Anti Roll Tanks in Pure Car and Truck Carriers

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Anti Roll Tanks in Pure Car and

Truck Carriers

B J Ö R N W I N D É N w i n d e n @ k t h . s e 0 7 3 - 7 0 1 7 2 5 6 Master thesis KTH Centre for naval architecture

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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

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i

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

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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

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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.

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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.

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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.

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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.

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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( e )

x =x ⋅

ω

⋅t [1]

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

0 sin( e )

M =M ⋅

ω

⋅ +t

ε

[2]

where M is the stabilising moment from the tank,

ε

should be 900 and M is the amplitude of 0

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

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.

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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 as

τ

, 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

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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,

ρ

_t is the density of the fluid and v is the flow speed at any point in the tank.

1 t v v P v Y t dy ρ y ∂ _{+ ⋅}∂ _{= −} _⋅∂ ∂ ∂ [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:

0 v 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 dP

Y

dt = −ρ ⋅ dy [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].

2

r

w

v = ⋅τɺ [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 t

w v⋅ ⋅ = ⋅ ⋅ρ n v 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

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2 r r r w v w w v 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 2 r t w w dP Y n τ ρ dy ⋅ ⋅ = − ⋅ ⋅ 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 x m forward of, _B₁ x m to starboard _B₂ of and x m above the centre of gravity can be described using the relative motions of the _B₃ ship using [Eq.10].

2 2 B3 4 B1 6 2 2 B3 4 B1 6

s = −x x ⋅ +x x ⋅x ⇒ ɺɺs = −ɺɺx x ⋅ +xɺɺ x ⋅xɺɺ [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 v Y 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 a_ij,b c_ij, _ij denotes coupling terms between

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2 2 4 4 4 4 6 6 pump valve/

a_τ ⋅ +xɺɺ a_τ ⋅ +ɺɺx c_τ ⋅ +x a_τ ⋅ +ɺɺx a_ττ ⋅ +τɺɺ b_ττ ⋅ +τɺ c_ττ⋅ =τ F [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⋅ + ⋅ + ⋅ =ɺɺ B x C xɺ F [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 F_τ in [Eq.14].

( , , )

A x⋅ + ⋅ + ⋅ = +ɺɺ B x C xɺ F F_τ

τ τ τ

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 aiτ and ciτ, depending on what degree of freedom is to be coupled with the tank angle.

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2 4 6 2 4 6 1 1 2 2 3 3 4 4 4 5 5 6 6 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 a B a a a a a b F x F x F x C F C c x x F F x F x F c c τ τ τ τ 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_τ⋅ + ⋅ +ɺɺ B x C_τ ɺ _τ⋅ =x F [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 forcesF_{1 6}₋ , which are wave induced and the internal forceF_{pump valve}_/ 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

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4 44 44 4 4 44 4 44 4 4 / 4 4 0 0 pump valve F a I a x b x c c x F a a b c c τ τ τ ττ

τ

ττ

τ

τ ττ

τ

+ − −               ⋅ + ⋅ + ⋅ =                            ɺɺ ɺ ɺɺ ɺ [16]

Tank coefficients are given by the dimensions of the tank according to Appendix 1. The hydrodynamic coefficientsa , ₄₄ b and ₄₄ c as well as the roll moment of inertia ₄₄ I are ₄₄ 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 η_ship 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 4 ship I B a I c g GM b

η

c I a ≈ ∇⋅ ⋅ ≈ ⋅ ≈ ⋅∇⋅ ≈ ⋅ ⋅ ⋅ + [17] 40 0 2 2 cos sin 2 2 k t W b k b k b F g L e k k

ρ

ζ

− ⋅ −  ⋅   ⋅  = ⋅ ⋅ ⋅ ⋅ ⋅_ ⋅ _ _+ ⋅ _ __       [18]

In [Eq.18],

ζ

₀ 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 6

a_ττ⋅ +

τ

ɺɺ b_ττ ⋅ +

τ

ɺ c_ττ⋅ = −

τ

a_τ ⋅ −ɺɺx a_τ ⋅ −xɺɺ c_τ ⋅ −x a_τ ⋅xɺɺ [19]

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equation of motion [Eq.20]. The equation of motion is used to calculate the natural frequency of a given configuration since a_ττ and c_ττis 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 a_ττ and b_ττ.

