The prediction of the ship's maneuverability in the design stage

ABSTRACT

THE PREDICTION OF THE SHIP'S MANEUVERABILITY

IN THE DESIGN STAGE

By: Dr. J.P. Hooft, member and Dr. U. Nienhuis, MARIN, Wageningen TECHNISCHE _UNIVERSITEIT LabOratoijurn voOt

8homecta

tchet Mekelweg _{2, 2628 CD Delft} TeL O15.78

This paper contains the evaluation of the ship's maneuverability in the design stage during, which

the underwater hull form and the äppendages are to be determined. For this purpose two basic aspects in the assessment of the ship's maneuverability are discerned, i.e. (1) the determination

of the influence of the hull form and of the appendages on the ship's hydrodynamic phenomena such as added mass and damping coefficients, and (2) the determination of the effect of the

hydrodynarnic aspects on the ship's maneuvering properties. This second aspect will be

elucidated by means of computer simulations which are applied for predicting the ship's

maneuverability in the design stage.

In the paper information will be given first about the hydrodynamic aspects by which the forces are generated on the hull in reaction to the ship's motions while maneuvering. After that the forces will be described which are caused by the rudders.

Having determined all relevant hydrodynamic quantities, then the ship's maneuverability can be

predicted by using computer simulations. Results from such simulations will be shown in.

1. INTRODUCTION

In the design stage of a ship all its dimensions and those of the appendages are considered with

respect to the consequences ori the various aspects of the ship's performance such as the

powering, the sealoeeping and the maneuverability. The result of the optimization of the design is a compromise of each of the components of the total performance.

In literature a lot of information is available to establish the influence of the hull forni on the ship's

powering (resistance and propulsion) and on the ship's dynamic behavior in a seaway (the

so-called seakeeping properties). Therefore in this paper the attention will be focused on the influence of the ship's hull form and its appendages on the maneuverability.

One of the serious problems in designing the hull form from a point of view of maneuverability

is caused by the uncertainty in specifying the maneuvering performances which have to be

satisfied. Depending on the ship's mission various types of maneuvers have to be performed each of which are difficult to define let alone to describe. In addition to this problem the assessment of a maneuver depends on the ship's type (defined by its form) which is affected largely by its mission and the environment in which it operates. Examples of completely different ship types which perform completely different maneuvering tasks are: freighters, tugboats, minesweepers, etc. In Section 2 the various aspects of the maneuvering performance will be discussed in detail.

Some of the above problems with respect to the definition of the ship's maneuverability in the design stage have been solved for a large category of ships by the 1993 1MO regulations; see

1MO 1993. In these regulations the following aspects were specified:

2. The limits within which the maneuvers have to be executed.

The 1MO regulations also require that in the design stage predictions are to be performed to prove

that the ship's maneuvering performance will comply with the 1MO criteria. lt is required also that

the predictions have to be validated by the results of sea thäls with the new ship. In this respect still some serious pròblems exist about (1) the reliability of the prediction method, and (2) the accuracy of the sea trial results. These two problems will b elucidated as follows:

Reliability of the, prediction method

Various types of prediction methods exist such as e.g.

extrapolation from past experience; in this method the maneuvering characteristics, are

predicted by means of empirical descriptions between the maneuvering properties measured

during sea trials and the ship's dimensions;

computer simulations using'hydrodynamic cOefficients derived from an empirical relation. with

the ship's dimensions;

experiments with self propelled ship models;

computér simulations using hydrodynamic coefficients derived from experiments with ship models restrained to a towing carriage (often referred to as PMM tests).

The reliability of a prediction method depends mainly on the following two items:

The method is defined to be reliable if the prediction result has proven to be in agreement with resUlts from other (mainly experimental) methods to ascertain' the ship's maneuverability,

especially so from sea trials.

lt shòuld be realized that there will always be some error bandwidth in the results of each of these prediction methods. That means that small changes in the input parameters will shov

a variation in the result of the method. The method is defined to be reliable if also the

The accuracy of the sea trials

lt is assumed that it wiN be possible nowadays to measure the various maneuvering properties (rudder angle, ship's position, speed and rate of turning) rather accurately. However the recording

of these items and of the conditions of the ship and the environment are often less

conscientiously performed. This can be indicated by the foliowing examples of problems in

recording precisely the needed information;

The ship's behavior before the start of a test maneuver (e.g a turning circle or zig-zag

maneuver) can have a large impact on the maneUver considered.

lt is therefore recommended to record the ship's behavior well in advance of the official beginning of the maneuver.

The ship's maneuverability will be influenced largely by its loading condition. Mostly only the

draft and the trim will be recorded. HoWever for various types of ships also the metacenter

height will affect the maneuverability. Therefore this item should also' be taken into

consideration.

The accuracy of the rudder angle and the ship's course angle should be given prime

attention. For an optimal recording of these items elaborate calibrations should be carried out.

The enviionmental conditions can have a significant influence on the tship's maneuvering properties. These conditions should therefore be recorded appropriately.

Even when due attention ¡s given to thè recording of all aspects relevant to maneuvering tests then still some degree of inaccuracy in the results of the sea trials have to be accepted because of the fact that the tests can never be executed under loo % ideal circumstances. This means that when considering the hypothetical case of, a large number of identical tests under identical circumstances then the measurements will be defined to be accurate ¡f and only if the deviation in the results óf all tests remains relatively small (e.g. 5%).'

BASIC MANEUVERING PROPERTIES

When considering the ship's maneuverability as an open loop system then the following aspects can be discerned:

1. The attainability by which the inherent maneuvering performance of the ship is defined. This

aspect describes the complete set of maneuvers that can be performed by all possible rudder

executions for all environmental conditions. Mostly one only defines the attainability for a subset of conditions such as for example the maneuverability in deep water at the service speed.

2.. The (coUrse) stability according tö the following definition; see also FIgure 1: "At a constant position of the steering systems (rudders, thrusters, etc.) the ship is defined to be course stable if after some short disturbance it will resume the original maneuver without any use of

the steering means". Mostly only the stability on a straight course is considered with the

rudder in the equilibrium position.

The sensitivity according to the following definilion: "The sensitivity S of the ship's

maneuverability to external variations is defined by the ratiò of the relative change óT/T of

the maneuvering performance T in comparison to the relative disturbance &/a of any

parameter a that has an effect on the maneuverability".

COURSE StASLE

Rate at turning

COURSE UNST.ABLE

Rate ot turning r

N

_\

Fig. 1. Definition of the course stability from the results of the spiral test.

width aihysteresis 4

height öt hysteresis

Table 1. 1MO maneuvering criteria for ships longer than loo m

From the turning circle maneuver:

*

TURNING ABILITY AT 35 ÓEG RUDDER ANGLE

- advance < 4.5 L

-

tactical diameter < 5 L

*

COURSE INITIATING ABILITY AT 10 DEG RUDDER ANGLE

- travelled distance < 2.5 L

at lo deg change of heading

From the zig-zag maneuver:

*

COURSE CHECKING ABILITY

-

lo/lo Z-maneuver

first overshoot < lo deg

if Lpp/U < i O sec

first overshoot < 20 deg if LpdU.> 30 sec

first overshoot: (5 + ½*L/U)

for

10 sec < LWU <30 sec

second execute <first overshoot + 15 deg

20/20 maneuver

first overshoot < 25 deg

*

STOPPING ABILITY track reach < 15

The attainability is öonsidered ¡n order to ascertain the area that is needed by the ship while executing its mission which consists for instance of accelerating, course keeping (and sometimes

also track keeping), course changing, course checking, stopping, etc.

