6 Roll Motion

8/3/2019 6 Roll Motion

1/22

1

Roll Motion of ShipsRoll Motion of Ships

ShipShip MotionsMotions

OscillatoryOscillatory shipship motionmotion ::

33 translatorytranslatory ((surgesurge,, swaysway andand heaveheave))

33 rotationalrotational ((rollroll,, pitchpitch andand yawyaw))

Metin Taylan, 2010 Metin Taylan, 2010

66 DoFDoF ShipShip MotionsMotions


Ship MotionsShip Motions



2/22

2

Ship MotionsShip Motions


Equation of MotionEquation of Motion

Equilibrium of all forces acting on the rigidEquilibrium of all forces acting on the rigidship in the 3 translatory directions, xship in the 3 translatory directions, x11, x, x22

33

Equilibrium of all moments acting on theEquilibrium of all moments acting on the

rigid ship in the 3 rotational directions, xrigid ship in the 3 rotational directions, x44,,

xx55 and xand x66


,,i

i

6,5,4i0Mi

i


ship reaction = external excitationship reaction = external excitation

,...,ijijjijjij

motionyoscillatorofonacceleratix

motionyoscillatorofvelocityx

motionshipyoscillatorx


tscoefficiencouplingthegiveijji

directionmotionj

momentorceorect on


inertiainertia forceforce/moment/momentdependingdepending onon thethexa

bodybody

dampingdamping forceforce/moment/momentdependingdepending ononthethe motionmotion velocityvelocity

restoringrestoring forceforce/moment/momentdependingdepending onon

xb

cx


ee par cu ar par cu ar osc a oryosc a ory mo onmo on xx

dd externalexternal excitationexcitation forceforce/moment/momentduedue totothethe seawayseaway


3/22

3


Coupled roll with heave and pitch :Coupled roll with heave and pitch :xcxbxa3jHeave 343343343

i=4i=4 moment equation for rollmoment equation for roll

jj direction (mode) of motiondirection (mode) of motion

In order solve the above cou led e uationIn order solve the above cou led e uation

dxcxbxa5jPitch 545545545

344444444


(= to estimate roll angle x), we must(= to estimate roll angle x), we mustadditionally solve the equation for heaveadditionally solve the equation for heave(i=3) and for pitch (i=5)(i=3) and for pitch (i=5)


If we rewrite the above equation;If we rewrite the above equation;dMxcxbxa c4344444444

MM4c4c :sum of all coupling moments for i=4 from:sum of all coupling moments for i=4 from

the motion directions j other than 4.the motion directions j other than 4.

Disregarding couplingDisregarding coupling MM4c4c=0=0 (uncoupled roll(uncoupled roll

motion):motion):


inertia termdamping term

restoring term

exciting term


Coefficients of the equation :Coefficients of the equation :

aa inertia coefficientinertia coefficient

bb damping coefficientdamping coefficientcc restoring coefficientrestoring coefficient

dd external roll excitation (wind, waves etc.)external roll excitation (wind, waves etc.)


Mass moment of inertiaMass moment of inertia

Inertia coefficientInertia coefficient aa is defined as :is defined as :

2'' TTT ''II'I

IITT Total mass moment of inertia of the rolling shipTotal mass moment of inertia of the rolling shipIITT mass moment of inertia of the shipmass moment of inertia of the ship

IITT added mass moment of inertiaadded mass moment of inertia

displacement massdisplacement mass

T


water densitywater density

iiTT roll radius of gyrationroll radius of gyration


4/22 4

Radius of GyrationRadius of Gyration

iiTT is the radius of a solid ring, which replacesis the radius of a solid ring, which replacesthe total mass of the ship as shown in thethe total mass of the ship as shown in the

..

This radius is enlarged by the inertia effect ofThis radius is enlarged by the inertia effect of

the surrounding water with respect to rollthe surrounding water with respect to roll

acceleration, the soacceleration, the so--called hydrodynamic masscalled hydrodynamic mass

moment or added mass momentmoment or added mass moment IITT


RadiusRadius ofof GyrationGyration


Linear Restoring MomentLinear Restoring Moment

For large heel, the static restoring moment is:For large heel, the static restoring moment is:

GZgM st


Linear Restoring MomentLinear Restoring Moment

For most ships at small heel up to about 5For most ships at small heel up to about 5

degrees the gradientdegrees the gradient GMGM is constantis constant

The parameterThe parametercc in the roll equation, is thein the roll equation, is the

)0(d

GM0

.deg5forGMGZ 0 00 GMsinGMGZ


0B0

0st GMFGMgGMgM

c


5/22 5

StiffStiff ShipShip vs. Tendervs. Tender shipship

When the initial stability is large (When the initial stability is large (GMGM00 isis

arge e s p s ca earge e s p s ca e ss .e s e s no.e s e s no

sensitive to small heeling moments.sensitive to small heeling moments.

