Evaluation of vulnerability to parametric rolling1057139/FULLTEXT01.pdfrolling and to evaluate the current standing criteria and give suggestions to further development. This has been

IN DEGREE PROJECT MECHANICAL ENGINEERING,SECOND CYCLE, 30 CREDITS

, STOCKHOLM SWEDEN 2016

Evaluation of vulnerability to parametric rolling

ANDERS SJULE

KTH ROYAL INSTITUTE OF TECHNOLOGYSCHOOL OF ENGINEERING SCIENCES

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 atKTH Centre for Naval Architecture in collaboration with Wallenius Marine. The main objective of thismaster thesis project have been to evaluate the vulnerability for parametric rolling for 3 generations ofPure Car/Truck Carriers (PCTC’s) from the Wallenius fleet. This has been done by using IMO’s currentversion of the second generation intact stability criteria for parametric rolling. The criteria is currentlyin 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 generationPCTC 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 criteriafor level 2 C2 is unable to capture sensibility ranking in a correct way.

Acknowledgements

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

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Contents

1 Introduction 4

2 Theory 62.1 Introduction to parametric roll resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.2 Stability variation in waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2.1 Coordinate system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.2.2 Calculating GZ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.2.3 Wave position in relation to ship in time . . . . . . . . . . . . . . . . . . . . . . . . 11

2.3 Damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.4 Inertia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.5 Time domain simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.5.1 Calculating Max roll angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.5.2 Time domain simulations of a C11 container ship . . . . . . . . . . . . . . . . . . . 13

3 IMO second generation intact stability criteria for parametric roll 173.1 Level 1 vulnerability criteria for parametric rolling . . . . . . . . . . . . . . . . . . . . . . 173.2 Level 2 vulnerability criteria for parametric rolling . . . . . . . . . . . . . . . . . . . . . . 18

3.2.1 Algorithm for assessing Level 2 C1 Criteria . . . . . . . . . . . . . . . . . . . . . . 183.2.2 Algorithm for assessing Level 2 C2 Criteria . . . . . . . . . . . . . . . . . . . . . . 19

4 Evaluation of Wallenius PCTC’s with IMO criteria for parametric roll 214.1 Wallenius PCTC and loading conditions for evaluation . . . . . . . . . . . . . . . . . . . . 214.2 Simulation results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224.3 Result discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

4.3.1 Load case 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224.3.2 Load case 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234.3.3 Comparing results with different speed steps . . . . . . . . . . . . . . . . . . . . . 24

4.4 Adjustments to bilge keel geometry to fulfill the criteria . . . . . . . . . . . . . . . . . . . 25

5 Observations made when implementing the current version of the IMO criteria 265.1 Now standing criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265.2 Suggested modifications of the criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

6 Conclusions 30

A Program Flowcharts 31A.1 Function flowcharts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31A.2 IMO criteria implementation flowcharts . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

B Results from program 35B.1 Sea state data from Level 2 C2 criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

B.1.1 Sea state data load case 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37B.1.2 Sea state data load case 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

B.2 Sea state data from Level 2 C2 criteria usin 21 speed steps . . . . . . . . . . . . . . . . . 45B.2.1 Sea state data load case 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46B.2.2 Sea state data load case 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

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C Program verification 54C.1 Verification of force equilibium calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

C.1.1 Heave equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54C.1.2 Calm water roll equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55C.1.3 Calm water pitch equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55C.1.4 Checking vulnerability calculations against IMO Calculations . . . . . . . . . . . . 57

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Nomenclature

In report In program Unit and description

LCG Ship.CoG(1) [m] Longitudinal center of gravityTCG Ship.CoG(2) [m] Transversal center of gravityKG Ship.CoG(3) [m] Vertical center of gravityx Ship.x [m] x-coordinates in ship fixed coordinate systemy Ship.y [m] y-coordinates in ship fixed coordinate systemz Ship.z [m] z-coordinates in ship fixed coordinate systemm Ship.m [kg] Mass of total displaced water at loading conditiond Ship.d [m] Draught corresponding to ship.mB Ship.B [m] Hull widthLpp Ship.Lpp [m] Length between perpendicularsL Ship.L [m] Ship lengthD Ship.D [m] Total hull depthUkn Ship.Ukn [Kn] Ship service speedGMc Ship.GM0 [m] Calm water GM0T0 Ship.T0 [s] Natural roll periodTe Te [s] Wave encounter periodCm Ship.CM [−] Midship area coefficientLBK Ship.LBK [m] Bilge keel lengthBBK Ship.BBK [m] Bilge keel breadthLBKtot [−] [m] Total bilge keel lengthBBKtot [−] [m] Total bilge keel breadth− Ship.dfull [m] Fully loaded draught− Ship.CW [−] Waterplane area coefficient− Ship.CP [−] Prismatic coefficient at draught5 ship.V [m3] Ship displacement volumeCB Ship.CBd [−] Block coefficient at draught− Ship.CMd [−] Midship section coefficient at draught− Ship.CPd [−] Prismatic coefficient at draught− Ship.CWd [−] Waterplane are coefficient at draughtxw State.xw [−] Wet offset points xyw State.yw [−] Wet offset points yxw State.zw [−] Wet offset points z− State.npw [−] Number of pointsXwl State.xwl [m] Waterline x-coordinatesYwl State.ywl [m] Waterline y-coordinatesZwl State.zwl [m] Waterline z-coordinatesBwl State.bwl [m] Waterline widthAs State.As [m] Wet section areaxB State.xB [m] Wet section area center of gravity x-coordinateyB State.yB [m] Wet section area center of gravity y-coordinatezB State.zB [m] Wet section area center of gravity z-coordinateLCB State.CB(1) [m] Longitudinal center of buoyancy in ship fixed coordinate systemTCB State.CB(2) [m] Transversal center of buoyancy in ship fixed coordinate systemKB State.CB(3) [m] Vertical cente of buoyancy in ship fixed coordinate systemS State.S [m2] Wet surface areaAWL State.AWL [m2] Waterplane areaLCB0 State.CB0(1) [m] Longitudinal center of buoyancy in global coordinate systemTCB0 State.CB0(2) [m] Transversal center of buoyancy in global coordinate systemKB0 State.CB0(3) [m] Vertical center of buoyancy in global coordinate system− Ship.ns [−] Number of sections in hull geometry− Ship.np [−] Number of points per section in hull geometryε Wave.epsvec [−] Wave position along the hullSw Wave.Sw [−] Wave steepnessλ Wave.lambda [m] Wave length

