1

SEAKEEPING

Summary

The concept of seakeeping is first explained and shipmotions and their effects are introduced. Waves in the sea areconsidered next. Regular waves and their properties are described.Irregular waves are then described and the concept of energyspectrum introduced. The statistical properties of irregular waves,some standard wave spectra and the Beaufort scale and the seastate code are mentioned. Ship motions in regular waves arediscussed, and expressions for the natural periods of heave, pitchand roll are derived. Forced heave, pitch and roll motions in wavesare considered briefly. Ship motions in irregular waves are thendescribed. The concepts of encounter frequency and motionspectrum are introduced and the statistical properties that can thenbe derived mentioned. Derived responses such as deck wetnessand slamming are described briefly. Motion stabilizers are thenconsidered. Seakeeping performance and criteria for itsassessment are then described. Ship design features that affectseakeeping are discussed briefly.

2

CONTENTS

1. Introduction

1.1 Waves and Seakeeping1.2 Ship Motions1.3 Effects of Ship Motions

2. Waves in the Sea

Origin of WavesRegular WavesIrregular WavesDescription of Sea Conditions

3. Ship Motions in Regular Waves

3.1 General Theory of Oscillations3.2 Ship Motions

4. Ship Motions in Irregular Waves

Encounter SpectrumResponse Amplitude OperatorMotion SpectrumDerived Responses

5. Motion Stabilizers

5.1 Introduction5.2 Roll Stabilizers

6. Seakeeping Performance

6.1 Assessment of Seakeeping Performance6.2 Performance Criteria6.3 Limiting Values of Responses6.4 Speed Polar Plot

7. Seakeeping and Ship Design

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1. INTRODUCTION

1.1 Waves and Seakeeping

The sea is never perfectly still and the waves in the sea affect the behaviour of the ship.The ability of a ship to carry out its mission in spite of the effects of the waves is a measure ofthe seakeeping qualities of the ship. The fundamental effect of the waves on the ship is to causeit to undergo various motions or oscillations.

1.2 Ship Motions

The motions that a ship is forced to execute due to the waves are conveniently dividedinto three linear oscillations along, and three angular oscillations about, axes oriented along thelength (x-axis), breadth (y-axis) and depth (z-axis) with the origin usually at the centre of gravityof the ship.

Oscillation Linear AngularAxis

x Surge Roll

y Sway Pitch

z Heave Yaw

1.3 Effects of Ship Motions

Excessive ship motions affect the comfort of the crew and passengers of the ship. Theforces and moments that arise due to ship motions affect the operation of various ship systems,making it impossible for the ship to carry out its mission in bad weather when the waves arehigh and the motions severe. Waves in the sea cause an increase in the resistance of the ship,resulting in a reduction in speed at constant power. High waves and the resulting motions maycause the bottom of the ship to emerge from the water momentarily and then fall into the waterwith a sharp impact. Water may also break on the deck causing damage. Oscillations of largeamplitude cause large forces to act on the structure of the ship. It may be necessary to reduce thespeed of the ship and to change its course to avoid the adverse effects of ship motions.

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2. WAVES IN THE SEA

2.1 Origin of Waves

The waves in the sea are usually caused by the action of wind on the surface of water dueto friction and local pressure variations. There are also interactions between different waves,and some waves break. The total wave system in a particular area may be assumed to be theresult of many independent waves distributed over space and time. The waves in the sea aregenerally irregular, i.e. they do not appear to have a fixed pattern. If the wave heights are small,it is possible to consider the waves in the sea to be the resultant of several independent systemsof regular waves. Regular waves are waves of constant shape moving at a fixed speed in a fixeddirection. Irregular waves can be regarded as a linear superposition of several independentregular waves.

2.2 Regular Waves

The equation of regular waves in water may be obtained from the hydrodynamicequations of motion for potential flow. The equation for a wave moving in the direction of thex-axis, with the z-axis positive upwards and the origin in the undisturbed surface of water is :

coshcos

coshak z d

k x c tkd

where :

elevation of constant pressure line originally at the level z

0.5a h = surface wave amplitude, h being the wave height

2k

“wave number”, being the wave length

d depth of water

c wave velocity or “celerity”

t time.