För the assessment of the maneuverability of a seagoing vessel longer than 100 m the 1MO 1993

i

The ability to change the heading of the ship by mean.of the available steering systems: the so-called "course changing ability" of thè ship. This aibillty is determined from the first part of

the turning circle maneuver (see Figure 2) at 35 deg and 10 deg rudder angle. lt will be

obvious that the safety of the ship is good if it is possIble to initiatô rapidly a change of

heading in order to avoid a suddén obstacle.

The ability to bring the ship on a straight course from the condition in which it is turning: the so-called "course chécking ability". This ability is deterrn med by the overshoot angle and overshoot time in the zig-zag maneuver; see Figure 3. In case that the ship is turning and will

continue to turn for some time in the same direction after the rudder has been put. in the opposite direction then it will be obvious that one considers the ship to be out of coñtrol if the

time of vershoot is too large. lñ that òase thé ship's rnaneuver during such an excessive overshoót time will seem to proceed independent of thé rudder action. Therefore, it will be

beneficiai for the ship's safety when it càn be kept under control properly by responding

quickly to the steering means without too much overshOot.

u C 'U Tactical diameter o Track of centro at Position et O

at rudder execute reference O

Dritt angle 6 i st Second A overshoot I añ9ló

'I

Period 2nd 3rd

First overshoot angie First overshoot lime

angle c

Time

Ruddorangle 6

Second overshoot Urne

4th rudder-execute

Various aspects of the ships maneuvering performance such as the ucourse keoping, the "course

changing" and the "çourse checkirig are influenced significantly by the ship's course stability according to the above given definition:

- _{Course keeping: According to the definition a course unstable ship with the rudder in the}

equilibrium position wiH immediately start to turn after any disturbance. Therefore, in order to keep the ship on a straight course continuous rudder actions are required to. correct for the continuous disturbances. A course stable ship with the rudder ¡n the equilibrium position will not continue to turn after a short limited disturbance. However, after the disturbance has

vanished the course of the ship will have been altere4. To restore the course deviation

continuous rudder actions are also needed to keep a course stable ship on a straight course.

No general conclusions can be drawn about the difference in rudder activity in the course keeping of a stable ship and that of a course unstable ship.

Course chanQing: In case of a course stable s.hip it will be m'ore difficult to introduce ¿ change

of course. Large rudder forces have to be applied to generate an adequate rate of turning frörn..a straight course. However, in case of a course unstable ship the ship will start to deviate from a straight course, easily at appreciably high rates of turning with only relatively small rudder forces.

Course checking: In case of a course stable ship the ship will have a natural tendency to stop turning and to resume to sail along a straight course in the absence of any disturbance. Therefore a course stable ship will have a good "course checking ability" irrespective o the rudder effectiveness. lt should be noted however that sometimes larger rudders are required

to rnak.e the ship course stable; see Section 6. In case of a course unstable ship it will be quite cumbersome to stop the turning rate once the ship has started to turn. Therefore, an insufficient course checking ability is mostly caused primarily by the course instability for two

will be required by the reversed rudder angle ¡n order to stop the ship's turning and bring her

on à straight course. Large rudders will be required in this case (1) to make the ship course

stable and (2) to achieve an acceptable course checking ability". If the ship is course

unstable while large rudders háve already been applied then there are two reasons to apply stabilizing fins instead of further increasing the rudders, Le.

with larger rudders the turning rate of the ship might become unacceptably large by which

it will become even more difficult to stop the turning of the ship, and

during the deceleration procedure the rudder effectiveness might be reduced drastically while fins will Still remain appropriately effective (see also Section 6).

Taking into account the above consideratjons then it sometimes happens that the maneuverability

of a course unstable ship may still be acceptable if the following two criteria are satisfied; see e.g.

the 1MO 1991 document:

The rate of turning at the equilibrium rudder position should remain within acceptable limits; see e.g. ITTC-'87. Such a limit is for instance about 30% of the turning rate at the maximum rudder angle. From Figure 1 it is seen that the rate of turning at the equilibrium rudder position corresponds to half the height of the hysteresis in the relation r(6) betweèn the

turning rate and the rudder angle if the ship is course unstable.

-The tüming at the equilibrium rudder position should be stopped by means of opposite rudder

angles of limited amount; see e.g. Nomoto (1977). Such a limit is for instance a rudder angle

of 5 degrees. From Figure 1 it is Seen that the rudder angle at which the turning can be reversed corresponds to half the width of the hysteresis ¡n the relation r(8) between the turning rate and the rudder angle if the ship ¡s course unstable.;

If these two criteria (which have not beén incorporated by the 1MO) are satisfied then it may happen that this typé of course unstable ship falls in the category of good controllable ships. lt is difficult to explain theoretically why such a ship is considered to be good còntroflable. However

from a practical point of view it is assumed that a ship with a limited course instability will respond

easily to any outside force and therefore also to any rudder action In Sections 3 and 4 a

derivation will be given about the hydrodynamic origin of the course stability..

In order to assess the ship's maneuverability aiso the sensitivity to variations according to the earlier given definition has tobe considered in additiòn to the above shown aspects of attainability

and stability. Examples of variable parameters which may affect the maneuverability are: - steering devices

- loading condition of the ship

- environmental conditions.

For a proper assessment of a system it is required that it is quite insensitive to variations of all kinds of nature. A system is considered more reliable when its sensitivity to variations of the influencing parameters is small. A low sensitivity, of the maneuverability of a ship will make, the maneuverability of that ship much more acceptable because of the increased reliability to the ship operator and to the certifying authority.

With respect to the certification of the ship's maneuverability:

If the maneuverability ¡s very sensitive to external disturbances then this will cause large

variations in the maneuvering performance measured during various trials Under nearly equal circumstances. This means that if the maneuverabIity is sensitive to the circumstances then the

maneuverability can not be measured with great accuracy. In that case orle will not only find that

there is no coherency in the trial results but also that the agreement will be small between the

full scale trial results and the predictions from eithér model tests or computer simulations.

With respect to the controllability of the ship:

If the maneuverability is insensitive to external disturbances then the ship's operator (either captain or pilot) will remain confident about the consequences of his contrôl actions. In that case

no continuous corrections have t be applied to realize the maneuver as anticipated.

For a large category of ships the maneuverability will be notably sensitive to variations in all kinds

of parameters. Two examples öf such sensitivities are:

- The maneuverability of containerships may vary with the metacenter height; see e.g. Oltmann

(1993). If the standard maneuvers were performed at a large MG. then the maneuverability may satisfy the 1MO criteria. However, it may then occur that the maneuverability, will be unacceptable in the normal operation condition at which the MG value will be much smaller.

The maneuverability may be influenced significantly by the trim of the. ship; seO e.g. Kijima (1993) whø presents some results frOm 10/10 and 20/20 zig-zag maneuvers as a function of

the trim conditiòn. lt is strongly recommended to determine the trim as accurately as possible.

In general the sensitivity of the maneuverability is independerit of the course stability. However, it appears that the maneuverability of long slender ships -which are course stable- are quite insensitive to disturbances. On thé opposite, the maneuverability of short fat ships -which are course unstable- appear to be quite sensitive to disturbances.

MATHEMATICAL.DESCRIPTION OF THE SHIPS MANEUVERABILITY

In

this section the set-up Will b

discussed of the mathematical model describing the

hydrodynamic phenomena which influence the ship's maneuverability. Such a mathematiçai model is used in computer programs by means of which the ship's maneuverability can be simulated; see e.g. Barr (1993).

When all relevant phenomena affecting the ship's maneuverability have to be taken into account

then a quite extensive mathematical model will be required. However, from a practical point of view, the mathematical modél should be kept as simple as possible. Generally two criteria are regarded to establish the extent of a. mathematical model with respect t the number of coupled

motions (degrees óf freedom) and the range of motions for Which the equations are still valid. These criteria are: -.

