Statistical analysis of the rolling motion of three coasters

CHALMERS UNIVERSITY OF TECHNOLOGY DEPARTMENT OF NAVAL ARCHITECTURE

AND MARINE ENGINEERING

GOTHENBURG - SWEDEN

DIVISION OF SHIP HYDROMECANICS REPORT NO. 24

STATISTICAL ANALYSIS OF THE ROLLING

MOTION OF THREE COASTERS

by

RALPH NORRBY and LARS ENGVALL

Reprinted from

"EUROPEAN SHIPBUILDING"

STATISTICAL ANALYSIS OF THE ROLLING MOTION

OF THREE COASTERS

by

Ralph Norrbyl and

Lars 0. Engvall2

Introduction

The value of

static stability calculations is

reduced owing to the uncertainty of the stability

requirements for various types of vessels. Stability criteria are called for in various countries, but the

great majority lack satisfactory theoretical basis.

Bad loading can quickly turn a good vessel into

a poor one as regards stability with catastrophic

results. Officers in command should be given the

opportUnity of checking the stability at a given

time in a practical way.

The problem of finding satisfactory stability for

ships is greatest on fishing vessels and coasters.

put it

is occurring with greater frequency on

larger cargo ships, the main reason being the

rationalization of cargo, handling. The size and

weight of hatches, masts and booms are increasing

and superstructures

are becoming shorter but

higher, which raises the ship's centre of gravity. Many accidents involving trawlers and coasters

indicate that nowadays the stability is often

inade-quate.

Extensive research in this field is now being carried out, particularly in Japan and Germany. But this type of investigation should be

under-taken by every large shipping nation, as the results

in the form of stability criteria, for example are influenced by local conditions such as weather,

wind, sea and type of vessel.

The Ship Hydrodynamics Division of the Chalmers University of Technology has been doing research for some years now on the stability of coastal vessels [I]. In order to obtain a guide to

the stability requirements and the checking of stability in this type of vessel, an instrument to

register rolling was installed in three coasters for a total period of about one year.

1 AB Karlstads Mekaniska Werlcstad, Sweden.

2 The Swedish Shipbuilders' Computing Centre.

The aim of the measurements was mainly to

study statistically the size of the angle of heel and the possibilities of calculating the probable maxi-mum angle of heel, the distribution of the rolling period, mean value etc. and to study the

connec-tion between the angle of heel and the rolling

period.

Acknowledgement is made to Professor C.

Fal-kemo who made this work possible, to the Swedish

Technical Research Council, which mainly finan-ced it and to the shipping companies and officers of vessels concerned in the investigation. Mr. S.

l3ramberg and Mr. B. Engvall helped with the

measuring equipment and the statistical

calcula-tions, r6ipectively.

Recording of rotting a. Scope

The recording was carried out on two shelter deckers and a single decker. Certain characteri-stics of the ships which are of interest and the

time for recording are given in the following table.

Vessel I sailed mainly between the Baltic and the Mediterranean, II in the North Sea and the Bay of Biscay, and III in the North Sea and the

Baltic.

A broad range was obtained with regard to

Vessel I

it

III

L_PP 65.80 59.25 43.00

L_PP/B m 6.09 5.93 4.62

Bm/ds 2.94 2.65 2.32

C B .671 .732 .650

No. of trips with

heeling recorded 11 8 13 Period of _{Dec. 60/ Dec. 61/ May 62/} recording May 81 Feb. 62 Aug. 62

wind, waves and their direction in relation to the vessel's course. Similarly, the type and amount of cargo varied considerably, giving measurements

with considerable differences in metacentric

height. The vessel therefore followed a varied and typical sailing programme for coasters during the

period of recording. These measurements can

therefore be considered highly representative. Owing to the large number of parameters involved,

however, it is not possible to give generally appli-cable distributions and connections with this

ma-terial.

b. Implementation

The recording of rolling was carried out on

board by the captain who received measuring

instructions from a «PM for recording heeling» and from staff of the Division when they visited

the vessel. The recording of rolling was made

TABLE I

External and internal conditions

° Figure after + sign gives direction and height of undercurrent.

** Se attached fig.

Regions: AG = AtlanticGibraltar; AP = AtlanticPortugal;

B = Bay of Biscay; E = English Channel; N North Sea;

Anch = at anchor. 0= Baltic;

Douglas scale: Dougl. 0 1 2 3

Wave height: 0 1-15 cm 15-60 cm 60-150 cm

4 5 6 7 8

1.5-3 m 3.0-4.5 m 4.5-7 m 7-11 m> 11 m

once or twice every 24 hours and the recording

period varied between 30 and 60 minutes a time. Before each measurement, the vessel's position, speed and draught and notes on the appearance of

the sea were given. These internal and external conditions are assembled in Table I. (The result in the form of tables and diagrams is only given for vessel I owing to space. The corresponding

figures and tables for vessels II and III can be

found in the Report No. 9 of the Division of Ship

Hydrodynamics.)

