Parametric rolling behaviour of azimuthing propulsion-driven ships

ARTICLE IN PRESS

Ocean Engineering 35 (2008) 1339– 1356

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Ocean Engineering

0029-80

doi:10.1

� Corr

E-m

journal homepage: www.elsevier.com/locate/oceaneng

Parametric rolling behaviour of azimuthing propulsion-driven ships

Osman Turan a,�, Zafer Ayaz b, Seref Aksu a, Jan Kanar c, Antoni Bednarek c

a Department of Naval Architecture and Marine Engineering, Universities of Glasgow and Strathclyde, Henry Dyer Building, 100 Montrose Street, Glasgow G4 0LZ, UKb SAIPEM UK Ltd., Saipem House, Station Road, Motspur Park, New Malden, Surrey KT3 6JJ (formerly of Department of Naval Architecture and Marine Engineering,

Universities of Glasgow and Strathclyde), UKc Centrum Techniki Okretowej S.A (Ship Design and Research Centre S.A.), Ul. Szczecinska 65, 80-392 Gdansk, Poland

a r t i c l e i n f o

Article history:

Received 5 March 2008

Accepted 1 June 2008Available online 10 June 2008

Keywords:

Extreme ship motions

Stability

Parametric rolling

Azimuthing podded drives

18/$ - see front matter & 2008 Elsevier Ltd. A

016/j.oceaneng.2008.06.001

esponding author. Tel.: +441415483211; fax:

ail address: [email protected] (O. Turan).

a b s t r a c t

The study investigates the experimental and numerical analysis of the occurrence of auto-parametric

rolling for large, high-speed pod-driven ships in waves. Considering unique design and performance

targets, the aim here is to exploit susceptibility to auto-parametric rolling behaviour and to identify

probable design and operational precautions. In order to achieve this aim, an existing non-linear time-

domain software to simulate capsizing and other critical manoeuvring behaviours of slow- to medium-

speed conventional and podded ships in waves is being enhanced for fast pod-driven vessels and then

compared against the dedicated model test conducted in long-crested regular and random waves for a

large, pod-driven containership model. This paper includes the presentation of current numerical

modifications for pod-driven ships and the verification analysis.

& 2008 Elsevier Ltd. All rights reserved.

1. Introduction

The introduction of the azimuthing pod drives into service onlarge commercial ships has brought many benefits in wide rangeof areas covering manoeuvring to hull-savings for extra passen-gers and cargo, noise and vibration, and consequent passengercomfort (Ayaz et al., 2005). The azimuthing pod-propulsionsystems have now been well proven in terms of their propulsionperformance in slow- to medium-speed range, whilst their low-speed and harbour-manoeuvring performance, especially forpassenger ships and ice-breakers are their biggest advantages(ITTC, 2005a). The challenge in modern transport now appears tofoster the application of this technology for very large and high-speed vessels, which are also growing in numbers, to meet thedemands of fast ship operators and the competitive marketconditions.

The investigation of these issues will require extensivenumerical and physical analysis in order to identify whetherlarge high-speed pod-driven ships could face similar dangers or ifthere are other important factors for the safe and effectiveoperation of these ships in terms of manoeuvring and control inwaves. As a result, the main focus of this study is the danger ofparametric resonance as a result of modifications in the design.

The stability characteristics of ships are greatly influenced bythe design approach in adopting their associated propulsion and

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+441415522879.

steering units. This has been a major concern since the introduc-tion of powerful and multi-functional azimuthing podded propul-sion and steering units within the last decade or so (FASTPOD,2005; Ayaz et al., 2005). While, the sizes and power output ofthese propulsion/steering units are getting larger along with thespeed and sizes of ships, the possible dangerous conditionsrelated to stability and safe operations of these vessels have notbeen properly addressed in spite of some concerns reported in theopen literature by some investigators (e.g. Van Terwisga et al.,2001; Toxopeus and Loeff, 2002) as well as by the last ITTC(2005a). Some of these concerns have been summarized withinformation from model tests and actual operational experiencesas follows:

�
Course keeping and directional stability in following andquartering seas resulting from extremely prammed aft hullshape. � Effect of large steering forces created by pod drives on roll
motion in heavy seas.
� Danger of parametric resonance as a result of roll period in
heavy seas.
� Excessive steering actions imposed by autopilot, causing wear
and tear of the bearings and steering engine in heavy seas.

These have not been thoroughly discussed even for relativelyslow speed current applications and therefore raise furtherquestions for the high-speed applications (ITTC, 2005a; VanTerwisga et al., 2001; Toxopeus and Loeff, 2002).

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Nomenclature

a amplitude of wave (m)aH interaction factor between hull and rudderaHpod interaction factor between hull and PODai, component wave amplitude (m)APOD effective pod area (m2)B beam (m)c phase velocity of wave (m/s)Cb block coefficientCð _jÞ damping force (N)df draught at fore (m)d mean draught (m)da draught at aftD depth (m)DP propeller diameter (m)fe wave frequency (Hz)FN rudder normal forces (N)Fn nominal Froude numberg gravitational acceleration (m/s2)GM metacentric height (m)H wave height (m)Hs significant wave height (m)Ixx roll moment of inertia (kg m2)Iyy pitch moment of inertia (kg m2)Izz yaw moment of inertia (kg m2)Jp advance coefficientJPOD POD advance coefficientk wave numberki component wave numberK0 roll external moment (N m)KG vertical position of centre of gravity from keel line (m)KP proportional gain (s)KR differential gainKT thrust coefficientLBP length between perpendiculars (m)LCG longitudinal position of centre of gravity from the

amidships (m)m ship mass (kg)M0 pitch external moment (N m)n propeller rate of rotation (rpm)N0 yaw external moment (N m)Nj normal vectorp pressure (N/m2)ps static wave pressure (N/m2)pd dynamic wave pressure (N/m2)P angular roll velocity (deg/s)Q angular pitch velocity (deg/s)R angular yaw velocity (deg/s)RT(u) total resistance force (N)S propeller slip ratiotp thrust deduction in forward motion and during a turn

at the propellertr thrust deduction in forward motion and during a turn

at the rudder position

tD time constant (s)Tj roll period (s)Tz modal period (s)U surge velocity (m/s)UPOD flow speed on pod unit (m/s)Urw effective wind speed (m/s)UR rudder inflow velocity (m/s), and angleV sway velocity (m/s)VR mean rudder inflow velocity (m/s)wp wake fraction in forward motion and during a turn at

the propellerwr wake fraction in forward motion and during a turn at

the rudder positionW heave velocity (m/s)X0 surge external force (Nt)xH longitudinal position of the point of action of the to

hull interaction force (m)xHpod longitudinal position of the point of action of the to

hull interaction force due to pod (m)xR longitudinal position of the rudder’s centre of pres-

sure (m)xpod longitudinal position of the pod’s centre of pressure

(m)Y0 sway external force (N)Z0 heave external force (N)zR vertical position of the rudder’s centre of pressure (m)zpod vertical coordinate of the pod’s centre of pressure (m)zy vertical coordinate of the centre of action of lateral

force (m)