The availability of accurate and extensive information about the hydrodynamic aspects

influencing the ship's behavior; only simple mathematical models (e.g. linear models) can be

arranged When little is known about the effect to be simulated.

Theôomplexity of the problem. to be solved by means of the simulation. The mathematical model can be kept simple if only rough descriptions of the ship's behavior (e.g. the initial stability or the controllability on a straight course) are needed or if. only rough indications are

required about the influence of some environmental óffect on the ship's maneuverability.

The simplest mathematical description of the ship's maneuverability was originally presented by Nomoto (1957) and consists of a single differential equation describing only the ship's turning

ability. Davidson and Schiff (1946) described the drifting and turning performance by a two

of the drifting and the turning on the longitudinal force can be so significant that the ship's Speed

in a turn might decrease to only 30% of the approach speed. This three degrees of freedom system appears to be quite eff ective to describe the maneuvèririg performance of a large category

of ships; see e.g. Norrbin (1971), ¡noue (1981), Ankudinov (1993) and many others. Ships with small MG values will roll during the maneuvering ¡ri the horizontal plane by which the surge and sway force and the yawing moment are affected arid therefore also the ship's maneuverability.

In that case the ship's maneuvering behavior should be described, by a set of four coupled

equations of motion; see e.g. Eda. (1 980), Hirano (1980) and Oltmann(1 993).

When also the ship's sinkage and trim variations affect th rudder force and the hull reaction forces while maneuvering (such as e.g. for SWATH ships) then the ship's maneuverability should

be déscribed by the complete set of six coupled equations of motions. This 6 DOF dscription is also required fór the description of the ship's maneuverability in waves; see e.g. Ankudinov (1983) and Hooft and Pieffers (1988).

Inmost time domain computer simulations a conti.nuoùs evaluation is perfòrmed of the dynamic variation of the various motion componentS. These motions are basically derived from Newton's

differential equations of motion; see e.g. Hóoft (1986):

dt2

IMI-1.

in which:

= à(t) = acceleration of the generalized motion § as a function of time

= (x,y,z.,4,O,)' = the generalized motiOn È _{= (X,Y,Z,K,M,N) = the generalized force}

1Ml =massmatrix

t

=time.

The force and mass in relation (3.1) depend on many factors such as the position s(t) and velocity

y ( ds/dt) óf. the ship, the actions of the steering devices, the environmental conditions, etc. lt Will be ôbvious. that all these factors may vary in time. Therefore proper knwlede abäut the relation between the influencing factors and the resulting force or mass components will bé required at any moment of time for an accurate derivation of the momentaty accelerations in Order

to reach a reliable simulation of the ship's performance.

When neglecting the inflUence of the heel angle then the description of the shIp's maneuverability

in still water follows from the following reduction of equation (3.1.):

These differential equations of the motions are defined relative to the ship fixed coordinate system

through th ship's center. of gravity. The force F (Xt,Y,N1), is partiy dependent on the ship's

acceleration. This part is, defined by the added mass coefficients m

= -X, m

=

m,

-Y, m. = -N and my,. = -N. lt is assumed that these coefficients remain independent ol other factors such as the ship's position and velocity or the environmental conditions of wind, waves and current.

In general the added mass coefficients can be predicted rather accurately. In the Initiai design stage empirical methods are availàble to determine these coefficients as a function of the main dimensions of the ship; see e.g. Motora (1960), Soding (1982) and hIC (1984). In a further stage

of the design these coefficients can be determined more accurately by means of two-dimensiónal.

m.(du/dt -r.v) =X

(3.2)

m. (dv/dt + r.u) .= Yt (3.3)

mass coefficients very accurately. This is due to the fact that the results of the computations will

not be influenced very much by variations in the amount of the added mass coefficients. From these considerations ¡t is concluded that no further attention is needed here to the derivation of the added mass coefficients.

In the case of constant added mass coefficients then equations (3.2) to (3.4) are rewritten as follows:

Thèse equations dèscribe the generation of the ship's motions by the generalized force

F (X,Y,N).

It consists of (1) the excitation force by the., rudder, the propeller and the

environmental phenomena such as wind, waves and current in addition to (2) the hydrodynamic reaction (damping) force on the hull. For a proper analysis of each of the above aspects on the ship's maneuverability during the design process use is made of the modular description:

-(X,Y,N) 'IH

+ +FR +FE . (3.8)

with the following notation:

FH - being the hull forces; see Sections 4 and 5 F - being the forces excited by the propeller(s)

FR - being thé forces excited by the rudder(s); see Sections 6 and 7 FE - being the forces generated by environmental phenomena.

With the equations (3.5) to (3.7) the simulation statts at an. initial position of the ship in which the

externalforce F(t1=O) is known. At this moment of time the acceleration a. (du/dt,dv/dt,dr/dt) is

(m + m)du/dt - m.r.v + X

(3.5)

(m + m)dv/dt + m.dr/dt = -m.r.0 +'(

(3.6)

determined from the solution of the above coupled equatiOns. From an integration one finds the velocity components u, v and r along the ship fixed coordinate System through the center of gravity:

u(t) = u(t-O) + j'.4t

(3.9)

v(t) = v(t»O) +

-th

(3,10)

r(t) r(t'O)

_{+ f..}

dt

(3.11)

When assuming that all force components are described in some reference point O (often taken

amidships) then the excitation force F(t2=t) at the next moment of time is determined as follows:

First the velocitycomponents u0 = u and y0= v-x.r are established in the reference point O with

XG being the distance of the center of gravity ¡n front of the reference point O. With these

velocities the force F0 ¡n the reference point O is determined using the formulations that hold for

this point. The fórce F in the center of gravity is then determined by:

XX0;

Y=Y0;

N =N0 -X0.YQ

(3.12)

With these forces at the next time step it will be possible to determine the accelerations at this

moment of time by solving the equations (3.5) to (3.7) again. And so the Whole process is

continued fOr äs long a period of time as is desired to complete a specific maheuver.

The track of the ship's path is described in an earth fixed system of cOordiñates XE and YE In this reference system the ship's heading j, (course angle) relative to XE ¡S determined at each moment

of time in the simulation by a continuous integration of the ship's rate of turn r(t) that was

The ship's velocity components in the earth fixOd coordinate system can now be transformed from

the ship fixed velocity components by:

UE «u.cos(ji) -v.sin()

(3.14)

-v.cos() +u.sin(w)

(3.15)

Integration of these velocities will provide the ship's position at each moment in the earth fixed system of coordinates:

XE(t) - XE(t"O) + fUE(t) dt (3.16)

yE(t) YE(t"O) + fVE(t) dt (3.17)

For the assessment of the ship's course stability use is made of equations (3.6) and 3.7) i.n which

the lateral force Y and yawing moment N are described by the following Iinearizations. Equation (3.5) can be neglected while assuming that the forward speed remains virtually unchanged by the

small deviations of the side velocity y and turning rate r.

(m +m)dv/dt + m,.dr/dt

(''ur-m).tIj + Y.u.v

(3.18)

mvr.dv/dt + ('

+ m.).dr/dt = N1.u.r +

(3.19)

Without the use of the rudder the development of the motions y and r after some small

disturbance will depend on the. following criteria. The ship will be course stable if the motions will

extinguish after the disturbance is vanished. This will be the case if the following relation holds:

course stable if:

or iñ a non-dimensiònal way:

course stable if:

N N "uv ("ur

N./

-<

Y'

(Y7' -m1)

(3.20) (3.21)

in which the forces have been made non-dimensional by dividig by (b.5pLTU2) while the

moments have been dividòd by (Q.5pL,2TU2). The non-dimensional mass m' corresponds to

2.CB.BIL.

