numerical simulation of parametric rolling in waves · system. coordinates in oxyz are written with...

Numerical simulation of parametric

rolling in waves

ER IK OVEGÅRD o v eg a r d@k th . s e 0 70 - 409 1 7 9 8

Master thesis

KTH Centre for Naval Architecture February 2009

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ABSTRACT This is a master thesis conducted at KTH Centre for Naval Architecture in collaboration with Seaware AB. Parametric rolling is a phenomenon where ships under certain conditions quickly develop large roll amplitudes due to parametric excitations. Ships with large bow and stern flare, such as container and PCTC ships, are especially sensitive to this phenomenon. Due to the non-linearities in the excitation forces ordinary linear strip theory in the frequency domain does not capture this phenomena. One way of predicting parametric rolling is by non linear simulations in the time domain. In July 2008 the EU Commission funded research project SAFEDOR made a call for participation for a benchmark study on “numerical simulation methods for the prediction of the Parametric Rolling of Ships in Waves”. This report describes a reimplementation of a method developed at KTH 1995 which has been used to generate a joint entry from KTH Centre for Naval Architecture and Seaware AB to the SAFEDOR benchmark study. The method used is a three degree of freedom time domain non-linear strip method, which is based on linear modeling of the inertia forces and diffraction forces, and the damping forces for heave and pitch, and non-linear modeling of the hydrostatic forces, Froude-Krylov forces and roll damping. The implementation is validated by comparison to linear strip theory and tank tests. The implementation show good agreement to linear theory in small waves and parametric rolling occurs in the simulations when it occurs in the tank tests. The tests for the benchmark study where calculated and submitted to SAFEDOR. The results, which are summarized in this report, are concluded to be reasonable. No tank test result from the study was available when the report was finished.

SAMMANFATTNING Det här är ett examensarbete utfört på KTH Marina System i samarbete med Seaware AB. Parametrisk rullning är ett fenomen där fartyg under vissa förutsättningar snabbt utvecklar stora rullningsamplituder på grund av parametrisk excitation. Fartyg med utsvept för och akter, såsom container och PCTC fartyg, är särskilt känsliga för detta fenomen. På grund av icke-linjäriteter i excitationskrafterna fångar inte den linjära strip-teorin i frekvensplanet detta fenomen. Ett sätt att prediktera fenomenet är med icke-linjära simuleringar i tidsplanet. I juli 2008 kallade det EU finansierade forskningsprojektet SAFEDOR till en jämförande studie i ”numerical simulation methods for the prediction of the Parametric Rolling of Ships in Waves”. Den här rapporten beskriver en återimplementering av en metod, utvecklad 1995 på KTH, som har använts för att generera ett gemensamt bidrag från KTH Marina System och Seaware AB till den jämförande studien. Den använda modellen är en tre-frihetsgrads-modell i tidsplanet, baserad på linjär modellering av tröghetskrafterna, diffraktionskrafterna och dämpningen i hävning och stampning, samt icke-linjär modellering av de hydrostatiska, Froude-Krylov- och rulldämpningskrafterna. Implementeringen är validerad mot linjär strip-teori och modellförsök. Implementeringen överensstämmer väl med linjär teori och parametrisk rullning uppstår i simuleringarna när det uppstår i modellförsöken. Testfallen för den jämförande studien är beräknade och sända till SAFEDOR. Inga resultat från modellförsök i studien fanns tillgängliga när rapporten färdigställdes.

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TABLE OF CONTENTS Nomenclature ................................................................................................................................................................4 1. Introduction ..........................................................................................................................................................6 2. Coordinate systems ..............................................................................................................................................6 3. Basic theory for ship motions in waves............................................................................................................7

3.1. Linear and non-linear solutions...............................................................................................................9 4. Forces...................................................................................................................................................................10

4.1. Restoring and Froude-Krylov forces ...................................................................................................10 4.2. Radiation forces .......................................................................................................................................11 4.3. Diffraction forces ....................................................................................................................................12

5. Solving the equation of motion .......................................................................................................................14 6. Validation of the code .......................................................................................................................................14

6.1. A box ship in the freuqency plane........................................................................................................14 6.2. Comparison of roll decay tests..............................................................................................................16 6.3. Ro-ro vessel, SSPA 2733........................................................................................................................18

7. SAFEDOR simulations and result ..................................................................................................................21 8. Future work, discussion and conclusion ........................................................................................................23 9. References ...........................................................................................................................................................24 Appendix 1, more about the coordiante systems...................................................................................................25 Appendix 3, integration of Kernel functions..........................................................................................................28 Appendix 4, hull geometry.........................................................................................................................................31 Appendix 5, roll damping ..........................................................................................................................................32 Appendix 7, comparison 0 knots..............................................................................................................................38 Appendix 8, comparison 10 knots............................................................................................................................40 Appendix 9, comparison with tank tests .................................................................................................................43 Appendix 10 SAFEDOR simulation results...........................................................................................................45

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NOMENCLATURE a wave.a Wave amplitude a ship.a Sectional added mass coefficient A ship.A Global added mass ax ax Particle acceleration in x-direction from undisturbed wave potential ay ay Particle acceleration in y-direction from undisturbed wave potential az az Particle acceleration in z-direction from undisturbed wave potential b ship.b Sectional damping coefficient B ship.Bwl Ship breadth at waterline B ship.B Global damping coefficients C C Matrix with restoring force coefficients CM Cm Amidship section coefficient F F Force or moment

restF Fc Restoring forces F KF − Ffk Froud-Krylov forces memF Fmem Memory forces dampF Fdamp Damping forces diffF Fdiff Diffraction forces

f f Force and moment acting on a section g env.g Gravitation constant Hi Wave height of wave component i I Moment of inertia k wave.k Wave number K ship.K Global kernel function k ship.k Local kernel function L ship.L Ship length at waterline LCG ship.LCG Longitudinal centre of gravity li l Euclidian length of segment i of a section Lpp ship.Lpp Length between perependiculars m ship.m Ship mass m ship.M Mass matrix of the ship n n Unit normal vector p p Pressure rxx ship.rxx Radius of mass inertia moment of roll ryy ship.ryy Radius of mass inertia moment of pitch rzz ship.rzz Radius of mass inertia moment of yaw s s Help variable described in Appendix 1 t t Time T ship.T Ship draught T wave.T Wave period Te wave.Te Wave encounter period Ts wave(i).T Period of wave component i U wave.U Ship speed u,v,w General wave particle velocities VCG ship.VCG Vertical centre of gravity vi v Vector to an offset point in a section vx vx Particle velocity in x-direction from undisturbed wave potential vy vy Particle velocity in y-direction from undisturbed wave potential vz vz Particle velocity in z-direction from undisturbed wave potential x x x-coordinate in oxyz X X X-coordinate in OXYZ x* x_ x*-coordinate in o*x*y*z* Y y_O Y-coordinate in OXYZ

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y* y_ y*-coordinate in o*x*y*z* z z z-coordinate in oxyz Z z_O Z-coordinate in OXYZ z* z_ z*-coordinate in o*x*y*z* ∆η delta_eta Difference in amplitude between two peaks in a roll decay test ∆ω The time between to max amplitudes in a roll decay test η eta Ship motion ηm eta_m Peak amplitude of peak m in a roll decay test κ kappa Fraction of the critical roll damping λ wave.lamda Wave length µ wave.mu Wave angle ρ env.rho Water density φ Wave potential functions ω wave.w Wave frequency ω0 w0 Natural roll frequency of the ship ωe we Wave encounter frequency ωeg w_eg Natural roll frequency

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1. INTRODUCTION Parametric rolling is a phenomenon where ships under certain conditions quickly develop large roll amplitudes due to parametric excitations. Ships with large bow and stern flare, such as container and PCTC ships, are especially sensitive to this phenomenon. Due to the non-linearities in the excitation forces ordinary strip theory in the frequency domain does not capture this phenomena. One way of predicting parametric rolling is by non linear simulations in the time domain. In July 2008 the EU Commission funded research project SAFEDOR made a call for participation [1] for a benchmark study on “numerical simulation methods for the prediction of the Parametric Rolling of Ships in Waves”. This report describes a reimplementation of a method developed at KTH 1995 [2] which has been used to generate a joint entry from KTH Centre for Naval Architecture and Seaware AB to the SAFEDOR benchmark study. The method is a three degree of freedom time domain non-linear strip method, which is based on linear modeling of the inertia forces and diffraction forces, and the damping forces for heave and pitch, and non-linear modeling of the hydrostatic forces, Froude-Krylov forces and roll damping. The aim has been to make a well documented implementation where the models for the different forces may be exchanged one at time to improve the result. The implementation should be possible to use in further research and master thesis work. The report will start with a description of the coordinate systems needed to model ship motion in waves and continue with an introduction to the modeling of ship motions in waves. Then follows a description of the modeling of the different forces acting on the hull and how to numerically solve the differential equation. It ends with a validation of the implementation and a presentation of the simulated result submitted to SAFEDOR and suggestions for further improvements.

