as constant. This was done by simply setting the acceleration in the surge to zero,

,

keeping a given initial velocity in surge constant. To avoid any confusion concerning the notation and purpose with respect to the velocity in surge which is used in the simulation, a clarification is made. The

notation U represents the constant velocity in surge that is used in the unified model, whereas is the difference in surge velocity with respect to a constant predetermined speed, caused by waves or other hydromechanical forces. In conclusion, the real or total velocity in surge is then given by the sum of these

velocities, . In addition to the changes made in order to achieve a working unified model, a unified main function was established where given simulation related initial could be initiated. .

Unified model

Wave induced model Manoeuvrability model

User

Hull geometry

Sea state Ship

condition

Figure 10. Unified model flow chart

Ship condition

Propeller and rudder

Added mass and damping coeff.

Results

Solver

Diffraction forces and moments

Memory forces and moments

Roll-damping forces and moments

Motion

Hull forces and moments

Rudder forces and moments

∑

Autopilot

Froude-Krylov forces and moments

Restoring forces and moments

EOM


4.3.3 The unified equation of motion

Including new couplings between the motions, degrees of freedom and an autopilot the unified equations of motion is written as,

[ ]

(13)

where the viscous roll-damping, memory, restoring, diffraction and Froude-Krylov are the wave-related forces and the rudder and hull the manoeuvrability model related forces.

4.3.3.1 The added mass matrix

The added mass matrix, expressed by

[

]

(14)

includes the added mass for and the added mass moment of inertia for . The

coupling terms for describe the added mass of the coupled degrees of freedom. The mass matrix

can be seen in equation 2.

Similar to the approach, which is taken in (Hua and Palmqvist 1995), the elements in the added mass matrix are calculated according to (de Jong 1973) with Lewis forms, as described in the wave ship induced model. This approach is seen as the more accurate method in comparison to the semi-empirical method used in the manoeuvring model. Moreover, the strip method, which is used in order to derive the added mass is frequency dependent in contrast to the semi-empirical method which depends on ship main particulars not considering the dynamic part.

4.3.3.2 The coupling

Although the vertical and lateral motions are mathematically decoupled in the model, as can be seen by looking at the elements in the added mass matrix, position dependent coupling between the lateral and vertical motion is established through the nonlinear approach of deriving the Froude-Krylov forces. The position dependent coupling is an important part in the models ability to simulate ship dynamic related phenomenon like parametric rolling.

19 5. Validation

5. VALIDATION

The unified mathematical model was validated, i.e. determining the level of accuracy of the model by comparing response amplitudes obtained from ship model tests, the already established wave-induced nonlinear ship motion model and a linear ship motion model with respect to different sea states and vessels. Two different vessels were used in the validation process, a box shaped vessel and a ro-ro type vessel. The verification of the model, which was done prior to the validation, is found in appendix B. In order to compare the response amplitudes obtained from the different models the response amplitudes in the translative degrees of freedom were normalized with the wave amplitude, a and the rotative degrees of freedom by

(15)

where a is the wave amplitude and

the wave number.

5.1 RO-RO VESSEL (SSPA 2733)

The model seakeeping experiments, conducted at the ship testing institute SSPA are based on differing speed, wave amplitude and frequency and heading angle with respect to the encountered waves. The model ship used in the experiments is based on a 135m, 11276 tonne ro-ro vessel and is scaled by 1:35. An accurate description of the test set-up and procedure is found in Garme (1997). The ship test main particulars are seen in table 1 and the test conditions are found in table 2. The normalized simulation results can be found in figures 9-16 and the results including response amplitudes are found in Appendix A. The mean wave amplitudes used as the simulation input values seen in table 2, where obtained by taking the mean of the wave amplitudes measured at the model ship aft and fore part in connection with the model experiments. These values, together with the result values are found in Appendix A. The hydrodynamic coefficients in the linear method are calculated according to the Lewis method where the roll damping of 10% of the critical roll damping was used. The viscous roll-damping forces, used in the 3 and 5 DOF models, are obtained by a semi-empirical method according to (Ikeda, Himeno, Tanaka (1978)), where a part of the damping coefficients are obtained through a roll decay test. The autopilot

coefficients, and , were chosen with respect to the ability of the ship to steer in waves with the highest frequency and waves used in the simulations. The rudder and propeller size were chosen with respect to steering ability and approximation throw a picture of the model ship and given scale found in Garme (1997). Table 1. Test ship main particulars.

Entity Value

Displacement 11276 ton

Draught 5.495 m

Trim 0

VCG 12.22 m

LCG 63.82 m

GM 1.54 m

Radius of inertia, roll 8.07 m

Radius of inertia, pitch 37 m

Radius of inertia, yaw 37 m

Rudder span 2 3.35 m

Rudder chord 2 2.025 m

Propeller 2 2 m

20

Table 2. Test conditions.

Test # Froudes number Heading Wave frequency [rad/s] Mean wave amplitude [m]

22 0.21 180 0.35 2.345 23 0.21 180 0.55 2.765 30 0.21 180 0.75 3.019 35 0.072 180 0.35 2.783

36 0.072 180 0.55 2.319 39 0.072 180 0.75 2.048 40 0.21 150 0.35 2.555 41 0.21 150 0.55 1.759 43 0.21 150 0.75 2.424

45 0.21 120 0.35 2.223 47 0.21 120 0.55 2.013 48 0.21 120 0.75 2.266 49 0.21 90 0.35 2.153 50 0.21 90 0.55 2.074

51 0.21 90 0.75 2.153 53 0.21 60 0.35 2.266 54 0.21 60 0.55 2.31 55 0.21 60 0.75 2.433 56 0.21 30 0.35 2.424

58 0.21 30 0.55 2.188 59 0.21 30 0.75 2.354 60 0.21 0 0.35 2.643 61 0.21 0 0.55 2.363 62 0.21 0 0.75 2.013

21

Figure 11. Heave, roll and pitch response as a function of frequency.


