Anti Roll Tanks in Pure Car and

Truck Carriers

B JÖRN WINDÉN w inden@k th . se 073 -7017256

Master thesis

KTH Centre for naval architecture Stockholm 2009

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Abstract This is a master thesis conducted at KTH Centre for Naval Architecture in collaboration with Wallenius Marine AB. Rolling motions is something that is undesired in all kinds of seafaring. In terms of propulsion resistance, comfort and route planning it would be desirable to reduce these motions. This thesis is an investigation on how different roll stabilising systems affect the performance of an 8000 unit PCTC vessel, special emphasis is put on investigating the performance of anti roll tanks. The ship in question has a recorded incidence of parametric rolling and the ability of the tanks to countervail this phenomenon is also investigated. The tank and fin stabilising systems are relatively equal when it comes to roll damping performance related to changes in the required forward propulsion power. The tanks however, have a higher potential for improvements, addition of features such as heeling systems and parametric roll prevention systems. The tank performance is also independent of the speed of the ship. The tanks are easier to retrofit and do not require the ship to be put in dry dock during installation. The conclusion of this thesis is that a combined anti roll and heeling system should be installed but that a further study has to be made on the performance of active rudder stabilisation. It is shown that passive tanks are efficient at preventing parametric rolling in some sea states. A proposal is made for a further study on a control system that could achieve the same performance for all sea states.

Sammanfattning Detta är ett examensarbete utfört på KTH Marina System i samarbete med Wallenius Marine AB. Rullningsrörelser är något som är oönskat i all form av sjöfart. Framsteg kan göras i både framdrivningsmotstånd, komfort och ruttplanering om dessa rörelser kunde minskas. Detta examensarbete består av en undersökning hur olika system för rulldämpning påverkar prestandan hos ett 8000 enheters PCTC-fartyg. Speciell vikt har lagts vid att undersöka prestandan hos antirulltankar. Det undersökta fartyget har en dokumenterad incident med parametrisk rullning och tankarnas förmåga att motverka detta fenomen undersöks. Tank- och fenstabilisatorer är i princip likvärdiga vad det gäller dämpningsprestanda relaterat till erforderliga ändringar i framdrivningseffekten. Tankarna har dock en större potential för förbättring och tillägg av ytterligare inslag som krängningshämmare och system för motverkan av parametrisk rullning. Tankarnas prestanda är också oberoende av fartygets fart. Tankarna är lättare att installera i efterhand och kräver inte att fartyget läggs i torrdocka under installationen. Slutsatsen av detta arbete är att en kombinerad antirull- och krängningshämmande tank bör installeras men att en vidare studie måste göras på prestandan hos aktiva roderstabiliseringssystem. Det visas att passiva tankar kan motverka parametrisk rullning i vissa sjötillstånd. Ett förslag om en vidare studie på reglersystem som skulle kunna ge samma prestanda vid alla sjötillstånd ges.

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Contents

ABSTRACT I

1 NOMENCLATURE 1

2 INTRODUCTION 3

2.1 BACKGROUND ..................................................................................................................................3 2.2 PURPOUSE.........................................................................................................................................4 2.3 PCTC FIDELIO ..................................................................................................................................4 2.4 TUNED DAMPERS..............................................................................................................................5

3 CHARACTERISTICS OF PASSIVE TANKS 7

3.1 EQUATION OF MOTION FOR THE FLUID IN A U-TUBE TANK ................................................................8 3.2 TANK CONTRIBUTIONS TO THE EQUATIONS OF MOTION OF THE SHIP...............................................11

4 ROLL DAMPING USING PASSIVE U-TUBE TANK 12

4.1 TUNING OF THE TANK.....................................................................................................................13 4.2 EVALUATION OF PERFORMANCE.....................................................................................................17

4.2.1 Roll-decay test ...........................................................................................................................18 4.2.2 Roll excitation ...........................................................................................................................18 4.2.3 Frequency response...................................................................................................................19

4.3 INCREASED PROPULSION POWER.....................................................................................................21

5 PARAMETRIC ROLL PREVENTION 26

5.1 SIMULATING PARAMETRIC ROLL WITH A 2 DOF MODEL .................................................................27 5.2 SIMULATING PARAMETRIC ROLL WITH A 4 DOF MODEL .................................................................28 5.3 EXTENDED STUDY WITH THE 4 DOF MODEL...................................................................................29 5.4 RISK ASSESSMENT...........................................................................................................................33

6 ACTIVE TANKS 34

6.1 PUMP SYSTEM.................................................................................................................................35 6.2 VALVE SYSTEM...............................................................................................................................36

6.2.1 Bias in the optimisation.............................................................................................................38 6.2.2 Further use ................................................................................................................................39

6.3 COMBINED SYSTEM.........................................................................................................................40

7 OTHER WAYS OF INCREASING THE ROLL DAMPING 40

7.1 STABILISING FINS............................................................................................................................41 7.1.1 Positioning of the fins................................................................................................................42 7.1.2 Increased propulsion resistance................................................................................................43

7.2 BILGE KEELS...................................................................................................................................44 7.2.1 Increased propulsion resistance................................................................................................44

7.3 ACTIVE RUDDER CONTROL..............................................................................................................45 7.3.1 Increased propulsion resistance................................................................................................46 7.3.2 Impact on manoeuvring.............................................................................................................47

8 COMPARISON BETWEEN DIFFERENT SYSTEMS 48

8.1 EFFICIENCY.....................................................................................................................................48 8.2 INSTALLATION ................................................................................................................................49

8.2.1 In newbuilds ..............................................................................................................................49 8.2.2 Retrofitting ................................................................................................................................49

9 CONCLUSIONS 51

10 REFERENCES 52

11 APPENDICES 53

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1 Nomenclature

A Inertia matrix in the equation of motion for a ship (6 DOF indexed as aij)

A Rudder area Aτ Inertia matrix in the equation of motion for a ship and tank in interplay (7 DOF indexed as aij) AR Rudder aspect ratio

B Damping matrix in the equation of motion for a ship (6 DOF indexed as bij)

Bτ Damping matrix in the equation of motion for a ship and tank in interplay (7 DOF indexed as bij) B Maximum beam of ship b Maximum beam of ship

b1-3 Coefficients in fin control system

bbk Extension of bilge keel

C Stiffness matrix in the equation of motion for a ship (6 DOF indexed as cij)

Cτ Stiffness matrix in the equation of motion for a ship and tank in interplay (7 DOF indexed as cij) c Flow velocity around rudder

CB Block coefficient

CL Rudder lift coefficient F External force vector in the equation of motion f Target function in active control system optimisiation

F4 External roll moment

F40W Amplitude of wave induced external moment

Fpump/valve 7:th term in the external force vector corresponding to moment applied to tank fluid.

Fτ4 Tank induced roll moment g Gravitational acceleration

g1-6 Coefficients in control system for active tanks and rudder stabilisers

GM0 Metacentric height

GMF Metacentric height corrected for free surface effect (fluid)

GMS Undisturbed (solid) metacentric height

gopt Optimum values of the coefficients g1-6 for a certain case h Vertical distance from keel to centre of gravity of added ballast

hd Height of duct in U-tube reservoir

hr Height to datum level undisturbed fluid in U-tube reservoir

ht Total height of U-tube reservoir

I44 Rotational inertia around longship axis

IWAx Area moment of inertia of waterline area k Wave number

K1-3 Coefficients in fin control system KB Vertical centre of buoyancy KG Vertical centre of gravity

KG Static gain of fin control system

KU Speed correction factor in fin control system

L Total length of ship (LOA) m Mass displacement

n Local width of channel (wr in the reservoirs and wd in the duct)

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P Local pressure in tank system

pret Pressure drop in air duct as a result of a valve or turbine

Pturbine Turbine power q Wall friction coefficient for tank fluid/wall interaction.

Qt Common magnitude term in tank equation of motion determined by the size of the tank s Laplace transform operator

s1-6 Absolute motions at a point in the hull due to geometry and global motions T Design draught (mean) u Ship velocity v Local flow velocity in tank system w Width between centres od U-tube reservoirs

wballast Mass of added ballast

wd Width of U-tube duct

wr Width of U-tube reservoir

x1-6 Global motions of the ship. Surge, sway, heave, roll, pitch and yaw.

x4m Measured roll angle used for fin control system

XB1 Longship distance from datum point to COG of the ship (positive forwards.)

XB2 Athwadship distance from datum point to COG of the ship (positive to starboard.)

XB3 Vertical distance from datum point to COG of the ship (positive up.) y Position variable in tank system (extending vertically in reservoirs and horizontally in the duct) Y Local external force in tank system

αd Desired fin angle obtained from control system δ Rudder deflection angle

ε Phase between roll angle x4 and tank angle τ

ζ0 Wave amplitude

ηship Equivalent linear roll damping coefficient ηt Tank oscillation decay coefficient (measured) η Tank oscillation decay coefficient (theoretical) Μ Metacentric height reduction factor ρ Density of the water in which the ship travels

ρt Density of fluid used in the anti roll tank τ Angle between water levels in two connected U-tube reservoirs

ω0,t Natural frequency of fluid oscillations i the tank

ω40 Natural roll frequency of ship

ωe Wave encounter frequency ∇ Mass displacement

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2 Introduction Designers of vessels that are to be subjected to the conditions at the open sea will always have to deal with the issue of how the ship will behave in waves. When rough weather produces large scale motions, the captain has to reduce speed or change the vessels course and choose a calmer route. The unpredictable behaviour of the weather systems makes optimal timetables and routes a rare occurrence rather than the common course of events, resulting in great losses for the ship-owners. It is therefore desirable to construct the hull to handle as severe weather conditions as possible without damaging the ship or harm the crew. Since the optimal hull form for dealing with rough weather doesn’t always concede the desired cargo capacity, a compromise between these and other criteria e.g. propulsion resistance is made when the lines of the hull are drawn. The oblong and relatively round bottomed shape of a ship makes rolling, i.e. rotation around the length axis, the motion that most often gets problematic proportions. A comparison can be made with a log floating in water; the log cannot be easily moved sideways, lifted or rotated. Making the log roll however is easy. Conventional ships have a more square like cross section than a log but the oblong shape remains which is the reason why the waves can easier make the ship roll than for example sway from side to side. If one could reduce the rolling motions this would ensure a safer passage through many weather conditions meaning less course deviations and time losses.

2.1 Background Wallenius operates a large fleet of PCTC-vessels (Pure Car and Truck Carrier), car transporters that frequents a large portion of the worlds oceans. This type of ship is relatively sensitive to rolling since the hull form concedes a relatively small damping of the rolling motion. Low damping causes the kinetic energy induced by the waves to remain longer in the hull making the total energy, and thereby the motions, build up to higher levels.

Figure 1 M/V Fidelio 8000 units PCTC

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The ships are also sensitive to so called parametric rolling that happens due to temporary stability variations in some wave conditions. If a passing of a wave puts parts of the hull above the waterline, those parts can no longer give a righting moment should the ship start to roll. If, on the other hand, the passing puts a larger part of the hull below the waterline, the ship would get a greater righting moment than usual resulting in powerful reverberation towards the neutral state. Since the waves commonly have a relative velocity to the ship, these stability losses and gains will happen in intervals. If the occurrence frequency coincides with the natural frequency of the ship it can induce the resonant phenomenon called parametric rolling with very large roll angles and velocities as a result. Severe parametric rolling may lead to large capital losses since the cargo and equipment onboard can be devastated. There are several systems for roll damping in rough weather available on the market today. No specific commercial system for parametric roll prevention is available as of 2009. One of the most efficient systems for roll damping is water filled anti rolling tanks that counteracts the rolling by moving water from side to side inside the hull. The motion of the sloshing water is tuned so that the mass of the water counteracts the rolling motion. In the latest newbuilds by Wallenius, the fuel tanks have been moved in such a way that it would be possible to install anti roll tanks in the bottom part of the hull.

2.2 Purpouse This thesis will summarily explain the theory behind stabilising (anti roll) tanks, evaluation of their performance onboard a Wallenius ship and a comparison with regard to efficiency and applicability with other systems for roll damping. The purpose is to give recommendations on the usage of roll damping systems on these ships with a special emphasis on stabilising tanks. A deeper analysis is also made on ways to reduce the risk of parametric rolling.

2.3 PCTC Fidelio The ship that all calculations are based on is the 8000 unit PCTC Fidelio. She is in the latest class of Wallenius ships and was delivered from the shipyard in September 2007. A general arrangement of the ship is shown in Figure 2 and the general particulars are defined in Table 1.