2. COORDINATE SYSTEMS To describe the modeling of ship motions in waves three coordinates systems and the vector describing the motion needs to be defined. The coordinate systems are similar to the ones in [2].

Figure 1, the coordinate systems used throughout the report.

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The coordinate systems are o OXYZ

Global earth fixes system with the XY-plane in the still water plane and the Z-axis pointing upwards (opposite direction of gravity). The X-axis is pointing in the opposite direction of the wave propagation. Coordinates in OXYZ are written with capital letters, X, Y and Z.

o o*x*y*z* o Ship fixed system with origin in the centre of gravity, CG, of the ship. Rotates with the ship with

the x-axis parallel to the keel line, the z-axis pointing upwards and the y-axis pointing to port. The ship geometry is defined in this coordinate system. Coordinates in o*x*y*z* are written with lower case letters, x,* y* and z*

o oxyz Local system with origin fixed in CG, only translating with the ship. In other words no rotations relative the X- and Y-axes are allowed. The z-axis is parallel to the Z-axis and the x-axis is pointing in the main direction of the ship. The equations of motion are expressed in this system. Coordinates in oxyz are written with lower case letters, x, y and z.

And the vector describing the motion is o η

η describes the oscillatory motions, i.e. the deviation from mean position of the ship, both in translative and rotative degrees of freedom. In other words the difference, including the rotation, between o*x*y*z and oxyz. η and its time derivatives are used in the equation of motions and are shown in Figure 2.

Figure 2, η, the variable describing the motion of the ship.

3. BASIC THEORY FOR SHIP MOTIONS IN WAVES In this chapter a short introduction to the modeling of ship motions in waves and the linear and non linear theory that is used is presented. As for all mechanical systems the motions of a ship in waves may be described by Newton’s second law,

i ij jF η= ⋅m ɺɺ , (3.1.1)

where mij is the ships mass matrix, jηɺɺ are the accelerations in all six degrees of freedom and Fi are the

forces and moments acting on the ship. Among these forces the hydromechanic ones are usually orders of magnitude larger than the aerodynamic which are ignored. Modeling of the hydromechanical forces on a body in a fluid requires a description of the fluid flow. For an ideal fluid, i.e. incompressible, irrotational, inviscid, the fluid flow can be described in terms of a

velocity potential, φ , which has the property

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( , , )u v w φ= ∇ (3.1.2)

where u, v, and w are the fluid velocities in the x-, y- and z-directions. From this the pressure acting on the ship hull is determined according to Bernoullis equation,

2

p gzt

∂φ ρρ ρ φ φ∂

= − − − ∇ ⋅∇ , (3.1.3)

and the forces by an integration over the wetted hull surface,

i SF pdS= ∫∫∫ . (3.1.4)

The velocity potential for a particular fluid flow is determined by finding a solution to the Laplace equation,

2 0φ∇ = (3.1.5)

with relevant boundary conditions. For simple flows, such as a steady flow past a fixed cylinder, relevant boundary conditions are quite easily determined. However, the flow around a ship travelling in wind generated waves is unsteady as well as unstationary. In addition parts of the boundaries are constituted by the moving wave surface and the moving wetted body surface. By assuming that the squared particle velocities are small, compared to the other terms in the Bernoulli equation, a completely linear model is found. The complete problem may then, due to the linearity, be divided into a set of sub problems, where one problem may be solved at a time. The total potential is divided into the radiation, restoring and excitation potential,

Radiation Restoring Excitationφ φ φ φ= + + . (3.1.6)

The excitation potential is divided in Froude-Krylov and diffraction, which leads to

Radiation Restoring FroudeKrylov Diffractionφ φ φ φ φ= + + + . (3.1.7)

Since it is assumed that the different potentials in (3.1.7) are independent problems, one problem may be solved at a time. By assuming that the ship and wave motions are small in relation to characteristic ship dimensions the excitation forces may be derived by looking at a fixed hull in waves. The restoring and radiation forces may be found by considering a hull oscillating with a given frequency in calm water. By assuming the variation of flow is much larger in the cross section plane than the variation of the flow in the longitudinal direction the ship may be divided into two dimensional strips for which the potential flow problems are solved. By solving these problems for a two dimensional cylinder and by using conformal mapping to an arbitrary section shape the forces acting on this two dimensional section may be found. In this implementation Lewis-forms according to de Jong [3], implemented by Seaware, are used to calculate the potentials. For linear theory in the frequency domain the equation of motion becomes

( )( ) ( ) ( ) ( )( ) ( )0 0 cosFroudeKrylov Diffractionij ij j ij j ij j i iM A B C F F tω η ω η η ω ω ω+ + + = + ⋅ɺɺ ɺ . (3.1.8)

This corresponds to a spring damping system with the Froude-Krylov and diffraction forces as the exciting forces, the radiation force as an added mass and damper, and the restoring force as a spring. Since

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most of the terms are frequency dependant the equations have to be solved for one frequency at a time, giving a response amplitude function for the ship.

3.1. LINEAR AND NON-LINEAR SOLUTIONS The main advantage of linear theory in the frequency domain is that it is possible to superimpose these solutions for different wave heights and wave lengths. This makes it possible to analyze ship motions in irregular seas in the frequency domain based on linear signal analysis theory. The main disadvantage of the linear calculations in the frequency domain is they cannot capture non linear effects. To simulate parametric rolling three different non-linear mechanisms are of certain concern. These are the hydrostatic, restoring forces, the Froude-Krylov forces and the viscous roll damping. The Froude-Krylov forces are related to the pressure from the undisturbed wave potential and the restoring forces are related to the hydrostatic pressure. These are calculated by integrating the pressure over the momentary wetted surface in every time step. To illustrate the importance of these forces the GZ curve for the hull ITTC A-1 [1] with a wave crest at the middle of the ship, the end of the ship and without waves are shown in Figure 3. If the encounter frequency is equal to half the natural roll frequency and the wave crest is at the end of the ship at the peak roll angle, the righting moment will be higher than usual and when the roll angle is close to zero and the ship should stop, the moment is smaller than usual. With an initial disturbance this will start a parametric rolling.

0 5 10 15 20 25 30 35 400

0.2

0.4

0.6

0.8

1

1.2

Roll angel [°]

GZ

[m

]

Flat wave surfaceCrest at midshipCrests at ends

Figure 3, the GZ curve for ITTC A-1 [1] with a wave crest at the middle of the ship, the end and without waves. The wave length

is 100 m and the wave height 6 m. By calculating the roll damping, restoring and Froude-Krylov forces non-linearly and transforming the other forces in the frequency domain to the time domain, according to linear signal theory, parametric rolling may be simulated without solving all the wave potential equations anew. With this exchange of forces and transformation the equation of motion then become,

( ) rest F K mem damp diffm A F F F F Fη −⋅ + = + + + +ɺɺ , (3.1.9)

in the time domain. The different components are, one at a time, explained in the next chapter.

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4. FORCES

4.1. RESTORING AND FROUDE-KRYLOV FORCES The hydrostatic restoring forces are the hydrostatic pressure,

restp gZρ= − (4.1.1)

integrated over the hull surface. The Froude-Krylov forces are the integrated pressure,

( ), , ,F K x y z t

pt

φρ− ∂

= −∂

, (4.1.2)

from the undisturbed wave acting on the hull, where φ is the wave potential for the undisturbed wave. The wave potential is described in Appendix 2. At every time step the two pressures are integrated over the instant wetted hull surface. The pressure is integrated along each section and then integrated along the length of the ship. This integration follows [2]. Let vi denote the vector to the offset point i and let ni denote the unit normal to the line segment between the offset point i and i+1. By assuming that the pressure, pi, from equation (4.1.1) and (4.1.2), varies linearly between the points, the sectional force and moment from the pressure becomes

( )

+

+

+= ⋅ ⋅ ≤ ≤

+= ⋅ ⋅ × ≤ ≤

∑

∑

1

1

1 32

4 62

i ij i i

i

i ij i i i

i

p pf l j

p pf l j

n

r n

. (4.1.3)

+= −1 2i i il v v (4.1.4)

where li is the Euclidian length of the segment i as described in equation (4.1.4) and ri is the position vector to the point on the segment where the force is acting as given in equation (4.1.5).

