Ship's yawing in waves

Introduction

This study has been completed through a

detailed analysis given to the contents and

experiments of Dr. Suyehiro's thesis.

That a small ship in waves tends to

be-come parallel to wave train in some case and

right angular in

the other case is a

phe-nomenon known old by any keen-eyed mari-ners. This is caused when gyrocouple changes the bow direction induced by the joint action of rolling and pitching of ship. From such

a viewpoint, Dr. Suyehiro attempted to give 1

Faculty WbMT

Dept. of Marine Technology Mekelweg 2, 2628CD Delft

The Netherlands

Ship's Yawing in Waves

By

Daikaku MANABE

Assistant Professor of Kyushu University

(Received December 8, 1959)

Contents

Page

Introduction 1

Chapter I Stability of Yawing When Wave Length is

Ex-tremely Large 1

Yawing motion induced by the rolling and pitching

in waves 1

Expression of three-dimensional oscillation of ship

due to rest coordinate system 4

Stability at the time of resonance and the condition

of yawing motion 6

Chapter II Stability of Yawing Motion When Wave Length

is Extremely Short 12

Fundamental equation of the general motion on the

waves 12

Stability diagram of square-ship 19

Chapter III Results of Water Tank Experiment 25

Model test 25

Conclusion 26

Bibliography 28

Appendix ₂₈

it a mathematical solution and found out a law in the relative magnitude of the period of wave as well as that of rolling and pitch-ing and the direction of yawpitch-ing, confirmpitch-ing it by a simple model experiment.

The author forwarded the analysis further on a general case where wave length corre-sponds to the size of ship to investigate the motion condition of yawing and conducted evaluation on the stable condition of various

types of ship by changing their natural period,

draft and wave length to compare it with the results of tank experiments.

The outline of the contents will be listed

4..

as follows. In Section TI,. the excerpts of Dr.

Suyehiro are quoted from Katsutada Sezawa's

book for the convenience of reference. Section

2 introduces the author's suggestion that

general motion of ship can be expressed by

taking a special limit of Eulerian angle,

pointing out that it can be explained well

availing rest system instead of moving system

as Dr. Suyehiro has done. In Section 3, the author who has found Dr. Suyehiro's record of experiments rather inclined to the one with longer period, explained it from the damping resistance at the time of resonance, and

re-vived theoretically Dr. Suyehiro's picturized

record by integrating yawing motion equation

graphically. The discussion so far has been confined to such a case where wave length is considerably large compared with ship, but in the following section, more detailed analysis were developed, taking the asymmetric yaw-ing couple of waves upon ship into

considera-tion. In Section 4, equations of general motion in waves were led newly by modifying

the drift and yawing on the basis of Kryloff's

theory. In case wave length is infinitely great, the result of it corresponds to the

equation of Dr. Suyehiro. In Section 5, the

property of stability discriminant equation for a box ship is explained as a simple example.

rolling:

pitching .

yawing :

Fig. 1 Yawing motion of ship due to pitching and rolling

LA6

I

dB dcb

+K

do + Dg,no Dgh0 cosacospt, di di di 1d20 +1dq do + K, dO _{+ Dgh'0=Dgh'0 sin a cos pt,} dt dt dt (PO dc6

de +K do _0.

dt dt dt

While in the sections so far the author has developed a general theory including the case where the scale of waves and the size of ship are nearly matching, in the following section its correspondence with theoretical value has been confirmed by towing tank experiment using a model ship. In Section 6,

explana-tions are made on the

results obtained on diversified types of ships in general.

Chapter I Stability of Yawing When Wave Length is Extremely Large 1. Yawing motion induced by the rolling

and pitching in waves

A brief statement will be made here on the

achievements done by Dr. Suyehiro as follows.

Taking axes x, y, z attached to ship passing the center of gravity G of ship, the rotation angles around these axes will be chosen as

0, cb, b as shown in the figure. Letting the

maximum gradient of wave slope be 0, wave

crest line will form the initial direction of

axis x and the angle of a of

ship. It will be specially assumed that the moment of inertia around a given line in yGz surface,

which is passing G, is constant. The equation

of motion about axes xyz will be as follows,

(1)

where, I, is the moment of inertia aroud axis x, I is that around axis z; h, h' the height of GM for rolling and pitching respectively, D displacement of ship ; K, K',K" damping

coef-ficients, 2711p the period of wave and g the gravitational acceleration.

I

Drawing a general solution of 0, 0 from the 1st and the 2nd equations

do I dt and substituting into the 3rd equation,

2 daCb _{_adp2 cos (8 0') adp2 cos (2pt 8 8')}

n de

provided

= a cos (pt 8) b cos (cat r), 0=a' cos (pt 8') b' cos (co' t r').

V(012-1,2)2+ k"p2

tan r tan ,9 ;

tan O' ₍₀₁₂k'p_p2 A V (0P2)2+ k2P2 kp tan ,

(i)22

to obtain dOldt, bpw[cos

p)t r +

cos {(w + p)t r 8'}]

ab'

[cos {(pa)t fi +r'} cos {(p+a)t--(3r}

+

aRo' [cos {(o a)t r+r'} cos{ ((o+ ()/)trr1].

. ... ... ... (4)

Since these depend on the periods of ship and wave, oscillatory force and initial condi-tions are too complicated to be discussed in general, following special case will be

studied with respect to its tendency.

( i ) When consideration is given to such a case where the period of wave is shorter than that of pitching, since the period of rolling is

longer than that of pitching as a

rule, we have P>co'>o) in this case. In the next place, h, k' may be assumed as

extreme-ly small.

.13

Collecting therefore the above results, they

may be rewritten as follows,

cPsb

=C sin a cos a -I- periodic terms ..(7)

dt2

C is plus constant and

C sin a cos a = n2 aa'p2

As a result, if a is plus, ship will have plus

yawing acceleration but if a is minus,

minus acceleration will be imposed, ultimate-ly setting to a balanced condition where a=0, namely, ship is in parallel to the crest wave.

(ii) In the next place, when the period of

wave has corresponded to that of pitching,

we will have as follows at the time of p= >a) for the same reason as before,

Therefore, the 1st term disappears, becoming

as follows, I- periodic terms c120 0 di' (5) (6) tan 7' = P,- tan (3' 1960) Ship's

In solving the above equations, consideration terms of the 1st and the 2nd terms as

great effect on the objective of this subject.

cP0 dO

+k+a)2/6= A

Yawing in Waves 3

will be forwarded, omitting the gyroscopic well as the 3rd term of damping force, having no

Then (rolling) cos pt,

dt

dO

+

°PO

B cos pt, (pitching) (2)

dt dq5 dO (yawing) dta dt dt K , K' Dgh k = ; 0), (0,2 _Dgh'

I

n=,

(3) A= (020 cos a, B =0/20 sin a. a' {(ca d20 + dt2 . -8

in which case,

it win never turn toward a

given direction.

When the period of wave fans upon the halfway of each period of the pitching and rolling of ship, we have ol>P>co,

.

accordingly,,

d2(/'C

sin rz-cos +periodic terms. 9

and the ship enters into a balanced condition at the position perpendicular to the crest of

wave.

When the period of Wave' corresponds

to that of the rolling of ship, we have al >p

= CO.

2 2

Therefore, 'there is no definite direction Simi-larly as in case of (ii).

When the period

of wave is

longer

than that

of the rolling of ship, we have

co'>o)>P,, when we have

-dk2 =Csin a cos,cr + periodic terms, (10)

dt'

where ship tends to be in parallel to the crest of wave similarly as in the case of (i).

So far are the contents of Dr. Suyehires

thesis.

2 Expression of three-diMenSional °sand-tion of ship due to rest coordinate'

system

As the theories in the preceding sections deal with phenomena extremely intuitive by deep insight, the author will hereby forward analysis according to order with a view to explaining them in details..

A general equation of rotational motion of a solid body with symmetrical axis will be-come as follows according to Routh. Letting the rest axes standing still in space be

OXYZ, 0 will be placed in the center of

gravity at the fixed point..

Letting moving system be

CAB and

A

Fig. 2 Coordinates system with symmet ricalt axis and Euler's angle (In accordance with Routh)

symmetrical axis correspond to 'OC, OB will rotate around OC.

In order to

relate the above

to the

reciprocal position with this

solid body, a certain plane of reference

in-cluding symmetrical axis will be considered, and assuming this as OCF, the angle formed

by that with surface OC A will be put as

x= L AO F. Assuming the components of ex-ternal couple acting upon the three axes of solid body as L, M, N and Euler's angle as

ZOC= 0, Z. XZC=-0, EC A-30 as shown in

the figure, we will have A (d:ti iddxt (A__r C)(02,03,= L, A (d dco; ddzt + (A --- C)(08c0,-- M, do", dt do dcb

= sn 0

_dti

dt sin0 cos 0, to, =Litcos'0+ cj-L' sin&sin 0,

dt dt

dz do , dst,

0)3 , cos LA.,

dt dt dl

where, may be chosen so as to simplify the geometrical condition equation of (12) or the,

dynamical equilibrium 'equation of (11). (i ) Putting especially as X=0, moving axes will become fixed to' solid body

com-pletely, and (11) will give' tti.e Well-known equation as follows, ...(10 .. ....(12) a ) L L +

1960) Ship's Yawing in Waves 5 A de) A - kJ-I.L./0)2(03= L, dl

Ad"

+ (A C)coow,--- M, dt

C d"

dt dx (ii) Putting as dl the plane of

reference is supposed to be rotating around

C at the rate equal to

the rotation rate of solid body and in direct opposition rate. Ac-cordingly,

if OC is

rested completely, this

plane of reference is supposed to be fixed in the space, and there appears such a charac-teristic that it may be substituted for the rest system 0 X Y Z. Then AdwI dt +C(02(03= L, Ada) Cwocoi=M, at

=N

... ...(13) ... ... .(14)

while inclining it right angle. Only the case of ship is different where rotation is extreme-ly slow and in both ways. In short, spinning,

neutation and precession are corresponding to

rolling, pitching and yawing respectively. Putting the above-mentioned Euler's angle

from the same viewpoint as follows,

7r

0 = ₂ + 6 0=71, = (15)

C d"

= N. dl

It is, however, natural that this advantage

is nullified when the moment of inertia of

three axes are various.

