Evaluation of vulnerability to parametric rolling

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IN

DEGREE PROJECT MECHANICAL ENGINEERING, SECOND CYCLE, 30 CREDITS

STOCKHOLM SWEDEN 2016,

Evaluation of vulnerability to parametric rolling

ANDERS SJULE

KTH ROYAL INSTITUTE OF TECHNOLOGY SCHOOL OF ENGINEERING SCIENCES

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This is a Master of Science thesis in Naval Architecture (course code SD271X)

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Abstract

This is a Master of Science thesis in Naval Architecture (course code SD271X). It was conducted at KTH Centre for Naval Architecture in collaboration with Wallenius Marine. The main objective of this master thesis project have been to evaluate the vulnerability for parametric rolling for 3 generations of Pure Car/Truck Carriers (PCTC’s) from the Wallenius fleet. This has been done by using IMO’s current version of the second generation intact stability criteria for parametric rolling. The criteria is currently in a developement stage so focus has also been on evaluation of the criteria itself.

The theory behind parametric roll and the IMO criteria is briefly explained followed by results of assess- ment using the current criteria on the Wallenius ships.

When comparing the ships under similar loading conditions the conclusion is that the third generation PCTC is the most sensitive, the second less sensitive than the third, and the first the least sensitive.

This is shown through the results from the IMO evaluation and simulation results.

After doing calculations on PCTC’s the conclusion is that the present 7 speed methodology in the criteria for level 2 C2 is unable to capture sensibility ranking in a correct way.

Acknowledgements

Special thanks to Anders Ros´en at the Center for Naval Architecture at KTH and Mikael Huss at Wallenius Marine for giving much help and guidance throughout the whole master thesis project.

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Introduction

Parametric roll is an event that can occur suddenly at sea. The outcome from this event can be loss of cargo, unsafe conditions to crew and in the most dramatic cases loss of lives and capsizing. An example of loss of cargo and unsafe conditions to crew actually happening was experiences in October 1998[1]

when a C11 type container ship lost a major amount of its cargo due to heavy rolling as an effect of parametric resonance.

Gaining knowledge of the phenomenon is important for avoiding dangerous situation for cargo and crew.

It is also important to take this into consideration when designing a ship and finding a balance between cost efficiency and safety.

IMO is currently undergoing work to establish new criteria for dynamic stability. One of these criteria is for parametric rolling. The Swedish company Wallenius have previously recorded experiences [2] of parametric rolling for ships in their fleet. It is in the company interest to deepen their knowledge about the phenomenon and gain knowledge of how their current fleet stand against the criteria. In this master thesis three generations of Pure Car/Truck Carriers (PCTC’s) from Wallenius have been evaluated against the current standing criteria, which is in a development stage at the moment of writing.

Parametric rolling is an event that can occur for ships in head or following seas. As the wave passes the ship it will have different stability properties at different times. For container ships and PCTC’s the aft and fore part of the ship is designed with a flared front and aft part. One of the reasons for this is to maximize the amount of cargo onboard. This will increase the stability of the ship when wave trough is amidship as the flared part will provide a larger waterline breadth, see figure 1.1. When the ship at another time has the wave crest at amidship the waterline breadth is smaller. At this state the ship has less stability compared to calm water stability, see figure 1.1 for comparison. It is this change in stability as the wave passes that leads to parametric rolling.

Figure 1.1: Difference in waterline breadth for wave trough amidship; calm water and wave crest amidship for a C11 typ container ship. Wave length equal to the ship length.

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The main aim of this master thesis has been to evaluate Wallenius PCTC’s for vulnerability to parametric rolling and to evaluate the current standing criteria and give suggestions to further development.

This has been done by creating a program that calculates IMO second generation intact stability criteria for parametric rolling level 1 and level 2 [3]. Three generations of Wallenius ships have been evaluated using the program. The criteria have also been evaluated and some improvements are suggested.

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Chapter 2

Theory

2.1 Introduction to parametric roll resonance

A transversally symmetric ship moving through pure head or following seas will experience heave, pitch and surge motion. In absence of external time varying forces from waves coming from starboard or portside no forced roll motions will occure. Even in these conditions the ship can experience roll motion referred to as parametric rolling [1]. Parametric rolling is caused by changes in stability as waves passes a ship. Stability variation in waves can be illustrated with how the GZ curve varies for the different wave positions for different wave heights, see figure 2.1. As seen in the figure stability increases with wave height when wave trough is amidship, while decreasing for wave crest amidship.

Figure 2.1: Example of GZ curves for calm water, wave crest amidship and wave trough amidship for a C11 type container ship. Wave length equal to the ship length

It is this change in stability as the wave passes that triggers parametric rolling. The bigger the stability variation is from wave crest amidship and wave trough amidship the bigger the roll amplitudes. When the wave crest is amidship the ship will have a reduced GM value and the ship will roll over to one side.

When the ship has rolled over to that side the wave trough reaches amidship and provides increased stability which will push the ship back into upright position. At the point in time when the ship has reached an upright position from the push back the ship carries momentum in roll and also the wave crest reaches amidship so the stability properties are again low. The ship will then fall over to the other side while the wave trough again moves amidship and provides a pushback again. This rolling motion rises in amplitude until the dampening forces are large enough to absorb the energy of the roll motion.

Parametric rolling occurs when the encounter period of the wave is close to half the natural roll period of the ship. As the wave will then provide the change in stability synchronous with the roll motion for the amplitudes to grow. Parametric roll can also occur when one wave passes for each natural roll

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period of the ship. The ship will then fall to one side to then be pushed back and fall to the side again.

For parametric resonance to develop a series of waves of equal wave length needs to pass in order for resonance amplitudes to grow.

When assesing if a ship is at risk for parametric roll one can look at some points to determine if the ship is vulnerable.

Operability

The presence of large enough waves and waves of certain wave height that causes big GM variation.

The direction of the ship in relation to the waves (head and following waves) Roll damping

Low roll damping due to small or no bilge keels, roll damping appendices and geometric shape of hull.

GM variation

Flared hull, as this increases waterline breadth when wave trough is amidship.

Large breadth/draft ratio.

Resonance

Encounter period of half that of the natural roll period, encounter period the same as natural roll period.

Sufficiently many encounters of similar type of waves for resonance to grow.

