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Title ON THE STATISTICAL DISTRIBUTION OF MEASUREMENTS MADE AT EQUAL INTERVAL OF TIME OFSHIP MOTION AT SEA

Author(s) KAWASHIMA, Rihei

Citation 北海道大學水産學部研究彙報, 18(4), 329-344

Issue Date 1968-02

Doc URL http://hdl.handle.net/2115/23332

Type bulletin (article)

File Information 18(4)_P329-344.pdf

Hokkaido University Collection of Scholarly and Academic Papers : HUSCAP

ON THE STATISTICAL DISTRIBUTION OF MEASUREMENTS MADE

AT EQUAL INTERVAL OF TIME OF SHIP MOTION AT SEA

Rihei KAWASHIMA.

Introduction

Generally speaking, the ship's motion induced by the action of ocean waves can be treated as a stationary stochastic process. Especially we know that the method of cross-spectral density functionsl) for analysis of ship response to waves is a practical way to study the sea-keeping quality in full scale ship experiment at sea.2 )

In the treatment at these data, the author has used digital computations in a spectram analysis. According to the author's observations 3) , the values were taken at one second interval from the recording charts; the number of observations used in this analysis is from 300 to 1,000, and the sum of the total time, (MAt), 300 to 1,000 seconds.

In this paper, the author discusses the statistical distribution of the above data and especially the two dimenstional distribution, with respect to the simplified method of calculation for correlograms.

Regarding the statistical distributions, it is well known that the most common and easily determined statistical parameters are the mean and variance (or mean square). The Gaussian (or Normal) and the Rayleigh probability distributions can be completely described by their means and variances.

The theory of statistical distributions of maxima of random functions has been discussed by D.E. Carghtright and M.S. Longuet-Higgins. 4)5) According to their theory, the maxima of the distribution can be used to measure the degree of agreement between many theoretial relations and actual observations.

However, in this paper, the author discusses the following two problems from a. ditferent point of view: (1) The statistical distributions of ship's motion and wave height; a.nd (2) Two dimensional distributions for time series.

Collection of Experimental data

Data for analysis were obtained on May 7, 1965, at Station No.1 (4:2°11'N., 145°45'E.) of Cruise No. 13 (east of Cape Erimo, south of Hokkaido) aboard Train­ing ship Oshoro Maru of Hokkaido University.t)

• FacuUy of Fisheries, Hokkaido UMversitJ

-3.29-

Bull. Fac. Fish., Iiokkaido Univ. C:XVIII,4

The wave heights were measured by means of a wave pole equipped with a specially designed radio telemetering apparatus7 ) ; t he angles of roll and pitch were measured by use of vertical gyro. Measurements of motion were taken for every run in which the ship 's course was 30 degrees to the estimated wave direction, the schematic diagram for relation of ship's head and wind direction in experiment is shown in Figure 2-d. The sea state of the time of the experiment was 4.

Typical brief series of values from the records are shown in Figure 1. The values used in the analysis are defined by the following constants; (1) .1t= 1 sec. and (2) M= 300, where .1t is the time interval between adjacent values ; M, the number of samples used ; and M iit, the total length of time for all observations.

Fig. 1 Typical!brief series of records used for analysis

The Probability Density Functions for Ship's Motion and Wave Height

The statistical distributions for wave height and angles of roll and pitch were estimated from the observations described above and t he corresponding histograms are shown in Figure 2. The means and variances for these distribut ions were computed according to Gaussian probability papers and these values are given in Figure 3 and Table l.

From the results it is concluded that the distributions are Gaussian in form ; t hen the probabili ty density functions can be represented by the following equation :

}) (Ao) = 1 e uy27t

(1)

where; m= mean valnc, and u 2=va.riance. Values for the variance and the means are given in Ta.ble 1.