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Test 35 36 39

180o Linear method

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Test 35 36 39

180o Linear method

180o 5 DOF

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Test 35 36 39

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22



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Test 40 41 43

150o Linear method

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150o Linear method

150o 5 DOF

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Test 40 41 43

150o Linear method

150o 5 DOF

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Test 45 47 48

120o Linear method

120o 5 DOF

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Test 45 47 48

120o Linear method

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Test 45 47 48

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120o 5 DOF

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23



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Test 49 50 51

90o Linear method

90o 5 DOF

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Test 49 50 51

90o Linear method

90o 5 DOF

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Test 49 50 51

90o Linear method

90o 5 DOF

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Test 53 54 55

60o Linear method

60o 5 DOF

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60o 5 DOF

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60o 5 DOF

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24



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Test 56 58 59

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o Linear method

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Test 56 58 59

30o Linear method

30o 5 DOF

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Test 56 58 59

30o Linear method

30o 5 DOF

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Test 60 61 62

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25

5.1.1 Results discussion

In this part of the thesis the motion responses in heave, roll and pitch obtained by the 5 DOF and 3 DOF models, linear method and the model experiment are analysed and discussed.

5.1.1.1 Test 22 23 and 30

Looking at figure 11 it is seen that the response results in heave and pitch are relatively similar, although a small differences between the mathematical models and model experiments can be found. Due to the relatively small response differences the reasons behind these could be found in the difference of the shape of the actual hull and the shapes used in the mathematical models, the difference in the DOF, measuring accuracy during the model experiments or numerical accuracy i.e. the integration step length of the solver. Now, looking at the roll motion in figure 11 no significant difference between the response results for test 22 and 30 are found. Looking at test number 23, on the other hand, it can be seen that the response is much larger in comparison to the mathematical models and the model experiment response. Regarding the response difference between the 5 DOF and the linear method it is seen as a result of the influence of the non-linear method in which the Froude-Krylov and roll damping forces are modelled. The difference in response between the 3 and 5 DOF however is seen as surprising, since the nonlinear parts of the two models are identical. On the other hand, the 5 DOF model forces do not only consist of the forces due to waves but also the rudder and hull forces. The large response from the 5 DOF is seen as parametric roll. Regarding the roll response, it is seen that the results are comparatively similar and equal to zero for test 22 and 30, with the exception of the model experiment response where a miner reaction can be detected. This small response is however no surprise, model experiments conducted in wave basins, where the vessel is allowed to move freely in 6 DOF and steered with an actual rudder as is the case with the model experiment in question, are bound to be influenced by the rudder movement. Thus, resulting in a rolling motion.

5.1.1.2 Test 35 36 39

No significant differences in the heave and pitch response are detected in figure 12. The explanation behind the response coherency between the mathematical models in the heave and pitch is seen as partly due to the relative wave heading of 1800, corresponding to head seas, which minimizes the coupling between roll, heave and pitch motions obtained through the method in which the Froude-Krylov forces are modelled in the 3 and 5 DOF models. The similarity between the model experiment results and the mathematical models for test 35 and 36 are however seen as surprising, since the standard deviation for test 35 and 36 with respect to the model heading are 1180, 1700 and 0.260 for test 39. Consequently, it could be said that the significance of test 35 and 36 with respect to the validation is very low. However, they were the only tests done in this particular speed and heading. On the other hand, test 39 has a very low standard deviation and is therefore seen as a valid value to compare to. The standard deviation and real model heading can be found in Garme (1997). The roll response seen in figure 12 for test 39 is classified as parametric according to Garme (1997) and is as such a good example of the difference between the linear and nonlinear models ability to simulate nonlinear phenomenon such as parametric rolling. The response difference between the 3 and 5 DOF and the model experiment is quiet large. This indicates that the roll-damping forces react differently in comparison to the model experiment.

26

5.1.1.3 Test 40 41 and 43

Looking at figure 13 it can be seen that no significant difference between the response in heave and pitch is obtained between the mathematical models. However, difference between the model experiment and the 5 DOF can be seen for test 43 and 41 in heave as well as for test 40 and 41 for pitch. The reason behind the difference could seen in the manner in which the waves where measured i.e. at the aft and stern part of the experiment model ship, which does not necessarily replicate the real wave encountered by the model ship. Regarding the roll response it can be seen that for test 41 and 43 no significant difference between the mathematical models or the model experiment is detected. The roll response for test 40, on the other, shows a large difference between the 5 DOF model, the model experiment and linear method whilst the 3 DOF response is close to the 5 DOF. The small response difference between the 3 and 5 DOF models can be seen as the difference due to the additional hull and rudder forces with respect to roll. The reason behind the large roll response obtained with the 5 DOF is mainly seen as the result of how the roll-damping forces are modelled. This, since the pressure integration associated with restoring and Froude-Krylov forces can be seen as reliable, hence the precise results in heave and pitch.

5.1.1.4 Test 45 47 and 48

Looking at figure 14 it is seen that the response results in heave and pitch are relatively similar, with the exception of test number 45 for pitch, where a small difference between the mathematical models and model experiments can be seen. Due to the relatively small difference the reason behind it can be said to fall into the category of measuring accuracy associated with the model experiments. Regarding the small response difference between the 5 DOF and the linear method it is seen as a result of the influence of the non-linear method in which the Froude-Krylov forces are modelled. In particular, the coupling effects, between the vertical and lateral motions. As to the cause of the difference between the 5 DOF and the 3 DOF models it is seen as the influence of the additional forces in sway, roll and yaw from the manoeuvrability model as well as the small change in the ship heading as the autopilot turns the rudder in order to maintain a steady course. Now, looking at the roll DOF in figure 14 it can be seen that for test 45 both the 3 and 5 DOF models response is far greater in comparison to the response obtained by the linear method and the model experiments. By analysing the curve obtained by the linear method it is noticed that a peak or global maximum response is attained at the corresponding wave frequency of 0.35 rad/s. This is interpreted as the Eigen frequency of the roll motion, hence the Eigen frequency is a linear, frequency dependent phenomenon and can as such be detected or simulated with the linear method. Although this explains the plausible cause of the response peak it does not answer the question of why the difference is of such extent. To answer this question the difference of the models in roll is addressed. The 5 DOF simulates ship motion with respect to time whereas the linear method simulate with respect to frequency. Furthermore, the Froude-Krylov and restoring forces are modelled nonlinearly as well as the roll damping forces in the 5 DOF model. Consequently, differences are expected. However, the size of the difference can not entirely be explained by the reasons given above.