Greek symbols

aPOD angle of attack of pod inflow (deg)w heading angle (deg)wc autopilot course from the wave direction (deg)d rudder/pod angle (deg)dR actual rudder angle (deg)f heel angle (deg)l wave length (m)y pitch angle (deg)r density (kg/m3)si random phase angle (deg)o wave frequency (rad/s)oe wave encounter frequency (rad/s)owi component circular frequency (rad/s)oG angular velocity (deg/s)x position of ship on the wave (m)c autopilot course (deg)cR desired heading angle (deg)z vertical position of ship (m)z(t) irregular wave elevation (m)D displacement (kg)FI potential associated with the incoming wave potentialFD potential of disturbed wave

O. Turan et al. / Ocean Engineering 35 (2008) 1339–13561340

Amongst a number of research studies in this area, two largeEuropean wide research projects have been undertaken to addressand produce solutions to aforementioned and many other designand operational problems of pod-driven ships (OPTIPOD, 2002;FASTPOD, 2005). As part of these research studies, an existing in-house 6-DOF non-linear numerical model has been enhancedwith inclusion of propulsion and steering actions of the pod drives

and used as an analysis tool to investigate some of the abovementioned stability problems of pod-driven ships (Ayaz et al.,2005, 2006a–c)

The effect of azimuthing pod drive has been investigated byInternational Towing Tank Conference (ITTC) through its propellerand manoeuvring committees and most recently a specialcommittee dedicated to these operation systems (ITTC, 2005a, b)

ARTICLE IN PRESS

With damping

Without damping

0.5 ω0/ωe

Θp

Fig. 1. Co-ordinate system and the relationship between axes systems.

O. Turan et al. / Ocean Engineering 35 (2008) 1339–1356 1341

in terms of propulsive efficiency and operational issues.The propulsion and steering systems, as mentioned, has directeffect on stability, yet the criteria for stability has been generallychosen for capsizing, therefore there are no clear instabilityidentifications under International Maritime Organization (IMO)intact ship stability rules for conventional or high-speed ships(IMO, 2000).

One of the aforementioned dynamic stability problems;parametric rolling, has recently gained wide attention due towell-publicized incidents and subsequent detailed experimentaland numerical investigations (France et al., 2003). However, thephenomenon itself has been well known and investigated bothanalytically and experimentally by numerous researchers for thelast 50 years (Grim, 1963; Paulling and Rosenberg, 1959; Chouet al., 1974; De Kat and Thomas, 1998). It has been initiallydescribed as a parametric change of metacentric height of avessel as a consequence of the travelling of the wave along theship in longitudinal waves in order of vessel length. It can bemodelled on the basis of parametrically induced swings of apendulum when the pivoting point is oscillating up and down(Tondl et al., 2000). The dynamic motion occurs due to nonlinearcoupling between vertical motions (heave-pitch) and rollmotion. This coupling becomes more significant in resonanceconditions for vertical motion in which energy is transferredto transversal motions induced by vertical accelerations. Insummary, this dynamic mode known as low-cycle resonance(parametric resonance) is caused by a Mathieu-type built-up oflarge roll motion. This large roll motion arises when a number offactors are combined, such as: the roll natural frequency ismultiple of half the encounter frequency, roll damping is lowand wave length is comparable to ship length. The resonancecondition, known as the region of principal resonance, isdescribed as

of ¼12oen ðn ¼ 1;2; . . .Þ (1)

where oe is the encounter frequency and oj is the natural rollfrequency. The region of principal resonance has been shown fromthe solution of linear Mathieu equation for unforced roll motionwhich can be written as

€fþ 22N

T

� �_fþo2

0 1�DGM

GM0ðcos oetÞ

� �f ¼ 0 (2)

where the term DGM/GM (YP) indicating fluctuating GM yields to

YP ¼DGM

GM¼

GMTROUGH � GMcrest

2GM0(3)

Here, GM0 represents the metacentric height in calm waterwhile N in (2) is non-dimensional roll damping coefficient. Therelationship in (1)–(3) has been illustrated by the well-knownInce–Strutt stability diagram of the solutions of Mathieu’sequation (Fig. 1).

It can be seen that propulsion and control systems do induceeffects on these parameters via altering design parameters (aft,bow hull form), loading conditions (metacentric height) andindeed creating external force which could contribute to theconditions in resonance described above.

There have been numerous recent researches during the lastdecade focusing on both theoretical and experimental analysis ofthe phenomena as well as its relation with ship design (Bulianet al., 2003; Bulian, 2005; Umeda et al., 2004; Belenky et al.,2006; Levadou and Veer, 2006). For instance, as a result ofthe aforementioned studies, recently design as well as operationalguidelines for the assessment of parametric roll in the designof container carriers has been issued by ABS (American Bureau ofShipping) (ABS, 2004) and many other classification societiesand national/international regulators are following suit. The

main purpose of the guidelines has been described as tosupplement the rules and the other design and analysis criteriathat the classification society issues for the classification ofcontainer carriers in relation to parametric roll resonancephenomenon. The ABS guidelines are based on a series ofnumerical and experimental studies undertaken by the societyin cooperation with other research and design organizations (ABS,2004).

Based on the above background, this paper presents theenhancement of an existing 6-DOF non-linear numerical model,which predicts the combined manoeuvring and seakeepingbehaviour of a ship, to include the effect of multiple number ofpod units in low- and high-speed operating conditions. Theenhanced numerical model has been validated using the extensivemanoeuvring and seakeeping model test data available for a high-speed container vessel developed in FASTPOD project (Ayaz et al.,2005, 2006a; FASTPOD, 2005). The main focus of this paper is toexploit susceptibility to auto-parametric rolling behaviour and toidentify probable design and operational precautions on thepossible threats to stability, in relation to the changes in design toaccommodate pod-structure. In order to achieve this aim, theenhanced non-linear time-domain software was comparedagainst the dedicated model test conducted in long-crestedregular and random waves for a large, pod-driven containershipmodel.

2. Mathematical model

In order to display the coupling between horizontal andvertical planes, the equations of motions are expressed in termsof the horizontal body axes system to handle large angles in pitchwhich may be realized during operation in steep waves. Thederivation of horizontal body axes system and its relation to otheraxes systems has been described in detail by Ayaz et al. (2006b)and Hamamoto and Kim (1993). Here, the directions of co-ordinate and the relationship of the horizontal body axes, which isfixed in the ship with the origin at G and defined by G-x0y0z0, toother axes system is illustrated in Fig. 2. The earth fixed system isdefined by O-xZz and general body axes, which is fixed in theship with the origin G being located at the centre of gravity of theship, is defined by G-xyz. By applying Newton’s second lawthe equation of motions are expressed in standard 6-DOF in termsof the linear (surge, sway, heave) motions and rotational(roll, pitch and yaw) motions, respectively, along with the external

ARTICLE IN PRESS

Fig. 2. Co-ordinate system and the relationship between axes systems (Hamamoto and Kim, 1993).


forces as below:

mð _U � VRÞ ¼ �

ZZS

pnX dSþ XH � FN sin d

þ ð1� tpÞrn2D4KT

mð _V þ URÞ ¼ �

ZZS

pnY dSþ rU

ZGX

FDnY dsþ YH

� ð1þ aHÞFN cos d

m _W ¼ �

ZZS

pnZ dSþ rU

ZGX

FDnZdsþ ZH (4)