Roll motion for a ship in pure head or following sea can be explained with Newtons second law for roll motion with no external forces acting on the system.

MIN( ¨φ) + MD( ˙φ) + MR(t, φ) = 0 (2.1) The first term represents a moment from the inertia of the ship around it’s longitudinal axis as a function of angular acceleration. The second term represent a moment due to the roll angle velocity of the system linked with damping. The third term represent a moment due to the restoring moment when a ship rolls. This moment is the force of buoyancy from the displaced water of the ship times the moment arm which is the transversal distance between the center of gravity and the center of buoyancy known as the GZ value. For parametric oscillations of a ship there are no external roll moments acting on the system [4]. There is just this change in GZ that give rise to the oscillating motion. This equation of roll motion can be expressed as:

φ + (2α + γ ˙¨ φ²) ˙φ + 1

I_x+ A₄₄· ρV g · GZ(t, φ) = 0 (2.2)

The third term in this equation 2.2 contains the change of GZ that gives rise to parametric resonance.

The second term in 2.2 contains the dampening term linked with angular roll velocity. This term affects how big the max roll amplitudes become for GZ variation. α and γ are dampening coefficients.

2.2 Stability variation in waves

The roll restoring moment in 2.1 can be expressed as:

MR= ρV g·GZ(t, φ) (2.3)

To calculate this moment one need to calculate how GZ varies in waves.

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2.2.1 Coordinate system

For doing calculations the ship geometry have been divided into strips, see figure 2.2 to represent the real geometry.

Figure 2.2: View of hull in global coordinate system. Hull: C11 type container ship.

When working with the ship geometry two coordinate systems have been used. A local, ship-fixed, and a global coordinate system. The x-axis and y-axis of the global coordinate system is the still water plane, the z-axis is perpendicular to the calm water plane, see figure2.3.

Figure 2.3: Global and local coordinate system

When positioning the ship in the local coordinate systems the center of gravity of the ship is positioned in the origin of the local coordinate system. Its x-axis is parallel with the keel of the ship, it’s y-axis is parallel with the transversal axis of the ship and the z-axis is parallel with the vertical axis of the ship.

This shift of position (from i.e. position of keel at AP) is done in program:

x = x − LCG; LCG = 0 y = y − T CG; T CG = 0 z = z − KG; KG = 0

The origin of the local coordinate system is positioned along the z-axis of the global coordinate system; at the same x and y coordinates as the origin of the x-y plane. The x-y plane in the global coordinate system is the calm water surface so the ship’s vertical center of gravity, origin of the local coordinate system,

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will initially be positioned higher or lower than the origin of the global coordinate system depending on the ship and loading condition.

Three degrees of freedom have been taken into consideration: heave, roll and pitch motion. Three degrees of freedom have been left out: yaw, sway and surge motion. Movements around these degrees of freedom occur in reality but have been left out for making the simulation work easier.

Hydrostatic properties are calculated from the geometry in the local coordinate system[5].

The volume displacement of the ship is calculated by integration of each wet section part over the ship length:

V =RL 0 Asdx

The longitudinal center of buoyancy is calculated as:

LCB =

RL 0 A_sx_Bdx

V

The transversal center of buoyancy is calculated as:

T CB =

RL 0 AsyBdx

V

The vertical center of buoyancy is calculated as:

KB =

RL 0 A_sz_Bdx

V

The lifting force from the displaced water is calculated as:

FB= ρV g

The gravitational force is calculated as:

F_g= mg

After the center of buoyancy in the local coordinate system is calculated the position of the center of buoyancy in the global coordinate system is calculated:

LCB₀= cos(η)LCB + (− sin(η)KB

T CB0= sin(η) sin(φ)LCB + cos(φ)T CB + cos(η) sin(φ)KB KB0= sin(η) cos(φ)LCB − sin(φ)T CB + cos(η) cos(φ)KB + ς

Where ς Heave [m]

φ Roll angle [Deg]

η Pitch angle [Deg]

In figure 2.4 the aft of the ship is viewed in the local and the global coordinate system for comparison.

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Figure 2.4: View of the aft part of the ship in the local (ship fixed) and global coordinate system. Hull:

C11 type container ship.

2.2.2 Calculating GZ

When calculating GZ in waves heave and pitch equilibrium is calculated at each wave position as the wave passes. First heave is adjusted to reach vertical force equilibrium as seen in figure 2.5. Then the vertical force is compared to accepted vertical force limit. Secondly pitch is adjusted until the center of gravity is aligned with center of buoyancy vertically see figure 2.6. Then pitch moment is compared with accepted limit for pitch moment. This is repeated until pitch moment and vertical force limit is acceptably low. For program implementation of force equilibrium calculations see appendix A.

Figure 2.5: Adjust heave until vertical force equilibrium.

Figure 2.6: Adjust pitch until moment equilibrium.

After the heave and pitch equilibrium calculations are done the transversal distance between the center of gravity and the position of the center of buoyancy in the global coordinate system is measured, see figure 2.7. This distance is the GZ value. As the ship heels it will rotate around its own longitudinal axis going through the center of gravity (it will rotate around the x-axis in the local coordinate system). In the global coordinate system it will rotate around the local x-axis positioned a distance in z over, under or trough the x-y plane in the global coordinate system. While moving along the z-axis in the global coordinate system (while doing heave, roll and pitch motion) the origin of the local coordinate system will remain fixed at the x and y coordinates at the origin of the global coordinate system. Because of this the transversal distance between the center of gravity in the global coordinate system and the center of buoyancy in the global coordinate system is the same as the transversal distance between the center of gravity in the local coordinate system and the center of buoyancy in the global coordinate system.

This is also the case when calculating pitch moment equlibrium when the longitudinal distance between the center of gravity and the center of buoyancy is measured. The GZ value is calculated as:

GZ = −T CB0

Where the sign depend on definition of positive direction for the y-axis in the global coordinate system.

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Figure 2.7: Distance between center of gravity and center of buoyancy. Hull: C11 type container ship.

When calculating a whole GZ curve for a specific wave position and wave height heave and pitch equilibrium is reached for each degree in roll, this is done with 5 degrees’ increment.