The designation «force» in Beaufort terms must be taken as an indication of the appearance of the

sea as seen by the captain from the ship. In such

a case the Beaufort figure cannot be directly

translated to wind force or height of waves. But

as a means of comparison, these Beaufort values are of more practical use than wind force or height

of waves, cf. Bennet [2].

No. of

measure-ment

Wind and waves°

Direction** Height

Region _{rel. to} ship

Force of wave GM d v

Beauf. Dougl. m m knots

AS B 4+1 6 4+5 1.0 3.28 13 A4 B 4+1 4 3+4 cc cc 13 A5 AP 5+3 4 2+3 q cc 13 C3 AP 7 4 5 0.9 cc 11 C4 AP 7 2 5 a q 12 C5 AP 6 5 5 c a 12 C6 B 4 7 5 cc c ₁₃ C7 C8 B E 5 5 7 10 5 7 cc c cc cc 13 11 C9 N 3 7 5 cc a 13 C10 N 5 6 4 cc cc 13 E3 AP 7 6 5 1.2 3.10 11 E4 AG 7+2 4 4+3 a « 12 Fl AG 7+6 1 0+4 1.4 2.84 13 F2 AG 7+6 3 0+4 cc _{c 13} G2

0

5- 5 4 2.0 2.49 14 13 B 3 2 4 1.3 3.30 13 K2 K4 AP Anch 2 0 3 1 5 0 1.5 a 2.72 cc 13 0 Li AG 7 2 4 1.8 2.44 13

Fig. 1. Measuring apparatus.

To obtain as objective estimates as possible the

captains were provided with a State of Sea Card

M. 0. 688 from the British Meteorological Office.

The card describes the appearance of the waves

for each force in the Beaufort scale. A photograph

accompanies each description showing a typical

view of the appearance of the sea from a ship.

The height of waves was expressed with the

help of the Douglas scale in the measurements on vessel I. Experience has shown, however, that the scale is too inexact for those wave heights which

are of interest. For vessels II and III the wave

heights were therefore expressed in metres.

The metacentric height GM was calculated

according to the qmoment method» [1] with the aid of existing loading plans and cargo figures

supplied by the captain. No correction was made for free water surfaces [1]. The calculation of GM

was made to provide a basis for calculating the

period of roll. It is not absolutely exact but should

be regarded as corresponding to calculations

which are often made by owners or captains

before a vessel leaves port. Information received

about the cargo and where it is stowed was very

inadequate and therefore the calculated GM values

are only approximate. Rough estimates have had

to be used for vessel III.

Measuring equipment

A measuring apparatus constructed by Mr. S. Bramberg was used for recording (Fig. 1). It

consists in principle of three units mounted on one

stand.

The power aggregate is

at the bottom. This

transforms DC or AC from the vessel's system to stabilised AC of 50 or 400 p/s.

The uppermost part of the stand houses a verti-cal sounding gyro which receives current for

mea-suring (400 p/s) from the power aggregate. The

gyro's points of suspension are fitted with adjust-able transformers which give a voltage

proportio-nal to the angle between the gyro and the stand.

The gyro was supplied to the Division by the

Royal Swedish Air Force.

Between the power aggregate and the gyro is a recorder of the self-balanced potentiometer

type. This receives current from the gyro and the

rolling is registered on a strip of paper 250 mm

wide.

The recorder is fitted with two measuring areas.

At its full extent it registers 12.5 or 50° heel. In

the sensitive area of measurement is was also

pos-sible to take clear measurements of heeling less

than 0.5°.

The apparatus proved to be reliable and easy to operate. It can be placed on board in an extra cabin, the engine casing or similar place but it

must be under cover. The dimensions of the

appa-ratus are 500 X 500 X 1,500 mm and it weighs

150 kg. There was some difficulty in placing it in

the small space available on the vessels concerned.

Angle of heel

a. Statistical distribution

The movements of a vessel which arise under

certain given external and internal conditions can be described by a distribution function. jasper [3]

has shown by a large number of measurements that the movements are in accordance with the

Rayleigh distribution.

The frequency function of the distribution is as follows:

29'

P (v) e

and the distribution function

P ()=1 e-f2/R

where p is the angle of roll and R a parameter

characteristic of every condition. The probability

of a certain angle r being exceeded is (1 P).

It appears from the functions that the Rayleigh

distribution is defined by only one parameter R.