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H Wave.H [m] Wave heightW Wave.W [−] Relative occurence of seastate combination wave.H wave.lambdaHs Wave.Hs [m] Significant wave heightTz Wave.Tz [s] Average zero-crossing periodHeff Wave.Heff [m] Effective wave height with wave length = ship.L for sea state combination of wave.HS and wave.TzVs Ship.Vs [Kn] Ship speedV ship.V [m3] Ship displacement volumeτ − [m] Trimr − [m] Roll radius of inertia including added mass4 − [kg] Mass displacementGZ GZ [m] Transversal distance between CoG and CoBGM GM [m] Distance between CoG and metacenterη state.eta(5) [Deg] Pitch angleφ state.eta(4) [Deg] Roll angleς state.eta(3) [m] Heave

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Chapter 1

Introduction

Parametric roll is an event that can occur suddenly at sea. The outcome from this event can be loss ofcargo, unsafe conditions to crew and in the most dramatic cases loss of lives and capsizing. An exampleof 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 ofparametric 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 betweencost efficiency and safety.

IMO is currently undergoing work to establish new criteria for dynamic stability. One of these criteriais for parametric rolling. The Swedish company Wallenius have previously recorded experiences [2] ofparametric rolling for ships in their fleet. It is in the company interest to deepen their knowledge aboutthe phenomenon and gain knowledge of how their current fleet stand against the criteria. In this masterthesis three generations of Pure Car/Truck Carriers (PCTC’s) from Wallenius have been evaluatedagainst 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 theship it will have different stability properties at different times. For container ships and PCTC’s the aftand fore part of the ship is designed with a flared front and aft part. One of the reasons for this is tomaximize the amount of cargo onboard. This will increase the stability of the ship when wave troughis amidship as the flared part will provide a larger waterline breadth, see figure 1.1. When the ship atanother time has the wave crest at amidship the waterline breadth is smaller. At this state the ship hasless stability compared to calm water stability, see figure 1.1 for comparison. It is this change in stabilityas the wave passes that leads to parametric rolling.

Figure 1.1: Difference in waterline breadth for wave trough amidship; calm water and wave crestamidship 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 parametricrolling 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 criteriafor parametric rolling level 1 and level 2 [3]. Three generations of Wallenius ships have been evaluatedusing 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, pitchand surge motion. In absence of external time varying forces from waves coming from starboard orportside no forced roll motions will occure. Even in these conditions the ship can experience roll motionreferred to as parametric rolling [1]. Parametric rolling is caused by changes in stability as waves passesa ship. Stability variation in waves can be illustrated with how the GZ curve varies for the different wavepositions for different wave heights, see figure 2.1. As seen in the figure stability increases with waveheight 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 aC11 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 stabilityvariation is from wave crest amidship and wave trough amidship the bigger the roll amplitudes. Whenthe 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 increasedstability which will push the ship back into upright position. At the point in time when the ship hasreached an upright position from the push back the ship carries momentum in roll and also the wavecrest reaches amidship so the stability properties are again low. The ship will then fall over to the otherside while the wave trough again moves amidship and provides a pushback again. This rolling motionrises 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 periodof the ship. As the wave will then provide the change in stability synchronous with the roll motionfor 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 forresonance amplitudes to grow.