2.2.3 Wave position in relation to ship in time

The relative speed of the wave to the ship speed can be written as c − U cos(µ) where c is wave speed and U is ship speed. µ is the angle in which the ship is meeting the waves. E.g. if the angle is 180 degrees (head seas) cos(µ) will be -1 and thus the relative speed will be larger than in e.g. µ = 0 degrees (following seas) where the relative speed will be lower. If µ = 90 degrees the term with cosine will have no effect.

c − U cos(µ) = _T^λ

e

From this equation the encounter period can be written as:

Te= _{c−U cos(µ)}^λ

The encounter frequency can accordingly be written as:

ω_e=^2π_T

e

The expression for the relationship between the encounter frequency and the global wave frequency ω:

ωe= ω −_{g cos(µ)}^ω²^U

For regular gravity waves the wave period is:

T =p(^2πλ_g )

The wave frequency can be written as:

ω =p(^2πg_λ )

The encounter period of the wave:

Te= √ ^λ

(^λg_2π−U cos(µ)

where U is ship speed in [^m_s]

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wave position along the hull:

W_pos= 2π(_T^t

e − f loor(_T^t

e)) − π

where f loor indicates the closest natural number down i.e. 2.67 has f loor(2.67) = 2.

2.3 Damping

The moment in newton’s second law for roll motion, see equation 2.1 contains a damping moment.

MD= (Ix+ A44)(2α + γ ˙φ²) ˙φ = B44(φa) ˙φ (2.4) B₄₄ is a damping coefficient as a function of roll amplitude[6]. The roll damping contains different parts: damping due to friction B_F; damping due to wave making B_W; damping due to the eddy B_E and bilge keel B_BK. When the ship is moving in forward speed a damping component due to lift is added B_L. B44= BF + BW + BE+ BL+ BBK

The dampening coefficient α and γ can be determined using Ikeda’s simplified method [6].

2.4 Inertia

The moment in newton’s second law for roll motion, see equation 2.1 contains an inertial moment due to the mass of the ship and added mass due to displaced water as it rolls.

MIN = (Ix+ A44) ¨φ

I_xis the ships roll moment of inertia and A₄₄ roll moment of inertia due to displaced water in roll.

2.5 Time domain simulations

2.5.1 Calculating Max roll angle

Max roll angle is calculated by solving the following equation in time.

φ + (2α + γ ˙¨ φ²) ˙φ + 1

I_x+ A₄₄· ρV g · GZ(t, φ) = 0 (2.5)

This equation can be solved by separating it into:

dx(1) = x(2)

dx(2) = −(2α + γ · x(2)²) · x(2) −_I^{ρV g}

x+A₄₄ · GZ(t, φ)

By using the solver ODE45 in Matlab a time domain simulation of the rolling motion of the ship at a certain speed and heading in waves with wavelength equal to the ship length at a specific wave height can be done.

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2.5.2 Time domain simulations of a C11 container ship

In this section time-series of simulated parametric roll for a C11 container ship is presented. The ship has the following load case data, see table 2.1.

Table 2.1: Ship data C11 LC2 Ship particulars unit Program

Lpp [m] 262

B [m] 40

D [m] 24.2

d [m] 11.5

τ [m] 0

ρ [_m^kg3] 1025

g [_m^N₂] 9.81

LCG [m] 125.517

C_b [−] 0.56012

C_m [−] 0.97097

GM_c [m] 1.9671

T0 [s] 25.586

V CG [m] 18.4

V s [^m_s] 12.165

LBK

LP P [−] 0.2857

BBK

B [−] 0.010

No resonance

In figure 2.8 roll motion is simulated. In the figure the roll motion of the ship is marked in blue and the wave motion marked in black. The x-axis indicates the simulation time in seconds and the y-axis indicates the wave amplitude and the ship roll motion amplitude in meters and degrees respectively. In this simulation the ship is going in 20.39 knots in head waves with wave height 2.62 meters and wave length equal to ship length. The ratio between wave encounter period and natural roll period is 0.33 which is under the resonance frequency at the ratio between wave encounter period and natural roll period of 0.5. The roll motion decreases after an initial disturbance of 5 degrees.

Figure 2.8: No resonance due to parametric resonance 1:2

In figure 2.9 roll motion is simulated for the ship going 0 knots in following waves with wave height 7.86 meters and wave length equal to ship length. After an initial disturbance of 5 degrees in roll the

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ship reaches roll amplitudes of close to 40 degrees before the damping forces are large enough to balance it out. The ratio between wave encounter period of the waves and natural roll period of the ship is 0.5 which means that 2 waves pass the ship for each roll period. This is referred to as 1:2 parametric resonance.

Figure 2.9: 1:2 Parametric roll 1:1

In figure 2.10 roll motion is simulated for the ship going 20.74 knots in following waves with wave height 7.86 meters and wave length equal to ship length. After an initial disturbance of 5 degrees in roll the ship reaches roll amplitudes of over 20 degrees before the damping forces are large enough to balance it out. The ratio between wave encounter period of the waves and natural roll period of the ship is 1 which means that 1 wave pass the ship for each roll period resulting in big roll motions to one side. This is refered to as 1:1 parametric resonance.

Figure 2.10: 1:1 Parametric roll 1.5:1

In figure 2.11 roll motion is simulated for the ship going 25.94 knots in following waves with wave height 7.86 meters and wave length equal to ship length. After an initial disturbance of 5 degrees in roll the ship reaches roll amplitudes of over 20 degrees before the damping forces are large enough to balance it out.

The ratio between wave encounter period of the waves and natural roll period of the ship is 1.5 which means that the ship rolls with 1.5 periods for each passing wave. This kind of parametric resonance appear for high ship speeds in following waves. This is referred to as 1:1.5 parametric resonance.

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Figure 2.11: 1.5:1 Parametric roll

Sensitivity to parametric roll differs when going in different speeds and heading. But also varies with wave height. Wave height have big influence over the change of GM. In figure 2.12 the sensitivity for parametric roll at different speeds is shown.

Figure 2.12: Areas where parametric roll over and under 25 Deg appear. Written in red in the plott is the position of the different types of parametric roll and no resonance case.

