On the swaying, yawing and rolling motions of ships in oblique waves

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ARÔf

ON THE SWAYING,

YAWING

AND ROLLING MOTIONS

OF SHIPS IN OBLIQUE

WAVES

MOVIMENTOS DE DERIVA, GUINADA

E JÔGO DOS NAVIOS EM MAR

DE VAGAS OBLIQUAS AO

RUMO

'rASAI

()

por

PRESENTATION

This Technical Contribution of Professor Fukuzo

Tasai, of the

Re-search Institute for Applied Mechanics Kyushu University,

Japan, has all

the characteristics of an excellent paper and one

of groat ob1ectivity

-Although. this is not the first time it is being published, as

it has already

appeared in the Bulletin of the above named Institute and in the wellknown magazine "International Shipbuilding Progress" However, we take pride in publishing the complete text in our magazine, although in smaller print

because of the quality of fhe lecture and because it will be new ti many

of aux readers

-CONTENTS

I. Introduction

U Exciting Force arid Moment in Oblique Waves

¡n.

The Coupled Equations of Sway. Yaw and Roll in Obli

que Waves

1V Numerical Calculation and Consideration

A The Roundhaul Netter of 79 tan type

The Submarine Chaser The Destroyer Conclusions References

APRESENTAÇAO

tste trahoiho do Professor FUKUZO TASAI. do Instituto de Pesqui.

sas de Mecánico Aplicada da

Universidade de Kyusbu. no Japâo,

apre-senta características de exceléncia técnica e de objetivulade incamum. Embora nba sejO inédito, pois já foi publicado pelo Boletim Oficial do mesmo Instituto e pela co-irmá "International Shipbuilding Progress". tasemos questbo de opresentá-lo integralmente em nosso revista, apenas em tipos menores, pelo quolidade e também par ser inédito para a maio-ria de noesos leitores.

RESUMO

Neste trabalho é apresentado um método aproximado para o

cOl

culo da fôrça de deriva, e momentos de quinada e jàgo, que agem sôbre o novio qucindo navego em mar de vagas obliquas ao ritmo. Em seguida. ¿ tratado o problema das equoçdes lineare. simultûneas que representarri o eleito combinado dc, deriva, quinada e jOga. no novio corn velocidode

finita, o firn de determinar um método aproximado para

resoluç&o do

problema das oscilaçäes farçadas.

Como primeiro passo para o estudo de taie movimentas

combina das, cam três graue de liberdade no plano lateral, o autoS investigou açóos mútuas reflexas entre a deriva, a gulnada e o jOga, quando o navia se acha corn velocidade zero. Isto é. o autor efetua o cálculo numérico para trés tipas do formas de carona, par meto da equaçoes lineares simultáneas, do

efeito combinado da deriva, guinada e bolanço. deduzidas por ¿le em

1965 paro a careno do novio cam velocidade zero.

Os resultados pedoni ser apresentados da seguinte moneira Na casa do cosco cam pOpo plana hO uma açòo notável canjunta do quinado cam o jOga.

Também, neste caso, se o quinada f Or impedida, a joga terá seu va lar máximo corn vagas transversais.

Dependendo da forma da carona, pode sor possível hover ogas

de

valôres máximos corn vagos transversale por causa da açäo

reflexo

da deriva e quinada.

Quando a seçbo transversal do navia apresenta um

valar elevado

pata o reloçäo bOca/calado, o coeficiente da estoico efetivo da vaga. Os vêzos. torna-se multo major que o encontrado pelo cálculo teórico do Dr. Watancibe.

Lr,eisr,r Of RessrOh Ifl.titOi5 for Appird

Sf5050-Sf0.. Kyu.50 Ouivrr.ity, Jps

t - N.' i - JAN. 1968

Lab. y.

SchepsboQkinth-Technische

Deli t

The author introduced in literature [tJ° certain meer coupled equations of swaying, yawing and rolling at beam seas and made an investigation, neglecting

yawing, on the cross coupling action of swaying and rolling. Then in [2] he showed that in the cose where the beam of a ship to wave length ratio is very small the following two solutions almost coincide: one is to solve rolling with the coupled equations of swaying and rolling derived from the calculatedvalues of hydrodynamic force and moment which act upon a restrainedbody and the other, with the ordinary equations of motion based upon the Frotade-Krilov's

theory.

So far as it is concerned with the rolling notion at beam seas, therefore, the difference of the solutions will be negligible whether the theory of restrainedbody is applied or ihat of Froude.Krilov is resorted to.

Regarding the coupled motions of swaying and yawing in oblique waves. we have Edo's method [3] for an approximate calculation, which gives us theresults

showing a good coincidence with experiments.

Lately, O. Grim and Takaishi 14] introduced. for a mathematical hull form of fore-and.aft symmetry, coupled equations of sway and roll in oblique waves and when ship velocity is zero. Here they calculate total rolling n'loment in

which the rolling moment based upon swaying inertia force is considered. They point out that, when G5M is small, the maximum value of rolling moment sometimes occurs at oblique waves of some 45 degrees inslead of beam

sea. It Itas been already known to a certain extent even with Froude'Krilov'a

theory that such phenomena can happen.

On the other hand, itis slated in literature [] that, when ship velocity is

zero, depending on hull shapes, the maximum value of rolling amplitude lu to be found not at beam sea butt at the slate of quartering seas. The validity of such a tendency is also seen in the resulto of experiment [6].

In this paper. discunnions are made on the coupled equations of swaying. yawing and rolling at oblique waves. Furthermore, she mutual coupling actions of thrse motions are investigated by means of numerical calculations made on

three kinds of hull shapes.

H. Exciting Force and Moment in Obliqae Waves

Assume in Fig. i that the ship is navigating at a constant velocity V in the direction at an angle to the propagating direction of a regular wave.

Numbers in brackets designate References at the end of the paper.

z, ;

Fig. 1. Wave und coordinates systems.

Wave

Let Ot'Ex,C and be space-fixed coordinates, G,'xyz (where G., is the center of gravity of the ship) hull-fixed coordinates, and O-'jC the coordi-nates with their origin at O (where O is the intersection of G, Z with the painted load water line) and in parallel with O,'Ex,C,.

Suppose the ship makes small motions around O-EtC. The equation of regular wave can be expressed by the formula

Ctv=és'C cos(Ke,wi) (2.1)

where h-wave amplitude, k=a,'/g-2tr/2, A-wave length. The orbital velocity in the t direction is

V.,.'hw sin ze"c cos(KE con2-Kqsiny-w.r)

os.=e(I-r cosy), r-aiV/g

(2,2)

Then the orbital acceleration is

i=,hat sine'5t sin(KE cosy-Xx siflZ-a?.t) (2.3)

In the above equation, E. 'i and C can be approximately replaced by X, y and z re-spectively since ship motion io small.

