The manoeuvrability of ships as influenced by environment and human behaviour

:2tj _97ß. PUBL. NO. Li OF THE N.S.H.B. fomthejournal?fNaigatioh, Vol. 27, No 3, JUly 1974

Lab. y. Scheepsbouwkw'd

Tech&sche Hogeschool

Deift

THE MANUVRABILITY OF SHIPS

AS INFLUENCED BY ENVIRONMENT

AND HUMAN BEHAVIOUR

Dr.

P. Hooft

THE ROYAL INSTITUTE OF NAVIGATION

AT THE ROYAL. GEOGRAPHICAL SOITY

The Manoéuvrability of Ships

s

Influenced. by Environment and

Human Behaviour

Dr J. P. Hooft

This paper by the Vice-President of the Netherlands Ship Model -Baaih presents the results of a study performed onthe N.S.M.B. ship manuvring simulatur aimed at the development of cntena for the manuvrabthty of ships sailing in a current Six pilots navigated four types of 250,000 d w t tankers in sixteen different conditions of

current.

I. INTRODÚ(TIÖN. Ships have grpwn so large. that jt is no longer taken

for granted that they can be manuvred as required The problem can be

split up in two aspects, the ship's inherent manoeuvring ability, which is

mainly a technical problem,1.2 and the thariner's ability to manoeuvre

theship, which is mainly a psychological problem.3.4

Various authorities have thus become interested in establishing criteria for ship manuvrabihty which depend on the circumstances under which the ship has to sail, even ships which can normally be kept under perfect control may fail in difficült circumstaìices. This means that

the ship designer as well as the designer of waterways and harbours is intereste4 not only in criteria for ship manoeuvrabihty but also in the criteria for environmental conditions Apart from this, port operators,

shipowners and governmental authorities are interested in these criteria

since the safety of a ship depends not only on manoeuvrability and en 'rironmental conditions but also on the rules of the road, the regulation

of traffic, the skill of mariners, navigational aids, &c.

To acqufre sorne insight into this very complex problem of human be-haviour a study has been performed on the ship manoeuvrmg simulator of the Netherlands Ship Model Basin,5 on which mariners can steer a ship according to the real life situation This permits accurate determination of

the conditions under which safe navigation becomes margmal or

im-possible, while the mariners themselves can also give an opinion on the

acceptability of the sailing conditions Research on a simulator is thus a big step forward compared to earlier studies based on techmcal aspects

only .

The setting up of manuvring criteria requires an extensive study involving a diversity of variables. In thó presentintroductory study, how-. ever, only a limited number of aspects were examined:

(i) A ship must be able to perform several types of manoeuvre, such as course keeping, altering course, stopping or avoiding ,a hazard.

368 J. P. HOOFT VOL. 27

In the present study only one type of manoeuvrethe recovery

from a disturbancehas beei analysed. In this experiment the pilot giving hehui orders had to sail the vèssel along a pre-determined straight line or as close to this line as possible. At the

beginning of each manoeuvre the ship. sailed in a disturbance

due to a cross current and, as the ship was initially sailing along a

straight line, mariners were not given the opportunity to bring the ship into such a position that the effect of the disturbance would have been counthracted. They could not anticipate the

disturbance; the task was to limit the effect of the disturbance as much as possible.

(ii) In practice ships are steered by several individuals, each with his

own degree of experience, and to minimize 'learning effects' it

was decided to operate the programme with pilots experienced in bringing 2So,000 d.w.t. tankers into Eúropoort. There will always be some scatter in the. takiig of deciions, choice of strategy &c , and the design of the experiment was such that each

pilot sailed m all the conditions, but in a different sequence (ui) As to navigational aids it was decided that the pilot would use

only a compass, a rate of turn indicator, a Decca plotter and a

leading line.

(iv) Since changes in a ship's form can considerably change her manuvring characteristics all the hydrodynanuc coefficients for ships of the same deadweight can have different values Nowa-days there is much concern for the hysterçsis fçund in the spiral

test of a ship (mdicating dynamic stability) while other aspects

such as time constants, rudder effectiveness, turning ability &c are overlooked To investigate the influence of the spiral test hysteresis on manoevrability, it was decided to compare the manuvrabihty of four ships for which the spiral test results differed only in the linear-drift and turning-damping moments and

damping forces.6 .

(y) Deep water operation was chosen since earlier studies in the literature mostly refer to ship characteristics in deep water only.