Figure 2 General arrangement of M/V Fidelio

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Table 1 General particulars for M/V Fidelio

Parameter Size Dim L 220 m B 32,2 m T 9,5 m

CB 0,617 -

GM0 1 m KG 14,48 m KB 5,34 m

There have been several requests by captains of the Fidelio class that the company should invest in stabilising systems in their ships. For example, in 2008 the captains of M/V Fidelio and her sister ship M/V Faust wrote to Wallenius Marine, “A stabilising system of type Intering or similar should be installed to save fuel and reduce damages to the cargo” [15]. Several other requests have been made by other captains that feel that the ships are somewhat unwieldy in rough weather because of their poor roll damping. On December 23rd 2008 Fidelio had a recorded incident of parametric rolling. The incident occurred in the Mediterranean Sea close to Cyprus. Rolling angles of up to 35 degrees were recorded as well as extensive damage to the cargo. Because of this, special emphasis has been put on investigating how such events could be prevented using roll stabilisation systems.

2.4 Tuned dampers The technology of using a tuned mass to reduce motions is often referred to as Tuned Mass Damper (TMD) or if the mass used is in liquid form Tuned Liquid Damper (TLD). The idea of using liquid filled tanks to damp motions dates back to the late 1800s when in 1862 it was suggested by William Froude that a tank partially filled with water could reduce the rolling motions of a ship if the frequency and phase of the oscillating water was tuned correctly. The principle was proven by a model test by and documented by Philip Watts in 1880 [5 & 6]. The technology has since spread to several sectors of application. In very tall buildings it is common to use tuned masses, liquid or solid to counteract the tilting force created by strong winds. These can consist of a large suspended mass or interconnected tanks at the top of the building, shifting the mass or the levels in the tanks makes the building lean against the external force reducing the discomfort of its residents. An example of applications of TMDs and TLDs in buildings is shown in Figure 3.

Figure 3 Tuned Mass Damper in the form of a 660 tonne suspended sphere installed in Taipei 101 in

Taiwan (left) and Tuned Liquid Damper installed in the Shin Yokohama Prince Hotel in Japan (right)

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In maritime applications the early tanks were simply free surface reservoirs fitted with baffles to tune the sloshing motion. In 1911 H. Frahm proposed a U-shaped system consisting of two reservoirs connected with a narrow duct [7]. The inertia of the fluid pushing through the narrow duct made it possible to tune this system to become a TLD. The fact that the only intrusion of the cargo space with this system is the duct, the reservoirs being placed in the ships sides, made this new system highly popular and is the one that is mainly used in ships today. Even though the concept was proven in 1911 the first instalments in commercial vessels did not happen until the mid 1950s. Ever since they were proposed, numerous technical investigations of the performance of stabilising tanks have been carried out. The main target of most of these studies has been to find a way to describe the fluid motions in the tanks in a way so that the effects of these motions can be incorporated into the performance of the entire ship. Almost all such studies has used a similar simplified fluid mechanics approach of calculating the fluid motions by integrating the function of state for a fluid particle over the length of the system. The integration is made from the top of one reservoir, via the duct, to the top of the opposite reservoir. The function of state for each method varies depending on what constraints the author of a particular study has set on the motions of both the ship and the fluid. The chosen method is an entry in a guide to seakeeping calculations written by Lloyd [3]. This method suits this study since it is more synoptic than most of the others. This is preferred since the purpose is not to give the exact performance of a certain design of tank but rather an initial overview. While other methods use more complex geometries to closely match the true shape of the tank, Lloyds method uses a more schematic geometry that is easy to scale when testing different sizes and designs of tanks. In general many of the methods described are validated and tested using a single design of the hull and the tank making them hard to apply to a more general case. Lloyd also presents entire response amplitude operators as opposed to single results. Consequentially the basic assumptions and formulations (Chapter 3) in this thesis are the same as in [3]. This provides a tool to make comparisons and give recommendations. If an actual design is to be conceived as a result of this study, a less schematic method should be chosen to evaluate its performance.

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3 Characteristics of passive tanks The simplest tank imaginable is the surface tank, or flume. This tank consists of a single rectangular compartment partially filled with fluid. The surface tank can be used to explain the basic principle of passive tank roll damping. The idea is to design the tank so that the motion of the water therein gives the most beneficial effect on the roll motion of the rest of the ship. To do this, the strategy is to make the water in the tank exert a maximum stabilising moment on the ship when this reaches its maximum roll rate. In other words, when the ship has the highest roll rate to starboard, the water in the tank should be positioned to give the maximum stabilising moment to port as explained in Figure 4. The motion is controlled by installing baffles in the tank or by narrowing the midsection.

Figure 4 Surface tank demonstrating the basic principle of passive tank roll damping. The desired behaviour of the fluid is thus a sinusoidal motion with the same frequency as the roll motion and the same phase as the roll rate. Since the objective is to be in phase with the roll rate the fluid motions should lead the rolling motion by 900. If the rolling motion of the ship is written as in [Eq.1]

4 40 sin( )ex x tω= ⋅ ⋅ [1]

the fluid motion and the resulting moment can be written as in [Eq.2]

0 sin( )eM M tω ε= ⋅ ⋅ + [2]

where M is the stabilising moment from the tank, ε should be 900 and 0M is the amplitude of

the moment. 0M is obviously determined by the size and shape of the tank and the amount

and type of fluid it is filled with. This is the general behaviour of all passive tanks. However the surface tank shown above is one of the more uncommon types since it requires much space in the middle of the ship, a space usually reserved for cargo. Furthermore, the surface tank has large free water surfaces which damage the initial stability of the ship. A more common application of passive tanks is the U-tube tank, consisting of one reservoir on each side of the ship connected with a narrow duct as pictured in Figure 5.

* Maximum roll rate to starboard * Maximum roll to starboard * Maximum roll rate to port * Maximum roll to port * Maximum stabilising moment to * Zero stabilising moment * Maximum stabilising moment * Zero stabilising moment port to starboard

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Figure 5 Different types of U-tube tanks. (a), simple (b), with throttle valve (c), with air duct and throttle valve (d), with active pump.

3.1 Equation of motion for the fluid in a U-tube tank To be able to analyze the behaviour of the fluid in the tank the equation of motion for the system must be assembled. The methods described below and the methods for tuning of the tank are mainly taken from the publication [3]. First, it would be convenient if the state of the tank could be described with one single variable that could constitute the equation of motion and be easy to relate to the rolling motion in terms of phase. Since the rolling motion is an angular quantity it would be convenient if the state of the tank could also be defined as an angle. If the reservoirs are said to be narrow compared to the beam of the ship, the effects of the angle of the water in each reservoir becomes negligible. On account of this assumption, the motion of the fluid in the ship-fixed coordinate system is quasi one dimensional, meaning that the only changing state is the water level in each reservoir. This can be simplified further to the angle between the levels in the starboard and port reservoir. This angle, henceforward referred to asτ , is the variable that will be used to express the equation of motion for the tank. The general dimensions of the tank used for calculations as well as a definition of τ are shown in Figure 6.

Figure 6 Definition of terms regarding tank geometry

a b c d

Pump

hd

wr wd wr

rd

hr

ht

Datum level τ

w

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Since the fluid in the tank is constantly in motion, fluid mechanics is required to fully evaluate the behaviour. No flow is said to go in the normal direction (fore to aft and vice versa), and no flow is said to go perpendicular to the prevailing flow direction (side to side in the reservoirs and vertically in the duct). The flow field can then be described with a simplified version of the Navier Stokes equations as expressed in [Eq.3]. The coordinate system is defined in Figure 7. In [Eq.3], the y-direction is the direction of the flow as described above, y extends horizontally in the duct and vertically in the reservoirs. Y denotes the external forces working on the fluid per unit mass, P is the pressure, tρ is the density of the fluid and v is the flow

speed at any point in the tank.

1

t

v v Pv Y

t dy yρ∂ ∂ ∂+ ⋅ = − ⋅∂ ∂

[3]

Figure 7 Definition of coordinate system for channel flow If the reservoirs and the duct are of constant cross sections, the flow speed will be constant along the entire length so that:

0v

dy

∂ = [4]

This is true everywhere except in the junction between reservoir and duct. If these corner effects are neglected [Eq.3] can be simplified to [Eq.5].

1

t

dv dPY

dt dyρ= − ⋅ [5]

If the width between the centres of each reservoir (where the angle τ is measured in Figure 6) is called w, the velocity in the reservoirs can be written as in [Eq.6].

2r

wv τ= ⋅ ɺ [6]

The velocity at any point in the tank can be obtained by saying that the fluid is incompressible. This yields that the mass flow rate must be constant at all times. r r t tw v n vρ ρ⋅ ⋅ = ⋅ ⋅ [7]

[Eq.7] implies that the mass flow at an arbitrary point, where the flow speed is v and the width of the duct is n, must be equal to the known mass flow in the reservoirs. This yields the expression for the flow speed at an arbitrary point as described in [Eq.8].

n

y

v

10

2

r r rw v w wv

n n

τ⋅ ⋅ ⋅= =⋅ɺ

[8]

With the expression for the flow speed in [Eq.8] the governing equation [Eq.3] can be rewritten as in [Eq.9].

1

2r

t

w w dPY

n dyτ

ρ⋅ ⋅ = − ⋅⋅ɺɺ [9]

The attention is now turned to the external forces Y acting on the fluid. These consist of gravitational forces, inertia of the accelerated fluid, wall friction and forces from throttle valves or pumps if any such features are present. The acceleration of a fluid element within the hull can be regarded as an absolute motion and calculated using a combination of the six variables that describes the motion of the ship. The inertia terms, and thereby the equation of motion itself will inevitably contain terms related to the movement of the ship itself. For example, the absolute lateral motions of a point located 1Bx m forward of, 2Bx m to starboard

of and 3Bx m above the centre of gravity can be described using the relative motions of the

ship using [Eq.10].

2 2 3 4 1 6 2 2 3 4 1 6B B B Bs x x x x x s x x x x x= − ⋅ + ⋅ ⇒ = − ⋅ + ⋅ɺɺ ɺɺ ɺɺ ɺɺ [10]

The wall friction terms would normally be considered to be proportional to the square of the local flow velocity. However in this case, where the length of the tank in the normal direction is much longer than the width of the local channel n, according to [3], it can be shown that the friction force can be expressed as a function of the local flow velocity v itself as shown in [Eq.11]. The coefficient q should be determined by experiment.

[ / ]friction

q vY N kg

n

− ⋅= [11]

The general equation of motion is then obtained by integrating [Eq.9] over the length of the tank along the y-axis, starting from the top of the starboard reservoir, along the duct and ending at the port reservoir. This path of integration of course neglects the difference in water levels in the two reservoirs since it is only going from one datum level to another. The effects of the height difference are however comparatively small and integration between datum levels gives a good approximate solution of the problem. [Eq.9], integrated over the entire tank becomes the equation of motion for the system.

The integrated form of [Eq.9] can be written in a convenient form using the same notation as the equation of motion for the ship where , ,ij ij ija b c denotes coupling terms between

acceleration, velocity and position and the corresponding applied force. For example, 35a is

the term that connects the pitch acceleration to applied heave force. With this notation the term 2aτ would denote the coupling between the sway acceleration and the applied force in

the tank and 2b τ would be the connection between the tank angular velocity (i.e. the velocity

of the fluid in the reservoirs according to [Eq.6]) and the applied sway force. With this notation the equation of motion can be simplified to [Eq.12] where the coupling terms are defined in Appendix 1.

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2 2 4 4 4 4 6 6 /pump valvea x a x c x a x a b c Fτ τ τ τ ττ ττ τττ τ τ⋅ + ⋅ + ⋅ + ⋅ + ⋅ + ⋅ + ⋅ =ɺɺ ɺɺ ɺɺ ɺɺ ɺ [12]

[Eq.12] is the general equation of motion of a U-tube tank.

3.2 Tank contributions to the equations of motion of the ship The same way the motion of the fluid in the tank is influenced by the state of the surrounding ship, the motion of the ship itself is naturally influenced by the state of the tank. The equations of motion for the ship are of the form A x B x C x F⋅ + ⋅ + ⋅ =ɺɺ ɺ [13] The matrix A in [Eq.13] contains all coupling terms between accelerations in all the six degrees of freedom and the external forces F. B couples the velocities and C relates the state of the ship to the applied force. The tank motions give rise to an addition to the external forces. These extra forces can be seen as the forces, or moments required to sustain a steady tank angle, angular velocity or angular acceleration and are called Fτ in [Eq.14].

( , , )A x B x C x F Fτ τ τ τ⋅ + ⋅ + ⋅ = +ɺɺ ɺ ɺɺ ɺ [14]

Since it is convenient to keep the force vector intact and work with a system more like the one in [Eq.13], the tank angle is often considered as an extra degree of freedom and the coupling coefficients between the tank motions and the added external force as additions to the matrices A, B, and C. Finally to make the system of differential equations complete the equation of motion for the tank itself [Eq.12] is added as a seventh equation to form the equations of motion for a ship with a stabilising tank in Figure 8. By considering the coupling terms between the tank motions and the applied external forces to the ship as the forces that are required to sustain a steady tank acceleration, velocity and angle, some coupling coefficients are, by definition set to zero. The nonzero coefficients are named ia τ and ic τ , depending on what degree of freedom is to be coupled with the tank angle.