When assesing if a ship is at risk for parametric roll one can look at some points to determine if the shipis 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 rollperiod.� 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 rollmotion 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 functionof angular acceleration. The second term represent a moment due to the roll angle velocity of the systemlinked with damping. The third term represent a moment due to the restoring moment when a shiprolls. This moment is the force of buoyancy from the displaced water of the ship times the moment armwhich is the transversal distance between the center of gravity and the center of buoyancy known as theGZ 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 motioncan be expressed as:

φ+ (2α+ γφ2)φ+1

Ix +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 affectshow 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 thereal 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 aglobal 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 positionedin the origin of the local coordinate system. Its x-axis is parallel with the keel of the ship, it’s y-axis isparallel 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 = 0y = y − TCG;TCG = 0z = z −KG;KG = 0

The origin of the local coordinate system is positioned along the z-axis of the global coordinate system; atthe same x and y coordinates as the origin of the x-y plane. The x-y plane in the global coordinate systemis 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 onthe ship and loading condition.

Three degrees of freedom have been taken into consideration: heave, roll and pitch motion. Three degreesof freedom have been left out: yaw, sway and surge motion. Movements around these degrees of freedomoccur 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 theship length:

V =∫ L

0Asdx

The longitudinal center of buoyancy is calculated as:

LCB =∫ L0AsxBdx

V

The transversal center of buoyancy is calculated as:

TCB =∫ L0AsyBdx

V

The vertical center of buoyancy is calculated as:

KB =∫ L0AszBdx

V

The lifting force from the displaced water is calculated as:

FB = ρV g

The gravitational force is calculated as:

Fg = mg

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

LCB0 = cos(η)LCB + (− sin(η)KBTCB0 = sin(η) sin(φ)LCB + cos(φ)TCB + cos(η) sin(φ)KBKB0 = sin(η) cos(φ)LCB − sin(φ)TCB + 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 thewave passes. First heave is adjusted to reach vertical force equilibrium as seen in figure 2.5. Then thevertical force is compared to accepted vertical force limit. Secondly pitch is adjusted until the centerof gravity is aligned with center of buoyancy vertically see figure 2.6. Then pitch moment is comparedwith accepted limit for pitch moment. This is repeated until pitch moment and vertical force limit isacceptably 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 centerof gravity and the position of the center of buoyancy in the global coordinate system is measured, seefigure 2.7. This distance is the GZ value. As the ship heels it will rotate around its own longitudinal axisgoing through the center of gravity (it will rotate around the x-axis in the local coordinate system). Inthe global coordinate system it will rotate around the local x-axis positioned a distance in z over, underor trough the x-y plane in the global coordinate system. While moving along the z-axis in the globalcoordinate system (while doing heave, roll and pitch motion) the origin of the local coordinate systemwill remain fixed at the x and y coordinates at the origin of the global coordinate system. Because ofthis the transversal distance between the center of gravity in the global coordinate system and the centerof buoyancy in the global coordinate system is the same as the transversal distance between the centerof 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 betweenthe center of gravity and the center of buoyancy is measured. The GZ value is calculated as:

GZ = −TCB0

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 equi-librium 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 andU 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 (follow-ing seas) where the relative speed will be lower. If µ = 90 degrees the term with cosine will have no effect.

c− U cos(µ) = λTe

From this equation the encounter period can be written as:

Te = λc−U cos(µ)

The encounter frequency can accordingly be written as:

ωe = 2πTe

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

ωe = ω − ω2Ug cos(µ)

For regular gravity waves the wave period is:

T =√

( 2πλg )

The wave frequency can be written as:

ω =√

( 2πgλ )

The encounter period of the wave:

Te = λ√(λg2π−U cos(µ)

where U is ship speed in [ms ]

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wave position along the hull:

Wpos = 2π( tTe− floor( t

Te))− π

where floor indicates the closest natural number down i.e. 2.67 has floor(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+ γφ2)φ = B44(φa)φ (2.4)

B44 is a damping coefficient as a function of roll amplitude[6]. The roll damping contains differentparts: damping due to friction BF ; damping due to wave making BW ; damping due to the eddy BE andbilge keel BBK . When the ship is moving in forward speed a damping component due to lift is added BL.

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 dueto the mass of the ship and added mass due to displaced water as it rolls.

MIN = (Ix +A44)φ

Ix is 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+ γφ2)φ+1

Ix +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)− ρV gIx+A44

·GZ(t, φ)

By using the solver ODE45 in Matlab a time domain simulation of the rolling motion of the ship ata certain speed and heading in waves with wavelength equal to the ship length at a specific wave heightcan 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 shiphas the following load case data, see table 2.1.

Table 2.1: Ship data C11 LC2

Ship particulars unit ProgramLpp [m] 262B [m] 40D [m] 24.2d [m] 11.5τ [m] 0

ρ [ kgm3 ] 1025g [ Nm2 ] 9.81LCG [m] 125.517Cb [−] 0.56012Cm [−] 0.97097GMc [m] 1.9671T0 [s] 25.586V CG [m] 18.4V s [ms ] 12.165LBKLPP

[−] 0.2857BBKB [−] 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 andthe wave motion marked in black. The x-axis indicates the simulation time in seconds and the y-axisindicates the wave amplitude and the ship roll motion amplitude in meters and degrees respectively. Inthis simulation the ship is going in 20.39 knots in head waves with wave height 2.62 meters and wavelength equal to ship length. The ratio between wave encounter period and natural roll period is 0.33which is under the resonance frequency at the ratio between wave encounter period and natural rollperiod 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 height7.86 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 balanceit out. The ratio between wave encounter period of the waves and natural roll period of the ship is0.5 which means that 2 waves pass the ship for each roll period. This is referred to as 1:2 parametricresonance.