( )++ +

+

++

+= + ⋅ − + ≠+

−= + =

2 113 3

1 1

1

11

if 0

if 02

i ii i i i i i

i i

i ii i i

p pp p

p p

p p

r v v v

v vr

. (4.1.5)

The intersection points are interpolated linearly and the pressure above the momentary wave surface is set

to zero. The sectional forces that are to give the forces restF and F KF − in (3.1.9) are then be transformed to o*x*y*z* and integrated along the x*-axis of the ship and transformed back to oxyz. The variables used for the integration are illustrated in Figure 4.

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Figure 4, variables used in the integration of the pressure, figure from [2].

To account for forces acting parallel to the x*-axis the same integration is done but on the water lines of the ship. These calculations are made in a similar way but ni is replaced with,

*1

0

0i

⋅

n , (4.1.6)

that is the normal of the waterlines projected onto the x*-axis.

4.2. RADIATION FORCES While it is the Froude-Krylov force that drives the motion and enables parametric rolling it is the damping that determines its magnitude. The damping consists of two parts. The memory forces, which are the inviscid damping in the frequency domain transformed to the time domain and the viscous damping. The only dimension where the viscous damping isn’t negligible is roll. While no analytical solutions for the viscous roll damping have been found [13] proposes a semi empirical method to determine the roll damping with and without speed with just a roll decay test at zero speed. This method is implemented as

described in detail in Appendix 5 and gives the force dampF in (3.1.9). By using signal theory for linear systems and the physical properties of the model Cummins [5] derived a way to transform the wave radiation related potential damping coefficient, Bij, from the frequency to the time domain according to equation (4.2.1) and (4.2.2) [16].

( ) ( ) ( )0

tmem

i ij jF t K t dτ η τ τ= − ⋅ − ⋅ ⋅∫ ɺ (4.2.1)

( ) ( ) ( )( ) ( )0

2cosij ij ijK t B B t dω ω ω

π

∞

= − ∞ ⋅ ⋅ ⋅∫ . (4.2.2)

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Since the wave potential damping includes an integral over the motion history the force is often called the memory force. The K in this equation is usually called the kernel function. The numerical calculations of equation (4.2.1) and (4.2.2) is described in Appendix 3. As mentioned earlier the local hydrodynamic coefficients are calculated with Lewis forms according to [3]. These are integrated along the x*-axis according to [4]. Cummins derivation [5] gives that the added mass

in the time domain is equal to the added mass when the frequency approaches infinity, ija∞. This leads to

the reduced form of the integrations from [4],

35 53 33

255 33

, 2,3, 4ij ij

L

L

L

A a dx i j

A A x a dx

A x a dx

∞ ∞

∞ ∞ ∞

∞ ∞

= ⋅ =

= = − ⋅ ⋅

= ⋅ ⋅

∫

∫

∫

. (4.2.3)

For the global damping coefficients no such simplification can be made. The full equations from [4], except for the non linear roll damping, are used and shown in equation (4.2.4) where index A denotes the value for the aft most section.

33 33 33

2

35 33 33 33 332

44 44 44

53 33 33 33

2 22

55 33 33 33 332 2

A

L

A A A

eL L

A

L

A A

L L

A AA a

e eL L

B b dx U a

UB x b dx U a dx U x a b

B b dx U a

B x b dx U a dx U x a

U UB x b dx b dx U x a x b

ω

ω ω

= ⋅ + ⋅

= ⋅ ⋅ + ⋅ ⋅ − ⋅ ⋅ − ⋅

= ⋅ + ⋅

= − ⋅ ⋅ − ⋅ ⋅ − ⋅ ⋅

= − ⋅ ⋅ + ⋅ ⋅ + ⋅ ⋅ + ⋅ ⋅

∫

∫ ∫

∫

∫ ∫

∫ ∫

(4.2.4)

For a pedagogical description of the calculations of the local and global coefficients in the frequency domain, [17] is recommended.

4.3. DIFFRACTION FORCES For high frequencies the diffraction forces are the dominant excitation forces. While the Froude-Krylov potential gives one flow through the wetted hull surface the diffraction potential gives another. Together they fulfill the body boundary condition that no water passes through the wetted hull surface. In this implementation the diffraction forces, just like the memory forces, are transformed from the frequency to the time domain according to Cummins [5]. But unlike the memory force these forces are derived by locking the hull in the wave system. Instead of the ship motions the undisturbed particle motions from the wave potential are used in equation (4.2.1). That gives, according to [2],

( ) ( )0

tDiffj ij i ij i

L

F a a t k t v d dxτ τ∞ = + − ⋅

∫ ∫ (4.3.1)

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( ) ( )0

cosij ijk t b t dω ω∞

= ⋅∫ . (4.3.2)

The resulting expressions for the diffraction forces are given in equation (4.3.3) to (4.3.7). Table 4.1 explains why different coefficients are included or not and the expressions for the water particle motions are found in Appendix 2.

( ) ( ) ( ) ( )2 22 22

0

tDiff

z z

L

F t a a t k t v d dxτ τ τ∞ = + −

∫ ∫ (4.3.3)

( ) ( ) ( ) ( )3 33 33

0

tDiff

z z

L


∫ ∫ (4.3.4)

( ) ( ) ( ) ( )4 24 24

0

tDiff

y y

L


∫ ∫ (4.3.5)

( ) ( ) ( ) ( )5 33 33

0

tDiff

z z

L

F t x a a t k t v d dxτ τ τ∞ = − ⋅ + −

∫ ∫ (4.3.6)

( ) ( ) ( ) ( )6 22 22

0

tDiff

z z

L

F t x a a t k t v d dxτ τ τ∞ = ⋅ + −

∫ ∫ (4.3.7)

Table 4.1, short motivation of why different components of the diffraction forces are included or not.

2 3 4 5 6 1 Symmetric hull

around centre line. Low speed. Symmetric hull

around centre line. Low speed. Symmetric hull

around centre line. 2 Used Hull symmetric

around centre line, low speed in sway.

Used Symmetric hull around centre line.

Symmetric hull around centre line.

3 Hull symmetric around centre line, low speed in sway.

Used Symmetric hull around centre line.

53 0a = Symmetric hull around centre line.

4 Used Symmetric hull around centre line.

4

4

0

0

v

a

==



5 Symmetric hull around centre line.

53 0a = Symmetric hull around centre line.

Used,

“ 55 33a x a∞ ∞= ⋅ ”


6 Symmetric hull around centre line.




Used,

“ 66 22a x a∞ ∞= ⋅ ”

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5. SOLVING THE EQUATION OF MOTION With all the different parts of the equation of motion described, the second order differential equation in the time domain,

( ) rest F K mem roll damp diffj ij ij j j j j jm A F F F F Fη ∞ − −⋅ + = + + + +ɺɺ , (5.1.1)

may be solved numerically. This is done by rewriting it as a first order differential equation as shown in equation (5.1.2).

where 1 6 and 7 12i

j i

j

y

y y i j

y

ηη

η

=

= = ≤ ≤ ≤ ≤=

ɺɺ

ɺɺɺ

(5.1.2)

This differential equation is solved with a 4th-order Adams-Bashforth-Moulton scheme, with Runge-Kutta 5 as a starting guess, as suggested by [7]. The scheme consists of Adams-Bashforth multistep method followed by a Moulton corrector. The scheme is described in Appendix 6.

6. VALIDATION OF THE CODE To validate the code, comparisons are made with Seaware’s and Tribons implementation of linear strip theory in the frequency plane. Comparisons are also made with experiments conducted by Garme [9]. To validate the non-linear viscous roll damping comparisons are made between the code, roll decay experiments and tank test with a wave direction of 90°. In all simulations the response functions are normalized with the wave amplitude, a, and the pitch and roll motions with the wave slope, κ , calculated according to equation (6.1.1).

k aκ = ⋅ (6.1.1)

6.1. A BOX SHIP IN THE FREUQENCY PLANE To verify the implementation of the model, simulations with a simple box ship are made with a wave amplitude of 0.1 m. At this small wave amplitude the simulations should agree with the linear theory in the frequency plane. The box ship is a closed box with dimensions according to Table 6.1.

Table 6.1, main particulars of the box ship.

Length 40 m Breadth 8 m Draught 2 m Freeboard 2 m Displacement 656 000 kg Block coefficient 1.0 LCG 20 m Origin at aft perpendicular VCG 2 m From keel line Radius of inertia, roll 3.2 m Radius of inertia, pitch 10 m Radius of inertia, yaw 10 m Trim 0 Viscous roll damping 10% Of critical roll damping

Simulations for the box ship where carried out in different speeds and regular waves frequencies to verify correspondence to linear theory in the frequency plane. Each simulation where carried out until a steady

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state solution was available and the mean of the last four of the peak values where used for the response function. The different simulations are shown in Table 6.2 and the result in Figure 5, Figure 6 and Appendix 7 and 8.