ðIxx cos2 yþ Iyy sin2 yÞ _P ¼ �ZZ

S

pðr� nYZÞdSþ rU

�

ZGX

FDðr� nYZÞdsþ KH

þ ð1þ aHÞzRFN cos d

Iyy_Q ¼ �

ZZS

pðr� nZXÞdS

þ rU

ZGX

FDðr � nZXÞdsþMH

ðIxx sin2 yþ Izz cos2 yÞ _R ¼ �ZZ

S

pðr� nXYÞdSþ rU

�

ZGX

FDðr� nXYÞdsþ NH

� ð1þ aHÞxRFN cos d (5)

where U, V, W are surge, sway, heave linear velocities, Q, P, R are roll,pitch, yaw angular velocities in horizontal body axes system and Ixx,Iyy, Izz are roll, pitch, yaw moments of inertias, respectively. In (1)and (2), the first terms in the right-hand side of the equationsincludes pressure term p, which corresponds to the incident (orFroude–Krylov) component of the wave-excitation forces/momentsincluding hydrostatic forces. These are calculated on the hull byintegrating pressure p up to the instantaneous wave surfacetogether with the kinematic relations involving n is normal vectorand r�n is vector fixed with respect to centre of gravity. The thirdterms represented by subscript H indicate hull (manoeuvring)forces, moments, which are obtained using well-known MMG(acronym for Japanese Manoeuvring Group, Inoue et al., 1981)method. The fourth terms involving FN and KT denote conventionalrudder forces and propulsion forces in surge motion, for the latterwhere: FN is the rudder normal force; aH the rudder-to-hullinteraction coefficient, xH the longitudinal coordinate of the pointof action of the rudder to hull interaction force, xR, zR are thelongitudinal and vertical coordinates of the rudder’s centre ofpressure, respectively; KT the thrust coefficient; D the propellerdiameter, n the propeller rate of rotation; and d the steering orrudder angle. Although, the above equations apply to a vessel to bedriven by podded propulsors, here, the notations for the steering/propulsion used follows the similar notations for the conventionalrudder/propulsor for the sake of convenience. The more detaileddescriptions of equations of motions and other components of themathematical model are given in Ayaz et al. (2005, 2006a–c).

The variables used in the equations of motion can besummarized as follows:

X: State vector xA9R00

X : ðza; xG; x0; y0; z0;U;V ;W ; P;Q ;R;f;y;c; dÞT (6)

ARTICLE IN PRESS


where x0, y0, z0 are the kinematics, d denotes the rudder angle, xG

and za represent the horizontal and vertical components of waveamplitude, respectively.

Following the previous research and experimental set-up usedin this study, the standard proportional-differential (PID) autopi-lot is employed in order to keep the vessel on course:

dR þ tr_dR ¼ KRðc� cRÞ þ KP

_cþ K i

Z t

0ðcðtÞ �cRÞdt (7)

where dR is the actual rudder angle, cR is the desired headingangle, Ki is the integral parameter, KR is yaw gain constant, KP is ayaw rate gain constant (Kp40, Kd40, Ki40)and tr is the timeconstant in rudder/pod activation. Here integral term Ki isintroduced to eliminate the steady-state error caused due toexternal effects such as wave drift, current and wind.

2.1. Calculation of wave forces including hydrostatic forces on a ship

The generalized Froude–Krylov force vector is given byintegration of pressure in the undisturbed wave system up tothe instantaneous wetted surface:

ZZS

pn dS ¼

ZZS

pðHSÞn dSþ

ZZS

pðHDÞn dS (8)

Here, the calculation of hydrostatic forces is of great importanceto identify the dangerous conditions presented in Section 1. Thehydrostatic forces and moments can be obtained by integratingthe pressure, p over the entire wetted surface of the ship. Thehydrostatic pressure p including that of a sinusoidal wave zW atany time and position x in the earth fixed axes is given by

p ¼ rgðzG � x0yþ z0Þ � rgae�kd

cosðkðxG þ x0 cosðcÞ � y0 sinðcÞ � ctÞÞ (9)

with

zW ¼ �zG þ x0yþ a cosðkðxG þ x0 cosðcÞ � y0 sinðcÞ � ctÞÞ (10)

where a is the amplitude of wave, c is the phase velocity of waveand d is the draught of the ship. Using the above expressions inGauss theorem, the Froude–Krylov forces can be described withrespect to the horizontal body axes in the form of heave, roll andpitch motion which include restoring terms as follows:

Z0F:K ðzG; y;c;fÞ � �ZZZ

V

qpðF;KÞ

qz0dV

ffi � rg

ZL

AðxÞdx� rg

�

ZL

FðxÞAðxÞ cosðkðxG þ x cosðcÞ

� ctÞÞdx (11)

K 0F:K ðzG; y;c;fÞ � �ZZZ

V

y0qpðF;KÞ

qz0� z0

qpðF;KÞ

qy0

� �dV

ffi � rg

ZL

y0BAðxÞdx� rg sinðcÞ

�

ZL

FðxÞAðxÞz0B sinðkðxG þ x cosðcÞ

� ctÞÞdx (12)

M0F:K ðzG;y;c;fÞ � �ZZZ

V

z0qpðF;KÞ

qx0� x0

qpðF;KÞ

qz0

� �dV

ffi rg

ZL

xAðxÞdxþ rg

�

ZL

FðxÞAðxÞx cosðkðxG þ x cosðcÞ

� ctÞÞdx (13)

where

FðxÞ ¼ aksinðkðBðxÞ=2Þ sinðcÞÞ

kðBðxÞ=2Þ sinðcÞe�kdAðxÞ (14)

Here Z0, K0 and M0 are heave, roll and pitch Froude–Kyrlov forcesincluding hydrostatic terms in horizontal body axes, respectively(F.K denotes Froude–Kyrlov), (y0b, z0b) the centre of buoyancy of theimmersed section at each instant. As shown in (6), these forces arecalculated as parameter of the Euler angles and vertical position ofcentre of gravity of ship on wave (zG) at each instant forinstantaneous wave surface. Therefore, Froude–Krylov forcescalculations are carried out using 2-D Gauss theorem followedthe method used in Hamamoto and Kim (1992).