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 height7.86 meters and wave length equal to ship length. After an initial disturbance of 5 degrees in roll theship reaches roll amplitudes of over 20 degrees before the damping forces are large enough to balance itout. The ratio between wave encounter period of the waves and natural roll period of the ship is 1 whichmeans that 1 wave pass the ship for each roll period resulting in big roll motions to one side. This isrefered 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 height7.86 meters and wave length equal to ship length. After an initial disturbance of 5 degrees in roll the shipreaches 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 whichmeans that the ship rolls with 1.5 periods for each passing wave. This kind of parametric resonanceappear 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 withwave height. Wave height have big influence over the change of GM. In figure 2.12 the sensitivity forparametric 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 isthe 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 differentvalues of the fraction between the wave encounter period and the natural roll period of the ship is pre-sented. The blue graph line represents the limiting significant wave height where the ship is sensitivefor parametric roll angles over 25 degrees. The natural roll period of the ship does not change but theencounter period changes depending on speed and direction of the ship. For large speeds in head seasthe encounter period is small. For large speeds in following waves the encounter period is long. Bysimulating for high speeds in head seas then decreasing the speed to zero in speed increments followedby simulations in following waves in increasing speeds one can get a picture of which speeds the ship issensitive to parametric roll and for which significant wave heights the ship is sensitive. The red text andnumbers 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 Atlanticscatter diagram presented in [3]. Each sea state is scaled to be equal in length to the ship length andhaving a corresponding effective wave height at each sea state, see [6] for explanation of calculations ofcorresponding effective wave height at each sea state. Max roll angles over 25 degrees are marked out ascolored in the sea-state diagram. The number in each cell is the probability for the sea state to occur inpercentage.

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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 theship and encounter period of the waves is indicated. In the second column the ratio between the naturalroll period of the ship based on GMmean and encounter period of the waves is indicated, see equation2.6 and 2.7. GMmean is calculated as the mean GM value of a whole wave with wave height 0.0167 · Lpassing at ten different positions along the hull. This wave height is choosen from Level 1 criteria forvulnerability for parametric rolling [3] as this wave height was used there to calculate GMmean. In thethird column the ship speed and heading is indicated. In the fourth column the calculated probabilitythat roll angles over 25 degrees occur due to parametric roll is presented. The lines in the table markedas colored means that at these ratios of TeT0

; at these speeds and headings parametric roll over 25 degreesoccur.

ω0mean =√

(ρV gGMmean

Ix +A44) (2.6)

T0mean =2π

ω0mean

(2.7)

16

Chapter 3

IMO second generation intactstability criteria for parametric roll

In this chapter the IMO second generation intact stability criteria for parametric roll is presented, allinformation of the criteria can be found in [3] and [6]. The article [7] have provided information about thecurrent state of the criteria. In this report the first two levels of the criteria are included. The first levelis more conservative than the second level. For program implementations of the criteria in flowchartssee 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<= RPR (3.1)

∆GM1 =GMmax −GMmin

2(3.2)

The ratio ∆GM1

GMcrepresent 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 = Swλ, where λ = length ofship, Sw = 0.0167

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

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

∆GM1 is then calculated as:

17

∆GM1 = GMmax−GMmin

2

RPR 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:

RPR0 > C1 or RPR0 > C2

RPR0 is preliminary set by IMO to 0.06. In the calculation of C1 and C2 wave statistics is used and theC 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 0.3502 0.001654 37.316 0.4953 0.02912 55.743 0.8574 0.092799 77.857 1.2955 0.199218 103.655 1.7326 0.248788 133.139 2.2057 0.208699 166.309 2.6978 0.128984 203.164 3.1769 0.062446 243.705 3.62510 0.024790 287.931 4.04011 0.008367 335.843 4.42112 0.0024790 387.440 4.76913 0.000658 442.723 5.09714 0.000158 501.691 5.37015 0.000034 564.345 5.62116 0.000007 630.684 5.950

C1 =

n∑i=1

WiCi (3.5)

Ci = 0 if:

GM(Hi, λi) > 0 and ∆GM(Hi,λi)GM(Hi,λi)

< RPR or VPRi > Vs

Ci = 1 if not

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

VPRi is calculated as:

18

VPRi = |2λiT0

√(GM(Hi, λi)

GMc)−

√gλi2π| (3.6)

Vs is the ship service speed and VPRi 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 φmax 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: ϕ0max(h, Fn = 0)

in three speeds in head seas: ϕheadmax (h, Fn,k)in three speeds in following seas: ϕfollowmax (h, Fn,k)

Where

F(n,k) = Vk√(Lg)

Vk = VsKk

with Kk according to Table 2.11.3.3 in [6]

Table 3.2: Values for Kk

k Kk

1 1.02 0.8663 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) theeffective wave is determined with wave length λij = L and wave height Heff,ij