In this paper, the author will calculate the exciting force by means oi the

3g

TRABALHOS PAPERS

H oqeschoo

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strip method. _{Following the method of Dr. Watanabe in his theory on pitch} and heave [7], we canput the swaying force acting upon the unit_length cross section of a shipapproximatelyas

F'*Fi'+m'+N'

V (2.4)

The first terni of (2.4) is the force based _{upon the Froude-Krilov's theory, the} second term orbital acceleration, the third the force based upon orbital velocity, and the fourth the influence of ship velocity. Here rn' and N0' signify the sway. ing added mass and the coefficient of damping of the unit length cross section

respectively. and V, signify the mean values of >, _{and V, respectively, and}

they can be approximated by the values at C-7/2on the centre plane of the

hull. That is to say, we can describe as follows:

',-hw' sinze''

sin(Xx cosz-wt)

V_4I.d sifle5 cos(Kecosz-w,l) (2.5)

where E-g and Tisthedraft of the cross aection.

Eda [3] used for

F,, F, and F,

in (2.4) the following approximation:

F,*pgSwøw Sin2et

sin(ICx cosz-w,I) (2.6)

where Sa is the immersed sectional area of the _{cross Section, and 9e, maximum} wave slope,

F,*m'Ft0-.pgSwK/9w sin Or_i/sta sin(Kxcus z-abt)

F,*N,'P...N,'he, sin zet

cos(Kx cosx-w,t) (2.8)

On the other hand, the pressure variation due to waves is

P'- - pghe cos(JCf cos z

-k

sin z -io.t) (2.9) Now integrating the pressure in (2.9) for a two.dimensional body with the Lewis form section. we will obtain the swaying force based upon the Froude-Krilovstheory asfollows:

F,',=pg8S,

sinzS ain(kx cos0-w:) (2.10)

where

S_SCsfr!.i.f00eaz sin(ky, sito2)dz, (2.51) and y, and r,are_{the coordinates of the contour of the Lewis form section with} oas the origin(Fig. 2).

Errorsfromthe approximate formula (2.6) will become large at a cross

section ofbroader beams. In this paper the formula (2.10)wilt be used through.

out the calculation in which the hull section is to be approximated by the Lewis

form Section.

On the other hand, there is a value exactly calculated by Tamura [8] con.

cerning the exciting force ofwaves actingupona two-dirttensional body under

restrained condition at beam sea. _{¡t must be noted here that an attempt to app.}

¡y the results of (8] to the case.

Z.0_n/2 and wwu,. is notappro. priate considering its boundary.

conditions. It can behowever used approximately for the cases where

z is nearly equal to rIZ, shipvelo.

city is low, and Ea is small. But the fourth term of (2.4) cannot beobtainedthroughthis method.

In this paper, therefore, the

author willalsoobtain a consistent formulaof approximate calculation

based upon the ideaof (2.4) con. cerning rollingmoment, indepen-dently of V. x and theperiodof encounter. That is to say, the

rolling moment on a restrained body can be written analogously to (2.4) as

follows

M0.'M00'+M,.'+M,,'+M,,' (2.12)

Now. M0' the rolling moment based upon the Frosde-Krilov's theory can be

obtained for the Lewis form section as:

M,i'[OG0-S+(P-.R)T[pgS,,9, sin sïn(.Kx cos5-w,f) (2.13)

where

P-R.

em sin(s, sinZ)Ydy,]

z Fig. 2.

e-5z sin(Hy, sin)Z, dz,

Substituting further (R-PiT/S-11,

we obtain

M0' pgS&,, sino .S(0G0-4)sin(Jfc coso -a,t) -F'51(ÖG5-11)

(2.7)

(2.16)

From the aboveequation itcan be seen thatS(OG0-Ii)rG0M is the coefficient

of effective wave Sloper. Also, (2.13) is one and the same withthe principal term of rolling_{moment (formula (20), p. 82, of [9]) which Dr. Watanabe used}

for calculatingr.

Both (2.11) and (2.14) can be calculated exactly. For example. Fig. 3 and Fig. 4 show the values of S and P-R for a broad beam section of H0=ß/2T-4.0 and e-S,,/BT'--0.8. while the values of S and P-R in the case of X-aI2

are indicated in Tables ¡ and II.

AS can be seen from these tables, P-R is negative for aSection ofsmall

H0; however, with H0 becomisg larger il becomes positive, and il growsvery

ta 05 60 5.0 ¿.0 30

2t

1.0 d-

fT

Fig. 3. _{Calculated valses of S for the section of H0='B/2T=4.O}_{and e=S,/ET0.8.}

Fig. 4. Calculated values of P-R for thesection ofHa=B/2T='4,0atsile=S./.ØT=O.8,

SWA FING. YAWING AND ROLLING MOTIONS OF SHIPSIN OBLIQUEWAVES 29

Fig. 5. Longitudinal distribution of the rolling moment based upon FroudeKrilov's

theory.

large for

¡f.3

and 4. Hence in the case of a ship with diversified cross section

inlongitudinaldirection.M00'- pgS,,sin

0.S(0-I1)

may change itssign isgoing

from fore to aft of Go(Fig.5).

Next, the moment of the second and the thied termsof (2.12)canbe. as

discussed in [2]. represented by the following approximateformulae:

M,,'*F,5'(0G5-I,) (2.17)

ad

2.f,,'*F,5'(O-I,,)

(2.18)

Also, similarly to the case of swaying, the fourth term can be written:

M,,'*V

_(m'(i-i,)I,

(2.19)

40

TECNOLOGIA NAVAL

d

(2.14)

where I, and 1, arc 15e momnt leverdue to swaying force.

(2.15) When theship_{oscillates with the circular frequency of encounter ou,. the values of}

K,', N,',I.andI,, willbe obtained as the function of io,frontthe TAMIJEAs table

[8].

Though we cannot check in detail the degeee of precision of the swaying force and rolling moment thus obtained, in the case of z,r/Z, we can investigate

it by_{comparing these with the results of Tamura [8]. One example is shown}_in Table Ill.

Through the comparison with Tamura [8] we can derive the following

points: the approximate values is this papershow a good coincidence with the

casesof fullsections. except for the casesoffine seclions. Thereason for this

descrepancy can be ascribed to the failure in the approximation of K,' _i

N,e_0.a of swayingforce. Itis natural, too, that with increasing Ea their ap-proximation should grow worse. Itis, however, possible to approximate total swaying force and total rolling moment, when EaO.4. with. an error of about10

01 02 0.3 04 05 06 07 0.8 09 1.0

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Table Ill.