For small initial metacentric height, theFor small initial metacentric height, the


. .. .

small heeling momentssmall heeling moments

StiffStiff ShipShip vs. Tendervs. Tender shipship


RollRoll MotionMotion


SpringSpring--MassMass DamperDamper SystemSystem andand RollRoll



6/22 6

Natural Roll PeriodNatural Roll PeriodCircular roll frequency :Circular roll frequency :

2 GMgGMgc

It is practical to refer to the natural roll period :It is practical to refer to the natural roll period :

2

T

2

T 'i'ia

1fandf2 0

2

f

1T


The natural roll period,The natural roll period, TT00 can be estimated with thecan be estimated with the

ship at free roll in still water condition using a stopwatchship at free roll in still water condition using a stopwatch

IMO requires the average of about 5 cycles be takenIMO requires the average of about 5 cycles be taken

Roll DampingRoll Damping

The oscillating free rolling motion eventually diesThe oscillating free rolling motion eventually diesout. Free roll transfers the roll energy to theout. Free roll transfers the roll energy to the

forces. The decay of the roll is due to dampingforces. The decay of the roll is due to damping



0cba

With the initial condition (at t=0)With the initial condition (at t=0) == 00 andand

dd/dt =0, the differential equation becomes :/dt =0, the differential equation becomes :

0a

c

a

b


ere;ere;

a

cand

a

b2 20


The solution of free rolling motion :The solution of free rolling motion :

02 20

For small damping, the frequencyFor small damping, the frequency ofof

the free roll can be approximated by thethe free roll can be approximated by the

tcostexp 00


natural frequencynatural frequency 00 from;from;

1Das)D1( 2022

0

2


7/22 7


From the solution;From the solution;tex

The ratio of 2 successive roll amplitudesThe ratio of 2 successive roll amplitudes nn andand

n+1n+1 at a distance of the natural period Tat a distance of the natural period T00 is :is :

0

nn0n TexpTt


1n1n01n TexpTt

)Texp()TT(expTexp

Texp0n1n

n0

1n0

n

1n


)Texp( 01n

n

The dimension ofThe dimension of is sis s--11. In order to define. In order to define

a dimensionless damping parameter;a dimensionless damping parameter;

1n

n

0

lnT

1


1s

sD

1

1

0


The dimensionless damping,The dimensionless damping,

ca2/bD

To estimate the damping paramater D,To estimate the damping paramater D,

1nn0

0ln2

1

2

T

D


successive roll amplitudes at one side aresuccessive roll amplitudes at one side are

to be measured and put into the equation.to be measured and put into the equation.

For most shipFor most ship DD 0.10 0.10

Rolling Period TestRolling Period Test

Rolling coefficient :Rolling coefficient :

'iC Tr

After necessary manipulations;After necessary manipulations;

.

GM

BC

GM

B5.0C2

GM

'i22T rrT0


GM

BCT r0


8/22 8


The practical importance of the aboveThe practical importance of the aboverelationship lies in estimating therelationship lies in estimating the

rolling period test.rolling period test.

2

0

r

T

BCGM


Weiss formula, 1953Weiss formula, 1953




The rolling period test should beThe rolling period test should beconducted with the ship in harbour inconducted with the ship in harbour in

interference from the wind and tide.interference from the wind and tide.

The ship can be made to roll by rhytmicallyThe ship can be made to roll by rhytmicallylifting up and putting down a weight orlifting up and putting down a weight orpeople running athwartships (people running athwartships (sallyingsallying


The initial roll amplitude for the measuredThe initial roll amplitude for the measuredroll decay should not exceed 5roll decay should not exceed 500


IMO allows estimating the stability byIMO allows estimating the stability by

means of rolling period tests for smallmeans of rolling period tests for small

..

IMO Resolution A.749(18) was adopted onIMO Resolution A.749(18) was adopted on

4 November 1993.4 November 1993.

However, the rolling period test must beHowever, the rolling period test must be


,,

other stability information is available.other stability information is available.


9/22 9


The Weiss formula gives GM as a function of;The Weiss formula gives GM as a function of;

Natural roll period, TNatural roll period, T00

Beam of the vessel, BBeam of the vessel, B

Rolling coefficient, CRolling coefficient, Crr



For Coasters of normal size, the observed CFor Coasters of normal size, the observed Crrvalues are;values are;

Empty ship or carrying ballastEmpty ship or carrying ballast 0.880.88

Ship fully loaded with liquids in tanksShip fully loaded with liquids in tanks 0.880.88

Comprising 20% of total loadComprising 20% of total load 0.780.78


..