The best estimate of R from material surveyed is the quadratic mean value of all values.

R

The angle of heel is measured from 0 to starboard or port. The 0 line in this case is that line around

which the vessel heels in a particular instance. This does not always coincide with F = 0 (ship in upright position) owing to list.

In all cases,

however, the angle of list was very small. There is

no noticeable difference between heeling to

lee-ward and windlee-ward.

It takes a long time to measure all values in a large number of recordings and in consequence several simplified methods have been found to determine the value of R. In this case a method according to Cartwright [4] was used, which is relatively simple and accurate and is carried out

as follows. (Owing to

the few observations in

measurement C8, the R value was established

cexactlyD by the quadratic average value). . Measuring is divided into 10 to 20 parts. The

number of heels (from 0 to the extreme at one

side) are to be equal a certain power of 2 (i.e. 2m) in all the parts. Only the highest' value in each area is measured, after which the Mean value cp max is calculated and R is cibtained thus

R()max) 2

2

Ma (first distribution moment) is tabulated as a

function of 2m and E [4] where E the width of

the spectrum.

The width of the spectruna is

calculated as

follows [5]

r2 = 1 (1 2r)2

r = the ratio between the .number of negative

maxima and the total number of maxima (Fig. 2). The width of the spectrum is based to some extent on an estimate as it is impossible to find an exact

reference line (Fig. 2). The values obtained,

however, Were low, e -LC. 0.4, and should not be of

any significance in the adjustment of the

amp-litude distribution to the Rayleigh distribution or for probable maximum values [5].

When calculating the value of R according to Cartwright it was assumed that e = 0 owing to

its slight influence on the result.

Fig. 3 shows an example of the distribution of the angle of heel for an arbitrary measurement, In Fig. 3a the measurements are recorded as a

histo-gram and the continuous curve is the theoretical

Rayleigh distribution for the R value concerned, calculated according to the above. Pig. . Mi shows

1 9 P05. MIN. 2 r2. Fig. .2. 1 EXACT 0=1411E, P 24

1111

R = 12.3 degrees Z following sea Measurement C 9 18

QN7 uBeaufortt/erai u fe r:i3nr tg

12 2 3 4 5 6 7 8 9 9 degrees Mien40441 Uwe.* 10,4474 010.04rfon as a: iD 2.0 BD 4.0 20 7.0 2 `2 ., 200 300 400 0.0 70.0 no sos 5 ice 2 92/

Fig. 3. Angle of heel distribution.

I

the cumulative distribution recorded in logarith-mic scale. The diagram gives the assumption of

F. as a function of the standardised variable

,,4

2 472

. The straight line is thus valid for all

' 1= R

Rayleigh distributions.

-As the Rayleigh distribution can nowadays be

considered generally accepted as a basis of cal,

culation for statistical investigations of vessel

motions, the suitability of the material obtained for theoretical distribution was not investigated

more closely.

A distribution which presents motions in a shoiter time with unchanged external and internal

conditions is called short-term distribution. The

revised distributions are of this type. With the aid Of these short-term distributions which are

deter-Mined by the parameter R, a long-term

distiibu-tion'can be constructed which applies to

thelves-sel's rolling motions over a longer period. The

various short-term distributions are tested for

theit relative occurrence on the vessel's normal voyages. This is easy to do using weather

sta-tistics. MEG. MAX.

b. Extreme value

For the Rayleigh -distribution the following applies according to [6] and [7]:

The most frequent size in the observations

is p.707 1/R.

The mean value for the size in the observa-tions is 0.866 VR.

The most probable extreme value from a

random sample of N observations is rm

k 1/R. Where N> 100,

k.=

1nN

ap-proximately, and therefore rm = 1/RlnN. r m was calculated for the recordings obtained (see Table II). N is the number of observations in

the measurements. If the weather and other in-fluential factors are taken as unchanged for six

hours, for example, the number of observations is

6N h, Where N h is the number of observations per

hour. com calculated for this period is also shown

in Table IL

corn calculated during one set of measurements was in relation to the maximum

angle Which is measured from the recordings.

In order to determine statistically with the aid

of measurements carried out, a maximum angle of

roll rm, which can be used as a limit value in

construction, the following is given [6]

TABLE II Extreme value

. .

r2m = Rm (y in N)

where Rm applies to the most adverse

circum-stances in which the measurements

were taken and which are also

as-sumed to be the most extreme the

vessel will have to operate in,

is a function of the risk f of a certain

angle being exceeded. Where f 01,

1

y = ln

(

From the available results of measurements it is now assumed for example, that the measurement

C8 is carried out in the most extreme weather

conditions. It was assumed for the calculations that the bad weather lasted for four or eight hours

and that the yessel was exposed to this twice or

three times a year in its lifetime, - which is assumed

to be 20 years. The risk (f) that the calculated

angle of heel will not be exceeded is put at 1/1000 and 1/10,000.