Table 6.2, different speeds and regular wave frequencies where simulations where conducted for the box ship, all wave amplitudes 0.1 m. Note that for zero speed the solutions are completely symmetric around the yz-plane.

Speed [knots] Wave angle (head=180°) [°]

Minimum frequency [rad/s]

Maximum frequency [rad/s]

0 0/180 0.40 2.0 0 45/135 0.40 2.0 0 90 0.40 2.0

10 0 0.40 2.0 10 45 0.40 2.0 10 90 0.40 2.0 10 135 0.40 2.0 10 180 0.40 2.0

0 0.5 1 1.5 2 2.5 3 3.50

1

2BoxShip ; T=2m ; KG=2m ; U=0kn ; µ=45deg

η 3 [no

n-di

m]

ω [rad/s]

0 0.5 1 1.5 2 2.5 3 3.50

0.5


η 5 [no

n-di

m]

ω [rad/s]

0 0.5 1 1.5 2 2.5 3 3.50

1


η 4 [no

n-di

m]

ω [rad/s]

Tribon VughtLewisTribon ScoresLewisTribon ScoresCloseFitSMC LewisSimulated

Figure 5, frequency response functions for 135° and 45° wave directions at zero speed. The simulated r esponse functions follow

Seawares response functions closely.

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0 0.5 1 1.5 2 2.5 3 3.50

1


η 3 [no

n-di

m]

ω [rad/s]

0 0.5 1 1.5 2 2.5 3 3.50

1


η 5 [no

n-di

m]

ω [rad/s]

0 0.5 1 1.5 2 2.5 3 3.5-1

0


η 4 [no

n-di

m]

ω [rad/s]


Figure 6, frequency response functions for 180° wav e directions at 10 knots.

As seen in Figure 5 and Figure 6 the correspondence between linear theory and the simulations is good. This shows that the numerical implementation of the convolutions and the pressure integration is working as expected. Since the box ship is completely symmetrical there is no difference between following and head seas at zeros speed. The differences between linear strip theory and the simulations are due to the fact that the diffraction forces are calculated with a mean velocity acting at the centroid of the wetted mean section instead of the line integral used in [3]. Another reason is that all forces are calculated on an actual square ship section instead of a Lewis from.

6.2. COMPARISON OF ROLL DECAY TESTS To verify that the non linear roll damping is correctly implemented roll decay tests corresponding to those done in [1] and [9] are simulated. The results are shown in Figure 7 and Figure 8.

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0 50 100 150 200 250 300 350-15

-10

-5

0

5

10

15

t [s]

η 4 [°]

Roll decay test, ITTC hull

SimulatedTank test

Figure 7, simulated and actual roll decay test of the ITTC-hull used in the SAFEDOR study.

0 50 100 150 200 250-8

-6

-4

-2

0

2

4

6

8

10

t [s]

η 4 [°]

Roll decay test, SSPA hull

SimulatedTank test

Figure 8, simulated and actual roll decay test of the SSPA hull used in [9].

18/45

The agreement is very good for the ITTC hull while it is less good for the SSPA hull. The change in natural frequency due to the damping differs between the simulation and the actual decay test. This

suggests that the some of the assumptions made when calculating 44,R D

iB − were not completely correct,

which should lead to a difference in damping and roll amplitude. There are three apparent sources of errors. The restoring force is not completely linear up to 8°, the dry moment of inertia is uncertain and the simulation is locked in sway while the model in the tank test was not. To get a better estimation of the roll damping models accuracy more roll decay tests are needed, preferably at different speeds. While there is a small deviation of the roll amplitude for the decay test with the SSPA hull the conclusion of the roll decay comparison is that the non linear damping is implemented correctly.

6.3. RO-RO VESSEL, SSPA 2733 With the implementations verified the code may be compared to model tests in large waves. Simulations corresponding to relevant tank tests in [9] are compared with the tests and linear strip theory. The main particulars of the ship in [9] are described in Table 6.3.

Table 6.3, main particulars of the ro-ro vessel, used in [9].

Length 135 m Breadth 24.15 m Draught 5.495 m Displacement 11 276 ton LCG 63.82 m Origin at aft perpendicular VCG 12.22 m From keel line Radius of inertia, roll 8.07 m Radius of inertia, pitch 37 m Radius of inertia, yaw 37 m Trim 0

The tank tests where conducted in scale 1:35 and then scaled to full scale by Froude’s scale laws. The complete ship geometry and roll decay-tests are found in [9]. Note that the roll radius of inertia given in [9] includes the added mass while the one in Table 6.3 does not. Simulations with wave directions of 180°, 150° and 0° where run since these correspond to the SAFEDOR tests [1]. To validate the non linear roll damping some cases with 90° wave direction where also chosen. The simulated tests are shown in Table 6.4.

Table 6.4 different speeds and regular waves where simulations where conducted for SSPA 2733.

Test # Froude number[-] Wave angle [°] Wave amplitude [m] Wave frequency [rad/s]

35 0.072 180 2.3 0.35

36 0.072 180 2.3 0.55

39 0.072 180 1.7 0.75

40 0.21 150 1.9 0.35

42 0.21 150 2.0 0.55

44 0.21 150 2.0 0.75

49 0.21 90 1.9 0.35

52 0.21 90 2.0 0.75

60 0.21 0 2.4 0.35

61 0.21 0 1.7 0.55

62 0.21 0 1.6 0.75 When normalizing the tank test and simulation results the mean wave amplitudes and wave heights are used but while running the simulations the complete vectors of wave amplitudes, phases and frequencies given in [9] are used. These vectors are a linear fitting of six wave components to the actual measured

19/45

wave surface in the tests. In Figure 9, Figure 10 and Appendix 9 the normalized transfer functions from Seaware AB, calculated with Lewis forms, are plotted together with simulations and model tests.

0 0.5 1 1.5 20

0.5

1

η3

35 36 39

SeawareTank testSimulation

0 0.5 1 1.5 20

0.5

1

Non

dim

ensi

onal

res

pons

e [-

]

η4

0 0.5 1 1.5 20

0.5

1

ω [rad/s]

η5

Figure 9, test 35, 36 and 39, head seas and a speed of Fn=0.072.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.5

1

η3

49 52


0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

2

4

Non

dim

ensi

onal

res

pons

e [-

]

η4

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.5

ω [rad/s]

η5

Figure 10, test 49 and 52, seas with a direction of 90° and a speed of Fn=0.21. The roll response in t he simulation is slightly

larger than the one in the tank test when ω=0.75.

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As seen in Figure 9 and Figure 10 the agreement between the tank test and the simulation is quite good for heave and pitch, with the exception from test 39 where no good explanation is found for the difference in heave. The largest error is found in the roll amplitudes. The viscous roll damping is acknowledged hard to determine [15] and is dependant of both speed and frequency. As seen in Figure 10 the roll damping is over estimated for the low frequency while it is underestimated at higher frequencies. According to [14] and [15] the roll damping increases with increased frequency. While test 49 and 52 agrees with this theory the differences in amplitudes seems to be larger than expected. Since the response function is very steep at this frequency a small deviation in natural frequency of the ship, or the waves in the tank test, will give a large difference in roll amplitude that could explain the unexpectedly large error. As seen in Figure 9 and Figure 11 the simulation captures the parametric rolling but the amplitude is far from correct. The faster growth is consistent with the lower damping for small amplitudes seen in the roll decay test in figure Figure 8. Since the tank test was ended before a steady state roll amplitude was found it is impossible to see how the larger amplitudes correlate between test and simulation.

0 10 20 30 40 50 60 70-10

-5

0

5

10

15

t [s]

η 4 [°]

Test 39

SimulatedTank test

Figure 11, roll amplitudes for test 39.

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7. SAFEDOR SIMULATIONS AND RESULT The SAFEDOR Call for participation [1] requested 22 tests to be simulated; 11 with one loading condition and 11 with another loading condition. The main particulars and the loading conditions are shown in Table 7.1 and the tests in Table 7.2. The differences between the loading conditions and the tests with parametric rolling are marked with grey.

Table 7.1, main particulars of the ITTC A-1 used in the SAFEDOR study. The differences between the loading conditions are marked with gray.

Particular Test 1-11 Test 12-22 Comment

Length 150 m 150 m Breadth 27.2 m 27.2 m Draught 8.5 m 8.5 m Displacement 23110 ton 23110 ton Block coefficient 0.667 0.667 LCG 73.0 m 73.0 m Origin at aft perpendicular VCG 10.20 m 10.58 m From keel line GM0 1.38 m 1.00 m Radius of inertia, roll 10.33 m 10.33 m Radius of inertia, pitch 37.5 m 38.2 m Radius of inertia, yaw 37.5 m 38.2 m Trim 0 m 0 m

Table 7.2, the simulations, tests where parametric rolling occurs are marked with gray.