-330-

1968]

-2 -,

R. Kawashima: Statisticai Distribution of Ship Motion

, 2

~. EU.JO.R-3 lIXP.IO.I-2

- --4- 8

~ -,6 - -4 - 4 jj

Fig. 2-a Angle of roll

Fig. 2-b Angle of pitch

Freq III

60

20

-, 0

o -120"~-30 0 30 III 10 121 em

Fig. 2-d Diagram for relative angles. between wind direction and ship's heads

Fig. 2-c Wave heights

I

~ I

GatSSi an probabili ty paper

·rrnITITITITITITITITrrrrrrrrrrrrmT~mTmTmmmn·~mnmnTITITITInn

EX.II0.1 1111111

EX.II0.2

EX.II0.3J

EX.II0.4 1-<++++-'.'.++++ EX.II0.5 H+t+t+H . ,

o .10.6 !±t

IIlIll • ".".'111 •..

lli.: tflT'H1 ... o ~ : N1Wfrl iRE:E:i . It • o • i.f-!,Iili~+i'flf,~ li~l!-ff.. " II ,cH?BttHiSL ttJ'li:!ih

tfu1H!t 'Utj () ~r!~hlh ~-:~i:d': ;t

~tf1U'llli . tt' 'RWm j Hll ~' ,il 'U:r;' +, il1Hf!i 1m, till' ,I In Iltlti,tll++H't++I+++t H+t++H++HH+t+++tftHttI

I #11.t~ ;f;:. ~H"1 J: R+~t-r .. IH ~/ H j;" \ f1tHi4 ji '!~j + It

I' ,t i1 tm~!ht i~1 ;. 111 "_ -.f '+ rn i:F ;_ 't:""j. t

. ttt + ill ttirlfl:ffi i Hi ! .i j,

10 -8 -6 -4 -2 0 2 4~---+6

deg Fig. 3-b Angle oj roll

0.01

0.1

20

30

'0 60

60

70

00

.0

.5

• 9

....

.UII

-I ...

.. •• 00

_0

70

110

OO

to ...

3.

20

10

0.1

lOI

-4

Gat..5Sian probability paper

0.01

0.1

II EX.II~.l

EX.II0.2 JTll1llTr

EX.II0.l Ili:III!111

1 EX.II0. 4 ! IIIP'I'II r,

XI~::{1: .110.6

~ 10 ::::

4+t-~

EX.II0.7

:.j.!"-+

+;t-... -h:l-t"t~ ...

20 ~ 30 i! '0 60

r 60 ~ 70 "'" 00 ~ .0 ~ 86

!i.

.. 88,11

-3 -2 -1 0 2 h ---+ 4 deg

III." i Fig. 3-a Angle of pitch ~

1968] R. Kawashima: Statistical Distribution of Ship Motion

~ '" g

.,; ~ .. 0 ~ .. Q ~ g g .. g gi ..

~ ~ ~ ~ '" "'= 0>

i d "" I- .... •

L.. ...

l .= ._.

.., ~ I

E ] .0 = u '- J ..... :cs ~ ..,

t :-c, 0 I ~

~ ~ '-~ t!)

I '" g: ::: .. .. ~ i .. ~ .. .. ~ .. Ii '" D ., ~ o. 0

Ii d

-333-

E

Bull. Fae. Fisk., Hokkaido Univ. [XVIII,4

Table 1 Variance of ship motions and wave htights = --

x.No·1 I Angles of roll* Angles of pitch'*

i ! I

I

I Means I ICo-var.1 Means S Var. ICo-var. S i Var.

1 1-0.250 2. 1251

4.818 4.752 0.480 1 I I 0.760 . 0.5776

1

0.5'30 ; 2 1-1.000 3.015 9.090 9.400 0.500 0.775' 0.6063 0.6575 I

3 1- 0 .608 2.