The maximum angles of heel which can be

Calculated

97'm

during N No. of 'Pm Fin Measured

V max

measure- during during

meat N

Nh R N 6Nh Fmax measured

degree2 degree degree degree

A3 228 342 32.7 13.3 15.8 14.4 0.92 A4 244 349 28.1 12.4 14.6 14.0 0.89 A5 173 346 5.3 5.2 6.3 6.0 0.87 C3 398 398 16.8 10.0 15.2 12.2 0.82 C4 192 384 14.4 8.7 103 8.8 0.99 C5 255 383 12.3 8.3 9.7 9.2 0.90 C6 222 380 25.0 11.6 13.9 13.4 0.87 C7 329 380 22.1 11.3 13.0 12.0 0.94 C8 65 390 43.6 13.5 18.4 20.0 0.68 C9 163 391 12.3 7.9 9.8 8.8 0.90 C10 208 405 6.8 6.0 7.3 7.0 0.86 E3 216 518 16.8 9.5 11.6 11.0 0.86 E4 241 482 20.3 10.5 12.7 10.0 1.05 Fl 320 491 12.3 8.4 9.9 9.0 0.93 F2 176 491 9.0 6.8 8.5 8.2 0.83 G2 194 523 3.2 4.1 5.1 5.6 0.73 13 320 450 4.4 5.1 5.8 5.2 0.98 K2 258 499 25.0 11.8 14.1 13.4 0.88 Li 528 528 7.8 6.7 7.9 7.2 0.87

The expression can then be written

=[

R

TABLE III

apected for a vessel in loaded condition on the

above assumptions are shown in Table II!.

The above assumed values of hour, year and times per year are approximations. To obtain a

more precise basis investigations should be made

into the weather conditions on the vessel's

tended route etc. Uncertainty of the initial values, four or eight honks and two or three times a year have, however, no very great effect on the

resul-tant rm. This is clear from the fact that the

quo-tient between rm for 20 years, 8 hours, three times

a year and 20 years, four hours, twice a year for

both values of f is not greater than 1-:03.

Using the calculated distribution (R value) for the severest Weather and the cumulative

distribu-tion in Fig. 3b, the probability of exceeding a

certain angle under these conditions can be cal-culated. By way of example the probability of heeling greater than 15° (measurement C8) is

found

2 ¶02 ₁₅₂

u =

_{R 43.6}

=5:16

From Fig. 3b by means of extrapolation

(1P)

0.6% is obtained

The probability of the vessel heeling more than

15° under prevailing conditions is 0.6%. In the storm C8 the vessel may heel more than 15° six times in 11/4 hours (1000 heelings, period 9.2

sec.).

It will appear from the above that this

calcula-tion of extreme values is based on Short-term

distribution. Refined statistical calculations of

extreme values which are based on long-term

distributions have arisen in investigatiOns of Stress

in vessels at sea. The following method is taken

from [8].

-In one and the same weather group (groups are

according to the Beaufort scale) it is possible to

take a large number of samples of the so value and

thus obtain as many versions of r. This estimate

and therefore r =mp u2sp.

rap and sp denote the mean value of the' r distri-bution and standard deviation in the p:te weather group. h (u2) has a mean value =-- 0 and standard deviation 1.

The primary question is the probability of a

Certain ri value being exceeded. h (r) can in this

case be taken as a function of weight. It should

be noted that the «whole weighb is included; i.e. the integral for h (r) in (1) is to apply to the whole

area of definition and not a special ri value. The

normal distribution is now defined from a) to

+00, but as r is positive some of the left-hand

side of the curve disappears. In other words a truncated normal distribution is obtained. If the ein-ve were far to the left in relation to the brig° it would have to be taken into account by

intro-ducing a truncating function into the

calcula-tions. In general, however, that part of the area to the left of the origo is only a small per mine

Of the total and therefore the truncating funiction

can be omitted. This is the case in the following

Time So ra f =1/1000 1=1/10,000 Measured in period of observation 20 20 Under 4 hours 24.9 26.9

cc8

« 25.5 27.4

20 yrs, 4 hrs, twice yearly 28.0 29.7

cc CC CC a 3 times « 28.3 30.0

« 8 a twice « 28.5 30,2

cc a CC « « 3 times a 28.8 30.5

is based on random selection of various numerical

values. The parameter r therefore has its own

distribution. Empirical studies of stress measure-ments have shown that this is approximately nor-mal.