In deep water, as d , the equation becomes :

coskza e k x ct

and on the surface of water,

cosa k x ct

which gives the surface wave profile.

5

The basic properties of these regular waves, which may be derived from the equation ofthe wave, are :

22kg

where is the “circular frequency”

cos cosa ak x ct k x t

0.5

2 2g gT gc

T

where T is the wave period

0.52 2T

c g

2 2

22 2

2c g gTg

.

The pressure at a level z is given by cos k zap g z k x t e .

The maximum wave slope is given by 2 aa

hk

.

The wave energy per unit area 1 22 aE g .

The foregoing is based on linear or first order wave theory. In the Stokes second ordertheory, the equation of the wave at time 0t is given by :

2

cos cos 2aa k x k x

This gives a wave profile that corresponds more closely than the linear wave with the profiles ofregular waves generated in a laboratory. The trochoidal wave theory also represents physicalwaves more closely than the linear wave theory. There are also higher order wave theories.However, the linear wave theory is almost always used in the study of seakeeping because it issimple to use and gives sufficiently accurate results.

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2.3 Irregular Waves

2.3.1 Component Waves

The waves in the sea, i.e. irregular waves, can be regarded as the resultant of a largenumber of regular waves of different frequencies, wave amplitudes, phases and directions ofpropagation. The phases of the component waves are assumed to have a completely randomdistribution and the entire “process” is assumed to be a stationary random process in which theelevations of the different waves have a “normal” distribution.

2.3.2 Long Crested Seas

It is convenient to first consider all the waves to be moving in the same direction,resulting in a “long crested sea”. Suppose there are n component waves, the elevation at a fixedpoint of the i-th wave being given by :

cosi ai i it

and its energy per unit area by :

1 22i aiE g .

The energy of all the waves is then given by :

1 2 2 21 22 ......a a anE g

If n , the energy of the waves for frequencies between and d can be written as :

1 22

1adE S d

g

S is a measure of the distribution of energy in the sea as a function of the (circular)

frequency , and is known as the wave spectral density. A curve of S as a function of is the wave spectrum. The total energy per unit area due to all the waves in the sea is given by :

2

0 0

12 aE g d g S d g

area under the S curve.

The moments of the wave spectrum are of great significance. The n-th moment of the wavespectrum (n is an integer) is given by :

0

nnm S d

7

0m is the area under the S curve.

If one looks at a record of the surface elevation of the irregular waves in the sea at apoint x as a function of time t or at a particular instant t over a length x in the direction ofwave propagation, one can determine the following parameters :

a apparent wave amplitude

h apparent wave height

zT apparent zero up-crossing period

cT apparent period based on adjacent crests

z apparent wave length based on adjacent zero up-crossings

c apparent wave length based on adjacent crests.

The average values of these apparent parameters are related to the moments of the wavespectrum, e.g. the average period between zero up-crossings or the average period based onadjacent crests have values :

0

22z

mTm

and 2

42c

mTm

.

The average wave length based on zero up-crossings is :

0

42z

mgm

The average period of the component waves is given by :

1

0 0 1

0

0 0

22

T S d S dmTm

S d S d

The average frequency of the waves is :

8

0 1

0

0

S dmm

S d

and the corresponding period is :

01

1

22 mTm

The frequency at which the waves have the highest energy, i.e. the value of at which S has the greatest value, is the “modal frequency” m and the corresponding period is the

“modal period” 2m mT .

The average wave amplitude and the average wave height are given by :

01.25a m 02.5h m .

The average amplitude and height of the one-third highest waves are :

1 03

2.0a m 1 03

4.0h m .

These are also called the “significant wave amplitude” and the “significant wave height”respectively.

Similarly, the average amplitudes and heights of the one-tenth and one-hundredth highestwaves are :

1 010

2.55a m 1 03

5.1h m

1 0100

3.335a m 1 0100

6.671h m .