5.1.1.5 Test 49 50 51

Looking at figure 15 it is seen that whilst the response results in heave are relatively similar, differences in the pitch DOF can be detected. The motion amplitudes behind the pitch responses are however in the order of magnitude 0.70 and therefore the response differences are seen as neglectable. The roll response for the 5 DOF model follows the pattern seen in the previous tests, i.e. it results in a greater response in comparison to the linear method and the model experiments.

27

5.1.1.6 Test 53 54 55

The results in figure 16 show that the heave and pitch responses are relatively similar, with the exception of test 53 for pitch where a difference the 5 DOF and the model experiment is quiet large. Looking at the roll DOF in figure 16 it can be seen that response obtained with 5 DOF and 3 DOF are larger in comparison to the linear method and the model experiments. Also, the peak response occurs at a lower frequency for the model experiment, 3 and 5 DOF models compared to the linear method. This indicates that the non-linear models capture this tendency better. What is more, the difference between the 3 and 5 DOF models is seen in test 53, 54 and 55. The added sway, roll and yaw seem to reduce the roll motion in test 53 but then again increase the roll in test 54 and 55. In conclusion it can be said that the 3 DOF responds more strongly for lower frequency and vice versa.

5.1.1.7 Test 56 58 59

The response in figure 17 for heave, pitch and roll follow the same pattern as in figure 16.

5.1.1.8 Test 60 61 62

Looking at figure 18 it can be seen that the compliance between heave and pitch response is very good. Regarding the roll response differences in can be seen between the mathematical models and the model experiment. One explanation behind the difference is seen as the added exciting wave force created as the waves come in contact the rudder and propeller on the model vessel. One plausible explanation could be found in the difference between the mathematical model and model vessel aft part as the model vessel is equipped with actual an actual rudder and propeller and therefore responds differently to waves coming from behind.

5.2 THE BOX VESSEL

Simulations with a box shaped ship were conducted with small wave amplitudes in order to analyse the linear behaviour of the simulation model. The main particulars of the box ship are found in table 3, the test input data can be seen in table 4 and the normalized simulation results can be found in figures 19-24. In the linear method the roll damping of 10% of the critical roll damping was used. The viscous roll-damping forces, used in the 3 and 5 DOF models, are obtained by a semi-empirical method according to (Ikeda, Himeno, Tanaka (1978)), where a part of the damping coefficients are obtained through a roll decay test. This particular part of the coefficient used in the box vessel simulations is the same as in the simulations used in the SSPA 2733 ship vessel. The chosen geometry of the box vessel is based on the manoeuvrability of a box shaped vessel. For example, a box shaped vessel with L=40, B=8 and D=4 that was simulated, was nearly impossible to steer. The rudder and propeller size were chosen on the basis of course keeping ability and with respect to the size of the box vessel. The autopilot coefficients used in the simulations are identical to the ones used in the SSPA 2733 vessel above.

28

Table 3. Box ship main particulars.

Entity Value Length 100 m Breath 10 m Draught 10 m

Displacement 307.5 ton Depth 3 m Trim 0 m VCG 3 m GM 1.27 m

Radius of inertia, roll 3.5 m Radius of inertia, pitch 25 m Radius of inertia, yaw 25 m Rudder, span 2 m

Rudder, chord 2 m Propeller, diameter 2 m

Table 4. Test input data.

Test # Speed [knots] Heading [degrees] Wave frequency [rad/s] Wave amplitude [m]

1 10 180 0.2-2 0.1 2 10 150 0.2-2 0.1 3 10 120 0.2-2 0.1 4 10 90 0.2-2 0.1 5 10 60 0.2-2 0.1

6 10 30 0.2-2 0.1 7 10 0 0.2-2 0.1


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31


5.2.1 Results discussion

Looking at the figures 19-22 it can be seen that the roll motion for both the nonlinear 5 DOF and the 3DOF are almost identical to each other whilst the linear model shows difference of results compared to the two models. The heave and pitch motion however between the three models do not show any difference with the exception of figure 21 where a small difference in the pitch degree of freedom can be detected. The difference in response seen in the roll degree of freedom shows that the reason behind the difference in roll, in general between these models, does not only depend on conditions where nonlinear behaviour is expected, but rather that the roll damping model differs for both linear and nonlinear conditions.

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32 6 Discussion and conclusion