All. infoiniation gained in thé present study should therefore be verified fOr water depth, according to real life situations and using ship characteristics refering tó shllôw water.7

(vi) Before each manoeuvre the pilots were informed of the direction and distribution of the current to be expected, but not of its

magnitude. Except for her size, no information wäs given about the type of ship, this reflects the real life ituation in part, since pilots board many ships whose manoeuvrability is unknown to them

2. SHIP CHARACTERISTiCS AND SAILING CONDITIONS. Ship manoeuvra-bility will be influenced by a large variety of parameters such as size and

N0..3 MANUVRABILITY.OF SHIPS 369

hull form,8,9 but in the present study only the influence of dynamic

stability (see Appendix I) on manuvrabiity has been analysed, and only one class of tanker of about 250,000 d.w.t. To change the stability index, only the turning and sway damping coefficients have been modified6 in such a way that a physical explanation can be given for the changes in the hydrodynamic coefficients.

The stability index o- follows from the linearized yaw and drift motion

of a ship after some disturbance from her initial heading. This situatibn

can be approximated by the condition that at the time t= o the ship's yaw and drift motions are zero while an external disturbance is exerted on the ship. The ship's motion in yaw () and drift (y) will change as time t

elapses:

=rlec1t+r2ec2I y =vietJlt+vze(72i

Since o-2 is always negative (see Appendix I) a ship is dynamically stable

when _{a is negative, because then db/dt and y will vanish after some}

time interval. When however o-1 is positive, the ship's motion will

increase to some maximum value governed, by non-linear forces. The ship is then said to be dynamically unstable.

In practice, ships are said to be dynamically stable when the spiral

test shows a unique relation between rudder angle and rate of turn. When a ship is dynamically unstable there may be two values for the rate of turn for the same steady rudder angle.

The manoeuvres were performed at an initial speed of 8 kt. and during

manoeuvre the engine setting was kept constant. This means that the ship's speed could decrease as a consequence of the manoeuvre. The

simulated situation corresponded to deep water without wind and waves.

The ship's initial position (heading 000°) was on a leading line. The distance travelled during each manoeuvre was ç km. (about 27 miles)

corresponding to about 20 min. cailing time.

3. CURRENT DISTRIBUTION. During each run of the test programme the current velocity was different, but its direction was always perpen-dicular to the leading line, setting the ship to the east. The current was composed to two parts:

The first component C1 increases rapidly to a constant value

C1 max

The second component C2 is a disturbance which increases to a

maximum value of C2 after i000 m and then decreases.

The effect of the current will be to set the ship east, but at the beginning

of the manoeuvre she will also change heading due to the current shear

(change of current velocity over the distance to be travelled).

370 - H J. P. HOOFT . VOL. 27

the study (developing ship handling criteria) before the programme was started and, after being made familiar with the simulator, were asked to command the' ship as they would -do in real life. They were required to keep the ship on the leading line as closely and smoothly as possible and were informed that their manoeuvres -would be analysed in order to judge and compare them with other manoeuvres. Before each manoeuvre pilots

were informed about the current distribution in rough figures; on the basis of this information, they could not however make accurate

pre-dictions. No information was given abóut the type of ship the pilot, was

to navigate and, as in ìeal life, the.pilot ga+e his orders to a helmsman

who then operated the rudder as instructed.

-ç. PRESENTATION OF RESULTS. During each manoeuvre the track of

the ship's centre of gravity was plotted on a paper chart while the cours

angle, rate of turn, rudder angle, lateral deviation from the leading line, propeller r.p.m. and ship's speed were recorded on a magnetic tape for

computer analysis. The present paper is based only on the analysis of the

lateral excursion of the centre of gravity from the leading line.

There is of course a scatter in the ndividual results; unless otherwise

specified all results refer to the average valüe obtained- from the manoeu-vres performed by the six pilots. This means that when ship -A (òr current condition A) is on the average better than ship -B (or current condition B) no conclusion can be drawn about the extreme values obtained, with ship

A or B. When however the variànce around the average value was im-portaht this has been taken into consideration An attempt has also beêñ made to correlate the. results of the analysis of lateral excursion with opinions expressed by pilots about the ship and the manoeuvre, each pilot being asked to rate the cönditions during his last manüvre from o (bad) to io (good), on the following c6nsideiations

(i) 'Cuid :the' ship. be kept under conirol? (il) How did you judge the manoeuvre? (iii) Did you find the ship easy to steer?