Now, since ship presents slender and

cZ C ct)/

cylindrical aspect and nearly symmetrical

dynamically with respect to the center line Fig. 3 Euler's angle for the_{oscillation of ship} passing through its bow and stern, it

corre-sponds to the case where a top revolving as well as $, 7.2c-O, the angular rate

corn-straight and at high rate is laid horizontally ponent will be

infinitestimal

sin cos $ cos '>2-=, (yawing)

a)2=:7- cos v+ cos $ sin 7.2:L.-$, (pitching) (16)

v sin $ (rolling)

Wave gradient component will be

0,-= Isin(:+ao)sin vcos(C+ adsin $ cos v}

u= (1) jsin(c+ ao)cos v + cos(C + adsin $ sin v}-4. -0 sin (C+ ao), (17)

0.,= cos (c + adcos $ '--70 cos (C+ ao).

Since the rate of pitching and rolling, therefore, is much larger than that of yawing and RI? I72.1>>R1, the gyroscopic terms of pitching and rolling will be omitted assuming it as much smaller than the gyro scopic term of yawing as well as O'H 1;24'1<<iE 2. 1, and

substituting oscillation force, restoring force and damping force into L, M, N respectively, the equation of ship's general oscillation will be,

24. -1- (A C)62= _{(yawing) I}

=- Dgh'e+ Dgh0 sin (c + a)cospt, _{(pitching) ... (18)}

Assuming in this equation that the yawing

angle is infinitestimal and a0- ca,,--=-(x, and

I,

0, 0 instead of A, C, v, c, all

corre-spond to Dr. Suyehiro's equations but for gyroscopic terms.

Since A>>C in case

of

ship, the absolute value is varied in moving

system, ii=4.1I being (A-C)IA with both in plus signs, there is no inconvenience in judg-ing yawjudg-ing direction or stability.

It has been testified thus that stability of yawing can be also discriminated sufficiently by a special Euler's angle with resting system.

It is not known clearly why Dr. Suyehiro forwarded researches choosing moving system,

giving special warning that he intended to

simplify the theory, but according to the

author's presumption, the reason is likely to

be as follows.

In the first place, it is obvious that Euler's equation represents both the equations of (11)

and (12). It is therefore natural that an

exact solution is obtained by starting from

this, but there is still a question of how to

express restoring force, resistance or wave compelling force accurately by the functions

such as co2, w3, 0, 0, in forming an

equa-tion. Expanding therefore from the case of pitching and rolling of primary order, it is likely that he formulated the equations

simp-ly by putting yr,, 0, 0 for the very small angles of pitching, yawing and rolling apart from the strict meaning of Euler's angle, from which it is detected that he had no intention

to express by taking the limit of Euler's

angle, as seen from the figures in the thesis. If Dr. Suyehiro had been aware of this fact, he could have managed to obtain an accurate result by means of rest axes without depend-ing upon such particular thdepend-ing as moving axes, for it is just a simple technic of

put-ting C for A C as far as the equation is

concerned and absolutely the same in carry-ing out integration as well as discriminatcarry-ing

stability.

In the second place, according to Dr. Suye-hiro, the said Routh's moving axes may be

considered as rest axes as the axes of

re-ference rest when symmetrical axes rest in the space. In actual case, since the

sym-metrical axes, namely, the longitudinal axes of ship, make pitching, it is considered to be

far from being absolute rest. However, as

ship makes no vibration of extremely short period, it may be tentatively looked upon as resting in case the object is laid in checking how yawing is carried out in each moment of pitching and rolling.

In short, forming first the motion equation by assuming three axes of ship simply as 0, sb separately from Euler's angle, the

solu-tion of pitching and rolling

will be easily obtained from experience, a clear judgement

can be made from the external couple

of yawing by getting gyroscopic couple for being

three-dimensional oscillation from this, but it

is feared whether or not it is representing the character of oscillation faithfully. He

might have availed Routh's assurance made in his book that moving axes can be looked upon as rest axes if ship is symmetrical.

So far is the author's inquiry made on Dr.

Suyehiro's thesis in accordance with Dr.

Wata-nabe's indication, but he would be happy to be given further informations on this problem from those who might have had the oppor-tunities to hear directly Dr. Suyehiro's

ex-planations on them.

3. Stability at the time of resonance and

the condition of yawing motion

The equation of motion on wave will be

co'e =0;260 sin (:+ao)cospt, ...(19) 72- -i-kv+0,2 =00 cos (c + adcos pt. )

but putting as I I, .c,1-. () and

evaluating its solution by assuming in the 1st and 2nd equations as constant, and

limit-ing the operation to yawing motion only in

differentiating it to substitute it into the right

side of the 1st equation, yawing motion will be integrated.

As it

will be seen by giving a close

ex-amination to the documentary film put at the end of the thesis, the change of course at the

tuning time when the period of wave has

corresponded to the particular period of

pitch-ing or rolling assumes the tendency of p>,01>(,) (case 1)

at p=a>,0 (case 2) and

the similarity of a >p>0) (case 3) at oi>co=P

(case 4).

Examining the case of tuning in

I,

cb,

f;'+k" +

1960) Ship's Yawing in Waves,

order to find out this reason, it will be as follows. Considering the damping term, i.e.,

free oscillation and taking the initial condition with E=)= O, 7,,= 0 as reference at t= 0,

co"(1)sin( c + a,)

(0),2_,e)2 kf2p2 p') cos pt + p sin pt- e- ("" /2)e{(w'2-p2)cos n' t +_2n (w"+ p2) sin n'

00 cos (c + ao)r, le .

p2)cos pt + kp sin pt- e- ("`{(.2 -p2)cos nt+

-

(a)4+Msin ntj}

(a? + li4P2L(D 2n

here, we have

0= co' -

k'2

4

and at the time of k, k'0, we can put n'-'---701, n-:co as 1> >1P/8a)", /e2/8co2'--70. The equation

of yawing will be put as IZ-I >> I. 1--'---.0 omitting the very small acceleration,

C a (02 (on 02

sin(C+ ao)cos(c, + an), _km {(d' - y)2 + ep} {(co'-p2)2+ rp2} x

sin cos

nt})x

(- (l)22)sin pt + k p2 cos pt + e-(kmtito2(1i2 + 40 +P (k2-40 .

r

1_(- (w"- pip sin pt + k' p2 cos pt + e-(k,mt w" 4n

IP + 4n") (22)

.

4nn."t) -.kp

though this may be given graphical

integra-tion by isoclinic line, further development will be made on the right side and dividing

it into two parts of constant term and the

periodic term including damping quality and

transforming as follows,

sin(C+a,) cos (c + ao)= A, {ao+ U(t)} . (23)

A -

(020x2p202

2k" {(.2 - p2)2 + {(0'2 -pay + k"p21-... (24)

a,= (0)2 - p2)(.'2 pl) + kk' (25)

which will be at once integrated

lncot(( + ao) Ao{aot U(t)dt}

cot au 0

where, c0= 0 is the value of C at the moment of t=0. Namely,

cot(c + ad =cot ao exp Au {ant + U(t)dt}. (26)

Since the 2nd term of the right side shows

the periodic repetitive motion, the general

tendency is expected to depend upon the sign

of a, exclusively. That is, putting simply as follows in order to see the general tendency,

2 k n2 =a) - -4-4n' sin t p2 cos n'tl.) (case 1)

P>a>co

at the time of : (case 5) w' > co> p .... ...(27) . -.Limcot(c + c +

where, the course ends paralleling to the crest

line of wave. And at the time of

(20)

Fig. 4 Period of wave and yawing direction (case 3) a >p > co : a,>0

Lim cot(c + a)) C+ (28)

2

k, k'=0, a,= (a)2 (w" p2) which shows that the course become rested in

(21) (w

[

. + p2)

right angle to the crest line.

From the above theory, in the case medium of these cases, namely at the time of

(case 2) p,=--co'>co

t(case

4) coi>o.)--p

an inference will be drawn that the above

60 45 30 7 4C 2C 9'0 75 a0=0"

is in amorphous or neutral condition.

Yawing condition has been thus expressed perfectly in numerical equation and the re-sult of graphical integration can also explain

the actual phenomena and the documentary film of Dr. Suyehiro well enough as shown in the attached figures.

Case 2 ,p,70/ > co 10 II 12 3 4 t sec '2 - 2 t, +2 -2 -+2 + 3

1111

1111L

F'

WoW

4 5 1,0 111. 12 13 14 t sec Fig. 6-1,, 6,2 Condition of yawing motion (when 'length of wave is extremery sma11)

it

.1 is' ite 1 .... Le i s tilt

64.

1

6" -so ...; . i .0. ii. 2 6 a 0 2 14 IF, 18 20 22 24 26 28

Fig. 5 Condition of yawing motion caused by rolfin,g and pitching Case 1 p> > co, +1 + 2 + 3 +5 5 75 45 30

0 75 60 45. 30 115 Case 4 co'>co=p

Fig. 6-3, 6-4 Condition of yaWing motion (when length of wave is large) Case 5

coa>co>p-10 i 112 13 4

t sec

Fig. 6-5 Condition of yawing Motion (when length of wave is large)

413 J14

t seci

Especially when the time of resonance is _{_(,02_p2)p { (012 _p2)p} considered in detail, putting as follows for a

considerably long period from the beginning '2(k''+ 472'2)'+p2(k'2-4n")}

of motion, 4n'

k,, k'=,0 exp(

_--,

k

_)t=

1 k''p2 (a _p2). > 0,

2 2 ₂

Collecting the coefficients of sin'pt and sin pt _{which shows the same tendency with (case 1)} sin n't, the constant term in question at the p>ai>oy.

time of (case 2). p=c0">(0,, will be as follows,

_{Then at}

_{the time of (case 4) co>co=P,i}

w'=1)

1960) Ship's Yawing in Waves :9

Case 3 co'>P>co 90 75 60 45 3 5 = 90 75 -60 45 -30

collecting similarly as above the terms of cos.' Pt and cos pt, cos nt, sin2 pt, sin pt, sin nt,

the former disappears leaving the latter only,

(w'2-p2) I,' +a)2(k2+4n2)+p2(k2-40} 4n k2 p' (a2 - p2) <0. 2

=

-which resembles to the tendency of (case 3)

co'>p>co.