The calculation of diffraction can be carried out similar to theradiation forces, however, the unique nature of the manoeuvringin waves due to low encounter frequency, has led to theadaptation of modified low encounter frequency slender-bodytheory proposed by Ohkusu (1986), in the development of theoriginal model. In this theory, emphasis is placed on taking intoaccount the effect of waves resulting from the disturbance of theincident waves by the ship, FD. It was stated in Ohkusu’s (1986)study that when the ship is advancing the waves with very lowencounter frequency; the incident waves appear as if they mightnot move relative to the ship while she advances some distance.Then that will approximate disturbance of the incident waves bythe ship when she keeps running quite a long time withoutchanging the position relative to the incident waves. FD might notbe called the diffraction waves in such situation. It is called thedisturbance to imply the disturbance of the incident waves andcan be written as

FD ¼sðxÞp logðk0ÞRþ FðxÞ

�X1n¼1

P2nðxÞ cosð2nÞy

R2nþ P2n�1ðxÞ sinð2n� 1Þ

yR2n�1

� �(15)

where R ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiy2 þ z2

py ¼ arctan(y/z) and k0 ¼ g/U2, s and Pn are to

be determined such that body condition is satisfied. According toOhkusu (1986), FD appears as if it might have been caused by linesingularities on the centre line of the ship, which in this case istaken to be sources and dipoles. F(x) then can be evaluated as

FðxÞ ¼ �1

2p

Zdx

dsdx

logð2k0Þjx� xjsgnðx� xÞh

�p2fH0ðk0ðx� xÞÞ � 2sgnðx� xÞY0ðk0jx� xjÞg

i(16)

where H0 is the Struve function and Y0 the zeroth-order Besselfunction of second kind FD in the near field has the form

FDðy; z : xÞ ¼ FS2Dðy; z : xÞ þ FðxÞ þFA

2Dðy; z : xÞ (17)

where FS2D and FA

2D are 2-D solutions symmetrical and anti-symmetrical with respect to the z-axis. By using conformalmapping techniques and dynamical pressure on the hull,general wave force equations are obtained as given by Ohkusu(1986). For the purpose of this study, the disturbance equationof Ohkusu’s (1986) theory which is used in numerical method,

ARTICLE IN PRESS


can be written as

FDIF ¼ rU

ZGX

FDNj ds (18)

Here j ¼ 1, 2, 3 denotes sway, heave and roll, respectively, and Nj isthe normal vector. For the pitch and yaw, values of the heave andsway at each cross section are multiplied by the distance betweenthe cross section and centre of gravity of the ship.

2.2. Implementation of frequency-dependent terms

The impulse response function for a unit can be written asfollows:

OðtÞ ¼Z t

�1

f ðtÞKðt � tÞdt ¼Z 1

0f ðt � tÞKðtÞdt (19)

where f(t) is the impulse function at time tot. Fouriertransformation of unit impulse response function.

Following the work by Cummins (1962), the real part of (11) orthe retardation (Kernel) function is written as the frequency-domain damping function:

KijðtÞ ¼2

p

Z 10

BijðoÞ cos ot do; i; j ¼ 1;2;3;4;5;6 (20)

where Bij is the damping, the notions i and j denote degrees-of-freedom and couplings. Finally, the radiation force in the timedomain is written as

Fij ¼ �aijð1Þ_Vj �

Z 10

KijðtÞVjðt � tÞdt; i; j ¼ 1;2;3;4;5;6 (21)

In a more common way, from general seakeeping models,equations of motions in Eqs. (4) and (5) can be written as (Ayazet al., 2006a–c)

ðMþ AÞ €XðtÞ þ BðXÞ _XðtÞ þ CðXÞXðtÞ

þ

Z 10

KijðtÞVjðt � tÞdt ¼ FðzW;XðtÞ; _XðtÞ; €XðtÞÞ (22)

where M is the inertia matrix, A the added inertia matrix, B thedamping coefficient matrix, C the restoring coefficient matrix, Fthe external force vector and zW the wave amplitude. Here, thefourth term in the left-hand side of (21) represents ‘‘so-called’’memory effect term incorporating the frequency-dependentvessel motion-related terms (radiation forces/moments) into(21). The impulse response function (Kij) will be solved fromadded mass and damping data and the convolution integral givenin (19) then evaluated for each term in the equations of motion ateach time step during the simulation. In this study, well-known 2-D strip theory is used for the calculation of radiation forces.

2.3. Numerical model for random waves

In the numerical calculation of motions in random seas, themodel presented by Ayaz et al. (2006b) was used. Here, therandom wave elevation in the earth fixed system can be written asa series of harmonic components and it is given as

zðtÞ ¼XN

i

ai cosðowit � kixþ siÞ (23)

where z(t) is the irregular wave elevation, ai, owi and ki are thecomponent wave amplitudes, circular frequencies and wavenumbers, respectively, and si are normally uniformly distributedrandom phase angles. N is the total number of component waves,taken as large as possible. If an infinite number of components are

utilized the wave will not to repeat itself:

zðtÞ ¼Z 1

0cosðowit � kiðoÞxþ siðoÞ

ffiffiffiffiffiffiffiffiffiffiffiffiffi2SðoÞ

p(24)

S(o) is the wave energy spectrum taken as JONSWAP in thisanalysis. As in the case of regular motions, equations of motionsshould be evaluated in ship-orientated system and the descriptionof this system in terms of the wave pressure with respect tohorizontal body-axis system yields

p ¼ rgðzG � x0yþ z0Þ

� rgXN

i

aie�kiz

0

cosðowit � kiðxG þ x0 cos c

� y0 sin cÞ þ siÞ (25)

The total incident wave excitation and diffraction forces due tothe action of confused seas can be written according to horizontalbody axes system as

Fwave_excðtÞ ¼XN

i

aiðFAFKþDIFÞi� cosðowit � kiðxG þ x0 cos c

� y0 sin cÞ þ siÞ þ ðFdFKþDIFÞiÞ (26)

where FAFKþDIF is the amplitude of the force in random seas and

FdFKþDIF is the phase lag of the force. Similar procedure is applied in

the calculation of Froude–Krylov forces at each instant. Theamplitude, ai, and phase angles, si, of each harmonic componentare obtained from a given wave energy spectrum and throughdigital random number generators.

2.4. Estimation of roll damping

In the prediction of ship motions in seaway, the accuraterepresentation of roll-damping characteristics becomes impor-tant. The non-linear damping motion could be described throughlinearized coefficients obtained from roll decay tests as repre-sented in (2). However, the terms will be constantly changedbased on the loading conditions and subsequent stabilitycharacteristics; such as non-linearity due to changes in geometrywith the free-surface effects. Therefore, in the numerical model,Ikeda’s (Himeno, 1981) pseudo-linearized terms which areobtained based on hull characteristics are used to calculate rolldamping which could be expressed as follows:

K _j ¼ ðBO þ BF þ BE þ BL þ BBKÞ � ð1� e�10Fn Þ (27)

where the damping coefficient B is the superposition of potential,friction, eddy, lift and bilge keel damping terms, denoted bysubscripts O, F, E, L, BK, respectively. Here the mean roll-angle isobtained from the slope of the roll curve in the numerical model.The second term in (27) represents a correction for forward speed.

2.5. Inclusion of pod effects in the mathematical model

An azimuthing podded drive is a highly attractive propulsionunit which combines the propulsion and steering actions of a ship,with a capability of 3601 azimuthing, using an electric motorfitted inside the gondola part of the pod unit as shown in Fig. 3.Electric motors are located outboard and power provided througheither diesel or turbo generators inside ship’s hull. In order toillustrate the hydrodynamic characteristics of pod-driven ships,the afterbody arrangements of ships driven by a conventionaltwin-screw propeller system and a podded propulsor system areshown in Fig. 4. Moreover, different aftbody applications for podor hybrid propulsion alternatives on large, high-speed container-ship have been illustrated in Fig. 5.

ARTICLE IN PRESS


When a pod unit is slewed, unlike a conventional rudder whichtakes advantage of the accelerated propeller flow, the propellerslipstream is parallel to the pod for the most part and hence thepod remains at a zero angle of attack. As a result, although the podpropulsor has the superior advantage of vectoring the propellerthrust in any direction, the pod body (housing) without itspropeller can only produce a lift proportional to the square of theship forward speed, except the ‘‘straight ahead’’ condition.