The rating given by the pilots was

2 unacceptable 6. marginally acceptable

3. very bad - 7.. good

.. bad 8. very, good

ç. marginally poor _{9. excellent}

io. superb

Although these were subjective ratings the opinions of the six pilots were very similar; the variance around the average values was used to

assess thè judgment of the ship or of the current condition: Unacceptable when 30 per ¿ent of the 'ratings is less than .

Acceptable when, 30 percent of the ratingsis less than 6 and more than

-.

Good when noneof the ratings s 1es. than ç

NO. 3 MANUVRABILITY OP SHIPS

31

6. DESCRIPTION OF THE MANOEUVRE. Based on the information that there was an east setting current the pilots set the rudder to port im-mediately after the start of the manoeuvre but, since they weré not in-formed of the type of ship they were handling, the rudder angle applied was irrespective of type of ship. Soon after the start the pilots ordered

such rudder angles as they considered necessary to correct for the course error and deviation from the leading line which they were sensing. They were not informed of the magnitude of the current velocity so that even when there was no current or disturbance peak, in anticipation of a current, they ordered port helm until they realized this.

In Fig. ithe ship's track for two different manoeuvres is indicated. It

may be assumed that during the left-Side manoeuvre in Fig. I an initially large current shear was inferred (large dC1/dx since in all cases the current

shear due to the current disturbance, C2, is zero at the starting point)

from a rather quick start of the ship's turn. For this reason the pilot will have applied too large a rudder angle to port if there is no disturbance (C2 = o). In the manoeuvre on the right Side of Fig. i a large deviation to

starboard will be experienced when there is a current disturbance only and the ship will mainly drift (which is difficult to perceive) while the turn is small. Due to this combination of errors the pilot will apply a

rudder angle too late to correct the ship's track.

The manoeuvres in Fig. i have been studied by considering two essen-tial values of the deviation of the track of the centre of gravity from the leading line:

The first maidmum excursion from the leading line after the ship has passed the. peak value of thecurrent disturbance is considered

since it is always important to krio* th sea room required when passing a disturbance. Since in some cases the disturbance was

zero, all first. maximum excursions were analysed after the first ì000 in travelled.

The second maximum excursion from the leading line after passing

the peak of the current disturbance is also considered, since it is important to know the ability Of the ship. to recover by human

control from a deviation due tò environmental disttfrbánce. First maximum excursion. After pilots have oiderd port ruçider at the start of the manoeuvre it is interesting to see how they keep the.

ex-cursion from the centre line as smallas possible and, how the. resulting

manoeuvre is mfluenced by the ship characteristics The maximum

excursions after the maximum value of the current disturbance was

passed are shown in Fig. .2. It can. be seçn that n influence due to ship

type can be distinguished. It will also be seen that with no current

disturbance pilöts gave so inuçh rudder to port that the first maximum

excursion after i 000 m (at which the currentdisturbance is at a

maxi-müm) will on an average be loo m to port. At increasing values of the

Y1 a first maximum devatjon of Ships, track from leadng linS after.

passing the peak value of the current.

second maximum deviation of shies track from leadiñg line after passing the peak value of' the urrent.

un n un un un n g. u C o e o C 1 g. 5 start of manOeuvre 372 j. P. HO OPT VOL. 27

cùrrent distribution .' the initial rudder initially too little

- angle 'applied is rudder angle to.

large relatie to 'port is ¡pplied te

the' turning effect . correct for the,.

of the current shear drifting of, the ship

H. i.

Schematic indication of track of centre of gravity of ship during two manoeuvres.

to tarboad. At a maximum current disturbance of 2 kt.. to starboard

the first.maximum deviation of th ship's track will be about 140 m to

starboard.

'It may .be qúestioned hów and why pilots 'manuvred ships in such a way that on avçrage no influencé of the ship type is experienced As to how they achieved 'this resûlt onè may conclude Trorn: the increased 'variátie around the average value, that with increaing instability index the résult +hich theilòt aims at is obtained less accuratély 'The amount of current disturbance' also influerices the scâtter 'of résuits

As, to why-pilots followed a strategy whichieùlted in a màñuvre

irrespective of ship type, it should bé ñòted that they had notiínetö get

t know' the ship during that specifià manoeuvre. To specify the strategy

in more 'detail it is importanttó take into considèration thé influence òf

dis-E u 0 staD,Ity indea _j a nttiSI nrea5e of current velocity peak of Current velocity FIG.5

FIG. 2. First amplitude of lateral excursion (overshoot) after passing the peak value of the current.

turbance (C2 m) The influence of both parameters on the first

over-shoot is plotted n Fig. 3. From this figure it may be seen that:

(i) With no current disturbance (C2 = o) so much rudder to port is given that the first maximum deviation after x000 m will be to

port of the leading line and the amount of this first maximum deviation to port will be up to some limit irrespective of the initial current shear. It may be assumed that this is due to an

increased application of rudder angle to port, as a result of the

reaction of the pilot to his perception of the increased current shear arising from themotion of the ship.