These coefficients, as the coefficients in the equation of motion for the tank, are only dependent of the dimensions of the tank itself.

12

2

4

6

2 4 6

11

22

33

444

55

66

4

0 0

0

0 0; 0

0 0

0

0 0 0 0 0 0 0 0 0

0

0

0; ;

0

0

0 0 0 0 0 pump

a

A B

A Ba

a

a a a a b

Fx

Fx

FxCFC x FxcFx

Fx

Fc c

τ

τ ττ

τ

τ τ τ ττ ττ

τ τ

τ ττ τ

= =−

= = =−

/valve

=Equation of motion for tank =Equation of motion for ship =Tank induced external forces

Figure 8 Structure of the equations of motion for a ship with a stabilising tank

The equation system is on the same form as [Eq.13], namely:

A x B x C x Fτ τ τ⋅ + ⋅ + ⋅ =ɺɺ ɺ [15]

In [Eq.15], F and x are defined as in Figure 8. Solving [Eq.15] will produce a complete model of how the ship with a tank will behave under the effects of the applied external forces1 6F − ,

which are wave induced and the internal force /pump valveF which is actually a moment since the

equation of motion for the tank is a moment equation.

4 Roll damping using passive U-tube tank To investigate the effects of a passive tank on roll motions, all one would have to do is solve the equations of motion for a ship with a passive tank [Eq.15] and compare the resulting roll response with one calculated using the equations of motion for the ship itself [Eq.13]. However, this requires all of the hydrodynamic coefficients in the A, B and C matrices to be calculated. This requires extensive calculations using strip theory. For an initial estimation of the effects on roll damping, a smaller model can be used. For simplicity, all degrees of freedom except roll- and tank motions are set to zero. This produces a simplified version of [Eq.15] as shown in [Eq.16].

13

444 44 4 44 44 44 4 4

/4 4

0

0 pump valve

Fa I a b c cx x x

Fa a b c cτ τ

τ ττ ττ τ τττ τ τ+ − −

⋅ + ⋅ + ⋅ =

ɺɺ ɺ

ɺɺ ɺ[16]

Tank coefficients are given by the dimensions of the tank according to Appendix 1. The hydrodynamic coefficients44a , 44b and 44c as well as the roll moment of inertia 44I are

estimated with simplifications suggested by [4], these simplifications are presented in [Eq.17] together with an estimation of the wave induced roll moment from the same publication. The displacement∇ in [Eq.17] is the mass displacement of the ship in kg, B is the maximum beam of the ship and shipη is taken from an estimation of the equivalent linear roll damping

coefficient of the hull in question using roll decay tests[13] as 0.023.

( )

( )

2

44

44 44

44 0

44 44 44 44

0.45

0.1

4ship

I B

a I

c g GM

b c I aη

≈ ∇ ⋅ ⋅≈ ⋅≈ ⋅∇ ⋅

≈ ⋅ ⋅ ⋅ +

[17]

40 0 2

2cos sin

2 2k t

W

b k b k bF g L e

k kρ ζ − ⋅ − ⋅ ⋅ = ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ + ⋅

[18]

In [Eq.18], 0ζ is the wave amplitude and k is the wave number.

4.1 Tuning of the tank Since the behaviour of the fluid in the tank is governed by its equation of motion [Eq.12], which in turn contains coefficients related to the dimensions of the tank itself, it is possible to tune the tank to the desired behaviour by changing these dimensions. As explained earlier the phase of the motion should differ 900 from the phase of the rolling motion. Since this is impossible to achieve for all frequencies, one frequency where this is true should be chosen. Since the roll motion is most violent close to the natural frequency of the ship, the tank should be tuned to give the maximum stabilising moment at that frequency. In practice, this means that the tank should be constructed so that its own natural frequency coincides with the one of the entire ship. This will produce large level variations in the reservoirs, corresponding to large stabilising moments at the natural frequency of the ship. Naturally the natural frequency should also be the frequency where the phase difference is 900 to achieve maximum performance of the tank. Observe that the equation of motion of the tank could be rewritten to separate tank angle terms from global motions. 2 2 4 4 4 4 6 6a b c a x a x c x a xττ ττ ττ τ τ τ ττ τ τ⋅ + ⋅ + ⋅ = − ⋅ − ⋅ − ⋅ − ⋅ɺɺ ɺ ɺɺ ɺɺ ɺɺ [19]

[Eq.19] could be seen as a damped spring-mass system with τ as the active variable and the right hand side acting as external forces. Setting these external forces to zero is equivalent with locking the ship and letting the fluid in the tank oscillate freely. This will yield the natural frequency of the tank system which can be obtained from the now homogenous

14

equation of motion [Eq.20]. The equation of motion is used to calculate the natural frequency of a given configuration since aττ and cττ is known from the geometry according to Appendix

1. The undamped natural frequency is assumed to be representative for the true damped natural frequency. The second use of this equation is to give a relationship between the decay coefficient of the tank motion, η and the coupling terms aττ and bττ .

20,

0,

0

2 0

2

t

t

a b c

or

where

c

a

b

a

ττ ττ ττ

ττ

ττ

ττ

ττ

τ τ τ

τ η τ ω τ

ω

η

⋅ + ⋅ + ⋅ =

+ ⋅ ⋅ + ⋅ =

=

=⋅

ɺɺ ɺ

ɺɺ ɺ

[20]

In the calculations in this thesis and in [3] a substitution for the internal damping η is done to

include the internal friction coefficient q in [Eq.11] as well as the stiffness coefficientcττ . In

practice, this is done by performing a scale model test of the tank oscillation, measure the true value of η and calculate the corresponding bττ according to Appendix 1. The used damping

coefficient is called tη .

The phase and the magnitude of the stabilising moment can be found by using the expressions for roll- and tank motions defined in [Eq.1] and [Eq.2] in the equation of motion for the tank [Eq.12]. [Eq.2] has also been changed to contain the tank angle τ instead of the resulting moment. This can be done since the stabilising moment is always in phase with the tank angle.

4 40

0

sin( )

sin( )e

e

x x t

t

ωτ τ ω ε

= ⋅ ⋅= ⋅ ⋅ +

[21]

The substitution in [Eq.21] inserted into [Eq.19] will yield an expression for the relation between 0τ and 40x in the frequency plane as well as the phase ε in [Eq.2]. The resulting

expressions are shown in [Eq.22].

( )( )

2

24 40

22 2 240

tan e

e

e

e e

b

c a

c a

x c a b

ττ

ττ ττ

τ τ

ττ ττ ττ

ωεω

ωτ

ω ω

− ⋅=− ⋅

− ⋅=

− ⋅ + ⋅

[22]

15

As stated before the tank motions contributions to the total excitation moment can be written as in [Eq.14], where the tank induced roll moment can be expressed as: 4 40 4 4sin( )eF F t a cτ τ τ τω ε τ τ= ⋅ ⋅ + = − ⋅ − ⋅ɺɺ [23]

Making the same substitution as in [Eq.21], the relation between the tank angle amplitude and the resulting stabilising moment in the frequency domain can be obtained.

( )( )

2404 4

0

224 440

22 2 240

e

e

e e

Fc a

c aF

x c a b

ττ τ

τ ττ

ττ ττ ττ

ωτ

ω

ω ω

= − ⋅

− ⋅→ =

− ⋅ + ⋅

[24]

The stabilising moment response to roll motion in the frequency plane as well as the phase of the roll motion (and stabilising moment) can be plotted to give an idea of how a certain tank will perform. In Figure 9, the natural frequency of the ship in which the tank is to be installed in ( 0.2≈ rad/s) has been marked. The data in Figure 9 has been created from the tank in Table 3.

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.40

20

40

60

80

100

120

140

160

180Phase of stabilising moment compared to phase of excitation

ω [rad/s]

ε [0 ]

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.40

0.5

1

1.5

2

2.5

3x 10

8 Transfer function for stabilising moment from tank

ω [rad/s]

Fτ 40

/x40

[N

m/r

ad]

Figure 9 Tank performance visualized as a phase and amplitude graph of the stabilising moment

Figure 9 shows the optimum performance, where the stabilising moment is at its maximum at the ships natural frequency and the phase difference is 900 at that same frequency. The fact that the peak of the stabilising moment coincides with the natural frequency of the ship indicates that the natural frequencies of the systems are the same.

16

As mentioned before the optimum performance is achieved by tailoring the dimensions of the tank and thereby the parameters in Figure 6. The two parameters that has the greatest influence on the performance is the height of the duct hd and the inner tank damping ηt. If these cannot be changed, attention should be turned to the width of the reservoirs wr and the width of the duct wd. The effects of an increase in each one of the parameters described in Figure 6 is shown in Table 2.

Table 2 Effects of increases of tank parameters

Increased parameter ω0t ε Fτ40

wr decreases decreases increases

ht not affected not affected not affected

wd decreases decreases increases

hd increases increases increases

rd not affected not affected not affected

hr not affected not affected not affected

xt not affected not affected increases

ηt not affected not affected at ω0t decreases

xB1 not affected not affected not affected Another thing that has to be concidered when designing an anti roll tank for a ship is the reduction of the metacentric height that the free water surfaces of the tank inevitably causes. This can be estimated by setting the ship at constant roll angle. This means that all velocity and acceleration terms in [Eq.15] are zero and the equation of motion for roll reduces to [Eq.25] where 4F is a steady applied moment required to sustain the constant roll angle.

44 4 4 4c x c Fτ τ⋅ − ⋅ = [25]

At a steady state, the tank angle τ will be equal to the roll angle 4x only with opposite sign.

The stiffness coefficient 44c is dependant on the current metacentric height which can be

written as a fraction of the original “solid” metacentric height SGM . This is done by

introducing a reduction term µ as described in [Eq.26] where FGM is called the “fluid” metacentric height.

(1 )F SGM GM µ= ⋅ − [26] [Eq.25] can now be rewritten to include the fluid metacentric height as well as the expression for 4c τ from given by the tank geometry. tQ is a scale term that is common for all terms in the

equation of motion, it is dependant on the size of the tank. The mass of the ship is denoted as m. 4 tc Q gτ = ⋅ [27]

2

2t r t

t

w w xQ

ρ ⋅ ⋅ ⋅= [28]

17

( ) 4 4(1 )S tm g GM Q g x Fµ⋅ ⋅ ⋅ − + ⋅ ⋅ = [29]

[Eq.29] can be rearranged to get an expression for the reduction term.

4

4

1 t

S S

Q F

m GM x m g GMµ = + −

⋅ ⋅ ⋅ ⋅ [30]

The last term in [Eq.30] is equal to one by definition, since the denominator represents the force needed to sustain a constant roll angle 4x which was also the definition of the

numerator. The expression for the reduction term then reduces to the one in [Eq.31]

t

S

Q

m GMµ =

⋅ [31]

The loss of initial stability is undesirable for this type of ship with relatively low SGM even without tanks since it will require the ship to be ballasted down to restore its stability. This increases the displacement and propulsion resistance of the ship. Ships with a higher value of

SGM however, might benefit from a reduction since it would make the motions smoother.

4.2 Evaluation of performance The methods described in the preceding section can be used to investigate the effects of a passive tank on a PCTC vessel undergoing steady roll excitation. Used data on M/V Fidelio is defined in Table 1. The dimensions of the tank that, by the method described above, has proven to fit the ship in Table 1 are defined in Table 3. The internal force of the tank, /pump valveF is set to zero in the

following calculations. The inner tank damping tη is dependant on the factor q in [Eq.11]

describing the internal friction resistance in the tank. As mentioned earlier this should be determined by experiment. Since this thesis does not include such an experiment. The value of the internal friction resistance and thereby the value of tη is taken from an example with a

similar sized tank (albeit with a smaller xt) in [3].

Table 3 Tank dimensions

Parameter Size Dim

wr 2 m

ht 5 m

wd 25 m

hd 0,125 m

rd -13,48 m

hr 2,5 m

xt 5 m

ηt 0,15 -

xB1 0 M

18

The result of the installation can be evaluated in several ways.

4.2.1 Roll-decay test To validate that the roll damping has actually increased, a simulated roll-decay test is conducted. The result is shown in Figure 10. The test is conducted by numerically solving [Eq.16] without any external moments and with a starting angle of 20 degrees. The result is compared to the decay behaviour in the real roll decay tests of the hull described in[13] and the time when the ship has reached a 5 degree peak amplitude differs with less than two seconds. Figure 10 also shows the tank angleτ . The tank angle obviously reaches its maximum allowed value (one reservoir is filled) during the first three amplitude peaks.

0 50 100 150 200 250 300 350-20

-15

-10

-5

0

5

10

15

20

Time [s]

Rol

l ang

le [

0 ]

Tank

Unstabilised

Stabilised

0 50 100 150 200 250 300 350-15

-10

-5

0

5

10

15

Tan

k an

gle

[0 ]

Time [s] Figure 10 Simulated roll-decay test with and without stabilising tank installed

Looking at Figure 10, it is obvious that the roll damping has increased. The system also shows a slight shift in its natural frequency since the system with an installed tank oscillates with a slightly longer period.