These results show that Dr. Suyehiro's theory slips uniformly toward the longer

length of wave as a whole, which is

also

detected pretty clarly in the attached figures. As seen in

Figs. 16 and 17, the facts that

the neutral or course-forking length of wave is seen so long as 72cm__,79cm, 83cm,90cm,

being 7cm longer, provide evidence.

Incidentally, checking for precaution the

stability of infinitestimal yawing at the ulti-mate position of a=0 or 7r/2, the result will

be as follows.

As considerable time has elapsed since the

initiation of motion, natural oscillation has

4(0) = 1, ea 0,

e-00'

00 ea 1, 22- 0.' 22- 00' o, 00 02-0o' 0, 0, ea 1, 02-0: 00 22-00' 0, 0, 0,

disappeared, and putting e-(kl*,

as well as k, k'->O and k".= 0 among (19), the

equation of yawing will be

p2,2,0,202

(1 COS 2pt)c= 0 ...(29)

(c02-P) (0)". -P)

here, when bow is in parallel to wave, we

have a=0, sin (a+ C, cos (a+ c)=-71 and

therefore, the double signs take plus signs,

whereas when bow is right angular to wave,

we have a0=70, sin (ao + cos(ao+ C)4

-taking minus signs. The above equation will by putting

h2,02a002

=ea, pt=z

... (30) 2(co' p2)(0)"p2)

and the reference form of Mathieu type tion will be

crc

dz2+00(1-cos 2z) (31)

The stability of the discrimination of this

system depends upon whether or not

satisfying

sin' pi)= 4(0) sin'E -1/0.

2 2

is purely imaginary number,

0, 0, 0, 0, 0, 0, 1, 0, ea 42- 00' ea 22- 01; 1, 4 00

( 00 y

00 0, 0, 00

e-00'

equa-..(32) z1(0):=:1, . pr.=-71/00

which is the discriminant of stability and will be as follows according to the size of co', co, P.

The 1st term of this is line 2 column 2 of the center, up to the 2nd term, line 3 column 3 of the center, and up to the 3rd term, line 4 column 4. Since the gradient angle of wave is

extremely small, assuming it as V-=-70,

4-0

=

)

1960) Ship's Yawing in Waves 11.

wave surface'

Fig. 7, 8 Rest and moving coordinates of general motion of 7-drifting 8-heaving on the waves, including surging.

To summarize it, when ship is stable in case it is rolling in parallel to the crest of wave, it is

the time when the period of wave is

shorter than that of pitching or when it is

longer than that of pitching, whereas when ship is stable in case it is pitching in right

angle to the crest of wave,

it is the time

when the period of wave is in the halfway of

the both.

Fig. 9, Moving and rest coordinates systems showing oscillation angle of yaw-ing,, rolling, and pitching.

case 0

42

( I ) p>01>o) co^>,0>p I 00>01. ' ,ft : -pure imaginary. stable. Bo>0.. it: real unstable

( II)

d>p>ca

J1:00<0real. unstable.

00>0

It: pure imaginary

stable (III) --.._ fy=a; ...--co I - oo.=0, it=0 neutral. .(33)

Fig. 10 Components for the moving coordinates of wave buoyancy

Chapter II Stability of Yawing Motion When Wave Length is Extremely Short 4. Fundamental equation of the general

motion on the waves

When the scale of wave is reduced so small

that waves of one or a few wave lengths

enter into the ship, the heeling force of wave does not work so effectively as in the preced-ing chapter but changes complicatedly, strong at a certain part of the bottom and weak at

other parts. Accordingly, for this asymmetri-cal character the condition of stability changes

according to the magnitude of the period of ship and wave as well as to the size of ship

and wave.

In this chapter, the author will investigate

the properties of yawing motion, assuming

the scale of wave as finite.

In investigating the amplitude of oscillation

in wave, we must always have general motion

equation of ship. Such a case of a fishing boat whose hull is very small compared to

the length of wave and which

is afloating with a long period with the swell of the

ocean has been well known since W. Froude

as the treatment of the above-mentioned rolling. Moreover, there is A. Kryloff's

epoch-making thesis in which he analysed general oscillation on waves for the first time, taking

the effect of the size of ship into

considera-tion. The author introduced yawing and

drift-ing motion on the basis of

this study and taking newly the Euler's angle indicating oscillation, obtained various equations as follows.

In the first place, as the coordinates

O-XYZ will be taken standard axis of rest

in space representing the position as well as the attitude of ship. Making OX correspond

to the traveling direction of wave, OY will be taken the perpendicular downward bottom

direction. OZ is in parallel to the crest of

wave and O-XZ represents stillwater surface. Then the moving axes that make translation

together with ship will be assumed as

G-XYZ, center of gravity as G, bow as GX, bottom as GY while GZ will

be taken in

starboard direction. The surface G-XZ is in

parallel to the stillwater draft surface of ship and the center of gravity is in the center of draft surface under an average load condition.

G-YZ is

the center transverse section and G-XY the center vertical section.

In the

next place, in order to express the form of

re-volves with ship will be considered. The

various axes of Gx, Gy, Gz will be assumed

as corresponding to each of the above-men-tioned axes GX, GY, GZ when ship is com-pletely at rest. Then Euler's angle showing oscillation angle will be chosen as follows.

Gaxbz is

the water surface and Gxyc

perpendicular surface, and

though Gcybz

surface is nearly perpendicular, it is a surface

right angular to Gx which

is slightly

in-clined.

On each surface, Gay and Gxc;

Gbc and Gyz; Gab and Gxz are

form-ing right angular axes, slippform-ing only by the infinitestimal angle of 0, 0, 0. 0= LaGX, i.e., the angle formed by the horizontal projection

of GX with the

longitudinal direction GX represents yawing, while 0= L bGZ, i.e., the

angle formed by the axes GX with water

surface in mid-ship section corresponds to

rolling and cb= LaGX i.e., the angle constitut-ing water surface becomes pitchconstitut-ing.

Assuming now that ship was at first at the position of 0, G XYZ must have

correspond-ed to ok- itZ at the origin of time.

The

coordinates (X, Yo, Z.) of G for 0, therefore,

give the distance of shifting drift of the

center of gravity in case of X11, Z, and verti-cal motion in case of Yo. The angle formed

by GX and OX is course or deflection angle, which generally gives the difference of ao

instead of passing

0, even

if GX were

stretched reversely toward the direction to

origin due to the drift from wave pressure. With respect to the relation of these four coordinates, transformation equations can be led naturally, a little troublesome as it is, by dealing with them one by one, shifting the

components of projection necessary for the

above-memtioned three surfaces of Gaxbz,

Gaxyc, Gcybz about oscillation and each

X= T cos ao= pg00 cos ao sin 27r(

r

Z Nsin = pg(9, sin a, sin27r( X t

A

surface

of GXYZ and OXY Z about

translation Since 0, 0, 0 are very little except a0, we can put as sin 0, sin 0, sin 0=0, 0, 0 ;

cos 0, cos 0, cos 0.'--- 1, 1, 1. Considering first

the angular velocity of ship,

w10

,--ip sin 0,

1

0)2=

sin 0+ip cos 0 cos 0,(34)

co, = cos 0 :I) cos 0 sin O. i.

and also

{X=

Xo- I - X cos a0-Z sin ao, 1

Z-=Zo+ X sin ao+ Z cos a, E (35)

Y= Y,, + Y. ₎

The fluid pressure acting upon ship is

periodic buoyancy due to trochoidal motion, and assuming that there occurs no destruction of wave motion by the existence of ship, the buoyancy per unit volume will be as follows

in the direction vertically upward to sub-surface,

N=pg[l

cos 27r -t

7

where, p is density of sea-water, g gravity

acceleration, A wave length, z- period. Letting

2r0 be wave height and 00=27rr0/A as wave gradient, the wave surface and its gradient

will be 17v, = rocos 27r(

-

t)

A 7 X

--" = eosin 27r( - -t - -).

...(37) dY dx 7

Here, the damping of wave motion

in the

direction of depth may be omitted as

within the

limit where we can put

(1-e-2'41A)127rhl putting draft as h.

Each horizontal and perpendicular

com-ponent of buoyancy N will be

(39)

1960) Ship's YawingYawing in Waves 13

T=Nsin

27r(_ -t-r

F= N cos e--N=pg[1+ 610 cos 27r( AX

-(38)

Expressing these two forces with respect to coordinate GXYZ,

=

Here (surging) (heaving) : (drifting) (rolling) (yawing) : (pitching) M,dU .Xdxdydz, dt Ydxdydz, dt M,04 Zdxdydz. dt

U = (X, cos ag+ Zo sin ao), dt

W= (X0 sin a, Z, cos ad, dt

Yo.

dt

Adah (BC)a),(a3=(Z,y Yz)dxdydz,

dt h

B da(C A)aa)1=

Z,x)dxdydz, dt

Cdo) (A B)ahoh= (17x X ,,y)dxdydz. dt

(40)

... ...

the next place, transforming the above three forces Y, Y, X into G xyz axes and

as-asuming as X,,

Z and then putting the moment of

inertia around G, and G, as

A, B, C,

... (42)

The motion equation on waves will be obtained, therefore, by conducting the integral

operation of the right side on the entire area of shipping space under water surface, but it should be expressed by moving axes G xyz which was obtained by fixing (X, Y, Z), (X2,, Y, ,

Z) to ship.