The highest expected wave height maxh in a sample of N successive waves is asfollows :

N maxh

100 06.5 m

1000 07.7 m

10000 08.9 m .

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2.3.3 Standard Wave Spectra

There are some wave spectra that are widely used, e.g. :

Pierson-Mosowitz Spectrum :

42

5 expw

g gSV

S spectrum ordinate in cm2 sec frequency in radians per sec 38.10 10 0.74g acceleration of gravity in cm per sec2

wV wind speed in cm per sec.

JONSWAP Spectrum :

This is based on data collected by the Joint North Sea Wave Observation Projectand is given by :

42

55exp4

r

p

gS

22 2exp

2p

pr

12 3100.076 UF g

3.3

0.07 p 0.09 p .

F is the “fetch”, i.e. the distance over which the wind blows with a constant velocity,and 10U is the wind velocity 10 m above the surface of water.

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Bretschneider Spectrum :

5 4expA BS

The two parameters A and B depend upon the modal frequency and the area under thewave spectrum.

ITTC Spectrum :

5 4expA BS

This is a form of the Bretschneider spectrum with :

134

1

173h

AT

41

691BT

.

13

h is the significant wave height and 1T the period corresponding to the average wave

frequency. If only 13

h is known, one may use the following values :

3 28.10 10A g 21

3

3.11Bh

.

2.3.4 Directional Spectra

If one is to consider the sea to have irregular waves moving in various directionsdistributed at random about a dominant direction of propagation, it is necessary to use a“directional spectrum” ,S , where is the angle that a particular wave component makeswith the reference direction. This is most conveniently done at present by writing :

,S S M

where S is the “point spectrum” and M is the “spreading function”. The spreadingfunction is usually taken as :

1 cos2

n

s sM

for s s

0 otherwise.

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angle between the direction of propagation of a wave component and thedominant wave direction.

s angular range of components on either side of the dominant wave direction.

At present, it is usual to take 2n and 90s , so that :

22 cosM

.

2.4 Description of Sea Conditions

There are two widely used methods to describe the condition of the sea – the BeaufortScale and the Sea State Code.

BEAUFORT SCALE

Scale Description Wind Speed, knots

0 Calm 1

1 Light air 1 – 3

2 Light breeze 4 – 6

3 Gentle breeze 7 – 10

4 Moderate breeze 11 – 16

5 Fresh breeze 17 – 21

6 Strong breeze 22 – 27

7 Near gale 28 – 33

8 Gale 34 – 40

9 Strong gale 41 – 47

10 Storm 48 – 55

11 Violent storm 56 – 63

12 Hurricane 64 and over

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SEA STATE CODE

Sea State Description of the Sea Significant Wave Height, m

0 Calm (glassy) 0

1 Calm (rippled) 0 – 0.10

2 Smooth (wavelets) 0.10 – 0.50

3 Slight 0.50 – 1.25

4 Moderate 1.25 – 2.50

5 Rough 2.50 – 4.00

6 Very rough 4.00 – 6.00

7 High 6.00 – 9.00

8 Very high 9.00 – 14.00

9 Phenomenal Over 14.00

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3. SHIP MOTIONS IN REGULAR WAVES

3.1 General Theory of Oscillations

3.1.1 Equation of Motion

The equation of motion for a system that is undergoing linear or angular oscillation maybe written as :

cosa ea x b x c x P P t

The terms in this equation represent the following :

ax inertia force or moment

bx damping force or moment

cx restoring force or moment

P exciting force or moment

aP amplitude of the exciting force or moment

e frequency of the exciting force or moment

t time

( x is the displacement from the mean position,2

2d xxd t

and d xxdt

).

The following cases arise :

(i) 0b and 0P : Free undamped oscillation

(ii) 0b and 0P : Free damped oscillation

(iii) 0b and 0P : Forced undamped oscillation

(iv) 0b and 0P : Forced damped oscillation.

3.1.2 Free Undamped Oscillation

This is also known as “natural oscillation” since it is not forced by an external force ormoment. The equation for free undamped oscillation is :

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0ax cx .