4.2.2 Roll excitation The second test is to exert a continuous sinusoidal moment on the hull by setting the external moment in [Eq.16] to4 40 sin( )W eF F tω= ⋅ ⋅ , where eω represents the encounter frequency of

the waves. The excitation is held at the natural frequency of the ship 40ω where reduction of

the motion is most desirable. The result is shown in Figure 11.

19

0 100 200 300 400 500 600 700 800 900 1000-10

-5

0

5

10

Time [s]

Rol

l ang

le [

0 ]

Tank

Unstabilised

Stabilised

0 100 200 300 400 500 600 700 800 900 1000-2

-1

0

1

2

Tan

k an

gle

[0 ]

Time [s] Figure 11 Simple sinusoidal excitation response of stabilised and unstabilised ship working at the natural

frequency

4.2.3 Frequency response To get an overview of how the ship responds to other frequencies than the natural roll frequency, a response amplitude operator is created for the system in [Eq.16]. The converged solutions for both the roll and fluid motions (solution when constant amplitude reached in Figure 11) are assumed to be sinusoidal, oscillating with the same frequency as the external moment.

40 44

0

sin( )

sin( )

x tx

t τ

ω ετ ω ετ

⋅ ⋅ + = ⋅ ⋅ +

[32]

To avoid extensive trigonometric calculus and problems with the phases 4ε and τε a complex

ansatz is made for the particular solution of the system as shown in [Eq.33] whereη̂ is the complex amplitude of the oscillations.

4ˆ( ) i t xt e whereωη η η

τ⋅ ⋅

= ⋅ =

[33]

Differentiation of this approach gives the expression for the velocities and accelerations.

ˆ i txi e ωω η

τ⋅ ⋅

= ⋅ ⋅ ⋅

ɺ

ɺ [34]

2 ˆ i txe ωω η

τ⋅ ⋅

= − ⋅ ⋅

ɺɺ

ɺɺ [35]

Inserting this in into [Eq.16] yields the complex form of the equation of motion.

20ˆ i t i te A B i C F eω ωη ω ω⋅ ⋅ ⋅ ⋅ ⋅ ⋅ − ⋅ + ⋅ ⋅ + = ⋅ [36]

20

Using the estimation of the wave induced roll moment in [Eq.18] and setting the tank internal moment to zero gives the complex amplitude η̂ as a function of the excitation frequencyω . The roll amplitude is obtained using the definition of the complex representation that states that the amplitude of the oscillation is the modulus of the complex amplitude, hence:

40

0

ˆx

ητ

=

[37]

The response amplitude 40( )x ω is shown together with the corresponding unstabilised

response in Figure 12. It is common to make the response amplitude non dimensional. In the roll response operator, this often means dividing the roll amplitude for each frequency with the corresponding wave slope for that particular wave. The wave slope is defined as 0 kκ ς= ⋅

where k is the wave number.

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.450

50

100

150

200

250

300

350

ω [rad/s]

x 40/k

ζ0 [

-]

Frequency response

Stabilised

Unstabilised

Figure 12 Frequency response for a ship with the tank in Table 3 together with the unstabilised response

Tests conducted by varying different tank dimensions and parameters, studying their impact on the obtained roll angle at different frequencies shows that, apart from the length of the tank

tx , the parameter that influences the roll behaviour the most is the internal tank decay

coefficient tη . tη is a measurement of how efficient fluid oscillations are damped within the

tank and is defined in [Eq.20]. The decay coefficient depends both on the geometry of the tank and the internal friction coefficient q in [Eq.11]. The frequency response for several internal decay coefficients are shown in Figure 13. The range of decay coefficients used is taken from an example in [3] which is based on a model test for a similar sized tank (albeit wit a smaller tx which does not influence the internal damping.) No new model test can be

incorporated into this thesis so the values used below are seen as attainable.

21

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.450

50

100

150

200

250

300

350

ω [rad/s]

x 40/k

ζ0 [

-]Frequency response

Unstabilised

Stabilised , ηt =0.1

Stabilised , ηt =0.3

Stabilised , ηt =0.5

Figure 13 Frequency response for a ship with tanks with different internal decay coefficents

For low decay coefficients the tank gives little amplitude reduction or in some cases an increase in roll amplitude compared the unstabilised ship away from the natural frequency. The unfavourable effects of the increased phase margin and reduced stabilising moment away from the natural frequency (see Figure 9) are enhanced by the “nervousness” of the fluid motions that are the result of low damping. However, the relative ease with which the tank can be set in motion increases the performance when operating close to the natural frequency. A tank with higher internal damping reduces the deterioration of performance away from the natural frequency but also allows for less optimal performance at design conditions.

4.3 Increased propulsion power As mentioned when discussing the tuning of the tank, a decrease in the metacentric height due to the free water surfaces in the tank is undesirable for this type of vessel with bad initial stability. Since loss of initial stability is something that is most often avoided at all costs, the consequence of lowering the metacentric height is that ballast is added in the lower parts of the ship to reduce KG and thereby, as [Eq.38] states, increasing GM0. Adding ballast increases the displacement of the ship, increasing the overall resistance.

0xWAI

GM KB KGV

= + − [38]

The required extra ballast weight can be calculated according to [Eq.39] where KGD is the desired new KG to keep the metacentric height constant, e.g. if the metacentric height has dropped by 10 cm, so should KG. The tanks themselves add mass below the centre of gravity and thus help to reduce the loss of the metacentric height. The geometry of the problem is described in Figure 14.

22

ballast

Dballast

KG h wKG

w

⋅∇ + ⋅=∇ + [39]

Figure 14 Effect on KG from added ballast To get an estimation of how the increased displacement wballast affects the resistance or, which is more relevant, the propulsion power, interpolated data from a towing tank experiment of the ship in question is used where the effects on the propulsion power from increased displacement and trim can be determined quite accurately. The increase in propulsion power must be related to the performance of the tank that caused it. To do this, the resistance of several tanks with increasing motion damping performances should be investigated. The increasing performance is achieved by increasing the obtained stabilising moment. Since the natural roll frequency of the ship as well as the original roll damping remains roughly the same for any tank configuration, the dimensions of the tank that affects the phase margin and peak value positioning according to Figure 9 and Table 2 cannot be changed. Of the parameters that remain, the most influential on the magnitude of the stabilising moment is the length of the tank in the longship direction, xt. Based on this, the increased performance is obtained by keeping all the parameters of the tank constant except for xt, which is set to increase from 0 to 20 m. The amplitude reduction at the natural frequency of the ship is obtained by taking the quotient of the obtained roll amplitude with and without installed tanks. This is once again done by solving [Eq.16] with a sinusoidal excitation and taking the quotient between the values of the stabilised and unstabilised response amplitude operators that corresponds to the natural frequency.

1Stabilised roll amplitude

Amplitude reductionUnstabilised roll amplitude

= − [40]

This quotient is calculated for each tank configuration as well as the increase in propulsion power required to compensate for the increase in displacement. This is calculated from towing

h

KG KGD

Current KG Desired KG

Ballast

23

tank data of the hull in question. The result is shown in Figure 15, which together with Figure 16 is created using the tank dimensions in Table 3.

0 10 20 30 40 50 60 70 80 90 1000

1

2

3

4

5

6

7

8

9

10

Decrease in roll amplitude at ω0 [%]

Pow

er in

crea

se [

%]

Figure 15 Power increase due to increased tank performance (increased xt) measured in decreased roll

amplitude at the natural roll frequency. Figure 15 is calculated for the same external moment as the simple roll test in Figure 11, that is, one that gives a roll amplitude of about 140 at the natural roll frequency of the ship without any extra damping features. Calculating the quotient at a different external moment produces different amplitude reductions. To illustrate this, Figure 15 is recreated for increasing values of the external moment, starting with near stillwater conditions. The maximum allowed tank length xt is set to infinity to get the entire span of the curves. The result is shown in Figure 16.

0 10 20 30 40 50 60 70 80 90 1000

1

2

3

4

5

6

7

8

9

Decrease in roll amplitude at ω0 [%]

Pow

er in

crea

se [

%]

Undamped roll amplitude : 00

Undamped roll amplitude : 50

Undamped roll amplitude : 100

Undamped roll amplitude : 150

Undamped roll amplitude : 200

Figure 16 Power increase due to increased tank performance at different values of the external moment

Unstabilised roll amplitude ≈0 0

Unstabilised roll amplitude 5 0

Unstabilised roll amplitude 10 0

Unstabilised roll amplitude 15 0

Unstabilised roll amplitude 20 0

24

As expected, the power loss for every percent the amplitude is reduced increases with the wave height. Another interesting phenomenon is that the graphs always end up following the same asymptotic curve corresponding to the curve for very small wave amplitudes. An explanation for the asymptotic nature is that the tank requires a certain degree of motion to be able to operate and give a stabilising moment. If the tank is efficient enough to reduce the roll motion to very small variations, it eliminates the very thing that fuels further damping, hence the tank can never reach the 100% reduction mark but stops at around 90%. The point where the graphs start following the asymptotic line can be deduced from the behaviour of the fluid in the tank. From Figure 6, a maximum possible tank angle can be calculated as:

( )2

tan( ) arctan

2

t rt rh hh h

w wτ τ

⋅ − −≤ ⇒ ≤

[41]

If this angle is reached, the reservoir on one side is completely filled. No more water can flow through the duct and no further stabilising moment can be obtained. As long as the fluid is allowed to oscillate freely, meaning that the maximum angle in [Eq.41] is never reached, the tank does not have to be longer to cope with larger roll motions. An increase in wave height will only result in tank oscillations with greater amplitude (giving a greater stabilising moment) and eventually the amplitude reduction will be the same. How great the reduction is depends on how much fluid is available, namely the length of the tank. A longer tank means a greater loss of metacentric height and thereby, as explained earlier, a greater increase in the resistance. This is the physical representation of the asymptotic curve where the added power depends only on how much the desired amplitude reduction is, not on how great the wave motions are. If the maximum possible angle is reached at any point, the stabilising moment will no longer grow with increasing roll amplitude. This will mean that less amplitude reduction is obtained which is why a point corresponding to a tank that cannot oscillate freely will be offset to the left (a lower reduction). If one looks at a given amplitude reduction, Figure 16 shows that a greater power loss is obtained for tanks that cannot oscillate freely since they have to be longer to archive a given reduction. The breaking point comes when the tank is long enough to be able to sustain an undisturbed fluid oscillation and still be able to deliver the desired amplitude reduction. The graph will then follow the asymptotic line that can be seen as the free oscillation curve. This means that, if the reservoirs are tall enough, the power increase would be the same regardless of how large the waves are (not including added wave resistance of course) and the graphs for all the wave amplitudes would follow the free oscillation curve. A taller reservoir also means more metacentric height lowering ballast. An increase in tank height thus decreases the needed ballast to compensate for the tank as well as increasing the wave height that it can cope with and still work with optimum performance. The tank performance shown in Figure 16 has significant resistance increases for constrained oscillating conditions. Therefore, a test is conducted where the height of the tank is increased. This is shown in Figure 17 where Figure 16 is recreated with a taller reservoir.

25

0 10 20 30 40 50 60 70 80 90 1000

1

2

3

4

5

6

7

8

9

10

Decrease in roll amplitude at ω0 [%]

Pow

er in

crea

se [

%]

Undamped roll amplitude : 00

Undamped roll amplitude : 50

Undamped roll amplitude : 100

Undamped roll amplitude : 150

Undamped roll amplitude : 200

Figure 17 Power increase due to increased tank performance at different values of the external moment

for a tank height of ht = 8 m As expected, the taller tank can cope with larger wave amplitudes better. However, space limitations in the hull and the disinclination towards intrusion of the cargo space must also be considered when determining the height of the reservoirs. The reasoning conducted above might be hard to grasp but the most important conclusions are listed below.

• A freely oscillating tank will achieve the same amplitude reduction for any wave amplitude since the tank motions and the produced stabilising moment will grow with increasing roll motions. This is due to the linearity of the used model.

• When the rolling motions increases so does the required tank motions to give a certain

amplitude reduction. If the tank is not tall enough to sustain these tank motions, the performance decreases and the tank will have to be made longer to achieve the same amplitude reduction. A longer tank gives more propulsion resistance which is why curves for higher wave amplitudes are lifted above the free oscillation curve.

• The diagrams in Figure 16 and Figure 17 can be interpreted in several ways. If one

looks at a fixed value on the amplitude reduction. The curves above tell how much power increase is needed to achieve that reduction for different sea states, in other words how long the tank has to be. If one looks on a fixed power increase corresponding to a fixed tank length, the curves to the right show how much amplitude reduction is achieved for different sea states.