(H,=B/2T, e=Stional area coefficient)

percent. Ors the other hand, there is Motoras study [tOJ on the approximate calculation of swaying force. The author however, will leave here the problems of such corrections in order to carry Out calculations by the approximate method introduced in this paper.

Now that we have obtained here the swaying force and rolling moment which act on any cross section of hull, we can obtain by means of the strip method the swayipg force, yawing moment and rolling moment, for Ilse whole ship, as foltowt.

Swaying force

F,.=Wø. sin z(F cosw,t-F, siflwJ) F,-.S,+S5±S5-VS1

(2.20)

F,+-+V3

Yawing moment:

M,.-W&,L sin (Y cosa,,:- Y, sinat,t) Y,_Ys+Y.-,-Y.-VY,

(2.21)

Y- l',+ Y- Y5+ Vr,

Rolling moment:

Ma,...W9,. sin2(M, cosa,-M, sinw,J)

M,F.ÖG,-(M,M.+M5- VM1)

_(2.22)

M.F,0G,-(M,+M,-M5+VM,)

The termsE,. E, M, in tise equations (2.20), (2.21) and (2.22) are to be given by the following equations with the substitution of K cot y-K,: (2.23)

S.,S.tin(K,x)dx1V,

SWA YING. YA WING AND ROLLING MOTIONS OF SHIPS IN OBLIQUE WA VES 31

=

J

sin(K,x)dx/Vo

si_J N'e"

s._J ()e.10a sin(K,x)dS/mow

S...Sx sin(K,x) dxl V,L S.K,enIstdX sin(K,x)dx/V,L

N,'e'x

sin(K1x)dx/m,wL

Y-

J ()e_n10Ux sin(K,x)dx/m,oL M1_J

S(R-P)T

S,K,'e" I. sin(K,x)dx/V,

M,

- J

N,en/d i,

nin(K,x)dx/mow

M7=j

(m'I,)e5'0 nin(K,x)dr/m,w

_j

The terms of even suffixes,E,, .5', M, are to be obtained by substituting sin(k,x) in the integrand ofE,, E, M, respectively with cos(k,x). In the equa. tions (2.24), V, and et, signify the volume of the displacement and mass of the

ship respectively.

With a hull shape which is symmetrical in fore and aft about the center of gravity G,,

E,=E,=E,=Y,=i'=T'=-M,=M1=M,-M,=o

(2.25)

With the hull shaped quite asymmetrical in fore and aft, the terms of (2.25) may sometimes become considerably large.

Also, referring to Fig. 1, the signs of E,, S, M, will be reversed

accord-md as y<90', (i.e. the state of following waves, k,>0), or z>90' (i.e. the state of head tea, k,.<0).

Now let us consider about rolling moment. In the first place, following the

VOL

I - N.° 1 - JAN. 1968

(2.24)

Froude-Krilov's theory. we get the terms M, and M, in equations (2.22) as the following equations:

M,,=S, 0G,- M,- J S{S.OGi+(P-R)T1sin(k,x)dx/V,

M,,-S5.OG,-

M1-J

S(S.OG,-i- (P-R)flcos(k,x)dx/V,

Between tise two states of "following wave" and "head sea ", while the sign of M,.,, changes, that of M,. doesn't. In this case, however, M,, will retain the same absolute value. On the ether hand, at beans sea where y=9.O', M,,-.0 will hold, As a result, according to hull shapes, Ms_/M..1+M,? may often become larger than in the case of y-90' when y-75' or y=IOS.

The values of M are equal regardless of whether y=75' or x-lOF. That is, M, has a symmetrical distribution about y-SO'. On tise other hand, as M,, changes its sign in going from y-75' to y-l05 or inversely, the phase difference between rolling moment and wave will change its sign too.

As can be seen from the equations, (2.20), (2.21) and (2.22), however, the

terms S,,, S,,, Y,,, Y,. M, and M,, do nor vanish even at V-0, and therefore sway. ing force, yawing moment and rolling moment in the case of y -75' must differ from the values in the case of x-b5'. This is due to the existence of the terms N;e_ro0d and N,'e-' 1,,. _{These values are large in sections of smaller 11s, and}

become relatively larger in comparison with K,'e-'° and X,'e'5rta i, with the increase of d

Hence in the case of a hull shape of fore-and-aft asymmetry, rolling moment has different magnitude in two cases of following wave and head sea, even when

V-0. However, since is generally small ir, the vicinity of rotlings resonance point, the asymmetry of rolling moment about y'9O' is naturally small. 111. The Coupled Equations of Sway, Yaw and Roll in Oblique Waves

Let s, and O be displacements of swaying, yawing and rolling motions re'

spectively.

In the case of V-0, the coupled equations of swaying, yawing and rolling in oblique waves can be written as follows as given in [IJ:

mo(l±K,Yr-hJ+m,+NÎ,r+moK,X,Ö-hN,2,,iI---F,,

(!,+I,)+N,,,u± m,K,+Ñ,XÒ+m,k,2,±NS,n=M,

O WG,M O +m,K, 2,' +N,, B,, + t,K,S,,Ç+N,X M1, (3.1) where,

msK_JP5.L'dx_Jm'dx. N=JNO'

S K,'

S+K,'."

K (Tamara) -F,', R7(Tamara) (P_R)_K,'e-'-Fl,/T -X?(l,,/T(Tamura) -E,(I.JTfTarnara 1.0 0.9699 1.9699 1.9699 0.2 0,917 1.314 2.126 2.061 -0.265 -0.248 -0. 253 -0.262 0.0 0.0315 0.4 0.830 1.175 1.823 I. 704 0.109 0.0941 0.7600.835_0.696 -0.643-0.637-a -0.238-0.195-0. -0.248--0.187-0,143 0.6 o. 7 0..' t.379 1. l7 0.133 0.1' 0.8 0.656 0.3691. 0.9952. 0.8382. 450 144j 0.119 0.076 r 1.0 0.2 0. 934 I.44 2.237 2.091 -0.159 -0. 141 0.514 0.

''

0.037 0,031 0.4 0. 869 1.404 2.019 1.735 -0.305 0.496 0.447 0.11 0,078 -0.409--0.498-0.467 0.6 0. 801 t. 190 1.550 1.331 -0. 296-0. 0.435 0.365 0.136, 0.079 0.8 0.734 0.988 1.393 1.023 174 0.376 0.299 0.130 0.051 m,,k,,=

Jm'

. . dx, Ñ,x2 - JN'x . dx

mox.fm(oGa-I)o.x N:Ss=JN'(0Gs_l)dx

¡._Jm'xzic.

N _JN',xi.dr

"uk,S',-

Jm'(b_z,)x

c, N,X2, =JN(OGo_I)x.dx (3.2)

/.-mass moment of inertia of yawing, ¡0+1, -'virtual mass moment of inertia of rolling, and N,,.-.equ,vslent linear damping coefficient of rolling.