0351

4.141 4.216 0.580 0.890 0.7921 0.7886 I 4 ,-0.850 2.800 7.840 7.613 0.640 1.045 1.0920 1.1(,63 5 1- 0

.720

I

1.515 2.295 2.334 0.750 I 0.975 0.9506 i 0.9478 . 6 -1.150 I ~:~~g I 5.063 5.019 0.680 0.933 0.8696. 0.82041 7 -1.300 I 8.703 8.853 0.750 1 0.810 0.6561 I 0.6512

* Angles ot roll; Ex. No.3 Roll ** Angles of pitch; Ex. No.5 Pitch

The Probability Density Functions for two Dimensional Distributions

w ave he

- !gh~

V ar.

O. 1764

The procedures for estimating two dimensional probability density functions can be illustrated as follows:

Use the data, x(i.::lt), i=l, 2, 3, .... , M, from the records. As described i 1

Figure 4-a, plot values of x (i.::lt) along the abcissa and x ((i+r).::It) along the ordinate, M-r points can then be determined from the Figure. The points will be scattered around lines x(i.::lt)=x(i+7).::It), and x(i.::lt)=-((i+r).::It). (equations (B) and (A), respectively)

For these lines, the distributions were calculated for the each meshes as described in Figure 4-a, the corresponding histograms are shown in Figure 4-b and

ROLL EX.NO.3

Fig. 4-a Two dimensional time series distributions for angle of roll

-334-

I

~ I

A

1 4 8 1~ -8" -4 () (Deg)

2 JL ~"-~=4~0-"i . 8~-12

(Deg)

B o I--=4T -2 6 2 4' 6(Deg)

" r;==;"1 -.- ••• -4 -2 0 2 4 6

(Deg)

2~ "-'4- • -2 • O"~ 2 • • ~6'

(Deg)

3 /L. ~.~ -8 -4 (,'''' 8 "12 F . ':4 . -2 . o· " 2 . 4 --6

4Li~' .f4 1\':' -8 -4 0 - 4 8 12 -4 -2 0 2 4 6

~ ~ :':". " 5 £b ,~, (Deg) - -u 2 4 ~6 -4 -~ ..

6!1'~" , - -4 0 4 8 12 7 (Deg)

~-~'4' "-812 8 (Deg)

• -b ':4' "0 :. 8' 12 (Deg)

6

7

A (Deg)

-4 -2 0 2 4 6 (Deg)

~ . -4 -2 0 ;1 - r~-6T 8 . (Deg)

':4 ' ':2' 0' ';; 4 6 (Deg)

Fig. 4-b Histogram for meshes in Fig. 4-a

80. FrelJ..

70

60

50

40

30

20

10

o -12 -8 -4 0 4

100 i FrelJ..

90

80

70

60

50

40

30

20

10

01 •• '7'=T', , , -6 -4 -2 0 2

A

8 12 (Deg)

B

4 6 (Ileg)

Fig. 4-c Histogram for beltlike area in Fig.4-a

,....

~

~

[ If Q>

i ~ e. i-f g" s,

l-f.

Gaussian probabili ty paper Gaussian probability paper

.... aG 'I 0.01 .... 111 0.01

.. 0,1 ",I 0,1

til til I

,j' I

)

.s

00 t 10

Fs t~

Fjl:

-+ -r_t1=f;· "

~'1F; H

, ! L

! -i: j ..

" ' il 1 +1 - .j. ... ,~

: "'

, , + ll, . 1-,' n

I

I I

eo

:J<J

II

III

20 110

30 70

eo

.0 70 :J<J

20

20 ~ 30

~ .0 ~ 60

~ 80

70

i 80

H t 90

~ 9C ~.

" t , , I ': I

II I t- ~ , 'h

go •• I ~ .. I It

+ltl· -d i

0,1 9 .• 0,1 89.8

'+ r

8 -6 -4 2 0 2 4 6 ... -+ 8' 0,01

(deg) 4 -3 -2 -1 0 2 , 3 ... --. 1

·UtI 0.01

(deg)

aG.aG >( ~ ..... ~

Fig. 5-a Angle of roll Fig. 5-b Angle of roll ~

1968] R, Kawashima: Statistical Distribution of Ship Motion

a Q Q 0 Q ..