The roll angle in each sea state ϕij(Heff , Fn) is then determined with interpolation from previous calcu-lated 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 presentedin Table 2.11.3.4.2 in [6]

at zero speed: ϕ0ij(Heff,ij , Fn = 0) = interp1(h, ϕ0

max, Heff,ij)in three speeds in head seas: ϕheadij (Heff,ij , Fnk) = interp1(h, ϕheadmax , Heff,ij)

in three speeds in following seas: ϕfollowij (Heff,ij , Fnk) = interp1(h, ϕheadmax , Heff,ij)

Cij is then determined as:Cij = 1, if ϕij > 25 degrees

19

Cij = 0, otherwise

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

Zero speed: C0ij VS ϕ0

ij

Three speeds in head seas: Cheadij VS ϕheadij

Three speeds in following seas: Cfollowij VS ϕfollowij

The components that calculates C2 is then determined. This consist of a C2 value for zero speed, 3 C2values for following waves and 3 C2 values for head waves:

at zero speed: C20(Fn = 0) =∑i

∑j

WijC0ij

in the three speeds in head seas: C2head(Fnk) =∑i

∑j

WijCheadij

in the three speeds in following seas: C2follow(Fnk) =∑i

∑j

WijCfollowij

Finally, C2 is determined as:

C2 = [

3∑k=1

C2head(Fnk) + C20(Fn = 0) +

3∑k=1

C2follow(Fnk)]/7 (3.7)

20

Chapter 4

Evaluation of Wallenius PCTC’swith IMO criteria for parametric roll

In this chapter Wallenius PCTC’s are evaluated using IMO second generation intact stability criteria forparametric 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 presentedin 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 PCTCd[m] 9.5 9.5 9.5τ [m] 0 0 0r[m] 14.8 14.8 14.8GM [m] 1.2 1.2 1.2T0[s] 27.1 27.1 27.1KG[m] 13.6 14.45 15.21KM [m] 14.8 15.65 16.41V [m3] 33700 34000 33800LBKtot[m] 109 109 109BBKtot[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 PCTCd[m] 9.5 9.5 9.5τ [m] 0 0 0r[m] 14.8 14.8 14.8GM [m] 2 2 2T0[s] 21.0 21.0 21.0KG[m] 12.8 13.65 14.41KM [m] 14.8 15.65 16.41V [m3] 33700 34000 33800LBKtot[m] 109 109 109BBKtot[m] 0.4 0.4 0.4

21

4.2 Simulation results

The results when using IMO second generation intact stability criteria for parametric roll for the 3 gen-erations of PCTC’s are shown in table 4.3. The ships in the different loading conditions passing or failingthe 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 LC2RPR 0.32116 0.32116 0.32076 0.32116 0.32116 0.32076C1 0.68211 0.8846 0.8846 0 0.68211 0.68211C2 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 LC2Level 1 passed No No No No No NoLevel 2 C1 passed No No No Yes No NoLevel 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 shippasses the level 1 criteria and the level 2 C1 criteria and all the ships pass the level 2 C2 criteria. The3:rd generation PCTC is judged as being the most vulnerable to parametric rolling having a probabilityof 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% proba-bility for parametric roll. When comparing these results with a plot for limiting significant wave heightfor roll angles over 25 degrees using 41 speed steps, see figure 4.1, the 3:rd generation PCTC in load case1 stand out as the most sensitive to parametric roll. It is sensitive in a larger domain of speeds than thetwo other generations of PCTC’s. The 2:nd generation PCTC is the second most vulnerable by beingvulnerable 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 rollingat significant wave heights of 5.5m and also being sensitive in the shortest speed interval compared tothe other generations of PCTC’s.

22

Figure 4.1: Comparison limiting significant wave height for roll angles > 25 degrees in parametric rollfor 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 noship passes the level 1 criteria. The 1:st generation PCTC passes the Level 2 C1 criteria by havingan operational speed that is under the speed for which parametric rolling can occur, see chapter 3 forexplanation of Level 2 C1 criteria. The 2:nd and 3:rd generation of PCTC’s fail level 2 C1. For level 2 C2the 1:st and 2:nd generation PCTC passes the criteria and 3:rd generation PCTC fails the criteria. The2:nd generation PCTC is judged as being the least vulnerable to parametric roll having the probabilityof 0.057%. The 1:st generation is judged as not vulnerable to parametric roll having the probability of0.42%. When comparing these results with the plot for limiting significant wave height for roll anglesover 25 degrees using 41 speed steps as seen in figure 4.2 the 2:nd generation PCTC is more vulnerableto parametric roll than the 1:st generation being sensitive for more speeds and for critical speeds; lowersignificant wave height. This is not obvious when comparing the value for C2 for the 1:st and 2:ndgeneration PCTC. The 3:rd generation PCTC is judged as being the most vulnerable to parametric rollhaving the probabilistic value 10,16%. Comparing this number to the C2 value for the 3:rd generationPCTC in load case 1 which was 3.6% the conclusion can be made that it is more sensitive in load case 2than in load case 1. When comparing the two plots in figure 4.1 and 4.2 and comparing with the numberof C2 in both load cases it gives another impression of vulnerability. From these plots the 3:rd generationPCTC is more sensitive to parametric roll in load case 1 than in load case 2. The 3:rd generation PCTCis sensitive to the same lowest significant wave height at critical speeds in both load case 1 and load case2. But is vulnerable in a larger range of speeds in load case 1 than in 2.