When setting tank dimensions, regard should be taken for all the factors mentioned in this chapter and the design should be based on the situation for every individual case. For example:

• What is the desired amplitude reduction for different sea states?

Free oscillation curve for 5 m tank

Unstabilised roll amplitude ≈0 0

Unstabilised roll amplitude 5 0

Unstabilised roll amplitude 10 0

Unstabilised roll amplitude 15 0

Unstabilised roll amplitude 20 0

26

• How much space is available for the tank in the hull?

• How does the cost of increased propulsion resistance, relate to decreased costs due to less voluntary speed reductions and other factors related to decreased roll motions?

5 Parametric roll prevention Of major interest in this analysis is whether or not stabilizing tanks can be used to reduce the risk if parametric rolling. Parametric excitation can occur when the water surface produces wave crest at the fore and aft and a depression amidships and vice versa as shown in Figure 18.

Figure 18 Changes in waterline area due to wave profile in following and head seas Due to the slender hull with large bow and stern flares, small variations of the waterline area amidships and a drastic increase at the end ships are the result of such wave crests and depressions. An increase in the total waterline area means a larger righting moment and a greater metacentric height. The metacentric height is derived from the moment if inertia of the waterline area around the longship x-axis

xWAI as shown in [Eq.38]. KB and KG are the

vertical centres of buoyancy and gravity of the ship and V is the total displaced volume. The ship will hence get a greater righting moment as long as this condition endures. If the situation is reversed and the ship has a crest amidships and depressions at the end ships the waterline area will still not change amidships but drop drastically at the endships. This will result in a smaller total waterline area and thereby a reduction in the metacentric height and the righting moment. As seen in Figure 18, the waterline area of the mid section remains relatively undisturbed by the wave while the area at the endships changes drastically. The dashed line in Figure 18 represents the waterline area at stillwater conditions. If the waves has a relative speed compared to the ship, this will mean that the waterline area and thereby the initial metacentric height will oscillate around the stillwater condition. The oscillation of the metacentric height is used to conduct an initial study of the effects of parametric excitation on a ship with installed tanks.

Wave and ship profile Waterline area

27

5.1 Simulating parametric roll with a 2 DOF model The initial investigation is conducted using a 2 DOF model (x4 andτ ) by setting the external roll moment to zero in [Eq.16], equivalent with following and head seas. Next the initial metacentric height of the ship is set to oscillate around its original value. The oscillation of the righting moment by itself is not enough to trigger parametric roll, the variation would have to occur with twice the ships natural roll frequency so that, given an initial disturbance the ship will loose its stability for one forth of a roll period (going from 00

roll angle to some peak value), making it heel over only to be forced back by a large increase in the righting moment for the next forth of a period going back from the peak value to 00 again where the stability is once again lost but now with a much larger initial velocity making the next heel more severe. This resonant phenomenon is the one called parametric excitation and, if the frequency endures, this will make the oscillation amplitude grow rapidly causing violent rolling. Parametric roll could hence be simulated by letting GM0 varying as in [Eq.42] where GM0 represents the initial value of the metacentric height and 0ξ the amplitude of the variation

determined by the size of the waves and the shape of the hull.

� ( )0 0 40 0sin 2GM t GMξ ω= ⋅ ⋅ ⋅ + [42]

A reasonable guess as to the magnitude of 0ξ for M/V Fidelio is half of the original

metacentric height of 1 m. Running the simulation by solving [Eq.16], keeping the wave induced forces at zero and varying GM0 as described in [Eq.42] and thereby varying the rotational stiffness coefficient 44c in [Eq.16] will produce a first simplified simulation of

parametric roll. The result of the simulation is shown in Figure 19 which also shows the tank angle. The tank dimensions in Table 3 are used.

0 20 40 60 80 100 120-20

-10

0

10

20

30

40

Time [s]

Rol

l ang

le [

0 ]

Tank

Unstabilised

Stabilised

0 20 40 60 80 100 120-5

0

5

Tan

k an

gle

[0 ]

Time [s] Figure 19 Simulation of parametric roll using variation of GM 0

28

The phenomenon where the roll amplitude increases steadily can be clearly seen in the unstabilised case where roll amplitudes of up to 300 are occurring after 100 seconds of simulation. The ship that has the installed tanks however shows almost no tendencies to be subjected to the phenomenon. According to the 2 DOF model, the passive tank is enough to greatly reduce the risk of parametric rolling. The effectiveness of passive tanks to prevent parametric rolling is not an established fact. On the contrary, there are investigations that support the above results by stating that the instalment of an anti rolling tank can greatly reduce or eliminate the phenomenon for certain sea states [8], however, others like an ongoing research project at NTNU [9] claims that it is useless and can sometimes make the phenomenon worse. Clearly the effect of the tanks depends on their design, the applied sea state and probably other unknown factors. To broaden the study and reduce the assumptions that might give faulty results, a more advanced approach is taken.

5.2 Simulating parametric roll with a 4 DOF model To get a better picture of passive tank behaviour in this particular vessel, the tank theory described by Lloyd is applied on an existing model capable of accurately capturing parametric excitation. This is essentially done by solving the entire system in Figure 8, including the added terms associated with the tank, but having the external forces, damping, stiffness and inertia of the ship calculated much more accurately than with the simplifications in [Eq.17] and [Eq.18]. These are now calculated using integration over time and over the surface of the hull. The entire system in Figure 8 is solved in the time plane. Furthermore, restrictions like the maximum allowed tank angle ([Eq.41]) are set. The model can simulate the wave-ship interaction accurately enough to capture phenomenon like parametric excitation. The used model was presented as a master thesis at the centre for naval architecture at KTH in 2009 [10] and is validated with a series of model tests. In this model a hull geometry and load case is specified. The hull used is the M/V Fidelio (Figure 1) with the load case she had on December 23rd 2008 when she had a recorded incident of parametric rolling which has been investigated by Seaware [11]. In the simulations conducted, the surge, sway and yaw degrees of freedom are locked. To get a first estimate of passive tank efficiency at parametric roll excitation using this model, the event on December 23rd 2008 is replicated. The wave period is set to 9.2 s, the wave height to 4 m and the waves are set to roll in from dead astern, these are the conditions that according to Seaware caused the incident [11]. The ship is set to move with 10 knots. In these conditions, a variety of tank configurations are tested and the result is shown in Figure 20.

29

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

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0

10

20

30

40η 4

[°]

t [s]

η4

η4 , ht = 5 m , xt = 10 m

η4 , ht = 11 m , xt = 10 m

η4 , ht = 5 m , xt = 5 m

η4 , ht = 11 m , xt = 5 m

η4 , ht = 11 m , xt = 15 m

η4 , ht = 5 m , xt = 15 m

Figure 20 Simulations at 10 knots, 4m waves from dead astern with a 9.2 s wave period with several tanks. The development of the dotted curve which represents the behaviour of the unstabilised ship closely resembles the real development recorded by the ships gyro. In this case, little difference is shown between 5 m and 11 m tall tanks. If the simulation is run further, and higher roll angles is achieved however, the tank height becomes significant. Longer simulations also show that, regardless of what configuration is chosen, the roll angle will eventually increase to above 20 degrees. The difference comes in how quick the development is. This is crucial since longer time series are purely theoretical. It is likely that the captain will change course or that the sea state will change enough for the parametric excitation to stop after a few minutes. The more time the captain has to react before the roll angles reach dangerous levels, the less risk for damage to the ship and cargo.

5.3 Extended study with the 4 DOF model Because of the somewhat unpredictable nature of passive tanks, one simulation is deemed as not enough to come to any conclusions about their efficiency. Therefore, a series of simulations with the 4 DOF model are ran over a large range of wave periods and encounter angles to get the whole picture on how the tanks cope with the phenomenon. The simulations are ran in a spectrum over wave periods of 4-15s in 4 m waves using a ship holding a constant speed of 10 knots throughout the simulation. The speed and wave height is consistent with the reigning conditions during the December 23rd incident. To get a reference a simulation is ran without any installed roll stabilising systems. The result is shown for all wave directions in Figure 21.

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Figure 21 Results of simulations at 10 knots, 4m waves and multiple wave periods and directions. The

elevation indicates the obtained maximum roll angle [0] after 150 s. If the roll angle for the exact heading and wave period where the incident happened is extracted from the data that Figure 21 is based on it differs with less than 1 degree compared to what roll angle was measured onboard. 34.4 degrees was calculated and 34.6 degrees was measured. Next a simulation with a 10 m long and 14 m tall tank is ran, the result is shown in Figure 22

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Figure 22 Results of simulations with a 10 long, 14 m tall tank, at 10 knots, 4m waves and multiple wave

periods and directions. The elevation indicates the obtained maximum roll angle [0] after 150 s.

31

The extent of the problematic area has been greatly decreased. However, the promising results of Figure 20 is shown to be somewhat lucky since only a small change in the wave period would have yielded much less reduction of the roll behaviour. The performance of the passive tank is analyzed by subtracting Figure 21 from Figure 22 producing the achieved amplitude reduction at the different sea states. The result is shown in Figure 23 where the sea states that correspond to the encounter frequency is equal to twice the natural frequency of the ship is shown (solid red line.) As shown in Figure 13, the ship with a tank installed will actually have two natural frequencies corresponding to the two peaks (the used tank has an internal damping coefficient of 0.15.) These two peaks are also shown in Figure 23 (dashed white lines.)

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Figure 23 amplitude reduction [0] achieved with 10 m long and 14 m tall passive tank, based on the data

from Figure 21 and Figure 22. There are two distinct areas where the amplitude reduction is significant. The steep incline at stern waves is moved to a longer wave period producing the large reduction around 9 seconds and 180 ± 45 degrees. This is where the simulations in Figure 20 are run and explains the promising results. Another area of large reduction is the area around 235 degrees, 11 seconds where a large dip is present. The tank also enhances the motions for longer periods close to 15 seconds for following seas. Why the performance of the tanks differs so much over the tested range of sea states has many possible reasons. One theory is that the frequency of the parametric rolling itself varies slightly around the natural frequency giving different performances for a tank that is tuned to that particular frequency. To get a better understanding of what causes these great differences in performance, the time series at 235 degrees, 11 seconds and 200 degrees, 15 seconds is extracted. The time series for when little or no reduction is achieved is also extracted at 200 degrees, 11 seconds. The result is shown in Figure 24.

32

0 50 100 150-40

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0

20

40µ = 2350 , T=11 s

Time [s]

η 4 [°]

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0

20µ = 2000 , T=15 s

Time [s]

η 4 [°]

0 50 100 150-40

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0

20

40µ = 2000 , T=11 s

Time [s]

η 4 [°]

With tanks

Without tanks

Figure 24 Time series extracted at points of improvement, deterioration and no effect of roll behaviour.

It seems that the greatest reduction is achieved when the parametric rolling itself happens close to the natural frequency of the ship itself (not any of the tank-ship interplay frequencies.) This can be seen in the first part of Figure 24 where the ship rolls at a higher frequency initially, without the tank the ship starts to roll with its natural frequency but since the tank is tuned to give maximum performance at this frequency (see Figure 9) it is very efficient at countervailing this particular behaviour. In the last part of Figure 24 on the other hand, the parametric rolling occurs at a slightly different frequency and since the performance curve is steep around the natural frequency, little or no reduction is achieved. The middle part of Figure 24 represents a case where the tank has actually made the phenomenon worse. The encounter frequency here is close to one of the coupled natural frequencies of the tank and ship (lower dashed line in Figure 23.) Without the tank, no parametric rolling occurs but since the ship-tank system is easily put into motion at this frequency a resonant phenomenon occurs where the amplitude grows beyond normal roll behaviour. The conclusion is that the amplitude reduction is the greatest when the parametric rolling happens at the natural frequency since this is the frequency where the tank is tuned to be at maximum performance. This most often corresponds to when the encounter frequency is twice the natural frequency. However, the complex wave-hull interplay gives rise to other effects that triggers parametric rolling at other frequencies and stops it from happening in some cases when the encounter frequency is “right”. This explains why most of the areas of large reduction lies along the solid red line of Figure 23 and why much of the plateau in Figure 21 remains untouched by the tanks. If the encounter frequency is close to any of the coupled natural frequencies, the system might trigger parametric excitation where none was present without the installation. As mentioned before, several similar investigations has been made but with far fewer simulation cases. The great differences in how the tanks perform in

33

relatively similar sea states might give an explanation as to why some studies supports passive tanks and some claim that they are useless for countervailing parametric excitation. This conclusion means that if a control system was installed that could capture the initial growth of parametric roll and tune the natural frequency of the tank to the frequency it occurs with, the same good performance as in Figure 20 could be achieved for all sea states. To test how an increase in the stabilising moment would affect the results another test is conducted where the length of the tank xt is increased to 15 m. The results can be seen together with the results for the 10 m tank in Appendix 5. Since the results of Figure 22 suggested that the motions could be enhanced due to the coupled natural frequencies, a test is ran where the internal damping coefficient is set to 0.4 to give a more even performance (see Figure 13). The result of this is a more even but lower reduction of the amplitude which is shown on the second page of Appendix 5 together with a comparison with a tank with lower internal damping.