# µ [°] H1 [m] T1 [s] λ [m] H2 [m] T2 [s] H3 [m] T3 [s] Description

1 - - - - - - - - Roll decay (in calm water)

2 180 3.6 10.63 177 - - - - Regular (1 harmonic)

3 180 5.7 10.63 177 - - - - »

4 180 3.6 10.63 177 - - - - »

5 180 5.7 10.63 177 - - - - »

6 180 2.4 10.63 177 2.4 9.66 2.4 11.55 Group (3 harmonics)

7 180 4 10.63 177 1 9.66 1 11.55 »

8 180 5 10.63 177 - - - - Irregular (JONSWAP spectrum, γ=3.3)

9 160 3.6 10.63 177 - - - - Regular (1 harmonic)

10 160 5.7 10.63 177 - - - - »

11 160 4 10.63 177 1 9.66 1 11.55 Group (3 harmonics)

12 - - - - - - - - Roll decay (in calm water)

13 0 3.6 8 100 - - - - Regular (1 harmonic)

14 0 6 8 100 - - - - »

15 0 3.6 8 100 - - - - »

16 0 6 8 100 - - - - »

17 0 2.4 8 100 2.4 7.11 2.4 8.89 Group (3 harmonics)

18 0 2.4 8 100 2.4 7.11 2.4 8.89 »

19 0 5 8 100 - - - - Irregular (JONSWAP spectrum, γ=3.3)

20 180 5 12.12 230 - - - - Regular (1 harmonic)

21 180 5 12.12 230 - - - - »

22 180 4 12.12 230 1 10.77 1 13.47 Group (3 harmonics)

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All simulations where allowed to reach a steady state solution and the max range, between the largest and smallest roll amplitude and the mean roll amplitude of the steady state periods where extracted. These results are shown in Table 7.3. Table 7.3, simulation results, tests where parametric rolling occurs are marked with dark gray, fields commented in the text are

marked with lighter gray.

# Range [°] Mean steady roll angle [°] Comments

2 10.7 10.68 Parametric rolling, converging after 400 s, aft flare never enters water.

3 0.04 0.18 Initial conditions creates small rolling, however decaying to zero.

4 15.5 15.50 Slowly starting parametric rolling, converging after 600 s, occasionally water on deck just aft of fore deck.

5 13.8 13.82 Parametric rolling, converging after 400 s, water on deck, fore deck occasionally submerged, rolling starts faster than for T04.

6 0.02 0.04 Initial conditions create small rolling, which might indicate that the condition is close to parametric. However finally decaying to zero.

7 14.7 16.22 Parametric rolling. Kind of converging after 500 s, approximately 1/3 of deck below water surface right after t=0, regularly water on deck just aft of fore deck.

8 0.00 0.00

Slight initial rolling due to initial conditions which however declines to zero. In addition to the here reported simulation another three 30 minutes simulations with different randomized waves phases has been made without initiation of parametric rolling. Simulations have also been made with larger initial roll velocities and equilibrium heel angles without development of parametric rolling.

9 14.1 14.09 Parametric rolling, converging after 600 s, stern flare never in water.

10 11.8 11.85 Parametric rolling, converging after 380 s, water on deck, port side of fore deck occasionally below water.

11 13.3 14.85 Parametric rolling, kind of converging after 600 s, water on deck, port side of fore deck deeper below water than in T10.

12 - Roll-decay test.

13 5.34 5.35 Small parametric rolling, converging after 600 s.

14 0.00 0.00 Initial parametric rolling tendency, may start with small change in encounter frequency, decays to zero.


16 32.3 32.34

Parametric rolling, very large and fast developing, converging after 200 s. Pitch and heave coupled with roll, larger heave but smaller pitch when roll amplitude is large, approximately half of the deck surface regularly submerged.

17 10.5 21.02 Parametric rolling with angles up to 20 degrees quickly developing and decaying at several of the wave groups.


19 0.0 0.00

Slight initial rolling due to initial conditions which however declines to zero. In addition to the here reported simulation another three 30 minutes simulations with different randomized waves phases has been made without initiation of parametric rolling. Simulations have also been made with larger initial roll velocities and equilibrium heel angles without development of parametric rolling.

20 11.5 11.51 Parametric rolling, converging after 450 s.

21 16.9 16.89 Parametric rolling, converging after 400 s, aft flare never enters water.

22 9.31 10.63 Parametric rolling slowly starting, kind of converging after 500-600 s.

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All test where started with a start guess in heave and pitch calculated from linear theory and an initial disturbance of 0.1°/s in roll. In test 8 and 19, marked with light gray, JONSWAP spectrums where simulated. These tests where repeated 4 times, each with a simulation time of 30 minutes with different randomized phases. 3 out of the four simulations where carried out with an unevenly loaded ship with 1.5° heel. This was done to enable parametric rolling to start after the initial disturbance had decayed. Despite this no parametric rolling started. While all results are reasonable some details may be noted before the tank test results are returned from SAFEDOR. In test two and three, marked with a lighter gray, the speed and wave frequency are the same but parametric rolling only starts with the smaller wave amplitude. The same phenomenon is visible at test 13 and 14. Both test 3 and 14 where simulated with a starting guess of 3°/s and with an uneven loading of 1° without any difference.

8. FUTURE WORK, DISCUSSION AND CONCLUSION The program is written in such a way that it is possible to increase the number of degrees of freedom just by adding expressions for the hydrodynamic coefficients and add the diffraction forces as described by [2]. This was left out of the current code simply because the SAFEDOR study was limited to three degrees of freedom. If simulation for all 6 degrees of freedom is implemented maneuvering according to [2] may be easily added. By having a main wave direction and course, instead of giving wave direction relative to the course, maneuvering simulation is easier to implement. Multiple wave directions could easily be added by making the wave number, k, a vector, as described by [11]. The wave potential then becomes,

( )( )

( ) ( )1 2

coshsin

coshx

x

k Z hgak X k Y t

k hφ ω ε

ω⋅ +

= − ⋅ ⋅ ⋅ + ⋅ + ⋅ +⋅

. (8.1.1)

From this equation pressure and particle velocities may be derived. Since all hydrodynamic coefficients are calculated with the ship at equilibrium without waves the error will increase with increasing deviation from this state. By calculating the coefficients for different drafts, as suggested by [10], this error may be decreased. To further improve accuracy the velocities of the diffraction forces could be calculated along the edge of the each instant cross section. By using boundary element methods, also called panel methods [18], instead of Lewis forms, asymmetrical hulls or hulls deviating far from Lewis may be simulated accurately. 2d panel methods may replace the Lewis forms without editing any other functions while 3d panel methods will require a new way of calculating the diffraction forces. Using methods that handles asymmetrical sections also opens the possibility to calculate new hydrodynamic coefficients for each time step, for the instant wetted sections. As seen in the comparison with linear theory the model give accurate result for low wave amplitudes. For larger wave amplitudes the model is better than linear theory while it still suffers from the hydrodynamic coefficients calculated at the mean position. The roll damping is working as expected for zero speed decay tests but the roll amplitude result is questionable if the roll frequency differs from the natural frequency or the speed is high. In summary the implemented model is a good platform to start further investigation of ship motion and parametric rolling. The model do capture parametric rolling, where linear theory does not, and with a speed of two to four minutes of computation per simulated minute is it a good compromise between time and accuracy.

24/45

9. REFERENCES 1. Papanikolaou, A: Call for Participation, International Benchmark Study on Numerical Simulation Methods

for the Prediction of Parametric Rolling of Ships in Waves. SAFEDOR, 14-08-2008. 2. Hua, Palmquist: A Description of SMS – A Computer Code for Ship Motion Calculation. Naval

Architecture, Department of Vehicle Engineering, KTH, Stockholm, 1995. 3. de Jong: Computation of the Hydrodynamic Coefficients of Oscillating Cylinders. Netherlands Ship Research

Centre TNO, Shipbuilding Departement, Delft, June 1973. 4. Salvesen, Tuck, Faltinsen: Ship Motions and Sea Loads. Trans. SNAME, Vol.78, 1970 5. Cummins, W.E: The Impulse Response Function and Ship Motions. Schiffstechnik, Forschungshefte für

schiffbau und schiffsmaschinenbau, Heft 47, June 1962. 6. Himeno, Y: Prediction of Ship Roll Damping – State of the Art. Departement of Naval Architecture

and Marine Engineering, The University of Michigan, Michigan, September 1981. 7. Kring, D. & Sclaounos P.D: Numerical Stability Analysis for Time-Domain Ship Motion Simulations.