= e :; ~ Jil ai '" g i d 0 ~ . ~ ~ " ex>

r " en "" ..... ..... co:

~ CD ..... = ....

~ = e .0 _ ...... .... '- 0

~ ,$

I '"' ~ = c

0 , lQ

~ , , ~ '-

~ .....

~ n i--i-' ~

11=, I i : ~ 0 il ~ i 0 ~ 0 g

"' 0" .. ~ n d • d

-337-

x(n + l) x(n + 2) x(n + 3)

x(n + 7) x(n + 8) x(n + 9)

'7 ~~ A B .!7.........,. . .. , . __ A ....... B .

'" r I "j 1~1.1t j I '" 00

x(n+13 x(n + 14)

x(n + 4l x(n + 5)

x(n + lO) x(n + ll)

rt)l !' Bl _ ' _0 ' '''' r ! /',..... A

/ ' . tj ;.:-fV 1 -', '''l~ Ii

Fig. 6-a Wave heights

x(n + 6)

x(n + 12) N ~ It::.

/ / ~

~ "l " >-"l ;;;. ~ ~ C ". ".

" ~ c

~ ~.

~ ~ >-< :-< >I>-

1968] R. Kawashima: Statistica l Distribution of Ship Motion

'" §' 0- ....

+ + + " " " >< >< >f

;:.:- l ' S .... .... + + + " " " ::r >f ><

~ 0 0-.... + .... + " + " >< " K ><

"0 '" .... 0

$ 00

" < G' .!:J

;;- C£ , .... <0 + + + .!:P " " " >< >f >< I"<

'" Cii ~ + + + " " " >< >< >f

" M .... L~ ;:-+ + .... "

on " +

>< i£; >< " >(

'" '" rn

-339-

x(n + 25) x(n + 26) x(n + 21) x(n + 28)

'" ... 0

I

x(n + 31) x(n+32) x(n + 33) x(n+34)

Fig. 6-b Angle of roll

x\n + 29) x(n+30)

x(n+35) x(n+36)

~ ";j

" " ";j :0;:'

!" i:l::: c "'" "'" " ~ c

c:: " ~.

>i ;:1 H !"" ..

1968) R. Kawashima: Statistica l Distribution of Ship Motion

;:::;- t ;:-G' .... .... + + + ,

* t::

t::

1< 1<

G' ~ .... + + -§ t:: t::

1< 1< ., 'p,

""' 0

" 00 t:: ..:

<.>

§: J:,

M oil + + fi; t:: t::

1< 1<

;::; ;:- §' + + + t::

== == 1< 1< 1<

-341-

· Bull. Fac. Fish., Hok1caido Univ. [XVIII,4

4-c. Also, the distributions are shown by Gaussian probability papers in Figure 5. However, from practical points of view, the method can be simplified by using distribution for each lines as shown by the beltlike, shaded area of Figure 4-a.

The distributions of these simplified method, based on Gaussian probability papers ~re shown in Figure 6. The alphabetical characters for lines in the Figures correspond to equations (A) and (B) noted above.

Sin~e the two dimensional distributi()ns for angles of roll and pitch and for wave heights in the sea are approximately Gaussian in form, then, the joint probability density functions can be given by the following equation:

p(X(iJt), X(i+T)Jt)=2 ~1 2exp [- 2(11 t) n'!Tl!TZ - P - P

X {( X (i J ~ - m): )2_ 2 p(X (i J t) - ml~~!} (i + 7) J t) - ma)

+ (~J(i + ~:t) -m2)2J

] (2)

where; mt; mean for x (iJt) , mt : mean for X«i+7)Jt), !Tt2 : variance for x(iJt), !T2

2 :

variance for X«i+T)Jt), i=1, 2, 3 .... , M., p: correlation coefficient for x(iJt) and x«i+,r)Jt).