IMO Resolution A.749(18) (1993) and IMOIMO Resolution A.749(18) (1993) and IMO

Circular 707 (1995) present an approximateCircular 707 (1995) present an approximate

100

L043.0

T

B023.0373.0C5.0 2


B

GMTC 0r

Different Modes of Roll ExcitationDifferent Modes of Roll Excitation

Roll excitation for a for a ship in a seaway:Roll excitation for a for a ship in a seaway:

1.1. Time varyingTime varying external excitationexternal excitation in the rightin the right--

hand side of the equationhand side of the equation2.2. Time varyingTime varyingparametric excitationparametric excitation in thein the

leftleft--hand side of the equationhand side of the equation



10/2210

Different Modes of Roll ExcitationDifferent Modes of Roll Excitation

A ship in beam seas can experience large rollA ship in beam seas can experience large rollwith large inertia forces acting on the cargo.with large inertia forces acting on the cargo.

Following and stern quartering seas at the sameFollowing and stern quartering seas at the samestability can be more dangerous with respect tostability can be more dangerous with respect tocapsizing and loss of the ship.capsizing and loss of the ship.

An excitation due to time variation of shipAn excitation due to time variation of shipreaction is called parametric. At parametricreaction is called parametric. At parametricresonance, the ship is in danger of capsizing.resonance, the ship is in danger of capsizing.

This is mostly seen in certain condition inThis is mostly seen in certain condition in


ong u na an s ern quar er ng seas.ong u na an s ern quar er ng seas. Both external and parametric excitations existBoth external and parametric excitations existsimultaneously in quartering seas.simultaneously in quartering seas.

Ship Rolling in Beam SeasShip Rolling in Beam Seas

There are only external excitation in beam seasThere are only external excitation in beam seas

written on the rigthwritten on the rigth--hand side of the equation.hand side of the equation.

dynamic reaction + static reaction = external excitationdynamic reaction + static reaction = external excitation

For small amplitudes,For small amplitudes,

Roll motion equation is a linear second orderRoll motion equation is a linear second order


differential equation.differential equation.

Ship Rolling in Beam SeasShip Rolling in Beam Seas

FB : bouyancy force

The amplitude of beam sea excitation;


AABAA GMgGMFcd

)tsin(dd A

)tsin(GMgd A

At the wave trough :

Wave Slope vs. Distance from CrestWave Slope vs. Distance from Crest

The wave slope is the first derivative of the wave ordinate with


direction of the wave.

)cos(5.0)( kxHx w

kxsinkH5.0x

)x( w

Wave ordinate :

Wave slope :


11/2211

Wave SlopeWave Slope

Wave slope amplitude :Wave slope amplitude :

)(25.0

5.0 radHH

kH wwwA

The exciting moment in beam seas:The exciting moment in beam seas:

ww

)tsin(L

HGMgd

w

w


Insert this into the roll motion equation:Insert this into the roll motion equation:

A

A3V

Transfer function

Amplitude of roll

Amplitude of wave slope

Solution of the EquationSolution of the Equation

The solution is the equation is given by theThe solution is the equation is given by thetransfer function Vtransfer function V33 which is the dynamicwhich is the dynamic

The dimensionless wave frequency withThe dimensionless wave frequency with

22223

4)1(

1)(

DV


tuning factortuning factor ;;

0w

0

T

T


TheThe dimensionless damping Ddimensionless damping D ::

nln1b

D

Rewite transfer function;Rewite transfer function;

1n0 ac

2222

0

2

0

3

4)()(

V


The resulting roll motion in beam seas:The resulting roll motion in beam seas:

)tsin(V 33A


TheThephase anglephase angle 33 between the exciting moment dbetween the exciting moment d

1

D2arctan23



12/2212

ResonanceResonance

Less sensitive to wave excitation


Very sensitive to wave excitation

Transfer Function of Roll in Beam SeasTransfer Function of Roll in Beam Seas


Important Results from the SolutionImportant Results from the Solution

1.1. The static heelThe static heel = 0 results from= 0 results fromconstant excitation independent of time:constant excitation independent of time:

10V

2.2. With the exciting wave frequency,With the exciting wave frequency, ,,

increasing there is a steady increase ofincreasing there is a steady increase ofthe roll response:the roll response:

Astat3


The dynamic response is always greaterThe dynamic response is always greaterthan the static heelthan the static heel VV33 > 1> 1

0w0

Important Results from the SolutionImportant Results from the Solution

3.3. There is dominant amplification in theThere is dominant amplification in the

region aroundregion around =1 (resonance).=1 (resonance).