Journal of Ship Research, Vol. 39, No. 4, pp. 313-320, December 1995. 8. Epperson, James F: An introduction to numerical methods and analysis. John Wiley & Sons, Inc. 2002. 9. Garme, K: Model Seakeeping Experiments Presented in the Time-Domain to Facilitate Validiation of

Computational Results. Naval Architecture, Department of Vehicle Engineering, KTH, Stockholm 1997.

10. Mikami, Shimada: Time-domain strip method with memory-effects function considering the body nonlinearity of ships in large waves. Journal of Marine Science and Technology, 11, 2006.

11. Kundu, P. K. and Cohen, I. M: Fluid Mechanics, Third Edition. Elsevier Academic Press, 2004. 12. Bodén, Ahlin, Carlsson: Signaler och mekaniska system. MWL, Department of Vehicle Engineering,

KTH, Stockholm, 2005. 13. Ikeda Y., Himeno Y., Tanaka N: A Prediction Method for Ship Roll Damping, Report of Department

of Naval Architecture, University of Osaka Prefecture, Osaka, Japan, No. 405, 1978. 14. Colbourne, B:The Effect of Forward Speed on Roll Damping. Massachusetts Institute of Technology,

1983. 15. Pesman, Bayraktar, Taylan: Influence of damping on the roll motions of ships. Proceedings, International

Conference on Marine Research and Transportation ’07, 2007 16. Lewandowski, E: The Dynamics of Marine Craft. World Scientific, 2004 17. Journée, J.M.J: Theoretical manual of SEAWAY. Delft University of Technology

Shiphydromechanics Laboratory, (Release 4.19, 12-02-2001), http://www.shipmotions.nl/DUT/PapersReports/1370-StripTheory-03.pdf.

18. Bertram, V: Practical ship hydrodynamics. Butterworth-Heinemann, Oxford, 2000.

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APPENDIX 1, MORE ABOUT THE COORDIANTE SYSTEMS In the implementation of the code the variable s, the rotation matrix T and the translation matrix t are used. These are defined in this appendix.

THE VARIABLE S s is mainly used as an extra variable to ease the implementation and includes both the instant motion, η, the constant motion associated with the ship speed, U, and the angle, µ, between the OXYZ and oxyz. s is the difference between OXYZ and o*x*y*z. If R and r* is the same point, described in OXYZ and o*x*y*z*, s is defined by

*s R r≡ − . (1.1.1)

This means that s, with VCG and T, in o*x*y*z*, as described in Figure 12, becomes

( )( )

cos

sin

0

0

U t

U t

VCG Ts

µµ

π µ

− ⋅ ⋅ − ⋅ ⋅ −

= +

− +

η . (1.1.2)

Figure 12, a box hull with the coordinate systems OXYZ and o*x*y*z* and VCG and T defined.

TRANSFORMATIONS To transform a coordinate, r, from one coordinate system to an other the rotation matrix T and the vector t is used. T corresponds to the differences in rotations of the axes between the coordinate systems and t to the distance between the different origins. R is transformed from OXYZ to oxyz by

*r R t= ⋅ +T (1.1.3)

and transformed back by

( )* 1R r t −= − ⋅T , (1.1.4)

y*

Y

T

VCG

Z

z*

26/45

where T is

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

1 2 3 2 1 3 1 3 2 3 1

3 2 3 2 1 3 1 3 2 3 1

2 2 1 2 1

cos cos cos sin sin sin cos cos sin cos sin cos

sin cos sin sin sin cos cos sin sin cos cos sin

sin cos sin sin sin

θ θ θ θ θ θ θ θ θ θ θ θθ θ θ θ θ θ θ θ θ θ θ θ

θ θ θ θ θ

− + = + − −

T

(1.1.5). The values for θ and T for different transformations are shown Table 0.1.

Table 0.1, values of t and θ for different transformations

Transformation between T θ OXYZ o*x*y*z* [ ]1 2 3s s s [ ]4 5 6s s s

OXYZ oxyz [ ]1 2 3s s s [ ]60 0 s

oxyz o*x*y*z* [ ]0 0 0 [ ]4 5 0s s

27/45

APPENDIX 2, THE WAVE MODEL In order to calculate the wave pressure a wave model has to be chosen. For all calculations linear wave theory, with long-crested regular waves, is used. In order to simulate wave spectrums the wave potential for a number of different, regular waves can be summed. Each wave potential is described in OXYZ by equation (2.1.1), where k is the wave number, a wave amplitudes, ω wave frequencies, h the depth of the sea and X, and Z coordinates in OXYZ. A detailed derivation of the wave potential function may be found in [11].

( )( )

( ) ( )cosh

sincosh

x

x

k Z hgak X t

k hφ ω ε

ω⋅ +

= − ⋅ ⋅ ⋅ + ⋅ +⋅

. (2.1.1)

With the wave potential in equation (2.1.1) the particle velocities become

( )( )( ) ( )

( )( )( ) ( )

( )( )( ) ( )

coshcos cos

cosh

coshcos sin

cosh

sinhsin

cosh

x

y

z

k Z hgakv k X t U

k h

k Z hgakv k X t U

k h

k Z hgakv k X t

k h

ω ε µω

ω ε µω

ω εω

⋅ += − ⋅ ⋅ ⋅ + ⋅ + −

⋅

⋅ += ⋅ ⋅ ⋅ + ⋅ + + ⋅

⋅ += − ⋅ ⋅ ⋅ + ⋅ +

⋅

, (2.1.2)

with the velocities in oxyz for the mean position of the ship. Since the accelerations are the time derivatives of the velocities they become

( )( )( ) ( )

( )( )( ) ( )

( )( )( ) ( )

coshsin cos

cosh

coshsin sin

cosh

sinhcos

cosh

x

y

z

k Z ha gak k X t

k h

k Z ha gak k X t

k h

k Z ha gak k X t

k h

ω ε µ

ω ε µ

ω ε

⋅ += ⋅ ⋅ ⋅ + ⋅ + ⋅

⋅ += − ⋅ ⋅ ⋅ + ⋅ + ⋅

⋅ += − ⋅ ⋅ ⋅ + ⋅ +

⋅

. (2.1.3)

Since the velocities and accelerations varies across the cross sections the mean velocities and accelerations for the box around each cross section is calculated by the integrals in equation (2.1.4) and (2.1.5).

( ) ( ) ( )

max max

min min

0

max min max min

1, , , ,

Y X

i i

T Y X

v v X Y Z dXdYdZ i x y zT Y Y X X −

= =⋅ − ⋅ − ∫ ∫ ∫ (2.1.4)

( ) ( ) ( )

max max

min min

0

max min max min

1, , , ,

Y X

i i

T Y X

a v X Y Z dXdYdZ i x y zT Y Y X X −

= =⋅ − ⋅ − ∫ ∫ ∫ (2.1.5)

Xmin, Xmax, Ymin and Ymax are the maximum and minimum coordinates of the cross sections. vi and ai in equation (2.1.4) and (2.1.5) are the velocities and accelerations used in the simulations.

28/45

APPENDIX 3, INTEGRATION OF KERNEL FUNCTIONS When in integrating the memory and diffraction forces it is important to consider the step size if the frequency in kernel functions. To avoid aliasing the step size in (3.1.1) has to be sufficiently small.

( ) ( ) ( )( ) ( )0

2cosij ij ijK t B B t dω ω ω

π

∞

= − ∞ ⋅ ⋅ ⋅∫ (3.1.1)

Nyquists theorem gives that the frequency step of a discretesized signal should be half the signal frequency. In this case the time t in equation (3.1.1) is the dependent variable and ω the independent variable that should have a sufficiently short step. The higher the value of t in (3.1.1) the lower the frequency step needed of ω, and the longer the computation time.

( ) ( )2

1

tmemj ij j

t

F K t dτ η τ τ= ⋅ − ⋅ ⋅∫ ɺ (3.1.2)

This t is dependant of the interval in the integral of the memory forces in equation (3.1.2) or (4.2.1) in the report. With the memory force in equation (3.1.2) t in equation (3.1.1) will become

2 1t t t= − . (3.1.3)

These relations give the minimum frequency step, ω∆ , according to equation (3.1.4)

2 1

/s rad st t

πω∆ ≥−

(3.1.4)

A sufficiently long integration interval in equation (3.1.2) may be found doing a simulation with t1=0 and then calculate the memory force of the last time step for different integration intervals. An example of the memory forces normalized with the memory force calculated with the full integration interval are shown in Figure 13. For the case in Figure 13 an integration time of 40 s is sufficient to get a numerical error of less than 1 % on the memory forces.