_+ .00Q81

O .000456

X .0001St .0.00112

NO. 3 MANUVB.ABILITY OF SHIPS 373

300. 200 100 o 100 200

west tea ding east

-max. velocity C2 01 CUrrent disturbance max Olcn 't 't

-/

/

/

_L_.

K I

\\\\\

k--

.- , , 374 J. P. HOOFT' VOL. .27 V 100 150 200 250m.

west leading _east

I,ne

FIG. 3. Influence of initial current shear and maximum current disturbance on the

first maximum deviation of the ship's track.

(ii) With increasing current distuibance the maximum overshoot,

after passing the peak value of the current, decreases to port and

shifts to starboard. This can be explained. on the assumption that the shift is noticed too late since a deviation from the leading line

cannot be accurately observed (a leading line is often called a

leading range). Thus rudder action will be applied .when the ship is already clriftingto starboard.

Sécond maximum excursión. After the first overshoot, when the maximum

value of the current impulse at boo m has been passed, the ship will come back to the leading Ime again By this tune pilots are trying to bring the ship into the final equilibrium position in the constant current

distribution, but they have first to overcome the ship motions initiated

by themselves to compensate for the currçnt disturbance. Figure ₄

shows the iliflüende of the initial current shear and maximum curtent disturbance on the second maximum deviation of the ship's track from

the leading line, after passing the peak value of the current.

Too great attention is paid to the initial current shear, since at the.

highest shear too much rudder is applied to counteract the ship's course error; hence the second overshoot of the ship's track is on an average too much to port, against the current direction.

On average the large drift to starboard (see the result at zero

initial current shear) due to the current disturbance results ina

o. E o o a a. 4I C a X 0.25 o o I' 4, C .4 C o

Q5 E o o U a u o o CC II 0.25 o 'n C. u C 'n C 'o -r max. velocity C2

of current disturbancemax

kn Okn ¡ I I t _t

--0.67 - 1.33kn

J--.-.-,- 2

/

\\

/ I

/

\'

I

NO.3 MANUVRABILITY OF SHIPS _3m

150 100 50 0 50 100 150m.

west leading east

line,

HG. 4. Influence of iìiitial current shear and maximum current disturbance on the second maximum deviation of the ship's track. -

-large fl±st overshoot to stárboä±danda reasonably good cómpen.

sation for this deviation, as mdicated by the small value of the

second overshoot to port.

: ''

r

Thus, up to the fitst overshoot the pila's action was mainly decided, on the basis of his perception of the initial current shear and the. amount

of current disturbance, there is little miluence of ship type on the amount

oftheflrstovershoot., _, '

From Fig, it can. however: be seen. that tie Second over oot

strongly ifluenced. byr the siiip's characteristiçs when,' due to pilQt's

actions, up to the first overshoot, the ship swings to the other side of-the leading line. Comparing Figs. 2 and ç it follows that:

-:

-(i) For the very unstable ship a flist overshoot of i.o, m'tO stärbóaM

m a 2 kt current disturbance is followed by a second overshoot of

2Ió in to port, before thé pi1ot'managés to redue thé hip's

motion to the eqi.uhbnum position

(u) For a ship that is easy to handle a first overshoot of 140 m to

-0.005 o 0.005 0.o0 current giSturbuflet O kfl 067 kfl 133 kn 2 Sn

-

-

i

-

-..---t t S-, \ I

V!

i.'

t'

¡

/1!

/1

1-/1

376 l.P.. HOOFT VOL. 27 300 200 100 o 100 200, West ediflg

PIG. _{ç. Second amplitude of the lateral eìcursion (overshoot) after passing the}

- _{- peak of the current}

disturance.

-overshoot of 30m to port, which is already more or less the final equilibrium position.