The solution of this differential equation is :

sinni ta a nx x e x t

ax is the amplitude of the oscillation and the phase angle, which depend upon the “initialconditions”. The “circular frequency” in radians per unit time with which the system oscillatesafter being disturbed momentarily from its position of equilibrium is the “natural (circular)frequency” given by :

nca

.

The natural frequency in oscillations per unit time and the natural time period of the oscillationare given by :

2n

nf

2

nn

T

.

3.1.3 Free Damped Oscillation

The equation for free undamped oscillation is :

0ax bx cx .

This has the solution :

ptax x e where

2

2 2b b cpa a a

.

There will be oscillation only if 2b ac , in which case the solution may be written as :

sinta dx x e t .

2ba

is the “decay constant” and 0.52 2d n is the damped natural frequency. If

nv , or2

2b ca a

, the system has “critical damping” and is on the boundary between

oscillating and not oscillating, i.e. oscillating with zero frequency.

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3.1.4 Forced Damped Oscillation

The equation for forced damped oscillation is :

cosa eax bx cx P t

which has the solution :

0 0sin sinta d a ex e x t x t .

The first term represents transient oscillations which die out with the passage of time dependingupon the value of the decay constant . Eventually, only the forced oscillations remain, asgiven by the second term which represents the steady state solution.

If a constant force or moment Pa were acting on the system, there would be a staticdisplacement from the position of equilibrium given by :

astatic

Pxc

.

The magnification factor of the oscillating system is defined as :

a

static

xx

and the tuning factor by :

e

n

.

As already indicated :

2ba

(decay constant) nca

(natural frequency)

The non-dimensional damping factor is given by n .

It can be shown that the phase angle and the magnification factor are related to the tuning factorand the non-dimensional damping factor as follows :

12

2tan

1n

16

0.52

22 2

1

1 4n

When the tuning factor 1 , i.e. the exciting frequency is equal to the natural frequency, thereis “resonance” and the magnification factor has a high value. If there were no damping, theamplitude of oscillation would become infinitely large when there was resonance.

3.2 Ship Motions

3.2.1 Motions with and without Restoring Forces or Moments

Surge, sway and yaw motions of a ship do not have restoring forces or moments, sincethe ship is in neutral equilibrium with respect to linear movements along the x- and y-axes andangular movement about the z-axis.. These motions can only be forced oscillations at theexciting frequency with an amplitude dependent upon the amplitude of the exciting force ormoment.

A ship is in stable equilibrium with respect to heel, trim and sinkage, e.g. if the heelangle of a ship in equilibrium is momentarily changed by a small amount, a righting momentacts to restore the position of equilibrium. Roll and pitch are thus ship motions with restoringmoments, and heave is a motion with a restoring force. These motions can therefore be freeoscillations with natural frequencies.

3.2.2 Heave

The equation for “free” heave motion is written as :

0zm a z bz cz

m is the mass (displacement) of the ship, za is the “added mass” in heave, b is the dampingcoefficient and c is the restoring force coefficient. The restoring force is the additionalbuoyancy due to a parallel sinkage z :

Wcz g A z

WA is the waterplane area at the equilibrium draught. The natural undamped frequency in heaveand the corresponding time period are :

Wz

z

g Am a

2 2 zz

z W

m aTg A

.

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The added mass and the damping coefficient may be determined by theoretical means orexperimentally. The added mass in heave is of the same magnitude as the mass of the ship.

The equation for the forced heaving of a ship in regular waves is :

cosz a em a z bz cz F t

The amplitude aF of the exciting force in regular waves of given length and height may bedetermined by calculating the pressure at points on the wetted surface of the ship and integratingthe pressure over the ship length.

3.2.3 Pitch

The equation for free undamped pitching motion may be written as :

' 0yyI c

'yyI is the “virtual mass moment of inertia” of the ship about a transverse axis through the centre

of gravity. The restoring moment in pitch is :

Lc g mGM

LGM is the longitudinal metacentric height. The natural pitch frequency and time period aregiven by :

'L

yy

g mGMI

' 22 2 2 2yy yy yy

L L L

I m k kT

g mGM gmGM g GM

yyk is the virtual mass radius of gyration about the transverse axis through the centre of gravity,and typically has a value of about one-fourth the length of the ship.