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0 20 40 60 80 100 120 140 160 1800.95

0.96

0.97

0.98

0.99

1

1.01

1.02

1.03

1.04

1.05

Integration time [s]

Nor

mal

ized

mem

ory

forc

e [-

]

Box ship,U=0 knots, µ=180°, ω=0.35, a=0.1 m

F

m,3

Fm,4

Fm,5

Figure 13, memory forces normalized with the memory force integrated from t=0 as a function of the integration time. The error

is in this example smaller than 1 % after 40 s.

It is reasonable to assume that this time is dependant of the encounter period, Te. By plotting the needed time as a function of the encounter period a relation may be found.

Figure 14, the required integration time, in wave encounter periods, to get the error of the memory force lower than 1 % as a

function the normalized frequency.

30/45

As seen in Figure 14, the frequency dependency is linear until a point is reached where the needed integration time increases rapidly. At this frequency the diffraction forces are dominant and the memory forces small. With an integration time 5 times the encounter period the error of memory forces will be sufficiently small. Since the diffraction forces are calculated by the same integral but with the particles motions instead of ship motions, the same integration time is used for these forces. If the natural roll period is longer than the encounter period the integration time has to be 5 times the natural roll period.

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APPENDIX 4, HULL GEOMETRY The hull is defined by three matrices containing the x, y and z-coordinates of the offset points. The numbers at the same position in the matrices describes the same point. Each column in the matrix describes one section. The first column describes the aft most section; the next column describes the section in front of the aft most and so on. The ship geometry is given in the coordinate system o*x*y*z. To ease the use of Matlabs built in plotting functions each section has got the same number of points. When inserting points at the intersection of the water surface the number of points may differ between each section. An extra point with the same position as the last one is then inserted.

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APPENDIX 5, ROLL DAMPING In all degrees of freedom except roll the viscous forces may be ignored. But for roll damping the viscous damping is a large part of the total damping. No analytical solutions for the roll damping have been found. In 1978 Ikeda et. al. [13] proposed a semi empirical method to determine the roll damping with and without speed with just a roll decay test at zero speed. If the a roll decay test is possible to describe by equation (4.1.1) B44,i may be determined by looking at the energy dissipation.

( ) 3

44 44,1 44,2 44,3 44

3 20

0

2 0

xxI A B B B Cη η η η η η

η δ η α η η β η ω η

+ + + + + = ⇔

+ ⋅ + ⋅ + ⋅ + =

ɺɺ ɺ ɺ ɺ ɺ

ɺɺ ɺ ɺ ɺ ɺ

(4.1.1)

If jη in equation (4.1.2) are the maximum and minimum amplitudes of the decay test, as seen in Figure

15, η∆ may be plotted as a function of mη .

1

1

2j j

m

j j

η ηη

η η η

+

+

+=

∆ = − (4.1.2)

0 50 100 150 200 250-8

-6

-4

-2

0

2

4

6

8

10

t [s]

η 4 [°]

RäknadModellförsök

ηm

ηj

ηj+1

Figure 15, roll decay test, with ηj and ηm shown.

To these points a third degree polynomial on the form

2 3 2 3

0 1 2 3

4 3

3 8m m m m m ma a aπη πκη αη βω η η η η∆ = + + ≡ + + (4.1.3)

may be fitted, as seen in Figure 16.

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0 0.02 0.04 0.06 0.08 0.1 0.12 0.140

0.005

0.01

0.015

0.02

0.025

ηm [rad]

∆η [

rad]

Roll decay testFitted polynomial

Figure 16, ∆η as a function of ηm and the fitted polynomial.

With the coefficients according to equation (4.1.4) and a comparison between (4.1.1) and (4.1.3) the roll

damping coefficients 44,iB may be calculated according to equation (4.1.5). In equation (4.1.4) δ is the

linear damping coefficient, 0ω the undamped natural frequency and ai the coefficients of the fitted

polynomial.

10

2

3 0

2 440

44

4

33

8

xx

a

a

a

C

I A

δπω

α

π βω

ω

=

=

=

=+

(4.1.4)

( ) ( )

( ) ( )

( ) ( )

44,1 44 44 44 1

44,2 44 44 2

1.5

4444,3 44 3

44

22

3

4

8

3

R Dxx xx

R Dxx xx

xxR Dxx

B I A C I A a

B I A I A a

I AB I A a

C

δπ

α

βπ

−

−

−

= ⋅ + = ⋅ + ⋅

= ⋅ + = ⋅ +

+= ⋅ + = ⋅

⋅ (4.1.5)

34/45

Care should be taken to only use decay test for angles where the hydrostatic restoring moment is linear. In

the calculations the C and xxI measured at the same time as decay test, and 44A from the linear strip

theory is used. The index R-D in equation (4.1.5) denotes that it is the experimentally determined damping coefficients from a roll decay test including all damping forces acting on the hull in the test. Since wave radiation damping is included in the memory forces that damping has to be subtracted from B44,1. Since the roll decay test consists of just one frequency, the convolution in equation (4.2.1) in the report, will correspond to a delta pulse at the ships natural roll frequency multiplied by B. Therefore B44(ωeg) at zero speed should be subtracted from B44,1. The damping at zero speed then becomes

( )( ) 344,1 44 0 4 44,2 4 4 44,3 4, 0R D

dampF B B U B Bω η η η η−= − = + +ɺ ɺ ɺ . (4.1.6)

Speed dependant roll damping

It is a well know fact that roll damping is speed dependant [6]. Usually there is just one roll decay test for zero speed available. Ikeda divided the damping into the different parts, wave potential, eddy, friction, lift and bilge keel damping according to equation (4.1.7). By looking at one part at a time and a roll decay test at zero speed, a semi empirical modification may be used to determine the roll damping at different speeds.

( ) ( ) ( ) ( )4 4 4 4 4 4,dampW E F L BKF B U B U B B U B Uω η η η η η η= ⋅ + ⋅ + ⋅ + ⋅ + ⋅ɺ ɺ ɺ ɺ ɺ ɺ (4.1.7)

The friction is generally orders of magnitude smaller than the rest of the components and its speed dependency negligible [6]. The wave potential damping, B44,W, may be divided into two parts, the wave potential damping from B44 and a 3d component, B44,S, determined semi-theoretically [6] as a fraction of the zero speed potential damping, B44(ωeg,U=0). This is described by equation (4.1.8) to (4.1.12).

( ) ( )44, 44 0, ,W SB B U B Uω ω= + (4.1.8)

( ) ( ) ( )( ) ( ) ( )( )244 150 0.25

44, 2 2 1 2

, 01 1 tanh 20 0.3 2 1

2eg

S

B UB A A A A e

ω − Ω−== ⋅ + + − ⋅ ⋅ Ω − + − − (4.1.9)

0

U

gωΩ = (4.1.10)

20D

T

gξ ω= (4.1.11)

21.21

21.02

1.0

0.5

D

D

D

D

A e

A e

ξ

ξ

ξξ

−−

−−

= +

= + (4.1.12)

The roll damping due to lift, B44,L may be described by

( )2

344, 2

0.151.0 1.4 0.7

2 0.5 0.15L N

VCG TVCG TB LT Uk

T Tρ

−−= ⋅ + +

(4.1.13)

( ) ( )2 3

24.1 0.045

0 if 0.91

106 0.91 700 0.91 if 0.91 1.00

0.35 if 1.00

N

M

M M M

M

T Tk

L L

C

C C C

C

π χ

χ

χχ

= + ⋅ −

= <

= ⋅ − − ⋅ − ≤ ≤= >

, (4.1.14)

35/45

where CM is amidships section coefficient. If no bilge keel is present the rest of the linear term B44,1 will, according to [6], be the frictional coefficient and since no term but the eddy component is non linear the non linear terms will be the eddy component. Ikeda derived the semi-empirical expression in equation (4.1.15) for the eddy speed dependency.

2 2

0 2 2 2

0.04

0.04E E

LB B

U L

ωω

⋅= ⋅+ ⋅

(4.1.15)

This is used to scale both B44,2 and B44,3. The total damping except for the wave potential damping in the memory forces will then become

( ) ( ) ( )ω ω

ωω

ωω

−

−

−

= − = + +

⋅= ⋅+ ⋅

⋅= ⋅+ ⋅

44,1 44 ,1 44 44 , 44 ,

2 2

44 ,2 44 ,2 2 2 2

2 2

44,3 44 ,3 2 2 2

, 0 ,

0.04

0.04

0.04

0.04

R D

eg S eg L

R D

R D

B B B U B U B U

LB B

U L

LB B K

U L

. (4.1.16)

The expressions in equation (4.1.16) are used throughout the simulation code. No model for bilge keel damping has been implemented.