It should be noted that these large overshoots can. bepari4y ascribed

to the fact that the pilots were not aware of the magnitude of the current and therefore executed. a strategy more or less independent of the type of

ship in. deteimining the width of a waterway one shoul4 therefore be aware of the situation of pilots who lack information about sailing

con-ditions and the ship's character. With a treater current disturbançe or a less stable ship there is, as before, an increased variance around the

average válues

7. JÚÍ)GMEÑT OPCURRENTS. From the analysis given'soiar it will be

obvious that the very first action of the pilot is strongly hfluenced by the

initial current shear Thereafter his main concern will be to manoeuvre

the ship through the current disturbance as well as possible In order to

judge the mfluence, both of the disturbance and the mitial current shear, an analysis was made óf the pilots! opinions abóút the manuvr they

lad performed No influence of ship characteristics on the rating of the

manoeuvre could be found. It can therefore, be: assumed that their

opinions were mainly influenced by the environment, while their rating

of the ship is given by their answers tO-the other questions. The average

rating of the six pilots is given in Fig 6 which therefore shows the limits

o

0 _knots 2 3

-max velocity of current disturbance .

-FIG. 6. Design chart of allowable initial current shear and maxim. m current

disturbance, according to the rating of the pilots.

-limited variety of conditions tested, such as type of manoeuvre, ship type and the distance (2000m) over which the current disturbance increases

and decreases. , . . , -,

8. JUDGMENT OF SHIP MANuvRABILrrY. From earlier results it was

found that the first overshoot was only influenced by the current

dis-tribution, the influence of ship characteristics could not be distinguished, and- that the second overshoOt was influenced both by the ship's dynamic stability and the current distribution.: , .. .

-. From these.findihgs one might expect that the total 'width of the lane

within which the- track of the ship's centre of gravity lies, would be

in-fluenced by the 'current as well as by the ship's dynamic stability and this

is confirmed by 'the results. In Fig 7 the maximum width of lane has

been plotted as;a function of the ship's dynamic stability and the current

disturbance. The answer to the question,. how easy the pilots found the ship to steer,. shows a striking agreement.:between the two results. In

easy ènvironmental conditions no difficulties arise for the pilot since only small corrections are required. This leads 'to the conclusion that a ship's

behaviour can only 'be judged in more difficult circumstances. The iii-fluence of the' current; disturbance on the width of lane required, or on

the rating assigned to -the ship, is least for the ship which is nearly stable: Figure 8 provides criteria.from which'the ship's dynaniic stability can

be deduced and from. Figs. and 8 one can compare the width of the lane

with the pilot's rating. When the track of the' ship's -centre of gravity

requires a lane of200m or m re, the pilot felt so unconfortabIe that the

conditions (determined by either ship or current) '.vere 'assessed as

un-acceptable This is important since until now. 'only, technical criteria for a ship's handling characteristics have been available;' a slip's,

manoeuvra-Unacceptable -average rt,ng 75 6 -. acceptable

---

--good

-___,

very good -

/

'/

,//_

/

/

/'

Sverae max wdth et lane

$U

:::

200m 4 F

7UR

RUV

--:111111

ii

378 J. P. HOOFT VOL. 27 unStable Stable Stability index

FIG. _{7. Design chart of maximum width of lane as an influence of current}

disturbance and dynamic stability.

bility was considered acceptable when the width of the lane was less than

the available sea room. Figures j and 8 provide criteria scaled down to

several categoiies which correspond also to conditions of safety;

un-acceptable conditions can be considered unsafe.

9. JUDGMENT OF SHIP CONTROLLABILITY. From the pilot's rating as to

how well they could keep the ship under control, it has been shown that,

besides current disturbance and the ship's dynamic stability, the initial

current shear is also of importance. Due to the limited extent of the

programme no systematic differences could be discerned between the

influence of each one of these three parameters but, generally speaking,

pilots gave a somewhat higher rating (i or 2 points) to controllability

than tó ease of steering. This can be explained by the fact that when they rated ease of steering they probably only judged the ship, while for rating

controllability environmental conditions were also considered; for

al-though a ship may be difficult to steer she may yet be kept under control

in the situations encountered. From the average of the ratings by each

pilot for all the conditions it can be concluded that the marginally stable

ship was judged to be best for the conditions experienced during the programme. From the maximum width of lane it can be concluded that

the ship that is nearly stable (but still unstable) is the best ship from a

manuvring point of view.

'J. _{o -0.005}

o 0

NO.3 MANUVRABILITY OF SHIPS 379

io. CONCLUSION. The striking agreement found between the

judg-ment of experienced pilots as to the manuvrability of the various ship types and the manoeuvres they carried out, leads to the conclusion that

valuable results of practical application can be obtained with the aid of a manuvring simulator. o

i

I.lFflT

_{i i"__II}

i

u,.

-ii

II-.--'.