23

Figure 4.2: Comparison limiting significant wave height for roll angles > 25 degrees in parametric rollfor 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 whichseems to correspond more with what is seen in figure 4.1 and 4.2. When using 21 speed steps the differentgenerations 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 probabilisticvalue for C2 have been calculated by summing all the probabilities for parametric roll angles over 25degrees at the different speeds and then dividing it by the number of speeds: 21. The results which canbe seen in table 4.5 shows that the 3:rd generation PCTC is the most vulnerable to parametric roll inload case 1 having a probability of 9,48%. In load case 2 where it is vulnerable for smaller speed intervalsit has a probability of 6.87%. The 2:nd generation PCTC is more sensitive in load case 1 than 2 havingthe probabilistic value 7.36% for load case 1 and 4.16% for load case 2. The 1:st generation PCTC hassimilar 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 0,073623:rd generation PCTC 0,036129 0,09483

Load case 2 1:st generation PCTC 0,0042209 0,010602:nd generation PCTC 0,00057071 0,041653:rd generation PCTC 0,1016 0,06870

In the current standing criteria the 3:rd generation PCTC would be judged as vulnerable to parametricroll in load case 2 and judged as not vulnerable in load case 1. The other PCTC’s at both loadingconditions would be judged as not vulnerable to parametric roll. If 21 speed steps where to be usedin the level 2 C2 criteria, and the limit value for being judges as vulnerable or not would remain as0.06, for load case 1 the 2:nd generation PCTC and the 3:rd generation PCTC would be evaluated to bevulnerable to parametric roll and for load case 2 the 3:rd generation PCTC would be judged as vulnerableto parametric roll. For load case 1 the 1:st generation PCTC would be evaluated as not vulnerable to

24

parametric roll and for load case 2 the 1:st and 2:nd generation PCTC’s would be evaluated as notvulnerable 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 changethe bilge keel geometry. When changing the bilge keel breadth of the 3:rd generation PCTC from 0.4meter to 0.8 meters, when using 7 speed steps, one get a C2 value of 0.0705 (Vs 0.1016 when having abilge keel breadth of 0.4 m). Making this adjustment would still not pass the current standing criteriahowever it will become less vulnerable to parametric roll. When using 21 speed steps one get a C2 valueof 0.0484 (Vs 0.0687 when having a bilge keel breadth of 0.4 m). As one can see from figure 4.3 changingthe bilge keel breadth reduces the vulnerability to parametric roll. The least significant wave height inwhich the 3:rd generation PCTC in load case 2 is sensitive to parametric roll with bilge keel breadth of0.4 meter is 2.5 meters. When adjusting the breadth of the bilge keel to 0.8 meters it’s increased to 3.5meters.

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

25

Chapter 5

Observations made whenimplementing the current version ofthe IMO criteria

5.1 Now standing criteria

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

When doing implementations of the now standing IMO second generation criteria for parametric rollsome questions about the current standing criteria came up. Most of the time of the master thesis havebeen 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 shipwith the load case presented in figure 2.1. Results from the simulation program was compared to theseresults for validation purpose. When following the instructions on how to calculate a probabilistic valueof vulnerability to parametric roll in this master thesis the value was 6.61%when simulating parametricroll 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 showsthe limiting significant wave height for different ratios of Te

T0 . For the seven speeds simulated parametricroll with roll angles over 25 degrees only occurred in 3 of them, this is marked with circles in 5.1. InAppendix C of the report, table C.5, one can see a comparison between the results from the programwritten in the master thesis compared with he IMO simulation results of a C11 container ship with thesame loading condition. The results for level 1 differs due to using different methods. But for level 2 theresults where similar. The slight difference in value for C2 in level 2 might be due to implementation ofthe effect of dynamic pressure along the hull, for different relative velocities between the hull of the shipand surrounding water, when calculating the damping coefficients α and γ.

26

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 tothe right the probability for roll angles over 25 degrees due to parametric roll is listed. When calculatingthe total probability for roll angles due to parametric roll over 25 degrees for these seven speeds, theprobability 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 percentageincrements in head and following sea. the following speeds would be covered in the diagram showingsensitivity to parametric roll. see figure 5.3. The ratios Te

T0for the 21 different speeds is marked with

circles in the plot.