-The cenier of application of the lateral force coefficient Y is défihOd by the static stability lever x aheàd of the center of gravity while the center of application of the lateral foçce coefficient

(Yam) is defined by the damping lever Xr ahead of the conter of gravity:

x,

N,/Y(J,

and Xr Nur/(Yur - m) (3.22)

In Figures 4 and 5 the static stability lever XHV respectively the damping lever XHr for the bare hull

are shown as a function of the ship's main parameters LIB, B/T and block coefficient CB for the even keel coñdition.

The difference between the damping lever Xr and static stability lever x is defined by the dynamic stability lever 1; see e.g. Rosemàn (1987):

dynamic stability lever: = X - X (3.23)

With these distances the relations (3.20) or (3.21) result into:

course stable if: or 'd' > 0 (3.24)

in which the distances have been made non-dimensional by dividing by the ship's length L Equation (3.24) leads to the conclusiòn that the ship is course stable if the center of application

of the force Y(v) due to drifting lies aft of the center of application of the apparent force

UB = 5 B/T- 2.5 B/T3.5 -0.6 0.7 0.8 Ca LIB = 6 0.9 I I 0.6 0.7 0.8 CB

Fig. 4. Influence of the. main parameters of Fig. 5. Influence of the main parameters of the bare hull in the even keel conditiOn the bare hull in the even keel condition

on the static stability lever XHV'. on the damping lever XH/.

0.9 0.6 _0.7 CB 0.8

,,.-"

/

0.9

/

/

/

1/

B/T-4.5 X 0.6 0.5 Hr 0.4 0.3 0.2 0.6 '-s' .-0.7 r 0.8 0.9 L/B = T BIT-25 CB

L/B =7

8/7=2.5 -B/T - 3.5 ---s- -s -s- _-.5-

-s-

_-s-

BIT 4.5 s

-I.

I___ 0.6 - ...Q,_ -. 0.8 0.9 0.6 0.7 0.8 CB 0.9 C9

L/B =5

0.5 0.4 _{B/T 2.5} 0.3 _{B/T -3.5} 0.2 B/T-4.5 0.5 B/T-4.5 0.4 I I I 0.6 XHV B/T2.5 0.6 J 0.5 XHr BIT 4.5 0.4 0.3 0.2 0.6

Pt

XHv 0.5 0.4 0.6

Î

XH, 0.5 04

Fig. 6. Schematic review of the forces on a çourse stable ship generated by the drift velOcity v turning rate r and ruddèr angle &

From equations (320) and (322) it is seen that the mass m of the ship has a significant influence

on the fact whether the ship will be course stable or not; see e.g. Figure 7. A slender ship (small

m') with small block coefficient and large length/beam ratio will tend to be more course stable than a short full ship (large m) irrespective of the hydrodynamic characteristics of the ship

course ùnstabie 1.0 -1.0 IJT-23.0 0.15 course stable

Fig. 7. nfluence of the ship's parameters Lir and m' = 2.C8.B/L on. the course stability of a bare

hull on even keel; from Jacobs (1964.).

L! 18.73 0.25

4. LINEAR HYDRÓDYNAMIC REACTION FORCES ON A BARE HULL

In reaction to the ship's motions hydrodynamic fOrces are generated on the hull of the ship and on itS appendages. In this section only the reaction forces on the bare hull will be discussed as a fünction of the ship's velocity components; the so-called damping forces. In general these forces

are described by a subdivision into linear effects (such as the forces in an Ideal" fluid Which can

be described by potentials) and non-linear effects (such as resistance forces in a viscous fluid). Experimentally the linear contributions are determined for small deviations from the equilibrium

position such as for small angles of attack at f rward speed of the ship. The non-linear

contributiôn is then derived from tests at larger angles of attack while assuming that the linear coefficients remain constant at the larger angles of attack.

In the initial design stage usually no model test results are available from which the hydrodynamic

damping forces can be derived. One then cän se computer simulations only if the hydrodynaniic forces cari be established otherwise.

Nowadays it is not'yet possible to determine the hydrodynaniic maneuvering forces analytically. Therefore, empirical methods have been developed by means of which in the initial design stage

the hydrodynamic coefficients can be éstirnated as a function of the ship's dimensions; see e.g. moue (1981), Kijima (1993), Ankudinov (1993) and others.

In empirical methods the linear hydródynamic coefficients are described rather accurately as a

function, of only a few aspects of the ship's dimensions. Two reasons can be given for the

achièved accuracy:

1. 'The linear hydrodynamic coefficients are most probably rather independent of local variations

in the ship's hull form (at the bow and/or the stem).

experimentally. This means that an acceptable level of: confidence has been achieved in the

description of the coefficients as a function of the main ship parameters.

lt appears that the non-linear contribution of the hydrodynamic characteristics can only roughly

be estimated by empirical methods. This is very inconvenient when tight turns have to be

predicted by means of computer simulations. In such maneuvers the non-linear contributions play a significant role. Three reasons are mostly giveh for this unfavorable aspect:

The non-linear hydrodynamic coefficients are rather sensitive to local form parameters of the ship. This means that much more information is reqUired as a function of the larger number of parameters.

Only for a limited number of ships the non-linear hydrodynamic coefficients have been

determined experimentally. This means that only a limited level of confidence has been achieved in the description of the coefficients as a function of the ship parameters.

Often only the hydrodynamic coefficients have been published without the actual model, test

results. Some of the authors use quadratic non-linear coefficients while others apply tertiary

non-linear coefficients. In this way the validity of a presented non-linear coefficient is limited and can not be compared with the corresponding coefficient for another hull form.

lt was thought that a better description of the non-linear component of the lateral force coUld be

achieved if one couÈd establish the local non-linear force component instead of the total force on

the ship; see Fedyayevski (1964). This local non-linear component is defined by the local cross flow drag coefficient; see e.g. Sharma (1982) and Hooft (1987 and 1993).

In the next section a description is given about the non-linear hydrodynamic component of the lateral hull force as derived from experiments with segmented models; see Matsumoto and

9 some examples of' the results of such experiments are 'shown in Which the transverse fOrces

are shown in a non-dimensional form according to:

Y1 «Y/(Ó.5pSU2)

(4.1)

in which V is the lateral force on the n-th of the N segments while Sn is the lateral area of the segment.

Pure drifting (r=O)

First the hull damping forces are considered in reaction t the ship's drift velocity y. From the

measurements with segmented models presented by Matsumoto and Suemitsu (1983) and

Beukelman (1988) the following results are derived. '

-lt ¡s assumed that the non-dimensional local lateral force Y,1 as a function of the drift, angle 13 can

be described by means of a linear coefficient Cy and a non-linear coefficient CD:

- Cy(n) .cos2(13).sin(13) - CD(n).sin(13). Isin(13) I (4.2)

in which Cy(n) is the local linear coefflcient and CD(n) the local cross flow drag coefficient.

lt can be shown that in equation (4.2) the linear contribution depends on the longitudinal velocity component u=U.cos(13) and not on thé resultant velocity U. This means that for zero forward speed (uO) no linear contribution in the lateral force exists.

In the application f equation (4.2) Use 'is made of the assutflpUøn that the linear coefficient Cy

is independent of the drift angle 13. This means that it is accepted that the local cross flow drag coefficient CD is a function of the drift angle 13. Various physical arguments can be given to prove

Y4, -0.4 -08 o -0.4 o Y10' -0.4 -0.8 o -08 o 10 o

wOoo

_a

lo 15 15 20 o Y2' -04 -0.8 o -0.4 -0.8 o Y7, -0.4 -0.8 o "8' -04 -08 -o a o 20

o'o

D

$3 o°o

_D o o o 5 lo 15 o 5 10 _f3 15 20 i ÖDO o jJ Q1J0 o

u

o lo l 0 15 20

woo

o _ID

o

o o D 13 15 20 lo 15 20

0.2 04 V 06 o 0.2 0.4

C

T oè

Fig. 9. Influence of the non-dimensional rate of tmingy on thé laterá! forces meàsured on each

of the la

segments; see MatsumotO ánd Suemitsu (1983).