5.4 Risk assessment To be able to give specific numbers on how much the risk of parametric rolling has decreased, the results of the simulations over many wave directions and periods must be related to wave statistics. The method used is similar to the method for calculating the total operability described in [4] where, instead of calculating the ships response at a certain sea state, the obtained roll angle from the simulations at that state is compared to a maximum allowed value. Statistics on what frequency waves with 4m amplitude occur in the North Atlantic where one of the trade routes of Wallenius vessels is located is used. All wave directions are said to be equally frequent and regular waves are assumed. The numbers of observations where the roll angle would exceed a certain value is compared to the numbers of observations where it does not. The threshold value is set to vary from 10 to 45 degrees to see how the definition of parametric roll affects the risk of experiencing it. The result is shown in Figure 25.

5 10 15 20 25 30 35 40 450

1

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3

4

5

6

Roll angle

Ris

c of

exc

eedi

ng r

oll a

ngle

[%

]

Without tanks

xt = 10 m

xt = 15 m

Figure 25 Risk of exceeding certain roll angles calculated from data in Figure 21 and Figure 22 as well as

for the 15 m tank.

34

The risk of exceeding angles below 7 degrees increases rapidly so a good definition of the border between regular rolling and parametric induced rolling is deemed to be between 5-10 degrees. In this area the reduction is about 30% for the shorter tank and 50% for the longer. However the risk of experiencing roll angles of over 30 degrees is reduced by 50% for the shorter tank and almost 100% for the longer. This is of course not the true risk of parametric rolling, only in 4 m waves in the North Atlantic; the results give a hint as to the magnitude of the reduction however.

6 Active tanks However convenient it is to work with the predictable, simple sinusoidal excitation associated with regular waves, the real conditions at sea are far from predictable. The idea of passive tank damping presupposes that the ship will roll with its natural frequency and that the fluid motions will follow the pattern described in the introduction to passive tank theory. In an irregular sea state the ship would mainly roll with its natural frequency since this is where the excitation would have the greatest impact on the roll angle. However, if multiple wave systems are present, these would also affect the roll angle, albeit with a smaller impact. The result is a ship that rolls with the natural frequency but gets unpredictable disturbances when waves with relatively large amplitudes, of different frequencies manage to influence the motion significantly. The result is that the harmonic tank motions are unsettled and when the disturbance is gone, the tank angle is out of phase with the roll angle resulting in poor performance and even increased rolling motions until the tank motions gets back into phase. Even if the overall roll motions are decreased with a passive tank, the system can be made somewhat more effective with the installation of an active control system. Such a system helps the fluid motions to be in phase with the roll and corrects for disturbances from the desired motion pattern.

35

6.1 Pump system The easiest way of defining a control system is to define a pump force that is directly controlled by several weighted factors giving the current state of ship and tank-fluid motions. The control system can then be written as shown in [Eq.43]. pumpF is the external force in the

tank equation of motion ([Eq.12]). ( )0 1 2 3 4 5 6pumpF F g x g x g x g g gτ τ τ= ⋅ ⋅ + ⋅ + ⋅ + ⋅ + ⋅ + ⋅ɺɺ ɺ ɺɺ ɺ [43]

The weighting factors 1 6g − can be obtained by running an optimization process, testing what

combination gives the best performance of the tank. This is done for a long time series of onboard-measured roll angles during a trip in rough weather. The roll angles are converted into the equivalent roll moment which would have produced these roll angles on the unstabilised ship. The roll moment is then applied as an external force on the tank stabilised ship whose motions are compared to the unstabilised roll series. Even if the tank will be somewhat optimised to cope with this particular sea state, if the state is representative enough for any given condition encountered at sea, the control system will be valid and close to optimum for the tank configuration. The optimum control system can be calculated with two different criteria:

• The system that gives the smallest total roll energy

• The system that produces the least number of peak angles above 50 The optimisation process with the different criteria is shown in [Eq.44].

max

41 6

1 6

1 6

01 6

0

1 6

( , )

( , )( )

( 5 )

(min( ( )))

t

peak

opt

FShip with control system x t g

g

x t g dtf g

sum x

g find f g

−−

−−

−

→ →

⋅

= >

=

∫ [44]

The optimum control system for both these criteria turns out to be identical and is shown in Table 4. The maximum allowed power that the pump can operate at is set as a constraint in the optimisation; the given power is set to 200 kW. The result of the addition of a control system is shown in Figure 26.

36

Table 4 Optimal values for the control coefficients in [Eq.43] for a 200 kW pump system

Parameter Optimal value

F0 4.4790e+007

g1 6.4395

g2 -9.5951

g3 0.7891

g4 5.5137

g5 4.4398

0 200 400 600 800 1000 1200 1400 1600-20

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-5

0

5

10

15

20

Time [s]

Rol

l ang

le [

0 ]

Tank

Active stabilised

Passive stabilisedUnstabilised

Figure 26 Irregular roll series with passive and active tank stabilisation

As shown in Figure 26, the difference between the active and the passive systems is hardly noticeable. The decrease in roll energy increases from about 50% with a passive system to 51% with the pump system. The small difference can be explained if one looks at the pump behaviour. For this example, the pump is almost constantly working at full capacity (200 kW) and is therefore unable to correct the motions fast enough to have any significant effect. The shear size of the tank needed for this ship thus makes it unfit for pump powered active tank damping. Although the pump system has been dismissed as a system for regular roll damping for this ship, a smaller pump might give good performance when it comes to preventing parametric rolling. As shown in the section on parametric rolling, the persistence of the critical frequency is critical to whether the ship will start to resonate or not. If a pump could be used to fill one reservoir completely when the initial growth is detected, the disturbance could put the ship out of sync before the resonance becomes dangerous. The pump would be better at effectively killing of the initial growth since both the passive and the valve systems require some degree of motion to be able to generate large tank angles and thereby large disturbing forces.

6.2 Valve system Instead of forcing the fluid to the desired position using a pump which, as described above, requires much energy, the fluid motions could be halted and thereby controlled by a series of valves. This method requires a closed system filled partially with fluid and partially with air

37

as shown in Figure 5 (c). If the air flow is halted by a closed valve, the water cannot move further without compressing the air on one side and decompressing it on the other. In practice this means that the water motions are completely halted when the valve is closed. If the valve is open partially the system can move, albeit more inertial than with an open valve. This can be used to gain the same tuning effect on the water motions as the duct with the advantage of being able to completely halt the motions at will, which is desired for longer rolling periods. A valve controlled tank most often has a duct that is optimised so that the passive operation of the tank (i.e. when the valve is completely open) is optimal at the lowest expected rolling period. The system is then made more inert or stopped entirely by closing the valve. The system that controls the valve actions is similar to the pump control system described above with the permeability of the valve as the target variable instead of the pump force. If the desired permeability is negative, i.e. the water is forced to flow the wrong way compared to what is optimal with respect to stabilising performance, the valves remain shut to halt the motions until they are back in phase. This can be compared to the pump working backwards to halt the water flowing the wrong way, although almost no energy is required to keep the valves shut. The valve system is optimised in the same way as the pump system, using the same irregular time series as a reference. The valve is modelled as a retarding force working in the opposite direction of the flow. The target function is the same as in [Eq.43] but the optimisation has the added constraints that the sign of the force and the flow must be opposite. Furthermore, if the force grows large enough to make the flow approach stagnant conditions, the tank motions are halted modelling a complete closure of the valve. The resulting system is shown in Table 5 and the resulting behaviour of the ship is shown in Figure 27.

Table 5 Optimal values for the control coefficients in [Eq.43] for a valve system

Parameter Optimal value

F0 100

g1 .0294

g2 -3.9408

g3 11.28851

g4 -5.8651

g5 -3.6978

38

0 200 400 600 800 1000 1200 1400 1600-20

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0

5

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20

Time [s]

Rol

l ang

le [

0 ]

Tank

Unstabilised

Stabilised

0 200 400 600 800 1000 1200 1400 1600-15

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-5

0

5

10

15

Tan

k an

gle

[0 ]

Time [s]

Figure 27 Irregular time series with valve controlled system

With the valve system the reduction in roll energy grows to about 56% compared to 50% for the passive system, a relatively small increase. The benefit of this system however is the reduction of major peaks. Since the valve system is able to halt the motions when they come out of synchronization, the system eliminates de deteriorated performance associated with a system out of phase. This can be seen in the behaviour of the tank angle in Figure 27 where the angle levels out at around 170 s where a disturbance occurs in the rolling moment. The retarding nature of the valve is also seen in that both coefficients related to tank motions (g4 and g5) in Table 5 are negative. The improvement can be put into perspective by looking at a similar comparison between passive and active tanks with similar results made in 2007 [12] that claims that “to achieve a specified roll reduction, the weight of a passive tank might be as large as five times that of an active tank.”

6.2.1 Bias in the optimisation The sea state that is used for the optimisation is said earlier to be representative for any sea state in rough weather. However, since the target of the optimisation is the reduction of roll energy, there is always a risk of optimising the system to remove certain key peaks and disturbances that has large impacts on the total energy. The risk of optimising the system for one particular sea state is investigated by putting the ships with the control systems in Table 4 and Table 5 into a completely different sea state and measure the reduction in the total roll energy. The pump controlled system reaches just below 51% reduction which is almost the same as for the target sea state. The valve system reaches 54% reduction. The fact that the systems behaves slightly worse in a different sea state than the one used for optimisation illustrates the need for longer more representative series to reduce the risk of optimising to reduce certain key peaks and disturbances. The fact that the systems work relatively well however proves that the method works and that the calculated control coefficients are sufficient in this early study of the concept. If they were to be optimised further the result would probably be a system that performs worse in any individual sea state compared to a system optimised for that state but performs better overall. For clarity,

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the passive system reaches almost the same reduction in all simulated sea states eliminating that factor from the bias.

6.2.2 Further use There are even more benefits in actively controlling the tank than the increased damping of irregular motions. As mentioned when discussing the effects of passive tanks on the impact of parametric excitation, it is desirable to stop the motions quickly before they are allowed to grow. Since the passive tanks gets a very small stabilising moment for small disturbances it might not be able to catch the initial growth of the parametric roll. It would therefore be better if the motions could be amplified using some control system, making the tank better at catching the phenomenon. The applicability of passive and active tanks on parametric roll damping as well as other related methods is discussed in chapter 5. In the valve system the retarding force comes from a pure energy loss due to turbulence in the closing valve. One possibility is to replace the valve with a turbine connected to a generator with variable resistance. In this way, the energy could be harvested and used to charge batteries and power appliances onboard. To get an idea of what the magnitude of the energy transfers are, the retarding pressure in the air duct is multiplied with the volumetric flow in the air tube which is equal to the volumetric flow in the reservoirs as shown in [Eq.45].

2turbine ret t r

wP p x wτ= ⋅ ⋅ ⋅ ⋅ɺ [45]

This represents what power the turbine would have to extract from the airflow in the tube and roughly what axial power would be transferred to the generator. Observe that the external influence /pump valveF is actually a moment working on the fluid around the centre of gravity and

not a force, it must thus be converted to an equivalent pressure drop in the air ductretp .

This is done by saying that the overpressure on one side of the valve also acts on the corresponding water surface in the reservoir with a lever arm of w/2 with respect to the centre of rotation, and the underpressure on the other side works on the water surface of the other reservoir so that:

//

22

2 2pump valveret t r

pump valve rett r

Fp x w wF p

w x w

⋅⋅ ⋅= ⋅ ⋅ ⇒ =⋅ ⋅

[46]

This is the pressure drop that would correspond to the correct moment in [Eq.43]. The turbine would naturally have to be reversible since the flow will shift directions. The turbine should also be complemented with a butterfly valve to stop the flow entirely if necessary. Something that would have to be investigated further in this concept is whether the turbine could stop the flow efficiently enough to achieve the same performance as the valve. Even if there is large rotational resistance from the generator, the air could still leak between the impeller blades keeping the flow in motion. The transferred power providing that the turbine could be as efficient as the valve is shown in Figure 28 and is evaluated in the same time series as in Figure 27.

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ardi

ng p

ower

[W

]

Time [s] Figure 28 Retarding power from valve or turbine required to achieve the performance in Figure 27

6.3 Combined system A combination of a pump and valve system would be the best solution since it could both efficiently damp regular rolling motions with very little added power. It could also be used to prevent parametric rolling and could be integrated with the ships anti heeling systems to reduce the number of tank systems installed. Such a system with combined anti roll and anti heel systems is available today from the manufacturer Rolls Royce and is marketed under the INTERING brand. This system does not contain a module for parametric roll prevention but contains both a pump and a valve system that could possibly be updated with software for this purpouse. A sketch of the system taken from a Rolls Royce product catalogue is seen in Figure 29.