X = x cos 0 cos 0+ y(sin 0 sing, cos 0 sin 0 cos 0) + z(cos sin + sin 0 sin 0 cos cp)

y0+ z(Y----,x,

Z=x cos 0 sin cb+Asin 0 cos ç!'+ cos 0 sin 0 sin 0) + z (cos 0 cos cbsin0sin 0 sin sb)

...(43)

.x,p+ y0+ z=iz,

Y---x sin 0+y cos 0 cos z sin 0 cos 0'--,x0+y-Now putting

u

27i x t )= 27, X0+ X co0- z sin ao t X0+ x cos ao- z sin ao t

2 r r A

then :

nsin

uo= (si7r.n

coscos u

A

2 x cos a, az sin an 27rx cos ao . 2 z sin a,

COS sin

A

, + sin 2rx cos ao . az sin a .

°)sinu, + (cos27rx cos aocos27rz sin a sin

A A A A

Y= F

= pg

[1+ 00 cos 2ir

"

Putting therefore the rates in the directions of X, Y, Z as well as effective inertia as (U, V, W),

(AlpAlp 114),

:,

sin

1950) Ship's Yawing in Waves 15

27rz sin ao

+sin 27rx cos x0 . 27rz sin ao

cos Uo= (cos 23". cos ao cos sin

A A A A

cos u

az sin au cos au nax

Sin2 z sin ao)

-A A A

.

(sin ax cos au_{cos -cos}

A

sin u

u=- 27r

( 4

-

t )

. ... ... ... ... ... ... ... ... ... ... ... .... ...(44)

A r

Regarding the condition of bottom (Lx B x h) as wall side and putting the outline of water

line as

z=z(x),

-

-

Sz< +

OS ySh,

2 2 2 2

the limit of integration will be expressed by

It

-L/2 hw -zdzdydx ,

where, h,, is the relative wave height seen from ship due to axes G - xyz, and putting

Y=

cos U0, y= II, to the above-mentioned transformation equation

Y=y0+ x0-y- z0 we will obtain hw= ro COS U0 - xcb + zO.

...

...

...

(45) Therefore, (surging) dU U2 Ch 1'Ll2 z I .M L

pge, cos a, sin Uodzdydx= - pg 0, cos a, [y] sin Uodzdx

dt -LI2 j hw -.I: -L/2, -: nw

L12

c

= pg 00 cos ao [ h-ro cos U,+yo+ x0- zO] sin Uodzdx -L/2 -:

and omitting the terms of wave surface and oscillation by assuming as h<<hy, and looking

upon them as second-order infinitestimal amount,

r-L

,

- pgh ex os ao sin Uodzdx,

/2 -:

considering further

c

sin27rz sina' dz -0 from bilateral symmetry

-. A

L/2

/./ _{= - pghNo cos ao}

[

_{sin 27rx c°s a° dx u}

-L/2 A

' cosaz sin ao dz.

+ Y/2 COS2nx cosa° dx sin u

-L/2 2

-,

A similarly (drifting)

dw

f L/2 f h a Ms ,t -1.12 '"w

S.

1,, pge, sin ao sin Uodzdydx

J j

L/2

= pglze sin exo sinax cosa" dx cos u Li-Liz

27rx cos a

cos - - odx sin u cos 2n..z sin a' dz. . ... ... ... (46)

-L/2 A

Putting next as ship mass,

2rx .

(L/2 f z

with pitching becomes extinct.

Next, integration of rotation couple will be by expressing axes GXYZ with Gxyz,

x=X cos 0 cos 0+ Y sin sin 0 cos 0-'--X +X0

y= X ( cos 0 sin 0 cos 0 +sin 0 sin O)+ Y cos 0 cos 0 + Z (cos 0 sin 0 + sin 0 sin 0 cos 0) X0+ Y+ ZO,

z=- X (cos 0 sin 0 sin 0 + sin 0 cos 0) Y cos 0 sin 0 + Z(cos 0 cos0 sin 0 sin 0 sin 0) =,--X0 YO + Z.

Letting (x, y, z) of the left side correspond to (X x, Yu, Z,) and (X, Y, Z) of the right sidf (x, y, z) on the side of force, we will have

(rolling)

Za Yyz-- (XV, YO + Z)y ( X0+ Y+ ZO)z = X(y,p+ zcb) Y (y0 + z) Z (y z0)

- pg(1-1- No cos U0)(y0 + z)+ pgeo sin a, sin Uoy-,---,pg[y0 + z (1 + 6), 0 cos U0) -1-3/6/0 sin a, sin Uo]

L/2 0 1.:

(Z,y Yyz)dzdydx

h Z(1+ eo COS Uo)idzdx

'`w

LIZ hto -z

=

pgr[

-L/2 -z (0 + 6), sin ao sin U0) +

[ ha (0+ On sin au sin Uo) z(hOo ro) cos Uo+ z(yo+ x0)z201dzdx

=

pg[

0 .Liz z2 dzdx

\

2

. 112 . 27rx cos a,

dx cos u+ *LIZ 27rx cos a,

+

2 sinaoi sin

dx sin it cos2rz sin ao dz

-LP' A -L12 A A

27rxos ao_dx

L12 Liz _{sin2nz sin a}

2 o dzi

+ (he() ro) ) sin nxcos aodx- cos u-f-

cos cin u

-L/2 2 -L/2 A

here z cosaz sin a0 dz=0. Especially at the time of A-0.0, we have

A = Pg[Yor L1 2 - L12 sin -L/2 z -zdzdx x. 27 cos ao A dx sin lf 2nh 27rx cosctod

)

- L/2 x cos u z 5. LIZ z az sin_{a° dz +} A

-l.

cos 0 1.12 --zxdzdx

here we have g zdz =O. In case of longitudinal symmetry, the term of 0 disappears, w hile

-,

'L/2 f h CL12 M= pi dzdydx= ph 2zdx -L/2 -L/2 be mass of ship, (heaving) : ML,

m

_{pg [1+ 00 cos Uo]dzdydx+ Mg,} dt -L/2 j"w

Pgr[Y]: [1+ 00 cos Uo]dzdx + Mg

- L/2 -z L12 1.:

= [h ro cos Uo+yo+ x0 z0] [1 + 6, cos Uojdzdx + Mg

-LIZ -2. L/2

pg [h+ (h00 ro) cos Uo+yo+ x0 z]dzdx + Mg

-L/2 (47) to -z

-1/2

-: L/2 pg + -L/2

1960) Ship's Yawing in Waves 17

= pg[ 0 sin at, sin 1,t]

(z'

dzdx,

2

where the 1st term corresponds to restoring force and also to the well-known form of oscillat-ing force of transverse wave by puttoscillat-ing as a0=n12.

In the next place,

(pitching)

17x _{= ( X.75+ Y+ ZO)x (X + YO .1p) y = X (x0 + y)+ Y(x yq5) + Z(x0 + ysb)}

-: -.pyry&,, cos a, sin U0 x(1+ or, cos U11) +y, _1

cLiz ch _{(y,x Xy)dzdydx} )-L./ 2)

L/2 f

[

2 h.

=pg 1 (01 cos a, sin U0+ 0) y x(1+ _{cos Uo)idzdx}

-Liz) 2

awlLw

1/2[ h--:

((9, cos at, cos U,, + (heo ro)x cos U11 xy x20+ xzOidzdx

pg[

L L/2

= (1.

(x2

-h2)dzdx

+ eo cos ao-1 si.nax cosa"dx os u

2 2 -L/2

cosax cos a"_{dx sin is)rz cos az sin a° dz}

A A

L/2 _{ax cos}

a"

L12

dx cos u x sin27rx cosaudx- sin ulc.- cosaz sin at, dz

(hoo r0) X cos

A -Liz A -z A

L/2

-L/2

-+y0\ xdzdxl.

Checking here the time of A-,00, we will have

CI-12 CL

= pg[(0+00 cos ao sin u)

(2

112

L/2

x dzdx + (y ro cos u)

-rXdillX1

where, the 1st term is _{the restoring couple of pitching, the 2nd term at the time of (20=0}

corresponds to the afore-mentioned oscillation couple of wave. The 3rd term is the related

term of heaving and it disappears at the time of longitudinal symmetry. Especially, since it

becomes as follows at the time of 2*00,

dV .12

-1'10 _dt = pgi (yor,, cos u)JSI dzdx.

it disappears. Lastly, (Yawing)

X,,z

= (X + zsb)z(Xsb YO + Z)x= X (z x0) + X (z0 + x0) Z(z0+ x). pg[(0, cos a, sin Uo+ 0)z+ (6),, sin ao sin Uo+ 0)x j

c.L/2 ch

j-112 jhw

N h

Pg Y [(e0 cos a sin IA, -.F- coz + (ea sin a, sin U0+ 0)x]dzdx

-.112 -z

."7 pgh[6,),,cos sin27rx cos a,dx

u

cos2x cosa" dx cos u} z sin

A A

dz

L12 L12

2LCZ sin ao L/2

sin a, x cos27rx cos ao_{dx sin u+ L/2x sin}

la cos

_{a" dx cos} cosaz sin_{a" dz}

-L/2 A A -z A -HO xdzdxl. -z (X z Z,x)dzdydx LIZ 2 -4./2

at the time of A>c>. , we have

Li

pgh[(6),sin ao sin u+ 0) xdzdxl, which will disappear by referring to rolling.