On the y-axis the values of significant wave heights in meters is presented. On the x-axis the different values of the fraction between the wave encounter period and the natural roll period of the ship is presented. The blue graph line represents the limiting significant wave height where the ship is sensitive for parametric roll angles over 25 degrees. The natural roll period of the ship does not change but the encounter period changes depending on speed and direction of the ship. For large speeds in head seas the encounter period is small. For large speeds in following waves the encounter period is long. By simulating for high speeds in head seas then decreasing the speed to zero in speed increments followed by simulations in following waves in increasing speeds one can get a picture of which speeds the ship is sensitive to parametric roll and for which significant wave heights the ship is sensitive. The red text and numbers in figure 2.12 indicates which kinds of parametric roll occur at these speeds.

In figure 2.13 The results from a simulation with 7 different speeds for sea states in a North Atlantic scatter diagram presented in [3]. Each sea state is scaled to be equal in length to the ship length and having a corresponding effective wave height at each sea state, see [6] for explanation of calculations of corresponding effective wave height at each sea state. Max roll angles over 25 degrees are marked out as colored in the sea-state diagram. The number in each cell is the probability for the sea state to occur in percentage.

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Figure 2.13: Roll angles over 25 degrees for different sea states

In the first column of the table above the matrix in figure 2.13 the ratio between natural roll period of the ship and encounter period of the waves is indicated. In the second column the ratio between the natural roll period of the ship based on GMmean and encounter period of the waves is indicated, see equation 2.6 and 2.7. GM_mean is calculated as the mean GM value of a whole wave with wave height 0.0167 · L passing at ten different positions along the hull. This wave height is choosen from Level 1 criteria for vulnerability for parametric rolling [3] as this wave height was used there to calculate GM_mean. In the third column the ship speed and heading is indicated. In the fourth column the calculated probability that roll angles over 25 degrees occur due to parametric roll is presented. The lines in the table marked as colored means that at these ratios of _T^T^e

0; at these speeds and headings parametric roll over 25 degrees occur.

ω0_mean=p

(ρV gGMmean

Ix+ A44

) (2.6)

T0_mean = 2π

ω₀_mean (2.7)

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Chapter 3

IMO second generation intact

stability criteria for parametric roll

In this chapter the IMO second generation intact stability criteria for parametric roll is presented, all information of the criteria can be found in [3] and [6]. The article [7] have provided information about the current state of the criteria. In this report the first two levels of the criteria are included. The first level is more conservative than the second level. For program implementations of the criteria in flowcharts see Appendix A. The type of ships that are being analyzed with the criteria is Pure Truck/Car Carriers (PCTC) from Wallenius Marine.

3.1 Level 1 vulnerability criteria for parametric rolling

A ship is considered to not be vulnerable to the parametric roll if:

∆GM₁ GMc

<= R_{P R} (3.1)

∆GM1= GMmax− GMmin

2 (3.2)

The ratio ^∆GM_GM¹

c represent the percentage of change of GM compared to calm water GM.

Calm water GM is calculated as:

GMc =GZ(φ = 0.5deg) − GZ(φ = 0deg)

(φ = 0.5deg) − (φ = 0deg) (3.3)

When calculating ∆GM introduce a sinusoidal wave with wave height h = S_wλ, where λ = length of ship, S_w= 0.0167

∆GM is quasi-statically calculated with the wave crest at 10 different positions along the length of the ship

ζ(x) =h

2cos(2πx λ ) = h

2cos(kx) (3.4)

where x = 2π^(1:11)−6₁₀

Gm is calculated for the 10 wave positions: GM : GM1, GM2...GM10

∆GM1is then calculated as:

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∆GM1=^GM^max^−GM₂ ^min

R_{P R} is calculated according to [3].

3.2 Level 2 vulnerability criteria for parametric rolling

A ship is considered to not be vulnerable to parametric rolling if:

R_{P R0}> C1 or R_{P R0}> C2

R_{P R0} is preliminary set by IMO to 0.06. In the calculation of C1 and C2 wave statistics is used and the C values are probabilistic units that presents the vulnerability for parametric roll.

3.2.1 Algorithm for assessing Level 2 C1 Criteria

The value of C1 is calculated as the weighted average from a set of waves.

Table 3.1: Wave data for evaluation of C1

Wave case number Weight Wi Wave length λi Wave Height Hi (m)

1 0.000013 22.574 0.350

2 0.001654 37.316 0.495

3 0.02912 55.743 0.857

4 0.092799 77.857 1.295

5 0.199218 103.655 1.732

6 0.248788 133.139 2.205

7 0.208699 166.309 2.697

8 0.128984 203.164 3.176

9 0.062446 243.705 3.625

10 0.024790 287.931 4.040

11 0.008367 335.843 4.421

12 0.0024790 387.440 4.769

13 0.000658 442.723 5.097

14 0.000158 501.691 5.370

15 0.000034 564.345 5.621

16 0.000007 630.684 5.950

C1 =

n

X

i=1

W_iC_i (3.5)

C_i= 0 if:

GM (H_i, λ_i) > 0 and ^{∆GM (H}_{GM (H} ⁱ^,λⁱ⁾

i,λi) < R_{P R} or V_{P Ri}> V_s Ci= 1 if not

GM (Hi, λi) is mean GM for a wave passing at 10 different positions at different wave heights and wave lengths. ∆GM (Hi, λi) is the GM variation amplitude for a specific wave height and wave length for a wave passing. This is calculated in the same way as for Level 1 criteria. The wave passing positions is the same as in level 1, but the wave height and wave length varies here.

VP Riis calculated as:

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VP Ri= |2λi

T₀

p(GM (Hi, λi) GM_c ) −

rgλi

2π| (3.6)

V_sis the ship service speed and V_{P Ri}is the speed in which parametric rolling can occur.

3.2.2 Algorithm for assessing Level 2 C2 Criteria

Two different alternative methods can be used for calculating the max roll angle [6]:

1 Degree of freedom time domain simulation method according to SDC3/WP5/Annex 4/App 3 in [6]

1 Degree of freedom averaging method according to SDC3/WP5/Annex 4/App 5 in [6]

The following describes an algorithm for assessing the level C2 criteria.