27

The ship is sensitive to parametric roll in 9 out of 21 of these speeds. In the table in figure 5.4 it isindicated in which speeds the ship is sensitive to parametric roll. By summarizing the rightmost columnfor speeds where roll angles over 25 degrees due to parametric roll occurs and dividing with the totalamount 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 beunder 6%. When simulating at 7 different speeds this value is 6.61% so it will fail the criteria. But whenchoosing to simulate for 21 different speeds it will be 4.75% and then it will pass the criteria. The criteriaas it is currently suggested with 7 speeds, can give a fair representation to vulnerability to parametricroll but only if it manages to cover speeds in which parametric resonance appear. When simulatingPCTC’s the experience was that some of these resonance speeds was missed for a load case that wasless sensitive to parametric roll than other. For example when comparing the 3:rd generation PCTC inload case 1 and load case 2. For load case 2 the 3:rd generation PCTC failed Level 2 C2 having 10.16percentage probability for parametric roll compared to 3.61% probability for parametric roll in load case1. When looking at the vulnerability curves for parametric roll for the 3:rd generation PCTC in loadcase 1 and 2 one can see that load case 1 is more sensitive to parametric roll due to being sensitive inmore speeds, see figure 5.5. The expected value for C2 should in turn be higher to represent this but itdoes not do this because when simulating parametric roll in 7 speeds, some speeds where the ship is themost sensitive to parametric roll, is missed. In the simulation for load case 2 a speed manages to matchthe most sensitive resonance speed and this will result in a higher probabilistic value than for load case 1where the most sensitive resonance speed is missed. The speeds are indicated by the circles in the figure.

28

Figure 5.5: Comparison of the sensitivity for parametric roll for the 3:rd generation PCTC in load case1 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 shipsthey might be the most sensitive to wave lengths equal to the total ship length. The total length of acontainer ship is a larger value than the length between perpendiculars. For PCTC’s the total lengthof the ship and length between perpendiculars are similar. So scaling the waves equal to the total shiplength 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 considerationwhen simulating parametric roll. This is also discussed in the chapter Evaluation of Wallenius PCTC’sin 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 sensitivityto parametric roll. The criteria for level 2 C2 now instruct that 7 different speeds should be taken intoconsideration when calculating the probability for parametric roll. In this thesis work this has proven togive some wrong indications to which ships are more sensitivity to parametric roll as this speeds can orcannot cover a speeds in which the ship is the most sensitive to parametric roll. The fact that the shipsdoes operate in some speeds more time than at other makes the solution of simply using more speedsteps to calculate vulnerability to parametric roll not right either. When calculating the probability forparametric roll one can weigh the probabilistic value at one speeds with a probabilistic value indicatinghow much time is spent in that speed during normal operation. For example, if the ship is operating in16 knots 60% of the time the probabilistic value during that speed should have a greater effect than foranother speed i.e. 19 knots in which the ship only operates in 10% of the time. Weighing each numbermakes this method not so general anymore. It’s highly individual how different ships operate so this canmake the criteria to complex.

29

Chapter 6

Conclusions

When comparing the different PCTC under similar loading conditions the conclusion is that the thirdgeneration PCTC is the most sensitive, the second less sensitive than the third, and the first the leastsensitive. 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 criteriafor level 2 C2 is unable to capture sensibility ranking in a correct way. A suggestion to further developthe criteria is to use more speed steps. In this thesis work 21 speed steps was tried which seemed tocorrespond well with the sensitivity for parametric roll for the ships at the different loading conditions.

30

Appendix A

Program Flowcharts

A.1 Function flowcharts

Figure A.1: Calculate GZ function

31

Figure A.2: Force equilibrium function

32

A.2 IMO criteria implementation flowcharts

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

33

Figure A.4: Flowchart level 2 (C2)

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

34

Appendix B

Results from program

35

B.1 Sea state data from Level 2 C2 criteria

36

B.1.1 Sea state data load case 1

37

3:rd generation PCTC load case 1

Figure B.1: Roll angles over 25 degrees at different sea states for different TeT0

1:st generation PCTCLC1

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

38

2:nd generation PCTC load case 1


2:nd generation PCTCLC1

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

39

1:st generation PCTC load case 1




40


41



3:rd generation PCTCLC2

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

42





43



1:st generation LC2

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

44

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

45


46

3:rd generation PCTC case 1




47





48





49


50





51





52





53

Appendix C

Program verification

C.1 Verification of force equilibium calculation

For verifying that the function ForceEquilibrium calculates right heave and pitch angles comparison havebeen 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 shipB = 8 [m] Breadth of shipM = 300000 [kg] Mass of shipρ = 1025 [kg/m3] Saltwater densityg = 9.81 [N/m2] Gravitational constantLCG = 20 [m] Longitudinal center of gravityKG = 2 [m] Vertical center og gravityTCG = 0 [m] Transversal center of gravity

C.1.1 Heave equilibrium

First hand calculations where done to determine the draft:

FB = TBLρgFG = Mg

Force equilibrium in heave gives:

Thand = mBLρ = 30000

8401025 = 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.