OE25 o Y1, -0.25 -0.5 0.25 o -0.25 -05 25 O Tanker D Container 0.25 o -025 -05 025 Y7. -025 -05 0.25 o O o 0.2 04 _Y 06 o 0.2 0.4 _Y 0.6 o o 0.2 0.4 _Y 0.6 o 0.2 OE4 y 0.6 o Y3, -0.25 -0.5 0.25 o Y4. -0.25 -0.5 0.25 o

o

o 'Y'8' -0.25 0.25 o Y9. -0.25 -05 0:25 B Y1 o 0.2 0.4 _'f 0.6 o 0.2 0.4 0:6

O41O

O O D 0.2 0.4 0.6 o 0.2 OE4

'f

0.5 o D O O o vio. o Y&. -0.25 -0.25 -05

From the test results on each segment as presented in Figure 8 one determines the local linear coefficient Cy(n) from the, range of small drift angles . Once this value of Cy(n) has been

established for each segment then the cross flow drag coefficient can de derived from the

measurements at higher drift angles.

From the above descriptions it is found that both the linear coefficient and the. cross flow drag coefficient vasies over the length of the shIp. In Figure 1 Oa the longitudinal distribution is given. of the linear coefficient Cy describing the linear contribution to the lateral force in reaction to a drift angle 13. In Table 2 the. main characteñstiçs are presented of the ships of which the Cy0 distribution is shown in Figura lOa.

Theoretically (see e.g. Jones (1946) and Jacobs (1964)) it'can be shown that the local linear force

per unit length Y(13) is determined by the instantaneous apparent acòeleration of the lOcal lateral

added mass of water alongside the ship:

Y(13) - -u.v.m

. (4.3)

with m being the lateral added mass of water per unit length of the ship and being the distance of the cross section from the forward perpendicular. The derivatives Y and are

deflned by: .

Y - dY/dT

m

_{= dm/d}

(4.4)

When the n-th segment ranges from '(n) behind the forward perpendicular to 'a(h1) then one finds

from equatión (4.3) that the lateral force on this segment is theoretically determined by:

(n) Y1,(13)

Table 2. Non-dimensional characteristics of the bare hull of two types of ships

course uñstable course unstable Ship 1: Container Ship 2: Tanker

Dimensions LIB 6.90Ô 5.730 LIT 20.550 15.310 BIT 2.979 2.673 CB 0.562 0.825 m' = 2 CB B/L 0.1629 0.2880 Trim 0.33° 0.0° Position of CG (forward of midships) -1.80% +2.80% Hydrodynamics

Y' = YJ(0.5 p

T) -0.2402 -0.2816

N' = NJ(0.5 p L,2 T)

-0.08707 -0.1325 XHV' = Xy/Lpp = NV/YV 0.3625 0.4705

= YJ(0.5 p

L2 T)

0.03732 0.0666 N,'

_{= N/(0.5 p}

L3 T)

-0.03227 -0.04325 XH,' = xHLpp = Nr'/(Yr'm') 0.2570 0.1953 'Hd = XHr' - XHV' -0.1055 -0.2752

AP 4 FP

0.40-0.30- m1

s' 0.20 0.10- Container, - Tanker 00

Fig. lOa. Longitudinal distributiön of the linear Fig. lOb. Longitudinal distribution of ifie lateral lateral force coefficient due to added mass..

drifting.

In Figure 1 Ob the longitudinal distribution of the non-dimensional lateral added mass

m/(O.5pLT) is preseñted of the two ships of which the linear coefficients are shown in Figure lOa. Comparing the results in Figures lOa and lOb it is seen that the tendencies in these figures correspond with the theory in equation (4.3), such as:

At the forward perpendicular there is a stepwise increment of the lateral added mass (large

positive value of m.,); see Figure lOb. This corresponds to the relatively large negative

measured lateral force coefficient on the first segment at the bow; see Figure lOa.

At the even keel condition the lateral added mass at the fore ship increases gradually

(moderate positive values of m) with increasing distance from the forward perpendicuiar which corresponds to a measuréd moderate negative coefficient Cy at the fore ship.

At the even keel condition thé lateral added mass remains approximately constant (m is zero) over the parallel midship while over this length of the ship a nearly zero value is

At the even keel condition the lateral added mass at the aft ship decreases gradually at increasing distance from the bow (rti is moderately negative) which corresponds to a moderate positive linear lateral force coefficient that was found from the experiments. It should be realized that a positive linear coefficient corresponds to a positive lateral force ¡n

reaction to a positive drift velocity y.

The resultant damping coefficients and NH follow from the summation of the lateral force coefficients Cy(n) on each of the segments:

Y'

- (Cy(n)S)/(LT)

and

NH'

_-

(4.6) At the even keel condition the positive reaction force in the aft part counteracts the negative reaction force in the fore ship and magnifies the negative yawing moment at the fore ship. Theoretically one would find that in an ideal fluid the resultant force coefficient _{H' would} be zero while the resultant yawing moment NH' would correspond to the theory of Munk

(1924).

At the even keel condition the diminished lateral äìè coefficient H' and the amplified

resultant yawing moment NH' causes a relatively large value of the distance xHV; see

equation (3.22). However, from equation (3.24) a small value of the distance x' is desired for the ship to be stable. In Figure 4 it can 'be seen in which way the static stability lever XHV'

¡s influenced by the 'ship's main parameters LiB, BIT and Ob.

In order to improve the course stability the static stability lever XHV' has to be reduced. This

can be realized by increasing the' absolute value of the hull force coefficient and

decreasing the absolute value of the hull moment coefficient NH'. This can be achieved by decreasing the positive reaction force coefficient Cy at the aft part of the ship while applying more added mass towards the stem (local higher values of' mfl), by which Im/ is reduced. Two examples of such applications are:

1. Elongation of a center skeg will result in larger added mass coefficients at the stèrn and thus of an increment of the absolute value of and a decrement of the 'absolute value

of NHP; see Figure 11. In the ¡TIC 1993 report of the Manoeuvring Cmmittee

information is given about the influence of the stem shape and skeg on the added mass at the stem.

2. At a positive blm (bow up) the added mass will increase more or less continuously

toWards the aft perpendicular (see Figure 1 2b) leading to a continuous negative vâlue of

the lateral force per segment over the whole length of the ship; see Êigure 12a.

A P.

N.

w (j .0 0.1 0.0 -0.1 -0.2 -0.3 -0.4 D, w u, o

Fig. 11. Influence of the center skg on the linear hydrodynamic coefficients accàrdìng to Jacobs (1964).

1.0

t

Cy -0.5 1.0 -1.5 .2.0 AP

'Ii

Positive trim Even keel Negative trim

4t;'

FP 0.10-0.0 AP 0.40-t Positive trim

-'

0.30-

-0.20-

m'

(bow out) ___. .

J

trim (bow in)

Even keel

Fig. 12a. InflUence of the trim on the lÖngitudilnal Fig. 12b Influence of the trim on the longitudinal

distribution of the linear lateral fOrce distribution of the lateral added mass.

coefficient Cy due to drifting.