The equation for pitching in waves is :

' cosLyy a eI b g mGM M t

aM is the amplitude of the pitch moment due to the waves and may be calculated from thepressure distribution along the length of the ship for waves of a given length and height.

The solution of this equation is :

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0 0cos costa d a ee t t

where :

'2

yy

bI

decay constant

0 0anda amplitude and phase of damped natural pitchmotion (transient phase)

2 2d damped natural pitch frequency

'

0.522 2 2 24

a yya

e e

M I

amplitude of forced pitch motion (steady state)

12 2

2tan e

e

phase of forced pitch motion.

It is usually necessary to consider heave and pitch together since both occursimultaneously in waves.

3.2.4 Roll

The equation for free undamped rolling is :

' 0xxI c

'xxI is the virtual mass moment of inertia about a longitudinal axis through the centre of gravity,

and the restoring moment is :c g m GM

where GM is the metacentric height. The natural roll period is given by :

'

2 2xx xxI kTg m GM g GM

The virtual radius of gyration about the longitudinal axis through the centre of gravity is givenapproximately by :

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0.33 to 0.45xxk B B

where B is the breadth of the ship. A large metacentric height is said to result in a low period ofroll (a “stiff” ship) and high accelerations and resulting inertia forces during rolling.

The equation for forced rolling in regular waves is :

' cosxx a eI b g mGM K t

The amplitude of the exciting moment is related to the wave slope :

aK g GM

is the displacement volume. is the maximum wave slope in the direction of the breadth ofthe ship and is given by :

sinh

where h is the wave height, the wave length and the angle between the ship centre line(x-axis) and the direction of propagation of the waves.

The solution of the equation for forced rolling is :

0 0cos costa d a ee t t

where :

'2

xx

bI

decay constant

0 0anda amplitude and phase of damped natural rolling(transient phase)

2 2d damped natural frequency

'xx

g mGMI undamped natural frequency

2

0.522 2 2 24a

e e

amplitude of forced rolling (steady state)

20

12 2

2tan e

e

phase of forced rolling.

It should be noted that the equations of rolling given in the foregoing apply only to smallroll amplitudes.

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4. SHIP MOTIONS IN IRREGULAR WAVES

4.1 Encounter Spectrum

Consider a ship moving at a speed V at an angle to waves of length and speed c .The relative velocity of the ship with respect to the waves is then cosc V , so that the“encounter period” is :

coseT c V

.

Recalling that :

gc

22 g

the encounter period can be expressed in terms of the wave frequency as :

2

22 2

cos coseg gT

g V g V

The encounter frequency is therefore :

2 22 cos2 cos2e

e

g V VT g g

and :2 cos1ed V

d g

The area under the wave spectrum represents the energy of all the waves in the sea, andthis remains the same whether the ship is standing at a fixed point or is moving. This allows thewave spectrum as encountered by the ship to be determined from the wave spectrum at a point,since the area under the “encounter spectrum” is equal to the area under the point spectrum ofthe irregular waves in the sea :

0 0

e eS d S d

so that :

1 12 cos1

ee

S S Sd Vgd

.

22

Thus, the encounter spectrum is obtained from the point spectrum of the sea by :

multiplying the abscissas by cos1e Vg

and

multiplying the ordinates by

12 cos1

eSVSg

.

Note that is zero for following seas and 180o for head seas.

4.2 Response Amplitude Operator

The response amplitude operator RAO for a particular “response” x , of the ship, e.g.heave or pitch, to encountered waves is defined as :

22Amplitude of response

Amplitude of regular wavesa e

x ea e

xxRAO

The response amplitude operators for the six basic ship motions – surge, sway, heave, roll, pitchand yaw – can be obtained from model experiments in regular waves or by theoretical means.