6 DISCUSSION AND CONCLUSION This master thesis report has given an account of and the reasons for the development of a ship simulation model in 5 DOF implemented in MATLAB script. Coordinate systems, simplifications, assumptions and the integration process as well as the verification and validation of the unified model have been presented. Simulation results show that the heave and pitch motions agree with the linear method, 3 DOF non-linear model as well as with the experimental results. The roll motion, however, differs from the linear method but also from the model experimental and the 3 DOF method at certain frequencies. The difference between the results obtained between the linear method and the unified model is seen as the result of the non-linear parts of the unified model i.e. the restoring, Froude-Krylov and viscous roll-damping, which enable the simulation of phenomenon, such as, parametric roll. Comparing the response obtained from the 3 DOF i.e. the wave induced model and the 5 DOF i.e. unified model no major difference is seen. This can be understood, hence the 5 DOF model is based on the 3 DOF model. The model experiment results differ from the 5 DOF model in that the 5 DOF model results in a larger roll motion in tests for relative wave direction differing from head and following seas. As the wave amplitude used in the simulations is obtained by calculating the mean value of waves measured at the fore and aft part of the ship model, it is uncertain what the amplitude of the actual encounter wave is. As a consequence the results include uncertainties regarding the input wave value used in the simulations. On the other hand, the response results are normalized with respect to the input values and therefore the possible difference caused by this uncertainty is seen as rather small. The assumption upon which the integration is based i.e. that no major interference between the turning motion of the ship and the incident velocity potential, radiation and diffraction waves arise and therefore the forces and moments of the two models can be superimposed, is seen as valid. This, because the turning motion of the ship is kept as small as possible during the simulations. However, the change of the ship draught due to waves is not taken into consideration in the manoeuvrability part of the unified model, even though the change of draught affects the ship hull forces. Regarding the question of the effect that the rudder and propeller geometry have on the simulation outcome, it is seen as quit small as can be seen in the simulation results. However, to answer this question with a greater certainty several tests with different rudder and propeller geometries should be made. The autopilot coefficients, which determine the amplitude of the autopilot output, were chosen with respect to the ability of the given ship to steer at the highest frequency and wave amplitude of the simulation series. When comparing the 5 DOF results to the 3 DOF it does not seem to have a great influence on the outcome, although in the case of test 22 it could be seen as the cause behind the large roll response. A conclusion that can be drawn from the results is that the roll motion tends to be quiet large in the tests, compared to the results obtained by the linear method and the model experiment. Furthermore, the expected, reducing the roll motion, due to the coupling effects of the added sway, roll and yaw degrees of freedom, in particular for relative wave directions differing from head and following seas could not be seen in the given results. Future research should therefore concentrate on the investigation of the equations that form the roll motion in the equations of motion.

33 References

REFERENCES Anon, (2002). The Specialist Committee on Esso Osaka, Final Report and Recommendations to the 23rd ITTC, Proceedings of the 23rd ITTC-Volume II. de Jong, (1973). Computation of the Hydrodynamic Coefficients of Oscillating Cylinders. Netherlands Ship Research Centre TNO, Shipbuilding Department, Delft. Fossen, T. I. (2005). A nonlinear unified state-space model for ship maneuvering and control in a seaway. Department of Engineering Cybernetics Norwegian University of Science and Technology. NO-7491 Trondheim, Norway. Garme, K. (1997). Model Seakeeping Experiments Presented in the Time-Domain to Facilitate Validation of Computational Tools. Naval Architecture, Departement of Vehicle Engineering, KTH, Stockholm. Garme, K. (2011). Marine Hydromechanics, Lecture notes for SD2703 Marine dynamics. Naval Architecture, Departement of Vehicle Engineering, KTH, Stockholm. Hua, J. and M. Palmqvist (1995). A Description of SMS – A Computer Code for Ship Motion Calculation. Naval Architecture, Departement of Vehicle Engineering, KTH, Stockholm. Hooft, J.P. and Pieffer J.B.M (1998). Manoeuvrability of Frigates in Waves, Marine Technology, Vol. 25. Ikeda, Y., Himeno and N. Tanaka (1978). A Prediction Method for Ship Rolling. Technical Report 00405. Department of Naval Architecture, University of Osaka Prefecture, Japan. Journèe, J.M.J. and J. Pinkster (2002). Introduction in ship hydrodynamics. Ship hydromechanics Laboratory, Delft University of Technology, The Netherlands. Kluwe, F. (2009). Development of a minimum stability criterion to prevent large amplitude roll motions in following seas. Technische Universität Hamburg-Harburg, Germany. ISBN 978-3-89220-678-4. Lee, T. et al (2003). On an Empirical Prediction of Hydrodynamic Coefficients for Modern Ship Hulls. Proceedings of MARSIM `03, Vol. III. Lewandowski, E.M. (2004). The dynamics of marine craft: Manoeuvering and seakeeping. Advanced Series on Ocean Engineering -Volume 22. ISBN 981-02-4755-9. Milchert, T. (2000). Handledning i fartygs projektering. Naval Architecture, Departement of Vehicle Engineering, KTH, Stockholm. Min-Guk, S. and K. Yonghwan (2011). Effects of Ship Motion on Ship Maneuvring in Waves. Departement of Naval Architecture and Ocean Engineering, Seoul National University. Ovegård, E. (2009). Numerical simulation of parametric rolling in waves. Centre for Naval Architecture, KTH, Stockholm. Perez, T. (2005). Ship Motion Control. Course Keeping and roll stabilisation using rudder and fins. Centre for Ships Structures (CeSOS), Norweigen University of Science and Technology (NTNU), Marine Technology Centre, NO-7491, Trondheim, Norway. Rosén, A. (2011). Introduction to seakeeping. Centre for Naval Architecture, KTH, Stockholm.

34

SHK, (2008). Loss of M/s Finnbirch between Öland and Gotland 1 november 2006. Swedish Accident Investigation Board, Report RS 2008:03e, Case S-130/06. Salveson, N., E.O. Tuck and O. Faltinsen (1970). Ship Motions and Sea Loads. Trans. SNAME, Vol. 78. Zachrisson, D. (2011). Manoeuvrability model for a Pure Car and Truck Carrier. Centre for Naval Architecture, KTH, Stockholm.

APPENDIX A - SIMULATION RESULTS AND INPUT DATA Table 5. Simulation results for the unified 5 DOF model (SSPA 2733).

Simulation 5DOF

W1 & W2 Heave Heave Heave Heave Heave Roll Roll Roll Roll Roll Pitch Pitch Pitch Pitch Pitch

Test # Froudes

num. Heading

Wave

frequency

[rad/s]

mean

amp [m]mean [m] min [m] max [m] amp [m]

nondim

mean(W1

&W2)

mean

[deg]min [deg] max [deg] amp [deg]

nondim

mean(W1

&W2)

mean


nondim

mean(W1

&W2)

1 22 0.21 180 0.35 2.345 0.10155 -2.11598 2.362864 2.239424 0.954978 -1.22E-06 -0.00018 0.000192 0.000188 0.000112 0.269667 -1.32154 2.074804 1.698171 1.012159

2 23 0.21 180 0.55 2.765 0.26319 -1.62822 2.061315 1.844767 0.667185 -0.04915 -16.2939 16.30713 16.30052 3.336778 0.305498 -3.64999 4.523536 4.086764 0.836576