Now, using F,,, M0, und M,, (putting V=0) as were given in thepreceding

chapter, we can calculate from (3.1) the motions of swaying, yawing and rolling in oblique waves for the case of V-0.

In the case of VO, each term on the left side of _(3.1) should be changed in general.

As a few examples from the

literatures on the calculation of coupled equations for swaying and yawing in the case V-mO, we can cite the works of Rydill [Il], Eda _13]. Motora 1121. etc. All these authors make their calculation on the assumption that the linear forces and moments which are gene. rally used for the theory of maneuverability in still water can simply be applied to the cane of periodic motion, too. Moreover, Eda [3] shows a good coinci-dence betwecn his calculation and experimental results. On the other hand, A. t. Reif [13] calculated swaying and yawing motions at the time of V'vO applying the force derivatives he had obtained regarding the periodic motion at the time of V-O here this author assumed that the values of these derivatives were al-most constant regardless of whether V'vO or V-O. He made further calculations on the lateral bending moment, the results of which show a general coincidence with Eda's calculation [3].

If steering is kept out of consideration, there will be no natural periods for swaying and yawing motions in waves; and in general there occur no such

reso-nance oscillations as are seen in the cases of pitching, heaving and rolling.

The-refore. the solutions of forced oscillation can be determined in this case chiefly

by exciting force of waves and such inertia forces as m,(l+K,)r and (J,I,)O; and the effects of damping force and other coupled terms remain only in the se' cond order. That the results of Eda [3] and Raff {13] almost coincide mutual-ly can be explained by the general coincidence of their estimations for both ex' ternal force and inertia force.

Little is known, however, about whether or not the coefficients on the left

sides of (3.1) can he represented by the sum of what is based upon the wave-making phenomenon as discussed in this paper and what is due to Ihe circulation around the hull [14].

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Concerning the case when the ship has a finite velocity (V.0) and is sway. ing and yawing periodically, experimental studies on the force and moment deri-vatives arc now being proceeded by both G. Van Leeuwen (15] and Motora [161 with the method of forced oscillation. Here Leeuwen [16] employed a model ship (Todd 60 Senes C8-0.7O), for which the author of this paper carried out S numerical calculation. Now let us compare the ressilts from both cases.

When the ship has a finite velocity, there arises the swaying inertia force

m,Vi,. Taking thit force into consideration, therefore, the fourth term of the first equation of (3.1) must be modified to the following form.

(rfloV+sT,2s),, (3.3)

The results of Leeuwen's experiment made under the conditions, F-0.20 and provided with neither rudder nor screw, are compared with the author's cal-culation, as shown in Table 1V.

As can be seen from the table, the tendency of variation of - N, and

Table 1V.

Ñ,., with respect to Ç is different from each other. Although 1,+1, and Ñ show an error of about 20% for w-5, all other terms arc roughly coincidental in the order of magnitude. That is to say, a fairly good approximation will be attained merely by introducing m,,Vr as the effect of ship velocity into the equ-ations (3.1). in which only wave-making resistance is considered.

The results of Lecuwen [15], however, make us assume that within the range, Ed<O'lO, it is better, as shown in [3], [11] and [12], to employ the derivatisrts for the case, w-.O, neglecting wave-making phenomena.

On the other hand, with the case when the ship velocity is not zero, we are almost ignorant about the hydrodynamic rolling moment to be induced from sway-ing and yawsway-ing motions, and everythsway-ing should be left for future studies.

If a ship has a finite velocity, therefore, one possible approximation at the present stage would be: to modify the forth term of the first equation of (3.1) as shown in (3.3), to calculate the solutions of forced oscillations, and to inves-tigate the mutual coupling influences of swaying. yawing and rolling.

IV. Numerical Calculatios and Consideration

In the case of V-0, numerical calculations are made, following the methods described in chapters It and III, on three kinds of hull shapes.

A, The Roundhaul Netter of 79 ton type

The main particulars of this boat are Lpfr=24.54 en, B-5,6 m and D2.5 m and the body plan is shown ¡n Fig. 6. The calculation was made for the full' loaded conditions of Ihr boat when it Starts the fishing-ground, that is:

W_l84.5m'. Ca"0.584,

G0--I.86

en, G0M=0.62en.

die-206m, df-l.23m, do-289m, J,-mo(0.242 L,,)5,

natural rolling period T*6.0 sec., equivalent linear damping coefficient a,-0.32 (deduced from the data of [9]).

In the first place, the values of F,,, Me. and M,, were calculated for the wave periods T.,-' 5.0, 6.0 and 6.5 sec, Here F,. and M,, were found to have maximum value at z-90'. while M showed, as in Fig. 7, two maximums in the vicinities of z-45' and 135'. Moreover, each of the curves F,.. M...I and M,, took almost symmetrical form in relation to z -90'.

Next, 6,ìø, shown in Fig. 8 is the solution of rolling obtained by Equations

(3.1). There is seen hardly any difference in its magnitude between two conditi-ons of head sea and following waves.

Also amplitudes of swaying and yawing showed, almost symmetrical dittribu. tion about z-90'. similarly as in Fig. 8.

Though this hull shape presents a large fore-and-aft asymmetry al a first glance, the rolling moment is distributed almost symmetrically in the x-direction

about G, and, further in this case, the value of IM, isvery large.

42

On the other hand, the coupled terms ,n,,K,x,. m,K,x. etc. are so small that yaw-ing motion can hardly have any influences upon rollyaw-ing motion, and therefore, it is quite natural that the rolling amplitude has maximum value at -9O'.

Let us next try a comparison between the numerical calculation based on Ta-muras [8] dala and the calculation of this paper about the external force in the

case, -.90' and Tw-6.0 sec., the results of which are shown in the following table:

l53

r- (1.335)

Here. the term (3) represents the rolling moment itt which swaying inertia force has been taken into consideration, namely

M M,,,

_"i+k"

F

On the other hand, the term (4) signifies the rolling moment of the Froude-Kri-by's theory obtained by employing the exact calculation values for the Lewis form section (Table I, II), namely.

M,- PJSo(S dG,+ (P- R)T]dx

As seen from this table, there is made about 1Ó%'s over-estimation on each of IF,,I and [M,,.] in the approximation of this paper.

Furthermore, the coefficient of effective wave slope r is larger than unity. From

a calculation after Dr. Watanabe's approximate method in which mean draft at trimmed conditions is used, we obtained r'-O.727. This value is about 56% of the values from the term (4). In Dr. Watanabe's method equivalent encan draft is adopted for calculation and there every cross section is assumed to be rectan-gular for simplification. While, the greater part of the stern of this boat has. with H,-B 2T-.3-4, generally a wider section. This seems tobo the reason why there has been caused such a heavy discrepancy between the results of this trea-tise and the results by Dr. Watanabe's method.