2 0, 0, 0 2 0 2 t t a b c or where c a b a ττ ττ ττ ττ ττ ττ ττ

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 coefficient c_ττ. 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 b_ττ according to Appendix 1. The used damping coefficient is called

η

_t.

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

ω

τ τ

ω

ε

= ⋅ ⋅ = ⋅ ⋅ + [21]

The substitution in [Eq.21] inserted into [Eq.19] will yield an expression for the relation between

τ

₀ and x in the frequency plane as well as the phase ₄₀

ε

in [Eq.2]. The resulting expressions are shown in [Eq.22].

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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 sin( e ) 4 4

F_τ =F_τ ⋅

ω

⋅ + = −t

ε

a_τ⋅ −

τ

ɺɺ c_τ ⋅

τ

[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.

(

)

(

)

2 40 4 4 0 2 2 4 4 40 2 2 2 2 40 e e e e F c a c a F x _c _a _b τ τ τ τ τ τ ττ ττ ττ

ω

τ

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 0.4 0 20 40 60 80 100 120 140 160 180

Phase 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 0.4 0 0.5 1 1.5 2 2.5 3x 10

8 _{Transfer function for stabilising moment from tank}

ω [rad/s] Fτ 4 0 /x4 0 [ N m /r a d ]

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

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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 F is a steady applied moment required to sustain the constant roll angle. ₄

44 4 4 4

c ⋅ −x c_τ ⋅ =

τ

F [25]

At a steady state, the tank angle

τ

will be equal to the roll angle x only with opposite sign. ₄ The stiffness coefficient c is dependant on the current metacentric height which can be ₄₄ written as a fraction of the original “solid” metacentric heightGMS. This is done by

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

metacentric height.

(1 )

F S

GM =GM ⋅ −µ [26]

[Eq.25] can now be rewritten to include the fluid metacentric height as well as the expression for c4τ from given by the tank geometry. Q is a scale term that is common for all terms in the t

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(

m g GM⋅ ⋅ S⋅ −(1

µ

)+ ⋅ ⋅ =Q gt

)

x4 F4 [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 GM

µ

= + − ⋅ ⋅ ⋅ ⋅ [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 x which was also the definition of the ₄ numerator. The expression for the reduction term then reduces to the one in [Eq.31]

t S Q m GM µ = ⋅ [31]

The loss of initial stability is undesirable for this type of ship with relatively low GMSeven

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

S

GM 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, F_{pump valve}_/ is set to zero in the following calculations. The inner tank damping η_t 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 η_t is taken from an example with a similar sized tank (albeit with a smaller xt) in [3].

Table 3 Tank dimensions Parameter Size Dim

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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 angle

τ

. 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] R o ll a n g le [ 0] Tank Unstabilised Stabilised 0 50 100 150 200 250 300 350 -15 -10 -5 0 5 10 15 T a n k a n g le [ 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

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0 100 200 300 400 500 600 700 800 900 1000 -10 -5 0 5 10 Time [s] R o ll a n g le [ 0] Tank Unstabilised Stabilised 0 100 200 300 400 500 600 700 800 900 1000 -2 -1 0 1 2 T a n k a n g le [ 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 4 4 0 sin( ) sin( ) x t x t _τ

ω

ε

τ

ω

ε

τ

⋅ ⋅ +     =    _⋅ _{⋅ +}     [32]

To avoid extensive trigonometric calculus and problems with the phases

ε

₄ and

ε

_τ a complex ansatz is made for the particular solution of the system as shown in [Eq.33] where

η

ˆ is the complex amplitude of the oscillations.

4 ˆ ( )t eiωt where x

η

_{= ⋅}

η

⋅ ⋅

η

₌ 

_τ

    [33]

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

ˆ i t x i

ω η

eω

τ

⋅ ⋅   = ⋅ ⋅ ⋅     ɺ ɺ [34] 2 ˆ i t x eω

ω η

τ

⋅ ⋅   = − ⋅ ⋅     ɺɺ ɺɺ [35]

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

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Using the estimation of the wave induced roll moment in [Eq.18] and setting the tank internal moment to zero gives the complex amplitude

η

ˆ as a function of the excitation frequency

ω

. 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 ˆ x

η

τ

  =     [37]

The response amplitude x₄₀( )

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

κ ς

= ⋅₀ k where k is the wave number.

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0 50 100 150 200 250 300 350 ω [rad/s] x4 0 /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

t

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0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0 50 100 150 200 250 300 350 ω [rad/s] x4 0 /k ζ0 [ -] Frequency response Unstabilised Stabilised , ηt =0.1 Stabilised , η_t =0.3 Stabilised , η_t =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.