Within the same context although the pod housing should beconsidered as a single lifting surface subjected to the incomingflow, its two distinct components, which are the strut andgondola, suggest that the hydrodynamic characteristics of theseparts can be developed separately and then combined by takinginto account the interaction between them as precise as possible.Bearing in mind the fact that the interaction between the poddedpropulsors and the hull is relatively weak, there is a need to takeinto account this effect properly by taking into account the

Strut

Pod main body (gondola)- motor

Fig. 3. Main body of an azimuthing pod drive.

4 pod units 2pods+2skegs in

0 1

0 1 2

Fig. 5. Different pod-propulsion and suitable

Fig. 4. Conventional and azimuthing propul

interaction amongst the pod housing, propeller and hull throughproperly selected interaction coefficients similar to the one forconventional rudders expressed in Eqs. (4) and (5).

Moreover, multitude number of the pods also requires takingthis interaction effect between the pod units properly intoaccount. In the following the motion equations given in (4) and(5) are modified to include the effect of the pod drives based uponthe above background and concentrating on the 4 pods applica-tion (i.e. 2 fixed pod units located at the forward of the 2 steerablepod units), as shown in Fig. 6. The following only states a briefsummary of these modifications, the reader is referred to Ayazet al. (2005) for the detailed derivations.

In Fig. 6, T indicates thrust vector of POD-induced forces, S

indicates the side forces created by the propeller and the pod-housing unit. These forces describe POD forces and momentswhich can be expressed as follows: These terms can be obtainedthrough lift and drag terms coefficients of the pod-unit.

CD;L ¼D; L

0:5rAPODU2POD

(28)

where APOD is the effective pod area and UPOD is the flow speedover the effective pod area. The pod drag and lift terms aredetermined based on the open water characteristics and can beexpressed as

L ¼ f ðbPOD; dPOD; JPODÞ ¼ f ðaPOD; JPODÞ

D ¼ f ðbPOD; dPOD; JPODÞ ¼ f ðaPOD; JPODÞ

aPOD ¼ bPOD � dPOD (29)

where the lift, L, and the drag, D, forces are determined withrespect to local drift angle of pod unit, bPOD, the pod’s deflectiondPOD, the angle of attack aPOD, and the advance coefficientJPOD. The pod angle of attack depends on whether a pod unit is a

CRP mode 2 pods+tandem propellers

2 0 1 2

aft body alternatives (FASTPOD, 2005).

sion arrangements at aft part of a ship.

ARTICLE IN PRESS

ypodxpod

δ

S

T

G

U

ψ

x’

y’

Fig. 6. Pod–hull interaction and thrust vector.


lee-ward or wind-ward one. The advance coefficient and sub-sequent thrust coefficient should be obtained from the total podand propeller thrust force by taking into account propeller bladedrag and pod-house unit resistance which can be expressed as

KTpod ¼ KTpodþprp þ DKblade þ DKpod-house (30)

By combining (28)–(30), lift and drag coefficients can be finallywritten in the polynomial forms as

CDðaÞ ¼ A1 cosðBaPOD � CÞ þ A2 cosð2ðBPOD � CÞÞ þ D

CLðaÞ ¼ KðJPODÞaPOD (31)

where constants A1, A2, C, D are obtained for different angle ofattack and drift angle based on regression analysis. From theexpressions given in (29)–(31) with a given local pod inflow angle(angle of attack), aPOD, lift and drag forces described in the flow-oriented coordinate system can be transformed into thrust, T, andside force, S, units described in a ship-fixed coordinate systems.The pod-unit side force, S, is written as

FN ¼ S ¼ 0:5rAPODU2PODf ðLÞ sinðaPODÞ (32)

where f(L) is the open water pod normal force coefficient which isa function of pod-strut aspect ratio. f(L) is given as follows:

f ðLÞ ¼6:13L

Lþ 2:25(33)

where L is the pod aspect ratio.Finally, the relationship between hull and pod unit is described

through the interaction coefficient, aHpod which is dependentupon advance coefficient JPOD, as seen from the previousformulations, and it is a significant parameter to describe sideforce and moment induced on hull by the pod units. The termcould be expressed as

aHpod ¼ aJPOD þ b; where JPOD ¼UPOD

nDP(34)

The terms a and b, here, were identified from a regression analysissimilarly to the ones in (10). In the absence of such informationthese terms can be taken based on open-water test results. Also,the inflow angle of POD unit, aPOD, is dependent upon the podlocation and motions (lee-ward, wind-ward). It could be givenbased on the experimental analysis (Ayaz et al., 2005):

aPOD ¼ ð0:55bPOD þ 3:4Þ � dPOD; wind-ward

aPOD ¼ ð0:04bPOD � 3:4Þ � dPOD; lee-ward (35)

With incorporation of the aforementioned terms and consideringvector representation of the pod-induced forces in the lateral andvertical planes, the conventional formulation in (4) and (5)becomes (36) and (37) as below. Here, subscript pod denotesforces and moments caused by pod drive. Here, the last terms inX0, Y0, K0, N0 (surge, sway, roll and yaw) are given in similar notationto the conventional MMG model in (1) and (2). In case of the fixed

pods, pod-induced forces will be modified for the local drift angleof pod, bPOD, instead of the pod deflection (or slewing) angle, dPOD:

mð _U � VRÞ ¼ �

ZZS

pnX dSþ XH � S sinðdÞ

þ ð1� tpÞT cosðdÞ

mð _V þ URÞ ¼ �

ZZS

pnY dSþ rU

�

ZGX

FDnY dsþ YH � ð1þ aHpodÞS cosðdÞ

þ X0pod sinðdÞ

m _W ¼ �

ZZS

pnZ dSþ rU

ZGX

FDnZ dsþ ZH (36)

ðIxx cos2 yþ Iyy sin2 yÞ _P ¼ �ZZ

S

pðr� nYZÞdSþ rU

�

ZGX

FDðr� nYZÞdsþ KH þ zRpodY 0pod

Iyy_Q ¼ �

ZZS

pðr� nZXÞdSþ rU

ZGX

FDðr� nZXÞdsþMH

ðIxx sin2 yþ Izz cos2 yÞ _R ¼ �ZZ

S

pðr� nXYÞdSþ rU

�

ZGX

FDðr� nXYÞds

þ NH � xpodY 0pod (37)

3. Model tests and validation of numerical model

In order to identify the occurrence of parametric rolling for apod-driven ship, dedicated model experiments have been carriedout for a high-speed large pod-driven containership and theenhanced numerical model has been validated using the results ofmodel tests. The experiments have been conducted, in accordancewith the specified conditions and methodologies presented inSection 1.1 in longitudinal following and head seas at model basinof CTO (Ship Design and Research Centre S.A.), Poland. CTO’s largetowing tank has 260 m�12 m�6 m size and is fitted out with atowing carriage of a maximum speed of 12 m/s. The wavegenerator can generate irregular waves corresponding to the seastate up to approx. 81 to a scale 1:25 and regular waves of amaximum height of 0.7 m and length of up to 7 m or of a lesserheight and length of up to 20 m. The towing carriage is equipped

ARTICLE IN PRESS

Fig. 7. Aft and fore hull view of large high-speed containership.