B. TheSubmarine Chaser

The profile of this ship is shown in Fig. 9. The main particulars are as

fol-lows: L,,-L,.-0-59.0 m, 8-7.1 en, d-2.33 en, Trim-0, W_480e C KG,-2.707 m, 0G,- -0.377 tu, G,M-0,736 m. It is assumed that the radius of gyration for ya-wing is -.0.25 L,. and o.-0.32.

(4) M,l/&

-1 5

Fig. 10 Longitudin.oI distribution ob the rolling moment derived from swaying mertia forer

Assuming T,7.0 sec., calculations were carried out for the case where the ship oscillates in the waves of which T-7 sec. i. e., A-7f en. ht Fig. 10 is

shown the distribution in the x-direction of the rolling moment due to swaying inertia force about G,(Fig. IO).

Next, the coefficients on the left side of the equations of motion (3.1)

be-come as follow:

9 7 5 1. 3 2 FP

In contrast with the case of the fishing-boat in the preceding section, 'n5K,x,. ntoK.x and N,x are very large.

The swaying force, yawing moment and rolling moment under restrained con-ditions arr shown in Fig.s bi, 12 and 13. The distributions of absolute values of both swaying forer and yawing moment have almost symmetric form in relation

to 9O'. The same tendency is observed in the distribution of M,, except that it is slightly double-peaked. However, Ihr curve of the rolling moment

TECNOLOGIA NAVAL

a,'05

,,0. 325 e,,=0. 832

u8

Lecuwen iii,- Ii- (kg nm'/rn) 13,4 & 7

Author 1.3 0.38

mo(l+K,) (kg. se.c'/m) 15.4 9.2

Leesawen

1.-N,

(kg. in. ate') 3. I 3.2 Author

1,,/.i. 1.44 0.53

1,+1. (kg. en. sec5) 4.9 3.1

Lcuwea - Y,, (rg. nec/rn) 27 47

Author N,V'j.L/W 1.8 3.2 N, (kg sec/rn) 25.2 44.8 L,,euwen

-N

(kg. m. sec) 8 13.5 Author Ñ./,L/WL 0.012 2I0 N (kg. rn-ate) 5.9 tS.0 Leeuwsn -N,. (kg. see) 2

-2

Author 0.041 0.106 Ñ,5, (kg. sec) 1.3 3.35

Lseuwen ,neV-Y (kg, sec) 2 I0.

Author ,noV+Ñ,2s (kg. sec) 7,6 9.65

Tamura's method this trealise Froude-Kriloy (1) F,.(/&,. 298 328 (2) M,,F/6o

224''

243'" 3

-

141'" 154'" (r-I-ZS)

(r-l.34)

ns0(I±K.)-99

N,-8-"",

rnJÇe,-234'

N,x-l9

Nx,=.4.7' ¡.+ 1.-21593

N,- 1209-"".

m,K,5',- -433'

N-.t°,' -43"'"',

J + J, ... 43710 N,,=80 (4.1)

(5)

based upon the Froude-Krilov's theory has the maximum at z 90. and besides,

itis of the single-peaked type. In Fig. 13 is shown by a dotted line the rolling moment M in which swaying inertia force is taken into consideration. Here the double.peaked type is emphasized.

Shown in Fig.s 14, 15 and 16 are the curves of the amplitudes and phase-differences of swaying, yawing and rolling calculated from the equations (3.1).

First of all. the swaying amplitude curve of Fig. 14 has the maximum at z «90'. In this ship, following waves cause greater influence upon swaying than brad sea. This is mainly due to the coupling efl'ecls of yawing.

Next, in the yawing amplitude curve of Fig. 15, the peak at head sea 135' is much greater than that at following waves 45'. Now, from the equations (3.1). we can obtain the yawing moment which has taken into consideration the coupling moments of swaying and rolling, as follows:

M,'Mw -mk,xs

(4.2)

Putting further M Y cos ut- Y, sin sat and inserting tite solutions of O and o into the equation (4.2) we can obtain the ?. and Y,. We cn find that each of nhKX.8 and mfC,x, has a great influence upon and Y. respective. 1y. Moreover Y, is much greater thaniy;L Therefore, due to the fact that m0

K,z5O is not only large but also rnjC5',<O. the value ofM,.in (4.2), and there-fore, the peak value of yawing amplitude s at head sea has become much gre-ater than that at following waves,

Now let us consider about rolling motion. Putting first Ø-0, and solving the coupled equation of swaying and rolling, we can obtain the amplitude curve, as

indicated by black circles in Fig. 16. in an emphasized dooble.peaked type. The

maximum is obtained here at quartering seas z 60'.

On the other hand the values of &o obtained from equations (31) are shown by white circles in Fig. 16. which yield the maximum at z 75'. However, in this cate the amplitude curve becomes flattened, and lite degree of asymmetry moderated.

This is due to the fact that, In this ship, although moK,,x', and Ñ,.x, are large and therefore the copuling action of yawing on rolltng la large too, the teru tnoK.x°s. etc. have negative sigas and therefore tend to leasen the coupling effect,

of swaying on rolling.

Next, the phase angle r,' at head sea becomes greater than the one at fol.

lowing waves. for either case of white and black circles. This shows us that, in the case of tocad sea, the deck edge is nearer to the surface of waves.

On the other hand, the curve of O based upon the Froude-Krilov's theory becomes that of the symmetrical single-peaked type. Moreover, at z=90'. we have r-O.94S. However, if Dr. Wataeabc's method is followed, T»O.823 will be obtained. The diffcrcnce between both results amounts to about 15%.

C. The Destroy«

The profile of this ship is shown in Fig. 9. The main particulars are as fol-lows: 115m, B-12m, d=4 m. W=2,890', Trim=0, G091 1.01 m. The cal. culaton was made, by assuming T,#10 sec., about the case of wave of which T, 10 sec. and À' 156m, with the assumption a,=0.30, and J,m0(0.25L)'. Shown in Fig. 17 are the calculation results of roiling amplitude and phase angle.

to the case of this hull shape, the same trend as in the case of the subchaser was observed on any motions of swaying, or rolling.

V CondumIos

The principal purpose of thin paper is to investigate whether, in the case of zero ship ship velocity, following waves participate in yielding the maximum value of rolling amplitude or not. In this paper, further discussion was made on the linear equations of motion of swaying, yawing and rolling in oblique waves. We are, however, stilt in the first stage in this field. The principal conclusions tobe obtained from tite calculations of this paper may be summarized as follows:

Regarding the swaying force, yawing and rolling moments acting on the Ship hull in oblique waves, the method discussed in Chapter tI gives considerably good approximation.