The roll angle φ_max is calculated in 11 different waves varying in wave height: λ = L , h = 0.01nL, n = 0, 1. . . , 10

The maximum roll angle is then calculated

at zero speed: ϕ⁰_max(h, F_n = 0) in three speeds in head seas: ϕ^head_max(h, F_n,k) in three speeds in following seas: ϕ^{f ollow}_max (h, F_n,k) Where

F_(n,k)= √^V^k

(Lg)

Vk= VsKk

with Kk according to Table 2.11.3.3 in [6]

Table 3.2: Values for K_k k Kk

1 1.0 2 0.866 3 0.50

A wave scattering diagram is chosen (standard case is for North Atlantic Table 2.11.3.4.2 in [3]) Each cell in this diagram has a row index: i, and a column index: j For each sea state (Hz,i, Tz,j) the effective wave is determined with wave length λ_ij = L and wave height H_{ef f,ij}

The roll angle in each sea state ϕ_ij(H_{ef f}, F_n) is then determined with interpolation from previous calculated max roll angles for waves with wave length equal to the ship length and for different wave heights.

This will produce seven 17x16 matrices containing max roll angles for each specific sea state presented in Table 2.11.3.4.2 in [6]

at zero speed: ϕ⁰_ij(Hef f,ij, F n = 0) = interp1(h, ϕ⁰_max, Hef f,ij) in three speeds in head seas: ϕ^head_ij (Hef f,ij, F nk) = interp1(h, ϕ^head_max, Hef f,ij) in three speeds in following seas: ϕ^{f ollow}_ij (Hef f,ij, F nk) = interp1(h, ϕ^head_max, Hef f,ij)

Cij is then determined as:

C_ij= 1, if ϕ_ij > 25 degrees

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Cij= 0, otherwise

The values for C_ij will be determined by if the roll angle ϕ_ij exceeds 25 degrees or not at:

Zero speed: C_ij⁰ VS ϕ⁰_ij

Three speeds in head seas: C_ij^head VS ϕ^head_ij Three speeds in following seas: C_ij^{f ollow} VS ϕ^{f ollow}_ij

The components that calculates C2 is then determined. This consist of a C2 value for zero speed, 3 C2 values for following waves and 3 C2 values for head waves:

at zero speed: C2⁰(F n = 0) =P

i

P

j

WijC_ij⁰ in the three speeds in head seas: C2^head(F nk) =P

i

P

j

WijC_ij^head in the three speeds in following seas: C2^{f ollow}(F n_k) =P

i

P

j

W_ijC_ij^{f ollow}

Finally, C2 is determined as:

C2 = [

3

X

k=1

C2^head(F n_k) + C2⁰(F n = 0) +

3

X

k=1

C2^{f ollow}(F n_k)]/7 (3.7)

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Chapter 4

Evaluation of Wallenius PCTC’s

with IMO criteria for parametric roll

In this chapter Wallenius PCTC’s are evaluated using IMO second generation intact stability criteria for parametric roll.

4.1 Wallenius PCTC and loading conditions for evaluation

The following ship and loading condition data was used for the simulation. Load case 1 data is presented in table 4.1. Load case 2 data is presented in table 4.2

Table 4.1: Load case 1 for 3 generations of PCTC Ship model 1:Gen PCTC 2:Gen PCTC 3:Gen PCTC

d[m] 9.5 9.5 9.5

τ [m] 0 0 0

r[m] 14.8 14.8 14.8

GM [m] 1.2 1.2 1.2

T0[s] 27.1 27.1 27.1

KG[m] 13.6 14.45 15.21

KM [m] 14.8 15.65 16.41

V [m³] 33700 34000 33800

LBK_tot[m] 109 109 109

BBK_tot[m] 0.4 0.4 0.4

Table 4.2: Load case 2 for 3 generations of PCTC Ship model 1:Gen PCTC 2:Gen PCTC 3:Gen PCTC

d[m] 9.5 9.5 9.5

τ [m] 0 0 0

r[m] 14.8 14.8 14.8

GM [m] 2 2 2

T0[s] 21.0 21.0 21.0

KG[m] 12.8 13.65 14.41

KM [m] 14.8 15.65 16.41

V [m³] 33700 34000 33800

LBKtot[m] 109 109 109

BBKtot[m] 0.4 0.4 0.4

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4.2 Simulation results

The results when using IMO second generation intact stability criteria for parametric roll for the 3 generations of PCTC’s are shown in table 4.3. The ships in the different loading conditions passing or failing the criteria is shown in table 4.4

Table 4.3: Values from simulation

1:Gen LC1 2:Gen LC1 3:Gen LC1 1:Gen LC2 2:Gen LC2 3:Gen LC2 RP R 0.32116 0.32116 0.32076 0.32116 0.32116 0.32076

C1 0.68211 0.8846 0.8846 0 0.68211 0.68211

C2 0.022936 0.03533 0.036129 0.0042209 0.00057071 0.1016 Table 4.4: IMO criteria fulfillment

1:Gen LC1 2:Gen LC1 3:Gen LC1 1:Gen LC2 2:Gen LC2 3:Gen LC2

Level 1 passed No No No No No No

Level 2 C1 passed No No No Yes No No

Level 2 C2 passed Yes Yes Yes Yes Yes No

4.3 Result discussion

4.3.1 Load case 1

The result from the evaluation of the criteria on the 3 generations of Wallenius PCTC’s is that no ship passes the level 1 criteria and the level 2 C1 criteria and all the ships pass the level 2 C2 criteria. The 3:rd generation PCTC is judged as being the most vulnerable to parametric rolling having a probability of 3.6% for parametric roll. The 2:nd generation PCTC is judged as almost as vulnerable with a 3.5%

probability for parametric roll. The 1:st generation is judged as the least vulnerable with a 2.3% probability for parametric roll. When comparing these results with a plot for limiting significant wave height for roll angles over 25 degrees using 41 speed steps, see figure 4.1, the 3:rd generation PCTC in load case 1 stand out as the most sensitive to parametric roll. It is sensitive in a larger domain of speeds than the two other generations of PCTC’s. The 2:nd generation PCTC is the second most vulnerable by being vulnerable for as small significant wave heights as the 3:rd generation but in a shorter speed interval.

The 1:st generation PCTC is the least sensitive to parametric roll being sensitive to parametric rolling at significant wave heights of 5.5m and also being sensitive in the shortest speed interval compared to the other generations of PCTC’s.