Pre turning (v=O)

The hull damping forces are now considered in reaction to the ship's rate of turning r. In Figure 9 the measured lateral forces on the various segments are shown as a function of the räte of

turning. These data have been preseñted by Matsumoto äpd. Suemitsu (1983). From these

measured values the following results are derived.

lt is assumed that the non-dimensional local lateral force Y' (see equation (4.1)) on the n-th

segment as a function of the rate of turning r can be descríbd by means of a linear coefficient Cy.,(n) and a non-lineär coefficient. CD(n):

Y(y)'

= Cy7(n).y - CD(n).(xfl'.y). Ixn,. (4.7)

In the application of equation (4.7) use ¡s made of the assumption that the linear cbefficient

Cy.,(n) does not change at increasing rate of tUrning. This means that ¡t is accepted that the local

From the test results on each segment as presented in Figure 9 one determines the local linear

coefficient Cy7(n) at the range of small turning rates r. Once this value of Cy7(n) has been

established for each segment then. the cross flow drag coefficient on each segment can be derived frOm the measurements at higher turning rates.

From the above descriptions, it is found that both the linear coefficient and the cross flow drag coefficient varies over the length ,f the ship. In Figure 13a the longitudinal distribUtion, of the linear coefficient Cy7 is given by which the linear component of the lateral force is described in reaction t a 'turning rate r. The main characteristics of the ships, used in this figure are shown ¡n Table 2.

Theoretically (see e.g Jacobs (1964)) one finds that the local linear component of the lateral fOrce

coefficient per unit length Y(r) ¡s proportional to the local moment per unit length Ç of the lateral added mass of water alongside the ship:

Y(r)

_-u.r.i

(4.8)

with Ç being the moment Of the lateral added mass of water pet unit length of the ship m and the distance Of the cross section from the forward perpendicular1 The derivatives Y and Ç are defined by:

dYIdT = dÇ/d with iv = x. (4.9)

When the n-th segment ranges from (n) behind the forward perpendicular to(n) then one finds from equation (4.8) that the lateral force on this segment is theoretically determined by:

(n)

Y(r)

0.5 0.25 0.0 Cyy .0.25 -0.5 -0.75 -1.0

Fig. 13a. Longitudinal distribution of the lineat lateral force coefficient y.1, due to

turning. 0.15- 0.10- 0.05-0.0 .005- .0.10-.0.15

f

= x'rn AP

0

/

Container

- - - Tanker

Fig. 13b. Longitudinal distnbution of the mcnent

of the lateral added mass.

FP

In Figure 13b the longitudinal distribütiön of the non-dimensional moment Ç' of the lateral added mass is presented of the two ships of whjch the linear coefficiént. Cy.), is shown in Figure 13a.

Comparing the results in Figures 13a and 13b it is seen that the tendencies in these figures

correspond with the theory in equatjon (4.8), such as:

From Figure 1 3b it, is seeñ that a stepwise increment of the moment öf the lateral added mass exists at the forward perpendicular. This corresponds to the rather large negative lateral

force measured on the first most forward segment; see Figure 13a.

At the even keel condition the moment Ç decreases continuously behind the forward.

perpendicular which corresponds to the positive lateral fórces measured on all segments behind the first one.

- The resultant damping coefficients ''Hr' and NHr' follow from the summation of the lateral force coefficients Cy(n) on. each of the segments:

- (E

Cy7(n) Sn)/(LppT) and NHr' ( X11'. Cy.1(n) S)I(LT) (4.11)

At the even keel conditiön the ñegative linear fOrce coefficient Cy on the first segment and the poiti'ìe force coefficients on the successive segments cancel each other; see equation

(4.11). As a consequence one finds a relatively small, value for the hull linear coefficient ''Hr

which can be either positive or negative.

The above described distribution of Cy7 for the even keel condition will also lead to a

relatively small value of the resultant damping moment coefficient NH(' at the even keel condition. According to equation (3.22) a small moment coefficient NH,.' will result into a relative small value of the damping lever XH,.'. However, from equation (3.24) the value of the damping lever XHr' should be as large as possible for the ship to be course stable.

In Figure 5 the damping lever XH,.' is shòwn as a function of the ship's main parameters LIB,

B/T and block coefficient Cb at the even keel condition. In order to ascertain the ship's course

stability in the design stage it is helpful to consider the effect of the ship's main characteistics

on the damping lever XHr' in relation to the static stability lever XHV' presented in Figure 4.

In order to improve the course stability the damping lever XH,.' has to be increased by

increasing the reaction force Hr' -by which (YH('-m') decreases- and increasing the absôlute

value of the yaw moment NH,.'. This can be achieved by increasing the added mass towards

the stem by which the moment Ç of the added mass m decreases more rapidly. This phenomenon can be elucidated by elongation of a center skeg by Which the moment of the

added mass coethcients at the stern are enlarged. Thus y! will increase and also the

absolute value of Nr'; seeFigure 11.

From the above considerations the following conclusions can be drawn with respect to the linear aspects of the hIl damping' forces on a bare hull:

1. In Figures 4 and 5 the static stability lever XHV' respectively the damping lever XH,.' are presented as a function of the main parameters LIB, B/T and Cb of the ship in the even keel

condition. Using these figures in the initiai design stage a clear indication is obtained about the amount of the course instability of the bare hull. It is rernatk le that the values derived

decrease the course iñstàbilit') by changing the hil form locally. In Figure 15 the results are shown of tests with three stem forms shown in Figurö 14: pram form, moderate pram form and conventional stern form. lt is seen that there is hardly any influence of the stem form on the bare hull force for small values of the drift angle.

pramform afterbody

moderate pramform afterbody

convential afterbody

0.01 0.002 o -0.002 -0.004 -0.006 -0.008 moderate conventional 10 15 20 25 30 35 40 pratntorm moderate pramtorni.-.., conventional_e',., o 5 10 15 20 25 30 35 40 13

Fig. 15. Influence of the stem form shown in Figure 14 on the lateral force and yaw moment on the bare hull of a short full ship.

From the findings in Item lit can be assessed how much effort has to be put iñ

limiting the instability by increasing the local non-linear contribution of the lateral force; see Section 5 or

- _{reducing the course instability by applying stabilizing appendages -such as ruddòrs}

and/or fins- to the hull; see Section 6.

From Figure 5 ¡t is seen that the damping lever XHr' of a slender ship with large LIB and small

block coefficient C8 will be larger than that of a short full ship while from Figure 5 it appears

-0.01 Y..

-0.02 -0.03 -0.04

course unstable than the full bodied ship as a result of the fact that the dynamic stability lever

1Hd' (being xHr'-xHv'; see equation 3.23) will be more pósitive respectively less negative for the

slender ship. This result is also confirmed by the data presented in Tablé 2 for the

containership (CB = 0.562 and LJB = 6.9) in comparison to the tanker (LIB = 5.73 and

5. NON-LINEAR HYDRODYNAMIC REACTION FORCES

ON A BARE HULL

In the previous section the linearized damping forces on the bare hull have been evaluated.

For the description of the hull forces in reaction to large hull velocities such as drifting and

turning more insight is needed in the origin of the non-linear contributions to the damping

forces. For this purpose again the results of segmented model tests are of essential need; see

Matsumoto and Suemitsu (1983) and Beukelman (1988). From these tests the transverse

force on each segment is determined as a function of the drift velocity y and turning rate r in relation to the forward speed component u:

3 =atan(v/u) and

y =r.LIu

(5.1)

with being the drift angle and y the non-dimensional rate of turning. In Figures 8 and 9

some results of the segmented model tests are presented in which the transverse forces Y on the n-th element are shown in a non-dimensional form according to equation (4.1).

In the previous section it was shown how the linear coefficients Cy(n) and Cy.(n) on each

segment were derived from the tests at small drift angles and small turning rates respectively.