Each r value has a certain Rayleigh distribution and for every probability of exceeding a certain ri

value there is a corresponding probability of

ex-ceeding a certain soi value. The multiplication (of the law of probability) gives

f (97,r) = g (so/r) h (r) where f (so,r) is the frequency function;

g (r/r) is the conditioned frequency

func-tion of so, which is identical with the Rayleigh distribution for a given r

value,

h (r) is the marginal distribution in r.

The probability of both exceeding a certain so

Value and a definite ri value is therefore

CO CO P (so >

r>

=

f f f (r,r)

clsodr cf,j ri CO CO

f

g (r/r) h (r) dsodr . (1) cpi ri

Where the normal distribution is tabulated it is

easy to introduce the standardised function h(u2) as

112 =- mP

ip 30 20 10 e

(9/2

(mp +

V2it P sP

If these probabilities Pjp are taken with weights for each weather group we obtain

P P (se > rj, for all groups)

3 °P (nip + u2sp)2 2 2 . e 2 du2 (Pp 5°3

m + u2

calculations. According to the above reasoning the

following expression for the probability is ob-tained:

Pjp=.-P(cp>pj, in the prte group)

ff g .1( so /r) h (r)dr dso 0 cpj 1 2 U2Sp2

)

1e

u2 ( <Pi (mp+ u2sp)2) du2 2 2 U2 2 du_{_2} Fig. 5. b) sp

= 2 Q p

jp ; where _{QP = 1 (3)} P=-1

P=1

Qp. is the weight of the p:te group.

Since the probability is [1 P(9)], to obtain a

value greater than or similar to f

, the. expected

number of values is similar to n [1 P(r)] in a

sample of n individuals. From this can be defined the characteristically greatest value xu (n) =-- xu by writing [9]

F (xu) = 1 for n 2 (4)

From (2) and (4) a relation 'between the size of

the sample and the probability of exceeding a 5, j,

value is obtained. This 9, j value will be

thecharac-teristally greatest value.

P Pj (so > so j) =t 1

[1 1/n]

1OlogP='°logn

(5)

The numerical calculations of the maximum heel

are best dealt with as follows.

The Beaufort Scale is first divided into a certain number of groups, e.g. five.

0 3 Beaufort

4 5

8. 6. 7

q

8 9

a

10 12

4C

Then the mean value of the r variable On p) and

the standard deviation (s p) for each group are

calculated. By inserting the quotient in (2) the

probability of a certain

_{value (NB not yj}

m.

ob-tamed. But through the relation (5) it is possible

to know n, i.e. the number of heelings on a given voyage in one year or in the life of the vessel, in order to obtain the probable maximum heel [dur-ing the period in question. The calculations have been programmed for Facit EDB by Mr. N. Nor-denstrom at the Division of Ship Design.

The result of an estimate for vessel I is shown in Fig. 4, where the probable maximum angle of heel is given as a function of the number of heel-ings.

A vessel of this type will heel during its lifetime about 108.5 times. Fig. 4 shows that a maximum

angle of heel of about 32° can be expected. The material collected is far too inadequate for

cal-culations of extreme values and therefore the

above only shows the method of calculation where the results are of little value.

Rolling period

a. Statistical distribution

The distribution of the rolling period has been

examined for all measurable recordings. Where

the diagram of recordings is jagged and irregular

and the periods very short it was not possible to

evaluate the registrations as intended (see Fig. 5a). Recordings of this type arise at times and only under conditions in which the vessel meets

waves within a forward sector of about 120°.

The length of the period was measured on the

recording rolls between the peaks of the readings.

About 100 periods were studied in each recording.

A sequence of cycles was measured and they are therefore not selected (see Fig. 5b).

The material was classified and set up in a

histogram, see Fig. 6.

B classified breadth = 0.75 sec.

Tm = mean value of period.

The following Characteristics wqre calculated for the distributions [10].

Tm = arithmetical mean value s standard deviation

V 100s the variation coefficient which Tm

gives the percentage relation between

s and Tm

gi = measure of asymmetry.

The positive value of gi distribution moves over

peak is on the left of the mean number

and the extended incline on the right (see Fig. 7a). For normal distribution

= 0 applies.

means that the

to the left. Its

b) 36_

24-12_

-4 -3 -2 Tm 1

2 34

Measurement K 4 Still water ( at anchor) Tm = 6.8 sec-$ = 0,8 sec. T sec. Measurement K 2 3 Beaufort Beam sea T = 7.1 sec. s m= 1.5 sec. V = 20.4% - + 0' 663 gl gz = +0,623 .71 T sec. -4-3-2-1 m1 2 3 4

Fig. 6. Rolling period distribution.

-g2 = measure of excess.