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Figure 4.1: Comparison limiting significant wave height for roll angles > 25 degrees in parametric roll for the 3 generations of PCTC in load case 1

4.3.2 Load case 2

The result from the evaluation of the criteria on the 3 generations of Wallenius PCTC’s is that no ship passes the level 1 criteria. The 1:st generation PCTC passes the Level 2 C1 criteria by having an operational speed that is under the speed for which parametric rolling can occur, see chapter 3 for explanation of Level 2 C1 criteria. The 2:nd and 3:rd generation of PCTC’s fail level 2 C1. For level 2 C2 the 1:st and 2:nd generation PCTC passes the criteria and 3:rd generation PCTC fails the criteria. The 2:nd generation PCTC is judged as being the least vulnerable to parametric roll having the probability of 0.057%. The 1:st generation is judged as not vulnerable to parametric roll having the probability of 0.42%. When comparing these results with the plot for limiting significant wave height for roll angles over 25 degrees using 41 speed steps as seen in figure 4.2 the 2:nd generation PCTC is more vulnerable to parametric roll than the 1:st generation being sensitive for more speeds and for critical speeds; lower significant wave height. This is not obvious when comparing the value for C2 for the 1:st and 2:nd generation PCTC. The 3:rd generation PCTC is judged as being the most vulnerable to parametric roll having the probabilistic value 10,16%. Comparing this number to the C2 value for the 3:rd generation PCTC in load case 1 which was 3.6% the conclusion can be made that it is more sensitive in load case 2 than in load case 1. When comparing the two plots in figure 4.1 and 4.2 and comparing with the number of C2 in both load cases it gives another impression of vulnerability. From these plots the 3:rd generation PCTC is more sensitive to parametric roll in load case 1 than in load case 2. The 3:rd generation PCTC is sensitive to the same lowest significant wave height at critical speeds in both load case 1 and load case 2. But is vulnerable in a larger range of speeds in load case 1 than in 2.

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Figure 4.2: Comparison limiting significant wave height for roll angles > 25 degrees in parametric roll for the 3 generations of PCTC in load case 2

4.3.3 Comparing results with different speed steps

When using more speed steps when evaluating the level 2 C2 criteria one reaches other results which seems to correspond more with what is seen in figure 4.1 and 4.2. When using 21 speed steps the different generations of PCTC’s have been simulated in head seas with 100% to 0% operational speed with 10%

speed decrements and in 10% to 100% speed in following sea with 10% increment. The total probabilistic value for C2 have been calculated by summing all the probabilities for parametric roll angles over 25 degrees at the different speeds and then dividing it by the number of speeds: 21. The results which can be seen in table 4.5 shows that the 3:rd generation PCTC is the most vulnerable to parametric roll in load case 1 having a probability of 9,48%. In load case 2 where it is vulnerable for smaller speed intervals it has a probability of 6.87%. The 2:nd generation PCTC is more sensitive in load case 1 than 2 having the probabilistic value 7.36% for load case 1 and 4.16% for load case 2. The 1:st generation PCTC has similar sensitivity in load case 1 and load case 2 with 1.12% and 1.06%.

Table 4.5: Value of C2 7 speed steps vs 21 speed steps

Load case Ship C2 (7 speed steps) C2 (21 speed steps) Load case 1 1:st generation PCTC 0,022936 0,01120

2:nd generation PCTC 0,03533 0,07362 3:rd generation PCTC 0,036129 0,09483 Load case 2 1:st generation PCTC 0,0042209 0,01060 2:nd generation PCTC 0,00057071 0,04165 3:rd generation PCTC 0,1016 0,06870

In the current standing criteria the 3:rd generation PCTC would be judged as vulnerable to parametric roll in load case 2 and judged as not vulnerable in load case 1. The other PCTC’s at both loading conditions would be judged as not vulnerable to parametric roll. If 21 speed steps where to be used in the level 2 C2 criteria, and the limit value for being judges as vulnerable or not would remain as 0.06, for load case 1 the 2:nd generation PCTC and the 3:rd generation PCTC would be evaluated to be vulnerable to parametric roll and for load case 2 the 3:rd generation PCTC would be judged as vulnerable to parametric roll. For load case 1 the 1:st generation PCTC would be evaluated as not vulnerable to

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parametric roll and for load case 2 the 1:st and 2:nd generation PCTC’s would be evaluated as not vulnerable to parametric roll.

4.4 Adjustments to bilge keel geometry to fulfill the criteria

As seen from the result the 3:rd generation PCTC in load case 2 did not fulfill the level 2 C2 criteria.

Modifications can be made in order to fulfill the criteria better. One of the things one can do is to change the bilge keel geometry. When changing the bilge keel breadth of the 3:rd generation PCTC from 0.4 meter to 0.8 meters, when using 7 speed steps, one get a C2 value of 0.0705 (Vs 0.1016 when having a bilge keel breadth of 0.4 m). Making this adjustment would still not pass the current standing criteria however it will become less vulnerable to parametric roll. When using 21 speed steps one get a C2 value of 0.0484 (Vs 0.0687 when having a bilge keel breadth of 0.4 m). As one can see from figure 4.3 changing the bilge keel breadth reduces the vulnerability to parametric roll. The least significant wave height in which the 3:rd generation PCTC in load case 2 is sensitive to parametric roll with bilge keel breadth of 0.4 meter is 2.5 meters. When adjusting the breadth of the bilge keel to 0.8 meters it’s increased to 3.5 meters.

Figure 4.3: Comparison vulnerability to parametric roll for the 3:rd generation PCTC in load case 2

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Chapter 5

Observations made when

implementing the current version of the IMO criteria

5.1 Now standing criteria

When simulating parametric roll 3 degrees of freedom have been taken into consideration: heave and pitch equilibrium in waves and roll motion around the ship longitudinal axis. 3 degrees of freedom have been left out: yaw motion, sway motion and surge motion. Movements around these degrees of freedom occur in reality but have been left out for making the simulation work easier. One other unrealistic aspect of simulating parametric roll at different wave heights and speeds is the fact that a ship in real life would not go in max operational speed i.e. 20 knots in 10-meter-high waves. So for even taking this into consideration when calculating a probabilistic value makes this method drift away from a realistic representation.

When doing implementations of the now standing IMO second generation criteria for parametric roll some questions about the current standing criteria came up. Most of the time of the master thesis have been spent on implementation and analysis of the level 2 C2 criteria.