When assuming that the linear coefficients of each segment remain constant for the larger

velocities then the non-linear contribution can be derived from equations (4.2) and (4.7)

according to:

Y11(n)

-

Y'(measured)

- Cy(n).sin(l3).cos2() - Cy(n).y

(5.2)

The non-linear contribution Y1'(n) is described by the cross flow drag coefficient C0(n) accord-ing to:

-Y'(n)

C0(n)

- 0.5pSvIvI =

_vn,.Ivn'I.

with the following parameters on the n-th segment: C0(n) = IoOal cross flow drag. coefficient

Sn = lateral area of the segment = local lateral velocity ( v+x.r)

Y1(n) = non-linear contribution of the transverse force on the segment.

As it is assumed that the linear coefficients in e uation (5.2) remain constant it will be obvious that the cross flow drag coeffiòient C0(n) of each segment will vary with the velocity conditiOn of the hull: CD(n) E CD(n;,'y as will be elucidated ¡n the following.

For pure transverse motions (zero forward speed and thus 900 drift angle ) one often finds a rather smooth longitudir!ai distribution of the cross flow drag coefficient C0 which increases or

decreases towards the ships ends depending on the form of the ends: For fine ships with

sharp ends the CD value increases towards the ends while for full bodied ships the CD vaJue

decreases towards the ends; see e.g. Faltinsen (1990). In Figure 16 the distribution of. the

cross flow drag coefficient C0U3=9O°) is shown for a container vessel and a tanker; see

Matsumoto and Suemitsu (1983). It is assumed that the cross flow drag coefficient at the

mid-ships section will depend mainly on the local form of the underwater hull (ratio B/T and

mids..ps section coefficient CM) and to a lesser degree on the local Reynolds number; see

e.g. Hoerner (1965) and Kapsenberg (1989). .

11.0

CD 1.5 0.5 0.0 AP FP

Fig. 16. Longitudinal distribution of the cross flow drag coefficient at 900 drift ángle; from

Matsumoto and Suemitsu (1983).

During maneuvèring at the service speed one will find that the drjft. angles remain smaller than

about 300. For these smaller drift angles the local cross flòw drag coeffiôients have been

determined according to the derivation in equation (5.2). The results of this derivation are

presented in Figure 17. From the results in this figure the. following conclusions can be drawn:

Thé cross flow dra coeffident on the first segment from the bow is most próbably caused by thé bow wave. Therefore it is assumed that this value depends on the forward speed of the ship; see also the findings by Matsumoto and Suemitsu (1983).

Aside of the cross flow drag coefficient on the most forward segment(s) it is seen from the

results in Figure 17 that for small drift angles the values of ÖD increase ät larder diS-tances from thé bow until sâme maximUm value is attained after which Cd decreases a little. At: ¡ncteásfrg drift angles this curve of Cd over the ship's length moves forward.

CD 1.5 0.5 1.5 0.5 o AP AP FP FP CD 1.5 0.5 1.5 0.5 o AP AP =12° FP FP

Fig. 17. Longitudinal distribution of the cross flow drag coefficient at drift angles up till 20°;

frOm Beuke/man (1988).

Lpp/T = 22.81

Todd6O model, even keel;

Fn = 0.15 Lpp/T = 17.50

LppíT= 14.20

1h 13 CCD 0.5 o (5.4) AP FP p 200 AP FP

Fig. 18. Longitudinal distribution of the corrected cross flow drag coefficient C0 derived from

The value of CD() at a given cross section varies at increasing drift angles from the

values shown in Figure 17 to the values shown in Figure 16 for the 900 drift angle.

There-fore, the relation

is considered between 0Dn(l3900) at an arbitrary drift angle and

CDfl(P=9O°) at a drift angle of 900 for the various ship types and ship conditions. For this purpose use is made of the corrected drag coefficient Ccon(f3) which is defined by:

CD(9O)

CcD()

- CD(9O)

leading to the fact that at 900 drift angle the coefficient CCD equals unity over the whole length of the ship.

Combining the results in Figures 16 and 17 according to equation (5.4) will yield the results

N

g0

0 0 0

a-10 ' -30 ' e-50 e-70

\

_\

\

N1

cc0

AP. _0.5 FP.

4

Fig. 19. Schematic indication of the forward shift of the dis fribution of the corrected cross flow drag coefficient C0 at increasing drift angles !3 for small forward speed (Fn is small).

From the results of the corrected drag coefficient 0CD() in Figure 18 it is concluded that this coefficient depends on the longitudinal locatioh behind the ÊP and on the drift angÍe as

shown schematically in Figure 19.

lt is assumed that the local cross flow drag coeffiòient for any arbitrary hull form can be

derived from using equation (5.4) in which:

-. the "corrected cross flow coefficient1 C from Figure 19 ¡s taken invariable for any arbi-trary hUH form, and

the cioss-flow drag coefficient CD«3=900) at 900 drift angle being a function of the hull

form as is shown for example in Figure 16.

When applying this method for a short full bodied ship then an acceptable agreement is found

between the resUlts from the calculätions based on the longitudinal distribution of the cross

flow drag coefficient on the one side and the results from model tests on the other; see Figure 20.

0.01 O Model test - Calculated O o 0.001

to

N -0.002 -0.004 -0.006 -0.008 o o o O O Model test - Calculated o 15 30 45

0

60 75 90

Fig. 20. Lateral force and yawing moment on a short full bodied hull from calculations based on the cross flow drag distribution in comparison with model test results.

From the longitudinal distribution of the cross flow drag coefficient at the service speed (see

Figure 17) it is seen that the non-linear contribution of the local transverse force leads to a

stabilization of the ships maneuverability which is explained as follows:

Up till about 3Ø0 drift angle the lateral force H() on the hull increases more than linearly

with increasing drift angles; see e.g. Figure 21. However, from this figure it ¡s also seen

that the total yawing moment NH(J3) increases almost linearly with increasing drift angle because of the fact that the non-linear transverse force contribution applies mainly at the

aft part of the hull. The result of these two tendencies ¡s that the center of application of

the lateral force Y(v) is shifted backwards as x decreases at increasing drift angle:

XHV = NH(v)/YH(v) (5.5) 75 45 60 90 o I5 30

to

Y.. -0.02 -0.04 -0.06

o 5 10 15

.--*

20 o O Tanker D Container O 5 10 15 20

Fig. 21. Influence of drift angle oli the láteral force and yawing moment for two hull forms

without appendages; from Matsumotô and Suerhitsu (1983,).

At increasing rate of turning the lateral fOrce on the tta hull increases älmost linearly with increasing rate of turning; see Figure 22. While turning only (vRO), the. local látéral veÊocity

v(x) = x.r is maximal at both ends of the hull having opposite directions. The resultant

transverse

hull force YH(r) therefore increases more than linéarly With r if the C0 value at the

stem

is larger than at the bw and vice versa. When YH(r) increases linearly with r

then (Y-m') will remain cônstant at increasing rates of turning. At increasing rate of

turning the reacting hull momént NH(r) increases mOre than linearly because óf the fact that th oppOsite non-linear force components àt fore and aft ship will both augment the linear moment component; see Figure 22. The result of these two tendencies is that the center of application of the lateral force. (Y(r)-m.u.r) is shifted fôrwards as x1, increases at increasing rate of turning:

.NH(r)/(YH(r) -m.u.r.) . (5.6) -0.005

o

D -0.001 O

o

D D

o

-0.002

o

-0.0 10

o

-0.003 O -0.0 15 I I _{-0 004} I-0.002

L. o

0.005 H" O Tanker D Container

o

0.001 -0.001 0.000' D O Tanker D Container NH 01 -0.0005 O Tanker D Container o D o B D O D o _U -0.002 _D o -0.0011.... 0 -0.003 o _.0.0015}.I o o o -0.004 I I I t I .0.002 I I I I I I o 0.1 0.2 0.3 0.4 0.5 0.6 _{0 0.1 0.2 0.3 0.4 0.5}

Ts.