Table 1Principal particulars of the full-scale and the model scale containership

Parameter Containership Model (1/51.3)

LBP (m) 275 5.360

B (m) 30.0 0.585

D (m) 21.65 0.422

df (design) (m) 10.3 0.201

da (design) (m) 10.3 0.201

df (scantling) (m) 12.5 0.244

da (scantling) (m) 12.5 0.244

Cb 0.569 0.569

r (m3) 10,725 4.075 (design)

LCG �7.213 (aft) m �0.141 (aft) m

Table 2Loading conditions of FASTPOD Cargo Ship in design and scantling conditions (full

scale)

Parameter Design Scantling

KG (m) 13.678 14.996

GM (m) 1.926 0.275

Tj (s) 19.05 40.1

kxx/B 0.38 0.37


with a multi-purpose computerized logging stand. The models aretested in different speeds, wave steepnesses, wave height to shiplength ratios (frequency). Lines and particulars of containershipand its model are given in Fig. 7 and Table 1.

The containership is, in its original design version, propelledwith four pod units all of which are equipped with 6.5 mpropellers. Each pod absorbs approximately 36 MW power withthe desired service speed approximately 35 knots. The model usedin this study is another version which consists of two azimuthingpod units both equipped with 6.5 m propellers and two tandempropellers (6.4 m diameter each) positioned between them (seeFig. 6). Each pod here also absorbs approximately 36 MW power atthe desired service speed of approximately 35 knots.

As can be seen from Table 1, the vessel has been tested for twodifferent loading conditions; design and scantling. Main hydro-static parameters for these two conditions have been presented inTable 2 in full-scale.

In order to analyse the vessel’s susceptibility to the parametricrolling behaviour, a number of runs with different parameters;wave height, speed and heading (from bow and aft) have beenconducted. Initially, as an indicator for possible parametric rollingbehaviour as explained in Section 1, the GZ curves in calm waterand different wave positions are plotted Figs. 8–10. The generaltrend is the typical behaviour of a vessel demonstrating thechange in vessel’s GZ between wave trough, crest and when vesselis at still water. The wave frequency and wave height for motionanalyses are selected in accordance with the GZ curves.

Before the motion analysis, as another important indicator, rolldamping of the pod-driven vessel in the shape of the roll decaycurves of design and scantling draught of the containership inzero speed and 21 knots at full scale are given in Figs. 11 and 12.

There is good agreement with the model test and the numericalmodel. Also, roll decay curve with respect to ship speed for rollangle of 101 is shown in Fig. 13.

The model tests have been carried out for different wave heightand speed in the longitudinal direction. The comparison ofnumerical model and model test have been categorized into twosea condition; steep regular waves and random waves (JONSWAPspectrum) and two environmental and operational parameter;wave height and speed for design and scantling draught condi-tions.

In the first part, the model was tested for different waveheights for f ¼ 0.525 Hz, c ¼ 1801, and Vm ¼ 1.22 m/s (Vs ¼ 17knots) in steep regular waves in Figs. 13–16. As can be seen, forthis resonance condition, parametric build-up occurs. There islinear relation between wave height and time to reach resonancebehaviour (maximum roll amplitude). The maximum roll ampli-tude also increases with wave amplitude and stabilizes where therestoring limit is achieved. This relation is also illustrated for rollamplitude as a function of wave height diagram in Vm ¼ 1.22 m/s(Vs ¼ 17 knots) and f ¼ 0.525 Hz. It should be noted the model hasbeen controlled in yaw motion to prevent capsizing during thetests. The agreement between model test and numerical model isgood, especially, quantitatively (Figs. 17 and 18).

In the second part, the model was tested for different speedsfor c ¼ 1801, relatively similar wave heights in steep regularwaves in Figs. 19–22. While, parametric roll occurs at theseconditions, as expected, roll amplitude reduces with increasingspeed (damping). The agreement between model test andnumerical model is also very good. However, due to the modelcontrolling method explained in the above diagram and also rpmcontrol difference between model tests and numerical modelresulted in different behaviour to reach resonance condition.During the model tests, model rpm has been accelerated to reachrequired speed and then it has been towed in the tank with yawcontrol. However, the numerical model has been started with thetarget rpm and model was free in 6-DOF. Therefore, the numericalmodel reaches the maximum roll at resonance condition morequickly while the model test displays slightly slower build-up. Therelation of roll amplitude and speed is also shown in Fig. 23.

Also, two runs are presented for different headings inFigs. 24 and 25 for similar operational condition. The parametricroll behaviour and resulting maximum amplitude is significantlydifferent for two cases. In head seas, the build-up of roll motion isfaster than following seas. This indicates that the transfer ofenergy between pitch and roll displacement is a more importantfactor here than in following seas where aft hull of the model dueto the positioning of azimuthing pods, dominates this behaviour.More importantly, however, is the amplitude of roll motion, ofwhich, in following seas the maximum amplitude is less thanhead seas amplitude despite higher wave amplitude. It has beennoted during tests that pitch motion becomes dominant infollowing seas cases while due to loss thrust, where pods aremore frequently out of water, the aforementioned energy transferpitch and roll does not extend more than 101 roll angle (Fig. 25).The similar behaviour also exists for the run in scantling draught

ARTICLE IN PRESS

2.5

2

1.5

1

0.5

00

-0.5

-1

Heel (deg)10 20 30 40 50 60 70 80 90

Still waterTroughCrest

2.5

2

1.5

1

0.5

0

-0.5

-1

0 10 20 30 40 50 60 70 80 90Heel (deg)


GZ

(m)

GZ

(m)

Fig. 8. GZ curve for containership in still water, wave trough and crest for l/Lpp ¼ 0.9 (left) and l/Lpp ¼ 1.1 (right) for a ¼ 4 m and c ¼ 01 in design loading condition.

3

2.5

2

1.5

1

0.5

0

-0.5

-1.5

-2

-1

0Heel (deg)

10 20 30 40 50 60 70 80 90


0.8

0.4

0

-1.2

-0.8

-0.4

2

-2.4

-1.6


GZ

(m)

GZ

(m)

0 10 20 30 40 50 60 70 80 90Heel (deg)

Fig. 9. GZ curve for containership in still water, wave trough and crest in design loading (left) and in scantling loading (right) conditions for l/Lpp ¼ 0.9, a ¼ 8 m and

c ¼ 1801.

1

0.5

0

-0.5

-2.5

-1.5

-1

-2

-3

0

Heel (deg)

10 20 30 40 50 60 70 80 90



GZ

(m)

GZ

(m) 0 10 20 30 40 50 60 70 80 90

Heel (deg)

1

0.5

0

-0.5

-1.5

-1

-2

Fig. 10. GZ curve for containership in still water, wave trough and crest for c ¼ 01 (left) and c ¼ 1801 (right) for l/Lpp ¼ 1.5, a ¼ 12 m and in scantling loading condition.


in following sea (Fig. 26). It indicates that the GM value does notaffect the occurrence of parametric rolling significantly for thishull model in the resonance conditions.