The explanatory notes [6] provided an example of criteria calculations on a C11 type container ship with the load case presented in figure 2.1. Results from the simulation program was compared to these results for validation purpose. When following the instructions on how to calculate a probabilistic value of vulnerability to parametric roll in this master thesis the value was 6.61%when simulating parametric roll in head and following sea at 100%, 86.6 % and 50% speed and also at 0% speed in both directions.

In figure 5.1 a plot of limiting significant wave height for a C11 container ship is presented, it shows the limiting significant wave height for different ratios of ^{T e}_{T 0}. For the seven speeds simulated parametric roll with roll angles over 25 degrees only occurred in 3 of them, this is marked with circles in 5.1. In Appendix C of the report, table C.5, one can see a comparison between the results from the program written in the master thesis compared with he IMO simulation results of a C11 container ship with the same loading condition. The results for level 1 differs due to using different methods. But for level 2 the results where similar. The slight difference in value for C2 in level 2 might be due to implementation of the effect of dynamic pressure along the hull, for different relative velocities between the hull of the ship and surrounding water, when calculating the damping coefficients α and γ.

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Figure 5.1: Sensitivity to parametric roll for C11 container ship

In figure 5.2 the max roll angles for different speeds and heading are presented. At the column furthest to the right the probability for roll angles over 25 degrees due to parametric roll is listed. When calculating the total probability for roll angles due to parametric roll over 25 degrees for these seven speeds, the probability for each speed is summarized and then divided with the total amount of speeds simulated.

For this case the probability for parametric roll is 6.61%.

If the ship was to be simulated for 21 different speeds instead with 0 to 100 percentage in 10 percentage increments in head and following sea. the following speeds would be covered in the diagram showing sensitivity to parametric roll. see figure 5.3. The ratios ^T_T^e

0 for the 21 different speeds is marked with circles in the plot.

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The ship is sensitive to parametric roll in 9 out of 21 of these speeds. In the table in figure 5.4 it is indicated in which speeds the ship is sensitive to parametric roll. By summarizing the rightmost column for speeds where roll angles over 25 degrees due to parametric roll occurs and dividing with the total amount of speeds simulated the total probabilistic value is 4,75%.

In order for a ship at a loading condition to pass the level 2 C2 criteria this probabilistic value has to be under 6%. When simulating at 7 different speeds this value is 6.61% so it will fail the criteria. But when choosing to simulate for 21 different speeds it will be 4.75% and then it will pass the criteria. The criteria as it is currently suggested with 7 speeds, can give a fair representation to vulnerability to parametric roll but only if it manages to cover speeds in which parametric resonance appear. When simulating PCTC’s the experience was that some of these resonance speeds was missed for a load case that was less sensitive to parametric roll than other. For example when comparing the 3:rd generation PCTC in load case 1 and load case 2. For load case 2 the 3:rd generation PCTC failed Level 2 C2 having 10.16 percentage probability for parametric roll compared to 3.61% probability for parametric roll in load case 1. When looking at the vulnerability curves for parametric roll for the 3:rd generation PCTC in load case 1 and 2 one can see that load case 1 is more sensitive to parametric roll due to being sensitive in more speeds, see figure 5.5. The expected value for C2 should in turn be higher to represent this but it does not do this because when simulating parametric roll in 7 speeds, some speeds where the ship is the most sensitive to parametric roll, is missed. In the simulation for load case 2 a speed manages to match the most sensitive resonance speed and this will result in a higher probabilistic value than for load case 1 where the most sensitive resonance speed is missed. The speeds are indicated by the circles in the figure.

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Figure 5.5: Comparison of the sensitivity for parametric roll for the 3:rd generation PCTC in load case 1 and 2

5.2 Suggested modifications of the criteria

In this section some suggestions to further develop the criteria is proposed.

Change of wave length

Not all ships are the most sensitive to wave length equal to ship length. For the case of a container ships they might be the most sensitive to wave lengths equal to the total ship length. The total length of a container ship is a larger value than the length between perpendiculars. For PCTC’s the total length of the ship and length between perpendiculars are similar. So scaling the waves equal to the total ship length will give different effects in comparison with a container ship and PCTC.

Change of speed steps

As discussed in the section: Now standing criteria. More speed steps can be taken into consideration when simulating parametric roll. This is also discussed in the chapter Evaluation of Wallenius PCTC’s in the section results discussion.

Other suggestions

The criteria as it stands now works as a good tool to compare different ships and load cases to sensitivity to parametric roll. The criteria for level 2 C2 now instruct that 7 different speeds should be taken into consideration when calculating the probability for parametric roll. In this thesis work this has proven to give some wrong indications to which ships are more sensitivity to parametric roll as this speeds can or cannot cover a speeds in which the ship is the most sensitive to parametric roll. The fact that the ships does operate in some speeds more time than at other makes the solution of simply using more speed steps to calculate vulnerability to parametric roll not right either. When calculating the probability for parametric roll one can weigh the probabilistic value at one speeds with a probabilistic value indicating how much time is spent in that speed during normal operation. For example, if the ship is operating in 16 knots 60% of the time the probabilistic value during that speed should have a greater effect than for another speed i.e. 19 knots in which the ship only operates in 10% of the time. Weighing each number makes this method not so general anymore. It’s highly individual how different ships operate so this can make the criteria to complex.

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Chapter 6

Conclusions

When comparing the different PCTC under similar loading conditions the conclusion is that the third generation PCTC is the most sensitive, the second less sensitive than the third, and the first the least sensitive. This is shown through the results from the IMO evaluation and simulation results

After doing calculations on PCTC’s the conclusion is that the present 7 speed methodology in the criteria for level 2 C2 is unable to capture sensibility ranking in a correct way. A suggestion to further develop the criteria is to use more speed steps. In this thesis work 21 speed steps was tried which seemed to correspond well with the sensitivity for parametric roll for the ships at the different loading conditions.