The covarianc~ and variance for the data are given in Tables 2 and 3.

Lag

1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20

Table 2 Means and variance

Angles of roU. Angles of pitch ••

A

-Means Iv~~:--

- 0.17 0.8464 -0.30 0.2304 -0.30 0.1225 -0.28 0.1806 -0.25 0.2756 -0.23 0.2256 -0.28 0.1936 -0.30 0.2162 -0.25 0.3025 -0.30 0.2601 -0.30 0.2401 -0.32 0.2304 -0.30 0.2916 -0.25 0.2025 -0.25 0.2209 -0.20 0:2401 -0.20 0.2500 -0.30 0.2401 -0.30 0.2500 -0.25 0.2500

-0.30 0.4225 I -0.60 7.6176 -0.52 1.2544 -0.30 0.3025 -0.80 3.6864 -0.80 4.0000 -0.30 0.3600 ~0.56 1.5625 -1.20 3.6864 -0.25 0.2500 -0.40 2.5600 -1.00 6.7600 -0.20 0.2256 -0.22 0.4489 -~.20 6.25QO -0.23 0.2256 -0.72 3.3856 -1.04 3.2400 -0.30 0.2500 -0.80 5.7600 -0.96 1.9600 -0.28 0.2304 -0.88 6.7600 -0.88 1.9600 -0.20 0.2601 -0.80 5.8564 -0.60 1.9600 -0.21 0.2401 -0.88 4.0000 -0.88 3.3856 -0.30 0.2209 -0.88 3.0976 -0.88 4.6656 -0.31 0.2401 -0.40 2.3716 -1.04 5.9536 -0.28 0.2500 -0.76 2.3104 -1.20 5.7600 -0.25 0.2025 -0.76 3.0976 -1.20 4.0000 -0.25 0.2209-1.08 3.5344 -1.00 3.8416 -0.22 .0.2401 -0.48 4.8400 -0.80 2.3104 -0.25 0.2304 ~0.80 5.7600 -1.00 1.9600 -0.30 0.2401 -1.28 4.8400 -1.00 2.6896 -0.22 0.2401 -0.72 4.0000 -1.08 4.4944 -0.25 0.2401 -0.88 3.5344 -1.00 4.0000

• Angles of roll; Ex. No.3 Roll .* Angles of pitch; Ex. No.5 Pitch

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1968] R. Kawashima: Statistical Distribution of Ship Motion

Table 3 Covariance for ship motions

Lag I Angles of roll liAngles of PitChl' Lag· I' Angles of rol1iAn~les of pitoh

Ex. No.3 Ex. No.5 I Ex. No.3 Ex. No.5 , 0 ,4.2157

·1 0.9478

I 19 0.0069 0.0942

1 2.9427' 0.3996 20 -1.1797 0.0545 .2 0.1312, -0.2484 21 -1.6892 -0.0921

3 -2.4437 -0.4828 22 -1. 2501 -0.2161 4 -3.3729 -0.2252· 23 -0.1345 -0.1648 5 -2.3825 0.0109 24 1.0128 0.0535 6 -0.3000 0.0291 25 1.5641 0.2496 7 1.6748 -0.0391 26 I 1.2650 0.2540 8 2.6040 0.0124 27 I 0.2994 0.0995 9 2.1817 0.0637 28 i -0.8197 -0.0350

10 0.7798 0.0927 29 I -1.5262 -0.0568 11 -0.8293 0.0807 30

I -1.4254 -0.0335

12 -1.8764 0.0706 31 -0.5195 -0.0401 13 -1.92® 0.0296 32

I

0.7289 -0.0,765 14 -1.0451 -0.0063 33 1.6029 -0.0703 15 0.2957 -0.0109 34 1.5611 -0.0322 16 1.4208 0.0028 35 0.6184 0.0551 17 1.7870 0.0185 36