4.3 Motion Spectrum

The motion spectrum for a particular motion is obtained by multiplying the encounterspectrum by the response amplitude operator, e.g. the pitch spectrum is obtained as :

2a e

e ea e

S S

From the motion spectrum, one can obtain information about that particular motion inthe given sea spectrum in the same way that the wave spectrum gives information about thewaves. For example, denoting the area under the motion spectrum curve by 0m (zero-thmoment) :

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Average amplitude of pitch motion 01.253a m

Significant pitch amplitude 1 03

2.000 m

Average amplitude of highest one-tenth waves 1 010

2.545 m

The greatest pitch amplitude expected on the average in N successive observations :

N Amplitude

100 03.25 m

1000 03.85 m

10000 04.45 m .

4.4 Derived Responses

From the six basic modes of ship motion in regular waves, it is possible to deriveresponse amplitude operators for responses such as :

- vertical and lateral motions, velocities and accelerations at specific points in the ship- relative motion between a point on the ship and the wave surface- shipping of seas on deck and slamming- added resistance and power in waves- wave bending moments.

This would require both amplitudes and phases to be known. Once the spectrum of theparticular response is obtained, its various statistical properties can be calculated.

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5. MOTION STABILIZERS

5.1 Introduction

Among the various motions that a ship undergoes in a seaway, only roll motion isrealistically capable of being reduced by devices that provide forces or moments opposing thatmotion.

5.2 Roll Stabilizers

There are basically three types of roll stabilizers used in ships : passive stabilizers,controlled passive stabilizers and active stabilizers.

Passive stabilizers are devices that require no power or control system. Bilge keels arepassive roll stabilizers that have no moving parts. Anti-rolling tanks are stabilizers in which themovement of water between tanks on either side of the ship provides a moment that opposes theroll motion. The water moves from side to side only in response to the rolling of the ship.Passive anti-rolling tanks may be free surface tanks, U-tube tanks or external tanks withopenings to the sea. A moving weight system may also be used as a passive roll stabilizer.

In controlled passive stabilizers, there is an arrangement to control the moment opposingthe roll motion. In anti-rolling tanks of this type, there may be a valve in the pipe or ductconnecting the tanks so that the rate at which water flows between the tanks can be controlled toproduce a time period and phase difference that will give the maximum stabilization. Instead ofcontrolling the flow of water, the tanks can be made air-tight and a pipe with a servo controlledvalve fitted to connect the air space on top of the tanks.

Active stabilizers not only have a control system to control the moment opposing the rollmotion but also require substantial power to operate. In active tank stabilizers, water is movedbetween the tanks on either side by a pump whose discharge is controlled by a control system.In fin stabilizers, there are fins projecting out of the hull near amidships which produce liftforces that oppose the roll motion. The angles of attack of the fins or their tail flaps arecontrolled to minimize the roll motion. In calm weather, the fins can be retracted into the hull toeliminate their resistance. Gyroscopic stabilizers have also been used.

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6. SEAKEEPING PERFORMANCE

6.1 Assessment of Seakeeping Performance

It is necessary to assess the seakeeping performance of a ship and to predict performancein given environmental conditions for the optimum performance of the ship. Seakeepingperformance can be related to three factors :

(i) Mission : The required performance depends upon the mission of the ship. Anaval ship may have several missions, e.g. transit, helicopter operation, andreplenishment at sea.

(ii) Environmental conditions : Performance depends upon the environmentalconditions, i.e. wind and waves. Environmental conditions are usually defined by“Sea State”, characterized by the significant wave height and the modal waveperiod.

(iii) Ship responses : These include amplitudes, velocities and accelerations of thedifferent motions and responses such as deck wetness, slamming and propelleremergence.

6.2 Performance Criteria

Two numerical measures of seakeeping performance have been developed. SeakeepingPerformance Index 1 (SPI-1) is based on mission effectiveness and is equal to the fraction of thetotal time that the ship can perform a specified mission for given ship speeds and headings in aspecified ocean area and season. Seakeeping Performance Index 2 (SPI-2) is based on transittime, and is the ratio of the ideal time needed to move between two points in calm weather to theactual time required in seas appropriate to a specified season or seasons. SPI-2 is also the actualaverage speed in specified conditions to the calm water speed.