0 x WA I GM KB KG V = + − [38]

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ballast D ballast KG h w KG 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.

1 Stabilised roll amplitude

Amplitude reduction

Unstabilised 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

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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 90 100 0 1 2 3 4 5 6 7 8 9 10

Decrease in roll amplitude at ω0 [%]

P o w e r in c re a s e [ % ]

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 90 100 0 1 2 3 4 5 6 7 8 9

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≈00

Unstabilised roll amplitude 50

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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 r t r h h h h w w

τ

≤ − ⇒

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.

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0 10 20 30 40 50 60 70 80 90 100 0 1 2 3 4 5 6 7 8 9 10

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≈00

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• 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

x

WA

I 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

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5.1 Simulating parametric roll with a 2 DOF model

The initial investigation is conducted using a 2 DOF model (x4 and

τ

) 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

ξ

₀ the amplitude of the variation determined by the size of the waves and the shape of the hull.

(

)

0 0 sin 2 40 0

GM = ⋅

ξ

⋅

ω

⋅ +t GM [42]

A reasonable guess as to the magnitude of

ξ

₀ 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 c in [Eq.16] will produce a first simplified simulation of ₄₄ 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] R o ll a n g le [ 0] Tank Unstabilised Stabilised 0 20 40 60 80 100 120 -5 0 5 T a n k a n g le [ 0] Time [s]

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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.

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0 20 40 60 80 100 120 140 160 180 200 -40 -30 -20 -10 0 10 20 30 40 η4 [ °] 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

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2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 10 10 10 10 10 10 ₁₀ 15 15 15 15 15 15 20 20 20 20 20 20 25 25 25 25 25 30 30 3₀ 30 30 35 35 35 35 35 4 4 4 4 3 3 3 3 3 3 3 3 2 2 2 2 2 2 2 2 40 40 4 4 44 4 4 5 5 3 5 5 3 3 3 5 2 10 ₁₀ 2 2 4 4 3 3 3 5 s 10 s 15 s 0 45 90 135 180 225 270 315

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 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 ₅ 5 5 5 5 10 10 10 10 10 10 10 15 15 15 15 15 2₀ 20 2₀ 20 20 2₅ 25 2 5 25 25 30 30 30 30 2 2 23 2 3 35 35 5 5 4 4 4 4 25 ₂₅ 30 30 20 20 4 4 15 15 5 5 30 5 ₅ 5 s 10 s 15 s 0 45 90 135 180 225 270 315

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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.) -35 -3 5 -35 -35 -35 -3₅ -3 5 -35 -35 -3 5 -35 -3₅ -3 5 -30 -3₀ -30 -30 -30 -30 -3 0 -30 -30 -3 0 -3 0 -3 0 -3 0 -25 -2₅ -25 -25 -25 -25 -2 5 -25 -25 -2 5 -2 5 -25 -2 5 -20 -2₀ -20 -20 -20 -2₀ -20 -20 -20 -20 -20 -2 0 -15 -1₅ -1 5 -15 -15 -15 -1 5 -15 -15 -1 5 -15 -15 -1 5 -10 -1₀ -10 -10 -10 -1₀ -1 0 -10 -10 -1 0 -10 -1₀ -1 0 -5 -5 -5 -5 -5 -5 -5 -5 -5 -5 -5 -5 -5 -5 -5 -5 -5 -5 -5 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 -10 -10 -10 -10 -10 -1₀ -10 -10 -15 -15 -1₅ 5 5 5 5 -20 -20 -15 -15 -15 0 0 -20 -20 0 0 -5 -5 0 0 -25 -2 5 -25 -25 0 0 0 0 -5 -5 5 5 -15 -15 -5 -5 -20 -20 0 0 0 0 -5 -5 -25 -2₅ 0 0 -10 -10 0 0 0 ₀ 5 s 10 s 15 s 0 45 90 135 180 225 270 315

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.

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0 50 100 150 -40 -20 0 20 40 µ = 235 0_{, T=11 s} Time [s] η4 [ °] 0 50 100 150 -40 -20 0 20 µ = 200 0_{, T=15 s} Time [s] η4 [ °] 0 50 100 150 -40 -20 0 20 40 µ = 200 0_{, T=11 s} Time [s] η4 [ °] 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.

Anti Roll Tanks in Pure Car and Truck Carriers