_g

average rating

uuuuuu

.0,01 0,005 o -0,005 unstable Stable Stability index

FIG. 8. Design chart of allowable dynamic stability and current disturbance,

according to the rating of the pilots.

In the present study the effect of the manoeuvring situation on human

behaviour was investigated by analysing the judgment of the pilot. The

human condition, as a consequence of his task, might be further analysed

by measuring physiological reactions such as heart beat, galvanic skin response and the electro-encephalogram.

lt may be questioned whether the above conclusions would still hold

if other manoeuvres had been performed under differing conditions; a lot

of research has still to be done to set up the criteria which the shipping world is in need of. However, the present study has shown that criteria on ship handling or sailing conditions cannot be established only from technical data gathered separately from the human element. Research programmes on a ship manoeuvring simulator, in which the practical experience of the mariner can be taken into consideration, are

indis-2 C s I, C a J a C a 9 o e a > a E

U : dx/dt V : dy/dt .USinp t(p). Y(p) z drift angle -Yin e,cr at CG. alOng 3 Circle - +/dt rate of torn torce CG. Centre et rCnity

FIG. 9. Definition of system coordinates

pensable for the final definition of manuvring criteria on ship handling

and sailing conditions.

The work described was carried out with the aid of six experienced

pilots of the District Rijnmond of the Dutch Pilotage and the author

is grateful to the Director-General of the Dutch Pilotage for allowing the pilots to participate in the programme. He is also indebted to the Director of the District Rijnrnond and the six participating piiots for their cooperation. The results described in this paper benefited greatly

from the considerable effort of the participating pilots as research subjects. REFERENCES

u

i Mandel, P. (x 961). Ship manuvring and control, Chapter VIII of Principles of Naval Architecture. S.N.A.M.E., New York.

NO.3 MANUVRABILITY OF SHIPS 381 2 Motora, S. (1972). Manuvrability, state of the art. N.S.M.B. 4oth Aflniversary Meeting, Wageningen.

3 Bilodeau, E. A. (1966). Acquisition of skill, New York, Academic Press.

4 Segel, L. (1960). Ship manuvrability as influenced by the transient response to the helm. First Symposium on Ship Manuvmbility; David Taylor Model Basin, Washington, Report 1461.

5 Hooft, J. P. and Oldenkamp, I. (1972). Construction, operation and capabilities of the N.S.M.B. ship manuvring simulator. N.S.M.B. publ. No. 382, Wageningen.

6 Berlekom, W. B. van and Goddard, T. A. (1972). Manuvring large tankers, Trans. S.N.A.M.E., New York.

7 Hooft, J. P. (1973). Manuvring large ships in shallow water. This Journai, 26, 189,

311.

8 Shiba, H. (196e). Model experiments about the manuvrability and turning of ships. First Symposium on Ship Manuvrabiity; David Taylor Model Basin, Washington, Report 1461.

9 Eda, H. and Crane, C. L. (1965). Steering characteristics of ships in calm water and waves, Trans. S.N.A.M.E., New York.

APPENDIX I

DETERMINATION OF THE STABILITY INDEX

The linearized equations of turning and drift motions for a ship in still water can

be written:

d2y ¿y d.&

(Mi-m1)-. Y-+ Y- -mur=Y .8

d2# dy dçb

6

By combining equations (s) and (2) one can eliminate the influence of drift. For this one rewrites equation (2):

dy

'-N '-N

_o-

¡ d24s After differentiation, the lateral acceleration is:

d2

i(

dS d2,/, d2

=--

- ,j(

+i)-Substituting equations () and (ii.) in equation (i) and remembering that

di//dt= r, one finds:

Ad2r dr d6

+B+Cr=D8+E-

_{dt2 dt dt}

When the rudder angle is zero and the rudder is not moved:

d2r dr

A - - + Cv

=o dt2 dt

After some initial disturbance the ship motion will then be: r=r. eatt

(5)

(I)

382 - J. P iiOOET VOL. 27

Substituting () into (6) gives:

Aa12 +Ba+ C = o ₍₈₎

frömwhich one findsthat there will be rOOts:

_B±./(B2_44C)

L (9)

It will be clear (smce A and B are positive) that one value of a 'ill always be

ilegative while the other may be either negative or positive. When both values are negative the iñitial motioti will vamsh after some time interval, the ship is then dflned as dynamically stable. When one of the vlies is positive the motion

PRÍNTFD Ñ GREAT BR!TAIN BY WILIAM CLOWES & SONS; iIMflEb