6.3 Limiting Values of Responses

The parameters that affect mission effectiveness may be grouped into three categories :

(i) Personnel : comfort, motion sickness, fatigue, task proficiency and safety

(ii) Operations : helicopter operations, shifting of cargo

(iii) Ship : damage to hull or deck equipment, loss of efficiency in ship systems.

Some important parameters and their typical values are :

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Roll amplitude 3o rmsPitch amplitude 1.5o rmsVertical displacement 1.25 m rmsVertical acceleration 0.2 g rmsLateral acceleration 0.1 g rmsMotion sickness 10 % in 4 hoursSlam acceleration 0.2 gSlam frequency 20 per hourDeck wetness frequency 30 per hour.

6.4 Speed Polar Plot

The speed polar plot is a method of presenting for a given sea state the boundaries ofspeed and heading between acceptable and unacceptable ship responses. The diagram consistsof lines radiating from a centre indicating the direction of the ship with respect to the dominantwave direction and concentric circles indicating different ship speeds. On this diagram, thelimits of speed and heading for the different ship responses (roll amplitudes, slam frequency etc)are marked, and the unacceptable combinations of speed and heading shaded. The unshadedregion shows the operable speed and heading combinations, and is called the “seakeepingoperating envelope”. A larger unshaded area indicates better seakeeping performance.

27

7. SEAKEEPING AND SHIP DESIGN

In considering ship motions, it is usual to treat the longitudinal motions (surge, pitch andheave) separately from the transverse motions (roll, sway and yaw). The effects of roll motionare minimized by not having an unnecessarily high metacentric height and by fitting rollstabilizers. If necessary, the ship’s heading may be changed to avoid heavy rolling in beam seas.The design features that affect pitch and heave motions are as follows :

- Ship size : The probability of a ship encountering waves of length equal to or greaterthan the ship length decreases for longer ships. Longer waves also have a relativelysmaller wave height. Larger ships thus usually have a better seakeepingperformance.

- Speed : A reduction in speed usually reduces pitch and heave motions in rough seas.

- Length-breadth ratio : This has a minor effect on seakeeping

- Length-draught ratio : High length-draught ratios sometimes lead to markedresonance in waves and to increased slamming. High length-draught ratios have alsobeen found to result in lower pitch and heave amplitudes in long waves but higheramplitudes in shorter waves.

- Block coefficient : Ships with high block coefficients have a greater speed loss inwaves but smaller motion amplitudes. However, the effect is small.

- Prismatic coefficient : A high prismatic coefficient may lead to reduced motions butgreater deck wetness. A high prismatic coefficient also leads to a lower speed loss athigh speeds but a higher speed loss at low speeds.

- Waterplane coefficient : A high waterplane coefficient leads to higher wave bendingmoments.

- Radius of gyration : A smaller radius of gyration about the transverse axis throughthe centre of gravity reduces ship motions in waves longer than the ship.

- Forward sections : U-shaped sections forward result in a smaller speed loss in wavesbut V-shaped sections result in smaller motion amplitudes. Above water flare in thesections forward reduces deck wetness.

- Freeboard : A high freeboard reduces deck wetness.

Many design features appear to have opposite effects in long waves and in short waves. Sincethe sea contains waves of varying lengths, hull form appears to have only a minor influence onship motions in a seaway.

28

ADDITIONAL READING

The preceding notes should be supplemented by the following :

1. Principles of Naval Architecture, Vol. III, Chapter 8.

2. Basic Ship Theory, Vol. II, Chapter 12.

Principles of Naval Architecture (three volumes), Edward V. Lewis, Editor, published bySociety of Naval Architects and Marine Engineers, 601 Pavonia Avenue, Jersey City, NJ, USA.Second Revision, 1988, ISBN No. 0-939773-01-5.

Basic Ship Theory (two volumes), K.J. Rawson and E.C. Tupper, published by Longman,London and New York, Second Edition, 1976, ISBN No. 0-582-44523-X and 0-582-44524-8.