It is ascribable to N,'e- tEj,, that the rolling moment acting on the ship hull in following waves at V-0 is different from the case in head sea. The coupled equations of motion for swaying. yawing and rolling in the case

V-0 are given by (3.1). tn the ease V.v0. however, it will be assumed that for the coefficients of the swaying and yawing motions, equations (3.2) can be employed approxiinatety. provided that m.,Vç is taken into consideration. In the case of an Roundhaul Netter, which apparently has a heavy fore-and. aft asymmetry, hardly any difference of rolling amplitude can exist between the two conditions of head sea and following waves. This is because M0, is very large, and the cross coupling lerma between swaying and yawing or yaw-ing and rollyaw-ing are ail very small.

In the cases of the subchaser and the destroyer, the amplitude curves for sway. ing. yawing and rolling showed asymmetric distributions about z = 90', due to the fact that m,k,i, mokA and Ñ,i are large. Yawing showed the maxi. mum peak in head sea du to the inertia couple of rolling. The degree of asymmetric property of rolling about z.=90'. however, decreased on account of yawing. lt is to be noted however, that a large peak of rolling would be produced in quartering seas should yawing be restrained by some methods. In the experiments [6J in which yawing was restrained, a rolling peak was produced in quartering seas. It seems to be possible that, according to hull shapes, the coupling actions of swaying and yawing on rolling might be unit. ed to produce a peak in quartering seas.

I - N.° i - JAN. 1968

We should be careful, as can be seen from the example of the flshing.boat and Table Ill, that the rolling moment of she Froude.Krilov's theory might become very large for a ship with sections of large H0 value.

The transverse bending moment, twisting moment, etc. of ships in oblique waves can be calculated from equations (3.1).

We heard Dr. Watanabe often referred in his lecture to a hull shape yielding the maximum rolling in quartering seas. Il is concluded that a reason thereof might consist in the coupling action of swaying and yawing on rolling, as dis.

cussed in this paper. It seems to br necessary hereafter to investigase, on various types of hull shapes, the coupling action by means of calculation and to examine

it experimentally loo.

In concluding this paper. the author heartily appreciates kind cooperasions of Mr. Tsuchiya of the Fishing.boat Laboratory, Mr. Kuwabara and Mr. Yoshimoto, both belonging lo the Usuki Iron Woorks. Mr. Kaneko, engineer of the Kure Do' ckyacd. who presented lines and other data to the author. Also thanks are due to Miss Okazaki and Miss Mori who assisted the author in numerical calculation.

VI. References

(I] F. Tasai: "Ship Motions in Beam Seas" Report of she Research Institute for Appli' ed Mechanics, Kyssshu University Vol. XIII, No. 45, 1965

(21 F. Tassi: "On the Equation of Rolling Motion of a Ship", Bulletin of the

Rese-arch Institute for Applied Mechanics, Kyushu University, Vol. 25, 1965

13] H. fida and C. Uncoln Crame: "Steering Characteristics of Ships in Calm Water

and Waves", S. N. A. M. E., 1965

4] 0. Grim and Y. Takaishi: "Dan Rollmoment in Schtagtaufender Welle". Schiff und

Hafen, Heft 10, 1965

(5] "Shipbuilding handbook, Vol. I" (published by Society of Naval Architects of Ja'

pan), page 570

(61 K. Taniguchi: "Experimental Results in Waves na a model of Fishing Boat", Journal

of SEIBU ZÖSENKAI, No. II, t957

('1 Y. Watanabe: "On the Theory of Heaving and Pitching Motions of a Ship".

Tech-nology Report of the Kyushu University, Vol. 31, No. t, Jane 1958

It] K. Tamara: "The Calculation of Hydrodynansic Furent and Moments acting on the Two-Dimensoonal BodyAccording to the Grim's Theory". Journnl of SEIBU ZÖSFNKAI, No. 26, Sept. 1963

(9] Y. Watanabe: "On the Motion of the Centre of Gravity of Ships and Effective

Wave Slope", Journal of Society of Naval Architects of Japan, Vol, 49. 1932

IO] S. Motora: "Stripwise Calculation of Hydrodynamic Forces Due to Beam Seas",

Journal of Ship Research, June 1964

Itt) L 3. Ryditl: 'A Lionne Theory for the Steered Motion of Ships in Waves", T. t. N, A.. t959

(12] S. Matura and T. Ksyarna: "On the Improvement of the Maneuverability of Ships

by means of Automatic Steering", J. S. N. A. of Japan, No. 116, 1964

(15) A, I. Ralf: "The Dynamic Calculation of Lateral Bending Moments on Ships in

Oblique Waves", T R G Report 147, 1964

(141 F. Tasai: "A Note on the Swaying, Yawing and Rolling Motions in Oblique

Wa-ves", Report in Technical Committee of SEIBU ZÖSENKAI, 1965

(IS] G. Van Leeuwen: "The Lateral Damping and Added Mass of a Horizontally

Oscil-lating Ship Model". T N O Report No. 65 5, 1964

116] S. Motora and M, Fauno: "On the Measurement of the Stability Derivatives by

means of Forced Yawing Techniqae". J. 5. N. A. of Japan, Na. 118, 1965

(Received September 13, 1966) 04 0.3 0.2 s.l o 15 30 45 60 75 90 im 120 13h 150 165 150 Z (deçreC) Fig, 7 Yawing moment

(6)

g

AP

L

Fig. 9 Profiles of the Submarine-chaser and the Destroyer

Swaying Porc, 15 60 75 90 105 120

X degree) Fig. 8 Rolling amplitude

Sub chazar Ils in 15 20 45 60 75 90 los 120 135

-X degree)

30 45 60 Fig. Il 75 go 80 / t05 120 135 150 ISO 165 165 180 180 E 500 1X0 300 200 too 0g 08 07 O6 y. Q5 03 02 o

i:

Roii,ng Moment 98.Sinl.at_Ce) Ñb IM nr.0,16) FteI/ - _{,.I/,_tProlas..sOSIocs Ten,aryl}

By Coupled Equation, at Sway, Yaw, Roll By Coupled Eguotions et Sony, Roil

-w- Frnude-Krllao'. Theory 300 20

b'

-t o' -3°. -55. -70' -t 100 -130 -150 -170' tao 120' 100 500 60 40' 20 5 30 ¿5 60 75 90 105 120 135 ISO 65 180 Xdegree) Fg. 04 Yawing Motion

'

Cow)uut- Ep) Sa o3 02 01

- Z degree)

IS 30 45 60 75 90 05 120 135 150 165 180 Fig. 15

Rolling Motion ) Subchaser I.. -.