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Appendix A

Program Flowcharts

A.1 Function flowcharts

Figure A.1: Calculate GZ function

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Figure A.2: Force equilibrium function

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A.2 IMO criteria implementation flowcharts

Figure A.3: Flowchart level 1 and level 2 (C1)

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Figure A.4: Flowchart level 2 (C2)

Figure A.5: Flowchart Level 2 (C2) continiued

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Appendix B

Results from program

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B.1 Sea state data from Level 2 C2 criteria

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B.1.1 Sea state data load case 1

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3:rd generation PCTC load case 1

Figure B.1: Roll angles over 25 degrees at different sea states for different ^T_T^e

0 1:st generation PCTC LC1

Figure B.2: Limiting significant wave height for sensitivity for parametric roll over 25 degrees 1:st generation PCTC LC1

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2:nd generation PCTC load case 1

0 2:nd generation PCTC LC1

Figure B.4: Limiting significant wave height for sensitivity for parametric roll over 25 degrees 2:nd generation PCTC LC1

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1:st generation PCTC load case 1

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B.1.2 Sea state data load case 2

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0 3:rd generation PCTC LC2

Figure B.8: Limiting significant wave height for sensitivity for parametric roll over 25 degrees 3:rd generation PCTC LC2

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0 1:st generation LC2

Figure B.12: Limiting significant wave height for sensitivity for parametric roll over 25 degrees 1:st generation LC2

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B.2 Sea state data from Level 2 C2 criteria usin 21 speed steps

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B.2.1 Sea state data load case 1

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3:rd generation PCTC case 1

Figure B.14: Limiting significant wave height for sensitivity for parametric roll over 25 degrees 3:rd generation PCTC LC1

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B.2.2 Sea state data load case 2

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Figure B.20: Limiting significant wave height for sensitivity for parametric roll over 25 degrees 3:rd

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Appendix C

Program verification

C.1 Verification of force equilibium calculation

For verifying that the function ForceEquilibrium calculates right heave and pitch angles comparison have been made with hand calculations. A box shaped geometry with the following geometrical properties, see table C.1, was used for comparison.

Table C.1: Data for comparison calculations

L = 40 [m] Length of ship

B = 8 [m] Breadth of ship

M = 300000 [kg] Mass of ship ρ = 1025 [kg/m³] Saltwater density g = 9.81 [N/m²] Gravitational constant LCG = 20 [m] Longitudinal center of gravity KG = 2 [m] Vertical center og gravity T CG = 0 [m] Transversal center of gravity

C.1.1 Heave equilibrium

First hand calculations where done to determine the draft:

FB= T BLρg F_G= M g

Force equilibrium in heave gives:

T_hand=_BLρ^m = _8401025³⁰⁰⁰⁰ = 0.9146341463[m]

With these data input the program produces:

Tprog= 0.9145m

Difference in draft for heave equilibrium:

∆ = Tprog− Thand= 1.3415 · 10⁻⁴≈ 0.13[mm]

This has an error margin of 10⁻⁴[m] , which is acceptable low.

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By increasing the weights with small increments ∆m one can compare the difference between hand calculations and program calculations, to finally calculate a maen difference for the draft.

T_hand=^m+∆m_BLρ

Table C.2: Data heave equlibrium calculations

∆m[kg] Thand[m] Tprog[m] ∆T [m]

1000 0.9177 0.9175 0.0002 2000 0.9207 0.9205 0.0002 3000 0.9238 0.9235 0.0003 4000 0.9268 0.9265 0.0003 5000 0.9299 0.9295 0.0004 6000 0.9329 0.9325 0.0004 7000 0.9360 0.9355 0.0005 8000 0.9390 0.9395 -0.0005 9000 0.9421 0.9425 -0.0004 1000 0.9451 0.9455 -0.0004

The mean difference in draft ∆Tmean= 0.00036[m], in the order 10⁻⁴[m] which is acceptable low.

C.1.2 Calm water roll equilibrium

Now comparison is made between GM₀ calculated by hand and then calculated by the program by calculating force equilibrium at different heel angles. By hand GM₀ can be calculated like this for the box geometry:

GM0= KB +^I^wax₅ − KG KB = ^T₂

IW AX=^LB₁₂³ 5 = LBT

This gives by hand calculations:

GM_0hand= 4.28864[m]

Using the program GM₀ is calculated as:

GM0prog= _RollAngle^GZ²^−GZ¹

2−RollAngle1

GM0prog= 4.2414[m]

The difference:

∆GM0 = GM0hand − GM0prog = 0.0472m ≈ 4.7cm. The difference is in the order 10⁻²[m] which is acceptably low.

C.1.3 Calm water pitch equilibrium

The results between hand calculations and program calculations are compared by calculating pitch angle after moving LCG.

Evaluation of vulnerability to parametric rolling

Evaluation of vulnerability to parametric rolling

ANDERS SJULE

Abstract

Acknowledgements

Contents

Nomenclature

Chapter 1

Introduction

Chapter 2

Theory

2.1 Introduction to parametric roll resonance

2.2 Stability variation in waves

2.2.1 Coordinate system

2.2.2 Calculating GZ

2.2.3 Wave position in relation to ship in time

2.3 Damping

2.4 Inertia

2.5 Time domain simulations

2.5.1 Calculating Max roll angle

2.5.2 Time domain simulations of a C11 container ship

Chapter 3

IMO second generation intact

stability criteria for parametric roll

3.1 Level 1 vulnerability criteria for parametric rolling

3.2 Level 2 vulnerability criteria for parametric rolling

3.2.1 Algorithm for assessing Level 2 C1 Criteria

3.2.2 Algorithm for assessing Level 2 C2 Criteria

Chapter 4

Evaluation of Wallenius PCTC’s

with IMO criteria for parametric roll

4.1 Wallenius PCTC and loading conditions for evaluation

4.2 Simulation results

4.3 Result discussion

4.3.1 Load case 1

4.3.2 Load case 2

4.3.3 Comparing results with different speed steps

4.4 Adjustments to bilge keel geometry to fulfill the criteria

Chapter 5

Observations made when

implementing the current version of the IMO criteria

5.1 Now standing criteria

5.2 Suggested modifications of the criteria

Chapter 6

Conclusions

Appendix A

Program Flowcharts

A.1 Function flowcharts

A.2 IMO criteria implementation flowcharts

Appendix B

Results from program

B.1 Sea state data from Level 2 C2 criteria

B.1.1 Sea state data load case 1

B.1.2 Sea state data load case 2

B.2 Sea state data from Level 2 C2 criteria usin 21 speed steps

B.2.1 Sea state data load case 1

B.2.2 Sea state data load case 2

Appendix C

Program verification

C.1 Verification of force equilibium calculation

C.1.1 Heave equilibrium

C.1.2 Calm water roll equilibrium

C.1.3 Calm water pitch equilibrium