I -0.6569 0.1448

18 1.2150 0.0552

Discussion

There are many applications in which knowledge of the frequency distributions of motion and wave height can be used to advantage. A few examples are (1) description of the statistical properties of ship motion, (2) estimation of the maxi­mum values of ship response to wave, and (3) the use of the method for simplified computation of correlograms. As for item (3), as long as the ship's motions are due to ocean wave action, the simplified method for calculations of correlograms can be used. The author has already used this method to solve ship's problems.S)

If the characteristics of the ship's response to waves are considered to be both a second order process and a Gaussian stationary process9), then use of the simplified method for calculation of correlograms will provide a fairly good fit for the data. This is an important characteristics in the studies of ship motion at sea.

Conclusion

These conclusions are based on observations of the relation of ocean waves to ship's motion at sea and from the above discussion, the following conclusions have been reached: (1) It is considered thatthe distributions for the da.ta. used in this experiment are Gawisian in form ;(2) For one sample' from 'these experiments, it is shown that the two dimensional distributions are Gaussian in form; and (3) From the statistical properties of ship motions due to wave action, the covariance and the mean can be used as parameters to represent the characteristics of random motions of a ship at sea.

B'Mll. Fat. FNIt •• HoklcGUlo Unit). [XVIII, "

Acknowledgements

The data presented in this paper were obtained as the results of the cooperative effort of the entire crew of T .S. Oshoro Maru. The author express his hearty thanks to Dr. M. Huzii and to Dr. Y. Yamanouchi for their helpful comments and advice. Thanks are also due to Mr. K. Amagai and Miss E. Ohtaki for processing for all of the needed to obtain the numerical results.

Summary

The main objective of this paper is to define the statistical properties of ship motion using records collected during a cruise of T.S. Oshoro Maru.

Data for analysis were sampled from the records of motion using the follow­ing constants: (1) The time interval between adjacent data values was 1 second, and (2) the number of observ&tions was 300.

From these data, (1), statistical distributions for various conditions are calculated according to Gaussian (or Normal) probability theory; (2) two dimens­ional time-series distributions for ship's motions are estimated according to Gaussian probability papers; (3) all diStributions can be represented by Gaussian probability density functions; and (4) the variance for ship motion and wave height defines the characteristics of a ship's motion at sea.

Literature

1) Yamanouchi, Y. (1964). Analysis of ship oscillations in waves. Ann. Inst. Stat. Math. SupplefMnt III. 19-32.

2) Kawashima, R. (1964). On the reeponlle function for the rolling motions of a fishing boat on ocean waves. ibid. Supplement III 33.-40.

3) Kawashima, R. (1965). On the response function for the motions of a fishing boat on ocean waves, Part II. Jour. Naut. Soc. Jap. 32, 49-55.

4) Longuet·Higginl, M.S. (1952). On the lltatiatieal distribution of the leighia of tea waves. Jour. Mar. Re&., 11, 24&-266-

5) Cartwright, D.E. and Longuet.Higgins, M.S. (1956). The statistical distribution of the maxima of a random function. Pro. Roy. Soc., Ser. A. 23, 212-232.

6) University of Hokkaido. (1966). Daa records of oceanographic observations and exploratory fishing. Faculty of Fisheries, University of Hokkaido, (10)

7) Kawashima, R. (1963). On the measurement of ocean waves. Brdl. 1"40. 1"i81. Holck. Unit). 14, 7-21.

8) Kawashima, R. (1962). On the remote monitoring of ship motions·II. Jour, J!lll.ut. Soc. Jap. 24, 11-17.

9) Hilmi, M. (1004). On a simplified method of the estimatiOn of the oorrelogram for a stationary GaUyian prooen·II. Kodai Ma.lMm«tical Semi_r liepo1'U"16; 199-212. Tokyo.

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