T s.

Fig. 22. Influence of non-dimensional turning rate y on the lateral force and yawing moment for two hull forms without appendages; from Matsumoto and Suemitsu (1983).

From the above it is seen that the static stability lever XHV decreases at increasing drift angle while the damping lever XHr increases at increasing turning rate. As a result of the non-linear

viscous contribution of the transverse hull force it is found that the dynamic stability lever

'Hd E XH(XHv will increase at increasing ship motions y and r.

Inì case that the ship is course unstable then it was found at negligibly small motions y and r

that the dynamic stability lever 'Hd is negative. From the above it is seen that at increasing

motions the dynamic stability lever will increase until the motions have reached some kind of equilibrium in which 'EId is zero at which the ship will continue to turn at some drift angle :

6. APPENDAGES IMPROVING THE COURSE STABILITY

The ship's maneuverability might be seriously deteriorated by the fact that the ship is very

course unstable. In that case it is observed for instance that the ship has a bad "course

checking ability" which follows from the large overshoot angles in the Z-maneuver. To improve

this type of bad maneuverability one tries in the first place to make the ship more course

stable (decreasing the còurse instability).

Improving the ship's stability can be realized by applying larger rudders (see e.g. Figures 23

and 24), fins and such at the stern of the ship. Basically the improvement is seen from the

following considerations. BARE HULL 1_0 o -1.0 -2.0

HULL WITH RUDDER AND PROPELLER

Fig. 23. Improving the course stability (more negative stability index a7) due tO the application of rudder nd propeller; from Jacobs (1964).

V

= atan(

-

-0.5 y

-rudder area as a percentage of the lateral area Lpp.T

Fig. 24. Improving the course stability (more negative stability index

) as a result of an

increasing rudder area; from Jacobs (1964).

In the neutral position of the rudder (rudder angle 6=0) the angle of incidênce of the flow 8H to- the -rudder, as a- consequence of the ship mötiön components u, y- arid r described

theoretically by: - -

-(6.1)

in which XR is the distance of the rudder relative to the center of reference. For small angles of incidence the effectiveness of the rudder or the fin is expressed by the coefflcient Y8 being positive as Will be shown in Section 7. One thus finds that due to the ship's motions the lateral

fOrce Yinduced by the rudderfollows from: -

-Y(ruddèr) -Y6 *3 - - (6.2)

From a combination of equations (6.1) and (6.2) -one fiñds the rudder force- YRU) due to the ship's drifting and YRy dúe to the ship's turning from (see also Figure 25):

2.0 1.0 O -2.0 LIB LIT

=

B/T CB = 7 18.75 2.68 0.6 o i Ó 2.0 3.0

Ht (r) - H (r)- mur

"H (r)

FIg. 25. Review of forceson the ship.

With these results one finds that:

1. With the rudder or fin the reactiOn force Y()' due to a drift angle iñcreases (more

nega-tive) because of the negative value of R'(); see also Figure 25:

H /

_'p

//

-

/

-

'JI

(6.4)

However, the reacting moment N()' due to drifting, decreases as the negative momeñt

NH()' is compensated by the positive moment NR( generated by the rudder:

N =N' +N' =NH' +O.5Y8'

(6.5)

Combining equations (6.4) and (6.5) shOws that the total reaction forcé n the hùll with

rudder Y(f3)' will shift backwards which means that the static stability lever x,' for the shjp with rudder Or fin will be smaller than that of the bare hull:

2. With the rudder or fin the reaction force Y(y)' = Y(y)'-m due to a turning rate decreases

(less negative) because of the positive value of _{R(7)'; see} also Figure 25:

-

(Y11'

IY./) -mt

'Ç4f/ +0.5V8t) -rn'

(6.7)

However, the reacting moment N(y)' due tò turning increaseS as the negative moment NH(y)' becomes more negative due to the negative moment _NR(y)' generated by the

rudder:

N./

-

N/ + ¿N./ N.

- 0.25

Y81 (6.8)

Combining equations (6.7) 'and (6.8) shows that. the total reaction force on the hull with rudder _Y(y)' will shift forwards which means that the damping lever xr' for the ship with

rudder or fin will be larger than that of the bare hull:

XI_

r

NH?' -0.25Y6

''Ht7' +0.5Y6

In Figures 26 and 27 two exaniles are given of the. influence of the rudder area on the linear hydrodynamic coefficients described in equations (6.4), (6.5), (6.7) and (6.8)

Combining equations (6.6) arid (6.9) one finds that the application of a rudder or a finj at the

stem will decrease the static 'stability lever and will increase the damping lever Xr'.. The application of a rudder or a' fin will therefore improve the course stability. The stabilizing ap-pendages for improving the course stability will be made so large as is needed to compensate for the instability of the bare huJl to such a degree that the instabilIty will become acceptable.

lt should be noted that from a pòint of view of maneuvering it is sometimes preferred to apply

0.1

-0.0

0.1

0.2

--0.3

-,

A.P.

Ni

Rudder area = 1.6 % LT

Fig. 26. Influence of the rudder on the linear hydrodynamic coefficients for a ship with a block coefficient of 0.6; from Jacobs (1964).

If the course instability of the bare hull is rather large then the ship will become quite course unstable fOr the condition in which the rudder has lost all its effecthieness (Y5'-40; see. Section

7) such as during stopping of the ship; see e.g. Yoshimura, (1993) who describes the

prob-lems which may occur during stopping of an unstable single screw ship by means of a CP

propeller. To overcome the problem of course instability during stopping one sometimes

applies fins and sometimes modifies the rudder configuration such that the rudder will remain effective dUring the stopping procedure.

A

0.1

-0.9 D w = D o-'Q. w Rudder area 1.6% LT base line

Fig. 27. Influence of the rudder on the linear hydrodynamic coefficients for a ship

with a block coefficient of 0.8; from Jacobs (1964).

LIB = 7

L/T = 18.75-B/T = 2.68 CB = 0.8

L/B = 7

LIT = 18.75 BIT 2.68 CB = 0.6

In case that the ship's response to the rudder is rather sensitive (a violent response of the

ship to the rudder) then it is not recommended to enlarge the rudder for decreasing the course instability. In that case the ship can only be made course stable by means of applying fins.

The results in Figures 26 and 27 show the effect of the rudder on the linear hydrodynarnic

coefficients. As indicated in the figures this effect applies to ships with a length/beam ratib of

L./B

7. However, for a short full bodied ship one may find that the rudder will have less

effect. In Figures 28 and 29 it is shown that for the = 5.48 hull of Figure 14 the rudder

will have n effect at all for the range of small motions. For the conventional hull form the

rudder does not have any effect for drift angles Up till about loo; see Figure 28. For the hull with the pram form stem one finds that the rudder does not have any effect on the transverse

force for drift angles up till about 10° and on the yawing moment up till 50 drift angles; see

Êigure 29. In his paper of 1993 Kose reports that he has tested hull forms for which in the

ballast condition the rudder did not have any stabilizing effect up till 150 drift angles.

For ¿Il these types of ships one finds that the application of the rudder did not improve the

course stability. If the ship is unacceptably course unstable then the stability can only be

improved by applying fins at thé sides of the stem where it is expected that the stabilizing

capacity (of the fins) will be more effective than near the center plane of the ship.

Also from the results by Róseman (1987) for full, bodied ships one finds that the rudder

stabilizing capacity is less effective than follows from the descriptions for óY' in equation

(6.4). ¿N' in equation (6.5),

in equation (6.7) and 4V; in equation (6.8). Roseman

gives for example values of betweén 0.6 and 0.685 for the ratio between and -'(5':