The measure applies to the degree of

accumulation of peaks around the centre

in comparison with a normal ,distribu-bution (see Fig. 7b). For normal

distri-bution g2 =- 0 applies.

The characteristics are collated in Table IV.

The tables show that

the distributions are

usually very «peakedD compared with normal

distributions, i.e. there is a crowding of periods

around the mean value.

The curves reveal no

clear tendency towards asymmetry. It varies

rela-tively little and assumes both positive and nega-tive values. The variation coefficient varies

irre-gularly but does increase with high Beaufort

figures.

Fig. 6a is the vessel in a calm sea, i.e. external conditions such as wind and waves have no

influ-ence of any consequinflu-ence on heeling. 6b applies

to the vessel under the influence of these external

V =11.6%

gi + 0,040

TABLE

Distribution of rolling period and size

_

conditions. The internal conditions (d, GM) are, however, in the main no different (same voyage)

from 6a.

(In two measurements in still water on vessel

III the accumulation of periods around the mean value was so marked that only two types could be

found; i.e. the width of the variation was only

about 1.5 sec in this case.)

A normal curve has been used for the measured distributions and is marked on Fig. 6 by crosses. In order to ascertain how far the distributions can

be regarded as normal they were x2 tested. The test indicates that on the 5% level (the risk of a

wrong conclusion

is 5%) about a third of the

number of measurements were normally

distri-buted. This means therefore, that it cannot be said

in general that the distribution

of the cycle is

normal. The claim is also supported by the fact that a )(2 test of all the material for each vessel

shows irregularities. The individual distributions (see Table IV) show, however, with the exception

of .those with extremely high peaks, taken from measurements in still water, that they are

rela-tively close to the normal distribution.

b. Mean value

The mean value Trn of the cycles are given in

Table. IV. Examination of the Tm values during one'voyage reveals a slight increase in Tm during the voyage. This can be explained by the decreas-ing fuel reserves in the bottom tanks and thereby

a decrease in GM.

Tm was drawn up as a function of GM (see Fig.

8). The values measured are Marked by dots. The

continuous curve is obtained with the aid of the

equation.

2711

T

V g GM

-where GM 0.20 m

The radius of gyration k was calculated according to Kato [11]. k -=BmV 0.125 [CB CD ± 1.1 0 CD 2 Hy.

(1 - CB)(2.20) +-4

M

where CB is the block coefficient

CD ii the area coefficient of the

upper-most continuous deck

HE is the effective height of the ship at

the side

d is the mean draught.

No. of measure-95% conf *dence ment d GM T m s V gt g2 TA u . T ill o

m m sec sec % sec sec

A3 3.28 1.0 9.9 1.4 14.0 -1.063 4.461 9.6 10.2 A4 « a 10.3 1.8 17.5 -0.775 1.958 9.9 10.7 A5 « « 10.5 1.7 15.7 0.748 2.321 10.2 10.8 C3 « 0.9 8.9 1.6 18.4 -0.057 1.285 8.6, 9.2 C4 CC CC 8.6 1.8 20.5 -0.505 1.418 8.2 9.0 C5 « a 8.9 1.7 19.3 0.366 1.335 8.6 9.1 C6 a 6C 9.4 1.7 18.4 0 1.610 9.1 9.7 C7 CC (( _9.1 . .1 23.3 0.448 0.982 8.7 9.5 C9 « « 9.2 1.8 19.6 0.178 0.455 8.8 9.6 C10 a CC _{8.8 1.5 16.6 ' -0.563 0.668 8.5 9.1} E3 3.10 1.2 6.9 1.4 20.7 -0.610 1.148 6.6 7.1 E4 (C (( 7.2 1.2 16.6 -0.433 0.664 7.0 7.4 Fl 2.84 1.4 7.3 1.1 15.1 0.298 1.907 7.1 7.5 F2 CC CC _{7.3 1.3 17.9 0.595 0.968 7.0 7.6} G2 2.49 2.0 6.5 1.2 18.1 -0.013 1.373 6.3 6.7 13 3.30 1.3 7.8 1.5 19.4 -0.129 1.498 7.5 8.1 K2 2.72 1.5 7.1 1.5 20.4 0.663 0.623 6.8 7.4 1C4 a a 6.8 0.8 11.6 0.040 2.817 6.6 7.0 Li 2.44 1.8 6.8 1.2 17.6 -0.080 0.674 10.2 10.6

In a survey of various types of vessels Kato has

shown that the maximum error in k is not more than 3%. The approximate calculations must be taken into account When considering the rela-tively poor agreement with the GM values in Fig. 8. There seems to be a systematic error

Where-by too high values of GM are obtained. Possibly

this is due in part to the fact that the moment

method gives too good a value of GM and partly that the 'constant 0.125 in Kato 's equation should

be modified for this type of ship. Fig..8 Shows that

Fig., 7.