In the last part, the motion tests in random seas have beenanalysed in scantling draught condition. Here, the model was testedfor two different conditions; two different speeds while one case has

been re-run in order to validate the recreation of parametric rollingbehaviour in random seas condition. In the first run, the hull wastowed in Hs ¼ 0.182 m (Hs ¼ 9.33 m in full scale), c ¼ 01, andVm ¼ 1.15 m/s (Vs ¼ 16 knots) (Fig. 27). In the second part, the vesselwas towed twice in Hs ¼ 0.220 m (Hs ¼ 11.29 m in full scale), c ¼ 01,and Vm ¼ 1.15 m/s (Vs ¼ 16 knots) (Figs. 28 and 29).

ARTICLE IN PRESS

-30

-25

-20

-15

-10

-5

0

5

10

15

20

25

5 10 15 20 25 30

Time (sec)

Rol

l (de

g)

-25

-20

-15

-10

-5

0

5

10

15

20

0 20 40 60 80 100 120

Time (sec)

Rol

l (de

g)

Exp. Num.Exp. Num.

Fig. 11. Roll decay tests and comparison with numerical model in zero speed for design loading condition (left) and scantling condition (right).

-30

-25

-20

-15

-10

-5

0

5

10

15

20

5 7 9 11 13 15

Time (sec)

Rol

l (de

g)

-15

-10

-5

0

5

10

15

20

90 200 110 120 130 140 150 160

Time (sec)

Rol

l (de

g)

Exp. Num.Exp. Num.

Fig. 12. Roll decay tests and comparison with numerical model in Vm ¼ 1.5 m/s (Vs ¼ 20.88 knots) for design loading condition (left) and scantling condition (right).

1.2

1

0.8

0.6

0.4

0.2

00 5 10 15 20 25 30 35 40

Vs (Knots)

N

Fig. 13. Roll decay coefficient with respect to ship speed for roll angle 101.


As can be seen, the short periods of parametric build-up of rollmotion can be seen for all three cases, the randomness of motioneffect to build-up time and indeed maximum amplitude.

The numerical model is in especially quantitative agreementwith the model tests in terms of roll amplitude. However, the

numerical model displays more pessimistic behaviour with higherroll amplitude periods in the analysis. Inevitably, wave generationdifference and model testing technique as explained above wouldcontribute to these differences.

In Figs. 28 and 29, the vessel was towed in the same condition,and the randomness of the waves results in differences in build-up of roll motion over the period of test time. There are also slightdifferences in maximum roll amplitude. The numerical model alsocaptures these differences between two runs in identical condi-tions with a higher duration of large-amplitude roll motions.

Finally, parametric rolling range for two different loadingconditions over environmental and operational parameters ana-lysed have been illustrated in Figs. 30 and 31. It is seen thatparametric rolling is experienced in slower speeds (Fig. 31) forscantling draught compared to design loading condition (Fig. 30).For higher speeds, larger pitch motions coupled with loss of thrustdue to the pods coming out of water, effect roll motion comparedto design loading condition. The GM value and related fore and afthull wetted surface area would also be important in creating thesedifferences as explained in Section 1. This also indicates thathigher GM would not necessarily mean better safety, more likelyit means phasing out of one dangerous resonance period in suchcases similar to analysis carried out by Ayaz et al. (2006c) forconventional container ships.

ARTICLE IN PRESS

20

15

10

5

0

-5

-10

-15

-20

0 10 20 30 40 50 60 70 80 90 100

Rol

l (de

g)

Exp.

Rol

l (de

g)

Num.

Time (sec)Time (sec)

20

15

10

5

0

-5

-10

-15

-20

0 10 20 30 40 50 60 70 80 90 100

Fig. 14. Comparison between model (left) and numerical (right) data in a ¼ 0.064 m (3.2832 m), fe ¼ 0.525 Hz, c ¼ 1801, Vm ¼ 1.22 m/s (Vs ¼ 17 knots) for design loading

condition.

30

20

10

0

-10

-20

-30

30

20

10

0

-10

-20

-30

0 10 20 30 40 50 60 70

Rol

l (de

g)

Exp.R

oll (

deg)

Num.


0 10 20 30 40 50 60 70


condition.

30

20

10

0

-10

-20

-30

30

20

10

0

-10

-20

-30

0 10 20 30 40 50 60 70

Rol

l (de

g)

Exp.

Rol

l (de

g)

Num.


0 10 20 30 40 50 60 70


condition.


ARTICLE IN PRESS

30

20

10

0

-10

-20

-30

30

20

10

0

-10

-20

-30

Rol

l (de

g)

Exp.

Rol

l (de

g)

Num.


0 20 40 60 80 1201000 20 40 60 80 120100

Fig. 17. Comparison between model (left) and numerical (right) data in a ¼ 0.170 m (8.72 m), fe ¼ 0.525 Hz, c ¼ 1801, f ¼ 0.525 Hz, Vm ¼ 1.22 m/s (Vs ¼ 17 knots) for design

loading condition.

Rol

l (de

g)

HA (m)

30

25

20

15

10

5

00 1 2 3 4 5 6 7 8 9

Fig. 18. Roll amplitude as a function of regular wave amplitude (Vs ¼ 17 knots, f ¼ 0.073 Hz).

Exp.

-30

-20

-10

0

10

20

30

0

Time (sec)

Rol

l (de

g)

Exp.

Num.

-30

-20

-10

0

10

20

30

0

Time (sec)

Rol

l (de

g)

Num.

10 20 30 40 50 60 70 10 20 30 40 50 60 70

Fig. 19. Comparison between model (left) and numerical (right) data in a ¼ 0.142 m (7.28 m), fe ¼ 0.572 Hz, c ¼ 1801, Vm ¼ 0.80 m/s (Vs ¼ 11.14 knots) for design loading

condition.


4. Conclusions

An existing 6-DOF non-linear numerical model has beenenhanced for the simulation of manoeuvring and seakeepingcharacteristics of pod-driven high-speed large ships. The mathe-

matical model had already been modified to accommodate amedium to relatively high-speed range. Further modifica-tions have been accomplished by introducing thrust andlateral force components of both azimuthing and fixed poddrives.

ARTICLE IN PRESS

Rol

l (de

g)

Time (sec)

Exp.

Rol

l (de

g)

Time (sec)

Num.

30

20

10

0

-10

0

-20

-30

10 20 30 40 50 60 70 60 90 100 110 120

30

20

10

0

-10

-20

-30

0 10 20 30 40 50 60 70 80 90 100 110 120


condition.

30

20

Rol

l (de

g)

10

0

-10

-20

-30

0 10 20 30 40 50 60

Exp .

70 80 90 100

Rol

l (de

g)

Time (sec)

30

20

10

0

-10

-20

- 30

0 10 20 30 40 50 60 70 80 90 100

Num .

Time (sec)


condition.

25

20

15

10

5

0

-5

-15

-10

-20

-25

-30

Exp .

0 10 20 30 40 50 60

25

Rol

l (de

g)

20

15

10

5

0

-5

-10

-15

-20

-25

-30

0 10 20 30 40 50


Num .

60

Rol

l (de

g)


condition.