36/45

APPENDIX 6, ABM AND RUNGE KUTTA 5 According to [8] the Adams-Basforth method of order p+1 is described by equation (5.1.2) and (5.1.3), which solves an equation on the form in equation (5.1.1).

( ),dy

f t ydt

= (5.1.1)

( )10

,p

n n k n k n kk

y y f t yλ+ − −=

= + ⋅∑ (5.1.2)

0

1

2

3

55

2459

2437

249

24

h

h

h

h

λ

λ

λ

λ

=

= −

=

= −

(5.1.3)

This step is then corrected with the Adams-Moulton corrector described by equation (5.1.4) and (5.1.5).

( )1

11

,p

n n n k n kk

y y f t yγ−

+ − −=−

= + ⋅∑ (5.1.4)

1

0

1

2

9

2419

245

241

24

h

h

h

h

γ

γ

γ

γ

− =

=

= −

=

(5.1.5)

To calculate the first three steps Runge-Kutta 5 is used. It is described by equation (5.1.6).

( )

5

11

1

2 1

3 1 2

4 1 2 3

5

25 1408 2197 10

216 2565 4104 5

1 1,

4 4

3 3 9,

8 35 35

12 1932 7200 7296,

13 2197 2197 2197

4,

n n n nn

n n

n n

n n

n n

n n

y y h b k

b

k f t y

k f t h y k h

k f t h y k k h

k f t h y k k k h

k f t h y

+=

= + ⋅

= −

=

= + + ⋅

= + + + ⋅

= + + − + ⋅

= + +

∑

1 2 3 4

39 3680 8458

216 513 4104k k k k h

− + − ⋅

(5.1.6)

37/45

step length, h,

1

10PPL

hg

≤ (5.1.7)

is, as suggested by [7], used throughout the simulations. To avoid aliasing and describe the full motion the Nyqvist theorem states that the step length should be less than half the period of the motion.

38/45

APPENDIX 7, COMPARISON 0 KNOTS

0 0.5 1 1.5 2 2.5 3 3.50

1


η 3 [no

n-di

m]

ω [rad/s]

0 0.5 1 1.5 2 2.5 3 3.50

0.05

0.1BoxShip ; T=2m ; KG=2m ; U=0kn ; µ=90deg

η 5 [no

n-di

m]

ω [rad/s]

0 0.5 1 1.5 2 2.5 3 3.50

2


η 4 [no

n-di

m]

ω [rad/s]


0 0.5 1 1.5 2 2.5 3 3.50

0.5


η 3 [no

n-di

m]

ω [rad/s]

0 0.5 1 1.5 2 2.5 3 3.50

1


η 5 [no

n-di

m]

ω [rad/s]

0 0.5 1 1.5 2 2.5 3 3.5-1

0


η 4 [no

n-di

m]

ω [rad/s]


39/45

0 0.5 1 1.5 2 2.5 3 3.50

1


η 3 [no

n-di

m]

ω [rad/s]

0 0.5 1 1.5 2 2.5 3 3.50

0.5


η 5 [no

n-di

m]

ω [rad/s]

0 0.5 1 1.5 2 2.5 3 3.50

1


η 4 [no

n-di

m]

ω [rad/s]


40/45

APPENDIX 8, COMPARISON 10 KNOTS

0 0.5 1 1.5 2 2.5 3 3.50

1


η 3 [no

n-di

m]

ω [rad/s]

0 0.5 1 1.5 2 2.5 3 3.50

1


η 5 [no

n-di

m]

ω [rad/s]

0 0.5 1 1.5 2 2.5 3 3.5-1

0


η 4 [no

n-di

m]

ω [rad/s]


0 0.5 1 1.5 2 2.5 3 3.50

1


η 3 [no

n-di

m]

ω [rad/s]

0 0.5 1 1.5 2 2.5 3 3.50

2


η 5 [no

n-di

m]

ω [rad/s]

0 0.5 1 1.5 2 2.5 3 3.5-1

0


η 4 [no

n-di

m]

ω [rad/s]


41/45

0 0.5 1 1.5 2 2.5 3 3.50

1


η 3 [no

n-di

m]

ω [rad/s]

0 0.5 1 1.5 2 2.5 3 3.50

1


η 5 [no

n-di

m]

ω [rad/s]

0 0.5 1 1.5 2 2.5 3 3.50

1


η 4 [no

n-di

m]

ω [rad/s]


0 0.5 1 1.5 2 2.5 3 3.50

1


η 3 [no

n-di

m]

ω [rad/s]

0 0.5 1 1.5 2 2.5 3 3.50

1


η 5 [no

n-di

m]

ω [rad/s]

0 0.5 1 1.5 2 2.5 3 3.50

5


η 4 [no

n-di

m]

ω [rad/s]


42/45

0 0.5 1 1.5 2 2.5 3 3.50

1


η 3 [no

n-di

m]

ω [rad/s]

0 0.5 1 1.5 2 2.5 3 3.50

1


η 5 [no

n-di

m]

ω [rad/s]

0 0.5 1 1.5 2 2.5 3 3.50


η 4 [no

n-di

m]

ω [rad/s]


43/45

APPENDIX 9, COMPARISON WITH TANK TESTS

0 0.5 1 1.5 20

0.5

1

η3

60 61 62


0 0.5 1 1.5 20

0.1

0.2

Non

dim

ensi

onal

res

pons

e [-

]

η4

0 0.5 1 1.5 20

0.5

1

ω [rad/s]

η5

0 0.5 1 1.5 20

0.5

1

η3

35 36 39


0 0.5 1 1.5 20

0.5

1

Non

dim

ensi

onal

res

pons

e [-

]

η4

0 0.5 1 1.5 20

0.5

1

ω [rad/s]

η5

44/45

0 0.5 1 1.5 20

0.5

1

η3

40 42 44


0 0.5 1 1.5 20

2

4

Non

dim

ensi

onal

res

pons

e [-

]

η4

0 0.5 1 1.5 20

0.5

1

ω [rad/s]

η5

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.5

1

η3

49 52


0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

2

4

Non

dim

ensi

onal

res

pons

e [-

]

η4

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.5

ω [rad/s]

η5

45/45

APPENDIX 10 SAFEDOR SIMULATION RESULTS TEST GM [m] Heading [°] Fn H1 [m] T1 (Tp) [s] H2(m) T2(sec) H3(m) T3(sec) Description RANGE

Roll Ampl. [°]

Mean roll ampl. [°]

T01 1.38 - 0 - - - - - - Roll decay (in calm water) -

T02 1.38 180 0.08 3.6 10.63 - - - - Regular (1 harmonic) 10.7 10.68 T03 1.38 180 0.08 5.7 10.63 - - - - » 0.04 0.18 T04 1.38 180 0.12 3.6 10.63 - - - - » 15.5 15.50 T05 1.38 180 0.12 5.7 10.63 - - - - » 13.8 13.82 T06 1.38 180 0.12 2.4 10.63 2.4 9.66 2.4 11.55 Group (3 harmonics) 0.02 0.04 T07 1.38 180 0.12 4 10.63 1 9.66 1 11.55 » 14.7 16.22 T08 1.38 180 0.12 5 10.63 Irregular (JONSWAP spectrum. γ=3.3) 0.00 0.00 T09 1.38 160 0.12 3.6 10.63 Regular (1 harmonic) 14.1 14.09 T10 1.38 160 0.12 5.7 10.63 » 11.8 11.85 T11 1.38 160 0.12 4 10.63 1 9.66 1 11.55 Group (3 harmonics) 13.3 14.85 T12 1 - 0 - - - - - - Roll decay (in calm water) - - T13 1 0 0.08 3.6 8.0 - - - - Regular (1 harmonic) 5.34 5.35 T14 1 0 0.08 6 8.0 - - - - » 0.00 0.00 T15 1 0 0.04 3.6 8.0 - - - - » 0.00 0.00 T16 1 0 0.04 6 8.0 - - - - » 32.3 32.34 T17 1 0 0.04 2.4 8.0 2.4 7.11 2.4 8.89 Group (3 harmonics) 10.5 21.02 T18 1 0 0.08 2.4 8.0 2.4 7.11 2.4 8.89 » 0.0 0.01 T19 1 0 0.08 5 8.0 - - - - Irregular (JONSWAP spectrum. γ=3.3) 0.0 0.00 T20 1 180 0.08 5 12.12 - - - - Regular (1 harmonic) 11.5 11.51 T21 1 180 0.12 5 12.12 - - - - » 16.9 16.89 T22 1 180 0.08 4 12.12 1 10.77 1 13.47 Group (3 harmonics) 9.31 10.63

numerical simulation of parametric rolling in waves · system. coordinates in oxyz are written with...

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