POS. g2

8 9 10 11

Fig. 8. Vessel no. I.

NORMAL DISTRIBUTION

. .

a).

oaC

the constant should be increased somewhat , for

Vessels I and II. On the other hand, for vessel III

it appears to be fairly

applicable. To alter the

value of the constant, however, more material is

heeded than available from the investigation. In order to obtain an interval evaluation of the

population mean value Tv., i.e.

the true Mean

value; a 95 per cent confidence interval has been

calculated;

Tin 1.96 vN T

Tm +1.96 v-K1

Fig. 8: Vessel no. III.

6 8 9 10 11 12 900C Fig. 8. Vessel no. II.

diagram the captain can easily judge the stability

of the ship [1].

The Peribd can easily be found by measuring,

for example, the time of 'ten consecutive rolls with

a stop watch. This is easy to do at sea as the ship rolls

considerably. At the quayside or in port

before sailing the vessel must be made to roll with the help of its cranes, by moving the rudder

etc. The rolling thus obtained is relatively slight but is easy to measure with a stop watch by

ob-serving the movement of the vessel's railings (or

some other fixed object on the ship) against a

distant fixed point on land.

The precision With which the captains measured

the rolling at sea is clear from Fig. 9a where T f

i.e. the mean value of about ten consecutive

pe-riods, is seen in relation to Tm ,

the true mean

value. The cycles measured agree in general with

the true ones. In most oases the deviation is less

than 1 sec., which must be considered accurate.

An error of a second in T f may of course mean a relatively large error in determining the GM value When this is great. With the value of GM where the stability can be critical, however, this accuracy is sufficient (see Fig. 8.).

Fig. 9b shows T10, i.e. ten consecutive periods

taken at random of recordings in relation to Tm.

The figure therefore shows the theoretically Pos-sible accuracy with which Tm can be determined from several groups of ten consecutive periods. A comparison with Fig. 9a shows that the scatter of T10 and Tf is of the same magnitude, which

indi-cates that the measurement of the period at sea with a stop watch is very reliable. The accuracy

of the measurement can naturally be increased by

repeating the measurement cycles as the mean value improves with the number of periods it is based on. A calculation of the mean value for

three series of ten, for example, chosen at random from the diagram of recordings shows that varia-tion from Tm is less than about -±- 0.5 sec. Connection between angle and period

In the measuring trials when the R value was calculated, i.e. with heelings greater than 5° (to

one side), the angle of roll was measured for each period. The angle F2 was taken in this case as the

mean value of the starboard and port readings

within the period.

The mean value of the largest third of the

heel-ings within each group of periods (breadth of

group is about 0.75 sec.) was calculated and can be called the significant angle of roll: The

maxi-mum angle in each group of periods was also

Fig. 9. Mean rolling periods measured at sea.

where N is the number of observations in the

measurement. The values of the upper and lower

limits are denoted by T ttu and T t, (see Table IV).

The significance of the above variance is that

in repeated random measurements the mean value

of the rolling period in 95 cases out of 100 is

within the limits and the remaining five are the

consequence of extreme or rare occurrences. On the basis of the measured results available,

it can be said that there is no appreciable

differ-ence in Tm between the natural rolling of the

vessel in still water on the one hand and rolling under the influence of wind and waves on the other. Several authors have confirmed that the mean value of the rolling period is practically independent of influence from various wave

spectrums (see references in [1]). This feature of Tm makes it a particularly suitable instrument for both builder and captain for judging stability. The

naval architect can draw up a diagram of the

rolling period on which the limit curve of the

lowest permissible metacentric height, i.e. the greatest permissible rolling period, is a function of

the draught. By comparing the period measured on board with the maximum permissible on the

measured. This value

is drawn over a period

distribution in Fig.

10. The results shown are

characteristic of all the measurements. The figures of the extreme value of the cycle are unreliable as in this case they are usually based on only one to

three readings.

The figures show that the maximum angles of

roll occur in the area where the cycles are most

frequent, i.e. immediately around the vessel's

natural rolling period. The rolling Motions are

regular within a period interval of about -!- 2 sec. 'around the mean value. Outside this regular belt

the motions are the result of external forces and

consequently 'both period and angle vary greatly. Conclusion

For several stability criteria the measured angle

of heel is

difficut to calculate [12], [13], [14].

Two methods of determining a vessel s maximum

angle of heel have been outlined, of whieh the

second, based on long-term clistribUtiori, is to be

preferred. To obtain a representative basis for

similar calculations it is preferable to measure

various types of ships over a long period.