General equation of motion on waves has been obtained so far, but in case when the condi

of water line is especially in longitudinal symmetry, we will have

c L/2 (

sin-2rx cos a,, x cosax cos ao, x)dx=

j -L/2 A A

Therefore, completing various equations assuming this as time of reference, but leaving da ing and resistance force for other occasion for consideration and reference,

dU

surging : M,

dt = pgh0, cos a0 D0 sin u, dW

drifting : M2

dt = pg12,00 sin ao- Da sin u,

heaving : M udV + pgS,y0=

pgr(1

-27th ) cos u. .... ... . ... ... ... .... ... .... ... ....

rolling : A dwl (BC)a2co3+ pgLo 0= pg[0,sinau)? Do+ (he, 0E01 sinu,

dt 2

pitching : C

dt kn. LP, /Wit/12+ pgNo

pg[

00 cos a

e

u .,0 -f- krt v.,, ro)Folsin uy

2

da) , A A_ ,

B dw2 (C

A)w3a),-- pgh [a, cos a, F0 00 sin a,- Ej cos u. dt

4Li2cos ax cos a, az sin a,

cos dzdx ; F0=4f./2 cos2nx cos a, rzsinaz sin a,dzdx,'

A A 0 A A

L/2 _{22-rx cos a 27rz sin a}

F0=4 x sin " cos ° dzdx,

S0= 4 .'"dzdx; Lo= 4 z (2 --ndzdx, N,= 4 Li.2 (22 ADdzdx.

0 0 o 2 0 o 2

d

1. d A

U = (xo cos ao+ zo sin ao), wi= 0,

dt di

W= (x, sin aozu cos ao), (56)

dt w' (55),

d dt

V=

dt y, .

w2

dt sb.

It is thus impossible to evaluate accurately general oscillation, as it is related mutually in

elliptic functions.

Now as it is plain from the right side of the equation of rolling and pitching, when wave length is finite, we have as follows in contrast to the case of infinite length

112

Co sin ao).Leosin ao --1/0+ 7-0)E,1±LO----= eosin ao rr,

2

0, cos ao-- Le,cos ao Do+ (heoro)Fol 00 cos a,- r, ,

2

where r, and r, are supplementary coefficients for wave surface gradient, of which value be obtained if the form of water line is known. Examining a square model referring tc readiness of integral operation for Dr. Suyehiro's circular model, the result will be as fol

yawing : ... ... ... ... will the ows. tion

mp--

_-z (he 0-. (52)

-

-(54) dzdx ...,

1960) Ship's Yawing in Waves 19 501--3S,S2Sa=-S,qx (I-XCetx) .0 4 30 20 HO 20 30 00 S.-s,n2-t 7rL cos a0 A I 70513 Sa= 2.2 ' 03 00 P I 5,08,2 04--3.141 S3 n-B sin a, - ; A 3 26033 3 S. S2S3 482773 f

Fig. 11 Direction of the couple of declination needle at the time of pitching right angular to wave erest. (when length of wave is small)

rd.., I ;

S,= sin' x,1

S2= 2 72 3 ,1 ..53= + (1 Xe)(xCOt X-1).

5. Stability diagram of square-type ship

Assuming the form of water line as

Z2

L L _{... ... ...}

then,

A A rcB sin a0

sin 7rL cos ao.

Do = 4

27r sin ao2n cos a,sin A A

A

AB

7 cB sin a0 A B sin a01sin 7rLircos a,

Eo = 4 cos ,

2r sin a, 27r cos

a,[

2 A 27r sin a,

in

A A

A A L TrL cos a, A ni, cos aol. 7rB sin a,

F, = 4 cos sin sin

2r sin a, 2r cos a,

[

2 A 27r cos a, A A

N0

4(1

L\ B

Sn= LB ;

L,-4( 1

133 112 B \ L 3 8 2 2 .1 2 ' 3 8 2 2 ) 2 Accordingly, ..(58) h h (59) (60) x siny 72' 2 cot y + (1 v cosec a0)y

(

\ 3 1 712 2 ) xy .Y2 35,S2 3-S1S2S3 147952 3 785398 94247$ 3 04313 =S1S2S3 2 }

(i_x secco,Xcxot2k -1 10 22 rz, 75 7C 65 90°

Fig. 12 Relation between draft and neutral wave length for a square ship

8.0

0

50I ao' 30t 20

Fig. 13 Figure of .direction stability for a. square ship,(LIB=-'7)

9 0;) 150 =tr cknoix 15- lTJC.),X7 I- 00 5(7'60'750

i

LW

El10' IIi x`' I

I\

750 600 0000 750 60030000 750 600 0 3 2 2 -3 85 30°0° V 0

+ + (,),0=r (96 sin ao cos pt, k'(*p + (0'4= r'e cos ao,--COSpt,

where gyro term will be

Fig. 14 Figure of direction stabifity of right square floating body (L/B=1)

x sin y E.2

+ (1 E sec ao)

x cot x-1 t.41_

xy 2 _/

\ a

2

Lim(r, r')= 1

A

The equations of rolling and pitching wilt ultimately . take the form of

(02,0,2p2e02

a0 cos ao

_{(w p) +kap -kw p2) +k pa}

2 2 2 2 / a pa 4194X

4=(0)2p2)(0'2p2)+kk'p2; .4,=rr', (64)

and when the wave length

is- infinite, the

direction of yawing will be known as

Lim zb, = 1 .(65)1

only by the sign of 4,'---7(w2p2)(w"p2),, which

corresponds to the case of Dr. Suyehiro

re-ferred to in the preceding chapter. Fig. 11. shows the result of examination given to the signs in case of pitching.

The boundary of stability or unstability the position of neutrality will be given by

47-p4x = (66)

As for 4=0 of these, of which the term

in-cluding y, i.e., that for rolling, will be

omit-ted, the wave length being so small as less than ship's width and choosing the conditions on the side where neutral wave, length match-ing ship's length, i.e., the term including x, we will have in the first place,

sin x=- 0 ... x=mir, m=-O, 1, 2 (67)

then as

2m+1 x cot x-1=01.

2

1960) Ships. Yawing in Waves 21

., .(62) i.e., . (61) sin ₍₆₃₎ ... ,

m=0, 1, 2... (68)

the wave length satisfying this condition will

be therefore

A,:=71-L cos al x=2L cos aln, n=0, 1. 2....(69)

In order to make 4A=0 or the domain of these plus and minus clear at a glance, it is

advantageous to take the following polar coordinate. Namely, radius r will be taken

to the reciprocal number 1/A of wave length, the unit of which will be chosen

appropriate-ly. Then declination will be set at the time

when the direction of bow has corresponded to the forward wave in case of a=0 and in

case of azimuthal angle a=42 when it has

corresponded to the direction of the crest of

wave. Then A, will be all laden on the

perpendicular line planted at the position of A=L/n, n=0, 1, 2 on the line of a=0. As for Ay, it is laid on the perpendicular line

plant-ed at the position of A=BIn on the line of of a=ir/2, but it become separated from origine.

At a place considerably separated, it does not always assume straight line, being subject to

e or 77, i.e., the effect of draft, but the case

of an ordinary ship L- - B=7 is shown in Fig. 13, while in Fig. 14 is shown an extreme case at the time of L=B.

As for z1=0 nextly, it is p=o7 or and assuming the periods of rolling or pitch-ing as rr, rp, it is expected to give the

posi-tion of

neutrality by drawing an arc with

this reciprocal, independently of

(AD, Ar)= (72, 72,)

...

(70) namely, a.

Lastly, if the signs of

(+), () are fixed

alternately in order from the center for each division obtained by overlapping these groups

of circles and straight lines so that the

domains of the same signs may not be con-tiguous one another, the diagram of stability discrimination will be obtained.

Conducting graphical integration by putting

as a0-1- C. =a, for the condition of yawing

motion similarly as in the preceding chapter, omitting the term of the acceleration of equa-tion of yawing and adding damping force, the

Fig. 15 Condition of yawing motion (when length of wave is small)

X COt x-1

+ (1 xe sec a) 2

0,

2

y cot y-1 0. 2 + (1y 72 cosec a) 2

If we put y*.x, a.(7r12) a in the latter, we will have the same result in the former,

and therefore, the equation of this equation will be examined for some time. Especially

at the time of we have

xn X'It 0 0.0000 0.0000 0.81650 1 4.4934 4.0781 0.49042 2 7.7253 7.4721 0.26766 3 10.9041 10.7227 0.18651 4 14.0662 13.9249 0.14363 5 17.2208 17.1050 0.11693 6 20.3713 20.2733 0.09865

result will be as seen in Fig. 15. It is Among the values of x, y that are

resposi-characterized where there is another boundary ble for 4,-0, the rest excluding sin x=0 and line at the position other than a=0 or 7r/2. sin y=-0 will be

1960) Ship's Yawing in Waves 23

x cot x-1 =,011 ...

ex

cot x-1 seca

=

which corresponds to well-known equation in the field of elasticity.

These root x.

is in table.

Solving the above equation for the general value of E, we have

Fissthlazinion Root of 141=e7

;Yaw Len9th of X ---371.

Woeli4thof Pitching.

24°,69f:5,14'i-0 calve i.engh} of X=211.

1" Rod =0 -Wave Lentith of

Rolling-ittivel.e.nyth of X=117

1/(x cot x 1

sec a x cot x-1 ,(73)

X2 '

and at the time of x=m7r, xcotx= °O.

Put-ting z

(x cot x 1)/x by taking the limit of

oti

kive iength ofx =air cs, _{1(k/fie of &est)}

Fig. 16-1 Result of model experiment (a square ship) LxBx 11=180 x 25 x 6 cm TR = 0.99 sec Tp = 0.68 sec \o / -6° No-I

/

/

0 00 o 0 '0 Ro///4i. goof Cit _ o c ,. h

"-.i ---

%.,

/ ±

t to r-- i 4p1 0 /, / 122o 8 I I 5Tobi1itY Diarconrn_. G4-9cteoilini),

Fig. 16-2, 16-3 Result of model experiment (a square ship) LxBxh=180x25x6 cm top : TR =0.856 sec bottom : = 0.824 sec.

-0 0 0 ,! 0 +-r o 0 RoYhV. .90o (72)

/

0 -0 0 / / / 0 I \ / :

90° 90° ad

/

a 20' I. a 90° II 89° 00 0 ft li I 1 ; ... ...."'--- ... /So /0 ./ 0 IC

\

9,5.0 0

the above equation,

Lim[zsec a ± z seca(1 2

)

2r-ioo X2 sec' a

/50

1/2

= 00, or x cos a

furthermore, when there are two equalt roots

(Z sec a)2 2-IZ=0'

*. x cot x=1 +2 cos'a;, 2 cos a . (75)

Q.)