The frequency of the peak response is :The frequency of the peak response is :

The resonant roll amplitude at the peak is:The resonant roll amplitude at the peak is:

0

22

0r 2


AA0

rD2

1

2


13/22


14/22

14

Roll in Beam Seas at Large AmplitudesRoll in Beam Seas at Large Amplitudes

a.a. CC22> 0 GZ over> 0 GZ over--linearlinear: curve bends to: curve bends to

largerlarger (right)(right)

b.b. CC22< 0 GZ under< 0 GZ under--linearlinear: curve bends to: curve bends to

smallersmaller (left)(left)


Roll in Beam Seas at Large AmplitudesRoll in Beam Seas at Large AmplitudesOver-linear roll response Under-linear roll response


GZ Variations in Longitudinal WavesGZ Variations in Longitudinal Waves

A ship in longitudinal waves experiences aA ship in longitudinal waves experiences a

completely different shape of thecompletely different shape of the

ship in still water and in beam seas.ship in still water and in beam seas.

The righting moment of the vessel variesThe righting moment of the vessel variesin time with the passing wave.in time with the passing wave.


motionmotion

GZ Variations in Longitudinal WavesGZ Variations in Longitudinal Waves

After-body midship Fore-body


Ship in longitudinal wave at different positions relative to the crest

Wave length = LWL

Draft : full load draft

Heel angle = 300


15/22

15

Comparison of GZ CurvesComparison of GZ Curves


Comparison of GZ CurvesComparison of GZ Curves

The change of GZ results from the changeThe change of GZ results from the changein the location of the center of bouyancy Bin the location of the center of bouyancy B

wave.wave.

Weight force, W and the center of gravity,Weight force, W and the center of gravity,

G remain constantG remain constant


GZ Changes in Longitudinal WaveGZ Changes in Longitudinal Wave


Wave Crest SituationWave Crest Situation

The freeboard amidships reduces considerably.The freeboard amidships reduces considerably.

It may even become negative.It may even become negative.

ue o ac o uoyancy a ove e ec s e aue o ac o uoyancy a ove e ec s e alarge heel, the center of buoyancy in heeledlarge heel, the center of buoyancy in heeled

condition Bcondition B shifts towards the center of gravityshifts towards the center of gravityG.G.

This shift of B reduces GZThis shift of B reduces GZ


Freeboards at sections 1 and 3 increase butFreeboards at sections 1 and 3 increase butcannot counteract the GZ reduction amidshipscannot counteract the GZ reduction amidships

Thus overall reduction in GZ resultsThus overall reduction in GZ results


16/22

16

Wave Trough SituationWave Trough Situation

The wave trough amidships results anThe wave trough amidships results an

ncrease o e r g ng ever .ncrease o e r g ng ever .

The effective freeboard of the midshipThe effective freeboard of the midship

section 2 is considerably increasedsection 2 is considerably increased

The overall GZ reduction in the crest isThe overall GZ reduction in the crest is


..

Influence of Wave Length on GZ in a Wave CrestInfluence of Wave Length on GZ in a Wave Crest


GZ reduction between still

water and wave crest

Wave HeightWave Height

Formula derived from wave statistics in the NorthFormula derived from wave statistics in the North

Atlantic:Atlantic:

)meterinL(L05.010L

w

ww

w


Effect of Speed on GZ CurvesEffect of Speed on GZ Curves

Blume and Hattendorf (1982) compared theBlume and Hattendorf (1982) compared thehydrostatic results with measurements onhydrostatic results with measurements onmodels of container ships in following seas.models of container ships in following seas.

For Froude Numbers between 0For Froude Numbers between 0 0.28 there0.28 therewas almost no difference in GZ.was almost no difference in GZ.

At FAt Fnn = 0.36 the reduction in the wave crest= 0.36 the reduction in the wave crestwas about half the value of the hydrostaticwas about half the value of the hydrostaticcalculationcalculation


At FAt Fnn = 0.36 the increase in the wave trough= 0.36 the increase in the wave troughwas about 10% less than the hydrostaticwas about 10% less than the hydrostaticresult.result.


17/22

17

Effect of Speed on GZ CurvesEffect of Speed on GZ Curves


C FactorC Factor

Blume and Hattendorff (1982, 1984) developedBlume and Hattendorff (1982, 1984) developeda soa so--called Ccalled C--Factor for usual merchant hullFactor for usual merchant hull

,,

in waves by a formula based on capsizingin waves by a formula based on capsizing

model experiments.model experiments.