3 30 0.21 180 0.75 3.019 0.288093 -0.34592 0.922735 0.634328 0.210112 -9.41E-06 -0.00028 0.000176 0.000228 2.30E-05 0.195277 -1.74343 2.258264 2.000848 0.201732

4 35 0.072 180 0.35 2.783 0.112197 -2.40782 2.715071 2.561447 0.92039 1.12E-06 -0.00137 0.001432 0.001338 0.000672 0.267306 -1.71326 2.50653 2.109897 1.05964

5 36 0.072 180 0.55 2.319 0.179068 -1.05632 1.433944 1.245133 0.536927 -4.82E-05 -0.00583 0.00492 0.005374 0.001312 0.275585 -2.63242 3.75458 3.193502 0.779449

6 39 0.072 180 0.75 2.048 0.097923 -0.23413 0.575903 0.405019 0.197763 0.149248 -25.1836 24.47254 24.19598 3.596144 0.465656 -1.191 2.190413 1.690707 0.251282

7 40 0.21 150 0.35 2.555 -0.01805 -2.29271 2.645298 2.469007 0.966343 0.583145 -12.5356 13.9774 13.23493 7.24004 0.462639 -1.09335 2.306013 1.699683 0.929795

8 41 0.21 150 0.55 1.759 0.126648 -1.176 1.418575 1.297288 0.737514 0.126642 -2.99362 3.987413 3.381667 1.088143 0.317344 -1.93895 2.707378 2.323165 0.747541

9 43 0.21 150 0.75 2.424 0.20289 -0.10114 0.492188 0.296664 0.122386 -0.61357 -1.33808 0.077904 0.641439 0.080546 0.204822 -1.73868 2.238533 1.988605 0.249712

10 45 0.21 120 0.35 2.223 -0.00831 -2.11483 2.332662 2.223745 1.000335 0.024365 -20.7016 21.61221 21.074 13.25006 0.483777 -0.49839 1.694223 1.096308 0.689292

11 47 0.21 120 0.55 2.013 0.156303 -1.79739 1.95867 1.87803 0.932951 1.620202 -6.9422 9.900321 8.401284 2.362234 0.441292 -1.77696 2.737436 2.257196 0.634668

12 48 0.21 120 0.75 2.266 0.245303 -1.05658 1.515068 1.285823 0.567442 1.863631 -1.91803 5.080815 3.160059 0.424482 0.395197 -2.89888 3.750728 3.324804 0.446612

13 49 0.21 90 0.35 2.153 0.019306 -2.17983 2.07462 2.127226 0.988029 0.525005 -22.4629 23.55255 22.94284 14.89407 0.355345 -0.15876 0.797502 0.478129 0.310393

14 50 0.21 90 0.55 2.074 0.061045 -1.95875 2.143459 2.051105 0.988961 1.796702 -15.8261 18.3705 17.02141 4.645237 0.614227 -0.11603 1.823207 0.969617 0.264614

15 51 0.21 90 0.75 2.153 0.104171 -1.93467 2.04927 1.99197 0.925207 1.830377 -9.40704 13.04658 10.92705 1.544837 0.61524 -0.83322 2.22752 1.53037 0.21636

16 53 0.21 60 0.35 2.266 0.035346 -2.06855 2.220307 2.14443 0.94635 0.386413 -13.0606 14.70627 13.60557 8.392031 0.211394 -0.11318 0.802166 0.457673 0.282296

17 54 0.21 60 0.55 2.31 -0.0472 -1.56827 2.091377 1.829824 0.792131 4.10617 -21.7968 30.08116 25.54221 6.258461 0.601936 -0.54802 2.244301 1.396158 0.342093

18 55 0.21 60 0.75 2.433 0.093486 -0.99514 1.298646 1.146895 0.471391 2.787311 -17.4649 23.63226 20.34423 2.545206 0.451797 -1.33705 2.737105 2.037076 0.254853

19 56 0.21 30 0.35 2.424 0.046733 -2.05572 2.296869 2.176295 0.897812 1.241166 -5.19611 8.120708 6.139444 3.540028 0.278184 -0.60923 1.226105 0.917667 0.529131

20 58 0.21 30 0.55 2.188 0.131078 -1.08537 1.23764 1.161504 0.530852 5.988385 -12.2198 18.87669 15.42626 3.990565 0.415133 -1.18953 1.936702 1.563117 0.404357

21 59 0.21 30 0.75 2.354 0.180559 0.046428 0.412806 0.183189 0.07782 -1.5234 -18.2488 12.15992 15.20435 1.966008 0.493284 -0.79848 1.772152 1.285316 0.166199

22 60 0.21 0 0.35 2.643 0.113016 -2.17874 2.461492 2.320114 0.877833 -8.71E-05 -0.00038 0.000347 0.00026 0.000138 0.306704 -0.90119 1.486844 1.194015 0.631427

23 61 0.21 0 0.55 2.363 0.126024 -0.91158 1.257692 1.084636 0.459008 -0.00011 -0.00093 0.000709 0.00082 0.000197 0.386674 -1.27642 2.075844 1.676132 0.401482

24 62 0.21 0 0.75 2.013 0.152179 0.040501 0.262495 0.110997 0.05514 0.000208 -0.00038 0.00063 0.000503 7.60E-05 0.436243 -0.36864 0.942319 0.655477 0.099115

35 Appendix B - Verification of the unified model

Table 6. Simulation results for the wave-induced 3 DOF model (SSPA 2733).

APPENDIX B - VERIFICATION OF THE UNIFIED MODEL

The purpose of the verification is to ensure that the computer code behaves according to the mathematical model in question, thus ensuring that the code meets the aim that it is intended for. The aim has been to widen the ability to simulate ship motion in waves by adding the sway and yaw degrees of freedom as well as forces and moments in the roll degree of freedom. The added moments and forces, now included in the unified model, include forces and moments due to rudder and hull. In order to keep the ship at a steady course an autopilot has been included. The approach of the verification is to test claims or expected outcomes that are in accordance with the mathematical model. The mathematical models used in the test are the wave-induces 3 DOF model (Ovegård 2009) and the unified 5 DOF model including the manoeuvring model (Zachrisson 2011).