90°

Fig. 16-6, 16-7 Result of model experiment (a square ship)

L x B= 180 x 25 cm in case of deep draft top : h= 10 cm Tp= 0.47 sec T1=0.60 sed

Ap= 34 cm ,R=56 cm bottom :: h =12 cm Tp=0.51 sec TE =0.95 sec

Ap= 40 cm -AR= 140 cm

When a= 0; x% and for above values are tabulated in the preceding table.

These relations are groups of curves in Tettet Z, as shown in the attached figure, Fig. 12. Though in the range of 0 the undulation is intense, this division is actually indispensable, and drawing horizontal line of e'=const. for the given draft, the value of x crossing curve is what is to be evaluated.

Fig. 16-4, 16-5 Result of model experiments (a square ship)

L x B=180 x 25 cm in case of shallow draft top : h=2 cm, Tp=0.48 cm TR=0.62 cm 36 cm AR= 80 cm bottom h =4 cm 7'p =0.50 cm Tp =0.85 cm = 38 cm AR= 110 cm (74) : :

1960) Ship's Yawing in waves 25 2- 4- 6-_ Fig. 17-1, 17-2 Si 4d, 3d 2d, tain.901d 90°

123 o 01106

0, 10 , \ - - -Jo I \ - -.0 -1-1 \ . _-.--- / I \ \ I \ \ 0 I \

'r

/

,

001 \ ,./ \ _0/

_,,

, I si...( \ 0 0... _, _{\\ 1'} ... 0\ 0 ,..,..-

\

\_\ o o \

o0

/tO

Chapter III Results of Water Tank Experiment

6. Model test

With a view to confirm the results of the

above theory, the author conducted experi-ment on three types of models ; square sircular and ship-shape.

To make a

brief

statement on it, the direction of yawing, the

position of neutrality or rate of yawing were observed by changing the period and draft of rolling and pitching variously while main-taining the position of the center of gravity

lest extreme drift should occur and steric

oscillation be hindered by floating models in any direction on the wave of various periods.

All detailed results are as shown in the

at-0 0 4P' 30 26'A 89.10. 15500 0

se

i o° / 0 t I I I

'

d IS , A , _/0/ ` .

01\ "

,

N_{s A} .... r -- 0'I . 0 \ , ._.

/

' ao. 1 . .

/

I. ...: , ,/ . I _I

.'

, ,.... 0 svss

Result of model experiment (an arc type ship), showing the rate of 45 yawing LxBxd---100x20x6.6 cm

left Tp=0.78 sec right Tp =0.74 sec

TR =0.88 sec TR= 1.00 sec

tached figures, in which the tendency of theo-retical analysis is proved rather well.

As general properties, the longer the natural period of model is, the more direction

stabili-ty depends only upon the

length of wave

period, as testified

in 4, i.e.,

Dr. Suyehiro's

theory, whereas the shorter period is or the

deeper draft is, the more the interference

from wave length and stability area gets com-plicated, which shows that the direction right angular to the crest of wave or the direction parallel to it is by no means ultimate, but

that even with any angles in the middle

between them, the direction can be stagnant. It is, however, extremely difficult for ship to be restored to its normalcy once it starts

rol-ling, for the attitude in parallel to wave is considered to be very stable, as plain from

Fig. 13. 6$6° 4- 2-h= 6 en Tp= 0.78sec TR = 088 -I 2 S 50' \ -Tp=

90°

X

Ut

o'

Conclusion

The stability of yawing motion at the time when ship is floating on waves largely de-pends upon the relative magnitude of the period of wave for the natural oscillation period of ship, which can be collectively

judg-ed by overlapping the measurement of ship and the condition due to the wave scale. When wave length is extremely long, the

ultimate attitude of ship is either in parallel

or in right angular to wave, but if wave

length is short, this clear-cut relation becomes

turiA 89 90° a 0 _....y. .1) 8

ambiguous, coming to rest in any direction. Before ending this paper, the author wishes to express his unbound gratitude to Messrs.

Sadao Uchino, Masayoshi Murakami and

Yasu-hiro Yukawa who are research assistants of

the Department of Naval Architecture, Kyushu University, for their consistent cooperation

and kind suggestions in conducting tank

ex-periments. He is also very thankful to Prof.

Watanabe for his guidance in analysis as well

as his painstaking perusal of the present

paper. (April 12, 1948) 8 h=4 O'R

6-

_{Tp= 0 58sec} 4 TR= 0 72s8c 2 0 8- 6-2 h= 4 (et Tp =0 62s86 In =0 94S" se! 2 4 6 I. 0 2 4 6

.5

.1 . . .

Fig. 17-3, 17-4 Result of model experiment (an arc type ship), showing the time required to 45' yawing

LxBxh--100x20 x4.0 cm

left Tp=0.58 sec right Tp=0.62 sec

TR= 0.72 sec = 0.94 sec 4 -I \ 0 0 --67 1,460 0

90C 219 6 0 -4 9da to4' A 6 \ 1,

\

/ % .:. 13,S \ "i S. \ ,

/

\ ; Ck ./ o I . I \ 01 .... --..0 N. 5 '"' 0 \ 5 A 5 1 cino _ 8 0 §9 1--1 , tayin'A Ii 8 15s 112 - 6' 2 2 4 6

ka. 3.o" _{TiA0.60e8C TTR=7.0 70S}

RC

I.

Fig. 17-5, 17-61 17-7

Result of model experiment (an arc type ship), showing the time Of tillnitig for varioua drafts

40 75.A-tit 1 8 69 IA 12 1 0 8 6 4 2 2 4 6 ht- 2 5"` 'fp= 0525" To= 0.60sec Ir 1. 4 _{12 I0} 6 2 0 2 4 h= 3 .0c "1 Tp* 0.62sec TR = O. 7 8sec .5 I 4 o Jr GO 14' Id 0 l80'- 60L1 .0 8 _{4 6}

30

Fig. 18 Result of model experiment (model of a cruiser)

Bibliography

.1) Kyoji Suyehiro : "Yawing of ships caused by

oscillation amongst waves ", Collected Papers of KyOji Suyehiro, pp. 157-170; Journal of the Society of Naval Architects 26 (1920) pp. 23-33; Memoirs of the Faculty of Engineering, Tokyo University to (1920) pp. 73-85; TINA 62 (1920) pp. 93-101.

2) K. Sezawa "Oscillation Science" (Shincli5-gaku)

(1932) in Japanese. pp. 464-468.

I 2.5

(0

Te = 0 74sec

Fig. 19 Result of model experiment (model of fishing vessel)

E.J. Routh : "Advanced rigid dynamics" (1920) pp 8-10.

A. Kryloff : "A general theory of the oscillation of a ship on waves ". TINA 40 (1898) pp. 135

-196.

Appendix

The values of equation of (73) are indicated in the following Tables.