I F AP IS 17 I.. -

1-Destroyer San,

'I

tu 17 I5 I X degree) X degree) Fig. 12 _{Fig. 16} 44

_{TECNOLOGIA NAVAL}

iS so us so 75 90 lOS 120 35 ISO las 1go

X

Fig. IS

LWL

(7)

Table I S (x=90)

Fig. 6 The body plan and profile oC the Roundhaul

netter

i:

_{--. By Coopl.d Eeaetione 01 Seo,. Roll}

-o- By Coapl.d Eqooton, oc Scot. Soc. Roll

15 30 45 60 75 90 105 120 135

- Z (degree)

Fig. 17 Table 11-1 ?-R (r=90 Table Il-2 P-R (x=90) 120

6

₁₀₀ 90 60 40 20 14=02

_IL

14=0.4 0 0.1 I 0.2 04 ₀₆₁ 1.0 I 0 _{0.1 I 0.2} 0.4 0.6 1.0 1.0 1.0 0.95401 091010.8291 O.75 0635 1.0 0.954 0.98810.8241 0.74 0620 09 5.0 0.956 0.91410.837j0.768 0.652 0.0 0.9560.913j0.8340.763 0659 0.8 1.0 0.959 0.92010.847 0.782 0.671I 1.0 0960 0.909 0.8451 0.777 0.660 0.7 1.0 0.962 0.9271 0.859 0.798 0,693 1.0 0.96 0.9251 085 0.793 0.682 06 1.0 0965 0.93410.873 0.816 0.719 0.0 0.9661 0.933 0.870 0.801 0.701 0.3 1.0 0.972 0.94410891 08411 0.735 1.0 097 0.94 0.88610.833 0.738

--- H=O.6' I -

fi

O 0.1 0.2 _{0.41 0.61} 1.0 0 I

all

'T 0.41 0.6 LO 1.0 1.00L953Ia906o.8igl 0.739f o.598 1.0 09S 9O 0802 0.72510569 0.9 1.0 0.9561 0.902 0.830 0.7541 0.619 1.0 09561 0.9101 0.824 0.7421 0.592 0.8 1.0 09591 0.918 0.841 O.77 0.642 1.0 0.9581 0.907 0.836 0.7591 0.617 0.7 1.0 0.962F 0.924 0.853 0.786 0.665F 1.0 0. 963 0953 0.948 0.7761 0.641 0.6 1.0 09681 0.933 0867 0,805 0.690F 1.0 0.9651 0.929 0.861 0.7941 0,667 0.5 1.0 0972f 0.939 0.880 0.823 0.718 F 1.0 0.969 0.937 0.873 0.0131 0 0.1 02 0.4 0.6 1.0 0 I 0.! 0.2 04 0.6 1.0 1.0 0.0 0.951 0901 0.803 0.7091 0531 1.0 0.949 0.8980.793 0.690 0.492 0.9 0.0 0.953 0. 908 0.817 0.726 0.560 1.0 0953 0906 08081 0.100 0.52! 0.8 1.0 0.957 0.915 0830 0.746 0.586 1.0 0.957 0.902 0.8 0.730 0.551 (07 1.0 0.961 0.921] 0.843 0,764 0.613 1.0 0961 0.926 0.836 0.750 0.579 0.6 0.0 0.964 0.92910.834 O.782 0.640 1.0 0.963 0925 0.84710.768 0.609 0.5 1.0 0.967 0.934f 0.869 0.801] 0.668 1.0 0.967 0.9321 0.861f 0.788 0.640

H.0.2

I

ffn04

O 0.1 02 0.4 0.6 I 1.0 0 0.1 02 I 04 I 0.6 1.0 1.0 -0.4720 -0.4433 -0.4165] -0.3677] -0.32591 _025601 -o,f -04109 -(03869 -03428 -03036 -0.2379 0.9 -0.4445 -0.4179 -0.39371 -034891 -0.30991 -o.2457] -0.40t -0.3777 -03559 -0.306! -0,2807 -0.2215 0,8 -04128 -0.3895 -0.36721 -0.32691 -0.2915] -023341 -O.36 ''03627 -0.3727, -O.287( -0.2355 -0.2229 07 -0.3760 -0.3556 -(033631 -0.50171 -02703 -O.2187 -0.3196 -0.3014 -(02845 -0.25371 -0.2263 -0.1816 0.6 -0.3303 -0.3132 -0.29791 -0.2689] -0.24331 -0.2601] -0.26841 -0.2533 -0.2396 -0.21* -0.1912 -01532 0.5 -0.2715 -OE2596 -0.249( -0.72771 _{-02082f -0l75} _O.2039f -0.0924-0,1826 -0.163 -0.1462 -0.1174 14=0.6

_He08

0 0.1 02 04 06 1.0 0 01 0.2 0.4 0.6 1.0 1.0 -0.37391 '-0.3539] -0,3349 -O.2987 -0.26541 -02070 -283 -0.27l -025751 -0.2336 -02111 -01845 0.9 -032771 -03099 -0.29241 -0.261

-23

-0.1820 -0,2246 _{-02143 -0.2643] -01842} _-0.1650 _-0.1293 0.8 -0.2795) -0.2633 -0.24851 -0.2208 -0.19791 -0.0536 -0.1626 -0.1535) -0.1460 -0.1288 -0.1139 -09079 0.7 -0.2267 -0.2111] -0.19801 -0.1745 -0.15331 -01189 -0.0912 -0.0843 -00766 -0,0644 -0.0537 -00368 0.6 -0. 0619 -0. 0506 - 1390 -0. 1204 -a 10371 -00767 -OE 0078 -0,0012 0.0043 0.0131 0.0199 00278 0.5 -0.0803 -0.0727 -0. 063 -0.0495 -00378f -0.01961 00987f 01024 01093, (01126 0.1151 0.1134 5e1. O I H01. 2 I 02 04 06 I 1.0 0 0.1 0.2 0.4 J 0.6 1.0.