TEST SHIP

The different tests were conducted with Ro-ro vessel (SSPA 2733). Information regarding the ship is found in the validation chapter.

VERIFICATION TESTS

Test title: The effects of differing ship heading with respect to relative wave direction.

Test # 1

Claim/expectation: (pitch motion>roll motion for ) AND (pitch motion<roll motion

for )

Procedure: Only the wave direction is changed, all other variables are kept constant.

Comments to result: (Figure 25-26) It can be seen that the roll is larger as the wave direction is 900 than with 1800 and that the pitch is larger for 1800 and smaller for 900.

Simulation 3DOF

W1 & W2 Heave Heave Heave Heave Heave Roll Roll Roll Roll Roll Pitch Pitch Pitch Pitch Pitch

Test # Froudes

num. Heading

Wave

frequency

[rad/s]

mean

amp [m]mean [m] min [m] max [m] amp [m]

nondim

mean(W1

&W2)

mean


nondim

mean(W1

&W2)

mean


nondim

mean(W1

&W2)

1 22 0.21 180 0.35 2.345 0.102 -2.116 2.362864 2.239424 0.954978 -1.87E-06 -5.73E-05 5.70E-05 5.64E-05 3.36E-05 0.269667 -1.32154 2.074804 1.698171 1.012159

2 23 0.21 180 0.55 2.765 0.281 -1.560 2.059667 1.810 0.655 0.000 -0.003 0.004 0.004 7.72E-04 0.251292 -3.53764 4.280885 3.909264 0.800241

3 30 0.21 180 0.75 3.019 0.102806 -0.14519 0.349093 0.247144 0.200311 -1.55E-05 -2.52E-05 -4.98E-06 9.97E-06 2.46E-06 0.303087 -0.52335 1.150347 0.836847 0.206455

4 35 0.072 180 0.35 2.783 0.112 -2.408 2.715071 2.561 0.920 0.000 -0.001 0.001 0.001 0.000317 0.267306 -1.71326 2.50653 2.109897 1.05964

5 36 0.072 180 0.55 2.319 0.179 -1.056 1.433944 1.245 0.537 0.000 -0.001 0.001 0.001 0.000182 0.275586 -2.63242 3.75458 3.193502 0.779448

6 39 0.072 180 0.75 2.048 0.10698 -0.235 0.587544 0.411 0.201 0.202 -23.421 24.199 23.810 3.538769 0.436351 -1.18725 2.147063 1.667157 0.247782

7 40 0.21 150 0.35 2.555 -0.01423 -2.308 2.638031 2.473 0.968 0.381 -11.857 12.884 12.371 6.767235 0.429283 -1.15577 2.161253 1.658512 0.907273

8 41 0.21 150 0.55 1.759 0.128039 -1.180 1.421452 1.301 0.739 -0.383 -3.474 3.442 3.458 1.112722 0.323688 -1.91113 2.671123 2.291125 0.737232

9 43 0.21 150 0.75 2.424 0.108767 -0.031 0.246234 0.139 0.097 -0.866 -1.485 -0.213 0.636 0.135934 0.287741 -0.90419 1.498608 1.201398 0.256803

10 45 0.21 120 0.35 2.223 -0.00342 -2.149 2.318896 2.234 1.005 -0.395 -20.402 20.337 20.369 12.80705 0.475248 -0.58268 1.444062 1.013373 0.637148

11 47 0.21 120 0.55 2.013 0.153208 -1.810 1.964007 1.887 0.937 0.308 -8.590 8.928 8.759 2.462803 0.382992 -1.57059 2.39482 1.982705 0.557488

12 48 0.21 120 0.75 2.266 0.213 -1.126 1.540602 1.333 0.590 -2.803 -6.934 1.550 4.242 0.571335 0.341583 -2.74637 3.357383 3.051879 0.411039

13 49 0.21 90 0.35 2.153 0.121 -2.224 2.041956 2.133 0.991 0.868 -24.155 23.277 23.716 15.39598 0.497912 -0.00621 0.962144 0.484176 0.314318

14 50 0.21 90 0.55 2.074 0.066 -1.949 2.151556 2.050 0.988 0.773 -15.797 16.221 16.009 4.368988 0.480571 -0.02596 1.266507 0.646236 0.176361

15 51 0.21 90 0.75 2.153 0.092 -1.911 2.030652 1.971 0.915 -0.247 -10.891 10.583 10.737 1.517974 0.441766 -0.24793 1.26434 0.756134 0.1069

16 53 0.21 60 0.35 2.266 0.030 -2.074 2.248339 2.161 0.954 -1.625 -16.499 15.276 15.888 9.799549 0.397874 -0.13983 0.780812 0.460321 0.28393

17 54 0.21 60 0.55 2.31 -0.022 -1.573 2.051605 1.812 0.784 2.509 -20.006 24.661 22.333 5.472235 0.519585 -0.46879 1.869606 1.169198 0.286482

18 55 0.21 60 0.75 2.433 0.134 -0.962 1.301798 1.132 0.465 3.612 -10.016 16.901 13.454 1.683248 0.424575 -1.11023 2.176067 1.643149 0.205569

19 56 0.21 30 0.35 2.424 0.048 -2.049 2.316514 2.183 0.901 -0.606 -6.669 6.130 6.399 3.689883 0.330312 -0.63265 1.271989 0.952318 0.54911

20 58 0.21 30 0.55 2.188 0.164 -1.074 1.335915 1.205 0.551 3.254 -8.941 12.065 10.499 2.715861 0.429524 -1.25537 1.945884 1.600626 0.41406

21 59 0.21 30 0.75 2.354 0.170 0.081 0.362534 0.141 0.060 -2.746 -15.934 10.496 13.215 1.708824 0.496513 -0.56928 1.577843 1.073564 0.138818

22 60 0.21 0 0.35 2.643 0.079 -2.096 2.367992 2.232 0.878 0.000 0.000 0.000 0.000 3.11E-05 0.333057 -0.85474 1.442089 1.148414 0.631193

23 61 0.21 0 0.55 2.363 0.126 -0.912 1.257692 1.085 0.459 0.000 0.000 0.001 0.000 0.00011 0.386674 -1.27642 2.075844 1.676132 0.401482

24 62 0.21 0 0.75 2.013 0.153 0.040 0.262625 0.111 0.055 0.082 -0.546 0.893 0.720 0.108848 0.424674 -0.36864 0.942318 0.65548 0.099115

36

Test title: Autopilot

Test # 2

Claim/expectation: The autopilot is keeping the yaw angle closer to zero then without it for

waves differing from

Procedure: Simulate over a longer time period (t=500s) with and without the autopilot for wave direction e.g.

and compare the results.