TR = 0 68SCC 150

TP = 0 50"C '9.5

i) a =0;' OX). 0.81650 - 0.81650 7.1 1.44032 0.15611 14.1 0.00659 - 0.00726 0.1 0.78409 - 0.85081 7.2 1.09620 0.15904 I 14.2 ' 0.05787 -0.32454 0.2 0.75347 - 0.88716 .7.3 0.79784 0.16538 14.3 0.06178 -0.53022 0.3 0.72451 - 0.92572 7.4 0.52353 0.18215 14.4 0.06349 -0.74049 0.4 0.69712 - 0.96667 7.51 14.5 0.06435 -0.96157 0.5 0.67119 - 1.01021 7:6 14.6 .0.06479 -1.19974 '0:6 0.64664 -1.05659 '7.7 14.7 0.06500 - 1.46287 0.7 0.62340 - 1.10606 7.8 - 0.23076 0.08242 14.8 0.06507 -1.76152 0.8 0.60138 - 1.15895 7:9 - 0.44375 0.09849 14.9 0.06505 -2.11099 0:9 0.58051 -1.21563 8.0 - 0.64894 0.10481 15:0 0.06496 -2.53476 10 0.56074 - 1.27655 8.1 - 0.85704 0.10791 15.1 0.06483 - 3.07129 1.1 0.54200 -1.34225 8.2 - 1.07447' 0.10952 .15.2 0.06467 -3.78892 1.2 0.52423 - 1.41334 8.3 - 1.30759. 0.11032 15.3 0.06448 - 4.82256 1.3 0.50739 - 1.49062 8.4 - 1.56400' 0.11063 15.4 0.06429 -.6.48182 1.4 0.49142 -1.57504 8.5 - 1.85380 0.11063 15.5 0.06409 -9.67116 1.5 0.47628 - 1.66779 8.6 -2.19135 0.11042 15:6 0.06388 - 18.64488 1.6 0.46192 - 1.77035 8.7 -2.59842 0.11007 15.7 0.06368 -248.79116 1.7 0.44832 - 1.88485 8.8, -3.11026, .0.10963 57r 0,06366 co 1.8 0.43540 -2.01312 8.9. - 3.78853, .0:10912 15.8 0106348 21.47900 1.9 0.42316 -2.15906 920 -4.75249' 0:10857 15.9 0.06329 10.09726 2.0 0.41156 - 2.32688 16.0. 0.06311 6.46454 9.1 - 6.26777 0:10800 2.1 0.40058 2.52265 _{9.2 -9.07211 0:10741} 16.1 0.06295 4.65031 2.2 0.39017 -2.75505 _{9.3 - 16.26708 0.10682 16.2 0.06282 3.54501} 2.3 2.4 0.38033 0.37102 - 3.03687 - 3.38775 _37r 81.01938,_co '0.10624_0:10610 16.3_16.4 0.062730:06269 2.78851 2.22858 2.5 0.36226 - 3.83956 _{9.5 26.01894} 0.10569 16.5 0.06273 1.78968 2.6 0.35400 - 4.44772 9:6 10.98354 0.10516 16.6 0.06289 1.42985 2.7 0.34625 - 5.31776 _{9.7 6.77157 0.10469 16.7 0.06325 1.12378} 2.8 0.33901 - 6.67871 _{9.8 4.76930 0.10427 16.8 0.06398 0.85505} 2.9 0.33229' - 9.13865 _{9:9 3.58091 0.10394 16.9 .0.06550 0.61210} 3:0 0.32610' - 15.02327 1010 2.78097 0.10373 17.0 0.06942 0.38536 3.1 re 3.2 3.3 3.4 3.5 3.8 3.7 3.8 3.9 4.0 0.32047 0.31831 0.31279 0.31037 0.30768 0.30526 0.30424 0.30530 0.30975 0.32080 .0.34955 - 49.02332 co 335.47042 12.81559 16.67075 4.46255 3.19316 2.35553 1.74939 1.27740 0:87783 10.1 10.2 10.3, 10.4 10.5, 10.6 1027 10.8, 10291 11.0' 2.19583 1.86396 1.37084 1.05724 0.78222 0.53181 '0.28238 -0.25788 0.10369 0.10348 0.10449 0.10577 0.10844 0.11468 0.13969 0.06721 17.1 17.2 17.3 17.4 17.5 17.6 17.7 17.8 17.9 18.0 0.09688 -020309 - 0.40901 - 0.61647 - 0.83236 - 1.06276 - 1.31466 - 1.59711 =1.92364 0.14754 0.04500 0.05039 0.05230 0.05319 0.05365 . 0.05388 0.05403 0.05400 4.1 11.1 -.0.46527 0.07548 18.1 -2.31322 0.05396 11.2 - 0.67203 0.07881 18.2 -=2.79707 0.05389 4.3 11.3 -0.88583 0.08046 18.3 3.42841 0.05379 4.4 4.5 4.6 4.7 4.8 - 0.06106 -0.34195 -055452 -0.75565 0104790 0.13290 0.15377 10.16331 11.4 11.5 11.6 11.7 11.8 -1.11279 -1.35977 - 1.66282 - 1.95259 -2.33212 0.08131 0.08173 0.08196 '0.08189 0.08177 18.4 18.5 18.6 18.7 18.8 -4.30728 5.80301 -8.00810 - 13.43329 -40.48491 0.05357 0.05356 '0.05340 0.05326 0.05312 4.9 5.0 5.1 -0.95603 -1.16226 - 1.38038 0.16818 '0.17064 0.17169 11.9 12.0 12.1 -2.79300 -3.05980 -4.21926 0.08158 0.08112 0.08106 67r 18.9 19.0 co 39.45543 13.03542 0.05305 0.05298 005284 5.2 - 1.61713 0.17187 12.2 - 5.45718 0.08075 19.1 7.66072 0.05272 5.3 -1.88104 0.17148 12.3 -7.57295 0.08044 19.2 5.31469 0.05260 5.4 -2.18373 0.17071 12.4 - 12.15164 0:08011 19.3 3.97950 0.05250' 5.5 -2.54219 0.16968 12.5 -'30.37782 0.07979 19.4 3.10332 0.05242 5.6 -2.98280 0.16848 47r oo 0.07958 19.5 2.47352 0.05237 5.7 -3.54959 0.16718 12.6 '5921117 0.07947 19.6 2.09265 0.05230 5.8 - 4.32258 0.16580 12.7 14.64093 0.07917 19.7 1.60144 '0.05242 5.9 -5.46479 0.16439 12.8 8.16660 '0.07888 19.8 127528 0.05259 6.0 - 7.36902 0.16298 12.9 5.53693 0.07862 19.9 '0.99262 0.05293 6.1 11.28496 0,16159 13.0 4.08721 0.078401 20.0 .0.84023 0:05316 6.2 -24.47004 0:16023 13.1 3.15432 0.07823 20.1 0.50981 0.05513 27r oo 0.15915 13.2 2.49260 0.07813 20.2 0,28999' 0.05972 6.3 118.45598 0:15894 13.3 1.99007 0.07814 20.3 6.4 16.57295 0.15774 13.4 1.58826 0.07831 2014 -0.08800 0:031481 6.5 8.61513 0.15664 13.5 1.25342 0.07873 20.5 - 0.29857 0:04193 6.6 5.64147 0.15570' 13.6 '0.96451 0:07960 20.6 -0.50280' 0:04427 6.7 4.06369 0.15494 13.7 0.70727 0.08139 20.7 - 0.71535 0.04525 6.8 3.07045 0.15446 13.8 0.47052 0.08566 20.8 - 0.95825 0.04578 6.9 2.37624 0.15434 13.9 023607 0.10348 20.9 -1.18099 0.04598 7.0 1.85454 0.15478 14.0 21:0 -1.45067 0.04611 29

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9.4

-4.2

-

--

_-

--

-0.0 0.81650 -0.81650 7.1 1.50316 0.14958 14.1 - 0.00728 0.00658 0.1 0.78297 - 0.85203 7.2 1.14760 0.15192 14.2 - 0.33256 0.05647 0.2 0.75131 - 0.88972 7.3 0.84015 0.15705 14.3 -0.54506 0.06010 0.3 0.72141 -0.92971 7.4 0.56041 0.17017 14.4 -0.76254 0.06165 0.4 0.69315 - 0.97221 7.5 14.5 -0.99129 0.06242 0.5 0.66644 -1.01742 7.6 14.6 - 1.23779 0.06280 0.6 0.64118 -1.06559 7.7 14.7 - 1.51014 0.06297 0.7 0.61730 -1.11698 7.8 - 0.20091 0.04733 14.8 - 1.81930 0.06301 0.8 0.59470 - 1.17194 7.9 - 0.45376 0.09632 14.9 -2.18108 0.06296 0.9 0.57333 - 1.23086 8.0 - 0.66552 0.10220 15.0 - 2.61978 0.06285 1.0 0.55311 -1.29417 8.1 - 0.88058 0.10503 15.1 - 3.17522 0.06271 1.1 0.53397 - 1.36244 _8.2 -1.10544 0.10645 15.2 - 3.91817 0.06253 1.2 0.51585 - 1.43632 8.3 - 1.34663 0.10712 15.3 - 4.98827 0.06234 1.3 0.49870 - 1.51661 8.4 -1.61198 0.10733 15.4 -6.70606 0.06214 1.4 0.48246 - 1.60430 8.5 -1.91193 0.10726 15.5 - 10.00790 0.06193 1.5 0.46709 -1.70062 8.6 -2.26134 0.10700 15.6 - 19.29818 0.06172 1.6 0.45253 - 1.80711 8.7 -2.68274 0.10661 15.7 -257.56314 0.06151 1.7 0.43875 - 1.92595 8.8 -3.21262 0.10614 57r

T.

0.06149 1.8 0.42568 -2.05906 8.9 - 3.91481 0.10560 15.8 22.24111 0.06130 1.9 0.41332 -2.21046 9.0 -4.91276 0.10503 15.9 10.45787 0.06111 2.0 0.40162 -2.38450 16.0 6.69700 0.06092 9.1 -6.48151 0.10444 2.1 0.39054 -2.58748 _{9.2 -9.38477 0.10383} 16.1 4.81878 0.06075 2.2 0.38006 -2.82837 _{9.3 - 16.83355 0.10323} 16.2 3.67450 0.06061 2.3 0.37015 -3.12040 _{9.4 - 83.87008 0.10263} 16.3 2.89132 0.06050 2.4 0.36079 - 3.48393 _37r rf _0.10249 16.4 2.31166 0.06044 2.5 0.35196 - 3.95192 _{9.5 26.94415 0.10206} 16.5 1.85731 0.06045 2.6 0.34365 - 4.58177 _{9.6 11.37835 0.10151 16.6 1.48484} 0.06056 2.7 0.33584 - 5.48272 _{9.7 7.01381 0.10101 16.7 1.16805} 0.06085 2.8 0.32853 - 6.89186 _{9.8 4.94492 0.10057} 16.8 0.88998 0.06147 2.9 0.32172 -9.43874 _{9.9 3.71464} 0.10020 16.9 0.63874 0.06277 3.0 0.31544 - 15.53107 _{10.0 2.88653 0.09994 17.0} 0.40472 0.06610 3 1 7r 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4.0 0.30969 0.30746 0.30211 0.29930 0.29624 0.29342 0.29183 0.29202 0.29502 0.30325 0.32419 - 50.73058 347.32624 13.28969 6.92837 4.64257 3.32894 2.46267 1.83677 1.35133 0.94648 10.1 10.2 10.3 10.4 10.5 10.6 10.7 10.8 10.9 11.0 2.28081 1.93728 1.42700 1.10262 0.81844 0.56048 0.30952 -0.26324 0.09982 0.09957 0.10038 0.10142 0.10364 0.10882 0.12744 0.06585 17.1 17.2 17.3 17.4 17.5 17.6 17.7 17.8 17.9 18.0 0.16792 - 0.20767 - 0.42031 - 0.63666 -0.85824 - 1.09670 - 1.35743 - 1.64977 - 1.98785 0.08512 0.04400 0.04904 0.05063 0.05158 0.05199 0.05218 0.05226 0.05225 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 5.0 0.43358 - 0.06132 - 0.34729 - 0.56563 -0.77290 -0.97975 -1.19287 0.51594 0.04770 0.13086 0.15074 0.15966 0.16411 0.16626 11.1 11.2 11.3 11.4 11.5 11.6 11.7 11.8 11.9 12.0 -0.47714 - 0.69081 -0.91194 -1.14677 - 1.40237 - 1.71565 -2.01601 -2.40889 - 2.88602 -3.16225 0.07360 0.07667 0.07815 0.07890 0.07925 0.07943 0.07931 0.07917 0.07895 0.07850 18.1 18.2 18.3 18.4 18.5 18.6 18.7 18.8 67r 18.9 19.0 - 2.39116 - 2.89207 - 3.54568 - 4.45565 - 6.00403 -8.28692 - 13.90348 -41.90938 co 40.84472 13.49894 0.05220 0.05212 0.05201 0.05188 0.05176 0.05161 0.05146 0.05132 0.05124 0.05118 0.05103 5.1 5.2 - 1.41841 - 1.60334 0.16709 0.16709 12.1 12.2 - 4.36455 - 5.64417 0.07836 0.07808 19.1 7.93464 0.05090 5.3 - 1.93644 0.16657 12.3 - 7.83457 0.07775 19.2 5.50586 0.05077 5.4 - 2.24974 0.16570 12.4 - 12.57478 0.07742 19.3 4.12357 0.05066 5.5 -2.62079 0.16459 12.5 - 31.44392 0.07708 19.4 3.21649 0.05057 5.6 - 3.07693 0.16333 47r ZF oo 0.07687 19.5 2.56448 0.05051 5.7 - 3.66371 0.16197 12.6 60.30544 0.07676 19.6 2.17018 0.05043 5.8 -4.46396 0.16055 12.7 15.16292 0.07644 19.7 1.66170 0.05052 5.9 6.0 - 5.64648 - 7.61790 0.15910 0.15766 12.8 12.9 8.46020 5.73778 0.07614 0.07587 19.8 19.9 1.32406 1.03150 0.05065 0.05094 13.0 4.23693 0.07563 20.0 0.87378 0.05112 6.1 -11.67200 0.15623 6.2 - 25.32220 0.15484 13.1 3.27115 0.07544 20.1 0.53205 0.05283 27r T co 0.15373 13.2 2.58611 0.07531 20.2 0.30535 0.05670 6.3 122.64569 0.15351 13.3 2.06590 0.07527 20.3 6.4 17.16862 0.15226 13.4 1.64997 0.07538 20.4 - 0.08948 0.03096 6.5 8.93009 0.15112 13.5 1.30344 0.07571 20.5 - 0.30654 0.04084 6.6 5.85157 0.15011 13.6 1.00452 0.07642 20.6 - 0.51770 0.04300 6.7 4.21819 0.14927 13.7 0.73854 0.07795 20.7 - 0.73762 0.04389 6.8 3.19000 0.14867 13.8 0.49377 0.08203 20.8 - 0.98901 0.04436 6.9 2.47145 0.14839 13.9 0.25617 0.09536 20.9 - 1.21957 0.04453 7.0 1.93159 0.14861 14.0 21.0 - 1.49874 0.04463 E