IMO implemented the CIMO implemented the C--factor for containerfactor for container

ships and fast ships with a small Cships and fast ships with a small C (0.554(0.554--


0.675) into IMO stability criteria (IMO, 1993)0.675) into IMO stability criteria (IMO, 1993)

C FactorC Factor

BP

2

w

B

2L

100

c

c

KG

T

B

'DTC

TT mean draft (m)mean draft (m)

BB moulded breadth of the ship (m)moulded breadth of the ship (m)

KGKG height of the center of gravity (m)height of the center of gravity (m)


no o e a en ess anno o e a en ess an

CCBB block coefficientblock coefficient

CCww waterplane coefficientwaterplane coefficient

C FactorC Factor

DD effective freeboard accounts for theeffective freeboard accounts for the

volume of the hatches above deck amidshipsvolume of the hatches above deck amidships

(from plus and minus L/4 of the main section).(from plus and minus L/4 of the main section).

Ship length is to beShip length is to be

100 m.

100 m. KG is to be larger than draft T.KG is to be larger than draft T.

The smaller the CThe smaller the C--factor , the larger are thefactor , the larger are the


the GZ values required.the GZ values required.

IMO asks for hydrostatic values in the form ofIMO asks for hydrostatic values in the form of

a required constant divided by Ca required constant divided by C


18/22

18

Intact stability Criteria Based onIntact stability Criteria Based on

C FactorC Factor


GZ Curve Required by IMO BasedGZ Curve Required by IMO Based

on C Factoron C Factor


EncounterEncounter PeriodPeriod ofof ShipShip andand

WavesWaves

TheThe encounter period, Tencounter period, TEE is the time elapsedis the time elapsed

the shipthe ship


EncounterEncounter PeriodPeriod ofof ShipShip andand WavesWaves



19/22

19

Encounter Period of Ship andEncounter Period of Ship and

WavesWaves

cosVcVrel

cosVc

L

V

LT w

rel

w

E

ww

w

w fLT

Lc


wT2

gc

Encounter Period of Ship andEncounter Period of Ship and

WavesWaves

2

ww Tg

L

2

TT

2w

E

g

w

2

gLc w

cosVL2

g

LT

w

w

E


The encounter frequency:The encounter frequency:

)Hz(T

1f

E

E )s/rad(f2 EE

Encounter Frequency

Wavedirection V

V

V

180 0

90

45


g

Vwwe

cos2

directionwavethe

torelativeangleheadingsship'

(m/s)speedshipV

frequencywave

frequencyencounter

w

e

Encounter Frequency

Example

ship speed = 20 knots, heading angle = 120 degree

wave direction : from north to south, wave period=12 seconds

Encountering frequency ?

Wave frequency : sradsT

w /52.012

22

Encountering angle : o60120180 120

N


V=20kts

60

Encountering freq. :

srad

g

Vw

we

/38.014.052.0

81.9

60cos)(10.29)52.0(52.0

cos

2

2

)/29.1020( smknotsV

S


20/22

20

Ship HeadingShip Heading

00 degreesdegrees following seasfollowing seas

9090 degreesdegrees starboard beam seasstarboard beam seas

180180 degreesdegrees head seashead seas

Metin Taylan, 2010 Metin Taylan, 2010 Metin Taylan, 2010 Metin Taylan, 2010


Passive Anti-Rolling Device

RollRoll MotionMotion ReductionReduction

Bilge Keel

- Very common passive anti-rolling device

- Located at the bilge turn

- Reduce roll amplitude up to 35 %.

Tank Stabilizer (Anti-rolling Tank)

- Reduce the roll motion by throttling the fluid Bilge keel


.

- Relative change of G of fluid will dampen the roll.

ThrottlingU-type tube


21/22

21

BilgeBilge KeelKeel LengthLength


BilgeBilge KeelKeel ConstructionConstruction


For large ships

Active Anti-Rolling Device

Roll Motion ReductionRoll Motion Reduction

Fin Stabilizer

- Very common active anti-rolling device

- Located at the bilge keel.

- Controls the roll by creating lifting force .

Roll moment


Lift

Anti-roll moment

Fin Stabilizer



22/22

22

Ship Operation

Roll Motion ReductionRoll Motion Reduction

Encountering frequency

g

Vwwe

cos2

roll

pitch


- ship speed

- heading angle or

- both

6 Roll Motion

Documents

Transcript of 6 Roll Motion