The rolling period concerned the measurements

indicated:

1. Distribution of the rolling period accumulates

around a mean value which corresponds to the

vessel's natural rolling period in cairn: water. The:distribution has only a slight' asymmetry. The mean value appears to be independent of external conditions such at wind and waves. Estimation of the vessel's rolling period can be

made on board with a stop .watch to check

stability with good accuracy.

Maximum angles of roll occur *hen the vessel .rolls in its natural rolling period.

The measurements which could hot be

evalu-ated are an important exception to this. lit such cases the vessel is sailing against the waves. The

rolling motions then appear to be coupled to the pitching and therefore the rolling angle and

pe-riod are relatively small. This condition should be

noted When measuring the period at sea at then the period is usuallj, estimated to be 'somewhat shorter than it actually is. The result is an

over-estimation of the GM and stability.

List of Symbols

BB/ = Breadth moulded of ship

CB Block coefficient of ship

GM Metacentric height of ship

Lpp Length between perpendiculars

N Number of observations (total)

111111...1

0.16006rnent A .1 besulut1 FolIowinS00 1106,, 0,001II) VESSEL III -2 1 VESSEL 25 u ... uuu. uI Ruaul 20 ChuuuLn._r.. 10.6 mec. 10 ° ID

Fig. 10. Connection between large angles of roll and

relevant rolling periods.

Histogram periOd distribution

x = significant angle of roll 0 = maximum angle of roll.

Norrby, R: The Stability of Coastal Vessels»,

'

Trans. RINA, 1962.

Bennet, R.: International Ship Structures

Con-gress Report of the Committee on «Response to Wave Loads», D'rMB Report 1537, 1961.

Jasper, N. H.: «Statistical Distribution of Ocean Wave-Induced Ship Stresses and Motions, with Engineering Applications», Trans. SNAME, Vol.

64, 1956

Cartwright, D. E.: gOn Estimating the Mean

Energy of Sea Waves from the Highest Waves in a Record», Proceedings of the Royal Society of London, Vol. 247, 1958.

Bennet, R.: gMlin.ingar av spanningar och,rOrelser i fartyg till sjoss» (Project S-1), SSF Report No.

13, 1958.

Jasper, N. H.: «Statistical Presentation of Motions and Hull Bending Moments of Essex-Class Air-craft Carriers», DTMB Report 1251, 1959.

REFERENCES

Birmingham, T. T., Brooks, R. L., Jasper, N. H.: «Statistical Preseniation of Motions and Mal

Bending Moments of Destroyers», DTMB Report

1298, 1959.

Bennet, R., Ivarsson, A., Nordenstrom, N.; aRe-snits from Full Scale Measurements and Predic-tions of Wave Bending Moments Acting on Ships»

(Pririjet S-15), SSF Report No. 32.

[ 91 Gumbel, E. J.: a Statistics of Extremes», New

York, 1960.

Hyrenius, H.: «Statistiska metoderA, 4th edition,

Stockholm, 1962.

Kato, H.: «ApproXimate Methods of Ciaduiating the Period of Roll of Ships», Journ. SNAJ, Vol. 89,

1956.

Yamagata, M.: aStandard of Stability Adopted in Japan», Trans. INA, Vol. 101, 1959.

Register der UDSSR: <Stabi/itatsvorschriften fiir See- und Hafenschiffe», 1959.

Rahola, J.: «The Judging of the Stability of Ships and the Determination of the Minimum Amount of Stability», Helsingfors, 1939.

N h -= Number of observations per hour

= Probability

Parameter of Rayleigh distribution

= Rolling period

T m = Mean value of rolling period

T f

= Mean value of rolling period

mea-sured on board

Tlo = Mean value of rolling period over 10 rolls

T p,o ,Upper limit of confidence for Tm T = Lower limit of confidence for Tm

V Variation coefficient

= Ship's draught at time of

Measure-ments

d s = Draught on summer load line

=-- Risk factor

g1 =- Measure of asyMmetry

g2 = Measure of excess

= Radius of gyration when rolling

-= Mean value Standard deviation

= Standardised variable

F

= Angle of roll

rn -= Calculated extreme value of angle- of

roll (0 port or 0 starboard)

r

max =

Maximum measured angle of roll

i02 =- Measured angle of roll

(port-starboard-port or starboard-(port-starboard-port-starboard)

DTMB = David Taylor Model Basin

ISP =- International Shipbuilding Progress

RINA

(INA) =- Royal Institution of Naval Architects

SNAJ = Society of Naval Architects in Japan.

SNAME--- Society of Naval Architects and Marine Engineers

SSF = The Swedish Shipbuilding Research