54

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

Thand = m+∆mBLρ

Table C.2: Data heave equlibrium calculations

∆m[kg] Thand[m] Tprog[m] ∆T [m]1000 0.9177 0.9175 0.00022000 0.9207 0.9205 0.00023000 0.9238 0.9235 0.00034000 0.9268 0.9265 0.00035000 0.9299 0.9295 0.00046000 0.9329 0.9325 0.00047000 0.9360 0.9355 0.00058000 0.9390 0.9395 -0.00059000 0.9421 0.9425 -0.00041000 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 bycalculating force equilibrium at different heel angles. By hand GM0 can be calculated like this for thebox geometry:

GM0 = KB + Iwax5 −KG

KB = T2

IWAX = LB3

12

5 = LBT

This gives by hand calculations:

GM0hand = 4.28864[m]

Using the program GM0 is calculated as:

GM0prog = GZ2−GZ1

RollAngle2−RollAngle1

GM0prog = 4.2414[m]

The difference:

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

C.1.3 Calm water pitch equilibrium

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

55

GM0L = KB + IWAY

5 −KG

Where

IWAY = BL3

12

By shifting LCG 0.1 m aft the following pitch angle for force equilibrium:

ηhand = arctan 0.1GM0L

= 7 · 10−4[rad]

In the program force equilibrium at the shift of LCG is:

ηprog = 6.8320 · 10−4[rad]

∆η = ηhand − ηprog = 0.168 · 10−4[rad]

By shifting LCG with ∆LCG one can calculate hos the pitch angle changes:

ηhand = arctan ∆LCGGM0L

Table C.3: Data pitch equilibrium calculations

∆LCG[m] ηhand[rad] ηprog[rad] ∆η[rad · 10−3

0.1 0.0007 0.00068320 0.01680.2 0.0014 0.0014 00.3 0.0021 0.0020 0.10.4 0.0028 0.0027 0.10.5 0.0035 0.0034 0.10.6 0.0042 0.0041 0.10.7 0.0049 0.0048 0.10.8 0.0055 0.0054 0.10.9 0.0062 0.0061 0.11.0 0.0069 0.0068 0.1

The mean difference of pitch angle is ∆ηmean = 0.0008168[rad], in the order of 10−4[rad], which isacceptably low.

56

C.1.4 Checking vulnerability calculations against IMO Calculations

The following data for a C11 container ship was calculated and compared with main particulars used byIMO: Table C.4: Ship data C11 LC2

Ship particulars unit IMO ProgramLpp [m] 262.2 262B [m] 40 40D [m] 24.45 24.2d [m] 11.5 11.5τ [m] 0 0

ρ [ kgm3 ] 1025 1025g [ Nm2 ] 9.81 9.81LCG [m] 125.52 125.517Cb [−] 0.559 0.56012Cm [−] 0.950 0.97097GMc [m] 1.965 1.9671T0 [s] 25.1 25.586V CG [m] 18.4 18.4V s [ms ] 12.165 12.165LBKLPP

[−] 0.292 0.2857BBKB [−] 0.010 0.010

The following results was reached by IMO using the method explained in section 2.11.2.1 in [3] for level1 and the simulation program using method the method explained in section 2.11.2.3 in [3] for level 1:

Table C.5: Calculations results Vulnerability for parametric roll criteria

Result IMO Calculated by program∆GMGMc

1.045 0.83379

RPR 0.3561 0.41286∆GM 2.0534 1.6487GMc 1.965 1.9655GMmean - 2.3549Level 1 Not passed Not passedLevel 2 C1 0.4368 0.4366Level 2 C2 0.07283 0.066095Level 2 Not passed Not passed

The results for level 1 differs due to using different methods. But for level 2 the results is similar. Theslight difference in value for C2 in level 2 might be due to implementation of the effect of dynamic pressurealong the hull, for different relative velocities between the hull of the ship and surrounding water, whencalculating the damping coefficients α and γ.

57

Bibliography

[1] W. N. France, M. Levadou, T. W. Treakle, P. J.Randolph, M. R.Keith, and C. Moore, “An inves-tigation of head-sea parametric rolling and its influence in container lashing systems, sname annualmeeting 2001 presentation,” Marine Technology, vol. 40, pp. 1–19, 2003.

[2] A. Rosen, M. Huss, and M. Palmquist, “Experience from parametric rolling of ships. chapter in thebook fossen t. i. and nijmeijer h. “parametric resonance in dynamical systems”, isbn 978-1-4614-1042-3, springer,” Parametric Resonance in Dynamical Systems, vol. 40, pp. 147–165, 2012.

[3] IMO, “Draft amendment to part b of the is code with regards to vulnerability criteria of levels 1 and2 for the parametric rolling mode,” SDC 2/WP.4, Annex 2, pp. 1–7, 2016.

[4] V. Belenky and N. B. Sevastianov, “Stability and safety of ships: Risk of capsizing, isbn-13: 978-0939773619, society of naval architects; 2 edition (30 jun. 2007),” 2007.

[5] M. Huss, Fartygs stabilitet. JURE forlag AB, 2007.

[6] IMO, “Draft explanatory notes on the vulnerability of ships to the parametric roll stability failuremode,” SDC 3/WP.5 Annex 4, page 1, pp. 1–34, 2016.

[7] N. Umeda and A. Francescutto, “Current state of the second generation intact stability criteriaachievements and remaining issues,” Proceedings of the 15th International Ship Stability Workshop,Stockholm, Sweden, 13-15 June 2016.

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Evaluation of vulnerability to parametric rolling1057139/FULLTEXT01.pdfrolling and to evaluate the current standing criteria and give suggestions to further development. This has been