Comments to result: (Figure 27-28) Comparing figure 3 and 4 it is seen that without the autopilot the ship gets instable. The conclusion is that the autopilot is working. The autopilot coefficients used in this example were: c1=0.88; c2=0.85;

Test title: Rudder coordinate system configuration

Test # 3

Claim/expectation: (When the rudder is set to a negative angle the ship yaw angle should turn to a negative angle) AND (When the rudder is set to a positive angle the ship yaw angle should turn to a positive angle) Furthermore, when the rudder angle is positive the sway motion shall move to the positive direction of its coordinate system. (Note! This is a consequence of the method in which the rudder angle is modelled. In the main.m of the manoeuvre model it can be read: positive rudder angle =starboard which means that a positive rudder angle results a sway direction in the positive direction.) Now, for the claim to be correct or true with respect to the unified model, the ship shall move to the port side for a positive rudder angle.

Procedure: Set the rudder to a constant positive angle (and vice versa) and

simulate for wave direction . Use small wave amplitudes.

Comments to result: (Figure 29-32)

As can be seen from the results everything works in accordance to the coordinate system i.e. when the yaw angle is positive the ship moves in the positive sway direction and vice versa.

Test title: The effect of the sway forces

Test # 4

Claim/expectation: Without the sway forces from the manoevrability model, that serve as a reaction force to the wave exciting forces, the ship will move a larger distance in the sway direction. ( NOTE! The manoeuvre sway forces also include lift forces )

Procedure: Simulate a case with and without the manoeuvre sway forces and compare the results.

Comments to result: (Figure 33-36) By comparing the sway motions in figure 33 and 35 it can be seen that the sway motion in figure 33 is much smaller than in figure 35 where sway forces are included.

37

Test title: The effect of the manoeuvring forces and moments on the roll motion

Test # 5

Claim/expectation: The roll motion shall be smaller with the manoeuvring forces and 5 DOF than 3 DOF and with only the seakeeping forces and moments.

Procedure: Test both the old and new model by simulating the

motion due to waves for . In addition a roll decay is to be made with and without the influence of the manoeuvring forces.

Comments to result: (Figure 37-40) Conversely to the expected the roll motion is larger for the unified 5 DOF model in comparison to the wave-induced 3 DOF model with respect to the relative wave direction of 1350. On the other hand, the roll decay in figure X shows that the roll motion declines faster for the 5 DOF unified model when compared to the wave-induced 3 DOF model. Consequently, the damping is larger for the unified model in the case of the roll decay test. This can be understood. Hence, a part of the motion energy that would have been transformed into roll motion is now partly transformed into sway motion.

Test title: Force and moment magnitude

Test # 6

Claim/expectation: (The exciting forces shall be greater or equal to the reaction forces) AND (the exciting and reaction forces shall not act in the same direction)

Procedure: Simulate different cases and point out the exciting and reaction forces and their magnitudes.

Comments to result: (Figure 41-45) Looking at the figures 41-45 it can be seen that the exciting forces are larger or equal to the damping forces for the different degrees of freedom.

38

VERIFICATION RESULTS

Figure 25.Test 1.

0 50 100 150 200 250 300 350 400-5

0

5x 10

-5 4 DOF - Roll - Angle

4 [

o]

time [s]

0 50 100 150 200 250 300 350 4000

1

25 DOF - Pitch - Angle

5 [

o]

time [s]

0 50 100 150 200 250 300 350 400-5

0

5x 10

-6 6 DOF - Yaw - Angle

6 [

o]

time [s]

39

Figure 26. Test 1.

Figure 27. without the autopilot

.

0 50 100 150 200 250 300 350 400-10

0

104 DOF - Roll - Angle

4 [

o]

time [s]

0 50 100 150 200 250 300 350 4000

1


5 [

o]

time [s]

0 50 100 150 200 250 300 350 400-0.2

0

0.26 DOF - Yaw - Angle

6 [

o]

time [s]

0 50 100 150 200 250 300 350 400 450 500-20

0


4 [

o]

time [s]

0 50 100 150 200 250 300 350 400 450 500-5

0


5 [

o]

time [s]

0 50 100 150 200 250 300 350 400 450 500-20

0

206 DOF - Yaw - Angle

6 [

o]

time [s]

40

Figure 28. with the autopilot

.

Figure 29. Positive rudder angle.

0 50 100 150 200 250 300 350 400 450 500-20

0


4 [

o]

time [s]

0 50 100 150 200 250 300 350 400 450 500-5

0


5 [

o]

time [s]

0 50 100 150 200 250 300 350 400 450 500-2

0


6 [

o]

time [s]

0 20 40 60 80 100 120 140 160 180 200-1

0

11 DOF - Surge - Position

1 [

m]

time [s]

0 20 40 60 80 100 120 140 160 180 200-20

0

202 DOF - Sway - Position

2 [

m]

time [s]

0 20 40 60 80 100 120 140 160 180 200-0.2

0

0.23 DOF - Heave - Position

3 [

m]

time [s]

41

Figure 30. Positive rudder angle.

Figure31. Negative rudder angle.