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--

--

-0.0 0.81650 -0.81650 7.1 1.71207 0.13133 14.2 - 0.36009 '0.05215 0.1 0.77916 - 0.85619 7.2 1.31705 0.13237 14.3 - 0.59588 0.05497 0.2 0.74404 - 0.89841 7.3 0.97721 0.13503 14.4 - 0.83785 0.05611 0.3 0.71099 -0.94333 7.4 0.67319 '0.14166 14.5 - 1.09265 '0.05663 0.4 0.67999 - 0.99125 7.5 0.37955 0.16596 14.6 -1.36738 0.05685 0.5 0.65064 -1.04211 7.6 14.7 -1.67102 0.05691 0.6 0.62312 0.59722 - 1.09648 -1.15454 7.77.8 -0.24798 0.07669 14.8,14.9 - 2.01575 -2.41922 .0.05686 0.05676 018 0.57284 - 1.21667 7:9 -0.48820 0.08952 15.01 - 2.90849 0.05661 019 0.54991 - 1.28328 8.0 -0.72245 0.09415 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 0.52832 0.50801 0.48888 0.47088 0.45392 0.43795 0.42289 0.40872 0.39534 0.38273 0.37083 - 1.35488 - 1.43205 -1.51554 -1.60621 - 1.70517 1.81378 - 1.93373 -2.06748 -2.06748 - 2.38717 -2.58245 8.1 8.2 8.3 8.4 8.5 8.6 8.7 8.8 8.9 9.0 9.1 -1196123 -1.21137 -1.47996 - 1,77565 -2.11004 - 2.49965 - 2.96961 - 3.56060 -4.34379 - 5.45689 - 7.20660 '0.09621 0.09714 1009747 0.09744 0.09719 0.09680 0.09631 '0.09577 0.09517 10.09456 0.09393 15.1 15.2 15.3 15.4 15.5 15.6 15.7 57r 15.8 15.9 16.0 - 3.52800 -4.35664 - 5.55019 - 7.46614 -11.14888 -21.51085 - 287.26091 co 24.82018 11.67767 7.48296 0.05644 0.05624 0.05603 0.05582 0.05559 0.05537 0.05515 0.05513 0.05493 0.05472 0.05452 2.1 0.35962 -2.80998 9.2 -1044484 0.09329 16.1 5.38808 005433 2.2 0.34904 - 3.07977 9.3 - 18.75292 0.09266 16.2 4.11181 005416 2.3 0.33988 -3.40738 9.4 -93.52249 10.09204 16.3 3.23831 0.05402 2.4 0.32965 - 3.81306 3z oo 0.09189 16.4 2.59183 0.05390 2.5 0.32078 - 4:33602 9.5 30.07469 0.09144 16.5 2.08514 0.05384 2.6 0.31244 - 5.03945 '9.6 12.71327 0.09086 16.6 1.66982 0_05385 2.7 0.30459 =, 6.04519 9.7 7.84671 0.09031 16.7 1.31668 0.05398 2.8 0.29723 - 7.61767 9.8 '5.53771 0.08980, 1618 1 00687 0.05433 2.9 0.29034 -10.45905 9.9 4.16554 0.08935 16.9 0.72730 0.05513 3.0 0.28393 - 17.25476 10.0 3.24199, 0.08898 17.0 0.46797 0.05717 3.1 0.27799 0.27566 - 56.51519 lOd 10.2 2.56654 2.18347 0.08871 0 08834 17.1 17.2 0.21608 0.06615, 3.2 0.27082 ,387.45823 10.3 1.61486 0.08890 17.3 0.22345, .0.04090 3.3 0.26714 14.88941 10.4 1.25373 0.08920 17.4 - 0.45900 0.04490 3.4 0.26332 7.79469 10.5, 0.93802 0.09043 17.5 -0.69761 0.04621 3.5 0.25968 5.24570 10.61 0.65312 0.09338 17.6 - 0.94648 0.04677 3.6 0.25691 .3.78155 10.7 0.38488 '0.10249 17.7 -1.21226 0.04703 3.7 0.25527 2.81719 10.8 17.8 - 1.50295 '0.04713 3.8 3.9 0.25532 0.25818 2112238 1.58727 10.9 -11.0' - 0.28170: 0.06153 17.918101 - 1.82893 - 2.20596, '0.047140.04709 4.0 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8' 4.9 5.0 0.26669 0.29141 - 0.06221 -0.36567 -0.60393 -0.83225 - 1.06124 - 1.29784 1.15057 0.76763 0.04701 0.12428 0.14119 0.14828 0.15151 0.15281 11.1 11.2 11.3 11.4 11.5 11:6 11.7 118 11.9 12.0 12.1 -0.51970 -0:75513 - 1.00116 - 1.26270, - 1.54756 -1.89725 -2.23175 -2.66990 -3.20204 - 3.51020 - 4.84892 0106781 10.07014 0.07119 0.07166 0.07181 0.07183 0.07164 0.07143 0.07116 0.07071 0.07053 18.1 18.2 18.3 18.4 18.5 18.6 18.7 18.8 67r 18.9 19.0 - 2.65577 - 3.21445 - 3.94344 - 4.95838 - 6.68538 - 9.23162 - 15.49609 46.73263 oo .45.56758 15.06731 .0,04700 0.04689 0.04676 0.04662 0.04649 0.04633 0.04617 0.04602 0.04594 0.04587 0.04572 5.1 -1.54870 0.15303 12.2 -6.27835 0.07019 5.2 - 1.82144 0.15259 12.3 - 8.72145 0.06984 19.1 8.86114 0.04557 5.3 -2.12577 0.15174 12.4 -14.00849 0.06949 19.2 6.15218 0.04544 5.4 -2.47506 0.15062 12.5 -35.05434 0.06915 19.3 4.61044 0.04531 5.5 - 2.888861 0.14932 47r co 0.06892 19.4 3.59874 0.04520 5.6 - 3.39761 0.14791 12.6 68.39414 0.06880 19.5 2.87153 0.04511 5.7 -4.05212 0.14644 12.7 16.92883 0.06847 19.6 2.43176 0.04500 5.8 - 494477 0.14494 12.8 9.45291 0.06815 19.7 1.86472 0.04502 5.9 -6.26379 0 14342 12.9 6.41643 0.06784 19.8 1.48823 0.04506 6.0 -8.46273. 0.14192 13.0 4.74247 0.06757 19.9 1.16209 0.04521 20.0 098631 0.04529 61 --12.98461 0.14044 13.1 3.66531 0.06732 6.2 - 28.20954 0.13899 13.2 2.90129 0.06713 20.1 0.605961 0.04638 2z Fc 0.13783 13.3 2.32117 0.06699 20.21 0.355031 0:04878 6.3 136.82711 0.13760 13.4 1.85742 0.06696 20.3 6.4 19.18265 0.13628 13.5 1.47116 0.06707 20.4 = 0.09456 0.02930 6.5 9.99374 0.13503 13.6 1.13818 0.06745 20.5 - 0.33385 0.03750 6.6 6.56010 0.13389 13.7 0.84233 0.06834 20.6 - 0.56861 0.03915 :6.7 4.73838 0.13288 13.g 0.57173 0.07049 20.7 -0.81355 0.03979 6.8 3.59176 0.13204 13.9 0.36610. 0.25979' 20.8 - 1.09373 0.04011 6.9 2.79064 0.13142 14.0 20.9 - 1.35079 0.04020 2.18903 0.13113 14.1 - 0.00732, 0.00654 21.0 -1.66210 0.04024 ifit a = 30° .31

-0.7

-11.0