Figure 29 INTERING combined anti heeling and anti roll system with valves and pump systems combined

7 Other ways of increasing the roll damping To get a better picture of how the benefits and disadvantages of passive tanks are related to other ways of achieving the same result. The two most common systems for roll damping used on the merchant fleet are stabilising fin systems and bilge keels. Another idea that will

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be investigated is whether an actively controlled rudder could be used for roll stabilisation. These three systems will be explained summarily and their performance compared to that of a passive tank.

7.1 Stabilising fins As with most roll stabilising systems, the idea of stabilising fins is to produce a moment that opposes the motion of the ship and thereby reduce the amplitude of heeling. The way a stabilising fin system does this is by changing its own angle of attack to generate a lifting force, the lifting force acts on the fin that is situated on the outside of the hull and thereby generates a moment around the centre of gravity. The source of the lift is naturally the forward speed of the ship generating a flow around the inclined fin. The general principle is shown in Figure 30. The theory for obtaining roll stabilisation from active fins is taken from [3].

Figure 30 Principle of stabilising fins To be able to oppose the rolling motion effectively, the angle of attack is controlled by a computerized system that takes the roll angle, given by a gyro as input and sends a signal to the fin servo stating what angle of attack is desired. The control system can have a transfer function on the form as shown in [Eq.47] where dα is the desired angle of attack and4mx is

the roll angle measured by a gyro.

( )

21 2 3

4 21 2 3

4( ) ( )

d m G U

d m

K K s K sx K K

b b s b s

or

s X s G s

α + ⋅ + ⋅= ⋅ ⋅ ⋅+ ⋅ + ⋅

Α = ⋅ [47]

The expression on the right hand side of [Eq.47] is the control system itself whose components and their purpose are explained in Table 6.

Water flow due to ships speed

Lift force opposing roll motion

Angle of attack of fin

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Table 6 Components of the fin control system

Component Purpouse

KG Overall gain setting controling the magnitude of the fin motions

KU Speed dependant gain setting to compesate for reduced lift at slow speeds

K1 Roll angle sensitivity

K2 Roll rate sensitivity

K3 Roll acceleration sensitivity

b1,b2,b3 Controller coefficients s Laplace transform operator

Theory of automatic control engineering[14] states that, if a control system like the one in [Eq.47] is fed with a sinusoidal input signal, in this case 4 ( )mx t , the output will also be

sinusoidal as shown in [Eq.48].

( ) ( )

( )( )

( ) ( ) ( )

: sin

: ( ) sin( )

arg

Y s U s G s

input u t t

output y t G i t

where G i

ω

ω ω φ

φ ω

= ⋅= ⋅

= ⋅ ⋅ ⋅ +

= ⋅

[48]

The input signal is indeed sinusoidal and oscillating with the wave encounter frequency. The angle of attack ( )d tα for any given time can now be calculated according to [Eq.48] and the

lift generated by the fin is added to the external forces in the equation of motion for the ship.

7.1.1 Positioning of the fins The positioning of the fin on the outside of the hull is another factor that affects its efficiency. The further from the centre of gravity the fin is placed, the greater the moment a certain lift force will generate. For most ships, this means that a positioning close to the turn of the bilge is optimal. The longship positioning is also important. The nature of the boundary layer flow around the hull changes along the length of the ship. Far aft, the boundary layer is thicker meaning that the fluid close to the hull where the fin operates moves at a slower relative speed. Wing profiles operate best at faster flow speeds so, by this line of thought, it would be best to place the fin close to the stem. The need to increase the lever arm of the fin places the optimum position somewhat closer to the middle section where the hull has a more full bodied shape. If the ship is fitted with more stabilising fins on the same side, care must be taken so that they do not interfere with each other. A downwash from a forward placed fin can push down on one that is placed further aft negating the positive addition to the stabilising moment. The same reasoning can be applied to ships with bilge keels where a fin placed forward of the bilge keel pushes the bilge keel up or down and thus negating its own effect. A fin placed aft of a bilge keel however, makes profits from the more laminar flow that usually follows a long plate aligned with the flow. If two fins are placed on each side of the bilge keel the total negative and positive effects are often said to add up to zero.

43

7.1.2 Increased propulsion resistance All wing profiles generate a lift as well as a drag force. While the lift is parallel to the ships vertical centreline, the drag is always directed aft adding to the propulsion resistance. This enables the same analysis of the damping efficiency as a function of the added resistance that was done for the passive tank. For the case with the tank, the curves were generated by increasing the tank length and noting the resulting power increase and amplitude reduction. For the fins there are several possibilities, either the gain of the control system KG is set at a constant value and the fins increased in size or vice versa. The aspect ratio of the fins could also be varied. Since wing profiles operating at a certain speed range most often have an optimal size and aspect ratio. It would not be wise to change this. Based on this, the procedure will be to keep the size of the fin constant at a preferred size for the operating speed and instead change the overall gain of the control system. This will result in small motions giving small amplitude reductions but also little drag for low gain settings and ever increasing values of both damping and resistance as the setting is increased. The dimensions of the chosen fin as well as the resistance curves for different sea states are shown in Figure 31. Figure 31 is calculated at 19 knots. The dimensions of the fin are taken from Appendix 3.

0 10 20 30 40 50 60 70 80 90 1000

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Decrease in roll amplitude at ω0 [%]

Pow

er in

crea

se [

%]

Undamped roll amplitude : 00

Undamped roll amplitude : 50

Undamped roll amplitude : 100

Undamped roll amplitude : 150

Undamped roll amplitude : 200

Figure 31 Power increase due to increased fin performance (fin placed 100 m from stem at the turn of the

bilge) The curves for all sea states start at about 0.7% increase in propulsion power. This represents the case where the fin has no angle of attack and the only resistance is the one generated by the shape of the profile and the skin friction. If a retractable fin was installed, the curve for stillwater conditions could be set to zero. The performance is similar to that of the passive tank with 0 to 6 percent increases in the propulsion power for roll damping of 0 to 90% at the natural frequency.

5.1 m 1.7 m

Unstabilised roll amplitude ≈0 0 Unstabilised roll amplitude 5 0 Unstabilised roll amplitude 10 0 Unstabilised roll amplitude 15 0 Unstabilised roll amplitude 20 0

44

7.2 Bilge keels The easiest and most common way of reducing roll motions is the instalment of bilge keels. These are long plates protruding near the turn of the bilge. Since the bilge keels most often are aligned with the streamlines around the hull, they give little propulsion resistance. When the ship is rolling however, the protruding plates act as paddles, shovelling water along increasing the added mass, creates eddies and thereby increases the roll damping. The principle is shown in Figure 32. The theory for obtaining roll stabilisation from bilge keels I taken from [3].

Figure 32 Principle of bilge keel eddy generation.

7.2.1 Increased propulsion resistance The added form resistance of the bilge keel is most often negligible but still exists, the majority of the added resistance comes from the added wet surface which causes an increase in the friction resistance. The only reasonable parameter to vary on the design to achieve more roll damping is the extension of the plate, bbk as shown in Figure 33.

Figure 33 Extension of the bilge keel The keel is set to extend over half of the total length of the ship, centred in the middle. This produces a resistance curve in the same manor as for the fins and the tank shown in Figure 34.

bbk

45

0 10 20 30 40 50 60 70 80 90 1000

1

2

3

4

5

Decrease in roll amplitude at ω0 [%]

Pow

er in

crea

se [

%]

Undamped roll amplitude : 00

Undamped roll amplitude : 50

Undamped roll amplitude : 100

Undamped roll amplitude : 150

Undamped roll amplitude : 200

Figure 34 Power increase due to increased bilge keel performance

Since the bilge keels are non retractable, large values of bbk is unwanted since it increases the beam or draught of the ship. This is the reason why the curves stop at 6% increase in power; this corresponds to a bilge keel extending 2 m from the hull which is more than is normally seen as acceptable. As expected the bilge keel gives little resistance for amplitude reductions of up to 10%. As opposed to the tank and the fins, the bilge keel gives better performance for larger wave amplitudes.

7.3 Active rudder control Almost all new ships have some sort of autopilot system installed. The autopilot controls the rudder motions to keep the ship on a steady course. By changing the angle of attack of the rudder an athwartships force is generated. Since the rudder is situated in the stern, away from the longship centre of gravity, a yawing moment is created. The yawing moment created by the rudder makes the ship rotate itself around the vertical axis making the bow point with a small angle relative to its own heading direction. This rotation gives the ship an angle of attack relative to the free stream pushing it sideways and thereby turning. The same way the rudder generates a yawing moment since it is offset from the longship centre of gravity, it also creates a rolling moment since it is offset from the vertical centre of gravity as shown in Figure 35.

Figure 35 Principle of rudder generated rolling moment

Rudder force

Roll moment from rudder

Unstabilised roll amplitude ≈0 0 Unstabilised roll amplitude 5 0 Unstabilised roll amplitude 10 0 Unstabilised roll amplitude 15 0 Unstabilised roll amplitude 20 0

46

If the rudder control system would be expanded to correct for rolling disturbances as well as yawing disturbances another system for roll damping could be achieved, one that does not require any additional equipment installation but merely a software update. The rudder angle δ is controlled by a servo that is in turn controlled by the autopilot computer. No explicit method similar to the one for the fin control system exists for a roll damping rudder at the moment. To achieve the most efficient roll damping in rough seas an optimisation process is ran on the control system. For simplicity and proof of concept, the yaw-correcting capabilities are initially neglected. The rudder angle is said to be controlled by three weighting factors and the roll angle, velocity and acceleration according to [Eq.49]. The optimisation process to obtain these factors via a simulation in an irregular sea state is the same as explained in the segment on active tank control. ( )0 1 2 3g x g x g xδ δ= ⋅ ⋅ + ⋅ + ⋅ɺɺ ɺ [49]

The rudder performance in an irregular sea state is shown in Figure 36.

0 200 400 600 800 1000 1200 1400 1600-20

-10

0

10

20

Time [s]

roll

angl

e [0 ]

StabilisedUnstabilised

200 400 600 800 1000 1200 1400 1600

-5

0

5

Rud

der

angl

e [0 ]

Figure 36 Rudder stabilisation performance in an irregular sea state.

The rudder system achieves a reduction of the total roll energy by 48% with a maximum deflection of about 40. If the simulation is made for a different time series than the one used for the optimisation, the rudder reaches a 36% reduction illustrating the need for longer series in the optimisation progress as explained in the section on biased optimisation in active tank control.In the simulation, a maximum turn rate of the rudder is set based on the performance of the rotary vane steering gear. This is set to 90 per second.

7.3.1 Increased propulsion resistance The rudder itself does not create much resistance. If the deflection endures and the ship begins to turn however, the resistance increases rapidly. Something that has to be investigated if a

47

further study on the rudder controlled roll damping is to be made is if the direction of the rudder force changes rapidly enough to avoid the ship starting a turn.

7.3.2 Impact on manoeuvring As an introduction to the earlier mentioned problem of how the ship will react to the rudder compensation, a brief manoeuvring study is made. The theory used is explained in [2] where the used rudder is defined in Table 7.

Table 7 Rudder dimensions and properties

Rudder area A 36 [m2]

Rudder aspect ratio AR 1.33 (x2) [ - ]

Lift curve slope LC δ∂ ∂ 5.6 [ - ]

Velocity increase over rudder c/u 1.3 [ - ] With these parameters and the methods described in [2], a simulation of how the ship behaves in stillwater conditions is made. The rudder angles generated by the control system for the simulation in Figure 36 is set as an input and the resulting course change of the ship is studied. This will give a hint of how the manoeuvring would be affected when the rudder stabilising system is active. The result of the simulation is shown in Figure 37.

0 200 400 600 800 1000 1200 1400 1600-4

-3

-2

-1

0

1

2

3

4Rudder simulation Speed=19.8 [knot]

Time [s]

Ang

le [

°]

Rudder angle [°]Yaw angle [°]

0 0.5 1 1.5 2 2.5 3

x 105

-4

-2

0

2

4

6x 10

4 Position

x-pos [m]

y-po

s [m

]

Figure 37 Impact on ships course from the rudder deflections generated by the roll stabilising control

system As seen in Figure 37, the rudder deflections has little impact on the yaw angle save for an initial growth. The explanation for this is the poor course stability of the ship. Even though a stabilising fin is used in the simulation, an initial disturbance like the first turn of the rudder endures long enough to make the ship come several kilometres of course. This behaviour however, is a positive sign since it indicates that the rudder changes direction quickly enough to avoid the ship moving in a zigzag pattern creating resistance and reducing speed. The

48

initial disturbance can be thwarted with a well working yaw regulating system. To get the yaw and roll correcting systems to work in interplay with each other could be a subject of further study. To verify that the majority of the course deflection is due to bad course stability, a test is ran where a small disturbance is given at t=0 and the rudder angle set to zero. The ship veers of course for about 650 seconds before beginning to turn back, returning to 00 yaw angle after about 1700 seconds. This is roughly the same behaviour as in Figure 37 indicating that the behaviour is related to course instability.