1.0 -0.1650 -0.16401 -0.16191 -01539 -0.04191 _o.1I191 -00193] -o.o3471 -00419 _O.0538 _-o.oses _-0.0519

09 0.8 -0.0124-00923 -00917]-001231 -0.01211-0.09191 -0.0669-0.0118

-00lI

-0.0810 -(08661_-00693 (00706 OEO58 00482 00316 0.0195 0.86152 01716 0. 1601 01499 0.1297 0. lilI 0,0792 07 00779 007761 O7 0.6746 00707 0.0605) 0. 0.2701 02675 0.2446 02094 01776 0.6 0. 1905 0. 19001 0.1880 0.1934 0. 174.9 01529 0.43 0. 4238 0.4138 0.3290 0. 3599 0.2927 0.5 0.3338 (03334 0.33091 0.3240 031091 02761f 0.622 06140 0.6050 05447' 0.4618 I H01. 4 H0=i.6 o J 0.1 J 0.2 0.4 I 0.6 I 1.0 0 01 02 I 0.4 06 1.0 1.0 I _0.0522 _0.1240 _0.1000 _0.06(8 f 0.0368 00123 0. 3460 0. 2288 0. 2586 0.1910 0.1400 0.0769 0,9 0.2611 ] 0.2356 0.2106 0.160! 0.1350 0. 0812 0.4049 0.4419 0.4006 0.3256 0.2607 0.1508 0,0 0.3880 ' 0.3640 0.3396 0.2936 j 0.2496 0.1712 0.6588 0.6169 0.5752 0.4930 0,4138 0.2711 0.7 0.5361 I (03143 0.4907 0.4422 0.3907 0,2866 0.8221 0.786! 0.7478 0.6656 0.5783 0.4016 0,6 07208 0.7010 0.6799 0.6292 ' 0,5708 0,4400 1.0550 1.0215 _0.9850 _0.9013 _0.8036 0.5865 0.5 f 0.9652 0,9492 0.9208 0.0700 f 0.8123 0.6518 1.3610 1.3360 1.3030 1.2173 1.1097 0.8456 14=2.0 _¡4=3.0

-o J 0.1 0.2 0.4 0.6 1.0 0 0.1 f 0.2 0.4 J 0.6 I 1.0 1.0 0.8295 I 0.7379 I 0.6525 I 0.5023 0.3783 0.2037 2.4948 2.2391 j 1.9768 1.4647 1.0073 03541 0.9 1.0176 0.9326 0.8484 i 0.6875 0.5.404 0.2998 2.6705 _2.6269 _2.3667 _1.8137 1.2674 04333 0.0 1.2386 1.1604 1.0783 0.9094 0,7400 0.4298 3.3232 3.0983 2.0365 2.2122 1.6182 0.5511 0.7 I 1.5079 j 1,4366 1.3584 1.1827 0.9914 0.6051 3.8904 3. 6830 3.4235 2.7833 I 2.8627 07238 0.6 1.0516 1.7898 0.7150 1.5335 1.3189 0,0455 4.6289 4.4449 4.1874 3.4976 ' 2.6578 0.9779 0.5 2.3170 2.2800 2.2090 2.0330 1.7707 1.1956 5.6435 5.4864 I 5.2329 4.4799 J 3.4906 1.3621 Ho=4.O j 0 0.1 0.2 0.4 06 1.0 i 1.0 I 4.8272 4.3148 3.7436 2.5422 1.4656 0.1640 0.9 ' 5.4951 4.9770 4.3882 _3.0665 _{t. 7920} 0, 1378 0.8 j 6.2433 5.7802 5.1774 3.7209 2.2166 _0.121! 0.7 7.2239 6.7897 6.1716 4.5597 2.7801 0.1196 06 8.5158 8.1150 7.4789 3.6738 3.5482 0.1429 05 IO. 3068 9.9486 9.2993 7.2237 4.6397 0.2100 o 0.1 _{021 0.41 0.61}

io

o 0.! 1.0 1.0 0.9491 0.895 O.783j O.668j 0.447 1.0 0.949 0.9 1.0 0.953 0.903 0.796 0.690 0.470 1.0 0.95 o. a 1.0 0.957 O.91 0.813 O.7I 0.508 1.0 0.956, 0.5 1.0 0.960 O.907j 0.827 0.733 0.540 1.0 0.959F 0.6 1.0 0.963 0.924 0.840 0.753 0.573 1.0 0. 96Z 05 1.0 0.966 0.930 0 8S4 O. 77 0.609 j 1.0 0.965 ß.=2. O O

0.10.2

0.41 0.6 1.0 0 0.1 1.0 1.0 0.947 0884 0.742 0.30 0296 1.0 0.938 0.9 1.0 0.951 0.692 0.760 0.615 0.322 1.0 0.943 0.8 1.0 0.954 0.901 0.778 0.643 0.365 1.0 0.948 0.7 1.0 0.958 0.909 0.796 0.669F 0.406 1.0 0.953 0.6 0.0 0.961 0906 0.813 0.6971 0.452 1.0 0.956 05 1.0 09630.9220829 0.729j ₀₅₀₄₁ 0.0 OE95 o oo 0.2 0.41 0.61 1.0 1.0 0.0 0.9271 0818 0.528 0230-0.1141 0.9 1.0 0.93. 0.831 0561 0269-0.092 08 1.0 0.940 0846 Sfl 0315-0.059 0.7 1.0 0945 0.860 0.629 0.3691-0.012 0.6 1.0 0950 0.875 0.670 0.4351 0.056 0.3 3.0 0.955 0894 0.720 0.5 0.155 0.2 _{0.41 0.61} 1.0 0.895 0.771 O.64 0.398 0.900 0.787 0.668 0,430 0.908 0.803 0.69! 0.463 0.915 0.818 0.7141 0.498 0.921 0 833 0736 0.535 0.927 0. 84 O. 759 0.376 0.2 0.4 0.6 1.0 855 0.647 0.41810.042 0.266 0.672 0.4521 0.074 0078 0.694 O.49 0.115 0.888 0.722 0.530 0.166 0.898 0.750 0.576 0.233 0908 0.78006300.322

VOL.

I - N.° 1 - JAN. 1968

RollIng Motion I Dentroyer I

eo _0Snit"tI

140

(8)

RESUMO

(1) Proreeor Emerttuß. Naval

Ar-chitecture and Marine

Engi-neering, University ot

Michi-gan.

Ann

Arbor,

Michigan

48104, USA.

During

siderable towing-tank re-

the past decade

con-search

and

mathematical

analysis have guided the hull

form design of certain types

of vessels, which reduce

bla-de

frequency

vibration

to

acceptable

levels.

However,

in cargo vessels where

maxi-mum deadweight capacity Is

essential, fining of the form

aft

Is

not

feasible.

High

powered work boats and

in-creased repowering of cargo

vessels, also suffer from

ex-cessive hull vibration.

The basic physical and

hy-draulic factors involved

In

the instigation of hull

vibra-tion; which is maximum at

the stern; with spaced nodes

forward; are as follows:

-AUnbalaned pitch, and or

weight of each propeller

blade.

B - Cavitation, which Is

a

function of blade mean

width

ratio

(MWR);

high tip speed and slip.

C - Excessive

blade

thrust

loading.

"'1

SUPPRESSION OF BLADE FREQUENCY VIBRATION

EL!MINAÇAO DA VIBRAÇAO DA FREQÜÊNCIA DA PA

by