ARTICLE IN PRESS

8

7

6

5

4

3

2

1

00 5 10 15 20 25

Vs (knot)30 35 40

ΘΘA

Fig. 23. Scattering of dimensionless roll amplitude as a function of ship speed for the conditions in parametrical rolling tests.

Exp.

-30

-20

-10

0

10

20

30

0

Time (sec)

Rol

l (de

g)

Exp.

Num.

-30

-20

-10

0

10

20

30

0

Time (sec)

Rol

l (de

g)

Num.

10 20 30 40 50 60 10 20 30 40 50 60


condition.

Exp.

-15

-10

-5

0

5

10

15

0

Time (sec)

Rol

l (de

g)

Num.

-15

-10

-5

0

5

10

15

0

Time (sec)

Rol

l (de

g)

20 40 60 80 100 120 140 160 20 40 60 80 100 120 140 160


condition.


The validation of the modified numerical model has beencarried out using the dedicated auto-parametric rolling experi-ments. Both manoeuvring and stability analysis for this particulartype of vessel had shown that the vessel’s stability can bejeopardized in terms of excessive heeling in calm water orparametric rolling in extreme waves due to low GM and dampingcharacteristics, which is compromised to accommodate the heavy

pod structures in the aft, despite very good manoeuvring andseakeeping characteristics in normal operational conditions. Thenumerical analysis has successfully re-created the auto-para-metric rolling, which resulted in the transfer of energy fromexcessive pitching phenomenon in resonance conditions, ob-served in the experiments for regular and random waves in pod-tandem propeller configuration.

ARTICLE IN PRESS

Num.

-15

-10

-5

0

5

10

15

0

Time (sec)

(deg

)

Exp.

-15

-10

-5

0

5

10

15

0

Time (sec)

(deg

)

10 20 30 40 50 60 70 80 90 100 10 20 30 40 50 60 70 80 90 100


condition.

Exp.

-15

-10

-5

0

5

10

15

0

Time (sec)

(deg

)

-15

-10

-5

0

5

10

15(d

eg)

Num.

Time (sec)

10 20 30 40 50 60 70 80 90 100 0 10 20 30 40 50 60 70 80 90 100

Fig. 27. Comparison between model (left) and numerical (right) data in Hs ¼ 0.182, c ¼ 01, Vm ¼ 1.14 m/s (Vs ¼ 15.87 knots) for scantling loading condition.

Exp.

-40

-30

-20

-10

0

10

20

30

40

0

Time (sec)

(deg

)

Num.

-40

-30

-20

-10

0

10

20

30

40

Time (sec)

(deg

)

10 20 30 40 50 60 70 80 90 100110 0 10 20 30 40 50 60 70 80 90 100



Although, the capsize was not a priority, the containership hasexperienced the parametric rolling in both a long-crestedfollowing and head seas for the conditions outlined concerning

the auto-parametric rolling phenomenon in the literature. It wasnoted that linear relation exists between amplitude of roll motionand wave amplitude. Increasing speed, therefore damping, did

ARTICLE IN PRESS

Exp.

-40

-30

-20

-10

0

10

20

30

0

Time (sec)

(deg

)

Num.

-40

-30

-20

-10

0

10

20

30

0

Time (sec)

(deg

)

10 20 30 40 50 60 70 80 90 100 110 120 10 20 30 40 50 60 70 80 90 100 110 120


0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0Vs (knots)

f (H

z)

fe=2/TΦ, ψ=180°

fe=1/TΦ, ψ=180°

fe=2/TΦ, ψ=0°

fe=1/TΦ, ψ=0°

λ=0.8~2.0Lpp

5 10 15 20 25 30 35 40

Fig. 30. Parametric rolling range observed in model test for full scale in design loading condition. Thick dashed line (l ¼ 0.8–2.0 Lpp) depicts values of Vs (ship speed) and f

(frequency) ensuring appearance of parametric roll.

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0Vs (knots)

f (H

z)

fe=2/TΦ, ψ=180°

fe=1/TΦ, ψ=180°fe=2/TΦ, ψ=0°

fe=1/TΦ, ψ=0°λ=0.8~2.0Lpp

5 10 15 20 25 30 35 40

Fig. 31. Parametric rolling range observed in model test for full scale in scantling loading condition. Thick dashed line (l ¼ 0.8–2.0 Lpp) depicts values of Vs (ship speed) and

f (frequency) ensuring appearance of parametric roll.


ARTICLE IN PRESS


also improvement motion behaviour although not significantly.The effect of GM or loading conditions due to the accommodationof azimuthing pods have been more apparent between runs indesign draught and scantling draught conditions. The shape of afthull as well as bow flare plays a significant part as most dangerousconditions have been observed for lower speeds in scantlingdraught while longer range of parametric rolling is observed fordesign loading condition with respect to vessel speed. Withincreasing speed, pitching motion coupled with loss of thrustwhen pods come out of water has led to lesser amplitude rollmotion and prevented building up of large-amplitude roll angles.

The modified numerical model has provided satisfactoryagreement with model test results especially qualitatively. Thebuilding up roll motion and the differences between steep regularwaves and random sea conditions have been captured bynumerical model. The differences have been mainly as a resultof model testing technique which used manual control in yawmotion to prevent possible capsizing as well as control of rpm toachieve desired speed. The numerical model has used fewerrestraints in these conditions. However, overall, the numericalmodel displays more pessimistic or a larger range of parametricalrolling behaviour compared to the model tests.

It should be acknowledged of course that wider range ofanalysis and application of risk control options i.e. changing thecourse of the vessel and the speed and different loadingconditions or employing ride control systems could be appliedto identify true operational limits for the occurrence of parametricrolling for large pod-driven container ships. This has been carriedout separately for this vessel and the results will be publishedfollowing this study.

These types of vessels have been subject to parametric rollingbehaviour, which is same as the container ships with conventionalthrust and control systems. It could be argued that positioning ofazimuthing pod drives and subsequent effects on the loadingconditions and wetted surface area, i.e. bow flare and aft hullform, affect the transfer between vertical motions and rolldisplacement in longitudinal seas.

Finally, the numerical model, based on the current validationanalysis could be used for the analysis of the dynamic stability ofhigh-speed, large podded ships in waves, and could be a usefultool during the preliminary design process for such ships.

Acknowledgments

This study was carried out under European Commissionresearch project FASTPOD (GRD2-2001-50063). The authors wishto thank all 17 partners (Newcastle University-School of MarineScience and Technology, Chantiers de L’Atlantique, CETENA S.p.A-Centro per gli Studi di Tecnica Navale, CTO- Ship Design ResearchCentre, Deltamarin, Fincantieri Cantieri Navali Italiani S.p.A.,Stocznia Gdynia S.A., Hamburgische Schiffbau VersuchsanstaltGmbH, Lloyd’s Register, Principia Marin, SSPA Sweden AB, TheShip Stability Research Centre, Universities of Glasgow andStrathclyde, Stocznia Szczecinksa Nowa Spolka z o. o., Stena RederiAB, Technicatome SA, BAe Systems Marine Ltd., VTT TechnicalResearch Centre of Finland) of FASTPOD project for the promptcollaboration during the whole development of the project.

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Parametric rolling behaviour of azimuthing propulsion-driven ships

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