8 Comparison between different systems To get a picture of the differences in performance and the strengths and downsides of each system, a comparing study is conducted.

8.1 Efficiency The study is made for waves that produces a 100 roll amplitude for an unstabilised ship and the result is measured in how the resistance curves relates to each other, the result is shown in Figure 38 where a 5 m tall tank is compared to a 5.1 m fin and a bilge keel extending over half the length of the ship.

Figure 38 Comparison between different roll damping systems in terms of power increase related to

amplitude reduction The tank and fin systems have equal performance if they are tuned to give between 40 and 70% amplitude reduction. The tank performs better for smaller amplitude reductions and the bilge keel performs worst for reductions above to 10%. When the ship is not operating in rough weather, the added resistance of both the fin and tank systems can be reduced. The tanks can be drained to increase the metacentric height and most fins on the market can be retracted into the hull to reduce drag. The bilge keel however gives an increase in the drag regardless of sea state.

0 10 20 30 40 50 60 70 80 90 1000

1

2

3

4

5

6

7

8

9

10

Decrease in roll amplitude at ω0 [%]

Pow

er in

crea

se [

%]

Tank

FinsBilge keel

49

The rudder stabilisation system gives little propulsion resistance compared to the others. It is also the one system that does not require any structural modifications on the hull. A study of how a control system for combined yaw and roll stabilisation should be made if this system is to be evaluated fully. All systems mentioned here has one advantage when it comes to propulsion power which has not been taken into consideration when creating the power increase curves. The hull shapes of all ships are optimised to have the smallest possible resistance when operating on an even keel with no roll angle. This means that a ship that is rolling will also get an increase in the propulsive power required to keep its speed. A reduction of the roll angles would hence give a decrease in the required propulsion power. The impact of this phenomenon is of course dependant on the individual hull form. However an estimate made in [1] states that the increase of the resistance is between 0.5 - 0.9 % for every degree of roll angle amplitude. For the example in Figure 38 where an unstabilised amplitude of 100 is used, a 40% reduction of roll amplitude would yield a 2% reduction of the required propulsion power using the lowest estimate of 0.5% per degree of roll. Both the tanks and fins gives an increase of about 1% at this level bringing the net difference in propulsion power to a 1% decrease.

8.2 Installation All treated systems require different installation processes. These are briefly summarized here for ships where the system is in the original plans and for ships that are fitted after their launcing.

8.2.1 In newbuilds When installing an anti roll tank in a newbuild, the bulkheads and decks are designed so that the reservoirs and duct are formed. These are properly sealed and valves are installed. If the tank is in the original plans little construction work is thus required. An example of how the tanks could be fitted in the hull of M/V Fidelio is seen in Appendix 2. This drawing is of an INTERING valve system and is created by the manufacturer Rolls Royce for Wallenius Marine. An overview of such a system is also shown in Appendix 4. Stabilising fins are most often delivered in pre assembled modules. They consist of; apart form the fin itself, a series of gears and hydraulic actuators. This is complemented with a system of lubricating tanks and hydraulic pumps to keep the system running. The module is roughly twice the length if the fin itself and houses all the control systems and a void into which the fin can be retracted when not in use. An example of such a module is seen in Appendix 3 and is taken from a product catalogue from the manufacturer Thyssen BV Industrietechnik GMBH. The modules are welded in place as a part of the hull structure.

8.2.2 Retrofitting A tank system can, in most cases, be installed while the ship is in dock. The hull of M/V Fidelio, as Appendix 1 shows already has space for a tank system so only minor changes to the internal structure would have to be made. These could be done without taking the ship out of operation for any significant amount of time. Retrofitting a stabilising fin system requires the ship to be brought into a dry dock. The hull has to be cut open and the pre assembled module fitted. The module would most likely be

50

placed at deck 1 (see Appendix 1) and as the product sheet in Appendix 2 shows this would mean an intrusion of about 4 meters into the hull. The width of the double plating here is between 2.5 and 3 meters meaning that some cargo space would be lost. A bilge keel is already installed. The rudder already has a computerized control system so an installation of a roll stabilising system would only mean a software update that can be done while the ship is in operation.

51

9 Conclusions The tank and fin systems are relatively equal when it comes to resistance increases for the interesting range of amplitude reductions. The performance of the tanks could be increased if a taller reservoir than the one the comparing Figure 38 shows was used while little change can be done to boost the performance of the fins. A taller reservoir than the one used and the one shown in Appendix 1 can be fitted. If the control valves are moved up to deck 4, a reservoir height of almost 14 m is possible. This leads to the conclusion that the passive tanks could offer a better performance for this ship than the fin system. An active valve system would further put the tanks in advantage with an increase in the installation cost as an effect. The increased installation work however, is not as extensive as when installing a fin module, especially if the ship is already launched which is the case with M/V Fidelio. Passive and active tanks are thus a relatively small investment that requires little maintenance and gives equal or better performance as comparable fin systems. If a complete system is bought, this can be merged with the heeling tank system to save space. This is done in the INTERING systems marketed by Rolls Royce where the tanks feature both a valve system for ordinary roll damping and a smaller pump to shift the transverse centre of gravity of the ship when loading asymmetrically. The pump could possibly be programmed to also deal with parametric rolling. It is shown that passive tanks give a noticeable effect on the risk of parametric rolling. Tanks in general also show great potential to possibly eliminate the risk altogether with the appropriate control system. Further studies have to be made on the performance of active rudder stabilisation. If this shows to have the same performance as tank and fin systems the minimal installation cost makes them superior from an economic point of view. With the knowledge obtained during this study the recommendation would be to install a combined anti roll and heeling system with valve controlled tanks and a small pump. Finally, if this study leads to the design of an actual tank in the ship in question, a less schematic model should be used to evaluate it.

52

10 References 1. Henschke. W, Schiffbautechnisches Handbuch vol. 1, p. 397, Berlin 1952 2. Kuttenkeuler, J, Surface Ship Controllability, KTH, Stockholm 2006 3. Lloyd, ARJM, Seakeeping: Ship behaviour in rough weather, Gosport United Kingdom 1998 4. Rosén, A , Introduktion till fartygs sjöegenskaper, KTH, Stockholm 2006 5. Watts, P, on a method of reducing the rolling of ships at sea, p 165 , United Kingdom 1883 6. Watts, P, The use of waterchambers for reducing the rolling of ships at sea ,United Kingdom 1885 7. Frahm, H, Neuartige schlingertanks zur abdämpfung von schiffsrollbewegungen und ihre erfolgreiche

anwendung in der praxis, Jahrbuch der schiffbautechnischen gesellschaft 12, p 283, Berlin 1911 8. Umeda, N , Hashimoto, H , Minegaki, S , Matsuda, A , An investigation of different methods for the

prevention of parametric rolling , J Mar Sci Technol 13:16–23 , Japan 2008 9. http://www.itk.ntnu.no/ansatte/Holden_Christian/prof/research.php , obtained on 090504. 10. Ovegård, E, Numerical Simulation of Parametric Rolling in Waves, KTH, Stockholm 2009 11. Palmquist, M, Analysis of heavy rolling incident on M/V Fidelio on 23rd Dec 2009, Technical report for

Wallenius Marine, Seaware AB, Stockholm 2009 12. Marzouk, O and Nayfeh, A , Mitigation of ship motion using passive and active anti-roll tanks , 2007

proceedings of the ASME International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, DETC 2007, v.1 Part A , p 215-229 , Las Vegas US 2008

13. Palmquist, M, Roll damping evaluation, Technical report for Wallenius Marine, Seaware AB, Stockholm 2009

14. Glad, T , Ljung, L , Reglerteknik Grundläggande teori 4:th edition p 81-83, Stockholm 2006 15. Letter to Wallenius Marine R&D from Captains of M/V Fidelio and M/V Faust, sent on 081201

53

11 Appendices Appendix 1 Coefficients in the equation of motion 54 Appendix 2 Proposal for installation of valve controlled tank system 55 Appendix 3 Fin module product sheet 56 Appendix 4 Overview of a combined anti roll and heeling system 57 Appendix 5 Results of parametric roll simulations 58

54

Appendix 1 Coefficients in the equation of motion The equation of motion for the tank [Eq.1] contains seven coupling terms that are dependant on tank dimensions. These are defined in [Eq.2].

2 2 4 4 4 4 6 6 /pump valvea x a x c x a x a b c Fτ τ τ τ ττ ττ τττ τ τ⋅ + ⋅ + ⋅ + ⋅ + ⋅ + ⋅ + ⋅ =ɺɺ ɺɺ ɺɺ ɺɺ ɺ [1]

The termsdr , rh , dh , rw , dw and tx are defined in Figure 1, 1Bx is the lateral distance from the

centre of gravity of the tank to the lateral centre of gravity for the entire ship.

( )2

4

4

6 1

2

2

2

22

[ ]2

t

t d r

t

t B

rt r

d r

rt r t

d r

t

t r tt

a Q

a Q r h

c Q g

a Q x

hwa Q w

h w

hwb Q q w c a

h w

c Q g

where

w w xQ kgm

τ

τ

τ

τ

ττ

ττ ττ ττ

ττ

η

ρ

= −= ⋅ += ⋅= − ⋅

= ⋅ ⋅ + ⋅

= ⋅ ⋅ ⋅ + = ⋅ ⋅ ⋅ ⋅

= ⋅

⋅ ⋅ ⋅=

[2]

Figure 1 Definition of terms regarding tank geometry

hd

wr wd wr

rd

hr

ht

Datum level τ

w

55

Appendix 2 Proposal for installation of valve controlled tank system

56

Appendix 3 Fin module product sheet

57

Appendix 4 Overview of a combined anti roll and heeling system

58

2

2

2

2

2

2

3

3

3

3 4

4

4

45

5

5

5

5

5

10

10

15

15

20

20

2525

30

30

35

35

4

4

3

3

33

2

2

2

2

40

40

44 4

5

3

55

3

35

2

5 s

10 s

15 s

0

45

90

135

180

225

270

315

No

stab

ilisa

tion

2

2

2

2

2

2

2

3

3

3

3 3

3

4

4

4

4

4

5

5

5

5

10

10

15

15

20

2025

25

2

2

3030

5

5

33 25

25

5

54

4

5 s

10 s

15 s

0

45

90

135

180

225

270

315

x t = 1

5 m

,

h

t = 1

4 m

,

η t= 0

.15

-20

-20

-20

-20

-20

-15

-15

-15

-15

-15

-10

-10

-10

-10

-10

-5

-5

-5

-5

-5 -5 -5

0

00

0

0

0

0

0

-10

-10

00

-15

-15

-10

0

0

0

-5

-5-15

0

0

55

5

5

00

00 5

s10

s15

s

0

45

90

135

180

225

270

315

2

2

22

2

2

2

3

3

3

3

3

3

4

4

4

4

45

55

5

5

10

10 15

15

20

20

25

25

30

30

2

23

3

3535

5

5

4

44

4

25

25

5 s

10 s

15 s

0

45

90

135

180

225

270

315

x t = 1

0 m

,

h

t = 1

4 m

,

η t= 0

.15

-35

-35

-35

-35-35

-30 -3

0

-30

-30

-30

-25

-25

-25

-25-25

-20

-20

-20

-20

-20

-15

-15

-15

-15

-15

-10

-10

-10

-10-10

-5

-5

-5

-5

-5

-5

-5

000 0 0

00

0

0

0

-10

-10

-10

-10

-15

-15

5

5

-20

-20

-150

-200

0 -50

0

-250

0

0

0

55

00

00 5

s10

s15

s

0

45

90

135

180

225

270

315

Appendix 5 Results of parametric roll simulations

59

2

22

2

2

22

3

3

3

3

3

444

4

45

5

5

5 5

55

1010

10

15

15

15

20

20

2025

25

25

30

30

35

35

4

4

3

3

3

3

2

2

22

4040

44

45

3

5

5

3

3

5

210

10

5 s

10 s

15 s

0

45

90

135

180

225

270

315

No

stab

ilisa

tion

2

2

2

2

22

223

3

3

3

3

33

4

44

4

4

4

5

5

5

5

5

5

10

1010

15

15

15

20

20

25

25

30

30

22

33

35

35

5

5

44 4

4

2525

3030

2020

5 s

10 s

15 s

0

45

90

135

180

225

270

315

x t = 1

0 m

,

h

t = 1

4 m

,

η t=

0.1

5

2

2

22

2

2

2

2

3

3

33

3

3

3

4

44

4

4

4

5

55

5

5

5

10

1010

15

15

20

20

25

25

3030

3535

2

2

5

55

s10

s15

s

0

45

90

135

180

225

270

315x t =

10

m

,

ht =

14

m

, η t=

0.4