0 20 40 60 80 100 120 140 160 180 200-0.5

0

0.54 DOF - Roll - Angle

4 [

o]

time [s]

0 20 40 60 80 100 120 140 160 180 2000

0.5


5 [

o]

time [s]

0 20 40 60 80 100 120 140 160 180 2000

10


6 [

o]

time [s]

0 20 40 60 80 100 120 140 160 180 200-1

0


1 [

m]

time [s]

0 20 40 60 80 100 120 140 160 180 200-20

0


2 [

m]

time [s]

0 20 40 60 80 100 120 140 160 180 200-0.2

0


3 [

m]

time [s]

42

Figure 32. Negative rudder angle.

Figure 33. With the sway manoeuvre forces.

.

0 20 40 60 80 100 120 140 160 180 200-0.5

0

0.54 DOF - Roll - Angle

4 [

o]

time [s]

0 20 40 60 80 100 120 140 160 180 2000

0.5


5 [

o]

time [s]

0 20 40 60 80 100 120 140 160 180 200-20

0


6 [

o]

time [s]

0 50 100 150 200 250 300 350 400 450 500-1

0


1 [

m]

time [s]

0 50 100 150 200 250 300 350 400 450 500-50

0


2 [

m]

time [s]

0 50 100 150 200 250 300 350 400 450 500-0.5

0


3 [

m]

time [s]

43

Figure 34. With the sway manoeuvre forces.

.

Figure 35. Without the sway manoeuvre forces.

.

0 50 100 150 200 250 300 350 400 450 500-2

0


4 [

o]

time [s]

0 50 100 150 200 250 300 350 400 450 500-2

0


5 [

o]

time [s]

0 50 100 150 200 250 300 350 400 450 500-0.5

0


6 [

o]

time [s]

0 50 100 150 200 250 300 350 400 450 500-1

0


1 [

m]

time [s]

0 50 100 150 200 250 300 350 400 450 500-1000

0


2 [

m]

time [s]

0 50 100 150 200 250 300 350 400 450 500-0.5

0


3 [

m]

time [s]

44

Figure 36. Without the sway manoeuvre forces.

.

Figure 37.

(Unified model)

0 50 100 150 200 250 300 350 400 450 500-5

0


4 [

o]

time [s]

0 50 100 150 200 250 300 350 400 450 500-2

0


5 [

o]

time [s]

0 50 100 150 200 250 300 350 400 450 500-20

0


6 [

o]

time [s]

0 20 40 60 80 100 120 140 160 180 200-1

0


1 [

m]

time [s]

0 20 40 60 80 100 120 140 160 180 200-1

0


2 [

m]

time [s]

0 20 40 60 80 100 120 140 160 180 200-2

0

23 DOF - Heave - Position

3 [

m]

time [s]

45

Figure 38.

(Unified model)

Figure 39. Test 5.

(Seaware old model)

0 20 40 60 80 100 120 140 160 180 200-10

0

10

X: 152.2

Y: 9.641


4 [

o]

time [s]

X: 113.9

Y: -8.819

0 20 40 60 80 100 120 140 160 180 200-2

0


5 [

o]

time [s]

0 20 40 60 80 100 120 140 160 180 200-0.5

0


6 [

o]

time [s]

0 20 40 60 80 100 120 140 160 180 200-1

0


1 [

m]

time [s]

0 20 40 60 80 100 120 140 160 180 200-1

0


2 [

m]

time [s]

0 20 40 60 80 100 120 140 160 180 200-2

0

23 DOF - Heave - Position

3 [

m]

time [s]

46

Figure 40. Test 5.

(Seaware old model)

Figure 41. Sway roll and yaw as the function of time in a roll decay test.

0 20 40 60 80 100 120 140 160 180 200-10

0

10

X: 113.6

Y: -7.627


4 [

o]

time [s]

X: 152.2

Y: 8.577

0 20 40 60 80 100 120 140 160 180 200-2

0


5 [

o]

time [s]

0 20 40 60 80 100 120 140 160 180 200-1

0


6 [

o]

time [s]

0 20 40 60 80 100 120 140 160 180 200-100

0

100Sway

2 [

m]

time [s]

0 20 40 60 80 100 120 140 160 180 200-20

0

20Roll

4 [

o]

time [s]

0 20 40 60 80 100 120 140 160 180 200-1

0

1Yaw

6 [

o]

time [s]

3 DOF

5 DOF

3 DOF

5 DOF

3 DOF

5 DOF

47

Figure 42. Roll motion as a function of time in a roll decay test.

Figure 43. Test 6.

0 20 40 60 80 100 120 140 160 180 200-10

-5

0

5

10

15Roll

4 [

o]

time [s]

3 DOF

5 DOF

0 20 40 60 80 100 120 140 160 180 200-1

-0.5

0

0.5

11 DOF - Surge - Force

X [

N]

time [s]

Fmemory

FDiffraction

FFroude-Krylov

Fmanoeuvre

0 20 40 60 80 100 120 140 160 180 200-1

-0.5

0

0.5

1x 10

6 2 DOF - Sway - Force

Y [

N]

time [s]

Fmemory

FDiffraction

FFroude-Krylov

Fmanoeuvre

48

Figure 44.

Figure 45.

0 20 40 60 80 100 120 140 160 180 200-2

-1

0

1

2x 10

8 3 DOF - Heave - Force

Z [

N]

time [s]

Fmemory

FDiffraction

FFroude-Krylov

Fg

0 20 40 60 80 100 120 140 160 180 200-2

-1

0

1

2x 10

7 4 DOF - Roll - Moment

K [

Nm

]

time [s]

Fmemory

FDiffraction

FFroude-Krylov

Froll-damp

Fmanoeuvre

0 20 40 60 80 100 120 140 160 180 200-2

0

2

4x 10

8 5 DOF - Pitch - Moment

N [

Nm

]

time [s]

Fmemory

FDiffraction

FFroude-Krylov

0 20 40 60 80 100 120 140 160 180 200-2

-1

0

1

2x 10

7 6 DOF - Yaw - Moment

M [

Nm

]

time [s]

Fmemory

FDiffraction

FFroude-Krylov

Fmanoeuvre

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